Equilibrium and stability of triple junctions in anisotropic systems

When a homophase or heterophase interface involves a crystalline solid, the interfacial energy is expected to depend upon the interface-plane orientation. In this paper, the equations governing equilibrium triple-junction configurations in anisotropic systems are reviewed. These equilibrium conditions were originally derived by considering the differential change in triple-junction interfacial energy associated with a differential change in triple-junction configuration and equating it to zero. However, the derived conditions do not distinguish between a triple-junction configuration that satisfies the equilibrium conditions by residing at a local energy minimum, a local energy maximum or at a saddle point. The present paper develops stability criteria for triple junctions with and without interfaces which have anisotropic energy, which can be used to determine whether triple-junction equilibrium conditions correspond to local energy minima (stable), maxima (unstable) or saddle points. For isotropic systems, there is a single solution to the triple-junction equilibrium conditions, and it is necessarily stable. For anisotropic systems, there are multiple solutions to the triple-junction equilibrium conditions, some of which may be stable, unstable or correspond to saddle points in triple-junction interfacial energy. Microstructure features and interface lengths/areas are expected to play a role in dictating the relative energies of stable configurations.     

Size effect on the Fe nanocrystalline phase transformation

A thermodynamics for the phase transformation from γ(fcc) to α(bcc) in nanocrystalline (NC) Fe is considered. Gibbs free energies of the interfaces in NC γ- and α-Fe particles were calculated, respectively, by means of a quasiharmonic Debye approximation, yielding a larger increase in the total Gibbs free energy of α-Fe than that of γ-Fe. This is attributed to the difference in their interfacial energies. As a result, the fcc NC Fe can be thermodynamically stable at room temperature when the grain size is sufficiently small. Taking into account the thermodynamic equilibrium condition, the critical grain size for the γ-Fe phase to exist in stable form at 300 K was quantitatively calculated for different excess volumes ΔV, a parameter describing the state of interface based on a dilated crystal model. The assumptions made in the present model and the factors influencing the critical grain size are discussed.     

Effect of phase fraction on the tri-junction in two-phase nanoparticle systems

The equilibrium dihedral angles at the solid–solid–vapor tri-junctions of two-phase Cu–Ag alloy nanoparticles 40–100 nm in diameter were measured as a function of phase fraction using transmission electron microscopy. The {111} solid–solid interface was cusp-oriented while the surface orientations of the Cu-rich and Ag-rich phases at the tri-junction were mostly free to vary. The dihedral angles at the tri-junction were found to vary with the phase fraction, due to the coupling between the relative amounts of each phase and the equilibrium conditions at the tri-junction. This equilibrium condition was used to measure the ratio of the Ag-rich and Cu-rich surface energies to the {111} interfacial energy, which were found as 2.1 and 3.5, respectively.     

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.     

Phase stability in Al/Ti multilayers

Structural stabilities in thin-film Al/Ti multilayers have been studied using transmission electron microscopy. It is shown that models based on interface-induced modifications to bulk stacking fault energies, and/or coherency-strain energy may not be used to explain the various phase transitions observed. Rather, phase stability may be rationalized by reference to a new model based on classical thermodynamics, involving the competition between bulk and interfacial energies. Thus, these phase transitions are shown to be driven by reductions in the overall interfacial energies. A biphase diagram (reciprocal of bilayer thickness vs composition) has been developed for the multilayers produced by magnetron sputtering. The influence of impurities possibly introduced during thin-foil preparation is considered and rationalized by reference to the biphase diagram for this system.     

Cu-Ni-Sn三元系相平衡的热力学计算

基于Cu-Ni-Sn三元系的相平衡和热力学的实验信息,采用亚正规溶体模型描述液相和fcc相的Gibbs自由能,为了预测该体系中bcc相的A2-B2有序-无序转变,bcc相的Gibbs自由能采用双亚点阵模型进行描述.利用CALPHAD(相图计算)方法评估了Cu-Ni-Sn三元系各相的热力学参数,计算的富Cu侧相图和热力学性质与实验数据比较一致.并对该三元系中bcc相的A2-B2有序-无序转变及fcc相的溶解度间隙进行了计算.这些计算结果对利用析出强化以及Spinodal分解开发高强度和高导电性的新型Cu基合金的组织设计具有一定的指导意义.     

