Understanding homogeneous nucleation in solidification of aluminum by molecular dynamics simulations

The 2NN MEAM interatomic potential of Al developed by Lee and Baskes [58] was used is this work to study solidification of Al by MD simulation. We recently tested this interatomic potential [38] which showed accurate prediction of solid–liquid coexistence properties (i.e. melting point, solid–liquid energy, melting point, specific heat etc) of Al as provided in table 1.

MD simulations of homogenous nucleation from pure Al melt were performed in a simulation box with size of 25 × 25 × 25 nm³ (64 × 64 × 64 unit cells, with 1000 188 atoms) and with the isothermal-isobaric (NPT) ensemble. Time step of 3 fs was used for all simulations. Temperature and pressure were controlled by Nose–Hoover thermostat and Parrinello–Rahman barostat [64] respectively. Periodic boundary conditions were employed in all three directions. All the MD simulations were performed in LAMMPS [65]. The 2NN MEAM interatomic potential of Al developed by Lee and Baskes [58] was used is this work; we recently tested this interatomic potential which showed accurate predication of solid–liquid coexistence properties of Al [38].

The OVITO visualization package was used to monitor the nucleation and solidification processes [66]. Within OVITO, common neighbor analysis (CNA) was used [67] to identify the local environment of atoms. Using CNA, one can distinguish atoms in different crystal structure regions by calculating the statistics of diagrams formed from the NN of each atom and comparing it with those previously known for standard crystals. For example, if a central atom and its 12 NN form a structure such as face-centered cubic (fcc), CNA identifies the central atom as fcc. Any such atom is considered an fcc atom. Atoms not identified as fcc, hexagonal close-packed (hcp), or any other crystal type implemented in OVITO are identified as amorphous liquid or amorphous solid atoms.

The predicted melting point of Al using a 2NN MEAM MD simulation is 925 K [38], 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 1325 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 figure 1(a). The percentage of amorphous liquid atoms keeps increasing with increasing the annealing temperature. Finally, the box had no crystalline atoms at 1325 K. The radial distribution function (RDF, g(r)) of the simulation box was calculated for all the temperatures, which is plotted in figure 1(b). There are no long-range peaks at 1325 K. The CNA analysis and RDF plots confirmed that Al was completely melted at 1325 K.

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For isothermal simulations, the Al melt was isothermally solidified at temperatures between the range of 300 and 800 K with 50 K intervals. Maximum nucleation rate was observed to be between 400 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 5.83 × 10¹⁰, 5.83 × 10¹¹ and 5.83 × 10¹² K s⁻¹. Different cooling rates were applied by changing the number of total time steps. The initial temperature of the melt was 1325 K, then the melt was cooled down to 450 K in 150 ps, 1500 ps and 15 000 ps (5000 000 time steps), which resulted in cooling rates of 5.83 × 10¹² K s⁻¹, 5.83 × 10¹¹ K s⁻¹ and 5.83 × 10¹⁰ K s⁻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 [42]. In experiments, cooling rates in rapid solidification of bulk Al lie between 10⁴ and 10⁷ K s⁻¹ [68–70], notably much slower than the rates used in MD.

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. Figure 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 figure 2(b) are extracted, and the measured distances from the surface atoms of the nucleus to the central atom show almost the same value in all directions. The magnified nucleus in figure 2(b) shows atoms with fcc (green) and hcp (red) crystal structures using CNA. The NN 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 figure 2(b) is ∼2.86 Å. It should be mentioned that few solid amorphous atoms get trapped inside the fcc/hcp crystalline phase nucleus, which do not have enough neighbors to be detected as a solid crystalline phase.

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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 [38]. 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 figure 2(b) around the 〈111〉 direction, the stacking fault plane can be observed (figure 2(c)), and the stacking fault is detected to have hcp crystal structure.

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 (100), y (010) and z (001) directions (figure 3(a)). In figure 3(a), one example is given to show how the nucleus size is determined in one of the three directions; the length in (001) direction is determined to be 15 Å for one of the nuclei. Measurements are taken in (100) and (010) directions as well. A similar process is applied to measure the size of all the nuclei at the different timesteps. Simulations are repeated 5 times to determine the uncertainty in the measurement in nucleus size. Five different nuclei in each simulations are chosen and the nuclei sizes are measured in three directions. Total of 75 measurements are done for each nucleus to determine the average and the error bar for each case (3-directions × 5-nuclei × 5-replicate simulations).

