Effect of elastic interaction on nucleation: I. Calculation of the strain energy of nucleus formation in an elastically anisotropic crystal of arbitrary microstructure

We analyze the anisotropic elastic interaction of a nucleating particle with an arbitrary pre-existing coherent microstructure. Under the assumption that the length scale of the microstructure is considerably larger than the size of the nucleus, their elastic interaction energy can be expressed as a linear function of the nucleus’s volume, and combined directly with the chemical nucleation driving force in the classical nucleation theory. Using cubic → cubic and cubic → tetragonal transformations as examples, we evaluate the elastic energy associated with the formation of a nucleus in a pre-existing microstructure. It is found that, similar to other stress-generating crystalline defects, coherent precipitates could have a significant effect on the spatial location of nuclei, resulting in correlated nucleation where the existing particles dictate where the new particles appear. This effect seems to be generic for nucleation in coherent solids and it could be responsible for the formation of self-organized morphological patterns during coherent transformations.     

Computer simulation of spinodal decomposition in constrained films

The morphological evolution during spinodal decomposition of a binary alloy thin film elastically constrained by a substrate is studied. Elastic solutions, derived for elastically anisotropic thin films subject to the mixed stress-free and constraint boundary conditions, are employed in a three-dimensional phase-field model. The Cahn–Hilliard diffusion equation for a thin film boundary condition is solved using a semi-implicit Fourier-spectral method. The effect of composition, coherency strain, film thickness and substrate constraint on the microstructure evolution was studied. Numerical simulations show that in the absence of coherency strain and substrate constraint, the morphology of decomposed phases depends on the film thickness and the composition. For a certain range of compositions, phase separation with coherency strain in an elastically anisotropic film shows the behavior of surface-directed spinodal decomposition driven by the elastic energy effect. Similar to bulk systems, the negative elastic anisotropy in the cubic alloy results in the alignment of phases along 〈1 0 0〉 elastically soft directions.     

Effect of substrate constraint on spinodal decomposition in an elastically inhomogeneous thin film

Spinodal decomposition in a binary thin film is studied by a three-dimensional (3D) phase field model. A cubic thin film with an (001) orientation is considered. The focus is on the effect of the types of substrate constraint on the morphological evolution during spinodal decomposition as compared to the corresponding bulk. The elastic strain effect is incorporated by solving the elasticity equations for an elastically inhomogeneous thin film with a free surface and constrained by a substrate. Temporal evolution for the composition field, and thus the morphological evolution, is obtained by solving the Cahn-Hilliard equation using a semi-implicit Fourier-spectral method. It is shown that a biaxial substrate constraint has essentially no effect on the spinodally decomposed two-phase morphology which is primarily controlled by the cubically anisotropic elastic interactions. The asymmetry in the strain components along the [100] and [010] directions from the substrate constraint results in the preferential alignment along one of the two directions. In the particular case of a harder phase whose lattice parameter increases with composition, a tensile substrate constraint along the film plane leads to the alignment of two-phase microstructures parallel to the tensile direction. This article is based on a presentation made in the 2003 Korea-Japan symposium on the “Current Issues on Phase Transformations”, held at Marriott Hotel, Busan, Korea, November 21, 2003, which was organized by the Phase Transformation Committee of the Korean Institute of Metals and Materials.     

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.     

共格沉淀析出过程的微观结构演化相场法模拟

用铜模吸铸法制备了Zr(65-x)Cu17.5Ni10Al7.5Fex(x=0,1,2,3,4,5,at％)块体合金,利用X射线衍射仪(XRD)、透射电镜(TEM)、同步热分析(DSC)、万能试验机和扫描电镜(SEM)研究了Fe对Zr基合金的非晶形成能力、热稳定性及力学性能的影响.结果表明:适量添加Fe元素有利于提高该合金的非晶形成能力和热稳定性,当Fe含量为x=2时,其内部结构为完全非晶结构,并且此成分具有较高的GFA和热稳定性(△Tx=58 K).Fe元素的适量加入也有利于提高该合金的强度和塑性,其中x=2的块体非晶合金的抗压强度σf、断裂应变εf和塑性应变εp分别高达2338 MPa、12.4％和2.0％.     

