Dynamics of Soft Matter: Neutron Applications

Neutron Scattering Applications and Techniques Series Editors: Ian S. Anderson Neutron Sciences Directorate Oak Ridge ...

Author: Victoria García Sakai | Christiane Alba-Simionesco | Sow-Hsin Chen

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Neutron Scattering Applications and Techniques

Series Editors: Ian S. Anderson Neutron Sciences Directorate Oak Ridge National Laboratory Oak Ridge, TN 37831-6477 USA [email protected] Robert L. McGreevy Neutron Sciences Directorate Oak Ridge National Laboratory Oak Ridge, TN 37831-6477 USA [email protected]

For further volumes: http://www.springer.com/series/8141

Alan J. Hurd Lujan Neutron Scattering Center at LANSCE Los Alamos National Laboratory PO Box 1663, MS H805 Los Alamos, NM 87545 USA [email protected]

Victoria Garc´ıa Sakai • Christiane Alba-Simionesco Sow-Hsin Chen Editors

Dynamics of Soft Matter Neutron Applications

123

Editors Victoria Garc´ıa Sakai CCLRC Rutherford Appleton Laboratory ISIS Facility Chilton OX11 0QX Didcot, Oxon United Kingdom [email protected]

Christiane Alba-Simionesco Laboratoire Léon Brillouin UMR 12 CEA-CNRS CEA Saclay, Gif sur Yvette Bâtiment 563 France [email protected]

Sow-Hsin Chen Department of Nuclear Science and Engineering Massachusetts Institute of Technology Massachusetts Avenue 77 Cambridge, Massachusetts 02139 USA [email protected]

ISSN 1868-0372 e-ISSN 1868-0380 ISBN 978-1-4614-0726-3 e-ISBN 978-1-4614-0727-0 DOI 10.1007/978-1-4614-0727-0 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011943091 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. c Rowan Hargreaves, ISIS Facility, UK Cover illustration: Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The broad field of Soft Matter (polymers, proteins, colloids, liquid crystals, and so on) has experienced an explosive growth in the last few decades and has never been more vibrant. The advances in technology and experimental methods, in theory and simulations, and the search for new “smart” materials to address social and global challenges, continues to expand the research in this field. The properties of Soft Matter systems lie across many disciplines – physics, biology, engineering, and chemistry – and the cross talk between scientists in all fields is of uttermost importance to gain a complete understanding of these systems. In addition, their characteristics span over a wide range of lengthscales and timescales which requires the combination of theoretical and simulation techniques, with a number of different experimental techniques. The importance of neutron scattering techniques was confirmed with the Nobel Prize award in 1994 to Shull for “. . . the development of the neutron diffraction technique . . . ” and Brockhouse for “. . . for the development of neutron spectroscopy . . . .” Neutrons tell us “where atoms are and how they move.” Three years earlier Pierre-Gilles de Gennes, a pioneer in polymer physics, was also awarded the Nobel Prize for “. . . discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers.” In his Nobel lecture on Soft Matter he addressed their two most important properties, complexity and flexibility. Neutron scattering is thus an ideal candidate for the characterization of soft matter systems. In fact, it was data from small angle neutron scattering measurements that provided evidence supporting the work of another Nobel Laureate, Paul Flory, in his prediction that polymer chains adopt self-avoiding random walks. Structural characterization of Soft Matter systems is the initial step to understanding such materials, but ultimately many of their macroscopic properties such as viscosity, conductivity, or enzymatic function, are directly related to their molecular motions. Thus, it is necessary to obtain a dynamical characterization as well. Neutrons offer the advantage that they provide temporal and spatial information simultaneously and especially for Soft Matter systems, are able to discriminate between H and D isotopes, allowing component selectivity in experiments. This v

vi

Preface

is of particular relevance to understand for example, the functionality of proteins in the presence/absence of water, the confinement of water in soft micellar systems, the mixing of two polymers, or the preparation of polymer nanocomposites to achieve new materials with tuneable properties. The aim of this book is to provide scientists, engineers, and advanced students with a reference on how neutrons are a key tool for the study of the dynamical processes in soft materials. It also hopes to highlight the importance of the complementarity of neutrons with other experimental techniques and with computational methods, and above all stimulate cross talk between research fields and collaboration between scientists of different backgrounds. This is of increasing importance with the trend in the field moving to the study of more and more complex systems, and with more difficult interpretation of neutron data. The book starts off by laying out the ground. First, the experimental techniques available to probe the wide range of dynamics in Soft Matter systems are presented. Existing computational methods are then reviewed, ranging from first-principle calculations to mesoscopic simulations. The core of the book is organized in three sections, mainly in increasing order of system complexity, but to some extent there is also a correlation with the evolution of neutron techniques. The first section deals with the dynamics in traditional macromolecules, i.e., in polymer systems. Gabrys and Kanaya (Chap. 3) introduce the vast range of motions possible and explain how neutrons have helped to distinguish them. Arbe and Colmenero (Chap. 4) move on to the unusual dynamical properties observed in polymer blends and relate these to the more fundamental phenomena of the glass transition in amorphous liquids. Chapter 5 treats the specific example of understanding the dynamics in solid polymer electrolytes as part of the move to greener and lighter batteries. This section finishes off with the dynamics at longer lengthscales, and provides a transition from polymers to the biological macromolecular world of proteins and lipids covered in the second section. Three of the chapters in this section deal with the active research field of the structure-dynamics-function relationship of proteins. Smith (Chap. 7) considers neutron data in combination with simulations, Longeville and Doster (Chap. 8) discuss the dynamic processes occurring in proteins, and Wood and Weik (Chap. 9) emphasize the role of hydration water for protein function. The section finishes with a chapter devoted to the dynamics of the lipid membranes which form an integral part of cells and living organisms. In the last section of the book we add extra parameters to the discussion, namely we highlight the importance of confinement, both soft and hard, and surface effects. Chapter 11 presents the emerging technique of time-resolved small angle scattering for studying kinetics. The effects of confinement in Soft Matter systems are discussed in detail with three examples: soft confinement using micellar systems (Chap. 12), nanoparticles in polymer matrices (Chap. 13), and “harder” confinement in nano/mesoporous materials (Chap. 14). The book finishes with examples in the field of shear dynamics in liquids and discusses the potential of an emerging technique, grazing incidence small angle neutron scattering (Chap. 15).

Preface

vii

The importance of neutrons in Soft Matter research is well understood within the scientific community and this is shown by the development directions of new neutron facilities. Not only are powerful new sources optimized for this type of research, with corresponding new instruments being built, but also the complementary tools such as the development of deuteration and computation laboratories are being set-up. There is huge scope for the new generation of scientists in the field of Soft Matter and dynamics, and neutrons will continue to play an important role in answering the many questions that will arise. Didcot, Oxon, UK Orsay, CX, France Cambridge, MA, USA

Victoria Garc´ıa Sakai Christiane Alba-Simionesco Sow-Hsin Chen

Contents

1

Experimental Techniques for Studies of Dynamics in Soft Materials Alexei P. Sokolov and Victoria Garc´ıa Sakai

2

Computational Tools to Understand Inelastic and Quasielastic Neutron Scattering Data .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Mark R. Johnson, Miguel A. González, Mohamed Zbiri, and Eric Pellegrini

Part I

1

25

Macromolecules: Polymers

3

Basic Modes of Motion in Polymers . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Barbara J. Gabrys and Toshiji Kanaya

4

Complex polymers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 Arantxa Arbe and Juan Colmenero

5

Solid Polymer Electrolytes .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 123 Janna K. Maranas

6

Future Perspectives: Moving to Longer Length and Time Scales, from Polymers to Biological Macromolecules . . . . . . . . . . . . . . . . . . 145 Dieter Richter

Part II

59

Bio-Macromolecules: Proteins and Lipids

7

Structure and Dynamics of Biological Systems: Integration of Neutron Scattering with Computer Simulation . .. . . . . . . . . . . . . . . . . . . . 189 Jeremy C. Smith, Marimuthu Krishnan, Loukas Petridis, and Nikolai Smolin

8

Protein Dynamics and Function . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 205 Stéphane Longeville and Wolfgang Doster

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Contents

9

Bio-Macromolecules and Hydration Water Dynamics . . . . . . . . . . . . . . . . . 247 Kathleen Wood and Martin Weik

10 Lipid Membrane Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 263 Maikel C. Rheinstädter Part III

Extra Complexity: Surface Effects and Confinement

11 Application of Time-Resolved Small Angle Neutron Scattering to Non-Equilibrium Kinetic Studies . . . . .. . . . . . . . . . . . . . . . . . . . 289 Jitendra P. Mata, William A. Hamilton, and Elliot P. Gilbert 12 Understanding the Stability of Micellar Systems of Interest for the Study of Glasses, Freezing and Soft Confinement . . . . 319 Tinka Spehr and Bernhard Frick 13 Structure and Dynamics of Polymer Nanocomposites Involving Chain-Grafted Spherical Nanoparticles... . . . . . . . . . . . . . . . . . . . 349 Peter F. Green, Hyunjoon Oh, Pinar Akcora, and Sanat K. Kumar 14 Surface and Confinement Effects in Nano/Mesoporous Materials .. . . 367 Jean-Marc Zanotti and Denis Morineau 15 Shear Dynamics: Understanding Boundary Slip and Anomalies in the Structural and Dynamical Properties of Liquids Under Flow . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 411 Max Wolff Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 439

Contributors

Pinar Akcora Department of Chemical Engineering, Columbia University, New York, NY, USA Arantxa Arbe Centro de F´ısica de Materiales (CSIC-UPV/EHU) – Materials Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain Juan Colmenero Centro de F´ısica de Materiales (CSIC-UPV/EHU) – Materials Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain and Donostia International Physics Center, Paseo Manuel de Lardizabal 3, 20018 San Sebastián, Spain Wolfgang Doster Technische Universität München, Physik Department E 13, James Franck Strasse 1, D-85747 Garching, Germany Bernhard Frick Institut Laue-Langevin, 6, rue Jules Horowitz, F-38042 Grenoble, France Barbara J. Gabrys Department of Materials, University of Oxford, Oxford, UK Elliot P. Gilbert Bragg Institute, Australian Nuclear Science and Technology Organisation, Menai, NSW, Australia Miguel A. González Institut Laue Langevin, 6, rue Jules Horowitz, F-38042 Grenoble, France Peter F. Green Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, USA William A. Hamilton Bragg Institute, Australian Nuclear Science and Technology Organisation, Kirrawee DC, NSW, Australia Mark R. Johnson Institut Laue Langevin, 6, rue Jules Horowitz, F-38042 Grenoble, France Toshiji Kanaya Division of Multidisciplinary Chemistry, Polymer Materials Science, Kyoto University, Uji, Kyoto-fu, Japan xi

xii

Contributors

Marimuthu Krishnan Oak Ridge National Laboratory, Oak Ridge, TN, USA Sanat K. Kumar Department of Chemical Engineering, Columbia University, New York, NY, USA Stéphane Longeville Laboratoire Léon Brillouin, CEA Saclay, F-91191 Gif sur Yvette Cedex, France Janna K. Maranas Department of Chemical Engineering, The Pennsylvania State University, University Park, PA, USA Jitendra P. Mata Bragg Institute, Australian Nuclear Science and Technology Organisation, Kirrawee DC, NSW, Australia Denis Morineau Institute of Physics of Rennes, CNRS-University of Rennes 1, Rennes, France Hyunjoon Oh Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, USA Eric Pellegrini Institut Laue Langevin, 6, rue Jules Horowitz, F-38042 Grenoble, France Loukas Petridis Oak Ridge National Laboratory, Oak Ridge, TN, USA Maikel C. Rheinstädter Department of Physics and Astronomy, McMaster University, 1280 Main Street, West Hamilton, ON L8S 4M1, Canada and Chalk River Laboratories, Canadian Neutron Beam Centre, Chalk River, ON, Canada Dieter Richter Institut für Festkörperforschung, Forschungszentrum Jülich, D52425 Jülich, Germany Victoria Garc´ıa Sakai ISIS Facility, Rutherford Appleton Laboratory, Harwell Science and Innovation Campus, Didcot, UK Jeremy C. Smith Oak Ridge National Laboratory, Oak Ridge, TN, USA Nikolai Smolin Oak Ridge National Laboratory, Oak Ridge, TN, USA Alexei P. Sokolov Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA and Department of Chemistry, University of Tennessee Knoxville, TN, USA Department of Chemistry, University of Tennessee, Knoxville, TN, USA Tinka Spehr Institut für Festkörperphysik, TU Darmstadt, Hochschulstr. 8, D-64289 Darmstadt, Germany Martin Weik Institut de Biologie Structurale, Grenoble, France Kathleen Wood Department of Biophysical Chemistry, University of Groningen, Groningen, The Netherlands Bragg Institute, Australian Nuclear Science and Technology Organisation, Menai, NSW, Australia

Contributors

xiii

M. Wolff Division for Materials Physics, Department of Physics and Astronomy Science, Uppsala University, 751 05 Uppsala, Sweden, Jean-Marc Zanotti Laboratoire Léon Brillouin, CEA-CNRS, Saclay, France Mohamed Zbiri Institut Laue Langevin, 6, rue Jules Horowitz, F-38042 Grenoble, France

Chapter 1

Experimental Techniques for Studies of Dynamics in Soft Materials Alexei P. Sokolov and Victoria Garc´ıa Sakai

1.1 Dynamics in Soft Materials The field of Soft Materials is experiencing an explosive growth during recent years due to a variety of current applications (including advanced plastics and elastomers, liquid crystals) and the essentially unlimited potential of future applications (various kinds of “smart” materials, bio-materials, etc.). The definition of Soft Matter includes broad classes of materials ranging from polymers, liquid crystals and colloidal systems, to biological systems. There are particular properties that differentiate Soft Materials from others: 1. The existence of a great variety of meta-stable states with comparable potential energies and separated by relatively small energy barriers (comparable to a few kT) 2. As a result, there is always a delicate balance between the Entropic and Enthalpic contributions to the free energy, both contributions playing an important role in the properties of soft materials (in strong contrast with hard materials) 3. Strong thermal fluctuations and high sensitivity of their structure to small external fields and perturbations 4. Macroscopic softness of the materials (the reason for the name) that reflects significant structural rearrangements under relatively small mechanical forces

A.P. Sokolov () Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN USA and Department of Chemistry, University of Tennessee Knoxville, TN, USA e-mail: [email protected] V. Garc´ıa Sakai ISIS Facility, Rutherford Appleton Laboratory, Harwell Science, and Innovation Campus, Didcot, OX11 0QX, United Kingdom e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 1, © Springer Science+Business Media, LLC 2012

1

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A.P. Sokolov and V. Garc´ıa Sakai 2 0 –2 –4

log(τ)

y

dar

on Sec in

a Ch

–6

al

ent

–8

m Seg

–10

Methyls

Fast Dynamics

–12 2.5

3.0

3.5

4.0

4.5

1000/T

5.0

5.5

6.0

6.5

7.0

[K –1]

Fig. 1.1 Relaxation map of polyisoprene (PIP) includes (1) chain and (2) segmental relaxation processes with strongly non-Arrhenius temperature variations; (3) a secondary relaxation; (4) methyl group rotations with Arrhenius temperature dependence; and (5) a fast relaxation with a rather weak temperature dependence of the characteristic relaxation time. Most of the data are from [6, 7]

The properties listed above emphasize that transitions between multiple metastable states under relatively small perturbations or due to equilibrium fluctuations are the distinct characteristics of soft materials. So, the dynamics, i.e., the motions of molecular (or other structural) units, are the key to the main macroscopic properties of Soft Matter. The dynamical processes that take place in soft materials are very complex and the microscopic mechanisms of many of these remain poorly understood. Cooperativity and dynamic heterogeneities are characteristic features of dynamics in soft materials [1–5]. However, even these concepts are still not well defined. Typically, the dynamics of soft materials include multiple relaxation processes on local and global scales. They span many (can be more than 15) decades in time (frequency) and are highly sensitive to temperature and external pressure. Figure 1.1 shows an illustrative example of the characteristic relaxation times found in the classical polymer polyisoprene (PIP), the major component of natural rubber. There are at least five relaxation processes that involve molecular motions at different length scales, from global chain relaxations to very local methyl group rotations. All of these relaxation processes reflect molecular motions in a very complex potential energy landscape with a large distribution of energy minima and energy barriers separating the energetic states. Studying the dynamics of soft materials thus can be rather challenging. The complexity of the dynamics in soft materials manifests itself not only through the large number of relaxation phenomena, but also by the nature of their behavior. Most of the relaxation processes in soft materials cannot be described by a single exponential decay. They are usually strongly stretched and can be described

1 Experimental Techniques for Studies of Dynamics in Soft Materials

3

in the time domain by the Kohlrausch–Williams–Watts (KWW) relationship: exp[−(t/τ )β ] [8], where τ is the characteristic relaxation time and β is the stretching parameter. In the frequency domain, the stretched processes are usually described by Cole–Cole, Cole–Davidson, or Havriliak–Negami distribution functions [9]. Secondary relaxations in most cases have symmetrically stretched shapes that are well described by the Cole–Cole distribution function, while the primary structural relaxation usually has an asymmetric shape which is strongly stretched from the high-frequency side [10]. In contrast, fast picosecond relaxation is stretched from the low-frequency side [11]. The reasons for the stretched spectra can be (1) dynamic heterogeneities that lead to a distribution of relaxation times, and/or (2) intrinsically non-exponential relaxation processes. Most of the relaxation processes also exhibit particular temperature dependences. Usually, only secondary relaxations follow an Arrhenius temperature dependence τ = τ0 exp(E/kT) (Fig. 1.1). However, chain and primary structural (segmental) dynamics exhibit much stronger temperature variations that are traditionally described by the Vogel–Fulcher– Tamman (VFT) equation: τ = τ0 exp[B/(T − T0 )]. The characteristic relaxation times of the fast picosecond process do not change significantly, down to very low temperatures [11]. This chapter provides a brief overview of the experimental techniques commonly used for analysis of the dynamics of soft materials. We realize that it is not possible to cover all techniques in a single chapter and thus will focus on the traditional and more broadly used techniques. We shall cover mechanical and dielectric relaxation spectroscopy, Nuclear Magnetic Resonance (NMR), light and X-ray scattering techniques. We will compare their advantages and disadvantages. The focus of this book is to highlight the importance of neutron scattering techniques in probing the dynamics in Soft Matter and so it will be put into context in this introductory chapter, but we will not give a detailed description here. In fact, given the complexity of the dynamics in soft materials, we wish to emphasize the complementarity between all the techniques discussed here, and that it is crucial to combine information from several of these techniques to fully understand the underlying mechanisms of molecular motions. For more in-depth reading on the basics of neutron scattering, we refer the reader to a detailed introduction on neutron scattering [12, and references there-in] and on neutron instrumentation [13] and to the more specific applications in the remit of Dynamics of Soft Matter that follow in other chapters of this book. There are of course other techniques, which will not be discussed, for example, time-resolved optical techniques (e.g., optical Kerr-effect [14]) that are used to monitor relaxation processes. A variety of techniques are based on fluorescence, measuring the kinetics of fluorescence decay and change of its polarization. Pump-probe techniques and forced Rayleigh scattering are actively used to study molecular diffusion and other relaxation processes. Also, optical microscopy has been actively used to study microscopic details of motions in colloidal systems. It has helped to visualize cooperativity and dynamical heterogeneities in these model soft materials [3]. Figure 1.2 presents a traditional view of the time (frequency) range accessible to different techniques. It is important to note that scattering techniques have a

4

A.P. Sokolov and V. Garc´ıa Sakai

Length [nm] 102

13

101

100

1011 109 107 105 103

IXS Light Scattering

Dielectric spectroscopy Mechanical Frequency [Hz]

10

10–1

Neutron Scattering

10–8 10–6 10–4 10–2

X-PCS 10

10–12 10–10

101 –2

10–14

Time [s]

103

–1

10

0

10

10

1

100 102

Q [nm–1] Fig. 1.2 Sketch showing the frequency (time) and wavevector (length) ranges accessible with traditional experimental techniques. Longer times (not shown) can be accessed by many techniques. IXS is Inelastic X-Ray Scattering and XPCS is X-ray Photon Correlation Spectroscopy. Dielectric and Mechanical relaxation spectroscopy do not have a particular length scale (Q), thus are placed outside of the main plot

significant advantage due to an additional variable – the scattering angle. It provides information on the geometry of the motion through the measurements of the scattering wavevector Q. Thus, in Fig. 1.2, the lengthscale (∼2π /Q) accessible to experimental techniques is shown as the x-axis. Mechanical and dielectric relaxation spectroscopies have no particular Q, and that is the reason they are outside of the main sketch presented in Fig. 1.2.

1.2 Neutron Scattering Spectroscopy The dynamics of molecules at a molecular level can be probed directly by neutron scattering spectroscopy. Like other scattering techniques, neutron spectroscopy simultaneously probes the timescale of the motion as well as the lengthscale over which the motion takes place. This enables us to obtain geometrical information and provides a deeper insight into the nature of the dynamical processes that lead to the viscoelastic and mechanical properties of Soft Matter systems. In this same context, neutron scattering spectroscopy is a powerful technique owing to the overlap of the time-length window it can access with that of computer simulations (molecular dynamics and lattice methods). Thus, there is a reciprocal advantage in combining data from these two methods: experimental neutron scattering data can be used to


5

validate theoretical models which are becoming increasingly elaborate and need to be more realistic given the increasing complexity of samples; and accurate theoretical models can be used to interpret dynamical data from neutron scattering experiments and create visualizations of the dynamical processes (see e.g., Chap. 2). The motions probed by neutrons cover a very broad range, from the measurement of momentum distributions on the fs time scale (eV), to slow dynamics such as the relaxation of whole polymer chains in the melt in the ∼100 ns (neV) timescale. The accessible scattering wavevector range allows analysis of geometry of molecular motions on length scales from below an Angstrom to dozens of nanometers. Traditional applications of neutron scattering spectroscopy to soft materials have mainly focused on inelastic (energy transfer peaks centers at finite energies – to study vibrational modes and complementing Raman or Infra-red spectroscopy) and quasi-elastic scattering, at energies close to the elastic line (small energy transfers centered at zero energy – to study rotational and diffusional processes which complement dielectric relaxation spectroscopy and NMR) [15]. In addition, the last 20 years have seen a significant increase in the number of experiments using Neutron Spin Echo (NSE), a technique which probes the longer time and length scales that are relevant for many soft materials [16]. Dynamical information from NSE nicely complements the structural data obtained from the very commonly used technique of small angle-neutron scattering. Furthermore, NSE has provided new insights into colloidal systems (Chap. 12) and offers new possibilities in surface science (Chap. 14). Neutrons have wavelengths and energies that are comparable with interatomic spacings and molecular motions, and thus allow us to probe motions at a molecular scale. The scattering cross-section is simple (in this energy range, it is just a constant) and can be measured on an absolute scale, allowing direct and quantitative comparison of neutron measurements with theory and modeling/simulation. In addition and in comparison to X-rays, neutrons are highly penetrating and nondestructive. Neutrons have no charge and a negligible dipole moment and so can travel large distances before being scattered or absorbed. This allows studies under external fields such as temperature, pressure, shear stress, etc. Due to their low energy (usually in meV range), neutrons are also nondestructive and samples can easily be irradiated for a long time (days) and reused after a neutron experiment. This is of particular importance for biological materials. In particular for Soft Matter studies, the large contrast achieved by the isotopic substitution of hydrogen (one of the main components of soft materials) with deuterium, without changing the intrinsic properties of the material, allows the selective study of specific units of a molecule (e.g., peptide in a protein) or components in a multicomponent system (e.g., polymer in solution). See for example Chap. 6, Section 2.1, for a “classical” example of using isotopic substitution (also known as contrast variation) to study single chain dynamics in liquid poly(ethyl ethylene). There are of course some disadvantages to the technique. First of all, it is not available to have on the bench of a university laboratory. Peer-reviewed beam-time proposals have to be approved to obtain neutron-measuring time at neutron sources.

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Despite continuing improvements at neutron facilities, in neutron instrumentation and optics, the brilliance of neutron beams is far lower than that from synchrotron X-rays. As a result, experiments require relatively large sample sizes from hundreds of mg’s to a few g’s which for many biological samples is very hard to achieve. All in all, neutron scattering spectroscopy has unique advantages over other spectroscopic techniques. However, its low accuracy (statistics), limited frequency (time) window, and limited access to spectrometers require extensive use of complementary techniques to get a more detailed and accurate picture of the underlying dynamics. In what follows, we present the measurable quantities of a neutron experiment in the context of dynamics. When a neutron is scattered by a nucleus, it can change its energy as well as its momentum such that: →2 → − h¯ − ΔE = Ef − Ei = ±¯hω = k f − k 2i , 2m − → → − → − h¯ k f − k i = h¯ Q ,

(1.1) (1.2)

where k = 2π /λ is the neutron wavevector (momentum) and λ is the neutron wavelength, m is the neutron mass, E is the energy, Q is the momentum transfer (Q = kf − ki ), and i and f refer to the initial and final states of the scattering process. In dynamical experiments, we measure the double differential cross-section, i.e., the intensity of scattered neutrons with an energy change ΔE into a solid angle Ω:

∂ 2σ kf ∝ N[σcoh Scoh (Q, ω ) + σincSinc (Q, ω )], ∂ Ω∂ E ki

(1.3)

where N is the number of scatterers, σcoh and σinc are called the coherent and incoherent cross-sections and Scoh (Q, ω ) and Sinc (Q, ω ) are the coherent and incoherent scattering laws, respectively (also sometimes known as dynamic structure factors). The coherent and incoherent cross-sections are calculated from a scattering length specific to each isotope of a particular chemical element. The dynamic structure factors are related to the intermediate scattering functions, Icoll (Q,t) and Iself (Q,t), which are the time Fourier transform of the “distinct” and “self” parts of the Van Hove correlation function: Scoh (Q, ω ) =

1 2π

Sinc (Q, ω ) =

1 2π

+∞

Icoll (Q,t) exp(−iω t) dt,

(1.4)

Iself (Q,t) exp(−iω t) dt,

(1.5)

Icoll (Q,t) =

1 ∑ exp{−iQrk (0)} exp{−iQrl (t)}, N∑ k l

(1.6)

Iself (Q,t) =

1 exp{−iQrk (0)} exp{iQrk (t)}. N∑ k

(1.7)

−∞

+∞ −∞


7

In the classical limit, the “distinct” part is the probability of finding a particle i at time t at a distance r from a position of a particle j at time t = 0, and the “self” part is the probability of finding a particle i at time t at a distance r from its position at time t = 0. Thus, incoherent scattering in a dynamical measurement gives information on self-dynamics (the dynamics of individual atoms) and coherent scattering gives information on collective motions. From hydrogenated samples, we mainly measure incoherent scattering, owing to the incoherent cross-section of hydrogen being 40 times larger than the coherent cross-section. The appropriate technique is quasi-elastic neutron scattering (QENS) which probes very small changes in the neutron energy, in the ueV range. Quasi-elastic neutron spectrometers cover timescales between 0.1 ps and 4 ns and ˚ probing motions such as vibrations (including the lengthscales between 1 and 30 A, Boson peak), rotations of small molecular units, localized motions, and structural relaxations. The dynamic structure factor can be analyzed in the frequency domain, or Fourier transformed to the time domain and modeled with an appropriate dynamical model. Not only temporal information such as diffusion coefficients and characteristic relaxation times can be extracted, but the so-called Elastic Incoherent Structure Factor (EISF) can also be calculated and fitted to models to obtain the geometry of the motion. The EISF is defined as the ratio of elastic scattering to total scattering. Finally, for the study of collective dynamics at the nanosecond timescale, the most efficient technique is NSE which measures the intermediate scattering function in the time domain directly. From deuterated samples, we mainly measure coherent scattering, which is more difficult to interpret since it involves the correlated motions of many atoms. Indeed, for systems as complex as most soft matter, a detailed interpretation requires the aid of some form of atomistic simulation. However, by selective hydrogenation of specific parts of an otherwise deuterated system, we can then selectively pick out the self-dynamics of those atoms. If the number of H atoms is sufficient then their incoherent scattering dominates the signal and this can be done directly. If the number of H atoms is small, then it may be necessary to measure the difference between the scattering from fully and partially deuterated systems.

1.3 Mechanical Relaxation Spectroscopy The mechanical properties of soft materials are one of the most important parameters for many practical applications. Knowledge of the mechanical properties and their time and temperature dependencies are crucial for these applications. Traditional mechanical relaxation spectroscopy measures stress under applied strain, or the reverse, strain under applied stress. The measurements may be in the time domain or in the frequency domain (dynamic mechanical measurements). Time domain measurements usually cover relaxation times from days to milliseconds. Traditional dynamic mechanical measurements cover frequencies

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G'(ω) and G''(ω) [Pa]

109 108 107

Segmental relaxation

Chain modes

106 105 104

G' G''

103

102 10–4 10–3 10–2 10–1 100 101 102 103 104 105

ωaT Fig. 1.3 Frequency dependence of the storage and loss shear moduli, G and G , respectively, for polystyrene of molecular weight 10, 000 g mol−1 . The spectra are constructed by combining measurements at different temperatures using the shift factor aT and assuming time-temperature superposition (TTS). Data are taken from [18]

between ∼10−6 Hz and ∼103 –104 Hz. However, there are also resonance and wave propagating techniques that can measure mechanical properties in the MHz–GHz frequency range. Brillouin light scattering probes mechanical properties in the GHz frequency range. In addition, achievements in inelastic X-ray scattering spectroscopy (IXS) have opened up the possibility of mechanical measurements even in the THz range [17] (though mostly for longitudinal waves). Thus, combining different techniques one can almost entirely cover the time range from picoseconds to many hours, albeit not necessarily continuously. Figure 1.3 shows mechanical storage and loss moduli for the polymer polystyrene. The modulus has a complicated spectrum with contributions from segmental dynamics and chain (Rouse) modes. The shear modulus of the segmental dynamics has an amplitude in the GPa range, while the modulus of chain modes is significantly lower in the MPa range. This difference is due to the variation in energetic (segmental) and entropic (chain) driving forces and is typical for polymers. Figure 1.3 clearly demonstrates that polymer dynamics span a very large frequency range. Unfortunately, it is not possible to cover such a broad frequency range continuously with any current mechanical relaxation technique. As a result, researchers measure the spectra in the accessible frequency range (usually covering 3 to 5 orders in time) at different temperatures and then construct a master curve similar to that shown in Fig. 1.3, assuming the time-temperature superposition (TTS) principle holds [19]. It is well-known, however, that TTS breaks down for most polymers [20–22]. Master curves should therefore be considered as qualitative and one must be cautious when performing a quantitative analysis.


9

The main disadvantage of mechanical relaxation spectroscopy is the absence of any microscopic information on the molecular motion underlying the relaxation process. The measured quantities (usually modulus and strain) have no direct relationship to molecular motions and many model assumptions are usually involved in the data interpretation. However, details of the real molecular motions are missing. Mechanical relaxation can measure bulk and shear moduli which provide information on mechanical relaxation under compression or deformation. This difference gives some additional hints as to the underlying mechanisms of the relaxation, but no microscopic details. Having said that, one can design some experiments where, by careful modification of the molecular structure, one can gain some microscopic insight even from mechanical measurements. An illustrative example of this is the work by Yee et al. on the nature of the secondary relaxation (γ -process) in polycarbonates [23, 24]. By designing tailored block co-polymers, the authors were able to demonstrate the existence of significant cooperativity along the chain involved in a single γ -relaxation event [23, 24]. This approach, however, requires significant effort and precise chemistry of materials. The advent of atomic-force microscopy (AFM) brings new possibilities to studies of the mechanical properties of materials. It provides information on the mechanical moduli, energy dissipation, and viscoelasticity of surfaces at the nanoscale [25]. Recent works involving pulling single molecules have also created significant interest in single molecule mechanical relaxation spectroscopy [26, 27]. This technique has already been applied to studies of proteins and other biological systems [26, 27]. Still, the use of AFM techniques in the analysis of dynamics in soft matter (even on surfaces) remains limited and is not well developed. However, there could be significant progress in this direction in the near future.

1.4 Dielectric Relaxation Spectroscopy One of the most powerful relaxation spectroscopy techniques is dielectric spectroscopy. This covers a very wide frequency range. Many laboratories now have dielectric spectroscopy instrumentation that can probe motions ranging from molecular vibrations (infra-red spectroscopy) to diffusion on the timescale of hours (micro-Hertz) [28]. Thanks to the great progress in electronics and radio-physics, dielectric spectroscopy provides measurements with very high accuracy covering a broad frequency range in a relatively short time. However, like mechanical relaxation measurements, its main disadvantage is the absence of microscopic information about the molecular motions. Dielectric spectroscopy measures the dielectric constant (time or frequency dependence), but no spatial scale information. In addition, the electrical conductivity can mask some of the signal. In the following, we present a general overview. Further details can be found in general textbooks on the subject [29, 30]. In simple terms, dielectric relaxation spectroscopy measures reorientation of dipoles and translational motion of charges. It is usually analyzed in terms of the

10


dielectric constant which in the frequency domain will have a real and an imaginary part (similar to the mechanical storage and loss modulus):

σ ε (ω ) = ε (ω ) − iε (ω ) + i εvac . ω

(1.8)

Here ε (ω ) is the complex permittivity, σ is the electrical conductivity, and εvac is the dielectric constant of vacuum. Application of an electric field leads to the reorientation of dipoles associated with molecular units. The complex permittivity can also be expressed as:

ε (ω ) − ε∞ 1 = 1 − iω ε0 − ε∞ π

∞ 0

Φ(t) exp(−iω t) dt,

(1.9)

where ε0 and ε∞ are the limiting values of the dielectric permittivity at 0 and infinite frequencies, respectively, and Φ(t) is a macroscopic relaxation function that for a single dipole μ can be expressed as the time autocorrelation function of its reorientation: Φμ (t) = μ (0)μ (t)/μ 2 = cos θ (t).

(1.10)

θ (t) is the angle between the dipole vector at t = 0 and t = t. In reality, systems have local field corrections and many dipoles; in this case, the contribution from all dipoles is measured as M(t) = ∑i μi (t). Then ∑ ∑ Pi (0)Pj (t) M(0)M(t) i j = , Φμ (t) = M(0)M(0) ∑ ∑ Pi (0)Pj (0)

(1.11)

i j

where Pi (t) is the instantaneous dipole moment of molecule i. As usual, the correlation function can be divided into “self” (i = j) and “distinct” (i = j). The major contribution is that from the “self” correlation term, and the cross-correlations (“distinct”) are neglected in many cases. Certain structures (e.g., some polymers or α -helixes in proteins) exhibit a persistent cumulative dipole moment along the structure. In this case, there may be a number of separate dielectric responses: (1) individual responses from structural units (e.g., monomers, residues) and (2) a collective response that is accumulated along the structure. Some polymers (e.g., polypropylene oxide, polyiosprene, polyoxybutylene) have the dipole moment accumulating along the chain contour together with a transverse dipole moment component of each monomer. Such polymers exhibit at least two dielectric relaxation processes, though there can be even more due to secondary relaxations: (1) a high-frequency process due to segmental motions of the transverse component μ ⊥ of the dipole moments of the monomers and (2) a low-frequency process due to the long-range motions of the cumulative dipole moment along the chain. In this case, the correlation function for the entire chain will be:


11

Chain

10–1 225K

290K

ε"(ω)

206K

10–2 Segmental Secondary relaxation 10–3

10–1

101

103

105

107

frequency [Hz] Fig. 1.4 Dielectric relaxation spectra of polyisoprene (PIP) with a molecular weight of 10 kg mol−1 at a few selected temperatures. The two strong peaks correspond to the chain (lower frequency) and segmental (higher frequency) dynamics [6]. The high-frequency shoulder of the spectrum at T = 206 K corresponds to the secondary relaxation. We emphasize that no TTS is required to measure dielectric relaxation spectra in such a broad frequency range

μ 2 Ri (0)Ri (t) + ∑ ∑ μi⊥ (0)μ ⊥ j (t) i j , Φμ (t) = μ 2 Ri (0)Ri (0) + ∑ ∑ μi⊥ (0)μ ⊥ j (0)

(1.12)

i j

where μ is the parallel component of the monomer’s dipole moment per unit length and is the correlation function of the end-to-end vector. Thus, dielectric spectroscopy can be used to measure the relaxation of the end-to-end vector, which is one of the main parameters in theories of polymer chain dynamics. However, as has been emphasized above, this is only possible for polymers that have dipole moments accumulating along the chains. Similar effects will appear in the dielectric relaxation spectra of proteins and polypeptides. The α -helical structure creates a cumulative dipole moment along its axis. As a result, the total dielectric response will have contributions from individual residues as well as from the whole α -helix [31]. Figure 1.4 shows the dielectric relaxation spectra of polyisoprene (PIP). The lowest frequency peak corresponds to the chain relaxation, while the higher-frequency peak is that due to the segmental relaxation. Comparing Figs. 1.3 and 1.4, it is clear that the relative amplitudes of the chain and segmental dynamics to the relaxation spectrum differ significantly between mechanical and dielectric spectroscopy. If in

12


mechanical relaxation these contributions are defined by the ratio of entropy-driven (chain) to energy-driven (segmental) forces and will be always low (∼10−3 ), the dielectric response depends on the amplitudes of the dipole moments (1.12). As a result, the ratio of the amplitudes of chain to segmental peaks in dielectric spectra varies significantly between polymers and the amplitude of the chain mode can be even higher than the segmental one (as for PIP in Fig. 1.4). The high-frequency shoulder of the segmental peak (Fig. 1.4) corresponds to the secondary relaxation which is dielectrically active in PIP. We want to emphasize that only molecular motions that affect dipole moments can be measured with this technique. For example, PIP has active methyl group dynamics (as seen in Fig. 2.1). However, since they have no dipole moment, it does not appear in the dielectric relaxation spectra. Another interesting example is poly(methyl methacrylate) (PMMA), which has a side group with a strong dipole moment. As a result, the amplitude of a secondary relaxation in PMMA is significantly higher than that of the segmental dynamics and thus dominates the dielectric spectra [32]. To conclude, we emphasize the power of dielectric spectroscopy in providing accurate measurements of relaxation spectra over a broad frequency range. However, the absence of any microscopic information significantly complicates interpretation. An example of this is for measurements of protein solutions. At least three additional peaks (compared to bulk water) appear in the dielectric relaxation spectra. The slowest of these is ascribed to protein rotation (tumbling). However, the assignment of the higher-frequency modes remains controversial [33–38]. In early works, both processes were assigned to relaxations of water molecules [33, 34]: the fastest process was ascribed to the dynamics of hydration water which is only ∼3–5 times slower than that in bulk water, while the slower process was assigned to tightly bound hydration water moving ∼100 times slower than bulk water. The assignment of the slower process, however, was questioned in later works [35– 39]. It was then ascribed to protein motions [36–38] and to a protein–hydration water cross-term [35]. In fact, even the assignment of the fast process was recently questioned [39]. Based on comparisons between neutron and dielectric spectroscopy data, the authors argued that this process is due to coupled protein–hydration water relaxation. This example clearly illustrates the difficulties in interpreting dielectric spectra: not only is a microscopic interpretation not feasible, but also the assignment of relaxation modes can be questionable. Neutron scattering measurements can alleviate some of these difficulties by providing a detailed microscopic picture to complement the relaxation times and spectral shapes obtained from dielectric spectroscopy, which can in turn be fed back into the analysis of neutron scattering data [39].


13

1.5 NMR Relaxation Spectroscopy NMR spectroscopy is a very powerful and well-developed technique for studying the structure and dynamics in a number of materials. It measures the motion of nuclear spins (i.e., almost directly atomic motions). There are many good textbooks describing the basics of NMR spectroscopy and here we will only present a brief overview of the main principles in the context of soft matter. Each nucleus has a spin I. Application of a magnetic field B leads to a split of the energy levels ΔE = μ B/I where μ is the magnetic momentum of the nucleus. Transitions between levels can be stimulated and monitored using resonance radiofrequency. Different nuclei have different resonance frequencies. There is proton NMR (for H-atoms), 13 C NMR, etc. Different isotopes of the same atom have different resonance frequencies and this effect is often used for selective studies of particular parts of the molecule. For example, regular isotopes 12 C and 16 O have I = 0 and do not contribute to the NMR signal, whereas 13 C and 17 O do, and thus can be used to selectively study a desired part of the molecule. For the same atom, NMR resonance frequencies depend on the neighboring atoms (chemical binding). For example, the resonance frequency of the H-atom is different for CH3 , CH2 , and OH groups. In addition, there is NMR fine structure that reflects differences in the nuclear resonance frequency due to spin–spin interactions of neighboring nuclei of the same type. All these details allow very selective studies of particular atoms of the molecule. Using NMR, one can study not only the average motion of all atoms, for example hydrogen, but also of particular atoms, such as H-atoms in methyl groups. Selectivity can also be achieved with neutron scattering, where the difference in scattering interaction between the hydrogen and deuterium atoms is exploited to highlight different parts of a molecule. Thus, information from both techniques can be used in a complementary fashion to obtain a more complete dynamical picture in soft materials. Spin systems have an equilibrium population of energy levels in a magnetic field. This population can be disturbed by a resonant radio-frequency. The resulting nonequilibrium spin system can relax back to its original equilibrium state through the so-called spin–lattice relaxation, characterized by a time T1 which is the average time taken for an individual nucleus to return to its equilibrium state. The “lattice” in this case refers to the other molecular units surrounding the nucleus that provide energy exchange through various molecular motions. The spin of each nucleus also has a precession about the direction of the magnetic field. This precession induces a magnetic field that might interact with the spins of other nuclei which are precessing at the same frequency. This spin–spin interaction does not change the overall energy of the spin system, but it shortens the lifetime of the spin states and broadens their spectra. The spin–spin relaxation is characterized by a time T2 . Molecular motions produce fluctuations of the local magnetic field. In this way, NMR relaxation can provide information about molecular motions on very local (atomic) scale. This relaxation depends strongly on the nucleus. Using 13 C, 15 N and H atoms provide a rather efficient method to study dynamics. There are

14


many different variations of the NMR spectroscopy technique that are used to study the dynamics of soft materials. Let us first consider the example of pulsedfield gradient NMR spectroscopy which is a tool for the study of molecular diffusion. Traditionally, this uses a magnetic field with the gradient along one axis. As a result, the nuclear resonance frequency is a function of position along the field gradient. Analyzing the NMR frequency as a function of time after the pulse provides a measure of molecular diffusion without using any markers. This technique is effective for the analysis of molecular diffusion in low-to-moderate viscosity materials [40]. The limitation is given by the time it takes to analyze the spin frequency before it relaxes and the strength of the field gradient. Multidimensional NMR, based on multiple pulses with varying frequencies and time sequence between pulses, has provided many interesting results for soft materials [41–45]. In particular, multidimensional NMR has emphasized the importance of dynamic heterogeneities in the structural relaxation of glassforming liquids [43, 44]. Unlike most other spectroscopic techniques used for dynamical studies, which measure two-point correlation functions and an average relaxation of the measured variable, 3- and 4-dimensional NMR can measure higher order correlation functions and can probe dynamic heterogeneities [43, 44]. These studies reveal the existence of dynamic heterogeneities (sub-ensembles of molecules relaxing with different relaxation times) in the structural relaxation at temperatures close to the glass transition temperature [43, 44]. The same studies demonstrate that these heterogeneities are short lived and faster relaxing molecules may become the slowest on time scales comparable to the average relaxation time [44]. Multidimensional NMR has been also used to estimate the length scale of the dynamic heterogeneities, which at temperatures close to Tg is between 1.5 and 3.5 nm [45]. Despite significant progress in NMR spectroscopy and the availability of a wide variety of NMR techniques, it suffers from the very strong localization of the probe – the spin of the nucleus. As a result, the dynamics probed by NMR are very localized, associated with the motion of specific atoms. This requires very strong model assumptions, so very sophisticated NMR measurements are needed to analyze largescale motions and/or cooperative motions of many structural units. In addition, the technique does not provide a broad and continuous frequency dynamical window and is usually effective in relatively narrow spectral ranges. However we should mention that very recent developments in field-cycling NMR [46] might open a possibility for a broadband NMR spectroscopy. Finally, interpretations of the measured T1 and T2 are often not straightforward and strongly model-dependent.

1.6 Light Scattering Light scattering is another experimental technique that is broadly used to study dynamics of soft materials. As with any scattering technique, it has the advantage of measuring relaxations and vibrational spectra as a function of the scattering angle θ .


15

The latter defines the scattering wavevector Q ≈ 2Qi sin(θ /2), where Qi is the wavevector of the incident light wave (note that change of the light photon energy during the scattering process is usually negligible). The analysis of light scattering spectra as a function of Q provides geometric details of molecular motion. The range of accessible Q is rather narrow, between 10−4 and 10−2 nm−1 , owing to the large wavelength of light (between 300 and 800 nm in the visible range). As a result, it does not allow studies of the geometrical details of motions at the molecular level (as is the case with neutron and X-ray scattering), but is very efficient for studies of colloidal systems. In most cases, high signal-to-noise ratio data are obtained with light scattering over a broad frequency (time) window and in a relatively short time (though in this context it still cannot in general compete with dielectric spectroscopy). An additional advantage of light scattering is the very small probe size. One can focus light to ∼1 μm and use it either for studies of extremely small samples (of the order of picograms) or for the analysis of the signal across the sample with resolution in space better than a micron. In most cases, the measured spectra do not require any significant corrections. Furthermore, light scattering has an additional parameter, the polarization of the scattered light. Scattered light can be polarized, I , having the same polarization as the incident light, or depolarized, I⊥ , with the polarization perpendicular to the incident light polarization. Analysis of the depolarization ratio I⊥ /I provides additional information about the type of molecular motion taking place: for example, a number of rotational processes will cause depolarized scattering while isotropic fluctuations such as those from density or chemical composition usually lead to polarized scattering. A significant disadvantage of the light scattering technique is that it measures fluctuations of the optical polarizability (in simple terms, fluctuations of the refractive index) caused by molecular motions, rather than the molecular motions directly. As a result, significant model assumptions are involved in the interpretation. In addition, transparent samples are required in many cases and a high fluorescence signal (either intrinsic or caused by impurities) usually destroys the accuracy of light scattering measurements. There are three major light scattering techniques: 1. Raman spectroscopy which usually covers a wavenumber range between 3–5 and ∼5,000 cm−1 (frequency range from ∼100 GHz up to ∼200 THz) and is used mostly for studies of vibrational modes and fast ps fluctuations 2. Interferometry (Brillouin spectroscopy) that covers a frequency range from ∼100 MHz up to ∼1 THz (∼30 cm−1 ) and is used for studies of acoustic vibrational modes (e.g., mechanical properties at GHz frequencies) and relaxation processes in the ps–ns time range 3. Photon Correlation Spectroscopy (PCS) (often called Dynamic Light Scattering) that works in the time domain and covers motions from about 10 ns (optimistic estimate) to hours. By combining these three techniques, one can essentially cover the entire frequency (time) range with a small gap remaining in the ns-regime [47]

16


A detailed theoretical description of light scattering can be found in [48]. In the following, we give only a short overview. In simple terms, light scatters from local fluctuations of the dielectric constant δ εyz (r,t) (indices yz refer to the two axes of the average light polarization). Light scattering measures the intensity of the scattered light I(Q, ω ): I(Q, ω ) ∝ Q4 I0

exp(−iω t)δ ε (Q,t)δ ε (−Q, 0) dt.

(1.13)

Here I0 is an incident intensity and δ ε (Q,t) is the Fourier transform of the dielectric constant fluctuations in space. We have omitted the polarization indices in the equation for simplicity. Due to extremely small variations of the value of the scattering wavevector Q, the term Q4 is usually neglected. The main challenge in the interpretation of light scattering spectra is to find the connection between the measured δ ε (Q,t) and the underlying molecular motion. The main approaches used to describe fluctuations in the dielectric constant can be divided into two groups: (a) Continuum approximation This approach considers δ ε (r,t) as a continuous function of r without taking into account molecular polarizability, based on the long wavelength of light relative to the characteristic molecular scales. In this case, δ εyz (r,t) for an isotropic one component system can be expressed in terms of an elasto-optical coefficient a and an local deformation tensor γyz (r,t) such that:

δ εyz (r,t) = a1 γiso (r,t) + a2γ˜yz (r,t),

(1.14)

where γiso (r,t) is the isotropic part of the deformation tensor and γ˜yz (r,t) = γyz (r,t) − δyz γiso (r,t)/3 represents the off-diagonal elements of the tensor. There are contributions due to isotropic compression and due to shear deformations. For isotropic fluctuations, the scattering intensity as a function of time is given by: Iiso (Q,t) ∝ δ εiso (Q,t)δ εiso (−Q, 0) = a21 γiso (Q,t)γiso (−Q, 0), 2 2 ∂ε ∂ε δ ρ (Q,t)δ ρ (−Q, 0) = S(Q,t), ∝ ∂ρ ∂ρ

(1.15)

where δ ρ are fluctuations in density ρ . Thus, for isotropic fluctuations, light scattering measures the same intermediate scattering function as in, for example, coherent neutron scattering, but multiplied by the elasto-optical coefficient. As usual, the frequency dependent intensity, I(Q, ω ), is the Fourier transform of I(Q,t). Now let us consider the example of isotropic fluctuations in a two-component system. In that case:

δε =

∂ε ∂ρ

δρ + T,C

∂ε ∂C

T,ρ

δC +

∂ε ∂T

ρ ,C

δ T,

(1.16)


17

where δ C and δ T correspond to fluctuations in concentration and temperature, respectively. The third term in the (1.16) is usually negligible. If the fluctuations of density and concentration are not correlated, then two contributions can be separated as: 2 2 ∂ε ∂ε 2 2 δ ρ + δ C2 . (1.17) δ ε = ∂ ρ T,C ∂ C T,ρ Concentration fluctuations are usually significant in solutions and polymer blends, and, in most cases, show diffusion-like behavior: I(Q,t) ∝ δ ε (Q,t)δ ε (−Q,t) ∝ S(Q) exp(−DQ2t).

(1.18)

Light scattering is often used for the analysis of diffusion coefficients and/or hydrodynamic radii of synthetic and biological molecules in solution. The traditional technique for this kind of study is photon-correlation spectroscopy [48]. However, this technique does not analyze I(Q,t), but the intensity–intensity correlation function: I(Q,t)I(Q,t + τ . (1.19) g2 (Q, τ ) = I(t)2 As a result, the measured PCS correlation time is half that of the real relaxation time of I(Q,t). (b) Molecular approach The second approach to describe light scattering is based on molecular polarizability. In this case, δ εyz (r,t) is expressed in terms of an optical polarizability tensor αyz (i,t), where i is the index of the molecule: 1 Iyz (Q, ω ) = 2π

∞ −∞

exp(−iω t) dt

∑ αyz (i,t)αyz ( j, 0) exp[iQ{ri (t) − r j (0)}]

.

i, j

(1.20) The correlation function in , as usual, can be decomposed into “self” (i = j) and “distinct” (i = j) correlation functions. In the particular case of polarized (y = z) scattering by a spherical molecule (assuming αzz (i,t) = α ):

α2 Izz (Q, ω ) = 2π

∞ −∞

exp(−iω t) dt

∑ exp[iQ{ri (t) − r j (0)}]

= α 2 S(Q, ω ).

i, j

(1.21) So, we again have a dynamical structure factor analogous to that obtained from neutron scattering, but weighted by the optical polarizability of the molecule. The latter depends on the molecular environment, density, etc. and thus the analysis is not as straightforward as in the case of neutron scattering where the nuclear scattering cross-section is the property of the nucleus independent of its surroundings.

18


Anisotropy of the optical polarizability of the molecules gives rise to depolarized light scattering (y = z). In the case of macromolecules, for example, optical anisotropy arises because of anisotropy of the molecular structural units (e.g., a monomer) and because of the nonspherical shape of the entire molecule (e.g., protein). In the first case, one can measure the internal dynamics of the macromolecule, whereas in the second case, motions of the entire molecule, such as a rotation, can be studied. To conclude this section, we want to emphasize that light scattering is the most commonly used scattering technique for studies of the dynamics of soft materials. As we showed above, it provides very similar information to that from neutron scattering, despite the more complex and less straightforward microscopic interpretation. However, it is limited by the rather low Q-range coverage as was shown in Fig. 1.2. As a result, information on the geometry of the motions on a molecular scale is not accessible. Nevertheless, it is very useful to have preliminary light scattering data before performing neutron scattering experiments. They serve as a good guide for the design of appropriate neutron scattering experiments since they provide information about the frequency and temperature range of interest, making the neutron experiment more efficient.

1.7 X-Ray Scattering IXS is a direct analog to light scattering (X-rays are very short wavelength light). However, it has a tremendous advantage because the wavelength of X-rays is comparable to the characteristic interatomic and intermolecular distances. Moreover, X-rays scatter essentially from atoms (although, to be precise, they scatter from charges) and provide direct information on atomic motions. Recent progress in synchrotron radiation has led to facilities that provide intense monochromatic and coherent X-ray beams with very high photon flux. These advantages make X-ray spectroscopy very attractive for studying the dynamics of soft materials. In addition, this technique requires only a small amount of sample and spectra can usually be measured in relatively short time (at least, in comparison to neutron scattering spectroscopy). There are, however, significant drawbacks to using X-rays. One of the main problems is the high energy of the X-ray photons which is typically in the 10–40 keV range. For comparison, neutron energies used for spectroscopy are in the meV range, i.e., a million times lower. The high energy of the photons has two negative implications. First, it is difficult to achieve good energy resolution. Even the best spectrometers can provide energy resolutions of only ∼10−7 of the incident energy, i.e., ∼1 meV (∼240 GHz, ∼1 ps), which is not sufficient for most of the studies of soft materials. This is currently the highest resolution achievable [17]. Secondly, X-rays of such energies can cause damage to samples since they are high enough to break bonds and alter the properties of biological and synthetic macromolecules. As a result, in many cases samples cannot be exposed to the X-ray beam for


INS

IXS

50 40

150

30 100

20

50 0

Neutrons [cts/h]

X-ray [cts/s]

200

19

10

-10 -8 -6 -4 -2

0

2

4

6

8

10

0

E [meV] Fig. 1.5 Comparison of inelastic X-ray (open circle) and neutron (open triangle) scattering spectra measured from a sample of polybutadiene at T = 140 K. The dash line shows the fit to the damped harmonic oscillator model and the solid line presents the inelastic peaks obtained from the fit. Data are taken from [50]. Note that, because of the different elemental scattering cross-sections for neutrons and X-rays, the spectra should not be identical

a long time. To overcome this, researchers either limit the measurement time or constantly move the sample (to illuminate different parts) during the measurements. This imposes additional limitations on the experiments that can be performed with X-ray spectroscopy. There are now two well-developed X-ray scattering techniques that have already found broad applications in studies of dynamics in soft materials. The first one is high resolution inelastic X-ray scattering (IXS) [17, 49, 50]. In some sense, it is analogous to light Brillouin scattering, but covers a much higher Q-range. It is the only direct way to measure the bulk mechanical properties of materials at the nanoscale. IXS provides the same information as inelastic neutron scattering (INS), and despite having poorer resolution, measurement statistics are much better (Fig. 1.5 shows hundreds of counts per second on the X-ray scale compared with only tens of counts per hour on the neutron scale). In addition, in contrast to neutrons, there are no kinematic limitations. (One can only measure Brillouin scattering using radiation that propagates faster than the speed of sound in the material being analyzed; for neutrons, this can mean working at unfeasibly small scattering angles or with insufficient energy resolution.) Many interesting results from studies of vibrational dynamics in glass-forming systems have been achieved using IXS [17, 49]. These results reveal that the broadening of the IXS modes varies essentially with Q2 , and in most cases, is temperature-independent [49]. This unusual Q-dependence (traditional hydrodynamic predicts a Q4 dependence) remains a subject of active discussion. IXS also provides interesting microscopic information on the nature of boson peak vibrations,

20


which are low-frequency modes that are general for the dynamics of small molecular glass-forming systems, polymers, and biological macromolecules. Unfortunately, the limited energy resolution prevents further significant progress in this direction. There are currently attempts to move to longer wavelengths (deep UV light) using synchrotron radiation in order to improve the energy resolution [51]. The other important technique is X-ray photon correlation spectroscopy (XPCS) [52, 53]. This is a complete analogy to usual light PCS, only using short wavelength photons. As a result, it provides the same information on the dynamics of soft materials, but on molecular length scales (Fig. 2.2). It has been demonstrated that XPCS can be used effectively for studies of thin polymer films and polymeric nanocomposite materials [52,53]. As for X-rays in general, but especially for XPCS, the drawback is the radiation damage caused to samples. As mentioned before, samples should either be exposed to the X-ray beam for just a few minutes or have to be in constant motion during the measurements. This results in the limited time range of the dynamics that can be measured currently with XPCS, those slower than ∼10−5–10−3 s. Future construction of X-ray sources with even higher brilliance might shift this limit to even shorter times [54].

1.8 Concluding Remarks In this chapter, we have presented an overview of the major experimental techniques traditionally employed for studies of dynamics in soft materials. Most of these are techniques that can be placed in any research laboratory (e.g., university laboratory). Unfortunately this is not the case with neutron scattering spectroscopy. In this case, the time gap between generating an idea and performing the experiment might be as long as 1 year. In addition, neutron scattering experiments suffer from a rather weak neutron flux per spectrometer in comparison to, for example, X-ray scattering. Despite these disadvantages, neutron scattering spectroscopy has unique properties that make it very attractive for studies of dynamics in soft materials. The most important advantage is that it measures atomic motions directly, on an absolute cross-section scale, because neutrons scatter directly from nuclei. Thus, the results of neutron scattering experiments can be compared directly and quantitatively to model predictions and to results of computer simulations. The wavelength range of neutrons, λ ∼ 0.1–1 nm, opens the scattering wavevector range to one that probes interatomic and molecular distances. Thus, it is a powerful technique to probe microscopic details of molecular motions (diffusive-like, rotation, etc.), their geometry and length scale, in addition to characteristic frequencies and relaxation times. The frequency (time) window accessible to neutron scattering is much broader than that accessible to X-rays (Fig. 1.2) and neutron beams do not damage samples, even those as sensitive as biological systems. Furthermore, most soft materials contain hydrogen atoms and, given the significant difference in scattering intensity from hydrogen and deuterium, isotopic substitution can be used to study the dynamics of a specifically labeled part of a molecule or a particular system


21

component. Coherent neutron scattering also provides important information on cooperativity (coherency) of atomic motions. This advantage, however, has not yet fully exploited in the soft matter field. All of these advantages make neutron scattering spectroscopy a unique tool for studies of dynamics in soft materials that cannot be matched or substituted by any other current experimental technique. In our personal view, one of the main obstacles to a wider and more efficient use of neutron scattering is the absence of a means to focus neutron beams to small enough spots (e.g., mm-size spots while maintaining divergence and hence resolution). As a result, neutron spectroscopy requires large amounts of sample which are unfeasible for many bioand nanotechnologies. Moreover, to achieve high enough signals, experimentalists have to sometimes work on the border of multiple-scattering effects which makes the data analysis harder; this approach is never used in, for example, light scattering, simply because there are enough incoming photons. Therefore, the ability to create a higher neutron flux per square millimeter (not even the total, just per surface area of the sample) without losing important characteristics of the neutron beam would significantly broaden the use of this technique, improve signal-to-noise ratio, and allow measurements with negligible multiple-scattering contributions. To conclude this chapter, we want to emphasize that the use of neutron scattering spectroscopy is most efficient when combined with other complimentary techniques. For example, dielectric spectroscopy provides accurate temperature dependencies, spectral shapes, which can be fed back into the analysis of neutron scattering spectra and provide accurate microscopic information on underlying molecular motions. The use of light scattering and molecular dynamics simulations as a guide to the accurate planning of a neutron scattering experiment is also very efficient. The combination of different experimental techniques is always the best strategy! Acknowledgments This work was sponsored by the Division of Materials Sciences and Engineering, DOE Office of Basic Energy Sciences.

References 1. Debenedetti PG, Stillinger FH (2001) Supercooled liquids and the glass transition. Nature 410:259–267 2. Berthier L et al (2005) Direct experimental evidence of a growing length scale accompanying the glass transition. Science 310:1797–1800 3. Weeks ER, Crocker JC, Levitt AC, Schofield A, Weitz DA (2000) Three-dimensional direct imaging of structural relaxation near the colloidal glass transition. Science 287:627–631 4. Russell EV, Israeloff NE (2000) Direct observation of molecular cooperativity near the glass transition. Nature 408:695–698 5. Qiu XH, Ediger MD (2003) Length scale of dynamic heterogeneity in supercooled D -sorbitol: comparison to model predictions. J Phys Chem B 107:459–464 6. Sokolov AP, Hayashi Y (2007) Breakdown of time-temperature superposition: from experiment to the coupling model and beyond. J Noncryst Solids 353:3838–3844

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7. Frick B, Fetters LJ (1994) Methyl group dynamics in glassy polyisoprene: a neutron backscattering investigation. Macromolecules 27:974–980 8. Williams G (1975) In: Davies M (ed) Dielectric and related molecular processes, vol 2. The Chemical Society, London, p 151 9. Angell CA, Ngai KL, McKenna GB, McMillan PF, Martin SWJ (2000) Relaxation in glass forming liquids and amorphous solids. Appl Phys 88:3113 10. Roland CM, Hensel-Bielowka S, Paluch M, Casalini R (2005) Supercooled dynamics of glassforming liquids and polymers under hydrostatic pressure. Rep Progr Phys 68:1405–1478 11. Surovtsev NV, Wiedersich J, Novikov VN, Rössler E, Sokolov AP (1998) Light scattering spectra of the fast relaxation in glasses. Phys Rev B 58:14888 12. Pynn R (2009) Neutron scattering – a non-destructive microscope for seeing inside matter. Neutron applications in earth, energy and environmental sciences. Springer, New York 13. Schober H (2009) Neutron scattering instrumentation. Neutron applications in earth, energy and environmental sciences. Springer, Berlin 14. Cang H, Novikov VN, Fayer MD (2003) Experimental observation of a nearly logarithmic decay of the orientational correlation function in supercooled liquids on the picosecond-tonanosecond time scales. Phys Rev Lett 90:197401 15. Garc´ıa Sakai V, Arbe A (2009) Quasielastic neutron scattering in soft matter. Curr Opin Coll Inter Sci 14:381–390 16. Farago B (2009) Recent developments and applications of NSE in soft matter. Curr Opin Coll Inter Sci 14:391–395 17. Sette F, Krisch MH, Masciovecchio C, Ruocco G, Monaco G (1998) Dynamics of glasses and glass-forming liquids studied by inelastic X-ray scattering. Science 280:1550–1555 18. Inoue T, Onogi T, Yao ML, Osaki KJ (1999) Viscoelasticity of low molecular weight polystyrene. Separation of rubbery and glassy components. Polymer Sci B Polymer Phys 37:389–397 19. Ferry JD (1980) Viscoelastic properties of polymers, 3rd edn. Wiley, New York 20. Plazek DJ (1965) Temperature dependence of the viscoelastic behavior of polystyrene. J Phys Chem 69:3480 21. Ngai KL, Plazek DJ (1995) Identification of different modes of molecular motion in polymers that cause thermorheological complexity. Rubber Chem Tech Rubber Rev 68:376 22. Ding Y, Sokolov AP (2006) Breakdown of time temperature superposition principle and universality of chain dynamics in polymers. Macromolecules 39:3322–3326 23. Liu Yee A (1998) Enhancing plastic yielding in polyestercarbonate glasses by 1,4Cyclohexylene linkage addition. Macromolecules 31:7865 24. Jho Yee A (1991) Secondary relaxation motion in bisphenol A polycarbonate. Macromolecules 24:1905 25. Mahaffy RE, Park S, Gerde E, Kas J, Shih CK (2004) Quantitative analysis of the viscoelastic properties of thin regions of fibroblasts using atomic force microscopy. Biophys J 86: 1777–1793 26. Rief M et al (1999) Single molecule force spectroscopy of spectrin repeats: low unfolding forces in helix bundles. J Mol Biol 286:553 27. Hugel T, Seitz M (2001) The study of molecular interactions by AFM force spectroscopy. Macromol Rapid Comm 22:989–1016 28. Lunkenheimer P, Schneider U, Brand R, Loidl A (2000) Glassy dynamics. Contemp Phys 41:15–36 29. Kremer F, Schönhals A (2003) Broadband dielectric spectroscopy. Springer, New York 30. Hill NE (1969) Dielectric properties and molecular behaviour. Van Nostrand Reinhold, London 31. Papadopoulos P, Floudas G, Klok HA, Schnell I, Pakula T (2004) Self-assembly and dynamics of poly(γ-benzyl-L -glutamate) peptides. Biomacromolecules 5:81–91 32. Shindo et al (1969) Dielectric properties of stereoregular poly(methyl methacrylates). J Polym Sci A 7:297–310 33. Miura N, Hayashi Y, Mashimo S (1996) Hinge-bending deformation of enzyme observed by microwave dielectric measurement. Biopolymers 39:183–187


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34. Hayashi Y, Miura N, Isobe J, Shinyashiki N, Yagihara S (2000) Molecular dynamics of hinge-bending motion of IgG vanishing with hydrolysis by Papain. Biophys J 79:1023–1029 35. Nandi N, Bhattacharyya K, Bagchi B (2000) Dielectric relaxation and solvation dynamics of water in complex chemical and biological systems. Chem Rev 100:2013–2045 36. Oleinikova A, Sasisanker P, Weingartner H (2004) What can really be learned from dielectric spectroscopy of protein solutions? A case study of ribonuclease A. J Phys Chem B 108:8467– 8474 37. Jansson H, Bergman R, Swenson J (2005) Relation between solvent and protein dynamics as studied by dielectric spectroscopy. J Phys Chem B 109:24134–24141 38. Khodadadi S, Pawlus S, Sokolov A (2008) Influence of hydration on protein dynamics: combining dielectric and neutron scattering spectroscopy data. J Phys Chem B 112:14273 39. Arbe A, Colmenero J, Frick B, Monkenbusch M, Richter D (1998) Investigation of the dielectric β-process in polyisobutylene by incoherent inelastic neutron scattering. Macromolecules 31:4926–4934 40. Price WS (1997) Pulsed-field gradient nuclear magnetic resonance as a tool for studying translational diffusion: part I. Basic theory. Concepts Magn Reson 9:299–336 41. Schmidt-Rohr K, Spiess HW (1994) Multidimensional solid-state NMR and polymers. Academic, London 42. Brown SP, Spiess HW (2001) Advanced solid-state NMR methods for the elucidations of structure and dynamics of molecular, macromolecular, and supramolecular systems. Chem Rev 101:4125–4155 43. Heuer A, Wilhelm M, Zimmermann H, Spiess HW (1995) Rate memory of structural relaxation in glasses and its detection by multidimensional NMR. Phys Rev Lett 75:2851–2854 44. Bohmer R, Chamberlin RV, Diezemann G, Geil B, Heuer A, Hinze G, Kuebler SC, Richert R, Schiner B, Sillescu H, Spiess HW, Tracht U, Wilhelm M (1998) Nature of the non-exponential primary relaxation in structural glass-formers probed by dynamically selective experiments. J Noncryst Solids 235:1–9 45. Qiu XH, Ediger MD (2003) Length scale of dynamic heterogeneity in supercooled D -sorbitol: comparison to model predictions. J Phys Chem B 107:459–464 46. Meier R, Kahlau R, Kruk D, Rossler EA (2010) Comparative study of the dynamics in viscous liquids by means of dielectric spectroscopy and field cycling NMR. J Phys Chem A 114: 7847–7855 47. Bergamn R, Borjesson L, Torell LM, Fontana A (1997) Dynamics around the liquid-glass transistion in poly(propylene) glycol investigated by wide-frequency-range light scattering techniques. Phys Rev B 56:11619 48. Berne BJ, Pecora R (2000) Dynamic light scattering with application to chemistry, biology and physics. Dover Publications Inc, Mineola 49. Dell’ Anna R, Ruocco G, Sampoli M, Viliani G (1998) High frequency sound waves in vitreous silica. Phys Rev Lett 80:1236–1239 50. Sokolov AP, Buchenau U, Richter D, Masciovecchio C, Sette F, Mermet A, Fioretto D, Ruocco G, Willner L, Frick B (1999) Brillouin and Umklapp scattering in polybutadiene: comparison of neutron and X-ray scattering. Phys Rev E 60:R2464 51. Masciovecchio C, Baldi G, Caponi S et al (2006) Evidence of a crossover in the frequency dependence of the acoustic attenuation in vitreous silica. Phys Rev Lett 97:035501 52. Li C, Koga T, Li C et al (2005) Viscosity measurements of very thin polymer films. Macromolecules 38:5144–5151 53. Bandyopdahyay R, Liang D, Harden JL, Leheny RL (2006) Slow dynamics, aging, and glassy rheology in soft and living matter. Sol State Comm 139:589–598 54. Grubel G (2008) X-ray photon correlation spectroscopy at the European X-ray free-electron laser (XFEL) facility. CR Physique 9:668–680

Chapter 2

Computational Tools to Understand Inelastic and Quasielastic Neutron Scattering Data Mark R. Johnson, Miguel A. González, Mohamed Zbiri, and Eric Pellegrini

2.1 Introduction Neutron scattering (NS) probes the way atoms move, giving an experimental signature of the dynamics. According to Newton’s laws of motion, an atom moves because a force has been exerted on it and it subsequently flies ballistically. In condensed matter, due to the high atomic density and the bonding between atoms, the ballistic regime is short and atoms are continually experiencing new forces and accelerations. The acceleration a depends linearly on the force F where the proportionality constant is the mass m of the atom: F = ma.

(2.1)

Ignoring zero point motion, there is no motion when all atoms are at minima of the potential energy surface (PES) and this arrangement of atoms constitutes an equilibrium structure. The most stable structure is the one with the lowest total energy, i.e. the global minimum of the PES. The equilibrium structure is the starting point for lattice dynamics (LD) calculations. Vibrational motion measured by inelastic neutron scattering (INS) explores the PES in the vicinity of the minima. Since the PES depends on all the atomic coordinates, displacing a single atom slightly from equilibrium will (a) induce a restoring force on that atom and (b) induce inter-atomic forces on all other atoms. In the case of a harmonic potential energy well, the magnitude of the restoring force depends linearly on the amplitude of the displacement. In real systems the ˚ The interharmonic approximation is valid for small displacements (e.g. < 0.05 A). atomic forces, measured by displacing atoms from equilibrium positions, are used to construct the dynamical matrix (DM), which is an expression of the equations of M.R. Johnson () • M.A. González • M. Zbiri • E. Pellegrini Institut Laue Langevin, BP 156, 6 rue Jules Horowitz, 38042 Grenoble, France e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 2, © Springer Science+Business Media, LLC 2012

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motion for all atoms. Diagonalising the DM gives a set of normal modes (NM) or harmonic oscillators, each one having a characteristic frequency and a displacement vector describing the relative motion of all atoms in that mode. The NM can be used to construct NS observables like the generalised vibrational density of states (GVDOS) and the scattering function S(Q, ω ) for powders or single crystals. In the case of an equilibrium structure, well-separated in coordinate space from other minima in the PES, the NM frequencies will all be bigger than or equal to zero. A larger atomic displacement, may however, push the structure towards another minimum of the PES, there being no restoring force on the displaced atom. In this case, the LD calculation will result in negative frequencies (or rather imaginary frequencies, see later) which are a signature of the instability. The equilibrium structure is also the starting point for a molecular dynamics (MD) simulation. In a MD simulation, the NS observables mentioned above are extracted from the time-dependent trajectories of the atoms via correlation functions, like the velocity auto-correlation function and the van Hove correlation functions. The system has to be displaced from equilibrium by an initial “impulse”, which is achieved with a Boltzmann distribution of initial velocities corresponding to the required simulation temperature, and thereafter the motion is perpetual. Each structure in the series of the trajectory of the MD simulation is calculated from the previous structure using the instantaneous forces and therefore the accelerations acting on each atom. In a MD simulation, the system will vibrate about its equilibrium geometry, unless the kinetic energy is sufficient to allow atoms to cross maxima in the PES, giving rise to large amplitude, atomic displacements. This motion can be local, for example rotations of molecular groups, or long-range, giving rising to liquid-like, translational diffusion. The experimental signature is quasielastic neutron scattering (QENS), which is a broadening of the elastic peak, as opposed to discrete, inelastic peaks. Whereas LD calculations correspond to a temperature of zero Kelvin, since the atomic positions are optimised to minima of the PES, MD simulations introduce temperature through the kinetic energy of the atoms. Forces are therefore central to LD and MD methods. They are calculated from an expression of the total energy of the system as the derivatives with respect to the appropriate atomic coordinates. While there are many methods and approximations used for calculating the energy of a system of atoms, two are widely available and enable a wide range of science and systems to be tackled. The energy of a set of atoms depends on how the valence electrons of the atoms organise themselves around the nuclei and the core electrons. A correct description of the electronic structure requires Schroedinger’s equation to be solved for the system of electrons and nuclei. Using traditional, Hartree-Fock methods [41, 42], these calculations can scale as badly as the seventh power of the number of atoms, i.e. doubling the number of atoms would increase the computational cost by a factor of 128 (N 7 where N is the number of atoms). One of the most computationally efficient methods is to calculate the one-electron density field around the nuclei rather than the many-electron wavefunction. This approach is embodied in density functional theory (DFT), which recovers the missing correlation in the Hartree-Fock approach,

2 Computational Tools to Understand Inelastic and Quasielastic...

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and codes based on this theory typically offer N 3 scaling. DFT-based methods can be applied to aperiodic (e.g. molecules) and periodic systems of hundreds of atoms and a timescale up to 100 ps can be reached, therefore enabling a wide range of materials studied by NS to be simulated accurately. The scope of DFT is further enhanced by linear scaling methods which allow 104 atoms to be handled, thereby increasing the overlap with classical methods. Larger systems like polymers and bio-molecules (the principal subject of this book) for which, in addition, slower dynamics and therefore longer timescales are of interest, generally require methods in which the computational cost per atom is much lower. This is achieved by treating electrons implicitly. The chemical bonds that are formed by electrons are described by springs. The net electron density around atoms is described by point charges. The long-range correlation of electrons (not treated in standard DFT-based methods) is described by the dispersive term in the Van der Waals interaction, which is typically proportional to r−6 , where r is the inter-atomic distance. Springs, point charges and interactions like electrostatic and VDW interactions constitute a force field (FF), which is a parameterized, analytical expression for the energy of a system. The complexity of the FF (the number of springs and interactions represented) and the parameters determine the precision and the extent to which a FF can be used on a range of systems. FF-based methods allow 105 atoms and a timescale up to 10 ns to be treated easily, with additional orders of magnitude in size and/or time available on more powerful computational resources. For larger systems, when atomic details are no longer relevant, for either the science under consideration or for the stability of the simulation, unified atom and coarsegraining techniques allow the time and length scales to be further increased. LD and MD simulations can equally well be performed with electronic structure or classical, FF-based energy calculations. The combination of simulation method and energy calculation depends on the physical effects that are sought. LD offers the most detailed information of how condensed matter systems vibrate, but this motion is described within the harmonic approximation for small amplitude displacements about an equilibrium structure. Large amplitude motion like diffusion at finite temperature requires MD simulations to be performed. If the large amplitude motion involves bond breaking, like proton migration in a hydrogen bond, then electronic structure-based energy calculations have to be used. Otherwise the size of the system and the (resulting) time scale of the associated dynamics dictate which type of energy calculation has to be used. Often, LD is used with DFT and MD is used with FFs, the second combination being common for soft matter studies. But this is an over-simplification, as we will demonstrate, as the scope of DFT increases from treating tens of atoms in the past to hundreds of atoms today and thousands of atoms in the very near future. In addition, increased computing resources enable ever larger matrices to be diagonalised, extending the scope of LD methods. Principal component analysis bridges the gap between LD and MD by enabling approximate NMs to be extracted from MD simulations. This chapter expands on the above ideas, illustrating them with recent examples, from both hard and soft matter. The chapter is presented in sections as follows. Section 2.2 relates total energy to forces acting on atoms, starting from the simplest

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example of a harmonic oscillator. Section 2.3 introduces LD starting from the equations of motion for a linear arrangement of beads. In Sect. 2.4, an overview of molecular dynamics is presented. LD and MD approaches are compared in Sect. 2.5. Total energy calculations are described for DFT and FF methods in Sect. 2.6, highlighting the increasing scope of DFT linear scaling methods. Section 2.7 presents a discussion of the relative merits of FF and DFT and the way in which these methods can be combined is highlighted. We have chosen to present simulation methods, LD and MD, before the energy calculations on which they are based because, to a certain extent, the choice when starting simulations should be made in this order, motivated by the physical phenomena of interest. We hope also, in this way, to decouple methods and energy calculations, avoiding established associations like “DFT can only be used for LD”. Section 2.8 discusses how to compare simulated and experimental data concerning, in particular, how the instrument resolution function and other artefacts can be included in the comparison. The chapter concludes with perspectives for the future. The basis for concepts and methods in this chapter originates in part in the field of hard matter, particularly for LD and DFT. Where appropriate, the extension to soft matter is discussed. A wide range of publications exists on simulations and we recommend the following books [1–4] and articles [5, 6] for those who seek more information after reading this chapter.

2.2 Energy to Forces One of the simplest, most pertinent examples of a potential energy well is the harmonic potential, described by a quadratic dependence of the energy E on the displacement x. kx2 . 2 The general expression for the force F acting on a particle in this well is, E=

F =−

dE = −kx, dx

(2.2)

(2.3)

which allows the force constant k to be defined. k=−

dF d2 E = 2. dx dx

(2.4)

The equilibrium position of a particle in this well is x = 0, where the energy is a minimum and the force acting on the particle is zero. If the particle is displaced from equilibrium, it will oscillate perpetually with a frequency ω where

ω2 = and m is the particle mass.

k , m

(2.5)


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This simple development allows us to introduce the “mass effect” which is used to interpret the vibrations of a material when one of the atom types (and possibly just one site) is replaced by another atom type. If the two materials are iso-structural and iso-electronic (e.g. in the case of isotopic substitution, like deuteration), then the “mass effect” on the frequency is ω 2

ω

=

m , m

(2.6)

since the PES, described here by k, is unchanged. The harmonic potential is so useful because it is the first non-zero term in the expansion of the PES about the equilibrium position, the linear term being zero at the equilibrium position. It transpires, in practice, that providing the temperature of a crystal is well below the melting point, the harmonic approximation is good.

2.3 Lattice Dynamics and INS In text books, LD are typically developed in terms of a linear chain of beads. For a monatomic chain, the only vibration is an acoustic phonon (see below). The next level of complexity involves two atoms of different mass, m and M, in a periodic cell of length a, so the distance between atoms is a/2 (see Fig. 2.1). The equations of motion for the two distinct particles are developed by combining foregoing expressions for the force F. F = ma = −kx

(2.7)

mx¨mn = k[xMn + xM(n−1) − 2xmn ]

(2.8)

M x¨Mn = k[xmn + xm(n+1) − 2xMn ].

(2.9)

a

unit cell ‘n-1’

unit cell ‘n’

unit cell ‘n+1’

Fig. 2.1 Diatomic chain of masses m (red) and M (yellow) and periodic cell length a

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ω

Optic branch

Acoustic branch

−π/a

0

π/a

q

Fig. 2.2 Dispersion curves for the diatomic chain

Assuming a plane wave solution for the periodic system of the form xmn (t) = Am ei(qna−ω t)

(2.10)

xMn (t) = AM ei(qna+qa/2−ω t) ,

(2.11)

gives a pair of simultaneous equations that can be expressed in determinant form. 2k − Mω 2 −2k cos qa 2 (2.12) 4 = 0. −2k cos qa 2k − mω 2 2 q is the wavevector corresponding to a wavelength λ = 2π /q. Solving the equations gives two solutions which depend on q, as shown in Fig. 2.2. The dependence of q on ω is called dispersion. The lowest frequency dispersion curve is the acoustic branch for which the frequency ω is zero at q = 0. The wavelength of this mode is infinite, which means all atoms move in phase and there is no restoring forcing (k = 0). A restoring force arises when there is relative motion of atoms, for example when q = π /a, which describes the shortest wavelength vibration in which only the heavy atoms move with frequency ω 2 = 2k/M. q = 0 is called the Brillouin zone centre, or the Gamma point, and q = π /a is the zone boundary. Extending q beyond the zone boundary (by convention q becomes Q) reproduces the dispersion of the first zone, so no new information is obtained, although these Q values are accessible in experiment and they determine, in part, the spectral intensity. The higher frequency dispersion curve is called the optic branch. At the zone centre, light and heavy atoms move in opposite directions and the centre of mass is stationary. At the zone boundary, only light atoms move with frequency ω 2 = 2k/m.


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In terms of simulation methods that will be presented below, it is important to note that a periodic model of length a is able to describe vibrations of all wavelengths. This is because the interaction range is a/2, so the inter-atomic interactions are contained within the periodic cell. In this model, long wavelength vibrations depend only on nearest neighbour interactions. For pedagogical purposes (no new information is obtained), the supercell can be extended to a length of 2a and the determinant can be expressed as 2k − Mw2 −keiqa/2 0 −ke−iqa/2 −ke−iqa/2 2k − mw2 −keiqa/2 0 −iqa/2 2k − Mw2 −keiqa/2 = 0, 0 −ke −keiqa/2 0 −keiqa/2 2k − mw2

(2.13)

showing the general form of the elements in the determinant. New information would be obtained if the extended cell was used to include second neighbour interactions, which remove the zero elements of the matrix. Generalising (2.13) further to three dimensions, the DM has matrix elements of the form, DMi j = √

ki j ei[q·(ri −r j )] , mi m j

(2.14)

where the indices {i, j} run over the N atoms, mass mi , and their Cartesian coordinates ri = {x, y, z}. The matrix kij is called the Hessian. Diagonalising the DM gives 3N q-dependent frequency branches which are composed of 3 acoustic branches and 3N-3 optic branches. The eigenvalues of the DM are ω 2 . The corresponding eigenvectors describe the relative displacements of the atoms/particles, the physical displacements being obtained by weighting by the mass, i.e. dividing each eigenvector component by the square-root of the corresponding atomic mass.

2.4 Direct Way for Calculating Phonons A general and conceptually simple method for calculating phonons is the direct method or supercell approach, which entails a complete determination of the DM. This approach is implemented in the PHONON code [7] used at ILL, France, and a related code, NMScatt [8], developed for bio-molecular systems [9]. The steps of the direct method are as follows: 1. Determine the equilibrium geometry of the periodic system (e.g. crystal cell). 2. Construct a supercell that will include all interatomic interactions (ideally a cube ˚ For soft matter systems, this condition will always be satisfied. of side ∼10 A). When FF are used for the underlying energy and force calculations, the supercell side should be twice the cut-off for long-range VDW and Coulomb interactions.

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3. Generate a set of structures in which each crystallographically inequivalent atom is displaced along the inequivalent Cartesian directions from equilibrium (typically 3n displacements where n is the number of inequivalent atoms) 4. For each structure, calculate the inter-atomic forces. 5. Construct the DM from these forces, the amplitude of displacement and symmetry. 6. Diagonalise the DM for any value of q. All the vibration frequencies obtained this way should be bigger than or equal to zero. Negative frequencies are actually imaginary frequencies since the eigenvalues of the DM are ω 2 and these are indicative of physical or numerical problems, as follows; The equilibrium structure is metastable and an atomic displacement drives the system towards another, stable structure. The equilibrium structure is not well-enough determined. Residual forces should be ˚ when using DFT. With a FF, which is an analytical expression for 108 elements) allowing complex systems to be treated in this way. However, the number of eigenmodes, matched by the dimension of the corresponding, atomistic eigenvectors, can no longer be analysed in a meaningful way by the methods used in smaller, traditionally hard matter systems, for example simple visualisation of modes and their symmetry analysis. Recent calculations on DNA have highlighted this problem [15]. The advantage of


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atomistic eigenvectors is that they can then be analysed on different length scales by summing the atomic displacements over user-defined beads and then projecting these bead displacements onto appropriate vectors. This approach was used to explore the base-pair opening character of vibrational modes, by projecting bead displacements onto the base-pair vector, see Fig. 2.5. By defining nucleotides as beads, the base-pair opening character peaks at 12–15 meV, well below the energy corresponding to the melting of DNA (Tm ≈ 370 K i.e. ∼32 meV). When base molecules are defined as beads, the maximum value of base-pair opening extends up to the energy of melting, showing that this type of vibrational mode can drive DNA melting and related processes. The lower frequency modes are delocalised over 4–5 base-pairs, whereas the higher frequency modes tend to be localised on a couple of base-pairs.

2.5 Molecular Dynamics for INS and QENS A MD simulation describes a perpetual, time- and temperature-dependent evolution of a system. The MD trajectory is a set of structures in which each atom has a position r and velocity v. The next structure in the trajectory is calculated from the instantaneous forces F acting on the atoms which gives the acceleration from (2.1). a=

F . m

(2.17)

According to the simplest integration of the equations of motion, the positions of the atoms in the next structure are a r(t + dt) = r(t) + v(t)dt + dt 2 , 2

(2.18)

v(t + dt) = v(t) + a(t)dt,

(2.19)

and their velocities are

where r, v, and a are vectors describing position, velocity and acceleration respectively. dt is the simulation time step which should be significantly (×5) shorter than the period of the highest frequency vibration. For systems containing H in which the highest frequency vibrations are ∼400 meV, dt is typically 1 fs. For systems composed of heavier atoms, or molecular systems treated as full or partially rigid units, dt can be as long as 5 fs. The total simulated time is equal to dt times the number of simulation steps, so the longer dt, the slower the dynamics that can be investigated (for the same number of steps). dt is, however, a time in which atoms fly ballistically, and if dt is too long, inter-atomic distances can become very short, forces and accelerations very large, and the simulation unstable. A long time step may also lead to problems of energy conservation during MD, which must be checked.

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12

simulation Xray experiment neutron experiment

energy [meV]

10 8 6

3 2.5 2

4

1.5

2

0.5

1 0

0 0

0 0.2 0.4 0.6 0.8 1

5

10

15 Q [nm-1]

20

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30

1 base molecule as bead

0.6

ENERGY GAP

0.4

DNA MELTING

projection

0.8

0.2 nucleotide as bead 0

0

10

20

30

40 50 60 energy [meV]

70

80

90

Fig. 2.5 (Upper panel) Dispersion curves (inset) and spectral dispersion measured by IXS and compared with simulation and neutron scattering data. (Lower panel) Base-pair opening character of eigenmodes as a function of frequency calculated for nucleotides as rigid bodies and base molecules as rigid bodies


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The MD simulation should be started from a structure for which the inter-atomic distances have been optimised, so the inter-atomic forces are close to zero. The system is set in motion by assigning atomic velocities randomly from a Boltzmann distribution of values, which corresponds to the temperature via the kinetic energy of the atoms. During a MD simulation, different combinations of thermodynamic quantities like, volume (V ), energy (E), enthalpy (H), temperature (T ), and pressure (P), can be conserved in simulations referred to as NVE, NVT, NPT or NPH. N is the number of atoms in the simulation box, which always remains constant. NPT simulations are the most realistic in that they correspond to experimental conditions in which pressure and temperature are fixed. In order for T and P to be controlled, the simulated system must be connected to a thermostat and a barostat, which typically act with a user-defined frequency and may perturb the dynamics of interest. The simplest and “safest” simulation ensemble is NVE since there is no coupling to an external bath. A NVE simulation corresponds to a micro-canonical ensemble. Once a system is set in motion, the initial phase of the trajectory is an equilibration phase in which the non-conserved quantities should reach and oscillate around stable values. While the initial structure can be regarded as an equilibrium structure, the initial velocities do not reflect the equilibrium dynamics of the system and the equilibration phase can be considered as allowing energy to be redistributed between NMs of the system. The equilibration phase and production phase can be separated into two simulations, otherwise the start, middle and end of a single simulation should be checked for the stabilisation of macroscopic (and microscopic) quantities. A simulation in equilibrium should give the same results regardless of the part of the trajectory analysed and the ensemble used.

2.5.1 Analysis: Extracting INS and QENS Observables INS observables are extracted from the equilibrated trajectory via time-dependent correlation functions. The VDOS is obtained from the Fourier transform of the velocity auto-correlation function V (t). V (t) = vi (t0 ) · vi (t0 + t).

(2.20)

The pVDOS can be calculated per atom and for each Cartesian direction, which amounts to extracting the NMs without having the phase relation between the atomic displacements. The GVDOS is therefore calculated, as in the LD approach, as the weighted sum of the pVDOS. GVDOS = ∑i σi pVDOSi .

(2.21)

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Fig. 2.6 A two-dimensional Q grid. δ Q = 2π /L where L is the dimension of the simulation box. The smallest ring Q(= q2x + q2y ) < 2π /L does not intercept any reciprocal lattice points. The second ring approximately intercepts 4 reciprocal lattice points, while the largest ring comes close to 8 points

qy

0

0

qx

The atomic mass does not enter this expression, unlike in equation (2.15), because the atomic masses are an integral part of the MD simulation and are therefore included in the pVDOS in this case. S(Q, ω ) is calculated from the atomic trajectories ri (t) via the intermediate scattering functions, (2.6) and (2.7). 1 e−iQ·ri (0) eiQ·r j (t) N ∑i ∑ j=i 1 Iinc (Q,t) = ∑i e−iQ·ri (0) eiQ·r j (t) . N

Icoh (Q,t) =

(2.22) (2.23)

Q can be chosen for a random set of directions, giving a powder averaged response, or a specific direction to probe a specific dynamic response, for example the longitudinal phonons along a DNA helix by aligning Q along the helix axis. For a simulation box of side L, the minimal accessible Q-value is 2π /L and at small Q only a few reciprocal lattice points are sampled (see Fig. 2.6). In practice, when sampling Q-space, the following information is required: initial Q (Qmin ), number of Q points (nQ ), Q-step (dQ, Qmax = Qmin + nQ .dQ), Q-width and the number of Q-points to be sampled in each Q range (nQSample ). For small Q, the number of reciprocal lattice points within the Q-range may be less than nQSample in which case all available points are sampled. If the simulation cell is the crystallographic unit cell, only the zone centre phonons are sampled. By constructing a supercell, longer wavelength phonons commensurate with the supercell are obtained. This approach is in contrast to LD in which all wavelength phonons can be calculated accurately, provided that the supercell contains all interatomic interactions. For QENS observables, the same correlation functions are exploited, the difference lying in the form of the calculated functions. In the case of diffusion, the VDOS is non-zero at zero frequency, this value being proportional to the diffusion constant. For S(Q, w), the elastic peak is (partially) broadened, the width reflecting the diffusion rate. The Q-dependence of the quasielastic signal is used to determine the local (e.g. rotational) or long-range, liquid-like nature of the diffusion.


39

For the analysis of MD trajectories, the code nMoldyn [16] is widely used and is freely available. In addition to the correlation functions indicated above, there are a number of standard tools or functions for analysing MD trajectories, particularly for QENS. For example, the mean square displacement, MSD(t) = [ri (t + t0 ) − ri (t0 )]2 ,

(2.24)

reveals to what extent atoms or groups of atoms move within a confined volume or diffuse freely. In the second case, the MSD shows a linear dependence with time (after an initial ballistic regime at short times), the gradient giving the diffusion constant. The MSD in (2.24) is a “running average” over all possible values of t0 . For a trajectory with NT points, spanning a total time Ttot with a time step dt, MSD(dt) will average over Ntot − 1 points and MSD(Ttot ) will be calculated from a single time difference. For statistical reasons, therefore, the MSD is often limited to t < Ttot /2. By calculating the MSD with respect to the initial structure, rather than a running average over the trajectory, the so-called RMSD indicates to what extent the structure deviates in time from the initial structure. Such a deviation could be interpreted as being due to slow dynamics or simply a simulation not being in equilibrium. It may also reflect the difference between the calculated equilibrium structure and an initial, experimental structure which is related to the accuracy of the energy calculation and the relevance of the initial structure. It is also common to separate the MSD and other correlation functions into rotational and translational components, the latter being derived from the centre of mass trajectories of user-defined units like molecules, monomers or residues in proteins. Structural parameters in a polymer or a protein can also be derived in terms of relative changes with respect to neighbouring monomers. For this purpose, quaternions are very useful and they have been used in the ScrewFit method [17] available in nMoldyn. While, as stated above, NMs cannot be extracted from MD simulations, it is possible to extract principal components [18], by analogy with the LD approach. The vibrational amplitude of an atom is inversely proportional to the force constant k that describes the (harmonic) potential in which the atom sits. From a MD trajectory, it is possible to determine the covariance matrix

σi j = (ri − ri )(r j − r j ),

(2.25)

which can be considered as the inverse of the Hessian of force constants. i and j run over the Cartesian directions. As in the LD approach, the covariance matrix must be mass-weighted and then diagonalised, the eigenfrequencies being the inverse of their LD counterparts. One advantage of principal components is that they can be determined over different lengths of MD trajectories and different time steps, giving access to the corresponding dynamical ranges, and the covariance matrix can be determined for atoms or groups of atoms, corresponding to the length scale of interest.

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Fig. 2.7 Layered structure of a doped poly-aniline system. Poly-aniline chains are at half height in the periodic cell

Example. QENS has been used to study the dynamics of doped poly-aniline (PANI) chains which are of interest for their electronic conductivity [19]. The dopants consist of sulphate head groups, which donate protons to the PANI chains, and longer hydrocarbon chains, which give rise to a layered structure as shown in Fig. 2.7. MD simulations were performed on the nanosecond timescale using the COMPASS FF [20] from which S(Q, w) was calculated using nMoldyn (Fig. 2.8). The excellent agreement between experimental and numerical data was found to be due to the accurate description of the chain dynamics, the MSD analysis showing an increasing amplitude of motion along the chain on moving away from the head group (Fig. 2.9). This dynamical model was also confirmed by the analysis of the Q-dependence of the QENS signal. The dynamics on the sub-nanosecond timescale are therefore dominated by the chain motion. Dynamics of the PANI chains, which may modulate the conducting properties, take place on a longer time scale.


41

Fig. 2.8 Measured (solid lines, IN6 ILL) and calculated (triangles) QENS spectra for PANIDB3EPSA for 4 temperatures ranging from 235 to 340 K. The grey, shaded curve is the resolution function

Fig. 2.9 MSD of protons along the chains of the DB3EPSA molecule, from ‘1’ close to the PANI chain to ‘9’, the maximum distance from the chain

Similar simulation methods have been used recently to study the dynamics of benzene on graphite to gain insight into molecular friction and whether molecules slide over a surface or hop between stable positions [21]. Experimental data in this case came from helium and neutron spin echo (NSE) measurements [22]. The combination of simulation and experiment demonstrates that the benzene molecules slide continuously over the graphite surface.

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2.6 Comparison of LD and MD for INS and QENS The following points present the relative merits of these two methods for calculating the dynamics of a system of atoms: Structure: LD is based on one equilibrium structure, whereas MD starts from an approximate equilibrium structure but may sample several structures corresponding to local minima of the PES. Temperature: The temperature corresponding to a phonon calculation is zero Kelvin since the atoms are at the minima of the PES. The effect of lattice variations with changing temperature can be obtained by repeating the phonon calculation for the corresponding lattice parameters. This is called the quasiharmonic approximation (QHA). In MD, the temperature is determined by the kinetic energy of the atoms. Harmonic approximation: This is strictly imposed in LD calculations whereas higher order contributions to the PES are explored in MD. NMs: They can be obtained for any wave-vector in a LD calculation allowing the full dispersion relations to be determined. In MD, the sampling of the Brillouin zone is limited at small Q and to Q values commensurate with the simulation cell. NMs cannot, in principle, be obtained from MD, although principal components can. Otherwise, the partial VDOS constitute the most detailed information. Imaginary modes: In a LD calculation, these may be physical or numerical in origin. Correlation functions used to analyse MD trajectories do not give rise to negative frequencies. Diffusion: Since LD is based on a single, equilibrium structure, diffusion cannot be handled by this method. MD explores configuration space and is well-suited to studying diffusion. If a reaction coordinate can be identified for the diffusion process, a series of energy calculations can be used to determine the PES. Example. In the case of 3,5-pyridine dicarboxylic acid [23] (see Fig. 2.10), both LD and MD methods were used to study the role of lattice vibrations in proton migration along short N–H..O hydrogen bonds with increasing temperature. LD calculations were performed to determine the molecular vibrations of the stable, low temperature phase of 3,5-PDCA (see Fig. 2.11). Based on the harmonic approximation, LD methods cannot give any insight into anharmonicand temperature-dependent potential energy wells. DFT MD simulations were therefore performed in the temperature range from 50 to 300 K and revealed the measured proton migration effect at 300 K (see Fig. 2.12). Fourier transforming the time-dependence of the N–H bond length showed that lattice modes at 120 cm−1 (15 meV) and 150 cm−1 (18 meV) drive the proton migration and the character of these modes was obtained from the LD calculations (see Fig. 2.11). The modes have


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Fig. 2.10 Crystal structure of 3,5-PDCA showing the short N–H..O bond (atoms: light graywhite-dark gray, respectively) and the normal O–H . . .O gen bond (dark gray-white-dark gray) as broken lines

frequencies which mean they can be activated at room temperature (the observed temperature of complete proton transfer) and they modulate the geometry of the short hydrogen bond. MD simulations predict a QENS signal measurable at high Q, due to the short effective jump distance of the proton. Such a QENS measurement would provide direct experimental evidence of the proposed proton dynamics in the fluctuating, single well potential of the short, strong hydrogen bond. Short hydrogen bonds are thought to play an important role in enzyme catalysis [24]. Mixed, quantum-classical MD simulations have shown a similar proton hopping effect on the picoseconds timescale in HIV-protease [25]. However, these numerical results cannot be tested experimentally on such a complex system, whereas crystalline molecular systems can be investigated by a range of experimental methods, allowing detailed understanding.

