Polymers, Liquids And Colloids In Electric Fields (Series in Soft Condensed Matter)

POLYMERS, LIQUIDS AND COLLOIDS IN ELECTRIC FIELDS Interfacial Instabilities, Orientation and Phase Transitions SERIES...

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POLYMERS, LIQUIDS AND COLLOIDS IN ELECTRIC FIELDS Interfacial Instabilities, Orientation and Phase Transitions

SERIES IN SOFT CONDENSED MATTER Founding Advisor: Pierre-Gilles de Gennes (1932–2007) Nobel Prize in Physics 1991 Collège de France Paris, France

ISSN: 1793-737X

Series Editors: David Andelman Tel-Aviv University Tel-Aviv, Israel Günter Reiter Universität Freiburg Freiburg, Germany

Published: Vol. 1

Polymer Thin Films edited by Ophelia K. C. Tsui and Thomas P. Russell

Vol. 2

Polymers, Liquids and Colloids in Electric Fields: Interfacial Instabilities, Orientation and Phase Transitions edited by Yoav Tsori and Ullrich Steiner

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Series in Soft Condensed Matter Vol.

POLYMERS, LIQUIDS AND COLLOIDS IN ELECTRIC FIELDS Interfacial Instabilities, Orientation and Phase Transitions Editors

Yoav Tsori

Ben-Gurion University of the Negev, Israel

Ullrich Steiner

University of Cambridge, UK

World Scientific NEW JERSEY

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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Series in Soft Condensed Matter — Vol. 2 POLYMERS, LIQUIDS AND COLLOIDS IN ELECTRIC FIELDS Interfacial Instabilities, Orientation and Phase Transitions Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4271-68-4 ISBN-10 981-4271-68-3

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World Scientific Review Volume - 9in x 6in

Foreword

The study of Soft Condensed Matter has stimulated fruitful interactions between physicists, chemists, and engineers, and is now reaching out to biologists. A broad interdisciplinary community involving all these areas of science has emerged over the last 30 years, and with it our knowledge of Soft Condensed Matter has grown considerably with the active investigations of polymers, supramolecular assemblies of designed organic molecules, liquid crystals, colloids, lyotropic systems, emulsions, biopolymers and biomembranes, among others. Taking into account that research in Soft Condensed Matter involves ideas coming from physics, chemistry, materials science as well as biology, this series may form a bridge between all these disciplines with the aim to provide a comprehensive and substantial understanding of a broad spectrum of phenomena relevant to Soft Condensed Matter. The present Book Series, initiated by the late Pierre-Gilles de Gennes, comprises independent book volumes that touch on a wide and diverse range of topics of current interest and importance, covering a large number of diverse aspects, both theoretical and experimental, in all areas of Soft Condensed Matter. These volumes will be edited books on advanced topics with contributions by various authors and monographs in a lighter style, written by experts in the corresponding areas. The Book Series mainly addresses graduate students and junior researchers as an introduction to new fields, but it should also be useful to experienced people who want to obtain a general idea on a certain topic or may consider a change of their field of research. This Book Series aims to provide a comprehensive and instructive overview of all Soft Condensed Matter phenomena. The present volume of this Book Series, edited by Yoav Tsori and Ullrich Steiner, impressively demonstrates that electric fields play an important role in Soft Condensed Matter phenomena. Due to their comparatively strong influence and long range, electric fields are particularly relevant when the system size becomes small like in block copolymer mesophases or at interfacial structures. Electric fields can induce phase transitions, provoke v

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Foreword

interfacial instabilities, govern wetting properties or allow tuning ordering processes in block copolymer systems. Within the next few years, our Series on Soft Condensed Matter will grow continuously and eventually cover the whole spectrum of phenomena in Soft Condensed Matter. We hope that many interested colleagues and scientists will profit from these endeavors.

David Andelman and G¨ unter Reiter Series Editors

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Preface

The aim of this book is to survey phenomena in soft matter systems that are triggered by electric fields. Strong electric fields are easily generated and cause stronger interactions in most materials compared to magnetic fields or to gravity. In confinement, electric fields exceeding 10 MV/m are readily produced by low-voltage sources. The manipulation of liquid and soft materials by electric fields is therefore well suited for nanotechnological and microfluidic applications. The topics covered in this book include field-induced phase transitions in simple liquids and polymers, liquid interfacial instabilities, electrowetting, and orientational and order-order phase transitions in blockcopolymers. The level of text is adequate for graduate students and researchers alike. The rich static and dynamical behavior described in the chapters are explained invoking simple physical mechanisms and physical quantities, such as the dielectric properties and conductivity of the liquids or polymers. The chapters are organized as follows. The first chapter, by D. Andelman and R. Rosensweig, is an introductory review of modulated phases. It surveys several examples of self-organizing materials, such as magnetic garnet films, two-dimensional ferromagnetic layers, and Langmuir dipolar films. It also describes in detail the well-known instabilities of ferrofluids subjected to magnetic fields (e.g. the Rosensweig instability). The second chapter by A. Onuki deals with solvation effects in polar fluids. By using a Ginzburg-Landau theory, he shows how to calculate the equilibrium ion and electric field distributions near an interface. The surface tension between two phases and the structure factor in the one-phase region near a critical point are given. The following chapter, by K. Orzechowski, is closely related. It gives a concise account of the changes occurring in the phase diagram of mixtures in uniform electric fields. The comparison with the theories of Landau and Lifshitz and the more recent theory by Onuki is also given. vii

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Preface

Chapters 4 and 5 describe the behavior of two immiscible liquids in electric fields. T. Russell and J. Bae describe the electrohydrodynamic interfacial instability which develops when a liquid film is subjected to a normal electric field. Here the instability occurs because the electrostatic energy is at a maximum when the external field is perpendicular to a dielectric interface. A fastest-growing wavelength is obtained by a linear stability analysis of pure dielectric liquids. According to the “leaky dielectric” model of G. I. Taylor, the existence of residual conductivity leads to the appearance of large viscous stresses, which lead to a faster dynamics and smaller values of the dominant wavelength. Related phenomena are discussed by F. Mugele in his review of electrowetting. Dissolved ions help to decrease the contact angle of a droplet placed on a solid substrate. The chapter presents the theory and experiments of contact angle saturation, the dynamics of droplets in microfluidics channels, droplet breakup, and various interfacial instabilities. Q. Tran-Cong-Miyata and H. Nakanishi’s chapter deals with phase separation transitions in polymer systems driven by light. They show that chemical reactions can be used to select the fastest-growing mode in the phaseseparation process of polymer mixtures. Hierarchical structures, morphologies with multiple length-scales, and spatio-temporal control of the system can also be obtained. Chapter 7, written by M. Schick, presents a fundamental approach to the thermodynamics of purely dielectric self-assembled phases in electric fields. This chapter first explains in detail how the electrostatic energy of such systems should be calculated. As examples, it discusses order-order, order-disorder, and orientational phase transitions which occur in blockcopolymers. A. Boeker and K. Schmidt describe the influence of electric fields on block-copolymers in solutions. Their unique experimental method allows them to record the dynamical orientation process, which is found to depend on the distance to the critical point. In addition, they describe an intriguing phenomenon, the reversible change in spacing of a lamellae-forming system induced by an electric field. The last chapter, by A. Zvelindosky and G. Sevink gives an account of the forefront of numerical methods used to calculate orientation and phases of block-copolymers in external fields. They show that dynamical density functional approaches can be used to obtain the dynamics of phase ordering as well as the long-time steady-state. We believe this book will be useful to people entering the field (no pun intended) as well as to active researchers. We hope the book will

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stimulate further innovation in this lively and fruitful interdisciplinary domain. We would like to thank the Series Editors, D. Andelman and G. Reiter, for their active and very positive role in bringing this book to life. Y. Tsori would like to express his gratitude to L. Leibler and P.-G. De Gennes, with whom he had numerous discussions and collaborations on the subject of electric fields in liquids and polymers.

Yoav Tsori, Beer-Sheva Ullrich Steiner, Cambridge

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contents

Contents

Foreword

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Preface

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Chapter 1 The Phenomenology of Modulated Phases: From Magnetic Solids and Fluids to Organic Films and Polymers D. Andelman and R. E. Rosensweig

1

Chapter 2 Solvation Effects of Ions and Ionic Surfactants in Polar Fluids A. Onuki

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Chapter 3 Change of Critical Mixing Temperature in a Uniform Electric Field K. Orzechowski

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Chapter 4 Electrohydrodynamic Instabilities of Thin Liquid Films T. P. Russell and J. Bae Chapter 5 Electrowetting: The External Switch on the Wettability and Its Applications For Manipulating Drops F. Mugele Chapter 6 Phase Separation and Morphology of Polymer Mixtures Driven by Light Q. Tran-Cong-Miyata and H. Nakanishi xi

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Contents

Chapter 7 Thermodynamics and the Phase Diagrams of Block Copolymers in Electric Fields M. Schick Chapter 8 Orienting and Tuning Block Copolymer Nanostructures with Electric Fields A. Boeker and K. Schmidt Chapter 9 Block Copolymers Under An Electric Field: A Dynamic Density Functional Approach A. V. Zvelindosky and G. J. A. Sevink Index

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Chapter 1 The Phenomenology of Modulated Phases: From Magnetic Solids and Fluids to Organic Films and Polymers∗ David Andelman The Raymond and Beverly Sackler School of Physics and Astronomy Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel Ronald E. Rosensweig 34 Gloucester Rd., Summit, NJ 07901, USA This chapter surveys aspects of patternings that occur in a wide array of physical systems due to interacting combinations of dipolar, interfacial, charge exchange, entropic, and geometric influences. We review well-established phenomena as a basis for discussion of more recent developments. While the materials of interest range from bulk inorganic solids and polymer organic melts to fluid colloids and granular suspensions, we note that often there are unifying principles behind the various modulated structures, such as the competition between surface or line tension and dipolar interaction in thermally reversible systems; their properties can be understood by free-energy minimization. In other cases, the patterns are determined by dissipative forces. In all these systems the patterning is modulated by the application of force fields. Another common feature of these disparate systems is that a phase diagram often emerges as a convenient descriptor. We also mention a number of interesting technological applications for certain of the systems under review.

1. Introduction A large number of diverse physical systems manifest some type of modulation in their structural properties.1 Examples of such structures in ∗ This

chapter is dedicated to the memory of Pierre-Gilles de Gennes, 1932-2007, a great scientist and close friend, who, with his characteristic gleefulness and insight, stimulated and supported us in our own studies. 1

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2

D. Andelman and R. E. Rosensweig

Table 1.

Table of Symbols

Symbol

Definition

Units

a B d d∗ d0 D E fp Fd fA g g(r) G(q) g1 H kB m M M Mq n NA NB N NB0 P q r rd t T Tc V µ µ0 ρm σ χ χAB ε γ φ(r)

microscopic length = µ0 (H + M ), magnetic induction particle diameter domain size of modulated phase domain size of BCP system demagnetization coefficient electric field free energy per particle dipolar layer free energy mole fraction of the A monomers acceleration of gravity kernel in Eq. (3) 2D Fourier transform of g(r) linear coefficient in the expansion of G(q) in Eq. (4) magnetic field magnitude Boltzmann constant = µ0 M V magnetic moment magnetization 2D magnetization as in Sec. 3 Fourier component of the 2D magnetization particle number density number of monomers of the A block on the chain number of monomers of the B block on the chain = NA + NB ; total length of polymer chain magnetic Bond number electric polarization wavenumber distance between dipoles center to center distance between particles sample thickness in the z-direction temperature critical (Curie) temperature volume = (∂nf /∂n)H,T ; chemical potential per particle permeability of vacuum mass density interfacial tension magnetic susceptibility Flory constant for polymers dielectric constant domain wall energy or line tension (in 2D) local volume fraction in an A-B di-block copolymer

m tesla m m m dimensionless m kg/s3 ampere joule joule dimensionless m/s2 1/m3 1/m dimensionless ampere/m joule/kelvin tesla m3 ampere/m ampere ampere m2 1/m3 dimensionless dimensionless dimensionless dimensionless ampere s/m2 1/m m m m kelvin kelvin m3 joule henry/m kg/m3 newton/m dimensionless dimensionless ampere2 s4 /m3 kg joule/m dimensionless

tesla=kg s−2 ampere; henry=m2 kg s−2 ampere2 ; joule=newton m; newton=kg m s−2

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Modulated Phases in Nature

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two-dimensional (2D) systems are elongated stripes and compact dropletlike domains as can be seen in Fig. 1. In the figure, domains in solid magnetic systems (garnet films) and in thin layers of ferrofluids (to be discussed in detail below) are shown side-by-side and exhibit striking similarity. In three dimensional (3D) systems, the domain morphology can be more complex and includes sheets, tubes, rods and droplets embedded in a three-dimensional matrix. The similarity between the resulting patterns in systems of different origins is quite surprising and may allude to a common unifying mechanism. An approach we adopt here is to view these systems as a manifestation of modulated phases, i.e. systems which, due to a competition between different interactions, achieve thermodynamic equilibrium in a state in which

(a)

(c)

(b)

(d)

Fig. 1. Domains in magnetic solids and fluids. (a) stripes and (b) bubble phase in ferromagnetic garnet film of 13 µm thickness grown on 111 face of gadolinium gallium garnet. Visualization in made using polarized optical microscopy (Faraday effect). Period d∗ ∼10 µm. Adapted from Ref. 2. (c) Ferrofluid confined between two glass plates exhibiting labyrinthine instability in a magnetic field.3,4 The period is d∗ ∼2 mm. (d) Bubble phase of a ferrofluid confined in a cell having a gap that increases from left to right. The mean bubble size is ∼1 mm. Adapted from Ref. 5.

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the appropriate order parameter shows a spatial modulation. Examples are abundant1 and include modulation of the magnetization field of ferromagnetic slabs6–8 and ferrofluids,3 polarization field in electric dipolar systems and certain liquid crystalline phases,9,10 the superconducting order parameter in the intermediate phase of type I superconductors,11 as well as the relative composition in block copolymer systems.12–14 Thus, in this chapter we review some of the interesting phenomena associated with modulated phases. We start by considering a simple example explaining the underlying mechanism of wavelength selection in a quasi two-dimensional dipolar system. We then review domains in related dipolar organic films on water/air interface (Langmuir monolayers) and magnetic garnet films. Two other examples of systems of current scientific interest having many applications are subsequently discussed: magnetic fluids (ferrofluids) and mesophases in block copolymers. We review how the competing interactions create interesting new phenomena when these systems are subjected to an external field (electric, magnetic) and describe their morphology, structure, phase separation, various instabilities and related phenomena. In addition, certain systems of granular suspensions are discussed having structures that are modulated by the application of forces such as magnetic attraction and viscous drag. A table of symbols is given at the beginning of this chapter. 2. Domains in Magnetic Solids Ferromagnetism15 is an important physical phenomenon associated with elements like nickel, iron and cobalt, as well as a large number of metallic alloys that show spontaneous magnetization M in the absence of external applied magnetic field. The reason for a macroscopic magnetization is deeply rooted in the existence of electronic spin Si associated with an atom at position i, and the strong direct exchange interactions of the type −JSi · Sj where J is the direct exchange integral. It is positive for ferromagnetic coupling and is related to the overlap in the charge distribution of the two neighboring atoms (i,j). The magnetization is temperature dependent. In the absence of an H field, as the temperature T is increased, the system gradually loses its magnetization, M (T, H=0), until at a special temperature, Tc , called the Curie temperature, the spontaneous magnetization drops to zero, M =0. In bulk magnetic systems, the uniform magnetization does not persist throughout the system but breaks up into spatial domains, each having a

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specific and distinct magnetic orientation. P.-E. Weiss in 1907 first introduced the concept of these magnetic domains.16 Study of the phenomenon has a long history including the notable analysis of Landau and Lifshitz in the 1930s17 [See also Refs. 18,19]. The domain size and its structure depend on competing interactions inside the magnet: the direct exchange interaction, the demagnetization fields and the crystal anisotropy. In order to explain on general grounds why dipolar systems prefer to break into domains of a well-defined size, we use the following simplified model, which gives the essential features without the need to review all the technical details.1,10 This model is applied in later sections to explain analogous features of magnetic garnet and films and dipolar Langmuir layers and is related to the labyrinthine instability of ferrofluids, as will be discussed below. 3. Domains in Two-Dimensional Ferromagnetic Layers Consider a monomolecular layer of atomic dipoles in the (x,y) plane, each having a magnetic dipole (electronic spin) that can only point along the perpendicular z–direction. We assume that the spins possess two possible values: S z = ±1/2, related to the two values of the atomic magnetic moment m = g0 µB S z , where g0 µB is the gyromagnetic factor, µB = e/2me c is the Bohr magneton and g0 ≈ 2 is the g-factor. The system can be described using an Ising model with nearest-neighbor-only ferromagnetic coupling, and the direct exchange interaction between adjacent spins minimized when two neighboring spins point in the same direction. Therefore, at low enough temperatures the magnetic order will be ferromagnetic; i.e. the spins prefer to be aligned in the same orientation even in the absence of an external field. By treating the Ising monolayer defined above at a coarse-grained level, we can perform the thermal average; namely, to sum with the proper Boltzmann weight factor over the microscopic spin degrees of freedom at finite temperatures. A local magnetization field for this 2D system, M(r) can be defined as a continuous function of the 2D position r. Close to the Curie temperature Tc and at zero applied magnetic field H, the magnetization is small and the ferromagnetic (M=0) to paramagnetic (M=0) transition can be described by an expansion of the free energy expressed in powers of M and its gradient. This is the starting point of the well-known GinzburgLandau theory.20 Because of the up-down spin symmetry in the absence of an orienting field, an expansion of the free energy has only even powers in

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M, and up to fourth order in M it can be written as: c α 2 β 4 2 2 FGL = d r |∇M| + M (r) + M (r) 2 2 4

(1)

The parameter c (related to the direct exchange interaction), α ∼ T − Tc and β are phenomenological parameters. The uniform state of the system, in which the magnetization is independent of position can be obtained from FGL by minimizing the integrand without the gradient term. This is known yields two possible as the Landau theory. For T < Tc , the minimization ferromagnetic states, M(T ) = ±M0 = ± |α|/β, while for T > Tc , the only solution is the paramagnetic state, M = 0. Any two magnetic spins also have a dipolar interaction leading to demagnetization terms which need to be included in the free energy. We consider the addition of these long-range interactions for Ising spins because these interactions have an important effect on the magnetic domain size. The dipolar interaction between any two colinear Ising spins, Siz and Sjz , that point in the z-direction with possible values ±1/2, are located in the (x,y) plane, and separated by a distance r is Ui,j =

mi mj (g0 µB )2 Siz Sjz = 4πµ0 r3 4πµ0 r3

(2)

where mi = g0 µB Siz is the atomic magnetic moment and µ0 the vacuum permeability. We recall that the energy for a parallel pair is repulsive (U > 0), while that of an anti-parallel pair is attractive (U < 0). The coarse-grained dipolar magnetic energy can be derived from Eq. (2) and after thermal averaging is written as: µ0 (3) d2 r d2 r M(r)g(r, r )M(r ) Fd = 8π where the double integral is taken over all possible dipole pairs. The 12 prefactor is included in order to avoid double counting of pairs. The kernel g(r, r ) = 1/|r − r |3 expresses the long-range nature of the dipole-dipole interaction, Eq. (2). The integral in Eq. (3) can subsequently be manipulated more conveniently in Fourier space. Using Mq and G(q) as the 2D Fourier transform of M(r) and g(r), respectively, we obtain µ0 µ0 2 g1 d2 q |q| Mq M−q (4) d q Mq G(q)M−q ≈ − Fd = 32π 3 32π 3 magnitude, the Because g(r) = 1/r3 , where r = |r| is the vectorial small q behavior of its Fourier transform G(q) = d2 r g(r) exp(−iq · r) is

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G(q) ≈ −g1 |q|, and a lower length cutoff, r = a, has to be introduced in the integration of Eq. (4) in order to take care of the diverging of g(r) = 1/r3 at r → 0. We note that this cutoff has no other effect on the small q dependence of Eq. (4). With H=0, a ferromagnetic state is described by a uniform magnetization, M = const., while a paramagnetic one by a zero magnetization, M = 0. These two states can be considered as the limit of zero q-mode. But is it possible to stabilize a non-zero q-mode in the layered system? Clearly the gradient square term in Eq. (1) opposes any such modulations. The Fourier transform of the |∇M|2 term yields a positive contribution that is proportional to q 2 M2q , whose minimum is always attained for q = 0 (uniform state). However, the dipolar-dipolar term in Eq. (4) favors short wavelength modulations (high q-modes) due to the reduction in dipolar energy when the spin pair is in an anti-parallel state. The combined free energy, Fd + FGL , includes the direct exchange, Eq. (1), as well as the long-range dipole-dipole interactions, Eq. (4). Representing the total free energy as an integral in Fourier space, its minimization with respect to q gives the most stable mode,10 q = q ∗ d µ0 G(q) + cq 2 = 0 3 dq 32π

⇒

q∗ = −

µ0 g1 µ0 dG/dq > 0 (5) 64π 3 c 64π 3 c

Some remarks are in order. In the derivation of q ∗ in Eq. (5) we neglected 4th order and higher terms in the free energy, Eq. (1). Estimating the free energy by its value at q ∗ is called the single-mode approximation. It can be justified for T Tc , where the most dominant q ∗ -mode contribution is a good approximation for the entire free energy.21 Note that for a single q-mode, the domain size by definition is d∗ = 2π/q ∗ . Up to a numerical prefactor, the domain wall width ξ is approximately equal to d∗ . This is indeed characteristic to domains close to the critical point. Their domain wall is not sharp and ξ can be substantially larger than atomic length scales. As the temperature is lowered and becomes considerably lower than Tc , the system cannot be described any longer within the single-mode approximation. Domains still prevail but their wall width ξ (of order of nanometers) becomes much smaller than the domain size d∗ (micrometers). The domain wall energy γ (per unit length) can be calculated and depends mainly on the short-ranged, direct exchange interaction. At low temperatures, an estimate of d∗ includes many q-modes and can be done for stripes, circular and other simple arrangement of domains. By considering an alternating arrangement of ±M stripe domains, the dipolar energy Eq. (3) can be

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calculated exactly. It includes logarithmic corrections related to magnetic fringe fields in 2D. This energy competes with the domain wall energy, which favors as few domains as possible. These two competing interactions10 give in 2D bγ 1 b −1 2π ∗ d = ∗ a exp = a exp NB0 (6) q 2 µ0 M2 2 where a is a microscopic length and b is dimensionless prefactor. The dimensionless number NB0 = µ0 M2 /γ is called the magnetic bond number and is also discussed in Sec. 7. We note that d∗ has a complicated dependence on temperature and magnetic field, but we do not further discuss it in this chapter. The same Bond number that fixes the domain size, is also instrumental in understanding various instabilities of isolated drop-like domains, such as domain division and elongation, and tip splitting.1 4. Dipolar Langmuir Films A manifestation of a 2D layer of dipoles can be achieved by spreading amphiphilic molecules at the water-air interface.9,10,22 Although the dipoles are electric ones, the treatment of the long-range dipole–dipole interaction is similar to the one discussed in the preceding section for dipoles having a non-zero contribution along the perpendicular z direction. We simply need to replace the magnetic field by an electric one, and the magnetization by the electric polarization. The variation in the polarization P = P ˆ z is related to the variation in the local concentration: P = µel n, where P is the polarization, µel the electric dipole moment of an individual molecule and n(r) the local number concentration of dipoles (per unit area). Amphiphilic molecules have a hydrophobic tail and a hydrophilic head that is either charged or dipolar. When these molecules are highly insoluble in the water, they form a Langmuir monolayer — a monomolecular layer that is spread at the air-water interface.23 The layer thermodynamics can be controlled by regulating the temperature or applying a surface (in-plane) pressure. Visualization of domains in the micrometer range is done by fluorescence optical and Brewster angle microscopies,22 while ordered lipid domains is studied using small angle X-ray scattering (SAXS).24 One predicts various thermodynamic states of the system as a function of temperature T and lateral pressure Π, in analogy with the gas, liquid, and solid phases in 3D

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systems. In some cases, domains of various shapes and morphologies appear to be stable over long time scales. When the molecular dipole is large, the selection of a preferred domain size can be largely attributed to the competition between dipolar interactions and the domain line tension γ. This selection is analogous with the pattern selection as discussed in the previous sections. For example, we show in Fig. 2 the domain structure of an alkyl lipid forming a Langmuir monolayer,25 and compared it with a fluoro-alkylated lipids where most of the alkyl groups are replaced by fluorinated ones, Fig. 3(a). Due to the large dipole associated with the CF3 chain extremity, the domain size of the fluorinated lipids is much smaller than that of hydrogenated lipids.

Fig. 2. Fluorescence microscopy of alkyl lipid monolayers at T = 20◦ C and area per molecule of 60˚ A2 showing 2D gas-liquid coexistence. The outer circle has a diameter of ∼ 240 µm. Adapted from Ref. 25.

We mention one set of experiments indicating that the observed patterns are due to an equilibrium q-mode selection as we have discussed above. In a Langmuir monolayer formed by the phospholipid DPPC,22 liquid-crystalline domains are seen. They take the shape of a network of elongated stripe-like structures embedded in a liquid-like background. It is known that cholesterol preferentially adsorbs to the domain perimeter and reduces the line tension γ between the domains and their liquid-like background.22 Indeed, when cholesterol was added to the DPPC monolayer, the system quickly reduced the domain width to another characteristic width. This experimental observation is in accord with the theoretical prediction, Eq. (6), where a reduction in γ strongly reduces the size of d∗ . The effect

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((a))

(b)

Fig. 3. (a) Partially fluoro-alkylated lipid monolayers at the same temperature and with area per molecule of 97˚ A2 showing stripe-like domains with stripe thickness around 5-8 µm. (b) Same setup as in (a) but with the addition of 0.1% mol of cholesterol. A noticeable thinning of the stripe is seen to about ∼ 1−3 µm. The outer circle has a diameter of ∼ 240 µm in all figure parts. Adapted from Ref. 25.

of cholesterol was studied also for fluorinated lipids in Ref. 25. Addition of a small amount of 0.1% mol of cholesterol thins the fluoro-alkylated stripes by a large factor, as can be seen in Fig. 3(b). Some of the problems in understanding the thermodynamics of Langmuir monolayers are related to their slow kinetics. In some cases, it is not clear whether the system reached its equilibrium state or is trapped in a long-lived metastable one. Thus, although dipoles play an important role in determining domain size and morphology, their precise role is not fully understood. 5. Magnetic Garnet Films A well-studied system that exhibits a domain structure arising from competing energies is a magnetic garnet film. The theoretical ideas date back to the 1930’s with the pioneering work17 of Landau and Lifshitz and their related work on the intermediate phase of type I superconductors.11 Garnet films had their days of glory in the 1960s and 1970s when they were used as magnetic storage devices (‘bubble memory’), but their larger size and slower speed compared to hard disk drives and flash memory devices made this application short-lived.7,19 However, even current research on meso– and nano–magnetism is largely inspired by the garnet films8 and is briefly reviewed below. Garnet films are ferromagnetic solid films grown so that the easy axis of magnetization is along the axis of growth.6 The magnetic spin can point

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‘up’ or ‘down’. Using polarized light microscopy coupled to the spin orientation (via the Faraday effect) to visualize details on the micrometer scale, it is observed that below the Curie temperature the film spontaneously forms domains with a disordered stripe morphology [see Fig. 1(a)]. It is quite evident that the stripe thickness (domain size) is well defined even though the stripes have no preferred orientation in the plane. Note that for larger external fields the garnet film morphology changes into a ‘bubble’ phase [see Fig. 1(b)], as will be discussed below in Sec. 5.1. The physics behind the creation of domains in garnet films, and in particular their preferred size d∗ is well understood,6,7,26 and closely related to the model 2D layered system presented in Sec. 3 above. The major difference between the two is that the garnet film has a slab geometry of finite thickness t. Any magnetized body of finite size produces magnetic charges or poles at its surface. This surface charge distribution, acting in isolation, is itself another source of a magnetic field, called the demagnetizing field. It is called the demagnetizing field because it acts in opposition to the magnetization that produces it. Consequently, the coarse-grained dipolar magnetic energy as in Eq. (3) can be calculated by mapping the system into a Coulomb interaction between two monolayers of opposite ‘charges’ separated by a distance t. The kernel appearing in Eq. (3) is now replaced by26 g(r) ∼

2 2 −√ 2 r r + t2

(7)

and the corresponding Fourier transform is G(q) ∼

4π (1 − exp(−qt)) qt

(8)

As in Sec. 3, the minimization of the free energy of Eq. (5) with the form of G(q) given by Eq. (8) yields an optimal value of the modulation wavevector, q ∗ . The connection between the finite thickness slab of the garnet and the 2D monomolecular dipolar layer can be seen by examining the qt 1 limit, where we find that G(q) ∼ −|q| as in Eq. (4). In the other limit of a thick slab, qt 1, G(q) ∼ 1/q, which also gives rise to a free–energy minimum at a non-zero value of q ∗ . The calculation of the demagnetizing field can also be done in another way. It is sensitive to the technique used to sum over the microscopic scale (lattice of atomic dipoles) and how the coarse-graining is done. The results in the small

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E External MagneticField

q limit should all converge to the same continuum description. For the opposite large q (small r) limit, different ways of introducing a microscopic cutoff are employed, but these have no qualitative effect on the free energy. Stripe-like domains can be stabilized even for zero applied magnetic field where there is a complete symmetry between the up and down spin orientations. In a slab of thickness t in the micrometer range, the resulting demagnetizing fields are strong enough to compete with the magnetic wall energy, and yield stable stripe-like domains with size d∗ in the 1-100 micrometer range. Beside its dependence on the slab thickness t, the stripe width d∗ depends on the temperature.

