Casimir Force between Nanoparticles Immersed in a Crosslinked Polymer Blend

We consider a crosslinked polymer blend made of two polymers of different chemical nature. We suppose that such a system incorporates small colloidal particles, which prefer to be attracted by one polymer, close to the spinodal temperature. This is the so-called critical adsorption. As assumption, the particle diameter, $d_0$, is considered to be small enough in comparison with the size of microdomains (mesh size) ξ* ~ $an^{1//2}$, with a - the monomer size and n - the number of monomers between consecutive crosslinks. The critical fluctuations of the crosslinked polymer mixture induce a pair-potential between particles located in the non-preferred phase. The purpose is the determination of the Casimir pair-potential, $U_2 (r)$, as a function of the interparticle distance r. To achieve calculations, use is made of an extended de Gennes field theory that takes into account the colloid-polymer interactions. Within the framework of this theory, we first show that the pair-particle is attractive. Second, we find for this potential the exact form: $U_2(r)//k_{B}T = - A_{H}(d_0//r)^2 exp(-r//ξ*) - B_{H}(d_0//r)^4 exp(-2r//ξ*)$, with the known universal amplitudes $A_{H}$ > 0 and $B_{H}$ > 0 (the Hamaker constants). This expression clearly shows that the pair-potential differs from its homologue with no crosslinks only by the two exponential factors exp(-r/ξ*) and exp(-2r/ξ*). The main conclusion is that the presence of reticulations reduces substantially the Casimir effect in crosslinked polymer blends.