Fluid dynamics in crystal growth: The good,the bad,and the ugly

Fluid dynamics are important in processes that grow large crystals from a liquid phase. This paper presents a primer on fluid mechanics and convection, followed by a discussion of the physics and scaling of flows in such processes. Specific examples of fluid flows in crystal growth systems are presented and classified according to their impact on outcomes, good or bad. Turbulence in crystal growth is discussed within the limited extent of our understanding, which is incomplete, or ugly.     

Solutal convection in crystal growth: effect of interface curvature on flow structuration in a three-dimensional cylindrical configuration

Due to temperature and concentration gradients in the molten phase, it is well known that convective flows can develop in the bulk under normal conditions of gravity. These motions modify the shape of the growing interface and the concentration distribution along it. This study will only focus on the case of pure solutal convection and the effect of a given interface curvature on the flow. In particular, the transition from a 3D-flow to a 2D one as a function of the interface curvature is examined in order to investigate possible values of the operating parameters and fluid properties. The main aim is to justify the use of 2D-simulation tools for predicting the convective transport in cylindrical crystal growth ampoules.     

On the inhomogeneity of the transition surface layer of the solid core of the earth

Different geophysical data and conclusions of theoretical models, which can give information about the behavior of the solid and liquid cores of the Earth as well as about the existence of a transition layer as a temperature-hysteresis region at a relatively weak first-order phase transition, are compared. It is concluded that liquid inclusions inevitably exist in this region; these inclusions are involved (due to the complex convective processes occurring in the liquid core) in the transport of light materials from some areas of the solid-core surface. The porosity and permeability of the transition layer determine the seismic acoustic inhomogeneities in these areas, which contact the convective flows in the liquid core. In particular, this explains the well-known ??east-west?? effect. Obviously, the model of the crystalline core is not the only possible alternative for a model of a core with a metallic glasslike structure.     

Crystal habit prediction ‐ Including the liquid as well as the solid side

Commercially available methods of morphology prediction utilize molecular dynamics to estimate the crystal growth rates but predominantly consider the solid side. For the extension of these methods to a multi‐component solid‐liquid system the diffusion coefficient is required. Since, the diffusion coefficient enables the calculation of crystal growth rates and the morphology in presence of additives and solvents. Modeling the diffusion coefficient is achieved by conducting MD on a system consisting of the crystal surface and the liquid phase. The achieved results match very well with the calculated diffusion coefficient (Wilke‐Chang). In this case study benzoic acid is used as model substance with water as solvent.     

Numerical investigation of crystal growth process of bulk Si and nitrides – a review

Heat and mass transfer during crystal growth of bulk Si and nitrides by using numerical analysis was studied. A three‐dimensional analysis was carried out to investigate temperature distribution and solid‐liquid interface shape of silicon for large‐scale integrated circuits and photovoltaic silicon. The analysis enables prediction of the solid‐liquid interface shape of silicon crystals. The result shows that the interface shape became bevel like structure in the case without crystal rotation. We also carried out analysis of nitrogen transfer in gallium melt during crystal growth of gallium nitride using liquid‐phase epitaxy. The result shows that the growth rate at the center was smaller than that at the center. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The numerical simulation of heat and mass transfer during growth of a binary system by the travelling heater method

The influence of convection and heat and mass transfer on the shape and position of melt/solid interfaces and on radial composition segregation is analysed numerically for the travelling heater method growth of a binary alloy in a vertical transparent ampoule. Results are presented for crystal and melt with thermophysical properties similar to Cd_xHg_1−xTe with the assumption that the pseudobinary CdTe-HgTe phase diagram is true. The two-dimensional axisymmetric heat transfer equation, hydrodynamical equation and convective diffusion equation are included in the mathematical model. The rates of crystal growth and dissolution are supposed to be proportional to the compositional supercooling in the melt near the interfaces. It is shown for the conditions when convection is absent that the interfaces are asymmetrically positioned respectively to the heater centre line. Intensive convection makes their position more symmetrical but the length of the liquid zone greater. The flow pattern in the melt appears to be greatly influenced by solutal gravitational convection. The nonlinear dependence of the melt density on the temperature and composition are used in the model. The cases when speed of the heater is antiparallel (stable density stratification) or parallel (unstable stratification) to the vector of gravitational acceleration are considered.     

Numerical optimization of czochralski sapphire single crystal growth using orthogonal design method

Crystal quality during Czochralski (Cz) growth is influenced significantly by the convexity of solid/liquid (S/L) interface, which is related to operating conditions, such as Radio‐Frequency (RF) coil position, crystal rotation and crucible rotation. Generally, a flat interface shape is preferred for high‐quality crystal growth. It is difficult to achieve the optimized conditions even from numerical modeling due to the large computational load from examining all of the affecting factors. Orthogonal design/test method, fortunately, provides an efficient way to organize the study of multiple factors with the minimization of computational load. In the paper, this method is adopted to investigate the affecting factors of Cz‐sapphire single crystal growth based on the coupled calculation of thermal field and melt flows. The orthogonal analysis clearly reveals the optimized growth conditions to achieve a relative flat S/L interface under the current ranges of the parameters.     

