Effect of elastic interaction on the formation of a complex multi-domain microstructural pattern during a coherent hexagonal to orthorhombic transformation

The effect of elastic interaction on the formation and dynamic evolution of multi-domain microstructures during a hexagonal to orthorhombic transformation in the absence and presence of an externally applied strain field is investigated numerically using the phase field model. In particular, three cases are considered, which include a single variant, two variants, and all three variants of the orthorhombic phase produced by the transformation. In each case, the morphology and spatial distribution of the orientation variants are characterized. It is shown that nucleation and growth of a single variant produces thin plates of the orthorhombic domains with definite habit planes. In the case of two variants, the domains developed at the initial stages are also platelets of well-defined habit planes, which is similar to the case of a single variant. However, the impingement and intersection of the platelets of different variants results in the formation of twin boundaries and “zig-zag” patterns. The overlap regions of the “zig-zag” cross sections remain untransformed which agrees very well with experimental observations. If all three variants are present, the hexagonal to orthorhombic transformation results in a number of unique multi-domain structures such as the star patterns, compound star patterns, fan patterns, etc., which have been frequently observed experimentally in systems undergoing hexagonal to orthorhombic or similar transformations. It is found that if the boundary of the system is constrained, e.g. a grain embedded in a polycrystalline material, the transformation can go to completion only when all three variants are present. In the presence of external strain field, the coupling between the applied strain field and the stress-free transformation strain associated with the domain formation leads to selective growth of variants.     

Simulation of the coarsening behavior of intermetallic precipitates in Ni75AlxV25-x alloys

The coarsening behavior of L12 and D022 in Ni75AlxV25-x （x,at.%） alloys including coherent strain was investigated using the microscopic phase-field model. The simulation results indicate that the shape transition and spatial correlation of L12 and D022 are caused by the morphological-dependent anisotropic elastic interactions in the system. The coarsening process of the particles is by means of neighbor particles impingement and aggregation into larger ones. For the strain-induced interactions between the precipitates,the LSW theory is altered for the coarsening behavior of L12 and D022. In addition,the simulation reveals that the growth and coarsening of D022 present two obvious stages at lower Al concentration regions and proceed simultaneously at high Al concentration regions. The growth and coarsening processes of L12 at the same regions is reverse to those of D022.     

Three-dimensional model of precipitation of ordered intermetallics

The development of the two-phase (f.c.c.+L1₂) coherent microstructure in the prototype Ni–Al superalloy is studied by using the three-dimensional computer simulation technique. The dynamics and morphology of the microstructure evolution are described by our three-dimensional version of the stochastic time-dependent kinetic equation which explicitly includes the coherency strain, elastic anisotropy and L1₂ ordering of the preciptate phase. The input parameters, the crystal lattice misfit, elastic moduli, interfacial energy and equilibrium compositions of the coexisting phases are taken from the published independent measurements. The simulation results demonstrate that the strain accommodation in the microstructure evolution results in the cuboidal-like precipitates faceted by the {100} planes. The size of the precipitates obtained in the simulation is of the order of 50 nm. The important conclusion is that the precipitates are always single-domain particles with no antiphase boundaries. This effect is associated with the ordered structure of precipitates. It causes the slowing down of the coarsening kinetics since it excludes the agglomeration of the out-of-phase precipitates in one particle. As has been shown previously, the latter is a very important coarsening mode in an absence of ordering.     

共格沉淀析出过程的模拟Ⅰ--微观结构演化

采用相场模型对共格沉淀析出过程微观结构演化进行模拟研究.模拟结果表明,应力场的存在将会对相变过程中析出相形态产生显著的影响,通过界面能与弹性应变能的相互竞争析出相在不同阶段呈现不同的形态如模量结构、网格结构、三明治多畴结构、沉淀宏观点阵结构以及板条状等;此外,点阵错配度中等(2%～4%)时,粒子的粗化过程将出现应力诱导反向粗化现象,这种粗化现象取决于粒子的点阵错配度与体积分数.     