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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 does not lose any atom back into the liquid. Figure 3 shows examples of nuclei size and number of atoms in the nuclei versus simulation time for isothermal and quenching cases. The arrows in figures 3(b) and (c) show when the nuclei reach the critical size. The number of atoms in the critical nucleus is ∼1400 atoms at 700 K (figure 3(b)) and for the case with a cooling rate of 5.83 × 10¹¹ K s⁻¹ in figure 3(c), 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 figures 3(b) and (c)). 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 figure 4(a). The average size of critical nuclei is between ∼0.82 and ∼4 nm (figure 4(a)) for all annealing temperatures in the isothermal process. The critical size of crystalline nuclei increases as the annealing temperature increases.

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In the quench processes, the average critical size of nuclei is found to be ∼1.8 nm for 5.83 × 10¹² K s⁻¹, ∼3.49 nm for 5.83 × 10¹¹ K s⁻¹, and ∼4.5 nm for 5.83 × 10¹⁰ K s⁻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 (figure 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 and 650 K is more than 40 nuclei in 25 nm³. 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 and 15 for different cooling rates, which is similar to that of 700 and 725 K isothermal cases. The maximum number of separable nuclei is seen at 715 K, 665 K and 655 K for 5.83 × 10¹⁰ K s⁻¹, 5.83 × 10¹¹ K s⁻¹ and 5.83 × 10¹² K s⁻¹ quench rates, respectively.

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The crystal nucleation from an undercooled Al melt predicted by MD simulations is shown in figure 6 for two isothermal cases (annealing at 475 and 700 K), and two quenching cases (cooling rates of 5.83 × 10¹⁰ and 5.83 × 10¹¹ K s⁻¹); only fcc Al atoms are shown for a better visualization of nuclei. The instantaneous time and temperatures during each quenching process are also shown in figures 6(c) and (d).

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As mentioned before, the number of nuclei in the system is reduced with increasing the annealing temperature above 500 K (figure 5); the same conclusion can be made by comparing figures 6(a) and (b). These figures also show that each nucleus can grow much bigger in size at a higher annealing temperature. Nucleation by quenching in figures 6(c) and (d) shows a very similar behavior to the nucleation of isothermal process at 700 K (figure 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.

As shown in figure 6, the number of nuclei differs at various temperatures, so the growth of some nuclei can be affected by the other nuclei in the system at particular time steps. This is more pronounced at lower temperatures. As shown in figures 7(a)–(c), in this frame initially two critical nuclei form simultaneously and later in figure 7(b) in a very quick succession (∼6 ps) two new critical nuclei form. Finally, within another ∼6 ps, these four nuclei come too close to each other. It should be noted that in lower temperatures the solid–liquid interfacial energy has large orientation dependence and formation of faceted nuclei is expected. Figure 7(c) shows some faceted nuclei.

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At higher temperatures, such as 700 K in figures 7(d)–(f), the free growth of nuclei is possible in most part of the solidification process. As shown in figures 7(d)–(f), the critical nucleus in the middle of the domain does not see significant influence from the neighboring nuclei, which from ∼20 ps later than first critical nucleus. Also because there is less effect from the solid–liquid interfacial energy anisotropy at higher temperatures, each nucleus can almost reach a spherical shape, figure 7(f).

The instantaneous temperature for crystal nucleation during quenching can be determined by plotting percentage of crystalline atoms versus temperature change (figure 8). Figure 8 shows that during quenching the nucleation process starts between ∼747 and ∼712 K for the slower cooling rates of 5.83 × 10¹⁰ (figure 8(b)) and 5.83 × 10¹¹ K s⁻¹ (figure 8(c)). For a high cooling rate of 5.83 × 10¹² K s⁻¹ (figure 8(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 figure 6). The number of fcc/hcp atoms is very low for 5.83 × 10¹² K s⁻¹ cooling rate that it does not reflect the nucleation starting temperature in figure 8(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 and 747 K.

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The percentage of atoms having different structures (amorphous, fcc or hcp) is plotted in figures 9(a)–(c) at three different annealing temperatures for the isothermal process. The percentage of fcc atoms for 700 and 725 K (∼60%–65%) is slightly higher than the percentage of fcc atoms generated at lower temperatures (∼50%–55%). At very low temperatures such as 350 K, the percentage of fcc atoms is very low. The lower amount of fcc atoms also causes a very low nucleation rate at 350 K. At lower temperatures (below 600 K), the accumulation of fcc atoms starts immediately. At the same time number of hcp stacking faults grows and number of amorphous atoms decreases. At 725 K the initial nucleation starts later than any other temperature. At temperatures, higher than 725 K, there is no nucleation in 600 ps.

