Crystal growth transport-controlled

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

We define the linear growth rate Vg as the linear velocity of displacement of a crystal face relative to some fixed point in the crystal. vg may be known as a function of c and c , derived from the theory of transport control, and as a function of c and cs as well, derived from the theory of surface control. Then c may be eliminated by equating the two mathematical expressions... 

From the above statements it follows that it should be possible to derive the growth kinetics and calculate the growth rate of uncontaminated electrolyte crystals when the following parameters are known molecular weight, density, solubility, cation dehydration frequency, ion pair stability coefficient, and the bulk concentration of the solution (or the saturation ratio). If the growth rate is transport controlled, one shall also need the particle size. In table I we have made these calculations for 14 electrolytes of common interest. For the saturation ratio and particle size we have chosen values typical for the range where kinetic experiments have been performed (29,30). The empirical rates are given for comparison. 

Dendritic. In electrodeposited films, dendritic grains result from mass-transport-controlled growth, and the individual crystals may vary in shape. 

The overall rate of growth depends upon the slowest step in this sequence. Crystal growth may be controlled either by the transport processes or by the chemical reaction at the surface. The mechanism of growth and the resulting crystal morphology... 

The adsorption of ions on iron oxides regulates the mobility of species in various parts of the ecosystem (biota, soils, rivers, lakes, oceans) and thereby their transport betv een these parts. Examples are the uptake of plant nutrients from soil and the movement of pesticides and other pollutants from soils into aquatic systems. In such environments various ions often compete with each other for adsorption sites. Adsorption is the essential precursor of metal substitution (see Chap. 3), dissolution reactions (see Chap. 12) and many interconversions (see Chap. 14). It also has a role in the synthesis of iron oxides and in crystal growth. In industry, adsorption on iron oxides is of relevance to flotation processes, water pollution control and waste and anticorrosion treatments. 

Nucleation is necessary for the new phase to form, and is often the most difficult step. Because the new phase and old phase have the same composition, mass transport is not necessary. However, for very rapid interface reaction rate, heat transport may play a role. The growth rate may be controlled either by interface reaction or heat transport. Because diffusivity of heat is much greater than chemical diffusivity, crystal growth controlled by heat transport is expected to be much more rapid than crystal growth controlled by mass transport. For vaporization of liquid (e.g., water vapor) in air, because the gas phase is already present (air), nucleation is not necessary except for vaporization (bubbling) beginning in the interior. Similarly, for ice melting (ice water) in nature, nucleation does not seem to be difficult. 

Figure 1-11 Interface- or transport-controlled crystal growth 51...

One-Dimensional, Diffusion-Controlled Crystal Growth. Neglecting bulk convection leads to an idealized picture of diffusion-controlled solute transport of a dilute binary alloy with the solute composition c0 far from an almost flat melt-crystal interface located at z = 0 (I, 21). 

All seven steps require time, resulting in a rate of incorporating clusters into the growing crystal surface, which is called crystal growth kinetics. The following two sections consider translation of such a rate into a macroscopic equation for correlation and prediction. It is difficult to say which of the steps control the process, or even if the conceptual picture is valid. However, the first step—species transport to the solid surface—is well established and a brief description is given in Section 3.2.1.2. 

One of the most controversial topics in the recent literature, with regard to partition coefficients in carbonates, has been the effect of precipitation rates on values of the partition coefficients. The fact that partition coefficients can be substantially influenced by crystal growth rates has been well established for years in the chemical literature, and interesting models have been produced to explain experimental observations (e.g., for a simple summary see Ohara and Reid, 1973). The two basic modes of control postulated involve mass transport properties and surface reaction kinetics. Without getting into detailed theory, it is perhaps sufficient to point out that kinetic influences can cause both increases and decreases in partition coefficients. At high rates of precipitation, there is even a chance for the physical process of occlusion of adsorbates to occur. In summary, there is no reason to expect that partition coefficients in calcite should not be precipitation rate dependent. Two major questions are (1) how sensitive to reaction rate are the partition coefficients of interest and (2) will this variation of partition coefficients with rate be of significance to important natural processes Unless the first question is acceptably answered, it will obviously be difficult to deal with the second question. 

The next step is to produce nearly perfect single-crystal boules of silicon from the ultrapure polycrystalline silicon. Many techniques have been developed to accomplish this, and they all rely on a similar set of concepts that describe the transport process, thermodynamically controlled solubility, and kinetics [8]. Three important methods are the vertical Bridgman-Stockbarger, Czochralski, and floating zone processes, fully described in Fundamentals of Crystal Growth by Rosenberger [9]. 

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