Radial segregation due to weak convection in a floating zone

The influence of weak convection, caused by surface tension forces, on radial dopant segregation occurring in crystals grown under microgravity conditions is studied numerically. The geometry considered corresponds to a floating-zone configuration with partially coated melt surfaces consisting of small evenly distributed spots of free surfaces. In order to distinguish dopant distribution due to weak convection clearly from distribution due to diffusion the spots only cover one quarter of the periphery. Thus, surface tension-driven convection is allowed only over one quarter of the floating-zone configuration resulting in an asymmetric dopant distribution. The percentage of free surfaces present is varied in order to alter the Marangoni flow rates. The maximum dopant concentration due to radial segregation is plotted as a function of a certain convection level. The results of the present numerical study are supposed to be used to design corresponding space experiments launched at the end of the year 2000.     

Influence of solidification of binary eutectic lamellar systems on surface tension driven convection in zero gravity conditions

The effect of solidification of the lamellar binary eutectic system on the onset of surface tension driven convection (so-called Marangoni instability) is studied in a zero gravity environment. Some main general conclusions concerning the possibility for onset of the Marangoni convection within the melt can be drawn from the analysis, viz., the curvature of the free surface has a destabilizing effect on the onset of the flow; the increased perturbed heat transfer rate from the system stabilizes the melt; the onset of the solutal Marangoni convection closely depends on the ratio of the lamellae half-widths and the solutal Marangoni instability is more sensitive against the perturbations than the temperature Marangoni one.     

Observation of concentration boundary layers around growing or dissolving KH2PO4 crystals in stagnant solutions

The solute concentration profiles around KH₂PO₄ crystals growing or dissoloving in stagnant aqueous solutions were studied by the schlieren method after Toepler. By this technique the thicknesses of concentration boundary layers around the crystals, formed by free convection of solution, were measured in dependence of crystal size and super (under) saturation. In the case of dissolution, theoretical estimation of the boundary layer thickness yielded a fair agreement with the observed value. For growth, the process is dominantly limited by surface kinetics, whereas volume diffusion plays a minor role.     

Influence of Gravitation on the Processes of Heat and Mass Transfer in Solution Crystal Growth by the Travelling Heater Method (THM) (II)

The two-dimensional quasi-stationary thermodiffusive hydrodynamic problem of the crystal PbTe growth from solution by the travelling heater method has been solved. Computation is given for the temperature field distribution in crystal and solution. The picture of fluxes in solution was obtained under free convection for different lengths of solvent zone. It is shown that in the absence of absence of gravitation at the growing interface in solution there occurs a constitutional supercooling, whereas the free convection removes it, which leads to greater morphological stability of a growing crystal face.     

Physical and physicochemical processes accompanying powder synthesis,growth of PbMoO<Subscript>4</Subscript> crystals,and their annealing in various media: III. Methods of controlling the crystallization front shape

Different methods of controlling the crystallization front shape are considered: variation in a crucible position in a heater, rotation of a crystal, introduction of impurities absorbing melt radiation, and formation of forced convection under dominant free convection in the melt.     

Large eddy simulation of Marangoni convection in Czochralski crystal growth

Large eddy simulation model is used to simulate the fluid flow and heat transfer in an industrial Czochralski crystal growth system. The influence of Marangoni convection on the growth process is discussed. The simulation results agree well with experiment, which indicates that large eddy simulation is capable of capturing the temperature fluctuations in the melt. As the Marangoni number increases, the radial velocity along the free surface is strengthened, which makes the flow pattern shift from circumferential to spiral. At the same time, the surface tension reinforces the natural convection and forces the isotherms to curve downwards. It can also be seen from the simulation that a secondary vortex and the Ekman layer are generated. All these physical phenomena induced by Marangoni convection have great impacts on the shape of the growth interface and thus the quality of the crystal. (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutocapillary convection in germanium‐silicon melts

Surface tension gradients in free crystal growth melts give rise to convective flow. If these gradients are due to thermal gradients, the well known thermocapillary (Marangoni) convection ensues. Concentration gradients due to segregation at the interface during growth can lead to additional solutocapillary convection. A system with large solutocapillary convection is Ge‐Si due to the pronounced segregation and the strong difference in surface tension; solutal buoyancy convection is also present due to the large density difference between Ge and Si. Solutocapillary convection will oppose thermocapillary convection in the Ge‐Si system since Si, having the higher surface tension, is preferentially incorporated into the crystal. A set of experiments directly proving and partially quantifying the effect has been conducted under microgravity during a parabolic flight campaign by recrystallizing Ge‐Si mixtures of different compositions, between 3% and 9% Si, in a crucible with tracers to visualize the movement. Solutocapillary flow with initial flow rates in excess of 5.5 cm/s at the onset of crystallization was measured. A slight dependence of the flow velocity on the initial Si content has been found. Experiments on the ground showed the same effect but with overall smaller speeds; this difference can be explained by the additional action of solutal buoyancy convection. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

