J. Comput. Chem. Jpn., Vol. 13, No. 5, pp. 257–262 (2014)
©2014 Society of Computer Chemistry, Japan
General Paper
Phase Diagram for a Lennard-Jones System Obtained through Constant-Pressure Molecular Dynamics Simulations Yosuke Kataoka1)* and Yuri Yamada2) 1) Department of Chemical Science and Technology, Faculty of Bioscience and Applied Chemistry, Hosei University, 3-7-2 Kajino-cho, Koganei, Tokyo 184-8584, Japan 2) Division of Liberal Arts, School of Science and Engineering, Tokyo Denki University, Hatoyama, Hiki-gun, Saitama 350-0394, Japan *E-mail:
[email protected] (Received: July 14, 2014; Accepted for publication: October 27, 2014; Online publication: December 6, 2014) The phase diagram for a Lennard-Jones system was estimated using conventional NPH molecular dynamics (MD) simulations. Standard periodic boundaries were assumed for unit cells containing 1000 molecules. An elongated unit cell with both solid and vacuum sections was found to be suitable for the NPH MD simulations when calculating both the melting and vapor pressure curves under pressures lower than the critical one. Under high pressures, a unit cell with both solid and liquid sections was used as the initial configuration to obtain the melting temperature. The results of these simulations were compared with the phase transition point given by the reported equations of state. Keywords: Melting Curve, Vapor Pressure, Sublimation Pressure, Triple Point, Critical Point
1 Introduction
calculations on the chemical potential [5].
The phase diagram for a Lennard-Jones (LJ) system [1] was
The simple image of the present method is the heating curve
estimated using conventional NPT molecular dynamics (MD)
obtained through thermal analysis. A similar curve can be ob-
simulations in a previous study [2], in which standard cyclic
tained through NPH MD for the condensed phase when the
boundaries were assumed. An elongated unit cell with both
initial temperature is changed over a reasonable range under
solid and fluid sections was found to be suitable for the NPT
a constant pressure p. If the obtained temperature T is almost
MD simulations when calculating both the melting and vapor
constant as a function of the initial temperature T0, and the mo-
pressure curves. The phase boundary between the solid and the
lecular configuration has the characteristics of a coexistence
gas-phases, however, was difficult to accurately determine us-
state, then this state (T, p) may be assigned to a phase boundary. The initial configuration is selected according to the magni-
ing NPT MD at low pressures. For this reason, NPH MD [3–5] is carried out using elongated
tude of the pressure. The sublimation pressure curve is calcu-
unit cells with either solid and vacuum or solid and liquid sec-
lated using a very long cell with a large vacuum section. Under
tions with periodic boundaries to estimate the phase diagram for
a moderate pressure, the number density will be less than that at
a LJ system. The coexistence states are obtained near the phase
the critical point. An initial solid-liquid configuration is used to
boundaries [1]. The summarized phase diagrams are compared
obtain the melting curve under high pressure. As for the crystal
with those for the LJ system obtained using the equations of
surface structure, the (100) surface is used for simplicity. The
state (EOS) [6–11]. As NPH MD is a conventional computa-
other surface like the (111) surface is a problem to be studied
tional method, it is far simpler than carrying out sophisticated
in future.
DOI: 10.2477/jccj.2014-0016
257
In this work, the molecular interactions of the molecules, modeled as spheres, were based on the LJ potential [1], which is a function of the interatomic distance r and is given by the
Table 1. Lennard–Jones parameters for argon [12]. (ε/k)/K 111.84
ε/10−21 J 1.54
σ/10−10 m (ε/σ3)/MPa (ε/σ3)/atm 3.623 32.5 320
following equation: σ 12 σ 6 = u (r ) 4ε − , r r (1) where ε is the depth of the potential well and σ is the separation at which u(σ) = 0. The constants ε and σ have units of energy and length, respectively; numerical values for these constants are provided in Table 1 for argon [12].
2 Sublimation Under a very low pressure, a solid-gas coexistence state is expected. For this reason, the initial configuration shown in Figure 1 was used.
Figure 1. Initial configuration and example of final configuration under very low pressure. T0 = 1.76 ε/k, p = 4.7 x10−4 ε/σ3. The right-hand side is a close-up of the region of high density.
The central part of the unit cell has an FCC structure with a number density N/V = σ−3 in the initial state. Here, N is the number of molecules per unit cell. The volume of the system is denoted by V. The conditions for the present MD calculations are summarized in Table 2. Examples of the time-variation of the temperature T, the potential energy Ep and the enthalpy H are shown in Figure 2 under a pressure p of 4.7 × 10−4 ε/σ3. The arrow in the Figure separates the transient and solid-gas coexistent states. The obtained temperature is plotted as a function of the initial temperature T0 in Figure 3. The temperatures of the solid-gas and liquid-gas coexistent states are not exactly constant; however, their variations are smaller than those in the solid state
Figure 2. Time-variations of temperature T, potential energy Ep
where the temperature changes systematically. For this reason,
and enthalpy H under a pressure of 4.7 x 10−4 ε/σ3. T0 = 1.76 ε/k.
the coexistent temperatures are calculated as average values.
