Understanding Homogeneous Nucleation in Solidification of Aluminum by Molecular Dynamics Simulations

Homogeneous nucleation from aluminum (Al) melt was investigated by million-atom molecular dynamics (MD) simulations utilizing the second nearest neighbor modified embedded atom method (MEAM) potentials. The natural spontaneous homogenous nucleation from the Al melt was produced without any influence of pressure, free surface effects and impurities. Initially isothermal crystal nucleation from undercooled melt was studied at different constant temperatures, and later superheated Al melt was quenched with different cooling rates. The crystal structure of nuclei, critical nucleus size, critical temperature for homogenous nucleation, induction time, and nucleation rate were determined. The quenching simulations clearly revealed three temperature regimes: sub-critical nucleation, super-critical nucleation, and solid-state grain growth regimes. The main crystalline phase was identified as face-centered cubic (fcc), but a hexagonal close-packed (hcp) and an amorphous solid phase were also detected. The hcp phase was created due to the formation of stacking faults during solidification of Al melt. By slowing down the cooling rate, the volume fraction of hcp and amorphous phases decreased. After the box was completely solid, grain growth was simulated and the grain growth exponent was determined for different annealing temperatures.


Introduction
In metal manufacturing processes involving solidification (e.g., casting 1 , welding 2 , and laser additive manufacturing 3 ), the crystal nucleation from the melt controls the formation and growth of nano-and micro-structures of metals. The solidification structures of materials significantly influence their mechanical and physical properties. If large undercooling can be achieved before crystal nucleation occurs (as in rapid solidification), different and potentially useful forms of crystalline metals may be produced 4 . To predict and control the solidification nano-and micro-structures in different manufacturing processes, a fundamental understanding of mechanisms of crystal nucleation and solidification is necessary.
The crystallization process during liquid to solid transformation can be monitored by using X-ray scattering 5, 6 , dilatometry 7 , differential scanning calorimetry 8 , or microscopic methods [9][10][11] . But there are several factors that limit the experimental studies of the nucleation process during solidification or crystallization, especially in pure materials (homogenous nucleation) 4 . There are difficulties in quantifying the surface free energy of liquid-solid interfaces and their anisotropy 12 . Also experiments are typically performed at temperatures that differ by hundreds of degrees from the actual nucleation conditions 4 . As a result, experimental a 혯 Ѕ Ą 瑹怴 咊 µ cleation rates in crystallization from the melt cannot provide reliable tests of the classical nucleation theory (CNT) 13,14 . Another fundamental problem with homogenous nucleation experiments, especially for metallic materials, is that it is difficult to purify a liquid to exclude all the impurities that can catalyze nucleation 15,16 . Homogenous nucleation from metallic melts is a very complex phenomenon. It starts from the interior parts of an undercooled liquid, and due to the opaque nature of metallic melts, it is very difficult to experimentally detect the nuclei 12,17 . Therefore, alternative theoretical or computational methods can be used to study homogenous nucleation in pure metals. The problem of nucleation from melt has been studied utilizing different approaches, including theoretical studies based on CNT 13,14 , density function theory (DFT) calculations 18 , solid-liquid coexistence by molecular dynamics (MD) simulations 17 , and other simulation studies based on phase-field [14], front tracking 19 , cellular automata 20 , and Monte-Carlo (MC) 21 used to provided necessary input information for higher scale models like phase-field models [24][25][26] .
There are few works on homogeneous nucleation during liquid-solid transformation 27,28 and liquid-vapor transformation 29 by MD simulations. Yasuoka et al. 29 investigated the dynamics of vapor phase homogeneous nucleation in a water system; their predicted nucleation rate was three orders of magnitude smaller than that of the CNT. In metals, it is not straightforward to observe the homogeneous nucleation and solidification processes at the atomistic scale. Shibuta et al. 27 utilized MD simulations and linked the empirical interpretation in metallurgy with the atomistic behavior of nucleation and solidification in pure iron (Fe). These major drawbacks of these works are the use of Finnis-Sinclair (FS) potential 30 and use of isothermal process for all the simulations. Utilizing an isothermal process in MD simulations does not resemble the experimental solidification process. In experiments with slow or fast cooling (quenching), the temperature change will affect the crystal nucleation and solidification processes. The utilized FS potential predicts the melting point of Fe to be 2,400 K, which is much higher than the experimental melting point of Fe (~1,811 K), and consequently results in inaccurate prediction of solid-liquid co-existence properties.
