Geometric view of the thermodynamics of adsorption at a line of three-phase contact

We consider three fluid phases meeting at a line of common contact and study the linear excesses per unit length of the contact line (the linear adsorptions Lambda_i) of the fluid's components. In any plane perpendicular to the contact line, the locus of choices for the otherwise arbitrary location of that line that makes one of the linear adsorptions, say Lambda_2, vanish, is a rectangular hyperbola. Two of the adsorptions, Lambda_2 and Lambda_3, then both vanish when the contact line is chosen to pass through any of the intersections of the two corresponding hyperbolas Lambda_2 = 0 and Lambda_3 = 0. There may be two or four such real intersections. It is required, and is confirmed by numerical examples, that a certain expression containing \Lambda_{1(2,3)}, the adsorption of component 1 in a frame of reference in which the adsorptions Lambda_2 and Lambda_3 are both 0, is independent of which of the two or four intersections of Lambda_2 = 0 and Lambda_3 = 0 is chosen for the location of the contact line. That is not true of Lambda_{1(2,3)} by itself; while the adsorptions and the line tension together satisfy a linear analog of the Gibbs adsorption equation, there are additional, not previously anticipated terms in the relation that are required by the line tension's invariance to the arbitrary choice of location of the contact line. The presence of the additional terms is confirmed and their origin clarified in a mean-field density-functional model. The additional terms vanish at a wetting transition, where one of the contact angles goes to 0

Thermodynamics of heterogeneous crystal nucleation in contact and immersion modes

One of most intriguing problems of heterogeneous crystal nucleation in droplets is its strong enhancement in the contact mode (when the foreign particle is presumably in some kind of contact with the droplet surface) compared to the immersion mode (particle immersed in the droplet). Many heterogeneous centers have different nucleation thresholds when they act in contact or immersion modes, indicating that the mechanisms may be actually different for the different modes. Underlying physical reasons for this enhancement have remained largely unclear. In this paper we present a model for the thermodynamic enhancement of heterogeneous crystal nucleation in the contact mode compared to the immersion one. To determine if and how the surface of a liquid droplet can thermodynamically stimulate its heterogeneous crystallization, we examine crystal nucleation in the immersion and contact modes by deriving and comparing with each other the reversible works of formation of crystal nuclei in these cases. As a numerical illustration, the proposed model is applied to the heterogeneous nucleation of Ih crystals on generic macroscopic foreign particles in water droplets at T=253 K. Our results show that the droplet surface does thermodynamically favor the contact mode over the immersion one. Surprisingly, our numerical evaluations suggest that the line tension contribution to this enhancement from the contact of three water phases (vapor-liquid-crystal) may be of the same order of magnitude as or even larger than the surface tension contribution

Thermodynamics of heterogeneous crystal nucleation in contact and immersion modes

One of most intriguing problems of heterogeneous crystal nucleation in droplets is its strong enhancement in the contact mode (when the foreign particle is presumably in some kind of contact with the droplet surface) compared to the immersion mode (particle immersed in the droplet). Many heterogeneous centers have different nucleation thresholds when they act in contact or immersion modes, indicating that the mechanisms may be actually different for the different modes. Underlying physical reasons for this enhancement have remained largely unclear. In this paper we present a model for the thermodynamic enhancement of heterogeneous crystal nucleation in the contact mode compared to the immersion one. To determine if and how the surface of a liquid droplet can thermodynamically stimulate its heterogeneous crystallization, we examine crystal nucleation in the immersion and contact modes by deriving and comparing with each other the reversible works of formation of crystal nuclei in these cases. As a numerical illustration, the proposed model is applied to the heterogeneous nucleation of Ih crystals on generic macroscopic foreign particles in water droplets at T=253 K. Our results show that the droplet surface does thermodynamically favor the contact mode over the immersion one. Surprisingly, our numerical evaluations suggest that the line tension contribution to this enhancement from the contact of three water phases (vapor-liquid-crystal) may be of the same order of magnitude as or even larger than the surface tension contribution

Monte Carlo Methods for Estimating Interfacial Free Energies and Line Tensions

Excess contributions to the free energy due to interfaces occur for many problems encountered in the statistical physics of condensed matter when coexistence between different phases is possible (e.g. wetting phenomena, nucleation, crystal growth, etc.). This article reviews two methods to estimate both interfacial free energies and line tensions by Monte Carlo simulations of simple models, (e.g. the Ising model, a symmetrical binary Lennard-Jones fluid exhibiting a miscibility gap, and a simple Lennard-Jones fluid). One method is based on thermodynamic integration. This method is useful to study flat and inclined interfaces for Ising lattices, allowing also the estimation of line tensions of three-phase contact lines, when the interfaces meet walls (where "surface fields" may act). A generalization to off-lattice systems is described as well. The second method is based on the sampling of the order parameter distribution of the system throughout the two-phase coexistence region of the model. Both the interface free energies of flat interfaces and of (spherical or cylindrical) droplets (or bubbles) can be estimated, including also systems with walls, where sphere-cap shaped wall-attached droplets occur. The curvature-dependence of the interfacial free energy is discussed, and estimates for the line tensions are compared to results from the thermodynamic integration method. Basic limitations of all these methods are critically discussed, and an outlook on other approaches is given