Interface equilibrium shape

The topic of capillarity concerns interfaces that are sufficiently mobile to assume an equilibrium shape. The most common examples are meniscuses, thin films, and drops formed by liquids in air or in another liquid. Since it deals with equilibrium configurations, capillarity occupies a place in the general framework of thermodynamics in the context of the macroscopic and statistical behavior of interfaces rather than the details of their molectdar structure. In this chapter we describe the measurement of surface tension and present some fundamental results. In Chapter III we discuss the thermodynamics of liquid surfaces. 

We noted in Section VII-2B that, given the set of surface tension values for various crystal planes, the Wulff theorem allowed the construction of fhe equilibrium or minimum firee energy shape. This concept may be applied in reverse small crystals will gradually take on their equilibrium shape upon annealing near their melting point and likewise, small air pockets in a crystal will form equilibrium-shaped voids. The latter phenomenon offers the possible advantage that adventitious contamination of the solid-air interface is less likely. 

In the case of supported metalhc particles, the construction is modified by introducing the adhesion energy (Wulff-Kaishew construction) [Henry, 1998]. The equilibrium shape is a Wulff polyhedron, which is truncated at the interface by an amount Ahs, according to the relation Ahs/hj = /3/(t where /3 is the adhesion energy of the crystal on the substrate. 

The Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal ... 

The equilibrium shape of a crystal is, as described above, a polyhedron where the size of the crystal facets is inversely proportional to their surface energy, ysg. In the present section we will consider other types of interfaces as well and we will show that the interface energies determine the equilibrium morphology of interfaces in general. 

We wish to prove by means of the Wulff construction (Section C.3.1) that the equilibrium shape of the grain boundary nucleus in Fig. 19.12 is indeed composed of two spherical-cap-shaped interfaces. 

H.M. Princen, The Equilibrium Shape of Interfaces, Drops and Bubbles. Rigid and Deformable Particles at Interfaces, in Surface and Colloid Science, E. Matijevic, Ed., Wiley-Interscience (1969), 1. (Analysis of a variety of shapes, including those around floating fluid or solid objects.)... 

The problem also occurs in other places in physics. One example is finding the energetically most favourable path that a moving object follows to get from one point to another under the influence of (conservative) external variables. An example from surface science is to find the spatial equilibrium shape of fluid interfaces under the constraint that the Interface is fixed at its extremities. 

Following the Wulff approach of the equilibrium shape of crystals, some authors have tried to include the presence of a substrate. The solution of this problem has been given by Kaishew [84] and Winterbottom [85]. It is known as the Wulff-Kaishew theorem. The crystal is now truncated at the interface by an amount Ahg, which is related to the adhesion energy of the crystal on the substrate (j ) and to the surface energy of the facet parallel to the interface (cr ) by the following relation ... 

However, these molecular dynamics calculations suffer some limitations the empirical nature of the potential (especially for the metal-support interaction) and the arbitrary separation between the metal-metal and metal-support interactions (the metal-metal potential is probably perturbed near the interface). Indeed, according to the type of potential used, very different results are obtained. In the case of Pd/MgO, a mean dilatation [91] or contraction [92] is observed. For finite-temperature molecular dynamics, the calculations are limited to very short times and it is not sure that the equilibrium shape is reached. As we have seen in the last section the cluster shape can be blocked for a long time on facetted metastable shapes. 

These results are in good agreement with the calculation of Molina and Hammer [79]. First, molecular oxygen reacts with CO, second, the fact that 1.5-nm particles are very active is compatible with the proposed model (see Fig. 3.17). Indeed, taking the truncated-octahedron equilibrium shape (see Fig. 3.18) observed by HRTEM with an aspect ratio of 0.6, the size giving the position of the edge in the second layer from the interface would be 1.4-1.6 nm. 

Of the three popular methods of v-determlnatlon the sessile drop Is the slowest, the pendant drop faster and the spinning drop the most rapid ( S). For commercial resin pairs the first two may require days before the drop reaches Its equilibrium shape. During this time there Is diffusion of the low molar mass fractions toward the interface gradually decreasing the value of the Interfacial tension coefficient (36). These two factors, the normal stress and the time scale, are generally responsible for the poor correlation between the predicted and measured droplet diameters in commercial blends. 

The process just described, by which a nonspherical drop would be driven toward the spherical equilibrium shape by pressure gradients associated with variations in interface curvature, is but one example of a large number of situations in which fluid motions are actually caused by pressure gradients that are produced by variations in the curvature of a fluid interface. 

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