Theories of Crystal Growth

32) provides a simple illustration of the growth of a face, however, it does not help us with two fundamental questions. 

Where do the steps come from and what is the rate controlling factor in determining the crystal growth rate The goal of crystal growth theories is to try to answer these two questions. 

An application of nucleation theory in two dimensions yields the expression for the critical size given below 

Employing nucleation theory it is possible to derive an expression for the rate of formation of critical size nuclei per unit surface area per unit time. A complete derivation can be found in Ohara and Reid (1973). A simplified version of the rate expression follows. 

Once a surface nuclei is formed, the next question is how does the nuclei spread to form a complete layer. The simplest crystal growth theory assumes that when a surface nuclei is formed, it spreads across the surface at an infinite velocity. The surface must then await the formation of another surface nuclei. Since the ratedetermining step in this model is the formation of a surface nuclei, the growth rate of the crystal can be expressed as 

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The theory of crystal growth accordingly starts usually with the assumption that the atoms in the gaseous, diluted, or hquid mother phase will have a tendency to arrange themselves in a regular lattice structure. We ignore here for the moment the formation of poly crystalhne solids. In principle we should start with the quantum-mechanical basis of the formation of such lattice structures. Unfortunately, however, even with the computational effort of present computers with a performance of about 100 megaflops... 

As with nucleation, classical theories of crystal growth 3 20 2135 40-421 have not led to working relationships, and rates of crystallisation are usually expressed in terms of the supersaturation by empirical relationships. In essence, overall mass deposition rates, which can be measured in laboratory fluidised beds or agitated vessels, are needed for crystalliser design, and growth rates of individual crystal faces under different conditions are required for the specification of operating conditions. 

Comprehensive reviews of theories of crystal growth have been presented by Garside(43), Nielsen(44), Pamplin145 and Kaldis and Scheei/46 . 

All real crystals deviate more or less from their equilibrium habits since all grow at finite velocities Rj. Hartman and Betmema (4) and Hartman (5.61 showed how the empirical law of Dotmay-Harker can be explained on the basis of current molecular theories of crystal growth. The energy required to split a crystal along the plane A--B parallel to the plane (hkl) is the sum... 

F. C. Frank, On the kinetic theory of crystal growth and dissolution process, in Growth and Perfection ofGrystals, eds. R. H. Doremus, B. W. Roberts, and V. Turnbull, New York, John Wiley Sons, 1958... 

Jackson, K. A. (1975) Theory of Crystal Growth, in Treatise on Solid State Chemistry, Vol. 5 (Ed. N.B. Hannay), Plenum Press, New York Janek, J., Majoni, S. (1994) Personal communication... 

The morphology of a microcrystal depends in a complex way on the thermodynamics and kinetics that determine the stabilities of the faces and their growth. Currently, an exhaustive theory of crystal growth in different atmospheres is not available nevertheless, a reasonable prediction of surface morphology based on the bulk crystalline structure of the solid is possible in many cases. 

Half-crystal position (kink position) — The term was introduced into the theory of crystal growth simultaneously and independently by W. Kossel [i] and - St ran-ski [ii,iii], who were the first to realize the necessity of a close consideration of the elementary acts of attachment and detachment of single particles (atoms, ions, or molecules) to and from a crystal surface. 

He started to work at the Chemical Faculty of Sofia University where he became a professor and the head of the Department of Physical Chemistry, in 1947. Kaishev founded the Institute of Physical Chemistry of the Bulgarian Academy of Sciences in 1958, and helped to establish the Central Laboratory of Electrochemical Power Sources [i]. Kaishev started to collaborate with - Stran-ski in Berlin in 1931 [iii] and became his assistant in Sofia in 1933. They laid the fundamentals of the crystal growth theory. They proposed the first kinetic theory of the two-dimensional nucleation and growth. The spiral type growth during electrocrystallization was first observed by Kaishev on silver [iii]. On the history of the creation of the molecular-kinetic theory of crystal growth see [iv]. 

Berlin, where he received his Ph.D. in 1925. In 1926 he habilitated for physical chemistry in Sofia, where he became extraordinary Professor in 1926, and full Professor in 1937. In 1930/31 he worked as a Rockefeller scholar at the TH Berlin, and in 1935/36 at the Technical Institute of the Ural in Sverdlovsk, USSR (now Ekaterinburg, Russia). He was Guest Professor in Breslau from 1941 to 1944 and then went to the Kaiser-Wilhelm-Institut of Physical Chemistry and Electrochemistry in Berlin. From 1945 until 1963 he was Professor at the Technical University of Berlin (West-Berlin). Stranski made fundamental contributions to the theory of crystal growth and surface chemistry (see - Stranski-Krastanov heteroepitaxial metal deposition) and [ii]. 

This chapter is concerned with the growth of crystals from aqueous systems. The approach is to present the fundamental concepts which are vital to the understanding of growth processes. This is followed by a discussion of the most important techniques available to the experimentalist and how the growth may be interpreted using current theories of crystal growth. Finally, the effects of impurities on the growth kinetics are examined. 

According to Stranski s theory of crystal growth, each component of the deposited product (D) must first find its way to a suitable site at the edge of a lattice plane before it can be accepted into the lattice. The additional overpotential observed when this reaction is inhibited is known as crystallization overpotential. 

Surface-growth coefficients. Much research on the growth of crystals by the interfacial reaction has been done and reported in standard monographs on crystallization. Although a coherent theory of crystal growth has evolved, numerical data on of a kind that can be used in design are scarce. 

Cahn, J. W. (i960). Theory of crystal growth and interface motion in crystalline materials. Acta Metall. 8, 554-62. [105, 108]... 

Hillig, W. Turnbull, D. (1956). Theory of crystal growth in undercooled pure liquids. J. Chem. Phys. 24, 914. [114]... 

Pfeiffer, H. Klupsch, T. Haubenreiber. W. Micrscopic Theory of Crystal Growth. Akadeinie-Verlag Berlin. 1989. ... 

So far there is no general acceptance of the surface energy theories of crystal growth, since there is little quantitative evidence to support them. These theories, however, still continue to attract attention, but their main defect is their failure to explain the well-known effects of supersaturation and solution movement on the crystal growth rate. 

Two processes are involved in the layer growth of crystals, viz. the generation of steps at some source on the crystal face followed by the movement of layers across the face. Consideration of the movement of macrosteps of unequal distance apart (BCF theory considers a regular distribution of monoatomic steps) led Frank (1958) to develop a kinematic theory of crystal growth. The... 

The diffusion theories of crystal growth cannot yet be reconciled with the adsorption layer and dislocation theories. It is acknowledged that the diffusion theories have grave deficiencies (they cannot explain layer growth or the faceting of crystals, for example), yet crystal growth rates are conveniently measured and reported in diffusional terms. The utilization of the mathematics of mass transfer processes makes this the preferred approach, from the chemical engineer s point of view at any rate, despite its many limitations. 

Frank, F.C. (1958) Kinematic theory of crystal growth and dissolution processes. In Doremus, Roberts and Turnbull (1958), 411-420. 

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