Large-scale simulations of Ostwald ripening in elastically stressed solids. II. Coarsening kinetics and particle size distribution

Ostwald ripening of misfitting second-phase particles in an elastically anisotropic solid is studied by large-scale simulations. The coarsening kinetics for the average particle size are described by a t^1/3 power law with a rate constant equal to its stress-free value when the particles are fourfold symmetric. However, the rate constant increases when the elastic stress is sufficient to induce a large number of twofold-symmetric particles. We find that interparticle elastic interactions at a 10% area fraction of particles do not affect the overall coarsening kinetics. A mean-field approach was used to develop a theory of Ostwald ripening in the presence of elastic stress. The simulation results on the coarsening kinetics agree well with the theoretical predictions. The particle size distribution scaled by the average particle size is not time invariant, but widens slightly with an increasing ratio of elastic to interfacial energies. No time-independent steady state under scaling is found, but a unique time-dependent state exists that is characterized by the ratio of elastic energy to interfacial energy.     

Coarsening of ordered intermetallic precipitates with coherency stress

The morphological evolution and coarsening kinetics of ordered intermetallic precipitates with coherency stress were studied using a diffuse-interface phase-field model in two dimensions (2D). The emphasis is on the effects of precipitate volume fraction. The average aspect ratio of the precipitates in the microstructure is found to increase with time and decrease with volume fraction. Contrary to all the existing coarsening theories but consistent with a number of experimental measurements on the coarsening kinetics of ordered γ′ precipitates in Ni-base superalloys, we found that the coarsening rate constant from the cubic growth law decreases as a function of volume fraction for small volume fractions (≲20%) and is constant for intermediate volume fractions (20–50%). From the simulation results, we infer that the two length scales in a stress-dominated coherent two-phase microstructure, the average precipitate size and average spacing between arrays of aligned precipitates, follow different growth exponents. It is demonstrated that as the volume fraction increases, the precipitate size distributions become broader and their skewness become increasingly positive.     

Effects of applied stress on coherent precipitates via a discrete atom method

Morphological evolution of coherent precipitates under an applied stress is analyzed by means of a discrete atom method, which is predicated upon Hookean atomic interactions and Monte Carlo diffusion, and mates no assumption of a specific shape. Precipitates having elastic constants different from those of the matrix phase art treated in dislocation-free, anisotropic elastic systems under a plane strain condition with a purely dilatational misfit. Under an applied tensile stress, soft particles with a positive misfit strain tend to become plates perpendicular to the applied stress axis, while hard particles elongate along the stress direction. If the elastic interaction between the applied stress and the coherency strain is strong enough, precipitates often split into smaller particles and then follow coarsening. Even in the absence of a coherency strain, particles are shown to undergo morphological evolution through Eshelby’s inhomogeneity effects. A particle shape depends on the following variables: the sign and magnitude of the coherency strain, the sense and magnitude of the applied stress, its stiffness relative to the matrix phase, and the magnitude of the interfacial energy.     

共格沉淀相粗化行为的计算机模拟研究

将微观弹性应变能理论和微观扩散方程相耦合，建立起时效过程微观晶格动力学模型。对溶质浓度为14at％的二元立方点阵模型合金的共格沉淀粗化行为进行模拟。研究发现：该合金沉淀机制以形核长大为主，兼有失稳分解特征；由于共格失配，沉淀相为片状，趋于沿弹性软化方向排列。其粗化作为伴随过程进行，位于软方向上的颗粒继续长大和粗化，位于软方向外的颗粒逐渐消失；而位于同一行或列上的颗粒则遵循：小颗粒溶解，大颗粒长大。     

Phase-field simulation of polarization switching and domain evolution in ferroelectric polycrystals

A phase-field model is developed for predicting the polarization switching and domain structure evolution under an applied electric field in ferroelectric polycrystals. The model takes into account realistic grain structures as well as various energetic contributions, including elastic energy, electrostatic energy, and domain wall energy. A hysteresis loop – average polarization as a function of applied electric field – is computed, and the detailed domain evolution process during switching is analyzed. In particular, the role of grain boundaries in the nucleation and growth of new domains is studied. It is shown that switching takes place through the nucleation of 90° domains at grain boundaries and subsequent growth into the grain interiors instead of direct 180° domain switching. A correlation between the domain structures in neighboring grains was observed, and polarization switching in one grain was found to affect the switching in neighboring grains.     

