Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies using Ultrasound, Volume 24, Second Edition (Studies in Interface Science)

Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies Using Ultrasound

Studies in Interface Science Vol. 1 Dynamics of Adsorption at Liquid Interfaces. Theory, Experiment, Application. By S.S. Dukhin, G. Kretzschmar and R. Miller Vol. 2 An Introduction to Dynamics of Colloids. By J.K.G. Dhont Vol. 3 Interfacial Tensiometry. By A.I. Rusanov and V.A. Prokhorov Vol. 4 New Developments in Construction and Functions of Organic Thin Films. Edited by T. Kajiyama and M. Aizawa Vol. 5 Foam and Foam Films. By D. Exerowa and P.M. Kruglyakov Vol. 6 Drops and Bubbles in Interfacial Research. Edited by D. Mo¨bius and R. Miller Vol. 7 Proteins at Liquid Interfaces. Edited by D. Mo¨bius and R. Miller Vol. 8 Dynamic Surface Tensiometry in Medicine. By V.M. Kazakov, O.V. Sinyachenko, V.B. Fainerman, U. Pison and R. Miller Vol. 9 Hydrophile-Lipophile Balance of Surfactants and Solid Particles. Physicochemical Aspects and Applications. By P.M. Kruglyakov Vol. 10 Particles at Fluid Interfaces and Membranes. Attachment of Colloid Particles and Proteins to Interfaces and Formation of Two-Dimensional Arrays. By P.A. Kralchevsky and K. Nagayama Vol. 11 Novel Methods to Study Interfacial Layers. By D. Mo¨bius and R. Miller Vol. 12 Colloid and Surface Chemistry. By E.D. Shchukin, A.V. Pertsov, E.A. Amelina and A.S. Zelenev Vol. 13 Surfactants: Chemistry, Interfacial Properties, Applications. Edited by V.B. Fainerman, D. Mo¨bius and R. Miller Vol. 14 Complex Wave Dynamics on Thin Films. By H.-C. Chang and E.A. Demekhin Vol. 15 Ultrasound for Characterizing Colloids. Particle Sizing, Zeta Potential, Rheology. By A.S. Dukhin and P.J. Goetz Vol. 16 Organized Monolayers and Assemblies: Structure, Processes and Function. Edited by D. Mo¨bius and R. Miller Vol. 17 Introduction to Molecular-Microsimulation of Colloidal Dispersions. By A. Satoh Vol. 18 Transport Mediated by Electrified Interfaces: Studies in the linear, non-linear and far from equilibrium regimes. By R.C. Srivastava and R.P. Rastogi Vol. 19 Stable Gas-in-Liquid Emulsions: Production in Natural Waters and Artificial Media Second Edition By J.S. D’Arrigo Vol. 20 Interfacial Separation of Particles. By S. Lu, R.J. Pugh and E. Forssberg Vol. 21 Surface Activity in Drug Action. By R.C. Srivastava and A.N. Nagappa Vol. 22 Electrorheological Fluids: The Non-aqueous Suspensions. T. Hao Vol. 23 Nanocomposite Structures and Dispersions - Science and Nanotechnology - Fundamental Principles and Colloidal Particles. Edited by: I. Capek

Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies Using Ultrasound 2nd edition

Andrei S. Dukhin and Philip J. Goetz

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2002 Copyright # 2010, Elsevier B.V. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333, E-mail: permissions@elsevier. com. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53621-1 ISSN: 1383-7303 For information on all Elsevier publications visit our Web site at www.elsevierdirect.com Print and bound in Great Britain 10 11 9

8 7 6 5

4 3 2 1

Contents

Preface to the Second Edition Preface to the First Edition List of Symbols

vii ix xi

Chapter 1. Introduction

1

Chapter 2. Fundamentals of Interface and Colloid Science 21 Chapter 3. Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

91

Chapter 4. Acoustic Theory for Particulates

127

Chapter 5. Electroacoustic Theory

187

Chapter 6. Experimental Verification of the Acoustic and Electroacoustic Theories

239

Chapter 7. Acoustic and Electroacoustic Measurement Techniques

261

Chapter 8. Applications for Dispersions

303

Chapter 9. Applications for Nanodispersions

343

Chapter 10. Applications for Emulsions and Other Soft Particles

369

Chapter 11. Titrations

401

Chapter 12. Applications for Ions and Molecules

425

Chapter 13. Applications for Porous Bodies

441

Bibliography

467

Subject Index

497 v

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Preface to the Second Edition

Ultrasound for Characterizing Colloids was first published in 2002. We decided to work on the second edition after two print runs were sold out and the book became unavailable. By the beginning of 2008, we gathered substantial justifications to prepare a new edition of the book instead of running an extra print of the first edition. First of all, about 40 papers had been published in various scientific journals by users of our instruments. There were, in addition, about a dozen of our own papers and several papers published by other scientists in the field. These papers demonstrated the wide spectrum of applications for characterization methods by ultrasound. Second, we discovered promising ways of using ultrasound that were known in specific scientific areas but not to us. These had not been mentioned in the first edition and we wanted to rectify the situation. One of the most striking examples was “seismoelectric current.” This electroacoustic and electrokinetic phenomenon had been known in geology for 70 years. However, there is practically no information about it in the major books on colloid and interface science despite the fact that it can be defined as “streaming current that is nonisochoric at ultrasound frequencies.” Frenkel’s theory of this effect is the first electroacoustic theory (1944) known to us, whereas Ivanov’s experiment is dated about the same time (1940) along with the first electroacoustic experiments by Hermans (1938). We present detailed descriptions, both theoretical and experimental, of this little known electrokinetic effect in Chapters 5 and 13. Seismoelectric current may become important in characterizing porous bodies. It clearly offers a simple way to measure z-potential in pores. In addition, it has the potential to characterize porosity and pore sizes. Another important example of discoveries in old works is “longitudinal rheology” and a host of related applications. Interpretation of acoustic data in rheological terms opens many new ways for applying ultrasound for characterizing complex systems. Mason predicted this 50 years ago for polymer solutions. There were other enthusiasts, but the lack of instruments and systematic reviews brought this field to a complete stop. We present here a review of the many works in “longitudinal viscosity” and provide some applications of our acoustic spectrometer as a “longitudinal rheometer.” One

vii

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Preface to the Second Edition

of the most important applications is the measurement of the “bulk viscosity,” an obscure second-viscosity coefficient of regular Newtonian liquids. Another example is Passynski’s theory (1938) that relates the speed of sound with the size and solvation number of ions. This theory was used by Bockris and others 40 years ago but had since been neglected. There is much discussion about measuring nanoparticles with sizes down to 1 nm using dynamic light scattering. This was done many decades ago using sound speed measurements with appropriate theoretical treatments. We discuss this in Chapter 12. There are several other new sections in the second edition. Most of them are on applications described in Chapters 8–13 but there are some new theoretical developments discussed in Chapters 3–5 as well. This field has become so wide-ranging in methods and applications that we have started talking more about the versatility of ultrasound for characterization purposes. This is explored in Chapter 1. This second edition is almost double the size of the first and has 50% more references. In preparing the first edition, we made the statement that it “marked the end of ultrasound’s ‘childhood’ in the field of colloids”. This second edition marks the beginning of adulthood. The number of groups working with our instruments has increased from 100 to 350. Geography has become much more diverse. In addition to groups in the United States, Germany, Japan, Taiwan, China, United Kingdom, Belgium, Finland, Singapore, South Korea, Mexico, and Canada, we have now users of our instruments in the Netherlands, Brazil, Thailand, Malaysia, India, Russia, Lithuania, Austria, Switzerland, Italy, Spain, France, Czech Republic, Columbia, Kuwait, Australia, and South Africa. We would also like to express our gratitude to our international agents: Quantachrome GmbH (Europe), Nihon Rufuto (Japan), Horiba USA, Acil Weber (Brazil), Advantage Scientific (China), and LMS (SE Asia). Without their dedication, we would not have been able to achieve so much in such a short time. They learned these new methods and now actively promote and teach them to others. We would like also to mention the contributions from our colleagues at Dispersion Technology: Betty Rausa, Lazlo Kovacs, Ross Parrish, Kenneth Schwartz, and Arthur Sigel. By taking proper care of everyday problems and the company’s needs, they gave us time to write this second edition. Andrei Dukhin President, Dispersion Technology, Inc.

Preface to the First Edition

The roots of this book go back 20 years. In the early 1980s Philip Goetz, then President of Pen Kem, Inc., was looking for new ways to characterize z-potential, especially in concentrated suspensions. His scientific consultants, Bruce Marlow, Hemant Pendse, and David Fairhurst, pointed towards utilizing electroacoustic phenomena. Several years of effort resulted in the first commercial electroacoustic instrument for colloids, the PenKem-7000. In the course of their work, they learned much about the potential value of ultrasound and collected a large number of papers published over the last two centuries on the use of ultrasound as a technique to characterize a diverse variety of colloids. From this it became clear that electroacoustics was only a small part of an enormous new field. Unfortunately, by that time Pen Kem’s scientific group broke apart: Bruce Marlow died, David Fairhurst turned back to design his own electrophoretic instruments, and Hemant Pendse concentrated his efforts on the industrial online application of ultrasound. In order to fill the vacuum of scientific support, Philip Goetz invited me to the United States, motivated by my experience and links to the well-established group of my father, Professor Stanislav Dukhin, then Head of the Theoretical Department in the Institute of Colloid Science of the Ukrainian Academy of Sciences. Whatever the reasons, I was fortunate to become involved in this exciting project of developing new techniques based on ultrasound. I was even luckier to inherit the vast experience and scientific base created by my predecessors at Pen Kem. Their hidden contribution to this book is very substantial, and I want to express here my gratitude to them for their pioneering efforts. It took 10 years for Phil and me to reach the point when, overwhelmed with the enormous volume of collected results, we concluded that the best way to summarize them for potential users was in the form of a book. There were many people who helped us during these years of development and later in writing this book. I would like to mention here several of them personally. First of all, Vladimir Shilov helped us tremendously with the electroacoustic theory. I believe that he is the strongest theoretician alive in the field of electrokinetics and related colloidal phenomena. Although he is very well known in Europe, he is less recognized in the United States. More recently, David Fairhurst joined us again. He brought his extensive expertise in the field of particulates, various colloids-related applications, and techniques; his comments were very valuable.

ix

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Preface to the First Edition

In our company, Dispersion Technology, Inc., Ross Parrish, Manufacturing and Service Manager, and Betty Rausa, Office Manager, extended their responsibilities last year, thereby allowing Phil and me the time to write this book. We are very grateful for their patience, reliability, and understanding. We trust that publishing this book will truly mark the end of ultrasound’s “childhood” in the field of colloids; it is no longer a “new” technique for colloids. Currently, there are more than 100 groups in the United States, Germany, Japan, Taiwan, China, United Kingdom, Belgium, Finland, Singapore, South Korea, Mexico, and Canada using Dispersion Technology’s ultrasound-based instruments. We do not know the total number of competing instruments from Malvern, Matec, Colloidal Dynamics, and Sympatec which are also being used in the field, but we estimate that there are over 200 groups working with them worldwide. We strongly believe that this is only the beginning. Ultrasonics has “come of age” and we hope that you will also share this view after reading this book and learning of the many advantages ultrasound can bring to the characterization of colloid systems. Andrei Dukhin President, Dispersion Technology, Inc.

List of Symbols

a b bsh C ci Cp Cdl Cs Cext Cabs Csca Di Du d e E hEi f F Fh Fhook k

particle radius cell radius radius of the shell in the core-shell model sound speed concentration of the ith ion species heat capacity at constant pressure DL capacitance electrolyte concentration extinction cross-section absorption cross-section scattering cross-section diffusion coefficient of the ith ion species Dukhin number particle diameter elementary electric charge electric field strength macroscopic electric field strength frequency of ultrasound Faraday constant hydrodynamic friction force Hookean force Bolzmann constant

kT ks K0 K Ir hIi I It Ii Is

thermal wavenumber shear wavenumber limiting conductances of cations and anions conductivity attributed with index local current in the cell macroscopic current intensity of the sound or light intensity of the transmitted sound or light intensity of the incident sound or light intensity of the scattered wave

xi

xii

List of Symbols

j jm (ka)

complex unit Bessel function

h H li L Lw mi M* NA Np N nm (ka) P pind

special function (see Special Functions) special function (see Special Functions) cell layer thickness gap in the electroacoustic chamber mass load mass of the solvated ions stress modulus Avogadro number number of particles number of the volume fractions Neumann function pressure induced dipole moment

R

gas constant

Robs R0 R1 r t t T Sexp v r Vdd Vhr Vcr Wext Wabs Wsca w u Um Udd

observation distance Rayleigh distance radius of the first Fresnel zone spherical radial coordinate transport numbers of cations and anions time absolute temperature measured electroacoustic signal volume of the solvated ions radial velocity corresponding to the dipole-dipole interaction radial velocity corresponding to the electro-hydrodynamic interaction radial velocity corresponding to the concentration interaction extinction energy absorption energy scattered energy width of the sound pulse speed of the motion attributed according to the index amplitude of the oscillation velocity energy of the dipole-dipole interaction

X

particle size

x

distance

List of Symbols

zi

valency of the ith ion species

z Z yi a b bp

valencies of the cations and anions acoustic impedance activity coefficient of the ith ion species attenuation specified with index thermal expansion attributed with index compressibility

d
dT dm e e0 f fc fT f
FIVI

viscous depth thermal depth scattering phase angles dielectric permittivity dielectric permittivity of the vacuum electric potential potential of the compression wave potential of the thermal wave potential of the shear wave phase of the Ion Vibration Current

Fe g gh gind
’ Fw k ks l m md mi m0i n n y r sd s sl t o ¼ 2pf

flow resistance hydrodynamic friction coefficient specific heat ratio Particle electric polarizibility dynamic viscosity volume fraction weight fraction reciprocal Debye length surface conductivity wave length electrophoretic mobility dynamic electrophoretic mobility electrochemical potential standard chemical potential kinematic viscosity dissociation numbers for cations and anions spherical angular coordinate density attributed according to the index electric charge of the diffuse layer electric surface charge standard deviation of the log-normal PSD heat conductance attributed according to the index frequency

xiii

xiv

oMW z c cdl (x) cd Ps Oe O S

List of Symbols

Maxwell-Wagner relaxation frequency electrokinetic potential angle in the polar coordinates Electric potential in the double layer Stern potential total scattered power by rigid sphere porosity drag coefficient entropy production

Indexes i p m s r y vis th sc int ef in out rod sur

index of the particle fraction or ion species particles medium dispersion radial component tangential component viscous thermal scattering intrinsic effective medium acoustic input acoustic output delay rod properties surface layer

Chapter 1

Introduction

1.1. Historical Overview 4 1.2. Versatility of UltrasoundBased Characterization Techniques 9 1.3. Comparison of UltrasoundBased Methods with Traditional Techniques 11 1.3.1. General Features 11

1.3.2. Particle Sizing 1.3.3. Measurements of z-Potential 1.3.4. Longitudinal and Shear Rheology 1.3.5. Characterization of Porous Bodies References

12 14 14 15 15

Several key words define the scope of this book’s second edition. All the words are mentioned in the title: ultrasound, liquids, nano- and microparticulates, and porous bodies. The key word, ultrasound, refers to characterization techniques described in this book, while all the others indicate the types of objects (such as dispersions and emulsions) that are studied by methods based on ultrasound. Each word is a key to a major scientific discipline. Ultrasound establishes acoustics as the main scientific basis for the measuring techniques presented here. The other key words define hydrodynamics, rheology, porosimetry, and colloidal and interfacial science as disciplines that deal with particulates and porous bodies. Historically, there has been curiously little real communication between acoustics and the scientific disciplines mentioned above. There is a large body of literature devoted to ultrasound phenomena in liquids, particulates, and porous bodies, but it has been mostly written from the perspective of scientists in the field of acoustics. There is limited recognition of ultrasound phenomena as of real importance for learning the properties of liquids, particulates, and porous bodies, and, in turn, developing applications. Scientists in these fields have not embraced acoustics as an important tool for characterizing their objects of interest. The lack of serious dialogue between these scientific fields is perhaps best demonstrated by the fact that there are no references to

Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23001-6 Copyright # 2010, Elsevier B.V. All rights reserved.

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CHAPTER

1

ultrasound or acoustics in the major handbooks on colloid and interface science [1, 2], rheology [3, 4], hydrodynamics [5–8], and porosimetry [9]. One may ask, “Perhaps this link does not exist because it is not important?” To answer this question, let us consider the potential place of ultrasound-related effects within an overall framework of nonequilibrium phenomena. It will be helpful to first classify nonequilibrium phenomena in two dimensions, as outlined in Table 1.1: the first is determined by whether the relevant disturbances are electrical, mechanical, or electromechanical in nature; the second is based on whether the time domain of that disturbance can be described as stationary, low frequency, or high frequency. The lowand high-frequency ranges are separated on the basis of the relationship between either the electrical or mechanical wavelength, l, and some system dimension, L. Light scattering clearly represents electrical phenomena in colloids at high frequency (the wavelength of light is certainly smaller than the system’s dimensions). However, until very recently, there was no mention in textbooks of mechanical or electromechanical phenomena in the region where l is shorter than the system’s dimensions. This would appear to leave two empty spaces in Table 1.1. Such mechanical wavelengths are produced by sound or, when the frequency exceeds our hearing limit of 20 kHz, ultrasound. Ultrasound wavelengths lie in the range from 10 µm to 1 mm, whereas the system’s dimensions are usually in the range of centimeters. For this reason, we consider ultrasound-related effects to lie within the high-frequency range. One of the empty spaces in Table 1.1 can be filled by acoustic measurements at ultrasound frequencies that characterize nonequilibrium phenomena of a

TABLE 1.1 Colloidal Phenomena Electrical nature

Electromechanical

Mechanical nature

Stationary

Conductivity, surface conductivity

Electrophoresis, electro-osmosis, sedimentation potential, streaming current/ potential, electro-viscosity

Viscosity, stationary colloidal hydrodynamics, osmosis capillary flow

Low frequency (l > L)

Dielectric spectroscopy

Electro-rotation, dielectrophoresis

Shear rheology

High frequency (l < L)

Optical scattering, X-ray spectroscopy

Empty? Electroacoustics!

Empty? Acoustics!

Introduction

3

mechanical nature at high frequency. The second empty space can be filled by electroacoustic measurements that permit characterization of electromechanical phenomena at high frequency. This book helps fill these gaps and demonstrates that acoustics (and electroacoustics) can provide useful knowledge to various scientific disciplines. As an aside, we do not consider the use of high-power ultrasound for modifying systems here. We only undertake the use of low-power sound as a noninvasive investigation tool that has unique capabilities. Several questions may come up when starting to read this book. We think it is important to deal with these questions right away, with some preliminary answers, which will be later clarified and expanded upon in the main text. Here are these questions and the short answers. Why should one care about acoustics if generations of scientists have successfully worked on a particular field without it? While it may be true that the usefulness of acoustics is currently not widely understood, it seems that earlier generations of scientists had a somewhat better appreciation. Many well-known scientists applied acoustics for characterization purposes, as will be described in a detailed historical overview in the next section. Briefly, we mention the names of Stokes, Rayleigh, Maxwell, Henry, Tyndall, Reynolds, and Debye. An interesting but not well-known fact is that Lord Rayleigh, the first author of a scattering theory, titled his major book as Theory of Sound. He developed the mathematics of scattering theory for both sound and light, but apparently considered sound to be more deserving of a book title than light. If acoustics is so important, why has it remained almost unknown as a characterization tool for such a long time? We think that the failure to exploit acoustic methods might be explained by a combination of factors: the advent of the laser as a convenient source of monochromatic light; technical problems with generating monochromatic sound beams over a wide frequency range; the mathematical complexity of the theory; and complex statistical analysis of the raw data. In addition, acoustics is more dependent on mathematical calculations than other traditional instrumental techniques. Many of these problems have now been solved mostly by the advent of fast computers and the development of new theoretical approaches. As a result, there are a number of commercially available instruments—developed by Matec, Malvern, Sympatec, Colloidal Dynamics, and Dispersion Technology—that utilize ultrasound for characterization of colloids. What information do ultrasound-based instruments yield? The versatility of ultrasound-based characterization methods deserves a special section in this chapter, as seen below. Over the last decade, we have personally thought on several occasions that we have exhausted new ideas and have finished developing a family of ultrasound-based characterization instruments. Reality has proved us wrong every

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1

time. New possibilities suddenly appear that we had not thought about earlier. One of the most astounding examples is the application of electroacoustics in studying porous bodies. We recently discovered a paper published 60 years ago that presents a theory of the electroacoustic effect in a porous body. This was the first electroacoustic theory that was published by Frenkel in 1944 [10]. What was even more surprising is that this old theory was initiated by an even older experiment of seismoelectric current, observed by Ivanov in 1940 [11]. These works create the basis for applying electroacoustics for studying properties of porous bodies. Where can one apply ultrasound? It can be applied for any liquid-state system that is Newtonian or nonNewtonian and as diverse as pure water to pastes. The following list gives some idea of the existing applications for which the ultrasound-based characterization technique is appropriate: aggregative stability, cement slurries, ceramics, chemical-mechanical polishing, coal slurries, coatings, cosmetic emulsions, environmental protection, flotation, ore enrichment, food products, latex, emulsions and micro emulsions, mixed dispersions, nanosized dispersions, nonaqueous dispersions, paints, photo materials.

This list is not complete. A table in Chapter 8 summarizes all experimental studies currently known to us. What are the advantages of ultrasound over traditional characterization techniques? There are so many advantages of ultrasound that we will discuss for a particular application in following sections of this chapter. Here is a striking example: ultrasound eliminates the need for sample dilution and special preparation for particle-sizing and z-potential measurements. Finally, we would like to stress that this book primarily targets scientists who consider ultrasound for characterization purposes. We will emphasize those aspects of acoustics that are important for these goals and neglect many others.

