Critical adsorption profiles around a sphere and a cylinder in a fluid at criticality: Local functional theory

We study universal critical adsorption on a solid sphere and a solid cylinder in a fluid at bulk criticality, where preferential adsorption occurs. We use a local functional theory proposed by Fisher, de Gennes, and Au-Yang ($[$C. R. Acad. Sci. Paris Ser. B {\bf 287}, 207 (1978)$]$ and $[$Physica {\bf 101}A, 255 (1980)$]$). We calculate the mean order parameter profile $\psi (r)$, where $r$ is the distance from the sphere center and the cylinder axis, respectively. The resultant differential equation for $\psi (r)$ is solved exactly around a sphere and numerically around a cylinder. A strong adsorption regime is realized except for very small surface field $h_1$, where the surface order parameter $\psi (a)$ is determined by $h_1$ and is independent of the radius $a$. If $r$ considerably exceeds $a$, $\psi (r)$ decays as $r^{-(1+\eta )}$ for a sphere and $r^{-(1+\eta )\mathrm{/}2}$ for a cylinder in three dimensions, where $\eta$ is the critical exponent in the order parameter correlation at bulk criticality.