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2005 | Buch

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Modeling of Soft Matter

herausgegeben von: Maria-Carme T. Calderer, Eugene M. Terentjev

Verlag: Springer New York

Buchreihe : The IMA Volumes in Mathematics and its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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This IMA Volume in Mathematics and its Applications MODELING OF SOFT MATTER contains papers presented at a very successful workshop with the same ti tle. The event, which was held on September 27-October 1, 2004, was an integral part of the 2004-2005 IMA Thematic Year on "Mathematics of Ma terials and Macromolecules: Multiple Scales, Disorder, and Singularities. " We would like to thank Maria-Carme T. Calderer (School of Mathematics, University of Minnesota) and Eugene M. Terentjev (Cavendish Laboratory, University of Cambridge) for their superb role as workshop organizers and editors of the proceedings. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Arnd Scheel, Deputy Director of the IMA PREFACE The physics of soft matter in particular, focusing on such materials as complex fluids, liquid crystals, elastomers, soft ferroelectrics, foams, gels and particulate systems is an area of intense interest and contemporary study. Soft matter plays a role in a wide variety of important processes and application, as well as in living systems. For example, gel swelling is an essential part of many biological processes such as motility mecha nisms in bacteria and the transport and absorption of drugs. Ferroelectrics, liquid crystals, and elastomers are being used to design ever faster switch ing devices. Experiments of the last decade have provided a great deal of detailed information on structures and properties of soft matter.

Inhaltsverzeichnis

Frontmatter

An Energetic Variational Formulation with Phase Field Methods for Interfacial Dynamics of Complex Fluids: Advantages and Challenges

Abstract

The use of a phase field to describe interfacial phenomena has a long and fruitful tradition. There are two key ingredients to the method: the transformation of Lagrangian description of geometric motions to Eulerian description framework, and the employment of the energetic variational procedure to derive the coupled systems. Several groups have used this theoretical framework to approximate Navier-Stokes systems for two-phase flows. Recently, we have adapted the method to simulate interfacial dynamics in blends of microstructured complex fluids. This review has two objectives. The first is to give a more or less self-contained exposition of the method. We will briefly review the literature, present the governing equations and discuss a suitable numerical schemes, such as spectral methods. The second objective is to elucidate the subtleties of the model that need to be handled properly for certain applications. These points, rarely discussed in the literature, are essential for a realistic representation of the physics and a successful numerical implementation. The advantages and limitations of the method will be illustrated by numerical examples. We hope that this review will encourage readers whose applications may potentially benefit from a similar approach to explore it further.

James J. Feng, Chun Liu, Jie Shen, Pengtao Yue

Non-Equilibrium Statistical Mechanics of Nematic Liquids

Abstract

The rotational diffusion of a general-shape object (a molecule) in a flow of uniaxial nematic liquid crystal is considered in the molecular field approximation. The full corresponding Fokker-Planck equation is derived, and then reduced to the limit of diffusion of orientational coordinates in a field of uniaxial nematic potential and the flow gradient. The spectrum of orientational relaxation times follows from this analysis. As a second main theme of this work, we derive a complete form of microscopic stress tensor for this molecule from the first principles of momentum balance. Averaging this microscopic stress with the non-equilibrium probability distribution of orientational coordinates produces the anisotropic part of the continuum Leslie-Ericksen viscous stress tensor and the set of viscous coefficients, expressed in terms of molecular parameters, nematic order and temperature. The axially-symmetric limits of long-rod and thin-disk molecular shapes allow comparisons with existing theories and experiments on discotic viscosity. The article concludes with more complicated aspects of non-linear constitutive equations, microscopic theory of rotational friction and the case of non-uniform flow and director gradients.

Chii J. Chan, Eugene M. Terentjev

Anisotropy and Heterogeneity of Nematic Polymer Nano-Composite Film Properties

Abstract

Nematic polymer nanocomposites (NPNCs) are comprised of large aspect ratio rod-like or platelet macromolecules in a polymeric matrix. Anisotropy and heterogeneity in the effective properties of NPNC films are predicted in this article. To do so, we combine results on the flow-processing of thin films of nematic suspensions in a planar Couette cell, together with homogenization results for the effective conductivity tensor of spheroidal inclusions in the low volume fraction limit. The orientational probability distribution function (PDF) of the inclusions is the central object of Doi-Hess-Marrucci-Greco theory for flowing nematic polymers. From recent simulations, the PDF for a variety of anisotropic, heterogeneous thin films is applied to the homogenization formula for effective conductivity. The principal values and principal axes of the effective conductivity tensor are thereby generated for various film processing conditions. Dynamic fluctuations in film properties are predicted for the significant parameter regime where the nematic polymer spatial structure is unsteady, even though the processing conditions are steady.