2.7 Total Energy Calculations The purpose of this section is to introduce the most useful methods for calculating the total energy of a system of atoms and the forces that act on each, which, we have seen in the foregoing, are central to the LD and MD methods. Figure 2.13 gives a simplified view of a range of computational methods, ranging from the most precise, high-level ab initio methods to mesoscale methods designed for very large

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Fig. 2.11 Measured and LD calculated INS spectra for 3,5-PDCA, fully protonated (upper-left) and hydrogen bond deuterated (lower left). The same Hessian (DFT force constants) is used for both isotopologues, only the atomic masses being different in the two DMs

systems. In ab initio and DFT methods, the electronic structure of the system of atoms is determined. These methods are referred to as first principles because they are constructed on the fundamental interactions of ions and electrons, or electronic structure calculations. In FF and mesoscale methods, the interactions between atoms and particles are simple, parameterised functions which implicitly represent the effect of the electrons in the system. DFT and FF are highlighted since they allow a wide range of systems, typically measured by NS, to be simulated. There are a large number of codes available for DFT (VASP [26], CASTEP [27], ABINIT [28],


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Fig. 2.12 Time-dependence and histograms of the N–H bond length (lower 4 panels). The onset of proton migration in the form of a tail in the histogram towards longer bond lengths is evident at 250 K. At 300 K, two peaks are seen in the histogram, showing two (meta-) stable positions for the proton. The time-dependence of the N–H bond length shows a quasi-periodic oscillation and the Fourier transform reveals two frequencies. The data for the normal O–H..O hydrogen bond is shown in the upper two panels for comparison

Ab initio; 200 cm−1 , 25 meV) comparable to those obtained by NM analysis. The MD spectra did not suffer from negative frequencies, but they did not give full NM information. For this reason, the NM calculation was repeated using the CHARMM FF, giving good agreement with the DFT MD approach and the experimental data at low frequency (see Fig. 2.15). The CHARMM NMs were used to assign the low frequency spectral features. We note, however, that the high frequency spectrum from CHARMM LD did not agree well with DFT and experimental results.

2.9 Comparing Simulated and Experimental Results Experimental data are normally corrected for instrument-related effects like scattering from the sample holder and environment and multiple scattering due to the sample shape and quantity. It is however difficult to deconvolute the measured signal with the instrument resolution function due to the finite energy transfer window of the instrument, so comparison of simulated and experimental data requires the simulated data to be convoluted with the resolution function.

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Fig. 2.15 Measured and calculated low frequency spectra of collagen, showing the contributions of collagen (PPG) and water from the DFT MD approach and the accurate calculation using the CHARMM force field

LD calculations give a set of delta functions for each wavevector. The VDOS obtained by summing over many wavevectors may show continuous intensity distribution due to dispersion, but there may also be sharp features or peaks that are not observed as such in the experimental spectra. Calculated spectra should therefore be convoluted with the known instrument resolution function, which often varies in width with energy transfer. In spectra calculated from MD simulations, the energy resolution is determined by the length of trajectory and the highest energy transfer is given by the time step, since the time-dependent correlation functions ((2.20), (2.22) and (2.23)) are Fourier transformed to obtain spectra. The time step cited here is not necessarily the time step of the simulation, which is typically 1 fs, but rather the time step between frames or structures that are saved in the trajectory file. The reason for this difference is that saving every frame from a 1-ns simulation would give ∼106 frames in an unreasonably big trajectory file, whereas ∼103 frames are usually adequate for analysis of MD output. Moreover, dynamics on the femtosecond timescale are normally not relevant when dynamics on the nanosecond timescale are being investigated. Two strategies are possible when Fourier transforming the time-dependent correlation functions. One is to use the window function associated with the Fourier


53

transform to reproduce the spectral resolution in the frequency domain. The other entails obtaining the highest possible resolution from the Fourier transform and then convoluting with the known or measured instrument resolution function. While including the instrument resolution may not be critical for comparing simulated and experimental INS data, it is very important for QENS data, especially when the width of the QENS signal is comparable to the width of the resolution function. We note that for diffusion processes the NSE technique measures the intermediate scattering function ((2.22) and (2.23) for coherent and incoherent scattering) directly, as introduced in Chap. 1, and there is no need to perform any Fourier transform. Rather than making any corrections to the experimental data, it is now possible, for powders and liquids, to combine a calculated S(Q, ω ) map with an instrument simulation in the McStas code [39] to perform complete virtual experiments. In this neutron, ray-tracing code, neutrons fly through optical elements (e.g. guides, monochromators, etc), arriving at the sample to be scattered. The probability of scattering at different energy transfers and in different directions is determined by the calculated S(Q, ω ) and the probability of a scattering event taking place is related to the path length of the neutron in the sample. It is possible, therefore, in virtual experiments to investigate, in detail, self-screening and multiple scattering [40]. Artefacts in the measured S(Q, ω ) may also arise from multiple scattering between the sample and its environment (e.g. cryomagnet) and these effects can also be simulated by treating the sample environment as a second sample.

2.10 Conclusion In the context of INS and QENS, we have presented two methods, LD and MD, for calculating S(Q, ω ) and related quantities based on total energy calculations, using DFT and FF methods. Four combinations of the methods and total energy calculations are possible and can be used in appropriate circumstances. However, LD tends to be applied to simpler systems in terms of having higher symmetry and a smaller number of atoms since the dispersion of phonon branches can be investigated, numerically and experimentally, in detail. In view of the typical system size, DFT methods can be applied and, indeed, are essential for correctly handling the electronic structure of materials composed of electron-rich atoms. FF methods tend to be reserved for larger systems which are beyond the scope of DFT. For these systems, complex by virtue of their size, dispersion relations at the atomic level are often not of primary interest and MD is used to extract dynamical quantities. To summarise, typical approaches to analysing INS data are LD + DFT and MD + FF, but this situation is evolving. QENS data can only be analysed with MD, combined with DFT or FF methods. DFT offers high accuracy across the frequency range, possibly having lower accuracy for low frequency vibrations, where the uncertainties in the numerical solution of Schrödinger’s equation are no longer negligible. FF methods, on the

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other hand, can offer good precision at low frequency since (1) the methods are analytical and (2) the collective modes depend on the average values of many interatomic force constants. Imprecision in specific force constants can become evident at higher frequency and, overall, FF methods offer much lower accuracy than DFT. The validity of a simulation is tested with experimental data, either by comparing reduced experimental data with numerical data or by including the calculated scattering function in an instrument simulation. Once good, or adequate, agreement has been obtained between simulation and experiment, the numerical data have to be analysed. In view of the number of atoms involved, DFT results are typically analysed at the atomic level, although the increasing scope of (linear scaling) DFT is changing this way of analysing data. For FF simulations, longer length scales are often more appropriate, requiring displacement vectors to be integrated over groups of atoms or translational and rotational correlation functions to be determined for such groups. Theoretical input may be required to define new correlation functions or, more generally, ways of analysing the numerical data so as to extract key aspects of structure and dynamics on the appropriate time and length scales. Using atomistic simulations to understand experimental data can be regarded as advanced data analysis. On the other hand, atomistic simulations should be performed before experiments and used to justify beam-time requests at central facilities. In this case, virtual experiments, which include a description of the instrument, allow a complete investigation of the proposed experiment, including the energy and momentum transfer ranges and resolution and sample geometry and environment. This type of simulation should become more widespread in the future.

References 1. Allen MP, Tildesley DJ (1987) Computer simulation of liquids. Clarendon, Oxford 2. Frenkel D, Smit B (2002) Understanding molecular simulation. Academic, San Diego 3. Cramer CJ (2004) Essentials of computational chemistry, theories and models. Wiley Ed, New York 4. Springborg M (2000) Methods of electronic structure calculations. Wiley, New York 5. Sutmann G (2002) Classical molecular dynamics in quantum simulations of complex manybody systems: from theory to algorithms, lecture notes. In: Grotendorst J, Marx D, Muramatsu A (eds) John von Neumann Institute for Computing, Julich, NIC Series, vol 10. ISBN 3-00009057-6, pp 211–254 6. Allen MP (2004) Introduction to molecular dynamics simulation in computational soft matter: from synthetic polymers to proteins, lecture notes. In: Norbert Attig, Kurt Binder, Helmut Grubmuller, Kurt Kremer (eds) John von Neumann Institute for Computing, Julich, NIC Series, vol 23. ISBN 3-00-012641-4, pp 1–28 7. Parlinski K (1999) Calculation of phonon dispersion curves by the direct method. Am Instr Phys Conf Proc 479:121–126 8. Merzel F, Fontaine-Vive F, Johnson MR (2007) NMscatt: a program for calculating inelastic scattering from large biomolecular systems using classical force field simulations. Comput Phys Commun 177:530–538 9. Meinhold L, Merzel F, Smith JC (2007) Lattice dynamics of a protein crystal. Phys Rev Lett 99:138101


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10. Plazanet M, Fontaine-Vive F, Gardner KH, Forsyth TV, Ivanov A, Ramirez-Cuesta AJ, Johnson MR (2005) Neutron vibrational spectroscopy gives new insight into the structure of poly (p-phenylene terephthalamide). J Am Chem Soc 127:6672 11. Plazanet M, Fukushima N, Johnson MR, Horsewill AJ, Trommsdorff HP (2001) The vibrational spectrum of crystalline benzoic acid: inelastic neutron scattering and density functional theory calculations. J Chem Phys 115:3241 12. Zbiri M, Johnson MR, Mutka H, Payen C, Schober H (2010) Phonon control of magnetic relaxation in the pyrochlore slab SCGO through rocking motion of the Kagome-plane triangles. Phys Rev B81:104414 13. Johnson MR, Koza MM, Capogna L, Mutka H (2009) Probing coupling between ‘rattling’ and extended lattice modes using time-of-flight neutron scatteringcombinedwith ab-initio calculations—introducing the PALD method. Nucl Instr Meth A 600:226–228 14. Koza MM, Johnson MR, Viennois R, Mutka H, Girard L, Ravot D (2008) Breakdown of phonon glass paradigm in La- and Ce-filled skutterudites. Nat Mater 7:805–810 15. Merzel F, Fontaine-Vive F, Johnson MR, Kearley GJ (2007) Atomistic model of DNA: phonons and base-pair opening. Phys Rev E 76:31917–31921 16. Rog T, Murzyn K, Hinsen K, Kneller GR (2003) nMoldyn: aprogram package for neutron scattering oriented analysis of molecular dynamics simulations. J Comp Chem 24:657–667. http://dirac.cnrs-orleans.fr/plone/software/nmoldyn/ (Accessed 14 December 2011) 17. Kneller GR, Calligari P (2006) Efficient characterization of protein secondary structure in terms of screw motions. Acta Cryst D62:302–311 18. Brooks BR, Janezic D, Karplus M (1994) Harmonic analysis of large systems. I. Methodology. J Comp Chem 16:1522 19. Sniechowski M, Djurado D, Bee M, Gonzalez MA, Johnson MR, Rannou P, Dufour B, Luzny W (2005) Force field based molecular dynamics simulations in highly conducting compounds of poly(aniline). A comparison with quasi-elastic neutron scattering measurements. Chem Phys 317:289–297 20. Sun H (1998) COMPASS: an ab initio force field optimised for condensed phase application – overview with detail on alkane and benzene compounds. J Phys Chem B 102:7338 21. Fouquet P, Johnson MR, Hedgeland H, Jardine AP, Ellis J, Allison W (2009) Molecular dynamics simulations of the diffusion of benzene sub-monolayer films on graphite basal plane surfaces. Carbon 47:2627 22. Hedgeland H, Fouquet P, Jardine AP, Alexandrowicz G, Allison W, Ellis J (2009) Measurement of single-molecule frictional dissipation in a prototypical nanoscale system. Nat Phys 5:561 23. Fontaine-Vive F, Johnson MR, Kearley GJ, Cowan J, Howard JAK, Parker SF (2006) Phonon driven proton transfer in crystals with short strong hydrogen bonds. J Chem Phys 124:234503 24. Cleland WW, Kreevoy MM (1994) Low barrier hydrogen bonds and enzymic catalysis. Science 264:1887–1890 25. Trylska J, Grochowski P, McGammon JA (2004) The role of hydrogen bonding in enzymatic reaction catalysed by HIV-1 protease. Protein Sci 13:513–528 26. VASP.http://cms.mpi.univie.ac.at/vasp/ 27. CASTEP. http://www.castep.org/ 28. ABINIT. http://www.abinit.org/ 29. Gaussian. http://www.gaussian.com/ 30. CHARMM. http://www.charmm.org/ 31. AMBER. http://ambermd.org/ 32. NAMD. http://www.ks.uiuc.edu/Research/namd/ 33. DL POLY. http://www.cse.scitech.ac.uk/ccg/software/DL POLY/ 34. Kohn W (1996) Density functional and density matrix method scaling linearly with the number of atoms. Phys Rev Lett 76:3168–3171 35. Siesta.http://www.xmarks.com/site/www.uam.es/departamentos/ciencias/fismateriac/siesta/ 36. Hine NDM, Haynes PD, MostofiAA, Skylaris C-K, Payne MC (2009) Linear-scaling densityfunctional theory with tens of thousands of atoms: expanding the scope and scale of calculations with ONETEP. Comput Phys Commun 180:1041. http://www2.tcm.phy.cam.ac. uk/onetep/ (Accessed on 3 December 2011)

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37. GULP.http://projects.ivec.org/gulp/ (Accessed 14 December 2011) 38. Fontaine-Vive F, Merzel F, Johnson MR, Kearley GJ (2009) Collagen and component polypeptides: low frequency and amide vibrations. Chem Phys 355:141–148 39. McStas.http://neutron.risoe.dk/ 40. Farhi E, Hugouvieux V, Johnson MR, Kob W (2009) Virtual experiments: combining realistic neutron scattering instrument and sample simulations. J Comp Phys 228:5251–5261 41. Hartree DR (1928) The wave mechanics of an atom with a non-coulomb central field. Part I. Theory and Methods. Proc Cambridge Phil Soc 24:89–110 42. Fock V (1930) Z Physik 61:126 43. Middendorf HD, Hayward RL, Parker SF, Bradshaw J, Miller A (1995) Vibrational neutron spectroscopy of collagen and model polypeptides. Biophys J 69:660–673

Part I

Macromolecules: Polymers

Chapter 3

Basic Modes of Motion in Polymers Barbara J. Gabrys and Toshiji Kanaya

In memory of Dr L Kay Nicholson

3.1 Introduction Neutron scattering came of age with the award of the Noble Prize to Shull and Brockhouse in 1994 “for pioneering contributions to the development of neutron scattering techniques for studies of condensed matter” [1]. Three years earlier, de Gennes was awarded the Nobel Prize in physics “for discovering that methods developed for studying order phenomena in simple systems can be generalised to more complex forms of matter, in particular to liquid crystals and polymers” [2]. The latter quotation can indeed be applied to studies of the structure and dynamics of polymers using neutron scattering. In this chapter, we outline the basic modes of motion for polymers and illustrate, using a historical approach, how neutron scattering has helped us learn about their complex dynamical characteristics. For more detailed considerations, the reader is referred to Higgins and Benoit [3]. Typical polymer motions range all the way from the local picosecond (ps) motions of small chemical groups to the long-range wholechain motions over several hundred nanoseconds (ns). We will provide examples of inelastic (INS) and quasielastic neutron scattering (QENS) experiments on pure polymers and refer the reader to other chapters in this book for the more complex polymeric systems such as blends (Chaps. 4 and 6), polymer electrolytes (Chap. 5) and polymer nanocomposites (Chap. 13).

B.J. Gabrys () Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK e-mail: [email protected] T. Kanaya Division of Multidisciplinary Chemistry, Polymer Materials Science, Kyoto University, Go-kasho, Uji, Kyoto-Fu 611-0011, Japan e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 3, © Springer Science+Business Media, LLC 2012

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3.2 Fundamentals of Neutron Scattering for Polymers In neutron scattering from the solid state, three types of spectra may be observed: elastic, inelastic (INS) and quasielastic (QENS) (see Fig. 3.1). The etymology of the word “quasi-elastic” goes back to the scattering from liquids where some correlations existing between the nearest neighbours appear as a broad peak in the measured spectra. Since there is no strictly elastic scattering present (atoms in a liquid having no defined average position over a long time), this feature has been named quasi-elastic. In solids, where there is elastic scattering, this term refers to the broadening of the elastic line due to stochastic inelastic processes such as localized motions.

3.2.1 Scattering Laws: Self- and Collective Dynamics As already presented in Chap. 1, the quantity measured in an inelastic neutron scattering measurement of dynamics resulting from an inelastic or a quasi-elastic event, is the double-differential scattering cross-section. This is related to the incoherent and coherent scattering laws Scoh (Q, ω ) and Sinc (Q, ω ) by:

Fig. 3.1 Anatomy of a neutron scattering spectrum, showing elastic, quasi-elastic and inelastic parts. The inset shows the inelastic component in more detail (figure courtesy of W. Zajac)

3 Basic Modes of Motion in Polymers

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Gd(r,t)

Gs(r,t) t=0

t=0

r=rj (0)

r=rj (0)

t=t

t=t

r=ri (t)

r=ri (t) t=0

t=0

r=ri (0)

r=ri (0) t=t

r=rj (t)

t=t

r=rj (t)

Fig. 3.2 Schematic representation of the van Hove time-space correlation function, G(r,t) = Gs (r, t) + Gd (r, t). The dotted lines represent the paths of two atoms (labelled i and j) as a function of time t (note: the lines do not represent polymer chains). Gs (r, t) is the self-correlation function, i.e. of an atom with itself; Gd (r, t) is the distinct correlation function, i.e. of an atom with other atoms

kf ∂ 2σ = N [σcoh Scoh (Q, ω ) + σinc Sinc (Q, ω )] , ∂Ω ∂E 4π h¯ ki

(3.1)

where σcoh and σinc are the coherent and incoherent scattering cross-sections, respectively, and N is the number of independent scatterers (single atoms). In the study of polymer dynamics, it is common to Fourier transform the scattering laws into the time domain, expressed as the distinct and self-intermediate scattering functions, Icoll (Q,t) and Iself (Q,t). These are then directly related to the van Hove correlation functions, which are expressed in real space, Gdistinct (r,t) (also known as Gd (r,t) and Gself (r,t), respectively). Figure 3.2 illustrates their meaning. The interpretation of these correlation functions is clear in the classical limit, where for all equivalent nuclei G(r, t) is the probability that, given a particle i at the origin (r = 0) at time t = 0, any particle j is found at r at a later time t. Correspondingly, Gs (r, t) yields the probability that the same particle is found at r at time t. Consequently, the measurement of Icoll (Q,t) and Iself (Q,t) in a neutron scattering experiment gives information on the motion of a particle with respect to any other particle in the system (collective dynamics), and the motion of a particle with respect to itself (self dynamics), respectively. In the Gaussian approximation, the distribution of particle positions r i (t) and rj (0) is described by a Gaussian distribution. Then the incoherent and coherent intermediate scattering functions can be written in terms of Q2 as: 1 Q2 2 (r exp − (t) − r (0)) , i j ∑ N∑ 6 i j 1 Q2 2 Iself (t) = ∑ exp − (ri (t) − ri (0)) . N i 6

Icoll (t) =

(3.2)

(3.3)

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Note that the time-dependent mean square displacement can be estimated from the incoherent scattering measurements. Also, the frequency-dependent mean square displacement can be obtained from the measurements of incoherent dynamic scattering law within the Gaussian approximation. In a neutron experiment, we typically measure the sum of incoherent and coherent scattering contributions and we use either deuterium labelling or polarised neutrons to separate the responses. In the first case, owing to the difference in scattering cross-sections between the hydrogen and deuterium isotopes H = 80.27 barn, σ D = 2.05 barn, σ H = 1.76 barn, σ D = 5.59 barn), the self(σinc inc coh coh intermediate scattering function can be accessed by measuring a fully hydrogenated sample (there being only a relatively small contribution from coherent scattering) and the collective dynamics by using deuterated samples. Isotopic labelling is of paramount importance for multi-components systems, when studying for example polymer blend dynamics (see Chap. 4). In the second case, since the neutron possesses a magnetic moment, it can interact with unpaired electron spins. This provides a means to align neutron spins with respect to the external magnetic fields, i.e. to polarise them. This can be done either as classical polarisation analysis for wide angle scattering [4] or by neutron spin-echo [5]. The separation of the coherent and incoherent parts of the scattering from a non-magnetic system with randomly oriented nuclear spins is of interest for the studies of both structure and dynamics in the same experiment, as in the studies of solid polymer electrolytes [6].

3.2.2 Polarisation Analysis and Neutron Spin-Echo Investigation of motions in hydrogen-rich polymer samples through QENS can be successfully analysed only when the elastic peak is free from any coherent component [7]. In crystalline samples, one achieves this by eliminating those scattering angles where Bragg peaks appear, either by judicious selection of detector positions or by using software. Poorly ordered materials or those with short range order give unresolved (smeared out) coherent spectra, and necessary assumptions and approximations are required to remove the coherent part, making quantitative analysis challenging. The magnitude of this problem for many interesting polymers can be seen from the example in Fig. 3.3. A decent quality QENS experiment with polarisation analysis would require long runs and is therefore rarely, if ever, performed. However, we can still benefit from the experimentally determined ratio of incoherent and coherent scattering as a function of the scattering angle. Furthermore, such experiments provide information on the structural correlations within the sample as is shown in Fig. 3.3, where two peaks are visible in the coherent ˚ −1 , signal, assigned to inter- and intra-molecular correlations (peaks at 0.9 and 2.2 A respectively). In wide angle diffraction with polarisation analysis, cross-sections are measured for scattering from an initial neutron state of specified momentum and spin to a final neutron state of specified momentum and spin. The cross-sections measured connect


63

Fig. 3.3 Experimental separation of coherent and incoherent scattering from partially deuterated poly(methyl methacrylate) (PMMA) [86], as measured on a wide angle neutron diffractometer with polarisation analysis at the Institut Laue-Langevin, France. Note the relative ratio of intensities: the highest intensity (top) is the sum of coherent and incoherent signals, the incoherent contribution is the nearly Q-independent signal (middle), and the coherent contribution is the lowest intensity signal (bottom) which contains the structural signatures (i.e. the peaks)

two neutron spin states [8]. Full polarization analysis (in all three xyz directions) is important in the study of magnetism, but for polymers, which are mostly nonmagnetic, we consider only polarization in the z-direction. The interaction of the ˆ I · σ ), can be expressed in terms of neutron spin I with the nuclear spin σ , B( the density matrix ρ [9] and the differential scattering cross-section dσ /dΩ. The magnetic field Bˆ is described in terms of the scattering matrix which describes the transformation of the neutron spin during the scattering process. The differential scattering cross-section can be written in terms of the scattered intensity with and without spin-flip, I ↑↓ and I ↑↑ , respectively, as: dσ 2 1 = I ↑↓ + I ↑↑ = |B|2 I(I + 1) + |B|2 I(I + 1). dΩ 3 3

(3.4)

Hence, the outgoing scattering from unpolarized nuclear spins is always one third without and two thirds with flip of the neutron spin from the polarized beam. Note that the polarization is not measured at all in this type of experiment, only the cross-sections connecting two neutron spin states I ↑↑ and I ↑↓ [9, 10]. Neutrons are thus scattered from the sample with or without spin-flip. The detectors register this change of spin direction through a “filter” of polarization analysers. The latter accept neutrons with the spin up only. During the experiment, the spin-flip (with the

64


spin-flipper on) and non-spin-flip (spin-flipper off) modes are alternated, counting neutrons separately for each mode. Then one third of spin-incoherent scattering is registered in the non-flip channel and the remaining two thirds in the spin-flip channel. The scattered intensities I ↑↑ and I ↑↓ have to be corrected for a finite flipping ratio R (measured separately using a quartz sample) and multiple scattering [11]: ↑↑ Icorr = I ↑↑ − (R − 1)−1(I ↑↓ − I ↑↑),

(3.5)

↑↓ Icorr = I ↑↓ + (R − 1)−1(I ↑↓ − I ↑↑).

(3.6)

The coherent scattering Icoh is related to the measured spin-flip and non-spin-flip scattering [12] by 1 ↑↓ ↑↑ Icoh = Icorr − Icorr 2

(3.7)

and the incoherent scattering Iinc to the spin-flip scattering only 3 ↑↓ . Iinc = Icorr 2

(3.8)

The final step is to normalise the coherent scattering cross-section to the incoherent one, which yields an absolute cross-section in barns:

∂ σcoh σinc Icoh . = ∂Ω 4π Iinc

(3.9)

In the first approximation, the respective Debye–Waller factors cancel out in this approach. In the neutron spin-echo (NSE) technique, first proposed by Mezei [5] and for which polarised neutrons are necessary, very small energy changes (ΔE) on scattering are detected by counting the difference in the number of precessions of the neutron spins while traversing identical magnetic guide fields before and after being scattered by a sample. Very small means ∼1 neV which is an improvement of nearly three orders of magnitude in energy resolution over traditional backscattering spectrometers (∼1 μ eV). The realisation of NSE as a technique provided a unique opportunity to probe the collective dynamics of polymers as well as allowing the confirmation of de Gennes’ predictions of tube-like confinement of polymer chains [13] and reptation dynamics at long times and large length scales. The basis of this technique and its applications has been described many times due to its importance for polymer science. We recommend reading one of the first, exceptionally clear articles [14] and a more recent comprehensive review of NSE and its application to polymer science [15]. The design of the first instrument, IN11, at the ILL was something of a conceptual novelty in time-of-flight techniques. It measures directly the change of neutron velocity during the scattering process via the Larmor precession of a neutron in an external magnetic guide field B.


65

It has a polariser and analyser placed in front of a movable detector(s). The measured quantity is the change in neutron polarisation as a result of scattering from the sample. If the dynamics are within the time-window of the spectrometer, the measured final neutron polarization Pz will be different to its initial value. The measured polarisation (for a quasimonochromatic beam and after appropriate corrections and normalization) is given by Pz = I(Q,t)/I(Q,t = 0),

(3.10)

where the Fourier time t is proportional to B. The quantity I(Q,t = 0) is the static structure factor, also designated S(Q), which would be measured in a diffraction experiment.

3.2.3 Assumptions Underlying the Theory and Models of Neutron Scattering for Polymers We are now in a position to review the approximations which are needed to extract information from neutron scattering experiments. In an idealized scattering experiment, an incident mono-energetic beam of particles (without spin) is represented by a plane wave being scattered from a target of N independent, bound particles; each of them is a scattering centre. A particle, localized at position r, can be represented by a δ -function potential, which on the jth nucleus, located at R j , is equal to V j (r − R j ). Assuming that contributions to the potential from all individual scattering centres are additive, the potential is given by V (R) ∝ ∑ fk j (r)δ (r − R j ).

(3.11)

all j

This type of potential falls to zero faster than 1/R; in the first Born approximation, the constant of proportionality in the above equation is given by 2¯h2 /m, and the potential V (R) is termed the Fermi pseudopotential. The scattering amplitude of the jth atom, fk j (r), depends on the wavevector of the scattered particle and is one of characteristic quantities measured. In the actual scattering experiment, there is a distribution of energies (wavelengths) and the scattering is treated as isotropic so that fk (r) does not show any angular dependence despite being labelled with k (whereas the equivalent quantity for X-ray scattering, for example, is k dependent). This holds under the assumption that the particle wavelength λ (r − R j ) for all values of r at which V (R) is sensibly finite. Such a model is clearly applicable to neutrons, whose wavelength of around 10−10 m is much greater than the nuclear radius of 10−15 m. We take a convenient assembly of atoms as scattering centres and assume that the neutron is scattered only once before leaving the sample. In reality, it can scatter more than once, and in order to correct for it, multiple scattering corrections (which

66


can be large if the experiment is not well designed) are applied [16]. In practice, polymer scientists avoid this situation by making the sample a 10% scatterer (i.e. 90% of the incident neutrons pass through the sample without scattering); the mean free path of neutron in such a sample then ensures that there is a high probability that it scatters only once. Polymer chains are very mobile even in the solid state. Therefore, the analysis of laws governing their dynamics usually relies on the assumption of the dynamical decoupling of various kinds of motions that chains may undergo. In consequence, translations, rotations and vibrations are treated as occurring independently, assuming that the timescales over which they occur are very different. The total scattering law is then a convolution of respective translational, vibrational and rotational laws and an experimental separation is obtained by choosing the proper energy and time domains and/or temperature at which some motions are not active. A trademark of translational motion in the observed spectrum is the broadening of the elastic peak, while rotational motion is characterized by a sharp central peak with a broad “skirt” (in solids). Vibrational motions produce well-defined peaks shifted away from the elastic peak by an energy corresponding to the corresponding vibrational transition. In order to have a closer look at this approximation, let us explore the expression for the incoherent intermediate scattering function (Equation 1.7): Iself (Q,t) =

1 exp{−iQrk (0)} exp{iQrk (t)}. N∑ k

(3.12)

Here we follow the reasoning for the case of the incoherent intermediate scattering function [3], which rests on the expression of the time-dependent position vector r in terms of three vectors: rk (t) = ck (t) + bk (t) + uk (t),

(3.13)

where the vector ck describes translation of the centre of mass, bk describes rotations about the centre of mass and uk gives the distance of the particle from its average position. Then (3.12) becomes: Iself (Q,t) =

1 exp(−iQ(ck (0) − ck (t))) exp(−iQ(bk (0) − bk (t))) N∑ k × exp(−iQ(uk (0) − uk (t))).

(3.14)

If these three vectors are assumed independent, we can replace the average value of the product in the above equation by the product of three averages: Iself (Q,t) =

1 exp −iQ(ck (0) − ck (t)) ⊗ exp−iQ(bk (0) − bk (t)) N∑ k ⊗ exp−iQ(uk (0) − uk (t)),

(3.15)


67

where the symbol ⊗ stands for convolution. In shorthand notation, the above equation can be written as Iself (Q,t) =

1 I tr (Q,t) ⊗ I rot (Q,t) ⊗ I vib(Q,t) N∑ k

(3.16)

with I tr (Q,t), I rot (Q,t) and I vib (Q,t) being contributions to the scattering of the translational, rotational and vibrational motions of scattering centres. If any of the motions is absent, for example vibration, then uk (t) is zero and the corresponding term is unity. The separation (3.16) is easily written for the coherent intermediate scattering function if we take a double sum over scattering centres k = l (Equation 1.6). The assumption that the intermediate scattering function can be treated as a convolution of three independent motions is crucial and translates into dynamical decoupling of three types of motion. It has to be examined carefully for each individual case since a priori it is not obvious that it holds. Hence, one needs a judicious choice of several factors: a polymer (for example one with a rotating group that is far away from the main chain, as is the case for the ester methyl group in PMMA), a temperature (where some motions are frozen) and frequency domain (that can be covered by an appropriate spectrometer).

3.3 Whole-Chain Dynamics: Polymer Solutions and Melts In the following sections, we will present the basic modes of motions in polymers, which are diverse and many, and show with examples the key role played by neutron scattering in understanding such dynamical modes. In this section, we treat the dynamics of polymers in solutions and melts, and then in Sects. 3.4–3.6, we present the dynamics in the solid state. There are several reasons why the motion of the whole chain was the initial focus of theoretical and experimental investigations. One is practical – polymers are of industrial importance and understanding their behaviour during processing as they melt ultimately helped a more efficient production. It is also recognized that macromolecules play a pivotal role in biology, and in a cell they “swim” in water. Polymers dissolved in solutions, especially dilute ones, allow for the validity of several theoretical approximations as the interaction between chains can be neglected. Moreover, with a clever combination of solvent and selective deuteration, a part of the polymer molecule can be highlighted or screened. Once the behaviour in dilute solutions is understood, one can then tackle semi-dilute and concentrated solutions, and eventually the bulk. Another link between the microand macro-world is the viscoelasticity of polymers. Hence, the seminal book of Doi and Edwards [17] on the theory of polymer dynamics focused on understanding the properties of polymeric liquids. As pointed out by the authors, it is easier to

68


understand the properties of such liquids rather than liquids composed of small molecules: in the case of polymers the multiplicity of interactions actually proves advantageous. Each polymer has only small, additive interactions with many other identical polymers, resulting in averaging of its environment. Therefore, it is deemed sufficient to deduce the mean properties; the macroscopic behaviour is dominated by the large-scale properties of the polymer molecule. This approximation works astonishingly well, and it was only with the progress of experimental techniques that it was possible to refine this fundamental model. In this chapter, we present the basic motions related to whole-chain dynamics, namely the Rouse Model and reptation dynamics. We use polymer melts as examples, but the general principles also apply to solutions. Data on more complex polymer systems can be found in Chaps. 4 and 6.

3.3.1 The Rouse Model The Rouse model was the first description of slow polymer motion in a liquid and a predecessor of the coarse-graining procedures which nowadays begin to dominate computer modelling of bio-macromolecules. In this model, a polymer chain is represented by NR beads of mass m connected by Hooke’s springs of length l. In the length scale between the segment length and the entanglement distance, polymer motions are driven by entropic forces. The effect of the surrounding solvent molecules is described by a frictional drag on the beads [18]. The modification of the Rouse model by Zimm [19] included hydrodynamic coupling between the beads, thus making it more realistic. The intermediate scattering functions were calculated for the Zimm model by Pecora (for light scattering) [20] and by de Gennes [13, 21]. In order to connect them with the outcome of a neutron spin-echo measurement, we take the following ratio for the incoherent scattering: 2 Iinc (Q,t) = exp − √ (ΩR (Q)t)1/2 . Iinc (Q, 0) π

(3.17)

The characteristic rate of the Rouse dynamics ΩR is given by ΩR =

1 l2 k B T Q4 , 12 ξ

(3.18)

where l and ξ are the segment length and the monomer friction coefficient, respectively. For the coherent scattering, we have Icoh (Q,t) = Icoh (Q, 0)

∞ 0

du exp(−u) exp −(ΩR (Q)t)1/2 h(u(ΩRt)−1/2 ) ,

(3.19)


69

Fig. 3.4 Incoherent (top) and coherent (bottom) intermediate scattering functions measured by NSE, for PDMS melts at T = 100◦ C. The data are scaled to the Rouse variable. Solid curves represent fits with the respective dynamics structure factors [23]

where

2 ∞ cos(xy) dx (1 − exp(−x2 )). (3.20) h(y) = π 0 x2 Both the coherent and incoherent intermediate scattering functions are scaled by the Rouse variable Q2 3kB T l 2 t Q2 l 2 √ 1/2 (ΩRt) = ≡ Wt, (3.21) 6 ξ 6 where W is the Rouse frequency [15]. It follows that, for polymer dynamics described in terms of the Rouse model, the intermediate scattering functions observed at different values of the momentum transfer vector Q fall on a master curve as can be seen from Fig. 3.4. The applicability of the Rouse model is limited to short chain polymers or to short time relaxation of long chains in the melt.

70


Fig. 3.5 Sketch of reptation motion

The polymer poly(dimethyl siloxane) (PDMS) in the melt is a good example of the Rouse motion. PDMS has little entanglement constraints, is highly flexible, has a low monomeric friction coefficient, and the time of motion falls within the range accessible by NSE [22, 23]. With this technique, one typically measures protonated polymers in deuterated solvents to provide contrast for a coherent scattering signal. Since most incoherence is due to the neutron spin, it is possible to select only the coherently scattered neutrons. In order to measure the incoherent scattering from a polymer melt, the following trick is used: measure deuterated polymer chains containing 12% of short protonated sequences at random intervals. The randomly distributed sequences give no constructive interference of partial waves scattered from different sequences, so that the observed signal was incoherent (see above). The results for such a sample of PDMS are shown in Fig. 3.4. There is good agreement between the experimental results and fits with theoretical functions. Moreover, these results demonstrated that the characteristic Rouse relaxation rate ΩR is proportional to Q4 [23] and the time dependence of the incoherent intermediate function Iinc(Q,t) can be described by a stretched exponential function with exponent of 0.5. Hence, the validity of the Rouse model for describing short chain dynamics in polymer melts was confirmed.

3.3.2 Reptation The concept of reptation which describes polymer motion in a tube, introduced by Edwards, was used by de Gennes [13] to solve the problem of a polymer moving through a mesh of entanglements. Since the polymer cannot cut across such a network, it can only move by wiggling around other polymer chains. The space allowed for this motion can be described by a tube of a given diameter: as a polymer chain moves a little, one can imagine its “head” moving forward in the front section of this tube and its tail disappearing from the end section, as schematically shown in Fig. 3.5.


71

Reptation can be divided into three regimes according to characteristic times of the motion: 1. Rouse regime: a short time region where chains do not feel the restriction due to a tube with a diameter d. This regime is limited by a time τe at which the mean square displacement of the segment reaches d 2 (t < τe ). 2. Local reptation regime: this is divided into two sub-regimes by the Rouse time τR . In the time interval of τe < t < τR , the Rouse modes of polymer chains equilibrate within the tube while for t > τR , the chain diffuses along the tube. 3. Diffusion regime: for times longer than the terminal time τd (t > τd ), at which chains completely creep out of the original tube, we expect normal diffusion behaviour. Thus, the reptation model predicts three (strictly speaking four) dynamic regimes for segmental diffusion. The coherent and incoherent intermediate scattering functions for Rouse and reptation dynamics have been calculated by a number of researchers. Firstly, de Gennes [24] calculated them for the long-time behaviour of the local reptation, neglecting the initial Rouse behaviour. Subsequently, Ronca [25] employed a generalised Langevin equation to describe the cross-over from the initial Rouse regime to the local reptation regime and calculated the intermediate scattering functions. Then des Cloizeaux [26] developed a “rubber-like” model for the transition regime to local reptation by considering infinite chains with spatially fixed entanglements at intermediate times. To test predictions from these theories, NSE measurements have been performed on a number of long chain polymer melts to observe reptation motion; we just mention some here. The first experiments were carried out on poly(tetrahydrofurane) (PTHF) [27]. Later measurements of the normalised coherent intermediate scattering function of polyisoprene (PI) melts [28] clearly showed a systematic deviation from the Rouse behaviour. NSE results on alternating (polyethylene-propylene) (PEP) copolymer [28] showed that both the local reptation theory of de Gennes and the theory of Ronca describe the local reptation regime well, but that the local reptation model could not describe the Rouse regime. All models presented could describe the local reptation mode at times 2π /Δω will give rise to scattering that cannot be distinguished from elastic, i.e. in practice the EISF equates motions that are slower than the resolution timescale with A0 (Q) and those that are faster with A1 (Q). In the next few paragraphs, we demonstrate the paradigm shift in thinking about polymers as big cousins of small molecules, obeying very similar (albeit modified) laws, through to full acknowledgment of their statistical nature. Initially, models adapted from small molecules were used to describe the local (side-chain) motion in polymers. For example, the motion of the ester methyl group in PMMA [62] was analysed in terms of the Barnes model [63] which is given by: rot Sinc (Q, ω ) = A0 (Q)δ (ω ) +

1 (2/3)τ A1 (Q) π 1 + ω 2 ((2/3)τ )2

(3.36)

√ √ with A0 (Q) = (1/3)[1 + 2 j0( 3)Qr] and A1 (Q) = (2/3)[1 − j0 ( 3)Qr], such that: rot Sinc (Q, ω ) = A0 (Q)δ (ω ) +

1 (2/3)τ [1 − A0(Q)] . π 1 + ω 2 ((2/3)τ )2

(3.37)

This model describes “instantaneous” jumps, with a correlation time τ , between three equidistant sites on a circle of radius r. The temperature dependence of the motion is assumed to follow the Arrhenius law. The EISF, which is equal to A0 (Q), is then a zero order Bessel function of Qr. However, there was a discrepancy between the calculated and measured EISF. In (3.36) and (3.37), it is assumed that the vibrational motions of protons about their average position and their rotations around an axis are dynamically uncoupled, and that the only vibrational contribution to the quasielastic scattering is through the Debye–Waller factor, exp (−Q2 u2 ). The term u2 is the mean square amplitude of the vibration of the atoms around their equilibrium position. The incoherent quasielastic scattering law (second term in (3.36)) is then a product of the Debye–Waller factor and the rotational law, (3.37). To experimentally study the ester methyl groups in PMMA, the main chain and α -methyl group protons were replaced by deuterons, resulting in a contribution from the coherent scattering of deuterons. A corrected EISF was necessary, A0 (Q), as adapted from methods developed for liquid crystals [64]. This was still not sufficient as the discrepancy between the calculated and measured EISF persisted.


87

The next step was to account for the contribution of coherent scattering estimated from neutron powder diffraction [65]: A0 (Q) =

A0 (Q) + x(Q) . 1 + x(Q)

(3.38)

In the above phenomenological expression, x(Q) is the ratio of the coherent to incoherent scattering determined from S(Q) and is valid for x < 0.1. While it worked for ambient temperature, it did not describe low (150 K) temperature measurements. Tunnelling at low temperatures was called to aid [66]; however, this could provide only hand-waving explanations. At that time, a unique instrument, a diffuse scattering diffractometer with spin polarization option (D7), was under construction in Grenoble [11]. This served to experimentally separate the coherent and incoherent parts of the scattering from PMMA and showed the extent to which the EISF was contaminated by coherent scattering, as shown in Fig. 3.3. The partial deuteration of the main chain and the α -methyl group helped to highlight the structural short-range order present in amorphous polymers. A simulation of the extent of the contamination of the EISF is shown in the figure below; its theoretical shape in both the “pure” case (where it is assumed that both components can be separated) and with the coherent component exactly as in Fig. 3.14 [45]. Two cases are considered: 1. Instantaneous jumps of a proton over three equidistant and equivalent sites on a circle (CH3 group) 2. Rotational diffusion for the same geometrical atomic arrangement These were interesting, though not comfortable findings, and led to different approach to methyl group dynamics. Due to the complexity of polymer structure and dynamics, it follows that the methyl groups in PMMA are not all in the same environment. This led to a successful interpretation of the H1 -NMR measurements on PMMA in terms of a Gaussian distribution of barriers to rotation and a combination of a “methyl thermometer” model, rather than a single barrier [67,68]. The “methyl thermometer” model is based on the assumption of single, independently reorienting methyl groups, all having the same environment. These groups then act as a “thermometer,” measuring different properties of the substance under investigation. The Gaussian distribution accounts for the variation in local environment of methyl groups, while maintaining the same principal value for the barrier heights and implicitly also for any effects of the cooperative motion of methyl groups in close proximity. Using distributions other than Gaussian gave the same quality of fit. Another description of the methyl group dynamics in PMMA, as observed by QENS, was sought by fast Fourier transform of the scattering law to the time domain and fitting a Kohlrausch–Williams–Watts (KWW) type function [69]: β t I(Q,t) = A0 (Q) + {1 − A0(Q)} exp − , τ

(3.39)

88

B.J. Gabrys and T. Kanaya 1.0

Observed EISF

0.8 A “pure” A contaminated B “pure” B contaminated

0.6

0.4

0.2

0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

1

Momentum transfer [Å ] Fig. 3.14 Contamination of the EISF by coherent scattering – simulation for (A) instantaneous jumps of a proton over three equidistant and equivalent sites on a circle (CH3 group) and (B) rotational diffusion for the same geometrical atomic arrangements

where β is the shape parameter and lies in the range 0 < β < 1. The fit was good and there were no problems reported with the EISF. The already mentioned RRDM [70] has also been used to account for the different rates of methyl group reorientation due to the different environments found in the amorphous polymeric state. The model considers the following scattering law for the methyl group rotations: N

S(Q, ω ) ≈ A(Q)δ (ω ) + 1 − A(Q) ∑ gi Li (ω ) ,

(3.40)

i=1

where Li (ω ) are Lorentzian functions, whose half-widths at half-maximum ωi correspond to the respective values of component characteristic times. The weight factors gi are obtained by sampling a log-Gaussian distribution of jumping rates g(log ωi ). Note that the Debye–Waller factor is left out for clarity here, on the basis of dynamical decoupling of different type of motions as mentioned above. The numbers of fit parameters in the above models differ and so it is left to interested readers to decide which model suits their needs best. However, it is worth noting that all these models assume the same EISF describing an averaged single methyl group rotation in a threefold potential. If the EISF is not corrected for the coherent scattering prior to data fitting, even the best fits leave a question mark! The above considerations reinforce the case for measuring the purely incoherent spectrum and coupling it with computer simulations. A good example of activity in this area is a molecular dynamics study of poly(vinyl methyl ether), PVME [71 and references therein].


89

Fig. 3.15 13 C nT 1 for s-PMMA (open circles) and si-PMMA (full circles) as a function of 1/T . Sample compositions are: 34.8% isotactic, 49.5% syndiotactic, and 15.7% heterotactic (si-PMMA); 4.9% isotactic, 72.5% syndiotactic, and 22.6% heterotactic (s-PMMA) [73]

3.5.2.3 Complementarity with NMR As mentioned in Sect. 3.5.2.1, the existence of a minimum in the spin-lattice relaxation time T1 in H1 -NMR points either to tunnelling or simply to a reorientation of a group of atoms. Other NMR techniques have been employed to detect and describe such motions, either to complement or to be used in conjunction with neutron scattering. C13 CP- MAS NMR is one of them: the carbon nucleus has a spin of 1/2 and is therefore sensitive to the application of a magnetic field. An advantage of using this technique over a wide temperature range is the observation of a chosen resonance line in the solid state. Moreover, both the 13 C spin-lattice relaxation time T1C and the chemical shift anisotropy serve as sensitive probes of molecular motion in localized environments. The quantities characterising rotational motion, the rotation rates and the activation energies can be determined with confidence from the nTlC plots as a function of temperature, provided that the TIC minimum can be detected (n denotes the number of protons directly bound to the carbon in question) [72]. The restricted motion of the α -methyl group in PMMA, which was not accessible by neutron scattering at that time, was detected by C13 CP- MAS NMR [73]. The results show both the influence of stereoregularity and the coupling of motions with those of the main chain motion, as expected due to the proximity of the α methyl group to the main chain. The samples used were predominantly syndiotactic PMMA (s-PMMA) of high molecular weight. Figure 3.15 shows 13 C spin-lattice

90


relaxation times plotted as nTlC as a function of 1/T for PMMA of two different tacticities. In both cases, broad but marked minima are visible, though the curve for s-PMMA is shifted to a higher temperature and has a more clearly defined minimum. This difference was attributed to the higher number of syndiotactic sequences in s-PMMA and slower reorientation of the methyl groups. The appearance of TlC minima in the T1 curves was assigned to the onset of α -methyl group rotation with a correlation time τc of the order of 1 ns. In order to explain the observed value of TlC minima in both PMMA stereoisomers, the authors proposed an extension of the 3-τ model [74]. This amounts to introduction of a motion with characteristic time τc , in the picosecond range, which is superimposed on the α -methyl group motion. Such a superposition describes a linking of local backbone and side-chain motions. The analogies between deuteron NMR and incoherent QENS have been exploited in studies of two-, four-, and sixfold jumps in small molecules including dimethylsulphone, hexamethylbenzene and benzene [75,76]. While neutron scattering probes correlation times typically less than 10−8 s, deuteron spin alignment used in 2 HNMR probes times typically longer than 10−4 s. Both methods rely on measuring single particle correlation functions and complement each other by virtue of accessing different time scales. In QENS, it is the EISF that describes the type of motion (time-averaged spatial distribution) of the molecule; A0 (Q) is directly measurable ((3.36) and (3.37)). In order to distinguish between different motions, one usually plots A0 (Q) as a function of the phase vector Q·R, where R is a jump vector. In macroscopic samples, a phase factor Q·a is used as different jump vectors have to be considered simultaneously. Deuteron NMR experiments exploit the electric quadrupole interaction of the deuteron nuclear spin I = 1 with an applied magnetic field B0 . The orientational information on the C-2 H bonds is coded in terms of frequency shift. In a spin alignment three pulse experiment, the alignment echo directly measures a single particle correlation function F(τ1 , τ2 ,t) = sin[τ1 ωQ (0)] sin[t ωQ (τ2 )],

(3.41)

where ωQ is the quadrupole frequency of a deuteron in a C–2 H bond at an angle θ relative to the direction of B0 . The sequence of pulses fulfils the relation τ1 τ2 < T1Q , with T1Q being the spin-lattice relaxation time of spin alignment. Figure 3.16 illustrates the above analogy. The quantity F∞ (τ1 ) is the normalized final state spin alignment echo height defined as: F∞ (τ1 ) =

F(τ1 , τ2 → ∞,t) . F(τ1 , τ2 = 0,t)

(3.42)

There are similarities and differences: e.g. for a twofold jump, both A0 (Q) and F∞ (τ1 ) curves converge to 0.5. However, A0 (Q) is equal to one at Q · a = 0, but this does not hold for the spin alignment echo height at δQ · τ1 = 0. This underlines the sensitivity of deuteron NMR for distinguishing between small phase factors of different motions.


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Fig. 3.16 Comparison of (a) the EISF obtained from QENS and (b) normalized spin alignment echo amplitude from deuteron NMR for two-site jump (dashed line), four-site jump (dashdot line), rotational diffusion on the circle (dotted line) and isotropic rotational diffusion (solid line) [75]

We believe that the combination of neutron scattering with other experimental techniques and modelling is here to stay and study materials such as liquid crystals are of growing technological importance (LC displays, beamers, etc). Recently, molecular dynamics in the nematic, smectic A and solid phases of (p, p-di-n-)heptyl-azoxybenzene (HAB) were studied by the quasielastic neutron scattering and 13 C CP-MAS NMR. QENS experiments have provided a deeper insight into the rotational dynamics in p, p-di-heptyl-azoxybenzene (HAB, 7AOB), a symmetric molecule with a dipole moment in its centre. Contrary to what is often believed, fast reorientation within the alkyl chains is possible even in the solid phase. In the smectic A and nematic phases, the motional complexity can be described within the so-called “stretched exponential” approach, pointing to a range of motional frequencies. For the first time, this data analysis has been validated by 13 C CP- MAS NMR data [77].

3.6 Beyond Harmonicity: Glass-Forming Polymers Up to now, we have discussed the basic modes of motion and illustrated, with simple polymer examples, how both INS and QENS have been useful for understanding them. We now move to the heavily researched topic of glass-forming polymers and the nature of the underlying mechanisms of the glass transition and the dynamical modes associated with it. Again, we will try to illustrate with examples how neutron spectroscopy has aided our understanding of this topic.

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3.6.1 Dynamics at Low Temperatures: Glassy State It is well known that glassy materials show anomalous thermal properties at low temperatures, such as in their heat capacity. As pointed out by Zeller and Pohl [78], the heat capacities of amorphous materials show an excess value compared with those of crystalline materials in two low temperature regions: one is below 1 K where heat capacity is proportional to temperature T , and the other is an excess heat capacity at around 10–20 K. The former has been interpreted in terms of tunnelling motion in an asymmetric double well potential [79, 80]. However, the excess heat capacity at around 10–20 K is less understood. In the last two decades, extensive studies have been performed by INS and Raman scattering [81], revealing that an excess excitation peak occurs at 2–3 meV in the spectra of all amorphous materials well below Tg , corresponding to the excess heat capacity at around 10–20 K. This is called the boson peak. In what follows we present neutron scattering investigations of the boson peak in amorphous polymers. Since thermal anomalies at 10–20 K are observed in all reported amorphous materials, this suggests that the boson peak is a generic feature. In order to confirm this hypothesis, INS investigations were carried out on various amorphous polymers and inorganic glasses [82]. The measured dynamic scattering laws S(Q, ω ) are shown for a range of systems in Fig. 3.17. All the samples, except highly crystalline polyethylene (h-PE), show a broad excitation peak in the energy range of 1.5–4.0 meV. These data are compelling evidence that the boson peak is a universal property of amorphous materials, but is absent in crystalline phases. The density of phonon states G(ω ) for the crystalline phase of polyethylene is proportional to ω 2 below ∼10 meV, meaning that the excitations in the crystalline phase can be well described by the Debye theory. For the amorphous phase, G(ω )/ω 2 shows a peak at about 2.5 meV and the value of G(ω ) is 7–8 times larger than that of the crystalline phase. The heat capacity C(T ) was calculated from the density of states G(ω ) for the amorphous and crystalline phases in a temperature range between 2 and 15 K; this agreed with the observed behaviour C(T )/T 3 , thus confirming that the boson peak at around 2–3 meV is the origin of the excess heat capacity of the amorphous phase.

3.6.2 Dynamics Near the Glass Transition Temperature 3.6.2.1 The Fast Process The temperature dependence of the intensity of the boson peak has been thoroughly studied for many glasses. As an example, the dynamic scattering law of polybutadiene (PB) is shown at various temperatures in Fig. 3.18 [83]. At temperatures below its Tg of 170 K, a broad boson peak is observed at 2–3 meV. The inelastic scattering intensity increases according to the calculated Bose population factor as shown by the dashed lines in the figure. On the other hand, an excess scattering


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Fig. 3.17 Measured S(Q, ω ) for various amorphous glass-forming systems [82]: (a) polyisobutylene (PIB) at 50 K; (b) PIB at 10 K; (c) cis-1,4-polybutadiene (PB) at 50 K; (d) cross-linked PB at 50 K; (e) trans-1,4-polychloroprene (PCP) at 50 K; (f) atactic polystyrene (PS) at 10 K; (g) semicrystalline polyethylene (s-PE) with degree of crystallinity of 0.46 at 10 K; (h) highly crystalline polyethylene (h-PE) with degree of crystallinity of 0.96 at 10 K; (i) poly(ethylene terephthalate) (PET) at 18 K; (j) atactic poly(methyl methacrylate) (PMMA) at 18 K; (k) epoxy resin (EPOX) at 50 K; (l) amorphous selenium (a-Se) at 150 K; (m) germanium (GeO2 ) glass at 295 K; (n) boric oxide (B2 O3 ) glass at 295 K

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˚ −1 for PB below and above Tg = 170 K Fig. 3.18 Dynamic scattering law S(Q, ω ) at Q = 1.54 A [83]. Dashed lines are the values expected from the Bose factor

intensity was observed at around Tg over the expected value. This excess scattering is attributed to the so-called fast process. Its characteristic time is 10−12 s and is a thermally activated process within the asymmetric double well potential. It is akin to the β process which is physically interpreted as a cage motion with the assistance of phonons. The onset temperature of the fast process is close to the Vogel–Fulcher temperature (VFT), which is ∼50 K below Tg , implying that the fast process is a precursor of the glass transition. 3.6.2.2 Segmental Relaxation: The α Process In the low-frequency region, below the fast process, the so-called α -process is observed; this governs the viscosity and is directly related to the glass transition. The relaxation time drastically increases, by more than several orders of magnitude, as the temperature is decreased towards Tg . Hence, it is necessary to combine a number of experimental techniques to cover the very wide time region where the dynamics associated with this process are active. Figure 3.19 shows the normalised intermediate scattering functions I(Q,t)/I(Q, 0) arising from the α -process of polybutadiene, measured by Richter and co-workers [84] using NSE, at temperatures above Tg = 181 K. Measurements were performed ˚ −1 of the static structure factor, S(Q), at both the first maximum position Q = 1.48 A ˚ −1 . The intermediate scattering function and close to the minimum at Q = 1.88A


95

Fig. 3.19 Scaling representation of normalised intermediate scattering functions of PB at temper˚ −1 [88] and (b) at Q = 1.88 A ˚ −1 [89] atures above Tg : (a) at Q = 1.48 A

is described by a KWW function (3.39), i.e. a stretched exponential relaxation. This type of behaviour is often observed in complex systems, including glassforming materials, and is interpreted in terms of a wide distribution of relaxation times. Using the distribution function g(ln τα ), the observed stretched exponential function can be described by I(Q,t) =

∞ −∞

g(ln τα ) exp(−t/τα )d(ln τα )

(3.43)

It is believed that the wide distribution of relaxation times is caused by the existence of a dynamic heterogeneity due to cooperative motion in the glass [85]. The data also showed that the intermediate scattering functions can be scaled to a master curve using shift factors aT determined from the temperature dependence of the viscosity, as shown in Fig. 3.19a. This result shows that neutron scattering can probe the same α -process as is observed in macroscopic viscosity measurements and that the relaxation mechanism does not change in the temperature range examined.

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Fig. 3.20 (a) Relaxation map for the dynamic behaviour of ortho-terphenyl showing data from six different experimental techniques, including neutron scattering [90]. (b) Relaxation map for PBD with data from QENS, NSE, viscosity measurements and 2 H-NMR [91]


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3.6.2.3 The β - Relaxation or Johari-Goldstein Process At smaller length scales, specifically near the Q−value of the minimum position in S(Q), the scaling does not work, as can be seen from the data in Fig. 3.19b. This is despite the fact that the relaxation curves fit nicely to a stretched exponential function. The relaxation times extracted deviate from the relaxation time of the α -process and the temperature dependence of these times follows an Arrhenius dependence, rather than VFT. The process observed is referred to as the β -relaxation or Johari-Goldstein (JG) process. The fact that the JG process is observed at intramolecular length scales suggests that the process is isolated rather than a cooperative motion. The Johari-Goldstein process, the α -relaxation and the fast process are not unique to glass-forming polymers. In fact, they are commonly observed in all sorts of glass-forming materials. This is shown in Fig. 3.20, which compares the relaxation time map of the small molecular glass former ortho-terphenyl (OTP) with that of PB. In fact, these maps summarize the relaxation dynamics observed in glass-forming polymers and in glass-forming materials in general. Specifically for polymers, the glass transition is related to chain flexibility. The glass transition and the dynamical processes discussed in this chapter will be expanded for more complex polymeric systems in the chapters to follow.

3.7 Summary The aim of this chapter was to provide an overview of the fundamentals relevant to the dynamical modes observed in polymers. More detailed examples in polymer dynamics research with neutrons are presented in the other chapters of this book. Here we have exposed the basic principles, how neutrons helped their early development, and have illustrated them with examples. The choice of examples is personal: either they were seminal or meaningful to us, but always exciting. Topics of interest in neutron scattering reflect changing research interests and instrumental developments at any given time. The first measurements on polymers using QENS looked at local motion through the lens of the whole-chain motion in solution and melts, being followed by the fashion for glass transition studies. Often the academic interest in the material was coupled with its potential for industrial applications. This is still the case, with for example some newly emerging biological studies aiming at a better understanding of drug delivery processes. As the problems posed become more demanding, the need for more sophisticated neutron scattering experiments increases. For bio-macromolecules, the dynamics are intimately linked with the structure: only some existing structural forms are biologically active, so it makes sense to investigate both structure and dynamics in the same experiment. This is possible if one makes use of polarized neutrons. Moreover, the data are complex, and its full understanding can come only through combination with modelling and complementary data from other experimental techniques.

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References 1. Brockhouse BN, Shull CG (1994) from http://www.nobelprize.org/nobel prizes/physics/ laureates/1994/ (Accessed 19 October 2011) 2. de Gennes PG (1991) The Nobel Prize in Physics 1991. From http://www.nobelprize.org/ nobel prizes/physics/laureates/1991/ (Accessed 19 October 2011) 3. Higgins JS, Benoit H (1994) Polymers and neutron scattering. Clarendon, Oxford 4. Moon RM, Riste T (1969) Polarization analysis of thermal-neutron scattering. Phys Rev 181(2):920 5. Mezei F (1972) Z Phys Neutron Spin-Echo: A new concept in polarized thermal neutron techniques. 255:146 6. Zajac W, Gabrys BJ et al (2002) Structure of poly(ethylene oxide) (PEO and PEO·LiSO3 CF3 ) studied with spin polarised neutrons. Solid State Ionics 147(3–4):213–223 7. Gabrys B, Higgins JS et al (1986) Contamination by coherent scattering of the elastic incoherent structure factor observed in neutron scattering experiments. J Chem Soc Faraday Trans 82:1923–1927 8. Schärpf O, Capellmann H (1993) The XYZ-difference method with polarized neutrons and the separation of coherent, spin incoherent, and magnetic scattering cross sections in a multidetector. Phys Stat Sol A 135:359 9. Schärpf O (1996) The spin of the neutron as a measuring probe. 10th International summer school of condensed matter physics, Bialowieza, Poland 10. Schärpf O, Anderson I (1996) Classical polarization analysis. J Neutron Res 4(1–4):227–240 11. Schärpf O (1985) Experience with spin analysis on a time-of-flight multidetector scattering instrument. Neutron scattering in the ‘nineties, Vienna, I.A.E.A 12. Schärpf O, Gabrys B et al (1990) Short range order in isotactic, atactic and sulphonated polystyrene measured by polarised neutrons. Institute Laue-Langevin, Grenoble 13. de Gennes PG (1971) J Chem Phys 55:572 14. Nicholson LK (1981) The neutron spin-echo spectrometer: a new high resolution technique in neutron spectroscopy. Contemp Phys 22(4):451–475 15. Richter D (2006) Neutron spin echo for the exploration of large scale macromolecular dynamics. J Phys Soc Japan 75(11):111004 16. Bée M (1988) Quasielastic neutron scattering: principles and applications in solid state chemistry, biology and materials science. Hilger, Bristol 17. Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon, Oxford 18. Rouse JPE (1953) A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J Chem Phys 21(7):1272–1280 19. Zimm BH (1956) Dynamics of polymer molecules in dilute solution: viscoelasticity, flow birefringence and dielectric loss. J Chem Phys 24(2):269–278 20. Pecora R (1968) Spectral distribution of light scattered from flexible-coil macromolecules. J Chem Phys 49(3):1032–1035 21. de Gennes PG (1967) Physics 3:37 22. Richter D, Ewen B (1989) Neutron-spin echo investigations on the dynamics of polymer systems. Prog Colloid Polymer Sci 80:53–62 23. Richter D, Ewen B et al (1989) Microscopic dynamics and topological constraints in polymer melts: a neutron-spin-echo study. Phys Rev Lett 62(18):2140 24. de Gennes PG (1981) Coherent scattering by one reptating chain. J Phys I France 42(5): 735–740 25. Ronca G (1983) Frequency spectrum and dynamic correlations of concentrated polymer liquids. J Chem Phys 79(2):1031–1043 26. des Cloizeaux J (1993) Dynamic form function of a long polymer constrained by entanglements in a polymer melt. J Phys I France 3(7):1523–1539 27. Higgins JS, Roots JE (1985) Effects of entanglements on the single-chain motion of polymer molecules in melt samples observed by neutron scattering. J Chem Soc Faraday Trans 2 Mol Chem Phys 81:757–767


99

28. Richter D, Butera R et al (1992) Entanglement constraints in polymer melts. A neutron spin echo study. Macromolecules 25(23):6156–6164 29. Schleger P, Farago B et al (1998) Clear evidence of reptation in polyethylene from neutron spin-echo spectroscopy. Phys Rev Lett 81(1):124 30. Denker JS (2009) Introduction to scaling laws. From http://www.av8n.com/physics/scaling. htm (Accessed on 28 November 2011) 31. Doi M (1996) Introduction to polymer physics. Clarendon, Oxford 32. Affouard F, Descamps M et al (2006) Proceedings of the 5th International Discussion Meeting on Relaxations in Complex Systems. Journal of Non-Crystalline Solids, 352, Lille, France 33. Kittel C (1976) Introduction to solid state physics. Wiley, New York 34. Twisleton JF, White JW (1972) Interchain force field and elastic constants of polytetrafluoroethylene. Polymer 13(1):40–42 35. Willis BTM (1973) Chemical applications of thermal neutron scattering [S.l.]. Oxford University Press, Oxford 36. Danner HR, Safford GJ et al (1964) Study of low-frequency motions in polyethylene and the paraffin hydrocarbons by neutron inelastic scattering. J Chem Phys 40(5):1417–1425 37. Myers W, Donovan JL et al (1965) Polyethylene frequency spectrum from “warm”-neutron scattering. J Chem Phys 42(12):4299–4300 38. Kitagawa T, Miyazawa T (1967) Inelastic-scattering cross section of neutron by crystal vibrations of polyethylene. J Chem Phys 47(1):337–338 39. Myers W, Summerfield GC et al (1966) Neutron scattering in stretch-oriented polyethylene. J Chem Phys 44(1):184–187 40. Kitagawa T, Miyazawa T (1973) Adv Polymer Sci 9:335 41. Braden DA, Parker SF et al (1999) Inelastic neutron scattering spectra of the longitudinal acoustic modes of the normal alkanes from pentane to pentacosane. J Chem Phys 111(1): 429–437 42. Alvarez F, Alegr´ıa A et al (2000) Origin of the distribution of potential barriers for methyl group dynamics in glassy polymers: a molecular dynamics simulation in polyisoprene. Macromolecules 33(21):8077–8084 43. Segall MD, Lindan PJD et al (2002) First-principles simulation: ideas, illustrations and the CASTEP code. J Phys Cond Matt 14:2717–2744 44. Smith J, Kuczera K et al (1990) Dynamics of myoglobin: comparison of simulation results with neutron scattering spectra. Proc Nat Acad Sci 87(4):1601–1605 45. Gabrys BJ, Zajac W et al (2001) QENS from “soft” systems: why use polarised neutrons? Physica B Cond Matt 301(1–2):69–77 46. Ediger MD (1991) Time-resolved optical studies of local polymer dynamics. Ann Rev Phys Chem 42(1):225–250 47. Schatzki TF (1962) Statistical computation of distribution functions of dimensions of macromolecules. J Polymer Sci 57(165):337–356 48. Monnerie L, Geny J (1969) J Chim Phys Phys Chim Biol 66:1691 49. Kanaya T, Kaji K et al (1991) Local motions of cis-1,4-polybutadiene in the melt. A quasielastic neutron-scattering study. Macromolecules 24(8):1826–1832 50. Kanaya T, Kawaguchi T et al (1999) Local dynamics of some bulk polymers above Tg as seen by quasielastic neutron scattering. Macromolecules 32(5):1672–1678 51. Colmenero J, Moreno AJ et al (2001) Neutron scattering investigations on methyl group dynamics in polymers. Prog Polymer Sci 30(12):1147–1184 52. Clough S, Heidemann A et al (1982) The rate of thermally activated methyl group rotation in solids. J Phys C Solid State Phys 15:2495–2508 53. Hewson AC (1978) The temperature dependence of inelastic neutron scattering in rotational tunnelling systems. I. Formulation and perturbation theory. J Phys C Solid State Phys 15(18):3841–3853 54. Van Hecke P, Janssens G (1978) NMR direct detection of tunnel splittings in solid SiH4. Phys Rev B 17(5):2124

100


55. Gabrys B, van Gerven L (1981) Rotational tunneling of methyl groups in pentamethylbenzene studied by NMR relaxation resonance. Chem Phys Lett 82(2):260–263 56. Allen PS, Cowking A (1968) Nuclear magnetic resonance study of hindered rotations in some methylbenzenes. J Chem Phys 49(2):789–797 57. Clough S, Horsewill AJ et al (1981) Neutron scattering study of methyl tunnelling in pentamethylbenzene. Chem Phys Lett 82(2):264–266 58. Colmenero J, Mukhopadhyay R et al (1998) Quantum rotational tunneling of methyl groups in polymers. Phys Rev Lett 80(11):2350 59. Bée M (1988) Quasielastic neutron scattering: principles and applications in solid state chemistry, biology and materials science. Hilger, Bristol 60. Bée M (1992) A physical insight into the elastic incoherent structure factor. Physica B Cond Matt 182(4):323–336 61. Kirov N, Dozov I et al (2006) Elastic incoherent neutron scattering of rotational and translational dynamics in liquid crystals. J Mol Struc 788(1–3):7–15 62. Gabrys B, Higgins JS et al (1984) Rotational motion of the ester methyl group in stereoregular poly(methyl methacrylate): a neutron scattering study. Macromolecules 17(4):560–566 63. Barnes JD (1973) Inelastic neutron scattering study of the “rotator” phase transition in n-nonadecane. J Chem Phys 58(12):5193–5201 64. Hervet H, Dianoux AJ et al (1976) Neutron scattering study of methyl group rotation in solid para-azoxyanisole (PAA). J Phys France 37:587 65. Richardson RM, Leadbetter AJ et al (1980) A self-consistent interpretation of nuclear magnetic resonance and quasi-elastic neutron scattering data from the smectic A and nematic phases of ethyl 4-(4 acetoxybenzylidine) aminocinnamate. Mol Phys 40(3):741–757 66. Gabrys B, Higgins JS et al (1985) Rotation of methyl groups in molecular solids and polymers at low temperatures: recent developments. Polymer 26(3):355–363 67. Gabrys B, Clark JN et al (1987) Dynamics of solid poly(methacrylate): an NMR study: polymer motion in dense systems. Springer, Grenoble 68. Connor TM (1964) Distributions of correlation times and their effect on the comparison of molecular motions derived from nuclear spin-lattice and dielectric relaxation. Trans Faraday Soc 60:1574–1591 69. Arrighi V, Higgins JS et al (1995) Rotation of methyl side groups in polymers: a Fourier transform approach to quasielastic neutron scattering. 1. Homopolymers. Macromolecules 28(8):2745–2753 70. Chahid A, Alegria A et al (2002) Methyl group dynamics in poly(vinyl methyl ether). A rotation rate distribution model. Macromolecules 27(12):3282–3288 71. Saelee C, Nicholson TM et al (2000) A molecular dynamics study of methyl group rotation in poly(vinyl methyl ether). Macromolecules 33(6):2258–2265 72. Kitamaru RZ (1990) Nuclear magnetic resonance: principles and theory. Elsevier, Amsterdam 73. Gabrys B, Horii F et al (1987) Carbon-13 NMR study of the α -methyl group rotation in solid poly(methyl methacrylate): detection of the 13 C T1 minimum. Macromolecules 20(1):175–177 74. Howarth OW (1980) 13C nuclear magnetic resonance study of molecular motions in natural rubber. JCS Faraday II 76:1219–1223 75. Fujara F, Wefing S et al (1986) Dynamics of molecular reorientations: analogies between quasielastic neutron scattering and deuteron NMR spin alignment. J Chem Phys 84(8): 4579–4584 76. Fujara F, Petry W et al (1988) Reorientation of benzene in its crystalline state: a model case for the analogy between nuclear magnetic resonance spin alignment and quasielastic incoherent neutron scattering. J Chem Phys 89(4):1801–1806 77. Zajac W, Urban S et al (2006) Stochastic molecular motions in the nematic, smectic-A, and solid phases of p, p -di-n-heptyl-azoxybenzene as seen by quasielastic neutron scattering and 13 C cross-polarization magic-angle-spinning NMR, Phys Rev E 73(5): 051704 78. Zeller RC, Pohl RO (1971) Thermal conductivity and specific heat of noncrystalline solids. Phys Rev B 4(6):2029 79. Phillips WA (1972) Tunneling states in amorphous solids. J Low Temp Phys 7(3–4):351–361


101

80. Anderson PW, Halperin BI et al (1972) Anomalous low-temperature thermal properties of glasses and spin glasses. Philos Mag 25(1):1–9 81. Kanaya T, Kaji K (2001) Dynamics in the glassy state and near the glass transition of amorphous polymers as studied by neutron scattering. Adv Polymer Sci 154:87–141 82. Inoue K, Kanaya T et al (1991) Low-energy excitations in amorphous polymers. J Chem Phys 95(7):5332–5340 83. Kanaya T, Kawaguchi T et al (1993) Low-energy excitation and fast motion near Tg in amorphous cis-1,4-polybutadiene. J Chem Phys 98(10):8262–8270 84. Frick B, Farago B, Richter D (1990) Phys Rev Lett 64:2921 85. Ediger MD, Lutz TR et al (2006) Dynamics in glass-forming mixtures: comparison of behavior of polymeric and non-polymeric components. J Noncryst Solids 352(42–49):4718–4723 86. Gabrys B, Schärpf O (1992) Scattering from polymers using polarised neutrons: a new development. Physica B 180 & 181:495–498 87. Adams MA, Gabrys BJ et al (2005) High-resolution incoherent inelastic neutron scattering spectra of polyisobutylene and polyisoprene. Macromolecules 38(1):160–166 88. Richter D, Frick B, Farago B (1988) Neutron Spin-Echo investigation on the dynamics of polybutadiene near the glass transition. Phys Rev Lett 61(21):2465–2468 89. Frick B, Zorn R et al (1992) Decoupling of time scales of motion in polybutadiene close to the glass transition. Phys Rev Lett 68:71–74 90. Rössler E, Warschewske U et al (1994) Indications for a change of transport mechanism in supercooled liquids and the dynamics close and below Tg. J Noncryst Solids 172–174 (Part 1):113–125 91. Kanaya T, Kawaguchi T et al (1999) Local dynamics of some bulk polymers above Tg as seen by quasielastic neutron scattering. Macromolecules 32(5):1672–1678

Chapter 4

Complex polymers Arantxa Arbe and Juan Colmenero

4.1 Neutrons, Simple Polymers and Complex Polymers The space/time resolution offered by inelastic and quasielastic neutron scattering has allowed characterization of the different dynamical processes taking place in polymers and, sometimes, tests of possible theoretical frameworks. The characteristic length and timescales of these processes span several orders of magnitude. At atomic and sub-picosecond level, the vibrational spectrum of a glassy polymer shows the Boson-peak feature which is typical for glasses [1]. Side-group motions occurring at the monomeric level reflect the disorder of the environment through a distribution of potential barriers; the direct observation of methyl group dynamics, including its tunnelling manifestation, has allowed us to establish this universal feature in the glassy state [2]. The structural relaxation has been directly followed by monitoring the decay of the intermolecular correlations at the first structure factor peak [3]; these kinds of study support the main predictions of the Mode Coupling Theory (MCT) for the glass transition. In the regime of the structural relaxation, the hydrogen motions have also been characterized by quasielastic neutron scattering (QENS). A subdiffusive behaviour has been found that crosses over towards Rouse-like dynamics at larger length and timescales [4]. Finally, the role of entanglements has been shown directly by neutron spin echo (NSE)

A. Arbe () Centro de F´ısica de Materiales (CSIC-UPV/EHU) - Materials Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain e-mail: [email protected] J. Colmenero Centro de F´ısica de Materiales (CSIC-UPV/EHU) - Materials Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain and Donostia International Physics Center, Paseo Manuel de Lardizabal 3, 20018 San Sebastián, Spain e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 4, © Springer Science+Business Media, LLC 2012

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for long enough polymer chains [3]. We can therefore say that our knowledge of polymer dynamics has substantially profited from the application of neutron scattering techniques and that the main bases for characterization of the different processes in linear homopolymers are well established. However, current technological demands require more versatile polymer-based materials that can be obtained in a number of ways. Applying advanced chemistry at the monomeric level, the complexity of homopolymer systems can be increased tremendously. For example: chains can be functionalized at specific sites; polymers can be synthesized with different architectures than linear (branched, combs, “H”, stars, rings,...) in order to profoundly change properties such as e.g. the rheological behaviour in homopolymers. Alternatively, an infinite number of systems can be obtained by combining polymeric chains of a given kind with other structural units or particles. For instance, the use of polymer blends is widely spread, and the inclusion of nanoparticles in polymeric matrices is nowadays a promising way to obtain materials with novel properties. In all of these complex systems, new features appear associated with nano-segregation, confinement effects, interactions at the surfaces / interfaces, etc., that are still very poorly understood. Neutron scattering offers a unique tool to unravel essential dynamic information in such complex materials, since combined with isotopic labelling it allows selectivity for the component dynamics. In this chapter we illustrate the potential of neutron scattering in the study of complex polymeric systems focusing on two kinds of materials: (i) polymer blends with different degrees of dynamic asymmetry and (ii) a family of homopolymers, poly(n-alkyl methacrylates), that show nanosegregation of the main chains and side groups. It is of particular interest to determine to what extent the increase of complexity in the system affects the different dynamical processes in the polymers. We will see that, under certain conditions, intriguing features emerge in both kinds of systems that might be attributed to confinement effects.

4.2 Dynamics of Polymer Blends Because of difficulties in commercializing new polymers, industry has increasing turned to blending existing polymers to optimize their end-use properties (e.g. mechanical and rheological). The design of tailor-made materials would be enormously facilitated by an understanding of the blending phenomenon at a molecular level. The key question to address is: How are the different dynamical processes of each component of the blend modified by blending? To answer this question, selective techniques obviously have to be considered. Neutron scattering is an excellent choice. Taking advantage of the different scattering lengths of hydrogen and deuterium we can label a given component. In particular, for studies of dynamics at local scales, deuterated materials are “invisible” to neutrons, and this probe therefore reveals the hydrogen motions of the protonated component. This allows the study of fast motions and processes active in the glassy state, as well as the

4 Complex polymers

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0.4

S(Q, ) (a.u.)

0.3

0.2

0.1

0 –2

–1

0

1

2

3

4

5

6

h (meV) Fig. 4.1 Incoherent scattering function showing the main-chain hydrogen-weighted density of states of PI in the bulk (filled symbols) and in a blend with PVE (50%/50%) (empty symbols)

dynamics at an interchain level, the segmental dynamics (α -relaxation) involved in the glass-transition phenomenon. Furthermore, hydrogen/deuterium substitution also allows labelling of specific chains in the system, whose motions can then be followed by, for example, NSE experiments on the single-chain dynamic structure factor.

4.2.1 Glassy Dynamics The neutron scattering experiments reported so far indicate that, if the interactions between the components are weak, blending does not appreciably affect vibrations, fast motions and processes in the glassy state. Let us consider some illustrative examples. Figure 4.1 shows a time-of-flight spectrum of polyisoprene (PI) with deuterated methyl groups in the glassy state. Methyl group deuteration removes the contribution of their rotation from the spectrum. Consequently, the data show the hydrogen-weighted vibrational density of states of PI, which shows the typical maximum around 1–2 meV known as the Boson peak. In the same figure, the results obtained for a blend of the same PI and deuterated poly(vinyl ethylene) (PVE) are shown. Since PVE is deuterated, this spectrum is dominated by the hydrogenweighted vibrational density of states of PI in the blend. As can be seen, the results

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1

Sel(T)/Sel(T= 0)

Tg(PVME) (PVME/PS) 0.5

0

0

100

200

300

400

T(K) Fig. 4.2 Temperature dependence of the elastic intensity measured by backscattering on PVME in the bulk (full symbols) and in a blend with PS (65% of PVME) (empty symbols) [6]

of both samples are almost indistinguishable, indicating that the Boson peak is unaffected by blending, even for concentrations of 50%/50%. Results on a blend head-to-head polypropylene (hhPP)/poly(ethylene propylene) (PEP) (50%/50%) [5] also corroborate that the sub-picosecond dynamics of both components remain the same upon blending. In a similar way, methyl group dynamics are not affected by blending, as shown by quasielastic neutron scattering on poly(vinyl methyl ether) (PVME)/polystyrene (PS) [6], poly(ethylene oxide) (PEO)/poly(methyl methacrylate) (PMMA) [7] and hhPP/PEP [5]. Figure 4.2 shows the temperature dependence of the elastic intensity measured on a bulk PVME sample and a sample composed by 65%PVME/35%PS, where PS is deuterated. In both cases the results are sensitive to PVME hydrogen dynamics. The decay of the elastic intensity reveals the occurrence of dynamical processes with timescales faster than or in the range of the instrumental resolution. The decrease of the elastic intensity at very low temperatures (T < 60 K) can be attributed to vibrational contributions; between approximately 60 and 150 K the additional decay is due to methyl-group dynamics [6, 8], and the final drop takes place only after the onset of the glass-transition in the system. As can be appreciated from this figure, the results in bulk and in the blend are indistinguishable in the glassy state. Only in the supercooled liquid state above the glass-transition temperature Tg are the hydrogen dynamics of PVME affected by blending: they are slowed down by the presence of the more slowly relaxing PS chains.

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107

S(Q,ω) / Smax (Q,ω)

100

T=Tg(PVE) 10−1

10−2

PVE PVE in PVE/PI 10−3

−12

−8

−4

0

4

8

12

E (μeV) Fig. 4.3 Comparison between the normalized scattering functions of PVE (full symbols) and ˚ −1 in a logarithmic representation [12] PVE/PI (50%/50%) (empty symbols) at 270 K and Q = 1.9 A as measured on a backscattering spectrometer. The dotted line represents the instrumental resolution

The effect of blending on the secondary relaxations, in particular on the β-process, has mainly been investigated by dielectric spectroscopy (DS). This is due to the wider dynamical window accessed by DS, that allows characterization of the secondary processes before they merge with the α-relaxation. We note that DS can be a selective technique for this kind of study if only one of the components is dielectrically active (or shows a much stronger dipole moment than the other). For the secondary process DS studies have found insensitivity upon blending for PVME/PS [9], PVME/poly(o-chlorostyrene) (PoClS) [10] and PEO/PMMA [11]. QENS has also contributed to this question in the case of PVE in a blend with PI (50%/50%) [12] as shown in Fig. 4.3: the full symbols correspond to data for bulk PVE at 270 K, i.e. at its glass-transition temperature. The broadening observed should thus be due to the dynamical process active in this dynamic window at such temperature – the secondary relaxation. Analogous measurements on a blend of PI/PVE where PI is deuterated reveal an identical spectrum for PVE in the blend to that for pure PVE, within error. This direct observation by neutron scattering confirms the DS results regarding the absence of effects due to blending on the secondary processes in the glassy state. Thus we can conclude that vibrations, methyl group rotations and secondary relaxation processes (i.e. the dynamical processes relevant in the glassy polymer state) are unaffected by blending. This behaviour is probably a consequence of the rather localized character of the motions involved in these processes.

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4.2.2 The Dynamics of the α-Relaxation The criterion used by most scientists to determine the miscibility of polymer blends is the observation of a single glass-transition temperature Tg as measured by e.g. differential scanning calorimetry (DSC). Instead of a rather abrupt step of the specific heat C p as measured for homopolymers, blends usually show a monotonous increase of C p that extends over a temperature range between the two Tg ’s of the pure substances. The position and broadening of such a step depends on blend composition. From its midpoint, an average glass-transition temperature might be deduced for the blend, Tg . Based on this observation it has traditionally been believed that miscibility implies a single glass-transition for both blend components – a concept that, as we will show in the following, has to be critically revised. During the last few years a large effort has been made to characterize and understand the effect of blending on the segmental dynamics in polymers [13]. Two main experimental observations have been established: (i) the broadening of the component response with respect to that of the homopolymer, which can be attributed to thermally activated concentration fluctuations and (ii) the dynamic heterogeneity. Dynamic heterogeneity refers to the observation of two different characteristic timescales for segmental relaxation, one for each component in the blend. This contradicts the a priori expectation of a single average relaxation time for the α-relaxation (corresponding to only one Tg measured by DSC) in a perfect and homogeneous blend at a molecular level. The observation of this heterogeneous behaviour has been realized by several techniques, mainly DS and NMR, in a large number of polymer blends [13]. Neutron scattering has also been used to confirm this feature at a molecular level. Figure 4.4 shows QENS results [14] for the system PI/PVE (50%/50%) at a temperature well above the glass transitions of both homopolymers [Tg (PVE) = 272 K and Tg (PI) = 213 K] and the average glass transition of the blend [Tg (50%PI/50%PVE) = 228 K]. By selectively deuterating each of the components we can isolate the signal from the other, revealing the hydrogen dynamics. The width of the scattering curve in comparison to the resolution is related to the inverse of the timescale of motions. Figure 4.4 reveals that this timescale is clearly different for PI and PVE in the blend. Dynamic heterogeneity is thereby elegantly demonstrated by this technique. Nowadays models based on the concept of self-concentration capture the main ingredients leading to two distinct timescales in miscible blends. The idea is that the local concentration around one segment of one of the blend components will always be richer in itself due to chain connectivity. As the average glass-transition in the blend depends on composition, both components experience different “effective glass-transitions” Tg,eff . These different glass-transitions, in fact, imply different relaxation times for both components. The most elaborated formulation of this family of models is the so-called Lodge&McLeish (LML) model [15], that allows us to quantify the deviations of the “effective glass-transitions” Tg,eff of the blend components with respect to Tg . QENS results on a blend of hhPP/PEP (50%/50%, difference in glass-transition temperature of the two homopolymers Δ Tg = 35 K)

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S(Q,ω) / S(Q,ω=0)

1

PI

0.1

PI /PVE PVE 0.01 resolution −15

−10

−5

0

5

10

15

20

E (μeV) ˚ −1 . From top to Fig. 4.4 Normalized QENS results at 330 K and momentum transfer Q = 1.1 A bottom: PI, PI in PI/PVE, PVE in PI/PVE, PVE, and instrumental resolution. The concentration of the blend is 50%/50% [14]

have recently provided support for the concept of self-concentration as a key ingredient leading to dynamic heterogeneity [5]. As can be seen in Fig. 4.5, the difference in the timescales observed for the blend components in this system, as well as their temperature dependencies, are perfectly predicted by the LML model.

4.2.3 Single-Chain Dynamics Neutron scattering allows the measurement of the single-chain dynamic structure factor of labelled chains. As an example, Fig. 4.6 shows results for the single-chain structure factor of PI and PVE homopolymers and as components in a 50%/50% blend [14]. The curves are shown in a scaling representation, where data from different temperatures have been scaled to the same temperature of 368 K. These experiments were performed by NSE on samples with a 10% labelled [protonated (h)] chain fraction in otherwise deuterated (d) matrices [hPI/dPI; hPVE/dPVE; hPI/(dPI /dPVE) and hPVE/(dPI/dPVE)]. As can be seen from the figure, the chain dynamics in pure PVE are slower than in pure PI, in accord with the differences in glass-transition temperatures of the two homopolymers. In both cases, the data can be well described by the Rouse model – at least at short times, where entanglement effects are not important. In the blend, the chain dynamics of the two components also follow the Rouse prediction very nicely. We can also see that the component dynamics are close to each other, pointing to a rather homogeneous

110


log[τw(Q=1Å-1,T)(s)]

−7 −8

hhpp

PEP T Tg,eff g,eff

−9

PEP

Tg

hhpp

Tg

−10 −11 200

250

300 T (K)

350

400

˚ −1 obtained by QENS for Fig. 4.5 Temperature dependence of the characteristic times at Q = 1 A hydrogens in hhPP, hhPP in hhPP/PEP, PEP in hhPP/PEP and PEP (from the top) [5]. The solid lines show the dependence expected from NMR and DS for hhPP and rheological measurements for PEP; the dashed lines are the prediction of the LML model for the blend components. The blend composition is 50%/50%. Solid arrows mark the glass-transition temperatures of the homopolymers and dashed arrows mark the effective glass transition temperatures of the polymer components 1

1

S(Q,t) / S(Q,0)

1 0.8

0.8

1 0.8 0.6 1 0.8 0.6 0.4

0.6 0.4

0.8 0.6 0.4 0.2

0.2

0.6 0.4 0.2 0 0.4 0.2 0

0

0.2 0

0

0 0.01

0

0.1

1

10

100

0 1000

t (ns) / a T (TR=368K) ˚ −1 vs. the rescaled time t/aT . From Fig. 4.6 Single-chain dynamic structure factors at Q = 0.15 A the top PVE, PVE in PI/PVE, PI in PI/PVE and PI [14]. The blend composition is 50%/50%. Different symbols refer to different temperatures in the interval between 305 and 422 K. Solid lines: Rouse structure factors

4 Complex polymers

a

A

111 AB

B

b T gA

A/AB T geff

B/AB T geff

−4 A/AB

B/AB

B

log [τ(s)]

A

log [τ(s)]

−2 −4 −6

PEO in PMMA PEO in PES PEO in PVAc

−2

T gB

2 0

0

g

−6

in PES

−8

pure PEO

−10

−8 −10 −12 cross Glass-forming liquid-like dynamics over

in PMMA

in PVAc

1/T Confined dynamics

Glassy dynamics

−12 2

2.5

3

3.5

4

4.5

5

−1

1000 / T(K )

Fig. 4.7 (a): “Cartoon” of the scenario for dynamic confinement in polymer blends. (b): Temperature dependence of pure PEO α-relaxation (solid line) and that expected from the LML model for the PEO component in the blend with PMMA (dashed) [7], poly(ether sulfone) (dasheddotted) [18] and PVAc (dotted) [19]. The process actually observed for PEO hydrogens in the three blends is shown as squares

dynamics in the blend in the high temperature range covered by these measurements. This homogeneity apparently contradicts the finding of dynamic heterogeneities in the system at more local length scales (at the level of the segmental relaxation) and lower temperatures (Fig. 4.4). The scenario proposed in [14] to explain these observations was a crossover from a universal entropy-driven dynamics, similar for both components, to more heterogeneous local motions specific for each of the two homopolymers [14].

4.2.4 Blends with Large Dynamic Asymmetry It has recently been found that dynamical asymmetric mixtures, i.e. mixtures of “slow” and “fast” components, exhibit unique properties, in particular concerning phase separation and network-like patterns (see, e.g. [16] and references therein). Miscible polymer blends of components with very different Tg ’s can be considered as dynamically asymmetric systems. It is easy to show that, in this case, the dynamic asymmetry Δ – which can be defined as the ratio between the relaxation times corresponding to the slow and fast components in the system – strongly depends on temperature. Let us consider a blend where the two components A and B have very different glass-transition temperatures, and, thereby, very different α-relaxation dynamics. In Fig. 4.7(a) the corresponding relaxation times are represented in an Arrhenius plot. The dashed lines show the behaviour expected

112


for their two relaxation times in the blend, as predicted e.g. from the LML model. In the high-temperature regime Δ is very small and both components in the blend behave dynamically as high-temperature polymer melts at equilibrium. However, as the temperature decreases τA dramatically increases and component A (slow) in the blend becomes more and more frozen. With regards to the dynamics of the B-component, this implies that there would be some kind of crossover from the high temperature behaviour – corresponding to the blend system in equilibrium – towards a low-temperature regime where this component will move within some kind of frozen network imposed by component A. There the dynamics of polymer B would be confined and thus consist of localized motions with low degree of cooperativity. The vast literature available concerning the confinement effects on the structural relaxation of glass-forming systems (see e.g. [17]), points to an Arrhenius law for the characteristic time in such situations. Therefore, for the characteristic time of B in the system we can expect a broad crossover from a normal liquid-like behaviour at high temperature [Vogel–Fulcher (VF) like] towards an Arrhenius-like temperature dependence at low temperature [dashed-dotted line in Fig. 4.7(a)]. Obviously, these effects should be more and more evident as soon as the concentration of the high-Tg slow component in the blend increases. Has this crossover been experimentally observed? The first evidence was provided by DS studies on the blend PVME/PS [20] [Tg (PVME) = 249K; Tg (PS) = 373 K]. By increasing the difference in Tg ’s of the two blend components the crossover is shifted to shorter times and may be observed by microscopic techniques such as QENS, which, in addition, provide space–time resolution. Hints of confined motions are revealed in the high momentum-transfer range (i.e. corresponding to small length scales) for PEO [Tg (PEO) = 220K] in blends with PMMA [Tg (PMMA) = 400K] (25%PEO/75%PMMA) [7] and also recently with poly(vinyl acetate (PVAc) [Tg (PVAc) = 315K] (20%PEO/80%PVAc) [19, 21] and poly(ether sulfone) (PES) [Tg (PES) = 382K] (25%PEO/75%PES) [18]. Figure 4.7b shows the expected behaviour for PEO in these blends starting from the pure PEO dynamics (dotted and dashed lines). As can be seen, in the neighborhood of Tg the processes observed for the hydrogen motions of the PEO component deviate from those expected in a standard system, following an Arrhenius-like behaviour characteristic of confined motions. These results provide microscopic evidence for the dynamic confinement effects. Furthermore, the QENS results corresponding to PEO dynamics in these blends suggest that the dynamic structure factors are very stretched in the crossover region and can only be fitted using extremely broad distributions of relaxation times (see also [22, 23]). As an example we can see in Fig. 4.8a the Fourier transformed data obtained for PEO in PMMA [24] by combining two neutron scattering spectrometers. However, due to the limited dynamic range of QENS, the exact form of the scattering functions cannot be unequivocally determined from the experimental data. Molecular dynamic simulations of PEO/PMMA show a clear crossover from convex to concave curvature of the PEO intermediate scattering function with increasing momentum transfer [7], passing through a logarithmic decay over more than 3 decades in time. This type of behaviour is certainly not typical for the atomic motions in a

4 Complex polymers

0.8

Q = 19 nm−1

0.6

Sinc (Q,t)

b

TOF

0.4

PEO in PMMA/PEO

0.2

0.6 Q (nm-1): 1 1.5 2 3

0.4 0.2

PEO 0 0.1

1 0.8

backscattering

S (Q,t)/S(Q)

a

113

1

10

t (ps)

100

1000

0

0

20

40

60

t (ns)

Fig. 4.8 (a) Fourier transformed data of pure PEO and 35% PEO/PMMA [24]. While the pure PEO data are well described by a typical stretched exponential with stretching exponent 0.5, for PEO in the blend an extremely broad distribution of relaxation times has to be used (solid line). A logarithmic decay (dotted line) also describes the long-time tail in the blend-data. (b): Singlechain dynamic structure factor of PEO in the same blend [24]. The dashed region represents the elastic contribution as calculated by dynamic random phase approximation. The dotted lines illustrate the Rouse theory with Rouse relaxation rate obtained from the backscattering data. The solid lines show the results of the Rouse model with random friction. All data correspond to T = 400 K

standard glass-forming polymer at temperatures above the glass-transition and has to be related to the motional restrictions imposed by the almost immobile PMMA component. Another intriguing question is the behaviour of the large-scale dynamics (Rouse, reptation, etc.) in asymmetric blends. In this direction, we can mention a recent neutron scattering study on PEO chain dynamics in blends with high contents of PMMA [24]. Strong deviations from the Rouse behaviour were found. These results were interpreted as a consequence of the heterogeneous environment imposed by the almost frozen PMMA matrix [24] (see Fig. 4.8(b)). Currently, the chain dynamics in such dynamically asymmetric systems is still an almost open field. On the other hand, simulation results corresponding to bead-spring models [25] suggest that the so-called renormalized Rouse models can be an adequate framework for describing chain dynamics in asymmetric polymer blends. These models are based on generalized Langevin equations which incorporate a memory function characterizing the density fluctuations of the matrix around the tagged chain. Understanding of the confinement effects produced by dynamic asymmetry in a system still requires a large effort from both experiment and simulation. Moreover, it is worth emphasizing that they are not yet taken into account by any of the existing models for blend dynamics, and a general theoretical framework is needed. In this direction, we may comment on a possible new route opened by a recent work involving simulation of a binary blend of bead-spring polymer chains [26] that showed an analogous phenomenology. It was proposed that this type of behaviour is present in a higher-order MCT scenario or fold transition with an exponent parameter close to 1 (see, e.g. [27]). In such a framework, the origin of this behaviour could be attributed

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to competition between two mechanisms for dynamic arrest: intermolecular packing by the neighbouring fast component and confinement effects due to the presence of the slow component. We may speculate that the emergence of logarithmic-like decays for the correlation functions might be a general consequence of confinement in the α-relaxation. We note that logarithmic-like decays could also be envisaged in results on diverse systems confined within well-defined geometries (e.g. salol [28], poly(dimethyl siloxane) [29], poly(methyl phenyl siloxane) [29], water in pores [30] and water in clays [31].

4.3 Polymers with Large Side-Groups: the Family of Poly(n-alkyl methacrylates) Self-assembly, confinement, dynamic asymmetry and nanophase separation could be considered as general phenomena ubiquitously found in multicomponent soft systems. In fact, many of the interesting properties for technological applications of soft materials emerge as a consequence of these phenomena. Interestingly, all these features are not exclusive to multicomponent systems but might also be present in some complex homopolymers, as is the case for the family of poly(n-alkyl methacrylates) (PnMA’s) (see chemical formula in Fig. 4.9) [32, 33]. Structural studies based on x-ray diffraction have pointed to aggregation of the side groups of different monomeric units, forming self-assembled alkyl nanodomains [called polyethylene (PE)-like nanodomains], the size of which is related to the length of the side group. From a dynamic point of view, dynamic heat capacity measurements indicate the presence of two glass-transition temperatures in these samples, that presumably correspond to the freezing of the motions within the alkyl nanodomains (PE-like, αPE ) and of the main-chain dynamics (α ). This scenario was proposed mainly based on calorimetric, dielectric and mechanical studies, which are not selective for the different processes at a molecular level, and on x-ray diffraction, where mostly carbons are highlighted without the possibility of distinguishing mainchain and side-group contributions to the diffraction patterns. It is clear that a selective technique providing space–time resolution at a molecular level could be essential for a deeper investigation into the structure and dynamics of PnMAs. Neutron scattering combined with isotopic labelling is thus naturally the right tool to separately study the different molecular groups in these systems and (i) provide a definite proof for the nano-segregated structure in PnMAs and (ii) selectively follow the dynamics of alkyl nanodomains (αPE ) and that of the main-chain system (α ).

4.3.1 Proof for the Nanodomain Structure Due to the very similar scattering lengths of the isotopes present in fully deuterated PnMA samples, coherent neutron scattering – clearly dominating the incoherent

4 Complex polymers

115

CH3 -[CH2-C-]n C=O CH2 - -

- ... -

-

L(Å)

14 12

PEMA

PBMA

10

nC

8 0

CH3

S(Q) (arb. units)

PHMA

16

PE

PMMA 1 2

3

nC

4

5

6

7

PHMA-d18 PHMA-d5 PBMA-d14

PEMA-d10

0

0.5

1 Q (Å-1)

1.5

2

Fig. 4.9 Neutron diffraction pattern for fully deuterated PnMA samples (full symbols) and a PHMA sample deuterated only along the main-chain (empty diamonds) [34]. The line shows the structure factor of bulk polyethylene [35] for comparison. Results have been shifted along the vertical axis for clarity. The inset shows the chain-length dependence of the inter-domain size deduced from the position of the first structure factor peak. The chemical formula of PnMAs is also shown

contribution – reveals the correlations of all pairs of atoms equally weighted, i.e. the true structure factor S(Q). Figure 4.9 shows the structural results obtained for three perdeuterated samples with different numbers of carbons along the side-chain nC : poly(ethyl methacrylate) (PEMA) (nC = 2), poly(butyl methacrylate) (PBMA) (nC = 4) and poly(hexyl methacrylate) (PHMA) (nC = 6) [34]. In all cases, two main maxima can be distinguished in S(Q): a common peak (peak II) centered at ˚ −1 and another peak (peak I) located at smaller Q-values, whose about Q = 1.3 A position shifts towards lower Q-values with increasing side-chain length. These patterns are rather similar to those reported from x-ray measurements [33]. Now let us exploit neutron scattering selectivity. For PHMA, Fig. 4.9 shows the resulting partial structure factor when the side chain is hydrogenated and the main-chain is deuterated (empty symbols). In this sample, the negative scattering length of hydrogens compensates that of the linked alkyl carbon and the side group becomes almost invisible for neutrons. Therefore, the disappearance of peak II in the coherent scattering of the partially deuterated PHMA sample elegantly demonstrates that such a peak must be produced by correlations involving the side-group atoms. Figure 4.9 also shows the structure factor obtained for pure PE on a fully deuterated

116

A. Arbe and J. Colmenero 1

a

S(Q,t)/S(Q,0)

0.8

b

c

0.6

PEMA −1 Q=0.6Å

0.4

400K 420K 450K PHMA 480K Q=0.4Å−1 510K 520K

PBMA −1 Q=0.5Å

0.2 0

10

−3

10

−2

−1

10

0

10

t/a T(TR=480K)

1

10

2

−4

10 10

−3

10

−2

10

−1

10

0

10

t/a T(TR=480K)

1

10

−4

10

−3

10

−2

10

−1

10

10

0

1

10

t/a T(TR=480K)

Fig. 4.10 Master curves obtained for PEMA (a) PBMA (b) and PHMA (c) at the inter-mainchain peak I by using the rheological (shear compliance data) shift factors [34]. The reference temperature is always 480 K. The solid lines are fits to KWW functions with β = 0.5

sample above the melting point [35]. This shows very much the same features as peak II in PHMA. Thus, the local environment of the side groups in PHMA is nearly identical to that in bulk PE. We can conclude then that the common peak ˚ −1 (peak II) in PnMAs reflects correlations between atoms centred at Q = 1.3 A located in side groups of different monomers in an environment of other alkyl side-chains – strongly supporting the existence of alkyl nanodomains. On the other hand, the comparison between fully and partially deuterated PHMA patterns shows that the remaining low-Q peak in the latter has to reflect main-chain correlations. From the Bragg approximation, distances between main chains of around 10, 13 ˚ can be deduced for PEMA, PBMA and PHMA, respectively (see insert and 16 A in Fig. 4.9). This points to an increasing size of nanodomains as suggested by the previous x-ray studies.

