Vapour-liquid equilibria for two centre Lennard-Jones diatomics and dipolar diatomics

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MOLECULAR PHYSICS, 1993, VOL. 80, No. 4, 997-1007 Vapour-liquid equilibria for two centre Lennard-Jones diatomics and dipolar diatomics By GIRIJA S. DUBEYt, SEAMUS F.

1 MOLECULAR PHYSICS, 1993, VOL. 80, No. 4, Vapour-liquid equilibria for two centre Lennard-Jones diatomics and dipolar diatomics By GIRIJA S. DUBEYt, SEAMUS F. O'SHEA Department of Chemistry, University of Lethbridge, Lethbridge, Alberta, Canada TIK 3M4 and PETER A. MONSON Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA (Received 4 January 1993; accepted 20 April 1993) Gibbs ensemble Monte Carlo simulations have been used to calculate vapour-liquid equilibria of diatomic and dipolar diatomic fluids. Dipolar two-centre Lennard-Jones diatomics with elongations L*= L/a between 0.33 and 1-00 have been studied for various values of dipole moments 1/~ ~--- p.d/(r 1/2. The results for phase equilibrium are in good agreement with recent theoretical predictions based on cluster-expansion perturbation theory. 1. Introduction A major advance in the use of molecular simulations in the study of vapourliquid equilibria has been the use of the Gibbs ensemble [1,2] method. This permits the direct simulation of bulk vapour and liquid phases satisfying the conditions of phase equilibrium. Gibbs ensemble Monte Carlo simulations have now been performed for a number of potential models, including pure Lennard-Jones fluids and mixtures [3-5], Stockmayer fluids and their mixtures, square-well fluids and polyatomic fluids and mixtures [6-9], the symmetrical non-additive hard-sphere system [10], the hard-core two-yukawa fluid [11], the Lennard-Jones fluids with a quadrupole interaction [12,13], hard-core Yukawa fluids [14], the non-spherical Gay- Berne fluid [15], square-well diatomics [16], more realistic models for alkanes and water [17,18], and hard ellipsoids [19]. The purpose of the present work is to apply the Gibbs ensemble method to some interaction site models of diatomic molecules. The models are two-centre Lennard- Jones potentials with point charge dipoles. These models can be used to investigate the influence of molecular shape and polarity upon vapour-liquid equilibria. Although the dipolar diatomic systems studied do not correspond directly to real polar diatomic fluids which would, of course, have heteronuclear repulsive cores, the parameters considered correspond roughly to moderately polar fluids, such as methyl chloride. Our study also affords the opportunity to test the accuracy of some recent theoretical predictions [20,21] obtained using a cluster-expansion perturbation theory developed for interaction site fluids [22]. This theory addresses the calculation t Present address: Department of Chemistry, New York University, New York, N.Y , USA /93 $ Taylor & Francis Ltd.

_ (~/~v2) 998 G.S. Dubey et al. of the contribution to the free energy arising from long-ranged contributions to the site-site potential, such as dispersion or Coulombic interactions.

