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We demonstrated numerical simulations of impurity-induced crystal growth hysteresis in spiral growth. The numerical scheme combines two different methods: the phase-field method for step dynamics and the Monte Carlo method for random impurity adsorption. We manually changed the supersaturation (down and up) and repeated this cycle to monitor the time variation of the normal growth rate. We found that the growth rate suddenly drops to almost zero as the supersaturation decreases, and it swiftly recovers as the supersaturation subsequently increases. We confirmed that the critical supersaturation at which the growth rate recovered is distinguishably larger than that at which the crystal growth stopped. The critical supersaturations were found to be consistent with those predicted by the mean field theory for spiral growth if the period of the down-and-up cycle of supersaturation was chosen appropriately. We also applied the mean field theory to the observed hysteresis loops to evaluate some physical quantit...

This paper presents the outline of a practical course on computer simulation that will be given at the 16th International Summer School on Crystal Growth (ISSCG-16). The aim of this course is to understand crystal growth processes from the molecular level to the macroscopic level through computer simulations. We will mainly study molecular-scale crystal growth and nucleation processes by using molecular dynamics simulations and macroscopic growth processes at crystal surfaces by using phase field simulations.

Growth hysteresis is one of the remarkable phenomena of crystal growth induced by impurities; namely, the growth rates of crystal take different values between when supersaturation is decreased and is increased. The appearance of hysteresis has been explained on the basis of the mean field theory, in which the physical quantities such as step velocity, step interval, and density of impurities adsorbed on the crystal surface are averaged over space and time. Here, we propose a new method for the numerical simulation of step dynamics with impurity adsorption–desorption kinetics on a growing crystal surface. We numerically reproduce the growth hysteresis in a more realistic situation, in which the physical quantities fluctuate. The calculated hysteresis agrees with the prediction of the mean field theory. We have also found that the supersaturation should be changed within a period that is similar to the impurity adsorption time scale or slightly longer in order to observe obvious growth hysteresis. Measurem...

This paper presents an extension of a classical model of the crystal growth, the Burton–Cabrera–Frank (BCF) model, to a system that includes impurity molecules and discusses the formulation of the amount of the impurity uptake into the crystal as a partition coefficient. The BCF model considers the following surface processes of host molecules: adsorption from a solution to a terrace, desorption from the terrace to the solution, surface diffusion on the terrace, incorporation at a step edge, and release from the step edge to the terrace. The above-mentioned processes have been formulated not only for the host molecules but also for the impurity molecules (Miura, 2020). In this paper, the author analytically solves the previous model and derives an analytic formula for the partition coefficient as a ratio of the uptake of the impurity molecule to that of the host molecule into the crystal. The dependence of the partition coefficient on supersaturation is systematically investigated. The analytic formula is applied to the data obtained in the experiments of protein crystal growth to discuss the behavior of impurity molecules on the growing crystal surface. This model provides a theoretical framework for eliciting quantitative data from the crystal growth experiments on impurity partitioning.

The step pinning hypothesis is widely accepted as an explanation for the inhabitation of crystal growth by impurity molecules. This hypothesis enables us to evaluate the average step velocity and the critical impurity density above which no step can move on the crystal surface polluted by impurities. The average step velocity and the critical impurity density were obtained analytically in some literature; however, they have not been verified with other approaches. Here, we propose a new method for the numerical simulation of the step pinning by impurities in the framework of a phase-field technique and report a quantitative study on the average step velocity and critical impurity density for the first time. Our results clarify the limitations of the physical assumptions of the analytical formulas and their applications. Our phase-field method provides a good numerical tool for the quantitative investigation of the effect of impurities on step dynamics if step pinning works as described in the Cabrera–Verm...

Cathodoluminescence (CL) of laboratory forsterite chondrules has been characterized to clarify the formation process of chondrules and related mechanism of the crystal growth in a supercooled melt. Color CL image of the experimentally grown forsterite exhibits significant blue luminescence in the main branches of the interior structure of lab-chondrule, which reflects to the anisotropy of crystallization. A new CL band centered at 450–525 nm (2.76–2.36 eV) in blue to green region might be assigned to a microdefect-related center, which is a diagnostic peak for the forsterite that was formed due to the rapid growth as high as ∼10 mm/s or higher from a supercooled melt.

The growth of crystal induces a change of ambient environment (temperature, concentration, etc.), and the environmental change gives some feedback to the growth of crystal. The interaction between the crystal growth and ambient environment is important to be taken into consideration, also in the crystallization process of cosmic crystals observed in chondritic meteorites. In this lecture, we will introduce the phase‐field simulation, which is one of the powerful numerical methods to treat the crystal growth and diffusion fields (temperature, concentration, etc.) simultaneously. Participants can experience some phase‐field simulations on their own laptop by using a newly developed Java program, which will be distributed at the school.

Chondrules are submillimeter-sized and spherical-shaped crystalline grains consisting mainly of silicate material observed in chondritic meteorites. We numerically simulated pattern formation of a forsterite (Mg2SiO4)-chondrule in the melt droplet using a phase-field method. Because of the large surface-to-volume ratio, the surface cooling term was introduced in the framework of this method. We reproduced an unique crystal growth pattern inside the droplet composed of two distinguishable parts; the rim that covers whole droplet surface, and dendrite inside the droplet. It was found that the rim was formed when there is a large temperature difference of ∼100 K between the center and surface of the droplet due to the large cooling flux at the surface. In order to obtain the temperature difference, we derived temperature distribution of the droplet analytically, and concluded that the rim was formed only when the droplet cools rapidly at a rate of Rcool∼103 K s−1. However, when the surface cooling was so lar...

A barred-olivine (BO) chondrule usually has an olivine rim that covers the chondrule surface. Numerous experiments have been carried out to reproduce the BO texture. However, the rim structure could be reproduced only in a few studies reported in the literature. The difficulty in reproducing the rim structure lies in the fact that its formation condition has not been constrained experimentally or theoretically. In the present paper, we have carried out numerical simulations of crystal growth of a highly-supercooled melt droplet of pure forsteritic composition (Mg2SiO4), and succeeded in reproducing the double structure, i.e. the rim and the dendrite. The droplet cools from the surface, the temperature of which should be cooler than the center of the droplet. Since a crystal grows faster along the cooler surface than across the hotter center, the rim was found to be formed when the temperature difference between the center of the droplet and its surface is large enough. From our results, both from numerical simulations and analytical consideration, we found that the double structure of rim and the dendrite could be formed only when the cooling rate is within a narrow range, which depends upon the degree of supercooling. Our results, for the first time, could explain why the formation of rim of BO texture was hardly reproduced in the previous experiments reported in the literature to date.