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

MOLECULAR PHYSICS, 1984, VOL. 53, No . 4, 849-863

Analytic studies of the hard dumbe l l fluid

III. Zero pole approximation for the hard dumbe l l Yukawa fluid

by P. T . C U M M I N G S

D e p a r t m e n t of Chemical Engineer ing, T h o r n t o n Hall, Univers i ty of Virginia, Charlottesville,

Virginia 22901 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 tran- sition 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 Orns t e in -Zern ike (SSOZ) equat ion for the hard dumbel l fluid ( H D F ) , a model fluid whose molecules are composed of two hard spheres each of d iameter a fused together so that the separation between the centres of the molecules is equal to L. Thus , the H D F is an example of an interact ion site model ( I S M ) fluid [3 -6] , the general t e rm given to fluids whose in termolecular pair potential u(12) (which depends on both the posit ions and orientat ions of molecules 1 and 2) can be decomposed into a sum of interact ions between sites (generally identified as the centres of the a toms in a po lya tomic fluid), viz.

u(12) = ~ u~p(r~#). (1.1) a,p=l

Here, m is the n u m b e r of sites in each molecule, r ~ is the distance between sites and fl in dist inct molecules, and u~(r) is the spherically symmet r i c si te-site pair potential be tween sites ~ and fl in dist inct molecules. For the H D F , there are only two sites, and each of the three possible interact ions are equal and given by

u l t ( r ) = u 1 2 ( r ) = u 2 2 ( r ) = u ( r ) = 0% r < a , l (1.2)

J ----- 0, r > (r,

850 P . T . Cummings

The usual measure of s tructure in I S M fluids is the site-site pair (or radial) distribution function g~#(r) proport ional to the probabil i ty density of finding sites a and fl in distinct molecules at separation r, and for the H D F the three possible pair distributions functions are, as with the pair potentials, all equal by sym- metry, so that gt 1(r) = gl 2(r) = g22(r) = g(r).

The SSOZ equation for the H D F , 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 three- dimensional Fourier t ransforms of the site-site total and direct correlation functions, h(r) = g(r) - 1 and c(r), respectively, while c0(k) is the Fourier t ransform of the intramolecular correlation function [3, 43 given by

c0(k) = 1 + sin ( k L ) / k L . (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, r < a,]. (1.5)

c(r ) = - - u ( r ) / k B T , r >

As detailed in I and I I , another approximation can be derived which is called the zero pole approximation (ZPA) and is given by

h(r) = - -1 , r < a , ] (1 .6)

c(r) = - - u ( r ) / k B T + F(r) , r > a,

where F(r) is a function described in detail in w of I I to which the reader is referred. For the H D F , the ZPA is analytically solvable due to the simplicity of th is closure f rom the particle-particle viewpoint discussed elsewhere [7, 4, 2]. The method used is a variant o f ' 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) = K a 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 H D F 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 H D F which is analogous to the generalized mean spherical approximation for the hard sphere fluid [13, 14]. F rom 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)

= - - D a e x p [ - - z ( r - - ~ ) ] / r , r > c r ,

Hard dumbell fluid 851

when K is identified with D/k B T, where D is the depth of the attractive well of the pa i r 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 H C Y F 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 non- spherical molecules, and the study of its propert ies within the ZPA is the subject of this paper.

In w the main equations f rom I I which are relevant to the ZPA for the H D Y F are given. The analytic solution reduces to a single quartic equation for a fundamental parameter fl which is studied as a function of density, tempera ture and elongation. In w results for the spinodal curve of the compressibil i ty 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 S S O Z 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 I I and a reiteration of the major equations.

T h e 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) f rom I I 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 ult imately 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 I I ; 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). Tha t 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 H C Y F [-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 H C Y F case.