Phase-field model study of the crystal morphological evolution of hcp metals

An expression for anisotropic interfacial energy of hexagonal close-packed metals has been formulated which is able to reproduce published data obtained using the modified embedded-atom method, covering the variation in interface energy as a function of orientation for a number of metals. The coefficients associated with the expression can be determined fully by measured or calculated interfacial energies of just three independent crystal planes. Three-dimensional phase-field model simulations using this representation of interfacial energy have been found to yield convincing crystal morphologies. The apparent rate of crystal growth as a function of orientation in the phase-field simulation agrees with predictions made by surface energy theory.     

A simulation study of the shape of β′ precipitates in Mg–Y and Mg–Gd alloys

The metastable β′ phase is a key strengthening precipitate phase in a range of Mg–RE (RE: rare-earth elements) based alloys. The morphology of the β′ precipitates changes from a faceted and nearly equiaxed shape in Mg–Y alloys to a truncated lenticular shape in Mg–Gd alloys. In this work, we study effects of interfacial energy and coherency elastic strain energy on the morphology of β′ precipitates in binary Mg–Y and Mg–Gd alloys using a combination of first-principles calculations and phase-field simulations. Without any free-fitting parameters and using the first-principles calculations, CALPHAD databases and experimental characterizations as model inputs (lattice parameters of the β′ phase, elastic constants and chemical free energy of Mg matrix and interfacial energies of the coherent β′/Mg matrix interfaces), the phase-field simulations predict equilibrium shapes of β′ precipitates of different sizes that agree well with experimental observations. Factors causing the difference in the equilibrium shape of β′-Mg₇Y and β′-Mg₇Gd precipitates are identified, and possible approaches to increase the aspect ratio of the β′ precipitates and thus to enhance the strength of Mg–RE alloys are discussed.     

Modeling and simulation of grain growth in Si3N4-II. The α–β transformation

The anisotropic Ostwald ripening model has been developed for completely faceted crystals. This model has been applied to the simulation of grain growth in β-Si₃N₄ with a highly anisotropic rod-like grain shape developed in the liquid phase. The reduction of aspect ratio after the phase transformation observed by previous studies is proved to be a consequence of the anisotropic Ostwald ripening. This model predicts a growth exponent n=3 for totally interfacial reaction controlled kinetics, and higher values when the diffusion constant approaches the interfacial reaction constants. This would explain the puzzling results reported by previous works that growth exponents n=3 or higher have been observed in the grain growth of faceted crystals. While the length distribution becomes wider with time, the reduced radius distribution approaches the shape that is known as the asymptotic distribution function derived from the LSW theory.     

Wetting anisotropy and oxygen activity dependency for oxides by liquid transition metals studied through shape changes of liquid Cu inclusions within MgO

The equilibrium shape of liquid metal inclusions in oxides is used to study anisotropy and oxygen partial pressure dependency of the metal–oxide specific interfacial free energy. Liquid copper inclusions are formed within single crystalline magnesia by internal reduction of mixed oxide (Mg,Cu)O. Their equilibrium shape is studied by TEM for various oxygen chemical potential. Precipitates always adopt cubo-octahedral morphology, but relative facet size strongly depends on the oxygen chemical potential. The shape evolution with oxygen chemical potential reveals important and anisotropic Gibbs' adsorption of excess oxygen to the interfaces at high oxygen activities. Gibbs' adsorption to crystallographically different MgO–liquid Cu interfaces is discussed in terms of a point defect model recently developed by the author (Backhaus-Ricoult, M., Phil. Mag., 2001, in press), where oxygen activity-dependent formation of interfacial charge transfer clusters between liquid metal and excess oxygen or magnesium is considered, while the relevant site and charge conservation laws in the two-phase system are respected and matter exchange with the surrounding gas atmosphere is allowed. Present experimental results are interpreted in terms of this interfacial defect model and compared with interpretation by other wetting models from the literature.     

Correlation between interfacial energy and phase diagram in ceramic-metal systems

Estimates of interfacial energies between ceramics (MX) and liquid metals (A) have been arrived at by employing an improved version of the Becker’s model for interfacial energy calculations at liquid metal-liquid metal interfaces to calculate the total energy of interatomic bonds across an interface. The results of such an approximation yield values that are very close to the experimental values; for instance, the estimated value for Al₂O₃/lFe is 2 J · m⁻², which compares well with the experimental data ranging from 1.9 to 2.3 J · m⁻². It is suggested that interfacial energies depend on two terms: (1) the formation energy of MX and (2) the pairwise interaction energies between M and A atoms and X atoms and vacant interstitial sites. It is also found that the calculated interfacial energies in eutectic systems (carbides-metal, etc.) are generally low, while those in the monotectic systems (oxides-metal, etc.) are generally high.     

