feature article - ACS Publications - American Chemical Society

J. Phys. Chem. 1983, 87, 4573-4585

(CHE-8119202) from the National Science Foundation. We thank Professor Joshua Jortner for sharing the results of the quantum yield measurements with us prior to

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publication and Dr. Willem Siebrand for helpful conversations. A full account of this work will be presented in a joint paper with Professor Jortner’s group.

FEATURE ARTICLE Fluid Phase Equilibria: Experiment, Computer Simulation, and Theory

Keith E. Gubblns,’t Katherlne S. Shlng,$ and Wllllam B. Streettt School of Chemical Engineerlng, Cornell University, Itheca, New York 14853 and Department of Chemlcal Engineering, University ot Southern Callfornla, Los Angeles. Calltorn& 90007 (Received: May 5, 1983)

We review recent advances in our understanding of the thermodynamics of fluid mixtures composed of fairly small, rigid molecules such as CHI, HC1, C2He, C2H4,etc. Phase equilibrium experiments at high pressures have shown the relationship between different types of critical lines and three-phase separations. Computer simulation techniques that yield the chemical potentials of the mixture components have been developed recently, and are reviewed. There have as yet been very few applications of these or other simulation techniques to mixtures, particularly for nonspherical molecules. Theoretical methods based on perturbation theory have advanced rapidly, and the present situation is reviewed, together with future needs. The complementary nature of experimental, theoretical, and simulation studies is emphasized. Research programs in which all three of these methods are combined are likely to prove particularly effective in the future. A key need in all work of this type at present is a better understanding of the details of the intermolecular forces for chemically complex mixtures.

1. Introduction Fluid behavior can be studied by three methods; experiment, computer simulation, or theory. In this review we consider the application of all three methods to thermodynamic properties and phase equilibria of mixtures of small, fairly rigid molecules such as H2, CHI, C02, CH,OH, HC1, etc. We do not deal with ionic fluids (plasmas, molten salts, electrolytes, liquid metals) or polymers, nor with the property behavior peculiar to aqueous solutions. Among the most significant advances that have occurred in recent years have been the rapid development of statistical mechanical perturbation theory and computer simulation methods; on the experimental side much new insight into phase equilibria and critical phenomena in nonideal mixtures has been gained, particularly through studies at high pressures. Computer simulations have been viewed with suspicion by some theorists and experimentalists, due largely to a misconception of the role that such simulations should play. At a Gordon Conference on Liquids a few years ago the question of whether computer simulators should be regarded as experimentalists or theorista was raised. An amusing but unedifying discussion School of Chemical Engineering, Cornel1 University.

* Department of ChemicaI Engineering, University of Southern

California.

0022-385418312087-4573$01.50/0

followed, which ended when a member of the audience suggested that such practitioners be termed “machinists”! Whether simulation is theory or experiment depends on the use that is made of the results (Figure 1). In this paper we shall be most interested in the use of simulations to provide “experimental” data on precisely defined model fluids that can be used to test theoretical approximations. Since the same intermolecular force laws are used in both the simulation and the theory an unambiguous test of the theoretical approximations is obtaiped. Such tests of a new theory are a very desirable prelude to comparisons with experimental data on real fluids, and provide much information about structure and motion at the molecular level. 2. Experimental Studies: Phase Equilibria at High Pressures Experimental studies of fluid phase equilibria at high pressures1-’ have shown that there are continuous tran(1) D. s. Tsiklis and L. A. Rott, Russ. Chem. Reu., 1967, 357 (1967). (2) G. Schneider, Adu. Chem. Phys., 17, 1 (1970). (3) G. Schneider, Ber. Bumenges. Phys. Chem., 76, 325 (1972). (4) G. Schneider in “Chemical Thermodynamics”,Vol. 2, Specialist Periodical Reports, The Chemical Society, London, 1978, Chapter 4. (5) W. B. Streett, Can. J . Chem. Eng., 52, 92 (1974). (6) W. B. Streett in “High Pressure Technology”, I. L. Spain and J. Paauwe, Ed., Marcel Dekker, New York, 1977, Chapter 12.

0 1983 Amerlcan Chemical Soclety

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Gubbins et al.

The Journal of Physical Chemistry, Vol. 87, No. 23, 1983

EXPERIMENT

Theory +

Potential

l------

II

THEoRY

r1-1

a Thewetical Model

Potentia I Model

m

I

I T

T

COMPUTER

SIMULATION

Figure 1. Computer simulation and theoretical studies are based upon the same intermolecular potential models; in comparing results from these two methods the simulation provides “experimental”data which can be used to make an unambiguous test of the theoretical approximations. In the comparison of simulation with experiment the simulation serves as “theory”4he comparlson tests the model used for the intermolecular potential.

sitions between phase diagrams that exhibit gas-liquid, liquid-liquid, and gas-gas phase separations. Critical lines are often observed to change continuously from one type of phase separation to another. When the lines representing a single degree of freedom (critical lines, threephase lines, pure-component vapor pressure curves, etc.) are plotted on P-T diagrams, the resulting graphs fall naturally into several different categories, providing a convenient basis for classification of fluid phase equilibria. A useful classification scheme has been devised by Van Konynenberg and Scott8vgwho used the van der Waals equation of state in a systematic study of fluid phase equilibria. On the basis of the P-T diagrams resulting from their calculations, Scott and Van Konynenberg grouped fluid phase diagrams into five classes, distinguished mainly by configurations of critical and threephase lines on P-T graphs. They recognized a sixth class that occurs in some aqueous systems, but was not among those predicted by the van der Waals equation. The basic classification scheme is shown in Figure 2. In classes I, 11,and VI, the gas-liquid critical line is continuous between the critical points of the pure components, C, and C,. In class 11, there is a liquid-liquid phase separation, bounded by the three-phase region LLG and the liquid-liquid critical line LL; these two lines intersect at the upper critical end point U. In class VI the liquid-liquid-gas three-phase region is bounded above and below by critical end points (U and L). In the class VI example illustrated in Figure 2 the liquid-liquid critical line has two branches, with a region of complete liquid miscibiity between; there are several other known configurations for the liquid-liquid critical lines in class VI systems, which have been described in detail by S ~ h n e i d e r . ~In - ~classes 111, IV, and V, the gas-liquid critical line is divided into two or three branches. In classes IV and V, the branch of the gas-liquid critical line originating in C, terminates in an upper critical end point while the branch originating in C, rises to a maximum pressure and passes continuously into a liquid-liquid critical line, terminating in a lower critical end point L. In class IV,there is a second liquid-liquid phase separation at lower temperatures, with a critical line ending in a second upper critical end point. In the final class, 111, the branch of the critical line originating in C, rises to high pressures, sometimes after passing through maximum and (7) J. S. Rowlinson and F. L. Swinton, ‘Liquids and Liquid Mixtures”, 3rd ed, Butterworth Scientific, London, 1982, Chapter 6. (8) P. H.Van Konynenberg, Ph.D. Thesis, University of California, Lo8 Angeles, 1968. (9) P . H. Van Konynenberg and R. B. Scott, Phil. Trans. R. SOC. London, 298,495 (1980).

I

T

T

P

r

T

I

T

Flgure 2. The six classes of fluid phase diagrams:*l9 -, pure component vapor pressure curves; -, critical line; - -, three-phase region; 0,critlcal point of pure component; A,crlticai end point. The subscripts a and 0 refer to the two components. The letters GL,LL, and LLG refer to gas-liquid, liquld-liquid, and liquid-liquid-gas, respectively. Examples of each class of behavior are listed in the figure. For other examples, see ref 2-7 and 10.

--

e

minimum pressures and/or a minimum in temperature. It is likely that these critical lines terminate at high pressures in a critical end point formed by an intersection with a three-phase region solid-liq~id-gas.~Such points lie at pressures in excess of 15000 atm in many systems. In some class I11 systems the critical line rises to supercritical temperatures at high pressures, leading to so-called gas-gas phase separations. Schematic three-dimensional drawings of P-T-X phase diagrams for the classes shown in Figure 2 can be found in publications of SchneiderF4 Streett and Gubbins,loand Rowlinson and S ~ i n t o n .Recent ~ phase equilibrium experiments for binary mixtures, carried out at Cornel1 include the following: simple nonpolar systems, such as krypton xenon,” krypton/methane,12 and krypton/ ethane;’$ systems containing polar and/or hydrogen bonded liquids such as carbon dioxideldimethyl ether,14 methanol/dimethyl ether,16and dimethyl ether/water;16 and a family of hydrogen binary mixtures including hydrogen/nitrogen,17hydrogen/methane,18 hydrogen/carbon (10) W.B.Streett and K. E. Gubbins, “Proceedings of the 8th Symposium on Thermophysical Properties”, Vol. 1, ASME, 1981, p 303. (11) J. C. G. Calado, E. Chang, and W. B. Streett, Physica A (Amsterdam). 117. 127 (1983). .~ (12) J.’C. G. Calado, U. Deiters, and W. B. Streett, J. Chem. Soc., Faraday Trans. I , 77, 2503 (1981). (13) J. C. G. Calado and W. B. Streett, unpublished. (14) C. Y. Tsang and W. B. Streett, J . Chem. Eng. Data, 26, 155 (1981). (15) E. Chana, J. C. G. Caledo, and W. B. Streett, J. Chem. Eng. Data, 27, 293 (1982). (16) M. E.Pozo and W. B. Streett, paper presented at Third International Conference on Fluid Properties and Phase Equilibria for Chemical Process Design, Callaway Gardens, GA, Ami1 10-15,1983; Fluid Phase Equilib., in press. (17) W.B.Streett and J. C. G. Calado, J . Chem. Thermodyn., 10,1089 (1978). I

~~

The Journal of Physical Chemistry, Vol. 87, No. 23, 1983 4575

Feature Article

F

P

T

Flgure 4. P-T diagrams for mixtures in which one pure component is a supercritical gas in the temperature range of interest. I n these diagrams A and C are the triple and critical points of the less-volatile component, and AD is its melting curve. Examples of systems that exhibit these types of phase diagrams are given in the text.

