7980
J. Phys. Chem. C 2007, 111, 7980-7985
Melting Point Determination from Solid-Liquid Coexistence Initiated by Surface Melting
Ali Siavosh-Haghighi and Donald L. Thompson*
Department of Chemistry, UniVersity of Missouri, Columbia, Missouri 65211
ReceiVed: January 11, 2007; In Final Form: March 22, 2007
A coexisting solid-liquid (s-l) system of nitromethane is created by surface-induced melting. A nitromethane
crystal with a free surface is simulated by molecular dynamics (MD) in the constant-volume and -energy
(NVE) ensemble for initial conditions generated by short MD simulations of the constant-volume and
-temperature (NVT) ensemble at temperatures slightly above the melting point. Melting starts at the surface,
initiating a solid-liquid interface, and the temperature drops as the system moves toward a state of equilibrium
in which the solid and liquid phases coexist. The temperature at which the coexisting solid and liquid reach
equilibrium is taken to be the melting point. The melting points of crystals with exposed (100), (010), and
(001) crystallographic faces are predicted to be 238, 245, and 242 K, respectively. The predicted melting
points are in good agreement with experiment (244.7 K) and previous simulations. The approach to equilibrium
during the NVE simulation is monitored by calculating the orientational order parameter, diffusion coefficient,
and density, which provide insights into the melting mechanism. The Sorescu-Rice-Thompson [J. Phys.
Chem. B 2000, 104, 8406] force field, which accurately describes the inter- and intramolecular motions, was
used.
1. Introduction
The melting point of a perfect crystal is higher than the
thermodynamic melting point as result of the free energy barrier
to the formation of a solid-liquid (s-l) interface. A straightforward way to model thermodynamic melting is to initiate a
constant volume and energy (NVE) simulation with the solid
in contact with the liquid. The solid and the liquid portions of
the system are separately prepared by performing simulations
near an estimate of the melting temperature; then the two are
joined in a single simulation with periodic boundary conditions.1,2 If the energy is too low there will be a net l f s change
and if it is too high then s f l will dominate. The temperature
and pressure change as the coexisting solid-liquid system comes
to dynamic equilibrium. While practical, this method requires
some effort to determine the conditions for equilibrium of the
coexisting solid-liquid system.
In the present study we have applied a method originally
developed by Ercoslessi et al.3,4 This approach is more
straightforward than the standard coexistence method1,2 because it avoids the separate preparations and then matching of solid and liquid simulation supercells. The basic idea is
to initiate molecular dynamics (MD) simulations for the
constant-volume-energy (NVE) ensemble for a solid in contact
with vacuum at a temperature slightly above the melting point.
Melting will begin at the free surface, forming a solid-liquid
interface which progresses inward. The temperature decreases
as a function of time as the melting progresses. If the initial
temperature is relatively close to the melting temperature, the
temperature of the system will asymptotically approach the
melting point Tmp.The temperature T, computed over the time
* Corresponding author.
of the simulation when both solid and liquid are present, can
be fit to
T ) T0 + ae-bt
(1)
where t is the simulation time, T0 is the asymptotic value of the
temperature, and a is a constant equal to 1 K. If a true s-l
interface is created T0 is an estimate of Tmp.
We have used this method to determine the melting point of
crystalline nitromethane with exposed surfaces (100), (010), and
(001), and to investigate the molecular-level mechanism of
melting. The NVE simulations provide more realistic results
for the time evolution of a melting system than do simulations
in which constraints must be used to maintain the ensemble
conditions, because applying a thermostat or barostat in a
simulation of the NPT ensemble affects the dynamics of the
system. We have monitored the molecular orientations and
mobility as melting occurs in the NVE simulations to determine
the details of the melting mechanism for a molecular solid.