M. Gregory Forest, Ruhai Zhou, Qi Wang, Xiaoyu Zheng, Robert Lipton

Non-Newtonian Constitutive Equations Using the Orientational Order Parameter

Abstract

Nonlinear hydrodynamic equations for non-Newtonian fluids are discussed. We start from the recently derived hydrodynamic-like nonlinear description of a slowly relaxing orientational order parameter tensor. The reversible quadratic nonlinearities in this tensor’s dynamics are material dependent due to the generalized nonlinear flow alignment effect that comes in addition to the material independent corotational convected derivative. In the entropy production these terms are balanced by linear and nonlinear orientational-elastic contributions to the stress tensor. These can be used to get a nonlinear dynamic equation for the stress tensor (sometimes called constitutive equation) in terms of a power series in the variables. A comparison with existing phenomenological models is given. In particular we discuss how these ad-hoc models fit into the hydrodynamic description and where the various non-Newtonian contributions are coming from. We also discuss the connection to the hydrodynamic-like description of non-Newtonian effects that employs a relaxing strain tensor.

Harald Pleiner, Mario Liu, Helmut R. Brand

Surface Order Forces in Nematic Liquid Crystals

Abstract

The notion of surface order force in nematic liquid crystals is presented and contrasted with the notions of similar forces already introduced in the literature. We illustrate how a surface order force could in principle be measured and how it would convey the mechanical signature of an intrinsically nanoscopic phenomenon, often referred to as order reconstruction. The relationship between this force and the occurrence of biaxial states of the nematic order tensor is further illuminated.

Fulvio Bisi, Epifanio G. Virga

Modelling Line Tension in Wetting

Abstract

Line tension can be viewed as the analogue, for three-phase contact, of surface tension. However, obtaining a coherent picture from the different avenues followed to model line tension is much harder than the analogous operation for surface tension. This essentially reflects the extreme sensitivity of line tension to the details of the model employed. Line tension has an impact on the equilibrium and stability of fluid droplets laid on a rigid substrate, in the presence of a vapor phase. In particular, the sign of line tension is a critical issue, that gave rise to conflicting interpretations. Here, we review the approaches to line tension from microscopic to macroscopic scales, stressing the mathematical problems involved. We also illustrate a stability criterion for wetting functionals to clarify the rôle of the sign of line tension. As an application, we discuss how stability of liquid bridges near the wetting or the dewetting transition mirrors the scaling laws for surface and line tension.

Riccardo Rosso

Variational Problems and Modeling of Ferroelectricity in Chiral Smectic C Liquid Crystals

Abstract

This article deals with modeling and analysis of chiral smectic C liquid crystals and their ferroelectric phases. The polarization field plays an important role in the problem. The total energy of the smectic C* contains the Oseen-Frank free energy of the nematic, together with smectic terms quadratic in the second order gradients of the complex wave function describing smectic layering, and expression for the surface energy. In addition, the polar self-interaction is taken into account, together with the electrostatic energy associated with an external electric field. The case of spatial variable electric fields is also addressed. Stability properties of the solutions are discussed to determine the interplay between the surface and electric energy terms. The physically relevant boundary conditions of the admissible fields bring out analogies to the problems of vortex tubes and vortex sheets in fluid mechanics.

Jinhae Park, M. Carme Calderer

Stripe-Domains in Nematic Elastomers: OLD and New

Abstract

Formation of stripe-domains has often been observed in nematic elastomers, starting from the pioneering work of Finkelmann and coworkers. One of the possible interpretations of this phenomenon is to view it as a material instability driven by energy minimization. This approach, first proposed by Warner and Terentjev, has been quite helpful in the analysis of stretching experiments of thin sheets, and in the modelling of soft elasticity associated with stripe-domain formation. Recently, complex stripe-domain patterns have been observed in nematic gels undergoing the isotropictonematic transition while being confined by two glass plates. We suggest that, once again, energy minimization can be seen as the driving mechanism for the formation of the observed patterns.

Antonio Desimone, Georg Dolzmann

Numerical Simulation for the Mesoscale Deformation of Disordered Reinforced Elastomers

Abstract

We study here the dynamical behavior of disordered elastic systems such as gels or filled elastomers, by dissipative molecular dynamics. We show that applied macroscopic deformations result in non-affine deformations at the scale of the filler particles. These non-affine deformations lead to slow meso-scale reorganizations, which could explain the long relaxation times measured in gels, and also in rubbers even at temperatures much above the glass transition temperature.

Didier Long, Paul Sotta

Stress Transmission and Isostatic States of Non-Rigid Particulate Systems

Abstract

The isostaticity theory for stress transmission in macroscopic planar particulate assemblies is extended here to non-rigid particles. It is shown that, provided that the mean coordination number in d dimensions is d + 1, macroscopic systems can be mapped onto equivalent assemblies of perfectly rigid particles that support the same stress field. The error in the stress field that the compliance introduces for finite systems is shown to decay with size as a power law. This leads to the conclusion that the isostatic state is not limited to infinitely rigid particles both in two and in three dimensions, and paves the way to an application of isostaticity theory to more general systems.

Raphael Blumenfeld

Backmatter

Titel: Modeling of Soft Matter
herausgegeben von: Maria-Carme T. Calderer
Eugene M. Terentjev
Verlag: Springer New York
Electronic ISBN: 978-0-387-32153-0
Print ISBN: 978-0-387-29167-3
DOI: https://doi.org/10.1007/0-387-32153-5