Casimir Forces between Spherical Particles in a Critical Fluid and Conformal Invariance

TL;DR

This paper investigates Casimir forces between spherical particles in a critical fluid using conformal invariance, providing explicit results and analyzing boundary conditions, with implications for colloidal systems.

Contribution

It applies conformal symmetry to compute Casimir interactions between spheres in critical fluids, including density profiles and boundary effects, extending previous methods to new geometries.

Findings

Explicit Casimir force calculations for spheres at criticality.

Profiles of thermodynamic densities near surfaces and small spheres.

Analysis of boundary condition effects on Casimir interactions.

Abstract

Mesoscopic particles immersed in a critical fluid experience long-range Casimir forces due to critical fluctuations. Using field theoretical methods, we investigate the Casimir interaction between two spherical particles and between a single particle and a planar boundary of the fluid. We exploit the conformal symmetry at the critical point to map both cases onto a highly symmetric geometry where the fluid is bounded by two concentric spheres with radii R_- and R_+. In this geometry the singular part of the free energy F only depends upon the ratio R_-/R_+, and the stress tensor, which we use to calculate F, has a particularly simple form. Different boundary conditions (surface universality classes) are considered, which either break or preserve the order-parameter symmetry. We also consider profiles of thermodynamic densities in the presence of two spheres. Explicit results are presented for an ordinary critical point to leading order in epsilon=4-d and, in the case of preserved symmetry, for the Gaussian model in arbitrary spatial dimension d. Fundamental short-distance properties, such as profile behavior near a surface or the behavior if a sphere has a `small' radius, are discussed and verified. The relevance for colloidal solutions is pointed out.