4.3 Equilibrium form 65

By putting

P Eatt/Ecr,

it is possible to express the following relationship with Jackson’s factor:

(1 P) .

When Esl is small and the interaction energy between liquid and solid is small (i.e. g is small), the roughening transition temperature becomes lower. Through this treatment, PBC analysis, which was originally criticized as being an arbitrary qualitative theory, is now correlated energetically to interface roughness.

Considerable difficulties arise during the analysis of the PBCs of crystals with complicated crystal structure, such as garnet or organic compounds. Bennema and van der Eerden [4] suggested a connected net model to analyze complicated structures and to provide a correlation with the roughening transition of an interface. Using this model, it became possible for the first time to construct a unified relation between PBC analysis and the roughening transition, and the structural form, the growth forms, and the effect of growth conditions predicted from computer calculations [5]. In the model, the centers of gravity of the atoms and molecules that constitute the crystal are connected, and nets are obtained. Connected nets correspond to two-dimensional crystals, and therefore depict the same thing as an F face in PBC analysis. Therefore, the net has a corresponding roughening transition temperature.

If we define a non-dimensional temperature 2kT/ , the connected net becomes a crystal face (an F face) on which two-dimensional layer growth takes place when

R c 0 and R c (from the relationship between R, corresponding to the R value, and c of the connected net). An S face consists of only one PBC with zero edge free energy, and R c 0; and a K face has zero edge free energy in all directions, and again R c 0. In other words, an S face is a net with no connection, and a K face is not a connected net (Fig. 4.4). An anomaly with PBC theory exists here, in that direct PBC theory (D-PBC theory), which satisfies stoichiometric composition, is presumed, whereas the connected net model allows a partial PBC (P-PBC) model. The importance of the connected net model is that it allows both D- PBC and P-PBC theories, by connecting networks.

4.3Equilibrium form

A droplet of liquid, which has a random structure and thus isotropic properties, takes a true spherical form at equilibrium. This is because a sphere is the form with minimum surface energy in the case of an isotropic material. Gibbs considered that a crystal, which has a regular structure for which anisotropy is the

66 Morphology of polyhedral crystals

Figure 4.4. The net in the connected net model [4]. (a) In a P lattice the connected net is the square, (100). (b) In a bcc lattice the connected net is (110), which is fundamentally the same as the connected net in (a). (c) An fcc lattice. The connected nets are (100), square, and (111), hexagonal.

essential property, should, at equilibrium, take a form such that the total surface area times the surface free energy is at a minimum. A crystal in this state is called the equilibrium form, which is unique under given temperature and pressure conditions.

Starting from Gibbs’ concept [6] that the total surface energy times the surface area is at a minimum, the following concepts emerged:

(1)Curie’s concept [7], which considered that the normal growth rates of crystal faces are proportional to the surface free energies; and

4.3 Equilibrium form 67

Figure 4.5. Wulff’s polar diagram based on ref. [8]. The equilibrium form is obtained by

drawing inscribed lines at the cusps.

(2)Wulff’s plot, indicating that the equilibrium form is obtained by connecting inscribed lines drawn at cusps on a raspberry-shaped polar diagram, as shown in Fig. 4.5, on which points at distances proportional to the surface free energy from the center of a crystal are plotted.

A closed system is required in order to investigate the equilibrium form experimentally. In the experiment, droplets of a supersaturated solution in gelatin are dispersed using a spray, and the morphological changes of the crystals formed in the droplets enclosed in the gelatin are recorded. The final morphology, i.e. that after which no further morphological changes occur, can be assumed to represent the equilibrium form. Taking NaCl as an example, the initial morphology is of dendritic form, but after transformation it eventually takes a simple cubic form, which is regarded as the equilibrium form under the given conditions. When a mother