The excludedhyphen;volume expansion in polymer chains: Evaluation of the Flory exponent in the Gaussian approximation

摘要

The selfhyphen;consistent excludedhyphen;volume chainhyphen;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 meanhyphen;square distance between atoms separated bykbonds (kGt;1) within an infinitely large chain is lang;r2(k)rang;prop;k6/5lcub;prod;infin;n=1Lnlsqb;ln(k/kmacr;)rsqb;rcub;2/5, whereLn(x) =1+anthinsp;lnthinsp;Lnminus;1(x),L1=1+a1x;kmacr; is a shorthyphen;range cutoff and theairsquo;s are suitable constants, generally comprised between 0 and 1, while limnrarr;infin;thinsp;an=1. Accordingly, the divergence associated with the shorthyphen;range cutoff is explicitly evaluated instead of being removed through renormalization techniques. For (k/kmacr;)rarr;infin; the above function is independent ofkmacr; and we have lnlang;r2(k)rang;prop;6/5thinsp;lnthinsp;k, thus reobtaining Floryrsquo;s asymptotic exponent. Numerical results obtained from the integral equation in the nonasymptotic regime give support to the above analytic solution.