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

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

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

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

ISSN: 1793-737X

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

Published: Vol. 1

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

Vol. 2

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

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Series in Soft Condensed Matter Vol.

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

Yoav Tsori

Ben-Gurion University of the Negev, Israel

Ullrich Steiner

University of Cambridge, UK

World Scientific NEW JERSEY

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

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

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

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Foreword

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

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

David Andelman and G¨ unter Reiter Series Editors

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Preface

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

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

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

Yoav Tsori, Beer-Sheva Ullrich Steiner, Cambridge

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Contents

Foreword

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Preface

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

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

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

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

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

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

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

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

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Table 1.

Table of Symbols

Symbol

Deﬁnition

Units

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

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

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

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

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

(a)

(c)

(b)

(d)

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

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

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

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

(1)

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

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

(2)

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

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

⇒

q∗ = −

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

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

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

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

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

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

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((a))

(b)

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

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

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

2 2 −√ 2 r r + t2

(7)

and the corresponding Fourier transform is G(q) ∼

4π (1 − exp(−qt)) qt

(8)

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

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E External MagneticField

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

P H

B H

S Temperature

TC

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

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

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

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(c)

(b)

(a)

t = 50 nm

t = 50 nm

t = 25 nm

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

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

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Iout

pinned

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Iin

conductor free S N N S

magnetic recording medium

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

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

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

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( )F (a) Ferrofluid fl id

(d) MR fluid

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(b) IInverse MR fl fluid id

( ) Inverse (c) I MSB

(e) Light trapping fluid

(f) Fluidized bed

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

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

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

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

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

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(a)

(b)

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

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

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

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

(12)

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

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

(14)

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

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Su urface deflection amplitud de

4 7 3

5 6

8

9 1

2 Magnetic field, H

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

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

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

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

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

(a) Ferrofluid, H0 = 0.035 tesla.

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

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

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

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

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

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

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

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

(16)

S

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

M H0 dV = − V

µ0 χH02 twf 2 1 + χD

(17)

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

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z

H0 x

P0

t

P

V wf

wl

M

FH

P0

F H0 1 FD

++++++++++

s

T

ͲͲͲͲͲͲͲͲͲͲ

wf

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

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

(18)

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

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

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

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

sinh ξ ξ

1 + nm2 λL2 (ξ) 2

(19)

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

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(a)

(b)

(c)

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

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

mH kB T

(20)

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

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

32

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27oC

I

1I

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

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

(21)

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

(22)

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

(23)

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

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. . . Particle volume concentration [%]

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

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

3χe 3 − χe

(25)

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

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Magnetic fieeld [dimensionless]

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

0.01

0.1

1

10

100

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

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

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

m2 (1 − 3 cos2 θ) 4πµ0 rd3

(26)

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

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

(27)

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

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

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

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

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(a)

(b)

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(c)

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

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

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

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

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

Unstably fluidized

u um

Stably fluidized

Unfluidized

M

p 1/2 2 p m

U u

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

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

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

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NFAB

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

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

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

3(d/2)2 + σΣ 2N a2

(28)

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

A

d0

B

A

B

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

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

(30)

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Y/d0

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

2

2

0

0

Ŧ2

Ŧ2

0

2

4

X/d

0

(a)

0

2

4

X/d0 (b)

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

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

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

(a)

(b)

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

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

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

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

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

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

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70. J. Rabinow, The magnetic ﬂuid clutch, AIEE Transactions 67, 1308–1315 (1948). 71. P. Weissler, Cadillac magnetic-rheological shocks, Popular Mechanics, May (2000); see also M. R. Jolly, J. W. Bender and J. D. Carlson, Properties and applications of commercial magnetorheological ﬂuids, Brochure. Lord Corporation, Cary, NC, USA. 72. W. I. Kordonsky et al., Magnetorheological polishing devices and methods, US Patent 6,503,414 (2003); see also, N. Umehara and K. Kato, Hydromagnetic grinding properties of magnetic ﬂuid containing grains at high speeds, J. Magn. Magn. Mater. 65, 397–400 (1987). 73. J. Popplewell, R. E. Rosensweig and J. K. Siller, Magnetorheology of ferroﬂuid composites, J. Mag. Mag. Mater. 149, 53–56 (1995); see also, J. Popplewell and R. E. Rosensweig, J. Phys. D 29, 2297–2303 (1996). 74. R. E. Rosensweig, On magnetorheology and electrorheology as states of unsymmetric stress, J. Rheology 39, 179–192 (1995). 75. M. Rasa, A. P. Philipse and D. Jamon, Initial susceptibility, ﬂow curves, and magneto-optics of inverse magnetic ﬂuids, Phys. Rev. E 68, 031402.1–16 (2003). 76. R. V. Mehta, R. Patel, B. Chudesama, H. B. Desai, S. P. Bhatnagar and R. V. Upadhyay, Magnetically controlled storage and retrieval of light from dispersion of large magnetic spheres in a ferroﬂuid, Current Science 93, 1071–1072 (2007); see also, J. N. Desai, Magnetic trapping of light, Current Science 93, 452–453 (2007). 77. P. W. Anderson, Absence of diﬀusion in certain random lattices, Phys. Rev. 109, 1492–1505 (1958). 78. S. John, Electromagnetic absorption in a disordered medium near a photon mobility edge, Phys. Rev. Lett. 53, 2169–2173 (1984). 79. R. V. Mehta et al., Eﬀect of dielectric and magnetic contrast on photonic band gap in ferrodispersion, Magnetohydrodynamics 44, 69 (2008). 80. M. Born and E. Wolf, Principles of optics: electromagnetic theory of propagation, interference and diﬀraction of light (Cambridge University, New York, 1999), 7th ed. 81. M. T. Lopez-Lopez, J. de Vicente, G. Bossis, F. Gonzalez-Caballero and J. D. G. Duran, Preparation of stable magnetorheological ﬂuids based on extremely bimodal iron-magnetite suspensions, J. Mater. Res. 20, 874–881 (2005). 82. R. E. Rosensweig, Theory for stabilization of magnetic colloid in liquid metal, J. Mag. Mag. Mater. 201, Special issue, Proceedings of ICMF8, 1–6 (1999). 83. R. E. Rosensweig, G. R. Jerrauld and M. Zahn, Structure of magnetically stabilized ﬂuidized beds in Continuum Models of Discrete Systems 4, O. Brulin and R. K. T. Hsieh, Eds.(North Holland, Amsterdam, 1981), pp. 137–144. 84. R. E. Rosensweig, Process concepts using ﬁeld-stabilized two-phase ﬂuidized ﬂow, J. Electrostat. 34, 163–187 (1995).

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85. R. E. Rosensweig, Magnetic stabilization of the state of uniform ﬂuidization, Ind. Eng. Chem. Fundam. 18, 260–269 (1979). 86. R. E. Rosensweig, M. Zahn, W. K. Lee and P. S. Hagan, Theory and experiments in the mechanics of magnetically stabilized ﬂuidized solids, in Theory of dispersed multiphase ﬂow, R. E. Meyer, Ed. (Academic, New York, 1983). 87. R. E. Rosensweig and G. Ciprios, Magnetic liquid stabilization of ﬂuidization in a bed of nonmagnetic spheres, Powder Tech. 64, 115–123 (1991). 88. V. Cabuil and S. Neveu, Complex magnetic systems derived from magnetic ﬂuids, In Magnetic ﬂuids and applications handbook, Eds. B. Berkovski and V. Bashtovoi (Begell House, New York, 1996), pp. 56–63. 89. F. Brochard and P. G. de Gennes, Theory of magnetic suspensions in liquid crystals, J. Phys. (Paris) 31, 691–708 (1974). 90. C. Y. Matuo, F. A. Torinho, M. H. Souza, J. Depeyrot and A. M. FigueredoNeto, Lyotropic ferronematic liquid crystals based on new Ni, Cu, and Zn ionic magnetic ﬂuids, Brazilian J. Phys. 32, 458–463 (2002). 91. W. Wen, X. Huang and P. Sheng, Electrorheological ﬂuids: structures and mechanisms, Soft Matter 4, 200–210 (2008). 92. F. S. Bates and G. H. Fredrickson, Physics Today 52, 32 (1999). 93. G. H. Fredrickson, The equilibrium theory of inhomogeneous polymers, (Oxford University, Oxford, 2005). 94. C. Park, J. Yoon and E. L. Thomas, Polymer 44, 6725 (2003). 95. D. G. Bucknall, Prog. Mat. Sci. 49, 713 (2004). 96. Y. A. Vlasov, X. Z. Bo, J. C. Sturm and D. J. Norris, Nature 414, 289 (2001). 97. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos and E. L. Thomas, Science 282, 1679 (1998). 98. A. K. Khandpur, S. Foerster, F. S. Bates, I. W. Hamley, A. J. Ryan, W. Bras, K. Almdal and K. Mortensen, Macromolecules 28, 8796 (1995). 99. L. Leibler, Macromolecules 13, 1602 (1980). 100. K. Binder, H. L. Frisch and S. Stepanow, J. Phys. II (France) 7, 1353 (1997). 101. G. H. Fredrickson and E. Helfand, J. Chem. Phys. 87, 697 (1987). 102. T. Ohta and K. Kawasaki Macromolecules 19, 2621 (1986). 103. K. R. Shull, Macromolecules 25, 2122 (1992). 104. M.W. Matsen and M. Schick, Phys. Rev. Lett. 72, 2660 (1994). 105. K. Binder, Acta Polymer 46, 204 (1995). 106. I. Szleifer, Curr. Opin. Colloid. Sci. 1, 416 (1996). 107. Y. Tsori and D. Andelman, Thin ﬁlm di-block copolymers in electric ﬁeld: Transition from perpendicular to parallel lamellae, Macromolecules 35, 5161–5170 (2002). 108. K. Amundson, E. Helfand, X. Quan and S. D. Smith, Macromolecules 26, 2698 (1993); K. Amundson, E. Helfand, X. Quan, S. D. Hudson and S. D. Smith, Macromolecules 27, 6559 (1994). 109. A. Onuki and J. Fukuda, Macromolecules 28, 8788 (1995). 110. G. G. Pereira and D. R. M. Williams, Macromolecules 32, 8115 (1999). 111. T. Thurn-Albrecht, J. DeRouchey and T. P. Russell, Macromolecules 33, 3250 (2000).

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112. A. V. Kyrylyuk, A. V. Zvelindovsky, G. J. A. Sevink and J. G. E. M. Fraaije, Macromolecules 35, 1473 (2002). 113. A. B¨ oker, A. Knoll, H. Elbs, V. Abetz, A. H.E. M¨ uller and G. Krausch, Macromolecules 35, 1319 (2002); A. B¨ oker, K. Schmidt, A. Knoll, H. Zettl, H. Hansel, V. Urban, V. Abetz and G. Krausch, Polymer 47, 849 (2006). 114. T. Xu, Y. Zhu, S. P. Gido and T. P. Russell, Electric ﬁeld alignment of symmetric di-block copolymer thin ﬁlms, Macromolecules 37, 2625–2629 (2004). 115. V. Olszowka, M. Hund, V. Kuntermann, S. Scherdel, L. Tsarkova, A. B¨ oker and G. Krausch, Large scale alignment of a lamellar block copolymer ﬁlm via electric ﬁelds: a time resolved SFM study, Soft Matter 2, 1089–1094 (2006). 116. K. Schmidt, H. G. Schoberth, M. Ruppel, H. Zettl, H. H¨ ansel, T. M. Weiss, V. Urban, G. Krausch and A. B¨ oker, Reversible tuning of a block-copolymer nanostructure via electric ﬁelds, Nature Matterials 7, 142–145 (2008). 117. M. W. Matsen, Electric-ﬁeld alignment in thin ﬁlms of cylinder-forming di-block copolymer, Macromolecules 39, 5512 (2006). 118. C.-Y. Lin and M. Schick, Self-consistent-ﬁeld study of the alignment by an electric ﬁeld of a cylindrical phase of block copolymer, J. Chem. Phys. 125, 034902 (2006). 119. T. Xu, A. V. Zvelindovsky, G. J. A. Sevink, O. Gang, B. Ocko, Y. Zhu, S. P. Gido and T. P. Russell, Electric ﬁeld induced sphere-to-cylinder transition in di-block copolymer thin ﬁlms, Macromolecules 37, 6980–6984 (2004); T. Xu, C. J. Hawker and T. P. Russell, Macromolecules 38, 2802 (2005). 120. Y. Tsori, F. Tournilhac, D. Andelman and L. Leibler, Phys. Rev. Lett. 90, 145504 (2003). 121. Y. Tsori, D. Andelman, C.-Y. Lin and M. Schick, Block copolymers in electric ﬁelds: A comparison of single-mode and self-consistent ﬁeld approximations, Macromolecules 39, 289 (2006). 122. C.-Y. Lin, M. Schick, and D. Andelman, Structural changes of di-block copolymer melts due to an external electric ﬁeld: a self-consistent ﬁeld theory study, Macromolecules 38, 5766 (2005). 123. J.-Y. Wang, T. Xu, J. M. Leiston-Belanger, S. Gupta and T. P. Russell, Ion complexation: a route to enhanced block copolymer alignment with electric ﬁelds, Phys. Rev. Lett. 96 128301.1–4 (2006).

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

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

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

(1)

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

µiK sol

iα ∆µiαβ = µiβ sol − µsol ,

(2)

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

(3)

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

n1α n1β

= ln

n2α n2β

=

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

(4)

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

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

(5)

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

(6)

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

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

Solvation interaction

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

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

(9)

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

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

(10)

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

(i = 1, 2).

(11) (12)

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

(13)

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

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

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

(15)

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

(17)

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

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Zi2 µim (z)/kB T > 1 at |z| ∼ ξ where ξ is the interface thickness. From Eq. (16) this condition is rewritten as ξ < Zi2 `Bc ε1 ∆φ/εc < κ−1 α .

(18)

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

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

(21)

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

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

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

3C . 4(`wa )2 kB T

(23)

(24)

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

(26)

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

(27)

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

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

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(29)

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

(31)

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

(32)

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

1 1 + − χion c . 2N1 φ 2N2 (1 − φ)

(33)

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

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

(34)

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

(35)

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

(36)

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

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

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

(38)

where Q1 = Z1 e and Q2 = −Z2 e are the ion charges and ξ = (Ceff /¯ r)1/2 is the correlation length. Among the ions of the same species (i = j), the second attractive term dominates over the first Coulomb repulsive term in the range a ∼ 4πCeff Zi2 e2 /εc (kB T )2 .

(39)

The right hand side is of order 4πZi2 (1 − n1 /n1L )`Bc /a for low molecularweight mixtures. Under the condition (39) there should be a tendency of aggregation of ions. This effect might explain a number of observations of micro-heterogeneities in near-critical binary mixtures containing salt.34,35 It is also of great interest how charged colloid particles interact in polar solvent. There can be an attractive interaction among them in the presence of ion-composition or ion-polarization coupling. Such theory will be reported shortly. 4. Equilibrium Conditions in One-Dimensional Cases and Surface Tension We consider the equilibrium conditions around an interface in our system. The total free energy is the sum, F = F0 + Fsol + Fim + Fam .

(40)

We define the chemical potentials µi = δF/δni (i = 1, 2, · · · ) of the ions and the chemical potential difference h = δF/δφ of the mixture. In equilibrium, these quantities are homogeneous constants.

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4.1. Two species of ions First we assume the presence of two species of ions. Some calculations yield µ1 = kB T [ln c1 + 1 + Z1 U − g1 ψ − ln Zam ] + Z12 µim ,

µ2 = kB T [ln c2 + 1 − Z2 U − g2 ψ] +

Z22 µim ,

(41) (42)

where ψ = φ − φc . If the first ion species has no amphiphilic character, we should set Zam = 1. The µim is given by Eq. (16) in the 1D case. Here we equate the functional derivatives δFim /δni with Zi2 µim in Eq. (16) neglecting the ion-density dependence of κ. To get simpler expressions, we introduce normalized ion densities and potential by c 1 = a 3 n1 ,

c 2 = a 3 n2 ,

U = eΦ/kB T.

(43)

Note that the ion sizes can be different from the size of the solvent molecules a. For example, the density of 10 mM salt is n = 6 × 1018 cm−3 . If a = 3˚ A, the normalized density c = a3 n is equal to 1.6 × 10−4 . The equilibrium ion densities are now expressed as c1 = c10 Zam exp[−Z1 U + g1 ψ − Z12 µim /kB T ], c2 = c20 exp[Z2 U + g2 ψ −

Z22 µim /kB T ],

(44) (45)

where c10 and c20 are constants. Some calculations also give h in the form, ε1 2 h = f00 (φ) − C∇2 φ − E − kB T (g1 n1 + g2 n2 ) + him + ham , (46) 8π where f00 = ∂f0 /∂φ and the third term (∝ E 2 ) is the functional derivative of the electrostatic energy with respect to φ.4 The ham = δFam /δφ is the contribution from the amphiphilic interaction. In the 1D case, him is given by the right hand side of Eq. (16) with dψ(z 0 )/dz 0 being replaced by P 2 0 0 i Zi dni (z )/dz , while ham is expressed as Z wa ` ham (z) dζX(z + ζ) ewa [φ(z+2ζ)−φ(z)] − ewa [φ(z)−φ(z+2ζ)] . (47) = kB T 2` −` Here we define X(z) = n1 (z)/Zam (z). In the 1D case, the charge neutrality should be satisfied as z → ±∞. If φ → φα as z → −∞ and φ → φβ as z → ∞, Eqs. (41) and (42) yield the general relations (3) and (4). Under Eq. (12) they are rewritten as g1 − g 2 e(Φα − Φβ ) = kB T ∆φ, (48) Z1 + Z 2 n1β n2β Z2 g 1 + Z 1 g 2 = = exp − ∆φ , (49) n1α n2α Z1 + Z 2

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where ∆φ = φα − φβ . The image and amphiphilic interactions vanish far from the interface and do not affect the above relations. 4.2. Three species of ions Performing x-ray reflectivity measurements, Luo et al.36 measured the ion distributions in the vicinity of an interface in water-nitrobenzene. They added two salts, tetrabutylammonium tetraphenylborate (TBA-TPB) and tetrabutylammonium bromide (TBA-Br). Then they realized a two-phase state with hydrophilic anion Br− , hydrophobic cation TBA+ , and hydrophobic anion TPB− . They detected strong accumulation or depletion of the ions on the two sides of the interface, which suggest a crucial role of the ion-solvent interactions dependent on the ion and solvent species. In our scheme, numerical analysis is needed to calculate such complicated ion distributions, as will be exemplified in Fig. 6 below. Here we examine how the potential difference ∆Φ in Eq. (3) is related to the bulk ion densities for three ion species.6 Let them have charges, Q1 = eZ1 , Q2 = −eZ2 , and Q3 = −eZ3 . If the anions 2 and 3 have no amphiphilic character, the normalized density of the third species c3 = a3 n3 is given in the form of Eq. (45) if the subscript 2 is replaced by 3. Again, α and β denote the more polar (water-rich) phase and the less polar (waterpoor) phase, respectively. To avoid cumbersome notation, we define the normalized chemical potential differences νi = ∆µiαβ /kB T (i = 1, 2, 3) and normalized potential difference ∆U = e(Φα − Φβ )/kB T . Then, c2β c3β c1β = eZ1 ∆U −ν1 , = e−Z2 ∆U −ν2 , = e−Z3 ∆U −ν3 , (50) c1α c2α c3α where νi = gi ∆φ with ∆φ = φα − φβ . The charge neutrality conditions are Z1 c1α = Z2 c2α + Z3 c3α and Z1 c1β = Z2 c2β + Z3 c3β in the bulk two phases. Im terms of the ion densities in the α phase, it follows the equation for ∆U in the form, (Z2 c2α + Z3 c3α )eZ1 ∆U −ν1 = Z2 c2α e−Z2 ∆U −ν2 + Z3 c3α e−Z3 ∆U −ν3 .

(51)

In the monovalent case Z1 = Z2 = Z3 = 1, the above equation is simplified to e2∆U = (c2α eν1 −ν2 + c3α eν1 −ν3 )/(c2α + c3α ),

(52)

which reduces to the expression (3) (independent of c2α ) for c3α = 0. However, in the experiment,36 the third species (TPB− ) was strongly hydrophobic such that c3α c3β was realized, while the second species

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(Br− ) was hydrophilic. Supposing such cases, let us assume g2 > 0 and g3 < 0 and choose c2α and c3β as control parameters. If we set X = exp[∆U − (ν1 − ν2 )/2],

(53)

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

(54)

R = e(ν1 +ν2 )/2 c3β /2c2α .

(55)

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

∆U ∼ = ν1 + ln(c3β /c2α ),

(57)

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

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

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

(59)

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

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

(63)

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

zint

(65)

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

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

(68)

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

(69)

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

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0.7

a 0.6

0

b c

U

-0.1

c

0.5

φ (z)

-0.2

a = 1.95 b = 1.92 c

0.4

0.3

0

b

χ=2.0

χ=2.0

a = 1.95 b = 1.92 c

-0.3

20

40

60

80

-0.4 0

100

20

a 40

60

z/a

80

100

z/a

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

0.02

1 C

a 0.015

4

1 0.5

10 ( c 1 - c2 )

a

b b 0.01

0.005

0

0

c

0

χ=2.0

a = 1.95 b = 1.92 c

20

-0.5

40

60

z/a

80

100

-1 0

c

χ=2.0

a

= 1.95 b = 1.92 c

20

40

60

80

100

z/a

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

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

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0.08 0.07

a

b

= 2.3 b = 2.05 c

0.04

a

0.05

0.03

g1= 4 g2= 2

0.02 0.01 0

0.01

0.02

0.03

0.04

a

< c1>= 0.001 g =10 g2= 5

b

1

0.03

0.04

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

c

a 2 γ / kBT

a 2 ∆γ / kBT

0.06

χ=3.0

< c > = 0.01 1

g1= 4 g2= 2 c

0.02

c b

0.01

a 0.05

0

1.92 1.94 1.96 1.98

2

2.02 2.04 2.06 2.08

2.1

χ

c1α

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

∆φ = φα − φβ ∼ = [3(χ − χc )/2]1/2 ,

(71)

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

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1.2 1.2 1

Ima ( z )

1

Ima( z )

0.8 0.6 0.4 0.2 0 38

n(z)/nα

0.8 -5

-3

c 1 α= 2.1x10

0.6

χ= 3 g1= 10

φ(z)

g2= 5

n(z)/nα

A =10 40

42

44

46

z/a

48

c 1 α= 10

0.4

χ= 2.3 g1= 4

0.2

g2= 2

φ(z)

A =4 50

52

54

0 40

42

44

46

48

50

52

54

56

z/a

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

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

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0

0.003

χ= 3 g1= 10 g2= -10

0.002

2

-0.02

c2

A=4

c1α=

∆γ1a / T

-0.01

2 x10

-Γ a

-0.03

-4

χ= 3

-0.04

g 1= 10 g2= -10

-0.05 0.001

c1

-0.06 -0.07

2

A=4

∆γ a / T 2

-0.08 0

30

40

50

z/a

60

70

0

0.0004

0.0008

0.0012

0.0016

0.002

c1α

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

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

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10

0.003

-2

χ= 3 g1= 10 g2= 10 g3= -13 A=4

c1 0.002

c2α=

2 x10

c3

-3

10

c3β= 0.001

10

10

35

40

45

50

55

60

65

z/a

70

c2

-4

c1 χ= 3.2

-5

c3β=

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

c2

0

4 x10

-4

-3

10

30

c2α=

-3

10

-6

30

35

-4 0.6 x10

c3

40

45

50

55

60

65

70

z/a

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

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

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

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0.06

5

χ= 3 b

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

0.04

1x10

c1α=

-3

= 2x10

-3

χ= 3

a

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

4

b

3.5

U

0.05

4.5

b 0.03

3

c1α=

1x10

-3

= 2x10

-3

a b

b

2.5 2

0.02

a

a

1.5 1

0.01

a 0.5

0 44

46

48

50

52

54

56

0

58

30

40

50

60

70

80

90

100

z/a

z/a

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

0.012

0

χ=3

c1 , c2

0.008

g1= 8 g2= -8 A= 4

c1α =10

b

-1

=8 b b

0.006

a

a

χ=3 A= 4 -3 c1α =10

-6

0 40

g1= 8 g2= -8

-5

0.002

45

50

55

z/a

60

=8 b

-3

-4 0.004

wa = 0 a

a

-2

-3

U

0.01

wa = 0 a

b

-7 40

45

50

55

60

z/a

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

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

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0.5

0.5

0.45

0.45

χ= 3

0.4

γ a2/ k BT

g1= 8 g2= -8 A= 4

0.35 0.3 0.25

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

0.35

γ a2/ k BT

0.3 0.25

-3

c1α = 10

0.2

χ= 3

0.4

0.2

0.15

>

0.1

Γ2 a >

0.05

Γ1 a >

0.15

Γ1 a2

2

0.1

2

Γ2 a2 >

0.05

0

0 0

2

4

6

wa

8

10

12

0

0.0005

0.001

c1α

0.0015

0.002

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

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

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

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

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

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

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

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

87

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K. Orzechowski

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

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

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

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

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

FE = F − 0.5ε oεE 2

(1)

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

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

∂2 p ∂ρ∂T

(2)

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

∆FEM = ∆F M + 0.5ε oε M E 2

(3)

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

 ∂ 2ε  2 ∆Tc ∝ − 2  E  ∂xv  c

(4)

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

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K. Orzechowski

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

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

( )

where

( )  2



(5)

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 ∂ (i )ε   i   ∂x  c

ε c(i ) = 

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

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

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

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

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

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

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

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

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

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

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

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

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

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K. Orzechowski

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

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

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

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

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

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

(

∆ε NDE / E 2 = ∆ε NDE / E 2

)

B

+ At −ψ

(6)

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

Tc = Tco + bE 2

(7)

and hence: 2

(

∆ε NDE / E = ∆ε NDE / E

2

)

B

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

−ψ

(8)

Assuming that bE2 << Tco, omitting the background term in the vicinity of the critical point and expanding the left hand side of the equation in a Maclaurin series against E2 the following was obtained:

∆ε NDE

 T − Tco   ≅ A o  Tc 

−ψ

Abψ E + o Tc 2

 T − Tco    o  Tc 

−ψ −1

E4

(9)

Equation 9 allows to estimate the ∆Tc(E2) shift from the analysis of the non-linearity of ∆εNDE(E2) dependence. In the performed experiments

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Orzechowski observed that for greater distances from the critical temperature the ∆εNDE was a linear function of E2. A detectable nonlinearity of ∆εNDE(E2) was observed when T − Tc is smaller than 0.5K. The Eq. (9) allows estimating the shift of the critical temperature dTc /dE2 equal to –0.87±0.2⋅10-15 [K.m2/V2]. This quantity was consistent with that obtained already by Debye and Kleboth in this same mixture but it contradicts the theoretical predictions. Orzechowski indicated some possible effects that could contribute the detected curvature of ∆εNDE(E2) and counterfeit the decrease in a critical temperature. As a possible explanation it was considered a non-linear deformation of fluctuations in a strong external electric field. According to the droplet model, fluctuations could be treated as real droplets. The electric field deforms droplets (spherical in the absence of electric field) resulting in an increase in a polarization vector and the NDE anomaly. Exploring the similarity between fluctuations and droplets very literally the non-linear deformation of droplets in a sufficiently strong electric field could be expected. According to estimations performed by Pyzuk92 the nonlinearity related to this effect could be observed very close to the critical point. In nitrobenzene + hexane for electric fields of order of 6⋅106 V/m Pyzuk estimated that nonlinear deformation of droplets could contribute at T − Tc < 0.01K. In experiment performed by Orzechowski such a temperature interval was not considered. Onuki69 gives an alternative explanation of the observed non-linearity of ∆εNDE(E2). He predicted that in a sufficiently strong electric field, very close to the critical temperature a saturation of ∆εNDE should be observed. The saturation is expectable below a cross-over temperature 0.8

τ e ≅ 4 ⋅ 10

−16

 ε 12    E 1.6 ε 

(10)

where ε is expressed as a series expansion of the order parameter

ε = ε o + ε 1 ( xv − xv ,c ) + 0.5ε 2 ( xv − xv ,c ) 2 + ⋯

(11)

xv – volume fraction, all quantities in SI units. In a case of nitrobenzene + isooctane τe ≅ 1.5⋅10−4 (ε, ε1 were calculated using the data presented by Debye and Kleboth59, an electric field was assumed to be 107 V/m). It

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allows expecting symptoms of saturation of the NDE effect in nitrobenzene + isooctane at T – Tc < 0.4K. This temperature range was investigated by Orzechowski and it is possible that the non-linearity of ∆ε(E2) was attributed not by the critical temperature shift but by the saturation predicted by Onuki. Both effects should result in a similar ∆ε(E2) dependence and it is difficult to discriminate between them. The presented experiments of Tc shift under influence of strong electric field generally proof very small effect. However, this is not the case of experiment performed by Reich and Gordon60 in the vicinity of LCST of a poly(vinyl methyl ether) – polystyrene mixture. The authors observed that the cloud point temperature was decreasing by 53 K under influence of a DC field of a strength of 2.7⋅107 V/m. Unfortunately, such an enormous effect was not supported by the presented by the authors mean-field model.60 3.2. Tc(E) shift deduced from electric permittivity anomaly The temperature dependence of dielectric permittivity (measured at low electric field strength) ε is predicted to be:93,94

ε / ρ = A1 + A2 t + A3t 1−α + A41−α + ∆

(12)

where ρ is the density, Ai are the constants, α is the critical exponent for the heat capacity at a constant pressure and composition, ∆ is the correction-to-scaling exponent.95 The three-dimensional Ising model predicts, α = 0.11, ∆ = 0.5.96 The dielectric constant is divided by the density, which has a critical anomaly described by the functional form similar to that of Eq. (12). According to Senger’s93 the sign of the critical amplitude (A3 in Eq. (12)) should be opposite to dTc/dE2. A similar function describing an electric permittivity anomaly as well as a relation of A3 amplitude to dTc /dE2 was obtained by Goulon and co-workers using the droplet model.72 Following that it seems that the measurements of the permittivity anomaly performed in a low electric field are a convenient way to estimate the shift of Tc under the influence of a strong electric field. Unfortunately, early measurements of permittivity anomaly led to a conflicting result. In some cases a sharp increase in |dε/dT|,98-100 while in

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105

Fig. 8. The dependence of electric permittivity in the close critical mixture obtainable in low and at high frequencies. The character of the anomaly depends on the frequency.

others a smooth character of ε and a small decrease in |dε/dT| derivatives101,102 close to the critical point were observed. The origin of those contradicting results becomes clear after precise measurements of an electric permittivity anomaly as a function of frequency.97,100,103–114 It was found that the increase in |dε/dT| is usually observed at low frequencies, whereas the decrease in this derivative in high ones. This phenomenon was explained106,109 in terms of the Maxwell-Wagner dispersion (characteristic of inhomogeneous systems). It was assumed that in a close vicinity of Tc the concentration fluctuations are large and long living, and the system could be treated as a really inhomogeneous mixture. An example of the temperature dependence of the electric permittivity obtained in ethanol + dodecane mixture108 is presented in Fig. 8. It is generally accepted, that the inherent anomaly of ε should be measured at high frequencies. The high frequency behaviors lead to the positive A3 amplitude (see Eq. (12)) and according to the droplet model72 as well as to thermodynamic arguments96 the derivative dTc /dE2 should be negative. In spite of confirmation of this result in direct measurements of Tc shift, this result contradicts most of the theoretical expectations. A possible reason for the discrepancy was offered by Onuki.69 His theory of the electric permittivity anomaly leads to the similar dependence as that in Eq. (12), but the different is a meaning of A3 amplitude. According to Onuki69 A3 is proportional to (ε12/ε – 1.5ε2), where εi is defined in Eq. (11). On the other hand, in the low field

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regime, the shift of the critical temperature caused by an electric field is predicted to be proportional to εε2/ε12. It means that there is not a simple proportionality between the critical amplitude describing permittivity anomaly (A3) and the shift of Tc. 4. Possible Reasons for Discrepancy between Predicted and Measured Shift of the Critical Temperature The situation presented in previous paragraphs is rather confusing. The theory predicts the upward shift of the critical temperature whereas most of the experiments demonstrate a downward shift. A careful examination of the experimental achievements allows showing some weaknesses of them but the question of a shift of Tc(E) remains. It could be interesting to show possible effects which could contribute to the critical temperature shift, which has not been included in the theoretical expectations or verified experimentally so far.

Fig. 9. The dependence of electric permittivity versus mole fraction in a close critical mixture expected at low and at high frequencies.

