Wedge covariance for two-dimensional filling and wetting

Abstract

A comprehensive theory of interfacial fluctuation effects occurring at 2D wedge (corner) filling transitions in pure (thermal disorder) and impure (random bonddisorder) systems is presented. Scaling theory and the explicit results of transfer matrix and replica trick studies of interfacial Hamiltonian models reveal that, for almost all examples of intermolecular forces, the critical behaviour at filling is fluctuationdominated, characterised by universal critical exponents and scaling functions that depend only on the wandering exponent ζ. Within this filling fluctuation (FFL) regime, the critical behaviour of the mid-point interfacial height, probability distribution function, local compressibility and wedge free energy are identical to corresponding quantities predicted for the strong-fluctuation (SFL) regime for critical wetting transitions at planar walls. In particular the wedge free energy is related to the SFL regime point tension which is calculated for systems with random-bond disorder using the replica trick. The connection with the SFL regime for all these quantities can be expressed precisely in terms of special wedge covariance relations which complement standard scaling theory and restrict the allowed values of the critical exponents for both FFL filling and SFL critical wetting. The predictions for the values of the exponents in the SFL regime recover earlier results based on random-walk arguments. The covariance of the wedge free energy leads to a new, general relation for the SFL regime point tension which derives the conjectured Indekeu-Robledo critical exponent relation and also explains the origin of the logarithmic singularity for pure systems known from exact Ising studies due to Abraham and co-workers. Wedge covariance is also used to predict the numerical values of critical exponents and position dependence of universal one-point functions for pure systems.

Interface unbinding in structured wedges

“…This transition is the analog of the critical filling transition in pure wedge geometry [5,14]. We have verified that for n = 2, 3, .…”

Interfacial structure at a two-dimensional wedge filling transition: Exact results and a renormalization group study

“…Fluid adsorption in wedge and cone-shaped nonplanar geometries has attracted much attention in the last few years [1][2][3][4][5]. Geometry plays an important role in the surface phase diagram, and new phase transitions as the filling transition arise.…”

Critical wetting in power-law wedge geometries

“…Note that by increasing the exponent γ the wall morphology can be changed smoothly from a planar substrate (γ = 0) to a linear wedge (γ = 1) and eventually to a parallel plate geometry (γ = ∞). The adsorption properties and interfacial fluctuation effects in each of these geometries, corresponding to wetting [20,21], filling [9,10,11] and capillary condensation [21], respectively, are very different to each other and have received considerable theoretical and experimental interest. The central question we ask here is, how do the wetting properties depend on the wall exponent γ?…”

“…Hereafter, we restrict our attention to the regime γ ≤ 1 and focus on the fate of the planar wetting transition as γ is increased from 0 to 1. The critical behaviour occurring at the limit γ = 1, corresponding to the filling of a linear wedge, is known in some detail [9,10,11]. Writing the wall-function z(x, y) = tan α |x|, with α the tilt angle, observe that the ratio r = sec α > 1.…”

“…Accordingly, the phase boundary for the wedge wetting (filling) is shifted and again thermodynamic arguments dictate that it occurs at a lower filling temperature T f , satisfying θ(T f ) = α rather than θ(T ) = 0 [1]. For planar substrates that undergo critical wetting transitions, the linear wedge filling transition is also continuous, but is characterised by critical exponents distinct to those at critical wetting [10,11]. Consider, for example, the divergence of the wedge mid-point interfacial height l w ∼ t ′−βw , mid-point roughness ξ ⊥ ∼ t ′−ν ⊥ and correlation length ξ y ∼ t ′−νy , where the latter is measured along the wedge and t ′ ∝ (T f − T )/T f .…”