Analytic studies of the hard dumbell fluid III. Zero pole approximation for the hard dumbell Yukawa fluid

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1 MOLECULAR PHYSICS, 1984, VOL. 53, No. 4, Analytic studies of the hard dumbell fluid III. Zero pole approximation for the hard dumbell Yukawa fluid by P. T. CUMMINGS Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia U.S.A. (Received 9 June 1984 ; accepted 30 June 1984) The analytic solution of the zero pole approximation (ZPA) for a homonuclear diatomic fluid interacting via site-site potentials having a hard core of diameter a and an attractive Yukawa tail is examined. Using the compressibility equation of state, the fluid is found to have a liquid-gas phase transition characterized by spherical model critical exponents. This behaviour is the same as that exhibited in the mean spherical approximation by the simple fluid analogue of the hard dumbell Yukawa fluid. The spinodal curve from the compressibility equation of state is discussed and found to exhibit a high degree of corresponding states behaviour for different elongations of the diatomic. The structure on the liquid and gas sides of the coexistence curve is calculated. 1. INTRODUCTION In the two previous papers in this series [I, 2; referred to as I and II respectively in this paper], attention has been focused on the analytic solution of the site-site Ornstein-Zernike (SSOZ) equation for the hard dumbell fluid (HDF), a model fluid whose molecules are composed of two hard spheres each of diameter a fused together so that the separation between the centres of the molecules is equal to L. Thus, the HDF is an example of an interaction site model (ISM) fluid [3-6], the general term given to fluids whose intermolecular pair potential u(12) (which depends on both the positions and orientations of molecules 1 and 2) can be decomposed into a sum of interactions between sites (generally identified as the centres of the atoms in a polyatomic fluid), viz. u(12) = ~ u~p(r~#). (1.1) a,p=l Here, m is the number of sites in each molecule, r~ is the distance between sites and fl in distinct molecules, and u~(r) is the spherically symmetric site-site pair potential between sites ~ and fl in distinct molecules. For the HDF, there are only two sites, and each of the three possible interactions are equal and given by ult(r) =u12(r)=u22(r ) =u(r) = 0% r<a,l , r > (r, J (1.2)

2 850 P.T. Cummings The usual measure of structure in ISM fluids is the site-site pair (or radial) distribution function g~#(r) proportional to the probability density of finding sites a and fl in distinct molecules at separation r, and for the HDF the three possible pair distributions functions are, as with the pair potentials, all equal by symmetry, so that gt 1(r) = gl 2(r) = g22(r) = g(r). The SSOZ equation for the HDF, introduced in a more general context by Chandler and Andersen [3], is a scalar equation given by /~(k) = (1 + co(k))2g:(k) + 2p(1 + co(k))~(k)l~(k), (1.3) where p is the number density of molecules, /~(k) and ~(k) are the threedimensional Fourier transforms of the site-site total and direct correlation functions, h(r) = g(r) - 1 and c(r), respectively, while c0(k) is the Fourier transform of the intramolecular correlation function [3, 43 given by c0(k) = 1 + sin (kl)/kl. (1.4) Here, L is the elongation of the diatomic molecule (the distance between the centres of the atoms). As discussed at length in II, a number of closure relations between h(r) and c(r) have been suggested: the original closure of Chandler and Andersen [3] for site-site potentials with hard cores is called the interaction site approximation (ISA) and for homonuclear diatomics is given by h(r)= --1, c(r) = --u(r)/kb T, r > r < a,]. (1.5) As detailed in I and II, another approximation can be derived which is called the zero pole approximation (ZPA) and is given by h(r) = --1, r < a,] c(r) = --u(r)/k B T + F(r), r > a, (1.6) where F(r) is a function described in detail in w of II to which the reader is referred. For the HDF, the ZPA is analytically solvable due to the simplicity of this closure from the particle-particle viewpoint discussed elsewhere [7, 4, 2]. The method used is a variant of'the Baxter factorization technique [8, 9] which was first applied to site-site problems by Morriss and coworkers [10-12]. In II, the analytic solution of the SSOZ equation subject to the closure relation h(r) = -1, r < a,~ (1.7 a) c(r) = Ka exp [-z(r - ~)]/r + F(r), r > 0,~ (1.7 b) was presented. The initial motivation for employing this closure is to model c(r) for the HDF outside the core using the exponentially decaying Yukawa so that equation (1.7 b) becomes an ansatz for the true site-site direct correlation function outside the core. By employing structural and/or thermodynamic consistency constraints to determine the Yukawa parameters K and z, this analysis yields a generalized zero pole approximation (GZPA) for the HDF which is analogous to the generalized mean spherical approximation for the hard sphere fluid [13, 14]. From equation (1.6), it is clear that the closure (1.7) can also be interpreted as the ZPA for a homonuclear diatomic fluid interacting with a site-site potential given by u(r) = ~, r < ~,'( (1.8) =--Daexp [--z(r--~)]/r, r>cr,