Beaglehole61 obtained a downward shift of Tc(E). He was conscious that it is inconsistent with the theory, which predicted that the shift of Tc should have the same sign as ∂ε/∂xv. This derivative was found positive in the investigated mixture. Beaglehole noticed that the mentioned derivative is positive if the permittivity measured at sufficiently high

Change of Critical Mixing Temperature in a Uniform Electric Field

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frequency is analyzed. The strong electric field is applied usually in a form of rectangular pulses and the permittivity measured at low frequency should be considered. As it was presented before, the critical anomaly of permittivity is different at low and high frequencies (see Fig. 8). Figure 9 presents the expected ε(x) dependence at low and at high frequencies. At low frequencies, sufficiently close to the critical temperature, the curvature of ε(x) was expected to be negative. In this way Beaglehole explained the obtained decrease in the critical temperature in a strong electric field. It is possible also to consider an electrostriction as an alternative reason for the discrepancy between the experiment and the theory.64 In the vicinity of the gas-liquid critical point the electrostriction was measured by Zimmerli and co-workers.115,116 The density of a fluid near the critical point strongly increases under the influence of an electric field, mainly because of a critical anomaly of isothermal compressibility. Compressibility of liquids is much smaller and the effect of a change of volume is probably different when the liquid is placed in the confined volume, or it could expand or suppress freely. The last case is expected when the cell is only partly filled by the liquid. Let’s concern the confined volume case. An electric field changes the volume of liquid, resulting in a change of pressure between electrodes. Because the critical temperature is very sensitive to pressure,44-50 electrostriction should influence the Tc. However, the change of Tc due to electrostriction is difficult to predict. Volume of a liquid sample could both increase and decrease in the presence of an electric field and depends on quantities having anomalous behaviors in a critical region (compressibility, permittivity, (∂ε/∂p)):79

 ∂ε  1 ∆VE = −0.5ε o E 2V  − (ε − 1)β  ≈ (13)  ∂p  ∆p E where p is the pressure, ∆pE is the change of pressure in result of electrostriction, β is the compressibility. According to measurements of Urbanowicz et al.45 in nitrobenzene + n-alkanes the ∂Tc/∂p derivative is negative for n-hexane, n-octane and n-decane and positive for longer hydrocarbons. Assuming that ∂Tc/∂p in nitrobenzene + isooctane is similar to that of nitrobenzene + n-octane (-0.039 K/MPa45), the decrease

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in the critical temperature by 0.08K (the decrease obtained by Orzechowski64 for the strongest electric field) could be equivalent to the increase by pressure of 2 MPa. The assumption that the electrostriction and the change of pressure is a reason for a change of temperature could be verified when the system with the positive ∂Tc/∂p derivative will be investigated. To test this assumption the NDE experiments were performed in nitrobenzene + tetradecane mixture. Urbanowicz et al. found that in this mixture the derivative ∂Tc/∂p is positive.45 According to the arguments presented before the shift of Tc should be demonstrated as a non-linearity of ∆εNDE(E2) dependence. A small, but detectable nonlinearity of ∆εNDE(E2) dependence was observed very close to Tc.117 Unfortunately, experiments in this region were difficult and conclusions not very certain. Nevertheless the experiments suggested again a small downward shift of the critical temperature, which seems to contradict the assumption that the electrostriction is responsible for the observed Tc shift. The next effect that could contribute to the experimentally observed shift of the critical temperature could be related to the gradient of an electric field expectable in each real experiment. When the electric field is non-uniform some increase in concentration of a polar component in the region of an elevated potential could be expected. On the other hand, the migration along the field gradient is diffusionally controlled and when the strong field is applied as short pulses (as in most experiments) the change of concentration is probably very small. Finally, it is necessary to mention that the effect of ions has been unexplored so far. Preliminary experiments90 and theoretical expectations69 confirm the importance of this aspect, but the definite answers have not been available yet. 5. Conclusions and Outlook The influence of the uniform electric field on the critical mixing temperature has attracted the attention of experimentalists and theoreticians for a long time. In spite of theoretical expectations and experimental attempts, the problem has not been definitely resolved yet. Most of the experiments show a decrease in an upper critical mixing

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temperature. These findings contradict the theory that in a frame of a mean field regime valuates an increase in UCST when the strong electric field is applied. There are a lot of additional effects that can contribute to the shift but up to now, they are not sufficiently elaborated theoretically and verified experimentally. Irrespectively of the troubles described in this chapter it seems that the shift of the critical mixing temperature in simple binary liquids is small and could be observed in the close vicinity of Tc only. The same conclusion is probably correct in complex binary mixtures, as for example polymers. Acknowledgments This work was supported by the Polish State Committee for Scientific Research, the Project No. 3 T09A 031 29. I would like to thank Professor Onuki and Professor Reich for reprints and to Professor Zimmerli who provided preprints as well as unpublished materials. References 1. Prigogine, R. Defay, in Chemical Therodunamics, Longmans Green and Co. London-New York-Toronto 1954. 2. C.A. Cerdeirina, C.A. Tovar, D. Gonzalez, E. Carballo and L. Romani, Fluid Phase Equilibria, 179, 101. 3. B.J. Hales, G.L. Bertrand and L.G. Helper, J. Phys. Chem. 70, 3970 (1966). 4. M. Misawa, K. Yoshida, K. Maruyama, H. Munemura and Y. Hosokawa, J. Phys. Chem. Solids, 60, 1301 (1999). 5. L. Rebelo, Phys. Chem. Chem. Phys., 1, 4277 (1999). 6. H.E. Stanley Introduction to Phase Transitions and Critical Phenomena, Clarendon Press. Oxford 1971. 7. N-C. Wong and C.M. Knobler, Phys. Rev. A, 24, 3205 (1981). 8. J.P. Schroeter, D.M. Kim and R. Kobayashi Phys. Rev. A 27, 1134 (1983). 9. L. Belkoura, F.P. Harnisch, S. Kolchens, Th. Muller-Kirschbaum and D. Woermann, Ber. Bunsenges. Phys. Chem., 91, 1036 (1987). 10. St. Fusenig and D. Woermann. Ber. Bunsenges. Phys. Chem., 97, 577 (1993). 11. W.A. Van Hook, H. Wilczura and L.P.N. Rebelo, Macromolecules, 32, 7299 (1999). 12. S.Z. Mirzaev and U. Kaatze, Chem. Phys. Lett. 328, 277 (2000). 13. U. Durr, S.Z. Mirzaev and U. Kaatze, J. Phys. Chem., 104, 8855 (2000). 14. R. Behrends, T. Telgmann and U. Kaatze, J. Chem. Phys. 117, 9828 (2002).

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47. A.A. Van Hook, H. Wilczura, A. Imre and L.P.N. Rebelo, Macromolecules, 32, 7299 (1999). 48. A.A. Van Hook, H. Wilczura, A. Imre and L.P.N. Rebelo, Macromolecules, 32, 7312 (1999). 49. D. Browarzik and M. Kowalewski, Fluid Phase Eq. 194-197, 451 (2002). 50. A.R. Imre, Chin. J. Polym. Science, 21, 241 (2003). 51. I.G. Grekova, E.T. Szymańska and Yu. I. Szymanski Ukr. Fiz. Zh. 26, 283 (1981). 52. L.A. Bulavin, A.V. Chalyi and A.V. Oleinikova, J. Mol. Liq. 75, 103 (1998). 53. D. Beysens and P. Gu, F. Perrotenoun, Phys. Rev. A, 38, 4173 (1998). 54. F.B. Hicks, T.C. Van Vechten and C. Franck, Phys. Rev. E, 55, 4158 (1997). 55. O.D. Alekhin, M.P. Krupsky, Y.L. Ostapchuk and E.G. Rudnikov, J. Mol. Liq., 105, 191 (2003). 56. O.D. Alekhin, and Y.L. Ostapchuk, Int. J. Thermoph. 24, 163 (2003). 57. J.L. Tveekrem and D.T. Jacobs, Phys. Rev. A, 27, 2773 (1983). 58. M. Misawa, K. Yoshida, K. Maruyama, H. Munemura and Y. Hosokawa, J. Phys. Chem. Solids 60, 1301 (1999). 59. P. Debye and K. Kleboth, J. Chem. Phys., 42, 3155 (1965). 60. S. Reich and J.M. Gordon, J. Polym. Sciences: Polym. Phys. Ed. 17, 371 (1978). 61. D. Beaglehole, J. Chem. Phys. 74, 5251, (1981). 62. D. Wirtz, K. Berend K and G.G. Fuller, Macromolecules 25, 7234 (1992). 63. D. Wirtz D and G.G Fuller Phys. Rev. Lett. 71, 2236 (1993). 64. K.Orzechowski, Chem. Phys. 240, 275 (1999). 65. Y. Tsori, F. Tournilhac and L. Leibler, Nature 430, 544 (2004). 66. Onuki, Erophys. Lett, 29, 611 (1995). 67. K.Y. Min, R.A. Wilkinson, G. Zimmerli and R.E. Kusner contribution presented in The American Physical Society meeating, (1997). Unpublished paper was send by Zimmerli. 68. M.D. Early, J. Chem. Phys. 96, 41 (1992). 69. Onuki, in Nonlinear Dielectric Phenomena in Complex Liquids, Ed. S.J. Rzoska, V.P. Zhelezny NATO Science Series. II. Mathematics, Physics and Chemistry vol. 157 Kulwer Academic Publishers pp.113. 70. L.D. Landau and E.M. Lifshitz in Electrodynamics of Continuous Media, Vol. 8 (Pergamon Press) 1984. 71. J.D. Jackson in Classical Electrodynamics, Third Edition, Willey &Sons, Inc. 72. J. Goulon, J-L. Greffe and D.W. Oxtoby, J. Chem. Phys., 70, 4742 (1979). 73. W. Oxtoby and H. Metiu, Phys. Rev. Lett. 36, 1092 (1976). 74. W. Oxtoby, Phys. Rev. A 15, 1251 (1977). 75. D. Beaglehole, J. Chem. Phys. 73, 3366 (1980). 76. J.K. Whitmer, S.B. Kiselev and B.M. Law, J. Chem. Phys. 123, 4720 (2005). 77. J.C. Le Guillou and Zain-Justin, J. Phys. Rev. Lett. 39, 95 (1997). 78. J. Des Cloiseaux and G. Jannik Polymers in Solution, Oxford University Press, Oxford, 1987. 79. Chełkowski in Dielectric Physics, Elsevier Scientific Publishing Company, Amsterdam-Oxford-New York 1980. 80. J. Ziolo Acta Phys. Pol. A52 51 (1977).

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Chapter 4 Electrohydrodynamic Instabilities of Thin Liquid Films

Thomas P. Russell and Joonwon Bae Department of Polymer Science & Engineering, University of Massachusetts 120 Governor’s Drive, Amherst, MA 01002 USA E-mail: [email protected] The interaction of an electric field with matter has been studied since 19th century. Extensive experiments have shown that an electric field can act on dielectric materials exerting an electrostatic force. Because the electrostatic interactions are relatively strong and long-ranged, they can be used to control structures on length scales that are difficult to achieve by any other means. There are techniques available to pattern materials to obtain features that are below the diffraction limit of light. Here some advances in the use of electric fields to achieve this end are reviewed, that demonstrate the simplicity and elegance of this approach.

1. Introduction The interface between two media is never infinitely sharp. There is always a gradient in the density at the interface. This gradient may be very small, several tenths of a nanometer in thickness, like at the surface of any material, or it can be much broader, as is the case between two fluids. Nonetheless, this density gradient corresponds to a gradient in the dielectric constant at the interface. When a voltage is applied across this interface, the gradient in the dielectric constant translates into a gradient in the electric field across the interface. Any field gradient, whether the origin is a electric field, a thermal field or concentration, translates into a corresponding pressure. Of relevance to this review is that the electrostatic pressure that occurs across the interface between two

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polymers. This electrostatic pressure serves to amplify fluctuations at the interface. This simple concept has proven to be an elegant means of extending the spatial resolution of lithographic processes beyond what is capable currently using standard photolithographic process. 2. Instabilities on Liquid/Air Interfaces — Single Layer Consider the schematic diagrams in Fig. 1.1 Here we have a thin polymer film between two planar electrodes with an air gap separating the surface of the film and the upper electrode. When a field is applied across the electrodes, capillary waves on the surface of the polymer film are amplified, since the physical location of the crests and troughs in the capillary waves results in a variation in the field strength across the film and, therefore, a variation in the electrostatic pressure. Enhancing the amplitude of the capillary waves increases the surface area of the polymer film and, as such, the Laplace pressure, which acts to keep the surface planar. A planar surface represents the minimum area of the surface and, hence, costs the least energy. In addition to the Laplace and electrostatic pressure, there are two additional pressures acting on the surface. One is hydrostatic pressure and the second is the disjoining pressure arising from the electronic nature of the substrate. In general, the hydrostatic pressure contribution is small and the disjoining pressure for films greater than a few tens of nanometers is also small. In comparison to the electrostatic and Laplace pressure, these forces can be a Planar Electrode (T>Tg

b Patterned Electrode (T>Tg

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

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

p = p0 − γ

∂2h + pel (h) + pdis (h) ∂x 2

(1)

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

U {ε p d − (ε p − 1)h}

(2)

pel = −ε 0ε p (ε p − 1) E p 2

(3)

Ep = is given by

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

h( x, t ) = h0 + ueiqx +t /τ

(4)

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

j=

h3  ∂p  −  3η  ∂x 

(5)

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

∂j ∂h + =0 ∂x ∂t

(6)

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

1

τ

=−

h03  4 ∂pel 2  γq + q  3η  ∂h 

(7)

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

∂pel <0 ∂h

(8)

all modes with

q < qc = −

1 ∂pel γ ∂h

(9)

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are unstable. With time, the fastest growing mode will eventually dominate, corresponding to the maximum in Eq. (7)2 3 − γU E 2 λ = 2π ε 0ε p (ε p − 1) 2 p

(10)

To compare the results of the dispersion relation to the experimental data, it is useful to introduce reduced variables:

λ0 =

ε 0ε p (ε p − 1) 2 U 2 γ

(11)

and

E0 =

U (12)

λ0

Equation (10) leads to

E  λ = 2π  p  λ0  E0 

−

3 2

(13)

λ0 is a characteristic length scale which is connected to the relative strength of the electrostatic and Laplace pressures. In addition, Eq. (7) gives a relation for the time constant of the instability:

 Ep  τ =π4  τ0  E0 

−6

(14)

Equation (14) shows that we may rescale our data in reduced coordinates. It should be noted that from Eqs. (13) and (14) data over a range of experimental parameters should superpose to a single master curve. Within the experimental scatter, the data is quantitatively described by Eq. (14), in the absence of any adjustable parameters. In the present analysis, it was assumed that interfacial charging caused by the

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finite conductivity of the polymer film is negligible. To experimentally verify that only polarization effects are responsible for the EHD instability, a rectangular alternating voltage with a frequency of 1 KHz was applied at the capacitor, yielding results which are in good agreement with the constant voltage experiments. As opposed to the gravity-controlled case, there is no lower threshold field for the EHD instability, due to the dissipative character of the viscous drag that opposes the destabilizing electrostatic force. While electrohydrodynamic instabilities can be generated on virtually any liquids surfaces, a unique advantage of polymers is the structure generated from the instability can be frozen in the polymer by quenching below the glass transition temperature. Steiner and Russell first realized this when a 93 nm thick polystyrene film was annealed for 18 h at 170°C with an applied voltage of 50 V.1 These studies on quenched films demonstrated the growth of surface fluctuations, the bridging of the fluctuations to the second electrode, the formation of a hexagonal array of PS posts spanning across the gap between the electrodes, and a quantitative agreement between the observed and calculated wavelength using the arguments described above. They emphasized that the structures were observed only with the application of a very high electric field (>107 V/m). No surface features were observed in the absence of an applied field. This was confirmed by a control experiment without an applied voltage and with the electrodes shorted. It should be noted that the gap between the electrodes is typically microns or less. 1 V applied to the electrodes separated by 1 µm produces a field of 1 V/µm. As the gap gets smaller the field strength increases, of course. Nonetheless, unless special precautions are taken, residual charge on the surface of an electrode can give rise to large fields that, in turn, can lead to the enhancement of surface instabilities. In addition to a hexagonal symmetry, second-order effects can be observed as well. As the fluctuations grow in amplitude, there is a lateral variation in the film thickness. This leads to a depletion area around the growing fluctuation as polymer is drawn in to the growing column. In fact, a series of rims (a damped wave) are observed around the growing column. Each rim experiences the electrostatic pressure and fluctuations around the rim with a characteristic period grow with time.

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

λ = 2π

3 − γU 2 E ε 0ε p (ε p − 1)2 p

(15)

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

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

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

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

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

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

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

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

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

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

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

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

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

EP =

U {ε p d − (ε p − 1) h0 }

E0 =

λ = 2π

(16)

U

(17)

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3 − γU 2 E P 2 ε 0 ε P ( ε P − 1)

(18)

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λ0 = 3γη τ = 2 3 4 ε 0 h0 U

ε 0 ε P ( ε P − 1) U 2 γ 6

(19)

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

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

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 1  1 −   εP 

(20)

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E  τ =π4 P  τ0  E0 

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−6

(22)

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

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

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

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

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

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

3

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

(23)

ε 2 ε1

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

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

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

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

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

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a

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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laterally in thin copolymer films. Thurn-Albrecht et al. found the threshold electric field strength, above which complete orientation of the cylindrical microdomains, using asymmetric diblock copolymers of polystyrene and poly(methyl methacrylate).26 The threshold field strength was independent of film thickness and could be described by the difference in interfacial energies of the components. At field strengths slightly below threshold, a coexistence of the domains parallel and perpendicular to the electrode surface was found. An extensive experiment was performed by Russell and colleagues to understand microdomain orientations under applied electric fields.27-32 It was revealed that the alignment of microdomains by electric field depends on the segmental interaction between two blocks and the difference in the interfacial energies of each block with the substrate. By modifying a surface with random copolymer brushes, interfacial energies were controlled and the influence of interfacial energy on the orientation of the copolymer microdomains by an electric field was examined.27 A complete alignment of the lamellar microdomains was achieved only when the interfacial interactions were balanced. In addition, the use of two orthogonal, external fields was shown to control the orientation of lamellar microdomains in three dimensions in diblock copolymer thin films.28 An elongational flow field was applied to obtain an in-plane orientation of the microdomains of the copolymer melt, and an electric field applied normal to the surface was used to further align the microdomains. A study on the effect of ionic impurities on the electric field alignment of lamellar microdomains in polystyrene-blockpoly(methyl mathacrylate) thin flims showed that the microdomains could be aligned in the direction of electric field regardless of the strength of interfacial interactions at lithium ion concentrations greater than 210 ppm.29 A more quantitative investigation on the microdomain alignment of symmetric diblock copolymer of polystyrene and poly(methyl mathacrylate) was performed as a function of film thickness and interfacial energy.30 For films with thickness t<10L, where L is the equilibrium period of the copolymer in the bulk, interfacial interactions are dominant, and the lamellar microdomains orient parallel to the substrate surface regardless of the applied field. If t>10L, interfacial

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

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

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

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

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

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

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

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

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

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

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

ε 0ε d 2 U = cos θ Y + η 2dσ lv

(1)

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

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

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

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

ε 0ε r 2d

Asl U 2 .

(3)

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

σ sleff (U ) = σ sl −

ε 0ε d 2d

U2

(4)

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

σ lv κ (r ) − Pel (r ) = ∆p = const.

(5)

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

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

Fx = ∑ ∫ Txk nk dA

(6)

k Σ

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

Fx =

ε 0ε d 2d

U 2 = σ lvη .

(7)

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

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

(a)

(b)

(c)

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

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

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thickness of the insulator. Using very thick insulators, they managed to visualize the region h<
3. Physical Challenges of EW Applications Apart from various application specific problems, there are number of general physical challenges that are more or less common to all applications. These challenges include a number of very fundamental problems of wetting and of general fluid dynamics. Figure 6 shows a schematic illustration of a typical EW-based lab-on-a-chip system. It consists of four reservoirs on the left-hand side typically containing different liquids. On the right of the reservoirs, there are a number of individually addressable electrodes, along which small drops that can be extracted from the reservoirs can be transported. Subsequently, drops can be merged, split-up, mixed, … [22] The progress of any biochemical process inside the drop is typically monitored optically via fluorescence.

Fig. 6. Schematic top view of an EW-driven lab-on-a-chip system. The square electrodes typically have lateral dimensions of 0.1 … 1mm.

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However, other detection techniques such as electrical measurements have also been applied [5]. All of these basic operations, drop generation, drop transport, splitting, merging, and mixing connect to challenging general problems in fluid dynamics, namely pinch-off dynamics (capillary breakup) [23, 24], dynamic wetting (contact line dynamics) [25], wetting of structured surfaces [26], and (chaotic) mixing in low Reynolds number flows [2] (see Fig. 6). Another important practical problem is evaporation and surface contamination. Owing to their high curvature small water drops in air usually loose a appreciable fraction of their volume on a time scale several minutes. Most applications therefore make use of oil as an ambient medium to allow for continued operation over long times. (Obviously, this is even more relevant for devices such as EW-driven lenses or displays than for lab-on-a-chip systems, where an experiment lasts only a short time. Furthermore, it is crucial to protect the substrate surface from adsorption of contamination. For an EW-based lab-on-a-chip system such surface contamination causes not only problems in terms of chemical purity, it also compromises the operation of the device. Hydrophilic spots on the substrate surface, e.g. caused by adsorbed proteins, act as pinning centers for the moving contact line and might increase the contact angle hysteresis to an extent that drop transport is no longer possible. It turned out, however, that these problems can be avoided in most practical cases, for instance by the combined action of AC electric fields (reducing net electrostatic attraction to the surface) [27] and ambient oil that forms a thin film effectively separating the substrate surface from the aqueous drop phase [28]. Before addressing some of the specific problems mentioned above, let us illustrate the general principle EW-driven drop actuation. We note first that the drops in these devices are typically large compared to the thickness of the insulator. Since we are only interested in their dynamics on this global scale, the local effects close to the contact line that were discussed in the previous section are not of interest. In that case, electrowetting can be simply considered as a means of tuning the wettability of the surface locally [29]. The behavior of the drops is therefore the same as on a chemically structured surface with patterned

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wettability. To manipulate drops it is thus essential to produce gradients in the surface wettability, which is achieved by patterning the electrodes on the substrate. The basic process of drop actuation is sketched in Fig. 7 along with the typical device geometry of EW-driven lab-on-a-chip systems. These devices are typically designed as a sandwich with two parallel surfaces at a distance of order 0.1mm. Apart from a reduction of evaporation losses in systems without oil, this geometry allows for providing electrical contact to the drop everywhere via the homogeneous electrode on the top surface. If only one of the electrodes on the bottom surface is actuated (the middle one in Fig. 7), then the contact angle is reduced only locally on top of that electrode. As a result, the drop experiences a net force towards the more wetting part of the substrate, i.e. towards the activated electrode. The ensuing dynamics is exactly the same as for the widely studied drops on surfaces with chemical wettability gradients including self-propelling drops (for the basic principles, see e.g. [30]). As in those cases, the dynamics of the drop are governed by a balance between the driving wettability gradient and the dissipative forces, which contain an important contribution due to contact line friction [5, 6, 31]. The latter statement is characteristic for any aspect of the dynamics of drops in electrowetting systems: the dynamics are fully determined by the hydrodynamics. The variation of the local contact angle upon applying a voltage is instantaneous on the time scale of the hydrodynamic response that is either given by the (Rayleigh) eigenfrequency of the drop in the inertial case or by some viscous relaxation time. Electrowetting abruptly changes the boundary condition at the contact line – the rest of the dynamics is hydrodynamic response, usually in the Stokes flow regime.

Fig. 7. Schematic side view of an EW-driven lab-on-a-chip system. The middle electrode on the bottom is activated, which leads to the asymmetric drop shape. (drawing not to scale; insulator thickness: O(µm); plate spacing: O(100µm)).

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4. Current Examples of Research on EW-Driven Drop Dynamics In this section, we will discuss two examples from the recent research in our laboratory, namely the effect of electric fields on the contact angle hysteresis for sessile drops in air and the dynamics of contact lines in ambient oil, where we observe the entrapment and subsequent break-up of thin oil films under the influence of the electric fields.

4.1. Contact angle hysteresis in AC and DC electrowetting If we look out the window on a rainy day, we see that small drops remain stuck whereas larger ones roll down. The origin of this daily life phenomenon is contact angle hysteresis. Due to chemical heterogeneity and topographic roughness the contact angle of sessile drops is not welldefined. Rather it can vary between two limiting values, the advancing contact angle θA and the receding contact angle θR. Only if the contact angle exceeds θA or if it is below θR the contact line can advance or recede, respectively. The fact that contact angle hysteresis is more efficient in pinning small drops rather than larger ones is due to the different scaling of the driving force (gravity), which scales with R3 in the present case, and the total pinning force due to contact angle hysteresis that scales with the length of the contact line, i.e. linearly with R. Minimizing contact angle hysteresis (as well as contact line dissipation) is therefore a crucial step for any drop-based microfluidic system. As with the example of the drop on the inclined window surface, contact angle hysteresis manifests itself in EW-driven devices in form of a threshold voltage for actuation. If the driving force is below that threshold, the contact line (and hence the entire drop) remains stuck (see [8, 32] for early examples). To study the behavior of contact angle hysteresis in EW and to explore whether it can be controlled whether we recently performed a systematic study of hysteresis measurements in the presence of both AC and DC electric fields [33]. Figure 8 shows typical contact angle hysteresis curves, both for DC (a) and for AC (b) voltage. The data show the contact angle as a function of time as the drop volume is first increased (from the first arrow up to the second) and the decreased (from the

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second (a) arrow on) at a slow constant (b) rate. For all data well defined increased (from the first arrow up to the second) and the decreased (from

Fig. 8. Contact angle vs. time in electrowetting (water on Teflon AF in air) for various voltages (top curve: zero voltage). (a) DC voltage; (b) AC voltage; AC frequency: 5 kHz.

(a)

(b)

Fig. 9. (a) Contact angle vs. EW number for AC (filled symbols) and DC (open symbols) voltage (water on Teflon AF in air). Solid and dashed lines display model result for AC and DC voltage, respectively. (b) schematic force balance at contact line including random pinning forces fp and the time-dependent electrostatic force fel.

second arrow on) at a slow constant rate. For all data well defined plateaus are visible both for θA and for θR. As the voltage is progressively increased from zero (top curve in both graphs) to higher voltage (lower curves) we note that both angles qualitatively decrease for both AC and DC voltage. However, the quantitative behavior is quite different: in the case of DC voltage both θA and θR decrease approximately at the same rate as U is increased. In contrast, AC voltage initially affects only the advancing contact angle, whereas θR remains more or less constant. Once θA has caught up with θR both angles decrease approximately at the same

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rate as we increase U further. This behavior is summarized in Fig. 9(a), where we show cosθ versus the EW number for both AC (full symbols) and DC (open symbols) voltage. To rationalize this behavior, we consider the force balance at the contact line as sketched in Fig. 9(b). In the ideal Young case, the contact angle is fully determined by the balance of the (horizontal component) of the interfacial tensions. The random pinning forces originating from surface heterogeneity give rise to the finite interval θR < θ < θA of values that θ can assume. Since we are interested only in the behavior of the system on a global scale, we can simply add the electrostatic fel (following Eq. 7) to this force balance — provided that the pinning forces and the electric forces are independent. For the DC case, this simply means that both θA and θR decrease in the same way, namely such that cos θ A / R (U ) = cos θ A / R (0) + η

(8)

This behavior is shown as the dashed lines in Fig. 9(a). For AC voltage, however, we note that the electrostatic force varies periodically as fel (t) = η (sin ω t)2 = η (1 – cos 2 ω t). (Here, we non-dimensionalized fel by dividing by σlv). Its contribution to the force balance varies between zero and 2η during an oscillation cycle of the AC field. Since fel is always directed towards the outward direction it can only assist the contact line along in advancing, but not in receding. Hence, θR is independent of U, whereas θA decreases with 2η, i.e. cos θ R (U ) = cos θ R (0) cos θ A (U ) = cos θ A (0) + 2η

(9)

for small voltage. As soon as the amplitude of the electric forces has reached the amplitude of the random pinning forces, i.e. for η = (cos θ R (0) − cos θ A (0)) / 2 , all pinning is wiped out by the electrical excitation and hence both θR and θA decrease following Eq. (1). This prediction is shown as solid lines in Fig. 9(a). While this simple analysis misses the subtleties of the depinning mechanism including its non-local character, the agreement with the experimental data shows that the picture captures the basic physics of the phenomenon.

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Using model inclined planes, we have indeed been able to mobilize drops above a certain threshold of AC voltage, whereas they remained stuck for DC voltage of even higher amplitude. Preliminary experiments with a EW-lab-on-a-chip device also showed a clear reduction of the threshold actuation voltage (for suitable addressing of the electrodes).

4.2. Oil film entrapment and break-up in ambient oil As mentioned above, it is frequently preferable to operate EW-driven devices in an ambient oil medium rather than in air. Amongst the aspects discussed above, operation in oil also tremendously reduces contact angle hysteresis. Essentially there is no true three-phase contact line if a an oil film is present between the drop and the substrate, as it is frequently the case (see [5], [6] and references there). However, ambient oil also complicates the dynamics of the system since the moving aqueous drop has to replace and squeeze out the ambient oil as it moves. The faster the drop moves, the more difficult it is to squeeze out the oil from underneath the moving drop. This problem is related to the wellknown Landau-Levich problem in fluid dynamics of depositing wetting liquid films by withdrawing solid plates from a bath.

Fig. 10. Video snapshots of the solid-liquid interfacial area during EW-driven spreading of a water drop in ambient silicone oil. Note the appearance of interference fringes behind the moving contact line and their subsequent breakup into small droplets (top left). Inset: schematic view of the setup allowing visualization through the transparent substrate. (adapted from Ref. [34])

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Recently, we addressed the problem of oil entrapment in two-phase electrowetting by following the EW-induced spreading dynamics of a salt water drop in ambient silicone oil using interference microscopy [34]. We imaged the solid-liquid interface (area Asl) of a through a transparent substrate as sketched in the inset of Fig. 10. The main panel of Fig. 10 shows views of the solid-liquid interface as the spreading process proceeds in clockwise direction. At t=0 (top right), the drop is held just above the substrate at zero voltage. As the voltage was increased (linearly in time), Asl increased as expected. The snapshots show clear interference fringes, indicating that an oil layer with a thickness comparable to the wavelength of light is entrapped by the moving contact line. As time proceeds, the contact line advances further. Simultaneously, the entrapped film becomes unstable and breaks up into a series of drops (see Fig. 10; top left). The characteristic size of these drops is seen to decrease from the center of the drop towards the contact line. The dynamics of the process can thus be split in two separate stages. First a thin oil-film is being entrapped under the spreading drop, second the entrapped layer becomes unstable and breaks up into droplets. The physical driving force that is responsible for both processes is the normal pressure exerted onto the water-oil film interface by the screening charges inside the aqueous drop (see Fig. 4). Far away from the contact line, this pressure reduces essentially to

p el0 ≈ ε d ε 0U 2 / 2d 2 ⋅ (1 − hε d / dε oil ) ,

(10)

where εoil is the dielectric constant of the oil. Obviously, this pressure increases when the thickness h of the oil layer decreases. As a consequence, the oil layer is unstable against small perturbations — and therefore it breaks up into the drops seen in Fig. 10. A full linear stability analysis of the oil film in the usual lubrication approximation yields that the wavelength of the fastest growing unstable mode, which turns out to be

λ m (U , h) = 2π

2 2σ ow ε oil d 3 (1 + ε d h ε oil d ) 3

ε 0 ε d3U 2

,

(11)

which amounts to values between 20 and 100µm, in agreement with typical size of the oil droplets observed in the experiments [34]. In

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particular, the decrease of the drop size from the center towards the edge of the drop is correctly reproduced by Eq. (11). For the typical experimental values, it is found that this decrease is primarily caused by the increasing voltage experienced by the oil film at larger r (remember that U~r in the present experiments). Let us now analyze the oil film entrapment process. Figure 11 shows the intensity of the interference fringes as a function of the distance from the drop center for various ramp speeds. Obviously the number of fringes (and thus the thickness of the entrapped oil film) increases with increasing ramp speed. This behavior is plausible since a faster motion of the contact line leaves less time to squeeze out the oil again the viscous forces. The dashed lines in Fig. 11 are fits to an algebraic dependence of the oil film thickness h ∝ r α . The resulting exponent of the fit is α = 1.35 ± 0.1 . It may seem surprising at first glance that the oil film thickness depends on the r. However, since U increases with time (and thus with r), the electrostatic pressure that is responsible for squeezing out the oil is also larger at later stages of the spreading process. Hence the entrapped film is thinner for larger r. To understand the radial dependence of the h quantitatively, we note that the dynamics of the moving contact line in the present case is very

Fig. 11. Radial intensity profiles (before break-up) for increasing ramp speeds of the voltage (top to bottom) corresponding to increasing contact line speeds. Dashed lines are fits to the data based on standard parallel beam interference (see text for details). Inset: cross-sections in radial direction of composite video snapshots for decreasing ramp speed (left to right).

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similar to the dynamics of a contact line in the dip coating process described by the model of Landau and Levich. A comprehensive explanation of this problem is given in Ref. [30]. Adapting that description to the present problem, one finds that the driving force for squeezing out the oil layer is indeed given by the electrostatic pressure p el ≈ ε d ε 0U 2 / 2d 2 . Balancing this pressure with the viscous dissipation in the liquid meniscus close to the advancing contact line and matching the curvature of the interfaces following Landau and Levich, one finds that the thickness of the entrapped film is given by

(

h= d R

2 / 3  Ca

)

   η 

2/3

∝

Ca 2 / 3 Ca 2 / 3 ∝ 4/3 U 4/3 r

(12)

where Ca = µ v / σ wo is the capillary number (v: contact line velocity). The exponent 4/3 is in close agreement with the value of 1.35 found above by fitting to the experimental data. The scaling with the capillary number follows the classical 2/3 law. Also this dependence is fairly well reproduced by the experimental data (see Fig. 12).

