Force between unlike star-polymers versus the solvent quality

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.