T;

T2

T

Figure 3. Transition from class I system (a) to one of the class I I I systems (b). Legend as in Figure 2, except that pure component triple points are also shown as triangles. The square labeled Q is a quadrupole point (see text).

monoxide,19 hydrogen/carbon dioxide,20 hydrogen/ ethylene?l and hydrogen/ethane.22 The hydrogen/X phase diagrams have been studied at pressures up to about 6000 atm. Together with earlier studies of helium/X and neon/X systems carried out at pressures as high as loo00 atm,17they provide a comprehensive picture of fluid phase behavior in binary systems in which one component is a supercritical gas at temperatures where the other is a liquid. These systems, which are discussed in the remainder of this section, form a subgroup of class I11 in the classification scheme of Figure 2. Almost all systems in which one of the components is helium, hydrogen, or neon (critical temperatures 5.6,33.2, and 44.2 K, respectively) fall into this category. The phase diagrams of these systems are related to those of classes I and I11 in Figure 2 through the sequence of changes shown in Figure 3. Figure 3a is a P-T projection of the principal boundary lines of a class I system in which two completely miscible liquids solidify to form partially miscible solid phases S1and Sa. The solid lines are the vapor pressure, sublimation, and melting curves of the pure components (aand p). Point Q is a quadruple point, where the four phases S1,Sa,L, and G are in equilibrium. Radiating from this point are four lines (in P-T-X space four triplets of lines) representing equilibria between the four possible combinations of three phases drawn from this group. If the temperature of C, lies far below that of A,, the three-phase region S2 + L + G (line Q-A,) curves upward and intersects the critical line at critical end points U1 and U2,as shown in Figure 3b. No liquid phase exists at temperatures between these points. ~

~~

~~~~

~~~~~

~

(18) C. Y. Tsang, P. Clancy, J. C. G. Calado, and W. B. Streett, Chem. Eng. Commun., 6, 365 (1980). (19) C. Y. Tsang and W. B. Streett, Fluid Phase EquiZib., 6, 261 (1981). (20) C. Y. Tsang and W. B. Streett, Chem. Eng. Sci., 36,993 (1981). (21) A. Heintz and W. B. Streett, J. Chem. Eng. Data, 27,465 (1982). (22) A. Heintz and W. B. Streett, Ber. Bunsenges. Phys. Chem., in press.

It is the right-hand portion of Figure 3b that is of interest here; the “triangular” region bounded by the vapor pressure curve A,-C,, the critical line C,-Ul and the three-phase region A,-Ul, is the region of gas-liquid separation. (In many cases the left-hand portion of the phase diagram is very nearly degenerate-that is point Q almost coincides with A, and point U2 with Cp) D ~ k o u p i lfor ,~~ example, has shown that the solubility of nitrogen in liquid hydrogen is of the order of to 10-lo mol % at temperatures from 20 to 30 K. With increasing difference between the critical temperatures of the pure components, the critical lines and three-phase lines solid-liquid-gas (SLG) can be arranged in the sequence shown in Figure 4a-e. In Figure 4a the three-phase region SLG has a minimum in temperature, and intersects the critical line at critical end point U (e.g., hydrogen/nitrogen,17 hydrogen/methane18). Details of phase behavior in the vicinity of this temperature $minimum have been discussed by Calado and Streett.24 In Figure 4b the slope of the SLG projection is everywhere positive; examples include n e ~ n / a r g o nand ~ ~ hydrogen/ carbon dioxide.20 In Figure 4c the critical end point lies at temperatures above C, and the separation into two fluid phases therefore extends to supercritical temperatures-a phenomenon commonly called gas-gas equilibrium. An example is the system helium/argon26where the critical end point lies at about 11000 atm and 200 K, about 50 K above the argon critical temperature. In parts d and e of Figure 4 the critical and three-phase lines diverge with increasing temperature and pressure, and the gas-gas phase separations persist to the highest pressures at which these systems have been studied (approximately 10000 atm). The systems helium/ hydrogen27and helium/nitrogen yield phase diagrams similar to Figure 4d, while helium/methane28 and h e l i u m / ~ e n o nyield ~ ~ diagrams similar to Figure 4e. If the critical line has a temperature minimum, as in parts c and d of Figure 4, the supercritical phase separation (23) Z. Dokoupil in “Advances in Cryogenic Engineering”,Vol. 6, K. D. Timmerhaus, Ed., Plenum Press, New York, 1961. (24) J. C. G. Calado and W. B. Streett, Fluid Phase Equilib., 2, 275 (1979). (25) W. B. Streett and J. L. E. Hill, J. Chem. Phys., 54,5088 (1971). (26) W. B. Streett and A. L. Erickson, Phys. Earth Planet. Inter., 5, 357 (1972). (27) W. B. Streett, Astrophys. J., 186,1107 (1973). (28) W. B. Streett, J. L. E. Hill, and A. L. Erickson, Phys. Earth Planet. Inter., 6, 69 (1972). (29) J. de Swam Arons and G. A. M. Diepen, J. Chem. Phys., 44,2322 (1966).

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The Journal of Physical Chemistry, Vu/. 87, No. 23, 1983

T P A

Figure 5. P-T-X phase diagram for a system similar to Figure 4b. (a) is a schematic threedimensional diagram, and (b), (c) and (d) are projections of lines and surfaces on the three principal coordinate planes (see text of discussion).

Gubbins et al.

less-volatile component (e.g., COPin the H2/C02system), and AD is its melting curve. The region of coexistence of three-phases, solid-liquid-gas, is the surface AFUEA (it has been left partially unshaded in the interest of clarity). This surface contains three lines, AE, AU, and AFU, representing solid, liquid, and gas phases, respectively. The parallel lines shading this surface are tie lines, connecting triplets of points that represent three coexisting phases. The critical end point U is a limiting point at which the gas and liquid phases become identical, in the presence of a solid phase (point E). Because P and T are field variables-that is, they are the same for phases in equilibrium-the surface AFUEA is a ruled surface, perpendicular to the P-T plane, that projects as a line, AFUE, on the P-T coordinate plane (Figure 5b). Four isotherms, Tl-T4, and one isobar, P I , are shown in Figure 5a. Tzis at the temperature of the critical end point U; the upper branch of this isotherm is a region of solid-liquid equilibrium that terminates at low pressures at a point on the pure component melting curve (AD). The lower branch of T2is a gas-liquid region that forms a dome (i.e., it has a maximum in pressure) at point U, where it is tangent to the upper branch. At higher temperatures (T3,T4)the gas-liquid and solid-gas regions are separate; the former diminishes in size with increasing T and vanishes at C,. The solid-gas phase separation extends to higher pressures, and is presumably bounded by another three-phase region solid-solid-gas, where the second component undergoes pressure-induced solidification. Figure 5c is a projection of the three-phase region and isotherm T3 on the P-X coordinate plane, and Figure 5d is a projection of isobar Pl and the three-phase region on the T-X coordinate plane. C,U is the gas-liquid critical line for the mixture. Additional three-dimensional drawings of high-pressure phase diagrams can be found in ref 2-7 and 18-20.

is usually referred to as gas-gas equilibrium of type 2, as distinguished from type 1in Figure 4e, where the slope of the critical line is everywhere positive; the distinction is 3. Computer Simulation of no fundamental significance. The term gas-gas equilibrium, although widely used to describe these supercriThe usual Monte Carlo (MC) and molecular dynamics tical phase separations at high pressures, is somewhat (MD) techniques30 used to simulate fluids can yield the misleading. The separations occur in fluid mixtures that internal energy and pressure with reasonable accuracy; have been compressed to liquidlike densities at supercrihowever, they do not give good results for the free energy tical temperatures. The elegant work of S ~ h n e i d e r ~ - ~ or chemical potential, and thus do not enable phase clearly shows that there is a logical continuity between equilibrium conditions to be calculated directly. Several these phase diagrams and those of more conventional special simulation techniques have been devised to oversystems. His experiments suggest that gas-gas phase come this difficulty. For mixtures the most useful of these separation at high pressure is a liquid-liquid-type sepatechniques are those that give the chemical potential diration, displaced to high temperatures at high pressures. rectly, rather than the free energy. In the subsection below These separations occur in mixtures that would probably we describe some recent advances in these methods.31 exhibit partially miscible liquid phases, but for the fact Only a few applications have yet been made to mixtures. that one freezes at temperatures above the critical temMost of the simulation studies so far (Section 3.2) have perature of the other. This tendency to form partially been by conventional methods, and so have not yielded miscible liquid phases asserts itself at high pressures, where the chemical potentials or phase equilibria. fluid mixtures have been compressed to liquidlike densi3.1. Methods for the Chemical Potential. Methods for ties. calculating the chemical potentials directly can be conNo detailed studies of solid-liquid-gas three-phase regions, of the types shown in Figure 4, have been reported (30)For a review of the conventional MC and MD methods see, for for mixtures of condensed gases. In our experiments we example: J. P. Valleau and S. G. Whittington, “Modern Theoretical have been able to detect the presence of a solid phase in Chemistry”, Vol. 5, “Statistical Mechanics”, Part A, B. J. Berne, Ed., equilibrium with gas and liquidz6and to measure the gas Plenum, New York, 1977,Chapter 4; also, W. G. Hoover and A. J. C. Ladd, “Molecular-Based Study of Fluids”, G. A. Mansoori and J. M. and liquid compositions, but we have not been able to Haile, Ed., American Chemical Society, Washington, DC, Adv. Chem. determine the composition of the solid. The quantitative Ser., 1983, Chapter 2, p 204. descriptions of these three-phase regions are therefore (31)For a more detailed discussion of the older methods used to calculate A or p see the following: J. P. Valleau and G. M. Torrie, incomplete; however, on the reasonable assumption that “Modern Theoretical Chemistry”, Vol. 5, “Statistical Mechanics“, Part the solid phase contains less dissolved gas than the liquid, A, B. J. Beme, Ed., Plenum, New York, 1977,Chapter 5;J. A. Barker and it is possible to deduce the qualitative features of the D. Henderson, Rev. Mod. Phys., 48,596(1976);N. Quirke, “Proceedings P-T-X diagrams. A schematic three-dimensional drawing of the N.A.T.O. Summer School on Superionic Conductors”, Odense, Denmark, Plenum, New York, 1980; K. S. Shing and K. E. Gubbins, of the diagram described by Figure 4b is shown in Figure “Molecular-Based Study of Fluids”, G. A. Mansoori and J. M. Haile, Ed., 5. In the schematic three-dimensional diagram of Figure American Chemical Society, Washington, DC, Adv. Chem. Ser., 1983, 5a, A and C, are the triple and critical points of the Chapter 4, p 204.

