Percus–Yevick Type of Integral Equation for the Excluded Volume Problem

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Abstract
A cluster expansion is written for the partition function of a polymer chain. An exact expression relating “nodal” and “elementary” graphs is presented. An analog of the Percus–Yevick approximation is made which leads to an integro-difference equation. This equation is solved exactly using a hard-core potential for the special case of the hard-core diameter equal to the polymer segment length (the “pearl-necklace” model). Results of numerical calculations are given for other values of this diameter ranging from zero to the segment length. This leads to values of γ ranging correspondingly from 1.0 to 2.0, where 〈ρ1N2〉Mγ with 〈ρ1N2〉 the mean-square end-to-end distance and M the molecular weight. The numerical results for 〈ρ1N2〉 as a function of chain length are in good agreement with the second-order perturbation theory of Fixman for small hard-core diameters.

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