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

Chasing the Critical Wetting Transition. An Effective Interface Potential Method

Wettablity is one of the important characteristics defining a given surface. Here we show that the effective interface potential method of determining the wetting temperature, originally proposed by MacDowell and Müller for the surfaces exhibiting the first order wetting transition, can also be used to estimate the wetting temperature of the second order (continuous) wetting transition. Some selected other methods of determination of the wetting temperature are also discussed.

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.

Orientation dependence of heterogeneous nucleation at the Cu-Pb solid-liquid interface

In this work, we examine the effect of surface structure on the heterogeneous nucleation of Pb crystals from the melt at a Cu substrate using molecular-dynamics (MD) simulation. In a previous work [Palafox-Hernandez et al., Acta Mater. 59, 3137 (2011)] studying the Cu/Pb solid-liquid interface with MD simulation, we observed that the structure of the Cu(111) and Cu(100) interfaces was significantly different at 625 K, just above the Pb melting temperature (618 K for the model). The Cu(100) interface exhibited significant surface alloying in the crystal plane in contact with the melt. In contrast, no surface alloying was seen at the Cu(111) interface; however, a prefreezing layer of crystalline Pb, 2-3 atomic planes thick and slightly compressed relative to bulk Pb crystal, was observed to form at the interface. We observe that at the Cu(111) interface the prefreezing layer is no longer present at 750 K, but surface alloying in the Cu(100) interface persists. In a series of undercooling MD simulations, heterogeneous nucleation of fcc Pb is observed at the Cu(111) interface within the simulation time (5 ns) at 592 K-a 26 K undercooling. Nucleation and growth at Cu(111) proceeded layerwise with a nearly planar critical nucleus. Quantitative analysis yielded heterogeneous nucleation barriers that are more than two orders of magnitude smaller than the predicted homogeneous nucleation barriers from classical nucleation theory. Nucleation was considerably more difficult on the Cu(100) surface-alloyed substrate. An undercooling of approximately 170 K was necessary to observe nucleation at this interface within the simulation time. From qualitative observation, the critical nucleus showed a contact angle with the Cu(100) surface of over 90°, indicating poor wetting of the Cu(100) surface by the nucleating phase, which according to classical heterogeneous nucleation theory provides an explanation of the large undercooling necessary to nucleate on the Cu(100) surface, relative to Cu(111), whose surface is more similar to the nucleating phase due to the presence of the prefreezing layer.

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.

Anisotropic interfacial tension, contact angles, and line tensions: a graphics-processing-unit-based Monte Carlo study of the Ising model

As a generic example for crystals where the crystal-fluid interface tension depends on the orientation of the interface relative to the crystal lattice axes, the nearest-neighbor Ising model on the simple cubic lattice is studied over a wide temperature range, both above and below the roughening transition temperature. Using a thin-film geometry L(x)×L(y)×L(z) with periodic boundary conditions along the z axis and two free L(x)×L(y) surfaces at which opposing surface fields ±H(1) act, under conditions of partial wetting, a single planar interface inclined under a contact angle θ<π/2 relative to the yz plane is stabilized. In the y direction, a generalization of the antiperiodic boundary condition is used that maintains the translational invariance in the y direction despite the inhomogeneity of the magnetization distribution in this system. This geometry allows a simultaneous study of the angle-dependent interface tension, the contact angle, and the line tension (which depends on the contact angle, and on temperature). All these quantities are extracted from suitable thermodynamic integration procedures. In order to keep finite-size effects as well as statistical errors small enough, rather large lattice sizes (of the order of 46 million sites) were found to be necessary, and the availability of very efficient code implementation of graphics processing units was crucial for the feasibility of this study.