Fig. 12. Dependence between contact line speed and thickness of the entrapped oil film (normalized for r-dependence.)

In summary, both the thickness and the (in)stability of the entrapped oil film are found to be controlled by the electrostatic pressure exerted on the interface by the screening charges inside the drop. The model presented here gives a clear explanation of the lubricating effect of ambient on the dynamics of drops in EW devices. While the presence of

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this oil film is desirable from the perspective of drop motion and of surface protection (e.g. from contamination by adsorbing proteins), it has also disadvantages from the perspective of sensing. In many microfluidic devices for biomedical purposes, sensing is performed using specific binding to surface-anchored sensor molecules. Obviously, this becomes impossible if an oil film (or an oil droplet) covers these molecules in the final state. For such applications it would thus be essential to control the oil-dewetting process in such a way that the sensing areas are in direct contact with the aqueous drop phase in the end.

Conclusions In summary, we showed that electrowetting is in essence an electromechanical process that deforms liquid surfaces by the action of the Maxwell stress on the drop surface. If one is only interested in the shape of the drop on a global scale (i.e. large compared to the thickness of the insulating layer), these electrostatic effects reduce to an effective local reduction of the contact angle above an activated electrode. From a static perspective, electrowetting then reduces to “ordinary” wetting of chemically patterned surfaces — with the important difference that the wettability is switchable on arbitrarily short time scales (compared to typical hydrodynamic time scales). With respect to dynamics, however, there are important differences between wetting of chemically patterned surfaces and electrowetting. We demonstrated that alternating electric fields and the resulting timedependent Maxwell stress lead to a substantial reduction of the contact angle hysteresis in ambient air, which greatly enhances the mobility of drops. In ambient oil contact angle hysteresis is virtually absent due to the lubricating effect of a thin oil layer trapped underneath aqueous drops. We showed that the thickness this oil layer is controlled dynamically by a competition between the electrostatic pressure and the viscous forces preventing squeeze-out of the oil. Once entrapped, the oil films are shown to be unstable and to break up into a series of individual drops. Both examples demonstrate thus that the details of the electric field close to the contact line play a crucial role for the dynamics of electrowetting devices.

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Acknowledgments This work was supported by the program of Dispersed Multiphase Flow of the Institute of Mechanics Process and Control Twente (Impact) and by the Micro- and Nanofluidics program of the MESA+ Institute for Nanotechnology.

References 1. Squires, T.M. and S.R. Quake, Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys., 77; 977-1026 (2005). 2. Stone, H.A., A.D. Stroock, and A. Ajdari, Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Annual Review of Fluid Mechanics, 36; 381411 (2004). 3. Teh, S.Y., et al., Droplet microfluidics. Lab on a Chip, 8; 198-220 (2008). 4. Song, H., D.L. Chen, and R.F. Ismagilov, Reactions in droplets in microfluidic channels. Angew. Chem. Int. Ed., 45; 7336-7356 (2006). 5. Fair, R.B., Digital microfluidics: is a true lab-on-a-chip possible? Microfluidics Nanofluidics, 3; 245-281 (2007). 6. Mugele, F. and J.-C. Baret, Electrowetting: from basics to applications. J. Phys. Cond. Matt. , 17; R705-R774 (2005). 7. Darhuber, A.A. and S.M. Troian, PRINCIPLES OFMICROFLUIDIC ACTUATION BY MODULATION OF SURFACE STRESSES. Ann. Rev. Fluid Mech., 37; 425-455 (2005). 8. Berge, B. and J. Peseux, Variable focal lens controlled by an external voltage: An application of electrowetting. European Physical Journal E, 3; 159-163 (2000). 9. Kuiper, S. and B.H.W. Hendriks, Variable-focus liquid lens for miniature cameras. Applied Physics Letters, 85; 1128-1130 (2004). 10. Krupenkin, T., S. Yang, and M. P., Tunable liquid microlens. Appl. Phys. Lett., 82; 316-318 (2003). 11. Hayes, R.A. and B.J. Feenstra, Video-speed electronic paper based on electrowetting. Nature, 425; 383-385 (2003). 12. Malloggi, F., et al., Electrowetting-controlled droplet generation in a microfluidic flow-focusing device. Journal of Physics-Condensed Matter, 19; (2007). 13. Lippmann, G., Relations entre les phénomènes électriques et capillaires. Ann. Chim. Phys., 5; 494-549 (1875). 14. Froumkine, A., Couche double. Électrocapillarité. Surtension. Actualités Scientifiques et Industrielles, 373; 1-36 (1936). 15. Berge, B., Electrocapillarite et mouillage de films isolants par l'eau. C.R.Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers., 317; 157-163 (1993).

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16. Landau, L.D. and E.M. Lifschitz, Elektrodynamik der Kontinua. Lehrbuch der Theoretischen Physik. Vol. VIII. 1985, Berlin: Akademie Verlag. 17. Jones, T.B., On the Relationship of Dielectrophoresis and Electrowetting. Langmuir, 18; 4437-4443 (2002). 18. deGennes, P.G., Wetting: statics and dynamics. Rev. Mod. Phys., 57; 827-863 (1985). 19. Buehrle, J., S. Herminghaus, and F. Mugele, Interface Profiles Near Three-Phase Contact Lines in Electric Fields. Phys. Rev. Lett., 91; 086101 (2003). 20. Buehrle, J. and F. Mugele, Equilibrium drop surface profiles in electric fields. J. Phys. Cond. Matt., 19; 375112 (2007). 21. Bienia, M., et al., Electrical-field-induced curvature increase on a drop of conducting liquid. Europhys. Lett., 74; 103-109 (2006). 22. Cho, S.K., H.J. Moon, and C.J. Kim, Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. Journal of Microelectromechanical Systems, 12; 70-80 (2003). 23. Eggers, J., Nonlinear Dynamics and Breakup of Free-Surface Flows. Rev. Mod. Phys., 69; 865-929 (1997). 24. Eggers, J. and E. Villermaux, Physics of liquid jets. Reports on Progress in Physics, 71; (2008). 25. Blake, T.D., The physics of moving wetting lines J. Coll. Interf. Sci., 299; 1-13 (2006). 26. Lipowsky, R., Morphological wetting transitions at chemically structured surfaces. Curr. Opinion Coll. Interf. Sci., 6; 40-48 (2001). 27. Yoon, J.Y. and R.L. Garrell, Preventing biomolecular adsorption in electrowettingbased biofluidic chips. Analytical Chemistry, 75; 5097-5102 (2003). 28. Srinivasan, V., V.K. Pamula, and R.B. Fair, An integrated digital microfluidic labon-a-chip for clinical diagnostics on human physiological fluids. LabChip, 4; 310315 (2004). 29. Mugele, F., et al., Electrowetting: a convenient way to switchable wettability patterns. Journal of Physics: Condensed Matter, 17; S559-S576 (2005). 30. deGennes, P. G., F. Brochard-Wyart, and D. Quéré, Capillarity and Wetting Phenomena. 2004, New York: Springer. 31. Ren, H., et al., Dynamics of electro-wetting droplet transport. Sensors and Actuators B, 87; 201-206 (2002). 32. Pollack, M.G., R.B. Fair, and A.D. Shenderov, Electrowetting-based Actuation of Liquid Droplets for Microfluidic Applications. Appl. Phys. Lett., 77; 1725-1726 (2000). 33. Li, F. and F. Mugele, How to make sticky surfaces slippery: contact angle hysteresis in electrowetting. Appl. Phys. Lett., (submitted); 2008). 34. Staicu, A. and F. Mugele, Electrowetting-Induced Oil Film Entrapment and Instability. Phys. Rev. Lett., 97; 167801 (2006).

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Chapter 6 Phase Separation and Morphology of Polymer Mixtures Driven by Light Qui Tran-Cong-Miyata and Hideyuki Nakanishi Graduate School of Science and Technology Department of Macromolecular Science and Engineering Kyoto Institute of Technology Matsugasaki, Kyoto 606-8585, Japan

1. Introduction Morphology control of multiphase polymeric materials has been one of the central research topics in polymer science because it provides a way to manipulate, on purpose, the strong correlations between morphology and physical properties of polymer materials.1,2 In the past five decades, numerous chemical as well as physical methods have been extensively developed for control of polymer morphology. From the chemical aspects, living polymerization has been exploited for “tailoring” the chemical structures of polymer chains. The immiscibility between different polymers has been well utilized in the morphology control of block copolymers by taking advantages of these sophisticated synthesis techniques. It has been demonstrated that di- or multiblock copolymers with well-defined chemical structures can be synthesized, giving rise to a wide variety of ordered structures such as spherical domains with body-centered cubic (bcc) arrangements, hexagonal cylinder, gyroids and lamellae.3 Since the chemical linkages between different polymer components of block copolymers do not allow polymer chains to diffuse freely, and also the radius of gyration of the molecule is actually limited by the chemical synthesis, the spatial length scale of the morphologies is constrained in the nanometer scales. Therefore, the terminology microphase separation is often used to describe phase separation in block copolymers. In contrast, the term macrophase separation is usually employed to describe phase separation phenomena in polymer mixtures. 171

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Because there is no constraint at the molecular level like block copolymers, phase separation of polymer blends is often terminated at a random twophase structure in the micrometer scales or even larger as a result of phase equilibrium. In practice, since most of polymer pairs are mutually immiscible, various mechanical techniques such as mixing under different stirring rates at high temperatures have been widely used to control the morphological length scales of these immiscible blends in the industry.4 However, with these practical methods one can only control the average length scales and cannot generate or modify the morphological regularity. External fields such as electric field,5 shear stress6 and recently magnetic field7 have been used to control morphology of polymer mixtures as well as block copolymers. Under these physical constraints, specific unstable modes of concentration fluctuations in the mixtures are selected, giving rise to mode-selection phenomena in phase separation. Particularly, phase separation under a shear field is a phenomenon far-from-equilibrium with the dynamics dictated by the shear rate. On the other hand, from the equilibrium viewpoint, there exists a similarity between morphology resulting from microphase separation of block copolymers and the Turing structures emerging from the reaction-diffusion mechanism.8 It is striking that these two kinds of structures with completely different origins are exactly the same, except that the characteristic length scales emerging from reaction-diffusion systems are about 105 times larger than those observed in microphase separation of block copolymers. Recently, it has been found that this similarity comes from the generic mechanism resembling the so-called “activator-inhibitor model”9 or the “competing interactions” phenomena observed in the wide range of physico-chemical systems.10,11 This resemblance suggests a novel method for morphology design of multi-component polymeric systems. At the nanometer length scales, a wide variety of highly ordered morphology has been designed for block copolymers by living polymerization techniques3,12 taking advantages of the competitions between the repulsion due to thermodynamically unfavored interactions and the attraction between different polymer components arising from the chemical linkages. In order to bridge the gap between the macro- and the microscales, we have tried to couple photochemical reactions to phase separation process. Here, we will show that manipulating the competitions between phase separation process and photochemical reactions could provide a promising way for morphology design of multi-component polymers.

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In this chapter, we will summarize the recent experimental results on morphology design obtained by coupling photochemical reactions to phase separation. As qualitatively predicted by recent theories on reactioninduced phase separation, it will be shown that, properly chosen chemical reactions can be efficiently used as a tool to select the unstable modes of concentration fluctuations in polymer mixtures. Particularly, to avoid the interference between the temperature of phase separation and the heat required for activating chemical reactions, photochemical reactions were utilized to induce phase separation. This selection allows us to initiate as well as terminate chemical reactions independently from thermodynamic variables such as temperature and/or pressure. The content of this chapter is as follows. First, theoretical aspects of phase separation in binary mixtures with and without chemical reactions will be summarized with emphasis on the role of chemical reactions. Introduced in the subsequent section are the experimental data of the mode selection process driven by photochemical reactions. The significant roles of photochemical reactions in phase separation of polymer mixtures will be experimentally verified in the following sections with the following examples: (1) emergence of hierarchical morphology in interpenetrating polymer networks (IPNs) as an example for systems with competing interactions; (2) phase separation induced by spatial and temporal modulation, (3) designing spatially graded morphology by non-uniform phase separation, (5) morphology with an arbitrary distribution of length scales by computer-assisted irradiation (CAI), (6) reversible phase separation driven by two UV wavelengths. Finally, concluding remarks and perspectives of reaction-induced phase separation in design of novel morphology will be discussed. 2. Fundamental Aspects of Phase Separation 2.1. Non-reacting systems There are two distinct mechanisms for liquid-liquid phase separation in binary mixtures: nucleation-and-growth and spinodal decomposition.13,19 The former begins with small nuclei in the metastable region of the mixture. Subsequently, these nuclei grow with time and approach phase equilibrium while their concentration remains saturated. On the other hand, in the spinodal decomposition mechanism, phase separation starts from concentration fluctuations in the unstable region. These periodic structures grow with time both in wavelength and composition. Gradually, the structure

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becomes coarsened with time and turns into droplets upon approaching phase equilibrium. The kinetic equation describing the spinodal decomposition process of binary mixtures was initially formulated by Cahn14 using the so-called square-gradient free-energy:15 Z F {φ(r, t)} = dr{f (φ) + κ[∇φ(r, t)2 } (1) Here, f (φ) is the free-energy of the mixture in the homogeneous miscible state. For metallic alloys or mixtures of small molecules, f (φ) can take the Landau’s double-minimum free energy. For polymer mixtures, FloryHuggins free-energy has been used for f (φ).16,17 The 2nd term on the RHS of (1) was proposed to express the effects of concentration fluctuations arising from the mixing process on the mixture free-energy. Furthermore, the square-gradient coefficient κ can be related to the Flory-Huggins binary interaction parameter χ.18 The kinetic equation describing phase separation for a conserved binary mixture (φA + φB = 1) can be derived by using the free-energy given in Eq. (1): δF {φ(r, t)} ∂φ(r, t) = M ∇2 ∂t δφ(r, t)

(2)

Here, M is the mutual mobility coefficient. Linearization of (2) around the average composition φ = φ0 of the mixture gives: # " 2 ∂(δφ) ∂ f (δφ) − 2κ∇2 (δφ) (3) = M ∇2 ∂t ∂φ2 φ=φ0 Solution of the linearized kinetic equation (3) in the reciprocal space q has the following form: δφ(q, t) = δφ(q, 0). exp[R(q)t]

(4)

Here, R(q) is the growth rate of concentration fluctuations with the wavelength ξ = 2π/q. In the context of square-gradient theory, R(q) is given by the following dispersion relation: " # ∂2F 2 2 R(q) = −M q + 2κq (5) ∂φ2 φ=φ0 The dispersion relation shown in Eq. (4) indicates that the mixture will be stable if R(q) < 0 and it becomes unstable, leading to phase separation if R(q) > 0. As time goes on, the morphology is gradually coarsened and eventually the mixture reaches phase equilibrium. The kinetics at later time

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belongs to the nonlinear regime of phase separation and is analytically intractable. Instead, different approaches such as scaling and numerical analysis have been used to analyze the late stage of phase separation kinetics. These details were recently reviewed by several research groups.19 – 22 2.2. Reacting systems Unlike the non-reacting cases, phase separation of reacting mixtures is a mode selection process where the chemical reaction plays the role of a selector for unstable concentration fluctuations. Due to this selection, phase separation in these particular cases does not proceed to phase equilibrium, but is spontaneously frozen by the chemical reaction. Consequently, stationary modulated structures emerge during the phase separation process until all the reactants are consumed. Coupling between autocatalytic reactions and phase separation of binary mixtures was initially studied by Huberman.23 Much later, Glotzer and co-workers studied phase separation of a binary mixture A/B accompanied by an A B reversible reaction24 using the following reaction-diffusion equation: δF {φ(r, t)} ∂φ(r, t) = M ∇2 + g(φ) ∂t δφ(r, t)

(6)

where the g(φ) term on the RHS expresses the contribution of the reaction to the phase separation process. For the A B reaction with the forward reaction rates k1 and the backward reaction rate k2 , the reaction term g(φ) is given by: g(φ) = −k1 φ + k2 (1 − φ)

(7)

The dispersion relation of Eq. (6) with (7) as a reaction term can be obtained by the conventional linear stability analysis: " # ∂2F 2 2 R(q) = −M q + 2κq − K (8) ∂φ2 φ=φ0 where K = (k1 + k2 ) is a function of reaction rates for the above-mentioned reversible reaction. The significance of the dispersion relation (8) is that the growth rate R(q) becomes negative as q → 0. This feature leads to the consequence that the phase separation process will proceed to some certain extent and is eventually frozen due to the suppression caused by the negative growth rate on the side of small wavenumber, i.e. large structures. For comparison, the dispersion relations (5) and (8) obtained respectively for the two cases

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with and without chemical reaction are illustrated in Fig. 1. The resulting length scales of the modulated structures for the case of reaction are determined by the reaction rate and the growth rate of phase separation. The modulated structures with well-defined characteristic length scales emerging from the coupling between the reaction and phase separation resembles the Turing structure25 in reaction-diffusion systems. In general, these modulated structures, known as modulated phases,26,27 which are generated by the competition between two (or more) antagonistic interactions, have been found in a wide range of chemical as well as physical phenomena including microphase separation of block copolymers.12 Experimentally, it has been shown that chemical reaction kinetics in a reacting polymer mixture can be strongly affected by the thermodynamics of the mixture, particularly in the vicinity of the critical point.28 Taking into account the correlations between chemical reactions and thermodynamics of the mixtures, Lefever and Carati reformulated, in a general way, phase separation phenomena driven by a general reversible reaction indicated below: a 1 X1 + a 2 X2 + · · · b 1 X1 + b 2 X2 + · · ·

(9)

By linear stability analysis, they found that there exist four possible cases depending on the forward and backward kinetics:29 (1) the reaction cannot induce phase separation, i.e. the mixture remains stable under reaction; (2) the spinodal decomposition is accelerated, instead of suppressed, by the reaction; (3) phase separation is spontaneously freezing

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due to the suppression of long-wavelength fluctuations (reproducing the results of Glotzer and coworkers); (4) the mixture is destabilized only by the chemical reaction. The last one resembles the case of Belousov-Zhabotinsky reaction8 where the nonlinear interactions between the two components are the only reason for the destabilization. The first three cases can be verified by experiments,30 whereas the last prediction has not yet been experimentally verified at this moment, though very interesting. Recently, chemical reaction-induced phase separation was extended into another direction by Mikhailov and co-workers who introduced into the reaction-diffusion equation an attractive lateral interaction potential between reactants. These non-local kinetic equations were proposed in order to explain the spatiotemporal behavior of traveling waves on the surface of platinum reacting with carbon monoxide.31 However, all the above-mentioned theoretical models were dealing with small molecule mixtures. Therefore, the unique effects of polymeric systems such as visco-elasticity were not considered. Recently, the contribution from the intrinsic properties of macromolecules such as visco-elasticity to phase separation of polymeric systems has been theoretically investigated by Doi and Onuki,32 Tanaka33 and more recently by Taniguchi and Onuki34 where it was shown that viscoelasticity of polymer plays an important role as a long-range effect in phase separation. However, these theories did not deal with reacting systems. Most recently, Ohta and co-workers have analyzed phase separation of polymerizing systems containing liquid crystals, taking into account the elastic effects of polymer.35 In general, as seen in the experimental section shown below, phase separation of reacting polymer systems is an extremely complicated phenomenon. The problem is likely theoretically intractable without tentative assumptions. The primary factor that makes the phenomena complicated is the continuous change in molecular weight and composition with reaction time. As a consequence, not only mobility, but thermodynamics of the mixtures is also continuously modified during the reaction. Furthermore, most of chemical reactions in polymer do not follow the mean-field kinetics and are highly nonlinear.36 Additional difficulty is caused by the coupling between the reaction kinetics and the concentration fluctuations in the reacting polymer systems.28 This coupling is again feedback on the thermodynamics of the mixture. For these reasons, no theory considering all these features of polymer is currently available. Shown below are our recent experimental data on phase separation of polymer blends driven by photochemical reactions. The primary purpose of these experiments is to

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elucidate the decisive factors controlling the phase separation kinetics of reacting polymer blends. The subsequent aim is to establish a novel method for design and control of polymer blends morphology in the mesoscopic scales. 3. Phase Separation of Polymer Blends Induced by Photochemical Reactions 3.1. Significance of photochemical reactions As described briefly above, light can be used as a tool to initiate and terminate a chemical reaction independently from thermodynamic conditions. In this work, two kinds of photochemical reactions, photodimerization of anthracene and photoisomerization of stilbene, were used for the control and design of polymer blend morphology. Polymer networks are generated in the former case, resulting in an increase in viscosity, whereas the local free-volumes of polymer segments and the thermodynamic interactions between the counterpart polymer and polymer segments bearing two isomers are modified in the later case. Both reactions can be used to thermo-dynamically destabilize miscible polymer mixtures.36,37 By combining these two photochemical reactions, the effects of viscosity as well as the viscoelasticity on phase separation were investigated. These two photochemical reactions are schematically illustrated in Fig. 2. To couple chemical reactions to phase separation of polymer blends, these photoreactive groups need to be chemically linked to either one or both polymer components of the blend depending on purposes. Details of these chemical syntheses are described in previous publications.30,38 3.2. Design of hierarchical morphology by using competing interactions in polymeric systems Interpenetrating polymer network (IPN) is a sort of molecular composites composed of two different polymer networks mutually entangled by cross-link during polymerization process.39 Because different polymers are usually immiscible upon mixing, phase separation takes place during the preparation process of IPNs, leading to phase equilibrium with random two-phase structures. The reason responsible for this behavior is that, for most cases, phase separation overcomes thermally activated cross-linking reactions. Here, we demonstrate that by taking advantages of photochemical reactions, the competition between phase separation and reactions can

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Fig. 2. Upper part 1: photodimerization of an anthracene derivative; Lower part 1: photoisomerization reaction).

be well controlled, resulting in various modulated phases with hierarchy in polymers. In these experiments, homogeneous mixtures of anthracene-labeled polystyrene (PSAF) dissolved in methyl methacrylate monomer were irradiated with ultraviolet light at 365 nm to induce photodimerization of anthracene, photopolymerization of MMA and photo-cross-link of PMMA. The chemical structure of these two polymers is shown in Fig. 3. To facilitate the morphological observation using the fluorescence mode of a laser confocal scanning microscope, the PSAF component was further labeled

(PMMA)

.

(PSAF)

Fig. 3. Chemical structure of anthracene-labeled polystyrene (PSAF) and poly (methyl methacrylate) (PMMA).

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q = 3.93x10-1 µm-1

q = 3.93x10-1 µm-1

q = 3.93x10-1 µm-1

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Fig. 4. Light scattering patterns and the corresponding 3D morphology of poly(crossstyrene)-inter-poly(cross-methyl methacrylate) IPNs obtained by varying the concentration of the photoinitiator Lucirine TPO and the light intensity at 30◦ C.

with fluorescein. This labeling provides a strong contrast between the two polymer components observed under laser confocal microscope. By changing the light intensity, the networking rates of both PSAF and PMMA chains can be regulated. On the other hand, changing the concentrations of Lucirin TPO and of the anthracene label content of PSAF enables the control of the molecular weights between cross-link junctions of PMMA as well as PSAF networks. It was found that upon changing these parameters, a variety of morphological transition took place. An example is illustrated in Fig. 4 for poly(cross-styrene)-inter-poly(cross-methyl methacrylate) IPNs prepared by varying the concentration of the photoinitiator Lucirin TPO and the light intensity at 30◦ .40 The 3-dimensional morphology of the IPNs was illustrated in the lower part, whereas the corresponding light scattering patterns are shown in the upper part of the figure. For low irradiation intensity (low rate of PMMA networking) and low concentration of Lucirin TPO (high molecular weight of PMMA), co- continuous structures were observed. In all the 3D morphologies shown in Fig. 4, the dark region imaged by fluorescence of the fluorescein marker corresponds to the PSAF-rich domains, whereas the transparent part indicates the PMMA-rich domains

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in the sample. Obviously, both the PSAF and PMMA components are 3dimensionally interconnected. For the same light intensity, increasing the initiator (Lucirin) concentration results in a phase inversion as observed in Fig. 4(b). The PSAF-rich co-continuous phase turns into dispersed (dark) domains inside the matrix of PMMA-rich phase. There also exist the minor phases of the counter components in both the PSAF-rich and PMMA-rich phases. At high concentrations of Lucirin and under high UV intensity, large scale spinodal structures of PSAF containing PMMA droplets were observed as seen in Fig. 4(c). These experimental results indicate that polymer materials with a wide variety of periodic, hierarchical structures in the micro- and sub-micrometer scales can be obtained by manipulating the competition between reaction and phase separation in multi-component polymers. Compared to phase separation induced by thermally activated reactions,41 the length scales of the morphology obtained by photochemical reactions have much larger characteristic length scales and is easily manipulated by varying the light intensity and the cross-linker concentrations. 3.3. Controlling phase separation of polymer blends by using temporal and spatial modulation When a polymer mixture becomes thermodynamically unstable, concentration fluctuations with various wavelengths and lifetimes emerge, and develop with time. For a given condition, some particular modes of these fluctuations become unstable and develop into initial morphology. Without any controls, the range of unstable modes gradually expands and the blend eventually approaches phase equilibrium with random two-phase structure. Since the final morphologies depend upon external conditions, it is expected that a particular range of the broad distribution of these unstable modes could be selected by using external perturbations with well-defined length scales and time scales. Here, we show that this selection is quite possible by using a light intensity distribution with specific length scales and time scales. 3.3.1. Morphology design by irradiation with spatial modulation As described in Sec. 2.2., for reaction-induced phase separation, periodic morphology with the initial length scale ξ0 corresponding to R(q) → 0 emerges when the reaction yield reaches a critical value. This critical length scale can be obtained by linear stability analysis of an appropriate kinetic equation describing the reaction-induced phase separation

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phenomena. From the practical viewpoint, it would be interesting to perform experiments where the length scales for phase separation are reduced to the order of this specific length ξ0 . Since it was established that photochemical reactions can be utilized to induce and control phase separation of polymer blends, irradiation through photo-mask with spacing in the micrometer range was carried out for this purpose. In order to restrict phase separation to micrometer domains, stilbenelabeled polystyrene (PSS) and poly(vinyl methyl ether) (PVME) (20/80) blends were irradiated through photo-masks with spacing in the range Λ = 1 cm (corresponding to the non-restricted case) to 20 µm (the maximal restriction). The blend with the thickness around 30 µm was strongly pressed against the mask to avoid the secondary effects coming from reflection or interference of incident light. It was found that for Λ = 1 cm, the morphology is isotropic, indicating that phase separation is not affected by this scale of the sample. However, as Λ reaches 50 µm, the morphology becomes slightly anisotropic. Eventually, phase separated structures turns into lamellae parallel to the direction of the photomask as the spacing Λ is 20 µm. To confirm that the blend has received UV light with the intensity distribution determined only by the grating of the photomask, we measured this light intensity distribution by using a poly (vinyl methyl ether) (PVME) film doped with a photobleachable dye NBD-chloride. The photobleachable sample was irradiated through the same photomask under the same optical alignment. The fluorescence intensity distribution measured under a laser scanning confocal microscope indicates that the blend under this irradiation condition was not influenced by secondary effects such as reflection or interference of the incident UV light source. Figure 5 shows the superlattice structure of a PSS/PVME (20/80) blend irradiated through a photomask with Λ = 20 µm. There exist two different characteristic length scales in the resulting morphology.42 One is the macro-lamellae with longer period (20 µm) which correspond to the refractive-index distribution of trans- and cis-isomers of stilbene generated in the sample by the modulated intensity distribution of the exciting light. Furthermore, inside the regions of the macro-lamellae illuminated by light, phase separation took place, resulting in micro-lamellae with orientation parallel to the direction of the photomask. The period of these micro-lamellae is ca. 1.8 µm which is in very good agreement with the Bragg spacing calculated from the wavenumber of the strong secondary diffraction spot seen on the right hand side of Fig. 5 (around 3.7 µm−1 ). The reason responsible for the emergence of these micro-lamellae would be originated from the anisotropy resulting

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6.3x10-1(µm-1)

Fig. 5. The superlattice structure resulting from phase separation of a PSS/PVME (20/80) blend induced by spatial modulation: morphology observed by phase-contrast laser microscope (top). The corresponding 2-D light scattering pattern obtained with a He-Ne laser (632.8 nm, middle); one-dimensional scattering profile converted from the 2-D data (bottome).

from the difference in the spatial restriction imposed on the unstable modes along the X- (perpendicular) and the Y-directions (parallel) of the grating. Most recently, anisotropic morphology was also found in experiments using computer-assisted irradiation (CAI) method where phase separation of polymer mixtures was induced by ON/OFF light stripes with different spacing.43 This morphological anisotropy greatly varies with the mobility of the reacting mixture. It was found that for mixtures in the liquid state, this anisotropy is driven by the osmotic pressure generated between the irradiated and unirradiated regions in the mixtures, whereas for mixtures in the bulk state, this morphological anisotropy originates from the elastic strain generated from the photo-cross-linked (hard) and non-cross-linked (soft) regions of the mixture. 3.3.2. Morphology design by using irradiation with temporal modulation As complementary to the case of spatial modulation, we show, in this section, that light intensity with temporal modulation can also be used to select

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unstable modes in reacting polymer blends. For mixtures in the bulk state, inhomogeneous kinetics of the cross-linking reactions generates a transient elastic strain field inside the reacting mixture. This reaction-induced strain can be relaxed under irradiation with a temporal modulation. As a consequence, the morphological length scale distribution is sharpening, resulting in an enhancement of morphological regularity.44 Phase separation of anthracene-labeled polystyrene (PSA)/PVME (15/85) blends was induced by irradiation with UV light modulated under various frequencies ranging from 0 Hz (corresponding to continuous irradiation) to 100 Hz. As described above, compared to thermally induced phase separation, the distribution of the morphological length scales Γ is much narrower when photochemical reactions are used to induce phase separation. Here, by irradiation with temporal modulation, it was found that Γ is even more sharpened, revealing the improvement of the wavelength selection process in reacting polymer blends by temporal modulation. Shown in Fig. 6 show the characteristic length scales distribution Γ and the contrast Cq max (the morphological regularity) obtained by circularly averaging the 2D-FFT (2-D fast Fourier transform) power spectra of the phase-contrast optical micrographs obtained for PSA/PVME (15/85) blends irradiated under different modulation frequencies. As a criterion for comparison, the cross-link density in all cases was kept constant by adjusting irradiation time under different frequencies. As seen from Fig. 6(a), the distribution Γ quickly decreases with increasing modulation frequency up to 5 Hz. Above this specific range of frequencies, Γ gradually increases and eventually approaches a constant. Furthermore, upon changing the ON/OFF time ratio of the irradiation process, the width Γ of the morphological length scales distribution decreases with increasing the OFF-time per one ON/OFF cycle of irradiation. This peculiar behavior would be due to the relaxation of the elastic stress accumulated in the blend under successive cross-link of the PSA component. Recently, this elastic stress and its relaxation process have been experimentally measured for the same cross-linked blends using Mach-Zehnder interferometry.45 Similar effects of elastic stress on the length scales distribution in the early state of phase separation have been reported for solutions of polymers with high molecular weights.46 The discussion on the effects of elastic stress on the morphological length scale distribution of reacting blends is given in terms of dispersion relation in the supplementary information of Ref. 44. Therefore, it can be concluded that the coupling between cross-link-induced elastic stress and concentration fluctuations in polymer blends leads to a decrease in regularity of phase

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log[C(qmax)/a. u. ]

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Fig. 6. (a) The characteristic length scales of distribution of a PSA/PVME (15/85) blend irradiated under various modulation frequencies at 95◦ C; (b) Dependence of the maximal Fourier intensity on the modulation frequency obtained under the conditions of Fig. 6(a).

separated structures. In contrary, the morphological regularity increases when this elastic stress is allowed to relax. These experimental results suggest a novel way of designing polymer blends with highly regular morphology. It should be noted that recent experiments show that for phase separation of mixtures with initially low viscosity such as solution precursors of IPNs (interpenetrating polymer networks) containing polystyrene dissolved in methyl methacrylate (MMA) monomer, modulation using a finite irradiation frequency contrarily decreases the morphological regularity.47 From the in situ observation under optical microscope, it was found that this disordering arises from the flow which takes place during the OFF time of irradiation. This situation greatly differs from the phase separation process induced by periodic irradiation in the bulk state where flow is prevented by the high viscosity of the reacting medium and only elastic deformation is a main factor for long-range interactions. 3.3.3. Spatially graded morphology designed by using strong light intensity As described in the previous sections, the feature of reaction-induced phase separation phenomena is the existence of a critical reaction yield beyond which the mixture becomes thermodynamically unstable and undergoes phase separation. This dynamic process can be controlled by changing the quench depth of the phase separation using light with high intensity.

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Fig. 7. Spatially graded morphology of poly (cross-styrene)-inter-poly(cross-methyl methacrylate) IPNs obtained by irradiation with 365 nm over 60 min at 30 ◦ C.

