Computational Polymer Physics

Lecturer(s)

Prof. Dr. Martin Kröger
PD Dr. Patrick Ilg

Hour(s) of lecture(s), exercise(s) and credit points

2L
2E
4CP

Teaching goals

The course offers an introduction to computer simulation methods and their foundations for the physics and material behavior of simple and complex materials and in particular polymeric and anisotropic liquids. The lecture is devised for students which have attended the course 402-0809-00L Introduction to Computational Physics. The goal is to i) introduce theoretical approaches used in soft matter physics, ii) motivate coarse-grained models, and ii) the numerical solution of many body problems for optical, mechanical, or electromagnetic applications.

Summary and outline

The lecture focuses on particle methods. Techniques such as Monte Carlo, equilibrium, beyond-equilibrium and nonequilibrium molecular dynamics, smoothed particle dynamics, dissipative particle dynamics, Brownian dynamics, embedded atoms, lattice Boltzmann will be introduced and applied. Master equations, Markov processes, Fokker-Planck equations, stochastic differential equations play a major role in the fundamental chapters. Substances: from simple towards structured fluids (gases, polymers, ferrofluids, liquid crystals, metals). The lectures and exercises will be structured as follows:

Contents

1. Monte Carlo integration, error estimates, scaling
   Monte Carlo in statistical physics, Quasi Monte Carlo
   Distributed random numbers, methods
   Accretion
   The Ballistic Deposition Model
   Diffusion limited aggregation (DLA) model
   Fractal dimensions, polymers
2. Random walk, theory, applications
   Monte Carlo simulation for the random walk
   Freely rotating polymers
   Wormlike polymers
   Self-avoiding walk (SAW)
   Slithering Snake, Pivot, Enriched samples algorithms
3. Master equation, introduction, applications
   Stationary solution of the Master equation
   Detailed balance
   Coupled equations for moments
   Metropolis Monte Carlo (MC)
   Thermostatted 1D harmonic oscillator via Metropolis MC
   Lennard-Jones system via Metropolis MC
   Gaussian integrals
4. Ising model and Phase transitions
   Metropolis MC algorithm for the 2D Ising model
   Finite size effects
   Phase transitions & percolation
   Percolation theory in the Ising Model
   Q-Potts model and foams
   Random site percolation model
5. Cellular Automata (CA), introduction, classification
   Moore models
   Traffic modeling
   Shock waves
   The Q2R Ising Model
6. Lattice gases, method, simulation and applications
   Lattice-Boltzmann, method, simulation and applications
7. Multiscale dynamics
   Molecular dynamics, implementation, applications
   Mesoscopic interaction potentials
   Finite difference methods
   Einstein frequency and configurational temperature
   Periodic boundary conditions
   Temperature control
   Lees-Edwards boundary conditions
8. Beyond-equilibrium molecular dynamics
   Long-range forces
   White and colored noise
   Random vector with given mean and covariance
   Whitening a random vector
   Brownian dynamics, theory and simulation
   Langevin equations
9. Rouse model for polymer solutions
   Reptation models for polymer melts
   Pompom-type models for polymer melts
   Flow channel, flow birefringence
   Dendronized polymers, simulations, experiments, applications
10. Kramers process
    Brownian dynamics of a FENE dumbbell
    Smoothed particle dynamics, soft fluid particles
    Dissipative particle dynamics
11. Ferrofluids, ferromagnetic chains, theory, simulation, applications
    Magnetorheological fluids, simulation methods
    Liquid crystals, theoretical approaches, simulation, experiments
    Liquid crystalline polymers, theoretical approaches, simulations, experiments
    Frank-Ericksen elasticity, theories, experiments
12. Wormlike micelles, living polymers, theory and simulation
    Plasticity, metals, embedded atoms simulations
    Phoretic forces
13. Coarse-graining procedures for polymeric systems, shortest paths
    Delaunay triangulation and Voronoi diagram
    Hyperbolically regularized hydrodynamics

A script, exercises, sample codes and other supplementary material will be available online.

Literature

• M.P. Allen, D.J. Tildesley, Computer simulation of liquids (Oxford, 1987).
• M. Rubinstein, R.H. Colby, Polymer physics (Oxford, 2003).
• H. Risken, The Fokker-Planck equation (Springer, Berlin, 1989).
• S. Hess, M. Kröger, P. Fischer, Einfache und disperse Flüssigkeiten, in: Bergmann-Schaefer, Band 5 (de Gruyter, Berlin, 2005).
• J. Honerkamp, Stochastische dynamische Systeme (VCH, Weinheim, 1990)
• H.C. Öttinger, Stochastic processes in polymeric fluids (Springer, Berlin, 1996).
• C.W. Gardiner, Handbook of stochastic methods (Springer, Berlin, 1985).
• M. Karttunen, I. Vattulainen, A. Lukkarinen (Eds.), Novel methods in soft matter simulations (Springer, Berlin, 2004).
• R.J. Gaylord, P.R. Wellin, Computer Simulations with Mathematica (Springer, Berlin, 1995).
• M. Kröger, Models for polymeric and anisotropic liquids (Springer, Berlin, 2005)

Further ..

The course is held simultaneously with two corresponding lectures building on the above-mentioned ‘Introduction to Computational Physics’: Computational Statistical Physics, 402-0812-00L (2L, 2E, 8 CP) by Prof. H.J. Herrmann (D-BAUG), and Computational Quantum Physics, 402-0810-00L (2L, 2E, 8 CP) by Prof. M. Troyer/Dr. P. De Forcrand.(D-PHYS).

This course is part of the area of specialization Materials Modeling and Simulation of the master degree program in Materials Science, and part of the batchelor and master degree programs in Computer Science/Theoretical Physics.