Theory of isotope diffusion in a material with multiple species and its implications for hydrogen-enhanced electrical conductivity in olivine

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The relationship between isotope diffusion coefficient and electrical conductivity is examined for a material where a dominant charge-carrying atomic species (e.g., hydrogen) is present as various forms with different diffusion coefficients (e.g., two protons trapped at M-site vacancy, one proton trapped at M-site vacancy etc.). It is shown that the isotopic diffusion occurs keeping the concentration ratio of each species fixed as determined by the thermo-chemical environment. Consequently, the isotope diffusion coefficient is the harmonic average of diffusion coefficients of individual species and is dominated by the slowest diffusing species. In contrast, when electric current is carried by charged species, the concentrations of individual species do not change. Therefore, electrical conductivity is related to the arithmetic average of individual diffusion coefficients dominated by the fastest diffusing species. The difference between these two cases can be large when different species have largely different diffusion coefficients. This model provides an explanation for the observed differences between experimental observations on isotopic diffusion (of H-D) and hydrogen-enhanced electrical conductivity and supports a hybrid model of hydrogen-enhanced electrical conduction where electrical conductivity is dominated by the fast moving hydrogen-related species. The species with the largest mobility may change with temperature leading to a change in anisotropy of conductivity. The degree of enhancement of electrical conductivity by hydrogen is high enough to explain most of the geophysically observed electrical conductivity of Earth’s upper mantle.