Simulation Of Crystal Growth Research Articles

Clear-air Seeding: Opportunities and Strategies

Clear-air seeding is the artificial creation or enhancement of cloud cover in clear air which is between saturation with respect to ice and saturation with respect to liquid water. Previous work has demonstrated that clear-air seeding is possible, and may be economically worthwhile through its effects on the radiation balance near the ground. Order-of-magnitude arguments, contrail observations and direct measurements are used to assess the frequency and horizontal extent of opportunities for clear-air seeding in the upper troposphere. Clear, ice-supersaturated air is a common occurrence in the mid-latitude upper troposphere at all times of year, typically appearing over 100 km-wide regions downstream of cyclonic or convective storm systems. The frequency of opportunities was also estimated for a particular application of clear-air seeding-cloud cover enhancement during winter nights to reduce space heating costs. Opportunities were studied for the middle and lower troposphere through a careful analysis of radiosonde humidity data, and for the upper troposphere through observalions of natural cirrus. Under suitable cloud cover and wind conditions, five to ten opportunities each season are estimated for locations in the northeastern United States, a frequency sufficient for economic advanlage based on a previous analysis. Two methods are discussed for seeding clear, ice-superssLurated air to create clouds. Seeding with ice crystals is studied using a numerical simulation of ice crystal cloud growth, which includes both ice physics and radiative transfer. Ice crystal seeding is most effective when the seeding rate is greatest and the moist layer seeded is deepest. Aircraft contrails generally utilize only a small fraction of the available water vapor in excess of ice saturation, before they settle out of the moist layer in which they were created. The development of clouds seeded with ice-nucleating aerosols, rather than ice crystals, depends greatly on the rate at which the seed material nucleates ice crystals. This type of seeding would produce a longer lasting, lower density cloud than would seeding with ice crystals.

Phase-field simulations of crystal growth with adaptive mesh refinement

In this paper, we propose the phase-field simulation of dendritic crystal growth in both two- and three-dimensional spaces with adaptive mesh refinement, which was designed to solve nonlinear parabolic partial differential equations. The proposed numerical method, based on operator splitting techniques, can use large time step sizes and exhibits excellent stability. In addition, the resulting discrete system of equations is solved by a fast numerical method such as an adaptive multigrid method. Comparisons to uniform mesh method, explicit adaptive method, and previous numerical experiments for crystal growth simulations are presented to demonstrate the accuracy and robustness of the proposed method.

Microscopic Rate Constants of Crystal Growth from Molecular Dynamic Simulations Combined with Metadynamics

Atomistic simulation of crystal growth can be decomposed into two steps: the determination of the microscopic rate constants and a mesoscopic kinetic Monte Carlo simulation. We proposed a method to determine kinetic rate constants of crystal growth. We performed classical molecular dynamics on the equilibrium liquid/crystal interface of argon. Metadynamics was used to explore the free energy surface of crystal growth. A crystalline atom was selected at the interface, and it was displaced to the liquid phase by adding repulsive Gaussian potentials. The activation free energy of this process was calculated as the maximal potential energy density of the Gaussian potentials. We calculated the rate constants at different interfacial structures using the transition state theory. In order to mimic real crystallization, we applied a temperature difference in the calculations of the two opposite rate constants, and they were applied in kinetic Monte Carlo simulation. The novelty of our technique is that it can be used for slow crystallization processes, while the simple following of trajectories can be applied only for fast reactions. Our method is a possibility for determination of elementary rate constants of crystal growth that seems to be necessary for the long-time goal of computer-aided crystal design.

Numerical computations of faceted pattern formation in snow crystal growth

Faceted growth of snow crystals leads to a rich diversity of forms with remarkable sixfold symmetry. Snow crystal structures result from diffusion-limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size, and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three spacial dimensions using a computational method recently introduced by the present authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. In our computations, surface energy effects, although small, have a pronounced effect on crystal growth. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns, and scrolls on plates. Although all these forms appear in nature, it is a significant challenge to reproduce them with the help of numerical simulations for a continuum model.