Dual nature of the orientational effect of ultrasound on liquid crystals

The new model of thresholdless distortion of the orientational structure in a homeotropic layer of nematic liquid crystal with free ends in ultrasonic field has been experimentally substantiated for the first time. The model is constructed within the concepts of nonequilibrium thermodynamics and statistical hydrodynamics of liquid crystals for the frequency range in which the elastic and viscous wavelengths are, respectively, longer and shorter than the layer thickness. The main regularities of the phenomenon, which relate the conditional effect threshold to the ultrasonic frequency and layer thickness, have been established based on the experimental data for (20–150)-μm-thick layers in the frequency range of 0.1–9 MHz. These data are compared with the results of numerical calculations, performed taking into account two mechanisms of liquid crystal structure distortion (convective and nonlinear relaxation ones).     

Study of the dissolution and growth of salt single crystals in inhomogeneous acoustic fields: II. A focused acoustic field

The growth and dissolution of the isolated (100) face of a KDP crystal at exposure of the phase boundary to focused acoustic fields have been investigated. Visualization tools have been developed and the scale and dynamics of the arising acoustic flows have been established. A quadratic dependence of the dissolution rate on the sound pressure has been obtained; such a dependence is in agreement with the theoretical concepts about the dependence of the mass exchange rate in the acoustic boundary layer on the field parameters. A significant growth response of the face under study, normally exposed to ultrasound along the acoustic axis in the prefocal and postfocal planes of a spherical concentrator (f = 1.4 MHz), has been revealed. It is shown that the mechanisms of mass exchange enhancement in these acoustic modes are radically different. The results obtained show a possibility of controlling growth and dissolution of crystals by varying the parameters of an inhomogeneous acoustic field.     

SiC晶体PVT生长系统的流体力学模型及其有限元分析

本文根据SiC晶体PVT生长炉的实际提出了生长系统温度场计算的流体力学模型,采用有限元法分析了生长腔内的热传导、辐射和对流对生长腔内和生长晶体中温度空间分布的影响.通过对生长腔内及生长晶体中温度瞬态和稳态分布的分析,得出在加热的初始阶段腔内气体对流对坩埚内的温度分布有较大影响,在系统热平衡后辐射对腔内温度分布起决定作用的结论.     

Investigations of the process of crystal growth from a liquid zone by Seebeck measurements

An arrangement for measuring the thermoelectric voltage (Seebeck signal) during the crystal growth from a liquid zone is described. Using the example of growing PbTe single crystals by THM it is shown that different equilibrium temperatures at both phase boundaries provide a differential Seekeck voltage depending on the crystal growth rate. Relaxation times which are needed to reach steady-state conditions with respect to the concentration difference between the growing and solving interface in the case of a start or sudden stop of the heater motion can be obtained.     

Fourier analysis based phase shift interferometric tomography for three-dimensional reconstruction of concentration field around a growing crystal

The present work is concerned with the three-dimensional reconstruction of concentration field around a crystal growing from its aqueous solution using Fourier analysis based phase shift interferometric tomography. Projection data of convective field around a growing NaClO₃ crystal have been recorded using a Mach–Zehnder interferometer. A new numerical method based on the concept of real time phase shift interferometry has been employed for the analysis of the interferograms. By interpreting the interferograms as projection data, the 3-D distribution of salt concentration has been determined using principles of tomography. Study shows that the Fourier analysis-based phase shift method is a novel approach for crystal growth studies as it requires only one interferometric image for the determination of full phase information as opposed to the conventional real time phase shift interferometry wherein at least three or more phase-shifted interferograms are needed. Results have been presented in the form of interferograms, path-integrated concentration contours and three-dimensional concentration profiles over select horizontal planes above the growing crystal. Based on the reconstruction results, distribution of salt concentration in the crystal vicinity is determined and an appropriate mechanism of buoyancy-induced fluid movement in the growth chamber is discussed.     

Polarization mechanism of LiNbO3 crystal due to thermoelectric power and current-induced electric fields during growth via micro-pulling-down method

The mechanism of polarization due to thermoelectric power and current-induced electric fields during the growth of LiNbO₃ crystals was studied using a micro-pulling-down method. With no applied electric current, a +c single-domain crystal was grown regardless of the domain orientation of the seed crystal. This +c domain growth was consistent with the direction of the electric field caused by the thermoelectric power in the liquid, despite an opposing electric field in the solid due to the opposite sign of the Seebeck coefficient. Thus, it was the electric field in the liquid that determined the domain structure of the growing crystal. On the other hand, when a current was applied from the melt to the crystal, a −c domain crystal was grown. The electric current required for this domain inversion to occur became larger as the temperature gradient in the solid phase decreased. This shows that the electric field in the solid phase became large enough to induce domain inversion from +c to −c through a combination of the thermoelectric power in the solid phase and current-induced electric field.     

