l l l l B I Emlllm

E L S E V I E R Fluid Phase Equilibria 112 (1995) 173-197

Vapour liquid equilibria of Lennard-Jones model mixtures from the NpT plus test particle method

Jadran V r a b e c a, A m a l Loff i ~'~, Jo h an n F i sche r b,,

a Institutfiir Thermo- und Flaiddynamik, Ruhr-Universitiit, D-44780 Bochum Germany b Institutfiir Land-, Umwelt- und Energietechnik, Unit,ersitiit~r Bodenkultur, A-1190 Wien Austria

Received 3 March 1995; accepted 26 May 1995

Abstract

The NpT + test particle method for the calculation of vapour liquid phase equilibria (VLE) is applied to ten binary Lennard-Jones mixtures which were studied previously with the Gibbs ensemble (Harismiadis et al., 1991). The present method uses one simulation for the description of the liquid. For the vapour two simulations are required at higher densities, whilst at lower vapour densities the Haar-Shenker-Kohler (HSK) equation of state is used. The chemical potentials from simulation are calculated by Widom's test particle insertion method (Widom, 1963) and adjusted with the results from the particle swap method, which yields the differences of the chemical potentials. Additionally, VLE results are obtained by using the Fischer-Lago-Bohn perturbation theory for the liquid and the HSK equation for the vapour. A comparison of the new simulation results with those of the Gibbs ensemble shows qualitative agreement but in most cases smaller error bars and considerably less scatter of the present data. Moreover, the simulations confirm the good to excellent performance of the perturbation theory.

Keywords: Theory; Computer simulations; Vapour-liquid equilibria; Lennard-Jones mixtures

1. Introduction

For the determination of vapour liquid phase equilibria (VLE) of pure substances by simulations different methods are existing for some time (Hansen and Verlet, 1969), among them the Gibbs ensemble (Panagiotopoulos, 1987) and the N p T + test particle method (M~511er and Fischer, 1990; Lotfi et al., 1992). VLE of mixtures have been calculated in several cases with the Gibbs ensemble

* Corresponding author. Institut f'0r Land-, Umwelt- und Energietechnik, Universit~it for Bodenkultur, Peter-Jordan-Str. 82, A-1190 Wien, Austria.

i Present address: Raab Karcher Energie Service GmbH, Nevinghoff 5, D-48147 Miinster, Germany.

0378-3812/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0378-3812(95)02795-5

174 J. Vrabec et aL / Fluid Phase Equilibria 112 (1995) 173-197

method, e. g. (Panagiotopoulos, 1989; Stapleton et al., 1989; Harismiadis et al., 1991). As an alternative we extended the NpT + test particle method to mixtures (Vrabec and Fischer, 1995), and presented results for the system argon + methane. The pure components of this mixture described by Lennard-Jones (LJ) spheres have similar size and energy parameters. Now we decided to investigate the performance of the new method in the case of L3 mixtures with larger size and energy ratios considering the accuracy of the results as a crucial item. Using the same model mixtures as Harismiadis et al. (1991) makes a direct comparison with the Gibbs ensemble results possible; these model mixtures are important for the development and testing of mixing rules to be used in equations of state. An additional interesting aspect is a comparison of VLE obtained by simulations with results from the Fischer-Lago-Bohn perturbation theory (Bohn et al., 1986).

The NpT+ test particle method as described (Vrabec and Fischer, 1995), requires as crucial quantities the chemical potentials and the partial molar volumes at prescribed values of temperature, pressure and concentration. So far, these quantities are obtained by Widom's test particle insertion method. An additional route for the determination of these quantities is the particle swap method (Sindzingre et al., 1987; Heyes, 1992). It seems worthwhile to study this method here in comparison with the test particle insertion method. Moreover, we investigated two different routes for the calculation of the isothermal compressibility which is required for the saturated densities.

2. Thermodynamic description of the method

For the determination of VLE data we consider a binary mixture consisting of the components A and B, with a given set of intermolecular potentials.

The basic idea of the NpT + test particle method is the calculation of the chemical potentials in the liquid at a prescribed temperature and a prescribed liquid concentration as functions of the pressure, and the calculation of the chemical potentials in the vapour as functions of the pressure and vapour concentration at the prescribed temperature. In this way one is able to find the VLE in the chemical potential vs. pressure ( tt~, p) plane, as the intersection of the liquid and the vapour branch of both chemical potentials at the same pressure. As will be explained below, in generai one simulation for the liquid and two for the vapour phase have to be performed; in most cases we find low vapour densities, so that it is possible to use instead of the vapour simulations some simple equation of state (e.g. virial expansion, perturbed virial expansion), which reduces the computational effort to one simulation per VLE.

Let us start with a summary of the method as presented recently (Vrabec and Fischer, 1995). First, we will give an outline of the procedure in terms of classical thermodynamics and subsequently describe the calculation of the required thermodynamic quantities from simulation.

The liquid branches of the chemical potential /x~i are approximated by a first order Taylor series

/./.,](p) =/..zli0 q- vl" ( p - plo) (1)

where the partial molar volumes v I

(0.,I = Op ]T,x Ui (2)

J. Vrubec el ul. /Fluid I’husr Equilihrm I I-7 (1995) 17.3-397 175

are the gradients. All quantities appearing in Eq. (1) can be determined in a single liquid simulation at the prescribed values of temperature and liquid concentration and an arbitrarily chosen pressure pl,.

Typically, the vapour branches have in the pi, p plane a logarithmical course. Therefore a first order Taylor series for these functions is unfavourable. A solution of this problem is to split the chemical potentials into the residual and ideal parts

pi=py’.+k*.lnll- kT +kT.lnyi+pid(T) (3)

with pid(T) being the merely temperature dependent part of the chemical potential, and

Only the residual part is expanded into a first order Taylor series treating the ideal parts analytically

4-Yh /-qPYY,) = & + -

i I , aP T,X

.(p--p;)+kT.In 5 -t i 1

+kT.ln JL i i Yil)

. h -Y*,,) T.p

(5)

The derivatives of the residual chemical potentials with respect to the pressure are given by

kT = [,; _ (‘id = py - - (6)

l‘.x P

The chemical potentials and the partial molar volumes are calculated by a vapour simulation at the prescribed temperature and arbitrary chosen values of pressure p1; and vapour concentration y,,,. In order to obtain the derivatives of the residual chemical potentials with respect to the concentration at constant temperature and constant pressure, a second simulation in the vapour phase at the same temperature and pressure but a different vapour concentration Y,, has to be performed. The derivatives are approximated by the difference ratios

From these two vapour simulations the chemical potentials in the vapour phase are ot;ained as functions of pressure and concentration

. ()‘A -Y,,,) T.,’

(8)

176 J. Vrabec et aL / Fluid Phase Equilibria 112 (1995) 173-197

Finally for the prescribed temperature and the prescribed liquid concentration, the vapour pressure p,, and the vapour concentration y at phase equilibrium are calculated using the equality of the chemical potentials in the two phases expressed by Eq. (1) and Eq. (8)

/'/qo" : /xI(P,~) = tzV(p~,Y), (i =A,B) (9)

where /xi~ denotes the chemical potential of component i at phase equilibrium. Other interesting thermodynamic properties are the bubble density p', dew density p", bubble

enthalpy h', and dew enthalpy h". We calculate them again with first order Taylor series inserting the vapour pressure and vapour concentration

p, = pg + i l l . p~. (p ,~_pg)

P" = Pg + fl'~ " P° " ( P'~-PV°) + ~YA V,p

h '=hZo+l-~p T,x " (P,~--PI)

io v) to v / ~- . ( p , ~ - p ~ o ) + • h" hV° + ~ Op T,y 10YA ] T , p

(Y --YAo)

(lO)

(11)

(12)

(Y--YA0) (13)

3. Determination of thermodynamic properties by simulation

According to the above thermodynamic description of the method, the set of thermodynamic quantities required from simulation at prescribed temperature, pressure and concentration, are density p, enthalpy h, chemical potentials /xi, partial molar volumes vi, isothermal compressibility fiT, and the derivative of the enthalpy with respect to the pressure at constant temperature and concentration (Oh/Op)T, x. Consequently the simulations are performed in the isobaric isothermal NpT ensemble. Density and enthalpy are determined simply by averaging the instantaneous volume and the instantaneous enthalpy. The quantity (Oh/Op)T, x is evaluated by

1( 1 ) T,x= ~ " - - £ - ~ ' [ < H V > - < H > < V > ] + < V > (14)

The quantities txi, vi, and fiT require special considerations, which follow.

