Homogeneous nucleation at high supersaturation and heterogeneous nucleation on microscopic wettable particles: A hybrid thermodynamic∕density-functional theory

Confinement Effects in Droplet Formation on a Solid Particle

Formation of a droplet around a spherical solid particle in supersaturated vapor is considered. The number and stability of equilibrium solutions in a closed small system are studied in the canonical ensemble in comparison to an open system in the grand canonical ensemble. Depending on the system's parameters, two modes exist in the canonical ensemble: the first one with only one solution and the second one with three solutions; the presence of the third solution is due to confinement. The analysis is conducted first on a macroscopic thermodynamic level of description, and then the results are supported by studies within two versions of classical density functional theory: the square-gradient approximation with the Carnahan-Starling equation of state for hard spheres on a completely wettable particle and the random-phase approximation with the fundamental measure theory on a poorly wettable particle. In the latter case, a solution breaking the spherical symmetry is observed at a small total number of molecules.

Heterogeneous nucleation of a droplet pinned at a chemically inhomogeneous substrate: A simulation study of the two-dimensional Ising case

Heterogeneous nucleation is studied by Monte Carlo simulations and phenomenological theory, using the two-dimensional lattice gas model with suitable boundary fields. A chemical inhomogeneity of length b at one boundary favors the liquid phase, while elsewhere the vapor is favored. Switching on the bulk field H_b favoring the liquid, nucleation and growth of the liquid phase starting from the region of the chemical inhomogeneity are analyzed. Three regimes occur: for small fields, H_b<H_b^crit, the critical droplet radius is so large that a critical droplet having the contact angle θ_c required by Young's equation in the region of the chemical inhomogeneity does not yet "fit" there since the baseline length of the circle-cut sphere droplet would exceed b. For H_b^crit<H_b<H_b^*, such droplets fit inside the inhomogeneity and are indeed found in simulations with large enough observation times, but these droplets remain pinned to the chemical inhomogeneity when their baseline has grown to the length b. Assuming that these pinned droplets have a circle cut shape and effective contact angles θ_eff in the regime θ_c < θ_eff < π/2, the density excess due to these droplets can be predicted and is found to be in reasonable agreement with the simulation results. On general grounds, one can predict that the effective contact angle θ_eff and the excess density of the droplets, scaled by b, are functions of the product bH_b but do not depend on both variables separately. Since the free energy barrier for the "depinning" of the droplet (i.e., growth of θ_eff to π - θ_c) vanishes when θ_eff approaches π/2, in practice only angles θ_eff up to about θ_eff^max≃70° were observed. For larger fields (H_b>H_b^*), the droplets nucleated at the chemical inhomogeneity grow to the full system size. While the relaxation time for the growth scales as τ_G∝H_b^-1, the nucleation time τ_N scales as lnτ_N∝H_b^-1. However, the prefactor in the latter relation, as evaluated for our simulations results, is not in accord with an extension of the Volmer-Turnbull theory to two-dimensions, when the theoretical contact angle θ_c is used.

Size-dependent contact angle and the wetting and drying transition of a droplet adsorbed onto a spherical substrate: Line-tension effect

The size-dependent contact angle and the drying and wetting morphological transition are studied with respect to the volume change for a spherical cap-shaped droplet placed on a spherical substrate. The line-tension effect is included using the rigorous formula for the Helmholtz free energy in the droplet capillary model. A morphological drying transition from a cap-shaped to a spherical droplet occurs when the substrate is hydrophobic and the droplet volume is small, similar to the transition predicted on a flat substrate. In addition, a morphological wetting transition from a cap-shaped to a wrapped spherical droplet occurs for a hydrophilic substrate and a large droplet volume. The contact angle depends on the droplet size: it decreases as the droplet volume increases when the line tension is positive, whereas it increases when the line tension is negative. The spherical droplets and wrapped droplets are stable when the line tension is positive and large.