P H

B H

S Temperature

TC

Fig. 4. Schematic phase diagram of modulated phases (garnet films). The 2D system exhibits stripe (S) and bubble (B) phases, along with the usual paramagnetic (P) phase in the temperature-field (T −H) plane. The lines indicate first-order transition lines from S to B and then from the B to P phase. Both lines merge at the Curie point Tc for H=0. Also indicated is the geometry of the stripe and bubble arrays for magnetic garnet films. Arrows indicate the magnetization direction. Adapted from Ref. 26.

5.1. Phase transitions The phase diagram of the garnet is shown in Fig. 4 and depends on temperature and external magnetic field (for a fixed slab thickness t). For H = 0, the up and down stripes are completely symmetric. When an external field

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is applied below the Curie temperature, the domains whose magnetization is parallel to the field direction grow at the expense of the oppositely oriented domains. But at some value of H, there is a first-order phase transition between the stripe morphology (S) and the so-called ‘bubble’ phase (B), as seen in Fig. 4. The bubble phase is composed of thin cylinders of up spin embedded in a background of down spins. Upon further increase of the magnetic field, the system has another first-order phase transition from the bubble phase into a paramagnetic (P) phase. Note that the two transition lines: S→ B and B→ P terminate at the Curie temperature Tc , for H = 0. Although the periodicity is by and large determined by such equilibrium considerations, the system shows a wide range of in-plane disorder [Fig. 1(a) and (c)]. This disorder is very sensitive to the sample history indicating that care be taken to avoid trapping the system in metastable states. A sample cooled in a non-zero H field which is then removed, shows different disorder compared with a sample annealed at the same temperature but at zero magnetic field. 6. Mesomagnetism and Nanomagnetism Mesomagnetism and nanomagnetism refer to domain structures in certain solid state magnetic materials and composites having small dimensions, e.g. thin magnetic films8 with thickness in the submicron range, see Fig. 5. These magnetic systems are to be compared with ferromagnetic garnet and ferrofluid films where the domain size and thickness are much larger, Fig. 1. The spatial modulations in these materials correspond as usual to minimum energy configurations. But in addition to the contributions of magnetic field and surface energy terms, one must also include the effects of the exchange and anisotropy energies. Exchange energy arises from the presence of electron spins as noted earlier, and anisotropy energy arises from the presence of a finite angle between magnetization and the crystalline axis. These energies govern the thickness of a domain wall, and when the sample size is small enough to be comparable with the wall thickness new phenomena arise including electron spin effects. A convenient method to control the sample size is by using thin films, in which only one dimension of the sample is small. Wires with two small lateral dimensions are also studied. Electrons carry charge and spin but conventional electronics employs only the transport of charge (current). In the newly developed field of

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(c)

(b)

(a)

t = 50 nm

t = 50 nm

t = 25 nm

Fig. 5. Examples of induced stripe domains ∼ 0.5 µm wide in epitaxial cobalt dots 50 nm thick. (a) Field applied parallel to the edge of the square dot, and; (b) along the diagonal as indicated by the H vector. (c) Circular stripe domains induced in ∼ 0.5 µm wide epitaxial cobalt dots 25 nm thick and demagnetized in the direction of the H vector. The dots were fabricated using X-ray lithography and ion–beam etching from continuous epitaxial hcp cobalt films in arrays of 5×5 mm2 . Visualization is done by a Magnetic Force Microscope (MFM). Adapted from Ref. 27.

spintronics (a neologism for spin-based electronics), the electron spin is transported from one location to another.28 The so-called giant magnetoresistive (GMR) effect is based on the field-dependent scattering properties of electron spin. In GMR and related devices having discrete (modulated) layers, the scattering that increases the electrical resistance can be tuned. Modulation of the structure in spintronic devices is achieved by design and manufacturing rather than as the result of a phase transition. The prototype device that is already in wide use, e.g. in most laptop computers, is a hard disk read head employing the GMR sandwich structure schematically shown in Fig. 6. This device called a spin-valve consists of thin ferromagnetic/nonmagnetic/ferromagnetic metal layers. One ferromagnetic layer has its magnetization latched by a fourth, permanently magnetic layer overlaid on it. Magnetic fringe field emanating from bits written on the hard disk change the direction of magnetization of the other, close by, ferromagnetic layer as they pass by. For ferromagnetic layers having parallel magnetization with that of the bits, the resistance to current flow is small, while antiparallel magnetization yields large resistance. At constant potential, the change in current passing through the films is sensed by an external electronic circuit to read out the bits (zeros and ones) of memory. The technology makes it possible to read out the information stored in the memory even though the physical size of a bit is very small.

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Iout

pinned

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Iin

conductor free S N N S

magnetic recording medium

Fig. 6. A GMR magnetic media reading head. As the magnetic recording medium passes beneath the GMR sensor, it switches the direction of magnetization of the adjacent soft magnetic film. When the direction of magnetization is the same in both magnetic sensor films, the resistance to current flow is least, whereas when the directions are opposite, the resistance is greatest.

A similar structure of thin parallel layers can be configured as a magnetic tunnel junction (MTJ) for the storing of bits of information.29 Latching of ferromagnetic films having parallel magnetization can represent a ‘0’ while antiparallel magnetization represents a ‘1’. Addressing an array of the junctions is accomplished with a cross grid of normal conductors. These memory devices require no power to preserve their magnetic state and could yield computers that boot up nearly instantaneously. 7. Ferrofluids and Other Dispersions of Magnetic Particles Much of the material of the following sections pertains to the magnetic fluids termed ferrofluids, suspensions of single-domain magnetic particles in a liquid carrier that are ultrastable against settling.3 The prototypical ferrofluid is made up of magnetite (Fe3 O4 ) colloidal particles having mean size (diameter) 10 nm, coated with a 2 nm monolayer of oleic acid, and suspended in a hydrocarbon carrier fluid such as kerosene. Many surfactants in addition to oleic acid are known that produce stable ferrofluids in a wide variety of liquid carriers such as other hydrocarbons, aromatics, esters, alcohols, fluorocarbon and water carriers. The particles are in rapid thermal

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or Brownian motion that prevents them from settling under gravity. Concomitantly the particles collide with each other and the coating prevents the particles from agglomerating together and settling out. The particles are said to be sterically stabilized. Another class of water-based ferrofluids are ionically stabilized with electric double layers.30 A ferrofluid worthy of the name is free of the chaining that results from the magnetic attraction and adherence of magnetic particles to each other with an energy that exceeds that of thermal displacements. Chaining is a topic in which there is much confusion in the literature. Using the typical colloidal particle size of 10 nm and the magnetization of the usual magnetic particle magnetite, the mean number of particles in a chain computed from the deGennes and Pincus theory31 is 1.36 in a strong H field, and 1.26 in zero field. Thus, the particles are essentially monodispersed. In simulations, the ferromagnetic particles are invariably taken to be larger and/or with stronger magnetic dipoles, resulting in particle chaining. The particle size does not have to be much larger than 10 nm before chaining becomes a practical problem in ferrofluids; at 13 nm, chains of magnetite are predicted to be infinite in length. The much larger (micrometers diameter) particles of a magnetorheological fluid (MR) chain easily, which is the basis for their applications, as discussed later. Ferrofluids based on elemental ferromagnetic particles of iron, nickel, cobalt and their alloys oxidize after days of contact with the atmosphere and are not suitable for long–term use, except in sealed systems. But other magnetic solids such as maghemite (Fe2 O3 ) and mixed metal ferrites yield ferrofluids that are long-term stable against oxidation in contact with the atmosphere. Ferrofluids are a solution of nanometer size colloidal particles in which thermal fluctuations are a governing influence in their behavior. Accordingly, statistical mechanical analysis permits definition of the magnetization law and other physical properties. This stands in contrast, for example, to the behavior of magnetorheological fluids containing particles in the micron size range which aggregate together when subjected to applied magnetic field and require mechanical force to become redispersed. Flow of magnetic fluid in a magnetic field is subject to polarization force and constitutes a discipline in itself (ferrohydrodynamics) comparable to but distinct from magnetohydrodynamics, i.e. the flow of conductive, nonmagnetic fluid (such as molten metals) in the presence of magnetic fields. An introduction to the science with an extensive treatment of the effects of flow fields is found in the monograph of Rosensweig.3

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( )F (a) Ferrofluid fl id

(d) MR fluid

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(b) IInverse MR fl fluid id

( ) Inverse (c) I MSB

(e) Light trapping fluid

(f) Fluidized bed

Fig. 7. Classification of particle suspensions in a fluid carrier prior to modulation by an applied magnetic field. (a) Nanometer single–domain magnetic particles in a nonmagnetic carrier; (b) Nonmagnetic micrometer size-range particles in a matrix of ferrofluid; (c) Nonmagnetic millimeter particles in ferrofluid; (d) Multi-domain magnetic particles in nonmagnetic carrier fluid; (e) Multi-domain magnetic particles in ferrofluid; (f) Multidomain magnetic particles suspended in a flowing stream of gas or liquid. MR denotes magnetorheological. MSB denotes magnetically stabilized (fluidized) bed. Particles indicate the ordering of sizes only.

Figure 7 is a schematic illustration of six types of fluid systems containing magnetic particles that will be discussed. Four out of the six are ferrofluid systems. Black particles indicate single–domain magnetic particles having size on the order of 10 nm typical of particles in a stable ferrofluid, i.e. one which remains free of chaining of particles whether subjected to an applied magnetic field or not. Gray denotes multi-domain magnetic particles, and white denotes particles that are nonmagnetic. Modulations (e.g. formation of particle chains) can take place when the composites are subjected to an applied magnetic field. The relative sizes of the particles are not shown to scale. For example, a one micrometer particle is 102 times larger and a one millimeter particle is 105 times larger in diameter than

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a 10 nm ferromagnetic particle typical of a ferrofluid. The illustrations indicate the random distribution of particles in the absence of an applied magnetic field. 7.1. Modulation of ferrofluid interfaces Modulation in the physical systems discussed up to this point refer to changes in the topology of systems having essentially constant volume and overall shape. This also is the case in certain phenomena of ferrofluids; for example, phase transition in applied magnetic field. In addition, in ferrofluids an important class of modulations concerns change in the geometry of the surface or interface, as in the normal field instability and labyrinthine instability. Other modulations concern steady motions induced within the fluid itself, such as occur in field-modulated convection. As indicated, ferrofluids exhibit a number of unique interfacial instabilities. These are phenomena occurring in ferrofluids of uniform temperature and colloidal composition. The number density of particles is on the order of 1023 per cubic meter, hence, the ferrofluid can be considered a continuum for most purposes. The modulations can be grouped into categories. Except where cited, the phenomena listed below are discussed in detail in Ref. 3. • Uniform steady magnetic field applied to motionless ferrofluid – – – – –

The normal–field instability Prevention of Rayleigh–Taylor instability Stabilization of a fluid column Droplet shape modulation Labyrinthine patterning

• Uniform steady magnetic field applied to ferrofluid in motion – Modulation of Kelvin-Helmholtz instability – Modulation of Saffman-Taylor instability • Modulations in time-varying magnetic field32 Additional modulations occur in ferrofluids supporting a temperature gradient when the magnetization is temperature dependent: • Convection of a plane layer in uniform applied magnetic field • Convection of a plane layer in a constant magnetic field gradient

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(a)

(b)

Fig. 8. Surface relief of normal-field instability of a ferrofluid reconstructed from X-ray images. (a) Oblique view. Each layer of a peak having a given color represents one millimeter of thickness; (b) Plan view. The containing vessel is 12 cm in diameter and the initial liquid depth is 3 mm. Adapted from Ref. 34.

• Convection in a spherical system with a radial magnetic field gradient33 The above is not an exhaustive list as systems can be rotated, concentration can be non-uniform, various instabilities can be combined, etc. In addition, in all the systems listed the magnetization is equilibrated, hence is collinear with the applied field. In comparison, in systems where the ferrofluid is subjected to rapid change in direction and magnitude of the applied field the magnetization lags the field, which excites additional forces. Modulation in this latter category is virtually unexplored. The normal-field instability is the best known, most studied one which many refer to as the Rosensweig instability, see Fig. 8. Accordingly, after a brief introduction, this overview highlights a selection of related works,

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many of which are concerned with non-linear aspects of this patterning phenomenon. 7.1.1. Normal field instability This is the first instability of a ferrofluid to be observed; it is striking, and it is the best known. Peaks form in a patterned array on the free surface of a pool of ferrofluid when the ferrofluid is exposed to a uniform, verticallyoriented magnetic field. This pattern persists under static conditions, in contrast to patterns such as Bénard cells produced in dissipative systems far from equilibrium. The ferrofluid pattern is sustained as a conservative system, i.e., in the absence of energy input or dissipation. The patterning can only onset in a ferrofluid having a magnetization that exceeds a critical value and was never seen until a ferrofluid having a sufficiently high magnetization was synthesized.35 The instability in its pristine form is realized in a horizontal pool of ferrofluid subjected to a uniform, vertically oriented, magnetic field, Fig. 8. The linear analysis and experimental validation were given by Cowley and Rosensweig36 valid for a non-linearly magnetizable fluid, where ‘nonlinearly’ refers to the functional dependence of magnetization on magnetic field H. The critical magnetization Mc is specified in dimensionless form by 1 µ0 Mc2 √ =2 1+ (9) rp g∆ρm σ where µ0 denotes the permeability of vacuum, g the gravitational constant, ∆ρm is the difference in mass densities of fluids across the interface, σ the interfacial tension, and rp is the dimensionless permeability ratio: 1/2 µc µt rp = (10) µ20 For non-linear media, the parameter rp depends on two permeabilities at the operating point: the chord permeability µc = B(H)/H, and the tangent permeability µt = ∂B(H)/∂H. Although the onset of instability depends crucially on the magnetic field via the critical magnetization Mc , the spacing between peaks λ at the onset is given by 1/2 σ (11) λ = 2π g∆ρm which is simply the capillary length between the two fluids. Note that it is the same as the wavelength at onset of Rayleigh-Taylor instability.3

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It is instructive to indicate the physics of the normal-field instability in a simple way. Normally oriented magnetization transforms the flat surface of a pool of the ferrofluid into a lower energy surface having an array of peaks that are spaced apart from each other. The onset of the instability is governed by two conditions: i) The undulating surface has the same energy as the free surface at the point of onset; ii) Its energy is minimized at the onset. The total energy is the sum of surface, gravitational, and magnetic terms. We will consider the energies associated with a wave train of sinusoidal form: h(x) = δ cos(2πx/λ), where λ is the wavelength and δ is amplitude of the disturbance, assumed small. The surface energy is proportional to the surface area. Distance along the surface between crests is given by s ≈ λ + (πδ)2 /λ. The length along the unperturbed interface is λ, hence the perturbation of surface energy is given by σ(πδ)2 /λ, where σ is the surface tension. The perturbation of gravitational energy along a wavelength corresponds to the work done in lifting ferrofluid from the trough region to the crest region. This is given by the product of the lifted fluid volume λδ/π with mass density of the ferrofluid ρm , the distance between the centroids πδ/4, and the gravitational acceleration g. Thus, the gravitational term is λρm δ 2 g/4. Rigorous formulation of the magnetic energy requires a separate computation of the magnetic field distribution to determine the energy density given by the integral of HdB over the system volume, before and after the perturbation of surface form. Magnetic energy density is reduced in a region occupied by the permeable ferrofluid as the fluid is more easily magnetized than empty space. Thus, because magnetization increases at the peaks of the waveform, and decreases at the troughs, the overall magnetic energy decreases with the formation of peaks and tends to offset the concomitant increase in gravitational and surface energies. At a critical value of magnetization the changes in energies balance and instability onsets. Here we assume the energy term depends on magnetic permeability of free space µ0 , magnetization M , and wave amplitude δ. From dimensional consideration the magnetic term is thus expressed as −αµ0 M 2 δ 2 where α is a proportionality factor, and the negative sign (see Sec. 3.6 of Ref. 3) corresponds to the reduction in field energy attendant to an increase of magnetization. Adding the three energy terms and factoring out δ 2 yields: σπ 2 /λ + λρm g/4 − αµ0 M 2 = 0

(12)

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The equality follows from the first governing condition. Differentiation with respect to λ and rearranging yields the result of the second governing condition in the form:

σ (13) λ = 2π gρm This is the expression for spacing between the peaks, the same as given by Eq. (11). Note that if λ were introduced along with or in place of δ, Eq. (13) would contain a dependence of λ on δ. This would be wrong because δ is of arbitrary size. Substituting for λ into the previous equation yields the relationship for the intensity of magnetization required for onset to occur. π µ0 Mc2 = √ gρm σ α

(14)

Comparison with Eq. (9) shows that π/α = 2(1 + rp )/rp . It is of interest to note that the onset of the normal-field instability of a ferrofluid bears analogy to the hexagonal patterns seen by direct observation in the transition of type II superconductors.37 In an incisive study, Gailitis38 using an energy method to investigate nonlinear aspects of the patterning showed that the instability is subcritical, i.e. onset could occur at a lower value of applied field provided the disturbance is sufficiently large, while at the critical point the onset is ‘hard’, i.e. the surface deformation onsets as a jump rather than in a continuous manner. In addition, the prediction was made that the pattern, which onsets as an hexagonal array, can transform to a square array at higher applied magnetic field, and that both transitions exhibit hysteresis. Some doubt remained, however, as the analyses are restricted to small values of the relative permeability. Subsequently, however, numerical analysis39 using the Galerkin finite element method confirmed the subcritical character of the instability and correctly predicted experimentally measured heights of the peaks, while conditions for the transition from hexagons to squares was studied by B. Abou et al.40 The sketch of Fig. 9 illustrates the predictions of Gailitis. 7.1.2. More recent work Interesting work has focused on the question of the shape and size of the peaks under various conditions. The most successful experimental results are obtained using the attenuation of X-rays directed vertically through

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Su urface deflection amplitud de

4 7 3

5 6

8

9 1

2 Magnetic field, H

Fig. 9. Schematic illustration of transformations in the normal–field instability according to Gailitis38 Field intensities from 1 to 2 are in the subcritical range; 2 is the onset field predicted by linear analysis; 2 to 3 represents the ‘hard’ transition to the hexagonal array of peaks; 4 to 5 depicts transition to a square array. Two regions of hysteresis can be seen on the curves. In decreasing field, 8 is known as the turning point. Adapted from Ref. 3.

the pool of ferrofluid.34 In Fig. 8, which displays the usual array of peaks over the entire surface, each color indicates a layer thickness of 1 mm. This technique was applied41 to study the surface shape generated by a local perturbation in the first subcritical hysteretic regime of the instability. This is the regime identified as 1-2-3-8-1 in Fig. 9. The perturbation is generated on a flat area of the fluid surface using a pulse of field from a small air coil placed below the center of the vessel. A single pulse produced a single peak of the hysteretic regime, and additional pulses generate additional peaks. A remarkable fact is that the peaks remain present after the pulse field is removed. The peaks are termed ‘solitons’ by these authors (although solitons are generally understood to refer to nonlinear traveling waves that can pass through others with no loss of form). These soliton peaks self-organize into molecule-like clusters of 2, 3, 4, 5, 6 and more peaks in symmetric arrays. Figure 10 illustrates a pattern of 9 solitons. Figure 11 illustrates the distribution of magnetic field and cusped shape of a peak in an hexagonal array in the normal field instability as determined by a numerical computation. The concentration of field and attendant increase in normal stress difference across the interface is mainly responsible for the formation of the peaks. A simple example of parametric stabilization is the inverted pendulum whose unstable upright position can be sustained by vertically vibrating its point of suspension. In fluid dynamics a most impressive example is the inhibition of Rayleigh-Taylor instability in which a horizontal fluid layer is

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Fig. 10. Nine ‘solitons’ (solitary structures), each generated by a transient, local pulse of magnetic field applied in the subcritical range 1-2 of Fig. 8. Peaks along rim of the container are an artifact due to the curved surface of the meniscus. The containing vessel is 12 cm in diameter and the liquid depth is 3 mm. Adapted from Ref. 41.

Fig. 11. Finite element computation of ferrofluid peak shape in the range 3-4 of Fig. 9 using µ/µ0 = 30. Adapted from Ref. 42.

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stabilized above another one of smaller density by vertically vibrating their container.43 Petrelis et al.44 experimentally demonstrated the parametric stabilization of the normal-field instability of a ferrofluid using vertical vibrations of the fluid container. The measurements were in good agreement with an analytical model. 7.1.3. Labyrinthine instability in polarized fluids Labyrinthine instability of a ferrofluid shown previously in Fig. 1(c) is shown again in Fig. 12(a) alongside its dielectric dual in Fig. 12(b). The ferrofluid is contained between closely spaced horizontal glass plates (Hele-Shaw cell) together with an immiscible nonmagnetic fluid that preferentially wets the glass allowing a clear view of the pattern. A magnetic field is applied normal to the cell faces producing a pattern of stripes. The system is governed by the interaction between magnetic dipolar and interfacial energies.4 Because thinner stripes have a fixed extent between the cell faces they possess a smaller demagnetizing field, resulting in a higher magnetization and a further reduction in magnetic energy. Concomitantly, the thinner stripes present a larger interfacial area and, hence, a larger interfacial energy which limits formation of ever thinner stripes.

(a) Ferrofluid, H0 = 0.035 tesla.

(b) Dielectric fluids, E0 = 16 kV/cm

Fig. 12. (a) Magnetic field applied to ferrofluid, and (b) electric field applied to dielectric oil yield labyrinthine patterning. Photos are 7 cm square. Adapted from Ref. 4.

Dielectric fluids are polarizable just as ferrofluids are magnetizable, and their response to applied electric fields provided free charge is absent is analogous mathematically and physically to the response and patterning of

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the ferrofluids. Although the high intensity of the requisite applied electric field tends to be difficult to achieve, a successful implementation is shown in Fig. 12(b) using lubricating oil (dark) paired with castor oil (transparent) in a specially insulated Hele-Shaw cell. Applied electric field of frequency 500 Hz insured the absence of free charge while the insulation insured similarity of field boundary conditions at the interface between the fluids and the electrodes. It should be noted that just as the magnetic garnet stripes in Fig. 1(a) are analogous to the ferrofluid stripes in Fig. 12(a), the stripe domains of the dipolar Langmuir monolayer shown in Fig. 3(a) are closely analogous to the dielectric labyrinth seen in Fig. 12(b). As implied previously, the magnetic systems are dual to the electric systems. Figure 13 illustrates stages of the onset of a related phenomenon when only a small amount of ferrofluid (a drop) is put into the cell. Numerical analysis has been successful in producing realistic simulations. In one approach the dipolar energy of the system is formulated as a function of its boundary.46 Another approach writes the free energy in terms of particle concentration expressed as a Landau expansion similar to Eq. (1), and combines these terms with a formulation expressing the surface energy.47 The latter study predicts a further transformation of the labyrinthine pattern into a bubble array when a rotating in-plane magnetic field is superposed on the steady, perpendicular magnetic field.

Fig. 13. Experimental transition of a circular cylindrical drop of ferrofluid in response to increasing magnetic field. Adapted from Ref. 45.

In a vertically oriented cell the two fluids form one layer over the other due to their difference in mass density, with the ferrofluid on the bottom when it is the denser. The flat interface between the fluids undergoes tran-

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sition that, in addition, depends on gravitational energy.3,35 A linearized theory predicts the onset condition45 and the same authors present experimental photographs of the early onset terming the behavior ‘comb’ instability. The highly convoluted labyrinthine patterns in a vertical cell in higher applied fields are reported in Ref. 35. Analysis using energy minimization predicts the width and spacing of stripes based on the demagnetizing field of dipoles assuming uniform magnetic surface charge on the stripe boundaries.4 In equilibrium the net force on a whole magnetized body is given by µ0 (M · ∇H) dV (15) V

which transforms by vector identities to ρV H0 dV + ρS H0 dS V

(16)

S

where ρS = µ0 M · n ˆ is surface density of equivalent magnetic poles and ˆ is the unit normal vector facing ρV = −µ0 ∇ · M is their volume density, n outward from the surface. A model stripe system is depicted in Fig. 14. Two glass plates with a spacing t in the z-direction bound an immiscible mixture of a ferrofluid and another, nonmagnetic fluid. In the model the two fluids are assumed to form a periodic pattern of infinitely long and straight stripes. The ferrofluid stripes have a width wf in the x-direction, while the nonmagnetic ones are of width wl . The total energy per cycle is a sum of magnetic and fluid interfacial energies, U = Um + Uσ , where the interfacial energy Uσ = 2σt depends on the interfacial tension σ and the magnetic energy is given by 1 Um = − µ0 2

M H0 dV = − V

µ0 χH02 twf 2 1 + χD

(17)

Thus, the energy per unit length along the interfacial direction is given by U/(wf + wl ). The problem reduces to calculating the magnetization M = χH = χH0 /(1 + χD) inside the magnetic stripes of finite cross-section, where χ is susceptibility and D = (H0 − H)/M is the demagnetization coefficient. In this system with putative spatially uniform magnetization the volume density of poles disappears and only surface poles remain. The demagnetization field of the surface poles is then computed from integration of the

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z

H0 x

P0

t

P

V wf

wl

M

FH

P0

F H0 1 FD

++++++++++

s

T

ͲͲͲͲͲͲͲͲͲͲ

wf

Fig. 14. The labyrinth (see Fig. 1(c)), modeled as a periodic system of infinitely long and parallel stripes in the x-direction: alternating between ferrofluid stripes with permeability µ, magnetization M and width wf , and nonmagnetic fluid stripes of width wl and permeability µ0 . The system has a thickness t in the z-direction which coincides with the direction of the applied field H0 . The interfacial tension between the two fluids is σ.

Coulomb expression for an infinitely long stripe, −M sin θdx/2πs where x2 + (t/2)2 is distance from the pole, x is the in-plane distance s = coordinate, and θ is angle subtended between s and the x coordinate (see Fig. 14). The integration generates an infinite series of terms due to contributions from opposite poles of all the stripes. Minimization of the total energy per unit length yields the governing expression for normalized stripe width W = wf /t. 2 χ2 NB0 ∂D − 2 =0 2 (1 + χD) ∂W W

(18)

and the magnetic bond number, NB0 = µ0 H02 t/2σ is the ratio of magnetostatic energy to interfacial energy Computation shows that stripe width decreases with increasing applied magnetic field H0 and susceptibility χ. The analysis above a priori assumes the existence of stripes and then computes their spacing. This procedure may be compared with that of sections 4 and 5, where the analysis aims at predicting both the onset and the spacing of stripe formation.

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Experiments in which wall spacing and applied field were systematically varied yielded stripe thicknesses in reasonable agreement with theory over a three decade range of the magnetic Bond number. Labyrinthine patterning in dielectric fluids was demonstrated experimentally as well.4 For the system of Fig. 13, the superposition of an in-plane rotating magnetic field with the perpendicular time-steady normal field yields intricate and varied patterns of which some rotate. Different patterns are formed depending on which field is applied first.32 Finally, we note that ferrofluid in a Hele-Shaw cell exhibits a liquid froth phase similar in appearance to soap bubbles confined between closely spaced walls when subjected to oscillatory magnetic field oriented perpendicular to the layer.48 7.1.4. Applications The normal–field instability sets a limit on certain applications where it is desired that the ferrofluid maintains a smooth surface. An illustrative example is the use of ferrofluid to produce inexpensive and versatile mirrors for astronomical optics and other uses.49 The application has been intensely studied and found to be feasible. A reflective colloidal film of silver particles is spread on a ferrofluid and forms a mirror surface. The surface can be shaped by the application of a magnetic field to yield adaptive mirrors. Local regions of the surface can be shaped in real time by application of magnetic fields to yield adaptive mirrors that compensate for atmospheric disturbances of refractive index that, otherwise, cause ‘twinkling’ of stars and reduction of the resolution of images. The shape can be rapidly varied in time with surface vertical displacements (‘strokes’) ranging from nanometers to several millimeters. Magnetization of the ferrofluid must be kept within limits to avoid formation of peaks. Beneficial use of peak formation is studied in a novel approach50 to electrospinning of polymer nanofibers. A two-layered system is employed with the lower layer being a ferromagnetic suspension and the upper layer a polymer solution. Vertical peaks perturb the interfaces so that when, in addition, an electrical voltage is applied, the perturbations of the free surface are drawn out as in ordinary electrospinning. As the desired result the production rate of fiber is higher. Electrostatic forces can be used to disrupt fluid interfaces for the production of droplets. Ferrofluid furnishes a convenient medium for the study of such electrostatic atomization as free charge is absent allowing study of the dipolar force effects in isolation.

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Magnetizable elastic materials (ferrogels) can be produced when, e.g. a water–base ferrofluid is used to swell a chemically cross-linked polymer.51 The magnetic nanoparticles attach to the flexible network of polymer chains by adhesive forces. The ferrogels can be used for switches, sensors, micromachines, biomimetric transducers, and controlled drug delivery systems that are remotely actuated with magnetic fields.52 The normal–field instability of ferrofluids has been extended53 to describe the deformation of these ferrogels.

