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[SMP29-08] Oscillatory compositional zoning in minerals induced by impurities: Catastrophic transitions in crystal growth rate
Keywords:Oscillatory compositional zoning, Theory of crystal growth, Impurities, Crystal growth inhibition, Numerical simulation
The chemical composition of minerals is generally inhomogeneous. The compositional distribution in which zones rich in one component and zones poor in that component are accumulated alternately is called an oscillatory compositional zoning (OCZ). One of the factors that causes OCZ is periodic changes in the crystal growth rate during mineral formation. We proposed a model that takes into account the inhibitory effect of impurities on crystal growth as a mechanism for the formation of OCZs [1]. On the crystal surface, there are molecular steps and the crystal surface is built up layer by layer (layer growth) as the steps advance. When impurities adsorb to the crystal surface, they inhibit the advance of the steps, as if pinning them in place (pinning effect). It has been theoretically shown that hysteresis appears in the dependence of the step velocity on the degree of supersaturation by considering the pinning effect of impurities that repeatedly adsorb and desorb on the crystal surface [2]. The OCZ formation model we proposed [1] was based on the mean field theory of the crystal growth hysteresis [2], and assumed that discontinuous changes in growth rate (catastrophic transitions) would occur in response to slight changes in supersaturation. However, it was not obvious whether catastrophic transitions would actually occur. In this study, we conducted numerical calculations of step dynamics assuming crystal growth in solution, with the aim of theoretically demonstrating the catastrophic transitions.
For numerical method, we used a phase-field method with step dynamics that takes into account the pinning effect of impurities [3, 4]. At the same time, we attempted to reproduce the periodic fluctuations in the crystal growth rate and the OCZ by solving the time variation of the solution concentration field around the crystal based on the diffusion equation.
As the initial state, we assumed a clean crystal surface with no impurities adsorbed, and started the calculation from a highly supersaturated state. As the time goes on, impurities begin to adsorb to the crystal surface, but the steps update the surface faster than the amount of adsorption increases, so the crystal surface is maintained in a largely clean state. At the same time, solute molecules in the solution are incorporated into the crystal, reducing supersaturation near the surface. When the supersaturation decreases, the step advance slows down, and the crystal surface is gradually contaminated with adsorbed impurities, causing a further decrease in the step velocity. Due to this positive feedback, step-advance almost completely stopped. The supersaturation near the crystal surface recovers due to diffusion transport from the bulk solution. However, before the supersaturation recovered, the number of adsorbed impurities increased and the surface became completely dirty (a state in which adsorption and desorption are in equilibrium). This change in step velocity is exactly the catastrophic transition that was predicted by mean field theory. After that, even when the supersaturation recovered, step-advance did not resume for a while, and only resumed when the supersaturation recovered to a certain level. When step-advance resumed, the adsorbed impurities were embedded in the new crystal layer, and a nearly clean surface was restored. Continuing the calculation further, the above cycle was repeated. This result shows that periodic fluctuations in crystal growth rate can occur even without assuming that catastrophic transitions. We also obtained the distribution of impurities in the crystal growth layer and found that the impurity concentration changed periodically, which is consistent with the characteristics of OCZ.
References: [1] H. Torii and H. Miura (2024), Sci. Rep. 14:13337. [2] H. Miura and K. Tsukamoto (2013), Cryst. Growth Des. 13, 3588. [3] H. Miura (2016), Cryst. Growth Des. 16, 2033. [4] H. Miura (2020), Cryst. Growth Des. 20, 245.
For numerical method, we used a phase-field method with step dynamics that takes into account the pinning effect of impurities [3, 4]. At the same time, we attempted to reproduce the periodic fluctuations in the crystal growth rate and the OCZ by solving the time variation of the solution concentration field around the crystal based on the diffusion equation.
As the initial state, we assumed a clean crystal surface with no impurities adsorbed, and started the calculation from a highly supersaturated state. As the time goes on, impurities begin to adsorb to the crystal surface, but the steps update the surface faster than the amount of adsorption increases, so the crystal surface is maintained in a largely clean state. At the same time, solute molecules in the solution are incorporated into the crystal, reducing supersaturation near the surface. When the supersaturation decreases, the step advance slows down, and the crystal surface is gradually contaminated with adsorbed impurities, causing a further decrease in the step velocity. Due to this positive feedback, step-advance almost completely stopped. The supersaturation near the crystal surface recovers due to diffusion transport from the bulk solution. However, before the supersaturation recovered, the number of adsorbed impurities increased and the surface became completely dirty (a state in which adsorption and desorption are in equilibrium). This change in step velocity is exactly the catastrophic transition that was predicted by mean field theory. After that, even when the supersaturation recovered, step-advance did not resume for a while, and only resumed when the supersaturation recovered to a certain level. When step-advance resumed, the adsorbed impurities were embedded in the new crystal layer, and a nearly clean surface was restored. Continuing the calculation further, the above cycle was repeated. This result shows that periodic fluctuations in crystal growth rate can occur even without assuming that catastrophic transitions. We also obtained the distribution of impurities in the crystal growth layer and found that the impurity concentration changed periodically, which is consistent with the characteristics of OCZ.
References: [1] H. Torii and H. Miura (2024), Sci. Rep. 14:13337. [2] H. Miura and K. Tsukamoto (2013), Cryst. Growth Des. 13, 3588. [3] H. Miura (2016), Cryst. Growth Des. 16, 2033. [4] H. Miura (2020), Cryst. Growth Des. 20, 245.