7156
J. Phys. Chem. 1988, 92, 7 156-7 160
The crossing between the P'B- and P'H- levels at large negative fields for the sequential mechanism predicts the unveiling of the B- intermediate in high external fields, which is, of course, unique for this sequential mechanism. The level crossing between the
P*B and P'B- levels at negative fields predicts the asymmetric field-dependent low-temperature retardation of the charge separation for the nonadiabatic/adiabatic mechanism. These qualitative predictions have to be subjected to experimental scrutiny.
Vapor-Liquid Equilibria in Flukls of Two-Center Lennard-Jones Molecules Sumnesh Gupta Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803- 7303 (Received: September 17, 1987; In Final Form: July 14, 1988)
Molecular dynamics simulationsare performed using 500 molecules for pure fluids of twecenter Lennard-Jones (126) molecules to obtain thermodynamic results, including the residual Helmholtz free energy, for several isotherms. Thermodynamic results obtained from these simulations are utilized to predict the vapor-liquid equilibria in these fluids. Simulation results have also been used to estimate the critical temperatures and critical densities of these fluids, and these are compared with the existing predictions from three theoretical methods. These comparisons show that the site-site Ornstein-Zernike equation, with the Percus-Yevick approximation, for these fluids overpredicts the critical temperatures, and a nonspherical reference potential based perturbation theory also shows small deviations whereas an approximate form of the zeroth-order Mayer function expansion cluster perturbation theory works well. These three methods predict well the critical densities of these fluids, within the combined uncertainties of the results.
Introduction
Sitesite potential models can be very useful in modeling fluids of nonspherical molecules.' Interaction sites in these models are usually considered to be spherical: and these represent either the chemical groups such as CH3 or the individual atoms such as N, Br, etc. The simplest of these site-site models is the two-center potential model with Lennard-Jones (12:6) spheres as interaction sites. Several computer simulations of pure dense fluids using this potential have been reportedw for thermodynamic properties and microscopic structure. Kabadi and Steele'O have also recently reported studies of translational and rotational dynamics in fluids of these two-center Lennard-Jones (12:6) molecules. Simulations involving mixtures have also been reported."-15 Fluids modeled by the two-center Lennard-Jones potential have also been studied through the integro-differential distribution function theories'*2including the recent work of Monson,16 who has studied the effect of molecular elongation on critical properties. Perturbation theories have also been applied.',2J7-19 The nonspherical reference based perturbation expansions by Fischer17 and also by Kohler et a1.l8 in the center frameI9 have worked well. A similar expansion by Quirke and TildesleyIg in the site frame also works well. These methods have also been extended to mixtures.20+21In spite of all this, simulation data on fluids modeled ( I ) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon: Oxford, 1984. (2) Streett, W. B.; Gubbins, K. E. Annu. Reu. Phys. Chem. 1977, 28, 373. (3) Streett, W. B.;Tildesley, D. J. Proc. R. Soc. London, A 1977,355,239. (4) Singer, K.; Taylor, A.; Singer, J. V. L. Mol. Phys. 1977, 33, 1757. (5) Cheung, P. S . Y.; Powles, J. G. Mol. Phys. 1975, 30, 921. (6) Romano, S.; Singer, K.Mol. Phys. 1979, 37, 1765. (7) Guillot, B.; Guissani, Y. Mol. Phys. 1985, 54, 455. (8) Barojas, J.; Levesque, D.; Quentrec, B. Phys. Reu. A. 1973, 7, 1092. (9) Wojcik, M.; Gubbins, K. E.; Powles, J. G. Mol. Phys. 1982, 45, 1209. (IO) Kabadi, V. N.; Steele, W. A. J. Phys. Chem. 1985, 89, 1467. (1 I ) Gupta, S.; Coon, J. E. Mol. Phys. 1986, 56, 1049. (12) Gupta, S . Fluid Phase Equilib. 1986, 31, 221. (13) Coon, J. E.; Gupta, S.; McLaughlin, E. Chem. Phys. 1987, 113.43. (14) Fincham, D.; Quirke, N.; Tildesley, D. J. J . Chem. Phys. 1986, 84, 4535. (15) Tildesley, D. J.; Enciso, E.; Sevilla, P. Chem. Phys. Lett. 1985, 100, 508.
(16) Monson, P. A. Mol. Phys. 1984, 53, 1209. (17) Fischer. J. J . Chem. Phys. 1980, 72, 5371. ( 1 8) Kohler, F.; Quirke, N.; Perram, J. W. J. Chem. Phys. 1979, 71,4128. (19) Quirke, N.; Tildesley, D. J. J . Phys. Chem. 1983, 87, 1972.
