Uphill diffusion

For solutions which present a mixing gap, the locus of points where the second derivative of G vanishes is called the spinodal. Within the spinodal, this second derivative is negative which results in negative diffusion coefficients or uphill diffusion (Figure 8.10). 

Even this more elaborated description of ion movements in response to gradients of chemical potential may turn out to be insufficient, in particular when uphill diffusion is active ... 

The mobile diluted species i has a normal behavior relative to diffusion, whereas the sluggish major species j undergoes uphill diffusion. 

Uphill diffusion in binary systems and spinodal decomposition... 

Uphill diffusion occurs in binary systems because, strictly speaking, diffusion brings mass from high chemical potential to low chemical potential (De Groot and Mazur, 1962), or from high activity to low activity. Hence, in a binary system, a more rigorous flux law is (Zhang, 1993) ... 

From the above analysis, it can be seen that D in Pick s first law J = -DVC (Equation 3-6) may be either positive or negative (accounting for uphill diffusion), and it can vary from positive to negative along a spinodal decomposition diffusion profile. If, on the other hand. Pick s law is modified as J = -(T>/y)Va (Equation 3-61), then V is always positive in a binary system. 

Uphill diffusion in a binary system is rare and occurs only when the phase undergoes spinodal decomposition. In multicomponent systems, uphill diffusion occurs often, even when the phase is stable. The cause for uphill diffusion in multicomponent systems is different from that in binary systems and will be discussed later. 

With each level of increased sophistication, the complexity of treating a diffusion problem increases tremendously, especially for natural silicate melts that have a large number of components. The simplest method (effective binary approach) is the most often used, but it cannot treat the diffusion of many components (such as those that show uphill diffusion). All the other methods have not been applied much to natural silicate melts. 

In all of the three cases above, components other than the discussed major components often cannot be treated as effective binary diffusion, especially for a component whose concentration gradient is small or zero compared to other components. Such a component often shows uphill diffusion. 

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.

The profiles are plotted in Figure 3-21. It can be seen that Si02 shows uphill diffusion. 

The components whose concentration gradient is small compared to other components cannot be treated as effective binary. They often show uphill diffusion. 

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. 

The effect of other concentration gradients sometimes leads to nonmonotonic profile in the trace element, a t5q)ical indication of uphill diffusion. 

Even in the absence of uphill diffusion, a trace element concentration profile often does not match that for a constant diffusivity by using the effective binary diffusion treatment. Hence, the effective binary diffusivity depends on the chemical composition, which is expected. 

Figure 3-24 Calculated diffusion-couple profiles for trace element diffusion and isotopic diffusion in the presence of major element concentration gradients using the approximate approach of activity-based effective binary treatment. The vertical dot-dashed line indicates the interface. The solid curve is the Nd trace element diffusion profile (concentration indicated on the left-hand y-axis), which is nonmonotonic with a pair of maximum and minimum, indicating uphill diffusion. The dashed curve is the Nd isotopic fraction profile. Note that the midisotopic fraction is not at the interface.

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. 

Behavior of trace element that can be treated as effective binary diffusion The above discussion is for the behavior of the principal equilibrium-determining component. For minor and trace elements, there are at least two complexities. One is the multicomponent effect, which often results in uphill diffusion. This is because the cross-terms may dominate the diffusion behavior of such components. The second complexity is that the interface-melt concentration is not fixed by thermodynamic equilibrium. For example, for zircon growth, Zr concentration in the interface-melt is roughly the equilibrium concentration (or zircon saturation concentration). However, for Pb, the concentration would not be fixed. 

If the diffusion of a minor or trace element can be treated as effective binary (not uphill diffusion profiles) with a constant effective binary diffusivity, the concentration profile may be solved as follows. The growth rate u is determined by the major component to be n D ff, and is given, not to be solved. Use i to denote the trace element. Hence, w, and Dt are the concentration and diffusivity of the trace element. Note that Di for trace element i is not necessarily the same as D for the major component. The interface-melt concentration is not fixed by an equilibrium phase diagram, but is to be determined by partitioning and diffusion. Hence, the boundary condition is the mass balance condition. If the boundary condition is written as w x=o = Wifl, the value of Wi must be found using the mass balance condition. In the interface-fixed reference frame, the diffusion problem can be written as... 

The coefficient Ln in Eq. 3.96 is positive and the equation therefore shows that the C flux will be in the direction of reduced C activity. Because the C activity is higher in the Si-containing alloy than in the non-Si-containing alloy at the same C concentration, the uphill diffusion into the non-Si-containing alloy occurs as observed. In essence, the C is pushed out of the ternary alloy by the presence of the essentially immobile Si. 

The solution for a diffusion couple in which two semi-infinite ternary alloys are bonded initially at a planar interface is worked out in Exercise 6.1 by the same basic method. Because each component has step-function initial conditions, the solution is a sum of error-function solutions (see Section 4.2.2). Such diffusion couples are used widely in experimental studies of ternary diffusion. In Fig. 6.2 the diffusion profiles of Ni and Co are shown for a ternary diffusion couple fabricated by bonding together two Fe-Ni-Co alloys of differing compositions. The Ni, which was initially uniform throughout the couple, develops transient concentration gradients. This example of uphill diffusion results from interactions with the other components in the alloy. Coupling of the concentration profiles during diffusion in this ternary case illustrates the complexities that are present in multicomponent diffusion but absent from the binary case. 

There are extensive reviews of the many measurements of the Ay, particularly in ternary systems [1]. Numerous systems exhibit uphill diffusion, due to strong particle-particle interactions, and efforts have been made to interpret the diffusivity behavior in terms of thermodynamic activity data and particle-particle interaction models. In many cases the diffusion behavior has been explained, and more details and discussion are found in Kirkaldy and Young s text [1]. 

The local composition changes during spinodal decomposition and precipitation by nucleation and growth are compared in Figure 11.24. It is interesting to note that spinodal decomposition requires uphill diffusion. The boundary between... 

When uphill diffusion can take place (because the directions of the gradients of chemical potential and of concentrations are opposed), the Pick approach fails, even at the qualitative level, to describe the mass transfer phenomena observed experimentally. Several known examples underly the shortcoming of the Pick s formulation [38]. 

The reader may find it helpful at this junction to consider the phenomenological model originally developed by Cahn (.2) to describe two phase metal alloys and more recently used in conjunction with polymer alloys. This model explains the appearance of isotropic, interdispersed domains in terms of spinodal decomposition. This may yield some insight into the reasons why "uphill" diffusion (that is, diffusion against the concentration gradient) occurs in phase inversion. The reader is also referred to the contribution of Strathmann in the present volume (.3). ... 

Lesher C (1994) Kinetics of Sr and Nd exchange in silicate liquids Theory, experiments and applications to uphill diffusion, isotopic equilibration, and irreversible mixing of magmas. J Geophys Res B... 

EFFECT OF UPHILL DIFFUSION IN SOLID ELECTROLYTES Zr02+8mol.%Sc203 AND Zr02 +8mol.% Y2O3... 

Key words Solid Electrolytes/Zirconia/ Uphill Diffusion. 

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