# Publications

### 2021

We are pleased to present a special issue of The Biophysicist on Teaching and Learning during COVID-19. The issue includes both Research Articles and shorter, informal Reports with papers on how formal learning and labs were taught remotely, as well as the even more difficult problem of how student research experiences were modified during this very challenging time. Other articles are related to the pedagogical aspects of the science behind COVID-19 that deal with aerosols and infection models. Finally, a Student Forum report describes the home-based experiments done by middle-school students who were learning remotely. We hope that this collection will be both enlightening and enjoyable to read, but that we do not have to organize such a special issue again!

The three-dimensional organization of chromatin contributes to transcriptional control, but information about native chromatin distribution is limited. Imaging chromatin in live Drosophila larvae, with preserved nuclear volume, revealed that active and repressed chromatin separates from the nuclear interior and forms a peripheral layer underneath the nuclear lamina. This is in contrast to the current view that chromatin distributes throughout the nucleus. Furthermore, peripheral chromatin organization was observed in distinct Drosophila tissues, as well as in live human effector T lymphocytes and neutrophils. Lamin A/C up-regulation resulted in chromatin collapse toward the nuclear center and correlated with a significant reduction in the levels of active chromatin. Physical modeling suggests that binding of lamina-associated domains combined with chromatin self-attractive interactions recapitulate the experimental chromatin distribution profiles. Together, our findings reveal a novel mode of mesoscale organization of peripheral chromatin sensitive to lamina composition, which is evolutionary conserved.

Maintaining homeostasis is a fundamental characteristic of living systems. In cells, this is contributed to by the assembly of biochemically distinct organelles, many of which are not membrane bound but form by the physical process of liquid–liquid phase separation (LLPS). By analogy with LLPS in binary solutions, cellular LLPS was hypothesized to contribute to homeostasis by facilitating “concentration buffering,” which renders the local protein concentration within the organelle robust to global variations in the average cellular concentration (e.g., due to expression noise). Interestingly, concentration buffering was experimentally measured in vivo in a simple organelle with a single solute, while it was observed not to be obeyed in one with several solutes. Here, we formulate theoretically and solve analytically a physical model of LLPS in a ternary solution of two solutes (ϕ and ψ) that interact both homotypically (ϕ–ϕ attractions) and heterotypically (ϕ–ψ attractions). Our physical theory predicts how the coexisting concentrations in LLPS are related to expression noise and thus, generalizes the concept of concentration buffering to multicomponent systems. This allows us to reconcile the seemingly contradictory experimental observations. Furthermore, we predict that incremental changes of the homotypic and heterotypic interactions among the molecules that undergo LLPS, such as those that are caused by mutations in the genes encoding the proteins, may increase the efficiency of concentration buffering of a given system. Thus, we hypothesize that evolution may optimize concentration buffering as an efficient mechanism to maintain LLPS homeostasis and suggest experimental approaches to test this in different systems.

We analytically predict the concentration profile of a long polymer chain in a poor solvent, confined between two walls where some of its monomers can form strong bonds with the walls. In contrast to adsorption boundary conditions, these concentration profiles can vary nonmonotonically. Boundary conditions that lead to a decrease in the monomer concentration at the walls (at fixed, total monomer concentration) show a transition between concentration profiles in which the polymer concentration is organized peripherally (near the walls) and profiles in which the polymer concentration is the highest in the center (away from the walls). We conclude by discussing the relevance of our findings to experimental observations and simulation studies of chromatin organization in the nucleus of living cells.

Intact-organism imaging of Drosophila larvae reveals and quantifies chromatin-aqueous phase separation. The chromatin can be organized near the lamina layer of the nuclear envelope, conventionally fill the nucleus, be organized centrally, or as a wetting droplet. These transitions are controlled by changes in nuclear volume and the interaction of chromatin with the lamina (part of the nuclear envelope) at the nuclear periphery. Using a simple polymeric model that includes the key features of chromatin self-attraction and its binding to the lamina, we demonstrate theoretically that it is the competition of these two effects that determines the mode of chromatin distribution. The qualitative trends as well as the composition profiles obtained in our simulations compare well with the observed intact-organism imaging and quantification. Since the simulations contain only a small number of physical variables we can identify the generic mechanisms underlying the changes in the observed phase separations.

### 2020

This article describes the redesign of a project-based course on soft and biological materials to include computational modeling. Including the construction of computational models in the course is described as a shift from constructivism-a theory that characterizes the development of formal reasoning, to constructionism-a theory that focuses on learning while constructing artifacts. This shift ameliorated two drawbacks in the original course: the limited conceptualization of entropy resulting from an unproductive use of the disorder metaphor, and the dependence of most students on the teacher for writing theoretical explanations for their final papers. In the redesigned curriculum, computer simulations provide concrete dynamic representations that students can draw upon for developing nuanced, formal reasoning on entropy and the 2nd law of thermodynamics. In addition, core computational models act as a flexible web that can be extended and modified, and allow a significant proportion of the students to build theoretical models on their own. We conclude that while the new design reflects a shift towards constructionism, we did not adopt a fully constructionist approach, rather a blend of constructivist and constructionist approaches.

Spontaneous contractions of cardiomyocytes are driven by calcium oscillations due to the activity of ionic calcium channels and pumps. The beating phase is related to the time-dependent deviation of the oscillations from their average frequency, due to noise and the resulting cellular response. Here, we demonstrate experimentally that, in addition to the short-time (1-2 Hz), beat-to-beat variability, there are long-time correlations (tens of minutes) in the beating phase dynamics of isolated cardiomyocytes. Our theoretical model relates these long-time correlations to cellular regulation that restores the frequency to its average, homeostatic value in response to stochastic perturbations.

Protein self-organization is a hallmark of biological systems. Although the physicochemical principles governing protein-protein interactions have long been known, the principles by which such nanoscale interactions generate diverse phenotypes of mesoscale assemblies, including phase-separated compartments, remain challenging to characterize. To illuminate such principles, we create a system of two proteins designed to interact and form mesh-like assemblies. We devise a new strategy to map high-resolution phase diagrams in living cells, which provide self-assembly signatures of this system. The structural modularity of the two protein components allows straightforward modification of their molecular properties, enabling us to characterize how interaction affinity impacts the phase diagram and material state of the assemblies in vivo. The phase diagrams and their dependence on interaction affinity were captured by theory and simulations, including out-of-equilibrium effects seen in growing cells. Finally, we find that cotranslational protein binding suffices to recruit a messenger RNA to the designed micron-scale structures.

Multivalent molecules can bind a limited number of multiple neighbors via specific interactions. In this paper, we investigate theoretically the self-assembly and phase separation of such molecules in dilute solution. We show that the equilibrium size (n) distributions of linear or branched assemblies qualitatively differ; the former decays exponentially with the relative size n/N (N = 〈n〉), while the latter decays as a power law, with an exponential cutoff only for n ≳ N2 ≫ N. In some cases, finite, branched assemblies are unstable and show a sol-gel transition at a critical concentration. In dilute solutions, non-specific interactions result in phase separation, whose critical point is described by an effective Flory Huggins theory that is sensitive to the nature of these distributions. This journal is

Recent experiments reveal that the volume of adhered cells is reduced as their basal area is increased. During spreading, the cell volume decreases by several thousand cubic micrometers, corresponding to large pressure changes of the order of megapascals. We show theoretically that the volume regulation of adhered cells is determined by two concurrent conditions: mechanical equilibrium with the extracellular environment and a generalization of Donnan (electrostatic) equilibrium that accounts for active ion transport. Spreading affects the structure and hence activity of ion channels and pumps, and indirectly changes the ionic content in the cell. We predict that more ions are released from the cell with increasing basal area, resulting in the observed volume-area dependence. Our theory is based on a minimal model and describes the experimental findings in terms of measurable, mesoscale quantities. We demonstrate that two independent experiments on adhered cells of different types fall on the same master volume-area curve. Our theory also captures the measured osmotic pressure of adhered cells, which is shown to depend on the number of proteins confined to the cell, their charge, and their volume, as well as the ionic content. This result can be used to predict the osmotic pressure of cells in suspension.

We review a theoretical, coarse-grained description for cardiomyocytes calcium dynamics that is motivated by experiments on RyR channel dynamics and provides an analogy to other spontaneously oscillating systems. We show how a minimal model, that focuses on calcium channel and pump dynamics and kinetics, results in a single, easily understood equation for spontaneous calcium oscillations (the Van-der-Pol equation). We analyze experiments on isolated RyR channels to quantify how the channel dynamics depends both on the local calcium concentration, as well as its temporal behavior ("adaptation"). Our oscillator model analytically predicts the conditions for spontaneous oscillations, their frequency and amplitude, and how each of those scale with the small number of relevant parameters related to calcium channel and pump activity. The minimal model is easily extended to include the effects of noise and external pacing (electrical or mechanical). We show how our simple oscillator predicts and explains the experimental observations of synchronization, "bursting" and reduction of apparent noise in the beating dynamics of paced cells. Thus, our analogy and theoretical approach provides robust predictions for the beating dynamics, and their biochemical and mechanical modulation.

### 2019

The original version of this Article contained an error throughout in which an incorrect symbol was used for the diffusion coefficient: it should be cambria math, italicized, and not bold. These have been corrected in both the PDF and HTML versions of the Article.

The classical Debye-Hückel (DH) theory clearly accounts for the origin of screening in electrolyte solutions and works rather well for dilute electrolyte solutions. While the Debye screening length decreases with the ion concentration and is independent of ion size, recent surface-force measurements imply that for concentrated solutions, the screening length exhibits an opposite trend; it increases with ion concentration and depends on the ionic size. The screening length is usually defined by the response of the electrolyte solution to a test charge but can equivalently be derived from the charge-charge correlation function. By going beyond DH theory, we predict the effects of ion size on the charge-charge correlation function. A simple modification of the Coulomb interaction kernel to account for the excluded volume of neighboring ions yields a nonmonotonic dependence of the screening length (correlation length) on the ionic concentration, as well as damped charge oscillations for high concentrations.

Actin filaments associated with myosin motors constitute the cytoskeletal force-generating machinery for many types of adherent cells. These actomyosin units are structurally ordered in muscle cells and, in particular, may be spatially registered across neighboring actin bundles. Such registry or stacking of myosin filaments have been recently observed in ordered actin bundles of even fibroblasts with super-resolution microscopy techniques. We introduce here a model for the dynamics of stacking arising from long-range mechanical interactions between actomyosin units through mutual contractile deformations of the intervening cytoskeletal network. The dynamics of registry involve two key processes: 1) polymerization and depolymerization of actin filaments and 2) remodeling of cross-linker-rich actin adhesion zones, both of which are, in principle, mechanosensitive. By calculating the elastic forces that drive registry and their effect on actin polymerization rates, we estimate a characteristic timescale of tens of minutes for registry to be established, in agreement with experimentally observed timescales for individual kinetic processes involved in myosin stack formation, which we track and quantify. This model elucidates the role of actin turnover dynamics in myosin stacking and explains the loss of stacks seen when actin assembly or disassembly and cross-linking is experimentally disrupted in fibroblasts.

During a first-order phase transition, an interfacial layer is formed between the coexisting phases and kinetically limits homogeneous nucleation of the new phase in the original phase. This inhibition is commonly alleviated by the presence of impurities, often of unknown origin, that serve as heterogeneous nucleation sites for the transition. Living systems present a theoretical opportunity: the regulated structure of living systems allows modelling of the impurities, enabling quantitative analysis and comparison between homogeneous and heterogeneous nucleation mechanisms, usually a difficult task. Here, we formulate an analytical model of heterogeneous nucleation of holes in the nuclear lamina, a phenomenon with implications in cancer metastasis, ageing and other diseases. We then present measurements of hole nucleation in the lamina of nuclei migrating through controlled constrictions and fit the experimental data to our heterogeneous nucleation model as well as a homogeneous model. Surprisingly, we find that different mechanisms dominate depending on the density of filaments that comprise the nuclear lamina.

Mechanical contraction in muscle cells requires Ca to allow myosin binding to actin. Beating cardiomyocytes contain internal Ca stores whose cytoplasmic concentration oscillates. Our theory explains observed single channel dynamics as well as cellular oscillations in spontaneously beating cardiomyocytes. The Ca dependence of channel activity responsible for Ca release includes positive feedback with a delayed response. We use this to predict a dynamical equation for global calcium oscillations with only a few physically relevant parameters. The theory accounts for the observed entrainment of beating to an oscillatory electric or mechanical field.

*Physics Teaching and Learning: Challenging the Paradigm*. p. 111-134 Abstract

### 2018

Meso-porous electrodes (pore width « 1 µm) are a central component in electrochemical energy storage devices and related technologies, based on the capacitive nature of electric double-layers at their surfaces. This requires that such charging, limited by ion transport within the pores, is attained over the device operation time. Here we measure directly electric double layer charging within individual nano-slits, formed between gold and mica surfaces in a surface force balance, by monitoring transient surface forces in response to an applied electric potential. We find that the nano-slit charging time is of order 1 s (far slower than the time of order 3 × 10−2 s characteristic of charging an unconfined surface in our configuration), increasing at smaller slit thickness, and decreasing with solution ion concentration. The results enable us to examine critically the nanopore charging dynamics, and indicate how to probe such charging in different conditions and aqueous environments.

An introduction to the physical properties of living active matter at the mesoscopic scale (tens of nanometers to micrometers) and their unique features compared with dead, nonactive matter is presented. This field of research is increasingly denoted as biological physics where physics includes chemical physics, soft matter physics, hydrodynamics, mechanics, and the related engineering sciences. The focus is on the emergent properties of these systems and their collective behavior, which results in active self-organization and how they relate to cellular-level biological function. These include locomotion (cell motility and migration) forces that give rise to cell division, the growth and form of cellular assemblies in development, the beating of heart cells, and the effects of mechanical perturbations such as shear flow (in the bloodstream) or adhesion to other cells or tissues. An introduction to the fundamental concepts and theory with selected experimental examples related to the authors' own research is presented, including red-blood-cell membrane fluctuations, motion of the nucleus within an egg cell, self-contracting acto-myosin gels, and structure and beating of heart cells (cardiomyocytes), including how they can be driven by an oscillating, mechanical probe.

Shape transitions in developing organisms can be driven by active stresses, notably, active contractility generated by myosin motors. The mechanisms generating tissue folding are typically studied in epithelia. There, the interaction between cells is also coupled to an elastic substrate, presenting a major difficulty for studying contraction induced folding. Here we study the contraction and buckling of active, initially homogeneous, thin elastic actomyosin networks isolated from bounding surfaces. The network behaves as a poroelastic material, where a flow of fluid is generated during contraction. Contraction starts at the system boundaries, proceeds into the bulk, and eventually leads to spontaneous buckling of the sheet at the periphery. The buckling instability resulted from system self-organization and from the spontaneous emergence of density gradients driven by the active contractility. The buckling wavelength increases linearly with sheet thickness. Our system offers a well-controlled way to study mechanically induced, spontaneous shape transitions in active matter.

One of the many ways cells transmit information within their volume is through steady spatial gradients of different proteins. However, the mechanism through which proteins without any sources or sinks form such single-cell gradients is not yet fully understood. One of the models for such gradient formation, based on differential diffusion, is limited to proteins with large ratios of their diffusion constants or to specific protein-large molecule interactions. We introduce a novel mechanism for gradient formation via the coupling of the proteins within a single cell with a molecule, that we call a "pronogen," whose action is similar to that of morphogens in multi-cell assemblies; the pronogen is produced with a fixed flux at one side of the cell. This coupling results in an effectively non-linear diffusion degradation model for the pronogen dynamics within the cell, which leads to a steady-state gradient of the protein concentration. We use stability analysis to show that these gradients are linearly stable with respect to perturbations.

Myosin II filaments form ordered superstructures in both cross-striated muscle and non-muscle cells. In cross-striated muscle, myosin II (thick) filaments, actin (thin) filaments and elastic titin filaments comprise the stereotypical contractile units of muscles called sarcomeres. Linear chains of sarcomeres, called myofibrils, are aligned laterally in registry to form cross-striated muscle cells. The experimentally observed dependence of the registered organization of myofibrils on extracellular matrix elasticity has been proposed to arise from the interactions of sarcomeric contractile elements (considered as force dipoles) through the matrix. Non-muscle cells form small bipolar filaments built of less than 30 myosin II molecules. These filaments are associated in registry forming superstructures (‘stacks’) orthogonal to actin filament bundles. Formation of myosin II filament stacks requires the myosin II ATPase activity and function of the actin filament crosslinking, polymerizing and depolymerizing proteins. We propose that the myosin II filaments embedded into elastic, intervening actin network (IVN) function as force dipoles that interact attractively through the IVN. This is in analogy with the theoretical picture developed for myofibrils where the elastic medium is now the actin cytoskeleton itself. Myosin stack formation in non-muscle cells provides a novel mechanism for the self-organization of the actin cytoskeleton at the level of the entire cell.This article is part of the theme issue ‘Self-organization in cell biology’.

We theoretically predict and compare with experiments, transitions from spontaneous beating to dynamical entrainment of cardiomyocytes induced by an oscillating, external mechanical probe. In accord with recent experiments, we predict the dynamical behavior as a function of the probe amplitude and frequency. The theory is based on a phenomenological model for a non-linear oscillator, motivated by acto-myosin contractility. The generic behavior is independent of the detailed, molecular origins of the dynamics and, consistent with experiment, we find three regimes: spontaneous beating with the natural frequency of the cell, entrained beating with the frequency of the probe, and a "bursting" regime where the two frequencies alternate in time. We quantitatively predict the properties of the "bursting" regime as a function of the amplitude and frequency of the probe. Furthermore, we examine the pacing process in the presence of weak noise and explain how this might relate to cardiomyocyte physiology.

Understanding the structural and thermodynamic properties of surfaces, interfaces, and membranes is important for both fundamental and practical reasons. Important applications include coatings, dispersants, encapsulating agents, and biological materials. Soft materials, important in the development of new materials and the basis of many biological systems, cannot be designed using trial and error methods due to the multiplicity of components and parameters. While these systems can sometimes be analyzed in terms of microscopic mixtures, it is often conceptually simpler to regard them as dispersions and to focus on the properties of the internal interfaces found in these systems. The basic physics centers on the properties of quasi-two-dimensional systems embedded in the three-dimensional world, thus exhibiting phenomena that do not exist in bulk materials. This approach is the basis behind the theoretical presentation of Statistical Thermodynamics of Surfaces, Interfaces, and Membranes. The approach adapted allows one to treat the rich diversity of phenomena investigated in the field of soft matter physics (including both colloid/interface science as well as the materials and macromolecular aspects of biological physics) such as interfacial tension, the roughening transition, wetting, interactions between surfaces, membrane elasticity, and self-assembly. Presented as a set of lecture notes, this book is aimed at physicists, physical chemists, biological physicists, chemical engineers, and materials scientists who are interested in the statistical mechanics that underlie the macroscopic, thermodynamic properties of surfaces, interfaces, and membranes. This paperback edition contains all the material published in the original hard-cover edition as well as additional clarifications and explanations.

### 2017

During migration of cells in vivo, in both pathological processes such as cancer metastasis or physiological events such as immune cell migration through tissue, the cells must move through narrow interstitial spaces that can be smaller than the nucleus. This can induce deformation of the nucleus which, according to recent experiments, may result in rupture of the nuclear envelope that can lead to cell death, if not prevented or healed within an appropriate time. The nuclear envelope, which can be modeled as a double lipid bilayer attached to a viscoelastic gel (lamina) whose elasticity and viscosity primarily depend on the lamin composition, may utilize mechanically induced, self-healing mechanisms that allow the hole to be closed after the deformation-induced strains are reduced by leakage of the internal fluid. Here, we present a viscoelastic model of the evolution of a hole nucleated by deformations of the nuclear lamina and estimate the herniation of chromatin through the hole and its relation to the lamin expression levels in the nuclear envelope.