1.1. HISTORICAL OVERVIEW When preparing the first edition of this book almost 10 years ago, we did an extensive search of the relevant literature. Since that time, we found only a single old article of great importance that was missing from our original review. It turns out that the first electroacoustic theory was developed 7 years earlier than thought by a completely different person in a completely different place. There are also recent developments that deserve to be mentioned. These changes are incorporated in the text below. The roots of our current understanding of sound go back to more than 300 years when Newton suggested the first theory for calculating the speed of sound [12]. Newton’s work is still interesting today because it demonstrates the importance of thermodynamic considerations when trying to adequately

5

Introduction

describe ultrasound phenomena. Newton assumed that sound propagated while maintaining a constant temperature, that is, an isothermal case. Laplace later corrected this misunderstanding by showing that it was actually adiabatic in nature [12]. This thermodynamic aspect of sound is a good example of the importance of keeping a historical perspective. In at least two incidences in the past 200 years, the thermodynamic contribution to various sound-related phenomena was initially neglected and only later found to be important. The first neglect of thermodynamics happened in the nineteenth century when Stokes’s purely hydrodynamic theory for sound attenuation [13, 14] was later corrected by Kirchhoff [15, 16]. The second neglect occurred in the twentieth century when Sewell’s hydrodynamic theory for sound absorption in heterogeneous media [17] was later extended by Isakovich [18] with the introduction of a mechanism for thermal losses. Table 1.2 lists the important steps in the development of our understanding of sound. From the beginning, sound was considered to be a rather simple

TABLE 1.2 Milestones in Understanding Sound in Relation to Colloids Year

Author

Topic

1687

Newton [12]

Sound speed in fluid, theory, erroneous isothermal assumption

Early 1800s

Laplace [12]

Sound speed in fluid, theory, adiabatic assumption

1820

Poisson [19, 20]

Scattering by atmosphere, arbitrary disturbance, first successful theory

1808

Poisson [19, 20]

Reflection from rigid plane, general problem

1845–1851

Stokes [13, 14]

Sound attenuation in fluid, theory, viscous losses

1842

Doppler [21]

Alternation of pitch by relative motion

1868

Kirchhoff [15, 16]

Sound attenuation in fluid, theory, thermal losses

1866

Maxwell [22]

Kinetic theory of viscosity

1870–1880

Henry, Tyndall, Reynolds [23–26]

First application to colloids—sound propagation in fog

1871

Rayleigh [27, 28]

Light scattering theory

1875–1880

Rayleigh [12, 29, 30]

Diffraction and scattering of sound, Fresnel zones in Acoustics Continued

6

CHAPTER

TABLE 1.2 Milestones in Understanding Sound in Relation to Colloids— Cont’d Year

Author

Topic

1878

Rayleigh [12]

Theory of Sound, vol. II

1910

Sewell [17]

Viscous attenuation in colloids, theory

1933

Debye [31]

Electroacoustic effect, introduction for ions

1936

Morse [32]

Scattering theory for arbitrary wavelengthsize ratio

1938

Hermans [33]

Electroacoustic effect, introduced for colloids

1938

Passynski [34]

Theory relating compressibility of liquid with solvating numbers of ions

1940

Ivanov [35]

Discovery of seismoelectric phenomena

1944

Frenkel [9]

First electroacoustic theory. Application— seismoelectric phenomena

1944

Foldy [36, 37]

Acoustic theory for bubbles

1948

Isakovich [18]

Thermal attenuation in colloids, theory

1946

Pellam, Galt [38]

Pulse technique

1947

Bugosh, Yaeger [39]

Electroacoustic theory for electrolytes

1951–1953

Yeager, Hovorka, Derouet, Denizot [40–43]

First electroacoustic measurements

1951–1952

Enderby, Booth [44, 45]

Electroacoustic theory for particulates

1953

Epstein and Carhart [46]

General theory of sound attenuation in dilute colloids

1958–1959

Happel, Kuwabara [47–49]

Hydrodynamic cell models

1962

Andreae et al. [50, 51]

Multiple frequencies attenuation measurement

1967

Eigen et al. [52, 53]

Nobel Prize, acoustics for chemical reactions in liquids

1971

Bockris, Saluja [54]

Measurement solvating numbers of ions with ultrasound

1972

Allegra, Hawley [55]

ECAH theory for dilute colloids Continued

1

7

Introduction

TABLE 1.2 Milestones in Understanding Sound in Relation to Colloids— Cont’d Year

Author

Topic

1973

Cushman [56]

First patent for acoustic particle sizing

1974

Levine, Neale [57]

Electrokinetic cell model

1978

Beck [58]

Measurement of z-potential by ultrasonic waves

1981

Shilov, Zharkikh [59]

Corrected electrokinetic cell model

1983

Marlow, Fairhurst, Pendse [58]

First electroacoustic theory for concentrates

1983

Uusitalo [60]

Mean particle size from acoustics, patent

1983

Oja, Peterson, Cannon [61]

ESA electroacoustic effect

1988

Harker, Temple [62]

Coupled phase model for acoustics of concentrates

1987

Riebel [63]

Particle size distribution, patent for the large particles

1988–1989

O’Brien [64, 65]

Electroacoustic theory, particle size, and z-potential from electroacoustics

1990

Anson, Chivers [66]

Materials database

1999

Shilov and others [67, 68]

Electroacoustic theory for CVI in concentrates

1990 to present

McClements, Povey [69–71]

Acoustics for emulsions

1996 to present

Dukhin, Goetz [72–74]

Combining together acoustics and electroacoustics for particle sizing, rheology, and electrokinetics

2002

Dukhin, Goetz [75]

First Edition of this book

2004

Shilov, Borkovskaya, Dukhin [76]

Electroacoustic theory for overlapped DLs, nonaqueous systems, nanoparticulates

2005

ISO committee on particle characterization

ISO standard 20998/1 for acoustic particle sizing

2009

Dukhin, Goetz, and Thommes [77]

Electroacoustics for characterizing porous bodies

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1

example of a wave on which a general theory of wave phenomena was developed. Later, the new body of understanding for sound was extended to other wave phenomena, such as light. Tyndall, for example, used references to sound to explain the wave nature of the light [23, 24]. Newton’s corpuscular theory of light was first opposed both by the celebrated astronomer, Huygens, and the equally celebrated mathematician, Euler. They both believed that light, like sound, was a product of wave motion. In the case of sound, the velocity depended upon the relation between elasticity and density in the body that transmitted the sound. The greater the elasticity of the body, the greater is the velocity; the lower the density of the body, the greater is the velocity of the sound. To account for the enormous velocity of propagation in the case of light, the substance that transmitted it was assumed to have both extreme elasticity and extreme density. The dominance of sound over light as examples of the wave phenomena continued even with Lord Rayleigh. He developed his theory of scattering mostly for sound and paid much less attention to light [12, 27–30]. At the end of the nineteenth century, sound and light parted ways because further investigation was directed more on the physical roots of each phenomenon instead of on their common wave nature. The history of light and sound in colloid science is very different. Light has been an important tool since the first microscopic observations of Brownian motion and the first electrophoretic measurements. It became more important in middle of the twentieth century because of the use of light scattering to determine particle size. In contrast, sound remained unknown in colloid science, despite a considerable amount of work in the field of acoustics where fluids that were essentially colloidal in nature were used. The goal of these studies was to learn more about acoustics than about colloids. This was the spirit in which the ECAH theory (Epstein-Carhart-Allegra-Hawley [46, 55]) for ultrasound propagation through dilute colloids was developed. Although acoustics was not specifically used for colloids, it was a powerful tool for other purposes [89, 90]. For instance, it was used to learn more about the structure of pure liquids and the nature of chemical reactions in liquids. These studies, associated with the name of Prof. Eigen who received a Nobel Prize in Chemistry in 1967 [52, 53, 78], are described in more detail in the chapter, “Fundamentals of Acoustics.” Curiously, the penetration of ultrasound into colloid science began with electroacoustics, which is more complex than traditional acoustics. An electroacoustic effect was predicted for ions by Debye in 1933 [31], and later extended to colloids by Hermans and, independently, Rutgers in 1938 [33]. Recently, we have learned that at about the same time, electroacoustic phenomena were observed in geology by Ivanov in 1940 [35]. Theory of this effect, called seismoelectric current, became the first ever electroacoustic theory by Frenkel in 1944 [9]. Early experimental electroacoustic work

Introduction

9

is associated with Yeager and Zana who conducted many experiments in the 1950s and 1960s with various coauthors [39–42]. This work was later continued by Marlow, O’Brien, Ohshima, Shilov, and the authors of this book [58, 64, 65, 67, 68, 79, 80, 91]. As already mentioned, there are now several commercially available electroacoustic instruments for characterizing the z-potential. Acoustics has only very recently attained some recognition in the field of colloid science. It was first suggested as a particle-sizing tool by Cushman and others in 1973 [56], and later refined by Uusitalo and others [50]. Acoustics for sizing was suggested for large particles by Riebel [63]. Development of a commercial instrument that could measure a wide range of particle sizes was begun by Goetz, Dukhin, and Pendse in the 1990s [72, 81–83]. At the same time, a group of British scientists including McClements, Povey, and others [66, 69–71, 84, 85] actively promoted acoustics, especially for the study of emulsions. As mentioned earlier, there are now several commercially available acoustic spectrometers, manufactured by Sympatec, Matec, and Dispersion Technology. In conclusion of this short historical review, we would like to mention a development that we consider of great importance for the future—the combination of both acoustic and electroacoustic spectroscopies. The synergism of this combination is described in the papers and patents by Dukhin and Goetz [68, 73, 74, 80–83, 86, 87].

1.2. VERSATILITY OF ULTRASOUND-BASED CHARACTERIZATION TECHNIQUES There are two distinctively different ultrasound-based measuring methods. One is called acoustics and the other one is known as electroacoustics. They measure completely different parameters. Acoustics is simpler because a single field is involved—field of the mechanical stress. The measured parameters are usually sound speed and attenuation coefficient. The electroacoustic method is more complicated because it is based on the coupling of two fields—electrical and mechanical. The measured parameters are the magnitude and phase of the electroacoustic signal. Four raw parameters can be used as a fingerprint of a particular liquid system. l l l l

Sound speed Attenuation coefficient (usually at multiple frequencies) Magnitude of the electroacoustic signal Phase of the electroacoustic signal

Alternatively, the raw data could be theoretically treated so that a multitude of other properties can be characterized. The theoretical treatment is the main reason for ultrasound’s versatility for characterization purposes. It turns out

10

CHAPTER

1

that there are two very different sets of calculated parameters that can be extracted from the raw data, depending on degree of our a priori knowledge about the system. If our prior knowledge is limited and we are forced to model a system as a homogeneous medium with unknown viscoelastic properties, then the ultrasound-based method allows us to calculate following parameters: l l l l l l l l l l l l

Viscosity longitudinal within the 1–100 MHz frequency range Viscous longitudinal modulus G00 Bulk viscosity for Newtonian liquids Elastic longitudinal modulus G0 Compressibility Newtonian liquid test in the megahertz range Isoelectric point—range of aggregative instability Optimum dose of surfactant Volume fraction of the dispersed phase from sound speed Kinetics of dissolution, crystallization Kinetics of sedimentation Verification of large particle presence in opaque systems

We would like to stress here that the system might be very complex and intuitively heterogeneous, but it should not prevent the application of the homogeneous mode to its description. For instance, milk can be treated as a homogeneous liquid when we ignore the fact that it is actually a collection of fat droplets, proteins, and sugars in water. Basically, any liquid system can be modeled as homogeneous or heterogeneous. These models are creations in our minds for adequate characterization of various physical and chemical properties. In the case of ultrasound-based techniques, we can first model a system as homogeneous and calculate the set of parameters presented above. Then, as the next step, we can apply the heterogeneous model and theoretically treat the same set of experimental raw data for extracting another set of parameters, given below: l l

l l l l l l l l

Particle size distribution of solid particles with known density Particle size distribution of soft particles (droplets) when thermal expansion coefficient is known Volume fraction of solids at the submicrometer range Hook’s parameter for particle bonds in structured systems Microviscosity x-Potential in particulates Surface conductivity Debye length when conductivity is known Ion size from compressibility in aqueous solution Ion size from electroacoustics in aqueous and nonaqueous solutions

Introduction

l l l

11

Electric charge of macromolecules Porosity, pore size, and x-potential in porous bodies Properties of deposits and sediments

Application of the heterogeneous model usually requires a priori information about the volume fraction of the dispersed phase. This parameter may be extracted from the raw data in some special cases, for instance when particles are rigid and their sizes range from 0.1 to 1 µm. Sometimes, it can be calculated from the sound’s speed as well. However, for the most reliable results, volume (weight) fraction of the dispersed phase must be treated as an independently known input parameter. In terms of traditional measuring techniques, ultrasound can combine the descriptions of rheology, particle size distribution, and z-potential into one unit instead of three independent instruments. This has substantial advantages over traditional methods. We discuss some of them below.

1.3. COMPARISON OF ULTRASOUND-BASED METHODS WITH TRADITIONAL TECHNIQUES The versatility of ultrasound-based methods makes them comparable with several traditional techniques, such as particle sizing by light scattering, electrophoretic light scattering, microelectrophoresis, and shear rheology.

1.3.1. General Features A major distinction of ultrasound-based techniques, in contrast to traditional light-based characterization methods, is that sound can propagate through concentrated, opaque liquid systems and even porous bodies. This peculiar feature opens up the opportunity for tremendous simplification in sample preparation. In particular, ultrasound-based methods applied to concentrated dispersions and emulsions eliminate the need for dilution. Systems can be characterized as they are. Dilution required by traditional techniques can destroy aggregates or flocs. The corresponding measured particle size distribution for the diluted system would not be correct for the original concentrated sample. Eliminating dilution is especially critical for z-potential characterization because the parameter is a property of both the particle and the surrounding liquid; dilution changes the suspension medium and the z-potential. This feature of eliminating dilution is applicable to both the measurements of particle size and z-potential. Sample handling for characterizing these parameters becomes as simple as in traditional rheological methods. Acoustic methods are very robust and precise [73, 74]. They are much less sensitive to contamination in comparison to traditional light-based techniques because the high concentration of particles in a fresh sample dominates any small residue left from the previous sample.

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Ultrasound-based methods are relatively fast. A single measurement can normally be completed within a few minutes. This feature, together with the ability to measure flowing systems, makes acoustic attenuation very attractive for on-line characterization.

1.3.2. Particle Sizing The numerous advantages of using ultrasound for characterizing particle size are summarized in Table 1.3. Detailed analysis of ultrasound-based techniques is given later in this book. The following is a short summary. There are several advantages of ultrasound-based over light-based instruments because of the longer wavelength used. The wavelength of ultrasound in water, at the highest frequency typically used (100 MHz), is about 15 µm. It increases even further to 1.5 mm at the lowest frequency (1 MHz). In contrast, light-based instruments typically use wavelengths on the order of 0.5 µm. If the particles are small compared to the wavelength, the Rayleigh long-wavelength requirement is said to be satisfied. Particle

TABLE 1.3 Features and Benefits of Acoustics over Traditional Particle-Sizing Techniques Feature

Benefit

No requirement of dilution

Less sensitive to contamination

No calibration with the known particle size

More accurate

Particle size range from 5 nm to 1000 µm with the same sensor

Simpler hardware, more cost effective

Simple decoupling of sound adsorption and sound scattering

Simplifies theory

Possible to eliminate multiple scattering even at high volume fractions up to 50% vol

Simplifies theory for large particle size

Existing theory for ultrasound absorption in concentrates with particles interaction

Possible to treat small particles in concentrates

Data availability over wide range of wavelengths

Allows use of simplified theory and reduces particle shape effects

Innate weight basis, lower power of the particle size dependence

Better for broad polydisperse systems, highly sensitive to nanoparticles

Particle sizing in dispersions with several dispersed phases (mixed dispersions)

Real-world, practical systems

Particle sizing in structured dispersions

13

Introduction

sizing in this long-wavelength range is more desirable than in the intermediateor short-wavelength ranges because of the lower sensitivity to shape factors and a simpler theoretical interpretation. So, applying the longer wavelengths available by acoustics allows us to characterize a greater range of particle sizes while still meeting the long-wavelength requirement. Nature has provided one more significant advantage of ultrasound over light which relates to the wavelength dependence. As the wave travels through the colloid, the combined effects of both scattering and absorption cause the extinction of both ultrasound and light [32, 88]. As most light scattering experiments are performed at a single wavelength, it is not possible to experimentally separate the two contributions of scattering and absorption to the total extinction. More often than not, the absorption of light is simply neglected in most light scattering experiments, which can lead to errors. In the case of ultrasound, the absorption and scattering are distinctively separated on the wavelength scale. Figure 1.1 illustrates the dependence of ultrasound attenuation as a function of relative wavelength, ka, as defined by: 2pa ð1:1Þ l where a is the particle radius and l is the wavelength of ultrasound. The attenuation curve has two prominent ranges. The lower frequency region corresponds to absorption; the higher frequency region corresponds to scattering. It is obvious from Figure 1.1 that both contributions can be easily separated because there is very little—indeed almost negligible—overlap. ka ¼

0.3 µm 0.5 µm 1 µm 5 µm 10 µm

9

attenuation [dB/cm/MHz]

8 7 6 5 4 3 2 1 0 10−4

10−3

10−2

10−1

ka

100

101

FIGURE 1.1 Scattering attenuation and viscous absorption of ultrasound.

102

103

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This peculiar aspect of ultrasound frequency dependence tremendously simplifies the theory. In the majority of cases, absorption and scattering can be considered separately. Except for the cases with very high volume fractions and some special nonaqueous systems with soft particles [49], this simplification is valid.

1.3.3. Measurements of z-Potential In contrast to acoustics, electroacoustics is a relatively new technique. In principle, it provides information for both particle-sizing and z-potential characterization. However, we believe that acoustics is better suited to particle sizing than electroacoustics. For this reason, as justified later in Chapters 4, 5, and 7, we consider electroacoustics to be primarily a technique for characterizing only the electric surface properties, such as the z-potential. In this sense, electroacoustics competes with microelectrophoresis and other traditional electrokinetic methods. However, electroacoustics has many advantages over traditional electrokinetic methods, which can be summarized as follows: l l l l l l l l l l

No dilution required, volume fraction up to 50% vol Less sensitive to contamination, easier to clean Higher precision ( 0.1 mV) Low surface charges (down to 0.1 mV) Electrosmotic flow not important Convection not important Measurement at high ionic strength exceeding 1 M level Measurement at very low ionic strength in nonpolar liquids Small sample volume, as little as 1 ml Faster

In addition, as explained later in Chapter 8, electroacoustic probes can be used for various titration experiments.

1.3.4. Longitudinal and Shear Rheology Ultrasound-based methods are similar to rheology in that they rely on applying stress to the system to learn about some of its properties. The difference is the type of stress applied. Traditional rheology uses shear stress, whereas ultrasound is a wave of longitudinal stress. Thus, there are two different rheological methods—shear rheology and longitudinal rheology. Chapter 3 presents detailed definitions and comparison of these two measurement techniques. Here we will only mention the main differences. First of all, the two different rheological methods work at very different frequency ranges. Shear rheology is applicable at a low frequency range up to roughly 100 kHz. Longitudinal rheology functions only at the megahertz frequency scale. This makes both methods complimentary, not competitive.

Introduction

15

Second, longitudinal rheology is nondestructive. Longitudinal stress does not break bonds between particles. It only causes the bonds to expand and collapse. Third, longitudinal rheology provides information about the bulk viscosity of Newtonian liquids. Bulk viscosity is a more sensitive probe of any structural feature in a Newtonian liquid but it is impossible to measure using shear-based techniques. Finally, measurement of the ultrasound attenuation at multiple frequencies helps in assessing whether a particular liquid can be described as Newtonian at the megahertz frequency range.

1.3.5. Characterization of Porous Bodies This is a relatively new field for ultrasound-based methods and still is under development. However, there is sufficient amount of data to be optimistic about the method’s future. Characterization of porous bodies usually includes measurements of the porosity, pore size, and the electric charging of the walls of the pores. The measurement of porosity and pore size can be achieved using gas adsorption and mercury porosimetry. The characterization of electric charging usually relies on streaming current/potential measurement. The propagation of ultrasound through a porous body generates a host of different effects that can be used for characterization purposes. Detailed analysis of the seismoelectric current may be the most promising: it simply is a streaming current at high ultrasound frequency in the nonisochoric mode. If this development can be confirmed, this method could compete with mercury porosimetry and minimize usage of this environmentally dangerous material. The seismoelectric current method would allow characterization of the electric surface properties of materials with very low hydrodynamic permeability because of small pore size. Many materials of this kind are impossible to be measured with classical electrokinetic methods.

REFERENCES [1] J. Lyklema, Fundamentals of Interface and Colloid Science, vol. 1, Academic Press, New York, 1993. [2] R.J. Hunter, Foundations of Colloid Science, Oxford University Press, Oxford, 1989. [3] H.A. Barnes, A Handbook of Elementary Rheology, University of Wales, Institute of NonNewtonian Fluid Mechanics, Aberystwyth, 2000. [4] C.L. Yaws, Handbook of Viscosity, vol. 4, Inorganic Compounds and Elements, Gulf Publishing Company, Houston, TX, 1997. [5] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1965. [6] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959. [7] H. Lamb, Hydrodynamics, sixth ed., Dover Publications, New York, NY, 1932.

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[8] J.O. Hirschfelder, C.F. Curtis, R.B. Bird, Molecular Theory of Gases and Liquids, Wiley, New York, NY, 1964. [9] S. Lowell, J.E. Shields, M.A. Thomas, M. Thommes, Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density, Kluwer, The Netherlands, 2004. [10] A.G. Ivanov, Bull. Aca. Sci. USSR, Ser. Geogr. Geophys. 5 (1940) 699. [11] M.C. Potter, D.C. Wiggert, Mechanics of Fluids, Prentice Hall, Englewood Cliffs, NJ, 1997. [12] L. Rayleigh, The Theory of Sound, vol. 2, second ed., Macmillan, New York, NY, 1896; first ed., 1878. [13] Stokes, On a difficulty in the Theory of Sound, Philos. Mag. Nov. (1848). [14] Stokes, Dynamic theory of diffraction, Camb. Philos. Trans. IX (1849). [15] Kirchhoff, Pogg. Ann. CXXXIV (1868) 177. [16] Kirchhoff, Vorlesungen uber Mathematische Physik, (1876). [17] C.T.J. Sewell, The extinction of sound in a viscous atmosphere by small obstacles of cylindrical and spherical form, Philos. Trans. R. Soc. Lond. 210 (1910) 239–270. [18] M.A. Isakovich, Zh. Exp. Theor. Phys. 18 (1948) 907. [19] Poisson, Sur l’integration de quelques equations lineaires aux differnces prtielles, et particulierement de l’equation generalie du mouvement des fluides elastiques, Mem., de l’Institut t.III (1820) 121. [20] Poisson, J. de l’ecole Polytech. t.VII (1808). [21] Doppler, Theorie des farbigen Lichtes der Doppelsterne, Prag (1842). [22] Maxwell, On the viscosity or internal friction of air and other gases, Philos. Trans. 156 (1866) 249. [23] J. Tyndall, Light and Electricity, D. Appleton & Comp., New York, NY, 1873. [24] J. Tyndall, Sound, Philos. Trans. third ed. (1874). [25] Henry, Report of the Lighthouse Board of the United States for the year 1874. [26] O. Reynolds, Proc. R. Soc. XXII (1874) 531. [27] J.W. Rayleigh, On the light from the sky, Philos. Mag. (1871). [28] J.W. Rayleigh, On the scattering of light by small particles, Philos. Mag. (1871). [29] J.W. Rayleigh, Acoustical observations, Philos. Mag. IX (1880) 281. [30] J.W. Rayleigh, On the application of the principle of reciprocity to acoustics, R. Soc. Proc. XXV (1876) 118. [31] P. Debye, J. Chem. Phys. 1 (1933) 13. [32] P. Morse, U. Ingard, Theoretical Acoustics, McGraw-Hill, New York, NY, 1968. [33] J. Hermans, Philos. Mag. 25 (1938) 426. [34] A. Passynski, Acta Physicochim. 8 (1938) 385. [35] J. Frenkel, On the theory of seismic and seismoelectric phenomena in a moist soil, J. Phys., USSR 3 (5) (1994) 230–241. Re-published in J. Eng. Mech. (2005). [36] L.L. Foldy, Propagation of sound through a liquid containing bubbles, OSRD Report No.6.1sr1130-1378, 1944. [37] E.L. Carnstein, L.L. Foldy, Propagation of sound through a liquid containing bubbles, J. Acoust. Soc. Am. 19 (3) (1947) 481–499. [38] J.R. Pellam, J.K. Galt, Ultrasonic propagation in liquids: application of pulse technique to velocity and absorption measurement at 15 Megacycles, J. Chem. Phys. 14 (10) (1946) 608–613. [39] J. Bugosh, E. Yeager, F. Hovorka, J. Chem. Phys. 15 (1947) 592. [40] E. Yeager, F.J. Hovorka, Acoust. Soc. Am. 25 (1953) 443. [41] E. Yeager, H. Dietrick, F. Hovorka, J. Acoust. Soc. Am. 25 (1953) 456. [42] R. Zana, E.J. Yeager, Phys. Chem. 71 (1967) 4241.

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[43] B. Derouet, F.C.R. Denizot, Acad. Sci. Paris 233 (1951) 368. [44] F. Booth, J. Enderby, On electrical effects due to sound waves in colloidal suspensions, Proc. Am. Phys. Soc. 208A (1952) 32. [45] J.A. Enderby, On electrical effects due to sound waves in colloidal suspensions, Proc. R. Soc. Lond. A207 (1951) 329–342. [46] P.S. Epstein, R.R. Carhart, The absorption of sound in suspensions and emulsions, J. Acoust. Soc. Am. 25 (1953) 553–565. [47] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Martinus Nijhoff, Dordrecht, The Netherlands, 1973. [48] J. Happel, Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles, AICHE J. 4 (1958) 197–201. [49] S. Kuwabara, The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers, J. Phys. Soc. Jpn. 14 (1959) 527–532. [50] J. Andreae, P. Joyce, 30 to 230 megacycle pulse technique for ultrasonic absorption measurements in liquids, Br. J. Appl. Phys. 13 (1962) 462–467. [51] J. Andreae, R. Bass, E. Heasell, J. Lamb, Pulse technique for measuring ultrasonic absorption in liquids, Acustica 8 (1958) 131–142. [52] M. Eigen, Determination of general and specific ionic interactions in solution, Faraday Soc. Discuss. 24 (1957) 25. [53] M. Eigen, L. deMaeyer, in: Weissberger (Ed.), Techniques of Organic Chemistry, vol. VIII, Part 2, Wiley, New York, NY, 1963. [54] J.O’M. Bockris, P.P.S. Saluja, Ionic salvation numbers from compressibilities and ionic vibration potential measurements, J. Phys. Chem. 76 (1972) 2140–2151. [55] J.R. Allegra, S.A. Hawley, Attenuation of sound in suspensions and emulsions: theory and experiments, J. Acoust. Soc. Am. 51 (1972) 1545–1564. [56] Cushman, et al., US Patent 3,779,070, 1973. [57] S. Levine, G.H. Neale, The prediction of electrokinetic phenomena within multiparticle systems. 1. Electrophoresis and electroosmosis, J. Colloid Interface Sci. 47 (1974) 520–532. [58] Beck, et al., Measuring zeta potential by ultrasonic waves, Tappi 61 (1978) 63–65. [59] V.N. Shilov, N.I. Zharkih, Yu.B. Borkovskaya, Theory of nonequilibrium electrosurface phenomena in concentrated disperse system. 1. Application of nonequilibrium thermodynamics to cell model, Colloid J. 43 (1981) 434–438. [60] S.J. Uusitalo, G.C. von Alfthan, T.S. Andersson, V.A. Paukku, L.S. Kahara, E.S. Kiuru, Method and apparatus for determination of the average particle size in slurry, US Patent 4,412, 451, 1983. [61] T. Oja, G. Petersen, D. Cannon, Measurement of electric-kinetic properties of a solution, US Patent 4,497,208, 1985. [62] A.H. Harker, J.A.G. Temple, Velocity and attenuation of ultrasound in suspensions of particles in fluids, J. Phys. D. Appl. Phys. 21 (1988) 1576–1588. [63] U. Riebel, et al., The fundamentals of particle size analysis by means of ultrasonic spectrometry, Part. Part. Syst. Charact. 6 (1989) 135–143. [64] R.W. O’Brien, Electro-acoustic effects in a dilute suspension of spherical particles, J. Fluid Mech. 190 (1988) 71–86. [65] R.W. O’Brien, Determination of particle size and electric charge, US Patent 5,059,909, Oct. 22, 1991. [66] L.W. Anson, R.C. Chivers, Thermal effects in the attenuation of ultrasound in dilute suspensions for low values of acoustic radius, Ultrasonic 28 (1990) 16–25.

