Inclusion of line tension effect in classical nucleation theory for heterogeneous nucleation: A rigorous thermodynamic formulation and some unique conclusions

Surface Heterogeneous Nucleation-Mediated Release of Beta-Carotene from Porous Silicon

We demonstrate that the release of a poorly soluble molecule from nanoporous carriers is a complex process that undergoes heterogeneous surface nucleation events even under significantly diluted release conditions, and that those events heavily affect the dynamics of release. Using beta-carotene and porous silicon as loaded molecule and carrier model, respectively, we show that the cargo easily nucleates at the pore surface during the release, forming micro- to macroscopic solid particles at the pores surface. These particles dissolve at a much slower pace, compared to the rate of dissolution of pure beta-carotene in the same solvent, and they negatively affect the reproducibility of the release experiments, possibly because their solubility depends on their size distribution. We propose to exploit this aspect to use release kinetics as a better alternative to the induction time method, and to thereby detect heterogenous nucleation during release experiments. In fact, release dynamics provide much higher sensitivity and reproducibility as they average over the entire sample surface instead of depending on statistical analysis over a small area to find clusters.

Do the contact angle and line tension of surface-attached droplets depend on the radius of curvature?

Results from Monte Carlo simulations of wall-attached droplets in the three-dimensional Ising lattice gas model and in a symmetric binary Lennard-Jones fluid, confined by antisymmetric walls, are analyzed, with the aim to estimate the dependence of the contact angle [Formula: see text] on the droplet radius [Formula: see text] of curvature. Sphere-cap shape of the wall-attached droplets is assumed throughout. An approach, based purely on 'thermodynamic' observables, e.g. chemical potential, excess density due to the droplet, etc, is used, to avoid ambiguities in the decision which particles belong (or do not belong, respectively) to the droplet. It is found that the results are compatible with a variation [Formula: see text], [Formula: see text] being the contact angle in the thermodynamic limit ([Formula: see text]). The possibility to use such results to estimate the excess free energy related to the contact line of the droplet, namely the line tension, at the wall, is discussed. Various problems that hamper this approach and were not fully recognized in previous attempts to extract the line tension are identified. It is also found that the dependence of wall tensions on the difference of chemical potential of the droplet from that at the bulk coexistence provides effectively a change of the contact angle of similar magnitude. The simulation approach yields precise estimates for the excess density due to wall-attached droplets and the corresponding free energy excess, relative to a system without a droplet at the same chemical potential. It is shown that this information suffices to estimate nucleation barriers, not affected by ambiguities on droplet shape, contact angle and line tension.

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.

Free-energy barriers for crystal nucleation from fluid phases

Monte Carlo simulations of crystal nuclei coexisting with the fluid phase in thermal equilibrium in finite volumes are presented and analyzed, for fluid densities from dense melts to the vapor. Generalizing the lever rule for two-phase coexistence in the canonical ensemble to finite volume, "measurements" of the nucleus volume together with the pressure and chemical potential of the surrounding fluid allows us to extract the surface free energy of the nucleus. Neither the knowledge of the (in general nonspherical) nucleus shape nor of the angle-dependent interface tension is required for this task. The feasibility of the approach is demonstrated for a variant of the Asakura-Oosawa model for colloid-polymer mixtures, which form face-centered cubic colloidal crystals. For a polymer to colloid size ratio of 0.15, the colloid packing fraction in the fluid phase can be varied from melt values to zero by the variation of an effective attractive potential between the colloids. It is found that the approximation of spherical crystal nuclei often underestimates actual nucleation barriers significantly. Nucleation barriers are found to scale as ΔF^{*}=(4π/3)^{1/3}γ[over ¯](V^{*})^{2/3}+const with the nucleus volume V^{*}, and the effective surface tension γ[over ¯] that accounts implicitly for the nonspherical shape can be precisely estimated.

Thermostability analysis of line-tension-associated nucleation at a gas-liquid interface

The influence of line tension on the thermostability of a droplet nucleated from an oversaturated vapor at the interface of the vapor and another immiscible liquid is investigated. Along with the condition of mechanical equilibrium, the notion of extremization of the reversible work of formation is considered to obtain the critical parameters related to heterogeneous nucleation. From the energetic formulation, the critical reversible work of formation is found to be greater than that of homogeneous nucleation for high value of the positive line tension. On the other hand, for high value of the negative line tension, the critical reversible work of formation becomes negative. Therefore, these thermodynamic instabilities under certain substrate wettability situations necessitate a free-energetics-based stability of the nucleated droplet, because the system energy is not minimized under these conditions. This thermostability is analogous to the transition-based stability proposed by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)]10.1021/j100009a041 in the case of partial wetting phenomena along with the positive line tension. The thermostability analysis limits the domain of the solution space of the present critical-value problem as the thermodynamic transformation in connection with homogeneous and workless nucleation is considered. Within the stability range of the geometry-based wetting parameters, three limiting modes of nucleation, i.e., total-dewetting-related homogeneous nucleation, and total-wetting-associated and total-submergence-associated workless nucleation scenarios, are identified. Either of the two related limiting wetting scenarios of workless nucleation, namely, total wetting and total submergence, is found to be favorable depending on the geometry-based wetting conditions. The line-tension-associated nucleation on a liquid surface can be differentiated from that on a rigid substrate, as in the former, the stability based on mechanical equilibrium and a typical case of workless nucleation with complete submergence are observed.

Exploring the effects and mechanisms of carbon nanomaterial diversity on the morphology of lysozyme crystals

A shift in the final size distribution and morphology was observed, and more pronounced X-ray diffraction peaks were achieved in lysozyme crystals with the addition of 3D CNMs.

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.

Effect of a triple contact line on the thermokinetics of dropwise condensation on an immiscible liquid surface

An extended thermokinetic model is developed for liquid-substrate-induced condensation by considering the collective influence of the line tension and the two mechanisms of molecular transport.

Thermokinetics of heterogeneous droplet nucleation on conically textured substrates

Within the framework of the classical theory of heterogeneous nucleation, a thermokinetic model is developed for line-tension-associated droplet nucleation on conical textures considering growth or shrinkage of the formed cluster due to both interfacial and peripheral monomer exchange and by considering different geometric configurations. Along with the principle of free energy extremization, Katz kinetic approach has been employed to study the effect of substrate conicity and wettability on the thermokinetics of heterogeneous water droplet nucleation. Not only the peripheral tension is found to have a considerable effect on the free energy barrier but also the substrate hydrophobicity and hydrophilicity are observed to switch over their roles between conical crest and trough for different growth rates of the droplet. Besides, the rate of nucleation increases and further promotes nucleation for negative peripheral tension as it diminishes the free energy barrier appreciably. Moreover, nucleation inhibition can be achievable for positive peripheral tension due to the enhancement of the free energy barrier. Analyzing all possible geometric configurations, the hydrophilic narrower conical cavity is found to be the most preferred nucleation site. These findings suggest a physical insight into the context of surface engineering for the promotion or the suppression of nucleation on real or engineered substrates.

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.