Predicting the distribution of a chemical species across
multiple
phases is of critical importance to environmental protection, pharmaceuticals,
and high added-value chemicals. Computationally, this problem is addressed
by determining the free energy of solvation of the species in the
different phases using a well-established thermodynamic formulation.
Following recent developments in sterically stabilized colloids and
nanocomposite materials, the solvation of polymer-grafted nanoparticles
in different solvents or polymer melts has become relevant. We develop
a Self-Consistent Field theoretical framework to determine the solvation
free energy of grafted particles inside a molten polymer matrix phase
at low concentrations. The solvation free energy is calculated based
on the notion of a pseudochemical potential introduced by Ben-Naim.
Grafted and matrix chains are taken to be of the same chemical constitution,
but their lengths are varied systematically, as are the particle radius
and the areal density of grafted chains. In addition, different affinities
between the nanoparticle core and the polymer (contact angles) are
considered. At very low or very high amounts of grafted material,
solvation depends on the adhesion tension between the bare particle
and the matrix or on the surface tension of the grafted polymer, respectively.
The dependence of the solvation free energy on molecular characteristics
is more complicated at intermediate grafting densities and high curvatures,
where the contribution of the entropy of grafted chains becomes significant.
In general, solvation is less favored in cases where the matrix chains
are much shorter than the grafted ones. The former tend to penetrate
and swell the brush, thus generating conformational and translational
entropy penalties. This effect becomes more pronounced when considering
large particles since the grafted chains have less available space
and extend more. For extremely low amounts of grafted material, we
observe the opposite trend, albeit weak. Based on our calculations,
we propose a generic model for estimating the solvation free energies
of grafted nanoparticles in polymer melts from their molecular characteristics.
The model and associated SCF formulation, illustrated here for chemically
identical grafted and matrix chains, can be extended to obtain partition
coefficients of grafted nanoparticles between different polymer melts.