Introduction Most colloidal systems are made of grains having a tendency to ¯occulate owing to the existence of certain long-range van der Waals attractions. The introduction of soluble polar head polymers leads to a grafting phenomenon such that two adjacent clothed grains repel each other until the stabilization of the colloids is achieved owing to the excluded-volume forces between monomers. Grafted polymers on solid walls or ¯uid interfaces are of considerable interest and ®nd many physicochemical applications [1]. We can quote, as typical examples, adhesion [2], wetting [3], chromatog- raphy [4] and colloidal stabilization [5]. A classic overview on steric stabilization can be found in the textbook of Napper [6] and in the review article of Halperin et al. [7]. From a theoretical point of view, grafted chains or simply polymer brushes on a plane surface were studied for the ®rst time by Alexander [8] and by de Gennes [9], using essentially scaling argu- ments. In the same way, Milner and coworkers [10] elaborated a self-consistent-®eld SCF) method for studying the end eects in polymer brushes. Some years ago, Witten and Pincus WP) [11] were interested in the determination of the repulsive potential between spherical particles clothed by long end-grafted ¯exible chains. According to these authors, the potential varies with the distance, r, between a pair of particles as Colloid Polym Sci 279:763±770 2001) Ó Springer-Verlag 2001 ORIGINAL CONTRIBUTION M. Badia M. Benhamou A. Derouiche J. L. Bretonnet Determination of the amplitude of the repulsive-pair potential between particles clothed by end-grafted polymers Received: 4 October 1999 Accepted: 2 January 2001 M. Badia á M. Benhamou á A. Derouiche Laboratoire de Physique des PolymeÁres et PheÂnomeÁnes Critiques, Faculte des Sciences Ben M'sik, B.P. 7955 Casablanca, Morocco J. L. Bretonnet &) Laboratoire de Physique de la MatieÁre Condense e, Universite de Metz, 57000 Metz, France Abstract We examine the problem of the determination of the repulsive potential between spherical particles clothed by long end-grafted ¯exible polymers. This potential varying with the distance according to a logarithmic law has a potential am- plitude 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 ®rst regarded as star polymers with small enough diameter and the same number of arms. Then, the ampli- tude potential is identi®ed to the critical exponent related to the con- tact probability between cores of these stars, which allows us to ®nd 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 renormal- ization theory to third order in e  4 ) d, where d is the critical dimension of the system. To deter- mine the best three-dimensional ex- pression for the potential amplitude, A L , use is made of the Pade±Borel transformation, which provides a closer form valid for small, inter- mediate and high values of L. This form of potential amplitude, consis- tent with the exact scaling asymp- totic value of Witten and Pincus [1986) Macromolecules 19:2509], allows us to ®nd the associated prefactor. The procedure is also extended to interacting stars of dierent numbers of arms.