7.2. Phase transitions in ferrofluids A phase transition of the gas-liquid type has been observed by a number of investigators in sterically stabilized ferrofluid.54–56 On applying an external magnetic field of critical intensity to the thin layer of the ferrofluid, highly elongated droplets of a concentrated phase of ferromagnetic colloid are formed having a clearly formed interface separating the drops from a surrounding dilute ferrofluid phase, see Fig. 15. When the applied field is removed, the elongated drops are unstable under interfacial tension and break up into smaller spherical drops that diffuse into the surroundings. The instability can also be initiated by adding a less compatible solvent to the ferrofluid. Ionically stabilized ferrofluids undergo this phase separation when electrolyte concentration is altered.58 Figure 16 shows the coexistence curve determined for this magnetic fluid at various dilutions. Below the curve the ferrofluid is spatially homogeneous, and above it exists the two-phase region where droplets of concentrated ferrofluid are in equilibrium with a surrounding phase of lower concentration. As previously mentioned, ionically stabilized ferrofluids can also exhibit phase separation, by applying magnetic field or, see Fig. 17, by changing the electrolyte concentration. A thermodynamic analysis of Cebers derives the free energy of a magnetic fluid per particle, fp , for sterically stabilized ferrofluids in the following form:59 nv0 fp = f0 + kB T ln − kB T ln 1 − nv0

sinh ξ ξ

1 + nm2 λL2 (ξ) 2

(19)

where f0 is a constant of integration, kB T is the thermal energy, n the number density of magnetic particles, v0 the coated volume of a particle, and λ = 1/3 is the Lorentz cavity constant. The Langevin function is

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(a)

(b)

(c)

50 Pm Fig. 15. (a) Droplets in a thin layer with field oriented normal to the layer of a kerosenebased sterically stabilized ferrofluid. (b) Elongated droplets induced by a 12.7 kA m−1 magnetic field oriented tangential to the layer. (c) Breakup into spherical droplets ∼0.8 s after removal of the field permits estimation of interfacial tension ∼ 8.1 × 10−4 mN m−1 based on viscous dominated instability as in Ref. 57; the droplets subsequently diffuse into the surrounding continuous phase. Part (b) and (c) are adapted from Ref. 56. Part (a) not previously published, is taken from a video recording.

defined as L(ξ) = coth(ξ) − ξ −1 and depends on a dimensionless variable ξ ξ=

mH kB T

(20)

m = µ0 vMd is the magnetic moment of a particle, where v is the volume of the magnetic core of a particle, Md the domain magnetization, and H is the applied external field. The chemical potential per particle is obtained

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Magnetiic field in the layerr, H [kA m-1]

32

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27oC

I

1I

Volume fraction of particles [%] Fig. 16. Experimental spinodal data for a kerosene–based sterically–stabilized ferrofluid having mean particle size 7.4 nm. Adapted from Ref. 56.

from the free energy per unit volume nfp : ∂nfp µ= ∂n T,H

(21)

Prediction is made of the spinodal and binodal curves on a plot of magnetic field versus concentration. The binodal curve defines the coexistence curves and corresponds to a pair of points having a common tangent on a line osculating the free-energy curve µ(n1 , T, H) = µ(n2 , T, H)

(22)

n1 [fp (n1 , T, H) − µ] = n2 [fp (n2 , T, H) − µ]

(23)

The spinodal curve corresponds to a pair of inflection points of the free energy where ∂µ =0 (24) ∂n H,T

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Electrolyte cooncentration [mol l-1]


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. . . Particle volume concentration [%]

Fig. 17. Experimental phase diagram of an ionic ferrofluid. The initial uniform volume fraction of the particles is 8%. Note the change in horizontal scale at high values. Adapted from Ref. 58.

Subscripts 1 and 2 denote the dilute and concentrated coexisting phases, respectively. In the zone between the spinodal and the binodal curves the ferrofluid is metastable. Some of the theoretical coexistence curves of the Cebers theory are qualitatively similar to the experimental curve of Fig. 16. That is, the coexistence curves are concave upward and above the curves the homogeneous solution separates into a concentrated and a dilute phase. At higher values of the ratio of particle-particle interaction to thermal-energy, phase separation at zero applied field is predicted. The theory also predicts that the concentrated phase is spontaneously magnetized in the absence of an external H field, though this is thought to be an extraneous prediction due to the use of mean-field theory. Thus, the effective field He acting to magnetize a particle of the ferrofluid is specified as the Lorentz relationship He = H + M/3 where H is the Maxwell field in the medium. A simple example shows the peril of this mean-field assumption. Assuming a linear medium, defining effective susceptibility χe = M/He , usual susceptibility χ = M/H, and solving for χ yields χ=

3χe 3 − χe

(25)

This relationship predicts spontaneous magnetization when χ=3 but the behavior is not seen experimentally. The lattice-gas model of Sano and

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Magnetic fieeld [dimensionless]

Doi60 yields coexistence curves that are qualitatively the same as those of the thermodynamic treatment of Cebers. The van der Waals interaction is specified as a constant value at particle contact. Interestingly, the Lorentz field condition is an outcome of the model, and not an initial assumption. A more recent treatment61 using two models for the fluid entropy yields similar results to those of Cebers,59 and Sano and Doi.60 Two other models predict an enhancement of field-induced magnetization but are free of spontaneous magnetization. One is based on the mean– spherical approximation,62 while the other uses a perturbation treatment.63 A recent Monte-Carlo treatment explicitly produces the phase diagram of a ferrofluid,64 see Fig. 18. The authors concluded that dipole-dipole and steric interactions alone can induce phase separation, and that additional attractive potentials need not be introduced.

0.01

0.1

1

10

100

Volume fraction of particles [%] Fig. 18. Calculated phase diagram of a ferrofluid. (1) Perturbation theory from Ref. 63; (2) Mean–spherical model from Ref. 62; The symbols (o) and (x) are from Monte–Carlo simulations of Ref. 64.

We would like to return to the issue of chaining of ferromagnetic particles that was discussed in the beginning of Sec. 7. Chaining of monodisperse magnetite particles in a 2D film is evident in the experimental images of Klokkenburg et al.65 obtained by cryogenic transmission electron microscopy. However, the particles have magnetic cores of 16 nm in one

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sample and 20 nm in another, hence will strongly chain under the deGennes and Pincus theory.31 The long chains have little thermal motion and hence sediment readily, especially in a strong magnetic gradient field, thus are not useful in ferrofluids. In another work, Tlusty and Safran66 treated phase separation of suspended dipolar particles in terms of the energy and entropy of chain free-ends and topological defects, yielding chain branching rather than the two-particle interactions of the aforementioned models. Their analysis is restricted to absence of an applied magnetic field and suggests that one phase consists of branched chains and the other of free chains. A zero-field phase-separated ferrofluid can be produced by adding a poor solvent. Such a preparation is used in Ref. 67 of Zhu et al. In conclusion, the importance of these studies, aside from their inherent interest, extends possibly to the understanding of the interactions of dipolar molecules. Such species include, for example, molecules such as hydrogen fluoride and even water. 7.3. Modulation of embedded objects Modulation of embedded objects in a ferrofluid has multiple interests: as a model for two dimensional melting of solids; for producing periodic structures of large molecules for analysis by scattering of waves; for self-organized manufacturing of microscopic arrays, etc. To a first approximation, when a spherical nonmagnetic particle is dispersed in a magnetized ferrofluid the void produced by the particle possesses an effective magnetic moment, m, equal in size but opposite in direction to the magnetic moment of the displaced fluid, i.e. m = −µ0 V χH where V is volume of the sphere, χ is the effective volume susceptibility of the ferrofluid, and H is the magnetic field. For relatively low fields (µ0 H < 0.01 tesla), χ is approximately constant and m increases linearly with H. The interaction energy between two spheres with a center-to-center separation distance rd is given by the dipolar relationship U=

m2 (1 − 3 cos2 θ) 4πµ0 rd3

(26)

θ is the angle between the line connecting the centers of the spheres and the direction of the field. A thermodynamic system is obtained by using sufficiently small spheres (of diameter d < 2 µm) having Brownian motion. The controlling parameter determining structural modulation is the ratio between the dipolar energy and the thermal energy. From Eq. (26), using

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the magnitude of U with θ = 0, rd = d, and ignoring a constant factor Dipolar energy m2 /µ0 d3 = Thermal energy kB T

(27)

7.3.1. Phase change model and alignment of particles A monolayer of equal size nonmagnetic spheres immersed in a thin layer of magnetic fluid yields an intriguing model exhibiting phase change properties of melting.68 This analog model utilizes micrometer-size polystyrene spheres that exhibit Brownian motion and can be viewed under a microscope as depicted in Fig. 19(a). Figure 19(b) shows that crystalline ordering of the spheres results from the application of a magnetic field oriented perpendicular to the layer. Each sphere is a hole in the magnetic fluid and acts as a magnetic dipole of reverse polarity repelled from its neighbors. Varying the field in this system changes the value of m and, hence, that of the ratio in Eq. (27), and can be considered as an adjustment of temperature in a molecular system. Thus, for example, melting or return to randomness is observed if the field is reduced. The system has been suggested for testing theories of two-dimensional melting via the vortex unbinding mechanism.20 Shown in Fig. 19(c) is the chain formation that results from a tangential orientation of the field. In this configuration the spheres attract each other. Ordering of dilute suspensions of macromolecules is attainable in magnetized ferrofluids for assemblies that are not amenable to conventional alignment techniques. Using this technique to obtain neutron-diffraction patterns permits study of the internal structure of macromolecules such as the tobacco mosaic virus (TMV) and tobacco rattle virus (TRV).69 TMV is a hollow cylindrical assembly of length 300 nm, external diameter 18 nm, and internal diameter 4 nm. Similarly, TRV is 23 nm in diameter, 5 nm in internal diameter, and shorter than TRV. TMV and TRV both align when dispersed in a ferrofluid and subjected to a modest level of magnetic field. The ability of the method to work with low concentrations makes the method of particular interest for aligning biological materials such as chromatin which are not easily obtainable in quantity. 7.3.2. Normal and inverse magnetorheological fluids Magnetorheological (MR) fluids are suspensions of magnetizable particles typically in the size range 2 to 10 µm in a nonmagnetic oil matrix, refer to Fig. 7(b) and (d). The particles are multi-domain and only produce a

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Fig. 19. (a) A side view of a single layer of nonmagnetic, micrometer-size spheres immersed in ferrofluid. (b) The layer organizes into a hexagonal lattice when field is oriented normal to the layer, and (c) into chains when field is tangential to the layer. A uniform volumetric magnetization is equivalent to a distribution of poles on the surface of the spheres. These poles confer strong diamagnetic character to the spheres. Adapted from Ref. 68.

net magnetization when a magnetizing field is applied. When subjected to a magnetic field the particles form chains resulting in the appearance of a yield stress and a large increase in viscosity. The technology dates back to the 1940s70 with a surge of scientific interest in the 1990s due to the availability of inexpensive computer control of applied field in real time, and significant commercialization has been realized in damping of vehicle shock absorbers,71 and production of complex optical surfaces, and other grinding and polishing applications.72 The suspension of nonmagnetic particles of 2 to 10 micrometer size in a matrix of ferrofluid yields an inverse magnetorheological fluid (IMR) having properties of yield stress and controllable increase of viscosity.73 Due to the absence of particle-particle contact-magnetization that is present in ordinary MR fluids the dependence of yield stress on particle concentration in an IMR predicted from a model of asymmetric stress shows good agreement compared to data.74

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Interest in magneto-optical effects in inverse MR fluids is found in Ref. 75. 7.3.3. Magnetic trapping of light This section deals with the system shown in Fig. 7(e), a suspension of MR particles in ferrofluid as the carrier. Mehta and co-workers76 observed that upon application of an external magnetic field on a dispersion of micrometer-sized magnetic spheres stably dispersed in ferrofluid, through which monochromatic, coherent light was passing, the light gets trapped inside the suspension for critical values of applied field. The photons remain trapped while the external magnetic field is acting. When the field is removed, photons are emitted from the medium with the same frequency as the incident light, but with lower intensity. As long ago as 1958, trapping of light or ‘localization’ was predicted in strongly scattering disordered media.77,78 The prediction received experimental confirmation first in the microwave and then in the visible using either disordered media or partially ordered structures such as liquid crystals. An interesting and potentially useful aspect of the new work is that one can tune the dielectric contrast of the micrometer-size carriers with respect to the ferrofluid by varying the applied magnetic field. A complex phenomenon is observed when the light beam is first switched off, and then with a delay, the applied magnetic field is switched off. After a total time delay of a second and a half, a flash of light appears both in the forward and backward directions,79 see Fig. 20. This is spectacularly long compared to nanosecond delays achieved in other materials. The simplicity of the phenomenon is of interest in photonics in optical memories, small threshold micro-lasers, fast optical switches, optical transistors, and other components that many believe will supercede conventional electronics. It is known that the micrometer-sized magnetic spheres form elongated chain-like structures under the influence of the external magnetic field, a true spatially modulated phase. The micrometer-size particles scatter light by the Mie mechanism, and the (much smaller) ferrofluid particles are Rayleigh scatterers.80 That is, Mie theory applies to the scattering of electromagnetic radiation by particles that are similar or larger than the wavelength of light, while Rayleigh scattering applies to particles that are much smaller than the wavelength of light. It is surmised that cavities are formed within the medium in which the light is trapped. When the field is removed the chains break up, somehow releasing the light.

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(a)

(b)

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(c)

Fig. 20. (a) Diffraction pattern of red light passes through the fluid with no external magnetic field applied. (b) The pattern is stopped by a particular magnetic field strength. (c) Photons appear again when the magnet is switched off. Adapted from Ref. 76.

This phenomenon is very surprising as even the best fiber optics dissipate transmitted light in time periods on the order of a millisecond, and the particle suspension is expected to be more lossy. The phenomenon has not been independently verified yet, its mechanism is uncertain and no theory or model has appeared in the literature. Hence its status is provisional. 7.3.4. Modulation of a nanoparticle cloud The micrometer–size metallic particles of an MR fluid tend to settle under gravity. The settling can be alleviated with the addition of an additive to form a gel network that is strong enough to suspend the particles, yet easy enough to yield to an applied shear stress. Another means has been reported based on a bimodal suspension of 2 µm iron particles in a matrix of magnetite ferrofluid having particles in the usual nanometer size range.81 This is another illustration of the morphology shown in Fig. 7(e). The mechanism preventing the sedimentation appears to be related to the presence of a diffuse cloud of the magnetite particles surrounding each particle of iron, see Fig. 21. Presumably the cloud forms because of dipole-dipole magnetic attraction due either to remanent or induced dipoles of the iron particles and their interaction with the permanent dipole moments of the ferrofluid particles. A model of nanoparticle cloud distribution under the influence of van der Waals forces treated as a variational problem is found in Ref. 82, and the same methodology should be useful in treating the magnetic problem. The model also predicts a repulsion between the larger particles when their associated clouds overlap, a mechanism that would help to alleviate sedimentation.

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Fig. 21. Transmission electron microscopy of bimodal particle mixture showing a diffuse cloud of magnetite nanoparticles surrounding micrometer-size iron particles. Adapted from Ref. 81.

7.3.5. Magnetically stabilized fluidized bed (MSB) This section concerns particulate systems of the type depicted in Fig. 7(f). Gas fluidized beds of particulates are industrially important as are liquid fluidized beds. In nature, quicksand is an example of the latter. When the particles are magnetizable and a magnetic field is applied, new modulational behavior arises, as will be discussed. If system conditions produce strong chaining, the desirable features of the MSBs can be lost and so must be avoided. Thus, the study of chaining can be valuable for defining the useful limits of operation. That useful MSBs are free of chaining is documented in a study83 wherein the bed was encased in polymer, sections taken and polished, and examined under a microscope. As the velocity of a gas flowing upward through a bed of particles is increased [see Fig. 7(f)], a point is reached where the bed becomes unweighted and is said to be fluidized. Any excess gas collects into bubbles having a sharp interface that are buoyant and rise, stirring the contents of the bed and back mixing the solids. These fluidized beds have industrial importance in processing petroleum vapors and chemicals where it is desired to achieve good contacting of vapors with solids while the back mixing maintains a constant temperature throughout. In other applications it would be

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desirable to prevent the formation of the bubbles, e.g. to create a moving bed contactor with continuous input and output of the solids.84 A stability analysis treating the bed as a continuum and determining the time evolution of a voidage wave passing through shows that if the particles are magnetizable and a uniform magnetic field applied, the formation of the bubbles can be totally prevented over a wide range of flow rates.85 The prediction is well confirmed by laboratory tests. Flow rates in excess of minimum fluidization velocity expand the bed. Here the boundary between stably and unstably fluidized regimes is similar to a coexistence curve. The operating regimes of the bed are depicted in the diagram of Fig. 22, which mirrors the appearance of a phase diagram of a molecular system. However, in this case the particles are typically in the sub-millimeter to millimeter size range and Brownian motion is negligible. An inverse composite, refer to Fig. 7(c), using one millimeter, hollow glass spheres, fluidized by upflow of a ferrofluid also exhibits magnetic

Unstably fluidized

u um

Stably fluidized

Unfluidized

M

p 1/2 2 p m

U u

Fig. 22. Predicted phase diagram of a magnetically stabilized fluidized bed (MSB). The unfluidized regime is the analog of the solid state, stably fluidized the liquid state, and unstably fluidized the gaseous state. Thus, velocity plays the role of temperature, and magnetization the role of pressure. In the stably fluidized state an object less dense than the expanded bed floats, and surface waves can propagate across the free interface at the top of the bed. u denotes superficial velocity, um minimum fluidization velocity, Mp particle magnetization, and ρp particle mass density. Mp is a function of the applied, modulating field H; when the field is removed Mp = 0 and the stably fluidized regime disappears. Adapted from Ref. 86.

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stabilization in an applied magnetic field. As a variant of the magnetically stabilized fluidized bed (MSB), the stabilization also prevents bed instability that usually results in back mixing, bypassing of the fluidizing fluid and chaotic flow.87 Quantitative agreement of the experimental coexistence curve with the theory is close compared to that for the ordinary MSB.87 This is believed due to the strong local polarization that occurs at the contact region of a pair of highly permeable magnetic particles. The resultant attractive force between such particles confers a Bingham-type rheology to the bed which is not modeled in the theory. This complication is absent in the inverse beds. 7.3.6. Other related phenomena A number of other multiphase ferrofluid systems are reviewed by Cabuil and Neveu.88 These include magnetic lamellar phases consisting of a periodic packing of alternate water and ferrofluid layers, mixture of ferrofluid with liquid crystalline carrier, magnetic vesicles, and magnetic emulsions. Modulating the alignment of a nematic liquid crystal by doping it with a small amount of ferrofluid, in principle, can be accomplished using just a weak magnetic field on the order of 10−2 tesla as was first suggested in the classic paper of Brochard and de Gennes.89 Recent work explores the synthesis of ferrofluids amenable to the doping.90 Ferroelectric analogs to ferrofluids conceptually employ a particle such as barium titanate that is permanently electrically polarized, and would respond to electric fields in the manner that ferrofluids respond to magnetic fields. However, attempt to produce such a dispersion have been unsuccessful. It is thought that the association of free charge neutralizes the polarity of the dispersed particles, a process that cannot occur in ferrofluids as magnetic monopoles are not found in nature. In contrast, there is much interest for technological applications of electrorheological (ER) fluids.91 In these systems larger particles are dispersed in a good insulator fluid. The polarization is not inherent to the particles but is induced by an external electric field and arises from the substantial dielectric difference between the particles and the carrier fluid. 8. Block Copolymers For our last example, we depart from magnetic colloids and granular magnetic systems to review the appearance of modulated phases in block

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copolymers (BCP). These are polymeric systems in which each polymer chain is composed of several homogeneous blocks. Block copolymers exhibit a fascinating variety of self-assembled nanoscale structures with various types of chain organization. We focus only on the simplest, linear A-B di-block chain architecture, in which a homogeneous and long polymer chain of type A is covalently bonded with a B chain.13,14,92–95 Composite materials made by mixing two or more different types of polymers are often incompatible and undergo phase separation. Such macrophase separation is hindered in BCP systems due to the chain connectivity. By properly choosing the polymer blocks, it is possible to design novel composite materials made of BCP chains with desired mechanical, optical, electrical and thermodynamical properties.13 For example, by joining together a stiff (rod-like) block with a flexible (coil) block, one can obtain a material that is rigid, but not brittle. Moreover, the interplay between flexibility and toughness can be controlled by temperature. More recently, BCPs are being explored in applications such as photonic band–gap materials, dielectric mirrors, templates for nano-fabrications and in other optoelectronic devices.96,97 Liquid melts of block copolymers or BCP-solvent liquid mixtures form spatially modulated phases in some temperature range. As an example we show in Fig. 23 the multitude of modulated phase in the well-studied polystyrene–polyisoprene (PS-PI) block copolymer system.98 The two important parameters that determine the phase diagram of the figure are the mole fraction of one of the two components, fA , and the product of two parameters, N χAB , where N = NA + NB is the BCP chain degree of polymerization (total number of monomers), and χAB ∼ T −1 is the Flory constant. The latter is a dimensionless parameter representing the ratio between the interaction energy to the thermal energy kB T , and quantifies the relative interaction between the A and B monomers. Typical values of χAB are small compare to unity (about χAB 0.1 for styrene-isoprene). At high temperatures (low value of N χAB ) the BCP melt is in a disordered liquid state in which the different chains show no particular organization. As the temperature is reduced below some critical value (N χc ≈ 10.5 in Fig. 23), the partially incompatibility between the A and B blocks causes a micro-phase separation into one of several modulated phases. These socalled meso-phases have spatial modulation in the 10 — 100 nm range and can have several symmetries. Figure 23 shows the lamellar (LAM), hexagonal (HEX), body centered cubic (of group symmetry Im¯3m) and bicontinuous gyroid (of group symmetry Ia¯3d) phases, as well as a perforated

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NFAB

Fig. 23. N χAB versus fPI phase diagram for PI-PS di-block copolymers, where fPI is the mole fraction of the isoprene block. The dash-dot curve is the mean-field prediction for the instability of the disordered phase. Solid curves are experimental ones and have been drawn to delineate the different phases observed but might not correspond to precise phase boundaries. Five different ordered microstructures (shown schematically) have been observed for this chemical system. Adapted from Ref. 98.

lamellar phase (HPL) that is believed to be a long-lived metastable state, but not a true, thermodynamic stable phase. 8.1. Modulated periodicity in BCP How can we understand the self-assembly and stability of various BCP modulated phases with definite periodicity? While sophisticated theories93,99–106 quite successfully reproduce complex phase diagrams such as

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Fig. 23, we present here a qualitative and heuristic explanation.92 Consider a symmetric di-block copolymer (fA = NA /N = 0.5) whose structure is that of a lamellar stack as depicted in Fig. 24. The uni-directional periodicity d is taken as parameter and its value will be determined later. We also assume that the two blocks have the same monomer size, a. If the A and B chains were not connected, the coarsening that usually occurs during phase separation would result in a macro-phase separation (theoretically with d → ∞). However, as the BCP periodicity d increases, the A-B chains start to stretch and lose entropy. The competition between coarsening and chain entropy results in a preferred domain size d; this is a characteristic of all BCP systems. For a lamellar phase with fA = 0.5, the free energy per copolymer chain can be written as a sum of two terms fchain = kB T

3(d/2)2 + σΣ 2N a2

(28)

The first term expresses the entropy cost of stretching an ideal chain (similar to a Gaussian random walk) of N monomers to span half of the lamellar period, d/2, from its unperturbed size =aN 1/2 . The second term is the is the surface tension (in units of interfacial energy per chain where σ ∼ χ1/2 AB energy per area), and Σ is the area per chain at the A/B interface. Because we consider a di-block polymer melt (i.e. with no solvent), the system is assumed to be incompressible: the volume occupied by each chain is fixed,

A

d0

B

A

B

Fig. 24. Schematic representation of a symmetric lamellar phase of di-BCP (fA = 0.5). The periodicity d0 is twice the thickness of each of the A and B lamellae.

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vchain = N a3 = Σd/2, where a3 is the volume of one monomer. Substituting the incompressibility condition, Σ ∼ d−1 , in Eq. (28), the chain free–energy is minimized with respect to the lamellar thickness d whose optimal value is 1/3 γAB σa2 aN 2/3 ∼ χ1/6 N 2/3 (29) d0 1.39 AB kB T where we used the simple scaling dependence of σ on χAB . Hence, from a simple free–energy minimization we find that the lamellae have a preferred periodicity d0 ∼ N 2/3 that scales as the two-thirds power of the BCP molecular weight; this should be compared with the unperturbed size ∼ N 1/2 . Hence, this means that the BCP chains in a lamellar phase are highly stretched due to their partial incompatibility. The prediction of novel structures using a simple free–energy minimization subject to structural and composition constraints is an essential element behind the more refined theories93 and is characteristic of all BCP systems. 8.2. Orientation of anisotropic phases by an electric field As noted previously, block copolymers form heterogeneous composite materials. Since most polymers are non-conducting dielectrics, a modulated phase of BCP is a heterogeneous dielectric, with spatially varying dielectric constant that depends on the dielectric constants of the two blocks, εA and εB . When an anisotropic BCP phase (such as a lamellar stack or an hexagonal arrangement of cylinders) is placed in a strong enough external electric field, E, the most apparent effect is an orientation of the BCP domains in the direction of the external field.107–117 The term in the free energy accounting for this effect is proportional to (εA − εB )2 E 2 . In coarse grained models of BCP melts, only the local relative A/B concentration is retained. It is represented by a continuous composition variable, φ(r), that varies between zero (pure B) and one (pure A). The dielectric constant can be taken as a linear interpolation of the local composition φ: ε(r) = φ(r)εA + (1 − φ(r))εB , and its spatial average is ε = fA εA +(1−fA)εB . In the weak segregation limit (χAB ≈ χc ), the electrostatic energy per unit volume was shown by Amundson and Helfand108 to have the form (εA − εB )2 (q · E)2 φq φ−q 2 ε q2 q

(30)

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Y/d0

where the sum is taken over all Fourier modes: φ(r) = q φq cos(q · r). Minimizing the sum of the electrostatic energy Eq. (30) and the nonelectrostatic BCP free energy produces an orientation transition107 shown in Fig. 25. At large enough E field, there is a first–order transition to a lamellar phase that is oriented in the direction parallel to the E field (the y axis in the figure). The modulations seen in Fig. 25(b) are typical of the weak segregation limit and disappear for larger E fields. In other works, such electric-field orientation was also reported for cylindrical phases.117,118

2

2

0

0

Ŧ2

Ŧ2

0

2

4

X/d

0

(a)

0

2

4

X/d0 (b)

Fig. 25. Calculated contour plots of a symmetric BCP lamellar phase between two planar electrodes and under external electric field. The electrode surfaces are at y = ±2d0 , and the field is in the y direction. The B monomers (colored black) are attracted to both surfaces. (a) For E–field slightly smaller than the critical field, E = 0.98Ec , the film exhibits a perfect parallel ordering. (b) For E-field just above the threshold, E = 1.02Ec , the film morphology is a superposition of parallel and perpendicular lamellae. Adapted from Ref. 107.

8.3. Phase transitions induced by electric fields It is well known that a drop of ferrofluid placed in a strong magnetic field elongates into a prolate ellipsoid and then, via a first-order phase transition, sharply transforms into a needle-like drop.3 This transition also occurs for charged or dipolar liquid drops. Quite recently a similar phenomenon was observed and modeled in bulk BCP systems.119–122 The starting point is a cubic (bcc) phase of isolated spherical drops rich in one of the blocks (say A), embedded in a background of the

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other component (B). As can be seen in Fig. 26, this situation occurs for asymmetric BCP (fA = 0.5). When such a cubic phase of spheres (bcc) is placed in an external E field, each of the spheres elongates in the same (1 1 1) direction of the E field [Fig. 26 (a)]. As the value of the E field increases, above a critical value Ec , the distorted cubic phase undergoes a first-order phase transition to an hexagonal phase of cylinders pointing also along the same E-field direction, Fig. 26 (b).

(a)

(b)

Fig. 26. Contour plots of a BCP phase in an electric field with fA = 0.37 and N χAB = 12. At zero E field, the stable phase is a cubic phase (bcc) (not drawn). (a) For E = 0.98Ec , just 2% below the critical field, and oriented along the (1 1 1) direction of the lattice, each of the spheres deforms into a prolate ellipsoid and the bcc phase changes continuously into a phase with an R¯ 3m space group symmetry. (c) For E = 1.02Ec , just 2% above the critical field, the system undergoes a first-order phase transition into an hexagonal array of cylinders, also pointing along the E–field direction. Adapted from Ref. 120.

Using two different computational techniques, the full phase diagram in the parameter space of fA , χAB and E can be calculated with semiquantitative agreement between the two methods. The resulting phase diagram for a fixed value of fA = 0.3 is shown in Fig. 27. The distorted cubic phase has an R¯ 3m group symmetry, and undergoes a phase transition to an hexagonal phase (hex) or a completely disorder phase (dis) depending on the initial value of the Flory constant, χAB . Although the full phase diagram has not yet been measured, some of the observations agree with the model presented here for the PS-PMMA (polystyrene-polymethylmethacrylate) system.119 We end this section by mentioning that mobile ionic impuri-

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Fig. 27. Phase diagram of block copolymers in an electric field, in the plane defined by ˆ0 , for fA = 0.3. The distorted the Flory constant χAB and the normalized electric field E bcc phase, denoted by its space group symmetry as R¯ 3m, is bounded by the hexagonal (hex) and disordered (dis) phases and terminates at a triple point where all three phases coexist. The solid line is the prediction of an analytical one-mode approximation, whereas the dashed lines are obtained by a more accurate, self-consistent numerical study. The ˆ0 at the triple point. axes are scaled by (χt , Et ), which are the values of χAB and E Adapted from Ref. 121.

ties can have an important effect on the phase transitions and alignment of modulated BCP phases, and is an active field of current investigations120,123 9. Conclusions This review considered modulated phases in a broad context encompassing scales ranging from the nano- to the macro-scale in materials as diverse as solid state metallics, inorganics and organics. Modulations in engineering systems such as fluidized beds, magnetorheological fluids, and block copolymers are also discussed. In equilibrated systems, the structure that develops is often due to a competition between the various energies associated with the structure and yields interesting visual patterning. Many of these patterns can be understood using an energy minimization that relies on Ginzburg-Landau type free energy expansions which preserve the system symmetry or simple geometric considerations. In addition, some dissipative structures are considered as well as certain solid state devices whose structure is fabricated by pattering. The modulating methods are

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complementary to each other with analysis of the dissipative structures requiring a mechanistic approach rather than an energy minimization. Some of the modulated phases are associated with significant technical applications, while others are related to biomaterials and biological systems.

