Analytic Solution of Emden-Fowler Equation and Critical Adsorption in Spherical Geometry

TL;DR

This paper derives an analytic solution to the Emden-Fowler equation relevant for critical adsorption phenomena in spherical geometries, providing insights into surface universality classes and reproducing known effects in limiting cases.

Contribution

It presents the first analytic solution for the order-parameter profile at the critical point in spherical geometry, advancing theoretical understanding of critical adsorption.

Findings

Analytic solutions for the Emden-Fowler equation in spherical shells.

Reproduction of the Fisher-de Gennes effect in the parallel-plate limit.

Foundation for perturbative calculations in critical phenomena.

Abstract

In the framework of mean-field theory the equation for the order-parameter profile in a spherically-symmetric geometry at the bulk critical point reduces to an Emden-Fowler problem. We obtain analytic solutions for the surface universality class of extraordinary transitions in $d=4$ for a spherical shell, which may serve as a starting point for a pertubative calculation. It is demonstrated that the solution correctly reproduces the Fisher-de Gennes effect in the limit of the parallel-plate geometry.