Quantifying microstructures in isotropic grain growth from phase field modeling: Methods

While digital microstructures can be produced using various numerical models, quantitative characterization of these microstructures is often limited to a few attributes, such as grain diameter, the number of faces per grain and their distributions. A large number of topological and geometric properties that are useful for both experimental and theoretical analysis of microstructure–property relations have yet to be explored. To address this problem, we developed a series of numerical methods to compute geometric and topological properties of the microstructures. To test these methods, we performed a quantitative analysis of the digital microstructures obtained from a phase field simulation during isotropic grain growth. The new characterization methods allow us to not only identify each individual microstructure entity, including grain cell, grain boundary interface, triple junction line and vertex point, but also calculate the interface area, the triple junction length, and the curvature of grain boundary interfaces and triple junction lines. The detailed time evolution of these microstructural attributes, particularly the distributions of interface area and the length of triple junction lines, is extracted during the grain-coarsening process. The quantitative information gives us a detailed picture of grain growth in the phase field model that has not been seen before. Both the quantitative microstructures and the dynamic behavior are expected to serve as a benchmark for comparison to the microstructures obtained from other methods, including experiments.     

The analytical modeling of normal grain growth

A fundamental understanding of grain-growth phenomena provides a means to predict and control many of the observed material properties. This article reviews the development and present state of analytical theories of grain growth. The major geometric, dynamic, and statistical factors that must be considered in a rigorous formulation of grain dynamics are outlined.     

Modified mean field models of normal grain growth

Models of normal grain growth can either start from a postulated kinetic law for individual grains and yield a distribution of grain sizes or they can start from a postulated distribution and the kinetic law may be derived. Both methods are studied and a whole family of distributions based on new kinetic laws are derived using the first method. Both methods have recently been applied using Onsager’s extremum principle but it is now shown that more classical procedures are sufficient. Kinetic laws give an indication of what physical factors govern the growth or shrinkage of individual grains. A Rayleigh’s distribution seems to indicate that large grains are surrounded by grains smaller than the critical size and small grains are surrounded by grains larger than the critical size. The effects of the new family of kinetic laws on the development of grain size distributions are studied by numerical simulations.     

相场法模拟球形和盘形第二相粒子对晶粒长大的影响

利用相场法模型,模拟研究含不同尺寸和面积分数的球形和盘形粒子的二维系统中晶粒的长大特征,揭示第二相粒子对晶粒长大的影响规律.结果表明:初始阶段晶粒长大符合长大指数n 为0.3～0.4的指数长大规律,其n与系统单位面积所含的粒子数量密切相关;晶粒长大过程中绝大多数粒子位于晶界处,其最终的平均晶粒半径可以用Zener关系表示;当粒子尺寸和面积分数一定时,粒子的形状对晶粒的长大过程没有明显影响.     

正常晶粒长大的遗传算法模拟

引入遗传算法模拟正常晶粒长大过程,基于正常晶粒长大动力学和能量最小原理,建立晶粒长大的对应遗传规则和能量最小的适应度函数。该算法模拟结果表明：系统的晶界总周长和系统晶界能随着遗传代数的传递不断减少,系统的热力学状态趋于稳定：在5000GAS前后阶段的晶粒长大指数”分别为0．477和0．414,与理论值0．500相近,与正常晶粒长大的动力学规律较符合;晶粒半径分布具有Weibull函数拓朴结构形式,平均晶粒边数为5．923。遗传算法可以根据实际晶粒长大过程的约束条件对遗传规则和适应度函数进行设置,因而具有良好的柔性。     

Quantitative evaluation of normal and abnormal grain growth of cemented carbides during liquid phase sintering

The liquid-phase sintering (LPS) of cemented carbides prepared from submicronic powders induces a micro-structural evolution generally ascribed to normal and abnormal grain growth. Such phenomena can be prevented by small additions of inhibitors (Cr, V). Presently, the mechanisms controlling either the grain growth or its inhibition are not strictly identified. In the present work, the effects of major parameters on grain growth (initial WC grain size, liquid composition, liquid fraction) are studied by image analysis of specimens sintered at 1450°C up to 8h.The evolution of the mean intercept and intercept distribution of WC grains is analysed in terms of the possible mechanisms involved.     

