Factors controlling heteroepitaxial phase formation at intermetallic-Al₃Sc/liquid interfaces

The ability to produce fine-grained solidified microstructures is of immense technological importance for producing structural alloys.^1,2 Grain refinement is also known to impact natural geological processes. For example, grain size provides insights to cooling rates and crystallization history and guides processes to extract metals from ore bodies.^3–5 Focusing on structural alloys, grain refinement in such materials is typically achieved by adding elements, e.g., Sc, Ti, and Zr to Al-based alloys, or extrinsic compounds, e.g., $α$-Al₂O₃ to Al-alloys and TiB to $β$-Ti alloys, to the liquid melt.^6–12 The elemental additions form compound-structures or substrates that are stable in a liquid phase, and, subsequently, they assist in the heterogeneous nucleation of primary/equilibrium phase.^{6,10,11,13,14} Extant literature indicates that atomic ordering at the substrate/liquid interface and substrate/solid misfit strains also determines the effectiveness of any substrate material in nucleating the primary phase. This behavior is also referred as the potency of substrate or nucleant.^13,15,16 Toward that end, using ab initio molecular dynamics (AIMD) simulations, we have investigated the formation-mechanism of face-centered-cubic (FCC)-Al on an intermetallic FCC-ordered Al₃Sc (strukturbericht L1₂) substrate. Intermetallic Al₃Sc is thought to produce grain-refinement in Al-Sc-based alloys.^6,8,10,12 Therefore, our simulations will help in evaluating the potency of the Al₃Sc substrate on the basis of interfacial ordering and misfit strains.

Notionally, misfit strains result from the disregistry between substrate and equilibrium phase crystal structures. The absolute values of such strains typically range $∼$0%–13%, and smaller magnitudes is indicative of a substrate’s potency.^{13,14,16–18} Misfit strains reported in the literature are usually computed using lattice parameters of bulk phase structures, and at temperatures close to the melting point of a material.¹³ These studies preclude estimation of misfit strains between ordered layers and substrate, because it very difficult to experimentally measure interatomic distances within such two-dimensional ordered structures. Experimental evidence of such interfacial ordering was first demonstrated by Oh et al. by performing in situ high-resolution transmission electron microscopy (HR/TEM) of the (0001)-Al₂O₃/liquid-Al interface at 1023 K.¹⁵ Insights from these state-of-the-art experiments were obtained by viewing the projection of substrate/liquid cross sections.^15,19 However, an estimation of substrate/ordered layer misfit strain requires characterization of the mating in-plane structures, which is difficult to obtain from HR/TEM. On the other hand, AIMD simulations of (0001)-Al₂O₃/liquid-Al interface have revealed the in-plane view of atomic structure/configurations within the ordered layer.²⁰ These computational studies demonstrate that AIMD simulations can be employed for estimating substrate/ordered layer misfit strains at temperatures that permit the formation of stable interfaces.

Furthermore, a careful examination of past AIMD results suggested that the 2D plane of ordered layer contained a recognizable threefold symmetry of (111)$Al$⁠,²⁰ which is a subset of sixfold symmetry shown by the (0001)-basal planes of $α$-Al₂O₃. Similar (111)-like ordering could be seen at the (111)-MgO/liquid-Al interface.²¹ Our recent AIMD simulations of Al₃Sc/liquid-Al interface indicated that ordered layers can nominally acquire the fourfold symmetry of the (001)-Al₃Sc substrate.²² Combined, these AIMD studies reveal two critical insights regarding the formation of an in-liquid ordered layer and its influence on equilibrium phase formation. First, ordered layer structure shares a symmetry-based “crystallographic” relationship with the substrate mating plane, or, conversely, a structurally stable substrate/liquid interface determines the “2D” crystallography of interfacial ordering. Second, disregistry between such 2D ordered layer and substrate mating plane may result in misfit strains and influence the subsequent formation of equilibrium FCC-Al.

To probe this matter, stability of (001)-Al₃Sc/liquid-Al and (111)-Al₃Sc/liquid-Al interfaces was evaluated, which allowed us to compare the formation of (001)- and (111)-like in-liquid ordering on the Al₃Sc substrate. Subsequently, the stable substrate/liquid interface was subjected to cooling profile using AIMD, where temperatures were systematically varied within the Al₃Sc-liquid coexistence regime of the binary Al-Sc phase diagram.²³ A key difference between this study and other AIMD investigations is that most utilized one or two temperatures, rather than probing structural changes over a well-defined temperature profile.^{20–22,24–26} Such a profile allowed us to systematically monitor structural changes and associated misfit strains as a function of temperature and propose a structurally informed mechanism for the formation of FCC-Al at the Al₃Sc/liquid interface.

Few comments are required regarding the application of AIMD for studying phase transition-related issues..^27–29 First, the presence of extensive database of pseudo-potentials allows us to probe phases in material systems whose semi-empirical counterparts does not currently exist, e.g., Al-Sc. Thus, AIMD results contribute toward the development of semi-empirical potentials required for classical MD simulations.³⁰ Second, it permits time-resolved examination of ultra-fast mechanisms, i.e., within femto-to-picosecond regime, that contributes to phase formation, e.g., structural changes within liquid phase, diffusion, chemical reactions,etc.^22,30–34 Third, a major reason for aforementioned advantages is that atomic dynamics are based on computing energy gradients from electronic densities, i.e., Helmann–Feynman’s theorem.^27–29,35 This electronic structure-based approach permits time-resolved monitoring of structural changes. However, they are computationally expensive and limits AIMD to probe structural variations to tens of picosecond.^22,34

Rest of the manuscript is organized into three broad sections: simulation details are described in Sec. II, results in Sec. III, and, finally, in Sec. IV we discuss our results within the context of extant theories and literature data.

Initial configurations used in our simulations are shown in Fig. 1, where panels Figs. 1(a1) and 1(b1) depict 384 atoms-(001)$Al3Sc$//(001)$Al$ and 480 atoms-(111)$Al3Sc$//(111)$Al$ interfacial supercells, respectively. The shaded regions in Figs. 1(a2) and 1(b2) show four- and threefold symmetry motifs on the interfacial (001)$Al3Sc$ and (111)$Al3Sc$ planes, respectively, whose symmetries commensurate with pure-Al planes. [Although not studied here, (011)$Al3Sc$ has a lower symmetry, i.e., twofold, compared to (001)$Al3Sc$ and (111)$Al3Sc$]. Together, these supercell configurations permitted us to compare the stability of (001)- and (111)-like interfacial ordering and examine the mechanism of equilibrium FCC-Al formation at the stable Al₃Sc/liquid-Al interface.

