Capillary condensation between nonparallel walls.

We study the condensation of fluids confined by a pair of nonparallel plates of finite height H. We show that such a system experiences two types of condensation, termed single and double pinning, which can be characterized by one (single-pinning) or two (double-pinning) edge contact angles describing the shape of menisci pinned at the system edges. For both types of capillary condensation, we formulate the Kelvin-like equation and determine the conditions under which the given type of condensation occurs. We construct the global phase diagram revealing a reentrant phenomenon pertinent to the change of the capillary condensation type upon varying the inclination of the walls. Asymptotic properties of the system are discussed and a link with related phase phenomena in different systems is made. Finally, we show that the change from a single- to a double-pinned state is a continuous transition, the character of which depends on the wetting properties of the walls.

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.

Critical-point wedge filling and critical-point wetting.

For simple fluids adsorbed at a planar solid substrate (modeled as an inert wall) it is known that critical-point wetting, that is, the vanishing of the contact angle θ at a temperature T_{w} lying below that of the critical point T_{c}, need not occur. While critical-point wetting necessarily happens when the wall-fluid and fluid-fluid forces have the same range (e.g., both are long ranged or both short ranged) nonwetting gaps appear in the surface phase diagram when there is an imbalance between the ranges of these forces. Here we show that despite this, the convergence of the lines of constant contact angle, 0<θ<π, to an ordinary surface phase transition at T_{c}, means that fluids adsorbed in wedges (and cones) always exhibit critical-point filling (wedge wetting or wedge drying) regardless of the range and imbalance of the forces. We illustrate the necessity of critical-point filling, even in the absence of critical-point wetting, using a microscopic model density functional theory of fluid adsorption in a right angle wedge, with dispersion and also retarded dispersionlike wall-fluid forces. The location and order of the filling phase boundaries are determined and shown to be in excellent agreement with exact thermodynamic requirements and also predictions for critical singularities based on interfacial models.

Critical behaviour of the contact angle within nonwetting gaps.

Recent density functional theory and simulation studies of fluid adsorption near planar walls in systems where the wall-fluid and fluid-fluid interactions have different ranges, have shown that critical point wetting may not occur and instead nonwetting gaps appear in the surface phase diagram, separating lines of wetting and drying transitions, that extend up to the critical temperatureTc. Here we clarify the features of the surface phase diagrams that are common, regardless of the range and balance of the forces, showing, in particular, that the lines of temperature driven wetting and drying transitions, as well as lines of constant contact angleπ>θ>0, always converge to an ordinary surface phase transition atTc. When nonwetting gaps appear the contact angle either vanishes or tends toπast≡(Tc-T)/Tc→0. More specifically, when the wall-fluid interaction is long-ranged (dispersion-like) and the fluid-fluid short-ranged we estimateπ-θ∝t0.16, compared withθ∝t0.77when the wall-fluid interaction is short-ranged and the fluid-fluid dispersion-like, allowing for the effects of bulk critical fluctuations. The universal convergence of the lines of constant contact angle implies that critical point filling always occurs for fluids adsorbed in wedges.

Surface Phase Diagrams for Wetting with Long-Ranged Forces.

Recent density functional theory and simulation studies of wetting and drying transitions in systems with long-ranged, dispersionlike forces, away from the near vicinity of the bulk critical temperature T_{c}, have questioned the generality of the global surface phase diagrams for wetting, due to Nakanishi and Fisher, pertinent to systems with short-ranged forces. We extend these studies deriving fully analytic results which determine the surface phase diagrams over the whole temperature range up to T_{c}. The phase boundaries, order of, and asymmetry between the lines of wetting and drying transitions are determined exactly showing that they always converge to an ordinary surface critical point. We highlight the importance of lines of maximally multicritical wetting and drying transitions, for which we determine the exact critical singularities.