0022-3654/88/2092-7156$01.50/0
by using the two-center Lennard-Jones potential is considered to be limited even for pure l i q ~ i d s ,especially ~ ~ , ~ ~ for the residual Helmholtz free energy. Here we report systematic NVT molecular dynamics simulations to obtain thermodynamic properties of fluids modeled using the two-center Lennard-Jones (12:6) potential. These results have also been used to predict the vapor-liquid equilibria in these fluids. In this regard, it may be worthwhile to point out that P o w l e ~ ~ ~ has already shown that thermodynamic results from simulations can be used to reasonably predict vapor-liquid equilibria in these fluids. Simulation results have also been used to estimate the critical densities and temperatures in these fluids which in turn are compared with earlier theoretical predictions. Simulation Methodology
For a pair of homonuclear molecules, 1 and 2, the two-center Lennard-Jones ( 1 2:6) potential model is given by'*2,25
Here w land w2 are the molecular orientations, i is an interaction site on molecule 1, j is an interaction site on molecule 2, r is the distance between the molecular centers, and t and u are the Lennard-Jones (12:6) interaction parameters.'v2 The site-site distances, rij, depend not only upon r, wl,and w2 but also upon the molecular elongation, I, which is a constant for rigid molecules. Thus, interaction between a pair of two-center Lennard-Jones (12:6) molecules usually is specified by t, u, and I* = l / u . Molecular dynamics simulations in the NVT ensemble have been performed for fluids of two elongation ratios, I* = 0.3292 (20) Fischer, J.; Lago, S . J. Chem. Phys. 1983, 78, 5750. (21) Enciso, E.; Lombardero, M. Mol. Phys. 1981, 44, 725. (22) Abascal, J. L. F.; Martin, C.; Lombardero, M.; Vazquez, J.; Baiion, A.; Santamaria, J. J. Chem. Phys. 1985. 82, 2445. (23) MacGowan, D.; Waisman, E. M.; Lebowitz, J. L.; Percus, J. K. J. Chem. Phys. 1984,80, 2719. (24) Powles, J. G. Mol. Phys. 1980, 41, 715. (25) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A. Intermolecular Forces: Their Origin and Determination; Clarendon: Oxford, 1981.
0 1988 American Chemical Society
Two-Center Lennard-Jones Molecules
The Journal of Physical Chemistry, Vol. 92, No. 25, 1988 7157
TABLE I: Second ( B ) and Third (C) Virial Coefficients for Two-Center Lennard-Jones Molecules from Simulation and Second Virial Coefficients, B *, Obtained from the f-Function Integration‘ I* T* B* C* B*/ 0.3292 0.3292 0.3292 0.3292 0.3292
2.9 3.2 3.3 3.4
3.5
-6.19 -5.20 -4.94 -4.66 -4.40
0.67 0.67 0.67 0.67 0.67
1.5 2.0 2.1 2.2 2.3
-14.6 -7.82 -7.15 -6.43 -5.71
7.67 7.00 6.87 6.40 6.30
TABLE II: Saturation Properties of Two-Center Lennard-Jones (126) Fluids for I * = 0.3292 and 0.67
T*
1 = //a, T* = k T / c , B* = Bfa’, C* = C/&
2.1 2.9 3.2 3.3 3.4
0.622 0.510 0.430 0.411 0.392
f 0.003 f 0.003 f 0.003 f 0.003 f 0.003
1.0 1.5 2.0 2.1 2.2
0.532 0.445 0.352 0.321 0.291
f 0.003 i 0.003 0.014 f 0.005 f 0.003 0.036 f 0.005 f 0.003 0.060 f 0.005 f 0.003 0.081 f 0.005
‘G/NkT = Z
and 0.67. The former represents nitrogen- and oxygen-like molecules, and the latter represents ethane-like molecules. All these simulations have been performed with systems of 500 molecules, and the potential was truncated at the site-site distance of rCut= 2 . 8 ~ .Long-range corrections to pressure and internal energy results were applied according to the method of Singer et al.4 for the site-site potential truncation. Simulations were performed at high densities for 10 000 time steps following an equilibration period of 2000 time steps. At lower densities, the system was allowed to equilibrate for 3000 time steps followed by production runs of 12000 time steps. Time steps of At* = 0.0015, t* = t/(ma2/c)1/2, were used. Results are reported in Tables IV and V. We refer the reader to our earlier papers (ref 11 and 12) for more details on our simulation methodology. Simulations were performed in such a manner that we could also calculate the residual Helmholtz free energy for the liquidphase region via the method of thermodynamic integration. Using experimental PVT data for nitrogen26 and ethanez7 as a guide along with the known potential parameters for these fluids, we estimated the temperatures, in reduced units, a t which no vapor-liquid equilibrium exists. Tests runs were then carried out on a CRAY-XMP, using 44-bit mantissa calculations, to find suitable temperatures at which simulations could be carried out for these fluids without going through the instability region for the complete isotherm. We found T* = 4.0 for I* = 0.3292 and T* = 2.5 for
