Dynamics of solidification in binary melts during rapid cooling I. The basic equations

In this paper we deal with the analysis of the general complex model which describes the solidification of the binary melt. Within the framework of this model we consider the mass and energy transport in the system and the kinetics of the phase transformation. The paper is divided into two parts. In the first part the set of equations is derived, which describe the evolution of the system within the framework of the theory of stochastic processes. The emphasis is laid on the rapid changes of temperature in the system. In the second part the analysis of the mentioned set of equations is presented. It results in the delimitation of the conditions under which the adiabatic approximation and the kinetic phase diagrams at the phase interface can be applied.     

Pressure effects in the planar solidification of a finite slab: convective cooling and prescribed heat flux boundary conditions

Summary The classical Stefan problem assumes a fixed melting temperature. However, when the solid phase is the one with lower density (e.g., water) the solidification of the system causes an overall volume increase that is often contrasted by the container walls. In that case the growing pressure determines a continuous lowering of the freezing point, and the temperature field as well as the interface motion are strongly affected. This paper is concerned with these aspects of the problem; the planar solidification of a slab of finite thickness, contrasted by an opposing elastic force, is numerically simulated. The effects of two different boundary conditions are analysed. When the solidification is driven by convective cooling, the continuous advancement of the melting front is replaced by an asymptotic behaviour, until thermal equilibrium is attained. When the boundary condition is specified in terms of a prescribed heat flow, the melting front velocity is slowed down by a growing adverse temperature gradient. The influence of various parameters on the process is presented and discussed.     

Ripening of surface phases coupled with oscillatory dynamics and self-induced spatial chaos through surface roughening

Some pattern formation processes on single-crystal catalytic surfaces involve transitions between alternative surface phases coupled with oscillatory reaction dynamics. We describe a two-tier symmetry-breaking model of this process, based on nanoscale boundary dynamics interacting with oscillations of adsorbate coverage on microscale. The surface phase distribution oscillates together with adsorbate coverage, and, in addition, undergoes a slow coarsening process due to the curvature dependence of the drift velocity of interphase boundaries. The coarsening is studied both statistically, assuming a circular shape of islands of the minority phase, and through detailed Lagrangian modeling of boundary dynamics. Direct simulation of boundary dynamics allows us to take into account processes of surface reconstruction, leading to self-induced surface roughening. As a result, the surface becomes inhomogeneous, and the coarsening process is arrested way before the thermodynamic limit is reached, leaving a chaotic distribution of surface phases. (c) 1999 American Institute of Physics.     

The melting and solidification of nanowires

A mathematical model is developed to describe the melting of nanowires. The first section of the paper deals with a standard theoretical situation, where the wire melts due to a fixed boundary temperature. This analysis allows us to compare with existing results for the phase change of nanospheres. The equivalent solidification problem is also examined. This shows that solidification is a faster process than melting; this is because the energy transfer occurs primarily through the solid rather than the liquid which is a poorer conductor of heat. This effect competes with the energy required to create new solid surface which acts to slow down the process, but overall conduction dominates. In the second section, we consider a more physically realistic boundary condition, where the phase change occurs due to a heat flux from surrounding material. This removes the singularity in initial melt velocity predicted in previous models of nanoparticle melting. It is shown that even with the highest possible flux the melting time is significantly slower than with a fixed boundary temperature condition.     

强磁场条件下金属凝固过程中第二相的迁移行为

理论分析表明,第二相的迁移行为可以通过迁移速度进行表征.影响迁移速度的因素包括第二相和熔体的物理性质、磁场强度和梯度大小、第二相的形状和体积等因素.强磁场下洛伦兹力的效果为促进第二相在基体中的均匀分布,其效率在磁场强度大于某一定值时逐渐降低.在梯度强磁场条件下,第二相迁移行为和分布状态的主要控制参数是梯度磁场下的磁化力.在磁场梯度较小时,因洛伦兹力的制约磁化力控制第二相迁移的效果不明显,随着磁场梯度的增加,磁化力的作用效果逐渐增强.通过研究强磁场下Al-Si合金、Al-Ni合金中原位自生第二相的迁移行为实 关键词： 强磁场 迁移 第二相 凝固     

Heat transfer and diffusion-controlled kinetics of liquid–solid phase in titanium matrix composite during selective laser melting

This study describes a self-consistent theoretical model of simulating diffusion-controlled kinetics on the liquid–solid phase boundary during high-speed solidification in the melt pool after the selective laser melting (SLM) process for titanium matrix composite based on Ti–TiC system. The model includes the heat transfer equation to estimate the temperature distribution in the melt pool and during crystallization process for some deposited layers. The temperature field is used in a micro region next to solid–liquid boundary, where solute micro segregation and dendrite growth are calculated by special approach based on transient liquid phase bonding. The effect of the SLM process parameters (laser power, scanning velocity, layer thickness and substrate size) on the microstructure solidification is being discussed.     