2 _ (~/~v2) 998 G.S. Dubey et al. of the contribution to the free energy arising from long-ranged contributions to the site-site potential, such as dispersion or Coulombic interactions. At present it is the only theory of interaction site systems with Coulombic site-site interactions capable of treating fluid properties over the range of densities required for prediction of vapour-liquid coexistence properties. However, it has not yet been subjected to extensive testing by comparison with molecular simulation results. Details of model potentials and simulations computational details are discussed in section 2 and the results of the Gibbs-ensemble simulations for phase-equilibria and critical behaviour of our simulation results as well as theoretical results are presented in section 3 and section 4 summarizes our conclusions. 2. Model potentials and computational details The focus of the present paper is diatomic and dipolar-diatomic molecules interacting with site-site Lennard-Jones 12-6 potentials and site-site Coulombic potentials. The total intermolecular pair potentials is thus of the form 2 2 u(1,2) = Z Z{qiqj/rij + 4e[(~r/rij )12 - (a/rij)6]}' (1) i j where rij is the distance between site i on one molecule and sitej on the other, qi is the charge on site i and ql = --q2 for charge neutrality. We used the Ewald summation method [23] to treat the Coulombic site-site interactions in the simulation so that li~lj~l [~ erfc (alrij + nl) Ew = qiqj Irij + n[ "= = I,,1=o +l/(~l3) Z qiqj (4~2/k2) exp (-k2/4a 2) cos (k. r/j)] kr / ~ ~ + (2~/(3L3)) ~ iq;rel2 + V~xlf, (2) i=1 i vself ~ / ~ ] (3) ex. = q2 where d is the intramolecular charge separation, here the internuclear distance, and a is a convergence parameter which controls the relative weights of the real space and reciprocal space sums, rfj is the vector between sites i and j, n is a vector whose components are (nxl, nyl, nzl); nx, ny and nz are integers, L is the length of the simulation cell and k = 2nn/L. When a partial charge formulation of the Ewald sum expression is used, as it is here, the reciprocal space terms include a contribution arising from the interaction between partial charges within an individual molecule and these must be removed from the total energy [24]; -ex.vself provides that correction. Our implementation of the Gibbs ensemble is essentially the same as that described by Panagiotopoulos and coworkers [3,4] and the reader is referred to these references for a more detailed description of the technique. Our calculations were performed mostly with N = 364 molecules, although some calculations with N = 512 particles were made. For the N = 364 molecule system we initialized the

Vapour-liquid equilibria 999 simulation with 256 molecules at a liquid-like density in one box and 108 molecules at a lower, vapour-like density for the other box.

3 Vapour-liquid equilibria 999 simulation with 256 molecules at a liquid-like density in one box and 108 molecules at a lower, vapour-like density for the other box. For the 512 molecule system there were 256 molecules in each phase initially. Generally the simulations were started either with the molecules of each box on the sites of an a-nitrogen lattice, or in an equilibrated state from a lower temperature. A simulation cycle in the Gibbs ensemble Monte Carlo method comprises three distinct types of moves: individual molecule displacement within a box, transfer of volume from one box to the other and particle transfers from one box to the other. The relative numbers of each type of move are chosen to optimize convergence. In these calculations a cycle consisted of a single attempted displacement of each molecule, followed by one trial volume transfer, and a number of attempted particle transfers. In the individual molecule displacements, the particles are chosen and displaced randomly, both translationally and orientationally (within the boxes) following the well established Metropolis scheme for canonical NVT simulations. Particle transfers were achieved by creating a particle at a random position in one subsystem and annihilating a randomly chosen particle in other region. The success rate for particle transfers was maintained in the range about 1-3%. The convergence and reliability of the results are very sensitive to the transfer rate; too low a transfer rate prevents equilibrium between the boxes, and too high a rate destroys equilibration within the boxes. The calculation required for the insertion step permitted the evaluation of the chemical potential by Widom's method [25] as generalized to the Gibbs ensemble [3]. The Lennard-Jones potential was truncated at an intermolecular separation equal to half the box length, and the usual long tail corrections were applied to energy, pressure and chemical potential. The simulations consisted of a minimum cycles of which were equilibration and Ncycle = 5000 for determination of ensemble averages. 3. Results and discussion The phase equilibrium results from our Gibbs ensemble Monte Carlo simulations are summarized in tables 1-7. The results are tabulated in reduced units defined as L* = L/o, T* = kt/e; E* = E/Nr p* =/9o3; P* = po'3/r /z~ :/.td/(r /Z] r = #l/nr and p~ = I~/N~. To avoid confusion we have used #~ for dipole moment, and labelled the chemical potential #r and/z~ for the liquid and vapour phases, respectively. The uncertainty reported with each result is the standard deviation of averages calculated over 100-cycle blocks during the production phase of the simulation and are only a rough guide to the true uncertainties which are more likely to be larger in most cases. In some cases simulations were repeated, starting from different initial conditions or with a different number of attempted molecule transfers per cycle. We found that the results obtained were the same within their combined uncertainties. Nevertheless these uncertainties must be treated with some care since, at best, they represent a 'local' estimate of the variability; the coexistence curves show roughness which is somewhat larger than the uncertainties but which is unlikely to be physically correct. Generally, we see that the pressures and chemical potentials calculated for the coexisting phases agree quite well for a given temperature. There is some disagree-