Ab inito molecular dynamics (AIMD) was performed using the Vienna ab initio simulation package (VASP) that employs the projector augmented plane-wave (PAW) method^36–39 and utilized Perdew–Burke–Ernzerhof (GGA-PBE) parameterization to describe the electron exchange correlation functional.^40,41 Electronic degrees of freedom was computed by using 360 eV plane-wave cut-off energy, global convergence of 0.01 eV, 0.2 Å$−1$ gamma k-points spacing, 0.2 eV Methfessel–Paxton smearing width, and $10−7$ eV Brillouin zone integration threshold. The AIMD simulations employed Born–Oppenheimer dynamics to compute atomic forces by using isobaric-isothermal ensembles, i.e., constant no. of particles (⁠$N$⁠), pressure (⁠$P$⁠), and temperature (⁠$T$⁠). Pressure and temperature were controlled using a Parrinello–Rahman barostat and Langevin thermostat, respectively.^28,42,43 All NPT simulations were performed at a nominal external pressure of $∼0$ GPa, and equations of motion were solved using time step $Δt=0.005$ picoseconds (ps) or $5×10−15$ s. It should be pointed out that the Parinello–Rahman barostat can dynamically change lattice shape and volume to achieve a desired pressure.^28,29,42,43 In VASP implementation, such dynamical changes are accomplished by adjusting the individual lattice vectors and angles for a given pressure and temperature; meaning, the coupled Parrinello–Rahman and Langevin dynamics can yield non-orthogonal supercells after equilibration. This matter is slightly different from classical MD approaches, which allows significantly larger supercell size and number of atoms than permissible in AIMD simulations.^28,44 In those classical MD approaches, pressure can be applied isotropically along lattice vectors to enforce orthogonality.^45–53 

Therefore, for analyses requiring orthogonal cells, AIMD was performed using the NVT canonical ensemble. Those simulations were carried out using $Δt=0.005$ ps, Nose-Hoover thermostat, and P $∼0$ GPa. Here, constant pressure is achieved by systematically varying the non-equal orthogonal lattice lengths, e.g., see Figs. 1(a1) and 1(b1). This approach requires performing several resource-intensive calculations for single temperature of interest.²² NPT simulations circumvent those extra steps by simultaneously varying the supercell shape and size at each iteration. Temperature equilibration in NPT and NVT runs was achieved within $∼20$ ps, and the average temperatures had a standard deviation of 30–50 K. Periodic boundary condition was enforced in all simulations.

NPT-AIMD was used to impose a step wise cooling profile, shown in Fig. 2(a), on the interfacial configurations. The initial simulation cells were first annealed at 1450 K, and, subsequently, cooled to 950 K in steps of $ΔT=100$ K. The resulting temperature profile performed sequential annealing at 1450, 1350, 1250, 1150, 1050, and 950 K for $δtdwell=50$ ps dwell time (or $104$ iterations). The correspondence of these temperatures to the binary Al-Sc phase diagram is shown in Fig. 2(b). (The phase diagram was adapted from Murray.²³) They ranged between the melting point of intermetallic L1₂-Al₃Sc (⁠$∼1597$ K) and melting point of aluminum (⁠$∼933$ K) or the eutectic line in Al-Sc [see Fig. 2(b)]. The cooling-profile parameters , i.e., $ΔT$ and $δtdwell$⁠, imposed a nominal cooling rate of $2×1012$ K/s $(=ΔTδtdwell)$⁠. It may be pointed out that the actual AIMD temperatures differed slightly from target values indicated in Fig. 2. In the manuscript, we will present results by referring to the target temperatures and invoke actual values as needed.

Analysis of simulation results was carried by computing pairwise distribution functions and atomic density profiles, and applying polyhedral template matching algorithm (PTM).⁵⁴ A pairwise partial distribution function (PDF) $gαβ$ between two types of particles, $α$ and $β$⁠, was computed from using^28,55

where $Nα$ and $Nβ$ are the number of $α$ and $β$-type particles present in the simulation box, respectively, $|ri→−rj→|$ is the distance between two particle types, $rshell$ is the distance of shell from a central particle, and $δ$ is the delta function. We have also computed radial distribution functions, which does not distinguish between atom types.^28,55 Ordering within Al-Al₃Sc and liquid-Al was quantified by computing atomic density profiles $ρ(z)$ using^20,22,56,57

where $Lx$ and $Ly$ are the in-plane dimensions along $x$ and $y$ axes, respectively, and $zhkl$ axis was perpendicular to the interface, e.g., along the [001] in Fig. 1(a1). $Nz(t)$ is the number of atoms sampled inside thin slices of volume (⁠$LxLyΔzhkl$⁠) at time $t$⁠. Here, $Lx$ and $Ly$ depended on the simulation temperature, while $Δz=0.15$ Å. PTM was employed to differentiate between $FCC$ coordinated and disordered/liquid atoms.⁵⁴ This algorithm uses root mean squared deviation (RMSD) to quantify the difference between an observed structure and a known coordination, e.g., FCC, HCP or BCC.⁵⁴ Our past studies showed that RMSD = 0.15 reasonably identified known coordination.^22,52,53 All visualizations were performed using Vesta and Ovito software packages.^58,59

Our results are divided into two subsections. In Sec. A, we will examine the structural stability of the (111)Al₃Sc/liquid-Al interface using the (111)$Al3Sc$//(111)$Al$ interfacial supercell [see Fig. 1(a1)]. Subsequently, in Sec. B, we present qualitative and quantitative data concerning the formation of equilibrium FCC-Al.

Figure 3(a) shows an AIMD snapshot after 50 ps of dwell at 1450K. Two key observations could be noted from this result: First, Sc atoms are randomly distributed within the supercell; implying that the prior, ordered Al₃Sc domain was completely disintegrated at 1450K [compare with Fig. 1(b1)]; and, second, Al atoms in the prior FCC-Al domain did not indicated the presence (111)-like ordering seen in (111)-Al₂O₃ and -MgO substrates.^20,21 Furthermore, PTM analysis showed that structures inside the supercell comprised of a disordered liquid without discernible trace of FCC— see Fig. 3(b). We also computed Al–Al (g$Al−Al$⁠) and Al–Sc (g$Al−Sc$⁠) partial distribution functions (PDFs) by using the atoms belonging to the prior Al₃Sc [see Fig. 1(b)], and plotted them in Fig. 3(c) using “□” and “○” symbols, respectively. [A radial distribution function (RDF) extracted from the entire supercell is also shown in Fig. 3(c)]. Both g$Al−Al$(r) and g$Al−Sc$(r) corresponded to an amorphous structure,^28,52,55 which confirmed that the dwelling at 1450 K for 50 ps resulted in a liquid phase.