微重力条件下Marangoni数对分离结晶过程的影响

为了了解微重力条件下新型分离结晶生长过程中熔体热毛细对流的基本特征,利用有限差分法进行了三维数值模拟.当熔体顶部分别为自由表面及固壁边界条件时,得到了新型分离结晶Bridgrnan生长过程中熔体热毛细对流的速度分布和温度分布.结果表明:熔体顶部为自由表面时,当Marangoni数较小时,在上自由表面和下部狭缝处自由表面的表面张力的驱动下,熔体内部产生了逆时针和顺时针两个流动方向相反的流胞,此时熔体内的流动状态为稳态;随着Marangoni数进一步的增大,流胞的流动逐渐增强并逐步向熔体内部扩展,熔体内部温度分布非线性增强,上自由表面和下部狭缝处自由表面处速度增大;当Marangoni数超过某一临界值后,流动转化为非稳态流动.当熔体顶部为固壁时,与熔体顶部为自由表面时相比,临界Marangoni数增大.流动失稳的物理机制是流速的变化和阻力的变化之间存在滞后.     

Mathematical modeling and experimental investigation of the effect of temperature gradients on crystallization processes under terrestrial and space conditions

Mathematical modeling of the processes of heat and mass transfer during directed crystallization under terrestrial and space conditions is performed on the basis of experimental data on the temperature distribution (boundary conditions). Convective processes are described by the system of Oberbeck-Boussinesq equations together with the heat-conduction equation (the Stefan problem). A dependence of the intensity of thermal gravitational convection on the radial and axial temperature gradients is established. It is shown that one of the necessary conditions for the growth of homogeneous semiconductor crystals under both terrestrial and zero-gravity (on board spacecraft) conditions is the absence of the free surface of a melt (the Marangoni convection) and optimization of the temperature gradients (first of all, the radial gradient).     

Measurement of two‐dimensional distribution of surface supersaturation over a sodium chlorate crystal surface using multidirectional interferometry

The two‐dimensional (2D) distributions of surface supersaturation of sodium chlorate crystals with and without solutal convection have been measured by means of a multidirectional interferometry (MDI) technique coupled with the principles of three‐dimensional (3D) computer tomography. When solutal convection was present over a top face, the supersaturation at the center of the face was depleted by a factor of >0.9 with reference to the value at the edges of the crystal. When the convection was suppressed using an upside‐down geometry, the depletion of supersaturation at the center of the face was much smaller, <0.4. Therefore, the supersaturation difference between the edges and the face center, which is responsible for the morphological stability due to volume diffusion for the solute, becomes less important compared to the effect of convection due to hydrodynamic reasons. This result should give us a key to solve why the crystal quality is sometimes better in convection‐free microgravity conditions because of improved stability of a crystal face caused by more homogeneous distribution of supersaturation over the crystal surface.     

On the significance of surface tension driven flow in floating zone melting experiments

The predominance of the surface tension driven (Marangoni) flow over the buoyancy driven free convection for small zone lengths in electron beam floating zone melting of doped molybdenum rods is pointed out. The observed different patterns of microsegregation and the critical conditions of their occurrence, i.e. B-cores below a critical zone length of l_c = 3.5 mm and B-striations above this value, are shown to be in qualitative agreements with theoretical predictions made from convection models.     

Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

We consider the instability of the steady, axisymmetric thermocapillary convection in cylindrical liquid bridges. Finite-difference method is applied to compute the steady axisymmetric basic solutions, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for liquid bridges of O(1) aspect ratio Γ (= length/radius) the first instability of the basic state is through either a regular bifurcation (stationary) or Hopf bifurcation (oscillatory), depending on the Prandtl number of the liquid. The bifurcation points and the corresponding eigenfunctions are predicted precisely by solving appropriate extended systems of equations. For very small Prandtl numbers, i.e. Pr < 0.06, the instability is of hydrodynamical origin that breaks the azimuthal symmetry of the basic state. The critical Reynolds number, for unit aspect ratio and insulated free surface, tends to be constant, Re_c → 1784, as Pr → 0, the most dangerous mode being m = 2. While for Pr 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is m = 3 for 0.1 Pr 0.8 and m = 2 for Pr 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent linear-theory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a nondeformable free surface.     

Influence of the surface layer of the solution on the nucleation of fibrillar trimolybdate crystals and their growth on air bubbles

The effect of the free surface area of solutions on the nucleation of fibrillar trimolydate crystals is described. The nucleation period is inversely proportional to the free surface area of solutions at a fixed volume. The observations of the growth of fibrillar monocrystals on air bubbles are reported.     