Acknowledgments We thank S. Safran for his comprehensive and incisive comments on the manuscript, and D. Ben-Yaakov, M. Schick, and Y. Tsori for additional useful comments. DA acknowledges support from the Israel Science Foundation (ISF) under grant no. 160/05 and the US-Israel Binational Foundation (BSF) under grant no. 2006055.

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Chapter 2 Solvation Effects of Ions and Ionic Surfactants in Polar Fluids Akira Onuki Department of Physics, Kyoto University, Kyoto 606-8502, Japan A Ginzburg-Landau theory is presented on polar binary mixtures containing ions. It takes account of electrostatic, solvation, image, and amphiphilic interactions among the ions and the solvent molecules. The ion distributions and the electric potential are calculated around an interface with finite thickness in equilibrium. The surface tension is increased for hydrophilic ion pairs, but is decreased for hydrophilic and hydrophobic ion pairs. Introducing the amphiphilic interaction, we also treat ionic surfactants, which aggregate at an interface and reduce the surface tension. A mesophase with periodic composition and ion modulations emerges for sufficiently large asymmetry between the cationic and anionic solvation strengths. Also, among ions, there arise long-range attractive interactions in the Ornstein-Zernike form, which are mediated by the composition fluctuations. Under strong solvation conditions, they can dominate over the Coulomb interaction in the range shorter than the correlation length. In the presence of three ion species, the ion distribution can be very complex.

1. Introduction In usual electrolyte theories, ions interact via the Coulombic potential in a fluid with a homogeneous dielectric constant ε. However, ε is strongly inhomogeneous particularly in the presence of mesoscopic structures in polar fluid mixtures (water and oil) or in polymer solutions. Furthermore, in most of the physics literature, the microscopic molecular interactions between ions and solvent molecules are not explicitly considered. Around a microscopic ion such as Na+ or Cl− in a polar fluid, the ion-dipole interaction gives rise to a solvation (hydration) shell composed of a number of solvent molecules (those of the more polar component for a mixture).1,2 The resultant solvation free energy per ion will be called the solvation 57

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chemical potential and will be written as µisol where i represents the ion species. It is important that µisol strongly depends on the solvent density or the composition for binary mixtures, with its typical values much larger than the thermal energy kB T . Thus the molecular interaction gives rise P to the free energy density i µisol ni ,3,4 where ni (i = 1, 2, · · · ) are the ion densities. It strongly influences phase transitions in polar fluids. Using the linear dielectric constant ε Born took into account the polarization around an ion to obtain his classic formula.5 It is the space integral of the electrostatic energy ε(∇Φ)2 /8π around the ion, where Φ = Zi e/εr is the potential with Zi e being the ion charge. The dominant contribution arises from the integral at small r as i i (µisol )Born = Zi2 e2 /2εRion = kB T Zi2 `B /Rion ,

(1)

i is called the Born radius1 . In terms of the where the lower cut-off Rion 2 i Bjerrum length `B ≡ e /εkB T , we can see (µisol )Born > kB T for Rion < `B . For mixtures of two fluid components, A and B, ε changes from the dielectric constant εB of the less polar component B to that εA of the more polar component A with increasing the volume fraction φ of the more polar component A, so the Born formula indicates strong dependence of µisol on φ. However, it is well-known that the Born formula neglects electrostriction (leading to the shell formation) and nonlinear dielectric saturation (due to strong electric field in the vicinity of an ion). We consider a fluid-fluid interface between a polar phase α and a less polar phase β with bulk compositions φα and φβ , across which there arises a difference of µisol because of its composition dependence:

µiK sol

iα ∆µiαβ = µiβ sol − µsol ,

(2)

(K = α, β) are the bulk values of the solvent chemical potential where of species i in the two phases. In electrochemistry,6,7 the difference of the solvation free energies ∆Giαβ between two phases has been called the standard Gibbs transfer energy. (Since ∆Giαβ is usually measured in units of kJ per mole, dividing it by the Avogadro number gives ∆µiαβ ). It is wellknown that if there are differences among ∆µiαβ (i = 1, 2, · · · ), an electric double layer emerges at the interface, giving rise to an electric potential jump ∆Φ = Φα − Φβ , called the Galvani potential difference, across the interface in equilibrium. In particular, if there are only two species of ions (i = 1, 2) with charges Z1 e and −Z2 e (Z1 > 0 and Z2 > 0), the Galvani potential difference is expressed as4,6 e∆Φ = (∆µ2αβ − ∆µ1αβ )/(Z1 + Z2 ),

(3)

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and the ion densities in the bulk two phases are related by ln

n1α n1β

= ln

n2α n2β

=

∆µ1αβ ∆µ2αβ Z1 Z2 + , Z1 + Z 2 Z1 k B T Z2 k B T

(4)

at sufficiently low ion densities. These relations readily follow from the continuity of the (total) ionic chemical potentials across the interface and the charge neutrality conditions in the bulk two phases. Note that Φ(z) near a thin interface changes on the scale of the Debye-H¨ uckel screening −1 length, κ−1 in the α phase and κ in the β phase. As a result, Φ changes α β −1 from Φα to Φβ on the spatial scale of κ−1 + κ , which becomes very long α β as the ion densities determined by Eq. (4) become very small in one of the two phases.4 Remarkably, ∆Φ in Eq. (3) is independent of the ion densities. It is typically of order 10kB T /e(∼ 0.1Volt) for ∆µ2αβ 6= ∆µ1αβ . It vanishes for symmetric ion pairs with ∆µ2αβ = ∆µ1αβ . As Eq. (4) indicates, the ion densities in the two phases are very different in many cases. For example, if ∆µ1αβ /kB T = ∆µ2αβ /kB T = 10 in the monovalent case, the common ratio n1β /n1α = n2β /n2α becomes e−10 = 2.4 × 10−4 . In the literature, data of ∆Giαβ on water-nitrobenzene at room temperatures are available,6,7 where the two components are strongly segregated. For aqueous mixtures with α being the water-rich phase, ∆Giαβ is positive for hydrophilic ions and negative for hydrophobic ions. For waternitrobenzene,6 we have ∆µiαβ /kB T = 13.6 for Na+ , 15.3 for Li+ , 26.9 for Ca2+ , 11.3 for Br− , and 7.46 for I− as examples of hydrophilic ions, while we have ∆µiαβ /kB T = −14.4 for BPh− 4 (tetraphenylborate) as an example of hydrophobic ions. Thus, the solvent chemical potential µisol (φ) strongly depends on the composition φ, as well as the dielectric constant ε(φ). As another consequence, ∆µiαβ is an important parameter dramatically influencing the nucleation process in polar fluids with ions. That is, when a polar fluid in a less polar phase β is brought into a metastable state, the solvation shell around an ion can serve as a seed of a critical droplet of a more polar phase α.8 For water at T = 0.6Tc ∼ = 390 K, for example, i ∆µαβ /kB T can be of order 50 − 100 between gaseous water (phase β) and liquid water (phase α).9 The nucleation barrier is much reduced in the presence of ions. The surface tension γ of a water-air interface has been examined extensively in the literature. Theoretically, Wagner10 found that ions in water are repelled away from the interface by the image charges in air, leading to an increase of the surface tension ∆γ > 0. Using Wagner’s idea, Onsager

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and Samaras11 obtained the limiting law for the excess surface tension, ∆γ = kB T nw DI [ln(1/DI κ) + EI ],

(5)

where DI = `Bw /4 and EI is a dimensionless constant of order unity with the Bjerrum length `Bw = e2 /εw kB T being 7˚ A in room-temperature water. The above expression is valid for κw `Bw 1 and in the thin limit of the interface width ξ. Levin and Flores-Mena12 argued that an ion-depletion layer is formed due to the finite size of the solvation shell radius1 (even without the image force). In these theories the interface thickness ξ is assumed to be infinitesimal. In our recent theory,4 the interface is diffuse and the composition-dependent solvation interaction serves to repel hydrophilic ions away from the interface (see discussions below Eq. (18)). Experimentally, at not extreme dilution, the linear behavior ∆γ = T nw λs

(6)

has been measured,13–15 where λs is the effective thickness of the iondepletion layer. For example, λs ∼ 3˚ A for NaCl, where the densities of Na+ and Cl− are nw /2. For very dilute salts around 1 mM in aqueous solutions, Jones and Ray13 detected a small negative minimum in ∆γ, which still remains an unsolved controversy.15,16 For water-oil interfaces, γ largely decreases with addition of hydrophilic and hydrophobic ions,17 while the linear law (6) holds for hydrophilic ion pairs. On the other hand, in the presence of surfactants, γ decreases due to excess adsorption of the surfactant molecules on an interface.18 In this chapter, we explain some fundamental solvation effects in a binary fluid mixture on the basis of our recent work.3,4 Similar approaches have also been proposed by other authors.16,19,20 We will assume that the dielectric constant in the more polar phase is twice larger than that in the less polar phase as an example, so we will consider water-oil systems like water-nitrobenzene (the dielectric constant of nitrobenzene is about 36). In Sec. 2, we will present a Ginzburg-Landau approach to the molecular interactions between ions and solvent molecules. In addition to the solvation and image interactions, we will propose a free energy contribution representing the amphiphilic interaction between surfactant and solvent molecules. We may then treat ionic surfactants and counterions which have charges and aggregate at an interface reducing the surface tension. In Sec. 3, we will calculate the composition structure factor and the ion interaction mediated by the composition fluctuations near the critical point of the mixture. They can be strongly affected even by a small amount of ions. In particular, we

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will discuss a mesophase with a charge density wave, which can appear for large solvation asymmetry between cations and anions. In Sec. 4, we will derive some fundamental relations on the equilibrium ion distributions and the surface tension. We will study the case of three ion species. In Sec. 5, we will give some numerical results on the ion distributions and the surface tension in various cases. 2. Ginzburg-Landau Theory 2.1.

Solvation interaction

Let us consider a polar binary mixture (water and oil) containing a small amount of salt. The volume fraction of the more polar component is written as φ. The other less polar component has the volume fraction 1 − φ. We neglect the volume fractions of the ions in this chapter. The ion densities are written as n1 , n2 , · · · . In our theory, we mostly suppose two ion species with charges Q1 = Z1 e and Q2 = −Z2 e, but we will also treat more complex cases in the presence of three ion species. In our scheme, φ, n1 , n2 , · · · are smooth space-dependent variables coarse-grained on the microscopic level. We present a Ginzburg-Landau scheme, where the interface thickness ξ is supposed to be longer than the molecular size a. The usual form of the free energy for a fluid mixture containing a small amount of ions is written as Z X ε(φ) 2 C 3 2 ni ln(ni a ) . (7) E + kB T F0 = dr f0 (φ, T ) + |∇φ| + 2 8π i As discussed in the first section, we introduce the solvation free energy,3,4 Z X Fsol = dr ni µisol (φ). (8) i

The free energy density f0 is dependent on φ and T . For low molecularweight mixtures we may use the Bragg-Williams form, f0 = a−3 kB T [φ ln φ + (1 − φ) ln(1 − φ) + χφ(1 − φ)].

(9)

Our theory can be used also for polymer solutions and blends if we use the Flory-Huggins form for f0 .18,21 We assume that the molecules of the two fluid components (the monomers for polymers) have a common volume a3 . The χ is the interaction parameter dependent on T , whose critical value is 2 for the free energy density in Eq. (9). The second term in the brackets of Eq. (7) is the gradient part, where we will set C = kB T χ/a in our numerical

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analysis.18 The third term is the electrostatic free energy with E = −∇Φ being the electric field. The electrostatic potential Φ satisfies −∇ · ε(φ)∇Φ = 4πρ,

(10)

where ε is the composition-dependent dielectric constant and ρ is the charge density. When there are only two species of ions, we have ρ = Z1 en1 − Z2 en2 . The last term in F0 in Eq. (7) is the entropic contribution valid at low ion densities. We do not know the functional forms of ε(φ) and µi (φ). We assume the linear composition dependence, ε(φ) = εc + ε1 ψ, µisol (φ) = µic − kB T gi ψ

(i = 1, 2).

(11) (12)

where ψ = φ − φc is the deviation from the critical composition φc . Then εc and µic are the critical values. We adopt these linear forms to obtain the physical consequences in the simplest manner. Empirically, ε can be approximated as a linear function of the composition in many mixtures investigated so far.2,22 The first term µic on the right hand side of Eq. (12) is an irrelevant constant when the ion numbers are conserved quantities without dissociation processes. It then follows the relation, ∆µiαβ = kB T gi ∆φ,

(13)

where ∆φ = φα − φβ is the composition difference. In aqueous mixtures, the coupling constants gi are positive for hydrophilic ions and negative for hydrophobic ions. The experimental values of the Gibbs transfer energy for water-nitrobenzene 6,7 suggest that the values of gi are typically of order 15 for monovalent hydrophilic ions and are even larger for multivalent ions such as Ca2+ or Al3+ . For the hydrophobic anion BPh− 4 , it is about −15, on the other hand. The resultant solvation coupling between the ions and the composition is very strong. The Born formula in Eq. (1) gives i near the critical point, but we do not use (gi )Born = Zi2 e2 ε1 /2kB T ε2c Rion it because of the defect of the Born theory. We rather treat gi as free parameters characterizing the strength of the solvation interaction. 2.2. Image interaction Inhomogeneous dielectric constant ε gives rise to an image chemical potential Zi2 µim acting on each ion (which is proportional to the square of its charge).4 We then construct the free energy contribution Fim =

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P dr i µiim ni . Note that Fim originates from the discrete nature of ions, while the electric field Φ determined by Eq. (10) is produced by the smoothly coarse-grained charge density ρ. Therefore, R the origin of Fim is different from that of the electrostatic free energy drεE 2 /8π in Eq. (7). In our previous work,4 assuming weak or moderate spatial variations of ε, we derived the following integral form valid to first order in ε1 , Z µim (r) = dr 0 ∇0 I(r, r0 ) · ∇0 φ(r 0 ), (14) R

where ∇0 = ∂/∂r0 . If the boundary effects are neglected, the function I(r, r0 ) depends only on the difference r − r 0 as I(r, r 0 ) = B0 e−2κ|r−r | /|r − r 0 |2 , 0

(15)

where B0 = e2 ε1 /8πε2c near the critical point and κ is the Debye-H¨ uckel 0 wave number. The factor e−2κ|r−r | arises from the screening of the image potential by the other ions, so the image interaction is weakened with increasing the ion density. In particular, around a planar interface, where all the quantities vary along the z direction, it follows the following Cauchy integral form, Z 0 ε1 dz 0 e−2κ|z−z | dφ(z 0 ) µim (z) = kB T Aa . (16) εc π z − z0 dz 0 We define the coefficient A by A = πe2 /4aεc kB T = π`Bc /4a

(17)

where `Bc = e2 /εc kB T is the Bjerrum length at ε = εc . Around an interface, where κ varies significantly, κ in Eq. (16) may be taken as the space-dependent local value [4πe2 m(r)/ε(r)kB T ]1/2 or the bulk value in the more polar phase. In our previous calculations of the surface tension4 there was no significant difference in these two choices. Let us consider the thin interface limit ξ → 0, where we set dφ(z)/dz = −∆φδ(z) with ∆φ = φα − φβ . Placing the interface at z = 0, we obtain µim (z) ∝ e−2κ|z| /z for |z| ξ in agreement with the original expression for the image potential.11,12 For finite ξ, µim changes from positive to negative in the interface region |z| 1 at |z| ∼ ξ where ξ is the interface thickness. From Eq. (16) this condition is rewritten as ξ < Zi2 `Bc ε1 ∆φ/εc < κ−1 α .

(18)

If the two ions are both strongly hydrophilic (or if g1 and g2 take large positive values), we may reproduce the formula (5) for ∆γ with DI = aAε1 ∆φ/πεc .4 For ∆φ ∼ 1 and ε1 /εc ∼ 1, Eq. (18) becomes ξ/a < Zi2 A < (aκα )−1 , where A is defined by Eq. (17). For finite interface thickness ξ, hydrophilic ions are repelled from an interface also due to the solvation interaction (due to the factor egi φ in Eqs. (44) and (45) below for gi > 0). Thus, even in the absence of the image interaction, a depletion layer of hydrophilic ions can be formed and the linear behavior ∆γ ∝ nα still follows. To draw qualitative results, the image interaction may be neglected for not very large A. See the discussions around Eqs. (5) and (6). 2.3. Amphiphilic interaction In our previous work,4 we calculated the ion distributions around an interface for simple structureless ions, including the solvation and image interactions. However, in many important applications, ions have an amphiphilic character and are preferentially adsorbed onto an interface. Here, extending our previous theory, we calculate the ion distributions when the first ion species is a cationic surfactant. The second species constitutes anionic counterions having no amphiphilic character. We suppose a rod-like shape of the molecules of the first species. The rod length 2` is considerably longer than a. We then introduce an amphiphilic free energy, Z Fam = −kB T drn1 ln Zam , (19) where Zam is the partition function of a rod-like dipole with its center at the position r. It is given by the following integral on the surface of a sphere with radius `, Z 1 dΩ exp − wa φ(r + ù) + wa φ(r − ù) , (20) Zam (r) = 4π R where the rod direction is along the unit vector u and dΩ represents the integration over the angles of u. The two ends of the rod are at r + ù and r − ù under the influence of the solvation potentials given by

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kB T wa φ(r + ù) and −kB T wa φ(r − ù), respectively. Our amphiphilic interaction is characterized by ` and wa . When the rod length 2` is longer than the interface thickness ξ, a surfactant molecule can be trapped at an interface with its hydrophilic (hydrophobic) end in the water-rich (waterpoor) region. The resultant chemical potential decrease is given by a = kB T wa ∆φ.

(21)

It is instructive to examine the case in which φ(r) varies slowly. If the expansion φ(r + ù) − φ(r − ù) = 2ù · ∇φ + · · · is allowable, we obtain

2 2 2 w ` |∇φ|2 + · · · . (22) 3 a The free energy Fam then provides a gradient contribution, so it may be combined with the original one (∝ C) in the total free energy. The new gradient term is of the form Ceff |∇φ|2 /2 with23 ln Zam =

4 Ceff = C − kB T wa2 `2 n1 = C(1 − n1 /n1L ). 3 We notice the presence of a Lifshitz point at n1 = n1L with n1L =

3C . 4(`wa )2 kB T

(23)

(24)

For n1 > n1L , a homogeneous electrolyte solution should be unstable at a finite wave number (see Eq. (30)), leading to a mesophase. If C ∼ kB T /a, we have a3 n1L ∼ (a/`wa )2 , so n1L can be very small for large wa 1. The above gradient expansion is invalid around an interface far from the critical point or for ξ < ∼2`. We should examine the behavior of Zam around a strongly segregated interface. In the one-dimensional (1D) case, where all the quantities vary along the z axis, Zam in Eq. (20) is simplified into the following 1D integral, Z 1 ` Zam (z) = dζ exp − wa φ(z + ζ) + wa φ(z − ζ) , (25) 2` −` where we have replaced φ(r ± ù) by φ(z ± ζ) with ζ = ùz . This form indicates that the adsorption should be significant for wa ∆φ = a /kB T 1. We assume this condition in the thin interface limit ξ ` for simplicity. The integrand in Eq. (25) can be large (= ewa ∆φ ) only if the interface is located between z + ζ and z − ζ. With the interface being placed at z = 0, we find Zam = 1 + (1 − |z|/`)[cosh(wa ∆φ) − 1]

(26)

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for |z| < `, while Zam ∼ = 1 for |z| > `. In the thin interface limit, we estimate the surfactant adsorption at the interface per unit area as Γ1 = (n1α + n1β )`[cosh(wa ∆φ) − 1]/2,

(27)

where the image interaction is neglected. The n1α and n1α are the bulk surfactant densities far from the interface. As a result, the surface tension γ decreases by 2kB T Γ1 from the Gibbs formula (see Eq. (64)), where the factor 2 accounts for the counterions. For wa ∆φ 1, the decrease of γ is considerable for n1α > e−wa ∆φ /a2 ` where we assume γ ∼ kB T /a2 ∼ without surfactant. It is worth noting that the proportionality relation Γ1 ∝ n1α ea /kB T for dilute surfactants was first found by Traube in his experiment.24 3. Structure Factor of Composition and Interactions Among Ions in One-Phase States In our previous papers,3,4 we examined the structure factor S(q) = h|φq |2 i of the composition fluctuations with wave number q in one-phase states. Here we consider a binary mixture near the critical point with a small amount of two ion species. As a generalization, we use the Flory-Huggins free energy density for f0 in Eq. (7) and include the amphiphilic interaction. We shall see that the ion-solvent interaction can strongly affect the composition fluctuations and the interaction among ions. We may neglect the image interaction near the critical point. 3.1. Composition fluctuations and mesoscopic phase We consider small plane-wave fluctuations with wave vector q in a homogeneous one-phase state near the critical point, where the inhomogeneity in the dielectric constant may be neglected (ε = εc ). The fluctuation contributions to F in the bilinear order are written as X |niq |2 X1 2π 2 ∗ (¯ r + Ceff q 2 )|φq |2 + |ρ | + k T − g n φ , δF = q B i iq q εc q 2 2ni q 2 i=1,2 (28) where φq , niq , and ρq are the Fourier components of φ(r), ni (r), and ρ(r) = e[Z1 n1 (r) − Z2 n2 (r)], respectively. The image interaction is neglected. The ni in the last term of Eq. (28) are the average ion densities satisfying Z1 n1 = Z2 n2 from the overall charge neutrality. We use the Flory-Huggins

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free energy (only in this subsection) to calculate the coefficient r¯ as ∂ 2 f0 kB T 1 1 r¯ = = + − 2χ ∂φ2 a3 N1 φ N2 (1 − φ)

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(29)

where N1 and N2 are the polymerization indices of the two components. The previous results3,4 can be obtained for N1 = N2 = 1. If the first ion species is a cationic surfactant, the coefficient of the gradient term becomes Ceff in Eq. (23), which is assumed to be positive here. We minimize δF with P respect to niq to obtain the free energy change δF = q kB T |φq |2 /S(q), where S(q) is the composition structure factor. Some calculations give γp2 q 2 kB T 2 2 , (30) = r¯ − ∆rion + Ceff q 1 − γp + 2 S(q) κ + q2 where κ = [4π(Z12 n1 +Z22 n2 )e2 /εc kB T ]1/2 is the Debye-H¨ uckel wave number and ∆rion is a constant shift of r¯. From Eq. (29) we may define the ioninduced shift of χ by χion = ∆rion a3 /2kB T c = (Z2 g1 + Z1 g2 )2 a3 n/2(Z1 + Z2 )2 .

(31)

where n = n1 +n2 . In our problem there appears a dimensionless parameter γp representing the ion asymmetry, γp = (kB T /4πCeff `Bc )1/2 |g1 − g2 |/(Z1 + Z2 ) p = |g1 − g2 |/[4(Z1 + Z2 ) χA(1 − n1 /niL )],

(32)

where `Bc = e2 /εc kB T is the Bjerrum length at ε = εc . The second line of Eq. (32) follows for the present choice C = kB T χ/a2, where A and n1L are defined by Eq. (17) and Eq. (24), respectively. Note that γp is independent of the ion density and increases with increasing the amphiphilic strength wa from Eq. (23). If γp < 1, S(q) is maximum at q = 0 and we predict the usual phase transition in the mean field theory. In terms of the shift in Eq. (31) the spinodal χ = χsp (φ, n) is given by χsp =

1 1 + − χion c . 2N1 φ 2N2 (1 − φ)

(33)

The critical value of χ at given n, denoted by χc (n), is the minimum of χsp 1/2 1/2 1/2 with respect to φ and is equal to χsp at φ = N2 /(N1 + N2 ). Thus, χc =

1 1/2 1/2 (N + N2 )2 − χion c . 2N1 N2 1

(34)

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For N1 = N2 = 1 we have χc = 2−χion c . See Eqs. (70) and (71) and the right panel of Fig. 3 for the critical behavior below the transition (χ > χc ). If we set r¯ = a0 (T − Tc0 ) at the critical composition, the critical temperature shift becomes ∆Tion = ∆rion /a0 , where a0 is a constant and Tc0 is the critical temperature in the absence of ions. For the monovalent case we find ∆Tion ∼ kB Tc (g1 + g2 )2 n/4a0 , where the factor (g1 + g2 )2 can be very large (∼ 100). This result is consistent with previous experiments of binary mixtures with salt,26–28 where the coexistence curve has been observed to shift greatly with doping of small amounts of hydrophilic ions. If γp > 1, the structure factor S(q) attains a maximum at an intermediate wave number qm given by qm = (γp − 1)1/2 κ.

(35)

The maximum of S(q) is written as S(qm ) = kB T /(¯ r − rm ) with rm = ∆rion + Ceff (γp − 1)2 κ2 ,

(36)

where we assume r¯ > rm . For r¯ < rm , a charge-density-wave phase should be realized. As long as γp > 1, this mesoscopic phase appears even for very small ion densities. For electrolytes, a mesoscopic phase was first predicted by Nabutovskii et al.,29 who assumed the bilinear coupling ρφ between the charge density ρ and the composition φ in the free energy. Recently, in their small-angle neutron scattering experiment, Sadakane et al.30 found periodic structures in a binary mixture of D2 O-trimethylpyridine containing sodium tetrarphenylborate (NaBPh4 ), where the scattered neutron intensity exhibited a peak at q ∼ 0.1/˚ A. Since this salt is composed of strongly hydrophilic and hydrophobic ions, we expect that the condition γp > 1 should have been satisfied. See the left panel of Fig. 5 for the ion distributions in such a case. In polyelectrolytes, electric charges are attached to polymers and the structure factor of the polymer takes a form similar to that in Eq. (30), leading to a mesophase at low temperatures, in the Debye-H¨ uckel approximation.31,32 In our theory, even a polymer solution consisting of neutral polymers and a polar solvent can exhibit a charge-density wave phase in the presence of a small amount of salt. This would explain a finding of a peak at an intermediate wave number in the scattering amplitude in polyethylene-oxide solutions with salt.33

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3.2. Effective interaction among charged particles In a homogeneous phase with γp < 1, we may eliminate the composition fluctuations in F assuming their Gaussian distributions and setP ting (¯r + Ceff q 2 )φq = kB T i gi niq . We then obtain attractive interactions among the ions mediated by the composition fluctuations. The resultant free energy of ions is written as Z Z Z X X 1 Vij (|r − r0 |)δni (r)δnj (r0 ), dr dr 0 kB T ni ln(ni a3 ) + Fion = dr 2 i,j i (37) where the deviations δni = ni − hni i need not be small. The effective interaction potentials Vij (r) are expressed as Vij (r) = Qi Qj

1 (kB T )2 e−r/ξ − gi gj , εc r 4πCeff r

(38)

where Q1 = Z1 e and Q2 = −Z2 e are the ion charges and ξ = (Ceff /¯ r)1/2 is the correlation length. Among the ions of the same species (i = j), the second attractive term dominates over the first Coulomb repulsive term in the range a ∼ 0 and g3 < 0 and choose c2α and c3β as control parameters. If we set X = exp[∆U − (ν1 − ν2 )/2],

(53)

Eq. (40) becomes a cubic equation, X − X −1 = 2R[1 − X 2 eν3 −ν2 ].

(54)

R = e(ν1 +ν2 )/2 c3β /2c2α .