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[67] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. New theory and CVI experiment, Langmuir 15 (20) (1999) 6692–6706. [68] A.S. Dukhin, V.N. Shilov, H. Ohshima, P.J. Goetz, Electroacoustics phenomena in concentrated dispersions. Effect of the surface conductivity, Langmuir 16 (2000) 2615–2620. [69] M. Povey, The application of acoustics to the characterization of particulate suspensions, in: V. Hackley, J. Texter (Eds.), Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, Am. Ceramic Soc., Ohio, 1998. [70] J.D. McClements, Ultrasonic determination of depletion flocculation in oil-in-water emulsions containing a non-ionic surfactant, Colloids Surf. 90 (1994) 25–35. [71] D.J. McClements, Comparison of multiple scattering theories with experimental measurements in emulsions, J. Acoust. Soc. Am. 91 (2) (1992) 849–854. [72] A.S. Dukhin, P.J. Goetz, Acoustic and electroacoustic spectroscopy, Langmuir 12 (19) (1996) 4336–4344. [73] A.S. Dukhin, P.J. Goetz, Acoustic and electroacoustic spectroscopy for characterizing concentrated dispersions and emulsions, Adv. Colloid Interface Sci. 92 (2001) 73–132. [74] A.S. Dukhin, P.J. Goetz, New developments in acoustic and electroacoustic spectroscopy for characterizing concentrated dispersions, Colloids Surf. 192 (2001) 267–306. [75] A.S. Dukhin, P.J. Goetz, Ultrasound for Characterizing Colloids, Elsevier, The Netherlands, 2002. [76] V.N. Shilov, Y.B. Borkovskaya, A.S. Dukhin, Electroacoustic theory for concentrated colloids with overlapped dls at arbitrary ka. Application to nanocolloids and nonaqueous colloids, J. Colloid Interface Sci. 277 (2004) 347–358. [77] A.S. Dukhin, P.J. Goetz, M. Thommes, Method for determining porosity, pore size and zeta potential of porous bodies, US Patent, pending, 2009. [78] L. DeMaeyer, M. Eigen, J. Suarez, Dielectric dispersion and chemical relaxation, J. Am. Chem. Soc. 90 (1968) 3157–3161. [79] V.N. Shilov, A.S. Dukhin, Sound-induced thermophoresis and thermodiffusion in electric double layer of disperse particles and electroacoustics of concentrated colloids, Langmuir (submitted for publication). [80] A.S. Dukhin, V.N. Shilov, Yu. Borkovskaya, Dynamic electrophoretic mobility in concentrated dispersed systems. Cell model, Langmuir 15 (10) (1999) 3452–3457. [81] A.S. Dukhin, P.J. Goetz, Characterization of aggregation phenomena by means of acoustic and electroacoustic spectroscopy, Colloids Surf. 144 (1998) 49–58. [82] A.S. Dukhin, P.J. Goetz, Method and device for characterizing particle size distribution and zeta potential in concentrated system by means of acoustic and electroacoustic spectroscopy, US Patent 09/108,072, 2000. [83] A.S. Dukhin, P.J. Goetz, Method and device for determining particle size distribution and zeta potential in concentrated dispersions, US Patent, pending. [84] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth study of ultrasound velocity and attenuation in suspensions: comparison of theory with experimental measurements, J. Colloid Interface Sci. 156 (1993) 261–269. [85] A.K. Holmes, R.E. Challis, D.J. Wedlock, A wide-bandwidth ultrasonic study of suspensions: the variation of velocity and attenuation with particle size, J. Colloid Interface Sci. 168 (1994) 339–348. [86] A.S. Dukhin, P.J. Goetz, Acoustic spectroscopy for concentrated polydisperse colloids with high density contrast, Langmuir 12 (1996) 4987–4997. [87] T.H. Wines, A.S. Dukhin, P. Somasundaran, Acoustic spectroscopy for characterizing heptane/water/AOT reverse microemulsion, JCIS 216 (1999) 303–308.

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[88] C. Bohren, D. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, NY, 1983. [89] L. Kinsler, A. Frey, A. Coppens, J. Sanders, Fundamentals of Acoustics, Wiley, New York, NY, 2000. [90] D. Blackstock, Fundamentals of Physical Acoustics, Wiley, New York, NY, 2000. [91] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65.

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Chapter 2

Fundamentals of Interface and Colloid Science

2.1. Real and Model Systems 2.2. Particulates and Porous Systems 2.3. Parameters of the Model Dispersion Medium 2.3.1. Gravimetric Parameters 2.3.2. Rheological Parameters 2.3.3. Acoustic Parameters 2.3.4. Thermodynamic Parameters 2.3.5. Electrodynamic Parameters 2.3.6. Electroacoustic Parameters 2.3.7. Chemical Composition 2.3.8. Electrochemical Composition of Aqueous and Nonaqueous Solutions 2.4. Parameters of the Model Dispersed Phase 2.4.1. Rigid Versus Soft Particles 2.4.2. Solid Versus Fractal and Porous Particles

23 25 26 26 26 27 28 28 30 30

31 34 36

2.4.3. Porous Body 39 2.4.4. Particle Size Distribution 40 2.5. Parameters of the Model Interfacial Layer 44 2.5.1. Flat Surfaces 46 2.5.2. Spherical DL, Isolated and Overlapped 47 2.5.3. Electric Double Layer at High Ionic Strength 49 2.5.4. Polarized State of the Electric Double Layer 51 2.6. Interactions in Colloid Interface Science 54 2.6.1. Interactions of Colloid Particles in Equilibrium: Colloid Stability 54 2.6.2. Biospecific Interactions 58 2.6.3. Interaction in a Hydrodynamic Field: Cell and CoreShell Models— Rheology 60

37

Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23002-8 Copyright # 2010, Elsevier B.V. All rights reserved.

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CHAPTER

2.6.4. Linear Interaction in an Electric Field: Electrokinetics and Dielectric Spectroscopy 66 2.6.5. Nonlinear Interaction in the Electric Field: Nonlinear

Electrophoresis, Electrocoagulation, and Electrorheology 2.7. Traditional Particle Sizing 2.7.1. Light Scattering: Extinction = Scattering + Absorption References

2

72 77

78 82

The goal of this chapter is to provide a general description of the objects that are referred to in the title of this book, and related phenomena. We introduce here some terms that will be used to characterize these objects, together with a brief overview of corresponding theories and traditional measuring techniques. There are several changes compared to the first edition. We added “porous” and “fractal” particles to the list of objects that can be studied with ultrasound. There is also a section on electrochemistry and double-layer structure in nonpolar liquids. We also included a section on “biospecific interactions.” It has gained major importance lately because of nanotechnology as well as ecological and human health concerns. There are also some minor modifications made to several other sections, such as electrophoresis of particulates with overlapped double layers, gravity as a factor of aggregative stability, etc. In writing this section, we follow Lyklema’s definition of a colloid as “an entity, having at least in one direction a dimension between 1 nm and 1 mm, that is, between 10–9 and 10–6 m. The entities may be solid, liquid, or, in some cases, even gaseous. They are dispersed in the medium” [1]. This medium may be also solid, liquid, or gaseous. We have found it more useful in this book to refer to these as “entities” instead as the “dispersed phase,” and the “medium” as the “dispersion medium.” There is an enormous variety of systems satisfying this wide definition of a colloid. Given three states for the dispersed phase and three states for the dispersion medium allows, altogether, nine possible varieties of colloids. For the purpose of this book, we restrict ourselves to just three of these possibilities, namely a collection of solid, liquid, or gas particles dispersed in a single liquid medium. All our colloids can, therefore, be defined as “a collection of particles immersed in a liquid”; the particles can be solid (a suspension or dispersion), liquid (an emulsion), or gas (a foam). These three types of dispersed systems play an important role in all kinds of applications: paints, latices, food products, cements, minerals, ceramics, blood, and many others. All these disperse systems have one common feature; because of their small size, they all have a high surface area relative to their volume. It is surface-related phenomena, therefore, that primarily determine their behavior in various processes and justify consideration of colloids as effectively a different state of matter.

Fundamentals of Interface and Colloid Science

23

What follows are the combined terms and methods that have been created over the course of two centuries to characterize these very special systems. In addition to Lyklema’s book, we also incorporate terminology from Nomenclature in Dispersion Science and Technology published by the US National Institute of Standards and Technology [2].

2.1. REAL AND MODEL SYSTEMS Real heterogeneous systems, by their very nature, are quite complex. Two of these complexities are the variation of the particle size and shape within a particular dispersion. Furthermore, the particles might not even be separate, but linked together to form some network, or the liquid might contain complex additives. This inherent complexity makes it difficult, perhaps impossible, to fully characterize a real heterogeneous system except in the simplest cases. However, we do not always need a complete, detailed characterization; a somewhat simplified picture can often be sufficient for a given particular purpose. There are two approaches to obtaining a simplified picture of any real system under investigation. The first approach is “phenomenological” [1], in that it relates only experimentally measured variables, and usually is based on simple thermodynamic principles. Usually, we model real system as “homogeneous” using this approach. For instance, we can characterize milk or blood as imaginary homogeneous liquids if we are looking for, let us say, viscosity of these liquids. This can be a very powerful approach, but, typically, it does not yield very much insight into any underlying structure within the system. The second approach is “modeling,” in which we replace the practical system with some approximation based on an imaginary “model system.” This model of the system is an attempt to describe a real liquid or colloid in terms of a set of simplified model parameters including, of course, the characteristics that we hope to determine. The model, in effect, makes a set of assumptions about the real world in order to reduce the complexity of the colloid, and thereby also simplifies the task of developing a suitable prediction theory. It is important to recognize that in substituting a model system for the real system, we introduce a certain “modeling error” into the characterization process. Sometimes, this error may be so large as to make this substitution inadequate or, perhaps, even worthless. For example, most particle-size measuring instruments substitute a model that assumes that the particles are spherical. In this case, a complete geometrical description of a single particle is specified using just one parameter, its diameter d. Such a model, obviously, would not adequately describe a dispersion of long, thin carpet fibers that have a high aspect ratio, and any theory based on this oversimplified model might well give very misleading results. This is an example of an extreme case in which the modeling error makes the model system completely inappropriate. It is important to acknowledge that some “modeling error” almost always exists. We can reduce it by adopting a more complicated model in response to the complexities of the real-world colloid, but we can seldom make it vanish.

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In turn, the selected model then limits, or increases, the complexity of the theory that must be brought into play to adequately predict any acoustic properties based on the chosen model. Having chosen a model for our colloid, one can then select and develop a more appropriate theory. In the course of such a development, certain theoretical assumptions, or approximations, which introduce additional “theoretical errors,” may be made. These theoretical errors reflect our inability to completely predict even the behavior of our already simplified model system. However, it would be pointless to use some overly elaborate and complex theory in an effort to reduce these theoretical errors to values much less than the already unavoidable modeling errors. In performing experimental tests to confirm various theories, it is of course necessary to minimize the difference between the real system used for the tests and the model system on which the proposed theory is based. For example, the model colloid mentioned by Lyklema [1] (a collection of spherical particles with a certain size in the liquid) describes quite precisely a real colloid consisting of a suspension of monodisperse latex particles. The design of any model liquid or colloid is also strongly related to the notion of a “dimensional hierarchy,” which we introduce here to generalize the rather widely accepted idea of macroscopic and microscopic levels of characterization. For instance, Lyklema described macroscopic laws as dealing on a large scale with “gigantic numbers of molecules.” In contrast, the microscopic level seeks an ab initio interpretation based on a very small scale or a molecular picture. In dealing with colloids, we further need to expand this simple picture of a merely two-dimensional hierarchy. Certainly, we can speak of a molecular, microscopic level, and the colloid as a whole can be viewed on a macroscopic level. But there are intermediate levels of importance as well. For instance, the size of a typical colloid particle, which lies in between the molecular and macroscopic dimension ranges, is also an important dimension in its own right, and we refer to this as the “single-particle level.” For many colloids, other key dimensions might also be important. For example, in a polymer system, we may wish to speak of a dimension level related to the length of the polymer chains. In a flocculated system of particles, a dimension related to the floc size may be useful. For a given colloid, we refer to the complete set of all important dimensions as its “dimensional hierarchy.” The design of a suitable “model colloid,” and the related theory, must somehow reflect the “dimensional hierarchy” of the real system. The model might place more emphasis on one dimension than on another; it depends on the final goal of the characterization. As an example, consider a dispersion of biological cells in saline. We might create an elaborate model at the single-particle level by modeling the cells as multilayer spheres with built-in ionic pumps, etc., while very simply at the “microscopic level” by modeling the saline dispersion medium as a continuum with a certain viscosity and density.

Fundamentals of Interface and Colloid Science

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However, there is one important dimension common to all colloids, namely the thickness of the interfacial layer. The behavior of this boundary layer plays a key role in many colloid-related phenomena and must also be adequately reflected in the design of the “model colloid.” The complexity of real colloids, as well as the corresponding diversity in the dimensional hierarchy necessary to depict these colloids, makes it practically impossible to provide a description of all the colloid models that have been proposed over the years. Historically, the more common models include the microscopic level to describe the dispersion medium, the single-particle level to characterize the disperse phase, and the interfacial level to describe the interface boundary layers. We present below some general models that have been developed during the last two centuries for each of these three levels. It is important to stress that there are two distinctively different models of liquids and colloids: “homogeneous” and “heterogeneous.” The differences between homogeneous and heterogeneous models are results of our perception and goals of the characterization procedure. Practically, any liquids and colloids can be homogeneous from some aspects and heterogeneous from the others. For instance, water becomes heterogeneous if we are interested in the interaction of water molecules between themselves. We will show that acoustics yields the possibility of describing complex systems using both homogeneous and heterogeneous models.

2.2. PARTICULATES AND POROUS SYSTEMS Complex heterogeneous systems represent the main class of objects that are being considered in this book. These systems have distinctively different phases—dispersion medium and dispersed phase. There is possibility of splitting this group of objects into two groups depending on the state of the continuous phase. If continuous phase is liquid, then corresponding systems could be referred to with a common name of “particulates.” Dispersed phase in this case is finitely divided on small pieces—particles, which float in the liquid dispersion medium. An alternative situation exists in porous bodies with a continuous solid matrix, which represents essentially a solid dispersion medium. This matrix could have voids with dimensions in the above-specified range. These voids (pores) can be filled with a gas or liquid. There are also systems that bear features of both particulates and porous bodies. Dispersion of porous chromatographic silica particles is an example of particulate dispersion with porous particles. There is also the possibility of smooth transition from a particulate system to a porous system. For instance, this occurs when sedimenting particles form a deposit. A deposit is essentially a porous body. Sedimentation transforms a particulate system into a porous one. Ultrasound offers the possibility of studying both of them simultaneously and learning important information from such a transition.

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2.3. PARAMETERS OF THE MODEL DISPERSION MEDIUM Although we have restricted our attention to colloids having only liquid media, we need not limit ourselves to pure liquids. The dispersion medium might be a mixture of different liquids, a liquid with many dissolved ion species, or some dissolved polymers. We can even include in our medium definition the possibility that the medium itself might be a colloid. For example, consider the case of kaolin clay dispersed in water and stabilized by the addition of latex particles. In reality, this suspension contains two types of dispersed particles: kaolin and latex. However, an alternative view would be to consider the kaolin particles dispersed in a new latex/water “effective dispersion medium.” This reduces a complex, three-phase colloid to a much simpler two-phase colloid, but the price that we must pay is a more complex model for this “effective dispersion medium.” Basically, we are playing here with two concepts: a homogeneous medium and a heterogeneous medium. Depending on the dimensional scale of the effect of interest, we can consider everything with lesser dimensions as the homogeneous medium. At the same time, the colloid system is described as a heterogeneous medium. Let us assume first that the dispersion medium is a simple, pure liquid. For clarity, we will organize the parameters that characterize this pure liquid into six classes: gravimetric, rheological, acoustic, thermodynamic, electrodynamic, and electroacoustic. To some extent, all of these parameters are temperature dependent; this dependence is strong for some parameters and negligible for others. Some of these parameters also depend on the frequency of the ultrasound, which we will discuss in the following text. The effect of the chemical composition of the liquid on these properties is considered in the last section of this chapter.

2.3.1. Gravimetric Parameters The density of the dispersing medium, rm, is an intensive (i.e., independent on the sample volume) parameter that characterizes the liquid mass per unit volume and is expressed in units of kg/m3. Since the volume of the medium changes with temperature, the density is also temperature dependent. There are some empirical expressions for this temperature dependency [3]. The density of the medium is assumed to be independent of the ultrasound frequency.

2.3.2. Rheological Parameters Viscosity and elasticity are rheological parameters of general interest for any material. Most pure liquids are primarily viscous, having little elasticity. Many traditional rheological instruments cannot measure the elastic

Fundamentals of Interface and Colloid Science

27

component, or, if possible, can measure it only at low frequencies. Elasticity yields information about structure. Small elasticity implies a weak structure. Acoustics provides a means to measure the elastic properties of liquids and to characterize their structure at much higher frequencies than those available with more traditional methods [4–6]. The elastic properties of the liquid are defined by a bulk storage modulus, Mm, which is the reciprocal of the liquid compressibility, bpm . This compressibility is the change in the liquid volume due to a unit variation of pressure. It is usually assumed that the liquid exhibits no yield stress. The SI unit of the bulk modulus is Pascal (Pa), which is equal to 0.1 dyne cm2 in CGS units. The viscous properties of the liquid are characterized by two dynamic viscosity coefficients: a shear dynamic viscosity,
m, and a volume dynamic viscosity, hvm [7]. The volume viscosity appears only in phenomena related to the compressibility of the liquid. All effects that can be described assuming an incompressible liquid are independent of the volume viscosity. This is why the volume viscosity is hard to measure and, therefore, is usually assumed to be zero. Ultrasonic attenuation is the only known way to measure volume viscosity [6]. It turns out that all liquids exhibit a volume viscosity. For example, the volume viscosity of water is almost three times larger than the shear viscosity. The volume viscosity depends on the structure of the liquid. Hence acoustic attenuation measurements potentially provide important information about the liquid structure. The SI unit for these dynamic viscosities is [Pa s], which is equal to 1000 centipoise (cP) in CGS units. Rheological parameters usually depend on temperature; some empirical expressions for this dependency can be found in the Handbook of Chemistry and Physics [3]. These parameters are independent of the frequency of the ultrasound for simple Newtonian liquids.

2.3.3. Acoustic Parameters Rheological and acoustic properties are closely related; each provides a different perspective of the viscoelelastic nature of the fluid. There are two key acoustic parameters: the attenuation coefficient, am, and the sound speed, cm. We have already described two key rheological parameters: the bulk storage modulus, Mm, and the dynamic viscosity,
m. The bulk modulus and sound speed both characterize the elastic nature of the liquid; the dynamic viscosity and attenuation both characterize its viscous, or dissipative, nature. Elastic and dissipative characteristics can be combined into one complex parameter. In acoustics, this complex parameter is referred to as a “compression complex wave number,” kc,m. The attenuation and sound speed are

28

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related to the imaginary and real parts of this complex wave number by the expressions: am ¼  Imkc,m cm ¼

o Rekc,m

ð2:1Þ ð2:2Þ

The unit of sound speed is [m/s]. The attenuation can be expressed in several different ways as discussed in Chapter 3. In all of the following text, we define attenuation in units of [dB/cm MHz]. Another complex parameter, also widely used in acoustics, is the acoustic impedance, Zm. It is defined as  
cm am  am l ð2:3Þ ¼ cm rm 1  j Zm ¼ cm rm 1  j o 2p where l is a wavelength of the ultrasound in the liquid. The attenuation is almost independent of variations in temperature, whereas the sound speed is very sensitive to temperature. For example, the sound speed of water changes 2.4 m/s per degree Celsius at room temperature. This is an important distinction between these two parameters. The frequency dependence of these two parameters is also quite different. For simple Newtonian liquids, the sound speed is practically independent of frequency, whereas the attenuation is roughly proportional to frequency, at least over the frequency range of 1–100 MHz (see Chapter 3 for details).

2.3.4. Thermodynamic Parameters There are three thermodynamic parameters that are important in characterizing the thermal effects in a colloid: thermal conductivity, tm [mW/cm  C], heat –4   capacity Cm p [J/g C], and thermal expansion bm [10 1/ C]. Fortunately, it turns out that for almost all liquids, except water, tm is virtually the same, and Cm p can be approximated by a simple function of the material density. Figures 2.1 and 2.2 illustrate the variation of both properties for more than 100 liquids; the data is taken from the paper of Anson and Chivers [8] as well as our own database. This reduces the number of required thermodynamic properties to just one, namely the thermal expansion coefficient. In principle, all thermodynamic parameters are temperature dependent, but independent of the ultrasound frequency.

2.3.5. Electrodynamic Parameters The two essential electrodynamic parameters that characterize the response of the liquid to an applied electric field are conductivity, Km, and dielectric permittivity, e0em.

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Fundamentals of Interface and Colloid Science

5 water

Cp [J/g k]

4 3 2 1 0 0.5

1.0

1.5

2.0 density [g/cu.cm]

2.5

3.0

3.5

FIGURE 2.1 Heat capacity at constant pressure for various liquids, Anson and Chivers [8].

thermal conductivity [mW/cm K]

7 water

6 5 4 3 2 1 0

0.5

1.0

1.5

2.0 density [g/cu.cm]

2.5

3.0

3.5

FIGURE 2.2 Thermal conductance for various liquids, Anson and Chivers [8].

The SI unit of conductivity is S/m [C/(s V m)] and of dielectric permittivity [C2/(Nm2)]. The relative dielectric permittivity, em, (dielectric constant) of the liquid is the dielectric permittivity of the medium divided by that of vacuum, e0, and is dimensionless. The dielectric permittivity of a vacuum, e0, is 8.8542  1012 C2/N m2. In principle, both the conductivity and dielectric permittivity are frequency dependent, even for a pure liquid. However, for the typical frequency range of acoustic measurements (1–100 MHz), both parameters are usually assumed to be independent of frequency.

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The temperature dependence of the dielectric permittivity for some liquids is given in The Handbook of Chemistry and Physics [3]. These two parameters can be combined into one complex number (either a  complex dielectric permittivity, em , or a complex conductivity, Km ). This simplifies the calculations with respect to an applied alternating electric field:  ¼ Km  joe0 em Km

  Km em ¼ em 1 þ j oem e0

ð2:4Þ ð2:5Þ

2.3.6. Electroacoustic Parameters There are two reciprocal electroacoustic effects that can be differentiated depending on the nature of the driving force. For ultrasound as the driving force, we speak of the ion vibration current as representative of the electroacoustic effect. The ion vibration current is an alternating current generated by the motion of ions in the sound field and is characterized by a magnitude, IVI, and phase, Fivi. The IVI and phase, when normalized by the magnitude and phase of the acoustic pressure, can be considered to be the electroacoustic properties of the dispersion medium. In the reverse case, when an electric field is the driving force, the resultant ultrasound signal, called electrosonic amplitude [9, 10], represents the electroacoustic effect. It is also characterized by a magnitude, ESA, and phase, Fesa. These also can be used as electroacoustic properties of the dispersion medium. The magnitude of both electroacoustic effects is much less sensitive to temperature than to phase. Both IVI and ESA can be considered to be frequency independent within the megahertz frequency range. Both electroacoustic effects are strongly dependent on the chemical composition of the liquid.

2.3.7. Chemical Composition The dispersion medium may contain many different chemical species. The collection of specified amounts of these species is called its chemical composition. Each species is characterized with an electrochemical potential, mi, given by: mi ¼ m0i þ 2RT lnyi ci þ zi Ff

ð2:6Þ

where m0i is a standard chemical potential that describes the interaction between species i and the liquid, ci is the concentration of that species, yi is an activity coefficient, which depends on the interaction between species, R is the Faraday constant, T is the absolute temperature, zi is the valence of that species if electrically charged, and f is an electric potential. Finally, there are three additional parameters that characterize the dynamic response of the solution. The best known is the diffusion coefficient, Di.

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In addition, we will need the mass, mi, and volume, Vi, of any solvated particles, especially for electrically charged species. These last two parameters play an important role in the electroacoustics of solvated ions. Although all properties of the dispersion medium depend on this chemical composition, the degree of this dependence is not the same for each property. In many cases, we can neglect the chemical composition. For instance, the viscosity of a liquid depends on the concentration of ions. The continuous adjustment of the hydration layer upon ion displacement leads to friction and hence to energy dissipation. This process is known as dielectric drag. For very small ions, this friction may well be stronger than for larger ions. Consequently, the viscosity,
s, of an aqueous electrolyte solution increases with ion concentration, c, and generally follows the Jones-Dole law:
s ¼ 1 þ Ac0:5 þ Bc þ   
w where
w is the viscosity of water. However, the parameters A and B are rather small and we can usually neglect this effect when the concentration of ions is low. For example, the viscosity of 1 M KCl (aq.) decreases only by 1% compared to that of pure water [3]. This does not mean that we can always neglect the influence of the chemical composition on the viscosity of the medium. Many chemicals strongly affect the viscosity of the medium, and the dimension of these viscositymodifying species is very important. In calculating particle size from acoustic data, the relevant viscosity to be used is the one experienced by the particle as it moves in response to the sound wave. In the case of gels, or other structured systems, this “microviscosity” may be significantly less than the “macroviscosity” measured with conventional rheometers. This example of viscosity thus highlights the complexity regarding the influence of chemical composition on a particular property of the dispersion medium. In general, there is no theoretically justified answer that will be valid in all cases. However, in many situations empirical experience does provide a guide. From our experience, the chemical composition is always important when considering sound speed, conductivity, and the electroacoustic signal. As a result, these are good candidates for studying the properties of ions, molecules, and other simple species in liquids. Conversely, attenuation is the least sensitive to chemical composition. This makes the attenuation measurement a very good candidate for a robust particle sizing technique.