To reliably study the crystal nucleation process from melt by MD simulations, the interatomic potentials used for MD simulations of solidification need to accurately predict the behavior of solid-liquid interfaces. In the early interatomic potentials, which were developed and used for MD simulations of Al such as Lennard-Jones (LJ) 31 and hard-sphere 32 36 and Lee et al. 43 .
To the best of our knowledge, there has been only one experimental study on homogenous crystal nucleation from pure Al melt based on the free boundary (also called the CNT method) and interacting boundary models 44 ; the incubation period (or the induction time) and small nuclei were undetectable in this study. There is only one work which used MD simulations 28 to study solidification of Al. However this study doesn't provide quantitative analysis on nucleation, critical nucleus formation, induction time, comparison of MD results to CNT, or details on solid state grain growth.
In this work, we studied the homogenous crystal nucleation from Al melt by MD simulations utilizing the second nearest-neighbor modified embedded atomic method (2NN MEAM) interatomic potential of Al 43 . Homogenous nucleation from Al melt was studied in both isothermal and quench processes. We also provide quantitative details of critical nucleus formation, and comparison of MD with CNT. The regimes of the crystallization process during quenching have been identified. In the last section we also provide detailed analysis of the solidstate grain growth mechanism of pure Al after solidification. The predicted melting point of Al using a 2NN MEAM MD simulation is 925 K 23 , which is in a very good agreement with the experimental value of 934 K. We found that at a temperature close to the melt temperature, the liquid has a fluctuating number of fcc atoms. We wanted to start the nucleation simulations with a pure liquid having no solid regions. In order to find the temperature at which a completely melted simulation box with no fcc crystal can be achieved in a relatively short simulation time (~150 ps), several simulations were performed by increasing the temperature of the simulation box higher than 925 K using 25 K intervals. After 16 intervals, when the temperature reached 1,325 K, we could obtain a completely melted simulation box in ~100 ps. The simulation is continued to 300 ps to make sure the initial melt is properly equilibrated. The CNA of the simulation box for very large time scale is provided in Fig. 1(a). The percentage of amorphous liquid atoms keeps increasing with increasing the annealing temperature. Finally, the box had no crystalline atoms at 1,325 K. The radial distribution function (RDF, g(r)) of the simulation box was calculated for all the temperatures,

Simulation Details
which is plotted in Fig. 1(b). There are no long-range peaks at 1,325 K. The CNA analysis and RDF plots confirmed that Al was completely melted at 1,325 K. For isothermal simulations, the Al melt was isothermally solidified at temperatures between the range of 300 K and 800 K with 50 K intervals. Maximum nucleation rate was observed to be between 400 K and 500 K, so we ran more simulations with 25 K intervals to more accurately determine the exact critical temperature of nucleation (when the nucleation rate is maximum). Each isothermal simulation was repeated five times to evaluate the possible errors.
Each isothermal simulation was run for a total of 500 ps (167,000 time steps) to simulate the crystal nucleation and solidification.
We also performed solidification by quenching with different cooling rates of 5.83x10 10 Ks -1 , 5.83x10 11 Ks -1 and 5.83x10 12 Ks -1 . Different cooling rates were applied by changing the number of total time steps. The initial temperature of the melt was 1,325 K, then the melt was cooled down to 450 K in 150 ps, 1,500 ps and 15,000 ps (5,000,000 time steps), which resulted in cooling rates of 5.83x10 12 Ks -1 , 5.83x10 11 Ks -1 and 5.83x10 10 Ks -1 , respectively. 450 K was chosen because it is lower than the critical temperature found in the isothermal process (see section 3.6). The quenching method was used to mimic the actual experimental procedure to produce undercooling where the temperature decreases from above the melting temperature with a certain cooling rate. This method of simulation is closer to what is performed experimentally and differs from the previous MD simulations of homogenous nucleation which usually utilized isothermal simulations 27 . In experiments, cooling rates in rapid solidification of bulk Al lie between 10 4 and 10 7 K/s [49][50][51] , notably much slower than the rates used in MD.

