The motion and growth behaviors of nucleuses in Al melt solidified under supergravity condition—molecular dynamics simulation

That supergravity can refine grains is verified in many materials. However, the underlying mechanism is still an open question. Although some convincing theories have been proposed, including the ‘crystal rain’ theory and the dendrite fragmentation theory, there is a lack of solid evidence, especially from the atomic scale. Based on the presetting nucleuses method, this study investigates the motion and growth behaviors of nucleuses during the solidification process of Al melt under supergravity condition with molecular dynamics simulation. It is found that supergravity builds a gradient pressure in the samples along the direction of supergravity, and the gradient pressure results in the gradient distribution of sample density. The preset nucleuses move directionally along the direction of supergravity forming ‘crystal rain’, while their directional moving velocity decreases due to the increase of buoyancy, which is caused by the increase of melt density in the motion path of the nucleuses. The supergravity-induced pressure not only decreases the critical size of nucleuses but also increases the growth velocity of nucleuses. The research results also indicate that larger nucleuses grow much faster than smaller ones at the same pressure. Owing to the gradient distribution of pressure, the nucleuses grow much faster along the direction of supergravity than other directions and evolve into an ‘inverted cone’ shape. Therefore, these findings show that supergravity can change the nucleation, motion and growth of nucleuses by establishing a gradient pressure in the melt, thus affecting the microstructure of the casting. Our results provide solid support for the ‘crystal rain’ theory and the nucleation rate rising theory from atomic scale.


Introduction
Microstructure control is an old and important subject in materials science. In order to control the microstructure of castings, a series of practical methods have been developed, such as directional solidification method, application of nucleating agents, electromagnetic stirring technology and so on. Large columnar crystals can be obtained by directional solidification [1]. With the help of nucleating agents, fine-grained materials can be produced [2]. Adopting electromagnetic stirring technology during solidification can not only significantly reduce defects, but also improve the component segregation [3].
As an important physical field, supergravity field can greatly promote the mass transport of melts. Therefore, researchers have been trying to use it to regulate the microstructure of materials for a long time. In 1993, Yeh et al studied the microstructure of a 7075 alloy solidified under supergravity condition, and found that the grains are refined compared with those obtained under normal gravity [4]. Then researchers successively investigated the microstructure of Al-Cu and Cu-Sn alloys solidified under different supergravity condition, the supergravityinduced grain refinement phenomenon was also observed. In addition, they found that grains become finer and finer along the direction of supergravity [5,6]. Zhao  Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. size of grains and supergravity, the result indicated that a second order exponential decay equation can well describe the decrease of the average grain size with gravity coefficient [7].
Although the supergravity-induced grain refinement phenomenon has been confirmed in different materials [8][9][10][11], the underlying mechanism is still an open question. To explain the observed phenomena, several theories have been proposed, including the nucleation rate rise theory [12], dendrite fragmentation theory [4,7] and 'crystal rain' theory [5]. After thermodynamic analysis, Chen et al thought that supergravity refines grains by raising the nucleation rate [12]. Yeh [4] and Zhao [7] et al proposed that supergravity can excite turbulence in melt, and turbulence can increase nucleuses by making nucleuses fall off the container wall and collide into many fragments. According to the principle of mechanics, under the condition of supergravity, the nucleuses with higher density than the parent melt will move directionally in its parent melt, forming 'crystal rain'. Thus Yang et al speculated that the nucleus densification caused by 'crystal rain' results in the grain refinement phenomenon [5].
Although all the theories sound reasonable, they lack solid evidence because the current experimental methods are difficult to detect the microstructure evolution of materials online. Some simulation methods, such as molecular dynamics simulation and phase-field method, can overcome this difficulty, thus they are suitable for exploring the grain refinement mechanism and finding solid evidence for the models mentioned above. Recently, we have simulated the microstructure evolution of Al melt solidified under different supergravity conditions with molecular dynamics method [13]. In our simulation, we clearly observed the similar phenomenon reported in experiment, that is, the grains are refined under supergravity condition and become finer and finer along the direction of supergravity. Moreover, the results clearly indicate that supergravity can increase the nucleation rate, which agrees well with Chen's model [12]. Using the phase-field method and phasefield crystal approach, Hu [14] and Zhang [15] et al studied the solidification under supergravity. Their results also reproduce the grain refinement phenomenon, and show that increasing the nucleation rate is a way of grain refinement by supergravity. Although these studies discussed the grain refinement mechanism, they did not involve the 'crystal rain' (motion of nucleuses) and growth of nucleuses. Investigating whether 'crystal rain' forms and how the grain growth velocity varies under supergravity condition is very important for constructing a panorama of the grain refinement mechanism under supergravity condition. However, as Hu said, it is difficult to investigate the motion of nucleuses in the framework of the current phase-field crystal model [14].