2.3.8. Electrochemical Composition of Aqueous and Nonaqueous Solutions On reviewing the existing literature, we have discovered that despite giving a quite detailed model of aqueous ionic solutions, modern electrochemistry still faces problem with low- and nonpolar liquids. For instance, the main reviews

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on this subject [11–18] do not provide a simple instruction on how to control ionic composition of the nonpolar liquids, how to measure ions concentration, and how to characterize ionic properties that would be needed for describing the double-layer structure. Complexity of these questions has been recognized for a long time. For instance, Lyklema wrote 40 years ago that “. . .electrolyte concentrations in apolar solvents are low and ill-defined. . ..” [17]. This statement remains valid to date to a certain degree. Uncertainty in the electrolyte concentration creates a profound problem for electrokinetcs in nonpolar liquids. For instance, it creates problem in calculating the Debye length, 1/k. Major differences in electrochemistry of aqueous and nonaqueous solutions are related to dielectric permittivity. It is known that water is an abnormal liquid in many aspects. It has very high dielectric permittivity, around 80. Conductivity of water-based solutions is also high. It varies from roughly 10–3 to 10 S/m depending on the concentration of the electrolyte. These peculiarities of water are related to the high polarity of its molecules. There are many other liquids with much lower dielectric permittivity, around 2, and much smaller conductivity, on the scale of 10–10 S/m. Here are some examples: diesel, kerosene, choroform, cyclohexane, decane, hexane, heptane, hexadecane, toluene, and all varieties of oils. These hydrocarbon liquids are practically nonpolar. Lyklema and van der Hoeven [12] suggested four classes of liquids based on the value of dielectric permittivity: l l l l

nonpolar if e < 5 low-polar 5 < e < 12 semipolar e > 12 polar e ffi 80

Dielectric permittivity affects strongly the ionization of various chemical species in liquid. Existence of ions in a liquid is the result of the balance between electrostatic attractive energy and thermal motion, as stated by the Bjerrum theory [14, 16]. This theory simply compares the attractive energy of the cation-anion interaction, E, with the energy of thermal motion, kT, where k is the Boltzman constant and T is the absolute temperature. This theory predicts that ions might exist separately only if their size aion exceeds a certain value, the so-called Bjerrum radius aB. Otherwise, they would build cationanion pairs because the thermal motion would not be able to prevent their aggregation at small distances. This condition can be formulated as the following nonequality: aion > aB ¼

e2 8pee0 kT

ð2:7Þ

Fundamentals of Interface and Colloid Science

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Statistically, some ions would exist even if their sizes are below the Bjerrum radius because there is some equilibrium distribution between free ions and pairs. There is a theory describing this distribution by Fuoss [15]. It yields the following equation for the dissociation constant aion:  2 2  4pd3 ze 2 n0 f exp ð2:8Þ aion ¼ 1  3 dee0 kT where d is the center-to-center distance in the ion pair, n0 is the number of ions per unit volume, z is valency, and f is the activity coefficient which is close to 1 in dilute systems. Simple calculations led Morrison [13] to the conclusion that the critical ion diameter is 0.7 nm in water and 28 nm in low-conducting media with a low dielectric permittivity of around 2. In both cases, the critical Bjerrum radius exceeds significantly the size of the low molecular weight atoms. In aqueous systems, the size of the ions builds because of the adsorption of water molecules. They are polar with substantial dipole moment, which can build a structure around the ion under the influence of its electric field. This hydration shell around the ions prevents them in coming close enough for building an ionic pair. They are sterically stabilized, as mentioned by Morrison [13]. A low-conducting nonpolar medium does not allow such steric stabilization because molecules of the liquid do not have dipole moment and are incapable of building protective shells around the ions. There are two opportunities for the ions to exist in these conditions. According to Kitahara [11], ions can be present in the nonpolar media if l l

soluble molecules dissociate into large ions; dissociation occurs with further solubilization of the simple, small ions in larger micelles.

Both approaches are used for controlling conductivity of nonaqueous solutions. The first one is usually associated with ionic surfactants. The second approach opens an interesting opportunity of using nonionic surfactants for building a protective solvating shell around ions. Figure 2.3 illustrates this idea. The surfactant’s polar parts would be attracted by the ions. This attraction builds up a surfactant shell around the ions. Ions become encapsulated into the reverse micelle. It is important to mention here that this micelle is quite different from a regular reverse micelle built by the attraction of the polar groups. It has an ion at the center. This could make this type of micelles quite different. It is possible also that there would be difference between micelles with cations and anions. They require opposite orientation of the polar group. It might happen that the surfactant structure would not allow either cations or anions reaching the required pole of the surfactant dipole moment. In this

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Cation micelle in oil

Ion in water

−

−

2

− + +

−

+ +

+

−

−

+

−

+

+

− −

+

Typical micelle in oil Anion micelle in oil

− −

+ + + +

−

−

− −

+ +

−

+ +

−

−

FIGURE 2.3 Various solvation structures in water and oil.

case, the surfactant would stabilize only one type of ions. The other type of ions remains bare. It would be possible for it to exist in small size because of the lack of accessible ions with the opposite charge. They are screened by a large shell of the surfactant. This hypothesis has found some experimental confirmation [19]. This field is far from complete in our understanding. It is clear that properties of ions in nonpolar liquids are rather difficult to get. There is practically no data on their diffusion coefficients. Very little is known about the means of controlling their concentration. It turns out that electroacoustics can be useful for shedding some light on this problem, as will be discussed in chapters dedicated to its applications.

2.4. PARAMETERS OF THE MODEL DISPERSED PHASE First, any model for particles in the dispersed phase must describe the properties of a single particle: its shape, physical and chemical properties, etc. The model must also reflect any possible polydispersity, that is, the variation in properties from one particle to another. This is a rather difficult task. Fortunately, at least with particle shape, there is one factor that helps— Brownian motion. Colloidal particles are generally irregular in shape, as illustrated in Figure 2.4. There are of course a small number of exceptions such as

35

Fundamentals of Interface and Colloid Science

+

Real System

- roughness - heterogeneous (mosaic) surface - hairness

-

-

-

+

-

-

- nonspherical

+

-

-

+

+

system modeling error

-

-

Model System

-

- spherical - smooth - homogeneous surface

-

-

-

FIGURE 2.4 Real and model spherical particle.

emulsions or latices. All colloidal particles experience Brownian motion which constantly changes their position. A natural averaging occurs over time; the resulting time-averaged particle looks like a sphere with a certain “equivalent diameter,” d. It is for this reason that a sphere is the most widely and successfully used model for particle shape. Obviously, a spherical model does not work well for very asymmetric particles. Here, an ellipsoidal model can yield more useful results [20–22]. This adds another geometrical property: an aspect ratio, the ratio of the longest dimension to the shortest dimension. For such asymmetric particles, the orientation also becomes important, bringing further complexity to the problem. Here we will simply assume that the particles can be adequately represented by spheres. This assumption affects ultrasound absorption much more difficult than ultrasound scattering, as is the case with light scattering. According to both our experience and preliminary calculations for ultrasound absorption [22], this assumption of a sphere is valid for aspect ratios less than 5:1. The material of the dispersed phase particles is characterized with the same set of gravimetric, rheological, thermodynamic, electrodynamic, acoustic, and electroacoustic parameters as the material of the dispersion medium described in the previous section. We will use the same symbols for these parameters, changing only the subscript to p. The fact that we define so many properties for both the medium and the particles does not mean that in order to derive useful information from the

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measured acoustic or electroacoustic measurement, we need to know this complete set for a particular dispersion. Later, in Chapter 4, we will outline the set of input properties that are necessary to constructively use ultrasound measuring techniques. The exact composition of this required set depends on the nature of the particles, particularly on their rheology. In this regard, it turns out that it is very useful to divide all particles into just two classes, namely rigid and soft.

2.4.1. Rigid Versus Soft Particles Historically, Morse and Ingard [5] introduced these two classes of rigid and nonrigid (soft) particles to the acoustic theory for describing ultrasound scattering. In the case of scattering, the notion of a rigid particle serves as an extreme but not real case. It turns out that this classification is even more important in the case of ultrasound absorption. In the case of ultrasound absorption, the notion of rigid particles corresponds to real colloids, such as all kinds of minerals, oxides, as well as inorganic and organic pigments: indeed, virtually everything except emulsions, latices and biological material. Rigid particles dissipate (absorb) ultrasound without changing their shape under either an applied sound field or an electric field. For interpreting ultrasound absorption caused by rigid particles, we need to know, primarily, the density of the material. There is no need to know any other gravimetric or acoustic properties or, in fact, any rheological, thermodynamic, electrodynamic, or electroacoustic properties at all. Furthermore, the sound speed in these rigid particles is, usually, close to 6000 m/s. Thus, in most cases, we can simply assume that this same value is appropriate. The sound attenuation inside the body of these particles is usually negligible. We do not need to know the electrodynamic properties, except in rare cases of conducting and semiconducting particles. It is important to mention here that submicrometer metal particles are not truly conducting. There is not much difference in electric potential between the particle poles to conduct an electrochemical reaction across the small distance between these poles. Such submicrometer particles represent a special case of the so-called “ideally polarized particles” [23]. These empirical observations reduce the number of parameters required to characterize rigid particles to just one, namely the density. It is also important to mention that most often the density of these particles significantly exceeds that of the liquid medium. The situation with soft particles is very different. First, the density of these softer particles quite often closely matches that of the dispersion medium. As a result, all effects that depend on the density contrast are minimized. Instead, the thermodynamic effects are much more important, and the thermal expansion coefficient of the particle and suspending medium replace the density contrast as a critical parameter for ultrasound absorption.

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Fundamentals of Interface and Colloid Science

So far, we have talked about particles that are internally homogeneous, constructed from the same material. However, there are some particles that are heterogeneous inside, consisting of different materials at different points within their bodies. Usually a “shell model” is used to describe the properties of these complex particles [24, 25]. Occasionally, a simple averaging of properties over the particle volume suffices, as for instance, using an average density.

2.4.2. Solid Versus Fractal and Porous Particles There are three different models that can be used for describing dispersed particles depending on their morphology. The most obvious and widely used is simply a model of “separate homogeneous solid particles.” This is the usual model for characterizing dispersions and emulsions. We simply assume that particles consist only of the dispersed phase material. This assumption yields a simple relationship between densities and weight fractions: rp ¼ rs wp ¼ ws where the index p corresponds to the particles and index s to the solid material. An alternative model of porous particles takes into account the possibility that part of the liquid would be trapped inside the particles with complex shape. This might occur either for naturally porous materials or for aggregate particles that are built up from some primary solid particles. This model introduces an additional parameter for characterizing particles—porosity, w. It is the volume fraction of the dispersion medium trapped inside of the particle. This model yields the following equations for the relationship between densities and weight fractions: rp ¼ ð1  Oe Þðrs  rm Þ þ rm fp ¼

fs w s rm ¼ 1  w ð1  wÞ½ws rm þ ð1  ws Þrs  

wrm wp ¼ w s 1 þ ð1  wÞrs

ð2:9Þ ð2:10Þ

 ð2:11Þ

The nature of this model eliminates possibility of advection—liquid motion through the particle, because we consider liquid inside of the pores trapped and moving with the particle. There is another model for porous particles that assumes certain symmetry in their structure. It is the “fractal model.”

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Fractals are aggregates of primary dispersed particles with a certain structural symmetry that reproduces itself from one spatial level to another. The notion of fractals was introduced almost 30 years ago by Mandelbrot [26–28] and is a widely accepted model for describing coagulation phenomena. There is a simple relationship between the radius of an aggregate that contains i particles (ai) and radius of the primary particles a1 given by ai ¼ kf i1=df a1

ð2:12Þ

The two parameters in this equation describe various types of aggregate structures. The parameter kf is less important, as its value is typically very close to 1, and we therefore omit considering it in our further analysis. The parameter df is more important and usually goes by the name of “fractal dimension.” For typical dispersions, this number varies from roughly 1 to 3. It equals 3 for coalescing emulsion droplets. The range of 2.1–2.2 corresponds to reaction-limited coagulation, which results in rather compact aggregates. Smaller values of this fractal dimension, less than 1.8, are typical of diffusion-limited coagulation with loose flocs. Finally, a fractal dimension equal to 1 represents linear chain aggregates [29–32]. Fractal aggregates contain a certain amount of trapped liquid inside. The amount of such trapped liquid can be quantified by one of three different parameters. The first way to describe this trapped liquid is by the porosity, w, which represents the volume ratio of the liquid and solid phases inside the aggregate:  3þdf ai w ¼ 1  fin ¼ 1  ð2:13Þ a1 where ’in is volume fraction of the solid phase inside the aggregate. An alternative parameter that can be used to describe the trapped liquid is the density of the aggregate, ragg, which is different from the density of the primary particles, rp, or the density of the liquid medium, rm, and is given by  3þdf ai ragg ¼ ðrp  rm Þ þ rm ð2:14Þ a1 A larger volume of aggregates leads to a larger volume fraction of the dispersed phase, ’agg, in the dispersion compared to the volume fraction of solids ’solids:  3df fagg ai ¼ ð2:15Þ fsolids a1 The weight fraction is given by the following equation: wp ¼

fp rp rm þ fp ðrp  rm Þ

ð2:16Þ

Fundamentals of Interface and Colloid Science

39

Of the three defined volume fractions, only ’solids is easily measurable with a pycnometer. Neither the aggregate volume fraction ’agg nor the volume fraction of solids inside the aggregate, ’in, is measurable with a pycnometer. The aggregates density, ragg, is also not easily measurable. However, there are known means for measuring the aggregates size. Light scattering offers one opportunity, as described in the literature [10, 20]. Acoustic attenuation spectroscopy is also suitable for this purpose as will be discussed in detail in this chapter. We will show that acoustics yields information on the aggregate size practically independently on the fractal number. There are many theoretical and experimental works dedicated to the hydrodynamic and mechanical properties of fractals, but very little is known about their electrical and, especially, their electrokinetic properties. This model ignores advection as well. This model requires also information on the primary size. It is possible to use this parameter as adjustable, instead of a standard deviation. This means that for the purpose of the fitting attenuation spectra we consider a monodisperse collection of aggregates with the same size and the same primary size. These two numbers are adjustable parameters instead of the median size and standard deviation of the lognormal particle size distribution (PSD).

2.4.3. Porous Body A porous body differs from a porous particle because of its large external dimension. It is assumed that it contains voids—pores with much smaller dimensions on micro and nanoscales. According to Lowell and others [33], pores can be classified as “micropores,” “mesopores,” and “macropores.” The internal width of micropores is less than 2 nm, mesopores between 2 and 50 nm, and macropores above 50 nm. This most known theory of sound propagation through a wet, porous body was developed by Biot in 1956 [34, 35], who mentioned the earlier work 10 years previously of Frenkel [36] as forming the essential scientific basis. The Biot theory is quite general and uses a number of input parameters. The total set consists of following physical properties of the solid matrix and liquid: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Density of sediment grains Bulk modulus of the grains Density of the pore fluid Bulk modulus of the pore fluid Viscosity of the pore fluid Porosity Pore size parameter Dynamic permeability Structure factor Complex shear modulus of the frame Complex bulk modulus of the frame

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The last four properties present the most problem in the application of the Biot theory. This was discussed in detail by Ogushwitz [37], who proposed several empirical and semiempirical methods for calculating these parameters. None of these methods is general. In some cases, it involves simply the substitution of one parameter with another unknown constant. This problem was also discussed by Barret-Guitepe et al. [38] in their study of compressibility of colloids. They talk about the importance of the “skeleton effect” and difficulty of measuring the required input parameters independently.

2.4.4. Particle Size Distribution Colloidal systems are generally of a polydisperse in nature, that is, the particles in a particular sample vary in size. Aggregation, for instance, usually leads to the formation of polydisperse sols, mainly because the formation of new nuclei and the growth of established nuclei occur simultaneously, and so the particles finally formed are grown from nuclei formed at different times. In order to characterize this polydispersity, we use the well-known concept of PSD. We define this here following Leschonski [39] and Irani [40]. There are multiple ways to represent the PSD depending on the physical principles and properties used to determine the particle size. A convenient general scheme proposed by Leschonski is reproduced in Table 2.1. The independent variable (abscissa) describes the physical property chosen to characterize the size, whereas the dependent variable (ordinate) characterizes the type and measure of the quantity. The different relative amounts of particles measured in certain size intervals form a so-called density distribution, qr(X), which represents the first derivative of the cumulative distribution, Qr (X). The subscript, r, indicates the type of the quantity chosen. Figures 2.5 and 2.6 illustrate the density and cumulative PSDs. The parameter, X, characterizes a physical property uniquely related to the particle size.

TABLE 2.1 Representation of PSD Data Type of quantity

Measure of quantity Cumulative Qr

r ðXÞ Density qr ðXÞ ¼ dQdX

r ¼ 0: number

Cumulative PSD on number basis

Density PSD on number basis

r ¼ 1: length

Figure 2.5

Figure 2.4

r ¼ 2: area

Cumulative PSD on area basis

Density PSD on area basis

r ¼ 3: volume, weight

Cumulative PSD on weight basis

Density PSD on weight basis

41

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Fundamentals of Interface and Colloid Science

qr (X)

DQr = qr (X) DX

DX

Xmin

Xmax

size [length]

FIGURE 2.5 Density particle size distribution.

1.0

Qr

Qr(X)

0.5

DQr(X1,X2) Qr(X2) Qr(X1)

0.0

DX Xmin

X1 X2 X50 size

Xmax

FIGURE 2.6 Cumulative particle size distribution.

It is assumed that a unique relationship exists between the physical property and a one-dimensional property unequivocally defining “size.” This can be achieved only approximately. For irregular particles, the concept of an “equivalent diameter” allows one to characterize irregular particles. This is the diameter of a sphere that yields the same value of any given

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physical property when analyzed under the same conditions as the irregularly shaped particle. The cumulative distribution Qr (X) in Figure 2.6 is normalized. This distribution determines the particles (in number, length, area, weight, or volume depending on r) that are smaller than the equivalent diameter X. Figure 2.7 shows the normalized discrete density distribution or histogram qr(Xi, Xiþ1). This distribution specifies the amount of particles having diameters larger than Xi and smaller than Xiþ1 and is given by: qr ðXi , Xiþ1 Þ ¼

Qr ðXiþ1 Þ  Qr ðXi Þ Xiþ1  Xi

ð2:17Þ

The shaded area represents the relative amount of the particles. The histogram transforms to a continuous density distribution when the thickness of the histogram column tends to zero. It is presented in Figure 2.7 and can be expressed as the first derivative from the cumulative distribution: dQr ðXÞ ð2:18Þ dX A histogram is suitable for presentation of the PSD when the value of each particle size fraction is known. It is the so-called full PSD. It can be measured using either counting or fractionation techniques. However, in many cases, full information about the PSD is either not available or not even required. For instance, Figure 2.8 represents the histograms with a nonmonotonic and nonsmooth variation of the column size. This might be related to the fluctuations and have nothing to do with the statistically representative PSD

qr

qr ðXÞ ¼

qr(Xi,Xi+1)

Xmin

Xi Xi+1

size

FIGURE 2.7 Discrete density particle size distribution.

Xmax

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Fundamentals of Interface and Colloid Science

qr (X)

modified log-normal distribution

histogram

Xmin

Xmax

size

FIGURE 2.8 Transition from the discrete to the continuous distribution.

for this sample. That is why in many cases histograms are replaced with various analytical PSDs. There are several analytical PSDs that approximately describe empirically the determined PSDs. One of the most useful, and widely used, distributions is the lognormal distribution. In general [40], it can be assumed that the size, X, of a particle grows, or diminishes, according to the following relationship: dX ðX  Xmin ÞðXmax  XÞ ¼ const  dt ðXmax  Xmin Þ

ð2:19Þ

This equation reflects the obvious fact that, when d approaches either Xmax, or Xmin, it becomes independent of time. Using the assumptions of the normal distribution of growth and destruction times, and combining with Equation (2.19), leads to: 0 2 1 ðX  Xmin ÞðXmax  Xmin Þ32 B 6 ln 1 ðXmax  XÞX 7 C pffiffiffi qr ðXÞ ¼ pffiffiffiffiffiffi ð2:20Þ expB 4 5 C @ A 2p lns 2 lns l

l

! where X and sl are two unknown constants that can have certain physical meaning, especially for the standard lognormal PSD (Equation 2.21). The distribution given by Equation (2.20) is called the modified lognormal distribution. It is not symmetrical on a logarithmic scale of particle sizes. One example of a modified lognormal PSD is shown in Figure 2.8. For the case when Xmin ¼ 0 and Xmax ¼ 1, the modified lognormal PSD reduces to the standard lognormal PSD: 8 2 39 X 2> > > ln = < 6 !7> 1 X 7 p ffiffi ffi ð2:21Þ exp 6 qr ðXÞ ¼ pffiffiffiffiffiffi 4 5 > 2 lnsl > 2p lnsl > > ; :

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! The parameters X and sl are then the geometrical median size and geometrical standard deviation. The median size corresponds to the 50% point on the cumulative curve. The standard deviation characterizes the width of the distribution. It is smallest (sl ¼ 1) for a monodisperse PSD. It can be defined as the ratio of the size at 15.87% cumulative probability to that at 50%, or the ratio at 50% probability to that at 84.13%. These points, at 16% and 84% roughly, are usually reported together with the median size at 50%. Lognormal and modified lognormal distributions require that we extract only a few parameters (two and four, respectively) from the experimental data. This is a big advantage in many cases when the amount of experimental data is restricted. These two distributions plus a bimodal PSD as the combination of two lognormals are usually sufficient for characterizing the essential features of the vast majority of practical dispersions.

2.5. PARAMETERS OF THE MODEL INTERFACIAL LAYER In the previous two sections we have introduced a set of properties to characterize both the dispersion medium and the dispersed particles. It is clear that the same properties might have quite different values in the particle as compared to the medium. For example, in an oil-in-water emulsion, the viscosity and density inside the oil droplet would be quite different from their values in the bulk of the water medium. However, this does not mean that these parameters change stepwise at the water-oil interface. There is a certain transition (or interfacial) layer, where the properties vary smoothly from one phase to the other. Unfortunately, there is no thermodynamic means to decide where one phase or the other begins. A convention suggested by Gibbs resolves this issue [1], but the details of this rather complicated thermodynamic problem are beyond the scope of this book. We wish to discuss here only the electrochemical aspects of this interfacial layer and these are generally combined as the concept of the “electric double layer” (DL). Lyklema [1] made a most comprehensive review of this concept, which plays a very important role in practically all aspects of colloid science. We present here a short overview of the most essential features of the DL, with particular emphasis on those that can be characterized using ultrasound-based technology. We will consider the DL in two states: equilibrium and polarized. We use term “equilibrium” as a substitute for the term “relaxed” used by Lyklema [1]. (The term relaxed is somewhat more general, and might include situations when the DL is created with nonequilibrium factors. For instance, the DL of living biological cells might have a component that is related to the nonequilibrium transmembrane potential [25, 41].) For the first equilibrium state, the DL exists in an undisturbed dispersion characterized by a minimum value of the free energy. The second “polarized” state reflects a deviation from this equilibrium state due to some external disturbance such as an electric field.

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Fundamentals of Interface and Colloid Science

According to Lyklema: “. . .the reason for the formation of a ‘relaxed’ (‘equilibrium’) double layer is the nonelectric affinity of charge-determining ions for a surface. . .” This process leads to the buildup of an “electric surface charge,” s, expressed usually in mC/cm2. This surface charge creates an electrostatic field which then affects the ions in the bulk of the liquid (Figure 2.9). This electrostatic field, in combination with the thermal motion of the ions, creates a countercharge and thus screens the electric surface charge. The net electric charge in this screening, diffuse layer is equal in magnitude to the net surface charge but has the opposite polarity. As a result, the complete structure is electrically neutral. Some of the counterions might specifically adsorb near the surface and build an inner sublayer, the so-called Stern layer. The outer part of the screening layer is usually called the “diffuse layer.” What about the difference between the surface charge ions and the ions adsorbed in the Stern layer? Why should we distinguish them? There is a thermodynamic justification [1], but we think a more comprehensive reason is kinetic (ability to move). The surface charge ions are assumed to be fixed to the surface (immobile); they cannot move in response to any external disturbance. In contrast, the Stern ions, in principle, retain some degree of freedom, almost as high as ions of the diffuse layer [42–46]. We give below some useful relationships of the DL theory from Lyklema’s review [1].

Stern layer Ψs

negatively charged diffuse layer Ψd

bulk of the liquid

ζ

positively charged particle

electric potential κ-1 Debye length

slipping plane Stern plane

FIGURE 2.9 Illustration of the structure of the electric double layer.