相场法模拟AZ31镁合金再结晶晶粒长大

建立一个相场模型,并对AZ31镁合金进行再结晶晶粒长大模拟,提出一系列法则以确定模型中参数的真实值,从而实现组织演化在工业应用范围内-的真实时空模拟.在300～400 ℃ 100 min内时,模拟结果与实验结果吻合得很好,在250 ℃时,模拟结果与实验结果有较大偏离,说明该合金体系的界面迁移激活能在低温时有所改变,定量模拟研究该合金组织的混晶程度.结果表明:合金在300～400 ℃时,随时效时间的延长,混晶特别严重.模型模拟相场法界面控制过程组织演变的真实时空,所确定的参数值也可以用于其他相似合金系统的模拟.     

Theory of normal grain growth in normalized size space

Simple mean-field deterministic theory of normal grain growth in both 3-dimension and 2-dimension cases is presented, predicting the Rayleigh size distribution. The main idea is an application of standard thermodynamics to normalized size space with unit length proportional to average size. In the normalized space, the change of normalized free energy caused by the size change of an arbitrary grain is independent of the reservoir, which consists of all the other grains. Such “decoupling” enables us to use the original Burke-Turnbull approach to relate driving force to velocity of grain growth in the normalized space.     

Increasing length scale of quantitative phase field modeling of concurrent growth and coarsening processes

We present an approach for increasing the length scale of quantitative phase field modeling of concurrent growth and coarsening processes with neither growth nor coarsening kinetics altered. We modify locally the free energy hump between the two equilibrium phases, which determines interface thickness, while keeping the driving forces for growth and coarsening unchanged. The approach is applied to precipitate growth and coarsening in a binary Ni–Al alloy.     

Modification of computer simulation of normal grain growth

A set of principles on transition probability was supplied for the physical process of grain growth. In accord with these principles, a modified transition probability considering the influence of temperature was put forward to simulate the normal grain growth relying on temperature and second phase particles. The modified transition probability correctly reflects the dependence of grain growth on the temperature. The effect of different shapes of second phase particles on the grain growth process was taken into account using the modified transition probability. The relationship between the area fraction of second phase particles and the limit of grain size of the matrix was given. The microstructural evolution patterns employed to 2-D were given. The results agree well with the real grain growth process. All these suggest that the modified transition probability is better than the conventional one.     

Phase field modeling of dendrite growth

Single dendrite and multi-dendrite growth for A1-2 mol pct Si alloy during isothermal solidification are simulated by phase field method. In the case of single equiaxed dendrite growth, the secondary and the necking phenomenon can be observed. For multi-dendrite growth, there exists the competitive growth among the dendrites dur-ing solidification. As solidification proceeds, growing and coarsening of the primary arms occurs, together with the branching and coarsening of the secondary arms.When the diffusion fields of dendrite tips come into contact with those of the branches growing from the neighboring dendrites, the dendrites stop growing and being to ripen and thicken.     

Phase field modeling of grain boundary migration with solute drag

The grain boundary (GB) motion in the presence of GB segregation is investigated by means of phase field simulations. It is found that the solute concentration at the moving GB may increase with increasing velocity and becomes larger than the equilibrium value, which is unexpected according to the solute drag theory proposed by Cahn, but has been observed in some experiments. A non-linear relation between the driving force (curvature) and the GB velocity is found in two cases: (1) the GB motion undergoes a transition from the low-velocity extreme to the high-velocity extreme; (2) the GB migrates slowly in a strongly segregating system. The first case is consistent with the solute drag theory of Cahn. As for the second case, which is unexpected according to solute drag theory, the non-linear relation between the GB velocity and curvature comes from two sources: the non-linear relation of the solute drag force with GB velocity, and the variation in GB energy with curvature. It is also found that, when the diffusivity is spatially inhomogeneous, the kinetics of GB motion is different from that with a constant diffusivity.     

On the theory of normal grain growth in two dimensions

A two-dimensional theory of grain growth has been derived based on the concept of curvature-driven migration of the grain boundaries. Grains with six neighbouring grains are of the critical grain size and will neither shrink nor grow. The rate equation is thus in accordance with the von Neumann–Mullins law. An analytical expression for the steady state distribution has been derived based on the same principles as Hillert's theory. The present theory contains both a minimum and a maximum grain size.     

