Spectral analysis of the collective diffusion of Brownian particles confined to a spherical surface

A more fundamental understanding of non-equilibrium phenomena involving fluids confined to restricted geometries presents indeed a difficult challenge. Here we explore a novel approach to describe the collective diffusion of Brownian particles confined to a spherical surface by adapting the dynamic density functional theory (DDFT) to this geometry. The ensuing diffusion equation is then transformed into a system of coupled ordinary differential equations by implementing spherical harmonics expansions of the relevant functions. This study is complemented with Brownian dynamics (BD) simulations performed with an innovative extension of the Ermak–McCammon algorithm, while employing conditional ensemble averages over initial non-equilibrium states. In both cases the relaxations processes are analyzed through the decay modes obtained from the spectral method. The simple DDFT approach considered here provides a fairly good description of the BD results. In particular, the theoretical predictions for the initial progression rates of the local density modes turn out to be almost exact, and we found that this can be explained in terms of the eigenvalues and eigenvectors corresponding to an initial renormalized potential. As an illustration, the model system has been tailored to the experimental conditions of Pickering emulsions stabilized by functionalized gold nanoparticles.