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2.5.1. Flat Surfaces The DL thickness is characterized by the so-called Debye length 1/k defined by: X ci zi k2 ¼ F2 ð2:22Þ e0 em RT i where the valences have the sign included; for a symmetrical electrolyte, zþ ¼ –z– ¼ z. Equation (2.22) allows the calculation of the Debye length only when the concentrations of the ion species are known. This information usually is not available. This justifies the introduction of approximate methods in evaluating this parameter. One suggests using the easily measurable conductivity of the liquid, Km, and then the following equation: Km ð2:23Þ e0 em Deff Conductivity of the dispersion can be used as an approximate value for the conductivity of the liquid. The main complicating error in this equation comes from the unknown effective diffusion coefficient Deff. However, this parameter varies over a limited range. For instance, the diffusion coefficients of most ions in aqueous solutions are similar and have values at room temperature in the range of 0.6  10–9 to 2  10–9 m2/s. The square root of this variation would lead to uncertainty in the scale of only a few tens of percent. The same equation can be used for estimating the Debye length in nonaqueous sultions as well. The diffusion coefficient of ions in nonaqeous systems usually is much smaller than in water as discussed in Section 2.3.8 due to the much larger ion size. It is possible to account for this variation using Equation (2.7) for Bjerrum size as an estimate of the ion size. This would imply that reverse ionic micelles form the diffuse layer in the particular nonaqueous solution. This might be not true if the hypothesis of the nonequal ion sizes in nonpolar liquids discussed in Section 2.3.8 is valid. For a flat surface and a symmetrical electrolyte of concentration, Cs, there is a straightforward relationship between the electric charge in the diffuse layer, sd, and the Stern potential, cd, namely: k2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fcd 8e0 em Cs RT sinh ð2:24Þ 2RT If the diffuse layer extends right to the surface, Equation (2.24) can then be used to relate the surface charge to the surface potential. Sometimes it is helpful to use the concept of a differential DL capacitance. For a flat surface and a symmetrical electrolyte, this capacitance is given by: sd ¼ 

Cdl ¼

ds Fcd ¼ e0 em k cosh dcc¼cs 2RT

ð2:25Þ

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Fundamentals of Interface and Colloid Science

For a symmetrical electrolyte, the electric potential, c, at the distance, x, from the flat surface to the DL is given by: zFcðxÞ 4RT expðkxÞ ¼ ð2:26Þ zFcd tanh 4RT The relationship between the electric charge and the potential over the diffuse layer is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   sd ¼ ðsigncd Þ 2e0 em Cs RT ðnþ ℓzþ cd þ n ℓz cd  nþ  n Þ1=2 ð2:27Þ tanh

where n is the number of cations and anions produced by dissociation of a  d is a dimensionless potential given by: single electrolyte molecule, and c d

 d ¼ Fc c ð2:28Þ RT In the general case of an electrolyte mixture, there is no analytical solution. However, some convenient approximations have been suggested [1, 42, 47, 48].

2.5.2. Spherical DL, Isolated and Overlapped There is only one geometric parameter in the case of a flat DL, namely the Debye length 1/k. In the case of a spherical DL, there is an additional geometric parameter, namely the radius of the particle, a. The ratio of these two parameters (ka) is a dimensionless parameter which plays an important role in colloid science. Depending on the value of ka, two asymptotic models of the DL exist. A “thin DL” model corresponds to colloids in which the DL is much thinner than the particle radius, or simply: ka  1

ð2:29Þ

The vast majority of aqueous dispersions satisfy this condition, except for very small particles in low ionic strength media. If we assume an ionic strength greater than 10–3 mol/l, corresponding to the majority of most natural aqueous systems, the condition ka  1 is satisfied for virtually all particles having a size above 100 nm. The opposite case of a “thick DL” corresponds to systems where the DL is much larger than the particle radius, or simply: ka 1

ð2:30Þ

The vast majority of dispersions in hydrocarbon media, having inherently very low ionic strength, satisfy this condition.

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These two asymptotic cases allow one to picture, at least approximately, the DL structure around spherical particles. A general analytical solution exists only for low potential: RT 25:8 mV ð2:31Þ F This so-called Debye-Hu¨ckel approximation yields the following expression for the electric potential in the spherical DL, c(r), at a distance, r, from the particle center: cd

a ð2:32Þ cðrÞ ¼ cd expðkðr  aÞÞ r The relationship between the diffuse charge and the Stern potential is then:   1 ð2:33Þ sd ¼ e0 em kcd 1 þ ka This Debye-Hu¨ckel approximation is valid for any value of ka, but this is somewhat misleading since it covers only isolated double layers. The approximation does not take into account the obvious probability of an overlap of double layers as in a concentrated suspension, that is, high volume fraction. A simple estimate of this critical volume fraction, ’over, is the volume fraction for which the Debye length is equal to the shortest distance between the particles. Thus: ’over

0:52

ð2:34Þ

1 3 1 þ ka

This dependence is illustrated in Figure 2.10. 0.5

vfr of DL overlap

0.4 0.3 0.2 0.1 0.0 0.01

0.10

1.00 ka

10.00

100.00

FIGURE 2.10 Estimation of the volume fraction of the overlap of the electric double layer.

Fundamentals of Interface and Colloid Science

49

It is clear that for ka  10 (thin DL) we can consider the DLs as isolated entities even up to volume fractions of 0.4. However, the model for an isolated DL becomes somewhat meaningless for small ka (thick DL) because DL overlap then occurs even in very dilute suspensions. The DL overlap becomes very important for electrophoresis of nanoparticulates and in nonaqueous systems. The electrophoretic theory which takes into account the overlap of DLs was formulated by Overbeek and coworkers [49, 50] and is discussed briefly in Section 2.5.3. Theoretical treatments have been suggested for these two extreme cases of DL thickness. Briefly they are: Thin DL (ka > 10): Loeb, Dukhin, and Overbeek [51–53] proposed a theory to describe the DL structure for a symmetrical electrolyte by applying a series expansion in terms of powers of 1/(ka). The final result is: d!  d 4 tanh zc 2FC z z c s 4 sd ¼  þ 2 sinh ð2:35Þ 2 ka k Thick DL (ka < 1): No theory for an isolated DL is necessary since DL overlap must be considered even in the dilute case. A theory that does include DL overlap has been proposed only very recently [54]: ! !! d  d 4 tanh zc d d 2FCs z zc 3B zc zc d 4 þ þ 2 sinh exp  sinh s ¼ 2 ka 2 2 k ka ð2:36Þ where A expðkðb  aÞÞ ðkb  1Þ 2ka The constant, A, is obtained by matching the asymptotic expansions of the long-distance potential distribution with the short-distance distribution and b is the radius of the cell. This new theory suggests, at last, a way to develop a general approach to the various colloidal-chemical effects in nonaqueous dispersions. B¼

2.5.3. Electric Double Layer at High Ionic Strength According to classical theory, the DL should essentially collapse and cease to exist when the ionic strength approaches 1 M. At this high electrolyte concentration, the Debye length becomes comparable with the size of the ions, implying that the counterions should collapse onto the particle surface. However, there are indications [55–62] that, at least in the case of hydrophilic surfaces, the DL still exists even at ionic strengths exceeding 1 M.

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The DL at high ionic strength is strongly controlled by the hydrophilic properties of the solid surfaces. The affinity of the particle surface for water creates a structured surface water layer, and this structure changes the properties of the water considerably from that in the bulk. Figure 2.11 illustrates the DL structure near a hydrophilic surface. Note the difference between the much lower dielectric permittivity within the structured surface layer, esur, compared to that of the bulk liquid, em. There are two effects caused by this variation of the dielectric permittivity [60]: l l

The structured surface layer becomes insolvent, at least partially, to ions; The structured surface layer may gain electric charge because of the difference in the standard chemical potentials of the ions.

Both effects influence the electrokinetic behavior of colloids at high ionic strength. The first effect is the more important one because it leads to a separation of electric charges in the vicinity of the particle. The insolvent structured layer repels the screening electric charge from the surface to the bulk and makes electrokinetics possible at high ionic strength if the structured layer retains a certain hydrodynamic mobility in the lateral direction. A theory of electro-osmosis for the insolvent and hydrodynamically mobile DL at high ionic strength has been proposed by Dukhin and Shilov in a paper which is not available in English. There is a description and

structured water layer

-

bulk

+

0

surface charge

particle

electroosmotic flow

+ ψ(X)

- +

ψsur

+ + +

-

εm

structured layer electric charge

εsur(x)

-

+ screening charge

-

+ insolvent hydrodynamically mobile

FIGURE 2.11 Electric double layer near hydrophilic surface with structured water layer.

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Fundamentals of Interface and Colloid Science

reference to this theory in an experimental paper that is available in English [61]. We discuss these effects in more detail in Chapter 11 with regard to the experimental study published in the article [63].

2.5.4. Polarized State of the Electric Double Layer External fields may affect the DL structure. These fields might be of various origins: electric, hydrodynamic, gravity, concentration gradient, acoustic, etc. Any of these fields might disturb the DL from its equilibrium state to some “polarized” condition [1, 42]. Such external fields affect the DL by moving the excess ions inside it. This ionic motion can be considered as an additional electric surface current, Is (Figure 2.12). This current is proportional to the surface area of the particle and to a parameter called the surface conductivity Ks [1]. This parameter reflects the excess conductivity of the DL due to the excess ions attracted there by the surface charge [143]. The surface current adds to the total current. But there is another component that depends on the nature of the particle, which has the opposite effect. For example, a nonconducting particle reduces the total current going through the conducting medium, simply because the current cannot pass through the nonconducting volume of the particle. In the important case of charged,

Electric field

Polarization electric field DL layer with higher conductivity

-

negatively charged particle

+

Bulk electric current

Surface current Polarization dipole charges

FIGURE 2.12 Mechanism of polarization of the electric double layer.

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nonconducting particles, these two opposing effects influence the total current value. The surface current increases the total current, whereas the nonconducting character reduces it. The amplitude of the surface current is proportional to the surface conductivity Ks. The amplitude current arising from the nonconductive character depends on the conductivity of the liquid, Km; the higher the value, the more the current lost, because the conducting liquid is replaced with nonconducting particles. The balance of these two effects depends on the dimensionless number, Du, given by: Du ¼

Ks aKm

ð2:37Þ

The particle size, a, is in the denominator of Equation (2.37) because the surface current is proportional to the surface area of the particle (a2), whereas the reduction of the total current due to the nonconducting particles is proportional to the volume of the particles (a3). The abbreviation, Du, for this dimensionless parameter was introduced by Lyklema. He called it the “Dukhin number” [1] after S. S. Dukhin, who explicitly used this number in his analysis of electrokinetic phenomena [23, 42]. The surface electric current redistributes the electric charges in the DL. One example is shown in Figure 2.12. It is seen that this current produces an excess of positive charge at the right pole of the particle and a deficit at the left pole. Altogether, this means that the particle gains an “induced dipole moment” (IDM) [23, 42] due to this polarization in the external field. A calculation of the IDM value for the general case is a rather complex problem. It can be substantially simplified in the case of the thin DL (ka > 10). There is a possibility to split the total electric field around the particle into two components: a “near-field” component and a “far-field” component. The near field is located inside the DL. It maintains each normal section of the DL in a local equilibrium. This field is shown as a small arrow inside the DL in Figure 2.12. The far field is located outside the DL, in the electro-neutral area. It is simply the field generated by the IDM. Let us assume now that the driving external field is an electric field. The polarization of the DL affects all the other fields around the particles: hydrodynamic, electrolyte concentration, etc. However, all other fields are secondary in relationship to the driving electric field applied to the dispersion. The IDM value, pidm, is proportional to the value of the driving force, which is the electric field strength, E, in this case. In a static electric field, these two parameters are related by the following equation: pind ¼ gind E ¼ 2pe0 em a3 where gind is the polarizability of the particle.

1  Du E 1 þ 2Du

ð2:38Þ

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Fundamentals of Interface and Colloid Science

The IDM value depends not only on the electric field distribution but also on the distribution of the other fields generated near the particle. For example, the concentration of electrolyte shifts from its equilibrium value in the bulk of the solution, Cs. This effect, called concentration polarization, complicates the theory of electrophoresis and is the main reason for the gigantic values found for dielectric dispersion in a low-frequency electric field. Fortunately, this concentration polarization effect is not important for ultrasound, since it appears only in the low-frequency range where the frequency, o, is smaller than the characteristic concentration polarization frequency, ocp: o ocp ¼

Deff d2

ð2:39Þ

where Deff is the effective diffusion coefficient of the electrolyte. Typically, this concentration polarization frequency is below 1 MHz, even for particles as small as 10 nm. The IDM value becomes a complex number in an alternating electric field, as well as polarizability. According to the Maxwell-Wagner-O’Konski theory [64–66], confirmed later by DeLacey and White numerical calculations [144], the IDM value for a spherical, nonconducting particle is: pidm ¼ gidm E ¼

ep  em E 2em þ ep

ð2:40Þ

where the complex permittivity and conductivity of the medium are defined by Equations (2.4) and (2.5). The complex permittivity of the charged nonconducting particle is given by: ep ¼ ep þ j

2Km Du oe0

ð2:41Þ

The influence of the DL polarization on the IDM decreases as a function of the frequency (following from Equation 2.40). There is a certain frequency above which the DL influence becomes negligible because the field changes faster than the DL can react. This is the so-called Maxwell-Wagner frequency, oMW, which is defined as: oMW ¼

Km e0 em

ð2:42Þ

So far, we have described the polarization of the DL mostly as it relates to an external electric field. Indeed, most theoretical efforts over last 150 years have been expended in this area. However, we have already mentioned that other fields can also polarize the DL. The major interest of this book is to describe in some detail the polarization caused not by this electric field but a hydrodynamic field, since this gives rise to the electroacoustic phenomena in which we are interested.

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2.6. INTERACTIONS IN COLLOID INTERFACE SCIENCE The characterization of interactions is strongly connected to the idea of a “dimension hierarchy.” We can speak of several specific dimensional levels in the colloid and each level has its own specific interaction features. Interactions that originate at the microscopic level, where we deal with molecules and ions, are the so-called chemical interactions. They are usually related to the exchange of electron density between various microscopic bodies. There are many indications that ultrasound techniques are well suited for characterizing such chemical interactions. Although not the main subject of this book, this aspect will be covered briefly in Chapter 3. We are most interested in effects that occur at the single-particle level or, in other words effects with a characteristic spatial parameter on the order of the size of a typical particle. This means that many thousands of molecules and ions are involved in this interaction. We observe the result of some averaging of individual chemical interactions. This averaging masks, and practically eliminates, any recognizable features of the original chemical interactions and microscopic objects. It justifies the introduction of a completely new set of concepts to describe effects at this higher structural level. We call all interactions that happen on this level “colloidal-chemical interactions,” and further subdivide these reactions into two categories: equilibrium and nonequilibrium reactions. Equilibrium interactions, as the name suggests, corresponds to interactions where the particles are in equilibrium with the dispersion medium. These interactions, subject to the classical DLVO theory, include: 1. Dispersion Van der Waals forces 2. Electrostatic forces of the overlapped double layers 3. Various steric interactions. Nonequilibrium interactions correspond to cases where the equilibrium between the particle and the dispersed phase is disturbed. For example, any external field, whether it is electric, hydrodynamic, gravitational or magnetic, might create such a disturbance. Living biological cells present a special case in which the equilibrium is disturbed by an electrochemical exchange between the cell and the dispersion medium. We briefly consider here both equilibrium and nonequilibrium colloidalchemical interactions. The purpose of this overview is merely to determine what features of these colloidal-chemical interactions may affect ultrasound in the dispersions and, consequently, to learn which of these features might possibly be characterized with ultrasound-based techniques.

2.6.1. Interactions of Colloid Particles in Equilibrium: Colloid Stability The affinity between the particles and the dispersion medium allows us to separate colloids into two broad classes. “Lyophobic” colloids represent systems with very low affinity between particle and the liquid, whereas for “lyophilic”

Fundamentals of Interface and Colloid Science

55

colloids this affinity is high. From this definition, it follows that particles of a lyophobic colloid tend to aggregate, in an effort to reduce the contact surface area between the particle and the medium. Such colloids are, usually, thermodynamically unstable in the strict sense, but might actually be kinetically stable for a long period. In contrast, lyophilic colloids are, often, thermodynamically quite stable, and consequently their kinetics are not as important. It is possible to convert one colloid class to the other by applying appropriate surface modifications. We can coagulate a lyophilic dispersion even though it was initially thermodynamically stable, or we can stabilize a lyophobic dispersion even though initially unstable. It is simply a question of our overall objective. We will show later that ultrasound-based techniques might be very helpful for optimizing these procedures. In order to explain the possibilities open to ultrasound characterization, we first need to give a short description of the particles’ interaction and coagulation. There are several mechanisms of particle-particle interaction which are described in detail in the DLVO theory [67–70], and these can be found in most handbooks on colloid science [1, 47, 71]. The presentation of the DLVO theory usually starts with the simplest case, in which the potential energy, Upp, on bringing two surfaces close together can be expressed as the sum of the Van der Waals attraction and DL repulsion. These two basic mechanisms exhibit quite different dependences with respect to the separation distance, xpp, between the surfaces. The Van der Waals attraction decays quite rapidly as the inverse power of the separation distance. The DL repulsion, on the other hand, typically decreases more slowly, more or less exponentially with this distance, and, typically, has a range on the order of the Debye length 1/k. As a result, the Van der Waals attraction is dominant for small separations, whereas the DL repulsion may be dominant at larger distances. A general potential energy plot illustrating qualitatively the main features of this interaction is given in Figure 2.13. The key feature of this potential energy plot is that there is often a welldefined potential energy barrier at a distance comparable to the Debye length. A particle that approaches this barrier, perhaps due to its random Brownian motion, must have sufficient energy to overcome this barrier if it is to fall into the primary minimum and result in an aggregated particle. This barrier is therefore one important key to achieving stability in a colloid system. This potential barrier exists when the DL repulsion exceeds the Van der Waals attraction. We cannot do much to change the attractive forces, since these are fixed by the Hamaker constants of the material. We can only manipulate the repulsive forces by changing either the potential at the Stern plane, or by changing the rate at which this potential decreases with distance (as determined by the Debye length). The Stern potential can be affected strongly by the addition of potential-determining ions to the medium. The Debye length can be decreased by increasing the ionic strength. Consequently, the height of this potential barrier depends on the Stern potential of the particles

56

CHAPTER

Upp

2

potential barrier

Xpp secondary minimum

primary minimum

FIGURE 2.13 Typical DLVO interaction energy.

and electrolyte concentration or ionic strength. An increase in the ionic strength reduces this barrier height by shifting it closer to the surface where the Van der Waals attraction is stronger. There is a critical concentration of electrolyte, Cccc, at which this barrier disappears altogether, and the particles might therefore coagulate. !4 d zc 1 tanh 4 3 5 5 2 9:85 10 e R T m Cccc ½moll1  ¼ ð2:43Þ !4 F6 A2 z6 d zc tanh þ1 2 A more detailed and accurate calculation of this critical coagulation concentration, as well as other stability criteria, can be found in the literature [1, 70]. In addition to ionic strength, aggregation stability is also sensitive to the concentration of DL potential-determining ions that control the value of the Stern potential. In principle, these two parameters (overall ionic strength and concentration of potential-determining ions) might be considered as independent parameters, which bring us to the conclusion that both aggregation stability and DL potentials should be considered in some multidimensional space. Contours of constant B-potential would then appear as a “fingerprint” of the particular colloid. Indeed, this idea of fingerprinting is the logical

57

6

10 12 14 16 18 20 22 24 26 28 30 32 34 0.0 0.5 % o, 1.0 rati olin -ka

7 8 pH

FIGURE 2.14 solution.

9 10

11 2.0

zeta [mV]

Fundamentals of Interface and Colloid Science

1.5 H SHP

Electrokinetic three-dimensional plot for kaolin slurry in sodium hexametaphosphate

way to characterize a colloidal system, and Marlow and Rowell have successfully applied this principle [72, 73] for various applications. We show, as one example, the three-dimensional plot of 40 wt% kaolin slurry that was titrated with sodium hexametaphosphate (Figure 2.14). From an examination of this, it is clear that the highest surface charge occurs at a pH value of 9.5 and at a hexametaphosphate to kaolin ratio of 0.6 wt%. This, then, is the optimum condition for gaining aggregative stability. Hexametaphosphate affects the stability of the kaolin particles through electrostatic interactions. There are different types of stabilizing agents that might enhance stability via other mechanisms, such as steric interactions. For instance, various polymers adsorb on the particle surface and change its affinity to the solvent from lyophobic to lyophilic. However, there is a danger, if the stabilizing agent is overdosed, of flocculating the colloid through the bridging effect. This is the reason why surface characterization is important. It allows one to determine the optimum dose of the stabilizing agent and thus eliminate the dangers of flocculation. There is one more factor that can affect strongly the stability of the heterogeneous system—gravitation. It is nonequilibrium and its principle will be mentioned in another section. However, it always exists and competes with other equilibrium factors mentioned here. Gravity affects both pair particle interaction and kinetics of the PSD evolution. There is a recent published review [74] that describes in detail several works dedicated to this subject. One of the main conclusions is that gravity becomes as important as the stability factor for particles with sizes roughly of 1 mm. It is a very

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approximate number that must be used to alert that gravity must be considered if the size exceeds this critical value.

2.6.2. Biospecific Interactions Intensive development of nanotechnology has generated concerns regarding nanoparticles’ impact on human health. The level of interest is so high that Nature published recently a review article on this subject [75]. Unfortunately, authors of this review missed one major peculiar feature of the living biological cells. These objects are not in equilibrium with the surrounding media. This prevents us from transposing directly knowledge collected in colloid science for describing these object and, in particular, for understanding their interaction with nanoparticles. As a result, the review [75] is applicable only for “dead” biological surfaces. Some very important aspects associated with the living biological cells’ functioning remained outside the scope of the review [75]. The authors of the review are not familiar with a large body of work that was carried out on cell-particle interactions during the last few decades of the twentieth century. The authors referred only two papers from among 96 citations that came from that time period. The earliest is dated 1995. This would not have been a problem had the more recent works properly reflected the observations and conclusions of the old studies. Unfortunately, it is not the case. The review overlooks some very important facts and mechanisms that may dominate cell-particle interactions. We hope to bring to your attention those forgotten papers. The earliest relevant work in this field, in my mind, was done by H.A. Pohl, the famous inventor of dielectrophoresis. He monitored the motion of small particles in the vicinity of biological cells for verifying the hypothesis that biocells generate an AC electric field. The earliest publication is dated 1980 [76]. His short essay published in 1987 [77] cited several more publications in major journals, and Ref. [78] is one of them. He observed indeed unusually long-range interactions between biological cells and dielectric particles, in scale of micrometers. In the 1990s, experiments on the same subject were performed by several groups in (the then) U.S.S.R. and some of them were published in international journals [79–84]. The Soviet investigators’ group led by Z. Ulberg and V.Karamushka had discovered some peculiar features of bacterial interactions with gold nanoparticles and dramatically distinguished them from typical interactions between nano- and micro-sized equilibrium objects. Some bacteria collected large numbers of gold nanoparticles on their surfaces (Figure 2.15). The interaction occurred in two stages: the nanoparticles were initially bound to the cell very weakly and reversibly, but the reversible aggregation gradually became irreversible over 1 h.

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Fundamentals of Interface and Colloid Science

FIGURE 2.15 Bacillus subtilis cells in the presence of gold particles. Scale bar 1.8 mm. (Adapted with permission from Ref. [79]).

The first stage of reversible aggregation was the most interesting and peculiar. It was controlled by the cell’s metabolism and, in particular, by the activity of the ion pumps and the transmembrane potential. Earlier authors proved this mechanism by using special chemicals that switched on and off specific ion pumps. Turning the ion pumps on and off created cycles of aggregation and release of the nanoparticles. Results of one of these experiments are shown in Figure 2.16. Pentachlorophenol, an inhibitor of metabolism, was injected at different times to release 1

+ +

Ct/Co

0.8

+

Cit

*

*

*

+

*

3 +

2

0.6 *

0.4

4

*

0.2 A 0

0

1

Ct 20

40

t, min

60

80

5

100

FIGURE 2.16 Kinetic curves of the gold particles adsorption by bacteria, where Ct is the relative concentration of gold particles in the solution. Curve 1 is uninterrupted adsorption. Curve 2 is adsorption when pentachlorophenol is added initially to the solution. Curves 3, 4, and 5 represent kinetics after injection of pentachlorophenol at different time moments. (Adapted with permission from Ref. [79]).

60

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gold particles back into the solution. The amount of released particles decayed with time, suggesting that the aggregation went from reversible to irreversible. This and other similar experiments indicated the existence of interaction mechanisms that solely existed for living cells as a result of their metabolism and ion exchange. We are not aware of the complete physicochemical theory for this interaction. Biospecific mechanism of a DL formation [83–85] was a first step in explaining these experiments as well as the observed metabolismdependent components of the cell electrophoresis [86–88]. The biospecific mechanism predicts a relationship between the living cell’s z-potential and its transmembrane potential DU. It postulates that transmebrane potential becomes redistributed between capacity of the membrane Cm and capacity of the cell DL, Cdl. This model yields the following expression for the fraction of the cell z-potential (Dz) that is associated with the transmembrane potential and cell metabolism: Dz ¼

Dse DUCm þ Cdl þ Cm Cdl þ Cm

ð2:44Þ

The first term in this equation takes into account possible change in the surface charge under the influence of the intramembrane field that can affect the conformation of intramembrane macromolecules. This additional component of the z-potential can be eliminated by reducing transmembrane potential. This correlation was observed experimentally in electrophoretic experiments [86–88]. The additional component of the z-potential would affect electrostatic component of the cell-nanoparticle interaction energy. This particular mechanism of cell-particle interactions is intriguing and opens new possibilities of controlling biological cell interaction with nanoparticles.