3 Hard dumbell fluid 851 when K is identified with D/k B T, where D is the depth of the attractive well of the pair potential. The fluid interacting with site-site pair potential (1.8) is the diatomic analogue of the hard core Yukawa fluid (HCYF), a simple fluid whose intermolecular potential has the form (1.8). In view of the fact that its pair potential has hard core repulsion as well as attraction which models the dispersion interaction, the HCYF has been studied extensively [-15-20] as a non-trivial model for simple fluids which is nevertheless analytically solvable in the mean spherical approximation. Since it is analytically solvable in the ZPA, the homonuclear diatomic fluid with site-site interaction (1.8), which we refer to as the hard dumbell Yukawa fluid (HDYF), can serve an analogous role for nonspherical molecules, and the study of its properties within the ZPA is the subject of this paper. In w the main equations from II which are relevant to the ZPA for the HDYF are given. The analytic solution reduces to a single quartic equation for a fundamental parameter fl which is studied as a function of density, temperature and elongation. In w results for the spinodal curve of the compressibility equation of state and pair distribution functions are presented for various elongations. The relevance of the results to the theory of corresponding states is discussed. Section 4 contains some concluding remarks about the significance of the results presented. 2. ANALYTIC SOLUTION OF THE SSOZ EQUATION The analytic solution of the SSOZ equation subject to the closure (1.7) was presented in II, and in this section our discussion is limited to a brief description of the results of II and a reiteration of the major equations. The Baxter factorization, when applied to the SSOZ subject to closure (1.7), yields a closed form analytic solution for the Baxter Q-function Q(r) given by (2.21) of II. (From this point onward, we find it convenient to denote equation (i.j) from II as (II.i.j).) Once the Baxter Q-function has been obtained, the site-site direct and pair correlation functions can be calculated immediately, since the Baxter factorization yields a linear integral equation connecting Q(r) and g(r) (equation (II.2.14), and an explicit expression for c(r) in terms of Q(r). The constants which appear in the various terms in the expression for Q(r) are all ultimately given in terms of a single parameter fl (defined in (II.2.6 b)), which in turn is the solution of a quartic equation X4/~ 4 + X3/~ 3 + X2/~ 2 + Xl/~ + X 0 = 0. (2.1) Expressions for the parameters X~, i = 0,... 4 are given in II; the key feature of these parameters is that they depend only on the elongation of the dumbell (L), the parameters in the Yukawa potential (D and z) and the state conditions (p and T). That is, Xi = Xi(L, z, K, p), (2.2) where we note that D and T enter through the dimensionless parameter K, while the dependence on L and z is through the dimensionless quantities L/a and za. The quartic equation (2.1) is completely analogous to that which arises in the solution of the mean spherical approximation for the HCYF [-17, 18, 20], and as we shall see, /~ exhibits qualitatively similar behaviour to the corresponding parameter (also called/~ in [17, 18]) which arises in the HCYF case.

