Theory of Excluded Volume in Coiled Polymers: Cumulant Expansion

Abstract

The excluded-volume effect in a randomly ooiled polymer is investigated by the method of cumulants. The spatial expansion factor a 2 of a polymer chain is expanded in a power series:where z is the usual excluded-volume parameter, and a general formula for Cn is given. By means of a "ladder" approximation C. is calculated to be 1.3438. Contributions of small and giant clusters in a polymer chain to a 2 are discussed. The above power series is predicted in the ladder approximation to converge for O~z~N, where N is the total number of segments constituting the polymer chain. We are unable, however, to dispose of the possibility that in general C .. for 5~n«N includes a term proportional to Nl/2 logN.

Statistical mechanical theory of semidilute polymer solutions

“…Figure 12. Diagrams contributing to Di(q',q") and D2(q',q")• range of q' and q" is taken into account, one may obtain approximately ( 2n \ 2n( 1 1 \2 a7-+a7'…”

“…So, let us define a factor S*(q) which is given by simply subtracting eq III. 12 from eq III.38; we have S*(q) _1_ (-p)ta0(q)¡2 2 1 + > 0( ) 2n2 V^l + opa0(qO 1 + upo0(q + q')^9 ' 9 + 9Í TV RT hen the left-hand side of eq IV.l is rewritten as pf L?2*(r) -1] dr = pS*{q = 0) (IV. 7) where the single-molecule term is also subtracted in the expression of a correlation function g*2(r).…”

“…(3) If we denote by grads the differentiation with respect to the configurational coordinates of solute molecules, we have -grad, U(N) = (II. 12) This means that U(N) is the average potential, i.e., the potential of mean force among N molecules in a solution of given solvent activity. (4) ZN* is given by C(T)N r r ( U(N)\ z»*"H^/'''^,pv^vrrJV| (IL13) where C(T) comes from the kinetic energy part of ZN* and d{N] is the volume element in configuration space.…”

The Gaussian Chain with Volume Exclusion

“…Here, direct use is made of the cluster function defined by eq VI. 1. The results for large N are independent of whether N is odd or even; since expressions for odd N are somewhat simpler than for even IV, only odd N will be treated.…”