Modeling of Soft Matter

September 27-October 1, 2004

Material from Talks

Martin Z. Bazant (Department of Mathematics, Massachusetts Institute of Technology) bazant@math.mit.edu http://math.mit.edu/~bazant

A Theory of Cooperative Diffusion in Dense Granular Flows (poster)

Co-authors: Chris Rycroft, Jaehyuk Choi, Ruben Rosales (Mathematics, MIT), Arshad Kudrolli (Physics, Clark University).

Many attempts have been made to describe fast, dilute granular flows starting from the kinetic theory of gases, but slow, dense flows require a fundamentally different approach, due to long-lasting, many-body contacts. In the case of slow drainage, various continuum models have been proposed for the mean flow, but no microscopic theory of fluctuations is available. Here, a new model is proposed which postulates that particles undergo cooperative random motion in response to diffusing "spots" of free volume. The Spot Model may be used in discrete simulations or analyzed in the continuum limit, where some new partial differential equations arise. It predicts spatial velocity correlations, slow cage breaking, and geometry-dominated diffusion in good agreement with particle-tracking experiments in our laboratory (http://math.mit.edu/~bazant/dryfluids).

Yuxing Ben (Department of Mathematics, Massachusetts Institute of Technology) yben@mit.edu

Nonlinear Electrokinetics in Microfluidic Devices (poster)

Co-Authors: Martin Bazant [1,2], Jeremy Levitan [1,3], Todd Thorsen [1,3], H. C. Chang [4].

[1] Institute for Soldier Nanotechnologies, MIT
[2] Applied Mathematics, MIT
[3] Mechanical Engineering, MIT
[4] Chemical Engineering, Notre Dame

We study nonlinear electrokinetic mechanisms for transporting fluid and particles in microfluidic devices with potential applications to biomedical chips. Nonlinear electrokinetics has many advantages, such as low voltage, low power, high velocity, and no significant gas formation in the electrolyte. It provides a challenge to theorists, however, because it involves new and complex mechanisms of interfacial charging and flow, which are not fully understood or explored.

Electrokinetics is the motion of liquid or particle under an applied electric field. In nonlinear electrokinetics, the flow has a nonlinear dependence on the applied field, which is useful in microfluidics because it allows steady fluid motion with AC forcing and much larger flow speeds than linear electrokinetics at high fields. We have been studied several such phenomena both theoretically and numerically, which include the nonlinear electrokinetics of an ion-exchange particle due to second-kind electrophoresis, AC electro-osmosis around patterned-surface electrode arrays, and induced-charge electro-osmosis around metallic or dielectric structures, including colloidal particles as well as wires and surface patterns in microfluidic devices. In all cases, we are working with collaborators on experiments to test and apply the theoretically predicted flows.

Fulvio Bisi (Dipartimento di Matematica, Istituto Nazionale di Fisica della Materia, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy) bisi@dimat.unipv.it

Nanomechanics of Order Reconstruction in Nematic Liquid Crystals (poster)

Co-authors: Epifanio G. Virga (Dipartimento di Matematica, Istituto Nazionale di Fisica della Materia, Università di Pavia, via Ferrata 1, 27100 Pavia, Italy) and Georges E. Durand (Laboratoire de Physique des Solides associ au CNRS (LA2), Universit Paris Sud, F91405 Orsay Cedex, France)

The occurrence of a classical bifurcation scenario in a nematic twist cell with directors prescribed at right angles at the cell's plates, as well as the unfolding of such bifurcation upon perturbing the boundary conditions, has already been shown (1). Within the Landau-de Gennes theory of liquid crystals, we present a model to describe the order reconstruction in a twist cell subject to uniaxial boundary conditions for the order tensor. A variational approach allows us to compute both the nanoforce and the nanotorque exerted on the plates of the cell by the intervening nematic liquid crystal; these mechanical quantities can in principle be measured (2). We show how the torque can indeed be used as a bifurcation parameter, giving a physically measurable quantity that could be used to classify the type of pattern for the director field across the cell. This means that the occurrence of the order reconstruction, obtained upon reducing the distance between the plates, could be revealed by an abrupt change in the torque measured. Furthermore, we study the dependence of the force on the distance between the plates for the different types of equilibrium solutions, and the condition under which such a dependence lacks monotonicity and unveils a hysteresis loop in a kind of molecular 'load diagram.'