4.3.2 Selective Dynamics Study: Anomalous Relaxation The previous structural study by neutron scattering provides a clear-cut proof for the existence of PE-like nanodomains in PnMAs, supporting the nanophase hypothesis. The correlations giving rise to peaks I and II in the structure factor would then reflect correlations between the structural subunits forming the mainchain system (peak I) and the alkyl-rich domains (peak II). The clear separation of these peaks in Q-space then allows selective study, at a molecular level, of the dynamics associated with each of the subsystems, i.e. their structural relaxations. This information can be obtained by NSE experiments at the two maxima. Focusing first on the relaxation of the main chains revealed by NSE on peak I, Fig. 4.10 shows that, independent of the length of the side-groups, this process exhibits the typical features of the α-relaxation in glass-forming systems: (i) the results at different temperatures collapse onto a single master curve when the timescale is scaled with the viscosity temperature dependence and (ii) the functional form can be described

4 Complex polymers

117

1 PEMA

a

PBMA

S(Q,t)/S(Q,0)

0.8

c

PHMA 330K 350K 372K 400K

b

0.6 0.4 0.2 0

300K 365K 420K 450K 480K

400K 450K 480K 520K 0.01

0.1

t (ns)

1

10

0.01

420K 450K 480K 0.1

t (ns)

1

10

0.01

0.1

1

10

t (ns)

˚ −1 ) for PEMA Fig. 4.11 Dynamic structure factor at the second (nanodomain) peak II (Q = 1.3 A (a), PBMA (b) and PHMA (c) [34]. Dotted lines are fits to KWW with β = 0.5 and solid lines to a logarithmic expansion

by a stretched exponential or Kohlrausch-Williams-Watts (KWW) function Φw (t) = DWF exp[−(t/τMC )β ], with a value close to 0.5 for the shape parameter β as is usually the case for polymers [3]. The amplitude is the Debye–Waller factor (DWF) accounting for the fast decay of the correlations in the microscopic regime below 2ps. Figure 4.10 shows that the three master curves differ only in the absolute timescale, which decreases with increasing length of the side-group, as a result of the plasticizing effect of the these groups. Thus, at the main-chain level, we observe completely standard α-relaxations. The same functional form describes very nicely the dynamic structure factor at peak II (alkyl subsystem) for PEMA, as shown in Fig. 4.11a. However, a qualitatively different relaxation pattern is revealed for the higher-order members investigated [Fig. 4.11b and c]: the decay of the intermolecular correlations within the alkyl nanodomains strongly deviates from the expected KWW with β ≈ 0.5 [dotted line in Fig. 4.11b as an example] and is nearly perfectly logarithmic (solid lines through the points). Thus a novel effect in the structural relaxation of these nanodomains emerges when the alkyl group contains at least about four hydrocarbons. The logarithmic functional form lacks a characteristic timescale. Therefore, for the higher-order PnMAs we can define τPE as the time where the dynamic structure factor at peak II decays a factor 1/e of its value at 2 ps (which takes into account the microscopic decay parametrized by the DWF). In this way, the τPE values are directly comparable with the respective τMC obtained at peak I. For the three polymers investigated τPE is faster than τMC . In PEMA the side chains are short and their motions are expected to be strongly coupled with those of the main chains. However, for the larger side-chains the dynamics strongly decouple. This is realized best for PBMA at 300K (approx. the glass-transition temperature of the main α-process) where the main-chain dynamics is frozen, while the side-chains exhibit the logarithmic decay in the NSE window. Thus, a dynamic asymmetry clearly develops between the main-chain system and the PE-like nanodomains for high-order PnMAs. This qualitatively different dynamics also persists towards high temperatures: for the investigated temperature range, the ratio τMC /τPE is ≈15 for

118


PEMA, while it becomes almost two orders of magnitude for PBMA and PHMA. Moreover, the comparison with the correlation time of pure PE [35] clearly shows that the dynamics of the PE-like domains differs more and more from that of bulk PE when the length of the side chain decreases. What could be the origin of the observed logarithmic decays? A possible explanation for the extremely stretched behaviour could simply rest on the presence of a strong gradient of mobilities along the side-chains. The dynamics of those side-chain atoms anchored to the main-chain would be strongly coupled to the main-chain atoms and thus would move relatively slow. On the other hand, those side-chain atoms further from the main chain would have more freedom and move as fast as PE-like atoms. In the framework of this qualitative picture we could invoke a broad distribution of relaxation times ranging from rather short times – similar to those found in PE melts – to the relaxation times characteristic of main-chain dynamics in each case. Similar ideas have been used to interpret the broad dielectric relaxation function observed in some diblock-copolymers (see, e.g. [36]). However, it is worthy of remark that it is not straightforward to deduce a distribution function of atomic mobilities / relaxation times from coherent data. Recent measurements of self-atomic motions via incoherent scattering [37] and coarse grained molecular dynamics simulations [38] have addressed this problem. On the other hand, the peculiar logarithmic relaxation function found probably demands more sophisticated models than those based on a simple “parallel” distribution of relaxation processes. For instance, it has been shown [39] that logarithmic decays naturally emerge in the framework of general kinds of model based on a hierarchically constrained dynamics, so that the relaxation of the system involves a sequential – rather than a parallel – series of correlated processes with an increasing characteristic time. An alternative explanation for the observation of logarithmic decays would emerge if we consider the structure of nano-domains for the alkyl groups together with the dynamic asymmetry between the collective dynamics of these groups (αPE ) and that of the main-chain subsystem (α). This situation might be similar to that found in some two-component miscible systems – for instance, the dynamically asymmetric polymer blends described in Sect. 12.3.4 – where the dynamics of the minority fast component is restricted (“confined”) by the slow one. It is noteworthy that in such systems a similar type of anomalous relaxation (i.e. logarithmic decay) has also been reported for the fast component. In that case, the observation was interpreted in terms of a high-order MCT transition, assuming the existence of two different arrest mechanisms for the fast component: bulk-like caging and some kind of “confinement effect” due to the presence of the slow component. We might speculate whether two-mechanisms also exist in PnMAs and a MCT scenario could be applied. We note however that in this case the effect of the anchorage between alkyl groups and the main chain probably needs to be taken into account. Obviously, the elucidation of the origin of the exotic dynamic behavior revealed by neutron scattering on these homopolymers demands further experimental and theoretical effort.

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4.4 Conclusions Due to the huge diversity of polymer-based materials and their technological potential, studying the dynamical behaviour of polymer systems with increasing complexity is an attractive and promising field. New phenomena emerge as a consequence of this complexity, which demands characterization at a molecular level from both basic and applied points of view. Thanks to the capability of deuterium labelling, neutron scattering is a unique experimental technique for selectively investigating the dynamics of the different components of complex polymer systems at different length scales – essential information within this framework. Here we have emphasized the role of neutron scattering studies for polymer blends and nanostructured homopolymers, where the concepts of dynamic asymmetry and confinement can play crucial roles. These are only a few representative examples from the wide field of soft matter [40] – there is still plenty of room to exploit the potential of neutron scattering in complex polymers! Acknowledgements We thank all our collaborators contributing to the research of our group in this subject. We also express our gratitude to Prof. D. Richter for the fruitful collaboration over many years addressing this kind of problems, and to our colleagues at the different neutron facilities. We acknowledge support by the European Commission NoE SoftComp, Contract NMP3CT-2004-502235, the “Donostia International Physics Center”, the projects MAT2007-63681, IT-436-07 (GV) and the Spanish Ministerio de Educación y Ciencia (Grant No. CSD2006-53).

References 1. Frick B, Richter D (1995) The microscopic basis of the glass transition in polymers from neutron scattering studies. Science 267:1939–1945 2. Colmenero J, Moreno AJ, Alegr´ıa A (2005) Neutron scattering investigations on methyl group dynamics in polymers. Prog Polym Sci 30:1147–1184 3. Richter D, Monkenbusch M, Arbe A, Colmenero J (2005) Neutron spin echo in polymer systems. Adv Polym Sci 174. Springer Verlag, Berlin Heidelberg New York 4. Richter D, Monkenbusch M, Willner L, Arbe A, Colmenero J, Farago B (2004) Direct observation of the crossover from α -relaxation to Rouse dynamics in a polymer melt. Europhys Lett 66:239–245 5. Pérez Aparicio R, Arbe A, Colmenero J, Frick B, Willner L, Richter D, Fetters LJ (2006) Quasielastic neutron scattering study on the effect of blending on the dynamics of head-tohead poly(propylene) and poly(ethylene-propylene). Macromolecules 39:1060–1072 6. Mukhopadhyay R, Alegr´ıa A, Colmenero J, Frick B (1998) Effect of blending on the methyl side group dynamics in poly(vinyl methyl ether). J Non-Cryst Solids 235–237:233 7. Genix A-C, Arbe A, Alvarez F, Colmenero J, Willner L, Richter D (2005) Dynamics of poly(ethylene oxide) in a blend with poly(methyl methacrylate): A quasielastic neutron scattering and molecular dynamics simulations study. Phys Rev E 72:031808(1)–(20) 8. Chahid A, Alegr´ıa A, Colmenero J (1994) Methyl group dynamics in poly(vinyl methyl ether). A rotation rate distribution model. Macromolecules 27:3282–3288 9. Cendoya I, Alegr´ıa A, Alberdi JM, Colmenero J, Grimm H, Richter D, Frick B (1999) Effect of blending on the PVME dynamics. A dielectric, NMR, and QENS investigation. Macromolecules 32:4065–4078

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10. Urakawa O, Fuse Y, Hori H, Tran-Cong Q, Yano O (2001) A dielectric study on the local dynamics of miscible polymer blends: poly(2-chlorostyrene)/poly(vinyl methyl ether). Polymer 42:765–773 11. Dion´ısio M, Fernandes AC, Mano JF, Correia NT, Sousa RC (2000) Relaxation studies in PEO/PMMA blends. Macromolecules 33:1002–1011 12. Arbe A, Alegr´ıa A, Colmenero J, Hoffmann S, Willner L, Richter D (1999) Segmental dynamics in poly(vinylethylene)/polyisoprene miscible blends revisited. A neutron scattering and broad-band dielectric spectroscopy investigation. Macromolecules 32:7572–7581 13. Colmenero J, Arbe A (2007) Segmental dynamics in miscible polymer blends: recent results and open questions. Soft Matter 3:1474–1485 14. Hoffmann S, Willner L, Richter D, Arbe A, Colmenero J, Farago B (2000) Origin of dynamic heterogeneities in miscible polymer blends: A quasielastic neutron scattering study. Phys Rev Lett 85:772–775 15. Lodge TP, Mcleish TCB (2000) Self-concentrations and effective glass transition temperatures in polymer blends. Macromolecules 33:5278–5284 16. Tanaka H (2000) Viscoelastic phase separation. J Phys Cond Matter 12:R207–R264 17. Proceedings of the 3rd international workshop on dynamics in confinement. In: Koza M, Frick B, Zorn R (eds.) Eur Phys J Special Topics 141 18. Genix AC, Arbe A, Arrese-Igor S, Colmenero J, Richter D, Frick B, Deen PP (2008) Neutron scattering investigation of a diluted blend of poly(ethylene oxide) in polyethersulfone. J Chem Phys 128:184901(1)–(11) 19. Tyagi M, Arbe A, Colmenero J, Frick B, Stewart JR (2006) Dynamic confinement effects in polymer blends. A quasielastic neutron scattering study of the dynamics of poly(ethylene oxide) in a blend with poly(vinyl acetate). Macromolecules 39:3007–3018 20. Lorthioir C, Alegr´ıa A, Colmenero J (2003) Out of equilibrium dynamics of poly vinyl methyl ether segments in miscible polystyrene–polyvinyl methyl ether blends. Phys Rev E 68:031805(1)–(9) 21. Tyagi M, Arbe A, Alegr´ıa A, Colmenero J, Frick B (2007) Dynamic confinement effects in polymer blends. A quasielastic neutron scattering study of the slow component in the blend poly(vinyl acetate)/poly(ethylene oxide). Macromolecules 40:4568–4577 22. Garc´ıa Sakai V, Maranas JK, Chowdhuri Z, Peral I, Copley JRD (2005) Miscible blend dynamics and the length scale of local compositions. J Polym Sci Part B Polym Phys 43: 2914–2923 23. Farago B, Chen C, Maranas JK, Kamath S, Colby RH, Pasquale AJ, Long TE (2005) Collective motion in poly(ethylene oxide)/poly(methylmethacrylate) blends. Phys Rev E 72:031809 (1)–(11) 24. Niedzwiedz K, Wischnewski A, Monkenbusch M, Richter D, Genix A-C, Arbe A, Colmenero J, Strauch M, Straube E (2007) Polymer chain dynamics in a random environment: heterogeneous mobilities. Phys Rev Lett 98:168301(1)–(4) 25. Moreno AJ, Colmenero J (2008) Entanglement-like chain dynamics in non-entangled polymer blends with large dynamic assymetry. Phys Rev Lett (1)–(4) 26. Moreno AJ, Colmenero J (2006) Is there a higher-order mode coupling transition in polymer blends? J Chem Phys 124:184906(1)–(6) 27. Götze W, Sperl M (2002) Logarithmic relaxation in glass-forming systems. Phys Rev E 66:011405(1)–(17) 28. Zorn R, Hartmann L, Frick B, Richter D, Kremer F (2002) Inelastic neutron scattering experiments on the dynamics of a glass-forming material in mesoscopic confinement. J Non Cryst Solids 307–310:547–554 29. Schönhals A, Goering H, Schick Ch, Frick B, Zorn R (2005) Polymers in nanoconfinement: What can be learned from relaxation and scattering experiments? J Non Cryst Solids 351: 2668–2677 30. Gallo P (2000) Single particle slow dynamics of confined water. Phys Chem Chem Phys 2:1607–1611

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31. Swenson J, Bergman R, Longeville S (2001) A neutron spin-echo study of confined water. J Chem Phys 115:11299–11305 32. Beiner M (2001) Relaxation in Poly(alkyl methacrylate)s: Crossover region and nanophase separation. Macromol Rapid Commun 22:869–895 33. Beiner M, Huth H (2003) Nanophase separation and hindered glass transition in side-chain polymers. Nat Mat 2:595–599 34. Arbe A, Genix A-C, Colmenero J, Richter D, Fouquet P (2008) Anomalous relaxation of self-assembled alkyl nanodomains in high-order poly(n-alkyl methacrylates). Soft Matter 4: 1792–1795 35. Arbe A, Colmenero J (2009) Characterization of the ‘simple-liquid’ state in a polymeric system. Self and pair correlation functions. Phys Rev E 80:041805(1)–(13) 36. Lorthioir C, Alegr´ıa A, Colmenero J, Deloche B (2004) Heterogeneity of the segmental dynamics of poly(dimethylsiloxane) in a diblock lamellar mesophase: dielectric relaxation investigations. Macromolecules 37:7808–7817 37. Arbe A, Genix A-C, Arrese-Igor S, Colmenero J, Richter D (2010) Dynamics in poly(nalkyl methacrylates): a neutron scattering, calorimetric, and dielectric study. Macromolecules 43:3107–3119 38. Moreno A.J, Arbe A, Colmenero J (2011) Structure and dynamics of self-assembled comb copolymers: comparison between simulations of a generic model and neutron scattering experiments. Macromolecules 44:1695–1706 39. Brey JJ, Prados A (2001) Slow logarithmic relaxation in models with hierarchically constrained dynamics. Phys Rev E 63:021108(1)–(4) 40. Garc´ıa Sakai V, Arbe A (2009) Quasielastic neutron scattering in soft matter. Curr Opin Colloid Interface Sci 14:381–390

Chapter 5

Solid Polymer Electrolytes Janna K. Maranas

5.1 Application and History Electrochemical devices contain three primary components: the anode, the cathode, and the electrolyte. In the lithium–ion battery used in portable electronic devices such as laptops and cell phones, the anode is carbon, the cathode is a metal oxide, and the electrolyte is a lithium salt dissolved in an organic solvent [1]. In this chapter, we consider an electrolyte where the organic solvent is replaced with a polymer. We refer to this as a solid polymer electrolyte and use the abbreviation SPE throughout. In 1973, a letter appeared in the journal Polymer reporting the first observation of ionic conductivity in complexes of salts in poly(ethylene oxide) (PEO) [2]. Such systems were proposed for use as electrolytes in lithium batteries in 1979 [3]. PEO remains one of the only polymers to solvate lithium salts, despite complications from its crystalline nature at room temperature. The suggestion of SPEs derived from the dendritic Li growth which occurred when a liquid electrolyte was used with a Li metal anode: a safety issue. Other solid electrolytes with high conductivity such as β-alumina did not lead to dendrite formation, but could not be cycled repeatedly due to volume changes during operation. This problem was ultimately resolved by replacing Li metal with the carbonaceous materials in use today. Regardless, the use of SPEs offers advantages other than the possibility of the higher-energy Li metal anode. Removal of the solvent means that a casing is not required: this impacts flexibility, weight, and thickness. Despite these advantages, SPEs do not have sufficient conductivity to power a portable device at room temperature. This is illustrated in Fig. 5.1, where the conductivity of 22 PEO-based SPEs is plotted as a function of temperature. The gray

J.K. Maranas () Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802 USA e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 5, © Springer Science+Business Media, LLC 2012

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Copolymer: No Zwitterion [4] Copolymer + Zwitterion [4] Copolymer + NP, no Zwitt. [4]

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PEG+MnZnAl2O3 [5] PEOLiTF+Al2O3 (37nm) [6]

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PEO-co-propylene oxide 100% [10] PEO-LiCF3SO3 + CO2 [11] PEO-LiCF3SO3 no CO2 [11]

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PEO6:LiTFSI - Crystalline [14]

Temperature (°C)

Fig. 5.1 The temperature dependence of ionic conductivity for 22 different PEO-based solid polymer electrolytes

box denotes the region of practical conductivity, as defined by that available from high-conductivity solid lithium conductors, such as β-alumina. This is only achieved by SPEs at temperatures above ∼80 ◦ C. Some of the curves are continuous, whereas others have a sharp decrease below the melting point of pure PEO: ∼60 ◦ C. This difference is related to the presence of crystallinity, which in most cases leads to a marked decrease in ionic conductivity. A wide variation is apparent: the identity and size of the anion, presence, identity and loading of nanoparticle fillers, and thermal history of the individual sample all influence ionic conductivity. The system PEO + LiClO4 was one of the first investigated, has been studied using various neutron techniques, and is the focus of work in our laboratory and this chapter. We denote the conductivity of PEO+ LiClO4 with the thick curve marked 1 in Fig. 5.1. We will also discuss the influence and mechanism of nanoparticles and the possibility of high conductivity through crystalline phases. The SPE denoted by curve 2 is filled with nanoparticle additives, and that in curve 3 is a low molecular weight, fully crystalline material. Both exhibit improved conductivity compared to curve 1. It is generally accepted that lithium motion is coupled to motion of the polymer backbone, or segmental motion (for more information see Chap. 4). The timescale of this motion is within the dynamic window of QENS spectrometers, and thus the technique is often used to study polymer motion in SPEs. In absence of crystallinity, the temperature dependence of conductivity and the temperature dependence of

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segmental motion are similar. When the polymer has crystallized, lithium movement is thought to be through the amorphous regions of the polymer, although this has been challenged by observations of high conductivity in fully crystalline systems. Both PEO mobility and crystallization are slowed upon salt addition, with the extent dependent on the salt concentration. Thus, an easily crystallizable polymer melt with high mobility becomes more amorphous and less mobile if used as a SPE.

5.2 Phase Behavior, Crystallinity, and Thermal History The conductivity of PEO-based SPEs decreases rapidly below 60 ◦ C, as illustrated in Fig. 5.1. Since this is the melting temperature (Tm ) of pure PEO, the decrease in mobility below 60 ◦ C is attributed to the formation of crystalline regions with restricted mobility [4–7]. While pure PEO readily crystallizes, the addition of lithium salt significantly disrupts crystallization [8–12]. Most PEOy LiX systems form two crystalline phases at a given salt concentration, as illustrated by the schematic phase diagram in Fig. 5.2. We indicate the salt concentration by the y in PEOy LiX, where y is the number of ether oxygens per lithium. Thus, the PEO8 LiClO4 system has ClO4 as the anion and an ether oxygen to Li ratio of 8:1. This frequently studied sample forms two crystalline phases at this concentration: PEO6 and PEO [13]. The PEO6 phase incorporates salt with an ether oxygen to Li ratio of 6:1, whereas PEO does not contain salt. As illustrated in Fig. 5.2, the PEO phase melts at 50 ◦ C and the PEO6 phase at ∼65 ◦ C. Both melting points vary with Li concentration, with their minimum values coincident at the eutectic point: PEO10 LiClO4 . At lithium contents less than this, the PEO phase is more stable, with PEO6 melting at 50 ◦ C.

150 °C

L 100 °C L + PEO6

L + PEO3

L +PEO

PEO6 + PEO3

50 °C PEO +PEO6 40:1

10:1

6:1

Fig. 5.2 Phase diagram for the PEOy LiClO4 system. The notation x : 1 indicates the EO:Li ratio

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Another contribution of neutron techniques in the area of SPEs is the identification that the structure reminiscent of the PEO6 phase persists into the liquid region. At temperatures up to 100 ◦ C, a random packing of PEO cylinders enclosing Li ions with coordination 5:1 has been observed using neutron diffraction with Li substitution [14]. The crystallization kinetics of PEO and PEO6 are significantly different: specifically the PEO phase crystallizes on the order of hours, whereas the PEO6 phase crystallizes on the order of weeks [9, 15, 16]. The time to crystallization of the PEO6 phase varies considerably as evidenced by comparing conductivity data obtained on first heatings available in the literature. Considerable variation is observed for the exact same system: for example, ref. [5] indicates that conductivity measured immediately after cooling for the PEO8 LiClO4 system reproduces that obtained during the first heating scan, indicating considerable recrystallization within a short time. This is unlikely to be the PEO phase because stoichiometry limits it to 15% and its formation has little influence on conductivity [15, 16]. Most data, including our own, indicate significant differences between data obtained on first and second heatings. There is also considerable disagreement between conductivity curves for the same sample reported by different groups, as confirmed by examining the three curves for PEOLiClO4 in Fig. 5.1. The phase diagrams [13, 17], crystallization kinetics [18–20], and connection to the liquid phase structures [21, 22] for other PEOy -LiX systems are available, with the phase diagram for the PEOy -LiTFSI system dependent on molecular weight [17, 20, 23]. PEO, like other polymers, is semicrystalline: this means that even within the solid regions of the phase diagram, PEOLiClO4 will have both crystalline and amorphous regions. Although it was suggested 20 years ago that ion motion along the helical structure of crystalline PEO may be the primary conduction mechanism [24], the majority of studies have supported conduction through the amorphous regions. The data in Fig. 5.1 represent a recent demonstration that crystalline SPEs conduct: curve 3, for fully crystalline PEO + LiTFSI, has the highest room temperature conductivity among the 22 curves presented. In addition to PEO + LiTFSI, the salts LiPF6 , LiAsF6 , and LiSbF6 also form crystalline complexes with high conductivity [25]. The crystalline phase with stoichiometry PEO6 :LiX, where X is the anion, forms by pairs of PEO chains that fold together in tunnels. The lithium cations are located within the tunnel, while the anions are positioned outside of the tunnel. Characteristic of these materials is the low molecular weight of the PEO: as the molecular weight is increased from 1 to 2 kg/mol, the conductivity drops by three orders of magnitude [26]. This suggests that as the material becomes semicrystalline, the limiting factor for conductivity is the morphology rather than the presence of crystalline regions. This position is supported by the observation that conductivity increases by a factor of 5–20 when SPEs are stretched [27]. While the crystal structures have been resolved using X-ray diffraction [13,17], and macroscopic crystal structure observed with optical measurements [9, 15, 18, 19], more limited information about the nanoscale structure of the crystalline regions in SPEs is available. We have recently studied the lamellar structure of the PEO phase in the SPE PEOy LiClO4 [28]. This is a precursor to study of the PEO6 structure; more difficult yet also more interesting because it can be conductive.


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Table 5.1 Connection between EO:Li coordination, the eutectic composition and maximum conductivity SPE system PEOy LiClO4 PEOy LiTFSI PEOy LiI PEOy LiAsF6

Coordination (EO:Li) Overall Crystalline y = 7.5 : 1 6:1 [13] y = 7.5 : 1 6:1 [20, 23] y=5:1 3:1 [29] – –

Concentration (EO:Li) Melt Eutectic 5:1 [14] 10:1 [13] 5:1 [21] 11:1 [20, 23] 3.5:1 [22] 11.5:1 [29] – 20:1 [13]

Max. conductivity 12:1 [13, 28] 11:1 [17] 15:1a [30, 31] 20:1 [13]

Observations made using neutron scattering are bolded a From computer simulation

The high conductivity through the PEO6 crystalline phase, along with the observation of similar local ordering with intermediate range structure in the melt, suggests that local regions of this structure may be important to conductivity within amorphous regions. The local melt lithium coordination resembles that of the lithium containing crystalline phase that would form if the temperature were lowered at that overall composition: this is illustrated in the first three columns of Table 5.1, which show the overall composition, and the coordination observed in the lithium containing crystalline phase and the melt. It is interesting to note that the highest conductivity is normally found near the eutectic composition in SPEs. This is illustrated in the last two columns of Table 5.1. Because the eutectic composition is that at which the two solid phases PEO and PEO6 have the same energy, it is reasonable to suggest that the liquid phase energies of local regions with these structures are also most similar at this composition. This is important because fluctuations in concentration would be maximized near the eutectic, providing a connection between these fluctuations and high conductivity. If a local PEO6 structure is formed via a fluctuation and either lasts longer or has larger spatial extent, this may improve conductivity. In order for these fluctuations to exist, a liquid–liquid spinodal line must be buried within the solid region of the phase diagram: this has not yet been investigated.

5.3 Dynamics in the Liquid State Most dynamic studies on SPEs have used neutrons because polymer motion at relevant temperatures is within the dynamic window of neutron instruments, and because ion movement complicates measurements like dielectric spectroscopy. In addition to decreasing the crystallinity of SPEs, the lithium salt also significantly decreases the PEO mobility, as demonstrated by QENS on various PEOy -LiX systems in the liquid phase [32–37]. The extent of the decrease depends on lithium concentration, anion identity and temperature. It is generally accepted that the lithium ions coordinate with the ether oxygens on the PEO chains, thus slowing the segmental motion of the polymer. In some systems, a second, slower process

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Table 5.2 The connection between dynamics measured using QENS, the most stable crystal phase, and the liquid phase coordination

System PEO7.5 LiClO4 PEO(15 or 20) LiI PEO7.5 LiTFSI PEO(15 or 20) LiBETI

Number of dynamic processes 2 [28, 32] 1 [34, 38] 2 [32, 33] 2 [35, 36]

Most stable crystalline phase PEO6 [13] PEO3 [29] PEO6 [20] PEO6 [34, 38]

Liquid phase coordination [EO:Li] 5:1 [14] 3.5:1 [22] 5:1 [21] Unknown

Observations made using neutron scattering are bolded

(evidenced by the requirement of two line shapes in fitting) emerges with the addition of salt: the commonality among them is that at the salt concentration where the measurements are made, PEO6 is the most stable crystalline phase. QENS measurements are made in the liquid state above the melting temperature. However, the observation of PEO6 phase remnants above the melting point suggests that the second process may be connected with this structure. Table 5.2 consolidates available experimental observations of dynamics, the most stable crystalline phase, and the liquid phase coordination. In all three systems where PEO6 is the most stable crystalline phase the second process is observed. In contrast, in the system PEO(15 or 20) LiI, which forms only the PEO3 and PEO crystalline phases, the second process is not observed. The origin of the extra process has been attributed to the formation and disruption of “crosslinks” between Li+ and ether oxygen atoms. Our investigation of polymer mobility in these systems shows that the segmental process of PEO slows with salt addition, in agreement with previous QENS measurements on this system. In addition, by extending our measurements to longer timescales, we observe the appearance of a second process. The data are taken using two neutron spectrometers at the NIST Center for Neutron research, in Gaithersburg, MD, USA: the disk chopper time of flight spectrometer (DCS) and the high flux backscattering spectrometer (HFBS). The data from each instrument are reverse Fourier transformed to the time domain, and combined in a single decay curve. This combination of spectrometers covers the time range between 1 ps and 4 ns. As noticed by others investigating SPEs using QENS, satisfactory fitting of the data requires two stretched exponential functions, and we assign stretching parameters and characteristic times for each. Once these parameters are available, the decay curves describing each process may be constructed individually, as presented in Fig. 5.3. The top portion of the figure shows the influence of salt on the segmental process. At the temperature illustrated, the SPE is in the liquid state with no crystallinity. As the salt concentration increases, this process slows down, with the difference between neat PEO and PEO4 LiClO4 being greater than that between PEO and a polymer with a much higher glass transition temperature (Tg ), such as PMMA. Clearly, the presence of salt has a large impact on the segmental dynamics.


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1 KWW1 T = 75°C q = 1.04 Å-1

0.9 0.8

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100

Time (ps) Fig. 5.3 Self intermediate scattering function obtained for QENS data and corresponding to the two dynamic processes observed in SPEs. Top: The segmental motion of PEO is increasingly slowed by ion addition. Bottom: Biased rotation of PEO6 cylinders is only observed when ions are present. The inset illustrates rotation of the PEO protons around a central lithium ion. The shaded circles at each position illustrate the probability of that rotational site: a darker shade indicates larger probability

Conductivity above the melting point originates from ion movement through the melt, assisted by segmental motion of the polymer. In the accepted picture, a lithium ion resides in a series of coordination sites, punctuated by hops between them. The hops are assisted by the segmental motions of the polymer, which present

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Higher mobility

1.E-03

0.8

1.E-05

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0.07

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4:1

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10:1

14:1

30:1

100:1

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Fig. 5.4 The trade-off between polymer mobility and ion content. The ion content is given as the ether oxygen:Li ratio on the x-axis. The mole fraction of lithium is plotted as filled circles with the value given to the right of each circle. As the ion content decreases, the segmental relaxation times of the polymer (right axis) also decrease. Segmental relaxation times are assigned from QENS data, as the characteristic time of the short time process. Because conductivity depends on both ion content and polymer mobility, intermediate ion contents provide the highest conductivity (left axis)

open sites or lower energetic barriers for hops out of the current site. At higher Li concentrations, more charge carriers are present, but the PEO motion assisting their movement is slow: as a result, an optimum concentration range for conductivity is observed (see Fig. 5.4). As mentioned above, this range includes the eutectic point, which for this system is 10:1, and fluctuations associated with this concentration may also contribute to its optimal conductivity. At temperatures below the melting point, the picture is more complicated than balancing polymer mobility with the number of charge carriers. Specifically, at 22 ◦ C, the same concentration range provides maximum conductivity, despite the fact that at 14:1, 30% of the sample has crystallized. Since the crystal formed was pure PEO, this has the effect of concentrating the amorphous region to a concentration of 8:1; compared to the 8:1 sample, there are fewer charge carriers and a portion of the sample is blocked by pure PEO crystal regions. Despite this, the 14:1 sample has comparable conductivity, suggesting that the presence of crystalline regions may have a stabilizing influence. Considering the potential importance of local structures similar to PEO6 above the melting point, it is possible that the presence of crystalline PEO stabilizes these structures, thus enhancing conductivity. The second, slower process does not decay to zero and is independent of concentration, as illustrated in the lower panel of Fig. 5.3. The appearance of an


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elastic incoherent structure factor (EISF) suggests that the motion is restricted. The characteristic relaxation times are Q-independent, which suggests the restricted motion is rotational. Given the link between the second process and the amorphous phase remnant of the PEO6 structure outlined above, it is likely that the rotation ˚ consistent involves this structure. The rotation occurs around a circle of radius 3 A, with the Li-ether oxygen distance observed in liquid state remnants of the PEO6 crystal structure. Since the PEO chains orient themselves around the Li atoms, it is reasonable to expect a rotation of this structure. The rotation is described by a 6-site weighted jump model, where the weighting restricts the rotation to an angle θ rather than tracing the entire circle, as indicated in the inset to the figure. The temperature in Fig. 5.3 is 50 ◦ C: the melting point of the PEO6 crystal. Although crystallization of this structure is slow and none is detected within the timescale of the measurement, it is likely that nucleation of small structures occurs, and it is the rotation of these structures that is detected. When the temperature is raised to 75 ◦ C, the rotation is less clear: at this temperature the rotating structures occur via fluctuations, which do not last long enough for complete rotation.

5.4 Influence of Nanoparticle Additives Nanoparticle fillers of TiO2 [5, 39–41], SiO2 [40], γ-LiAlO2 [42], and Al2 O3 [5, 43–49] [γ−, α−, β-, and amorphous forms] have been added to solid polymer electrolytes. When reduced to a few microns or smaller, these fillers increase the conductivity [50,51]. Adding nanoparticles influences the conductivity in two ways: the sharp drop associated with crystallinity is avoided and the entire curve is shifted to larger values. This is observed for various filler types, anions LiX, and salt concentrations, although the majority of measurements are of the PEO8 LiClO4 system with Al2 O3 . Active and passive particles (active particles contain Li and thus serve as conducting pathways) have similar influences, and optimal effects are obtained at particle loadings of 5–15 wt. %. Available studies are consistent in finding that crystallinity is reduced when nanoparticles are present [4, 48, 49]. This is true in regions of the phase diagram where PEO6 and PEO are the crystalline phases (the majority of studies), but also when PEO3 and PEO are the crystalline phases, as occurs with PEOy LiI [52] and PEOy NaI [53]. For PEOy LiClO4 , it appears that the phase most influenced by the presence of nanoparticles is not the fastforming PEO phase, but the slower-forming PEO6 phase. Our group has studied the influence of nanoparticles on crystallization of the PEO phase in PEOy LiClO4 [54]. The pure PEO phase is isolated via thermal history: it forms within hours, whereas the PEO6 phase requires weeks. We find marginal influence on the rate or extent of the pure PEO crystalline phase with the addition of nanoparticles. The time to re-crystallize PEO6 with nanoparticles is estimated to be around 1 month, with the crystal structure differing from that normally found in PEOy -LiX electrolytes [49]. The conductivity of nanofilled SPEs is sensitive to heat treatment, with conductivity increasing even above the melting point with annealing at moderate temperatures

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before the first heating [51]. The authors of ref. [5] suggest that the ether oxygen atoms are attracted to the nanoparticle surface, lowering the tendency of PEO to crystallize and creating amorphous regions that promote lithium-ion transport. The second effect of nanoparticles, the enhancement of the conductivity above the melting point, implies that a reduction in crystallinity cannot be the sole mechanism by which nanoparticles improve conductivity. One possibility is the interaction of the particles with the ether oxygens on the polymer chain, the lithium ions or the anions. Depending on the interaction, the effect will be different. If the anions interact substantially with the particles, more charge carriers are created, thus increasing the transference number (the fraction of cations contributing to the total conductivity). This is observed but varies with particle type [49]. If the ether oxygens or lithium ions interact with the particles, it will alter their interactions with each other, creating “pathways” for fast ion transport near the particle surface independent of polymer mobility. Some of these hypotheses have been tested, for example by comparing acidic, basic and neutral particles [45, 55, 56]. Different conclusions have been reached regarding the type of particle that most influences conductivity, thus the way in which these interactions increase conductivity above the melting point remains unclear. One commonality among all possible mechanisms regarding surface interactions is the direct connection to the amount of surface present. The particles could also alter the dynamics of the polymer host, potentially enabling faster ion mobility promoted by the polymer. In contrast to the effect that lithium salt has on PEO dynamics, the change in mobility of a SPE upon addition of nanoparticles is not well characterized. In one study using QENS, the change in polymer host dynamics is separated from changes in crystallinity by using a completely amorphous sample: a cross-linked copolymer of PEO with poly(propylene oxide) (PPO) [57]. In this study the authors conclude that the mobility of the polymer host is not responsible for the increased conductivity based on the fact that the only change with nanoparticle addition is a larger fraction of relaxation processes that occur outside the timescale of the instrument (>275 ps). We used QENS to measure the PEOy LiClO4 + Al2 O3 system, and the results confirm this suggestion, despite the differences between the two systems. We studied y = 8, 10, and 14 SPEs with 11 nm γ-Al2 O3 particles at loadings of 5, 10, and 25 wt. %. In all cases the segmental dynamics of PEO are not influenced by the presence of nanoparticles. This is consistent with the observation that Tg does not change when nanoparticles are introduced. The slower rotational process is not influenced by nanoparticles, with the exception of the y = 8 sample, illustrated in Fig. 5.5. Segmental motion is unaffected but the addition of nanoparticles further restricts the angle of rotation compared to the PEO8 LiClO4 sample without nanoparticles. The difference between the y = 8 SPE and the others can be appreciated by examining Fig. 5.2. This salt concentration is the only one where PEO6 is the most stable crystalline structure. The y = 10 SPE is at the eutectic composition, where both are equally favored, and the pure PEO crystal is favored in the y = 14 SPE. The γ-Al2 O3 particles have both acidic and basic sites and the acidic (electron accepting)


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Fig. 5.5 The use of QENS to determine the influence of nanoparticles on PEO dynamics for PEO8 LiClO4 . This is the only lithium content where dynamics are affected, and only the slow rotational process changes. The inset to the figure illustrates that rotation becomes more restricted in the presence of nanoparticles. The shaded circles at each position illustrate the probability of that rotational site: a darker shade indicates larger probability

sites can coordinate with the ether oxygens in PEO. This is more likely when the PEO chains are not coordinated with lithium, and thus we expect that PEO will interact with the particle surface more readily than PEO6 or PEO coordinated with lithium outside of this structure. Since the rotation underlying the second process is associated with the PEO6 structure, the invariance of this process to nanoparticle addition for y = 10 and 14 indicates that these structures do not interact with the nanoparticle surface. This is consistent with the idea that pure PEO is the most stable crystal structure for PEO14 LiClO4 . At the eutectic, PEO and PEO6 are equally favored energetically, but PEO is also more likely to interact with the particle surface than PEO5 . In the y = 8 sample there are more PEO6 structures, both because this is the most stable crystal and because the lithium concentration is higher. In this case, the increased restriction for rotation of the PEO6 structure suggests that it interacts with the nanoparticle surface. One difference between our study and previous investigations of nanocomposite SPEs is our consideration of a range of salt concentrations. Normally, nanocomposites are studied at the salt concentration that provides optimum conductivity without particles: for the PEOy LiClO4 system the normal target is y = 8. Varying the salt concentration led to an interesting observation: the conductivity is improved only at the eutectic composition, as illustrated in Fig. 5.6. This differs from other studies showing improvement for PEO8 LiClO4 . While the reasons for the difference

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Fig. 5.6 The influence of nanoparticle fillers on conductivity of solid polymer electrolytes. The 11 nm alumina particles significantly influence conductivity at the eutectic point of 10:1, whereas only modest changes are observed at other ion contents

are uncertain; thermal history, water content, and extent of particle aggregation all influence the conductivity of filled systems. Our results for y = 8 were repeated on three occasions, including two separately prepared batches. The results did not change within error, indicating that we were successful in maintaining the same set of conditions across samples. It is useful to consider the reasons conductivity enhancement via nanoparticles is localized at the eutectic composition. Above we considered that the eutectic composition typically presents the highest conductivity in unfilled SPEs, and suggested a possible link between this observation and concentration fluctuations. If the PEO6 structure is particularly efficient for lithium movement, its formation via fluctuations to a larger spatial extent and/or for longer times at the eutectic composition would play a role in increasing conductivity. We consider that nanoparticles may stabilize fluctuating structures via the interaction of these structures with the nanoparticle surface. If this is the case, fluctuations producing the PEO6 structure would be stabilized at the eutectic. However, we observe the opposite: the PEO6 structures do not interact with the particle surface for y = 10, as QENS reveals no change in their rotation. In contrast the PEO6 structures in the y = 8 sample do interact with the particle surface, yet the conductivity in unaffected. One explanation for this is that although PEO6 structures are stabilized by interaction with the particle surface, they also become directed along particle surfaces. As will be discussed in the following section, nanoparticles are aggregated under typical preparation methods for SPE samples. At the eutectic, the interaction of PEO, rather than PEO6 with the particle surface could allow for stabilization of fluctuations without directing the PEO6 structures. This discussion pertains to conductivity measurements above the melting point, in which the relevant structures appear via fluctuation. Our QENS results are made at the eutectic temperature, 50 ◦ C, before allowing enough time for complete crystallization. Under these conditions, we expect the rotating structures are crystalline nuclei. At higher temperatures, the observed motion is not pure


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rotation; as discussed above, at higher temperatures, it appears that the time scale of fluctuations is less than that of rotation, and thus the rotating structures break up before the rotation is complete.

5.5 Nanoparticle Dispersion The level of nanoparticle dispersion impacts both the available surface area and the average distance between nanoparticles. Depending on particle loading and aggregation, it may be reasonable to compare the nanofilled SPE with a SPE in confinement. Considering the 10 wt. % SPE filled with 11 nm particles discussed above, if the nanoparticles are evenly dispersed, the average interparticle spacing will be 18 nm. If the particles aggregate into 200 nm clusters, the average spacing will be 325 nm. Within some of this range, the confinement introduced by addition of nanoparticles to a Li salt/polymer complex can affect both crystallization and polymer mobility. Nanoparticles without Li salts can aggregate significantly in polymer nanocomposites. Often transmission electron microscope (TEM) micrographs reveal clusters several times larger than the primary particles, separated by distances larger than the cluster size [58–60]. These clusters grow in number as the particle loading is increased and eventually form percolating networks. Small angle neutron scattering (SANS) measurements confirm this picture: in latex-silica nanocomposites, aggregates containing a constant number of particles co-exist with single particles [61]. Nanocomposites of PEO with silica particles are reported to produce good particle dispersion, although aggregation occurs on annealing [62]. The extent of particle aggregation depends on many factors including the method of sample preparation, particle size and loading [61, 63], polymer molecular weight [59], and pH [64]. With respect to SPEs, few studies address particle aggregation. In one study using 3.5 nm ZnO particles, larger aggregates are significant above 5% particle loading [63]. In another, SANS was performed in order to assess particle dispersion in a polymer/Li salt/particle system at 10 and 20% particle loadings [65]. The data in this study were consistent with strings of aggregates, two particles wide, which formed a fractal network. Although the state of aggregation is unknown in most studies reporting increased conductivity in nanofilled SPEs, we can examine the influence of the particles on quantities that are normally reported, such as the glass transition temperatures. The Tg ’s of PEO/Li-ion salts doped with Al2 O3 [47, 66], TiO2 [67], and SiO2 [68] nanoparticles do not change upon nanoparticle addition. Other studies have reported increases [69, 70] or decreases [71, 72] of Tg in the range of 2–7 ◦ C. Among results in the literature, we have found no correlation between changes in Tg and salt or nanoparticle identity, or the extent of conductivity improvement. In samples measured in our lab, a Tg difference of 1 ◦ C between PEOy LiClO4 filled and unfilled with Al2 O3 nanoparticles was observed. The small changes and lack of correlation with improvement in conductivity suggests that either the nanoparticles

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Fig. 5.7 The dispersion of alumina nanoparticles in SPEs. Top panel SANS data indicating the primary particle size and significant clustering for pure PEO with 10 wt. % nanoparticles, and 10:1 PEO + LiClO4 with 5, 10, and 25 wt. % nanoparticles. Although not shown, the curves for all lithium contents and nanoparticle loadings are similar. Curves are shifted on the y-axis for clarity. Bottom panel: field emission scanning electron microscopy image of PEO with 10 wt. % alumina nanoparticles. Images with lithium are similar but have less contrast. The box indicates the size scale accessible with SANS relative to the scale of the image

aggregate or they do not impact polymer dynamics. This is consistent with the invariance of segmental motion to nanoparticle addition. To assess aggregation, we studied nanocomposites of the PEOy LiClO4 system with varying lithium concentration and particle loading [54]. The samples were prepared using the method most often described in the SPE literature: dissolving the particles and polymer in anhydrous acetonitrile and sonicating. All the nanocomposites we studied showed similar features, indicative of significant particle aggregation. Representative data is shown in Fig. 5.7. Field emission scanning electron microscopy (FE-SEM) images show aggregation with a wide


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range of cluster sizes. Some are as large as a micron in size and others less than 50 nm. The clusters less than 50 nm appear spherical, with ellipsoidal clusters apparent at intermediate sizes. While this method images a specific area of the sample, SANS measurements confirm that this picture is representative. Above the melting point, the only feature in the SANS data is the nanoparticle structure: the primary particle size appears, in combination with spherical clusters of ∼20 nm and 40 × 140 × 200 nm ellipsoidal clusters. Approximately 60% of the aggregates are larger than the SANS size window. The intensity but not the shape changes with particle loading, indicating similar nanoparticle structure regardless of particle loading. Although it appears that under conditions typically used to prepare SPEs the particles aggregate to such an extent that confinement effects are not relevant, it is useful to consider the expected influence if a dispersed nanocomposite could be prepared. For the purposes of this discussion, we consider a well-dispersed SPE with 11 nm particles at 10 wt. % loading; above we cited the typical particle spacing in this case as 18 nm. As evidenced by a large amount of literature on the effect of confinement in pure polymer thin films, the Tg and thus dynamics of a polymer confined to this extent can be significantly affected, but this observation depends on the nature of the confinement [73, 74], which may or may not include surface interactions. Many studies have appeared on the dynamics of polymer thin films both at freestanding interfaces and those supported by a substrate. Free-standing polymers are not influenced by substrate interactions and show the largest changes in Tg [75]. In supported polymer thin films, the Tg can either increase or decrease, depending on the nature of the polymer–substrate interaction. For example, the Tg of PMMA at a silica substrate interface increased [76], the Tg of PS on silica decreased, and the Tg of PMMA on alumina decreased [77]. It has been shown that the Tg ’s of capped samples in which the polymer wets the supporting surfaces do not change [78], but will vary if the polymer is nonwetting, in which case the polymer–substrate interface will be more like a free-standing interface [79]. Formally, nanocomposites can be viewed as confined, capped systems. Within nanocomposite systems, the Tg has been observed both to change and to remain the same, depending on the polymer and the nanoparticle. For example, polyurethanes [80] and PDMS [81] blended with SiO2 nanoparticles show little change in Tg . In contrast, the Tg of PMMA doped with Al2 O3 nanoparticles at filler concentrations greater than 0.5 wt. % decreases by 25 ◦ C [82]. We measured the Tg and aggregation of pure PEO without salt. The Tg increase is negligible (3 ◦ C) with 10 wt. % Al2 O3 nanoparticles, and the extent of aggregation is similar to the SPE. This means that the influence of confinement is uncertain: the aggregated system consists mainly of bulk polymer that is not confined. Because of the association expected between the PEO ether oxygens and the particle surface, the dynamics of PEO confined by alumina would likely decrease. This is suggested by simulation results on similar systems (PEO doped with TiO2 [83] and SiO2 [84]), where dynamics slow down with particle addition.

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When a Li salt is present, the competition between Li ions and the nanoparticles for interaction with the PEO ether oxygens introduces an additional complication. In this case, the segmental dynamics of the polymer may be influenced indirectly by the surface, which could interact with the ionic species and reduce the polymer– surface interactions that might otherwise slow the dynamics. In this case, the polymer would behave more like a pure polymer at a nonwetting interface. The way in which confinement and surface interactions influence the modes of motion not present in pure polymers (such as the rotational motion described above) is also not clear. A few studies have appeared that examine PEO confined to ˚ within intercalated hydrophobic silicates such as lithium montmorillonite 5–10 A [85–87]. The segmental or α-relaxation of PEO is replaced with an Arrhenius process that remains active below Tg . While the used temperature for SPEs is above the glass transition of PEO or PEO/salt complexes, the high temperature extrapolation of this process may be important with respect to conductivity in confined SPEs. Interestingly, when the ionic conductivity of the confined system is compared to that of a different unmodified SPE, conductivity is improved below the melting temperature of PEO, but at high temperature, conductivity is lowered [86], indicating that extreme confinement both slows polymer mobility and limits crystallization in SPEs. The latter is reasonable, given that confinement influences the morphology and crystallization kinetics of crystallizable polymers. For PEO in particular, the isothermal growth rate of crystallization is significantly reduced in thin films [88]. Confinement influences growth for films up to 150 nm, and for 13 nm films, the crystal growth rate is reduced by a factor of 40. Although crystallization of neat PEO in confinement has been studied [89–94], no studies have yet appeared that consider the influence of confinement on crystallization of various phases in PEO-based SPEs. If the effect of confinement on crystallization of PEO6 and PEO in SPEs is similar to that of pure PEO, it will be fruitful to pursue ways to evenly disperse nanoparticles.

5.6 Past and Future Role of Neutrons Neutron scattering techniques have played an important role in the study of solid polymer electrolytes. Neutron diffraction in combination with lithium labeling has been used to determine the lithium coordination and local structure of various SPEs [14,21,22]. SANS has provided information on the nanoscale structure of crystalline regions [28] and the dispersion of nanoparticles [54,65] in SPEs. In the former case, the difference in lithium content of the various crystalline phases enhances contrast and makes study of crystallinity more feasible than in neat polymers. QENS has provided most of what we know about polymer mobility in SPEs [32–37]. Based on these measurements, we know that although increasing ion content provides more charge carriers, it also slows the mobility of the polymer. In the future, we can use neutrons to understand the nature of polymer mobility when confining surfaces are present. The influence of confining surfaces in SPEs


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is not clear, and thus we do not have a good contact point with nanofilled SPEs. A connection between polymers confined in thin films and nanocomposites has been made for neat polymers, but not for polymers with added salt [79]. Because changes in dynamics in confined systems are often understood based on interactions between the confining surface and the polymer, and the presence of ions introduces new interactions that must be taken into consideration, such as particle-cation and particle-anion interactions, the response of neat polymers cannot be expected to resemble that of polymer/salt systems. Neutron reflectivity can be used to determine the location of ion-rich regions relative to surfaces, such as alumina, commonly used as nanoparticle additives. This can be combined with QENS measurements on samples confined by the same surfaces, enabled by new instruments with high flux. Together, they will provide a benchmark for the influence of confinement, from which we will understand whether pursuing well dispersed nanofilled SPEs is desirable. Below the melting points of PEO and PEO6 , we also need to understand the influence of confinement on crystallinity.

References 1. Tarascon J-M, Armand M (2001) Issues and challenges facing rechargeable lithium batteries. Nature 414:359–367 2. Fenton DE, Parker JM, Wright PV (1973) Complexes of alkali metal ions with poly(ethylene oxide). Polymer 14:589 3. Armand MB, Chabagno JM, Duclot MJ (1979) In: Vashishta P (ed) Fast ion transport in solids. North Holland, New York, p 131 4. Singh TJ, Mimani T, Patil KC, Bhat SV (2002) Enhanced lithium-ion transport in PEGbased composite polymer electrolyte with Mn0.03 Zn0.97 Al2 O4 nanoparticles. Solid State Ionics 154:21–27 5. Croce F, Appetecchi GB, Persi L, Scrosati B (1998) Nanocomposite polymer electrolytes for lithium batteries. Nature 394:456–458 6. Chiu C-Y, Chen H-W, Kuo S-W, Huang C-F, Chang F-C (2004) Investigating the effect of miscibility on the ionic conductivity of LiClO4/PEO/PCL ternary blends. Macromolecules 37:8424–8430 7. Imrie CT, Ingram MD, McHattie GS (1999) Ion transport in glassy polymer electrolytes. J Phys Chem B 103:4132–4138 8. Tominaga Y, Izumi Y, Kwak G-H, Asai S, Sumita M (2003) Effect of supercritical carbon dioxide processing on the ionic association and conduction in a crystalline poly(ethylene oxide)-LiCF3 SO3 complex. Macromolecules 36:8766–8772 9. Choi B-K (2004) Optical microscopy study on the crystallization in PEO-salt polymer electrolytes. Solid State Ionics 168:123–129 10. Rocco AM, da Fonseca CP, Pereira RP (2002) A polymeric solid electrolyte based on a binary blend of poly(ethylene oxide), poly(methyl vinyl ether-maleic acid) and LiClO4 . Polymer 43:3601–3609 11. Lin J-H, Woo EM, Huang Y-P (2006) Effects of lithium salt and poly(4-vinyl phenol) on crystalline and amorphous phases in poly(ethylene oxide). J Polymer Sci B Polymer Phys 44:3357–3386 12. Bamford D, Reiche A, Dlubek G, Alloin F, Sanchez J-Y, Alam MA (2003) Ionic conductivity, glass transition, and local free volume in poly(ethylene oxide) electrolytes: single and mixed ion conductors. J Chem Phys 118:9420–9432

140

J.K. Maranas

13. Robitaille CD, Fauteux D (1986) Phase diagrams and conductivity characterization of some PEO-LiX electrolytes. J Electrochem Soc 133:315–325 14. Mao G, Saboungi M-L, Price DL, Badyal YS, Fischer HE (2001) Lithium environment in PEO-LiClO4 polymer electrolyte. Europhys Lett 54:347–353 15. Marzantowicz M, Dygas JR, Krok F, Nowiński JL, Tomaszewska A, Florjańczyk Z, ZygadłoMonikowska E (2006) Crystalline phases, morphology and conductivity of PEO;TFSI electrolytes in the eutectic region. J Power Sources 159:420–430 16. Henderson WA, Passerini S (2003) Ionic conductivity in crystalline-amorphous polymer electrolytes–P(EO)6 :LiX phase. Electrochem Commun 5:575–578 17. Lascaud S, Perrier M, Vallée A, Besner S, Prud’homme J (1994) Phase diagrams and conductivity behavior of poly(ethylene oxide)-molten salt rubbery electrolytes. Macromolecules 27:7469–7477 18. Marzantowicz M, Dygas JR, Krok F, Łasińska A, Florjańczyk Z, Zygadło-Monikowska E (2006) In situ microscope and impedance study of polymer electrolytes. Electrochim Acta 51:1713–1727 19. Marzantowicz M, Dygas JR, Krok F, Florjańczyk Z, Zygadło-Monikowska E (2007) Influence of crystalline complexes on electrical properties of PEO:LiTFSI electrolyte. Electrochim Acta 53:1518–1526 20. Edman L, Ferry A, Doeff MM (2000) Slow recrystallization in the polymer electrolyte system poly(ethyleneoxide)n -LiN(CF3 SO2 )2 . J Mater Res 15:1950–1954 21. Mao G, Saboungi M-L, Price DL, Armand MB, Howells WS (2000) Structure of liquid PEOTFSI electrolyte. Phys Rev Lett 84:5536–5539 22. Londono JD, Annis BK, Habenschuss A, Borodin O, Smith GD, Turner JZ, Soper AK (1997) Cation environment in molten lithium iodide doped poly(ethylene oxide). Macromolecules 30:7151–7157 23. Labrèche C, Lévesque I, Prud’homme J (1996) An appraisal of tetraethylsulfamide as plasticizer for poly(ethylene oxide)-LiN(CF3 SO2 )2 rubbery electrolytes. Macromolecules 29:7795–7801 24. Armand MB (1986) Polymer electrolytes. Ann Rev Mater Sci 16:245–251 25. Gadjourova Z, Martin y Marero D, Andersen KH, Andreev YG, Bruce PG (2001) Structures of the polymer electrolyte complexes PEO6:LiXF6(X = P, Sb), determined from neutron powder diffraction data. Chem Mater 13:1282–1285 26. Staunton E, Andreev YG, Bruce PG (2007) Factors influencing the conductivity of crystalline polymer electrolytes. Faraday Discuss 134:143–156 27. Golodnitsky E, Livshits E, Rosenberg Y, Peled E, Chung SH, Wang Y, Bajue S, Greenbaum SG (2000) A new approach to the understanding of ion transport in semicrystalline polymer electrolytes. J Electroanal Chem 491:203–210 28. Fullerton-Shirey SK, Maranas JK (2009) Effect of LiClO4 on the structure and mobility of PEO-based solid polymer electrolytes. Macromolecules 42:2142–2156 29. Fateux D (1987) Phase equilibria. In: MacCallum JR, Vincent CA (eds) Polymer electrolyte reviews. Elsevier Applied Science, New York 30. Borodin O, Smith GD (2000) Molecular dynamics simulations of poly(ethylene oxide)/LiI melts 2. Dynamic properties. Macromolecules 33:2273–2283 31. Müller-Plathe F, van Gunsteren WF (1995) Computer simulation of a polymer electrolyte: lithium iodide in amorphous poly(ethylene oxide). J Chem Phys 103:4745–4756 32. Mao G, Perea RF, Howells WS, Price DL, Saboungi M-L (2000) Relaxation in polymer electrolytes on the nanosecond timescale. Nature 405:163–165 33. Mao G, Saboungi M-L, Price DL, Armand M, Mezei F, Pouget S (2000) a-Relaxation in PEOLiTFSI polymer electrolytes. Macromolecules 35:415–419 34. Mos B, Verkerk P, Pouget S, van Zon A, Bel G-J, de Leeuw SW, Eisenbach CD (2000) The dynamics in polyethyleneoxide-alkali iodide complexes investigated by neutron spin-echo spectroscopy and molecular dynamics simulations. J Chem Phys 113:4–7 35. Triolo A, Arrighi V, Triolo R, Passerini S, Mastragostino M, Lechner RE, Ferguson R, Borodin O, Smith GD (2001) Dynamic heterogeneity in polymer electrolytes. Comparison between QENS data and MD simulations. Physica B 301:163–167


141

36. Triolo A, Lo Celso F, Passerini S, Arrig´ı V, Lechner RE, Frick B, Triolo R (2002) Segmental dynamics in polymer electrolytes. Appl Phys A 74:S493–S495 37. Carlsson P, Zorn R, Andersson D, Farago B, Howells WS, Börjesson L (2001) The segmental dynamics of a polymer electrolyte investigated by coherent quasielastic neutron scattering. J Chem Phys 114:9645–9656 38. Capiglia C, Imanashi N, Takeda Y, Henderson WA, Passerini S (2003) Poly(ethylene oxide) LiN(SO2CF2CF3)2 polymer electrolytes IV. Raman characterization. J Electrochem Soc 150:A525–A531 39. Scrosati B, Croce F, Persi L (2000) Impedance spectroscopy study of PEO-based nanocomposite polymer electrolytes. J Electrochem Soc 147:1718–1721 40. Ahn JH, Wang GX, Liu HK, Dou SX (2003) Nanoparticle dispersed PEO composites for Li batteries. J Power Sources 119–121:422–426 41. Forsyth M, MacFarlane DR, Best A, Adebahr J, Jacobsson P, Hill AJ (2002) The effect of nano-particle TiO2 fillers on structure and transport in polymer electrolytes. Solid State Ionics 147:203–211 42. Borghini MC, Mastragostino M, Passerini S, Scrosati B (1995) Electrochemical properties of polyethylene oxide-Li[(CF3 SO2 )(2)N]-gamma-LiAlO2 composite polymer electrolytes. J Electrochem Soc 142:2118–2121 43. Dissanayake MAKL, Jayathilaka PARD, Bokalawala RSP, Albinsson I, Mellander B-E (2003) Effect of concentration and grain size of alumina filler on the ionic conductivity enhancement of the (PEO)9 LiCF3 SO3 : Al2 O3 composite polymer electrolyte. J Power Sources 119: 409–414 44. Weston JE, Steele BCH (1982) Effects of inert fillers on the mechanical and electrochemical properties of lithium salt-poly(ethylene oxide) polymer electrolytes. Solid State Ionics 7:75–79 45. Croce F, Persi L, Scrosati B, Serraino-Fiory F, Plichta E, Hendrickson MA (2001) Role of the ceramic fillers in enhancing the transport properties of composite polymer electrolytes. Electrochim Acta 46:2457–2461 46. Jayathilaka PARD, Dissanayake MAKL, Albinsson I, Mellander B-E (2002) Effect of nanoporous Al2 O3 on thermal, dielectric and transport properties of the (PEO)9 LiTFSI polymer electrolyte system. Electrochim Acta 47:3257–3268 47. Tambelli CC, Bloise AC, Rosário AV, Pereira EC, Magon CJ (2002) Characterisation of PEOAl2 O3 composite polymer electrolytes. Electrochim Acta 47:1677–1682 48. Bloise AC, Tambelli CC, Franco RWA, Donoso JP, Magon CJ, Souza MF, Rosario AV, Pereira EC (2001) Nuclear magnetic resonance study of PEO-based composite polymer electrolytes. Electrochim Acta 46:1571–1579 49. Croce F, Curini R, Martinelli A, Persi L, Ronci F, Scrosati B, Caminiti R (1999) Physical and chemical properties of nanocomposite polymer electrolytes. J Phys Chem B 103:10632–10638 50. Quartarone E, Mustarelli P, Magistris A (1998) PEO-based composite polymer electrolytes. Solid State Ionics 110:1–14 51. Kumar B, Scanlon LG (2000) Composite electrolytes for lithium rechargeable batteries. J Electroceramics 5(2):127–139 52. Dai Y, Greenbaum S, Golodnitsky D, Ardel G, Strauss E, Peled E, Rosenberg Y (1998) Lithium-7 NMR studies of concentrated LiI/PEO-based solid electrolytes. Solid State Ionics 106:25–32 53. Wieczorek W, Florja’nczyk Z, Stevens JR (1995) Composite polyether based solid electrolytes. Electrochim Acta 13–14:2251–2258 54. Fullerton-Shirey SK, Maranas JK (2010) Structure and mobility of PEO/LiClO4 sOLID polymer electrolytes filled with Al2O3 nanoparticles. J Phys Chem C 114:9196–9206 55. Bloise AC, Tambelli CC, Franco RWA, Donoso JP, Magon CJ, Souza MF, Rosario AV, Pereira EC (2001) Nuclear magnetic resonance study of PEO-based composite polymer electrolytes. Electrochim Acta 46:1571–1579 56. Marcinek M, Bac A, Lipka P, Zalewska A, Zukowska G, Borkowska R, Wieczorek W (2000) Effect of filler surface group on ionic interactions in PEO-LiClO4 -Al2 O3 composite polyether electrolytes. J Phys Chem B 104:11088–11093

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57. Karlsson C, Best AS, Swenson J, Howells WS, Börjesson L (2003) Polymer dynamics in 3PEG-LiClO4-TiO2 nanocomposite polymer electrolytes. J Chem Phys 118:4206–4212 58. Cole DH, Shull KR, Baldo P, Rehn L (1999) Dynamic properties of a model polymer/metal nanocomposite: gold particles in poly(tert-butyl acrylate). Macromolecules 32:771–779 59. Guyard A, Persello J, Boisvert J-P, Cabane B (2006) Relationship between the polymer/silica interaction and properties of silica composite materials. J Polymer Sci B Polymer Phys 44:1134–1146 60. Chen Y, Zhou S, Yang H, Wu L (2005) Structure and properties of polyurethane/nanosilica composites. J Appl Polymer Sci 95:1032–1039 61. Oberdisse J, Deme B (2002) Structure of latex-silica nanocomposite films: a small angle neutron scattering study. Macromolecules 35:4397–4405 62. Zhang Q, Archer LA (2002) Poly(ethylene oxide)/silica nanocomposites: structure and rheology. Langmuir 18:10435–10442 63. Xiong H-M, Zhao X, Chen J-S (2001) New polymer-inorganic nanocomposites: PEO-ZnO and PEO-ZnO-LiClO4 films. J Phys Chem B 105:10169–10174 64. Oberdisse J, El Harrak A, Carrot G, Jestin J, Boué F (2005) Structure and rheological properties of soft-hard nanocomposites: influence of aggregation and interfacial modification. Polymer 46:6695–6705 65. Karlsson C, Best AS, Swenson J, Kohlbrecher J, Börjesson L (2005) A SANS study of 3PEGLiClO4-TiO2 nanocomposite polymer electrolytes. Macromolecules 38:6666–6671 66. Johansson P, Ratner MA, Shriver DF (2001) The influence of inert oxide3 fillers on poly(ethylene oxide) and amorphous poly(ethylene oxide) based polymer electrolytes. J Phys Chem B 105:9016–9021 67. Best AS, Ferry A, MacFarlane DR, Forsyth M (1999) Conductivity in amorphous polyether nanocomposite materials. Solid State Ionics 126:269–276 68. Tominaga Y, Asai S, Sumita M, Panero S, Scrosati B (2005) A novel composite polymer electrolyte: effect of mesoporous SiO2 on ionic conduction in poly(ethylene oxide)-LiCF3 SO3 complex. J Power Sources 146:402–406 69. Krawiec W, Scanlon LG, Fellner JP, Vaia RA, Vasudevan S, Giannelis EP (1995) Polymer nanocomposites: a new strategy for synthesizing solid electrolytes for rechargeable lithium batteries. J Power Sources 54:310–315 70. Capiglia C, Mustarelli P, Quartarone E, Tomasi C, Magistris A (1999) Effects of nanoscale SiO2 on the thermal and transport properties of solvent-free, poly(ethylene oxide) (PEO)-based polymer electrolytes. Solid State Ionics 118:73–79 71. Wieczorek W, Stevens JR, Florjańczyk Z (1996) Composite polyether based solid electrolytes. The Lewis acid-base approach. Solid State Ionics 85:67–72 72. Choi B-K, Kim Y-W, Shin K-H (1997) Effects of ceramic fillers on the electrical properties of (PEO)16 LiClO4 electrolytes. J Power Sources 68:357–360 73. Ellison CJ, Torkelson JM (2003) The distribution of glass-transition temperatures in nanoscopically confined glass formers. Nat Mater 2:695–700 74. Fryer DS, Nealey PF, de Pablo JJ (2000) Thermal probe measurements of the glass transition temperature for ultrathin polymer films as a function of thickness. Macromolecules 33: 6439–6447 75. Forrest JA, Mattsson J (2000) Reductions of the glass transition temperature in thin polymer films: probing the length scale of cooperative dynamics. Phys Rev E 61:R53–R56 76. Priestley RD, Ellison CJ, Broadbelt LJ, Torkelson JM (2005) Structural relaxation of polymer glasses at surfaces, interfaces, and in between. Science 309:456–459 77. Ash BJ, Siegel RW, Schadler LS (2004) Glass-transition temperature behavior of alumina/PMMA nanocomposites. J Polymer Sci B 42:4371–4383 78. Sharp JS, Forrest JA (2003) Free surfaces cause reductions in the glass transition temperature of think polystyrene films. Phys Rev Lett 91:235701-1–235701-4 79. Bansal A, Yang H, Li C, Cho K, Benicewicz BC, Kumar SK, Schadler LS (2005) Quantitative equivalence between polymer nanocomposites and thin polymer films. Nat Mater 4:693–698


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80. Chen YC, Zhou SX, Yang HH, Gu GX, Wu LM (2004) Preparation and characterization of nanocomposite polyurethane. J Colloid Interface Sci 279:370–378 81. Winberg P, Eldrup M, Maurer FHJ (2004) Nanoscopic properties of silica filled polydimethylsiloxane by means of positron annihilation lifetime spectroscopy. Polymer 45:8253–8264 82. Ash BJ, Schadler LS, Siegel RW (2002) Glass transition behavior of alumina/ polymethylmethacrylate nanocomposites. Mater Lett 55:83–87 83. Borodin O, Smith GD, Bandyopadhyaya R, Byutner O (2003) Molecular dynamics study of the influence of solid interfaces on poly(ehthylene oxide) structure and dynamics. Macromolecules 36:7873–7883 84. Barbier D, Brown D, Grillet A-C, Neyertz S (2004) Interface between end-functionalize PEO oligomers and a silica nanoparticle studied by molecular dynamics simulations. Macromolecules 37:4695–4710 85. Elmahdy MM, Chrissopoulou K, Afratis A, Floudas G, Anastasiadis SH (2006) Effect of confinement on polymer segmental motion and ion mobility in PEO/layered silicate nanocomposites. Macromolecules 39:5170–5173 86. Wong S, Vaia RA, Giannelis EP, Zax DB (1996) Dynamics in a poly(ethylene oxide)-based nanocomposite polymer electrolyte probed by solid state NMR. Solid State Ionics 86–88: 547–557 87. Vaia RA, Sauer BB, Tse OK, Giannelis EP (1997) Relaxations of confined chains in polymer nanocomposites: glass transition properties of poly(ethylene oxide) intercalated in montmorillonite. J Polymer Sci Polymer Phys 35:59–67 88. Dalnoki-Veress K, Forrest JA, Massa MV, Williams A (2001) Crystal growth rate in ultrathin films of poly(ethylene oxide). J Polymer Sci B 39:2615–2621 89. Floudas G, Tsitsilianis C (1997) Crystallization kinetics of poly(ethylene oxide) in poly(ethylene oxide)-polystyrene-poly(ethylene oxide) triblock copolymers. Macromolecules 30:4381–4390 90. Taden A, Landfester K (2003) Crystallization of poly(ethylene oxide) confined in miniemulsion droplets. Macromolecules 36:4037–4041 91. Sun L, Zhu L, Ge Q, Quirk RP, Xue C, Cheng SZD, Hsiao B, Avila-Orta CA, Sics I, Cantino ME (2004) Comparison of crystallization kinetics in various nanoconfined geometries. Polymer 45:2931–2939 92. Huang P, Zhu L, Guo Y, Ge Q, Jing AJ, Chen WY, Quirk RP, Cheng SZD, Thomas EL, Lotz B, Hsiao BS, Avila-Orta CA, Sics I (2004) Confinement size effect on crystal orientation changes of poly(ethylene oxide) blocks in poly(ethylene oxide)-b-polystyrene diblock copolymers. Macromolecules 37:3689–3698 93. Hsiao M-S, Chen WY, Zheng JX, Van Horn RM, Quirk RP, Ivanov DA, Thomas EL, Lotz B, Cheng SZD (2008) Poly(ethylene oxide) crystallization within a one-dimensional defect-free confinement on the nanoscale. Macromolecules 41:4794–4801 94. Massa MV, Dalnoki-Veress K, Forrest JA (2003) Crystallization kinetics and crystal morphology in thin poly(ethylene oxide) films. Eur Phys J E 11:191–198

Chapter 6

Future Perspectives: Moving to Longer Length and Time Scales, from Polymers to Biological Macromolecules Dieter Richter

6.1 Introduction The physics of condensed matter is in the process of a radical shift of emphasis. The main focus in the twentieth century was the study of crystalline matter. This meant that the crucial issues were related to structure on the atomic scale and the dynamics of elementary excitations, i.e. they involved the exploration of length and time scales around 0.1–1 nm and 10−14–10−12 s, respectively. Neutron scattering has played a very important role in this endeavour, as for example emphasized by the award of the 1994 Nobel Prize to Brockhouse and Shull. The trends in recent years indicate a paradigm shift in condensed matter research, both fundamental and applied, towards complex matter and mesoscopic length and time scales. For example, the new programs of the US Department of Energy in the science of materials concentrate on “Complex and Collective Phenomena” and “Nanoscale Science”. This implies the exploration of structures on the 1–100 nm length scale in systems which do not support well-defined elementary excitations and whose behaviour is strongly determined by dissipative dynamical processes. Dynamics are expected to play a central role in understanding the nature and functionality of complex matter and the most relevant time scale associated with dissipative and nanoscale phenomena ranges from ps to macroscopic times. Neutron scattering has the unique capability to directly explore a large part of these nanoscopic time and length scales; the neutron spin-echo technique allows us to cover the largest fraction of this space and time domain simultaneously. Neutron spin echo (NSE), small-angle neutron scattering (SANS) and neutron reflectometry instruments will form the backbone for research with neutrons in a number of these fields of growing interest. The NSE method is unique in providing a means to measure dynamical processes in

D. Richter () Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 6, © Springer Science+Business Media, LLC 2012

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the SANS regime. For larger momentum transfers – as required, for example, for the study of glasses – it is complemented by backscattering spectrometers. Unlike classical spectroscopic methods such as dielectric or mechanical response, which yield the temporal development of macroscopic averages, the neutron spectroscopic methods – especially in combination with selective labelling – have the potential to identify the spatio-temporal nature of specific relaxations at a molecular level. Furthermore, neutron scattering when combined with computer simulations will have a very large impact on many future scientific endeavours to understand old and new soft matter systems, including bio-macromolecules. Simulations provide the atomic positions, possibly as a function of time, from which the expected neutron scattering spectrum may be calculated directly and compared to experimental results, on an absolute scale. Neutron scattering therefore provides the best method for testing and benchmarking computational methods, e.g. Monte Carlo (MC) and molecular dynamics (MD) simulations (see also Chaps. 2 and 7). Neutron scattering, and in particular NSE, in combination with computer simulations can thus importantly advance our understanding of a huge class of soft matter problems and also of the relationship of molecular motion to biological function. Polymeric materials are commonly used because they are durable, cheap to produce and easy to process and because they exhibit very favourable mechanical properties. Polymers may behave like a viscous fluid or rubber elastic and are very tough with high impact strength or even brittle. In the simplest case, polymers are long linear chain molecules built from a repeating unit: the monomer. In contrast to synthetic polymers, biopolymers are highly complex molecular structures with a sharp molecular weight and a uniquely defined sequence of building blocks. They are the molecular manifestations of life with a high variety of structures and specific functions. Nevertheless, at least in their denaturated state, they exhibit similarities with synthetic materials, while the specific function evolves in their folded state. The general principles of self-assembly in structure formation are also common to both species. The diffusive motions of long flexible polymers, on the one hand, show fascinating fundamental physics, and on the other, represent one of the great challenges of modern materials science. The drive towards a molecular understanding of the complex viscoelastic properties of polymer liquids is the focal point of rheology and connects the classical chemical engineering approach with modern physics [1]. Here the tube model invented by Doi and colleagues [2, 3] and de Gennes [4] has shown itself to be the most successful molecular model describing the topological confinement imposed by the mutually interpenetrating polymer chains in the melt. A theory of viscoelasticity that describes the main features of polymer melt rheology has been developed in terms of this reptation model which was introduced earlier in Chap. 4. Large-scale protein motions are critical for proteins to coordinate precise biological function. Such dynamics are invoked in regulatory proteins, motor proteins, signalling proteins and structural proteins. Structural studies have documented the conformational flexibility in proteins that accompanies their activity. Results from macroscopic studies such as biochemical kinetics and single molecule detection have also shown the importance of conformational dynamics and Brownian thermal

6 Future Perspectives: Moving to Longer Length and Time Scales...

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fluctuations within proteins or protein complexes. However, the time-dependent dynamic processes that facilitate such protein motions remain poorly understood and experimentally nearly untouched [5, 6]. We start this chapter by discussing aspects of the large-scale molecular dynamics of polymer melts [7]. These have been previously introduced in Chap. 4. At the beginning, the standard model of polymer dynamics – the Rouse model – is briefly described and some experimental results displaying the basic correctness of this approximation are presented. Then we turn to reptation and present the molecular observation of the confining tube as well as the predicted crossover in the meansquare displacement from Rouse motion to local reptation along the tube. We also address the centre of mass motion of the chain and show that interchain coupling is important, leading to deviation from the predicted linear time dependence of the centre of mass diffusion coefficient below the Rouse time. In Sect. 6.3, we discuss the molecular foundations of rheology. We commence with a molecular observation of CLFs – a process which is important as a correction for reptation but essential for the hierarchical relaxation of branched polymers. We then turn to non-linear rheology and first present a direct observation of chain retraction after step strain of a linear polymer. Thereafter, we briefly address the hierarchical relaxation of a branched model polymer. In Sect. 6.4, we turn to more complex systems and present three examples. First, we review some results on the dynamics of soft interfaces in a lamellar mesophase of a block copolymer demonstrating the surface tension as a driver for the interface dynamics. Then we turn to the polymer dynamics under random hard confinement where multi-scale dynamics is observed. Finally, protein diffusion in polymeric gels is addressed showing the potential of neutron spin-echo spectroscopy to measure the dynamics of proteins in crowded environments. In Sect. 6.5, we turn to the large-scale dynamics of proteins and present as a first example the observation of the functional cleft opening dynamics in alcohol dehydrogenase (ADH). Finally, Sect. 6.6 concludes and gives an outlook for the future.

6.2 Large-Scale Polymer Dynamics 6.2.1 The Rouse Model The standard model of polymer dynamics, the Rouse model [8], treats the dynamics of a Gaussian chain in a heat bath. Thereby it implicitly assumes that, on the length and time scales considered, all forces from local potentials which are related to the individual chemical nature of a given polymer have already decayed. Then only entropic forces originating from the conformational chain entropy drive the dynamics. At scales QRE > 1, where RE is the chain end-to-end distance, the normalized single-chain dynamic structure factor S(Q,t)chain for this model can be

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√ written as a function of a scaling variable u = Q2 W 4t combining spatial and temporal scales [9]: ∞ u ∞ cos(6xs/u) 2 ds exp −s − × [1 − exp(−x )]dx , (6.1) Schain (u) = 3π 0 x2 0 W = 3kB T /(ζ 2 ) is the elementary Rouse frequency, given by the ratio of the entropic force 3kBT /2 and the friction coefficient ζ . 2 is the mean-squared segment length. Equation (6.1) is only valid in the asymptotic Q−2 tail of the scattering function. A full treatment [1] yields the dynamic structure factor also in the low Q limit. At t = 0, this immediately leads to the well known Debye function S(Q) = Nf Debye (Q2 R2g ) with fDebye (x) = 2/x2 [exp(−x) − 1 + x] and N the chain length. For t → ∞, all internal chain correlations are lost and the structure factor displays the Gaussian density profile within a polymer coil. We note that the polymer coil is also subject to translational diffusion in addition to the Rouse modes. We now turn to the self-motion in the Gaussian approximation. The structure factor is described by: Q2 (6.2) Sself (Q,t) = exp − r2 (t) , 6 where r2 (t) is the mean-square segment displacement (MSD). In the Rouse model, r2 (t) assumes the form [7]: r (t) = 6Dcmt + 2 2

2

3kBT t πζ 2

1/2 ,

(6.3)

where Dcm = kB T /N ζ is the centre of mass diffusion coefficient. Figure 6.1 shows the time-dependent mean-square displacement r2 (t) obtained from a high molecular weight (Mw = 80 kg mol−1 ) monodisperse polyethylene– propylene (PEP) melt at 492 K [10]. r2 (t) has been calculated, following (6.2), as: r2 (t) = −

6 nSself(Q,t). Q2

(6.4)

The thus obtained mean-square displacement follows the predicted square root law in time with high accuracy (for the high Mw polymer, the translational diffusion does not play any role). Since quasielastic neutron scattering resolves dynamic processes in space and time, these measurements give direct information on the ˚ 2 , i.e., the segment displacement at a given time. For example, r2 (10 ns) = 620 A ˚ during this time interval. average proton has travelled about 25 A The pair correlation function arising from the segment motion within a single given chain can be observed if some protonated chains are dissolved in a deuterated matrix. Figure 6.2 shows the observed spectra from poly(ethyl ethylene) (PEE)


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Fig. 6.1 Data for h-PEP represented as 6n[Sinc (Q,t)]/Q2 vs. time for T = 492 K. The solid line describes the asymptotic power law

Fig. 6.2 Single-chain structure factor for a PEE melt at 473 K. The numbers above the curves ˚ −1 . The solid lines are a joint fit with the Rouse model (6.1) represent the experimental Q values in A

(90% dPEE Mw = 24.5 kg mol−1 and 10% hPEE Mw = 21.5 kg mol−1 , both with narrow molecular weight distributions) [11]. The solid lines give the prediction of the dynamic structure factor from (6.1). Obviously, very good agreement is achieved.

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6.2.2 Reptation For long chains, topological chain-chain interactions in terms of entanglements become important and dominate the dynamical behaviour. In the famous reptation model, these constraints are described by a virtual tube which localizes a given chain and limits its motion to a one-dimensional Rouse motion inside the tube (local reptation) and a slow diffusive creep motion out of the tube (reptation) [3, 4]. By using NSE, it has become possible to observe the dynamic structure factor S(Q,t) associated with tube confinement and local reptation [12–14]. de Gennes [4] and Doi and Edwards [3] have formulated a tractable analytic expression for the dynamic structure factor. They neglected the initial Rouse regime, i.e. the derived expression is valid for t > τe once confinement effects become important (τe is the entanglement time, the Rouse relaxation time of an entanglement strand). The dynamic structure factor is composed of two contributions, Sloc (Q,t) and Sesc (Q,t), reflecting local reptation and escape processes (creep motion) from the tube. Schain (Q,t) Q2 d 2 Q2 d 2 loc S (Q,t) + exp − Sesc (Q,t). (6.5) = 1 − exp − Schain (Q) 36 36 The local reptation part was calculated as [15]: Sloc (Q,t) = exp

t τ0

erfc

t τ0

1/2 ,

(6.6)

where τ0 = 36/(W4 Q4 ). For short times, Schain (Q,t) decays mainly due to local reptation (first term of (6.5)), while for longer times (and low Q) the second term resulting from the creep motion dominates. The ratio of the two relevant time scales τ0 and τd is proportional to N 3 . Therefore, for long chains at intermediate times τe < t < τd , a pronounced plateau in Spair (Q,t) is predicted. Such a plateau is a signature of confined motion and is also present in other models for confined chain motion. The dynamic structure factor data for a Mw = 36 kg mol−2 polyethylene (PE)-melt in Fig. 6.3 very clearly show the tendency to form plateaus at long times [14]. In the spirit of (6.5), and neglecting the ongoing decay of Schain (Q,t) due to local reptation, we may obtain a first estimate for the degree of confinement from the heights of the achieved plateaus. Identifying the plateau levels with a ˚ a value which is Debye-Waller factor describing the confinement, we get d = 45 A, a lower estimate for the true tube diameter since Sloc (Q,t) is not fully relaxed. The dashed lines in Fig. 6.3 are the predictions from this Debye-Waller factor estimate. The confinement effect also expresses itself in terms of the MSD. If this reaches the order of the tube diameter, then motional restrictions are expected. For the


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1,00 0.86

S(Q,t) / S(Q)

0,75

0.71

0,50

0.46 0.28

0,25

0,00 0

50

100

150

t / ns Fig. 6.3 The dynamic structure factor from a Mw = 36 kg mol−1 PE melt at 509 K as a function of time. The solid lines are a fit with the reptation model (6.5). The Q values are (from the top) ˚ −1 . The horizontal dashed lines display the prediction of the Q = 0.05, 0.077, 0.115 and 0.145 A Debye-Waller factor estimate for the confinement size

crossover time τe , we may take the relaxation time of a polymer section, spanning the tube diameter, as

τe =

1 d 4 ζ0 1 d4 = 2 . 2 2 3π kB T π W 4

(6.7)

For times t > τe , one-dimensional curvilinear Rouse motion along the tube has to be considered. We may distinguish two regimes of curvilinear motion: (a) The dynamics within the tube is dominated by Rouse relaxation – we are at times t < τR . Then the mean-square displacement along the tube should follow the t 1/2 law discussed above. (b) We are at times t > τR ; then diffusion like creep is dominating and the meansquare displacement should be proportional to t. In order to arrive at the behaviour in laboratory space, we have to consider that, due to the Gaussian conformation of the tube, a mean-square displacement along the tube gives rise to only the square root of this displacement in lab-space. With that we arrive at ⎧ 1/4 2 ⎪ ⎨ 2d kBζTπ t , τe < t < τR , 0 r2 (t) =

1/2 ⎪ ⎩ 2d kB Tt , τR < t < τd . Nζ

(6.8)

0

The first process, where the chain performs Rouse motion along the tube, is called “local reptation”. The creep-like diffusion along the tube which eventually leads to a complete tube renewal is termed pure reptation.

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Fig. 6.4 Incoherent scattering data from PE represented as −6n[Sself (Q,t)]Q2 , which is the mean-square displacement r2 (t) as long as the Gaussian approximation holds. Solid lines: the asymptotic power laws r2 (t) ∝ t 1/2 ; t 1/4 . Dotted lines: prediction from the Gaussian approximation. Dashed lines: see text

In the Gaussian approximation, the self-correlation function of a reptating chain would directly relate to the above-calculated mean-square displacements. However, as Fatkulin and Kimmich [16] have shown, diffusion along the one-dimensional tube contour is not a Gaussian process in the laboratory frame – the corresponding self-correlation function becomes non-Gaussian for t > τe . 4 2 2 Q d r (t) Q2 d r2 (t) Sself (Q,t > τe ) = exp × erfc √ , (6.9) 72 3 3 6 2 √ 2 where erfc is the complementary error function erfc(x) = 2/ π x∞ e−t dt. We 2 2 note that (6.9) is strictly valid only for t > τe when r (t) ≈ d . The effect on the scattering function is that if (wrongly) interpreted in terms of the Gaussian approximation, the crossover to local reptation appears to occur at significantly lower values of τe . The general asymptotic t 1/4 law remains untouched. The expected crossover from a t 1/2 to a t 1/4 law has been recently observed by NSE spectroscopy [10]. Experiments were performed on polyethylene (PE) samples at 509 K. The observed spectra were converted via (6.4) to mean square displacements. Thereby the Gaussian assumption is implied (Fig. 6.4). ˚ 4 ns−1 into (6.3), Inserting the known Rouse rate W 4 (509 K) = (7 ± 0.7) × 104 A the solid line ∼t 1/2 is obtained. It quantitatively corroborates the correctness of the Rouse description at short times. The data also clearly reveal a transition to a t 1/4 law, though (6.8) would predict the dotted line. The discrepancy can be explained by considering the non-Gaussian character of the curve-linear Rouse


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Fig. 6.5 Extracted centre of mass mean square displacement for Q = 0.3 nm−1 for PE chains of −1 different Mw : 0.5 (squares), 1.5 (circles), 2 2.5 (triangles) and 5.1 kg mol (diamonds). Solid lines: ∼ Dt. Arrows: τR (N) Guenza (see text). Dashed lines: rCM

motion. Fixing W 4 and d to the values obtained from single-chain structure measurement (Fig. 6.3), the dashed lines in Fig. 6.4 reveal the prediction of the nonGaussian treatment.

6.2.3 Chain Diffusion at Short Times The Rouse model predicts a centre of mass MSD of a diffusing chain which is strictly linear in time. Exploiting the space time sensitivity of neutron scattering, a measurement of S(Q,t)chain at QRg < 1, where the internal chain modes are not contributing to the signal allows access to Dcm at short times on the scale of the chain. For times shorter than the longest internal chain relaxation time, the Rouse time τR , recent experiments showed that the Rouse prediction for the centre of mass displacement is not fulfilled [17]. Figure 6.5 shows results for the centre of mass MSD for different short chains lengths. For each chain, the corresponding Rouse time τR (N) = N 2 /W π 2 is indicated. With increasing length, an increasingly important sub-diffusive behaviour at short times becomes evident. At longer times, a crossover to a centre of mass displacement proportional to t with a crossover time around the corresponding Rouse time takes place. The dashed lines in Fig. 6.6 extrapolate the long time behaviour towards shorter times.

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Fig. 6.6 Principles of contour length fluctuations (CLFs): (a) The surrounding chains confine a given chain inside a virtual tube of diameter d. (b) The chain ends perform fluctuations retracting into the original tube, thus “forgetting” the initial confinement of the vacated tube parts. (c) The chain ends may then explore the surroundings

While local intermolecular interactions are ignored in the Rouse theory, more realistically in a polymer melt, where a chain spans a volume V ≈ R3g , it may interact √ with the n ∼ = N chains that on average fill the volume of a given chain. Guenza [18] has derived a generalized Langevin equation (GLE) for the resulting cooperative dynamics of such interacting polymers in the melt. Thereby an effective potential acting between the centres of mass of a pair of molecules has been derived as: √ 3r2 27 2 1 108 exp − 2 , W (r) = √ √ 1− 4Rg 4π π N ρ ∗ π 2 ρ ∗2 N

(6.10)

where ρ ∗ = ρ 3 with ρ the density. The potential is Gaussian with a range of the order of Rg . An approximate solution of the GLE leads to a sub-diffusive centre of mass MSD. The solid lines in Fig. 6.5 show the result of the GLE approach. Thereby the soft interchain potential W (r) (6.10) was employed using the pre-factor as the fitting parameter and introducing the proper Rg into the Gaussian potential. With this approach, an adequate description of the experimental data is achieved (solid lines in Fig. 6.5). We note that for the translational diffusion, and following the calculations of Guenza also for the lowest relaxation modes, the interchain coupling through the soft interaction potential of (6.9) plays a non-negligible role.

6.3 Molecular Rheology The application and industrial processing of many soft condensed matter systems strongly depend on their rheological properties, which are determined by the interactions and motions of the constituent structural units such as chain molecules, aggregates, colloidal particles, and surfactants. Understanding of these factors is one of the great challenges of basic soft condensed matter science and would facilitate the molecular design of new materials. Reptation, as the basic mechanism behind the linear rheology of linear chains, was discussed above. Here we will concentrate


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on the molecular basis of non-linear rheology. We will start with the CLFs of a chain in its tube, a process important as a correction for reptation in the case of linear chain, but essential for the hierarchical relaxation of branched polymers where CLF are considered to be the fundamental processes facilitating the release of side branches. Then we will turn to the molecular foundations of non-linear rheology, discussing first the chain retraction of a linear chain in its tube after a step strain and then presenting some results on the relaxation behaviour of a branched so-called H-polymer.

6.3.1 Contour Length Fluctuations (CLF) The CLF effect evolves from the participation of the chain ends in the local reptation process. Any chain retraction and subsequent expansion will lead to a loss of memory of the original confinement of the tube. The process is depicted schematically in Fig. 6.5. Mathematically, the problem is treated as a first passage problem. Whenever a tube contour s is visited by the free end, it ceases to exist. The functional form of the tube survival probability μ (t) has been derived from scaling arguments [19], yielding 1/4 Cμ t μ (t) = 1 − , (6.11) Z τe where the numerical constant Cμ = 1.5 ± 0.02 is obtained from stochastic simulations, Z = N/Ne is the number of entanglements and Ne the number of segments forming an entanglement strand. Equation (6.11) provides a quantitative knowledge of the chain fraction which is still confined at a time t. All parameters are known from the NSE experiments on the dynamics of asymptotically long chains, where the CLF effect does not play a role. With this knowledge, an experiment was designed where the dynamic structure factor of a chain which is subject to CLF was compared with that of an identical chain where the contrast of those segments which are affected by CLF within the experimental time frame was matched. The experimental idea is displayed in Fig. 6.7. Case (a) is realized by performing an experiment on a fully protonated chain in a deuterated matrix. In this case, the full chain dynamics, including the CLF, is observed. Case (b) is realized by a chain where the inner part is protonated, while the two outer chain sections of a length which would be affected by CLF are deuterated and thus are not visible in the deuterated matrix. Then the dynamics should be equal to those of an asymptotically long fully confined chain. With the known parameters for PE, (6.11) yields that, on average on each side, 220 monomers are released during the observation time of 190 ns. The contrasting experiments were performed on two different chains of molecular weight 25 kg mol−1 , one of which was fully hydrogenated and the other had deuterated labels of about Mw ∼ = 4 kg mol−1 corresponding to 260 monomers on each end. Both were studied in a deuterated matrix of the same molecular weight [20].

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Fig. 6.7 (a) For a fully protonated chain in a deuterated matrix, local reptation and CLFs are visible. (b) For a centre-labelled (protonated) chain with deuterated chain ends in a deuterated matrix, only local reptation is visible, the CLFs of the chain ends are masked out

Fig. 6.8 Dynamic structure factor of a centre-labelled 25 kg mol−1 PE chain (dark symbols) compared to a fully labelled chain (light symbols) of the same overall molecular weight. Q values (nm−1 ): 0.5 (squares), 0.96 (circles), 1.15 (triangles). Lines: for centre-labelled chain, pure reptation model [(6.5), dark]; for fully labelled 25 kg mol−1 chain (light), CLF was considered [21]

In Fig. 6.8, the measured normalized dynamic structure factor S(Q,t)/S(Q) is plotted as a function of time t for different Q values. For a better visibility, not all Q values are shown. Figure 6.8 presents the experimental results for the two chains. Comparing the levels of decay in Fig. 6.8, we realize that S(Q,t)/S(Q) from the fully labelled chain decays significantly more strongly than that from the corresponding centre-labelled counterpart. The constraints are apparently stronger than for the chain where the ends are visible. We also note that, in the case where


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the ends were masked, the chain centre part shows exactly the same structure factor as a very long chain [14], signifying directly the action of CLF at the chain ends and the remaining full confinement in the centre. Concluding this paragraph, we emphasize that CLF also affect the macroscopic melt properties in a significant way, as alluded to in the beginning. 1. The dynamic modulus G(ω ) directly relates to the total tube survival probability as expressed by Sesc (Q,t) in (6.5). CLF modify the tube survival and this should be mirrored by G(ω ). It has been shown that CLF introduce an ω −1/4 regime into the spectrum of G (ω ) [22]. 2. The translational chain diffusion is necessarily also affected. Firstly, the terminal time defining the diffusion “step” is reduced, since reptation has only to relax the not-yet-released central parts of the tube. Secondly, the diffusive length is reduced since only the displacement of the central part counts. Both effects do not cancel and the net effect gives D ∼ Z −2.4 up to Z ∼ = 300, in very good agreement with experiment [23]. 3. The anomalous power law exponent of 3.4 for the viscosity mass relation has also been attributed to CLF. The treatment of Millner and McLeish even predicts the crossover to pure reptation within the proper range of Z [24]. It is the virtue of NSE experiments that they can confirm the CLF mechanism quantitatively on a molecular level in space and time. Recently, constraint release processes were also probed on a molecular scale. NSE experiments on bimodal melts proved their molecular existence, and in particular, showed that constraint release facilitated by CLF of the confining chains may contribute significantly [25].

6.3.2 Chain Retraction We now turn to the molecular processes behind non-linear rheology and address the question of initial chain relaxation after an applied step strain. A large step extensional elongation is expected to affinely deform the chain contour. This will cause the chain radius of gyration parallel to the flow to increase and, conversely, that in the perpendicular direction to decrease. After cessation of the strain, one expects the first relaxation process to be a retraction of the chain within the still affinely deformed tube. The longest time scale of this process should be the equilibration time τR (Rouse time) of the chain along the tube. This mechanism is related by a fluctuation dissipation theorem to the chain CLFs. Chain retraction in the elongated tube should reduce the radius of gyration in all directions. Therefore, perp Rg is expected to attain a minimum, manifested as an increase in the perpendicular scattered intensity at some time after the deformation, determined by the Rouse time, before diffusive mechanisms return it to equilibrium. Such non-equilibrium experiments obviously cannot be carried out by NSE, which measures the equilibrium thermal fluctuations, but can be done using the quenched SANS technique. In such measurements, a deformation device, suitable

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b

0

Intensity/ I0

Intensity/ I0

10–1

10–2

10–3

c

10

t=0

Parallel Perpendicular Affine Theory

10–1

–1

q [(nm) ]

100

100

10–2

10–3

d

t = 3.2τR

10–1

10–1

q [(nm)–1]

100

0

t = 0.4τR

10–1 10–2

10–3

Intensity/ I0

Intensity/ I0

10

–1

10

q [(nm)–1]

100

100 t = 15.7τR

10–1 10–2

10–3

10–1

q [(nm)–1]

100

Fig. 6.9 Comparison of measured and predicted normalized structure factors after 0, 0.4, 3.2 and 15.7 τR (a–d, respectively). The solid lines are predictions of the theory of Graham et al. [26] and the dotted lines correspond to a completely affine deformation on all length scales

for the investigation of low Tg materials, is used in situ at a SANS instrument. After the step strain, the sample is rapidly quenched and well-defined relaxation times are achieved by controlled annealing close to Tg . High molecular weight (∼250 kg mol−1 ) polyisoprene mixtures of hydrogenated and deuterated molecules were uniaxially stretched and rapidly quenched. The SANS pattern of the initial deformed state served as a reference to assess the subsequently annealed samples at relaxation times of 0.4, 3.2, 8.4 and 15.7 times the Rouse time τR . To perform the experiments, the melt was quenched, studied, and thereafter relaxed stepwise in a controlled way. After each relaxation, the sample was quenched again and studied by SANS. The plots of scattered intensity against scattering vector (Fig. 6.9) demonstrate that, throughout the relaxation regime, the anisotropy is strong at large length scales and weakens as the scale of the tube diameter is reached at higher Q. As relaxation progresses, features at higher Q relax to isotropy faster than at lower Q. At t = 3.22τR , the parallel scattering shows significant relaxation relative to the fully affine curves (dotted lines). However, throughout this time interval, the perpendicular scattering remains close to the affine curve over almost the entire range of scattering vectors (Fig. 6.9a–c) [27]. Note, in particular, the small but significant increase of the scattered intensity between t = 0 and 0.4τR that is visible in the perpendicular component of


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Fig. 6.10 Comparison between experimental data (symbols) and theoretical prediction for the relaxation of the radius of gyration perpendicular to the stretch direction after deformation. The experimental data highlight the non-monotonic region of Rperp g

the scattering. It indicates the presence of a shrinking Rg in the perpendicular direction. These data were quantitatively evaluated in the frame of the model of Graham et al. [26], who formulated a stochastic microscopic evolution equation for the dynamics of the space curve describing the tube contour. This approach encompasses reptation, CLFs, retraction and constraint release phenomena. In order to include chain fluctuation at the level of the tube, the theory was extended in terms of the Warner Edwards model for polymer networks [28], where an isotropic localization potential is assumed. The solid lines in Fig. 6.9 present the results of this modelling. A plot of the deformed radius of gyration as a function of time portrays the SANS signature of chain retraction. Figure 6.10 clearly shows the experimental observation of a pronounced minimum in Rperp g (t), providing a microscopic confirmation of the tube retraction concept. The theory demonstrates that the time of occurrence and depth of this minimum are consistent with the retraction process, although the model systematically under-predicts the degree of deformation by ∼4% of the total deformation. In conclusion, using well-controlled quench SANS experiments, the chain relaxation in the non-linear regime can be elucidated over a broad range of length scales, including the experimental manifestation of the long predicted chain retraction process. Modelling of these data provides a significant quantitative test for molecular theory and illustrates the possibility of linking molecular structure and processing history.