Table 2. Simulation conditions used in this study. N is the number of molecules in the unit cell and a and c are the dimensions of the unit cell; a rectangular, a = b < c. The cut off, the time increment dt and the total MD length are shown for the case of argon. The Hernandez method [13] is used for high temperature vaporization. # 1 2 3
target sublimation pressure melting & vaporization melting
ensemble = NPH
258
c/a unit cell 100 10 2.09 N=
initial configu-
number density
20000 20000 1000 integration
ration solid/vacuum solid/vacuum solid/liquid Gear and
(N/V)/σ−3, intial 0.010 0.100 1.091 MD time step
method
Hernandez
dt/fs = 1
cut off/Å
MD length/ps
5441 272 500 1000
J. Comput. Chem. Jpn.
Figure 4. Initial configuration and example of the final conFigure 3. Temperature of each state as function of initial temperature under pressure of 4.7x10−4 ε/σ3.
figuration under a pressure of 0.00312 ε/σ3. The central section of the middle Figure is enlarged on the right-hand side. T0 = 3.2 ε/k.
The main features of this Figure are as follows: The liquid state and the solid-liquid coexistence state are not present. The gas state is observed at very high initial temperatures (T0 > 4.4 ε/k). There are many large clusters in the gas state configurations. The initial kinetic energy is not enough to disperse the liquid structures into monomers. The temperature obtained in the gas state is lower than that in the liquid-gas coexistence state because of the large enthalpy of vaporization [1]. The average coexistence temperature of the solid-gas state TSG and the liquidgas state TLG satisfy the following inequality:
TSG ≥ TLG
(2)
Figure 5. Temperature and volume as functions of time. T0 = 3.2 ε/k, p = 0.00313 ε/σ3.
These features indicate that the phase boundary is determined by TSG. The liquid state that appears in the liquid-gas
In Figure 6, three stable phases (solid, liquid and gas states)
coexistence state is that of a supercooled liquid. Only the solid
are observed. The reason why the temperature of the gas state
and gas states are stable under this pressure.
decreases as T0 increases is due to the large vaporization en-
3 Melting and vaporization Under moderate pressures, melting and vaporization are expected. To investigate such conditions, the initial configuration shown in Figure 4 was used. An example of the final configuration for the case where the initial temperature was 3.2 ε/k is also depicted for a pressure of 0.00312 ε/σ3. This final configuration is assigned to the liquidgas coexistent state.
thalpy [1]. Figure 7 shows the temperature as a function of the initial temperature, plotted near the critical pressure. The transition state between the liquid and gas states is assigned to that shown by Figure 7, although the interface is not clear in the liquid-gas coexistence state.
4 Melting under high pressure In order to obtain the solid-liquid coexistent state under high
Figure 5 shows that the temperature is stable after a short
pressure, an initial configuration was used in which the solid
relaxation time. The volume is larger than that at the critical
and the liquid phases are combined as shown in Figure 8. An
point of argon [1].
example of the time-variation of the temperature is shown in
DOI: 10.2477/jccj.2014-0016
259
Figure 8. Initial, intermediate and final configurations. T0 = 2.62 ε/k, p = 7.11 ε/σ3. Figure 6. Average temperature as function of initial temperature. p = 0.00313 ε/σ3.
Figure 9. Temperature as function of time. The time variation of the volume V is also shown. T0 = 2.62 ε/k, p = 7.11 ε/σ3. Figure 7. Temperature as function of initial temperature. p = 0.118 ε/σ3.
Figure 9. In this Figure, the solid-liquid coexistent state persisted until the time indicated by the arrow. The T vs. T0 plot is shown in Figure 10 for a pressure of 7.11 ε/σ3. Only the solid, liquid and solid-liquid coexistent states are observed under this pressure, which should be higher than the critical pressure [1].
5 Phase diagram and triple point The phase diagram is obtained in the summary of the coexistence temperature as a function of pressure, as shown by the
Figure 10. Temperature as function of initial temperature.
circles in Figure 11. This is compared with the transition point
p = 7.11 ε/σ3.
determined by the published equations of state [6–11]. The agreement between the two is satisfactory. An enlarged view of
termined with a relatively high accuracy as shown in Table 3. On
the region around the triple point is shown in Figure 12.
the contrary, the triple point pressure is not easily determined.
The solid-liquid coexistence temperature curve is very steep in
It was estimated from the point at which the melting curve and
p-T space, meaning that the triple point temperature can be de-
the boiling point curve cross using the plot shown in Figure
260
J. Comput. Chem. Jpn.
Figure 11. Phase diagram for Lennard-Jones system obtained through NPH molecular dynamics simulations. The phase transition points given by the published equations of state [6–11] are also shown.
Figure 12. Phase diagram around triple point.
12. This crossing point is also close to the point at which the
the triple point temperature [20]. The 12–6 LJ function may not
melting curve and the sublimation curve cross. The estimated
always be suitable to express the low temperature properties
triple point pressure is shown in Table 3 and compared with that
of argon.
found in previous work [2,17]. The present result is consistent
Several additional Figures have been included to depict the
with free energy calculations [16–18]. The experimental results
states near the triple point. Figure 13 describes the initial and
on argon [1] are also compared with the calculated properties.
time dependent configurations under the triple point pressure.
