We develop a three-dimensional self-consistent field-theoretic
approach (3D-SCFT) for polymer matrix nanocomposites of arbitrary
geometries, e.g., polymer-grafted nanoparticles (PGNPs), polymer brushes,
and particle solids. The spatio-temporal discretization of Edwards’s
diffusion equation is realized with the finite element method (FEM).
Unidimensional implementations for PGNPs (1D-SCFT) invoke the smearing
approximation (SA), which treats the grafting points as being delocalized
across a spherical shell. By conducting detailed comparisons between
1D-SCFT and 3D-SCFT, we assess the accuracy of the SA in terms of
reproducing key structural and thermodynamic properties of dilute
grafted silica/polystyrene NPs in molten polystyrene. The SA yields
accurate radially averaged structural features such as the mean brush
thickness and its scaling with grafting density, chain length, and
particle size. The free energy is reproduced accurately as well, albeit
noticeable deviations are observed when transitioning toward the mushroom
regime. In the SA, the stretching free energy is a function of the
radial distance of the free end of a grafted chain from the particle
surface. In 3D-SCFT, the grafting points are fixed in space, and thus
chain stretching is described more accurately. 3D-SCFT offers direct
access to the spatial distribution of the segment density of a chain
and affords detailed visualization of the mushroom-to-dense brush
transition, at the levels of both the whole system and individual
grafted chains. By taking advantage of the single-chain representation
in 3D-SCFT, we explore NPs with arbitrary grafting distributions (e.g.,
rings, tadpoles, and dual-poles) and the corresponding variation of
the free energy and structural properties. To the best of the authors’
knowledge, this is the first time that the grafting distribution is
examined as an additional degree of freedom in a 3D field-theoretic
framework. Our work constitutes a step toward the computational design
of nanocomposites with tailor-made self-assembly properties, achieved
by controlling the interactions of nanoparticles through modulation
of the distribution of points of attachment of grafted chains on their
surface (e.g., Janus particles).