Simulation of Dendritic Growth in Solidification of Al-Cu Alloy by Applying the Modified Cellular Automaton Model with the Growth Calculation of Nucleus within a Cell

Rather than designated directly as solid if the micromesh (or cell) larger than a nucleus is chosen as the nucleation site, the growth of a nucleus in the cell is considered in the application of the modified cellular automaton model to simulate the evolution of dendritic microstructures in the solidification of Al-Cu alloy. The growth velocity of a nucleus or a dendrite tip is calculated according to the KGT (Kurz-Giovanola-Trivedi) model, which is the function of the undercooling. In this study, the dendritic microstructures, such as the free dendritic growth in an undercooled melt and the dendritic growth in the directional solidification, are simulated with the modified growth algorithm in the nucleation cell. The simulated results for the temporal and final morphologies are shown and are in agreement with the experimental ones.

to the mesh instead of regarding it as completely solidified. The directional solidification processes for different solutal concentrations are simulated and compared with the experimental results. Mathematical models Nucleation. The continuous nucleation condition is considered in the present work where different Gaussian distributions account for the nucleation both on the mold wall and in the bulk liquid. The increase of grain density d n is induced by an increase in the undercooling d(ΔT) according to the following Gaussian distribution [3]: where ΔT mn is the mean nucleation undercooling, ΔT σ is the standard deviation, and n max is the maximum density of nuclei given by the integral of this distribution from 0 to ∞. Thus, the density of grains n(ΔT) formed at any undercooling ΔT is given by Growth kinetics and orientation. The growth velocity of a dendrite tip under a certain undercooling ΔT is calculated according to the KGT model [8], which is v(ΔT) = k 1 ΔT 2 +k 2 ΔT 3 . k 1 and k 2 are the functions of the initial concentration. The total undercooling at the dendrite tip is given by the sum of the various contributions to undercooling [3]: where ΔT c , ΔT t , ΔT r , ΔT k are the solutal, thermal, curvature, and kinetic undercooling, respectively. The kinetic undercooling is small for the metallic alloy under normal solidification, so that the effect is neglected. Nuclei formed on the mold wall or in the bulk liquid grow along the preferential growth direction, which corresponds to〈10〉for cubic metals in the present two-dimensional model.
Solute Transport. The local equilibrium at the solid-liquid interface is assumed to balance the interfacial concentrations that are described as follow.
where k is the partition coefficient, * s C and * l C are the interface equilibrium concentrations in the solid and liquid phases, respectively.
As the solidification proceeds, the solidified region rejects solute to the neighboring liquid according to Eq. (4). Diffusion within the entire liquid domain is redistributed by the following equation.
where D l is the solute diffusion coefficient in the liquid phase, f s is the solid fraction, and k is the partition coefficient. The last term of the right hand side of the equation indicates the amount of solute rejected at the solid-liquid interface. The governing equation for diffusion in the solid phase is given by where D s is the solute diffusion coefficient in the liquid phase. Thermal transport. The governing equation for two-dimensional transient heat conduction is where ρ is the density, C p is the specific heat, λ is the thermal conductivity, and ΔH is the latent heat.