Solutal convection around growing protein crystals and diffusional purification in Space

Convective and diffusional mass transport to an isolated crystal growing from solution, with slow linear interface kinetics, is considered analytically as a generic scaling model. We focus on the interface kinetics which is slow as compared to the diffusion mass transport which is typical of protein crystal growth. Independently, full-scale numerical solutions of transport equations around a cylindrical crystal, at the center of the bottom of a cylindrical cell filled with the solution, are found. The two approaches give results that agree over a wide range of parameters, providing dimensionless relationships that allow predictions of the contribution of convection and diffusion to mass transport. Requirements for microgravity level in Space experiments to achieve diffusional mass transport are estimated on the basis of these relationships. Coefficients of impurity distribution between a growing crystal and its solution, under the influences of convection and diffusion around the crystal, were numerically evaluated as functions of time. The results provide further support for the hypothesis concerning the role of the impurity depletion zone in the purification of a growing crystal. They also reveal that in general, the impurity distribution within the crystal is not homogeneous due to convection. The effects of various factors on growth kinetics and crystal purity are considered.     

The Effect of Free Convection on Solid/Liquid Interface in the Czochralski Crystal Growth Method

This paper regards the interaction of free and forced convection in the Czochralski configuration. A relationship between the crystal rotation rate and growth parameters at which the crystallization front remains flat, at which the crystallization front remains almost flat, has been studied. This relationship is compared with experimental data.     

Mass transport effect within the boundary layers in solution crystal growth: Holographic study of KTP and KDP crystal growth

Using holographic phase-contrast interferometric microphotography, we have carried out real-time investigations of the mass transport processes taking place during the high temperature solution growth of KTiOPO₄ (KTP) crystal and low-temperature solution growth of KH₂PO₄ (KDP) crystal. Our experiments demonstrate that a mere diffusion boundary layer is not existing. The mass transport process within the boundary layers is a result of the coupled effect of diffusion and convection actions, no matter whether it is high-temperature or low-temperature solution growth. Under free convection state, the influence of bulk supersaturation on the thickness of solute boundary layer exists in the two different regions. The solute concentration distribution within the layer is an exponential function of the position.     

Physical and physicochemical processes accompanying powder synthesis,growth of PbMoO<Subscript>4</Subscript> crystals,and their annealing in various media: II. Shape of the crystal-melt interface caused by melt evaporation and supercooling

The temperature distribution in a melt of PbMoO₄ is investigated using a modified thermocouple and a scanning device. It is shown that the PbMoO₄ melt whose thermal field is significantly inhomogeneous at different depths demonstrates a strong (below the melting temperature) supercooling at the surface due to the evaporation. The position of the isotherm corresponding to the melting temperature of PbMoO₄ in the melt is determined by the degree of the melt supercooling at different growth parameters. The results obtained show that the conditions of mass and heat exchange in the melt are mainly determined by convection. The free convection remains dominant even at intense rotation of a crystal and the most thorough thermal isolation of the crystallization unit.     

Helical impurity distribution produced by a hydrodynamic instability

Temperature oscillations have been recorded in the liquidn phase during vertical solidarification of salt solutions and dilute aqueous suspensions. The oscillations produce a characteristic solidification pattern consisting of a helical segragation of impurities. The pitch of the helix is correlated with the temperature period of the oscillation. A model is proposed based on unstable thermal convection. It involves the rotation of a convective roll around its own axis. It is shown that the solidification technique provides a new method for the determination of the characteristics of the convective behaviour.     

Marangoni Convective Effect during Crystal Growth in Space

GaSb:Te and GaInSb samples have been solidified under microgravity conditions during the D2 Spacelab mission. Experimental design and parameters are described. Analysis of the thermal data taken during the flight, associated to numerical simulations of heat transfer in the experiment, with the help of FIDAP, gave the experimental conditions (thermal gradients and growth rate). Quantitative chemical analyses of the samples show a chemical segregation characteristic of strong mixing in the melt during crystal growth. Silica crucibles with an internal screw thread groove on the inner wall were used in order to get dewetting of samples from the crucible. It was therefore supposed that Marangoni convection on the free surface associated to the groove might have been the source of convection. This hypothesis has been studied by numerical simulation using FIDAP and the velocity field obtained is in agreement with a strong perturbation of the solutal boundary layer ahead the solid-liquid interface. This can explain the observed chemical segregation.     

Growth and properties of dyed KDP crystals

Potassium dihydrogen phosphate (KDP) crystals doped with xylenol orange (XO) and methylthymol blue (MTB) are grown from aqueous solutions by the method of solvent evaporation at room temperature under the conditions of natural convection and by the method of temperature lowering. Studied is the influence of the mother solution acidity on the character of the crystal coloration. The color and coloration intensity of the grown crystals are shown to strongly depend on the solution's pH. It is revealed that the crystal habit changes in the presence of organic dyes. The optical transmission spectra and the luminescence spectra of KDP:XO solutions and of the grown crystals are measured. The effect of thermal treatment and UV‐irradiation on the coloration stability of the crystals is studied. It is found that the laser damage threshold in the prismatic impurity rich colored sectors of KDP:XO is the same as that in the prismatic sectors of pure KDP crystals. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)