Equilibrium crystallization growth theories

In order to fully understand such technologically important phenomena as near-equilibrium crystal growth and homogeneous nucleation, it is necessary to have detailed information as to the microscopic properties of the equilibrium interface between a crystal and its melt. Unfortunately, there is an almost total lack of experimental data on such systems as the interface lies between two condensed phases of similar density, making study difficult. Thus, computational methods, such as computer simulation or density functional theory, go beyond their usual roles in the interpretation of experimental data and become important in determining the generic phenomenology of such systems. 

The layout of this article is as follows Section 2 considers the equilibrium aspects of the crystals whilst Sects. 3 and 4 explain the growth theories, divided into nucleation and non-nucleation theories, respectively. Finally, Sect. 5 provides an overview and suggests future lines of investigation. 

In the next section we describe a very simple model, which we shall term the crystalline model , which is taken to represent the real, complicated crystal. Some additional, more physical, properties are included in the later calculations of the well-established theories (see Sect. 3.6 and 3.7.2), however, they are treated as perturbations about this basic model, and depend upon its being a good first approximation. Then, Sect. 2.1 deals with the information which one would hope to obtain from equilibrium crystals — this includes bulk and surface properties and their relationship to a crystal s melting temperature. Even here, using only thermodynamic arguments, there is no common line of approach to the interpretation of the data, yet this fundamental problem does not appear to have received the attention it warrants. The concluding section of this chapter summarizes and contrasts some further assumptions made about the model, which then lead to the various growth theories. The details of the way in which these assumptions are applied will be dealt with in Sects. 3 and 4. 

Having discussed some equilibrium properties of a crystal, we now outline and contrast the bases of the growth theories which will be dealt with in more detail later. The theories may be broadly split into two categories equilibrium and kinetic. The former [36-42] explain some features of the lamellar thickness, however the intrinsic folding habit is not accounted for. Therefore, at best, the theory must be considered to be incomplete, and today is usually completely ignored. We give a brief summary of the approach and refer the interested reader to the original articles. The kinetic theories will be the topic of the remainder of the review. 

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... 

The following discussion is excerpted from Mullin (1993) and Elwell and Scheel (1975). Diffusional boundary theory is well-established (see e.g., Bird et al., 1960) and the concept of a boundary unstirred layer was introduced a century ago. Noyes and Whitney (1897) proposed that the change in the rate of crystal growth (dm/dt) was controlled by diffusion from the bulk concentration to the crystal (equilibrium) interface. 

The BCF theory discusses crystal growth in terms of the physical features of the molecular processes rather than in terms of the chemical changes which occur. If the adsorption interaction between the bulk ions and the crystal surface is slow, this may be the rate-determining step and the surface layer or adsorption layer may be considered to be in equilibrium with the crystal. In this situation, it is the chemical reaction between the solvated ions and the crystal surface which determines the rate of growth. 

Theories that assume equilibrium among atoms at various surface sites and among different adsorbates have been successful in explaining the nature of evaporation [39], crystal growth [40], and adsorption [41], in many systems. 

Growth theories can be divided into equilibrium theories and kinetic theories. Equilibrium theories explain some features of the crystal thickness. They predict the existence of two minima in free energy, one at a finite crystal thickness and the other at infinite thickness. The crystal thickness associated with the... 

CAI s that were once molten (type B and compact type A) apparently crystallized under conditions where both partial pressures and total pressures were low because they exhibit marked fractionation of Mg isotopes relative to chondritic isotope ratios. But much remains to be learned from the distribution of this fractionation. Models and laboratory experiments indicate that Mg, O, and Si should fractionate to different degrees in a CAI (Davis et al. 1990 Richter et al. 2002) commensurate with the different equilibrium vapor pressures of Mg, SiO and other O-bearing species. Only now, with the advent of more precise mass spectrometry and sampling techniques, is it possible to search for these differences. Also, models prediet that there should be variations in isotope ratios with growth direction and Mg/Al content in minerals like melilite. Identification of such trends would verify the validity of the theory. Conversely, if no correlations between position, mineral composition, and Mg, Si, and O isotopic composition are found in once molten CAIs, it implies that the objects acquired their isotopic signals prior to final crystallization. Evidence of this nature could be used to determine which objects were melted more than once. 

Fig. 5.34 Schematic of the growth rate in the three regimes of the Lauritzen Hoffman theory. Here AT = 7 J, - T where J 2, is the equilibrium melting point and Tc is the crystallization temperature,...

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