Crystal Structure of Nuclei
The primary observation of nucleation in MD simulations shows the formation of nuclei from the melt. CNA and visual inspection are used to study the structure of the nucleus throughout the quenching and annealing. The formation of crystal structures and stacking faults occurred in the same way for both the isothermal and quenching processes. Fig. 2(a) shows that the crystalline nuclei form in different parts of the melt; the unstructured melt is removed from the simulation box so the crystalline nuclei can be seen. The spherical nature of the nuclei is observed visually at first. The atomic coordinates of the cluster atoms in the specified nucleus in shows atoms with fcc (green) and hcp (red) crystal structures using CNA. The nearest neighbor distance for fcc Al should be 2.86 Å as the lattice constant is 4.05 Å. The distance between two nearest atoms within the fcc (green) atoms in Fig. 2 We calculated the difference between formation energies of fcc and hcp Al to be only 0.03 eV, whereas the difference between formation energies of fcc and bcc Al was determined to be 0.12 eV 23 . Since there is a random thermal fluctuation of energy during solidification, this thermal fluctuation of energy can cause formation of hcp stacking faults in the Al system, but it is not enough to promote formation of bcc atoms. By rotating the simulation box in Fig. 2 around the <111> direction, the stacking fault plane can be observed ( Fig. 2(c)), and the stacking fault is detected to have hcp crystal structure.

Critical Nucleus Size
The minimum size required for continuous growth of a crystalline nucleus is known as the critical nucleus size. In this study, the size of a nucleus is taken as the average, as discussed below, of the maximum length of the nucleus in x, y and z directions; this is assumed to be equivalent to the diameter of a spherical shaped nucleus. The nucleus size and number of atoms in the nucleus are determined by direct observations. Before a nucleus reaches its critical size, for a short period of time (nucleus origin time, discussed in Section 3.9), the nucleus gains and loses atoms. We assume that a crystalline nucleus reaches its critical size when it doesn't lose any atom back into the liquid. Fig. 3 shows examples of nuclei size and number of atoms in the nuclei versus simulation time for isothermal and quenching cases. The arrows in Fig. 3

(a) and (b)
show when the nuclei reach the critical size. The number of atoms in the critical nucleus is ~1400 atoms at 700K ( Fig. 3(a)) and for the case with a cooling rate of 5.83x10 11 Ks -1 in Fig.   3(b), the critical sized nucleus has ~1000 atoms. After reaching the critical size, the crystalline nuclei grow in size and gather more fcc and hcp atoms. The evolution of critical nucleus can also be monitored by potential energy change with time and the visualization snapshots (Inset Fig. 3ab). The crystalline nuclei reach the critical size slightly before the sudden change of slope. At that point the nucleus has become large enough to overcome the free energy barrier for phase separation. A critical nucleus does not become smaller after it reaches the critical size. The critical nuclei are found to be quiet stable against the mobility of liquid phase, structural change, i.e. fcc-hcp, or continuously changing shape. As it was mentioned before, each simulation was performed 5 times; after the nuclei reach the critical size, a set of 5 random critical nuclei are chosen at each annealing temperature from each isothermal simulation; total of 25 critical nuclei were selected for each annealing temperature. The average critical size and its standard deviation versus annealing temperature are plotted in Fig. 4(a). The average size of critical nuclei is between ~0.82 nm and ~4 nm ( Fig. 4

(a))
for all annealing temperatures in the isothermal process. The critical size of crystalline nuclei increases as the annealing temperature increases.
In the quench processes, the average critical size of nuclei is found to be ~1.8 nm for 5.83x10 12 Ks -1 , ~3.49 nm for 5.83x10 11 Ks -1 , and ~4.5 nm for 5.83x10 10 Ks -1 cooling rate. With a slower cooling rate the nucleation occurs at higher temperatures, which results in a larger critical size for nuclei. It should be noted that nucleation rates decrease with slower cooling rates (discussed in section 3.7), and the nuclei can grow larger before the whole simulation box becomes solid. During the isothermal process, the maximum number of crystalline nuclei in the system varies ( Fig. 5) with annealing temperature. When a relatively low annealing temperature is applied (below 600 K), the nucleation starts instantly, and since the driving force for solidification is very high, fcc crystalline atoms form all over the simulation box in the early stages of simulation. This will result in formation of multiple critical nuclei simultaneously, and a higher number of nuclei will form but they grow to a smaller size compared to the nuclei in higher annealing temperatures. The maximum number of separable nuclei in the simulation box for annealing temperatures between 350 K and 650 K is more than 40 nuclei in 25 nm 3 . Above 650 K the maximum number of separable nuclei is reduced; for example, at 700 K and 725 K, 12 and 9 nuclei are detected, respectively. While the number of nuclei is reduced by increasing the annealing temperature above 500 K, each nucleus can grow to a much larger size before the simulation box is completely solid. In the quenching process, the maximum number of separable nuclei varies between 9 to 15 for different cooling rates, which is similar to that of 700 K and 725 K isothermal cases. The maximum number of separable nuclei is seen at 715 K, 665 K and 655 K for 5.83x10 10 Ks -1 , 5.83x10 11 Ks -1 and 5.83x10 12 Ks -1 quench rates, respectively. As mentioned before, the number of nuclei in the system is reduced with increasing the annealing temperature above 500 K (Fig. 5); the same conclusion can be made by comparing Fig. 6(a) and Fig. 6(b). These figures also show that each nucleus can grow much bigger in size at a higher annealing temperature. Nucleation by quenching in Fig. 6(c) and Fig. 6(d) shows a very similar behavior to the nucleation of isothermal process at 700 K ( Fig. 6(b)). This indicates that in the quenching process nucleation starts at high temperatures with a small number of nuclei which can grow in size. Fig. 6. Snapshots of nuclei formation and growth during solidification for two isothermal processes at annealing temperatures of (a) 475 K and (b) 700 K, and for two quenching processes with cooling rate of (c) 5.83x10 11 Ks -1 and (d) 5.83x10 10 Ks -1 . Fcc are shown with green color; hcp and amorphous atoms are ignored for a better visibility of nuclei.