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 ß_{osc} characterizing the scale of the interfacial height ℓ_{0}∝R_{w}^{ß_{osc}} at osculation, for large R_{w}, falls into two regimes representing fluctuation-dominated and mean-field-like behavior, respectively. These two regimes are separated by an upper critical dimension, which is determined here explicitly and depends on the range of the intermolecular forces. In the fluctuation-dominated regime, representing the universality class of systems with short-range forces, the exponent is related to the value of the interfacial wandering exponent ζ by ß_{osc}=3ζ/(4-ζ). In contrast, in the mean-field regime, which was not previously identified and which occurs for systems with longer-range forces (and higher dimensions), the exponent ß_{osc} takes the same value as the exponent ß_{s}^{co} for complete wetting, which is determined directly by the intermolecular forces. The prediction ß_{osc}=3/7 in d=2 for systems with short-range forces (corresponding to ζ=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 ß_{osc}≈ß_{s}^{co}≈0.326 close to the predicted value of 1/3 appropriate to the mean-field regime for dispersion forces.

Phase behavior of fluids in undulated nanopores.

The geometry of walls forming a narrow pore may qualitatively affect the phase behavior of the confined fluid. Specifically, the nature of condensation in nanopores formed of sinusoidally shaped walls (with amplitude A and period P) is governed by the wall mean separation L as follows. For L>L_{t}, where L_{t} increases with A, the pores exhibit standard capillary condensation similar to planar slits. In contrast, for L

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.

Capillary condensation and depinning transitions in open slits.

We study the low-temperature phase equilibria of a fluid confined in an open capillary slit formed by two parallel walls separated by a distance L which are in contact with a reservoir of gas. The top wall of the capillary is of finite length H while the bottom wall is considered of macroscopic extent. This system shows rich phase equilibria arising from the competition between two different types of capillary condensation, corner filling, and meniscus depinning transitions depending on the value of the aspect ratio a=L/H and divides into three regimes: For long capillaries, with a<2/π, the condensation is of type I involving menisci which are pinned at the top edges at the ends of the capillary. For intermediate capillaries, with 2/π1, condensation is always of type II. In all regimes, capillary condensation is completely suppressed for sufficiently large contact angles which is determined explicitly. For long and intermediate capillaries, we show that there is an additional continuous phase transition in the condensed liquid-like phase, associated with the depinning of each meniscus as they round the upper open edges of the slit. Meniscus depinning is third-order for complete wetting and second-order for partial wetting. Detailed scaling theories are developed for these transitions and phase boundaries which connect with the theories of wedge (corner) filling and wetting encompassing interfacial fluctuation effects and the direct influence of intermolecular forces. We test several of our predictions using a fully microscopic density functional theory which allows us to study the two types of capillary condensation and its suppression at the molecular level for different aspect ratios and contact angles.

Edge Contact Angle, Capillary Condensation, and Meniscus Depinning.

We study the phase equilibria of a fluid confined in an open capillary slit formed when a wall of finite length H is brought a distance L away from a second macroscopic surface. This system shows rich phase equilibria arising from the competition between two different types of capillary condensation, corner filling and meniscus depinning transitions depending on the value of the aspect ratio a=L/H. For long capillaries, with a<2/π, the condensation is of type I involving menisci which are pinned at the top edges at the ends of the capillary characterized by an edge contact angle. For intermediate capillaries, with 2/π1, condensation is always of type II. In all regimes, capillary condensation is completely suppressed for sufficiently large contact angles. We show that there is an additional continuous phase transition in the condensed liquidlike phase, associated with the depinning of each meniscus as they round the upper open edges of the slit. Finite-size scaling predictions are developed for these transitions and phase boundaries which connect with the fluctuation theories of wetting and filling transitions. We test several of our predictions using a fully microscopic density functional theory which allows us to study the two types of capillary condensation and its suppression at the molecular level.

Breaking Cassie's Law for Condensation in a Nanopatterned Slit.

