Statistical Physics of Semidilute Polymer Systems (And); Soft Billiard Systems
Baldwin, Philip Rupert
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https://hdl.handle.net/2142/77411
Description
Title
Statistical Physics of Semidilute Polymer Systems (And); Soft Billiard Systems
Author(s)
Baldwin, Philip Rupert
Issue Date
1987
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Physics, Fluid and Plasma
Language
eng
Abstract
This thesis consists of two disjoint parts. In the first part, we carefully study the semidilute regime of polymer solutions. The renormalization group (RG) method using the $\varepsilon$-expansion has been a systematic method giving good semiquantitative agreement with both real and computer experiments for the static and dynamical properties of polymers in dilute solution. However, polymer theory in the semidilute regime is less sound. We first review RG results on statics, using the so-called direct method and employing the Edwards Hamiltonian. We present a careful RG account of arbitrarily polydisperse semidilute polymer solution, and give a clear account of semidilute diagrammatics. We provide results for the osmotic pressure and scattering function for homopolymer and ternary polymer solutions. We discuss effective medium theory as well as the limitations of our modelling. Next we build a dynamical model for polymers. We present a RG-exposition of polymer theory under the Kirkwood approximation, and clearly demonstrate that no screening results in this model for the calculation of the cooperative diffusion constant and the initial decay rate of the dynamic scattering function. Our results for these quantities show good agreement with experiment for moderately dense systems. Finally, we discuss the mode-coupling method, in order to study tracer diffusion constant, viscosity and the dynamic screening length.
The second part of the thesis deals with a problem related to the foundations of statistical physics. We study a variation of a Sinai-type billiard system, where circular scatterers of radius $R$ are centered on the lattice sites of a 2-d square lattice and given a finite potential height $U$ (measured in units of kinetic energy). For 1 $>$ $U >$ 0, the Kolmogorov-Sinai entropy of a point particle travelling on the lattice is numerically calculated as a function of $U$ and $R$ and empirically seen to be a universal function of $U/R\sp2$ for $U,R$ sufficiently small. For 0 $> U > U\sb{c}(R)$ we prove the non-ergodicity of the system and discuss the structure of the phase space; $U\sb{c}(R)$ is also explicitly given. The $U \to 0$ and $U \to U\sb{c}$ transitions are also discussed. Using tools of number theory, algorithmic aspects of the billiard problem are investigated. The entropy of the billiard system for small $R$ is shown to be related to how difficult it is to find the next scatterer in a billiard simulation.
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