Diffusive crystal dissolution

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. 

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 ...

Under diffusion-controlled dissolution conditions (in the anodic direction) the crystal orientation has no influence on the reaction rate as only the mass transport conditions in the solution detennine the process. In other words, the material is removed unifonnly and electropolishing of the surface takes place. 

Mechanisms of dissolution kinetics of crystals have been intensively studied in the pharmaceutical domain, because the rate of dissolution affects the bioavailability of drug crystals. Many efforts have been made to describe the crystal dissolution behavior. A variety of empirical or semi-empirical models have been used to describe drug dissolution or release from formulations [1-6]. Noyes and Whitney published the first quantitative study of the dissolution process in 1897 [7]. They found that the dissolution process is diffusion controlled and involves no chemical reaction. The Noyes-Whitney equation simply states that the dissolution rate is directly proportional to the difference between the solubility and the solution concentration ... 

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). 

The purpose of most experimental studies of diffusion is to obtain accurate diffusion coefficients as a function of temperature, pressure, and composition of the phase. For this purpose, the best approach is to design the experiments so that the diffusion problem has a simple anal3hical solution. After the experiments, the experimental results are compared with (or fit by) the anal3hical solution to obtain the diffusivity. The method of choice depends on the problems. The often used methods include diffusion-couple method, thin-source method, desorption or sorption method, and crystal dissolution method. 

To use this method to obtain diffusivity, the dissolution must be diffusion controlled. The diffusion aspect was discussed in Section 3.5.5.1, and the heterogeneous reaction aspect is discussed later. The melt growth distance (L, which differs from the crystal dissolution distance by the factor of the density ratio of crystal to melt) may be expressed as (Equation 3-115d)... 

Tracer diffusivities are often determined using the thin-source method. Self-diffusivities are often obtained from the diffusion couple and the sorption methods. Chemical diffusivities (including interdiffusivity, effective binary diffusivity, and multicomponent diffusivity matrix) may be obtained from the diffusion-couple, sorption, desorption, or crystal dissolution method. 

Zhang et al. (1989) treated the interplay between diffusion and interface reaction during the initial and transient stages of crystal dissolution in a silicate melt. Using the interface reaction rate of diopside, they found that the period for... 

Note that because the diffusion equations are for the melt phase, the rate is also that for melt motion. Therefore, during crystal growth, instead of crystal growth rate, melt consumption rate is used in the diffusion equation. During crystal dissolution, instead of crystal dissolution rate, the melt growth rate is used. Equation 4-39c may be applied to convert the rates. 

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. 

In the case of one-dimensional crystal dissolution with u = Uq, if the reference frame is fixed at the faraway melt (x = oo), the melt does not flow even though the melt is generated at the interface at velocity u. (The interface moves as a rate of u.) Hence, the diffusion equation is Equation 3-9 without a velocity term ... 

If temperature or pressure varies during crystal dissolution, the problem becomes more complicated because both the diffusivity and the interface melt concentration vary, causing the dissolution rate to vary. Although the diffusivity dependence on time is not difficult to tackle anal3dically, the variation in the interface condition and the consequent change in dissolution rate cannot be treated simply. Hence, the treatment here is for constant temperature and pressure. Numerical method is necessary to handle crystal dissolution with variable temperature and pressure. 

One-dimensional diffusive dissolution With the above general discussion, we now turn to the special case of one-dimensional crystal dissolution. Use the interface-fixed reference frame. Let melt be on the right-hand side (x > 0) in the interface-fixed reference frame. Crystal is on the left-hand side (x < 0) in the interface-fixed reference frame. Properties in the crystal will be indicated by superscript "c". For simplicity, the superscript "m" for melt properties will be ignored. Diffusivity in the melt is D. Diffusivity in the crystal is D. The concentration in the melt is C (kg/m ) or w (mass fraction). The initial concentration in the crystal is or simplified as or if there would be no confusion from the context. It is assumed that the interface composition rapidly reaches equilibrium. In the following, diffusion in the melt is first considered, and then diffusion in the crystal. 

Diffusion in the melt. Ignoring melt density variation, the diffusion equation in the melt during crystal dissolution is... 

For the profile in the melt during crystal dissolution (melt growth) or crystal growth, the absolute value of a is small. Hence, the diffusion... 

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