2.6.3. Interaction in a Hydrodynamic Field: Cell and Core-Shell Models—Rheology A hydrodynamic field exists when a particle moves relative to the liquid medium. There are a number of effects caused by this motion (see Ref. [89]). When a particle moves relative to a viscous liquid, it creates a sliding motion of the liquid in its vicinity. The volume over which this disturbance exists is usually referred to as the hydrodynamic boundary layer. As the volume fraction of the sample is increased, the average distance between particles becomes smaller. When this distance becomes comparable to the thickness of the hydrodynamic layer, these layers overlap and, as a result, one moving particle affects its immediate neighbors. The particles interact through their respective hydrodynamic fields, and such interactions can be defined mathematically. The motion of the liquid in the hydrodynamic layer surrounding a particle is described in the terms of a local liquid velocity, u, and a hydrodynamic

Fundamentals of Interface and Colloid Science

61

pressure, P. Liquids are usually assumed to be incompressible, described mathematically as: Divu ¼ 0

ð2:45Þ

We will see later that this assumption can be used even for ultrasound-related phenomena in the long-wavelength range. The assumption of fluid incompressibility then allows one to use a simplified version of the NavierStokes’ equation [89], namely: du ¼
rot rot u þ grad P ð2:46Þ dt Equation (2.46) neglects terms related to the volume viscosity, hvm , of the liquid. These last two equations require a set of boundary conditions in order to calculate the liquid velocity and hydrodynamic pressure. The boundary conditions at the surface of the particle are rather obvious: rm

ur ðr ¼ aÞ ¼ up  um

ð2:47Þ

uy ðr ¼ aÞ ¼ ðup  um Þ

ð2:48Þ

where r and y are spherical coordinates associated with the particle. The other set of boundary conditions must specify the velocity of the liquid somewhere in the bulk. For a dilute system this presents no problem. Particles do not interact and this point can be selected infinitely far from the particles. This yields, for instance, the well-known Stokes’ law [90] for the frictional force, Fh, exerted on particles moving relative to the liquid: Fh ¼ 6p
aOðup  um Þ

ð2:49Þ

where O is the drag coefficient. In the case of a concentrated colloid, this approach is not viable because the particles interact and the hydrodynamic field of one particle affects that of its neighbors. The concept of a “cell model” [91, 92] allows one to take into account this interaction and resolve the hydrodynamic problem for a collection of interacting particles. The basic feature of a cell model is that each particle in the concentrated system is considered separately inside the spherical cell of liquid that is associated only with the given individual particle (Figure 2.17). In the past, the cell model has been applied only to monodisperse systems. This restriction allows one to define the radius of the “cell.” Equating the solid volume fraction of each cell to the volume fraction, ’, of the entire system yields the following expression for the cell radius, b: a b¼p ffiffiffiffi 3 ’

ð2:50Þ

62

CHAPTER

2

cell surface

particle

b, cell radius fraction of the liquid associated with the particle

FIGURE 2.17 Illustration of the cell model.

The cell boundary conditions formulated on the outer boundary, at r ¼ b, of the cell reflect the particle-particle interaction. The two most widely used versions of these boundary conditions are named after their authors: the Happel cell model [91] and the Kuwabara cell model [92]. Both of them are formulated for an incompressible liquid. For the Kuwabara cell model, the cell boundary conditions are given by the following equations: rot ur¼b ¼ 0

ð2:51Þ

ur ðr ¼ bÞ ¼ 0

ð2:52Þ

In the case of the Happel cell model they are: uy @ 1 @ur ðr ¼ bÞ ¼ þr r ¼0 ry @r r @y

Y

ur ðr ¼ bÞ ¼ 0

ð2:53Þ ð2:54Þ

The general solution for the velocity field contains three unknown constants C, C1, and C2:    Z b b3 x3 ð2:55Þ ur ðrÞ ¼ C 1  3 þ 1:5 1  3 hðxÞdx r r r

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Fundamentals of Interface and Colloid Science

   Z b b3 x3 uy ðrÞ ¼ C 1 þ 3  1:5 1 þ 3 hðxÞdx 2r 2r r

ð2:56Þ

b~ ~  C2 h2 ðbÞÞ ~ C ¼  ðC1 h1 ðbÞ 3

ð2:57Þ

where hðxÞ ¼ C1 h1 ðxÞ þ C2 h2 ðxÞ 2 0 13 expðxÞ 4x þ 1 x þ 1 h1 ðxÞ ¼ sinx  cosx þ i@ cosx þ sinxA5 x x x 2 0 13 expðxÞ 4x þ 1 1  x sinx  cosx þ i@ cosx þ sinxA5 h2 ðxÞ ¼ x x x

ð2:58Þ

The drag coefficient can be expressed, in a general form, for both Kuwabara and Happel cell models: O¼

  a~2 dðC1 h1 þ C2 h2 Þ C1 h1 þ C2 h2 4j~ a2  þ a~ 3 9 dx x¼~ a

ð2:59Þ

a=a; and x is normalized in the same manner where a~2 ¼ a2 wrm =2hm ; b~ ¼ b~ as a˜. Coefficients C1 and C2 are different in the two cell models (see Table 2.2). There are several special functions used in this theory. They are defined as follows: ~  Ið~ I ¼ IðbÞ aÞ I1, 2 ¼ j

ℓ

xð1þjÞ

x

TABLE 2.2 Parameters of the Cell Model Solutions Kuvabara

Happel

C1

~ h2 ðbÞ I

~  2I23 ~ 2 ðbÞ bh ~ bI þ 2ðI2 I13  I1 I23 Þ

C2



~ h1 ðbÞ I



~  2I13 ~ 1 ðbÞ bh ~ bI þ 2ðI2 I13  I1 I23 Þ

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2

" !# x2 3x 1 xð1þjÞ 3ð1  xÞ ~ IðxÞ ¼ h1 ðbÞℓ þ j 3  ... 3  3 x 2b~ b~ 2b " !# 2 3ð1 þ xÞ x 3x 1 xð1þjÞ ~ℓ h2 ðbÞ þj 3þ 3 3 x b~ 2b~ 2b~

I13 ¼ 

I13 ¼ 

ℓ

xð1þjÞ



3 b~ ℓ

xð1þjÞ



  3 3 2 ð1 þ xÞ þ j x þ x 2 2

  3 3 ð1 þ xÞ þ j x2 þ x 2 2

3 b~ The Happel cell model is better suited to acoustics because it describes energy dissipation more adequately, whereas the Kuwabara cell model is better suited to electroacoustics because it automatically yields the Onsager relationship [42, 47]. In the case of a polydisperse system, the introduction of the cell is more complicated, because the total liquid can be distributed between size fractions in an infinite number of ways. However, the condition of mass conservation is still necessary. Each fraction can be characterized by particles having radii ai, cell radii bi, thickness of the liquid shell in the spherical cell li ¼ biai, and volume fraction ’i. The mass conservation law ties these parameters together as:  N  X li 3 1þ ’i ¼ 1 ð2:60Þ ai i¼1

This expression might be considered as an equation with N unknown parameters, li. An additional assumption is still necessary to determine the cell properties for the polydisperse system. This additional assumption should define for each fraction the relationship between the particle radii and shell thickness. We have suggested the following simple relationship [47]: li ¼ lani

ð2:61Þ

This assumption reduces the number of unknown parameters to only two, related by: N X

ð1 þ lan1 Þ3 ’i ¼ 1 i

ð2:62Þ

i¼1

The parameter n is referred to as a “shell factor.” Two specific values of the shell factor correspond to easily understood cases. A shell factor of 0 depicts the case in which the thickness of the liquid layer is independent of the

65

Fundamentals of Interface and Colloid Science

particle size. A shell factor of 1 corresponds to the normal “superposition assumption,” giving the same relationship between particles and cell radii as in the monodisperse case, that is, each particle is surrounded by a liquid shell which provides each particle with the same volume concentration as the volume concentration of the overall system. In general, the “shell factor” might be considered to be an adjustable parameter because it adjusts the dissipation of energy within the cells. However, our experience using this cell model with acoustics for particle sizing [93, 94] indicates that a shell factor of unity is almost always more suitable. There is another approach to describe the hydrodynamic particle interaction; it is the so-called effective media approach. It was used originally by Brinkman [95] for hydrodynamics, and later by Bruggeman for dielectric spectroscopy [23]. This approach and its relationship to the cell model are fully described in a recent review [96]. The effective media approach is very useful for interpreting acoustic spectroscopy results, and will be discussed later in Chapter 8. Recently, a new “core-shell model” has become quite popular [96]. It attempts to combine the cell model and “effective media” approaches as illustrated in Figure 2.18. Here, each particle is surrounded by a spherical layer of liquid, as in the regular cell model. However, the colloid beyond this spherical layer is modeled as some effective media. The properties of the effective media and the liquid shell thickness depend on the particular application. There are applications for this core-shell model in acoustics, in particular with the theory of the thermodynamic effects associated with sound propagation through the colloid (see Chapter 4).

effective media

particle

FIGURE 2.18 Illustration of the core-shell model.

liquid shell

66

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2

Theoretical developments of the hydrodynamic particle-particle interaction are the backbone of rheological theories that describe reactions of colloids to mechanical stresses. Such reactions might be very complex because colloids possess features of both liquids and solids. They are able to flow and exhibit viscous behavior as liquids. At the same time, they have a certain elasticity as with solids. This is why colloids are referred to as viscoelastic materials. In the simplest linear case, the total stress of the colloid, sshear, in a shear flow of shear rate, Rshear, can be presented as the combination of a Newtonian liquid (stress is proportional to the rate of strain) and a Hookean solid (stress is proportional to the strain) [71]: Ms ðoÞ ð2:63Þ Rshear cosot o where o is the frequency of the shear oscillation, and
s and Ms are, respectively, the macroscopic viscosity and elastic bulk modulus of the colloid. The macroscopic viscosity and bulk modulus depend on the microscopic properties of colloids, such as the PSD and the nature of the bonds that bind the particles together. These links are usually modeled as Hookean springs. There are a number of theories [71, 97–101] that relate the macroscopic viscosity and bulk modulus to the microscopic and structural properties of particular colloids using the Hookean spring model. Later (Chapter 4), we will use this model to describe ultrasound attenuation in structured colloids. This is possible because of the similarities between rheology and acoustics; both deal with responses to mechanical stress: shear stress in rheology and longitudinal stress in acoustics. This parallelism is explored in more detail in the same chapter. sshear ¼
s ðoÞRshear sinot 

2.6.4. Linear Interaction in an Electric Field: Electrokinetics and Dielectric Spectroscopy Applying an electric field to a colloid system generates a variety of effects. One group of such effects is called “electrokinetic phenomena.” Electrophoresis is the best known member of this group, which also includes electro-osmosis, streaming potential, and sedimentation potential [102–106]. These electrokinetic effects depend on the first power of the applied electric field strength, E. There are other effects induced by an electric field which are nonlinear in terms of the electric field strength, but they are usually less well known and less important. They are not well known because the electric field under normal conditions in aqueous systems is rather weak. It becomes especially clear when the electric field is presented in the natural normalized form as follows: EaF E½V=cm E~ ¼ RT 250 a¼1 mm

ð2:64Þ

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Fundamentals of Interface and Colloid Science

In aqueous colloids, the electric field strength is limited by high conductivity and is typically no more than 100 V/cm. A stronger applied electric field creates thermal convection and this masks any electrokinetic effects. Accordingly, the dimensionless electric field strength is small and linear effects are dominant. There are still some peculiar nonlinear effects (described in the next section). The most widely known and used electrokinetic theory was developed a century ago by Smoluchowski [107] to describe the electrophoretic mobility, m, of colloidal particles. It is important because it yields a relationship between the z-potential of particles and the experimentally measured parameter, m: m¼

em e0 z
m

ð2:65Þ

There are two conditions restricting the applicability of Smoluchowski’s equation. The first restriction is that the DL thickness must be much smaller than the particle radius a: ka  1

ð2:66Þ

where k is the reciprocal of the Debye length. In the situation where the DL is not thin compared to the particle radius, Henry (see Ref. [108]) applied a correction to modify Smoluchowski’s equation: m¼

e0 em B fH ðkaÞ
m

ð2:67Þ

where fH ðkaÞ is called the Henry function. A simple, approximate expression has been derived by Ohshima [109]: fH ðkaÞ ¼

2 þ  3 3 1þ

1 2:5 ka½1 þ 2 expðkaÞ

3

ð2:68Þ

Figure 2.19 illustrates the form of the Henry function values as a function of ka. The second restriction on the Smoluchowski equation is that the contribution of surface conductivity, ks, to the tangential component of electric field near the particle surface is negligibly small. This condition is satisfied when the dimensionless Dukhin number, Du, is sufficiently small: Du ¼

ks 1 Km a

ð2:69Þ

where Km is the conductivity of medium outside the double layer. Equation (2.67) is termed the Smoluchowski electrophoresis equation and is usually derived for a single particle in an infinite liquid. In this derivation,

68

CHAPTER

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Henry function, Ohshima approximation

1.0

0.9

0.8

0.7

0.6 0

1

ka

10

100

FIGURE 2.19 Henry function calculated according to the Ohshima expression.

the electrophoretic mobility, m, is defined as the ratio of field-induced particle velocity, V, to the homogeneous field strength, Eext , in the liquid at a large distance from the particle: V ð2:70Þ m¼ Eext Smoluchowski’s equation is applicable to a particle with any geometrical form: a convenient property that makes it applicable to any arbitrary system of particles, including a cloud of particles or a concentrated suspension. This property was experimentally tested by Zukoski and Saville [110]. However, in order to apply Smoluchowski’s equation to a concentrated suspension, one should take into account the difference between the external electric field in the free liquid outside the suspension, Eext, and the averaged electric field inside the dispersion, hEi. The static mobility m in Equation (2.65) is determined with respect to the external electric field, that is, the field strength in the free liquid outside the suspension. For concentrated suspensions, an alternative electrophoretic mobility, hmi, (or electro-osmotic velocity), is defined with respect to the field hEi: hmi ¼

ee0 z Ks n Km

ð2:71Þ

where Ks is the macroscopic conductivity of the dispersed system and hmi ¼

V hEi

ð2:72Þ

An equation similar to Equation (2.71) but with opposite sign is termed Smoluchowski’s electro-osmosis equation, and is usually expressed in this

69

Fundamentals of Interface and Colloid Science

form (Kruyt and Overbeek [47], Dukhin [42], and O’Brien [111]). It is valid under the same conditions as given above for electrophoresis, that is, a thin DL and negligible surface conductivity. The same absolute values and opposite signs of electrophoretic and electro-osmotic velocities reflect the difference in the frames of reference between these two phenomena. In the case of static electrophoresis, the frame of reference is the liquid, whereas in the case of electro-osmosis it is the particle matrix. These two definitions of the electrophoretic mobility are identical if we take into consideration the well-known [23] relationship between Eext and hEi: hEi Km ¼ ð2:73Þ Ks Eext Both expressions (Equations 2.65 and 2.71) specify an electrophoretic mobility (m or hmi) which is independent of both the volume fraction and the system geometry. This happens because the hydrodynamic and electrodynamic interactions of particles have the same geometry [42] when conditions 2.67 and 2.70 are valid. From this viewpoint, Smoluchowski’s equation is unique. Surface conductivity and the associated concentration polarization of the DL destroy this geometric similarity between the electric and hydrodynamic fields and, consequently, the electrophoretic mobility becomes dependent on the particle size and shape; a correction is required and several theories have been proposed. One numerical solution for a single particle in an infinite liquid (for very dilute colloids) was derived by O’Brien and White [112]. However, experimental tests by Midmore and Hunter [113] indicate that the Dukhin-Semenikhin theory [108] is probably the most adequate among the several approximate formulas [48, 114], because it has the advantage of distinguishing between the z-potential and the Stern potential, making it more useful for characterizing surface conductivity. The Dukhin-Semenikhin theory yields the following expression for electrophoretic mobility: " # 4Msh2B þ 2G1 þ lnBchB ð2Msh2B  12mB þ 2G2Þ ee0 z ð2:74Þ 1 m¼
ka þ 8Msh2B  24m lnchB þ 4G1 z2

where

! !

d d   B B D

D 3m c c G1 ¼ s ch  ch ; G2 ¼ s sh  sh ; M ¼1þ 2 ; 2 2 D 2 D 2 z m¼

0:34ðzþ Dþ  z D Þ ; Dþ D ðzþ  z Þ

d

 d ¼ zFc ; B ¼ zFB ; c RT RT

In general, there is a great difference between the theory in dilute systems and that in concentrated systems. Particle interaction makes the theory for concentrated systems much more complicated. Unfortunately, the simplicity and

70

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2

wide applicability of Smoluchowski’s equation is a rather unique exception. The simplest and the most traditional way to resolve the complexity of the electrokinetic theory for concentrated systems is to employ a cell model approach, in a manner similar to that used in the case of hydrodynamics. For instance, Overbeek and coworkers derived an expression for the electrophoretic mobility that takes into account that DLs overlap [49], which later was experimentally verified by Ross and Long [50].   kb  1 a kb  1 expð2kðb  aÞÞ  1þ expðkðb  aÞÞ 1þ 2s kb þ 1 b kb þ 1 m¼ kb  1 3
a ðka þ 1Þ expð2kðb  aÞÞ  ka þ 1 kb þ 1 ð2:75Þ where b is the radius of an imaginary shell around the particles calculated assuming that the volume fraction of solid inside of the shell equals that of the dispersion. This (Overbeek) theory must be used for concentrated dispersions of nanoparticles and even to rather dilute nonpolar dispersions. The first electrodynamic cell model was applied for pure electrodynamic problem by Maxwell, Wagner, and, later, O’Konski [64–66] for calculating conductivity and dielectric permittivity of concentrated colloids comprised of particles having a given surface conductivity. This theory gives the following relationship between complex conductivity and dielectric permittivity: ep  em es  em ¼ ’ es þ 2em ep þ 2em

ð2:76Þ

  Kp  Km Ks  Km ¼  ’    Ks þ 2Km Kp þ 2Km

ð2:77Þ

where the index, s, corresponds to the colloid, m corresponds to the media, and p to the particle. Complex parameters are related to the real parameters as:  ¼ Km  joe0 em Km

ð2:78Þ

  Km em ¼ em 1 þ j oem e0 ep ¼ ep þ j

2Km Du oe0

According to the Maxwell-Wagner-O’Konski theory, there is a dispersion region where the dielectric permittivity and the conductivity of the colloid are frequency dependent. Two simple interpretations exist for this MaxwellWagner frequency, oMW. From DL theory, it is the frequency of the DL

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relaxation to the external field disturbance. From general electrodynamics, it is the frequency at which active and passive currents are equal. Thus, oMW can be defined by two expressions: k2 Deff ¼ oMW ¼

Km e0 em

ð2:79Þ

The Maxwell-Wagner-O’Konski cell model is applicable to the pure electrodynamic problem only. The other cell model mentioned above (Happel or Kuvabara cell model, Section 2.5.3) covers only hydrodynamic effects. An electrokinetic cell model must be valid for both electrodynamic and hydrodynamic effects. It must specify the relationship between macroscopic, experimentally measured electric properties and the local electric properties calculated using the cell concept. There are multiple ways to do this. For instance, the Levine-Neale cell model [115] specifies this relationship using one of the many possible analogies between local and macroscopic properties. The macroscopic properties are current density, hIi, and electric field strength, hEi. According to the Levine-Neale cell model, they are related to the local electric current density, I, and electric field, rf as: hIi ¼

Ir b cosyr¼b

hEi ¼ 

1 @f cosy @rr¼b

ð2:80Þ

ð2:81Þ

Relationships (Equations 2.80 and 2.81) are not unique. There are many other ways to relate macroscopic and local fields [146]. This means that we need a set of criteria to select a proper cell model. One set has been suggested in the electrokinetic cell model created by Shilov and Zharkikh [116]. Their criteria determine a proper choice of expressions for macroscopic “fields” and “fluxes.” The first criterion is the well-known Onsager relationship which constrains the values of the macroscopic velocity of the particles relative to the liquid, hVi, the macroscopic pressure, hPi, the electric current, hIi, and the field hEi: hVi hEi ¼ hIi hrPi¼0 hrPihIi¼0

ð2:82Þ

This relationship requires a certain expression for entropy production, S: 1 ðhIihEi þ hVihrPiÞ ð2:83Þ T It turns out that the expression derived by Shilov and Zharkikh for the macroscopic field strength is different from that of the Levine-Neale model. It is: S¼

hEi ¼

f b cosyr¼b

ð2:84Þ

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However, the expression for the macroscopic current is the same in both models. It is important to mention that the Shilov-Zharkikh electrokinetic cell model condition (Equation 2.84) fully coincides with the relationship between macroscopic and local electric fields of the Maxwell-Wagner-O’Konski electrodynamic cell model. The Shilov-Zharkikh cell model, allowing for some finite surface conductivity but assuming no concentration polarization, yields the following expressions for the electrophoretic mobility and conductivity in a concentrated dispersion [145]: hmi ¼

e0 em B 1’
m 1 þ Du þ 0:5’ð1  2DuÞ

Ks 1 þ Du  ’ð1  2DuÞ ¼ Km 1 þ Du þ 0:5’ð1  2DuÞ

ð2:85Þ

ð2:86Þ

Identical electrodynamic conditions in the Shilov-Zharkikh and MaxwellWagner-O’Konski cell models yields the same low-frequency limit for the conductivity (Equation 2.86). In addition, for the extreme case of Du ! 0, Equations (2.85) and (2.86) lead to Equation (2.71) for electrophoretic mobility. Consequently, the Shilov-Zharkikh cell model reduces to Smoluchowski’s equation in the case of negligible surface conductivity. This is an important test for any electrokinetic theory because Smoluchowski’s law is known to be valid for any geometry and volume fraction. Simultaneous measurement of conductivities Ks and Km makes possible calculation of the Dukhin number. This method was used by Lyklema and Minor [117], but with another conductivity theory [118], which seems to be the linear version of the Maxwell-Wagner theory. In the case of dielectric spectroscopy, there is a method that is similar to the “effective media” approach used by Brinkman for hydrodynamic effects. It was suggested first by Bruggeman [119]. We will use both these approaches for describing particle interaction in acoustic and electroacoustic phenomena.

2.6.5. Nonlinear Interaction in the Electric Field: Nonlinear Electrophoresis, Electrocoagulation, and Electrorheology A strong electric field disturbs DL, which in turn leads to the various deviations in classical linear theories of electrokinetic phenomena. The most obvious is the appearance of a nonlinear component in the electrophoretic mobility. This effect has been known for several decades. A historical overview can be found in a recent article [120]. It turns out that the most pronounced nonlinearity should be observed for conducting particles with a nonconducting shell. Such type of particles is usually referred to as ideally

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polarized particles. A simple theory of this nonlinear electrophoresis has been developed [121]. It yields the following expression for electrophoretic mobility: m¼

em e0 z 9em e0 a2 @Cdl E2  8
Cdl @’ ’¼z


ð2:87Þ

There are several other theories of nonlinear electrophoresis reviewed in the paper mentioned [120]. This effect has attracted serious attention recently because of the development of microfluidics. Induced electric charge can be used for creating wellcontrolled electro-osmotic flow is small channels. Another effect of the strong electric field is associated with the particleparticle interaction. In general, the contribution of the electric field to particle-particle interaction is nonlinear with the electric field strength, E. This follows from simple symmetry considerations. If such a force proportional to E exists, it would depend on the direction of the electric field, or on the sign of E. This implies that, in changing the direction of E, we would convert attraction to repulsion and vice versa. This is clearly impossible. Any interaction force must be independent of the direction of the electric field. This requirement becomes valid if the interaction force is proportional, for example, to the square of the electric field strength, E2. A linear interaction between particles affects their translational motion in an electric field. A mechanism exists for a linear particle interaction with an electrode in a DC field [122], which leads to a particle motion either towards the electrode or away from it [123]. In contrast, a nonlinear interaction induces a mutual or relative motion on the particles. We now give a short description of the theory and experiments related to this nonlinear particleparticle interaction. This nonlinear particle-particle interaction is the cause of two readily observed phenomena that occur under the influence of an electric field. The first phenomenon is the formation of particle chains oriented parallel to the electric field, as shown in Figure 2.20. These chains affect the rheological properties of the colloid [125, 126, 147]. The second effect is less well known but was first reported by Murtsovkin [124]. He observed that under certain conditions the particles build structures perpendicular, or at some other angle, to the electric field, as shown in Figure 2.20. These experiments clearly indicate that the application of an electric field can affect the stability of colloids, and that is why this condition is of general importance for colloid science. We have already mentioned that the volume of the dispersion medium adjacent to a colloid particle in an external electric field differs in its characteristics from those of the bulk dispersion medium. Such intensive

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FIGURE 2.20 Chains of particles in parallel with the electric field strength. Obtained from Murtsovkin [124].

parameters as the electric field, the chemical potential, and the pressure vary from the immediate vicinity of a particle to distances on the order of the dimension of the particle itself. If the volume fraction of the particles is sufficiently large, the particles will frequently fall into these regions of inhomogeneity of the intensive parameters. Naturally, the particles’ behavior in these regions will differ from that in the free bulk. Historically, it happened that the influence of each intensive parameter on the relative particle motion was considered separately from the others. As a result, three different mechanisms of particle interactions in an electric field are known. A short description of each follows. The oldest, and best known, is the mechanism related to the variation of the electric field itself in the vicinity of the particles. The particles gain dipole moments, pind, in the electric field, and these dipole moments change the electric field strength in the vicinity of the particles. This, in turn, creates a particle-particle attraction. This mechanism is usually referred to as a dipole-dipole electrodynamic interaction [127, 128]. This dipole-dipole interaction is responsible for the formation of particle chains and the phenomenon of electrorheology [125]. The dipole-dipole interaction is potential, which means that the energy of interaction, normally characterized with a potential energy, Udd, is independent of the trajectory of the relative motion: Udd ¼

p2ind ð1  3 cos2 yÞ 4pe0 em r 3

ð2:88Þ

where r and y are the spherical coordinates associated with one of the particles. The trajectory of the relative particle motion is shown as (A) in Figure 2.21, where rstart and ystart are the initial particle coordinates.