Line tension and morphology of a sessile droplet on a spherical substrate

The effects of line tension on the morphology of a sessile droplet placed on top of a convex spherical substrate are studied. The morphology of the droplet is determined from the global minimum of the Helmholtz free energy. The contact angle between the droplet and the spherical substrate is expressed by the generalized Young's formula. When the line tension is positive and large, the contact angle jumps discontinuously to 180^{∘}, the circular contact line shrinks towards the top of the substrate, and the droplet detaches from the substrate, forming a spherical droplet if the substrate is hydrophobic (i.e., the Young's contact angle is large). This finding is consistent with that predicted by Widom [J. Phys. Chem. 99, 2803 (1995)JPCHAX0022-365410.1021/j100009a041]; the line tension induces a drying transition on a flat substrate. On the other hand, the contact angle jumps to 0^{∘}, the circular contact line shrinks towards the bottom of the substrate, and the droplet spreads over the substrate to form a wrapped spherical droplet if the substrate is hydrophilic (i.e., the Young's contact angle is small). Therefore, not only the drying transition of a cap-shaped to a detached spherical droplet but also the wetting transition of a cap-shaped to a wrapped spherical droplet could occur on a spherical substrate as the surface area of the substrate is finite. When the line tension is negative and its magnitude increases, the contact line asymptotically approaches the equator from either above or below. The droplet with a contact line that coincides with the equator is an isolated, singular solution of the first variational problem. In this instance, the contact line is pinned and cannot move as far as the line tension is smaller than the critical magnitude, where the wetting transition occurs.

Nucleation of liquid droplets and voids in a stretched Lennard-Jones fcc crystal

The method of molecular dynamics simulation has been used to investigate the phase decay of a metastable Lennard-Jones face-centered cubic crystal at positive and negative pressures. It is shown that at high degrees of metastability, crystal decay proceeds through the spontaneous formation and growth of new-phase nuclei. It has been found that there exists a certain boundary temperature. Below this temperature, the crystal phase disintegrates as the result of formation of voids, and above, as a result of formation of liquid droplets. The boundary temperature corresponds to the temperature of cessation of a crystal-liquid phase equilibrium when the melting line comes in contact with the spinodal of the stretched liquid. The results of the simulations are interpreted in the framework of classical nucleation theory. The thermodynamics of phase transitions in solids has been examined with allowance for the elastic energy of stresses arising owing to the difference in the densities of the initial and the forming phases. As a result of the action of elastic forces, at negative pressures, the boundary of the limiting superheating (stretching) of a crystal approaches the spinodal, on which the isothermal bulk modulus of dilatation becomes equal to zero. At the boundary of the limiting superheating (stretching), the shape of liquid droplets and voids is close to the spherical one.

Line-tension-induced scenario of heterogeneous nucleation on a spherical substrate and in a spherical cavity

Line-tension-induced scenario of heterogeneous nucleation is studied for a lens-shaped nucleus with a finite contact angle nucleated on a spherical substrate and on the bottom of the wall of a spherical cavity. The effect of line tension on the free energy of a critical nucleus can be separated from the usual volume term. By comparing the free energy of a lens-shaped critical nucleus of a finite contact angle with that of a spherical nucleus, we find that a spherical nucleus may have a lower free energy than a lens-shaped nucleus when the line tension is positive and large, which is similar to the drying transition predicted by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)]. Then, the homogeneous nucleation rather than the heterogeneous nucleation will be favorable. Similarly, the free energy of a lens-shaped nucleus becomes negative when the line tension is negative and large. Then, the barrier-less nucleation with no thermal activation called athermal nucleation will be realized.

Line-tension effects on heterogeneous nucleation on a spherical substrate and in a spherical cavity

The line-tension effects on heterogeneous nucleation are considered when a spherical lens-shaped nucleus is nucleated on top of a spherical substrate and on the bottom of the wall of a spherical cavity. The effect of line tension on the nucleation barrier can be separated from the usual volume term. As the radius of the substrate increases, the nucleation barrier decreases and approaches that of a flat substrate. However, as the radius of the cavity increases, the nucleation barrier increases and approaches that of a flat substrate. A small spherical substrate is a less active nucleation site than a flat substrate, and a small spherical cavity is a more active nucleation site than a flat substrate. In contrast, the line-tension effect on the nucleation barrier is maximum when the radii of the nucleus and the substrate or cavity become comparable. Therefore, by tuning the size of the spherical substrate or spherical cavity, the effect of the line tension can be optimized. These results will be useful in broad range of applications from material processing to understanding of global climate, where the heterogeneous nucleation plays a vital role.

Density functional theory of a curved liquid-vapour interface: evaluation of the rigidity constants

It is argued that to arrive at a quantitative description of the surface tension of a liquid drop as a function of its inverse radius, it is necessary to include the bending rigidity k and Gaussian rigidity k in its description. New formulae for k and k in the context of density functional theory with a non-local, integral expression for the interaction between molecules are presented. These expressions are used to investigate the influence of the choice of Gibbs dividing surface, and it is shown that for a one-component system, the equimolar surface has a special status in the sense that both k and k are then the least sensitive to a change in the location of the dividing surface. Furthermore, the equimolar value for k corresponds to its maximum value and the equimolar value for k corresponds to its minimum value. An explicit evaluation using a short-ranged interaction potential between molecules shows that k is negative with a value around minus 0.5-1.0 kBT and that k is positive with a value that is a bit more than half the magnitude of k. Finally, for dispersion forces between molecules, we show that a term proportional to log(R)/R(2) replaces the rigidity constants and we determine the (universal) proportionality constants.