4 852 P.T. Cummings In figure 1, the behaviour of the parameter fl as a function of density p is shown for the HDYF with an elongation L = o-/2, Yukawa parameter z = 2o--1 and along four isotherms indicated by four different values of K (0"2, 0-4, 0'6 and 0-8). (Since K = D/kB T, K is equivalent to an inverse reduced temperature; thus the four values of K correspond to successively lower temperatures.) The first observation which may be made is that although the quartic (2.1) has in principle four roots, for attractive Yukawa site-site potentials there are at most two real roots. For K below a particular value Kt , there are two real results for fl (shown as the upper and lower curves in the figure) for all physically meaningful densities. The lower curve is the physically correct branch of the solutions of equation (2.1), since this solution correctly approaches zero as K---~ 0 while the other diverges to infinity. For K > K t (or, equivalently, sufficiently low temperature), there is an interval of density over which the quartic (2.1) has no real roots. Mathematically, the switch from two real roots to no real roots is characterized by the discriminant of the quartic becoming zero, and the two real roots becoming equal. Thus, we can plot the locus of points along which the discriminant of (2.1) is zero: this is shown as the line L 1 in the figure. The line L 1 clearly distinguishes between the physically correct and incorrect branches of the /3 curves, since the correct value of fl lies below L1 on any given isotherm. In addition to the existence of real solutions, there is another constraint which the value of/~ must satisfy : the compressibility equation of state is given by 1 OP = p f c(r) dr = a2/2, (2.3) kb T ~-fip r where a is the parameter in the analytic solution defined in (II.2.10), so that the limit of metastability (called the spinodal curve) is given as the locus of points along which a = 0. At low density, a is positive, so that physically meaningful results are obtained only when a > 0. The curve L 2 in figure 1 shows the locus of points along which a = 0, and solutions of equation (2.1) for which a > 0 lie below the curve L z. From figure 1, it is clear that for values of K below a critical value K c = 0" (temperatures above the critical temperature To) there is no density at which a = 0, while for values of K above K c (T > To), there are two densities, pg and Pl, at which a = 0. The lower value, pg, corresponds to the gas side of the spinodal curve, while the large value, Pl, corresponds to the liquid side. At K--Kc, the equation a = 0 has a double root in density at pg = Pl = Pc, the critical density. Thus, the liquid-gas critical point of the HDYF from the compressibility equation of state is given by T = To, po-3= pc o-3= The qualitative behaviour of the parameter fl--including the relative positions of the two curves L 1 and L2--is the same as that for the HCYF in the mean spherical approximation [-17, 18, 20]. As in the ease of the latter fluid, an analysis of the near vicinity of the compressibility equation critical point of the HDYF in the ZPA reveals that the critical exponents have spherical model values: that is, 7 = 2, 6 = 5 where ~ and 6 are defined by c~p ~ (T - To) -~ at p = Pc' T/> To, (2.4) kb T'3-fi r 3p[ ~ [p_pc{_~+l at T=T c. (2.5) k B T "o-fi I r

5 Hard dumbell fluid ' I ' I ' I '. : ~, ~.." po 3 Figure 1. The parameter /Y as a function of density pa ~ for the HDYF with elongation L = a/2 and Yukawa potential parameter z = 2a -1 along the isotherms g = 0-2 (... ), 0.4 (---), 0-6 ( ) and 0"8 (... ). Also shown is the locus of points along which the discriminant of equation (2.1) is zero (LI) and the locus of points along which the compressibility diverges (L2). Figure 2 shows the behaviour of the parameter fl for the HDYF with elongation L = a/3 and Yukawa parameter z = 2a-1 at inverse temperatures K = 0"2, 0"4, 0"6 and 0"8. For this elongation, K t = , K c = 0" and pea 3 = 0" In order to compare the results for the ZPA applied to the HDYF to the mean spherical approximation (MSA) for the HCYF, it is necessary to examine the L--~ 0 limit of the HDYF and the SSOZ equation. Physically, when L---~ 0, the four equal site-site interactions become centred at the same site : the centre of the sphere that is obtained as L is taken to zero. Thus, the HCYF which is obtained in this limit has an intermolecular potential Uncvv(r) given by UHCYF(r) = 0% r < G,] -- -4D~exp [-z(r--a)]/r, r>g, (2.6) since the intermolecular potential has four site-site contributions. The function F(r) reduces to a delta function at r = 0 in the L---~ 0 limit, so that the ZPA for the HDYF reduces to h(r) - -1, r < cr,~ (2.7) c(r) = K~ exp E-z(r - ~)]/r, r > G, 3 with K = D/ks T as before. Since e)(k)---~ 1 as L--~ 0, the SSOZ equation becomes /~(k) = 4~(k) + 4p~(k.)h(k), (2.8) where p is now the density of the atoms formed as L ~ 0.

