# CA Modeling of Microsegregation and Growth of Equiaxed Dendrites in the Binary Al-Mg Alloy

## Abstract

**:**

## 1. Introduction

## 2. Description of the Model

_{S}, preferential angle of crystallographic orientation—θ

_{0}, interface curvature—K, direction normal to the solidification front—φ. During solidification, each cell changed its phase state from liquid (f

_{S}= 0) to transitional (0 < f

_{S}< 1) and finally to solid state (f

_{S}= 1). Initially, the temperature and concentration distribution across the domain was homogeneous and consistent with the values T

_{0}and C

_{0}. The model uses the von Neumann neighbourhood to solve the mass and energy transport equations, and the Moor neighbourhood to determine the remaining quantities. In order to reduce the artificial anisotropy of dendritic growth induced by the CA grid, the procedure of following the preferred growth directions was applied [30]. The quantitative and qualitative description of the growth kinetics of dendritic crystals is represented by three basic fields: solid fraction field, concentration field and temperature field. The instantaneous interface location and the instantaneous shape of dendritic structures are recreated by conjugate solving of the system of model equations.

#### 2.1. Modelling of Component Diffusion and Solid Phase Growth

_{ef}depends on the phase state of the central cell and the configuration of the cells of the immediate surroundings at the moment t and takes homo- and heterogeneous neighbourhoods into account:

_{S,i,j}, f

_{S,Ne}are solid phase fractions in the central cells i,j and in neighbouring cells (von Neumann neighbourhood), respectively, D

_{S}, D

_{L}are the diffusion coefficients of the component in the liquid and solid phases, respectively, and k

_{0}is the component separation coefficient.

_{0}—preferential angle of crystallographic orientation, δ—surface tension amplitude, m

_{s}—crystal symmetry coefficient (m

_{s}= 4), K—interface curvature, φ—direction normal to the solidification front, m

_{L}—direction coefficient of the liquidus line, G—Gibbs-Thomson coefficient, T

^{F}—solidification front temperature, T

_{L}—liquidus temperature, W

_{0}—the initial component concentration.

_{S}= 0, pseudo-initial condition). As a result, the component concentration in the cells of the interface decreases to the value:

_{S}= 0 and f

_{S}= 1.

^{f}the normal growth rate υ

_{n}is determined from dependence [15,18,19]:

#### 2.2. Temperature Field Approximation and Interpolation

^{−9}s in the case of modelling the Al alloys solidification on a CA grid with a dimension of 1 µm. In order to overcome this inconvenience, implicit methods are used, or a temperature that varies in time, but is homogeneous, is assumed for the entire calculating area [4,12,13,14]. Another variant that was adapted to the model is the determination of the temperature field using two grids. A sparse grid was superimposed on the main CA grid, the constant of which (Δx

_{B}) is several times larger than the automaton cell dimension a. A block of cells with dimension n

_{B}× n

_{B}on the main grid corresponds to a single cell of the sparse grid. Feedback occurs between the grids. On the sparse grid, for each time interval, the heat transport equation is solved, and the determined temperature field is interpolated into the main grid of the automaton. In turn, the instantaneous increments of the solid fraction calculated in CA unit cells are summed up on blocks and transferred to the sparse grid.

_{B}× n

_{B}met the relation 2

^{n}× 2

^{n}elementary cells (n—natural number). The temperature interpolation from the central cells to the entire main grid is carried out in 2n steps. In the first step, the temperatures in the cells $(\pm {2}^{n-1}a\sqrt{{2}^{}},\pm {2}^{n-1}a\sqrt{{2}^{}})$ located from central cells are calculated. The obtained values together with the rewritten ones constitute the input data for the second step, in which the temperatures in the cells from the environment are determined $(\pm {2}^{n-1}a,\pm {2}^{n-1}a)$. In the third and fourth steps, interpolation is performed based on the neighbourhoods $(\pm {2}^{n-2}a\sqrt{{2}^{}},\pm {2}^{n-2}a\sqrt{{2}^{}})$ and $(\pm {2}^{n-2}a,\pm {2}^{n-2}a)$. The procedure is continued by systematically reducing the distance between cells in the main and intermediate directions by half (according to the wind rose). Moving from the distant to the nearest neighbourhood, the temperatures in all cells of the automaton are determined. Since the distance between the cells is the same in each step, the interpolation problem comes down to calculating the mean of the four surrounding cells. The interpolation scheme on a fragment of the main grid with blocks with a dimension of 2

