Non-equilibrium isotopic and elemental fractionation during diffusion-controlled crystal growth under static and dynamic conditions

Abstract

Rapid crystal growth can lead to disequilibrium uptake of growth-medium components whose diffusivities limit their dispersal near an advancing crystal interface. The recent documentation of an isotope mass effect on diffusion raises the possibility that even isotope ratios in crystals may be subject to this effect. Building upon existing 1-dimensional treatments, we describe a numerical modeling approach in which a spherical grain grows at the center of an infinite spherical medium of predetermined composition. Local equilibrium at the interface between the crystal and the growth medium is assumed, but the concentration of the species of interest in the growth medium is allowed to vary near the interface as a consequence of slow diffusion combined with rejection from (or incorporation within) the growing crystal. The disequilibrium uptake of elements and isotopes depends upon the ratio of crystal growth rate ( R) to diffusivity in the growth medium ( D). Conditions of fast mineral growth in a viscous magma—e.g., in lava lakes or small igneous bodies—result in accumulation of elements with K << 1 (or depletion of elements with K >> 1) near the growing mineral interface, forming a compositional boundary layer in the growth medium. In a static system, the magnitude of this compositional perturbation depends critically upon the diffusivity of the element or isotope of interest in the growth medium. If the system is dynamic—i.e., experiencing free or forced convection—then the vigor of convection also affects behavior. Significant fractionation of elements and isotopes is predicted to occur within the boundary layer during progressive crystal growth because diffusion rates of individual elements vary with size and charge and those of isotopes of the same element depend on their masses. Local equilibrium at the interface between the crystal and its growth medium means that a fast-growing crystal will record this fractionation in its resulting radial concentration profile. If the boundary-layer thickness, BL, is small (say, < 100 μm) and the equilibrium partition coefficient, K, is < 0.5, then a first-order estimate of the steady-state isotopic fractionation in a growing crystal is given by δ ( ‰ ) = 1000 ⋅ ( 1 − D A D B ) ⋅ ( R ⋅ B L D A ) ⋅ ( 1 − K ) , where D A and D B are the diffusivities of the faster and slower species in the growth medium and δ is the deviation from the equilibrium isotope ratio in parts per thousand. For isotopes of a single element, D A and D B will generally differ by < 1%, but plausible R/ D ratios can nevertheless lead to deviations from equilibrium between the crystal and the growth medium of up to ~ 3‰. The model may bear on disequilibrium crystal-growth phenomena in a variety of geologic settings—including element- and isotopic profiles in crystals of both igneous and metamorphic rocks. It is suggested that compositional core to rim profile of a crystal may be a proxy for the near surface composition of the growth medium during crystal growth. Isotopic effects are discussed in detail because these have not been addressed previously; igneous systems are emphasized because higher crystal growth rates are more conducive to disequilibrium (including in the compositions of melt inclusions).