The instantaneous temperature for crystal nucleation during quenching can be determined by plotting percentage of crystalline atoms versus temperature change (Fig. 7). Fig. 7 shows that during quenching the nucleation process starts between ~747 K and ~712 K for the slower cooling rates of 5.83x10 10 Ks -1 (Fig. 7 (b)) and 5.83x10 11 Ks -1 (Fig. 7 (c)). For a high cooling rate of 5.83x10 12 Ks -1 ( Fig. 7 (a)) the nucleation starts below 700 K, and the exact temperature of formation of first nucleus is found to be ~586 K from dumps (per atom data) available from LAMMPS (such as Fig. 6). The number of fcc/hcp atoms is very low for 5.83x10 12 Ks -1 cooling rate that it doesn't reflect the nucleation starting temperature in Fig. 7(a). The most solid atoms for this cooling rate remain at amorphous configuration, and the change in overall crystal structure is not significant until about 575 K. The number of nuclei is very high, but they grow much less in size compare to the nuclei in slower cooling rates. Overall the quenching simulations suggest that the nucleation starting temperature is between 586 K -747 K.

Crystallization during nucleation
The percentage of atoms having different structures (amorphous, fcc or hcp) is plotted in

Temperature-dependent nucleation regimes
As it was shown in the previous section 3.3, nucleation is a temperature driven phenomenon, and a change in temperature affects the rate and behavior of nucleation. Potential energy is one of the fundamental quantities that correlates temperature with nucleation and solidification processes. Fig. 10(a) shows the potential energy versus simulation time for different isothermal annealing temperatures. When the Al melt is brought directly to a low annealing temperature, there is a very sharp drop in the initial potential energy due to the specific heat of the liquid.
Below 600 K, the Al melt starts solidifying immediately within the first few time steps.
For higher annealing temperatures (such as 650 K, 700 K and 725 K) the solidification doesn't happen immediately. The time required to form the first critical nucleus (or nuclei) after starting the annealing is ~40 ps, ~75 ps and ~ 250 ps for annealing temperatures of 650 K, 700 K and 725 K, respectively.
In isothermal cases, we can roughly see from   solid, and solid-state grain growth starts; this region can be easily identified for low cooling rates (e.g., < 710 K for 5.83x10 10 Ks -1 ). For the low cooling rates, the almost vertical slope line ( Fig.   10(c)) signifies the release of latent heat due to crystallization (or solidification). This event represents the fast and spontaneous formation of solid nuclei during solidification. Fig. 10(c) shows nuclei formation can only happen in a temperature range depending on the cooling rate.
This finding shows the drawbacks of isothermal simulations and clearly shows the existence of different temperature regions in the solidification process.
Initially atoms attempt to crystallize from the melt by formation of small clusters of fcc and hcp atoms, but as the simulation progresses most of these clusters of atoms dissolve back into the liquid phase. This region can be identified as the sub-critical (unstable) nucleation regime in Fig. 10(d) where there are small fluctuations in the total number of crystalline atoms.
