Tricritical Wedge Filling Transitions with Short-ranged Forces

We show that the 3D wedge filling transition in the presence of shortranged interactions can be first-order or second order depending on the strength of the line tension associated with to the wedge bottom. This fact implies the existence of a tricritical point characterized by a short-distance expansion which differs from the usual continuous filling transition. Our analysis is based on an effective one-dimensional model for the 3D wedge filling which arises from the identification of the breather modes as the only relevant interfacial fluctuations. From such analysis we find a correspondence between continuous 3D filling at bulk coexistence and 2D wetting transitions with random-bond disorder. PACS numbers: 68.08.Bc, 05.70.Np, 68.35.Ct, 68.35.Rh Tricritical wedge filling transitions with short-ranged forces 2 α x y l0 ξ lW (y) ξ y Figure 1. Schematic illustration of a typical interfacial configuration and relevant lengthscales for a fluid adsorption in a 3D wedge. Fluid adsorption in micropatterned and sculpted geometries has become the subject of intense study over the last decade. Highly impressive technological advances which allow the tailoring of micro-patterned and structured solid surfaces on the nanometer to micrometer scale [1] are a landmark in the development of the emerging microfluidic industry [2] which aims at minituarizing chemical synthesis plants or biological analysis equipment in much the same way the silicon chip brought about the electronics revolution. However, the theoretical understanding of this phenomenon is far from being complete. Recent studies of filling transitions for fluids in 3D wedges show that interfacial fluctuations are greatly enhanced compared with wetting at flat substrates [3, 4]. The control of such enhanced interfacial fluctuations is crucial for the effectiveness of the microfluidic devices. Fortunately, there are simple theoretical approaches which take into account these effects. For example effective Hamiltonian predictions for the critical exponents at continuous (critical) wedge filling with shortranged forces have been confirmed in large scale Ising model simulation studies [5]. Similar experimental verification of the predicted geometry-dominated adsorption isotherms at complete wedge filling [6] raise hopes that the filling transition itself and related fluctuation effects will be observable in the laboratory. We further develop the theory of wedge filling in this paper, focussing on the emergence of a new type of continuous filling: tricritical filling. First we briefly review the fluctuation theory of 3D wedge filling. Consider the interface between a bulk vapour at temperature T and saturation pressure with a 3D wedge characterised by a tilt angle α. Macroscopic arguments dictate that the wedge is partially filled by liquid if the contact angle θ > α and completely filled if θ < α [7]. The filling transition refers to the change from microscopic to macroscopic liquid adsorption as T → Tf , at which θ(Tf ) = α, and may be first-order or continuous (critical filling). Both of these transitions can be viewed as the unbinding of the liquidvapour interface from the wedge bottom. Characteristic length scales are the mean interfacial height above the wedge bottom lW , the roughness ξ⊥ and the longitudinal correlation length ξy, measuring fluctuations along the wedge (see Figure 1). The relevant scaling fields at critical filling are θ − α and the bulk ordering field h (which is proportional to the pressure difference with the saturation value). At coexistence (h = 0) we define critical exponents by lW ∼ (θ − α)−βW and ξy ∼ (θ − α)−νy . The roughness can be related to ξy by the scaling relationship ξ⊥ ∼ ξW y , where ζW is the wedge wandering exponent. For short-ranged forces, ζW = 1/3. For shallow wedges, i.e. α ≪ 1, the free energy of an interfacial configuration can be modelled by an effective Hamiltonian based on the capillary wave model for wetting of planar substrates [8]. However, an analysis of this model [3] shows that the liquidvapour interface across the wedge is aproximately flat and soft-mode fluctuations arise Tricritical wedge filling transitions with short-ranged forces 3 from local translations in the height of the filled region along the wedge. These breather modes are the only relevant fluctuations in the continuous filling phenomena, and can be taken into account by the following effective Hamiltonian [3]