A phase-field model for evolving microstructures with strong elastic inhomogeneity

An efficient phase-field model is proposed to study the coherent microstructure evolution in elastically anisotropic systems with significant elastic modulus inhomogeneity. It combines an iterative approach for obtaining the elastic displacement fields and a semi-implicit Fourier–spectral method for solving the time-dependent Cahn–Hilliard equation. Each iteration in our iterative numerical simulation has a one-to-one correspondence to a given order of approximation in Khachatuyran's perturbation method. A unique feature of this approach is its ability to control the accuracy by choosing the appropriate order of approximation. We examine shape dependence of isolated particles as well as the morphological dependence of a phase-separated multi-particle system on the degree of elastic inhomogeneity in elastically anisotropic systems. It is shown that although prior calculations using first-order approximations correctly predicted the qualitative dependence of a two-phase morphology on elastic inhomogeneity, the local stress distributions and thus the driving force for microstructure evolution such as coarsening were in serious error quantitatively for systems with strong elastic inhomogeneity.     

Erratum

Abstract

Basic equations relating to the anisotropic theory of elasticity for multilayer heterostructures are presented. These equations account for the plastic deformation and the layer lattice parameters mismatch with respect to the substrate. Special attention is given to take into account the variation in the components of the elastic stiffness tensor in the layers of various compositions. A solution of the one-dimensional problem of elastically and plastically deformed state is cited for heterostructures of the cubic system. Particular cases of a two-layer heterostructure with variable components of elastic stiffness tensor were studied in the absence of plastic deformations, as well as with constant components of elastic stiffness tensor in the presence of dislocations in the heterostructure.

The most promising non-destructive double and triple-crystal X-ray diffraction methods for the measurement of components of the strain tensor and heterostructure curvature are discussed. Particular attention is paid to the determination of a number of structural features of epitaxial systems: viz., the critical thicknesses of pseudomorphous layers, the strain gradient in the films of heterogeneous solid solutions, recovering of proper deformation profiles on the basis of measured results for heterostructure curvature with its layer by layer elimination, determination of lattice parameter mismatch, determination of concentration in heterostructures of four-component solid solutions, and determination of the thickness of the transition width on the heteroboundary of a two-layer heterostructure. A calculation of stresses in the layers of a five-layer double heterostructure with two consistent layers of solid solutions, and in the layers of epitaxial superlattice, has been carried out.

In conclusion, X-ray diffraction methods allowing one to measure deformation in ultrathin surface layers are discussed.     

Three-dimensional phase-field modeling of martensitic microstructure evolution in steels

In the present work a 3-D elastoplastic phase-field (PF) model is developed, based on the PF microelasticity theory proposed by A.G. Khachaturyan and by including plastic deformation as well as anisotropic elastic properties, for modeling the martensitic transformation (MT) by using the finite-element method. PF simulations in 3D are performed by considering different cases of MT occurring in an elastic material, with and without dilatation, and in an elastic perfectly plastic material with dilatation having isotropic as well as anisotropic elastic properties. As input data for the simulations the thermodynamic parameters corresponding to an Fe-0.3%C alloy as well as the physical parameters corresponding to steels acquired from experimental results are considered. The simulation results clearly show autocatalysis and morphological mirror image formation, which are some of the typical characteristics of a martensitic microstructure. The results indicate that elastic strain energy, anisotropic elastic properties, plasticity and the external clamping conditions affect MT as well as the microstructure.     