The sudden change in the slope shows the regime change from sub-critical to super-critical (stable) nucleation, and the temperature at which this transition occurs is named sc T . The exact value of sc T depends on the cooling rate, and remains between ~715-725 K for 5.83x10 10 Ks -1 and 5.83x10 11 Ks -1 . Multiple super-critical nuclei are formed in the system following the formation of the first critical nucleus and they grow until the whole simulation box is solid and solid-state grain growth starts.
The second sudden change in the slope shows another regime change from super-critical nucleation to solid-state grain growth, and the temperature at which this transition occurs is named gg T . The difference between sc T and gg T , is very small (only 9 K for 5.83x10 10 Ks -1 cooling rate, sc T = 724 K start and gg T = 715 K end) for the slowest cooling rate. Super-critical nuclei will grow until the whole box is solid, and then solid-state grain growth occurs. This solid-state grain growth and end of nucleation are the same temperature ( gg T ). The grain growth is not a part of nucleation process, but an essential part of solidification. So overall the solidification process can be divided into three temperature based thermodynamics regimes, i) the sub-critical (unstable) regime, ii) the super-critical (stable) nucleation regime when, multiple critical nuclei form along with the growth of the previously formed critical nuclei, and iii) solid-state grain growth regime.
In any experimental method, the Al melt needs to release heat to go down to a specific annealing temperature. Even if the Al melt can be brought to a constant temperature environment instantly, it is practically impossible that the Al melt will go down to the lower annealing temperature immediately. In other words, the quench rate in isothermal processes are infinite. In quenching we showed, the sc T for 5.83x10 10 Ks -1 , 5.83x10 11 Ks -1 , 5.83x10 12 Ks -1 are 724 K, 715 K and 586 K respectively. So, we can see as the cooling rate decreases the sc T increases. In MD, the quench rates are very high. If we assume in bulk experiments the cooling rate to be 1-100 Ks -1 the nucleation temperature should be higher (above ~725 K). So, in the real world there is no nucleation at all at lower annealing temperature (i.e. 650 K for Al). We also observed the slope in the super-critical region (Fig. 10(d)) is getting more vertical as the cooling rate is reduced. So, in experiments when the cooling rate is much slower, the sc T and gg T can be almost the same temperature.

Nucleation rate versus annealing temperature: isothermal solidification
It is evident from the previous section that annealing temperature certainly affects the nucleation process, so it is expected that it would also affect the nucleation rate. The nucleation rate for each annealing temperature is calculated by fitting a line to the data on number of nuclei versus time, where the slope of the line is the nucleation rate (Fig. 11). The nucleation rate increases as the annealing temperature increases from 300 K to 475 K (Table 1). At room temperature (300 K) very few separable crystalline nuclei can be found; for higher annealing temperatures, the kinetic energy of atoms increases, which helps liquid atoms overcome the activation or free energy barrier to produce critical sized crystalline nuclei.
The critical temperature of nucleation is the temperature at which the nucleation rate is a maximum. Our initial simulations were done for undercooling temperatures between 300 K and 800 K with an interval of 100 K. We found that the nucleation rate is a maximum between 400 K and 500 K; therefore to find the exact critical nucleation temperature, more simulations were performed between annealing temperatures of 400 K to 500 K with an interval of 25 K. From the slopes of the fitted lines in Fig. 11, the maximum nucleation rate of 5.74x10 35 m -3 s -1 occurs at the annealing temperature of 475 K ( Table 1). The typical nucleation rate for the homogeneous nucleation of a pure metal near the critical temperature has been estimated previously from experiment to be in the order of 30 10 and 40 10 m −3 s −1 52 , which is comparable to our MD results.
Since the nucleation rate is maximum at 475 K, we can come to a conclusion that ~475 K is the critical temperature of nucleation for Al. The calculated critical temperature from MD is ~2 m T , where m T is the melting temperature. Once the solidification progresses the distance between different nuclei is reduced, and the simulation box eventually transforms into the bulk solid crystalline Al with hcp solidification defects and grain boundaries.   Fig. 12 and Table. 2, the nucleation rates are obtained in the same way it was obtained for the isothermal process. The time in Fig. 12 is a small part of time among the whole time steps. Most of the nucleation happens between this part, so it is chosen to study the nucleation rate in quenching. It is shown in Table. 2 that the nucleation rate goes down from 1.12x10 35 m -1 s -1 at cooling rate of 5.83x 10 12 Ks -1 to 8x10 33 m -1 s -1 at cooling rate of 5.83x 10 10 Ks -1 . Fig. 12. The number of nuclei as a function of time for various quench rates. The slope of these curves is the nucleation rate (see Table 2). The x axis shows the time between the start and finish of nucleation. The isothermal simulations in the previous section showed that the nucleation rate is temperature dependent. In quenching crystallization begins by formation of small clusters of atoms at high temperatures. In section 3.5 (Fig. 10) the nucleation regimes for quenching show that the crystallization generally occurs between 586 K and 725 K. In a slower cooling rate, the crystallization occurs at a higher temperature.