6 854 P.T. Cummings 12 ' I ' I ' I ' 4 9.,, ~..._.. f ' ~. ~..-" -- "" "~*~'~'~ ~ ~ L 0, I I I i I--'-- "-~ pa 3 Figure 2. The parameter /3 as a function of density p6 3 for the HDYF with elongation L = a/3 and Yukawa potential parameter z = 2or -1 along the isotherms K = 0.2 (... ), 0"4 (---), 0"6 ( ) and 0"8 (... ). The lines L a and L z have the same significance as in figure 1. Additionally, we need to consider the relation between the atomic fluid correlation functions of the HCYF and the site-site correlation functions of the HDYF in the L--* 0 limit. Denoting the atomic fluid total and direct correlation functions by h*(r) and c*(r) respectively, then the usual Ornstein-Zernike (OZ) equation for a simple fluid such as the HCYF is given by l~*(k) = ~*(k) + p~*(k)l~*(k), (2.9) where p is the number density of atoms. Thus, by comparing (2.8) and (2.9), it is clear that h*(r) = h(r) (i.e. the site-site and atomic total correlation functions are the same) while c*(r)= 4c(r) (i.e. the atomic direct correlation function is four times the site-site correlation function). Now consider the MSA for the HCYF with interaction potential (2.6): application of the MSA yields the following conditions on the h*(r) and c*(r) h*(r) = --1, r < ~,] c*(r) = K*~ exp [-z(r -- ~)]/r, r > ~, (2.10) where K*= 4D/k B T = 4K. This closure on c*(r) is completely consistent with the closure (2.7) on c(r) and the relationship between c*(r) and c(r). Thus, we find that the ZPA for the HDYF reduces to the MSA for the HCYF in the limit L--~ 0. This situation is to be contrasted with other, non-linear approximations such as the site-site version of the Percus-Yevick approximation discussed in detail by Monson [-21]. One consequence is that the results of the MSA for the HCYF--in particular, the behaviour of the parameter /~ [17, 18, 20]--can be directly compared with those of the ZPA for the HDYF presented in this paper, keeping in mind that the parameter K (denoted K* here) used in [17, 18, 20] is four times the parameter K of the HDYF.

7 Hard dumbell fluid THERMODYNAMIC AND STRUCTURAL PROPERTIES OF THE HDYF In figure 3, we present the spinodal curve of the HDYF (the locus of points in the dimensionless temperature 1/K and density plane along which a = 0) with z = 2a-1 and for three elongations L = 0, a/3 and a/2. The critical temperature and dimensionless critical density Pc a3 are both higher for shorter elongations. In comparing critical densities, it is perhaps more appropriate to compare the critical volume fractions given by [21] fc = (~z/h)pc d3 = (z~/6)pca3( 1 + 3L*/2 -- L'3/2), / (3.1) where d is the diameter of the sphere occupying the same volume as the dumbell and L*= L/0-. For L = 0, a/3 and 0-/2, fc = , and respectively. While noting that the 9 per cent difference between Pc for L = 0-/3 and L = a/2 has been reduced to a difference of only 3 per cent in fr we see that fc for the HCYF (L = 0) is significantly lower than the result for either non-zero elongation. Despite this, it is clear that critical volume fraction is not a very strong function of elongation (over a 50 per cent range in elongation, the critical volume fraction changes by only 17 per cent), suggesting that molecular volume may be more useful than the reduced density p0-3 as a correlating parameter for homonuclear diatomics. In considering the dependence of the results for T c on elongation, we note that our comparison for different values of L may not be very meaningful. That is, comparing results in the form used in figure 3 is tantamount to assuming that site-site pair potentials are equal. However, the intermolecular potential depends not only on the site-site potentials, but also on the elongation of the molecule, and ' I ' I ' I ' 3 I/K 2 L-O I L : 1/2 0, I ~ I f I L o.o Figure 3. The spinodal curve (--) and the limit of real solutions (---) [corresponding to I-'2 and L 1 respectively in figures 1 and 2] shown in the form of temperature (l/k) versus density for the HDYF with Yukawa potential parameter z = 2a-1 at three elongations: L = 0, a/3 and a/2. Notice that at L -- 0, the HDYF reduces to the HCYF. po-3

