## 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.

## Few interesting reads

### August 5, 2009

From the latest PNAS:

[1] The elastic modulus, percolation, and disaggregation of strongly interacting, intersecting antiplane cracks

P M Davis and L Knopoff

We study the modulus of a medium containing a varying density of nonintersecting and intersecting antiplane cracks. The modulus of nonintersecting, strongly interacting, 2D antiplane cracks obeys a mean-field theory for which the mean field on a crack inserted in a random ensemble is the applied stress. The result of a self-consistent calculation in the nonintersecting case predicts zero modulus at finite packing, which is physically impossible. Differential self-consistent theories avoid the zero modulus problem, but give results that are more compliant than those of both mean-field theory and computer simulations. For problems in which antiplane cracks are allowed to intersect and form crack clusters or larger effective cracks, percolation at finite packing is expected when the shear modulus vanishes. At low packing factor, the modulus follows the dilute, mean-field curve, but with increased packing, mutual interactions cause the modulus to be less than the mean-field result and to vanish at the percolation threshold. The “nodes-links-blobs” model predicts a power-law approach to the percolation threshold at a critical packing factor of p c = 4.426. We conclude that a power-law variation of modulus with packing, with exponent 1.3 drawn tangentially to the mean-field nonintersecting relation and passing through the percolation threshold, can be expected to be a good approximation. The approximation is shown to be consistent with simulations of intersecting rectangular cracks at all packing densities through to the percolation value for this geometry, p c = 0.4072.

From the latest issue of Phil. Mag.:

The effect of a high magnetic field on the morphology of the MnBi primary phase during the directional solidification has been investigated experimentally and the results show that an application of a high magnetic field has enhanced the faceted growth and the coarsening of the MnBi primary phase. This may be attributed to the effect of a high magnetic field on the diffusion of the solute Mn and the growth anisotropy of the MnBi crystal.

[2] A new counter-example to Kelvin’s conjecture on minimal surfaces

R Gabbrielli

A new counter-example to Kelvin’s conjecture on minimal surfaces has been found. The conjecture stated that the minimal surface area partition of space into cells of equal volume was a tiling by truncated octahedra with slightly curved faces (K). Weaire and Phelan found a counter-example whose periodic unit includes two different tiles, a dodecahedron and a polyhedron with 14 faces (WP). Successively, Sullivan showed the existence of an infinite number of partitions by polyhedra having only pentagonal and hexagonal faces that included WP, the so-called tetrahedrally close packed structures (TCP). A part of this domain contains structures with lower surface area than K. Here, we present a new partition with lower surface area than K, the first periodic foam containing in the same structure quadrilateral, pentagonal and hexagonal faces, in ratios that are very close to those experimentally found in real foams by Matzke. This and other new partitions have been generated via topological modifications of the Voronoi diagram of spatially periodic sets of points obtained as local maxima of the stationary solution of the 3D Swift-Hohenberg partial differential equation in a triply periodic boundary, with pseudorandom initial conditions. The motivation for this work is to show the efficacy of the adopted method in producing new counter-examples to Kelvin’s conjecture, and ultimately its potential in discovering a periodic partition with lower surface area than the Weaire-Phelan foam. The method seems tailored for the problem examined, especially when compared to methods that imply the minimization of a potential between points, where a criterion for neighboring points needs to be defined. The existence of partitions having a lower surface area than K and an average number of faces greater than the maximum value allowed by the TCP domain of 13.5 suggests the presence of other partitions in this range.

[3] The cross-slip energy unresolved

G Schoeck

Recent progress in dislocation dynamics modeling of work hardening has reawakened the interest in cross-slip, which can lead to dynamic recovery in fcc crystals. It is pointed out that neither continuum theory nor atomic modeling at present are able to reliably derive the reaction path and the activation energy of cross-slip. Classical continuum theory with the concept of Volterra dislocations fails, because during the nucleation process the effective Burgers vectors of the partials are not conserved and the specific atomic misfit energy changes. Atomistic modeling fails, because the ad hoc potentials used at present are unable to reliably predict the energies for atomic displacements far from equilibrium. It is, however, possible to derive the stress conditions necessary in order that cross-slip can spread. An important contribution to the driving force results from the ‘Escaig stress’ acting on the edge components of the partials forming a dissociated screw dislocation and changing their separation. Contrary to the widely held assumption, the driving force is however independent of whether the dislocation in the cross-slip plane will be expanded or compressed.

## Solidification and precipitation in Mg alloys

### July 8, 2009

M Wang et al

A formulation of solid-liquid interfacial thermodynamic and kinetic anisotropic characteristics for hexagonal close-packed metals is proposed. The two- and three-dimensional dendritic growth of primary Mg in undercooled Mg-Al alloy melts are modeled using the phase-field method, with consideration of the integration of crystallographic lattice symmetry and experimental observations. The morphologies of three-dimensional dendrites are obtained and the calculated results have shown intricately hierarchical branched structures. The excess free energy of solution system is based on the Redlich-Kister model.

