Temperature and Degrees of Freedom of Potential Energy vs Its Kinetic Counterparts and Their Roles in Decoding Phase Transition

doi:10.21203/rs.3.rs-4000521/v1

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Temperature and Degrees of Freedom of Potential Energy vs Its Kinetic Counterparts and Their Roles in Decoding Phase Transition

https://doi.org/10.21203/rs.3.rs-4000521/v1

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Temperature, a fundamental metric in thermal energy characterization, encounters intriguing exceptions during phase transitions, where it maintains a constant value despite significant internal energy alterations. Equipartition theorem also failed in interpreting these phenomena. In this study, we introduce a novel framework termed "potential energy temperature (T_pot)" and associated degrees of freedom (D_pot) to provide deeper insights into phase transitions. Our investigations reveal that T_pot diverges considerably from conventional temperature (T_kin) defined by kinetic energy, and D_pot is influenced not only by dimensions in Cartesian coordinate but also by the number of interacting atoms. A noteworthy finding is the correlation between phase changes and increased D_pot, which explains the observed increase in potential energy using the equipartition theorem. Additionally, we identify a sudden change in T_pot during the phase transition, diverging from conventional descriptions. Furthermore, our study unveils unconventional concepts, such as the potential energy of an atom being significantly lower at higher temperatures than at absolute zero. These findings offer a fresh perspective on the phase changes of matter, challenging existing paradigms and providing insights into this complex yet fundamental natural process.

Physical sciences/Chemistry/Physical chemistry/Thermodynamics

Physical sciences/Chemistry/Theoretical chemistry/Statistical mechanics

Physical sciences/Physics/Statistical physics thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena

The concept of temperature serves as a measure of the kinetic energy exhibited by atoms or molecules within a system. This definition is closely tied to the kinetic energy of these particles but is also used to characterize the thermal energy present in the system, encompassing transitional, rotational, and vibrational energies.

In a thermodynamic equilibrium, temperature is defined through the Boltzmann distribution law,¹ associated with the conventional concept of temperature measured by a thermometer adhering to the zeroth law of thermodynamics. The equipartition theorem, applicable to the internal thermal energy within a system, states that each degree of freedom contains 1/2kT of thermal energy, where k is the Boltzmann constant and T is the kinetic energy-based conventional temperature, including translational and rotational energy. For vibrational energy, each mode's contribution tends toward zero at low temperatures due to discrete energy levels in quantum mechanics. At high temperatures or classical mechanics, the vibrational energy may reach kT with half from kinetic and the other half from potential contributions.

During phase transitions, many system's temperature remain constant despite internal energy changes, challenging the equipartition theorem^2,3 and add complexity to the understanding and interpretation of phase transition. The Clapeyron equation defines changes in pressure and temperature in association with entropy and volume of the system during phase transitions, while molecular dynamics simulations often use abrupt changes in potential energy to signify phase changes. ^4–8 Various models, like KTHNY theory ^9–13 for two dimensional materials,¹⁴ the Lindmann index¹⁵, and Trouton’s rule¹⁶ aid in understanding phase transition mechanisms. There have been several excellent review articles about the phase transition with different applications^14,17–21 and we will not repeat the work here.

There should be no argument for the importance in understanding phase transitions which are crucial for designing materials with desirable properties, such as nanoparticles for catalysts,^22,23 energy,^24,25 pharmaceutical applications²⁶, and high entropy alloys.²⁷ Despite the common use of potential energy to investigate phase changes of bulk and nano-materials, ^4–8 there is limited focus on a thorough examination of the potential energy distribution of atoms within a system.^5,8

Our molecular dynamics simulations with a 9 − 6 Lennard Jones (LJ) potential²⁸ explored the distribution of potential energy in one, two, and three-dimensional crystals and during melting. Our aim was to uncover changes in physical properties during phase transition and determine the applicability of the equipartition theorem and Boltzmann distribution in this context. We would like to gain a deeper understanding of the physical insights behind the abrupt changes in potential energy. Our findings indicate that the Boltzmann distribution remains applicable but is distinct from kinetic energy distribution, with the coupling among atoms introducing more complexity.

The studies show that during the phase transition process, there is a significant increase in the degrees of freedom of potential energy (D_pot), and the temperature of potential energy (T_pot) undergoes a sudden variation, contrasting with the constant temperature defined by kinetic energy (T_kin) during melting. The equipartition theorem can be applied to explain potential energy variation using the defined D_pot and T_pot even it is challenging when using T_kin. We also examined the effects of the number of coupling atoms near an atom and found that it has a significant effect on the fitted D_pot, playing an essential role in maintaining a stable liquid phase.