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FIGURE 2.21 Structure of particles perpendicular to the electric field strength. Obtained from Murtsovkin [124].

Dipole-dipole interaction is the only potential mechanism of particle interaction in an electric field. The two others are not potential, and this means that the energy spent on the relative motion depends on the trajectory of the motion. In order to compare the intensities of these different mechanisms, it is convenient to define them in terms of the radial component of the relative motion velocity, Vr. For the dipole-dipole interaction, Vr is given by: r ¼ Vdd

e0 em a5 E2 ð1  DuÞ2 ð1  3 cos2 yÞ 2
ð1 þ 2DuÞr4

ð2:89Þ

where the value of the dipole moment is substituted according to Equation (2.38). The second, “electrohydrodynamic,” mechanism was suggested by Murtsovkin [124, 125, 129–131]. He observed that an electric field induces an electro-osmotic flow that has quadruple symmetry around the particle. The radial component of the relative (nonconducting) particle motion velocity is: Vhr ¼

3e0 em a3 E2 Dujzjð1  3 cos2 yÞ 2
ð1 þ 2DuÞr2

ð2:90Þ

The trajectory of this relative motion is shown in Figure 2.19B. The third, “concentration” mechanism [132] is related to the concentration polarization of the particles. It is defined as: Vcr ¼

3a3 Cs FE2 Duð1  cos2 yÞ djmj 2ð1 þ 2DuÞRTr 2 dCs

ð2:91Þ

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Dipole-dipole electrodynamic interaction

-

+ r2 r 2start

A

=

-

+

sin2q cosq cosqstart

sin2qstart

Electro-hydrodynamic interaction

1

r(r 2start − a2) r start (r 2 − a2)

B

=

2

sin2q cosq sin2qstart cosq start

Concentration polarization interaction C(r)

ζ+Δζ C0 1

C

2 r sinq = rstart sinq start

FIGURE 2.22 Three mechanisms for the interaction of particles in an electric field.

where Cs is the concentration of the electrolyte in the bulk of the solution. Again, the corresponding trajectory is given in Figure 2.22(C). Comparison of Vr for various mechanisms indicates that the electro2 and hydrodynamic and concentration mechanisms decay with distance as rpp 4 are longer ranged than the dipole-dipole mechanism which decays as rpp . This explains why, in some cases, particles build structures perpendicular to the electric field. These effects are especially pronounced for systems of biological cells and for metal particles [25]. The electrohydrodynamic and concentration mechanisms for nonconducting colloidal particles are only appropriate for static or low-frequency electric fields. They do not operate above the frequency of concentration polarization.

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This means that, at the high frequency of ultrasound, only the dipole-dipole interaction can affect the stability of the colloid.

2.7. TRADITIONAL PARTICLE SIZING There are three groups of traditional particle sizing methods that can be applied to characterize the size of colloidal particles [133, 134]. These groups are (1) counters, (2) fractionation techniques, and (3) macroscopic fitting techniques. Particle counters yield a full PSD weighted by number. This group includes various microscopic image analyzers [135], electrozone counters (Coulter counters), and optical zone counters. Fractionation techniques also yield a full PSD, but weighted by mass. They achieve this by separating particles into fractions with different sizes. This group includes sieving, sedimentation, and centrifugation [148]. The last group, the “macroscopic technique,” includes methods related to the measurement of various macroscopic properties that are particle size dependent. All of these methods require an additional step in order to deduce the PSD from the measured macroscopic data. This is the so-called ill-defined problem [136, 137]. We further discuss this problem later with regard to acoustics in Chapter 7. Each of these three groups has advantages and disadvantages over the others. There is no universal method available that is able to solve all particle sizing applications. For example, the “counters group” has an advantage over “macroscopic techniques” in that it yields full PSD. However, statistically representative analysis requires the counting of an enormous number of particles, especially for polydisperse systems. The technique can be very slow. Sample preparation is rather complicated and might affect the results. Transformation of a number-weighted PSD to a volume-based PSD is not accurate in many practical systems. Fractionation methods are, statistically, more representative than counters, but they are not suitable for small particles. It can take hours to perform a single particle size analysis, particularly of submicrometer particles, even using an ultracentrifuge. In these methods, hydrodynamic irregularities can dramatically disturb the samples and consequently affect the measured PSD. Macroscopic methods are fast, statistically representative, and easy to use. However, the price to be paid is the necessity to solve the ill-defined problem, and that limits particle size information. Fortunately, in many cases there is no need to know the exact details of the full PSD. Median size, PSD width, or bimodality is sufficient for an adequate description of many practical colloids and colloid-related technologies. This is why macroscopic methods are very popular and widely used in many laboratories and plants for routine analysis. Acoustics, as a particle sizing technique, belongs to the third group of macroscopic methods. It competes with the most widely used macroscopic

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method—light scattering [138, 139]. It can be considered also as an alternative to neutron scattering for characterizing microemulsions and polymer solutions. The advantages of acoustics over light scattering were given in the Introduction. There are a number of points of similarity between light and ultrasound that are important in the discussion of the application of acoustics to particle sizing analysis. To this end, a short description of light scattering follows. A full review can be found in the book by Bohren and Huffman [138].

2.7.1. Light Scattering: Extinction = Scattering + Absorption This section is a selection of various important statements and notions presented in the book by Bohren and Huffman [138]. The presence of the particles results in the extinction of the incident beam. The extinction energy rate, Wext, is the sum of the energy scattering rate, Wsca, plus the energy absorption rate, Wabs, through an imaginary sphere around a particle. Wext ¼ Wabs þ Wsca

ð2:92Þ

In addition to reradiating electromagnetic energy (scattering), the excited elementary charges may transform part of the incident electromagnetic energy into other forms (e.g., thermal energy), a phenomenon called absorption. Scattering and absorption are not mutually independent processes, and although, for brevity, we often refer only to scattering, we will always as well imply absorption. Also, we will restrict our treatment to elastic scattering: that is, the frequency of the scattered light is the same as that of the incident light. If the particle is small compared to the wavelength, all the secondary wavelets are approximately in phase; for such a particle, we do not expect much variation of scattering with direction. Single scattering occurs when the number of particles is sufficiently small and their separation sufficiently large so that in the neighborhood of any particle the total field scattered by all particles is small compared to the external field. Incoherent scattering occurs when there is no systematic relation between the phases of the waves scattered by the individual particles. However, even in a collection of randomly separated particles, the scattering is coherent in the forward direction. The extinction cross section Cext may be written as the sum of the absorption cross section, Cabs, and the scattering cross section, Csca: Cext ¼

Wext ¼ Cabs þ Csca Ii

ð2:93Þ

where Ii is the incident irradiance. There is an important optical theorem [138] that defines extinction as depending only on the scattering amplitude in the forward direction.

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This theorem, common to all kinds of seemingly disparate phenomena, involves acoustic waves, electromagnetic waves, and elementary particles. It is anomalous because extinction is the combined effect of absorption and scattering in all directions by the particle. Extinction cross-section is a well-defined, observable quantity. We measure the power incident at the detector with and without a particle interposed between the source and the detector. The effect of the particle is to reduce the detector area by Cext. Hence the description of Cext is as an area. In the language of geometrical optics, we would say that the particle “casts a shadow” of area Cext. However, this “shadow” can be considerably larger or smaller than the particle’s geometric shadow. Importantly, for a particle that is much larger than the wavelength of the incident light, the light scattered tends to be concentrated around the forward direction. Therefore, the larger the particle, the more difficult it is to exclude scattered light from the detector. The optical theorem neglects receiving the scattered light, but it creates a problem for large particles [138]. By observing only the transmitted light, it is not possible to determine the relative contribution of absorption and scattering to extinction; to do so requires an additional, independent observation. Multiple scattering aside, the underlying assumption is that all light scattered by the particles is excluded from the detector. The larger the particle, the more the scattering envelope is peaked in the forward direction and hence the greater the discrepancy between the measured and calculated extinction. Thus, a detection system has to be carefully designed in order to measure an extinction that can be legitimately compared with the theoretical extinction. This is particularly important for large particles. An extinction efficiency, Qext, can be defined as: Qext ¼ Cext = pa2

ð2:94Þ

As the particle radius, a, increases, Qext approaches a limiting value of 2. This is twice as large as that predicted from geometrical optics. This puzzling result is called the extinction paradox, since it seemingly contradicts geometrical optics. Yet geometrical optics is considered to be a good approximation, especially when all particle dimensions are much larger than the incident wavelength. Moreover, the result contradicts “common sense”; we do not expect a large object to dissipate twice the amount of energy that is incident upon it! Although for large objects geometrical optics is a good approximation for the exact wave theory, even for very large objects geometrical optics is still not exact. This arises because all practical materials are anisotropic; they posses “edges.” The edge deflects rays in its neighborhood, rays that, from the viewpoint of geometrical optics, would have passed unimpeded. Irrespective of how small the angle is through which they are deflected, rays are counted as having been removed from the incident beam and this

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contributes, therefore, to the total extinction (roughly speaking, we may say that the incident wave is influenced beyond the physical boundaries of the obstacle). These edge-deflected rays cause diffraction that can be described in terms of Fresnel zones [138, 140]. It is convenient to describe diffraction by considering light transmission through an aperture instead of the scattering by the particle. For instance, we may ask how the intensity changes when the point of observation, Pobs, moves along the axis of an aperture of fixed dimensions which models the particle. Since the radii of the Fresnel zones depend on the position of Pobs, we find that the intensity goes through a series of maxima and minima, occurring, respectively, when the aperture includes an odd or an even number of Fresnel zones. A special case is when the source is very far away, so that the incident wave may be regarded as a plane wave. The radius of the first Fresnel zone, R1, increases indefinitely as the distance from the aperture Robs of Pobs goes to infinity. Thus, if the point of observation is sufficiently far away, the radius of the aperture is certainly smaller than that of the first Fresnel zone. As the distance, Robs, gradually decreases, that is, Pobs approaches the aperture, the radius, R1, decreases correspondingly. Hence an increasingly large fraction of the first Fresnel zone appears through the aperture. Again, as Pobs moves closer and closer to the aperture, maxima and minima follow each other at decreasing distances. When the distance Robs of Pobs from the aperture is not much larger than the radius of the aperture itself, R, then the distances between successive maxima and minima approach the magnitude of the wavelength. Under these conditions, of course, the maxima and minima can no longer be practically observed. From our previous discussion, when the distance of the source and the point of observation from the aperture is large, compared with the square of the radius of the aperture divided by the wavelength (Rayleigh distance, R0), it follows that only a small fraction of a Fresnel zone appears through the aperture. The diffraction phenomenon observed in these circumstances is called Fraunhofer diffraction. When, however, the distance from the aperture of either the point of observation or the source is R0 or smaller, then the aperture uncovers one or more Fresnel zones. The diffraction phenomenon observed is then termed Fresnel diffraction. The exact solutions for scattering and absorption cross-sections of spheres are given in the Mie theory [141]. It might appear that we are faced with the straightforward task of obtaining quantitative results from the Mie theory. However, the number of terms required for convergence can be quite large. For example, if we were interested in investigating the rainbow created by 1-mm water droplets, we would need to sum about 12,000 terms. Even for smaller particles, the number of calculations can be exceedingly large. Computers can greatly reduce this computation time, but problems still remain related to the unavoidable need to represent a number having an infinite

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number of digits by a number with a finite number. The implicit round-off error accumulates as the number of terms required to be counted increases. Unfortunately, there is no unanimity of opinion in defining the condition under which round-off error accumulation becomes a problem. If we were interested only in scattering and absorption by spheres, we would need to go no further than the Mie theory. But physics is, or should be, more than just a semi-infinite strip of computer output. In fact, great realms of calculations often serve only to obscure from view the basic physics that can be quite simple. Therefore, it is worthwhile to consider approximate expressions, valid only in certain limiting cases, that point the way toward approximate methods to be used to tackle problems for which there is no exact theory. If a particle has a nonregular geometrical shape, then it is difficult, if not impossible, to solve the scattering problem in its most general form. There are, however, frequently encountered situations when the particles and the medium have similar optical properties. If the particles, which are sometimes referred to as “soft,” are not too large (but they might exceed the Rayleigh limit), it is possible to obtain relatively simple, approximate expressions for the scattering matrix. This is expressed as the Rayleigh-Gans theory. Geometrical optics is a simple and intuitively appealing approximate theory that need not be abandoned because an exact theory is at hand. In addition to its role in guiding intuition, geometrical optics can often provide quantitative answers to small-particle problems: solutions that are sufficiently accurate for many applications, particularly in light of the precision, accuracy, and reproducibility of actual measurements. Transition from single-particle scattering to scattering by a collection of particles can be simplified using the concept of an effective refractive index. This concept is meaningful for a collection of particles that are small compared with wavelength, at least as far as transmission and reflection are concerned. However, even when the particles are small compared to the wavelength, this effective refractive index should not be interpreted too literally as a true refractive index on the same footing as the refractive index of, say, a homogeneous medium. For example, attenuation, in a strictly homogeneous medium, is the result of absorption and is accounted for quantitatively by the imaginary part of the refractive index. In a particulate medium, however, attenuation may be wholly or in part the result of scattering. Even if the particles are nonabsorbing, the imaginary part of the effective refractive index can be nonzero [138]. When an incident beam traverses a distance, x, through an array of particles, the irradiance is attenuated according to It ¼ Ii expðaxÞ

ð2:95Þ

a ¼ Np Cext ¼ Np Cabs þ Np Csca

ð2:96Þ

where

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Now, the exponential attenuation of any irradiance in particulate media requires that: ax 1

ð2:97Þ

This condition might be relaxed somewhat if the scattering contribution to total attenuation is small: Np Csca x 1

ð2:98Þ

An amount of light, dI, is removed from a beam propagating in the x direction through an infinitesimal distance between x and x þ dx in an array of particles dI ¼ aIdx

ð2:99Þ

where I is the beam irradiance at x. However, light can get back into the beam through multiple scattering; that is, light scattered at any other position in the array may ultimately contribute to the irradiance at x. Scattered light, in contrast to absorbed light, is not irretrievably lost from the system. It merely changes direction, is lost from a beam propagating in a particular direction, but then contributes in other directions. Clearly, the greater the scattering cross-section, the number density of particles, and thickness of the array, the greater will be the multiple scattering contributions to the irradiance at x. Thus, if Np Csca is sufficiently small, we may ignore multiple scattering and can integrate Equation (2.99) to yield the exponential attenuation Equation (2.95). The expression for the field, or envelope, of light scattered by a sphere is normally obtained under the assumption that the beam is infinite in lateral extent; such a beam, however, is difficult to produce practically in a laboratory. Nevertheless, it is physically plausible that scattering and absorption, by any particle, will be independent of the extent of the beam provided that the beam is large compared to the particle size; that is, the particle is completely bathed in the incident light. This is supported by the analysis of Tsai and Pogorzelski [142], who obtained an exact expression for the field scattered by a conducting sphere when the incident beam is cylindrically symmetric with a finite cross-section. Their calculations show no difference in the angular dependency of the light scattered by a conducting sphere between infinite and finite beams provided that the beam radius is about 10 times larger than the sphere radius.

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[3] R. Weast (Eds.), Handbook of Chemistry and Physics, 70th ed., CRC Press, Florida, 1989. [4] S. Temkin, Elements of Acoustics, first ed., Wiley, New York, 1981. [5] P.M. Morse, K. Uno Ingard, Theoretical Acoustics, McGraw-Hill, New York, NY, 1968. Princeton University Press, New Jersey, 1986, p. 925. [6] T.A. Litovitz, Lyon, Ultrasonic hysteresis in viscous liquids, J. Acoust. Soc. Am. 26 (1954) 577–580. [7] Stokes, On a difficulty in the Theory of Sound, Philos. Mag. Nov. (1848). [8] L.W. Anson, R.C. Chivers, Thermal effects in the attenuation of ultrasound in dilute suspensions for low values of acoustic radius, Ultrasonic 28 (1990) 16–25. [9] D.W. Cannon, New developments in electroacoustic method and instrumentation, in: S.B. Malghan (Ed.), Electroacoustics for Characterization of Particulates and Suspensions, NIST, 1993, pp. 40–66. [10] R.J. Hunter, Review. Recent developments in the electroacoustic characterization of colloidal suspensions and emulsions, Colloids Surf. 141 (1998) 37–65. [11] A. Kitahara, Nonaqueous systems, in: A. Kitahara, A. Watanabe (Eds.), Electrical Phenomena at Interfaces, Marcel Decker, New York, 1984. [12] Ph.C. van der Hoeven, J. Lyklems, Electrostatic stabilization of suspensions in non-aqueous media, Adv. Colloid Interface Sci. 42 (1992) 205–277. [13] I.D. Morrison, Electrical charges in non-aqueous media, Colloids Surf. A 71 (1993) 1–37. [14] N. Bjerrum, A new form for the electrolyte dissociation theory, Proceedings of the 7th International Congress of Applied Chemistry, Section X, London, 1909, pp. 55–60. [15] R.M. Fuoss, Ionic association. III. The equilibrium between ion pairs and free ions, J. Am. Chem. Soc. 80 (1958) 5059–5060. [16] J. Bockris, A.K. Reddy, Modern Electrochemistry, vol. 1, A Plenum/Rosetta ed., Plenum Press, New York, 1977. [17] J. Lyklema, Principles of the stability of lyophobic colloidal dispersions in non-aqueous media, Adv. Colloid Interface Sci. 2 (1968) 65–114. [18] D.N.L. McGown, G.D. Parfitt, E. Willis, Stability of non-aqueous dispersions. 1. The relationship between surface potential and stability in hydrocarbon media, J. Colloid Interface Sci. 20 (1965) 650–664. [19] A.S. Dukhin, P.J. Goetz, How non-ionic “electrically neutral” surfactants enhance electrical conductivity and ion stability in non-polar liquids, J. Electroanal. Chem. 588 (2006) 44–50. [20] R. Treffers, M. Cohen, High resolution spectra of cool stars in the 10- and 20-micron region, Astrophys. J. 188 (1974) 545–552. [21] R. Fuchs, Theory of the optical properties of ionic crystal cubes, Phys. Rev. B11 (1975) 1732–1740. [22] H.P. Pendse, T.C. Bliss, W. Han, Particle shape effects and active ultrasound spectroscopy, in: V.A. Hackley, J. Texter (Eds.), Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, Amer. Ceramic Soc., Westerville, OH, 1998. [23] S.S. Dukhin, V.N. Shilov, Dielectric Phenomena and the Double Layer in Dispersed Systems and Polyelectrolytes, Wiley, New York, NY, 1974. [24] P.V. Zinin, Theoretical analysis of sound attenuation mechanisms in blood and erythrocyte suspensions, Ultrasonics 30 (1992) 26–32. [25] A.S. Dukhin, Biospecifical mechanism of double layer formation in living biological cells and peculiarities of cell electrophoresis, Colloids Surf. 73 (1993) 29–48. [26] B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman Company, New York, 1982.

84 [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

[39] [40] [41] [42] [43] [44] [45] [46]

[47] [48] [49] [50] [51] [52]

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Chapter 3

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

3.1. Longitudinal Waves and the Wave Equation 92 3.2. Acoustic Impedance 95 3.3. Propagation Through Phase Boundaries: Reflection 97 3.4. Longitudinal Rheology and Shear Rheology 99 3.5. Longitudinal Rheology of Newtonian Liquids: Bulk Viscosity 103 3.5.1. In Hydrodynamics 103 3.5.2. In Acoustics 103

3.5.3. In Analytical Chemistry 3.5.4. In Molecular Theory of Liquids 3.5.5. In Rheology 3.6. Attenuation of Ultrasound in Newtonian Liquid: Stokes Law 3.7. Newtonian Liquid Test Using Attenuation Frequency Dependence 3.8. Chemical Composition Influence References

104 104 104

108

110 114 121

This chapter has been significantly modified with a more specific description of the relationship between acoustics and rheology. The purpose of this chapter is to provide a general overview characterizing sound wave propagation in homogeneous liquids. It is important to stress here that practically any liquid system can be modeled as “homogeneous.” For instance, we can treat milk as a homogeneous liquid, forgetting that it is actually a collection of fat droplets in water with sugars and proteins. This approach would provide only limited information about the system. However, it is useful when we deal with very complex systems and some degree of abstraction helps us learn at least something. This approach is essentially identical to a rheological description of the complex liquids. That is why Studies in Interface Science, Vol. 24. DOI: 10.1016/S1383-7303(10)23003-X Copyright # 2010, Elsevier B.V. All rights reserved.

91

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rheological aspect of acoustics plays an important role in this chapter. The relationship between acoustics and rheology was recognized long ago by such prominent scientists in the field of acoustics as Rayleigh [1, 2], Morse [3, 4], and Temkin [5]. Unfortunately, there is still very little appreciation of the possibilities that this opens in the rheological field. We will start with introducing some basic information on acoustics in homogeneous media. Then, we will present some general concepts of longitudinal rheology and its relationship with shear rheology. Then, we provide some examples of longitudinal rheology for Newtonian liquids. The last section is dedicated to the influence of the chemical composition on the acoustic properties of Newtonian liquids.

3.1. LONGITUDINAL WAVES AND THE WAVE EQUATION We call traveling compression waves in liquids “longitudinal waves,” in contrast to “transverse waves” typified by a vibrating string. The direction that the material moves, relative to the direction of wave propagation, makes the difference. For a longitudinal wave, the liquid molecules move back and forth in the direction of the wave propagation, whereas for a transverse wave this motion is perpendicular to the direction of propagation. Longitudinal waves, in the most general case, are three dimensional with a potentially very complex geometry. Here, we will consider only two-dimensional plane waves, which have the same direction of propagation everywhere. Longitudinal waves can propagate in a fluid because the fluid has a finite compressibility that allows the energy to be transported across space. In addition to these compression waves, the sound wave itself may induce shear and thermal waves at the boundaries with other surfaces. Compression waves are able to propagate over long distances in the liquid, whereas shear and thermal waves exist only in the close vicinity of phase boundaries. This chapter covers only compression longitudinal waves. Shear and thermal waves play a major role in colloidal systems because of the extended surface area. This aspect is dealt with in detail later, in Chapter 4. We consider the liquid itself to be in equilibrium, and having a certain density, rm, at a given static pressure, P, and temperature, T. The relationship between these three parameters can be described by two partial derivatives, one at constant temperature and one at constant pressure. Together, these equations constitute the so-called equations of state, namely:
 1 @rm p b ¼ ð3:1Þ rm @P T b¼


 1 @rm rm @T P

ð3:2Þ

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

93

We commonly refer to the coefficient bp as the isothermal compressibility and the coefficient b as the thermal expansion coefficient. The presence of a sound wave at a particular instant of time, and point in space, causes an incremental variation of the density, pressure, and temperature about their equilibrium values rm, Peq, and T. We designate the instantaneous values of these incremental changes as drm, dP, and dT, respectively. The pressure variation is the easiest to detect and is in fact a typical output of an acoustic experiment. The sound wave pressure is usually expressed in decibels, abbreviated as dB, and defined as 20 times the logarithm to the base 10 of the ratio of the pressure amplitude, in dynes per centimeter squared, and a reference level of 1 dyne/cm2. The variation of the temperature in the liquid, dT, due to the presence of the sound wave is usually very small, because typically the high thermal conductivity of the liquid rapidly smoothes out any local perturbations. Consequently, the density variation, drm, is adequately described considering only the isothermal compressibility bp:
 @rm P ¼ rm þ rm bp P ð3:3Þ rm þ drm ¼ rm þ @P T The variation of the density is independent of the equilibrium pressure. The variation of the pressure in the case of a plane sound wave is described with a wave equation: @2P 1 @2P ¼ 2 2 2 @x cm @t

ð3:4Þ

where the sound speed cm is found from: c2m ¼

1 b p rm

ð3:5Þ

The values for sound speed and density of several liquids are given in Table 3.1 using the data published in Ref. [6]. The sound speed through most solid materials is much higher than that of fluids and is typically between 5000 and 6000 m/s. Compressibility is a measure of the liquid elasticity. Precise sound speed measurement provided by modern acoustic spectrometers allows fast, nondestructive, and precise calculation of both storage modulus and compressibility as presented in Table 3.1 for all 12 liquids. These values agree well with independent data from the Handbook of Chemistry and Physics [7] for some of the liquids. The speed of sound is temperature dependent. The magnitude of this dependence is usually several meters per second per degree, as shown in Table 3.1 according Ref. [6]. Figure 3.1 illustrates this dependence for water.

