## Simulation of dendritic microstructures

### February 24, 2011

[1] Phase-field simulation of micropores constrained by the dendritic network during solidification

H Meidani and A Jacot

A phase-field model has been developed to describe the morphology of pores constrained by a dendritic solid network, and are forced to adopt complex non-spherical shapes. The distribution of the solid, liquid and gas phases was calculated with a multiphase-field approach which accounts for the pressure difference between the liquid and the gas. The model considers the partitioning of the dissolved gas at interfaces, gas diffusion and capillary forces at the solid/liquid, liquid/gas and gas/solid interfaces. The model was used to study the influence of the dendrite arm spacing (DAS) and the solid fraction on the state of a pore. The calculations show that a pore constrained to grow in a narrow liquid channel exhibits a substantially higher mean curvature, a larger pressure and a smaller volume than an unconstrained pore. Comparisons with simple geometrical models indicate that analytical approaches show a good trend but tend to underestimate the pore curvature, in particular at high solid fractions, where pores have to penetrate the thin liquid channels. For pores spanning over distances larger than the average DAS, the simulations showed that the radius of curvature can vary between two limits, which are given by the size of the narrowest section that the pore needs to pass in order to expand and by the largest sphere that can be fitted in the interdendritic liquid. The pore curvature is therefore a complex non-monotonic function of the DAS, the solid fraction, the hydrogen content and statistical variations of the liquid channel width.

[2] Simulation of a dendritic microstructure with the lattice Boltzmann and cellular automaton methods

H Yin et al

A new modeling approach combining the lattice Boltzmann method (LB) and the cellular automaton technique (CA) was developed to simulate solidification at the microscale. The LB method was used for the coupled calculation of temperature, solute content and velocity field, while the CA method was used to compute the liquid/solid phase change. To validate the accuracy of the LB–CA model and its efficiency for the simulation of dendrite growth under convection, comparisons of the tip characteristics and dendrite morphologies under various simulation conditions were made with those obtained by analytical means and by a finite element model coupled with the cellular automaton technique (FE–CA model). The results show that the LB–CA model is computationally much more efficient than the FE–CA model for simulations of dendritic microstructures under convection. The tip splitting phenomenon was captured for high cooling rates and with comparatively coarse grids due to mesh-induced anisotropy and thermal instabilities. The simulated dendrite morphologies obtained with various anisotropy and Gibbs–Thomson coefficients were discussed. The results show that the dendrite growth direction does not always follow the crystallographic direction and high branching phenomena can occur with small anisotropy and/or Gibbs–Thomson coefficients.

[1] Modeling the overall solidification kinetics for undercooled single-phase solid-solution alloys. I. Model derivation

H Wang et al

Departing from the volume-averaging method, the equiaxed solidification model was extended to describe the overall solidification kinetics of undercooled single-phase solid-solution alloys. In this model, a single grain, whose size is given assuming site saturation, is divided into three phases, i.e. the solid dendrite, the inter-dendritic liquid and the extra-dendritic liquid. The non-equilibrium solute diffusion in the inter-dendritic liquid and the extra-dendritic liquid, as well as the heat diffusion in the extra-dendritic liquid, is considered. The growth kinetics of the solid/liquid interface is given by the solute or heat balance, where a maximal growth velocity criterion is applied to determine the transition from thermal-controlled growth to solutal-controlled growth. A dendrite growth model, in which the nonlinear liquidus and solidus, the non-equilibrium interface kinetics, and the non-equilibrium solute diffusion in liquid are considered, is applied to describe the growth kinetics of the grain envelope. On this basis, the solidification path is described.

[2] Interactions between carbon solutes and dislocations in bcc iron

H Hanlumyuang et al

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.