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6.3.3 Hierarchical Relaxation in Branched Polymers The rheological properties of polymer melts are strongly modified by chain branching. Branched polymers exhibit a significantly broader spectrum of relaxation times compared with their linear counterparts. Furthermore, their non-linear rheology is significantly altered. Given its great importance for polymer processing, a quantitative molecular understanding is one of the great challenges of polymer science. So far, attempts to create a molecular rheology for branched polymers on the basis of the tube concept have only been partially successful [22]. We briefly describe SANS experiments on the controlled relaxation of H-polymers (polymers with an H shape) where, for the first time, relaxation experiments on branched polymers on the molecular scale were attempted [29]. Figure 6.11 displays the morphology of an H-polymer, including the confining tube for two different stages of relaxation. In the framework of the tube model, the relaxation of an H-polymer has to take place in a hierarchical fashion. Since the crossbar can move only if the arms have retracted from their confining tubes, CLF play a key role in the relaxation. A number of relaxation stages were proposed: 1. At early times, CLFs of the arms take place in a similar way as for linear chains. 2. For deeper arm fluctuations, in which larger sections of the arm contract closer to the branching point, an increasingly important entropic barrier has to be crossed, slowing down the process exponentially; at the same time, the liberated arm segments dilute the tube confinement of the crossbar. 3. After complete arm retraction, the crossbar dynamics take over, starting with CLF of the branch points, where all of the friction is concentrated. 4. The crossbar dynamics continue with reptation within the diluted tube. In these terms, a quantitative theory for G(ω ) was worked out [22]. In the time regime relevant for the hierarchical relaxation processes, direct dynamic neutron scattering experiments are impossible. Again a solution lies in SANS experiments on polymers relaxing from a well-defined non-equilibrium state (e.g. after step strain). Recently, a study has been performed on a melt of H-polymers where the arm tips have been labelled using deuterium,. The polymer was stretched by different amounts (twofold, threefold: λ = 2, 3, where λ is the chain extension factor). As for the case of the arm retraction, the melt was studied after controlled annealing and re-quenching in several steps. In parallel, the dynamic modulus was studied. An attempt was made to describe both G (ω ) and the SANS data in the same theoretical frame. From rheology at the reference temperature of T = 20 ◦ C, stages (1) and (2) were expected within a time frame of 10−5–102 s, while the crossbar motion was predicted to take place between 101 and 105 s. Figure 6.11 presents SANS results for λ = 2. The studied relaxation times corresponded to stages (1) and (2) in the early stages of polymer relaxation. The lines display a fit with the predictions of the tube theory. Although the data showed a good fit with the theory, a number of important discrepancies between the scattering data and the rheological reference were manifest.


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Fig. 6.11 (a) Small-angle neutron scattering intensities for a stretched H-polymer sample (twodimensional detector image). (b) Intensity cuts along the directions parallel to the deformation (open circles) and perpendicular to the deformation (open squares) compared with theoretical calculations (see [29]) for the sample strained to λ = 2 after annealing times “0” (top graph) and 6 × 102 s (bottom graph) at 25 ◦ C (the sample was quenched to −85 ◦ C). The term dΣ/dΩ(q) represents the absolute macroscopic cross-section. (c) H-polymer in its confining tube immediately after deformation (top schematic), with the arms fully confined, and after 6 × 102 s (bottom schematic) when the arms have relaxed by 12%

1. Rheology overestimated the amount of arm relaxation by a factor of 3 or 4. 2. The undiluted tube diameter of d = 3.3 nm is much smaller than the rheological value of d = 5.1 nm. 3. The best fits were obtained by disregarding any arm retraction or branch point withdrawal within the stretched tube – both should occur in non-linear response. The discrepancies between theoretical prediction and experiment at later stages (not shown here) are even more important. In particular, a growing anisotropy with increasing relaxation time was observed that cannot be accounted for theoretically. These first experimental results demonstrate the ability of neutron scattering, together with careful synthesis and new theoretical developments, to take up the challenge and to scrutinize the molecular process underlying the complex rheology of branched polymers. Further experiments and theoretical progress in dealing with the very difficult treatment of a combination of annealed and quenched variables will eventually lead to a true molecular rheology.

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6.4 More Complex Systems With the performance increase enabled by neutron optics, neutron instrumentation and eventually neutron sources, more complex systems come into focus. In this section, we present a few examples of what can be done at present, indicating the way forward when new sources become fully operational.

6.4.1 Dynamics of a Soft Interface in Polymeric Mesophases Depending on composition incompatible diblock-copolymers, assume a number of different ordered phases reflecting the corresponding minimal free energy of the system. At a composition around 50/50 volume fraction, the lamellar phase prevails. Under these circumstances, both components may assume viscoelastic states. In the following, we present an NSE experiment aiming on the fluctuation dynamics of the interfaces, where the two blocks meet. To visualize the interface, the experiments were carried out on a 68,000 molecular weight triblock copolymer of deuterated polyethylethylene (PEE) and polypropylene (PEP), where 24 monomers of PEP at the junction point were hydrogenated. In this way, the junction points were labelled with the label forming a sheet in the lamellar phase. Figure 6.12 presents the dynamic structure factors obtained at 433 and 473 K in the short-time regime (t < 10 ns). We observe a strong Q-dependent decay of the structure factor. In a first qualitative evaluation, the NSE data have been parameterized in terms of

Fig. 6.12 Dynamic structure factor from the junction-labelled triblock copolymer for different ˚ −1 , open triangles ˚ −1 , filled squares: Q = 0.18 A Q values. T = 433 K: filled circles: Q = 0.20 A −1 −1 ˚ , open triangles up: Q = 0.114 A ˚ , open circles: Q = 0.08 A ˚ −1 , open squares: down: Q = 0.14 A −1 −1 ˚ . T = 473 K: filled circles: Q = 0.20 A ˚ , open triangles up: Q = 0.10 A ˚ −1 , open Q = 0.05 A −1 −1 ˚ . The solid lines are result of the fit with the ˚ , open squares: Q = 0.05 A circles: Q = 0.08 A complete structure factor for surface undulations and Rouse motion


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Table 6.1 Relaxation times from an initial slope evaluation of the NSE spectra compared to the corresponding predictions from the Rouse model Q 0.05 0.08 0.1 0.114 0.14 0.18 0.2

T = 433 K τRouse (ns) 1,440 220 – 53 23 8.6 5.6

τexp (ns) 61 ± 10 31 ± 7 – 10 ± 2 6±1 2.2 ± 0.6 1.7 ± 0.5

T = 473 K τRouse (ns) 384 58 24 – – – 1.5

τRouse /τexp 23 7 – 5.3 3.8 3.9 3.3

τexp (ns) 52 ± 10 18 ± 3 4 ± 0.5 – – – 1 ± 0.5

τRouse /τexp 6.4 3 6 – – – 1.5

initial slopes. The resulting relaxation times are displayed in Table 6.1. Table 6.1 also contains the corresponding Rouse relaxation times of an unrestricted Rouse chain at the given Q values [30]. Obviously, the chain dynamics close to the junction point are greatly accelerated. This strong enhancement of the relaxation is evidence for the existence of additional forces beyond entropic forces acting on the junction points. If the PEP-PEE triblock chains are arranged in micro-domains separated by well-defined interfaces, then the additional fast relaxation process observed by NSE would result from the undulation motions of the interface. The thermal motions of the interface are driven by the surface tension and are overdamped. With an effective viscosity ηeff in the interfacial region, the relaxation rate τ (Q)−1 follows a linear dispersion τ (Q)−1 = Q · (γ /4ηeff ). In the Q-range of the NSE experiment, the scattering is dominated by the singlechain scattering of the labels which tag the local interface motions in an incoherent way. The corresponding single-chain motion is affected by two processes: (1) the h-PEP section undergoes the entropy-driven Rouse motion active in polymer melts and (2) in addition, it follows the undulations of the interface. The two processes were approximated as independent and are modelled by a Rouse chain with an anchor in the interface that follows the undulations. The time correlation function for the displacements due to undulations assumes the form kB T f (t) = u − u(t)u(0) = 4πγ 2

+ Ei 1,t/τ 2ξπ , −Ei 1,t/τ 2π

ln

ξ

(6.12)

u2 = 1/(2π )2 d 2 Qu2 (Q) denotes the mean-squared amplitude of the surface fluctuations with |u2 (Q)| = kB T /Q2 γ . Since the integral is logarithmically divergent, the integration has to be restricted to the interval between a minimal and a maximal Q. They are related to the finite domain size ξ and a molecular size . With that we arrive at u2 = (kB T /2πγ )n(ξ /). The time-dependent contributions are given by the exponential integral function Ei(1, x) = 1∞ e−xt /t dt. Applying the Gaussian approximation and performing an angular average of the one-dimensional

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undulation motion perpendicular to the local interface plane, the self-correlation function is obtained: 1 √ erf Q f (t) self (Q,t) = π Sint . (6.13) 2 Q f (t) In the approximation of independent motion, the total structure factor is given by self self self (Q,t) · SRouse (Q,t). (SRouse has to be calculated for a chain with the product of Sint one end fixed to a surface.) Figure 6.12 displays a fit with the complete dynamic structure factor, where the Rouse relaxation rate was taken from an experiment on a corresponding homopolymer. The fit parameters were the surface tension and the effective local viscosity of the short-labelled PEP segments. The data are well described by this approach. Values of γ = 2.3 × 10−3 N m−1 at 433 K and γ = 2.4 × 10−3 N m−1 at 473 K are obtained for the surface tension. These compare well with the theoretical values of γ = 1.5 × 10−3 N m−1 at 433 K and γ = 1.4 × 10−3 N m−1 at 473 K deduced from a relation developed by Helfand and Wassermann [31] for the strong-segregation regime. For the local viscosity, we obtain ηeff = 1.4 × 10−3 Ns m−2 at 433 K. This could be compared to the local viscosity which one obtains, e.g. for a monomeric unit in the frame of the Rouse model, ηeff = (3kB T /W 4 )ρ Na /36M0 = 2.7 × 10−3 Ns/m2 (Na : Avogardo number, M0 monomeric weight, ρ : polymer density). We note that via u2 , which depends only on γ , and τ (Q), which depends on ηeff /γ , both quantities are separately evaluated by the fitting procedure. The mean-square amplitudes of the undulations ˚ at 433 K and 15.5 A ˚ at 473 K. The contribution of the interface u2 are 15 A undulations to the scattering intensity decreases with increasing Q and temperature. This feature is particularly obvious in the short-time limit through the variation of the ratio τRouse /τexp . Summarizing, the combination of NSE experiments with advanced chemical synthesis makes it possible to isolate the surface dynamics of an interface in a mesophase-separated block copolymer melt. Modelling the surface forces by an expression for the surface tension, it is possible to describe the NSE spectra consistently. The experimental surface tension agrees reasonably well with the Helfand predictions for the strong-segregation limit.

6.4.2 Polymer Dynamics Under Random Confinement In miscible polymer blends with greatly different component glass transition temperatures, it appears that locally the dynamics of both components are decoupled. A particular prominent case is the system PMMA/PEO (Tg PEO ≈ 200 K and Tg PMMA ≈ 400 K), which has already been discussed in Chap. 4 and for which NMR studies found up to 12 orders of magnitude different local relaxation times [32]. In such a situation, the low Tg component moves in the random environment created by the frozen high Tg component and constitutes a show case for multi-scale dynamics. The resulting hard confinement is very nicely shown in an NSE experiment looking


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1.1

S (Q,t)/S(Q)

1.0 0.9 0.8 0.7 0.6 0.5

0

4

8

12

16

20

24

t /ns

Fig. 6.13 Dynamic structure factor for a hPEO/dPMMA blend at 400 K

at a 75/25% mixture of hydrogenated PEO in a deuterated PMMA matrix. Under these circumstances, the collective fluctuations of the PEO component with respect to the PMMA matrix are scrutinized [33]. As Fig. 6.13 shows, on the NSE time scale these fluctuations are completely frozen – the signal is entirely elastic. Thus, we deal with motions which are entirely restricted to the PEO component. To make these motions visible, (1) the incoherent scattering originating from the PEO protons and (2) the single-chain dynamic structure factor from a mixture of deuterated and protonated PEO chains in a deuterated PMMA matrix were investigated [34]. The self-motion at intermediate length ∼1 nm and time scales ∼ns was studied by neutron backscattering. Using the Gaussian approximation, (6.2), the obtained data were transformed into MSDs. Figure 6.14 displays results for a 35% hPEO/dPMMA sample. The figure includes data points obtained at the two smallest Q values, Q = 2.4 and 3.2 nm−1 . In all cases, the Fourier-transformed data superimpose showing the validity of the Gaussian √ approach in the investigated Q and time regime. Plotting the obtained MSDs against t, good agreement with the Rouse prediction of (6.3) was obtained. Depending on temperature, themaximum displacement of a PEO segment observed during about 1 ns amounts to r2 (t) < 1 nm. The values for the Rouse variable W 4 are given in Table 6.2, including the result for pure PEO. Seemingly, at intermediate scales the PEO dynamics within the frozen PMMA matrix may be characterized by Rouse dynamics featuring a reduced relaxation rate compared to pure PEO. Figure 6.15 displays measurements of the single-chain dynamic structure factor of PEO chains in the PMMA matrix. The dotted lines in Fig. 6.15 are the result of a Rouse description based on the average Rouse relaxation rates obtained from the backscattering data at the 1 ns scale. Obviously, this description fails

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Fig. 6.14 Mean-square displacements obtained from Fourier-transformed backscattering spectra from the 35% PEO/PMMA sample at different Q and temperatures. The solid lines give the prediction of (6.3) using W 4 values from a direct fit to the backscattering spectra

Table 6.2 Rouse rates, as deduced from incoherent data, and distribution width σ for all samples, obtained from the analysis of the dynamic structure factor

Sample Pure PEO (T = 400 K) 25% PEO/PMMA (T = 400 K) 35% PEO/PMMA (T = 400 K) 35% PEO/PMMA (T = 375 K) 35% PEO/PMMA (T = 350 K) 50% PEO/PMMA (T = 350 K)

W 4 (nm4 ns−1 ) 1.29 0.1 0.22 0.12 0.05 0.13

σ – 2.1 1.6 1.8 1.9 1.6

grossly, predicting a decay that is far too fast. Apparently, the motion is strongly slowed down towards longer times or larger length scales indicating the multi-scale character of the dynamics. Fitting an effective Rouse rate quantified, the slowing down to factors of 4–20 compared to the 1 ns scale. Forced Rayleigh scattering diffusion measurements on dilute short PEO dynamics in a PMMA matrix [36] on the micrometer scale displayed 5 orders of magnitude, further slowing down of the motion compared to the NSE results. Thus, the global dynamics of chain diffusion on the μm scale are apparently not related to the chain modes on a 80 ns time and about 1.5 nm spatial scale that are measured using neutron scattering. The data were interpreted in terms of a Rouse model where the beads undergo random fraction with a distribution of friction coefficients following a log-normal


167

Fig. 6.15 Single-chain dynamic structure factor of the 35% PEO/PMMA system at T = 400 K. Q values are 1, 1.5, 2 and 3 nm−1 from top to bottom. The dashed region represents the elastic contribution of 0.265 as calculated by dynamic RPA [35]. The dotted lines illustrate the Rouse theory with a Rouse relaxation rate obtained from the backscattering data. The solid lines show the results of the Rouse model with random friction

distribution. This fit introduced one new parameter, the width of the distribution σ . Its centre was fixed to the average friction obtained from the backscattering results. The solid lines in Fig. 6.15 show the best result which could be achieved for the 35% PEO/PMMA sample at T = 400 K using σ = 1.6. The three lowest Q values are in very good agreement with the experimental data; however, the highest Q data are reproduced at short times only. The origin of this significant deviation at the highest Q and longer times is still unclear, though it could be related to additional confinement effects imposed by the matrix. The values of σ obtained by applying the same procedure to the other compositions and temperatures are given in Table 6.2. As expected, the width of the distribution function increases with increasing PMMA concentration (in the limit of zero PMMA concentration, it should go to zero) and within one blend σ increases with decreasing temperature. We note that the model also reproduces the short-time Rouse behaviour with the observed average Rouse rate in a very consistent way (Fig. 6.14). Coherent neutron scattering at high momentum transfer observes local processes and reveals information on the relaxation rate distribution. Figure 6.16 compares Fourier-transformed spectra from pure PEO and 35% PEO/PMMA obtained from the TOF instrument FOCUS at T = 400 K. Fourier-transformed backscattering data are also included for the blend. The blend data are strongly broadened compared to the spectrum from pure PEO, demonstrating a broad distribution of local relaxation

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Fig. 6.16 Fourier-transformed TOF data of pure PEO and 35% PEO/PMMA at T = 400 K. For the blend system, Fourier-transformed backscattering data are also included. The solid line through the PEO data describes a fit with a stretched exponential (β = 0.5). Line through the blend data, see text

processes in the random PMMA environment. Both spectra merge at Sself (Q,t = 0.4 ps) = 0.4, indicating the transition to microscopic dynamics which is not altered by blending. The solid line through the blend data displays the convolution of the lognormal distribution function from the random Rouse model with the PEO-relaxation function, resulting in a perfect agreement with the observed spectrum. Having in mind that the log-normal distribution was obtained from the NSE and backscattering data, this agreement which combines data over 5 orders of magnitude in time is remarkable. Finally, we note that extended MD simulations were carried out in parallel to these experiments. A Rouse analysis of these simulations revealed strongly stretched Rouse mode correlators, in disagreement with the RRM which maintains single exponential correlators. Seemingly, the RRM misses chain interaction effects which are present in coarse-grained simulations on bead spring models [37].

6.4.3 Protein Diffusion in Polymeric Gels To understand transport mechanisms in complex materials such as living tissue, or to improve diffusion-controlled processes, a knowledge of the molecular diffusion processes is essential. In this example, a study on protein diffusion in a polymer

6 Future Perspectives: Moving to Longer Length and Time Scales... Table 6.3 Sample compositions with resulting mesh size, ξ , and size ratio, R/ξ

Sample, concentration (wt.) 2% PEO 10% PEO 20% PEO 1% protein in 2% dPEO 1% protein 10% dPEO 1% protein 20% dPEO

169 ˚ Mesh size, ξ (A) 91 26 14 91 26 14

R/ξ

0.21 0.76 1.4

gel of various mesh sizes is presented [38]. The experiments were carried out on α -lactalbumin as a globular protein in polyethyleneoxide (PEO)-networks of various mesh sizes. The diffusion of a protein within a polymer network or gel depends on three parameters. 1. The ratio between the size of the diffusing protein and the mesh size of the polymer gel. 2. The fractal dimensions of the protein. 3. The time for the network to rearrange relative to the time for the macromolecules to diffuse over the length of a mesh size [39]. The study addressed both the network dynamics of the PEO molecules in water and the protein diffusion made visible by matching deuterated PEO with D2 O. The molecular weight of the PEO was 216 kg mol−1 , assuming a stable highly entangled network at the concentration and time scales investigated. The correlation length ξ (mesh size) can be estimated for good solvents by:

ξ = Rg (c∗ /c)3/4

(6.14)

with the overlap concentration c∗ , the Avogadro constant Na and the concentration c. 3/5 Using c∗ = 3Mw /4π Na R3g in (6.14), with Rg ∝ Mw for a good solvent, we found that the mesh size is independent from molecular weight. Table 6.3 gives the mesh sizes ˚ and the ratio R/ξ with R the hydrodynamic radius of α -lactalbumin R = 19.7 A. 2 −1 ˚ The average effective diffusion coefficients were 11.9 A ns for a 2% PEO ˚ 2 ns−1 for the 10 and 20% PEO solutions, respectively solution and 5.4 and 2.1 A ˚ 2 ns−1 . (see Fig. 6.17). The corresponding value for free diffusion D0 is 16.5 A This reduction of the diffusion rate needs to be compared with macroscopic measurements using fluorescene recovery after photobleaching (FRAP) [39]. The latter results were described in terms of a phenomenological relation Deff /D0 = ˚ 2 ns−1 for 2, exp[−β (R/ξ )δ ] with β = 2.2 and δ = 0.95 yielding 9.9, 3.0 and 0.83 A 10 and 20% PEO solutions. The NSE results suggest a decrease of the macroscopic diffusion measured on μm scales compared to the NSE values obtained on the scale of the mesh. Furthermore, sub-diffusive behaviour was observed for the 20% sample which was characterized by a line shape parameter ε = 0.5(S(Q,t) ∼ exp(t/τ )ε ). Obviously, strong interaction takes place between the polymer relaxation and the protein diffusion. These experiments gave a first indication that the diffusion of nanosized objects through polymer gels on the scale of the mesh displays interesting

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Fig. 6.17 Presents the effective diffusion coefficient of the protein in the deuterated network resulting from a fit with a simple exponential decay ∼ exp −Deff Q2 τ

and rich phenomena which need to be clarified in the future. A better understanding will contribute, for example, to the design of release processes in drug carriers or the diffusion processes of proteins in the crowded environment of a cell, to name a few examples.

6.5 Large-Scale Dynamics of Proteins Proteins are dynamic flexible objects undergoing different types of motions on a large range of time scales. Table 6.4 lists the most important dynamic regimes. In this section, we are interested in the large-scale domain motions occurring on the nano- to microsecond scale and we want to find out to what extent these dynamics play a role in biological functions. Commonly, the biological view of protein function is governed by the structure function paradigm, stating that the function is determined by the protein structure. Such a view is, for example, expressed in the “lock and key” model where the substrates enter the protein in a rigid reactive conformation like a key in a lock. Other assumptions relate to an induced fit where the substrates induce conformational changes in the partially flexible protein, or to the idea of pre-organization, where fluctuations lead to the reactive confinement. While the study of the large-scale dynamics of synthetic polymers has reached some maturity, corresponding studies of large-scale motions in biopolymers are

6 Future Perspectives: Moving to Longer Length and Time Scales... Table 6.4 Motions in proteins ˚ Timescale (s) Length scale (A) 10−15 –10−12 0.001–0.1 10−12 –10−9

0.1–10

10−9 –10−6

1–100

10−6 –10−1

10–100

171

Type Bond stretching, angle bending Constant dihedral motion Side chain motion Loop motion, collective motion Folding in small peptides Helix coil transition Domain motions Protein folding

still a major challenge. Some insight has been gained on more local scales, e.g. by time-dependent crystallography [40], while at larger scales fluorescent resonant energy transfer (FRET) experiments on proteins carrying fluorescent labels give some insight into fluctuation ranges [41]. Large-scale motions of protein domains are critical for protein function as nanomachines. Understanding the underlying processes requires a simultaneous access to the proper length and time scales at close to natural environmental conditions in solution. This can be realized by the combination of SANS providing structural characterization and neutron spin echo (NSE) furnishing space time resolution on nm and ns scales. Here we discuss a novel recent study on the functional domain motion in ADH [6]. ADH is a class of enzymes that are important for many organisms, allowing the inter-conversion between alcohols and ketones. In humans, ADH is present as a dimer and catalyzes the oxidation of ethanol, thereby allowing the consumption of alcohol in beverages. In yeast, on the other hand, it is at the basis of the fermentation process converting acetaldehyde into ethanol. In the process, the cofactor nicotinamide adenine dinucleotide (NAD) is needed, assisting the oxidation reaction at the zinc catalytic side. Figure 6.18 displays a schematic structure of the dimer based on crystallographic data. The two monomeric units are clearly visible. Each monomer is built from two domains, the catalytic and the binding domain, with a small opening in between where the cofactor NAD needs to be placed and the catalytic reaction takes place. The experiments were undertaken on ADH from yeast forming a tetrameric structure. Figure 6.19a presents the SANS scattering intensities in solution at different ADH concentrations. The coherent scattering intensity in a solution of equal particles is given by I(Q) ∝ NS(Q)F(Q) where the structure factor S(Q) results from the inter-particle interactions of N scatterers and depends on concentration, while the concentration-independent form factor F(Q) = ∑ b j bk exp(iQ(r j − rk )) j,k

(6.15)

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Fig. 6.18 Dimer of alcohol dehydrogenase. The molecule presented by spherical caps is the NAD cofactor used in the chemical reaction

1

b

a

I(q)/a.u.

0.1

c 0.01

crystall structure

0.5% 1% 2% 6% 0 0

0.1 0.2 0.3 0.4 0.5

15% expanded

1

2 q/nm–1

Fig. 6.19 (a) SANS intensity from ADH with bound cofactor at 0.5% wt. concentration (circles) and protein form factors (lines, see text). The shoulder around 1 nm−1 is characteristic for the tetrameric structure. (b) Illustration of the overall protein structure with two crossed dimmers. The cofactor-binding cleft (arrow) separating the inner cofactor-binding domain and the exterior catalytic domain. (c) Concentration scaled, low Q scattering intensity of the protein solution. The dashed line represents an expanded configuration which would be needed in order to explain the dynamic data in terms of rigid body motion


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Fig. 6.20 Dynamic light scattering results for the translational diffusion of tetrameric ADH in water solution at different concentrations. The characteristic rates 1/τ = D∗ Q2 are plotted vs. Q2

describes the scattering of the single tetramer. b j are the atomic scattering lengths and r j the atomic position vectors. The coordinates of the crystal structure from the protein data bank were used for the form factor calculation. As may be seen from Fig. 6.19a, the structural model based on the crystal structure (solid line) is in very good agreement with the SANS results. Figure 6.19c presents the scattering intensities after scaling with concentration. The observed decrease of the scattering intensity at low Q is caused by the structure factor S(Q), which is extracted by dividing by the lowest concentration data. The dynamics in solution are significantly influenced by the overall translational diffusion of the molecular aggregate. Therefore, in the next step the translational diffusion coefficient needs to be studied. Dynamic light scattering is the proper tool since it investigates the overall dynamics on the proper length scale. Figure 6.20 displays light scattering results for different ADH concentrations as a function of momentum transfer squared. We note that we apparently observe identical translational diffusion coefficients for all concentrations. At 5 ◦ C, this is equal to DDLS = 2.35 ± 0.2 × 10−2 nm2 ns−1 . The corresponding hydrodynamic radius RH = 4.5 ± 0.2 nm compares well with the radius of gyration RG = 3.4 nm of the crystal structure, assuming RG ≈ 3/5RH = 3.5 nm for a sphere. The dynamics on the scale of the protein were investigated with NSE, providing the intermediate dynamic structure factor I(Q,t)/I(Q) in a time range from about 100 ps to 170 ns. Figure 6.21 displays the NSE results for a large number of

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Fig. 6.21 Intermediate scattering function measured by NSE for a protein concentration of 5% wt. with the bound cofactor. The broken line is a fit to Q = 0.68 nm−1 for times above 75 ns

different momentum transfers Q. The data are presented in a log linear fashion showing directly the single exponential decay observed in all cases. Fits with single exponential decays are shown by straight lines. We note, however, that at intermediate Q (Q = 0.68 nm−1 ), small but systematic deviations appear indicating a two-component structure. Comparing the NSE Q-range with the SANS data (Fig. 6.9), we realize that the range of the structure factor where intermolecular interactions are important is covered as well as the regime of internal structure. Best estimates for the first cumulant ∂ /∂ tn(I(Q,t))|t=0 = −Γ(Q) = Q2 Deff (Q) were extracted from the spectra, with the relaxation rate Γ(Q) and the effective diffusion coefficient Deff (Q) carrying the essential information on the observed dynamics. The experimental results (Fig. 6.22a) show a strong Q modulation with a maximum around Q = 0.1 nm−1 . Furthermore, we realize that at low Q the data are in agreement with the concentration-independent light scattering results. We also see that beyond the statistical error in the low Q flank of the 5% data, the relaxation without the cofactor NAD is faster than that including the cofactor. We may conclude that in the dynamics of the ADH tetramer, on the scale of the aggregate itself, we observe significant contributions beyond translational diffusion. We now want to interpret the data and commence with the low Q data which are affected by the interactions between the molecules. In this regime, the effective diffusion coefficient relates to the diffusion coefficient D0 at infinite dilution through Deff (Q) = D0

H(Q) , S(Q)

(6.16)


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Fig. 6.22 (a) Effective diffusion coefficient Deff (Q) for three different solutions of ADH. (b) comparison of Deff (Q) corrected for S(Q) and H(Q) with different models (see text)

where H(Q) is the hydrodynamic factor and S(Q) is the inter-particle structure factor. The structure factor may be extracted from the SANS data by dividing the concentration-dependent results by the lowest concentration result (C = 0.25%). With this experimental structure factor, the data may be corrected. This correction removes the low Q increase of the 5% data, but leaves the results at higher Q unaffected (see Fig. 6.22b). The hydrodynamic factor cannot be measured directly. A first approximation in terms of a Percus-Yevick model shows that (1) at a 1% level, the correction factor H(Q)/S(Q) leaves the experimental data practically untouched and (2) at 5%, the correction is somewhat weaker than the experimentally observed effect. Nevertheless, beyond Q = 0.6 nm−1 , the ratio of H(Q)/S(Q) remains constant.

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The prime reason for a Q-dependent structure in Deff (Q) is rotational motion of the molecule. In a first cumulant approximation, the effective diffusion coefficient of a rigid body undergoing translational and rotational diffusion has the form [42]

Deff (Q) =

kB T × Q2

↔ Q Q iQr j −iQr K H bK e ∑ jK b j e Q×rj Q × rK ∑ jK b j eiQr j bK e−iQrK

.

(6.17)

Here ri and rk are the atomic coordinates, bi and bk the corresponding neutron ↔

scattering lengths and H the mobility tensor. The sum runs over all atoms of the molecule or molecular aggregate and the pointed brackets indicate an ensemble average. The denominator is the form factor of the aggregate. The mobility matrix ↔ ↔ ↔ H is a 6 × 6 tensor involving translational (T ) and rotational (R) parts, including a ←→

translational rotational coupling (T R). For the simplest case of an isotropic particle, T = Dtrans /kB T and R = Drot /kBT . The evaluation of (6.16) is complicated by the hydrodynamic interaction between the different parts of the molecule. In the biophysical literature, one can find the computer code HYDROPRO, which was developed by the group around de la Torre et al. [42]. In this code, a complicated molecule is approximated by a rigid aggregate of small spheres. Its diffusive motion, including the hydrodynamic interaction, is then calculated by a proper superposition of the motions of the rigidly connected spheres. Calculations with HYDROPRO need the crystallographic coordinates of all atoms as an input. All data sets for Deff (Q) at the different concentrations with and without cofactor are compared in Fig. 6.22b. After rescaling with the diffusion coefficients at different concentrations in the region above Q = 0.6 nm−1 , all data sets are consistent. We furthermore note that the data at 5% concentration are of significantly higher statistical accuracy. The solid line in Fig. 6.22b displays the result of the HYDROPRO calculations for a rigid molecule. While the line describes the general form of the effective diffusion coefficient data reasonably well, significant deviations are evident on the low Q flank. The assumption of a significant expansion of the protein structure (≈15%) would shift the flank to match the position of the data points, but also decrease DQ eff (Q) by a factor of 0.89 and rescale the SANS form factor by a factor 1/1.15 to lower Q (Fig. 6.9a dashed line), all contradicting the experimental results. Thus, the deviations indicate additional dynamics not described by the rigid body translational and rotational diffusion. The deviations are shown in Fig. 6.23 with a pronounced peak around 0.8 nm−1 . In the case of the bound cofactor, the peak weakens and loses its high Q part. A first interpretation of this result may be carried out in terms of a normal mode analysis. For this purpose, an elastic network model is used where the complicated bonded and non-bonded interactions are replaced by a pair-wise Hookian potential controlled by a single parameter. As has been shown, such a simple formulation is sufficient to describe the anomalous low frequency motion of large proteins [43].


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Fig. 6.23 Difference between the corrected diffusion coefficients and the calculated translational or rotational diffusion coefficient

In this model, two close atoms are connected by a spring if the equilibrium distance between them is sufficiently small. The elastic forces acting on the particles obey Hookes law and depend only on the change in distance between them. If Ri0 are the equilibrium positions of the particles, and Ri (t) are their actual coordinates, then the Hookean Potential has the form 2 EH = f ∑ Ai j |Ri − R j | − |R0i − R0j | (6.18) i, j

A is the adjacency matrix with Aij = 1 if |R0i − R0j | < 0 and Aij = 0 otherwise. In principle, the dynamics of such an elastic network are non-linear because the distances |Ri − R j | are non-linear functions of the coordinates. Close to equilibrium, the equations of motions can however be linearized to yield an equation of motion mi rï = − ∑ Λi j r j ,

(6.19)

j

where Λ is the 3N × 3N dynamical matrix obtained by the linearization process. In this linear approximation, the motions can be described as a sum of independent oscillating normal modes. ri (t) = ∑ kα eαi sin(ωα t + φα )

(6.20)

α

with ωα and eαi representing non-zero eigenvalues and the respective eigenvectors of the matrix Λ and kα being the amplitude factor of the mode. ϕα is a phase angle. The large-scale slow motions that we are interested in are dominated by the soft modes with small eigenvalues. In the case of the overdamped modes which are seen in the

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Fig. 6.24 (a) Diffusion form factor of the normal modes 7 (top two lines) and 11 (bottom two lines) for the protein configuration with and without the cofactor. (b) Motional pattern of mode 7: without cofactor, the exterior domain (catalytic domain) tilts outwards and opens the cleft. The inner domain with connection points between the monomers remains stiff. (c) Motional pattern of mode 11: with and without the bound cofactor, the monomers within a dimer exhibit a torsional motion around the long dimer axis (in the image plane), which is more pronounced with the cofactor

experiment, the oscillating part needs to be replaced by an exponential (exp[−λα t]). The relaxation rates λα contain the unknown friction factors within the molecule and with the surrounding water molecules. A first approximation for the dynamic structure factor may be obtained in terms of a small displacement approximation of the cross-section. In this approximation, the dynamic structure factor reads S(Q,t) ≈ ∑ bk bl eiQr k eiQr e + ∑ k,l

α

k B T − λα t bk bl iQrk iQre α α e e e Qek Qel . √ ∑ 2 ωα mk ml α (6.21)

In the first cumulant approximation, we have to take the logarithmic derivative with respect to time at time equal zero. Subtracting the translational and rotational parts, we finally obtain √ ∑α λα kB T /ωα2 ∑k, (bk be / mk ml )eiQrk e−iQre Qeαk Qeα ΔDeff (Q) = (6.22) Q2 ∑k, bk be eiQr k eiQr e describing the dynamic form factor of the eigenmodes. An evaluation of the mode form factors ΔDαeff (Q)/λα shows that the most prominent contributions to ΔDαeff (Q) at low Q arise from the first non-trivial mode 7 and mode 11, which are displayed in Fig. 6.24a with and without the cofactor. Figure 6.24b, c display the corresponding motional patterns. We note the


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astonishingly close resemblance of the predicted ΔD0eff (Q) with the observation displayed in Fig. 6.23, though the experimental Q-range is somewhat lower than that calculated. Concerning molecular function, mode 7 is of highest interest. Here the outside catalytic domains move with respect to the relatively rigid tetrameric core part, thereby changing the width of the cleft [44, 45]. With the cofactor, the mode 7 form factor loses its high Q flank, as observed in the experiment, and the vibrational pattern indicates a less flexible cleft region, though similar. While the original crystal structure only displays a small narrow pocket, where the cofactor has to fit in, the opening motion of mode 7 provides more configurational freedom, thus improving the ability to accept or release the cofactor for biological function. The other contributing mode (11) also shows a pronounced low Q peak. This is of more mechanistic nature, describing the torsion of the dimer – the cleft is not opened. Finally, the amplitude and force constant for the modes relevant for the low Q dynamic feature were estimated by re-evaluating NSE curves at Q = 0.68 nm−1 (Fig. 6.11). By describing the long time part (t > 75 ns) using the time constant prediction for combined translational rotational diffusion at 5% conc. (τ = 112 ns) and extrapolating to t = 0, an amplitude of about 0.9 was found. The additional short time part (t < 75 ns) decays with a time constant of τ ≈ 30 ns, much smaller than time constants for translation (τ ≈ 100 ns) or rotation (τ > 100 ns) which are not distinguishable experimentally. Via a Debye-Waller factor estimate, we get the relevant mean-square displacement of the internal mode as Q2 r2 /3 = n0.9 giving r2 = 0.7 nm2 . From equipartition, a spring constant f = kB T /r2 = 5.6pN/nm is obtained – a value well within the range of spring constants considered for biomolecules [45]. In summary, using NSE it becomes possible to study the internal dynamics of tetrameric ADH on the space time scale of the protein. The large-scale correlated domain motions revealed through a normal mode analysis are understood as a result of the moving exterior catalytic domains relative to the stiff core of the tetrameric protein. The motional amplitude of about 0.8 nm is significant in order to assist the functionally necessary incorporation of the cofactor.

6.6 Conclusions and Outlook This chapter summarizes the status of high-resolution neutron studies of the dynamics of macromolecules, polymers and proteins. It focuses on the large-scale dynamics of such molecules and has disregarded the multitude of results on more local motions, such as those associated with the glass transition, which are covered in other chapters of this book. In the case of synthetic polymers, recent results on the universal dynamics of flexible polymers are presented, from the entropy-driven Rouse dynamics to the chain confinement and reptation, including reptation limiting processes. Emphasizing the quenched SANS technique, aspects of the molecular motions of branched polymers including those in the non-linear regime of rheology

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have been discussed. For the case of biopolymers, some first experiments on the domain motion of ADH, an important enzyme for fermentation and the oxidation of ethanol, are discussed. An attempt is made to transmit the flavour of what can be achieved with high-resolution neutron spin-echo spectroscopy and with timeresolved SANS, permitting an access to molecular motions in space and time. Some of the major future challenges in the field of macromolecular dynamics as may be addressed by high-resolution neutron scattering techniques including timeresolved (quenched) SANS are discussed in what follows. Polymer melts and molecular rheology. Polymer melts exhibit a rich and complex rheological behaviour that depends sensitively on molecular architecture, such as the nature and amount of branching, the composition of the polymers, or both. The ultimate goal, to relate the constitutive equation to molecular properties, is both a major challenge for basic science and a matter of great practical importance for the processing of polymer materials. The tube model of de Gennes and Doi and Edwards was extended to non-linear rheology and to take account of the hierarchical relaxation in branched systems. Experiments with partially labelled chains will allow us to focus on the behaviour of particular points along the chains, such as branching points or interfaces in mesophase-separated materials. Further progress will result from detailed observations of the motions of the constituents of polymer molecules with various architectures, e.g. stars, H-polymers, combs or polymeric dendrimers. Self-healing smart materials, supramolecular polymers and networks and gels pose another challenge for microscopic understanding. Specific labelling will provide a better understanding of multi-component systems like block copolymers and will allow, e.g. study of the interactions of different polymers at close contact and the dynamical properties of inner interfaces. Worm-like micelles are another example of linear chains that become entangled beyond a sufficient length. Branched worm-like micelles also exist. These self-aggregating micellar systems are constantly breaking and reforming. Their viscoelastic behaviour may be described in a similar way to that of polymer solutions. Due to the much larger scattering intensity of these micelles, experiments are easier but will need a much longer time scale of observation. By mixing different types of surfactants, the properties of these micelles may be tuned. The dynamics of this system are interesting because external influences, such as counter ion concentration and composition change, considerably alter the structural flexibility and interactions and consequently alter also the rheological behaviour. In addition, an increasing number of applications rely on controlled viscosity introduced by this type of micelles in complex fluids. Complex fluids under constraints. Complex fluids containing mesoscopic structures, like polymers in solution, molecular aggregates such as micelles, or microemulsion structures, interact in a non-trivial way with structures of comparable size in porous media, or with added objects. Besides, the purely geometrical restrictions on a mesoscopic scale, there are hydrodynamic, electrostatic and


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osmotic pressure-related interactions. These determine the physical chemistry of polymer solutions, surfactant systems or aggregates covered with polymeric brushes in solution. The interactions are important for stabilization of the colloidal systems that are found in a huge variety of applications ranging from cosmetics and pharmaceuticals, food stuffs, paints and agrochemicals, to fire fighting and oil recovery agents and fuel additives. Neutron spin-echo spectroscopy, in particular at larger length and time scales, will be a very important tool for disentangling these interactions and for directly accessing the molecular rheology of such materials. Linear non-charged polymers in solution have been studied extensively by light scattering and neutron spin-echo spectroscopy. More complex geometries in the form of chemically bound aggregates in solution such as stars and dendrimers or micelles are of growing interest. The influence of their connectivity and overall shape yields specific dynamic properties which can be probed by neutron spinecho, e.g. collective density fluctuations may be discriminated from caging effects by different labelling. The huge possible variation of architectures may thereby lead to unknown effects. Finally, the role of flexibility fluctuations in molecular recognition, protein folding and association is also a topic for which we expect valuable contributions from the advanced use of NSE in the future. Molecular networks. Rubbers of various kinds are indispensable in daily life; transportation relies on tires, shock-absorbing elements are ubiquitous in smallto-large machines and buildings, and flexible protective enclosures and seals are further applications. These applications need tailored properties, for example rubber tires contain small particle fillers to achieve the right combination of elasticity damping friction and wear resistance. Neutron scattering contributes significantly to our understanding of the underlying physics of the molecular network; NSE will augment this by dynamical information such as the local mobility of molecular structures. Fillers are vital for the different application of rubbers. They interact by space filling, adsorption of the network chains, chemical binding and possible modification of hydrodynamic interaction. All of these interactions will leave finger prints on the polymer dynamics in the space and time domain of NSE instruments. One of the scientific challenges which polymer physics has had to cope with in the last few decades was to improve understanding of the basic mechanisms and structure-property relationships between mechanical performance and molecular details under external load. The so-called quenched disorder in a melt of polymers, through covalent cross-linking or equivalently the thermal quenching of nonequilibrium melt structures of linear and branched polymers, basically tells us that the entanglement topology of chains is the key component. The NSE technique provides a unique means of independently probing the fluctuation ranges of both cross-links and polymer segments in the presence of uni-axial deformation fields. The NSE technique could also be used to clarify and separate the mechanisms in the reinforcement process which is typical for filled elastomers. The presence of fillers definitely must affect the dynamics of the chains at the filler surface or produce a local confinement effect in between filler particulates.

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Gels and polyelectrolytes. Gels are very soft solids that share properties with swollen networks and with semi-dilute and concentrated solutions. The networkforming cross-links may be chemical bonds or reversible physical associations. The dependence of the latter on external parameters such as temperature, pH or ion concentration has led to a number of applications as functional materials. Diffusion of mesoscopic objects, e.g. proteins, may be controlled by the mesh size. Such diffusive motions, and the local and global mesh dynamics including their material dependencies, are accessible by NSE and dynamic light scattering. Charged molecules, e.g. polyelectrolytes, exhibit properties that are quite different from non-ionic systems, electrostatic forces determining the intermolecular interactions. Properties may be modified by variation of the ion composition in solution. Gels of polyelectrolytes show a large response of the swelling on the electrolyte quality or external forces, e.g. chemo-mechanical electro drives may be made; applications include diapers, drug delivery, cholesterol control, paper making, sludge flocculation and electrolyte systems for batteries. However, only a very few inelastic neutron experiments on polyelectrolytes in solution exist. They have been hampered to a large extent by intensity restrictions that may be overcome with new instrumentation. Glasses. Our life would be difficult to imagine without the ubiquitous presence of glass-forming polymers. Nevertheless, despite the long history of research with classical spectroscopic techniques, the molecular nature of the glass transition and the associated relaxation processes are still not well understood. Investigations using neutrons are, in principle, able to reveal the underlying molecular motions. Since relaxations influence the stability, ductility and impact strength of materials, their understanding will be a basis for the design of new materials with improved properties. Though there have already been quite a number of high-resolution neutron studies, there are still a larger number of challenges to be solved in the future: 1. One of the most important problems relates to the much invoked possible length scale of the α -relaxation. 2. At intermediate scales, the collective dynamics are determined by the stress relaxation in the materials. Their relation to the molecular motion is largely unclear. 3. The question how local relaxations couple to the stress relaxation is still unclear. 4. The extent of collectivity in the β-process is still not solved. 5. In polymer blends, the issue of dynamic miscibility is a highly discussed topic; this is connected to the question of a length scale for the dynamic heterogeneity. Biophysics. The continuous performance improvements in high-resolution neutron spectroscopy are leading to neutron studies on the cooperative large-scale function motions in biopolymers. With rapidly developing computing techniques, the time range of simulations is also growing and the system size is increasing. Thus, the new opportunities to use neutron spectroscopy to gain insight into the function of biological molecules will be boosted by the dramatic increase in size, time range


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and realism of computer simulations; these are needed to guide the experiments and to interpret the results. Furthermore, the ability to produce sufficient amounts of specific proteins by genetic engineering will increase the power of such “sample amount demanding” techniques as NSE. Together with the enhanced instrument capabilities, these facts increase the expectation that biophysics with NSE will strongly develop, from the currently still small activities, into a flourishing branch. Proteins serve as nano-machines, such as catalysts for chemical reactions, molecular transport or many other functions. Today, we know that the dynamics of proteins are essential for many of their functions – practically all catalytic reactions are accompanied by drastic conformational changes. Recent NSE experiments have shown how to address the thermal fluctuations which lie behind the functional dynamics and a large increase of the corresponding studies is therefore expected. These works might also be extended in the future to the function-enabling dynamics in protein complexes. Thus, high-resolution NSE together with extended simulations will reveal essential information about the relationships between biological function and motion. The diffusional motion of proteins in crowded environments will be another field where neutron spin-echo will contribute significantly. After some pioneering studies [7], the door is now open to study protein motion in membranes, the interactions of their dynamics with that of the lipids, the motion of proteins through meshes, the interaction of the protein with these meshes and so on. Nanostructured materials. Nanotechnology is based on the specific properties of materials which are determined by their structure on the nanometre scale. Designing and controlling composite materials on the nanoscale can be promoted by the self-assembly and self-organization of compartmented macromolecules. The nanostructuring of materials by non-equilibrium preparation or by self-organization is a new means for combining or creating new material properties. Many of these new materials contain soft components whose dynamics are accessible to NSE. Functional microgels, filled rubbers, magnetic fluids and block-copolymers are a few examples. In these cases, the soft interphase leads to forces and friction that cause fluctuation dynamics in the NSE regime. NSE, in combination with SANS, will allow the extraction of the structural parameters and the elastic and viscous contributions that determine both the microscopic dynamics and the macroscopic properties. Nanoparticles have both an important effect on our life (protein complexes, colloid particles, etc.) and play an important role in industry (formulation of pigments, production of catalysts, pharmaceuticals, etc.). It is necessary to understand at a molecular level how such nanoparticulate systems are formed. For example, many active organic compounds are poorly soluble in water, or even not soluble. Aqueous forms of applications therefore require special formulation techniques to utilize or optimize the physiological or technical action. NSE will provide information on their mesoscopic diffusion properties and their mutual interactions. Other examples for nanostructures are polymer nanocomposites, which we have discussed above. To summarize, in this chapter we have first given some examples of the applications of high-resolution neutron techniques in the field of macromolecular

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systems. These examples showed the state of the art and tried to pinpoint the future opportunities – ranging from the molecular foundations of rheology to the functional dynamics of biopolymers. Then, trying to give an outlook, we have addressed a number of fields in the science of macromolecules where we expect an increasing impact of high-resolution neutron techniques and in particular neutron spin-echo. Based on this evidence, we expect a bright future and an ever-growing importance for these techniques in the context of soft matter science and biological physics.

References 1. Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon Press, Oxford 2. Edwards SF, Grant JWV (1973) The effect of entanglements of diffusion in a polymer melt. J Phys A 6:1169 3. Doi M, Edwards SF (1978) Dynamics of concentrated polymer systems. Part 1 – Brownian motion in the equilibrium state. J Chem Soc Faraday Trans 2 74:1789; Dynamics of concentrated polymer systems. Part 2 – Molecular motion under flow. J Chem Soc Faraday Trans 2 74:1802; Dynamics of concentrated polymer systems. Part 3 – The constitutive equation. J Chem Soc Faraday Trans 2 74:1818 4. de Gennes PG (1971) Reptation of a polymer chain in the presence of fixed obstacles. J Chem Phys 55:572 5. Bu Z, Biehl R, Monkenbusch M, Richter D, Callaway DJE (2005) Coupled protein domain motion in Taq polymerase revealed by neutron spin-echo spectroscopy. PNAS 102:17646 6. Biehl R, Hoffmann B, Monkenbusch M, Falus P, Preost S, Merkel R, Richter D (2008) Direct observation of correlated domain interdomain motion in alcoholdehydrogenase. Phys Rev Lett 101:138102 7. Richter D, Monkenbusch AA, Colmenero J (2005) Neutron spin echo in polymer systems. Adv Polymer Sci 174:1–221 8. Rouse PR (1953) A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J Chem Phys 21:1272 9. de Gennes PG (1967) Quasi-elastic scattering of neutrons by dilute polymer solutions. I. Freedraining limit. Physics (USA) 3:37 10. Wischnewski A, Monkenbusch M, Willner L, Richter D, Farago B, Kali G (2003) Direct observation of the transition from free to constrained segment motion in entangled polymer melts. Phys Rev Lett 90:058302 11. Montes H, Monkenbusch M, Willner L, Rathgeber S, Fetters LJ, Richter D (1999) Neutron spin echo investigation of concentration fluctuation dynamics in melts of diblock copolymer. J Chem Phys 110:10188 12. Richter D, Farago B, Fetters LJ, Huang JS, Ewen B, Lartigue C (1990) Direct microscopic observation of the entanglement distance in a polymer melt. Phys Rev Lett 64:1389 13. Butera R, Fetters LJ, Huang JS, Richter D, Pyckhout-Hintzen W, Zirkel A, Farago B, Ewen B (1991) Microscopic- and macroscopic evaluation of fundamental facets of the entanglement concept. Phys Rev Lett 66:2088 14. Schleger P, Farago B, Lartigue C, Kollmar A, Richter D (1998) Clear evidence of reptation in polyethylene from neutron spin echo spectroscopy. Phys Rev Lett 81:124 15. de Gennes PG (1981) Coherent scattering by one reptating chain. J Phys (Paris) 42:735 16. Fatkullin N, Kimmich R (1995) Theory of field-gradient NMR diffusometry of polymer segment displacements in the tube-reptation model. Phys Rev E 52:3273 17. Zamponi M, Wischnewski A, Monkenbusch M, Willner L, Richter D, Falus P, Farago B, Guenza MG (2008) Cooperative dynamics in homopolymer melts: a comparison of theoretical predictions with neutron spin echo experiments. J Phys Chem B 112:16220


185

18. Guenza MG (2003) Cooperative dynamics in semiflexibile unentangled polymer fluids. J Chem Phys 119:7568 19. Likhtman AE, McLeish TCB (2002) Quantitative theory for linear dynamics of linear entangled polymers. Macromolecules 35:6332 20. Zamponi M, Monkenbusch M, Willner L, Wischnewski A, Farago B, Richter D (2005) Contour length fluctuations in polymer melts: a direct molecular proof. Europhys Lett 72:1039 21. Wischnewski A, Monkenbusch M, Willner L, Richter D, Likhtman AE, McLeish TCB, Farago B (2002) Molecular observation of contour length fluctuations limiting topological confinement in polymer melts. Phys Rev Lett 88:058301 22. McLeish TCB (2002) Tube theory of entangled polymer dynamics. Adv Phys 51:1379 23. Lodge T (1999) Reconciliation of the molecular weight dependence of diffusion and viscosity in entangled polymers. Phys Rev Lett 83:3218 24. Millner ST, McLeish TCB (1998) Reptation and contour-length fluctuations in melts of linear polymers. Phys Rev Lett 81:725 25. Zamponi M, Wischnewski A, Monkenbusch M, Willner L, Richter D, Likhtman AE, Kali G, Farago B (2006) Molecular observation of constraint release in polymer melts. Phys Rev Lett 96:238302 26. Graham RS, Likhtman AE, McLeish TCB, Millner ST (2003) Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J Rheol (New York) 47:1171 27. Blanchard A, Graham RS, Heinrich M, Pyckhout-Hintzen W, Richter D, Likhtman AE, McLeish TCB, Read DJ, Straube E, Kohlbrecher J (2005) Small angle neutron scattering observation of chain retraction after a large step deformation. Phys Rev Lett 95:166001 28. Warner M, Edwards SF (1978) Neutron scattering from strained polymer networks. J Phys A 11:1649 29. Heinrich M, Pyckhout-Hintzen W, Allgaier J, Richter D, Straube E, Read DJ, McLeish TCB, Groves DJ, Blackwell RJ, Wiedenmann A (2002) Arm relaxation in deformed H-polymers in elongational flow by SANS. Macromolecules 35:6650 30. Montes H, Monkenbusch M, Willner L, Rathgeber S, Richter D, Fetters LJ, Farago B (2002) Direct observation of domain wall excitations in symmetric diblock copolymer melts at and above the order disorder transition. Europhys Lett 58:389 31. Helfand E, Wassermann ZR (1976) Block copolymer theory. 4. Narrow interphase approximation. Macromolecules 9:879 32. Lutz TR, He Y, Ediger MD, Ca H, Ling G, Jones AA (2003) Rapid poly(ethylene oxide) segmental dynamics in blends with poly(methyl methacrylate). Macromolecules 36:1724 33. Niedzwiedz K (2007) Polymer dynamics in miscible polymer blends. PhD Thesis, University of Mnster/Germany 34. Niedzwiedz K, Wischnewski A, Monkenbusch M, Richter D, Genix AC, Arbe A, Colmenero J, Strauch M, Straube A (2007) Polymer chain dynamics in a random environment: heterogeneous mobilities. Phys Rev Lett 98:168301 35. Akcasu AZ (1993) In: Brown W (ed) Dynamic light scattering, the method and some application. Clarendon, Oxford 36. Haley JC, Lodge TP (2005) Dynamics of a poly(ethylene oxide) tracer in a poly(methyl methacrylate) matrix: remarkable decoupling of local and global motions. J Chem Phys 122:234914 37. Moreno AJ, Colmenero J (2006) Anomalous dynamic arrest in a mixture of large and small particles. Phys Rev E 74:021409; Relaxation scenarios in a mixture of large and small spheres: dependence on the size disparity. J Chem Phys 125:164507 38. Biehl R, Guo X, Prud’homme PK, Monkenbusch M, Allgaier J, Richter D (2004) Diffusion of compact macromolecules through polymer meshes: mesh dynamics and probe dynamics. Physica B 350:76 39. Cheng Y, Prud’homme PK, Thomas JL (2002) Diffusion of mesoscopic probes in aqueous polymer solutions measured by fluorescence recovery after photobleaching. Macromolecules 35:8111

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40. Schotte F, Lim M, Jackson T, Smirnov AV, Sonan J, Olsen JS, Phillips GN Jr, Wulff M, Aninfrud PA (2003) Watching a protein as it functions with 150-ps time-resolved X-ray crystallography. Science 300:1944 41. Haran G, Haas E, Szipkowska BK, Mas MT (1992) Domain motions in phosphoglycerate kinase: determination of interdomain distance distributions by site-specific labeling and timeresolved fluorescence energy transfer. Proc Natl Acad Sci (USA) 89:11764 42. de la Torre JG, Huertas ML, Carrasco B (2000) Calculation of hydrodynamic properties of globular proteins from their atomic-level structure. Biophys J 78:719 43. Tirion MM (1996) Large amplitude elastic motions in proteins from a single-parameter, atomic analysis. Phys Rev Lett 77:1905 44. Luo J, Bruice TC (2007) Low-frequency normal modes in horse liver alcohol dehydrogenase and motions of residues involved in the enzymatic reaction. Biophys Chem 126:80 45. Marshall BT, Sarangapani KK, Wu J, Lawrence MB, McEver RP, Zhu C (2006) Measuring molecular elasticity by atomic force microscope cantilever fluctuations. Biophys J 90:681

Part II

Bio-Macromolecules: Proteins and Lipids

Chapter 7

Structure and Dynamics of Biological Systems: Integration of Neutron Scattering with Computer Simulation Jeremy C. Smith, Marimuthu Krishnan, Loukas Petridis, and Nikolai Smolin

7.1 Introduction As already alluded to in the previous chapter, the characterization of the structure and internal dynamics of biomolecules such as proteins is essential to understanding the mechanisms of their biological functions. A wide range of experimental techniques has been deployed to elucidate the nature of the functional dynamics of these biomolecules [1]. Among these, the combination of molecular dynamics (MD) simulation and neutron scattering techniques has emerged as a highly synergistic approach to elucidate the dynamics of biological systems. Neutron scattering can be used to test molecular simulation models by direct comparison. This can be done in two ways. The first involves directly comparing experimental and calculated scattering intensities. Secondly, one can indirectly compare experiment and simulation by examining “derived” quantities, such as the radius of gyration and the fractal dimensions of the biomolecules. However, obtaining these derived quantities from experiments and simulation requires approximations and modeldependent data interpretation. Havingmade the comparison with experiments, computer simulation can be then used in the theoretical interpretation of experimental data. For example, data from small-angle neutron scattering (SANS) experiments on heterogeneous, multicomponent biological systems produce complex neutron scattering patterns that can be difficult to interpret, especially when the scattering length densities of the different components are similar. This problem is circumvented experimentally by the use of contrast variation techniques that make it possible to separate scattering resulting from the different components, through the controlled replacement of hydrogen with deuterium. The complex task of identifying scattering contributions

J.C. Smith () • M. Krishnan • L. Petridis • N. Smolin Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge TN 37831-6164, USA e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 7, © Springer Science+Business Media, LLC 2012

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from the various components is simplified with the use of computer simulations. Once the MD simulation has been performed, it is straightforward to compute scattering intensities of the whole system or of the individual components. In other words, the simulation can be considered as a “virtual contrast variation” technique. A general introduction to the simulation methods available for combination with dynamical neutron scattering data was given in Chap. 2. This chapter provides an overview of combined applications of neutron scattering experiments and computer simulations to understand molecular motions of specifically proteins and biopolymers. This chapter covers developments in the applications of dynamic neutron scattering to understand the protein glass transition and role of solvent interactions in controlling protein dynamics, and SANS to develop atomic resolution models of large biomolecular complexes.

7.1.1 Neutron Scattering Functions and Atomic Fluctuations This chapter begins with a brief introduction to some basic equations relating neutron scattering to atomistic dynamics that are relevant to computer simulation. In a neutron scattering experiment, a neutron beam of suitable wavelengths and energies is directed at a sample. When neutrons interact with the sample, they are scattered by the atomic nuclei of the sample, which results in changes in the energy and momentum of the incident neutrons. By measuring these changes, information about the structure and internal dynamics of atoms/molecules that make up the sample can be inferred [2]. In a neutron scattering experiment, one measures the number of neutrons scattered within a solid angle between Ω and Ω + dΩ with a change in energy h¯ ω and momentum h¯ Q as was explained in Chap. 1. This number is proportional to the double-differential cross-section δ2 σ /δΩδω , which in turn is proportional to the dynamic structure factor, S(Q, ω ) [2]:

∂ 2σ ∝ S(Q, ω ). ∂ Ω∂ ω

(7.1)

The dynamic structure factor can be written in terms of the van Hove function, G(r,t), which characterizes the space-time correlation of individual atoms as well as between pairs of atoms, as shown in Fig. 7.1. It is evident from the figure that Fourier transformation of physical quantities determining atomic dynamics in the “Molecular Dynamics Space” leads to information determined in “Neutron Scattering Space” and vice versa. These quantities contain both dynamical and structural information.

7 Integration of Neutron Scattering with Computer Simulation

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Fig. 7.1 Schematic representation of the relationship between neutron scattering functions (determined from experiments) and the space-time correlation function (calculated using MD simulations) is shown here. Here, Ri (t) denotes the atomic coordinates of ith atom at time t

7.1.2 Small-Angle Neutron Scattering In SANS experiments, one measures the flux of neutrons of wavelength λ scattered into an angle θ . The scattering intensity I(Q) is a function of the scattering vector: I(Q) ∝ P(Q) · S(Q),

Q=

θ 2π , sin λ 2

(7.2)

where P(Q) is the particle form factor and S(Q) the particle structure factor. The proportionality sign in (7.2) accounts for instrument-specific factors. SANS intensity profiles are obtained by examining scattering bodies, ensembles of particles that scatter in identical way, for example a scattering body may be a solid sphere that is made of tightly packed atoms. The form factor P(Q) takes into account interference effects between scattering by different atoms of the same scattering body and is given by: 1 P(Q) = n(r) exp[iQ · r]dr, (7.3) Np where Np is the total number of atoms in the scattering body and the scattered Np density n(r) = ∑i=1 δ (r − ri ), (ri is the coordinate of the ith atom). P(Q) is determined by the overall shape of the scattering body.

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In this work, we will focus on the coherent scattering structure factor S(Q) that is given by: sin(Qr) dr, (7.4) S(Q) = 4πρ r2 [g(r) − 1] Qr where g(r) is the pair-correlation function and ρ is the density of the scattering particles. The structure factor describes how I(Q) is modulated by interference effects between radiation scattered by different scattering bodies and can be used to gain information about the relative positions of these scattering bodies. The combination of simulations and SANS is commonly employed to construct structural models of biological systems [3]. One application is rigid body modeling, where a high-resolution structure is fit to a model that can describe correctly the experimental SANS data. Rigid body modeling can be improved by allowing configurational flexibility to the structures via the use of coarse-grained molecular dynamics simulations [4]. In the absence of high resolution structures macromolecular shapes can be reconstructed ab initio, by placing dummy “solvent” and “sample” atoms and employing Monte-Carlo-based switching between solvent and sample atoms [5–7]. In this chapter, we focus on a third application, where SANS profiles are directly calculated from MD trajectories [8–11].

7.1.3 Protein Glass Transition Neutron scattering has been a major technique in protein glass transition research. Following the first neutron scattering experiment reporting glass transition behavior on hydrated myoglobin powders [12, 13], it was demonstrated that the same transition is present in protein MD simulation [14]. Subsequently, a large number of neutron scattering and other experimental and computer simulation studies on various biological systems have revealed that many proteins exhibit this dynamical transition around at temperatures around 180–220 K [12, 13, 15–31]. Below this transition, the dynamics are similar to that of a glassy material, while at temperatures above the transition, protein atoms exhibit liquid-like dynamics [32]. The evidence so far points to the glass transition being a general phenomenon among proteins. Furthermore, in some proteins correlations have been observed between the onset of protein function, such as ligand binding or proton pumping, and the onset of the transition. Thus, it has been suggested that these proteins function only when the temperature is above the dynamical transition, although enzyme function below the transition has been demonstrated [23] (see Chaps. 8 and 9). The 180–220 K transition is sensitive to changes in solvent conditions. For example, proteins immersed in viscous solvents, such as trehalose, exhibit no transition [33]. The above observations led to the following questions concerning the microscopic dynamical details of the protein glass transition: (1) Is the dynamical transition in a solvated protein controlled by the solvent or does the intrinsic


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anharmonicity of protein dynamics also play a role? (2) Do proteins exhibit intrinsic anharmonic dynamics below the glass transition temperature? The change in gradient of mean-square displacement (MSD) vs. temperature is consistent with the dynamics of a protein changing from harmonic to anharmonic across the transition. However, neutron scattering experiments and MD simulations have shown signatures of anharmonic dynamics well below the ∼220 K dynamical transition [24, 29–31]. The dynamic processes associated with this low-temperature anharmonicity, and how these motions may be related to global dynamical changes at the dynamical transition, have yet to be fully understood. A recent neutron scattering study on hen egg-white lysozyme showed the existence of a low-temperature onset of anharmonicity at around 100 K, the origin of which was suggested to be methyl group rotation [30,31]. Indeed, in neutron scattering experiments, the main contribution to the scattered protein intensity arises from the nonexchangeable hydrogen atoms and significant fraction of nonexchangeable hydrogens in proteins resides on CH3 groups: 26% in lysozyme, for example. Thus, the CH3 groups contribute significantly to the scattered intensity. Also, it has been suggested that a dominant contribution of the relaxation observed in dry myoglobin neutron scattering is due to methyl dynamics [19]. 1 H NMR relaxation studies have also investigated the reorientational dynamics of C–H bond vectors of methyl groups, and 1 H NMR experiments on dry lysozyme have shown that 70% of the total proton relaxation is due to methyl dynamics [34]. The MSD of atoms of hydrated myoglobin (structure shown in Fig. 7.2a) as a function of temperature calculated by MD simulations is shown in Fig. 7.2b [35]. The MSD increases linearly at low temperatures, then exhibits two slope changes: one at around 150 K and the other at around 220 K. The change at around 220 K is the solvent-driven dynamical transition as observed in many biological systems. At 150 K, rotational excitations of methyl groups were observed in the MD and these jump-like motions of methyl protons will lead to quasielastic neutron scattering of the type that has been observed experimentally at ∼150 K in proteins [30, 31, 35]. In lysozyme, the low-temperature anharmonicity was observed at 100 K and was attributed to the onset of methyl dynamics. It was also demonstrated that the anharmonic dynamics observed at around 100 K is independent of hydration level, while the dynamical transition at 200–220 K is observed only at hydration levels greater than 0.2 g water per gram protein [36]. Thus, recent neutron scattering experiments and MD simulations have demonstrated the nonnegligible role of intrinsic anharmonicity of protein dynamics in protein glass transition.

7.1.4 Protein-Solvent Dynamical Coupling Water plays a crucial role in determining the structures, dynamics, and function of biomolecules. Water molecules in the hydration layer of biomolecules (biological water) are important not only for the thermodynamic stability of proteins, but also play a central role in several biomolecular functionalities, such as interaction,

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Fig. 7.2 (a) Hydrated myoglobin is shown (new cartoon and thin lines representation) together with water molecules (small spheres: water oxygen atoms), heme (thick lines), and carbon monoxide (big spheres). Water molecules that are present in the first two hydration shells around the protein are shown. (b) Time-averaged MSD of myoglobin as a function of temperature from MD. The straight lines are fits to the data for different temperature ranges (solid line, 0–150 K; dashed line, 150–220 K; dotted line, above 220 K) and are shown as a guide to the eye (reproduced with permission from ref. [35]. Copyright 2008 American Chemical Society)


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catalysis, recognition, etc. H/D substitution is often used to decompose the dynamics of proteins from that of solvent. Via this contrasting technique, water and protein motions can be probed separately by measuring deuterated protein samples hydrated in H2 O and natural-abundance proteins in D2 O, respectively [37–39]. A number of experiments and simulations have indicated that when a protein is solvated, the dynamical transition is strongly coupled to the solvent [20, 33, 40–44]. To determine the driving force behind the protein glass transition, a set of molecular dynamics simulations of myoglobin surrounded by a shell of water was performed using a dual heat bath method, in which the protein and solvent are held at different temperatures, at various temperatures of the components [27]. The results show that the protein transition is driven by a dynamical transition in the hydration water that induces increased fluctuations primarily in sidechains in the external regions of the protein. The water transition involves activation of translational diffusion and occurs even in simulations where the protein atoms are held fixed [27, 45]. Figure 7.3a presents the protein fluctuations calculated from a control set of simulations (in each simulation the protein and solvent are at the same temperature), together with those obtained by fixing the temperature of one component at a temperature below the dynamical transition while varying the temperature of the other. In the control set, the experimentally known dynamical transition is reproduced, with nonlinearity starting at 220 K. Fixing the solvent temperature at 80 or 180 K suppresses the dynamical transition and the protein MSD increases linearly with temperature up to 300 K. Therefore, low-temperature solvent cages the protein dynamics. Figure 7.3a also shows that holding the protein temperature constant at 80 or 180 K and varying the solvent temperature also abolish the dynamical transition behavior in the protein. In summary then, Fig. 7.3a demonstrates that holding either component at a low temperature suppresses the protein dynamical transition. Figure 7.3b,c show the effect of holding one component above the transition temperature while varying the temperature of the other. Holding the solvent temperature at 300 K (Fig. 7.3b) leads to increased protein fluctuations at most temperatures relative to the other simulation sets. However, there is again no clear deviation from linearity, i.e., no dynamical transition behavior. In contrast, fixing the protein at 300 K and varying the solvent temperature (Fig. 7.3c) recover the dynamical transition behavior in the protein, incipient at ∼200 K, a slightly lower temperature than in the control set. When fixing the solvent at 300 K, only effects due to changes with temperature in the sampled region of the protein energy landscape appear. The absence of a dynamical transition indicates, then, that these changes do not control the transition. However, when the protein is held at 300 K, variations with temperature in the sampled solvent landscape trigger the protein transition Figure 7.4 shows the side-chain fluctuations in the control simulations as a function of distance from the protein center of mass. The dynamical transition is seen to be most pronounced in the outer parts of the protein, i.e., those close to the solvent shell – above the transition the outer shells exhibit both stronger

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Fig. 7.3 Mean-square fluctuations of the protein nonhydrogen atoms for different sets of simulations. (a) Filled squares, control set with protein and solvent at same temperature; filled triangles, protein held at 80 K; filled circles, solvent held at 80 K; open triangles, protein held at 180 K; open circles, solvent held at 180 K. (b) Solvent held 300 K. (c) Protein held at 300 K. Figure reproduced from ref. [27]

fluctuations and a larger change in gradient (see inset in Fig. 7.4) than the inner atoms. The solvent transition drives dynamical transition behavior primarily in the side-chain atoms of the external protein regions, i.e., those closest to the solvent. It is believed that hydration is crucial for enzyme catalytic function, that dry enzymes are nonfunctional [36, 46], and that, below a threshold hydration level, enzymes are inactive. Experimental work on the protonic conductivity of protein powders is consistent with a two-dimensional percolation transition of hydration water at the surfaces of various proteins upon increasing of the hydration level [36]. Computer simulations have shown that, upon increasing the hydration level,


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Fig. 7.4 Mean-square fluctuations of the protein side-chain heavy atoms for five different shells, ˚ thick (except for the inner and outer shells which are 8 and 6 A, ˚ respectively). The inset each 4 A shows the difference in the slopes of lines fitted below and above 220 K as a function of distance from the protein center of mass. Linear fits to the data above and below 220 K are also shown for the outermost shell. Figure reproduced from ref. [27]

water molecules form a spanning hydrogen-bonded network enveloping protein [47–51]. Formation of this water network may play a role in collective dynamics, as a hydrogen-bonded network of water molecules may in principle exhibit dynamics which are not present in disconnected groups. Recently, the relationship between enzyme dynamics and activity at low hydration was examined [24,52,53]. It was found that significant intraprotein quasielastic scattering exists even below the dynamical transition. Furthermore, measurements have demonstrated enzyme activity at hydrations as low as 3% [53]. The temperature dependence of the dynamics of mesophilic and thermophilic dihydrofolate reductase has been examined using elastic incoherent neutron scattering [54]. It was demonstrated that the distribution of atomic displacement (MSD) amplitudes can be derived from the elastic scattering data by assuming a (Weibull) functional form [54] that resembles distributions seen in molecular dynamics simulations. A particular advantage of using the Weibull model is that the distribution of atomic fluctuation amplitudes can be estimated. Although the Weibull model has only two adjustable parameters, the combination of a power law and an exponential function confers versatility on the distribution profile. The thermophilic enzyme has found to have a significantly broader distribution than its mesophilic counterpart. Furthermore, although the rate of increase with temperature of the atomic MSDs extracted from the dynamic structure factor

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was found to be comparable for both enzymes, the amplitudes were found to be slightly larger for the thermophilic enzyme. Therefore, these results imply that the thermophilic enzyme is the more flexible of the two. A physical characterization of protein–protein interactions is very important for understanding the mechanics of cell function. Recent all-atom lattice-dynamical calculations for a crystalline protein, ribonuclease A, showed that the sound velocities, density of states, heat capacity (CV ), and thermal diffuse scattering are all consistent with available experimental data [55]. The CV was found to be proportional to T −1.68 for T < 35 K, significantly deviating from a Debye solid. In the vicinity of the Bragg peak, inelastic scattering of X-rays by phonons was found to originate from acoustic mode scattering. The results suggest an approach to protein crystal physics combining all-atom lattice-dynamical calculations with experiments on next-generation neutron sources. Interprotein motions in low and fully hydrated carboxymyoglobin crystals have also been investigated using MD simulations [56]. Below ≈240 K, the calculated dynamic structure factor showed a peak arising from interprotein vibrations. Above ≈240 K, the intermolecular fluctuations of the fully hydrated crystal increase drastically, whereas the low-hydration model exhibits no transition. Autocorrelation function analysis demonstrated the transition to be dominated by the activation of diffusive intermolecular motion. The potential of mean force for the interaction remains quasiharmonic. Furthermore, it was shown that above 240 K, the intermolecular fluctuations of the fully hydrated protein crystal increase drastically, whereas a low-hydration model exhibits no transition. Finally, we draw attention to work designed to derive simplified analytical models for diffusive protein dynamics. MD simulations of oligopeptide chains reveal configurational subdiffusion at equilibrium extending from 10−12 to 10−8 s. Trap models, involving a random walk with a distribution of waiting times, cannot account for the subdiffusion mechanism, which has been found rather to arise from the fractal-like structure of the accessible configuration space [57]. These conceptual approaches will hopefully be of use in analyzing quasielastic scattering.

7.1.5 Density of the Protein Hydration Shell We now present an example of the use of experimental SANS data used to test computational models. Several studies have indicated that it is necessary to take into account hydration effects in small-angle scattering (SAS) studies, X-ray or neutron [9, 58, 59]. MD simulations provide an interpretation of neutron solution scattering data in terms of the density of water on the surface of lysozyme [9]. The simulationderived scattering profiles are in excellent agreement with the experiment. In the ˚ thick first hydration layer was found to be 15% denser than bulk simulation, the 3 A water. About two thirds of this increase is the result of a geometric contribution that would also be present if the water was unperturbed from the bulk. The remaining third arises from modification of the water structure and dynamics (Fig. 7.5).


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Fig. 7.5 Comparison of MD calculated (solid lines) with experimental X-ray and neutron SAS profiles (taken from ref. [9]). In the main figure, the y axis of different profiles is shifted for clarity. Inset shows calculated SAS intensities with common origin

7.1.6 Structural Analysis of Lignocellulose SANS is especially well suited for studying disordered polymers. Theoretical approaches to interpret the scattering include treating the disorder polymers as fractals [60]. Fractals are self-similar objects whose structure remains invariant under magnification [61]. Various biological systems are known to be fractals, such as biomembranes and antibody aggregates [62]. The use of analogy with fractals in the context of SANS originates from the experimental observation that the scattering intensity I(Q) is often proportional to a negative power of the scattering vector, Q: I(Q) ∝ Q−α ,

(7.5)

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where α is the power-law exponent. This power-law behavior is typical of fractal systems and useful information about the system can be deduced from the value of the exponent α . When discussing SANS, it is important to consider two types of fractals: mass fractals and surface fractals [61]. Mass fractals are objects for which the scaling relation gives the mass, M, inside a sphere of radius, R: M(R) ∝ RDm ,

(7.6)

where Dm ≤ 3 is the mass-fractal dimension. In contrast, surface fractals are better understood by considering the space filling ability of a curve or surface [63]. Their properties are determined by the surface-fractal dimension, Ds , which takes values 2 < Ds < 3. The surface-fractal dimension is best understood by imagining that the surface of a molecule is covered with a monolayer of spheres of radius r. Let N(r) be the number of such spheres required to fully cover the surface of the molecule, then: N(r) ∝ r−Ds .

(7.7)

The above two fractal dimensions provide examples of derived quantities that can be used to compare simulation with experiments. Ds and Dm can be obtained using two separate approaches: firstly, by analyzing the scattering intensities obtained experimentally or calculated from MD trajectories. As mentioned earlier, SANS intensities often display a power-law dependence on the wavevector, see (7.5), and the exponent α is directly related to the fractal dimension of the biomolecule. If 1 < α < 3, then mass-fractal scattering is observed and α = Dm . If 3 < α < 4, then surface scattering is observed and now α = 6 − Ds . The power-law exponent α can be measured directly either from the slope of ln I vs ln q, or, more accurately, using the so-called “unified approach” to analyze SANS data [64]. The unified approach describes scattering from complex systems that contain many structural levels: one level described by a Guinier and an associated power-law regime. A function is derived that models both the Guinier exponential and structurally limited powerlaw regimes. For more information, see ref. [64]. The second way to obtain values for the fractal dimensions of biopolymers is through analysis of the atomic-coordinate information of MD trajectories, i.e., without computing SANS spectra. Polymer theory provides a computationally straightforward way to obtain the mass-fractal dimension by linking the easily calculated radius of gyration of a polymer with the total number of monomers N comprising the polymer: (7.8) Rg ∝ N 1/Dm . Depending on the interactions between the solvent and the solute, the theory predicts only three possible values for Dm = 5/3, 2, and 3, for good-, theta-, and bad solvents, respectively. To obtain the surface-fractal dimension, one computes the surface accessible surface area (SASA). This is done by rolling a sphere of radius r on the surface of the molecule and then using the points visited by the center of


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the sphere to define a surface. The observed surface area is a function of the size of the probe radius, as using a smaller probe detects more surface details leading to a larger surface area. For a surface fractal covered with N such spheres of radius r, the SASA is given by: A(r) ∝ N · r2 ∝ r−Ds +2 (7.9) the second relation obtained using (7.7). It is hoped that, in the future, the mass fractal and surface fractal approaches will be useful in understanding lignocellulostic biomass. Lignocellulose is a complex biomaterial made of cellulose microfibrils embedded in a matrix of polysaccharides (hemicellulose and pectins) and lignin [65]. Its structural analysis requires characterization techniques capable of spanning many length scales (from angstroms to micrometers) while differentiating between the components, such as lignin, hemicellulose, and cellulose. SANS is ideally suited to this task because it can probe the length scales appropriate for lignocellulose characterization, and furthermore, neutron scattering enables contrast variation techniques that make it possible to separate scattering results for different compounds within intact lignocellulose through the controlled replacement of hydrogen with deuterium. Deuteration of lignocellulose, however, is a difficult task because of the toxic effect of D2 O on the germination, growth, and development of plants [66]. Computer simulation can potentially bypass this experimental hurdle, since it can act as a “virtual contrast variation” technique. This is accomplished by selecting a component (cellulose, lignin, or hemicellulose) of the model and calculating the SANS profile for this component alone. However, this approach faces many challenges, including the construction of an accurate model and the mismatch between the length scales probed by experiments (up to micrometers) and simulation (up to hundreds of nanometers). This is where the fractal nature of biopolymers becomes useful: experiments show that these materials scatter as mass or surface fractals over a wide range of length scales (wavevector range ˚ −1 < Q < 0.05 A ˚ −1 ), the lower limit of which is accessible with atomistic 0.01 A MD simulations. One can then use the computational methods described above to evaluate the mass- and surface-fractal dimensions. We hope to be able to present results combining SANS with MD in the analysis of lignocellulose in the near future.

7.2 Conclusions In this chapter, we have provided an overview of synergistic applications of various neutron scattering techniques and computer simulation to unravel atomistic details of many dynamical phenomena of physical and biological interest that occur on the subnanosecond time scale. The advent of next-generation neutron sources (such as the Spallation Neutron Source, SNS, at Oak Ridge National Laboratory, USA), together with advanced deuteration facilities and continuing rapid increase

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in computing power, will open up new vistas for further high-resolution insights into large length scale and long time-scale biomolecular structure and dynamics. Acknowledgements This research is funded in part by the Genomics:GTL Program, Office of Biological and Environmental Research, U.S. Department of Energy, under the BioEnergy Science Center. The BioEnergy Science Center is a U.S. Department of Energy Bioenergy Research Center supported by the Office of Biological and Environmental Research in the DOE Office of Science. The research is also funded in part by FWP ERKP704 “Dynamic Visualization of Lignocellulose Degradation by Integration of Neutron Scattering Imaging and Computer Simulation” funded by the DOE office of Bioscience and Environmental Research. Finally, JCS acknowledges funds from the DOE ORNL Laboratory Directed Research and Development funds (grant no: Systems Biology L00044).

References 1. Brooks CL III, Karplus M, Pettitt BM (1987) Proteins: a theoretical perspective of dynamics, structure, and thermodynamics. Wiley, New York 2. Bee M (1987) Quasielastic neutron scattering: principles and applications in solid state chemistry, biology and materials science. Adam Hilger, Philadelphia 3. Neylon C (2007) Small angle neutron and X-ray scattering in structural biology: recent examples from the literature. Eur Biophys J 37:531–541 4. Yang SC, Park S, Makowski L, Roux B (2009) A rapid coarse residue-based computational method for X-ray solution scattering characterization of protein folds and multiple conformational states of large protein complexes. Biophys J 96:4449–4463 5. Svergun DI (1999) Restoring low resolution structure of biological macromolecules from solution scattering using simulated annealing. Biophys J 76:2879–2886 6. Walther D, Cohen FE, Doniach S (2000) Reconstruction of low-resolution threedimensional density maps from one-dimensional small-angle X-ray solution scattering data for biomolecules. J Appl Crystallogr 33:350–363 7. Heller WT, Abusamhadneh E, Finley N, Rosevear PR, Trewhella J (2002) The solution structure of a cardiac troponin C-troponin I-troponin T complex shows a somewhat compact troponin c interacting with an extended troponin I-Troponin T component. Biochemistry 41:15654–15663 8. Svergun D, Barberato C, Koch MHJ (1995) CRYSOL – a program to evaluate x-ray solution scattering of biological macromolecules from atomic coordinates. J Appl Crystallogr 28: 768–773 9. Merzel F, Smith JC (2002) Is the first hydration shell of lysozyme of higher density than bulk water? Proc Natl Acad Sci USA 99:5378–5383 10. Tiede DM, Zhang RT, Seifert S (2002) Protein conformations explored by difference highangle solution X-ray scattering: oxidation state and temperature dependent changes in cytochrome C. Biochemistry 41:6605–6614 11. Tjioe E, Heller WT (2007) ORNL SAS: software for calculation of small-angle scattering intensities of proteins and protein complexes. J Appl Crystallogr 40:782–785 12. Doster W, Cusack S, Petry W (1990) Dynamic instability of liquid-like motions in a globular protein observed by inelastic neutron-scattering. Phys Rev Lett 65:1080–1083 13. Doster W, Cusack S, Petry W (1989) Dynamical transition of myoglobin revealed by inelastic neutron-scattering. Nature 337:754–756 14. Smith J, Kuczera K, Karplus M (1990) Dynamics of myoglobin – comparison of simulation results with neutron-scattering spectra. Proc Natl Acad Sci USA 87:1601–1605 15. Knapp EW, Fischer SF, Parak F (1983) The influence of protein dynamics on Mossbauerspectra. J Chem Phys 78:4701–4711


203

16. Knapp EW, Fischer SF, Parak F (1982) Protein dynamics from Mossbauer-spectra – the temperature-dependence. J Phys Chem 86:5042–5047 17. Diehl M, Doster W, Petry W, Schober H (1997) Water-coupled low-frequency modes of myoglobin and lysozyme observed by inelastic neutron scattering. Biophys J 73:2726–2732 18. Reat V, Patzelt H, Ferrand M, Pfister C, Oesterhelt D, Zaccai G (1997) Dynamics of different functional parts of bacteriorhodopsin: H-H-2 labeling and neutron scattering. Proc Natl Acad Sci USA 95:4970–4975 19. Doster W, Settles M (2005) Protein-water displacement distributions. Biochim Biophys Acta 1749:173–186 20. Reat V, Dunn R, Ferrand M, Finney JL, Daniel PM, Smith JC (2000) Solvent dependence of dynamic transitions in protein solutions. Proc Natl Acad Sci USA 97:9961–9966 21. Daniel RM, Dunn RV, Finney JL, Smith JC (2003) The role of dynamics in enzyme activity. Annu Rev Biophys Biomol Struct 32:69–92 22. Daniel RM, Finney JL, Reat V, Dunn R, Ferrand M, Smith JC (1999) Enzyme dynamics and activity: time-scale dependence of dynamical transitions in glutamate dehydrogenase solution. Biophys J 77:2184–2190 23. Daniel RM, Smith JC, Ferrand M, Hery S, Dunn R, Finney JL (1997) Enzyme activity below the dynamical transition at 220 K. Biophys J 75:2504–2507 24. Kurkal V, Daniel RM, Finney JL, Tehei M, Dunn RV, Smith JC (2005) Enzyme activity and flexibility at very low hydration. Biophys J 89:1282–1287 25. Brown KG, Small EW, Peticola WL, Erfurth SC (1972) Conformationally dependent lowfrequency motions of proteins by laser Raman spectroscopy. Proc Natl Acad Sci USA 69: 1467–1469 26. Tournier AL, Smith JC (2003) Principal components of the protein dynamical transition. Phys Rev Lett 91(20):208106 27. Tournier AL, Xu JC, Smith JC (2003) Translational hydration water dynamics drives the protein glass transition. Biophys J 85:1871–1875 28. Hayward JA, Finney JL, Daniel RM, Smith JC (2003) Molecular dynamics decomposition of temperature-dependent elastic neutron scattering by a protein solution. Biophys J 85:679–685 29. Hayward JA, Smith JC (2002) Temperature dependence of protein dynamics: computer simulation analysis of neutron scattering properties. Biophys J 82:1216–1225 30. Roh JH, Curtis JE, Azzam S, Novikov VN, Peral I, Chowdhuri Z, Gregory RB, Sokolov AP (2006) Influence of hydration on the dynamics of lysozyme. Biophys J 91:2573–2587 31. Roh JH, Novikov VN, Gregory RB, Curtis JE, Chowdhuri Z, Sokolov AP (2005) Onsets of anharmonicity in protein dynamics. Phys Rev Lett 95(3):038101 32. Kneller GR, Smith JC (1994) Liquid-like side-chain dynamics in myoglobin. J Mol Biol 242:181–185 33. Cordone L, Ferrand M, Vitrano E, Zaccai G (1999) Harmonic behavior of trehalose-coated carbon-monoxy-myoglobin at high temperature. Biophys J 76:1043–1047 34. Andrew ER, Bryant DJ, Cashell EM (1980) Proton magnetic-relaxation of proteins in the solidstate—molecular-dynamics of ribonuclease. Chem Phys Lett 69:551–554 35. Krishnan M, Kurkal-Siebert V, Smith JC (2007) Methyl group dynamics and the onset of anharmonicity in myoglobin. J Phys Chem B 112:5522–5533 36. Rupley JA, Careri G (1991) Protein hydration and function. Adv Protein Chem 41:37–172 37. Wood K, Frolich A, Paciaroni A, Moulin M, Hartlein M, Zaccai G, Tobias DJ, Weik M (2007) Coincidence of dynamical transitions in a soluble protein and its hydration water: direct measurements by neutron scattering and MD simulations. J Am Chem Soc 130:4586–4587 38. Jasnin M, Moulin M, Haertlein M, Zaccai G, Tehei M (2007) Down to atomic-scale intracellular water dynamics. EMBO Rep 9:543–547 39. Tehei M, Franzetti B, Wood K, Gabel F, Fabiani E, Jasnin M, Zamponi M, Oesterhelt D, Zaccai G, Ginzburg M, Ginzburg BZ (2007) Neutron scattering reveals extremely slow cell water in a dead sea organism. Proc Natl Acad Sci USA 104:766–771 40. Fitter J, Lechner RE, Dencher NA (1997) Picosecond molecular motions in bacteriorhodopsin from neutron scattering. Biophys J 73:2126–2137

204

J.C. Smith et al.

41. Ferrand M, Dianoux AJ, Petry W, Zaccai G (1993) Thermal motions and function of bacteriorhodopsin in purple membranes – effects of temperature and hydration studied by neutron-scattering. Proc Natl Acad Sci USA 90:9668–9672 42. Paciaroni A, Cinelli S, Onori G (2002) Effect of the environment on the protein dynamical transition: a neutron scattering study. Biophys J 83:1157–1164 43. Fitter J (1999) The temperature dependence of internal molecular motions in hydrated and dry alpha-amylase: the role of hydration water in the dynamical transition of proteins. Biophys J 76:1034–1042 44. Teeter MM, Yamano A, Stec B, Mohanty U (2001) On the nature of a glassy state of matter in a hydrated protein: relation to protein function. Proc Natl Acad Sci USA 98:11242–11247 45. Tournier AL, Xu JC, Smith JC (2003) Solvent caging of internal motions in myoglobin at low temperatures. Physchemcomm 6:6–7 46. Finney JL (1996) Hydration processes in biological and macromolecular systems. Faraday Discuss 103:1–17 47. Brovchenko I, Krukau A, Smolin N, Oleinikova A, Geiger A, Winter R (2005) Thermal breaking of spanning water networks in the hydration shell of proteins. J Chem Phys 123:10 48. Oleinikova A, Brovchenko I, Smolin N, Krukau A, Geiger A, Winter R (2005) Percolation transition of hydration water: from planar hydrophilic surfaces to proteins. Phys Rev Lett 95:4 49. Oleinikova A, Smolin N, Brovchenko I (2006) Origin of the dynamic transition upon pressurization of crystalline proteins. J Phys Chem B 110:19619–19624 50. Oleinikova A, Smolin N, Brovchenko I (2007) Influence of water clustering on the dynamics of hydration water at the surface of a lysozyme. Biophys J 93:2986–3000 51. Smolin N, Oleinikova A, Brovchenko I, Geiger A, Winter R (2005) Properties of spanning water networks at protein surfaces. J Phys Chem B 109:10995–11005 52. Kurkal V, Daniel RM, Finney JL, Tehei M, Dunn RV, Smith JC (2005) Low frequency enzyme dynamics as a function of temperature and hydration: a neutron scattering study. Chem Phys 317:267–273 53. Kurkal-Siebert V, Daniel RM, Finney JL, Tehei M, Dunn RV, Smith JC (2006) Enzyme hydration, activity and flexibility: a neutron scattering approach. Biophys J 89:4387–4393 54. Meinhold L, Clement D, Tehei M, Daniel R, Finney JL, Smith JC (2007) Protein dynamics and stability: the distribution of atomic fluctuations in thermophilic and mesophilic dihydrofolate reductase derived using elastic incoherent neutron scattering. Biophys J 94:4812–4817 55. Meinhold L, Merzel F, Smith JC (2007) Lattice dynamics of a protein crystal. Phys Rev Lett 99(13):138101 56. Kurkal-Siebert V, Agarwal R, Smith JC (2007) Hydration-dependent dynamical transition in protein: protein interactions at approximate to 240 K. Phys Rev Lett 100(13):138102 57. Neusius T, Daidone I, Sokolov IM, Smith JC (2007) Subdiffusion in peptides originates from the fractal-like structure of configuration space. Phys Rev Lett 100(18):188103 58. Hubbard SR, Hodgson KO, Doniach S (1987) Small-angle X-ray-scattering investigation of the solution structure of troponin-C. J Biol Chem 263:4151–4157 59. Grossmann JG, Abraham ZHL, Adman ET, Neu M, Eady RR, Smith BE, Hasnain SS (1993) X-ray-scattering using synchrotron-radiation shows nitrite reductase from achromobacterxylosoxidans to be a trimer in solution. Biochemistry 32:7360–7366 60. Schmidt PW (1991) Small-angle scattering studies of disordered, porous and fractal systems. J Appl Crystallogr 24:414–435 61. Mandelbrot BB (1983) The fractal geometry of nature. Freeman, New York 62. Dewey TG (1997) Fractals in molecular biophysics. OUP, New York 63. Pfeifer P, Avnir D (1983) Chemistry in noninteger dimensions between 2 and 3.1. Fractal theory of heterogeneous surfaces. J Chem Phys 79:3558–3565 64. Beaucage G (1995) Approximations leading to a unified exponential power-law approach to small-angle scattering. J Appl Crystallogr 28:717–727 65. Cosgrove DJ (2005) Growth of the plant cell wall. Nat Rev Mol Cell Biol 6:850–861 66. Boudet A, Humphrey TJ, Davies DD (1975) Measurement of protein turnover by density labeling. Biochem J 152:409–416

Chapter 8

Protein Dynamics and Function Stéphane Longeville and Wolfgang Doster

8.1 Introduction Proteins were discovered by the Dutch chemist G. Mulder as early as 1838. They were named by J. J. Berzelius from the Greek word Protos, which signifies first in importance, probably because they constitute more than 50% of the dry weight of the cells. Another explanation suggests that proteins (like protean) arise form the Greek god Proteus who has the capability to appear under different forms. This refers to the very strong variety of protein structures but such an approach is an anachronism because protein structures were only studied during the twentieth century. Proteins are ubiquitous in cells and serve all types of metabolism and function: they can be structural proteins, help transport or catalyse reactions and also be involved in regulation and signal pathways or act as molecular motors. Proteins are biological macromolecules composed of one or several chains of amino acids whose amino acid sequences are coded by the genome and constitute the primary structure of the proteins. To achieve their function, most of them must reach a unique structure by a folding mechanism, which is not fully understood and is a very active field of structural biology.

S. Longeville () Laboratoire Léon Brillouin, CEA Saclay, F-91191 Gif sur Yvette Cedex, France e-mail: [email protected] W. Doster Technische Universität München, Physik Department E 13, James Franck Strasse 1, D-85747 Garching, Germany e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 8, © Springer Science+Business Media, LLC 2012

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The structure–function relationship alone does not account for protein activity, since structural fluctuations and conformational adjustments are required to adapt to the conversion of substrate molecules and regulation. Biological physics is based on soft interactions dominated by kB T at ambient temperature. Thus “cold” neutrons ˚ and energies of a few meV allow the study of with typical wavelengths of 5–10 A, atomic and molecular structures together with their motions. In contrast, the typical energies of photons at atomic wavelengths, X-rays, are in the range of keV far above kB T . It is also interesting to notice that low-temperature measurements on biological objects will be necessary to unravel the different dynamical mechanisms by time separation or study of the activation energies. Therefore, high energy resolution of some neutron spectrometers will be relevant for biological physics. Small angle neutron scattering has now been used for more than 40 years [1] to unravel the structure of proteins in solution or in crystal form. Inelastic and quasielastic neutron scattering investigations of proteins really started with the study of dynamical transition in hydrated powders [2] and has developed over the past 20 years to measurements of protein diffusion in vivo [3]. The information on the structure and dynamics of macromolecules obtained from a neutron experiment are fully described by the dynamical structure factor S(Q, ω ) in the frequency domain, or equivalently the intermediate scattering function, I(Q,t), in the time domain. These have been described in the previous chapters. These two functions are generally very difficult to manage since they are composed of contributions from the different dynamical processes weighted by the atomic scattering length. A given scattering centre inside a protein performs different types of motions including Brownian translational and rotational motions, large-scale internal and small group fluctuations and high-frequency oscillations. This leads to a complicated dynamical function (or intermediate scattering function depending on the type of measurement). To separate the different types of motion it is necessary to use different types of samples (powder, solutions, in-cell measurements, etc.) as well as employ various spectrometers with different wave vector and frequency ranges. In addition, the large difference in scattering length densities between hydrogen and deuterium, can be used to either hide the contribution of some molecules (for example, the water) or selectively probe self- or collective dynamics. The signal is separated into incoherent scattering functions which measure the self-correlation of all atoms in the sample and coherent ones, probing the pair-correlation function of all centres. For a solution of biomacromolecules such as proteins, the motions are generally separated into global and internal motions. The former include generally translational and rotational Brownian diffusion, which depends at very low concentration on the temperature, a friction term as a function of the solvent viscosity and the shape of the macromolecule. The latter includes all low-frequency and highfrequency modes of motions together with possible large amplitude domain motions overdamped due to the friction with the solvent.

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8.2 Protein-Internal Motions Most proteins contain an active site, which is generally shielded from the solvent, thus providing a unique environment for chemical reactions. However, the substrate and product molecules must enter and leave the protein site, which involves the solvent. The main goal of this chapter is to demonstrate that protein-internal motions and functional processes can be classified into two types: Class 1 are those solvent-decoupled processes which represent truly internal motions. In contrast, the rates of class 2 motions vary with the viscosity of the solvent near the protein surface.

8.2.1 Protein-Internal Structural Motions and Biological Function We start the chapter on protein-internal motions with a detailed look at protein function. Each protein has its unique function, so there is no general definition, that applies to all proteins. The notion of “biological function” denotes an overall process, which is composed of several elementary steps. For instance, the loss of proteolytic activity of the protein “lysozyme” below a critical degree of hydration (0.25 g water/g protein), less than a monolayer, is often attributed the loss of watermediated motion at the active site [4]. A more likely reason for the loss in apparent activity is the transfer of substrate molecules, which is arrested at low hydration. In the following, we define protein function as a protein-assisted multi-step process involving a small ligand molecule: Several elementary steps contribute to the overall reaction, which defines the protein activity: • The ligand in solution is transfered across the protein-solvent interface • The ligand migrates through the protein structure to the active site and • The ligand binds to the active site, where it is chemically transformed To complete the functional cycle, also the reverse reactions must happen: the dissociation of the possibly modified ligand from the active site, migration through the protein matrix and release to the solvent. Since proteins are close-packed structures, the incorporation of a ligand may induce a structural reorganization assisted by small-scale fluctuations. Here, packing defects in the otherwise dense protein structure play an important role. This issue has been extensively studied with the heme protein, myoglobin. It reversibly binds gas ligands at the heme site, which is buried from the solvent in the protein matrix. The heme–iron binding site, blue in Fig. 8.1, is thus not directly accessible to the ligands.The “arms” of the heme group, however, the polar propionic acid side chains, reach out into the solvent and couple the heme displacements to motions in the solvent [5].

