Crystal macroscopic growth rates

In the analysis of crystal growth, one is mainly interested in macroscopic features like crystal morphology and growth rate. Therefore, the time scale in question is rather slower than the time scale of phonon frequencies, and the deviation of atomic positions from the average crystalline lattice position can be neglected. A lattice model gives a sufiicient description for the crystal shapes and growth [3,34,35]. 

Experimental justification for specification of the angle at the point of three-phase contact comes from the results of Surek and Chalmers (139), which verify that a particular value of < )0 measured macroscopically can be associated with the crystal growth of a material in a specific crystallographic orientation and that < >o is roughly independent of growth rate. 

The preceding treatment is, undoubtedly, an oversimplification. For example, many diatomic molecules dissociate upon adsorption (e.g., H2, SiH, GeH). Each atom from the dissociated molecule then occupies its own distinct surface site and this naturally changes the rate law expression. When these types of details are accounted for, the Langmuir-Hinshelwood mechanism has been very successful at explaining the growth rates of a number of thin-film chemical vapor deposition (CVD) processes. However, more important, our treatment served to illustrate how crystal growth from the vapor phase can be related to macroscopic observables namely, the partial pressures of the reacting species. 

AFM is now utilized to relate microscopic measurement of step velocity to macroscopic face growth rates (Malkin et al. 1996). Such data can be collected at a very rapid rate but does require some familiarity with the technique and access to a research caliber AFM. Likewise, microcalorimetry may be utilized to extract crystal growth rates at a very rapid rate, provided the protein s heat of crystallization is sufficient to yield a measurable signal (Darcy and Wiencek, 1998). Both of these techniques can provide growth rates over a wide range of conditions within days, as opposed to months by more traditional video microscopy techniques. 

The very complex nature of the macroscopic and microscopic structures as they affect strength and the behavior of diamond abrasive/particles still requires extensive work to elucidate fully. However, from a crystallization point of view, gaining control over the crystallization behavior is the key to the production of optimal diamond abrasives. This, of course, may be achieved by choice and manipulation of the pressure/temperature conditions, source carbon structure and solvent/catalyst metal type, leading to control over nucleation and growth rates. 

Figure 9.14 Growth rate of small fragments of crystals after the process of secondary nucleatlon. The rate is compared with that of macroscopic crystals. Note the considerable dispersion in the rates Graph taken from Wang, PhD dissteration TL) Munchen.

The growth rate of fragments/secondary nuclei immediately after their formation is found to be generally lower than that of a macroscopic unhurt crystal of the same species (Figure 9.14). 

There is an additional phenomenon, namely random fluctuations in growth rates of the different crystals around a mean value. For example, one crystal will display several growth rates during its growth in the crystallizer for the time period fres however, there may be a mean, as shown in Figures 6.4.5(e) and (f). The governing macroscopic population balance equation for an MSIVIPR crystallizer, where /Zp) = 0, B = He = 0, may be written as... 

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