More importantly, our AIMD simulations demonstrated that the (111)$Al3Sc$/liquid-Al interface is structurally unstable at 1450 K, and, could not have participated in the heterogeneous nucleation of FCC-Al at lower temperatures—consistent with experimental observations of {100}$Al3Sc$//{100}$Al$ orientation relationship seen in Al-Sc alloys.^6,10 Consequently, the (111)$Al3Sc$//(111)$Al$ interfacial supercell was not subjected to the complete temperature profile shown in Fig. 2(a). [The instability of the (111)$Al3Sc$/liquid-Al interface will be further discussed in Sec. IV.] In the remainder, we will focus on results obtained from the (001)$Al3Sc$//(001)$Al$ interfacial supercell.

We first examine the effect of (001)$Al3Sc$/liquid–Al interface on crystallization by comparing and contrasting PDFs and RDFs at 1450, 1350, 1250, 1150, 1050, and 950 K (see Fig. 4). Specifically, we have evaluated the temperature variation in g$Al−Sc(r)$ [Fig. 4(a)] and g$Al−Al(r)$ [Fig. 4(b)] by extracting atoms belonging to the prior L1₂-Al₃Sc and pure-Al domains, respectively (similar to that reported in the previous section). These PDFs were also compared with the global, supercell RDF, i.e., g$(r)$ [Fig. 4(c)]. In Fig. 4, all g$Al−Sc(r)$⁠, g$Al−Al(r)$⁠, and g$(r)$ were computed from snapshots recorded after 50 ps of dwell time at each temperature.

Plots in Fig. 4(a) showed multiple g$Al-Sc(r)$ peaks at every target temperatures, and these peaks became noticeable sharper upon cooling to 950 K. These observations reveal that L1₂-Al₃Sc retained crystallinity at 1450 K, and the “degree of crystallinity” within Al₃Sc increased upon cooling. Thus, the thermal stability of (001)Al₃Sc/liquid–Al interface is in stark contrast to the structure obtained after annealing (111)$Al3Sc$//(111)$Al$ at 1450 K for 50 ps [see Figs. 3(a) and 3(b)]. These differences in high temperature structural stability may be attributed to stronger inter-planar bonds between ${002}Al3Sc$ planes than ${222}Al3Sc$ (their crystallography are compared in Fig. 1). An indirect evidence of stronger bonds between (002)$Al3Sc$ can be obtained by comparing the locations of the first nearest neighbor (NN) peak in g$Al-Sc(r)$ of (001)$Al3Sc$/liquid–Al and (111)$Al3Sc$/liquid-Al interfacial supercells at 1450 K. In the Former, a 1NN peak was located at 2.86 Åand the latter at 3.1 Å[shown with a dotted line in Fig. 4(a)]; meaning stronger bonds caused smaller interatomic spacing. Results related to the bonding character between ${002}Al3Sc$ planes will be presented later in Sec. III  C.

The g$Al−Al(r)$ plots in Fig. 4(b) indicated the presence of “extra” peaks that were clearly noticeable at 1050 and 950 K (marked with black colored arrows). They indicate that a fraction of the liquid phase have transformed in crystalline solid. Those peaks were even more prominent in the global RDFs or g$(r)$ presented in Fig. 4(c), which also captured newer peaks (marked with red arrows). It is likely that such additional structural signatures result from g$(r)$ sampling the Al–Al and Al–Sc pair-correlations across the liquid–Al/Al₃Sc interface; in addition to Al₃Sc and bulk liquid domains. Consequently, g$(r)$ also captured extra peaks at 1250 and 1150 K [marked with red colored arrows in Fig. 4(c)], which were not observable in the plots obtained from the bulk Al₃Sc and liquid-Al domains. More importantly, these extra peaks point toward in-liquid ordering,^20,22,57 and potentially, captured FCC-Al formation near the liquid–Al/Al₃Sc interface. To better understand the change in the local structure during cooling, we have further examined the liquid-Al/Al₃Sc interfacial supercells at 1450, 1250, and 950 K in greater detail.

Figure 5(a) shows the atomic arrangement after dwelling at 1450, 1250, and 950 K for 50 ps. Visual inspection of the three snapshots indicated an increased atomic ordering with decreasing temperature. Such an ordering can be qualitatively identified as a reduction in “off-lattice” atomic distortions within L1₂-Al₃Sc, and noticeable “layering” or in-liquid ordering within the pure-Al domain [marked with arrows in Fig. 5(a)]. The observed reduction in L1₂-Al₃Sc lattice distortions is also corroborated by g$Al−Sc(r)$ plots in Fig. 4(a) that manifested sharper peaks with cooling. This ordering was also quantified by computing atomic density profiles along [001]$Al3Sc$ by using Eq. (2) (see Sec. II). These results are plotted in Fig. 5(b) that compares atomic densities at 1450, 1250, and 950 K. In these plots, peaks inside the shaded region correspond to (002) planes of Al₃Sc, while the peaks outside indicate ordering within pure-Al (marked with arrows). We find that the number of peaks within pure-Al increases with decreasing temperature. Combined, Figs. 5(a) and 5(b) qualitatively and quantitatively demonstrate that stepwise cooling from 1450 to 950 K caused layer-wise ordering near the (001)$Al3Sc$/liquid-Al interface.

We have also examined the structure of an in-liquid ordered layer at 1450K and their correspondence with {001} planes of Al₃Sc. Figure 5(c) shows that the ordered layer contained distorted quadrilateral motifs (shaded), and a comparison with Fig. 1(a1) indicated that constituent Al atoms in that layer overlay on top of the (002) planes of Al₃Sc [indicated with shaded circles—middle figure in panel (c)]. We find that the structure of the ordered layer closely resembled intermetallic Al₃Sc at 1450 K and manifested (002)$FCC$-like structure.