(1) F. Bisi, E. C. Gartland, R. Rosso and E.G. Virga, Phys. Rev. E, 68, (2003)

(2) R. G. Horn, J. N. Israelachvili, E. Perez, J. Physique 42, 39 (1981)

Raphael Blumenfeld (Cavendish Laboratory) rbb11@phy.cam.ac.uk http://www.poco.phy.cam.ac.uk/~rbb11

Stress Field Equations in Granular Solids - A Shift of Paradigm
Slides:   pdf

A key concept to the understanding of stress transmission in granular materials is the Marginally Rigid State and an experiment is described which establishes the relevance of this state. The MRS can be regarded as a critical point wherein a particular lengthscale diverges. Granular matter at the MRS is statically determinate (isostatic), obviating elasticity theory. A new "isostaticity theory" is formulated for stress transmission at the MRS. It explains the force chains, frequently observed in experiments, and makes it possible to predict their individual trajectories.

The insight from the static theory makes it possible to formulate a theory for the yield and failuer of granular solids and a new set of equations is presented for the flow of granular matter in this regime.

Zhenlu Cui (Department of Mathematics, Florida State University) zcui@math.fsu.edu

A Second Order Tensor Theory for Flows of Cholesteric Liquid Crystal Polymers (poster)

Joint work with Qi Wang.

We derive a second order tensor based hydrodynamic theory for flows of cholesteric liquid crystal polymers following the continuum mechanics formulation of McMillan's second order tensor theory for liquid crystals. We present some equilibrium solutions with various symmetries. Then, we study the permeation modes in weak shear and Poiseuille flows using a coarse-grained method. Our results extend those obtained by LE theory.

Antonio DeSimone (International School of Advanced Studies (SISSA), Trieste, Italy) desimone@sissa.it

Soft Elasticity of Nematic Elastomers and Wetting Properties of Rough Surfaces
Slides:    pdf

Bridging across length scales is one of the fundamental challenges in the modelling of soft material systems whose mechanical response is driven by rough energy landscapes. We will illustrate the use of relaxation and homogenization techniques through two case studies.

Brian DiDonna (IMA Postdoc) didonna@ima.umn.edu

Elasticity of Random Spring Networks (poster)

Some microstructured materials, such as sparse crosslinked polymer networks (e.g. actin networks in living cells) or open cell metal foams, can be described mechanically as networks of elastic rods with various bending potentials and force extension relations.We study the fundamental problem of how a ball and spring network with a known degree of randomness, either in geometry or in spring constants, reacts to an externally applied shear. Namely, how uniform is the local shear and displacement field on different length scales, and how does relaxation from an affine (stickly uniform) shear field soften the bulk elastic constant. Scaling of spatial correlation functions with distance, applied shear, and degree of randomness is derived and demonstrated through numerical simulation.

Masao Doi (Tokyo University) doi@ap.t.u-tokyo.ac.jp

Masao Doi (Tokyo University) doi@ap.t.u-tokyo.ac.jp

OCTA: Open Computational Tool for Soft Matters (poster)

OCTA is an intergrated simulation programs for soft materials developed by a joint project of industry and academia supported by Japanese governemnt. It is a free software and can be downloaded at [1]. OCTA consists of several simulators specialized to certain phenomena of polymers (molecular dynamics, rheology, interfacial phenomena, micro fluidics) and a simulation platform. The role of the simulation platform is to facilitates the collaboration of various simulation programs. The overview and the demonstrationo of the OCTA system will be given [2].

[1] OCTA - integrated simulation system for soft materials- , http://octa.jp.

[2] M. Doi, J. Comp. App. Math. 149 13-25 (2002)

James J. Feng (Department of Chemical & Biological Engineering, University of British Columbia) jfeng@chml.ubc.ca http://faculty.chml.ubc.ca/jfeng

Simulating Two-Phase Complex Fluids Using a Diffuse-Interface Model (poster)

In two-phase complex fluids such as emulsions and polymer blends, the components are often microstructured complex fluids themselves. To model and simulate the fluid dynamics of such systems, one has to deal with the dual complexity of non-Newtonian rheology and evolving interfaces. Recently, we developed a diffuse-interface formulation which incorporates complex rheology and interfacial dynamics in a unified framework. This presentation will describe recent results on drop deformation, coalescence and self-assembly in a liquid crystalline matrix.