(55)

where we define ∼ 1 for large |g3 | 1 to obtain X = We may √ well assume X e 2 R + 1 + R or p 1 e ∆Φ ∼ (56) = (ν1 − ν2 ) + ln(R + 1 + R2 ), T 2 If ν1 + ν2 1, we readily reach the regime R 1 even at small c3β , where X∼ = 2R and X 2 eν3 −ν2 ∼ = (c3β /c2α )2 eν1 +ν3 . That is, we find 2 ν3 −ν2

∆U ∼ = ν1 + ln(c3β /c2α ),

(57)

for e−(ν1 +ν2 )/2 c3β /c2α e−(ν1 +ν3 )/2 . In this case, c1α ∼ = c2α , c1β ∼ = c3β , −(ν1 +ν2 ) 2 ν1 +ν3 2 ∼ ∼ c3β /c2α c2α . c2β = e c2α /c3β c3β , and c3α = e 4.3. Surface tension As stated in Sec. 1, the surface tension γ of a water-air interface has been observed to increase with increasing the amount of small hydrophilic ions in water. However, for a pair of hydrophilic and hydrophobic ions, it can decrease with increasing the salt density even without amphiphilic interaction (see the right panel of Fig. 5).4 It is well-known that the surfactant molecules accumulate at the interface and γ decreases with increasing the surfactant density.18 We here examine the behavior of γ for the case γp < 1, where γp is the asymmetry parameter defined by Eq. (32). To calculate γ we introduce the grand potential by Z Z X Ω = drω = F − dr(hφ + µi ni ). (58) i

For given constant h, µ1 , µ2 , · · · , Ω is minimized as a functional of φ, n1 , n2 , · · · in equilibrium, yielding Eqs. (41), (42), and (46). In our system the grand potential density ω is expressed as ω = f0 (φ) +

ε 2 C |∇φ|2 − hφ − kB T n − ρΦ + E , 2 8π

(59)

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P where n = i ni . Use has been made of Eqs. (41) and (42) to eliminate µi . We suppose a planar interface perpendicular to the z axis; then, ω(z) should tend to a common constant ω∞ as z → ±∞. To show this, we calculate the space derivative, X dµam d dµim + n1 − (him + ham )ψ 0 , (60) ω − C(ψ 0 )2 + ρΦ = ni Zi2 dz dz dz i where ψ 0 = dψ/dz and µam = −kB T ln Zam (see Eq. (25)). The him and ham are defined by Eqs. (46) and (47). Note that Fim and Fam are invariant with respect to a small displacement δζ of the interface position or with respect to the change of ψ and ni to ψ − ψ 0 δζ and ni − n0i δζ, respectively, where n0i = dni /dz. Then we find Z Z X i 0 0 µim ni ] = 0, dz[ham ψ 0 + µam n01 ] = 0, (61) dz[himψ + i

where we have pushed the lower and upper bounds of the integrals to ∓∞. Owing to these relations, the z integration of the right hand side of Eq. (60) vanishes, leading to ω(z) R → ω∞ as z → ±∞. It is convenient to express the surface tension γ = dz[ω(z) − ω∞ ] as Z Z ε(φ) 2 C E , (62) γ = dz f0 (φ) + |∇φ|2 − kB T n − hφ − Cα − dz 2 8π where Cα = f0 (φα ) − kB T nα − hφα and h = [f0 (φα ) − f0 (φβ ) − kB T (nα − nβ )]/∆φ

(63)

with nα and nβ being the bulk values of n and the integrand of the first term vanishes as z → ±∞. RFrom Eq. (10) the electric field is expressed as −1 z 0 0 0 E(z) −∞ dz ρ(z ). There should be no net charge or R ∞ = −Φ (z) = 4πε(z) −∞ dzρ(z) = 0 around the interface if E(z) → 0 far from the interface. Away from the ion-induced critical point, we are interested in the excess surface tension ∆γ = γ − γ0 , where γ0 is the surface tension without ions. Let φ(z) → φ0 (z) and h → h0 as n → 0. We expand ω(z) in Eq. (59) with respect to the deviation δφ = φ − φ0 . Neglecting the terms of order (δφ)2 , we obtain h − h0 ∼ = −kB T (nα − nβ )/∆φ and4 Z ε(φ) 2 E , (64) ∆γ ∼ = −kB T Γ − dz 8π where Γ is the adsorbed ion density defined by Z ∞ Z zint dz[n(z) − nβ ]. dz[n(z) − nα ] + Γ= −∞

zint

(65)

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The interface position zint is determined by the Gibbs construction,25 Z L φα zint + φβ (L − zint ) = dzφ(z), (66) 0

in a finite system in the region 0 < z < L with L ξ. In Eq. (64) the lower and upper bounds far from the interface are pushed to infinity. If the last term in Eq. (64) is neglected, we obtain the Gibbs equation ∆γ = −kB T Γ at relatively low adsorption.25 The adsorbed density Γ can be much enhanced for surfactants, while it is negative for low-density hydrophilic ions at an water-air interface. The last negative term in Eq. (62) or Eq. (64) arises from the last two terms in Eq. (59), which is written as γe . In all the examples in our previous work,4 it was at most a few percents of ∆γ. To roughly estimate it, let us employ the Poisson-Boltzmann equation dU (z)2 /dz 2 = κ2β sinh(U (z) − Uβ ) in the region z > zint and dU (z)2 /dz 2 = κ2α sinh(U (z) − Uα ) in the region z < zint , where κα and κβ are the Debye-H¨ uckel wavenumber. Here we take the thin interface limit ξ → 0 and neglect the image potential. We impose the continuity of U and εdU/dz at the interface. Then γe is approximated by the Poisson-Boltzmann result, nα p 1 + b2 + 2b cosh(∆U/2) − 1 − b , (67) γePB = −2kB T κα where b = εβ κβ /εα κα = (εβ nβ /εα nα )1/2 , with εα and εβ being the dielectric constants in the bulk phases, and ∆U = Uα −Uβ = (g1 −g2 )∆φ/2 is the normalized potential difference. Here γePB = 0 for g1 = g2 , so we assume (g1 −g2 )∆φ > 1. Typical behaviors of γePB in the monovalent case are as follows. (i) If g1 and g2 are both considerably larger than unity with g1 > g2 , we have b ∼ e−(g1 +g2 )∆φ/4 1 and be∆U/2 = e−g2 ∆φ/2 1, so that a2 |γePB |/kB T ∼ aκα e−g2 ∆φ/2 /A 1, where A is defined by Eq. (17). This result is consistent with our previous results.4 (ii) In the case g1 = −g2 1, we have ∆U = g1 ∆φ and b ∼ 1. Then a2 |γePB |/kB T ∼ aκα eg1 ∆φ/4 /A grows with increasing g1 . This case will be numerically examined in the right panel of Fig. 5. 1/2 Note the relation γePB ∝ nα , leading to the square root dependence of ∆γ at low ion densities. In Fig. 5, we shall see this dependence for hydrophilic and hydrophobic ion pairs. On the other hand, for hydrophilic pairs, we propose the following form, ∆γ/T ∼ = −As (nα /`Bα )1/2 + λs nα ,

(68)

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where `Bα is the Bjerrum length in the α phase, The coefficient As is small, while the second term is of the well-known form accounting for the ion depletion. With this form, ∆γ should exhibit a small minimum given by (∆γ)min = −T λs nm

(69)

at n = nm = (As /2λs )2 /`Bα . As an example, let the ion concentration giving this minimum be 1 mM in the water-rich phase. Then we obtain As = 1.2 × 10−2 by setting λs = 3˚ A and `Bα = 7˚ A. For water-air interfaces, Jones and Ray13 found a negative minimum in ∆γ of order −10−4 γ0 . We notice that their data can well be fitted to Eq. (68) with As ∼ 10−2 . However, we have assumed appreciable ion densities even in the less polar β region. That is, our one-dimensional calculations are justified only when the screening length κ−1 β in the β region is much shorter than any characteristic lengths in experiments, which are the inverse curvature of the meniscus or the wavelength of capillary waves, for example. In the literature10–12,16 ions are treated to be nonexistent in the air region, so we do not still understand the Jones-Ray effect. 5. Numerical Results of Ion Distributions In the one-dimensional geometry, we display equilibrium profiles of the composition and the ion densities and calculate the surface tension for various parameter values. In the monovalent case, we set A = 4 (except in the left panel of Fig. 4) and 1 /c = 0.8. The dielectric constant of the α phase is twice larger than that of the β phase at χ = 3, so the inhomogeneity of the dielectric constant is rather mild. The condition γp < 1 is satisfied in all the examples (see the discussion around Eq. (32)). In the following figures, we give profiles along the z axis, where the α phase is on the left and the β phase is on the right. 5.1. Including solvation and image interactions We first assume no amphiphilic interaction. In Fig. 1, we show the composition φ(z) and the normalized electric potential U (z) = eΦ(z)/kB T (taken to be zero in the α phase) near an interface for three values of χ with g1 = 4 and g2 = 2, where we fix the product c1α c1β = 10−4 . As χ is decreased with γp < 1, we approach the critical point dependent on the salt density, as discussed in Sec. 3. The critical value of χ is given by Eq. (34). The electric potential jump is given by Eq. (48). In Fig. 2, we display the ion density

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0.7

a 0.6

0

b c

U

-0.1

c

0.5

φ (z)

-0.2

a = 1.95 b = 1.92 c

0.4

0.3

0

b

χ=2.0

χ=2.0

a = 1.95 b = 1.92 c

-0.3

20

40

60

80

-0.4 0

100

20

a 40

60

z/a

80

100

z/a

Fig. 1. Composition φ(z) (left) and normalized electric potential U (z) (right) on approaching the criticality as χ = 2, 1.95, and 1.92 with c1α c1β = 10−4 .4 The ions are both hydrophilic with g1 = 4 and g2 = 2. The critical value of χ is 1.91 from Eq. (34). From A. Onuki, Phys. Rev. E 73 (2006) 021506.

0.02

1 C

a 0.015

4

1 0.5

10 ( c 1 - c2 )

a

b b 0.01

0.005

0

0

c

0

χ=2.0

a = 1.95 b = 1.92 c

20

-0.5

40

60

z/a

80

100

-1 0

c

χ=2.0

a

= 1.95 b = 1.92 c

20

40

60

80

100

z/a

Fig. 2. Normalized ion density c1 (z) (left) and normalized charge density c1 (z) − c2 (z) (multiplied by 104 ) (right) with varying χ with the same parameter values as in Fig. 1.4 From A. Onuki, Phys. Rev. E 73 (2006) 021506.

c1 (z) and the normalized charge density c1 (z) − c2 (z). The ion densities are reduced in the β phase as in Eq. (49). We can see that an electric double layer at the interface diminishes as the critical point is approached. It is convenient to choose n ¯ ≡ (nα nβ )1/2 (= 2(c1α c1β )1/2 /a3 in the monovalent case) in two phase states as the parameter representing the degree of ion doping.4 Over a rather wide parameter region, the bulk concentrations in the two phases are expressed as 1 1 1 (φα + φβ ) − ∼ [(g1 Z2 + g2 Z1 )/(Z1 + Z2 )]3 a3 n ¯, = 2 2 32

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0.08 0.07

a

b

= 2.3 b = 2.05 c

0.04

a

0.05

0.03

g1= 4 g2= 2

0.02 0.01 0

0.01

0.02

0.03

0.04

a

< c1>= 0.001 g =10 g2= 5

b

1

0.03

0.04

< c1>= 0.001 g = 4 g2= 2 1

c

a 2 γ / kBT

a 2 ∆γ / kBT

0.06

χ=3.0

< c > = 0.01 1

g1= 4 g2= 2 c

0.02

c b

0.01

a 0.05

0

1.92 1.94 1.96 1.98

2

2.02 2.04 2.06 2.08

2.1

χ

c1α

Fig. 3. Left: Normalized excess surface tension a2 ∆γ/kB T versus c1α , obeying Eq. (6).4 Right: Normalized surface tension a2 γ/kB T as a function of χ. It tends to zero as χ → χc . The two ion species are both hydrophilic in these cases.

∆φ = φα − φβ ∼ = [3(χ − χc )/2]1/2 ,

(71)

where use has been made of the Landau expansion of the free energy density f0 in Eq. (9). The first line is the shift of the critical composition. The second line is the usual mean field expression for the average order parameter difference. These relations are consistent with Figs. 1 and 2. The surface tension γ can be calculated numerically from Eq. (62) using Eq. (63). In Fig. 3, we show the excess surface tension ∆γ = γ − γ0 versus c1α when the two ion species are both hydrophilic, in accord with the experiments.13–15 Without ions, we find a2 γ0 /kB T = 0.498, 0.103, and 0.0773 for χ = 3, 2.3, and 2.05, respectively. Though these γ0 values are very different, ∆γ increases roughly linearly with increasing the ion density. The right panel of Fig. 3 displays the surface tension γ itself for three cases, where the interface is located at the middle of the cell and the space average hc1 i is fixed. For each curve the critical value χc is given by Eq. (34) and γ tends to zero as χ → χc . We illustrate how the two terms in Eq. (65) can be interpreted graphically, where the two ion species are both hydrophilic and the electrostatic term, the third term in Eq. (64), is negligible. We also demonstrate relevance of the image interaction at small ion densities. In Fig. 4, we display the normalized ion density n(z)/nα , the composition φ(z), and the image factor defined by Ima(z) = exp[−µim (z)/kB T ],

(72)

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1.2 1.2 1

Ima ( z )

1

Ima( z )

0.8 0.6 0.4 0.2 0 38

n(z)/nα

0.8 -5

-3

c 1 α= 2.1x10

0.6

χ= 3 g1= 10

φ(z)

g2= 5

n(z)/nα

A =10 40

42

44

46

z/a

48

c 1 α= 10

0.4

χ= 2.3 g1= 4

0.2

g2= 2

φ(z)

A =4 50

52

54

0 40

42

44

46

48

50

52

54

56

z/a

Fig. 4. Normalized ion density n(z)/nα , composition φ(z), and image factor Ima(z) for χ = 3, g1 = 10, g2 = 5, and c1α = 2.1 × 10−5 (left) and for χ = 2.3, g1 = 4, g2 = 2, and c1α = 10−3 (right).4 The gray regions correspond to the two terms in Γ in Eq. (65). From the curves of Ima(z), the image interaction serves to repel the ions from the interface on the left, while it is not important on the right. From A. Onuki, Phys. Rev. E 73 (2006) 021506.

in the monovalent case. This factor appears in c1 and c2 in Eqs. (44) and (45). In Fig. 4, the areas of the left and right gray regions multiplied by nα a are equal to the first term and the minus of the second term in Eq. (65), respectively. The ion density n(z) is shifted to the left of the interface at z = zint (∼ = 50a). This means that the ions are repelled from the interface in the α phase. We furthermore mention detailed characteristic features. (i) In the left panel, we set χ = 3, c1α = 2.1 × 10−5 , g1 = 10, g2 = 5, and A = 10, where the ion density is very small and 1/2κα = 6.7a is rather long. The first term of Eq. (65) is 103.5% of the total a2 ∆γ/kB T = 7.32c1α . The formula (6) can be used in this example. See the discussion around Eq. (18). (ii) In the right panel, we set χ = 2.3, c1α = 10−3 , g1 = 4, g2 = 2, and A = 4, where the ion density is relatively large and the ion reduction factor in Eq. (49) (∼ e−1.8 ) is not very small. The first term in Eq. (65) is then 158% of the total ∆γ/kB T = 2.03a−2c1α . In this case Ima(z) ∼ = 1 at any z, so the image interaction is not important. The ion distributions for a pair of strongly hydrophilic and hydrophobic ions are very singular. In the left panel of Fig. 5, we show √ c1 (z) and c2 (z) for g1 = −g2 = 10 at χ = 3, where Eq. (32) gives γp = 5/4 3 < 1. Notice that c1α and c1β coincide from Eq. (49) and is set equal to 2 × 10−4 . Here ∆γ = −0.041T a−2 and Γ = 0.014a−2. We can see a marked growth of the electric double layer and a deep minimum in the ion density c1 + c2 at the interface position. In our previous work,4 we obtained milder ion profiles for g1 = −g2 = 4. In the right panel of Fig. 5, we examine how ∆γ = γ − γ0 and ∆γ1 = γ1 − γ0 are decreased with increasing c1α for g1 = −g2 = 10,

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0

0.003

χ= 3 g1= 10 g2= -10

0.002

2

-0.02

c2

A=4

c1α=

∆γ1a / T

-0.01

2 x10

-Γ a

-0.03

-4

χ= 3

-0.04

g 1= 10 g2= -10

-0.05 0.001

c1

-0.06 -0.07

2

A=4

∆γ a / T 2

-0.08 0

30

40

50

z/a

60

70

0

0.0004

0.0008

0.0012

0.0016

0.002

c1α

Fig. 5. Left: Normalized ion densities, where χ = 3, A = 4, g1 = −g2 = 10, and c1α = c1β = 2 × 10−4 . For this hydrophilic and hydrophobic ion pair, a microphase separation forming a large electric double layer is apparent. Right: a2 ∆γ/T and a2 ∆γ1 /T as functions of c1α , where γ1 is the first term in Eq. (62) and ∆γ1 R= γ1 − γ0 is very close to −Γa2 . This shows that the electrostatic part γe = γ − γ1 = − dzεE 2 /8π dominates over ∆γ1 ∼ = −kB T Γ.

where γ1 is the first term on the right hand side of Eq. (62). We notice the 1/2 following. (i) The changes ∆γ and ∆γ1 are both proportional to c1α at 1/2 small c1α . Here |∆γ|/c1α is of order unity, so ∆γ R is appreciable even for very small c1α . (ii) The electrostatic part γe = − dzεE 2 /8π is known to be important in this case from comparison between ∆γ1 and ∆γ = ∆γ1 + γe . (iii) We confirm that the modified Gibbs relation ∆γ1 ∼ = −kB T Γ holds excellently. In Fig. 6, we display the ion distributions in the presence of three ion species in the monovalent case with Q1 = e, Q2 = −e, and Q3 = −e. Since the absolute values of gi are taken to be large, we can see steep and complex variations of the ion distributions around the interface. In the left panel, the first and second species are both hydrophilic but the third one is hydrophobic as g1 = g2 = 10 and g3 = −13, where χ = 3, e(Φα −Φβ )/T = 7.92, and γ = 0.446T /a2. This is the case discussed around Eqs. (56) and (57), since X 2 e(g3 −g2 )∆φ = 1.5 × 10−2 and R = 2.7 × 103 . In the right panel, the first and third species are hydrophobic but the second species is hydrophilic as g1 = −10, g2 = 12, and g3 = −15, where χ = 3.2, e(Φα − Φβ )/T = 7.01, and γ = 0.600T /a2 (with γ0 = 0.620T /a2). The ion distributions in this latter case can be compared with those in the experiment by Luo et al.,36 so the curves are written on a semilogarithmic scale as in their paper. The adopted parameter values are inferred from their experimental data.

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10

0.003

-2

χ= 3 g1= 10 g2= 10 g3= -13 A=4

c1 0.002

c2α=

2 x10

c3

-3

10

c3β= 0.001

10

10

35

40

45

50

55

60

65

z/a

70

c2

-4

c1 χ= 3.2

-5

c3β=

g1= - 10 g2= 12 g3= - 15 A=4

c2

0

4 x10

-4

-3

10

30

c2α=

-3

10

-6

30

35

-4 0.6 x10

c3

40

45

50

55

60

65

70

z/a

Fig. 6. Left: Normalized ion densities c1 (z), c2 (z), and c3 (z) in the presence of three ion species with g1 = g2 = 10 and g3 = −13. The second species does not penetrate into the β region. Right: Those for g1 = −10, g2 = 12, and g3 = −13 on a semi-logarithmic scale, resembling to those in the experiment.36 The third species does not penetrate into the α region.

5.2. Including amphiphilic interaction in addition to solvation and image interactions We give numerical results including the amphiphilic interaction in addition to the solvation and image interactions. We set 2` = 5a and change wa . In Fig. 7, we show c1 , c2 , and U with wa = 12 for two ion densities, c1α = 10−3 and 2 × 10−3 . The other parameter values are χ = 3, g1 = 4, and g2 = 8, so the counterions are more strongly repelled from the interface into the α phase. We can see marked adsorption of the ions at the interface. For niα ≥ niβ we define the areal densities of adsorbed ions by Z > Γi = dz[ni (z) − niα ] (i = 1, 2), (73) ni >Ath n1α

where the integration is in the region with ni (z) > Ath n1α . We set Ath = 2 > −3 1.05 here. In Fig. 7, (a2 Γ> 1 , a Γ2 ) is given by (0.034, 0.029) for c1α = 10 −3 and by (0.091, 0.085) for c1α = 2 × 10 . We have γ = 0.497, 0.426, and 0.336 for c1α = 0, 10−3 , and 2 × 10−3, respectively, in units of kB T /a2 . The distribution of the counterions is wider than that of the ionic surfactant. The normalized potential U (z) has a peak at z = zp near the interface and slowly relaxes in the β phase on the scale of the screening length κ−1 β . In Fig. 8, we examine the case of hydrophilic cations and hydrophobic anions with g1 = −g2 = 8 for the two cases of wa = 0 and 8 by setting c1α = 10−3 and χ = 3. The adsorbed densities defined by Eq. (73) are 2 > (a2 Γ> 1 , a Γ2 ) = (0.043, 0.040) for wa = 8. The surface tension is 0.497

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0.06

5

χ= 3 b

g1= 4 g2= 8 A= 4 wa =12

0.04

1x10

c1α=

-3

= 2x10

-3

χ= 3

a

g1= 4 g2= 8 A= 4 wa =12

4

b

3.5

U

0.05

4.5

b 0.03

3

c1α=

1x10

-3

= 2x10

-3

a b

b

2.5 2

0.02

a

a

1.5 1

0.01

a 0.5

0 44

46

48

50

52

54

56

0

58

30

40

50

60

70

80

90

100

z/a

z/a

Fig. 7. Normalized cationic-surfactant density c1 (bold line) and counterion density c2 (broken line) in the left panel, and normalized electric potential U in the right panel, where (a) c1α = 10−3 and (b) 2 × 10−3 with g1 = 4 and g2 = 8. The excess surfactant −2 in (a) and 0.091a−2 in (b). density accumulated on the interface is Γ> 1 = 0.034a

0.012

0

χ=3

c1 , c2

0.008

g1= 8 g2= -8 A= 4

c1α =10

b

-1

=8 b b

0.006

a

a

χ=3 A= 4 -3 c1α =10

-6

0 40

g1= 8 g2= -8

-5

0.002

45

50

55

z/a

60

=8 b

-3

-4 0.004

wa = 0 a

a

-2

-3

U

0.01

wa = 0 a

b

-7 40

45

50

55

60

z/a

Fig. 8. Normalized cation density c1 (bold line) and anion density c2 (broken line) in the left panel and normalized electric potential U in the right panel, where g 1 = −g2 = 8, and c1α = 10−3 . Here (a) wa = 0 (no amphiphilic interaction) and (b) wa = 8 (surfactant). −2 in (a) and The excess cation density accumulated on the interface is Γ> 1 = 0.005a −2 0.040a in (b). The surface tension decreases with increasing wa as in the right panel of Fig. 9.

and 0.402 for wa = 0 and 8, respectively, in units of kB T /a2 . In this case, the electric double layer is enlarged and ∆U is a monotonically decreasing function of z. In Fig. 9, we examine how the surface tension γ decreases to zero and how the adsorption increases with increasing wa (left) and c1α (right) in our

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0.5

0.5

0.45

0.45

χ= 3

0.4

γ a2/ k BT

g1= 8 g2= -8 A= 4

0.35 0.3 0.25

g1= 4 g2= 2 A= 4 wa =12

0.35

γ a2/ k BT

0.3 0.25

-3

c1α = 10

0.2

χ= 3

0.4

0.2

0.15

>

0.1

Γ2 a >

0.05

Γ1 a >

0.15

Γ1 a2

2

0.1

2

Γ2 a2 >

0.05

0

0 0

2

4

6

wa

8

10

12

0

0.0005

0.001

c1α

0.0015

0.002

Fig. 9. Normalized surface tension a2 γ/kB T , normalized adsorbed cationic-surfactant 2 > and counterion densities a2 Γ> 1 and a Γ2 as functions of wa (left) and of c1α (right). Here g1 = −g2 = 8, and c1α = 10−3 (left). and g1 = 4, g2 = 2, and wa = 12 (right).

1D calculations. In the left panel, the accumulation occurs rather abruptly for wa ∼ >5, where g1 = −g2 = 8, χ = 3, and c1α = 10−3 . This behavior is consistent with Eq. (27). In the right panel, where g1 = 4, g2 = 2, > χ = 3, and wa = 12, the adsorbed densities Γ> 1 and Γ2 increase linearly as 87c1α a−2 and 64c1α a−2 , respectively, and the surface tension decreases as γ = (0.497 − 177c1α )kB T /a2 at small c1α . We notice the following. (i) −1 > We find Γ(∼ = Γ> 1 + Γ2 ) is proportional to nα = 2v0 c1α . In the literature, the Langmuir adsorption isotherm Γ = Γmax Kn/(1 + Kn) is well-known, where n is the surfactant density far from the interface with Γmax and K being constants. It predicts Γ ∝ n for n K −1 . (ii) Our results are roughly in accord with the Gibbs adsorption equation ∆γ = −kB T Γ at low surfactant densities, where a rather small discrepancy arises from the third electrostatic term in Eq. (64) in the present case. 6. Summary and Remarks We summarize our results. (i) We have introduced the composition-dependent solvation chemical potentials in Eq. (8). The solvation chemical potential µsol in Eq. (12) is bilinear with respect to φ and ci and is characterized by the parameters gi dependent on the ion species i. For aqueous mixtures, gi > 0 for hydrophilic ions, while gi < 0 for hydrophobic ions. In the asymmetric case g1 6= g2 , the Galvani potential difference arises at an interface. (ii) The image chemical potential stems from the composition-dependence of the dielectric constant ε(φ) in Eq. (11), as expressed in the integral form (12). It can be important around an interface where ε changes abruptly.

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For a pair of strongly hydrophilic cations and anions, the ions are significantly repelled from the interface when the screening length κ−1 α is longer than the Bjerrum length `Bα in the strongly segregated case. (iii) When hydrophilic and hydrophobic ions coexist, there appears a tendency of microphase separation at an interface, as is evident in Fig. 5. The surface tension decreases drastically for large |g1 | and |g2 | (even without amphiphilic interaction), where the electrostatic contribution to the surface tension, the last term in Eq. (62) or Eq. (64), is important. Though not yet well studied, a mesophase can appear in near-critical binary mixtures with addition of such salt.30 (iv) To describe ionic surfactants, we have presented the amphiphilic interaction Fam in Eq. (19) characterized by the parameter wa and the rod length 2`. As Figs. 7–9 demonstrate, Fam serves to induce adsorption of ionic surfactants onto an interface reducing the surface tension γ, which becomes significant for large wa . In our 1D calculations in Fig. 9, γ decreases to zero with increasing the surfactant density or the parameter wa . In real 3D systems,18 micelles are formed from the interface beyond a critical surfactant density before vanishing of γ. In addition, we are neglecting the steric effect due to the finite size of the surfactant molecules. (v) In one-phase states of near-critical mixtures, a peak can appear at an intermediate wave number in the structure factor of the composition fluctuations for γp > 1, where γp (∝ g1 − g2 ) defined by Eq. (32) is the parameter representing the solvation asymmetry. Below the transition, a mesoscopic phase can emerge, as observed recently.30 It can occur for strongly asymmetric salt (say, salt composed of hydrophilic cations and hydrophobic anions).3 It should be induced more easily for ionic surfactants, as Eq. (32) indicates. (vi) We have derived the attractive interactions among ions mediated by the critical fluctuations as in Eq. (38). We should then investigate how they can produce large-scale structures near the criticality.34,35 It is also of interest how charged colloidal particles interact in polar fluids in the presence of strong solvation effects. (vii) To understand the experiment,36 we have examined the situation where three ion species are present. As shown in Fig. 5, the ion distributions around an interface can be much more complex than in the case of two ion species. We mention future problems. (1) There can be an electric double layer and a potential difference at an interface in general charged systems, including polymers, surfactant systems,

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and gels. Results on the structure factor in Sec. 3 can be used for polymer solutions and blends. The solvation coupling between the ion densities and the composition should generally be present in such systems. From the viewpoint of the solvation effects, as delineated in this chapter, more experiments with addition of ions are informative in soft matters. (2) We have assumed mild heterogeneity of the dielectric constant of a mixture. However, it can be very strong in aqueous mixtures including polymeric systems. For example, we may assume εA ∼ 100 and εB ∼ 1. We have not yet understood electric field effects in such extreme (but common) situations. (3) The steric effect due to a finite volume fraction of ionic surfactants becomes crucial with increasing its density, leading to saturation of the adsorption onto an interface, though it has been neglected in this chapter. We will soon report on this effect. (4) Wetting should be greatly influenced by ions. Wetting on colloid surfaces can give rise to attraction among colloids.37 If the wetting layer is more polar than the outer fluid, ions can even be confined within the layer.38 (5) Phase separation in ionic systems should also be investigated. Ions are more strongly segregated than the fluid components for large gi . We already examined the effect of a very small amount of ions on nucleation.9 (6) Ion dynamics near an interface is also intriguing, when ionic surfactants are present or when electric field is applied. (7) We examined solvation effects of charged particles in liquid crystals.39 In nematic states the dielectric tensor anisotropically depends on the director orientation and ions distorts the orientation order over long distances and sometimes create manometer scale defects. References 1. J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 1991). 2. Y. Marcus, Ion Solvation (Wiley, New York, 1985). 3. A. Onuki and H. Kitamura, J. Chem. Phys. 121, 3143 (2004). 4. A. Onuki, Phys. Rev. E 73, 021506 (2006). 5. M. Born, Z. Phys. 1, 45 (1920). 6. Le Quoc Hung, J. Electroanal. Chem. 115, 159 (1980); ibid. 149, 1 (1983). 7. T. Osakai and K. Ebina, J. Phys. Chem. B 102, 5691 (1998). 8. J.J. Thomson, Conduction of Electricity through Gases (Cambridge University Press, Cambridge, 1906), Sec. 92. 9. H. Kitamura and A. Onuki, J. Chem. Phys. 123, 124513 (2005). 10. C. Wagner, Phys. Z. 25, 474 (1924).