We theoretically predict the nonlinear elastic responses of polydisperse biopolymer gels to uniaxial compression. We analyze the competition between compressive stiffening due to polymer densification by out-going solvent flow and compressive softening due to continuous polymer buckling. We point out that the polydispersity in polymer lengths can result in an intrinsic, equilibrium mode of nonaffine compression: nonuniform strain but with uniform force distribution, which is found to be more energetically preferable than affine deformation. In this case, the gel softens significantly after the onset of polymer buckling at small compression, but as compression increases, densification-induced stiffening becomes important and a modulus plateau should be observed for a large range of strain. We also relate our results to recent compression experiments on collagen gels and fibrin gels.

Many of the most important molecules of life are polymers. In animals, the most abundant of the proteinaceous polymers are the collagens, which constitute the fibrous matrix outside cells and which can also self-assemble into gels. The physically measurable stiffness of gels, as well as tissues, increases with the amount of collagen, and cells seem to sense this stiffness. An understanding of this mechanosensing process in complex tissues, including fibrotic disease states with high collagen, is now utilizing ' omics data sets and is revealing polymer physics-type, nonlinear scaling relationships between concentrations of seemingly unrelated biopolymers. The nuclear structure protein lamin A provides one example, with protein and transcript levels increasing with collagen 1 and tissue stiffness, and with mechanisms rooted in protein stabilization induced by cytoskeletal stress. Physics-based models of fibrous matrix, cytoskeletal force dipoles, and the lamin A gene circuit illustrate the wide range of testable predictions emerging for tissues, cell cultures, and even stem cell-based tissue regeneration. Beyond the epigenetics of mechanosensing, the scaling in cancer of chromosome copy number variations and other mutations with tissue stiffness suggests that genomic changes are occurring by mechanogenomic processes that now require elucidation.

In the version of this Letter originally published, the numbering of Supplementary Video files and the references to those files in the text did not match. All online versions of the Letter have been corrected so that the Supplementary Videos are numbered sequentially from 1–17.

Although myosin II filaments are known to exist in non-muscle cells(1-2), their dynamics and organization are incompletely understood. Here, we combined structured illumination microscopy with pharmacological and genetic perturbations, to study the process of actomyosin cytoskeleton self-organization into arcs and stress fibres. A striking feature of the myosin II filament organization was their 'registered' alignment into stacks, spanning up to several micrometres in the direction orthogonal to the parallel actin bundles. While turnover of individual myosin II filaments was fast (characteristic half-life time 60s) and independent of actin filament turnover, the process of stack formation lasted a longer time (in the range of several minutes) and required myosin II contractility, as well as actin filament assembly/disassembly and crosslinking (dependent on formin FmnI3, cofilin1 and alpha-actinin-4). Furthermore, myosin filament stack formation involved long-range movements of individual myosin filaments towards each other suggesting the existence of attractive forces between myosin II filaments. These forces, possibly transmitted via mechanical deformations of the intervening actin filament network, may in turn remodel the actomyosin cytoskeleton and drive its self-organization.

### 2016

Mechanobiological studies of cell assemblies have generally focused on cells that are, in principle, identical. Here we predict theoretically the effect on cells in culture of locally introduced biochemical signals that diffuse and locally induce cytoskeletal contractility which is initially small. In steady-state, both the concentration profile of the signaling molecule as well as the contractility profile of the cell assembly are inhomogeneous, with a characteristic length that can be of the order of the system size. The long-range nature of this state originates in the elastic interactions of contractile cells (similar to long-range "macroscopic modes" in non-living elastic inclusions) and the non-linear diffusion of the signaling molecules, here termed mechanogens. We suggest model experiments on cell assemblies on substrates that can test the theory as a prelude to its applicability in embryo development where spatial gradients of morphogens initiate cellular development.

*International Perspectives on Science Education for the Gifted:Key issues and challenges*. p. 43-56 Abstract

Science apprenticeship programmes offer gifted and highly capable high school students the opportunity to participate in research projects in a university setting. Science apprenticeship programmes can serve the needs of these talented and curious youngsters, which are often not met in the regular school science curriculum. Such students are characterized by their motivation to apply their creativity, critical thinking and reasoning capabilities to explore in depth topics that interest and challenge them (Gardner, 2008). These needs can be accommodated via research apprenticeships that embody the complexity of authentic scientiﬁc investigations (Taber, 2007). It is thus expected that participation in cutting-edge scientiﬁc research projects foster these students’ enthusiasm for learning science.

### 2015

The Gerhard M. J. Schmidt Center for Supramolecular Architectures was founded to promote interdisciplinary research in the fields of molecular chemistry, condensed matter chemistry and physics, and biomaterials/biophysics.

We present a model of biopolymer gels that includes two types of elastic nonlinearities, stiffening under extension and softening (due to buckling) under compression, to predict the elastic anisotropy induced by both external as well as internal (e.g., due to cell contractility) stresses in biopolymer gels. We show how the stretch-induced anisotropy and the strain-stiffening nonlinearity increase both the amplitude and power-law range of transmission of internal, contractile, cellular forces, and relate this to recent experiments.

We theoretically predict the rate of transcription (TX) in DNA brushes by introducing the concept of TX dipoles that takes into account the unidirectional motion of enzymes (RNAP) along DNA during transcription as correlated pairs of sources and sinks in the relevant diffusion equation. Our theory predicts that the TX rates dramatically change upon the inversion of the orientation of the TX dipoles relative to the substrate because TX dipoles modulate the concentrations of RNAP in the solution. Comparing our theory with experiments suggests that, in some cases, DNA chain segments are relatively uniformly distributed in the brush, in contrast to the parabolic profile expected for flexible polymer brushes.

We present a generic and unified theory to explain how cells respond to perturbations of their mechanical environment such as the presence of neighboring cells, slowly applied stretch, or gradients of matrix rigidity. Motivated by experiments, we calculate the local balance of forces that give rise to a tendency for the cell to locally move or reorient, with a focus on the contribution of feedback and homeostasis to cell contractility (manifested by a fixed displacement, strain or stress) that acts on the adhesions at the cell boundary. These forces can be either reinforced or diminished by elastic stresses due to mechanical perturbations of the matrix. Our model predicts these changes and how their balance with local protrusive forces that act on the cell's leading edge either increase or decrease the tendency of the cell to locally move (toward neighboring cells or rigidity gradients) or reorient (in the direction of slowly applied stretch or rigidity gradients).

Recent experiments show that both striation, an indication of the structural registry in muscle fibres, as well as the contractile strains produced by beating cardiac muscle cells can be optimized by substrate stiffness. Here we show theoretically how the substrate rigidity dependence of the registry data can be mapped onto that of the strain measurements. We express the elasticity-mediated structural registry as a phase-order parameter using a statistical physics approach that takes the noise and disorder inherent in biological systems into account. By assuming that structurally registered myofibrils also tend to beat in phase, we explain the observed dependence of both striation and strain measurements of cardiomyocytes on substrate stiffness in a unified manner. The agreement of our ideas with experiment suggests that the correlated beating of heart cells may be limited by the structural order of the myofibrils, which in turn is regulated by their elastic environment.

### 2014

We calculate the line tension between domains in phase separated, ternary membranes that comprise line active molecules (linactants) that tend to increase the compatibility of the two phase separating species. The predicted line tension, which depends explicitly on the linactant composition and temperature, is shown to decrease significantly as the fraction of linactants in the membrane increases toward a Lifshitz point, above which the membrane phase separates into a modulated phase. We predict regimes of zero line tension at temperatures close to the mixing transition and clarify the two different ways in which the line tension can be reduced: (1) The linactants uniformly distribute in the system and reduce the compositional mismatch between the two bulk domains. (2) The linactants accumulate at the interface with a preferred orientation. Both of these mechanisms have been observed in recent experiments and simulations. The second one is unique to line active molecules, and our work shows that it is increasingly important at large fraction of linactants and is necessary for the emergence of a regime of zero line tension. The methodology is based on the ternary mixture model proposed by Palmieri and Safran [Palmieri, B.; Safran, S. A. Langmuir 2013, 29, 5246], and the line tension is calculated via variationally derived, self-consistent profiles for the local variation of composition and linactant orientation in the interface region.

We review recent theoretical efforts that predict how line-active molecules can promote lateral heterogeneities (or domains) in model membranes. This fundamental understanding may be relevant to membrane composition in living cells, where it is thought that small domains, called lipid rafts, are necessary for the cells to be functional. The theoretical work reviewed here ranges in scale from coarse grained continuum models to nearly atomistic models. The effect of line active molecules on domain sizes and shapes in the phase separated regime or on fluctuation length scales and lifetimes in the single phase, mixed regime, of the membrane is discussed. Recent experimental studies on model membranes that include line active molecules are also presented together with some comparisons with the theoretical predictions. (C) 2014 Elsevier B.V. All rights reserved.

When a system exchanges energy with a constant-temperature environment, the entropy of the surroundings changes. A lattice model of a fluid thermal reservoir can provide a visualization of the microscopic changes that occur in the surroundings upon energy transfer from the system. This model can be used to clarify the consistency of phenomena such as crystallization or similar phase transitions with the second law of thermodynamics; in those phenomena, students intuitively grasp that the system entropy decreases, but may not have a clear picture of how it is compensated by an increase in the reservoir entropy. The model may be used in the classroom to visually demonstrate how processes in which the entropy of the system decreases can occur spontaneously; specifically, it shows how the reservoir temperature affects the magnitude of the entropy change that occurs upon energy transfer from the system.

### 2013

We analyze the development in students’ understanding of fundamental principles in the context of learning a current interdisciplinary research topic—soft matter—that was adapted to the level of high school students. The topic was introduced in a program for interested 11th grade high school students majoring in chemistry and/or physics, in an off-school setting. Soft matter was presented in a gradual increase in the degree of complexity of the phenomena as well as in the level of the quantitative analysis. We describe the evolution in students’ use of fundamental thermodynamics principles to reason about phase separation—a phenomenon that is ubiquitous in soft matter. In particular, we examine the impact of the use of free energy analysis, a common approach in soft matter, on the understanding of the fundamental principles of thermodynamics. The study used diagnostic questions and classroom observations to gauge the student’s learning. In order to gain insight on the aspects that shape the understanding of the basic principles, we focus on the responses and explanations of two case-study students who represent two trends of evolution in conceptual understanding in the group. We analyze changes in the two case studies’ management of conceptual resources used in their analysis of phase separation, and suggest how their prior knowledge and epistemological framing (a combination of their personal tendencies and their prior exposure to different learning styles) affect their conceptual evolution. Finally, we propose strategies to improve the instruction of these concepts.

A recently proposed ternary mixture model is used to predict fluctuation domain lifetimes in the one phase region. The membrane is made of saturated, unsaturated, and hybrid lipids that have one saturated and one unsaturated hydrocarbon chain. The hybrid lipid is a natural linactant which can reduce the packing incompatibility between saturated and unsaturated lipids. The fluctuation lifetimes are predicted as a function of the hybrid lipid fraction and the fluctuation domain size. These lifetimes can be increased by up to three orders of magnitude compared to the case of no hybrids. With hybrid, small length scale fluctuations have sizable amplitudes even close to the critical temperature and, hence, benefit from enhanced critical slowing down. The increase in lifetime is particularly important for nanometer scale fluctuation domains where the hybrid orientation and the other lipids composition are highly coupled.

One of the most unique physical features of cell adhesion to external surfaces is the active generation of mechanical force at the cell-material interface. This includes pulling forces generated by contractile polymer bundles and networks, and pushing forces generated by the polymerization of polymer networks. These forces are transmitted to the substrate mainly by focal adhesions, which are large, yet highly dynamic adhesion clusters. Tissue cells use these forces to sense the physical properties of their environment and to communicate with each other. The effect of forces is intricately linked to the material properties of cells and their physical environment. Here a review is given of recent progress in our understanding of the role of forces in cell adhesion from the viewpoint of theoretical soft matter physics and in close relation to the relevant experiments.

A ternary mixture model is proposed to describe composition fluctuations in mixed membranes composed of saturated, unsaturated, and hybrid lipids (with one saturated and one unsaturated hydrocarbon chain). The hybrids are line-active and can reduce the packing incompatibility between the saturated and unsaturated lipids. We introduce a lattice model that extends previous studies by taking into account the dependence of the interactions of the hybrid lipids on their orientations in a simple way. A methodology to recast the free energy of the lattice model in terms of a continuous, isotropic field theory is proposed and used to analyze composition fluctuations in the one phase region (above the critical temperature). The effect of hybrid lipids on fluctuation domains rich in saturated/unsaturated lipids is predicted. The correlation length of such fluctuations decreases significantly with increasing amounts of hybrids; this implies that nanoscale fluctuation domains are more probable compared to the case with no hybrids. Smaller correlated fluctuation domains arise even when the temperature is close to a critical point, where very large correlation lengths are normally expected. This decrease in the correlation length is largest as the hybrid composition tends toward a crossover value above which stripelike fluctuations are predicted. This crossover value defines the Lifshitz line. The characteristic wavelength of the stripelike fluctuations is large close to the Lifshitz point but decreases toward a molecular size in a membrane that contains only hybrids. Micrometer size, stripelike domains have recently been observed experimentally in giant unilamelar vesicles (Guys) made of saturated, unsaturated, and hybrid lipids. These results suggest that the line activity of hybrid lipids in such mixtures may be significant only at large hybrid fractions; in that regime, the interface between domains can be diffuse and several hybrid molecules with correlated orientatio

Cells probe their mechanical environment and can change the organization of their cytoskeletons when the elastic and viscous properties of their environment are modified. We use a model in which the forces exerted by small, contractile acto-myosin filaments (e. g., nascent stress fibers in stem cells) on the extracellular matrix are modeled as local force dipoles. In some cases, the strain field caused by these force dipoles propagates quickly enough so that only static elastic interactions need be considered. On the other hand, in the case of significant energy dissipation, strain propagation is slower and may be eliminated completely by the relaxation of the cellular cytoskeleton (e. g., by cross-link dissociation). Here, we consider several dissipative mechanisms that affect the propagation of the strain field in adhered cells and consider these effects on the interaction between force dipoles and their resulting mutual orientations. This is a first step in understanding the development of orientational (nematic) or layering (smectic) order in the cytoskeleton. We use the theory to estimate the propagation time of the strain fields over a cellular distance for different mechanisms and find that in some cases it can be of the order of seconds, thus competing with the cytoskeletal relaxation time. Furthermore, for a simple system of two force dipoles, we predict that in some cases the orientation of force dipoles might change significantly with time, e. g., for short times the dipoles exhibit parallel alignment while for later times they align perpendicularly. DOI: 10.1103/PhysRevE.87.042703

We present design guidelines for using Adapted Primary Literature (APL) as part of current interdisciplinary topics to introductory physics students. APL is a text genre that allows students to comprehend a scientific article, while maintaining the core features of the communication among scientists, thus representing an authentic scientific discourse. We describe the adaptation of a research paper by Nobel Laureate Paul Flory on phase equilibrium in polymer-solvent mixtures that was presented to high school students in a project-based unit on soft matter. The adaptation followed two design strategies: a) Making explicit the interplay between the theory and experiment. b) Re-structuring the text to map the theory onto the students' prior knowledge. Specifically, we map the theory of polymer-solvent systems onto a model for binary mixtures of small molecules of equal size that was already studied in class.

### 2012

A dynamical model of a soft, thermally fluctuating two-dimensional tube is used to study the effect of thermal fluctuations of a confining environment on diffusive transport. The tube fluctuations in both space and time are driven by Brownian motion and suppressed by surface tension and the rigidity of the surrounding environment. The dynamical fluctuations modify the concentration profile boundary condition at the tube surface. They decrease the diffusive transport rate through the tube for two important cases: uniform tube fluctuations (wave vector, q = 0 mode) for finite tube lengths and fluctuations of any wave vector for infinitely long tubes.

Contractile function of striated muscle cells depends crucially on the almost crystalline order of actin and myosin filaments in myofibrils, but the physical mechanisms that lead to myofibril assembly remains ill-defined. Passive diffusive sorting of actin filaments into sarcomeric order is kinetically impossible, suggesting a pivotal role of active processes in sarcomeric pattern formation. Using a one-dimensional computational model of an initially unstriated actin bundle, we show that actin filament treadmilling in the presence of processive plus-end crosslinking provides a simple and robust mechanism for the polarity sorting of actin filaments as well as for the correct localization of myosin filaments. We propose that the coalescence of crosslinked actin clusters could be key for sarcomeric pattern formation. In our simulations, sarcomere spacing is set by filament length prompting tight length control already at early stages of pattern formation. The proposed mechanism could be generic and apply both to premyofibrils and nascent myofibrils in developing muscle cells as well as possibly to striated stress-fibers in non-muscle cells.

The aggregation of inhomogeneously charged colloids with the same average charge is analyzed using Monte Carlo simulations. We find aggregation of colloids for sizes in the range 10-200 nm, which is similar to the range in which aggregation is observed in several experiments. The attraction arises from the strongly correlated electrostatic interactions associated with the increase in the counterion density in the region between the particles; this effect is enhanced by the discreteness and mobility of the surface charges. Larger colloids attract more strongly when their surface charges are discrete. We study the aggregation as functions of the surface charge density, counterion valence, and volume fraction.

We show how strain stiffening affects the elastic response to internal forces, caused either by material defects and inhomogeneities or by active forces that molecular motors generate in living cells. For a spherical force dipole in a material with a strongly nonlinear strain energy density, strains change sign with distance, indicating that, even around a contractile inclusion or molecular motor, there is radial compression; it is only at a long distance that one recovers the linear response in which the medium is radially stretched. Scaling laws with irrational exponents relate the far-field renormalized strain to the near-field strain applied by the inclusion or active force.

Adherent cells exert active forces and elastically deform their substrate. We consider a simple model geometry of a spread cell adhered to a thick substrate layer. We show how anisotropic cell shapes and active cell contractility induce cytoskeletal shear within the cell in a substrate-stiffness dependent manner. This cytoskeletal shear represents a possible mechanical guidance cue for cell force polarization, and could, for example, trigger initial nematic alignment of nascent stress fibers at early stages of cytoskeletal organization. Cell induced substrate strains propagate a depth into the substrate that is comparable to the linear dimension of the spread cell. As a consequence, cellular strains depend on the lateral dimensions of the spread cell. We employ Fourier techniques and a mean-field coupling approximation, which allows both for analytical progress and qualitative insight.

We present a theory that predicts that cholesterol molecules phase separate from multicomponent lipid bilayer membranes, but not from monolayers of the same composition. In our model, the tilting of cholesterol molecules in monolayers is not energetically favorable because their exocyclic chains are very short. However, in bilayers, two correlated cholesterol molecules in the opposing leaflets of bilayers can form "dimers'', whose chains are effectively twice as long and can tilt cooperatively. Our theory predicts that the cooperative tilting of these dimers drives the phase separations of cholesterol in bilayers.

We describe an elective course on soft matter at the level of introductory physics. Soft matter physics serves as a context that motivates the presentation of basic ideas in statistical thermodynamics and their applications. It also is an example of a contemporary field that is interdisciplinary and touches on chemistry, biology, and physics. We outline a curriculum that uses the lattice gas model as a quantitative and visual tool, initially to introduce entropy, and later to facilitate the calculation of interactions. We demonstrate how free energy minimization can be used to teach students to understand the properties of soft matter systems such as the phases of fluid mixtures, wetting of interfaces, self-assembly of surfactants, and polymers. We discuss several suggested activities in the form of inquiry projects which allow students to apply the concepts they have learned to experimental systems. (C) 2012 American Association of Physics Teachers. [DOI: 10.1119/1.3647995]

בשלוש השנים האחרונות מתקיימת תכנית ייחודית בשם "חומר רך ומבולגן" המיועדת לתלמידים הלומדים במגמת כימיה או פיזיקה ברמת 5 יחידות לימוד. התכנית, בהיקף 2 יח"ל, היא תוצר של שתוף פעולה בין המחלקה להוראת המדעים והמחלקה לחומרים ופני שטח במכון ויצמן, ובית הספר למדע עכשווי במכון דוידסון לחינוך מדעי. במסגרת התכנית התלמידים מכירים מודלים מתחום התרמודינמיקה הסטטיסטית המסבירים תופעות מורכבות, החל מהפרדת פאזות בתערובות בינאריות, המשך להתארגנות עצמית של מולקולות סבון, וכלה במודלים לתמיסות של פולימרים. להלן נספר מעט על חומרים רכים, נציג את המסגרת התאורטית המשמשת להבנת הפרדת פאזות בתערובות בינאריות והתארגנות עצמית של חומרים פעילי שטח כמו סבון, ונציג ניסוי המאפשר לאשר את המודל התיאורטי.