Eliot Fried (Department of Mechanical Engineering, Washington University in St. Louis) efried@me.wustl.edu http://www.me.wustl.edu/ME/faculty/efried/

Universal States in Nematic Elastomers
Slides:   pdf

For a particular material model, a state that can be maintained in equilibrium for by the action of surface tractions alone is called controllable. If a state is controllable for all material models in a particular class, that state is called universal. Universal states are of central importance in the design of experiments for the determination of constitutive relations, as such experiments should not be based on specific model expressions but, rather, should produce states sustainable by the full range of material models belonging to a broadly relevant class. Universal states have been used with benefit for the study of elastomeric solids, viscoelastic fluids, and uniaxial nematic liquid crystals. In addition to background material, this talk will discuss recent results and open problems related to universal states in nematic elastomers.

Francois Graner (Shannon Laboratory, Department of Physics, Université Joseph Fourier) francois.graner@ujf-grenoble.fr http://graner.net/francois/

Models for Elastic, Plastic, Fluid Materials
Slides:   pdf

It is difficult to predict the constitutive relations of foams, emulsions, granular materials or gels from first principles. Experimentally, the mechanical behaviors of such viscoplastic materials do not appear to change discountinuously. However, mathematical singularities appear as soon as a solid exhibits plastic deformation, or a liquid a non-zero restoring force. The continuum elasticity theory and the Navier-Stokes equations break down.

Visco-elasto-plastic theories require an interpolation between these apparently orthogonal descriptions. We use physical quantities which exist and are measurable in all regimes. We define generalized stress and strain tensors as statistical averages over microscopical details (avoiding the use of a microscopic reference state). They recover each correct limiting behaviors when either classical theory applies.

We perform local and averaged measurements on foam flowing past an obstacle or through a constriction, or under Couette shear. Applications include discrete mechanics, where the sample size (upper cut-off) is not significantly larger than the individual size (lower cut-off): granular materials, nano-fluidics.

Alexander Grosberg (Department of Physics and Astronomy, University of Minnesota) grosberg@physics.umn.edu

Knots in Polymer Physics

We report a series of computer simulations and related analytical studies of the role of topological constraints in the equilibroium properties of polymeric loops. We show that no-knots loops exhibit swelling similar to that of self-avoiding chains even when polymer has negligible excluded volume. We further show that global topology of the loop has a profound effect on the local fractal geometry of the polymer. For compact polymers, this leads to locally shrunken conformations. We discuss implications of these findings for a number of polymer systems.

Andrei A. Gusev (Institute of Polymers, Department of Materials, ETH Hoenggerberg, HCI H527, CH-8093 Zurich, Switzerland) gusev@mat.ethz.ch

Finite Element Mapping for Spring Network Representations of the Mechanics of Solids
Slides:   pdf

We present a general finite element mapping procedure for defining spring network representations of solid mechanics. The procedure is rigorous and equally suitable for setting regular and unstructured spring network models of generally anisotropic solids. We use the procedure to define close-packed triangular and simple cubic lattice spring models of isotropic 2D and 3D elastic media, respectively. We extend the study to heterogeneous solids and show that the mapped spring network approach constitutes an appealing route for incorporating subelement level constitutive equations.

Cheng-Cher Huang (School of Physics and Astronomy, University of Minnesota) huang001@umn.edu

Experimental and Theoretical Studies of Liquid crystal SmC* Variant Phases
Slides:   pdf

In 1989, the discovery of antiferroelectric response in a liquid crystal mesophase (SmCA*) is an important landmark in soft condensed matter physics. Soon after, at least, three new mesophases (i.e. SmC.*, SmCFI2*, and SmCFI1*) were identified. Collectively, all these four new mesophases and SmC* are called SmC* variant phases. Since then, enormous experimental and theoretical effort has been aimed at addressing the following two important questions. Experimentally, our research group has accomplished remarkable tasks to identify the molecular arrangements in the SmC* variant phases [1]. Theoretically, one would like to figure out the origin or mechanism of such a rich phase sequence within a temperature window less than 50K. As a starting point, a phenomenological model based on mean-filed approaches will be presented [2].

