Modified Kelvin Equations for Capillary Condensation in Narrow and Wide Grooves

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 β_{osc} characterizing the scale of the interfacial height ℓ_{0}∝R_{w}^{β_{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 β_{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 β_{osc} takes the same value as the exponent β_{s}^{co} for complete wetting, which is determined directly by the intermolecular forces. The prediction β_{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 β_{osc}≈β_{s}^{co}≈0.326 close to the predicted value of 1/3 appropriate to the mean-field regime for dispersion forces.

Enhancing Boiling Heat Transfer on a Superheated Surface by Surfactant-Laden Droplets

Boiling, one of the most common phase-change heat transfer methods, is widely used in nuclear power plants, spacecraft, integrated circuits, and other situations, where rapid and efficient heat transfer is crucial. However, boiling heat transfer is efficient only in a specific surface temperature range when a droplet impacts a superheated surface. Here, we enhance the boiling heat transfer and extend this temperature range by adding a tiny amount of surfactant. We find that surfactants can weaken the Kelvin effect of boiling bubbles, and thus reduce the onset of boiling driven temperature and significantly enhance the maximum vaporization rate of the droplet effectively. In particular, different from previous studies, we find that the surfactants at lower concentrations can increase the Leidenfrost temperature of the droplets. All the above effects jointly expand the temperature range of effective boiling heat transfer. This study sheds new light on the role of surfactants in the boiling process and offers a new medium to promote heat-transfer applications.

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<L_{t}, the condensation occurs in two steps, such that the fluid first condenses locally via bridging transition connecting adjacent crests of the walls, before it condenses globally. For the marginal value of L=L_{t}, all the three phases (gaslike, bridge, and liquidlike) may coexist. We show that the locations of the phase transitions can be described using geometric arguments leading to modified Kelvin equations. However, for completely wet walls, to which we focus on, the phase boundaries are shifted significantly due to the presence of wetting layers. In order to take this into account, mesoscopic corrections to the macroscopic theory are proposed. The resulting predictions are shown to be in a very good agreement with a density-functional theory even for molecularly narrow pores. The limits of stability of the bridge phase, controlled by the pore geometry, is also discussed in some detail.

Intrusion of liquids into liquid-infused surfaces with nanoscale roughness

We present a theoretical study of the intrusion of an ambient liquid into the pores of a nanocorrugated wall w. The pores are prefilled with a liquid lubricant that adheres to the walls of the pores more strongly than the ambient liquid does. The two liquids are modeled as a binary liquid mixture of two species of particles, A and B. The mixture can decompose into a liquid rich in A particles, representing the ambient liquid, and another one rich in B particles, representing the liquid lubricant. The wall is taken to attract the B particles more strongly than the A particles. The ratio w-A/w-B of these interaction strengths is changed in order to tune the contact angle θ_{AB} formed by the A-rich/B-rich liquid interface between the two fluids and a planar wall, composed of the same material as the one forming the pores. We use classical density functional theory in order to capture the effects of microscopic details on the intrusion transition, which occurs as the concentration of the minority component or the pressure in the bulk of the ambient liquid is varied, moving away from bulk liquid-liquid coexistence within the single-phase domain of the A-rich bulk ambient liquid. These liquid structures have been studied as a function of the contact angle θ_{AB} and for various widths and depths of the pores. We also studied the reverse process in which a pore initially filled with the ambient liquid is refilled with the liquid lubricant. The location of the intrusion transition, with respect to its dependence on the contact angle θ_{AB} and the width of the pore, qualitatively follows the corresponding shift of the capillary-coexistence line away from the bulk liquid-liquid coexistence line, as predicted by a macroscopic capillarity model. Quantitatively, the transition found in the microscopic approach occurs somewhat closer to the bulk liquid-liquid coexistence line than predicted by the macroscopic capillarity model. The quantitative discrepancies become larger for narrower cavities. In cases in which the wall is completely wetted by the lubricant (θ_{AB}=0) and for small contact angles, the reverse transition follows the same path as for intrusion; there is no hysteresis. For larger contact angles, hysteresis is observed. The width of the hysteresis increases with increasing contact angle. A reverse transition is not found inside the domain within which the ambient liquid forms a single phase in the bulk once θ_{AB} exceeds a geometry-dependent threshold value. According to the macroscopic capillarity theory, for the considered geometry, this is the case for θ_{AB}>54.7^{∘}. Our computations show, however, that nanoscale effects shift this threshold value to much higher values. This shift increases strongly if the widths of the pores become smaller (below about ten times the diameter of the A and B particles).

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/π<a<1, depending on the value of the contact angle the condensation may be of type I or of type II, in which the menisci overspill into the reservoir and there is no pinning. For short capillaries, with a>1, 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.

Capillary condensation under atomic-scale confinement

Capillary condensation of water is ubiquitous in nature and technology. It routinely occurs in granular and porous media, can strongly alter such properties as adhesion, lubrication, friction and corrosion, and is important in many processes used by microelectronics, pharmaceutical, food and other industries^1-4. The century-old Kelvin equation⁵ is frequently used to describe condensation phenomena and has been shown to hold well for liquid menisci with diameters as small as several nanometres^1-4,6-14. For even smaller capillaries that are involved in condensation under ambient humidity and so of particular practical interest, the Kelvin equation is expected to break down because the required confinement becomes comparable to the size of water molecules^1-22. Here we use van der Waals assembly of two-dimensional crystals to create atomic-scale capillaries and study condensation within them. Our smallest capillaries are less than four ångströms in height and can accommodate just a monolayer of water. Surprisingly, even at this scale, we find that the macroscopic Kelvin equation using the characteristics of bulk water describes the condensation transition accurately in strongly hydrophilic (mica) capillaries and remains qualitatively valid for weakly hydrophilic (graphite) ones. We show that this agreement is fortuitous and can be attributed to elastic deformation of capillary walls^23-25, which suppresses the giant oscillatory behaviour expected from the commensurability between the atomic-scale capillaries and water molecules^20,21. Our work provides a basis for an improved understanding of capillary effects at the smallest scale possible, which is important in many realistic situations.

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.

Effect of line tension on axisymmetric nanoscale capillary bridges at the liquid-vapor equilibrium

The effect of line tension on the axisymmetric nanoscale capillary bridge between two identical substrates with convex, concave, and flat geometry at the liquid-vapor equilibrium is theoretically studied. The modified Young's equation for the contact angle, which takes into account the effect of line tension, is derived on a general axisymmetric curved surface using the variational method. Even without the effect of line tension, the parameter space where the bridge can exist is limited simply by the geometry of substrates. The modified Young's equation further restricts the space where the bridge can exist when the line tension is positive because the equilibrium contact angle always remains finite and the wetting state near the zero contact angle cannot be realized. It is shown that the interplay of the geometry and the positive line tension restricts the formation of capillary bridge.

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 μ→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 ¯]μ∼D^{-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.