The effect of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of two-dimensional shear flow on the stability of a crystal interface in the supercooled melt of pure substance. Similar to the three-dimensional shear flow as described in our previous paper, the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude. However, compared with the case of Laplace equation for steady state thermal diffusion field, due to the existence of time partial derivatives of the temperature fields in diffusion equation the absolute value of the gradients of the temperature fields increases, therefore destabilizing the interface. The circular interface is more unstable than in the case of Laplace equation without time partial derivatives. The critical stability radius of the crystal interface increases with shearing rate increasing. The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

Linear stability analysis on a spherical particle growing from a binary melt under the far-field flow

The solutions of temperature and solute fields around a spherical crystal growing from a binary melt under the far-field flow are obtained.Based on the results,a linear stability analysis on the spherical interface growing from the binary melt under the far-field flow is performed.It is found that the constitutional supercooling effect ahead of the spherical crystal interface under the far-field flow is enhanced compared with that without the flow.The growth rate of the perturbation amplitude at the up-wind side of the spherical crystal interface is larger than that at the down-wind side.The critical stability radius of the crystal interface decreases with the increasing far-field flow velocity.Under the far-field flow,the whole spherical interface becomes more unstable compared with that without the flow.     

远场来流条件下过冷熔体球晶生长的稳定性

建立了远场来流条件下过冷熔体球晶生长的温度场和浓度场稳态模型,分析了对流对球晶周围温度场和浓度场的影响,并以Trivedi的纯扩散球晶稳定性判据为基础,推导出远场来流条件下过冷熔体球晶生长的临界稳定性判据. 研究表明:远场来流条件下,迎流面的扰动振幅增加速率明显大于背流面的扰动振幅增加速率. 振幅增加速率最大值对应的扰动阶次从迎流面到背流面逐渐减小,随着球晶半径增加而增大. 对流使迎流面的稳定性降低,背流面的稳定性增大. 随着流速的增加,球晶的临界稳定半径减小. 关键词： 球晶 远场来流 界面形态稳定性 Trivedi判据     

Linear and non-linear stability of melt flows in magnetic fields

This review considers the stability of melt motion in two simplified models of semiconductor crystal growth by either vertical gradient freeze (VGF) or Czochralski (Cz) processes under the influence of various magnetic fields. In VGF the crystal is grown at the bottom of the crucible, resulting in a stable thermal stratification of the melt. The presence of a stabilizing temperature gradient surprisingly decreases the stability of the flow driven by a rotating magnetic field (RMF). The instability of the travelling magnetic field (TMF)-driven flow, in contrast, is significantly delayed by thermal stratification in VGF. The TMF may, thus, be used in VGF to control the shape of the solidification interface or the radial dopant distribution without causing undesirable flow oscillations. The crystal is pulled out from the melt in the Cz process, producing an unstable temperature gradient below the crystal. The RMF is able to force the resulting unstable buoyant flow into a state of small-scale, high-frequency turbulence that may be regarded as stable for practical purposes. This effect is experimentally observed over a wide range of Grashof numbers, up to 10⁹, characteristic for a large Cz system.     

Behavior of the Melt in the Vertical Bridgman Method with a Low Axial Temperature Gradient

The thermal conditions for the growth of Ge crystals with a diameter of 50 mm by the vertical Bridgman method in the case of low thermal-gravitational convection are studied using model experiments. Distilled water being hydrodynamically similar to the Ge melt is used as the model liquid. When modelling by means of the light cut method, it is established that mixture particles move along the heat flow direction from top to bottom. It is shown that an axial temperature gradient of 2 K/cm or more increases the contribution of thermal diffusion to mass transfer at a vertical flow rate of 0.09 mm/s or more. The numerical simulation of thermal convection in the Oberbeck–Boussinesq approximation confirms the absence of convection under the given thermal conditions. However, the deviation of the container axis from the vertical by 0.5° during the process of crystal growth contributes to the increase in the flow rates in the liquid phase up to 0.55 mm/s.     