1. A. Cady, et al., Phys. Rev. E 64, 050702 (2001) and Ref found therein.

2. D. A. Olson, et al., Phys. Rev. E 66, 021702 (2002).

Cheng-Cher Huang (School of Physics and Astronomy, University of Minnesota) huang001@umn.edu

Optical Investigations on the Biaxial Smectic A Phase of a Bent-Core Compound (poster)

We report an antiferroelectric biaxial Smectic A liquid crystal phase from one bent-core molecule 1g14. We also extracted the critical exponent associated with the biaxiality, which is different from the one that is associated with the uniaxial - antiferroelectric biaxial SmA phase transition.

Antal Jákli (Liquid Crystal Institute, Kent State University) jakli@lci.kent.edu http://www.lci.kent.edu/jakli.html

Electric Field-Induced Motion of Solid Particles in Smectic Liquid Crystals (poster)

Joint work with G. Liao, J. R. Kelly, O.D. Lavrentovich (Liquid Crystal Institute, Kent State University, Kent, OH 44242, USA).

Accurate characterization of rheological properties of smectic liquid crystals is a very difficult task due to the formation of defects during flow processes. The study of motion of colloid particles in smectic liquid crystals is important not only from the standpoint of the rheology of smectics, but is also of biological relevance, as similar phenomena might be expected for proteins in cell membranes. We studied electric field-induced rotational and translational motion of spherical and cylindrical glass particles embedded in the SmA slab of octyl cyanobiphenyl (8CB) and in SmA* and SmC* slab of ferroelectric liquid crystal CS2003 (from Chisso Co.). Above a threshold electric field the particles rotate about their symmetry axes analogous to the Quincke observed in isotropic fluids.[LINK][i] At higher fields some of them also begin to move along the smectic layers perpendicular to the electric field. Studying the movement of the particles allows us to study the flow behavior of the smectic A and smectic C liquid crystals around the spheres. We find that the flow is purely viscous and does not involve any permeation at sufficiently large speeds in accordance with the theory of de Gennes.[LINK][ii] We also observed a very interesting new phenomenon: when the sample contains air bubbles, the moving spheres stick to the bubbles and rotate collectively with a field- dependent (linear) speed that is independent of the radius of the bubbles. At higher fields even the bubbles can move and tend to stick together. The details of the motion and the underlying physical mechanism will be discussed.
[LINK] [i] T.B. Jones, "Electromechanics of Particles", Cambridge Univ. Press., New York, (1995)
[LINK] [ii] P.G. de Gennes, Phys. Fluids, 17, 1645 (1974)

Sookyung Joo (IMA Postdoc) sjoo@ima.umn.edu

The Phase Transition Between Chiral Nematic and Smectic Liquid Crystals (poster)

We study the Chen-Lubensky model to investigate the phase transition between chiral nematic and smectic liquid crystals. First, we prove the existence of the minimizers in an admissible set where the order parameter vanishes on the boundary. The splay, twist, and bend Frank constants are considered to diverge near N* -- C* phase transition based on physical observation, while only twist and bend constants can be assumed to diverge near N* -- A* transition. Then we describe the transition temperatures for both smectic A side and smectic C side when a domain is a considerably large liquid crystal region confined in two plates.

Randall D. Kamien (Department of Physics & Astronomy, University of Pennsylvania ) kamien@physics.upenn.edu http://www.physics.upenn.edu/~kamien/

Bending The Rules
Slides:   pdf

We discuss the ordering of liquid crystalline phases which possess both cubic symmetry and smectic-like, lamellar ordering. We will show that there is a fundamental frustration in this system. We propose an ansatz based on triply-periodic minimal surfaces. We discuss more general constructions based on topological field configurations and tesselation of the hyperbolic plane.

David Kinderlehrer (Department of Mathematical Sciences, Carnegie Mellon University) davidk@andrew.cmu.edu http://www.math.cmu.edu/people/fac/kinderlehrer.html

Cooperative Effects in a Dye/Liquid Crystal System

We discuss the dichroic dye/liquid crystal interaction known as the Janossy effect and studied by Palffy-Muhoray, Kosa and E. A consistent variational principle is offered that takes advantage of Monge-Kantorovich mass transport ideas and some consequences, like whether or not such a formulation can actually predict the observed Janossy effect, are discussed. As time permits, we shall discuss the general issue of diffusion mediated transport, the interaction of transport mechanisms and diffusion at extremely small scale with hints to other systems. This is joint work with Stuart Hastings and Michael Kowalczyk.

