Polymer chains in four dimensions

Abstract

We investigate analytically the asymptotic logarithmic behavior in four dimensions ( d = 4) of very long continuous polymer chains with excluded volume, described by the two-parameter Edwards model. Using renormalization theory, we calculate directly all the connected partition functions L N of N polymer chains in this model, and obtain their logarithmic expression, including the subleading order. To first order, they are found in the universal form L N ∼ S 2( N−1) ( ln S) h N with h N = 1 4 (2− N) , and where S is the Brownian area of the chains, given in terms of their Brownian radius ° R, by ° R 2 = dS. The partition functions L 1, L 2, L 3, L 4, and the averageend-to-end square distance R 2, and square radius of gyration R G 2 are calculated up to logarithmic sub-subleading order. The virial expansion in 4d of the osmotic pressure is obtained. For semi-dilute solutions, the logarithmic correction to Edwards' tree approximation for the osmotic pressure is calculated, and we find the new result Πβ ∝ C 2/√ln C, where C is the monomer concentration. We give in 4d a set of geometrical quantities, characterizing a single chain: the form factor, the average 2 nth power R [2 n] of the end-to-end distance, the average 2 nth radius of gyration R G [2 n] , and their universal ratio. We also consider in 4d the probability distribution for the internal distances inside a chain.