金属熔体近壁面流动剪切模型及其对金属凝固影响的理论研究

本文建立了金属熔体近壁面流动剪切模型, 并分析了流动剪切对金属凝固的影响. 针对A356合金计算结果表明:层流流动的熔体内部剪应力随垂直斜板表面距离的增大而减小, 随着流动长度的增加先急剧下降之后趋于稳定; 紊流流动的熔体所受的剪应力随着垂直倾斜板表面距离的增大先急剧下降之后趋于稳定, 随着流动长度的增加而不断增大; 斜板倾角越大, 斜板上相同位置的熔体层受到的剪应力越大; 熔体垂直斜板表面距离越小, 柱状晶所承受的弯曲应力越大; 斜角越大, 斜板上相同位置的柱状晶的弯曲应力越大; 随着熔体在倾斜板表面流动长度的增加, 在层流阶段, 倾斜板表面柱状晶根部所受的弯曲应力先急剧下降之后趋于平稳, 而在紊流阶段, 弯曲应力是缓慢增加的; 理论分析表明柱状晶在熔体近壁面流动过程受到的最大弯曲应力低于αup -Al晶粒的屈服强度, 所以斜板上熔体流动产生的弯曲力不能将柱状晶折断, 只能将晶粒冲刷游离到熔体中使晶粒增殖, 与实验结果相符合. 所以本模型可以很好地解释熔体近壁面流动过程中的剪切本构关系以及剪应力对凝固组织的影响.     

界面动力学对过冷熔体中球晶生长界面形态的影响

考虑了在非平衡凝固条件下球晶生长过程中界面动力学系数随界面温度的变化,利用渐近分析方法求出了在过冷熔体中球晶生长温度场和界面的近似解析解,研究了非线性界面动力学过冷对于过冷熔体中球晶界面形态和生长速度的影响.研究表明,界面动力学系数越大,球晶的生长速度越快; 反之,表明界面动力学系数越小,球晶的生长速度越慢.与忽略界面动力学的情形比较,在球晶生长过程中依赖于界面温度变化的界面动力学显著地减缓了晶体生长的速度. 关键词： 球晶 界面形态 渐近分析     

Free convection flow of nanofluid over infinite vertical plate with damped thermal flux

In this article, the unsteady free convection flow and heat transfer of nanofluid past over an infinite vertical plate is considered. The fractional generalized Fourier's law with Caputo time derivatives with power-law model to describe the influence of memory on the nanofluid behavior. The analytical solutions for dimensionless temperature and velocity fields and dimensionless thermal flux are obtained by means of Laplace transformation. The fluid is water based nanofluid containing nanoparticles of CuO or Ag. The effects of fractional and physical parameters are discussed graphically.     

直拉硅单晶生长过程中工艺参数对相变界面形态的影响

为改善晶体相变界面形态,提高晶体品质,提出了一种融合浸入边界法（immersed boundary method,IBM）和格子Boltzmann法（lattice Boltzmann method,LBM）的二维轴对称浸入边界热格子Boltzmann模型来研究直拉法硅单晶生长中的相变问题.将相变界面视为浸没边界,用拉格朗日节点显式追踪相变界面;用LBM求解熔体中的流场和温度分布;用有限差分法求解晶体中的温度分布.实现了基于IB-LBM的动边界晶体生长过程研究.得到了不同晶体生长工艺参数作用下的相变界面,并用相变界面位置偏差绝对值的均值和偏差的标准差来衡量界面的平坦度,得到平坦相变界面对应工艺参数的调整方法.研究表明,相变过程与晶体提拉速度、晶体旋转参数和坩埚旋转参数的相互作用有关,合理地配置晶体旋转参数和坩埚旋转参数的比值,能够得到平坦的相变界面.     

Thermal and chemical diffusion in the rapid solidification of binary alloys

Solidification of binary alloys is characterized by the necessity to reject away from the advancing front two conserved quantities: the latent heat released at the solid-liquid interface and the solute atoms that cannot be accommodated in the solid phase. As thermal diffusion is much faster than chemical diffusion, the latter is generally assumed to be the rate limiting mechanism for the process, and the problem is addressed through the isothermal approximation. In the present paper we use the phase-field model to study the planar growth of a solid germ, nucleated in its undercooled melt. We focus on the effects of a noninstantaneous thermal relaxation. The steady growth predicted at large supersaturation in the isothermal limit is prevented. Depending on the value of the Lewis number the growth rate is limited by either mass or heat diffusion; in the latter case we observe a sharp transition between two different regimes, in which originates a nonmonotonic time dependence of the interface temperature. The effects of this transition reflect in the composition of the solidified alloy.     