Correlation between interfacial energy and phase diagram in ceramic-metal systems

Estimates of interfacial energies between ceramics (MX) and liquid metals (A) have been arrived at by employing an improved version of the Becker’s model for interfacial energy calculations at liquid metal-liquid metal interfaces to calculate the total energy of interatomic bonds across an interface. The results of such an approximation yield values that are very close to the experimental values; for instance, the estimated value for Al₂O₃/lFe is 2 J · m⁻², which compares well with the experimental data ranging from 1.9 to 2.3 J · m⁻². It is suggested that interfacial energies depend on two terms: (1) the formation energy of MX and (2) the pairwise interaction energies between M and A atoms and X atoms and vacant interstitial sites. It is also found that the calculated interfacial energies in eutectic systems (carbides-metal, etc.) are generally low, while those in the monotectic systems (oxides-metal, etc.) are generally high.     

Barrier-free heterogeneous grain nucleation in polycrystalline materials: The austenite to ferrite phase transformation in steel

The process of grain formation during structural solid-state phase transformations has commonly been analysed in terms of the classical nucleation theory. In polycrystalline materials a new phase generally forms at grain boundaries in the parent phase microstructure. Under certain conditions the net interfacial energy to form a new phase can be relatively small due to the release of grain boundary energy from the parent phase. The nucleation process is then governed by cluster dynamics. We propose a simple model to predict the cross-over between different heterogeneous nucleation regimes and apply it to recent synchrotron X-ray diffraction experiments on ferrite nucleation in steel.     

Prediction of surface and interfacial tension based on thermodynamic data and CALPHAD approach

In this article, following a brief introduction concerning experimental measurements of surface and interfacial tensions, methods for calculating surface tension and surface segregation for binary, ternary, and multicomponent high-temperature melts based on Bulter＇s original treatment [ 1] and on available physical properties and thermodynamic data, especially excess Gibbs free energies of bulk phase and surface phase versus temperature obtained from thermodynamic databases using the calculation of phase diagram （CALPHAD） approach, with special attention to the model parameter β, have been described. In addition, the geometric models can be extended to predict surface tensions of multicom- ponent systems from those of sub-binary systems. For illustration, some calculated examples, including Pb-free soldering systems and phase-diagram evaluation of binary alloys in nanoparticle systems are given. On the basis of surface tensions of high-temperature melts, interracial tensions between liquid alloy and molten slag as well as molten slag and molten matter can be calculated using the Girifalco-Good equation [2]. Modifications are suggested in the Nishizawa＇s model [3] for estimation of interracial tension in liquid metal （A）/ceramics （MX） systems so that the calculations can be carried out based on the sublattice model and thermodynamic data, without deliberately differentiating the phase of MX at high temperature. Finally, the derivation of an approximate expression for predicting interfacial tension between the high-temperature multicomponent melts, employing Becker＇s model [4] in conjunction with Bulter＇s equation and inteffacial tension data of the simple systems is described, and some examples concerning pyrometallurgical systems are given for better understanding.     

枝晶生长的数值模拟

对朱、洪已发展的一种微观尺度cellular automaton(CA)模型做了进一步改进．在改进的模型中用二元合金的Gibbs-Thomson方程建立固／液界面的平衡关系．考虑了动力学和表面能各向异性对枝晶择优生长方向的影响．应用改进的模型模拟了不同择优取向的单枝晶在过冷熔体中的自由生长、定向凝固过程中柱状晶的竞争生长以及等轴晶的演变过程．模拟结果表明,改进后的模型成功地模拟出各种不同择优取向的单枝晶和多枝晶的生长形貌。     

Thermodynamic Assessment of the Al-Mn and Mg-Al-Mn Systems

The binary Al-Mn system has been critically evaluated based upon available phase equilibrium and thermodynamic data, and optimized model parameters have been obtained giving the Gibbs energies of all phases as functions of temperature and composition. The liquid solution has been modeled with the modified quasichemical model to account for short-range ordering. The results have been combined with those of our previous optimizations of the Al-Mg and Mg-Mn systems to evaluate and optimize the Mg-Al-Mn system. All available data for the ternary system are reproduced with only one small ternary model parameter for the liquid phase.     