### 2011

Recognition of external mechanical signals is vital for mammalian cells. Cyclic stretch, e. g. around blood vessels, is one such signal that induces cell reorientation from parallel to almost perpendicular to the direction of stretch. Here, we present quantitative analyses of both, cell and cytoskeletal reorientation of umbilical cord fibroblasts. Cyclic strain of preset amplitudes was applied at mHz frequencies. Elastomeric chambers were specifically designed and characterized to distinguish between zero strain and minimal stress directions and to allow accurate theoretical modeling. Reorientation was only induced when the applied stretch exceeded a specific amplitude, suggesting a non-linear response. However, on very soft substrates no mechanoresponse occurs even for high strain. For all stretch amplitudes, the angular distributions of reoriented cells are in very good agreement with a theory modeling stretched cells as active force dipoles. Cyclic stretch increases the number of stress fibers and the coupling to adhesions. We show that changes in cell shape follow cytoskeletal reorientation with a significant temporal delay. Our data identify the importance of environmental stiffness for cell reorientation, here in direction of zero strain. These in vitro experiments on cultured cells argue for the necessity of rather stiff environmental conditions to induce cellular reorientation in mammalian tissues.

The discovery that adherent tissue cells actively sustain internal tension and exert mechanical forces on their surroundings has opened new vistas in the field of cell and tissue mechanics. Cellular forces, generated by acto-myosin contractility, play a central role in numerous aspects of cell behavior and function. Apart from the various specific functions that cells perform by applying forces (e.g., wound healing. remodeling of the extracellular matrix and muscle contraction), cells also apply stresses as a generic means for sensing and responding to the mechanical nature of their environment. In addition, the internal tension plays a role in actively controlling the elastic moduli and shape stability of the cell. In this review, we survey recent theoretical and experimental studies of the physical consequences of cell mechanical activity including its role in cell morphology, adhesion strength, stress-fiber polarization, and the elastic properties of cells. We also discuss the role of cell mechanics in orienting cellular assemblies and in the response of cells to external loads. (C) 2011 Elsevier Ltd. All rights reserved.

The remarkable striation of muscle has fascinated many for centuries. In developing muscle cells, as well as in many adherent, nonmuscle cell types, striated, stress fiberlike structures with sarcomere-periodicity tend to register: Based on several studies, neighboring, parallel fibers at the basal membrane of cultured cells establish registry of their respective periodic sarcomeric architecture, but, to our knowledge, the mechanism has not yet been identified. Here, we propose for cells plated on an elastic substrate or adhered to a neighboring cell, that acto-myosin contractility in striated fibers close to the basal membrane induces substrate strain that gives rise to an elastic interaction between neighboring striated fibers, which in turn favors interfiber registry. Our physical theory predicts a dependence of interfiber registry on externally controllable elastic properties of the substrate. In developing muscle cells, registry of striated fibers (premyofibrils and nascent myofibrils) has been suggested as one major pathway of myofibrillogenesis, where it precedes the fusion of neighboring fibers. This suggests a mechanical basis for the optimal myofibrillogenesis on muscle-mimetic elastic substrates that was recently observed by several groups in cultures of mouse-, human-, and chick-derived muscle cells.

Despite their neutrality, surfaces or membranes with equal amount of positive and negative charge can exhibit long-range electrostatic interactions if the surface charge is heterogeneous; this can happen when the surface charges form finite size domain structures These domains can be formed in lipid membranes Inhere the balance of the different ranges of strong but short ringed hydrophobic interactions and longer ranged electrostatic repulsion result in a finite, stable domain size. If the domain size is large enough, oppositely charged domains in two opposing surfaces or membranes can be strongly correlated by the elecrostatic interactions, these correlations give rise to an attractive interaction of the two membranes or surfaces over separations on the order of the domain size. We use numerical simulations to demonstrate the existence of strong attractions at separations of tens of nanometers. Large line tensions result in larger domains but also increase the charge density within the domain: This promotes correlations and, as a result, increases the intermembrane attraction. On the other hand, increasing the salt concentration increases both the domain size and degree of domain anticorrelation, but the interactions are ultimately reduced due to increased screening. The result is a decrease in the net attraction as salt concentration is increased.

*IMAGING, MANIPULATION, AND ANALYSIS OF BIOMOLECULES, CELLS, AND TISSUES IX*. Farkas DL., Nicolau DV. & Leif RC.(eds.). Vol. 7902. (1Proceedings of SPIE). Abstract

We have quantitatively and systemically measured the morphologies and dynamics of fluctuations in human RBC membranes using a full-field laser interferometry technique that accurately measures dynamic membrane fluctuations. We present conclusive evidence that the presence of adenosine 5'-triphosphate (ATP) facilitates nonequilibrium dynamic fluctuations in the RBC membrane and that these fluctuations are highly correlated with specific regions in the biconcave shape of RBCs. Spatial analysis reveals that these nonequilibrium membrane fluctuations are enhanced at the scale of the spectrin mesh size. Our results indicate the presence of dynamic remodeling in the RBC membrane cortex powered by ATP, which results in nonequilibrium membrane fluctuations. (This conference proceeding paper is primary based on our recent publication - please refer to YK Park et al., Proc. Nat. Acad. Sci., 107, 1289 (2010) for details.)

We predict spontaneous nematic order in an ensemble of active force generators with elastic interactions as a minimal model for early nematic alignment of short stress fibers in non-motile, adhered cells. Mean-field theory is formally equivalent to Maier-Saupe theory for a nematic liquid. However, the elastic interactions are long-ranged (and thus depend on cell shape and matrix elasticity) and originate in cell activity. Depending on the density of force generators, we find two regimes of cellular rigidity sensing for which orientational, nematic order of stress fibers depends on matrix rigidity either in a step-like manner or with a maximum at an optimal rigidity. Copyright (C) EPLA, 2011

Hybrid lipids (with one saturated tail and one unsaturated tail) have been proposed as agents that can reduce the line tension between domains. We use a liquid crystal model to predict the effect of the degree of unsaturation in the unsaturated tails of hybrid lipids on the miscibility phase diagram and line tension between domains in membranes comprising saturated lipids (with two saturated tails), hybrid lipids, and cholesterol (SHC membranes). In contrast to SHC membranes containing hybrid lipids with a single (or small) degree of unsaturation, for which the phase diagram shows two distinct two-phase regions, SHC membranes containing hybrid lipids with degrees of unsaturation larger than a threshold value exhibit a single two-phase region. This enables hybrid lipids with larger degrees of unsaturation to steadily increase their chain order with decreasing temperature and enhances the reduction of the line tension. This theory predicts that the line tension between domains in SHC membranes is sensitive to the degrees of unsaturation of hybrid lipids.

### 2010

We use a liquid-crystal model to predict that hybrid lipids (lipids that have one saturated and one unsaturated tail) can stabilize line interfaces between domains in mixed membranes of saturated lipids, hybrid lipids, and cholesterol (SHC membranes). The model predicts the phase separation of SHC membranes with both parabolic and loop binodals depending on the cholesterol concentration, modeled via an effective pressure. In some cases, the hybrid lipids can reduce the line tension to zero in SHC membranes at temperatures that approach the critical temperature as the pressure is increased. The differences in the hybrid saturated tail conformational order in bulk and at the interface are responsible for the reduction of the line tension. Copyright (C) EPLA, 2010

The shape and differentiated state of many cell types are highly sensitive to the rigidity of the microenvironment. The physical mechanisms involved, however, are unknown. Here, we present a theoretical model and experiments demonstrating that the alignment of stress fibres within stem cells is a non-monotonic function of matrix rigidity. We treat the cell as an active elastic inclusion in a surrounding matrix, allowing the actomyosin forces to polarize in response to elastic stresses developed in the cell. The theory correctly predicts the monotonic increase of the cellular forces with the matrix rigidity and the alignment of stress fibres parallel to the long axis of cells. We show that the anisotropy of this alignment depends non-monotonically on matrix rigidity and demonstrate it experimentally by quantifying the orientational distribution of stress fibres in stem cells. These findings offer physical insight into the sensitivity of stem-cell differentiation to tissue elasticity and, more generally, introduce a cell-type-specific parameter for actomyosin polarizability.

The active regulation of cellular forces during cell adhesion plays an important role in the determination of cell size, shape, and internal structure. While on flat, homogeneous and isotropic substrates some cells spread isotropically, others spread anisotropically and assume elongated structures. In addition, in their native environment as well as in vitro experiments, the cell shape and spreading asymmetry can be modulated by the local distribution of adhesive molecules and topography of the environment. We present a simple elastic model and experiments on stem cells to explain the variation of cell size with the matrix rigidity. In addition, we predict the experimental consequences of two mechanisms of acto-myosin polarization and focus here on the effect of the cell spreading asymmetry on the regulation of the stress-fiber alignment in the cytoskeleton. We show that when cell spreading is sufficiently asymmetric the alignment of acto-myosin forces in the cell increases monotonically with the matrix rigidity; however, in general this alignment is non-monotonic, as shown previously. These results highlight the importance of the symmetry characteristics of cell spreading in the regulation of cytoskeleton structure and suggest a mechanism by which different cell types may acquire different morphologies and internal structures in different mechanical environments.

The response of cells to shear flow is primarily determined by the asymmetry of the external forces and moments that are sensed by each member of a focal adhesion pair connected by a contractile stress fiber. In the theory presented here, we suggest a physical model in which each member of such a pair of focal adhesions is treated as an elastic body subject to both a myosin-activated contractile force and the shear stress induced by the external flow. The elastic response of a focal adhesion complex is much faster than the active cellular processes that determine the size of the associated focal adhesions and the direction of the complex relative to the imposed flow. Therefore, the complex attains its mechanical equilibrium configuration which may change because of the cellular activity. Our theory is based on the experimental observation that focal adhesions modulate their cross-sectional area in order to attain an optimal shear. Using this assumption, our elastic model shows that such a complex can passively change its orientation to align parallel to the direction of the flow.

A simple model of the line activity of a hybrid lipid (e.g., POPC) with one fully saturated chain and one partially unsaturated chain demonstrates that these lipids preferentially pack at curved interfaces between phase-separated saturated and unsaturated domains. We predict that the domain sizes typically range from tens to hundreds of nm, depending on molecular interactions and parameters such as molecular volume and area per headgroup, in the bulk fluid phase. The role of cholesterol is taken into account by an effective change in the headgroup areas and the domain sizes are predicted to increase with cholesterol concentration.

Membranes containing highly charged biomolecules can have a minimal free-energy state at small separations that originates in the strongly correlated electrostatic interactions mediated by counterions. This phenomenon can lead to a condensed, lamellar phase of charged membranes that coexists in thermodynamic equilibrium with a very dilute membrane phase. Although the dilute phase is mostly water, entropy dictates that this phase must contain some membranes and counterions. Thus, electrostatics alone can give rise to the coexistence of a condensed and an unbound lamellar phase. We use numerical simulations to predict the nature of this coexistence when the charge density of the membrane is large, for the case of multivalent counterions and for a membrane charge that is characteristic of biomolecules. We also investigate the effects of counterion size and salt on the two coexisting phases. With increasing salt concentration, we predict that electrostatic screening by salt can destroy the phase separation.

We review the characteristics and differences of the adhesion of living cells to surfaces compared to the adhesion of inert matter. In particular, we show that while mimicking cell adhesion by vesicles sheds light on the role of membrane fluctuations and ligand phase separation, it neglects the role of active processes that in live cells control the number of adhesion proteins and their spatial organization. Recent models that incorporate active cell contractility are shown to predict some of the features. In this review, we mainly focus on thermodynamic modeling approaches and outline challenges for future theoretical work. Special Issue on Surface and Interfacial Aspects of Cell Adhesion, Part 2

The remarkable deformability of the human red blood cell (RBC) results from the coupled dynamic response of the phospholipid bilayer and the spectrin molecular network. Here we present quantitative connections between spectrin morphology and membrane fluctuations of human RBCs by using dynamic full-field laser interferometry techniques. We present conclusive evidence that the presence of adenosine 5'-triphosphate (ATP) facilitates non-equilibrium dynamic fluctuations in the RBC membrane that are highly correlated with the biconcave shape of RBCs. Spatial analysis of the fluctuations reveals that these non-equilibrium membrane vibrations are enhanced at the scale of spectrin mesh size. Our results indicate that the dynamic remodeling of the coupled membranes powered by ATP results in non-equilibrium membrane fluctuations manifesting from both metabolic and thermal energies and also maintains the biconcave shape of RBCs.

*IUTAM Symposium on Cellular, Molecular and Tissue Mechanics*. Arruda E. M. & Garikipati K.(eds.). Vol. 16. p. 91-101 Abstract

We present a theoretical treatment of the orientational response to external stress of active, contractile cells embedded in a gel-like elastic medium. The theory includes random forces as well as forces that arise from the deformation of the matrix and those due to the internal regulation of the stress fibers and focal adhesions of the cell. We calculate both the static and high frequency limits of the orientational response in terms of the cellular polarizability. For systems in which the forces due to regulation and activity dominate the mechanical forces, we show that there is a non-linear dynamical response which, in the high frequency limit, causes the cell to orient nearly perpendicular to the direction of the applied stress.

*Nuclear Mechanics And Genome Regulation*. p. 143-175 (1Methods in Cell Biology). Abstract

Recent discoveries have established that mechanical properties of the cellular environment such as its rigidity, geometry, and external stresses play an important role in determining the cellular function and fate. Mechanical properties have been shown to influence cell shape and orientation, regulate cell proliferation and differentiation, and even govern the development and organization of tissues. In recent years, many theoretical and experimental investigations have been carried out to elucidate the mechanisms and consequences of the mechanosensitivity of cells. In this review, we discuss recent theoretical concepts and approaches that explain and predict cell mechanosensitivity. We focus on the interplay of active and passive processes that govern cell-cell and cell-matrix interactions and discuss the role of this interplay in the processes of cell adhesion, regulation of cytoskeleton mechanics and the response of cells to applied mechanical stresses.

### 2009

The nonlinear dependence of cellular orientation on an external, time-varying stress field determines the distribution of orientations in the presence of noise and the characteristic time, tau(c), for the cell to reach its steady-state orientation. The short, local cytoskeletal relaxation time distinguishes between high-frequency (nearly perpendicular) and low-frequency (random or parallel) orientations. However, tau(c) is determined by the much longer, orientational relaxation time. This behavior is related to experiments for which we predict the angle and characteristic time as a function of frequency.

We present a theoretical model to explain recent observations of the orientational response of cells to unidirectional curvature. Experiments show that some cell types when plated on a rigid cylindrical surface tend to reorient their shape and stress fibers along the axis of the cylinder, while others align their stress fibers perpendicular to that axis. Our model focuses on the competition of the shear stress-that results from cell adhesion and active contractility-and the anisotropic bending stiffness of the stress fibers. We predict the cell orientation angle that results from the balance of these two forces in a mechanical equilibrium. The conditions under which the different experimental observations can be obtained are discussed in terms of the theory.

Using a surface force balance with fast video analysis, we have measured directly the attractive forces between oppositely charged solid surfaces (charge densities sigma(+), sigma(-)) across water over the entire range of interaction, in particular, at surface separations D below the Debye screening length lambda(S). At very low salt concentration we find a long-ranged attraction between the surfaces (onset ca. 100 nm), whose variation at D <lambda(S) agrees well with predictions based on solving the Poisson-Boltzmann theory, when due account is taken of the independently-determined surface charge asymmetry (sigma(+) not equal vertical bar sigma(-)vertical bar).

Cell membranes contain small domains (on the order of nanometers in size, sometimes called rafts) of lipids whose hydrocarbon chains are more ordered than those of the surrounding bulk-phase lipids. Whether these domains are fluctuations, metastable, or thermodynamically stable, is still unclear. Here, we show theoretically how a lipid with one saturated hydrocarbon chain that prefers the ordered environment and one partially unsaturated chain that prefers the less ordered phase, can act as a line-active component. We present a unified model that treats the lipids in both the bulk and at the interface and show how they lower the line tension between domains, eventually driving it to zero at sufficiently large interaction strengths or at sufficiently low temperatures. In this limit, finite-sized domains stabilized by the packing of these hybrid lipids can form as equilibrium structures.

This special issue of the Journal of Physical Chemistry contains research articles on the properties of soft matter—polymers, colloids, liquid crystals, surfactant self-assembly, wetting, and biologically inspired materials. As mentioned in the biography of de Gennes, our understanding of these systems was, in many cases, formed and, in all cases, vastly informed by the concepts and analogies that he introduced. Some of the papers published here are directly related to de Gennes’ particular contributions, and we mention a few representative examples of such papers.

### 2008

We present a comprehensive, theoretical treatment of the orientational response to external stress of active, contractile cells embedded in a gel-like elastic medium. The theory includes both the forces that arise from the deformation of the matrix as well as forces due to the internal regulation of the stress fibers and focal adhesions of the cell. We calculate the time-dependent response of both the magnitude and the direction of the elastic dipole that characterizes the active forces exerted by the cell, for various situations. For static or quasistatic external stress, cells orient parallel to the stress while for high frequency dynamic external stress, cells orient nearly perpendicular. Both numerical and analytical calculations of these effects are presented. In addition we predict the relaxation time for the cellular response for both slowly and rapidly varying external stresses; several characteristic scaling regimes for the relaxation time as a function of applied frequency are predicted. We also treat the case of cells for which the regulation of the stress fibers and focal adhesions is controlled by strain (instead of stress) and show that the predicted dependence of the cellular orientation on the Poisson ratio of the matrix can differentiate strain vs stress regulation of cellular response.

Recent experiments have measured attractive interactions between two surfaces that each bear two molecular species with opposite charge. Such surfaces form charged domains of finite size. We present a theoretical model that predicts the dependence of the domain size, phase behavior and the interlayer forces as a function of spacing and salt concentration for two such interacting surfaces. A strong correlation between two length scales, the screening length and the surface separation, at the spinodal is shown. Remarkably, the first-order phase transition to infinite sized domains depends logarithmically on the ratio of the domain size to the molecular size. Finally, we fit the predicted pressure with experiments.

Cell focal adhesions are micrometer-sized aggregates of proteins that anchor the cell to the extracellular matrix. Within the cell, these adhesions are connected to the contractile, actin cytoskeleton; this allows the adhesions to transmit forces to the surrounding matrix and makes the adhesion assembly sensitive to the rigidity of their environment. In this article, we predict the dynamics of focal adhesions as a function of the rigidity of the substrate. We generalize previous theories and include the fact that the dynamics of proteins that adsorb to adhesions are also driven by their coupling to cell contractility and the deformation of the matrix. We predict that adhesions reach a finite size that is proportional to the elastic compliance of the substrate, on a timescale that also scales with the compliance: focal adhesions quickly reach a relatively small, steady-state size on soft materials. However, their apparent sliding is not sensitive to the rigidity of the substrate. We also suggest some experimental probes of these ideas and discuss the nature of information that can be extracted from cell force microscopy on deformable substrates.

We predict theoretically the steady-state orientation of cells subject to dynamical stresses that vary more quickly than the cell relaxation time. We show that the orientation is a strong function of the Poisson's ratio, P, of the matrix when cell activity is governed by the matrix strain; if cell activity is governed by the matrix stress, the orientation depends only weakly on V. These results can be used to differentiate systems in which the strain or the stress determine the setpoint for the mechanosensitivity of cells.

