Effective force between confined colloids clothed by end-grafted polymer chains in solution

We consider colloidal particles clothed each by f end-grafted polymer chains, and immersed in a good solvent. The clothed particles are small enough to be regarded as star polymers. The purpose is a quantitative study of the influence of the confinement on the expression of the effective force between a pair of particles. Such a force originates from the swelling of grafted chains due to the presence of solvent. To simplify, we chose as confining geometry a thin space of thickness D delimited by two parallel plates. The problem depends on two natural lengths, which are the gyration radius, R3∼af1/5N3/5, of an isolated star polymer and the thickness D. Thus, there exist two regimes depending on whether R3 is smaller or greater than D. The system is confined only when D is below R3. We first present a detailed study of conformations of a confined star polymer (R3≫D). The confinement imposes to the star polymer to adopt a configuration parallel to the plates, which is characterized by a parallel radius, denoted by R∥. We show that the latter increases when D is decreased according to R∥∼af1/4D−1/4N3/4. The main result is that the star polymer becomes two dimensional, and where the grafted chains have blobs of size D as new units. The second purpose is a quantitative investigation of the effective force between two confined star polymers. For the unconfined regime (R3≪D), it is found that the force, F3(r), decays with interparticle distance r as F3(r)/kBT≃θ3(f )/r, with the amplitude θ3(f )≃(5/18)f3/2. Hence, for this regime, the force is similar to that relative to a three-dimensional colloidal solution of infinite extent. For the confined regime (R3≫D), however, a drastic change of the force expression occurs. For this regime, we show that the force is similar to the one governing the interaction between two-dimensional star polymers, and is given by F2(r)/kBT≃θ2(f )/r, with the exact amplitude θ2(f )=(2+9f2)/24. The two forces then decrease with distance according to the same decay, but with different amplitudes. As a conclusion, the effective force between two confined star polymers is θ2(f )/θ3(f )≃(3/20)(2+9f2)/f3/2 times more intense than that between unconfined ones. Finally, we say that the confinement may be a mechanism that reinforces the colloidal stabilization.