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A SPECTER is haunting the scientific world-the specter of com­ puters. All the powers of traditional science have entered into a holy alliance to exorcise this specter: puristic theoreticians and tradition­ alistic experimentalists, editors and referees of prestigious journals, philosophers of science and mathematicians. Where is a pioneering computer simulation that has not been decried as unreliable by its opponents in power? The Computer Manifesto As a result of the enormous progress in computer technology made during the last few decades, computer simulations have become a very powerful and widely applicable tool in science and engineering. The main purpose of this . book is a comprehensive description of the background and possibilities for the application of computer simulation techniques in polymer fluid dynamics. Mod­ eling and understanding the flow behavior of polymeric liquids on the kinetic theory level is not merely a great intellectual challenge but rather a matter of immense practical importance, for example, in connection with plastics manu­ facture, processing of foods, and movement of biological fluids. The classical computer simulation technique for static problems in statis­ tical mechanics is the Monte Carlo method developed in the early 1950s. The name of this method underlines how unusual and strange the idea of using ran­ dom numbers in the exact sciences is at first glance. However, the Monte Carlo method is a rigorous and efficient means for evaluating moments and static spa­ tial correlation functions for given probability distributions.

Inhaltsverzeichnis

Frontmatter

Stochastic Processes, Polymer Dynamics, and Fluid Mechanics

1.. Stochastic Processes, Polymer Dynamics, and Fluid Mechanics

Abstract
One can attempt to achieve a theoretical understanding of polymer fluid dynamics on two different levels: continuum mechanics and kinetic theory. Continuum mechanics deals with the formulation and solution of a system of macroscopic equations for the density, velocity, temperature, and possibly other fields describing the fluid structure, which are related to conservation laws for mass, momentum, energy, and maybe other quantities associated with the additional fields. In order to obtain a closed system of macroscopic equations one needs to supplement the fundamental conservation laws by certain empirically or microscopically founded equations of state for the fluxes of the conserved quantities. Such equations of state, which are characteristic for a given material, are often referred to as constitutive equations. The formulation of suitable constitutive equations and of general admissibility criteria for such constitutive equations is a central part of continuum mechanics. In polymer kinetic theory, one attempts to understand the polymer dynamics, the constitutive equation for the momentum flux or stress tensor, and eventually polymer fluid dynamics by starting from coarse-grained molecular models. Excellent reviews of the state of the art in both continuum mechanics and kinetic theory are given in the two volumes of the comprehensive introductory textbook Dynamics of Polymeric Liquids by R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager [1, 2]. The more recent literature in continuum mechanics and in kinetic theory has been reviewed in [3] and [4], respectively.
Hans Christian Öttinger

Stochastic Processes

Frontmatter

2.. Basic Concepts from Stochastics

Abstract
Stochastics is concerned with the mathematical analysis of intuitive notions like “chance,” “randomness,” “fluctuations,” or “noise.” In order to develop a sound mathematical theory of stochastics it is necessary to give rigorous definitions for the following basic concepts, which have an intuitive meaning based on everyday experience, especially from games of chance: event, probability, random variable, expectation, stochastic independence, conditioning, convergence, and stochastic process. The formal definitions of these objects in this chapter are made clear by many examples. The examples and exercises of this chapter play a most important role in this book because they not only illustrate the meaning of abstract concepts, but also contain important information required in the subsequent chapters.
Hans Christian Öttinger

3.. Stochastic Calculus

Abstract
The theory of stochastic processes provides the framework for describing stochastic systems evolving in time. Our next goal is to characterize the dynamics of such stochastic systems, that is, to formulate equations of motion for stochastic processes.
Hans Christian Öttinger

Polymer Dynamics

Frontmatter

4.. Bead-Spring Models for Dilute Solutions

Abstract
The idea of establishing a relationship between diffusion equations and stochastic differential equations has a long tradition in the kinetic theory of dilute polymer solutions. In 1969, R. Zwanzig [2] described the rigorous equivalence between the two approaches for bead-spring chains with hydrodynamic interaction, and he pointed out the fundamental importance of this equivalence for computer simulations. In particular, Zwanzig rearranged the diffusion tensor in the Fokker-Planck equation so that he obtained the Stratonovich-version of the stochastic differential equation. In 1978, M. Fixman [3] showed in two landmark papers how simulations of polymer dynamics can be constructed and applied (Brownian dynamics simulations).Fixman discussed the relationship between Fokker-Planck equations and stochastic differential equations for bead-spring models with hydrodynamic interaction, even in the presence of constraints. Moreover, he presented numerical integration schemes for stochastic differential equations, in particular, the explicit Euler scheme and the corresponding implicit and predictor-corrector treatments of the diffusion term (which requires an intuitive understanding of the difference between the Itô- and Stratonovich approaches). Fixman also pointed out that the usual integration schemes may not be useful for stochastic differential equations. In the same year, D. L. Ermak and J. A. McCammon carried out a computer simulation for short chains in order to study the effect of hydrodynamic interaction on dynamic equilibrium properties. Their paper [4], which contains a very clear presentation of their explicit Euler scheme for integrating the Itô-version of the stochastic equations of motion (in the mathematical literature, the validity of this most basic integration scheme has been known since 1955) and an explicit procedure for carrying out the Cholesky decomposition of the diffusion tensor, has most frequently been cited in papers on Brownian dynamics. We here show how the rigorous theory of stochastic differential equations can be exploited to obtain more efficient simulation algorithms, even in more general cases where a formulation of the stochastic equations of motion by direct physical arguments is not so straightforward.
Hans Christian Öttinger

5.. Models with Constraints

Abstract
Mechanical models of polymers involving constraints have a long tradition in polymer kinetic theory (see Chap. 11 of [1]). The Kirkwood-Riseman chain proposed in 1948 consists of N beads and N — 1 rigid rods of length L with a fixed angle θ between successive links; links can rotate freely, that is, no rotatory potentials are included. This model is also called the freely rotating bead-rod chain. The Kramers chain proposed in 1944 is a freely jointed bead-rod chain with N beads and N — 1 rigid rods of length L.
Hans Christian Öttinger

6.. Reptation Models for Concentrated Solutions and Melts

Abstract
Concentrated polymer solutions and polymer melts are extremely complex many-particle systems. For that reason, it is unquestionably important to make both ingenious and far-reaching assumptions in describing the dynamics of polymers in such undiluted systems, and to make use of computer simulations. A widely and successfully applied class of molecular models for the polymer dynamics in concentrated solutions and melts relies on the notion of reptational motion [1]. The first reptation theory for the rheology of undiluted polymers was developed in a series of papers by Doi and Edwards published in 1978 and 1979 [2-5]. The Doi-Edwards model is based on the assumption that each polymer in a highly entangled system moves (“reptates”) in a tube formed by other polymers. Several further assumptions need to be made, and several different representations of the reptating polymer are used in order to derive the final diffusion equation describing the dynamics of the polymers. For the stress tensor required in deriving rheological properties, the Doi-Edwards model employs a formula from rubber elasticity.
Hans Christian Öttinger

Backmatter

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