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11. L. Onsager and N. N. T. Samaras, J. Chem. Phys. 2, 528 (1934). 12. Y. Levin and J. E. Flores-Mena, Europhys. Lett. 56, 187 (2001). 13. G. Jones and W. A. Ray, J. Am. Chem. Soc. 59, 187 (1937); ibid. 63, 288 (1941); ibid. 63, 3262 (1941). 14. N. Matubayasi, H. Matsuo, K. Yamamoto, S. Yamaguchi, and A. Matuzawa, J. Colloid Interface Sci. 209, 398 (1999). 15. P. B. Petersen and R. J. Saykally, J. Am. Chem. Soc. 127, 15446 (2005). 16. M. Manciu and E. Ruckenstein, Adn. Colloid Interface Sci. 105, 10468 (2003). 17. J. D. Reid, O. R. Melroy, and R.P. Buck, J. Electroanal. Chem. 147, 71 (1983). 18. S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes (Westview Press, 2003). 19. J.E. Guyer, W.J. Boettinger, J.A. Warren, G.B. McFadden, Phys. Rev. E 69, 021603 (2004); ibid. 69, 021604 (2004). 20. Y. Tsori and L. Leibler, Proceedings of the National Academy of Sciences of the United States of America 104, 7348 (2007). 21. A. Onuki, Phase Transition Dynamics (Cambridge University Press, Cambridge, 2002). 22. P. Debye and K. Kleboth, J. Chem. Phys. 42, 3155 (1965). 23. M. Laradji, H. Guo, M. Grant, and M. Zukermann, J. Phys. A24, L629 (1991). 24. J. Traube, Ann. Chem. Liebigs. 265, 27 (1891). 25. J.W. Gibbs, Collected works, vol.1,pp.219-331 (1957), New Haven, CT: Yale University Press. 26. E.L. Eckfeldt and W.W. Lucasse, J. Phys. Chem. 47, 164 (1943). 27. B.J. Hales, G.L. Bertrand, and L.G. Hepler, J. Phys. Chem. 70, 3970 (1966). 28. V. Balevicius and H. Fuess, Phys. Chem. Chem. Phys. 1 ,1507 (1999). 29. V.M. Nabutovskii, N.A. Nemov, and Yu.G. Peisakhovich, Phys. Lett. A, 79, 98 (1980); Sov. Phys. JETP 52, 111 (1980) [Zh.Eksp.Teor.Fiz. 79, 2196 (1980)]. 30. K. Sadakane, H. Seto, H. Endo, and M. Shibayama, J. Phys. Soc. Jpn., 76, 113602 (2007). 31. V. Yu. Boryu and I. Ya. Erukhimovich, Macromolecules 21, 3240 (1988). 32. J.F. Joanny and L. Leibler, J. Phys. (France) 51, 545 (1990). 33. I.F. Hakim and J. Lal, Europhys. Lett. 64, 204 (2003). 34. J. Jacob, M.A. Anisimov, J.V. Sengers, A. Oleinikova, H. Weing¨ artner, and A. Kumar, Phys. Chem. Chem. Phys. 3, 829 (2001). 35. A. F. Kostko, M. A. Anisimov, and J. V. Sengers, Phys. Rev. E 70, 026118 (2004). 36. G. Luo, S. Malkova, J. Yoon, D. G. Schultz, B. Lin, M. Meron, I. Benjamin, P. Vanysek, and M. L. Schlossman, Science, 311, 216 (2006). 37. D. Beysens and T. Narayanan, J. Stat. Phys. 95, 997 (1999). 38. N.A. Denesyuk and J.-P. Hansen, J. Chem. Phys. 121, 3613 (2004) 39. A. Onuki, J. Phys. Soc. Jpn. 73, 511 (2004).

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Chapter 3 Change of Critical Mixing Temperature in a Uniform Electric Field

Kazimierz Orzechowski Faculty of Chemistry, University of Wroclaw Joliot-Curie 14, 50-383 Wroclaw, Poland E-mail: [email protected] The influence of an external uniform electric field on the critical mixing temperature is still a subject of a debate. In this chapter the available experiments and theoretical expectations will be presented and discussed.

1. Introduction The liquids can mix together in all proportions or they can give mutually saturated phases. Formation of the phases in a given temperature and a pressure range is possible when the Gibbs free energy of mixing (∆GM = ∆HM - T∆SM) is positive.1,2 ∆GM is a function of temperature, pressure, external fields; it depends also on an addition of impurities.3,4 In ideal solutions the ∆GM is negative because the entropy of mixing is positive and the enthalpy of mixing is equal to zero. A lack of mutual mixing is a symptom of non-ideality of a mixture and a separation into phases is a very common observation. It is expected that each real mixture has to separate into phases, in a given temperature and a pressure range. Certainly, in many cases it is not possible because components may crystallize or vaporize before the parameters necessary to separation into phases are attained. Characteristic of regular solutions1 is an upper critical mixing point. An example of such a phase diagram is presented in Fig. 1.

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Fig. 1. Binary liquid mixture with an upper critical solution temperature. At the maximum of the coexistence curve the phase transition is classified as a second order (continuous). Binodal curve separates stabile one-phase region and metastabile twophase region, spinodal curve separates meta-stabile and stabile two phase area.

The temperature of the maximum of the curve it is an upper critical solution temperature (UCST). In temperatures higher than Tc the components mix in all proportions. At temperatures lower then the critical one two mutually saturated phases are formed. The composition of the coexisting phases is described by the binodal curve which separates the region of a stabile one-phase region and a two-phases metastable region. The spinodal curve (dotted line in Fig. 1) separates the metastable and stable two-phase area in the phase diagram. Also a situation is possible when the liquids mix in all proportions in low temperatures, but separate into phases in high temperatures. Then the coexistence curve is bent down and has a critical point at the minimum (lower critical solution temperature LCST). Sometimes, a single mixture has both UCST and LCST and forms a closed loop of two phases or two separate two-phase regions. The upper and lower critical mixing point could collapse in a single point and form a double critical point. A description of different possibilities of such phase diagrams could be found in literature.1,5 The phase diagrams with an upper critical solution temperature are the most frequently observed example and in this chapter we will refer mainly to this case.

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The phase transition at the critical point is a continuous one (second order) and belongs to the (3,1) universality class.6 Phase transitions in other points of the coexistence curve are classified as a first order. In the vicinity of the critical mixing point when temperature and concentration are approaching the critical values (Tc and xc) many macroscopic properties demonstrate critical behaviors. The most spectacular one is a critical opalescence7-17 a strong increase of sound attenuation12-15,18-22 an anomaly of the heat capacity23-26 compressibility27 a non-linear dielectric effect,28-33 the Kerr effect.34,35 The reason for these anomalies is the increase of a correlation length and a lifetime of concentration fluctuations when the system approaches the critical point. In one phase region, large and long-living concentration fluctuations in the vicinity of the critical mixing point cause similarity between the critical mixture and a really inhomogeneous system as emulsion or suspension. In this chapter the influence of a uniform electric field on the critical mixing temperature will be presented. Simple binary liquids as well as complex systems, as for example mixtures of polymers, belong to the same universality class. According to the universality hypothesis the critical divergence of macroscopic properties of phase transitions that belong to the same universality class should be similar. However the universality concerns critical exponents, not critical amplitudes.6 It means that the observation of some phenomena in simple liquids – for example a shift of critical temperature under the influence of an electric field, could not be applied to the complex system without any experimental verification. However, we can expect, that the properties observed in simple and in complex systems could be similar, qualitatively at least. Critical temperature is very sensitive to many perturbations as shear,36-43 pressure,44-50 gravity,51-56 content of impurities.57,58 It should depend also on the strength of an external electric field. Experiments on this influence are difficult and only few experimental results have been published so far.59-65 Unfortunately there are discussions concerning the correctness of the performed experiments.66-69 The expected shift of Tc is small, which forces a use of very strong electric fields. Main troubles in measurements of a shift of critical temperature in a strong electric field are: a/ possible electrode reactions, b/ heating caused by current

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conduction. The experimental status quo of the critical temperature shift under the influence of an electric field is still the subject of a debate.67-69 Theoretical expectations seem to be more clear, however also in this case conflicting opinions could be found in literature.59,61,63,66,69,70 This chapter is devoted to a description and an evaluation of main theories and available experiments describing this subject. A plan of the chapter is as follows: at first theoretical predictions of a shift of the critical temperature under the influence of an electric field will be presented, then experimental data and finally some remarks on possible reasons for the discrepancy between theory and experiments. 2. Shift of Tc under the Uniform Electric Field. Theoretical Predictions In literature it is possible to find several papers dealing with the prediction of Tc(E) shift. However, it is convenient to start from very simple arguments, which could be helpful to predict the discussed shift qualitatively. It is a generally known rule that “similar dissolves similar”. Let’s suppose that one of the components of a binary liquid mixture is polar, the second one non-polar. The differences between components consist in the presence of dipolar interactions between polar molecules and the absence of them for non-polar ones. The polar component interacts much stronger with the electric field than the non-polar one. It means that the electric field increases an energy difference between components, which should result in an increase of temperature necessary for infinitive miscibility. Following that the electric field should increase an upper critical solution temperature of mixtures of polar + non-polar liquids. Elements of this “naive” attempt can be found in more sophisticated calculations of Tc(E) shift. Most experiments in binary liquid mixtures were performed in constant pressure, which means that the change of Gibbs free energy (G) is a correct potential. However, in a case of condensed matter the difference between Gibbs free energy and Helmholtz free energy (F) is negligible and most authors use F for a description of the systems in question. To investigate the influence of an electric field on the critical parameters the electric field contribution to the free energy should be


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included. Unfortunately this simple starting point for the prediction of Tc(E) shift is different in different papers, which leads to contradicting conclusions. Landau and Lifshitz70 for one component system used the equation:

FE = F − 0.5ε oεE 2

(1)

where εo is the absolute permittivity of vacuum, ε - is the relative permittivity, F – free energy in the absence of the electric field The first and the second derivative of the chemical potential µi=(∂F/∂xi)T,V over concentration (expressed as mole fraction, volume fraction or molarity) should vanish in critical temperature and concentration. Simple calculations give the following Tc(E) shift:

 ∂ 2ε  ∆Tc = 0.5ε o ρE 2  2   ∂ρ  T

∂2 p ∂ρ∂T

(2)

where ρ is the density. Assuming applicability of the Clausiuss-Mossotti relation the critical temperature of gas-liquid critical point should increase under the influence of the electric field. Debye and Kleboth59 considered the binary liquid mixture in the vicinity of UCST. In the calculations they used Helmholtz energy of mixing in a form similar to that used by Landau and Lifshitz, but the sign before the term responsible for the interaction with electric field was reversed:

∆FEM = ∆F M + 0.5ε oε M E 2

(3)

where the superscript M refers to the mixing quantity. When the first and the second derivative of the chemical potential over volume fraction equated to zero, the obtained shift of the critical temperature was found opposite to the second derivative of permittivity over volume fraction:

 ∂ 2ε  2 ∆Tc ∝ − 2  E  ∂xv  c

(4)

Similar equation was obtained also by Wirtz and Fuller.61,62 Because in most liquids mixtures of polar + non-polar components the curvature of ε(xv) dependence is positive, the Eq. (4) allows to expect a decrease of Tc under influence of an electric field. The expression for the free energy

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used by Debye and Kleboth was criticized by many authors.61,68,69 Such an electric field dependence of free energy is acceptable in a case of conductors in constant charge.61 The equation in the form (3) is apparently applicable to dielectrics in a case of constant potential, however, free energy defined in this way contains not only the free energy of the system in question, but also the work that has to be done by an external voltage source so that the potential could remain constant (when the permittivity is the fluctuating quantity).71 Onuki69 carefully reanalyzed the case of constant charge and constant potential for the fluctuating dielectric medium. He concluded that the correct term describing the influence of the electric field on free energy should have a form of that used by Landau and Lifshitz70 independently if the constant charge or constant potential is considered. It means that in the final equation derived by Debye and Kleboth the sign should be reversed. Analyzing both constant current and constant potential case, Onuki obtained66,69 two positive contributions to the critical temperature shift (if only (∂2ε/∂x2) > 0). The first one is similar to that obtained by Debye, but the sign is reversed. The second contribution is related to dipolar interactions between fluctuations. In the strong-field regime the fluctuations are elongated in the field direction but are suppressed perpendicularly to the field. As a result the dimensionality is reduced, which gives rise to the additional upward shift of Tc. A critical temperature shift under the influence of the electric field was analyzed also by Goulon and co-workers72 on the base of the droplet model.73,74 In this model large and long living concentration fluctuations are treated as droplets. For a description of the micro-inhomogeneous system methods constructed for really inhomogeneous mixtures were used. In the strong electric field the large droplets are perturbed, which alternates a correlation length. The authors predicted that the shift of the critical temperature is:

1 1 (1) ∆Tc E 2 ∝ − ε c(2 ) − εc 3ε c 2

( )

where

( )  2



(5)


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 ∂ (i )ε   i   ∂x  c

ε c(i ) = 

The equation derived on the basis of the droplet model differs from that obtained by Debye and Kleboth because the temperature shift is related not only to the second derivative of permittivity, but also to the (εc(1))2/εc quantity. Using the Eq. (5) and data published by Debye and Kleboth an upward shift of Tc is expected. In summary, most theories predict an upward shift of the critical mixing temperature under the influence of the electric field. 3. Shift of Tc under the Uniform Electric Field. Experimental Results The question concerning the influence of electric field on the critical mixing point was a subject of a relatively small number of experiments. Unfortunately, the conclusions of this experiments are not clear, both as concerns the magnitude of the effect and even the character of the shift (increase or decrease under the influence of the field). Independently of the method used it is generally accepted that in systems where the electric permittivity is a scalar quantity the shift of Tc should be independent of the field direction. Following that as a measure of the critical temperature shift is the quantity dTc /dE2. 3.1. Direct measurements of Tc(E) shift Measurements of the Tc(E) shift are apparently simple. It is necessary to detect the influence of the external electric field on some macroscopic quantity, which has a very strong divergence in the vicinity of Tc. The most spectacular evidence of approaching the critical point is the strong increase in the intensity of the scattered light. This phenomenon is most frequently used for estimation of Tc shift. To the best knowledge of the author, the first experiment of a shift of Tc under the influence of the electric field was published by Debye and Kleboth.59 The authors used turbidity measurements to detect the discussed effect. They investigated a mixture of nitrobenzene + 2,2,4trimethylpentane (iso-octane). The selection was dictated by low specific

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conductivity and a large difference between electric permittivity of the components. The theoretical expectations presented by the authors lead to the conclusion that the shift of Tc should be opposite to the (∂2ε/∂x2) derivative. The large difference between permittivities of constituents should enhance the expectable effect. In measurements under the influence of a strong electric field the main trouble is related to the heating because of current conduction. In a case of a mixture with UCST the increase in temperature may results in an increase in a distance from critical temperature, which apparently simulates a decrease in UCST under the field influence. Taking into account the conductivities of the constituent liquids, Debye and Kleboth estimated that the field of intensity up to 5⋅106 V/m could be applied as pulses of duration of 0.1 – 0.2 ms in order that the increase in temperature would be smaller than 0.01K. Short pulses protect also from electrode reactions. In the measuring cell (Fig. 2) the parallel-plate platinum electrodes have small slits and the light was passing the sample parallel to the applied field

Fig. 2. A scheme of experimental set-up used by Debye and Kleboth for the investigation of Tc shift.

Debye and Kleboth measured intensity of light passing the sample. Temperature of the sample was decreased step by step from one phase region towards Tc (UCST). When the temperature was approaching the critical value a very strong decrease in the intensity of light passing the sample was observed (light scattering). The intensity of the transmitted light was treated as a sensitive probe of a distance of the actual


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temperature from the Tc. When the strong electric field was applied (as short pulses) the intensity of the light passing through the sample was increasing, which was interpreted as an increase in a distance of actual temperature and the critical one. Analysis of the change of the light intensity allowed estimating that under the influence of a field of intensity 4.5⋅106 V/m the critical temperature was decreasing by 0.015K (dTc/dE2 = −7.4⋅10−16 Km2/V2). According to the mean-field model, described in the previous paragraph the shift of Tc should be proportional to E2 and related to the second derivative of electric permittivity over the volume fraction. The electric permittivity of the mixture was measured in a function of concentration for close critical temperature (T-Tc = 0.5K) at 0.5 MHz. On the basis of these experiments the second derivative of permittivity was obtained and the shift of the critical temperature calculated. The calculated shift and that obtained from optical measurements were found consistent. Apparently, the calculations and experiments were compatible. However, many authors criticized both theoretical expectations66-68 and experiments.67,69 Min and co-workers67 reanalyzed the data presented by Debye and Kleboth and concluded that the decrease of Tc was an apparent effect. According to Min and coworkers Debye and Kleboth did not consider the effect of dipolar interactions66 between fluctuations in a presence of the electric field. The field polarizes fluctuations, which leads to additional interactions. This effect does not change a critical temperature but suppresses longitudinal fluctuations and increases the intensity of transmitted light measured parallel to the field. The problem of the influence of the electric field on the critical temperature was studied also by Beaglehole.61 The author explored a discontinuity of the adsorption coefficient at the liquid-vapor interface at the critical temperature. A scheme of the experimental set-up is presented in Fig. 3. Two configurations of the measuring cell were used. In the first one the electrodes were oriented parallel to the surface of a liquid mixture (Fig. 3, on the left) in the second cell vertically to the surface (Fig. 3 on the right). A circularly polarized light beam was reflected off the surface. Analysis of ellipticity (ρ’) of the reflected light allows estimating the

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Fig. 3. A scheme of experimental set-up used by Beaglehole for investigations of adsorption in a presence of the electric field in near critical binary mixtures.

adsorption which was found to have a sharp discontinuity at the phase transition temperature.75 In the experiments a mixture of aniline and cyclohexane was used. The author observed that the discontinuity of ρ’(T) is shifted upwards when the electrodes were oriented vertically, and downwards when electrodes were parallel to the surface of a liquid. The shift was roughly proportional to E2. The author pointed out that parallel orientation of the electrodes could be treated as equivalent to that used by Debye and Kleboth and hence both experiments give the same character of Tc shift. The shift of the critical temperature estimated on the basis of the presented data is dTc/dE2 = −8⋅10−13 Km2V−2, which is much larger than that obtained by Debye and Kleboth (in nitrobenzene + isooctane). The experimental section of the paper was supplemented by the mean-field theory. The author used the same method as that of Debye and Kleboth. However, Beaglehole used the correct form of the free energy dependence on the electric field. As a result, he predicts the increase in the critical temperature under the influence of the electric field, which contradicts his experimental finding. The absolute change of Tc measured by Beaglehole is very big, both when compared with the theory and previous experiments.59 In the description of the experiments performed by Beaglehole there is no information if the electric field was applied as AC, DC or pulses. If DC or low-frequency AC electric field was applied the strong Joule heating effect is expected, which may strongly disturb experiments. The author discussed the influence of the electric field on the adsorption and concluded that the absorption is not disturbed by E field and the only expectable effect is a shift of Tc. However, this assumption is not necessarily correct. The discontinuity of


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ρ’ is a result of the preferable absorption of one of the components on the liquid-vapor surface. An application of the strong electric field perpendicularly to the surface has to result in a decrease in free energy of a more polar component. In the investigated mixture the aniline has a bit higher electric permittivity and considerably larger surface tension then cyclohexane. In a presence of a strong electric field there are two opposite effects: the first one is the accumulation of the component of lower surface tension on the liquid-vapor surface (cyclohexane), the second the increase in stability of more polar component (aniline) because of interaction with E-field. An additional trouble in an interpretation of ellipticity experiments is a question concerning a prewetting transition76 and the influence of the electric field on this phenomenon. Wirtz and co-workers62-63 performed small angle light scattering measurements of the structure factor in polymer solutions and in simple liquids. Measurements were performed close to the critical mixing point under the influence of an external field. The experiments preformed in one phase region shows that the electric field anisotropically distorted concentration fluctuations, which induces electric birefringence and shifts the critical temperature. Measurements were supplemented by the theoretical background. In the presented theory they considered an influence of the electric field on the structure factor describing the intensity of the scattered light. According to the theory, in the absence of the electric field the structure factor is isotropic77 and the iso-intensity scattering patterns should be concentric circles in the screen perpendicular to the accident light. When the electric field in the direction perpendicular to the light is applied the iso-intensity scattering patterns should be elliptical. The ellipticity of the structure factor allows calculating a correlation length of the fluctuations in the direction parallel and perpendicular to the electric field. The theory predicts that the parallel correlation length strongly increases, whereas the perpendicular correlation length slightly diminishes under the influence of the external electric field. The free energy was analyzed in a framework of the continuous model78 with an additional term due to the electric field. The term related to the field increases the free energy (as in the case of Debye and Kleboth attempt). Mean-field calculations lead to

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the conclusion that the shift of the critical temperature suppresses the two-phases region: decreases UCST and increases LCST. The shift is expectable for the bimodal and spinodal curve. The model was addressed mainly to polymers of low polarity however, the authors extended the interpretation to simple binary liquids as nitrobenzene + hexane. In order to check the predicted properties small-angle light scattering measurements were performed. In the experiments a Kerr cell was used and the scattered light was detected perpendicularly to the light beam. A scheme of an experimental set-up is presented in Fig. 4.

Fig. 4. A scheme of an experimental set-up used by Wirtz and Fuller for investigations of electric field induced anisotropy of light scattering in near critical binary mixtures.

In conformity with the theoretical expectations the intensity of the scattering pattern was elliptical in a presence of the electric field when temperature is close to Tc. The elongation of fluctuations in the direction of the electric field was visible, whereas the suppression perpendicular to the field was very small and often not detected. On the basis of these results the authors doubt if the effect predicted already by Onuki66 and related to the perpendicular suppression of fluctuations could be observed. The model presented by Wirtz and co-workers predicts the Tc(E) shift and “remixing” of the initially two-phase system. Indeed, for the electric field intensity of 5000 V/cm and Tc – T ≤ 0.05K (two phases region) they observed predicted remixing.63 Under the influence of the electric field the picture characteristic of the close-critical one-phase region was recovered. Authors interpreted it as a decrease of UCST. Measurements performed in polystyrene + cyclohexane shows that the


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decrease of Tc under the influence of the electric field is proportional to E2 and the dTc/dE2 ≅ −1.8⋅10−10 [Km2/V2] and for nitrobenzene + hexane dTc/dE2 ≤ –2⋅10−14 [Km2/V2]. Both the presented theory and the performed experiments seem to be mutually consistent. However, the electric field of intensity up to 5⋅105 V/m (in the case of nitrobenzene + hexane 106 V/m) was applied for a relatively long time (of order of seconds) and tone can expect that the heating because of the current conduction disturbs the experiments. Following that it is possible that the announced remixing under the influence of the electric field is an apparent effect. The non-linear dielectric effect (NDE) offers a suitable way to investigate the Tc(E) shift. The NDE increment consists in measurements of a difference of the electric permittivity caused by a strong external field.79 Appropriate analysis of the experimental results obtained in the vicinity of the critical point should allow estimating the shift of Tc in a strong electric field.64 The NDE increment is defined as: ∆εNDE = ε(E) – ε(E→0) where ε(E) is the permittivity measured in a strong electric field, ε(E→0) is the permittivity measured in a low intensity field. Figure 5 presents the dependence of the electric polarization (P) versus electric field as predicted by the Debye-Langevin theory.79 Electric polarization is a macroscopic dipole moment of a unit volume. In liquids containing

Fig. 5. The dependence of the electric polarization versus internal field predicted by the Debye-Langevin theory.

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dipolar molecules polarization is a consequence of partial ordering of molecules in the direction of an external field. For extremely strong fields the polarization should attain a saturation value (Psat.), however, in simple liquids it is impossible because of an electrical break down. According to the Debye-Langevin theory in simple liquids containing rigid, polar molecules, the NDE increment should be negative and proportional to the square of the electric field strength. Electric permittivity is related to the derivative of polarization over the electric field: ε = εo–1 (∂P/∂E)S (where εo is the absolute permittivity of vacuum, S is the entropy). Figure 5 illustrates the decrease in permittivity when the electric field grows up. In simple liquids the change of permittivity under the influence of a strong electric field is usually very small ∆ε ≅ 10–5 – 10–3, which requires constructing special equipment to detect this effect.79 The main trouble in NDE experiments consists in heating a sample under the influence of a strong electric field. Because in liquids the dε/dT derivative is usually negative, the heating gives a stepwise decrease in the permittivity. To avoid or to minimize this effect the NDE experiments are performed using a short high field (HV) polarizing pulses modulated by low amplitude, high frequency (usually of order of MHz) measuring field. The scheme of the field sequence applied usually in NDE experiments is presented in Fig. 6.

Fig. 6. Upper plot — the sequence of fields applying in NDE experiments. In Orzechowski’s experiments polarizing field had amplitude up to 107 V/m, duration time 1 ms, measuring field had amplitude 103V/m and frequency 5 MHz. Lower plot — the NDE response when the NDE increment is negative and the heating is observed.


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The lower plot in Fig. 6 presents the change of permittivity under the influence of a strong electric field pulse when the NDE increment is negative and the heating effect is observed. Measurements performed in a function of time allow separating the NDE increment related to molecular orientation and that of heating. The NDE increment, according to the Debye-Langevin theory should be negative. However, when the strong field changes the dipole moment of molecules or complexes, the NDE increment could be positive.79 The positive NDE increment is also promoted by fluctuations, both thermodynamic80,81 as well as critical.81,82 In the vicinity of the critical mixing point very large, positive NDE increment was observed.28-33,82-88 The temperature dependence of the NDE increment (expressed as ∆εNDE/E2 which should be field — independent quantity) close to Tc is very sharp and strongly depends on the distance from Tc (see Fig. 7). Orzechowski64 proposed to explore it for the estimation of a shift of the critical temperature under the influence of E.

Fig. 7. The NDE increment (as ∆ε/E2) in a function of temperature close to UCST.

Both in simple and in complex liquids the ∆εNDE was found to be a linear function of E2.79,84 Only in a case of macromolecules with a very large dipole moment the non-linearity was observed and it is linked with a higher order of the Taylor expansion of the Debye-Langevin function. Orzechowski assumes that the non-linearity of the ∆εNDE = f(E2) dependence should be observed when the strong electric field shifts the

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K. Orzechowski

critical temperature. A negative curvature of ∆εNDE = f(E2) is expected in the case of a downward shift of Tc, the positive curvature in a case of the upward one. For the purpose of experiments the mixture of nitrobenzene with 2,2,4-trimethylopentane was selected — the system investigated already by Debye and Kleboth.59 Considerable differences in permittivities of the constituent liquids provide large and easy to measure NDE increment, while low conductivities reduce the Joule heating effect after applying a strong electric field. The equation describing the critical divergence of the NDE increment (expressed as ∆ε/E2) has the following form:

(

∆ε NDE / E 2 = ∆ε NDE / E 2

)

B

+ At −ψ

(6)

where (∆εNDE/E2)B is the background term, A is the critical amplitude, t is the critical reduced temperature t = (T − Tc)/Tc and ψ is the critical exponent. Theoretical expectations predict ψ = 0.59.72,89,90 Some experiments confirm this value87,90 but sometimes the fitted ψ exponent value is smaller.85,86,88 The decrease in ψ was explained by Rzoska91 by a crossover from non-classical to classical conditions in a system with large fluctuations elongated in the direction of an external field. According to theoretical expectations the shift of the critical temperature should be proportional to the square of an electric field:

Tc = Tco + bE 2

(7)

and hence: 2

(

∆ε NDE / E = ∆ε NDE / E

2

)

B

 T − Tco − bE 2   + A o 2  Tc + bE 

−ψ

(8)

Assuming that bE2 Tg

b Patterned Electrode (T>Tg

Fig. 1. Schematic representation of the electrode device for electrohydrodynamic instability formation.

Electrohydrodynamic Instabilities of Thin Liquid Films

115

ignored. So, in the case of planar thin films, the electrostatic pressure is countered by the Laplace pressure. These fluctuations grow in amplitude until the polymer films hits the upper electrode, producing columns of polymer; hexagonally pack, with a characteristic separation distance and a diameter that is defined by the volume occupied by the polymer between the electrodes. These two competing pressures results in the growth of fluctuations with a characteristic wavelength. Rather than having the wavelength be dictated by the electrostatic pressure and the sample geometry, a topography can be placed on the upper electrode, for example lines or dots raised from the electrode surface (shown schematically in Fig. 1). Here the field strength is higher for the areas where there is a closer approach of the electrode to the film surface. Consequently, those areas are drawn to the upper electrode replicating the topography of the upper electrode. Shown schematically in Fig. 1(b) is the case where lines are placed on the electrode surface by electron beam or photolithographic processes. This topography causes a variation in the electrostatic pressure at the film surface, preferentially drawing the polymer under the topographic features to the electrode surface. Schaeffer et al. suggested that the overall pressure distribution at the film surface is expressed by,2

p = p0 − γ

∂2h + pel (h) + pdis (h) ∂x 2

(1)

with p0 being the atmospheric pressure. The second term, the Laplace pressure, stems from the surface tension γ and the fourth term, the disjoining pressure pdis, arises from dispersive Van der Waals interactions. The electrostatic pressure for a given electric field in the polymer film,

U {ε p d − (ε p − 1)h}

(2)

pel = −ε 0ε p (ε p − 1) E p 2

(3)

Ep = is given by

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T. P. Russell and J. Bae

For high enough values of Ep, only the Laplace and electrostatic terms need to be considered. In a stability analysis, a small sinusoidal perturbation of the interface with wave number q, growth rate τ -1, and amplitude u is considered:

h( x, t ) = h0 + ueiqx +t /τ

(4)

The modulation of h gives rise to a lateral pressure gradient inside the film, inducing a Poiseulle flow j

j=

h3  ∂p  −  3η  ∂x 

(5)

where η is the viscosity of the liquid. A continuity equation enforces mass conversation of the incompressible liquid:

∂j ∂h + =0 ∂x ∂t

(6)

Equations (1), (5), and (6) establish a differential equation that describes the dynamic response of the interface to the perturbation. In a linear approximation (to order O(u)), a dispersion relation is obtained:

1

τ

=−

h03  4 ∂pel 2  γq + q  3η  ∂h 

(7)

As opposed to the inviscid, gravity-limited case (τ -1 ∝ q), the viscous stresses lead to a q2-dependence of in the long-wavelength limit, typical for dissipative systems. Fluctuations are amplified if τ > 0. Since

∂pel 107 V/m). No surface features were observed in the absence of an applied field. This was confirmed by a control experiment without an applied voltage and with the electrodes shorted. It should be noted that the gap between the electrodes is typically microns or less. 1 V applied to the electrodes separated by 1 µm produces a field of 1 V/µm. As the gap gets smaller the field strength increases, of course. Nonetheless, unless special precautions are taken, residual charge on the surface of an electrode can give rise to large fields that, in turn, can lead to the enhancement of surface instabilities. In addition to a hexagonal symmetry, second-order effects can be observed as well. As the fluctuations grow in amplitude, there is a lateral variation in the film thickness. This leads to a depletion area around the growing fluctuation as polymer is drawn in to the growing column. In fact, a series of rims (a damped wave) are observed around the growing column. Each rim experiences the electrostatic pressure and fluctuations around the rim with a characteristic period grow with time.


