Diffusion, crystallization

The pharmaceutical industry has taken great interest of late in the study of polymorphism and solvatomorphism in its materials, since a strong interest in the phenomena has developed now that regulatory authorities understand that the nature of the structure adopted by a given compound upon crystallization can exert a profound effect on its solid-state properties. For a given material, the heat capacity, conductivity, volume, density, viscosity, surface tension, diffusivity, crystal... 

Figure 1-11 Concentration profile for (a) crystal growth controlled by interface reaction (the concentration profile is flat and does not change with time), (b) diffusive crystal growth with t2 = 4fi and = 4t2 (the profile is an error function and propagates according to (c) convective crystal growth (the profile is an exponential function and does not change with time), and (d) crystal growth controlled by both interface reaction and diffusion (both the interface concentration and the length of the profile vary).

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. 

Diffusive crystal growth at a fixed temperature would not result in a constant crystal growth rate (see below). However, under some specific conditions, such as continuous slow cooling, or in the presence of convection with diffusion across the boundary layer, time-independent growth rate may be achieved. Similarly, time-independent dissolution rate may also be achieved. 

One-dimensional diffusive crystal dissolution During crystal dissolution, the surface concentration of the crystal may be treated as constant. Define u = a(Dlt) to be the melt growth rate (instead of melt consumption rate). Then the concentration profile is... 

Diffusive crystal dissolution means that crystal dissolution is controlled by diffusion, which requires high interface reaction rate and absence of convection. In nature, diffusive crystal dissolution is rarely encountered, because there is almost always fluid flow, or crystal falling or rising in the fluid. That is, crystal dissolution in nature is often convective dissolution, which is discussed in the next section. One possible case of diffusive crystal dissolution is for crystals on the roof or floor of a magma chamber if melt produced by dissolution does not sink or rise. For these... 

Although diffusive crystal dissolution is seldom encountered in nature, its theoretical development is instructive for understanding convective crystal dissolution, and it is often encountered in experimental studies. Such experiments are easy to conduct, and can be applied to infer diffusion coefficients, to establish equilibrium conditions, and to investigate the rate of diffusive crystal dissolution. Furthermore, the interface-melt composition and diffusivity obtained from diffusive crystal dissolution experiments are of use to estimate convective crystal dissolution rates (Section 4.2.3). 

Mathematically, diffusive crystal dissolution is a moving boundary problem, or specifically a Stefan problem. It was treated briefly in Section 3.5.5.1. During crystal dissolution, the melt grows. Hence, there are melt growth distance and also crystal dissolution distance. The two distances differ because the density of the melt differs from that of the crystal. For example, if crystal density is 1.2 times melt density, dissolution of 1 fim of the crystal would lead to growth of 1.2 fim of the melt. Hence, AXc = (pmeit/pcryst) where Ax is the dissolution distance of the crystal and Ax is the growth distance of the melt. 

The diffusion equation for three-dimensional diffusive crystal dissolution in the spherical case (Eq. 4-90) is rarely encountered and too complicated. Hence, such problems will not be treated here. 

The results above have the following applications (i) estimation of diffusive crystal dissolution distance for given crystal and melt compositions, temperature, pressure, and duration if diffusivities are known and surface concentrations can be estimated and (ii) determination of diffusivity (EBDC) and interface-melt concentrations. Those diffusivities and interface concentrations can be applied to estimate crystal dissolution rates in nature. 

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. 

The boundary layer thickness 5. For convective crystal dissolution, the steady-state boundary layer thickness increases slowly with increasing viscosity and decreasing density difference between the crystal and the fluid. It does not depend strongly on the crystal size. Typical boundary layer thickness is 10 to 100/rm. For diffusive crystal dissolution, the boundary layer thickness is proportional to square root of time. 

By the above definition, b is positive for crystal dissolution, and negative for crystal growth. During convective crystal dissolution, the dissolution rate u is directly proportional to b. During diffusive crystal dissolution, the dissolution rate is proportional to parameter a, which is positively related to b. Hence, for the dissolution of a given mineral in a melt, the size of parameter b is important. The numerator of b is proportional to the degree of undersaturation. If the initial melt is saturated, b = 0 and there is no crystal dissolution or growth. The denominator characterizes the concentration difference between the crystal and the saturated... 

Crystai growth distance and behavior of major component This problem is similar to diffusive crystal dissolution. Hence, only a summary is shown here. Consider the principal equilibrium-determining component, which can be treated as effective binary diffusion. The density of the melt is often assumed to be constant. The density difference between the crystal and melt is accounted for. 

Figure 4-23 Trace element diffusion profiles during diffusive crystal growth for (a) various Df/D ratios and (b) various Ki values.

Zhang Y., Walker D., and Lesher C.E. (1989) Diffusive crystal dissolution. Contrib. Mineral. Petrol. 102, 492-513. 

Figure 3-32 Extracting diffusivity from diffusive crystal dissolution ...

Figure 4-23 Trace element diffusion profiles during diffusive crystal growth...

These interactions are vital to understanding the dynamics of many surface phenomena, such as cluster formation and diffusion, crystal growth, surface reactivity, adsorption and desorption, and many others. 

Diffuse crystal/crystal interfaces often appear in systems subject to incipient chemical or structural instabilities associated with phase separation, long-range ordering, or displacive phase transformations [2], Examples of interfaces associated with the first two types are shown in Fig. 18.7. 

A fixed amount of condensed phase enclosed by an interface will undergo essentially the same process, except that the time scales may differ greatly. For solid phases, the interfaces will reduce gradients in curvature by diffusional processes such as interface diffusion, crystal diffusion, and vapor transport. At similar time scales (in the case of crystal diffusion) interfaces will move because atoms will experience differences in diffusion potential across an interface arising from differences in the curvature according to Eq. 3.76. 

FIGURE 2.10 An array of commercially available and commonly used plastic plates for both sitting and hanging drop vapor diffusion crystallization. Also in the picture is a box of silicone coated cover slips for hanging drops. Courtesy of Hampton Research. 

In practice, vapor diffusion crystallization trials are generally carried out using 24-well Linbro plates with each well sealed by a microscope cover slip of appropriate size (See Figure 2.2b). Each well of the plate contains 500 (rl of a precipitant solution. A 2- to 4- tl drop, containing equal parts protein solution (-10 mg/ml) and precipitant solution (taken from the well), is then placed on a siliconized microscope cover slip that is carefully inverted and placed over the well. A bead of high vacuum grease previously applied to the lip of each well provides the seal between the well and the cover slip. 

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