Force between unlike star-polymers versus the solvent quality

Abstract

We re-examine here the computation of the effective force between two star-polymers A and B of different chemical nature, which are immersed in a common solvent. This force originates from the excluded-volume interactions and chemical segregation. We assume that the solvent quality may be different for the two unlike star-polymers, that is the solvent can be 1) a good solvent for A and B, 2) a good solvent for A and a Theta-solvent for B, or 3) a Theta-solvent for the two polymers. The purpose is a quantitative study of the effect of the solvent quality on the effective force, which is a function of the center-to-center distance. Calculations are achieved using the renormalization theory applied to the Edwards continuous model. We first show that, when the mutual interactions are present, the effective force decays as the inverse of distance, but with a universal amplitude depending on the solvent quality. Second, we demonstrate the existence of three kinds of forces related to situations 1), 2) and 3) described above, and give the third-order epsilon-expansions (epsilon = 4 - d, 4 is the critical dimension) of the corresponding amplitudes. These series can be resummed using the Borel-Leroy techniques to obtain the best three-dimensional values for the expected force amplitudes. Finally, this work must be regarded as a natural extension of a published one which dealt with the same problem, but where the solvent was assumed to be good for the two unlike star-polymers.

Soft matter with soft particles

“…6. The logarithmic law for short distances as well as the value of the numerical prefactor, 5/18f 3/2 were rederived by using field-theoretical techniques by von Ferber et al 91 Additional corroboration for the correctness of the pair potential of eqn (39) was offered by Benhamou et al, 92,93 who calculated theoretically the effective interaction potential using scaling and field-theoretical arguments and extending thereby the calculation to stars composed of different (incompatible) polymer chains. Another extension of the pair interaction, this time to two stars of different functionalities, f 1 and f 2 has been presented in ref.…”

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.…”