3.1. Chemical potentials

Widom's test particle insertion yields directly the absolute values of the chemical potentials by

/./w = /./id(T ) + kT" In x i - - kT. ln(V. exp{-fl~bi})/N (15)

where ~ denotes the potential energy of the test particle of the species i ( i = A , B), V the instantaneous volume and the brackets averaging in the NpT ensemble. It is known, that the statistics of test particle insertion in fluids with high density becomes poor. In the case of mixtures with larger

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197 177

size ratios one can expect that successful insertion of test particles of the larger species becomes even harder. Therefore we used as an additional route the particle swap method (Sindzingre et al., 1987; Heyes 1992), which yields the differences of the chemical potentials A/, = /ZA --/*B

x A AlL s = kT. ln(exp{-/3A~b}) + kT. I n - - (16)

Xu

where Zig,, is the energy change on removing a particle of species A and replacing it by a particle of species B. In the present paper only particles of the smaller species are swapped to the large species. It is not possible to obtain a second independent set of chemical potentials with the swap method, because the total Gibbs free energy can not be evaluated. Therefore we invented a statistical adjustment of the chemical potentials using both the results of the test particle insertion and of the particle swap.

The starting point for the statistical adjustment method is the fact, that the average values of the chemical potentials (/.£A) , ( / Z B ) and the average (A/x) of the difference A# = /ZA - - /ZB are available together with their statistical uncertainties 6 A, 6 u, and 6 a. We assume now for the probability distribution of the chemical potentials /*A, /XU and the difference A/Z Gaussian distribu- tions

1 2 -~

WA(/ZA) = 2V~-~..6Aexp{--(/ZA--{/.*A)) /62} (17)

WB(/ZB) = 2V/~-~. 8Bexp{-- ( /~B- ( /ZB))2/82} (18)

1 w~(A/~) = 2 x / ~ . 6 a e x p { - ( A / Z - ( A / Z ) ) 2 / 6 2 } (19)

The crucial point is now, that the simulation results will in general not fulfil the condition

(A/Z) = (P'A) - (/ZB) (20)

and hence we are looking for a readjustment of ( /x A) and (/ZB). For that purpose, an intuitive procedure is the following. We form the product of all three distributions W = WA(/ZA) " WB(/ZB) " wa(Ap,), which, as A/~ = /ZA --/'*U, is a function of the two independent variables /ZA and /ZB. The adjusted averages /Z,~ and /x~ are those values of /ZA and /.% which yield the maximum of the function W. A straightforward calculation then yields

82 /z~= {/z A) - 6A+ 62 + 82 • [( /Z A) -- ( /Z B) -- (A/z)] (21)

82 /zh = (/ZB) + 62 + 82 + 62 " [(/ZA) - ( / * , ) - ( a / z ) ] (22)

In all liquid simulations Eq. (15) and Eq. (16) were evaluated, so the chemical potentials could be adjusted by Eqs. (21) and (22). Reviewing the results we can find three different cases. In the first case, the adjusted values are practically identical with the chemical potentials obtained by test particle insertion. The first two state points in Table 1 show such a behaviour. We found this for almost all

178 J. Vrabec et aL / F l u i d Phase Equilibria 112 (1995) 173-197

Table 1 Chemical potentials from test particle insertion /zA,w /z Bw and adjusted values tz~,, /x]~ obtained with the help of the particle swap difference A/z s = / z A - - /x B. The bracketed numbers indicate the statistical uncertainties in the last digits; (T * = k T / E A ;

p , =po.A~/eA; p~ = p ( x ~ e r ~ + ZXAXBerA~ + Z XBO'/~); ~i = ( tXi -- t z i a ( T ) ) / kT ) * * * ~ w ~ a T p x A Pm /2~ /z B /2~, /z B Al 2s

o -B /e r a = 1.0; E B / e A = 0.5 0.75 0.065 0.50000 0 .69454(82) - - 5 . 9 6 4 ( 2 3 ) - - 2 . 7 3 0 ( 1 8 ) - -5 .966 - -2 .728 - -3 .238 (4 )

erB/erA = 1.5; E B / e A = 0.66 0.75 0.008 0.69922 0.73521 (52) - -5 .850 (14) - -5 .060 (140) - -5 .850 - -5 .046 - -0 .810 (100)

O'B/er A = 1.5; ~ ' B / e A = 1.0 1.00 0.020 0.90000 0.68890 (120) - 3.977 (15) - 7.590 (140) - 3.975 - 7.757 3.880 (110)

erB/OA = 0.5; e B / e A = 0.75 1.00 0.150 0.50000 0.60871 (97) - -4 .626 (44) --2.191 (3) - -4 .598 --2.191 --2 .339 (69)

erB /OrA = 1.5; e B / e A = 0.75 1.00 0.025 0.89844 0.66376 (93) -- 3.941 (12) -- 6.072 (109) -- 3.943 -- 5.933 1.966 (45)

O'B/erA = 0.5; e B / e A = 0.66 0.75 0.150 0.10000 0.65540 (58) -- 12.000 (2700) -- 1.487 (8) -- 10.172 -- 1.487 - -8 .570 (660)

simulations of mixtures with a size parameter ratio O'B/O k = 1.0. In the second case one chemical potential is adjusted by about the order of its statistical uncertainty (c. f. Table 1, state points three and four). This happened, when either the size ratio is o-B/o- a = 1.5 and the concentration of the larger component is low, or when the size ratio is o-B/o- A = 0.5. The third case is distinguished by a strong adjustment of one chemical potential, which is actually calculated from the other by adding A/.zs; see the last two state points in Table 1. This occurs when the mixture has a large size ratio, a low concentration of the large component, and a high fluid density, so that insertion of the large test particles is almost impossible. For state point 5 of Table 1, the running averages /z~, /z~ and A/z s are shown in Fig. 1. Actually this behaviour can be found rarely in our simulations, because according to our results, the uncertainty of the difference A/x s mostly is in the range of the uncertainty of the chemical potential for the larger component. So it looks like, that in a dense fluid consisting of components with a large size ratio, it is similarly difficult to insert test particles as to swap particles from small to large.

Test particle insertion offers the possibility to increase the number of test particles to get more accurate results without increasing the number of time steps, whereas the number of swapped particles is limited by the number of real particles involved in the simulation.

3.2. Partial molar volumes

The test particle insertion allows us to calculate directly the absolute values of the partial molar volumes (Sindzingre et al., 1987; Heyes, 1992) by

( V 2" exp{ -/3~bi})

vw = (V-exp{-/3~9i}) - ( V ) (23)

J. Vrabec et a l . / Fluid Phase Equilibria 112 (19951 173-197 1-79

-3.6

-4.0

- . 1 .,t

-5.6

. . . . . . . . . Z Z Z Z Z Z Z Z Z Z ~ Z Z - - - - - - _ . . . . .