Simulation results reveal that, at temperatures higher than 0K, the distribution of potential energy can extend well below the energy levels observed at T_kin=0K due to multiple-body interactions. Although intriguing, this phenomenon may be attributed to the classical treatment of the system using the LJ potential. Nonetheless, our study offers an alternative perspective, providing insights into the physical aspects of phase changes, fostering ongoing discussions within the scientific community.

We utilized the in-house program package CRYSTL for all simulations, as detailed in the method section. An example of the simulation at T0 = 10K is displayed in Figure S1. T0 represents the applied constant temperature to the system. Our choice of the 9 − 6 LJ potential of Au, V(r), is based on reference²⁸ including an extra cutoff function, f_c(r).

$$V\left(r\right)={f}_{c}\left(r\right){\epsilon }_{0}\left[2{\left(\frac{{r}_{0}}{r}\right)}^{9}-3{\left(\frac{{r}_{0}}{r}\right)}^{6}\right]$$

$${f}_{c}\left(r\right)=\left\{\begin{array}{cc}1& r\le {r}_{min}\\ 1-6{x}^{5}+15{x}^{4}-10{x}^{3}& {r}_{min}\le r\le {r}_{max}\\ 0& {r}_{max}\le r\end{array}\right.$$

$$x=\frac{r-{r}_{min}}{{r}_{max}-{r}_{min}}$$

where, r symbolizes the distance between two atoms, ${\epsilon }_{0}$ signifies the depth of potential energy at the distance r = r₀. Cut-off functions, commonly employed in various models, play a crucial role in fine-tuning short- and long-range interactions among atoms.²⁹ We implemented the function f_c(r) defined by parameters r, r_max, and r_min, ensuring its first and second derivatives are zero at both ending positions for the sake of stable simulations. We set r_min=3.0 Å in all the simulations.

In an equilibrium system, the distribution of the kinetic energy of atoms can be expressed following the Boltzmann distribution.

$$f\left(E\right)={E}^{\frac{{n}_{d}-2}{2}}{e}^{-\frac{E}{kT}}$$

Here, E represents the kinetic energy of an atom, T denotes the temperature defined by kinetic energy, k is the Boltzmann constant, and n_d (with values 1, 2, 3 in Cartesian coordinate) indicates the degrees of freedom of the atom in space. We utilized the same equation when analyzing distributions of potential and total energies, however both the degrees of freedom and temperature are different.

We initiated the analysis with a one-dimensional chain comprising 503 atoms and when each atom only couples with its nearest two neighbors. The distance distribution function among atoms at T0 = 100K is depicted in Figure S2 with an interatomic distance of about 3 Å. To ensure that each atom experiences coupling only from its two immediate neighbors, we set r_max to 5 Å in the initial exploration. We concentrate on the solid state at temperatures T0 from 50 to 100K where the bonding between atoms remains intact.

Figure 1. (a, b, and c) Simulated and fitted distributions of kinetic (E_kin/kT0), potential (E_pot/kT0), and total (E_total/kT0) energy of atoms in a one dimensional chain at temperatures (T0) of 50 and 100K, (d) averaged energy, (e) fitted dimension, and (f) temperature of kinetic, potential, and total energy over temperatures (T0) ranging from 50 to 100K, with r_max set at 5 Å where each atom will couple dominantly with two nearest neighbors. (g, h, and i) Simulated and fitted distributions of kinetic (E_kin/kT0), potential (E_pot/kT0), and total (E_total/kT0) energy of atoms in a one-dimensional chain at temperatures (T0) of 50 and 100K, (j) averaged energy, (k) fitted dimension, and (l) temperature of kinetic, potential, and total energies of temperature (T0) from 50 to 100K when r_max is set to 12 Å.

Figure 1(a, b, and c) presents the simulated and fitted kinetic (E_kin/kT0), potential (E_pot/kT0) and total (E_total= E_kin +E_pot) energy distributions of atoms at T0 of 50 and 100K. The distribution functions are calculated using data of about 500 steps at constant temperature regime. For the total energy distribution in Fig. 1c, the agreement between simulated and fitted curves is encouraging but not as strong as observed for kinetic and potential energies in Fig. 1a and b.

