Growth of equiaxed dendritic crystals settling in an undercooled melt,Part 2: Internal solid fraction

Experiments are conducted to measure the internal solid fraction evolution of equiaxed dendritic crystals that are freely growing and settling in an undercooled melt using the transparent model alloy succinonitrile–acetone. The internal solid fraction is determined from the measured settling speed and crystal envelope shape and size. Depending on the melt undercooling and acetone concentration, the internal solid fraction is found to vary between 0.55 and 0.1. In all experiments, the internal solid fraction decreases continually during settling. Based on heat and solute balances, a model is developed for predicting the internal solid fraction evolution under convective conditions. Nusselt and Sherwood number correlations are obtained that allow for the calculation of the thermal and solutal boundary layer thicknesses at the crystal envelope. The measured and predicted internal solid fraction evolutions are found to be in good agreement.     

Measurements of dendrite tip growth and sidebranching in succinonitrile–acetone alloys

Experiments are carried to investigate free dendritic growth of succinonitrile–acetone alloys in an undercooled melt. The measurements include the steady dendrite tip velocity and radius, the non-axisymmetric amplitude coefficient of the fins near the tip, and the envelope width, projection area, and contour length of the sidebranch structure far from the tip. It is found that the measured dendrite tip growth Péclet numbers agree well with the predictions from a stagnant film model that accounts for thermosolutal convection in the melt. The measured tip selection parameter, σ^?, is verified to be independent of the alloy composition, but shows a strong dependence on the imposed undercooling. The universal amplitude coefficient, A₄, is measured to be equal to 0.004, independent of the undercooling, but the early onset of sidebranching prevents its accurate determination for more concentrated alloys. For the self-similar sidebranch structure far from the tip, scaling laws are obtained for the measured geometrical parameters. While melt convection causes some widening of the sidebranch envelope, and the early onset of sidebranching for alloy dendrites results in a 25% upward shift of the envelope width, the projection area and, hence, the mean width of a sidebranching dendrite, as well as its contour length in the sidebranch plane, obey universal power laws that are independent of the convection intensity and the alloy composition.     

Effect of solid–liquid density change on dendrite tip velocity and shape selection

Phase-field simulations are used to examine tip velocity and shape selection in free dendritic growth of a pure substance into an undercooled melt in the presence of a density change between the solid and liquid. The dendrite is assumed to grow two-dimensionally inside a Hele-Shaw cell. The phase-field model is coupled with a previously developed two-phase diffuse interface model to simulate the flow in the liquid that is induced by the density change. The predicted dependence of the dendrite tip growth Péclet number on the relative density change is compared with an available analytical solution and good agreement is obtained. The simulations verify that the dendrite tip selection parameter, modified to account for the different densities of the solid and liquid phases, is independent of the relative density change.     

Columnar grain growth pattern with fluid flowing in molten pool

A coupled model was used to simulate columnar grain growth in TIG (tungsten inert‐gas) molten pool of nickel base alloy. The cellular automaton algorithm for dendritic growth is incorporated with solute transport model to take fluid flow into consideration. The results indicate that shear flow changes the solute distribution at the S/L (Solid/Liquid) interface, leading to asymmetrical growth of columnar grains. The dendrite arms on the upstream side grow fast, while the growth of dendrite arms on the downstream side is much delayed. However, dendrite arms on both sides are not as well‐developed as the grain growth without flow. With inlet flow velocity increasing, the phenomenon becomes more obvious. In addition, shear flow also results in more severe coring segregation. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dendritic growth with interfacial energy anisotropy

A theory for dendritic growth of needle shaped crystals growing in supercooled pure melts and dilute alloys is proposed. This theory predicts the growth velocity and tip radii as functions of supercooling and alloy concentrations without introduction of additional conditions such as marginal stability. A key ingredient of the theory is introducing the effect of the interfacial energy anisotropy. Deviation of the shape of the dendrite from a paraboloid of revolution is permitted, consistent with a small interfacial energy anisotropy. The radius of curvature and the growth rate as functions of melt supercooling and alloy concentrations are determined in terms of the anisotropy, and are compared with experimental results. The predictions are in agreement with the experimental results, especially at large supercoolings. The deviation at lower supercoolings can be attributed to the neglect of natural convection in the present theory.     