8 856 P.T. Cummings to compare different elongation molecules meaningfully probably requires adjusting the site-site potentials in some way. (For example, one might demand that the second virial coefficients be as close as possible or that the coefficient of the longest ranged term in the spherically symmetric part of u(12) be equal.) Seeking such a relationship between site-site potentials for different elongations amounts to seeking a microscopic basis for a two-parameter corresponding states theory (CST) [23, 24] for diatomic molecules of the HDYF type. While we have not identified a microscopic basis for a two-parameter CST, we are able to verify that CST holds for our class of diatomic fluids by considering the spinodal curves for different elongations plotted as functions of the usual CST reduced variables: TIT c (=KJK) and P/Pc. This is illustrated in figure 4 where, in these reduced units, we find that the spinodal curves for the three elongations L = 0, a/2 and o'/3 are indistinguishable on the scale of the graph. Remarkably, the locus of the limit of real solutions also exhibits a high degree of corresponding states. We now turn to an examination of the structure of the HDYF along a subcritical isotherm (K > K~). Specifically, we consider the HDYF with L = a/2, z = 2a -1 and K= 0"6. At this temperature, pga3= 0"1264 and pl 0"3= 0"3415. The pair distribution function g(r) is shown in figures 5 to 9 at two gas densities (pa 3= 0-08, 0.126) and three liquid densities (pa3= 0.342, 0"45 and 0"6). The pair distribution function for the HDF at the same density and elongation (corresponding to K--~ 0, or high temperature) is shown for comparison. On the gas side of the spinodal region (also known as the limit of metastability), the pair distribution is monotonic beyond the cusp at a + L, is increasingly long-ranged as p--~ p, and is much larger than the HDF correlation function at the same I-2 ' I ' ] ' I-0 Kc/K I 2 3 P/Pc Figure 4. The spinodal curve ( ) and the limit of real solutions (,--) plotted in the form of reduced temperature (KJt 0 against reduced density (P/Pc) for the HDYF with Yukawa potential parameter z = 2a- 1 for the three elongations L = O, 6/3 and a/2. On the scale of this figure, these curves are indistinguishable for the different elongations.

9 Hard dumbell fluid 857 I ' I ' I g(r) // /!!! / 0 0 ] P ] ~ r /(Y Figure 5. The pair distribution function g(r) for the HDYF with elongation L = a/2 and Yukawa potential parameter z = 2a- 1 at dimensionless inverse temperature K = 0-6 and density per 3 = 0"08. The dashed curve is the HDF result at the same reduced density. I ' I ' I I g(r) / / / / /, I, I F r/o- Figure 6. The pair distribution function g(r) for the HDYF with elongation L = a/2 and Yukawa potential parameter z ~ 2a- 1 at dimensionless inverse temperature K = 0-6 and density pa 3 = 0'126. The dashed curve is the HDF result at the same reduced density.

10 858 P.T. Cummings ' I ' I ' I ' g(r) r ] i I 2 3 r/o" Figure 7. The pair distribution function g(r) for the HDYF with elongation L = a/2 and Yukawa potential parameter z = 2a- 1 at dimensionless inverse temperature K = 0.6 and density pa3 = The dashed curve is the HDF result at the same reduced density. I I ~ [ I I ' g(r) l I, I I 2 3 r/o- Figure 8. The pair distribution function g(r) for the HDYF with elongation L -~ a/2 and Yukawa potential parameter z = 2a- t at dimensionless inverse temperature K = 0-6 and density pa3 = 0'45. The dashed curve is the HDF result at the same reduced density.