-6.0

- 6 . 4

2.4

~x~ s

2.0

1.6 -T - - ll0 2; 3; 4r0 5~0 6;

time steps/103

Fig. 1. Running averages of the chemical potentials /2~ and /2~ obtained by test particle insertion and the difference a/2 ~ from the particle swap method, for the Lennard-Jones LB mixture O-u/O'a = 1.5, e u / e A = 0 . 7 5 at the state point kT/eA = 1.00, p~x~/eA = 0.025, x A = 0.89844.

Similar problems as for the chemical potentials can be expected. The swap method yields in a first step the differences of the partial molar volumes Av = c A - v B

{ V - e x p { - / 3 A O } )

AL . . . . ( e x p { - / 3 A O } } + ( V ) (24)

Table 2

,~ of w and particle swap t i Comparison of simulation results for the partial molar volumes obtained by test particle insertion t i

the LB I J mixture o ' u / t r A = 1.5, f fB /EA = 1.0 at k T / e A = 1.00; (P* = Ptr2/EA; Pm = p(.l'~CrA + -X\~ r~{(r/{ B~ + x~cri~); ,,,* = , , / o x b

P XA Pm t~ A ~'~ * t A t B

0.01 0.10156 0.69344 (96) 1.23 (15) 4.68 (65) I).64 (94) 0.01 0.30078 0.68369 (90) 1.38 (14) 5.47 (711 0.33 (76) 0.01 0.50000 0.67734 (83) 1.30 (15) 4.58 (95) 1.68 (37) 0.01 0.69922 0.67790 (86) 1.47 (14) 2.88 (92) 1.38 (22) 0.02 0.90000 0.68890 (120) 1.42 (21) 6.60 (200) 1.64 (12)

4.891111 5.22 (32) 4.43 (37) 4.73 (50) 2.61/(110)

180 J. Vrabec et aL / Fluid Phase Equilibria 112 (1995) 173-197

Table 3 Comparison of simulation results for the isothermal compressibility f i t evaluated by Eq. (27) and Eq. (28) for different LB

* 2 3 . * LJ mixtures and different state points; (T = kT / EA; p* =pox~/EA; Pm = p(x2 0"~ + 2XA XBO'3AB + XB~t~), f i t =

I~T EA / O ' 2 )

O 'B/Or A IEB//E A T * p * X A p~ f iT Eq. (27) f l ; Eq. (28)

1.0 0.50 0.75 0.080 0.10156 0.2113 (32) 37.0 (39) 42.8 (66) 1.0 0.50 0.75 0.075 0.35156 0.6249 (15) 0.776 (64) 0.765 (59) 1.5 0.66 0.75 0.008 0.80078 0.75819 (66) 0.205 (13) 0.205 (13) 1.5 0.75 1.00 0.030 0.19922 0.4485 (81) 49.4 (81) 33.5 (63) 1.5 1.00 1.00 0.010 0.50000 0.67734 (83) 0.525 (33) 0.520 (32) 0.5 0.75 1.00 0.600 0 . 0 1 9 5 3 0.12785 (56) 3.22 (12) 3.42 (14) 0.5 1.00 0.75 0.010 0.50000 0.74763 (39) 0.0588 (16) 0.0587 (16)

but as the total volume v is also known, we obtain immediately

v~ = v +xB" ~lv (25)

s ~ o v B v - x A A v (26)

which enables us to determine a second independent set of partial molar volumes. In all our liquid simulations both Eq. (23) and Eq. (24) were evaluated, so that for all state points

two sets of partial molar volumes were available. Reviewing the results, c.f. Table 2, we mostly find an agreement between these sets within their statistical uncertainties.

The partial molar volumes obtained by test particle insertion v w show the same problems as the chemical potentials obtained by this method. In general the particle swap method has the same problems, but according to Eqs. (25) and (26) the partial molar volumes v~, for x A ~ 0 and v~ for x B ~ 0 are dominated by the total volume v, which has very small statistical uncertainties in the liquid. Therefore for these cases the results f rom particle swap become very accurate, as can be seen from Table 2.

For the calculation of VLE data we used always the partial molar volumes from particle swap.

Table 4 VLE data were calculated for ten Lennard-Jones mixtures using the LB combining rule. The models are the same as introduced by Harismiadis et al. (1991); (T * = k T / ~ A)

O'B / O" A ~n/eA T *

1.0 0.50 0.75; 1.00 1.0 0.66 0.75; 1.00 1.0 0.75 0.75; 1.00 1.5 0.66 0.75; 1.00 1.5 0.75 0.75; 1.00 1.5 1.00 1.00 0.5 0.50 0.75; 1.00 0.5 0.66 0.75; 1.00 0.5 0.75 0.75; 1.00 0.5 1.00 0.75; 1.00

J. Vrabec et al. / F l u i d Phase Equilibria 112 (1995) 173-197 181

Table 5 Vapour liquid equilibria for different LJ mixtures using the LB combining rule. For the vapour phase always the HSK equation (Fischer and Bohn, 1986) was used except for the state points indicated by a; VLE data obtained from simuations

, , , 9 3 ~' 3 with a system size of 500 particles are marked with b; (T = k T / E A ; P = p~rA3/eA; Pm = P(XTxO'/~ + 2XAXBO'2B + XBO'I3))

T* x A YA P~ P~* P~*

o 'a /o" A = 1.0; eB/E A = 0.5 0.75 ~b 0.30000 0.0666 (15) 0.0798 (16) t).5966 (16) 0.75 ~b 0.33000 0.0656 (12) 0.0761 (15) [I.6121 (14) 0.75 a 0.35156 0.0665 (17) 0.0754 (21) I).6251 (18) 0.75 0.39844 0.0643 (13) 0.0685 (16) 0.6444 (14) 0.75 0.44922 0.0657 (15) 0.0682 118) [I.6735 (10) 0.75 0.50000 0.0680 (19) 0.0671 (22) tl.69501 (96) 0.75 0.55078 0.0677 (24) 0.0615 (19) 11.71182 (80) 0.75 0.60156 0.0702 (22) 0.0598 (24) 11.72988 (93) 0.75 0.69922 0.0763 (30) 0.0520 (22) [I.75819 (69) 0.75 0.73828 0.0775 (36) 0.0436 (19) 0.76662 (63) 0.75 0.80078 0.0883 (44) 0.0377 (15) 0.78266 (50) 0.75 0.82031 0.0931 (53) 0.0346 (17) [I.78620 (57) 0.75 0.92969 0.164 (13) 0.0170 (10) 0.80826 (46) 0.75 1.00000 1.00000 0.00264 (7) [I.82158 (38) 1.00 ab 0.60000 0.353 (13) 0.1388 (40) [I.5372 (45) 1.00 ~ 0.69922 0.3354 (42) 0.1115 (24) [I.5844 (21) 1.1)0 0.80078 0.3741 (45) 0.0898 (18) 0.6336 (17) 1.00 0.89844 0.4759 (59) 0.06012 (98) /).6692 (11 ) 1.00 0.94922 0.6137 (56) 0.04287 (63) 0.68636 (91) 1.00 1.00000 1.00000 0.02505 (22) Ik70081 (38) o-B/o- A = 1.0; EB/EA = 0.66 0.75 0.00000 0.00000 0.03676 (26) 0.6162 (10) 0.75 0.10156 0.02034 (31) 0.03296 (66) 0.6532 (23) 0.75 0.30078 0.0556 (14) 0.02657 (68) 0.70472 (90) 0.75 0,50000 0.0937 (32) 0.01944 (52) [I.74583 (64) 0.75 0.69922 0.1582 (84) 0.01438 (60) 0.78037 (57) 0.75 0.89844 0.3572 (68) 0.00684 (26) 0.80863 (29) 0.75 1.00000 1.00000 0.00264 (7) 0.82158 (38) 1.00 a 0.44922 0.2556 (18) 0.0903 (14) I).5336 (26) 1.00 0.50000 0.2783 (21) 0.0838 (15) 11.5557 (28) 1.00 0.60156 0.3270 (27) 0.07202 (93) I).5963 (16) 1.00 0.69922 0.3900 (39) 0.06034 (84) [I.6271 (15) 1.00 0.80078 0.4887 (49) 0.04927 (68) 0.6560 (11) 1.00 0.89844 0.6479 (47) 0.03741 (50) /).67856 (88) 1.00 1.00000 1.00000 0.02505 (22) 0.70081 (38)

