J. Phys. II France
Volume 4, Numéro 3, March 1994
Page(s) 533 - 539
DOI: 10.1051/jp2:1994143
J. Phys. II France 4 (1994) 533-539

"Fluctuating cluster" equations for a polymer to $O(\epsilon^2)$. (Toward exact Flory equation?)

Z. Alexandrowicz

Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel and Departamento de Fisica, CCE-UFRN, Campus Universitario, 59072-970 Natal-RN, Brasil.

(Received 8 November 1993, accepted 17 November 1993)

Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact $\epsilon$ expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly as a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by b phantom bifurcations. Analogous description of the polymer leads to a modified Flory-like equation, which describes r and b, jointly, and to two more equations for r and b in separate. These give very accurate exponents $\nu$ and $\gamma$, correct to $O(\epsilon^2)$. In addition, the approach can be extended to a many polymer system, and to other critical clusters.

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