11 Hard dumbell fluid 859 ' I ' I ' I ' g(r) 0 I o, 1, I, 2 r/o" 3 4 Figure 9. The pair distribution function g(r) for the HDYF with elongation L = a/2 and Yukawa potential parameter z = 2a-1 at dimensionless inverse temperature K = 0-6 and density pa 3= 0-6. The HDF result at the same reduced density is indistinguishable from the HDYF curve on the scale of this figure. density. On the liquid side of the spinodal region, the pair distribution function displays oscillations characteristic of liquid structure. As the density p is increased, the difference between the HDF and the HDYF diminishes as the repulsive interaction increasingly dominates the structure of the fluid. At moderate liquid densities, the primary effect of the attraction is to increase the value of g(r) between contact (r = a) and the cusp (r = a + L). The structure factor for a homonuclear diatomic fluid is given by [25] S(k) = 1 + co(k) + 2p/~(k). (3.2) Results for S(k) for the HDYF are displayed in figures 10 to 13. The solid curve in each figure corresponds to the subcritical isotherm K = 0"6 for the HDYF with L = a/2 and z = 2a -1, so that these structure factors correspond to the pair distribution functions shown in figures 5 to 9. The figures additionally contain the structure factors for the HDF at the same density (shown by the broken curve) and the HDYF with elongation L = a/3 at the corresponding state (i.e. at the same Kc/K and P/Pc as the HDYF with elongation L = a/2) shown by the dotted curve. The comparison between the HDYF and the HDF (both with L = a/2) shows that S(k) is dominated at large wavenumber by repulsive interactions, while, except at high density, the attraction significantly effects the low wavenumber part except at high density. In comparing the HDYF with L = a/2 and a/3 at thermodynamically corresponding states, we note that structure factors differ both in amplitude and phase. The latter difference no doubt arises in part due to the different length scales present on the two fluids, which in turn implies that a corresponding scaling of wavenumber k would be necessary to compare structure factors more meaningfully. (For example, rescaling the wavenumber using the factor (pr = a/3)/pc(l = 6/2)) 1/3 might be appropriate.) However, we

12 860 P.T. Cummings ' I ' I ' S(k),,.~ "o "o,,o,'. //.~ ~ "~176./~ %~ ODO~ ~ ~ ~ I I I [, ~k_, 0 5 I0 1,5 20 ko- Figure 10. potential parameter z=2cr -1 at temperature K=0'6 The structure factor S(k) for the HDYF with elongation L = a/2 and Yukawa and density p~3=0.126 ( ) (Kc/K---0"8673, P/Pc = 0'573). The structure factors for the HDF at the same elongation and density (---) and the HDYF with elongation L = a/3 and at the same reduced temperature and density (... ) are shown for comparison. e -'" sckl -, ; - Figure 11. o I, I, I, 0 5 I ko- The structure factor S(k) for the HDYF with elongation L = a/2 and Yukawa potential parameter z = 2or -1 at temperature K = 0"6 and density pa3= ( ) (Kc/K = , P/Pc = 1-555). The dashed and dotted curves have the same meaning as the corresponding curves in figure 10.

13 Hard dumbell fluid 861 ' I ' I ' t : S(k) "--_..._... ;.'"...,. o,"~ I I i I, i i 0 5 I Figure 12. The structure factor S(k) for the HDYF with elongation L = 0/2 and Yukawa potential parameter z -- 2o- 1 at temperature K = 0-6 and density pa 3 = 0-45 ( ) (Kr 0"8673, P/Pc = 2-046). The dashed and dotted curves have the same meaning as the corresponding curves in figure 10. S(k) 2 ' I ' I ' I ~ t o 5, I ~ I I to t5 20 kcr Figure 13. The structure factor S(k) for the HDYF with elongation L = or/2 and Yukawa potential parameter z x at temperature K = 0-6 and density per 3 = 0'6 ( ) (Kc/K= , P/Pc = 2-729). The dashed and dotted curves have the same meaning as the corresponding curves in figure 10.