2. Methods
The initial configuration of the crystal structure was based
on the neutron diffraction scattering data reported by Trevino
and Rymes.5 The simulations were performed using the
DL_POLY (v. 2.15) code.6 Periodic boundary conditions were
applied in all directions, thus two exposed surfaces are necessary
as illustrated in Figure 1. The integration step size was 0.75 fs,
the cutoff range for the potential 11 Å, and the verlet shell
0.5 Å. Ewald sum method was used to calculate electrostatic
interactions in periodic system. Equilibration in the NVE runs
was determined by monitoring the density, orientational order
parameter, diffusion coefficient, and temperature as functions
of time; these quantities were computed every 500th integration
10.1021/jp070242m CCC: $37.00 © 2007 American Chemical Society
Published on Web 05/15/2007
Melting Point Determination from s-l Coexistence
J. Phys. Chem. C, Vol. 111, No. 22, 2007 7981
Figure 1. Top panel: initial configuration of the simulation cell with the (100) surface exposed to vacuum. Bottom panel: illustration of the
rectangular fixed-volume slabs (defined by the dashed vertical lines) approximately one-molecular size in width (2.56 Å) for which quantities were
computed to monitor the progression of the melting through the crystal. Because of the symmetry of the simulation cell, the properties of the
molecules in slabs n and n′ were combined to compute averages of quantities as functions of the distance from the exposed surface or the center
of the simulated crystal.
TABLE 1: Simulations
NVT simulation
run
1
2
3
4
5
6
7
8
9
10
11
simulated crystal
supercell (Å)a
exposed
surface
10 × 4 × 3
157.32 × 25.28 × 26.19
10 × 4 × 3
157.32 × 25.28 × 26.19
10 × 4 × 3
157.32 × 25.28 × 26.19
10 × 4 × 3
157.32 × 25.28 × 26.19
10 × 4 × 3
157.32× 25.28 × 26.19
10 × 4 × 3
157.32 × 25.28 × 26.19
5×8×3
26.22 × 151.68 × 26.19
5×4×6
26.22 × 25.28 × 157.14
5×4×3
78.66 × 25.25 × 26.19
5×4×3
26.22 × 75.84 × 26.19
5×4×3
26.22 × 25.25 × 78.75
NVE simulation
time
(ps)
T
(K)
Eb
time
(ps)
T0
(K)c
100
15
255
-23.248
1700
233 ( 6
100
15
260
-23.235
1700
238 ( 6
100
15
265
-23.229
1700
237 ( 8
100
15
270
-23.221
1700
238 ( 8
100
15
276
-23.214
1700
240 ( 8
100
15
280
-23.202
1700
242 ( 10
010
15
270
-23.206
1700
245 ( 8
001
15
270
-23.210
1700
242 ( 10
100
3.75
260
-23.174
900
237 ( 15
010
3.75
260
-23.195
900
245 ( 10
001
3.75
260
-23.203
900
241 ( 11
a
Crystal dimensions (unit cells) and simulation box size. b The energy zero is for isolated molecules at 0 K. Units: kJ mol-1 molecule-1. c The
uncertainties are the root-mean-square fluctuations.
step. The density, orientational order parameter, and diffusion
coefficient were calculated within rectangular fixed-volume
slices approximately one-molecular size in width (2.56 Å) as
illustrated in Figure 1. Because of the symmetry of the
simulation cell, the properties of the molecules in slabs n and
n′ were combined to compute averages of quantities as functions
of the distance from the exposed surface or the center of the
simulated crystal.
The simulations that were performed are summarized in
Table 1. In one series of simulations a large supercell (10 ×
4 × 3 unit cells) was used to ensure sufficient time for the
formation of a true solid-liquid interface. These simulations
7982 J. Phys. Chem. C, Vol. 111, No. 22, 2007
Siavosh-Haghighi and Thompson
Figure 2. Plot of the temperature as a function of time calculated for
NVE simulations of supercells of dimensions 10 × 4 × 3 unit cells
for initial conditions prepared by running short NVT simulations at
260 and 280 K. The temperature approaches the melting point as the
system approaches equilibrium.
were done for the (100) crystallographic face exposed to
vacuum. Another series of simulations were done using smaller
supercells. Melting initiated at the (010) face was simulated
using a supercell of dimensions 5 × 8 × 3 and for the (001)
exposed surface a 5 × 4 × 6 supercell was used. The predicted
melting points are statistically the same for the various sizes of
simulation supercells. The larger supercell allowed for a more
detailed analysis of the mechanism.
