Numerical simulation of the effect of hypergravity on the dendritic growth characteristics of aluminum alloys

Abstract

The cellular automata-lattice Boltzmann method is used to simulate the dendritic growth process of aluminum alloys under the action of hypergravity by performing coupling heat and mass transfer, solidification and flow. The dendrite arm spacing, growth rate, and dendrite morphology vary greatly with the size and direction of hypergravity, and solute segregation occurs. Compared with the gravity of the earth (1 g), hypergravity strongly strengthens the buoyancy-driven flow and considerably affects the morphology of the solidified grain. The dendritic growth rate is also accelerating. According to the direction of hypergravity in relation to the dendritic growth direction, there exist different flow states that show stable or unstable dendritic growth dynamics. For columnar crystal growth, when the hypergravity and growth direction are identical, the dendrite tip undergoes downward melt flow, and the dendrite grows in a stable manner. When the hypergravity and the growth direction are opposite, the dendrite tip undergoes upward melt flow, the dendrite grows in an unstable manner, and the primary dendrite spacing decreases. For the growth of equiaxed crystals, the convection induced by hypergravity causes the equiaxed crystals to be asymmetric, and the solute segregates in the direction of gravity. Channel segregation occurs in the mushy zone in the presence of equiaxed crystal chains.

1. Introduction

One of the most crucial steps in the creation of aluminum alloys is solidification. The unstable dendritic growth phenomenon at the liquid-solid interface, which is brought on by the disturbance of heat and solute, typically occurs during the solidification process of alloys [1,2]. The process of dendrite formation during the solidification of aluminum alloys is intricate and involves multiple processes such as heat diffusion, solute diffusion, melt flow, and migration of the liquid-solid interface. The shape, orientation, size determine the characteristics of castings [3]. As an illustration, the primary dendrite arm spacing (PDAS) may largely determine the ultimate tensile strength [4,5]. Therefore, the study of dendrite growth characteristics has important application value in order to prepare superior castings. Dendrite growth is generally known as a complex phase transformation process. In the previous several decades, an abundance of numerical research has been committed to studying dendrite growth [[6], [7], [8], [9], [10]]. The liquid phase convection caused by gravity plays a crucial role [[11], [12], [13]] in the change in dendrite morphology, which can lead to constitutional undercooling and solute distribution changes, thus causing a large number of microstructure changes during solidification, including interface instability and solute segregation.

In the process of strong flow, such as centrifugal casting, hypergravity strongly strengthens the flow driven by buoyancy forces. Centrifugal casting is a significant technique to generate high-quality castings and an efficient means to establish a stable hypergravity field on the earth. In this process, the effective gravity reaches dozens of earth gravity levels, and buoyancy-driven liquid convection is powerfully enhanced, which changes the heat and mass transfer processes in the melt and considerably affects the microstructure of the solidification structure [14].

Some numerical simulations and experimental research have been recently conducted on the solidification structure in the gravity field. Fernández [15] et al. found that hypergravity completely changes the liquid flow state in the solidification process and produces the macrosegregation phenomenon, and the degree of segregation increases with increasing hypergravity level. Yang [16] et al. experimentally found that the heavy solute Cu in the Al–Cu alloy can produce macrosegregation in the hypergravity field. Nelson T [11] et al. concluded experimentally that macrosegregation and remarkably developed big dendritic stems were observed during upward solidification, and finer dendrite growth was observed during downward solidification. Grugel RN [17] et al. concluded experimentally that the PDAS decreased with increasing hypergravity level when the gravity level was downward and the dendrites were growing upward.

However, these simulations mainly focus on macrosegregation, while only a few scholars have theoretically discussed the influence of hypergravity on dendrite growth at the mesoscopic scale. Steinbach [18] found that melt convection introduced a transport mode of symmetry destruction, which was determined by the growth direction relative to the gravity vector, and the stable state and unstable state could be distinguished. Viardin [19,20] et al. studied the effect of hypergravity on the microstructure of columnar crystals via phase field (PF). They found that the segregation of Al at the front of dendrite growth led to convection near the dendritic tip and produced the phenomenon of tip splitting.

The reviewed literature presented above shows that hypergravity significantly affects the structure of solidification. However, to date, only a few studies have been conducted on microstructures under hypergravity conditions. Solidification knowledge in the field of hypergravity is relatively poor. The detailed morphology of equiaxed crystal and columnar crystal growth under the condition of hypergravity still needs to be clarified.

This study's primary goal is to explore the impact of a hypergravity field on the dendrite growth characteristics of aluminum alloys at the mesoscopic scale, which can provide useful information for material process optimization and advanced alloy design. However, the solidification process is difficult to observe and detect because of the melt's high temperature and opacity. Numerical simulation provides an effective method to research the dendritic growth of the solidification process, which can reveal the complex interactions in the solidification process with exceptional precision.

In recent decades, the PF method has always been the preferred calculation method for simulating dendrite growth in numerical simulations [21]. However, given the high calculation cost, small calculation area, complex programming technology [22] and the need for advanced algorithms and huge computing resources [23], the cellular automata (CA) method was adapted to simulate the microstructure and the dendrite growth process with a lower calculation cost [24] and a larger calculation area. At the same time, this study combined the lattice Boltzmann method (LBM) to show the effect of convection caused by adding hypergravity on dendrite growth. Several research results [25,26] have shown that LBM benefits from excellent numerical steadiness and high computational effectiveness when studying the microstructural evolution of metals.

This work uses the coupled CA-LBM to simulate the solidification process of Al‒4.0%Cu alloy under the influence of buoyant convection in hypergravity at the mesoscopic scale. The method takes heat transfer, liquid flow, and solute diffusion into account, shows the detailed morphology of equiaxed crystal and columnar crystal growth in a hypergravity field, and lays a foundation for future theories.

2. Numerical method

The following presumptions were taken into account throughout the simulation.