### 2007

The equilibrium microstructures in microemulsions and other self-assembled systems show complex, connected shapes such as symmetric bicontinuous spongelike structures and asymmetric bicontinuous networks formed by cylinders interconnected at junctions. In microemulsions, these cylinder network microstructures may mediate the structural transition from a spherical or globular phase to the bicontinuous microstructure. To understand the structural and statistical properties of such cylinder network microstructures as measured by scattering experiments, models are needed to extract the real-space structure from the scattering data. In this paper, we calculate the scattering functions appropriate for cylinder network microstructures. We focus on such networks that contain a high density of network junctions that connect the cylindrical elements. In this limit, the network microstructure can be regarded as an assembly of randomly oriented, closed packed network junctions (i.e., the cylinder scattering contributions are neglected). Accordingly, the scattering spectrum of the network microstructure can be calculated as the product of the junction number density, the junction form factor, which describes the scattering from the surface of a single junction, and a structure factor, which describes the local correlations of different junctions due to junction interactions (including their excluded volume). This approach is applied to analyze the scattering data from a bicontinuous microemulsion with equal volumes of water and oil. In a second approach, we included the cylinder scattering contribution in the junction form factor by calculating the scattering intensity of Y junctions to which three rods with spherical cross section are attached. The respective theoretical predictions are compared with results of neutron scattering measurements on a water-in-oil microemulsion with a connected microstructure. (c) 2007 American Institute of Physics.

The fluctuation spectra and the intermembrane interaction of two membranes at a fixed average distance are investigated. Each membrane can either be a fluid or a solid membrane, and in isolation, its fluctuations are described by a bare or a wave-vector-dependent bending modulus, respectively. The membranes interact via their excluded-volume interaction; the average distance is maintained by an external, homogeneous pressure. For strong coupling, the fluctuations can be described by a single, effective membrane that combines the elastic properties. For weak coupling, the fluctuations of the individual, noninteracting membranes are recovered. The case of a composite membrane consisting of one fluid and one solid membrane can serve as a microscopic model for the plasma membrane and cytoskeleton of the red blood cell. We find that, despite the complex microstructure of bilayers and cytoskeletons in a real cell, the fluctuations with wavelengths lambda greater than or similar to 400 nm are well described by the fluctuations of a single, polymerized membrane (provided that there are no inhomogeneities of the microstructure). The model is applied to the fluctuation data of discocytes ("normal" red blood cells), a stomatocyte, and an echinocyte. The elastic parameters of the membrane and an effective temperature that quantifies active, metabolically driven fluctuations are extracted from the experiments.

Several cell types, among them red blood cells, have a cortical, two-dimensional (2D) network of filaments sparsely attached to their lipid bilayer. In many mammalian cells, this 2D polymer network is connected to an underlying 3D, more rigid cytoskeleton. In this paper, we consider the pressure exerted by the thermally fluctuating, cortical network of filaments on the bilayer and predict the bilayer deformations that are induced by this pressure. We treat the filaments as flexible polymers and calculate the pressure that a network of such linear chains exerts on the bilayer; we then minimize the bilayer shape in order to predict the resulting local deformations. We compare our predictions with membrane deformations observed in electron micrographs of red blood cells. The polymer pressure along with the resulting membrane deformation can lead to compartmentalization, regulate in-plane diffusion and may influence protein sorting as well as transmit signals to the polymerization of the underlying 3D cytoskeleton.

Recent experiments have shown that salt solutions containing surfaces with two oppositely charged species show stable, possibly equilibrium, structures with finite domain sizes. The short-range interactions between the two species would normally result in phase separation that is driven by the line tension with macroscopically large domains of each species. In this paper, we show that, when at least one of the charged species is mobile, finite domains can occur in equilibrium. The domain size is determined by a competition of the electrostatic free energy that promotes charge mixing and small domains, with the line tension that promotes macroscopic phase separation. We calculate the equilibrium patch size as a function of the surface charge and the concentration of dissolved monovalent salts in the bulk phase. An important finding is the prediction of a first-order transition from finite patches to macroscopic phase separation of the two charge species as the salt concentration is increased.

Many physiological processes depend on the response of biological cells to mechanical forces generated by the contractile activity of the cell or by external stresses. Using a simple theoretical model that includes the forces due to both the mechanosensitivity of cells and the elasticity of the matrix, we predict the dynamics and orientation of cells in both the absence and presence of applied stresses. The model predicts many features observed in measurements of cellular forces and orientation including the increase with time of the cellular forces in the absence of applied stress and the consequent decrease of the force in the presence of quasi-static stresses. We also explain the puzzling observation of parallel alignment of cells for static and quasi-static stresses and of nearly perpendicular alignment for dynamically varying stresses. In addition, we predict the response of the cellular orientation to a sinusoidally varying applied stress as a function of frequency.

Mechanical forces generated by contractile cells allow the cells to sense their environment and to interact with other cells. By locally pulling on their environment, cells can sense and respond to mechanical features such as the local stress (or strain), the shape of a cellular domain, and the surrounding rigidity; at the same time, they also modify the mechanical state of the system. This creates a mechanical feedback loop that can result in self-polarization of cells. In this paper, we present a quantitative mechanical model that predicts the self-polarization of cells in spheroidally shaped domains, comprising contractile cells and an elastic matrix, that are embedded in a three-dimensional, cell-free gel. The theory is based on a generalization of the known results for passive inclusions in solids to include the effects of cell activity. We use the active cellular susceptibility tensor presented by Zemel [Phys. Rev. Lett. 97, 128103 (2006)] to calculate the polarization response and hence the elastic stress field developed by the cells in the cellular domain. The cell polarization is analyzed as a function of the shape and the elastic moduli of the cellular domain compared with the cell-free surrounding material. Consistent with experiment, our theory predicts the development of a stronger contractile force for cells in a gel that is surrounded by a large, cell-free material whose elastic modulus is stiffer than that of the gel that contains the cells. This provides a quantitative explanation of the differences in the development of cellular forces as observed in free and fixed gels. In the case of an asymmetrically shaped (spheroidal) domain of cells, we show that the anisotropic elastic field within the domain leads to a spontaneous self-polarization of the cells along the long axis of the domain.

We predict the nature (attractive or repulsive) and range (exponentially screened or long-range power law) of the electrostatic interactions of oppositely charged, planar plates as a function of the salt concentration and surface charge densities (whose absolute magnitudes are not necessarily equal). An analytical expression for the crossover between attractive and repulsive pressure is obtained as a function of the salt concentration. This condition reduces to the high-salt limit of Parsegian and Gingell where the interaction is exponentially screened and to the zero salt limit of Lau and Pincus in which the important length scales are the inter-plate separation and the Gouy-Chapman length. In the regime of low salt and high surface charges we predict-for any ratio of the charges on the surfaces-that the attractive pressure is long-ranged as a function of the spacing. The attractive pressure is related to the decrease in counter-ion concentration as the inter-plate distance is decreased. Our theory predicts several scaling regimes with different scaling expressions for the pressure as a function of salinity and surface charge densities. The pressure predictions can be related to surface force experiments of oppositely charged surfaces that are prepared by coating one of the mica surfaces with an oppositely charged polyelectrolyte. Copyright (C) EPLA, 2007.

### 2006

Cylindrical micelles are known to exhibit two types of morphologies: branched networks and linear, worm-like (or thread-like) micelles. These structures correspond to two types of topological defects: end-caps and junction points. Although either type of defect increases the micelle energy (when compared to the cylindrical sections), they are stabilized by an increase in the translational (end-caps) or configurational (junctions) entropy. End-caps reduce the length of the cylindrical micelles, resulting in a suspension of linear, worm-like micelles. Y-junction branch points cause the formation of a network structure that may percolate and coexist thermodynamically with a "sol" of finite cylinders with end-caps. In this paper, we review current experimental and theoretical studies of non-ionic cylindrical micelles in aqueous solutions. We focus on single and multicomponent amphiphiles, and consider both small molecules and macromolecules (polymers), in order to identify the driving forces that determine the type of topological 'defect' and the resulting system morphology. (c) 2006 Elsevier B.V. All rights reserved.

Cells play an active role in the maintenance of mechanical homeostasis within tissues and their response to elastic forces is important for tissue engineering. We predict the collective response of an ensemble of contractile cells in a three-dimensional elastic medium to externally applied strain fields. Motivated by experiment, we model the cells as polarizable force dipoles that change their orientation in response to the local elastic strain. The analogy between the mechanical response of these systems and the dielectric response of polar molecules is used to calculate the elastic response function. We use this analogy to evaluate the average cell orientation, the mean polarization stress, and the effective elastic constants of the material, as a function of the cell concentration and matrix properties.

Cell/matrix adhesions are modulated by cytoskeletal or external stresses and adapt to the mechanical properties of the extracellular matrix. We propose that this mechanosensitivity arises from the activation of a mechanosensor located within the adhesion itself. We show that this mechanism accounts for the observed directional growth of focal adhesions and the reduction or even cessation of their growth when cells adhere to a soft extracellular matrix. We predict quantitatively that both the elasticity and the thickness of the matrix play a key role in the dynamics of focal adhesions. Two different types of dynamics are expected depending on whether the thickness of the matrix is of order of or much larger than the adhesion size. In the latter situation, we predict that the adhesion region reaches a saturation size that can be tuned by the mechanical properties of the matrix.

Focal adhesions are micrometer-sized protein aggregates that connect actin stress fibers to the extracellular matrix, a network of macromolecules surrounding tissue cells. The actin fibers are under tension due to actin-myosin contractility. Recent measurements have shown that as the actin force is increased, these adhesions grow in size and in the direction of the force. This is in contrast to the growth of condensed domains of surface-adsorbed molecules in which the dynamics are isotropic. We predict these force-sensitive, anisotropic dynamics of focal adhesions from a model for the adsorption of proteins from the cytoplasm to the adhesion site. Our theory couples the mechanical forces and elasticity to the adsorption dynamics via force-induced conformational changes of molecular-sized mechanosensors located in the focal adhesion. We predict the velocity of both the front and back of the adhesion as a function of the applied force. In addition, our results show that the relative motion of the front and back of the adhesion is asymmetric and in different ranges of forces, the adhesion can either shrink or grow in the direction of the force.

We show that the exponential length distribution that is typical of actin. laments under physiological conditions dramatically narrows in the presence of i) crosslinker proteins or ii) polyvalent counterions or iii) depletion-mediated attractions. A simple theoretical model shows that, at equilibrium, short-range attractions that are known to enhance the tendency of. laments to align parallel to each other, lead to an increase in the average. lament length and a decrease in the relative width of the distribution of. lament lengths.

The concept of network immunity, i.e., the robustness of the network connectivity after a random deletion of edges or vertices, has been investigated in biological or communication networks. We apply this concept to a self-assembling, physical network of microemulsion droplets connected by telechelic polymers, where more than one polymer can connect a pair of droplets. The gel phase of this system has higher immunity if it is more likely to survive (i.e., maintain a macroscopic, connected component) when some of the polymers are randomly degraded. We consider the distribution p(sigma) of the number of polymers between a pair of droplets, and show that gel immunity decreases as the variance of p(sigma) increases. Repulsive interactions between the polymers decrease the variance, while attractive interactions increase the variance, and may result in a bimodal p(sigma).

### 2005

Motivated by recent attempts to confine biochemical processes inside water-in-oil microemulsions, we studied the composition and stability of mixed-amphiphile water-swollen micelles in oil from a theoretical point of view. A novel adsorption model demonstrates how the micellar contents (DNA, proteins, etc.) can dramatically affect the composition of the amphiphilic film and the resulting distribution of micelles. Special attention is given to the effect of electrostatic interactions within the micelles as well as between different ones. Since in a low dielectric medium charge fluctuations can lead to long-range intermicellar attractions, we suggest that the presence of amphiphilic polymers in the surfactant film may be needed to stabilize such microemulsions.

We review recent theoretical work that analyzes experimental measurements of the shape, fluctuations and adhesion properties of biological cells. Particular emphasis is placed on the role of the cytoskeleton and cell elasticity and we contrast the shape and adhesion of elastic cells with fluid-filled vesicles. In red blood cells (RBC), the cytoskeleton consists of a two-dimensional network of spectrin proteins. Our analysis of the wavevector and frequency dependence of the fluctuation spectrum of RBC indicates that the spectrin network acts as a confining potential that reduces the fluctuations of the lipid bilayer membrane. However, since the cytoskeleton is only sparsely connected to the bilayer, one cannot regard the composite cytoskeleton-membrane as a polymerized object with a shear modulus. The sensitivity of RBC fluctuations and shapes to ATP concentration may reflect topological defects induced in the cytoskeleton network by ATP. The shapes of cells that adhere to a substrate are strongly determined by the cytoskeletal elasticity that can be varied experimentally by drugs that depolymerize the cytoskeleton. This leads to a tension-driven retraction of the cell body and a pearling instability of the resulting ray-like protrusions. Recent experiments have shown that adhering cells exert polarized forces on substrates. The interactions of such "force dipoles" in either bulk gels or on surfaces can be used to predict the nature of self-assembly of cell aggregates and may be important in the formation of artificial tissues. Finally, we note that cell adhesion strongly depends on the forces exerted on the adhesion sites by the tension of the cytoskeleton. The size and shape of the adhesion regions are strongly modified as the tension is varied and we present an elastic model that relates this tension to deformations that induce the recruitment of new molecules to the adhesion region. In all these examples, cell shape and adhesion differ from vesicle shape a

Two oppositely charged surfaces or membranes show a long-range attraction, even in the presence of neutralizing counterions. We predict, in the Poisson-Boltzmann limit of very small salt concentrations, the distance and charge dependence at which the intervening counterions are released to a reservoir with a small amount of salt or identical counterions. Surprisingly, this depends on two length scales: the Gouy-Chapman (GC) length, lambda(0), inversely proportional to the surface charge, and the Debye-Huckel (DH) screening length, lambda(D), in the reservoir. The characteristic distance at which the counterions are released scales with lambda(0) log(lambda(D)/lambda(0)). Since the amount of salt (or the concentration of counterions) in the reservoir can be very small, this logarithmic factor can greatly enhance the range over which the attractions between the surfaces can be significant. The attractive interaction between the surfaces is long-ranged, the pressure scaling inversely with the square of the distance for a wide range of length scales-much smaller than the DH screening length, but much greater than the GC length.

We show theoretically how adenosine 5'-triphosphate (ATP)-induced dynamic dissociations of spectrin. laments ( from each other and from the membrane) in the cytoskeleton network of red blood cells (RBC) can explain in a unified manner both the measured fluctuation amplitude as well as the observed shape transformations as a function of intracellular ATP concentration. Static defects can be induced by external stresses such as those present when RBCs pass through small capillaries. We suggest that the partially freed actin at these defect sites may explain the activation of the CFTR membrane-bound protein and the subsequent release of ATP by RBCs subjected to deformations. Our theoretical predictions can be tested by experiments that measure the correlation between variations in the binding of actin to spectrin, the activity of CFTR, and the amount of ATP released.

*Mechanics of the 21st Century*. Kowalewski T. A. & Gutkowski W.(eds.). p. 329-342 Abstract

We review recent theoretical work that analyzes experimental measurements of elastic interactions of biological cells with their environment. Recent experiments have shown that adhering cells exert polarized forces on substrates. The interactions of such "force dipoles" in either bulk gels or on surfaces can be used to predict the nature of self-assembly of cell aggregates and may be important in the formation of artificial tissues. Cell adhesion strongly depends on the forces exerted on the adhesion sites by the tension of the cytoskeleton. The size and shape of the adhesion regions is strongly modified as the tension is varied and we present an elastic model that relates this tension to deformations that induce the recruitment of new molecules to the adhesion region.

### 2004

We predict the conditions under which two oppositely charged membranes show a dynamic, attractive instability. Two layers with unequal charges of opposite sign can repel or be stable when in close proximity. However, dynamic charge density fluctuations can induce an attractive instability and thus facilitate fusion. We predict the dominant instability modes and time scales and show how these are controlled by the relative charge and membrane viscosities. These dynamic instabilities may be the precursors of membrane fusion in systems where artificial vesicles are engulfed by biological cells of opposite charge.

Cellular adhesions are modulated by cytoskeletal forces or external stresses and adapt to the mechanical properties of the extracellular matrix. We propose that this mechanosensitivity can be driven at least in part by the elastic, cell-contractility-induced deformations of protein molecules that form the adhesion. The model accounts for observations of anisotropic growth and shrinkage of focal adhesions in the direction of the force and predicts that focal adhesions only grow within a range of force that is determined by the composition and matrix properties. This prediction is consistent with the observations of a force threshold for the appearance of elongated focal adhesions and the disruption of adhesions into fibrils on a mobile extracellular matrix. The growth dynamics is calculated and the predicted sliding of focal adhesions is consistent with several experiments.

We calculate the hydrodynamic interaction Lambda(k) (Oseen interaction kernel) and relaxation frequency omega(k) for the fluctuations of a membrane that is harmonically bounded to a permeable or impermeable wall. We show that due to the confining wall there is an increase in the effective viscosity of the fluid surrounding the membrane. This has been observed in experiments on confined membranes, such as lamellar phases and the red-blood cell membrane. Our results allow a quantitative analysis of these experiments, in terms of the strength of the membrane confining potential and dislocations.

We calculate the elastic response of thin films grafted to a solid substrate whose upper surface is subject to a stress. This issue is addressed in the context of biological cell adhesion where adhesive junctions consist of a thin layer of proteins grafted to the extracellular matrix and sheared by the cell contractility apparatus. We show that the finite thickness of the layer limits stress-induced deformations to short ranges proportional to the thickness of the film. In addition, we show that the attachment boundary condition creates an effective shear response to surface stresses that couples all the directions, even for fluidlike layers. We predict that perturbations with wavelengths of order of the film thickness induce resonancelike responses for isotropic rubberlike materials or anisotropic media with high shear moduli. We use these results to predict the elastic deformations of a layer of proteins under shear stress and propose that the resulting, polarized elastic response to local surface forces can explain the observed, anisotropic growth of cell-substrate junctions when subject to external stresses.

We predict theoretically the thermodynamics and relaxation kinetics of solutions of cylindrical branched micelles. Using a recently developed theory in combination with the experimental data, we explain the unusual, inverted temperature dependence of the phase separation observed in wormlike micelles and dilute microemulsions. We extend the model to treat the temperature dependence of the relaxation kinetics and explain the observations.

Anchorage-dependent cells collect information on the mechanical properties of the environment through their contractile machineries and use this information to position and orient themselves. Since the probing process is anisotropic, cellular force patterns during active mechanosensing can be modeled as anisotropic force contraction dipoles. Their buildup depends on the mechanical properties of the environment, including elastic rigidity and prestrain. In a finite sized sample, it also depends on sample geometry and boundary conditions through image strain fields. We discuss the interactions of active cells with an elastic environment and compare it to the case of physical force dipoles. Despite marked differences, both cases can be described in the same theoretical framework. We exactly solve the elastic equations for anisotropic force contraction dipoles in different geometries (full space, half space, and sphere) and with different boundary conditions. These results are then used to predict optimal position and orientation of mechanosensing cells in soft material.

We analyze the thermal fluctuations of fluid membranes in the presence of periodic confining harmonic potentials. This is a simple model of the biologically important, inhomogeneous attachment of the cytoskeleton to the external, fluid membrane of the cell. We study a two-dimensional checkerboard potential as well as one-dimensional, sinusoidal and periodic highly localized, delta function potentials. The membranes are described by an energy functional that includes the curvature bending modulus of the membrane and the harmonic external potential. We predict the magnitude of the membrane shape fluctuations. The sinusoidal potentials give a spontaneous surface tension, and an emergent intermediate-range order in the membrane undulations. The delta function potentials induce a renormalization of the curvature modulus, with perfect pinning at the delta potential sites. After spatial averaging, the delta-function potentials also give rise to an effective surface tension. Finally, we compare these results with measurements of the fluctuations of the red-blood cell membrane, which shows the effects of cytoskeleton attachment to the cellular membrane.