O. 1796 (66) O. 1636 (60) 0.1611 (81) 0.1347 (53) O. 1340 (58) o.13o8 (70) O. 1144 (53) O. 1100 (66) 0.0907 (54) 0.0720 (39) 0.0602 (30) 0.0544 (32) 0.0247 (16) 0.00363 (10) 0.252 (16) 0.1649 (55) 0.1232 (35) 0.0760 (17) 0.05202 (99) 0.02964 (32)

0.0663 (10) 0.0577 (17) 0.0437 (14) 0.02997 (94) 0.02136 (99) 0.00965 (39) O.00363 (10) 0.1466 (45) 0.1311 (43) 0.1045 (22) 0,0820 (17) 0.0637 (12) 0.04603 (81) 0.02964 (32)

O'B/O" A = 1.0; e B / e A = 0.75 0.75 0.00000 0.00000 0.01879 (17) 0.70081 (38) 0.02964 (32) 0.75 0.10156 0.02414 (76) 0.01563 (42) 0.71500 (91) 0.02386 (74) 0.75 0.30078 0.0763 (29) 0.01355 (40) 0.74573 (63) 0.02033 (68) 0.75 0.50000 0.1446 (52) 0.01064 (29) [).77121 (57) 0.01557 (48) 0.75 0.69922 0.259 (13) 0.00837 (34) 11.79340 (52) 0.01204 (52) 0.75 0.89844 0.542 (16) 0.00433 (14) 0.81191 (29) 0.00603 (21) 0.75 1.00000 1.00000 0.00264 (7) [).82158 (38) 0.00363 (10) 1.00 a 0.14844 0.0969 (20) 0.0885 (23) 0.4854 (69) 0.171 (18) 1.00 " 0.30078 0.1754 (13) 0.07755 (91) [).5516 (24) 0.1172 (25)

182 J. Vrabec et a l . / Fluid Phase Equilibria 112 (1995) 173-197

Table 5 (continued)

T* x A YA P~ P~* P~*

1.00 0.50000 0.2939 (27) 0.05959 (82) 0.6082 (18) 1.00 0.69922 0.4550 (46) 0.04416 (58) 0.6470 (13) 1.00 0.89844 0.7351 (39) 0.03168 (44) 0.68443 (83) 1.00 1.00000 1.00000 0.02505 (22) 0.70081 (38) ~B/~A = 1.5; e B /EA= 0.66 0.75 0.00000 0.00000 0.01089 (8) 0.6162 (10) 0.75 0.10156 0.0666 (15) 0.01001 (25) 0.6298 (18) 0.75 0.30078 0.1798 (50) 0.00896 (29) 0.6649 (11) 0.75 0.50000 0.267 (14) 0.00808 (48) 0.69662 (95) 0.75 0.69922 0.296 (31) 0.00811 (97) 0.73524 (57) 0.75 0.80078 0.392 (65) 0.0059 (11) 0.75787 (67) 0.75 1.00000 1.00000 0.00264 (7) 0.82158 (38) 1.00 a 0.55078 0.4910 (56) 0.03929 (55) 0.4856 (51) 1.00 0.60156 0.5306 (56) 0.03760 (61) 0.5145 (37) 1.00 0.69922 0.5962 (64) 0.03527 (59) 0.5683 (18) 1.00 0.80078 0.682 (11) 0.03220 (75) 0.6097 (14) 1.00 0.89844 0.775 (16) 0.03003 (93) 0.6536 (11) 1.00 1.00000 1.00000 0.02505 (22) 0.70081 (38) CB/~A = 1.5; E B / E A = 0.75 0.75 0.00000 0.00000 0.00557 (5) 0.70081 (38) 0.75 0.10156 0.0858 (24) 0.00506 (15) 0.7015 (10) 0.75 0.30078 0.2334 (92) 0.00494 (20) 0.71476 (87) 0.75 0.50000 0.371 (18) 0.00448 (22) 0.72902 (58) 0.75 c 0.69922 0.500 (37) 0.00410 (32) 0.75284 (47) 0.75 c 0.80078 0.556 (59) 0.00375 (42) 0.76923 (56) 0.75 c,b 0.90000 0.752 (61) 0.00315 (27) 0.79060 (38) 0.75 1.00000 1.00000 0.00264 (7) 0.82158 (38) 1.00 0.19922 0.2003 (31) 0.02972 (46) 0.442 (13) 1.00 0.30078 0.3013 (38) 0.02895 (44) 0.4957 (39) 1.00 0.50000 0.4771 (66) 0.02905 (48) 0.5568 (27) 1.00 0.69922 0.6644 (86) 0.02765 (48) 0.6061 (15) 1.00 0.89844 0.860 (14) 0.02624 (64) 0.664009 (95) 1.00 1.00000 1.00000 0.02505 (22) 0.70081 (38) ~ B / ~ A = 1.5; E B / f fA = 1.0 1.00 0.00000 0.00000 0.00742 (7) 0.70081 (38) 1.00 0.10156 0.1934 (43) 0.00815 (18) 0.69236 (96) 1.00 0.30078 0.4889 (99) 0.01056 (22) 0.68394 (90) 1.00 0.50000 0.705 (11) 0.01347 (23) 0.67859 (83) 1.00 0.69922 0.8707 (95) 0.01675 (26) 0.67993 (87) 1.00 b 0.90000 0.9713 (41) 0.02196 (41) 0.6893 (12) 1.00 1.00000 1.00000 0.02505 (22) 0.70081 (38)

~B / ~ A = 0.5; E B / ~ A = 0.5 0.75 a,b 0.10000 0.00286 (63) 0.5845 (62) 0.5641 (12) 0.75 b 0.20000 0.00421 (60) 0.3391 (20) 0.61126 (52) 0.75 0.30078 0.0095 (12) 0.2144 (15) 0.64199 (65) 0.75 0.50000 0.0169 (16) 0.09647 (35) 0.69101 (39) 0.75 0.69922 0.0512 (43) 0.04311 (44) 0.74257 (37) 0.75 0.89844 0.185 (14) 0.01320 (26) 0.79347 (52) 0.75 1.00000 1.00000 0.00264 (7) 0.82158 (38) 1.00 a 0.21875 0.0898 (96) 0.815 (13) 0.4718 (26)

0.0844 (19) 0.0568 (10) 0.03845 (68) 0.02964 (32)

0.0663 (10) 0.0556 (20) 0.0426 (18) 0.0342 (26) 0.0332 (50) 0.0205 (43) 0.00363 (10) 0.1410 (47) 0.1174 (38) 0.0952 (28) 0.0725 (27) 0.0565 (26) 0.02964 (32)

0.02964 (32) 0.02432 (85) 0.02039 (99) 0.01574 (91) 0.0123 (11) 0.0104 (13) 0.00664 (63) 0.00363 (10) 0.1465 (60) 0.1184 (40) 0.0911 (29) 0.0625 (17) 0.0414 (14) 0.02964 (32)

0.02964 (32) 0.02698 (74) 0.02556 (67) 0.02482 (56) 0.02404 (48) 0.02698 (61) 0.02964 (32)

0.1395 (26) 0.06894 (52) 0.04178 (34) 0.01828 (8) 0.00925 (10) 0.00440 (9) 0.00363 (10) 0.2046 (62)