AIMD configurations obtained from the cooling profile were further subjected to PTM-based analysis to better understand the structural changes along the cooling profile (see Sec. II). Figure 6(a) shows the temperature-variation of the average FCC fraction within the entire supercell (“○” symbol and denoted as “Total FCC”) and pure-Al regions outside of Al₃Sc (“◊” symbol). A total of 1000 AIMD snapshots, between 45 and 50 ps, were utilized to compute the average FCC fractions at each temperature. A variation in the pure-Al FCC fraction was obtained by subtracting the total FCC at each temperature (“○” symbols) with that of 1450 K, because only Al₃Sc contained FCC coordinated atoms at that temperature. (Note that the total FCC fractions at 1350 and 1450 K are comparable). These plots show that the total FCC fraction steadily increases after 1350 K and experiences a steep increase around 1150 K. However, such a temperature-dependent increase in FCC coordination occurs predominantly within the pure-Al domain, i.e., outside of Al₃Sc. At 950 K, the FCC fraction in pure-Al is comparable to Al₃Sc at 1450 K, i.e., approximately twofold increase. More importantly, Fig. 6(a) demonstrates that the (001)Al₃Sc substrate causes FCC coordinated layering at temperatures greater than the melting point of aluminum.

Even though NPT-AIMD simulations produced non-orthogonal cells (see Sec. II), the simulation results are comparable to our past studies that used orthogonal cells.²² Our simulations correctly reproduced in-liquid ordering at the L1₂-Al₃Sc/liquid–Al interface (compare panels corresponding to 1450 K in Figs. 5(a) and 5(b), and Fig. 3 in Ref. 22), and the presence of distorted square motifs in a such ordered layer [compared Fig. 5(c) and Fig. 5 in Ref. 22]. This consistency between current and past studies points toward the efficacy and reliability of employing NPT-AIMD simulations for studying solid/liquid interfaces.

Extant MD studies show that liquid-to-solid transformation results in a sharp or abrupt change in the total energy, i.e., a first-order transition.^{2,45,46,52,53} The onset of that sharp change is called nucleation or crystallization temperature.^{45,46,48,52,60,61} Furthermore, such crystallization temperature appears several degrees below the melting point and corresponds to thermal undercooling.^{45,46,48,52,60,61} Classical and ab initio MD studies have also shown the presence of metastable in-liquid ordering at temperatures greater than the crystallization temperature.^22,46,52 At slower cooling rates, those in-liquid ordering facilitates bulk, homogeneous nucleation of new crystal structures below the crystallization temperature. On the other hand, faster cooling rates substantially reduce atomic mobility and freeze the in-liquid ordered structures within an amorphous matrix after solidification.^46,48,52 The latter transformation process cannot be categorized as first-order, because energy vs temperature is continuous (without any sharp changes) and is best described as either a second-order, or higher order phase transition.^{2,45,46,48,52} Also worth noting is that such in-liquid ordering forms homogeneously within the liquid phase irrespective of the order of transition. The present work differs from those MD studies due to the presence of a prior solid substrate inside the liquid environment. Therefore, even if cooling will introduce structural changes within the liquid Al, and they are expected to be influenced by the solid-Al₃Sc substrate.

Figure 6(c) depicts a continuous, linear trend in energy vs temperature within 950–1450 K (also see supplementary material). In the context of extant literature,^{45,46,48,52,53,60} this trend suggests second- or higher-order transition within 1450–950 K, rather than first-order. The fast cooling rate (⁠$2×1012$ K/s) imposed by the simulations did not allow crystallization via homogeneous in-liquid ordering “far” from the Al₃Sc substrate. However, a different type of ordering is noted: (001)$Al3Sc$/liquid-Al interface facilitated (001)-type FCC coordinated ordering at lower temperatures [see Fig. 6(a)]. Insight into this mechanism is obtained by comparing Figs. 5 and 6(b1) and 6(b2). They revealed that the transformation of the prior ordered layer to FCC coordination resulted in an FCC-Al/liquid-Al interface. During cooling, that interface “moved” in an atomic, layer-by-layer manner, while leaving behind FCC coordination layers in its wake. In other words, there appears to be an temperature dependent “interfacial trapping” mechanism, which constrains the movement of atoms near the (001)$Al3Sc$/liquid-Al interface at higher temperatures and (001)Al/liquid-Al at lower temperatures. Next, we probe this mechanism in detail by examining the stability of the (001)$Al3Sc$/liquid-Al interface.

To better understand the stability of the (001)$Al3Sc$/liquid-Al interface, we have computed the excess electronic charge densities (⁠$ΔρT$⁠) within the interfacial cells using⁶¹ 

where $ρTAl−Sc$ and $ρTAl$ are the charge densities at a temperature (⁠$T$⁠) of the original supercells, and those obtained after replacing Sc with Al, respectively. Equation (3) allowed us to calculate the spatial distribution of $ΔρT$ at $T=1450$ and 950 K and helped determine lattice-level regions with charge localization. Several DFT studies have shown that $ΔρT$ localization between crystallographic planes and directions is a signature of covalent bond character, which increases inter-atomic bond strength compared to metallic bonds.^61–71 Such enhanced bonding was shown to stabilize bulk and interfacial structures, improve resistance to lattice deformation, and increase defect migration barrier.^61–71 

Figure 7 shows the $ΔρT$ distribution at 1450 and 950 K using $10−3$ $e−$Å$−3$ isocharge surfaces (⁠$e−$ denotes electronic charge), where regions manifesting charge localization are indicated with arrows. We find that $ΔρT$ is roughly oriented along $⟨001⟩Al3Sc$⁠, and its localization connected: (i) (002) planes of intermetallic Al₃Sc; and (ii) in-liquid ordered layering of Al atoms and (001)$Al3Sc$⁠. Such anisotropic $ΔρT$ localization is indicative of enhanced bonding between (001)$Al3Sc$ planes and (001)$Al3Sc$-layer and ordered layers. Discussion regarding the relative stability of (001)$Al3Sc$/liquid-Al and (111)$Al3Sc$ / liquid-Al interfaces will be discussed until Sec. IV.