A two phase elastic–plastic self-consistent model for the accumulation of microstrains in Waspaloy

A two phase self-consistent scheme for the accumulation of elastic lattice strains in the nickel-base superalloy Waspaloy is presented. The microstructure is idealised as a set of randomly orientated anisotropic grains which are assumed to be spherical and embedded within a homogenous effective medium which is assigned the properties of the bulk. Each grain is modelled as a medium with the properties of identically aligned cubic γ-Ni with a spherical inclusion of γ′-Ni₃Al. The γ′ is treated as an elastic anisotropic solid, and the γ is modelled as an elastic–plastic single crystal according to the Taylor–Bishop–Hill plasticity theory. The diffraction elastic constants and microstrains accumulated are compared with those found experimentally using neutron diffraction, as previously reported by Stone et al. The predictions are found to be adequate, and in particular the shift of load, as reflected in the microstrain, from the matrix to the precipitate between 3 and 10% plastic strain is reproduced. The implications of these results for the development and use of Reitveld-derived schemes for the measurement of bulk residual stress at spallation sources are discussed.     

Simulation of the evolution of GP zones in Al–Cu alloys: an extended Cahn–Hilliard approach

A kinetic model, based on the approach given by Cahn and Hilliard, was applied for a simulation of the formation and growth of Guinier–Preston (GP) zones in Al–Cu alloys. Thermodynamic data were provided by the CALPHAD method for the metastable phases to warrant for simulation of real Al–Cu alloys. The classic Cahn–Hilliard equation was extended to include the anisotropic elastic strain energy, which determines the morphology of GP zones. The model was applied to a simulation of spinodal decomposition c_Cu⁰=0.06 and precipitation of an alloy with concentration c_Cu⁰=0.02 which is beneath the maximum solubility. The latter simulation was carried out assuming pre-existing nuclei since the Cahn–Hilliard model does not include any mechanism for nucleation. The results (length and thickness of the precipitates) were in good agreement with experiments.     

Three-dimensional phase-field modeling of spinodal decomposition in constrained films

The morphological evolution during spinodal decomposition in binary alloy thin films elastically constrained by substrates is studied. Elastic solutions, derived for both elastically isotropic and anisotropic thin films subject to mixed stress-free and constraint boundary conditions, are employed in a three-dimensional phasefield model to investigate the effect of coherency strain and substrate constraint on microstructural evolution. The temporal evolution of the Cahn-Hilliard equation under thin film boundary conditions is solved with a semi-implicit Fourier-spectral method. The phase separation with coherency strain in an elastically anisotropic film shows the behavior of surface-directed spinodal decomposition driven by the elastic energy effect. Negative elastic anisotropy in the cubic alloy causes the alignment of the phases along <100> elastically soft directions. This article is based on a presentation made in the 2002 Korea-US symposium on the “Phase Transformations of Nano-Materials,” organized as a special program of the 2002 Annual Meeting of the Korean Institute of Metals and Materials, held at Yonsei University, Seoul, Korea on October 25–26, 2002.     

Interactions between carbon solutes and dislocations in bcc iron

Carbon solute–dislocation interactions and solute atmospheres for both edge and screw dislocations in body-centered cubic (bcc) iron are computed from first principles using two approaches. First, the distortion tensor and elastic constants entering Eshelby’s model for the segregation of C atoms to a dislocation core in Fe are computed directly using an electronic-structure-based the total energy method. Second, the segregation energy is computed directly via first-principles methods. Comparison of the two methods suggests that the effects of chemistry and magnetism beyond those already reflected in the elastic constants do not make a major contribution to the segregation energy. The resulting predicted solute atmospheres are in good agreement with atom probe measurements.     

Debye temperature, self-diffusion and elastic constants of intermetallic compounds

The Debye temperature of a material is a suitable parameter to describe phenomena of solid-state physics which are associated with lattice vibrations. It basically depends on the elastic constants. In recent work a simple method was put forward that allows one to derive precise Debye temperatures of crystals with cubic, hexagonal and tetragonal symmetry from the elastic constants. The type of chemical binding does not play any role. It is one aim of the present work to apply this method to various intermetallic compounds, i.e. to critically analyse published Debye temperatures and to calculate hitherto unknown values. It is a further aim to show that the activation energy of self-diffusion is also connected with the elastic constants by a simple law at least for the cubic B2 and L1₂ intermetallics, as it was recently found for face-centred cubic metals. Some consequences for high-temperature plasticity are discussed.     