14 862 P.T. Cummings note that rescaling in wavenumber space will not reduce the difference in structure factor amplitude, so that it is clear that the structure factor (and hence the' structure in general) does not follow a simple corresponding states rule--a result which is perhaps to be expected, since the structure fundamentally involves the two length scales a and L. The enhanced peaks of S(k) for the L = a/3 HDYF compared to the L = a/2 HDYF can be rationalized on the following basis: in a simple fluid, the position kp of the principal peak of the structure factor generally corresponds in r-space to the position r r of the first peak in the pair distribution function approximately by the relation kp ~ 2n/rf. Thus, for the hard sphere fluid, the first peak in S(k) occurs at ka ~ 2n. This follows by simple inspection of the expression for S(k) for a simple fluid. In a homonuclear diatomic fluid, the structure most easily identified with the first peak in a simple fluid g(r) is smeared out over the range (a, ~r + L). Thus, for the shorter elongation, the principal r-space structure is more localized, with the result that one can expect to see a stronger signal in the corresponding first peak of the structure factor, as we do indeed observe in figures 10 to CONCLUSION The HDYF represents a model for real homonuclear diatomic molecules because it incorporates both a hard core, diatomic shape and a model for an attractive dispersion interaction. Moreover, it is analytically solvable within the ZPA, making it an attractive vehicle for the qualitative study of various equilibrium properties of diatomic fluids, such as thermodynamic and structural properties of pure diatomic fluids and their mixtures and inhomogeneous molecular systems. The ZPA has the virtue of reducing to the MSA for the HCYF as the elongation goes to zero, making it possible (within the framework of the same approximation) to compare molecular and simple fluids. We are currently investigating the thermodynamics of the HDYF both from the ZPA energy equation of state and from Monte Carlo simulation (since the accuracy of the ZPA for the HDYF has not yet been established) and extensions of the present work to mixtures of simple and diatomic fluids are being explored. The author is indebted to G. P. Morriss, G. Stell and P. A. Monson for very helpful discussions. This research has been supported by the Camille and Henry Dreyfus Foundation through the award to the author of a Grant for New Faculty in the Chemical Sciences; this support is gratefully acknowledged. REFERENCES 1"1] MORRISS, G. P., and CUMMtNCS, P. T., 1983, Molec. Phys., 49, ] CUMM~NCS, P. T., MormIss, G. P., and STELL, G., 1984, Molec. Phys., 51,289. 1"3] CHANDLER, D., and ANDERSEN, H. C., 1972, ff. chem. Phys., 57, ] CUMM~NCS, P. T., and STELL, G., 1982, Molec. Phys., 46, 383. [5] GRAY, C. G., and GUBmNS, K. E., 1984, Theory of Molecular Fluids (Oxford University Press). [6] CHANDLER, D., 1982, The Liquid State of Matter : Fluids, Simple and Complex, edited by E. W. Montroll and J. L. Lebowitz (North-Holland). 1"7] H~bYE, J. S., and STELL, G., 1977, S.U.N.Y. at Stony Brook College of Engineering and Applied Science Report No [8] BAXTER, R. J., 1968, Aust.ff. Phys., 21,563.

15 Hard dumbell fluid 863 [9] An equivalent set of factorized equations was derived by: WERTHEIM, M. S., 1964, J. math. Phys., 5, 643, using Laplace transform techniques. [10] MORRISS, G. P., PERRAM, J. W., and SMITH, E. R., 1979, Molec. Phys., 38, 465. [11] MORRISS, G. P., and SMITH, E. R., 1981,J. statist. Phys., 24, 611. [12] CUMMINGS, P. T,, MORRISS, G. P., and WRIGHT, C. C., 1981, Molec. Phys., 43, [13] WAISMAN, E., 1973, Molec. Phys., 25, 45. [14] HOYE, J. S., LEBOWITZ, J. L., and STELE, G., 1974, J. chem. Phys., 61, [15] HOYE, J. S., and BLUM, L., 1977, J. statist. Phys., 16, 399. [16] HCYE, J. S., and STELL, G., 1976, Molec. Phys., 32, 195. [17] CUMMINGS, P. T,, and SMITH, E. R., 1979, Molec. Phys., 38, 997. [18] CUMMINGS, P. T., and SMITH, E. R., 1979, Chem. Phys., 42, 241. [19] HENDERSON, D., WAISMAN, E., LEBOWITZ, J. L., and BLUM, L., 1978, Molec. Phys., 35,241. [20] CUMMINGS, P. T., and STELL, G., 1983,o7. chem. Phys., 78, [21] MONSON, P. A., 1984, Molec. Phys. (in the press). [22] STREETT, W. B., and TILDESLEY, D. J., 1978,07. chem. Phys., 68, [23] PITZER, K. S., 1939, J. chem. Phys., 7, 583. [24] GUGGENHEIM, E. A., 1945,07. chem. Phys., 13, 253. [25] LOWDEN, L. J., and CHANDLER, D., 1973,07. chem. Phys., 59, 6587.