• (1)The physical characteristics of the material were assumed to be constant;
• (2)The melt was considered an incompressible fluid, i.e., $\partial \rho /\partial t=0$;
• (3)The motion of dendrites was ignored [27];
• (4)The solid-liquid interface was in a state of regional balance, i.e., $k=C_s^{*}{/C}_l^{*}$;
• (5)The sum of the curvature, thermal, and constitutional undercooling at the dendritic tip equals the total undercooling [28,29].

2.1. CA method

In the present CA method, temperature, solute concentration, fraction of solid, and crystallographic orientation characterize each cell. The solidification degree of grid points is expressed by the fraction of solid. According to the solid fraction $f_s$ of grid points, the cell is divided into liquid (L), solid (S) and interface (S/L) [30]. The solid fraction $f_s$ increases until the cell is totally solidified. The process of increasing the solid fraction $f_s$ is the liquid-solid transformation process and the growth process of grains. The $f_s$ is determined by Eq. (1):

$$f_s\left(t+\delta t\right)=\frac{L\left(t+\delta t\right)}{\delta x\left(\left|\mathit{sin}\theta \right|+\mathit{cos}\theta \right)}\text{,}$$ (1)

where $\delta t$ is the time step; $\delta x$ is the space step; $\theta$ is the angle formed by the x-axis and the interface norm, which is calculated by Eq. (2)

$$\theta =\arccos \left(-\frac{{\partial }_x{f}_s}{\sqrt{{\left({\partial }_x{f}_s\right)}^2+{\left({\partial }_y{f}_s\right)}^2}}\right)\text{,}$$ (2)

The $L\left(t+\mathrm{\delta }t\right)$ in Eq. (1) is related to the decentered square algorithm(Fig. 1) used in the cell capture process. The solid fraction $f_s$ of this cell increases continuously when a cell in the calculation domain nucleates. A square with an inclined angle ${\theta }_0$ is arranged at the center of the nucleating cell according to the solid fraction of the cell. As shown in Fig. 1 (a), the diagonal length of the square increases with grain growth, and the semidiagonal length of the square is calculated by Eq. (3):

$$L\left(t+\delta t\right)=L\left(t\right)+{\overrightarrow{V}}_n\bullet \delta t\text{,}$$ (3)

Fig. 1.

Decentered square algorithm.(a) the increase in diagonal length of a square with grain growth; (b) new squares recreated in the cell.

where ${\overrightarrow{V}}_n$ is the solidification interface's normal growth rate [31], which is calculated by Eq. (4):

$${\overrightarrow{V}}_n={\overrightarrow{V}}_x{\overrightarrow{n}}_x+{\overrightarrow{V}}_y{\overrightarrow{n}}_y\text{,}$$ (4)

where ${\overrightarrow{n}}_x$ and ${\overrightarrow{n}}_y$ are the components of the interface normal phase $\overrightarrow{n}$, and ${\overrightarrow{V}}_x$ and ${\overrightarrow{V}}_y$ are the velocity components of the solid‒liquid interface in the x and y directions, which are determined by Eq. (5) and Eq. (6):

$${\overrightarrow{V}}_x=\frac{1}{\delta x\left(1-k\right)C_l^{*}\left(i,j\right)}\left\{D_s\left[\begin{array}{c}\left(C_s^{*}\left(i,j\right)-C_s\left(i-1,j\right)\right)f_s\left(i-1,j\right)\\ +\left(C_s^{*}\left(i,j\right)-C_s\left(i+1,j\right)\right)f_s\left(i+1,j\right)\end{array}\right]+D_l\left[\begin{array}{c}\left(C_l^{*}\left(i,j\right)-C_l\left(i-1,j\right)\right)\left({1-f}_s\left(i-1,j\right)\right)\\ +\left(C_l^{*}\left(i,j\right)-C_l\left(i+1,j\right)\right)\left({1-f}_s\left(i+1,j\right)\right)\end{array}\right]\right\}\text{,}$$ (5)

$${\overrightarrow{V}}_y=\frac{1}{\delta x\left(1-k\right)C_l^{*}\left(i,j\right)}\left\{D_s\left[\begin{array}{c}\left(C_s^{*}\left(i,j\right)-C_s\left(i,j-1\right)\right)f_s\left(i,j-1\right)\\ +\left(C_s^{*}\left(i,j\right)-C_s\left(i,j+1\right)\right)f_s\left(i,j+1\right)\end{array}\right]+D_l\left[\begin{array}{c}\left(C_l^{*}\left(i,j\right)-C_l\left(i,j-1\right)\right)\left({1-f}_s\left(i,j-1\right)\right)\\ +\left(C_l^{*}\left(i,j\right)-C_l\left(i,j+1\right)\right)\left({1-f}_s\left(i,j+1\right)\right)\end{array}\right]\right\}\text{,}$$ (6)

where $k$ is the partition coefficient; $D_s$ and $D_l$ are the solid and liquid diffusion coefficient; $C_s$ and $C_l$ are the actual solute concentrations; $C_s^{*}$ and $C_l^{*}$ are equilibrium solute concentrations, and $C_s^{*}=k{C}_l^{*}$. Furthermore, $C_l^{*}$ is expressed as Eq. (7):

$$C_l^{*}=C_0+\frac{1}{m_l}\left(T^{*}-T_l+\Gamma \kappa f\left(\theta ,{\theta }_0\right)\right)\text{,}$$ (7)

where $C_0$ is the initial solute concentration; $\Gamma$ is the Gibbs-Thomson coefficient; $T^{*}$ and $T_l$ are the interface temperature and equilibrium liquidus temperature; $ϵ$ is the anisotropy parameter; $m_l$ is the liquidus slope; and $f\left(\theta ,{\theta }_0\right)$ is the interface anisotropy function [32], which is expressed as Eq. (8), as follows:

$$f\left(\theta ,{\theta }_0\right)=1-15ϵ\cos \left[4\left(\theta -{\theta }_0\right)\right]\text{,}$$ (8)

where ${\theta }_0$ is the preferred growth angle.