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Fig. 8.1 Secondary structure of the oxygen storage protein myoglobin with helices A–F, the heme group (blue), which binds oxygen, the xenon-binding cavities 1–4 (orange), the distal His64 (E7), the distal cavity docking site B and the proximal docking site C. The arrow denotes the most probable ligand escape pathway into the solvent. via the distal gate of His64

Because gas ligands must find their way to the heme by migrating through dense protein structure, structural fluctuations and internal cavities are required to facilitate ligand binding. With X-ray scattering various cavities in myoglobin were discovered, which act as xenon binding sites. These Xe-docking sites turned out to be crucial to the tranfer of small gas ligands like dioxygen or CO inside the protein [6, 7]. The four major Xe-binding sites are indicated in Fig. 8.1. Timeresolved X-ray crystallography of the photolysed Mb–CO complex and molecular dynamics simulations have established a series of ligand dockings sites and their time sequence [8–12]: After photolysis from the Fe-binding site A, the CO-molecule occupies the distal pocket B (Fig. 8.1) from which it can rebind to A, or escape to the solvent S via the distal gate of HisE7. Rebinding from the solvent S →A is the slowest process. An alternative route from B involves the transfer via cavity Xe4 to the proximal site of the heme, Xe1 or Xe2. In the latter case, rebinding occurs from the kinetic state C →A. Further exit pathways involving Xe3 have been identified by simulations [10–12]. The kinetic results can be represented by Gibsons four state model [7]: SBC A


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Fig. 8.2 The recombination kinetics of carbon monoxide (CO) to the heme iron of myoglobin after dissociation by a laser flash in various solvents with different viscosity: 60% (by weight) ethylenglycol/water, 90% glycerol–water, 80% sucrose–water and 98% sucrose–water. Three kinetic intermediates according to the scheme are indicated: S (solvent), B (distal pocket), C (protein matrix and proximal site). The arrow emphasizes the constant time position of the C-intermediate with increasing solvent viscosity

In the following, we discuss the action of the solvent, and of the solvent viscosity in particular, on elementary steps of CO-binding to myoglobin. For this purpose, we have reevaluated the early flash-photolysis experiments of Kleinert et al. according to Gibsons kinetic scheme [4, 13, 14]. Figure 8.2 displays the recombination kinetics of CO + myoglobin after a nanosecond laser flash in solvents of varying viscosity. The three kinetic intermediates, B, C and S, are also indicated. The arrow points towards a drastic increase in the solvent viscosity. The main effect of viscosity on the kinetics is the reduction in the amplitude of the slowest process, the external rebinding from the solvent, S→A. The amplitude of S→A is equal to the escape fraction of ligands, Nout ≤1, which leave the protein after photolysis instead of rebinding internally. A decrease of Nout indicates, that the ligand exit rate across the protein–solvent interface decreases with the external viscosity. At 90% sucrose-water, the solvent is in a glassy state at 290 K. Below the glass temperature Tg ≈ 325 K, the viscosity is infinite, thus Nout ≈ 0. As Fig. 8.2 shows, that the internal (geminate) recombination process and ligand migration involving the decay of the intermediates B and C are still operating in spite of a glassy external environment. The rate of C → A seems to be rather independent of the solvent viscosity. The amplitude of the fast geminate recombination from the heme pocket B → A, increases with the solvent viscosity in compensation to a decreasing Nout . Figure 8.3 shows several internal transition rates versus the external

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Fig. 8.3 Internal transition rates of CO in myoglobin at 290 K versus the external solvent viscosity according to the four state model

viscosity, evaluated according to the kinetic four-state model: The formation of the covalent bond at the heme iron (B→A) and the crossing rate from the distal to proximal site of the heme kBC are independent of the viscosity. Since the intercavity migration of the CO molecule requires structural adjustments, one has to invoke the existence of class 1 structural fluctuations, which are decoupled from the solvent. On the other hand, the exit and entry rates belong to class 2 fluctuations, which are strongly coupled to the solvent [14]. The viscosity of various biosolvents versus the temperature is given in Fig. 8.4 [13]. The data were obtained from a combination of viscosity and specific heat spectroscopy experiments. The relaxation rate and viscosity are related by the Maxwell equation [13, 14]:

η = g∞ · T · τs ,

(8.1) −1 −1 cP · K s [13]. where g∞ denotes the high frequency bulk Figure 8.5 displays CO-exit rates from horse myoglobin versus the viscosity in several solvents. The viscosity in a given solvent is modified by varying the temperature. In addition to a dependence of viscosity, the exit rates also depend on a protein intrinsic barrier of HBS ≈ 25 kJ/mol [13, 14]. For this reason, the absolute values of ks and kBS differ by a factor of 100. To compare the exit rates between different solvents, the data were corrected to a common temperature of 290 K based on Kramer’s law of activated escape [14]: modulus, ≈109

kBS =

A · exp(−HBS /RT ) ηs

(8.2)


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Fig. 8.4 Solvent relaxation rates ks ∝1/η derived from viscosity and specific heat spectroscopy experiments ref. [13]. Acronyms: S(sucrose), W(water), G(glycerol), E(ethylene glycol). Myoglobin hydration water: neutron scattering (full circles) dielectric relaxation (full triangles), Mössbauer spectroscopy (triangle top down) and calorimetry (full and open diamond) [14]. Lines are fits to a super-Arrhenius VFT law, (8.13)

A is a prefactor and ηs denotes the viscosity near the protein surface. The results are also shown in Fig. 8.5. In glycerol-water, the CO-exit (and entry) rates vary with the inverse of the bulk viscosity kBS ∝1/η . Moreover, the respective values of kBS (η ) fall right on top of each other, at least at high viscosity. This shows that the bulk viscosity is the essential parameter, while the chemical composition plays a minor role. These processes thus belong to class 2. In contrast, in the 80% sucrosewater solution, the exit rates vary less than the reciprocal bulk viscosity and may exceed ks at high viscosity. Such reduced viscosity effects have been interpreted as indicating a fractional solvent exposure of the reaction, leading a sublinear power law [16]. However, the cosolvent sucrose is known to be partially excluded from the protein domain. The thermodynamic experiments of Timasheff and collaborators have shown that the concentration of cosolvents near the protein surface can be different from the bulk [17]. Some cosolvents like sucrose are more excluded than others from the protein domain, leading to a reduced viscosity near the surface as compared to the bulk [5, 13, 14, 17]. This exclusion can explain the observed sublinear viscosity effect and the dependence like: ηs ∝ ηbulk κ with the exponent κ ≤1. At lower viscosities the CO-exit rates in glycerol-water and in dilute aqueous solution deviate however from a 1/η law. In this regime, the escape of CO, which requires the displacement of solvent molecules to create a cavity, becomes less and less rate-limiting. Ansari et al. report on a similar viscosity dependence of a conformational transition rate of myoglobin, which nearly coincides with the CO-exit rate as shown in Fig. 8.4 [15]. The onset of a plateau, which they observe at low viscosity, has been attributed to the influence of an protein-internal viscosity.

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Fig. 8.5 CO-exit rates kBS versus viscosity in several solvents, reevaluated experiments of ref. [13] corrected to 290 K: open triangle: in aqueous solution, filled triangle: 75% glycerol-water, full squares: 90% glycerol-water, full circles: 80% sucrose-water, open squares: rate of conformational change of myoglobin in 56 and 79% glycerol-water [15], full line: solvent relaxation rate ks in 75% glycerol-water [14]

We thus arrive at the simple and basic conclusion, that the kinetics of ligand binding to myogobin can be decomposed into elementary steps, which belong either to class 1 (internal) or to class 2. It was anticipated, that the CO migration requires the assistance of structural fluctuations. We thus turn to the question, whether a similar classification into two classes can be performed with structural relaxation processes in myoglobin. An important class of fluctuations refers to displacements of the heme group, which is the active site. With Mössbauer spectroscopy, one can derive displacement fluctuations of the heme iron on a nanosecond time scale [5,18,19]. Results obtained with different solvents are shown in Fig. 8.6a: The displacements follow a linear temperature dependence reflecting vibrational motions independent of the protein environment. Above about 200 K a dynamical transition occurs, the displacements increase, because structural relaxation processes are being increasingly resolved within the pico- to nanosecond window of the spectrometer. Most interesting, the onset of the transition depends on the solvent composition: While myoglobin


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a

b

Fig. 8.6 (a) Dynamcial transition and mean square displacements of the heme of myoglobin from Mössbauer spectroscopy in myoglobin crystals (open circles), 75% glycerol-water (open squares) and 80% sucrose-water (triangles) [5] (b) Solvent escape fraction Nout of CO after photolysis from myoglobin in 75 and 90% glycerol-water [14]

crystals and myoglobin in a 75% glycerol-water solution exhibit a common onset temperature of TΔ ≈ 210 K, with the more viscous solvent 80% sucrose-water one observes that the onset is shifted to a higher temperature of TΔ = 240 K. The motion of the heme group thus belongs to class 2 fluctuations. One type of heme-solvent coupling could occur directly via its polar side chains or indirectly via proteinmatrix fluctuations. Figure 8.6 also displays the respective bulk glass temperatures. The structural relaxation time at Tg is in the range of seconds and the viscosity approaches 1013 Poise. The same α -process is presumably probed by Mössbauer spectroscopy on a much faster time scale, the respective time resolution is the nuclear life time of the 57 Fe nucleus of 140 ns. This time shift leads to an upshift in the onset temperature of recorded relaxational displacements from Tg to TΔ . One should expect that the

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bulk solvent viscosities at the onset temperatures are equivalent. The sucrose-water system however has a much higher bulk viscosity at 240 K than glycerol-water at 210 K. This discrepancy suggests as above, that the surface viscosity is drastically lower with protein-sucrose-water than the bulk value due to preferential hydration [5]. Fig. 8.6b relates the structural coordinate of heme displacements to a functional parameter, the ligand escape fraction Nout . The kinetic onset shows a similar upshift in TΔ in the more viscous solvent. Nout is also a dynamic quantity, reflecting the partitioning between ligand escape and internal rebinding. In the four state model, it can be approximated by: Nout = kBS (η ) /(kBA + kBC + kBS (η )).

(8.3)

At the onset temperature of 250 K with 75% glycerol-water, kBA is approximately 2 × 106 s−1 and kBS amounts to 2 × 105 s−1 , while kBC can be neglected, yielding for Nout ≈ 0.1. This is close to the observed value of Nout at 250 K. In this case, the relevant biological ”resolution” time is set by the solvent-independent internal binding rate kBA . Biological function turns on, when the escape rate starts to exceed the internal binding rate. On this biologically relevant time scale, the “dynamic transition” occurs at 250 K with a 75% glycerol-water solvent. In the next section, we investigate other structural processes of myoglobin probed by elastic and quasielastic neutron scattering.

8.2.2 Dynamical Structural Distributions in Proteins In the following we show how the displacement distribution can be reconstructed from experimental neutron scattering functions based on a moment expansion [4,20,21]. We again use myoglobin in various environments. As with kinetic experiments, one can discriminate between class 1 and class 2 type fluctuation according to their coupling to the solvent. The neutron scattering cross-section of D2 Ohydrated proteins is dominated (95%) by the non-exchangeable hydrogen atoms and thus incoherent scattering. The corresponding self-intermediate scattering function, Is (Q,t) records displacements of individual hydrogen atoms (j) [4, 21]: Is (Q,t) = exp(iQr j (0)) · exp(−iQr j (t))

(8.4)

averaged over all atoms j. The scattering vector Q is the instrumental parameter to modify the spatial scale probed by the scattering process. From Is (Q,t) one derives by a Fourier transform the displacement distribution function Gs (r,t): Gs (r,t) =

d3Q exp(−iQr) · Is(Q,t) (2π )3

(8.5)


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It denotes the probability density, that atom (j), which is initially at r0 moves to a position r within a time interval t, averaged over all atoms j: For a classical system, this is equivalent to: Gs (r,t) =

d 3 r0 p(r0 + r, r0 ,t) · p0 (r0 )

(8.6)

with the equilibrium distribution p0 (r) = p(r, r0 ,t = ∞)

(8.7)

The displacement distribution is defined as the long time value of equ. 8.6: Gs (r,t → ∞) =

d3Q exp(−iQr) · Is(Q,t → ∞) (2π )3

(8.8)

The long-time value of the intermediate scattering function is the so-called elastic incoherent structure factor EISF(Q). The displacement distribution is thus the Fourier transform of the EISF(Q), which represents the fraction of the elastic scattering component in the frequency domain at infinite instrumental resolution. Due to limitations of the experimental Q (and time)-range, a direct transform according to equ. 8.8 is rarely possible. Approximations such as a model-independent moment expansion of the G(r,t) can be useful [4] or specific dynamical models, which account for the molecular structure. The experimental elastic fraction is generally convoluted with the resolution function and is thus not identical with the EISF(Q). In the following, we discuss models of the following intermediate scattering function, I (Q,t) = EISF (Q) + {1 − EISF (Q)} · Φ (t) ,

(8.9)

which separates the time- and Q-dependence of the spectrum, since for local molecular processes the time correlation function Φ (t) is independent of Q. Also Φ (t→ ∞) = 0. The EISF(Q) contains information on the geometry of the motion, which is a fingerprint of a molecular process. We focus on experiments performed with myoglobin, embedded in three different environments: (a) water: fully hydrated (0.35 g D2 O / g protein) (b) vacuum (lyophilized to less than 0.05 g/g) and (c) a glassy perdeuterated glucose matrix, Tg = 325 K. The backscattering spectrometer IN13 (ILL, Grenoble) provides an unusually large Q-range of up to 5 A˚ −1 . Figure 8.7 shows representative scattering data approximating the intermediate scattering function I(Q, t = 50 ps) at fixed time at various temperatures in the three environments. Dehydrated and glassy myoglobin display similar scattering functions, while hydration leads to an additional decrease in the scattering function at high Q.

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Fig. 8.7 Experimental I(Q, τ = 50 ps) of myoglobin in three different environments, vacuum (dehydrated), vitrified (perdeuterated glucose glass) and D2 O-hydrated (0.35 g/g) using the thermal backscattering spectrometer IN13 at the ILL in Grenoble and fits to a three-fold rotation (methyl group). The data are corrected for multiple scattering

The 2data could be well represented by a combination of Gaussian displacements, Δ xtrans and three-fold rotatational jumps according to:

√ 2 sin 3Q · r a 1+2· √ (8.10) EISF(Q) = exp −Q Δ xtrans · 1 − a + 3 3Q · r This choice is motivated by the fact that methyl groups in proteins carry almost 30% of the total neutron cross-section. This fact is accounted for by the factor a ≈ 0.27 in equ. 8.10. Figure 8.8a shows the elastic fraction of Fig. 8.7 (hydrated myoglobin) now separating the two components, gaussian and rotational, according to equ. 8.10. In particular the rotational component approximates the EISF(Q) of a three-fold rotation quite well, which is assigned to methyl group torsional transitions. Here the wide Q-range of the back-scattering spectrometer IN13 (at the ILL, France) is quite essential. Also shown in Fig. 8.8 are data from the backscattering spectrometer HFBS at the NCNR, NIST (USA). With such a limited Q-range, no assignments are possible. The interpretation of these data by Roh et al. [22] is thus at best an educated guess. The transition rates are shown in the Arrhenius plot of Fig. 8.8b. Both quantities, the EISF and the torsional rates, are required to establish a meaningful intermediate scattering function. The full lines in Fig. 8.7 represent such fits at fixed resolution time, τΔ , based on the data in Fig. 8.8. To transform the apparent EISF(Q) to the spatial domain (8.8), we approximate the data in Fig. 8.7 by a sum of Gaussian functions. Figure 8.9 displays the resulting displacement distribution functions at various temperatures referring to a fixed instrumental time window of ≈ 50–100 ps.


a

217

b

Fig. 8.8 (a) Elastic intensity of hydrated myoglobin (IN13) at 300 K, separated into a Gaussian translational component (line with dots) and a non-Gaussian rotational contribution (open circles, IN13, ILL) [4] and and EISF(Q) of methyl group rotation with 25% cross-section (solid line), full circles and triangles: hydrated lysozyme with HFBS according to [22], (b) Arrhenius plot of methyl group rotation rates, derived from quasielastic spectra (IN5, Grenoble) with dehydrated myoglobin and alanine dipeptide crystals [21], the full line was derived from elastic scan data (IN13, Grenoble) on alanine dipeptide crystals

A change in temperature shifts the effective time scale of molecular motions with respect to the instrumental window. At low temperature only vibrational motions are resolved, which implies a Gaussian distribution of displacements with a maximum ˚ at r = 0.2 A. The maximum broadens slightly when increasing the temperature up to 240 K. Then a distinct shift and a further broadening of the peak occurs. This effect originates from small-scale diffusive motions, which become resolved above 240 K. This feature is absent in dry and vitrified samples and belongs to solvent-coupled or class 2 motions. By contrast,the displacements on a scale of 1.5 A˚ are observed in all three environments above 200 K. On such a scale only proton displacements due to rotational jumps are plausible, which is demonstrated in Fig. 8.9. Thus, rotational transitions of side chains, in particular of methyl groups, in the protein interior are not strongly coupled to the properties of the environment [4]. Torsional transitions of methyl groups occur also in solid environments such as molecular crystals. This component, which represents the solid aspect of structural fluctuations, belongs to class 1. Figure 8.10 shows the temperature evolution of the second moment of the distribution with respect to the three environments. Below 20 K, only zero point vibrations contribute to the displacements (0.014 ± 0.003 A˚ 2 ), the vibrational

218


Fig. 8.9 Displacement distribution, 4π r2 G(r,t = 50 ps), of hydrated myoglobin at fixed time with ˚ vibrational and water-induced librational motions, peak increasing temperature, peak at r = 0.25 A: ˚ side-chain rotational transitions above 1.0 A:

Fig. 8.10 Second moment of the displacement distribution at fixed time in three different environments, vacuum (dehydrated, open squares), vitrified (perdeuterated glucose glass, open triangles) and D2 O-hydrated (0.35 g/g, full circles) using the thermal backscattering spectrometer IN13 at the ILL in Grenoble. The data are corrected for multiple scattering


219

component follows a coth-function with temperature. The displacements of the vitrified sample are slightly lower than those of the hydrated or dry sample. Anharmonic enhancements become significant above 120 K with all three samples, reflecting rotational (class 1) motions of side chains. The onset is gradual and consistent with an Arrhenius temperature dependence of rotational jump rates as shown in Fig. 8.8b. The assignment to methyl group rotation is facilitated by the observation that the class 1 transition is absent in perdeuterated proteins [23]. Water induces additional translational motions of side-chains, which become relevant above TΔ = 240 K. The onset of water-assisted protein dynamics is abrupt due to a super-Arrhenius temperature dependence of the structural relaxation rate [4,24]. The dynamical transition observed with neutron scattering at TΔ = 240 K for the nonexchangeable protein hydrogens should be compared with the onset of the heme displacements in fig. 6a at TΔ = 210 K in myoglobin crystals. The solvent is water in both cases. Apart from different reporter groups, the spatial scale and the time scale probed by the two methods are quite different. Mössbauer spectroscopy is sensitive to small-amplitude motions of the heme iron, the effective wavevector is Q = 7.2 A˚ −1 , which are faster than about 1 μ s. With the neutron back-scattering spectrometer IN13 Qmax = 5 A˚ −1 , and only motions faster than 200 ps are resolved. Assuming that both methods record the same type of solvent-coupled (class 2) fluctuations, the difference in TΔ can be attributed to the different time windows. With neutron scattering, one can clearly detect class 1 motions in proteins: internal processes such as ligand migration between cavities are most likely assisted by rotational transitions of side chains. Class 2 motions apply to lateral motions of the heme in its cleft (Fig. 8.1), watercoupled librations of side chains and ligand entry and exit transitions. One important result of the analysis of the hydrogen displacement distribution is the identification of two distinct molecular processes, associated with rotational transitions and translational–librational motion of side chains. Only the latter depend on the solvent. Another important result is that the corresponding correlation functions are not additive, instead their composition is a product of the following form: I(Q,t) = Irot (Q,t) · Itrans (Q,t, ηs )

(8.11)

The heterogeneity of dynamics sites can lead to a sum of Gaussian displacement distributions, which may account for the observed non-Gaussian shape of the EISF(Q). However the non-Gaussian Q dependence persists in the dehydrated case, when the translational component is arrested. The non-Gaussian nature of the dynamic process, which is active also in the dry state, is thus intrinsic. Transitions between distinct sites, such as rotational transitions, are intrinsically non-Gaussian [4]. The addition of water to myoglobin leads to a further decrease in the elastic scattering function (Fig. 8.7 at 300 K), by enhancing the Gaussian factor. Thus only the Gaussian displacements increase due to class 2 motions. This is very different from just adding another process due to a heterogeneity of reporter groups. This striking result suggests, that a dominant single reporter group exists for neutron scattering in proteins, which performs two kinds of motions simultaneously, rotation

220


and libration–translation. Since the methyl groups in proteins comprise typically 25–30% of the incoherent neutron cross-section, one arrives at the remarkable result, the neutron scattering spectra of D2 O-hydrated proteins reflect essentially three independent types of motion of the methyl side chains: Vibration, rotation and libration–translation. Only translation is coupled to the solvent (class 2). One could imagine as a plausible model, that the axis of the methyl group moves or the group is translated along with side chain to which it is attached.

8.2.3 The Dynamical Transition from Elastic Scattering Experiments The protein-solvent glass transition is a dynamical effect revealed by the abrupt onset of class 2 protein structural displacements. The onset is observed at TΔ , depending on the instrumental resolution Δ ω [2, 25]. In 1989, two transitions were identified and assigned later to class 1 and class 2. Only the second onset at TΔ = 240 K depends on the degree of hydration. The first onset of anharmonic motion at TΔ ∗ ≈ 160–180 K persists even with dry or vitrified proteins as Fig. 8.10 shows. This effect was attributed above to class 1 rotational transitions of side chains [4, 25]. In the following, we introduce a concept of the dynamical transition which unambiguously defines TΔ . We show how dynamic information can be deduced from elastic neutron scattering experiments at a fixed instrumental resolution. A more detailed treatment referred to as “elastic resolution spectroscopy” is presented elsewhere. In this method, the instrumental resolution is varied continuously [26, 27]. Figure 8.11 shows the intermediate scattering function of hydration water of myoglobin. The original data were collected in the frequency domain by subtracting the spectra of H2 O and D2 O-hydrated myoglobin and a subsequent Fourier transform to the time domain [28]. On a short time scale, librational motions of water result in a fast β-process. The second slower decay, the α-process, varies with Q and involves reorientation and translation of water molecules along the protein surface [4, 14]. The time window is limited by the instrumental resolution at τΔ ≈ 12–15 ps. At τΔ the correlation function has decayed to a finite value depending on the temperature, I(Q, t = τΔ ) = Iel (T). For longer times there is no further decay observable due to instrumental limitations. The plateau value beyond τΔ appears in the frequency domain as a delta component with elastic amplitude Iel (τΔ ,T)δ (ω ). The respective elastic intensity in Fig. 8.11b exhibits a step-like decrease. The associated “dynamical transition” implies that the structural plasticity coupled to this molecular process is fully available beyond τΔ , when Iel is sufficiently small, ≈350 K. The term “dynamical transition” is justified only because it refers to a collective structural process, the α-relaxation, which determines the viscosity of the liquid (see 8.4). This is not the case for local processes like rotational transitions of side-chains. Structural arrest on a microscopic scale leads to a macroscopic freezing of the liquid, which turns into a glass. The transition is discontinuous in the


a

221

b

Fig. 8.11 (a) Intermediate scattering function of hydration water, I(Q,t), derived by Fourier transforming IN6 time-of-flight spectra of hydrated myoglobin (h = 0.35 g/g) [4, 14], (b) Elastic intensity derived from the value of the intermediate scattering function at the instrumental resolution time τΔ : Iel = I(Q,t = τΔ )

dynamics, but continuous with respect to the molecular structure. It is a dynamical transition, which depends on the relevant experimental time scale. A liquid, if probed on a short enough time scale, looks like a solid even on a macroscopic scale. Relaxation processes in complex systems are generally non-exponential in time. A very useful model of heterogeneous processes involving a distribution of relaxation times is the Kohlrausch stretched exponential function, which is commonly used to describe dynamics in polymer systems (see for example Chap. 4): Φ (t) = exp − (t/τc )β

(8.12)

where β ≤1 is the stretching parameter. Figure 8.12a shows this function for β = 1 (exponential), 0.5 and 0.35. With decreasing β , the decay broadens, involving both fast and slower components compared to the monoexponential case. However, independent of β , the correlation functions coincide at t = τc , which defines both the time scale and the characteristic temperature of the dynamical transition. The effect of the instrumental resolution function (dotted line) creates a long-time tail, which will cause further stretching in the high temperature tail of the elastic intensity. The elastic fractions in Fig. 8.12b were derived based on the following assumptions: (1) the correlation time τc (T) varies with the temperature according to an Arrhenius law with an activation energy of 17 kJ/mole, a prefactor of 10−13s, and (2) τΔ = 2 ns (for classical reactor backscattering spectrometers like HFBS at NIST, USA, SPHERES at FRMII, Germany or IN16 at the ILL in France). All curves coincide at TΔ independent of β at Iel = 0.368, while the onset temperatures are quite different.

222


Fig. 8.12 (a) correlation function Φ (t/τc ) approximated by a Kohlrausch stretched exponential with three dynamic exponents β = 0.35, 0.5, 1. (b) Resulting elastic intensity at τΔ = 2 ns, assuming an Arrhenius law for τc (T ) = A · exp(Ea /RT ) with Ea = 17 kJ/mol and a prefactor A = 10−13 s. The effect of a Gaussian instrumental resolution is also shown for β = 1 (dotted line). The arrow indicates the location of the dynamical transition temperature TΔ at τΔ = τc , independent of β

The dynamical transition temperature is thus defined by TΔ , where τc = τΔ and not by the low temperature deviation from harmonic behavior. The relaxation rates of glass-forming liquids, like those shown in Fig. 8.4, display a super-Arrhenius temperature dependence, which is phenomenologically characterized by a VogelFulcher-Tamman law (VFT):

τc−1 = τ0−1 · exp{(−H/ (T − T0 ))} ,

(8.13)

where τ0−1 denotes a prefactor. H is the high-temperature activation energy [in K] and T0 denotes a critical temperature, where the correlation time reaches infinity. The parameters of VFT-fit are given in [13]. In the context of the ligand transfer rates of Fig. 8.5, it was mentioned that solvent-coupled protein processes are generally slower than solvent relaxation rates due to protein-intrinsic barriers. Also we have emphasized that the viscosity near the protein surface can differ drastically from the bulk value. In the limiting case that the relaxation rate of the bulk solvent coincides with the protein relaxation rate τc ≈ τs , we can deduce from the data in Fig. 8.4 a corresponding dynamical transition at a given instrumental resolution τΔ . This is shown in Fig. 8.13. For β , we adopt the plausible value of 0.5 and we ignore the slight distortions resulting from the shape of the resolution function (see Fig. 8.12a). The curves refer to an elastic resolution of 2 ns except if indicated otherwise. The dynamical transition onsets


223

Fig. 8.13 Calculated elastic fraction according to (8.15) and (8.16) using the data in Fig. 8.4. It was assumed that τc (T) = τs for all solvents. TΔ is defined by the dashed line crossing Iel (T). The instrumental resolution was set to τΔ = 2 ns, except if indicated otherwise, the full circles refer to elastic neutron scattering experiments with hydrated lysozyme, τs = 0.1 ns [29]

occur in the observed temperature range, 200–300 K. The dashed line indicates our new definition of TΔ . Hydrated myoglobin and myoglobin crystals, observed with Mössbauer spectroscopy at τΔ = 140 ns, exhibits the lowest onset temperature of 200 K. This is quite close to the observed onset, around 200–210 K in Fig. 8.6. While the onset is quite sharp, the transition itself is significantly broader than the calculated one. Also shown is water as the solvent at a resolution of 0.1 ns superimposed with data on hydrated lysozyme, which fits quite well [29]. However, very similar results were obtained with lysozyme in 90–100% glycerol at 0.1 ns, which should be shifted to higher temperatures. This would suggest a lower effective viscosity near the surface than in the bulk. Such a comparison can provide dynamic information about the state of the solvent near the protein surface. To analyse real data one has to account for the vibrational Debye–Waller factor and the EISF(Q) of the relevant process corrected for the finite resolution. Instead of considering the elastic intensities, it is more popular to focus on a derived quantity, the atomic mean square displacements. In [30], a phenomenological relationship between apparent mean square displacements and the bulk solvent viscosity ηb was suggested:

Δ x2

app

≈ 1/ log(ηb )

(8.14)

Neither the EISF nor the instrumental resolution were explicitly taken into account. With the analysis based on (8.15) and (8.16), we can test this relationship starting

224


from the intermediate scattering function of equs. 8.9 and 8.11. For this purpose, it is sufficient to consider the low-Q range, since the displacements were derived assuming a Gaussian approximation. Second, we are essentially interested in the onset of anharmonic motion. This is the regime, where the structural relaxation time is still larger than τΔ , τc τΔ typically by a factor of 5 [4]. Expanding both, the Gaussian EISF(Q) at small Q is then: EISF(Q → 0) ≈ 1 − Q2 · Δ xtrans

(8.15)

and the stretched exponential at t = τΔ τc yields:

τΔ Φ (t = τΔ ) ≈ 1 − τc

β (8.16)

which together with:

yields:

I(Q → 0,t = τΔ ) ≈ 1 − Q2 Δ x2 (τΔ )

(8.17)

τΔ β 2 C Δ x τ ≈ Δ x2trans = β Δ τc η

(8.18)

The second equality in equ. 8.18 assumes the Maxwell relation between solvent relaxation rate and bulk viscosity: τc ∝ η . Equation 8.18 is certainly quite different from the one proposed above with equ. 8.14 [30]. Figure 8.14 shows the analysis of real elastic scattering data in the present context. To emphasize the contribution of water, a perdeuterated protein phycocyanin, hydrated with 0.3 g/g H2 O was investigated with the back-scattering spectrometer SPHERES at τΔ ≈ 2 ns [31]. The data are roughly compatible with the dashed line (W) in Fig. 8.13. A complete analysis requires to account for the harmonic component as well as for the EISF(Q), the step height of the transition. At TΔ = 255 K, the correlation time of water τc equals the resolution time τΔ ≈ 2 ns. The various curves refer to different values of the exponent β and correspondingly, different activation energies of the assumed Arrhenius law. The full line represents the choice of β = 0.5, an activation energy of 4000 K (dotted 4200 K) and the preexponential of 10−15s. Assuming β = 0.35, one obtains instead H = 7600 K and a preexponential of 10−22 s. These differences emphasize the important influence of the relaxation time distribution on activation parameters. For a given β , one can derive relevant dynamic information from the transition curve. To investigate Iel (Q,T) directly it is better to start with derived mean square displacements. So far we have only discussed the effect of the main structural relaxation (α ) on the elastic scattering intensity. The intermediate scattering function in Fig. 8.11 shows however a two-step decay. The rate of the fast component varies little with temperature and momentum exchange Q. Only its amplitude increases with the


225

Fig. 8.14 Analysis of elastic neutron scattering data of perdeuterated phycocyanin hydrated with 0.3 g/g H2 O and fits to the stretched exponential model. The value of TΔ is indicated. The nominal ˚ −1 . The experiment was performed energy resolution was 0.65 μeV and the data refer to Q = 1.5A with the back-scattering spectrometer SPHERES at the FRMII facility in Munich [31] .

temperature above Tg . To emphasize its local nature, we call it the “fast β-process”. The increase of the β-amplitude above Tg gives rise to a first onset in the mean square displacements at about 180 K, which is displayed in Fig. 8.15. Since the β-correlation time ≈1 ps is always shorter than the resolution time of current spectrometers, independent of the temperature, there is no effect of the chosen time window of observation on the apparent displacements. Analysis of high-frequency vibrational spectra of the hydrogen bond network suggests that the β-process originates mainly from fast hydrogen bond fluctuations of water molecules in the cage of nearest neighbours, bonded either to other water molecules or polar protein residues [14]. Figure 8.15 compares the displacements of protein hydration water observed for two time windows of τΔ = 2 ns and 15 ps. The first onset is independent of the instrumental resolution, while the second onset shifts with decreasing time window to higher temperatures. The second onset at TΔ , originates from collective structural fluctuations giving rise to the α-process. The α-time scale increases strongly with decreasing temperature. This is the reason, why the second onset at TΔ depends on the observation time scale, τΔ . Since protein class 2 displacements are tightly coupled to the water of hydration, protein motions also display a two-step decay in their density correlation function on the same time scale. This two-step scenario was developed in the original analysis of hydrated myoglobin with neutron- time-of-flight and back-scattering spectroscopy [2].

226


Fig. 8.15 Mean square displacements of protein hydration water from elastic intensities, triangles: myoglobin, data from Fig. 8.11b at 15 ps (time-of-flight spectrometer IN6 at the ILL), circles: perdeuterated c-phycocyanin, H2 O-hydrated, 0.3 g/g at 2 ns (SPHERES, FRMII Munich), and thermal expansion of hydration water (myoglobin) derived from the O-H stretching vibration. The glass temperature Tg and the resolution dependent onset of the α -process, TΔ are indicated [14,31]

8.2.4 Conclusion on Protein-Internal Structural Motions and Biological Function The main conclusion of the first part of this chapter is the discrimination of class 1 and class 2 processes according to their coupling to the solvent. Intercavity migration of the ligand belongs to class 1 as well as the rotational transition of internal side chains. The ligand transfer across the protein–solvent interface and translational motion of surface side-chains are coupled to the solvent and belong to class 2. The main dynamic coupling parameter is the solvent viscosity at the protein surface ηs which can be different from the bulk value. Even small globular proteins can thus provide a unique chemical environment, which is also dynamically isolated from the solvent. Even though the heme group, the active site of myoglobin, performs translational motions, which are modulated by the external solvent, the binding of the ligand to the heme iron is a class 1 reaction. One of the unsolved puzzles of the field is that the final binding reaction kinetics of B→A is nonexponential at all temperatures and independent of the solvent [32]. A distribution of activation enthalpies, reflecting the conformational heterogeneity of the protein structure was invoked, but the molecular origin of the disorder is still obscure. In [5], it was suggested that the sliding motion of the heme in its cleft, modulates the effective force on the heme iron, which is covalently attached to the imidazole side chain of His F8. This effect also modulates the barrier of


227

0.5

100 K 200 K 210 K 220 K 230 K 240 K

B

g (H)

0.4 C 0.3

S

0.2

0.1

10

20

30

40

H (kJ / mol) Fig. 8.16 Normalised distribution function of activation energies referring to elementary steps of the CO-binding kinetics to horse myoglobin. B, C and S denote the partial enthalpy distributions of rebinding from the respective docking sites to final state A according to the four state model and Fig. 8.1 [13]

B→A when the covalent bond between heme iron and CO is formed. If the crossing of the barrier at the heme site occurs on a time scale that is faster than the visco-elastic sliding, an apparent static distribution of barrier heights will result. The above analysis indicates, that the structural relaxation of the solvent is the main factor that determines the rate of heme sliding. It follows that the observed barrier distribution should change significantly, when the rate of visco-elastic sliding becomes comparable to the rate of covalent bond formation at the heme iron. In the case of CO-myoglobin in 75% glycerol-water, the crossover takes place between 210 and 220 K, where the solvent relaxation rate ks (T) becomes comparable to the rebinding rate kBA at about 106 s−1 (Figs. 8.3 and 8.4). Figure 8.16 shows, that the activation enthalpy spectrum of B→A is constant between 100 and 200 K. However, above 200 K the B-distribution shifts towards higher values, suggesting relaxation to a new product state B*→B [33]. This suggests, that heme sliding does affect the rate of bond formation. However, the distribution does collapse to a delta-function. The C- and S- distributions change in relative weight with respect to B, as a result of the temperature dependent population of kinetic states. In particular the escape fraction Nout increases with the temperature above 200 K. In the second part of the chapter, a more precise definition of the dynamical transition is suggested. We start from the time domain with a distribution of relaxation times and switch to the frequency domain by defining the elastic intensity as the long-time value of the intermediate scattering function limited by the

228


resolution of the spectrometer. The dynamical transition is thus defined by the temperature, where the elastic intensity has decreased to its 1/e value. This value is independent of the detailed relaxation time distribution, while the onset temperature is strongly affected by the shape of the spectrum. This concept allows to test various coupling models, for instance whether the solvent relaxation rate τs of various solvents equals the protein structural relaxation rate τc . Several misconceptions exist about the nature and the mechanism of the dynamical transition and its relation to the glass transition [14, 25]. It is often claimed that the dynamical transition can be suppressed by vitrifying the protein in a glassy matrix [23, 23a]. While class 2 motions can be entirely suppressed at infinite external viscosity, class 1 motions (and the related anharmonic onset) are active irrespective of the environment (see Fig. 8.9). One likely origin of the discrepancy derives from the fact that the glassy matrix dominates the elastic scattering function unless it is perdeuterated. In Figs. 8.7–8.10 a perdeuterated glucose matrix was used. Also the work of Tsai et al. [34] on lysozyme in glycerol (see figure 11.3 in the Chapter of Wood and Weik) is in strong contradiction with Paciaroni et al. [29]. While Tsai et al. derive an onset temperature of 388 K, the onset in 90–100% glycerol occurs at 240 K as Fig. 8.13 shows. The dynamical transition is certainly not related to an energy landscape or a molecular resilience [23]. Instead it reflects the activation energy spectrum of molecular processes probed by a spectrometer on a finite time window. It does not even make sense to ask, what is actually “driving” the dynamical transition, protein motions, rotation or translation of water molecules. There is no driving force, just collective fluctuations of the proteinwater system, that will be structurally arrested at the glass transition. The glass transition, which implies by definition the freezing of translational motions on a macroscopic time scale of 100 s, occurs at about 170 K for the protein-water system. At Tg = TΔ (100 s), however the protein-water systems starts to soften due to an increasing amplitude of fast hydrogen bond fluctuations. On a molecular scale, the coupling mechanism between protein and its solvation shell is the hydrogen bond network. A molecular transfer necessitates the simultaneous breaking of several hydrogen bonds. Fluctuations at this basic level may be considered as the main driving force of the dynamical transition [14, 24, 35].

8.3 Global Diffusion of Macromolecules 8.3.1 Dynamic Light Scattering of Colloids As early as 1908 and 1910, with the works of Smoluchovsky and Einstein, respectively, it was clear that density fluctuations in condensed matter lead to local inhomogeneities that could scatter beams. The progress of such studies was slow due to the unavailability of high intensity light beams, until the introduction of the laser at the beginning of 1960s. Then began an intense activity on experimental


229

studies on colloidal systems by dynamical light scattering (DLS) together with theoretical papers aiming at the interpretation of the experimentally measured quantities. Phillies [36, 37] introduced the generalized Stokes–Einstein relaxation that determines the mutual diffusion coefficient measured by DLS: Dm =

(1 − Φ ) f

∂π ∂c

(8.19) P,T

−1 is the osmotic compressibility of the solution, Φ the hydrowhere −1 c (∂ π /∂ c) dynamic fraction of the molecules in solution and f is a friction coefficient. This diffusion coefficient being extracted from the pair-correlation function is different in nature from the self-diffusion coefficient measured by, for example, tracer diffusion experiments: Ds . For very low concentrations, the interactions between molecules can be neglected and Ds and Dm are equal to Do the diffusion coefficient of a unique macromolecule in a solvent (0 refer to zero concentration). Do given by the Einstein’s law : Do = kB T / f where T is the temperature in K, kB the Boltzmann constant and f is a friction term that Stokes has shown to be equal to 6πη R for a sphere of radius R in a continuous solvent of viscosity η . When the concentration of macromolecules in solution increases, the interactions cannot be neglected and Dm (Φ ) begins to be significantly different to Ds (Φ ) and D0 . Moreover and adding to the different nature of the diffusion coefficients, one should distinguish between temporal regimes. One usually introduces three time domains: (1) t τB with τB is the time the correlation of the velocity needs to relax, that is, the necessary time to reach a Brownian regime for the macromolecules; (2) for τB t τi (τi ) is the time needed by the particle to interact with its neighbors) one defines a short time diffusion coefficient: the macromolecules only experience hydrodynamic interactions; and (3) in the time domain t τi , we observe the diffusion of the particle in solution, the hydrodynamic and direct interactions are established and we define a long time diffusion coefficient. In the short time regime, memory effects can be neglected and the dynamical functions can be expressed as a function of the equilibrium distributions; the theory is now relatively complete [38–40]. Ackerson has derived a relation for the apparent diffusion that can be split into contributions from the direct and hydrodynamic interactions: H(Q) D = D0 (8.20) S(Q)

Beenakker and Mazur [41, 42] have calculated the hydrodynamic factor, H(Q) for solutions of concentrated hard spheres. In particular, they have shown the characteristic wave vector dependance of H(Q) for different volume fractions Φ . The theoretical predictions for H(Q, Φ ) can be compared to experimental results. The dependence of the self-diffusion coefficients at short and long times, Dss and DLs , on the volume fraction, cannot be calculated exactly, but several

230


theoretical approximations have been introduced. Some authors have decoupled the contribution from hydrodynamic and direct interactions, as Beenakker and Mazur [41, 42]. Médina–Noyola [43] introduced a way of calculating DLs from Dss and the structure factor S(Q) of the solution. Tokuyama and Oppenheim [44] have developed a more systematic way of calculating Dss and DLs that includes both hydrodynamic and direct interactions for hard sphere solutions. The structure of charged particles in solution was studied theoretically by Hayter and Penfold [45], who derived an analytical structure factor for charged spheres interacting with a screened electrostatic potential. This theory (Mean Spherical Approximation) can be applied at rather small global macromolecular charges if the particle concentration is not too high, and if the contribution of counterions to the scattered intensity can be neglected. The latter only affects the screening of the Coulombic potential by reducing the Debye length. The potential no longer decreases with 1/r but rather with a Yukawa-type function. Hansen and Hayter [46] introduced a renormalization method of the effective radius of the particle in order to extend the theory to low concentration of highly charged colloids. In such a case the contact potential is much higher than kB T , and thus the contact probability of the particles is almost zero g(r > 2a) 0. The hard sphere potential doesn’t play a physical role. Their method involves rescaling the radius of the particle. Finally in 1986, Belloni [47] included the colloidal concentration in the calculation of the Debye length and thus the screening of the electrostatic forces.

8.3.2 Protein Structure Factor and Diffusion The cytoplasm topology directly influences protein diffusion. Beyond transport mechanisms, the kinetics of a biochemical reaction will be affected by the mobility reduction, which can become diffusion limited inside the complex interior of cells. The transport inside cells can generally be divided into three types : 1. The active transport which requires energy (ATP hydrolysis). 2. Simple diffusion (or Brownian) in which the mean-square displacement of the molecules is a linear function of time : r2 Dt. 3. Anomalous diffusion, in contrast to simple diffusion, in which the mean-square displacement of the molecules is NOT a linear function of the time : r2 Dt α . It can be subdiffusive (α < 1) and superdiffusive (α > 1), the former being mostly encountered in cells. It is now widely accepted that the dominant mechanism is diffusion [48] (nonactive transport), although there is one group suggesting a combined transport in the nucleus [49]. The nature of the diffusion process (Brownian or non-Brownian) still remains a matter of discussion. In what follows, only diffusive transport will be discussed because neutron scattering is probably less suited for studying activated transport.


231

The motion of a particle, at infinite dilution and in the absence of field, can be described by a random walk, which leads to: r2 = 6Dot,

(8.21)

where r2 is the mean square displacement of the particle during a time t. The presence of other macromolecules in the cytoplasm will strongly modify the value of Do , which is dependent on the excluded volume (the space which is physically not accessible to the center of mass molecule due to the presence of others). For a total macromolecular volume fraction Φ of the order of that present in the cytoplasm (Φ 0.3 − 0.4), the diffusion coefficient Ds (Φ ) can be reduced by more than one order of magnitude. A significant number of experimental and theoretical studies aim to measure and understand the mechanisms that lead to the reduction in Ds (Φ ). As mentioned before, this problem is very complex because one single diffusion coefficient at infinite dilution is replaced by a number of different ones at higher crowding fractions. The discovery of fluorescence proteins, such as the Green Fluorescence protein [86], together with improvements in imaging technology, has allowed the study of protein diffusion inside cells. Fluorescence techniques [50] like FRAP [51] (fluorescence recovery after photobleaching), FRET (fluorescence resonance by energy transfer) and FCS (fluorescence correlation spectroscopy), complement those usually used to study molecular diffusion (DLS, NMR, tracer methods ..) in vivo. All these methods probe the diffusion of molecules over distances typically of the order of the cell size or smaller, but in any case much higher than the inter molecular distances. In contrast, neutron scattering allows the study of interactions and molecular motions over typical protein–protein distances in protein solutions and in cells. Thus, we can test models developed to describe the statics and dynamics of micellar and colloidal suspensions. These objects generally have a size significantly higher than proteins where the first interaction peak can be probed by light scattering.

8.3.3 Protein Structure Factor Proteins in solution with their well-defined tertiary structure provide an excellent model system to study the interaction of simple charged molecules. Polydispersity arises only from protein aggregation and not from distributions in particle size. For some proteins, aggregates can be avoided under certain biochemical conditions or eliminated by centrifugation of the solution. The first attempt to study the structure and interparticle interactions in protein solutions using the analysis of Hayter and Penfold [45] was performed on Bovine Serum Albumin (BSA) in 1983 [52]. BSA is a prolate ellipsoidal-shape protein ˚ and b = 20 A, ˚ thus the analysis of the data were performed (a,b,b) with a = 70 A using a form factor of an ellipsoid with a structure factor calculated for equivalent

232


charged hard sphere. Later the same analysis was performed on hemoglobin inside red cells [53] and in solution [54], a protein that has a nearly spherical shape and therefore leads to more satisfactory results. Hayter and Penfold [45] calculated the structure factor of a colloidal solution that experiences an interaction potential composed of a hard sphere part plus an electrostatic tail using the Mean Spherical Approximation (MSA). Mathematically, −κ (r−2r ) the electrostic tail follows a Yukawa-type function β V (r) = Vi j (2r0 )2r0 e r 0 for r > 2r0 and V (r) = ∞ for r < 2r0 . β = 1/kBT and Vi j (2r0 ) is the contact potential which depends on the global protein charge Zp . κ is the inverse of the Debye length LD , which reflects the screening of the potential due to counterions in solutions. The neutron scattered intensity by a solution of monodisperse macromolecules in solution with a spherical symmetry can be described I(q) = Φ v0 (Δ ρ )2 F 2 (Q).S(Q),

(8.22)

where Φ is the volume fraction occupied by the macromolecules, v0 their volume, Δ ρ is the scattering length density difference between the solvent and the macromolecules (in cm−2 ), F(Q) is the form factor of the particle and S(Q) denotes the interparticle structure factor. The form factor of the protein can be measured at very low concentration where S(q)∼1, and it is then possible to access to the structure factor by a simple division of the scattered spectra by the form factor at each protein volume fraction. The spectra are then refined using the calculated structure factor of Hayter and Penfold [45] or with the corrections introduced by Belloni [47]. The free parameters of the refinements are the volume fraction Φ , the radius of the protein r0 and the net protein charge Zp . The Debye length is generally computed from the ionic strength of the solution and is implemented in the model. For myoglobin ˚ and Zp 2e for each volume fraction, the radius is solutions [55] one gets r0 16 A a little bit smaller than the real hard sphere radius of the protein but the small charge is compatible with the fact that the protein is at high concentration and will impose the pH of the solution close to its isoletric point. For hemoglobin solutions [53, 54] the experimental results can be satisfactory compared to theoretical calculations, although the volume fraction extracted from the analysis is slightly lower than the real ones, which has been interpreted as being due to the limited aggregation of the hemoglobin tetramers in solution.

8.3.4 Protein Diffusion as a Function of the Concentration The first step to understand the physical mechanism that leads to the mobility reduction in crowded media is to look at the evolution of the diffusion coefficient of a protein solution of as a function of the concentration. Riveros-Moreno and Wittenberg have measured the concentration dependence of the self-diffusion coefficient in myoglobin and hemoglobin solutions [56] up to volume fractions


233

of Φ 0.2 and Φ 0.26, respectively. They found, a plateau at low protein concentration followed by an exponential decrease of Ds down to more than one order of magnitude at higher protein concentration. Alpert and Banks [57] showed that the mutual diffusion coefficient Dm , measured by dynamical light scattering, has a much weaker dependance then the self-diffusion coefficient Ds . This was soon confirmed by Hall et al. [58] who compared the self- and the mutual-diffusion coefficient evolution as a function of the concentration in hemoglobin solutions. The self-diffusion coefficients are long-time ones because the techniques used for the measurements probe the diffusion process over much longer than the intermolecular ones. The analysis of the diffusion properties of proteins using a combined approach by SANS or small-angle X-rays scattering (SAXS) to get the structure factor, and neutron spin-echo spectroscopy (NSE) to obtain the apparent diffusion coefficient has been performed by different authors. On the one hand, myoglobin and hemoglobin solutions were studied [3, 4, 55, 59, 60] motivated by the aim to understand whether protein dynamics could be described by models developed for colloids and if protein diffusion can assist oxygen diffusion. On the other hand, experiments were performed on Ferritin solutions, to study the dynamics at high concentrations where paracrystalline order occurs [61, 62], to study the hydrodynamic interactions in perfectly monodisperse spherical macromolecular solutions [63], or more recently to study diffusive dynamics in solution [64]. Neutron spin echo spectroscopy gives access to full intermediate scattering function S(Q,t): 1 N −iQ.[ri (0)−r j (t)] S(Q,t) = , (8.23) bi b j e N ∑ i, j where i and j run over all the scattering centers N, of respective scattering lengths bi and b j and position ri (0) at time t = 0 and r j (t) at time t. In the wave vector range 2π /Q d (where d is the average distance between two scattering centers), usually referred to as the small angle limit, the intermediate scattering function of a solution of almost spherical macromolecules in solution, reads: S(Q,t) 1

2 2 Φ v p (Δ ρ ) F (Q) N

N

∑e

−iQ.[ri (0)−r j (t)]

(8.24)

i, j

where Φ is the volume fraction occupied by macromolecules of volume vp , i and j run over all the molecule centers N’ whose positions are ri (0) at time t = 0 and r j (t) at time t, respectively. F(q) is the form factor and Δ ρ is the scattering length density contrast between the macromolecules and the solvent. The normalized intermediate S(Q,t) scattering function I(Q,t) = S(Q) obtained for myoglobin solutions are presented in Fig. 8.17.

234


1.0

S(Q,τ)/S(Q,τ = 0)

0.8

0.6

Q=0.076 Å–1 –1

Q=0.120 Å

–1

0.4

Q=0.207 Å

0.2

Myoglobin 14.7 mM + D2O TS = 37°C

0.0 100

1000

10000

Spin Echo times (ps)

Fig. 8.17 Intermediate scattering function I(q,t) measured on the spin echo spectrometer G1 bis (Laboratoire Léon Brillouin, Saclay, France) on myoglobin solution of protein volume fraction Φ 0.2

There is no significant departure from a single relaxation decay whatever the concentration of the solution. We can consequently extract a wavevector-dependent diffusion coefficient by refining the curves using the relation: I(Q,t) ∼ e−D(Q)Q t . 2

(8.25)

Figure 8.18 shows the evolution of the apparent diffusion coefficient D(Q) as a function of the wave vector for three different protein concentrations. Independent of protein concentration, the wave vector evolution of D(Q) is similar, an increase at low-wave vector and a plateau at high Q. The extrapolation of D(Q) to Q = 0 leads to the mutual diffusion coefficient measured by light scattering Dm , whereas the value of the plateau corresponds to a self-diffusion coefficient Ds . The plateau is observed in the wave vector range where S(Q) ∼ 1 which generally corresponds to the incoherent approximation of coherent scattering. In this Q range, a small change in the position of the proteins, ri (0) − r j (t) will induce a strong variation of the phase term Q.[ri (0) − r j (t)] and the cross terms i = j of (8.24) will vanish leaving only the self-correlation terms. 8.3.4.1 Concentration Dependance of the Self-Diffusion Coefficient The concentration dependence of the self-diffusion coefficient Ds (Φ ) obtained for a myoglobin solution is shown in Fig. 8.19. The theoretical calculation by Tokuyama and Oppenheim [44] for the short- and the long-time self-diffusions for a hard sphere solution are also shown in the figure.


235

0.14

Myoglobin

14.7 mM: φ = 0.18 21.7 mM: φ = 0.28 30.0 mM: φ = 0.4

0.10

2

cm s–1

0.12

Dc(Q) 10

–5

0.08 0.06 0.04 0.02 0.00 0.00

0.05

0.10

0.15

0.20

0.25

–1

wavevector Q (A )

Fig. 8.18 Apparent diffusion coefficient measured on myoglobin solution of protein volume fraction ranging from Φ 0.18 to Φ 0.4

For globular proteins like myoglobin, the Brownian time can be estimated using Do τB Mf = M N kB T 10 ps. The interaction time can be estimated in one of two ways: (1) some authors [65] define it as the time necessary to diffuse by a distance 2 equal to its own hydrodynamic radius such that τi Da o 30 ns, where a is the hydrodynamic radius of the protein. This calculation, which is not dependent on the quantity of protein in solution, holds for moderate concentrations but can differ significantly at high values of Φ ; (2) assume that it corresponds to the time necessary to diffuse over the mean surface to surface distance of two proteins, which is clearly 1/3 d2 concentration dependent. Thus τi 2 Do with d = a 34Φπ − 2 . This leads for Φ 0.2 an interaction time τi 34 ns very similar to the first estimation, but decreases down to τi 1 ns for Φ 0.4. Nevertheless, this interaction time remains a rather crude approximation, because, for example, the diffusion coefficient which is used for the calculation should not be the infinite dilution one Do but rather the short time one which depends on the concentration. The relaxation times measured in myoglobin solutions extracted from the Fig. 8.17 are of the order of 1 ns, which falls in between the Brownian and the interaction time. The measured diffusion coefficient should consequently experience only hydrodynamic interactions. Under these conditions, it is a short time diffusion coefficient DSs (Φ ). In Fig. 8.19 the refinement of the diffusion coefficients measured by tracer methods [66], which by definition corresponds to the long time limit, is represented by a continuous black line. The correspondence between our NSE data and the value DLs (Φ ) is rather good. To understand which type of diffusion coefficient we are indeed measuring (shortor long- time), it is interesting to compare the theoretical results. Also added in

236


Fig. 8.19 Self-diffusion diffusion coefficients of myogoblin solutions measured by neutron spinecho of myoglobin solutions as a function of the concentration (Full circle), best refinement of the long-time self diffusion coefficient measured by tracer methods (full line), theoretical calculation for short- (dot) and long-time self diffusion coefficient obtained by Tokuyama and Oppenhein [44]. The full squares are the neutron spin echo results after including the water hydration shell in the calculation of the hydrodynamic volume (see text)

figure 8.19 are the results of the computation of Tokuyama and Oppenheim [44] for the short- and the long-time diffusion coefficients for hard sphere solutions. Our results together with the ones obtained by tracer methods are significantly lower than the theoretical predictions also for the long-time diffusion coefficient. In what follows, we try to explain this discrepancy. In 1977, Ross and Minton [67] noticed the need to include the volume of the hydration water shell in the computation of the hydrodynamic volume, in order to describe the viscosity of hemoglobin solutions by a hard quasi-spherical model. To compute the volume fraction on Fig. 8.19, we used the protein concentration (determined with a high precision by UV absorption) and the specific volume of the protein as vp 0.741 cm3 .g−1 : this corresponds to the dry volume fraction of the protein. The full squares in Fig. 8.19 are the spin echo results where the dry volume was replaced by the hydrodynamic one by using Φh = cp vh with vh = vd + δ vs . vs is the specific volume of the solvent and δ (0 ≤ δ ≤ 1) is the weight fraction of the hydration shell contributing to the volume, which is commonly assumed to be δ = 0.35. There is a quite good agreement between the experimental results and the calculation of Tokuyama and Oppenheim [44] for the long-time diffusion coefficient for hard sphere solutions, which supports the idea that the self-diffusion coefficient measured by spin echo is a long-time one.


237

S(q,τ)

1

0.1

0

20

40

60

80

100

Spin Echo time (ps) −1

˚ ,t) measured on the spin echo spectrometer Fig. 8.20 Intermediate scattering function I(Q = 1 A G1 bis (Laboratoire Léon Brillouin, Saclay, France) on hemoglobin solution of protein volume fraction Φ 0.25 (circle) and on water (square). The contribution of the protein is proportional to its ratio of protons in solution and is represented as the dot curve

8.3.4.2 Search for the Short-Time Diffusion Coefficient In hard sphere suspensions at high volume fraction (Φ 0.4), it was shown by the time evolution of the mean square displacement that particles typically achieve their asymptotic long time behaviour after diffusing over distances equal to only a few tenths of its diameter [68, 69]. In protein solutions, at slightly lower concentration (350 mg.ml−1 ), one expects to observe the transition from long-time to smalltime behaviour at high wavevectors. the mixture of coherent and spin incoherent scattering leads to a strong decrease of the scattered beam polarisation, which is highly unfavourable for NSE measurements. Therefore, such measurements must be performed on fully incoherent samples (H-protein in H2 O). Figure 8.20 shows the intermediate scattering function of a hemoglobin solution (Φ 0.25) and that ˚ −1 . The protein contribution is proportional to the ratio of pure water at q = 1 A of the number of hydrogens in the protein with respect to the total number in solution, which can be estimated around 25%. A clear signature of the protein can be separated from the water contribution, which is represented by the dotted app curve. The apparent diffusion, which can be extracted from the signal is Dmes = −7 2 −1 14.9(±3) ∗ 10 cm s , and is more than twice the Stokes–Einstein one at low concentration, which should normally be the highest limit for both DLs and DSs . It is thus necessary to introduce an additional dynamical phenomenon to explain this relaxation time. Proteins undergo aleatory rotational motions, referred to as rotational Brownian motions which are similar in nature to the translation ones. Their physical origin is an aleatory torque which results from unbalanced collisions

238


of the solvent molecules with the protein. At high momentum transfer, these motions will contribute to the translational one as an additional term D˜ r R2g Dr . Taking the values given in the literature [70] and assuming that for a solution of concentration ∼350 mg.ml−1 one gets Dss D0s /2 (corresponds to the theoretical predictions), we −1 get a new apparent diffusion coefficient Dapp = D˜ rrbc + Ds 6.2 ∗ 10−7 cm2 .s which remains much above the measured one. Thuis the relaxation phenomena which are measured by NSE probably differ in nature from a Brownian motion. Most likely this effect originates from internal motions of the protein [71]. 8.3.4.3 Wave Vector Dependence of D(Q) As was seen in Fig. 8.18, the wavevector dependence of the apparent diffusion coefficient is always the same whatever the protein concentration is solution. This behaviour is characteristic of the collective nature of D(Q). In molecular liquids the interactions arise from direct forces. In 1959, P.-G. de Gennes calculated a relationship between the second moments of the coherent and incoherent scattering 2 = ω 2 /S(Q). In colloidal or protein solutions, we have to account peaks [72] : ωcoh inc for additional interactions mediated by the solvent namely the hydrodynamic interactions, which appear in relation 8.3.1. In the limit of zero scattering wave vector, this value must lead to the one measured by light scattering. It is weakly dependent on the protein concentration because the increase of the friction value is partially compensated by the strong variation of the osmotic compressibility. When the scattering wave vector increases, the structure factor increases, which accounts for the decrease of D(Q). Formally equation 8.3.1 is only valid in the limit of short times, when the neighboring molecules can be considered as immobile. This means that the diffusion coefficient is the one measured at short times. But we have shown that we only have access to the long-time diffusion coefficient DLs , and not DSs , which would exclude any possibility to compute the hydrodynamic factor H(Q). The top panel in figure 8.21 plots the product D(Q) ∗ S(Q)/Do deduced from the neutron scattering data. In accordance with the theoretical predictions for H(Q) [42], this product oscillates in phase with the structure factor, and after renormalisation by a constant in order to account for the difference between the short- and long-time diffusion coefficient the agreement between experimental results and theoretical results is satisfactory. P. Segrè and P. Pusey showed a similar relation between the short- and long-time apparent diffusion coefficient, DS (Q) and DL (Q), in colloidal suspensions up to volume fractions of 30% [84]. This observation has currently no theoretical explanation, but Fig. 8.21 tends to indicate a strong contribution of hydrodynamic interactions in protein mobility. 8.3.4.4 Measurements Inside Cells The measurement of hemoglobin diffusion inside red blood cells (RBC) is not straightforward. Dynamic light scattering is dominated by membrane fluctuations


239

Fig. 8.21 Apparent diffusion coefficient, structure factor and product D(Q) ∗ S(Q)/Do obtained for myoglobin solution at a concentration of 14.7 mM (Φ 0.2)

[73], whereas NMR measures protein motions over hundred of nanometers where cell confinement effects can become important [74, 75]. Krueger an Nossal [53] have used neutron scattering to study the structure factor of hemoglobin solutions inside red blood cells. They have especially shown, using contrast matching, that the membrane and hemoglobin contributions occur at different length scales and although membrane scattering can not be eliminated stricto-sensus because of its inhomogeneity (the membrane contains lots of different protein) both signals can be easily separated. The dynamics of hemoglobin was studied inside red blood cells by neutron spin-echo spectroscopy [3] and more recently using timeof-flight spectrometry [76]. For contrast reasons, the first study was performed using red blood cells, which have been dialysed against D2 O, at the physiological ˚ −1 to Q ∼ 0.13 A ˚ −1 . This wave vector range temperature of 37◦ C from Q ∼ 0.02A

240


surrounds the protein structure factor maximum, which means, that it ranges from typically intermolecular distances up to few tens of nanometers. The results can be summarised as follows [3]: • The diffusion coefficient of hemoglobin in the red blood cells is equal to Ds = −1 1.75(±0.2).10−7cm2 .s , in heavy water and at T = 37◦ C. When corrected for temperature and the viscosity difference between H2 O and D2 O, this leads to an −1 equivalent of Ds = 1.1(±0.2).10−7cm2 .s in water at T = 20◦ C. • The diffusion of hemoglobin at high concentration can be understood on the basis of theoretical concepts developed for colloidal suspensions. The main difference is that the effective hydrodynamic volume fraction of the protein must include the hydration shell because of the highest surface over volume ratio of the proteins. • The protein–protein friction in the RBCs is mainly controlled by hydrodynamic interactions. This conclusion is based on the wave vector dependence of the apparent diffusion coefficient, and cannot be deduced only from the volume fraction dependance of Ds (Φ ), because it can be reproduced by Brownian dynamics simulations of protein without hydrodynamic interactions [85]. The time-of-flight spectra were measured on D2 O-exchanged solutions (in order ˚ −1 to reduce the contribution of the solvent) in the wave vector range Q = 0.5 A −1 ˚ for different temperatures [76]. The analysis was similar to the one to Q = 1.6 A developed previously for protein solutions [77] in order to separate the contributions from Brownian diffusion and internal motions. The resolution of the spectrometer was set to a minimum of 41 μev and reaches more than 60 μev (Full-Width Half Maximum) at high angles. The line width of the Brownian diffusion contribution follows a characteristic Q2 dependence at intermediate wave vectors but seems to ˚ −2 . At high wave vectors, the saturate to a plateau around 4 μev for Q2 ≤ 0.75 A authors claim that the curves show an inflection as is predicted theoretically for a jump diffusion models [78] (figure [76]). The diffusion coefficient extracted from the Q2 regime is similar to the one measured at infinite dilution Do , by DLS and by macroscopic methods [56]. Perez et al. [77] estimated, in a different way to that described above, the contribution of rotational Brownian motions to the apparent diffusion coefficient in low concentration protein solutions measured by incoherent scattering. They used numerical integration of the Sears calculation for the contribution to the dynamical structure factor of the rotational diffusion of protons on a sphere [79, 80]. They concluded that rotational diffusion leads to an apparent increase of the translational diffusion coefficient and that Ds Dapp /1.27. This calculation is valid for low protein concentration when the Stokes–Einstein laws occur for both translational and rotational Brownian diffusion, and reads for a sphere: Do =

kB T 6πη RH

(8.26)

Dr =

kB T 8πη R3H

(8.27)


241

One should notice that the dimensions of both diffusion coefficients is not the same since r2 = 6Dst, whereas δ θ 2 = 4Drt. Generally for diffusion in solution −1 Ds is given in cm2 .s and Dr in s−1 . The application to high concentration solutions is not straightforward because in general the Stokes–Einstein relations break down at high concentrations. For example in red blood cells the rotational diffusion coefficient is reduced by a factor of 2 [70], whereas the long-range translational one decreases by a factor of 7 [3]. The two type of motions do not experience the same influence of protein–protein interactions. In [76] the same procedure was applied to the short-time diffusion coefficient which is theoretically reduced (Dss 0.56Do ) by as much as the rotational diffusion time, although it is closer to 0.34 when including the water hydration shell in the computation of the hydrodynamic volume. The plateau at small wave vectors and the apparent saturation at high Q have been interpreted as a confinement due to the neighbouring molecules and a jump diffusion mechanism. The plateau at small wave vectors would mean that up to a certain distance the protein is freely diffusing with a coefficient DSs , then would be confined for a certain time, and after a time higher than the interaction time would diffuse over long range but with a decreased diffusion coefficient DLs . In terms of intermediate scattering function, I(Q,t), this would lead to a two time decay function with a plateau corresponding to the apparent EISF of the protein’s centre of mass. In fact, to get such a function the two characteristic times should be clearly time separated (more than one to two orders of magnitude), which is clearly not the case because the short- and long-time diffusion coefficients differ only by a factor from 2 to 3 at this concentration. Second, the full decay of I(Q,t) was measured by NSE [3] and did not show such a behaviour. The jump diffusion model assumes that the jumping time can be neglected with respect to the residence time; the authors calculate both times as τr 280 ps for the residence time and τ j ≈ 50 ps. In fact, the picture is certainly closer to the traditional I(Q,t) for colloidal suspension at this volume fraction, a stretched decay function, the short time diffusion coefficient being obtained from the first cumulant analysis of the function, whereas the full decay occur with the long-time diffusion coefficient. Measuring such complicated decay functions is easier in the time domain, the convolution by the resolution function in the energy range could lead to erroneous conclusions.

8.3.4.5 Coupling of Internal and Diffusive Motions Recently, a method was developed that allows to study the protein domain dynamics by an analysis of the departure from the DQ2 law that is generally observed when only translational motions contribute to the signal [81, 82]. The method is not straightforward and involves sophisticated data treatments, since the effect of structure factor, hydrodynamic factor and rotational motions, must first be eliminated from the measured signal (i.e. the apparent diffusion coefficient Deff (Q)), although the calculation is simplified by the low concentrations of protein in solution.

242


In [82], the authors studied the interdomain motion in Alcohol dehydrogenase (ADH). From the effective diffusion coefficient Deff (Q), a single tetramer effective diffusion coefficient is deduced using the classical Ackerson formula [39]: Doeff (Q) = Deff (Q) ∗ S(Q)/H(Q). The prime reason for the Q modulation Doeff (Q) is the rotational Brownian diffusion of the aspherical tetramer, which can be calculated using the computer code HYDROPRO [83]. Finally the difference between the calculated and the measured single effective diffusion coefficient Δ Doeff (Q) is compared with the non trivial lower frequency mode normal calculation and some motions prevailing can be identified. (See figure do we insert a figure of prl 101 138102?).

8.3.5 Conclusion Dynamics are fundamental for proteins to achieve their functions. Stochastic processes, driven by Brownian noise are of primary importance from a molecular level, acting as plasticizers, up to the cell where their unbalanced effects lead to transport and protein motions. These processes span from the picoseconds, as for example, water diffusion or small group internal motions, up to long time large domain fluctuations, associated to functions and protein diffusion. Neutron scattering is a valuable tool for the investigation of the dynamics of proteins and hence the correlation between these motions and protein function. Internal and global motions can be separated by appropriate sample choice (for example hydrated powder to study internal dynamics without translational diffusion) or spectrometer configurations. A general rule is to extend the energy (or time) range of investigation as far as possible by combining different types of spectrometer. Inelastic or quasielastic, neutron scattering can span over 7 or 8 orders of magnitude in time. A particular difficulty is to combine measurements in the time domain, as is the case with neutron spin-echo, with broader bandwidth measurements in the energy domain (Time-of-flight, backscattering). The transformation from energy (¯hω ) to the time domain is not straightforward and needs a careful account of the h¯ ω − Q windows of the spectrometer as well as their resolution shapes or even coherent to incoherent ratios. It is sometimes interesting if not necessary to refine models to spectra obtained with different spectrometer configurations or even different types of spectrometer. Using the refined parameters obtained at low resolution as input for the refinements of the high resolution and so on. Such procedure has the advantage of allowing the test of models over a wide range of wave vector and energy transfer. In any case, strategies must be developed to study a specific type of motion over the entire domain it spans and eventually to study the coupling with other motions.


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References 1. Jacrot B (1976) Study of biological structures by neutron scattering from solution. Rep Prog Physics 39:911–953 2. Doster W, Cusack S, Petry W (1989) Dynamical transition of myoglobin revealed by inelastic scattering. Nature 337:754–756 3. Doster W, Longeville S (2007) Microscopic diffusion and hydrodynamic interactions of hemoglobin in red blood cells. Biophys J 93:1360–1368 4. Doster W, Settles M (2005) Protein–water displacement distributions. Biochim Biophys Act 1749:173–186 5. Lichtengegger H, Doster W, Kleinert T, Birk A, Sepiol B, Vogl G (1999) Heme-solvent coupling, a mossbauer study of myoglobin in sucrose. Biophys J 76:595–603 6. Tilton RF, Kunz ID, Petsko GA (1984) Cavities in proteins. Biochem 23:2849–2857 7. Brunori M, Gibson QH (2001) Cavities and packing defects in the structural dynamics of myoglobin. EMBO reports 2, 8:674–679 8. Srajer V, Teng TY, Ursby T, Pradervand C, Ren Z, Adachi S, Schildkamp W, Bourgeois D, Wulff M, Mofat K (1996) Photolysis of the carbon monoxide complex of myoglobin: nanosecond time-resolved crystallography. Science 274:1726–1729 9. Schotte F, Lim M, Jackson TA, Smirnov AV, Soman J, Olson JS, Phillips GN, Wullff M, Anfinrud PA (2003) Watching a protein as it functions with 150 ps time-resolved X-ray crystallography. Science 300:1944–1947 10. Tomita A, Sato T, Nozawa S, Ichiyanagi K, Ichikawa H, Chollet M, Kawai F, Park S-Y, Yamato T, Tsuduki T, Koshihara S-Y, Adachi S-I (2009) Visualizing breathing motion of internal cavities in concert with ligand migration in myoglobin. Proc Natl Acad Sci 106: 2612–2616 11. Elber R, Karplus M (1990) Enhanced sampling in molecular dynamics: use of the time dependent hartree approximation for a simulation of carbon monoxide diffusion through myoglobin. J Am Chem Soc 112:9161–9175 12. Cohen J, Arkhipov A, Braun R, Schulten K (2006) Imaging the migration pathway for o2 , co, no and ce inside myoglobin. Biophys J 91:1844–1857 13. Kleinert Th, Doster W, Leyser H, Petry W, Schwarz V, Settles M (1998) Solvent composition and viscosity effects on the kinetics of co-binding to horse myoglobin. Nature 37:717–733 14. Doster W (2010) The protein-solvent glass transition. Bioch Biophys Act 1804:3–14 15. Ansari A, Jones C, Henry ER, Hofrichter J, Eaton MA (1992) The role of solvent viscosity in the dynamics of protein conformational-changes. Science 256:1796–1796 16. Beece D, Eisenstein L, Frauenfelder H, Good D, Marden M, Reinisch L, Reynolds A, Sorenson L, Yu K (1980) Viscosity and protein dynamics. Biochem 19:5147–5157 17. Timasheff SN (2002) Protein hydration, thermodynamic binding and preferential hydration. Biochem 41:13473–13482 18. Parak F, Knapp EW (1984) A consistent picture of protein dynamics. Proc Natl Acad Sci USA 81:7088–7092 19. Parak F, Achterhold K (2005) Protein dynamics on different timescales. J Phys Chem Solids 66:2257–2262 20. Doster W (2005) Brownian oscillator analysis of molecular motions in biomolecules. In: Fitter J, Gutberlet T, Katsaras J (eds) Neutron Scattering in Biology. Springer Series in Biological and Medical Physics, Biomedical Engeneering. p 461–482 21. Doster W (2006) Dynamic structural distibutions in proteins. Physica B 385–386:831–834 22. Roh RH, Novikov VN, Gregory RB, Curtis JE, Chaowduri Z, Sokolov AP (2005) Onset of harmonicity of protein dynamics. Phys Rev Lett 95:038101–038103 23. Zaccai J (2000) How soft is a protein? a protein force constant measured by neutron scattering. Science 288:1604–1607 23a. Cordone L, Ferrand M, Vitrano E, Zaccai G (1996) Biophys J 76:1043–1047

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24. Doster W (1998) Dynamical transition of proteins, the role of hydrogen bonds. In: MarieClaire Bellissent-Funel (ed) Hydration Processes in Biology. (Les Houches Lectures) IOS Press 25. Doster W (2008) The dynamical transition of proteins, concepts and misconceptions. Eur Biophys J 37:591–602 26. Doster W, Diehl M, Petry W, Ferrand M (2001) Elastic resolution spectroscopy: a method to study molecular motions in small biological samples. Physica B 301:65–68 27. Doster W, Diehl M, Gebhardt R, Lechner, RE, Pieper J (2003) Elastic resolution spectroscopy: a method to study molecular motions in small biological samples. Chem Phys 292:487–494 28. Settles M, Doster W (1996) Anomalous diffusion of protein hydration water. Faraday Discussions 103:269–279 29. Paciaroni A, Cinelli S, Onori G (2002) Effect of the environment on the protein dynamical transition, a neutron scattering study. Biophys J 83:1157–1164 30. Onori G, Cornicchi E, Paciaroni A (2005) Picosecond time scale fluctuations of proteins in glassy matrices: the role of viscosity. Phys Rev Lett 95:158104 31. Doster W, Busch S, Appavou MS, Gaspar A, Wuttke J, Scheer J (2010) The dynamical transition of protein hydration water. Phys Rev Lett 104:098101–098104 32. Austin RH, Beeson KW, Eisenstein L, Frauenfelder H, Gunsalus IC (1975) Dynamics of ligand binding to myoglobin. Biochem 14:5355–5373 33. Post F, Doster W, Karvounis G, Settles M (1993) Structural relaxation and nonexponential kinetics of ligand binding to horse myoglobin. Biophys J 64:1833–1844 34. Neumann DA, Tsai AM, Bell LN (2000) Molecular dynamics of solid state lysozyme as affected by glycerol and water: a neutron scattering study. Biophys J 79:2728–2732 35. Demmel F, Doster W, Petry W, Schulte A (1997) Vibrational frequencies as a probe of hydrogen bonds: thermal expansion and glass transition of myoglobin in mixed solvents. Eur Biophys J 26:327–335 36. Phillies GDJ (1974) Effects of intermolecular interactions on diffusion. 1. 2-component solutions. J Chem Phys 60:976–982 37. Phillies GDJ (1975) Continuum hydrodynamic interactions and diffusion. J Chem Phys 62:3925–3932 38. Pusey PN (1975) Scaled particle theory of fluid mixtures. J Phys A Math Nucl Gen 8: 1433–1440 39. Ackerson BJ (1976) Correlations for interacting brownian particles. J Chem Phys 64:242–246 40. Ackerson BJ (1978) Correlations for interacting brownian particles. 2. J Chem Phys 69: 684–690 41. Beenakker CWJ, Mazur P (1983) Self-diffusion of spheres in a concentrated suspension. Physica 120A:388–410 42. Beenakker CWJ, Mazur P (1984) Diffusion of spheres in a concentrated suspension 2. Physica 126:349–370 43. Médina-Noyola M (1988) Long-time self-diffusion in concentrated colloidal dispersions. Phys Rev Lett 60:2705–2708 44. Tokuyama M, Oppenheim I (1994) Dynamics of hard sphere suspension. Phys Rev E 50:R16– R19 45. Hayter JB, Penfold J (1981) An analytical struture factor for macroion solutions. Mol Physics 42:109–118 46. Hansen JP, Hayter JB (1982) A rescaled msa structure factor for dilute charged collodal dispersions. Mol Phys 46:651–656 47. Belloni L (1986) Electrostatic interactions in collodal solutions – comparaison between primitive and one-component model. J Chem Phys 85:519–526 48. Dix JA, Verkman AS (2008) Crowding effects on diffusion in solutions and cells. Annu Rev Biophys 37:247–263 49. Carmo-Fonseca M, Platani M, Sweldow JR (2002) Macromolecular mobility inside cell nucleus. Trends Cell Biol 12:491–495


245

50. Lippincott-Schwartz J, Snapp E, Kenworthy A (2001) Studying protein dynamics in living cells. Nature 2:444–456 51. Phair RD, Misteli T (2000) High mobility of proteins in the mamalian cell nucleus. Nature 404:604–609 52. Bendedouch D, Chen S-H (1983) Structure and interparticle interactions of boviner serum albumin in solution studied by small-angle neutron scattering. J Phys Chem 87:1473–1477 53. Krueger S, Nossal R (1988) Sans studies of interacting hemoglobin in intact erythrocytes. Biophys J 53:97–105 54. Krueger S, Chen S-H, Hofrichter J, Nossal R (1990) Small-angle neutron-scattering studies of hba in concentrated solutions. Biophys J 58:745–757 55. Longeville S, Doster W, Kali G (2003) Myoglobin in crowded solutions: structure and diffusion. Chem Phys 292:413–424 56. Riveros-Moreno V, Wittenberg JB (1972) Self-diffusion coefficients of myoglobin and hemoglobin in concentrated solutions. J Biol Chem 247:895–901 57. Alpert SS, Banks G (1976) The concentration dependence of the hemoglobin mutual diffusion coefficient. Biophys Chem 4:287–296 58. Hall RS, Johnson CS (1980) Experimental-evidence that mutual and tracer diffusioncoefficients for hemoglobin are not equal. J Chem Phys 72:4251–4253 59. Longeville S, Doster W, Diehl M, Gaehler R, Petry W (2003) Neutron resonance spin echo: Oxygen transport in crowded protein solutions. In: Mezei F, Pappas C, Gutberlet T (eds) Neutron spin echo spectroscopy, basic trends and applications. Lecture notes in physics, vol 601. Springer, Berlin, pp 325–335 60. Le Coeur C, Demé B, Longeville S (2009) The compression of random coils due to macromolecular crowding. Phys Rev E 79:031910 61. Haeussler W (2003) Structure and dynamics in apoferritin solutions with paracrystalline order. Chem Phys 292:425–434 62. Haeussler W, Farago B (2003) Diffusive dynamics in ordered solutions of apoferritin solutions near the structure factor peaks. J Phys Condens Matter 15:S197–S204 63. Banchio AJ, Gapinsky J, Patkowski A, Haeussler W, Fluerasu A, Sacanna S, Holmqvist P, Meier G, Lettinga MP, Naegele G (2006) Many-body hydrodynamic interactions in chargedstabilized suspensions. Phys Rev Lett 96:138303 64. Haeussler W (2008) Neutron spin echo studies on ferritin: free-particle diffusion and interacting solutions. Eur Biophys J 37:563–571 65. Gapinski J, Wilk A, Patkowski A, Häußler W, Banchio AJ, Pecora R, Nägele G (2005) Diffusion and microstructural properties of solutions of charged nanosized proteins: experiment versus theory. J Chem Phys 123:054708 66. Muramatsu N, Minton AP (1988) Tracer diffusion of globular-proteins in concentrated protein solutions. Proc Natl Acad Sci USA 85:2984–2988 67. Ross PD, Minton AP (1977) Hard quasispherical model for the viscosity of hemoglobin solutions. Biochem Biophys Res Comm 76:971–976 68. van Megen W, Underwood SM, Snook I (1986) Tracer diffusion in concentrated colloidal suspension. J Chem Phys 85:4065–4072 69. Pusey PN (1991) Colloidal suspensions. In: Hansen JP, Levesque D, Zinn-Justin J (eds) Liquids, freezing and glass transition. Les Houches Summer School Proceedings, vol. 1, Elsevier Science Ltd, Oxford, p 504 70. Lavalette D, Tetreau C, Tourbez M, Blouquit Y (1999) Microscopic viscosity and rotational diffusion of proteins in a macromolecular environment. Biophys J 76:2744–2751 71. Le Coeur C, Longeville S (2008) Microscopic protein diffusion at high concentration by neutron spin-echo spectroscopy. Chem Phys 345:298–304 72. de Gennes P-G (1959) Liquid dynamics and inelastic scattering of neutrons. Physica 25:825 73. Jones CR, Johnson CS, Penniston JT (1978) Photon correlation spectroscopy of hemoglobin – diffusion of oxy-hba and oxy-hbs. Biopolymers 17:1581–1593 74. Everheart CH, Johnson CS (1982) Pulse field gradient nmr determination of the temperature dependence of the tracer diffusion coefficient of hemoglobin. Biopolymers 21:2049–2054

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75. Kuchel PW, Chapman BE (1991) Translational diffusion of hemoglobin in human erythrocytes and hemolysates. J Magnet Res 94:574–580 76. Stadler AM, Digel I, Artmann GM, Embs JP, Zaccai G, Bueldt G (2008) Hemoglobin dynamics in red blood cells: correlation to body temperature. Biophys J 95:5449–5461 77. Perez J, Zanotti J-M, Durand D (1999) Evolution of the internal dynamics of two globular proteins from dry powder to solution. Biophys J 77:454–469 78. Singwi KS, Sjoelander A (1960) Resonance absorption of nuclear gamma rays and the dynamics of atomic motions. Phys Rev 120:1093–1102 79. Sears VF (1966a) Cold neutron scattering by homonuclear diatomic liquid: 1 free rotation. Can J Phys 44:1279 80. Sears VF (1966b) Cold neutron scattering by homonuclear diatomic liquid: 2 hindred rotation. Can J Phys 44:1999 81. Bu Z, Biehl R, Monkenbusch M, Richter D, Callaway DJE (2005) Coupled protein domain motion in taq polymerase revealed by neutron spin-echo spectroscopy. Proc Natl Acad Sci 102:17646–17651 82. Biehl R, Hoffmann B, Monkenbusch M, Falus P, Préost S, Merkel R, Richter D (2008) Direct observation of correlated interdomain motion in alcohol dehydrogenase. Phys Rev Lett 101:138102 83. Huertas ML, de la Torre JG, Carrasco B (2000) Calculation of hydrodynamic properties of globular proteins from their atomic-level structure. Biophys J 78:719–730 84. Segre PN, Pusey PN (1996) Phys Rev Lett 77:771–774 85. Dwyer JD, Bloomfield VA (1993) Brownian dynamics simulations of probe and self-diffusion in concentrated protein and DNA solutions. Biophys J 65:1810–1816 86. Martin Chalfie, Yuan Tu, Ghia Euskirchen, William W. Ward, Douglas C. Prasherf (1994) Science 263:802–805

Chapter 9

Bio-Macromolecules and Hydration Water Dynamics Kathleen Wood and Martin Weik

9.1 Introduction–Neutrons as a Relevant Probe for Bio-Molecular and Water Dynamics Neutron scattering allows us to explore the microscopic properties of matter on timescales from ps to hundreds of ns by combining several different instrument types. The scattering function measured by neutron spectroscopy can be divided into two parts: coherent and incoherent. Coherent scattering probes atomic positions and collective motions within the sample, as neutrons scattered by different atoms at the same time in the sample interfere. Incoherent scattering, which probes “self” or diffusive dynamics, corresponds to the interference of neutrons scattered by the same atom at different times. In this chapter, we will concentrate on the dynamics of bio-molecules and hydration measured by incoherent neutron scattering, but coherent water dynamics is now also studied near soluble proteins [1] and lipid bilayers [2,3]. The interested reader is also referred to a special edition of [62] containing the proceedings from “Biomolecular dynamics and protein–water interactions – a neutron scattering workshop”, which contains many relevant papers. Two main types of measurements are extensively used to characterise macromolecular and hydration water dynamics: extracting atomic mean square displacements (MSD) from the elastically scattered neutrons as has been shown in Chap. 7, and the analysis of quasi-elastic neutron scattering (QENS) data, which we will

K. Wood () Department of Biophysical Chemistry, University of Groningen, Groningen, The Netherlands and Bragg Institute, Australian Nuclear Science and Technology Organisation, Menai NSW, Australia e-mail: [email protected] M. Weik Institut de Biologie Structurale, Grenoble, France e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 9, © Springer Science+Business Media, LLC 2012

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Fig. 9.1 Schematic representation of scattering intensities as a function of wavevector (Q) and energy transfer (E). A represents the elastic peak centred at zero energy transfer and B indicates the quasi-elastic broadening

explain briefly here. A fuller description is given in [4], which also contains a summary of models that can be applied to neutron scattering from bio-molecules. Scattered neutrons can be measured both as a function of wavevector (Q) and energy transfer (E). The resulting spectrum is schematically drawn in Fig. 9.1. As can be seen in the figure, an “elastic peak” (A in Fig. 9.1) is observed centred at zero energy transfer. The way the elastic peak varies as a function of wavevector can be used to extract atomic MSD of the sample in the Gaussian approximation: ln

Iel (Q) Inorm (Q)

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Where Iel is the elastically scattered intensity, Inorm is the intensity used to normalise the data (either scattering from the sample at very low temperature or from a vanadium sample), representing the instrument’s resolution function, a is a factor depending purely on the definition used, generally either 1/3 or 1/6, and < u2 > is the MSD. The equation given above is valid only at low Q(Q2 < u2 >≤ 1 if a = 1/3 or Q2 < u2 >≤ 2 if a = 1/6). Generally, to extract the MSD from the elastic scattering, full energy scans are not performed, but only the elastic peak is measured. Recording the elastic peak can then be done for a wide temperature range, and the dynamics as a function of temperature for the sample can be characterised relatively quickly. This type of measurement is typically referred to as a “fixed-window” or “elastic” scan and is commonly performed on backscattering spectrometers.

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At the foot of the elastic peak, quasi-elastic broadening can be seen (B in Fig. 9.1). Quasi-elastic broadening is generally fit with one or several Gaussian or Lorentzian functions, the widths and heights of which can be interpreted in the context of different models to describe the microscopic properties of protein or water dynamics. In biological samples, incoherent neutron scattering is strongly dominated by hydrogen, as all other atoms found have very low incoherent scattering crosssections. At the time-scales studied, hydrogen atoms reflect the dynamics of the groups they are attached to (e.g. side-chains or water molecules). Deuterium is a weak incoherent scatterer and can therefore be used to mask the contribution of one component of a complex system by specifically replacing hydrogen atoms. As was shown in Chap. 7, the timescales accessible by neutron scattering are also those of molecular dynamics simulations, which have played an important role in the field of bio-molecular hydration. They have been used extensively to interpret experimental results and led to the design of new experiments. Several pieces of work reviewed in this chapter combine simulations with scattering.

9.2 Protein Dynamics Characterised by Temperature Dependence of Atomic Mean Square Displacements 9.2.1 Protein Dynamical Transition A fundamental goal of studying protein dynamics is to gain theoretical insight into the energy landscape underpinning macromolecular structure and dynamics. By varying an external parameter, such as temperature, the aim is to “tease-apart” the range of motions that occur: at lower temperature, the range of dynamics proteins explore is simplified. As already mentioned, measuring the MSD as a function of temperature by neutron scattering is a relatively fast way to characterise a sample’s dynamics. Such a technique was first applied to a protein sample, myoglobin, by Doster et al. [5]. The famous work showed that at 180 K, a strong onset of dynamics takes place in the protein and was named the “protein dynamical transition”. The results were interpreted in the context of energy levels in the protein’s energy landscape: at lower temperature, the protein behaves as if trapped in a harmonic energy well, and at higher temperatures above the dynamical transition, excitation of an additional motion occurs and the protein can jump to wells of higher energy. An effective environmental force constant can be calculated from the slope of measured MSD as a function of temperature that quantifies the molecular resilience of a protein structure [6]. Subsequent work showed it to be a general phenomenon of biological molecules and strongly dependent on hydration – the transition does not occur if the sample is not hydrated to a sufficient degree. All bio-molecules measured to date – soluble

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proteins [7–10], membranes [11–13], RNA [14, 15] and DNA [16] – undergo a solvent-dependent transition in the range of temperatures from 150 to 250 K. Since both hydration and dynamics are crucial for macromolecular biological function and the dynamical transition does not occur when sufficient hydration is not present, it has become a much studied phenomenon and still generates debate [17, 18]. An example of atomic MSD measured as a function of temperature at several hydration levels, recorded on the purple membrane, is given in Fig. 9.2 [19]. The purple membrane is composed predominantly of a single protein type, bacteriorhodopsin, and various lipid species. The data reproduced in Fig. 9.2 show that the protein dynamical transition is observed only in samples hydrated to a sufficient degree. Below the transition temperature, dynamics at the timescale measured is independent of the hydration level, whilst above it, increasing hydration allows larger MSD.

9.2.2 Influence of Hydration on Protein Dynamics 9.2.2.1 Tuning Protein Dynamics by Modifying Their Solvent As was already mentioned in Chap. 7, measuring the dynamics as a function of temperature and solvation has shown that changing the nature of the solvent itself influences the dynamic response from the protein. By dissolving proteins in a mixture of trehalose, an effective bio-protectant, Cordone et al. [20] showed that it is possible to suppress the protein dynamical transition entirely. Saccharide glasses such as trehalose are known to protect proteins at low hydration levels and at high temperatures. The authors describe that the mechanism behind this protection is the trapping of the protein in their low temperature harmonic potential. Later, Tsai et al. [21] measured lysozyme in different amounts of glycerol. As seen in Fig. 9.3, replacing hydration water with glycerol shifts the protein transition temperature to a higher temperature when compared to the protein hydrated in water. Glycerol does, however, have the same “plastifying” effect on proteins as water, with more glycerol allowing increased dynamics above the transition temperature. Cornicchi et al. [22] measured lysozyme in glycerol–D2O and glucose–D2 O mixtures at varying concentrations. Having a number of data sets allowed the authors to verify that there is a relationship between the amplitude of the atomic MSD and solvent viscosity. As can be seen in Fig. 9.4 data from their work, this relation is valid for a number of samples. Measurements of the enzyme xylanase in DMSO and methanol with pure DMSO and methanol showed that the enzyme dynamics follow that of the pure solvents [23]. Phosphate buffers are often used for proteins since they have a stabilising effect, and this was shown by neutron scattering to be reflected in a rigidification of human butyrylcholinesterase dynamics [24].


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9.2.2.2 Theoretical Aspects and Molecular Dynamics Simulations All of the above-mentioned studies pointed to a tight coupling between the dynamics of a biological macromolecule and the dynamics of its solvent, which has even been

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Fig. 9.3 Atomic MSD measured on lysozyme in different solvents [from ref. 21]

Fig. 9.4 Relationship between solvent viscosity and atomic MSD of lysozyme in different amounts of water, glucose, and glycerol [from ref. 22]

termed a “slaving” of one to the other [25,26]. Since then, it has also been proposed that protein motions are “solvent-slaved” and “hydration-shell coupled” depending if their temperature dependence follows that of bulk or hydration water [27]. Molecular dynamics simulations with protein and solvent at different temperatures have also shown that solvent mobility determines amplitudes of atomic protein


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fluctuations [28]. In particular, the onset of hydration-water translational diffusion has been identified as the main trigger of the protein dynamical transition [29, 30]. Molecular dynamics simulations have reproduced a rigidification of protein dynamics in glycerol-trehalose glasses [31] and their analysis suggested it may be due to inertial effect from the bulk glass and not due to specific protein–solvent hydrogen bond interactions.

9.3 Measurements of Hydration Water Dynamics 9.3.1 Water Hydrating Macromolecules 9.3.1.1 Subtracting Spectra to Extract Water Dynamics One method of studying hydration water dynamics is to prepare two samples of a protonated biological macromolecule, one in H2 O and one in D2 O. If the samples are prepared and measured in an identical manner, the subtraction of the spectra from the D2 O sample from that of the H2 O sample effectively removes the contribution from the solute and reflects the hydration water dynamics. The procedure is based on the assumption that proteins display identical dynamics in both H2 O and D2 O and should be used cautiously, since evidence exists that the assumption may not be strictly valid [32]. A detailed QENS study using the subtraction of spectra method was performed on lysozyme hydration water at the NIST backscattering spectrometer from 180 to 270 K [33]. The authors fit the quasi-elastic broadening observed using a relaxing cage model for the hydration water dynamics. The model describes the translational water dynamics as the product of two functions, one representing the short-time dynamics of a water molecule within a cage, and the second a translational relaxation time. Figure 9.5b shows the extracted translational relaxation time as a function of temperature from [33]. The figure shows that hydration water displays an Arrhenius temperature dependence at low temperature, and then crosses over to super-Arrhenius behaviour at 220 K. The authors assigned the change in hydrationwater dynamics to a fragile-to-strong dynamic cross-over and correlated it to the protein dynamical transition that occurs in the same temperature range. Molecular dynamics simulations have corroborated this assignment [34]. Based on dielectric relaxation spectroscopy experiments, however, the occurrence of a fragile-to-strong cross-over has been questioned [35, 36]. Chen and co-workers performed similar neutron scattering studies on RNA [15] and DNA [16], which intriguingly showed that they display the same behaviour as the protein lysozyme, with a dynamic cross-over in the hydration water for all three bio-molecules at approximately 220 K. The data for DNA hydration water are plotted in Fig. 9.5a and can be seen to share the same temperature dependency

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Fig. 9.5 Average translational relaxation time of hydration water. (a) Is DNA hydration water; and (b) lysozyme hydration water. A cross-over from Arrhenius to non-Arrhenius behaviour is seen in both [from ref. 16]

as lysozyme hydration water, pointing to the general nature of the hydration water transition. As mentioned in the introduction, temperature is extensively used as a parameter for probing features of the protein energy landscape. Recently, protein dynamics were also measured using neutron scattering at different pressures [37]. Figure 9.6 shows the atomic MSD extracted for both hydration water and lysozyme as a function of temperature at different pressures between ambient and 1,600 bar. The authors note that both the protein and its hydration water display the same dynamics and transition temperatures, and the MSD of both can be superimposed if the protein MSD are multiplied by a scaling factor of 4.2. By obtaining QENS


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data and fitting it with the relaxation cage model, a translational relaxation time was calculated for the hydration water, and it was found that structural relaxation times are shorter at higher pressures, for the same temperature. The shorter relaxation time of the water is thought to allow the protein to maintain its flexibility under the high pressures measured. 9.3.1.2 Direct Measurement of Water Dynamics Using Deuterated Bio-Molecules It is also possible to directly probe hydration water dynamics without the need for subtracting spectra from the bio-molecule by taking advantage of the difference in

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neutron scattering cross-sections for hydrogen and its heavier isotope deuterium. After suitable adaptation, bacteria can be grown on a deuterated carbon source in D2 O and express deuterated proteins, which can then be purified [38, 39]. If the deuterated protein is then resuspended in H2 O, the dynamics of hydration water will be probed in incoherent neutron scattering experiments, and the dynamics of the deuterated component masked. Several neutron scattering studies have made use of deuterated proteins. QENS experiments at room temperature on per-deuterated C-phycocyanin were the first to be performed [40]. They determined the average diffusion coefficient of protein hydration-water to be 10 times slower than that of bulk water. Elastic incoherent neutron scattering experiments between 235 and 320 K on deuterated Cphycocyanin in the presence of trehalose also evidenced a 10- to 15-fold reduction in hydration-water dynamics with respect to bulk water [41]. Deuterated maltose-binding protein (MBP), a 41 kDa sugar-binding soluble protein, was produced and hydrated in H2 O to study the temperature-dependent water dynamics of a soluble protein [42]. In order to assess if the protein dynamics are indeed “slaved” to its solvent, the dynamics of the MBP was also probed using a natural abundance (protonated) sample hydrated in D2 O. The results are shown in Fig. 9.7a. As can be seen in the figure, both the protein and its hydration water display the same dynamics at low temperature, and a strong onset of dynamics occurs in both at the same temperature (at around 200 K). A sudden change in the elastic intensity (at the origin of a sudden change in MSD) has also been seen in neutron scattering experiments on per-deuterated C-phycocyanin hydrated in H2 O at 210 K and has been attributed to a conventional glass transition in the hydration water [43]. In [43], the experimental data were combined with molecular dynamics simulations on the same system, which qualitatively reproduced the experimental MSD. Analysis of the simulation showed that the transition in the protein occurred at the same temperature as the onset of translational diffusion in the hydration water and implied that translational water motions drive the protein dynamical transition as suggested earlier [29,30]. Other authors have on the contrary argued that the protein transition is driven by rotational water motions [44]. Analysis of the protein–water hydrogen bond network relaxation rates showed a transition in the relaxation rate at the same temperature as that of the protein, explaining the coupling observed between the two entities [43]. Whilst all the data reviewed in the preceding paragraphs point to a direct correlation between macromolecule and solvent dynamics, when a membrane system was studied a surprisingly different picture emerged. In [45] and [46], deuterated purple membranes hydrated in H2 O were measured: data are reproduced in Fig. 9.7b. An onset of solvent dynamics was found at about 200 K, which was supported by diffraction [47] and dielectric spectroscopy and calorimetric measurements [48]. In parallel, a natural abundance (protonated) purple membrane sample, hydrated in D2 O, was measured to observe the membrane MSD. Surprisingly, no dynamical transition or onset in purple membrane dynamics is observed at 200 K, as can be seen in Fig. 9.7b. At the ps–ns timescale, the data suggest that membrane and water dynamics are not coupled around the 200 K temperature region. Molecular


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Fig. 9.7 Comparison of hydration water dynamics and a soluble protein dynamics (a) and membrane dynamics (b). Deuterated maltose-binding protein (MBP) hydrated in H2 O is labelled “water” in (a), hydrogenated MBP in D2 O labelled “MBP”. In (b), deuterated purple membrane hydrated in H2 O is labelled “water” and hydrogenated purple membrane in D2 O labelled “PM” [from ref. 49]

dynamics simulations were also performed in [44] to interpret the experimental results. The simulations showed that the transition observed in the hydration water corresponded to the onset of translational diffusion, which does not trigger one in the membrane. The results suggest that lipids may play an important role in the dynamical behaviour of membrane proteins [49].

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0.25

G T (meV)

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0.15

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Fig. 9.8 Full width at half maximum of a Lorentzian function assigned to translational motion as a function of wavevector squared at 281 K (squares) and 301 K (triangles). Data were measured on the IRIS spectrometer with a resolution of 17 μeV [from ref. 52]

9.3.2 “In Cellulo” Measurements of Water Dynamics Compared to other techniques, neutron scattering is often prohibitive for biomolecules as large sample quantities are required. Studying whole cells is therefore suitable – even deuterated cells can be easily grown in extremely large quantities. Neutrons are a non-destructive probe, meaning that the cells can be studied in a state close to that found in vivo. The feasibility of such studies was proven first by studying the dynamics of proteins in cells adapted to different temperatures [50]. More recently, studies of in cellulo water have been performed by analysing quasi-elastic spectra. Jasnin et al. [51] have performed a detailed study of water in Escherichia coli cells on two spectrometers, IRIS (at ISIS, UK) and IN6 (at ILL, France), with resolutions of 17 and 90 μeV, respectively (full width half maximum). The authors grew the bacteria in deuterated media to obtain deuterated cells and minimise scattering from cell components. The cells were then resuspended in H2 O buffer: the incoherent signal from such a sample is dominated by the scattering from the cytoplasmic water. Subtracting spectra of deuterated bacteria resuspended in D2 O buffer allowed the authors to eliminate the contribution of cell components to the signal. The observed quasi-elastic broadening from IRIS was fit with a single Lorentzian, the half-width-half-maximum of which is plotted as a function of wave vector transfer in Fig. 9.8. The data were fit with a jump diffusion model, where water molecules are assumed to reside for a mean residence time, then


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diffuse to the next site. The work showed that water in the cytoplasm had similar diffusion coefficients to that of bulk (1.5 × 10−5 cm2 s−1 for E. coli cell water and 1.7 × 10−5 cm2 s−1 for the buffer), but with residence times that are twice as long in the cells (2.6 ps compared to 1.5) [51]. The authors postulate that the longer residence times could be due to longer times spent by water molecules in the hydration layer, which then diffuse at the same rate as bulk water. Using the lower resolution spectrometer IN6, it was possible to observe the rotational contribution of water molecules to dynamics, which was found to be similar in cytoplasmic water and in bulk. The results indicated a high mobility of cellular water, in contradiction to what others had postulated on the nature of the cell’s interior [52]. Interestingly, a similar study on the archaea Haloarcula marismortui [53] that live in high salt conditions revealed a residence time of cytoplasmic water of 243 ps. The archaea studied can thrive in environments up to 4 M salt, and the result was interpreted in the context of specific interactions involving macromolecules, water, and salt ions. The dynamics of water contained in human red blood cells (RBC) was also directly probed by neutron scattering [54]. RBC consist mostly of cytoplasmic water and haemoglobin at a concentration of ∼330 mg mL−1 . The study reported similar findings to those from bacterial cytoplasm from Jasnin et al.: 90% of the water in RBC has the same diffusion coefficient as that of the buffer in which the cells are suspended, and 10% of the water is seen to be immobile on the time scale of the experiments (7–40 ps) and is identified as dynamically bound water in the hydration layer of haemoglobin. The dynamics of whole cells using neutron scattering has recently been reviewed in [55].