4 1000 G.S. Dubey et al. Table 1. Summary of the results of the Monte Carlo simulation of the Gibbs ensemble for the 12-6 diatomic for bond length L* = The symbols are defined in the text. L* = /4i* = 0.0 Vapour phase Liquid phase N T* Ncycl e /~ P* E~ ~ Pl* /)1" El* /4* 364 2" " ' '73 4-0" I " '506 0" " ' ' " t " " ' ' " ment at the lowest temperatures studied, but we believe this is principally a consequence of significant uncertainty in the calculated values of the virial pressures of liquids at those temperatures. We have used these vapour-liquid equilibrium data to estimate the critical temperatures and densities. For this purpose we have assumed that the coexistence densities follow the law of rectilinear diameters [26] and a power law near the critical point, i.e. (Pl* + p'v)~2 = AT* + B. (4) pl* - p* = C 1 T* fl (5) where p* and ~* are the saturation vapour and liquid densities at temperature T*,/3 is the critical exponent (we have assumed that/3 = 1/3), and A, B, C are constants. The values of the critical constants are presented in tables 8 and 9. Table 2. Vapour-liquid coexistence data from the Gibbs ensemble Monte Carlo simulation for L* = 0"6. The symbols are defined in the text. L* = 0'6 /~ = 0'0 Vapour phase Liquid phase N T* Ncycl r /9~ Pv* E~ /~ f~* P1* El* /zl* 364 2" ' "37 0"396 0" "02 -I t-0' "03 4-0" "95-8"32 0'377 0'038-8"96-8"29 4-0" " ' " ' "11 0"340 0" "06 4-0' " "01 +0" " " ' "02 4-0" "03 0"004 -t " " " " " "79-8' " " " "096-2' '36-8'01 4-0" ' '13 4-0' ' " " "07 4-0"03 4-0" " "13 4-0"02

5 Vapour-liquid equilibria 1 O T* ~- 2.0 \ \ \ \ \ \ 1.5 \, \ 1.0 \ 0.0 0' I J 4 i ~.7 Figure 1. Vapour-liquid coexistence curves for bond length L* = , 0-6 and 1"00 (from top to bottom, respectively). The lines are the theoretical results of McGuigan et al. [21], the symbols are our Gibbs ensemble results. The 9 and "k represent the critical points T* predicted by theory and Gibbs simulation, respectively. 2.4~ <> 2.o~ ~. T* 1.8~ r~ I Io 1" <> t 0~4 0.5 Figure 2. Vapour-liquid coexistence curve for bond length L*= 0-67, where <> are the molecular dynamic results of Gupta [27], x are our Gibbs ensemble results and 9 and represent the critical points T* predicted by MD and Gibbs simulation, respectively.

6 1002 G.S. Dubey et al / 0.25 / P* 0.20 / 0.10 / "0q ,0 T* Figure 3. Vapour pressure curves for diatomic fluid with reduced bond length L* = l-0, 0.6 and (from left to right, respectively). The lines are the theoretical results of McGuigan et al. [21] and the symbols are our Gibbs ensemble results. The coexisting densities for the non-polar 12-6 diatomics are plotted against temperature in figures 1,2. For the case of L*= 0-67, results from molecular dynamics [27] calculations are also shown and the agreement is seen to be excellent. For the other three cases we have also plotted predictions from cluster perturbation theory [21,22]. The agreement is again very good, with some deterioration for the shortest bond length at the highest temperature. The agreement is especially good for L* = 1.0. The somewhat poorer agreement at lower values of L* can be understood at least in part in terms of the cluster diagrams at each order in the density contributing to the free energy which are neglected by the theory of Lupkowski and Monson [20]. These diagrams make diminishing contributions to the free energy as L* increases. Figure 3 shows the comparison of simulation and theoretical results for the vapour pressures. Very good agreement is again apparent. Table 8 shows a Table 3. Vapour-liquid coexistence data from the Gibbs ensemble Monte Carlo simulation for L* = The symbols are defined in the text. L* /z~ = 0"0 Vapour phase Liquid phase N 7"* Ncycl e /9~ P* E~' /~ /~* PI* El* 14" " I "07 +0" " "039 0" l -I " "061-1"53-7" '53-7" I " I-0"07 i0.02