### 2003

One approach to the understanding of fusion in cells and model membranes involves stalk formation and expansion of the hemifusion diaphragm. We predict theoretically the initiation of hemifusion by stalk expansion and the dynamics of mesoscopic hemifusion diaphragm expansion in the light of recent experiments and theory that suggested that hemifusion is driven by intramembrane tension far from the fusion zone. Our predictions include a square-root scaling of the hemifusion zone size on time as well as an estimate of the minimal tension for initiation of hemifusion. Whereas a minimal amount of pressure is evidently needed for stalk formation, it is not necessarily required for stalk expansion. The energy required for tension-induced fusion is much smaller than that required for pressure-driven fusion.

This reply discusses the problem of the interpretation of neutron spin-echo experiments on micromulsion droplets. It is explained how shape fluctuations and polydispersity are coupled in these systems in which the total surface area is a fixed quantity. Other interpretations by Lisy and coworkers do not agree with ours on this particular point. This is problably the reason why they obtain unrealistic values of the bending elastic contants, whereas the description that we use leads to excellent agreements with independent determinations of these elastic constants. (C) 2003 Elsevier Science B.V. All rights reserved.

We study theoretically a model system of a transient network of microemulsion droplets connected by telechelic polymers and explain recent experimental findings. Despite the absence of any specific interactions between either the droplets or polymer chains, we predict that as the number of polymers per drop is increased, the system undergoes a first-order phase separation into a dense, highly connected phase, in equilibrium with dilute droplets, decorated by polymer loops. The phase transition is purely entropic and is driven by the interplay between the translational entropy of the drops and the configurational entropy of the polymer connections between them. Because it is dominated by entropic effects, the phase behavior of the system is extremely robust and is independent of the detailed properties of either polymers or drops.

We predict the thermodynamic and structural behavior of solutions of self-assembling chains, cross-links and end-cap molecules. We find that at the mean-field level, the entropy of self-assembled junctions induces an effective attraction that can result in equilibrium between a sol phase and a connected network. A connected network can also be formed in a non-thermodynamic transition upon increase of the monomer or cross-link density, or with decreasing temperature. For rigid rods, at low temperatures, we predict a transition from an isotropic network to anisotropic bundles of rods linked by cross-links, that is triggered by the interplay between the configurational entropy of the cross-link distribution, and the rotational and translational entropy of the rods.

We analyze theoretically both the static and dynamic fluctuation spectra of the red blood cell in a unified manner, using a simple model of the composite membrane. In this model, the two-dimensional spectrin network that forms the cytoskeleton is treated as a rigid shell, located at a fixed, average distance from the lipid bilayer. The cytoskeleton thereby confines both the static and dynamic fluctuations of the lipid bilayer. The sparse connections of the cytoskeleton and bilayer induce a surface tension, for wavelengths larger than the bilayer persistence length. The predictions of the model give a consistent account for both the wave vector and frequency dependence of the experimental data.

We propose an explanation for the recently measured slow fluctuations and "blinking" in the surface-enhanced Raman scattering spectrum of single molecules adsorbed on a silver colloidal particle. We suggest that these fluctuations may be related to the dynamic relaxation of the surface roughness on the nanometer scale and show that there are two classes of roughness with qualitatively different dynamics. The predictions agree with the measurements of surface roughness relaxation. Using a theoretical model for the kinetics of surface roughness relaxation in the presence of charges and optical electrical fields, we predict that the high-frequency electromagnetic field increases both the effective surface tension and the surface diffusion constant and thus accelerates the surface smoothing kinetics and time scale of the Raman fluctuations in a manner that is linear with the laser power intensity, while the addition of salt retards the surface relaxation kinetics and increases the time scale of the fluctuations. These predictions are in qualitative agreement with the Raman experiments.

Mechanical force is known to play an important role in the regulation of cellular behaviour, including adhesion, motility, differentiation and proliferation. For stationary, mechanically active cells like fibroblasts, adhesion to flat substrates occurs mainly at sites of focal adhesions, which are micron-sized protein aggregates at the plasma membrane, which on the cytoplasmic side connect to the actin cytoskeleton. In recent years, evidence has been growing that focal adhesions act as mechanosensors which convert mechanical force into biochemical signalling. We have investigated the relationship between force and aggregation at focal adhesions by a new method which combines elastic micro-patterned substrates (to record substrate deformation), fluorescence labelling of focal adhesion proteins (to monitor aggregation) and numerical solution of the inverse problem of linear elasticity theory (to calculate forces at focal adhesions). We have found that force correlates linearly with lateral size of aggregation, with a stress constant of 5.5 nN/mum(2). This finding indicates that mechanosensing involves regulation of aggregation. (C) 2002 Elsevier Science B.V. All rights reserved.

Fluids that respond to magnetic fields are predicted to display complex behaviour and morphologies. New experiments with colloidal dispersions provide direct evidence for self-assembled chains and networks in such systems.

Self-assembly in soft-matter systems often results in the formation of locally cylindrical or chain-like structures. We review the theory of these systems whose large-scale structure and properties depend on whether the chains are finite, with end-caps or join to form junctions that result in networks. Physical examples discussed here include physical gels, wormlike micelles, dipolar fluids and microemulsions. In all these cases, the competition between endcaps and junctions results in an entropic phase separation into junction-rich and junction-poor phases, as recently observed by electron microscopy and seen in computer simulations. A simple model that accounts for these phenomena is reviewed. Extensions of these ideas can be applied to treat network formation and phase. separation in a system of telechelic (hydrophobically tipped, hydrophilic) polymers and oil-in-water microemulsions, as observed in recent experiments.

### 2002

We predict theoretically that long-wavelength surface charge modulations universally reduce the pressure between the charged surfaces with counterions compared with the case of uniformly charged surfaces with the same average surface charge density. The physical origin of this effect is the fact that surface charge modulations always lead to enhanced counterion localization near the surfaces, and hence, fewer charges at the midplane. We confirm the last prediction with Monte Carlo simulations.

We study a generic model of self-assembling chains that can branch and form networks with branching points (junctions) of arbitrary functionality. The physical realizations include physical gels, wormlike micelles, dipolar fluids, and microemulsions. The model maps the partition function of a solution of branched, self-assembling, mutually avoiding clusters onto that of a Heisenberg magnet in the mathematical limit of zero spin components. As for the calculation of thermodynamic properties as well as the scattering structure factor, the mapping rigorously accounts for all possible cluster configurations, except for closed rings. The model is solved in the mean-field approximation. It is found that despite the absence of any specific interaction between the chains, the presence of the junctions induces an effective attraction between the monomers, which in the case of threefold junctions leads to a first-order reentrant phase separation between a dilute phase consisting mainly of single chains, and a dense network, or two network phases. The model is then modified to predict the structural properties at the mean-field level. Independent of the phase separation, we predict a percolation (connectivity) transition at which an infinite network is formed. The percolation transition partially overlaps with the first-order transition, and is a continuous, nonthermodynamic transition that describes a change in the topology of the system. Our treatment that predicts both the thermodynamic phase equilibria as well as the spatial correlations in the system allows us to treat both the phase separation and the percolation threshold within the same framework. The density-density correlation has the usual Ornstein-Zernicke form at low monomer densities. At higher densities, a peak emerges in the structure factor, signifying the onset of medium-range order in the system. Implications of the results for different physical systems are discussed.

Forces exerted by stationary cells have been investigated on the level of single focal adhesions by combining elastic substrates, fluorescence labeling of focal adhesions, and the assumption of localized force when solving the inverse problem of linear elasticity theory. Data simulation confirms that the inverse problem is ill-posed in the presence of noise and shows that in general a regularization scheme is needed to arrive at a reliable force estimate. Spatial and force resolution are restricted by the smoothing action of the elastic kernel, depend on the details of the force and displacement patterns, and are estimated by data simulation. Corrections arising from the spatial distribution of force and from finite substrate size are treated in the framework of a force multipolar expansion. Our method is computationally cheap and could be used to study mechanical activity of cells in real time.

Using both small-amplitude and singular-perturbation theories we predict theoretically that the presence of surface charge modulations gives rise to an enhancement of the counterion density near the surface above and beyond that of a uniform, charged surface. We confirm these predictions with Monte Carlo simulations. Our study focuses on the weak-to moderate-coupling regime which is complementary to a similar investigation performed by Moreira and Netz (Europhys. Lett., 57 (2002) 911) in the strong-coupling case.

We predict a condensation phenomenon in an overall neutral system, consisting of a single charged plate and its oppositely charged counterions. Based on the "two-fluid" model, in which the counterions are divided into a "free" and a "condensed" fraction, we argue that for high surface charge, fluctuations can lead to a phase transition in which a large fraction of counterions is condensed. Furthermore. we show that depending on the valence, the condensation is either a first-order or a smooth transition.

We review the continuum, statistical thermodynamics of surfaces and interfaces in soft matter where both the energy and entropy of the surface are comparable. These systems include complex fluids that are dominated by either surface tension or the interfacial curvature, such as: fluid and solid interfaces, colloidal dispersions, macromolecular solutions, membranes, and other self-assembling aggregates such as micelles, vesicles, and microemulsions. The primary focus is on the theoretical concepts, their universality, and the role of fluctuations and inhomogeneities with connections to relevant experimental systems. (C) 2001 Elsevier Science B.V. All rights reserved.

Recent experimental results demonstrate that it is possible to grow a variety of different multiphase, nested nanotube structures. This paper predicts the structure and energetics of such multiphase nanotubes. There are several distinct contributions to the energetics: the internal and external surface energies, the energy of the interface between the different phases, the long-range (van der Waals) interactions between interfaces, and the elastic bending energy. We perform energy minimizations to compare the energies of two- and three-layer films and nanotubes. We present physical guidelines, quantitative theory, and structure maps that show how materials and geometric parameters influence the stability of competing structures.

Biological cells in soft materials can be modeled as anisotropic force contraction dipoles. The corresponding elastic interaction potentials are long ranged (similar to1/r(3) with distance r) and depend sensitively on elastic constants, geometry, and cellular orientations. On elastic substrates, the elastic interaction is similar to that of electric quadrupoles in two dimensions and for dense systems leads to aggregation with herringbone order on a cellular scale. Free and clamped surfaces of samples of finite size introduce attractive and repulsive corrections, respectively, which vary on the macroscopic scale. Our theory predicts cell reorientation on stretched elastic substrates.

### 2001

Experiments have shown that the depletion of polymer in the region between two apposed (contacting or nearly contacting) bilayer membranes leads to fusion. In this paper we show theoretically that the addition of nonadsorbing polymer in solution can promote lateral contraction and phase separation of the lipids in the outer monolayers of the membranes exposed to the polymer solution, i.e., outside the contact zone. This initial phase coexistence of higher- and lower-density lipid domains in the outer monolayer results in surface tension gradients in the outer monolayer. Initially, the inner layer lipids are not exposed to the polymer solution and remain in their original "unstressed" state. The differential stresses on the bilayers give rise to a Marangoni flow of lipid from the outer monolayers in the "contact zone" (where there is little polymer and hence a uniform phase) to the outer monolayers in the "reservoir" (where initially the surface tension gradients are large due to the polymer-induced phase separation). As a result, the low-density domains of the outer monolayers in the contact zone expose their hydrophobic chains, and those of the inner monolayers, to the solvent and to each other across the narrow water gap, allowing fusion to occur via a hydrophobic interaction. More generally, this type of mechanism suggests that fusion and other intermembrane interactions may be triggered by Marangoni flows induced by surface tension gradients that provide "action at a distance" far from the fusion or interaction zone.

We predict the elastic properties of mixed amphiphilic monolayers in the swollen state within the blob model using scaling arguments. First the elastic moduli and the spontaneous curvature of a bimodal brush are determined as a function of the composition and the relative chain length, We obtain simple and useful scaling functions which interpolate between the elastic moduli of a pure short-chain brush and a pure long-chain brush. By using the analogy between block copolymer interfaces and polymeric brushes, the effect of mixing on self-assembled diblock copolymer monolayers is investigated in the swollen state. We calculate various interfacial properties, such as the equilibrium surface coverage, interface curvature, and the mixing free energy as a function of the composition. In general, we find a nonlinear dependence on the composition, which deviates from the simple linear averaging of the properties of pure components. Our results are used to discuss a recent experiment on the effect of amphiphilic block copolymers on the efficiency of microemulsions.

Mechanical forces play a major role in the regulation of cell adhesion and cytoskeletal organization. In order to explore the molecular mechanism underlying this regulation, we have investigated the relationship between local force applied by the cell to the substrate and the assembly of focal adhesions. A novel approach was developed for real-time, high-resolution measurements of forces applied by cells at single adhesion sites. This method combines micropatterning of elastomer substrates and fluorescence imaging of focal adhesions in live cells expressing GFP-tagged vinculin, Local forces are correlated with the orientation, total fluorescence intensity and area of the focal adhesions, indicating a constant stress of 5.5 +/- 2 nN mum-2. The dynamics of the force-dependent modulation of focal adhesions were characterized by blocking actomyosin contractility and were found to be on a time scale of seconds. The results put clear constraints on the possible molecular mechanisms for the mechanosensory response of focal adhesions to applied force.

A critical phase separation in self-assembling colloidal dispersions is predicted. The coexisting phases are a dilute gas of ends that coexists with a high-density liquid of branching points. Our model provides a unified explanation for the branched structures, the unusually low critical temperature and density, and the consequent two-phase coexistence 'islands' in both cylindrical microemulsions and dilute dispersions of dipolar particles (e.g. ferrofluids).

We discuss the structure and the interaction of telechelic brushes. We show that the association of functionalized chain ends is capable of giving rise to attractive interactions between telechelic brush-covered surfaces, in contrast to conventional repulsion. Our predictions for the interaction free energy are in agreement with experimental data.

We calculate the static properties of macroion density fluctuations in both bull; and in confined, strongly interacting, macroion suspensions (macroions interact via the Derjaguin-Landau-Verwey-Overbeek potential) in terms of a simple density-functional ansatz. We show how to map a strongly interacting suspension to a weakly interacting one and obtain the renormalized charge, diameter and scattering structure factors analytically. The model is extrapolated to predict crystalline order in terms of a Hansen-Verlet-type of criterion as well as an effective hard-sphere crystallization condition, and good agreement with simulations is found. The increase in correlations observed in recent experiments in two, confined layers is demonstrated.

When sheets of layered material like C, WS2 or BN are restricted to finite sizes, they generally form single- and multi-walled hollow nanoparticles in order to avoid dangling bonds. Using continuum approaches to model elastic deformation and van der Waals interactions of spherical nanoparticles, we predict the variation of mechanical stability, adhesive properties and phase behavior with radius R and thickness h. We find that mechanical stability is limited by forces in the nN range and pressures in the GPa range. Adhesion energies scale linearly with R, but depend only weakly on h. Deformation due to van der Waals adhesion occurs for single-walled particles for radii of few nm, but is quickly suppressed for increasing thickness. As R is increased, the gas-liquid coexistence disappears from the phase diagram for particle radii in the range of 1-3 nm (depending on wall thickness) since the interaction range decreases like 1/R.

### 2000

Effective pair potentials for hollow nanoparticles such as those made from carbon (fullerenes) or metal dichalcogenides (inorganic fullerenes) consist of a hard core repulsion and a deep, but short-ranged, van der Waals attraction. We investigate them for single-walled and multiwalled nanoparticles and show that in both cases, in the limit of large radii the interaction range scales inversely with the radius R, while the well depth scales linearly with R. We predict the values of the radius R and the wall thickness h at which the gas-liquid coexistence disappears from the phase diagram. We also discuss unusual material properties of the solid, which include a large heat of sublimation and a small surface energy.

A defect-induced, critical phase separation in dipolar fluids is predicted, which replaces the usual Liquid-gas transition that is driven by the isotropic aggregation of particles and is absent in dipolar fluids due to strong chaining. The coexisting phases are a dilute gas of chain ends that coexists with a high-density Liquid of chain branching points. Our model provides a unified explanation for the branched structures, the unusually Low critical temperature and density, and the consequent two-phase coexistence "islands" that were recently observed in experiment and simulation.

Multi-walled hollow: nanoparticles made from tungsten disulphide (WS2) show exceptional tribological performance as additives to liquid lubricants due to effective transfer of low shear strength material onto the sliding surfaces. Using a scaling approach based on continuum elasticity theory for shells and pairwise summation of van der Waals interactions? we show that van der Waals interactions cause strong adhesion to the substrate which favors release of delaminated layers onto the surfaces. For large and thin nanoparticles, van der Waals adhesion can cause considerable deformation and subsequent delamination. For the thick WS2 nanoparticles, deformation due to van der Waals interactions remains small and che main mechanism for delamination is pressure which in fact leads to collapse beyond a critical value. We also discuss the effect of shear flow on deformation and rolling on the substrate.

We predict theoretically the gradual formation of fluctuating, connected microemulsion networks from disconnected globules as the spontaneous curvature is varied, in agreement with recent direct measurements of these topological transitions. The connectivity induced instability together with emulsification failure of the network relate the ultralow tensions and wetting transition to the changing microstructure.

We predict theoretically the gradual formation of fluctuating, connected microemulsion networks from disconnected cylinders as the spontaneous curvature and the radius are varied, in agreement with recent direct measurements of these topological transitions. We discuss the rob of the topological defects, the network junction and the end-cap of the disconnected cylinders, in the connectivity transition. The optimal shapes and curvature energies of the junctions and end-caps are calculated numerically and compared with analytic approximations.

The effect of rigid inclusions on the phase behavior of a film containing a mixture of lipid molecules is investigated. In the proposed model the inclusion-induced deformation of the film, and the resulting energy cost are strongly dependent upon the spontaneous curvature of the mixed film. The spontaneous curvature is in turn strongly influenced by the composition of film. This coupling between the film composition and the energy per inclusion leads to a lateral modulation of the composition, which follows time local curvature of the membrane. In particular, it is shown that inclusions may induce a global phase separation in a film which would otherwise be homogeneously mixed. The mixed film is then composed of patches of different average composition, separated by the inclusions. This process may be of relevance to explain some aspects of lipid-protein association in biological membranes.

### 1999

We predict the fluctuation contribution to the interaction between two surfaces with both mobile layer charges and delocalized counterions. The correlation (coupling) between the layer-charge fluctuations and the counterion fluctuations (around a piecewise homogeneous mean-field density profile) is taken into account in the Gaussian approximation. We find that this correlation significantly increases the magnitude of the: interlayer fluctuation attraction. The counterion fluctuation pressure is calculated as a function of the intersurface distance and we show how the large and small distance limits correspond to three-dimensional (3D) and 2D fluctuations, respectively. In addition, we predict the charge density-density correlation functions. Experimental implications of the model are discussed. [S1063-651X(99)13911-4].

Gradual disruption of the actin cytoskeleton induces a series of structural shape changes in cells leading to a transformation of cylindrical cell extensions into a periodic chain of "pearls." Quantitative measurements of the pearling instability give a square-root behavior for the wavelength as a function of drug concentration, We present a theory that explains these observations in terms of the interplay between rigidity of the submembranous actin shell and tension that is induced by boundary conditions set by adhesion points. The theory allows estimation of the rigidity and thickness of this supporting shell. The same theoretical considerations explain the shape of nonadherent edges in the general case of untreated cells.

Direct evidence by cryogenic temperature transmission electron microscopy shows the existence of networks in microemulsions near the two-phase closed loops. Networks formed by interconnected oil-swollen cylinders were observed in the water-rich regions of the phase diagram of the C12E5/water/n-octane system. The coexisting phases within the loops were shown to be concentrated and dilute networks. Similar micellar networks were also found in the binary system. These observations substantiate the suggested theoretical link between the structural bicontinuity and the unique phase separation and criticality of microemulsions: All these regimes are governed by the entropic attraction between network junctions.

A tutorial review of the theory of curvature elasticity of thin films is presented with an emphasis on the physical origins of the bending energy. We begin with a discussion of surface curvature and focus on the role of special surfaces of curvature to show how such surfaces can be defined to eliminate either the coupling of the compressibility and bending terms (neutral surface) or the saddle-splay (Gaussian curvature) modulus. Next, we consider phenomenological models for curvature elasticity and discuss the coupling of the curvature degrees of freedom with other properties of the system such as the packing area and the number of molecules at the interface. The pressure distribution in the film is related to the bending moduli. We then connect the elastic moduli to the physical properties of both solid and liquid thin films with a detailed discussion of the role of solid elasticity (including defects), electrostatic interactions (applicable to polar head groups and chain packing (using a block copolymer model of amphiphilic molecules). Finally, we demonstrate the effects of fluctuations and inhomogeneities in these systems in a discussion of the role of thermal undulations in renormalizing the bending moduli and of mixtures of amphiphiles of different chain lengths in fluid films. The article is concluded with a brief review of experimental characterizations of curvature elasticity in self-assembling systems.