Issac Klapper (Department of Mathematical Sciences, Montana State University) klapper@math.montana.edu

Biofilms As Soft Materials (poster)

Biofilms are complex polymeric materials consisting of microorganisms protected within a self-secreted extracellular polymeric matrix. They are ubiquitous - for example, it has been estimated that approximately %75 of bacteria are found in the biofilm phenotype. Material properties of biofilms play an important role in their extraordinary resistance to mechanical and chemical stresses. This poster is intended as a brief introduction to biofilms including some discussion of their rheological properties and attempts to model them as continuum materials.

Frédéric Legoll (IMA Postdoc)

High-Order Averaging Schemes for Molecular Dynamics Simulations (poster)

Many thermodynamical properties of chemical systems are defined as averages of observables over the phase space of the underlying microscopic system. A way to compute these ensemble averages is to use Molecular Dynamics and to compute time averages on long trajectories. In this work, we propose high-order averaging formulae to compute time averages. In some cases, these formulae significantly improve the convergence rate with respect to the simulation time. Many numerical examples will be provided, as well as error bounds.

This is joint work with Eric Cancès, Claude Le Bris and Gabriel Turinici (CERMICS ENPC and INRIA Rocquencourt, France), and with François Castella, Philippe Chartier and Erwan Faou (INRIA Rennes, France).

Didier Long (Laboratoire de Physique des Solides, Université Paris-Sud, Bat. 510) long@lps.u-psud.fr legoll@ima.umn.edu

Non-Linear Behavior of Filled Elastomers by Molecular Dissipative Dynamics
Slides:   pdf

Whereas non-reinforced rubbers have a low resistance to tear and wear, reinforced elastomers can be used under very demanding conditions. The corresponding effects are very important since the energy of tearing can be hundred of times larger than that for the non-reinforced material. Recent progress regarding the glass transition mechanisms in the vicinity of interfaces have allowed for understanding some aspects of the reinforcement, such as the high shear modulus and some dissipative properties. By implementing this picture in dissipative molecular dynamics simulations, we show here how one can account for some of the striking non-linear properties of these systems, such as the Payne effect (strong drop of the shear modulus at a few percents deformation) and the Mullins effect (hysteresis of mechanical properties, partially recoverable after waiting or thermal treatment of the system). We show also that the associated mechanisms lead to some specific plastic properties.

Robert B. Meyer (Martin Fisher School of Physics, Brandeis University) meyer@brandeis.edu

Electrically and Mechanically Driven Instabilities in Thin Layers of Nematic Liquid Crystal Gel
Slides:   pdf

A layer of nematic liquid crystal is aligned as a single crystal between rigid plates, and gelled into a soft solid by formation of a crosslinked polymer network within the nematic phase. Changing the temperature produces changes in the nematic order parameter, resulting in stresses in the gel, due to its confinement. Likewise, an applied electric field produces a torque on the nematic director, either stabilizing or destabilising its initial orientation. The combination of elastic and electric stresses leads to buckling instabiltites in the sample, which are observed by polarized light microscopy. Experimental results and theoretical analysis will be presented.

Robert Pelcovits (Department of Physics, Brown University) pelcovits@physics.brown.edu

Visualization of Topological Defects in Liquid Crystals
Slides:   pdf

I will discuss our ongoing work on the visualization of topological defects in numerical simulations of liquid crystals. We are collaborating with a computer scientist who has developed techniques which allow easy visualization of the features of tensor fields, originally in the context of MRI scans of the brain. In our case we wish to visualize the nematic order parameter tensor in order to locate and characterize defects in our data set. Our data set is produced by quenching a Gay-Berne nematic fluid of 65K particles. I will discuss the challenges faced in the study of defects in fluid (as opposed to lattice) models of liquid crystals and the significant progress we have made to date.

Harald Pleiner (Max Planck Institute for Polymer Research, D 55021 Mainz, Germany) pleiner@mpip-mainz.mpg.de http://www.mpip-mainz.mpg.de/~pleiner

A Physicists' View on Constitutive Equations

Joint work with M. Liu (Inst. f. Theoret. Physik, Universitaet Tuebingen, D 72076 Tuebingen, Germany), and Helmut R. Brand (Theoret. Physik III, Universitaet Bayreuth, D 95440 Bayreuth, Germany)

Hydrodynamic equations for various kinds of complex fluids (simple liquids, binary mixtures, liquid crystals, superfluids, crystals, etc.) can be derived rigorously using general physical laws and principles. This hydrodynamic method is generalized to include slowly relaxing quantities, in particular those describing viscoelasticity. By this procedure it is guaranteed that the resulting equations are in agreement with basic physical laws and requirements.