Vapour-liquid equilibria 1003 Table 4. Vapour-liquid coexistence data from the Gibbs ensemble Monte Carlo simulation for L* = 1-0. The symbols are defined in the text. L* = 1.0 /~ = 0.

7 Vapour-liquid equilibria 1003 Table 4. Vapour-liquid coexistence data from the Gibbs ensemble Monte Carlo simulation for L* = 1-0. The symbols are defined in the text. L* = 1.0 /~ = 0.0 Vapour phase Liquid phase N 7"* Ncycle P~ P* E~ /~ PI* el* E~* /~* : / t t comparison of the critical densities and temperatures estimated from our Gibbs ensemble results together with predictions from cluster-expansion perturbation theory, [22], the first-order perturbation theory of Fischer et al. [28] and from a recently developed semi-empirical equation of state due to Sowers and Sandier [29]. For the one value of L*, 0"3292, for which data are available from both theories and from simulation, the predictions from the cluster perturbation theory for the critical temperature represent an improvement over those from the first-order perturbation theory which does not include the influence of attractive forces upon the fluid structure at low and moderate density. The two levels of perturbation theory provide equally accurate preductions of the critical density, but the corrections to perturbation theory have a bigger effect on the critical temperature than on the critical density. The predictions from the semiempirical equation of state are also reliable for these systems. Table 5. Vapour-liquid coexistence data from Gibbs ensemble Monte Carlo simulation for dipolar diatomics with bond length, L* = 0.6 and/~ = The symbols are defined in the text. L* = 0-6 /~ = 2.0 Vapour phase Liquid phase N T* Ncycl e /gv* Pv* ~ /d~ ill* P~ ~ //'1" 364 2" "012 0"020-0"57-9' '007-11"96-8" "07 -t t ' '040-1" "392 0"047-10" " " "033 0"052-1"33-9' "86-9" ' " " " " " " " " "83-8"75 0"320 0" "73 +0"006 +0"003 +0" "

8 1004 G.S. Dubey et al od ~ D o [] 2.6 T* I i 1" p* Figure 4. Vapour-liquid coexistence curve for L*= 0.60 and /~ = 3-00, L*= 0.60 and /z~ = 2.00, and L* = 1.00 and/z~ = 3-00 (from top to bottom, respectively). The lines are the theoretical results of Lupkowski and Monson [22], the symbols are our Gibbs ensemble results. The 9 and ~r represent the critical points T'predicted by theory and Gibbs simulation, respectively. Figure 4 shows the coexisting densities against temperature for the dipolar diatomic systems considered. The predictions from the cluster perturbation theory are also shown [22]. For L* = 0.6 the agreement is best for the smaller of the two dipole moments considered. For the L* = 1.0 the agreement is qualitatively good but the theory consistently underestimates the liquid density. Figure 5 shows the corresponding results for the vapour pressures. Table 9 shows a comparison of the critical Table 6. Vapour-liquid coexistence data from Gibbs ensemble Monte Carlo simulation for dipolar diatomics with bond length, L* = 0-6 and/~ = The symbols are defined in the text. L* = 0.6 ~* = 3'0 Vapour phase Liquid phase N 7* Ncycl e /9~ Pv* E~' /~ Pl* Pl* El* Pt* ' " t " " "341 0" " " " "03 +0" " "78 0"305 0" " " ' " "006 +0" " '74-10"71 0" ' " " "06 4-0'01 +0" t "68 0" " ' "02+0" "