We show how phase separation, in the form of a redistribution of impurities (dopants in a semiconductor), can occur at impurity concentrations that are more than one order of magnitude lower than hitherto observed. This phenomenon results from the balance between long-range electrostatic repulsion and the elastic attraction of the dopants, which deforms the anisotropic host lattice. We observed such a phase separation for Ag in (Cd, Hg)Te at Ag concentrations <0.02 at. %. This also leads to the formation of a thermodynamically (as opposed to kinetically) stable p-n junction in the a-phase region. Searching for phase separation at such low concentrations requires highly sensitive analyses, here made possible because of the difference in conductivity type between the phases.

### 1998

We consider the effects of lateral charge fluctuations on the linear shear response of thin films. These fluctuations break the in-plane symmetry of the system and at short enough times cause the interaction energy to depend on the relative positions of the top and bottom surfaces of the film. This gives rise to a shear stress which can be significant depending on the time scale of charge reequilibration and on the charge and thickness of the film. The results have implications for the shear of charged membranes as well as the shear and frictional properties of electrolytes between two charged surfaces. [S0031-53007(98)07669-8].

The activity of embedded proteins is known to vary with lipid characteristics. Indeed, it has been shown that some cell-membrane proteins cannot function unless certain non-bilayer-forming lipids (i.e., nonzero spontaneous curvature) are present. In this paper we show that membranes exert a line tension on transmembrane proteins. The line tension, on the order of 1-100 kT/protein, varies with the lipid properties and the protein configuration. Thus, membranes composed of different lipids favor different protein conformations. Model predictions are in excellent agreement with the data of Keller et al, (Biophys. J. 1993, 65:23-27) regarding the conductance of alamethicin channels.

We discuss the properties of fluid membranes, whose fluctuations are restricted due to lack of area. The additional free-energy cost of the difference between the actual and optimum areas of the membrane is derived, and two ways in which the membrane can lower this free-energy cost are discussed. Firstly, it is shown that the area dependence of the surface tension can surprisingly lead to a stable hole of finite size in the membrane. We then discuss the surface-tension-driven exchange between two membranes, and conclude that this process is likely to be observed between a tense and a fluctuating membrane. The relevance of these results for recent observations on systems of vesicles is briefly discussed.

We propose that phase separation in a mixed, two-component, substrate in the presence of adsorbing molecules can be incomplete, with an equilibrium finite domain size. The free energy of the-system is shown to decrease near interfaces between the two substrate components, as a result of mismatch between the natural spacing of the adsorbate molecules and the periodic potential of the substrate lattice. These systems are of current experimental interest in the context of surface induced freezing, where ice nucleation df varying efficiency is achieved by mixed monolayers of two types of amphiphiles above supercooled water.

We consider the fluctuations of charges in membranes and their effect on the interactions between two fluid membranes a distance h apart. For the case where the counterions are highly localized in the membrane planes we find that the attraction scales as 1/h(3) at large distances and as 1/h when the membranes are closer than a typical screening length. W-hen the counterions are delocalized between the membranes a simple ideal-gas approximation indicates that a primary contribution to the membrane attraction are the fluctuations of this gas; in some cases, these attractions can exceed the repulsive interactions found for uniform charge distributions (the Poisson-Boltzmann or mean-field repulsions). These results may be relevant to understanding the role of charge fluctuations in membrane attraction and adhesion.

Recent experiments that probe the effect of confinement on the dynamical properties of thin liquid films have shown that these films undergo an abrupt transition to a solid below a critical thickness. To model this behavior, we present a mean-field theory for confinement-induced first-order phase transitions based on a Ginzburg-Landau type expression for the free-energy. We show how the equilibrium melting temperature increases as the film thickness is decreased; for thick films, the melting temperature approaches its bulk value. The predicted phase diagram includes three phases: an ordinary liquid (where both the surface layer and the bulk of the film are liquid), a crystalline solid phase (where both the surface layer and the bulk are ordered) and a quasi-liquid phase in which the bulk of the film is liquid-like but the surfaces are ordered. Transitions between the phases are always first order.

We present a new approach to probing single-particle dynamics that uses dynamic light scattering from a localized region. By scattering a focused laser beam from a micron-size particle, we measure its spatial fluctuations via the temporal autocorrelation of the scattered intensity. We demonstrate the applicability of this approach by measuring the three-dimensional force constants of a single bead and a pair of beads trapped by laser tweezers. The scattering equations that relate the scattered intensity autocorrelation to the particle position correlation function are derived. This technique has potential applications for measurement of biomolecular force constants and probing viscoelastic properties of complex media.

### 1997

We show that the local binding of a membrane to another membrane or to a surface has a long-range effect on the membrane profile far from the adhesion site. For systems dominated by long range repulsive interactions between the membranes this results in a linear profile with a small overshooting of the profile in the vicinity of the binding. For systems with only short-range repulsive interactions this results in a logarithmic membrane profile with a maximum slope at the periphery of the adhesion site. Interactions between two such patches are dominated by the membrane's response to pinning and are important in understanding aggregation properties of systems of many pinning sites.

Bound membranes respond to pinning sites by locally unbinding and overshooting their equilibrium spacing. This nonlinear response can lead to long-ranged interactions between these sites. We introduce a theory which incorporates bending elasticity, fluctuations, and intermembrane interactions to calculate the profiles of bound membranes subject to local pinning. This theory predicts several scaling regimes where the overshoot scales both with the pinning strength and site size. We also calculate the effective, membrane induced interaction between the sites, and predict aggregation properties. Aggregation and reversibility of pinning sites leads to interesting collective phenomena.

We introduce a model for microemulsions whose basic building blocks are cylindrical tubes connected by spherical junctions forming a network. The model predicts analytic scaling laws which quantitatively reproduce several prominent experimental features of the phase diagram, including the closed loops of 2-phase coexistence and the 3-phase body. The interfacial nature of our model, which takes into account only the curvature energy and the entropy of the interface, explains the observed water/oil symmetry and the collapse of the experimental data onto a single universal scaling curve.

We developed an experimental technique which probes the dynamics of a single colloidal particle over many decades in time, with spatial resolution of a few nanometers. By scattering a focused laser beam from a particle observed in an optical microscope, we measure its fluctuations via the temporal autocorrelation function of the scattered intensity g(t). This technique is demonstrated by applying it to a single Brownian particle in an optical trap of force constant k. The decay times of g(t), which are related to the particle position autocorrelation function, scale as k(-1), as expected from theory.

### 1996

We predict the magnitude of fluctuations of two-dimensional, supercrystal stripe phases of Langmuir monolayers, composed of polar molecules, in the low temperature regime. Our model includes both the microscopic line tension and the interdomain, long-range dipolar interactions. We calculate (in the long wavelength approximation) the elastic energy of the stripes and show that the stripes exhibit long-range orientational order. We predict that the stabilization of the stripe width by the dipolar interactions tends to decrease the thermal roughness of the domain walls compared with systems with only short-range interactions. In the case of crystalline stripes our results suggest a possible finite-temperature first-order roughening transition.

We calculate the membrane-induced interaction between inclusions, in terms of the membrane stretching and bending moduli and the spontaneous curvature. We find that the membrane-induced interaction between inclusions varies nonmonotonically as a function of the inclusion spacing, The location of the energy minimum depends on the spontaneous curvature and the membrane perturbation decay length, where the latter is set by the membrane moduli. The membrane perturbation energy increases with the inclusion radius. The Ornstein-Zernike theory, with the Percus-Yevick closure, is used to calculate the radial distribution function of inclusions. We find that when the spontaneous curvature is zero, the interaction between inclusions due to the membrane deformation is qualitatively similar to the hard-core interaction, However, in the case of finite spontaneous curvature, the effective interaction is dramatically modified.

We review the theory of curvature elasticity of thin, fluid films starting with a phenomenological formalism motivated by microscopic, physical examples. The effects of finite compressibility of the layer, exchange of molecules between the film and the solution, and the intrinsic bending stiffness of the film are taken into account; these degrees of freedom account for all the important molecular modes of the system. The effects of fluctuations of the number of molecules in the film (which is in equilibrium with surfactant molecules or micelles in solution) and in the area per molecule about its equilibrium value is shown to soften the curvature elastic moduli. The relationship of the bending moduli to the pressure distribution in the film is discussed and it is demonstrated that isotropic liquid films have no bending modulus, in the continuum limit. The bending moduli for charged membranes and for block copolymers are estimated. Finally, some recent experimental measure of curvature elasticity are discussed and compared with the theory.

We present a simple model of surface modified first-order phase transitions, based on the density-functional theory of freezing. Motivated by recent experiments of surface induced freezing, we show how supercooling may in fact be inhibited below a certain temperature which depends on the lattice mismatch between the monolayer and the nucleated crystal, as well as on the macroscopic strength of the surface treatment. We also apply the model to systems which tend to surface-freeze above the melting point, and correlate their surface-freezing and supercooling temperatures.

### 1995

Loosely bound membranes exhibit an unusual elastic response when pinched together by optical tweezers, locally unbinding to a large intermembrane distance. Tweezing a stack of many bound membranes produces extreme local swelling in the vicinity of the tweezing point. We introduce a model that incorporates bending elasticity, fluctuations, and intermembrane interactions to calculate the membrane profiles subject to a local pinch. Theoretically, we find strongly overshooting profiles in agreement with experiment. We predict scaling behavior of the overshoot with the pinch strength and size.

We present a coherent structural and thermodynamic approach to sponge phases of surfactant solutions based on the curvature energy of the surfactant film characterized by the bending moduli kappa, , the surfactant concentration phi(S) and the inside/outside ratio phi. We find that an entropy-driven transition between symmetric (S) and asymmetric (A) sponge phases with a critical phi(B) similar to kappa(-1/3) is possible. Film and bulk scattering are given consistently, in terms of the correlation length xi(kappa, , phi(S)) and the structural wave vector k(0)(kappa, , phi(S)). The film scattering at small wave vectors has approximately an arctangent form and-depending on the value of k(0) xi- more or less pronounced irregularity at about 2k(0).

In a recent paper Gompper and Goes [Phys. Rev. E 50, 1325 (1994)] use the random-interface approach to discuss film scattering from sponge and microemulsion phases. We show that their calculations of the film scattering lack the integration measure so that subsequent conclusions about the random-level-surface model have to be revised. Furthermore we find in contrast to that paper that a sponge-lamellar transition in the random-interface approach is possible.

The solubility of proteins in membranes depends both on the protein properties and membrane characteristics. In this paper, we calculate the concentration of membrane inclusions as a function of their concentration in solution and the physical properties of the membrane. We find that the membrane deformation plays a key role in determining the adsorption. When there is no thickness mismatch between the inclusions and the membrane, the effective adsorption energy decreases with the lipid spontaneous curvature and hence strongly favors protein adsorption into the membrane.

### 1994

We investigate theoretically the effect of embedded inclusions on membrane structure, and the corresponding membrane-induced interactions between inclusions. We find that the membrane thickness, which is perturbed from its equilibrium value by the coupling to the embedded inclusions, decays non-monotonically with distance from the inclusion boundary. As a result, the membrane-induced interactions between inclusions vary non monotonically with spacing. The periodicity of the perturbation profile, as well as the strength and range of the induced interactions, are proportional to the ratio of the amphiphile bending modulus and compressibility. In systems where the inclusions impose a thickness-matching constraint, the induced interactions are attractive. However, the presence of an energy barrier at a finite spacing may hinder aggregation. In systems where the inclusions impose a specific contact-angle, the interaction energy is minimal at a finite inclusion spacing.

A self-consistent field model is used to study the effect of mixing on self-assembled diblock copolymer monolayers and bilayers, in the strong segregation limit. The mixtures contain chains of identical chemistry but different molecular weights or asymmetries. The interactions between such chains are shown to be attractive, so that segregation is unfavorable. The equilibrium curvature and surface density of the monolayer vary nonlinearly with mixture composition, due to interactions between the components. In bilayers, the spontaneous curvature energy promotes demixing, i.e., nonuniform distribution of the components between the two monolayers composing the bilayer. While a homogeneously mixed system adopts a lamellar configuration, in the demixed bilayer equilibrium vesicles are favorable. The stability boundaries of the lamellar, vesicular, and saddle-shaped phases are calculated as a function of chain characteristics and mixture composition.

We present a unified theory of the bending of crystalline films that accounts for both elastic effects and crystal defects. Our theory predicts a transition from a bent coherent film with no dislocations to an incoherent, dislocated one as the film thickness or curvature is increased. The presence of the dislocations serves to renormalize the bending modulus of the system to smaller values. The degree to which the dislocations relax the elastic bending energy is found by calculating the equilibrium dislocation density and bending energy as a function of elastic constants, curvature, and film thickness. We demonstrate that at critical values of the curvature or thickness, there is a second-order phase transition between the undislocated and dislocated film. Generalizing these results to anisotropic elastic systems shows that weak bonding between crystal planes (such as in graphite) leads to a significant decrease in the critical curvature or thickness. An analysis of the case where the relaxation of the bending energy occurs by the formation of grain boundaries is also presented. We find that the introduction of grain boundaries can relieve the energy of the curved crystal more effectively than can the introduction of a uniform array of dislocations. Nonetheless, dislocation formation may be the dominant relaxation mechanism for very thin films (thin compared to the dislocation spacing in the grain boundary) and/or when dislocation migration kinetics are slow. Examples based upon nested fullerenes and bilayer surfactants are discussed.

Sponge-like phases of amphiphilic mixtures are characterized by mesoscopic domains of water and/or oil, separated by surfactant films. These systems are analysed using a mean-spherical model to approximate the free energy of an interfacial system with a bending Hamiltonian for the case of zero spontaneous curvature. Both the scattering structure factor and die phase diagram are determined as functions of concentrations and bending moduli. The scaling properties of the structure factor distinguish a typical microemulsion regime with a well defined structural length scale from a regime unstable with respect to phase separation. The phase diagram shows the multi-phase sequence observed in experiment.

Recent experiments that probe the effect of alcohol monolayers on the freezing of water are an example of well-characterized surface nucleation, where one has control over the instability by systematic surface modification. We present a simple theory of surface-modified, first-order phase transitions and show how supercooling may in fact be inhibited below a minimal supercooling temperature which is dependent on the macroscopic strength and spatial extent of the surface treatment. The results show that the temperature range where supercooling is possible can indeed vanish for strong enough surface treatments, in qualitative agreement with the experiments.

### 1993

The properties of membranes containing inclusions, such as proteins or colloidal particles, are calculated as a function of the bilayer interfacial energy and bending coefficients. We find that the inclusion-imposed perturbation leads to damped oscillations in the membrane profile and, hence, to nonmonotonic short-ranged, membrane-induced interactions between inclusions. The preferred spacing between inclusions is predicted to depend on the spontaneous curvature of the amphiphile and the magnitude of the perturbation at the inclusion boundary.

We apply the continuum theory of van der Waals interactions to a four-component, layered system, composed of a bulk vapor phase, a thin hydrocarbon film, and a water film which is in equilibrium with bulk ice, at the triple point. We rind that for thin hydrocarbon films, these interactions result in a finite film of water at the ice surface with a thickness which increases with the hydrocarbon film thickness, d, reaching a value of about 1000 angstrom for d almost-equal-to 300 angstrom. This corresponds to incomplete surface melting of ice but with a relatively thick wetting layer. However, for larger d, we predict a discontinuous wetting transition as the water film thickness jumps to infinity, indicating complete surface melting of ice.

The Gaussian model of random interfaces is related to the statistical mechanics of microemulsions by a variational approximation to the free energy. This allows the prediction of the structure factor as a function of surfactant and water/oil concentrations and the bending modulus. Different scaling regimes for the behaviour of the structure factor (and the real-space structure) are identified and we predict how the structure changes from one with a well-defined length scale to one with no well-defined domain size as the surfactant concentration or the bending energy are decreased. Upon a further decrease in these quantities, the correlation length of the exponential decay diverges, signifying an instability of this system.

The definition of a, appearing in the third line of page 977, should be written as:a is equal to [formula].

The surfaces of thin, liquid films can be unstable due to thinning van der Waals interactions, leading to the formation of holes in the initially uniform film. These instabilities can be greatly retarded in viscoelastic materials (and completely inhibited in elastic materials) even when the finite frequency shear modulus, E, is small compared to the infinite frequency modulus, G. This occurs when E/G much greater than (a/h0)5 much less than 1 where a is a molecular size and h0 is the film thickness. We relate the growth rate of the instability to the dynamic viscosity, eta (omega), with examples for the cases of a polymer brush, an elastic fluid (gel), and a transient polymer network, described by reptation dynamics.

Modification of the surface of a metastable, supercooled phase can induce the transition to the equilibrium phase. The minimal supercooling temperature of the bulk phase is determined by both the strength and spatial extent of the surface treatment. Our results show that the temperature range where supercooling is possible can indeed vanish for strong enough surface treatments, in qualitative agreement with recent experiments which probe the effect of alcohol monolayers on the freezing of water.

We investigate the effect of mixing two diblock copolymers of identical chemical structure but different molecular weight on self-assembled bilayers, using theoretical concepts of curvature elastic energy. Entropy and chain-chain interactions are found to favour ideal mixing in the two monolayers composing the bilayer, while curvature energy favours an inhomogeneous distribution of chains among the layers and, therefore, vesicle formation. The vesicle phase boundaries are determined as a function of mixture composition and molecular weight. We find that adding a small fraction of shorter chains to a copolymer bilayer is much more effective in destabilizing lamellar bilayers than the addition of a small fraction of longer chains.

Phase separation of two-component mixtures in fluid bilayers is shown to result in a stable one-phase vesicle region near the critical composition for the mixture. The resulting phase diagram exhibits tricritical behavior: The critical point for phase separation lies on the line of continuous transitions between lamellar and vesicle phases. Near the continuous transitions, the polydispersity of the resulting vesicles is large. Possible implications for experiments on two-component surfactant systems are also discussed.

### 1992

Using a coarse-grained statistical model of microemulsions, we show that curvature-induced interactions result in enhanced water-oil correlations for an equilibrium ensemble of surfactant films. The predicted domain size and osmotic compressibility vary with bending modulus and spontaneous curvature, and can be tested with scattering measurements.

### 1991

We calculate the structure factor for microemulsions with finite spontaneous curvature by using a statistical mechanics model of the interface ensemble. The morphological transition between hard spheres and the random bicontinuous phase is treated in a unified fashion, and the implications for experiment are discussed.

Surfactant monolayers at water-oil interfaces and amphiphilic bilayers in either water or oil exhibit large-scale, isotropic structures in the neighborhood of ordered, lamellar phases. A model for the structure, based upon a treatment of bicontinuous microemulsions, is used to compare the phase stability of the lamellar and bicontinuous L3 phases.

The saddle-splay modulus KBAR of a surfactant monolayer determines the stability of a spherical droplet microemulsion phase. Negative values of KBAR increase the droplet-size polydispersity, but inhibit shape fluctuations. Large values of the observed polydispersity may therefore not be indicative of small values of the bending modulus K, but of a ratio of KBAR/K approximately -2.

Although large, spherical surfactant vesicles are generally unstable to either lamellar or micellar phases, mixtures of two surfactants can lead to spontaneous vesicle formation. We show theoretically how the energetic stabilization of mixed vesicles can occur by considering the curvature elasticity of the surfactant bilayer. Interactions between the two species (of the proper sign and magnitude) are crucial to stabilizing these vesicles. These interactions lead to composition asymmetries and effective spontaneous curvatures of the inner and outer layers that are of equal and opposite signs. The vesicles have a Gaussian distribution about an average size determined by the effective spontaneous curvature; the width of the distribution is calculated as a function of concentration. The stability of these vesicles with respect to a flat lamellar phase is estimated. The predictions of the ranges of stability of the various phases as a function of the three concentrations (solvent, e.g., water-and the two amphiphiles) are in qualitative agreement with recent experiments.