The interface curvature $\kappa$ is calculated as Eq. (9), as follows:

$$\kappa =\left[{\left(\frac{\partial f_s}{\partial x}\right)}^2{{\left(\frac{\partial f_s}{\partial y}\right)}^2]}^{-\frac{3}{2}}[2\frac{\partial f_s}{\partial x}\frac{\partial f_s}{\partial y}\frac{{\partial }^2{f}_s}{\partial x\partial y}-{\left(\frac{\partial f_s}{\partial x}\right)}^2\frac{{\partial }^2{f}_s}{\partial y^2}-{\left(\frac{\partial f_s}{\partial y}\right)}^2\frac{{\partial }^2{f}_s}{\partial x^2}\right]\text{,}$$ (9)

When the surrounding liquid-phase cell is touched by the square's four vertices, the touched liquid-phase cell is marked as an interface cell. As shown in Fig. 1 (b), a new square is recreated in the cell, its size also changes with the solid phase fraction of the cell. As solidification proceeds, the original interface cells are continuously transformed into solid-phase cells, while the new interface cells continue to capture the surrounding liquid-phase cells according to the above rule until the end of solidification. While solidifying, solute and temperature redistribution occur, and solute and temperature are rejected by the liquid at the interface. The quantity of solute and energy rejected at the interface is decided according to Eqs. (10), (11).

$$\Delta C_l=C_l\left(1-k\right)\Delta f_s\text{,}$$ (10)

$$\Delta T{c}_p=\Delta f_{s}L\text{,}$$ (11)

where $L$ is the latent heat and $c_p$ is the specific heat of liquid phase.

2.2. LBM method

The flow, temperature and solute concentration field were all computed in this work using the LBM method. The flow field uses the LBM evolution formula [33] as Eq. (12):

$$f_{\alpha }\left(\overrightarrow{r}+{\overrightarrow{e}}_{\alpha }\Delta t,t+\Delta t\right)-f_{\alpha }\left(\overrightarrow{r},t\right)=-\frac{1}{{\tau }_f}\left[f_{\alpha }\right(\overrightarrow{r},t)-f_{\alpha }^{eq}(\overrightarrow{r},t\left)f_{\alpha }\right]+F_i(\overrightarrow{r},t)\text{,}$$ (12)

where $f_{\alpha }$ is the distribution function; ${\overrightarrow{e}}_{\alpha }$ is the discrete moving velocity, which is calculated by Eq. (13). Furthermore, ${\tau }_f$ is the relaxation time, which is calculated by Eq. (14).

$${\overrightarrow{e}}_{\alpha }=\left[\begin{array}{c}0\\ 0\end{array}\begin{array}{c}1\\ 0\end{array}\begin{array}{c}0\\ 1\end{array}\begin{array}{c}-1\\ 0\end{array}\begin{array}{c}0\\ -1\end{array}\begin{array}{c}1\\ 1\end{array}\begin{array}{c}-1\\ 1\end{array}\begin{array}{c}-1\\ -1\end{array}\begin{array}{c}1\\ -1\end{array}\right],$$ (13)

$${\tau }_f=3\frac{\vartheta }{c^2\Delta t}+0.5\text{,}$$ (14)

where $c$ is the lattice speed, $\vartheta$ is the kinematic viscosity. In Eq. (12), $f_{\alpha }^{eq}\left(\overrightarrow{r},t\right)$ is the equilibrium distribution function as in Eq. (15):

$$f_{\alpha }^{eq}\left(\overrightarrow{r},t\right)=w_{\alpha }\rho \left(\overrightarrow{r},t\right)\left(1+\frac{{\overrightarrow{e}}_{\alpha }\bullet \overrightarrow{u}}{{c_s}^2}+\frac{{\left({\overrightarrow{e}}_{\alpha }\bullet \overrightarrow{u}\right)}^2}{{{2c}_s}^4}-\frac{{{\displaystyle \overrightarrow{u}}}^2}{2c_s^2}\right)\text{,}$$ (15)

where $\rho \left(\overrightarrow{r},t\right)$ represents the lattice density; $c_s=\frac{c}{\sqrt{3}}$ is the sound speed of particles; $\overrightarrow{u}$ is the fluid velocity; $w_{\alpha }$ is a lattice constant; and $w_0=\frac{4}{9},w_{1-4}=\frac{1}{9},w_{5-8}=\frac{1}{36}\text{.}$ In Eq. (12), $F_i\left(\overrightarrow{r},t\right)$ is the source term experienced by fluid particles because of gravity. Discrete lattice effects can be avoided by utilizing a representation of the forcing term proposed by Guo [34] et al, for the external force F, which can be calculated by Eq. (16):

$$F_i\left(\overrightarrow{r},t\right)=\left(1-\frac{1}{{2\tau }_f}\right)w_{\alpha }\left(3\frac{{\overrightarrow{e}}_{\alpha }-\overrightarrow{u}}{c^2}+9\frac{{\overrightarrow{e}}_{\alpha }\mathbf{⬝}\overrightarrow{u}}{c^4}{\overrightarrow{e}}_{\alpha }\right)F⬝\Delta t\text{,}$$ (16)

where $F$ is the combined force of fluid particles in the hypergravity field given the existence of a temperature gradient and concentration gradient. $F$ is calculated as Eq. (17):

$$F=n\overrightarrow{g}{\rho }_0{\beta }_T\left(T-T_0\right)+n\overrightarrow{g}{\rho }_0{\beta }_C\left(C-C_0\right)\text{,}$$ (17)

where $\overrightarrow{g}$ is gravity's acceleration; $n$ represents the size of hypergravity; ${\beta }_T$ and ${\beta }_C$ are the temperature and concentration expansion coefficient; $T$ and C are the temperature and concentration of the fluid; ${\rho }_0$ is the liquid density; $T_0$ is the initial temperature.

