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

TL;DR

This paper investigates universal critical adsorption profiles around a sphere and a cylinder in a fluid at criticality using local functional theory, deriving exact and numerical solutions for the order parameter decay.

Contribution

It provides exact and numerical solutions for the critical adsorption profiles around spheres and cylinders, extending understanding of surface critical phenomena.

Findings

Order parameter decays as r^{-(1+η)} around a sphere.

Order parameter decays as r^{-(1+η)/2} around a cylinder.

Strong adsorption occurs except for very small surface fields.

Abstract

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)/2} $ for a cylinder in three dimensions, where $\eta$ is the critical exponent in the order parameter correlation at bulk criticality.