^{3}× 2

^{3}for the next two steps is illustrated in Figure 1c,d. The diagram shows the solution using the periodic boundary condition at the ends of the cellular automaton. The sparse grid size is determined by the ratio Δx

_{B}= 2

^{n}a.

#### 2.3. Nucleation and Initial Growth Period under Transient Diffusion Conditions

_{N}which is less than T

_{L}. Temperature T

_{N}depends directly on the rate of cooling and diffusion of the component in the liquid phase. The idea of the algorithm is as follows. Once the liquidus temperature is reached, a randomised location of nucleus cells is determined in the CA area, and it is assumed that the proportion of the solid phase in them is one. The formation of nucleus cells is related to the “production” of an excess of the alloy component ΔW = W

_{L}− W

_{S}. It is assumed that the component is evenly rejected to all cells in the Moore vicinity and its concentration in each of these cells is increased by the amount ΔW/8. From that moment on, the procedure for solving the diffusion equation is started. The initial increase in the component concentration around nucleus cells makes them inactive for a certain period of time. Only as a result of channelling the component deep into the bath and dissipating superheat, they are able to connect their neighbours. This occurs as soon as neighbouring cells reach temperature T

^{F}, determined by Equation (3). Temperature T

^{F}is considered the nucleation temperature T

_{N,}and the time needed for lowering the temperature from T

_{L}to T

_{N}an incubation period.

#### 2.4. Optimisation of the Time Step for Model Equations

_{stab}which ensures the stability of the calculations. The stability condition for the numerical solution of the diffusion equation is expressed by the dependence:

_{B}:

_{stab}is also limited by an additional criterion that relates to the speed at which the solidification front moves through the interface cell. In order to ensure high “precision” of calculations, the increment of the solid fraction in one step of the calculations must be sufficiently small. The construction of the automaton assumes that the cell changes its state from transient to solidified at the moment when the solid fraction reaches one, and exceeding this value is omitted. Moreover, due to the use of an explicit scheme to solve the diffusion equation, excessive solid phase growth can lead to overestimated values of the component concentration in the interface cells. It was established in [27] that the elementary increment of the solid phase ∆f

_{el}in one-time step should be in the range (0.01–0.1). Following these guidelines and preliminary simulations, it was assumed that the maximum increment in one step of the calculations was 0.02. By inserting this value into Equation (8) and taking dependencies (13) and (14) into account, the time step limitation is obtained in the form of the formula:

_{stab}= ∆t

^{T}is solved. After the liquidus temperature is exceeded, optimization of the time step is done with regard to quantity ∆t

^{f}, comparing it with ∆t

^{T}and ∆t

^{W}. For high solids growth rates, upper limit ∆t

_{stab}is related to the second term of formula (15), while for medium and low velocities, it is related to the stability of the explicit scheme of the thermal conductivity equation. Due to step ∆t

^{T}, despite the use of the multi-grid technique, it can be several dozen times smaller than steps ∆t

^{f}or ∆t

^{W}; in order to shorten the calculating time, the algorithm uses a separate iteration procedure for the thermal problem. In this procedure, the number of iterations based on the ratio ∆t

^{f}/∆t

^{T}or ∆t

^{W}/∆t

^{T}are determined. The isolation of the energy equation means that the remaining model equations can be solved with the limitation for ∆t

^{f}or ∆t

^{W}. Optimization is performed for each iteration, and a set new value ∆t

_{stab}is valid for the next step. No calculations are updated for the current step.

