Critical effects and scaling at meniscus osculation transitions

We propose a simple scaling theory describing critical effects at rounded meniscus osculation transitions which occur when the Laplace radius of a condensed macroscopic drop of liquid coincides with the local radius of curvature $R_w$ in a confining parabolic geometry. We argue that the exponent $\beta_{\rm osc}$ characterising the scale of the interfacial height $\ell_0 \propto R_w^{\beta_{\rm osc}}$ at osculation, for large $R_w$, falls into two regimes representing fluctuation-dominated and mean-field like behaviour, respectively. These two regimes are separated by an upper critical dimension, which is determined here explicitly and which depends on the range of the intermolecular forces. In the fluctuation-dominated regime, representing the universality class of systems with short-ranged forces, the exponent is related to the value of the interfacial wandering exponent $\zeta$ by $\beta_{\rm osc}=3\zeta/(4-\zeta)$. In contrast, in the mean-field regime, which has not been previously identified, and which occurs for systems with longer ranged forces (and higher dimensions), the exponent $\beta_{\rm osc}$ takes the same value as the exponent $\beta_s^{\rm co}$ for complete wetting which is determined directly by the intermolecular forces. The prediction $\beta_{\rm osc}=3/7$ in $d=2$ for systems with short-ranged forces (corresponding to $\zeta=1/2$) is confirmed using an interfacial Hamiltonian model which determines the exact scaling form for the decay of the interfacial height probability distribution function. A numerical study in $d=3$, based on a microscopic model Density Functional Theory, determines that $\beta_{\rm osc} \approx \beta_s^{\rm co}\approx 0.326$ close to the predicted value $1/3$ appropriate to the mean-field regime for dispersion forces.