9 Vapour-liquid equilibria ~ ~ ~ "... I l 0.15 P* 0.10k / / i o/ u / / / /.: / 0.05! / ; /, /; / 1.o ~15 22o z~ a.0 a T" Figure 5. Vapour pressure curves for dipolar diatomics fluid with reduced bond length L* = 1.00 and M' = 3.00, L* = 0.60 and ~ = 2.00 and L* = 0-60 and ~ = 3.00 (from left to fight, respectively). The lines are the theoretical results of Lupkowski and Monson [221 and symbols are our Gibbs ensemble results. densities and temperatures for dipolar diatomic fluids estimated from our Gibbs ensemble results together with predictions from cluster perturbation theory. The theory tends to underestimate the critical density and overestimate the critical temperature but the overall agreement is quite good. 4. Conclusions In this paper we have applied the Gibbs ensemble to the study of vapour-liquid Table 7. Vapour-liquid coexistence data from Gibbs ensemble Monte Carlo simulation for dipolar diatomics with bond length, L* = 1-00, and p~ = The symbols are defined in the text. L*= 1-0 /h~ = 3"0 Vapour phase Liquid phase N T* Ncycl e /~ P* E~ P-~ g* Pl* Ei* /4" 364 1' "317 0" " :0-07 5: :0"05 5:0" " " "009 5:0"001 5:0-07 5:0-06 5: :0"009 5:0-08 5:0" ' "11 0"263 0" " : :0-05 5: "08 5: " " " :0"007 5:0"003 5: : " " " " " : :0"001

10 1006 G.S. Dubey et al. Table 8. Comparison of the Gibbs simulation estimates of the critical temperature T* and critical density p'for various bond lengths L* with those of duster perturbation theory 9 [21], molecular dynamics of [27], the first-order perturbation theory of Fischer et al. [28], and semiempirical equation of state theory of Sowers and Sandier [29]. Reference L* T* p* Simulation (this work) Simulation [27] Cluster perturbation theory [21] First-order perturbation theory [28] Equation of state [29] Simulation (this work) Cluster perturbation theory [21] Simulation (this work) Simulation [27] First-order perturbation theory [28] Equation of state [29] Simulation (this work) Cluster perturbation theory [21] equilibria of diatomic and dipolar diatomic fluids. The principal focus of the work has been to test the predictions of the cluster expansion perturbation theory [20-22] which incorporates the influence of the attractive forces upon the fluid structure which is important at moderate and low densities, and the influence of multipolar interactions which is important at all densities. For the case of 12-6 diatomics the simulation results are in very good agreement with predictions from cluster perturbation theory. These predictions are comparable in accuracy to those obtained from the optimized cluster theory for the atomic 12-6 potential [30]. The result for 12-6 diatomics with a point-charge dipole are also very good for the case of low dipole strength. For high dipole strength the results are still in quite good agreement with the theory although there is tendency for the theory of underestimate the saturated liquid density. Overall we believe that this comparison is an important demonstration that prediction of vapour-liquid equilibria for systems with both non-spherical molecular shape and polarity lies within the grasp of currently available theoretical methods. Table 9. Comparison of the Gibbs simulation estimates of the critical temperature T'and critical density p* for various bond lengths L* and dipolar strengths/~ with those from cluster perturbation theory. Reference L* p~ T* /9* Simulation (this work) 0"60 2" Cluster perturbation theory [22] '73 Simulation (this work) Cluster perturbation theory [22] ' "150 Simulation (this work) ' Cluster perturbation theory [22] ' 124

Vapour-liquid equilibria 1007 Research on this problem at the University of Lethbridge was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. S. F.