SiO gas emission and triple line dynamics of small silicon droplets on quartz

Periodic spreading motions of small silicon melt droplets on quartz (SiO2) have been observed by a video microscope and mass signals of the emitted molecules were thereby recorded simultaneously. The triple line of the droplet expands slowly and retracts spontaneously. The intensity of a mass signal at 44amu is clearly correlated to these changes of the triple line circumference and seems to stem from SiO molecules that are emitted as reaction products from the vicinity of the triple line. The velocity of the triple line has been studied as a function of temperature. Results can be described by an Arrhenius-like behavior with an activation energy of 111±14kJmol−1. For a detailed analysis of the experimental results a model for the oxygen transport through the interfaces and its diffusion through the droplet is presented. It can be used to predict etch profiles on the substrate in the vicinity of the triple line by using well known thermophysical data of silicon and quartz. These theoretical profiles are in good agreement with the experimental profiles that have been measured ex-situ by atomic force microscopy and provide a value for the diffusion constant of D=3.3×10−4cm2s−1 for oxygen in liquid silicon which fits well with the values used in crystal growth simulations.

Phase Field Method Simulation of Dendrite Crystal Growth of Metal Nickel Based on Fractal Theory

Using fractal theory simulation of dendrite crystal DLA growth model of pure substance, the undercooling during solidification process of crystal nucleation is simulated; and then in the crystal nuclei are formed on the basis of a pure substance, the phase field model and combined with the finite difference method further differentiation simulation of dendrite crystal growth. According to MATLAB programming, the simulation results obtained by field and temperature field can be seen in the DLA growth, growth model with random premise, for the same kind of material simulated dendrite crystal have both similarities and differences exist. Then, we can get the conclusion, through fractal growth of DLA model with phase field model of dendrite nucleation, growth process is carried out the simulation results, a simple by phase field model is more accord with the dendrite crystal in the experiment.

Simple evolution of complex crystal species

Cairns-Smith has proposed that life began as structural patterns in clays that self-replicated during cycles of crystal growth and fragmentation. Complex, evolved crystal forms could then have catalyzed the formation of a more advanced genetic material. A crucial weakness of this theory is that it is unclear how complex crystals might arise through Darwinian evolution and selection. Here we investigate whether complex crystal patterns could evolve using a model system for crystal growth, DNA tile crystals, that is amenable to both theoretical and experimental inquiry. It was previously shown that in principle, the evolution of crystals assembled from a set of thousands of DNA tile types under very specific environmental conditions could produce arbitrarily complex patterns. Here we show that evolution driven only by the dearth of one monomer type could produce complex crystals from just 12 monomer types. When a monomer type is rare, crystals that use few of this monomer type are selected for. We use explicit enumeration to show that there are situations in which crystal species that use a particular monomer type less frequently will grow faster, yet to do so requires that the information contained in the crystal become more complex. We show that this feature of crystal organization could allow more complex crystal morphologies to be selected for in the right environment, using both analysis in a simple model of self-assembly and stochastic kinetic simulations of crystal growth. The proposed mechanism of evolution is simple enough to test experimentally and is sufficiently general that it may apply to other DNA tile crystals or even to natural crystals, suggesting that complex crystals could evolve from simple starting materials because of relative differences in concentrations of the materials needed for growth.

A simple anisotropic surface free energy function for three-dimensional phase field modeling of multi-crystalline crystal growth

Phase field modeling has emerged as a powerful tool in crystal growth simulation, but the description of the anisotropic function for three-dimensional (3D) interfacial free energy without miss-orientation remains quite difficult, especially for the multi-crystalline cases. In this paper, a simple function for keeping the convexity in the reciprocal surface free energy is proposed, and the simulated equilibrium shapes are in good agreement with the analytical ones. Some examples on the dendritic growth in a supercooled melt for mono- and multi-crystalline crystals are also illustrated.