Stochastic modeling of columnar dendritic grain growth in weld pool of Al‐Cu alloy

A multi‐scale model is used to simulate columnar dendritic growth in TIG (tungsten inert‐gas) weld molten pool of Al‐Cu alloy. The grain morphologies at the edge of the weld pool are studied. The simulated results indicate that the average primary dendrite spacing changes during the solidification process in the weld pool because of the complicated thermal field, solute diffusion field and competitive growth. And it is shown that the secondary dendrite arms grow insufficiently in the space between dendrite trunks if the primary dendrite spacing is small. And the phenomenon has been explained by analyzing the influence of the solute accumulation on the constitutional undercooling and undercooling gradient when there are two different opposite solute diffusion fields. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of surface kinetics on the dendritic growth of ice in supercooled water

The dendritic growth of ice in supercooled water was investigated experimentally. It has been found that, with an increase in the degree of initial supercooling of water, the data of measuring the growth rate of the dendrite tip v _t, the average distance between the tip and the first side branch z? _SB, and the fractal dimension d _f of the dendrite contour diverge from the analytically and numerically calculated parameters of dendritic growth. It is shown that the discrepancy between experiment and theory is due to the transition from the diffusional growth to the growth controlled by the surface kinetics. The nature of the anisotropic surface kinetics of ice growth in supercooled water is discussed.     

Investigation of directional solidified Al–Ti alloy

Al–1 wt% Ti alloy was directionally solidified upwards under argon atmosphere under the two conditions; with different temperature gradients (G = 2.20–5.82 K/mm) at a constant growth rate (V = 8.30 μm/s) and with different growth rates (V = 8.30–498.60 μm/s) at a constant temperature gradient (G = 5.82 K/mm) in a Bridgman furnace. The dependence of characteristic microstructure parameters such as primary dendrite arm spacing (λ₁), secondary dendrite arm spacing (λ₂), dendrite tip radius (R) and mushy zone depth (d) on the velocity of crystal growth and the temperature gradient were determined by using a linear regression analysis. A detailed analysis of microstructure development with models of dendritic solidification and with previous similar experimental works on dendritic growth for binary alloys were also made.     

Growth Rate Effects during Indium-Antimony Crystal Growth

Local interface velocities are tracked radioscopically in the III-V semiconductor compound indium-antimony grown in a vertical Bridgman-Stockbarger furnace. Comparisons are made of interface velocities from five different compositions (40, 49, 50, 55, and 60 at.% Sb). Under specific growth conditions, the growth velocity for stoichiometric melts was comparatively constant and very close to the translation velocity. Measured chemical homogeneity was excellent, though polycrystallinity could occur when concentration boundary layers formed ahead of the interface. Off-stoichiometric melts exhibited initial supercooling, resulting in transient interface velocities and polycrystallinity. The observed supercooling is governed by chemical segregation in the melt. Thus, local growth velocity fluctuations are unambiguously attributed to a coupling of ompositional effects in the melt and crystal facetting kinetics.     

CFD modelling of single crystal growth of potassium dihydrogen phosphate (KDP) from binary water solution at 30 °C

Crystal growth rate depends on both diffusion and surface reaction. In industrial crystallizers, there exist conditions for diffusion-controlled growth and surface reaction-controlled growth. Using mathematical modelling and experimental information obtained from growth studies of single crystals, it is possible to separate these phenomena and study how they are affected by concentration, slip velocities of particles, temperature and finally estimate the parameters for crystal growth models.In this study, a power-law growth model using activity-based driving force is created. Computational fluid dynamics (CFD) was used to evaluate the thickness of a diffusion layer around the crystal. Parameters of the crystal growth model were estimated using a non-linear optimization package KINFIT. Experimental data on growth rate of the (1 0 1) face of a potassium dihydrogen phosphate (KDP) single crystal and simulated data on the thickness of a diffusion layer at the same crystal face were used in parameter estimation. The new surface reaction model was implemented into the CFD code. The model was used to study the effect of flow direction on growth rate of the whole crystal with various slip velocities and solute concentrations.The developed method itself is valid in general but the parameters of crystal growth model are dependent on the system. In this study, the model parameters were estimated and verified for KDP crystal growth from binary water solution.     