The enhanced inter-planar/layer bonding also impacted atomic trajectories at 1450 and 950 K. Panels in Fig. 8 show trajectories obtained from snapshots within 45–50 ps of annealing at both temperatures. At 1450 K, the distribution of atomic trajectories could be visually categorized into two regions [see panel Fig. 8(a)]: First, those confined within and very near Al₃Sc (marked with arrows), and, second, “far away” regions that corresponded to random atomic motion in a liquid phase. The in-plane confinement of atomic trajectories was even more distinct at 950 K [see panel Fig. 8(a)], because such confinement extended to several layers away from Al₃Sc (marked with arrows) compared to 1450 K. Thus, the localization of electronic charge along $⟨001⟩Al3Sc$ resulted in in-plane confinement of atomic trajectories, which introduced layer-wise ordering adjacent to the Al₃Sc/liquid-Al interface.

This observation can also be interpreted on the basis of local reduction in temperature or undercooling near the (001)$Al3Sc$/liquid-Al interface. (Here, the term “local” refers to few angstroms near the interface rather than the bulk molten material). In principle, such an interfacial temperature can be roughly estimated by applying kinetic theory of mono-atomic gases, which assume that forces between atoms are only due to mutual collisions instead of interatomic potentials. For that purpose, atomic trajectories were first converted to velocities, and, subsequently, the average kinetic energy (computed from the atomic velocities) was equated to the equipartition energy (comprising three degrees of freedom of translatory motion^29,72,73). This classical statistical thermodynamics-based notion results in the following expression:

where $TKinEstimate$ is the temperature estimate based on kinetic theory, $mAl$ and $mSc$ are the masses of Al and Sc, respectively, $vAl$ and $vSc$ are the computed velocities of each Al and Sc atoms, respectively, $NAl$ and $NSc$ are the number of Al and Sc in the supercell, respectively, $N$=$NAl+NSc$⁠, and $kB$ is the Boltzmann’s constant. For sake of completion, we have derived Eq. (4) in  Appendix A1.

Note that the NPT simulations were performed using Parinello–Rahman–Langevin dynamics, and such a Barostat–thermostat combination resulted in non-orthogonal simulation boxes (see Sec. II). However, non-orthogonal axes are numerically less tractable when applying Eq. (4) than orthogonal axes. Therefore, NVT simulations were performed at 1450 K for 50 ps, because it preserved the orthogonality between supercell axes.²² Figure 8(b1) depicts an NVT snapshot after 50 ps, which yielded atomic trajectories comparable to NPT [see 1450 K panel in Fig. 8(a)]. Atomic velocities were computed from 45–50 ps of NVT trajectories, and, $TKinEstimate$ was estimated using Eq. (4). Plots presented in Fig. 8(b2) plots variation in the imposed “global” simulation temperatures (⁠$TBath$⁠) and the computed $TKinEstimate$ as a function of AIMD iterations. $TBath$ refers to the temperature of external bath used by a Nose–Hoover thermostat to integrate equations of motion for an NVT ensemble;^28,29,74 while, $TKinEstimate$ was determined from the atomic trajectories [see Eq. (4) and  Appendix A1].²⁹ The former requires pseudo-Hamiltonian formulation containing kinetic, potential, and dissipation terms that helps in maintaining a steady temperature. On the other hand, $TKinEstimate$ is obtained from kinetic energy, by assuming that atoms as solid spheres and forces on atoms result only from collisions. Difference between the two temperatures is further elucidated on  Appendix A2. Notwithstanding, $TKinEstimate$ has allowed us to quantify the constrains imposed by enhanced bonding (see Fig. 7) on the atomic motion [see Figs. 8(a) and 8(b1)].

Average temperatures obtained from Fig. 8(b2) were: $⟨TKinEstimate⟩=122.6±19.1$ K and $⟨TBath⟩=1436.3±61.7$ K. The resulting kinetic undercooling was $ΔTKinLocal(⟨TBath⟩−⟨TKinEstimate⟩)=1313.7±80.8$ K. Furthermore, we note that Eq. (4) is primarily applicable to liquid or gaseous states.^72,73 It underestimates the value of $⟨TKinEstimate⟩$⁠, because the atoms inside solid-Al₃Sc and the adjoining ordered layers are relatively stationary compared to the surrounding liquid. The actual value of $⟨TKinEstimate⟩$ may be greater, i.e., $ΔTKinLocal$ is smaller, than the computed value. Notwithstanding, using this simplified approach, we have notionally correlated the contribution of enhanced interfacial bonding to temperature—an intensive thermodynamic parameter. This correlation qualitatively demonstrated that such undercooling can facilitate ordering near the solid/liquid interface; even at global temperatures significantly higher than the melting point of aluminum.

Our detailed AIMD investigation of Al-Sc showed: (i) structural stability of the Al₃Sc/liquid-Al interface is governed by interfacial crystallography (Figs. 3 and 5), i.e., (001)$Al3Sc$/liquid-Al vs (111)$Al3Sc$/liquid-Al; (ii) (001)$Al3Sc$/liquid-Al was stabilized by enhanced interfacial bonding (Fig. 7); and, (iii) the stable (001)$Al3Sc$/liquid-Al initiates formation of FCC-coordinated atomic-scale layering along $⟨001⟩FCC−Al$ (Figs. 5 and 6). Herein, we will discuss the relative stability of the two solid/liquid interfaces; analyze FCC-Al formation in the context of extant theories;^{1,13,16,18,75,76} compare and contrast the potency of L1₂-Al₃Sc with other nucleants;¹³ and propose a structurally informed mechanism for the formation of FCC-Al on the Al₃Sc substrate.

The AIMD simulations at 1450 K showed that the supercells containing (001)$Al3Sc$/liquid-Al retained the L1₂-Al₃Sc structure, while the (111)$Al3Sc$/liquid-Al interface did not (compare Figs. 3 and 5). We further learned that electron charge localization along $⟨001⟩Al3Sc$ have contributed to the structural stability of (001)$Al3Sc$/liquid-Al (Fig. 7). It is worth pointing out that our results are also consistent with extant experimental data that showed $⟨001⟩FCC$//$⟨001⟩Al3Sc$ orientation relationship between FCC-Al and Al₃Sc within the as-solidified microstructures of Al-Sc alloys.^6,10

