Exact effective force between star-polymers in a $\Theta$ -solvent

Abstract

We re-examine here the computation of the effective force between two star-polymers of respective numbers of branches f(1) and f(2), immersed in a common Theta-solvent. Such a force originates essentially from the repulsive three-body interactions. To achieve this, we take advantage of some established results using renormalization theory for three-dimensional star-polymers, or conformal invariance for two-dimensional ones. We first show that, in dimension d = 3, the force, F(r), decreases with the center-to-center distance r as F(r)/kappa BT congruent with Af1f2 x [r ln (R2/r2]-1 (r<R), with the exact universal amplitude Af1f2 = f1f2(f1 + f2 - 2)/22. Second, in dimension d = 2, we find that the force decays more slowly as F(r)/kappa BT congruent with Bf1f2 x r-1 (r<R), with the exact universal amplitude Bf1f2. For high distances compared to the gyration radius, R approximately a square root of N, of a single polymer chain at the Theta-point, an exponential decay of the force is expected.

Star Polymers in Solvents of Varying Quality

“…23,27 More importantly, the interactions and correlations between star polymers in solvents of varying quality have hardly received any attention, at least not by means of a microscopic approach. Benhamou et al 28,29 have employed field-theoretical methods to determine the effective center-to-center force F(r) between two stars exactly at the Θ point. They found a functional form…”

“…First, it is desirable to have a quantitative description of ρ( r ) for the whole r range, including the diffuse polymer layer for r ≫ R , in a fashion similar to that of athermal stars. , More importantly, the interactions and correlations between star polymers in solvents of varying quality have hardly received any attention, at least not by means of a microscopic approach. Benhamou et al , have employed field-theoretical methods to determine the effective center-to-center force F ( r ) between two stars exactly at the Θ point. They found a functional form F ( r ) = k B T A f 1 f 2 [ r ln( R 2 / r 2 )] −1 for r ≪ R , where k B is the Boltzmann constant and A f 1 f 2 = f 1 f 2 ( f 1 + f 2 − 2)/22 is a prefactor that depends on the functionalities of the two interacting stars, which are generally different from each other.…”