The use of magnetic fields in vertical Bridgman/Gradient Freeze-type crystal growth

This paper outlines advanced vertical Bridgman/Gradient Freeze techniques with flow control using magnetic fields developed for the growth of semiconductor crystals. Low-temperature flow modelling, as well as laboratory-scaled crystal growth under the influence of rotating, travelling, and static magnetic fields are presented. Experimental and numerical flow modelling demonstrate the potential of the magnetic fields to establish a well-defined flow for tailoring heat and mass transfer in the melt during growth. The results of the growth experiments are discussed with a focus on the influence of a rotating field on the segregation of dopants, the influence of a travelling field on the temperature field and thermal stresses, and the potential of rotating and static fields for a stabilization of the melt flow.     

磁场对水平温度梯度下双层流体热对流的影响

在水平温度梯度下,双层流体交界面的表面张力会出现梯度,驱动热毛细对流运动,造成热剪切层内的扰动.本文数值模拟了不同重力条件下,双层流体内的对流现象,得出了在微重力时,对流运动将引起热剪切层内强烈的扰动.为了减弱这种扰动,我们利用磁场对流体的运动进行控制.为此,又对微重力条件下,不同方向应用磁场下的热剪切层内扰动行为进行了数值研究,结果显示,磁场对热剪切层稳定性有促进作用,加入法向的应用磁场最为有效.     

Estimation of temperature elevation generated by ultrasonic irradiation in biological tissues using the thermal wave method

In most previous models,simulation of the temperature generation in tissue is based on the Pennes bio-heat transfer equation,which implies an instantaneous thermal energy deposition in the medium.Due to the long thermal relaxation time τ(20 s-30 s) in biological tissues,the actual temperature elevation during clinical treatments could be different from the value predicted by the Pennes bioheat equation.The thermal wave model of bio-heat transfer(TWMBT) defines a thermal relaxation time to describe the tissue heating from ultrasound exposure.In this paper,COMSOL Multiphysics 3.5a,a finite element method software package,is used to simulate the temperature response in tissues based on Pennes and TWMBT equations.We further discuss different factors in the bio-heat transfer model on the influence of the temperature rising and it is found that the temperature response in tissue under ultrasound exposure is a rising process with a declining rate.The thermal relaxation time inhibits the temperature elevation at the beginning of ultrasonic heating.Besides,thermal relaxation in TWMBT leads to lower temperature estimation than that based on Pennes equation during the same period of time.The blood flow carrying heat dominates most to the decline of temperature rising rate and the influence increases with temperature rising.On the contrary,heat diffusion,which can be described by thermal conductivity,has little effect on the temperature rising.     

弱熔体对流对定向凝固中棒状共晶生长的影响

利用渐近方法求出在弱对流熔体中定向凝固棒状共晶生长的浓度场的渐近解,研究了弱熔体对流对定向凝固中棒状共晶生长的影响.结果表明,弱熔体对流对定向凝固中棒状共晶生长有显著的作用;平均界面过冷度不仅与棒状共晶的棒间距、生长速度有关,还与流动强度有关;当生长速度一定时,随着流动强度增大,棒状共晶的平均界面过冷度减小.利用最小过冷原则,获得棒间距与生长速度和流动强度的关系.结果表明,当生长速度比较小时,随着流动强度增大,棒状共晶的棒间距增大;当生长速度比较大时,随着流动强度增大,棒状共晶的棒间距变化减弱;棒状共晶的生长速度越小,流动对棒状共晶生长的影响越大.利用本文的解析结果计算在对流条件下Al-Cu共晶的棒间距,结果显示随着转速增大或径向距离增大,共晶的间距增大,这与Junze等的实验结果相符合.     

远场来流对过冷熔体中球状晶体生长的影响

利用渐近分析方法研究了远场来流引起的对流对过冷熔体中球状晶体的生长形态的影响.结果表明,由远场来流导致的对流使得正在生长的球状晶体的界面在向着来流的前部朝来流方向相反的方向生长, 并且提高了朝来流的相反方向的生长速度, 在背风方向衰减；正在衰减的球状晶体的界面在向着来流的前部加速衰减, 在背风方向减缓衰减. 关键词： 球状晶体 远场来流 对流 界画形态     

Rheological Behavior of Blends of Poly(Phenylene Sulfide) with a Thermotropic Liquid Crystalline Aromatic Poly(Ether Ester)