In this study, we examined supercells contained one unit-cell wide L1₂-Al₃Sc (see Fig. 1). This limiting choice was guided by two factors: first, AIMD is computationally more expensive than classical MD and requires a longer computation time even for small systems; and, second, lack of semi-empirical potentials for Al-Sc systems, which would have allowed the application of classical MD to examine substantially larger simulation boxes. Therefore, it is quite possible that the (111)$Al3Sc$/liquid-Al interface with a thicker L1₂-Al₃Sc substrate, e.g., more than 2-unit-cell wide, may have withstood the extreme temperatures employed here. Therefore, the relative stability of the two interfaces noted in our AIMD results may have influence from the substrate size (specifically thickness) and interface-type. This matter can be qualitatively examined using the Gibbs–Thompson equation, which relates the depression in melting point (⁠$ΔTM$⁠) caused by a particle size (⁠$rGT$⁠),^2,77–80

where $γsolid/liquid$ is the solid/liquid interfacial energy, $ΔHF$ enthalpy of fusion, and $TM$ is the melting point. $ΔHF$ and $TM$ are material properties that are independent of orientation, while the initial thickness of the L1₂-Al₃Sc structure, i.e., $rGT$ was comparable in both cases (see Fig. 1). However, our simulations varied $γsolid/liquid$⁠, by exposing two different crystallographic planes of Al₃Sc to liquid-Al and creating the interfaces: (001)$Al3Sc$/liquid-Al and (111)$Al3Sc$/liquid-Al. The extraordinary stability of (001)$Al3Sc$/liquid-Al at 1450 K and the existence of the in-liquid ordered layer indicated substantial wetting of (001)$Al3Sc$ by liquid-Al. (This matter is further discussed in the next section.) This suggested a smaller $γsolid/liquid$ value for the (001)$Al3Sc$/liquid-Al interface and comparatively larger value for (111)$Al3Sc$/liquid-Al. Therefore, the latter interface experiences a greater depression in $TM$ [Eq. (5)] and melts at 1450 K (see Fig. 3). This interplay between substrate size and interface stability is currently being investigated. Notwithstanding, our AIMD simulations show that (001)$Al3Sc$/liquid-Al interfaces are energetically more favorable than (111)$Al3Sc$/liquid-Al; at least within one L1₂-Al₃Sc unit-cell limit. Hereafter, all analyses will focus on the results related to (001)$Al3Sc$/liquid-Al interface.

We will first invoke classical heterogeneous nucleation theory (CHNT), which assumes that secondary phase embryo forms as a spherical cap on a flat substrate.¹ This assumption allows us to analytically express the activation/critical free energy (⁠$ΔGhetero$⁠) using a well-known form

where $θ$ is the contact angle between the embryo and substrate, $ΔT$ is the thermal undercooling, $ΔHliquid→FCC$ is the heat of fusion, $γsolid/liquid$ is the interfacial energy, $To$ is the freezing temperature, and $η$ is a shape factor. In Eq. (6), terms inside the first bracket are mostly material-specific parameters that are independent of the substrate, while, those inside the second and third brackets—specifically $ΔT$ and $θ$—are locally influenced by the Al₃Sc substrate. This dependence allows $ΔT$ and $θ$ to substantially reduce $ΔGhetero$ as follows: First, Figs. 7 and 8 showed that enhanced interfacial bonding can locally cause significant undercooling (⁠$ΔTKinLocal$⁠) that will reduce free energy due to its inverse square contribution, i.e., $ΔT≈ΔTKinLocal$⁠; and, second, Fig. 5(a) shows that there was a pre-existing wetting layer in the form of in-liquid ordering at 1450 K prior to FCC-Al formation at lower temperatures; meaning $θ≈0$⁠, which, according to Eq. (5), imply $ΔGhetero≈0$⁠. Thus, classical theory suggests that the formation of FCC-Al on L1₂-Al₃Sc requires minimal activation/critical free energy, if at all.

Our qualitative CHNT-based analysis notionally correlates well with our AIMD result presented in Fig. 6(c), i.e., total energy vs $T$⁠, which indicated (001)$Al3Sc$-mediated FCC-Al formation best characterized as higher order liquid$→$solid transformation rather than first-order transition process (also see the supplementary material). This assertion are also consistent with extant literature on heterogeneous nucleation;^13,75 it is a barrier-less transformation, and this process is dominated by growth of the equilibrium phase following in-liquid interfacial ordering, e.g., see Fig. 5. Furthermore, Quested and Greer suggested that epitaxial growth is facilitated by a critical undercooling, which is a function of $γsolid/liquid$ and entropy of fusion.⁷⁵ Based on the observed trajectory confinement near (001)$Al3Sc$/liquid-Al [Fig. 8(b)], we believe that $ΔTKinLocal$ contributes to that critical undercooling required for epitaxial growth. This matter is currently under investigation.

Podmaniczky et al. categorized epitaxial growth on the basis of $Δγ=γsolid/liquid+γsolid/substrate−γliquid/substrate$⁠.¹⁶ (The interfacial energies are related to the contact angle via $cos⁡θ=(γliquid/substrate−γsolid/substrate)/γsolid/liquid$^1,75.) Categories of epitaxial growth are¹⁶ 

• Case-I—$Δγ>0$⁠: Island formation, or Volmer–Weber mechanism

• Case-II—$Δγ<0$⁠: Layer-by-layer growth, or van der Meerwe mechanism

• Case-III—$Δγ≈0$⁠: Layer-by-layer growth but limited to a critical thickness, or Stranski–Krastanov mechanism

The presence of interfacial in-liquid ordering (⁠$θ≈0$⁠) and limited spatial extent of FCC coordination within the pure-Al domain at each temperature [“□” symbol in Figs. 6(a) and 6(b)] indicated that epitaxial growth on (001)$Al3Sc$ corresponded to case-III, i.e., $Δγ≈0$⁠. Here, the critical thicknesses ranged from one-to-three atomic layers within $950<T<1450$ K [see Figs. 5(b) and 6(b)]. Next, we turn our attention toward quantifying the effectiveness of such epitaxy on forming FCC-Al on the (001)Al₃Sc substrate.

Misfit strain (⁠$fmisfitT$⁠) between a substrate and equilibrium solid phase crystals is often used to gauge the effectiveness of a heterogeneous nucleant; an effective or potent nucleant must have small absolute $fmisfitT$ values.^13,18,76 Typically, such strains are computed using the characteristic dimensions of substrate/phase mating planes, e.g., inter-planar distances, where crystallography of those planes is determined by a substrate/phase orientation relationship.¹³ Furthermore, $fmisfitT$ estimations are primarily based on a single temperature, e.g., near the melting point of Al, or $T=933$ K.¹³ However, the temperature profile employed in our AIMD simulations allowed us to compute $fmisfitT$ over a range of temperatures (⁠$950<T<1450$ K).