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The Journal of Physical Chemistry, Vol. 87, No. 23, 1983 4577

veniently divided into those based on the test particle equations (usually used in the canonical or microcanonical ensembles) and those that use the grand canonical ensemble. Test Particle Methods. Consider a mixture of components, A, B, ..., a,...)M in which there are NA molecules of A, NBof B, etc. in volume V at temperature T . The configurational chemical potential of component a,pa,, is given by

‘ “ ” ‘

where the last form follows because the N , are large, 8,(N,) is the (canonical) configurational partition function when there are N , molecules of component a,and Q,(N, - 1 ) is the corresponding function when one a molecule is removed. It is s t r a i g h t f o ~ a r dto~show ~ that ( 1 ) reduces to pac

= -kT In (exp(-pU,))N-l

+ kT In p a

(2)

where /3 = l / k T , ( ...)N-l is a canonical ensemble average over the N - 1molecules interacting with potential energy UN-.l (excluding molecule 1 of component a, the “test particle”), and V, is the intermolecular potential energy of the test particle due to interactions with all the other ( N - 1) molecules (2,3, ...,N) in the mixture. If UNis the total intermolecular potential energy for the N molecule system, we can write U,(X,X~...x,) = U,(X~;X~...XN) + U,-~(X~X~...XN) (3) where xirepresents the coordinates for molecule i on which the intermolecular potential depends (center-of-mass location, orientation, etc.). It is convenient to introduce the residual chemical potential par = pOc- pa:d, where paidis the ideal gas value at the same temperature, density, and composition as the real fluid. For an ideal gas U, = 0, so that from (2) we see that the pacid= kT In p a and par = -kT

In (exp(-PU,))N-l

(4)

Equation 4 was first derived by W i d ~ mand , ~ ~has been called by various names, including the Widom equation, the test particle equation, and the potential distribution theorem.33 An alternate form of (4), useful for computer simulation work, is

(5) where f(U,) dU, is the probability that U, lies in the range U, to U, + dU,. Equations 4 and 5 can be interpreted as follows. An array of test particles of component a are placed at fixed locations and orientations in the fluid of N - 1 molecules, and exp(-PU,) is obtained as a function of the coordinates of all of the N - 1 molecules; a Boltzmann-weighted average over the coordinates xP..xN is then taken. We note that (a) the averages in (4) and ( 5 ) are taken over an ensemble in which the test particles do not interact with the N - 1 solvent molecules-the test particle is a “ghost”; (b) these equations are general-they do not rely on any approximation of pairwise additivity nor on any assumption concerning the type of molecules (rigid, spherical, etc.); ( c ) attractive configurations will be important in determining the average in (4) and (5),because of the exp(-PU,) term, but these will be sampled less and less frequently as the density (32)B. Widom, J. Chem. Phys., 39, 2808 (1963). (33)B. Widom, J.Phys. Chem., 86,869 (1982).

1.0:

0.1-

f

tB

C

t A

Flgure 8. The distribution functions f and g, and the quantities f exp(-@U,) and f exp(-DUt)lg for a Lennard-Jones fluid at k T l t = 1.2, p 8 = 0.85 kom MC simulation. The shaded area gives exp(-/3& and hence the chemical potential (see eq 5). The part of fexp(-/3Ut) that is shown as a solid curve (A to B) is obtained directly in the simulation, while the part shown as a dashed curve (B to C) is obtained by the f-g sampling method (from Shing and G ~ b b i n s ~ ~ ) .

increases (or as the a-particle diameter increases, for mixtures). In principle (4) can be used to calculate par in a simulation for any density. In practice, the length of run needed becomes prohibitive at high density because of the difficulty of sampling the attractive regions; at high density any “holes” in the liquid make a large contribution to par, but the sampling of such holes is inefficient by conventional methods. This is illustrated in Figure 6. In practice it is that for Lennard-Jones fluids this problem becomes serious for pa3 > -0.5-0.65 ( p is the number density, CJ is the molecular diameter). Two methods are available to overcome this high density problem. The first makes use of an “inverse test particle” equation, while the second uses umbrella sampling. The inverse test particle equation is obtained by noting that (exp(-/3U,))N-1 = (exp(PU,)),-l, where ( . . . ) N is now a canonical average over all N molecules interacting with potential energy, U,, including molecule 1 of component a;this relation is easily d e r i ~ e d ~by~ B writing ~ out the ensemble averages involved and using (3). Thus an alternative equation to (4) is37 par =

kT ln (exp(Put,))~

(6)

or

where g(U,,) dU,, is the probability that U , lies in the range U , to U , + dU, when the test particle is a real (34)K.S. Shing and K. E. Gubbins, Mol. Phys., 43,717 (1981). (35)K. S. Shing and K. E. Gubbins, Mol. Phys., 46, 1109 (1982). (36)J. G. Powles, W. A. B. Evans, and N. Quirke, Mol. Phys., 46,1347 (1982). (37)This equation was derived independently by M. J. de Olivera (private communication to J. S. Rowlinson, 1979) and by K.S. Shing (1980). See ref 33 and 35 for derivation.

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The Joumal of Physical Chemistry, Voi. 87, No. 23, 1983

molecule and interacts with its neighbors. Repulsive configurations will be important in determining the average in (6) and (7) because of the exp(PU,) term. Such configurations will only occur rarely at normal temperatures, because the test particle is a real molecule. Thus (6) is not directly useful in simulations, except perhaps at high temperatures. It i s useful when combined with the test particle equation, (4), however. From (5) and (7) it can be shown35that for a given U, the functions f and g are related hy = exp(8pcl,,)exp(-PUdf(Ud

(8)

Thus, if there is a range of U , where both f and g can be calculated reasonably accurately, (8) can be used to estimate the chemical potential. A more accurate procedure, known as ”f-g sampling”, is the following. The chemical potential is given by the area beneath the f exp(-/3Ut) curve-the shaded area in Figure 6 (we omit the subscript a for pure fluids). The distribution function f can be obtained with reasonable accuracy only over the range of Ut from A to R. The remaining Ut region (B to C) will not be sampled by the test particle method alone. However, if g is also calculated we can make an estimate of exp(@p,) = f exp(-PU,)/g from the region of overlap of the f and g distributions. g is obtained accurately over the range B to C; we can then use (8) to estimate f exp(-j3Ut) in this region, and hence obtain the remaining part of the integral. By suitably choosing the point B it is possible to maximize the accuracy of the calculation. The f-g sampling method works for pure fluids of spherical molecules up to the highest liquid densities, and for most mixtures encountered in practice. It will fail for cases where there is insufficient overlap between the f and g curves to use eq 8. This will occur for highly nonideal mixtures when the test particle is much larger than the solvent molecules. This problem can be partially solved by using restricted umbrella sampling as described below. The conventional test particle method, based on (4), has been used to study hard sphere,%spherical Lennard-Jones (LJ),35shifted-force L e n n a r d - J ~ n e s and , ~ ~ two-center Lennard-Jones39*40 fluids. The f-g sampling technique has been applied to pure LJ fluids and their m i ~ t u r e s . ~The ~~’ test particle method offers three important advantages. First, only minor changes are required to conventional MC or MD programs, in contrast to the methods described below. Second, it can be applied to MD as well as MC. Third, it does not involve any “unnatural” sampling procedures (in contrast to umbrella sampling), so that other properties (thermodynamic properties, correlation functions, etc.), in addition to the chemical potential, can be obtained in the usual way. The principal limitation of the method is it.. possible failure for very large solutes or high densities. Test Particle Method with Umbrella Sampling.34 In the umbrella sampling method42one generates a biased chain of molecular configurations by assigning a weight w(U,) to each value of the configurational energy U, of the test particle. This weighting function is chosen to be large for those configurations which contribute significantly to the property of interest. Thus,configurations are chosen with probability proportional to w( U,) exp(-/3UN-,) rather than exp!-BIIN.,). For the chemical potential the ensemble (38) D. J. Adams, Mol. Phys.,28, 1241 (1974). (39)S. Romano and K. Singer, Mol. Phys., 37, 1765 (1979). (40)J. G. Powles, Mol. Phys.,41, 715 (1980). (41)K. S. Shing and K. E. Gubbins, Mol. Phys.,49, 1121 (1983). (42)G.M.Torrie and J. P. Valleau, Chem. Phys.Lett., 28,578(1974); d. Comp. Phys., 23,187 (19’77);see also J. P. Valleau and G. M. Torrie, ref 31.