The simulation supercell was constructed by placing the
crystal in an empty larger orthorhombic box; see column 2 of
Table 1. Conditions corresponding to 1 atm and temperatures
255, 260, 265, 270, 276, and 280 K were assigned to the atoms,
and short NVT simulations were run to obtain initial conditions
for the NVE simulations; see column 4 and 5 of Table 1. The
NVE simulations were run until equilibration was achieved,
generally in less than 1700 ps; see column 6 of Table 1. The
total energies for the NVE runs are given in column 7 of
Table 1. The temperature was calculated as a function of the
simulation time and fit to eq 1. The values obtained for T0 are
given in the last column of Table 1 with root-mean-square
fluctuations in the temperature in the NVE simulations; these
results will be discussed in the next section.
The density, orientational order parameter, and diffusion
coefficient were computed to determine the state of the system
during the simulations. The orientational order parameter is
defined as the ensemble average:
〈|cos θ|〉 )
1
N
∑i |ei‚k|
(2)
where ei is the unit vector along C-N bond and k is the unit
vector along (i.e., along the [001] direction) of the unit cell.
The angle between the C-N bond and edge c is close to 0°
and 180° in the crystal; thus, 〈|cos θ|〉 ) 0.93 in the crystal and
decreases as the crystalline order diminishes. The diffusion
coefficient D was calculated using
〈∆r2〉 ) 6Dt
(3)
where r is the position vector of a molecule relative to the center
of mass of the system and t is the time.
The density was calculated in each of the slices as defined
in Figure 1, except for slices 10 and 10', by considering a
molecule to be within the slice in which its center of mass was
Figure 3. The (a) orientational order parameter and (b) density as
functions of time computed for all the molecules in NVE simulations
of supercells of dimensions 10 × 4 × 3 unit cells for initial conditions
corresponding to 260 K (blue curve), and 280 K (red curve). (c) Extent
of melting in simulations with initial temperatures 260 K (blue curve)
and 280 K (red curve) illustrated by the percent solid as a function of
time.
located. The average density of whole slab was obtained by
averaging the densities of all slices.
The force field is that developed by Sorescu et al.7 It has
been shown to accurately predict the solid- and liquid-state
properties of nitromethane.8,9 It was used with minor modifications by Agrawal et al. 10 to study the melting of nitromethane,
giving results in good agreement with experiment. The intermolecular forces are described by the Buckingham (exp -6)
potential plus Coulombic interactions with fixed partial charges.
The intramolecular interactions are represented by a sum of
Morse potentials for the bond stretches, harmonic oscillators
for the bond angles, and cosine sum for the torsion angles. A
complete description of the force field7 and values of the
constants10 are available elsewhere.
3. Results and Discussion
The density, order parameter, extent of melting, diffusion
coefficient, and temperature as functions of time were computed
in NVE simulations of crystalline nitromethane for energies
corresponding to initial temperatures over the range 255-
Melting Point Determination from s-l Coexistence
J. Phys. Chem. C, Vol. 111, No. 22, 2007 7983
Figure 4. Fits of NVE simulation results to eq 1 for initial conditions
corresponding to 255, 260, 265, 270, 276, and 280 K (see column 5 of
Table 1 for corresponding E values) for the 10 × 4 × 3 supercell. The
values of T0, which we take to be the melting point, for the range of
temperature between 260 and 280 K are in the range 237.1-241.8 K;
the uncertainties are in the range 5.7-10.1 K for NVE; thus, the T0
values are statistically the same. The initial temperature 255 K is too
close to actual melting point for a true solid-liquid interface to be
established; thus, it results in a quasi-liquid layer that does not have
the properties of liquid nitromethane. The quasi-liquid surface is
influenced by the immediate solid phase.
280 K; see columns 6 and 7 of Table 1. These quantities were
used to determine the state of the system. The temperature at
which the equilibrium between the solid and liquid was achieved
is taken to be the melting point. Analyses of the behavior of
the order parameter and diffusion coefficient were used to study
the mechanism of melting.