A comparative study between MEAM and Tersoff potentials on the characteristics of melting and solidification of carborundum

Molecular dynamic simulations of bulk melting, surface melting and crystal growth of SiC are carried out. The atomic interactions in SiC are calculated by MEAM and Tersoff potentials separately. The results show that the bulk melting of SiC with MEAM potential exhibits its relations to temperature similar to that with Tersoff potential, while can be indicated by the mean atomic energy, Lindemann index and structure order parameter. The difference between them is the bulk melt point: MEAM is 4250 K, while Tersoff is 4750 K. At the same superheat degree, the velocities of surface melting of SiC separately, with MEAM and Tersoff potentials are in substantial agreement. But at the same absolute temperature, the surface melting of SiC with MEAM potential is faster than that which the Tersoff potential, which is due to the difference in thermodynamic melting point. The Measured value of the thermodynamic melting point of MEAM is 3338 K compared with 3430 K of Tersoff. On the crystal growth side, the crystal growth velocity of SiC with MEAM potential is related to the undercooling. The fastest velocity corresponds to the undercooling of 400 K. However, the crystal of SiC with Tersoff potential cannot grow in the undercooling of 0 K1000 K. Overall, the MEAM potential is better than Tersoff potential in the sense of describing the melting and solidification of carborundum.

Numerical simulation of Cz crystal growth in rotating magnetic field with crystal and crucible rotations

Three-dimensional numerical simulations were carried out to analyze the effect of crystal and crucible rotating directions on heat and oxygen transport in Czochralski crystal growth system under rotating magnetic field. A low-Reynolds number k-ε model was used for the description of the turbulent processes. The computational results show that when rotations of crystal, melt and crucible are in the same direction, the oxygen concentration in the melt is the lowest and a low concentration crystal can be obtained. Under rotating magnetic field, the heat transport is dominated by lorentz force induced streaming field and the influence of crystal and crucible rotations on heat transport is weak. Moreover, the effect of crystal rotation on the melt turbulent flow and oxygen transport is stronger than that of crucible.

Phase field modeling of crystal growth with nonlinear kinetics

The phase field model has emerged as a powerful tool for the simulation of crystal growth. With the thin-interface approximation, the interface thickness could be greatly relaxed to the length scale of microstructures. So far, the model is used only for linear kinetics. However, nonlinear kinetics, where the kinetic coefficient is a function of temperature or velocity, is often encountered in crystal growth, particularly for faceted growth. In this paper, we propose numerical methods to the thin-interface phase field model for nonlinear kinetics. Besides the benchmarking with the available solutions, an example on a facet growth that develops into different morphologies with different undercoolings is simulated for the first time.

A new constraint for chondrule formation: condition for the rim formation of barred-olivine textures

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.

Fixed and Deforming Grid Solutions of Solidification in an Undercooled Melt: A Benchmark Problem

A problem involving the solidification of a solid seed in an undercooled melt is posed. A sharp interface between solid and liquid is assumed, and the equilibrium solidification temperature at this interface is controlled by both curvature and kinetic undercooling. Numerical solutions based on a fixed grid and on a deforming grid are developed. In the limits of a vanishing surface energy and large molecular mobility, these solutions are verified by comparing with known analytical solutions. In general applications, with curvature and kinetic undercooling present, predictions from the fixed and deforming grid solutions are in close agreement. The results obtained highlight the role of the surface curvature and kinetic undercooling in controlling solidification speed and acceleration. In addition, both the fixed and deforming solution methods provide verification tools for more general 2-D and 3-D crystal growth simulations.

A fast, robust, and accurate operator splitting method for phase-field simulations of crystal growth

In this paper we propose a fast, robust, and accurate operator splitting method for phase-field simulations of dendritic growth in both two- and three-dimensional space. The proposed method is based on operator splitting techniques. We split the governing phase-field equation into three parts: the first equation is calculated by using an explicit Euler's method. The second is a heat equation with a source term and is solved by a fast solver such as a multigrid method. The third is a nonlinear equation and is evaluated using a closed form solution. We also present a set of representative numerical experiments for crystal growth simulation to demonstrate the accuracy and efficiency of the proposed method. Our simulation results are also consistent with previous numerical experiments.

