Determination of the amplitude of the repulsive-pair potential between particles clothed by end-grafted polymers

We examine the problem of the determination of the repulsive potential between spherical particles clothed by long end-grafted flexible polymers. This potential varying with the distance according to a logarithmic law has a potential amplitude that depends on the number, L, of grafting chains per particle. The purpose of this work is to compute such a potential amplitude. The clothed particles are first regarded as star polymers with small enough diameter and the same number of arms. Then, the amplitude potential is identified to the critical exponent related to the contact probability between cores of these stars, which allows us to find a universal function for the expected potential amplitude depending on L and d-space dimension only. In two-dimensional space, conformal invariance is used to extract the potential amplitude as a function of L. For dimensions greater than 2, the potential amplitude is obtained within the framework of renormalization theory to third order in epsilon = 4 - d, where d is the critical dimension of the system. To determine the best three-dimensional expression for the potential amplitude, A(L), use is made of the Pade-Borel transformation, which provides a closer form valid for small, intermediate and high values of L. This form of potential amplitude, consistent with the exact scaling asymptotic value of Witten and Pincus [(1986) Macromolecules 19:2509], allows us to find the associated prefactor. The procedure is also extended to interacting stars of different numbers of arms.