The temperature as a function of time during NVE simulations
of supercells of dimension 10 × 4 × 3 with initial conditions
prepared by running short NVT simulations at 260 and 280 K
are shown in Figure 2. The density, orientational order
parameter, and percent solid as functions of time for NVE
simulations initially equilibrated at 260 and 280 K are shown
in frames a, b, and c, respectively, of Figure 3. The order
parameter, given by eq 2, is 0.93 for the crystal, and decreases
as the melting progresses (see Figure 3a). It levels off at an
equilibrium value, which is determined by the extent of melting.
The magnitude of the order parameter for liquid nitromethane,
calculated in a separate simulation, is 0.78. The state of the
system can also be determined by monitoring the density. The
average density as a function of time is shown in Figure 3(b)
for initial temperatures 260 K (blue curve) and 280 K (red
curve). To ascertain that a true liquid phase is formed we
calculated the density of the melted portion; at 239 K the density
of the liquid portion is 1.13 ( 0.12 g/cm3, which is essentially
the value (1.17 g/cm3) calculated by Sorescu et al.9 These results
show the same behavior as the order parameter. Because there
is incomplete melting at equilibrium, the final equilibrium values
shown in Figure 3b are more than that of liquid nitromethane,
which for this force field at 240 K is 0.0118 Å-3.9 The extent
of melting is shown in Figure 3c. The percent solid is determined
by characterizing the solid layers (as defined in Figure 1) as
either solid or liquid based on the order parameter. The fraction
of the system that remains solid at the end of the simulations
range from about 37% to 79%.
The temperature as a function of time in the NVE simulations
for initial temperatures over the range 255-280 K (see column
5 of Table 1) were fit to eq 1; the resulting curves are shown in
Figure 4 and the asymptotes T0 are given in the last column of
Table 1. If the initial temperature is too low the result is the
formation of an amorphous surface layer, not a solid-liquid
Figure 5. (a) The temperature as a function of time for the NVE
simulations of supercells of dimensions 10 × 4 × 3 unit cells for initial
conditions corresponding to 270 K for exposed face (100), 5 × 8 × 3
unit cells for exposed surface (010), and 5 × 4 × 6 unit cells for
exposed face (001). (b) Fits of the results in panel a to eq 1. The
temperature decreases faster for the crystal with the (100) free surface.
The rates of decrease in temperature for the (010) and (001) exposed
surfaces are similar and result in higher asymptotic values, 244.7 and
242.5 K, respectively, compared to that (237.7 K) for the exposed (100)
face.
interface. For instance, the diffusion coefficient for the outermost
layer of the slab initially equilibrated at 255 K is equal to
0.03 Å2/ps, much lower than that of liquid (∼0.1 Å2/ps).14 The
NVE simulation for initial temperature 255 K does not result
in a solid-liquid equilibrium, rather in a solid with a disorder
surface. The NVE simulations for the other initial temperatures
reach equilibrium temperatures in the range 237-242 K, all
within the range of uncertainties thus statistically the same. The
predicted melting point is thus in the range 237-242 K. We
did not observe any evaporation in the simulations. The pressure
remained close to zero; always less than the fluctuation. For
example, the pressure of the equilibrated system for the highest
initial temperature studied, 280 K, was calculated to be 0.22 (
0.37 kbar.
The values of the melting point predicted by this method are
14-19 K lower than the value, 256 K, obtained by the standard
solid-liquid coexistence method used by Agrawal et al.10 The
pressure was 1 atm in the coexisting solid-liquid simulations
performed by Agrawal et al., while in current study there is no
external pressure applied to the simulation cell. We believe that
the solid and liquid phases are not very sensitive to variations
of 0 to 1 atm of pressure; thus, we would assume the differences
in the computed melting points obtained by the different
approaches should be comparable except for the differences
resulting from the types of ensembles. Thus, we ascribe the
difference to the differences in the thermodynamic averaging.