The macrofluid density $\rho$ and macrovelocity $\overrightarrow{u}$ are calculated from the distribution functions as Eqs. (18), (19):

$$\rho =\sum _{\alpha =0}^8{f}_{\alpha }\text{,}$$ (18)

$$\overrightarrow{u}=\frac{\left(\sum _{\alpha =0}^8{f}_{\alpha }{\overrightarrow{e}}_{\alpha }+F\bullet \frac{\Delta t}{2}\right)}{\rho }\text{,}$$ (19)

Similar to the LBM evolution formula for the flow field, the evolution formula for the temperature field and the solute concentration field can be expressed as follows:

$$h_{\alpha }\left(\overrightarrow{r}+{\overrightarrow{e}}_{\alpha }\Delta t,t+\Delta t\right)-h_{\alpha }\left(\overrightarrow{r},t\right)=-\frac{1}{{\tau }_h}\left[f_{\alpha }\left(\overrightarrow{r},t\right)-h_{\alpha }^{eq}\left(\overrightarrow{r},t\right)\right]+H_i\left(\overrightarrow{r},t\right)\text{,}$$ (20)

$$g_{\alpha }\left(\overrightarrow{r}+{\overrightarrow{e}}_{\alpha }\Delta t,t+\Delta t\right)-g_{\alpha }\left(\overrightarrow{r},t\right)=-\frac{1}{{\tau }_g}\left[g_{\alpha }\left(\overrightarrow{r},t\right)-g_{\alpha }^{eq}\left(\overrightarrow{r},t\right)\right]{+G}_i\left(\overrightarrow{r},t\right)\text{,}$$ (21)

where $H_i\left(\overrightarrow{r},t\right)$ and $G_i\left(\overrightarrow{r},t\right)$ are the source terms, which are expressed as Eq. (22) and Eq. (23):

$$H_i\left(\overrightarrow{r},t\right)=\frac{w_{\alpha }\Delta f_{s}L}{c_p}\text{,}$$ (22)

$${G_i\left(\overrightarrow{r},t\right)=w}_{\alpha }{C}_l\left(1-k\right)\Delta f_s\text{,}$$ (23)

In Eqs. (20), (21), the equilibrium distribution functions for temperature $h_{\alpha }^{eq}\left(\overrightarrow{r},t\right)$ and solute concentration $g_{\alpha }^{eq}\left(\overrightarrow{r},t\right)$ are given by Eqs. (24), (25), as follows:

$$h_{\alpha }^{eq}\left(\overrightarrow{r},t\right)=w_{\alpha }T\left(\overrightarrow{r},t\right)\left(1+\frac{{\overrightarrow{e}}_{\alpha }\bullet \overrightarrow{u}}{{c_s}^2}+\frac{{\left({\overrightarrow{e}}_{\alpha }\bullet \overrightarrow{u}\right)}^2}{{{2c}_s}^4}-\frac{{{\displaystyle \overrightarrow{u}}}^2}{2c_s^2}\right)\text{,}$$ (24)

$$g_{\alpha }^{eq}\left(\overrightarrow{r},t\right)=w_{\alpha }C\left(\overrightarrow{r},t\right)\left(1+\frac{{\overrightarrow{e}}_{\alpha }\bullet \overrightarrow{u}}{{c_s}^2}+\frac{{\left({\overrightarrow{e}}_{\alpha }\bullet \overrightarrow{u}\right)}^2}{{{2c}_s}^4}-\frac{{{\displaystyle \overrightarrow{u}}}^2}{2c_s^2}\right)\text{,}$$ (25)

The relaxation time of the temperature field ${\mathrm{\tau }}_{\mathrm{h}}$ and the solute concentration field ${\mathrm{\tau }}_{\mathrm{g}}$ are given by Eqs. (26), (27), as follows:

$${\tau }_h=3\frac{\alpha }{c^2\Delta t}+0.5\text{,}$$ (26)

$${\tau }_g=3\frac{D}{c^2\Delta t}+0.5\text{,}$$ (27)

The corresponding macrotemperature $T$ and macroconcentration $C$ can be calculated as Eq. (28) and Eq. (29):

$$T=\sum _{\alpha =0}^8{h}_{\alpha }\text{,}$$ (28)

$$C=\sum _{\alpha =0}^8{g}_{\alpha }\text{,}$$ (29)

2.3. Computational procedure

The actual physical unit is used in the CA method, and the dimensionless lattice unit is used in the LBM method. The following Eq. (30) is used for the conversion of physical units and lattice units:

$${\left(\vartheta \frac{\Delta t}{\Delta x^2}\right)}_{LB}={\left(\vartheta \frac{\delta t}{\delta x^2}\right)}_{CA}\text{,}$$ (30)

In the LBM method, $\Delta t=\Delta x=1$.

The dimensionless thermal diffusion coefficient ${\alpha }_{LB}$ and solute diffusion coefficient $D_{LB}$ can be obtained from $Pr$ and $Sc$ as Eqs. (31), (32):

$${\alpha }_{LB}=\frac{{\vartheta }_{LB}}{Pr}\text{,}$$ (31)

$$D_{LB}=\frac{{\vartheta }_{LB}}{Sc}\text{,}$$ (32)

In the initial state, place one or more crystal nucleates in the calculation area as the starting point of crystal growth, and determine the preferred growth angle of the crystal nucleate, which is filled with a certain degree of undercooling $\Delta \mathrm{T}$ and a certain concentration of ${\mathrm{C}}_0$ aluminum alloy liquid. Fig. 2 shows the computational flowchart of the CA-LBM method. The boundary conditions of this study were determined using a second-order precision non-equilibrium extrapolation approach. The computational domain and boundary condition design used in this work are displayed in Fig. 3. During solidification, the equiaxed dendritic grains nucleated in the whole zone, while the columnar grains initiated from the bottom and evolved upward.

Fig. 2.

Computational flowchart of the CA-LBM method.

Fig. 3.

Design of the computational domain and boundary conditions.