9.4 Perspectives Much controversy still exists about the microscopic properties of water, even in its bulk state [56], and we are still at the beginning of the characterisation of its interaction with bio-molecules. Neutrons are an important probe to study water dynamics and structure, due in part to their scattering cross-sections. As we have re-iterated in this chapter and following on from Chap. 7, simulations are playing a crucial role in the field and are often used to interpret neutron results. They are a necessary tool to progress the modelling of water and protein behaviour. New instruments with increased flux will make sample quantities become less prohibitive for neutron scattering, and easier access to user deuteration facilities facilitates experiments requiring labelling (e.g. http://www.ill.eu/sites/deuteration/). In the chapter, we have not discussed the results from other techniques making great progress in the field. However, we would like to give the interested reader a few starting points. NMR [57], fluorescence measurements [58], terahertz spectroscopy [59], and dielectric relaxation spectroscopy [60, 61] are all suitable for studying hydration water dynamics, and undoubtedly, it is only through a combination of techniques that we will complete our picture of the interaction between biomolecules and their hydration water that is essential for their function.

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References 1. Orecchini A, Paciaroni A, De Francesco A, Petrillo C, Sacchetti F (2009) Collective dynamics of protein hydration water by brillouin neutron spectroscopy. J Am Chem Soc 131(13): 4664–4669 2. Rheinstadter MC, Seydel T, Demmel F, Salditt T (2005) Molecular motions in lipid bilayers studied by the neutron backscattering technique. Phys Rev E Stat Nonlin Soft Matter Phys 71(6 Pt 1):061908 3. Rheinstadter MC, Seydel T, Salditt T (2007) Nanosecond molecular relaxations in lipid bilayers studied by high energy-resolution neutron scattering and in situ diffraction. Phys Rev E Stat Nonlin Soft Matter Phys 75(1 Pt 1):011907 4. Gabel F, Bicout D, Lehnert U, Tehei M, Weik M, Zaccai G (2002) Protein dynamics studied by neutron scattering. Q Rev Biophys 35(4):327–367 5. Doster W, Cusack S, Petry W (1989) Dynamical transition of myoglobin revealed by inelastic neutron scattering. Nature 337(6209):754–756 6. Zaccai G (2000) How soft is a protein? A protein dynamics force constant measured by neutron scattering. Science 288(5471):1604–1607 7. Fitter J (1999) The temperature dependence of internal molecular motions in hydrated and dry alpha-amylase: the role of hydration water in the dynamical transition of proteins. Biophys J 76(2):1034–1042 8. Kealley CS, Sokolova AV, Kearley GJ, Kemner E, Russina M, Faraone A, Hamilton WA, Gilbert EP (2010) Dynamical transition in a large globular protein: macroscopic properties and glass transition. Biochim Biophys Acta 1804(1):34–40 9. Roh JH, Curtis JE, Azzam S, Novikov VN, Peral I, Chowdhuri Z, Gregory RB, Sokolov AP (2006) Influence of hydration on the dynamics of lysozyme. Biophys J 91(7):2573–2588 10. Wood K, Caronna C, Fouquet P, Haussler W, Natali F, Ollivier J, Orecchini A, Plazanet M, Zaccai G (2008) A benchmark for protein dynamics: ribonuclease A measured by neutron scattering in a large wavevector-energy transfer range. Chem Phys 345(2-3):305–314 11. Ferrand M, Dianoux AJ, Petry W, Zaccai G (1993) Thermal motions and function of bacteriorhodopsin in purple membranes: effects of temperature and hydration studied by neutron scattering. Proc Natl Acad Sci USA 90(20):9668–9672 12. Fitter J, Lechner RE, Dencher NA (1997) Picosecond molecular motions in bacteriorhodopsin from neutron scattering. Biophys J 73(4):2126–2137 13. Natali F, Castellano C, Pozzi D, Congiu Castellano A (2005) Dynamic properties of an oriented lipid/DNA complex studied by neutron scattering. Biophys J 88(2):1081–1090 14. Caliskan G, Briber RM, Thirumalai D, Garcia-Sakai V, Woodson SA, Sokolov AP (2006) Dynamic transition in tRNA is solvent induced. J Am Chem Soc 128(1):32–33 15. Chu XQ, Fratini E, Baglioni P, Faraone A, Chen SH (2008) Observation of a dynamic crossover in RNA hydration water which triggers a dynamic transition in the biopolymer. Phys Rev E Stat Nonlin Soft Matter Phys 77(1 Pt 1):011908 16. Chen SH, Liu L, Chu X, Zhang Y, Fratini E, Baglioni P, Faraone A, Mamontov E (2006) Experimental evidence of fragile-to-strong dynamic crossover in DNA hydration water. J Chem Phys 125(17):171103 17. Daniel RM, Finney JL, Smith JC (2003) The dynamic transition in proteins may have a simple explanation. Faraday Discuss 122:163–169; discussion 171–190 18. Lee AL, Wand AJ (2001) Microscopic origins of entropy, heat capacity and the glass transition in proteins. Nature 411(6836):501–504 19. Lehnert U, Reat V, Weik M, Zaccai G, Pfister C (1998) Thermal motions in bacteriorhodopsin at different hydration levels studied by neutron scattering: correlation with kinetics and lightinduced conformational changes. Biophys J 75(4):1945–1952 20. Cordone L, Ferrand M, Vitrano E, Zaccai G (1999) Harmonic behavior of trehalose-coated carbon-monoxy-myoglobin at high temperature. Biophys J 76(2):1043–1047


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21. Tsai AM, Neumann DA, Bell LN (2000) Molecular dynamics of solid-state lysozyme as affected by glycerol and water: a neutron scattering study. Biophys J 79(5):2728–2732 22. Cornicchi E, Onori G, Paciaroni A (2005) Picosecond-time-scale fluctuations of proteins in glassy matrices: the role of viscosity. Phys Rev Lett 95(15):158104 23. Reat V, Dunn R, Ferrand M, Finney JL, Daniel RM, Smith JC (2000) Solvent dependence of dynamic transitions in protein solutions. Proc Natl Acad Sci USA 97(18):9961–9966 24. Gabel F, Weik M, Doctor BP, Saxena A, Fournier D, Brochier L, Renault F, Masson P, Silman I, Zaccai G (2004) The influence of solvent composition on global dynamics of human butyrylcholinesterase powders: a neutron scattering study. Biophys J 86(5):3152–3165 25. Fenimore PW, Frauenfelder H, McMahon BH, Parak FG (2002) Slaving: solvent fluctuations dominate protein dynamics and functions. Proc Natl Acad Sci USA 99(25):16047–16051 26. Iben IE, Braunstein D, Doster W, Frauenfelder H, Hong MK, Johnson JB, Luck S, Ormos P, Schulte A, Steinbach PJ, Xie AH, Young RD (1989) Glassy behavior of a protein. Phys Rev Lett 62(16):1916–1919 27. Fenimore PW, Frauenfelder H, McMahon BH, Young RD (2004) Bulk-solvent and hydrationshell fluctuations, similar to alpha- and beta-fluctuations in glasses, control protein motions and functions. Proc Natl Acad Sci USA 101(40):14408–14413 28. Vitkup D, Ringe D, Petsko GA, Karplus M (2000) Solvent mobility and the protein “glass” transition. Nat Struct Biol 7(1):34–38 29. Tarek M, Tobias DJ (2002) Role of protein-water hydrogen bond dynamics in the protein dynamical transition. Phys Rev Lett 88(13):138101 30. Tournier AL, Xu J, Smith JC (2003) Translational hydration water dynamics drives the protein glass transition. Biophys J 85(3):1871–1875 31. Curtis JE, Dirama TE, Carri GA, Tobias DJ (2006) Inertial suppression of protein dynamics in a binary glycerol-trehalose glass. J Phys Chem B 110(46):22953–22956 32. Jasnin M, Tehei M, Moulin M, Haertlein M, Zaccai G (2008) Solvent isotope effect on macromolecular dynamics in E. coli. Eur Biophys J 37(5):613–617 33. Chen SH, Liu L, Fratini E, Baglioni P, Faraone A, Mamontov E (2006) Observation of fragile-to-strong dynamic crossover in protein hydration water. Proc Natl Acad Sci USA 103(24):9012–9016 34. Kumar P, Yan Z, Xu L, Mazza MG, Buldyrev SV, Chen SH, Sastry S, Stanley HE (2006) Glass transition in biomolecules and the liquid-liquid critical point of water. Phys Rev Lett 97(17):177802 35. Pawlus S, Khodadadi S, Sokolov AP (2008) Conductivity in hydrated proteins: no signs of the fragile-to-strong crossover. Phys Rev Lett 100(10):108103 36. Swenson J, Jansson H, Bergman R (2006) Relaxation processes in supercooled confined water and implications for protein dynamics. Phys Rev Lett 96(24):247802 37. Chu XQ, Faraone A, Kim C, Fratini E, Baglioni P, Leao JB, Chen SH (2009) Proteins remain soft at lower temperatures under pressure. J Phys Chem B 113(15):5001–5006 38. Leiting B, Marsilio F, O’Connell JF (1998) Predictable deuteration of recombinant proteins expressed in Escherichia coli. Anal Biochem 265(2):351–355 39. Teixeira SC, Ankner J, Bellissent-Funel MC, Bewley R, Blakeley MP, Coates L, Dahint R, Dalgliesh R, Dencher N, Dhont J, Fischer P, Forsyth VT, Fragneto G, Frick B, Geue T, Gilles R, Gutberlet T, Haertlein M, Hauss T, Haussler W, Heller WT, Herwig K, Holderer O, Juranyi F, Kampmann R, Knott R, Kohlbrecher J, Kreuger S, Langan P, Lechner R, Lynn G, Majkrzak C, May R, Meilleur F, Mo Y, Mortensen K, Myles DA, Natali F, Neylon C, Niimura N, Ollivier J, Ostermann A, Peters J, Pieper J, Ruhm A, Schwahn D, Shibata K, Soper AK, Straessle T, Suzuki UI, Tanaka I, Tehei M, Timmins P, Torikai N, Unruh T, Urban V, Vavrin R, Weiss K, Zaccai G (2008) New sources and instrumentation for neutrons in biology. Chem Phys 345(2–3):133–151 40. Dellerue S, Bellissent-Funel MC (2000) Relaxational dynamics of water molecules at protein surface. Chem Phys 258(2-3):315–325

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41. Gabel F, Bellissent-Funel MC (2007) C-phycocyanin hydration water dynamics in the presence of trehalose: an incoherent elastic neutron scattering study at different energy resolutions. Biophys J 92(11):4054–4063 42. Wood K, Frolich A, Paciaroni A, Moulin M, Hartlein M, Zaccai G, Tobias DJ, Weik M (2008) Coincidence of dynamical transitions in a soluble protein and its hydration water: direct measurements by neutron scattering and MD simulations. J Am Chem Soc 130(14):4586–4587 43. Doster W (2010) The protein-solvent glass transition. Biochimica Et Biophysica Acta-Proteins and Proteomics 1804(1):3–14 44. Zanotti JM, Gibrat G, Bellissent-Funel MC (2008) Hydration water rotational motion as a source of configurational entropy driving protein dynamics. Crossovers at 150 and 220 K. Phys Chem Chem Phys 10(32):4865–4870 45. Wood K, Plazanet M, Gabel F, Kessler B, Oesterhelt D, Tobias DJ, Zaccai G, Weik M (2007) Coupling of protein and hydration-water dynamics in biological membranes. Proc Natl Acad Sci USA 104(46):18049–18054 46. Wood K, Plazanet M, Gabel F, Kessler B, Oesterhelt D, Zaccai G, Weik M (2008) Dynamics of hydration water in deuterated purple membranes explored by neutron scattering. Eur Biophys J 37(5):619–626 47. Weik M, Lehnert U, Zaccai G (2005) Liquid-like water confined in stacks of biological membranes at 200 K and its relation to protein dynamics. Biophys J 89(5):3639–3646 48. Berntsen P, Bergman R, Jansson H, Weik M, Swenson J (2005) Dielectric and calorimetric studies of hydrated purple membrane. Biophys J 89(5):3120–3128 49. Frolich A, Gabel F, Jasnin M, Lehnert U, Oesterhelt D, Stadler AM, Tehei M, Weik M, Wood K, Zaccai G (2009) From shell to cell: neutron scattering studies of biological water dynamics and coupling to activity. Faraday Discuss 141:117–130; discussion 175–207 50. Tehei M, Franzetti B, Madern D, Ginzburg M, Ginzburg BZ, Giudici-Orticoni MT, Bruschi M, Zaccai G (2004) Adaptation to extreme environments: macromolecular dynamics in bacteria compared in vivo by neutron scattering. EMBO Rep 5(1):66–70 51. Jasnin M, Moulin M, Haertlein M, Zaccai G, Tehei M (2008) Down to atomic-scale intracellular water dynamics. EMBO Rep 9(6):543–547 52. Pollack GH, Cameron IL, Wheatley DN (2006) Water and the cell. Springer, Dordrecht, The Netherlands 53. Tehei M, Franzetti B, Wood K, Gabel F, Fabiani E, Jasnin M, Zamponi M, Oesterhelt D, Zaccai G, Ginzburg M, Ginzburg BZ (2007) Neutron scattering reveals extremely slow cell water in a dead sea organism. Proc Natl Acad Sci USA 104(3):766–771 54. Stadler AM, Embs JP, Digel I, Artmann GM, Unruh T, Buldt G, Zaccai G (2008) Cytoplasmic water and hydration layer dynamics in human red blood cells. J Am Chem Soc 130(50): 16852–16853 55. Jasnin M (2009) Atomic-scale dynamics inside living cells explored by neutron scattering. J R Soc Interface 6(Suppl 5):S611–S617 56. Ball P (2008) Water: an eduring mystery. Nature 452(7185):291–292 57. Halle B (2004) Protein hydration dynamics in solution: a critical survey. Philos Trans R Soc Lond B Biol Sci 359(1448):1207–1223; discussion 1223–1204, 1323–1208 58. Li T, Hassanali AA, Kao YT, Zhong D, Singer SJ (2007) Hydration dynamics and time scales of coupled water-protein fluctuations. J Am Chem Soc 129(11):3376–3382 59. Born B, Kim SJ, Ebbinghaus S, Gruebele M, Havenith M (2009) The terahertz dance of water with the proteins: the effect of protein flexibility on the dynamical hydration shell of ubiquitin. Faraday Discuss 141:161–173; discussion 175–207 60. Frauenfelder H, Chen G, Berendzen J, Fenimore PW, Jansson H, McMahon BH, Stroe IR, Swenson J, Young RD (2009) A unified model of protein dynamics. Proc Natl Acad Sci USA 106(13):5129–5134 61. Rupley JA, Siemankowski L, Careri G, Bruni F (1988) Two-dimensional protonic percolation on lightly hydrated purple membrane. Proc Natl Acad Sci USA 85(23):9022–9025 62. Doster W, Gutberlet T (2010) Protein-water interactions. Biochimica Et Biophysica ActaProteins and Proteomics 1804(1):1–242

Chapter 10

Lipid Membrane Dynamics Maikel C. Rheinstädter

10.1 Introduction One of the major challenges of modern physics is how it can contribute to biology and “life sciences,” which are considered as the key sciences of the twenty-first century. High-tech life sciences include the emerging biotechnology and biomedical device industries, functional foods, and nutraceuticals, and also the development of new biomaterials and pharmaceuticals. Modern physical techniques and concepts represent the most highly developed tools to access and manipulate material properties, but they are certainly underrepresented in the life sciences. Clifford Shull and Bertram Brockhouse developed elastic and inelastic neutron scattering techniques for studying structure and dynamics in condensed matter, and neutron and X-ray scattering techniques were developed and very successfully applied to investigate structure and dynamics in crystalline systems. The challenge we face is how to apply these powerful techniques to soft matter and biology, i.e., systems with a high degree of static and dynamic disorder. Even though biological membranes were studied for decades, very few biologically relevant processes were revealed on a molecular level. The reason is the combination of very small nanometer length scales and very fast dynamics of pico- and nanoseconds, which pose particular experimental challenges. The dynamics involve processes from fast motions of lipid molecules and small protein groups to slow conformational changes of large membrane areas and of membrane-embedded proteins. Neutron and X-ray scattering can be used as microscope to study structure and dynamics in these systems, because they give access to the relevant length and time scales.

M.C. Rheinstädter () Department of Physics and Astronomy, McMaster University, 1280 Main Street, West Hamilton, ON L8S 4M1, Canada and Chalk River Laboratories, Canadian Neutron Beam Centre, Chalk River ON, KOJ 1JO, Canada e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 10, © Springer Science+Business Media, LLC 2012

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Often, the structure of biomolecular systems is relatively well known and the corresponding experimental techniques, such as X-ray crystallography, nuclear magnetic resonance (NMR), and also atomic force microscopy (AFM), are welldeveloped standard techniques. Dynamical properties are often less well understood, but are important for many fundamental biomaterial properties such as elasticity and interaction forces. They may determine or strongly affect certain functional aspects, such as diffusion and transport through a membrane, and be relevant for protein function. Because real biological materials lack an overall crystal structure, in order to fully characterize the fluctuations and to compare experimental results with membrane theories, the measurements must cover a very large range of length and time scales [1]. The combination of various inelastic scattering techniques, such as inelastic neutron, inelastic X-ray scattering, and dynamic light scattering (DLS), enlarges the window of accessible momentum and energy transfers – or better: accessible length and time scales – and allows one to study structure and dynamics from the nearest atomic neighbor distances in lipid molecules to more than 100 nm (3 orders of magnitude), covering time scales from about one-tenth of a picosecond to almost 1 s (12 orders of magnitude). Figure 10.1 depicts the length and time scales accessible by the aforementioned techniques and more details are discussed in Chap. 1. An emerging technique is high-speed AFM which currently can deliver nanometer resolution real space images with a time resolution of milliseconds [2]. Figure 10.1 shows that combining different techniques allows us to explore the multiscale character of biological membranes [1, 3–5]. In addition and as discussed in detail in Chap. 7, Molecular Dynamics (MD) simulations are an invaluable tool to develop models for molecular structure and dynamics in membranes and proteins. Because of increasing computer power and optimized algorithms, large system sizes and long simulation times and also more and more complex systems can be addressed [7–12]. The dashed rectangle in Fig. 10.1 marks the dynamical range accessed by computer simulations. The elementary time scale in simulations is in the order of femtoseconds. The field is currently boosted by interdisciplinary interest in life sciences. New developments and improvements in scattering instrumentation, sample preparation and environments and, eventually more powerful sources, give quantitative access to the molecular properties of bilayers on lateral length scales ranging from ˚ Often, phospholipid bilayers are used as micrometers down to a few Angstroms (A). model systems to study fundamental properties and the structure of these systems is often well established by X-ray and neutron scattering techniques [13–19]. But the trend is to study more complex and relevant systems, decorated with cholesterol, peptides, and proteins, to eventually develop a better understanding of biological membranes. Scattering techniques are particularly powerful in spatially ordered systems and were very successfully applied to study structure and dynamics (such as phonons) in crystals. In soft matter and biology, a periodic structure is often missing or not well developed and new concepts and approaches are required to study structural and

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Fig. 10.1 Accessible length and time scales, and corresponding energy and momentum transfer, for some spectroscopic techniques covering microscopic to macroscopic dynamics. Light scattering techniques include Raman, Brillouin, and dynamic light scattering (DLS). Inelastic Xray and neutron scattering access dynamics on Angstrom and nanometer length scales. Dielectric spectroscopy probes the length scale of an elementary molecular electric dipole, which can be estimated by the bond length of a C–O bond (about 140 pm). The area marked by the dashed box is the dynamical range accessible by computer simulations. High-speed atomic force microscopy is an emerging technique, which allows imaging in real space. For further details on some of these techniques, see Chap. 1

dynamical properties. In contrast to computer simulations where different molecular components and dynamics can be directly accessed and extracted, in experiments there is usually a superposition of signals of different molecular components and interactions, and also single-molecule and collective dynamics. By using different sample geometries, these signals can be separated by their position in reciprocal and frequency space. In particular with neutrons, selective deuteration is used to distinguish between self- and collective dynamics. In addition, because of the availability of more powerful spectrometers, smaller system dimensions and less material can be used in experiments. However, experiments are still orders of magnitude away from single-molecule spectroscopy of, for example, membraneembedded proteins.

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10.2 Structure of Membrane Systems Membranes are disordered systems and thus it is not possible to use crystallographic measurements for a high-resolution structural characterization. The elastic X-ray and neutron scattering techniques which are mostly employed for membrane studies are small angle scattering (SAS), Small and Wide Angle Diffraction, and Specular Reflectivity. SAS is used to gain information on shape, size, and interactions of lipid lamellar systems and vesicles: it gives radial averaged information in all directions of space. SAS experiments provide information about the size, shape, and orientation of the different components of a sample for distances ranging from ˚ 50 to 5,000 A. Diffraction experiments are mainly used to determine the in-plane and normal structure of stacked lipid lamellar systems and can for instance locate an object within the lipid bilayer, such as a cholesterol molecule or peptides. Reflectivity is used to study the structure of bilayers in the direction perpendicular to the plane of the layers in a planar configuration and allows the determination of the structure and composition of a membrane with a very high spatial resolution. At continuous, reactor sources, collimated, monochromatic beams of wavelength λ are used to obtain a good Q and with it good spatial resolution. Spallation sources use broad wavelength bands and time of flight techniques to achieve high resolution. In both cases, the scattered beam is measured by a detector (punctual, mono, or bidimensional) and the intensity of the outgoing over that of the incoming beam recorded as a function of the wave vector transfer, Q. Scattered neutrons are not energy discriminated, but considered as scattered elastically. This assumption is usually well justified as the inelastic scattering cross-section is orders of magnitude weaker than the elastic cross-section and the intensity of inelastically scattered neutrons weak in diffraction experiments. The angle between the incoming and outgoing beams is defined as the scattering angle, 2θ . The scattering vector Q is then given by Q = 4π sin(θ )/λ . Because only the intensities of the scattered waves can be measured and not the phase shifts, a direct transformation back into real space is usually difficult.

10.3 Dynamics in Complex Membranes Motions in proteins and biological membranes occur on various length and time scales [3, 4]. The functional behavior of membrane proteins is likely to depend on the lipid bilayer composition and physical properties, such as hydrophobic thickness and elastic moduli. Dynamics in complex membranes involve interactions between the different constituents, such as lipids, cholesterol, peptides, and proteins. How the variety of inter- and intraprotein motions, occurring over different time and length scales, interact to result in a functioning biological system remains an open field for those working at the interface of physics and biology. The dynamical coupling between proteins is thought to be important for the understanding of


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macromolecular function in a cellular context because it can lead to an effective interprotein communication. The commonly assumed interaction mechanism between inclusions in membranes is a lipid-mediated interaction due to local distortions of the lipid bilayer [20–24], with a strong dependence on the bilayer properties, in particular elastic properties. Motions in lipid bilayers range from the long wavelength undulation and bending modes of the bilayer, with typical relaxation times of nanoseconds and lateral length scales of several hundreds lipid molecules, to the short-wavelength density fluctuations in the picosecond range on nearest neighbor distances of lipid molecules [5, 6, 13, 25–31]. Local dynamics in lipid bilayers, i.e., dynamics of individual lipid molecules such as vibrations, rotations, librations (hindered rotation), and diffusion, have been investigated by, e.g., incoherent neutron scattering [25–29] and NMR [32, 33] to determine the short-wavelength translational and rotational diffusion constant. Collective undulation modes were investigated using neutron spin-echo (NSE) spectrometers [6, 28, 29, 34] and DLS [35–37]. In contrast to other spectroscopic techniques, such as dielectric spectroscopy, inelastic X-ray and neutron scattering give a wave vector resolved access to molecular dynamics. Excitation frequencies and relaxation rates are measured at different internal length scales of the system. A typical dynamical scattering experiment measures (Q, ω ) pairs, it delivers a frequency together with a corresponding length scale and possibly also corresponding direction, such as parallel or perpendicular to the protein axis. This additional information is very important to assign the measured frequencies to certain molecules or molecular components.

10.3.1 Local and Collective Molecular Motions in Membranes and Proteins Atomic and molecular motions in membranes and proteins can be classified as local or self-correlated, and collective or pair-correlated dynamics. Local dynamics refer to the motion of molecules or functional groups in local energy potentials. The force and time constants involved are determined by the local friction and restoring forces. These types of dynamical processes are incoherent because the particles move independently in their local environments. Examples include vibrations, rotations, librations (hindered rotation), and diffusion of individual lipid molecules. In a neutron experiment, this is given by the incoherent scattering function, Sinc (Q, ω ). In contrast, the coherent scattering function Scoh (Q, ω ) gives details of the coupled, collective molecular motions which involve interactions between neighboring particles or functional groups (pictured as a spring). Coupled particles basically behave like coupled pendulums. Collective molecular motions determine, for instance, elasticity of membranes, interactions between membrane-embedded proteins, and are important for transport processes. In biology, any dynamics will most likely show a mixed behavior of particles moving in local potentials, but with a more or less pronounced coherent character.

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Fig. 10.2 Schematic of some elementary dynamical modes occurring in lipid bilayers. (a) Local modes include diffusion and vibrations, rotations, and librations (hindered rotations) of single lipid molecules, (b) coupling of membrane-embedded proteins in a hexagonal arrangement, taking bacteriorhodopsin in purple membrane as an example. The interaction between protein trimers is depicted as springs with effective spring constant k, (c) motion of molecules or functional groups in local energy potentials, which gives rise to incoherent dynamics, and (d) interaction between particles as probed via coherent dynamics measurements

Figure 10.2 depicts some of the local and collective modes occurring in a phospholipid bilayer. Rotational and lateral diffusion, vibrations, and rotations of the single lipid molecules can be investigated by, e.g., incoherent inelastic neutron scattering, NMR, or dielectric spectroscopy (Fig. 10.2a, c). Only coherent probes, such as coherent inelastic neutron scattering or inelastic X-ray scattering, can elucidate collective molecular motions, depicted as a system of coupled membraneembedded proteins in Fig. 10.2b, d.


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Table 10.1 Coherent and incoherent scattering lengths and cross-sections for selected elements as provided by the NIST Center for Neutron Research (http://www.ncnr.nist.gov/resources/n-lengths/) Nuclide bcoh (fm) binc (fm) σcoh (b) σinc (b) 1H −3.7406 25.274 1.7583 80.27 2 H 6.671 4.04 5.592 2.05 3H 4.792 −1.04 2.89 0.14 14 N 9.37 2.0 11.03 0.5 15 N 6.44 −0.02 5.21 0.00005 12 C 6.6511 0 5.559 0 13 C 6.19 −0.52 4.81 0.034 16 O 5.803 0 4.232 0 17 O 5.78 0.18 4.2 0.004 18 O 5.84 0 4.29 0 Note that 1 b = 1barn = 10−28 m2 = 100 fm2

10.3.2 Selective Deuteration and Labeling Techniques In neutron scattering, the total scattering of a sample will consist of coherent and incoherent scattering contributions. The fraction of coherent and incoherent scattering depends on the atomic composition and the respective scattering lengths. Substitution of certain elements in a compound by their isotopes may increase contributions of certain molecules or functional groups to the coherent or incoherent scattering contribution. The coherent and incoherent neutron cross-sections, bcoh and binc , of an element can be illustrated by two extreme cases. If all the nuclei in a sample have different b’s, there is no interference between the different atoms. The incoherent scattering therefore depends only on the correlation between the positions of the same nucleus at different times. Incoherent scattering therefore probes the local atomic or molecular environment. If all scattering lengths are the same, i.e., all nuclei are identical for the neutron probe, the coherent scattering still depends on the correlation between the positions of the same nucleus at different times, but also on the correlation between the positions of different nuclei at different times. It therefore gives interference effects and allows to measure interaction forces. In general, every element has coherent and incoherent scattering lengths. Table 10.1 lists the scattering lengths for selected elements, which are common in biological materials. Most noticeable are the very large incoherent cross-section for hydrogen, 1 H, and the relatively large coherent cross-section for deuterium, 2 H. This is important for selective deuteration, as will be discussed below. The incoherent cross-section of hydrogen atoms is about 40 times larger than that of deuterium and of all other atoms present in biological macromolecules. Consequently, hydrogen atoms dominate the incoherent scattering signal of biological samples. The hydrogen atoms reflect the movements of larger groups to which they are attached, such as amino acid side chains. Deuteration, i.e., the substitution of protons by deuterium (2 H), is often

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Fig. 10.3 Schematic example of how selective deuteration can be used to highlight molecules or functional groups in neutron scattering experiments. Depicted are lipid vesicles (ring) with embedded proteins (shows are rectangles) in a solvent: (a) hydrogenated membrane and embedded proteins in a deuterated solvent, thus visible in incoherent neutron scattering experiments, (b) solvent and membrane are deuterated, allowing the investigation of the local dynamics of the embedded proteins, and (c) deuterated proteins and solvent allow investigating membrane dynamics

used to suppress the incoherent scattering contribution of certain functional groups to the total scattering. While in protonated samples the incoherent scattering is usually dominant and the time self-correlation function of individual scatterers is accessible, (partial) deuteration emphasizes the coherent scattering and gives access to collective motions by probing the pair correlation function. We try to explain this with the example of a protein embedded in a lipid membrane dissolved in a common solvent. This is shown schematically in Fig. 10.3. In a sample where both the membrane and the proteins are hydrogenated but the solvent is deuterated, self-correlated, diffusive motions of membranes and embedded proteins are probed (Fig. 10.3a). Diffusive dynamics of the proteins are highlighted when solvent and membrane are deuterated, and hydrogenated proteins are used (Fig. 10.3b). The effect of protein insertion on membrane dynamics can be studied in Fig. 10.3c, with deuterated solvent and proteins. Note that at the same time, the interfaces between protonated and deuterated areas scatter coherently. The preparation in Fig. 10.3c can therefore be used to study possible protein–protein interactions. A typical neutron spectrum, measured by inelastic X-ray or neutron scattering at a particular Q-vector, is shown in Fig. 10.4. The total signal typically consists of a sharp peak at energy transfer zero, which marks the instrumental resolution. The resolution is often well described by a Gaussian peak shape. Quasielastic scattering is observed as a broadening of the instrumental resolution. In addition, inelastic


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Fig. 10.4 Typical excitation spectrum for X-rays and neutrons. The signal consists of a sharp central line, the instrumental resolution, a broad component centered at energy transfer zero (quasielastic scattering), and pairs of excitations

peaks may be observed. Excitations always occur as pairs of inelastic peaks at frequencies of −ω0 and +ω0 , caused by creation and deletion of dynamical modes. Starting from the simple models for local and collective modes in Fig. 10.2c, d, assumptions for peak shapes of the corresponding modes can be developed. Typical local modes, such as diffusion and libration (hindered rotation), are examples for relaxators, i.e., overdamped oscillations. The characteristic frequency of a relaxator is ω0 = 0, so that the response is centered on the elastic line. The width of the response is determined by the relaxation time τ0 , Δ¯hω = 2π/τ0 . In the time domain, a relaxation is described by an exponential decay e−t/τ0 , the Fourier transform of which is a Lorentzian function. So local, incoherent dynamics give rise to a quasielastic broadening in the energy domain and are described by Lorentzian line shapes, centered at energy transfer zero. Coherent dynamics probe the elastic springs between molecules or atoms. They can be considered as particles bound by an isotropic harmonic force and damped through hybridization with a phonon bath. Those propagating (acoustic) or oscillating (optical) modes have well-defined eigenfrequencies and lead to inelastic excitations at energy values ±¯hω0 . They are described by a damped harmonic-oscillator model. The width of the inelastic peak is related to the lifetime of the excitation. A strict derivation of the peak profiles can be found in the books by Squires [38] and Lovesey [39]. The origin of quasielastic and inelastic scattering is purely coherent for X-rays. In the case of liquids, for instance, the spectrum can be evaluated by a generalized effective eigenmode theory [40] in terms of a heat mode (a collective diffusion process) and sound modes. For neutrons, the quasielastic part of the spectrum can have contributions from coherent and incoherent scattering, but is most likely dominated by the incoherent scattering of hydrogen atoms. The corresponding dynamical modes are relaxating local modes, such as diffusion of molecules or functional groups. The relaxation times can be determined from the width of

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a Lorentzian peak shape. Excitations stem from coherent scattering and involve dynamics between different particles from interference effects. So by preparing a hydrogenated or deuterated sample and tuning the spectrometer to quasi- or inelastic scattering (QENS or INS respectively), a neutron scattering experiment can be made sensitive to local or collective dynamics. Collective molecular motions impact on, e.g., properties and functionalities of artificial and biological membranes, such as elasticity, transport processes, and determine interaction between proteins. Two examples will be discussed in detail in what follows with emphasis on the dynamics-property and dynamics-function aspects of the experiments: 1. Mesoscopic shape fluctuations in aligned multilamellar stacks of 1,2dimyristoyl-sn-glycero-3-phoshatidylcholine (DMPC) bilayers were studied using the NSE technique. From the dispersion relation in the fluid phase, values for the bilayer-bending rigidity κ , the compressional modulus of the stacks B, and the effective sliding viscosity η3 were determined (Sect. 10.4.2). This technique offers a novel approach to quantify the elasticity parameters in membranes by direct measurement of dynamical properties and also the impact of collective molecular motions on membrane properties [6]. 2. Experimental evidence for a protein-protein interaction between the bacteriorhodopsin proteins in purple membrane (PM) was found by inelastic neutron scattering (Sect. 10.4.3). The dynamics were quantified by measuring the spectrum of the acoustic phonons in the two-dimensional (2D) Bacteriorhodopsin (BR) protein lattice. In this case, INS was used to study interactions between constituents of a biological membrane [68].

10.3.3 Techniques in Brief: Inelastic Neutron Scattering Instruments Only recently, the first inelastic scattering experiments in phospholipid bilayers to determine collective motions of the lipid acyl chains were performed using inelastic X-ray [41] and neutron [42] scattering techniques. In the case of single membranes, the scattering signal is usually not sufficient for a quantitative study of the inelastic scattering. To maximize the scattering signal, multilamellar samples composed of stacks of several thousands of lipid bilayers separated by layers of water are necessary. A high orientational order of the samples which gives rise to pronounced Bragg peaks and excitations is a prerequisite to a proper analysis of the corresponding correlation functions. The use of well-oriented samples leads to welllocalized elastic and inelastic signals in reciprocal space and allows distinguishing motions in both the plane of the membrane (q|| ) and perpendicular to the bilayers (qz ). During the experiments, the membranes were kept in a humidity chamber to control temperature and humidity and were hydrated with D2 O from the vapor


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phase. The collective motions of the lipid acyl chains were probed by using partially, chain-deuterated lipids (DMPC-d54). The experiments discussed here were conducted on two different types of neutron spectrometers, namely tripleaxis and spin-echo spectrometers. Technical details and accessible length and time scales of the spectrometers are discussed in more detail in [6, 43]. Here only a brief description of the techniques is given. NSE spectrometry. Fluctuations on the mesoscopic scale are determined by the elasticity parameters of the bilayers, i.e., the compressibility of the stacked membranes, B, and the bending modulus κ . The relaxations in this regime are in the nanosecond time-range with accompanying small Q-vales. NSE spectrometers are highly suited for these experiments. The NSE technique offers extremely highenergy resolution from Larmor tagging the neutrons: the neutron’s spin is used as an individual label to measure the velocity and energy of the neutron by counting the number of Larmor precessions in a well-defined magnetic field. A neutron spinecho measurement is in essence a measurement of neutron polarization [43]. The polarized neutron beam passes through a magnetic field perpendicular to the neutron polarization and neutron spin precesses before arriving at the sample, acquiring a precession angle ϕ1 . At the sample, the beam is scattered before passing through a second arm, acquiring an additional precession angle ϕ2 in the reversed sense. For elastic scattering, when the velocity in the two arms is identical, the total precession angle is Δϕ = ϕ1 − ϕ2 = 0 for all incoming neutron velocities. If the neutron scatters inelastically by gaining or losing a small energy transfer h¯ ω (the velocity v2 in the second arm is faster or slower than v1 ), there will be a linear change Δϕ = τω with τ being a real time in the case of quasielastic scattering. The spin-echo technique thus works in the time domain and measures the intermediate scattering function S(Q, t) in contrast to the three-axis technique. For a quasielastic response, assumed to have Lorentzian lineshape with half-width Γ, the polarization will then show a single exponential decay PNSE = Ps e−Γ t . Triple-axis spectrometry. Fast motions in the picosecond time-range due to sound propagation in the plane of the bilayer are best measured on triple-axis spectrometers. The energy of the incident and scattered neutrons is determined by Bragg scattering from crystal monochromators (graphite in most cases). Advantages of triple-axis spectrometers are their relatively simple design and operation and the efficient use of the incoming neutron flux to the examination of particular points in (Q, ω ) space. By varying the three axes of the instrument, the axes of rotation of the monochromator, the sample, and the analyzer (as depicted in Fig. 10.5a), the wave vectors ki and kf and the energies Ei and Ef of the incident and the scattered neutrons, respectively, can be determined. The momentum transfer to the sample and the energy transfer, h¯ ω , are then defined by the laws of momentum and energy conservation to Q = kf − ki and h¯ ω = Ei − Ef . The accessible (Q, ω ) range is limited by the range of incident neutron energies offered by the neutron guide, as well as by mechanical restrictions of the spectrometer.

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Fig. 10.5 (a) Schematic of a triple-axis spectrometer. (b) Short-wavelength dispersion relations in the gel (Pβ ) and fluid phase Lα of the phospholipid (DMPC) bilayers [42]

10.4 Collective Molecular Dynamics in Membranes and Proteins 10.4.1 Dispersion Relations Collective molecular motions in membranes and proteins are attracting increasing attention as they are speculated to be responsible for certain functionalities in model membranes, such as transport of small molecules, pore opening, and membrane fusion processes. Collective motions of functional groups may drive protein function. In a cellular context, they may contribute to the understanding of macromolecular function because they can lead to an effective coupling and communication between the different constituents.


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Collective motions can be modeled by interacting particles, as sketched in Fig. 10.2b, d. The corresponding dynamics is described by a set of coupled pendulums (particles bound by an isotropic harmonic force). In inelastic scattering experiments, excitation energies are probed at different internal length scales and different directions. The Q-dependence of the excitation energies is usually not a trivial function. The energies will depend on the interaction constant, k, but also on the geometrical arrangement of the objects. It will depend on the length scale, the Q-value, but critically also on the coupling path, i.e., the direction, which is probed, as depicted in Fig. 10.2b. A typical experiment to measure collective dynamics records excitation frequencies at different Q-values (different internal length scales) over a continuous Q-range. The corresponding curve h¯ ω (Q) is called a dispersion relation. It describes the complex dynamics of a (oriented) system of coupled oscillators. Technically, it describes the way waves with a certain wavelength propagate through the system. Only coherent probes are capable to elucidate coherent dynamics. Chain-deuterated lipids (DMPC-d54) hydrated by heavy water (D2 O) were used to enhance the contribution of the collective motions of the lipid acyl chains over other contributions to the total scattering. The samples were kept in a closed temperature and humidity-controlled aluminum chamber. Hydration of the lipid membranes was achieved by separately adjusting two heating baths, connected to the sample chamber and to a heavy water reservoir, hydrating the sample from the vapor phase. Temperature and humidity sensors were installed close to the sample. ˚ were achieved. The corresponding relative humidity (RH) can be d-spacings of 53 A determined from d-RH curves (for instance from Figure 3 in ref. [44]) to be 99.5%. The main transition in DMPC occurs at 23.4 ◦C [45]. The transition temperature, Tm , is slightly lowered in the deuterated compound (DMPC-d54) to about 21◦ C [46]. Figure 10.5b shows the short-wavelength dispersion relation in a DMPC membrane. The experiments were performed using the IN12 cold triple-axis spectrometer at the Institut Laue Langevin (ILL), Grenoble, France. The particular shape of the corresponding dispersion relation can qualitatively be explained as follows: at small q|| , longitudinal sound waves in the plane of the bilayer are probed and give rise to a linear increase of ω ∼ q|| . The low q-range of the dispersion is difficult to access by inelastic neutron scattering because of the kinematic restriction. Because of the distinct particle character of neutrons, the corresponding dispersion relation is proportional to h¯ 2 /2mn q2 (with neutron mass mn ). The parabolic shape of the neutron dispersion prevents inelastic experiments at small q-values and at the same time high-energy transfers. This is called the kinematic restriction. The lipid dispersion saturates at some maximum value, before a pronounced ˚ −1 , the average nearest neighbor distance of two minimum is observed at q0 ≈ 1.4 A −1 ˚ q0 can be interpreted as the center of the first ˚ ≈ 4.5 A. lipid chains of 2π/1.4 A Brillouin zone of a two-dimensional liquid. Collective modes with a wavelength of the average nearest neighbor distance 2π/q0 are energetically favorable leading to a minimum in the dispersion relation. In highly ordered crystalline systems, the energy of the longitudinal phonon branches goes to zero at the zone centers. Here, the high degree of static and dynamic disorder in the lipid bilayers eventually

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1.8 1.6

q (Å–1) ||

1.4 1.2

a

subgel

Lb

P b

L a

1.8 1.6 1.4 1.2 b 250

260

271

280 285

295 300

310

Temperature (K)

Fig. 10.6 Chain correlation peak in DMPC as measured with (a) energy-integrated neutron diffraction for temperatures 250 < T < 310 K to cover gel, ripple, and fluid phase. (b) “True” elastic scattering, measured with an energy resolution of 0.9 μeV using a neutron backscattering spectrometer (from ref. [47]). No elastic intensity is observed in the fluid phase of the lipid bilayer

leads to a minimum at finite energy values (soft-mode). Two comments are in order: q0 is usually the position of the first maximum in the static structure factor S(q|| ) measured in neutron and X-ray diffraction experiments. When measuring diffraction, the energy of the scattered neutrons or X-rays is not discriminated. However, in a study of in-plane diffraction using a high-energy resolution of 0.9 μeV (corresponding to a timescale of about 1 ns) backscattering spectrometer [47], there was no truly elastic intensity observed in the fluid phase of a DMPC bilayer, as shown in Fig. 10.6. The structures usually observed in diffraction experiments in fluid phases of lipid bilayers are therefore dynamic with lifetimes in the order of nanoseconds, only. Secondly, the in-plane structure of lipid bilayers is often characterized in experiments by the position of the average nearest neighbor lipid tail correlation peak. However, the parameter used in molecular dynamics simulations is usually the area per lipid, rather than the nearest neighbor distance. The determination of exact areas per lipid from experiments is a very active field of research and important to compare experiments and simulations. While it is not possible to relate the area per lipid straightforwardly to the inter chain distance [48], precise lipid areas could be determined by combining neutron and X-ray scattering techniques [49]. A quantitative theory which predicts the absolute energy values of maximum and minimum on the basis of molecular parameters is absent so far. However, the dispersion relation can be extracted from molecular dynamics simulations by temporal and spatial Fourier transformation of the molecular real space coordinates [9] which shows excellent agreement with the data. Note that the dispersion relation


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found is similar to those in ideal liquids, e.g., liquid argon [50,51], liquid neon [52], or liquid helium [53]. The interior of the lipid bilayer, the C-atoms or C–D groups of the lipid acyl chains, behave like a quasi liquid. In contrast to real liquids, the chain atoms of the lipid molecules are chemically bound to each other, leading to smaller mobility and diffusion and, as a consequence, pronounced excitations. To establish possible dynamics-function relations, such as a connection with transmembrane transport, the dispersion relation in a membrane-containing drug enhancers will be studied. It is speculated that the soft-mode in such a system drop down to small energy values because enhanced fluctuations of the lipid chains may be the driving transport mechanism.

10.4.2 Membrane Properties According to linear smectic elasticity theory [54, 55], thermal fluctuations in the fluid phase of the membrane are governed by the free energy functional (Hamiltonian): H=

d2 r A

N−1

∑

n=1

2 1 1B , (un+1 − un )2 + κ ∇2|| un 2d 2

(10.1)

where κ denotes the bilayer-bending rigidity, A the area in the xy-plane, N the number of bilayers, and un the deviation from the average position nd of the nth bilayer, d is the lamellar spacing. B and K = κ /d are elastic coefficients, governing the compressional and bending modes of the smectic phase, respectively. A fundamental √ length scale in these systems is given by the smectic penetration length Λ = K/B. Aligned lipid bilayers allow a separate determination of both parameters K and B [56, 57]. Figure 10.7 shows the intermediate scattering function S(q|| ,t) for selected q|| values. The experiments were performed on the spin-echo spectrometers IN11 and IN15 at the ILL. Data have been taken at three different temperatures, at 19 ◦ C, in the gel (ripple, Pβ ) phase of the phospholipid bilayers, at 22 ◦ C, just above the temperature of the main transition, and at 30 ◦ C, in the fluid Lα phase of the membranes and above the regime of the so-called anomalous swelling. Two relaxation processes, one at about 10 ns (τ1 ) and a second, slower process at about 100 ns (τ2 ), were observed. The relaxation rates τ1 −1 and τ2 −1 in the gel and the fluid phase are shown in Fig. 10.7a, b. Both relaxation branches are dispersive. The fast process shows a q|| 2 increase at small q|| values and a bend at about ˚ −1 . The dispersion in the gel phase and close to the phase transition in q|| ∼0.015 A Fig. 10.7b appears to be more pronounced as compared to the fluid 30 ◦ C dispersion. A soft-mode appeared in the T = 22 ◦ C dispersion, indicating a significant softening of the bilayer at a well-defined wave number. The slow branches at T = 19 and 30 ◦ C also show increasing relaxation rates with increasing q|| values, but with a distinct nonpolynomial behavior.

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Fig. 10.7 (a) Dispersion relations in the fluid phase of the phospholipid bilayers, at T = 30 ◦ C. The solid line is a fit to Equation (10.2) [6]. (b) Dispersion relations in the gel (19 ◦ C) and in ˚ −1 at the range of critical swelling (22 ◦ C). A pronounced soft-mode is observed at q0 = 0.015 A 22 ◦ C marked by the dotted vertical line. (c) Fluctuation spectrum of solid-supported multilamellar membranes [58], and corresponding analytical dispersion relation (d). (e) Surface mode which is responsible for the slow relaxation branch in part (a) [59, 60]

The dispersion relation of the fast branch with relaxation rates between 1 and 10 ns can be attributed to undulation dynamics. Qualitatively, at very small q|| values, the membranes behave as liquid films and their dynamics are basically determined by the viscosity of the water layer in between the stacked membranes (film regime). With increasing q|| , there is a transition into a bulk-elasticity regime where the dynamics depend on the elastic properties of the lipid bilayers. At this point, the dispersion bifurcates in two relaxation branches. The faster one, which is out the experimentally accessible time window of the NSE spectrometer, is mainly determined by the compressional module B. The fluctuation spectrum of solidsupported membranes was modeled analytically [58] and is depicted in Fig. 10.7c. Quasiacoustic and optical modes were found (quasi because the modes are not


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propagating due to the solid support) with increasing fluctuation amplitudes towards the center of the membrane stack. The calculated dispersion of the quasiacoustic branch is shown in Fig. 10.7d, and the shape qualitatively agrees well with the measured data. The slower branch, which is observed here, can be assigned to the bending modulus κ . The additional slow dispersion branches in Fig. 10.7a, b with relaxation rates of about 100 ns could be attributed to a surface relaxation mode [59, 60]. The mode is depicted in Fig. 10.7e and particular to a stack of membranes. Following the idea of Ribotta et al. [61], the relaxation rates of the undulations can be described by:

τ −1 (q|| ) =

4 2 κ /d 2 q|| + (π /Λ D) q|| 4 , η3 q|| + (π /D)2 /μη3

(10.2)

(where η3 is the layer sliding viscosity), and the following results were obtained: ˚ η3 = 0.016 Pa s. B was calculated to be B = 1.08 × κ = 14.8kB T, Λ = 10.3 A, ˚ The resulting effective sliding viscosity of the membrane 107 J/m3 (d = 54 A). system η3 was found to be 16 times higher than that of water, what points to different properties of the interstitial hydration water, as compared to bulk water. Equation (10.2) does not describe a pure undulation mode, which is probed at qz values of qz = 2π/d, only. If the scattering is probed at finite components Δqz := (qz − 2π/d) or measured with a relaxed qz resolution, there is a mixing of baroclinic modes, which are distinctly slower than a pure undulation because they involve a relocation motion of the water layer. The parameter D describes an effective finite-size cutoff-length, which was related to the instrumental resolution D = π/Δqz . From the soft-mode in the fluid dispersion at T = 22 ◦ C, it can be concluded that the well-known softening of phospholipid membrane upon approaching the main phase transition temperature from the fluid phase, i.e., the regime of critical swelling or anomalous swelling [44], occurs on a well-defined length scale, only ˚ The elastic parameters of membranes can also be determined by (2π/q|| ∼ 420 A). analysis of the diffuse nonspecular scattering. The most important contribution of inelastic measurements and the determination of a dispersion relation is that the stiffness of the bilayers is determined as a function of internal length scale.

10.4.3 Interactions in Membranes Recently, interprotein motions in a carboxymyoglobin protein crystal were reported from a molecular dynamics simulation [12,72]. Experimentally, phonon-like excitations of proteins in hydrated protein powder were reported [73]. Finally, Protein–protein interactions in a biological membrane, the purple membrane (PM), were reported from inelastic neutron scattering [68].

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PM occurs naturally in the form of a two-dimensional almost crystal-like structure, consisting of 75% (wt./wt.) of a single protein, bacteriorhodopsin (BR), that functions as a light-activated proton pump, and 25% various lipid species (mostly phospho- and glyco-lipids) [62]. BR is a proton-transporting membrane protein, formed of seven transmembrane alpha-helices arranged around the photosensitive retinal molecule. The protein in the lipid matrix is organized in trimers that form a ˚ as depicted in highly ordered 2D hexagonal lattice with lattice parameter a ∼ 62 A, Fig. 10.2b. The structure of PM is well established by electron microscopy, neutron, and X-ray diffraction experiments as reviewed for instance in [62–67]. The dynamic neutron scattering experiments were performed on the IN12 cold triple-axis spectrometer at the ILL. IN12 turned out to be highly suited for elastic and inelastic investigations in oriented biological samples because of its flexibility, good energy resolution, and extremely low background. It allows the measurement of diffraction and inelastic scattering in the same run without changing the set-up, which is crucial to assign dynamical modes to structural properties and molecular components. IN12 was equipped with a vacuum box to avoid air scattering at small scattering angles and with vertically focusing monochromator and analyzer to increase the neutron flux at the sample position. All scans were done with ˚ −1 resulting in a Q resolution of ΔQ = 0.005 A ˚ −1 and an energy fixed kf = 1.25 A resolution of Δ¯hω = 25 μeV. Deuterated PM was produced and hydrated by H2 O in order to suppress the contribution of the membrane hydration water to the phonon spectrum in the coherent inelastic neutron scattering experiments. Correlations and motions in membranes are often well separated in reciprocal space because of the largely different length and time scales involved. The prominent distances in PM, such as lipid–lipid distances, BR-BR monomer, and trimer distances, lead to correlation ˚ for the peaks. Because the length scales involved span a wide range from 4.5 A ˚ for the trimer-trimer distance, the corresponding signals lipid chain distance to 62 A appear well separated in reciprocal space. The same holds for the different time scales involved from the picoseconds (molecular reorientations) to the nano- or microsecond (membrane undulations, large protein motions). The use of oriented samples further allows separating correlations in the plane of the membranes, and perpendicular to the bilayers. Figure 10.8a depicts energy scans at constant-q|| scans ˚ −1 and q|| = 0.45 A ˚ 1 . The total inelastic signal at selected values between q|| = 0.35 A consists of a Gaussian central peak due to instrumental resolution, a quasielastic broadening, which is described by a Lorentzian peak shape, and pairs of excitations, described by damped harmonic-oscillator peak profiles. The excitation spectrum of the 2D protein lattice was modeled analytically for comparison with the experimental data. The protein trimers were taken as the centers ˚ and the spectrum of a primitive hexagonal lattice with lattice constant a = 62 A, of the acoustic phonons was calculated. The model is depicted in Fig. 10.2b. The basic hexagonal translations are marked by arrows. The interaction between the protein trimers is contained in springs with an effective (longitudinal) spring constant k (Fig. 10.2b). The calculated longitudinal spectrum Cl (q|| , ω ), defined by Cl (q|| , ω ) = (ω 2 /q2|| )S(q|| , ω ), is shown in Fig. 10.8b. The statistical average in


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˚ −1 . The Fig. 10.8 (a) Energy scans at q|| values of q|| = 0.35, 0.37, 0.39, 0.41, 0.43, and 0.45 A total inelastic signal consists of a central peak (Gaussian), a Lorentzian quasielastic contribution, and the excitations, fitted using damped harmonic-oscillator peak profiles. (b) Calculated excitation spectrum Cl (q|| , ω ) in the range of the experimental data. Data points mark the positions of excitations, as determined from the fits in part (a). The errorbars give the uncertainty in determining the peak position. The horizontal line at h¯ ω = 0.45 meV marks the position of a possible optical phonon mode, not included in the calculations (from ref. [68])

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the plane of the membrane leads to a superposition of the different phonon branches, which start and end in the hexagonal Bragg peaks (at h¯ ω = 0). The positions of excitations, as determined from the fits in Fig. 10.8a, are marked by the data points. The absolute phonon energies cannot be determined from the model, but depend on the coupling constant k. So the energy of the phonon spectrum in Fig. 10.8b was fitted to best match the experiment. The experimentally determined excitation energies occur at the intersections of the inelastic scans with the calculated phonon branches and well reproduce the features of the theoretical excitation spectrum within the given errors. Note that because the proteins trimers were treated as dots with an effective mass of Mtr , the calculations do not include any contributions from intramonomer or intratrimer dynamics, i.e., possible optical modes. The excitations ˚ −1 (corresponding to a length observed in the experiment at q|| values around 0.42 A ˚ and energy value of 0.45 meV (marked by the solid line in scale of about 15 A) Fig. 10.8b) have no equivalent in the theoretical spectrum and possibly stem from optical phonon branches. Because the distance agrees well with the monomer– monomer distance in the protein trimer, this mode can tentatively be assigned to optical-like monomer dynamics (the corresponding form factor is likely to be maximum at the corresponding nearest neighbor distance). The commonly assumed interaction mechanism between inclusions in membranes is a lipid-mediated interaction due to local distortions of the lipid bilayer [20–24], with a strong dependence on the bilayer properties, in particular elastic properties. The PM may, however, be a special case because there are very few lipids between neighboring BR proteins [69]. While the nature of the interaction still will be mainly elastic, it is not likely to be purely lipid-mediated but partially a direct protein–protein interaction. The strength of the interaction can be determined from the data in Fig. 10.8. The energy of the zone-boundary phonon at the M˚ −1 ) point of the hexagonal Brillouin zone (for instance, at a q|| value of 0.35 A relates to the coupling constant by Mtr ω 2 = 6k. Because the energy is determined to h¯ ω = 1.02 meV, the effective protein–protein spring constant k is calculated to about k = 54 N m−1 . The amplitude of this √ √ mode of vibration can be estimated from ˚ and the interaction force the equipartition theorem to x2 = √ kB T /k = 0.1 A, between two neighboring trimers to F = k x2 = 0.5 nN. Using the same approach, the spring constant for graphite for comparison is calculated to 27,000 N m−1 for the in-plane interaction, and 3.5 N m−1 for out-of-plane interactions. The force constant that we measure in PM is 1–2 orders of magnitude larger than the effective vander-Waals force constant in graphite, but 2–3 orders of magnitude weaker than a C–C bond. The mechanism reported here may be relevant to model the photo cycle in PM. It was reported [70] that the BR proteins undergo structural changes during the ˚ Because of the elastic coupling photo cycle, involving displacements of up to 1.7 A. of the BR proteins, those displacements can propagate to neighboring proteins. ˚ yields a On the microscopic level, displacing or distorting a BR trimer by 1.7 A force between neighboring trimers of 9 nN (using the model presented here). It can therefore be speculated that there is a protein–protein communication during the photocycle in PM.


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10.4.4 Single Particle vs Collective Dynamics The “softness” of the BR proteins in PM was also probed by elastic incoherent neutron scattering [71], and force constants of 0.1 N m−1 were reported. The shear modulus in stacks of PM was determined by X-ray reflectivity to μ = 0.02 N m−1 ˚ respective 3.8 A. ˚ [65]. The corresponding amplitudes of vibration were 2.2 A, These techniques were mainly sensitive to diffusion of proteins (and lipids), and the force and time constants involved are determined by local friction and restoring forces. The corresponding force constants are about two orders of magnitude smaller than the one reported here. Energies of collective, phonon-like excitations were also reported for proteins in a protein crystal (from computer simulations in refs. [8, 72]) and hydrated protein powder (from inelastic X-ray scattering [73]), and energies of about 1 meV for inter- and even 10 meV for intraprotein motions were found. The reported energies cannot be quantitatively compared to the excitations in this study because the systems are distinctly different. However, it seems that the energies found for protein interactions are consistently higher than the force constants from experiments, which probe the local energy landscapes. The amplitudes of diffusive, self-correlated motions are therefore about one magnitude larger than those of collective, pair-correlated motions. “Motional coherence,” i.e., a structurally coherent state, was reported recently in fluid phospholipid membranes [74]. The authors reported that lipids move as coherently coupled clusters of about ˚ rather than individual lipids. As a consequence, micrometer lipid diffusion 30 A may appear slow when studied with mesoscopic techniques such as fluorescent light microscopy because the movement of clusters is measured rather than the motion of individual lipids. Diffusion constants from incoherent neutron scattering are usually faster because they measure fast motions of lipids over much shorter, nanometer distances. So while for a long time, motions in biological materials were considered as thermally activated vibrations or librations in local potentials, at least part of the fluctuation spectrum stems from interactions. The future challenge is to understand the impact of collective molecular motions in membranes and proteins on biological function. Acknowledgments It is my pleasure to thank all my colleagues who were involved in the original work and publications. The data presented here were collected at the high flux reactor of the Institut Laue-Langevin in Grenoble

References 1. Rheinstädter MC, Seydel T, Häußler W, Salditt T (2006) Exploring the collective dynamics of lipid membranes with inelastic neutron scattering. J Vac Sci Technol A 24:1191–1196 2. Vo¨ıtchovsky K, Contera SA, Ryan JF (2009) Lateral coupling and cooperative dynamics in the function of the native membrane protein bacteriorhodopsin. Soft Mat 5:4899

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3. Frauenfelder H, Sligar S, Wolynes P (1991) The energy landscapes and motions of proteins. Science 254:1598–1603 4. Fenimore P, Frauenfelder H, McMahon B, Young R (2004) Bulk-solvent and hydration-shell fluctuations, similar to α - and β -fluctuations in glasses, control protein motions and functions. Proc Natl Acad Sci U S A 101:14408–14413 5. Bayerl T (2000) Collective membrane motions. Curr Opin Colloid Interface Sci 5:232–236 6. Rheinstädter MC, Häussler W, Salditt T (2006) Dispersion relation of lipid membrane shape fluctuations by neutron spin-echo spectrometry. Phys Rev Lett 97:048103 7. Smith JC (1991) Protein dynamics: comparison of simulations with inelastic neutron scattering experiments. Q Rev Biophys 24:227–291 8. Hayward JA, Smith JC (2002) Temperature dependence of protein dynamics: computer simulation analysis of neutron scattering properties. Biophys J 82:1216–1225 9. Tarek M, Tobias D, Chen S-H, Klein M (2001) Short wavelength collective dynamics in phospholipid bilayers: a molecular dynamics study. Phys Rev Lett 87:238101 10. Tarek M, Tobias D (2002) Role of protein-water hydrogen bond dynamics in the protein dynamical transition. Phys Rev Lett 88:138101 11. Wood K, Plazanet M, Gabel F, Kessler B, Oesterhelt D, Tobias DJ, Zaccai G, Weik M (2007) Coupling of protein and hydration-water dynamics in biological membranes. Proc Natl Acad Sci U S A 104:18049–18054 12. Meinhold L, Smith JC, Kitao A, Zewail AH (2007) Picosecond fluctuating protein energy landscape mapped by pressure—temperature molecular dynamics simulation. Proc Natl Acad Sci U S A 104:17261–17265 13. Salditt T (2000) Structure and fluctuations of highly oriented phospholipid membranes. Curr Opin Colloid Interface Sci 5:19–26 14. Krueger S (2001) Neutron reflection from interfaces with biological and biomimetic materials. Curr Opin Colloid Interface Sci 6:111–117 15. Salditt T (2005) Thermal fluctuations and stability of solid-supported lipid membranes. J Phys Condens Matter 17:R287–R314 16. Kuˇcerka N, Nieha M-P, Pencera J, Harroun T, Katsaras J (2007) The study of liposomes, lamellae and membranes using neutrons and X-rays. Curr Opin Colloid Interface Sci 12:17–22 17. Krueger S, Meuse C, Majkrzak C, Dura A, Berk NF, Tarek M, Plant AL (2001) Investigation of hybrid bilayer membranes with neutron reflectometry: probing the interactions of melittin. Langmuir 17:511–521 18. Fragneto G, Rheinstädter M (2007) Structural and dynamical studies from bio-mimetic systems: an overview. C R Phys 8:865–883 19. Tanaka M, Sackmann E (2005) Polymer-supported membranes as models of the cell surface. Nature 437:656–663 20. Kralchevsky PA (1997) Lateral forces acting between particles in liquid films or lipid membranes. Adv Biophys 34:25–39 21. Bohinc K, Kralj-Iglic V, May S (2003) Interaction between two cylindrical inclusions in a symmetric lipid bilayer. J Chem Phys 119:7435–7444 22. Biscari P, Bisi F (2002) Membrane-mediated interactions of rod-like inclusions. Eur Phys J E 6:381–386 23. Lagüe P, Zuckermann MJ, Roux B (2001) Lipid-mediated interactions between intrinsic membrane proteins: dependence on protein size and lipid composition. Biophys J 81:276–284 24. Dan N, Pincus P, Safran S (1993) Membrane-induced interactions between inclusions. Langmuir 9:2768–2771 25. König S, Pfeiffer W, Bayerl T, Richter D, Sackmann E (1992) Molecular dynamics of lipid bilayers studied by incoherent quasi-elastic neutron scattering. J Phys II (France) 2:1589–1615 26. König S, Sackmann E, Richter D, Zorn R, Carlile C, Bayerl T (1994) Molecular dynamics of water in oriented DPPC multilayers studied by quasielastic neutron scattering and deuteriumnuclear magnetic resonance relaxation. J Chem Phys 100:3307–3316 27. König S, Bayerl T, Coddens G, Richter D, Sackmann E (1995) Hydration dependence of chain dynamics and local diffusion in L-alpha-dipalmitoylphosphatidylcholine multilayers studied by incoherent quasi-elastic neutron scattering. Biophys J 68:1871–1880


285

28. Pfeiffer W, Henkel T, Sackmann E, Knorr W (1989) Local dynamics of lipid bilayers studied by incoherent quasi-elastic neutron scattering. Europhys Lett 8:201–206 29. Pfeiffer W, König S, Legrand J, Bayerl T, Richter D, Sackmann E (1993) Neutron spin echo study of membrane undulations in lipid multibilayers. Europhys Lett 23:457–462 30. Lindahl E, Edholm O (2000) Mesoscopic undulations and thickness fluctuations in lipid bilayers from molecular dynamics simulations. Biophys J 79:426–433 31. Lipowsky R, Sackmann E (eds) (1995) Structure and dynamics of membranes. Elsevier, Amsterdam 32. Nevzorov A, Brown M (1997) Bilayers from comparative analysis of 2H and 13C NMR relaxation data as a function of frequency and temperature. J Chem Phys 107:10288–10310 33. Bloom M, Bayerl T (1995) Membranes studied using neutron scattering and NMR. Can J Phys 73:687–696 34. Takeda T, Kawabata Y, Seto H, Komura S, Gosh S, Nagao M, Okuhara D (1999) Neutron spin echo investigations of membrane undulations in complex fluids involving amphiphiles. J Phys Chem Solids 60:1375–1377 35. Hirn R, Bayerl T, Rädler J, Sackmann E (1999) Collective membrane motions of high and low amplitude, studied by dynamic light scattering and micro-interferometry. Faraday Discuss 111:17–30 36. Hirn RB, Bayerl TM (1999) Collective membrane motions in the mesoscopic range and their modulation by the binding of a monomolecular protein layer of streptavidin studied by dynamic light scattering. Phys Rev E 59:5987–5994 37. Hildenbrand MF, Bayerl TM (2005) Differences in the modulation of collective membrane motions by ergosterol, lanosterol, and cholesterol: a dynamic light scattering study. Biophys J 88:3360–3367 38. Squires G (1978) Introduction to the theory of thermal neutron scattering. Dover, New York 39. Lovesey S (1984) Theory of neutron scattering from condensed matter. Clarendon Press, Oxford 40. Liao C, Chen S, Sette F (2000) Analysis of inelastic X-ray scattering spectra of lowtemperature water. Phys Rev E 61:1518–1526 41. Chen S, Liao C, Huang H, Weiss T, Bellisent-Funel M, Sette F (2001) Collective dynamics in fully hydrated phospholipid bilayers studied by inelastic X-ray scattering. Phys Rev Lett 86:740–743 42. Rheinstädter MC, Ollinger C, Fragneto G, Demmel F, Salditt T (2004) Collective dynamics of lipid membranes studied by inelastic neutron scattering. Phys Rev Lett 93:108107 43. Mezei F (ed) (1980) Neutron spin echo. Springer, Heidelberg 44. Chu N, Kuˇcerka N, Liu Y, Tristram-Nagle S, Nagle JF (2005) Anomalous swelling of lipid bilayer stacks is caused by softening of the bending modulus. Phys Rev E 71:041904 45. Jenskea R, Lindström F, Gröbner G, Vettera W (2008) Impact of free hydroxylated and methylbranched fatty acids on the organization of lipid membranes. Chem Phys Lipids 154:26–32 46. Aussenac F, Laguerre M, Schmitter J-M, Dufourc EJ (2003) Detailed structure and dynamics of bicelle phospholipids using selectively deuterated and perdeuterated labels. 2 H NMR and molecular mechanics study. Langmuir 19:10468–10479 47. Rheinstädter MC, Seydel T, Salditt T (2007) Nanosecond molecular relaxations in lipid bilayers studied by high energy resolution neutron scattering and in-situ diffraction. Phys Rev E 75:011907 48. Hub JS, Salditt T, Rheinstädter MC, de Groot BL (2007) Short range order and collective dynamics of DMPC bilayers. A comparison between molecular dynamics simulations, X-ray, and neutron scattering experiments. Biophys J 93:3156–3168. doi:dx.doi.org 49. Kuˇcerka N, Liu Y, Chu N, Petrache HI, Tristram-Nagle S, Nagle JF (2005) Structure of fully hydrated fluid phase DMPC and DLPC lipid bilayers using X-ray scattering from oriented multilamellar arrays and from unilamellar vesicles. Biophys J 88:2626–2637 50. de Schepper I, Verkerk P, van Well A, de Graaf L (1983) Short-wavelength sound modes in liquid argon. Phys Rev Lett 50:974–977

286

M.C. Rheinstädter

51. van Well A, Verkerk P, de Graaf L, Suck J-B, Copley J (1985) Density fluctuations in liquid argon: coherent dynamic structure factor along the 120-K isotherm obtained by neutron scattering. Phys Rev A 31:3391–3414 52. van Well A, de Graaf L (1985) Density fluctuations in liquid neon studied by neutron scattering. Phys Rev A 32:2396–2412 53. Glyde H (1994) Excitations in liquid and solid helium. Clarendon Press, Oxford 54. Caillé A (1972) X-ray scattering by smectic-A crystals. C R Acad Sci Ser B 274:891–893 55. Lei N, Safinya C, Bruinsma R (1995) Discrete harmonic model for stacked membranes: theory and experiment. J Phys II 5:1155–1163 56. Lyatskaya Y, Liu Y, Tristram-Nagle S, Katsaras J, Nagle JF (2000) Method for obtaining structure and interactions from oriented lipid bilayers. Phys Rev E 63:011907 57. Salditt T, Vogel M, Fenzl W (2003) Thermal fluctuations and positional correlations in oriented lipid membranes. Phys Rev Lett 90:178101 58. Romanov V, Ul’yanov S (2002) Dynamic and correlation properties of solid supported smecticA films. Phys Rev E 66:061701 59. Bary-Soroker H, Diamant H (2007) Nanoscale surface relaxation of a membrane stack. Phys Rev E 76:042401 60. Bary-Soroker H, Diamant H (2006) Surface relaxation of lyotropic lamellar phases. Europhys Lett 73:871–877 61. Ribotta R, Salin D, Durand G (1974) Quasielastic rayleigh scattering in a smectic-A crystal. Phys Rev Lett 32:6–9 62. Haupts U, Tittor J, Oesterhelt D (1999) Closing in on bacteriorhodopsin: progress in understanding the molecule. Annu Rev Biophys Biomol Struct 28:367–399 63. Zaccai G (2000) Moist and soft, dry and stiff: a review of neutron experiments on hydrationdynamics-activity relations in the purple membrane of Halobacterium salinarum. Biophys Chem 86:249–257 64. Koltover I, Salditt T, Rigaud J-L, Safinya C (1998) Stacked 2D crystalline sheets of the membrane-protein bacteriorhodopsin: a specular and diffuse reflectivity study. Phys Rev Lett 81:2494–2497 65. Koltover I, Rädler J, Salditt T, Safinya C (1999) Phase behavior and interactions of the membrane-protein bacteriorhodopsin. Phys Rev Lett 82:3184–3187 66. Neutze R, Pebay-Peyroula E, Edman K, Royant A, Navarro J, Landau E (2002) Bacteriorhodopsin: a high resolution structural view of vectorial proton transport. Biochim Biophys Acta 1565:144–167 67. Lanyi J (2004) Bacteriorhodopsin. Annu Rev Physiol 66:665–688 68. Rheinstädter MC, Schmalzl K, Wood K, Strauch D (2009) Protein-protein interaction in purple membrane. Phys Rev Lett 103:128104 69. Baudry J, Tajkhorshid E, Molnar F, Phillips J, Schulten K (2001) Molecular dynamics study of bacteriorhodopsin and the purple membrane. J Phys Chem 105:905–918 70. Luecke H, Schobert B, Richter H-T, Cartailler J-P, Lanyi JK (1999) Structural changes in bacteriorhodopsin during ion transport at 2 angstrom resolution. Science 286:255–260 71. Zaccai G (2000) How soft is a protein? A protein dynamics force constant measured by neutron scattering. Science 288:1604–1607 72. Kurkal-Siebert V, Agarwal R, Smith JC (2008) Hydration-dependent dynamical transition in protein: protein interactions at ≈240 K. Phys Rev Lett 100:138102 73. Liu D, Chu X-Q, Lagi M, Zhang Y, Fratini E, Baglioni P, Alatas A, Said A, Alp E, Chen S-H (2008) Studies of phononlike low-energy excitations of protein molecules by inelastic X-ray scattering. Phys Rev Lett 101:135501 74. Rheinstädter MC, Das J, Flenner EJ, Brüning B, Seydel T, Kosztin I (2008) Motional coherence in fluid phospholipid membranes. Phys Rev Lett 101:248106

Part III

Extra Complexity: Surface Effects and Confinement

Chapter 11

Application of Time-Resolved Small Angle Neutron Scattering to Non-Equilibrium Kinetic Studies Jitendra P. Mata, William A. Hamilton, and Elliot P. Gilbert

11.1 Introduction In thischapter, we address the application of small-angle neutron scattering (SANS) to the study of kinetics. SANS is a routine technique used for probing the structure of soft materials at the nanometre scale. In the absence of a satisfactory definition to describe the difference between dynamics (the general subject of this book) and kinetics, we attempt to do so here, at least in the parlance of neutron scattering. Kinetics is concerned with measurement of the change from one spatial configuration to another, where each spatial configuration represents a quasi-static structure. The evolution of the structure is expressed quantitatively by time. We distinguish this from “dynamics” in which one is specifically concerned with the motions associated with the evolution from one quasi-static state to another. The latter information is obtained directly from inelastic or quasielastic neutron scattering techniques, with the data typically obtained either at fixed wavevector transfer, q, or averaged over a range of q (see e.g. Chap. 1). In a SANS experiment, it is generally structural information that is sought and data are simultaneously collected over a broad range of q but with no energy resolution; inelastic and quasielastic processes are largely ignored. However, inelastic scattering measurements can yield structural information, e.g. the radius of a rotating methyl group can be obtained from quasielastic neutron scattering, while dynamic information can be obtained from SANS, e.g. diffusion coefficients from application of a Cahn-Hilliard-Cook model for spinodal decomposition [1]. It is also worthy noting that devices can be installed on existing SANS instruments to obtain dynamic information using, for

J.P. Mata • W.A. Hamilton • E.P. Gilbert () Bragg Institute, Australian Nuclear Science and Technology Organisation, Locked Bag 2001, Kirrawee DC, NSW 2232, Australia e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 11, © Springer Science+Business Media, LLC 2012

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Fig. 11.1 Comparison of dynamic and kinetic studies. Note the temporal gap between dynamic studies approaching the microsecond time domain from shorter times and quasi-static SANS studies approaching from longer times (adapted from Gähler [49])

example, MIEZE techniques [2]. Figure 11.1 illustrates the separation of dynamic and quasi-static studies. For studies of dynamics, neutron spin echo represents the neutron scattering technique with the finest (i.e. longest) time resolution, being able to measure correlation times up to approximately 1 μs. Limitations of this technique include the need for strongly scattering systems and the absence of magnetic fields. Considering the problem from the other direction, TISANE represents the SANS technique with the finest time resolution for kinetic studies, but this technique also has its limitations [3]. The challenge is to bridge the gap in time between these two techniques. In this chapter we deal specifically with kinetics measured using the SANS technique. We will illustrate the range of approaches that have been employed to maximise time resolution. There is an abundance of literature on the measurement of kinetics with SANS extending to minutes and hours (and beyond) but these are outside the scope of this review. Here we have limited ourselves to examples in which low supra-second, millisecond and microsecond kinetics have been reported. We start the chapter by briefly describing the sample environments and methods currently available to enable kinetic investigations, many of which are not particular to neutron scattering. We then describe a few examples of kinetic studies using synchrotron-based small angle X-ray scattering (SR-SAXS) as an introduction to the type of science achievable with neutrons and to why neutron scattering provides essential complementarity to SAXS despite inherently lower flux. We address broad areas, namely responses to periodic perturbation (stress, temperature, pressure, fields), mobility in colloidal systems, macromolecules and micelles, and the kinetics of chemical reactions, phase transitions and growth processes. Finally, we provide a summary of current activities in kinetics studies using SANS with the state-of-theart approach that provides the greatest time resolution, namely Time Involved SANS (TISANE). This stroboscopic approach, in combination with the application of a periodic stimulus, allows kinetic processes to be studied on a time scale extending down to microseconds [3].

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11.2 Methods for Enhancing Time-Resolution in SANS Irrespective of the technique, there are several factors that must be addressed to maximise the time resolution of the signal obtained from a sample undergoing a rapid change. The first factor is the shortest timeframe that may provide a statistically viable data point. The brute force approach is to use the maximum flux available – the so-called single-shot kinetics experiment – but this immediately places synchrotron studies at a significant advantage to those with neutrons. The flux issue can be circumvented to some extent if the system responds completely reversibly to a stimulus, without degrading. In this case, one can repeat the experiment many times to accumulate sufficient statistics within each time bin [4]. In conventional time-slicing, data are collected for some time interval, Δt, in a block of histogramming memory, and then collected in another block of memory for the next Δt and so forth. A series of such measurements may be repeated many times to build up reasonable statistics in every time interval. With this approach, time resolution down to 50–100 ms is straightforward. In modern computer data acquisition systems “time-stamping” of individual neutron detection events allows this histogramming to be achieved after the fact, to allow the optimisation of binning schemes that serve to balance statistics and time resolution. Thus a single sample could be studied many times if it responds in the same way to an external effect. In this way, if a single 100-s snapshot could be collected with appropriate statistics for data analysis, then 1,000 repetitions could provide, in principle, a time resolution of 100 ms. Such is the case for studies of photoactive yellow protein (PYP), a phototaxis photoreceptor from Ectothiorhodospira halophila. The experimental setup used to investigate the response of the protein to cyclic flash-light illumination is shown in Fig. 11.2 [5]. Hamilton and co-workers [4] conducted studies to investigate the phase transition in cetylpyridinium-hexanol/dextrose-brine under cycled shear conditions in which the 2D SANS detector output was multiplexed into several memory modules, rather than through measurement of sequential runs. The system was sheared until a steady Lα state had been reached; it was then stopped and allowed to relax fully to the L3 state. This was repeated several thousand times. Figure 11.3 is a schematic view of typical time cycled stroboscopic data collection used in their study, demonstrating typical features of the method. Each data acquisition cycle was synchronised at a delay to a stop signal on a Couette shear cell [4, 6, 7]. To maximise the time resolution down to the millisecond range, the process under investigation has to be repeatable to enable the collection of sufficient statistics. Therefore, either the process must be reversible so that one is able to restore the initial condition or, in the case of irreversible processes such as a reaction, enough sample material must be available to allow the experiment to be repeated many times. Irreversible systems can be easily studied by the “stopped-flow environment.” In such an experiment, two or more liquids are mixed rapidly, in about 10 ms, to investigate fast real-time kinetics. Depending on technique and sample environment, two different modes of operation are possible. In the first option one

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Fig. 11.2 Time resolved (TR) studies of the photoactive yellow protein response to cyclic flashlight illumination (image courtesy of R. May, IL and W. Crielaard, University of Amsterdam [5])

Fig. 11.3 (a) Schematic of typical time cycled stroboscopic data collection. TR data bins are synchronised with a SANS sample control parameter (here Couette shear rate applied to a L3 sponge phase). (b) Typical data indicating relaxation of binned SANS signal and corresponding tracking of control parameter (after ref. [4, 7])


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Fig. 11.4 The intrinsic wavelength spread in conventional SANS results in a range of times-ofarrival of neutrons at the detector (bottom triangle) that consequently imposes a limit on the finest time resolution available. TISANE requires the synchronisation of a periodic stimulus at the sample position with an upstream chopper to pulse the neutron beam. This generates a condition in which the neutrons arriving at the detector have been scattered from the sample at the same time within the periodic sample stimulus but from different neutron pulses (top triangle). This TISANE condition is shown in more detail in Fig. 11.5 (adapted from Gähler [49])

can supply a signal to restart the data acquisition cycle. Slightly more sophisticated communication between sample environment and electronics can be achieved in the second option by readjusting the initial condition of the system. To study such systems, data processing software is required to collect timestamped data effectively. The intrinsic wavelength spread in conventional SANS studies imposes the limits on time-resolved (TR) measurements of a few milliseconds (Fig. 11.4). This results in a range of times-of-arrival of neutrons at the detector that may be comparable to the time domain of interest. Similar, but smaller, timeresolution uncertainties apply in the case of spallation sources where this results from finite neutron moderation times. While there have been improvements in detector acquisition electronics to enable time resolutions to be measured down to 1 μs, these much greater intrinsic uncertainties clearly indicate that the response time of the electronics is not a significant limitation. Dynamical studies of nanosized inhomogeneities by SANS are usually limited to slow processes where the system remains in a quasi-steady state during the data acquisition time.

11.2.1 Time Involved Small-Angle Neutron Scattering (TISANE) The finest time-resolution achievable using SANS techniques to date is via an approach proposed by Roland Gähler and recently used for the first time, namely

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time t

Periodic sample properties are seen at another time base at the detector

Chopper period TC

LCS

Sample: periodic changes of scattering properties period TS

LSD

Detector signal period TD

Fig. 11.5 Distance-time diagram for the TISANE setup at a SANS instrument. A broad wavelength band passes a chopper with repetition time TC . The scattering properties of the sample are modulated with time period TS , represented by the periodic change in grey-scale. Neutron trajectories are shaded in the same way upon passing the sample. Shortly after the sample the shades mix, but at certain distance, LSD , where the detector is placed the shades re-order. In this way the time response of the sample scattering properties may be imaged at the detector (after ref. [3])

TISANE or TISANS with the acronyms referring to time-involved small-angle neutron experiment or scattering respectively [3]. This approach overcomes the time-resolution limits discussed above (Fig. 11.4). We briefly describe the physical requirements for conducting such an experiment and its limitations. However, the reader is strongly encouraged to read the original reference [3] in which this is covered in extensive detail. In typical time dependent SANS experiments, the resolution Δt is limited by frame overlap, i.e. the spread in neutron arrival times at the detector. The wavelength spread in the incident beam, typically 10–20%, therefore provides a lower limit on the time resolution that can be achieved. For example, for a sample˚ only detector distance, L, of 20 m and a neutron wavelength band of 10% at 10 A, experiments above Δt ≈ 5 ms are usually feasible. In TISANE, this severe frame overlap limitation is overcome so that it is no longer a hindrance but actually enhances the intensity without sacrificing time resolution. There are three essential requirements for conducting a TISANE measurement: 1. A fast chopper near the beginning of the collimator of the SANS instrument 2. A device which modulates the scattering properties of the sample 3. A detector with suitably fine time resolution


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The principle of TISANE is shown in Fig. 11.5. The lines represent the neutron trajectories of different wavelengths from successive chopper openings. The fast chopper is located before the sample to pulse the beam in such a way that the pulsing is correlated with the cycling of the external stimulus of the sample. The aim is to have a one-to-one correlation between the scattered neutron arrival time and stimulus of the sample from different pulses. In this way, neutrons arrive at the same point of the periodic stimulus cycle and are then added together. It can be seen that by the time the neutrons reach the sample position, there is extreme frame overlap. The time modulation of the sample scattering properties is illustrated by a grey-scale (colour) sequence. For example, the distance from one “blue” region of the sample to the consecutive “blue” region corresponds to the time period of the sample modulation. The shades of the neutron trajectories are chosen to match the shade of the sample modulation at the instant of time when the neutron passes the sample. One can see that at a certain distance downstream from the sample all the shades reorder themselves and it is at this position that one strategically places the detector. The distance at which this occurs is obtained geometrically via TS /TC = LSD /LCD and TD /TS = LCD /LCS . In effect, each individual shade represents a physical observation of the sample in a specific configuration during one full time period. Recording the pattern at the detector as function of time (say 10–20 detector frames within one cycle) reveals the response function of the sample to the external field. Note that in general the frequency of the detector signal differs from the sample modulation frequency and the chopper frequency. The re-ordering at the detector is not affected by the angular spread of the neutron trajectories in the space–time diagram and thus TISANE works for a very broad wavelength band. For a system that can be repeatedly stimulated so that sufficient statistics may be obtained, and with the available technology, 10 μs time-resolution is achievable using this approach. We will discuss the only experimental example of TISANE in the last section of the chapter; in this case a conventional 3 He detector was employed for measuring the response of a magnetic nano-fluid to the application of a minimum 2 ms period magnetic field. However, scintillation detectors or MHz readout gas detectors are required to reach the full potential of TISANE as they can accommodate much finer time resolution, in the range of 1 μs. The relationships between time and distance, given above, set corresponding constraints on the frequency of the high resolution signal at the detector with respect to the chopper and sample frequencies. For example, LCD = 20 m, LSD = 11 m, a chopper cycle time of 60 μs, a neutron velocity of 500 ms−1 with 10% velocity spread and a sample stimulus period of 33 μs, yields a TISANE condition in which oscillations are observed on the detector plane with a period of 73 μs. A chopper duty cycle of between 5 and 20% is optimal for such studies as long as the sample responds linearly to the applied modulation (assumed here to be cosine-like). In the case of a non-linear response (e.g. a magnetic field), the signal pattern may become non-sinusoidal and its resolution may require a smaller duty cycle. For this reason, counter-rotating double disk choppers are appropriate so that the duty cycle may be varied from 0 to 15% by proper phasing. Other conditions are required to

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perform TISANE, such as phase locking, to ensure that there are no significant phase shifts between the chopper, sample and detector frequencies during a complete experimental run. In some cases this phase locking stability may be required for many hours in order to ensure sufficient statistics. Variations away from the TISANE condition, e.g. changing the sample period or equivalently the ratio of LCD to LSD , also obviously impact on signal quality. However, errors in length in the 1 cm range should produce reliable data for a 10% wavelength spread. For the standard distances discussed above, a minimum time period TC ≈ 50 μs is achievable with double choppers for beam widths in the 1 cm range and a 10% duty cycle, with a maximum period of ≈2 ms being feasible. Employing the TISANE condition, these limits convert to sample modulation times of between 5 μs and 1 ms. At the finest time resolution, the sample itself is constrained to a thickness of less than or equal to 0.6 mm for neutron speeds of 400 m s−1 , assuming that the time the neutrons spend in the sample is at least three times lower than TS . With somewhat relaxed conditions on the maximum frequency, comfortable sample thicknesses in the range of several mm are possible. Evidently an intrinsic limitation to the application of TISANE, is to systems that behave reversibly. The time resolution is only achievable via periodic processes in which the SANS response of the sample can be measured as a function of the frequency and amplitude of the modulated conditions (e.g. pressure, flow or applied fields). Thus experiments on singular events, e.g. non-reversible fast reactions, are excluded. Nonetheless, this approach represents the finest time resolution achievable for SANS and provides significant opportunities for the study of a broad range of systems. A detailed presentation of the technique is available in Kipping et al. [3].

11.3 Sample Environments Sample environments can be used to create pseudo-environments for sample processing, or to mimic the conditions in which the material behaves in nature. Bras and Ryan [8] have reviewed sample environments for SR-SAXS and many of these are also suitable for conducting time-resolved SANS measurements (TR-SANS) measurements, albeit with some modification – for example, to accommodate larger beam sizes or, through the strategic selection of alternative materials, to minimise parasitic scattering or neutron activation.

11.3.1 Stopped-Flow Stopped-flow apparatus is one of the most frequently used devices for studying rapid kinetics. This allows one to trigger a rapid mixing of two or more solutions at various volume ratios and flow rates, in a small volume and with short dead time, in order to follow the very early stages of a reaction. The device is suited to TR studies down


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Fig. 11.6 Schematic view of the path of the solutions filling the three syringes through the delay lines and mixing chambers up to the observation cell (after ref. [9])

to between 10 and 90 ms; this is determined by the finite time required for precursor mixing and flow to the sample position. The overall time resolution is, of course, affected by the actual time resolution of the SANS instrument in the ways discussed earlier in the chapter. The broad applications of stopped-flow in both SAXS and SANS have recently been described in [9]. The general stopped-flow apparatus consists of two or three vertical stainless steel syringes, each of ∼20 mL and driven by independent stepping motors, and one or two mixing chambers (Fig. 11.6). These syringes can be filled from the top to avoid air bubbles that may form during filling or the setup process. In the experiment, a delay line is installed soon after mixer 1 (mixing chamber connected with syringes 1 and 2) whose volume determines the evolution time tET between the two mixing chambers. After mixing solution 3 (in mixer 2) with the mixed solution from the delay line, the final mixture (1 + 2 + 3) is stopped in the observation cell (Hellma cell) for measurement. The minimal time to begin the observation after mixing, tET + tDT (the time between mixer 2 and the middle of the cell), is of the order of 50–200 ms depending upon the flow rate and the configuration of the syringes. The sequence of mixing defines the required volumes and times. Due to the volume of observation cell being 0.2 mL and given its geometry and a 1 mm beam path (Fig. 11.6), at least 3 times the cell volume is necessary to completely empty the old solution and replace it by a fresh one. The data acquisition can be triggered at any time during the mixing sequence by sending a signal from the device together with the sequence of acquisition, defining the number of spectra and their associated times.

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An alternative system has been discussed by Akiyama et al. [10] for SAXS studies of the conformational change of Cytochrome c during folding. In this case a unique T-shaped mixing point apparatus has been constructed, which allowed a change of the distance between the mixing and observation points [10].

11.3.2 Temperature-and Pressure-Jump (T-Jump, P-Jump) Phase changes can result from sudden variations in temperature and pressure; the associated structural changes can be studied through the use of T-jump and P-jump cells respectively. Balsara et al. have reported the early and intermediate stages of nucleation and growth in a block copolymer blend, as a result of applying T- and P-jumps separately [11, 12].

11.3.2.1 Temperature Jump Temperature-jump devices are designed to induce rapid temperature-driven responses or reactions in a sample. A technical problem to overcome is the ability to deposit energy uniformly into the sample, or extract energy from it, in addition to enabling rapid temperature changes. Various approaches have so far been developed to achieve this. Electrical resistive heating was the most popular option in the early stages of temperature jump development [13]. The fastest temperature jumps can obviously be achieved through the use of good thermal conductors like metals and alloys. Pullen et al. have developed a rapid heat and quench cell (RHQC) for the Australian SANS instrument, Quokka, to enable samples to be rapidly heated or quenched and to allow the simultaneous measurement of SANS [14]. The rapid temperature change is achieved by translating the sample from one chamber, controlled at a specified (and typically elevated) temperature, to another chamber at a different (typically sub-ambient) temperature (Fig. 11.7). The sample is cooled by maintaining a constant flow of nitrogen through a heat exchanger which is submerged in liquid nitrogen. This gas subsequently flows through a temperatureregulating nichrome heater coil prior to entering the cold chamber, to provide control over the temperature of the gas. This heater coil is controlled via a Lakeshore 340. Similarly, the sample is heated using nitrogen gas in a separate line, with the gas passing through an alumina tube containing a nichrome wire wound to form a heater coil which is powered by a programmable power supply. Both chambers of the RHQC are allowed to equilibrate before the sample is translated to prevent temperature overshooting and minimise thermal lag. Tests have been conducted showing that the sample can be subjected to temperatures up to 600 K and then quenched down to 150 K or rapidly heated from a low temperature. Measured


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Fig. 11.7 Image of the RHQC sample environment with the sample in the neutral position. Inset: location of the thermocouple measuring sample temperature in situ (sectional view) (after ref. [14])

cooling rates of up to 11 K s−1 and heating rates up to 19 K s−1 have been recorded during the testing stages. The device has been used to investigate the early stage demixing kinetics of hydrogenated/deuterated paraffin blends quenched from the melt [14]. More recently the same system has been quenched from the molten state (389 K) to 300 K with data collection as a single file of 6,000 frames with 100 ms resolution. Clear phase changes are evident from the variation in temperature of the sample as a function of time in the form of inflexion points. It is thus possible that structural changes in the system can be correlated with the thermal response of the sample, similar to that observed in simultaneous differential scanning calorimetry and SAXS [15]. There are a range of other methods that use electromagnetic radiation to administer energy to a sample. Massive pulsed lasers have been utilised to achieve fast heating rates (10 K in 2 ms) without temperature gradients [16]. This method has been applied to study the phase transitions in a phospholipid system. Caffrey et al. [17] studied the kinetics of the lamellar and hexagonal phase transitions in phosphatidylethanolamine by TR-SAXS with a microwave sample heating device. With a capillary tube sample cell, rates of ∼30 K s−1 with 120 W of microwave power have been achieved [17]. While the time resolution of T-jump can reach a few microseconds, due to heat dissipation, the kinetics cannot be recorded within less than a few milliseconds. Narrow temperature jumps, and the irreversibility in performing cooling jumps, are some of the limitations of classical T-jump instruments, but some of these have apparently been overcome by improvements in commercial T-jump instruments. For example, Bio-Logic’s mT-jump is a commercial device based on stopped-flow methods that claims millisecond resolution with a 40 K jump possible either in cooling or heating.

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11.3.2.2 Pressure Jumps Pressure jumps can also be used to initiate phase transitions, but the changes in thermodynamic state which can be achieved this way tend to be fewer than those observed by other methods. However, an advantage is that damage to the sample is also small owing to the intrinsically weaker gradient driving any transition. Another advantage of this method over all the others considered here is the rapidity, uniformity and reproducibility of the change which can be made throughout the sample – in particular pressure jumps from high to low pressure. This type of behaviour is essentially impossible to achieve by the other phase forcing parameters considered here – such as temperature, applied shear or mixing – which depend unavoidably on transport across the dimensions of the sample cell. A distinct disadvantage, however, is the change of temperature and acoustic speed that results when a pressure jump is applied. This can make it somewhat difficult to obtain a “clean” thermodynamic picture of transitions driven by this method. Pressure jump methods have been employed to examine [18] the gelation mechanism in aqueous solutions of β-lactoglobulin by dynamic light scattering (DLS) and SANS. Osaka et al. have investigated a P-jump from 0.1 to 315 MPa and the resultant evolution of gel structure by using DLS and SANS as a function of time. Pressure-dependent SANS experiments were conducted in a pressure-jump device constructed from two concentric cells [18]. The inner cell is made from aluminium with quartz and sapphire windows with a sample thickness of 2.0 mm, connected through a rubber diaphragm to transmit the pressure; the outer chamber is filled with D2 O and the pressure is controlled by pressurising D2 O by a double-cylinder hand pump at constant temperature. The pressure-induced gels were found to form fractal aggregates consisting of primary particles of β-lactoglobulin monomers. Pressureinduced denaturation is attributed to a decrease of the hydrophobic interaction at high pressure, followed by gradual denaturation. Osaka et al. have reported temperature-jump SANS measurements using the same pressure cell without any modification [18].

11.4 Synchrotron Time-Resolved SAXS (SR-TRSAXS) The advent of high intensity synchrotron radiation (SR) X-ray sources has accelerated the development of TR-SAXS, with the study of the dynamics of muscle contraction in real time being one of the main driving forces [19]. The technique has subsequently been extended to a variety of other macromolecular systems. It is wellaccepted that the sample preparation required for imaging techniques like electron microscopy, often involving evacuated sample chambers, can impact greatly on interpretation. However, conformational changes occurring in biological processes in environments that mimic native conditions can be studied by SR-TRSAXS experiments. More recently the emphasis has shifted towards the simultaneous measurement of TR-SAXS with other techniques. There are, of course, a large number


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Fig. 11.8 Evolution of SAXS intensity upon mixing the cationic TTAB and anionic LPFO micelles, Spherical vesicles (fitted lines) are formed within the mixing process, which grow over a few seconds. For clarity, successive scattering curves have been multiplied by the factor indicated in the legend (after ref. [21])

of examples in the literature using TR-SAXS. Here we present two examples with time resolution in the millisecond range and in which SANS has been, or could be, used as a complementary tool. A detailed introduction to the development of SR-TRSAXS, and to its applications, has recently been published [20]. The self-assembly of amphiphilic molecules represents an active area of study in soft matter; the rich phase behaviour produces a range of equilibrium structures such as micelles, vesicles, lipid bilayers and liquid crystalline phases. In addition to purely measuring the kinetics of self-assembly, another area of interest relates to the pathways for self-assembly [21]. Various external factors such as solvent selectivity [22], pressure [23] and/or temperature [24] jumps can bring about these transitions. One example is the micelle to vesicle transformation in mixed surfactant solutions. Weiss et al. [21] initiated the self-assembly process of anionic with zwitterionic surfactant micelles by rapidly mixing them in an equimolar ratio using a stoppedflow device (Fig. 11.8). Within the mixing time, the original micelles transformed to disk-like micelles which evolved further to a critical size, and then subsequently closed to form monodisperse unilamellar vesicles within a second [21]. Structural information on the formation and growth of poly(ethylene-altpropylene)-poly(ethylene oxide) (PEP-PEO) block copolymer micelles has been studied by means of SR-SAXS with millisecond time resolution. Using a

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Fig. 11.9 (a) The time evolution of the distribution of the micellar ensemble in terms of the aggregation number, Pmean , corresponding to the fit results of the 0.5% micellar system. (b) Correlation between unimer consumption and micellar growth: time evolution of the unimer concentration (line) compared to the associated increase of Pmean (circles). The relative Gaussian width of the micellar distribution σrel = σ /Pmean is also shown (squares). (c) Schematic view of the nucleation and growth type process of diblock copolymers leading to the final micellar structures (after ref. [22])

quantitative model, Lund et al. [22] show that the self-assembly process can be viewed as a nucleation and growth process where the elemental growth mechanism involves the exchange of single molecules. The resulting time evolution of the micellar size distribution, g(P,t), is illustrated for the 0.5% micellar system in Fig. 11.9a. It shows that the initial free unimers are consumed rapidly in a primary micellisation event that resembles a classical over-nucleation leading to metastable micelles (region I). These micelles can only grow further if other micelles dissolve by emitting unimers. This is particularly clear from Fig. 11.9b, where one can see that the micellar size, represented by Pmean , exhibits a shoulder at the time when the unimers are consumed below the critical micelle concentration (region II). Interestingly this shoulder is also characterised by a peak in the width of the Gaussian micellar distribution function. The broader distribution signifies micellar rearrangement processes where small micelles shrink and large micelles grow. At later times the distribution narrows again as the equilibrium is approached (region III). The scenario is summarised and depicted schematically in Fig. 11.9c [22].


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Quantitative modelling revealed that the formation and growth of micelles is well described by a nucleation and growth type process governed by the elemental unimer insertion or expulsion mechanism. The kinetic pathway, including the initial aggregation as well as the kinetic stability and the growth rate, is controlled by a system specific micellisation potential. By appropriately estimating this potential for a given system, e.g. by means of structural investigations or theoretical models, the desired equilibrium or non-equilibrium properties may be predicted [22]. For experiments requiring microsecond or sub-microsecond time resolution, laser pump-probe methods offer a wide range of opportunities [23]. These experiments utilise the full fundamental or harmonic of an undulator peak, to provide X-ray intensities of the order of 109 photons per pulse. So far this technique has primarily been employed for probing ultrafast structural dynamics in photosensitive simple molecules and proteins. With further advances in instrumentation, the SAXS range will become accessible by this method allowing access to sub-microsecond to nanosecond range kinetics. The state-of-the-art in laser pump-probe TR-SAXS, for the study of molecular structural dynamics down to the sub-nanosecond range, has recently been reviewed [23].

