Crystal growth diffusional

Rate of Growth Crystal growth is a layer-by-layer process, and since growth can occur only at the face of the crystal, material must be transported to that face from the bulk of the solution. Diffusional... 

Heterogeneous reactions of the type A+B = AB can, in principle, occur in two ways. 1) The product molecule AB is formed from A and B in the surrounding solvent or immediately at the surface of the AB crystal. These AB molecules are then added to the crystal on its external surface. This is additive crystal growth. 2) The solid product AB forms between A and B and separates the reactants spatially. Further reaction is possible only via (diffusional) transport across the reaction layer AB. This is reactive crystal growth [H. Schmalzried (1993)]. The moving AB interfaces in additive crystal growth are inherently unstable morphologically (see Chapter 11). 

All modern pictures and models of hydrate crystal growth include mass transfer from the bulk phases to the hydrate. Unfortunately, some confusion arises due to the fact that two interfaces are usually considered, and the driving forces may not be intuitive for those not familiar with the area. In order to provide a basis for the modeling section, a brief overview of the diffusional boundary layer is given. 

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 over-all process of crystal growth in a seeded solution is analogous to other mass transfer situations encountered in chemical engineering and may be treated as a diffusional step in series with a surface reaction step. Solution supersaturation provides the driving force required for each step, as portrayed schematically in Fig. 12. First, solute molecules or ions diffuse through the solution to the growing crystal. Second, upon reaching the surface, the molecules or ions must be accepted and incorporated into the crystal lattice. 

Kinetic phenomena can also be used to delimit the timescales of magmatic processes and, unlike radiometric ages, do not require absolute constraints on the timing of eruption. Two important kinetic controls on crystal properties are crystal growth and diffusional relaxation of compositional heterogeneities in minerals. Rates of crystal settling are not described here but have also been used to delimit crystal storage times (e.g., Anderson et al., 2000 Resmini and Marsh, 1995). 

Rate of Growth Crystal growth is a layer-by-layer process, and since growth can occur only at the face of the crystal, material must be transported to that face from the bulk of the solution. Diffusional resistance to the movement of molecules (or ions) to the growing crystal face, as well as the resistance to integration of those molecules into the face, must be considered. As discussed earlier, different faces can have different rates of growth, and these can be selectively altered by the addition or elimination of impurities. 

Nielsen, A. (1961) Diffusional controlled growth of a moving sphere. The kinetics of crystal growth in potassium perchlorate precipitation. The Journal of Physical Chemistry, 65, 46-49. 

The introductory section of this chapter cites texts showing full development of the crystal growth models described above. Those models describing continuous growth, two-dimensional nucleation, and screw dislocation (BCF and variants), with or without diffusional limitation, predict values of kinetic order (exponent r) between 1 and 2 in... 

Crystal growth is a diffusional process, modified by the effect of tlie solid surfaces on which the growth occurs. Solute molecules or ions reach the growing faces of a crystal by diffusion through the liquid phase. The usual mass-transfer coefficient kjf applies to this step. On reaching the surface, the molecules or ions must be accepted by the crystal and organized into the space lattice. The reaction occurs at the surface at a finite rate, and the overall process consists of two steps in series. Neither the diffusional nor the interfacial step will proceed unless the solution is supersaturated. 

Figure 3.7 Important diffusional processes (volume and surface) affecting crystal growth. (Reproduced with permission from Rosenberger 1986.)...

If crystal growth and dissolution, which occur in a parallel way are both purely diffusional, all crystal faces should at equal distance from equilibrium, grow and dissolve at the same rate. In reality growing planes are planar, while dissolving pianes become pitted and eroded (3,4), indicating that in the latter case an increased rate of mass transfer is taking place. Such pitted surfaces may again initiate secondary nucleation. 

The diffusion theories of crystal growth cannot yet be reconciled with the adsorption layer and dislocation theories. It is acknowledged that the diffusion theories have grave deficiencies (they cannot explain layer growth or the faceting of crystals, for example), yet crystal growth rates are conveniently measured and reported in diffusional terms. The utilization of the mathematics of mass transfer processes makes this the preferred approach, from the chemical engineer s point of view at any rate, despite its many limitations. 

The approach that has been irsed to calculate ice crystal growth starts with an analogy between the governing eqnations and the boundary conditions for electrostatic and diffusion problems. Poisson s eqrration in electrostatics and Green s theorem lead to the following eqrration for the ice crystal diffusional mass growth rate. 

Crystallization Process Systems Nucleation followed by diffusional growth... 

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