Volume 101, Issue 4

In the classical description of surface diffusion, transport on a curved interface is associated with the Laplace – Beltrami operator acting on a chemical potential (difference). An early attempt to model surface diffusion via the phase-field approach goes back to Cahn, Elliott and Novick-Cohen; they use a scalar mobility approaching zero in the bulk. Similar models have been proposed first on the basis of heuristic ideas and then underpinned by asymptotic analysis. As it turns out, most of these analyses suffer from a subtle flaw, due not to a miscalculation but rather to early termination of the calculation. The asymptotic analysis provides all the equations desired for the correct sharp-interface limit. Unfortunately, it provides an additional equation, which is one restriction too many. Consequences for dynamical simulations of this kind of model are explored numerically. We construct two models based on the introduction of a tensorial mobility that approximate known sharp-interface equations without adding undesired constraints. Numerical simulations suggest superior performance of the new models in at least some situations.

We discuss a model of diffusion limited growth in solid-state transformations, which are strongly influenced by elastic effects. Density differences and structural transformations provoke stresses at interfaces, which affect the phase equilibrium conditions. We study the growth of a stable phase from a metastable solid in a channel geometry, and perform phase field simulations. Extensions to plastic models are discussed.

In this paper, numerical and implementation aspects of the phase-field crystal model with elastic interactions are addressed. This model leads to a time-dependent, sixth-order system of differential equations which yields a continuous density field, from which individual atom positions can be extracted and tracked in time. We solve the system using high-order finite elements, and reconstruct the elastic properties of the crystal from the atomic displacements with a meshless interpolation scheme. Numerical simulations are performed of benchmark experiments, which include deformation and dislocation dynamics.

The phase-field model of Wheeler, Boettinger and McFadden is extended to the case of fast solidification in which local non-equilibrium phenomena occur in the bulk phases and within the diffuse solid – liquid interface. Such an extension leads to the characteristic diffusion speeds of atoms (both within the diffuse interface and inside the bulk phases) and to the speed of the interface propagation. As a result, the model is described by a system of hyperbolic equations for the atomic diffusion transport as well as for the phase-field. This model is applied to the problem of solute trapping, which is accompanied by the entrapment of solute atoms beyond chemical equilibrium by a rapidly moving interface. The model predicts the beginning of complete solute trapping and diffusionless solidification at a finite solidification velocity.

Multiple junctions between different grains of a pure material are investigated using the multi-phase field model in 2D and 3D. According to theoretical considerations of the multi-phase field model equations, there may be quasi-static solutions that depend on the ratio of the interface mobilities. Numerical calculations in 2D and 3D indicate that the system always converges to the static solution as described by Young's law independent of the interface mobilities. No quasi-static solutions are found, which is attributed to the necessity of continuous solutions within the interface region of a phase-field model. The effect of interface mobility on the dynamics of the interface angle is discussed.

Stress generation and evolution in a cellular microstructure observed in the directional solidification process of a binary alloy system were simulated using a phase field model. The Ni–Cu system was chosen as a typical alloy, and two-dimensional simulations were carried out. The elastic stress induced by the volumetric contraction due to solidification was considered, and stress distribution in the solidified region was calculated. Results showed that a complex stress state is generated in the interfacial region, while it is homogeneous in the bulk solid. Under a condition causing the growing cells to coalesce, remarkably large stress was observed at the tip of the decayed cell, leading to a stress concentration around the liquid droplets and grooves subsequently generated. In order to show the effect of binary composition on the stress distribution, the dependence of Cu concentration on the elastic coefficient was considered, and simulations were carried out. Consequently, stress distribution in the bulk solid was observed along the cell boundaries, while no stress distribution was generated when this dependence was not taken into consideration.

The stable polarization distributions of freestanding ferroelectric nanotubes with different surface charges are investigated numerically using a phase field model based on the time-dependent Ginzburg – Landau equation. For a nanotube without surface charges, a pure vortex structure is formed as a result of the energy minimization. When the uniform surface charges are applied on the upper and lower surfaces, the out-of-plane components become nonzero. The out-of-plane components of polarizations are found to be proportional to the density of surface charge. When the density of surface charge exceeds a critical value, the in-plane components of polarizations disappear. This makes ferroelectric nanotubes experience an unusual transition from a vortex state to a single-domain state. Due to the different geometries, the critical charge density for the transition in nanotubes is larger than that of nanodots.

In this work auto-adaptive finite-element techniques are presented that allow for quantitatively reliable numerical computation of phase field models. These techniques are based on goal oriented error estimation for finite elements with dual weighted residuals and algorithms for auto-adaptive mesh adaptation in space and in time. It is discussed for which kind of problems the computational overhead of such methods is expected to be justified by a sufficient reduction of the problem size.

A phase-field model is presented which is specially tailored for engineering-oriented application. It addresses multiphase, multicomponent alloys and considers lattice specific crystallographic anisotropy. Versatile application examples to magnesium alloys demonstrate the model's capability of handling important aspects of microstructure formation during solidification. Thermodynamic data is derived from a Calphad database for the Mg – Al – Zn – Ca – Mn system. First, orientation selection and texture evolution is studied under directional growth conditions. A new approach is proposed to model the anisotropy of the hexagonal close-packed magnesium phase. Subsequently, grain structure formation is simulated for technical Mg – Al-based alloys under equiaxed growth conditions. It is shown how the cooling rate and the addition of specific alloy elements affect the grain size. Finally, the solidification path and its effect on the precipitation of the secondary phase MgAl has been investigated.