Influence of external magnetic field,finite-size effects and chemical potential on the phase transition of a complex scalar field

A scalar model is built, as a quantum field theory defined on a toroidal topology, to describe a phase transition in films subjected to periodic boundary conditions and influenced by an external and constant magnetic field. Criticality is studied and the relations between the critical temperature, the film thickness, the magnetic field strength and the chemical potential are investigated. Since the model describes a second-order phase transition a comparison with the Ginzburg–Landau theory is made.     

非平衡凝固与固态相变中有关形核和长大的竞争研究

非平衡凝固和固态相变作为同热力学和动力学紧密相关的复杂理论体系,在物理冶金和凝聚态物理研究领域占据重要的地位。上述非平衡过程包含形核、生长和碰撞三个微观过程,其发生和发展均由一些竞争过程伴随、辅助甚至驱动。正确理解这些竞争过程决定了能否准确描述整个非平衡过程;材料制备中,其力学和物理/化学性能的提高也归根于这些竞争的发生与发展。因此,很有必要对非平衡凝固和固态相变中的竞争现象进行研究。本综述第一部分从形核/生长类相变角度对非平衡凝固和固态相变进行了概述,并强调了伴随其发生的竞争现象。第二部分立足于经典形核理论,对有关形核热力学、动力学以及形核方式的竞争现象进行了描述。第三部分对凝固中有关热扩散和溶质扩散以及固态相变中有关界面控制和扩散控制的生长方式竞争进行了描述。第四部分对本综述进行了总结性分析和概括,并对本领域的未来发展方向进行了预测。     

考虑晶粒分布的多晶体材料超声散射统一理论

现有超声散射统一理论可通过多晶体材料的微观结构和力学特性,实现全频域范围内衰减和相速度的正演建模,但其忽略晶粒尺寸分布的影响,进而降低了正演模型的计算精度.本文对不均匀介质的波动方程进行二阶Keller近似,用全频域格林函数推导介质中的平均波;以截断对数正态分布描述晶粒分布,构建加权的空间相关函数;结合材料的弹性模量协方差,建立含晶粒分布的超声散射统一理论,揭示晶粒分布对超声散射的影响规律;制备304不锈钢试块并开展超声散射实验.结果表明考虑晶粒分布特性后,纵波衰减谱和相速度谱相对于实验结果的相异性降低约49%和64%,横波衰减谱和相速度谱相对于实验结果的相异性降低约12%和4%.可见,本文的统一理论模型能有效修正晶粒分布导致的衰减谱和相速度谱偏差,为晶粒分布反演评价提供理论基础.     

Dynamics of Faceted Grain Boundary Grooves

The dynamics of dislocation-free crystal facets is examined in the context of grain boundary grooves at the junction between two crystallites of a solid and the liquid phase. The geometry and thermal conditions of grain boundary grooves allow a detailed analysis of facet morphology during solidification in terms of the nucleation and spreading rates of elementary crystal planes. Observations on the freezing of water in a two-dimensional cell reveal several dynamical features which are treated by the theory. Additional observations provide indications for the stiffness and premelting of grain boundaries.     

Dynamics of solidification in binary melts during rapid cooling II. Analysis of basic equations

In the first part of our paper we have derived a set of stochastic differential equations which describe the solidification of binary melts. The equations have been derived within the framework of the model in which the mass and heat transport and the kinetics of the phase transition is considered. In the second part of our paper we present the analysis of the set of general equations. On the basis of this analysis it will be obvious which approximations can be used for the solution of the basic equations in a particular regime of solidification. The adiabatic approximation is one of them. Another situation occurs when the thermodynamic conditions of the phase transformation change fast with time. The system not only moves away from thermodynamic equilibrium but also we can observe inertia of the system, which results in a delay of the evolution of the system respecting the steady-state regime (e.g. the nucleation processes) and the adiabatic approximation cannot be assumed. Concluding this paper, the method used in paper [13] to describe the nucleation in binary systems is presented as an example of the solution of the set of general equations in the case where the adiabatic approximation cannot be adopted.     

