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This section offers basic recipes and sample applications which allow the reader to immediately start his/her own simulation project on topics we dealt with in this book. Concerning molecular dynamics and Monte Carlo simulation there are, of course, several useful books already available which describe the ‘art of simulation‘ [141, 156, 256] in an exhaustive way. The reason we print some simple codes is that we skipped algorithmic details in the foregoing chapters. Simulations are always performed using dimensionless numbers, and all dimensional quantities can be expressed in terms of reduced units, cf. Sect. 4.3 for conventional Lennard-Jones units. In this chapter, we concentrate on the necessary, and skip anything more sophisticated. Codes have been used in classrooms, they are obviously open for modifications and extensions, and offer not only an executable, but all necessary formulas for doing simulations in the correct (which is often essential) order. The overall spirit is as follows: codes are short, run without changes, demonstrate the main principle in a modular fashion, and are thus in particular open regarding efficiency issues and extensions. Algorithms are presented in the MatlabTM language, which is mostly directly portable to programming languages like fortran, c, or MathematicaTM. For an introduction we refer to [423]. Additional commands needed to visualize the results are given in the figure title for each application. Simulation codes, in a less modular fashion, are also available online at www.complexfluids.ethz.ch. Functions are shared over sections, for that reason we begin with an alphabetic list of all (nonbuiltin) functions in this chapter. [ABSTRACT FROM AUTHOR]

In Chap. 9 we summarized methods for analyzing the equilibrium statistics of physical system, using Monte Carlo methods. While at equilibrium complex fluids may be isotropic or anisotropic, under the influence of external fields, all systems tend to become anisotropic (isotropic external fields are seldom). The anisotropy is induced by the external fields (flow field, magnetic field etc.) and must reflect the tensorial symmetry of the applied field. We have to deal with the coupling of tensors of different ranks, with important implications on Onsager reciprocal relationships, a topic discussed elsewhere. This chapter is concerned with the dynamics, in particular, the orientational dynamics of structured fluids subjected to orienting fields. The dynamics and anisotropy is properly modeled by using orientational distribution functions, their equations of change, and the corresponding equations of change for the moments (here, alignment tensors) of the distribution function. We restrict ourself to discuss the case of one-particle (single-link) orientational distribution functions. [ABSTRACT FROM AUTHOR]

In this chapter we summarize definitions and properties of cartesian, anisotropic, irreducible and isotropic tensors and related tensor operators. In several aspects more exhaustive, detailed treatments, eventually using a different notation, can be found in [4, 82, 417-420]. We present rules and properties using tensor product notation and tend to avoid component notation (except in footnotes). [ABSTRACT FROM AUTHOR]

Monte Carlo methods use random numbers, or ‘random' sequences, to sample from a known shape of a distribution, or to extract distribution by other means. and, in the context of this book, to (i) generate representative equilibrated samples prior being subjected to external fields, or (ii) evaluate high-dimensional integrals. Recipes for both topics, and some more general methods, are summarized in this chapter. It is important to realize, that Monte Carlo should be as artificial as possible to be efficient and elegant. Advanced Monte Carlo ‘moves', required to optimize the speed of algorithms for a particular problem at hand, are outside the scope of this brief introduction. One particular modern example is the wavelet-accelerated MC sampling of polymer chains [406]. [ABSTRACT FROM AUTHOR]

One of the major issues raised by the Boltzmann equation is the problem of the reduced description. Equations of hydrodynamics constitute a closet set of equations for the hydrodynamic field (local density, local momentum, and local temperature). From the standpoint of the Boltzmann equation, these quantities are low-order moments of the one-body distribution function, or, in other words, the macroscopic variables. The problem of the reduced description consists in deriving equations for the macroscopic variables from kinetic equations, and predicting conditions under which the macroscopic description sets in. The classical methods of reduced description for the Boltzmann equation are: the Hilbert method, the Chapman-Enskog method, and the Grad moment method, reviewed in [98]. The general approach to the problem of reduced description for dissipative system was recognized as the problem of finding stable invariant manifolds in the space of distribution function. The notion of invariant manifold generalizes the normal solution in the Hilbert and in the Chapman-Enskog method, and the finite-moment sets of distribution function in the Grad method. A generalization of the Grad moment method is the concept of the quasiequilibrium approximation, cf. Sect. 2.6 and [94, 98]. Boltzmann's kinetic equation has been expressed in GENERIC form [347], cf. Sect. 8.3, demonstrating that no dissipative potential is required for representing these equations. [ABSTRACT FROM AUTHOR]