11.5 X-Ray Versus Neutron The flux of the most intense source of neutrons in the world currently available for SANS, ∼108 neutrons cm−2 s−1 , is small when compared to that of a 1 mW red laser of ∼1017 photons cm−2 s−1 [24]. Neutron sources are therefore relatively “dim.” In addition, neutron beams are typically large (perhaps tens of millimetres or more) and require similarly “large” samples to minimise the data collection time [25]. However, this can be seen as advantageous in some senses, as large samples may more closely represent a realistic situation of the material and can also enable one to probe the bulk phase [25]. In contrast, for some SR-SAXS measurements, the sample size may be too small to be truly representative of the bulk although the fine beam dimensions open up the possibility of studying structural variations across a material by rastering [26]. The potential for radiation damage with fluxes of ∼2 × 1013 photons s−1 mm−2 at 12.5 keV, and the corresponding beam heating, are significant issues for many biomolecular systems. Neutron scattering presents several advantages over X-rays, e.g. a greater sensitivity to light atoms (H, C, etc.) and the possibility to employ contrast variation; anomalous SAXS provides an X-ray alternative approach in some cases. Moreover, SANS experiments offer the possibility of solvent or molecular labelling to follow only the structural arrangements of a particularly chosen compound [9, 27]. SANS measurements lend themselves more readily to absolute scaling and thus can be quantitative. Due to the highly penetrating nature of the radiation, neutron scattering can also be applied to the study of materials contained within thick or complicated sample environments so that one may measure the structural changes of a material during a process, for example the influence of shear on the formation of micelles in

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complex fluids by transmitting a neutron beam through a Couette shear cell [4]. One can mimic different processes and carry out real TR experiments studying the structural changes that occur as a consequence. Finally, new stroboscopic methods such as TISANE allow dynamic processes to be studied on a time scale extended to microseconds, thus overcoming intrinsic time resolution limits due to wavelength spread [3].

11.6 Current Kinetic Studies with SANS Technical advances in source performance, instrument optics, and sample environment design have had a great impact on the fields of research that can be undertaken by TR-SANS. To illustrate its potential, we will present here several examples of in situ studies taken from different fields of research. There are many other interesting examples that have deliberately been omitted due to the very coarse time resolution involved. However, we have endeavoured to cover examples over different time scales as the time resolution for measurements often depends on the system of interest and its implicit kinetics, rather than intrinsic limitations of accessing fine time resolutions. Indeed, as described above, cyclic data acquisition opens up the ability to measure sub-second kinetics without any significant modification in the instrument. In Sect. 11.6.1 we describe kinetic studies in surfactant and vesicle systems; Sect. 11.6.2 covers biomolecular systems including protein gelation and fibril formation. We describe mesophase precipitation and self-assembly in templating systems in Sect. 11.6.3. The last section of the chapter, Sect. 11.6.4, is devoted to TR investigations highlighting the efficiency of TISANE, and to investigations of ultra fast kinetic processes. We exclude from our examples several more specialised approaches. For example, van den Brandt et al. have investigated TR-SANS from polarised proton domains in frozen glycerol-water solutions containing cobalt paramagnetic complexes [28]; Wyslouzil and coworkers have measured aerosol nucleation rates where a position scan across a supersonic jet was used to extract the time evolution of nanodroplets as well as soot development in one-dimensional, laminar premixed ethylene flames [29]. Both these studies represent pseudo-kinetic measurements [30].

11.6.1 Surfactant and Vesicle Systems Surfactants exhibit rich structural behaviour. Their stability and microstructure result from the interplay of attractive and repulsive forces between colloids. A small variation in the intermolecular force balance (pH, temperature or ionic strength variation, dilution, mixing, or application of a magnetic field) may induce a reorganisation of the system, such as phase separation or a structural phase transition. The equilibrium states of many surfactant systems are largely characterised and


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Fig. 11.10 (a) Illustration of a SANS experiment to determine the exchange kinetics of unimers in a micellar system. (b) Corresponding scattering curves of the system PEP5-PEO15 in DMF. Top: scattering curves of the protonated (circles) and the deuterated (triangles) micellar solutions and of a 50/50 mixture (diamonds) of both (polymers were completely mixed in a good solvent) representing the condition at t = infinite. Bottom: three-dimensional plot of the scattering intensities as a function of scattering vector Q and time t at T = 70 ◦ C (after ref. [31])

generally well-understood; however, models of intermediate phases and structures under non-equilibrium conditions have been far less characterised. Obviously, an understanding of the phase evolution and possible intermediate species is crucial to enable reaction rates and final state characteristics to be modified, particularly for technologically relevant materials. The 2001 study by Willner et al. [31] of unimer exchange kinetics in block copolymer micelles is an excellent example of the relative strengths and technical requirements of SANS and the exploitation of the large difference in scattering length density of protonated and deuterated species. In these measurements they tracked the evolution of the SANS signal following the mixing of two polymeric micellar solutions of equal concentration made using protonated and deuterated species of poly(ethylene-alt-propylene)–poly ethylene oxide (PEP-PEO) and dimethylformamide (DMF). Each polymer solution was prepared in a mixture of light and heavy water contrast matched against the average mixed micellar scattering length density (Fig. 11.10). This resulted in two initial solutions with comparable

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coherent scattering signal (the signal for deuterated micelles was 35% higher due to a somewhat larger aggregation number than for the protonated micelles) and a fully mixed final system with minimal signal. This approach maximises the signal difference between the initial and fully mixed states. While at room temperature, the mixing proceeded very slowly, the authors found that, at temperatures between 60 and 80 ◦ C, the exchange of species between micelles was sufficiently rapid that it could be conveniently accessed by repeated (cycled) SANS measurements of mixing series of about 1 h total duration and with a 3 min integration interval for individual data sets. The statistics of the integrated SANS signals gathered in this way enabled them to identify the relative contribution of the two processes to the mixing. A rapid process with a time constant of the order of minutes was consistent with a single unimer insertion/expulsion scaling mechanism predicted by Halperin and Alexander [32]. A much slower process with a time constant about two orders of magnitude longer remained unexplained. However, from the structural information available from their scattering data, the authors were able to rule out both competing “micellar collision,” which had been invoked to explain fluorescence quenching decays observed in comparable systems, and isotope effects on micellar surface tension, the latter having been considered as the cause of the differing aggregation numbers between deuterated and hydrogenous micelles in the pre-mixture solutions as shown in Fig. 11.10. Some dilute ionic surfactants thicken markedly under applied shear, with steady state viscosities increasing by more than an order of magnitude. Oda et al. [33] used TR measurements to tackle the problem of determining the unknown structures giving rise to this behaviour in one such system, the dimeric surfactant ethanediyl1,2-bis(dodecalmethylammonium bromide), by tracking the SANS signal from solutions as they returned to equilibrium after the cessation of flow. They observed a relaxation of local anisotropy over time scales of approximately 100 s, but also observed changes in the structure that persisted over at least several hours. They interpreted their scattering and rheological results by postulating that the shearinduced viscous phase consists of a loosely connected network of long-lived aggregates with a typical size of 1 μm and where counter-ion mediated attraction between rod-like micelles could explain the formation of the aggregates and their long lifetime. Vesicles are closed bilayer shells; they have been the subject of increased interest during the past decades due to their use as models for proto-cells and their potential as microreactors for drug encapsulation and delivery. The challenge in their broader application is the formation of inexpensive stable vesicles with welldefined sizes. Some common, but energetically costly, methods to obtain vesicles are sonication, dialysis, extrusion of lamellar suspensions or reverse-phase evaporation; however, in most cases these methods form aggregates that are unstable and highly polydisperse [9]. In 2007 Nakano et al. [34] measured the time evolution of their SANS signal to extract the exchange mobility of lipids in bilayer vesicles. In a similar study to that of Willner above, large unilamellar vesicles of deuterated and hydrogenous dimyristoylphosphatidylcholine (DMPC) were prepared and mixed.


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Scattering patterns from a series of such mixtures at different temperatures were collected at 3 min intervals over several hours. In this case the contrast, and therefore the signal, decays as a result of the intermixing of the micelles and could be represented by a simple model with two kinetic parameters – the rate of transbilayer lipid migration between vesicles and the rate of inter-bilayer exchange between the inner and outer leaflets of the individual vesicle bilayers. The authors noted the suitability of the technique for the generic testing of equilibrium lipid exchange kinetic models as applied to lipid migration in biological processes, such as translocation and lipid transfer proteins, and for evaluation of the importance of relevant factors. Grillo used “stopped-flow” scattering to investigate the progress of structural changes that can be triggered in amphiphilic systems, such as vesicles and micelles, as a result of mixing them with other surfactants or additives [9]. They followed the formation and growth of spontaneous anionic vesicles after addition of a range of salts to micellar solutions of the anionic surfactant sodium bis(2-ethyl hexyl)sulfosuccinate (AOT) in D2 O from ∼500 ms to 5 h after mixing. SANS data acquisition started 90 ms after initiation of the mixing, with data being summed over ten repetitions to provide adequate statistics. They found a rapid transition from micelles to vesicles which was complete within the first 500 ms, which precluded observation of any intermediate structures. Over longer times, the vesicle radii and the growth rate depended on the salt concentration and decreased with increasing ionic strength. The driving force for this transition is the screening of the electrostatic repulsion between adjacent AOT headgroups which favours the formation of a locally planar bilayer. Assuming that the aggregation is controlled and limited in equilibrium by micelle diffusion, a simple kinetic treatment predicted that the average radius should increase with the power law R ∝ t 1/6 , in close agreement with their experimental data. In further stopped-flow mixing studies, using both SANS and SAXS [35], the same group obtained neutron exposure times of 50 ms using cycled data collection whereas in SAXS studies, in which mixing was carried out in smaller cells, the time resolution was increased to ∼5 ms. They investigated the formation of unilamellar vesicles upon mixing of cationic and anionic surfactant solutions (a “catanionic” mixture) and also in an ionic surfactant solution upon the addition of a co-surfactant. For both systems, the micelles that were initially present disappeared quickly and the slow formation of monodisperse unilamellar vesicles followed. Although the latter process is purely governed by diffusion and the final states were similar, they observed that vesicle formation proceeded via disk-like intermediates in the first case and rod-like intermediates in the second. As part of the same study [35] the solubilisation of methyl heptanoate into a zwitterionic surfactant was investigated; the emulsion droplets that were initially present were slowly transformed into welldefined microemulsions. The static phase diagram indicates that a lamellar phase is present at intermediate concentrations. The expected evolution from the emulsion to microemulsion, via a similar intermediate, was not observed. The solubilisation apparently proceeds directly via the formation of micro-emulsion droplets. Finally, they followed the disintegration of micelles when mixing a micellar solution with a

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bad solvent for the micelles. It is observed that the disintegration process does not occur in a simple dissolution step; instead, smaller micellar structures are reformed after passing through a minimum aggregation stage. In each case these results indicate the rich complexities and surprises which often emerge in kinetic solution studies. In addition to vesicles, self-assembled bilayer systems exhibit a rich variety of extended mesophases with correspondingly varied properties: the stacked lamellar Lα phases are anisotropic, viscous and birefringent, cubic ordered phases are sometimes stiff enough to ring like bells when struck, while the isotropic disorder of the passages connecting L3 “sponge” phases allow them to be free flowing and optically inactive. Even though they may not exhibit long range order, the membrane makes connections over long distances. In the sponge phases, for example, one convoluted membrane sheet spans the entire solution volume separating it into two equivalent sub volumes. Such a structure would be expected to have consequences in the pathways and kinetics of phase transitions. Gotter et al. recently performed an extensive TR-SANS study of temperature-induced phase sponge to lamellar transitions, and vice versa, in the water–n-decane-C12E5 membrane system [36]; their results were interpreted in the light of findings from a prior 2 H-NMR study. SANS T-jump measurements were made on a series of different bilayers, both dilute and more concentrated, and with varying solvent viscosities (by the addition of sucrose) with a time resolution of approximately 2 min. The progression of the topological phase change in each direction had markedly different characters. L3 to Lα phase formation occurred through a nucleation and growth process over minutes; however, no coexistence of the phases was observed as might have been expected from the equilibrium phase diagram. L3 phase development from Lα proceeded in a mechanistically different and more rapid manner involving uncorrelated passage formation events which re-create the random connections of the isotropic sponge structure. The manipulation of neutron contrast in these investigations, to create film (membrane) contrast samples, allowed the group to monitor the bilayer thickness through the transitions showing that there was no evidence for changes in the short range order of the bilayer structure. Such behaviour may be considered possible due to the very different topologies and local curvatures in the two structures. Time cycled SANS techniques with multiplexing with resolution ∼0.1 s were applied by Hamilton et al. to extract dynamic parameters of mesophase relaxation in shear-induced steady state systems after the cessation of hydrodynamic stress in two complex fluid systems (Fig. 11.11). The first of these was a mixed counterion surfactant (70:30 cetyltrimethylammonium 3,5 dichlorobenzoate and CTABromide in ∼1 wt. % solution in D2 O) threadlike micellar system [37]. The authors had earlier identified the existence of a highly oriented hexagonal near-surface phase when flowing past a solid surface using SANS in reflection geometry. The alignment of the phase under bulk shear was previously observed in Couette cell SANS measurements. The highly ordered surface state has obvious rheological implications as it represents a minimum interference state for the micelles as they pass by the surface and each other. More interestingly, from the point of TR-measurements, the relaxation time of the surface phase is far faster than the relaxation observed for


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Fig. 11.11 (a) Schematic of SANS scattering geometries for shear-induced Lα to equilibrium L3 relaxation and TR-data, with postulated “stalk” path for L3 passage reformation. (b) Comparison of membrane contact times estimated from shear rates and DLS measurements compared with SANS relaxation times for full structural relaxation vs. both membrane volume fraction φ (simple dilution along L3 phase region) and membrane CpCl to hexanol composition ratio h, which varies the intrinsic membrane curvature across L3 phase stability region (after ref. [7])

bulk aligned micellar phase to an equilibrium entangled phase. Multiplexed SANS measurements were performed on this system, over a thousand repeated cycles of shear alignment and relaxation, covering the first few seconds in two series with 0.1 and 0.5 s time binning. The TR-multiplexed measurements revealed that, on cessation of shearing flow, the hexagonal diffraction pattern decayed to a ring over about 1 s (actually an arc of scattering in this reflection geometry), with the ratio of scattering vector values indicating that the lattice was undergoing a two dimensional melting transition. As expected, higher order peaks decayed more rapidly, while the rate of decay of the individual peak intensities allowed the determination of the two dimensional diffusion constant for the micelles. After the initial rapid relaxation, the sample continued to relax with the scattering arc intensity continuing to weaken slowly, consistent with previously observed bulk entanglement relaxation times of about 30 min. The same technique was later used by this group to measure the relaxation of another steady state phase in bilayer membranes: cetylpyridinium chloride (CpCl) with hexanol cosurfactant as a bulk anisotropic stacked lamellar Lα phase induced by Couette shear to an equilibrium isotropic L3 sponge phase [7]. In this case the relaxation process requires a topological phase transition as the initially separated

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membranes re-establish the isotropic network of passages (topologically handles) that characterise the sponge phase. The aim of this study was to compare this structural relaxation time to the interval between diffusive membrane contacts to determine the activation energy barrier to the membrane fusion process that reestablishes the solution channel handles of the sponge phase. Significantly, this provides hard values for a simple system akin to the biological (lipid) membranes in which similar processes operate, such as cell mitosis and vesicle budding and transport. The study indicates the interplay of a number of factors that may need to be considered in a TR-hydrodynamic study. The structural relaxation time was simply estimated by measuring the time required to re-establish the isotropic L3 scattering signal after the cessation of shear with a shift correction of ∼0.05 s; this was identified as the time required for the Couette shear cell to slow down enough for the applied shear to fall below the threshold of the shear-induced lamellar phase transition. To bring the relaxation time into the range accessible by the technique, both in terms of the relaxation time (∼0.1–10 s) and lowering the initial applied shear rate required to reach the shear-induced Lα phase, the authors increased the viscosity of the heavy brine solvent in the sponge phase (used for neutron contrast against the hydrogenous membrane surfactants) by over an order of magnitude, from ∼1 to 16 cP, by adding 40 wt. % dextrose. This inert thickener slowed the diffusion of the membranes thus increasing the time between contacts of membrane sheets; this allows the possibility of establishing a passage between membranes provided that an activation energy barrier could be overcome. The latter means that the applied shear rate necessary to achieve the shear-induced state can be used as an estimate of the membrane contact frequency as it represents the rate of strain required to disrupt the geometry of any passage (re-)formation event. The contact rate was also determined from DLS measurements that indicated simple cooperative diffusion of the membranes. The contact interval can then be estimated as the time for diffusion over the average membrane separation. This compared extremely well with the reciprocal of the critical shear rate for inducing sponge transitions to the lamellar phase. The re-emergence of the isotropic sponge signal after the Couette rotation was stopped, proceeded as Avrami relaxation indicating random re-creation of sponge passages throughout the sample volume. As one would expect from the argument above, this time scaled with the critical shear rate for inducing the sponge to lamellar transition as the reciprocal of the cube of the membrane volume fraction, from 2 s for 7 vol. % down to ∼0.15 s for a 3 vol. % sponge and is close to the resolution limit of these measurements. The diffusive contact interval throughout a constant CpCl/hexanol composition dilution series was lower by a factor of ∼820 allowing the authors to extract the activation energy barrier to fusion and passage formation in this membrane system of 6.7 kB T . In a second sample series, varying the CpCl/hexanol membrane composition to cross the stability range of the L3 phase, they observed a linear change in relaxation time and therefore activation energy with mass ratio increasing to a maximum of 10 s at the boundary


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to the adjacent L3 /Lα coexistence region; the activation energy of 10.3kBT falls to 5.8kB T at the opposite boundary where the sponge phase coexists with isotropic solvent rich regions. Since membrane fusion will necessarily depend strongly on membrane curvature, as does the equilibrium phase behaviour, these last values are, in principle, measurements of Gaussian curvature energy changes in the CanhamHelfrich Hamiltonian, constraining theories of intermediate structures in fusion, such as the “stalk” connection which has been postulated to form between proximal membrane leaflets on contact (see Fig. 11.11). A common technical factor in TR-measurements is the way in which macroscopic considerations limit the precision to which the measurement can be initialised; for example, mixing times in stopped-flow cell studies, thermal conduction times in T-jump measurement, pressure ramp times in P-jumps, and in this study in hydrodynamic flow limits. In addition to the braking time of the Couette shear cell mentioned above, the authors pointed out the necessity of taking into account the hydrodynamic stopping time required for the fluid itself to come to rest which is inversely proportional to the solvent viscosity. For typical dilute sponge phases, this stopping time would have been longer than the observed relaxation time which scales with the viscosity. In this case the measurement itself is made possible by a change in the ratio of these effects: the stopping time decreases with viscosity, while the relaxation time increases, resulting in a relative gain factor of more than two orders of magnitude. Consideration of such effects is important to ensure that scattering measurements aimed at microscopic structure are not compromised by macroscopic fluid effects. This may have a bearing on the limiting mixing time in stopped-flow measurements.

11.6.2 Biomolecular Systems SANS has proved to be a suitable method to obtain information about the structure of biomolecules such as proteins and lipids [38–40]. TR-SANS is able to trace the time-dependent interaction of these molecules in the second and millisecond range. A protein gel can be characterised by a fractal structure whose dimension may increase due to enhanced branching of the aggregates with increase in temperature. Chodankar et al. [38] have shown the effect of pH and protein concentration on the temperature-induced protein gelation of bovine serum albumin (BSA) using phase diagrams, SANS and TR-SANS. The author has reported that above the isoelectric pH, the gelation temperature of the protein increases considerably with increasing pH. However, the increase in protein concentration leads to a decrease in gelation temperature in the low-protein-concentration regime and shows a small increasing trend at higher protein concentrations. The growth of the temperaturedependent fractal structure can be delayed by the combination of the increase in solution pH and protein concentration. With the help of TR-SANS and temperaturejump measurements, the kinetics of the formation of protein gels and subsequent transformation within a time of 1 min has been recorded. Differences are observed

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between gradual heating and temperature jump in such systems. In particular, at higher temperatures where gradual heating shows a fractal structure, there is a collapse of the fractal structure during temperature-jump heating [38]. Arnaudov et al. [39] studied the kinetics of heat-induced fibrillar aggregation of bovine β-lactoglobulin at pH 2.0 and 80 ◦ C by TR-SANS. Similar to the first example, this experimental set-up does not require any special cycling data acquisition system due to the time resolution on the order of minutes. TR-SANS data was collected to understand the kinetics of fibril formation from free monodisperse monomeric β-lactoglobulin and the process of irreversible fibrillar aggregation [39]. Using deuterated proteins, Rossle et al. [40] investigated the reaction of GroE (chaperonin from Escherichia coli) with the nucleotide analogue AMP-PNP under different contrast conditions with stopped-flow. Two conformational intermediate states of the chaperonin pathway were identified with 1 s time resolution [40].

11.6.3 Mesophase Precipitation Due to their unique structural properties, mesoporous solids have been used for a range of environmental and technological challenges. Considerable effort has been dedicated to developing these materials with pore sizes in the 2–50 nm range. Most mesoporous solids have irregularly spaced pores with a wide pore size distribution. The discovery of ordered mesoporous materials with regular channel systems, long range order, high thermal stability and the potential for isomorphous substitution, have lead to rapid advances in this area and given rise to new opportunities in materials chemistry and catalysis [41–44]. To gain a better understanding of the templating and growth mechanism in these materials, TR-SANS has been widely used. Né et al. [45] have reported the in situ TR-SAXS and SANS to investigate the mechanism of bulk precipitation of a hexagonally ordered ZrO2 /surfactant mesophase. The precipitate has been produced by the addition of sulphate to an oxychloride zirconium/hexadecyltrimethylammonium bromide CTAB water solution. CTAB acts as a structure-directing agent in the precipitation reaction. The precipitate obtained after a few seconds is organised in a 2D hexagonal network (Fig. 11.12). Figure 11.12 shows the SANS obtained after the two precursor solutions of CTAB/ZrOCl2 and MgSO4 are mixed via a stopped-flow system. The data shows the consumption of micelles and the growing of grains. There are no pre-existing cylindrical micelles in the precursor solution and the spherical micelles act as a reservoir for the monomer to feed the grain growth. The precipitation occurs through a two-step mechanism: a random nucleation/growing process and a first-order reorganisation from locally ordered cylinders to a hexagonal lattice in the grain at its final size [45].

11 Application of Time-Resolved Small Angle Neutron Scattering... I (cm–1)

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In a related example, Zholobenko et al. [46] explored the synthesis of mesoporous material (SBA-15 and SBA-15-TMB) using SANS. The study reports three major stages of SBA-15 synthesis using a block copolymer P123 as the templating agent. In situ TR-SANS revealed that only spherical micelles of P123 block copolymer were present in the synthesis mixture during the first stage, within the first 5 min of the reaction. Due to very high contrast between the organic micelles and the deuterated solvent, neutron scattering is the most appropriate technique to provide insight into the synthesis process. The formation of hybrid organic–inorganic micelles in the second stage of synthesis is observed between 5 and 20 min of the reaction; this can be seen from the transformation from spherical to cylindrical micelles which takes place before the precipitation of the ordered SBA-15 material. After 20–25 min (the third stage), these micelles begin to aggregate into a two-dimensional hexagonal structure. Zholobenko et al. [46] shows that the precipitation takes place as the result of self-assembly of the hybrid cylindrical micelles into the two-dimensional hexagonal structure of SBA-15. Initially, the cylinders are only weakly linked via their coronas in the hexagonal matrix. As the synthesis proceeds, the voids between the cylinders are increasingly filled with the silicate species which undergo condensation reactions resulting in cross-linking and covalent bonding between coronas of the cylindrical micelles. After about 1 h of the reaction, the scattering density in the spaces between the cylinders is very close to that in the micelle corona. The condensation of the silicate species continues for the duration of the experiment over approximately ∼20 h at 40 ◦ C. At this stage, the diameter of the

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inner core of the resultant SBA-15 structure remains equal to the diameter of the PPO core in the surfactant-silicate cylindrical micelles. Subsequent exposure of the synthesis mixture to higher temperatures, typically 80–100 ◦C under hydrothermal conditions and about 500 ◦ C during calcination, leads to further condensation and densification of the structure, accompanied with a decrease in the unit cell parameter and an increase in the pore size, resulting in the formation of a highly ordered and thermally stable SBA-15 mesoporous phase [46].

11.6.4 Research with TISANE The only experimental demonstration of the TISANE method thus far has been conducted by Wiedenmann et al. [47] on three magnetic nanoparticle systems, two of which were solutions of ferrofluids (Fe3 O4 and Co) and the other a dispersion of cobalt nanoparticles in metallic copper [47, 48]. It is known that magnetic singledomain nanoparticles in colloidal dispersions spontaneously form dipolar chains in head-to-tail conformation. The presence of an external magnetic field not only can induce the growth and alignment of these chains but may also introduce lateral interactions resulting in the formation of elongated magnetic sheets with a range of symmetries, e.g. hexagonal, that can coexist with the chain phase. To investigate the kinetics of ordering and relaxation, all three systems were studied in an oscillating applied magnetic field (amplitudes up to 40 mT and frequencies between 50 and 500 Hz). There are two processes that control the thermal reorientation of a magnetic moment in a magnetic colloid: Brownian rotation of the particle and and/or Néel relaxation of the particle moment. In the case of the ferrofluids, dipolar interactions between particles influence both mechanisms, whereas, in the case of the single-domain superparamagnetic Co precipitates embedded in a solid Cu matrix, magnetic particle moments can be reoriented only via the Néel mechanism. Thus, the ferrofluids were also investigated as a function of temperature below the melting point of the carrier fluid (where Brownian motion effects are removed) and above the melting temperature. The frequency and temperature dependence therefore allows the relevant relaxation mechanisms to be established. The authors demonstrated that the dynamics of the field-induced ordering in the ferrofluid systems is governed by the fast Brownian rotation of individual nanoparticles and small aggregates, while the magnetic relaxation of longer dipolar chains and local hexagonal domains is much slower. In both ferrofluid systems, the magnetic field oscillations induce fluctuations in the scattering intensity. Once the oscillating field is switched on, larger structures grow that become partially aligned when the field reaches its maximum value. Subsequently, these structures either completely disintegrate when the field direction changes orientation or, if the frequency is high enough, the magnetic moments will change orientation and the particle positions will remain essentially unchanged. The formation of these locally ordered domains in both dispersions is mainly governed by magnetic interactions


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between single dipolar chains, while the disintegration of these structures upon reducing the field is predominantly determined by diffusion. Correspondingly, at temperatures far below the melting point of the solvent, the inter-particle contribution is close to unity, manifesting as only relatively small structures if cooling is performed in zero field. Upon increasing the temperature, the scattering increases and reaches a maximum just above the melting point in the direction perpendicular to the field. Subsequently, when the temperature, is increased to room temperature, there is a corresponding decrease again as a result of growing thermal fluctuations. In contrast, no field-induced inter-particle correlations occur in the diluted solid CuCo alloy where the moment relaxation is purely of fast Néel type and which follows the oscillating field at all temperatures.

11.7 Conclusions and Outlook There is evidently a vast array of science that operates within the sub-second timedomain. However, it is only now with the advances in the intensy of neutron sources, optimised instrument design, more sophisticated data acquisition protocols, fast detection electronics, synchronised control techniques, advanced sample environments and the development of new real-time techniques, that SANS is rightly taking its place in the arsenal of methods for fine time-resolution studies in soft condensed matter. Indeed, the SKIN conference, now in its second meeting, is focused in extending the utilisation of SANS and neutron reflectometry to ever improved time resolution. In the coming decade, we will no doubt see SANS extended into studies of systems with both enormous technological importance (such as electro-rheological fluids and complex fluids induced by oscillatory shear) or of biological importance (e.g. muscle contraction and the conformational changes of light-responsive organisms). In addition, the new high intensity spallation sources, namely SNS, J-PARC, ISIS-TS2 and, in the near future, the ESS, with their intrinsically better time resolution, are likely to have a major impact in this area. Given these advances, there has never been a better opportunity for their detailed investigation.

References 1. Elder KR, Rogers TM, Desai RC (1988) Early stages of spinodal decomposition for the CahnHilliard-Cook model of phase separation. Phys Rev B 38:4725–4739 2. Bleuel M, Littrell K, Gähler R, Lal J (2005) MISANS, a method for quasi-elastic small angle neutron scattering experiments. Phys B 356:213–217 3. Kipping D, Gähler R, Habicht K (2008) Small angle neutron scattering at very high time resolution: principle and simulations of ‘TISANE’. Phys Lett A 372:1541–1546 4. Porcar L, Hamilton WA, Butler PD, Warr GG (2004) Relaxation of a shear-induced lamellar phase measured with time-resolved small-angle neutron scattering. Phys B 350:e963–e966

316

J.P. Mata et al.

5. Gilbert EP (2001) Report on small-angle neutron scattering for the Australian Replacement, Research Reactor. http://www.ansto.gov.au/data/assets/pdf file/0015/3804/w7report.pdf. 16–17 6. Porcar L, Hamilton WA, Butler PD, Warr GG (2002) A vapor barrier Couette shear cell for small angle neutron scattering measurements. Rev Sci Instrum 73:2345–2354 7. Porcar L, Hamilton WA, Butler PD, Warr GG (2004) Topological relaxation of a shear-induced lamellar phase to sponge equilibrium and the energetics of membrane fusion. Phys Rev Lett 93:198301 8. Bras W, Ryan AJ (1998) Sample environments and techniques combined with small angle X-ray scattering. Adv Colloid Interface Sci 75:1–43 9. Grillo I (2009) Applications of stopped-flow in SAXS and SANS. Curr Opin Colloid Interface Sci 14:402–408 10. Akiyama S, Takahashi S, Kimura T, Ishimori K, Morishima I, Nishikawa Y, Fujisawa T (2002) Conformational landscape of cytochrome c folding studied by microsecond-resolved smallangle X-ray scattering. Proc Natl Acad Sci U S A 99:1329–1334 11. Balsara NP, Lin C (1996) Early stages of nucleation and growth in a polymer blend. Phys Rev Lett 77:3847–3850 12. Hammouda B, Balsara NP, Lefebvre AA (1997) Small-angle neutron scattering from pressurized polyethylbutylene/polymethylbutylene blends. Macromolecules 30:5572–5574 13. Tate MW, Gruner SM, Shyamsunder E, d’Amico KL (1992) Kinetics of the lamellar-inverse hexagonal phase transition determined by time-resolved X-ray diffraction. Biochemistry 31:1081–1092 14. Pullen SA, Gilbert EP, Olsen SR, Lang EA, Doolan KR (2008) In-situ rapid heat – quench cell for small angle neutron scattering. Meas Sci Technol 19:65707–65714 15. Gilbert EP, Sutton D, Nelson A, Terrill N, Martin C, Lal J, Lang E (2005) Phase separation in the organic solid state: the Influence of quenching protocol in unstable n-alkane blends. In: Proceedings of the 16th international conference on the chemistry of the organic solid state molecular crystals and liquid crystals, vol 440. pp 93–105 16. Laggner P, Kriechbaum M, Rapp G (1991) Structural intermediates in phospholipid phase transitions. J Appl Crystallogr 24:836–842 17. Caffrey M, Magin RL, Hummel B, Zhang J (1990) Kinetics of the lamellar and hexagonal phase transitions in phosphatidylethanolamine. Time-resolved X-ray diffraction study using a microwave-induced temperature jump. Biophys J 58:21–29 18. Osaka N, Takata S, Suzuki T, Endo H, Shibayama M (2008) Comparison of heat- and pressureinduced gelation of ß-lactoglobulin aqueous solutions studied by small-angle neutron and dynamic light scattering. Polymer 49:2957–2963 19. Rosenbaum G, Holmes KC, Witz G (1971) Synchrotron radiation as a source for X-ray diffraction. Nature 230:434–437 20. Narayanan T (2009) High brilliance small-angle X-ray scattering applied to soft matter. Curr Opin Colloid Interface Sci. doi:10.1016/j.cocis.2009.05.005 21. Weiss TM, Narayanan T, Wolf C, Gradzielski M, Panine P, Finet S, Helsby WI (2005) Dynamics of self-assembly of unilamellar vesicles. Phys Rev Lett 94:038303 22. Lund R, Willner L, Monkenbusch M, Panine P, Narayanan T, Colmenero J et al (2009) Structural observation and kinetic pathway in the formation of polymeric micelles. Phys Rev Lett 102:188301 23. Ihee H (2009) Visualizing solution-phase reaction dynamics with time-resolved X-ray liquidography. Acc Chem Res 42:356–366 24. Higgins JS, Benoit H (1993) Polymers and neutron scattering. Oxford Science, Oxford 25. Lopez-Rubio A, Gilbert EP (2009) Neutron scattering: a natural tool for food science and technology research. Trends Food Sci Technol. doi:10.1016/j.tifs.2009.07.008 26. Lemke H, Burghammer M, Flot D, Rossle M, Riekel C (2004) Structural processes during starch granule hydration by synchrotron radiation microdiffraction. Biomacromolecules 5:1316–1324


317

27. Zhao Y, Tanaka H, Miyamoto N, Koizumi S, Hashimoto T (2009) Combined SANS, SEC, NMR, and UV–Vis studies of simultaneous living anionic copolymerization process: simultaneous elucidation of propagating living chains at three different length scales. Macromolecules 42:1739–1748 28. van den Brandt B, Glättli H, Grillo I, Hautle P, Jouve H, Kohlbrecher J, Konter JA, Leymarie E, Mango S, May RP, Stuhrmann HB, Zimmer O (2003) Neutron scattering from polarised proton domains. Phys B Condens Matter 335:193–195 29. Heath CH, Streletzky KA, Wyslouzil BE, Wölk J, Strey R (2003) Small angle neutron scattering from D2 O–H2 O nanodroplets and binary nucleation rates in a supersonic nozzle. J Chem Phys 118:5465–5473 30. Wang H, Zhao B, Wyslouzil B, Streletzky K (2002) Small-angle neutron scattering of soot formed in laminar premised ethylene flames. Proc Combust Inst 29:2749–2757 31. Willner L, Poppe A, Allgaier J, Monkenbusch M, Richter D (2001) Time-resolved SANS for the determination of unimer exchange kinetics in block copolymer micelles. Europhys Lett 55:667–673 32. Halperin A, Alexander S (1989) Polymeric micelles: their relaxation kinetics. Macromolecules 22:2403–2412 33. Oda R, Weber V, Lindner P, Pine DJ, Mendes E, Schosseler F (2000) Time-resolved smallangle neutron scattering study of shear-thickening surfactant solutions after the cessation of flow. Langmuir 16:4859–4863 34. Nakano M, Fukuda M, Kudo T, Endo H, Handa T (2007) Determination of interbilayer and transbilayer lipid transfers by time-resolved small-angle neutron scattering. Phys Rev Lett 98:238101 35. Gradzielski M, Grillo I, Narayanan T (2004) Dynamics of structural transitions in amphiphilic systems monitored by scattering techniques. Progr Colloid Polymer Sci 129:32–39 36. Gotter M, Sottmann T, Baciu M, Olsson U, Wennerström H, Strey R (2007) A comprehensive, time-resolved SANS investigation of temperature-change-induced sponge-to-lamellar and lamellar-to-sponge phase transformations in comparison with 2H-NMR results. Eur Phys J E 24:277–295 37. Hamilton WA, Butler PD, Magid LJ, Han Z, Slawecki TM (1999) Fast relaxation of a hexagonal Poiseuille shear-induced near-surface phase in a threadlike micellar solution. Phys Rev E 60:R1146–R1149 38. Chodankar S, Aswal VK, Kohlbrecher J, Vavrin R, Wagh AG (2009) Small-angle neutron scattering study of structure and kinetics of temperature-induced protein gelation. Phys Rev E 79:021912 39. Arnaudov LN, de Vries R, Cohen Stuart MA (2006) Time-resolved small-angle neutron scattering during heat-induced fibril formation from bovine ß-lactoglobulin. J Chem Phys 124:084701 40. Rössle M, Manakova E, Holzinger J, Vanatalu K, May RP, Heumann H (2000) Time-resolved small-angle neutron scattering of proteins in solution. Phys B 276–278:532–533 41. Kresge CT, Leonovich ME, Roth WK, Vartuli JC, Beck JS (1992) Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism. Nature 359:710–711 42. Beck JS, Vartuli JC, Roth WJ, Leonwicz ME, Kresge CT, Schmitt KD et al (1992) A new family of mesoporous molecular sieves prepared with liquid crystal templates. J Am Chem Soc 114:10834–10843 43. Inagaki S, Fukushima Y, Kuroda K (1993) Synthesis of highly ordered mesoporous materials from a layered polysilicate. J Chem Soc Chem Commun 8:680–681 44. Zhao D, Feng J, Huo Q, Melosh N, Fredrickson GH, Chmelka BF, Stucky GD (1998) Triblock copolymer syntheses of mesoporous silica with periodic 50 to 300 angstrom pores. Science 279:548–552 45. Né F, Testard F, Zemb Th, Grillo I (2003) How does ZrO2 /surfactant mesophase nucleate? Formation mechanism. Langmuir 19:8503–8510 46. Zholobenko VL, Khodakov AY, Impéror-Clerc M, Durand D, Grillo I (2008) Initial stages of SBA-15 synthesis: an overview. Adv Colloid Interface Sci 142:67–74

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47. Wiedenmann A, Keiderling U, Meissner M, Wallacher D, Gähler R, May RP, Prévost S, Klokkenburg M, Erné BH, Kohlbrecher J (2008) Low-temperature dynamics of magnetic colloids studied by time-resolved small-angle neutron scattering. Phys Rev B 77:184417 48. Wiedenmann A, Keiderling U, Habicht K, Russina M, Gähler R (2006) Dynamics of field-induced ordering in magnetic colloids studied by new time-resolved small-angle neutronscattering techniques. Phys Rev Lett 97:057202 49. Gähler R (2009) Personal communication

Chapter 12

Understanding the Stability of Micellar Systems of Interest for the Study of Glasses, Freezing and Soft Confinement Tinka Spehr and Bernhard Frick

12.1 Introduction This chapter focuses on neutron scattering studies of microemulsions in the droplet phase (L2 -phase) at low tempertaures, in fact at such low temperatures that the liquid constituents of the microemulsions tend to freeze or form glasses. One may argue that the freezing or the glass transition phenomena are already sufficiently complex, so why would we want to study these phenomena in microemulsions? There are several motivations. First, microemulsions consist of self-organized micro- or mesoscopic structures whose dimensions can be easily tuned. For scientists who study freezing, supercooling or glass-forming phenomena this is a very attractive feature, because the existence of a correlation length related to molecular rearrangements or at least its importance for the glass transition is disputed. Therefore, it is important to know if microemulsions allow for creating compartments of variable size in which glass-forming liquids can be confined. This would be an ideal means for studying the influence of finite size, i.e. spatial confinement, on the structure and dynamics of confined liquid. We also ask if these confining mesoscopic structures can easily be converted from three to two dimensions, e.g. by organization from droplet into lamellar structures. If the answer is yes, then this could be used to study the influence of the dimensionality of the confinement on the dynamics. Furthermore, understanding the dynamics of especially water in confined spaces is important in other fields such as biology, as has been discussed in previous chapters of this book. Another interesting question

T. Spehr () Institut für Festkörperphysik, TU Darmstadt, Hochschulstr. 8, D-64289 Darmstadt, Germany e-mail: [email protected] B. Frick Institut Laue-Langevin, 6, rue Jules Horowitz, F-38042 Grenoble, France e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 12, © Springer Science+Business Media, LLC 2012

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concerns the role of wall interactions: are the dynamics different when a liquid is enclosed in a porous hard host and is in contact with rigid walls, compared with the same liquid in a “soft confinement” of microemulsion compartments and is in contact with highly mobile walls. Finally, confined liquids in microemulsions play an important role in other fields of science and in applications [52]. In order to answer any or all of these questions, we need to understand first the complex dynamics of microemulsions themselves. This leads to the second reason for studying microemulsions at low temperatures. Like for many complex systems it is important to separate the dynamic processes of the multi-constituent systems. This can be conveniently done by freezing out certain types of motion, which then hopefully helps for a better understanding of the global dynamics. In determining the structural thermal stability of microemulsions neutron small angle scattering (SANS) plays a key role here since it probes the length scales of interest and makes use of contrast matching using H/D selectivity. This contrast matching tool can also be used to highlight certain types of motions and characterize the dynamics of these systems. Of course, neutrons can be used to unravel only a part of the complex structure and dynamics scenario, thus as usual a combination of different experimental methods as was discussed in Chap. 1, is the most fruitful strategy. This chapter is structured as follows: in Sect. 12.2 we will briefly introduce aspects of microemulsion structures and dynamics. We mainly consider water-in-oil droplet microemulsions and their behaviour at low temperatures, as probed by neutrons and other techniques. Section 12.3 is then devoted to a short review of neutron scattering experiments near room temperature. In Sect. 12.4 we discuss scattering experiments at low temperatures focussing on confined water systems (reverse water-swollen micelles). In Sect. 12.5 we review experiments on microemulsions for studying glass formation. We hope to illustrate the possibilities offered by neutron scattering in this complex field, with the the examples chosen in Sects. 12.3 to 12.5.

12.2 Microemulsion Characteristics 12.2.1 What Are Microemulsions? When water and oil are mixed they separate into two macro-phases of oil and water. Upon addition of surfactants, thermodynamically stable, nano-phase separated mixtures, called microemulsions, are formed. The term microemulsion was first used by Schulman and Hoar to name the spontaneously forming isotropic solution of water, a non-polar liquid, surfactant and alcohol [26], more than 60 years ago. Together with Winsor they systematically investigated the rich phase behaviour of such mixtures [27]. Nowadays microemulsions are more generally defined as

12 Understanding the Stability of Micellar Systems...

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an isotropic and macroscopically homogenous mixture of a polar and a non-polar liquid mediated by one or several surfactants. The term surfactant is an English coinage for surf ace active agents and it denotes amphiphilic molecules that unite both a hydrophilic and a hydrophobic part in one molecule. For pure water (or pure oil) the addition of surfactants in very low concentrations changes the surface tension and one observes an enrichment of the surfactant at the water–air (or oil– air) interface. With increasing surfactant concentration (above the critical micelle concentration) the surfactant molecules self-organize into nano-sized objects. The objects formed are called micelles or reverse micelles, depending if their tails or heads aggregate into the interior of the micelle (see sketch in Fig. 12.1 below for a reverse micelle). The shape of the micelles depends on the type of surfactant; often nearly spherical. At very high surfactant concentrations the surfactants may even form liquid crystalline phases. Adding surfactants to oil–water mixtures one observes an enrichment of surfactant molecules at the oil–water interface where the hydrophobic tails can dive into the oil and the polar hydrophilic heads into the water. Though not limited to oil–water mixtures we consider now only these. With increasing concentration the oil–water interface can be increased by nano-structuring the liquid, i.e. nanocompartments of oil or of water are formed which are separated by surfactant monolayers. Classified after their major component one speaks of water-in-oil (w/o) or oil-in-water (o/w) microemulsions. Depending on the nature of the surfactant, the composition and other parameters like temperature or pressure, a variety of different structures are formed. These include lamellar, droplet or bi-continuous phases. In what follows we consider the so-called L2 -phase of a ternary microemulsion consisting of water, the anionic surfactant AOT and oil. Figure 12.1 shows a phase diagram for this water/AOT/oil system around room temperature. The L2 -phase forms in the highlighted area: the microemulsion consists of spherical water droplets that are coated by a mono-layer of surfactant, dispersed in the continuous oil matrix. A reverse water-swollen micelle is shown on the right of Fig. 12.1. Two parameters are commonly used to describe the composition of this type of microemulsion: • The molar ratio of water to AOT, ω = n[water] n[AOT] , which defines the size of the droplets. V , which determines the droplet density • The volume fraction of droplets, φ = Vdrop tot in the oil matrix. For a more detailed discussion of microemulsions we refer the reader to a recent book covering the basics of microemulsion formulation as well as their applications [52]. Before starting the discussion of the dynamics of microemulsions let us recall the relation between the properties of the surfactant film and the microemulsion structure. In Fig. 12.1 we show the chemical structure of the surfactant AOT (sodium bis[ethylhexyl] sulfosuccinate). With its two hydrocarbon tails attached to the anionic headgroup the AOT molecule has a conical shape. When many of these molecules arrange at the oil–water interphase this results in a spontaneous

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Fig. 12.1 Top: Ternary phase diagram of the water/AOT/decane system at 25◦ C from [10], reproduced in [9]. In the L2 -area the microemulsion consists of reverse water swollen micelles as shown in schematic drawing to the right. Bottom: Chemical structure of an AOT molecule, showing its conical shape: two hydrophobic hydrocarbon tails are attached to the hydrophilic head group. An AOT-surfactant film at the oil–water interface has thus a spontaneous curvature towards water which favours the formation of water-in-oil droplets

curvature of the surfactant film towards water. In general, the formation of a certain microemulsion structure is closely related to the spontaneous curvature C0 of the surfactant film. After Helfrich the bending energy per unit area reads: F=

κ (C1 + C2 − 2C0 )2 + κ˜ C1C2 , 2

(12.1)

where C1 and C2 are the two principal curvatures of the surfactant membrane and κ , κ˜ are bending moduli [23]. For C0 = 0 a flat surfactant film is favoured. In the following we will focus on water-in-oil droplets where the surfactant film is curved towards the water; C0 is in that case per definition negative.


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Microemulsions are highly dynamic objects with characteristic times spread over a wide range, from ms to ps (e.g. [55]). At the long times dynamical processes include the uptake of oil (water) into (reverse) micelles, the diffusion of the micellar nano-objects, their collision and separation/coalesence – the mechanism for exchange of (water) oil between (reverse) micelles –, the rotation of the micellar nano-objects and their shape fluctuations. A number of experimental techniques can be used to probe these motion (see review in [55]), including nuclear magnetic resonance (NMR), ultrasound, fluorescence quenching in combination with stopped flow or electron paramagnetic resonance (EPR). Neutron scattering methods can access the long timescales via kinetic experiments (e.g. SANS with stopped flow), and NSE, which can probe motions to t < 10 μs like diffusion and shape fluctuations. On the shorter timescales, t < 10 μs, diffusion and rotational motions of the smaller constituents (oil, water and surfactant molecules) are observed. In this time range we are mainly concerned with the dynamics of the confined liquid phases, which is expected to be a function of the spatial constraint in nanometersize compartments. In contrast, the dynamics of the continuous phase should be very similar to that of the bulk liquid. It is important to note that liquids can be geometrically confined only on a relatively short timescale compared with the micellar exchange. The dynamics at this short timescale can be probed only by a few experimental methods but can also be studied by molecular dynamic simulations (see e.g.: [1,8,14,34]). Here we focus mainly on results obtained by inelastic neutron scattering studies. Moreover we will discuss only experimental results obtained at low temperatures, in particular below the freezing point of bulk water T = 273 K. Reverse water swollen AOT micelles dispersed in oil have been the subject of various studies near room temperature but rarely below the freezing point of bulk water. Similarly, investigations on the dynamics of microemulsions in general were carried out mainly near room temperature, employing a large number of experimental techniques. As already mentioned the intention of our own recent neutron scattering studies was to characterize if reverse microemulsions were suitable systems for the study of freezing and supercooling of water in soft confinement. The first low temperature study on a water-in-oil droplet microemulsion was carried out 30 years ago. By means of photon correlation spectroscopy the diffusion coefficient of reverse water-swollen micelles in water/AOT/iso-octane with 0 < ω < 60 was measured for temperatures still above T = 273 K. Assuming that the spherical shape of the reverse micelles was maintained their radii were calculated using the Stokes-Einstein relation [60]. Microemulsions with ω > 10 were found to be unstable below about T = 291 K and showed a size increase at temperatures above this stability boundary. For the smaller micelles (ω < 10) a constant radius was observed between T = 291 K and about 313–323 K and for nominally water free micelles, ω = 0, the radius did not change between T = 253 K and T = 368 K. In addition, for ω = 0 no cloud point was detected down to T = 188 K. A series of studies using different experimental techniques have focused on the freezing behaviour of the water confined to reverse micelles. Results from differential scanning calorimetry DSC [7, 21, 44, 46], fluorescence probing [38],

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infrared (IR) spectroscopy [39], NMR [21, 41] and neutron backscattering [47] showed contradictory results: whereas Nucci and coworkers [39] reported a freezing temperature of the water core independent of droplet size, most other groups found a strong correlation between freezing temperature and droplet size. All experiments agreed in revealing a number of closely bound water molecules attached to the AOT surfactant layer, also confirmed by more recent studies performed around room temperature using mid-infrared ultrafast pump-probe spectroscopy [12] and IR spectroscopy [40]. Experimental results indicated that the water core of the reverse micelles is composed of a water layer strongly immobilized by the anionic surfactant head groups and a core where the water behaviour approaches that of bulk water. Whereas there seems to be consensus that water fractions with different behaviour exist, the actual amount of bound water and the exchange properties of these different fractions is still the subject of discussion [40]. Published numbers for bound water molecules range from 2 to 10 per AOT molecule [47].

12.3 Scattering Results on Reverse Micelles Above Room Temperature 12.3.1 Neutron Instruments for the Study of Microemulsions The static structure can be measured using SANS instruments and neutron diffractometers. Two of the most advanced instruments of each type are D22, a SANS instrument and D7, a diffuse scattering spectrometer which offers the unique possibility to separate coherent and incoherent scattering with energy analysis. Both instruments are located at ILL in France. SANS is probably the best known technique in neutron scattering, applied to determine mesoscopic structures: it gives information about shape, size and spatial organization of scattering objects with ˚ to hundreds of nanometers. The length scales probed are sizes in the order of a few A ideal for soft condensed matter systems including polymers, colloids or surfactant aggregates. A listing of worldwide accessible SANS instruments can be found under [62]. Wide-angle neutron diffractometers can be used in combination with H/D exchange to determine the local structures in the sample. When using polarized neutrons like on D7, the coherent scattering, which contains the structural information, can be separated from the typically high but weakly angular dependent incoherent background and can be measured even for disordered systems in absolute units if the incoherent scattering cross section is known. The separation of coherent and incoherent scattering by means of polarization analysis is rarely applied in SANS. To measure dynamics, neutron spin echo (NSE), introduced by Mezei [36], offers the best energy resolution and a wide dynamic range. In contrast to other QENS spectrometers, NSE uses the neutron spin to determine the energy transfer.


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By measuring the spin polarization of the neutron beam the intermediate scattering function I(Q,t) is deduced directly in time and space. The energy resolution of NSE is given by the longest measurable time, which at present reaches Fourier times of 1 μs with both the IN15 instrument (ILL) and the NSE spectrometer at the SNS (USA). NSE is used primarily to probe coherent scattering, although it is feasible to measure dominantly incoherent scattering samples with a somewhat higher background. More difficult is the measurement of mixed coherent and incoherent dynamics, which can only be reliably determined if the dynamics of both components are sufficiently separated in time at the Q-range of interest, because coherent and incoherent scattering result in a beam polarization of opposite sign. The other type of spectrometers, backscattering (BSS) [61], time-of-flight (TOF) and spallation source TOF-BSS measure the dynamic scattering law S(Q, ω ) in energy space. These spectrometers all have different dynamic ranges (energy spanned between resolution width and maximum energy transfer) and energy resolution. Typically improving energy resolution involves a decrease in count rate. The dynamic range covered by the spectrometer is either limited by the mechanics of a Doppler drive (for BSS) or by the time separation of neutron pulses (for TOF and TOF-BSS). The best energy resolution is obtained for BSS where one uses perfect crystals (mostly Si(111)) in backscattering reflection (Bragg angle of 90◦ ). This enables investigations of high momentum transfers with high energy resolution. For TOF high energy resolution results from a high time resolution of the flight time, which is achieved for long flight paths and/or long neutron wavelengths. A high neutron wavelength will severely limit the momentum transfer range according to the kinematic conditions. The limited dynamic range of each technique makes it necessary to combine data from several instruments in order to cover a wide energy (time) range. A detailed discussion of theoretical basics and applications of QENS can be found in [5]. The analysis of data collected on different instruments involves the convolution of a theoretical model function with the respective instrumental resolution function and by fitting to the measured data in the energy domain. The alternative is to Fourier transform the measured spectra, obtaining thus I(Q,t) and then dividing by the measured resolution (usually the sample at very low temperatures where all motions are frozen and elastic on the timescale of the spectrometer). Resolution corrected I(Q,t) from different spectrometers can then be visualized directly and evaluated together in time space over a wide time range (3–4 decades). The aforementioned QENS spectrometers are most frequently used for incoherent measurements taking advantage of the large cross section of hydrogen. Coherent scattering can be investigatet as well, since the signal obtained is the sum of the coherent and incoherent signals. A separation of the contributions is only possible by measuring reference samples which enable a correct “background correction” or by spectrometers using polarized neutrons like D7 or DNS (at FRMII, Germany), which provide separate coherent and incoherent inelastic contributions, though with modest energy resolution and at the expense of counting statistics.

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Fig. 12.2 Typical neutron scattering instruments suitable for the investigation of the static structure and dynamics of microemulsions. D22 (ILL) is a small-angle instrument and D7 (ILL) a polarized, diffuse scattering spectrometer with wide-angle diffraction option. The lower half shows inelastic spectrometers: for longest times (lowest frequencies) the NSE spectrometer with nearly 1 μs time resolution and then three complementary spectrometers operating in frequency space and covering together more than 3 decades in time. Lower left: IN16 (ILL) reactor BS spectrometer with resolution dE < 10−3 meV, middle BASIS (SNS) with dE ∼ 3.2 · 10−3 meV – the highest flux spallation source spectrometer and the TOF spectrometer IN5 (ILL) with variable wavelength dependent resolution dE ∼ 10−2 − 3 · 10−1 meV

A schematic of several of the instruments discussed here is shown in Fig. 12.2. For updated and more detailed information on neutron instruments the reader should visit facility webpages.1

12.3.2 SANS on Microemulsions There exists a vast literature on SANS data for reverse micelles and it is certainly beyond the scope of this chapter to review all of it. For a more comprehensive treatise of SANS as a tool to investigate microemulsions we refer to [19]. We merely choose a representative example which shows the pertinence of SANS

1 For example: ILL: www.ill.eu; FRM2:www.frm2.tum.de; SNS: http://neutrons.ornl.gov/, ISIS: www.isis.stfc.ac.uk.


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to structurally characterize microemulsions. We will constrain ourselves to the theoretical description of SANS intensity from dilute droplet microemulsions. For interacting poly-disperse droplets of number density n, where the position of the droplets is uncorrelated with their size, the scattering cross section per unit volume I(Q) can be written as the product of the averaged droplet form factor F(Q) and a structure factor S(Q) accounting for droplet–droplet interactions. In dilute systems where the positions of the individual scatterers are uncorrelated, the structure factor describing the interference of waves scattered by different scatterers will be S(Q) = 1. The intensity then reads: I(Q) = n · F(Q) + Iinc

(12.2)

For a microemulsion consisting of spherical water cores coated by a mono-layer of AOT dispersed in oil, the form factor F(Q) for a spherical core-shell particle holds: j1 (QRc ) j1 (Q(Rc + d)) 2 , F(Q) = 4π (ρc − ρs)R3c + 4π (ρs − ρm )(Rc + d)3 QRc Q(Rc + d) (12.3) where Rc is the radius of the core, d the thickness of the shell and ρc , ρs , ρm denote the coherent scattering length densities of core, shell and matrix respectively. j1 is the spherical Bessel function of first order. The size polydispersity of the droplets is accounted for by using in (12.2) a form factor F(Q) that is averaged over Schultz–Zimm distributed radii [33]: f (R, z) =

z+1 Rc

z+1 Rzc

1 z+1 Rc ; exp − Rc Γ (z + 1)

with the Gamma function Γ (x) = droplet size polydispersity through: p=

z > −1

(12.4)

∞ −t x−1 dt. The parameter z relates to the 0 e t

R2c − Rc 2 1 =√ Rc 1+z

(12.5)

Contrary to the Gauss distribution the Schultz-Zimm distribution is asymmetric and the unphysical case of negative radii is excluded. The big advantage of neutron scattering is that one can adjust the scattering length densities ρ of the different components by means of selective deuteration. For structural investigations the most appropriate mixture for our chosen example would be a mixture of deuterated oil and water, and protonated surfactant. This is called shell-contrast and the scattering is optimized as the incoherent flat background is minimal while the coherent contrast difference between shell and core/matrix is maximum. In Fig. 12.3 we show SANS curves for shell-contrast water/AOT/oil

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Fig. 12.3 SANS curves of four D2 O/AOT/d-oil microemulsions in the droplet phase. From top to bottom the molar ratio of water/AOT decreases from ω = 12 to ω = 3. Note that while all curves look qualitatively similar the characteristic minimum shifts to lower Q with increasing ω shown by the dotted arrow indicating that with increasing ω the droplets are becoming bigger. Solid lines are fits to a core shell-form factor and a structure factor (figure adapted from [50])

microemulsions in the L2 droplet phase around room temperature. From top to bottom the curves belong to samples with decreasing droplet radii with a molar ratio ω of water to AOT decreasing from 12 to 3. The solid lines correspond to fits with (12.2) using the above described core-shell form factor, (12.3). These fits yield the size of the droplet core Rc , the thickness of the surfactant shell d and the size polydispersity p of the droplets. Droplet core radii Rc deduced from SANS and small-angle x-ray scattering (SAXS) are shown in Fig. 12.4 as a function of ω . Note that the core radius increases linearly with ω . This result agrees with the theoretical prediction based on the assumption of a constant ω -independent head-group area of the AOT molecule [11]. For this microemulsion system one may thus control the ˚ size of the water confinement (=water core radius Rc ) between a few Angstroems and several nanometers.

12.3.3 Neutron Diffraction To our knowledge there are nearly no investigations using wide-angle neutron diffraction to study local structures in microemulsions, probably because it is expected that both the majority and minority phases maintain their local structures


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ω Fig. 12.4 Linear relationship between droplet core radius Rc and molar ratio ω . The dotted line shows the theoretical prediction based on simple geometrical considerationsassuming a constant volume of the confined water molecules and a constant AOT headgroup area [11] (figure from [50])

compared with the bulk. However, for low temperature investigations it might be interesting to check the local structure of the confined liquid. In Fig. 12.5 we show how a combination of SANS and wide-angle diffraction (here using polarized neutrons on D7 to separate the incoherent scattering and to measure Sloc (Q) in absolute units) can in principle reveal information on the droplet shape via the form factor F(Q), the droplet–droplet interaction via S(Q) and the local structure factor from neutron diffraction Sloc (Q).

12.3.4 NSE As already mentioned microemulsions are highly dynamic. Over time microemulsion droplets undergo translational diffusion in the oil matrix and their shape fluctuates. These thermally activated small fluctuations around the time-averaged spherical droplet shape are related to the bending energy. These fluctuations can be expressed by an expansion into spherical harmonics [37, 42] in which mainly the 0th and 2nd order terms are important. The thermal fluctuations corresponding to the 0th order term correspond to the size fluctuations (polydispersity), measured, e.g. by SANS, and are related to the the bending moduli of the surfactant membrane. The second order deformation modes are among other factors (like the viscosities

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Fig. 12.5 The combination of SANS and neutron diffraction allows for determining the form factor of the microscopic objects (here reverse micelles), the structure factor arising from the mesoscopic arrangement of the micelles as well as the local structure of the confined liquid (contrary to the shell contrast shown here a core contrast might be considered for the local structure)

ηw , ηt on both sides of the membrane, entropy of mixing h(φ )) related to the bending moduli (12.2) which influence the relaxation rates λ2 and the mean square amplitudes of fluctuation |u2 |2 as shown in the following: −1 R 3kB T |u2 |2 = kB T 16 κ − 12 κ˜ − h(φ ) Rs 4π

(12.6)

Based on the theoretical model for viscoelasticity of vesicle dispersions proposed by Seki and co-workers [45], the associated relaxation rate λ2 calculates with [30]: R 24 κ κ˜ 3kB T λ2 = 3 4 − 3 − h(φ ) (12.7) κ 4πκ 23ηw + 32ηt R0 Rs Here the viscosities inside ηw and outside ηt the fluctuating shell are explicitly considered. These fluctuations of the shell contribute to the decay of the intermediate scattering function as: I(Q,t) ∝ exp(−Ddrop Q2t) 4π [ j0 (QR)]2 + 5F2(QR)u2 (t)u2 (0)

(12.8)


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with the weighting factor F2 (QR) = [4 j2 (QR) − QR j3 (QR)]2 . The exponential term describes the translational diffusion of the entire droplet, the first term in the bracket is the static form factor of the shell, and the time dependence of the fluctuations is contained in the autocorrelation function of the fluctuation amplitude u2 . By evaluating the initial slope of the normalized intermediate scattering function one can therefore determine by NSE the bending elasticity κ through the measurements of the time constants 1/λ 2 of these fluctuations. Rather than evaluating this sum in exponentials one may take the initial slope of the normalized intermediate scattering function [37]. The measured normalized intermediate scattering function S(Q,t)/S(Q) can then be evaluated in terms of a single exponential: S(Q,t) = exp(−Deff (Q) · Q2t). S(Q)

(12.9)

The effective diffusion coefficient Deff (Q) has a characteristic Q-dependence: it contains the Q-independent translational diffusion coefficient Dt of the droplets and a Q-dependent term Ddef (Q) representing the droplet shape fluctuations [31]: Deff (Q) = Dt + Ddef (Q).

(12.10)

The deformation coefficient Ddef is defined as: Ddef =

1 λ2 |u2 |2 5 F2 (QR) . · 2 Q 4π [( j0 (QR)]2 + |u2 |2 5 F2 (QR)

(12.11)

Ddef has a peak centered at about the Q-value which corresponds to the minimum of the static form factor of the droplets and the peak form can now be evaluated to yield the dimensionless mean squared fluctuation amplitude |u2 |2 and the relaxation rate λ2 of the (l = 2) mode. Kawabata and co-workers derived the following expression for the bending modulus κ , where κ˜ and h(φ ) have cancelled out [30]: 1 κ= 48

23 ηw + 32 ηt kB T + λ 2 R3 2 πp 3

(12.12)

Equation (12.12) expresses the bending modulus κ in terms of parameters that can be accessed experimentally: polydispersity p and droplet radius R measurable by SANS and λ2 measurable by NSE. For a comprehensive and instructional treatise of NSE applied to study microemulsions we refer the reader to [24]. Here we will only point out a few recent studies. More than 20 years ago NSE was for the first time used to access the bending elasticity κ of surfactant films [15, 28] and since then it was many times successfully applied to measure shell fluctuations of many different microemulsion systems. Deduced values for κ range typically in the order of kB T . Recent studies

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tried to elucidate for example the impact of various parameters like charge density, concentration, co-surfactants or solvated macromolecules on surfactant film bending elasticity. No measurable difference was found between the elasticity of charged and uncharged microemulsion droplets [16]. Kitchens and co-workers varied the oil matrix that surrounds the droplets and studied the effect of bulk oil properties on the AOT bending elasticity [32]. Upon introducing proteins in water swollen AOT micelles it was found that the shell fluctuations were enhanced by the presence of the protein [25] The influence of temperature and pressure was investigated by Kawabata and co-workers [31]. As usual temperature and pressure were seen to have an opposite effect: whereas increasing temperature was found to decrease the membrane rigidity, pressure was found to increase its rigidity [31]. In summary, the cited studies evidence that the elastic properties can be altered and NSE allows for measuring the induced changes. Regarding confinement this means that one can, to a certain degree, tune the softness of the wall.

12.3.5 Time-of-Flight and Backscattering Results Incoherent QENS techniques are, besides NMR, possibly the most appropriate to directly study molecular motions in microemulsions. They are well adapted to measure molecular motions as they probe appropriate length and timescales and moreover contrast variation by selective deuteration offers the possibility to highlight motions of a particular component. But nevertheless while dynamics in water/AOT/oil microemulsions have extensively been studied by many different methods the number of QENS studies is rather small. The first QENS measurements on microemulsions date back about 25 years. Tabony and co-workers applied TOF spectroscopy to investigate monomer motions in water-in-oil droplet microemulsions [53, 54]. They studied the water/AOT/ d-cyclohexane system with ω ranging from 4.5 to 16 at room temperature. Evaluating spectra of samples containing H2 O and D2 O they accessed the dynamics of the confined water and described it by free translational diffusion, using a single Lorentzian with a Q2 -dependent linewidth. The resulting water diffusion ˚ 2 /ns was found to correspond to that of bulk aqueous coefficient of Dt = 120 A electrolytic solutions (half of its value in bulk water). Contrary to more recent studies no significant effect of the size of the micelles was observed. Furthermore they studied the AOT dynamics and its dependence on the core viscosity [17]. The AOT motion was found to be of a diffusive nature over the whole investigated Qrange, contrary to the rotational behaviour at low Q that was deduced from prior studies. When changing the polar core from water to the more viscous glycerol a considerable slowing down of the AOT molecules close to glycerol was observed. Altogether they came to the conclusion that there is no correlation between local AOT mobility and kinetics of droplet exchange. They also studied the influence on the water dynamics when solubilizing chymotrypsin enzymes inside the water core.


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In that case, a certain fraction of the water, around 250–500 molecules per enzyme, appeared to be immobilized whereas the dynamical behaviour of the other fraction seemed unaffected [17]. Despite the success of these pioneering experiments it took more than 10 years until AOT-based microemulsions were studied again by QENS. Comparing water in lecithin reverse micelles with water confined in water/AOT/cyclohexane microemulsions Aliotta and co-workers observed a stronger dependence of water dynamics on the micelle size for AOT than for lecithin [3]. Freda and co-workers applied BS to study the overall mobility of AOT reverse micelles in cyclohexane [18]. For “dry” reverse AOT micelles the QENS signal was entirely attributed to the overall motion of the micelle. With increasing water content ω > 2.5 they observed the onset of micelle intrinsic AOT dynamics. In a recent investigation, combining MD simulations and QENS experiments, Harpham and co-workers focused on the water motion in swollen d-AOT reverse micelles dissolved in isooctane-d8 [20]. For the smallest micelles with ω < 2.5 the translational component was too narrow to be resolved. For micelles containing 5 molecules of water per AOT the translational diffusion was found to be about five times slower than in bulk water at the same temperature. The deduced rotational diffusion coefficients increased with increasing droplet size and were slightly smaller than those of bulk water. Altogether the rotational mobility was found to be less affected than the translational mobility of the water. Results from MD simulations allowed the conclusion that the decoupling approximation for the different kinds of proton motions is justified [20].

12.4 Scattering Results on Micelles at Low Temperatures 12.4.1 Water-in-Oil Micelles at Low Temperatures SANS at low temperature: As described before water/AOT/oil systems form waterin-oil droplets over a wide range of compositions. By choosing the appropriate molar ratio ω , the size of the droplets can be varied in a controlled way between ˚ and several nm, see Fig. 12.4. This linear relationship between droplet a few A size and molar ratio ω has in the past only been confirmed experimentally around room temperature. But besides the requirement to offer a tunable size, an ideal system for the investigation of soft confined liquids should also allow variation of the temperature under otherwise unchanged conditions. For this reason, we have investigated the water/AOT/oil-microemulsion in the droplet phase using SANS to characterize the low temperature stability and structure of the water swollen reverse micelles [49]. Our recent study extended the phase diagram – well known around room temperature – into the region where the confined water is supercooled. SANS allowed us to determine the temperature dependent structure of the micelles in a direct way. Prepared under shell-contrast, four microemulsions with ω = 3, 5, 8 and 12 were measured as they were cooled between T = 300 K and 220 K in steps of Δ T = 1 K. Figure 12.6 shows three exemplary SANS curves of the ω = 8 sample

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Fig. 12.6 SANS intensity of D2 O/AOT/toluene-d8 (ω = 8) microemulsion at three different temperatures. The error indicates the shifting of the form factor minimum to higher Q-value with lowering of the temperature while the qualitative shape of the scattering curve remains unchanged: the droplets decrease with decreasing temperature (figure from [49])

(D2 O/AOT/toluene-d8) at temperatures of T = 290, 250 and 230 K. The abovementioned core-shell model describes all data very well over the whole investigated temperature range. The reverse micellar structure seems to be preserved, only the size of the droplets changes. With decreasing temperature the droplet size shrinks which can already be seen by a clear shift of the form factor minimum to higher Q as emphasized by the arrow in Fig. 12.6. Fig. 12.7 plots the determined core radii Rc of all investigated samples as a function of temperature. The initial (room temperature) droplet size remains unchanged with decreasing temperature even below the freezing point of the bulk deuterated water. At a well-defined temperature Ts the radii Rc abruptly start to decrease and then tend to a plateau value at the lowest temperatures. Whereas the shrinking temperature Ts and the kinetics of shrinking (slope of the curve) depend on the water content of the droplets, Rc seems to converge to a similar low temperature value for all compositions. For ˚ is all samples investigated a remaining finite core radius of about Rc = 5–6 A observed. The temperature Ts is found to depend on the molar ratio ω , thus the size of the droplets, Ts increases with 1/Rc . Our studies could thus prove that the microemulsion structure remains intact over a wide temperature range, down to temperatures far below the freezing point of bulk water. The findings suggest that the droplet instability is related to the formation of ice via homogeneous nucleation [49]. They may also confirm an earlier assumption that water is expelled from the droplets at low temperatures [48].


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Fig. 12.7 Droplet core radius Rc as a function of temperature T for four D2 O/AOT/d-oil microemulsions with a molar ratio ranging between 3 ≤ ω ≤ 12. The smaller the radius at room temperature the lower the temperature down to which the droplet size remains unchanged. For all four investigated samples the droplets remain stable down to temperatures below the freezing point of bulk water. Below they shrink to a size where they still contain a number of about 2 to 4 water molecules per AOT molecule (figure adapted from [49])

The structural data do now point to this water/AOT/oil microemulsion in the droplet phase as an appropriate model system for the investigation of water in soft confinement where confining size and temperature can be varied over a wide range. Backscattering – elastic temperature scans: Elastic temperature scans with neutron backscattering spectrometers allow us to follow the temperature dependence of the dynamics on a timescale of a few nanoseconds. As we have already mentioned, the technique takes advantage of the large incoherent signal from hydrogen atoms and so a measurement of H2 O-filled reverse micelles in deuterated oil (and if possible with deuterated surfactant) will probe the slowing down and finally the freezing of water as the temperature is decreased (in addition to the freezing of any other microemulsion constituents). Figure 12.8 shows an example for two microemulsions with different droplet sizes and different oils. These dynamic measurements are the first to evidence the low temperature instability or freezing and also the existence of a pronounced hysteresis between cooling and heating, the area of which depends on the droplet size [47]. Figure 12.9 shows how the water freezing temperature depends on droplet size. Similarly to SANS an approximate 1/Rc dependence was found. In addition, the data show that reversibility is achieved only after heating up to about the melting temperature of ice. Independently of the droplet size, the full dynamics is only recovered around T = 273 K.

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Fig. 12.8 Backscattering elastic fixed window scans of microemulsions with large droplets of ˚ (top figure) and small droplets of radius Rc ≈ 18 A ˚ (bottom figure). Heating and radius Rc ≈ 46 A cooling cycles are indicated by arrows, dotted vertical lines show the freezing points of bulk water and the used oil [47]. Note that upon cooling the integrated elastic intensity increase, corresponding to the freezing of water on the ns-timescale, is clearly shifted to lower temperatures compared with the freezing point of bulk water. The amount of water supercooling depends on droplet size: with decreasing micelle size the freezing temperature of the enclosed water decreases


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Fig. 12.9 Water supercooling Δ T as a function of micelle core radius Rc . Hollow symbols are results from elastic fixed window scans on backscattering (determined as the middle of the intensity step attributed to the water freezing, see Fig. 12.8), full symbols refer to SANS results. The solid line shows the Gibbs–Thomson dependency assuming that homogeneous nucleation leads to ice formation inside the droplets (figure from [49])

NSE at low temperatures – shape fluctuations: Our aim is to investigate the behaviour of the liquid (water) enclosed in microemulsion droplets over the entire temperature range of structural stability. Therefore, we need to know first the temperature dependence of the droplet translational diffusion coefficients. This is important, as we will see later, because we will have to distinguish between diffusion of molecules inside the droplets and the global diffusion of the molecules together with the entire droplet. Second, we not only have to know if the confining structure is preserved but also if the hardness of the confinement changes within the temperature range of interest. For the reverse micelles studied here the hardness of the confinement is determined by the bending elasticity of the surfactant membrane. By means of NSE we may access both of these quantities. Recent low temperature results on water swollen reverse AOT micelles will be briefly outlined in the following. To investigate the relationship between the bending elasticity κ of the surfactant shell and the measured structural stability of the droplets we have previously investigated the AOT/D2 O/toluene-d8 microemulsion system with ω = 8, a reverse micelle system with quite small water cores of about Rc ≈ 12 A˚ [49]. As shown before the SANS experiments prove that for this composition the room temperature structure is preserved down to a temperature of T = 255 K [49]. For the NSE experiment the temperature was therefore varied between T = 300 and 250 K, thus covering the entire stability range.

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Fig. 12.10 Normalized intermediate scattering function S(Q,t)/S(Q) for D2 O/AOT/Toluene-d8, (ω = 8) at T = 265 K for a selection of scattering vectors Q. Solid lines are fits to a single exponential function (figure from [49])

In Fig. 12.10 we show intermediate scattering functions for this sample at a temperature of T = 265 K, well within the low temperature stability range of the droplets. The data are fitted by single exponential relaxation functions in order to determine the effective diffusion coefficient Deff (see (12.9)). The resulting values of Deff at all temperatures are then shown in Fig. 12.11. The determined effective diffusion coefficient contains the Q-independent droplet translational diffusion coefficient Dt which corresponds to the baseline of Deff . The characteristic peak for Deff (Q) – located at the minimum of the form factor of the droplets – reveals the contribution of the droplet fluctuations. Following the model proposed by Kawabata and co-workers the entire Qdependence of Deff is fitted yielding the translational diffusion coefficients Dt , the fluctuation time constants and the dimensionless amplitudes of the shell fluctuations (see (12.10)) [31]. Knowing the radius of the droplets and their size polydispersity from SANS we may then further calculate the bending elasticity κ . For the AOT/D2 O/toluened8 microemulsion the bending elasticity is constant κ = 0.3kbT over the whole investigated temperature range. The temperature dependence of the translational diffusion coefficient of the droplets could be perfectly described by the Stokes– Einstein equation giving a hydrodynamic droplet radius being 1.3 times bigger than the geometric radius. Overall the results from NSE – applied to probe the dynamics of the droplets – were very consistent with the structural results obtained by SANS. This shows that


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Fig. 12.11 Effective diffusion coefficient for water swollen reverse micelles (D2 O/AOT/Toluened8, ω = 8) obtained by fitting the intermediate scattering function measured on NSE to a single exponential. The solid line shows the SANS intensity of the investigated microemulsion around room temperature. Note the peak of Deff near the form factor minimum (modified figure from [49])

by combining both methods one may draw an overall picture of the mesoscopic micellar structure and dynamics. This knowledge is an indispensable precondition for the study of the enclosed liquids. Backscattering and Time-of-Flight studies of the local dynamics: Having characterized the structure (SANS) and micellar self-dynamics (NSE) we now move to experiments that probe the dynamical behaviour of liquids confined in the droplets. For these we again have a hydrogenated liquid confined within the droplet with the oil and surfactant deuterated, in order to probe the dynamics of the confined liquid. Unfortunately, due to the low volume fractions of droplets used (∼10%) to avoid droplet interactions, and despite the large incoherent scattering signal from H atoms, the signal is dominated by unwanted scattering contributions. In order to correct for this one can then measure under equal conditions a sample with D2 O swollen droplets and subtract the resulting signal as a background from the H2 O swollen droplets. Subtraction of a reference where the target phase is deuterated corrects for all unwanted incoherent contributions, coherent self- (i.e. intra-constituent) contributions, but not for the coherent cross-contributions (coherent inter-constituent contributions). This procedure leaves, within a small error, the incoherent scattering of the target phase (here H2 O). A similar methodology was presented in Chap. 9 for the study of hydration water in protein solutions. A more detailed discussion of this contrast variation procedure can be found elsewhere [58].

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Fig. 12.12 Intermediate scattering function of water confined to ω = 8 reverse d-AOT micelles ˚ −1 ). Combining data from instruments IN5, dispersed in toluene-d8 (T = 270 K, Q = 0.3 and 0.5 A BASIS and IN16 we can access water dynamics from ps to ns [50]. Due to crystallization the normalization with the low temperature measurement (= resolution) contains coherent contributions (modified figure from [51])

Figure 12.12 shows I(Q,t) data obtained by combining Fourier-transformed spectra from three different spectrometers: TOF spectra from IN5, TOF-BSS spectra from BASIS and BSS spectra from IN16. All data were measured in the frequency domain and Fourier transformed to time space. The normalization was done through division of sample spectra by the measured resolution function. In our case this was unfortunately not the sample measured at low temperatures to avoid Bragg contamination resulting in a loss of the absolute normalization. The obtained I(Q,t) data were then evaluated simultaneously for the two Q-values shown. Overall using these three complementary neutron spectrometers gives access to a time range of more than 3 decades ranging from ps to ns. These data evidenced a clear slowing down of the average water dynamics with decreasing ω [50].

12.5 Glass-Forming Liquids Confined in Microemulsions The general aim in studies of dynamics in confinement is to learn more about the importance of a possible correlation length involved in the glass transition [13], which would change the dynamics of a confined supercooled liquid compared with


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the bulk liquid. However, many experiments have proved that in hard confinement, surface effects and constraints imposed by the rigid walls of the host are extremely important and may even dominate the dynamics of the liquid in small pores. The interaction of the confined liquid with the confining walls is often the reason why the glass transition temperature changes in hard confinement [35], instead of – or in the best case together with – the expected decrease of Tg with a decreasing spatial dimension of the confinement [13]. Similarly, the characteristic Vogel–Fulcher-like temperature dependence of the structural relaxation times are seen to sometimes approach an Arrhenius law in hard confinement. On theoretical grounds this is expected because of a growing correlation length with decreasing temperature (or a growing size of cooperative rearranging regions in the Adam–Gibbs picture), which in turn leads to a dramatic increase in relaxation time. However, such a crossover to an Arrhenius temperature dependence could also be a result of the increasing influence of surface dynamics. Thus the first motivation for studying glass-forming liquids confined in microemulsion micelles is to find systems for which the geometrical confinement is the most relevant factor influencing the dynamics in confinement. Also a change of the type of the confining wall from rigid to soft should give new insights. The second motivation is to be able to tune the confinement size easily. We have discussed above that for microemulsion droplets surface interaction is an important issue, at least in the case of ionic surfactants. The interaction might be of different nature, but the influence of the liquid in contact with the surfactant monolayer can by no means be neglected. Assuming stability of microemulsion droplets upon cooling, we need to consider the relative timescale of the confining wall dynamics, i.e. the surfactant dynamics (and its possible modification by the dynamics of the continuous phase); will the confined liquid dynamics be accelerated or slowed down near the surface? Thus even though surface effects can most probably not be neglected, soft confining systems allow tuning with the temperature dependent viscosity of the constituents and thus to go from a soft confinement to a hard confinement situation upon decreasing the temperature. Furthermore there is evidence that the density in hard confinement is not the same as in the bulk, which further complicates the comparison of the dynamics in bulk and under confinement. For the soft microemulsion droplets such a density difference is expected to be less important, though a small Laplace pressure inside the droplets does exist. Microemulsions have been studied in the context of glass transition by a number of techniques. Here we point out two key works. Angell et al. determined the phase diagram of a pseudoternary microemulsion system and studied glass formation using DSC from very low temperatures up to room temperature [4]. Later they applied SANS to investigate the stability down to T = 100 K for some pseudoternary microemulsion systems [2]. Angell states, maybe somewhat exaggerated, that “in microemulsion form, crystallization of molecular liquids during normal cooling

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will tend to be an unusual phenomenon”, but also that “micromulsions frequently become unstable and separate on cooling”.2 We highlight also the work by Richert and co-workers since it inspired us to perform the neutron experiments that we describe below. They investigated propylene glycol (PG) confined in a microemulsion system (described below) by solvation dynamics and dielectric spectroscopy [22, 56]. They realized soft confinement, meaning the glass transition of the enclosed PG appears at temperatures where the oil phase is still unfrozen and mobile. The observed correlation times for the confined PG decreased with decreasing droplet size and compared to the bulk PG correlation time. Hard confined PG in Vycor showed increasing correlation times and this was interpreted as being mainly due to surface effects. Concerning inelastic neutron scattering there is to our knowledge few, yet very recent, activity. Zorn and co-workers [57] studied the same microemulsion system just described by inelastic neutron scattering. The polar glass-forming liquid propylene glycol (Tg from DSC is 167 K) was confined in microemulsion droplets, found to have only about 1.7 nm diameter (about 22 PG molecules). Only one droplet size was investigated. The ternary microemulsion system consisted of 10.2% PG (protonated to enhance contrast), 53.7% decahydronaphthalene (deuterated cis/trans mixture of decalin with Tg = 135 K) as the hydrophobic component, and 36.1% deuterated AOT as surfactant. The investigations were carried out in the temperature range 2 < T < 300 K. At temperatures where PG freezes, decalin is still liquid and thus soft confinement is realized. The microemulsion structure of PG/decalin/AOT was investigated by SANS at temperatures between 249 and 303 K. The characteristic form factor minimum for the droplets (visible for the water/AOT/toluene microemulsions in Fig. 12.5) was not detected due to the onset of the molecular structure factor at large Q. Data fitting revealed that the droplet size remained constant within error in the investigated temperature range and low temperature stability was assumed based on information from literature [56]. The polydispersity increased slightly with decreasing temperature. Zorn et al. combined data from two spectrometers to cover a wide dynamic range. The PG was protonated and the surfactant and oil were deuterated. Furthermore, a reference sample with same composition but with deuterated PG was measured and subtracted. As described earlier, this method leaves essentially the incoherent scattering from PG. The outcome of the experiment was – like for the solvation dynamics results – that the alpha relaxation of PG in confinement was faster than that for bulk PG. Surprisingly, this is valid up to quite high temperatures. One may speculate that this observation is due to interactions with the AOT surfactant, thus due to wall effects. Furthermore the neutron data showed that methyl group rotations

2 Angell

already at that time pointed to the interest of such small confined systems: “... properties of molecular liquids which can be obtained in microemulsion form... can be investigated at temperatures below the glass transition of the .... matrix, in isolated non-communicating microsystems containing of the order of 1000 molecules or less. The dimensions of such microsystems are of special interest since they fall in the “gray” range between localized and collective phenomena . . . ”


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were about a factor 8 faster in confinement, ascribed to lower energy barriers and a higher attempt frequency. The analysis took also into account the droplet diffusion which fell within the spectrometer’s energy window. The intermediate scattering function was obtained by combining TOF and BSS data and it was found that at least in the intermediate time range where neither methyl group rotation nor droplet diffusion are close in relaxation time, there is a clear acceleration of dynamics in confinement. For completeness we mention that the authors also studied the influence of soft confinement on the low-frequency vibrational modes of the so-called Boson peak. One finds that the low-frequency wing of Boson peak is enhanced, in contrast to hard confinement where these low-frequency modes are strongly suppressed [59]. Very recently another system (glycerol/AOT/(xylene/toluene)) was investigated with SANS and QENS by Blochowicz and co-workers [6]. In this study, droplets of diameters between 2 and 9.2 nm were measured to study the influence of confinement size on the glycerol dynamics. The volume fraction of droplets was 40% (60% for the largest droplet). Stability of the microemulsion droplets at low temperatures was proven using shell-contrast for SANS. For the QENS experiments the optimum contrast (protonated glycerol in otherwise deuterated environment) could not be achieved due to the unavailability of d-AOT. Instead the AOT dynamics was measured with d-glycerol/p-AOT/d-oil as a reference, and was used to correct the data. Again a wide dynamic range was achieved by combining Fourier-transformed data from two spectrometers. The I(Q,t) data allowed then for time-temperature superposition and thus for a better characterization of the spectral shape. All spectra showed stretched exponential relaxations, where the 9.2 nm droplets could be described like the bulk glycerol with β ≈ 0.6. A slightly more stretched relaxation curve was found for confinement size of 4.8 nm and for 2 nm a smaller stretching parameter β = 0.48 was determined. The effect of confinement on the dynamics was also seen in the average relaxation time. Bulk and 9.2 nm confined glycerol follow the same Vogel–Fulcher temperature dependence known from literature, whereas 4.8 and 2.0 nm confined glycerol exhibit an increasing acceleration with decreasing confinement size. Furthermore temperature dependence of the relaxation time tends towards an Arrhenius T -dependence for small confinement sizes. This is expected for purely geometrical confinement and similar effects have been observed with dielectric and neutron spectroscopy for liquids in hard confinement (e.g. [29,43]). However, again the relevance and amount of the surface interaction on the observed Arrhenius behaviour and stretching could not clearly be resolved since neutron scattering gives averaged data. Currently this group is performing neutron experiments on a microemulsion system for which it is possible to go from soft to hard confinement within the temperature range of the supercooled confined liquid.

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12.6 Summary and Conclusion In this chapter we have discussed microemulsions as model systems for the study of freezing and the dynamical behaviour of liquids under soft confinement. We have mainly concentrated on a ternary water-in-oil (w/o) microemulsion based on the anionic surfactant AOT. Over a wide range of compositions the structure of this microemulsion consists of spherical water droplets coated by a mono-layer of surfactant (so-called water-swollen reverse micelles), which are thermodynamically stable when dispersed in the continuous oil matrix. We have shown how neutron scattering experiments have added to the field. SANS and neutron diffraction were shown to be ideal tools for the characterization of the microemulsion structure and the local structure of the liquid phases. The structural data is nicely complemented by dynamical information obtained from QENS measurements, covering motions occurring on timescales ranging from sub-picoseconds to fractions of microseconds with the spatial resolution matching the length scales of interest for studying diffusional processes in soft condensed matter. All the discussed methods profit from the sensitivity of neutrons to light elements and the possibility of highlighting selected sample compounds by means of H/D exchange. We have shown that this microemulsion system offers the possibility to study water (or other polar liquids) in a well-defined soft confinement, where both, confinement size as well as temperature, may be varied over a large range. In particular, we emphasized the fact that the droplet phase is stable for certain compositions at temperatures lower than T < 270 K. The reverse micelles are therefore also suited for the study of water in the supercooled state. Despite all the aforementioned unique advantages of microemulsions as model systems for the systematic study of soft confinement, we do not want to conceal that their complexity renders them also challenging. From both structural and dynamical points of view microemulsions need to be carefully characterized prior to investigating possible confinement effects.

References 1. Abel S, Sterpone F, Bandyopadhyay S, Marchi MJ (2004) Molecular modeling and simulations of AOT-Water reverse micelles in isooctane: structural and dynamic properties. J Phys Chem B 108:19458–19466 2. Alba-Simionesco C, Teixeira J, Angell CA (1989) Structural characterisation of glass forming oil/water microemulsions by neutron scattering. J Chem Phys 91:395 3. Aliotta F, Fontanella M, Lechner R, Pieruccini M, Rufe B, Vasi C (2002) Percolative phenomena in lecithin reverse micelles: the role of water. Colloid Polym Sci 280:193–202 4. Angell CA, Kadiyala RK, MacFarlane DR (1984) Glass forming microemulsions. J Phys Chem 88:4593 5. Bee M (1984) Quasielastic neutron scattering: principles and applications. In: Solid state chemistry, biology and materials science. Adam Hilger, Philadelphia, USA (1988) 6. Blochowicz T, Gouirand E, Fricke A, Spehr T, Stühn, B, Frick B (2009) Accelerated dynamics of supercooled glycerol in soft confinement. Chem Phys Lett 475:171–174


345

7. Boned C, Peyrelasse J, Moha-Ouchane M (1986) Characterization of water dispersion in water/AOT microemulsions using differential scanning calorimetry. J Phys Chem 90:634–637 8. Brodskaya E, Mudzhikova G (2006) Molecular dynamics simulation of aot reverse micelles. Mol Phys 104:3635–3643 9. De T, Maitra A (1995) Solution behaviour of aerosol OT in non-polar solvents. Adv Colloid Interface Sci 59:95–193 10. Delord P, Larche FC (1984) In: Luisi P, Straub BE (eds) Reverse micelles – Biological and technological relevance of amphiphilic structures in apolar media. Plenum Press, New York and London, p 137 11. Van Dijk M, Joosten J, Levine Y, Bedeaux D (1989) Dielectric study of temperature-dependent aerosol OT/water/isooctane microemulsion structure. J Phys Chem 93:2506–2512. 12. Dokter A, Woutersen S, Bakker H (2006) Inhomogeneous dynamics in confined water nanodroplets. PNAS 103:15355–15358 13. Dynamics in Confinement (2000) In: Frick B, Büttner H, Zorn R (eds), vol. 10 (EDP SCIENCES, Les Ulis, France, 2000). Frick B, Koza M, Zorn R (2003) Euro Phys J E 12, 3. Special Topics on Dynamics in Confinement (2007). In: Koza M, Frick B, Zorn R (eds), Eur Phys J–ST 141, Grenoble, 2007. Special Topics on “Dynamics in Confinement” (2010). In: Zorn R, van Eijck L, Koza M, Frick B (eds), Eur Phys J 189(1). 14. Faeder J, Ladanyi BM (2000) Molecular dynamics simulations of the interior of aqueous reverse micelles. J Phys Chem 104:1033–1046 15. Farago B, Richter D, Huang JS, Safran SA, Milner ST (1990) Shape and size fluctuations of microemulsion droplets: the role of cosurfactant. Phys Rev Lett 65:3348–3351 16. Farago B, Gradzielski M (2001) The effect of charge density of microemulsion droplets on the bending elasticity of the amphiphilic film. J Chem Phys 114:10105–10122 17. Fletcher P, Robinson B, Tabony J (1986) A quasielastic neutron scattering study of water-in-oil microemulsions stabilised by aerosol-OT. J Chem Soc Faraday Trans 1(82):2311-2321 18. Freda M, Onori G, Paciaroni A, Santucci A (2003) Hydration-dependent internal dynamics of reverse micelles: a quasielastic neutron scattering study. Phys Rev E 68:021406 19. Grillo I (2008) Small angle neutron scattering and applications in soft matter. In: Borsali R, Pecora R (eds) Soft matter characterization, Springer 20. Harpham MR, Ladanyi BM, Levinger NE, Herwig KW (2004) Water motion in reverse micelles studied by quasielastic neutron scattering and molecular dynamics simulations. J Phys Chem 121:7855–7868 21. Hauser H, Haering G, Pande A, Luisi P (1989) Interaction of water with sodium bis(2-ethyl-1hexyl) sulfosuccinate in reverse micelles. J Phys Chem 93:7869–7876 22. He F, Wang LM, Richert R (2007) Confined viscous liquids: Interfacial versus finite size effects. Euro Phys J – Special Topics 141 23. Helfrich W (1978) Steric interaction of fluid membranes in multlayer systems. Z Naturforsch Sect A-A J Phys Sci 33:305–315 24. Hellweg T (2009) Scattering techniques to study the microstructure of microemulsions. In: Stubenrauch C (eds) Microemulsions, Wiley 25. Hirai M, Kawai-Hirai R, Iwase H, Kawabata Y, Takeda T (2002) Effects of proteins on dynamics of water-in-oil microemulsions. Appl Phys A 74:1254–1256 26. Hoar T, Schulman J (1943) Transparent water in oil dispersions: the oleopathic hydromicelle. Nature 152:102–103 27. Holmberg K, Jönsson B, Kronberg B, Lindman B (2003) Surfactants and polymers in aqueous solution. John Wiley & Sons, Ltd., West Sussex, England 28. Huang J, Milner S, Farago B, Richter D (1987) Study of dynamics of microemulsion droplets by neutron spin-echo spectroscopy. Phys Rev Lett 59:2600–2603 29. Huwe A, Kremer F, Behrens P, Schwieger W, Molecular dynamics in confining space: From the single molecule to the liquid state. Phys Rev Lett 82:2338 30. Kawabata Y, Seto H, Nagao M, Takeda T (2002) Temperature- and pressure-dependences of shape fluctuations in a ternary microemulsion system. J Neutron Res 103:131–136

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31. Kawabata Y, Nagao M, Komura S, Takeda T, Schwahn D, Nobutou H (2004) Temperature and pressure effects on the bending modulus of monolayers in a ternary microemulsion. Phys Rev Lett 925:056103 32. Kitchens M, Bossev D, Roberts C (2006) Solvent effects on AOT reverse micelles in liquid and compressed alkanes investigated by neutron-spin-echo spectroscopy. J Phys Chem B 110:20392–20400 33. Kotlarchyk C, Stevens R, Huang J (1988) Study of Schultz distribution to model polydispersity of microemulsion droplets. J Phys Chem 92:1533–1538 34. Levinger NE (2002) Water in confinement. Science 298:1722 35. Le Quellec C, Dosseh G, Audonnet F, Brodie-Linder N, Alba-Simionesco C, Haeussler W, Frick B (2007) Influence of surface interactions on the dynamics of the glass former orthoterphenyl confined in nanoporous silica. Euro Phys J – Special Topics 141:11–18 36. Mezei F (1980) Neutron Spin Echo. Lecture Notes in Physics, vol. 128, Springer, Berlin, Heidelberg, New York 37. Milner S, Safran S (1987) Dynamical fluctuations of droplet microemulsions and vesicles. Phys Rev A 36 38. Munson CA, Baker GA, Baker SN, Bright FV (2004) Effects of subzero temperatures on fluorescent probes sequestered within aerosol-OT reverse micelles. Langmuir 20:1551 39. Nucci N, Vanderkooi J (2005) Temperature dependence of hydrogen bonding and freezing behavior of water in reverse micelles. J Phys Chem B 109:18301 40. Piletic I, Moilanen D, Spry D, Levinger N, Fayer M (2006) Testing the core-shell model of nanoconfined water in reverse micelles using linear and nonlinear IR spectroscopy. J Phys Chem A 110:4985 41. Quist P-O, Halle B (1988) Water dynamics and aggregate structure in reversed micelles at sub-zero temperatures. J Chem Soc Faraday Trans 184:1033 42. Safran S (1983) Fluctuations of spherical microemulsions. J Phys Chem 78:2073–2076 43. Schönhals A, Göring H, Schick C, Frick B, Zorn R (2003) Glassy dynamics of polymers confined to nanoporous glasses revealed by relaxational and scattering experiments. Euro Phys J E 12:173 44. Schulz P-C (1998) DSC analysis of the state of water in surfactant-based microstructures. J Therm Anal Calorim 51:135–149 45. Seki K, Komura S (1995) Viscoelasticity of vesicle dispersions. Phys A 219:235–289 46. Senatra D, Zhou Z, Pieraccini L (1987) A study of the properties of water-in-oil microemulsions in the subzero temperature range by differential scanning calorimetry. Prog Colloid Polym Sci 73:66–75 47. Spehr T, Frick B, Grillo I, Stühn B (2008) Supercooling of water confined in reverse micelles. J Phys Condens Mat 20:104204 48. Simorellis A, VanHorn W, Flynn P (2006) Dynamics of low temperature induced water shedding from AOT reverse micelles. J Am Chem Soc 128:5082–5090 49. Spehr T, Frick B, Grillo I, Falus P, Müller M, Stühn B (2009) Structure and dynamics of reverse micelles containing supercooled water investigated by neutron scattering. Phys Rev E 79:031404 50. Spehr T (2010) Water dynamics in soft confinement – Neutron scattering investigations on reverse micelles. Ph.D. thesis, Technische Universität Darmstadt (2010) 51. Spehr T, Frick B, Zamponi M, Stühn B (2011) Dynamics of water confined to reverse AOT micelles. Soft Matter 7:5745–5755 52. In: Stubenrauch C (ed) Microemulsions. Background, new concepts, applications, perspectives. John Wiley & Sons, UK 53. Tabony J, Llor A, Drifford M (1983) Quasielastic neutron scattering measurements of monomer molecular motions in micellar aggregates. Colloid Polym Sci 261:938–946 54. Tabony J (1985) Quasielastic neutron scattering measurements of molecular motions in micelles and microemulsions. Chem Phys Lett 113:75–81


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55. Tondre C (2005) Dynamic processes in microemulsions. In: Zana R (eds) Dynamics of surfactant self-assemblies: micelles, microemulsions, vesicles and lyotropic Phases. CRC Press, Hoboken 56. Wang LM, He F, Richert R (2004) Intramicellar glass transition and liquid dynamics in soft confinement. Phys Rev Lett 92:95701 57. Zorn R, Mayorova M, Richter D, Frick B (2008) Inelastic neutron scattering study of a glassforming liquid in soft confinement. Soft Matter 4:522–533 58. Zorn R (2010) Neutron spectroscopy for confinement studies. Eur Phys J Spec Top 189(1):65–81 59. Zorn R (2010) Boson peak in confined disordered systems. Phys Rev B 81:054208 60. Zulauf M, Eicke HF (1979) Inverted micelles and microemulsions in the ternary system H2 O/Aerosol-OT/Isooctane as studied by photon correlation spectroscopy. J Phys Chem 83:480–486 61. https://www.ill.eu/other sites/BS-review/index.htm 62. http://www.ill.eu/instruments-support/instruments-groups/groups/lss/more/world-directoryof-sans-instruments

Chapter 13

Structure and Dynamics of Polymer Nanocomposites Involving Chain-Grafted Spherical Nanoparticles Peter F. Green, Hyunjoon Oh, Pinar Akcora, and Sanat K. Kumar

13.1 Introduction The class of materials known as polymer nanocomposites (PNCs), i.e., polymer hosts filled with nanoparticles, possess properties that typically differ significantly from the pure polymer, even at low nanoparticle concentrations [1, 2]. While this class of materials has been the focus of intense study for the last 50 years, especially in the context of filled rubbers, they have become of even more industrial and scientific interest since the early 1990s when it was first demonstrated that the thermal and mechanical properties of Nylon-6 were improved by the addition of 2–4 wt. % micatype layered silicates to the extent that it could be used in an engine compartment [3, 4]. Significant property enhancements have also been documented in polymers to which various nanoparticles (e.g., graphene, quantum dots, nanorods, fullerenes, and metallic nanocrystals) have been added, rendering them useful for a range of applications, from structural and biomedical to electronic and optical [1, 2, 5–7]. In contrast to macroscale composites, where micron-sized colloidal particles are used as a filler, the unusual property enhancements exhibited by PNCs are related, in part, to the significant surface to volume ratio associated with the size of the nanoparticles [1, 2]. For a given volume fraction of nanoparticles, the number of polymer segment/particle interactions with particles of volume 1 nm3 is many orders of magnitude larger than the number of such interactions with particles of volume 1 μm3 . There is an additional effect: the influence of the particle on the local structure, extending beyond the immediate length-scale associated with the P.F. Green () • H. Oh Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI, USA e-mail: [email protected] P. Akcora • S.K. Kumar Department of Chemical Engineering, Columbia University, New York, NY, USA e-mail: [email protected] V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0 13, © Springer Science+Business Media, LLC 2012

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point of contact. The length-scale over which the local polymer segmental packing is affected depends on the nature of the segment/particle interaction [8]. Such interactions have important implications regarding the dynamics and the mechanical properties, as we will show later in this chapter. In light of the diversity of polymers and nanoparticles, the potential for use of PNCs is nearly limitless. However, to realize the full potential of PNCs, significant challenges must be surmounted, primarily on two fronts. PNCs generally possess nonequilibrium morphologies due to the complex interplay of enthalpic and entropic interactions leading to particle aggregation, particle bridging interactions, and phase separation at various length scales [9]. The second challenge is associated with understanding and predicting property enhancements of these materials, which are intimately connected to their morphology. A promising strategy for controlling the dispersion, and morphology, of PNCs is to graft polymer chains onto the nanoparticles to form a brush layer [7, 10– 13]. The free chain/brush interfacial interactions may be “tuned” by controlling grafting density, σ , the degrees of polymerization of the grafting chains, N, and of the polymer host, P, the nanoparticle size, D, and shape. A primary goal of this chapter is to show that changes in all these parameters profoundly influence the phase behavior and properties of this class of PNCs, specifically the dynamics and glass transition. Another important goal is to illustrate the role in neutron techniques toward elucidating the dynamics of PNCs. The chapter is organized in two major sections, morphology and dynamics, which are discussed in Sects. 13.2 and 13.3, respectively. The discussion of the morphology of chain-grafted nanoparticle/polymer mixtures is divided into two parts to highlight different microstructures that develop, depending on the grafting density. When the grafting density is low, various microstructures, e.g., short strings, 2-dimensional sheets of varying topographies, are typically encountered. At high grafting densities, the structures are typically more isotropic. In Sect. 13.3, a distinction is made between the long-range and short-range dynamics. Implications of the connection between structure and the dynamics are addressed. The document is concluded with a discussion of the future outlook as it pertains to the use of neutrons to address some of the challenges associated understanding the structure and dynamics of PNCs.