Ni75Cr16.4Al8.6合金粗化过程标度行为的微观相场研究

基于微观相场模型,研究Ni75Cr16.4Al8.6的粗化后期演化过程。结果表明：L12相和DO22相的标度函数曲线图表明合金沉淀在粗化阶段均表现出标度行为,不加应变能时DO22相结构函数因子和标度函数峰值均大于L12相;加入应变能后,两相的结构函数因子和标度函数的峰值均变小,反映了应变能对两相沉淀粗化过程的阻碍影响;相对于L12相,DO22相减小幅度更大,L12相结构函数因子和标度函数峰值大于DO22相,应变能对DO22相的阻碍作用更大。     

应力场对相分解过程影响的计算机模拟研究（Ⅱ）——非对称成分及拐点以下的相分离模拟

采用扩散界面场模型对非对称成分点及拐点以下成分点同构两相分解过程中应力场（包括外场）对析出相的形态及动力学过程的影响作了比较系统的研究.模拟结果表明：对于各向异性系统在相分解过程中由于弹性应变能与界面能间的相互作用将会导致析出相在不同阶段呈现不同的形貌如网格结构、三明治多畴结构、沉淀宏观点阵结构以及板条状等;对于非均匀弹性模量系统在外加应力场作用下,沉淀相的析出过程发生重大的变化,粒子沿弹性软方向呈各向异性析出.此外,模拟结果还发现,在应力场（包括外加场情况）存在条件下,相析出的中前期长大过程中小粒子沿着弹性软方向相互联结成大粒子,在相变后期孤立的小粒子被大粒子的长大所消耗.     

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.     

Phase-field simulation of coarsening of γ precipitates in an ordered γ′ matrix

The evolution of the microstructure and coarsening kinetics of disordered γ precipitates in an ordered monovariant γ′ matrix was studied using a diffuse-interface phase-field model in two dimensions. Specifically, we studied the morphological evolution, average precipitate size and size distribution as a function of time for a given temperature and for different concentrations. It was found that the coarsening kinetics for the average particle size are described by a t^1/3 power law with a rate constant that increases with the volume fraction of the γ phase. The particle size distribution scaled by the average particle size is time invariant. We find that the particle shapes are perturbed by elastic interaction and local distributions of γ particles. Comparison of the phase-field simulation results with experiments shows good qualitative agreements.     

Pure orthorhombic zirconia islands grown on single-crystal sapphire substrates

This study reports on the occurrence of pure orthorhombic zirconia obtained for the first time at ambient pressure. Thin films of pure zirconia are composed of isolated islands which generally present heteroepitaxial relationships with the underlying sapphire substrate. Epitaxial growth develops at high temperature in the stability domain of the tetragonal phase of zirconia. On cooling, most of the tetragonal islands transform into monoclinic, but numerous islands present an orthorhombic structure with Pbc2₁ and Pbca space groups. It is first suggested that these two orthorhombic phases are formed because the tetragonal → monoclinic reconstructive phase transition was impeded due to the pre-existing heteroepitaxial relationships that developed between the tetragonal zirconia phase and the substrate. However, the orthorhombic phases are much better interpreted as alternative phases that arise as a consequence of some constraints produced by the substrate. Thus, these phases represent metastable structures when appropriate external anisotropic constraints are applied.     

Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations

The first Phase Field model of evolution of a multi-dislocation system in elastically anisotropic crystal under applied stress is formulated. The model is a modification and extension of our Phase Field Microelasticity approach to the theory of coherent phase transformations. The long-range strain-induced interaction of individual dislocations is calculated exactly and is explicitly incorporated in the Phase Field formalism. It also automatically takes into account the effects of “short-range interactions”, such as multiplication and annihilation of dislocations and a formation of various metastable microstructures involving dislocations and defects. The proposed 3-dimensional Phase Field model of dislocations does not impose a priori constraints on possible dislocation structures or their evolution paths. Examples of simulation of the FCC 3D system under applied stress are considered.     