Agrawal et al. determined the melting point by direct simulation
of the solid-liquid coexistence system in the NVT ensemble
at various temperatures. The calculation of a thermodynamic
quantity requires long simulations (ergodicity theorem).11
Furthermore, the slow rate of energy transfer from the liquid to
7984 J. Phys. Chem. C, Vol. 111, No. 22, 2007
Siavosh-Haghighi and Thompson
Figure 6. The order parameter, density, and diffusion coefficient as functions of time for the core (left column) and the surface (right column) for
NVE simulations of 5 × 4 × 3 crystals of nitromethane with exposed (100) surfaces for initial conditions corresponding to 260 K. The core is
approximately one unit cell in width, slices 1 and 1′ in Figure 1, and surface layer is defined by 5 and 5′ in Figure 1; thus, the average of the surface
layer is effectively over one-half the width of a unit cell. The dashed lines indicate the values of the quantities calculated in separate liquid-state
simulations. Orientational disorder of the molecules precedes translational freedom.
the solid phase in NVE simulations requires long simulation
times (on the order of nanoseconds).12,13 The length of the
solid-liquid coexistence simulation performed by Agrawal et
al. was 225 ps., while the trajectories in the present study were
integrated for 1700 ps, and the temperature versus time curves
were extrapolated to infinity to determine the temperatures at
equilibrium. Also, we note that in another study14 of nitromethane using the same force field we calculated the melting
point to be 251 K using 1200 ps simulations of the NVT
ensemble, which is in close agreement with current results. In
an earlier study,14. we calculated 239 ( 5 K for the melting
point by fitting the thickness of the melted layer of a slab of
nitromethane computed in an NVT simulations at different
temperatures to the equation l(T) ) K(Tmp - T)-1/3, which was
suggested by Mori et al.15 and Pluis et al.16 for van der Waals
solids. The present results are consistent with that result.
Simulations were performed for exposed (100), (010), and
(001) crystallographic surfaces using supercells of dimensions
10 × 4 × 3, 5 × 8 × 3, and 5 × 4 × 6 unit cells, respectively,
for initial conditions corresponding to 270 K. Figure 5a shows
the changes in the temperature as functions of time for the three
cases and the curves obtained by fitting the results to eq 1 are
shown in Figure 5b. The rate of decrease in the temperature is
faster for the melting initiated at the (100) free surface. Taking
the values of T0 in eq 1 to be the melting points predicts 238,
245, and 242 K, respectively, for the (100), (010), and (001)
cases. The present results are in accord with results of an earlier
study,14 using a different approach, in which we calculated
values of 251, 266, and 266 K, respectively, for the melting
points of (100), (010), and (001) surface-initiated melting.
Figure 6 shows the variation of the order parameter, density,
and diffusion coefficient of the core (left column) and the surface
(right column) calculated in a NVE simulation for the total
energy corresponding to 260 K of a 5 × 4 × 3 crystal with the
(100) face exposed. There was complete melting in the case of
the exposed (100) surface, however, the predicted melting point
is still a good estimate. The dashed lines in each plot indicate
the value of the quantity for the liquid state, which we calculated
in separate single-phase simulations of liquid nitromethane.
Orientational disordering of the molecules precedes mobility.
The translational freedom of molecules was determined by
calculating the diffusion coefficient using eq 3 within the volume
elements (illustrated in Figure 1); the results are shown in Figure
6. The diffusion coefficient was averaged over 18.75 ps. While
orientational disorder of molecules at the core reaches the liquidstate value within about 550 ps, the values of the density and
diffusion coefficient at the core do not reach the liquid-state
values until about 620 ps. The time gap between onsets of
molecular orientational disorder and translational freedom at
the surface is even larger. The orientational disorder reaches
that of the liquid state at about 70 ps, while translational freedom
is not reached until about 160 ps. Since the surface molecules
are coordinated to fewer neighboring molecules than those in
the core, the energy barriers to reorientation are lower.
