#### Faculty Advisor

George D.J.Phillies

#### 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.

#### Publisher

Worcester Polytechnic Institute

#### Degree Name

MS

#### Department

Physics

#### Project Type

Thesis

#### Date Accepted

2002-05-01

#### Copyright Statement

All authors have granted to WPI a nonexclusive royalty-free license to distribute copies of the work. Copyright is held by the author or authors, with all rights reserved, unless otherwise noted. If you have any questions, please contact wpi-etd@wpi.edu.

#### Accessibility

Unrestricted

#### Repository Citation

Merriam, Susan Carol, "Direct Demonstration of Self-Similarity in a Hydrodynamic Treatment of Polymer Self-Diffusion" (2002). *Masters Theses (All Theses, All Years)*. 608.

https://digitalcommons.wpi.edu/etd-theses/608

#### Subjects

self-similarity, polymer self-diffusion, hydrodynamic, Polymers, Hydrodynamics