Some Topics Concerning the Radius of Gyration of Linear Polymer Molecules in Solution

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Abstract
Fixman's integral for the distribution function P(S 2 ) obeyed by S 2 of the unperturbed polymer chain was evaluated analytically, and its asymptotic forms valid for t ≪ 1 and t ≫ 1 were derived. Here S is the radius of gyration of the chain and t is the ratio of S 2 to its statistical average 〈S 2 〉 . The values of P(S 2 ) calculated from the analytic solution agreed with Koyama's computer values over the entire range of t . The leading term of the asymptotic form for t ≪ 1 confirmed the previous result of Fixman. The asymptotic form for t ≫ 1 agreed not with Fixman's but with that of Forsman. The Flory–Fisk theory for the excluded volume effect in polymer chains was reformulated with the exact form being substituted for P(S 2 ) in their basic equation for α s 2 . Here α s is the linear expansion factor of a polymer molecule defined in terms of 〈S 2 〉 . The calculated curve for α s 3 as a function of the interaction parameter z differed markedly from that derived by Flory and Fisk who had employed an approximate expression for P(S 2 ) . Our curve followed quite closely the one recently deduced by Yamakawa and Tanaka from an entirely different treatment. However, both are different in the limiting behavior at indefinitely large z , our curve tending to the fifth power type as represented by α s 5 = 1.668z , whereas Yamakawa–Tanaka's approaches α s 4.35 = 1.11z .

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