Molecular dynamics simulation is suitable for tracing the motion of nucleuses, while the 'crystal rain' was not observed in our recent work [13]. We guess that there are two factors that may directly lead to the difficulty of observing the 'crystal rain' in our simulation. The first factor is that the direction of crystallization is opposite to that of supergravity, the second one is the high nucleus density. These two factors are caused by the large supergravity adopted for reproducing the experimental phenomena within limited simulation time, and make nucleuses difficult to move, so it is difficult to observe the 'crystal rain'. In this work, we explore whether the 'crystal rain' can form in Al melt solidified under supergravity condition by studying the motion of the preset nucleuses embedded in the top side of the Al melt, this method can circumvent the two obstacles mentioned above. At the same time, the growth behavior of the preset nucleuses is discussed in detail.

Simulation details
To study the motion and growth of nucleuses, five Al melt samples containing a preset nucleus are constructed at 750 K. First, a FCC Al sample with the size of 200a 0 × 70a 0 × 70a 0 (a 0 is the lattice constant) is melted at 1600 K within 0.2 ns, and immediately relaxed at 750 K for another 0.2 ns. In this process, the NPT (constant pressure constant temperature) ensemble and three-dimensional boundary condition are used. Then, five spherical Al nucleuses with different sizes (R = 15, 20, 25, 30, 35 Å) cut from a perfect Al crystal are embedded in the top side of the Al melt respectively, producing five samples. Figure 1(a) shows the initial profile of the Al melt sample embedded a R = 15 Å spherical nucleus. Before applying supergravity, under the three-dimensional periodic boundary condition, the samples are relaxed 0.2 ns at 750 K with the NPT ensemble. Subsequently, the samples are simulated with the NVT ensemble (constant volume constant temperature). In the NVT ensemble stage, change the boundary condition of the x-axis to the non-periodic shrink-wrapped boundary, fix a 10 Å thick atom layer at the bottom of the sample, and impose supergravity on all of the atoms (except the fixed ones) along the negative direction of the x axis. The supergravity condition is generated with a method which imitates the means commonly adopted to generate supergravity condition in experiments, and this method has already been applied in our recent work [13]. Specifically, imagine the samples rotating around an axis which is perpendicular to the xy plane (shown in figure 1(b)). In this method, the 'centrifugal force' plays the role of supergravity, thus atoms are subjected to different supergravity if the distances between them and the rotation axis are different. As the supergravity is not a uniform value, the rotation angular velocity ω is used to label the intensity of supergravity. In this work, the rotation angular velocity ω is set to 2.0 × 10 10 rad s −1 . As the sizes of the samples along the y and z axes are much smaller than the distances from atoms to the rotation axis, the atoms with a same x-coordinate are treated with a same distance to the rotation axis.
