Isotopic Constraints on Earth System Processes. Группа авторов

      Onsager (1945) showed that a way to deal with transport in a multi‐component system is to express the diffusive flux of a component as a linear combination of the concentration gradients of all the independent components in the system. The one‐dimensional diffusive flux of component i in an n‐component liquid is then given by

(1.3)

where D_ij are elements of a (n‐1) by (n‐1) diffusion matrix with the n^th component taken as the dependent component. For example, equation 1.3 for the ternary system CaO‐Al₂O₃‐SiO₂ can be written in matrix form as

      (1.4)

      when SiO₂ is taken as the dependent component. The flux of the SiO₂ can be calculated using

or depending on the reference frame, by requiring that the sum of all the volume fluxes ν _i J_i ( ν _i is the molar volume of species i) must add up to zero (see de Groot & Mazur, 1962).

Another issue that is especially relevant in natural silicate liquids is that the driving force for chemical diffusion is chemical potential gradients ∇ μ _i ( μ _i now denotes chemical potential, not reduced mass). The more general version of equation 1.1 is

      (1.5)

      where the L_i is called the phenomenological coefficient for diffusion and

is now identified as the diffusion coefficient for a non‐ideal binary system. The extension of this to a multi‐component system governed by a phenomenological diffusion matrix L_i,j was studied experimentally by Liang et al. (1966a; 1966b; 1967) with the non‐ideal molten CaO‐Al₂O₃‐SiO₂ system.

      1.2.2. Effective Binary Diffusion Coefficients

When the formulation of the diffusive flux given previously is applied to multi‐component geologically relevant silicate materials, one is faced with the problem that to date only one complete diffusion matrix for an igneous melt (an 8‐component basaltic melt) has been reported (Guo & Zhang, 2018). When there is a lack of a full diffusion matrix for a specific system of interest, what is commonly done is to represent the multi‐component system as if it were a binary system made up of the component of interest with all the other components taken together as a second component. The mass flux in such a pseudo‐binary system will involve a single diffusion coefficient (i.e., n = 2 in equation 1.3) and thus given a diffusion profile of a component i one can in most cases fit the data using equation 1.2 to determine the effective binary diffusion coefficient

for component i. Cooper (1968) showed how the effective binary diffusion coefficient of a given component is related to the full diffusion matrix of the system, and he also pointed out that effective binary diffusion coefficients depend not only on the local composition and thermodynamic state (e.g., temperature and pressure) but also on the direction of the diffusive flux in composition space. It follows from this that in order to determine the appropriate effective binary diffusion coefficient for modeling a natural diffusion profile, one needs to run an experiment with a diffusion couple juxtaposing compositions that are as close as possible to those of the far‐field values of the natural system.

      1.2.3. Self‐Diffusion Coefficients

A very basic type of diffusion coefficient is the self‐diffusion coefficient that measures of the intrinsic mobility of a species. A common experimental approach for measuring self‐diffusion coefficients is to locally alter the isotopic composition of the element of interest in an otherwise homogeneous medium and observe the rate at which the isotopic difference spreads. In effect, this eliminates all diffusive coupling to the other species because their gradients are all zero. The self‐diffusion coefficient

of elements in silicate liquids can be very different from each other. For example, Liang et al. (1996a) reported a large set of experimentally determined self‐diffusion coefficients as function of composition in molten CaO‐Al₂O₃‐SiO₂ showing that

~ 10×

,

~2×

, and

~1 to 2×

. These differences in self‐diffusion are in marked contrast to the very similar magnitude of the effective binary diffusion coefficients in a molten rhyolite‐basalt diffusion couple. Fig. 1.1 shows this by overlaying diffusion profiles measured by Richter et al. (2003) in a diffusion couple in which natural rhyolite liquid and a natural basaltic liquid were juxtaposed and annealed in a piston cylinder experiment. The remarkably similar diffusive behavior the major oxides