Matrix diffusion

Note from (5.18) that only one of the three transformed scalars is reacting. Note also from (5.19) that the diffusion matrix is greatly simplified when Ti = f2 = f3.17... 

This expression does not determine the mixing model uniquely. However, by specifying that the diffusion matrix in the resulting FP equation must equal the conditional joint scalar dissipation rate,88 the FP model for the molecular mixing term in the form of (6.48)... 

In terms of the equivalent stochastic differential equation (6.106), p. 279, this choice yields a diffusion matrix of the form B() = S -(i(), where the matrix G does not depend on the molecular-diffusion coefficients. 

Property (ii) is also controlled by the behavior of Sg(0)Cg(0). In general, the diffusion matrix should have the property that it does not allow movement in the direction normal to the surface of the allowable region.100 Defining the surface unit normal vector by n(0 ), property (ii) will be satisfied if Sg(0 )Cg(0 )n(0 ) = 0, where 0 lies on the surface of the allowable region. This condition implies that (e 10 )n(0 ) = 0, which Girimaji (1992) has shown to be true for the single-scalar case. Thus, the FP model satisfies property (ii), but the user must provide the unknown conditional joint scalar dissipation rates that satisfy (e 0 )n(0+) = 0. 

Choosing the drift coefficient to be linear in U and the diffusion matrix to be independent of U ensures that the Lagrangian velocity PDF will be Gaussian in homogeneous turbulence. Many other choices will yield a Gaussian PDF however, none have been studied to the same extent as the LGLM. 

The full treatment of multicomponent diffusion requires a diffusion matrix because the diffusive flux of one component is affected by the concentration gradient of all other components. For an N-component system, there are N-1 independent components (because the concentrations of all components add up to 100% if mass fraction or molar fraction is used). Choose the Nth component as the dependent component and let n = N 1. The diffusive flux of the components can hence be written as (De Groot and Mazur, 1962)... 

In the general case of three-dimensional multicomponent diffusion in an anisotropic medium (such as Ca-Fe-Mg diffusion in pyroxene), the mathematical description of diffusion is really complicated it requires a diffusion matrix in which every element is a second-rank tensor, and every element in the tensor may depend on composition. Such a diffusion equation has not been solved. Because rigorous and complete treatment of diffusion is often too complicated, and because instrumental analytical errors are often too large to distinguish exact solutions from approximate solutions, one would get nowhere by considering all these real complexities. Hence, simplification based on the question at hand is necessary to make the treatment of diffusion manageable and useful. 

When one refers to the diffusion equation, it is usually the binary diffusion equation. Although theories for multicomponent diffusion have been extensively developed, experimental studies of multicomponent diffusion are limited because of instrumental analytical error and theoretical complexity, and there are yet no reliable diffusivity matrix data for practical applications in geology. Multicomponent diffusion is hence often treated as effective binary diffusion by treating the component under consideration as one component and combining all the other components as the second component. 

Below, the effective binary approach and the concentration-based diffusivity matrix are introduced. The modified effective binary approach (Zhang, 1993) has not been followed up. The approach using the activity-based diffusivity matrix, similar to activity-based diffusivity T> (Equations 3-61 and 3-62), is probably the best approach, but such diffusivities require systematic effort to obtain. 

Choosing an appropriate reference frame such as a = 1 for all k, the diffusivity matrix can be written as... 

Using the diffusivity matrix, the diffusion equation for component i can be written in the following form ... 

Therefore, in the transformed components, the diffusion is decoupled, meaning that the diffusion of one component is independent of the diffusion of other components. The equation for each w, can be obtained given initial and boundary conditions using the solutions for binary diffusion. The final solution for C is C = Tw. When the diffusivity matrix is not constant, the diffusion equation for a multicomponent system can only be solved numerically. 

Values of a diffusion coefficient matrix, in principle, can be determined from multicomponent diffusion experiments. For ternary systems, the diffusivity matrix is 2 by 2, and there are four values to be determined for a matrix at each composition. For quaternary systems, there are nine unknowns to be determined. For natural silicate melts with many components, there are many unknowns to be determined from experimental data by fitting experimental diffusion profiles. When there are so many unknowns, the fitting of experimental concentration... 

Solving the diffusion profiles given the diffusivity matrix (ternary systems)... 

Below is a numerical example. Consider a three-component system MgO-A1203-S102 with MgO being component 1, AI2O3 being component 2, and Si02 being the dependent component. The diffusivity matrix (in arbitrary unit) is... 

Figure 3-21 Calculated diffusion profiles for a diffusion couple in a ternary system. The diffusivity matrix is given in Equation 3-102a. The fraction of Si02 is calculated as 1 - MgO - AI2O3. Si02 shows clear uphill diffusion. A component with initially uniform concentration (such as Si02 in this example) almost always shows uphill diffusion in a multi-component system.

Similar to binary diffusivities, each element in the diffusivity matrix is expected to depend on composition, sometimes strongly, especially for highly nonideal systems. If the nonideality is strong enough to cause a miscibility gap, the eigenvalues would vary from positive to zero and to negative. If there is no miscibility gap, the eigenvalues are positive but can still vary with composition. 

Similar to binary systems, the intrinsic diffusivity matrix is expected to be much less dependent on the composition of the system, which is the main advantage of using the activity-based diffusivity matrix. To solve the above equation, it is necessary to know the relations between the activity of every component and all concentrations. 

The set of diffusion equations based on the activity-based diffusivity matrix is expected to be complicated and solvable only numerically. Because of its complexity, no attempt has been made using this formulation to obtain the "intrinsic" diffusivity matrix. 

For a complete description of the diffusion process, the diffusion matrix approach is necessary as in the following examples ... 

Given a diffusion couple of different melts (e.g., basalt and andesite), if one wants to predict the diffusion behavior of all major components, the diffusion matrix approach is necessary. 

For the dissolution of a crystal into a melt, if one wants to predict the interface melt composition (that is, the composition of the melt that is saturated with the crystal), the dissolution rate, and the diffusion profiles of all major components, thermodynamic understanding coupled with the diffusion matrix approach is necessary (Liang, 1999). If the effective binary approach is used, it would be necessary to determine which is the principal equilibrium-determining component (such as MgO during forsterite dissolution in basaltic melt), estimate the concentration of the component at the interface melt, and then calculate the dissolution rate and diffusion profile. To estimate the interface concentration of the principal component from thermod5mamic equilibrium, because the concentration depends somewhat on the concentrations of other components, only... 

Similarly, for the general problem of crystal growth in silicate melts, it is necessary to use the diffusion matrix approach. 

In principle, the diffusion matrix approach can be extended to trace elements. My assessment, however, is that in the near future diffusion matrix involving 50 diffusing components will not be possible. Hence, simple treatment will still have to be used to roughly understand the diffusion behavior of trace elements the effective binary diffusion model to handle monotonic profiles, the modified effective binary diffusion model to handle uphill diffusion, or some combination of the diffusion matrix and effective binary diffusion model. 

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. 

The diffusion behavior of components that are not the principal equilibriumdetermining component is difficult to model because of multicomponent effect. Many of them may show uphill diffusion (Zhang et al., 1989). To calculate the interface-melt composition using full thermod3mamic and kinetic treatment and to treat diffusion of all components, it is necessary to use a multicomponent diffusion matrix (Liang, 1999). The effective binary treatment is useful in the empirical estimation of the dissolution distance using interface-melt composition and melt diffusivity, but cannot deal with multicomponent effect and components that show uphill diffusion. 

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