As a consequence, the characteristic length scales, their distribution as well as the rate of phase separation can be manipulated by coupling a spatio-temporal distribution of light intensity to the critical phenomena of polymer blends. The simplest example is taking advantages of a spatial gradient of the light intensity generated by the Lambert-Beer law along the propagating direction inside the photoreactive samples. Illustrated in Fig. 7 is the morphology with a gradient of characteristic length scales obtained for a poly (cross-styrene)-inter-poly(cross-methyl methacrylate) IPNs. This structure was obtained by irradiation with the light intensity 0.06 mW/cm2 and in situ observed under a laser scanning confocal microscope (LSCM).48 It should be noted that co-continuous morphology with uniform length scales was obtained with the same mixture upon irradiation with the light intensity below 0.03 mW/cm2 . The contrast of the 3D image in Fig. 7 was generated by using fluorescein covalently labeled on the PS component. For this particular sample, both the polymer components were independently cross-linked by taking advantages of photo-dimerization of the anthracene moieties labeled on the PS component and photoaddition of ethylene glycol dimethacrylate as a cross-linker to the photopolymerization of MMA monomer. Under these experimental conditions, the characteristic length scale of these co-continuous morphologies is smaller on the side

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of incident light and is gradually larger toward the backside of the samples, reflecting the effects of the intensity gradient on the quench depth of the blend. These results suggest that morphology with an arbitrary distribution of characteristic length scales can be designed and controlled if an appropriate distribution of light intensity is used to induce phase separation of photo-reactive polymer blends. Careful investigation of the corresponding photo-polymerization and photo-cross-link kinetics suggests that the heat associated with polymerization of MMA in the mixture also additionally contributes to the phase separation kinetics.49 Compared to a number of experimental methods that have been utilized so far to produce spatially graded structures by polymerizing a guest polymer distributed by sorption in a host polymer matrix50 – ,53 the method using light described here provides a simpler and easier way to control the gradient structure of polymers. 3.3.4. Morphology with an arbitrary distribution of characteristic length scales designed by photochemical reactions: The computer-assisted irradiation (CAI) method As described in the previous sections, the characteristic length scales emerging in reaction-induced phase separation are determined by the rate of quenching the blend from the one-phase into the two-phase region of the blend. This quenching rate strongly depends on the incident light intensity. In order to generalize the method of generation and control of polymer morphology using UV light irradiation described in the previous sections, we have developed the new techniques, the so-called computer-assisted irradiation (CAI) method.54 The block diagram of this CAI instrument is shown in Fig. 8 where a light pattern I(ξ, τ ) with a characteristic length scale ξ and a characteristic time scale τ is first generated on a computer and is then transferred to a digital projector via which this computer-designed light pattern is projected onto a photo-reactive blend placed under a brightfield optical microscope. The morphology emerging from this irradiation method is in situ monitored by a CCD camera that is connected to a second computer for data storage and analysis. By this CAI method, phase separation of a photoreactive polymer blend can be spatially and temporally induced by visible light at a given temperature. For this purpose, polymer blends that can undergo phase separation under irradiation with visible light were chemically designed and utilized for this CAI experiment.54 Namely, polystyrenes labeled with a trans-cinnamic acid derivative (PSC) were syn-

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Fig. 8. The block diagram

The block diagram of the computer-assisted irradiation (CAI) apparatus.

thesized and blended with poly(vinyl methyl ether) (PVME). The resulting PSC/PVME mixtures can undergo phase separation upon irradiation with visible light. In the presence of 5-nitroacenaphthene, the photosensitizer for trans-cinnamic acid group, the PSC component undergoes photo-crosslinked upon irradiation with 405 nm, providing a chemical system to study reaction-induced phase separation driven by visible light. An example is demonstrated in Fig. 9 where a PSC/PVME (20/80) blend was irradiated with a concentric light pattern produced by 405 nm visible light. As shown in Fig. 9 obtained by low magnification, the blend exhibits morphology with bright and dark concentric patterns in response to the irradiation intensity for 60 min of irradiation. These dark and bright regions in the irradiated blend correspond respectively to the regions illuminated by strong and weak light intensity. As a result, the transmission of the blend changes in response to the characteristic length scales of the morphology in these regions. As obviously seen in the insets where the morphology emerging in the regions irradiated with different irradiation intensities is illustrated, the characteristic length scales vary in response to the light intensity: stronger light, larger length scales. This example reveals a strong potential of designing and controlling the morphology as well as the physical properties of polymer blends by using this CAI method.

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B

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Fig. 9. Phase separation of a PSC/PVME blend irradiated with a concentric visible light (405 nm) pattern observed under an optical microscope. Shown in each inset is the corresponding 2D-FFT power spectra of the morphology observed at different positions in the irradiated blend.

3.3.5. Reversible phase separation of polymer blends driven by two UV wavelengths: Effects of reaction-induced deformation on morphology As described in Sec. 3.3.2, modulation of light can be used as a tool to select a particular range of unstable modes, i.e. the wavelengths range of the unstable concentration fluctuations, in the reacting polymer blends. Furthermore, it was also found that the wavelength distribution of the modulated structures originated from this mode-selection process is strongly affected by the elastic strain accompanying the cross-linking reaction as revealed by both Mach-Zehnder interferometry (MZI)45 and phase-contrast optical microscopy.44 In order to actively control this cross-link induced strain and also to elucidate the effects of its relaxation on the phase separation kinetics, reversible photo-cross-link of anthracene-labeled polystyrene (PSA) component in a PSA/PVME (20/80) blend was performed by using two UV wavelengths 365 and 297 nm. The former was used to promote the formation of PSA networks via anthracene photodimerization, consequently

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inducing phase separation, whereas the shorter wavelength was used to photodissociate the anthracene photodimers, therefore homogenizing the phase-separated mixture.55 Shown in Fig. 10 is the change in the light scattering profiles with irradiation time by using these two wavelengths.

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Obviously, in the cross-linking process using 365 nm UV light, a PSA/ PVME (20/80) mixture becomes unstable and undergoes phase separation via the spinodal decomposition as revealed by the scattering peak that gradually moves to the side of larger wavenumber. Apparently this behavior seems to be in conflict with the conventional scattering data obtained for phase separating mixtures where the scattering peak is expected to move to the side of small wavenumbers corresponding to an increase in the characteristic length scales of the morphology. This time-evolution of phase separation has been widely observed in a large number of studies including the reverse temperature quench from the spinodal decomposition into the miscible region of polymer blends.? The mechanism responsible for this particular behavior was elucidated by in situ measurements of the elastic strain and its relaxation process taking place during irradiation cross-link by using Mach-Zehnder interferometry.45 These relaxation data are then compared to the cross-link kinetics observed under the same experimental conditions. It was found that there exists a strong correlation between these two processes, cross-linking and deformation.55 On the other hand, irradiation of a phase-separated blend with 297 nm reveals the reversibility of the phase separation process. These experiments suggest a possibility of processing polymer mixtures as thermoplastics and use them as a thermoset by taking advantage of reversible phase separation driven by two UV wavelengths at 365 and 297 nm. These results also indicate the potential for photorecycling of polymer blends by alternatively using these two UV wavelengths. 4. Concluding Remarks So far, we have demonstrated various examples of using photochemical reactions as a mode-selector to control and design morphology of multicomponent polymers. As pointed out recently by Antonietti and Ozin,57 chemical reactions alone cannot produce structure with long-range order. On the other hand, materials design requires generation and control of shape and form. To fulfill this demand, it is necessary to couple additional mechanism to chemical reactions. One way to satisfy this demand would be either introducing non-linearity into reaction kinetics as already seen in Belousov-Zhabotinsky reaction8 or coupling photochemical reactions to critical phenomena as demonstrated here. For controlling phase separation of polymer blends using chemical reactions, the conclusions may be summarized as follows:

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(1) Compared to thermally activated reactions, photochemical reactions provide an efficient selection tool for phase separation because by using light, their initiation and termination can be performed independently from thermodynamic variables such as temperature or pressure. (2) From the viewpoint of chemistry and materials science, using light to drive phase separation would be limited by the Lambert-Beer law for thick samples. However, for samples with adequate thickness or low content of photo-reactive groups, unique features of photochemical reactions such as reversible cross-link for reversibly driving phase separation,55 polarizationselective cross-linking reaction for directional phase separation,58 selective reactions between specific sites or between particular polymer components and so on can be utilized for designing morphology ranging from nano- to micrometer scales. (3) From the physical viewpoint, reaction-induced phase separation is closely related to the phenomena of competing interactions where two or more antagonistic phenomena compete with each other. By manipulating these competitions by using light, a wide variety of ordered structures can be produced. Dynamically, spatio-temporal response of polymeric systems under periodic and/or chaotic forcing by light can be induced and examined. Because of the close relation with practical problems such as polymer processing, understanding these fundamental processes could provide a great help to design highly functional polymer materials in the near future. Acknowledgments This work is financially supported by the Ministry of Education (MONKASHO) Japan through Grant-in-Aid for Scientific Research on Priority-Research-Area Dynamic Control of Strongly Correlated Soft Materials (2001–2003), Molecular Nano Dynamics (2004–2006) and Soft Matter Physics (2007–2008). We also greatly appreciate the collaboration with Dr. Tomohisa Norisuye and all the graduate students in our Polymer Molecular Engineering Laboratory at the Department of Polymer Science and Engineering, Kyoto Institute of Technology, Kyoto, Japan. This work would not be possible without their great efforts and elaboration. References 1. For example, see Polymer Blends, D.R. Paul and C.B. Bucknall, eds., John Wiley, New York (1999), Vols. 1 & 2.

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2. For example, see Structure and Properties Relationship in Multiphase Polymeric Materials, T. Araki, Q. Tran-Cong and M. Shibayama eds., Marcel Dekker, New York (1998). 3. N. Hadjichristidis, S. Pispas and G.A. Floudas, Block Copolymers: Synthetic Strategies, Physical Properties, and Applications, Wiley-Interscience (2002). 4. For example, see A. Echte “Rubber-Toughened Styrene Polymers — A Review” in Rubber Toughened Plastics, C. Keith Riew ed., Advances in Chemistry Ser. No. 222, American Chemical Soceity, Washington DC, 1989, pp. 15–64. 5. S. Moriya, K. Adachi and T. Kotaka, Langmuir 2, 155 & 161 (1986). 6. A.I. Nakatani and C.C. Han, Shear Dependence of the Equilibrium and Kinetic Behavior of Multicomponent Systems in Ref. 2., Chapter 7, pp. 233–267. 7. S. Sakurai, Polymer (2008), in press. 8. For example, see Chemical Waves and Patterns, R. Kapral and K. Showalter, eds. (Kluwer Academic Publishers, Dordretch, 1995). 9. T. Ohta, A. Ito and A. Tetsuka, Phys. Rev. A 42, 3225 (1990). 10. C. Roland and R.C. Desai, Phys. Rev. B 42, 6658 (1990). 11. M. Seul and D. Andelman, Science 267, 476 (1995). 12. Y. Matsushita, Block and Graft Copolymers, in Ref. 2., Chapter 4, pp. 121– 154; For recent developments, see Y. Matsushita, Macromolecules 40, 772 (2007) and references cited therein. 13. See, for example, J.S. Kirkaldy and D.J. Young, Diffusion in the Condensed State (The Institute of Metals, London, 1987). 14. J.W. Cahn, Acta Met. 9, 795 (1961); J.W. Cahn, J. Chem. Phys. 42, 93 (1965). 15. J.W. Cahn and J.E. Hilliard, J. Chem. Phys. 28, 258 (1958); ibid. 31, 688 (1959). 16. P.J. Flory, Principles of Polymer Chemistry (Cornell University Press, Ithaca, NY, 1953), Chapters 12 and 13. 17. P.-G. de Gennes, J. Chem. Phys. 72, 4756 (1980). 18. G.H. Fredrickson, Theoretical Methods for Polymer Surfaces and Interfaces, Chapter 1, in Physics of Polymer Surfaces and Interfaces, ed. Isaac C. Sanchez (Butterworth-Heinemann, 1992), pp. 1–28. 19. J.D. Gunton, M. San Miguel and P.S. Sahni, Phase Transitions 8, 267 (1983). 20. T. Hashimoto, Phase Transitions 12, 47 (1988). 21. K. Binder, Adv. Polym. Sci. 112, 181 (1994). 22. H. Furukawa, Dynamics of Phase Separation and Its Application to Polymer Mixtures in Ref. 2, Chapter 2, pp. 35–66. 23. B.A. Huberman, J. Chem. Phys. 65, 2013 (1976). 24. S.C. Glotzer, E.A. DiMarzio and M. Muthukumar, Phys. Rev. Lett. 74, 2034 (1995). 25. A.M. Turing, Phil. Trans. Roy. Soc. B237, 37 (1952). 26. E.Y. Vedmedenko, Competing Interactions and Pattern Formation in Nanoworld, Wiley-VCH, Weinheim (2007). 27. M. Seul and D. Andelman, Science 267, 476 (1995).

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28. Q. Tran-Cong, A. Harada, K. Kataoka, T. Ohta and O. Urakawa, Phys. Rev. E 55, R6340 (1997). 29. R. Lefever, D. Carati and N. Hassani, Phys. Rev. Lett. 75, 1674 (1995); D. Carati and R. Lefever, Phys. Rev. E 56, 3127 (1997). 30. T. Ohta, O. Urakawa and Q. Tran-Cong, Macromolecules 31, 6845 (1998); Q. Tran-Cong, J. Kawai and K. Endoh, Chaos 9, 298 (1999) and Chapter 5 in Ref. 2., pp. 155–194. 31. For review, see: A.S. Mikhailov, M. Hildebrand and G. Ertl, Nonequilibrium Nanostructures in Condensed Reactive Systems in: Coherent Structures in Classical Systems, D. Reguera, L.L. Bonilla and J.M. Rub´ı, eds., Lecture Notes in Physics, Vol. 567, Springer, New York (2001), pp. 252–269. For the discussion on the relation between the spatial length scales of the patterns induced by chemical reactions and by competing interactions, see M. Hildebrand, A.S. Mikhailov and G. Ertl, Phys. Rev. E 58, R5483 (1998). 32. M. Doi and A. Onuki, J. Phys. II (France) 2, 1631 (1992). 33. H. Tanaka, J. Phys.: Condens. Matter 12, R207 (2000). 34. A. Onuki and T. Taniguchi, J. Chem. Phys. 106, 5761 (1997). 35. H. Nakazawa, S. Fujinami, M. Motoyama, T. Ohta, T. Araki, H. Tanaka, T. Fujisawa, H. Nakada, M. Hayashi and M. Aizawa, Comp. Theor. Polym. Sci. 11, 445 (2001). 36. K. Kataoka, A. Harada, T. Tamai and Q. Tran-Cong, J. Polym. Sci. Polym. Phys. 36, 455 (1997). 37. O. Urakawa, O. Yano, Q. Tran-Cong, A.I. Nakatani and C.C. Han, Macromolecules 31, 7962 (1998). 38. A. Harada and Q. Tran-Cong, Macromolecules 30, 1643 (1997). 39. For example, see IPNs Around the World - Science and Engineering, S.C. Kim and L.H. Sperling, eds., John Wiley & Sons, New York (1997). 40. H. Nakanishi, M. Satoh, T. Norisuye and Q. Tran-Cong-Miyata, Macromolecules 37, 8495 (2007). 41. M. Weber, W. Heckmann and A. Goeldel, Macromol. Symp. 233, 1 (2006). 42. Q. Tran-Cong-Miyata, S. Nishigami, S. Yoshida, T. Ito, K. Ejiri and T. Norisuye, in Nonlinear Dynamics in Polymeric Systems, J. A. Pojman and Q. Tran-Cong-Miyata, eds., Chapter 22, ACS Symposium Series No. 869, Am. Chem. Soc., Washington DC, (2004), pp. 276–290. 43. A. Masunaga, S. Ishino, H. Nakanishi and Q. Tran-Cong-Miyata, Kobunshi Ronbunshu 64, 249 (2007). Master Dissertation, Department of Polymer Science and Engineering, Kyoto Institute of Technology, Kyoto, March 2007. 44. Q. Tran-Cong-Miyata, S. Nishigami, T. Ito, S. Komatsu and T. Norisuye, Nature Materials 3, 448 (2004). 45. K. Inoue, S. Komatsu, X.-A. Trinh, T. Norisuye and Q. Tran-Cong-Miyata, J. Polym. Sci. Polym. Phys. 43, 2898 (2005). 46. N. Toyoda, M. Takenaka, S. Saito and T. Hashimoto, Polymer 42, 9193 (2001). 47. K. Noma, M. Chiyama, T. Norisuye and Q. Tran-Cong-Miyata, to be published.

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48. H. Nakanishi, N. Namikawa, T. Norisuye and Q. Tran-Cong-Miyata, Soft Matter 2, 149 (2006). 49. H. Nakanishi, M. Satoh, T. Norisuye, Q. Tran-Cong-Miyata Macromolecules 39, 9456 (2006). 50. C.F. Jasso, J.J. Martinez, E. Mendizabal and O. Laguna, J. Appl. Polym. Sci. 58, 2207 (1995). 51. G. Akovali, K. Biliyar and M. Shen, J. Appl. Polym. Sci. 20, 2419 (1976). 52. G. Akovali, J. Appl. Polym. Sci. 73, 1721 (1999). 53. Y. Agari, M. Shimada, A. Ueda and S. Nagai, Macromol. Chem. Phys. 197, 2017 (1996). 54. S. Ishino, H. Nakanishi, T. Norisuye, Y. Awatsuji and Q. Tran-Cong-Miyata, Macromol. Rapid Commun. 27, 758 (2006). 55. X.-A. Trinh, J. Fukuda, Y. Adachi, H. Nakanishi, T. Norisuye and Q. TranCong-Miyata, Macromolecules 40, 5566 (2007). 56. Z.A. Akcasu, I. Bahar, B. Erman, Y. Feng and C.C. Han, J. Chem. Phys. 97, 5782 (1992). 57. M. Antonietti and G. A. Ozin, Chem. Eur. J. 10, 28 (2004). 58. K. Kataoka, O. Urakawa, H. Nishioka and Q. Tran-Cong, Macromolecules 31, 8809 (1998).

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Chapter 7 Thermodynamics and the Phase Diagrams of Block Copolymers in Electric Fields M. Schick Department of Physics Box 351560 University of Washington, Seattle WA 98195-1560, USA Basic electrostatics and some less familiar thermodynamics is reviewed, and useful Claussius-Clapeyron-like equations are derived. They are applied to predict the form of phase diagrams of two systems. The first is a bulk system of body-centered-cubic phase which undergoes a phase transition either to a hexagonal or disordered phase on the application of an electric field. The second is a surface film of cylindrical phase which can be oriented perpendicular to the substrate by the application of a field.

1. Introduction The genesis of this chapter is my collaboration with David Andelman who introduced me to interesting problems in which electric fields were applied to block copolymers, a system with which I was familiar. The first problem considered was that of a bulk, block copolymer system of body-centeredcubic (bcc) phase in a field. Because of the accumulation of polarization charge on the spheres, the free energy of the system increases in an electric field with respect to that of neighboring phases, like the hexagonal phase of cylinders. Eventually a phase transition occurs between them. It took quite a while, and an intensive calculation by Chin-Yet Lin, before the phase diagram became clear to me. After I understood it, I realized that by the use of some elementary thermodynamics I could have, and should have, understand the nature of the phase diagram before I began a difficult calculation. When encountering a second problem concerning a polymer film, the usual form of such systems, I turned again to thermodynamics. I had been used to surface thermodynamics, but no text had prepared me for the 197

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“surface excess displacement field”, a concept that threw me at first until I realized that it was simply another excess surface density like those I had encountered previously and reviewed in a series of Les Houches lectures.1 Again, the thermodynamics provided the general form of the phase diagram of the system in question, a cylindrical phase in this case. A fortuitous element in these studies is that I have had the pleasure of teaching a junior-level course in Electrostatics and Electrodynamics at the University of Washington for the last couple of years. This has provided me the opportunity to understand the subtleties of polarizable systems which had eluded me when I had first encountered them. It is in the hope that insights from electrostatics and thermodynamics will prove as useful, and beautiful, to someone else as they are to me that I agreed to write this chapter. 2. Review of Basic Electrostatics in Polarizable Materials The two equations which determine the electric field E(r) when all charges are at rest, that is, the regime of electrostatics, are Gauss’ law ρc (r) , 0

(1)

∇ × E(r) = 0,

(2)

∇·E= which is always valid, and

which is only valid in the regime of electrostatics. Together, the two equations are simply a statement of Coulomb’s law for they have the solution Z 0 0 0 (r − r ) 1 dr ρc (r ) , (3) E(r) = 4π0 |r − r0 |3 which, again, is only valid in electrostatics. The great difficulty with either Eq. (1) or Eq. (3) is that the local charge density, ρc , in a polarizable medium depends upon the electric field itself so that the equations are self-consistent ones. The standard way to proceed is to observe that the largest contribution to the electric field, after any free charges which may be around, is from the induced dipoles of the medium. Let the local dipole moment per unit volume be P(r). As one knows the electric field produced by an electric dipole, one readily shows that the electric field produced by P(r) is the same as if there were a local bulk charge density ρb (r) ≡ −∇ · P(r)

(4)

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and a local surface charge density ˆ, σb (r) ≡ P(r) · n

(5)

ˆ is a unit normal to the surface. These charges are said to be where n “bound”, as opposed to “free”, hence the subscript b. Note that if the dipole density, P, were known then the bound charge density, ρb could be obtained from Eq. (4). The reverse is not true, however, for ρb only gives us the divergence of P, and a vector field is only completely determined by its divergence and its curl. We do not know the latter. One next separates the total charge density, which appears on the right hand side of Eq. (1), into a density ρf of free charge, which one controls, and bound charge, which one does not: ρc = ρ f + ρ b = ρ f − ∇ · P

(6)

Substituting this into Eq. (1) and rearranging we can write Gauss’ law in the form ∇ · D(r) = ρf (r),

(7)

where we have defined the displacement field D ≡ 0 E + P.

(8)

The apparent advantage of this manoeuvre is that the divergence of the displacement is given only by the free charge. We have buried the bound charge by our definitions. But this is just so much hand waving and buys us nothing for at this point we have two equations; the new version of Gauss’ law, Eq. (7) above and the defining equation of electrostatics, ∇ × E = 0. But neither of the two fields E and D are well defined because we know only the divergence of one and only the curl of the other. This impasse should not be a surprise. We said that the problem was difficult because the charge density, the source of the electric field, depends upon the electric field itself, and we have not hazarded a guess as to the nature of this relationship. Doing so relates the dipole density P to the electric field E. The choice for most materials, including the ones of interest here, is to make the reasonable assumption that the relation is a linear one; that is P = 0 χe E,

(9)

where the dimensionless number χe is the electric susceptibility. Upon substitution of this into the defining equation for the displacement, Eq. (8),

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one obtains D = 0 (1 + χe )E, = 0 κE,

(10)

where κ ≡ (1 + χe ) is the dielectric constant of the material. The fields are now well defined. We satisfy the defining equation of electrostatics ∇ × E = 0 by introducing the electric potential, V , E(r) = −∇V (r).

(11)

This leaves only Gauss’s law to be satisfied. Substitution of D = κE = −κ∇V into Eq. (7) yields ∇2 V +

∇κ ρf · (∇V ) = − , κ κ0

(12)

where I have allowed for the fact that the dielectric constant can vary spatially. Just how the dielectric constant varies spatially is not obvious. Suppose that there are two distinct monomers, A and B, joined in a diblock copolymer with polymerization index N of which a fraction, fA , is A monomer. Let the volume of all monomers be v. Then the local volume fraction of A monomers, φA (r), has the average value φ¯A = fA , and the local volume fraction of B monomers, φB (r), has the average value φ¯B = 1 − fA . If we denote the deviations of the local volume fractions from their average values by δφA (r) and δφB (r) = −δφA (r) respectively, with −fA ≤ δφA ≤ 1 − fA , then one can write quite generally the local dielectric constant as κ(r) = κA fA + κB (1 − fA ) + (κA − κB )g(δφA ),

(13)

where κA and κB are the dielectric constants of the pure A and B systems respectively. The function g(δφA ) is such that the dielectric constant in Eq. (13) above is greater than unity, and g(−fA ) = −fA , g(1−fA ) = 1−fA , which guarantees that κ = κA or κB when the system is pure. Other than these small restrictions, little can be said about the function g(δφA ). One obvious choice is simply g(δφA ) = δφA ,

(14)

κ(r) = κA φA (r) + κB φB (r).

(15)

so that Eq. (13) becomes

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It must be emphasized that this is simply a choice, and is not derived from anything basic. One might employ any of an infinite number of other possibilities, such as g(δφA ) = −fA + sin2

π(δφA + fA ) , 2

(16)

in which the effect of adding a different monomer would only begin to affect the dielectric constant in second order in the local volume fraction. 3. Basic Thermodynamics Let us recall the basics of thermodynamics. An excellent book on the subject is that of Callen.2 There are two equations for the energy, U , of a system consisting of Ni molecules of type i in a volume Ω. The first is the Euler form X µi Ni − pΩ, (17) U = TS + i

where S is the entropy, T the temperature, p the pressure, and µi the chemical potential of the i’th component. The content of this equation is that the energy is extensive; that is, if one doubles the volume, the number of molecules of each component, and the entropy of the system, then one has also doubled the system energy. The second is the statement of the first law, X dU = T dS + µi dNi − pdΩ. (18) i

By differentiating the Euler form directly and comparing the result with the first law, one obtains the Gibbs-Duhem relation X SdT + Ni dµi − Ωdp = 0, (19) i

or, defining the entropy per unit volume, s ≡ S/Ω, and the number densities ρi ≡ Ni /Ω, X dp = sdT + ρi dµi . (20) i

It is convenient to introduce the energy per unit volume, X u ≡ U/Ω = T s + µi ρi − p, i

(21)

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whose differential is found immediately from the first law, Eq. (18), X dU = T dS + µi dNi − pdΩ, i

= (T s +

X

µi ρi − p)dΩ + Ω(T ds +

X

µi dρi ),

i

i

= udΩ + Ωdu,

(22)

to be du = T ds +

X

µi dρi .

(23)

i

Because the temperature is more easily controlled than the entropy, one introduces the Helmholtz free energy via the Legendre transformation F ≡ U − T S, dF = dU − T dS − SdT, X µi dNi − pdΩ = −SdT +

(24)

i

or the Helmholtz free energy per unit volume f ≡ u − T s, df = du − T ds − sdT, X = −sdT + µi dρi ,

(25)

i

where Eqs. (18) and (23) have been used. When the system, which consists of a polarizable material, is in the presence of an electric field, the differential contribution to the energy per unit volume is shown in any standard text, such as Griffiths,3 to be E · dD so that Eq. (23) becomes X du(r) = T ds + µi dρi + E(r) · dD. (26) i

For linear dielectrics, the electrostatic contribution to the energy per unit volume is κ0 E 2 (r) uelec (r) = . (27) 2 One now observes that while the above is useful if one controls the free charge, changes of which are related to changes in the displacement, dD, via Eq. (7), more often one controls the voltage. Changes in the voltage are directly related to changes in the electric field, dE. Hence it is convenient to

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make a Legendre transformation both with respect to the entropy, as before, but also with respect to the displacement, to produce a thermodynamic potential which is a function of temperature and electric field, f (r) ≡ u − T s − E(r) · D(r), X µi dρi − D(r) · dE. df (r) = −sdT +

(28) (29)

i

Essentially one is now including the battery in the system, and the last term is the decrease in internal energy per unit volume of the battery due to the work it must do to keep the voltage on the plates constant. The net contribution to the free energy per unit volume of the whole system, which includes the increase in energy of the material and decrease in energy of the battery, is κ0 E 2 (r) , 2 D(r) · E(r) =− . (30) 2 It should be noted that in the case in which the electric and displacement fields vary over the sample, as they do in the cases of interest here, the free energy density of Eq. (28) must be averaged over the sample. Similarly, the change in free energy that one wants is the change in free energy, averaged over the sample, when the voltage on the capacitor plates is changed from V to V + dV , or equivalently, when the electric field between the plates is changed from E0 to E0 + dE0 where Z 1 dEz (r) dV =< dE >= dr. (31) dE0 ≡ ` A dz felec (r) = −

where the brackets denote a spatial average and ` is the distance between the plates of area A which are perpendicular to the z axis. The amount by which the spatially averaged free energy density changes due to the change in field, dE0 is −D0 dE0 with D0 ≡

< D dE > , < dE >

(32)

X

(33)

so that df = −sdT +

µi dρi − D0 dE0 ,

i

where df is the change in the spatially averaged free energy density. I have assumed that the system is isotropic so that the averaged displacement is in

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the z direction as is E0 . For a system of a single species of block copolymer the number of which is fixed, as will be considered below, this becomes df = −sdT − D0 dE0 .

(34)

The electrostatic contribution to the spatially averaged free energy density from the linear dielectric can now be written felec = −

D0 E 0 . 2

(35)

Now let us consider the coexistence of two phases. They must be at the same temperature T and are between the same capacitor plates and so are subject to the same E0 . Hence the Helmholtz free energy per unit volume is the same in each phase. Let one phase be denoted a and the other b. Consider one point, (T, E0 ) on the coexistence curve. The free energies of the two phases are equal, fa (T, E0 ) = fb (T, E0 ). If we move along the coexistence curve to a point (T + dT, E0 + dE0 ), the free energies of the two phases are again equal. Thus the changes in free energies dfa = −sa dT − D0,a dE0 , dfb = −sb dT − D0,b dE0 , are equal. If we subtract them, we obtain the Claussius-Clapeyron equation for the slope of the phase boundary ∆s dE0 =− , dT ∆D0

(36)

where ∆s ≡ sa − sb , and ∆D0 ≡ D0,a − D0,b . This equation will play an important role in our analysis. It is convenient to recast it slightly. In the polymer system, temperatures usually enter via the dimensionless Flory parameter χN = c/T,

(37)

where c is a constant. Further we introduce dimensionless electric and displacement fields 1/2

ˆ0 ≡ E 0 E

0 v p kB T

ˆ 0 ≡ D0 D

vp 0 k B T

,

1/2

(38) ,

(39)

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where vp = N v is the volume per polymer chain. In terms of these dimensionless quantities, the Claussius-Clapeyron equation becomes ˆ0 ˆ0 vp ∆(s/kB ) dE E = . + ˆ0 d(χN ) χN ∆D 2χN

(40)

4. Electric Field Induced Bulk Phase Transition I want to consider now an interesting situation in which the application of an external field brings about a phase transition. In particular, I will consider a block copolymer system in which the architecture is such that at low temperatures the system is in the body-centered-cubic state. Thus its free energy is lower, inter alia, than that of the disordered phase or the hexagonal, cylindrical phase. However, as an electric field is turned on, its free energy increases relative to that of these two phases and, at some value of the electric field, a first-order transition occurs. This is less obvious than one might expect, so let me go through the argument. The contribution to the free energy from the electric field is that of Eq. (35) fel = −D0 E0 /2. As I said above, E0 is fixed by the voltage, so we have to evaluate D0 in the various phases. This quantity was defined above and is repeated here < D(r)δE(r) > , < δE(r) > < D(r)δE(r) > = < D(r) > + − < D(r) > , < δE(r) >

D0 =

(41) (42)

where δE(r) is a small electric field whose spatial average is a small change in voltage divided by the distance between plates. In the disordered phase, this evaluation is simple because there are no correlations. Thus the square bracket above vanishes and D0,dis = < D(r) >= 0 < κ(r)E(r) >, = 0 < κ(r) >< E(r) >, = 0 κ0 E 0 ,

(43)

where κ0 is the spatial average of the dielectric constant. In an hexagonal phase oriented with the normals to the cylinder axes perpendicular to the field, the displacement D0 takes the same value. This follows from the fact that the electric field must be uniform inside and outside the cylinders, and in fact, takes the same value E0 . If one thinks of a sharp boundary between inside and outside the cylinders, then the basic equation ∇×E = 0 ensures that the components of the fields on either side of this boundary

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are the same, in accord with the above. As the field is constant, there is no correlation between the displacement and the electric field or the dielectric constant and the field, so one finds that D0,cyl = D0,dis = 0 κ0 E0 .

(44)

The above argument also applies to a lamellar phase in which the lamellae are aligned so that the normals to the planes are perpendicular to the field. For an arbitrary arrangement, or for the bcc arrangement in particular, the average value of the displacement D0 differs from the above value as the correlations, the square bracket in Eq. (42), are non-zero. That D0 should, in general be smaller, than its value in the disordered, hexagonal, and lamellar phases can be understood as follows. In asking for an average of the displacement, we are asking for a average of the dielectric constant. In a conductor, all mobile charges move to the surface of the conductor. There is maximum separation of charge, and the dielectric constant is infinite. If the media were not polarizable at all, there would be no separation of charge so that the electric susceptibility would vanish and the dielectric constant would be unity. Polarizable materials are between these two extremes. In the lamellar and hexagonal phases, the separation of the polarization charge is as large as it can be, appearing on the surfaces of the dielectric facing the capacitor plates, and the dielectric constant is κ0 > 1. In the bcc phase, much of the polarization charge is confined to the surfaces of the spheres so that the separation of charge is reduced. Hence one expects that the dielectric constant is less than that of the lamellar and hexagonal phases. The conclusion of this reasoning is easily verified in perturbation theory.4 A simple example of the reduction brought about when the polarization charge can not separate maximally is provided by the case of a lamellar phase where there equal amounts of monomers A and B.5 As noted above, if the lamellae are oriented so that the normals to the planes are perpendicular to the field, then the electric field is the same in all layers. There are no correlations so that D0 =< D(r) > and takes the maximum value D0,parallel = 0

κA + κ B E0 , 2

(45)

where I have assumed a strong segregation limit for simplicity. If the lamellae are oriented with their normals parallel to the field, then, it follows from Gauss’ law in the absence of free charge that the displacement is the same in each layer, so again there are no spatial correlations and D0 =< D(r) >.