The Mobility of Water Molecules through Gas Hydrates

Crystal growth simulations of gas hydrates have suggested that hydrate cages may occasionally be occupied by H(2)O rather than guest molecules, leaving interstitial defects within the hydrate crystal. Further inspection of the behavior of these interstitial H(2)O molecules has revealed that they are relatively highly mobile entities within a gas hydrate. In this paper, we report these observations and examine the molecular mechanisms responsible for the transport of these interstitial molecules through hydrate crystals. Four distinct pathways for the H(2)O molecule transport between cages are found, each facilitated by the presence of empty cages. The relative richness of the observed behavior of interstitial defects suggests that interstitial diffusion could be an important mechanism for the mass transport of H(2)O molecules through gas hydrates.

Phase-field investigation of microstructure evolution under the influence of convection

Numerical simulation of solidification and crystal growth has attracted industrial attention as a powerful engineering tool for processes and alloy optimization. In this work, we present a detailed study of the influence of flow on microstructure evolution during solidification. First, we include convection effects into an existing quantitative phase-field model that is meant to simulate dendritic growth of a binary alloy. We apply it to investigate the solidification of a Fe-Mn alloy under external flow conditions. In addition, we present an extension of the quantitative phase-field model of two-phase growth which includes natural and forced convection effects in the melt phase. We use this extension to investigate directional solidification of eutectic lamellae under the influence of convection as well as microstructure formation of peritectic growth in the presence of convection.

Crystal growth simulations of methane hydrates in the presence of silica surfaces

We present a molecular dynamics simulation study of the crystal growth of methane hydrates in the presence of model silica (SiO(2)) surfaces. The crystal growth under apparent steady-state conditions shows a clear preference for bulk solution. We observe rather disordered water arrangements very close to the silica surface within about 5 Å in both liquid and crystalline regions of the system. These disordered structures have dynamic and structural properties intermediate between those exhibited by molecules in bulk liquid and crystalline phases. The presence of methane molecules appears to help stabilize these structures. We observe that under appropriate conditions, the hydroxylated silica surfaces can serve as a source of methane molecules which can help promote hydrate growth near the surfaces.

Multi-Scale Molecular Simulation of Crystal Growth and Mass Transport in Solution in Copper Electrodeposition

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A volume of fluid approach for crystal growth simulation

A new approach to simulating the dendritic growth of pure metals, based on a recent volume of fluid (VOF) method with PLIC (piecewise linear interface calculation) reconstruction of the interface, is presented. The energy equation is solved using a diffuse-interface method, which avoids the need to apply the thermal boundary conditions directly at the solid front. The thermal gradients at both sides of the interface, which are needed to obtain the front velocity, are calculated with the aid of a distance function to the reconstructed interface. The advection equation of a discretized solid fraction function is solved using the unsplit VOF advection method proposed by López et al. [J. Comput. Phys. 195 (2004) 718–742] (extended to three dimensions by Hernández et al. [Int. J. Numer. Methods Fluids 58 (2008) 897–921]), and the interface curvature is computed using an improved height function technique, which provides second-order accuracy. The proposed methodology is assessed by comparing the numerical results with analytical solutions and with results obtained by different authors for the formation of complex dendritic structures in two and three dimensions.

Numerical simulations of bulk crystal growth on different scales: silicon and GeSi

AbstractNumerical modelling has become an important tool for improving or introducing new processes in bulk crystal growth. For a complete description, scales ranging from atomistic ones up to those of industrial furnaces have to be considered. This article presents methods used on different scales for the Czochralski growth of silicon as well as of Ge1–xSix alloys and for ingot casting of silicon. It shows the recent developments for the different scales and the attempts at coupling the approaches.