Addressing Nanocomposite Systems via 3D-SCFT: Assessment of Smearing Approximation and Irregular Grafting Distributions

We develop a three-dimensional self-consistent field-theoretic approach (3D-SCFT) for polymer matrix nano composites of arbitrary geometries, e.g., polymer-grafted nano particles (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).