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The electric fields do differ, of course, EA = D/κA , EB = D/κB . Then the average electric field, 1 1 D , (46) + < E >= E0 = 20 κA κB so that D0,series =

2κA κB 0 E 0 , κA + κ B

(47)

which is easily seen to be less than the maximum value D0,parallel of Eq. (45). The conclusion of this argument is that the average displacement in the bcc phase is less than that in either the disordered or hexagonal phases. Hence the dielectric contribution to the free energy, fel = −D0 E0 /2, will not be such a large negative number, and the difference of free energies between these phases will decrease. For a sufficiently strong field, a transition will occur, as noted earlier. We can now determine the general nature of the phase diagram of this system in the electric field, temperature plane as follows. We have chosen an architecture such that for E = 0 and T = 0, the bcc phase is the stable one. As the temperature is increased at zero field, the system undergoes a first-order transition to the disordered phase at some temperature, or equivalently, some value of the Flory temperature, χN (E = 0). At zero electric field, the difference in displacements between the bcc and disordered phases is zero, of course, but the difference in their entropies is non-zero. The Claussius-Clapeyron Eq. (40) tells us that the slope of the boundary ˆ0 /d(χN ), becomes infinite between these two phases in the E, T plane, dE as the E = 0 axis is approached. For non-zero electric fields, there is a non-zero difference in displacement fields between the two phases. Further we have argued above that the value of displacement field is larger in the disordered phase than in the bcc phase. The entropy density is also larger ˆ 0 are of in the disordered phase than in the bcc phase. Hence ∆s and ∆D the same sign. Therefore the Claussius-Clapeyron Eq. (40) says that the ˆ0 /d(χN ), will be positive. slope of the boundary, dE The behavior of the boundary between bcc and hexagonal phases is also easily understood. At zero temperature and zero field, the bcc phase was chosen to be the phase of lowest energy. As the electric field increases, the system eventually make a first-order transition to the hexagonal phase. The slope of the boundary between these phases is zero, from Eq. (40), as 1/χN = 0. To determine the slope of this boundary at non-zero temperatures, we need to know the sign of the difference of entropy densities. We

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recall from mean field theory that, on lowering the temperature, the system passes from the disordered phase to the bcc phase to the hexagonal phase. Hence the entropy of the bcc phase is greater than that of the hexagonal phase. But the displacement is less in the bcc phase than in the hexagonal ˆ 0 are opposite, and Eq. (40) predicts phase. Thus the signs of ∆s and ∆D a negative slope. The range of temperature over which the bcc is stable decreases as the electric field increases, and finally vanishes at a triple point, at which the disordered, bcc, and hexagonal phases coexist. For larger fields, the disordered and hexagonal phases can coexist. As the entropy densities of these two phases differ but their electric displacements do not, the ClaussiusClapeyron Eq. (40) tells us that the slope of the boundary between them is infinite. We now know what the phase diagram should look like in the electric field, temperature plane. The rest is simply calculation. An example of the result of such a calculation6 for the architecture fA = 0.1 is shown in Fig. 1. Because application of the electric field lowers the Im3m symmetry of the bcc phase to R3m, the region of this phase is so labeled in the figure.

3

Ê0

Dis

Hexagonal

2

1

0

R ¯3 m 50

55

60

65

70

75

80

χN Fig. 1. Calculated phase diagram of a diblock copolymer in the presence of an external ˆ0 and the electric field. It is shown as a function of the dimensionless electric field E dimensionless Flory parameter χN assumed to be inversely proportional to temperature. The fraction of A block in the copolymer is 0.1. For details, see Lin, Schick, and Andelman.6

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5. Basic Surface Thermodynamics We now consider a system of volume Ω which is bounded by a surface of area A. For the moment, the electric field is zero. It is convenient to separate the total energy, Utot , entropy, Stot , etc. into bulk and surface pieces. One does this by defining the bulk densities Utot , Ω Stot , sb ≡ lim Ω→∞ Ω

ub ≡ lim

Ω→∞

(48) (49)

and surface excess densities Utot − Ωub , A Stot − Ωsb ss ≡ lim . Ω,A→∞ A

us ≡

lim

Ω,A→∞

(50) (51)

Then the total energy, entropy, and so on have the form Utot = Ωub + Aus + ...+,

(52)

= Ub + Us + ...,

(53)

where the terms not written out correspond to edge terms, point terms, etc. which will be ignored. The first law for the total system, which in the absence of the surface had been given by Eq. (18), now reads X dUtot = T dStot + µi dNi,tot − pdΩ + σdA, (54) i

where σ is the surface tension. If we write dUtot = dUb + dUs ,

(55)

X

(56)

use dUb = T dSb +

µi dNi,b − pdΩ,

i

and decompose Stot and Ni,tot as in Eq. (53), we obtain X dUs = T dSs + µi dNi,s + σdA.

(57)

As the excess surface free energy is extensive, X Us = T S s + µi Ni,s + σA.

(58)

i

i

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Just as in the bulk case it was convenient to define an energy per unit volume, here we define an excess energy per unit area Us , A Ss X Ni,s µi + + σ, =T A A i X ≡ T ss + µi ni,s + σ,

us ≡

(59) (60) (61)

i

whose differential is easily found, from Eq. (57), to be X µi dni,s . dus = T dss +

(62)

i

In the presence of an electric field, the system develops a displacement field Dtot and we define a bulk average displacement field, < Db > and surface displacement field Ds according to R Dtot (r)dr , (63) < Db > ≡ lim Ω→∞ Ω Z Ds (t) ≡

(Dtot (t, z)− < Db >)dz,

(64)

where t is the position vector in the plane of the film. The differential of the surface excess energy now contains a contribution from the electrostatic interactions X dus = T dss + µi dni,s + E(t) · dDs . (65) i

We again introduce a Legendre transform X gs (t) ≡ us − ss T − µi ni,s − E(t) · Ds (t),

(66)

i

= σ − E(t) · Ds (t), X dgs (t) = −ss dT − ni,s dµi − Ds (t) · dE.

(67) (68)

i

As in the bulk case, we must average this free energy over the film to produce gs =< gs (t) > with a differential X dgs = −ss dT − ni,s dµi − Ds,0 dE0 , (69) i

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where Ds,0 is an average over the film completely analogous to its definition in bulk, Eq. (41), Ds,0 ≡

< D(s)δE(s) > , < δE(s) >

(70)

where now the average is over the position vector s. The field Ds,0 is the surface excess displacement. For a one component system, the above differential reduces to dgs = −ss dT − ns dµ − Ds,0 dEext .

(71)

From this, one again derives a Claussius-Clapeyron equation by noting that as one moves along a boundary of coexistence between phases a and b, the change in free energy, g, must be the same in either phase. Hence if one plots the phase diagram at fixed temperature in the electric field, chemical potential plane, the slope of the boundary is given by ∆ns dEext =− . dµ ∆Ds,0

(72)

6. Electric Field Induced Surface Phase Transition The system of interest is one of technical application. One makes a cylindrical phase of block copolymer. Because the substrate invariably prefers one block over the other, the cylinders will lie flat, parallel to the substrate. For technological applications, one would like the cylinders to be aligned perpendicular to the substrate. One way to do this is to apply a field perpendicular to the substrate. As we saw earlier, the electrostatic contribution to the free energy will be larger, negative, if the cylinder axes are parallel to the field. Eventually this electrostatic energy will outweigh any surface contribution to the system and a surface transition will take place. In fact this simple argument makes a prediction: because the gain in electrostatic free energy if the cylinders align is on the order of E 2 d0 A, with d0 and A the film thickness and area respectively, and because the surface energy gained when the cylinders lie flat is a constant independent of the field or thickness, a transition should occur when E 2 d0 A attains some critical, constant value. Thus the value of the electric field at the transition is expected to vary as Ec ∼

1 1/2 d0

.

(73)

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We consider the phase diagram at constant temperature in the field, chemical potential plane. As the thickness of the film is expected to be a monotonic function of the chemical potential, we can consider the phase diagram equivalently in the field, thickness plane. At zero electric field, as one increases the film thickness, one encounters surface phases corresponding to m layers of cylinders lying parallel to the substrate within the film, with m increasing by an integer as one passes from one phase to the next. In between these surface phases there is one in which the cylinders are perpendicular to the substrate.7 It is clear that as one turns on the electric field, the phase space of the perpendicular phase increases, and that of the parallel phases decreases. Furthermore, the displacement field Ds,0 is expected, by identical arguments to those given above for the bulk case, to be less than that in the perpendicular phase. Hence as one increases the thickness, Ds,0 will decrease on going from the perpendicular phase into one of the parallel phases, and will increase as the parallel phase is left and the perpendicular phase re-entered. As the excess surface density, ns increases monotonically as the chemical potential or the thickness is increased, the Claussius-Clapeyron Eq. (72) tells us that the slope of

0.65

0.8 Ê0

Perp. 0.6

0.55 0.45 0.35

Ê0

0.3

0.4

0.5

0.6

(N1/2a/do)1/2 0.4

II

0.2

0

2

3

III

4

IV

5

6

V

7

do/(N1/2a)

VI

8

9

VII

10

Fig. 2. Calculated surface phase diagram of a diblock copolymer adsorbed on a surface in the presence of an external electric field. It is shown at constant temperature as a ˆ 0 and the dimensionless thickness d0 /N 1/2 a function of the dimensionless electric field E with a the identical Kuhn lengths of the A and B components. Parallel phases are denoted by a roman numeral corresponding to the number of cylinders in the film. The perpendicular phase is marked “Perp”. For details, see Lin and Schick.9

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the boundary between a parallel phase and the surrounding perpendicular phase will be positive as one enters the parallel phase and negative as one exits it. Thus one expects a phase diagram to appear as a series of wickets, with the region inside each wicket labeled by the number of parallel layers in it. Furthermore, these wickets must approach the E = 0 axis perpendicularly. This follows from the fact that, at E = 0, there is no difference in the displacement of the two phases, while there is a non-zero difference in surface densities. Thus the Claussius-Clapeyron equation predicts that the slopes of the phase boundaries are infinite. Finally, Eq. (73) predicts that the height of these wickets will decrease with thickness. Again, all that remains is a calculation, but the form of the results is anticipated. Such a result is shown in Fig. 2. The inset shows that the peaks of the wickets do 1/2 indeed decrease as 1/d0 . Thus far we have considered phases in which cylinders are aligned either parallel or perpendicular to the substrate. There is one more phase to consider. If the interaction with the substrate is strong, then one would expect that any cylinders that were perpendicular to the substrate in most of the film would be cut off before they reached the substrate itself so that the latter could be covered with the monomer that it preferred. Such a phase is called an intermediate phase and was discussed in the context of films of lamellar-forming diblocks by Pereira and Williams5 and Tsori and 1.2

Perp.

1

Ê0

0.8

Intermed.

0.6 0.4

II

0.2 0

2

3

III 4

IV 5

6

V 7

do/(N1/2a)

VI 8

9

VII 10

ˆ0 and thickness Fig. 3. Phase diagram as a function of dimensionless electric field E d0 /N 1/2 a for the same system as that in Fig. 2 but with a stronger surface field. The new intermediate phase is labeled “Intermed.” For details, see Lin and Schick. 9

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Andelman.8 A phase diagram for such a system is shown in Fig. 3. One anticipates that the slope of the boundary between intermediate and perpendicular phases is negative because the effect of surface fields becomes less important as the film becomes thicker. One also knows that the displacement is smaller in the intermediate phase than in the perpendicular phase because the cylinders are cut off and there is less separation of charge. The negative slope of the phase boundary and the Claussius-Clapeyron Eq. (72) tells me that as the perpendicular phase has the larger displacement, it must also have the larger thickness. That is, the very narrow coexistence region in Fig. 3 between perpendicular and intermediate phases, one whose finite thickness is too small to be seen in the figure, has tie lines which have a positive slope. This is something which would not have occurred to me, and is an example in which information that one can intuit can be combined with thermodynamic relations to obtain answers to questions one would not even have thought to ask. That is very good indeed! Acknowledgment I wish to thank David Andelman for introducing me to these problems, and for many years of very enjoyable collaboration. Many thanks are owed to my former student Chin-Yet Lin for his spirited and successful pursuit of these problems. Lastly, the support of the U.S.-Israel Binational Science Foundation (BSF) under Grant 287/02 and of the National Science Foundation under Grant No. DMR-0503752 is gratefully acknowledged. References 1. Schick, M., in Les Houches, Session XLVIII; Liquids at Interfaces (Elsevier, 1990, Amsterdam). 2. Callen, H.B., Thermodynamics (Wiley, 1960, New York). 3. Griffiths, D.J., Introduction to Electrodynamics (Prentice Hall, 1999, Upper Saddle River). 4. Admundson, K., Helfand, E., Quan, X. and Smith, S.D., Macromolecules 1993, 26, 2698. 5. Pereira, G.G. and Williams, D.R.M., Macromolecules 1999, 32, 8115. 6. Lin, C.-Y., Schick, M. and Andelman, D., Macromolecules 2005, 38, 5766. 7. Matsen, M. W., J. Chem. Phys. 1997, 106, 7781. 8. Tsori, Y. and Andelman, D., Macromolecules 2002, 35, 5161. 9. Lin, C.-Y. and Schick, M., J. Chem. Phys. 2006, 125, 034902.

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Chapter 8 Orienting and Tuning Block Copolymer Nanostructures with Electric Fields Alexander B¨ oker and Kristin Schmidt Lehrstuhl f¨ ur Physikalische Chemie II, Universit¨ at Bayreuth Universit¨ atsstrasse 30, 95440 Bayreuth, [email protected] In the first part of this chapter, we discuss the origin of the electric field driven alignment of block copolymer nanostructures in solution. We followed the reorientation kinetics of various model block copolymer solutions exposed to an external electric d.c. field. The characteristic time constants follow a power law indicating that the reorientation is driven by a decrease in electrostatic energy. Moreover, the observed exponent suggests an activated process in line with the expectations for a nucleation and growth process. When properly scaled, the data collapse onto a single master curve spanning several orders of magnitude both in reduced time and in reduced energy. In the second part, we show that electric d.c. fields can be used to tune the characteristic spacing of a block copolymer nanostructure with high accuracy by as much as six percent in a fully reversible way on a time scale in the range of several milliseconds. We discuss the influence of various physical parameters on the tuning process and study the time response of the nanostructure on the applied field. A tentative explanation of the observed effect is given based on the anisotropic polarizabilities and permanent dipole moment of the monomeric constituents.

1. Introduction In the past, electric fields have successfully been used to achieve long-ranged order in block copolymer nanostructures [1–3]. From theoretical models, there are two possible major driving forces for the orientation to take place. First, there is the dielectric mechanism, which is based on the dielectric contrast between the copolymer blocks leading to a minimum in electrostatic free energy whenever the dielectric interfaces are oriented parallel to the electric field vector [4]. The gain in free energy should be proportional to 215

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E 2 (εA − εB ) /hεi with εA,B being the dielectric constants of the polymer blocks and hεi being the average dielectric constant for the system. Due to the quadratic dependence on the electric field strength, the dielectric mechanism is expected to work both in d.c. and a.c. electric fields. Another approach recently discussed is the theoretical possibility of mobile ions contributing to the reorientation process as they may lead to an effective polarization of the anisotropic block copolymer structure thus allowing for microdomain alignment at field strengths much lower than required by the dielectric mechanism [5]. An electric field will then exert a torque on the microdomains until a parallel orientation of such dipoles along the field direction is established. If mobile ions are present in a certain block copolymer system, the ionic effect is expected to dominate the dielectric driving force in d.c. fields or at low frequency a.c. fields (< 50 Hz) as the effect is inversely proportional to the frequency. Recently, many parameters governing the reorientation process (e.g. polymer concentration, temperature, defect density and electric field strength) have been studied experimentally both with in situ [6–9] and ex-situ methods [1–3, 10–12], leading to increasing control over the domain orientation in block copolymers rendering them into highly valuable systems for various nanotechnological applications [13]. However, the underlying driving forces have not yet been quantified. In the first section of this chapter, the issues outlined above are addressed and a quantitative study of the reorientation kinetics in solutions of various model block copolymers exposed to an electric field is described with respect to the dielectric contrast of the respective blocks. For many potential applications, however, besides the overall orientation of the microdomains, the dimensions of the nanostructures need to be tuned precisely as well. Therefore tools for the systematic variation of the characteristic spacing of the nanostructures in a predictable and simple manner are indispensable. For microphase separated copolymers, tuning of the morphology and size of the nanoscopic patterns formed is typically achieved by changing the molecular weights or the block ratio of the polymers used. However, this approach only allows control of the characteristic spacing on coarse scales, whereas precise adjustment of the spacing is impossible. The addition of a homopolymer corresponding to one or both of the polymer blocks or the addition of a non-selective solvent has successfully been used to fine-tune block copolymer nanostructures [14–16]. However, an exact adjustment to within a percent of the characteristic spacing seems barely possible. Moreover, these approaches are not reversible. Therefore,

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Orienting and Tuning Nanostructures with Electric Fields Table 1. Molecular weight, composition, and polydispersity of the studied block copolymers. Block copolymera S46 I54 108 S50 I50 100 S55 I45 51 S58 I42 48 S64 I36 47 S50 V50 78 S50 T50 100 S47 H10 M43 82

Mn [kg/mol]

Mw /Mn

φS [%]

wS [%]

108 100 51 48 47 78 100 82

1.05 1.02 1.04 1.04 1.03 1.05 1.03 1.04

43 48 52 55 61 52 53 50

46 50 55 58 64 50 50 47

a The

subscripts denote the weight fraction of the respective blocks and the superscript gives the number-average molecular weight in kg/mol.

in the second section of this chapter, we will turn to the use of electric fields to tune the characteristic spacings of block copolymer nanostructures in a highly precise and reversible way.

2. Experimental 2.1. Synthesis In the following chapter block copolymers with different compositions were studied. All consist of polystyrene as the first block linked to either polyisoprene, poly(2-vinyl pyridine), poly(2-hydroxyethyl methacrylate)-bpoly(methyl methacrylate)a or poly-(tert-butyl methacrylate) as a second block, respectively. The block ratio and overall molecular weight were determined by 1 H-NMR in combination with the gel permeation chromatography (GPC) results of the corresponding polystyrene precursor. GPC of the final block copolymers yielded the polydispersities. From small angle x-ray scattering (SAXS) measurements the bulk structures of all polymers were identified as lamellar phases. Table 1 summarizes the molecular weight, composition, and polydispersity of all materials. a The two blocks, PHEMA and PMMA, are known to form a mixed phase and can be treated as a single block [3].

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2.2. Sample preparation and capacitor setup Block copolymer solutions of different concentrations in toluene or tetrahydrofuran (THF) were prepared. The alignment experiments were performed in a home built capacitor. The capacitor consists of two planar gold electrodes separated by a variable distance between 1 and 2 mm embedded in a Teflon block with two glass windows (perpendicular to the electrodes in the direction of the X-ray beam). The sample solution is introduced into the capacitor via a syringe inserted at one side of the cell with an intake diameter corresponding to the electrode distance. After filling, the capacitor is closed with two Teflon screws. A d.c. voltage of up to 12.5 kV or an a.c. voltage up to 2 kV/mm and 5 kHz was applied across the capacitor resulting in a homogeneous electric field pointing perpendicular to the X-ray beam direction (see Fig. 1). Both the voltage at the electrodes and the current through the sample were monitored during the course of the experiment. Within the sensitivity of the setup (I ≈ 0.01 mA), no leakage currents were detected after the electric field was applied.

90° j E

0°

q X-ray V Fig. 1.

Experimental setup for the in-situ SAXS measurements.

2.3. Synchrotron small-angle X-ray scattering The in situ SAXS measurements were carried out at the ID2 beamline at the European Synchrotron Radiation Facility (ESRF, Grenoble, France). The beam size was 100 µm × 100 µm with divergence of 20 µrad × 40 µrad. The maximum photon flux at the sample position was of the order of 3.0 · 1013 photons/s/100 mA with ∆λ/λ = 0.015% at 12.4 keV. The beam energy was set to 12.4 keV, corresponding to a wavelength of 0.1 nm.

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The detector system is housed in a 10 m evacuated flight tube. The detector is a fiber optically coupled FReLoN (Fast-Readout, Low-Noise) CCD based on Kodak KAF-4320 image sensor. It has an input field of 100 mm × 100 mm, nominal dynamic range of 16 bit and full frame rate of 3 frames/s (2048 × 2048). With 2 × 2 binning, the readout rate is about 6 frames/s and at still higher binning (8 × 8) up to 20 frames/s can be obtained. The spatial resolution determined by the point spread function is about 80 µm. Prior to data analysis the raw data have to be corrected. First corrections for detector artifacts, i. e. subtraction of dark current and readout noise, division by detector response, and spatial distortion were applied. Then, the data were normalized to absolute scattering intensities. After this step all detector and beamline dependent features are corrected and the sample background is subtracted. 2.4. Data evaluation Due to shear forces occurring during the filling procedure, the lamellar microstructure is prealigned. The microdomains are subjected to two competing external fields of different symmetry, i. e. the interfacial field between polymer solution and the electrode surface and the external electric field. In order to quantify the microdomain alignment, the order parameter P2 was calculated by integrating the scattering intensity I(q, ϕ) over the azimuthal angle ϕ from 0◦ to 360◦ .

3 cos2 ϕ − 1 (1) P2 = 2 with

2

cos ϕ =

R 2π 0

dϕ(I(q, ϕ) cos2 ϕ |sin ϕ|) R 2π dϕ(I(q, ϕ) |sin ϕ|) 0

(2)

Depending on the position of the maxima of the scattering intensity the calculation yields two different ranges of the order parameter. For lamellar alignment parallel to the electrodes (maximum at ϕ = 0◦ ), P2 ranges from 0 to 1 with P2 = 1 corresponding to perfect lamellar alignment where all lamellar normals are oriented perpendicular to the surfaces, i. e. the electrodes. For alignment of the lamellae along the electric field direction (maximum at ϕ = 90◦ ), P2 ranges from 0 to −0.5 with P2 = −0.5 corresponding to the case where all lamellae are aligned parallel to the electric

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field vector, however, with the lamellar normals being isotropically oriented in the plane of the electrodes. In the following step, to quantify the orientation kinetics, the orientational order parameter P2 was calculated for each single scattering pattern acquired during the course of the experiment. The behavior of P2 as a function of time t has been fitted by a single exponential as described by t

P2 (t) = P2 ,∞ + (P2 ,0 − P2 ,∞ )e− τ

(3)

with P2 ,0 and P2 ,∞ being the limiting values of the order parameter before application of the electric field and at later times, respectively, and τ being the time constant. An exemplary data evaluation is shown in Fig. 2. In order to determine the exact peak position and thus the characteristic spacings for the lamellae oriented along the field lines and for those oriented perpendicular to the field, i. e. parallel to the electrodes, the corrected two-dimensional SAXS data were averaged over a 30◦ opening angle in horizontal direction (perpendicular to the electric field lines) and vertical direction (parallel to the field lines) as shown in Fig. 3. The peak position in q was analyzed with a Voigt based fitting model and the lamellar spacing d was then calculated according to d = 2π/q. 2.5. Computer simulation We employ the dynamic self-consistent field theory (SCFT), which describes the dynamic behavior of each molecule (modeled as Gaussian chains) in the mean-field of all other molecules [17, 18]. The phase separation can be monitored by the scalar order parameter Ψ(~r, t), which is the normalized deviation of the density of a polymer component from its average value. In the case of an incompressible diblock copolymer melt the system is described by only a single order parameter. Simulating a diblock copolymer solution requires an extra order parameter for the solvent; however, we use a simplified model with only one order parameter in the present study. It was shown theoretically that a block copolymer melt can serve as a good approximation to describe general features of phase behavior of concentrated block copolymers solutions with nonselective good solvents [19]. As we have shown recently, such description is well justified and gives an excellent agreement with experiments in the case of a nonselective or almost nonselective solvent [20]. The time evolution of the order parameter in the simplest case follows a diffusion type equation [21] ˙ = M ∇2 µ + η Ψ

(4)

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(a)

0 0sec sec

34 sec 34 sec

6 6sec sec

(b) Intensity [a.u.]

34 sec

0 sec

6 sec

0

90

180

270

360

[°]

(c)

−0.4

P2

−0.2

0

0.2 0.4 0

20

40 60 Time [sec]

80

Fig. 2. Example of SAXS data evaluation. (a) Two-dimensional scattering pattern of a 37.5 wt % solution of S50 I50 100 dissolved in toluene before (t = 0 s) and after (t = 6 s, t = 34 s) applicaton of an electric field. The arrow indicates the direction of the electric field vector. (b) Azimuthal intensity distribution at first-order reflection. (c) Time dependence of the orientational order parameter P2 . The solid line represents a leastsquares fit to the data according to Eq. (3) with P2 ,0 = 0.52, P2 ,∞ = −0.32, and τ = 5 s. Reprinted with permission from Macromolecules [6]. Copyright (2003) American Chemical Society.

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E

Fig. 3. Two-dimensional scattering pattern of a 50 wt % solution of S 55 I45 51 dissolved in THF after applicaton of an electric field. The white and the red sectors correspond to lamellae oriented along the field direction and to those oriented perpendicular to the field, respectively. The arrow indicates the direction of the electric field vector.

with the constant mobility M , and the thermal noise η [18]. The ~ has the form chemical potential of an electric field E E 2in the presence ∂ε 0 0 µ = µ − ∂Ψ T · 8π [22], where µ is the chemical potential in the absence of the electric field, and ε is the dielectric constant of the polymeric material, which can be approximated as ε ≈ ε0 + ε1 Ψ for small Ψ. The electric field ~ inside the material deviates from the applied electric field E ~ 0 = (0, 0, E0 ) E ~ ~ and can be written via an auxiliary potential as E = E0 − ∇ϕ. The po~ = 0. Keeping only tential is related to Ψ via the Maxwell equation div εE leading terms, one can rewrite Eq. (4) in the form [23]. ˙ = M ∇ 2 µ0 + α ∇ z 2 Ψ + η Ψ

(5)

with α ≡ M E02

ε1 2 4π ε0

(6)

The chemical potential without the electrostatic contribution µ0 is calculated using self-consistent field theory for the ideal Gaussian chains with the mean field interactions between copolymer blocks A and B, described by a parameter εAB [23]. The model system we study in the following is a symmetric A4 B4 copolymer melt. The simulations have been performed in a two-dimensional box with 256 × 256 grid points and periodic boundary conditions [17]. For the simulations, the electric field strength is parameterized by α ˜ ≡ α/(kT ·M v) [23–25], where v is a polymer chain volume. The samples were shear-aligned with the dimensionless shear rate γ˜˙ = 0.001, for details please see [26].

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Table 2. Dielectric contrasts of the different diblock copolymers in THF solutions at room temperature. Block copolymer system S50 I50 100 S50 T50 100 S47 H10 M43 82 S50 V50 78

Dielectric contrast (εA − εB )b 0.25 1.24 2.00 4.90

± ± ± ±

0.10 0.10 0.25 0.39

b The

ε-values for PS, PI, PtBMA, and PMMA were measured by dielectric spectroscopy [27], εPHEMA and εP2VP are taken from [28] and [29], respectively.

3. On the Physical Origin of Block Copolymer Alignment In previous investigations on solvent based systems [3], the interplay between viscosity (i. e. chain mobility) and dielectric contrast (i. e. electric driving force) turned out to be critical. In order to study the influence of the dielectric contrast on the reorientation, four different lamella forming diblock copolymers, S50 V50 78 , S47 H10 M43 82 , S50 T50 100 , and S50 I50 100 of comparable chain lengths, however, with different dielectric contrast were studied. Details about these systems can be found in Table 1. The polymers were dissolved in THF, which is a rather neutral solvent for the block copolymer systems under investigation with concentrations slightly above the order-disorder concentration. Among the four materials the dielectric contrast (εA − εB ) varies between 0.25 and 4.9 (see Table 2). The measured dielectric contrasts in THF after correction for the pure solvent are in agreement with the values in the melt. The ion content of all samples is estimated to range between 1-10 ppm. The alignment experiments were performed at room temperature in a capacitor as described in Sec. 2.2. The order parameter P2 and the characteristic time τ for reorientation were calculated according to Eq. (1) and Eq. (3), respectively. 3.1. Scaling behavior It is expected that the rate of alignment 1/τ is proportional to the driving force for reorientation. Therefore, the discussion is started with the rate dependence on the electric field strength E and on the dielectric contrast (∆ε = εA − εB ). In Fig. 4 the alignment rate as a function of the electric field strength for the four different block copolymer systems is shown. Systematically, a power-law dependence with a nearly identical exponent

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1 / t [1/s]

of around 2.7 for all four systems is observed. As outlined above, a simple theoretical consideration leads to an exponent of 2 for the dielectric effect. The fact that consistently an exponent significantly larger than two is observed is in accordance with earlier predictions by Amundson et al. [10]. The authors pointed out that an exponent larger than 2 is to be expected if the alignment process is characterized by an activation step involving an energy barrier. Such an activation energy is anticipated, e. g. for the creation of undulations serving as nuclei for domains aligned in the preferred direction [30, 31]. Indeed, under the experimental conditions chosen throughout the present study, the block copolymer solutions are only weakly phaseseparated and microdomain reorientation is dominated by nucleation and growth [3].

10

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-1

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10

-3

10

-4 -5

10 -1 3 10

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0

3 10

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Fig. 4. D.c. electric field dependence of the rate of alignment 1/τ for N 40 wt % S50 V50 78 , • 40 wt % S47 H10 M43 82 , 40 wt % S50 I50 100 , and F 50 wt % S50 T50 100 in THF. The solid lines represent least squares fits of the power law 1/τ = a · E x to the data points yielding xS V 78 = 2.73, xS H M 82 = 2.73, xS I 100 = 2.63, and 50 50 47 10 43 50 50 xS T 100 = 2.73. Reprinted with permission from Soft Matter [9]. Copyright (2007) 50 50 Royal Society of Chemistry.

Aside from the power law itself the data in Fig. 4 also indicate the relevance of the dielectric contrast. The system with the largest dielectric contrast (S50 V50 78 , ∆ε = 4.9) exhibits the fastest reorientation kinetics while the two systems with the smallest dielectric contrast (S50 T50 100 , ∆ε = 1.24; S50 I50 100 , ∆ε = 0.25) exhibit the slowest reorientation behavior. For a quantitative comparison between the different block copolymers the viscosities of the respective solutions need to be taken into account. There-

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De /<e> E [kV /mm ] Fig. 5. Same data as in Fig. 4 with the x- and y-axis scaled by E 2 (εA − εB )2 /hεi and η, respectively for N 40 wt % S50 V50 78 , • 40 wt % S47 H10 M43 82 , 40 wt % S50 I50 100 , and F 50 wt % S50 T50 100 in THF. Reprinted with permission from Soft Matter [9]. Copyright (2007) Royal Society of Chemistry.

fore, the shear viscosities η at 1 rad/s of the samples were determined. Assuming furthermore the predicted dependence of 1/τ on the dielectric properties one can try to create a master curve by plotting the data of 2 Fig. 4 as η/τ versus E 2 (εA − εB ) /hεi. The result of this procedure is shown in Fig. 5. Three of the four block copolymers fall onto a single curve, covering three orders of magnitude in field energy and five orders of magnitude in η/τ . This scaling behavior is a strong indication that the dielectric contrast of the pure block copolymers constitutes the major driving force for the reorientation process in electric fields. Interestingly, the most polar system (S50 V50 78 ) deviates towards smaller values. It is assumed that this deviation is due to an electrorheological effect [32] leading to a larger viscosity under electric field influence. This effect is expected to be more pronounced for polar materials. 3.2. Computer simulations The results outlined above are strongly corroborated by analysis of the real space data provided by numerical calculations. H. G. Schoberth used the dynamic self-consistent field (DSCF) theory to calculate the structure evolution in a lamella forming diblock copolymer melt with parameters chosen to yield a reorientation mechanism similar to the one observed in the experiments [6, 33].

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1/t

0.1

0.01 0.2

0.3

0.4

0.5

0.6

0.7 0.8

~

a Fig. 6. Dependence of the rate of reorientation on the parameter α ˜ (α ˜ ∝ E 2 ∆ε2 / hεi) calculated from dynamic self consistent field simulations for a ◦ 32 x 32 x 32 and a • 64 x 64 x 64 grid points box. The solid line represents a least squares fit of the power law 1/τ = a · α ˜ x to the data points yielding x = 1.45, i. e. 1/τ ≈ E 2.9 . Reprinted with permission from Soft Matter [9]. Copyright (2007) Royal Society of Chemistry.