The macroscopic triple point temperature of argon is higher
Figure 14 is an example of a temperature vs. time plot under the
than that in the LJ system by about 10%. The critical tempera-
triple point pressure. These Figures show the solid-gas coexis-
ture Tc is estimated by MD as follows [19].
tence state and the liquid state. The T vs. T0 plot for the triple
Tc= ( MD) 1.3207 ε /k = 147.71 K (3) Tc (exp) = 150.72K
In this comparison, the MD result with the LJ parameters in Table 1 is close enough to the experimental value on argon. Another LJ parameter set chosen such that both methods give the same critical temperature [19,20] gives a 6% difference in
point pressure is given in Figure 15. In the liquid-gas transition region, it is difficult to determine whether to assign the liquidgas coexistence state or the gas state, because the cell is too long and very narrow. Some examples of the initial configurations and animations are shown in the appendix of this journal.
Table 3. Comparison of the calculated temperature (Ttr), pressure (ptr), liquid density (ρL) and solid density (ρS) values at the triple point with published data. NPH MD NTP MD [2] NEV MD [2] Ladd & Woodcock [14] Hansen & Verlet [15] Agrawal & Kofke [16] Barroso & Ferreira [17] Ahmed & Sadus [18] Experiment [1]
DOI: 10.2477/jccj.2014-0016
Ttr/(ε/k) 0.683 0.686 0.683 0.67 0.68 0.687 0.692 0.661 0.749
ptr/(ε/σ3) 0.0012 0.001 0.0006 −0.47 0.0011 0.0012 0.0018 0.0021
rL/σ−3 0.845 0.845 0.847 0.818 0.850 0.850 0.847 0.864 1.030
rS/σ−3 0.960 0.960 0.961 0.963 0.960 0.962 0.978 1.147
261
starts with a rectangular unit cell containing both solid and vacuum/liquid sections depending on the pressure. The triple point may be estimated from NPH MD calculations using a rectangular unit cell that consists of a solid section in conjunction with a large vacuum portion. The authors would like to thank the Research Center for ComFigure 13. Initial and final configuration under triple point pressure. T0 = 1.78 ε/k.
puting and Multimedia Studies of Hosei University for the use of computer resources.
References
Figure 14. Temperature and volume as functions of time. T0 = 1.78 ε/k, p = 1.2x10−3 ε/σ3.
Figure 15. Temperature as function of initial temperature. p = 1.2x10−3 ε/σ3. In the liquid-gas transition region, the liquid-gas coexistence state was not assigned.
6 Conclusions A simple method for obtaining the phase diagram for a LJ system governed by NPH MD has been developed. This approach
262
[1] P. W. Atkins, Physical Chemistry, Oxford Univ. Press, Oxford (1998). [2] Y. Kataoka, Y. Yamada, J. Comput. Chem. Jpn., 13, 115 (2014). [CrossRef] [3] H. C. Andersen, J. Chem. Phys., 72, 2384 (1980). [CrossRef] [4] M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford (1992). [5] R. J. Sadus, Molecular Simulation of Fluids: Theory, Algorithms and Objective-Orientation, Elsevier, Amsterdam (1999). [6] J. K. Johnson, J. A. Zollweg, K. E. Gubbins, Mol. Phys., 78, 591 (1993). [CrossRef] [7] J. Kolafa, I. Nezbeda, Fluid Phase Equilib., 100, 1 (1994). [CrossRef] [8] Y. Tang, B. C.-Y. Lu, Fluid Phase Equilib., 165, 183 (1999). [CrossRef] [9] W. Okrasinski, M. I. Parra, F. Cuadros, Phys. Lett. A, 282, 36 (2001). [CrossRef] [10] M. A. van der Hoef, J. Chem. Phys., 113, 8142 (2000). [CrossRef] [11] M. A. van der Hoef, J. Chem. Phys., 117, 5092 (2002). [CrossRef] [12] F. Cuadros, I. Cachadina, W. Ahumada, Mol. Eng., 6, 319 (1996). [CrossRef] [13] E. Hernández, J. Chem. Phys., 115, 10282 (2001). [CrossRef] [14] A. J. Ladd, L. V. Woodcock, Mol. Phys., 36, 611 (1978). [CrossRef] [15] J.-P. Hansen, L. Verlet, Phys. Rev., 184, 151 (1969). [CrossRef] [16] R. Agrawal, D. A. Kofke, Mol. Phys., 85, 43 (1995). [CrossRef] [17] M. A. Barroso, A. L. Ferreira, J. Chem. Phys., 116, 7145 (2002). [CrossRef] [18] A. Ahmed, R. J. Sadus, J. Chem. Phys., 131, 174504 (2009). [Medline] [CrossRef] [19] H. Okumura, F. Yonezawa, J. Phys. Soc. Jpn., 70, 1990 (2001). [CrossRef] [20] Y. Kataoka, Y. Yamada, J. Comput. Chem. Jpn., 13, 146 (2014).
J. Comput. Chem. Jpn.