In Fig. 13 we show that the nucleation rates in the quenching process and the isothermal cases with high annealing temperatures are almost similar. At the highest cooling rate of 5.83x 10 12 Ks -1 the nucleation rate is 1.12x10 35 m -3 s -1 ( Table 2) lies between the nucleation rates isothermal cases at annealing temperatures of 650 K (1.12x10 35 m -3 s -1 ) and 700 K (0.07x10 35 m -3 s -1 ). The rate of nucleation is calculated using the same procedure used for isothermal process.
Nucleation rate at cooling rate of 5.83x 10 10 Ks -1 is very close to the nucleation of 725 K.
where T is the temperature, B k is the Boltzmann constant, and 0 I is a coefficient that depends on temperature and the interface free energy, SL  56 . * W is defined by 57 , where, A is a constant that depends on the solid-liquid interface energy and enthalpy. Eq. 3 also suggests that homogeneous nucleation rate strongly depends on the undercooling or the annealing temperature. The nucleation rate is maximum at the critical temperature. The critical temperature can be derived from Eq. 3 by setting its first derivative to zero. This suggests that the critical temperature is 3 5 m cr T T  (~550 K). As it was mentioned before, the calculated critical temperature from MD is ~2 m T (475 K), which is a reasonable estimation from MD simulations and close to the CNT and experimental values of critical temperature of nucleation, which lies between 0.5-0.6 times of the melting temperature 60,61 .
We can also find the critical radius from CNT, which is suggested to be: We previously calculated SL  ,the specific free energy of the critical nucleus formation is estimated to be the interface the solid-liquid interface free energy of 172.6 mJ-m -2 and m H  to be 11.50 kJmol -1 for Al 23 . The atomic volume in solidification is available from isothermal simulation. By utilizing Eq. 2 and considering the normalized temperature for annealing, is calculated for different annealing temperatures. So according to CNT the calculated critical radius (size/diameter) lies between 1.25 (2.5) nm and 2.0 (4.0) nm for different annealing temperatures.
The prediction of critical size from CNT is dependent on the annealing temperature (Fig.   14). In section 3.2, we showed at that the critical size calculated by MD simulations is between ~0.82 nm and 4 nm in the isothermal cases, and it is between ~1.8 to ~4.5 nm for quenching cases. CNT predicts almost similar critical sizes to MD simulations from 650 K. But CNT estimates the critical size to be higher than the values obtained from both isothermal and quenching simulations for lower annealing temperatures. In MD simulations of quenching, the size of the critical nucleus increases as the quench rate decreases. As shown before in Section 3.3 with a lower quench rate the nucleation starts at higher temperature, and this results in formation of a larger size critical nucleus. CNT overestimate the critical nucleation size (Fig. 14)   decreases with lower annealing temperature (or increasing undercooling) 65,66 . Therefore the numerator of Eq. (4) should also decrease with lower annealing temperature, making the critical size from CNT closer to MD simulation results.

Determination of induction time
In sections 3.6 and 3.7, nucleation rates are calculated for both isothermal and quenching cases which show how frequently nucleation events occur in the superheated melt of Al. For higher nucleation rates, a system can escape the metastable superheated liquid state and form the crystalline phase. The ability of a system to sustain small thermal fluctuations while in a metastable equilibrium state is characterized by the induction time, which is defined as the time elapsed between the establishment of supercooling and the appearance of persistent, stable nuclei 17 . The theory of homogenous nucleation suggests that the induction time is closely related to the nucleation rate, and the relationship depends on whether the system escapes the metastable state 17,67,68 . Nucleation can be divided into mono or polynuclear mechanisms 17 . When the system undergoes a phase transformation under conditions allowing the formation of many statistically independent nuclei it is called polynuclear mechanism, and for single nucleus it is called mononuclear mechanism. The formulations for the induction time for mononuclear, polynuclear, and combination of both mechanisms are given by Kashchiev et al. 68 . When the system volume is small, similar to our cases, polynuclear formulation reduces to that of the mononuclear case.