First-principles calculations of β″-Mg5Si6/α-Al interfaces

The metastable β″-Mg₅Si₆ phase is often the most effective hardening precipitate in Al-rich Al–Mg–Si alloys. Two important factors that control the precipitate morphology are the strain energy and the interfacial energy between the precipitate and the matrix. By means of a first-principles supercell approach and density functional theory calculations, we have studied the interfacial properties between β″-Mg₅Si₆ and α-Al. We carefully construct a large number of interfacial cells in order to elucidate preferred interfacial terminations and orientations, as well as atom alignment and intermixing across the interface. Each of the low-energy interfaces we found possesses two key attributes: a high number of Al–Si bonds across the interface, and a face-centered cubic topological alignment of atoms across those interfaces. Our first-principles results yield quantitative values for the interfacial energies, lattice mismatches and strain energies that can be used in future predictions of precipitate morphologies as a function of size.     

Misorientation texture development during grain growth. Part I: Simulation and experiment

Grain growth in two and three dimensions with anisotropic interfacial properties was simulated using the Monte Carlo method. The relative effects of grain boundary energy and mobility anisotropy on number- and area-weighted misorientation distribution functions (MDFs) were compared. Results indicate that energy anisotropy has a measurable effect on misorientation texture development, while mobility anisotropy does not. Qualitatively similar results are obtained in all simulations regardless of dimensionality or crystal symmetry. Microstructures with random orientation texture appear to evolve steady-state MDFs, while those with a preferred orientation do not. Experimentally measured number- and area-weighted MDFs in polycrystalline magnesia are shown to be comparable to those measured in our simulations.     

Effects of coherency stress and vacancy sources/sinks on interdiffusion across coherent multilayer interfaces – Part I: Theory

A phase field model corresponding to vacancy-mediated interdiffusion in coherent multilayers with completely miscible constituents is developed to explore the effects of several factors on interdiffusion across coherent multilayer interfaces, such as: (1) the dependence of diffusion potentials and mobilities on coherency stress; (2) the dependence of diffusion potentials and mobilities on composition; (3) the elastic constant inhomogeneity resulting from a inhomogeneous composition distribution; and (4) the properties of vacancy sources/sinks. The Gibbs free energy of the system consists of chemical and elastic energies. The gradient energy is neglected as the multilayers under consideration can be chemically well approximated by an ideal substitutional solution model. Elastic energy is a function of the stress-free strain and inhomogeneous elastic moduli distributions, while the stress is solved by anisotropic phase field microelasticity theory. The diffusion potentials are obtained straightforwardly as functional derivatives of the free energy with respect to composition and are in keeping with previous derivations that involved many mathematical manipulations or quite advanced theories. The diffusion mobilities are affected by the stress through modification of the vacancy formation and migration energies. Two limiting cases of vacancy sources/sinks are taken into account: ideal vacancy sources/sinks are uniformly and densely distributed, or not present at all, so the vacancy concentration is in equilibrium all the times, as determined by the local stress and composition in the former case, but deviates from the equilibrium concentration in the latter. The model can be conveniently extended to consider the non-ideal activity of vacancy sources/sinks by introducing a general kinetic relation for the vacancy creation rate.     

Effects of anisotropy on the equilibrium shape of nanoscale pores at grain boundaries

Molecular dynamics simulations have been performed to study the interaction between a faceted pore and an anisotropic grain boundary (GB). Nickel was chosen as a convenient model system. In order to establish the equilibrium crystal shape (ECS) of the pore, studies were also conducted on isolated pores. Isolated pores were found to be subject to the nucleation inhibition of equilibration that has been predicted by Rohrer et al. (J Am Ceram Soc 2000;83:214, 2001;84: 2099). This work shows that configurations close to the ECS can be obtained if supersaturation within a pore is artificially increased by adding mobile adatoms to the internal surfaces of the pores. In the case of pores located at GBs, the nucleation energy barriers to facet displacement are not present for facets in contact with the GB at the triple line, but may still persist for facets that have no contact with the GB. This problem can be overcome by approaching the equilibrium shape from different initial configurations. The configuration of the GB in the vicinity of the pore has been found to be essentially planar, indicating that GB puckering in the vicinity of anisotropic pores is not generally necessary. The present calculations show that incompatibilities between misoriented pore facets that meet at the triple line with the GB are easily accommodated by local atomic rearrangements at the disordered region of intersection with the GB.