Figure 1(d, e, and f) illustrates the averaged energy, dimension, and temperature of kinetic, potential, and total energy of atoms. In Fig. 1d, E_total equals kT0 with half from E_kin and the other half from E_pot contributions. The fitted dimension of kinetic energy (D_kin=1) in Fig. 1e aligns with the one-dimensional chain and temperature of kinetic energy (T_kin/T0) is also close to one. Notably, the fitted D_pot in Fig. 1e is two, twice that of D_kin.

The increased dimension can be attributed to the coupling between atoms and their two closest neighbors. In classical mechanics, the increasing D_pot indicates an increased configuration probability of potential energy for each atom. As the temperature rises above 0K, the thermal vibration of atoms creates an asymmetric condition at two sides of the atom, resulting in two different configurations (or dimensions) for atoms to maintain the same energy level. Consequently, the potential energy of atoms exhibits a dimension of two instead of one. It's noteworthy that even though D_pot exhibits a higher dimension, the fitted T_pot reduces to 1/2T0. Moreover, E_pot remains at 1/2kT0.

The fitted D_total is 2.8, which coincidentally is close to the sum of D_kin and D_pot at 3. The fitted T_total is approximately 0.63T0. In accordance with the equipartition theorem, the calculated total energy from D_total×T_total is 0.88 kT0, lower than kT0, as indicated in Fig. 1d. These results highlight the challenges associated with convergence in molecular dynamics simulations. While we achieved satisfactory convergence of the total energy, maintaining all distribution functions close to the theoretical distribution remains a complex task.

In the subsequent studies, we adjusted r_max to 12 Å, the value used by the original authors to truncate atom interactions.²⁸ When r_max = 12 Å, each atom couples with 8 nearby atoms approximately. The results are displayed in Fig. 1g-l. The kinetic energy distribution in Fig. 1g remains largely unchanged in comparison to r_max = 5 Å. However, the potential energy distribution deviates from the monotonically decreasing trend in Fig. 1b. A peak appears at E_pot=0.13kT0. A more intriguing observation is the extension of the energy distribution tail well below the energy at 0K. The presence of a tail below zero implies that as the temperature T0 rises above zero, the relaxation of atomic coordinates from their lattice positions not only results in higher potential energy but also leads to an unusual phenomenon where the potential energy of some atoms becomes lower than their energy at 0K. This phenomenon resembles symmetry breaking, ³⁰ where the symmetry of a lower energy state is lower than that of a higher state.

We suspect that the non-zero distribution at negative energy might be a consequence of the classical treatment of atom interactions using a Lennard Jones (LJ) potential and addressing potential energy individually instead of grouping atoms in each vibrational mode. Classically, this result is understandable. At zero temperature in classical mechanics, all atoms are fixed at lattice positions. When an atom interacts with multiple nearby atoms (e.g., 8 when r_max=12 Å), the attractive nature of long-term interactions applies constraints among atoms. At higher temperatures, atoms relax and oscillate near their lattice positions due to increased kinetic energy, reducing constraints, and potentially leading to even lower energy for an atom under certain conditions.

Figure 1h indicates that we can still use Eq. 4 to approximately fit the distribution curve by disregarding the distribution at negative potential energy. The same challenge persists when fitting the distribution function of total energy in Fig. 1i and subsequent two- and three- dimensional arrays.

Future work will explore more robust mathematical derivations to account for these non-zero distributions at negative energy. This will either confirm that these anomalies are artifacts of classical treatment and will vanish with the application of quantum mechanics theory, or it will necessitate the development of a more suitable distribution function within the framework of classical mechanics.

The averaged energy, fitted temperature and dimension of kinetic, potential, and total energies from T0 of 50 to 100K are shown in Fig. 1j-l when r_max=12 Å. E_kin/kT0 is well conserved. E_pot is close to E_kin, however, there is a slight uphill trend with increasing T0 for E_pot/kT0. The same is for E_total/kT0.

In Fig. 1k, D_kin remains constant across all temperatures. D_pot increases from 2 to 2.7 when r_max is varied from 5 to 12 Å at T0 = 50K. The value gradually drops to 2.6 at 100K. However, we have yet to identify a logical reason for the drop in D_pot with increasing T0. Neither for D_total which exhibits the opposite trend, steadily increasing with T0. We posit that the change in the volume of the chain may influence the fitted dimension, and anharmonicity in the potential energy may also play a role. Further exploration is necessary to uncover the physical origin of these phenomena.