From the simulation the order parameter P2 can be calculated. By analyzing P2 with a single-exponential fit the rate of reorientation can be extracted. In Fig. 6 the rate dependence on the parameter α ˜ is shown for small and large simulation boxes. The parameter α ˜ is quadratic in the electric field and the dielectric contrast and hyperbolic in the average dielectric constant. Thus, Fig. 6 can be compared with Fig. 5. Again a power law dependence of the reorientation rate is observed. The power law dependence of α ˜ to the power of 1.45 corresponds to an exponent of 2.9 for E. For the given governing mechanism of nucleation and growth, this exponent is independent of the size of the simulation box as well as the initial orientational order of the system, e. g. initial lamellar tilt, as can be seen from the different initial states of alignment in Fig. 7 (a)–(d). Large differences in initial alignment after shear influence the mechanism of the process [7], however, as long as an activation step is involved, the scaling of 1/τ versus E is not affected. Therefore, one may regard this behavior as an universal property of the specific reorientation process. As mentioned above an exponent larger than 2 indicates an activated state. Indeed, if the results of the simulations in real space (shown in Fig. 7) were followed an activated step is observed. As soon as the electric field is applied the lamellae start to undulate and eventually disrupt. These undulations serve as nuclei which merge in the electric field direction as has been shown earlier [6]. This nucleation process is rather fast. However, the achievement of

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(a)

(b)

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x z

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x z

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~

~

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~

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Fig. 7. Initial (a), (c) and final (b), (d) three dimensional structure of the 32 x 32 x 32 (a), (b) and 64 x 64 x 64 (c), (d) grid points boxes. The black rectangle indicates the detailed representation in (e). The snapshots are taken at the dimensionless times t˜ = 0, 35, 45, 55, 70, 1000. The arrows indicate the direction of the electric field vector. Reprinted with permission from Soft Matter [9]. Copyright (2007) Royal Society of Chemistry.

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perfect long ranged order requires the successive annihilation of low energy defects, which requires much longer times. 3.3. Estimation of the threshold electric fields While the above considerations were solely concerned with the field dependence of the reorientation rate, it is interesting to examine the energy gain in the capacitor once alignment parallel to the electric field has been achieved. It was pointed out earlier that the effect of an electric field will compete with the influence of the boundary surfaces (i. e. the capacitor plates), which are known to induce an alignment of lamellar microdomains parallel to the boundary surfaces due to reduction in interfacial energy. This interplay leads to a threshold field strength Et below which no reorientation can be achieved [11]. Above this threshold the gain in electrostatic energy after reorientation is sufficient to compensate for the energetic penalty associated with the formation of T-junctions [34]. If the field strength is reduced below a critical value of E, no reorientation is observed on the time scale of the experiment (ca. 3600 s). The threshold field Et can be approximated by fitting a hyperbolic dependence τ = α(E − Et )a to the data in Fig. 4 (see Fig. 8 for S50 V50 78 and S47 H10 M43 82 ). By this, Et for the four block copolymer systems is obtained (Table 3).

300 250

t [s]

200 150 100 50 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

E [kV/mm] Electric field dependence of the time constant τ for 4 40 wt % S50 V50 78 and S47 H10 M43 82 in THF. The solid lines represent least squares fits of the power law τ = α(E − Et )a to the data points yielding the treshold fields Et(S V 78 ) = 0.185 kV/mm and Et(S H M 82 ) = 0.315 kV/mm. Fig. 8.

◦ 40 wt % 50

50

47

10

43

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Table 3. Threshold fields and dielectric contrasts for the different diblock copolymers in THF solutions at room temperature. Block copolymer system S50 I50 100 S50 T50 100 S47 H10 M43 82 S50 V50 78

Threshold field Et [V/mm] 487 434 315 185

± ± ± ±

9 29 45 23

Dielectric contrast (εA − εB )c 0.25 1.24 2.00 4.90

± ± ± ±

0.10 0.10 0.25 0.39

c The

ε-values for PS, PI, PtBMA, and PMMA were measured by dielectric spectroscopy [27], εPHEMA and εP2VP are taken from [28] and [29], respectively.

Qualitatively, a clear correlation between Et and ∆ε is found: the larger the dielectric contrast the smaller the threshold field needed to overcome the defect energy described above. This result seems reasonable assuming that the defect energy will be of the same order of magnitude for all four block copolymers studied. Since in the framework of the dielectric effect the energy gain is expected to increase with the dielectric contrast, this finding, again, indicates the importance of dielectric contributions to the overall process. 3.4. Kinetics in a.c. electric fields While the results discussed so far are in line with the predictions for a dielectrically driven microdomain orientation one shall now turn to a critical experimental test of possible contributions of mobile ions. An experiment allowing a.c. electric fields to be applied to the sample in situ at the X-ray beamline was setup. According to the considerations by Tsori et al. [5] ionic contributions should no longer play a role once the frequency of the a.c. electric field is considerably higher than the inverse time needed by the ions to move to the domain boundaries. First this characteristic frequency for a 40 wt % solution of S50 V50 78 in THF is estimated. The drift velocity s of an ion carrying a charge Ze in an electric field of strength E is given by s=

Ze E 6πηa

(7)

where η is the viscosity of the solution and a is the radius of the ion. Since Li+ ions are expected to be the main contamination due to the ionic

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polymerization procedure, a quantitative estimate for this ionic species is carried out. With ZLi+ = 1, aLi+ = 59 pm [35], η40 wt % S50 V50 78 = 170 Pa s and E ≈ 0.7 kV/mm a drift velocity of s ≈ 0.6 µm/s is obtained. For an estimate of the characteristic time the smallest characteristic distance of the system, i. e. the lamellar spacing of d = 32 nm was used. Given the above drift velocity the ion needs at least 53 ms to travel this distance. Hence, a characteristic frequency of around 20 Hz is found for this system. Considerably larger distances will be involved for the ion to travel to the outer limits of the grains, which would translate into considerably lower frequencies. Yet, in order to ensure that the system is in the high frequency regime, the frequency was increased by more than two orders of magnitude above this estimate and measurements at 5 kHz with an effective electric field strength of around 0.4 kV/mm were performed. In Fig. 9(a) the time evolution of the azimuthal angular dependence of the scattering intensity during the reorientation of a S50 V50 78 solution in the presence of an a.c. electric field is shown. The reorientation process is dominated by nucleation and growth of domains: the initial peaks at ϕ = 0◦ and 180◦ decrease and grow at the final position at ϕ = 90◦ and 270◦ . This finding is in line with earlier observations in d.c. electric fields [7]. From a single-exponential fit to P2 the time constant of τ = 75 s for the reorientation process is extracted (see Fig. 9(b)). The fact that reorientation in a.c. electric fields at a frequency considerably above what is expected to be the high frequency limit [5] is observed points to the fact that for the system under study any contribution from mobile ions on the microdomain reorientation can be excluded. A quantitative comparison of the time constant with the corresponding d.c. value is barely possible as the exact field strength acting on the sample is difficult to assess due to unknown a.c. resistance of other parts of the set-up. 4. Tuning the Periodicity of Block Copolymer Microdomains with Electric Fields In this section, the reversible tuning of the block copolymer microdomain spacing over a relative range of six percent without changing the molecular weight or the block ratio, and without chemically modifying the polymer or including any additives is shown. The effect on a lamellar polystyrene-bpolyisoprene diblock copolymer was studied in detail. A significant decrease (slight increase) of the lamellar distance was found for lamellae oriented along (perpendicular to) the electric field direction. The influence of the

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360

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180 90 0 -0.2

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P2

-0.1 0 0.1

0

20

40

60

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time [s] Fig. 9. (a) Time evolution of the azimuthal angular dependence of the scattering intensity for a 40 wt % solution of S50 V50 78 in THF exposed to an a.c. electric field with E ≈ 0.4 kV/mm and f = 5 kHz. (b) Evolution of the orientational order parameter P 2 for the measurement in (a). The solid line represents a least squares fit of an exponential to the data yielding a time constant τ of 75 s. Reprinted with permission from Soft Matter [9]. Copyright (2007) Royal Society of Chemistry.

effective segregation power, the overall molecular weight, the composition, the choice of the solvent and the effect of ions was investigated. Furthermore, the kinetics of the lamellar deformation and the relaxation process was monitored. 4.1. Effect of an electric field on the polymer chains For the first investigation of the general effect of an electric field on the microphase separation a 50 wt % solution of S55 I45 51 in THF was studied. A strongly anisotropic scattering pattern as shown in Fig. 10 is observed for high electric fields. The position of the first order Bragg peak is different for lamellae which are aligned in the field direction and those which are aligned perpendicular to the field direction. Figure 11 shows the effect of the electric field strength on the lamellar spacing calculated from the position of the first-order Bragg peak for the lamellae aligned in field direction and the lamellae aligned perpendicular. The lamellar distance for the lamellae oriented along the field lines decreases

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0 kV/mm

6 kV/mm

12 kV/mm

Fig. 10. Two-dimensional scattering patterns of a 50 wt % solution of S 55 I45 51 in THF for different electric field strengths. Reprinted with permission from Nature Materials [36]. Copyright (2008) Macmillan Publishers Ltd.

27.0 26.8

d [nm]

26.6 26.4 26.2 26.0 25.8 25.6 25.4 0

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4

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E [kV/mm] Fig. 11. Dependence of the lamellar distance d of • parallel and ◦ perpendicular to the electric field lines aligned lamellae on the electric field strength E for a 50 wt % solution of S55 I45 51 in THF. Reprinted with permission from Nature Materials [36]. Copyright (2008) Macmillan Publishers Ltd.

rapidly with increasing electric-field strength, whereas the lamellar distance for the lamellae oriented perpendicular to the field only slightly increases. This behavior may be explained by stretching of the polymer chains. The only theoretical prediction of a chain stretching of block copolymers under the influence of an electric field was described by Gurovich [37, 38]. His results based on SCFT calculations indicate that the electric field polarizes the monomers, interacts with the induced polar moments, and eventually orients them. In consequence, chains are elongated parallel or perpendicular to an applied field depending on the anisotropic polarizability of the monomers. The equilibrium microphase structures of block copolymers result from a competition between entropic and enthalpic contributions to the free energy. The former accounts for the entropic losses due to stretching or compression

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of the polymer chains. The latter accounts for the interfacial energy which depends on the segregation power. While the interfacial energy favours large lamellar spacings, the entropic stretching favours smaller ones where the chain is in its most probable conformation [39]. The interfacial energy of a phase separated block copolymer solution can be estimated by considering the polymer concentration, polymer composition, interaction parameter of the blocks among each other, and the interaction parameter between the blocks and the solvent [40]. It is given 1 Finterfacial = χAB fA fB + χAS fA fS + χBS fB fS kB T

(8)

with kB is Boltzmann’s constant, T is the temperature, χ is the interaction parameter and f the volume fraction of A-block, B-block and solvent. For a 50 wt% solution of S55 I45 51 in THF (χPS−PI = 0.17, χPS−THF = 0.32, χPI−THF = 0.40, fPS = 0.245, fPI = 0.231, fTHF = 0.524 [41]) we get Finterfacial = 245.58 J/mol corresponding to a lamellar distance of 26.7 nm which was determined experimentally, and for a 57 wt % solution, an interfacial energy of Finterfacial = 251.74 J/mol corresponding to a lamellar distance of 27.8 nm can be determined. It was shown earlier that the lamellar distance is proportional to the concentration of the polymer solution in a concentration range between 35 wt % and 70 wt % [6]. Therefore, for the relatively small changes observed here (namely about 6 % change in lamellar spacing corresponding to concentrations of 50 wt % versus 57 wt %), a linear approximation for the relationship between lamellar distance and interfacial energy can be used. Hence, the reduction of the interfacial energy due to the electric field can be estimated. For the 50 wt % solution of S55 I45 51 in THF, a decrease of the lamellar distance from 26.7 nm without field to 25.6 nm under an electric field of 12 kV/mm is observed, corresponding to an interfacial energy of Finterfacial = 245.58 J/mol and Finterfacial = 239.35 J/mol, respectively. Thus, the interfacial energy is reduced by 6.23 J/mol due to the electric field. If a lamella is aligned parallel to the field direction (Fig. 12), the following effect is expected: the chains will be stretched along the field direction, i.e. parallel to the phase boundary. Therefore, to a small extent, the electric field counteracts the stretching induced by the microphase separation. In consequence, the chain conformation entropy increases and the lamellar distance decreases. On the other hand, if the lamellae are oriented perpendicular to the electric-field direction, the chain stretching induced by the electric field acts perpendicular to the lamellar plane and therefore adds

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PS PI without field

electric field

Fig. 12. Schematic illustration of the proposed chain stretching effect for aligned lamellae. Reprinted with permission from Nature Materials [36]. Copyright (2008) Macmillan Publishers Ltd.

to the stretching already induced by the microphase separation. In consequence, the lamellar spacing is increasing and the conformational entropy further decreases. Owing to the associated entropic penalty, this effect is indeed expected to be weaker than in the other direction in agreement with the experimental observation (Fig. 11). Such a chain stretching is associated with a reorientation of the monomeric units of the polymer chain. For the overall process to be energetically favorable, the decrease in the electric energy associated with the monomer reorientation has to compensate the energy penalty for the chain stretching. The electric energy could be decreased by two different effects: anisotropic polarizability of the polymer and reorientation of permanent dipole moments of the chain monomers. Owing to its symmetry and rotational freedom, the phenyl side group is not expected to have a net influence on the chain orientation. Hence, a stronger response of the polyisoprene chains is expected because of the high polarizability and dipole moment of the 1,4-polyisoprene backbone C=C double bonds [42]. In the following, the gain in electric energy compensating the energetic penalty for the resulting chain stretching is considered. A decrease of the lamella thickness by 1.1 nm due to the electric field was observed. The length of one polyisoprene monomer is 0.68 nm [43]. As the monomers can be reoriented from two different orientations perpendicular to the electric field lines (in-plane and out-of-plane with respect to the interface), it can be estimated that four polyisoprene monomers have to change their orientation from perpendicular to the field lines to parallel to the field to cause the observed decrease in the lamellae thickness. The change of electric energy associated with this reorientation is given by the gain in polarizability. Per polyisoprene chain, it is w = 1/2nαE 2

(9)

with α being the difference in polarizability of the polyisoprene monomer in the direction parallel and perpendicular to the backbone. It amounts to

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α = 2.42·10−40 Cm2 /V [44]. With the effective experimental electric field of Eeffective = E/ε = 2.28 · 106 V/m and the four monomers calculated above, it amounts w = 0.0015 J/mol. If the chains are stretched owing to different polarizabilities along and perpendicular to the backbone of the chain as described by Gurovich, the gain in energy from reorientation is about three orders of magnitude too small to solely induce the observed effect. However, when considering the electric energy due to reorientation of the permanent dipole moment of the polyisoprene monomers, it is adequate to account for the experimental observations. For this per polyisoprene chain can be calculated w = npk × E

(10)

with pk = 10 · 10−31 Cm [42] it amounts w = 5.49 J/mol which is in the right order of magnitude and would yield a sufficient driving force for the observed effect. 4.2. Influence of different physical parameters To assess the influence of several physical parameters the relative change in the lamellar distance was quantified as ∆d =

d⊥ − d k d0

(11)

where d⊥ and dk are the lamellar distances of lamellae aligned perpendicular and parallel to the electric field lines, respectively. For all solutions a linear dependence of ∆d on the electric field strength was found. However, a different behavior was observed for different degrees of phase separation, different compositions, and different solvents. Influence of the degree of phase separation A common approach for block copolymer solutions is to adopt the dilution approximation, which states that the phase diagram of a copolymer solution can be obtained from the corresponding melt phase diagram by replacing χAB with φP χAB , where φP is the polymer volume fraction in the solution. Thus, the degree of phase separation can be changed by varying the polymer concentration. Comparing a 50 wt % S55 I45 51 and a 57 wt % solution of S55 I45 51 in THF (Fig. 13) the chain-stretching effect is stronger for systems with a lower concentration, i. e. a lower degree of phase separation. As mentioned above, the lamellar distance is determined by the balancing of the interfacial energy and the entropic energy. The degree of phase separation has an influence

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E [kV/mm] Fig. 13. Dependence of the relative change in the lamellar distance ∆d on the electric field strength E for a ♦ 50 wt % and a 57 wt % solution of S55 I45 51 in THF. Reprinted with permission from Nature Materials [36]. Copyright (2008) Macmillan Publishers Ltd.

only on the interfacial energy, i. e. a lower degree of phase separation leads to a lower interfacial energy. Thus, as the electric energy of the field is the same, the entropic energy has a stronger influence at lower concentrations and the relative change is higher for the solutions with lower interfacial energy. Influence of the solvent Furthermore, an influence of the solvent is shown by comparing the copolymer S55 I45 51 dissolved in toluene and in THF (see Fig. 14). The THF solution is more strongly influenced by the electric field than the toluene solution. Both THF and toluene are good solvents for PI (χtoluene−P I = χT HF −P I = 0.40) and PS; however, the solubility of PS is slightly higher in THF than in toluene (χtoluene−P S = 0.44, χT HF −P S = 0.32) [41]. Thus, the interfacial energy is lower for the THF solutions. Because the same arguments as noted above can be applied to this case, the THF solutions are more affected. Influence of ions It is interesting to investigate the influence of ions. Since ions can form complexes with the monomers and thus increase the dielectric constant of the respective block, an increasing chain stretching effect is possible. However, a complexation of the ions with either polystyrene or polyisoprene is not expected. Indeed by comparing an ion free solution of S55 I45 51 in THF and an ion contaminated solution, no significant difference can be observed (Fig. 15).

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E [kV/mm] Fig. 14. Dependence of the relative change in the lamellar distance ∆d on the electric field strength E for a 50 wt % solution of S55 I45 51 in ♦ THF and ♦ in toluene. Reprinted with permission from Nature Materials [36]. Copyright (2008) Macmillan Publishers Ltd.

2.5

Dd [%]

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E [kV/mm] Fig. 15. Dependence of the relative change in the lamellar distance ∆d on the electric field strength E for a 50 wt % of S58 I42 48 in THF ♦ with and without ions.

Influence of composition In addition, a significant influence of the block copolymer composition on the magnitude of the chain stretching effect is found. This is illustrated in Fig. 16, where four different block copolymers with varying PS content (all 50 wt % solutions in THF) are compared by calculating the slope of the ∆d versus E curves. As the composition determines the entropic energy, a symmetric copolymer is expected to exhibit a lower entropic energy than an asymmetric one. Hence, the increase of the entropic energy by electric field induced chain stretching is more efficient for a symmetric copolymer, leading to a more pronounced effect for the symmetric block copolymer in this study (φP S ≈ 0.5).

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1.0 0.8 0.6 0.4 0.2 0.0 44 46 48 50 52 54 56 58 60 62 64 66 68

fstyrene [%] Fig. 16. (a) Dependence of the relative change in the lamellar distance ∆d on the electric field strength E for different block copolymer compositions: S46 I54 108 , 5 S50 I50 100 , ♦ S55 I45 51 , ◦ S58 I42 48 , and 4 S64 I36 47 , all 50 wt % solutions in THF. (b) Dependence of the strength of the chain stretching effect δ∆d/δE on the styrene volume fraction φ S for the same solutions.

In summary, a stronger chain stretching effect is found for lower incompatibility or higher symmetry of the block ratio. The molecular weight as well as possible ions have no influence on this effect. 4.3. Kinetic measurements As the change in the characteristic spacing of the microdomain structure can be highly useful with respect to possible technological applications, it is of great interest to quantify the kinetics and the reversibility of this process. Therefore, the time evolution of the lamellar distance upon application of the electric field was followed. To be able to monitor the time dependence

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of this process, the lowest possible resolution of the CCD detector was used that allows images with the highest possible time resolution, i.e. 45 ms per image. Fig. 17(a) shows a plot of the time dependence of the lamellar distance for lamellae oriented in the field direction when the electric field is switched on. As soon as the electric field is applied, the copolymer structure responds and the periodicity of the lamellae is decreased immediately. This process is faster than the time resolution of the detector, i.e. the process has a time constant considerably smaller than 45 ms. To assure that the process is reversible, the relaxation of the domain spacing after the applied field was switched off was followed. Figure 17(b) shows the time evolution of the lamellar distance of the aligned lamellae when the field strength is decreased (owing to the high voltage applied, it takes several seconds for the capacitor to unload). The domain size closely follows the decreasing electric field strength. This very rapid response of the concentrated solutions suggests a reasonable kinetic behavior in the melt. However, for future applications, there is also the possibility to quickly quench the solutions under an applied electric field. This would freeze the structure at any desired microdomain spacing. 5. Outlook In this chapter, it was shown that the electric-field-induced alignment of block copolymer microdomains from solution is predominantly driven by the dielectric contrast between the respective blocks. When properly scaled, the kinetic data (rates of orientation, 1/τ ) collapse onto a single master curve spanning several orders of magnitude both in reduced time and in

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reduced energy. The observed exponent suggests an activated process in line with the expectations for a nucleation and growth process. These observations are corroborated by dynamic self-consistent simulations. The knowledge of this scaling behavior now allows the custom-made synthesis of block copolymers and turns electric fields into a powerful and wellcontrolled tool to direct block copolymer self-assembly. Moreover, experimental evidence pointing to the molecular origin of the orientation becomes increasingly significant. The interaction of each single monomer unit with the electric field not only seems to lead to the orientation of the whole chain, but also causes a significant stretching of dielectrically active Gaussian chains. As a consequence, a suffiently strong electric field can be used to tune the dimensions of the block copolymer nanostructures. Here, one may speculate that further experiments may lead to a confirmation of the theoretical prediction, that a strong electric field may induce a shift in the order-disorder transition temperature (TODT ) [37, 38]. Such effects have already been found to be induced by shear fields in lamellar block copolymer solutions [45]. For electric fields, Wirtz and Fuller reported on field-induced structures in homopolymer solutions near the critical point [46, 47]. Tsori et al. recently found that electric field gradients of moderate field strength (4 kV/mm at 1 kHz) lead to demixing of blends consisting of low molecular weight PDMS and squalane [48]. Thus, the future investigation of the above described phenomena and their impact on the morphology formation could shed light on the question if an electric field is able to influence the phase behavior of a block copolymer nanostructure. All these investigations will contribute to a deeper understanding of the interactions of polymer nanostructures with external electric fields thus enhancing the high potential of the directed self-assembly processes for future technological applications of block copolymer templates such as data storage, nanoelectronics or lab-on-a-chip devices [13]. Acknowledgments We would like to thank all coworkers from the University of Bayreuth and the European Synchrotron Radiation Facility (ESRF) who have contributed to the work presented in this chapter, in particular: Franz Fischer, Helmut H¨ ansel, Georg Krausch, Heiko G. Schoberth, Frank Schubert, Volker Urban, Thomas M. Weiss and Heiko Zettl. In addition, we would like to acknowledge Heiko G. Schoberth, Agur Sevink and Andrei Zvelindovsky for performing the MesoDynTM simulations.

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This work is financially supported by the ESRF and the Deutsche Forschungsgemeinschaft (DFG) in the framework of the Sonderforschungsbereich SFB 481 (Teilprojekt A2). A. B¨ oker acknowledges support by the Lichtenberg-Programm of the VolkswagenStiftung. Finally, we thank Wolfgang H¨ afner, Heiko G. Schoberth and Thomas M. Weiss for proof-reading the manuscript. References [1] K. Amundson, E. Helfand, X. N. Quan, S. D. Hudson, and S. D. Smith, Alignment of lamellar block copolymer microstructure in an electric field 2. mechanisms of alignment, Macromolecules. 27(22), 6559–6570, (1994). ISSN 0024-9297. [2] T. Thurn-Albrecht, J. DeRouchey, T. P. Russell, and R. Kolb, Pathways toward electric field induced alignment of block copolymers, Macromolecules. 35(21), 8106–8110, (2002). ISSN 0024-9297. [3] A. B¨ oker, A. Knoll, H. Elbs, V. Abetz, A. H. E. M¨ uller, and G. Krausch, Large scale domain alignment of a block copolymer from solution using electric fields, Macromolecules. 35(4), 1319–1325, (2002). ISSN 0024-9297. [4] L. Landau, E. Lifshitz, and L. Pitaevskii, Landau and Lifshitz course of theoretical physics: electrodynamics of continuous media. (Pergamon Press, New York, 1984), 2nd edition. [5] Y. Tsori, F. Tournilhac, D. Andelman, and L. Leibler, Structural changes in block copolymers: coupling of electric field and mobile ions, Phys. Rev. Lett. 90(14), 145504/1–145504/4, (2003). ISSN 0031-9007. [6] A. B¨ oker, H. Elbs, H. H¨ ansel, A. Knoll, S. Ludwigs, H. Zettl, A. V. Zvelindovsky, G. J. A. Sevink, V. Urban, V. Abetz, A. H. E. M¨ uller, and G. Krausch, Electric field induced alignment of concentrated block copolymer solutions, Macromolecules. 36(21), 8078–8087, (2003). ISSN 0024-9297. [7] K. Schmidt, A. B¨ oker, H. Zettl, F. Schubert, H. H¨ ansel, F. Fischer, T. M. Weiss, V. Abetz, A. V. Zvelindovsky, G. J. A. Sevink, and G. Krausch, Influence of initial order on the microscopic mechanism of electric field induced alignment of block copolymer microdomains, Langmuir. 21(25), 11974–11980, (2005). ISSN 0743-7463. [8] A. B¨ oker, K. Schmidt, A. Knoll, H. Zettl, H. H¨ ansel, V. Urban, V. Abetz, and G. Krausch, The influence of incompatibility and dielectric contrast on the electric-field-induced orientation of lamellar block copolymers, Polymer. 47(3), 849–857, (2006). ISSN 0032-3861. [9] K. Schmidt, H. G. Schoberth, F. Schubert, H. H¨ ansel, F. Fischer, T. M. Weiss, G. J. A. Sevink, A. V. Zvelindovsky, A. B¨ oker, and G. Krausch, Scaling behavior of the reorientation kinetics of block copolymers exposed to electric fields, Soft Matter. 3(4), 448–453, (2007). ISSN 1744-683x. [10] K. Amundson, E. Helfand, X. Quan, and S. D. Smith, Alignment of lamellar block copolymer microstructure in an electric field 1. alignment kinetics, Macromolecules. 26(11), 2698–2703, (1993). ISSN 0024-9297.

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[11] T. Thurn-Albrecht, J. DeRouchey, T. P. Russell, and H. M. Jaeger, Overcoming interfacial interactions with electric fields, Macromolecules. 33(9), 3250–3253, (2000). ISSN 0024-9297. [12] T. Xu, Y. Q. Zhu, S. P. Gido, and T. P. Russell, Electric field alignment of symmetric diblock copolymer thin films, Macromolecules. 37(7), 2625–2629, (2004). ISSN 0024-9297. [13] C. Park, J. Yoon, and E. L. Thomas, Enabling nanotechnology with self assembled block copolymer patterns, Polymer. 44(22), 6725–6760, (2003). ISSN 0032-3861. [14] K. I. Winey, E. L. Thomas, and L. J. Fetters, Swelling a lamellar diblock copolymer with homopolymer — influences of homopolymer concentration and molecular weight, Macromolecules. 24(23), 6182–6188, (1991). ISSN 0024-9297. [15] K. I. Winey, E. L. Thomas, and L. J. Fetters, Ordered morphologies in binary blends of diblock copolymer and homopolymer and characterization of their intermaterial dividing surfaces, J. Chem. Phys. 95(12), 9367–9375, (1991). ISSN 0021-9606. [16] H. Tanaka, H. Hasegawa, and T. Hashimoto, Ordered structure in mixtures of a block copolymer and homopolymers 1. solubilization of low molecular weight homopolymers, Macromolecules. 24(1), 240–251, (1991). ISSN 00249297. [17] G. J. A. Sevink, A. V. Zvelindovsky, B. A. C. van Vlimmeren, N. M. Maurits, and J. G. E. M. Fraaije, Dynamics of surface directed mesophase formation in block copolymer melts, J. Chem. Phys. 110(4), 2250–2256, (1999). ISSN 0021-9606. [18] B. A. C. van Vlimmeren, N. M. Maurits, A. V. Zvelindovsky, G. J. A. Sevink, and J. G. E. M. Fraaije, Application of dynamic mean-field density functional theory, Macromolecules. 32(3), 646–656, (1999). ISSN 0024-9297. [19] G. H. Fredrickson and L. Leibler, Theory of block copolymer solutions — nonselective good solvents, Macromolecules. 22(3), 1238–1250, (1989). ISSN 0024-9297. [20] A. Knoll, A. Horvat, K. S. Lyakhova, G. Krausch, G. J. A. Sevink, A. V. Zvelindovsky, and R. Magerle, Phase behavior in thin films of cylinderforming block copolymers, Phys. Rev. Lett. 89(3), 035501/1–035501/4, (2002). ISSN 0031-9007. [21] A. Onuki, Phase Transition Dynamics. (Cambridge Univ. Press, 2002). [22] L. Landau and E. Lifshitz, Electrodynamics of Continuous Media. (Pergamonn, Oxford, 1960). [23] A. V. Zvelindovsky and G. J. A. Sevink, Comment on ”microscopic mechanisms of electric-field-induced alignment of block copolymer microdomains”, Phys Rev Lett. 90(4), 049601, (2003). ISSN 0031-9007. [24] K. S. Lyakhova, A. Horvat, A. V. Zvelindovsky, and G. J. A. Sevink, Dynamics of terrace formation in a nanostructured thin block copolymer film, Langmuir. 22(13), 5848–5855, (2006). ISSN 0743-7463. [25] K. S. Lyakhova, A. V. Zvelindovsky, and G. J. A. Sevink, Kinetic pathways of order-to-order phase transitions in block copolymer films under an electric field, Macromolecules. 39(8), 3024–3037, (2006). ISSN 0024-9297.

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[26] A. V. Zvelindovsky, G. J. A. Sevink, B. A. C. van Vlimmeren, N. M. Maurits, and J. G. E. M. Fraaije, Three-dimensional mesoscale dynamics of block copolymers under shear: The dynamic density-functional approach, Phys. Rev. E. 57(5), R4879–R4882, (1998). ISSN 1063-651x. [27] F. Kremer and A. Sch¨ onhals, Broadband dielectric spectroscopy. (Springer, Berlin, 2002). [28] R. Yamaguchi and S. Sato, Relationship between film thickness and electrooptical properties in polymer-dispersed liquid-crystal films, Jpn. J. Appl. Phys. 1. 33(7A), 4007–4011, (1994). ISSN 0021-4922. [29] S. Negi, J. Li, S. M. Khan, and I. M. Khan, High dielectric constant (microwave frequencies) polymer composites, Abstr. Pap. Am. Chem. S. 214, 223, (1997). ISSN 0065-7727. [30] J. Fukuda and A. Onuki, Dynamics of undulation instability in lamellar systems, J. Phys. II. 5(8), 1107–1113, (1995). ISSN 1155-4312. [31] M. W. Matsen, Stability of a block copolymer lamella in a strong electric field, Phys. Rev. Lett. 95(25), 258302/1–258302/4, (2005). ISSN 0031-9007. [32] C. Price, N. J. Deng, F. Lloyd, H. Li, and C. Booth, Studies of poly(styrene) solutions in an electric field: viscosity and dynamic light scattering, J. Chem. Soc.– Faraday Trans. 91(9), 1357–1362, (1995). ISSN 0956-5000. [33] A. B¨ oker, V. Abetz, and G. Krausch, Comment on ”Microscopic mechanisms of electric-field-induced alignment of block copolymer microdomains” — Reply, Phys. Rev. Lett. 90(4), 049602, (2003). ISSN 0031-9007. [34] Y. Tsori and D. Andelman, Thin film diblock copolymers in electric field: Transition from perpendicular to parallel lamellae, Macromolecules. 35(13), 5161–5170, (2002). ISSN 0024-9297. [35] D. R. Lide, Handbook of Chemistry and Physics. (CRC Press, 1996). [36] K. Schmidt, H. G. Schoberth, M. Ruppel, H. Zettl, H. H¨ ansel, T. M. Weiss, V. Urban, G. Krausch, and A. B¨ oker, Reversible tuning of a block copolymer nanostructure via electric fields, Nature Materials. 7(2), 142–145, (2008). ISSN 1476-1122. [37] E. Gurovich, On microphase separation of block copolymers in an electric field — 4 universal classes, Macromolecules. 27(25), 7339–7362, (1994). ISSN 0024-9297. [38] E. Gurovich, Why does an electric field align structures in copolymers, Phys. Rev. Lett. 74(3), 482–485, (1995). ISSN 0031-9007. [39] M. W. Matsen and F. S. Bates, Block copolymer microstructures in the intermediate-segregation regime, J. Chem. Phys. 106(6), 2436–2448, (1997). ISSN 0021-9606. [40] P. de Gennes, Scaling concepts in polymer physics. (Cornell University Press, Cornell, 1991), 4th edition. [41] C. I. Huang, B. R. Chapman, T. P. Lodge, and N. P. Balsara, Quantifying the ”neutrality” of good solvents for block copolymers: Poly(styrene-b-isoprene) in toluene, benzene, and THF, Macromolecules. 31(26), 9384–9386, (1998). ISSN 0024-9297. [42] K. Adachi and T. Kotaka, Dielectric normal-mode relaxation, Prog. Polym. Sci. 18(3), 585–622, (1993). ISSN 0079-6700.

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[43] T. P. Lodge and T. C. B. McLeish, Self-concentrations and effective glass transition temperatures in polymer blends, Macromolecules. 33(14), 5278– 5284, (2000). ISSN 0024-9297. [44] J. Furukawa, S. Yamashit, T. Kotani, and M. Kawashim, Stiffness of molecular chain of synthetic rubber, J. Appl. Polym. Sci. 13(12), 2527–2540, (1969). ISSN 0021-8995. [45] N. P. Balsara, B. Hammouda, P. K. Kesani, S. V. Jonnalagadda, and G. C. Straty, in-situ small angle neutron scattering from a block copolymer solution under shear, Macromolecules. 27(9), 2566–2573, (1994). ISSN 00249297. [46] D. Wirtz, K. Berend, and G. G. Fuller, Electric-field-induced structure in polymer solutions near the critical point, Macromolecules. 25(26), 7234– 7246, (1992). ISSN 0024-9297. [47] D. Wirtz and F. G. G., Phase transitions induced by electric fields in nearcritical polymer solutions, Phys. Rev. Lett. 71(14), 2236–2239, (1993). ISSN 0031-9007. [48] Y. Tsori, F. Tournilhac, and L. Leibler, Demixing in simple fluids induced by electric field gradients, Nature. 430(6999), 544–547, (2004). ISSN 0028-0836.