## 3. Numerical Simulations Results

#### 3.1. Free Growth of a Single Dendrite

^{2}. The side length of the cell was selected empirically. If the value “a” of the cells is selected too high, the shape of the growing main arms of the dendrite ceases to depend on their preferential crystallographic orientation and begins to depend only on the symmetry of the CA grid. For the cooling rates used in the simulations, the growth of dendrites in the direction corresponding to their assumed orientation is obtained for a cell side length of up to 2 µm. The diffusion equation was solved assuming a periodic boundary condition at all ends of the CA grid. For the energy equation, it was assumed that the heat dissipation from the modelled area occurs perpendicular to its surface and at a constant value of the heat flux—the boundary condition of the second type. After exceeding the temperature T

_{L}one nucleus cell was placed in the centre of the calculating domain. This cell was assigned an orientation angle that is inconsistent with both of the major directions of the CA grid. Preferential angle of crystallographic orientation θ

_{0}was 25°.

_{0}.

_{N}where the growth of the dendrite begins. However, due to the very small number of interface cells, the amount of generated solidification heat is still very small and the temperature is further lowered. At the same time, the degree of concentration undercooling and the growth rate of the solid fraction increase. Over time, the dendrite surface (interface) becomes larger and the growth rate is intensified, resulting in the release of more solidification heat and the suppression of the temperature drop. The local minimum for which the solidification heat is equal to the cooling rate is revealed in the solidification curve. The minimum temperature also corresponds to the maximum degree of undercooling of the alloy and the maximum growth rate of the solid fraction.

#### 3.2. Validation of the Numerical Model

^{2}reaches a value above 0.99. Figure 6a shows the calculation results for a dendrite with the preferred crystallographic orientation 0°, growing with undercooling 4.2 K. Based on the obtained approximating functions, the curvature and then the diameter of the dendrite front were determined using dependence K = d

^{2}y/dx

^{2}× [1 + (dy/dx)

^{2}]

^{−3/2}and the relation R = 1/K. The comparative characteristics of the dendrite tip diameter calculated from the numerical model and the LGK analytical model for different undercooling are shown in Figure 5b. The LGK model prediction was performed assuming stability parameter σ = 1/4π

^{2}. When evaluating the test results, it can be observed that both the obtained parameter value range R as well as the tendency of its changes depending on undercooling are satisfactory. The second real parameter that allows for validating the numerical model is the dendrite front growth rate under the set diffusion conditions. Simulated growth rate (Figure 6c) also presents a good agreement with the results predicted by the LGK model. However, it should be noted that the values calculated from the numerical model are always below the LGK curves. This fact was found in other papers [18,19,36] and is explained by the sensitivity of the LGK model to parameter σ, which changes depending on undercooling and the initial alloy composition.

#### 3.3. Multiple Dendrite Growth Simulation

^{2}using a cellular automaton with elementary cells sized 408 × 408. The diffusion equation was solved assuming the same boundary conditions as in the previous simulations. On the other hand, for the heat conduction equation on the lower and upper walls, a periodic condition was assumed, and on the right wall—thermal symmetry q

_{b}= 0, and on the left wall—Newton’s boundary condition:

_{∞}—the bulk temperature of the cooling media.

_{0}equal to 670 °C. After reaching the liquidus temperature, 12 nucleus cells with the initial composition kC

_{0}and the preferred crystallographic orientation ranging from 0° up to 90° were placed randomly in the calculating domain. Coupled motion and interaction of equiaxial dendritic crystal growth during solidification of Al + 5 wt.% Mg alloy is presented in Figure 7.

^{CA}(f

_{S}), the average temperature from the entire calculating domain was assumed.