Study of the initial transient in the one-dimensional analytical models of impurity segregation during melt crystallization in the presence of convection

The well-known one-dimensional Burton-Prim-Slichter and Ostrogorsky-Müller analytical models obtained for the stationary mass transfer regime describe in a simple form the dependence of the effective impurity segregation coefficient on the ratio of the crystal growth and convective flow rates. Solutions for the initial transient regime are found in both models. It is shown that the formulas obtained make it possible to determine both the crystal growth rate and the convective mixing intensity on the basis of the analysis of impurity segregation in crystal.     

Preferred crystallographic orientation of nanocrystals embedded inside nanopores

The preferred crystallographic orientation of nanocrystals plays a significant role in determining their properties. From the wide variety of nanocrystal growth techniques, we focus in this paper on crystal growth by precipitation from liquid solutions inside porous substrates, and discuss the progress that has been made during the last decade concerning the control of crystal growth direction through this method. In this overview, the motivation and principal mechanisms of achieving highly oriented nanocrystals are presented. Moreover, different experimental challenges within the described growth technique are probed. The paper presents the thermodynamic and kinetic considerations for favoring crystal growth inside pores rather than bulk growth. A special focus is made on the origin of obtaining preferred crystallographic orientations in various types of materials, including varying perspectives of thermodynamic and kinetic driving forces. The paper ends with technological application of crystal growth with preferred crystallographic orientation inside nano-pores.     

Some Phase Boundary Region Correlations in the Liquid Phase Epitaxy Growth Process

In the present work the behavior of n-component liquid phase during the liquid phase epitaxial growth process is considered. As it follows from these considerations, in the liquid phase epitaxy growth one must distinguish two types of components - with a positive concentration gradient and with negative concentration gradient. A possible mechanism for deposition of the components having a negative concentration gradients, for which, in general, the liquid phase is superheated, is presented.     

晶体生长时的固液相结构变化分析

晶体的溶解、熔化以及结晶成核生长时的固液相原子结构是怎样变化的,晶体生长时的生长基元是原子还是原子团.本文根据各种材料液相结构的最新研究结果,提出不饱和配位结构转换模式,并以此模式对各种常见晶体材料从溶解、熔化到结晶生长时的液态母相结构变化以及晶体成核过程进行了描述和分析,认为晶体生长时的界面结构和液相结构十分接近,溶解、熔化主要是晶体表面的不饱和配位原子(离子)转换到液相结构的过程,晶体生长主要是液相中的不饱和配位原子(离子)转换到固液生长界面的位错位置,使配位结构更饱和的过程.随着液相过饱和度的增大,液相结构单元的原子数越来越多,吸附到晶体生长界面若来不及转换回液相,就形成新的位错生长中心,形成晶体生长缺陷.     

X-ray diffraction studies of protein crystal disorder

Protein crystals contain many kinds of disorder, but only a small fraction of these are likely to be important in limiting the diffraction properties of interest to crystallographers. X-ray topography, high-angular-resolution reciprocal space measurements, and standard crystallographic data collection have been used to probe three factors that may produce diffraction-limiting disorder: (1) solution variations during crystal growth, (2) macromolecular impurities, and (3) post-growth crystal treatments. Variations in solution conditions that occur in widely used growth methods may lead to variations in equilibrium protein conformation and crystal packing as a crystal grows, and these may introduce appreciable disorder for sensitive proteins. Tetragonal lysozyme crystals subjected to abrupt changes in temperature, pH, or salt concentration during growth show increased disorder, consistent with this mechanism. Macromolecular impurities can have profound effects on protein crystal quality. A combination of diffraction measurements provides insight into the mechanisms by which particular impurities create disorder, and this insight leads to a simple approach for reducing this disorder. Substantial degradation of diffraction properties due to conformation and lattice constant changes can occur during post-growth crystal treatments such as heavy-atom compound and drug binding. Measurements of the time evolution of crystal disorder during controlled crystal dehydration – a simple model for such treatments – suggest that structural metastability conferred by the constraints of the crystal lattice plays an important role in determining the extent to which the diffraction properties degrade.     

Gaseous control of vapour concentration in crystal synthesis from the vapour phase

The gaseous control in the crystal synthesis from the initial components is discussed. The dependences of the vapour concentrations in the growth region on inert gas flows have been calculated. It is shown that the use of additional inert gas flows conducted directly to the entries into the reaction zone makes it possible to change the vapour concentrations in the growth region rapidly and independently one from another, and these concentrations are linear with the main transport gas flows.     

Three-dimensional computations of solution hydrodynamics during the growth of potassium dihydrogen phosphate I. Spin up and steady rotation

A novel, massively parallel implementation of the Galerkin finite element method is used to study three-dimensional, time-dependent flows which occur during the rapid growth of potassium dihydrogen phosphate crystals from solution in a system employed by researchers at Lawrence Livermore National Laboratory. Computations for the hydrodynamics of system spin up and steady rotation indicate the importance of time-dependent flow phenomena and emphasize the significant role played by the support and crystal geometry in forming the complicated flows in this system. Predicted flow structures correlate well with experimental observations of inclusion formation.