Numerical method
The modified cellular automaton model used to predict the evolution of dendritic structures is based on the coupling of the macroscopic heat and microscopic mass transport. The designation of a nucleus site in a mesh (cell) is followed by applying the growth mechanism to the mesh instead of regarding it as completely solidified. Nucleation and growth algorithm. The cellular automaton model for the nucleation and growth simulation consists of (1) the geometry of a cell, (2) the state of a cell, (3) the neighborhood configuration, and (4) the transition rule that determines the state of a given cell during any time step. The calculation domain is divided into uniform square arrangement of micromeshes, called cells. Each cell is characterized by different variables such as temperature, concentration, crystallographic orientation, solid fraction and state (i.e. solid or liquid). Two kinds of cells are necessary according to Eq. (5) and (6). One is the liquid cell including the solid-liquid interface and the other is the solid cell. If a cell is a predetermined nucleation cite according to Eq. (1) by the local undercooling, the nucleus in the cell will grow along the preferential direction until the growth touches the boundaries of the cell. Then the state of the cell changes from liquid to solid with a reference index. The cell will grow continuously along the preferential direction corresponding to its crystallographic orientation if the undercooling of any adjacent liquid cell is sufficient to initiate the growth. The growth length L at a given time t n from the nucleus to the boundaries of nucleation cell or the solid cell to its liquid neighbors can be calculated by where Δt i is the time step size, θ is the angle of the cell's preferential growth direction〈10〉with respect to the line between the solid cell to its liquid neighbors or the nucleus to each boundary of the nucleation cell, and n indicates the time step number. r 0 is the critical radius forming a nucleus only effective in the nucleation cell and is zero elsewhere.
( ) ⎦ are the growth rates at time t 1 and t i . The growth rate in any liquid or nucleation cell can be calculated by using the KGT model depending upon the local undercooling ΔT(t i ). The neighborhood of 8 cells around a liquid cell is considered to estimate the liquid-to-solid state transition of the liquid cell. The local undercooling ΔT(t i ) is given by where T l is the equilibrium liquidus temperature, m is the liquidus slope, C 0 is the initial concentration, and Γ is the Gibbs-Thomson coefficient. s / l K , C s/l (t i ) and T s/l (t i ) are the mean curvature, the concentration and the temperature of the solid-liquid interface in the liquid cell at time t i . Then, the solid fraction f s (t n ) contributed from a solid cell to the liquid cell at a given time can be expressed as where l is the length between the solid cell and the liquid cell or half the size (dx/2) of the nucleation cell. Since the neighborhood of 8 cells around a liquid cell is considered, for example, dx = l if the solid cell is located at one of the four nearest east, west, south and north neighbors, and 2dx = l if the solid cell is located at one of the corner neighbors. When f s (t n )≧1, which means that the growth front of the solid cell touches the center of the liquid cell.
The transformation of state from liquid to solid depends on the total solid fraction The primary dendrite will grow and coarsen along the preferential 〈10〉direction by the means of the algorithm described above. As the growing and the coarsening processes of the primary trunk proceed, the solute will be enriched in the liquid near the solid-liquid interface due to the solute redistribution, which will destroy the interface stability and therefore form the side branching or the secondary arms along the primary trunk. Calculation of the interface curvature. The interface curvature in a cell with the solid fraction f s is calculated by the counting cell method [9], which is expressed as ( ) where dx is the cell size and n is the number of the neighboring cells. The values of the curvature vary from 1/dx to 0 for convex surfaces and from 0 to -1/dx for concave surfaces. Calculation of the solute transport. An explicit finite difference scheme is applied for calculating the solute diffusion in both the solid and liquid phases, and the zero-flux boundary conditions are used for the cells located at the surface of the calculation domain. The mesh arrangement is the same as the growth algorithm described above. When a cell transforms its state from liquid to solid by nucleation or growth, its concentration is changed according to Eq. (4). The rejected amount of solute * * l s (C kC ) − will accumulate in the neighboring liquid cell(s). Solution scheme for macroscopic thermal transport. As the thermal diffusivity of metallic alloys is 3-4 orders of magnitudes greater than the solute diffusivity, the kinetics for microstructure evolution can be assumed to be solute transport-controlled, and therefore the thermal diffusion can be considered to be complete in the microscopic scale. Thus the thermal transport is calculated through the arrangement of macroscopic meshes. The governing equation (Eq. (7)) and related boundary conditions are solved by the explicit finite difference algorithm. The value of temperature simulated is located in the center of the square mesh. The latent heat effect in the last term of Eq. (7) is dealt with by using the temperature recovery method depending on the total variation of the solid fraction in the macroscopic mesh, which is contributed from the change of the liquid-to-solid state in the microscopic cells included. Therefore, the equivalent temperature ΔT L recovered from a liquid cell due to the latent heat release during nucleation or growth is evaluated by where ΔH v is volumetric latent heat release, ρC p is the volumetric specific heat, and ΔX and dx are the mesh sizes for the macroscopic and microscopic schemes, respectively. Temperature of the macroscopic mesh is then recovered based on Eq. (13) by the newly solidified macroscopic cells. Using these updated temperatures, macroscopic heat transfer calculation can be continued. Integration of the macroscopic and microscopic schemes. The present model consists of two schemes: the combination of solute transport in FDM and modified cellular automaton model for simulating the evolution of dendritic structures and the macroscopic heat transport in FDM. Based on the calculated temperature profile in the cells, the calculation of nucleation and growth as well as the solute redistribution are carried out by the modified cellular automaton model as described above.
For two different schemes, two kinds of time step are used; one for the macroscopic heat transfer calculation based on the macroscopic mesh, and the other for the modified cellular automaton model, based on microscopic cells, as follows: Macroscopic time step: where λ is the thermal conductivity. Microscopic time step: where V max is the maximum growth velocity within all liquid cells during each time step and D l is the solute diffusion coefficient in the liquid phase. Within a Δt macro , the relationship between both time steps is ΣΔt micro ≦Δt macro .

