Diffusion trace element

Whitmer, A.M., Ramenofsky, A.F., Thomas, J., Thibodeaux, L., Field, S.D. and Miller, B.J. 1989 Stability or instability. The role of diffusion in trace element studies. Archaeological Method and Theory 1 205-273. 

U-series elements are unusual in that, although they are trapped as one species at the time of crystal growth, they will decay to a different species with time. Consequently, the possibility exists for a highly compatible parent to decay to a highly incompatible daughter nuclide. With time the daughter will be expelled from the lattice, typically by diffusion. This creates a balance between uptake and decay that is not a consideration for stable trace elements, and so deserves brief mention here. 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

For kinetic disequilibrium partitioning of trace elements, equation (9.6.6) after Burton et al. (1953) is commonly presented as an alternative to equation (9.6.5) due to Tiller et al. (1953) (e.g., Magaritz and Hofmann, 1978 Lasaga, 1981 Walker and Agee, 1989 Shimizu, 1981). However, the relative values of viscosity and chemical diffusivity in common liquids and silicate melts make the momentum boundary-layer (i.e., the liquid film which sticks to the solid) orders of magnitude thicker than the chemical boundary layer. It is therefore quite unlikely that, except for rare cases of transient state, liquid from outside the momentum boundary-layer may encroach on the chemical boundary-layer, i.e., <5 may actually be taken as infinite. As a simple description of steady-state disequilibrium fractionation, the model of Tiller et al. (1953) has a much better physical rationale. A more elaborate discussion of these processes may be found in Tiller (1991a, b). 

The trace elements are introduced into seawater by river runoff, atmospheric transport, hydrothermal venting, groundwater seeps, diffusion from the sediments, and transport from outer space, usually as micro meteorites. The magnitudes of the first three of these fluxes, which are considered to be the major ones, are given in Table 11.1. Anthropogenic activities have significantly increased some of these fluxes, as discussed later. 

Sinking particles transport trace elements to the sediments. Once in the sediments, chemical reactions can resolubilize a significant fraction of the particulate metals. This process is termed diagenetic remobilization and is the subject of the next chapter. The resolubilized elements can diffuse across the sediment-water interface into the deep zone. 

When the concentration levels are higher, such as Ni diffusion between two olivine crystals with the same compositions except for Ni content (e.g., one contains 100 ppm and the other contains 2000 ppm), it may be referred to as either tracer diffusion or chemical diffusion. Tracer diffusivity is constant across the whole profile because the only variation along the profile is the concentration of a trace element that is not expected to affect the diffusion coefficient. 

If both chemical concentration gradients and isotopic ratio gradients are present (e.g., basaltic melt with Sr/ Sr ratio of 0.705 and andesitic melt with Sr/ Sr ratio of 0.720), the homogenization of isotopic ratio is referred to as isotopic diffusion (Lesher, 1990 Van Der Laan et al., 1994), although some prefer to call it isotopic homogenization. If there are concentration gradients in both major and trace elements, the diffusion of the trace elements is referred to as trace element diffusion (Baker, 1989). Isotopic diffusion and trace element diffusion are really part of multicomponent diffusion, which is complicated to handle. Isotopic diffusion should not be confused with self-diffusion, and trace element diffusion should not be confused with tracer 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.

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... 

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

In the other limiting case of D/ > D, the trace element profile approaches uniform concentration because the trace element diffuses much more rapidly than the equilibrium-determining component. 

Suppose crystal growth rate is constant with growth rate u = 0.1 /im/s. For a trace element, the initial concentration in the melt is 10 ppm, and the partition coefficient between the crystal and the melt is 2.5. The diffusion coefficient in the melt is 10 /im /s. 

Baker D.R. (1989) Tracer versus trace element diffusion diffusional decoupling of Sr concentration from Sr isotope composition. Geochim. Cosmochim. Acta 53, 3015-3023. 

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

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