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The relationship between the dynamics and structure of amorphous thin films and nanocomposites near their glass transition is an important problem in soft-matter physics. Here, we develop a simple theoretical approach to describe the density profile and the a-relaxation time of a glycerol-silica nanocomposite (S. Cheng et al., J. Chem. Phys., 2015, 143, 194704). We begin by applying the Derjaguin approximation, where we replace the curved surface of the particle with the planar one; thus, modeling the nanocomposite is reduced to that of a confined thin film. Subsequently, by employing the molecular dynamics (MD) simulation data of Cheng et al., we approximate the density profile of a supported liquid thin film as a stationary solution of a fourth-order partial differential equation (PDE). We then construct an appropriate density functional, from which the density profile emerges through the minimization of free energy. Our final assumption is that of a consistent, temperature-independent scaled density profile, ensuring that the free volume throughout the entire nanocomposite increases with temperature in a smooth, monotonic fashion. Considering the established relationship between glycerol relaxation time and temperature, we can employ Doolittle-type analysis (‘‘naı ¨ ve’’ free-volume model), to calculate the relaxation time based on temperature and film thickness. We then convert the film thickness into the interparticle distance and subsequently the filler volume fraction for the nanocomposites and compare our model predictions with experimental data, resulting in a good agreement. The proposed approach can be easily extended to other nanocomposite and film systems. 

We present a rigorous analysis of the transient evolution of nearly circular bilayer interfaces evolving under the thin interface limit, ε ≪ 1, of the mass preserving L2-gradient flow of the strong scaling of the functionalized Cahn–Hilliard equation. For a domain Ω ⊂ R2 we construct a bilayer manifold with boundary comprised of quasi-equilibria of the flow and a projection onto the manifold that associates functions u in an H2 tubular neighborhood of the manifold with an interface Γ embedded in Ω. The linearization of the flow about the manifold does not present a clear spectral separation of modes normal and tangential to the manifold. The dimension of the parameterization of the interfaces and the bilayer manifold controls both the normal coercivity of the manifold and the coupling between normal and tangential modes, both of which increase with this dimension. The key step in the analysis is the identification of a range of dimensions in which coercivity dominates the coupling, permitting the closure of the nonlinear estimates that establish the asymptotic stability of the manifold. Orbits originating in a thin, forward invariant, tubular neighborhood ultimately converge to an equilibrium associated to a circular interface. Projections of these orbits yield interfacial evolution equivalent at leading order to the regularized curve-lengthening motion characterized by normal motion against mean curvature, regularized by a higher order Willmore expression. The curve lengthening is driven by absorption of excess mass from the regions of Ω away from the interface, leading to high dimensional dynamics that are ill-posed in the ε → 0+ limit. 

Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter \begin{document}$$ \varepsilon\ll1 $$\end{document}, we show that this induces undulated bilayer solutions whose width perturbations decay on an \begin{document}$$ O\!\left( \varepsilon^{-1/2}\right) $$\end{document} inner length scale that is long in comparison to the \begin{document}$ O(1) $$\end{document}$ scale that characterizes the bilayer width.