In Fig. 1l, T_kin/T0 remains close to one. T_pot/T0 slightly increases with rising T0, aligning with the observed trend in E_pot/kT0 in Fig. 1j. There is no significant change in T_total/T0.

Figure 2 displays variations in E_pot/kT0, D_pot, and T_pot/T0 for atoms at T0 of 50, 80, and 100 K for different values of r_max. A consistently observed turning point at r_max = 6 Å is notable. Both E_pot/kT0 and dimension increase with increasing r_max. T_pot/T0 increases initially and drops after turning point. The data robustly support the conclusion that the number of coupling atoms significantly influences the fitted values. As shown in Figure S2, the average distance between two neighboring atoms is approximately 3 Å. When r_max is less than six, each atom experiences coupling only with its two closest neighbors, resulting in a dimension consistently less than two. As r_max surpasses six, each atom couples with more nearby atoms, leading to a rapid growth in dimension that reaches a plateau when r_max = 10 Å.

From the studies of a one-dimensional array, we learned that D_pot is significantly different from D_kin and increases when an atom is coupled with more atoms nearby. This lays the foundation for investigating the melting process of crystals in two or three dimensions using D_pot. In the case of a two-dimensional crystal, it can withstand much higher temperatures than the one-dimensional case to maintain a solid state. We will focus on temperatures (T0) higher than 300 K. Figure S3 illustrates the radial distribution among atoms in a two-dimensional array including 506 atoms, along with three snapshots at different temperatures (T0) and r_max.

The results for two-dimensional arrays are presented in Fig. 3. Figures 3(a, b, and c) correspond to r_max=4.5 Å to keep each atom primarily coupling with its six nearest neighbors. r_max is reduced from 5 to 4.5 Å since the second nearest distance shown in Figure S3a is reduced to 5 Å in two dimensional arrays. In Fig. 3a, E_pot/kT0 is shown at different T0 and simulation steps. Data of more temperatures are displayed in Figure S4. The relative energy E_pot/kT0 at constant temperature regime after step of 80,000 experiences a rapid increase at T0 = 1100 K. The abrupt change is attributed to bond breaking among atoms. With r_max=4.5 Å, atoms dominantly couple with their six nearest neighbors only. Once the bond breaks, there is no restoring force to keep the atoms connected, leading the array to evaporate directly into a dissociated (gas) state as indicated in Figure S3c.

Figure 3b illustrates the change in pressure within the array at different temperature (T0) and simulation steps. When T0 is below 1000K, the pressure initially increases as the temperature gradually rises from 0K to T0 but drops and stabilizes around the applied one atm pressure. At T0 = 1100K, there is a drastic pressure increase after step = 75,000, caused by the transition of atoms from an ordered lattice position to a relatively random arrangement. After the rearrangement, the pressure gradually drops, instead of returning to about one atm, the pressure is leveling off at about 0.25 GPa even at the end of the simulations. The periodic distances of the array in Figure S5a continues to rise at T0 = 1100K indicating the array is separating apart. Eventually, the periodic distance will keep increasing until the pressure of the system gradually drops to one atm. The distribution function of potential energy in Fig. 3c reinforces this conclusion. The distribution functions smoothly follow the Boltzmann distribution when T0 is below 1000K, but multiple oscillation peaks appear when T0 = 1100K due to atoms at the separated boundaries.

In Fig. 3(d, e, and f), the outcomes are illustrated at r_max of 12 Å. Notably, at T0 = 940K, there is a sudden surge of E_pot/kT0 from 1.4 to 2.2 at constant temperature region and then stabilized around the value. Supportive data at more T0 can be found in Figure S4b. The surge signifies that the crystal undergoes a phase change around 940 K, transitioning into a liquefied state. Moving to Fig. 3e, at T0 = 940 K, the pressure of the array exhibits a rapid increase at step around 75,000, followed by a gradual decline and consistently oscillates around one atm, indicating that the system reaches equilibrium and maintains a stable liquid state. The periodic distance in Figure S5b supports the same conclusion with a noticeable sudden change at T0 = 940K. The distribution function of potential energy in Fig. 3f smoothly adheres to the Boltzmann distribution across the entire studied temperature range. It is noteworthy that, at T0 = 300K and r_max=12 Å, there exists a tail of energy distribution extending well below zero, although the population's magnitude is not particularly significant

In Fig. 3g, E_pot/kT0 of atoms is depicted as T0 increases from 300 to 1200K for r_max=4.5 and 12 Å. For r_max=4.5 Å, E_pot/kT0 exhibits a gradual increase with rising T0, reaching a point at 1100K where the system disintegrates when kinetic energy surpasses the lattice energy. In contrast, for r_max=12 Å, E_pot/kT0 shows a steeper incline as T0 increases, particularly experiencing a sharp spike at around T0 = 900 K, signifying a phase change from a solid to a liquid state. Beyond T0 = 1000K, E_pot/kT0 continues to rise, albeit with a smaller slope.

