Crystal growth rate control

Because interface reaction and mass/heat transfer are sequential steps, crystal growth rate is controlled hy the slowest step of interface reactions and mass/heat transfer. For a crystal growing from its own melt, the growth rate may be controlled either by interface reaction or heat transfer because mass transfer is not necessary. For a crystal growing from a melt or an aqueous solution of different composition, the growth rate may be controlled either by interface reaction or mass transfer because heat transfer is much more rapid than mass transfer. Different controls lead to different consequences, including the following cases ... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

One-dimensional crystal growth at constant growth rate The crystal growth rate may be controlled by factors other than the diffusion process itself. In such a case, the growth rate may be constant. Assume constant D and uniform initial melt. The diffusion problem can be described by the following set of equations ... 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

No study has been made to discover which of the several resistances is important, but a simple rate equation can be written which states that the rate of the over-all process is some function of the extent of departure from equilibrium. The function is likely to be approximately linear in the departure, unless the intrinsic crystal growth rate or the nucleation rate is controlling, because the mass and heat transfer rates are usually linear over small ranges of temperature or pressure. The departure from equilibrium is the driving force and can be measured by either a temperature or a pressure difference. The temperature difference between that of the bulk slurry and the equilibrium vapor temperature is measured experimentally to 0.2° F. and lies in the range of 0.5° to 2° F. under normal operating conditions. 

The structure of a experimental fluidized bed crystallizer (FBC) is shown in Fig. 12.4, where the crystallizer is actually a universal equipment for the measurement of crystal-growth rate. The solution enters the FBC at its bottom, and leaves the FBC by overflow. All the other parts of the experimental system are the same as shown in Fig. 12.3, and so are not shown in Fig. 12.4. The operation procedure for the FBC is the same as for the ISC. For convenience of comparison, the corresponding conditions, temperature and concentration of the solution, operated in the ISC and the FBC are rigorously controlled to be the same, with the deviation of the operating temperature no greater than 0.05 °C. 

As described in previous chapters in Part II, liquid-continuous impinging streams (LIS) has the feature of very efficient micromixing and can provide a uniform and controllable supersaturation environment for the crystallization process, favoring the production of uniformly large-sized crystalline also, it has been proved experimentally that, to an extent, LIS can enhance crystal-growth rate. For industrial application, Wu [237] designed and patented the impinging stream crystallizer, the structure of which is shown in Fig. 17.3. 

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. 

Early investigators (N3, N4) assumed that crystal growth was controlled by solute diffusion and that the solution contacting the crystal surface was saturated. Berthoud (B4) was the first to suggest that the rate of the surface reaction must be taken into account. Valeton (VI) later added further support to this theory, and the following equations were developed for rate of crystal growth ... 

The size of the crystalites produced will depend on the nucleation rate and crystal growth rate at the solidification front. Both are controlled by the local supersaturation. The rate expressions for homogeneous nucleation (equation (6.15)) and heterogeneous nucleation (equation... 

FIG. 146. Temperature schedule of controlled crystallization a — crystal growth rate, h — nucleation rate, c — temperature schedule (after Pavlushkin, 1970). 

Note. A small response could indicate that other system properties were controlling (i.e., inherent crystal growth rate or nucleation rate). A large response would indicate sensitivity to secondary nucleation and/or crystal cleavage and require additional experimentation and evaluation of scale-up requirements. The laboratory results should be evaluated relative to each other since scale-up can be expected to make additional changes in PSD, especially when a large response is experienced in these simple experiments. 

Goals. To demonstrate the impact of controlled crystal growth rate on crystal morphology Issues. Crystal morphology, continuous crystallization... 

Continuous crystallization with control of the crystal growth rate can significantly improve the crystal... 

As shown in Fig. 10-11, the occluded residual solvent was found to be a sti ong function of time constants for crystal growth rate/nucleation rate over the entire range of parameters studied. Specifically, it was shown that the higher the ratio of (fi//xo)/(G/4> which means more rapid nucleation rate, the higher the residual solvent, and vice versa. This finding cleai ly shows the importance of properly controlling the nucleation rate and the crystal growth rate in the process. 

The crystal growth rates of PVDF, PA-6, and POM amount to at least lOpm/min in the temperature range where their crystallization steps occur (6,52,67). A dispersed particle, therefore, once nucleated, crystallizes promptly and the primary rather than the secondary nucleation is the rate-controlling factor of the crystallization kinetics of the dispersed phase. Thus, the crystallization temperatures as observed in the DSC-cooling run agree roughly with the nucleation temperatures. 

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. 

Looking at Eqs. (2.50) and (2.51), we see that when kd ki, the crystal growth rate will be diffusion controlled and Kg = kd. When ki < kd, the crystal growth rate will be controlled by the rate of solute incorporation into the crystal. 

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