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995 ~ 173-197 183

Table 5 (continued)

T * x A YA P~ Pm* P~*

1.00 '~ 0.30078 1.00 0.50000 1.00 0.69922 1.00 0.89844 1.00 1.00000

~rB/<r A=0.5; EB/E A= 0.75 0.00000 0.75 t, 0.10000 0.75 t, 0.20000 0.75 ~ 0.30000 0.75 0.50000 11.75 0.69922 0.75 0.89844 (7.75 1.00000 1.110 ~.b 0 .10000

1.00 ~,.b 0.20000 1.00 0.30078 1.00 0.50000 1.0l) 0.69922 1.00 0.89844 1.00 1.00000

orb/~r~ ~ = 0.5; e~ /E A = 0.75 0.00000 0.75 b 0.20000 0.75 )' 0.30000 0.75 0.50000 0.75 0.69922 0.75 0.89844 0.75 1.00000 1.00 ,,b 0 .10000

1.00 h 0.20000 1.00 0.30078 1.00 0.50000 1.00 0.69922 1.00 0.89844 1.00 1.00000

c % / ( r a = 0.5; %/CA = 0.75 0.00000 0.75 0.50000 0.75 0.69922 0.75 0.89844 O.75 1.00000 1.00 0.00000 1.00 ~,b 0 .10000

1.00 b 0.20000 1.00 0.30078 1.00 0.50000 1.00 /).69922 1.00 0.89844 1.00 1.00000

0.66

0.75

1.00

0.0634 (36) 0.5536 (57) 0.4979 (18) 0.1083 (19) 0.1026 (36) 0.2746 (23) 0.5650 (13) 0.05865 (68) 0.2007 (40) 0.13650 (97) 0.62030 (90) 0.03883 (37) 0.4745 (60) 0.05485 (55) 0.67299 (75) 0.02899 (39) 1.00000 0.02505 (22) 0.70081 (38) 0.02964 (32)

0.00000 0.2941 (32) 0.61616 (90) 0.0663 ( 111 0.0006 (161 0.2041 (23) 0.65796 (60) 0.04106 (611 0.0034 (39) 0.12665 (87) 0.66727 (34) 0.02385 (25) 0.0082 (49) 0.0849 (11) 0.67903 (35) 0.01577 (25) 0.0462 (78) 0.04229 (41) 0.71186 (63) 0.009l)2 ( 11 ) 0.1038 (97) 0.()2013 (30) 0.75246 (36) 0.00523 (8) 0.336 (12) 0.00760 (14) 0.79698 (36) 0.0037(I (7) 1.00000 0.00264 177 0.82158 (38) 0.00363 1101 0.01318 (90) 0.7708 (611 11.4784 (22) I). 1483 1211 0.0287 (12) 0.5037 (30) 0.5256 (12) 0.09163 179) 0.0426 (30) 0.3307 (25) 0.5474 (13) 11.05885 (69) 0.1153 (45) 0.1718 (14) 0.5912 (13) 0.03838 (44) 0.2485 (54) 0,09152 (68) 0.63361 (81) 0.02974 (32) 0.5649 (59) 0.04196 (52) 0.67549 (87) 0.02616 (41) 1.00000 0.02505 (22) 0.7l)081 (38) 0.02964 (32)

0.00000 0.1503 (13) 0.70081 (38) 0.02964 (32) 0.024 (30) 0.0749 (48) 0.69247 (33) 11.0154 1171 0.031 (16) 0.05091 (93) 0.69790 (38) 1t.01034 (26) 0.064 (15) 0.02773 (46) 0.72235 (52) 0.00630 (13) 0.132 (15) 0.01383 (25) 0.75814 (39) 0.00394 (8) 0.403 (161 0.00560 (15) 0.79752 (37) 0.00313 (9) 1.00000 0.00264 (7) 0.82158 (38) 0.00363 (10) 0.00650 (68) 0.5398 (49) 0.5446 (101 0.0886 (22) 0.0288 (16) 0.3523 (171 0.56244 (94) 0.06012 (48) 0.0451 (30) 0.2448 1121 0.5751 (111 0.04350 (37) 0.1242 (59) 0.1336 (10) 0.6(i1670 (98) 0.03056 (36) 0.3060 (83) 0.07684 (83) 0.64277 (83) 0.02915 (47) 0.6168 (55) 0.03781 (52) 0.67776 (87) 0.02579 (45) 1.00000 0.(12505 (22) 0.70081 (38) 0.02964 (32)

0.00000 0.02112 (56) 0.82158 (38) 0.00363 (10) 0.162 (41) 0.(10788 (39) 0.74753 (39) 0.00246 (13) 0.349 (44) 0.00580 (39) 0.77154 (34) 0.00291 1211 0.667 (20) 0.00357 (21) 0.80226 (40) 0.00316 (19) 1.00000 0.00264 (7) 0.82158 (38) 0.00363 (10) 0.00000 0.2004 (181 0.70081 (38) 0.02964 (32) 0.030 (12) 0.1553 (24) 0.66371 (42) 0.02569 (53) 0.060 (14) 0.1249 (21) 0.64647 (47) 0.02297 (64) 0.072 1121 0.0949 (141 0.63887 (62) 0.01771 (37) 0.1923 (98) 0.06434 (82) 0.64438 (68) 0.01771 (30) 0.400 (11) 0.04401 (77) 0.66146 (85) 0.02004 (45) 0.7478 141) 0.03052 (46) 0.68358 (83) 0.02559 (48) 1.00000 0.02505 (22) 0.70081 (38) 0.02964 (32)

The uncertainty of the chemical potential /x B, i.e. 6 B for the calculation of the VLE was calculated by ( b[~): = (SA~) 2 + (~a) 2.

184 J. Vrabec et al. /Fluid Phase Equilibria 112 (1995) 173-197

3.3. Isothermal compressibility

The motivation to examine the isothermal compressibility is, that this quantity, needed here to determine the saturated densities, shows large statistical uncertainties for simulations in the region around the critical point. Therefore it is calculated in this paper with two different fluctuation formulae

1 1 - - . [ ( V 2) - ( V ) 2] (27)

~ T = k-"T " < V >

and

1 )311 1 ] /3T • < 7 > - < V > (28)

According to Hill (1956), Eqs. (27) and (28) should yield identical results. In all simulations we evaluated the fiT by both formulae. In Table 3 comparison for some state points is given. Actually the results show for state points with high density a very good agreement, their difference is much smaller than their statistical uncertainty. The results in the vapour phase close to the critical region show less agreement, but their statistical uncertainties are growing up to 17%, so that their error bars overlap in all cases. According to our investigation the results of both formulae show the same quality. For the calculation of VLE data we used always Eq. (27).