In [292] we provided a statistical interpretation of the director theory of Ericksen and Leslie (EL) [293-295] for nematic liquid crystals. Starting from a Fokker-Planck equation of the type (6.1) supplemented by a mean-field plus external potential, and using an expression for the stress tensor derived for structural theories of suspensions, we interpreted the EL viscosity coefficients and molecular fields in terms of the parameters characterizing a suspension, i.e., particle geometry, particle concentration, degree of alignment, solvent viscosity, and the potential. It turned out that the theory of Kuzuu and Doi [73] for concentrated suspensions of rod-like polymers, the affine transformation model by Hess and Baalss [296], the results by Hand [297] and Sin-Doo Lee [298], were contained as special cases. In distinction to Kuzuu and Doi in [292] we also obtained an expression for the tumbling parameter in terms of order parameters and particle shape, which had been confirmed independently by Archer and Larson [299]. Here, in order to review the highly coarse-grained models depicted at the top of Fig. 1.2 we summarize the macroscopic framework developed by EL. We give an example on how the microscopic quantities such as an anistropic gyration tensor for polymeric chains, or the shape of suspended ellipsoidal (colloidal) particles enter the anisotropic viscosities. [ABSTRACT FROM AUTHOR]

Having discussed the range of applicability for various FENE chain models mostly listed in the upper part of Table 14.1 one may expect that we also review the FENE models in the lower part of this table. Forttunately, several reviews exist summarizing the constitutive equations following from the approximations involved in the FENE-P. and FENE-L. models, cf. [4, 35, 66] such there is no need to summarize them - and there usefulness in micro-macro applications - here. Rather, we turn to simple low dimensional models depicted in the upper part of Fig. 1.2, i.e. tube models and elongated particle models for the dexcription of complex fluids. One may ask how these levels of descriptions are related. This will be discussed in Sect. 8.10.1. [ABSTRACT FROM AUTHOR]

In order to be prepared for the analysis of the flexible FENE-C (FENE model which allows for scission/recombination), FENE-B (which allows for bending stiffness) and FENE-CB fluids (both bending stiffness and scission) to be discussed below, we summarize results for the configurational statistics of wormlike chains (WLC) in external fields by using the method of functional integrals (FI) in quasimomentum space. From the correlation functions, statistical properties of WLCs, such as gyration radius and scattering functions can be obtained. By varying the bending rigidity the WLC exhibits a crossover from an ideal Gaussian chain to a rodlike chain. Simulations on the WLC model are widely available, see e.g. [185-188]. [ABSTRACT FROM AUTHOR]

A dense collection of repulsive FENE chains serves as a suitable microscopic model for both entangled and unentangled polymer melts. We will consider once more linear and monodisperse chains although FENE models are immediately applicable to polydisperse polymers with aribtrary architectures. Besides its success for the study of polymer melts at equilibrium [13,139-141], the nonlinear viscoelastic and structural properties of FENE chain models such as viscosities and scattering patterns are in accordance with experimental results for shear- and elongational flows [17, 18, 142-146]. Due to the computational demands caused by the strong increase of relaxation time with molecular weight (M) only recently it has been observed, that the basic model also exhibits the experimentally observed rheological crossover, certainly related to the ability of polymers to form knots (topological constraints) between macromolecules which is further discussed in [5, 147-154]. The crossover manifests itself in a change of power law for the zero shear viscosity at a certain M. [ABSTRACT FROM AUTHOR]