Effect of the Temperature Fluctuation of the Melt on β‐BaB2O4 (BBO) Crystal Growth

In this paper the effect of the growth temperature fluctuation, for instance, the transient furnace temperature variation due to a short‐term electric power supply interruption on BBO crystal growth was investigated based on the theory of temperature wave transmitting in melt and the boundary layer theory of melt. It was found that the critical width of the temperature pulse to avoid the temperature wave penetrating through the boundary layer and reaching to the growth interface at a constant rotation speed (9∼4 r/min) is 69∼150 s and the corresponding amplitude of the temperature pulse is high more than 60 °C due to the large thickness of the velocity boundary layer of the melt. This result indicates that a small transient temperature fluctuation has no significant effect on the crystal quality, and therefore implies that not only transport processes but interface growth kinetics, a two‐dimensional nucleation growth mode at the interface may also dominate the crystal growth.     

ø300 mm直拉硅单晶生长过程中的变晶现象及其影响因素

直拉法生长直径300 mm硅单晶过程中,直径均匀是获得高品质硅单晶的关键。在生产实践中发现,当硅晶体进入等径生长阶段,过高的提拉速度会引起晶体发生扭晶现象,导致晶线断裂随即变晶,对等径生长不利。本文采用数值模拟和理论相结合的方法分析了ø300 mm硅单晶生长过程中扭晶现象的成因,建立了不同提拉速度下晶体直径与熔体温度分布的关系,分析了晶体发生扭晶的影响因素。结果表明,随着提拉速度的增加,熔体自由表面产生过冷区且该过冷区随提拉速度的增加不断扩大,过冷区的产生是导致晶体发生扭晶的主要原因。提出了一种基于有限元热场数值模拟的最大稳定提拉速度的判别方法,并给出了通过改变晶体旋转速度来改善熔体自由表面温度分布的工艺措施建议,从而避免晶体扭晶现象的发生。研究结果对设计大尺寸硅单晶生长热场具有一定的指导作用。     

Modulation of dendrite growth of cuprous oxide by electrodeposition

Cuprous oxides with different dendrite morphologies were formed on F-doped tin oxide (FTO) covered glass substrates by potentiostatic deposition of cupric acetate. The effects of pH value (varied from 5.00 to 5.80) of electrolytes on the crystal morphologies of cuprous oxide were studied. Different crystal morphologies of cuprous oxides were obtained by the change of velocity of vertical growth and lateral growth through varying the pH value of electrolyte. The processes of Cu₂O dendrite crystal growth were analyzed through SEM images at different deposition times. Photoelectrochemical properties of the Cu₂O thin films prepared in the system are also studied.     

Numerical simulation of thermal and mass transport during Czochralski crystal growth of sapphire

The thermal and flow transport in an inductively heated Czochralski crystal growth furnace during a crystal growth process is investigated numerically. The temperature and flow fields inside the furnace, coupled with the heat generation in the iridium crucible induced by the electromagnetic field generated by the RF coil, are computed. The results indicate that for an RF coil fixed in position during the growth process, although the maximum value of the magnetic, temperature and velocity fields decrease, the convexity of the crystal‐melt interface increases for longer crystal growth lengths. The convexity of the crystal‐melt interface and the power consumption can be reduced by adjusting the relative position between the crucible and the induction coil during growth. (© 2010 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.     