Equilibrium particle morphologies in elastically stressed coherent solids

We determine the three-dimensional equilibrium shapes of particles with a purely dilatational misfit in an elastically anisotropic medium with cubic symmetry. We have identified a succession of cuboidal shapes with four-fold rotational symmetry that minimize the total energy of the system. In the process of determining these equilibrium morphologies, we have also developed a computationally efficient approach to determine the equilibrium shape which is many orders of magnitude faster than a standard implementation of Newton's method. For small elastic stress a (100) cross-section of the three-dimensional equilibrium shape agrees well with the two-dimensional calculation. However, for larger values of the elastic stress, the agreement is not as good. Elastic-stress-induced configurational forces are identified as the reason for the non-spherical equilibrium shapes.     

Hf含量对镍基粉末高温合金中γ′相形态的影响

研究了含Hf镍基粉末高温合金在长期时效处理过程中γ′相形态的演化过程.结果表明:合金在高温长期处理过程中立方状γ′相发生分裂、呈现出二重平行状和八重小立方体组态.八重小立方体组态作为择优形态不再发生分裂,处于低能稳定状态.不同Hf含量合金的错配度发生明显变化,γ′相的长大或粗化过程可以粗略地分为"界面控制"和"应变控制"2个阶段,γ′相间弹性相互作用能对合金中析出相的形态变化起着重要作用.     

Solidification structure formation in undercooled Fe–Ni alloy

The Fe alloy melts containing 7.5, 15, 22.5 and 30 at% Ni were bulk undercooled to investigate the structure evolution. When the undercooling of the four melts is lower than the critical value 110, 125, 175 and 325 K, respectively, only the stable face-centered cubic phase crystallizes. In this case a grain refinement caused by solid superheating is observed in all the alloys, but another grain refinement induced by recrystallization can merely occur in the Fe–30 at%Ni alloy undercooled by 190–325 K. Alternate crystallization of the metastable body-centered cubic phase occurs above the critical undercooling. It is indicated that the subsequent heterogeneous nucleation of the stable phase in the metastable solid and remaining liquid coexisting system is influenced not only by the morphology and surface area of the metastable solid, but also by the effective undercooling of the remaining liquid. On the basis of the experimental results and the theoretical analyses, a structure evolution map for bulk Fe–Ni system is constructed.     

三维枝晶生长数值模拟及实验验证(英文)

建立了用于模拟立方晶系合金三维枝晶生长的改进元胞自动机模型。该模型将枝晶尖端生长速率、界面曲率和界面能各向异性的二维方程扩展到三维直角坐标系,从而能够描述三维枝晶生长形貌演化。应用本模型模拟在确定温度梯度和抽拉速度条件下三维柱状晶生长过程的一次臂间距调整机制和不同择优取向柱状晶之间的竞争生长。使用NH4Cl?H2O透明合金进行凝固实验,模拟结果和实验结果吻合较好。     

Simulations of stress-induced twinning and de-twinning: A phase field model

Twinning in certain metals or under certain conditions is a major plastic deformation mode. Here we present a phase field model to describe twin formation and evolution in a polycrystalline fcc metal under loading and unloading. The model assumes that twin nucleation, growth and de-twinning is a process of partial dislocation nucleation and slip on successive habit planes. Stacking fault energies, energy pathways (γ surfaces), critical shear stresses for the formation of stacking faults and dislocation core energies are used to construct the thermodynamic model. The simulation results demonstrate that the model is able to predict the nucleation of twins and partial dislocations, as well as the morphology of the twin nuclei, and to reasonably describe twin growth and interaction. The twin microstructures at grain boundaries are in agreement with experimental observation. It was found that de-twinning occurs during unloading in the simulations, however, a strong dependence of twin structure evolution on loading history was observed.