4. Phase equilibria from simulations and perturbation theory

Phase equilibria were calculated for ten different model mixtures considered already by Harismi- adis et al. (1991), for which the model parameters are compiled in Table 4. For the unlike interactions

Table 6 Comparison of densities and chemical potentials from (a) PT and (b) HSK with simulation results for the model mixture o" a / t r A = 1.0 and E B / E A = 0.5 at k T / e A = 0.75. The phase equilibrium for this system is shown in Fig. 2; ( p * = pA 3 /CA;

• 2 3 Or. = (X2 CrA 3 + 2XA XBtrA3B + XBO'B); /2i = (/Zi -- ~ i d ( T ) ) / k T )

(a) P* XA Pm,PT Pro,sire ]ZA,PT ~A,sim f£ B,PT ~t~ B,sim 0.04001 0.820 0.78969 0.78679 (33) - 5.745 - 5.742 (32) - 3.168 0.06931 0.600 0.74125 0.73133 (57) - 5.846 - 5.884 (22) - 2.709 0.08885 0.400 0.6801 0.6568 (10) - 5.938 - 5.980 (14) - 2.541 (b) P* XA Pm,HSK Pm,sim ]~A,HSK ~A,sim /~ B,HSK

- 3 . 2 0 4 (24) - 2 . 7 6 9 (17) - 2 . 6 2 1 (13)

~B,sim 0.04001 0.094 0.06505 0.06494 (38) - 5 . 7 4 9 6 - 5 . 7 5 1 8 (69) - 3 . 1 6 7 8 0.06931 0.078 0.14030 0.14155 (58) - 5 . 8 4 9 4 - 5 . 8 4 9 1 (67) - 2 . 7 0 8 4 0.08885 0.096 0.2402 0.2679 (28) - 5 . 9 4 0 - 6 . 0 2 1 (20) - 2 . 5 4 1 3

- 3 . 1 7 2 2 ( 6 1 ) - 2 . 7 1 7 9 (48) - 2 . 5 6 6 1 (87)

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197 185

the Lorentz-Bethelot (LB) mixing rule was used throughout

1 (JAB = "2( O'A "4- OrB) (29 )

CAB = V/CA • % (30 )

The present VLE data for the mixtures, compiled in Table 5 were obtained with liquid simulations in the NpT ensemble. Details about the simulations are given in the Appendix. For the vapour in all cases the Haar-Shenker-Kohler (HSK) equation of state (Fischer and Bohn, 1986) in combination with appropriate hard sphere mixture equations (Boublik, 1975; Mansoori et al., 1971) was used,

0.16 a

p;

0.12

0.08

0.04

0.00 0.0 0.2

0.16

p;

0.12

0.08

0.04

0.00

+

b +

ar~

I E

0.0 012

4 u-q-4

i I I 04 0.6 0.8

.r A, Y.4

+

o14

@

I I

0.6 0.8

Fig. 2. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture CB/O'A = 1.0, % / E A = 0 . 5 . Present results ( O ) are given in comparison with Harismiadis et al. (1991) ( v ) and FLB perturbation theory ( - - ) at k T / E A = 0.75. For k T / E A = 1.0 results are given from present calculations ( O ) in comparison with Harismiadis et al. ( ,7). The bars indicate the simulation uncertainties.

186 J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197

except for the indicated state points in the region around the critical point. There, also for the vapour simulations were performed.

For all averages from simulation, statistical uncertainties were determined with the method of Fincham et al. (1986). As the statistical uncertainties of the adjusted chemical potentials, we took, except for three cases indicated in Table 5, the statistical uncertainty of the chemical potentials from Widom's test particle method, which was believed to be a pessimistic choice. Now the uncertainties of all values in the phase equilibrium were obtained by the error propagation law.

The vapour pressure, bubble, and dew density of the pure components at the considered tempera- tures were calculated with the correlations given by Lotfi et al. (1992), the bubble and dew enthalpy (Lotfi, 1994) as well.

For all mixtures additionally VLE data at the lower temperature, mostly k T / e A = 0.75, were

0.12

p;

0.08

p;

0.08

a

', v 1 7 ' ,

,+ 0 + 18 IIl

0.04

0.00 0.2 0.4 0.6 0.8 1.0

XA, YA 0.0

0.12 b

I I

I 0.04

Y 0.00

0.0

i J

o12 o14

. +

e

o:6 o:8 o;

Fig. 3. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture O'B/O'A = 1.0, e B / e A = 0.66. Present results ( 0 ) are given in comparison with Harismiadis et al. (1991) ( v ) and FLB perturbation theory ( - - ) at k T / e A = 0.75. For k T / e A = 1.0 results are given from present calculations (C)) in comparison with Harismiadis et al. (x7). The bars indicate the simulation uncertainties.

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197 187

calculated by using the Weeks-Chandler-Anderson (Weeks et al., 1971) type Fischer-Lago-Bohn perturbation theory (PT) (Bohn et al., 1986) for the liquid side. The reason of applying it only to the lower temperature is its restriction to high liquid densities. PT yields the Helmholtz free energy in the form a(T, p, x). For prescribed values of the temperature and concentration and different values of the density the pressure

p2 p = 2A----~[a( p + A p , x i ) - a( p - A p , x i ) ]

and the chemical potentials (Lotfi and Fischer, 1989)

t x i - 2 A p 1 + - - a p + ~ p , - p 1 + A p / p

(31)

a p - A p , P

,-aP/P)l (132)

0.10

PT,

0.08

0.06

0.04

0.02

0.00

0.10

P;

0.08

a

L

i 0.0 0.2 0.1 (}.6 0.8 1.0

;rA~ yA

b

0,06 -

i 0.04

i ®

o

0.02

0.00 0.0 0.2 0 4 O.6

±

®

o

1

016 pT,~

Fig. 4. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture erB/erA = 1.0, E B / e A = 0.75. Present results ( 0 ) are given in comparison with Harismiadis et al. (1991) ( • ) and FLB perturbation theory ( - - ) at k T / e A = 0.75. For k T / E A = 1.0 results are given from present calculations ',C)) in comparison with Harismiadis et al. ( v ) . The bars indicate the simulation uncertainties.

188 J. Vrabec et al.// Fluid Phase Equilibria 112 (1995) 173-197

were calculated, where Ap denotes a small variation of the density. With the help of these data one is able to fit two simple polynomials for the chemical potentials as functions of the pressure for the prescribed temperature and concentration. The polynomials are used in the calculation of VLE instead of Eq. (1) for the two components. Because of the restriction of perturbation theory to high densities, the vapour in equilibrium has low densities and can therefore be described by the HSK equation.

For one model mixture for which VLE data have been calculated below, densities and chemical potentials from PT and HSK are compared with simulation results in Table 6. We see that the PT results for the density and the chemical potentials are excellent at high densities and become worse for lower densities. At the contrary, the HSK results are excellent for low densities and become worse with increasing density. Both findings fit into the general picture.

The results of the NpT + test particle method for the ten mixtures are presented together with the

0.05

p;

0.04

0.03

0.02

0.01

0.00 0.0 012 014 016 ols ,0

XA~ ~]A

0.05 b

p;

0.04

mll

0.03 -

0.02 •

0.01 y

0.00 0.0

i |

! b • I

t - , v i • I

o12 o14

D A ,

0.6 018 p;,

Fig. 5. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture or B / t r A = 1.5, c a / c A = 0.66. Present results ( O ) are given in comparison with Harismiadis et al. (1991) ( • ) and FLB perturbation theory ( - - ) at k T / e A = 0.75. For k T / E A = 1.0 results are given from present calculations ( © ) in comparison with Harismiadis et al. ( v ) . The bars indicate the simulation uncertainties.

J. Vr abec et al. / Fluid Phase Equilibria 112 (1995) 173-197 189

Gibbs ensemble results of Harismiadis et al. (1991) and with the results from perturbation theory in Figs. 2-11. In comparing the results we should keep in mind that the present NpT+ test particle method prescribes the liquid concentration and hence simulation uncertainties occur for the vapour pressure p,~, the vapour concentration y, and the saturated densities p' and p". On the other hand, the Gibbs ensemble NpT-modification prescribes the pressure p,~ and therefore simulation uncertainties occur in the concentrations x and y as well as in the saturated densities p' and p"; the Gibbs ensemble NVT-modification which had to be used in half of the cases, because it was difficult to specify the correct coexistence pressure in advance (Harismiadis et al., 1991), has uncertainties for all phase equilibrium data.

A survey of Figs.2-11 shows firstly that the phase equilibria for the mixtures correspond very well with the VLE for the pure components. We also observe a qualitative agreement of the results from

0.04

p;

0.03

0.02

0.01

0.00 0.0

0.04

p;

0.03

0.02

0.01

0.00 0.0

a

@ 0

012 , , -- ,

0.,1 0.6 0.8 l.O : r A , Y A

b

+

- - - - - E - - 4

p~

Fig. 6. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture O'B/O" A = 1.5, e B / e A = 0.75. Present results ( 0 ) are given in comparison with Harismiadis et al. (1991) ( • ) and F L B

perturbation theory ( - - ) at k T / e A = 0.75. Fo r k T / e A = 1.0 results are given from present calculations (©) in comparison with Harismiadis et al. ( v ). The bars indicate the simulation uncertainties.

190 J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197

both simulation methods and from perturbation theory. A general finding, however, is that the N p T + test particle method yields considerably smaller uncertainties in the saturated liquid densities than the Gibbs ensemble. A more detailed discussion of all model mixtures is given in the following.

In Fig. 2, component B is for both temperatures supercritical, the bubble and dew lines meet in a critical point of the mixture. For the lower temperature 13 VLE points are calculated, so that the scatter of our method can be estimated. The uncertainties of the present data for this system are throughout smaller than the uncertainties of the Gibbs ensemble. The PT shows its typical behaviour, in yielding for lower densities too high pressures.

In Fig. 3, the pure component B has also a VLE at the lower temperature. The uncertainties and the scatter of the present data for this system are very small. The Gibbs ensemble results seem to fail the vapour pressure of pure component B considerably. The agreement between simulation and PT

0.03

P;

0.02

0 . 0 1

a

0.00 0.0 012 014 016 018

XA~ YA

0.03 b

P;

0.02

0.01

0.00 0.0

1.0

i

& o16 o18 p2

Fig. 7. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture O'B/O'A = 1.5, eB/eA=l.0. Present results (O) are given in comparison with Harismiadis et al. (1991) (,7) and FLB perturbation theory (--) at k T / e A = 1.0. The bars indicate the simulation uncertainties.

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-107 1 t} I

for the vapour concentrations and vapour pressures is good, whereas the liquid densities from PT for x A ~ 0 are 3% too high.

In Fig. 4, component B is for the higher temperature (kT/e A = 1.00) just above its own critical temperature (kT/EB= 1.333 > To*= 1.310). The agreement between present results and PT is excellent. Here we can observe the typical behaviour, that the uncertainties of the vapour concentra- tion increase, when the gradient of the dew line decreases.

In Fig. 5, it was extremely difficult to obtain the VLE for this system at the lower temperature by simulations; especially for x A ~ 1 both test particle insertion and particle swap performed very poorly.

In Fig. 6, the calculation of the chemical potentials for the lower temperature was like for the system before extremely difficult. Here, for three state points indicated in Table 5. the statistical

P;

1.0 -J

0.8-

{}.6

0.4

0,2

0.0 0.0 0.2 O.4

o

°

0 .6 0 .8 1.11

• r 4 . !14

p;

~.ol b I

0.8

0.6 ~ •

{1.4

0.2

0.0 ~ - 0.0 012

|

- - - - r . . . .

014

:7; ~ \

o.{~ o.,~ PF,,

Fig. 8. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture O'B/O" A = 0.5, eB /e A = 0.5. Present results ( 0 ) are given in comparison with Harismiadis el al. (1991) ( • ) and FLB perturbation theory ( - - ) at k T / E A = 0.75. For k T / e A = 1.0 results are given from present calculations (O) in comparison with Harismiadis et ai. ('7). The bars indicate the simulation uncertainties.

192 J. Vrabec et al. /Fluid Phase Equilibria 112 (1995) 173-197

uncertainty of the chemical potential of the larger component was calculated by the addition of the uncertainties of the chemical potential of the smaller component ~B from test particle insertion and the uncertainties of the difference 6,a obtained by particle swap. The VLE of the higher temperature show an azeotropic behaviour (Table 5), in the pressure composition diagram the dew line touches the bubble line.

In Fig. 7, the uncertainties and the scatter of the present data are throughout reasonably smaller than for the Gibbs ensemble results. PT yields again vapour pressures which are 5% too high.

In Fig. 8, the dew line in the pressure composition diagram for the lower temperature has a very large gradient, therefore the uncertainties for the vapour concentration are extremely small. PT shows

1 . 0

p;

0.8-

0.6-

0.4-

0.2-

0.0 0.0

a

t4N ~

o o

Q O

~ B ~ O

I I I

0.2 0.4 0.6

b 1.O-

p;

0.8

0.6

• o

0.8 1.0 X A , YA

I t7 I I

lit

• •

0.4 w ~w • •

0.0 I T I -

0.0 012 0'.4 °.6 0.s p;,

Fig. 9. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture o" n / % = 0.5, E B / % = 0.66. Present results ( 0 ) are given in comparison with Harismiadis et al. (1991) ( v ) and FLB perturbation theory ( - - ) at k T / % = 0.75. For k T / % = 1.0 results are given from present calculations (©) in comparison with Harismiadis et al. ( '7) . The bars indicate the simulation uncertainties.

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197 193

0.8

p;

0.6

0.4 • @

0.2 ~ ~-~

I q I

0 . 0 , -- : : " 0 . 0 0.2 0.4 016

I

0.8 XA~ YA

0.8

1.0

p;

0.6

@

0.4 o

o

0.2 •

0.0 0.0 012 o:4

0

o

0.6 0.8

Fig. 10. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture o" B /o" A = 0.5, ~B/EA = 0.75. Present results ( O ) are given in comparison with Harismiadis et al. (1991) ( • ) and FLB perturbation theory ( - - ) at kT/EA=0.75. For kT/EA= 1.0 results are given from present calculations (C)) in comparison with Harismiadis et al. ( v ) . The bars indicate the simulation uncertainties.

a very good agreement with simulation results. The uncertainties of the liquid densities from the Gibbs ensemble are remarkable.

In Fig. 9, for the higher temperature we can observe good agreement with Gibbs ensemble results, which have small uncertainties and scatter in the pressure composition diagram, whereas NpT + test particle results have still smaller error bars.

In Fig. 10, for the lower temperature at x a ~ 0 the present vapour concentrations have large uncertainties, resulting from the poor performance of test particle insertion and particle swap. The agreement of PT with simulation results is excellent.

In Fig. 11, for the lower temperature test particle insertion and particle swap for x A < 0.5 performed very poor, therefore no VLE data are given there. The agreement of PT with simulation results is excellent.

194 J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197

0.20

p;

0.15

0.10

0.05

a

0.00 0.0

b 0.20 ±

p;

0.15

!1

0.10 1

o

0.05

0.00 0.0

o

o G o

0:2 014 0'.6 0'.8 XA, YA

1.0

o12 014

o

o

o16 o18 p;

Fig. 11. (a) Vapour pressure vs. composition and (b) vapour pressure vs. density diagram for the Lennard-Jones (LB) mixture tr B / t r A = 0.5, e B / e A = 1.0. Present results (D) are given in comparison with Harismiadis et al. (1991) ( • ) and FLB perturbation theory ( - - ) at k T / e A = 0.75. For kT /e .A= 1.0 results are given from present calculations ( O ) in comparison with Harismiadis et al. ( v ) . The bars indicate the simulation uncertainties.

5. Conclusions

In the present paper firstly different routes for the calculation of the thermodynamic quantities needed in the NpT + test particle method for mixtures were investigated. The chemical potentials and the partial molar volumes were calculated with Widom's test particle insertion method and the particle swap method, the isothermal compressibilities were obtained either as fluctuations of the volume or as fluctuations of the density. A methodological progress is seen in the adjustment method for the chemical potentials, Eq. (21) and Eq. (22), which combines results from Widom's test particle insertion and particle swap by a weighting procedure based on the simulation uncertainties. Compari- son of test particle results and the particle swap results for larger size ratios showed only in some few cases advantages of the particle swap method.