3. Model validation

To confirm the correctness of the established CA-LBM coupling method, assuming that the static melt is unaffected by the fluid flow, that is, the pure diffusion state, the growth behavior of free dendrites under a given melt undercooling is quantitatively described with the Lipton-Glicksman-Kurz (LGK) analytical method [35]. For the 2D LGK method, the total undercooling $\Delta \mathrm{T}$ at the dendritic tip is provided by Eq. (33):

$$∆T=\frac{∆H}{c_p}Iv\left(P_t\right)+\left|m\right|C_0\left\{1-\frac{1}{1-(1-k)Iv\left(P_c\right)}\right\}+\frac{2\Gamma }{R},$$ (33)

$$R=\frac{\Gamma /{\sigma }^{*}}{\frac{P_{t}L}{C_p}-\frac{2P_c{m}_l{c}_0\left(1-k_0\right)}{1-\left(1-k_0\right)I_v\left(P_c\right)}}\text{,}$$ (34)

where $\Delta H$ is the latent heat; $P_c$ and $P_t$ are the solutal and thermal Peclet number; $P_c=\frac{RV}{2\mathrm{D}}$, $P_t=\frac{\text{RV}}{2\alpha }$, $Iv\left(P\right)$ is an Ivantsov function; $R$ in Eq. (34) is the tip radius; and $\mathrm{V}$ is the tip velocity. From the above mentioned two formulas, the LGK theoretical analytical solution can be obtained.

The simulation takes the Al-4.0% Cu alloy as the study item. The physical property parameters used are displayed in Table 1. The calculating zone is divided into 500 × 500, grid size $\Delta x$ = 2.5 μm. When $t=0$, selecting the preferred growth angle ${\theta }_0=45^0$, the nucleation point with the initial undercooling $\Delta T$ is 10 K in the calculation area's center, and the change rule of the steady-state tip growth rate with the undercooling is simulated.

Table 1.

Physical property parameters of Al-4.0%Cu alloy.

Physical Parameter	Symbol	Value	Unit
Kinematic viscosity	$\vartheta$	1.4 × ${10}^{-3}$	Pa $\bullet \mathrm{s}$
Thermal diffusivity	$\alpha$	4.313 × ${10}^{-5}$	${\mathrm{m}}^2/s$
Solid diffusivity	$D_s$	3 × ${10}^{-13}$	${\mathrm{m}}^2/s$
Liquid diffusivity	$D_l$	3 × ${10}^{-9}$	${\mathrm{m}}^2/s$
Liquidus slope	$m_l$	−2.6	$\mathrm{m}·\mathrm{K}/\%$
Liquidus temperature	$T_l$	648.0	$\mathrm{K}$
Gibbs–Thomson constant	$\Gamma$	2.4 $\times {10}^{-7}$	m $\bullet$ K
Partition coefficient	$k$	0.17	–
Specific heat of liquid phase	$c_p$	1070.0	J/(kg $\bullet$ K)
Interfacial energy anisotropy	$ϵ$	0.0267	–
Latent heat	$L$	397.0	KJ/kg

Fig. 4 (a) to (c) show the dendrite growth morphology under pure diffusion conditions when the undercooling is 10 K. The findings indicate that the dendrite morphology, solute concentration field and temperature field are almost completely symmetrical, which is in line with the literature's findings [[36], [37], [38]]. The findings of the simulation and the LGK analytical method are quantitatively compared, which is shown in Fig. 4(d). The comparison demonstrates that the simulated values are in excellent agreement with the analytical values, demonstrating that the CA-LBM coupled method can be successfully utilized for the analysis of microstructural evolution in the solidification process.

Fig. 4.

CA‒LBM method simulating dendrite growth morphology under pure diffusion conditions at $\Delta T$ = 10 K: (a) dendrite morphology; (b) solute concentration field; (c) temperature field; (d) contrasting the analytical and numerical results.

4. Simulation results and discussion

In this research, only the size and direction of the gravity vector are changed, as the solidification parameters and other numerical parameters remain unchanged.

4.1. Growth of a single columnar crystal in a hypergravity field

In this section, the morphological changes in the Al-4.0%Cu alloy during the growth of a single columnar crystal in the range of $\pm 100\mathrm{g}$ hypergravity are simulated, and the simulation is performed in a rectangular area of 500 × 750 $\mathrm{\mu m}$. At t = 0, place a nucleation point with a preferred growth angle ${\theta }_0=0^0$ at the bottom center of the rectangular area. Fig. 5 shows the growth morphology and solute distribution at different times (t = 0.025 s, t = 0.25 s, t = 0.57 s, and t = 0.76 s) at different gravity levels.

Fig. 5.

Growth process of a single columnar crystal under different gravity levels..(In positive gravity, the gravity vector's direction is identical to the primary dendritic growth direction, while in negative gravity, the gravity vector's direction is opposite to the primary dendritic growth direction.)

The primary dendritic phase is blue, and the liquid rich in copper is green or red. The direction and value of the gravity vector are marked on the top of each column, and the time is marked on the left of each row. When solidification begins, the solute-rich element Cu is discharged from the solid into the liquid and diffuses into the whole melt. When gravity is positive, the gravity vector's direction is consistent with the primary dendritic growth direction, and when gravity is negative, the gravity vector's direction is opposite to the primary dendritic growth direction. The simulation results reveal that the evolution of dendrite morphology is related to the size and direction of the gravity vector. This study defines positive gravity along the positive direction of the y-axis and negative gravity along the negative direction of the y-axis. Under different sizes and directions of the gravity vector, the dendrite morphology is different.

The microstructure maintains a stable dendrite state [18] at lower negative ($-1\mathrm{g})$ and positive gravity ($0\mathrm{g}\sim +100\mathrm{g}$) levels, which is mainly represented by a primary dendrite ‘A’ growing at the bottom center nucleation position. With increasing gravity level, the length of the primary dendrite ‘A’ becomes increasingly longer, while the length of secondary and other dendrites becomes increasingly shorter. At $+50\mathrm{g}$, the growth of secondary and other dendrites at the bottom of the computational domain is completely inhibited. At the level of large negative gravity (-$10\mathrm{g}\sim -100\mathrm{g}$), the microstructure maintains an unstable dendrite state [18]. With increasing negative gravity level, the length of primary dendrite ‘A’ becomes increasingly shorter, while the length of secondary and other dendrites becomes longer and more obvious, and the number of branches increases; that is, the PDAS decreases. A similar conclusion was also drawn by the PF method in the literature [39].