Gubbins et al.

average required f,(U,) that given by (4), (exp(-@Ub)) (we omit the subscript N - 1for convenience here), and this can be written as j d x N - ’ w-’ exp(-PU,)w (exp(-PU,)) =

exp(-PVN-J

dxN-’ w-lw exp(-@UN-,)

= (exp(-@U,) / w )/,

( 1/ w ),

(9)

where w is an arbitrary function of U,, and (...), denotes an ensemble average over the weighted chain of configurations. Equation 9 can also be written as ( exp(-P V,) ) =

(l/w),-’

1.’” f,(U,)

exp(-PUt,)w(UJ’

dU, (10)

where f,(U,) d(U,) is the probability that U, lies in the range U, to U , + dU, in the weighted system. The weighting function w must be chosen so as to give a distribution f, that is relatively flat (or “umbrella-shaped”) over the region A to C of Figure 6. This can be achieved in practice by first carrying out a short run to estimate f(U,) of eq 5, and then making w( U,) proportional to f-’. In cases where the region BC of Figure 6 is large (e.g., very high density or large solute molecules) the statistical errors in f i n this region may be so large that it is not possible to estimate suitable weighting functions over the whole region. In such cases it may be necessary to use several stages of umbrella sampling to cover the full range A to C. This method works for pure LJ fluidsMup to the highest densities (two umbrella sampling stages are needed for the highest densities and low temperature), and has been used for LJ mixtures35 when the solute molecule is large, ( U - / C T B ~ ) ~ > -1.5. It has also been used to calculate the free energy in LJ mixtures as a function of comp0sition,4~ and hence determine the phase equilibrium conditions. The advantage of the umbrella sampling method is that it works at high densities and for very nonideal mixtures. Among the disadvantages are that it is restricted to MC and cannot be used in MD calculations, and it requires more program modification than the test particle method. The determination of a suitable weighting function requires some trial and error, particularly if several stages are needed. Also, although the method yields the chemical potential efficiently, other properties (pressure, energy, molecular correlation functions, etc.) cannot usually be obtained with sufficient accuracy, and a separate simulation is required. Although analogous equations to (9) can be written for these other properties (one simply replaces exp(-@U,) there by the appropriate dynamical variable for the property of interest), the weighting functions required to give p may not be suitable for these other properties. This last drawback can be overcome at intermediate densities (up to pa3 0.75 for LJ fluids) by using a technique known as restricted umbrella sampling5 in which the weighting function acts only on the test particle, and so does not influence the structure of the fluid; the test particle is now a “smart ghost” (the solvent molecules are unaware of the test particle, but the test particle seeks out the “holes”). The use of restricted umbrella sampling has the same effect as using a very large number of unbiased test particles, and results in improved overlap between the distribution functions f and g. For a LJ fluid at p 2 = 0.7 this technique increases the number of configurations that contribute to the integral in eq 5 by

-

(43) G. M. Torrie and J. P. Valleau, J. Chem. Phys., 66,1402 (1977).

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The Journal of Physical Chemistry, Vol. 87, No. 23, 1983 4578

TABLE I: Computer Simulation Studies a factor of 30. The method fails at very high densities, of Fluid Mixtures however. Grand Canonical Monte Carlo. In this ensemble, the references chemical potential p, temperature T, and volume V are potential model MC MD fixed and the density fluctuates. The mean density is hard spheres (additive diameters) 55-59 60, 61 found as an ensemble average at the end of the simulation. hard spheres (nonadditive 62 63 This simulation method has been implemented in somediameters) what different ways by Norman and F i l i n ~ v , ~ ~hard spheres t square well 64 ad am^,"^' Rowley, Nicholson, and Parsonage,@Mezei,Q hard disks 65,66 67,68 Yao et al.,60and Coker and Watts.51 The method involves 69-7 2 soft spheres [ u = e ( o / r ) ’ ’ ] Lennard-Jones (bulk) 35, 41, 52, 84-104 essentially two steps. The first is the displacement of the 73-83 particles in the system and is identical with the procedure Lennard-Jones (gas-liquid interface) 105 105, 106 used in the canonical ensemble simulation. The second exponential-6 107 step involves adding or removing particles according to the Bo betic-Barker 91 requirements of the grand canonical ensemble weighting 108, 109 linear molecule in Lennard-Jones function, so that the number density in the system flucsolvent nonlinear molecule in Lennard110-11 2 tuates. Jones solvent Grand canonical Monte Carlo works best at high temLennard-Jones molecule in 113 peratures and low densities, since the addition of molecules Stockmayer solvent is then allowed with sufficient frequency for adequate 114 Stockmayer (LJ + dipole) sampling of the density fluctuations relevant to this enhard sphere plus multipole 115 semble. Recently, M e ~ e has i ~ ~modified the sampling hard spherocylinders 116-1 1 8 hard diatomics (dumbells) 119,120 procedure so that states of higher densities can be studied. two-center Lennard-Jones 121,122 In his cavity biased &TV) Monte Carlo method, a network diatomics of uniformly distributed test points was generated ir, the hard diatomics plus electrostatic 123 fluid and the fraction of these points in a suitable cavity interactions was found. Insertion of a new particle was attempted at two-center Lennard-Jones 51 this cavity instead of at randomly selected points. The diatomics plus electrostatic interactions fraction of test points in the chosen cavity also allows the proper normalization of the ensemble averages. For the Lennard- Jones fluid, Mezei found that the efficiency of then also be angle dependent. Furthermore, since it is p the insertion process was increased by a factor of 8 while that is specified, the density (and also the composition in the required CPU time increased by a factor of 2.5. the case of mixtures) is not known until the end of the Since the number of particles in the system is allowed simulation; this is inconvenient in practice. This method to fluctuate, the ensemble permits density fluctuations and is restricted to Monte Carlo calculations, and does not also permits concentration fluctuations in the case of appear to be useful for molecular dynamics. mixtures. Therefore, this ensemble should be more suit3.2. Simulation Studies of Fluid Mixtures. There is able for studying systems close to phase transitions and now a large body of literature on computer simulations of for systems close to the critical region; also this method pure fluids, but studies of mixtures are much more limited. is particularly suited to studies of adsorption@and chemA convenient, though rather arbitrary, distinction can be ical r e a ~ t i o n s .Grand ~ ~ canonical Monte Carlo simulations made between studies of simple model fluids, in which the are more complex and more time consuming than canonaim is to provide data suitable for testing theories, and ical ensemble Monte Carlo simulations. Adams noted that studies aimed at simulating real experimental mixtures the results are sensitive to errors in the random-number (e.g., much of the work on aqueous solutions). It is the generators used. At low temperatures and high densities, former category on which we focus attention here. A it is very difficult and time consuming to sample the summary of the existing studies of which we are aware is density fluctuations adequately, even with Mezei’s imgiven in Table I. There have been extensive studies of proved cavity biased version. This problem will become more severe if a more complex, angle-dependent potential is used, because the success of the insertion attempts will (44) G. S.Norman and V. S. Filinov, High Temp. Res. USSR, 7, 216 (1969). (45) D. J. Adams, Mol. Phys., 29, 307 (1975). (46) D. J. Adams, Mol. Phys., 32, 647 (1976). (47) D. J. Adams, Mol. Phys., 37,211 (1979). (48) L. A. Rowley, D. Nicholson, and N. G. Parsonage, J. Comput. Phys., 17,401 (1975);D. Nicholson, N. G. Parsonage, and L. A. Rowley, Mol. Phys., 44,629 (1981); W. Van Megan and I. K. Snook, ibid., 45,629 (1982); D. Nicholson and N. G. Parsonage, ‘Computer Simulation and the Statistical Mechanics of Adeorption”,Academic Press, London, 1982. (49) M. Mezei, Mol. Phys., 40, 901 (1980). (50) J. Yao, R. A. Greenkorn, and K. C. Chao, Mol. Phys., 46, 587 (1982). (51) D. F. Coker and R. 0. Watts, Mol. Phys., 44, 1303 (1981). (52) I. R. McDonald in “Statistical Mechanics”,Vol. 1, K. Singer, Ed., Chemical Society, London, 1973, pp 138-48. (53) D. Henderson, Annu. Reu. Phys. Chem., 25, 461 (1974). (54) D. Levesque, J.-J. Weis, and J.-P. Hansen in ‘Monte Carlo Methods”, K. Binder, Ed., Springer-Verlag, Berlin, 1979, Chapter 2. (55) E. B. Smith and K. R. Lea, Nature (London), 186, 714 (1960); Trans. Faraday Soc., 59, 1535 (1963). (56) A. Rotenberg, J. Chem. Phys., 43,4377 (1965). (57) L. L. Lee and D. Levesque, Mol. Phys., 26, 1351 (1973).

(58) C. Seiter and B. J. Alder, J. Solution Chem., 7, 73 (1978). (59) P. H. Fries and J.-P. Hansen, Mol. Phys., 48, 891 (1983). (60) B. J. Alder, J. Chem. Phys., 40, 2724 (1964); B. J. Alder, W. E. Alley, and J. H. Dymond, ibid., 61, 1415 (1974). (61) D. L. Ermak, B. J. Alder, and L. R. Pratt, J. Phys. Chem., 85, 3221 (1981). (62) D. J. Adams and I. R. McDonald, J.Chem. Phys., 63,1900 (1975). (63) T. W. Melnyk and B. L. Sawford, Mol. Phys., 29, 891 (1975). (64) B. J. Alder, W. E. Alley, and M. Rigby, Physica, 73,143 (1974). (65) A. Bellemans, J. Orban, and E. de Vos, Chem. Phys. Lett., 1,639 (1968). (66) J. A. Zollweg, J. Chem. Phys., 72, 6712 (1980). (67) A. A. Clifford and E. Dickinson, Mol. Phys., 34, 875 (1977). (68) E. Dickinson, Mol. Phys., 33, 1463 (1977). (69) D. J. Evans and H. J. M. Hanley, Phys. Rev. A , 20,1648 (1979). (70) H. J. M. Hanley and D. J. Evans, Mol. Phys., 39, 1039 (1980). (71) H. J. M. Hanley end D. J. Evans, Int. J. Thermophys., 2 , l (1981). (72) H. J. M. Hanely, D. J. Evans, and S. Hess, J. Chem. Phys., 7 8 , 1440 (1983). (73) I. R. McDonald, Chem. Phys. Lett., 3, 241 (1969). (74) K. Singer, Chem. Phys. Lett., 3 , 164 (1969). (75) J. V. L. Singer and K. Singer, Mol. Phys., 19, 279 (1970).

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16

I

I

I

0.2

0.4 7)

Figure 7. The compressibility factor for pure hard spheres (HS), an equlmolar mixture of spheres and spherocylinders (length to diameter ratio I = / l o = 1, where the length excludes the hemlspherlcai caps and cr is the diameter) of equal volumes (HS SC), and pure spherocylkrders(I' = / / a = 1). Here 9 = p ~ ~ uwhere v pu = N / V is number density, x , is mole fraction, and v, is molecular volume of component a (from Monson and Rigbylie).