^{CA}(f

_{S}) towards lower temperatures is caused by concentration and capillary undercooling, which determine the dendritic growth rate. Location of the end of solidification temperature (T

_{S}

^{CA}) in the CA model depends mainly on the rate of diffusion in the solid phase and for the given cooling conditions it occupies an intermediate place (532 °C) between the equilibrium solidus temperature (T

_{S}) and the temperature of the eutectic transformation (T

_{E}).

## 4. Experimental Tests

#### 4.1. Tests of the Solidification Process Using the DDTA Method

_{N}—the temperature of the onset of solidification, δ

_{T}= T

_{3}− T

_{1}—temperature range of recalescence, T

_{1}, T

_{3}—temperatures at which, respectively, maximum and minimum undercooling of the alloy occur in the initial solidification period T

_{2}—maximum thermal effect of solid phase growth, T

_{K}—temperature of the solidification end.

_{N}and T

_{1}calculated from the numerical model. The actual end of solidification temperature is 538 °C and compared with the temperature T

_{S}from the CA model is higher by 6 °C. It should be noted here that the difference between the equilibrium and actual solidus temperature is 55 °C. The total time needed to dissipate the heat of overheating, solidification and cooling down to the temperature of 500 °C in the thermal centre of the casting in real conditions is equal to 40 s. This time, calculated on the basis of the CA solidification model, is slightly shorter by 1.5 s. In order to determine the compliance of the numerical simulation results with the DDTA experimental measurements, the percentage error of the numerical simulation was calculated using the dependence:

_{S}(Table 1) indicate that the values of the actual characteristic solidification points are very close to the simulated values. The maximum error of numerical simulations does not exceed 8%, and its average value, taking all the assessed values into account, is 1.92%. Further assessment of the adequacy of the numerical model was performed based on the kinetics of the growth of the solid phase. In the DDTA method, the temperature is measured in the thermal centre of the casting, for which the energy balance is determined by the dependence:

_{Z}and characteristics T(t). The values of these parameters determined on the basis of DDTA curves (Figure 9b) are: α

_{L}= 336 Wm

^{−2}K

^{−1}, ξ

_{L}= 19.1 × 10

^{−3}s

^{−1}for T > T

_{L}, and α

_{S}= 268 Wm

^{−2}K

^{−1}, ξ

_{S}= 15.6 × 10

^{−3}s

^{−1}for T < T

_{S}, In the solidification temperatures range T

_{L}–T

_{S}their change is assumed to be linear. Based on the thermal parameters of the metal mould, the “base curve” (red line in Figure 9b), which shows the change in the casting cooling rate in the absence of internal heat sources, was determined. The area limited by the blue and red curves (Figure 9b) shows the thermal effect related to the heat of solidification, and integration of this area determines the kinetics of the solid phase growth (Figure 10).

_{S}(t), it can be concluded that, in terms of solidification kinetics, the results of computer simulations also correspond to the experimental results of DDTA. The difference between the calculated solidification time and the actual solidification time is 2.5 s, and the average growth rate of the solid fraction (determined from the slope of the curves f

_{S}(t) in the middle solidification period (0.2–0.8) f

_{S}) is equal to 0.035 s

^{−1}for CA model and 0.039 s

^{−1}for DDTA tests.

#### 4.2. Results of Microstructural Tests

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Diagram of temperature interpolation on a fragment of the main grid with blocks of size 2

^{3}× 2

^{3}for two consecutive steps of calculations. (

**a**) calculation of temperature field on coarse lattice; (

**b**) Transfer of temperatures from a coarse lattice to central cells of CA blocks; (

**c**) step 1. calculation of average temperature on the basis of neighborhood $(\pm {2}^{n-1}a\sqrt{{2}^{}},\pm {2}^{n-1}a\sqrt{{2}^{}})$; (

**d**) step 2. calculation of average temperature on the basis of neighborhood $(\pm {2}^{n-1}a,\pm {2}^{n-1}a)$.

**Figure 2.**Instantaneous Mg concentration fields and free dendrite growth sequences in the Al + 5 wt.% Mg alloy cooled at the rate: (

**a**) 5 K/s, (

**b**) 25 K/s, (

**c**) 45 K/s.

**Figure 3.**The results of the simulation of the equiaxial dendrite growth for the cooling rate of 45 K/s: (

**a**) the morphology of the dendrite with orientation θ

_{0}= 45°, (

**b**) Mg concentration profiles during dendrite growth determined along line 1, (

**c**) magnesium distribution along the section starting from the base of the dendrite and crossing the secondary arms (line 2) − f

_{S}= 0.1.