铝合金应变诱导晶粒异常长大行为的晶体塑性-相场法耦合模拟（英文）

提出一种预测变形铝合金在退火过程中应变诱导晶粒异常长大行为的介观耦合建模方法。基于晶体塑性有限元法(CPFE)计算塑性变形中的位错密度和变形储能分布;以储能和局部界面曲率为晶界迁移驱动力,建立基于连续场法的改进相场(PF)模型,并采用插值法将CPFE计算结果向PF模型进行映射,实现CPFE-PF的耦合模拟。采用理想双晶组织模拟初步校准PF模型,然后将该CPFE-PF耦合模型应用于变形AA3102铝合金退火过程的多晶组织演化模拟,进一步验证模型有效性。结果表明,退火时变形基体中的低储能形核点优先长大;而当变形量在临界塑性应变附近时,由于结晶形核数量的有限性和储能分布的不均匀性,将产生晶粒异常长大现象。     

Pinning effect of second-phase particles on grain growth in polycrystalline films studied by 3-D phase field simulations

Three-dimensional simulations of grain growth in thin films containing finely dispersed second-phase particles were performed using a phase field model. The simulations show that although the growth behavior of the columnar grain structures in thin films is essentially two-dimensional, the interaction between the particles and the grain boundaries is three-dimensional. Grain boundaries can therefore more easily break free from the particles than in purely two-dimensional systems, resulting in fewer grain boundary–particle intersections and a larger final grain size. For a given volume fraction f_V and size of the particles r, the final grain size ${\overline{R}}_{\text{lim}}$ increases with film thickness. Moreover, it was found that particles located in the middle of the film are most efficient in pinning grain boundaries. A classical Zener type relation ${\overline{R}}_{\text{lim}}/r=K(1/f_{\text{V}}^b)$ cannot describe these effects.     

相场法模拟多个空间取向的棒状第二相粒子对晶粒长大的影响

采用相场法研究多个空间取向的棒状第二相粒子以及圆形第二相粒子对基体晶粒长大的影响.结果表明:在晶粒长大过程中,绝大部分棒状第二相粒子位于晶界处并与晶界方向一致,圆形第二相粒子大多位于三晶交点处;第二相粒子表现出强烈的钉扎晶界的作用,极限晶粒半径可以用Zener关系表示;在第二相粒子面积分数和粒子尺寸相同的情况下,当第二相粒子面积分数较小(＜5%)时,棒状与圆形第二相粒子对晶粒长大的钉扎作用没有明显差别;当粒子面积分数较大(＞5%)时,棒状第二相的钉扎效果好于圆形第二相的钉扎效果.     

Phase field modeling of the tetragonal-to-monoclinic phase transformation in zirconia

The allotropic phase transformation in zirconia from the tetragonal to monoclinic double lattices is known to occur by a martensitic twinning mechanism which shows a complex dependence on temperature, stress and environment. This paper is concerned with the development of a phase field model which accounts for the main metallurgical mechanisms governing this martensitic transition. The symmetry reduction and orientation relationship between the parent and product phases were simulated using several non-conserved order parameters representing different transformation paths. Inhomogeneous and anisotropic elastic properties were considered to determine the resultant elastic stresses. Governing equations of the tetragonal-to-monoclinic transformation were solved in a finite element framework under a variety of initial and boundary conditions. It was shown that applying different initial conditions, such as seed embryo or random, did not change the twinning patterns or the final volume fractions of the parent and product phases after the relaxation period. On the other hand, enforcing different boundary conditions resulted in completely different twinning patterns and phase volume fractions. The model was able to predict both the “V” shape morphology of twinning and the surface stress relief with “gable roof” patterns, which were observed by transmission electron microscopy and atomic force microscopy to be characteristic of the tetragonal-to-monoclinic transition.     

Topological and metrical analysis of normal grain growth in three dimensions

Grain growth theories normally describe a grain using the concept of “grain radius”. However, this assumption bypasses all topological information relating to the grain, such as the number of its faces and edges. This study, by contrast, introduces a new methodology, treating normal grain growth in three dimensions in terms of both metrical and topological properties of the grains.