The mechanical properties of aluminum cast products depend to a large extend on their specific material microstructure, but can be impaired by defects which appear at the solidification stage, e. g. hydrogen pores. In the present work, we focus on specific issues for the phase-field modelling of microporosity defect formation in aluminium casting connected to hydrogen bubbles. Because of the large differences in the specific volumes of the melt and gas phase, large volume changes have to be handled by the model. In order to take into account different molar volumes, a modification of a multicomponent/multiphase-field model is proposed. It is based upon volume fraction considerations including additional mass transport that allow for the pressure dependent volume expansion of hydrogen bubbles growing in the melt. Furthermore, it also considers the pressure dependency of the hydrogen solubility. A simulation of the microstructure formation for an Al7 % Si0.30 % Mg alloy using the standard phase-field model is shown to be in agreement with experiment. The modified phase-field formulation has been applied to pure aluminium containing hydrogen. The equilibrium state for a gas bubble in liquid aluminium and bubble formation during solidification have been simulated for different pressures.

Fission product accumulation and gas bubble microstructure evolution in nuclear fuels strongly influence their thermo-mechanical properties such as thermal conductivity, gas release, volume swelling and cracking, and hence fuel performance. In this paper, a general phase-field model is developed to predict gas bubble formation and evolution. Important materials processes and thermodynamic properties including the generation of gas atoms and vacancies, sinks for vacancies and gas atoms, elastic interaction among defects, gas re-solution, and inhomogeneity of elasticity and diffusivity are accounted for in the model. The results demonstrate the potential applications of the phase-field method in investigating: 1) heterogeneous nucleation of gas bubbles at defects; 2) effect of elastic interaction, inhomogeneity of material properties, and gas re-solution on gas bubble microstructures; and 3) effective properties from the output of phase-field simulations such as distribution of defects, gas bubbles, and stress fields.

It is very important to understand the development of the microstructure in the interdiffusion zone between coatings and superalloy substrates for designing bond coat materials. In this study, the reaction-diffusion phenomena in the Ir-coating/Ni – Al binary substrate system are simulated using the phase-field model. The thermodynamic database developed based on the CALPHAD method is directly incorporated in the simulation. The effect of coating thickness on the growth of the secondary precipitated -phase is discussed. In the case of a semi-infinite system, the thickness of the -phase increases proportionally with the square root of time. In the case of a thin (1 thick) coating, inconstant -phase growth is observed. It is suggested that the phase growth is related to the change in the -phase fraction, which is derived from a thermodynamic line calculation from the substrate composition to pure Ir.

The calculation of γ(A1) + γ′(L1 2 ) or γ(A1) + γ′′(D0 22 ) two-phase microstructure changes in Ni-base superalloys has been one of the main subjects in the phase-field method, because the morphology of these two-phase microstructures crucially influences the high temperature mechanical properties of Ni-base super alloys. In this study, more complicated phase decomposition behavior, in which three phases, γ (A1), γ′(L1 2 ) and γ′′(D0 22 ) are simultaneously included, is modeled on the basis of the conventional phase-filed method. The results obtained are as follows: The present simulation model reasonably reproduces the complex feature of microstructure changes including γ, γ′ and γ′′ three phases, e. g. the characteristic morphology of the γ′ particles covered with γ′′ phase is simulated realistically. The morphology of the microstructure is mainly controlled by the elastic strain energy induced from the lattice mismatches among these three phases. In particular, the magnitude relation among the lattice parameters of the three phases is a key factor to control and understand the morphological changes of the complex microstructure including γ, γ′ and γ′′ three phases.

In this paper, austenite formation in an Fe – C system was simulated with a phase field approach. The model deals with a detailed description of morphological changes during austenite formation from ultrafine ferrite – cementite aggregates isothermally annealed in the intercritical region to form dual phase microstructures. Long-range diffusion of carbon is explicitly considered. The model is capable of resolving carbide particle sizes of about 100 nm to simulate the morphological complexity during austenite formation. Simulations were carried out in two- and three-dimensions. It was observed that morphological aspects of austenite formation depend significantly on spacing and distribution of cementite particles that provide suitable nucleation sites for austenite. This dependency can primarily be attributed to overlapping diffusion fields and curvature effects.

Phase field modelling is used to simulate austenite grain growth in the heat affected zone (HAZ) of an X80 linepipe steel. The HAZ experiences a very steep temperature gradient during welding which restricts grain growth. In addition to this phenomenon known as thermal pinning, austenite grain growth is affected by pinning due to precipitates and their potential dissolution. Grain growth has first been simulated for bulk samples subjected to rapid heating conditions to replicate thermal cycles at various positions in the HAZ. Effective grain boundary mobilities are introduced that are consistent with strong pinning at lower temperatures and weak pinning at higher temperatures. These two temperature regimes are separated by the estimated dissolution temperature of fine NbC precipitates. These mobility relationships are then used to predict austenite grain growth in the HAZ using typical time–temperature profiles.

The austenite grain growth of a microalloyed steel was investigated via annealing experiments and phase-field simulations using the phase-field code Micress. The technique described in a previous work was enhanced and applied to an Nb, Ti microalloyed linepipe steel for the case of isothermal heat treatment between 1 050 and 1 200 °C. The input parameters for the phase-field simulations were deduced from physical models based on the results of isothermal holding experiments. A further improvement was the use of the software package MatCalc to simulate at a lower scale the coarsening of the pinning particles. The results of these simulations showed good agreement with the experimental results.

The effects of grain boundary segregation and solute drag on grain size distributions were studied by employing the multi-phase-field method. It is shown that for normal grain growth the solute drag effect results in narrow steady state grain size distributions compared with the ideal grain growth. Besides influencing the normal grain growth, the solute drag can also induce abnormal grain growth. In some cases of abnormal grain growth self-similar grain size distributions are found and these distributions can be fitted by the lognormal function, which is consistent with the experimental observations.