We start with the nonlinear hydrodynamic equations for elastic media derived from basic physical principles. For the Eulerian strain tensor the lower convected time derivative is obtained, unambiguously [1-3]. Adding a relaxation term the permanent elasticity is transformed into viscoelasticity [1,2], where both, the short time and the long time limit, are given correctly. The dynamic equation for the strain tensor obtained that way still shows the lower convected derivative universally. It covers the usual non-Newtonian effects, like shear thinning, strain hardening, stress overshoot, normal stress differences and Weissenberg effect, non exponential stress relaxation, etc. [4]. When brought into the more familiar form of a dynamic equation for the stress tensor ("constitutive equation"), it comprises most of the well-known ad-hoc models (Maxwell, Oldroyd, Giesekus), and is more general in structure than those, but is in disagreement with some of them (Johnson-Segalman, Jeffries) [5]. It imposes some restrictions on, and reveals some interdependencies of, the various non-Newtonian contributions that are otherwise introduced heuristically. It is shown how these contributions originate from (nonlinear) elasticity, viscosity, strain relaxation and convection. The time derivative for the stress tensor is no longer of the lower convected type, but is material dependent. We also discuss the connection to those descriptions of viscoelasticity that utilize an orientational (or configurational) order parameter [6].

[1] H. Temmen, H. Pleiner, M. Liu, and H.R. Brand, Phys. Rev. Lett. 84 (2000) 3228; 86 (2001) 745.

[2] H. Pleiner, M. Liu, and H.R. Brand, Rheol. Acta 39 (2000) 560.

[3] M. Grmela, Phys. Lett. A, 296 (2002) 97.

[4] O. Mueller, M. Liu, H. Pleiner, and H.R. Brand, to be published.

[5] H. Pleiner, M. Liu, and H.R. Brand, Rheol. Acta, DOI: 10.1007/s00397-004-0365-8 (2004).

[6] H. Pleiner, M. Liu, and H.R. Brand, Rheol. Acta 41 (2002) 375.

Riccardo Rosso (Dipartimento di Matematica, Università di Pavia) rosso@dimat.unipv.it

Periodic Saddle-Splay Freedericksz Transition in a Nematic Cell (poster)

By use of a local stability analysis, we predict the existence of a Periodic Saddle-Splay Freedericksz (PSSF) transition, supplementing the existing set of external-field-driven Freedericksz transitions in a nematic cell. The onset of the PSSF transition requires a weaker field than the classical, aperiodic, splay Freedericksz transition, provided that the saddle-splay elastic constant is large enough, and the anchoring strengths at the plates of the cell differ from one another. We determine the threshold condition for the PSSF transition and the structure of the associated unstable mode.

André M. Sonnet (Department of Mathematics, University of Strathclyde) ams@maths.strath.ac.uk http://www.maths.strath.ac.uk/~aas02102

Defect Dynamics in Liquid Crystals (poster)

The dynamics of topological defects is of interest in many different branches of physics. Liquid crystals are of particular importance because they allow to conduct easily accessible experiments for many effects. One example that has recently attracted a lot of attention is the annihilation of defect pairs in nematic and smectic liquid crystals. Somewhat surprisingly, it was found that the speed with which a defect moves depends strongly on its topological charge. Numerical explorations for nematic disclination lines and defects in smectic films have established that material flow plays a major role in the pair annihilation process. We present an analytical argument to show that the origin of the asymmetry can be retraced to the opposite parity of elastic and viscous forces under topological charge change.

Eugene M. Terentjev (Cavendish Laboratory, University of Cambridge) emt1000@cam.ac.uk http://www.poco.phy.cam.ac.uk/~emt1000

Kinetic Theory of Rotational Diffusion and Anisotropic Viscosity of Liquid Crystals
Slides:   pdf

We shall discuss the molecular-statistical approach to describing the rotational diffusion of anisotropic molecules (rods or disks) in a mean field of liquid crystalline order (nematic or smectic-C). The issues of microscopic stress tensor and its averaging, of the spectrum of relaxation times, of the role of more delicate ordering (e.g. smectic layering or biaxiality), and of the route to the full pair-correlation theory will be considered.