Kinetics, crystal growth transport

The scope of kinetics includes (i) the rates and mechanisms of homogeneous chemical reactions (reactions that occur in one single phase, such as ionic and molecular reactions in aqueous solutions, radioactive decay, many reactions in silicate melts, and cation distribution reactions in minerals), (ii) diffusion (owing to random motion of particles) and convection (both are parts of mass transport diffusion is often referred to as kinetics and convection and other motions are often referred to as dynamics), and (iii) the kinetics of phase transformations and heterogeneous reactions (including nucleation, crystal growth, crystal dissolution, and bubble growth). 

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. 

At least two resistances contribute to the kinetics of crystal growth. These resistances apply to (1) integration of the crystalline unit (e.g., solute molecules) into the crystal surface (i.e., lattice), and (2) molecular diffusion or bulk transport of the unit from the surrounding solution to the crystal surface. As aspects of molecular diffusion and mass transfer are covered elsewhere, the current discussion will focus only on surface incorporation. 

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]. 

Because of the importance of microstructure on dielectric and ferroelectric properties, the transformation pathway associated with conversion of the amorphous film into the crystalline state has been studied extensively. The basic mechanism involved is one of nucleation and growth, although the formation of intermediate phases that can impact the thermodynamic driving forces associated with the transformation frequently occurs. " Another key aspect of CSD films is that crystallization occurs well below the melting point of the materials. Therefore, compared to standard mixed-oxide processing of bulk materials, the thermodynamic driving forces associated with the transformation are much greater and the kinetics of mass transport are much less. 

Apart from the dimensions of critical nuclei under varying conditions, it is also important to consider the kinetics of nucleation, especially relative to the kinetics of crystal growth. The nucleation rate (/) is commonly estimated by means of transition state theory. With the aid of the best available values for transport properties of undercooled water,it has been estimated that in the neighbourhood of —40°C, J increases rapidly with decreasing temperature, by about a factor 20 per degree nucleation is thus a well-defined event that is hardly affected by the rate of cooling. ... 

The kinetics of growth processes will be dependent on the slowest step in the following series of events (Fig. 3.5) (i) bulk transport of ions or molecules to the crystal surface (diffusion), (ii) dehydration of ions at the surface, (iii) adsorption of ions or molecules at the surface, (iv) two-dimensional diffusion across the surface to active sites,... 

Singh and Coble (406) have reported epitaxial growth of UO single crystal on a uranium dioxide substrate, using chlorine gas as transport agent, and discussed the kinetics of the crystal growth. 

Thus, it appears from the evidence cited that transport by diffusion in the liquid layer is not the rate-limiting step in zeolite crystal growth, but that the incorporation of solute by surface integration kinetics may well be. Data from [22] suggests that a first order surface reaction is the rate limiting step for silicalite crystal growth, under the conditions studied. 

These initial kinetic studies (Holand 1996) have shown that this process is a case of Ostwald ripening. In short, this means that the mass transport to the location of the crystal growth, that is, the diffusion process from the glass matrix, is the rate-determining... 

---