The evolution of these microstructures is closely related to the melt flow pattern caused by hypergravity. In Al–Cu alloys, the solute (Cu) is heavier than the solvent (Al). In the presence of gravity convection, the heavy solute Cu can sink/float, thus forming a convective plume [40]. Fig. 6 presents a detailed diagram of the liquid phase flow when the streamline is superimposed on the concentration and temperature field when the gravity level is +50 g. For every level of positive gravity, the flow direction caused by the density difference is in a downward direction, and the Cu-rich liquid is transported from the tip of the primary dendrite ‘A’ to the root, resulting in solute accumulation near the root (Fig. 6(a)). At the same time, the downward flow releases the latent heat downstream, resulting in the temperature increasing at the root, as shown in Fig. 6(b). This phenomenon leads to an increase in the concentration and temperature gradient at the primary dendrite tip ‘A’, thus encouraging the growth of primary dendrite ‘A’ and preventing the growth of other dendrites near primary dendrite ‘A’. The larger the concentration gradient and temperature gradient are, the larger the growth rate of the dendrite tip [41].

Fig. 6.

Detailed diagram of the liquid phase flow in the mushy zone (+50 g): (a) superposition of streamline and concentration field; (b) superposition of streamline and temperature field.

When the gravity level is the earth's gravity (±1 g; Fig. 5), the primary dendrite ‘A’ still dominates; however, at this time, the flow is very weak. Furthermore, the density difference caused by the concentration and temperature gradient is extremely small. Compared with pure diffusion, the length of branch ‘A’ slightly changed, but the number of branches next to branch ‘A’ changed.

For all the higher negative gravity levels, taking −50 g as an example, as shown in Fig. 7, the flow direction generated is upward close to the primary dendrite tip. The upward flow can transport Cu-rich liquid from the interdendritic region to the dendrite tip, and also transport Cu-rich liquid from the tip to the whole melt volume. This flow pattern causes the dendrite growth to be highly unstable, leading to the solute concentration and temperature enrichment at the central dendrite tip ‘A’, as demonstrated in Fig. 7(a) and (b). Thus, the growth of central dendrite tip ‘A’ is inhibited. Given the high gravity, the concentration gradient and temperature gradient in the melt are enhanced, resulting in the density difference in the melt, forming a vortex distributed symmetrically around the centerline. Under the action of the vortex, the solute is brought to the center, and the constitutional undercooling is higher at the side than in the center. The dendrites at the side grow faster than those at the center, and the primary dendrite spacing decreases because of the growth of new dendrites. With increasing time, the length of the side branches is gradually longer than that of the central dendrite, thus forming a multidendritic structure. The growth of the microstructure and its relationship with flow in the hypergravity environment are accurately reproduced by this mesoscopic method.

Fig. 7.

Detailed diagram of the liquid phase flow in the mushy zone (−50 g): (a) superposition of streamline and concentration field; (b) superposition of streamline and temperature field.

Fig. 8 shows the change in central dendrite length with gravity level in the range of −100 g to +100 g. The central dendrite length increases with increasing gravity level. Compared with pure diffusion when the gravity level is 0, the growth of the central dendrite arm is encouraged when the gravity level is greater than 0 g, and the growth of the central dendrite arm is prevented when the gravity level is less than 0 g. This agrees with the analysis's earlier findings.

Fig. 8.

Relationship between central dendrite length and gravity (t = 0.76 s).

The change in the maximum dendritic tip growth rate with the gravity level is shown in Fig. 9. Within the gravity range of stable dendritic growth (−1 to 100 g), the maximum dendritic tip growth rate increases with increasing gravity level, showing a minimum value that appears near 0 g. In the gravity range of unstable dendritic growth (−100 to −10 g), the maximum dendritic tip growth rate increases with increasing negative gravity level, and the increased amplitude is smaller than that of the positive gravity level.

Fig. 9.

Relationship between the maximum dendrite tip growth rate and gravity.

4.2. Growth of multiple columnar crystals in a hypergravity field

In this part, the growth of multiple columnar grains in the Al-4.0%Cu alloy under different gravity levels was simulated. According to the difference between the gravity direction and the growth direction, stable or unstable dendrite growth is formed [42]. Fig. 10(a) shows the distribution of solutes and the shape of the dendrites during the growth of multiple columnar crystals at different gravity levels at different times (t = 0.025 s, t = 0.25 s, t = 0.57 s, and t = 0.76 s). Fig. 10(b) shows the magnified view of part A in Fig. 10(a), while Fig. 10(c) shows the magnified view of part B in Fig. 10(a). The morphology of dendrites is completely different at different gravity levels.

Fig. 10.

(a) Growth process of multiple columnar crystals under different gravity levels; (b) enlarged view of part A; (c) enlarged view of part B.

Under the earth's gravity conditions, the dendrites grow at the same growth rate and are evenly distributed. In the negative hypergravity field, the mushy zone's macroscopic density gradient is opposite to the gravity direction, and the fluid dynamics are unstable. With an increase in the gravity coefficient, the primary dendrites become increasingly dense, resulting in a sharp decline in the PDAS. Fig. 11 shows the connection between the number of primary dendrite arms ${\mathrm{N}}_{\mathrm{p}}$ and the average PDAS in the negative hypergravity field with the change in gravity. The larger the negative hypergravity coefficient is, the greater the number of primary dendrite arms monotonously increases with the increase in gravity absolute value, and the smaller the PDAS is, which is consistent with the results of Steinbach's [18] research. The corresponding conclusions were also reached experimentally by Grugel RN [17] et al. When the negative supergravity is large, lateral tertiary branches appear, as shown in Fig. 10(b).