Stability of the crystal-melt system in the growing of large lithium niobate crystals by the low melt level Czochralski technique

The Lyapunov stability of the crystal-melt system and the solidification rate is studied for the growing of lithium niobate crystals by the low melt level Czochralski technique. The relaxation time is calculated as a function of growth conditions. The curves of transient processes are plotted at various variable perturbations. It is shown that, for optically homogeneous crystals to be produced, the stability of the solidification rate rather than the crystal-melt system as a whole should be ensured. The stability of the solidification rate is found to be maximal at the maximum temperature gradient at the solidification interface and the minimum melt level in a crucible.     

Monte Carlo computer experiments on critical phenomena and metastable states

Ising and Heisenberg models are studied by the Monte Carlo method. Several hundred up to 60 000 spins located at two- and three-dimensional lattices are treated and various boundary conditions used to elucidate various aspects of phase transitions. Using free boundaries the finite size scaling theory is tested and surface properties are derived, while the periodic boundary condition or the effective field-like ‘self-consistent’ boundary condition are used to derive bulk critical properties. Since Monte Carlo averages can be interpreted as time averages of a stochastic model, ‘critical slowing down of convergence’ occurs. The critical dynamics is investigated in the case of the single spin-flip kinetic Ising model. Also non-equilibrium relaxation processes are treated, e.g. switching on small negative fields the magnetization reversal and nucleation processes are studied. The metastable states found can be understood in terms of a scaling theory and the droplet model. Using a spin exchange model the phase separation kinetics of a binary alloy is simulated.     

Modelling of laser surface alloying and dispersing of ceramics

Laser‐supported processes can be used to modify the electrical and thermal properties of ceramic substrates locally. These processes are characterized by a strong thermal interaction between the laser beam and the ceramic surface that leads to localized melting. During the dynamic melting process an additive material is injected into the melt pool in order to modify the physical properties. The heat and mass transfer during this dynamic melting and solidification process has been studied numerically in order to identify the dominating process parameters. Simulation tools based on a finite‐volume method have been developed to describe the heat transfer, fluid flow and the phase change during the melting and solidification of the ceramic. The results of the calculation have been validated against experimental results.     

A mathematical model of scandium ore deposit formation in liquating magmatic melts

Solidification of liquating silicate magmatic melts may lead to formation of rare earth mineral deposits. By the example of quasi-binary system SiO₂–Sc₂O₃, the processes of cooling and directional solidification of the melt in an intrusive chamber have been studied, and velocities of the phase fronts and the width of the phase separation field have been calculated. Using the fluctuation approach, the physical and mathematical model of the formation and growth of dispersed phase in the continuous cooling of liquating melt was developed, and the conditions of incorporating the dispersed inclusions by solidified matrix phase were determined. The proposed model allows obtaining quantitative estimates of the size and number of inclusions per unit of hardened rock, depending on the solidification conditions and the initial chemical composition of the melt.     

Convective instability of solidification with a phase transition zone

The morphological instability of solidification is analytically studied in the presence of an anisotropic and heterogeneous phase transition zone with allowance for a liquid flow and convective heat-andmass transfer in this two-phase zone. The mechanism of breaking the stability of solidification is considered; it consists in a convective heat and impurity transfer during a liquid flow along channels in the phase transition zone. The morphological instability is subjected to linear analysis with allowance for a liquid flow in the liquid phase of the system, impurity diffusion in the two-phase zone, and the dependence of the transfer coefficients on the phase composition. The perturbation evolution parameter is determined for an anisotropic and heterogeneous two-phase zone, and neutral stability curves of the process are obtained. It is shown that taking into account impurity diffusion and an increase in the heterogeneity of the phase transition zone broaden the instability region and that a decrease in the anisotropy narrows this region. A new criterion of convective morphological instability of solidification with a two-phase zone is found, and it substantially broadens the instability region when the liquid flow velocity increases.     

On the multicomponent surface phase balance equations

The system of the field equations is formulated at the phenomenologically modelledn-component surface phase (e.g. interface, solidification front, shock wave, membrane etc.). The surface phase balance equations then constitute a closed set of the general boundary conditions of the appropriate balance equations of volume phases (as, for example, the heat conduction or diffusion equations etc.).     

Twodimensional stochastic theory of crystal growth

In the paper the twodimensional model of the stochastic theory of crystal growth is presented and used for the study of the morphology of the solidification front under different external conditions and for different rates of kinetic processes on the solidification front. The results show that the stability of the solidification front depends on the kinetic processes, which thus must be taken into account in the stability conditions.     

Free Fermions and the Classical Compact Groups

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: (i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; (ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions.