^{1,2,3,4,a)}, C. Cazorla

^{5}and M. J. Gillan

^{1,3,4}

### Abstract

Molecular dynamics simulation is used to study the time-scales involved in the homogeneous melting of a superheated crystal. The interaction model used is an embedded-atom model for Fe developed in previous work, and the melting process is simulated in the microcanonical (*N*, *V*, *E*) ensemble. We study periodically repeated systems containing from 96 to 7776 atoms, and the initial system is always the perfect crystal without free surfaces or other defects. For each chosen total energy *E* and number of atoms *N*, we perform several hundred statistically independent simulations, with each simulation lasting for between 500 ps and 10 ns, in order to gather statistics for the waiting time τ_{w} before melting occurs. We find that the probability distribution of τ_{w} is roughly exponential, and that the mean value 〈τ_{w}〉 depends strongly on the excess of the initial steady temperature of the crystal above the superheating limit identified by other researchers. The mean 〈τ_{w}〉 also depends strongly on system size in a way that we have quantified. For very small systems of ∼100 atoms, we observe a persistent alternation between the solid and liquid states, and we explain why this happens. Our results allow us to draw conclusions about the reliability of the recently proposed Z method for determining the meltingproperties of simulated materials and to suggest ways of correcting for the errors of the method.

The work of D.A. was conducted as part of a EURYI scheme award as provided by EPSRC (see www.esf.org/euryi). Calculations were performed on the UCL Legion service, the HECToR national service (UK) and used resources of the Oak Ridge Leadership Computing Facility, located in the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DE-AC05-00OR22725 (USA). We acknowledge helpful comments from one of the referees on the uncertain theoretical basis of the Z method.

I. INTRODUCTION

II. TECHNIQUES

III. RESULTS

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Liquid solid interfaces
- 31.0
- Melting
- 12.0
- Solid liquid phase transitions
- 12.0
- Free energy
- 8.0
- Ab initio calculations
- 7.0

## Figures

Time-dependent temperature and pressure in four independent simulation runs, showing homogeneous melting from superheated hcp solid Fe in a system of 7776 atoms. All four simulations were initiated from perfect crystal positions, with initial random velocities corresponding to the same temperature *T* _{m} = 15 600 K, the mean quasi-steady temperatures of the superheated solid and the final liquid being *T* _{sol} = 7590 K and *T* _{liq} = 6315 K.

Time-dependent temperature and pressure in four independent simulation runs, showing homogeneous melting from superheated hcp solid Fe in a system of 7776 atoms. All four simulations were initiated from perfect crystal positions, with initial random velocities corresponding to the same temperature *T* _{m} = 15 600 K, the mean quasi-steady temperatures of the superheated solid and the final liquid being *T* _{sol} = 7590 K and *T* _{liq} = 6315 K.

Histograms of waiting times τ_{w} before the transition to liquid constructed from repeated simulations at two initial temperatures for the system of 7776 atoms. Histograms shown by dashed (red) and solid (black) lines result from initial temperatures of *T* _{i} = 15 800 and 16 000 K, respectively, the quasi-steady solid and liquid temperatures in the two cases being *T* _{sol} = 7640 and 7740 K and *T* _{liq} = 6410 and 6505 K. Dashed and dotted curves show exponential functions fitted to histograms (see text).

Histograms of waiting times τ_{w} before the transition to liquid constructed from repeated simulations at two initial temperatures for the system of 7776 atoms. Histograms shown by dashed (red) and solid (black) lines result from initial temperatures of *T* _{i} = 15 800 and 16 000 K, respectively, the quasi-steady solid and liquid temperatures in the two cases being *T* _{sol} = 7640 and 7740 K and *T* _{liq} = 6410 and 6505 K. Dashed and dotted curves show exponential functions fitted to histograms (see text).

Dependence of mean waiting time 〈τ_{w}〉 on final liquid temperature *T* _{liq} for systems of *N* = 7776 (black circles), 976 (red squares), 392 (green diamonds), 150 (blue triangles) and 96 (brown stars) atoms. Quantity plotted is 〈τ_{w}〉^{−1/2} as function of *T* _{liq}. Straight lines are linear least-squares fits to the data for each *N* value.

Dependence of mean waiting time 〈τ_{w}〉 on final liquid temperature *T* _{liq} for systems of *N* = 7776 (black circles), 976 (red squares), 392 (green diamonds), 150 (blue triangles) and 96 (brown stars) atoms. Quantity plotted is 〈τ_{w}〉^{−1/2} as function of *T* _{liq}. Straight lines are linear least-squares fits to the data for each *N* value.

Scaling of mean waiting times 〈τ_{w}〉 with system size specified by number of atoms *N*. Quantity plotted is (*N*〈τ_{w}〉)^{−1/2} as function of final liquid temperature *T* _{liq}.

Scaling of mean waiting times 〈τ_{w}〉 with system size specified by number of atoms *N*. Quantity plotted is (*N*〈τ_{w}〉)^{−1/2} as function of final liquid temperature *T* _{liq}.

Alternation between solid and liquid states: temperature as function of time in one of the constant-energy MD simulations on the system of 96 atoms, with total energy such that mean liquid-state temperature is *T* _{liq} = 6760 K, showing alternation between mean temperatures *T* _{sol} and *T* _{liq}.

Alternation between solid and liquid states: temperature as function of time in one of the constant-energy MD simulations on the system of 96 atoms, with total energy such that mean liquid-state temperature is *T* _{liq} = 6760 K, showing alternation between mean temperatures *T* _{sol} and *T* _{liq}.

Histograms of temperature distribution at different constant total energies *E* in the system of 96 atoms. The histogram at each *E* was obtained by sampling over typically 128 simulations, each having a typical duration of 5 ns. Instead of giving *E* directly, we specify each histogram by the liquid-state temperature *T* _{liq}. Histograms shown by solid (black), dashed (red), dotted (green), chain (blue), and dotted-chain (black) lines are for *T* _{liq} = 6935, 6760, 6590, 6565, and 6473 K.

Histograms of temperature distribution at different constant total energies *E* in the system of 96 atoms. The histogram at each *E* was obtained by sampling over typically 128 simulations, each having a typical duration of 5 ns. Instead of giving *E* directly, we specify each histogram by the liquid-state temperature *T* _{liq}. Histograms shown by solid (black), dashed (red), dotted (green), chain (blue), and dotted-chain (black) lines are for *T* _{liq} = 6935, 6760, 6590, 6565, and 6473 K.

Fraction of time spent by the system in the liquid state for different liquid-state temperatures *T* _{liq} in simulations of 96-atom system.

Fraction of time spent by the system in the liquid state for different liquid-state temperatures *T* _{liq} in simulations of 96-atom system.

Z plot from a sequence of constant-energy AIMD simulations on system of 150 atoms, duration of simulations being 50 ps. Black filled circles with error bars show final mean temperature and pressure, with upper-left branch corresponding to energies for which the system remains solid, and lower right branch to energies for which homogeneous melting occurs. Dashed (red) line shows the *ab initio* melting curve obtained in earlier work using the free energy technique, and green filled square with error bar shows point on *ab initio* melting curve obtained with *ab initio* MD simulations on a system of 980 atoms containing coexisting solid and liquid.

Z plot from a sequence of constant-energy AIMD simulations on system of 150 atoms, duration of simulations being 50 ps. Black filled circles with error bars show final mean temperature and pressure, with upper-left branch corresponding to energies for which the system remains solid, and lower right branch to energies for which homogeneous melting occurs. Dashed (red) line shows the *ab initio* melting curve obtained in earlier work using the free energy technique, and green filled square with error bar shows point on *ab initio* melting curve obtained with *ab initio* MD simulations on a system of 980 atoms containing coexisting solid and liquid.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content