Stochastic Processes in Polymeric Fluids: Tools and Examples for Developing Simulation Algorithms

ISBN-13: 9783540583530 / Angielski / Miękka / 1995 / 362 str.

Hans Christian Vttinger; Hans C. A-Ttinger; Hans C. Ottinger

Opis

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."

Spis treści

1 Stochastic Processes, Polymer Dynamics, and Fluid Mechanics.- 1.1 Approach to Kinetic Theory Models.- 1.2 Flow Calculation and Material Functions.- 1.2.1 Shear Flows.- 1.2.2 General Extensional Flows.- 1.2.3 The CONNFFESSIT Idea.- References.- I Stochastic Processes.- 2 Basic Concepts from Stochastics.- 2.1 Events and Probabilities.- 2.1.1 Events and ?-Algebras.- 2.1.2 Probability Axioms.- 2.1.3 Gaussian Probability Measures.- 2.2 Random Variables.- 2.2.1 Definitions and Examples.- 2.2.2 Expectations and Moments.- 2.2.3 Joint Distributions and Independence.- 2.2.4 Conditional Expectations and Probabilities.- 2.2.5 Gaussian Random Variables.- 2.2.6 Convergence of Random Variables.- 2.3 Basic Theory of Stochastic Processes.- 2.3.1 Definitions and Distributions.- 2.3.2 Gaussian Processes.- 2.3.3 Markov Processes.- 2.3.4 Martingales.- References.- 3 Stochastic Calculus.- 3.1 Motivation.- 3.1.1 Naive Approach to Stochastic Differential Equations.- 3.1.2 Criticism of the Naive Approach.- 3.2 Stochastic Integration.- 3.2.1 Definition of the Itô Integral.- 3.2.2 Properties of the Itô Integral.- 3.2.3 Itô’s Formula.- 3.3 Stochastic Differential Equations.- 3.3.1 Definitions and Basic Theorems.- 3.3.2 Linear Stochastic Differential Equations.- 3.3.3 Fokker-Planck Equations.- 3.3.4 Mean Field Interactions.- 3.3.5 Boundary Conditions.- 3.3.6 Stratonovich’s Stochastic Calculus.- 3.4 Numerical Integration Schemes.- 3.4.1 Euler’s Method.- 3.4.2 Mil’shtein’s Method.- 3.4.3 Weak Approximation Schemes.- 3.4.4 More Sophisticated Methods.- References.- II Polymer Dynamics.- 4 Bead-Spring Models for Dilute Solutions.- 4.1 Rouse Model.- 4.1.1 Analytical Solution for the Equations of Motion.- 4.1.2 Stress Tensor.- 4.1.3 Material Functions in Shear and Extensional Flows.- 4.1.4 A Primer in Brownian Dynamics Simulations.- 4.1.5 Variance Reduced Simulations.- 4.2 Hydrodynamic Interaction.- 4.2.1 Description of Hydrodynamic Interaction.- 4.2.2 Zimm Model.- 4.2.3 Long Chain Limit and Universal Behavior.- 4.2.4 Gaussian Approximation.- 4.2.5 Simulation of Dumbbells.- 4.3 Nonlinear Forces.- 4.3.1 Excluded Volume.- 4.3.2 Finite Polymer Extensibility.- References.- 5 Models with Constraints.- 5.1 General Bead-Rod-Spring Models.- 5.1.1 Philosophy of Constraints.- 5.1.2 Formulation of Stochastic Differential Equations.- 5.1.3 Generalized Coordinates Versus Constraint Conditions.- 5.1.4 Numerical Integration Schemes.- 5.1.5 Stress Tensor.- 5.2 Rigid Rod Models.- 5.2.1 Dilute Solutions of Rod-like Molecules.- 5.2.2 Liquid Crystal Polymers.- References.- 6 Reptation Models for Concentrated Solutions and Melts.- 6.1 Doi-Edwards and Curtiss-Bird Models.- 6.1.1 Polymer Dynamics.- 6.1.2 Stress Tensor.- 6.1.3 Simulations in Steady Shear Flow.- 6.1.4 Efficiency of Simulations.- 6.2 Reptating-Rope Model.- 6.2.1 Basic Model Equations.- 6.2.2 Results for Steady Shear Flow.- 6.3 Modified Reptation Models.- 6.3.1 A Model Related to “Double Reptation”.- 6.3.2 Doi-Edwards Model Without Independent Alignment.- References.- Landmark Papers and Books.- Solutions to Exercises.- References.- Author Index.

Blurb

This book consists of two strongly interweaved parts: the mathematical theory of stochastic processes and its applications to molecular theories of polymeric fluids. The comprehensive mathematical background provided in the first part should be equally useful in many other branches of engineering and the natural sciences. As a benefit from the second part one gains a more direct understanding of polymer dynamics, one can more easily identify exactly solvable models, and one can develop efficient computer simulation algorithms in a straightforward manner. In view of the many examples and exercises, on the one hand, and the numerous applications to problems from the front line of science, on the other hand, this volume may be used equally well as a basic textbook or as an up-to-date reference book.