**Figure 4.**(

**a**) changes in the concentration of magnesium at the solidification front depending on the share of the solid phase (dendrite size) and different cooling rates, (

**b**) changes in the average temperature in the initial period of crystal growth.

**Figure 5.**Kinetic characteristics of the initial solidification period: (

**a**) the growth rate of the dendrite front, (

**b**) the effect of the cooling rate on the concentration undercooling value and the transition time from the transient state to the steady state.

**Figure 6.**Comparison of the predictive parameters of the LGK model and the developed CA model: (

**a**) the shape of the dendrite front and its parabolic approximation, (

**b**) the effect of concentration undercooling on the dendrite front diameter, (

**c**) the effect of concentration undercooling on the velocity of the dendrite front.

**Figure 7.**Multi-dendritic growth sequence in an solidifying Al + 5 wt.% Mg alloy: (

**a**) 0.02, (

**b**) 0.1, (

**c**) 0.2, (

**d**) 0.45, (

**e**) 0.7, (

**f**) 0.9 of solid fraction, (

**g**) actual microstructure of the Al + 5 wt.% Mg alloy cast into a metal mold (Section 4).

**Figure 8.**Solidification curves determined from the numerical CA model, equilibrium model and Scheil model.

**Figure 9.**DDTA test results: (

**a**) appearance of a metal mold with a thermocouple, (

**b**) solidification and cooling curves of the Al + 5 wt.% Mg alloy.

**Figure 10.**Comparison of the growth kinetics of the solid fraction calculated in the CA solidification model with DDTA tests.

**Figure 11.**Microstructure test results: (

**a**) die casting (

**b**) magnesium concentration profile in the selected area of the die casting, (

**c**) surface Mg distribution calculated from the CA model (f

_{S}= 0.99) (

**d**) magnesium concentration profile determined from the CA model.

Characteristic Quantity | Marking | DDTA Tests | The Numerical Model | Simulation Error, % |
---|---|---|---|---|

The temperature of the onset of solidification, °C. | T_{N} | 627 | 629 | 0.32 |

Temperature of maximum undercooling of the alloy, °C. | T_{1} | 624 | 625 | 0.16 |

Temperature range of recalescence, °C. | δ_{T} = T_{3} – T_{1} | 1 | 1 | 0.00 |

Maximum thermal effect of solid phase growth, °C. | T_{2} | 625 | 626 | 0.16 |

The end of solidification temperature, °C. | T_{K} | 538 | 532 | 1.12 |

Solidification time, s | t_{K} – t_{P} | 31.5 | 34 | 7.94 |

The total time of solidification and cooling to the temperature of 500 °C, s | t_{500}–t_{0} | 40 | 38.5 | 3.76 |

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Zyska, A.
CA Modeling of Microsegregation and Growth of Equiaxed Dendrites in the Binary Al-Mg Alloy. *Materials* **2021**, *14*, 3393.
https://doi.org/10.3390/ma14123393

**AMA Style**

Zyska A.
CA Modeling of Microsegregation and Growth of Equiaxed Dendrites in the Binary Al-Mg Alloy. *Materials*. 2021; 14(12):3393.
https://doi.org/10.3390/ma14123393

**Chicago/Turabian Style**

Zyska, Andrzej.
2021. "CA Modeling of Microsegregation and Growth of Equiaxed Dendrites in the Binary Al-Mg Alloy" *Materials* 14, no. 12: 3393.
https://doi.org/10.3390/ma14123393