The induction time for the mononuclear mechanism is given by, where V is the volume of the system and I is the nucleation rate. Through this relationship, the induction time *  can be calculated from the previously obtained nucleation rate. It is worth mentioning that the role of I is weaker in the polynuclear case than in the mononuclear case 17 .
As *  refers to the time required for the system to escape from the metastable to a stable crystalline state, we can also assume that it is the minimum time required for the first crystalline nucleus to form.
Mullin 67 alternatively defined the induction time as The definition of induction time is valid for the quenching cases. But for isothermal processes superheated melt is kept at an annealing temperature directly and the nucleation occurs immediately. The time difference between first and second critical nucleus is very small until 600 K. Only at higher annealing temperatures such as 700 K or 725 K, there is a detectable time between formation and growth of the first critical nucleus and the formation of a secondary nucleus. This is evident by comparing the snapshots of nuclei formation and growth during solidification for isothermal and quenching processes in Fig. 6(a) and (b). But as it was discussed before, in an isothermal process the whole process of crystallization happens without any change in temperature. It is not possible to generalize the induction time for isothermal processes, as we cannot get all the quantities for the Mullin's formulation for all the annealing temperatures. The isothermal process is equivalent to CNT which also assumes constant temperature for nucleation.
Overall it is more meaningful to calculate the induction time for the quenching process.
During quenching solid atoms start gathering and attempt to form an initial nucleus before it reaches the critical size. The number of atoms and the size of the initial nucleus fluctuate for a few picoseconds before reaching the critical size. We refer to the time between the initial attempt to form a nucleus (20-25 clustered solid atoms) at a site and the formation of a critical size nucleus (1000-1500 clustered solid atoms, shown before in Fig. 3(b)) as the nucleus origin time ( o t ). In Fig. 15, o t and g t are shown for the quench process at the cooling rate of 5.83x10 11 Ks -1 .
We first determined o t , n t and g t for different cooling rates by utilizing snapshots of MD simulations. The induction times calculated by using Eq. (5) and the Mullin's definition 67 are presented in Table 3. The initial relaxation time for the melt at 1,325 K (150 ps) is not included in the reported induction times. The problem with calculating induction time from Mullin's original formula is related to n t . n t is dependent on the superheat temperature and the nucleation rate. As it was shown previously in section 3.3, the first nucleus (nuclei) occurs between 586 K and 725 K for Al for different cooling rates, but n t will be significantly different for different cooling rates. In this work, the induction time is assumed to be the combination of

Grain growth and microstructural evolution
Grain growth is usually defined as an increase in the mean grain size in polycrystals with an increase in annealing time. As discussed in Section 3.5, solid-state grain growth occurs as soon as the simulation box is completely solid (the third regime), and this phenomenon is interesting both from the experimental and theoretical points of view, as it affects the mechanical properties of materials.
To study solid-state grain growth, the simulation box is quenched from 1,325 K to 450 K, and then the resulting nanostructure is annealed at temperatures between 300 K and 725 K for 3,000 ps. The average grain size before starting the annealing process was ~5 nm.
Insignificant grain growth is observed for annealing temperatures lower than 450 K, (such as at 400 K and lower in Fig. 16(a)). At higher annealing temperatures, the grain boundary motion results in formation of larger grains ( Fig. 16(b) and Fig. 16(c)). The effect of temperature on grain growth is related to the mobility of atoms. This is also very relevant to experimental observations where more grain growth is generally detected at higher annealing temperatures [69][70][71][72] . The average grain size versus simulation time is shown for different temperatures in Fig.   17. The grain growth starts immediately for annealing temperatures higher than 600 K. For annealing temperatures below 450 K, the grain size remains below 10 nm at the end of 3,000 ps of annealing, whereas at 600 K the grains become as large as 15-20 nm. At 600 K and higher annealing temperatures no separate grains remains at the end of 3,000 ps of annealing, and the simulation box turned into a large single crystal, with a few stacking faults.  Table 4.
The temperature dependent grain growth can also be explained using a grain growth exponent ( n ). Grain growth can be described by a power law [73][74][75] , where 0 D is the initial average grain size before annealing (at 0 t= ), D is the average grain size after a period of annealing, t is the time, and K is the overall rate constant. n is the grain growth exponent which depends on various factors such as grain boundary area, surface area, grain volume, and number of grains. The parameters are determined by fitting Eq. 6 to the simulation data (see Fig. 17) and the results for n and K are presented in Table 4.