[2] Phase transformations in QE22 Mg alloy

G Barucca et al

The precipitation sequence in a QE22 Mg alloy is followed by differential scanning calorimetry, microhardness, electrical resistivity, positron annihilation spectroscopy and transmission electron microscopy (TEM) observations, after different thermal treatments. The decomposition of the supersaturated solid solution occurs via the formation of nanosized coherent structures (GP zones) followed by the co-precipitation of two metastable phases responsible for the peak ageing condition. The stable phase (Mg, Ag)12Nd appears at the highest annealing times, leading to over-ageing and hardness reduction. TEM observations provide information on the crystallographic structure of the forming phases, allowing some inconsistencies present in the literature to be clarified. Activation energies are derived from both calorimetric and resistometric measurements at different scanning rates.

[1] L2 droplet interaction with α-Al during solidification of hypermonotectic Al–8 wt.% Bi alloys

P L Schaffer, R H Mathiesen and L Arnberg

Studies of Al-based hypermonotectics have so far focused mainly on droplet motion and coagulation dynamics, with limited attention given to the interaction between droplets and the advancing solidification front which is decisive for the final distribution of the second phase within the α-Al matrix. The current work presents results from directional solidification experiments with Al–8 wt.% Bi alloys. It was found that droplets with large radii were frequently pushed and small droplets were engulfed. This is contradictory to the many models that have been proposed to explain pushing/engulfment of solid particles and can in part be ascribed to the fact that while solid-particle models only consider single, non-interacting particles that remain unaffected by solutal gradients ahead of the advancing solidification front, droplet–droplet interaction and local solute gradients have been found to be critical for droplet pushing/engulfment behaviour in hypermonotectic alloys.

P C Millett, and Y U Wang

A novel mesoscale simulation approach to modeling the collective interactions of charged colloidal particles allowing investigation of complex self-assembly processes is presented. Diffuse interface field variables are used to describe the shape, size, location and orientation of each individual particle within the computational domain. In addition, these field variables are used to determine the spatially resolved particle charge density distributions as well as the magnitude and direction of the electric field throughout the medium. Individual particle positions and rotations are updated in time as a result of long-range electrostatic and short-range repulsive forces and torques. Illustrative results of the model’s capability to evolve both monodisperse and binary distributions of charged particles of various shapes and charge characteristics are presented.

[3] Kinetics of diffusion-controlled transformations: Application of probability calculation

H Wang et al

An analytical model has been developed to describe the overall kinetics of diffusion-controlled transformations assuming site saturation or continuous nucleation, in combination with one-dimensional growth. On the basis of linear approximation of the concentration gradient, the method of probability calculation is adopted to model the transformed fraction. First, the so-called geometrical model was re-derived, assuming that the diffusion-controlled growth, according to the parabolic growth law, stops due to geometrical impingement of grains plus their diffusion fields. Then, the transformation kinetics subjected to soft impingement was described, following an analogous approach. The effect of soft impingement, depending on the degree of supersaturation, has been interpreted by evolution of the transformed fraction and the growth exponent. The model was applied to describe the isothermal austenite–ferrite transformation of 0.37C–1.45Mn–0.11 V microalloyed steel, and a good agreement between model predictions and experimental results has been obtained.

[4] Pattern formation in constrained dendritic growth with solutal buoyancy

I Steinbach

Competing self-organization between the solidification pattern and the convection pattern in a directional solidification environment is investigated theoretically and by phase-field simulations. Melt flow introduces a mode of transport with broken symmetry dependent on the direction of growth relative to the vector of gravity. A stable and an unstable regime can be distinguished. The interaction between spacing selection and convection leads to a new type of scaling that explains results from phase-field simulations and solidification experiments under enhanced gravity.

[5] Study of twinned dendrite growth stability

M A Salgado-Ordorica, J Vallonton and M Rappaz

Under certain thermal conditions (G ≈ 1 × 104 K m−1, νs ≈ 1 × 10−3 m s−1), left angle bracket1 1 0right-pointing angle bracket twinned dendrites appear in aluminum alloys and can overgrow regular columnar dendrites, provided that some convection is also present in the melt. In order to check the stability of such morphologies, directionally solidified twinned samples of Al–Zn were partially remelted in a Bridgman furnace and then resolidified under controlled conditions, with minimal convection. It was found that, although twin planes remain stable during partial remelting, non-twinned dendrites regrow during solidification. They have a crystallographic orientation given by those of the twinned and untwinned “seed” regions, and grow along preferred directions that tend to be those of normal specimens.

[6] Atomistic considerations of stressed epitaxial growth from the solid phase

N G Rudawski and K S Jones

A dual-timescale model of stressed solid-phase epitaxial growth is developed to provide a basis for the atomistic interpretation of experiments where the macroscopic growth velocity of (0 0 1) Si was studied as a function of uniaxial stress applied in the plane of the growth interface. The model builds upon prior empirical modeling, but is a significant improvement as it provides solid physical bases as to the origin of growth being dual-timescale and more accurately models growth kinetics.