The excluded-volume expansion in polymer chains: Evaluation of the Flory exponent in the Gaussian approximation

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Abstract
The self-consistent excluded-volume chain-expansion problem is reexamined, in the Gaussian approximation. The integral equation resulting from minimization of the free energy is solved analytically on the basis of a conjecture that is shown to be logically consistent. The mean-square distance between atoms separated by k bonds (k≫1) within an infinitely large chain is 〈r2(k)〉∝k6/5{∏∞n=1Ln [ln(k/k̄)]}2/5, where Ln(x) =1+an ln Ln−1(x), L1=1+a1x; k̄ is a short-range cutoff and the ai’s are suitable constants, generally comprised between 0 and 1, while limn→∞ an =1. Accordingly, the divergence associated with the short-range cutoff is explicitly evaluated instead of being removed through renormalization techniques. For (k/k̄)→∞ the above function is independent of k̄ and we have ln〈r2(k)〉∝6/5 ln k, thus reobtaining Flory’s asymptotic exponent. Numerical results obtained from the integral equation in the nonasymptotic regime give support to the above analytic solution.