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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\left(r\right) $ , decreases with the center-to-center distance r as $F\left(r\right) /k_{\rm B}T\simeq A_{f_1f_2} \cdot \left[ r\ln \left(R^2/r^2\right) \right]^{-1}$ $ \left(r < R\right) $ , with the exact universal amplitude $ A_{f_1f_2} = f_1f_2\left(f_1 + f_2-2\right) /22$ . Second, in dimension d = 2, we find that the force decays more slowly as $F\left(r\right) /k_{\rm B}T\simeq B_{f_1f_2} \cdot r^{-1}$ $ \left(r < R\right) $ , with the exact universal amplitude $B_{f_1f_2} = \left(2 + 4f_1f_2\right) /21$ . For high distances compared to the gyration radius, $R\thicksim a\sqrt{N}$ , of a single polymer chain at the $\Theta$ -point, an exponential decay of the force is expected. |
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