11 Vapour-liquid equilibria 1007 Research on this problem at the University of Lethbridge was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. S. F. O'Shea and P. A. Monson are grateful to Professor J. S. Rowlinson and the staff of the Physical Chemistry Laboratory, Oxford University for their hospitality during the period when this project was initiated, and to Professor D. J. Tildesley for his help and encouragement with the Gibbs ensemble. References [1] GUBmNS, K. E., 1989, Molec. Simul., 2, 223. [2] PANAGtOrOPOULOS, A. Z., and STAPLE'rON, M., 1989, Fluid Phase Equil., 53, 133. [3] PANAGIOrOPOULOS, A. Z., 1987, Molec. Phys., 61, 813; Ibid., 62, 701. [4] PANAGIOTOPOULOS, A. Z., QUIRKE, N., STAPLErON, M., and TILDESLEV, D. J., 1988, Molec. Phys., 63, 527. [5] SMIT, B., WILLIAMS, C. P., HENDRIKS, E. M., and OE LEEUW, S. W., 1989, Molec. Phys., 68, 765. [6] DE LEEVW, S. W., S~IT, B., and WILLrAX4S, C. W., 1990, J. chem. Phys., 93, [7] VE6A, L., OE MIGUEL, E., RULL, L. F., JACKSON, G., and McLuRE, I. A., 1992, J. chem. Phys., 96, [8] PANAGIOTOPOULOS, A. Z., 1989, Molec. Phys., 62, 701. [9] SINOH, R. P., PITZER, K. S., DE PABLO, J. J., and PRAUSNITZ, J. M., 1990, J. chem. Phys., 92, [10] AMAR, J. G., 1989, Molec. Phys., 67, 739. [11] RtJDISILL, E. N., and CUMMINGS, P. T., 1989, Molec. Phys., 68, 629. [12] STAPLETON, M. R., T~LDESLEV, D. J., PANAGIOTOPOULOS, A. Z., and QUmKE, N., 1989, Molec. Simul., 2, 147. [13] SUIT, B., and WILLIAM, C. P., 1990, J. Phys.: Condens. Matter, 2, [14] SUIT, B., and FRENKEL, D., 1991, Molec. Phys., 74, 35. [15] DE MIGVEL, E., RULL, L. F., CHALAM, M. K., and GUaBINS, K. E., 1990, Molec. Phys., 71, [16] YETHIaAJ, A., and HALL, C. K., 1991, Molec. Phys., 72, 619. [17] DE PAaLO, J. J., and Pa_~USNIrZ, J. M., 1989, Fluid Phase Equil., 53, 177. [18] DE PAaLO, J. J., PRAUSNITZ, J. M., STRAUCn, H. J., and CUMMINGS, P. T., 1990, J. chem. Phys., 93, [19] DE MIGUEL, E. and ALLEN, M. P., 1992, Molec. Phys., 76, [20] LUPKOWSKI, M., and MONSON, P. A., 1987, J. chem. Phys., 87, [21] McGtJ~GAN, D. B., LUPKOWSKI, M., PAQUET, D., and Mo~soN, P. A., 1989, Molec. Phys., 67, 33. [22] LupKowsgI, M. and MONSON, P. A., 1989, Molec. Phys., 67, 53. [23] EWALD, P. P., 1921, Ann. Phys., 64, 253. [24] HEYES, D. M., 1983, CCP5 Quarterly, 8, 29. [25] WIDOM, B., 1963, J. chem. Phys., 39, [26] GUGGENHEIM, E. A. J., 1945, J. chem. Phys., 13, 253. [27] Gu~rA, S., 1988, J. phys. Chem., 92, [28] FISCHER, J., LUSTING, R., BREITENFELDER-MANSKE, H., and LEMMING, W., 1984, Molec. Phys., 52, 485. [29] SowEP.S, (3. M., and SANDLER, S. I., 1992, Molec. Phys., 77, 351. [30] StrNG, S. H., and CHANDLER, D., 1974, Phys. Rev. A, 9, 1688.

Phase transitions of quadrupolar fluids

Phase transitions of quadrupolar fluids Seamus F. O Shea Department of Chemistry, University of Lethbridge, Lethbridge, Alberta, Canada, T1K 3M4 Girija S. Dubey Brookhaven National Laboratory, Upton, New