D_pot in Fig. 3h exhibits a consistent increasing trend with rising T0 when r_max is 4.5 Å. The D_pot value is close to six, correlating with the number of nearby atoms in a hexagonal array. Beyond T0 = 1100K, a temporary drop in dimension occurs during the system crash, followed by a sharp increase. This momentary decrease is attributed to the transformation of lattice positions of atoms to random positions, where the drop in the number of coupling atoms outpaces the increase in the volume of the array.

For r_max=12 Å, the increasing slope of D_pot is significantly larger than when r_max=4.5 Å. At T0 = 900K, a slight drop in D_pot is followed by a sharp increase. The simulations suggest that the transition from a solid crystal to a liquid state introduces random distribution of atoms, increasing the number of configuration or the dimension of potential energy (D_pot). In other words, the randomness of atom positions in a liquid enhances the array's configurational diversity, allowing the system to accommodate more potential energy with higher dimension at a given temperature during the phase transition.

Analyzing the fitted T_pot/T0 in Fig. 3i reveals that T_pot/T0 remains nearly constant when r_max=4.5 Å and T0 < 1080K similar to the one-dimensional case when r_max=5.0 Å. There is a sharp oscillation of T_pot/T0 when T0 is larger than 1080K and the array starts to dissociate. Conversely, for r_max=12 Å, T_pot/T0 decreases with rising T0 during the solid state and experiences a sharp increase during the phase transition process around T0 = 940 K. This increase is attributed to the coexistence of liquid and solid phases. As liquid and solid phases have distinct potential energy, introducing liquid phase to a solid phase elevates the relative internal energy, resulting in a higher T_pot/T0. As more liquid contributes to the coexisting phase, its fitted dimension will increase sharply and T_pot/T0 starts to drop. However, when the array is completely liquefied, T_pot/T0 begins to slightly rise with increasing T0.

We also explored three-dimensional arrays including 500 atoms, and the results are presented in Fig. 4. The findings align closely with those observed in two-dimensional cases. We further reduced r_max to 4.0 angstrom to keep each atom coupling with its nearest 12 neighbors due to the second nearest inter-atom distance is reduced to 4.3 Å in three dimensions. For r_max=4.0 Å, the crystal undergoes dissociation when T0 is larger than 1860K as indicated in Fig. 4a. Notably, the fitted dimension, as shown in Fig. 4b, is close to 10, deviating from the expected 12, the number of neighboring atoms, when the temperature (T0) is below 1000K. D_pot increases and does surpass 12 when T0 is over 1700K. There is no significant change for the fitted T_pot/T0 when T0 is less than 1840K.

For r_max=12 Å, both E_pot/kT0 and D_pot increase as T0 rises, with a pronounced spike at about T0 = 1800K. The fitted T_pot/T0 exhibits a steeper decline compared to the two-dimensional arrays, extending into the temperature range when the crystal melts into a liquid. There is a slight increase in T_pot/T0 at T0 = 1800K, followed by a rapid decrease with further increasing T0. Even in the liquid phase, T_pot/T0 continues to decline with increasing T0, presenting a contrasting trend to the two-dimensional case. The distance distribution function, the simulated and fitted distribution functions of potential energy, pressure, periodic distance, and the average potential energy at different T0 and various simulation steps are displayed in supplementary materials Figure S6-10, respectively. It is worth noting that the population of atoms with negative energy becomes very prominent at T0 = 300K even though they disappear when T0 is further increased over 1000K at r_max=12 angstrom shown in Figure S7b. The calculated melting temperature can be tuned to be close to the experimentally measured value of 1337K by reducing ε₀ in Eq. 1. ³¹ Considering the possible supercooling and superheating in the simulated phase transition, ⁴ and the fine-tuning does not alter the general conclusion, we did not change ε₀ in the presented data.

