Three-dimensional wedge filling in ordered and disordered systems

Abstract

We investigate interfacial structural and fluctuation effects occurring at continuous fillingtransitions in 3D wedge geometries. We show that fluctuation-induced wedgecovariance relations that have been reported recently for 2D filling and wetting havemean-field or classical analogues that apply to higher-dimensional systems. Classicalwedge covariance emerges from analysis of filling in shallow wedges based on asimple interfacial Hamiltonian model and is supported by detailed numericalinvestigations of filling within a more microscopic Landau-like density functionaltheory. Evidence is presented that classical wedge covariance is also obeyed forfilling in more acute wedges in the asymptotic critical regime. For sufficientlyshort-ranged forces mean-field predictions for the filling critical exponents andcovariance are destroyed by pseudo-one-dimensional interfacial fluctuations. Weargue that in this filling fluctuation regime the critical exponents describing thedivergence of length scales are related to values of the interfacial wandering exponentζ(d) defined for planar interfaces in (bulk) two-dimensional(d = 2) andthree-dimensional (d = 3) systems. For the interfacial height , with θ the contactangle and α the wedgetilt angle, we find βw = ζ(2)/2(1−ζ(3)). For pure systems (thermal disorder) we recover the known resultβw = 1/4 predicted by interfacial Hamiltonian studies whilst for random-bond disorder we predictthe universal critical exponent even in the presence of dispersion forces. We revisit the transfer matrix theory ofthree-dimensional filling based on an effective interfacial Hamiltonian model and discussthe interplay between breather, tilt and torsional interfacial fluctuations. We show that thecoupling of the modes allows the problem to be mapped onto a quantum mechanicalproblem as conjectured by previous authors. The form of the interfacial height probabilitydistribution function predicted by the transfer matrix approach is shown to be consistentwith scaling and thermodynamic requirements for distances close to and far from the wedgebottom respectively.