Isothermal precipitation and growth process of perovskite phase in oxidized titanium bearing slag

The isothermal precipitating behavior of perovskite phase in oxidized titanium bearing slag was studied by quenching method. The kinetics of precipitating process and crystal growth of perovskite phase was analyzed. The results show that the precipitating and growth of perovskite are non-equilibrium process at the beginning of isothermal treatment. There are two factors influencing the growth rate of perovskite phase on non-equilibrium condition, one is the supersaturation concentration of perovskite and the other is the coarsening arising from the growth of larger perovskite at the expense of smaller ones. The precipitation kinetics of perovskite phase can be nearly described by the JMAK equation.     

极端不可压缩Re2P的电子结构、化学键及弹性性质的第一性原理研究

利用基于密度泛函理论（DFT）的广义梯度近似（GGA）,研究Co2P类型结构的极端不可压缩Re2P的电子结构、化学键和弹性性质。能带结构显示Re2P为金属性材料;态密度和分态密度的计算结果表明,费米能级附近的态密度主要来自Re-5d态;布居分析表明Re2P中的化学键具有以共价性为主的混合离子一共价特征。计算得到Re2P的晶格参数、体模量、剪切模量和单晶的弹性常数,由此导出弹性模量和泊松比。结果表明,Re2P是力学稳定的,且具有一定的脆性。     

Analytical description of phase coarsening at high volume fractions

An analytical theory of volume fraction effects in diffusion-controlled phase coarsening is developed. It is based on the application of the Lifshitz–Slyozov–Wagner procedure of Ostwald ripening theory to a self-similar particle volume evolution equation which is approximated by a quadratic polynomial of the scaled particle size. A family of analytical particle size distributions with the scaled maximum particle size as a parameter is derived, which contains the size distribution of normal grain growth as a genuine limit distribution at ultra-high volume fraction without changing the t^1/3-Ostwald ripening kinetics. This reflects an important characteristic feature of particle coarsening at nearly space filling, which was first noted by Ardell and which is in principle in agreement with recent observations. The obtained analytical expressions for the maximum particle size, the particle size distribution and the coarsening rate constant are easy to use for evaluating coarsening data. The results compare well with a number of experimental and simulation results at intermediate and high volume fractions of the second phase.     

Diffuse-interface description of strain-dominated morphology of critical nuclei in phase transformations

A diffuse interface model combined with the minimax technique is implemented to predict the morphology of critical nuclei during solid to solid phase transformations in both two and three dimensions. It takes into account the anisotropic interfacial energy as well as the anisotropic long-range elastic interactions. It is demonstrated that the morphology of critical nuclei in cubically anisotropic solids can be efficiently predicted by the computational model without a priori assumptions. A particular example of cubic to cubic transformation within the homogeneous modulus approximation is considered. The effect of elastic energy contribution on the size and shape of a critical nucleus is studied. It is shown that strong elastic energy interactions may lead to critical nuclei with a wide variety of shapes, including plates, needles and cuboids with non-convex interfaces.     

Microstructure characterization of Cu-rich nanoprecipitates in a Fe–2.5 Cu–1.5 Mn–4.0 Ni–1.0 Al multicomponent ferritic alloy

The evolution of precipitates in a Fe–2.5 Cu–1.5 Mn–4.0 Ni–1.0 Al multicomponent ferritic alloy during annealing at 500 °C was systematically investigated by aberration-corrected scanning transmission electron microscopy. The atomic-scale structure and chemistry characterization reveal that primary precipitates with enriched Cu, Ni, Mn and Al originate from continuous growth of B2 ordered domains in the as-quenched alloy. The formation of a Cu-rich body-centered cubic (bcc) phase takes place by the decomposition of the B2 ordered primary phase, which forms a Cu-rich bcc core and ordered B2-Ni(Al,Mn) shell. The B2 shells serve as a buffer layer to moderate the coherent strain and to prohibit the inter-diffusion between the Cu-rich precipitates and bcc-Fe matrix, giving rise to a low coarsening rate of the precipitates. The Cu-rich precipitates experience a structural transformation from bcc to 9R at a critical size of ～6 nm during long time annealing, corresponding to obvious coarsening of the precipitates and dramatic loss in hardness of the alloy.