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Percentage of atoms having different structures (amorphous, fcc or hcp) for different quench rates are also plotted in figures 10(a)–(c). In quenching the process of nucleation starts after first regime when mostly sub-critical nucleus/nuclei forms and dissolves. Slowly the temperature reduces to nucleation regime and then the nucleation happens very fast. However the rate of formation of fcc atoms differs for different cooling rates. The slower the cooling rate is the more time the melt has to solidify the fcc/hcp atoms and this lowers the number of amorphous atoms. With a slower cooling rate number of hcp stacking faults also decreases.

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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. Figure 11(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.

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Below 600 K, the Al melt starts solidifying immediately within the first few time steps. For higher annealing temperatures (such as 650, 700 and 725 K) the solidification does not 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 figure 11(a) that the behavior of melt changes with temperature. At 750 K and above, the potential energy is flat for the entire simulation time indicating that no nucleus forms. Between 600 and 725 K three distinct regimes can be observed. We initially see a flat region similar to 750 K where no nucleation occurs. Then the curvy decay of the potential line indicates that nucleation happens at this stage. This indication is verified against per atom data (i.e. LAMMPS dumps), and it is found that the decay of potential energy and nuclei formation happen simultaneously. The final flat region shows the start of the solid-state grain growth. From 350 to 575 K the sharp decrease in potential energy starts immediately and continues until the end of the simulation, which shows that the nucleation starts immediately, and towards the end of the simulation the potential energy curves becomes parallel to each other. At lower annealing temperatures (below 350 K), even though the Gibbs free energy difference between fcc/hcp and amorphous liquid atoms is large, the low mobility of atoms at low temperatures affects the kinetics of nucleation and not all the liquid atoms can form crystalline structures. An amorphous solid structure is retained. At the final stage of all cases, curves become slowly parallel to the time axis with an offset from each other. This offset is due to the specific heat of the solid phases, which results in lower potential energy at lower temperature. Overall, the isothermal simulations do not give the temperature range for the different stages of nucleation; these simulations only indicate that liquid Al melt has to be below 725 K for nucleation to occur.

Figure 11(b) shows the potential energy versus time for the quenching process with different cooling rates from 1325 to 450 K. It is not possible to identify the nucleation regimes by this figure, but the plot of the potential energy versus temperature, figure 11(c), reveals three temperature-dependent regimes very clearly. As shown in figure 11(c), the potential energy decreases when temperature decreases during quenching simulations. During this period (>725 K), sub-critical nucleus/nuclei form and dissolve back into the melt. For the low cooling rates (5.83 × 10¹⁰ and 5.83 × 10¹¹ K s⁻¹), the sharp change in the slope (figure 11(b)) indicating a sudden decrease in potential energy occurs at temperatures range of 715–725 K. The beginning of the sharp change in slope shows the start of formation of critical nuclei (at ∼725 K), and once the first sub-critical nucleus reaches the critical size the crystallization happens very fast at low cooling rates. The super-critical nuclei grow bigger in size until the simulation box is completely solid, and solid-state grain growth starts; this region can be easily identified for low cooling rates (e.g., <710 K for 5.83 × 10¹⁰ K s⁻¹). For the low cooling rates, the almost vertical slope line (figure 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. Figure 11(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 figure 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 T_sc. The exact value of T_sc depends on the cooling rate, and remains between ∼715–725 K for 5.83 × 10¹⁰ K s⁻¹ and 5.83 × 10¹¹ K s⁻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 T_gg. The difference between T_sc and T_gg, is very small (only 9 K for 5.83 × 10¹⁰ K s⁻¹ cooling rate, T_sc = 724 K start and T_gg = 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 (T_gg). 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 T_sc for 5.83 × 10¹⁰ K s⁻¹, 5.83 × 10¹¹ K s⁻¹, 5.83 × 10¹² K s⁻¹ are 724 K, 715 K and 586 K respectively. So, we can see as the cooling rate decreases the T_sc increases. In MD, the quench rates are very high. If we assume in bulk experiments the cooling rate to be 1–100 K s⁻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 (figure 11(d)) is getting more vertical as the cooling rate is reduced. So, in experiments when the cooling rate is much slower, T_sc and T_gg can be almost the same temperature.