During the whole simulation process, the preset nucleuses are regarded as rigid bodies, and the thermostat only acts on the atoms not in the preset nucleuses and fixed regions. In the all simulations, temperature and pressure are controlled by langevin thermostat and parrinello-rahman barostat respectively [16,17], and the timestep is set to 1 fs. The interatomic potential adopted in this work was developed by Sheng [18] and has been well tested in our recent work [13]. The software OVITO [19] is used to analyze microstructure. In order to analyze the motion characteristics of the preset nucleuses, the time-dependent dx, dy and dz (shown in figure 1 except the dz) are calculated when the samples are under supergravity condition. To study the pressure and density distribution along the direction of supergravity, the samples are sliced into many slabs (about 3.5 Å) along this direction, and then the atom number density and average value of the z component of atomic stress of the slabs are calculated. Figure 2 presents the solidification process of the sample containing a R = 15 Å spherical preset nucleus, the green region in the initial configuration (0.0 ns) is the preset nucleus. As observed in our recent work [13] and Hu's phase-field crystal simulation [14], crystallization advances gradually from bottom to top against the direction of supergravity, the average size of the grains roughly increases along the same direction in the final configuration (2.0 ns), and these phenomena were well explained in our recent work [13]. In short, a gradient pressure has been established along the direction of supergravity (shown in figure 3), and larger pressure in the lower part makes nucleation much easier and enhances the nucleation rate to a much higher value. Which is consistent well with the theoretical analysis from Chen [12] and phase-field crystal simulation results reported by Hu [14] and Zhang [15] et al It is worth noting that the pressure at each inner point gradually increases and converges to a steady value, this phenomenon implies that supergravity needs a period of time to play its effect. Figure 4 exhibits the distribution of atom number density along the direction of supergravity at different times. It clearly shows that the density presents a gradient distribution and gradually increases with time under the influence of supergravity, which agrees well with Hu's results [14], and is obviously caused by the distribution and variation characteristics of pressure. In addition, we also observed the growth of the preset nucleus and the contraction of the sample in the direction of supergravity. All the above mentioned phenomena were also observed in other samples (shown in figures S1-S4).

Results and discussions
To confirm whether the 'crystal rain' formed during the solidification process, the motion of nucleuses should be analyzed in detail. As the samples gradually contract along the direction of supergravity (the x axis), the time-dependent x-coordinate of nucleuses is not a good physical quantity to characterize the 'real motion' of the    nucleuses in this direction. Considering that the bottoms of the samples are fixed, the contraction of the samples is actually realized by the downward motion of the top side, thus the distance dx (shown in figure 1(b)) between the top boundary of the samples and the mass center of nucleuses can roughly describe the 'real motion' of the nucleuses along the direction of supergravity. For the sake of specification, dy and dz, are used to analyze the motion of the nucleus along the y and z axes respectively, where dy and dz are the distance between the mass center of nucleuses and the upper boundary in the y and z directions (shown in figure 1(b)). Figure 5 illustrates the time-dependent dx, dy and dz of the preset nucleuses in different samples, and there are some significant differences among these three quantities. dx presents similar variation characteristics for preset nucleuses with different sizes, that is, dx increases at a gradually decreasing rate and gradually approaches a saturation value, which implies that the nucleuses move directionally along the direction of supergravity at a reduced velocity. However, dy and dz oscillate in an irregular way. Obviously, the 'crystal rain' proposed by Yang formed along the direction of supergravity [5], while the nucleuses showed random Brownian motion in the other two directions. Through theoretical and experimental research, Song et al reported the directional motion of inclusion phases in molten Aluminum under supergravity condition [20]. Interestingly, Zimmermann et al directly observed the cascade-like settling of growing dendrites in Neopentylglycol-(D)Camphor alloy under high level of supergravity [21], which strongly supports our result. We believe that the increase of the density of Al melt resulted in the decrease of the directional moving velocity of the preset nucleuses. It is well known that the directional motion of a nucleus is driven by the resultant force of supergravity and buoyancy. According to Archimedes principle, if the density of Al melt increases, the buoyancy of nucleuses will increase. Therefore, the preset nucleuses move more and more slowly.
The derivative of the time-dependent dx is equal to the instantaneous velocity of the preset nucleus in the direction of supergravity. As dx increases near linearly in the first stage, the data points in that stage are linearly fitted to obtain the average moving velocity. Figure 6 shows the relation between the initial size of the preset nucleuses and its average moving velocity. It clearly shows that the nucleuses move at a very high velocity, and the larger nucleuses move faster than the smaller ones. The high moving velocity is due to the large rotation  angular velocity adopted in this simulation. The difference in moving velocity should be attributed to the different initial positions of the preset nucleuses. The larger nucleuses are placed at a lower position to maintain the same distance between the top boundaries of the different nucleuses and the top boundaries of the samples. Therefore, the larger nucleuses are subjected to larger initial supergravity.
