Title page for ETD etd-0501102-142449

Document Type thesis

Author Name Merriam, Susan Carol

URN etd-0501102-142449

Title Direct Demonstration of Self-Similarity in a Hydrodynamic Treatment of Polymer Self-Diffusion

Degree MS

Department Physics

Advisors
George D.J.Phillies, Advisor

Keywords
self-similarity
polymer self-diffusion
hydrodynamic

Date of Presentation/Defense 2002-04-29

Availability unrestricted

Abstract
The self-diffusion coefficient of a polymer in solution may be expanded in the concentration of the polymer, as seen in equation 1. The linear term would represent a perturbation due to the presence of another polymer; the c^{2} term would represent a perturbation due to interactions
of trios of polymers.

Phillies determined the c^{2} term of a virial expansion of the self-diffusion coefficient for trios of polymers interacting via a ring. Here I determine a correction to the c^{2} term due to trios of polymers interacting via a figure-eight scattering diagram: the equivalent of four polymers interacting in a ring where the second polymer and the fourth polymer are the same.

D_{s}(c) = D_{0}(1+ alpha D_{0} c + beta D_{0}^{2}c^{2}+...) 1
or,
D_{s}(c) = D_{0}(1+ alpha D_{s}(c)c). 2

A D_{0} may be replaced by D_{s}(c) in equation 1 to arrive at equation 2. The left-hand-side of equation 2
is the final self-diffusion coefficient, and the D_{s}(c) on the right-hand-side of this equation is that due to the question of self-similarity. If the D_{s}(c) on the right-hand-side is given by equation 1, resulting in
beta=alpha^{2}, it may be said that the system exhibits self-similarity. I demonstrate self-similarity quantitatively for a polymer solution using a generalized Kirkwood-Riseman model of polymer dynamics.

The major physical assumption of the model I utilize to derive equation 2 is that, in solution, polymer motions are dominantly governed by hydrodynamic interactions between the chains.

First, I review the Kirkwood-Riseman model for intrachain hydrodynamic interactions. I then discuss Phillies' extension of this model to interchain interactions for duos or trios of polymers in a ring. I analytically calculate the hydrodynamic interaction tensor from a multiple
scattering picture T_{54321}, for five polymers in solution and verify this tensor by numerical differentiation.

Finally, I perform the ensemble average of the self-interaction tensor b_{1232} appropriate to the figure-eight scattering diagram both analytically and with a Monte Carlo
routine, thereby verifying equation 2 to second order in concentration.

Files
merriam.pdf

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Document Type	thesis
Author Name	Merriam, Susan Carol
URN	etd-0501102-142449
Title	Direct Demonstration of Self-Similarity in a Hydrodynamic Treatment of Polymer Self-Diffusion
Degree	MS
Department	Physics
Advisors	George D.J.Phillies, Advisor
Keywords	self-similarity polymer self-diffusion hydrodynamic
Date of Presentation/Defense	2002-04-29
Availability	unrestricted
Abstract The self-diffusion coefficient of a polymer in solution may be expanded in the concentration of the polymer, as seen in equation 1. The linear term would represent a perturbation due to the presence of another polymer; the c^{2} term would represent a perturbation due to interactions of trios of polymers. Phillies determined the c^{2} term of a virial expansion of the self-diffusion coefficient for trios of polymers interacting via a ring. Here I determine a correction to the c^{2} term due to trios of polymers interacting via a figure-eight scattering diagram: the equivalent of four polymers interacting in a ring where the second polymer and the fourth polymer are the same. D_{s}(c) = D_{0}(1+ alpha D_{0} c + beta D_{0}^{2}c^{2}+...) 1 or, D_{s}(c) = D_{0}(1+ alpha D_{s}(c)c). 2 A D_{0} may be replaced by D_{s}(c) in equation 1 to arrive at equation 2. The left-hand-side of equation 2 is the final self-diffusion coefficient, and the D_{s}(c) on the right-hand-side of this equation is that due to the question of self-similarity. If the D_{s}(c) on the right-hand-side is given by equation 1, resulting in beta=alpha^{2}, it may be said that the system exhibits self-similarity. I demonstrate self-similarity quantitatively for a polymer solution using a generalized Kirkwood-Riseman model of polymer dynamics. The major physical assumption of the model I utilize to derive equation 2 is that, in solution, polymer motions are dominantly governed by hydrodynamic interactions between the chains. First, I review the Kirkwood-Riseman model for intrachain hydrodynamic interactions. I then discuss Phillies' extension of this model to interchain interactions for duos or trios of polymers in a ring. I analytically calculate the hydrodynamic interaction tensor from a multiple scattering picture T_{54321}, for five polymers in solution and verify this tensor by numerical differentiation. Finally, I perform the ensemble average of the self-interaction tensor b_{1232} appropriate to the figure-eight scattering diagram both analytically and with a Monte Carlo routine, thereby verifying equation 2 to second order in concentration.
Files	merriam.pdf