We can also relate the self-diffusion coefficient to the nucleation regime (figure 11(e)). The increasing self-diffusion coefficient with increasing temperature results in higher mobility of the Al atoms or vice versa. As previously shown in figure 11(a), the potential energy indicates that no nucleation occurs after 725 K. The reason for this phenomena is the higher mobility of atoms, with self-diffusion coefficient of 1.35 × 10⁻⁹ m² s⁻¹ at 750 K. The self-diffusion coefficient in MD is comparable to experimental finding [71].

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 (figure 12). The nucleation rate increases as the annealing temperature increases from 300 to 475 K (table 2). 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.

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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 and 800 K with an interval of 100 K. We found that the nucleation rate is a maximum between 400 and 500 K; therefore to find the exact critical nucleation temperature, more simulations were performed between annealing temperatures of 400–500 K with an interval of 25 K. From the slopes of the fitted lines in figure 12, the maximum nucleation rate of 5.74 × 10³⁵ m⁻³ s⁻¹ occurs at the annealing temperature of 475 K (table 2). 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 10³⁰ and 10⁴⁰ m⁻³ s⁻¹ [72], 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 ∼$\tfrac{{T}_{{\rm{m}}}}{2},$ where T_m 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.

In quenching crystallization begins by formation of small clusters of atoms at high temperatures. Many of them form and dissolve back into the liquid; a few will survive. The nuclei starts forming after sometime. The beginning time for the nuclei depends on the cooling rate, slower the cooling rate the later the nucleus (nuclei) forms. From figure 13 and table 3, the nucleation rates are obtained in the same way it was obtained for the isothermal process. The time in figure 13 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 3 that the nucleation rate goes down from 1.12 × 10³⁵ m⁻¹ s⁻¹ at cooling rate of 5.83 × 10¹² K s⁻¹ to 8 × 10³³ m⁻¹ s⁻¹ at cooling rate of 5.83 × 10¹⁰ K s⁻¹.

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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 (figure 11) the nucleation regimes for quenching show that the crystallization generally occurs between 586 and 725 K. In a slower cooling rate, the crystallization occurs at a higher temperature.

In figure 14 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 5.83 × 10¹² K s⁻¹ the nucleation rate is 1.12 × 10³⁵ m⁻³ s⁻¹ (table 3) lies between the nucleation rates isothermal cases at annealing temperatures of 650 K (1.12 × 10³⁵ m⁻³ s⁻¹) and 700 K (0.07 × 10³⁵ m⁻³ s⁻¹). The rate of nucleation is calculated using the same procedure used for isothermal process. Nucleation rate at cooling rate of 5.83 × 10¹⁰ K s⁻¹ is very close to the nucleation of 725 K.

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Utilizing the isothermal MD simulation data, the time required to crystallize at least 40% of fcc and hcp atoms from undercooled liquid Al is determined as a function of temperature and plotted as a TTT curve in figure 15. The TTT curve has the typical nose shape for crystallization, which has been observed in various experiments for different metallic materials and metallic glasses [73–75]. The TTT diagram was also replicated by some MD simulations previously [76–78].

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In figure 14, the top horizontal line (dotted red line) presents the TTT diagram for normal liquid Al. This is also the highest possible T_sc for Al. The bottom horizontal line (dotted green line) is the normal solid or it can be also referred to the minimum possible T_gg for Al. The circular points (with error bars) on the TTT diagram are the times taken to crystallize at least 40% of fcc and hcp atoms at different temperatures. The vertical line on the right side of the nose (dotted pink line) is the cooling with the highest quench rate of 5.83 × 10¹² K s⁻¹ applied in our study. This line remains on the right side of the nose that suggests occurrence of crystal nucleation at 5.83 × 10¹² K s⁻¹. The tip of the nose is where the nucleation rate is the highest as the fastest crystallization occurs. As mentioned before in section 3.6, the critical temperature for nucleation is determined to be around 475 K, and the nose area in figure 15 also belongs to a similar temperature range.

A higher quench rate of 10¹³ K s⁻¹ is also applied, which is shown by the vertical line on the left side of the nose (dotted purple line) in figure 15. The final structure obtained at this quench rate is a glassy structure. This result is consistent with the prediction of other TTT diagrams from MD simulations. In the work by Lu et al [78], it was theoretically shown that the quench rate for forming glass structures is generally above 10¹² K s⁻¹. Other MD simulation studies on bulk metallic glasses [79] also showed the critical cooling rate for glass formation is 10¹¹–10¹² K s⁻¹. Bulk metallic glasses are generally multi metallic compounds, so it is more difficult to promote homogeneous nucleation in bulk metallic glasses than in single element metals. This happens due to different sizes of atoms and also different thermal vibration at higher temperatures. Therefore, we suggest that the cooling rate for formation of glassy structure from pure Al melt is 5 × 10¹²–10¹³ K s⁻¹.