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Chapter 9 Block Copolymers under an Electric Field: A Dynamic Density Functional Approach A. V. Zvelindovsky∗ and G. J. A. Sevink∗∗ ∗

Centre for Materials Science, University of Central Lancashire, Preston, PR1 2HE, United Kingdom ∗∗

Leiden Institute of Chemistry, PO Box 9502, 2300 RA Leiden, The Netherlands

We review the transition pathways found between different block copolymers structure-types, upon confinement in a thin film and application of an uniform external electric field. The method considered is the dynamic density functional theory (DDFT), which is a dynamic version of self-consistent field theory. We disregard the screening due to free ions and consider a dielectric mechanism to study the transitions. The structure types vary from spheres to lamellae. The transition dynamics depends on the degree of phase separation, and takes detours across phase boundaries depending on the position with regard to the phase boundary.

1. Introduction Block copolymers are fascinating materials capable of forming a large variety of soft nanostructures depending on the chemical composition of blocks and the architecture of relatively simple molecules.1 A challenging task for the application of block copolymers in nano-manufactured devices is to form perfect arrays of nanostructures. One possible way to do so is by the application of an external field to a block copolymer sample. Examples of such fields commonly used in experimental practice are confinement,2–6 shear flow,1,7 electric field8–13 and temperature gradients.14 In the presence of an electric field, an orientation of microdomains parallel to the external electric field is energetically favored due to the different dielectric properties of the two blocks. For lamellar and cylindrical systems this effect counteracts any interfacial alignment due to preferential 245

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attraction to the electrode surfaces. For spherical systems, this effect leads to an elongating of the spheres in the direction of the electric field. Several theoretical groups have studied electric field induced orientational and order-to-order phase transitions15–23 and compared with experiments.24–29 Most of the theoretical analysis is based on the calculation of an extra contribution to the free energy due to electrostatics for different orientation of structures, and then comparing the energies of the static patterns. As to the dynamic behavior, Onuki and Fukuda examined the dynamics of lamellar undulation instability induced by the electric field.17 To answer the question how a phase transition proceeds further in time one needs to develop an approach to examine the dynamics of the pattern evolution. Recently, such a method has been proposed30 for a block copolymer system under an electric field. It employs an approximation to the model in Ref. 31. The authors of Ref. 31 were the first who numerically solve the Maxwell equation for polymer melts under electric field. In contrast to most of the static theoretical studies, experimental studies have intrinsically considered the dynamics pathways of electric field induced phase transitions.32–35 It was observed that the initial stage of the process history prior to the application of an electric field is an important factor in the transition kinetics. When a block copolymer system is not or poorly microphase separated prior to the application of the electric field, composition fluctuations can already be oriented by the electric field,24 and the orientation of microdomains takes place directly after cooling through the order-disorder temperature. In an initially ordered system the rearrangement dynamics depends on the defect density25 and on the degree of microphase separation,28 which determine the growth of undulation instability to a great extent. In the present work the degree of microphase separation simply denotes the strength of A-B blocks interaction in A-B block copolymers, leading to various microstructures like spheres, cylinders, lamellae, etc. Electric field induced thin film sphere-to-cylinder (S-to-C) transition was observed experimentally by Ting et al.29 for a sphere forming system. In this system, cross-sectional TEM snapshots of the intermediate stages of the alignment process indicated that spherical microdomains were first deformed in ellipsoids under an electric field and then interconnect in cylindrical microdomains oriented in the direction of electric field. Electric field induced reorientation of a cylinder forming block copolymer (C|| -to-C⊥ ) was studied by Thurn-Albrecht et al.27,34 for different initial situations. The electric field was found to orient composition

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fluctuations in the disordered state, resulting in a preferred growth of oriented nuclei. An electric field applied to a microphase separated copolymer led to a disruption of grains into smaller pieces that were able to rotate. This disruption occurred via the growth of undulations at the cylinder interfaces. For lamellae forming block copolymer the electric field induced alignment-kinetics has been studied by many groups. Amundsen et al.25,26 proposed two mechanisms: selective electric-field-induced disordering and alignment through movements of defects. Only the latter mechanism was supported by their experiments, and the fundamental process was found to be the movement of edge dislocations.26 In Ref. 35 an in-situ SAXS alignment study of an ordered lamellar block copolymer melt in an electric field was found to indicate that reorientation of lamellar domains goes via a disruption or disordering of the original structure, leading to a substantial loss of long range order. In the intermediate stage the system tends to reduce the grain sizes so the rotation of domains can occur. Similar Ting et al.33 showed that the dominant mechanism is one where lamellae are locally disrupted and reappear in the direction of the applied field. B¨ oker et al.32 identified two distinct microscopic mechanisms of electric field induced L|| -to-L⊥ transition in a concentrated diblock copolymer solution: nucleation and growth of domains, and grain rotation. This behavior was initially attributed to the difference in viscosity due to the different solvent concentrations. It was later argued28,30,36 that viscosity may play a role, but that these mechanisms are intrinsic to the degree of microphase separation of the system. In the present chapter we describe behaviour of all basic structures (spheres, cylinders, lamellae and gyroid) under electric field in the framework of the dielectric mechanism. Recently, a possible influence of mobile ions on reorientation was discussed in the literature as well.22,37 2. Model Dynamic self consistent field theory is used to describe the block copolymer system.38,39 The time evolution of the system is modeled by diffusion dynamics ∂ρI = M ∇ 2 µI + η I , (1) ∂t where ρI is the concentration of blocks of type I, which has dimensions of inverse volume; µI is chemical potential, M is constant mobility and

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ηI is thermal noise, which is related to M via the fluctuation-dissipation theorem.38 An extensive description of relevant dynamical models can be found in the book Ref. 40. We use the most simple model for M , which is just a constant. A more elaborate model can be found in Ref. 41. The chemical potential µI = δF/δρI can be found from the free energy38,39: Z Φn X F 0 [ρ] = −kT ln − UI (r)ρI (r)dr n! V I Z 1X εIJ (|r − r0 |)ρI (r)ρJ (r0 )drdr0 + 2 2 I,J V XZ + εIM (|r − r0 |)ρI (r)ρM (r0 )drdr0 V2

I

κ + 2

Z

V

X I

ρI (r) − ρ0I

!2

dr ,

(2)

where nought at F indicates that the free energy is calculated for the system in the absence of any external fields. The Boltzmann constant is denoted by k, T is the temperature, n is the number of polymer molecules in the volume V occupied by the system, and Φ is the intra-molecular partition function for ideal polymer chains. The parameter κ determines the compressibility of the system and ρ0I is the mean concentration of the I-block (where the average is taken over the sample volume V ). For κ → ∞ the system becomes incompressible. The external potentials UI and the concentration fields ρI are related via the density functional.38 The polymer chains are modeled as so-called Gaussian chains, where the single chain Hamiltonian describes a set of connected harmonic springs: H = kT

N 3 X (Rs − Rs−1 )2 2a2 s=2

(3)

with a being the Gaussian chain bond length, N the number of beads in the chain, Rs the coordinate of bead s. The time evolution of the system as described by Eq. (1) assumes that the system is in quasi-equilibrium, so that there exists a free energy functional, Eq. (2). This free energy functional does not explicitly depend on time. On every time instance of the systems evolution, Eq. (1), all intra-molecular degrees of freedom are assumed to be equilibrated. This is a good approximation in case that the internal dynamics of a single chain is faster than the collective dynamics of the ensemble of chains. Although the free energy description, Eq. (2),

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incorporates spatial resolution on the level of beads of chains, the time evolution described by Eq. (1) is coarse grained, and the smallest physical volume entering it is the volume of a single polymer chain v. We use this volume to introduce a dimensionless variable ψ = v(ρA −ρ0A ). Here A stands for A-blocks. One variable is sufficient in case of incompressible copolymer melts with only two block types, which we will consider in the remainder. Therefore we will drop the subscripts for block types. The variable ψ is an order parameter, which is the local deviation of the volume fraction of blocks A from their uniform distribution. It is small for weakly segregated systems. In the presence of an electric field E(r) the chemical potential can be split into two terms42 : µ = µ0 + µel ,

ξ=

E 2 ∂ µel =− . v 8π ∂ψ

(4)

where E = |E|, (r) is the dielectric constant of the dielectric, and the CGS system of units is used. The electric field inside the material can be expressed via an auxiliary potential ϕ: E = E0 − ∇ϕ, where E0 = (0, 0, E0 ) is the uniformly applied electric field along the z-axis. We model the dielectric constant by (r) ≈ ¯ + 1 ψ(r), which can be seen as the first two terms of a series expansion of (ψ(r)) about ψ(r) = 0. The electric part of chemical potential, Eq. (4), can be written as ξ = −1 (E02 − 2E0 ∇z ϕ + (∇ϕ)2 )/8π. It is assumed that 1 ψ/¯ << 1. Therefore, |∇ϕ|/E0 << 1 and we omit the last term in our treatment. Note that the second term vanishes if ∇z ϕ = 0. This happens in a structure that is fully aligned with the field direction (the very last stages of alignment process), which is of no interest here. Using the Maxwell equation divE = 0 we find in the leading powers of ∇ϕ and ψ: ¯∇2 ϕ(r) = E0 1 ∇z ψ. Finally we arrive at E02 21 2 ∇ ψ, 4π ¯ z and for the time evolution of the order parameter ∇2 ξ =

ψ˙ = b∆µ0 + α∆z ψ + ηe,

(5)

(6)

where b = M v is the mobility in Einstein sense,43 associated with the volume of one polymer chain; α = bvge , ∆ is the Laplacian, ∆z ≡ ∂ 2 /∂z 2 , and ηe is the properly redefined noise term. The coefficient ge describes the strength of the electrostatic contribution to the free energy of the system and can be also found in Ref. 17: ge = (4π¯ )

−1 2 2 1 E 0

(CGS), ge = 0 ¯−1 21 E02 (SI)

(7)

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where 0 is the dielectric constant of vacuum. In the numerical simulations the electric field strength is parameterized by the dimensionless variable α e=

α 0 21 E02 v = (SI), kT b ¯ kT

(8)

where ¯ = |ψ=0 and 1 = ∂/∂ψ|ψ=0 . The purpose of the present simulation is to determine general features of the kinetic pathway of the transitions, and not to search the parameter space for exact values. Nevertheless it is instructive to estimate whether our parameters are in reasonable experimental reach. We choose the experimental system from our previous study,29 for which estimates of the electric field strength also have been made in several other theoretical studies.22,23 The experimental system was a PS-b-PMMA diblock copolymer melt with molecular weight 151K and average PMMA volume fraction f¯ = vρ0P M M A =0.1.29 Using the values for the molecular weight of the monomers MP M M A = 100g/mol, and MP S = 104g/mol, and polymer densities 1.1g/cm3 for PMMA and 1.04g/cm3 for PS, one finds the volume occupied by one polymer chain equal to v = 240nm3.23 We model the dielectric constant by = P M M A f + P S (1 − f ) (where f = vρP M M A = f¯ + ψ), ¯ and 1 = P M M A −P S . With respect which gives ¯ = P M M A f¯+P S (1− f) to the values of the dielectric constants the experimental literature is not very clear. We consider the values for dielectric constants of pure components from Refs. 22 and 23 P M M A = 6, and P S = 2.5, which are the same as the ones stated in Ref. 29; in Ref. 44 these constants are given by 5.2 and 3, respectively. Using the expression for the parameter α e in SI units, Eq. (8), we find that for the experimental temperature T=170 o C the electric field is 8V /µm for a typical value of α e = 0.1. The experiments in Ref. 29 were performed at 40V /µm. If we alternatively consider the values of earlier work of Amundson,26 P M M A = 3.8 and P S = 2.5, we obtain 20V /µm (for α e = 0.1), which is a factor of two lower than in the experiments.29 The theoretical work of Ref. 23 reports a value 79V /µm as minimal required for sphere-to-cylinder transition at 157 o C (a calculation for the experimental temperature T=170 o C of Ref. 29 would result in 80V /µm, and therefore not make much difference). We conclude that our calculated electric field underestimates the experimental electric field of Ting et al.29 In turn, this experimental value is overestimated by the theoretical predictions of Lin et al.23 The origin of these differences remains a challenge for future work. However, some comments can be made here. (i) Independent investigations of Feng et al.45 report orientational transitions in lamellar and cylindrical

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systems under combined (steady shear and electric) external fields for the same range of electric fields as in our study. Their model is a combination of Oono’s Cell Dynamics a lgorithm and our electrostatic model; their dimensionless electric field parameter (k in Ref. 45) was taken equal to 0.2. (ii) A direct comparison with the static calculations in Ref. 23 is not possible as our structures are often not in equilibrium. In some cases the starting structures (prior to the application of an electric field) are frozen-in metastable states, just like in experimental situations, giving rise to a decrease of the electric field necessary for the transition. In the remainder, this distinction is often highlighted. Moreover, we argue in the results section that this will only affect the electric field necessary for the transition, but not the peculiarities of the transition pathway itself, which are the subject of this study. (iii) As discussed in Ref. 23 a full solution of the Maxwell equation gives a somewhat higher value for the electric field required. That could bring our values a bit closer to the experiments. However, we believe that the approximations made in the derivation of our model are well justified in most cases. (iv) As we showed, the estimates are very sensitive to the values of dielectric constants used. The model with dielectric constants of pure components might be not good enough for block copolymer melts. (v) Our model has essentially one free parameter — the physical volume of the problem, which we took as the polymer chain volume. The twofold increase of the electric field (from 20V /µm to 40V /µm), keeping α e constant, would lead to a decrease of effective radius of the physical volume by 37%. Physically, this means that the polymer chain configuration is affected by the electric field on a characteristic size comparable with polymer blocks rather than the chain as a whole, and the theory proposed by Gurovich should be reexamined.15,16 The dimensionless time step is ∆τ = kT bh−2∆t = 0.5, where h is the numerical grid spacing and the value 0.5 is chosen to ensure stability of the numerical scheme.38 The Gaussian bond length parameter a, Eq. (3), and the grid spacing h are related via a so-called grid scaling parameter ah−1 = 1.15, which value is chosen from numerical considerations as well. As b and t enter the simulations only as the product, all simulations can be done in dimensionless time without specifying either of them explicitly. Determining the exact value of the time step is a difficult problem which is out of scope of the present paper. By comparing details of experimentally observed kinetics of phase transition in block copolymers with simulations we have shown earlier, that one simulation time step is about 6 seconds for the specific system studied.4 Nevertheless it is instructive to estimate this value

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here. Using the notation for the diffusion coefficient D = kT b, we get for the time step introduced above ∆t = 0.378a2/D. We give an estimate for the same system as above.29 As the majority component (90%) is PS, this block will be the limiting block in the molecular diffusion of the PS-PMMA block copolymer. We take the value for the diffusion of hydrogenated PS (hPS) reported in Ref. 46 by means of dynamic secondary ion mass spectrometry. For hPS material with molecular weight of 150K at temperature scaled to 150 o C the bulk diffusion was found as D ≈ 5 · 10−15 cm2 /sec. However, as the temperature in Ref. 29 was 170 o C, the diffusion would be somewhat higher, while, on another hand, the block copolymer molecule diffusivity would decrease due to presence of microphase boundaries. Moreover, there will be a difference in diffusion between hPS and the PS used in Ref. 29. On top of all this, the diffusion in thin films (the systems of our study) slows down by about one order of magnitude as reported in Ref. 46. The above value is therefore only a rough estimate, which we shall employ in the derivation of the timestep. Our model in the work29 is a Gaussian chain A2 B10 , which gives a volume fraction of the A-block equal to f¯ = 0.167 and chain length N = 12. To estimate the Gaussian bond length a we compare the experimental value for the first-order peak position29 q ∗ ≈ 0.174 nm−1 with the analytic expression for f¯ = 0.167: q ∗ ≈ 2.19/R (where R2 = a2 N/6).47 Substitution leads to a ≈ 8.9 nm. Again, this is a rough estimate, as Leibler’s expression is not a good approximation for such short chains. Finally all estimates combined in the expression for the timestep gives ∆t ≈ 60 sec, which is well within the experimental range. We model confined systems by positioning two solid surfaces in the simulation box, representing the electrodes. Both surfaces span the box in the x and y direction completely; in the vicinity of surfaces rigid wall boundary conditions apply.39 The energetic interactions are fully parameterized by the interaction parameter ε0IJ (bead-bead) and ε0IM (bead-surface), which are the weights of the Gaussian kernels in the expressions for εKL in Eq. (2), and are given in kJ/mol as in all our previous works.39 We omit zero-superscripts in the following for simplicity. These parameters are directly related to the more familiar Flory-Huggins interaction parameter χ = ε/(NA kT ), where NA is Avogadro number.38 Although we will operate with ε in the remainder, we will provide the corresponding χN value for all systems under consideration. For the surface interaction in AB and ABA systems the strength is characterized by the value of εAM and εBM . In line with our previous work, we introduce an effective interaction param-

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eter εM = εAM − εBM since only the difference counts in the calculation of the chemical potential.48 For all simulations we set εBM = 0. The sign of εM shows the preferential attraction of A-beads (negative) or B-beads (positive) to the surface. 3. Objective of the Study 3.1. Systems of interest The same ABA triblock copolymer A3 B12 A3 as in our previous studies was chosen to study the electric field induced transitions in spherical micellar (S) and cylindrical (C) systems. The behavior of the cylindrical morphologies in confinement was excessively studied for both the symmetric film-surface interactions3,5 and the asymmetric case;6 we can use this knowledge to our advantage. Moreover, the bulk diagram of A3 B12 A3 was simulated in detail,5 and we found the following structures as a function of the energetic interaction εAB (in kJ/mol): εAB ≤ 5.75 — disorder, for 5.8 ≤ εAB ≤ 6.0 — A-rich spheres (S), for 6.1 ≤ εAB ≤ 6.5 — hexagonally packed A-rich cylinders (C) and for εAB ≥ 6.6 — A-rich bicontinuous structure (Bic). For electric field induced transitions in a lamellar system we have chosen the same symmetric diblock A4 B4 which was already considered in Refs. 28 and 30. For this diblock we observe a disordered state in bulk for εAB = 6, while at εAB ≥ 7 we observe well defined lamellar domains. In Table 1 we present a list of chain architecture, εAB , χN , and bulk morphology for the systems studied in this article. 3.2. Transitions of interest In the presence of surfaces (electrodes), microstructures will experience surface effects due to commensurability and surface energetics. For lamellar systems the selective surfaces (εM 6= 0) induce a parallel orientation and Table 1. system N o 1 2 3 4 5 6

architecture A3 B12 A3 A3 B12 A3 A3 B12 A3 A3 B12 A3 A 4 B4 A 4 B4

Systems of study. εAB , kJ/mol 5.8 5.9 6.1 6.5 7.0 8.0

χN 30.4 31 32 34 16.3 25.7

morphology S S C C L L

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non-selective (εM = 0) a perpendicular orientation of the lamellae in the vicinity of the electrodes.2 For cylindrical structures, the same is true albeit that the perpendicular orientations are found at a non-zero εM value, due to a different balance of energetic and entropic contributions to the free energy.48 For spherical systems, the presence of surfaces leads to the formation of layers of spheres.49 In the presence of an electric field, an orientation of microdomains parallel to electric field lines (perpendicular to the electrodes) is energetically favored due to the different dielectric properties of the two blocks. For lamellar and cylindrical systems this effect counteracts the interfacial alignment. In consequence, a minimum electric field strength (threshold electric field strength) is required to overcome the parallel interfacial alignment.28 Above this threshold, L and C undergo an orientational transition, a L|| -toL⊥ and C|| -to-C⊥ transition respectively. For spherical systems, the system responds to the applied electric field by elongating in the direction of the electric field, the amplitude of which depends on the strength of the electric field. For spheres the threshold value of interest is the electric field strength at which a S-to-C⊥ transition occurs. For all film simulations, the electrode surfaces have a preference to the B-block; the strength of which depends on the value of the effective surface interaction parameter εM . In sphere forming systems we consider two cases, the thin film system and a bulk system. The phase behavior for a bulk spherical system under an electric field is summarized in Fig. 1. The initial structure was obtained after 6000 timesteps of simulation without the electric field, starting from homogeneous mixture of components. The spheres are well developed but poorly ordered in space. The simulation time is not sufficient to achieve perfect ordering (see Ref. 38 for discussion). We see that a S-to-C transition exists in bulk. The threshold electric field strength for the electric field inducing this transition is around α e = 0.02. For small values of α e the spheres elongate. Upon an increase of the electric field strength, the spheres merge into short cylinders. We found coexisting morphologies of spheres and cylinders (an example is shown as a crop of the simulation box for α e = 0.018 in Fig. 1), where the cylinders are not precisely aligned along the field lines. Even higher values of electric field strength lead to ordered cylindrical domains along the field direction. The same type of transition was observed in Refs. 22 and 23. The difference with those works is that their transition is abrupt: below the threshold electric field strength there are only elongated spheres, above this value only cylinders.

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0.01

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Fig. 1. Diagram of structures as a function of the electric field strength α e for a bulk A3 B12 A3 block copolymer melt with εAB = 5.9 (kJ/mol). The box size is 32 × 32 × 32 grid points (shown completely only for the lower row). The upper row are details taken from the total simulation volume. All simulations were performed for 6000 timesteps in absence of an electric field and consequently 6000 timesteps with electric field. Each square corresponds to one simulation. Structures here and everywhere in the remainder are isodensity surfaces for f = 0.5. Copyright (2006) American Chemical Society.

The static phase behavior of block copolymer systems confined in a thin film is described in detail in Ref. 50. We found in particular that the threshold value of electric field depends on the strength of the surface interaction: the stronger the preference of one of the blocks to the surface, the larger the strength of electric field required for the order-to-order transition. We note that the diagrams presented in Ref. 50 are essentially not phase diagrams, as the structures are pathway dependent. In our case, the electric field is applied to an already well-developed microstructure. Even the situation where in all simulations the electric field is applied from the beginning would not be the same as in the case of the static phase diagram,21 as the surface induced microphase separation (initiated by the presence of the electrodes) will play a role as well. This is a fundamental difference between static calculations aimed at deriving equilibrium morphologies, and our dynamic simulations. The phase boundaries in our diagrams exhibit hysteresis. The boundaries depend on the history of the sample: the sequence of applying an electric field and the temperature quench. In the following sections we concentrate on the kinetics of the transitions.

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4. Sphere Forming System System 1 with εAB = 5.8 (χN = 30.4) is a spherical system close to the order-disorder transition (ODT) while system 2 with εAB = 5.9 (χN = 31) is further away from ODT. Although the morphology is spherical, the influence of surface field on the spheres in layers next to the surface can be seen from the fact that these spheres are slightly flattened in the direction parallel to the electrodes. The electrode surfaces have a preference to the B-block. 4.1. Results 4.1.1. S-to-C⊥ transition: close to disorder We start with a spherical system close to the ODT boundary (system 1 in Table 1). The box was chosen to be large enough in both lateral directions to avoid the influence of periodic boundary conditions in these directions. First the system was quenched from a homogeneous melt for 4000 timesteps to form a film of three layers of spheres (see Fig. 2(a), left, timestep 4000). At this point an electric field was applied for another 6000 timesteps. The final stage after 10000 timesteps is shown in Fig. 2(a) (right). We see that on its way to the final cylindrical state, the system passes through the disordered state. The formation of cylindrical structures proceeds via a nucleation and growth mechanism, as there is still a rather large disordered region (grey in Fig. 2(a)) in the middle of simulation box at timestep 10000. A more detailed look at the top view of the system after 4000 timesteps (Fig. 2(a), left) shows equally sized spheres (dots) with grain boundaries between different patches; the inset (detail of the larger box) shows that the structure is indeed spherical in all three layers. The top view of the system after 10000 timesteps (Fig. 2(a), right) shows thinner dots (indicating perpendicular cylinders, as seen from the inset) with a much better lateral ordering than the initial spheres, and fewer grain boundaries. Due to limitation of computational time we stopped our simulation when the tendency of this transition was clear. The details of transition are shown in Fig. 2(b). From this figure we see that the layers next to the electrodes melt almost completely upon application of the electric field while the middle layer melts only partly, leaving a small patch. The C⊥ structure grows from the remaining and partially melted spheres in the middle layer. Several of these seeds (like the one showed in detail in Fig. 2(b)) remain in the full simulation box after

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melting, being the source of the grain boundaries where the emerging C⊥ clusters meet. The overall structure shows a hexagonal ordering. 4.1.2. S-to-C⊥ transition: far from disorder The second example is a sphere forming system further away from ODT (system 2 in Table 1). Figure 3(a) shows details of the pathway of S-to-C⊥ transition. When the electric field is applied on a microphase separated

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a

b

Fig. 3. (a) Structural evolution of a A3 B12 A3 block copolymer film with εAB = 5.9 and εM = 4 in a 32 × 32 × 20 box, with electrodes at z = 1, 20. An electric field (e α = 0.04) was applied after an initial 2000 timesteps. The right column shows side views at corresponding timesteps. (b) Aspect ratio of a sphere as function of electric field strength. The spheres were taken from the middle layer of the simulation box. The cylinder length b for α e = 0.035, 0.04 is finite due to confinement. Copyright (2006) American Chemical Society.

melt, the spheres first elongate and become ellipsoidal in shape. Subsequently the elongated spheres merge to form undulating cylinders. The interconnections are not oriented in the field direction but, rather, are dictated by the proximity of the ellipsoids. The undulations in the newly

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formed cylinders decay. Short cylinders rotate as a whole by diffusion. At the end the cylinders are perpendicular to the electrodes (C⊥ ). The slight tilt of the cylinders in the final stage is due to the presence of walls. It does not change in time any more. A similar effect is also found in cylindrical systems; we therefore postpone a detailed discussion of its origin to the section about cylinder forming systems. Just below the threshold electric field strength the elongation in the direction of the electric field is not sufficient to form perpendicular cylinders. We analyzed the degree of elongation for a constant surface field by plotting the aspect ratio of one sphere as function of the electric field strength (Fig. 3(b)). The selected sphere is taken from the middle layer where all spheres are equally elongated. We note that no coexistence of spheres and cylinders is observed in any of the confined simulations. The aspect ratio of spheres is approximately linear for the values of α e below the threshold. The aspect ratio drops when the S-to-C⊥ transition takes place, indicative of a discontinuous process. The difference with the bulk sphere simulation, where a coexisting morphology is observed (Fig. 1), can be attributed to the confinement and the small thickness of the film accommodating 3 layers of spheres. The abrupt transition seen in Fig. 3 is very similar to the one earlier reported by static calculations in Ref. 22 (compare their Fig. 1; see also the discussion of our Fig. 1 in Sec. 3.2 above). 4.2. Discussion of the kinetic pathways of S-to-C⊥ transition When a system is close to ODT (system 1 in Table 1), the S-to-C⊥ transition proceeds via partial melting into the disordered phase in thin film, as illustrated in Fig. 2. In the vicinity of surfaces there are two opposite tendencies: the surface field tends to elongate spheres along the surface, and the electric field tends to elongate spheres along the field lines. Close to ODT, this competition leads to partial disordering (melting) of the system. For the thin film the remaining or partially melted spheres in the middle layer serve as a nucleus for a new C⊥ phase. The overall mechanism can be characterized as a nucleation and growth mechanism. As the surface field is of short range,3,48 its influence on the microstructure is the strongest in the vicinity of electrodes. This explains why we observe melting in the layers next to the surfaces. To verify this statement we performed a simulation for a twice thicker film and smaller lateral dimensions (32 × 32 × 40, results not shown here). After application of the electric field the layers next to the

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surface melt while spheres in the middle of the film elongate and connect into perpendicular cylinders. No melting was observed in the middle of the film. In a system further away from ODT (system 2 from Table 1), the competition of surface field and electric field does not lead to melting, but to the elongation of spheres in the direction of the electric field; the shape depends on the balance between the ponderomotive force and surface tension. Experimentally, electric field induced sphere-to-cylinder transition in thin films was observed in detail for a diblock system.29 In Ref. 29 a direct comparison of our method with experimental TEM and AFM snapshots for a diblock copolymer melt and one value of block-block interaction showed a 1-1 correspondence of the transition pathway, including the coexistence of spheres and cylinders during the merging process. Here we show that these details are also observed for an ABA triblock. We conclude that the transition pathway is most sensitive to the degree of microphase separation, and rather independent of the polymer architecture. 5. Cylinder Forming System In the phase diagram, the cylinder forming system is surrounded by the spherical (lower χN ) and the bicontinuous (higher χN ) phase. System 3 from Table 1 with εAB = 6.1 (χN = 32) is close to the phase boundary with the spherical phase while system 4 with εAB = 6.5 (χN = 34) is close to the boundary with bicontinuous phase. All simulations were performed in a simulation box that accommodates three parallel layers of cylinders. The electrodes have an energetic preference to the B block. 5.1. Results 5.1.1. C|| -to-C⊥ transition: close to spheres For system 3 the electric field was applied between 2000 and 6000 timesteps only. The transition kinetics is shown in detail in Fig. 4. Upon application of an electric field the parallel cylinders start to undulate with a wavelength roughly equal to the distance between the cylinders (timestep 2200). At the next stage the system breaks up into well-distinguished spheres in the whole box (timestep 2800). Upon comparing the time evolution of the sphere forming system from Fig. 3 (timesteps 2200-4200) and the cylinder forming system from Fig. 4 (timesteps 3000-3400) we see that the next

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stages in the C|| -to-C⊥ transition follows exactly the same route as the S-to-C⊥ transition (Sec. 4.1.2). The detailed snapshots in Fig. 4 (right) show that the spheres merge with each other to form undulating cylinders in a tilted direction with respect to the direction of the electric field; the origin is similar to described in Sec. 4.1.2. At the end the cylinders are perpendicular to the electrodes. In order to ensure that the final cylinders are not a kinetically driven metastable phase in the region of the phase diagram where spheres are

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the actual stable equilibrium structure, we have continued our simulation with the electric field switched off for another 4000 timesteps to see if they eventually break up into spheres. Figure 4 (timestep 10000) shows that the structure remains cylindrical, but experiences a slight tilt with respect to the direction perpendicular to the electrodes. We attribute this tilt to the interplay of energetic and entropic factors associated with the surfaces. The surface prefers the B-blocks; the A-rich cylinder caps close to the surface represent an energy penalty. In a search for the least frustrated situation the system finds a slightly tilted orientation. In the presence of an electric field this tilt is suppressed by the tendency of the system to avoid dielectric interfaces under an angle to the field lines. 5.1.2. C|| -to-C⊥ transition: close to bicontinuous Next we focus on a system 4 from Table 1. We compare two systems with the same χN but different defect density. The system in Fig. 5(a), has no defects in the initial structure prior to applying an electric field, while that in Fig. 5(b) has a defect in the form of an open end of a cylinder before the electric field was applied. In both cases the electric field is close to the threshold electric field strength. In the system without defects (Fig. 5(a)) the transition goes via undulation instabilities, leading to a break up of the system in the whole box almost instantly. The system with a defect (Fig. 5(b)) follows a defectmediated transition: the open end of the cylinder serves as a nucleus for the formation of the new C⊥ phase. We provide additional insight on the pathway of the defect-free system, by considering the top view of the structure (right column in Fig. 5(a)). It confirms that the transition takes place via undulations with primary wavelength roughly equal to the distance between the cylinders, and that the breakup takes place in whole box almost simultaneously. The cylinders not only undulate in shape, but also at larger scale adapt their position and curvature in space. The latter leads to a complete rearrangement of the undulations into a hexagonal lattice even before the cylinders actually break and reconnect to form the new perpendicular cylinders. The transition times in Fig. 5(a) (undulation and break-up at once) and Fig. 5(b) (defect-mediated) can not be directly compared to each other due to the difference in both electric field strengths and surface interactions. Figure 6 shows the free energy without electrostatic contribution for a range of electric field strengths (where C⊥ is the equilibrium structure) and con-

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stant effective interaction with surface. We have chosen to consider the free energy without electrostatic contribution, as this reduced free energy will provide us insight into the change of the degree of microphase separation in time and changes in shapes of the cylinders, like undulations. The simulation, of course, has accounted for the total free energy, as described in Sec. 2. In all cases, at 2000 timesteps the electric field was applied which is seen in the graphs as a small jump down, after which the systems reaches a plateau. This plateau corresponds to undulated cylinders (see Figs. 5(a) and 15(b)). One may consider the plateau as an excited state with a lifetime that depends on the strength of the applied electric field. Consequently, the

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Fig. 6. Time evolution of free energies without the electrostatic contribution for simulated systems with εM = 4 and varying electric field strengths α e = 0.082 (1), 0.084 (2), 0.086 (3) and 0.088 (4) respectively. The electric field was applied at 2000 timesteps. The orientation transition time ∆τ as function of electric field strength α e is shown as an inset. Copyright (2006) American Chemical Society.

length of the plateau is an indication of the characteristic lifetime of this exited state. This length (in timesteps) is plotted as an inset in the Fig. 6 as a function of electric field parameter. It decays as a function of the electric field strength. Cylinders break up and the recombination is fast: after the plateau, the free energy without the electric field contribution decreases quite rapidly to the value that corresponds to a perpendicular orientation of cylinders. The defect-mediated transition can be seen most clearly in a laterally larger box (see Fig. 7). In the absence of an electric field, the system forms three layers of parallel cylinders with many typical defects like open ends of cylinders (timestep 1700). After the electric field was applied, nuclei of the C⊥ phase form at the position of defects in the structure. At their edges the newly formed perpendicular clusters grow in the lateral direction leading to the formation of several grains of hexagonally packed cylinders perpendicular to the electrodes. The electric field cannot help in eliminating the grain boundaries, as the free energy of the hexagonal pattern is invariant with respect to rotation around the cylinder axis (the electric field direction). Figure 7(b) shows the details of the transition kinetics, focusing on a nucleus of the C⊥ phase still surrounded by C|| . The open end of a cylinder initiates the appearance of open ends in the neighboring layers. Consequently the open ends swell and eventually connect with each other in a characteristic three-arm connection. Due to these connections between neighboring cylinder layers, the system becomes bicontinuous: a three-arm connection is characteristic for the gyroid phase. We observe rows of such

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Fig. 7. Structural evolution of a A3 B12 A3 block copolymer film with εAB = 6.5 and εM = 4 in a 256 × 256 × 20 box, with electrodes at z = 1, 20. The electric field (e α = 0.1) was applied between 1700 and 4000 timesteps. (a) Top view at 1700, 2100, 2500 and 4000 timesteps and side view at 1700 and 4000 timesteps. Details of the total simulation box are shown as insets. (b) Time evolution of a single defect in the system from (a) shown via selected regions of the total simulation box. (c) Time evolution of the free energy without the electrostatic contribution for the system from (a). Copyright (2006) American Chemical Society.

