Excluded-Volume Effects in Linear Polymer Chains: A Hierarchy of Differential Equations

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Abstract
A hierarchy of differential equations for the expansion factor of a linear polymer chain is derived in a purely formal way by successive differentiation of the general equation of Fixman with respect to the binary-cluster integral for a pair of chain segments. The hierarchy is truncated by a physically reasonable assumption to obtain an approximate solution. The assumption introduced is discussed in detail. In order to obtain numerical results, it is necessary to develop at least the third-order perturbation theory, and the third expansion coefficient is evaluated to be 6.459 for the mean-square end-to-end distance. Assuming that the third coefficients are the same for the end-to-end distance and the radius of gyration, there is obtained the equation for the expansion factor αS for the latter, αS2=0.541+0.459(1+6.04z)0.46, where z is the well-known excluded-volume parameter. Although this equation has the asymptotic form αS4.35=constant z at large z, a definite conclusion concerning the value of the exponent of αS cannot be drawn from the present formulation. From the above equation and the detailed analysis, however, it may be concluded that the correct values of αS must be between those predicted by the Flory theory and the Ptitsyn theory.

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