CNT can provide some insights on the homogeneous nucleation process. CNT suggests that there is a free (activation) energy barrier, W*, for formation of a solid nucleus with a critical size of r*. The nucleation typically happens when the probability of energy fluctuation is sufficient to overcome the activation barrier. The probability of energy fluctuation is given by the Arrhenius type equation and the rate of homogeneous nucleation is [37, 80–82],

$$\begin{eqnarray}&&I={I}_{0}\exp \left(-\displaystyle \frac{{W}^{* }}{{k}_{{\rm{B}}}T}\right),\end{eqnarray} \tag{ 1 }$$

where T is the temperature, k_B is the Boltzmann constant, and I₀ is a coefficient that depends on temperature and the interface free energy, σ_SL [82]. W* is defined by [83],

$$\begin{eqnarray}&&{W}^{* }=\left(\displaystyle \frac{16\pi }{3}\displaystyle \frac{{\sigma }_{{\rm{SL}}}^{3}}{{({\rm{\Delta }}{G}_{{\rm{V}}})}^{2}}\right),\end{eqnarray} \tag{ 2 }$$

ΔG_v is the difference between the free energies of liquid and solid crystal per unit volume. If the change in molar heat capacities is constant, ΔG_v according to Hoffman is equal to ${\rm{\Delta }}{H}_{{\rm{m}}}(T{\rm{\Delta }}T/{{T}_{{\rm{m}}}}^{2})$ [84, 85], where ΔT is the undercooling (ΔT = T_m − T), and ΔH_m is enthalpy of melting. By combining equations (1) and (2), the homogeneous nucleation rate becomes:

$$\begin{eqnarray}&&I={I}_{0}\exp \left[\left(-\displaystyle \frac{16\pi {\sigma }_{{\rm{SL}}}^{3}{{T}_{{\rm{m}}}}^{4}}{3{k}_{{\rm{B}}}{({\rm{\Delta }}{H}_{{\rm{m}}})}^{2}}\right)\displaystyle \frac{1}{{T}^{3}{({\rm{\Delta }}T)}^{2}}\right]={I}_{0}\exp \left(-\displaystyle \frac{A}{{T}^{3}{({\rm{\Delta }}T)}^{2}}\right),\end{eqnarray} \tag{ 3 }$$

where, A is a constant that depends on the solid–liquid interface energy and enthalpy. Equation (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 equation (3) by setting its first derivative to zero. This suggests that the critical temperature is ${T}_{{\rm{cr}}}=\tfrac{3{T}_{{\rm{m}}}}{5}$ (∼550 K). As it was mentioned before, the calculated critical temperature from MD is ∼$\tfrac{{T}_{{\rm{m}}}}{2}$ (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 and 0.6 times of the melting temperature [86, 87].

We can also find the critical radius from CNT, which is suggested to be:

$$\begin{eqnarray}&&{r}^{* }=2\displaystyle \frac{{\sigma }_{{\rm{SL}}}}{{\rm{\Delta }}{G}_{{\rm{V}}}}.\end{eqnarray} \tag{ 4 }$$

We previously calculated σ_SL to be 172.6 mJ m⁻² and ΔH_m to be 11.50 kJ mol⁻¹ for Al [38]. The atomic volume in solidification is available from isothermal simulation. By considering the normalized temperature for annealing, ${T}_{{\rm{normalized}}}=T/{T}_{{\rm{m}}},$ ΔG_v 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 (figure 16). In section 3.2, we showed at that the critical size calculated by MD simulations is between ∼0.82 and 4 nm in the isothermal cases, and it is between ∼1.8 and ∼4.5 nm for quenching cases. CNT predicts almost similar critical sizes to MD simulations from 650 K. As it can be noticed in figure 16, MD simulation results are closer to CNT predictions at higher temperatures (700 and 725 K) than at lower temperatures. Unlike at large undercooling temperatures (e.g., solidification at 450 K) where multiple critical nuclei form simultaneously, at lower undercooling temperatures (e.g., solidification at 700 and 725 K) the critical nuclei form one after another (figure 6). This is why each critical nucleus can grow in size without any influence from neighboring nuclei. So the nuclei can grow as much as predicted by CNT. Similarly, in MD simulations with slower quench rates of at 5.83 × 10¹⁰ and 5.83 × 10¹¹ K s⁻¹, the size of the critical nucleus matches well with the CNT predictions because the nucleation starts at temperature higher than 725 K. In figures 6(c) and (d) this phenomenon can be clearly observed.