We study the phase transitions of a fluid confined in a capillary slit made from two adjacent walls, each of which are a periodic composite of stripes of two different materials. For wide slits the capillary condensation occurs at a pressure which is described accurately by a combination of the Kelvin equation and the Cassie law for an averaged contact angle. However, for narrow slits the condensation occurs in two steps involving an intermediate bridging phase, with the corresponding pressures described by two new Kelvin equations. These are characterised by different contact angles due to interfacial pinning, with one larger and one smaller than the Cassie angle. We determine the triple point and predict two types of dispersion force induced Derjaguin-like corrections due to mesoscopic volume reduction and the singular free-energy contribution from nanodroplets and bubbles. We test these predictions using a fully microscopic density functional model which confirms their validity even for molecularly narrow slits. Analogous mesoscopic corrections are also predicted for two-dimensional systems arising from thermally induced interfacial wandering.

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 δµâ†’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.

Three-Phase Fluid Coexistence in Heterogenous Slits.

We study the competition between local (bridging) and global condensation of fluid in a chemically heterogeneous capillary slit made from two parallel adjacent walls each patterned with a single stripe. Using a mesoscopic modified Kelvin equation, which determines the shape of the menisci pinned at the stripe edges in the bridge phase, we determine the conditions under which the local bridging transition precedes capillary condensation as the pressure (or chemical potential) is increased. Provided the contact angle of the stripe is less than that of the outer wall we show that triple points, where evaporated, locally condensed, and globally condensed states all coexist are possible depending on the value of the aspect ratio a=L/H, where H is the stripe width and L the wall separation. In particular, for a capillary made from completely dry walls patterned with completely wet stripes the condition for the triple point occurs when the aspect ratio takes its maximum possible value 8/π. These predictions are tested using a fully microscopic classical density functional theory and shown to be remarkably accurate even for molecularly narrow slits. The qualitative differences with local and global condensation in heterogeneous cylindrical pores are also highlighted.

Scaling of wetting and prewetting transitions on nanopatterned walls.

We consider a nanopatterned planar wall consisting of a periodic array of stripes of width L, which are completely wet by liquid (contact angle θ=0), separated by regions of width D which are completely dry (contact angle θ=π). Using microscopic density functional theory, we show that, in the presence of long-ranged dispersion forces, the wall-gas interface undergoes a first-order wetting transition, at bulk coexistence as the separation D is reduced to a value D_{w}∝lnL, induced by the bridging between neighboring liquid droplets. Associated with this is a line of prewetting transitions occurring off coexistence. By varying the stripe width L, we show that the prewetting line shows universal scaling behavior and data collapse. This verifies predictions based on mesoscopic models for the scaling properties associated with finite-size effects at complete wetting including the logarithmic singular contribution to the surface free energy.

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}^{-}

Bridging of liquid drops at chemically structured walls.

Using mesoscopic interfacial models and microscopic density functional theory we study fluid adsorption at a dry wall decorated with three completely wet stripes of width L separated by distances D_{1} and D_{2}. The stripes interact with the fluid with long-range forces inducing a large finite-size contribution to the surface free energy. We show that this nonextensive free-energy contribution scales with lnL and drives different types of bridging transition corresponding to the merging of liquid drops adsorbed at neighboring wetting stripes when the separation between them is molecularly small. We determine the surface phase diagram and show that this exhibits two triple points, where isolated drops, double drops, and triple drops coexist. For the symmetric case, D_{1}=D_{2}≡D, our results also confirm that the equilibrium droplet configuration always has the symmetry of the substrate corresponding to either three isolated drops when D is large or a single triple drop when D is small; however, symmetry-broken configurations do occur in a metastable part of the phase diagram which lies very close to the equilibrium-bridging phase boundary. Implications for phase transitions on other types of patterned surface are considered.

Symmetry-breaking morphological transitions at chemically nanopatterned walls.

