Die Gleichgewichtsform von Kristallen und die Keimbildungsarbeit bei der Kristallisation

Abstract

Chap. 1 gives a historical review of most important papers, dealing with the equilibrium form of crystals and the free energy of formation of a critical nucleus.

The equilibrium form (Chap. 2) has been determined thermodynamically by means of the Gibbs-Curie-equation (Eq. 2.1) and the Gibbs-Wulff-theorem (Eq. 2.2). The kinetic derivation leads to the expression of Stranski and Kaischew (Eq. 2.5) concluding that the mean work of separation for the outermost lattice-layer of all three-dimensional equilibrium crystal faces is of the same value and decreases with increasing supersaturation. This is valid also for the two-dimensional nucleus (Eq. 2.7–2.10).

One can estimate the free energy of nucleation following Gibbs and Volmer (Eq. 2.13) by means of the specific free surface energy and the supersaturation or from the amount of the work of separation of a single unit and the mean work of separation following Stranski and Kaischew (Eq. 2.14).

In a one-component system (Chap. 3) it has to be distinguished between crystals with non-polar-bond, ionic crystals and crystals with a mixed type of bond. The surface energies can be computed vectorial, if the radius of action of the binding-force is limited. With that method the coordination numbers for the single lattice-layer can be evaluated.

The equilibrium forms of crystals with non-polar bonds (Chap. 3a) have been summarized in Table 1a for the simple-cubic, face-centered-cubic, body-centered-cubic, diamond-, β-tin- and Se-Te-lattice and also for the hexagonal close packing of spheres or ellipsoids. It is distinguished between the equilibrium forms for ϕ₁>0, ϕ₂, ϕ₃ ... = 0; ϕ₁, ϕ₂>0; ϕ₃, ϕ₄ ... = 0; ... (ϕ_i = work of separation for the ith-nearest neighbour). The number of the equilibrium-form-faces increases with the radius of action of the binding-force. For jodine it is shown that the faces of the equilibrium-form can be calculated by the vector products of bond vectors. Evidence for the Stranski and Kaischew-conclusion can be gained for crystals with non-polar bonds through the bond vectors: Every face of the equilibrium form of an ideal crystal grows via the two-dimensional nucleus. The surface energies for crystals of the NaCl-type (Chap. 3b) are calculated by means of the NaCl-octupol-lattice (Fig. 3), which permits to evaluate the surface energies of coarsened {h k l}-surfaces. In Table 1 b the faces of equilibrium form of the ionic crystals are compiled.

On the basis of arsenolite, which crystallizes in the diamond-lattice, it was shown in Chap. 3 c, that the work of separation from special neighbours may be less than zero for crystals with polar intramolecular bonds. In this way it is possible to interpret the growth-form, which consists of {111} and {100}.

Concerning surface structures (Chap. 4) it may be distinguished between roughness and coarsening. A small number of surface vacancies (n _L ) and of Ad-units (n _Ad) on the surface compared to the total number (n _o) will be characterized as roughness. The thermodynamic treatment of coarsening is given for the model of the two dimensional simple square lattice. The free ledge energy ρ_h1 (Eq. 4.12; Fig. 10) and the degree of equilibrium coarsening x _g (Eq. 4.13; Fig. 10) are computed. If the ledge energy increases with coarsening (crystals with non polar bonds), x _g will also increase with temperature. For crystals with polar bonds the ledge energy of non-equilibriumform-ledges decreases with the coarsening and x _g with the temperature.

For the two-component systems (Chap. 5) the equilibrium forms and the free energies of nucleation have been investigated for a one-component crystal on a heterogeneous substrate, for a one-component crystal with adsorption of impurities on the faces and for a mixed crystal.

On the basis of Stranski-Krastanow-mechanism in chapter 5a a simple cubic crystal on a structureless surface in a heterogeneous substrate is considered. The crystal is situated on a flat substrate (Index 1K, Fig. 12), in a concave edge (Index 2K), in a concave corner (Index 3K), at a step (Index StK; Fig. 13) or in a concave corner of two crossing steps (Index DStK). The relations for the nucleus dimension (ledge dimension a _ik, b _ik, c _ik) and the free energy of formation of a nucleus (A _ik, i = 1,2,3, St, DSt; Eq. 5.2–5.6) are given. It is shown, that in this case, too, the mean work of separation is the same for all faces.

The calculations for the two-dimensional free energy of nucleation (Index K2) lead to four districts (Fig. 14), where either two-dimensional nuclei grow in monomolecular layer (A), or only three-dimensional nuclei (B), or two- and three-dimensional nuclei (C), or only two-dimensional nuclei (D) are possible. The dependence of the free energy of nucleation and of the effect of decoration (F _i = A _ik/A _K; A _K = free energy of nucleation, homogeneous) upon the contact angle α is investigated for a spherical nucleus (Eq. 5.8a, A.1–A.4; Fig. 17), a cube (Eq. 5.10, 5.13, 5.14; Fig. 17, 19, 20), an octahedron (Eq. 5.11; Fig. 18) and a cube-octahedron (Eq. 5.12; Fig. 18).

The adsorption of impurities (Chap. 5b) is treated by means of the Stranski-model (Fig. 21). Equations for the specific free surface energies (Eq. 5.18) and the specific free ledge energies (Eq. 5.19) for the simple cube lattice in dependence on the coverage degree ϑ are given. By these equations the conditions (5.20) of the stability of the single faces are determined. The dependence of coarsening (Eq. 5.22; Fig. 26) and its influence on the free-ledge-energy (Eq. 5.21; Fig. 27) and the adsorption-isotherms (Eq. 5.23; Fig. 25) have been investigated.

For the determination of the equilibrium form of mixed crystals (Chap. 5c) the conditions of a regular mixture are set up. The free energy of the crystal (Eq. 5.26) with N₀ surface sites, n surface layers and N_v units in the volume phase, the composition of the surface layers (Eq. 5.30, 5.32; Fig. 32), and the free surface energy (Eq. 5.34, 5.35; Fig. 28) are calculated. In the case of the cubic face centered lattice it is shown, that at interaction between first and second next neighbours besides {100}, {110} and {111} (the faces of the equilibrium form in a one-component-crystal) also {210}, {310}, {311}, {321}, {331}, {531}, {731} and {931} are present in the equilibrium form of a mixed crystal (Fig. 33). In a regular mixture not only the composition of the surface layers with unoccupied coordination sites but also the next lattice-layers are different from the composition of the volume (Fig. 32). If 2ϕ_AB ≧ ϕ_AA + ϕ_BB, in all surface layers enrichment of the surface active component happens. In the opposite case in a part of the layers a defect of the surface-active-component occurs. While for 2ϕ_AB ≦ ϕ_AA + ϕ_BB the edges and corners are sharp in the equilibrium form, they are rounded off for 2ϕ_AB> >ϕ_AA+ϕ_BB, because besides the above mentioned faces all other ones — if only in small extension — are present in the equilibrium form.

In the appendix (A.I–A.IV) some mathematical derivations are compiled.