Equilibrium shapes

The equilibrium shape of a macroscopic crystal is an old problem first addressed by Wulff [37], who showed the equilibrium shape at OK to be a polyhedron. At the equilibrium, the surface energy is given by the famous Wulff s theorem  

The transition to the bulk structure appears for several metals at relatively large sizes, 17,000 atoms for Ni [39] and about 50,000 atoms for Cu [42]. It is important to notice that the energy difference between these structures is often very small. This explains the very rapid fluctuations between different structures observed by HRTEM [43] that we call quasi-melting [44]. 

If the crystal is lying on a support, the equilibrium shape is modified by the interaction with the substrate. This problem has been solved independently by Kaichew [45] and by Winterbottom [46]. The equilibrium shape is expressed by the Wulff-Kaichew theorem represented by the following equation  

In this case, the Wulff shape is truncated at the interface by an amount Ah, which is proportional to the adhesion energy [3. The latter represents the work to separate the supported crystal from the substrate at an infinite distance, hg and 7s are the central distance and the surface energy of the facet parallel to the interface, respectively. In particular, this theorem shows that the stronger the particle-substrate interaction (given by / ) is the flatter is the supported particle. Equation (3.7) offers a simple way for determining the adhesion energy of a supported crystal from TEM pictures of supported particles observed in a profile view [47]. 

Another factor, which has a strong effect on the equilibrium shape of crystals is the adsorption. Indeed, from Gibbs we know that adsorption reduces the surface energy (the number of broken bonds decreases)  

FIGURE 6.17 Equilibrium crystal shape as described by WuUTs theorem in this case, yj 72. Redrawn with permission from N sen [2]. 

FIGURE 6J8 Hypothetical three-dimensional ciystal presenting the three main types of possible faces flat (F), step (S), kink (K) faces. Redrawn, with permission from ElweU and Scheel [28]. 

Gibbs notes that for macroscopic crystals, the free energy associated with the volume of the crystal will be larger than changes in free energy, due to departures from its equilibrium shape. For these crystals, their shape will depend on kinetic factors, which are affected by crystal defects, surface roughing, and impurities in the solvent. 

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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. 

Returning to equilibrium shapes, these have been determined both experimentally and by solution of the Young-Laplace equation for a variety of situations. Examples... 

The equilibrium shape of a liquid lens floating on a liquid surface was considered by Langmuir [59], Miller [60], and Donahue and Bartell [61]. More general cases were treated by Princen and Mason [62] and the thermodynamics of a liquid lens has been treated by Rowlinson [63]. The profile of an oil lens floating on water is shown in Fig. IV-4. The three interfacial tensions may be represented by arrows forming a Newman triangle ... 

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. 

The surface tensions for a certain cubic crystalline substance are 7100 = 160 ergs/cm, 7110 = 140 eigs/cm, and 7210 = 7120 = 140 ergs/cm. Make a Wulff construction and determine the equilibrium shape of the crystal in the xy plane. (If the plane of the paper is the xy plane, then all the ones given are perpendicular to the paper, and the Wulff plot reduces to a two-dimensional one. Also, 7100 = 7010. etc.)... 

An enlarged view of a crystal is shown in Fig. VII-11 assume for simplicity that the crystal is two-dimensional. Assuming equilibrium shape, calculate 711 if 710 is 275 dyn/cm. Crystal habit may be changed by selective adsorption. What percentage of reduction in the value of 710 must be effected (by, say, dye adsorption selective to the face) in order that the equilibrium crystal exhibit only (10) faces Show your calculation. 

There have been some studies of the equilibrium shape of two droplets pressed against each other (see Ref. 59) and of the rate of film Winning [60, 61], but these are based on hydrodynamic equations and do not take into account film-film barriers to final rupture. It is at this point, surely, that the chemistry of emulsion stabilization plays an important role. 

O. Shochet, K. Kassner, E. Ben-Jacob, S. G. Lipson, H. Miiller-Krumbhaar. Morphology transitions during non-equilibrium growth I. Study of equilibrium shapes and properties. Physica A 757 136, 1992. 

This shows the time evolution of the distribution over J when it was a (5-function at t = 0. The distribution spreads and shifts to J = 0, until it finally takes the equilibrium shape (pe- Gradual transformation of the distribution is typical for correlated change in J t) caused by weak collisions. 

The inner envelope of these surfaces defines the equilibrium shape. 

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 potential energy V of the elastomer is presumed to be given as a function of the atomic coordinates x (lwell-defined equilibrium shape, there must be equilibrium positions x for all atoms that are part of the continuous network. Expand the potential in a Taylor series about the equilibrium positions, and set the potential to zero at equilibrium, to obtain... 

The equilibrium shape of the droplet may be addressed within the framework of Gibbs free-energy arguments. Analysis and estimates are significantly eased by... 

For a reversible change at constant temperature and volume of both phases and for a constant number of moles of the components, the equilibrium shape can be... 

Here /, is the surface energy of the crystal surface i. The equilibrium shape of a crystal is thus a polyhedron where the area of the crystal facets is inversely proportional to their surface energy. Hence the largest facets are those with the lowest surface energy. 

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 ... 

Creep recovery response is due to freezing in local deformation of the polymer molecules when the polymer cools rapidly. The polymer molecules are frozen into shapes that distort their Gaussian spherical equilibrium shape. If the polymer is heated or allowed to relax over a very long time there will be dimensional changes as the polymer molecules assume their thermodynamic equilibrium states (Gaussian spherical equilibrium shape). 

Where are the nuclei . This is nob just a question of equilibrium shape as measured by n.m.r./ x-ray or neutron spectroscopy/ but also concerns what possible shapes the molecule can asscune as it interacts with its partner in general/ what flexibility it possesses. Flexibility is clearly a property of both small molecules and the protein binding sites. 

For formation or evaporation of a Hquid from vapor, the equilibrium shape of the Hquid is a spherical drop so the sphere approximation is more appropriate than for a solid particle, where we approximate it as a sphere to make the mathematics simple. 

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