Casimir forces between spherical particles in a critical fluid and conformal invariance

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 ${\mathit{R}}_{\mathrm{-}}$ and ${\mathit{R}}_{+}$. In this geometry the singular part of the free energy δscrF only depends upon the ratio ${\mathit{R}}_{\mathrm{-}}$/${\mathit{R}}_{+}$, and the stress tensor, which we use to calculate δscrF, 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 ε=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.