Logarithmic finite-size effects on interfacial free energies: phenomenological theory and Monte Carlo studies

The computation of interfacial free energies between coexisting phases (e.g., saturated vapor and liquid) by computer simulation methods is still a challenging problem due to the difficulty of an atomistic identification of an interface and interfacial fluctuations on all length scales. The approach to estimate the interfacial tension from the free-energy excess of a system with interfaces relative to corresponding single-phase systems does not suffer from the first problem but still suffers from the latter. Considering d-dimensional systems with interfacial area L(d-1) and linear dimension L(z) in the direction perpendicular to the interface, it is argued that the interfacial fluctuations cause logarithmic finite-size effects of order ln(L)/L(d-1) and order ln(L(z))/L(d-1), in addition to regular corrections (with leading-order const/L(d-1)). A phenomenological theory predicts that the prefactors of the logarithmic terms are universal (but depend on the applied boundary conditions and the considered statistical ensemble). The physical origin of these corrections are the translational entropy of the interface as a whole, "domain breathing" (coupling of interfacial fluctuations to the bulk order parameter fluctuations of the coexisting domains), and capillary waves. Using a new variant of the ensemble switch method, interfacial tensions are found from Monte Carlo simulations of d = 2 and d = 3 Ising models and a Lennard-Jones fluid. The simulation results are fully consistent with the theoretical predictions.

Degenerate Ising model for atomistic simulation of crystal-melt interfaces

One of the simplest microscopic models for a thermally driven first-order phase transition is an Ising-type lattice system with nearest-neighbour interactions, an external field, and a degeneracy parameter. The underlying lattice and the interaction coupling constant control the anisotropic energy of the phase boundary, the field strength represents the bulk latent heat, and the degeneracy quantifies the difference in communal entropy between the two phases. We simulate the (stochastic) evolution of this minimal model by applying rejection-free canonical and microcanonical Monte Carlo algorithms, and we obtain caloric curves and heat capacity plots for square (2D) and face-centred cubic (3D) lattices with periodic boundary conditions. Since the model admits precise adjustment of bulk latent heat and communal entropy, neither of which affect the interface properties, we are able to tune the crystal nucleation barriers at a fixed degree of undercooling and verify a dimension-dependent scaling expected from classical nucleation theory. We also analyse the equilibrium crystal-melt coexistence in the microcanonical ensemble, where we detect negative heat capacities and find that this phenomenon is more pronounced when the interface is the dominant contributor to the total entropy. The negative branch of the heat capacity appears smooth only when the equilibrium interface-area-to-volume ratio is not constant but varies smoothly with the excitation energy. Finally, we simulate microcanonical crystal nucleation and subsequent relaxation to an equilibrium Wulff shape, demonstrating the model's utility in tracking crystal-melt interfaces at the atomistic level.

Monte Carlo tests of nucleation concepts in the lattice gas model

The conventional theory of homogeneous and heterogeneous nucleation in a supersaturated vapor is tested by Monte Carlo simulations of the lattice gas (Ising) model with nearest-neighbor attractive interactions on the simple cubic lattice. The theory considers the nucleation process as a slow (quasistatic) cluster (droplet) growth over a free energy barrier ΔF(*), constructed in terms of a balance of surface and bulk term of a critical droplet of radius R(*), implying that the rates of droplet growth and shrinking essentially balance each other for droplet radius R=R(*). For heterogeneous nucleation at surfaces, the barrier is reduced by a factor depending on the contact angle. Using the definition of physical clusters based on the Fortuin-Kasteleyn mapping, the time dependence of the cluster size distribution is studied for quenching experiments in the kinetic Ising model and the cluster size ℓ(*) where the cluster growth rate changes sign is estimated. These studies of nucleation kinetics are compared to studies where the relation between cluster size and supersaturation is estimated from equilibrium simulations of phase coexistence between droplet and vapor in the canonical ensemble. The chemical potential is estimated from a lattice version of the Widom particle insertion method. For large droplets it is shown that the physical clusters have a volume consistent with the estimates from the lever rule. Geometrical clusters (defined such that each site belonging to the cluster is occupied and has at least one occupied neighbor site) yield valid results only for temperatures less than 60% of the critical temperature, where the cluster shape is nonspherical. We show how the chemical potential can be used to numerically estimate ΔF(*) also for nonspherical cluster shapes.