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995 J 173-197 I t)5

Comparison of the VLE results from the NpT + test particle method with those from the Gibbs ensemble and from perturbation theory were given for ten model mixtures at mostly two temperatures. We observed qualitative agreement of the results from all three methods. In peculiar we want to point out the good to excellent performance of perturbation theory• Comparing the simulation results for the pressure vs. composition behaviour, the present method shows considerably less scatter than the Gibbs ensemble results and in general smaller error bars for each individual point• In addition, a general finding is that the NpT+ test particle method yields for the saturated liquid densities uncertainties which are by an order of magnitude smaller than those from the Gibbs ensemble.

The accuracy of the phase equilibria from the NpT+ test particle method together with new equations of state (Kolafa and Nezbeda, 1994; Mecke et al., 1995) offers the possibility for the development and testing of mixing rules•

6. List of symbols

a

h H k N

P T l '

U i

V W

X

Y YA

Helmholtz free energy enthalpy instantaneous enthalpy Boltzmann constant number of particles pressure temperature volume partial molar volume of component i instantaneous volume Gaussian distribution liquid concentration in phase equilibrium vapour concentration in phase equilibrium vapour concentration of component A

6.1.1. Greek letters /3 /3T

/zi

A/z

P Pm Ap O"

Boltzmann factor isothermal compressibility statistical uncertainty Lennard-Jones size parameter chemical potential of component i reduced chemical potential of component i; ~£i = ( /'2'i - - Izid(T))/kT difference of chemical potentials density

• * 2 3 r e d u c e d density; Pm = P ( x~ O'A ~ + 2 x A x BOV2B + X B O'13 )

small variation of the density Lennard-Jones size parameter potential energy of a test particle of component i energy change of a swapped particle

196 J. Vrabec et al. /F lu id Phase Equilibria 112 (1995) 173-197

6.1.2. Subscripts A component A B component B c critical i component i sim obtained by simulation o- in phase equilibrium

6.1.3. Superscripts ' on the bubble line ,, on the dew line * reduced by Lennard-Jones parameters a obtained by statistical adjustment id perfect gas contribution 1 liquid res residual s obtained by particle swap v vapour w obtained by test particle insertion

Acknowledgements

The authors thank Dr. V. I. Harismiadis for having made available his tabulated VLE data. Moreover, they thank Dipl.-Ing. D. Heene for his interpretation of our VLE data, and cand.-ing. S. Bracke for his work with the perturbation theory. The simulation runs were performed on the Siemens $600 computer at the Rechenzentrum der RWTH Aachen.

Appendix A

This Appendix summarizes some details of our molecular dynamics simulations. All runs were performed in the NpT ensemble with 256 or 500 particles. Periodic boundary conditions and the minimum image convention were used, with a cut-off radius between 2.8 and 4.5 O'A; the long range corrections were considered. The translational equations of motion were solved with a fifth-order predictor-corrector method. The time step was taken to be 0.005 in usual units. Each production run was preceded by 5000 equilibration time steps, starting from a fcc-lattice and had a length of 66000 time steps. The pressure was kept constant by the method of Andersen (1980), where the membrane mass was set to 10 - 6 in the vapour and 5 × 10 - 4 in the liquid at low temperatures; the temperature was kept constant by momentum scaling after each time step. The uncertainties of the results were calculated according to the method of Fincham et al. (1986). Widom's test particle insertion was applied using normally 256 or 500 test particles in every time step, corresponding to the number of real particles, in some cases the number of test particles was increased to 1000. For the particle swap method, in every time step all particles of the smaller species were swapped to large.

J. Vrabec et al. / Fluid Phase Equilibria 112 (1995) 173-197 lt~7

References

Andersen, H.C., 1980. Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys., 72: 2384-2393.

Bohn, M., Fischer, J. and Kohler, F., 1986. Prediction of excess properties for liquid mixtures: results from perturbation theory for mixtures with linear molecules. Fluid Phase Equilibria, 31: 233-252.

Boublik, T., 1975. Hard convex body equation of state. J. Chem. Phys., 63: 4084. Fincham, D., Quirke, N. and Tildesley, D.J., 1986. Computer simulation of molecular liquid mixtures. I. A diatomic

Lennard-Jones model mixture for CO 2 / C e l l 6. J. Chem. Phys., 84: 4535-4546. Fischer, J. and Bohn, M., 1986. The Haar-Shenker-Kohler equation. A fundamental equation of state for gases. Molec.

Phys., 58: 395-399. Hansen, J.-P. and Verlet, L., 1969. Phase transitions of the Lennard-Jones system. Phys. Rev., 184: 151-161. Harismiadis, V.I., Koutras, N.K., Tassios, D.P. and Panagiotopouios, A.Z., 1991. How good is conformal solutions theory

for phase equilibrium predictions? Gibbs ensemble simulations of binary Lennard-Jones mixtures. Fluid Phase Equilibria, 65: 1-18.

Heyes, D.M., 1992. Chemical potential, partial enthalpy and partial volume of mixtures by N p T molecular dynamics. Molec. Simulation, 8: 227-238.

Hill, T.L., 1956. Statistical Mechanics. McGraw-Hill Book Company Inc., New York, 1956. Kolafa, J. and Nezbeda, I., 1994. The Lennard-Jones fluid: An accurate analytic and theoretically-based equation of state.

Fluid Phase Equilibria, 100: 1-34. Lotfi, A., 1994. private communications. Lotfi, A. and Fischer, J., 1989. Chemical potentials of model and real dense fluid mixtures from perturbation theory and

simulations. Molec. Phys., 66: 199-219. Lotfi, A., Vrabec, J. and Fischer, J., 1992. Vapour liquid equilibria for the Lennard Jones fluid from the NpT plus test

particle method. Molec. Phys., 76: 1319-1333. Mansoori, G.A., Carnahan, N.F., Starling, K.E. and Leland, Jr., T.W., 1971. Equilibrium thermodynamic properties of the

mixture of hard spheres. J. Chem. Phys., 54: 1523-1525. Mecke, M., Miiller, A., Winkelmann, J., Vrabec, J., Fischer, J., Span, R. and Wagner, W., 1995. An accurate van der Waals

type equation of state for the Lennard-Jones fluid. Int. J. Thermophys., in press. M611er, D. and Fischer, J., 1990. Vapour liquid equilibrium of a pure fluid from test particle method in combination with

NpT molecular dynamics simulation. Molec. Phys., 69:463-473 and 75: 1461-1462. Sindzingre, P., Ciccotti, G., Massobrio, C. and Frenkel, D., 1987. Partial enthalpies and related quantities in mixtures from

computer simulation. Chem. Phys. Lett., 136: 35-41. Panagiotopoulos, A.Z., 1987. Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a

new ensemble. Molec. Phys., 61: 813-826. Panagiotopoulos, A.Z., 1989. Exact calculations of fluid-phase equilibria by Monte Carlo simulation in a new statistical

ensemble. Int. J. Thermophys., 10: 447-457. Stapleton, M.R., Tildesley, D.J., Panagiotopoulos, A.Z. and Quirke, N., 1989. Phase equilibria of quadrupolar fluids by

simulation in the Gibbs ensemble. Molec. Simulation, 2: 147-162. Vrabec, J. and Fischer, J., 1995. Vapour liquid equilibria of mixtures from the NpT plus test particle method. Molec. Phys.,

in press. Weeks, J.D., Chandler, D. and Andersen, H.C., 1971. Role of repulsive forces in determining the equilibrium structure of

simple liquids. J. Chem. Phys., 54: 5237-5247; Andersen, H.C., Weeks, J.D. and Chandler, D., 1971. Relationship between the hard-sphere fluid and fluids with realistic repulsive forces. Phys. Rev., A 4: 1597-1607.

Widom, B., 1963. Some topics in the theory of fluids. J. Chem. Phys., 39: 2808-2812.