Exact effective force between star-polymers in a Theta-solvent.

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.