Toward that end, we extracted atomic configurations from the mating (001)Al₃Sc and (002)-Al planes at each temperature [e.g., see Fig. 9(a)], computed their first nearest neighbor (1NN) $g(r)$-RDFs, and located peak positions in $g(r)$⁠. Temperature dependent misfit strain was computed from those peak positions by using the expression

where $rPeak,(002)−AlT$ and $rPeak,(001)Al3ScT$ are peak positions in the 1NN RDFs of (002)$Al$ and (001)$Al3Sc$ at a temperature $T$⁠. Figure 9(b) plots the 1NN RDFs from both planes at 950, 1050, 1150, 1250, 1350, and 1450 K, and the lines denote their corresponding peak positions, i.e., $rPeak,(002)−AlT$ and $rPeak,(001)Al3ScT$⁠. We note that cooling discernibly reduced $|rPeak,(002)−AlT−rPeak,(001)Al3ScT|$⁠, which suggested improved lattice-correspondence between the mating planes at lower temperatures. Using Eq. (7), the misfit strains were computed and plotted in Fig. 9(c) as a function of $T$⁠. Literature values^13,17 for Al₃Ti/Al, TiB₂/Al, ZrB₂/Al and Al₃Sc/Al systems were also compared in Fig. 9(c).

These $fmisfitT$ vs $T$ plots highlighted three points regarding the Al₃Sc/Al system, and our AIMD methodology. First, the NPT-AIMD computed misfit strain at 950 K is in remarkable agreement with experiments, i.e., $fmisfitT≈0.5%$¹⁷—see Fig. 9(c). This agreement provided a firm basis for comparing our AIMD results with other nucleants. For example, Al₃Sc and Al₃Ti (space group $I4/mmm$ and strukturbericht D0$22$⁠) have comparable misfit strains within $933<T<950K$⁠. In other words, Al₃Sc and Al₃Ti have similar potency as heterogeneous nucleant; Al₃Ti is a highly effective nucleant.^13,76,81,82 We also note that Al₃Sc is more effective at high temperatures (⁠$1150<T<1450K$⁠) than TiB₂ and ZrB₂. Combined, these comparisons motivate a need for high-throughput screening of elements that will stabilize L1₂-Al₃Sc-like structures within liquid Al-based alloys. Second, the magnitude of substrate/solid misfit strain reduced with temperature, i.e., from 7.4% to 0.5% during cooling from 1450 to 950 K, respectively. Such a temperature dependent change in $fmisfitT$ can be rationalized on the basis of enhanced interfacial bonding (Fig. 7) and related kinetic undercooling (Fig. 8). Furthermore, a careful comparison of trends in $fmisfitT$ [Fig. 9(c)] and FCC coordinated fraction in the pure-Al domain [Fig. 6(a),“□” symbol] indicated a sharp rise in both quantities after $∼1200$ K. This similarity underscores the linkage between misfit strains and the phase growth initiated by heterogeneous nucleation. Third, the NPT ensemble generated cooling profiles can reliably probe the potency of nucleants in metallic alloys, albeit within limitations (see Sec. II). These AIMD simulations provide crucial insights into in-liquid structural transformations that often require elaborate in situ experiments.

Finally, detailed analyses of atomic configurations along the AIMD-cooling profile helped in formulating a structurally informed mechanism for the formation of FCC-Al on the Al₃Sc substrate. Figure 10 schematically shows the proposed structural changes as a function of imposed “global” temperature (defined by the cooling profile).

The process initiates with the formation of faceted Al₃Sc particles within the alloy melt at $T<TMAl3Sc∼1597$ K [see Fig. 2(b)]. Our results indicate that such facets will be dominated by (001)$Al3Sc$ planes [compare Figs. 3(a) and 5(a)]. The resulting (001)$Al3Sc$/liquid-Al interface form a (002)-like ordered layer of Al atoms within the melt (Fig. 5). The presence of the (002)-like ordered layer at 1450 K suggested that its formation is possibly near-instantaneous and is guided by the local bonding character (Fig. 7). Notwithstanding, the enhanced interfacial bonding between the (001)$Al3Sc$ and (002)-Al layer, i.e., via localization of electron charge densities (Fig. 7), restricts the trajectories of interfacial Al atoms (see Fig. 8) and locally introduces kinetic undercooling. As the global temperature is lowered below 1450 K, that undercooling facilitates the transformation of the (002)-Al layer to FCC coordinated (002)$FCC−Al$ planes in a layer-by-layer manner. This epitaxial transformation is characterized by the reduction in misfit strains [Fig. 9(c)], and formation of a bounding (002)-Al order layer at the (002)$FCC−Al$/liquid-Al interface [see Figs. 5(b) and 6(b2)]. Below $T<TMFCC−Al$⁠, the liquid phase completely transforms to FCC-Al, and the (002)-crystallography developed during in-liquid transformations at $T>TMFCC−Al$ is reflected as $001Al$//$001Al3Sc$ orientation relationship in the solid state, i.e., in the as-cast microstructure.^6,10 Combined, the process of FCC-Al formation at the (001)$Al3Sc$/liquid-Al interface can be characterized as a heteroepitaxial nucleation mechanism, where the crystallography of the FCC-Al “nuclei” is uniquely determined by (001)$Al3Sc$⁠.

Formation of equilibrium face-centered-cubic (FCC)-Al on an intermetallic Al₃Sc substrate was investigated using ab initio molecular dynamics (AIMD) simulation. First, we compared (001)$Al3Sc$/liquid-Al and (111)$Al3Sc$/liquid-Al interfaces at 1450 K, and found that (001)$Al3Sc$/liquid-Al is comparatively more stable. Subsequently, this interface was subjected to a stepwise cooling from 1450 to 950 K. Examination of structural changes near the (001)$Al3Sc$/liquid-Al interface yielded the following observations and inferences:

1. At 1450 K, the (001) plane of L1₂Al₃Sc ordered the neighboring liquid into a mono-layer of Al atoms. Localization of electron density along $[001]Al3Sc$ chemically bonded (001)$Al3Sc$ to the (002)ordered-layer, and the structure of that layer was comparable to (002) planes of FCC-Al.