We employed the molecular dynamics simulation software package CRYSTAL, developed in-house by our research group. The simulations utilized the Leapfrog integrator, implemented through the following pseudocode:

Loop i = 1, nloop

! Update acceleration

get acceleration (a(i))

! Update velocity

v(i + 0.5) = v(i-0.5) + a(i) × dt

! Update coordinate

x(i + 1) = x(i) + v(i + 0.5) × dt

End loop

Velocity and space scaling are applied using constant temperature and pressure algorithm proposed by Andersen³², Heyes³³, and Haak³⁴ with equations 5–7.

$${T}_{scal}=\sqrt{1+\frac{dt}{{\tau }_{T}}(\frac{{E}_{kin0}}{{E}_{kin}}-1)}$$

Where T_scal denotes the scaling factor for temperature, dt is the time step which is 2 fs in the simulations, τ_T represents the coupling constant of the system with environmental bath, E_kin0 =0.5kT0, refers to the kinetic energy at T0, where k is the Boltzmann constant and T0 is the value set for constant temperature algorithm, E_kin is the kinetic energy of atoms in the system at each step and is calculated using $\sum _{i=1}^{N}0.5{m}_{i}{v}_{i}^{2}$ for N atoms with mass, m_i, and velocity v_i, respectively.

Constant pressure of one atm is applied to the system. The pressure of the system including both kinetic and virial contribution is calculated using Eq. 6³³

$$P={P}_{kin}+{P}_{viral}$$

$$=\frac{\sum _{i=1}^{N}{m}_{i}{v}_{i}^{2}}{{n}_{d}V}-\frac{\sum _{i=1}^{N-1}\sum _{j>i}^{N}{r}_{ij}{F}_{ij}}{{n}_{d}V}$$

where, r_ij represents the distance between atoms i and j, and F_ij denotes the force acting exclusively between atoms i and j only. The variable n_d indicates the dimension of the system in Cartesian coordinates. For one- and two-dimensional arrays, the width or thickness of the arrays is determined by the interparticle distance between two nearest neighbors when calculating their volume.

The scaling factor for pressure is calculated using Eq. 7.

$${P}_{scal}=1-\frac{compressibility\times dt}{3{\tau }_{p}}({P}_{0}-P)$$

Where P_scal represents the scaling factor applied to the system's volume (periodic distance), while compressibility denotes the compressibility of the system, τ_p is the coupling constant. The absolute value of the system's compressibility is not critical; instead, the crucial factor is the ratio of compressibility to τ_p. For the sake of simulation stability, we employed a ratio of compressibility/τ_p = 10 in the simulations. P₀ signifies the applied constant pressure of one atm, and P represents the calculated pressure of the system at each simulation step, determined using Eq. 6. In the simulations, we averaged pressure over 1000 steps to avoid big fluctuations of the calculated pressure. This average has no significant effect on our conclusions.

In the simulations, we specifically excluded transitional and rotational motions of the system, focusing solely on vibrational modes for the contribution to thermal energy. We excluded both translational and rotational motion of the system by removing the translational and rotational velocity of the system with a coupling coefficient of 0.001.

For the potential energy of atoms, we employed the 9 − 6 Lennard Jones (LJ) potential, V(r), as described in reference.²⁸ We introduced a cutoff function to investigate the impact of long and short range coupling among atoms on phase transitions. The choice of the LJ potential is motivated by its clarity in elucidating coupling effects among atoms, even though it only considers two-body interactions rather than multiple-body interactions.^35,36 We specifically opted for the 9 − 6 LJ potential over the 12 − 6 LJ potential, as the force field generated fusion entropy closer to experimental values,³¹ despite the simulated melting temperature being much higher than the melting temperature of Au at 1337K. ³¹ While we explored alternative LJ potentials with results not shown, resulting in variations in the calculated internal energy at different temperatures, the melting temperature and latent heat values, the overall conclusion remains consistent.

Competing interests

The authors declare no competing interests.

Author Contribution

A.W. carried out simulations, wrote and revised manuscript. S.Z. developed the software package, carried out simulations, wrote and revised manuscript.

Acknowledgements

SZ is grateful for the support from NSF (CBET #2230729 & #2230891) and the services provided by the OSG Consortium, which is supported by the National Science Foundation awards #2030508 and #1836650.

Data availability

All data generated or analysed during this study are included in this published article and its Supplementary Information files.

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Temperature and Degrees of Freedom of Potential Energy vs Its Kinetic Counterparts and Their Roles in Decoding Phase Transition

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