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three-arm connections forming a front. This front propagates inside the C|| phase, thus forming a cluster of perpendicular cylinders. Figure 7(b) shows that the three-arm connections break and transform into banana-shape cylinders, which slowly straighten. A whole box of these banana-shape cylinders can be already seen in Fig. 5(a) (timesteps=3300), suggesting that newly formed cylinders are never perfectly aligned in the direction of the electric field. We now illustrate the remark about metastability (comparison to the equilibrium theory of Lin et al.;23 remark (ii) in the Model section) as well as our repeated notion that our diagrams in Ref. 50 are essentially no phase diagrams. Earlier in Ref. 5 we calculated a diagram for varying surface energetics εM and thickness H for the same A3 B12 A3 triblock copolymers (εAB = 6.5) in the absence of an electric field (Fig. 8 in that article). From this diagram we observe that the system of Fig. 7(a) (εM = 4 and H = 18) lies within the C|| region, but is close to the boundary with C⊥ . If we now switch the electric field off after all cylinders have reoriented themselves into perpendicular ones, we observe that they remain in this perpendicular orientation, and do not transform back to the parallel orientation. We can rationalize this observation by considering in Fig. 7(c) the free energy without electrostatic contribution of the system from Fig. 7(a). After 4000 timesteps the electric field was switched off. In the absence of the electric field, i.e. before 1700 timesteps and after 4000 timesteps, the free energy shown in Fig. 7(c) is the total free energy of the system, and it is lower for the perpendicular cylinders. We conclude that the starting configuration of parallel cylinders was a kinetically trapped state caused by the surface induced phase separation. Without external help, like the electric field in this case, the system will never change its orientation, as the thermal fluctuations are too small to overcome the energetic barrier imposed by the presence of selective surfaces preferring the parallel orientation. This is the fundamental reason for the previously mentioned ‘hysteresis’, meaning, that the location of phase boundaries depends on the kinetic history of the system. To rationalize this observation even more, we shortly discuss the structural evolution of an initially homogeneously mixed confined system in the presence of a much stronger surface field (the figure is omitted here, but can be found in50 ). The system is located much more towards the centre of the C|| region, εM = 7 and H = 18, in the structure diagram of Fig. 8 in Ref. 5. A twice stronger electric field was applied for a relatively short period of time to an already phase separated structure, in order to induce

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(c) 2200

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Fig. 8. (a) Structural evolution of a A4 B4 block copolymer film with εAB = 7 and εM = 3 in a 32 × 32 × 20 box, with electrodes at z = 1, 20. The electric field (e α = 0.12) was applied at 2000 timesteps. (a) Left — 3D views, right — side view on a vertical slice. (b) Time evolution of Euler characteristic for the system from (a). (c) Left — horizontal slices (z = 6) at different time steps for the system shown in (a), right — structural evolution of the same block copolymer in a 2D box (32 × 32), in the absence of an electric field. (d) Time evolution of the Euler characteristic for the systems shown in (c), solid line — left column and dashes line — right column. In order to enable a comparison we shifted the starting time for the system in the right column of (c) by 2000 timesteps. Copyright (2006) American Chemical Society.

an orientational transition. As the position in the structure diagram (see Fig. 8 in Ref. 5) is relatively close to the surface-field induced perforated lamellae (PL) region, the system forms three layers of parallel cylinders with typical ring-like defects rather fast in the absence of the electric field. After switching on the electric field, these defects serve as nuclei of the new C⊥ phase (rather early along the transition pathway) just like in Fig. 7(a).

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The difference between these two systems is that these defects are now ring-like (patches of the neighbouring PL phase) instead of cylinder ends. These newly formed perpendicular clusters grow in the lateral direction, leading to a rather swift formation of several grains of hexagonally packed cylinders perpendicular to the electrodes. The details of the transition are very much alike the ones illustrated by Fig. 7(b). Only the kinetics of the process is accelerated by the stronger electric field. In contrast to the system of Fig. 7, the perpendicular cylinders transform back into parallel cylinders after switching the electric field off. This back-transition from C⊥ to C|| takes place via bicontinuous intermediates, just like the C|| -C⊥ orientational transition under the influence of the electric field. At the end of this back-transition the free energy reaches the same value as before the electric field was switched on and levels off to a constant value, indicative of a C|| structure being the most stable structure for this system. The final stage is three layers of parallel cylinders, again with ring-like defects. The situation before and after the electric field is switched on is roughly comparable in terms of defect density; only the position of the defects is different. 5.2. Discussion of kinetic pathways of C|| -to-C⊥ transition We have considered two types of cylinder forming systems with different degree of microphase separation depending on which the system undergoes the C|| -to-C⊥ transition via different pathways. The C|| -to-C⊥ transition via an intermediate S phase was found experimentally.51 We will confirm our earlier statements more quantitatively using the evolution of the Euler characteristics.52,53 As an example of a cylinder forming system with lower degree of microphase separation (system 3 in Table 1) we considered the system from Fig. 4. From visual inspection it is clear that spheres are an intermediate structure for the C|| -to-C⊥ transition in this system. As an example of a cylinder forming system with higher degree of microphase separation (system 4 in Table 1) we considered two systems from Figs. 5(a) and 5(b). Visual inspection and Minkowski functional analysis50 confirm that the intermediate structure in the C|| -to-C⊥ transition for defected and defect-free structures is a bicontinuous one. In all observed transitions in cylinder forming systems undulation instability plays an important role. In the C|| -C⊥ transition for lower degree

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of microphase separation the undulations enable the system to break up to spheres (see Fig. 4). The growth of undulation instability enables the C|| to-C⊥ transition in a defect free structure as well (see Figs. 5(a) and 5(c)). Undulation instability also plays a role in the defect mediated transition (see Figs. 5(b), 7(a) and 7(b)). In such systems undulations are particularly important for the formation of nuclei of the C⊥ phase at the location of defects in the parallel microstructure. This finding is in agreement with an earlier suggested mechanism34 where transitions in cylinder forming thin film was enabled by a developing undulation instability, leading to the disruption of an oriented domain and followed by reorientation or rotation of small grains. 6. Lamellae Forming System We showed earlier for 2D system28,30 that nucleation (via undulations and local structure breakage) and growth is the governing mechanism for lamellar systems with lower degree of microphase separation, while rotation of grains (via movement of defects and annihilation of opposite sign defects) is the main mechanism for lamellar systems with a high degree of microphase separation. Here, we focus on the detailed kinetics of these orientational transitions in three dimensions. We have chosen a diblock A4 B4 melt, the same as used in our previous studies in 2D.30 6.1. Results 6.1.1. L|| -to-L⊥ transition: close to disorder Figure 8(a) shows the details of the dynamics of the transition from lamellae parallel to the electrodes (L|| ) to lamellae perpendicular to the electrodes (L⊥ ) for system 5 (Table 1). We observe lamellar undulations appear after the electric field was applied. The wavelength of the undulations is roughly equal to the inter lamellar spacing, which can be seen from the 2D slices (side view, right column in Fig. 8(a), 2200 and 2300 timesteps). Undulated lamellae evolve into a highly connected bicontinuous structure (timestep 2300). This intermediate structure quickly changes to a L⊥ structure (timestep 2600) with some remaining defects. The defects in the L⊥ are persistent even after a long time: only a small defect rearrangement is visible between 2600 timesteps and 8000 timesteps. At stronger electric fields defect free structures are easily obtained. Our intermediate stage of undulated lamellae is visually similar to the morphology found in static cal-

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culations in Ref. 21 as a superposition of L|| and L⊥ close to the threshold electric field strength (compare their Fig. 4(b) and our Fig. 8(a) at 2200 and 2300 timesteps). The nature of the intermediate stage of the L|| -to-L⊥ transition in Fig. 8(a) is clarified by the time evolution of Euler characteristics presented in Fig. 8(b). The zero Euler characteristic value at timestep 2000 corresponds to perfect lamellae. After the electric field is applied the Euler characteristic rapidly drops to very negative values, corresponding to a highly interconnected (bicontinuous) structure. At the final stage the Euler characteristic has a small negative value reflecting the presence of long-living defects in the final L⊥ structure (see Fig. 8(a), timestep 8000). The left column in Fig. 8(c) shows details of L|| -to-L⊥ transition by means of horizontal slices through the lamellae shown in Fig. 8(a). Initially, the density of A-component is homogeneous, corresponding to well microphase separated lamellae, parallel to the xy plane. After the electric field is switched on, several brighter and darker patches appear. These patches reflect the undulations in the lamellae. The regions of perpendicular structures that are formed when time progresses, rearrange themselves quickly into parallel stripes. It is remarkable that the whole process of orientational phase transition as seen in the 2D slices is visually similar to a microphase separation of a 2D lamellar system following a temperature quench in the absence of an electric field (Fig. 8(c), left). Starting from a homogeneous mixture, fluctuations lead to the formation of spheres, followed by their merging into elongated micelles, and finally to the formation of lamellar stripes. The naive conclusion would be that the paths through the energy landscapes of these two phenomena are similar, although they are very different in nature. In order to challenge the presumed similarity from this visual inspection, we compare the evolution of the Euler characteristic for both simulations in Fig. 8(d). From this comparison we see that the behavior is quite different: the 2D slices of the 3D system (solid line) have a rather positive Euler characteristic in the initial stages, indicating many disconnected micelles. The system from right column of Fig. 8(c) is somewhat negative, indicating a bicontinuous structure. From this comparison, we see that it can be dangerous to compare different systems based on visual inspection alone. 6.1.2. L|| -to-L⊥ : far from disorder Finally we discuss system 6 which was extensively considered in our earlier simulation studies30,50 and the associated experimental/simulation work.28,37,54

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We observe that rotation by defect movement is the dominant mechanism of realignment, both in 3D and 2D.30,50 In the middle of the film lamellae orient in the electric field direction, while close to the substrates, regions of predominantly parallel orientation remain. At the positions where perpendicular and parallel lamellae meet, a connection in the shape of a T-junction is formed.50 This structure is very similar to the mixed morphology in static calculations21 (their Fig. 5). In experiments with much thicker films of polymer solutions and relatively strong polymer-electrode energetic interaction, a mixed morphology was also found to be persistent.55 During this transition, the system remains lamellar. No intermediate structures with other symmetries have been observed. 6.2. Discussion of kinetic pathways of L|| -to-L⊥ transition For systems with a weak A/B interface the L|| -to-L⊥ transition takes place via a bypass in the neighboring bicontinuous phase. We do not observe any significant melting of the structure into a disordered state. Therefore, the present pathway is not trivial to guess. The naive suggestion would be that the application of an electric field just shifts the system along the temperature axis in phase space. This would lead to (partial) melting and consequent buildup of new lamellae from disorder, as the symmetric block copolymer melt can only form lamellar or disordered phases. Our results suggest that the pathway in phase space is much more elaborated. The system develops undulation instability. It was already shown by Onuki and Fukuda17 that this process can happen in a certain range of the parameters controlling the lamellar interfacial strength and the electric field value. Such interfaces are considered to be weak (this term should not be mixed up with weak segregation, which is indifferent to the presence or absence of an electric field). Another important conclusion from our results is that the new L⊥ phase emerges immediately from the intermediate bicontinuous phase. Therefore, the system can in principle have clusters of only two lamellar orientations: decaying L|| and growing L⊥ . Although we do not perform very large scale simulations here, our earlier 2D study in very large simulation boxes supports this conclusion.30 As we showed in Ref. 30, the new L⊥ phase appears via nucleation at the position of defects. It was shown in Refs. 28 and 30 that nucleation and growth is the main mechanism in the orientational transition in lamellar systems with low degree of microphase separation. For the systems with stronger A/B interface (higher χN parameter) the transition follows another pathway. During the whole transition, the

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system remains completely in the lamellar phase. Due to a stronger interfacial tension, undulation instability will not grow for electric field strengths sufficient to initiate undulations in systems with weaker interfaces. The presence of defects is therefore a prerequisite in this transition. Typical defects are open ends of lamellae. Under the influence of an electric field, defects propagate perpendicular to the lamellar stripes. Defects of different sign propagate in opposite direction, and occasionally annihilate when two of them meet. This leads to local rotation of a small piece of lamellae. The process continues until all defects are gone. 7. Conclusions We reviewed the behavior of confined systems of sphere, cylinder and lamellae forming block copolymers under an applied electric field by means of dynamic self-consistent theory. We summarize our results here. Threshold value of electric field strength. When no electric field is applied, the majority of cylinder and lamellae forming systems experiences alignment parallel to the electrodes. In sphere forming systems, the presence of electrodes leads to the formation of layers of spheres. When a sufficiently strong electric field is applied, the orientation of microdomains switches to perpendicular to the electrodes (parallel to external electric field lines). Sphere forming systems respond to the applied electric field by elongating spheres in the direction of the electric field, the amplitude of which depends on the strength of the electric field. Lamellar and cylinder forming systems undergo an orientational transition while spheres undergo a phase transition with a change of symmetry. The threshold value of the electric field strength α e (quadratic in E0 ) is found to be approximately linear in the effective surface interaction parameter εM . For spheres surface energetics modifies the behavior similar to the cylindrical and lamellar case. Kinetic pathways of phase transitions. The major take-home message of this chapter is that the kinetic pathway taken by a system near a phase transition can be very different from one that is far from a phase transition (illustrated by Fig. 9). Sphere forming systems close to ODT undergo a sphere-to-cylinder transition with partial disordering of a system in a transient state (schematically shown as pathway 1 in Fig. 9). Sphere forming systems further away from ODT transform into cylinders via elongation and merging of spheres (pathway 2 in Fig. 9). Cylinder forming systems close to the boundary with

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6

L ||

L^ 5

Bic C ||

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C^

3

S

2 1

Dis Fig. 9. Schematic diagram summarizing kinetic pathways (denoted by arrows) for sphere (S), cylinder (C) and lamellae (L) forming systems under an electric field. Open squares denote initial positions; black squares denote positions after the phase transition. ‘Dis’ denotes disordered phase, and ‘Bic’ — a bicontinuous phase. Copyright (2006) American Chemical Society.

spheres transform into the spherical phase on their way to the C⊥ structure (pathway 3 in Fig. 9). Cylinder forming systems close to the boundary with the bicontinuous phase transform via an intermediate bicontinuous structure (pathway 4 in Fig. 9). In a lamellae forming system we found two distinctly different mechanisms for the L|| -to-L⊥ transition. In a system relatively close to the ODT the transition goes via a transient bicontinuous phase (pathway 5 in Fig. 9). The transition via a local rotation of lamellar grains caused by defect movement was observed in a system further away from ODT (pathway 6 in Fig. 9). The system remains in the lamellar phase during the whole transition. Very recently a more complex morphology, gyroid, was studied under an electric field as well. Using the dynamic density functional approach it was found that the gyroid structure transforms to cylinders via a nontrivial fivefold interconnected intermediate structure, Fig. 10.56 The same transition was observed in Ref. 57 using Oono’s Cell Dynamics simulation algorithm instead, which was earlier adapted for electric fields.58 Defects and undulations. Structural defects and undulation instability play an important role in the phase transition under an electric field, and the associated mechanisms depend on the initial ordering as well as compete on different time scales. Undulation instability is important in the transi-

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Fig. 10. Gyroid-to-Cylinder transition under an electric field. From left to right: initial gyriod, five-fold interconnected intermediate, hexagonally packed cylinders. Copyright (2007) American Chemical Society.

tion kinetics in systems close to any phase boundary as shown for sphere (Sec. 4.1.2), cylinder (Sec. 5.1.2) and lamellae forming systems (Sec. 6.1.1). In the only system far from any phase boundary (lamellae forming system of section VII.A2) defect movement is the only mechanism of alignment. In other systems defects may serve as nuclei of a new phase. We show an example of such a defect-mediated transition for a cylinder forming system (Sec. 5.1.2). For spherical systems close to ODT (Sec. 4.1.1) we observe partial melting as an intermediate stage. Acknowledgments The authors would like to thank Toshihiro Kawakatsu, Takashi Honda, Dung Q. Ly, and Kateryna Lyakhova for contributing to the content of this chapter. The authors acknowledge the Stichting Nationale Computerfaciliteiten (NCF), supercomputer centre SARA (Amsterdam), and the High Performance Computing Facility at UCLan (UK). References 1. Hamley, I.W. The Physics of Block Copolymers, Oxford University Press: Oxford, 1998. 2. Fasolka, M.J. and Mayes, A.M., Ann. Rev. Mat. Res. 2001, 31, 323. 3. Knoll, A., Horvat, A., Lyakhova, K.S., Krausch, G., Sevink, G.J.A., Zvelindovsky, A.V. and Magerle, R., Phys. Rev. Lett. 2002, 89, 035501. 4. Knoll, A., Lyakhova, K.S., Horvat, A., Krausch, G., Sevink, G.J.A., Zvelindovsky, A.V. and Magerle, R., Nat. Mater. 2004, 3, 886.

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5. Horvat, A., Lyakhova, K.S., Sevink, G.J.A., Zvelindovsky, A.V. and Magerle, R., J. Chem. Phys. 2004, 120, 1117. 6. Lyakhova, K.S., Sevink, G.J.A., Zvelindovsky, A.V., Horvat, A. and Magerle, R., J. Chem. Phys. 2004, 120, 1127. 7. Keller, A., Pedemonte, E. and Willmouth, F.M., Nature (London) 1970, 225, 538. 8. Thurn-Albrecht, T., Schotter, J., K¨ astle, G.A., Emley, N., Shibauchi, T., Krusin-Elbaum, L., Guarini, K., Black, C.T., Tuominen, M.T. and Russell, T.P., Science 2000, 290, 2126. 9. Mansky, P., DeRouchey, J., Russell, T.P., Mays, J., Pitsikalis, M., Morkved, T. and Jaeger, H., Macromolecules 1998, 31, 4399. 10. Xu, T., Wang, J. and Russell, T.P., In: Nanostructured Soft Matter, Zvelindovsky, A.V. (Ed.), Springer: Dordrecht, 2007, p. 171. 11. Morkved, T.L., Lu, M., Urbas, A.M., Ehrichs, E.E., Jaeger, H.M., Mansky, P. and Russell, T.P., Science 1996, 273, 931. 12. B¨ oker, A., In: Nanostructured Soft Matter, Zvelindovsky, A.V. (Ed.), Springer: Dordrecht, 2007, p. 199. 13. Thurn-Albrecht, T., Steiner, R., DeRouchey, J., Stafford, C., Huang E., Bal, M., Tuominen, M; Hawker, C.J. and Russell, T.P., Adv. Mater. 2000, 12, 787. 14. Hashimoto, T., Bodycomb, J., Funaki, Y. and Kimishima, K., Macromolecules, 1999, 32, 952. 15. Gurovich, E., Macromolecules 1994, 27, 7339. 16. Gurovich, E., Phys. Rev. Lett. 1995, 74, 482. 17. Onuki, A. and Fukuda, J., Macromolecules 1995, 28, 8788. 18. Pereira, G.G. and Williams, D.R.M., Macromolecules 1999, 32, 8115. 19. Ashok, B., Muthukumar, M. and Russell, T.P., J. Chem. Phys. 2001, 115, 1559. 20. Tsori, Y. and Andelman D., Int. Sci. 2003, 11, 259. 21. Tsori, Y. and Andelman, D., Macromolecules 2002, 35, 5161. 22. Tsori, Y., Tournilhac, F., Andelman, D. and Leibler, L., Phys. Rev. Lett. 2003, 90, 145504. 23. Lin, C.-Y., Schick, M. and Andelman, D. Macromolecules 2005, 38, 5766. 24. Amundson, K., Helfand, E., Davis, D.D., Quan, X. and Patel, S.S., Macromolecules 1991, 24, 6546. 25. Amundson, K., Helfand E., Quan, X. and Smith, S.D. Macromolecules 1993, 26, 2698. 26. Amundson, K., Helfand, E., Quan, X., Hudson, S.D. and Smith, S.D. Macromolecules 1994, 27, 6559. 27. Thurn-Albrecht, T., DeRouchey, J., Russell, T.P. and Jaeger, H.M., Macromolecules 2000, 33, 3250. 28. B¨ oker, A., Elbs, H., H¨ ansel, H., Knoll, A., Ludwigs, S., Zettl, H., Zvelindovsky, A.V., Sevink, G.J.A., Urban, V., Abetz, V., M¨ uller A.H.E. and Krausch, G., Macromolecules 2003, 36, 8078. 29. Xu, T., Zvelindovsky, A.V., Sevink, G.J.A., Gang, O., Ocko, B., Zhu, Y.Q., Gido, S.P. and Russell, T.P., Macromolecules 2004, 37, 6980.

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30. Zvelindovsky, A.V. and Sevink, G.J.A, Phys. Rev. Lett. 2003, 90, 049601. 31. Taniguchi, T., Sato, K. and Doi, M., In Statistical Physics, CP 519, Tokuyama, M. and Stanley, H.E., Ed., American Institute of Physics: Melville, New York, 2000, pp. 581–583. 32. B¨ oker, A., Elbs, H., H¨ ansel, H., Knoll, A., Ludwigs, S., Zettl, H., Urban, V., Abetz, V., M¨ uller, A.H.E. and Krausch, G., Phys. Rev. Lett. 2002, 89, 135502. 33. Xu, T., Zhu, Y., Gido, S.P. and Russell, T.P., Macromolecules, 2004, 37, 2625. 34. Thurn-Albrecht, T., DeRouchey, J., Russell, T.P. and Kolb, R., Macromolecules 2002, 35, 8106. 35. DeRouchey, J., Thurn-Albrecht, T., Russell, T.P. and Kolb R., Macromolecules 2004, 37, 2538. 36. B¨ oker, A., Abetz, V. and Krausch, G. Phys. Rev. Lett. 2003, 90, 049602. 37. Schmidt, K., Schoberth, H.G., Schubert, F., H¨ ansel, H., Fischer, F., Weiss, T. M., Sevink, G. J. A., Zvelindovsky, A.V., B¨ oker, A. and Krausch, G., Soft Matter 2007, 3, 448. 38. van Vlimmeren, B.A.C., Maurits, N.M., Zvelindovsky, A.V., Sevink, G.J.A. and Fraaije, J.G.E.M., Macromolecules, 1999, 32, 646. 39. Sevink, G.J.A., Zvelindovsky, A.V., van Vlimmeren, B.A.C., Maurits, N.M. and Fraaije, J.G.E.M., J. Chem. Phys. 1999, 110, 2250. 40. Onuki, A., Phase Transition Dynamics, Cambridge Univ. Press: Cambridge, 2002. 41. Kawakatsu, T., Phys. Rev. E 1997, 56, 3240; 1998, 57, 6214. 42. Landau, L.D. and Lifshitz, E.M. Electrodynamics of Continuous Media, Pergammon: Oxford, 1960. 43. Landau, L.D. and Lifshitz, E.M. Fluid mechanics, Pergammon: Oxford, 1987. 44. Lin, Zh., Kerle, T., Russell, T. P., Sch¨ affer, E. and Steiner, U., Macromolecules 2002, 35, 3971. 45. Feng, J. and Ruckenstein, E., J. Chem. Phys., 2004, 121, 1609. 46. Pu, Y., Rafailovich, M.H., Sokolov, J., Gersappe, D., Peterson, T., Wu, W.L. and Schwarz, S.A., Phys. Rev. Lett., 2001, 87, 206101. 47. Leibler, L., Macromolecules 1980, 13, 1602. 48. Huinink, H.P., van Dijk, M.A., Brokken-Zijp, J.C.M. and Sevink, G.J.A., Macromolecules 2001, 34, 5325. 49. Yokoyama, H., Mates, T.E. and Kramer, E.J., Macromolecules 2000, 33, 1888. 50. Lyakhova, K.S., Zvelindovsky, A.V. and Sevink, G.J.A., Macromolecules 2006, 39, 3024. 51. Xu, T., Zvelindovsky, A.V., Sevink, G.J.A., Lyakhova, K.S., Jinnai, H. and Russell, T.P., Macromolecules 2005, 38, 10788. 52. Sevink, G.J.A., In: Nanostructured Soft Matter, Zvelindovsky, A.V. (Ed.), Springer: Dordrecht, 2007, p. 269. 53. Sevink, G.J.A. and Zvelindovsky, A.V., J. Chem. Phys. 2004, 121, 3864. 54. Schmidt, K., B¨ oker, A., Zettl, H., Schubert, F., H¨ ansel, H., Fischer, F., Weiss, T. M., Abetz, V., Zvelindovsky, A. V., Sevink, G. J. A. and Krausch, G., Langmuir 2005, 21, 11974.

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55. B¨ oker, A., Knoll, A., Elbs, H., Abetz, V., M¨ uller, A.H.E. and Krausch, G., Macromolecules, 2002, 35, 1319. 56. Ly, D.Q., Honda, T., Kawakatsu, T. and Zvelindovsky, A.V., Macromolecules 2007, 40, 2928. 57. Pinna, M. and Zvelindovsky, A.V., Soft Matter 2008, DOI: 10.1039/b706815h. 58. Pinna, M., Zvelindovsky, A.V., Todd, S. and Goldbeck-Wood, G. J. Chem. Phys. 2006, 125, 154905.

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Index

AC electric field, 96 activator-inhibitor, 172 amphiphilic, 57, 64, 66, 67, 70, 80, 83 aniline, 96

cell dynamics algorithm, 251, 273 chain stretching, 232 kinetics, 238 chemical potential, 58, 248 chemical reactions, 173, 176, 177, 191, 194 cholesterol, 9 Claussius-Clapeyron, 197, 207, 208, 211–214 computer simulations, 220 block copolymers in electric fields, 220 comparison with experimental data, 225 conductor, 153 contact angle, 149–152, 154, 155, 158–160, 162, 163, 167, 169 contact line dissipation, 160 contact line friction, 159 counterions, 60, 64, 80 critical point, 66, 76, 88, 89, 91, 93, 99, 102, 103, 105, 107 critical temperature, 68 Curie temperature, 4, 11 cusps, 129 cyclohexane, 96, 98

Beaglehole, 96, 106, 107, 111 Belousov-Zhabotinsky reaction, 191 Berge, 151, 168 Bjerrum length, 60, 67, 75, 83 block copolymer orientation electric field a.c. electric field, 230 capacitor, 218 d.c. electric field, 215 dielectric mechanism, 216 governing parameters, 216 mobile ions, 216 threshold electric field, 228 time constant, 220 power law, 223 block copolymers, 43 E-field phase transitions, 47 electric field, 46 lamellar phase, 45 modulation, 44 orientation in E field, 46 body-centered cubic phase, 197 bond number, 8, 28 Born radius, 58

Debye, 91–94, 96, 97, 99, 101, 103, 111 Debye-Huckel, 59, 68, 74 defect, 246, 262, 264, 273, 274 dewetting, 167 diblock copolymer, 200 dielectric bilayers, 144 dielectric breakdown, 143

Cahn-Hilliard, 121 calculation of orientational order parameter, 219 capillary number, 166 capillary waves, 114, 121 279

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dielectric constant, 58, 62, 119, 200, 206 dielectric contrast, 121, 143, 144 dielectric contrast in block copolymers, 223 dipolar interactions, 6 dipoles, 8 disjoining pressure, 114, 115, 155 dispersion relation, 116, 117, 174, 175, 184 dissipative structures, 49 domains, 4, 5 bubble phase, 13 stripe phase, 11, 12 electric, 149, 150, 153–156, 158, 160, 162, 167, 169 electric double layer, 83 electrocapillary, 151, 154 electrohydrodynamic, 113, 114, 118, 119, 121, 124, 125, 128–130, 133, 134, 137, 139–144 electrostatic pressure, 165–167 electrostaticpressure, 113–115, 118, 119, 121, 123, 125, 128, 129, 132 electrostriction, 58 electrowetting, 149–153, 158–161, 164, 167–169 elongation, 246 ER, electrorheological fluids, 42 Euler characteristic, 270 fastest growing mode, 117 fastest growing unstable mode, 164 ferroelectrics, 42 ferrofluid, 3, 15 alignment of particles, 36 embedded objects, 35 Labyrinthine Instability, 18 light trapping, 38 normal–field instability, 18, 20 parametric stabilization, 23 phase diagram, 34 phase transition, 30 solitons, 23 ferrogels, 30

Index

ferromagnetism, 4 fingering instability, 133 first-order phase transition, 205 Flory constant, 43 Flory-Huggins, 174 fluctuation-dissipation theorem, 248 Fourier transform, 11 free energy, 11 free energy of mixing, 87 Galvani, 82 Galvani potential, 58 garnet films, 10 Gauss’ law, 199, 200, 202 Gaussian chains, 248 Gibbs transfer energy, 58 Gibbs-Duhem relation, 201 Ginzburg-Landau, 57, 60, 61 Ginzburg-Landau theory, 5 GMR, giant magnetoresistance, 14 gradients, 245 grain boundaries, 264 growth exponents, 142, 143 hierarchical morphology, 173 hydrodynamics, 159 hydrophilic, 158 hydrostatic pressure, 114 hysteresis, 149, 150, 158, 160, 163, 167, 169 IMR, inverse magnetorheological fluids, 37 instability, 114, 117–119, 122, 128–130, 134–137, 140 iso-octane, 93 lab-on-a-chip, 149, 157, 159, 163, 168, 169 labyrinthine instability, 25 Landau, 91, 92, 111 Landau-Levich, 163 Langevin, 99, 101 Langmuir monolayer, 8, 10 Laplace, 151, 154 Laplace pressures, 117, 123

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World Scientific Review Volume - 9in x 6in

Index

LCST, 88, 98, 104 line tension, 9 Lippmann, 151, 153, 154, 168 localization, 38 lubrication approximation, 164 Mach-Zehnder, 184, 189, 191 magnetically stabilized fluidized bed, 40 magnetization, 5 Maxwell, 105, 149, 154–156, 167 mesomagnetism, 13 metastable, 88 microfluidics, 149, 168 monovalent, 79 MR, magnetorheological fluids, 16, 36 MTJ, magnetic tunnel junction, 15 nanomagnetism, 13 nitrobenzene, 59, 60, 62 normal field instability, 18 Onuki, 92, 104, 109, 111, 112 order disorder temperature, 256 order order phase transitions, 246 order order transition, 255 order-disorder temperature, 246 orientation, 246 Ornstein-Zernike, 57 partition function, 248 patterning, 1 phase diagram, 12 phase inversion, 181 phase separation, 171–178, 181–185, 187, 189, 191, 192 photochemical reactions, 172, 173, 177, 178, 181, 182, 184, 191, 192 photodimerization, 178, 179 pinning, 158, 160–162 Poiseulle flow, 116 polystyrene, 98, 104

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281

Reynolds number, 158 self consistent fields, 247 single-mode approximation, 7 Small-Angle X-ray Scattering Synchrotron, 218 solvation, 57–62, 64, 75, 80, 83, 84 chemical potential, 82 sphere to cylinder transition, 246, 254, 260, 261, 272 spinodal, 67 spintronics, 14 square-gradient free-energy, 174 stress tensor, 155 superconductivity intermediate phase, 10 surface tension, 59, 66, 72, 73, 115, 119–121, 127, 129, 133, 143, 260 T-junction, 271 threshold electric field, 228 TMV, tobacco mosaic virus, 36 topographic features, 115 topological defects, 35 triblock copolymer, 253, 260 tuning the dimensions of nanostructures, 216, 230 chain stretching, 232 influence of composition, 237 influence of degree of phase separation, 235 influence of ions, 236 influence of solvents, 236 UCST, 88, 91, 94, 98, 101, 109 undulation instability interfacial instability, 246, 262, 272 Van der Waals, 115 viscoelasticity, 178 Viscous stresses, 116, 133