Interaction force between incompatible star-polymers in dilute solution

We consider a low-density assembly of spherical colloids, such that each is clothed by L end-grafted chemically incompatible polymer chains either of types A or B. These are assumed to be dissolved in a good common solvent. We assume that colloids are of small size to be considered as star-polymers. Two adjacent star-polymers A and B interact through a force F originating from both excluded-volume affects: and chemical mismatch between unlike monomers. Using a method developed by Witten and Pincus (macromolecules 19. 2509 (1986)) in the context of star-polymers of the same chemical nature, we determine exactly the force F as a function of the center-to-center distance h. Vl;e find that this force is the sum of two Continuous Fe and F,. The former, that results from the excluded volume, decay as F-e similar to A(L)h(-1), with the L-dependent universal amplitude A(L) similar to L-3/2. While the second, which comes from the chemical mismatch, decays more slowly as F-e similar to chiB(L)h(-1-tau), where tau is a critical exponent whose value is found to tau congruent to 0.40; and chi is the standard Flory interaction parameter. We find that the corresponding L-dependent universal amplitude is B-L similar to L(3+r)/2. Theses forces are comparable near the cores of two adjacent star-polymers, i.e. for h similar to h(c) similar to a chi (1/tau)rootL (a is the monomer size). Finally, for two star-polymers of the same chemical nature (A or B), the force F that simply results from excluded-volume effects coincides exactly with F-e, and then the known result is recovered.