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TABLE 3.1 Sound Speed and Compressibility of 12 Newtonian Liquids Sound Sound speed T Storage speed at coef. at 25  C modulus G0long  25 C (m/s) (m/s/ C) at 25  C (109 Pa)

Liquid

Compressibility at 25  C (1010 Pa1)

Cyclohexane

1256

6.66

1.23

1.15

Cyclohexanone

1407

4.20

1.88

0.75

1-Pentanol

1273

2.88

1.31

1.07

Toluene

1308

4.70

1.53

0.92

Methanol

1106

3.65

0.97

1.46

Hexane

1078

4.65

0.77

1.84

2-Propanol

1143

4.40

1.03

1.37

Butyl acetate

1193

4.44

1.26

1.12

Acetone

1166

4.97

1.08

1.31

Ethanol

1147

4.26

1.03

1.36

Pyridine

1417

4.57

1.98

0.72

Water

1496

þ2.40

2.23

0.63

1522 1520

Sound speed (m/s)

1518 1516 1514 1512 1510 1508 1506 1504 1502 1500 28.

29.

30.

31.

32.

33.

34.

35.

36.

Temperature (C) FIGURE 3.1 Speed of sound in water measured at different temperatures using DT-600 [6].

3

Fundamentals of Acoustics in Homogeneous Liquids: Longitudinal Rheology

95

The simple wave equation (Equation 3.4) is valid for uniform, homogeneous liquids having negligible attenuation due to viscous and thermal effects and, furthermore, only if the acoustic pressure is small compared to the equilibrium pressure. This simplified model of acoustic phenomena turns out to be quite adequate in practice, since the typical ultrasound pressure levels used for acoustic characterization are thousands of times weaker than the normal equilibrium pressure of 1 atm. We do not consider any nonlinear effects that might occur with the very high intensity ultrasound sometimes used for various modifications of liquids and colloids. The general solution of the above wave equation describes a pressure wave traveling in the positive, x, direction as a function P(xcmt): Pðx, tÞ ¼ P0 expðam xÞexp½ jðot  FÞ

ð3:6Þ

where am is the attenuation of the medium, o is the frequency of ultrasound, and P0 is the initial amplitude of the pressure. A similar equation can be written for the fluid velocity, v. The ratio between the acoustic pressure and the fluid velocity is called the acoustic impedance, which will be discussed now.

3.2. ACOUSTIC IMPEDANCE There is a convenient analogy between electric and acoustic phenomena. It is based on the linear relationship between the driving force and the corresponding flow. In the case of an alternating electric field, it is electric voltage and electric current. According to Ohm’s law, the current is proportional to the voltage; the coefficient of proportionality is called the electric impedance. The acoustic impedance, Z, is introduced in a similar way, as a coefficient of propotionality between pressure, P, and velocity of the particles, v, in the sound wave: P ð3:7Þ v As with the electric impedance, the acoustic impedance is a complex number. It depends on the other acoustic properties of the media. To derive this relationship, we consider the propagation of a plane longitudinal wave through the medium along the x-axis. This medium is characterized by a given density, rm, sound speed, cm, and longitudinal wave attenuation, am. The particle’s displacement, u, for a plane sound wave is a simple harmonic function: Z¼

uðx, tÞ ¼ u0 expðam xÞexp½ jðot  FÞ

ð3:8Þ

where F is the phase, F ¼ kx with k ¼ 2p/l, l is the wavelength in meters; o ¼ 2pf, where f is frequency in hertz. The attenuation coefficient, am, has dimension of neper per meter. Attenuation expressed in decibels (dB) is

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related to the attenuation expressed in nepers (np) through the following: dB/m ¼ 8.686 np/m. These equations allow us to calculate the wavelength for different fluids. For instance, the wavelength in water varies roughly from 15 mm at 100 MHz up to 1.5 mm at 1 MHz. It is important to note that the wavelength of ultrasound in the megahertz range is much larger than the wavelength of light; for example, the wavelength of green light is about 0.5 mm. This difference in wavelength between light and ultrasound results in a substantial variance between these two wave phenomena, which otherwise have many similarities. Combining together the two exponential functions and expressing the frequency and phase through the wavelength, we can rewrite Equation (3.8) as: 

 2pj am l uðx, tÞ ¼ u0 exp cm t  x 1  j ð3:9Þ l 2p To derive an expression for the pressure, we use Hooke’s law, which relates a force exerted on an element of the medium, Fhook, and a strain caused by the sound wave, @u/@x: Fhook ¼ MS

@u @x

ð3:10Þ

where S is the cross-sectional area. The pressure is simply the opposite of the force per unit area of the surface:
 @u am l 2 @u 2 2pj P ¼ M ¼ rm c ¼ rm cm 1j uðx, tÞ ð3:11Þ @x @x l 2p We express the velocity of the particles’ motion as a time derivative of the displacement: @uðx, tÞ 2pjcm uðx, tÞ ð3:12Þ ¼ l @t Replacing displacement with the particle velocity in the expression for the pressure gives:
 am l v ð3:13Þ P ¼ rm cm 1  j 2p vðx, tÞ ¼

It directly follows by analogy with this expression that the acoustic impedance, Z, is:
 am l Zm ¼ rm cm 1  j ð3:14Þ 2p where the attenuation is in (np/m), wavelength in (m), sound speed of the longitudinal wave in (m/s), and density is in (kg/m3).

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The attenuation does not contribute significantly to the acoustic impedance. Even in highly concentrated colloids at high frequency, when the attenuation becomes about 1000 dB/cm, it still contributes only 2% to the acoustic impedance. This allows us to approximate the acoustic impedance as a real number equal to the product of the density and sound speed.

3.3. PROPAGATION THROUGH PHASE BOUNDARIES: REFLECTION Acoustic impedance is a very convenient property for characterizing effects that occur when the sound wave meets the boundary between two phases. There are certain similarities between longitudinal ultrasound and light reflection and transmission through the phase boundaries. For instance, the ultrasound reflection angle from a plane surface is equal to the incident angle which is the same as for light (see Figure 3.2): yi ¼ yr

ð3:15Þ

where the index i corresponds to the incident wave and index r corresponds to the reflected wave. The transmitted wave angle must satisfy wavefront coherence at the border. Again, this yields the same relationship as with light transmission: sinyi c1 ¼ sinyt c2

ð3:16Þ

where indexes 1 and 2 correspond to the different phases.

reflected wave

incident wave

phase 1

qr

qi

qt phase 2

transmitted wave

FIGURE 3.2 Illustration of the sound propagation through a phase boundary.

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Propagation of the sound wave through the phase border should not create any discontinuities in pressure or the particle’s velocity. This condition yields the following relationships for the pressure in the reflected and transmitted waves: Pr Z2 cosyi  Z1 cosyt ¼ Pi Z2 cosyi þ Z1 cosyt

ð3:17Þ

Pt 2Z2 cosyi ¼ Pi Z2 cosyi þ Z1 cosyt

ð3:18Þ

In the case of normal incidence, when yi ¼ yt ¼ 0, these equations simplify to: Pr Z2  Z1 ¼ Pi Z2 þ Z1

ð3:19Þ

Pt 2Z2 ¼ P i Z2 þ Z 1

ð3:20Þ

and

From these equations, an important relationship is derived between the phases of the reflected and incident waves. If Z2 > Z1, then the reflected pressure wave is in phase with the incident wave; otherwise, it is 180 out of phase. The pressure value determines the intensity of the ultrasound, I: I¼

P2 2rc

ð3:21Þ

For normal incidence, we can use Equations (3.19) and (3.20) to obtain the ultrasound intensity of the reflected and transmitted waves: Ir ðZ2  Z1 Þ2 ¼ Ii ðZ2 þ Z1 Þ2

ð3:22Þ

It 4Z2 Z1 ¼ Ii ðZ2 þ Z1 Þ2

ð3:23Þ

and

At normal incidence, the reflected wave interferes with the incident wave. This leads to the buildup of standing waves. For a perfect reflector, the particle displacement in reflected and incident waves compensate each other completely when they are out of phase. They add together when they are in phase. This leads to a repeating pattern of nodes and maxima. Standing waves do not transmit any power, since the power coming back equals the power going out.

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Standing waves can superimpose with the traveling waves when the reflection is not perfect. This effect occurs when ultrasound propagates through a multilayer medium. The case of three phases is important and well characterized. A standing wave appears in the first and the second layers. They superimpose here with traveling waves if reflection at the phase boundaries is not perfect. In the case of normal incidence, it is possible to derive [5] an analytical expression for the intensities of the incident and transmitted waves: It3 ¼ Ii1

4Z3 Z1
2 2pl Z3 Z1 2pl2 2 ðZ3 þ Z1 Þ2 cos2 þ Z2 þ sin2 l2 Z2 l2

ð3:24Þ

where l2 is the thickness of the second layer. One important conclusion follows from Equation (3.24). There are two cases when the second layer becomes transparent for ultrasound propagation. The first one is rather obvious; it happens when the thickness of the second layer is much less than the wavelength ðl2  l2 =4Þ. The second case is related to the standing waves built up in the intermediate layer, when l2 ¼ nl2/2. These conditions are important for designing acoustic and electroacoustic devices.

3.4. LONGITUDINAL RHEOLOGY AND SHEAR RHEOLOGY As noted in rheological handbooks, there are two distinctively different types of rheology—“shear rheology” and “extensional rheology” [8]. Basically, they differ in the geometry of the applied stress. In shear rheology, the stress is tangential to the surface from which the stress is applied. This causes a sliding relative motion of the liquid layers. In contrast, in extensional rheology, the stress is normal to the surface applying the stress. If the liquid is considered incompressible, the volume is unchanged and we say this is an isochoric process. The rheology of oscillating extensional stress differs substantially from traditional steady-state extensional rheology. The isochoric condition might be violated when the extensional stress varies in time. Extensional nonisochoric rheology with oscillating extensional stress is called “longitudinal rheology.” There is an old review by Litovitz and Davis [9] that presents in parallel both shear and longitudinal rheological theories. Both shear and extensional stress cause displacement in the liquid. In the case of a step change in the shear stress, the liquid displacement propagates with a certain speed from the interface where it has been generated into the bulk. Eventually, it reaches some steady-state condition within the entire system. Many modern rheometers, instead of applying a steady-state stress, apply stress that varies in time. It is achieved by moving the interface backward and forward.

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When the frequency of such an oscillation is low, the displacement originated at the interface has enough time to fill the entire system during the first half of the cycle. All parts of the system would then experience displacement in one direction. Increasing the frequency would eventually alter this situation. The direction of the interface motion would change before the displacement can fill the entire system. Consequently, there would be areas in the system with opposite displacements. Such oscillating stress creates a “wave” of displacement. This is the point where rheology comes into contact with acoustics. Acoustics is a science that studies propagation of mechanical waves. There is a spatial characteristic of the waves—“wavelength.” It is the distance of wave propagation during a single period of oscillation. If wavelength is much shorter than size of the system, the subjects of rheology and acoustics would merge. This is situation that we consider in this paper. We will be dealing with waves of oscillating stresses, both shear and extensional. There are several different ways of generating these oscillations: rotating cylinders, shaking plates, vibrating surfaces of piezo-crystals, etc., see Barnes [8], Dukhin and Goetz [10], Portman and Markgaf [11], Soucemarianadin et al. [12], and Williams et al. [13–15]. There is a set of notations introduced in acoustics for characterizing these oscillating stresses. One of these notations is the frequency of oscillation, usually denoted as o. The other one is pressure P(x,t) at given moment t and given point x. In the case of monochromatic oscillation at a single frequency, this pressure is usually expressed in terms of complex numbers, see Equation (3.6). The attenuation coefficient, a, determines how quickly the pressure amplitude decays with distance due to various dissipative effects. An alternative way to characterize the pressure decay is to use the notion of “penetration depth” x0 where: 1 ð3:25Þ a The pressure amplitude decays by a factor ℓ from its initial value P0 when the distance traveled equals the “penetration depth,” where ℓ  2.71, the base of the natural logarithm. The penetration depth differs from the wave length l, which depends mostly on frequency: x0 ¼

l¼ where f is frequency in hertz.

2pV V ¼ o f

ð3:26Þ

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The attenuation coefficient and the penetration depth are phenomenological and experimentally measurable parameters that are used in acoustics for characterizing dissipative processes. The speed of the wave propagation is related to elastic properties of the liquid. Rheology operates with completely different set of notions for characterizing the viscoelastic properties of liquids. In the case of shear stress, it is a “complex shear modulus” G*, which has a real component “storage modulus” G0 and an imaginary component “loss modulus” G00 : G ¼ G0 þ jG00

ð3:27Þ

In a manner similar to shear rheology, Litovitz and Davis [9] introduce a “complex longitudinal modulus” M*, which corresponds to the longitudinal rheology case, where M ¼ M0 þ jM00

ð3:28Þ

However, it turns out that these two modulii are in fact identical if one makes a suitable change in subscript. Therefore we will use the same symbol G* for both of them, with the subindex “shear” for shear stress, and subindex “long” for longitudinal stress. Similarly, we will use the same indices for both the attenuation coefficient and penetration depth for distinguishing between these two types of stress. The above-mentioned similarity appears in the general relationship between acoustic and rheological parameters for both stresses. Functionally, these relationships are the same for both stresses: 0 1 o  2

G0shear ¼ ro2 V 2 2 long

a2 shear long

4o2 þ a2

shear long

V

B4p2 1 C C ro2 B @ l2  x2 A 0shear

2

V25

¼ 2ro V 2 3

2

long

4o2 þ a2

shear long

where r is density.

12

B4p2 1 B @ l2 þ x2

ð3:29Þ

C C A

0shear long

ro2

a shear V G00shear long

long

32 ¼ 0

32 ¼ 0 V25

lx

4p 0shear long

B4p2 1 B @ l2 þ x2

0shear long

12 C C A

ð3:30Þ

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It is possible to introduce shear viscosity and longitudinal viscosity for non-Newtonian liquids using the same expression for both parameters: G ¼ G0 þ iG00 ¼ G0 þ io shear ðoÞ

ð3:31Þ

long

The derivation of Equations (2.29) and (3.30) for the shear rheology case is from the paper by Williams and Williams [13–15], who in turn make reference to a much earlier book by Whorlow [16]. The derivation of the identical equations for the longitudinal rheology case is taken from the review by Litovitz and Davis [9]. They, in turn, make reference to much earlier works by Miexner [16, 17, 99], who apparently derived the general thermodynamic theory of stress-strain relationship in liquids. Together, Equations (3.29) and (3.30) provide general phenomenological link between acoustics and rheology. One can use them for calculating the shear loss modulus G00 shear and shear storage modulus G0 shear from the acoustically measured shear penetration depth x0shear (or shear wave attenuation ashear) and shear wave sound speed. The same can be done for longitudinal parameters. The longitudinal loss modulus G00 long and longitudinal storage modulus G00 long are linked to the sound attenuation coefficient along. The whole difference between shear rheology and longitudinal rheology is comprised of the specific differences in the penetration depth. It is possible to make some estimates of each penetration depth using well-known theories. The penetration depth of the shear stress xoshear can be estimated using following well-known equation for viscous boundary layer, see for instance Dukhin and Goetz [10]: rffiffiffiffiffiffiffi  ð3:32Þ x0 long  ro Most liquids do not support shear waves very well, which means that the penetration depth is rather short. In water, for instance, the shear penetration depth is approximately 0.1 mm at a frequency of 100 Hz. It quickly decreases with frequency. In the case of ultrasound frequency of 1 MHz, it is only about 10 mm and becomes just 1 mm at 100 MHz. On the other hand, longitudinal waves are able to penetrate much further into the liquid. We can use the Stokes’s law, Equation (3.43), for estimating the longitudinal penetration depth, x0long. It yields the following simple expression for estimating x0long: x0 long 

rV 3 o2 

ð3:33Þ

The longitudinal penetration depth is much longer than the shear penetration depth. In water, the longitudinal penetration depth is about 1000 m at

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1 MHz, and 10 cm even at 100 MHz. Longitudinal waves propagate long distances in liquids. It is easy to generate and measure them. One of the peculiarities of the longitudinal viscosity is the small value of the parameterðaV=oÞ2 . As a result, general equations for longitudinal viscoelastic modulus could be simplified, and longitudinal viscosity is: 2along rV 3 ð3:34Þ o2 The possibility of studying a much higher frequency range is a clear advantage of using longitudinal stress waves over shear stress waves. long ¼

3.5. LONGITUDINAL RHEOLOGY OF NEWTONIAN LIQUIDS: BULK VISCOSITY The dynamics of Newtonian liquids is described by the Navier-Stokes equation, which can be found in most books on hydrodynamics, for example, Happel and Brenner [18], Landau and Lifshitz [19], and the fundamental work on theoretical acoustics by Morse and Ingard [3, 4]. For a compressible Newtonian liquid, the Navier-Stokes equation can be written as:  
 @v 4 r þ ðvrÞv ¼ grad P þ Dv þ b þ  grad div v ð3:35Þ @t 3 The last term on the right-hand side of this equation takes into account the compressibility of the liquid and becomes important only for such effects where liquid compressibility cannot be neglected. It contains two viscosity coefficients: one is the familiar shear viscosity ; the other one, b, is much less known. We use term “bulk viscosity” for this parameter throughout this study, but the same property has been referred to by many different names in various scientific disciplines as described below. A rose by any other name. . .

3.5.1. In Hydrodynamics The earliest known mention of this parameter was made by Sir Horace Lamb in his famous work Hydrodynamics first published in 1879 [20, 100]. The sixth edition of this book contains reference to a “second viscosity coefficient.” Other more modern books on hydrodynamics by Happel and Brenner [18], Landau and Lifshitz [19], as well as Potter and Wiggert [21] use the same name.

3.5.2. In Acoustics Theoretical Acoustics by Morse and Ingard [3, 4] refers to this parameter as “bulk viscosity,” and we follow here their lead. This same name is used by

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Kinsler et al. [22], in Fundamentals of Acoustics, Bhatiain Ultrasonic Adsorption [101], Shortley and Willians in Elements of Physics [102]. However, at the same time Temkin [5] in Elements of Acoustics uses instead the term “expansion coefficient of viscosity.” To obfuscate things a bit further, Litovitz and Davis [9] use yet another term “volume viscosity.”

3.5.3. In Analytical Chemistry Kaatze et al. [23] in an extensive review on ultrasonic spectroscopy for characterizing the chemistry of liquids, uses again the term “volume viscosity” as did Carroll and Patterson in their study using Brillouin spectroscopy [24].

3.5.4. In Molecular Theory of Liquids Several authors calculate this parameter using various molecular theories of liquids [21, 25–32], and following the lead of Morse and Ingard universally adopt the term “bulk viscosity.”

3.5.5. In Rheology Hurtado-Laguna et al. [33] in their studies of molten polymers, so far the most important application associated with volume viscosity, again adopt the term of “volume viscosity.” There is also a historical paper by Graves and Argrow [34] that applies the term “coefficient of bulk viscosity.” Bulk viscosity is critical for understanding the longitudinal rheology of Newtonian liquids. At the same time, it is negligible for traditional lowfrequency shear rheology which operates with incompressible liquids where div v ¼ 0

ð3:36Þ

and the Navier-Stokes equation reduces to the much more familiar form:   @v þ ðvrÞv ¼ grad P þ Dv ð3:37Þ r @t If the liquid compressibility is unimportant, we are then, and only then, justified in using the simpler approximation of Equation (3.35). Indeed, many textbooks just assume the liquid to be incompressible and bulk viscosity therefore plays no role. It is not surprising that bulk viscosity is little known when the modern Handbook of Viscosity by C. Yaws [35] and the Handbook of Chemistry and Physics [7] do not mention it at all. Interestingly, it was noted by Stokes, and indeed known as the Stokes’s hypothesis [22, 36], that there are a few compressible fluids, such as monatomic gases, where this bulk viscosity is zero. This hypothesis raised some question about the very existence of bulk viscosity, even in liquids. Several papers [34, 37] discuss this issue and take up the task of proving that bulk

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viscosity does indeed exists for liquids. Experimental justification for the existence of a bulk viscosity for liquids remained elusive with a paucity of meaningful data. Modern measurement techniques now allow precise determination of bulk viscosity as will be shown here. In fact, we have found that the bulk viscosity is not even close to zero for any of the liquids that we have measured and, indeed, the bulk viscosity is often larger than the dynamic viscosity, as we will show later. One might ask why one should bother to learn about this seemingly obscure bulk viscosity. A general answer was given by Temkin [5], who presented a very clear physical interpretation of both dynamic and bulk viscosities in terms of molecular motion. He points out that molecules have “translational,” “rotational,” and “vibrational” degrees of freedom in liquids and gases. The classical dynamic viscosity, , is associated only with the “translational” motion of the molecules. In contrast, the bulk viscosity b reflects the relaxation of both rotational and vibrational degrees of freedom. This leads one to conclude that, if one wants to study rotational and vibrational molecular effects in complex liquids, one must learn how to measure bulk viscosity. A more specific answer is that measuring the bulk viscosity, which is a typical output parameter of molecular theory models, allows one to then test the validity of such models. As one important example, Enskog’s theory [38] yields an analytical expression for both dynamic and bulk viscosities.
 1 þ 0:8 þ 0:761Y ð3:38Þ  ¼ b0 Y b ¼ 1:002b0 Y

ð3:39Þ

where b0 is second virial coefficient, and Y is relative collision frequency. Another classical theory by Kirkwood [25] and Kirkwood, Buff, and Green [26] presents integral expressions for both dynamic and bulk viscosities. However, at the end of the paper they declare that numerical calculation of the integrals is possible only for dynamic viscosity, stating that a numerical procedure for calculating bulk viscosity is not possible due to “. . .extraordinary sensitivity to the equilibrium radial distribution function . . ..” There are several recent theories for calculating bulk viscosity for Lennard-Jones liquids using the Green-Kubo model. They are described in the theoretical papers by Hoheisel, Vogelsang, and Schoen [27], Okumura and Yonezawa [28, 29], Dyer, Pettitt, and Stell [31], Meier, Laesecke, and Kabelac [30], and Tani and Bertolini [32]. Verification of these theories requires experimental data for the bulk viscosity of Newtonian liquids, but strangely enough there are only a few studies known to us that report experimental bulk viscosities for such Newtonian liquids.

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The first is a 50-year-old review by Litovitz and Davis [9] that reports a bulk viscosity for water of 3.09 cP at 15  C and for methanol 2.1 cP at 2  C. This reported value for the bulk viscosity for water can serve as a reference point for new studies on this subject and validate the applied experimental procedure. It is noted that this review also presents bulk viscosity values for molten salts, molten metals, and some other exotic liquids at very low temperatures, but this is not particularly helpful data in the current context. The second, more recent study by Malbrunot, Boyer, Charles, and Abashi [37] presents bulk viscosity data for liquid argon, krypton, and xenon near their triple point. This article includes a very clear account of the experimental technique, and we will follow a similar procedure with some additional steps that provide better precision. There are also some reports of the bulk viscosity in liquid metals, such as by Jarzynski [39]. This paucity of bulk viscosity data is surprising as expressed by Temkin [5] in his book Elements of Acoustics as well as by Graves and Angrow in their paper “Bulk viscosity: Past and Present” [34]. Recently, more data on bulk viscosity became available thanks to the development of a modern acoustic spectrometer DT-600 by Dispersion Technology, Inc. Bulk viscosity data for 12 Newtonian liquids were published in Ref. [6]. Table 3.2 presents these results. The only one independent data, that is, bulk viscosity of water from the Litovitz-Davis review [9], agrees well with this new result. They report a value of 3.09 cP at 15  C, whereas we obtained a somewhat smaller value of 2.43 cP at a somewhat higher temperature of 25  C. An even better verification would be to compare the ratio of bulk to shear viscosity at a given temperature. Litivitz-Davis report a ratio of 2.81 at 15  C, whereas we obtained a value 2.73 at 25  C. This is the only room-temperature-independent verification of our bulk viscosity measurement that we managed to find. It is interesting to note that the bulk viscosity for these 12 liquids varies over a much wider range than the dynamic viscosity, by almost an order of magnitude. This seems to confirm that bulk viscosity is more sensitive to the molecular structure of the liquid. However, there is no correlation established between this parameter and other intensive properties of these liquids, as stated in Ref. [6]. The only known method of characterizing bulk viscosity is measurement of ultrasound attenuation. This is discussed in Section 3.6. There are at least three different methods of measuring attenuation coefficient that have been applied for characterizing liquids: Brillouin spectroscopy [24, 40], laser transient grating spectroscopy [41–43], and acoustic spectroscopy [10]. Each of these has advantages and disadvantages with regard to this particular task. The first two methods are usually performed at gigahertz frequencies, whereas acoustic spectroscopy is typically employed in the megahertz range.

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TABLE 3.2 Attenuation at 100 MHz, Temp. Coefficient, and Dynamic and Bulk Viscosity for 12 Newtonian Liquids Liquid

Attenuation at 100 MHz (dB/cm/MHz)

Attenuation T coefficient aT

Dynamic viscosity (cP)

Bulk viscosity (cP)

Linear variation coefficient

Cyclohexane

2.07

0.05

0.894

“17.4”

“0.082”

Cyclohexanone

0.63