Measuring the multicomponent diffusion matrix: Experimental design and data analysis for silicate melts
The extended form of Fick's Law, which allows the diffusive flux of a chemical species to be a function of all the concentration gradients, provides an accurate and useful description of chemical diffusion in isothermal, multicomponent systems. For an ( N + 1)- component system, N 2 diffusion coefficients are required, however. Although recent attempts to measure the multicomponent diffusion matrix for natural silicate compositions and simpler analogs have been unsuccessful, this does not mean that the extended Fick's Law is inapplicable. We show that the diffusion matrix cannot be measured unless the experiment is carefully designed. The optimal experiment is a set of 2N isothermal interdiffusion runs using at leastN distinct diffusion couples. The concept of orthogonal couples i.e., mutually perpendicular composition directions, provides a practical guide for choosing distinct couples. The number of couples required depends on the size of the errors in the concentration measurements. Chi-square fitting is an appropriate technique for analyzing diffusion data, because it allows one to directly apply thermodynamic constraints on the eigenvalues of the diffusion matrix. We discuss the details of implementing chi-square fitting for isothermal interdiffusion experiments, including Jacobian and Hessian matrices for both the finite and infinite diffusion couple models. We also apply this method to data from the literature, extracting the diffusion matrix for a Na 2O-CaO-SiO 2 composition and making preliminary observations about some dacite-rhyolite and CaMgSi 2O 6-CaAl 2Si 2O 8 experiments.