2. Cooling below 1450 K caused layer-by-layer, epitaxial growth of (002)$FCC−Al$ on the Al₃Sc substrate. At a given temperature, the terminating (002)$FCC−Al$ was always in contact with an (002)-ordered layer. At 950 K, i.e., above the melting point of aluminum, liquid-Al near Al₃Sc did not completely transform into FCC, rather formed a mixture of two-three layer thick planes of (002)$FCC−Al$ and in-liquid ordered layers. A qualitative analysis using classical heterogeneous nucleation theory indicated that the formation of FCC-Al on (001)$Al3Sc$ requires a minimal activation energy.

3. AIMD-computed interfacial misfit strains—at (001)$Al3Sc$/(002)$FCC−Al$ and (001)$Al3Sc$/(002)-ordered layer—ranged from 0.5% to $−7.4%$ within 950–1450 K, respectively. The magnitude of such misfit strains reduced significantly between 1250 and 950 K, which also coincided with a sharp increase in FCC coordination. This observation underscored the correlation between misfit strains and formation of equilibrium phases.

4. Comparison of AIMD-computed misfit strains and literature data showed that the potency of FCC-ordered L1₂ Al₃Sc is similar to well-known tetragonal D0$22$-Al₃Ti and better than several other nucleants. Our results motivate the need for discovering other elements that can stabilize L1₂-Al₃Sc-like structures in liquid Al-based alloys.

See the supplementary material for the AIMD simulation of FCC formation during dwell at 950 K. Analysis involves a comparison of change in energy with the FCC fraction.

D.C. and B.S.M. acknowledge that this research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement No. W911NF-20-2-0190. D.C. is also thankful for computation time on the Pittsburgh Supercomputing Center’s new Bridges-2 cluster that was allocated through No. NSF-XSEDE MAT200006.

The authors have no conflicts to disclose.

Hunter Wilkinson: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Writing – original draft (equal). Brianne Boyd: Visualization (equal); Writing – review & editing (equal). John O'Connell: Software (equal); Visualization (equal). Reilly Knox: Software (equal); Visualization (equal). Alex J. Rinehart: Writing – review & editing (equal). Bhaskar S. Majumdar: Conceptualization (equal); Funding acquisition (equal); Writing – review & editing (equal). Deep Choudhuri: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal); Visualization (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.

Since Eq. (4) in Sec. III C is a classical formulation, we will invoke classical statistical mechanics to derive that expression. We start with the classical virial theorem (e.g., see Ref. 29, p. 82)

where $xm$ and $xn$ are the $m$th and $n$th components of the phase space variable $x$ corresponding to a particle, $H$ is the Hamiltonian or total energy of the system, $⟨⋯⟩$ represents an ensemble average, e.g., NVT, $T$ is the ensemble temperature, $k$ is the Boltzmann constant, and $δmn$ is the Kronecker delta. For a given atom type, Eq. (A1.1) can be reformulated using atomic momentum and setting $m=n$⁠,

where $pm$ is the $m$th component of the momentum of a particle. Subsequently, both sides of Eq. (A1.2) can be summed over the total number of atoms $Natoms$ present in an ensemble such that

where the factor “3” represents three degrees of freedom available to a particle‘s momentum in phase-space. We will use Eq. (A1.3) to derive a relationship between ensemble temperature (⁠$T$⁠) and the average kinetic energy for a two particle system, e.g., a system containing Al and Sc atoms shown in Fig. 8(b1). Toward that end, we first write the classical Hamiltonian of a generalized system comprising atom types “1” and “2” with masses $m1$ and $m2$⁠, respectively,

where $i=1,…,3N1$ and $j=1,…,3N2$⁠, $N1$ and $N2$ are the number of “1” and “2” atom types in the system, $p1,i$ and $p2,j$ are $i$th and $j$th components of momenta of atom types “1” and “2,” respectively, $r1,i→$ and $r2,j→$ are the position vectors, and $U(r1,i→;r2,j→)$ is the interatomic potential. By substituting Eq. (A1.4) into Eq. (A1.3), we obtain two equations

where $T1$ and $T2$ are the temperatures of collections of atom types “1” and “2,” respectively.

However, since both atom types reside in the same ensemble, we can assume that the temperature is same across the system, i.e., $T1=T2=TKinEstimate$⁠. Therefore, adding Eqs. (A1.5) and (A1.6), and substituting $p1,i=m1v1,i$⁠, $p2,j=m2v2,j$⁠, and $N=N1+N2$⁠, we get

where $v1,i$ and $v2,j$ are the $i$th and $j$th components of particles “1” and “2,” respectively, and the two $⟨⋯⟩$ contain kinetic energy contributions from the same ensemble average. Equation (A1.7) can be further reformulated absorbing three components of velocities into the summations, which allows us to sum over number of particles instead, i.e.,

where $α=1,…,N1$ and $β=1,…,N2$ represent atoms of type “1” and “2,” respectively, while $v→1,α$ and $v→2,β$ are their velocity vectors. These velocities could be computed using atomic trajectories obtained from MD simulations and substituted into Eq. (A1.4) in order to estimate the temperature for a collection of atoms.

For a system maintained at a constant no. of particles (⁠$N$⁠) and volume (⁠$V$⁠), i.e., canonical ensemble, the temperature is defined by

where $T$ is the system temperature, $E$ is the internal energy, and $S$ is the entropy. A Nose–Hoover thermostat controls the temperature of a canonical ensemble [indicated in Eq. (A2.1)] by “attaching” an external thermal bath to the system under study.^28,29 The equation of motion describing such a thermostat essentially conserves the total energy represented by a pseudo-Hamiltonian (⁠$H$⁠),²⁸ 

where $Tbath$ [=$T$ in Eq. (A2.1)] is the desired temperature of the thermal bath, $Pdissipation$ is a dissipation term that helps in maintaining$Tbath$⁠, $η$ is the dynamical friction coefficient related to the dissipation term in Eq. (A2.2), $K(p→$ and $V(r→)$ denotes the kinetic and potential energy contributions, and $g=3N−3$ is the number of degrees of freedom.

Thus, Eq. (A2.2) demonstrates that multiple energy terms contribute toward maintaining a “steady” system temperature, $Tbath$⁠. On the other hand, $TKinEstimate$ in Eq. (A1.8) is based on classical mechanics and was extracted from atomic velocities; meaning, it samples only the kinetic properties of atoms present inside a system maintained at a near-constant temperature, $Tbath$⁠. Consequently, it is expected that $Tbath$ is always greater than $TKinEstimate$⁠.