## Specific surface area during solidification

### May 11, 2010

Evolution of specific surface area with solid fraction during solidification

L Ratke and A Genau

The specific surface area varies with solid fraction during phase transformation from liquid to solid. The few measurements available show a non-linear dependence of the specific surface area on the solid fraction, with an initial increase as the amount of solid increases, followed by a decrease as the system moves toward complete transformation. We derive a simple model for this behaviour assuming a combination of growth and coalescence. We obtain a relation exhibiting an increase with the square root of fraction solid at low volume fractions, independent of a growth law, and a decrease at higher volume fractions which depends on the model chosen to describe the coalescence of dendrites. By choosing an appropriate constant, the model accurately describes recent data presented by Limodin and co-workers.

## Diffusivities in ternary melt

### March 31, 2010

Diffusivities of an Al–Fe–Ni melt and their effects on the microstructure during solidification

Lijun Zhang et al

A systematical investigation of the diffusivities in an Al–Fe–Ni melt was presented. Based on the experimental and theoretical data about diffusivities, the temperature- and composition-dependent atomic mobilities were evaluated for the elements in Al–Ni, Al–Fe, Fe–Ni and Al–Fe–Ni melts via an effective approach. Most of the reported diffusivities can be reproduced well by the obtained atomic mobilities. In particular, for the first time the ternary diffusivity of the liquid in a ternary system is described in conjunction with the established atomic mobilities. The effect of the atomic mobilities in a liquid on microstructure and microsegregation during solidification was demonstrated with one Al–Ni binary alloy. The simulation results indicate that accurate databases of mobilities in the liquid phase are much needed for the quantitative simulation of microstructural evolution during solidification by using various approaches, including DICTRA and the phase-field method.

## Grain refinement during solidification

### March 7, 2010

Grain refinement in highly undercooled solidification of Ni85Cu15 alloy melt; direct evidence for recrystallisation mechanism

T Zhang et al

The grain refinement occurring upon rapid solidification of undercooled Ni85Cu15 alloy melts has been studied. Applying theoretical calculations for stress accumulation in dendritic skeletons, the grain refinement occurring with ΔT >not, vert, similar180K is ascribed to the plastic deformation of dendrites and the subsequent recrystallization. This has been evidenced directly using high resolution transmission electron microscopy (HRTEM) observation for the as-solidified granular crystals.

## Fluid flow in 3D single crystal dendrites

### February 15, 2010

Modeling fluid flow in three-dimensional single crystal dendritic structures

J Madison et al

Convection during directional solidification can cause defects such as freckles and misoriented grains. To gain a better understanding of conditions associated with the onset of convective instabilities, flow was investigated using three-dimensional (3D) computational fluid dynamics simulations in an experimentally obtained dendritic network. A serial-sectioned, 3D data set of directionally solidified nickel-base superalloy measuring 2.3 × 2.3 × 1.5 mm was used to determine the permeability for flow parallel and normal to the solidification direction as a function of solid fraction (fS). Anisotropy of permeability varies significantly from 0.4 < fS < 0.6. High flow velocity channels exhibit spacings commensurate with primary dendrite arms at the base of the mushy zone but rapidly increase by a factor of three to four towards dendrite tips. Permeability is strongly dependent on interfacial surface area, which reaches a maximum at fS = 0.65. Results from the 3D simulation are also compared with empirical permeability models, and the microstructural origins of departures from these models are discussed.

## Phase field model of polycrystalline solidification

### January 25, 2010

A quantitative multi-phase field model of polycrystalline alloy solidification

N Ofori-Opoku and N Provatas

A multi-phase field model for quantitative simulations of polycrystalline solidification of binary alloys is introduced. During the free-growth stage of solidification, the model exploits the thin-interface analysis developed by Karma [3] in order to realistically capture bulk phase diffusion and the sharp interface corrections predicted by traditional models of solidification. During grain boundary coalescence, the model is constructed to reproduce the properties of repulsive grain boundaries described by Rappaz et al. [29]. The model provides a very simple mechanism for decoupling of solute and concentration fields at steady state, an important feature for calculating grain boundary energies.