Fig. 11.

Connection between the number of primary dendrite arms and the average PDAS in the negative hypergravity field with the change in gravity.

In the positive hypergravity field, the macroscopic density gradient in the mushy zone is the same as the gravity direction, the fluid dynamics are stable, and the primary dendrite spacing is unchanged. However, with the increase in gravity value, the dendrite gap increases, and the dendrite growth is faster. Fig. 12 shows the connection between the average growth rate and gravity. The average growth rate appears to be the minimum value near 0 g, and with the increase in the positive/negative hypergravity value, the growth rate gradually increases. The growth range in the positive hypergravity field is larger than that in the negative hypergravity field. When the positive hypergravity value is sufficiently large, given the serious solute enrichment in the interdendritic liquid (Fig. 10(c)), dendrite remelting becomes prominent in the middle of the columnar dendrite.

Fig. 12.

Connection between average growth rate and gravity.

Fig. 13 shows the superimposed diagram of the flow field and concentration field in the (a) negative hypergravity field (−50 g), (b) earth gravity field (+1 g), and (c) positive hypergravity field (+50 g). The flow pattern in the melt can be clearly seen. In the negative hypergravity field (Fig. 13(a)), the upward flow brings the solute out of the interdendritic region, and the solute concentration in the interdendritic region decreases, which reduces the solidification length and spacing of the mushy zone, thus reducing the PDAS. The heavy solute Cu escapes upward and destroys the stability of growth [18]. Moreover, the convection vortices gather around the dendrite tips. In the earth gravity field (Fig. 13(b)), the recirculating flow is weak, and convection is not apparent. In the positive hypergravity field (Fig. 13(c)), the heavy solute Cu sinks downward, and the solute gathers in the interdendritic area. The friction between the floating melt and the fixed dendrite decreases, and the dendrite gap increases. At the same time, given the high flow intensity, two large convective vortices can be clearly seen in the melt, which in turn act as the driving force of solute transport, increasing the growth rate of the dendrite tip.

Fig. 13.

Superposition of the flow field and concentration field (a) in the negative hypergravity field (−50 g); (b) in the earth's gravity field (+1 g); (c) in the positive hypergravity field (+50 g).

In summary, positive hypergravity increases the dendrite gap, macrosegregation and large dendrite trunks were observed, and negative hypergravity decreases the PDAS. The dendrite grows finer, which can obtain a fine microstructure with no obvious macrosegregation. The finer microstructure is advantageous for enhancing mechanical properties. This difference is due to the regulation of solute flow and convection by buoyancy caused by hypergravity. This finding is consistent with T. Nelson's [11] experimental results and the proposed theoretical method. Moreover, the effect of gravity on spacing is noticeably greater in the case of negative hypergravity than in the case of positive hypergravity.

4.3. Growth of a single equiaxed crystal in a hypergravity field

In this section, morphological changes in the single equiaxed crystal growth of the Al‒4.0% Cu alloy in the range of ±100 g were simulated. Place a preferred growth angle ${\theta }_0=0^0$ at the center of the simulation area at t = 0. As shown in Fig. 14, the morphology (Fig. 14(a)), solute concentration field (Fig. 14(b)) and temperature field (Fig. 14(c)) of a single equiaxed crystal under different gravity levels are shown. The results show that in the absence of gravity, the dendrite presents a fourfold symmetrical shape, the dendrite morphology, temperature field and concentration field are completely symmetrical along the four growth directions, and well-developed secondary dendrite arms can be observed on the four primary dendrite arms. Under the condition of the earth's gravity (g = ±1), the degree of convection in the melt is weak, and the dendrite is close to a fourfold symmetrical shape. Under the effects of hypergravity, the symmetry of dendrites disappeared.

Fig. 14.

Single equiaxed crystal at different gravity levels: (a) dendrite morphology; (b) solute concentration field; (c) temperature field.

The length of the dendrite arm was quantified and compared to further quantitatively illustrate the asymmetry of dendrites. Fig. 15 shows the proportion of the upper dendrite arm L₂ to the lower dendrite arm L₁ vs. gravity, which represents the degree of asymmetry. With increasing gravity, the asymmetry of dendrites became more obvious. The asymmetry of the dendrite arm is called “anisotropy induced by hypergravity convection” [43]. With increasing hypergravity, the anisotropy caused by hypergravity convection increases, which increases the density difference in the melt and thus the velocity difference, causing the dendrites to increase asymmetrically.

Fig. 15.

L₂/L₁ vs. gravity. (Inset illustrates the definition of arm length.)

The evolution of the microstructure under different gravity values is related to the melt flow pattern caused by hypergravity and buoyancy forces. The existence of convection can change the buildup of solutes in front of the interface, thus affecting the fourfold symmetry of dendrites. For all negative gravity levels, the solute and latent heat discharged from the periphery of the dendrite tip flow upward along both sides of the lower arm, and the solute and heat near the lower arm are washed to the upper arm, leading to the solute's enrichment and heat near the upper arm. Based on the phase diagram, solute enrichment reduces the liquidus temperature, thus reducing the local undercooling. It also lessens the solidification's pushing power, resulting in shorter branches of the upper arm, which are less developed and have smaller lateral branches.

Fig. 16 is a detailed diagram of the flow of the liquid phase, where the streamline is superimposed on the concentration field (Fig. 16(a)) and temperature field (Fig. 16(b)) when the gravity level is −50 g. The upward flow produces two symmetrically distributed vortices on the dendrite's left and right sides, and the vortices's rotation directions are opposite. Moreover, solute and heat accumulate in the upper dendrite. Fig. 16(a) produces a “T”-shaped solute distribution, which is the solute plume described by Shevchenko [40] et al. For all positive gravity levels, Fig. 17 is a detailed diagram of the flow, where streamlines are superimposed on the concentration (Fig. 17(a)) and temperature fields (Fig. 17(b)). The flow direction in the melt is downward, resulting in solute concentration and heat at the lower dendrite, resulting in a shorter dendrite arm at the lower dendrite, it is in line with the Al–Cu alloy's experimental findings [44,45].

