We consider a system consisting of very small colloidal particles clothed each by f end-grafted flexible polymer chains we regarded as star polymers, and hard spherical colloidal particles in a good solvent. Our main objective is to determine the expression of the interaction force between a spherical colloid and a star polymer as a function of distance between them. We limit ourselves to the case where the star polymer is smaller than the colloid. In the first part, the system is dissolved in a melt of short linear chains of polymerization degree P<N, where N denotes the polymerization degree of grafted chains. To compute the expected force, we consider two regimes: (1) high-grafting density [Formula: see text] and (2) small-grafting density (f < f*). For (f > f*), we show that the expression of the expected force coincides exactly with that of the case of a small molecular weight solvent. For (f < f*), we show that there is a change in behavior. In the second part, we assume that the lengths of the f grafted chains were randomly distributed and there is talk of a polydisperse star polymer. We show that the computation of the expected force depends on the relative values of the polymerization degree of longest grafted chain, N, when it is compared to the typical one Nc ~ f1/(α-1). Here α is the polydispersity exponent. We distinguish two regimes depending on whether N < Nc or N > Nc. For the regime with N < Nc, and comparing the expression of the force obtained for the monodisperse case, we can say that the polydispersity of grafted chains induce a drastic change of the force expression. For the regime with N > Nc, we found the existence of two distance-ranges. For small distances, the effective force expression is identical to that relative to the above case (N < Nc). But for high distances, the effective force expression is similar to the monodisperse case.

We consider a dilute solution of very small spherical particles clothed each by f end-grafted flexible polymer chains, we regard as star-polymers, and immersed in a Θ-solvent. The presence of a Θ-solvent induces an effective force between adjacent star-polymers. We are interested in the investigation of the polydispersity effects on this interaction force. This work is a natural extension of that where we have considered the same system but in the presence of a good solvent. Consequently we use the same theoretical approach to determine the structure of our system, and particularly the extension of polydisperse star-polymer in a Θ-solvent. We assume that the lengths of the f grafted chains are randomly distributed, with a distribution [Formula: see text]. The latter is the number of chains having more than n monomers and which generally is related to the probability to have a grafted chain of n monomers. To make explicit calculation of the expected force, FΘ(h), as a function of the interparticle-distance h we choose the particular form for the distribution [Formula: see text] used in the good solvent case. This distribution is a power law in the n-variable that is, [Formula: see text], if n < N, and [Formula: see text], if n > N. Here, 1 ≤ α < 2 is the polydispersity exponent and N is the polymerization degree of the longest grafted chain. To obtain the force expression, we use some method developed by Witten and Pincus for monodisperse star-polymers in a good solvent. We show that the computation of the force depends on the polymerization degree N compared with the typical number Nc ~ f1/(α-1).