+

mixtures of hard spheres and of U atoms, but little work has yet appeared on fluids of nonspherical molecules. This (76)I. R. McDonald, Mol. Phys., 23,41 (1972). (77)J. V. L. Singer and K. Singer, Mol. Phys., 24,357 (1972). (78)I. R. McDonald, Mol. Phys., 24,391 (1972). (79)G. G.Devjatich, V. M. Stepanov, V. V. Kulinich, and M. J. Shirobokov, J. Phis. Chim., 50, 1811 (1976). (80)G. M. Torrie and J. P. Valleau, J. Chem. Phys., 66,1402(1977). (81)G. Fiorese and G. Pittion-Rossillon, Chem. Phys. Lett., 77,562 (1981). (82)K. Nakanishi, S. Okazaki, K. Ikari, T. Higuchi, and H. Tanaka, J. Chem. Phys., 76,629 (1982). (83)C. Hoheisel, U. Deiters, and K. Lucm, Mol. Phys., 49,159(1983). (84)A. M. Evseev, A. V. Chelovskii, and G. P. Misvurina, Russ. J. Phys. Chem., 47,944 (1973). (85)I. Ebbsjo, P. Schofield, K. Skold, and I. Waller, J.Phys. C, 7,3891 (1974). (86) K. C. Mo, K. E. Gubbins, G. Jacucci, and I. R. McDonald, Mol. Phys., 27,1173 (1974). (87)G. Jacucci and I. R. McDonald, Physica A (Amsterdam), 80,607 (1975). (88)K. Toukubo and K. Nakanishi, J . Chem. Phys., 65,1937(1976). (89)K. Toukubo and K. Nakanishi, J . Chem. Phys., 67,4162(1977). (90)C. Hoheisel, Ber. Bunsenges. Phys. Chem., 81, 462 (1977);C. Hoheisel and U. Deiters, ibid., 81,1225 (1977). (91)A. L. Tsykalo, N. I. Kartseva, V. I. Los, and J. F. Doroshenko, 'Proceedings of the 5th Symposium on Computers in Chemical Engineering", Oct. 1977,VisokC Tatry, Czechosolvakia, 1977,p 940. (92)K. Toukubo and K. Nakanishi, Ber. Bunsenges. Phys. Chem., 81, 1046 (1977). (93)K.Toukubo, K. Nakanishi, and N. Watanabe, J. Chem. Phys., 67, 4162 (1977). (94)K. Nakanishi, K. Toukubo, and N. Watanabe, J.Chem. Phys., 68, 2041 (1978). (95)C. Hoheisel and U. Deiters, Mol. Phys., 37,95 (1979). (96)C. Hoheisel, Mol. Phys., 38,1243 (1979). (97)K. Nakanishi and K. Toukubo, J.Chem. Phys., 70,5848(1979). (98)K. Nakanishi, H. Narusawa, and K. Toukubo, J. Chem. Phys., 72, 3089 (1980). (99)D. L. Jolly and R. J. Bearman, Mol. Phys., 41, 137 (1980). (100)C. Hoheisel, Phys. Chem. Liquids, 9,245,265 (1980). (101)H. Narusawa and K. Nakaniihi, J. Chem. Phys., 73,4066(1980). (102)R. J. Bearman and D. L. Jolly, Mol. Phys., 44,665 (1981). (103)C. Hoheisel, Ber. Bunsenges. Phys. Chem., 85,1054 (1981). (104)M.Schoen and C. Hoheisel, Mol. Phys., submitted for publication. (105)G. A. Chapela, G. Saville, S. M. Thompson, and J. S. Rowlinson, J. Chem. Soc., Faraday Trans. 2, 73,1133 (1977). '

Gubbins et ai.

is an area that clearly merits attention. The early work (up to 1978) on mixtures of spherical molecules has been reviewed before,52-Mand we only briefly mention the main conclusions here. For hard molecules (spheres, spherocylinders, dumbells, etc.) the results yield the excess volume P,and hence the excess Gibbs energy GE,which is simply the integral of VE over pressure in this case. Both the simulation and theoretical results for such hard particles show several features in common: (a) GE is always negative (at least, for moderate en1ongations)hence there is no fluid-fluid phase transition (except possibly for very elongated molecules, where an isotropic-nematic transition may occur); (b) both the size ratio of the two molecules and the pressure have a very pronounced effect on the magnitude and shape of the VE curve; and (c) molecular shape has virtually no effect on the qualitative shape of the VE curve, and only a minor effect on the magnitude of VE. Molecular shape has somewhat more influence on the compressibility factor P/pkT (see Figure 7), but the effect is still quite small. U mixtures exhibit a much wider variety of behavior than hard molecules and can yield the first five classes of phase '~ diagrams shown in Figure 2. Early ~ o r k ~ *concentrated on fairly ideal mixtures ( ~ u / t g and g ou/agg ratios close to unity), but later work has included highly nonideal mixtures (see, e.g., ref 35,41, and 43). Calculations have been made in the canonical, isobaric, and grand canonical ensembles by using MC, and also by MD. Several studies have used the methods described in the previous subsection to obtain the free energy or chemical potential (see, e.g., ref 35, 41, 43, and 82). The work reported so far on mixtures of nonspherical molecules has been too limited to draw general conclusions, and these studies have not included calculations of the chemical potential. Initial studies123on hard diatomics with added point charges show that shape effects on the thermodynamics are larger when strong electrostatic forces are present. This is to be expected, since molecules with a nonspherical core can no longer easily align themselves into orientations that are the most favorable ones for the electrostatic forces. Several types of fluid mixtures have been excluded from Table I, including ionic mixtures of various types" (molten salts,124electrolyte solutions,lZ6 liquid metals) (106)D. J. Lee, M. M. Telo da Gama, and K. E. Gubbins, to be submitted for publication. (107)F. H. Ree, J . Chem. Phys., 78,409 (1983). (108)E. F. O'Brien, Mol. Phys., 26,453 (1973). (109)D. Frenkel and J. van der Elsken, Chem. Phys. Lett., 40, 14 (1976). (110)T. W. Zerda and J. Zerda, J . Phys. Chem., 87,149 (1983). (111)D. W. Rebertus, B. J. Berne, and D. Chandler, J . Chem. Phys., 70,3395 (1979). (112)J. P. Ryckaert, Ph.D. Thesis,Univeraiti Libre de Bruxelles, 1976. Translated, J. M. Haile and P. W. Haile, University Microfilms International, Ann Arbor, 1980. (113)C.Hoheisel, Mol. Phys., in press. (114)M. Neumann, F.J. Fesely, 0. Steinhauser, and P. Schuster, Mol. Phys., 36,841 (1978). (115)B. J. Costa Cabral, D. Rinaldi, and J. L. Rivail, Chem. Phys. Lett., 93,157 (1982). (116)P. A. Monson and M. Rigby, Mol. Phys., 39,977 (1980). (117)T. Boublik and I. Nezbeda, Czech. J. Phys., B30, 121 (1980). (118)I. Nezbeda and T. Boublik,&ech. J. Phys., B30, 953 (1980). (119)I. Aviram and D. J. Tildesley, Mol. Phys., 35,365 (1978). (120)M. C.Wojcik and K. E. Gubbins, Mol. Phys., 49,1401 (1983). (121)E. Enciso, P. Seville, and D. J. Tildesley, Chem. Phys. Lett., in press. (122)K.Nakanishi, H. Tanaka, and H. Touhara in 'Chemical Engineering Thermodynamics", S. A. Newman, Ed., Ann Arbor Science, Ann Arbor, 1982. (123)M. C.Wojcik and K. E. Gubbins, Mol. Phys., in press. (124)B. Hafskjold and G. Stell in "Studies in Statistical Mechanics", Vol. VIII, 'The Liquid State of Matter", E. W. Montroll and J. L. Lebowitz, Ed., North-Holland, Amsterdam, 1982,p 175.