Melting Point Determination from s-l Coexistence
J. Phys. Chem. C, Vol. 111, No. 22, 2007 7985
4. Conclusions
References and Notes
We have tested a simple and straightforward method suggested by Ercoslessi et al.3,4 for calculating melting points for
molecular solids. This approach is more straightforward than
the standard coexistence method1,2 because it avoids the separate
preparations and then matching of solid and liquid simulation
supercells. A coexisting nitromethane solid-liquid (s-l) equilibrium is created by surface-induced melting. A crystal with a
free surface is simulated by MD in the constant-volume-energy
(NVE) ensemble for initial conditions generated by short MD
simulations of the constant-volume-temperature (NVT) ensemble
at temperatures slightly above the melting point. In the NVE
simulation melting begins at the surface and the temperature
decreases as the system moves toward the solid-liquid coexistence equilibrium. The temperature at which the coexisting solid
and liquid are at equilibrium is taken to be the melting point.
The computed melting points are 238, 245, and 243 K,
respectively, for the crystals with exposed (100), (010), and
(001) crystallographic faces. The predicted melting points are
in good agreement with experiment17 (244.7 K) and previous
simulations.10,14,18 By monitoring the orientational order parameter, diffusion coefficient, and density as the system
approaches equilibrium during the NVE simulation insight into
the melting mechanism is obtained. The molecules first gain
rotational freedom followed by translational freedom.
(1) Morris, J. R.; Wang, C. Z.; Ho, K. M.; Chan, C. T. J. Phys. ReV.
B 1994, 49, 3109.
(2) Morris, J. R.; Song, X. J. Chem. Phys. 2002, 116, 9352.
(3) Ercolessi, F.; Tomagnini, O.; Iarlori, S.; Tosatti, E. In Nanosources
and Manipulation of Atoms Under High Fields and Temperature: Applications; Binh, V. T., Garcia, N., Dransfeld, D., Eds.; Nation-ASI Series E;
Kluwer: Dordrecht, The Netherlands, 1993; Vol. 235, p 185.
(4) For a review of the method and some applications, see: Di Tolla,
F. D.; Tosatti, E.; Ercolessi, F. In Monte Carlo and Molecular Dynamics
of Condensed Matter Systems, Binder, K., Ciccotti, G., Eds.; SIF: Bologna,
1996; Conference Proceedings Vol. 49, Chapter 14, p 346.
(5) Trevino, S. F.; Rymes, W. H. J. Chem. Phys. 1980, 73, 3001.
(6) Smith, W. F.; Leslie, M.; Forester, T. R. DL POLY 2.15; :
Daresbury Laboratory at Daresbury: Warrington, U.K., 2003.
(7) Sorescu, D. C.; Rice, B. M.; Thompson, D. L. J. Phys. Chem. B
1997, 101, 798.
(8) Sorescu, D. C.; Rice, B. M.; Thompson, D. L. J. Phys. Chem. B
2000, 104, 8406.
(9) Sorescu, D. C.; Rice, B. M.; Thompson, D. L. J. Phys. Chem. 2001,
105, 9336.
(10) Agrawal, P. M.; Rice, B. M.; Thompson, D. L. J. Chem. Phys.
2003, 119, 9617.
(11) Reichl, L. E. A Modern Course in Statistical Physics, 2nd ed.; John
Wiley & Sons: New York, 1997; p 296.
(12) Nada, H.; van der Eerden, J. P.; Furukawa, Y. J. Cryst. Growth
2004, 266, 297.
(13) Fernández, R. G.; Abascal, J. L. F.; Vega, C. J. Chem. Phys. 2004,
124, 144506.
(14) Siavosh-Haghighi, A.; Thompson, D. L. J. Chem. Phys. 2006, 125,
184711.
(15) Mori, H.; Okamoto, H.; Isa, S. Physica 1974, 73, 237.
(16) Pluis, B.; Denier, van der Gon, A. W.; Frenken, J. W. M.; van der
Veen, J. F. Phys. ReV. Lett. 1987, 59, 2678.
(17) Jones, W. M. and Giauque, W. F. J. Am. Chem. Soc. 1947, 69,
983.
(18) Zheng, L.; Luo, S.-N.; and Thompson, D. L. J. Chem. Phys. 2006,
124, 154504.
Acknowledgment. We thank Dr. Thomas D. Sewell for
reading and commenting on the manuscript. This work was
supported by a Multidisciplinary University Research Initiative
(MURI) grant managed by the U.S. Army Research Office.