A new type of thermotropic liquid crystalline aromatic poly(ether ester) (PEE) was prepared from 1,3-bis(4′-carboxyphenoxy)benzene, 1,4-diacetoxybenzene, and p-acetoxybenzoic acid through a melt transesterification process. The rheological behavior of blends of poly(phenylene sulfide) (PPS) with PEE was studied using a high-pressure capillary rheometer with the shear rate range of 50 s^?1 to 3000 s^?1. The results show that according to the range of shear rate, the flow curves of PEE/PPS blends can be divided into three zones: a first shear-thinning zone (n < 1, “n” represents non-Newtonian indexes), a shear-thickening zone (n > 1), and a second shear-thinning zone (n < 1), and the former two zones are more obvious with the increase of PEE content or elevated temperature. In the second shear-thinning zone, the PPS melt is close to a Newtonian fluid at high temperature and high shear rate; meanwhile the non-Newtonian behavior of the PPS melt at high temperature is enhanced with the addition of PEE. The apparent viscosity of PPS melts sharply dropped after adding PEE, especially at relatively low temperature and low shear rate. The curve of apparent viscosity vs. shear rate starts to flatten out after adding PEE, suggesting that the addition of PEE lowers the sensitivity of PPS to shear rate. As the content of PEE increases, the activation energy of the viscous flow, ΔE_η, of PPS decreases, which means that adding PEE weakens the temperature sensitivity of the apparent viscosity of the PPS melt. It can clearly be seen that the addition of PEE is beneficial to the processing of PPS.     

The effect of two-dimensional of a crystal interface in shear flow on the stability a supercooled melt

A model is developed based on the time-related thermal diffusion equations to investigate the effects of twodimensional shear flow on the stability of a crystal interface in the supercooled melt of a pure substance. Similar to the three-dimensional shear flow as described in our previous paper, the two-dimensional shear flow can also be found to reduce the growth rate of perturbation amplitude. However, compared with the case of the Laplace equation for a steady-state thermal diffusion field, due to the existence of time partial derivatives of the temperature fields in the diffusion equation the absolute value of the gradients of the temperature fields increases, therefore destabilizing the interface. The circular interface is more unstable than in the case of Laplace equation without time partial derivatives. The critical stability radius of the crystal interface increases with shearing rate increasing. The stability effect of shear flow decreases remarkably with the increase of melt undercooling.     

Heat and mass transfer characteristics during rapid solidification of Fe-Cu peritectic alloys

The viscose flow and microstructure formation of Fe-Cu peritectic alloy melts are investigated by analyzing the velocity and temperature fields during rapid solidification, which is verified by rapid quenching experiments. It is found that a large temperature gradient exists along the vertical direction of melt puddle, whereas there is no obvious temperature variation in the tangent direction of roller surface. After being sprayed from a nozzle, the alloy melt changes the magnitude and direction of its flow and velocity rapidly at a height of about 180 μm. The horizontal flow velocity increases rapidly, but the vertical flow velocity decreases sharply. A thermal boundary layer with 160–300 μm in height and a momentum boundary layer with 160–240 μm in thickness are formed at the bottom of melt puddle, and the Reynolds number Re is in the range of 870 to 1070 in the boundary layer. With the increase of Re number, the cooling rate increases linearly and the thickness of thermal boundary layer increases monotonically. The thickness of momentum boundary layer decreases slowly at first, then rises slightly and decreases sharply. If Re < 1024, the liquid flow has remarkable effects on the microstructure formation due to dominant momentum transfer. The separated liquid phase is likely to form a fiber-like microstructure. If Re>1024, the heat transfer becomes dominating and the liquid phase flow is suppressed, which results in the formation of fine and uniform equiaxed microstructures. Supported by the National Natural Science Foundation of China (Grant Nos. 50121101 and 50395105)     

Liquid/solid interface profile of melt grown oxide crystals II. Crystal quality

Temperature gradients and rotation rates influence the quality of Czochralski grown oxide crystals. Decreasing the heat transfer from the melt level and increasing the rotation rate increases the optical homogeneity and structural perfection of the crystal, due to convex interface becoming flattened. Growth striations due to temperature fluctuations in the melt during growth are affected by the heat losses from the melt level but are practically unaffected by the rotation rate as long as a narrow ring of centrifugally streaming melt in the neighbourhood of the growing crystal is not formed. The parts of the crystals grown in the presence of this ring contain no striations.