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*r

4. Theory

The most promising theoretical approach for thermodynamic properties is perturbation theory, and we discuss the current status of this theory below. We give only the physical ideas of such theories; a detailed treatment has been given by Gray and Gubbins.lB Important alternative approaches to the theory of liquid mixtures, which we do not cover here, include the Kirkwood-Buff theory,130the decorated lattice gas model,131and renormalization group theory.132 We also omit any discussions of many of the older theories (cell and lattice theories, regular solution theory, random mixture theory, and so forth), and theories that apply only to specialized states (for example, the critical region) or particular sorts of mixtures (polymer solutions, aqueous mixtures, fused salts, and so forth). 4.1. Mixtures of Hard Molecules. The simplest nonideal mixtures are those composed of hard particles. Their study provides valuable insight into the effects of molecular size and shape on the thermodynamic properties, in the absence of complications from attractive forces. In addition to integral equation and perturbation theory, it is possible to study such mixtures by using scaled particle theory. In that theory one calculates the work required to add a hard molecule to the fluid by first adding a point molecule and then scaling this molecule up to ita fullsize.13 The derivation of scaled particle theory is valid only for mixtures of molecules of the same shape, although the final expression obtained seems to work quite well even when the molecules have different shapes. Studies of mixtures of hard bodies have been made by Perm-Yevick theory,'% scaled particle theory,'% various modified forms of scaled particle theory,1s138 and computer simulation136(see Table I). These studies are in general agreement and the main conclusions have been summarized in section 3.2. The rather small influence of molecular shape on the compressibility factor is shown in Figure 7 for a mixture of hard (125)See, for example: K. Heinzinger and P. C. Vogel, Z. Naturforsch. A , 29,1164 (1974);P. C. Vogel and K. Heinzinger, ibid., 30,789 64, 22 (1977);G. (1975);D.J. Adams, Faraday Discuss., Chem. SOC., Palinkas, W. 0.Riede, and K. Heinzinger, 2.Naturjorsch. A , 32, 1137 (1977). (126)E. Pollock and B. J. Alder, Phys. Reu. A , 15,1263 (1977);B. J. Alder, E. L. Pollock, and J. P. Hansen, Roc. Natl. Acad. Sci. U.S.A.,77, 6272 (1980). (127)D.W. Wood in "Water: A Comprehensive Treatise", Vol. 6,F. Franks, Ed.,Plenum, New York, 1979,Chapter 6. (128)D. L. Beveridge, M. Mezei, S.Swaminathan,and S. W. Harrison, ACS Symp. Ser., No.86,191(1978);P. K. Mehrotra and D. L. Beveridge, J. Am. Chem. SOC.,102,4287 (1980);D.L.Beveridge, M. Mezei, P. K. Mehrotra, F. T. Marchese, V. Thinunalai, and G. Ravi-Shanker,Ann. N . Y. Acad. Sci., 367,108 (1981);M. Mezei and D. L. Beveridge, J. Chem. Phys., 74,622(1981);S. Okazaki, K.Nakanishi, H. Touhara, N. Watanabe, and Y. Adachi, ibid., 74,5863(1981);W. L. Jorgensen, ibid., 77,4156 (1982);W. L. Jorgensen, ibid., 77,5757 (1982);J. M. Goodfellow, J. L. Finney, and P. Barnes, Proc. R. SOC.London, Ser. B,214,213(1982);J. M.Goodfellow, Proc. Natl. Acad. Sci. U.S.A.,79,4977(1982);S.Okazaki, K. Nakanishi, and H. Touhara, J. Chem. Phys, 78,454 (1983). (129)C. G.Gray and K. E. Gubbins, "Theory of Molecular Fluids", Oxford University Press, London, in press. (130)J. G. Kirkwood and F. P. Buff, J. Chem. Phys., 19,774 (1951); J. P. OConnell, Mol. Phys., 20, 27 (1971). (131)J. C.Wheeler, Annu. Reo. Phys. Chem., 28, 411 (1977). (132)J. S. Walker and C. A. Vause, Phys. Reu. Lett., 79A 421 (1980); "Proceedinga of the Eighth Symposium on Thermophysical Properties", American Society of Mechanical Engineering, New York, 1982,p 411. (133)R. M.Gibbons, Mol. Phys., 17,81 (1969);18,809 (1970). (134)J. L. Lebowitz and J. S. Rowlinson, J. Chem. Phys., 41, 133 (1964). (135)J. PavliEek, I. Nezbeda, and T. Boublik, Czech. J . Phys., 27B, 1061 (1970). (136)T. Boublik and I. Nezbeda, Czech. J. Phys., 30B, 121 (1980). (137)T.Boublik and I. Nezbeda, Czech. J. Phys., 30, 953 (1980). (138)T. Boublik, Mol. Phys., 42,209 (1981).

Flgure 8. Variation of the reduced Henry constant KA with the ratio of W diameters u,/a, for W mixtures with eAA = tAB= ,e, kT/t,, = 1.2, paw3 = 0.7, xA = 0. The Mc simulation results were obtained with the f-g sampling method for ((rAB/(rBe)3 I 1.5, and by umbrella sampling for (u,/u,)~ = 2.0. The curves labeled VDW1, VDWq, and VDW3, are the vdW-1-, -2-, and -3-fluid theory results, wlth the mixing carried out at constant volume; when the mixing is carried out at constant pressure the results are unchanged for vdW-1-, but are poorer for the vdW-2- and -3-fluid theories. The curve labeled LL is the Lee-Levesque theory result, based on a hard sphere mixture as reference system.

spheres and spherocylinders of equal volumes. Similar results are found for hard dumbells.'20 The values of VE for these mixtures are very small, because the nonideality arises entirely from the difference in shape of the two species, their volumes being the same. 4.2. Perturbation Theory for Spherical Molecules. Until 1971, work on the theory of liquid mixtures focused almost exclusively on simple, spherical molecules. Perturbation and conformal solution theories, which relate the free energy of the real mixture to a hard-sphere mixture and to an ideal mixture, respectively, are successful for spherical molecules (neon, argon, krypton, xenon) and are quite good even for weakly nonspherical molecules (e.g., nitrogen, methane). These theories give a good account of systems with phase diagrams of classes I and I11 of Figure 2, provided that the molecules are spherical and not too different in size. They can predict qualitative behavior of classes 11, IV,and V, but this generally requires the use of potential parameters that are physically unrealistic. They cannot predict behavior of class VI. We give only a brief account of these theories here; for details, reviews are a ~ a i l a b l e . ~ ~ , ~ ~ Conformal Solution Theory. In this approach it is assumed (1)that the molecules are conformal; i.e., they obey the same intermolecular force law, differing only in the values of the potential parameters ea@ and a#@,and (2) the values of en@ and a,@ for the molecular pairs are not too different from each other. Of the various forms of this theory the van der Waals one-, two-, and three- (vdW-1, -2, -3) fluid seem to be the best. In these (139)T. W.Leland, J. S.Rowlinson, and G. A. Sather, Trans. Faraday

Soc., 64,1447 (1968);W. R. Smith, Can. J. Chem. Eng., 50,271 (1972).

(140)T.W.Leland, J. S.Rowlinson, G. A. Sather, and I. D. Watson, Trans. Faraday Soc., 65,2034(1969);D.Henderson and P. J. Leonard, Proc. Natl. Acad. Sci. U.S.A., 67,1818 (1970);68,632 (1971).

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theories the free energy of the (binary) mixture of interest is equated to that for an ideal mixture composed of one, two, or three "pseudocomponents", whose potential parameters are obtained from mixing rules given by the theory. For the vdW-1 and vdW-2 theories the mixing rules are VdW-1: ux3 = x,xp,; tX4,3 = x,x@t,p,; (11)

c

c aB

aB

vdW-2:

ax2 =

B

xp,?

txaux2 =

C x,+,p,~ B

(12)

The parameters tup,aUBneeded in the vdW-3 fluid theory are simply the parameters for the AA, AB, and BB pairs in the real mixture. The subscript x that appears in (11) and (12) serves to remind us that these pure component properties depend on the mixture composition. These theories have been extensively tested against computer simulation results.35*41~52~53 The vdW-1 theory is found to be the best overall, while vdW-3 is generally poorest. All of these theories give quite good results over wide ratios of e (e.g., 0.25 I tAA/tgg I 4) when the molecules are of the same size. When the molecules differ much in size all of these theories fail; this is illustrated in Figure 8 for the Henry constant. Of the vdW-n-fluid theories the vdW-1 has the firmest theoretical foundation, since it can be derived from perturbation theory, as shown by Smith.139 The vdW-1 theory has been extended to second order.86J42 The extended theory gives better results for molecules that differ in size, but numerical calculations are more involved. Perturbation about a Hard-Sphere Fluid. In this approach the properties of the mixture are related to those for a mixture of hard spheres through a first-order expansion in powers of ul/kT, where u1 is usually the attractive portion of the intermolecular pair potential. A second expansion is needed to relate the properties of the reference system of soft spheres to those of hard spheres. Three forms of the theory have been proposed, based on the extension to mixtures of Barker-Hender~on,'~~ variational,lMand Weeks-Chandler-Ander~on~~ perturbation theories for pure fluids. Of these, the extension of the Weeks-Chandler-Anderson theory to mixtures by Lee and L e ~ e s q u eappears ~~ to be the most successful. These theories are a great improvement over the vdW-n-fluid theories when the molecules differ in size, because the size difference is incorporated into the reference system of hard spheres; this improvement can be seen clearly in Figure 8. Efficient methods for calculating the hard-sphere diameters and pair correlation functions needed in the theory are described by Lee and Le~esque.~'Nevertheless, the calculations require about lo3 times more computer time than the n-fluid theories, so that the latter are still useful within their range of validity. Comparisons of theory and experiment have included studies of a variety of simple liquid mixture^,^^^^^ mixtures of argon and n e ~ p e n t a n e , ' ~ ~ and the properties of dissolved gases in a variety of solvent~.~*~ (141)R. L. Scott, J. Chem. Phys., 25, 193 (1956). (142)W. R. Smith, Mol. Phys., 22, 105 (1971). (143)P. J. Leonard, D. Henderson, and J. A. Barker, Trans. Faraday SOC., 66,2439 (1970). (144)G. A. Mansoori and T. W. Leland, J . Chem. Phys., 53, 1931 (1970);56,5335 (1972). 67,3474 (145)B.L. Rogers and J. M. Prausnitz, Trans. Faraday SOC., (1971). (146)S. Goldman and T. R. Krishnan, J. Solution Chem., 5, 693 (1976);S.Goldman, ibid., 6,461 (1977);J. Phys. Chem., 81,608(1977); J. Chem. Phys., 67,727 (1977);69,3775 (1978);Acc. Chem. Res., 11,409 (1979).