We study the curvature elastic properties of monolayers of diblock copolymers adsorbed at the interface of two incompatible solvents which are also selective solvents for the two blocks. At saturation, the interfacial free energy is minimized with respect to contributions from the chain conformation free energy, the interfacial tension, and the two-dimensional translational entropy of the chains. For a curved interface, this minimization leads naturally to curvature elasticity. The three elastic coefficients, the spontaneous radius of curvature, the bending modulus, and the Gaussian bending modulus, as functions of the molecular weights, the interfacial tension, the interaction parameters, etc., are obtained for a number of cases. Our study employs the theory for grafted chains recently developed by Milner et al. to obtain the chain conformation free energy which takes into account the nonuniformity of the chain-end distribution. This improvement not only affects the overall prefactor of the free energy but it changes the relative values of the three elastic coefficients as well. We consider the cases of both a swollen monolayer and a monolayer consisting of a melt of copolymer chains, as well as an interesting case where one of the blocks is in the swollen condition and the other block is in the melt condition. Because the chains in the melt and the swollen conditions have distinctively different scaling behaviors, the mixed case displays some features that are different from either the swollen and the melt cases.

### 1990

Neutron spin-echo and small-angle neutron-scattering techniques were employed to study both the size and the shape fluctuations of the di-2-ethylhexyl sulfosuccinate water-in-oil microemulsion system. We observe a softening of the bending rigidity of the interface with the addition of butanol as cosurfactant. Comparison of both techniques with the theory implies that for the simultaneous description of the shape and size fluctuations both the splay and saddle-splay bending moduli are the key parameters, and the spontaneous curvature has little influence.

*Trends in Colloid and Interface Science IV*. Zulauf M., Lindner P. & Terech P.(eds.). Germany: . Vol. 81. p. 60-63 Abstract

Neutron spin echo (NSE) and small angle neutron scattering (SANS) technique were employed to study the shape fluctuations of di-2-ethylhexyl sulfosuccinate (AOT)/water/decane microemulsions. We show that NSE can directly measure the shape fluctuations of these water-in-oil droplets. A comparison of the dynamics and static measurements yields both the splay modulus, as well as the saddle-splay modulus. With the addition of butanol as cosurfactant, softening of the interface was observed.

We examine theoretically the effect of electrostatics on the self-assembly of charged cylindrical micelles which behave as living polymers. The growth of micelles as a function of increasing surfactant and/or electrolyte concentration exhibits three distinct regimes. The most striking feature of the growth law is the existence of a dilute regime, (i), in which the average micelle size varies slowly with concentration. At higher concentrations, regimes (ii) and (iii) are characterized by more rapid growth than for neutral micelles. This may be responsible, in part, for the anomalous scaling of rheological properties, as observed in recent experiments.

The wetting behavior of two-phase systems confined inside cylindrical pores is studied theoretically. The confined geometry gives rise to wetting configurations, or microstructures, which have no analog in the well-studied planar case. Many features observed in experiments on binary liquid mixtures in porous media, previously interpreted in terms of random fields, are shown to be consistent with wetting in a confined geometry with no randomness.

We examine theoretically the effect of electrostatics on the self-assembly of charged, cylindrical micelles. The growth of micelles as a function of increasing surfactant and/or electrolyte concentration exhibits three distinct regimes. The most striking feature of the growth law is the existence of a dilute regime, i), in which the average micelle size varies slowly with concentration. At higher concentrations, regimes ii) and iii) are characterized by more rapid growth than for neutral micelles. This may be responsible, in part, for the anomalous scaling of rheological properties, as observed in recent experiments.

We show that the Marangoni effect drives the fingering instability observed at the edge of an aqueous surfactant drop spreading on a thin film of water. A calculation of the unperturbed flow profile demonstrates that the spreading of the drop is controlled by the dynamics of a thin layer which develops in front of the drop. The surface-tension gradient in this region leads to the fingering instability via a mechanism mathematically similar to that in Hele-Shaw flow despite the very different underlying physics.

We develop a microscopic‐level formulation for the curvature elasticity of monolayer and bilayer systems of typical surfactant molecules. It is argued that both the bending and saddle‐splay force constants k and k̄ are determined primarily by the conformational entropy of the flexible hydrocarbon chain rather than by the electrostatic interactions associated with hydrophilic head groups. A priori estimates of the chain contributions are made for the first time, without the use of any adjustable parameters. Both k and k̄ are shown to be calculable wholly from the conformational statistics describing the planar film. In particular, these constants are expressed in terms of the derivatives and moments of the lateral pressure profile characterizing chain packing in the unbent layers. By considering the dependence of the curvature elasticity on chain length, area per molecule, and composition in mixed films, we are able to account for the order‐of‐magnitude variations in k observed in a variety of different surfactant systems. The replacement of long chain molecules by short ones is shown to be especially efficient in lowering the bending energy from 10’s of kBT to kBT. The effect of ‘‘free’’ vs ‘‘blocked’’ exchange are also presented and contrasted with the case of fixed area‐per‐molecule bending deformation. Finally, monolayer vs bilayer results are compared and the calculated signs and magnitudes of k and k̄ are discussed in the context of planar bilayer stability.

The curvature elastic energy of bilayer vesicles formed by a mixture of two surfactants, which individually form either micelles or lamellar bilayer phases is described theoretically. In the limit of large bending elastic modulus K being much greater than the temperature T, the free energy is minimized by vesicles with different concentrations of the two surfactants in each monolayer of the bilayer. Vesicles are more stable than lameliar structures only when interactions or complexing of the two surfactants is taken into account.

We consider the electrostatics of charged, cylindrical micelles with spherical end-caps. In the semidilute regime (volume fraction Φ << 1), and with no added salt, a model calculation which includes nonlinearities suggests that Coulomb interactions result in an additional contribution to the free energy of an end-cap that modifies the growth law for the average micelle size. In some cases, the micelle size varies approximately as Φ (1/2)(1+Λ), where Λ > 0 depends on the renormalized coulomb charge of an end cap. These results may help explain anomalies, seen at low added salt, in recent experiments on the dynamics of worm-like micelles.

The curvature free energy of an interfacial membrane is used to predict the relative stability of cylindrical, lamellar and ordered, bicontinuous, saddle-shaped structures. Because they are surfaces of constant mean curvature, the saddle-shaped structures have a finite range of stability as a function of concentration even when the saddle-splay elastic constant bar K < 0, favoring structures where the two radii of curvature have the same sign.

The onset and morphology of various equilibrium emulsion phases for an A and B homopolymer mixture with diblock A-B copolymers as the compatibilizer, is investigated. Attention is focused on the relation between the relative lengths of the blocks and the elastic properties of the interfacial layer which determines the phase behavior. The curvature elastic moduli K and K, as well as the spontaneous radius of curvature, R 0, are obtained. The situation studied here differs from the conventional interfacial approach to microemulsions with short-chain surfactants, in that (1) the area per copolymer Σ on the interface is not fixed, but is determined by a balance between the elastic energy of the chain and the interfacial tension ; thus Σ also depends on the curvatures ; (2) as a result of (1), the elastic coefficients K, K and R0 are interdependent, and in contrast to the short chain systems, the saddle-splay modulus K is always negative (see text for the sign convention). The phase diagram as a function of these parameters is therefore different from that obtained by conventional interfacial descriptions for short surfactants.

### 1989

We show that a thin film with small dynamic contact angle and driven by an external body force is unstable to the formation of fingers in the direction perpendicular to the main flow. The instability is largest in the capillary region near the contact line, where the force due to surface tension is comparable to the viscous and gravitational forces. The fastest growing wavelength is calculated in the limit of small-amplitude disturbances. These instabilities may be related to finger patterns observed in gravitational flows and spinning drops.

We report on a new hydrodynamic instability which occurs at the spreading edge of a thin wetting film. A drop of aqueous surfactant solution placed on a glass surface moistened with a thin layer of water spreads by propagating fingers, whose velocity and shape depend on the thickness of the ambient water layer and on the surfactant concentration. The two fluids are miscible and show negligible viscosity difference, ruling out a Saffman-Taylor instability. We propose that the Marangoni effect, which is fluid flow induced by gradients in surface tension, drives the instability.

*Phase Transitions in Soft Condensed Matter*. SHERRINGTON D. & RISTE T.(eds.). p. 245-248 (1NATO ADVANCED SCIENCE INSTITUTES SERIES, SERIES B, PHYSICS). Abstract

Most studies of the dynamics of wetting have focused on the spreading of pure non-volatile drops on smooth and dry surfaces. For the most part, such spreading processes are well understood, with good agreement between experiment 1,2 and theory 3,4. In all such studies of spreading on smooth substrates, the macroscopic wetting process, whether spontaneous or forced, proceeds by the movement of a uniform and circular contact line (the boundary where the air, liquid and solid meet). As first discovered by Marmur and Lelah 5, some unusual wetting behavior can result from the addition of surfactant to the spreading liquid. Anionic and non-ionic aqueous surfactant solutions spread on smooth glass by developing fingers, which branch as they grow and leave behind a dendritic-like pattern. The wetting process with surfactant proceeds at a rate which is approximately an order of magnitude faster than the spreading of a pure liquid, although the surface coverage is quite non-uniform. We summarize below some results of an experimental 6 and theoretical study 7 of the effect of adding surfactant to a spreading drop. The interested reader is referred to refs. 6 and 7 for all further details.

### 1988

We study the micellization of associating polymers (hydrocarbon chains with dipolar head groups in hydrophobic solvents) for the case where the polar heads form a linear array. The head interaction is assumed to be short ranged and the chain conformation (free) energy is assumed to follow spherical scaling in the small aggregation number limit and to have cylindrical scaling in the large aggregation number limit. In the small aggregation number limit, the aggregation number is only weakly dependent on the chain length. Numerical solutions as well as analytical studies show that the size distribution displays features of both the spherical and cylindrical aggregates: namely the absence of a sharp CMC transition and a rather sharp distribution function with a width that increases slowly with the mean aggregation number. In the large aggregation limit, the aggregation number can be a strong function of the chain molecular weight. We discuss the importance of the ‘‘end cap’’ effect in determining the mean size as a function of the chain lengths.

The correlation functions and structure factor of microemulsions are calculated from a statistical thermodynamic approach which has been previously used to obtain the equilibrium phase diagrams. The structure factor, S (q ), has a peak at a wavevector qmax ~ π/ξ where ξ is the domain size of the oil and water regions. The physical origin of this peak are the correlations induced by the presence of the curvature energy of the surfactant film ; the renormalization of the bending modulus by thermal fluctuations plays an important role in stabilizing these fluctuations. Real space representations of these correlations show that there is significant strength in S(q) from several different wavevectors.

We present a simple model for the anomalous (flow-birefringent) isotropic phase, known as L3, that is seen in certain surfactant solutions at volume fractions of a few percent. The proposed structure consists of locally sheetlike sections of semi-flexible surfactant bilayer, connected up at larger distances into a multiply connected random surface, having a preferred structural length scale of order the persistence length of the bilayer. A first-order transition between this isotropic sheetlike phase and the nearby swollen lamellar phase is described.

A thermodynamic model for microemulsions (previously described) is used to predict some further aspects of the phase behavior. The stability of the microemulsion under variation of molecular parameters (such as the bending constant and spontaneous curvature of the surfactant monolayer) is discussed. An appropriate cut through the multidimensional parameter space yields the characteristic “fish” phase diagram, as foundexperimentally. Our model exhibits a phase behavior close to the experimental one, not only in terms ofthe variation with surfactant concentration but also in terms of the sensitivity of the phase diagram tothe nature of the surfactant, particularly its spontaneous curvature.

### 1987

We present a simple phenomenological model to describe the phase equilibria and structural properties of microemulsions. Space is divided into cells of side ξ; each cell is filled with either pure water or oil. Surfactant molecules are presumed to form an incompressible fluid monolayer at the oil–water interface. The monolayer is characterized by a size‐dependent bending constant K(ξ), which is small for ξ≥ξK, the de Gennes–Taupin persistence length. The model predicts a middle‐phase microemulsion of structural length scale ξ≊ξK which coexists with dilute phases of surfactant in oil and surfactant in water. (These phases have ξ≊a, a being a molecular length.) On the same ternary phase diagram, we find also two regions of two‐phase equilibrium involving upper‐ and lower‐phase microemulsions that coexist with either almost pure water or oil. At low temperatures and/or high values of the bare bending constant, K0≡K(a), the middle‐phase microemulsion may be entirely precluded by separation to a lamellar phase, whereas at high temperature and/or low values of K0, there is a first‐order transition between a disordered microemulsion and a lamellar phase. In the absence of spontaneous curvature the phase diagram is oil–water symmetric. It may be asymmetrized by: (i) spontaneous curvature in the middle phase or (ii) a difference between the free energy of the two dilute phases. If the asymmetry is sufficiently large, the three‐phase region disappears.

The hydrodynamic fluctuations of nearly spherical vesicles and microemulsion droplets are considered. The incompressibility of the enclosed fluid and surfactant or lipid layer imposes a constraint of constant droplet volume and area on the fluctuations. These overdamped modes, driven by bending energy and damped by viscosity of the surrounding fluids, change the shape of the surface and may scatter neutrons or light. A dynamical structure factor S(q,t) is computed and a first frequency moment at fixed wave number ω char (q) obtained. In the limit of a stiff droplet at fixed ‘‘excess area,’’ a new mode is obtained —an overdamped oscillation among ellipsoidal shapes about the minimum-energy (usually prolate) shape. Prospects for observing this fluctuation are discussed.

We examine the stability of spherical, micellelike aggregates composed of long‐chain, polymeric surfactants in dilute solution. The associating polymers consist of long, unbranched hydrophobic chains with one polar end‐group per chain and are in a good solvent for the chains. Although the head groups prefer to aggregate in the nonpolar solvent and thus stabilize the spherical structure, the energy associated with the chain repulsions tends to destabilize the aggregates as the chain length increases. For a liquid‐drop model of the head interactions, the average aggregation number evolves smoothly towards zero as the chain length increases. An alternate model of the head region, which assumes a preferred head‐packing geometry, results in the ultimate destabilization of the spherical aggregate as the chain length is increased beyond a certain threshold. Both models yield a critical micelle concentration (CMC) at low values of the polymer concentration.

A universal mechanism which results in short‐range attractive interactions between globules in micellar, vessicular, or microemulsion systems is described. The perturbation of the surfactant concentration profile by the globules is the physical origin of this attraction.

Publisher SummaryThe chapter presents a study on stage ordering in intercalation compounds. The chapter presents an analysis on the staging phenomenon. While the point of view is theoretical, connections and applications to experiment are emphasized. Although staging is observed in many types of intercalation compounds, the most numerous experiments and models have focused on graphite intercalation compounds. While some of the theory depends on particular electronic properties of graphite, the staging phenomenon is discussed in general terms. Most of the experimental examples are drawn from studies of intercalated graphite with a few examples from measurements on intercalated layered dichalcogenides. Recent reviews of the properties of intercalated graphite have emphasized the experimental consequences of the staging phenomenon. This chapter presents a discussion on the following conceptual and theoretical issues: the microscopic origin of the interactions responsible for staging, the phenomenon of staging phase transitions, and the kinetics of staging and intercalation. The discussion of these topics is preceded by a brief introduction to intercalation compounds (with emphasis on intercalated graphite) and an outline of the effects of stage ordering on the lattice, electronic, and transport properties of intercalated graphite.

### 1986

We consider the kinetic evolution of perturbations to thin films. Since all small (nonsubstrate intersecting) perturbations to the film surface decay, we consider the evolution of large perturbations, in the form of a single hole which exposes the substrate. For large holes, the hole radius increases at a constant rate under the assumption of evaporation/condensation kinetics. When the dominant transport mode is surface diffusion, large holes grow with a rate proportionaltot -3/4[1og3(t / p:)]. Small holes with a radii less than Pc shrink, where Pc is the film thickness divided by the tangent of the equilibrium wetting angle. The growth of these holes eventually leads to hole impingement which ruptures the film, creating a set of disconnected islands. The relaxation time for these islands to go to their equilibrium shape and size ( Peq ) scales as p~ or P:q for evaporation/condensation or surface diffusion kinetics, respectively.

A stability theory is presented which describes the conditions under which thin films rupture. It is found that holes in the film will either grow or shrink, depending on whether their initial radius is larger or smaller than a critical value. If the holes grow large enough, they impinge to form islands; the size of which are determined by the surface energies. The formation of grooves where the grain boundary meets the free surface is a potential source of holes which can lead to film rupture. Equilibrium grain boundary groove depths are calculated for finite grain sizes. Comparison of groove depth and film thickness yields microstructural conditions for film rupture. In addition, pits which form at grain boundary vertices, where three grains meet, are another source of film instability.

We consider the instabilities of a solid film perturbed by a single hole which exposes the substrate. We predict growth for holes larger than a critical size, proportional to pe, the ratio of the film thickness and the equilibrium contact angle. The asymptotic velocity of the hole decays to zero with time t as t-'[lo2 (t/p$)]. The growth of these holes eventually ruptures the film, creating a set of disconnected islands.

A simple model is used to calculate the phase behavior of microemulsions. The surfactant film at the oil-water interface is treated as an incompressible layer whose bending constant is renormalized by thermal fluctuations. Two- and three-phase equilibria involving upper-, lower-, and middle-phase microemulsions are found. The structure of the middle phase is characterized by the persistence length of the film, beyond which the bending constant is renormalized to smaller values.

We study the extent to which excluded volume determines the percolation threshold for permeable elements in the continuum. An expansion due to Coniglio, De Angelis, Forlani, and Lauro exploits a similarity between the statistical mechanics of hard particles and statistics of percolation of permeable objects. This expansion shows that the expectation value of the excluded volume completely determines the threshold at lowest order in element density. Permeable rods in the continuum may be analyzed with the help of Onsager’s treatment of virial coefficients for hard rods. Systems of rods provide cases in which higher-order terms will alter the proportionality of threshold to the inverse of the expected excluded volume and cases in which this proportionality remains exact.

We study a model for dynamic percolation relevant to the electrical conductivity of water in oil microemulsions. The charge carriers reside on percolation cluster sites and can propagate by hopping between nearest-neighbor sites. The cluster sites also undergo diffusion, so that the clusters continuously rearrange themselves. The conductivity below the percolation threshold pc is finite and, as p→pc, increases as ‖pc-p∥−s˜where s̃ differs from the static exponent s. In the neighborhood of pc, the conductivity depends as a power law on the rate of cluster rearrangement.

The energy of a monolayer of surfactant on a surface is calculated as a function of the orientation of the surfactant tails. The structure is determined by a competition between the spacings preferred by the polar heads and nonpolar tails. As a function of the head-tail mismatch, we find a region of no molecular tilt, and one of uniform tilt. In the tilted state, solitonlike defects resembling micellar clusters are stabilized by entropy or finite-size effects.

### 1985

The equilibrium shape of the pit where a dislocation intersects a free surface is calculated in a variational formalism. The constant dislocation line tension approximation, made by previous workers, is shown to be clearly inadequate for all realistic cases. Employment of a variable dislocation line tension model for the case of finite dislocation density shows the existence of two regimes of behaviour. In the first, pits formed at dislocation-surface intersections have a well defined shape and are of finite depth. However, a second regime exists, in which the pit depth is infinite and the dislocation core size exceeds the bulk value. This may result in the formation of open-core dislocations, as discussed by Frank.

We determine the aspect-ratio dependence of the critical percolation threshold for various systems of rods. An exact expansion, due to Coniglio et al., tests the conjecture that the threshold is proportional to the inverse of the expected excluded volume. We confirm the conjecture, and show that the proportionality becomes equality, for isotropic rods in three dimensions, in the slender-rod limit. In this limit, the critical region in which nonclassical exponents are valid vanishes.