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When the upper electrode is replaced by a topographically patterned master, the separation distance between the electrode and the surface of the film varies. The electrostatic forces are strongest for smallest electrode spacing and the time needed for the instability to form is much shorter for smaller values of spacing. Consequently, the variation in the electrostatic pressure can be used to replicate the topography of the master electrode. In a study by Shaffer et al.,1 a line pattern was used and a high fidelity replication of the electrode pattern was obtained. Remarkably, features ~140 nm in width could be achieved using this simple replication process. These experiments did not push the limits of the technique and were not done under stringent clean room conditions. Even still, the feature size resolution obtained in these studies was comparable to the best result being obtained using photolithographic processes. However, the electrohydrodynamic instability approach is done without the use of solvents and represents a very simple route to pattern polymer surfaces. Further studies by Steiner et al. investigated the influence of polymer molecular weight, glass transition temperature (Tg), surface tension, and dielectric constant on the spatial characteristics and growth kinetics of EHD’s.2 Thin films of polymers, polystyrene (PS) (Mw=108 kg/mol), poly(methyl methacrylate) (PMMA) (Mw=99 kg/mol), and poly(styrene bromide) (PSBr) (Mw=127 kg/mol) with an average thickness of 100 nm were annealed at 170 °C under an applied electric fields of 107~108 V/m. These polymers have similar physical properties with the exception of their dielectric constants (εPS=2.6, εPMMA=3.6, and εPSBr=5.5). In all cases, the growth of fluctuations occurred with the application of an applied field, as would be expected, and the general characteristics of the fluctuations were similar. With increasing field strength the characteristic wavelength of the EHD’s decreased. In this study it was shown that by modifying the above arguments, a general equation describing the wavelelength of EHD’s is given by

λ = 2π

3 − γU 2 E ε 0ε p (ε p − 1)2 p

(15)

120


where U is the applied voltage, ε0 and εp are the permittivity in vacuum and the liquid dielectric constant, γ is surface tension of the polymer, and Ep is electric field strength in the polymer. A comparison of the data with this slightly modified EHD argument led to reduction of all the data to a master curve that quantitatively described the experimental observations without any adjustable parameters, as shown in Fig. 2.

Fig. 2. Variation of wavelength vs. the electric field in the polymer film Ep in reduced coordinates. The different symbols correspond to four data sets:  PS with h0=93 nm, d=450–1000 nm, U=30 V; ∇ PS with h0=120 nm, d=600–1730 nm, U=50 V; ο PMMA with h0=100 nm, d=230–380 nm, U=30 V; ◊ PBrS with h0=125 nm, d=400–620 nm, U=30 V. The crosses correspond to an AC experiment (rectangular wave with a frequency of 1 kHz and an amplitude U=37 V) using a PMMA film with h0=100 nm, d=230–360 nm. The inset shows some of the data in non-reduced coordinates vs. d.

Structure formation at a polymer-air interface in an electric field is analogous to phase separation in polymer blends. In the latter system, phase domains develop with time with a periodicity equal to the dominant wavelength of concentration fluctuations in the system. The wavelength of these fluctuations is dictated by a balance between thermodynamics and dynamics. Thermodynamics, which is governed by the interfacial tension produced by formation of phases, favors the


121

growth of large domains, while dynamics favor the growth of smaller phases. The dynamics of phase separation in polymer blends, i.e. the rate at which fluctuations grow at early stages of the phase separation, is characterized by an exponential growth, in accordance with the linearized Cahn-Hilliard arguments.3 However, at later stages, deviations from this behavior are seen and the hydrodynamics associated with the flow of the polymers must be considered. The growth of fluctuations at the air-polymer interface in an electric field should follow a similar pattern, as it is similarly governed by a balance between surface tension and electrohydrodynamic flow. The linearized theory1,2 of Schaeffer et al. predicts that the rate of growth in the amplitude of the dominant wavelength should be exponential. However, deviations from this should be expected as the flow of the polymer in the thin films becomes dominant. When the polymer film is above its glass transition temperature, a spectrum of capillary waves is present at the liquid-air interface, due to thermal fluctuations. However, surface tension suppresses the amplitudes of these waves. Therefore, as shown by the interference optical micrograph in Fig. 3(a), the film surface is initially featureless. Slight variations in the intensity over the field of view are possible due to positioning of the sample at a slight angle to the imaging plane. If there is a significant gradient in gap spacing, i.e. the gap is wedge-shaped, another set of fringes, parallel lines running perpendicular to the gradient, will be evident in the micrograph, even when the film is smooth. In the experiments no fluctuations were observed when no electric field was applied, implying that PDMS behaves as a perfect dielectric. When a voltage is applied between the substrate and opposite electrode, electrostatic pressure opposes surface tension. If there is sufficient voltage and dielectric contrast at the interface, electrostatic pressure overcomes surface tension and causes a roughening of the film due to the amplification of surface waves. A continuous wave laser was used here to avoid heating; consequently, no temperature gradient induced instabilities should occur. Within seconds of application of electrostatic pressure to the PDMS-air interface, a lateral pattern of intensity peaks is apparent in the confocal image, as shown in Fig. 3(b).

122


Fig. 3. Various stages of structure development by electric field-amplified instability, as a function of time, of a PDMS film on silicon. (a) [0.0 s] The film is initially featureless. (b) [36.6 s] Fluctuations appear as lateral variations in intensity over the plan of the surface. (c) [75.8 s] Field-amplified peaks exhibit reflection interference fringes as height increases. (d) [109.7 s] Peaks are encircled by fringes as height increases further. (e) [129.3 s] As peaks grow, the number of fringes around each peak increases. (f ) [144.1 s] When peaks span the two planar electrodes, cylindrical structures are formed. The arrow in (f ) indicates a pillar which has shifted laterally after electrode contact. Laser scanning confocal micrographs were acquired by reflection imaging through a transparent electrode. Laser wavelength is 458 nm; image dimension is 740 × 740 µm2.

The reflective interference of light is dependent on the refractive index of the film, n, and the incident wavelength, λ, such that the change in height, ∆h, between intensity maxima or minima is given by ∆h = λ/2n. For these experiments, ∆h ranges from 160 nm to 200 nm. Thus, as the intensity increases from a minimum to a maximum in Fig. 3(b), thickness


123

variations are less than ∆h/2, i.e. less than 80 nm. The fluctuations have a typical spacing of one hundred microns or more in the plane of the film. Consequently, these fluctuations represent only minor height variations of the film surface and, as such, conform to the linearized theoretical framework which assumes the lubrication approximation. The amplitude of height variations increases with time due to electrostatic pressure. The intensity of bright regions of Fig. 3(b) is increased in Fig. 3(c). However, in some areas the center of the bright spot becomes darkened in the center. Here, a decrease in intensity results from destructive interference of reflected light, as the height approaches a thickness for which an intensity minimum is observed. Comparison of Fig. 3(c) to Fig. 3(f ), in which the final column morphology is shown, confirms that the characteristic spacing of the final columnar morphology corresponds to the wavelength of the undulations observed at early times. As peak height continues to increase with time, the undulations in the film surface are characterized by rings of interference fringes, as shown in Fig. 3(d). At the base of each feature, the fringes are noticeably broader and less axially symmetric than at the center, due to the sensitivity of the position of the peak to flow in the plane of the film resulting from the mass transport from the surrounding flat film, and the smooth peak shape arising from the Laplace pressure normal to the film surface. This micrograph illustrates that the characteristic distance between growing peaks is determined by the competition between electrostatic and Laplace pressures. For example, in the lower left quadrant of Fig. 3(d), a row of three peaks is visible, of which the middle peak eventually decreases in height while the other two peaks grow, the result of which is shown in Fig. 3(e). Figure 3(d) also shows that disparities in height between peaks which, though slight at first, become amplified with time as a result of the exponentially increasing growth rate. When amplitude of the fluctuations increases sufficiently to span the air gap between the film and upper electrode, the polymer fluid and the electrode come into contact. First contact is made by the center of the

124


peak, followed by an equilibration of the structure to a columnar shape. At the lower right edge of Fig. 3(e) and in Fig. 3(f ), columns of PDMS appeared as circular contacts between the film and the upper electrode. There is a strong driving force towards the alignment of air-polymer surfaces parallel to the electric field in order to minimize an effective torque acting on the interface. Furthermore, spreading of PDMS on the ITO-glass results in a reduction of surface free energy at the upper electrode.

240

(a)

1000

(b)

220 800

180

Height (nm)

Intensity (A.U.)

200

160 140 120 100 80

600

400

200

60 40

0

20

40

60

80

100 120 140 160 180

X coordinate (µm)

20

40

60

80

100

120

140

160

180

X coordinate (µm)

Fig. 4. (a) Cross-sectional intensity profile of a peak showing interference fringes. (b) Calculation of peak shape from fringe spacing (squares) and Gaussian fit to points.

The growth in amplitude of the surface waves was determined from the lateral variation in film thickness from the interference fringes that encircle each peak. Since the period of the fringes is dictated by the slope of the film, they can be use to calculate the three dimensional shape of the surface fluctuations. Figure 4 shows an example of a surface topography cross-section calculated from interference fringes. The peak height was determined by fitting the resulting topographic profile to a Gaussian peak function, weighted with fitting constraints at the center of the peak to most accurately estimate the height in the center of the feature. Though it does not arise from the model for electrohydrodynamic flow, the Gaussian peak function served as a useful mathematical fit on which to map individual peaks.


125

Fig. 5. Development of peak height (natural log scale) with time for a typical feature. line indicates best fit for data up to 320 s. Image dimension is 198 × 384 µm2.

From a frame-by-frame calculation of peak height, the growth in the amplitude of the waves was found to depend exponentially on time. A typical growth curve is shown in Fig. 5 in a semi-log plot. A single exponent could be used to describe the early stages of structure formation. However, the growth in some cases was found to accelerate towards the upper electrode at the final stages of structure formation and the rate became faster than the initial exponential dependence. The linearized theory takes into account the increased electrostatic pressure at the peaks, which is the reason for the initial phase of exponential growth. In the later stages, however, the shape of the fluctuation cannot be described by a simple smooth function, since it is becoming increasingly pointed. The electric field at the peak will be much higher than elsewhere across the surface, and field lines will no longer be parallel. The peak shape at these late stages could not be measured directly due to the acceleration in growth and rapid change in the shape. In the final stages of pillar formation, the shape more closely resembles an electrohydrodynamic spout, seen by Oddershede and Nagel.4 They

126


observed a divergence in tip curvature as a conical surface of a charged liquid approached an opposing electrode. The experimental data consistently show exponential growth of peak height with time (Fig. 5). For thinner films and at lower voltages, the exponent is smaller, corresponding to slower structure formation. For thicker films and higher voltages, the exponent was larger, corresponding to faster structure formation. To enable comparison between these experiments, the electric field in each experiment was reduced to the same dimensionless parameter used by Schäeffer et al. and the initial growth exponent from each curve was similarly reduced to a dimensionless characteristic time τ0,

EP =

U {ε p d − (ε p − 1) h0 }

E0 =

λ = 2π

(16)

U

(17)

λ0

3 − γU 2 E P 2 ε 0 ε P ( ε P − 1)

(18)

2

λ0 = 3γη τ = 2 3 4 ε 0 h0 U

ε 0 ε P ( ε P − 1) U 2 γ 6

(19)

 εP   1  d − h0   1 −    ε P −1   εP 

3γηU 8  ε ε ( ε − 1)  τ0 = 4 2 3  0 P P  π ε 0 h0  γ 

6

2

 1  1 −   εP 

(20)

2

(21)


E  τ =π4 P  τ0  E0 

127

−6

(22)

In the above expressions, γ is polymer surface tension, U is applied voltage, η is viscosity, h0 is initial polymer film thickness, d is separation distance between the two electrodes, ε0 is permittivity in a vacuum, εP is polymer dielectric constant. E0, λ0, and τ0 are characteristic parameters used to reduce experimental results to dimensionless values. A log-log plot of τ/τ0 as a function of EP/E0 is shown in Fig. 6, along with predicted values according to Eqs. (20) and (21). No fitting parameters are used to scale the data. The error bars in the figure are due to the uncertainty in the exponential fit. However, the scatter in the data may arise from the uncertainty with which the fastest growing wavelength is chosen from within the field of view in the microscope. It is not possible to know which peak corresponds precisely to the fastest growing wave in the system. Note that the values and trends of the experimental results correspond remarkably well to the predicted values for the three samples of PDMS with different viscosities that were studied. No systematic deviations from the predicted characteristic time values were seen as a function of viscosity.

Fig. 6. Variation of dimensionless characteristic time with dimensionless electric field and comparison with predicted values. Both axes have logarithmic scales.

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3. Instabilities on Liquid/Liquid/Air Interfaces — Double Layer The electrohydrodynamic instabilities of liquid/liquid interfaces have also been studied extensively by Russell and Steiner.5 The reduction in the difference in the dielectric constants and the reduction in the interfacial tension results in a change in the characteristic wavelength of EHD instabilities at a liquid/liquid interface. The reduced wavelength and the similarity in the viscosity of the two liquids (polymer/polymer) translate into the electrostatic pressure to act on a different time scale. Consequently, the interfaces are destabilized at different times, leading to a lateral redistribution of both liquids. Thin films of poly(methylmethacrylate) (PMMA) (Mw = 90 kg/mol, 150 nm) and polystyrene (PS) (Mw = 100 kg/mol, 100 nm) were spin-coated from toluene solutions. The PS film was floated onto a pool of deionized water and then transferred onto the PMMA layer to form a PS/PMMA bilayer. The capacitor assembly containing bilayer was heated to 170°C for 24 h at an applied voltage of 50 V (~108 V/m).

Fig. 7. Model of hierarchic structure-formation process. (a) The polymer-air surface is destabilized and the initial instability results in the column formation. (b) The bottom polymer layer is deformed. (c) In a secondary instability, the deformation of lower layer is enhanced, driving the polymer upward. (d) Finally, the polymer of the lower layer has formed a mantle around the primary columns.


129

In this system, the lateral distribution of the instabilities in the polymer is similar to that seen in single-layer polymer films. But columns produced with a bilayer showed a distinct rim — a subtle difference from those prepared from a single-layer film. The columns after removal of PS revealed that the composition of the columns consisted of a cylindrical PS core, surrounded by PMMA. The structure of the columns can be understood in terms of a sequential electrostatic destabilization of the polymer bilayer, as depicted in Fig. 7.5 The PS layer destabilizes initially, forming columns that span to the upper electrode. During the formation of the columns, the lower layer is deformed at the liquid/liquid/air contact region. These deformations nucleate a secondary instability that causes the PMMA to be drawn upward around the PS columns. More specifically, the balance of forces at both interfaces has to be considered. Because there are two deformable dielectric interfaces (that is, the PS–PMMA interface and the PS–air interface), both interfaces are destabilized by the electric field. The electrostatic pressure at the PS–air surface is greater than at the PS– PMMA interface, but this difference in the destabilizing force is compensated by a larger surface tension at the PS–air surface. The growth of the instabilities at the PS–PMMA interface is damped more strongly and is slower in comparison to the free surface, which destabilizes after a few hours. Hence the PS layer is destabilized by the electric field during the initial phase of the film instability. Initially, fluctuations of the PS–air surface develop, leading to the formation of PS columns spanning to the top electrode. During the PS column-formation, PS moves over the PMMA and deforms the interface between the two polymers. The destabilizing driving force is largest at the peaks of the PMMA cusps, which are adjacent to the PS columns. As a consequence, the electrostatic force leads to an increase in the PMMA cusp height, that is, the PMMA is drawn upward along the perimeter of the PS columns. The final morphology consists of PS columns coated by a PMMA layer. The electrohydrodynamic instabilities of PMMA/PS/air were further examined. Lin et al. investigated the PMMA (Mw=95 kg/mol, 228nm)/PS (Mw=96 kg/mol, 284 nm)/air interface under different electric field strengths where the electrode spacing was varied.6 The growth of surface fluctuations and column formation in the upper PS layer on the

130


underlying PMMA layer could be seen by selective removal of the PS layer with selective solvent. After removing the PS layer with cyclohexane, a line scan of the PMMA surface, corresponding to the interface between the PS and PMMA, was featureless. Consequently, the PS/air interface deforms much more readily than the PMMA/PS interface. Qualitatively, the PS/air surface is expected to deform much more rapidly than the PMMA/PS interface, and the viscous damping for the deformation of a free surface is much smaller than that of a polymerpolymer interface. On the other hand, the PS/PMMA interfacial tension is lower, facilitating interfacial deformation. As an extension of the previous study, Lin et al. characterized the structure formation at the interface of liquid/liquid bilayers.7 A good agreement between theory and experiment was found over many orders of magnitude in reduced wavelength and field strength using no adjustable parameters regardless of the polymers. Three types of bilayer assemblies were employed. These systems are PS (Mw=30 kg/mol, 550 nm)/PDMS (η=10,000 cSt, 570 nm)/air, PS (Mw=96 kg/mol, 730 nm)/PMMA (Mw=27 kg/mol, 290 nm)/air, and PMMA (Mw= 27 kg/mol, 290 nm)/PDMS (η=10,000 cSt, 730 nm)/air, respectively. The PS/PDMS bilayer after 1 day at 170°C under a 50V showed columns of PS through the upper PDMS layer. The distribution of the center-to-center distances of adjacent columns was determined and the average separation distance was 12.9 µm with a full width at half maximum (fwhm) of 1.86 µm. The electrohydrodynamic instability at the interface between two polymers under an applied voltage U across two electrodes separated by a distance d causes an amplification of fluctuations of a characteristic wavelength λ. The wavelength is given by 1

3

 γ 12  2  d - h0 h0  2 2π λ= -     1 1  ε 0   ε1 ε 2  U

(23)

ε 2 ε1

where d and h0 are the thickness of polymer 1 and polymer 2 with dielectric constants ε1 and ε2, respectively. γ12 is the interfacial tension between polymer 1 and 2, and ε0 is the dielectric permittivity in a


131

vacuum. Substituting the parameters for the PDMS/PS bilayer into Eq. (23) yielded a characteristic distance of 15.8 µm, which agreed well with the 12.9 µm value measured experimentally. In a second set of experiments, the upper PDMS layer was replaced by a PMMA layer to form PS/PMMA bilayer. Since the interfacial tension between PS and PMMA (1.7 mN/m at 170°C) is smaller than that for a PS/PDMS bilayer (6.1 mN/m at 170°C) at any given temperature, it would be expected that the characteristic wavelength would decrease. PMMA columns formed within the PS layer. Experiments on PMMA/PDMS bilayers showed essentially the same behavior as that of the PS/PDMS bilayers. The film thickness of underlying PMMA layer was much thinner than that of the upper PDMS layer, therefore the PMMA columns were obtained. Most importantly, there was excellent agreement between experiment and theory over 4 orders of magnitude in the reduced wavelength and reduced field strength. A master curve (Fig. 8) was shown to describe the results from a wide range of systems over many orders of magnitude in reduced field strength and distance with no adjustable parameters. Using the liquid/liquid/air system, it was possible to create three dimensional hierarchical structures. The formation of PS pillars in the PMMA/PS/air trilayer was accompanied by a smaller deformation in the PMMA film, resulting in a rim at the PS/PMMA contact line at the base of the pillars. This approach to layered, hierarchically ordered structures has the advantage that it results in significantly reduced feature sizes in the pattern after the PS component is removed with a selective solvent. Leach et al. investigated the inverted configuration of a PMMA film on a PS coated substrate.9 One might expect the dynamics of the PS/PMMA/air trilayer system to mimic the PMMA/PS/air system, with PMMA forming an array of pillars on top of the underlying PS, followed by concentric growth of PS around the PMMA pillars. Surprisingly, though, the resultant structure produced resembled that of the PMMA/PS/air system. However, the mechanism by which the pillars formed was different due to changes in interfacial forces in the inverted configuration. The change in system dynamics resulting from a placement of PMMA in the middle of the trilayer is a downward electrostatic force at the PMMA/PS interface. At the PMMA/air interface, there is a force toward the upper electrode, since PMMA has a

132


Fig. 8. Master curve of the typical distance λ in varieties of thin film and bilayer experiments as a function of the electric field strength in layer 1 and layer 2. The different symbols corresponded to 10 data sets: (♦) PS/PMMA bilayer with hPS=730 nm; (d - hPS)PMMA=290 nm, U ) 30 V; () PMMA/PDMS bilayer with hPMMA=180-290 nm, (d - hPMMA)PDMS=690–1030 nm, U=19–50 V; () PS/PDMS bilayer with hPS=305 nm, (d - hPS)PDMS=400 nm and 720 nm respectively, U=50 V; () PS/PDMS bilayer with hPS=550 nm, (d - hPS)PDMS=570 nm and 700 nm respectively, U=50 V; (ο) PSBr/air single layer with hPSBr=740 nm, d=1.66–1.98 µm, U=20–60 V; () dPS/air single layer with hdPS=530 nm, d=1.06–1.85 µm, U=30 V. E1 and E2 are the electric field strength in layer 1 and 2. Other data sets (, ∇, , ) are measurements from ref. 8.

dielectric constant higher than air. This is akin to the force acting on the PS layer in the non-inverted system. The downward electrostatic pressure at the PMMA/PS interface is due to PMMA having a larger dielectric constant than PS. In addition to altering the electrostatic forces, the inverted trilayer system generated an interface that can dewet. The PMMA/PS/air trilayer has one dewetting interface, PS/PMMA. In contrast, the inverted system of PS/PMMA/air has two interfaces that energetically favor dewetting, the PS/PMMA interface and the PS/silicon substrate with 2 nm thick oxide layer. Thus, the dewetting of the bottom PS layer is highly favored. The electrostatic forces and the interfacial energies work in tandem to generate the structures. The structures formed when the electric field was applied across a PS (Mw=8 kg/mol, 500 nm)/PMMA (Mw=32 kg/mol, 500 nm)/air trilayer were such that PS formed the core that was sheathed by PMMA.9 Since PMMA has a higher dielectric constant than both PS and air, the


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PMMA/air interface is pulled toward the upper electrode and the PS/PMMA interface is pulled toward the lower. When the molecular weight of PMMA was increased to 99 kg/mol to slow down the kinetics in the growth of the instabilities, it was found that fluctuations at the PS/PMMA interface were increasing in magnitude and some were fully grown. When the molecular weight of polystyrene was changed to 157 kg/mol, increasing the viscosity 300 times, while keeping the molecular weight of the PMMA at 32 kg/mol, the kinetics of dewetting of PS on silicon surface under the electric field was retarded. Along with the electrostatic forces, the dewetting forces at two of the interfaces played a major role. The size and spacing of the structures were controlled by the electrostatic forces, while the dewetting kinetics dictated the interface at which the instabilities grew. The electrohydrodynamic instabilities on PMMA/PS/air bilayer system generated a very unique cage-type structure (Fig. 9(a)) as reported by Russell and coworkers.10 Around the ring of PMMA strands, there was an elevated rim of PMMA that corresponded to the diameter of the columns formed by the PMMA (Mw=25 kg/mol, 1000 nm)/PS (Mw=200 kg/mol, 700 nm) bilayer. A cross-sectional SEM image of a cleaved cage-type structure is shown in Fig. 9(b) where the diameter of the cage is 16.5 µm with a height of 4 µm. The structure can be understood in the context of electrohydrodynamic instabilities. As shown previously, the upper PS layer formed columnar structures on the lower layer of PMMA. After the growth of the PS structures, the contact line at the PS/PMMA interface was locally deformed, forming a rim on the PMMA surface. In addition to surface tension, viscous stresses, due to lateral movement of the contact line during growth of the PS structures, led to a deformation of the contact line. Subsequently, the electric field strength at the top of the rim was enhanced by the curvature of the structures. This led to fluctuations along the rim to grow with a characteristic wavelength around the rim. The fluctuations were amplified causing the formation of columns around the PS columns. They observed the growth of a fingering instability along the circumference of the rim. This fingering instability arose from the flow of the thin film of PMMA on the PS pillar under the influence of electric field stresses. Because PMMA has a small contact angle (23°) with PS,

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a

b

Fig. 9. (a) SEM image of a single “cage”. (b) Cross-sectional SEM image of a cleaved single “cage”.

flow of PMMA induced an instability and led to the formation of fingers perpendicular to the direction of flow as shown in Fig. 3(b). Once the fingering instability formed, the electric field amplified the instability and caused the formation of strands around the circumference of the existing PS columns. 4. Pattern Formation in Thin Polymer Films under Electric Field So far, extensive experimental and theoretical approaches have been reported for the understanding of electrohydrodynamic instabilities of polymer thin films. Recently, Voicu et al. conducted a study on polymer pattern formation induced by electrohydrodynamic instabilities.11 The study investigated development of polymer morphologies as a function of time. It was found that the initial phase of the pattern formation process was a sinusoidal surface undulation, irrespective of the sample parameters. The later stage depended on the relative amount of polymer in the gap between the electrodes. The onset of the instability is a low-amplitude sinusoidal undulation with a wavelength shown in Fig. 10. As the instability evolves, other mechanisms come into play. In this case, an important parameter is the ratio of film thickness to gap spacing. Figure 10 shows the case for a


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Fig. 10. Electrohydrodynamic pattern formation in a homogeneous electric field. (a) Schematic representation of the capacitor setup. A brominated PS film (125 nm) was deposited onto an indium tin oxide covered glass slide. (b) Optical microscope images of an instability in a 125 nm film at 164°C. (c) Computer simulation study by Verma et al. for filling ratio of 0.5.

polymer (brominated PS) film with a thickness h ≈ 125 nm and a plate spacing d ≈ 255 nm, corresponding to a filling ratio of f = 0.49 ± 0.1. In Fig. 10(b), the film initially develops a wave pattern, which is visible in the wave evolution time of 682 min (t = 0 corresponds to the earliest time a surface wave can be optically discerned). The wave pattern is amplified and, with time, columns are formed. Figure 10(c) shows the simulation results by Verma et al.13 for f = 0.5. While the initial process is very similar, the late-stage distribution of the columns is different, showing a more irregular distribution of the columns. The sample after

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t = 270 min shows that some of the columns have coalesced to form larger and elongated structures. This is reproduced by the simulation results. With time, coarsening occurs and the columns fully disconnected from each other at a later stage. The amount of column coalescence during pattern formation increases with increasing filling ratio. Electrohydrodynamic pattern formation can be spatially controlled by introducing a heterogeneity into the electric field. One simple case is a single elevated point protruding from the top electrode. Figure 11 shows the progress of an instability initiated at a single point. The nucleated column is surrounded by a radial wave propagating outward from the nucleation point (0 min). The rim develops a lateral undulation, leading to the formation of columns along the rim (7–70 min). Once the first shell of columns is complete, this process is continued radially outward: rims surrounding the columns lead to the nucleation of further columns. For sufficiently long times, this nucleated column formation process competes with the pattern formation caused by the homogeneous field.

Fig. 11. Nucleated pattern formation in an electric field. The images show a 125 nm PSBr film (filling ratio 0.5) at 10 V at 164°C.