Thermodynamic properties of a symmetrical binary mixture in the coexistence region

A three-dimensional symmetric binary fluid is studied, as a function of temperature, in the two-phase (liquid-liquid) coexistence region via Monte Carlo simulations. Particular focus has been in the understanding of curvature-dependent interfacial tension, which is observed to vary as σ(R) = σ(∞)/[1+2(ℓ/R)(2)], implying that a Tolman length is zero in the limit R → ∞. The length ℓ is found to have a critical divergence the same as the correlation length, but its amplitude is significantly larger (ℓ ~/= 4ξ). Our findings hence imply that the barrier against homogeneous nucleation is significantly reduced (in comparison with the classical nucleation theory) in the critical region. We also report results for the critical behavior of the flat interfacial tension σ(∞) and the concentration susceptibility, as well as the amplitude ratios involving these thermodynamic quantities. Noting that the interatomic potential in our model is described by the Lennard-Jones form that decays faster that 1/r(3), all of our results for critical phenomena are expectedly consistent with the Ising universality class of three spatial dimensions.

Diffusive domain coarsening: early time dynamics and finite-size effects

We study the diffusive dynamics of phase separation in a symmetric binary (A + B) mixture with a 50:50 composition of A and B particles, following a quench below the demixing critical temperature, both in spatial dimensions d=2 and d=3. The particular focus of this work is to obtain information about the effects of system size and correction to the growth law via the appropriate application of the finite-size scaling method to the results obtained from the Kawasaki exchange Monte Carlo simulation of the Ising model. Observations of only weak size effects and a very small correction to scaling in the growth law are significant. The methods used in this work and information thus gathered will be useful in the study of the kinetics of phase separation in fluids and other problems of growing length scale. We also provide a detailed discussion of the standard methods of understanding simulation results which may lead to inappropriate conclusions.

Impact of negative line tension on the shape of nanometer-size sessile droplets

The sign and value of the line tension has been measured from the size dependence of the contact angle of nanometer-size sessile fullerene (C60) droplets on the planar SiO2 interface, measured with atomic force microscopy (AFM). Analysis according to the modified Young's equation indicates a negative line tension, with a magnitude between -10{-11} and -10{-10}  N/m, in good agreement with theoretical predictions. The experiments also indicate that droplets with contact area radii below 10 nm are in fact two-dimensional round terraces.

Domain coarsening in two dimensions: conserved dynamics and finite-size scaling

We present results from a study of finite-size effect in the kinetics of domain growth with conserved order parameter for a critical quench. Our observation of a weak size effect is a significant and surprising result. For diffusive dynamics, appropriate scaling analysis of Monte Carlo results obtained for small systems using a two-dimensional Ising model also shows that the correction to the expected Lifshitz-Slyozov law for the domain growth is very small. The methods used in this work to understand the growth dynamics should find application in other nonequilibrium systems with increasing length scales.

Method for wettability characterization based on contact line pinning

We demonstrate an efficient and reliable method for wettability characterization by determining the contact angle theta which a liquid-vapor interface makes with a solid wall. The purpose is to overcome the difficulties, related to the curvature of the liquid-vapor interface, which make measurements of theta rather uncertain, especially on the micro- and nanoscale. The method employs a specially designed slitlike channel in contact with a reservoir whereby the wettability of one of the slit walls is to be examined whereas the other (auxiliary) wall is separated by half into a lyophilic and a lyophobic part so as to pin the incoming fluid and fix the one end of the liquid-vapor interface. In the present work, the physical background of the method is elucidated theoretically while the method's applicability is demonstrated by molecular-dynamics simulation of a typical Lennard-Jones fluid, in contact with an atomistic wall. The wettability of the latter, as described by the corresponding contact angle theta, is accurately determined by variation of the liquid-wall interaction in a very broad interval.