A Monte Carlo study of spherical particles shows that increased interaction strength may either raise or lower the volume fraction required for percolation. The sense of the change depends on the distance at which two particles are considered connected, the dimensionality, and the proximity to the critical temperature. An on-lattice simulation supports the continuum result.

*Symposium C – Thin Films: The Relationship of Structure to Properties*. p. 53-60 Abstract

The primary focus of instability discussed in this paper is beading, which is the process by which an initially uniform film which has been deposited on a substrate, breaks up into an array of beads or islands. The author examine the thermodynamics of the beading process and the effects of small perturbations on an initially uniform, flat film. They find that small perturbations do not lead to beading and that perturbations with amplitudes sufficient to bring the film surface into contact with the substrate are required. The physical origin of these types of pertubations are elucidated, and it is shown that these perturbations may be associated with microstructural defects. The dynamical aspects of beading are also discussed.

The percolation behavior of spherical particles with attractive interactions is studied with use of Monte Carlo simulations. These systems differ from lattice Ising systems, which have been previously analyzed, in the necessity to define a shell parameter δ to specify a connected cluster. For small values of δ, correlations due to the attractive interactions drastically lower the percolation threshold in the vicinity of the gas-liquid critical point. For larger values of δ, these shifts are smaller, but the effects of long-range correlations show up as enhanced finite-size effects. The simulation results are discussed in the light of recent experiments which measure the temperature and concentration dependence of the conductivity in interacting microemulsions.

### 1984

Monte Carlo simulations have been carried out to study the effect of temperature on the growth kinetics of a circular grain in two dimensions (20) and a spherical grain in three dimensions (3~) for the Ising model. The domain radius shrinks as R 2(t) = R ~ - at, where Ro and R (t) are the initial and instantaneous radii, respectively. The slope a is strongly temperature dependent for T> 0.6Tc in 20 and for T> 0.2Tc in 3D, in contrast to earlier theories which neglect fluctuations. An analytical theory for d = 2 which considers the thermal fluctuations which give rise to roughening of the domain walls agrees with the T dependence of the Monte Carlo data in this regime.

The regions of stability of spherical, cylindrical, and lamellar phases of microemulsions are calculated within mean-field theory. In the cylindrical phase, thermal fluctuations determine a temperature dependent persistence length below which the cylinders are rigid (rod-like) and above which the cylinders are flexible (polymer-like). The length of the polymer-like chains depends on concentration and temperature. The radii of gyration of these flexible microemulsions are also calculated.

The orbital magnetic susceptibility of two coupled graphite layers is calculated as a function of temperature and Fermi energy (μ). The susceptibility is shown to be diamagnetic at small values of μ, and paramagnetic at large values of μ, with a paramagnetic peak at μ≅γ1, where γ1 is the interlayer coupling.These results explain the increase in the susceptibility as a function of stage which has recently been observed in KC12n.

We have used small-angle neutron scattering to study surfactant interactions in an inverted micellar and a related microemulsion system. The structure-factor fits imply an attractive interaction between micelles with a short-ranged potential due to the overlapping of the surfactant hydrocarbon tails. The linear increase in this interaction with size accounts for the critical transition observed in the microemulsion. This analysis is confirmed by Monte Carlo simulations.

### 1983

A phenomenological mean-field model of globules for a three-component microemulsion is presented and typical phase diagrams are obtained. Two types of two-phase coexistence are found as well as a three-phase equilibrium. The topology of the phase diagram is determined by the ratio of the critical globule radius to the natural bending radius and by the radius dependence of the attractive interactions.

The domain growth kinetics of magnetically ordering systems with p-fold degenerate ground states is analyzed. We present Monte Carlo results for the approach to equilibrium of systems with multigenerate ground states which have been rapidly quenched from T å Tc to T « Tc. We show that pinning center are effective in slowing the dynamics of domain growth in systems with p ⩾ 3.

A statistical model for microemulsions is used to study the stability of a single phase of spherical droplets to thermal fluctuations. The amplitude of size (l=0 spherical harmonics) and shape (l=2) fluctuations are calculated as a function of the concentration and temperature. The divergences in these fluctuations are shown to define a range of stability for the spherical droplet phase.

Monte Carlo simulations have been carried out to study the effect of temperature on the growth kinetics of a circular grain. As predicted by d=3 theories of domain kinetics, the circular domain shrinks linearly with time as A(t)=A0−αt, where A0 and A(t) are the initial and instantaneous areas, respectively. However, in contrast to d=3, the slope α is strongly temperature dependent for T>~0.6Tc. An analytical theory which considers the thermal fluctuations agrees with the T dependence of the Monte Carlo data in this regime.

An efficient Monte Carlo procedure is applied to the study of the kinetics of low- and high-Q Potts models on surfaces quenched from a high temperature (T≫Tc) to low temperatures (T<Tc). The kinetics of these models are analyzed in simulations which yield the time dependence of the size and shape of the domains. After an initial transient period, the mean domain size R increases with time as R∼tn. For the triangular lattice at all temperatures and for the square lattice at temperatures T>0.5Tc, the exponent n decreases from 1/2 for Q=2 (Ising model) to approximately 0.41 for large Q. The distribution of grain sizes and shapes is also calculated from the results of the simulations. For large Q, these distributions are time independent. On the square lattice at low temperatures the growth exponent is approximately equal to 0, due to the nucleation of pinned domains. The Monte Carlo results are discussed in terms of experimental studies of the ordering of adsorbed atoms on surfaces and of the growth of grains in polycrystalline materials.

The approach to equilibrium of ordered superlattice structures on surfaces is analyzed. Both Monte Carlo methods as well as analytic models of interacting domain walls are used to study the kinetics of quenched, two-dimensional systems with Q degenerate equilibrium states. In particular, the growth of long-range order in the magnetic analogs (Q-component Potts models) of the commensurate superlattice structures found on surfaces is analyzed in a study of the kinetics of typical domain geometries that are found when systems are quenched from high (disordered state) to low temperatures. Calculations and simulations of the time and temperature dependence of the domain sizes for the model system of isolated domains indicate that for two dimensions roughening fluctuations of the domain walls strongly influence the grain growth kinetics. Strip-shaped domains approach equilibrium almost entirely through these fluctuations, while circular domains shrink due to deterministic curvature, but with a rate that is strongly temperature dependent even for temperatures outside the critical region. Pinning effects in systems with Q≥3 lead to extremely slow kinetics for particular domain geometries. Low-temperature quenches on a triangular lattice indicate that these pinned geometries are rarely nucleated, while quenches on a square lattice equilibrate much more slowly due to pinning. The application of the theory to superlattice grain growth on surfaces is discussed in the following paper which presents results for the kinetics of quenched systems in terms of the simple domain geometries examined here.

### 1982

A phenomenological amplitude equation introduced earlier to calculate steady states of convection in a finite container, is studied numerically as a function of space and time. The effect of sidewalls on wave number selection is demonstrated by observing the response of the system to the sudden imposition of boundary conditions. The wave vectors of stable steady states are studied as a function of Rayleigh number near threshold, and it is shown that more than one state can be stable for fixed Rayleigh number.

The approach to equilibrium of an initially strip-shaped domain in d=2 is calculated using both a model for interacting domain walls and Monte Carlo simulations of kinetic Ising systems. The dynamics are dominated by fluctuation effects, which result in time scales for equilibrium much faster than those predicted by a deterministic theory, but slower than those of an initially curved interface.

### 1981

High-resolution measurements have been performed of the convective heat current as a function of time when a Rayleigh- Bénard cell is swept through its threshold with a specified time-dependent heat input. The results are interpreted in terms of the amplitude equation which exactly describes the slow variations in space and time of hydrodynamic quantities near the threshold. A phenomenological forcing field is added to this equation, and its form and magnitude are fitted to the onset time of the convective heat current. A deterministic model in which the field is an adjustable constant yields a good fit to the data for both a step and a linear ramp in the heat input. An alternative stochastic model, in which the field is a Gaussian variable with zero mean and a white-noise spectrum, is adequate for the ramp experiments, but cannot fit the step data for any value of the mean-square field. The systematics of the field and onset time versus ramp rate are studied in both the deterministic and stochastic models, and attempts are made to interpret the field in terms of physical mechanisms. When the data for long times are analysed in terms of the amplitude equation, it is found that the state first excited at onset is not the roll pattern which is stable in steady state. Instead, the system goes first to an intermediate state, which we tentatively identify as a hexagonal configuration. The decay of this state is governed by a further adjustable field in the amplitude equation.

A model, self-consistent band structure is calculated for a thin film of n graphite layers bounded by two, partially ionized intercalant layers for stages n=2-8. The quantum mechanics of the electrons in the graphite layers is modeled using a variant of the three-dimensional linear combination of atomic orbitals Hamiltonian whose parameters have been determined for pure graphite. The effects of the nonhomogeneous distribution of electrons in the n layers (screening) are taken into account by adding a self-consistently determined layer-potential term to the tight-binding Hamiltonian. The layer charge densities, potentials, and total energies are presented for n=2-8 along with representative band structures for charge transfer per intercalant (f) of f=1 and 14 (referred to C12nX). The stage dependence of the total energy in this model is related to the stage dependence of the chemical potential (intercalant vapor pressure) in an intercalation reaction. Comparison of theory and experiment indicates the significance of the electronic energy in stabilizing the high-stage structures.

The domain-coarsening kinetics of ordering systems with p-fold—degenerate equilibrium states, quenched from high temperatures, is analyzed. For long times (t) and low temperatures, the domain sizes equilibrate as a power law in

t for p<d+1 and as a logarithmic function of t for p>~d+1, where d is the spatial dimensionality of the system. The relation of these slowly equilibrating, kinetically disordered systems to glasses and suggestions for simulations and experiments are discussed.

### 1980

A review is presented of some examples of cooperative phenomena in graphite intercalation compounds with an emphasis on the origin and role of staging. A phenomenological model of interactions between intercalant atoms is treated to yield phase diagrams as a function of stage, temperature, chemical potential, and concentration. The range of stoichiometries as a function of temperature and chemical potential is related to concurrent changes of stage. The cooperative behavior of the intercalant atoms is further studied in a calculation of the time dependence of the structure factor describing the instantaneous density distribution of intercalant. Upon quenching from a homogeneous, nonstoichiometric stage-1 phase, the intercalant atoms tend to cluster within the plane to form islands whose positions in different planes are correlated as a result of the interlayer interaction. The microscopic origin of these interlayer interactions is examined from the point of view of both elastic and electrostatic interactions. Collective effects in the distribution of charge donated to the graphite layers by the intercalants manifest themselves in the screening of the intercalant layers. The results of a quantum-mechanical, self-consistent model for the nonhomogeneous c-axis charge distribution are presented. The theoretical results are related to measurements of both the lattice and electronic structure of these materials.

The nonlinear Thomas-Fermi equations which describe the screening of intercalant layers in graphite intercalation compounds are solved analytically in the approximation that the graphite c-axis band dispersion is neglected. The total energy of the system of charged intercalant layers and the self-consistent charge distribution are calculated analytically for general configurations of the intercalant layers. Because of electrostatic screening, pure stage configurations minimize the total (electrostatic plus band) energy for fixed chemical potential. Estimates of the stage dependence of both the electrostatic and elastic interactions between intercalant layers are presented. Experimental tests of these mechanisms for staging are suggested.

The phase diagram for a staged intercalation compound is calculated in mean-field theory. At low temperatures, the dependence of the staging on the chemical potential is related to the interlayer interactions responsible for staging. At high temperatures, a dilute stage-1 phase is predicted for all values of the chemical potential. The stage-2-stage-1 transition is second order for temperatures greater than the tricritical temperature. The predicted phase diagrams are related to recent experiments.

We examine an elastic mechanism for staging based on the domain model for graphite intercalation compounds first proposed by Daumas and Hérold. The attractive elastic interactions between intercalant atoms are shown to result in the formation of two-dimensional intercalant islands. The interaction energies of these islands are calculated as a function of island size and separation in a simplified strip geometry and related to the domain model. Because of these long range (range ∼ island size) coherency strains, the free energy of mixed or randomly staged crystals is greater than that of a pure stage intercalation compound. The implications of both the elastic and electronic interactions on the thermodynamics of intercalation (stage dependence of the equilibrium vapor pressure) are discussed. Experiments are suggested where the kinetics of the staging process are related to the microscopic interactions involved.

The observed magnetic susceptibility (xc) of alkali graphite intercalation compounds (gic) for H parallel to the hexagonal c-axis, is paramagnetic. In K gic the molar value of xc varies rapidly with stage (n), increasing by an order of magnitude from C8K (n = 1) to C48K (n = 4). Calculations of xc as a function of Fermi energy (μ) for a single layer model of gic are presented and indicate that the orbital contribution (Xcor) to Xc is paramagnetic for μ ≳ 4kT in contrast to the usual Landau-Peierls diamagnetism. Furthermore, Xcor is shown to be a sensitive function of the conduction electron charge distribution from layer to layer for n≥3, while the spin (Pauli) contribution to Xc and the specific heat coefficient are less sensitive to this distribution. These results suggest that measurements of the stage dependence of Xc can be used to estimate the c-axis screening length in gic.

### 1979

We have measured the magnetic susceptibility of alkali-graphite intercalation compounds for H→ parallel to the hexagonal c→ axis (χc) and for H→c→ (χa) from 4.2 to 300 K. We find (1) χc is large and paramagnetic and (2) the ratio of the molar values of χc and the Pauli contribution to the susceptibility (χP) as inferred from the specific heat increases rapidly as a function of stage, varying from about 2.4 in stage 1 to 17 in stage 4. The ratio χcχP is essentially independent of which alkali metal is intercalated (at least at stage 1). The stage dependence of χcχP is in qualitative agreement with the results of a calculation of the π-electron orbital contribution to the susceptibility of graphite intercalation compounds. The orbital part of the susceptibility is shown to be sensitive to the conduction-electron charge distribution from layer to layer in stage 3 and higher compunds, while the spin susceptibility and specific-heat coefficient are less sensitive to this distribution.

The orbital magnetic susceptibility (χorc) of graphite intercalation compounds is calculated for H→c→ using a tight-binding model for the π-electron energy bands and the compact expression for χorc derived by Fukuyama, which includes both intraband and interband terms. The results are presented as an approximate analytic expression for χorc as a function of Fermi level μ (0<μ3 eV) and temperature. For μkBT1 the susceptibility is large and diamagnetic (due to interband transitions), while for kBTμ3 eV, χorc is paramagnetic (mainly due to intraband transitions) in contrast to the usual Landau-Peierls diamagnetism of conduction electrons in parabolic bands. Furthermore, χorc is shown to be a sensitive function of μ and hence of the conduction-electron charge distribution in each graphite layer. We show that this theory accounts for the main features of the experimental data (stage and temperature dependence of χorc) and suggest that measurements of the stage dependence of χorc can be used to estimate the c-axis screening length in graphite intercalation compounds.

We investigate the effects of elastic coherency strains in graphite intercalation compounds. The long-range interaction energies of two-dimensional islands of intercalant are calculated. Using the domain model of Daumas and Hérold, we show that these strains drive a mixed-stage or randomly staged crystal to pure-stage ordering.

First- and second-order Raman scattering are described for EuSe near TN = 4.6 K. The second-order spectral features are interpreted using previous mode assignments for EuSe. Spin fluctuation effects are described for T ≳ TN and related to the phonon dispersion relations.

Magnetic phase dependent, inelastic light scattering from phonons in the europium chalcogenides is reviewed with an emphasis on the behavior of the spectra near magnetic phase transitions. As a result of the one phonon-one spin Raman scattering mechanism which gives rise to these spectra, both broad and sharp first-order Raman lines are observed. The calculated temperature dependence of the broadline Raman spectra is related to spin fluctuations and compared with experiment.

The inelastic light scattering spectra of the europium chalcogenides are analyzed in terms of a one phonon-one spin Raman process. Calculations are presented for the temperature dependence of the integrated intensity of the “spin-disorder” spectrum of EuS and compared with experiment. The observed temperature dependence of the Raman spectrum of EuSe in the two-sublattice antiferromagnetic phase is presented and examined in terms of the one phonon-one spin mechanism. Experiments are suggested in which the Raman spectra can be used to measure the magnetic field dependence of the order parameter exponent β in EuTe.

The ordering and fluctuations of the spins in the magnetic semiconductors EuSe and EuTe are correlated with the selection rules and intensities of inelastic light scattering from one phonon-one spin excitations in these materials. Calculations of the Raman line shape in EuSe are presented for T > >Tc and T ≈ Tc.

### 1978

Magnetic field dependent correlation functions for antiferromagnets near the spin flop‐paramagnetic phase boundary have been calculated using an RPA technique. The exponents describing the field dependence of the magnetization, the staggered magnetization and the correlation length, which are obtained from these calculations, are compared to those obtained using renormalization groups results. Here, we use our recent results relating the symmetry forbidden one‐phonon Raman scattering in magnetic semiconductors to the two‐spin correlation function, to calculate the field dependence of the Raman integrated intensity in EuTe. Light scattering experiments are suggested where the exponents related to the spin flop‐paramagnetic phase boundary, Hc(T), and the field dependence of the staggered magnetization can be measured from the one phonon–one spin Raman spectrum in EuTe.

### 1977

Inelastic light scattering from non-zone-center phonons in the magnetic phases of the europium chalcogenides is interpreted in terms of two-spin correlation functions. Calculations of the Raman line shape in paramagnetic EuS are presented in addition to predictions of the temperature dependence of the centroid and integrated intensities of the first-order Raman spectra of all four europium chalcogenides. In the critical region, analytical expressions which relate the temperature dependence of the integrated intensity to the correlation length κ are derived. The calculations make use of a formalism which relates the Raman and conductivity tensors. The symmetry operations of time reversal and spatial inversion are used to show that the scattering tensor arising from a one-phonon-one-spin mechanism is symmetric under the interchange of the incident and scattered photons. The results are compared with existing experiments and the temperature dependence of the characteristic spin frequency in paramagnetic EuS is inferred.

The first-order Raman scattering experiments in the europium chalcogenides are analyzed in terms of both the symmetries of the various magnetic phases and the resonant enhancements associated with the electronic structure. Both a one-electron band model and a localized magnetic exciton model are used for a calculation of the Raman scattering intensity. The parameters of these models are chosen to be consistent with the optical reflectivity and with the observed resonant Raman enhancement. Comparison with experiment is given in terms of a dimensionless plot of the frequency dependence of the Raman intensity for several europium chalcogenides in the ferromagnetic phase. Comparison with the experimental data suggests that the excitonic model for the electronic transitions is responsible for the observed resonant enhancement behavior.

### 1976

First-order Raman scattering from phonons in the magnetic phases of the europium chalcogenides is analyzed in terms of symmetry for both electric-dipole and electric-quadrupole virtual electronic transitions. The magnetic ordering increases the size of the unit cell and lowers the point group symmetry of the various phases. The magnetic symmetry is used to derive the allowed Raman transitions and polarization selection rules. A phonon dispersion curve for EuSe along the [111] axis is inferred from recent experimental data by applying the selection rules derived on the basis of the magnetic symmetry.

### 1975

Information about the anharmonic properties of crystals can be obtained from a study of the temperature dependence of the linewidth and frequency of their Raman spectra. We have calculated the frequency shift of the zone-center optical phonon in silicon and germanium as a function of temperature assuming that the lifetime and self-energy of the k = 0 optical phonon is governed only by decay into two LA phonons. We find that our model not only fits the experimental data for the frequency shift, but is also highly consistent with calculations and experimental data for the linewidth. The consistency in the fitting of the theoretical curves thus obtained to the experimental data, determining the same, single adjustable parameter for both the shift and the width seems to corroborate the view that there is a selection rule, aside from momentum conservation, which limits the optical phonon decay into two LA phonons only.

### 1974

Measurements of the current transported when an intense, relativistic electron beam penetrates a low-pressure gas show that the net current exceeds the primary beam current. As the pressure is increased, a gradual transition to conditions of magnetic neutralization is observed. Computations of the trajectories of secondary electrons in partially neutralized, uniform beams suggest that secondary electron drift may be responsible for the current amplification.