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Steiner and colleagues also conducted another novel experiment. It was shown that annealing of polymer films in a solvent vapor during pattern formation is a versatile and robust approach.12 In particular, the low viscosity of the swollen polymer significantly reduced the time required for pattern replication. By tuning the interplay of the intrinsic wavelength and the periodicity of the template patterns, two different replication modes could be selected. By superposing a lateral variation of the electrode spacing with the pattern-selection process, it was possible to switch between the two pattern-replication modes in a single set of experiments. It was shown that the filling ratio was a critical parameter. 5. Electrohydrodynamic Instabilities with Patterned Electrode The preferred induction of electrohydrodynamic instabilities at locations of highest electric field lies at the base of the electrohydrodynamic lithography. A common methodology is the use of topographically patterned top electrodes. To explore the process of pattern replication, Voicu et al. studied electrohydrodynamic instabilities induced by a topographic line grating as the top electrode.11 Figure 12 shows the instability produced by an electrode with an array of lines. After annealing for 108 min (t=0) undulations appear under the lines protruding from the electrode surface. With time the columns make contact with the electrode, resulting in linear arrays of columns. Figure 12(b) shows, in comparison, the simulations by Verma et al.13 for a system with filling ratio of 0.5. The simulations are qualitatively similar to the experimental results. Since the wavelength of the intrinsic instability is larger by a factor of two compared to the periodicity of the grating in Fig. 12(b), only every second line of the grating is replicated. An intriguing study was conducted by Wu and Chou on the effect of non-Newtonian fluid motion of polymer on the electrohydrodynamic instabilities under patterned mask.14 Lithographically induced selfassembly (LISA), is an electrohydrodynamic instability process, in which a thin layer of melted polymer (usually PMMA) self-assembles into well organized pillar arrays that bridge the lower substrate and the

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Fig. 12. Nucleation of columns induced by an electrode with an array of lines. (a) A 125 nm PSBr film (filling ratio 0.25) annealed under 30 V at 170°C. (b) Simulation results of Verma et al. for filling ratio of 0.5.

upper mask. The electrostatic forces, which are accumulated at the air– polymer interface, place the interface under tension. Under certain conditions, the instabilities are well organized and micrometer sized hexagonal pillar arrays form. Previous studies assumed the fluid motion to be Newtonian, ignoring the non-Newtonian effect of the polymer melt. Wu and Chou investigated the effects of polymer elasticity on the instability. They assumed the fluid motion to be non-Newtonian and to obey the Oldroyd B constitutive equation. The Oldroyd B model is one


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of the simplest models capable of describing, at least qualitatively, the rheological behavior of dilute polymer solutions. The main findings are that viscoelasticity plays a significant role in the instability process and the polymer elasticity destabilizes the system. Extensive research on the electrohydrodynamic instabilities under topographically patterned electrode has been done by Pease and Russel.15-16 They created diverse types of patterned electrode such as circular, triangular, and square geometry. Here, the mask was held in close proximity to the polymer surface, leaving an air gap, and the system was heated above the glass transition temperature. By using a patterned mask, the location and domain orientation of the structures can be well controlled. While the fundamentals of this patterning process under homogeneous electric field are reasonably well understood, recent cylindrically symmetric structures challenge the existing theory. The literature contains several examples of these structures. Chou detailed two examples of concentric rings with four to five rings each surrounding a central pillar. Schäeffer et al. have shown a rosette in which 12 pillars circumscribe a central pillar.1 They suggested that a locally amplified electric field produced the rosette. Subsequent work by the same group indicated that a competition between hydrodynamic flow and dewetting may play a role in the rosette features. Pease and Russell examined the experiments and experimental conditions that gave rise to some of the ringlike features. Those results were compared to an electrohydrodynamic model of the process capable of describing these cylindrical structures.15 Thin films of poly(methyl methacrylate) (Mw=2 kg/mol, 90 nm) were annealed under an electric field. In this system, the mask was patterned with protrusions ranging in height from 10 to 40 nm using photolithography. Each protrusion was cylindrical with a diameter of 3 mm. The protrusions, which do not initially contact the polymer surface, act as a nucleation points for ring development. Completed concentric rings, in which the central pillar was clearly visible, were observed after exposure to electric field. The annular width of the rings was 1 mm and the ring-to-ring spacing was 3 mm. The rings were 170 nm high and took 60 min to form at 95°C. Up to 10 fully formed rings have been seen around a single central pillar. More rings form as

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the dwell time increased. An array of single rings formed under a mask with 40 nm high protrusions spaced 40 mm apart. The rings took 30 min to form at 90°C. Each set of rings centered on a protrusion patterned in relief on the mask, but did not show the central pillar that would be expected beneath the mask. The polymer tends to climb the sidewalls of the protrusion and is torn away when the mask is removed. Theoretical considerations confirmed that ring diameters and annular widths followed predicted trends as a function of the ring number and mask-substrate separation. The ring-to-ring distances were constant for each concentric cluster. However, the ring-to-ring spacing did not vary systematically with the measured height. The other meaningful finding was that complete rings formed when the temperature was between the glass transition temperature and ~120°C. For temperatures ≥120°C, closed rings did not result. Instead ring segments near the center and pillar arrays were observed farther out. In summary, they have presented sets of concentric rings obtained experimentally and developed an electrohydrodynamic description of ring and ringlike structures based on the perfect dielectric model in cylindrical coordinates. Subsequently, Russel and coworkers performed a systematic study on the large-scale alignment of electrohydrodynamic patterning with a geometrically controlled mask electrode.16 With a featureless mask, hexagonal arrays of pillars generally form since their nonlinear growth rate is faster than that of other patterns. A significant challenge to the implementation of bottom-up approaches to patterning surfaces via selfassembly or instabilities is the natural tendency to form multidomain structures. Like other patterns induced by hydrodynamic or chemical instabilities, the spatial ordering of the pattern is generally very hard to extend over large areas. It was suggested that an alternative approach to overcoming this limitation was to use patterned masks. When periodic protrusions in the form of gratings are scribed into the mask, the resulting alternation of the electric field shifts the spacing from the natural wavelength of the instability and guides the polymer into structures that can conform completely the pattern on the mask. To take full advantage of the capability of the electrohydrodynamic patterning under featureless masks and retain the goal of creating ordered patterns


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over large areas, they proposed a strategy for arranging relatively simple patterns on the mask. Figure 13 shows the surface patterns obtained after using a triangular mask with a polymer film (PMMA Mw=2 kg/mol, 45 nm), annealing for 3 h with an applied voltage of 10 V. Regular rows of pillars form under the ridges, and ordered triangular arrays are generated within each individual triangular domain bounded by the ridges. The optical microscope image in Fig. 13(c) shows an ordered pattern spanning more than 100 periods in both the x and y directions, which is the largest array of ordered pillars from electrohydrodynamic patterning available in the literature. The formation of pillars guided by ridges spaced widely enough to allow the natural growth of pillars, but close to the correlation length to preserve uniform arrays within each unit cell, was investigated.

Fig. 13. Optical microscope images of a 45 nm PMMA (2K) film annealed for 3 h at 130°C under 10V. The array contains pillars in a larger triangular network within which three (a), six (b), and ten (c) triangular packed pillars are enclosed.

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The same experimental strategy was applied for the case of square patterned mask. More than 1000 pillars were aligned into square arrays. This is particularly significant because the natural domains are hexagonal and typically preserve order over only seven periods. It was demonstrated that by proper design of the mask, the limitation of the natural domain size can be overcome to achieve ordered patterns over large areas. 6. Theoretical Approaches to Electrohydrodynamic Instabilities Along with the remarkable progress in the experimental investigation of electrohydrodynamic instabilities of thin films, theoretical advances have also been made. The most prominent work was conducted by Pease and Russel.17-22 A part of the research effort was devoted to the identification of mechanism that is responsible for the pattern formation. The previous results of theoretical research indicated that: (1) the wavelength decreases inversely with the square root of electric field strength in accord with the linear stability analysis, (2) the pattern conforms to the geometry of the mask, and (3) replacing air with a nonconducting liquid accelerates the process and decreases the period by reducing the interfacial tension. In a subsequent study, Pease and Russel addressed the need for further basic understanding of the mechanism.17 Computations based on a lubrication approximation, which assumes the wavelength of the fluctuations to be large relative to the gap, provide a convenient means of exploring the coupling between hydrodynamic stresses and the electric fields. The forces guiding the evolution of the film into the periodic microstructures are still not well understood. The other question is how important conductivity in the polymer is. Given the apparent importance of free charge conductivity in the film, the “leaky dielectric model” was adopted, which allowed for accumulation and redistribution of charges on the interfaces. They undertook a linear stability analysis to ascertain the dependence of the growth exponent and characteristic wavelength on the conductivity and film thickness. The analysis showed that the growth exponents and characteristic wavelengths were much larger than those for the perfect dielectric model. These moved the resulting estimates for length scales and growth


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times closer to those observed experimentally, though a discrepancy still existed between observations and predictions for the length scales. Pease and Russel found that the lubrication approximation failed when the surface tension was small and electric fields were large, typical of experiments with a polymer/organic liquid instead of air in the gapprecisely the conditions that generate the smallest pillar arrays. They adopted a general linear stability analysis to predict conditions where pillars/holes pack more tightly, have smaller diameters, greater aspect ratios, and larger growth exponents for both perfect and leaky dielectric films, in which the smallest features reach deep into the submicron length scales.18 In order to substantiate their findings, they cited experimental results reported by Lin et al.8 about polyisoprene/air (140 nm, dielectric constant 2.37, surface tension 32 mN/m, 20 V). The analysis highlighted two approaches to smaller pillars by either making the polymer film conducting or choosing polymers with high dielectric contrast. After examining the initial stages of electrohydrodynamic process under pattern-free masks by deriving a generalized linear stability analysis not restricted to the lubrication approximation, Pease and Russel compared this model to experimental data from the literature and found good agreement over a wide range of conditions including applied voltages and oxide layers on the mask and substrate.19 A significant discrepancy at the highest fields was seen due, possibly, to dielectric breakdown, suggesting that the minimum feature size may be limited. Viscous effects may also limit the effectiveness of large decreases in surface tension or large increases in the electric field, leading to lower limits of the feature sizes. Long-range ordering seems to decrease as surface tension decreases and the potential increases, indicating that smaller pillars come with decreased quality. The study presented a detailed comparison of generalized model, which contained only measurable parameters, with data from the literature and discussed the implications regarding the minimum feature size, long-range order, and process time scales. The analysis led to the understanding that dielectric breakdown limits the applied potential, and that a reduction in the surface tension might not decrease spacing much below mask-substrate separations. In addition, large fields might also disrupt long-range order.

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Based on this theoretical analysis, Pease and Russell also performed simulations on the structural evolution under a planar patterned mask.20 In addition, they examined the thermodynamics and kinetics of electrohydrodynamic instabilities in dielectric bilayers.21 They constructed a parametric map, depending on the dielectric contrast and ratio of two film thicknesses, that described the conditions under which hexagonally ordered pillars or holes formed when the viscosity of the upper layer is negligible. It was shown that the distinct formation of arrays of pillars and holes resulted from nonlinear interactions among different modes and, hence, was governed by the kinetics. The dynamic structures of pillars or holes continued to evolve and individual pillars or holes coalesced in a coarsening process until a stable state was reached in the form of a localized pillar, hole, or a roll structure. The selection of the pillar or hole at the final steady state represented a thermodynamic preference that could be predicted qualitatively. All experiments indicated that the dielectric contrast between neighboring fluids, the film thickness ratio of the lower and upper layers, and their viscosities could significantly influence the patterns formed. The coupling of kinetics and thermodynamics produced intriguing patterns and phenomena of both theoretical importance and practical interest.22 7. Block Copolymers under Electric Field: Electrohydrodynamic Instabilities and Microdomain Orientation Thin films of block copolymers can be manipulated on two different length scales simultaneously by use of an electric field. Electrostatic pressure generated at the surface of a block copolymer film between two electrodes with an air gap, as in homopolymer films, an array of hexagonally ordered columns, tens of microns in size. Within each column the diblock copolymer microphase separated into hexagonally packed cylindrical microdomains, tens of nanometers in size. The orientation of these microdomains was controlled by the interfacial energies of each block with the surfaces of the electrode and the direction of the applied field. Microdomain alignment parallel to and normal to the applied field could be controlled by the strength of the interfacial interactions.


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Xiang et al. employed three types of block copolymers polystyreneblock-poly(2-vinylpyridine) (PS-b-P2VP, 70 kg/mol-30 kg/mol), polystyrene-block-polyisoprene (PS-b-PI, 26 kg/mol-10 kg/mol), and polystyrene-block-polybutadiene (PS-b-PBD, 16 kg/mol-38 kg/mol), respectively.23 Thin films of block copolymers (600 nm) were annealed for 24 h at 170°C under 30 V. A hexagonally packed array of columns was observed by optical microscopy, which protruded from an underlying thin PS-b-PVP film. The columns were well-ordered, and no defect in the packing of the columns was found over a 500 µm × 500 µm area. The ordering of the column array originates from an electrostatic repulsion between the columns as they grow from fluctuations on the film surface. In order to study the effect of electric field on the microdomain orientation of block copolymer, the microtomed TEM study was conducted. Sections for PVP cylindrical domains indicated that the PVP cylinders were oriented parallel to the substrate surface rather than parallel to the electric field direction. Such a domain orientation was also observed for the PS-b-PI columns grown under the same experimental condition but at 125°C. In the case of PS-b-PBD columns, however, the influence of the electric field on the microdomain alignment was observed. The characteristic times required to produce columns bridging the two electrodes are normally less than several hours. While the microphase separation of the block copolymer occurs rapidly, achieving domain alignment required a much longer time to reach a fully aligned state. External forces such as those induced by flow and surfaces have been shown to cause alignment of block copolymer microstructure. Alignment of microdomain by an electric field offers the possibility of producing block copolymers with tailored anisotropic properties; e.g., spatiallyspecific anisotropic properties could be imposed by a localized field. In addition, the response of block copolymer microstructure to a electric field should reveal large scale properties of block copolymers like defect mobility. In a series of papers, Amundson reported the kinetics and mechanism of block copolymer microdomain alignment.24-25 Since then, electric fields have become an effective mean to orient nanoscopic domains

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laterally in thin copolymer films. Thurn-Albrecht et al. found the threshold electric field strength, above which complete orientation of the cylindrical microdomains, using asymmetric diblock copolymers of polystyrene and poly(methyl methacrylate).26 The threshold field strength was independent of film thickness and could be described by the difference in interfacial energies of the components. At field strengths slightly below threshold, a coexistence of the domains parallel and perpendicular to the electrode surface was found. An extensive experiment was performed by Russell and colleagues to understand microdomain orientations under applied electric fields.27-32 It was revealed that the alignment of microdomains by electric field depends on the segmental interaction between two blocks and the difference in the interfacial energies of each block with the substrate. By modifying a surface with random copolymer brushes, interfacial energies were controlled and the influence of interfacial energy on the orientation of the copolymer microdomains by an electric field was examined.27 A complete alignment of the lamellar microdomains was achieved only when the interfacial interactions were balanced. In addition, the use of two orthogonal, external fields was shown to control the orientation of lamellar microdomains in three dimensions in diblock copolymer thin films.28 An elongational flow field was applied to obtain an in-plane orientation of the microdomains of the copolymer melt, and an electric field applied normal to the surface was used to further align the microdomains. A study on the effect of ionic impurities on the electric field alignment of lamellar microdomains in polystyrene-blockpoly(methyl mathacrylate) thin flims showed that the microdomains could be aligned in the direction of electric field regardless of the strength of interfacial interactions at lithium ion concentrations greater than 210 ppm.29 A more quantitative investigation on the microdomain alignment of symmetric diblock copolymer of polystyrene and poly(methyl mathacrylate) was performed as a function of film thickness and interfacial energy.30 For films with thickness t10L, interfacial


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interaction became less important, and lamellar microdomains in the center of the films could be oriented in the direction of electric field. For a asymmetric polystyrene-block-poly(methyl mathacrylate), an electric field induced sphere-to-cylinder transition was observed.31 Under electric field, the asymmetric diblock copolymer formed spherical microdomains that were deformed into ellipsoids and, with time, interconnected into cylindrical microdomains oriented in the direction of applied field.32 References 1. E. Schaeffer, T. Thurn-Albrecht, T. P. Russell, U. Steiner, Nature, 874 (2000). 2. E. Schaeffer, T. Thurn-Albrecht, T. P. Russell, U. Steiner, Europhys. Lett., 518 (2001). 3. K. A. Leach, Z. Lin, T. P. Russell, Macromolecules, 4868 (2005). 4. L. Oddershede, S. R. Nagel, Phys. Rev. Lett., 1234 (2000). 5. M. D. Morariu, N. E. Voicu, E. Schaeffer, Z. Lin, T. P. Russell, U. Steiner, Nature Mater., 48 (2003). 6. Z. Lin, T. Kerle, T. P. Russell, E. Schaeffer, U. Steiner, Macromolecules, 6255 (2002). 7. Z. Lin, T. Kerle, T. P. Russell, E. Schaeffer, U. Steiner, Macromolecules, 3971 (2002). 8. Z. Lin, T. Kerle, S. M. Baker, D. Hoagland, E. Schaeffer, U. Steiner, T. P. Russell, J. Chem. Phys., 2377 (2001). 9. K. A. Leach, S. Gupta, M. D. Dickey, C. G. Wilson, T. P. Russell, Chaos, 047506 (2005). 10. M. D. Dickey, S. Gupta, K. A. Leach, E. Collister, C. G. Wilson, T. P. Russell, Langmuir, 4315 (2006). 11. N. E. Voicu, S. Harkema, U. Steiner, Adv. Funct. Mater., 926 (2006). 12. S. Harkema, U. Steiner, Adv. Funct. Mater., 2016 (2005). 13. R. Verma, A. Sharma, K. Kargupta, J. Bhaumik, Langmuir, 3710 (2005). 14. L. Wu, S. Y. Chou, J. Non-Newtonian Fluid Mech., 91 (2005). 15. P. Deshpande, L. F. Pease III, L. Chen, S. Y. Chou, W. B. Russel, Phys. Rev. E, 041601 (2004). 16. N. Wu, L. F. Pease III, W. B. Russel, Adv. Funct. Mater., 1992 (2006). 17. L. F. Pease III, W. B. Russel, J. Non-Newtonian Fluid Mech., 233 (2002). 18. L. F. Pease III, W. B. Russel, J. Chem. Phys., 3790 (2003). 19. L. F. Pease III, W. B. Russel, Langmuir, 795 (2004). 20. N. Wu, L. F. Pease III, W. B. Russel, Langmuir, 12290 (2005). 21. N. Wu, W. B. Russel, Ind. Eng. Chem. Res., 5455 (2006). 22. L. F. Pease III, W. B. Russel, J. Chem. Phys., 184716 (2006).

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23. H. Xiang, Y. Lin, T. P. Russell, Macromolecules, 5358 (2004). 24. K. Amundson, E. Helfand, X. Quan, S. D. Smith, Macromolecules, 2698 (1993). 25. K. Amundson, E. Helfand, X. Quan, S. D. Hudson, S. D. Smith, Macromolecules, 6559 (1994). 26. T. Thurn-Albrecht, J. DeRouchey, T.P. Russell, H. M. Jaeger, Macromolecules, 3250 (2000). 27. T. Xu, C. J. Hawker, T. P. Russell, Macromolecules, 6178 (2003). 28. T. Xu, J. T. Goldbach, T. P. Russell, Macromolecules, 7296 (2003). 29. T. Xu, J. T. Goldbach, J. Leiston-Belanger, T. P. Russell, Colloid Polym. Sci., 927 (2004). 30. T. Xu, Y. Zhu, S. P. Gido, T. P. Russell, Macromolecules, 2625 (2004). 31. T. Xu, A. V. Zvelindovsky, G. J. A. Sevink, O. Gang, B. Ocko, Y. Zhu, S. P. Gido, T. P. Russell, Macromolecules, 6980 (2004). 32. T. Xu, C. J. Hawker, T. P. Russell, Macromolecules, 2802 (2005).

Chapter 5 Electrowetting: The External Switch on the Wettability and Its Applications for Manipulating Drops

Frieder Mugele Department of Science and Technology, University of Twente P.O. Box 217,7500AE Enschede, The Netherlands E-mail: [email protected] In this chapter we discuss the basic principles of electrowetting in equilibrium conditions as well as two examples of dynamic electrowetting. We show that the electrowetting phenomenon is caused by the balance between the electrostatic stresses – the Maxwell stress – acting on the drop surface and the surface tension forces, which can be viewed as a reduction of the effective solid-liquid interfacial tension on a large scale. With respect to dynamics, we show that AC electric fields substantially reduce the contact angle hysteresis typically encountered on any solid surface in ambient air. For electrowetting in an ambient oil environment, we discuss the displacement of the oil by the aqueous drop that is moving under the influence of the electric field. Furthermore, we explain the basic principles used for the operation of various electrowetting-driven microfluidic devices, in particular lab-ona-chip systems.

1. Introduction The generation and manipulation of liquid drops has become a new paradigm of research in the area of microfluidics in recent years (for recent reviews, see [1-6]). Compared to conventional continuous flow microfluidic systems, droplet based systems have the advantage of offering large numbers of discrete container of liquid with wellcontrolled composition. Slowly varying that composition (or other reaction conditions) from drop to drop allows for mapping large 149

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parameter spaces in combinatorial chemical or physical problems, such as drug screening or protein crystallization. Alternatively, in bio-medical applications, one may want to perform a large variety of tests by combining small drops of a body fluid with various test reagents. Two main physical approaches have been used to achieve the manipulation of drops. The first one makes use of closed microfluidic channels. The forces applied to generate and manipulate the drops are exclusively on hydrodynamic. Two immiscible liquids flow through the more or less complex network of microfluidic channels, driven by external pressure. Typically, the continuous phase consists of oil and the drop phase is the aqueous solution of interest. While these systems offer the advantage of a very high throughput, the degree of control over the individual drops is rather limited. Alternative platforms are based on the manipulation of individual drops by switchable external forces such as surface acoustic waves, temperature gradients [7], and electric fields, in particular electrowetting [5, 6]. These systems have the advantage of working with open flat substrates without requiring the explicit definition of channels (notwithstanding the fact that actual devices frequently consist of sandwiches of two parallel substrates in order to reduce evaporation.) This offers the opportunity of reconfiguring a specific device for several applications by simply reprogramming the way in which the actuators (e.g. electrodes) are addressed. These systems offer exquisite control over each individual drop, however, the high throughput capabilities are limited, which implies somewhat different target applications [5], also outside the classical area of microfluidics [8-11]. Recently, first attempts were presented that combine the strength of both platforms by integrating electrical functionality into continuous flow microfluidic channels [12]. In this chapter, we will focus on the physical principles underlying the electrowetting effect and its applications. We will give first an introduction into the origin of electrowetting (Sec. II), then discuss some physical challenges involved in typical EW devices (Sec. III), and finally (Sec. IV) present two recent examples from our laboratory addressing physical problems related to the dynamics of contact lines in electrowetting, namely contact angle hysteresis (in ambient air) as well as contact line dynamics in ambient oil.

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2. Origin of the Electrowetting Effect 2.1. Historical background Historically, the origin of EW can be traced back to the work of Gabriel Lippmann in the late 19th century, although strictly speaking wetting not the topic of his frequently cited work from 1875 ([13], see for [6] an English translation). Lippmann was performing capillary rise experiments using mercury and sulphuric acid as the two liquids competing to fill a glass capillary. Both liquids in the tube were connected to bulk reservoirs in glass beakers, into which he immersed electrodes. Lippmann noted that the height of rise (or depression) of the Hg-H2SO4-meniscus inside the capillary varied systematically as he applied a voltage between the two liquids. Since the Hg always formed a perfect 180° contact angle with the glass, he could simply use Laplace law and the density difference to extract a voltage-dependent interfacial tension σ (U). He found that σ displayed a maximum at what we call now the potential of zero charge Upzc, and that it decreased approximately parabolically upon increasing or decreasing the voltage away from Upzc. This is the electrocapillary effect that Lipmmann discovered. 2.2. Modern electrowetting Modern electrowetting experiments make use of Lippmann’s ideas. However, there are also important differences. While it is possible to vary the contact angle in systems with a direct metal-electrolyte contact [14], these systems have the disadvantage of being rather delicate to handle because they are electrochemically unstable and prone electrolytic decomposition of the aqueous phase if the applied voltage exceeds a threshold that is typically below 1V. To avoid this problem, Berge suggested in 1993 [15] to separate the aqueous drop from the metallic counter electrode by introducing an insulation dielectric layer. This was the birth of “electrowetting on dielectric”, the basis of all modern applications of EW. The generic configuration (see Fig. 1) thus consists of a flat electrode covered by an insulating layer that is either intrinsically hydrophobic (such as the popular Teflon AF layers with a typical thickness d of a few micrometer) or that is covered by a thin

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Fig. 1. A schematic illustration of a generic EW setup.

Fig. 2. Typical electrowetting curve, cosine of contact angle vs. applied voltage. (specific system: aqueous drop in ambient silicone oil; substrate: Teflon AF).

hydrophobic top coating (e.g. a hydrophobic self-assembled monolayer or a nanometer thin Teflon AF layer). At zero voltage, a sessile drop on such a surface displays a large Young contact angle or order 120° in air — or close to 180° in systems with ambient oil. In the simplest way, electrical contact is provided to the drop by immersing a Pt wire. When a voltage is applied between the drop and the electrode on the substrate the contact angle decreases. Provided that the voltage is not too high, the decrease in contact angle is such that the cosine of θ increases quadratically with U as shown in Fig. 2. At higher voltage, the voltagedependence gradually saturates until θ eventually becomes independent of U. Qualitatively, this saturation behavior is related to non-linearities and instabilities in the materials properties at high voltage. Quantitatively, however, it is poorly understood and several competing explanations were proposed (see [6] for a discussion). In the following, we will focus on the low-voltage behavior, which is well-understood and which forms the basis of all applications of EW.

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The parabolic increase in cosθ is described by the electrowetting equation (sometimes also referred to as Lippmann-Young equation): cos θ = cos θ Y +

ε 0ε d 2 U = cos θ Y + η 2dσ lv

(1)

Here, we have introduced the dimensionless electrowetting number η = ε 0ε rU 2 / (2dσ lv ) , which measures the strength of the electrostatic energy compared to surface tension. (σlv denotes the interfacial tension between the drop and the ambient medium; ε0 is the vacuum permittivity and εd the dielectric constant of the dielectric layer.) There are two ways to understand the origin of the EW equation. Obviously, the equilibrium state corresponds to a minimum of the total free energy of the drop at constant volume and constant voltage. Under these conditions, the total free energy is given by 1 G = ∑ σ i Ai − ∫ D ⋅ EdV (2) 2 i

where the σi and Ai denote the specific energies and the areas of the interface i=lv (liquid-ambient med.), sl (solid-liquid), and sv (solidambient med.). E and D denote the electric field and the electric displacement, respectively. The integral extends over the entire system bounded by the bottom electrode and by other boundaries at a distance much larger than the drop size. The drop is considered a perfect conductor (i.e. the electric field is completely screened from the interior of the drop), held at a fixed potential by an external battery. For the most popular approach to the problem, we first note that the insulator thickness d is typically much smaller than the drop size (see Fig. 3). In

Fig. 3. Electric field distribution in EW. The field energy is mostly concentrated in the parallel plate capacitor formed by the drop and the counter electrode (left). Electric fringe fields pull on the contact line (right).

F. Mugele

154

that case, the electric field is mostly concentrated in the insulating layer underneath the drop: the drop and the electrode on the substrate form a parallel plate capacitor. Ignoring the electrostatic energy in the stray fields along the edge of the capacitor (which are only linear in the drop size), we find thus G ≈ ∑ σ i Ai − i

ε 0ε r 2d

Asl U 2 .

(3)

Since the electrostatic energy is now reduced to a contribution that is proportional to Asl, it can be combined with σsl to an effective interfacial tension

σ sleff (U ) = σ sl −

ε 0ε d 2d

U2

(4)

which is indeed similar to Lippmann’s electrocapillary equation. (However, one should note that the electrostatic energy is distributed over the dielectric layer rather than being located within the Debye screening layer, as in Lippmann’s case. Inserting Eq. (4) into Young’s equation, one obtains directly the EW equation (Eq. (1)). From this perspective, EW thus leads to a modified version of Young’s equation). If we approach the minimization of Eq. (2) more systematically — or if we ask more physically — what are the forces that pull on the contact line to induce the observed reduction of the contact angle, we arrive at a more refined microscopic picture of the EW effect. Along the threephase contact line there will be strong local electric fields (the fringe fields of the parallel plate capacitor discussed above; see Fig. 3). These electric fields give rise to a Maxwell stress pulling on the liquid surface along the outward normal. For a perfect conduct, the electric fields are oriented normal to the surface, and hence the Maxwell stress, given by Pel = ε 0 E 2 / 2 , will appear as an additional contribution in the Laplace equation. The latter then reads

σ lv κ (r ) − Pel (r ) = ∆p = const.

(5)

Here κ is the local curvature of the drop surface. From this perspective, EW thus appears as a modification of Laplace’s equation rather than a modification of Young’s equation.

Electrowetting

155

Both approaches are of course equivalent; however, they refer to different scales. The equivalence can be rationalized best by considering the total force exerted by the electric fields. To calculate it, consider a closed box Σ around the contact line, as sketched in Fig. 4. According to its basic definition, the Maxwell stress tensor expresses not only the components of a stress acting on a surface, but also the momentum flux density of the electromagnetic field [16]. Hence, if we integrate the stress tensor over a closed surface, we obtain the net force exerted by the electromagnetic field on the material inside that box. The net force in the x-direction is given by the following integral

Fx = ∑ ∫ Txk nk dA

(6)

k Σ

Here, nk is the k-th component of the outward normal vector along Σ. If we choose the box sufficiently large such that the electric stray fields, which are localized somewhere around the contact line have decayed to zero along most of the segments of Σ. It is easy to see that the only remaining term originates from the short section along Σ between the solid-liquid interface and the electrode (see Fig. 4). That contribution is exactly equal to

Fx =

ε 0ε d 2d

U 2 = σ lvη .

(7)

as first shown by Jones [17]. This is again exactly the same value as in Eq. (4). Hence, if we consider the system on a scale that is sufficiently large for the fringe fields to be decayed on its boundaries, i.e. on a scale of a few times the insulator thickness d, we find that the net contribution of the electric field is identical. Hence, on that scale, both pictures are indeed equivalent, as they should be. In particular, knowledge about the details of the surface shape on small scales is not required to describe the macroscopic response of the system. (Note that this situation is similar to the one in conventional wetting, where molecular forces also lead to a distortion of the surface profile within the range of the disjoining pressure [18]. However, the macroscopic contact angle can still be obtained by simply balancing the surface tensions in the classical Young picture without knowing about the detailed shape on small scales.)

F. Mugele

156

Fig. 4. Integration box for calculating the net horizontal component of the electrostatic force. The thick section on the bottom left of Σ produces the only non-zero contribution.

(a)

(b)

(c)

Fig. 5. Water drop in silicone oil at zero voltage (a) and at a voltage corresponding to η ≈1 for an insulator thickness of 10µm (b) and 150µm (c), respectively.

The advantage of the Maxwell stress picture is, however, that it does provide information about the fine structure of the surface profile within the range of order d from the contact line. To obtain the detailed surface profile within this region, Buehrle et al. [19, 20] adopted a numerical scheme, in which they iteratively calculated the electric field distribution for a given surface profile and then adapted the surface profile to satisfy Eq. (5) actually was actually equal to Young’s angle independent of the applied voltage. Despite a weak algebraic divergence of the electric field, the net force on the contact line thus vanishes and the local contact angle is only determined by the balance of the chemical forces included in the interfacial tensions. This important finding, which was also supported by approximate analytical developments [21], was confirmed in a recent experimental study [20]. In that work the authors, made use of the fact that the characteristic length scale of the problem is given by the

Electrowetting

157

thickness of the insulator. Using very thick insulators, they managed to visualize the region h

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