Various experimental observations reveal an important aspect of the behavior of polymer solutions which is not captured by FENE dumbbell models. When the experimental data for high molecular weight systems is plotted in terms of appropriately normalized coordinates, the most noticeable feature is the exhibition of universal behavior. By this it is meant that curves for different values of a parameter, such as the molecular weight, the temperature, or even for different types of monomers can be superposed onto a single curve. For example, when the reduced intrinsic viscosity is plotted as a function of the reduced shear rate, the curves for polystyrene in different types of good solvents at various temperatures collapse onto a single curve [4]. There is, however, an important point that must be noted. While polymers dissolved in both theta solvents and good solvents show universal behavior, the universal behavior is different in the two cases. An example of this is the observed scaling behavior of various quantities with molecular weight. The scaling is universal within the context of a particular type of solvent. The term universality class is used to describe the set of systems that exhibit common universal behavior [99]. Thus theta and good solvents belong to different universality classes. [ABSTRACT FROM AUTHOR]

Dumbbell models are very crude representations of polymer molecules. Too crude to be of much interest to a polymer chemist, since it in no way accounts for the details of the molecular architecture. It certainly does not have enough internal degrees of freedom to describe the very rapid motions that contribute, for example, to the complex viscosity at high frequencies. On the other hand, the elastic dumbbell is orientable and stretchable, and these two properties are essential for the qualitative description of steady-state rheological properties and those involving slow changes with time. For dumbbell models one can go through the entire program of endeavor - from molecular model to fluid dynamics - for illustrative purposes, in order to point the way towards the task that has ultimately to be performed for more realistic models. According to [4], dumbbell models must, to some extend then, be regarded as mechanical playthings, somewhat disconnected from the real world of polymers. When used intelligently, however, they can be useful pedagocically and very helpful in developing a qualitative understading of rheological phenomens. [ABSTRACT FROM AUTHOR]

We hope that the complexity of the world is neither in contrast with the simplicity of the basic laws of physics [1] nor with the simple physical models to be reviewed or proposed in the following. However, physical phenomena occurring in complex materials cannot be encapsulated within a single numerical paradigm. In fact, they should be described within hierarchical, multi-level numerical models in which each sub-model is responsible for different spatio-temporal behavior and passes out the averaged parameters to the model, which is next in the hierarchy (Fig. 1.1). Polymeric liquids far from equilibrium belong to the class of anisotropic liquids.1 This monograph is devoted to the understanding of the anisotropic properties of polymeric and complex fluids such as viscoelastic and orientational behavior of polymeric liquids, the rheological properties of ferrofluids and liquid crystals subjected to external fields, based on the architecture of their molecular constituents. The topic is of considerable concern in basic research for which models should be as simple as possible, but not simpler. Certainly, it is also of technological relevance. Statistical physics and nonequilibrium thermodynamics are challenged by the desired structure-property relationships. Experiments such as static and dynamic light and neutron scattering, particle tracking, flow birefringence etc. together with rheological measurements have been essential to adjust or test basic theoretical concepts, such as a ‘linear stressoptic rule' which connects orientation and stress, or the effect of molecular weight, solvent conditions, and external field parameters on shape, diffusion, degradation, and alignment of molecules. [ABSTRACT FROM AUTHOR]

The development of constitutive relationships which connect strain or strain rate with stress and material behavior is at the heart of a successful macroscopic modeling of complex fluids, and soft matter in general. We reviewed simple physical models which allow to find such relationships for the case of neutral bulk polymeric fluids, nematic fluids, ferrofluids, colloidal suspensions. We have shown that the simplest approximate treatments (Chap. 2) such as the Peterlin approximation turn out to be insufficiently precise. More detailed chain models which allow to capture molecular architecture, finite extensibility, bending stiffness and interchain interactions without approximation, on the other hand, are computationally expensive while remaining conceptually simple (Chaps. 3-5). In order to predict rheooptic behaviors on time and length scales relevant for applications chemical details are shown to be not essential. These models serve to make progress towards appropriate decoupling approximations for stochastic differential equations, and a reduced description using relevant (slow) variables (Chaps. 6-8). Most interestingly, they provide deep insight into the microscopic origins of viscoelastic behavior. [ABSTRACT FROM AUTHOR]