Macro-segregation, dynamics of interface and stresses in high pressure LEC grown crystals

High dislocation density and strong dopant inhomogeneities have been found in high pressure liquid-encapsulated Czochralski (HPLEC) grown crystals. The origin and underlying mechanisms of these defects are attributed to the complex nature of transport phenomena in the HPLEC system. Our integrated computer model (MASTRAPP) can simulate this process by calculating the flow and heat transfer in both the melt and the gas, and thermal-elastic stress in the crystal. In this work, this model has been further extended to investigate the development of thermal stress in the growing crystal and the redistribution of dopant in the melt. The results for InP growth show complex gas flow and heat transfer pattern in the system. Two large stress spots are predicted by the model, one at the edge of the crystal just above the encapsulant layer and the other in the top corner of the crystal. Although the stress always remains largest at the first location, its value decreases as the crystal grows, due to the enhanced cooling of the crystal. A curved crystal/melt interface is also found to introduce high thermal stresses in its vicinity, which may be dangerous because of a high temperature at the interface and thus a low strength of the crystal. The model also predicts both radial and longitudinal dopant segregation in the growing crystal, and shows that the dopant redistribution in the melt is caused by the complex flow pattern in the melt. This is the first time, that a strong radial dopant segregation has been predicted based on a comprehensive flow model for a HPLEC growth.     

Shape evolution of detached Bridgman crystals grown in microgravity

Detached (or dewetted) Bridgman crystal growth defines that process in which a gap exists between a growing crystal and the crucible wall. In microgravity, the parameters that influence the existence of a stable gap are the growth angle of the solidifying crystal, the contact angle between the melt and the crucible wall, and the pressure difference across the meniscus. During actual crystal growth, the initial crystal radius will not have the precise value required for stable detached growth. Beginning with a crystal diameter that differs from stable conditions, numerical calculations are used to analyze the transient crystal growth process. Depending on the initial conditions and growth parameters, the crystal shape will either evolve towards attachment at the crucible wall, towards a stable gap width, or inwards towards eventual collapse of the meniscus. Dynamic growth stability is observed only when the sum of the growth and contact angles exceeds 180°.     

Influence of growth conditions on microstructure and properties of GaSe crystals

GaSe crystals have been grown from melt. There are several reasons why it is difficult to meet ideal demands for nonlinear optic material, GaSe single crystal. First, these crystals have a tendency towards lamination because of great difference in a and c crystal lattice parameters and very weak Vander der Waals forces in c direction. Next, there is a great difference in saturation vapor pressure of the components, which can cause nonstoichiometry of a melt-grown crystal composition. Another obstacle in the growth of perfect GaSe crystals is dendrite formation caused by instability of the growth front. To overcome this obstacle we used Bridgman technique and have found the temperature and pressure conditions, and growth velocity which provide growth of perfect bulk single crystals of about 100 mm in length and 20 mm in diameter. Sharp Laue patterns and a rocking curve confirm perfect structure of the grown crystals. Electron-probe X-ray microanalysis shows stoichiometric composition of GaSe crystals and X-ray phase analysis reveals presence of single-phased hexagonal structure.     

Stability effects in dendritic crystal growth

We propose that the maximum-velocity principle conventionally used in theories of dendritic crystal growth be replaced by a stability criterion of the form ?ρ² = constant, where ? is the growth velocity and ρ is the tip radius. Our argument is based on a linear stability analysis of the nearly-paraboloidal steady-state solution in the case of small but nonvanishing capillary. We find that dendrites which are too broad and slow suffer tip-splitting instabilities, whereas those which are too sharp and fast tend to broaden and slow down because of a side-branching instability. Our calculated operating point (with no free parameters) determines a curve of growth versus undercooling which is in substantial agreement with the data of Glicksman, Schaefer and Ayers.     

Diffusion-controlled crystal growth in K2OSiO2 compositions

The growth of cristobalite dendrites in two K₂OSiO₂ glasses containing 10.3 and 15.0 mol% K₂O has been studied over a wide range of temperature. The kinetics of crystallization were linear; i.e. the growth rates were independent of time. The observed crystallization rates were consistent with a diffusion-controlled mechanism. Measurements of the tip radii of curvature of the dendrites are consistent, at low undercoolings, with the predictions of Horvay and Cahn for the growth of an isolated dendrite. However, at lower temperatures at which the dendrites are closely spaced, the predicted radii are seriously in error. A model is described which takes account of the overlapping diffusion fields of neighboring dendrites and gives improved agreement with the observed growth rates.The presence of a maximum in each of the crystal growth rate versus temperature curves has not been explained. This phenomenon is apparently related to changes in the size, shape and spacing of the dendrites, which occur in the same temperature range.