13.2 Structure and Morphology of Homopolymer/ Chain-Tethered Nanoparticle PNCs From a practical perspective, polymer–nanoparticle enthalpic interactions are invariably so strong (and unfavorable) that the nanoparticles aggregate at very low concentrations and form a separate phase. In polymer/C60 mixtures, for example, the C60 particles exhibit a strong tendency to aggregate, even as low as φ = 1 wt.% [14]. In fact, one can measure diffraction patterns from the C60 –C60 aggregates in

13 Structure and Dynamics of Polymer Nanocomposites Involving Chain-Grafted...

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the polymer. This problem is so ubiquitous to PNCs that novel means have been developed to facilitate the spatial dispersion of nanoparticles into a polymer matrix [10, 15–19]. In addition to the translational entropy of the nanoparticles and the conformational entropic penalties the host chains experience in order to accommodate the nanoparticles, the interfacial interactions between the chains grafted to the particles and the free chains in the surrounding melt have a profound impact on the morphological development in these PNCs. While there are several strategies that can facilitate nanoparticle dispersion, a very popular one today is to graft chains to the nanoparticles [11–13, 17, 20–29]. Grafting enables control of the host chain/nanoparticle interactions, which can play a dominant role in the morphological development of PNCs [12, 16, 17, 20, 30–35]. Small angle neutron scattering (SANS) has been used to determine quantitatively the extent of aggregation, and approximate average number of particles per aggregate, of chain-grafted silica nanoparticles as a result of the grafting reaction. Additionally, time-resolved SANS (which was discussed in detail in Chap. 11) has been used to study the kinetics of the reaction. SANS measurements enable rational control of the reaction process and the eventual structure of grafted nanoparticle systems. Depending on the grafting chain/host chain chemistries, the grafting density, particle size and shape, the grafting chain degree of polymerization, and the host chain degree of polymerization, very different morphologies develop: homogeneous dispersed phases, short particle strings, anisotropic 2-dimensional sheets, and separate 3-dimensional nanoparticle-rich phases [21]. At low grafting densities, when the bare nanoparticle surfaces can come into contact with each other and with the free homopolymer chains, anisotropic structures, e.g., short strings, and 2-dimensional sheets of varying topographies are encountered [21]. In contrast, particles densely grafted with chains are sterically well stabilized and hence either show good dispersion or crystallanity when they are placed in a homopolymer matrix. To distinguish between these very different scenarios, the discussion of chain-grafted nanoparticle/homopolymer mixtures below is divided into two sections, high and low grafting densities, thereby highlighting the differences in the microstructural features that form.

13.3 Mixtures Containing Particles with High Grafting Densities Since the particle surfaces are well shielded by the grafted chains, the thermodynamics of these mixtures are basically governed by brush/melt chain interactions [2]. In the simplest case, the chemistries of the host chains and the grafted chains are identical and hence the interactions that determine the structure and phase behavior are primarily entropic in origin [12, 17, 28, 36]. Parenthetically, when the grafting density of chains on a flat surface is low, i.e., when σ < 1/N, the grafted layers

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Fig. 13.1 Pictorial depiction of the case where the grafted nanoparticles tend to aggregate and the case where they are well dispersed

form “mushrooms” that are readily interpenetrated by the host (or matrix) chains [12, 17, 20, 27, 29, 37–40]. With increasing grafting density, the grafted chains are forced to stretch (entropic penalty) to accommodate each other, and the host chains are essentially completely expelled from the grafted layer. This is the so-called “dry” brush condition. This transition from the so-called “wet” brush to “dry” brush regime occurs when: σ N = (P/N)−2 (for flat interfaces) [41, 42]. Qualitatively, in mixtures involving sufficiently large particles, where the particle radius is given by R N 1/2 a (where a is the statistical segment length), the melt/brush interactions may be understood by approximating the particles as flat surfaces [29, 36]. The curvature of a particle becomes particularly important when R < N 1/2 a. As the radius decreases, the grafted chains became less stretched for a constant σ and P [43]. In other words, at constant σ , the wet brush/dry brush transition occurs for higher grafting densities as one considers particles with higher curvature. This result is in accord with intuition and with recent experimental results [44]. At sufficiently high grafting densities, where the melt chains are expelled from the brush, aggregation of the nanoparticles is favored. The driving force is due to the finite interfacial tension that exists between the grafted layers and the free melt chains when the brush is “dry” [12, 41, 42]. Consequently, the system minimizes its free energy by nanoparticle aggregation, in order to minimize the unfavorable contact between the brush and the host and depicted in Fig. 13.1. Note that for particles with the same grafting density, the particle–particle attraction decreases with decreasing particle size because the wet brush/dry brush transition only occurs for larger grafting density with increasing particle curvature. There are additional considerations that influence particle–particle dispersion. If the particles reside in proximity, with free melt chains confined between them, the melt chains experience a loss of conformational entropy. This cost is relieved only with the exclusion of the chains from between the particles [9, 17, 43, 45]. This entropic cost of confinement increases with the length of the free melt chains.


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Fig. 13.2 Scanning transmission electron microscopy images of PS/Au-PS10 PNC containing 1 wt. % nanoparticles (scale bar 100 nm)

Secondly, the entropy of mixing favors dispersion of the nanoparticles: for a spherical particle of radius R in a melt, the entropy of mixing is Fmix ∼ (φ /R3 )ln φ , indicating that dispersion is favored with decreasing particle size (φ is the particle concentration). The elastic energy, Fs , associated with stretching of the host chains to accommodate the particles, Fs ∼ (R/Rg )2 , where Rg is the radius of gyration of the host chains, is an additional cost for particle dispersion, specifically when Rg < R. It is now worthwhile to consider a real example to illustrate the role of melt/brush interactions on nanoparticle morphology. Consider gold nanoparticles of radius R = 5 nm, onto which polystyrene (PS) chains of N = 10 are grafted (at σ = 0.7 chains nm−2 ), which reside in a melt of PS chains of P = 50. We know from separate experiments that the “dry” brush condition is met [11]. At very low concentrations, the grafted nanoparticles are well dispersed; in fact they remain well dispersed in the PS host of P = 50 for concentrations up to 1.5 wt. %. The implication here is that even though the situation involves the “dry” brush condition, dispersion is possible for low enough particle loadings, with this being driven by the favorable entropy gain on mixing. However, aggregation is significant at concentrations greater than 3 wt. %. Of course, as N increases, the system becomes increasingly miscible and the particles are dispersed, even at much higher concentrations. Figure 13.2 shows scanning transmission electron microscopy images of nanoparticles dispersed in a sample at a 1 wt. % concentration. In contrast, C60 particles show a strong tendency to aggregate in a PS host, even for concentrations lower than 1 wt. %, which reflects the influence of the C60 –C60 interactions [14,46]. As a related example, we point out that sufficiently small nanoparticles with the same grafting density and same N, such as R = 2 nm gold core nanoparticles, remain well dispersed into a higher molecular weight host [2, 12]. We conclude this section by pointing out that in the case of thin films, interfaces play an important role toward the development of the final morphology. Typically, the nanoparticles segregate towards the interfaces (free surface or polymer/substrate interface) when phase separation occurs. It is known from the theory of star

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molecules that chains tethered to a central point suffer less of an entropic loss near an interface than linear chains of comparable size. A natural consequence of this effect is that tethered nanoparticles have a tendency to exhibit some degree of surface enrichment, even when the system is miscible. In fact, near the surface they can exhibit some degree of order, reflecting the strong particle–particle interactions, which are van der Waals in origin. As an example, consider a system where dodecane thiols are grafted to gold nanoparticles of radius R = 3 nm. The dodecane/PS enthalpic interactions are unfavorable so the nanoparticles exhibit a tendency to phase separate from a PS melt and segregate to the free surface (dodecane has a lower surface energy than PS) [47]. It is thus evident that a balance between the melt and brush interfacial interactions, the translational entropy of the nanoparticles, and the elastic energy of the chains determines the nanoparticle distribution in this class of PNCs. Clearly, control of R, N, P, and σ enables control of the nanoparticle distribution in nominally athermal mixtures of chain-tethered nanoparticles with homopolymers [2].

13.4 Mixtures Containing Particles with Low Grafting Densities As discussed above, experimental and theoretical reports suggest that particles with grafted chains can be readily “mixed” with polymer melts with the same chemical structure as the brush [2,17,19,20,27–29,32,36,40,43,48–60]. A growing consensus is that the composite will have well-dispersed nanoparticles if the matrix polymer chains are shorter than those of the brush. While variables such as the graft density and particle radius are also predicted to play a role, the overall point is that only phase separation phenomena are observed/predicted in this context. Thus, all of the works to date suggest that we can obtain the extremes, either the particles are aggregated into large spheres or form uniform particle dispersion. No other, intermediate, structures appear to have been obtained by any of these workers. Exceptions to this statement are recent findings that densely grafted particles apparently crystallize (presumably into an face-centered cubic lattice) when present in large concentrations in a lower molecular weight matrix [60–63]. In contrast to these past works, there is simulation and experimental work which considers the behavior of fullerene nanoparticles grafted with polyethylene oxide (PEO) chains. When a fullerene is functionalized with only one PEO chain and placed in water, it behaves analogous to a traditional amphiphile since the fullerene is hydrophobic while the PEO is hydrophilic [64]. Experimentally, such “linear” amphiphiles can assemble into cylinders, sheets, etc. following well-developed ideas in this field [65, 66]. For fullerenes with multiple PEO grafts which are uniformly distributed on the particle surface, assembly into string-like and branched polymers has been found by computer simulations [65, 67–70]. These simulations suggest the following physical picture: even though the particles are uniformly grafted with the


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Fig. 13.3 TEM micrographs of nanocomposites created with silica particles with 100 kg mol−1 polystyrene brushes at a grafting density of 0.05 chains nm−2 . The matrix PS molecular weight is listed in the figure as well as the annealing time in days. The scale bar in each figure is 0.5 μm

chains, there is a strong propensity for the hydrophobic particle cores to contact each other in order to shield themselves from the water. Due to the relatively low grafting density, this process is achieved by forcing the PEO chains “out of the way,” resulting in an anisotropic effective interparticle potential, and hence anisotropic self-assembly. Experimentally, in the limit of low grafting densities, it was found that spherical silica nanoparticles isotropically grafted with polymers self-assemble into a range of superstructures when they are dispersed into the corresponding homopolymer [1, 10, 17, 18, 20, 71]. The images in Fig. 13.3 [31, 68] show that this assembly is driven by the microphase separation between the immiscible, inorganic particle core and the (organic) grafted chains – a process analogous to the self-assembly of block copolymers (or amphiphiles) [72]. To gain a better understanding of these results, an analytical theory was developed which has the following two essential ingredients.

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It was assumed that there is an extremely short-ranged (“point”) interparticle attraction. This is counteracted by the entropy of distorting the polymer brush chains when two particles approach each other. The minimization of the resulting free energy yields a “morphology” diagram which is similar to the simulations. Both theory and simulation thus show that the polymer-grafted particles readily assemble into “block copolymer-like” morphologies, which balance core–core attractions and brush entropy.

13.5 Dynamics Polymer segments have been shown to exhibit reduced mobility in the vicinity of the particles. In one limit, polymer chains may become permanently attached (“bound” polymer layer), whereas in the other the attachment may be temporary, with characteristic attachment times shorter than the longest relaxation time of the chain. Either of these scenarios have important implications on the dynamical and mechanical properties of the PNC [14, 18, 31, 46, 68, 73–80]. The effects are manifested in changes in the relaxation spectrum, as measured by a range of techniques, including neutron scattering and dielectric spectroscopy. Changes in the frequency dependencies and/or magnitudes of the relaxation spectrum, as measured by loss and storage moduli G (ω ) and G (ω ) (ω is the angular frequency), respectively, are well documented [76]. In addition to changes in the moduli, changes in the glass transition accompany changes in the dynamics, though the connection is very different than that encountered in homogeneous homopolymer melts. Below we provide an overview of the long-range dynamics, giving specific examples involving chain-grafted nanoparticle/polymer mixtures. In Sect. 13.3.3, experiments on the short-range dynamics are described.

13.6 Long-Range Chain Dynamics: An Overview To understand the role of nanoparticles on the dynamics of polymer chains, it is worthwhile to briefly discuss salient features of the dynamics of homogeneous polymer melts [81–83] The frictional drag experienced by a moving chain in a dense melt is assumed to be distributed uniformly throughout the chain, with each monomer experiencing an average frictional drag of ζ (T ). The center of mass diffusion coefficient of an unentangled chain composed on N monomers is DRO ∝ T (N ζ )−1 , where T is the temperature. The longest relaxation time associated with the correlation function of the end-to-end vector of the unentangled chain is τRo ∝ ζ N 2 . For long, entangled, chain melts, a chain executes translational motion along its own contour within the confines of a virtual tube formed due to the constraints imposed by its neighbors. This is the basis of the Doi-Edwards model which has enjoyed enormous success at describing many of the dynamical


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features of entangled polymer melts [81]. The longest relaxation time, associated with the diffusion of the chain into an entirely new “tube,” i.e., the Reptation time, is τREP ∝ (ζ /T )N 3 . The associated translational diffusion coefficient is D ∝ (T /ζ )N −2 . These time scales are readily assessed using a combination of techniques, including viscoelasticity, dielectric spectroscopy, and neutrons (as was shown in Chaps. 4 and 6). The question we ask here is what influence do nanoparticles have on chain dynamics? In one scenario, different segments of a diffusing polymer chain may experience transient interactions with a particle on time scales much smaller that the longest relaxation time. This would lead to an increase in the longest relaxation time, i.e., ζ will increase. The shape of the relaxation spectrum may remain constant, but the average time scales would shift, reflecting the increase of the friction factor. This phenomenon is typified in the following example. Quasielastic neutron scattering (QENS) measurements of PS/C60 PNCs reveal evidence of immobilization of polymer segmental motions that occur over nanosecond time scales at the particle surfaces [46]. Rheological studies of the same system reveal that the viscoelastic spectrum remains constant; only a shift of the longest time scale is observed. The longest relaxation time τ increased, as did the viscosity, and the plateau modulus increase. Additionally, time–temperature superposition continued to be valid [14, 46]. In PNCs, the rheological measurements show evidence that the nanocomposites exhibit behavior that ranges from a viscous response at low frequencies and low particle concentrations to solid-like (elastic) behavior at high nanoparticle concentrations, for a wide range of frequencies [75,76,84,85]. For low particle concentrations, as discussed above, the viscoelastic spectra of PNCs are qualitatively similar to that of the pure polymer, though they may be shifted. At larger particle loadings, the magnitudes of the spectra increase, showing increasing evidence of solid-like behavior. Simulations show that the transition from viscous to solid-like behavior is not necessarily connected to a percolation threshold [74, 75]. An additional feature of these PNC systems is that the local strain field in the vicinity of the particle is distorted. This effect is manifested in the high frequency response of G’(ω ), and particularly at high particle loadings. Specifically, this strain field distortion leads to an enhancement of G”(ω ) at high ω . Particle jamming effects also enhance the modulus at high particle loadings, thus compounding the interpretation of the effect on the dynamics in a diverse range of systems. There is evidence of more than one glass transition temperature in some systems [78, 79]. This effect is particularly prominent for smaller particles (comparable to the size of the chain) where the chains are permanently attached to the nanoparticles and a layer of “bound” polymer chains surrounds the particle. Long and coworkers proposed that there would be a gradient in the glass transition in the vicinity of each particle [86, 87]. They rationalized this in terms of a dynamical heterogeneity, percolation model. Briefly, the model proposes that the dynamics in glassy polymers are heterogeneous, characterized by local regions of fast and slow dynamics and that the heterogeneity arises from thermal fluctuations of the density. The onset of the glass transition, and the dynamics, is necessarily associated with percolation of these

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slow domains in the framework of this model. It follows that the glass transition temperature (Tg ) will increase locally in the vicinity of a strongly interacting particle. Such a shift of the Tg will lead to local variations of the mechanical modulus of the sample. This would lead to an increase in the average modulus of the sample containing particles in which the polymer–particle interactions are strong, provided the particles are well dispersed.

13.7 Long-Range Dynamics of Chain-Grafted/Homopolymer Mixtures It is clear from the foregoing that the influence of nanoparticles on the dynamics of chains can be quite diverse and complex, where the effects range from a slowing down of the single chain dynamics at low φ , to a permanent attachment to the particles leading to different Tg ’s, to jamming or strain field effects that induce changes in the overall viscoelastic spectrum. Much less work has been done on the polymer/chain-grafted nanoparticle systems, but the findings are of particular interest. The melt/brush interactions and nanoparticle size, core size, D, and brush thickness play a critical role in the dynamics. The effect of melt/brush interactions on the glass transition temperature, Tg (φ ), of PNCs is illustrated in Fig. 13.4 where the Tg is seen to decrease dramatically with φ , for very small amounts of nanoparticles, φ 0, ∑ AeiKr , ∑ aeikr + aeik r , outside z < 0,

(15.3)

where k, k’, K denote the momentum of the neutron before and after reflection outside and inside the medium, respectively. If h¯ 2 (k sin θ )2 /2m, where θ denotes the incident beam angle with respect to the surface, is smaller than V0 waves cannot penetrate into the material and are totally externally reflected. For larger incident beam angles, the waves are partially reflected and partially transmitted. Snell’s law n = cos θ / cos Φ , with Φ the angle of the refracted beam with respect to the sample surface, gives the critical angle of total external reflection: cos θc = n.

(15.4)

Generally this angle θC is small ( < 1◦ ) for thermal neutrons and a Taylor expansion gives: 1 V0 ρN bcoh 2 =λ , (15.5) sin θC ≈ θC = E π

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with λ the wave length of the neutron. The amplitude of the refracted intensity can be calculated as: A = aei(ω t−kx−k(θ

1 2 − λ 2 ρ b) 2 z) 2π N

.

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The refracted beam is partially absorbed. The penetration depth (depth, at which the beam is attenuated to 1/e) close to total external reflection is [79]: 1

2

2 λ 2 2 z1/e = √ θ − θc2 + β 2 − θ 2 − θc2 . 4 2π 1

(15.7)

where β = λ ρN σabs /4π , with σabs the absorption cross section for neutrons, describes the absorption. Figure 15.2 depicts the penetration depth plotted for different β . For strong absorption, the penetration depth above the critical angle is small. For weak absorption, the penetration depth jumps to relatively large values at the critical angle. For angles larger than the critical angle this expression rapidly approaches the known Lambert–Beer expression [79]: ρN σabs I(z1/e ) = e−ρN σabs d1/e ≈ e− θ −θc z1/e . I(0)

(15.8)

Here d1/e denotes the path length of the neutrons inside the material along the direction of k, where the intensity drops to 1/e. As the incoherent scattering is independent of the angle and the phase information is lost, in most cases, it can be treated as an additional absorption term, or more precisely as flat background in a diffraction experiment. σabs = σabs + σinc . (15.9) For the reflected intensity one obtains similar to optics the Fresnel reflectivity: RF = kc4 /Q4z .

(15.10)

15 Shear Dynamics: Understanding Boundary Slip and Anomalies...

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The intensity reflected under equal incident and exit angles is called specular. In the case of neutrons the absorption for most materials, including liquids, is low and the penetration depth at the critical angle jumps from about 20 nm to at least several μm. This implies that an experiment with particular surface sensitivity has to be done below the critical angle of total reflection and probes at most a sample thickness of 20 nm. Considering a surface area of 5 × 5 cm2 this results in a scattering volume of 0.05 mm3 , or 0.05 mg for water, which is rather small and at the limit with respect to signal-to-noise ratio on current neutron sources.

15.3.3 In-plane Correlations Using Grazing Incidence SANS To address in-plane ordering for many reflectivity studies a position sensitive detector (PSD), see e.g. [123], is used for data collection to detect the so called offspecular scattering. Figure 15.3 shows the scattering geometry for grazing incidence SANS (GISANS) and neutron reflectometry (NR) studies. Neutrons are reflected at an interface and registered by a two-dimensional PSD. The z-direction is defined as the normal to the interface, the x-direction is in the scattering and interface planes, and the y-direction is perpendicular to x and z. To elucidate the peculiarities of GISANS and NR, Fig. 15.4 shows a comparison between the GISANS scattering pattern (left panel) taken at an incident angle of 0.3◦, for micelles cubic close packed at an interface, and a map where the reflectivity data, including the off-specular scattering, is plotted over Qx and Qz (right panel) for similar experimental conditions [122]. The black line, plotted in both panels of Fig. 15.4, indicates where the two hyper surfaces intersect. The three components of Q, with respect to the coordinate system defined in Fig. 15.3, can be written as follows:

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Fig. 15.4 GISANS and NR: Comparison of GISANS (left) and NR (right) data for a cubic closed packed sample in contact to a silicon interface [121]

2π (cos θ − cos θ cos θ⊥ ), λ 2π sin θ⊥ , Qy = λ 2π Qz = (sin θ + sin θ ), λ

Qx =

(15.11) (15.12) (15.13)

where θ and θ are the incident and exit beam angles out of the interface plane, respectively, and θ⊥ is the scattering angle in the plane of the sample surface relative to the x–z plane. It turns out that Qz and Qy show a sine dependence and increase linearly with the scattering angle (θ + θ ) and θ⊥ , respectively, for small angles. Qx is calculated from (15.11). For small angles the cos shows a θ 2 dependence and is nearly 1, meaning a much smaller component of Q along the x-direction. This is an important observation as it implies that at small scattering angles a highly anisotropic coherence volume (nm along y and z, whereas μm along x) is probed in the plane of the interface [121].

15.3.4 Comparison of SANS and GISANS Figure 15.5 gives an overview of how anisotropy in a crystalline micellar block copolymer system forming a faced centred cubic (fcc) lattice may develop under shear and close to an interface. The left panel shows the intensity taken in transmission SANS geometry. The intensity along the Debye–Scherrer rings is constant confirming a three-dimesional powder structure. The circles in the map


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Fig. 15.5 Anisotropy in scattering: The left panel depicts the SANS pattern for an fcc micellar powder structure and the middle panel the SANS pattern for an fcc micellar crystal under shear whereas the right panel shows the GISANS pattern taken for the fcc micellar crystal at rest [122]

mark the position of the first five Debye–Scherrer rings as expected for a fcc lattice ˚ [118, 122]. with a lattice constant of 295 A The middle panel shows the transmission SANS pattern for the sample with an applied shear rate of 100 s−1 . The shear gradient was perpendicular to the scattering plane along the z-direction and the flow was approximately along the x-direction. The angular intensity distribution with six-fold symmetry shows that the [220] direction is oriented parallel to the flow velocity [62]. As seen from the angular width of the reflections the crystal is not very well ordered and intense from the (111) and (311) reflections, e.g. is projected into the detector plane [21]. The scattering pattern with six-fold symmetry is explained by a highly twinned fcc crystal structure, ABCABC and ACBACB stacking, as often found in colloidal or micellar systems [62]. The right panel shows data taken at rest in GISANS geometry. Several peaks become visible arising from a two-dimensional powder structure with the [34] direction of the fcc lattice parallel to the surface normal (cubic dense packing). Note that in this case, next to the confining interface, a two-dimensional powder develops, in contrast to the twinned ABCABC and ACBACB layer sequence observed under shear. Reflections indexed with a prime result from crystallites with an orientation that is tilted by 180◦ around the surface normal with respect to the unprimed indices. To obtain the real peak positions the Q-component parallel to the surface normal is corrected according to refractive effects Q = Q2z,m − Q2c . Qz,m and Qc denote the measured and critical momentum transfer, respectively. By evaluating the relative intensities of the Bragg reflections in the GISANS data, information on texture with respect to the in-plane ordering, can be extracted. By taking additional reflectivity measurements along Qz , and diffuse scattering data along Qx , the long range inplane correlations on a micrometer length scale become visible [127].

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15.4 Inelastic Scattering In the following, the energy transfer resulting from Doppler scattered neutrons is discussed and also how it can be related to the velocity profile of flowing liquids. The limits at which a velocity profile and a diffusion constant can be extracted in one measurement are evaluated.

15.4.1 Doppler Effect If flowing liquids are measured using neutron spectroscopy, in addition to the quasielastic scattering, the neutrons become inelastically Doppler scattered, which may be visible in the spectra [65]. The mass of the scattering particles is assumed to be much bigger than that of the neutron. By the use of the conservation of momentum and energy the energy transfer can be calculated using:

Δ E = h¯ Qus ,

(15.14)

where us denotes the velocity of the scatterer. For an angle ε between the center of mass velocity and the transfer of momentum, where θ is half the scattering angle and Q = 4λπ sin θ , Δ E is:

Δ E = 2pus sin θ cos ε =

2h us sin θ cos ε . λ

(15.15)

15.4.2 Combination of Inelastic and Quasielastic Scattering Combining quasielastic scattering from the diffusion of molecules with the inelastic scattering from a flowing liquid, the following terms have to be considered in the scattering function [115]: 1. The molecular diffusion given by a Lorenztian function that is convoluted with an assumed Gaussian instrumental resolution function [9] 2. A velocity distribution function for the inelastic part of the spectrum 3. A smearing of the Q-value according to the relaxed Q-resolution of the instrument, since neutron spectrometers generally use a rather divergent beam in order to gain in intensity In order to evaluate the extent to which the quasielastic and inelastic contributions to the scattering function can be separated in a neutron data set, the scattering function was simulated for different scenarios. The results are show as contour


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Fig. 15.6 Inelastic and quasielastic scattering: Contour maps of the difference in intensity distributions between simulated quasi- and inelastic spectra. The x- and y-axes are normalized to the position of the inelastic peak from the moving surface, maximum velocity. (a) and (b) show the comparison between a linear velocity distribution and one with surface slip and stick, respectively. The velocity distributions are illustrated in (d). The difference between the linewidth for a velocity distribution with surface slip and a linewidth of one for a linear distribution is shown in (c) [115]

plots in Fig. 15.6. The x-axis represents the energy transfer and the y-axis scales with the quasielastic linewidth for the internal diffusion. Both axis are in units of the inelastic peak position from the moving surface confining the liquid. This corresponds to the maximum velocity and energy transfer to the neutrons from inelastic scattering. The greyscale represents the difference in scattering patterns which is shown as local percentage values. By plotting the spectra in this particular way it is easy to determine the resolution with respect to quasielastic and inelastic scattering since the calculated scattering functions for different physical scenarios are directly compared. Figure 15.6a shows the difference in the energy spectrum between a linear velocity distribution in a liquid confined between a fixed and a moving surface and a velocity distribution with surface slip; a liquid layer with no internal shear gradient and a velocity that is half of that of the moving surface. Figure 15.6b is a difference plot, for a linear velocity profile and one with the liquid sticking to each boundary; half the liquid is at rest and half of it moves at the speed of the moving surface (equivalent to an infinitely steep gradient at a discontinuity point in

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the middle of the liquid layer). The velocity distributions that have been used for the calculation are depicted in (d). For a statistical error smaller than 10% in each data point a distinction is possible if the quasielastic linewidth is not larger than the energy value of the inelastic peak position. Figure 15.6c shows the difference between a linear velocity distribution with a quasielastic linewidth of 1 (in units of the inelastic peak position) and a distribution with surface slip and quasielastic linewidths between 0 and 2, and demonstrates that the linewidths can be determined from the wings of the experimental spectra [115]. The important result from this simulation is that the diffusivity can be determined in all cases, but a macroscopic velocity profile can only be extracted if the energy of Doppler scattered neutrons is larger than the quasielastic line width.

15.4.3 Microscopic Diffusion and Macroscopic Flow The local shear rate γ˙ = ∂∂ dv is the determining time scale for the hydrodynamic motion in liquids. The important time scale for Brownian motion is the relaxation time τ . The ratio of the two time scales provides a dimensionless number which allows us to estimate the influence of macroscopic flow on the microscopic diffusivity. This number is called Peclet-number: ˙ . Pe = γτ

(15.16)

Considering the Peclet number, two scenarios can be distinguished: • For Pe 1 the diffusive motion of the molecules is too fast to be affected by flow. • For Pe 1 the shear gradient is large enough to induce alignment of molecules or even anisotropic diffusion. The time scales probed by QENS range from about 1 ps to around 100 ns. For a fully protonated sample of 0.1 mm thickness (to achieve less than 10 % scattering and avoid multiple scattering effects) sheared between a fixed and a moving surface, the minimum velocity of the moving plate in order to expect an effect on diffusion is 103 ms−1 and thus not possible to realize experimentally. Moreover, at high velocities, the neutrons experience a large Doppler shift which smears the energy spectrum over a large range of energies. The above considerations hold for simple liquids but have to be corrected for entanglement effects in polymer systems. One possibility to overcome the above restriction is to probe the diffusivity perpendicular to the macroscopic flow with no inelastic energy transfer to the neutrons. However, to minimize inelastic Doppler scattering a high Q-resolution is required. The situation changes if surface slip with a slip length of up to several μm is considered. At present the microscopic origin of surface slip is not well explored, but it is assumed that the anomaly extends only a few molecular diameters from the solid surface. Considering a slip length of 1 μm decaying to the bulk velocity profile


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over 1 nm we get an enhancement of the shear gradient in the near surface region by a factor of 103 . As a result, possible changes in diffusivity may become significant. On the other hand, the sample volume is rather small and difficult to probe and it is questionable whether the intensity on existing neutron spectrometers is sufficient to extract meaningful data. For simple liquids with relaxation times in the ps range and with rather small slip length, almost no effect of the macroscopic flow field on the diffusivity can be expected, even in the regions near the surface.

15.5 Experimental Results The use of neutron scattering techniques for the investigation of liquid flow is rather new and thus a very limited number of experiments have been carried out. Here we present results for Newtonian liquids and polymer systems, and although we cannot draw a final and conclusive picture of surface slip, we hope to demonstrate the potential of the different methods and how they can open new routes for the investigation of flow in liquids.

15.5.1 Microscopic Diffusion and Macroscopic Flow in Newtonian Liquids Figure 15.7a shows the quasielastic line width extracted from the data as a function of Q2 for hexadecane at rest taken at IN16 (Institut Laue Langevin, Grenoble, France) [33] inside a flow cell [119]. From the slope the diffusion constant 3.95 ∗ −1 10−6 ± 1 ∗ 10−7 cm2 s can be calculated which is close to the literature value of −1 3.86 ∗ 10−6cm2 s [59]. Figure 15.7b shows a quasielastic spectrum taken at one particular Q value and orientation of the flow cell, in reflection and transmission geometry as given in Fig. 15.8. For transmission and reflection geometries, Q is parallel and perpendicular to the macroscopic flow respectively. Two data sets are shown corresponding to the liquid at rest and under shear. It turns out that the line width for both orientations without flow is similar, as expected for an isotropic diffusion. With flow the spectra taken in reflection geometry do not change proving an unchanged diffusion along the shear gradient. In contrast, in transmission geometry the shift of the peak in the inelastic regime is clearly visible. As a flow cell with no moving parts of the cell itself was used, all the inelastic scattering can be attributed to the liquid. From a quantitative analysis of the scattering law, which includes inelastic scattering with a parabolic velocity distribution, no impact of the flow on the translational diffusion constant is found. In Fig. 15.7c, the velocity calculated from the energy transfer according to (15.15), is plotted as a function of Q. A maximum velocity of 1.77 ms−1 was found [119].

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a 35

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25 20 15 10 5

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Q [Å ] Fig. 15.7 Newtonian liquids: (a) Quasielastic line width extracted from the data taken at IN16 (at the ILL, France) for hexadecane and plotted over Q2 . The lines show the linear fit to the data. (b) Quasielastic spectra taken for hexadecane at IN16 in transmission and reflection geometry without and with flow. In transmission geometry the inelastic Doppler scattered neutrons clearly shift the spectrum. (c) Velocity of hexadecane calculated from the inelastic energy transfer of Doppler scattered neutrons at different Q values. The lines indicate the mean value of the velocity [119]


Momentum transfer Q

ow Fl

Incoming neutrons

Outgoing neutrons

Transmission geometry: Incoming neutrons

Momentum transfer Q

Fl ow

Reflection geometry:

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Outgoing neutrons

Fig. 15.8 Scattering geometry for inelastic neutron spectroscopy: For the reflection geometry (left panel) Q is parallel to the shear gradient whereas for the transmission geometry (right panel) it is parallel to the macroscopic flow [120]

15.5.2 Commercial Lubricants The motion of molecules in a commercial motor oil was monitored using the highresolution backscattering spectrometer IN16 (at the ILL, France) [33]. For this study, a plate–plate shear cell was used [113] and like in the previous example, the motion of the sheared liquid was investigated in the two scattering geometries shown in Fig. 15.8. For both geometries data was recorded at a constant Q of 1.4 ˚ −1 . In an earlier study it was shown that the velocity profile of lubricant changes A qualitatively for different temperatures [65]. In Fig. 15.9 the dots represent the detected neutron intensity as a function of energy transfer. The first row compares the results in reflection and transmission with the liquid at rest, whereas the second row compares the spectra with one disc rotating with us = 1.8 ms−1 (at the radial position of the neutron beam), corresponding to a shear rate of 5,000 s−1 , for the two different scattering geometries. The liquid at rest exhibits a broadened, quasielastic spectrum, and the peaks are well fitted by a Lorentzian line shape shown by the solid lines in Fig. 15.9. The identical line shapes and widths confirm that the liquid at rest has isotropic dynamical properties. Setting one disc in rotation leads to a broadening in the QENS signal in reflection geometry. This might be explained by a shear induced accelerated molecular diffusion in the direction of the shear gradient. An alternative explanation for the widening would be turbulent flow with a velocity component along the surface normal. Figure 15.10 depicts this scenario for a velocity along the surface normal of 0.35 ms−1 and unchanged diffusivity. There is only a very small difference between the two line shapes, and within experimental error no discrimination between the two models is possible. In transmission geometry, the liquid under shear gives a peak which is smeared out due to inelastic scattered neutrons from the macroscopic flow of the liquid. This

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Fig. 15.9 Commercial lubricants: Quasielastic spectra taken for ANTAR motor oil at IN16 [116]. (a, c and b, d) show data taken in reflection and transmission geometry, respectively. The panels on top a and b were taken at rest whereas for the lower ones c and d a shear rate of 5000 s−1 was applied

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Energy [μeV] Fig. 15.10 Turbulent flow: Comparison of the line shape for an increased quasielastic line width and an unchanged line width with additional macroscopic flow along the shear gradient


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Fig. 15.11 Surface sensitive shear experiments: Sketch of the experimental set-up with the rheometer mounted on the sample stage of V6. Neutrons enter a single crystalline block of silicon on the narrow side and get scattered from the bottom of the solid-liquid interface [126]

profile translates directly to a specific macroscopic velocity distribution. The data cannot be described by a linear velocity profile. The best fit shown by the red line is based on a linear velocity distribution starting from 0.15 * vdisc up to 0.7 * vdisc . From the wings of the spectrum the quasi-elastic Lorentzian line width is found unchanged with respect to the sample at rest.

15.5.3 Near Surface Relaxation of Micelles Hamilton et al. [42] have shown that after the cessation of shear the relaxation of a micellar mixed counterion system is much faster close to a flat quartz substrate than in the bulk. Upon cessation of shear, although the micelles remain aligned, the off-specular scattering falls by about an order of magnitude within seconds and a correspondingly rapid loss of orientational order results in a ring of scattering. Only over a longer time period consistent with the bulk decay time from going from an aligned to a fully entangled micellar state in bulk scattering measurements, does the observed off-specular ring scattering fall to a weaker background level of scattering from fully entangled micelles in the static solution. More recently, the relaxation of Pluronic F127 micelles was studied for surfaces with different surface energies [126]. In contrast to the counterion system which is in a more disordered liquid-like state, the F127 micelles form a crystalline structure at rest. The scattering geometry is shown in Fig. 15.11. The experiment was performed using a commercially available rheometer (Bohlin CSR-10) that was mounted on the sample stage of the V6 reflectometer (at HMI, Germany). In this study, information on the re-crystallization of the long range orientational correlations (or layering) after cessation of shear at different interfaces was extracted. Figure 15.12 depicts the integrated intensity of the (111) reflection, first order reflection on the specular line, normalized to the peak intensity before shearing the sample, plotted versus time. The open and closed symbols represent data for a hydrophilic and a hydrophobic

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Order parameter

1.0 0.8 Hydrophobic surface Hydrophilic surface

0.6 0.4 0.2 0.0 0

2

4 Time [h]

6

8

Fig. 15.12 Relaxation of micelles: Specular intensity plotted versus time after cessation of shear taken with the sample in contact with a hydrophilic and a hydrophobic substrate [126]

interface, respectively. For the hydrophobic interface the recrystallization time is two hours whereas for the hydrophilic interface it is 5 hours. In addition, the extrapolation for infinite times shows that only 25% of the initial (before shearing) intensity is recovered for the hydrophilic interface whereas the crystal relaxes completely at the hydrophobic interface [126].

15.5.4 Complex Liquids As shown in the previous section, the interface has an influence on the relaxation of polymer micelles. For this system the effect extends to the local dynamics under shear. A similar micellar system, Pluronic P85, was studied in the crystalline phase in contact to a hydrophilic and a hydrophobic silicon wafer [120]. The polymer was dissolved at 33% in weight in deuterated water. Measurements of the sample at the time-of-flight spectrometer NEAT (at HMI, Germany), shows a Q2 dependence of the Lorentzian line width, characteristic of translational diffusion. The slope of the line width translates into a diffusion constant of 7.27 * 10−6 cm2 s−1 , which is a little smaller than the literature data for deuterated water of 2.94 * 10−5 cm2 s−1 [70]. This could be due to a restricted mobility of the water molecules bound to the hydration shell of the polymer micelles. The system under shear was measured on the triple axis instrument IN3 (at the ˚ −1 . The extracted linewidths are ILL, France) for Q-values between 0.3 and 1.7 A shon in Fig. 15.13. The left and right panel display data taken with the sample in reflection and transmission geometries, at rest and with an applied shear rate of 6000 s−1 , respectively. The Q2 fit to the time-of-flight data is shown for comparison as solid line. Two thing should be noted: (1) The diffusion of the water molecules in the sample is isotropic, and (2) The diffusion is not affected by the applied shear.

15 Shear Dynamics: Understanding Boundary Slip and Anomalies... 0.30

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Fig. 15.13 Water dynamics: Line width for the faster mode with the sample in reflection (left panel) and transmission (right panel) geometry at rest (closed circles) and for a shear rate of 6,000 s−1 (open circles) in contact with machined aluminum surfaces [120]

Figure 15.14 shows QENS spectra taken with a 0.6 mm thick sample (10 % ˚ −1 on IN16 (ILL), which has a higher energy scattering) under shear at Q = 1.4 A resolution than NEAT. For the measurements under shear the velocity of the moving surface was 1.3 m/s, corresponding to a shear rate of 2,200 s−1 . The left and right panels show data taken with two hydrophilic and hydrophobic interfaces as wall material of the moving and the fixed disk, respectively. The two upper panels correspond to data taken with the sample at rest. In addition to the elastic line in reflection (first row) as well as in transmission geometry (second row), a quasielastic broadening of similar width is observed. For the sample in contact with the hydrophilic interface the line narrows under shear in reflection geometry but remains unchanged in transmission geometry. This shows that under shear the diffusion constant is reduced in the direction of the shear gradient. To fit the data in transmission geometry under shear the scattering law included the inelastic part from Doppler scattered neutrons, resulting in a distribution of Lorentzian lines shifted in energy with respect to each other. The calculations revealed no change in the diffusivity along the flow direction [120]. For the hydrophobic interface shear has no impact on the diffusivity.

15.6 Models To provide a model for describing surface, slip two vastly different scenarios have been proposed. In the case of real slip the liquid slides over the solid interface on an atomic scale. This implies either very high shear rates [60] or a roughness of the solid boundary which is laterally on a vastly different length scale in comparison to the size of the liquid molecules [35]. The second scenario for moderate shear is apparent slip over molecularly smooth surfaces. In this case, a microscopic boundary layer is assumed, which is different structurally and/or dynamically to the bulk liquid. The boundary layer leads to a different viscosity and is observed as surface slip on a larger length scale, although the no-slip boundary condition is still valid microscopically.

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Fig. 15.14 Polymer dynamics: Quasielastic spectra taken with a crystalline micellar sample at a hydrophilic (left) and a hydrophobic interface (right). The top panels (a–d) show data taken with the sample at rest either in reflection or in transmission geometry and the bottom panels (e–g) show data taken at a shear rate of 2,200 s−1 [120]

To explain apparent slip two models are under discussion. The first one assumes a microscopic layer with depleted density of the liquid near the solid interface [108]. Depletion layers at liquid/solid interfaces have been observed in various cases [64] using x-ray (XRR) and neutron reflectometry, but have only recently been recorded


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in situ under shear load [40, 125] with NR. An alternative explanation suggests the alignment of molecules near the surface as a consequence of shear stress [6]. This interpretation relates well to a larger slip length observed for hexadecane compared with squalane, which is a more symmetric molecule [92]. Surfaceinduced molecular layering of flowing liquids has been observed in confined propanol [103] using a surface force apparatus, and in eicosane [25] using an atomic force microscope after quenching the flowing alkane. However, these techniques are either indirect or require direct contact with the interfacial liquid and thus may perturb the system.

15.7 Summary In this chapter, the usefulness of QENS on the investigation of liquids under flow has been presented. The research in this field is relatively new and requires a combination of quasielastic and inelastic Doppler scattering. Due to the relatively low intensity of current neutron beams the experiments are still rather challenging. Nevertheless, it has been shown that diffusivity and surface properties are correlated, just like for structural properties. In addition, velocity profiles for liquids in motion were successfully extracted from inelastically Doppler scattered neutrons and the velocity was determined. In the same measurements the molecular diffusion constants under flow were evaluated. It was shown that a commercial motor oil may slip at the solid–liquid interface, and in the same time either the diffusivity becomes anisotropic or alternatively the flow becomes turbulent. The relaxation of polymer micelles after cessation of shear is sensitive to the surface energy of the solid substrate. Moreover the molecular dynamics under shear is slowed down along the shear gradient only for a more hydrophilic silicon interface.

15.8 Outlook In this final section, we would like to discuss some very recent experiments which may offer completely new possibilities for neutrons to contribute in the fundamental understanding of the solid/liquid boundary. As already seen for the investigation of surface slip, QENS could play an important role if applied under grazing incident beam geometry, allowing surface sensitive measurements. Up to now surface dynamics by neutrons has been done by for example, probing the collective dynamics in membranes [87, 89] or membrane water dynamics [88] in stacks of membranes/layers in order to increase the scattering volume. Local dynamics in lipid bilayers and the dynamics of individual lipid molecules were investigated by incoherent neutron scattering [55–57,82,83]. The use of membranes allows bulk samples exhibiting a large interface area of the membranes enhancing the useful signal. Recently also large interest was attracted by investigating the

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dynamics of e.g. water or glassy materials in confinement [34, 67, 105, 133]. It is well-established that the hydrogen bonding network of water is perturbed by the interaction with a hydrophilic substrate, and these structural changes may have a pronounced effect on the thermophysical properties of water confined in narrow pores or cavities [1, 2]. In particular, it was found that freezing and melting of water in nanometer-sized pores occurs at temperatures much lower than in the bulk [47, 53, 72, 73, 94, 95]. The classical theories of wetting phenomena [61] predict that in the case of full wetting a thin liquid-like layer will intervene between the solid and the wall, at temperatures around and somewhat below the solid/liquid coexistence temperature. For ice, against a macroscopically flat silica surface, it was found [27] that interfacial premelting starts at a temperature 15 K below the melting point and that the thickness of the quasi-liquid layer strongly increases as the melting point is approached. However, when adressing a single surface in grazing incident geometry the sample volume is small but penetrated under a small incident beam angle over a large surface area. At first one may assume that multiple scattering effects in this geometry will make quasielastic measurements impossible, since the path length of the neutrons inside the sample is long. However, incoherent scattering is isotropic and accordingly for a sufficiently thin sample multiple scattering effects can be neglected, as the same arguments as in transmission geometry apply. Neutron scattered once has a low probability to get scattered again as it is highly likely that it may leave the sample. Nevertheless, below the critical angle of total reflection the scattering volume probed is typically on the order of 0.05 mm3 . Experiments above the critical angle are impossible, at least in the case of incoherent scattering, if surface sensitivity has to be retained as the absorption of neutrons is low. This implies that only a very restricted cross section of the incoming neutron beam can be used, resulting in a further reduction of the signal. In spite of this, a couple of recent experiments have provided new information on the solid/liquid interface. As a first step the density profile of a liquid in contact with a solid substrate has to be extracted under shear [124]. It appears possible to evalute the liquid structure factor [39] below the critical angle of total external reflection and it seems that the results are consistent with MD simulations [37] and complementary AFM measurements [24]. The liquid is expected to precrystallize close to the interface. Even more demanding is the attempt to apply additional energy resolution and extract the local diffusivity in the region near the interface. Recent triple axis experiments demonstrate that a diffusion constant for water close to a silicon surface can be extracted below the critical angle of total reflection [38]. However, these experiments remain time consuming as the intensity is very low and just above the background level, but the diffusivity appears changed in the interfacial layer. One way of increasing the amount of useful signal is to use neutron spin echo techniques [112]. In addition, future neutron reflectometers will use the neutron spin for angular encoding allowing measurements with more divergent beams [13, 36, 66, 85, 86]. These techniques can be applied to implement additional energy resolution


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Acknowledgments The author acknowledges financial support of the DFG (MA801/12-2 and ZA161/18-2) within the priority program (SPP) 1164 and the BMBF (ADAM 03ZA7BOC). The author wants to thank Bernhard Frick for the excellent support during the measurements on IN16 and him as well as Andreas Magerl and Hartmut Zabel for fruitful discussions. Additionally, thanks to Nicole Voss for her help in finalizing the manuscript.

References 1. Special section on water in confined geometries (2004) J Phys Condens Matter 16(45). Institute of Physics and IOP Publishing Limited 2. Alba-Simionesco C, Coasne B, Dosseh G, Dudziak G, Gubbins KE, Radhakrishnan R, Sliwinska-Bartkowiak M (2006) Effects of confinement on freezing and melting. J Phys Condens Matter 18:R15 3. Almdal K, Bates FS, Mortensen K (1992) Order, disorder, and fluctuation effects in an asymmetric poly(ethylene-propylene)-poly(ethylethylene) diblock copolymer. J Chem Phys 96:9122 4. Anastasiadis SH, Russel TP, Satija SK, Majkrzak CF (1989) Neutron reflectivity studies of the surface-induced ordering of diblock copolymer films. Phys Rev Lett 62:1852 5. Baalss D, Hess S (1986) Nonequilibrium molecular-dynamics studies on the anisotropic viscosity of perfectly aligned nematic liquid crystals. Phys Rev Lett 57(1):86–89 6. Barrat J, Bocquet L (1999) Influence of wetting properties on hydrodynamic boundary conditions at a fluid-solid interface. Faraday Discussions pp 119–127 7. Barrat JL, Bocquet L (1999) Large slip effect at a nonwetting fluid-solid interface. Phys Rev Lett 82(23):4671–4674 8. Bauer T, Oberdisse J, Ramos L (2006) Collective rearrangement at the onset of flow of a polycrystalline hexagonal columnar phase. Phys Rev Lett 97:258,303 9. Bee M (1988) Quasielastic neutron scattering. Hilger, Bristol 10. Berret JF, Gamaz-Corrales R, Oberdisse J, Walker LM, Lindner P (1998) Flow-structure relationship of shear-thickening surfactant solutions. Europhys Lett 41:677 11. Bocquet L, Barrat JL (2007) Flow boundary conditions: from nano- to micro-scales. Soft Matter 3:685 12. Boue F, Lindner P (1994) Semi-dilute polymer solutions under shear. Europhys Lett 25:421 13. Bouwman WG, Kruglov TV, Plomp J, Rekveldt MT (2005) Spin-echo methods for SANS and neutron reflectometry. Phys B 357:66 14. Bhushan B, Israelachvili J, Landman U (1995) Nanotribology: friction, wear and lubrication at the atomic scale. Nature 374:607 15. Campbell SE, Luengo G, Srdanov VI, Wudl F, Israelachvili J (1996) Very low viscosity at the solid-liquid interface by adsorbed C60 monolayers. Nature 382:520–522 16. Cipelletti L, Ramos L (2005) Slow dynamics in glassy soft matter. J Phys Condens Matter 17:R253 17. Cottin-Bizonne C, Cross B, Steinberger A, Charlaix E (2005) Boundary slip on smooth hydrophobic surfaces: intrinsic effects and possible artifacts. Phys Rev Lett 94:056102 18. Craig V, Neto C, Williams D (2001) Shear-dependent boundary slip in an aqueous newtonian liquid. Phys Rev Lett 87(5):054,504 19. Debye P, Cleland RL (1959) Flow of liquid hydrocarbons in porous Vycor. J Appl Phys 30:843 20. Diat O, Roux D, Nallet F (1993) Effect of shear on a lyotropic lamellar phase. J Phys II France 3(9):1427–1452 21. Diat O, Porte G, Berret JF (1996) Orientation and twins separation in a micellar cubic crystal under oscillating shear. Phys Rev B 54(21):14,869–14,872

434

M. Wolff

22. Doshi D, Watkins E, Israelachvili J, Majewski J (2005) Reduced water density at hydrophobic surfaces: effect of dissolved gases. Proc Natl Acad Sci USA 102(27):9458–9462 23. Dowson D (1979) History of tribology. Longmans, New York 24. Drummond, Alcantar L, Israelachvili J (2002a) Shear alignment of confined hydrocarbon liquid films. Phys Rev E 66:011705 25. Drummond C, Alcantar N, Israelachvili J (2002b) Shear alignment of confined hydrocarbon liquid films. Phys Rev E 66:1–6 26. Eiser E, Molino F, Porte G, Diat O (2000) Nonhomogeneous textures and banded flow in a soft cubic phase under shear. Phys Rev E 61(6):6759–6764 27. Engemann S, Reichert H, Dosch H, Bilgram J, Honkimaeki V, Snigirev A (2004) Interfacial melting of ice in contact with SiO2 . Phys Rev Lett 92:205,701 28. Escalante J, Gradzielski M, Hoffmann H, Mortensen K (2000) Shear-induced transition of originally undisturbed lamellar phase to vesicle phase. Langmuir 16(23):8653–8663. 29. Faber TE (1995) Fluid dynamics for physicists. Cambridge University Press, Cambridge UK 30. Feynman R (1965) Lectures on Physics. Addison Weseley Publishing Company, Reading, MA, USA 31. Förster S, Konrad M, Lindner P (2005) Shear thinning and orientational ordering of wormlike micelles. Phys Rev Lett 94:017803 32. Fredrickson GH (1987) Surface ordering phenomena in block copolymer melts. Macromolecules 20:2535 33. Frick B, Gonzales M (2001) Five years operation of the second generation backscattering spectrometer IN16-a retrospective developments and plans. Physica B 301:8 34. Frick B, Koza M, Zorn R (eds) (2003) The Euro Phys J E: Special Issue: Dynamics in Confinement. EDP Sciences 35. Galea TM, Attard P (2004) Molecular dynamics study of the effect of atomic roughness on the slip length at the fluid-solid boundary during shear flow. Langmuir 20(8):3477–3482. 36. Georgii R, Böni P, Janoschek M, Schanzer C, Valloppilly S (2007) MIRA-A Flexible Instrument for VCN. Physica B 397:150 37. Gupta SA, Cochran HD, Cummings PT (1997) Shear behavior of squalane and tetracosane under extreme confinement. II. Confined film structure. J Chem Phys 107:10,327 38. Gutfreund P, Wolff M, Gerth S, Hies A, Magerl A, Zabel H (2009a) Institut laue-langevin, exp. report no.: Test-1566, 6-02-436 39. Gutfreund P, Wolff M, Zabel H (2009b) Institut laue-langevin, exp. report no.: 6-02-452 40. Gutfreund P, Wolff M, Maccarini M, Gerth S, Ankner JF, Browning J, Halbert CE, Wacklin H, Zabel H (2011) Depletion at solid-liquid interfaces: flowing hexadecane on functionalized surfaces. J Chem Phys 134:064711 41. Hamilton WA, Butler PD, Baker SM, Smith GS, Hayter JB, Magid LJ, Pynn R (1994) Shearinduced hexagonal ordering observed in an ionic viscoelastic fluid in flow past a surface. Phys Rev Lett 72(14):2219–2222 42. Hamilton WA, Butler PD, Magit LJ, Han Z, Slawecki TM (1999) Fast relaxation of a hexagonal Poiseuille shear-induced near-surface phase in a threadlike micellar solution. Phys Rev E 60:R1146 43. Hanley HJM, Rainwater J, Hess S (1987) Shear-induced angular dependence of the liquid pair correlation function. Phys Rev A 36(4):1795–1802 44. Huang DM, Sendner C, Horinek D, Netz RR, Bocquet L (2008) Water slippage versus contact angle: a quasiuniversal relationship. Phys Rev Lett 101:226,101 45. Huang E, Russell TP, Harrison C, Chaikin TM, Register RA, Hawker CJ, Mays J (1996) Using surface active random copolymers to control the domain orientation in diblock copolymer thin films. Macromolecules 31:7641 46. Israelachvili J (1991) Intermolecular and surface forces. Academic Press, London 47. Jaehnert S, Chavez FV, Schaumann GE, Schreiber A, Schoenhoff M, Findenegg GH (2008) Melting and freezing of water in cylindrical silica nanopores. Phys Chem Chem Phys 10:6039 48. Jones RAL, Kramer EJ, Rafailovich MH, Sokolov J, Schwarz SA (1989) Surface enrichment in an isotopic polymer blend. Phys Rev Lett 62:280


435

49. Joseph P, Cottin-Bizonne C, Benoˆıt JM, Ybert C, Journet C, Tabeling P, Bocquet L (2006) Slippage of water past superhydrophobic carbon nanotube forests in microchannels. Phys Rev Lett 97(15):4 50. Kataoka DE, Troian SM (1999) Patterning liquid flow on the microscopic scale. Nature 402:794 51. Keller A, Pedemonte E, Willmouth FM (1970) Macro-lattice from segregated amorphous phases of a three block copolymer. Nature 225:538 52. Kellogg GJ, Walton DG, Mayes AM, Lambooy P, Russel TP, Gallagher PD, Satija SK (1996) Observed surface energy effects in confined diblock copolymers. Phys Rev Lett 76:2503 53. Kittaka S, Ishimaru S, Kuranishi M, Matsuda T, Yamaguchi T (2006) Enthalpy and interfacial free energy changes of water capillary condensed in mesoporous silica, MCM-41 and SBA-15. Phys Chem Chem Phys 8(3223) 54. Klein J, Kumacheva E, Mahalu D, Parahia D, Fetters LJ (1994) Reduction of frictional forces between solid surfaces bearing polymer brushes. Nature 370:634–636 55. Koenig S, Pfeiffer W, Bayerl T, Richter D, Sackmann E (1992) Molecular dynamics of lipid bilayers studied by incoherent quasielastic neutron scattering. J Phys II 2:1589 56. Koenig S, Sackmann E, Richter D, Zorn R, Carlile C, Bayerl T (1994) Molecular dynamics of water in oriented DPPC multilayers studied by quasielastic neutron scattering and deuteriumnuclear magnetic resonance relaxation. J Chem Phys 100:3307 57. Koenig S, Bayerl T, Coddens G, Richter D, Sackmann E (1995) Hydration dependence of chain dynamics and local diffusion in l-a-dipalmitoylphosphtidylcholine multilayers studied by incoherent quasi-elastic neutron scattering. Biophys J 68:1871–1880 58. Krim J (1996) Friction at the Atomic Scale. Sci Am 275:74–80 59. http://www.springermaterials.com/navigation/ (accessed on 28 November 2011) 60. Lauga E, Brenner MP, Stone HA (2005) Microfluidics: the no-slip boundary condition. Springer. In: Foss J, Tropea C, Yarin A (eds) Ch. 15 in Handbook of Experimental Fluid Dynamics. Springer, New-York 61. Loewen H (1994) Melting, freezing and colloidal suspensions. Phys Rep 237:249 62. Loose W, Ackerson BJ (1994) Model calculations for the analysis of scattering data from layered structures. J Chem Phys 101(9):7211–7220 63. Maccarini M, Steitz R, Himmelhaus M, Fick J, Tatur S, Wolff M, Grunze M, Janecek J, Netz RR (2007a) Density depletion at solid-liquid interfaces: a neutron reflectivity study. Langmuir 23(2):598–608 64. Maccarini M, Steitz R, Himmelhaus M, Fick J, Tatur S, Wolff M, Grunze M, Janecek J, Netz RR (2007b) Density depletion at solid-liquid interfaces: a neutron reflectivity study. Langmuir 23:598–608 65. Magerl A, Zabel H, Frick B, Lindner P (1999) Flow dynamics of sheared liquids explored by inelastic neutron scattering. Appl Phys Lett 74:3474 66. Major J, Dosch H, Felcher GP, Habicht K, Keller T, te Velthuis SGE, Vorobiev A, Wahl M (2003) Combining of neutron spin echo and reflectivity: a new technique for probing surface and interface order. Physica B 336:8 67. Malardier-Jugroot C, Head-Gordon T (2007) Separable cooperative and localized translational motions of confined water. Phys Chem Chem Phys 9:1962–1971 68. McLeish TCB, Allgeier J, Bick DK, Bishko G, Biswas P, Blackwell R, Blottier B, Clarke N, Gibbs B, Groves DJ, Hakiki A, Heenan RK, Johnson JM, Kant R, Read DJ, Young RN (1999) Dynamics of entangled H-polymers: theory, rheology, and neutron-scattering. Macromolecules 32:6734 69. Mezger M, Reichert H, Schoeder S, Okasinski J, Schroeder H, Dosch H, Palms D, Ralston J, Honkimaki V (2006) High-resolution in situ x-ray study of the hydrophobic gap at the water– octadecyl–trichlorosilane interface. Proc Natl Acad Sci USA 103(49):18401–18404 70. Mills R (1973) Self-diffusion in normal and heavy-water in range 1-45 degrees. J Phys Chem 77:685 71. Morfin I, Lindner P, Boue F (1999) Temperature and shear rate dependence of small angle neutron scattering from semidilute polymer solutions. Macromolecules 32:7208

436

M. Wolff

72. Morishige K, Iwasaki H (2003) X-ray study of freezing and melting of water confined within SBA-15. Langmuir 19:2808 73. Morishige K, Uematsu H (2005) The proper structure of cubic ice confined in mesopores. J Chem Phys 122:044711 74. Mueller-Buschbaum P, Gutmann JS, Cubitt R, Petry W (2004) Grazing incidence small-angle neutron scattering-an advanced scattering technique for the investigation of nanostructured polymer films. Physica B 350:207 75. Mueller-Buschbaum P, Metwalli E, Moulin JF, Kudryashov V, Haese-Seilller M, Kampmann R (2009) Time of flight grazing incidence small angle neutron scattering - A novel scattering technique for the investigation of nanostructured polymer films. Eur Phys J Special Topics 167:107 76. Neto C, Evans DR, Bonaccurso E, Butt HJ, Craig VSJ (2005) Boundary slip in Newtonian liquids: a review of experimental studies. Rep Prog Phys 68(12):2859–2897 77. Oberdisse J, Harrak AEE, Carrot G, Jestin J, Boue F (2005) Structure and rheological properties of soft-hard nanocomposites: influence of aggregation and interfacial modification. Polymer 46:6695 78. Olmsted PD (2008) Perspectives on shear banding in complex fluids. Rheol Acta 47:283–300 79. Parratt LG (1954) Surface studies of solids by total reflection of x-rays. Phys Rev 95: 359–369 80. Persson BNJ (1993) Theory of friction and boundary lubrication. Phys Rev B 48:18140 81. Persson BNJ (1998) Sliding Friction. Springer, Berlin, Heidelberg, New York 82. Pfeiffer W, Henkel T, Sackmann E, Knorr W (1989) Local dynamics of lipid bilayers studied by incoherent quasi-elastic neutron scattering. Europhys Lett 8:201 83. Pfeiffer W, Henkel T, Sackmann E, Knorr W (1993) Neutron spin echo study of membrane undulations in lipid multibilayers. Europhys Lett 23:457 84. Pit R, Hervet H, Leger L (2000) Direct experimental evidence of slip in hexadecane: solid interfaces. Phys Rev Lett 85(5):980–983 85. Plomp J, de Haan VO, Delgliesh RM, Langridge S, Well AA (2007) Neutron spin-echo labelling at offspec, an ISIS second target station project. Thin Solid Films 515:5732 86. Rekveldt MT (1997) Neutron reflectometry and SANS by neutron spin echo. Physica B 234:1135 87. Rheinstaedter MC, Ollinger C, Fragneto G, Demmel F, Salditt T (2004) Collective dynamics of lipid membranes studied by inelastic neutron scattering. Phys Rev Lett 93:108107 88. Rheinstaedter MC, Seydel T, Demmel F, Salditt T (2005) Molecular motions in lipid bilayers studied by the neutron backscattering technique. Phys Rev E 71:061908 89. Rheinstaedter MC, Seydel T, Haeussler W, Salditt T (2006) Exploring the collective dynamics of lipid membranes with inelastic neutron scattering. J Vacuum Sci Tech A 24:1191 90. Sandrin L, Manneville S, Fink M (2001) Ultrafast two-dimensional ultrasonic speckle velocimetry: a tool in flow imaging. Appl Phys Lett 78(8):1155–1157 91. Schmatko T, Hervet H, Leger L (2005a) Friction and slip at simple fluid-solid interfaces: the roles of the molecular shape and the solid-liquid interaction. Phys Rev Lett 94:244501 92. Schmatko T, Hervet H, Leger L (2005b) Friction and slip at simple fluid-solid interfaces: the roles of the molecular shape and the solid-liquid interaction. Phys Rev Lett 94:980–983 93. Schmidt G, Nakatani AI, Butler PD, Karim A, Han CH (2000) Shear orientation of viscoelastic polymer-clay solutions probed by flow birefringence and SANS. Macromolecules 33:7219 94. Schmidt R, Hansen EW, Stoecker M, Akporiaye D, Ellestad OH (1995) Pore size determination of MCM-51 mesoporous materials by means of 1H NMR spectroscopy, N2 adsorption, and HREM. A Preliminary Study. J Am Chem Soc 117:4049 95. Schreiber A, Ketelsen I, Findenegg GH (2001) Melting and freezing of water in ordered mesoporous silica materials. Phys Chem Chem Phys 3:1185 96. Schwartz JA, Vykoukal JV, Gascoyne PRC (2004) Droplet-based chemistry on a programmable micro-chip. Lab on a chip 4:11


437

97. Schwarzl JF, Hess S (1986) Shear-flow-induced distortion of the structure of a fluid: Application of a simple kinetic equation. Phys Rev A 33(6):4277–4283 98. Sears VF (1989) Neutron optics: An introduction to the theory of neutron optical phenomena and their applications. Oxford University Press Inc, USA 99. Sedlmeier F, Janecek J, Sendner C, Bocquet L, Netz RR, Horinek D (2008) Water at polar and nonpolar solid walls (Review). Biointerphases 3:FC23 100. Seo Y, Satija S (2006) No intrinsic depletion layer on a polystyrene thin film at a water interface. Langmuir 22(17):7113–7116 101. Singer IL, Pollock HM (eds) (1992) Fundamentals of friction: macroscopic and microscopic processes. Kluwer, Dordrecht 102. Steitz R, Gutberlet T, Hauss T, Klosgen B, Krastev R, Schemmel S, Simonsen A, Findenegg G (2003) Nanobubbles and their precursor layer at the interface of water against a hydrophobic substrate. Langmuir 19(6):2409–2418 103. Sun G, Bonaccurso E, Franz V, Butt H (2002) Confined liquid: simultaneous observation of a molecularly layered structure and hydrodynamic slip. J Chem Phys 117:10311–10314 104. Tabor D (1973) Friction. Doubleday, New York 105. Takahara S, Sumiyama N, Kittaka S, Yamaguchi T, Bellissent-Funnel M (2005) Neutron scattering study on dynamics of water molecules in MCM-41. 2. Determination of translational diffusion coefficient. J Phys Chem B 109:11,231 106. Thompson PA, Troian S (1997) A general boundary condition for liquid flow at solid surfaces. Nature 389(6649):360–362 107. Thorsen T, Maerkel SJ, Quake SR (2002) Microfluidic large-scale integration. Science 298:580 108. Tretheway DC, Meinhart CD (2004) A generating mechanism for apparent fluid slip in hydrophobic microchannels. Physics of Fluids 16(5):1509–1515 109. Tyrrell J, Attard P (2001) Images of nanobubbles on hydrophobic surfaces and their interactions. 87:176104 110. Urbakh M, Klafter J, Goudon D, Israelachvili J (2004) The nonlinear nature of friction. Nature 430:525 111. Vinogradova O (1995) Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir 11(6):2213 112. Walz M, Falus P, Wolff M, Voss N, Zabel H, Magerl A (2007) Institut laue-langevin, exp. report no.: 9-11-1269, 9-11-1322, 9-11-1354 113. Wolff M, Magerl A, Hock R, Frick B, Zabel H (2002) Investigation of sheared liquids by neutron backscattering and reflectivity. Appl Phys A A74:374 114. Wolff M, Magerl A, Frick B, Zabel H (2003a) Quasielastic neutron scattering for the investigation of liquids under shear. Chem Phys 292(2–3):283–287 115. Wolff M, Magerl A, Frick B, Zabel H (2003b) Sheared liquids explored by means of neutron scattering. J Phys Condens Matter 15:S337–S342 116. Wolff M, Magerl A, Frick B, Zabel H (2003c) Understanding of lubrication with-neutrons. Schmierung verstehen mit Neutronen. Mat-wiss u Werkstofftech 34(6):568–570 117. Wolff M, Scholz U, Hock R, Magerl A, Leiner V, Zabel H (2004a) Crystallization of micelles at chemically terminated interfaces. Phys Rev Lett 92(25):255501 118. Wolff M, Scholz U, Hock R, Magerl A, Leiner V, Zabel H (2004b) Crystallization of micelles at chemically terminated interfaces. Phys Rev Lett 92(25):255501. doi 10.1103/PhysRevLett. 92.255501 119. Wolff M, Frick B, Magerl A, Zabel H (2005a) Flow cell for neutron spectroscopy. Phys Chem Chem Phys 7(6):1262–1265 120. Wolff M, Magerl A, Zabel H (2005b) Dynamics and structure in complex liquids under shear explored by neutron scattering. Phys Rev E 71(1):13 121. Wolff M, Magerl A, Zabel H (2005c) Structure of polymer micelles close to the solid interface – A grazing incidence small angle neutron scattering study. Euro Phys J E 16(2):141–145 122. Wolff M, Magerl A, Zabel H (2007a) NS-SANS for the investigation of micellar systems. Thin Solid Films 515:5724

438

M. Wolff

123. Wolff M, Zhernenkov K, Zabel H (2007b) Neutron reflectometery with ADAM at the ILL: present status and future perspectives. Thin Solid Films 515:5712 124. Wolff M, Akgun B, Walz M, Magerl A, Zabel H (2008a) Slip and depletion in a Newtonian liquid. Eur Phys Lett 82:36001 125. Wolff M, Akgun B, Walz M, Magerl A, Zabel H (2008b) Slip and depletion in a newtonian liquid. EPL 82(3):1–5 126. Wolff M, Steitz R, Gutfreund P, Voss N, Gerth S, Walz M, Magerl A, Zabel H (2008c) Grazing incidence small-angle neutron scattering-an advanced scattering technique for the investigation of nanostructured polymer films. Langmuir 24:11331 127. Wolff M, Magerl A, Zabel H (2009) Crystallization of soft crystals. Langmuir 25:64 128. Yoshizawa H, Chen YL, Israelachvili J (1993) Fundamental mechanisms of interfacial friction. 1. Relation between adhesion and friction. J Phys Chem 97:4128 129. Zeytseva NV, Goral VN, Montagna RA, Baeumner AJ (2005) Development of a microfluidic biosensor module for pathogen detection. Lab on a chip 5:805 130. Zhu Y, Granick S (2001) Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys Rev Lett 87(9):096,105 131. Zipfel J, Lindner P, Richtering W (1998) Shear-induced orientations in a lyotropic defective lamellar phase. Europhys Lett 43(6):683 132. Zipfel J, Lindner P, Tsianou M, Alexandridis P, Richtering W (1999) Shear-induced formation of multilamellar vesicles (“Onions”) in block copolymers. Langmuir 15:2599 133. Zorn R, Moyorova M, Richter D, Frick B (2008) Inelastic neutron scattering study of a glassforming liquid in soft confinement. Soft Matter 4:522

Index

A Ackerson, B.J., 229 Adam, G., 378 Adams, M.A., 78 Akcora, P., 349 Akiyama, S., 298 Alba-Simionesco, C., 377, 383 Alexander, S., 306 Aliotta, F., 333 Alpert, S.S., 233 α-process, 94–95 Alvarez, F., 78 Angell, C.A., 341 Ansari, A., 211 Arbe, A., 103 Arnaudov, L.N., 312

B Balsara, N.P., 298 Banks, G., 233 Beenakker, C.W.J, 229, 230 Belloni, L., 230, 232 Benoit, H., 59 Berzelius, J.J., 205 β-relaxation, 95–97 Biomacromolecules. See Hydration water dynamics Biomolecular systems, 311–312 Blochowicz, T., 343 Boson peak, 92 Branched polymers, 160–161 Bras, W., 296 Brillouin spectroscopy, 15 Brockhouse, B.N., 59, 145, 263 Busselez, R., 384

C Caffrey, M., 299 Chain diffusion, 153–154 Chain-grafted/homopolymer mixtures, 358–360 Chain retraction deformed radius of gyration, 159 diffusive mechanisms, 157 normalized structure factors, 158 Chen, S.H., 253 Chodankar, S., 311 Classical force fields, 48–49 Coherent inelastic scattering, 74–75 Coherent scattering length density, 373 Collective dynamics, 60–62 Colmenero, J., 103 Complex liquids, 428–429 Complex polymers blends dynamic asymmetry, 111–114 fast motions and processes, 104 glassy dynamics, 105–107 α-relaxation dynamics, 108–109 single-chain dynamics, 109–111 poly(n-alkyl methacrylates) anomalous relaxation, 116–118 dynamic heat capacity, 115 nanodomain structure, 115–116 neutron diffraction pattern, 114 Complex systems polymer dynamics, confinement, 164–168 protein diffusion, polymeric gels, 168–170 soft interface dynamics, polymeric mesophases, 162–164 Confinements large-scale polymer dynamics, 164–168 mesoporous solids

V. Garc´ıa Sakai et al. (eds.), Dynamics of Soft Matter: Neutron Applications, Neutron Scattering Applications and Techniques, DOI 10.1007/978-1-4614-0727-0, © Springer Science+Business Media, LLC 2012

439

440 Confinements (cont.) bioprotectant solutions, 384 bulk and confined salol, 379–380 confined orthoterphenyl, 380–381 cooperative dynamics, 382 excluded volume effects, 382 glass transition temperature, 378–379 multicomponent glass-forming systems, 385 neutron structure factor, bulk benzene, 383 radial dependence, diffusion coefficient, 380–381 super-Arrhenius behavior, 377 temperature dependence, bulk and confined toluene, 382–383 temporal fluctuations and dynamic heterogeneities, 378 Contour length fluctuations (CLF) dynamic structure factor, 156 fully protonated and centre-labelled chain, 155–156 melt properties, 157 process, 154, 155 Cordone, L., 250 Cornicchi, E., 250 Corset effect, 398

D de Gennes, P.G., 64, 68, 70, 71, 146, 150 de la Torre, J.G., 176 Density functional theory (DFT), 46–48 des Cloizeaux, J., 71, 72 Deuterated maltose-binding protein (MBP), 256 Dielectric relaxation spectroscopy dielectric constant, 9 dipolar reorientation, 10 poly(methyl methacrylate) (PMMA), 12 proteins and polypeptides, 11 Dielectric spectroscopy, 356–357 Diffusion concentration dependance, 232–234 internal and diffusive motion coupling, 241–242 measurements inside cells, 238–241 self-diffusion coefficient, 234–236 short-time diffusion coefficient, 237–238 wave vector, 238 dynamic light scattering, colloids, 228–230 liquid flow, 422–423 protein structure factor biochemical reaction, 230

Index fluorescence techniques, 231 in solutions, 231–232 types, 230 Doi, M., 67, 146, 150 Donth, E., 378 Doppler effect, liquid flow, 420 Dosseh, G., 379, 381 Doster, W., 205, 249 Dynamical transition, proteins activation enthalpy spectrum, 227 anharmonic motion, 218, 220 correlation functions, 221–222 coupling parameter, 226 elastic intensity, 221–222 elastic neutron scattering analysis, 224–225 fast β-process, 225 glass-forming liquids, 222 glass transition, 228 hydration water, intermediate scattering function, 220–221 mean square displacements, 225–226 α-relaxation, 220 vibrational Debye-Waller factor, 223 Dynamic asymmetry, polymer blends, 111–114 Dynamic light scattering, colloids, 228–230

E Edwards, S.F., 67, 70, 150 Einstein, 228 Elastic incoherent structure factor (EISF) contamination, 87 definition, 84 mesomorphic systems, 85 methyl group dynamics, PMMA, 88 motion in polymer, 86 thermometer, 87 Elastic properties, membranes, 277–279 Electrolytes. See Solid polymer electrolytes Eutectic point, 130

F Fatkullin, N., 152, 399 Fixed-window scan, 248 Flowing liquids. See Liquid flow Freda, M., 333 Frick, B., 319 Fukuda, K., 371

G Gähler, R., 293 Garc´ıa Sakai, V., 1 Gibbs, J.H., 378

Index

441

Gilbert, E.P., 289 Glass-forming liquids Boson peak, 343 hard confinement, surface effects and constraints, 340 propylene glycol, 342 pseudoternary microemulsion system, 341 scattering function, 342 Vogel–Fulcher-like temperature dependence, 341 Glass-forming polymers, 91 Glass transition, proteins, 192–193 Glass transition temperature fast process, 92–94 segmental relaxation, 94–95 Glassy dynamics, 105–107 Glassy state, 92 Gonzalez, M., 25 Gotter, M., 308 Grafted nanoparticle-based nanocomposites, 360–361 Graham, R.S., 158, 159 Grazing incidence SANS, 417–418 Green, P.F., 349 Grillo, I., 307 Guégan, R., 392–394

lattice dynamics (LD) diatomic chain, 29 dispersion curves, 30 dynamical matrix (DM), 31 optic branch, 30 powder, 33–34 lipid membrane dynamics, 272–274 molecular dynamics (MD) simulation two-dimensional Q grid, 38 VDOS, 37 Inelastic scattering, flowing liquids Doppler effect, 420 microscopic diffusion and macroscopic flow, 422–423 quasielastic scattering combination, 420–422 Inelastic X-ray scattering (IXS), 19 Interfacial water, 387–389 Ionic conductivity, 123–124

H Halperin, A., 306 Hamilton, W.A., 289, 291, 308, 427 Hansen, J.P., 230 Harpham, M.R., 333 Hayter, J.B., 230–232 Helfand, E., 164 Higgins, J.S., 59 Hoar, T., 320 Hohenberg, 46 Homopolymer/chain-tethered nanoparticle polymer nanocomposites, 350–351 Hydration shell, proteins, 198–199 Hydration water dynamics direct measurement, deuterated biomolecules, 255–257 neutron scattering, 257–259 quasi-elastic broadening, 249 scattering intensities, 248 subtracting spectra method, protonated biomacromolecule, 253–255 Hydrogen-deuterium labelling, 256–257

K Kanaya, T., 59, 80 Kawabata, Y., 331, 332, 338 Keller, A., 412 Kimmich, R., 152, 399 Kipping, D., 296 Kitagawa, T., 76 Kitchens, M., 332 Kityk, A.V., 390, 391 Kleinert, T., 209 Kohn, W., 46 Kolesnikov, A.I., 386 Krishnan, M., 189 Krueger, S., 239 Kumar, S.K., 349 Kutnjak, Z., 390

I Incoherent inelastic scattering, 75–76 Inelastic neutron scattering (INS)

J Jackson, C.L., 374, 375 Jasnin, M., 258 Johari-Goldstein (JG) process, 95–97 Johnson, M.R., 25

L Lagrené, K., 399, 402 Lal, J., 400, 402 Large-scale polymer dynamics chain diffusion, 153–154 complex systems confinement, 164–168 protein diffusion, 168–170 soft interface dynamics, 162–164

442 Large-scale polymer dynamics (cont.) molecular rheology chain retraction, 157–159 contour length fluctuations (CLF), 155–157 hierarchical relaxation, branched polymers, 160–161 H-polymer, 155 reptation, 154 protein dynamics, 170–179 reptation, 150–153 Rouse model, 147–149 Lattice dynamics (LD) diatomic chain, 29 dispersion curves, 30 dynamical matrix (DM), 31 optic branch, 30 powder, 33–34 Lechner, R.E., 370 Lefort, R., 381, 393, 394, 396 Light scattering continuum approximation, 16–17 molecular approach, 17–18 polarization, 15 relaxations and vibrational spectra, 14 techniques, 15 Lignocellulose, 199–201 Lipid membrane dynamics collective molecular dynamics coherence, 283 dispersion relations, 274–277 interactions, 279–282 properties, 277–279 collective undulation modes, 267 cooperative protein dynamics, 266 inelastic neutron scattering instruments, 272–274 lipid-mediated interaction, 267 local and collective molecular motions, 267–268 scattering technique, length and time scales, 264–265 selective deuteration and labeling techniques, 269–272 structure, 266 Liquid crystals complex phase diagram, 391–392 correlation lengths, 392–393 dynamical correlation length, 395 fixed window neutron backscattering, 393–394 intermediate scattering functions, 393–394 mesophases, 389 nematic phase, 389

Index neutron scattering structure factor, 391–392 optical birefringence, 390–391 smectic phase, 390 Liquid flows commercial lubricants, 425–427 complex liquids, 428–429 inelastic scattering Doppler effect, 420 microscopic diffusion and macroscopic flow, 422–423 quasielastic scattering combination, 420–422 models, 429–430 near surface relaxation, micelles, 427–428 Newtonian liquids, 423–424 surface anomalies, 412–414 surface sensitive scattering techniques in-plane correlations, grazing incidence SANS, 417–418 reflectivity, 415–417 SANS vs. GISANS, 418–419 small-angle scattering, 414–415 Liquid water interfacial, 387–389 nanotube, 386–387 Lithium ion batteries, 123 Lodge&McLeish (LML) model, 108 Longeville, S., 205 Long-range chain dynamics chain-grafted/homopolymer mixtures, 358–360 frictional drag, 356 glass transition temperature, 358 grafted nanoparticle-based nanocomposites, 360–361 jamming effects, 357 rheology, 357 Lovesey, S., 271 Lund, R., 302

M Malikova, N., 370 Maranas, J.K., 123 Masuda, H., 371 Mata, J.P., 289 Mazur, P., 229 McKenna, G.B., 374, 375 McLeish, T.C.B., 157 Mechanical relaxation spectroscopy atomic-force microscopy (AFM), 8 disadvantage, 8 storage and loss moduli, polystyrene, 8–9 strain under applied stress, 7

Index Médina-Noyola, M., 230 Membrane dynamics collective molecular dynamics coherence, 283 dispersion relations, 274–277 interactions, 279–282 properties, 277–279 collective undulation modes, 267 cooperative protein dynamics, 266 inelastic neutron scattering instruments, 272–274 lipid-mediated interaction, 267 local and collective molecular motions, 267–268 selective deuteration and labeling techniques, 269–272 Mesoporous solids classification, 368 confinement and glass transition, 377–384 freezing and melting, organic phases crystallization, 377 melting temperature depression, 374–375 neutron structure factor, benzene, 376–377 phase diagram, confined benzene, 375–376 hard confinement, supramolecular aggregates MCM, 369 metal organic frameworks (MOF), 369 translational diffusion coefficient, 369 urea inclusion compounds, 370 Vycor, 368–369 homogeneous confinement, 373–374 interfacial water, 387–389 liquid crystals, 389–395 nanotube water, 386–387 polymers, 395–402 powder averaging limitation, 370–371 scanning electron microscopy (SEM), 371–373 Mezei, F., 64, 324 Micellar systems. See Microemulsions Microemulsions characteristics definition, 320 diffusion coefficient, 323 parameters, 321 phase diagram, water-AOT-oil system, 321–322 reverse micelles, 324 structure, 322 surface active agents, 321 techniques, 323

443 glass-forming liquid confinement, 340–343 neutron diffraction, 328–329 neutron instruments, 324–326 neutron spin echo (NSE), 329–332 SANS, 326–328 time-of-flight and backscattering, 332–333 water-in-oil micelles, low temperatures backscattering and time-of-flight, local dynamics, 339–340 elastic temperature scans-neutron backscattering, 335–337 NSE, shape fluctuations, 337–339 SANS, 333–335 Millner, S.T., 157 Minton, A.P., 236 Miyazawa, T., 76 Mode coupling theory (MCT), 103 Molecular dynamics (MD) simulation equilibration phase and production phase, 37 INS observables two-dimensional Q grid, 38 VDOS, 37 myoglobin, 195 protein dynamics, 251–253 QENS observables layered structure, doped poly-aniline system, 40 PANI chains, 40–41 Q-dependence, 38 RMSD, 39 structure, 35 thermodynamic quantities, 37 Molecular rheology chain retraction, 157–159 contour length fluctuations (CLF), 155–157 hierarchical relaxation, branched polymers, 160–161 H-polymer, 155 reptation, 154 Morineau, D., 367, 379, 382, 383 Mulder, G., 205 Multidimensional NMR, 14 Myoglobin structure, 193, 194

N Nakano, M., 306 Nanocomposite, 135–137 Nanoporous materials. See Mesoporous solids Nanosegregation, 104 Nanostructured homopolymers. See Poly(n-alkyl methacrylates) Nanotube water, 386–387 Né, F., 312

444 Neutron coherent scattering length, 373 Neutron scattering, 138–139 energy to forces, 28–29 force field-DFT method combination, 49–51 lattice dynamics (LD), 29–31 lattice dynamics vs. molecular dynamics, 42–43 molecular dynamics (MD), 35–41 phonons direct method, 31–32 INS observables extraction, 32–33 large-scale phonon calculations, 34–35 powder-averaged lattice dynamics, 33–34 simulation vs. experiment, 51–53 total energy calculations classical force fields, 48–49 computational methods, 43, 45 density functional theory (DFT), 46–48 first principles methods, 44 Neutron scattering spectroscopy dynamic structure factors, 6 elastic incoherent structure factor (EISF), 7 molecular dynamics and lattice methods, 4 rotational and diffusional processes, 5 Neutron spin echo (NSE), 5, 171 Newtonian liquids, 423–424 NMR relaxation spectroscopy chemical binding, 13 molecular diffusion, 14 spin-lattice and spin-spin relaxation, 13 Nossal, R., 239 Nucci, N., 324

O Oda, R., 306 Oh, H., 349 Oppenheim, I., 230, 234, 236 Osaka, N., 300

P Paciaroni, A., 228 Peclet-number, 422 Pecora, R., 68 Pellegrini, E., 25 Penfold, J., 230, 231 Perez, J., 240 Petridis, L., 189 Phillies, 229 Phonons direct method, 31–32

Index INS observables extraction, 32–33 large-scale phonon calculations, 34–35 powder-averaged lattice dynamics, 33–34 Photon correlation spectroscopy (PCS), 15 P-jump. See Pressure jump (P-jump) PNC. See Polymer nanocomposites (PNC) Pohl, R.O., 92 Polarisation analysis, polymers, 62–65 Poly(n-alkyl methacrylates) anomalous relaxation, 116–118 dynamic heat capacity, 115 nanodomain structure, 115–116 neutron diffraction pattern, 114 Poly(ethylene oxide) (PEO)-LiClO4 . See Solid polymer electrolytes Polymer blends fast motions and processes, 104 glassy dynamics, 105–107 large dynamic asymmetry, 111–114 α-relaxation dynamics, 108–109 single-chain dynamics, 109–111 Polymer dynamics. See Large-scale polymer dynamics; Polymer blends Polymer nanocomposites (PNC) long-range chain dynamics chain-grafted/homopolymer mixtures, 358–360 frictional drag, 356 glass transition temperature, 358 grafted nanoparticle-based nanocomposites, 360–361 jamming effects, 357 rheology, 357 mechanical properties, 356 mixtures containing particles high grafting densities, 351–354 low grafting densities, 354–356 structure and morphology, 350–351 Polymers chain dynamics, liquid poly(ethyl ethylene), 5 characteristic relaxation times, polyisoprene, 2 conformational transitions, 80–82 correlated mechanisms, 395 dielectric relaxation spectra, polyisoprene, 11 dynamics, 396 elastic incoherent structure factor (EISF), 84–88 glass-forming, 91 glass transition temperature fast process, 92–94 segmental relaxation, 94–95

Index glassy state, 92 inelastic scattering, 73–74 inelastic X-ray vs. neutron scattering spectra, polybutadiene, 19 Johari-Goldstein process, 95–97 mechanical storage and loss moduli, polystyrene, 8–9 neutron scattering spectrum, 60 NMR, 88–91 phonon modes density of states, polyethylene crystals, 75–76 dispersion curve, polytetrafluoroethylene (PTFE), 74–75 vibrations, 77–79 polarisation analysis and neutron spin-echo, 62–65 poly(methyl methacrylate) (PMMA), relaxation, 12 quasi-elastic scattering, 79–80 reptation, 70–72 Rouse model, 68–70 scaling laws, 72–73 self-and collective dynamics, 60–62 side-chain motion, methyl groups, 82 solutions and melts, 67–68 time-resolved SANS, 397–398 tunnelling, 82–84 zero average contrast and neutron spin-echo, 398–402 Polymethyl methacrylate (PMMA) nanocomposites, 360–361 Polystyrene, 353 Pressure jump (P-jump), 300 Protein dynamical transition, 249–250 Protein dynamics atomic mean square displacements, 249–251 characteristics, 205 collective molecular dynamics coherence, 283 dispersion relations, 274–277 interactions, 279–282 membrane properties, 277–279 collective undulation modes, 267 cooperative dynamics, 266 diffusion concentration dependance, 232–242 dynamic light scattering, colloids, 228–230 protein structure factor, 230–232 dynamical structural distributions

445 displacement distribution functions, 216–218 elastic incoherent structure factor, 215 elastic intensity, 216–217 intermediate scattering function, 215–216 internal processes, 219 rotational transitions, 219 temperature evolution, second moment, 217–218 transition rates, 216–217 types of motion, 220 dynamical transition activation enthalpy spectrum, 227 anharmonic motion, 218, 220 correlation functions, 221–222 coupling parameter, 226 elastic intensity, 221–222 elastic neutron scattering analysis, 224–225 fast β-process, 225 glass-forming liquids, 222 glass transition, 228 hydration water, intermediate scattering function, 220–221 mean square displacements, 225–226 α-relaxation, 220 vibrational Debye-Waller factor, 223 history, 205 inelastic neutron scattering instruments, 272–274 internal structural motions biosolvent viscosity vs. temperature, 210–211 CO-exit rates vs. viscosity, 210–212 conformational transition rate, 211 dynamcial transition and mean square displacements, 212–213 internal transition rates vs. external solvent viscosity, 209–210 multi-step process, 207 packing defects, 207 recombination kinetic, carbon monoxide, 209 secondary structure, myoglobin, 207–208 solvent escape fraction, 213–214 lipid-mediated interaction, 267 local and collective molecular motions, 267–268 molecular dynamics simulations, 251–253 selective deuteration and labeling techniques, 269–272 solvent modification, 250, 252

446 Protein glass transition, 192–193 Protein hydration shell, 198–199 Protein-internal structural motions biosolvent viscosity vs. temperature, 210–211 CO-exit rates vs. viscosity, 210–212 conformational transition rate, 211 dynamcial transition and mean square displacements, 212–213 internal transition rates vs. external solvent viscosity, 209–210 multi-step process, 207 packing defects, 207 recombination kinetic, carbon monoxide, 209 secondary structure, myoglobin, 207–208 solvent escape fraction, 213–214 Protein motion ADH, 171 alcohol dehydrogenase dimer structure, 171–172 diffusion form factor, 178 dynamic light scattering, 173 dynamic regimes, 170–171 effective diffusion coefficient, 174–175 HYDROPRO, 176 intermediate scattering function, 173–174 motional pattern, 178–179 SANS scattering intensities, 171–172 Protein–protein interactions, 279–282 Proteins glass transition, 192–193 hydration shell density, 198–199 lignocellulose structural analysis, 199–201 neutron scattering functions and atomic fluctuations, 190–191 small-angle neutron scattering, 191–192 solvent dynamical coupling, 193–198 Protein-solvent dynamical coupling hydration, 196 interprotein motions, 198 mesophilic and thermophilic dihydrofolate reductase, 197 nonhydrogen atom fluctuations, 195–196 protein-protein interactions, 198 side-chain heavy atom fluctuations, 195, 197 water network, 197 Pullen, S.A., 298 Pure reptation, 151 Pusey, P.N., 238

Index Q Quasielastic neutron scattering (QENS) molecular dynamics (MD) simulation layered structure, doped poly-aniline system, 40 PANI chains, 40–41 Q-dependence, 38 RMSD, 39 polymer nanocomposites, 357 water hydrating macromolecules, 253–254

R Radhakrishnan, R., 375 Rapid heat and quench cell (RHQC), 298–299 Relaxation spectroscopy dielectric (see Dielectric relaxation spectroscopy) mechanical (see Mechanical relaxation spectroscopy) NMR (see NMR relaxation spectroscopy) Reptation motion dynamic structure factor, 150–151 incoherent scattering, 152 polymers normalized intermediate scattering functions, melts, 71–72 regimes, 71 schematic representation, 70 regimes, curvilinear motion, 151 Rouse motion, 152–153 self-correlation function, 152 Rharbi, Y., 397 Rheinstädter, M.C., 263 Ribotta, R., 279 Richert, R., 342 Richter, D., 94, 145 Riveros-Moreno, V., 232 Roh, R.H., 216 Ronca, G., 71, 72 Rossle, M., 312 Ross, P.D., 236 Rouse model, 68–70 Ryan, A.J., 296

S SANS. See Small-angle neutron scattering (SANS) Scaling laws, 72–73 Scheidler, P., 379 Schleger, P., 71 Schulman, J., 320

Index Segmental relaxation, 94–95 Segré, P.N., 238 Seki, K., 330 Self-diffusion coefficient, 234–236 Self-dynamics, 60–62 Sham, 46 Shear dynamics. See Liquid flow Shull, C.G, 59, 145, 263 Side-chain motion, 82 Small-angle neutron scattering (SANS) dynamic vs. kinetic, 290 liquid flow, 414–415 microemulsions, 326–328 proteins, 191–192 SR-TRSAXS (see Synchrotron timeresolved SAXS (SR-TRSAXS)) TISANE (see Time involved small-angle neutron scattering (TISANS)) TRSANS (see Time-resolved small angle neutron scattering (TRSANS)) water-in-oil micelles, low temperatures droplet core radius, 334–335 soft confinement, 335 stability and structure, 333 structure, 334 X-ray vs. neutron, 303–304 Smith, J.C., 189 Smolin, N., 189 Smoluchovsky, 228 Soft confinement, 335 Soft matter, dynamic of classical polymer polyisoprene (PIP), 2 dielectric relaxation spectroscopy dielectric constant, 9 dipolar reorientation, 10 poly(methyl methacrylate) (PMMA), 12 proteins and polypeptides, 11 light scattering continuum approximation, 16–17 molecular approach, 17–18 polarization, 15 relaxations and vibrational spectra, 14 techniques, 15 mechanical relaxation spectroscopy atomic-force microscopy (AFM), 8 disadvantage, 8 storage and loss moduli, polystyrene, 8–9 strain under applied stress, 7 neutron scattering spectroscopy dynamic structure factors, 6

447 elastic incoherent structure factor (EISF), 7 molecular dynamics and lattice methods, 4 rotational and diffusional processes, 5 NMR relaxation spectroscopy chemical binding, 13 molecular diffusion, 14 spin-lattice and spin-spin relaxation, 13 properties, 1–2 X-ray scattering advantages and disadvantages, 18 high resolution inelastic X-ray scattering (IXS), 19 vs. neutron scattering spectra, 19 XPCS, 20 Sokolov, A.P., 1 Solid polymer electrolytes crystallization kinetics, 126 dynamic, liquid state, 127–131 eutectic composition, 127 ionic conductivity, 123–124 lamellar structure, 126 nanoparticle additives, 131–135 phase diagram, 125 role of neutrons, 138–139 segmental motion, 124 semicrystalline material, 126 Solvent-slaving, 251 Soper, A.K., 382 Spehr, T., 319 Squires, G., 271 Stopped-flow, 296–298 Surface anomalies, liquid flow, 412–414 Surface sensitive scattering techniques in-plane correlations, grazing incidence SANS, 417–418 reflectivity, 415–417 SANS vs. GISANS, 418–419 small-angle scattering, 414–415 Surface slip, 429 Surfactant and vesicle systems Avrami relaxation, 310 cell mitosis, 310 2D diffusion constant, 309 fluorescence quenching decays, 306 mesophase relaxation, steady state systems, 308–309 microemulsions, 307 microreactors, 306 stopped-flow scattering, 307 structural phase transition, 304

448 Surfactant and vesicle systems (cont.) temperature-induced phase transitions, 308 unimer exchange kinetics, 305 Synchrotron time-resolved SAXS (SR-TRSAXS) conformational changes, 300 kinetic pathway, 303 micellar ensemble distribution, 302 self-assembly process, micelles, 301

T Tabony, J., 332 Temperature jump (T-jump), 298–299 Timasheff, S.N., 211 Time involved small-angle neutron scattering (TISANS) phase locking, 296 principle of, 295 requirements, 294 Time-resolved small angle neutron scattering (TRSANS) biomolecular systems, 311–312 Brownian rotation and Nëel relaxation, 314 inter-particle contribution, 315 mesophase precipitation hexagonally ordered ZrO2 /surfactant, 312–313 porous material, 313 structure, 314 two-step mechanism, 312 methods intrinsic wavelength spread, 293 irreversible systems, 291 nanosized inhomogeneities, 293 photoactive yellow protein (PYP), 291–292 single-shot kinetics experiment, 291 time cycled stroboscopic data collection, 291–292 pressure jump (P-jump), 300 stopped-flow apparatus, 296–298 surfactant and vesicle systems Avrami relaxation, 310 cell mitosis, 310 2D diffusion constant, 309 fluorescence quenching decays, 306 mesophase relaxation, steady state systems, 308–309 microemulsions, 307 microreactors, 306 stopped-flow scattering, 307 structural phase transition, 304 temperature-induced phase transitions, 308

Index unimer exchange kinetics, 305 temperature jump (T-jump), 298–299 TISANE. See Time involved small-angle neutron scattering (TISANS) T-jump. See Temperature jump (T-jump) Tokuyama, M., 230, 234, 236 Tracht, U., 378 Transmission electron microscopy, nanocomposites, 353 Tsai, A.M., 228, 250 Tunnelling inelastic and quasielastic spectra, poly(vinyl acetate), 84, 85 NMR field-cycling technique, 82 pentamethylbenzene, rotational tunnel splittings, 83 rotational rate distribution model (RRDM), 84 Twisleton, J.F., 74

V van den Brandt, B., 304

W Wassermann, Z.R., 164 Water-in-oil micelles, low temperatures backscattering and time-of-flight, local dynamics, 339–340 elastic temperature scans-neutron backscattering droplet sizes and oils, 335–336 supercooling, 335, 337 NSE, shape fluctuations bending elasticity, 337 effective diffusion coefficient, 338–339 hardness, 337 normalized intermediate scattering function, 338 structure and dynamics., 339 SANS droplet core radius, 334–335 soft confinement, 335 stability and structure, 333 structure, 334 Weik, M., 247 Weiss, T.M., 301 White, J.W., 74 Whole-chain dynamics, polymer solutions and melts, 67–68 Wiedenmann, A., 314 Willner, L., 305, 306

Index Wittenberg, J.B., 232 Wolff, M., 411 Wood, K., 247 Wyslouzil, B.E., 304

449 vs. neutron scattering spectra, 19 XPCS, 20

Y Yee, 8 X Xia, Y.D., 376 X-ray photon correlation spectroscopy (XPCS), 20 X-ray scattering advantages and disadvantages, 18 high resolution inelastic X-ray scattering (IXS), 19

Z Zanotti, J.-M., 367 Zbiri, M., 25 Zeller, R.C., 92 Zholobenko, V.L., 313 Zimm, B.H., 68 Zorn, R., 342, 380

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