We study the structure and morphological changes of fluids that are in contact with solid composites formed by alternating and microscopically wide stripes of two different materials. One type of the stripes interacts with the fluid via long-ranged Lennard-Jones-like potential and tends to be completely wet, while the other type is purely repulsive and thus tends to be completely dry. We consider closed systems with a fixed number of particles that allows for stabilization of fluid configurations breaking the lateral symmetry of the wall potential. These include liquid morphologies corresponding to a sessile drop that is formed by a sequence of bridging transitions that connect neighboring wet regions adsorbed at the attractive stripes. We study the character of the transitions depending on the wall composition, stripes width, and system size. Using a (classical) nonlocal density functional theory (DFT), we show that the transitions between different liquid morphologies are typically weakly first-order but become rounded if the wavelength of the system is lower than a certain critical value L_{c}. We also argue that in the thermodynamic limit, i.e., for macroscopically large systems, the wall becomes wet via an infinite sequence of first-order bridging transitions that are, however, getting rapidly weaker and weaker and eventually become indistinguishable from a continuous process as the size of the bridging drop increases. Finally, we construct the global phase diagram and study the density dependence of the contact angle of the bridging drops using DFT density profiles and a simple macroscopic theory.

Continuous condensation in nanogrooves.

We consider condensation in a capillary groove of width L and depth D, formed by walls that are completely wet (contact angle θ=0), which is in a contact with a gas reservoir of the chemical potential µ. On a mesoscopic level, the condensation process can be described in terms of the midpoint height ℓ of a meniscus formed at the liquid-gas interface. For macroscopically deep grooves (D→∞), and in the presence of long-range (dispersion) forces, the condensation corresponds to a second-order phase transition, such that ℓ∼(µ_{cc}-µ)^{-1/4} as µâ†’µ_{cc}^{-} where µ_{cc} is the chemical potential pertinent to capillary condensation in a slit pore of width L. For finite values of D, the transition becomes rounded and the groove becomes filled with liquid at a chemical potential higher than µ_{cc} with a difference of the order of D^{-3}. For sufficiently deep grooves, the meniscus growth initially follows the power law ℓ∼(µ_{cc}-µ)^{-1/4}, but this behavior eventually crosses over to ℓ∼D-(µ-µ_{cc})^{-1/3} above µ_{cc}, with a gap between the two regimes shown to be δ[over ¯]µâˆ¼D^{-3}. Right at µ=µ_{cc}, when the groove is only partially filled with liquid, the height of the meniscus scales as ℓ^{*}∼(D^{3}L)^{1/4}. Moreover, the chemical potential (or pressure) at which the groove is half-filled with liquid exhibits a nonmonotonic dependence on D with a maximum at D≈3L/2 and coincides with µ_{cc} when L≈D. Finally, we show that condensation in finite grooves can be mapped on the condensation in capillary slits formed by two asymmetric (competing) walls a distance D apart with potential strengths depending on L. All these predictions, based on mesoscopic arguments, are confirmed by fully microscopic Rosenfeld's density functional theory with a reasonable agreement down to surprisingly small values of both L and D.

Demixing, surface nematization, and competing adsorption in binary mixtures of hard rods and hard spheres under confinement.

A molecular simulation study of binary mixtures of hard spherocylinders (HSCs) and hard spheres (HSs) confined between two structureless hard walls is presented. The principal aim of the work is to understand the effect of the presence of hard spheres on the entropically driven surface nematization of hard rod-like particles at surfaces. The mixtures are studied using a constant normal-pressure Monte Carlo algorithm. The surface adsorption at different compositions is examined in detail. At moderate hard-sphere concentrations, preferential adsorption of the spheres at the wall is found. However, at moderate to high pressure (density), we observe a crossover in the adsorption behavior with nematic layers of the rods forming at the walls leading to local demixing of the system. The presence of the spherical particles is seen to destabilize the surface nematization of the rods, and the degree of demixing increases on increasing the hard-sphere concentration.