4.3. Perturbation Theory for Nompherical Molecules. The reference fluid used in perturbation theory should meet two requirements; its properties should be well understood, and it should be close to the fluid of interest. By "close" we mean that the structure of the two fluids, as given by the pair correlation function, should be as similar as possible. In the limit where these structures become the same, first-order theory should be exact. For atomic fluids it is possible to satisfy both of these requirements by using the repulsive branch of the potential as the reference fluid potential. For molecular fluids the situation is more complex, and no reference fluid is yet available which meets both of the above requirements. Two types of theory have been proposed those that adopt a reference fluid of spherical molecules, and those in which the reference molecules are nonspherical. The spherical molecule reference fluid is well understood-the equation of state and structure are known-but it is not close to the fluid of interest, so that the series convergence is often poor. Reference fluids of nonspherical molecules are generally closer to the real fluid, but are more poorly understood. Spherical Reference Molecules. Two different choices of spherical reference potentials have been proposed; we refer to the resulting theories as the "u expansion" and the " f expansion". The u expansion was first proposed over 30 years ago by Barker147and Pople,"@and has been extensively compared with experiment. The pair potential is divided into an isotropic (reference) part uo(r)and an anisotropic (perturbation) part u,(rw1w2),where wi (= &0ixL or for nonlinear or linear molecules, respectively) is the orientation of molecule i, and uo is defined by

uo(r.)= ( u ( r w 2,,,),)

(13)

where (..),, denotes an unweighted average over the orientations. bsually some simple model, such as the LJ, is chosen for uo(r),so that the reference fluid properties are known from computer simulation studies149and theory. With this choice of reference the fiist-order term Al in the series for the Helmholtz free energy vanishes, and the second- and third-order terms take relatively simple forms. However, the series converges slowly, and even the thirdorder theory is insufficient for strong polar or quadrupolar forces. This led Stell et a1." in 1974 to propose a simple Pad6 approximant to the series:

A = A0

+ Az(1 - A,/A2)-'

(14)

This equation is found to agree well with computer simulation results for fluids in which the molecules have spherical or near-spherical cores, even when strong electrostatic forces are It is less satisfactory for molecules with highly nonspherical shapes.155 General expressions have been g i ~ e n ' ~for~ A2 J ~and A,; they in(147)J. A. Barker, J. Chem. Phys., 19, 1430 (1951);Proc. R . SOC. London, Ser. A , 219,367 (1953). (148)J. A. Pople, h o c . R. SOC. London, Ser. A , 215,67(1952);Discuss. Faraday SOC.,15,35(1953);Proc. R.SOC.London, Ser. A, 211,498,508 (1954). (149)J. J. Nicolas, K. E.Gubbins, W. B. Streett, and D. J. Tildesley, Mol. Phys., 37, 1429 (1979). (150)G. Stell, J. C. Rasaiah, and H. Narang, Mol. Phys., 23, 393 (1974). ~ .- . S. S. Wang, C. G. Gray, P. A. Egelstaff, and K. E. Gubbins, Chem. Phys. Lett., 21, 123 (1973);24, 453 (1974). (152)L.Verlet and J. J. Weis, Mol. Phys., 28,665 (1974). (153)I. R. McDonald, J. Phys. C, 7,1225 (1974). (154)G. N. Patey and J. P. Valleau, J. Chem. Phys., 61,534 (1974); 64,170 (1976). (155)C.G. Gray, K. E.Gubbins, and C. H. Twu, J . Chem. Phys., 69, 182 (1978). (156)M. Flytzani-Stephanopoulos,K.E. Gubbins, and C. G. Gray, Mol. Phys., 30,1649 (1975).

(cl)

The Journal of Physlcal Chemistry, Voi. 87, No. 23, 1983

Feature Article

1

I

I

I

4503

1

P/bar

'120

170

220

270

320

500

600

700

T/K

Figure 9. PTdiagram for a mixture of W molecules (a) with molecules interacting with W plus dlpole-dipde potential (b); ebb/e, = 1.6, abb/um = 1, e& and a, given by the Lorentz-Eerthelot rules. Here solki lines are vapor pressure curves for the pure components, dashed lines are crltica~h i , and p' _= pb+= ~ / ( e , a , ~ ) " ~ .he dlpole Is added to the less volatile component b here, leading to class 111 behavior. Gas-gas immiscibility occurs for large p * .

P/bar

volve two- and three-body integrals over the correlation functions for the reference system. These integrals have Figure 10. Comparison of theory and experiment for Xe/HCI, using been evaluated for a variety of potential forms and fitted combining rules for ei2 and ai* due to K ~ h l e r . ' ~The * solid lines are to simple functions of temperature and d e n ~ i t y . ' ~ ~ J ~from ~ the Pad6 approxlmant of (14) using an Intermolecular potential Details of the calculation procedure have been remodel that includes dipolar, quadrupolar, and shape terms. The dashed vie~ed.'~~J~~ lines are frm vdw-1-fluid theory using a W model. The RedlicbKwong equation gives results that are indistinguishable from vdW-1 theory on In calculations based on eq 14, the anisotropic potential the scale of the plot (from ~ u c a s ' ~ ~ ) . u, is written as a s u m of electrostatic, induction, anisotropic dispersion, and anisotropic overlap terms. The anisotropic found that quadrupolar forces are particularly effective in dispersion term is usually approximated by the London increasing the dissolving power. expression, and the remaining anisotropic potential conDetailed comparisons of (14) with experimental data tribution@are represented by the first few terms in an have been made for a variety of mixtures of small rigid expansion in generalized spherical harmonics. Equation molecules, such as CO, COz,NzO, C2H6, C2H4, C2H2,HC1, 14 has been used to explore the relationship between inHBr, CHI, CF4, etc. Work up to 1981 has been retermolecular forces and the resulting phase diaviewed.10J62J63More recent studies have included those gram.156157J597160Thus, if one or both of the components by Clancy et al. on Hz/CH4,164AI-/CO,'~~ Kr/C2H6,166 interacts with a Lennard-Jones plus a dipole-dipole term, N20/CzH4,16' and CzH6/C2H4,'6s by Lucas" on CO/CH4, it is possible to obtain any of classes I-V of Figure 2 by Xe/HBr, Xe/HCl, and Xe/CzH4,and by Goldman170on suitable adjustment of the parameters in the potential (see hydrophobic effects in water. GibbPOhas examined Figure 9). Similar results are obtained if a quadrupolemixtures of the n-alkanes up to C8 and the first four quadrupole term is usedin place of the dipole-dipole one. primary alcohols, and Winkelmann17' has recently applied If, instead, the anisotropic part of the potential consists the theory to mixtures of chloroform, acetone, diethyl only of an overlap term designed to simulate the shape of ether, dimethyl formamide, water, and methanol. These a linear molecule, then only classes I and I11 are obcomparisons of theory and experiment involve the fitting tained.'% The anisotropic overlap and dispersion parts of of several potential parameters to experimental data for the potential seem to have a relatively small effect on the both the pure fluid and the mixture, so that definitive phase diagram, whereas the effect of electrostatic forces conclusions about the success of the potential model is is large. A detailed study of the effect of various potential often difficult. An exception is the work of Lucas,158J69 terms on systems showing gas-gas immiscibility has been made by Gibbs." Jonah et al.161 have recently carried out a study of the influence of various types of intermolecular (162)C. H.Twu and K. E. Gubbins, Chem. Bng. Sci., 33,879(1978). potential terms on the dissolving power of solvents used London, Sect. C,77,101 (163)K.N. Marsh, Annu. Rep. Chem. SOC. in supercritical extraction of liquids and solids. They (1980). (157)K.E.Gubbins and C. H. Twu, Chem. Eng. Sci., 33,863(1978). (158)B.Moser, K.Lucas, and K. E. Gubbins, Fluid Phase Equilib., 7, 153 (1981). (159)C. H.Twu, K. E. Gubbins, and C. G. Gray, J . Chem. Phys., 64, 5186 (1976). (160)G. M. Gibbs, "Chemical Thermodynamic Data on Fluids and Fluid Mixtures", National Physical Laboratory Conference, No. 149,IPC Scientific and Technical Press, Guildford, 1979; Thesis, University of Oxford, 1979. (161)D. A. Jonah, K. S. Shing, V. Venkatasubramanian, and K. E. Gubbins in 'Chemical Engineering at SupercriticalFluid Conditions", M. E. Paulaitis, J. M. L. Penninger, R. D. Gray, Jr., and P. Davidson, Ed., Ann Arbor Science, Ann Arbor, 1983,p 221.

(164)P. Clancy and K. E. Gubbins, Mol. Phys., 45, 1 (1981). (165)P. Clancy, E.J. S. G. Azevedo, L. Q.Lobo, and L. A. K. Staveley, Fluid Phase Equilib., 9,267 (1983). (166)J. C. G. Calado, E. Chang, P. Clancy, and W. B. Streett, J. Phys. Chem., to be submitted. (167)L. Q.Lobo, L. A. K. Staveley, P. Clancy, K. E. Gubbins, and J. R. S. Machado, J . Chem. Soc., Faraday Trans. 1, in press. (168)J. C. G. Calado, E. J. S. Gomes de Azevedo, P. Clancy, and K. E. Gubbins, J . Chem. Soc., Faraday Trans. 1 , in press. (169)K. Lucas, "Proceedings of the Eighth Symposium on Thermophysical Properties",American Society of Mechanical Engineering. - New York, 1981,p 314. (170)S. Goldman, J. Chem. Phys., 75, 4064 (1981). (171)J. Winkelmann, Fluid Phase Equilib.,7,207(1981);ibid., 11, 207 (1983).