From Table 4, the grain growth exponent remains less than the ideal value (0.5 for parabolic growth). At 600 K the growth is almost parabolic until ~1,000 ps. After 1,000 ps the simulation box becomes a single crystal and the model does not apply. For lower annealing temperatures the smaller values of n signify slower grain growth.

Conclusions
Homogenous nucleation from Al melt was investigated by million-atom MEAM-MD simulations. The main challenge of experimental studies of homogenous nucleation from pure Al is to observe the formation and growth of nuclei inside the melt during the solidification period, and the current work has enabled overcoming this challenge. We used both visual analysis such as direct observation of nuclei, and quantitative analysis of the data such as nucleation rate, induction time, fcc/hcp volume fraction, etc., to study the homogeneous nucleation process. Our MD simulations of homogenous nucleation utilizing a 3D simulation box with maximum of 5 million time steps allowed investigating the isothermal solidification process for 0.5 nanosecond and the quenching solidification process up to 15 nanoseconds.
Inspections by CNA showed that each nucleus had mainly fcc atoms with some hcp atoms. As the solidification process progressed, the hcp crystalline atoms aligned themselves to form stacking faults.
The average size of critical nuclei was determined to be between ~0.82 nm and ~4 nm in the isothermal processes, and between 1. Utilizing the potential energy and percent crystalline atoms versus temperature data for quenching simulations (Fig. 10(c) and Fig. 10(d)), the solidification process can be divided into three temperature based thermodynamics regimes, where the specific temperatures ( sc T and gg T ) depend upon the quench rate: i. Sub-critical unstable nucleation regime above sc T , ii. Super-critical Stable nucleation regime between sc T and gg T , and iii. Solid-state grain growth regime below gg T .
These regions were not clearly seen for isothermal cases with low annealing temperatures. Only at high temperature annealing of 650 K, 700 K and 725 K, could these three distinct regions be observed. The change in instantaneous temperature during nucleation (i.e. solidification) indicated that quenching is more realistic simulation procedure to study a nucleation process. As cooling rate decreases, the sc T moves towards the melting point.
We also determined the percentage of different type atoms for both isothermal and quenching cases. In the isothermal cases with higher annealing temperatures such as 700 K and 725 K, the percentage of fcc atoms (~60-65 %) was higher compared with that of the cases with lower annealing temperatures (~50-55 %). At very low annealing temperatures such as 300K and 350 K, the percentage of fcc atoms was very low (< 45%). In the quenching cases, by decreasing the cooling rate from 5.83x 10 12 Ks -1 to 5.83x 10 10 Ks, the percentage of fcc atoms increased from ~20% to ~80%.
To determine the critical temperature for homogenous nucleation in the isothermal cases, the nucleation rate was calculated by plotting the number of nuclei versus time. The critical temperature of Al was determined to be ~475 K, with a maximum nucleation rate of 35 5.74 10  m -3 s -1 . The nucleation rate in quenching simulations was determined to be one to two orders of magnitude lower than that in isothermal cases with annealing temperatures lower than 600 K.
This was attributed to the fact that in the quenching cases the nucleation occurred only between ~747 K to ~586 K, however in the isothermal cases with low annealing temperatures the nucleation and solidification occurred almost instantly. The nucleation rates for the isothermal cases with annealing temperatures of 700 K and 725 K are almost the same as those for quenching cases. Since nucleation during quenching occurs at much higher temperature than the critical temperature, it is not clear that the critical temperature and maximum nucleation rate has any significance for the actual nucleation process.
The critical nucleus size and the critical temperature for nucleation determined by MD simulations were compared to the CNT predictions. The critical temperature for nucleation obtained from CNT was close to the results obtained by MD simulations for the isothermal cases.
The calculated critical size of nucleus using CNT increases with increasing annealing temperature, and is very close to the values obtained from MD simulations above 650 K. But, CNT estimates the critical size to be higher than MD simulations for lower annealing temperatures, and this is because we have assumed SL  is independent of temperature. Since the solid-liquid interface energy is expected to decrease with decreasing temperature, using a temperature dependent SL  will result in an additional decrease in the critical size of the nucleus at lower temperatures, confirming the MD simulation results.
The induction time, which is closely related to the nucleation rate, was also calculated by MD simulation results. In theory (Eq. (5) Significant grain growth occurred in a temperature region above 500 K and below 650 K.
At lower annealing temperatures, low mobility of atoms results in a very low grain growth rate.
Grain growth exponent (n) increased by increasing the annealing temperature, and it reached the ideal value of 0.5 at 600 K.