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The difference between CNT and MD at lower annealing temperatures can be explained by analyzing equation (4). To calculate the critical nucleation size at different annealing (or undercooling) temperatures by equation (4), σ_SL at the melting point is used similar to the most of the other works in the literature [88–90]. This means the numerator of equation (4) is kept constant for calculating the critical size nucleus at different temperatures. However σ_SL decreases by lowering the annealing temperature (or increasing the undercooling) [91, 92]. Therefore the numerator of equation (4) should also decrease with lowering the annealing temperature, making the critical size predicted from CNT to become closer to the MD simulation data.

In section 3.5 we divide the overall solidification process into three different parts. In the unstable nucleation regime, T > T_sc, and the grain growth regime, below T_gg, nucleation does not happen and CNT is not applicable. CNT is only applicable in the super-critical stable nucleation regime between T_sc and T_gg. As shown in figure 16, CNT is in good agreement with MD simulations at higher temperatures, so CNT is mostly valid for the temperature range right below the super critical temperature (T_sc).

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 [19]. 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 [19, 93, 94]. Nucleation can be divided into mono or polynuclear mechanisms [19]. 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 [94]. 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,

$$\begin{eqnarray}&&{\tau }^{* }=\displaystyle \frac{1}{IV},\end{eqnarray} \tag{ 5 }$$

where V is the volume of the system and I is the nucleation rate. Through this relationship, the induction time τ* 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 [19]. As τ* 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 [93] alternatively defined the induction time as ${\tau }^{* }={t}_{{\rm{r}}}+{t}_{{\rm{n}}}+{t}_{{\rm{g}}};$ the induction time is divided into three periods. t_r is the relaxation time required for the system to achieve a quasi-steady-state distribution of molecules in the system; t_n is the time required for the formation of the first stable nucleus (critical sized); and t_g is the time between formation of the first stable nucleus (or nuclei) and the second stable nucleus (or nuclei) inside the melt, and after this time the cluster of crystalline atoms do not dissolve back into the liquid phase.

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 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 figures 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 figure 3(b)) as the nucleus origin time (t_o). In figure 17, t_o and t_g are shown for the quench process at the cooling rate of 5.83 × 10¹¹ K s⁻¹.

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We first determined t_o, t_n and t_g for different cooling rates by utilizing snapshots of MD simulations (figure 17). The induction times calculated by using equation (5) and the Mullin's definition [93] are presented in table 4. The initial relaxation time for the melt at 1325 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 t_n. t_n 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 and 725 K for Al for different cooling rates, but t_n will be significantly different for different cooling rates. In this work, the induction time is assumed to be the combination of t_o and t_g. These two quantities must be minimum for the nucleation rate to be maximum and vice versa. The results show a pattern of gradually increasing induction time with slower cooling rate.

t_o can be compared with the results of theory for τ* in equation (5). However the theoretical values are much lower, because in theory, the induction time is based on the fact that nucleation is stationary [94]. Stationary means the temperature is constant throughout the solidification and a supersaturation is imposed on the system [94]. Stationary nucleation is the simplest case of nucleation. In a realistic nucleation system such as in quenching of a superheated melt, none of these conditions hold true.

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 1325 to 450 K, and then the resulting nanostructure is annealed at temperatures between 300 and 725 K for 3000 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 figure 18(a)). At higher annealing temperatures, the grain boundary motion results in formation of larger grains (figures 18(b) and (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 [95–98].

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The average grain size versus simulation time is shown for different temperatures in figure 19. 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 3000 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 3000 ps of annealing, and the simulation box turned into a large single crystal, with a few stacking faults.

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The temperature dependent grain growth can also be explained using a grain growth exponent (n). Grain growth can be described by a power law [99–101],

$$\begin{eqnarray}&&{(D/{D}_{0})}^{1/n}-1=Kt,\end{eqnarray} \tag{ 6 }$$

where D₀ is the initial average grain size before annealing (at t = 0), 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 equation (6) to the simulation data (see figure 19) and the results for n and K are presented in table 5.

From table 5, the grain growth exponent remains less than the ideal value (0.5 for parabolic growth). At 600 K the growth is almost parabolic until ∼1000 ps. After 1000 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.