Citation
Curro, John Gillette (1969) Theoretical Investigation of the Effect of Intramolecular Interactions on the Configuration of Polymeric Chains. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8KEPR512. https://resolver.caltech.edu/CaltechETD:etd10072002145049
Abstract
A theoretical investigation of the effect of intramolecular interactions on the configurational statistics of a polymer molecule is presented. This problem has been studied by many authors and is known as the "excluded volume problem" in the literature. A statistical mechanical approach is used. Many of the similarities between the theory of "classical fluids" and the excluded volume problem are exploited.
The configurational statistics of 2 and 3 segment chains are computed exactly for the "hard sphere potential". The integrations were performed by introducing bipolar and tripolar coordinate systems. It was found that the mean square endtoend distance for these cases was n^{1.33} where n is the number of segments. These results are of no practical use in predicting the properties of real polymer chains which are much longer. It is instructive, however, to compare these exact results with approximate theories in the limit of short chain length.
A "cluster expansion" is written for the partition function of a polymer chain with the ends of the chain fixed. This is analogous to the cluster expansion for the partition function of an imperfect gas. The firstorder term in this expansion is evaluated for the hard core potential. In the limit of small hard core diameters, the firstorder term leads to the wellknown firstorder perturbation theory for the mean square endtoend distance. The exact results of this firstorder correction term are used to construct higherorder terms of a specified "isolated topology". If only these terms are used in the cluster expansion, incorrect results are obtained for the mean square endtoend distance. This indicates that higherorder terms of complicated topology are significant for longer chain length.
Various approximate integral equations for the restricted partition function of a polymer chain are presented. The most promising of these equations is the analog of the wellknown PercusYevick equation in the theory of liquids. In deriving this equation two topologically distinct types of graphs are defined. These are the "nodal and elementary" graphs. An exact equation relating these types of graphs is presented. The analog of the PercusYevick approximation is made which leads to an integrodifference equation. This equation is solved exactly using the hard core potential for the special case of the hard core diameter equal to the polymer segment length. Results of numerical calculations are given for other intermediate values of this diameter ranging from zero to the segment length (the "pearl necklace" model). This leads to values of γ ranging correspondingly from 1.0 to 2.0 where <r^{2}_{1N} ∝ M^{γ}> with <r^{2}_{1N} the mean square endtoend distance and M the molecular weight. The numerical results for <r^{2}_{1N} as a function of chain length are in good agreement with the secondorder perturbation theory of Fixman for small hard core diameters.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  (Materials Science and Chemical Engineering) ; Polymeric Chains 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Materials Science 
Minor Option:  Chemical Engineering 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  27 April 1969 
Record Number:  CaltechETD:etd10072002145049 
Persistent URL:  https://resolver.caltech.edu/CaltechETD:etd10072002145049 
DOI:  10.7907/8KEPR512 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  3958 
Collection:  CaltechTHESIS 
Deposited By:  Imported from ETDdb 
Deposited On:  07 Oct 2002 
Last Modified:  26 Apr 2024 20:49 
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