Critical adsorption and Casimir torque in wedges and at ridges

Abstract

Geometrical structures of confining surfaces profoundly influence the adsorption of fluids upon approaching a critical point ${T}_{c}$ in their bulk phase diagram, i.e., for $t=(T\ensuremath{-}{T}_{c})∕{T}_{c}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}0$. Guided by general scaling considerations, we calculate, within the mean-field theory, the temperature dependence of the order parameter profile in a wedge with opening angle $\ensuremath{\gamma}<\ensuremath{\pi}$ and close to a ridge $(\ensuremath{\gamma}>\ensuremath{\pi})$ for $T\ensuremath{\gtrless}{T}_{c}$ and in the presence of surface fields. For a suitably defined reduced excess adsorption ${\overline{\ensuremath{\Gamma}}}_{\ifmmode\pm\else\textpm\fi{}}(\ensuremath{\gamma},t\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}0)\ensuremath{\sim}{\overline{\ensuremath{\Gamma}}}_{\ifmmode\pm\else\textpm\fi{}}(\ensuremath{\gamma}){\ensuremath{\mid}t\ensuremath{\mid}}^{\ensuremath{\beta}\ensuremath{-}2\ensuremath{\nu}}$ we compute the universal amplitudes ${\overline{\ensuremath{\Gamma}}}_{\ifmmode\pm\else\textpm\fi{}}(\ensuremath{\gamma})$, which diverge as ${\overline{\ensuremath{\Gamma}}}_{\ifmmode\pm\else\textpm\fi{}}(\ensuremath{\gamma}\ensuremath{\rightarrow}0)\ensuremath{\sim}1∕\ensuremath{\gamma}$ for small opening angles, vary linearly close to $\ensuremath{\gamma}=\ensuremath{\pi}$ for $\ensuremath{\gamma}<\ensuremath{\pi}$, and increase exponentially for $\ensuremath{\gamma}\ensuremath{\rightarrow}2\ensuremath{\pi}$. There is evidence that, within the mean-field theory, the ratio ${\overline{\ensuremath{\Gamma}}}_{+}(\ensuremath{\gamma})∕{\overline{\ensuremath{\Gamma}}}_{\ensuremath{-}}(\ensuremath{\gamma})$ is independent of $\ensuremath{\gamma}$. We also discuss the critical Casimir torque acting on the sides of the wedge as a function of the opening angle and temperature.