4504

The Journal of Physical Chemistty, Vol. 87, No. 23, 1983

Gubbins et ai.

in which there are no fitted parameters for the mixture. The comparison with experiment is therefore a test of the mixing rules in the theory, and of the combining rules used for the unlike-pair parameters. When the mixture contains polar molecules the molecular treatment based on (14) 263.15K gives much better results than theories which assume the molecules to be spherical (e.g., vdW-1-fluid theory); an example is shown in Figure 10. The theory has been extended'73 to include the effects of a nonaxial quadrupole, i.e., molecules whose lack of symmetry leads to two independent components (e.g., Q,,, Q,, - Q, ) of the quadrupole moment. The effects are appreciabe for N20/C2H4 mixture^,'^^ and are expected to be larger for mixtures involving asymmetric top polar 234.15 molecules such as H20 and MeOH. Other extensions of the theory have included the incorporation of quantum correctionsla (important in H,/X mixtures), three-body dispersion f o r c e ~ , ' ~and J ~ multibody ~ induction f o r ~ e s . ' ~ ~ J ' ~ Both the ~imulation'~~ and the~retical'~~ calculations show 0 0.2 0.6 I .o that multibody induction forces have a significant effect on thermodynamic properties. calculation^'^^ for pure HF, Flgure 11. Vapor-liquid equilibria for COP (l)-C2H6(2) mixtures at for example, indicate that the induction forces contribute 234.15 and 263.15 K from experiment (points) and theory (lines) of about 20% to the total configurational energy, with roughly BoubYk using a reference fluid of hard convex molecules: ---, spherical half of this amount due to the pair induction forces and core Kihara model, k 1 2 = 0.12; spherical core, k , , = 0;-, the other half due to multibody induction. The renornonsphericalcore, k , , = 0.11. Here k , , = t12/(~,,t,,)"2 (from P a v l i k malization theory of Wertheim,'75 which is phrased in the and B o ~ b i i k ' ~ ~ ) . form of the Pad6 of eq 14,should be particularly useful of knowledge of the properties of suitable reference sysfor numerical calculations, and has recently been extended tems. Quite accurate equations are now available for the to mixtures.17$ thermodynamic properties of fluids of hard nonspherical The above calculations were based on the u expansion. molecules such as dumbells, spherocylinders, etc.; apThe f expansion was first proposed in 1957,'79 but was not proximate theoretical expressions are known for the siteapplied to molecular liquids until 1974." The reference site correlation functions for dumbells and similar interpotential is defined by action site model (ISM) fluids from the reference interexp[-Du&)l = (ex~[-Du(rw~w~)l),,,, (15) action site model (RISM) theory.'@ Most nonspherical reference perturbation theories can be divided into two in place of (131, and the expansion is made in terms of the classes: (a) those that divide the pair potential u(rwlw2) perturbation in exp(-pu) rather than in u itself. This into a repulsive and an attractive branch, as in Weekschoice of reference potential has the advantage that it Chandler-Andersen (WCA) theory for spherical molecules; includes a contribution from the anisotropic intermolecular this splitting is of course different for different molecular forces; the expansion is better suited to potentials having orientations; and (b) those that are restricted to ISM poa hard nonspherical core, and is exact in the limit of low tentials, for which the potential between sites a and p, density. The f expansion usually gives better results than u&), is divided into repulsive and attractive branches; the u expansion for the structure of the fluid, particularly this splitting is independent of the molecular orientations. for the centers correlation function g(r),but seems to be In both approaches the reference fluid properties are repoor for thermodynamic properties. For appreciably anlated to those of a fluid of hard molecules, e.g., dumbells. isotropic molecules the convergence is poor, and since In theories of type (a)182J83J85-1w the reference fluid pair higher order terms in the series are intractable it is not correlation function g,(rwlo2) is unknown, and is usually possible to overcome this by a Pad6 approximant. The f calculated via the f expansion. Such theories have the expansion does not seem to have been applied to mixtures. advantage of applying to general potentials, and have been Nonspherical Reference Molecules. Much of the recent applied to fluids of two-center U,1*187 Kihara,'83J88J89and effort in perturbation theory has been directed at finding Gaussian overlaplWmolecules. The principal weakness in suitable reference systems of nonspherical molecules. theories of this type at present seems to be in the evaluExpansions about such a reference fluid offer the possiation of the reference correlation function g,(rwlw2), since bility of more rapid series convergence for highly anisothe f expansion converges poorly for significantly anisotropic forces, but early was hindered by a lack have the tropic molecules. Theories of type (b)'uJ91-194 (172) F. Kohler, Monatsh. Chem., 88, 857 (1957). (173) K. E. Gubbins, C. G. Gray, and J. R. S. Machado, Mol. Phys., 42, 817 (1981). (174) K. P. Shukla, S.Singh, and Y. Singh, J. Chem. Phys., 70,3086 (1979). (175) M. S. Wertheim, Mol. Phys., 37, 83 (1979). (176) G. N. Patey, G. M. Torrie, and J. P. Valleau, J. Chem. Phys., 71, 96 7RI - -l l R ~ (177) G. N. Patey, M. L. Klein, and I. R. McDonald, Chem. Phys. Lett., 73, 375 (1980). (178) C. Joslin, C. G. Gray, K. E. Gubbins, and V. Venkatasubramanian, to be submitted for publication. (179) J. A. Barker, Proc. R. SOC.London, Ser. A , 241, 547 (1957). (180) J. W. Perram and L. R. White, Mol. Phys., 28, 527 (1974); W. R. Smith, Can. J. Phys., 52, 2022 (1974). (181) S . I. Sandler, Mol. Phys., 28, 1207 (1974).

(182) K. C. Mo and K. E. Gubbins, Chem. Phys. Lett., 27, 144 (1974);

J. Chem. Phys., 63, 1490 (1975). (183) T. Boublik, Collect. Czech. Chem. Commun., 39, 2333 (1974). (184) D. Chandler and H. C. Andersen, J. Chem. Phys., 57, 1930 (1972). (185) F. Kohler, N. Quirke, and J. W. Perram, J.Chem. Phys., 71,4128 (1979). (186) J. Fischer, J. Chem. Phys., 72, 5371 (1980). (187) N. Quirke and D. J. Tildesley, J. Phys. Chem., 87, 1972 (1983). (188) T. Boublik, Mol. Phys., 32, 1737 (1976); Fluid Phase Equilib., 3, 85 (1979);Collect. Czech. Chem. Commun., 46, 1355 (1981). (189) J. PavliEek and T. Boublik, Fluid Phase Equilibria, 7, 1, 15 (1981). (190) P. A. Monson and K. E. Gubbins, J. Phys. Chem., 87, 2852 (1983). (191) D. J. Tildesley, Mol. Phys., 41, 341 (1980).

J. Phys. Chem. 1983, 87, 4585-4588

4585

advantage that one only needs to know the site-site cornot been tested against computer simulation results. relation functions gabfor the reference fluid, and not the 5. Conclusion full pair correlation function. The RISM theory gives a In the past decade, phase equilibrium experiments on reasonable account of these functions. fluid mixtures at high pressures have shown that there are Few comparisons of theory and experiment have yet continuous transitions between binary phase diagrams that been reported for these theories. B o ~ b l i k ' ~has J ~com~ exhibit different types of critical lines and liquid-liquidpared his theory (based on the Kihara potential model gas three-phase separations. There is a need for further expanded about a fluid of hard convex molecules) with systematic studies of mixtures of small, relatively simple experimental data for both the excess properties and vamolecules, including polar, nonpolar, and hydrogen-bonded por-liquid equilibria for mixtures involving Ar, N2,02,CS2, molecules, to provide data for testing and refining moC02, CHI, C2H4,C2H6,and cyclopentane. Agreement lecular theories of dense fluid mixtures. ranged from quite good to rather poor (see Figure 11). A There have also been significant advances in our ability theory based on the ISM model (site-site LJ) has been to make theoretical calculations for mixtures in which compared with experimental excess properties for Ar-N2 anisotropic intermolecular forces or quantum effects occur. and Ar-02 mixtures by Enciso and Lombardero.lg3 The principal limitation in all these calculations at present Agreement with experiment is rather poor. is the uncertainty concerning the form of the intermoleA major problem in the development of these theories cular potential function. This is particularly the case for is how to include electrostatic forces when the molecules polar and hydrogen-bonded fluids, where the existing are also significantly nonspherical. The use of hard dummodels are clearly inadequate. Further theoretical debells as a reference for a fluid of dumbells with an added dipole or quadrupole moment gives poor c o n v e r g e n ~ e ; ~ ~ velopment ~ ~ ~ ~ ~ is needed in order to deal with shape and electrostatic forces simultaneously, with hydrogen-bonded an approximate treatment of higher order terms and the systems, and with multibody induction effects. There is use of a Pad6 approximant to resum the series has been also a need for computer simulation studies of model suggested1%as a way to overcome this problem, but has mixtures of nonspherical molecules, both to provide a test of new theories and to explore the relation between (192)J. Lombardero, L. F. Abascal, and S. Lago, Mol. Phys., 42,999 thermodynamic behavior and the underlying intermole(1981). cular forces. (193)E.Enciso and M. Lombardero, Mol. Phys., 44,725 (1981). (194)M. Lombardero and J. L. F. Abascal, Chem. Phys. Lett., 86,117 (1982). (195)J. 0.Valderrama, S. I. Sandler, and M. Fligner, Mol. Phys., 42, 1041 (1981). (196)E.Martina, G.Stell, and J. M. Deutch, J. Chem. Phys., 70,5751 (1979);74,3636 (1981).

Acknowledgment. Our work in this area has been supported by the National Science Foundation under Grants CPE-8209187 and CPE-8104708, and by the Gas Research Institute under Grant 5083-260-0780.

ARTICLES Excited Electronic States of Ethylene Sulfide. Circular Dichroism Study of ( + ) - ( R ,R)-2,3-Dimethylthiirane D. Cohen,+M.

6.S. Green,* R. Arad-Yellin,' Harold Basch,*t and A. Gedankenft

Department of Chemlstry, Bar-Iian Unlverslty, Ramat-an, Israel 52 100; Department of Structural Chemistry, Welzmann Institute of Science, Rehovot, Israel; and Israel Institute of 8iobglcal Research, Ness-Ziona, Israel (Recelved: September 2, 1962)

The excited electronic states of an optically active substituted ethylene sulfide ((+)-(R,R)-2,3-dimethylthiirane, DMT) is discussed in light of vacuum-UV absorption and circular dichroism (CD) spectra and ab initio self-consistent-fieldcalculations. Experimental and calculated excitation energies and oscillator and rotational strengths are compared and provide the basis for the assignment of the excited electronic states. In the energy region below 2100 A all the observed bands are assigned as an n, Rydberg transitions.

-

Introduction The optical absorption spectrum of ethylene sulfide has been recorded by several groupsl-3 and different electronic Bar-Ilan University.

* Weizmann Institute of Science.

* Israel Institute of Biological Research. 0022-365418312087-4585$01.50/0

transition assignments have been proposed to interpret the results. The absorption spectrum of ethylene sulfide reveals two broad and diffuse bands in the region of 23G260 (1)R.E.David, J . Org. Chem., 23,216 (1958). (2)L.B. Clark and W. T. Simpson, J. Chem. Phys., 43,3666 (1965). (3) N. Basco and R. D. Morse, Chem. Phys. Lett., 20, 404 (1973).

0 1983 American Chemical Society