Filling and wetting transitions on sinusoidal substrates: a mean-field study of the Landau-Ginzburg model

Kelvin equation for bridging transitions

We study bridging transitions between a pair of nonplanar surfaces. We show that the transition can be described using a generalized Kelvin equation by mapping the system to a slit of finite length. The proposed equation is applied to analyze the asymptotic behavior of the growth of the bridging film, which occurs when the confining walls are gradually flattened. This phenomenon is characterized by a power-law divergence with geometry-dependent critical exponents that we determine for a wide class of walls' geometries. In particular, for a linear-wedge model, a covariance law revealing a relation between a geometric and Young's contact angle is presented. These predictions are shown to be fully in line with the numerical results obtained from a microscopic (classical) density functional theory.

Meniscus osculation and adsorption on geometrically structured walls

We study the adsorption of simple fluids at smoothly structured, completely wet walls and show that a meniscus osculation transition occurs when the Laplace and geometrical radii of curvature of locally parabolic regions coincide. Macroscopically, the osculation transition is of fractional, 7/2, order and separates regimes in which the adsorption is microscopic, containing only a thin wetting layer, and mesoscopic, in which a meniscus exists. We develop a scaling theory for the rounding of the transition due to thin wetting layers and derive critical exponent relations that determine how the interfacial height scales with the geometrical radius of curvature. Connection with the general geometric construction proposed by Rascón and Parry is made. Our predictions are supported by a microscopic model density functional theory for drying at a sinusoidally shaped hard wall where we confirm the order of the transition and also an exact sum rule for the generalized contact theorem due to Upton. We show that as bulk coexistence is approached the adsorption isotherm separates into three regimes: A preosculation regime where it is microscopic, containing only a thin wetting layer; a mesoscopic regime, in which a meniscus sits within the troughs; and finally another microscopic regime where the liquid-gas interface unbinds from the crests of the substrate.

Height of a liquid drop on a wetting stripe

Adsorption of liquid on a planar wall decorated by a hydrophilic stripe of width L is considered. Under the condition that the wall is only partially wet (or dry) while the stripe tends to be wet completely, a liquid drop is formed above the stripe. The maximum height ℓ_{m}(δμ) of the drop depends on the stripe width L and the chemical potential departure from saturation δμ where it adopts the value ℓ_{0}=ℓ_{m}(0). Assuming a long-range potential of van der Waals type exerted by the stripe, the interfacial Hamiltonian model is used to show that ℓ_{0} is approached linearly with δμ with a slope which scales as L^{2} over the region satisfying L≲ξ_{∥}, where ξ_{∥} is the parallel correlation function pertinent to the stripe. This suggests that near the saturation there exists a universal curve ℓ_{m}(δμ) to which the adsorption isotherms corresponding to different values of L all collapse when appropriately rescaled. Although the series expansion based on the interfacial Hamiltonian model can be formed by considering higher order terms, a more appropriate approximation in the form of a rational function based on scaling arguments is proposed. The approximation is based on exact asymptotic results, namely, that ℓ_{m}∼δμ^{-1/3} for L→∞ and that ℓ_{m} obeys the correct δμ→0 behavior in line with the results of the interfacial Hamiltonian model. All the predictions are verified by the comparison with a microscopic density functional theory (DFT) and, in particular, the rational function approximation-even in its simplest form-is shown to be in a very reasonable agreement with DFT for a broad range of both δμ and L.

Filling, depinning, unbinding: Three adsorption regimes for nanocorrugated substrates

We study adsorption at periodically corrugated substrates formed by scoring rectangular grooves into a planar solid wall which interacts with the fluid via long-range (dispersion) forces. The grooves are assumed to be macroscopically long but their depth, width, and separations can all be molecularly small. We show that the entire adsorption process can be divided into three parts consisting of (i) filling the grooves by a capillary liquid; (ii) depinning of the liquid-gas interface from the wall edges; and (iii) unbinding of the interface from the top of the wall, which is accompanied by a rapid but continuous flattening of its shape. Using a nonlocal density functional theory and mesoscopic interfacial models all the regimes are discussed in some detail to reveal the complexity of the entire process and subtle aspects that affect its behavior. In particular, it is shown that the nature of the depinning phenomenon is governed by the width of the wall pillars (separating grooves), while the width of the grooves only controls the location of the depinning first-order transition, if present.

When does Wenzel's extension of Young's equation for the contact angle of droplets apply? A density functional study

The contact angle of a liquid droplet on a surface under partial wetting conditions differs for a nanoscopically rough or periodically corrugated surface from its value for a perfectly flat surface. Wenzel's relation attributes this difference simply to the geometric magnification of the surface area (by a factor r_w), but the validity of this idea is controversial. We elucidate this problem by model calculations for a sinusoidal corrugation of the form z_wall(y) = Δ cos(2πy/λ), for a potential of short range σ_w acting from the wall on the fluid particles. When the vapor phase is an ideal gas, the change in the wall-vapor surface tension can be computed exactly, and corrections to Wenzel's equation are typically of the order σ_wΔ/λ². For fixed r_w and fixed σ_w, the approach to Wenzel's result with increasing λ may be nonmonotonic and this limit often is only reached for λ/σ_w > 30. For a non-additive binary mixture, density functional theory is used to work out the density profiles of both coexisting phases for planar and corrugated walls as well as the corresponding surface tensions. Again, deviations from Wenzel's results of similar magnitude as in the above ideal gas case are predicted. Finally, a crudely simplified description based on the interface Hamiltonian concept is used to interpret the corresponding simulation results along similar lines. Wenzel's approach is found to generally hold when λ/σ_w ≫ 1 and Δ/λ < 1 and under conditions avoiding proximity of wetting or filling transitions.

Geometry-induced interface pinning at completely wet walls

We study complete wetting of solid walls that are patterned by parallel nanogrooves of depth D and width L with a periodicity of 2L. The wall is formed of a material which interacts with the fluid via a long-range potential and exhibits first-order wetting transition at temperature T_{w}, should the wall be planar. Using a nonlocal density functional theory we show that at a fixed temperature T>T_{w} the process of complete wetting depends sensitively on two microscopic length scales L_{c}^{+} and L_{c}^{-}. If the corrugation parameter L is greater than L_{c}^{+}, the process is continuous similar to complete wetting on a planar wall. For L_{c}^{-}<L<L_{c}^{+}, the complete wetting exhibits first-order depinning transition corresponding to an abrupt unbinding of the liquid-gas interface from the wall. Finally, for L<L_{c}^{-} the interface remains pinned at the wall even at bulk liquid-gas coexistence. This implies that nanomodification of substrate surfaces can always change their wetting character from hydrophilic into hydrophobic, in direct contrast to the macroscopic Wenzel law. The resulting surface phase diagram reveals a close analogy between the depinning and prewetting transitions including the nature of their critical points.

Filling transitions on rough surfaces: Inadequacy of Gaussian surface models

We present numerical studies of wetting on various topographic substrates, including random topographies. We find good agreement with recent predictions based on an analytical interface-displacement-type theory, except that we find critical end points within the physical parameter range. As predicted, Gaussian random surfaces are found to behave qualitatively different from non-Gaussian topographies. This shows that Gaussian random processes as models for rough surfaces must be used with great care, if at all, in the context of wetting phenomena.