Fig. 16.

Detailed diagram of the liquid phase flow in the mushy zone (−50 g): (a) superposition of streamline and concentration field; (b) superposition of streamline and temperature field.

Fig. 17.

Detailed diagram of the liquid phase flow in the mushy zone (+50 g): (a) superposition of streamline and concentration field; (b) superposition of streamline and temperature field.

4.4. Channel segregation in a hypergravity field

During the solidification of aluminum alloys, segregation of solutes is inevitable, this causes fissures and the second phase to occur [46]. In the as-cast microstructure of superalloys, equiaxed grains with arbitrary orientations easily form freckles [47,48]. Freckles are channel-like defects characterized by equiaxed grain chains [49]. Given the existence of hypergravity, the convection of hot melt [50] caused by the high density difference between the interdendritic melt and the whole liquid is amplified, which easily causes macrosegregation. The convection of solute enriched melt inter dendritics is the cause of channel segregation [51].

While solidifying, the solid-liquid interface becomes unstable due to disturbance, thus forming a mushy zone. The mushy zone is regarded as a permeable porous media composed of a dendritic network filled with liquid channels. The flow of the aluminum alloy liquid phase in the mushy zone can be regarded as the percolation behavior of porous media. The mushy zone is made up of a solid skeleton and numerous randomly distributed small pores. The permeability (K) of the mushy zone is based on the liquid phase's volume percentage ($g_1$) and the characteristic length ($l_c$) of the dendritic structure, which is given by Eq. (35):

$$K=\frac{l_c^2{g}_1^3}{180{\left(1-g_1\right)}^2}\text{,}$$ (35)

In a hypergravity field, the percolation rate of the liquid phase between dendrites is proportional to the gravity coefficient; that is, hypergravity greatly increases the percolation rate of the solute-rich liquid phase between dendrites, resulting in the generation of element segregation and a completely different dendritic morphology from that in a normal gravity field. Fig. 18(a) and (b) show the dendritic morphology of the earth's gravity field and the hypergravity field. Compared with the earth gravity field, channel segregation along the gravity direction occurs in the hypergravity field due to the existence of hypergravity. Channel segregation is a defect caused by local convection driven by density inversion due to the density difference between dendrites caused by component segregation in the mushy zone. Fig. 19(a) shows the velocity vector plots in the earth gravity field and in the hypergravity field; Fig. 19 (b) shows the streamlines in the earth gravity field and in the hypergravity field. The results of localized flow fields in the earth's gravity field and in the hypergravity field can be clearly seen. The hypergravity effect is mainly due to the intensification of natural convection caused by fluctuations in the solute concentration field and temperature field, etc., which results in the formation of a vortex-like localized flow field in the dendritic gaps. As a result, the morphology of the dendrites in Fig. 18 (b) undergoes segregation along the gaps in the channel.

Fig. 18.

(a) Earth gravity field; (b) hypergravity field; (c) enlarged view of the red framed part.

Fig. 19.

(a) Velocity vector diagrams in the earth gravity field and hypergravity field; (b) Streamline diagrams in the earth gravity field and hypergravity field.

The black circles in Fig. 18(a) and (b) illustrating parts A and A′, B and B′, C and C′, D and D’ represent the difference between the morphology of dendrites in the hypergravity field and in the earth gravity field. In the earth gravity field, the dendrites are close to the fourfold symmetrical shape, while in the hypergravity field, the dendrites grow faster in the direction of gravity, which is near the channel segregation diagram proposed by Yang [16] et al. In the direction opposite to the arrow direction in Fig. 18(b), solute segregation is caused by heat solute convection. A magnified section of the red box in Fig. 18(b) is shown in Fig. 18(c), and solute segregation can be clearly seen. The study of channel segregation can provide accurate predictions for solute segregation and liquid feeding flow in the process of hypergravity casting.

5. Conclusions

The CA‒LBM coupling technique was applied to simulate the microstructural evolution and solute segregation of the Al-4.0%Cu alloy during equiaxed and columnar solidification under a hypergravity field. The direction and degree of hypergravity led to changes in the PDAS and grain morphology. The purpose of this method was to forecast the microstructure of casting parts under different gravity circumstances.

During the solidification process of columnar crystals, positive hypergravity increased the dendrite gap. Furthermore, macrosegregation and dendrite trunks were observed. With increasing hypergravity, the growth rate of dendrites increased significantly. When the positive hypergravity was sufficiently large, dendrite remelting occurred. The negative hypergravity reduced the PDAS, and a refined microstructure was obtained. Moreover, macrosegregation was not obvious. Lateral tertiary branches were found under larger negative hypergravity, and the effect of gravity on the spacing was noticeably greater under negative hypergravity than under positive hypergravity. The difference could be attributed to hypergravity, which changed the flow in the melt, causing different degrees of flow around the dendrite. The different gravity directions took the form of extremely different flow states and led to either stable or unstable dendritic growth regimes.

In the process of equiaxed crystal solidification, hypergravity changed the distribution of solutes' symmetry, changing the morphology of dendrites. With increasing hypergravity, the asymmetry of dendrites increased. In addition, hypergravity greatly increased the flow rate of the solute-rich liquid phase between dendrites, resulting in channel segregation.

The present method can predict the effect of the hypergravity field on the growth characteristics of equiaxed and columnar crystals at the mesoscopic scale. The solute segregation and liquid phase feeding flow in the hypergravity casting process can also be accurately predicted.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Yanying Zhang: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision. Ruifeng Dou: Writing – original draft, Validation, Supervision, Resources, Methodology. Junsheng Wang: Methodology. Xunliang Liu: Investigation, Funding acquisition. Zhi Wen: Funding acquisition, Formal analysis.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.