In addition to motion, the growth of nucleuses under supergravity condition is also an interesting problem, which will have an important influence on the microstructure of the final casting. In order to clearly present the growth characteristics of the preset nucleus, some snapshots of the sample containing a R = 15 Å spherical preset nucleus are shown in figure 7 after deleting the atoms not in the crystalline structure and coloring the original preset nucleus with magenta. As shown in the snapshots, the preset nucleus starts growing after experiencing a period of relaxation time, which can be further verified from the time-dependent solidified atoms on the preset nucleus (shown in 8). In addition, the preset nucleus presents some extent of growth anisotropy, and the growth anisotropy makes the growing preset nucleus evolve into an 'inverted cone' shape. All these typical characteristics are also observed in other samples (shown in figure S5-S8) and will be explained in detail in the following.
The existence of relaxation time indicates that the sizes of the preset nucleuses are smaller than the critical sizes required in the paths of the preset nucleuses before growth, but the preset nucleuses are not melted because they are treated as rigid bodies during the simulations. In fact, the relaxation time is the time required for the preset nucleuses to reach the critical growth pressure. Three factors, including the initial size and moving velocity of the preset nucleuses and the pressure rising speed in the samples, jointly influence the length of the relaxation time. In terms of theoretical analysis, a smaller nucleus require higher critical growth pressure [22,23]. For all samples, the pressure rises up with a same speed from zero. Thus, the initial size and directional moving velocity determine when the preset nucleuses begin to grow. From the previous discussion, we know that smaller nucleuses require higher critical growth pressure and move more slowly towards the high-pressure region than the larger ones, thus the smaller nucleuses start to grow much later, which agrees well with the result obtained from the number variation of atoms solidified on the preset nucleus (shown in figure 8). After a period of relaxation, the preset nucleuses all have entered a rapid growth stage.
Because the pressure increases along the direction of supergravity, for simplicity, the critical growth pressure of the preset nucleus is defined as the pressure at the height of its centroid when the preset nucleus begins to grow. The calculated results are shown in figure 9, which clearly shows that the critical growth pressure decreases with the size of nucleus as revealed by thermodynamic analysis, and a good linear relation can be observed. From the perspective of nucleation, this result reflects that pressure can reduce the critical size of nucleuses, which is in line with the results obtained from the phase-field simulation [15]. Therefore, increasing pressure can not only increase the nucleation rate, but also reduce the critical size of nucleuses. This finding is in good agreement with the observed grain refinement phenomenon. Combining with the characteristics of pressure distribution, the gradient distribution of grain size can be reasonably explained.
Supergravity changes the pressure state of the samples, and exploring how supergravity influences the growth velocity of nucleuses is actually to clarify how pressure influences the growth velocity. However, it is difficult to study this problem in samples with uneven pressure, thus some simulations were carried out at 750 K under different even pressures to uncover the relation between pressure and growth velocity of different sizes of preset nucleuses. To obtain accurate results, the growth velocity is calculated after a relatively short relaxation time (0.1 ns) to eliminate the effect of size change. Figure 10 presents the simulation results, the growing velocity increases near linearly with pressure for a certain size nucleus, this pressure accelerated growth phenomenon was also observed in the Mg-Al-Sn system using the phase-field simulation method [24]. At first glance, this result seems to imply that larger grains will form in the lower part of the samples solidified under supergravity condition, which conflicts with the grain size distribution characteristics exhibited in figure 2. These facts suggest that the increase of growth velocity caused by pressure is completely covered by the increase of nucleation rate caused by pressure, that is to say, although the growth of nucleus is accelerated, there is no enough space for nucleuses to grow due to high nucleus density. This finding was also reported by Shang et al in the Mg-Al-Sn system [24]. In addition, figure 10 shows that larger nucleuses grow much faster than smaller ones at a same pressure, which also exists in the growth process of many naturally formed minerals [25].
Based on the discovery that pressure promotes the growth of nucleuses, it is easy to explain why the spherical preset nucleuses present a certain degree of growth anisotropy and gradually evolve into an 'inverted cone' shape under supergravity condition. As mentioned at the beginning, supergravity establishes a gradient pressure in the Al melt, thus the lower part of the nucleuses grows much faster than the upper part due to the higher pressure nearby. Because of this growth characteristic, the spherical preset nucleuses naturally grow unevenly and become an 'inverted cone' shape after a period of time.