A perspective on the interfacial properties of nanoscopic liquid drops

The structural and interfacial properties of nanoscopic liquid drops are assessed by means of mechanical, thermodynamical, and statistical mechanical approaches that are discussed in detail, including original developments at both the macroscopic level and the microscopic level of density functional theory (DFT). With a novel analysis we show that a purely macroscopic (static) mechanical treatment can lead to a qualitatively reasonable description of the surface tension and the Tolman length of a liquid drop; the latter parameter, which characterizes the curvature dependence of the tension, is found to be negative and has a magnitude of about a half of the molecular dimension. A mechanical slant cannot, however, be considered satisfactory for small finite-size systems where fluctuation effects are significant. From the opposite perspective, a curvature expansion of the macroscopic thermodynamic properties (density and chemical potential) is then used to demonstrate that a purely thermodynamic approach of this type cannot in itself correctly account for the curvature correction of the surface tension of liquid drops. We emphasize that any approach, e.g., classical nucleation theory, which is based on a purely macroscopic viewpoint, does not lead to a reliable representation when the radius of the drop becomes microscopic. The description of the enhanced inhomogeneity exhibited by small drops (particularly in the dense interior) necessitates a treatment at the molecular level to account for finite-size and surface effects correctly. The so-called mechanical route, which corresponds to a molecular-level extension of the macroscopic theory of elasticity and is particularly popular in molecular dynamics simulation, also appears to be unreliable due to the inherent ambiguity in the definition of the microscopic pressure tensor, an observation which has been known for decades but is frequently ignored. The union of the theory of capillarity (developed in the nineteenth century by Gibbs and then promoted by Tolman) with a microscopic DFT treatment allows for a direct and unambiguous description of the interfacial properties of drops of arbitrary size; DFT provides all of the bulk and surface characteristics of the system that are required to uniquely define its thermodynamic properties. In this vein, we propose a non-local mean-field DFT for Lennard-Jones (LJ) fluids to examine drops of varying size. A comparison of the predictions of our DFT with recent simulation data based on a second-order fluctuation analysis (Sampayo et al 2010 J. Chem. Phys. 132 141101) reveals the consistency of the two treatments. This observation highlights the significance of fluctuation effects in small drops, which give rise to additional entropic (thermal non-mechanical) contributions, in contrast to what one observes in the case of planar interfaces which are governed by the laws of mechanical equilibrium. A small negative Tolman length (which is found to be about a tenth of the molecular diameter) and a non-monotonic behaviour of the surface tension with the drop radius are predicted for the LJ fluid. Finally, the limits of the validity of the Tolman approach, the effect of the range of the intermolecular potential, and the behaviour of bubbles are briefly discussed.

Heterogeneous nucleation at a wall near a wetting transition: a Monte Carlo test of the classical theory

While for a slightly supersaturated vapor the free energy barrier ΔF(hom)(*), which needs to be overcome in a homogeneous nucleation event, may be extremely large, nucleation is typically much easier at the walls of the container in which the vapor is located. While no nucleation barrier exists if the walls are wet, for incomplete wetting of the walls, described via a nonzero contact angle Θ, classical theory predicts that nucleation happens through sphere-cap-shaped droplets attracted to the wall, and their formation energy is ΔF(het)(*) = ΔF(hom)(*)f(Θ), with f(Θ) = (1-cosΘ)(2)(2+cosΘ)/4. This prediction is tested through simulations for the simple cubic lattice gas model with nearest-neighbor interactions. The attractive wall is described in terms of a local 'surface field', leading to a critical wetting transition. The variation of the contact angle with the strength of the surface field is determined by using thermodynamic integration methods to obtain the wall free energies which enter Young's equation. Obtaining the chemical potential as a function of the density for a system with periodic boundary conditions (and no walls), the droplet free energy of a spherical droplet in the bulk is obtained for a wide range of droplet radii. Similarly, ΔF(het)(*) is obtained for a system with two parallel walls. We find that the classical theory is fairly accurate if a line tension correction for the contact angle is taken into account.

Influence of vapor depletion on nucleation rate

During condensation in finite systems part of molecules is transformed from supersaturated mother phase to a new one and depletion of the mother phase occurs. Kinetic equations describing homogeneous nucleation process including decrease of supersaturation are solved numerically. It is shown that dependency of nucleation rate on nucleus size reaches some maximum, which decreases with time and moves to higher nucleus sizes. Nucleation rate is negative for undercritical size of nuclei.