Results and discussion
The Al-Cu alloy is chosen to simulate the several microstructures of dendritic growth in the present work with the material properties listed in table 1. Free dendritic growth in an undercooled melt. In order to simulate free dendritic growth in an undercooled melt, the calculating domain is divided into 320×320 cells with a cell size of 0.2 μm, which is fine enough to resolve a dendrite tip radius [5]. The size is larger than the critical radius r 0 of a crystal, which is approximated by the following equation [10] ( ) ( ) where Γ is the Gibbs-Thomson coefficient and 0 r T Δ is the undercooling for the occurrence of nucleation. Based on the calculation of the Gaussian distribution in the bulk liquid, the nucleation will appear if the simulated temperature is lowered by the amount of ΔT mn . Thus, for simplicity, the mean nucleation undrecooling ΔT mn in the bulk is used for 0 r T Δ . In the beginning of simulation, one nucleus with the preferential growth direction of 0 o or 45 o with respect to the horizontal direction is assigned in the center of the area. The initial concentration of the calculation domain is assumed to be C 0 . Table 1. Thermal and physical properties of Al-Cu alloy [5,10]  2.4×10 -7 The simulated dendrite morphology of an Al-2.0mass%Cu alloy solidified into an undercooled melt is shown in Fig. 1 for three stages: (a) the initial growth stage before emitting the side branch, (b) the initial of the secondary arms emitting from the primary trunk, and (c) the dendrite morphology with well-developed secondary and even tertiary arms. The different levels of darkness in the figure indicate the concentration profiles in both the solid and liquid phases. Within the solid region, along the centerline of primary trunks or side arms, there exists a spine with lower concentration, which is considered as the result of the combined effect of curvature and interface kinetics [11]. The concentration in solid near the solid-liquid interface shows the higher concentration where the final solidification occurs. It could be seen from Fig. 1 (c) that the tertiary arm branching occurs only at one side of the secondary arms. These phenomena have been consistent with the observation of experiments and also simulated by the modified cellular automaton and phase field models [5,11]. The results of Fig. 1 illustrates the similar capability of Prediction of the Structures formed in the directional solidification. The present model is applied to predict the evolution of dendritic grain structures of Al-2.5mass%Cu and Al-4.5mass%Cu alloys unidirectionally solidified over a copper chill plate with a constant temperature of 298 K. the nucleation parameters are listed in table 2. The symbols indexed "s" and "b" correspond to nucleation parameters on the mold surface and in the bulk liquid, respectively. Fig. 2 indicates the simulated and experimental macro-and micro-structures of Al-Cu alloys unidirectionally solidified with a pouring temperature of 1013 K. Fig. 2(a)-2(d) indicate the simulated structures and Fig. 2(e)-2(h) the experimental ones. The left two figures of either simulated or experimental ones indicate the case of the Al-2.5mass%Cu and the others for the case of the Al-4.5mass%Cu. Fig. 2(a) and 2(c) indicate the grain structures simulated by the classical cellular automaton model and Fig. 2(e) and 2(g) are obtained experimentally [5]. Fig. 2(b) and 2(d), indicating the dendritic structures simulated by the proposed method, are also in good agreement with 2(f) and 2(h). The case of Al-4.5mass%Cu has smaller undercooling in the bulk liquid, resulting in the nucleaction of equiaxed structures. For this simulation, a calculation domain is divided into 400×400 cells with a cell size of 3 μm, which is larger than the critical radius calculated by Eq. (16). It is seen from the figures that the proposed method can be applied to predict both the columnar and equiaxed dendritic morphology in a casting process.  Fig. 2. Comparison between predicted and experimental results in directional casting of Al-Cu alloys: (a), (b), (c) and (d) indicate the simulated results; and (e), (f), (g) and (h) indicate the experimental results in [5]. (a) and (c) indicate the macrostructures obtained by the classical cellular automaton model in [5], and (b) and (d) indicate the microstructures obtained by the proposed model.

Conclusions
The proposed model, which is based upon the coupling of a modified cellular automaton model with the growth calculation of a nucleus in a given nucleation cell, has been developed to simulate the evolution of the dendritic structure in solidification of alloys. For a free dentritic growth in the undercooled melt, it is found that the proposed model can quantitatively describe the evolution of dendritic growth features, including the growing and coarsening of the primary trunks, the branching of the secondary and tertiary dendrite arms, as well as the solute segregation patterns. Moreover the directional solidification with the columnar and equiaxed grains is simulated by the proposed method and the evolution from nucleation to impingement between grains is observed.