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E-International Scientific Research JournalISSN: 2094-1749 Volume: 2 Issue: 4, 2010is a scalar field depending on the position vectormixture in the presence of colloids.The Hamiltonian governing the system is given byof the representative point of theWhere T is the absolute temperature and the Boltzmann's constant. The first term in theright-hand side of the above equality accounts for the (reduced) bulk Hamiltonian of the hostmixture [54]where d is the space dimensionality andrepresents the distance from thecritical point. Here, denotes the Flory interaction parameter [54,55], which is inverselyproportional to temperature, and is its critical value. There, is thecoupling constant, where N is the common polymerization degree of chains, andisa constant (a being the monomer size). The first two terms in the right-hand side of formula(1a) are just the expansion to fourth order of the standard Flory-Huggins (FH) free energy[54,55] around the critical composition, or equivalently around . There,are the values of field on the surfaces of colloids. The quantity accounts forthe elementary surface on colloid i, c stands for the surface coupling constant [38,39,56],which measures the interaction strength between polymer mixture and surfaces of colloids, andfor the (reduced) surface chemical potentials difference of the two polymers[6]. In fact, and play the role of the local magnetization and surface magnetic field in thecontext of magnetic materials in the presence of surfaces [38,39,56].It is easy to see that the order parameter satisfies the boundary conditions318
E-International Scientific Research JournalISSN: 2094-1749 Volume: 2 Issue: 4, 2010where the notation means the normal derivative of field on the surface of colloid i.For small enough particles, the Hamiltonian governing the system reduces toThis total Hamiltonian then describes a critical binary polymer mixture in the presence of Mcolloids (boundaries). Therefore, the interactions between polymer mixture and colloidsdepend only on two surface microscopic parameters . We emphasize that the quadraticsurface term, in Eq. (2), tends to increase the temperature parameter t appearing in the bulkHamiltonian, relation (1a). This means that such a term reduces the chemical interactions in thehost system. The surface linear term that breaks the symmetry is introduced in orderto take into account the preference of colloidal particles for one polymer of the mixture. Thecritical adsorption emerges in the limit , at fixed [ 38,39].Furthermore, it was shown [27] that the scale-invariance leads to some relationsbetween the surface microscopic parameters and the amplitudes of composition profilesand bulk two-point correlation functions of the free mixture, constructed with the fieldcomposite operator. More precisely, it was found [27] thator theand represent the amplitudes of the mean-values of fields and relative to onesphere of radius R that is immersed in a polymer blend [27], that is319
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