852 P . T . C u m m i n g s

In figure 1, the behaviour of the parameter fl as a funct ion of densi ty p is shown for the H D Y F with an elongat ion L = o-/2, Yukawa parameter z = 2o--1 and along four isotherms indicated by four different values o f K (0"2, 0-4, 0 '6 and 0-8). (Since K = D/kB T, K is equivalent to an inverse reduced t empera tu re ; thus the four values of K cor respond to successively lower temperatures . ) T h e first observat ion which m a y be made is that a l though the quart ic (2.1) has in principle four roots, for attractive Yukawa site-site potentials there are at mos t two real roots. For K below a part icular value K t - 0.533358, there are two real results for fl (shown as the upper and lower curves in the figure) for all physical ly meaningfu l densities. T h e lower curve is the physical ly correct b ranch of the solutions of equat ion (2.1), since this solution correct ly approaches zero as K---~ 0 while the o ther diverges to infinity. For K > K t (or, equivalently, sufficiently low temperature) , there is an interval of densi ty over which the quart ic (2.1) has no real roots. Mathemat ica l ly , the switch f rom two real roots to no real roots is character ized by the d iscr iminant of the quart ic becoming zero, and the two real roots becoming equal. Thus , we can plot the locus of points along which the d iscr iminant of (2.1) is zero : this is shown as the line L 1 in the figure. T h e line L 1 clearly dist inguishes between the physical ly correct and incorrect branches of the /3 curves, since the correct value of fl lies below L1 on any given isotherm.

In addit ion to the existence of real solutions, there is another const ra int which the value of/~ mus t satisfy : the compress ibi l i ty equat ion of state is given by

1 = 1 -- 4p f c(r) dr = a2/2, (2.3)

OP

kB T ~-fip r

where a is the parameter in the analytic solut ion defined in ( I I .2 .10) , so that the limit of metastabil i ty (called the spinodal curve) is given as the locus o f points along which a = 0. At low density, a is positive, so that physical ly meaningfu l results are obta ined only when a > 0. T h e curve L 2 in figure 1 shows the locus of points along which a = 0, and solutions of equat ion (2.1) for which a > 0 lie below the curve L z .

F r o m figure 1, it is clear that for values of K below a critical value K c = 0"520389 ( temperatures above the critical t empera tu re To) there is no densi ty 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. T h e lower value, pg, cor responds to the gas side of the spinodal curve, while the large value, Pl, cor responds to the l iquid side. At K - - K c , the equat ion a = 0 has a double root in densi ty at pg = Pl = Pc, the critical density. T h u s , the l iqu id-gas critical point o f the H D Y F f rom the c o m - pressibili ty equat ion of state is given by T = To, po-3= pc o-3= 0-219851. T h e qualitative behaviour of the parameter f l - - inc lud ing the relative posit ions of the two curves L 1 and L2- - i s the same as that for the H C Y F in the mean spherical approximat ion [-17, 18, 20]. As in the ease of the latter fluid, an analysis of the near vicinity of the compressibi l i ty equat ion critical point of the H D Y F in the Z P A reveals that the critical exponents have spherical mode l 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 _ p c { _ ~ + l at T = T c. (2.5) k B T "o-fi I r

H a r d dumbel l f lu id 853

12 ' I ' I ' I ' .

: ~, ~.."

0.0 0.2 0.4 0.6 0.8 pO 3

Figure 1. The parameter /Y as a function of density pa ~ for the H D Y F 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 discr iminant of equation (2.1) is zero (LI) and the locus of points along which the compressibil i ty diverges (L2).

F i g u r e 2 shows the b e h a v i o u r of the p a r a m e t e r fl for the H D Y F w i t h e lon - ga t i on L = a/3 a n d Y u k a w a p a r a m e t e r z = 2 a - 1 at i nve r se t e m p e r a t u r e s K = 0"2, 0"4, 0"6 a n d 0"8. F o r th i s e l o n g a t i o n , K t = 0 .424133 , K c = 0"413381 a n d p e a 3 =

0"241711. I n o r d e r to c o m p a r e the resu l t s for the Z P A a p p l i e d to the H D Y F to the m e a n

spher ica l a p p r o x i m a t i o n ( M S A ) for the H C Y F , it is n e c e s s a r y to e x a m i n e the L--~ 0 l im i t o f the H D Y F a n d the S S O Z e q u a t i o n . Phys i ca l ly , w h e n L---~ 0, t he f o u r e q u a l s i t e - s i t e i n t e r a c t i o n s b e c o m e c e n t r e d at t he same site : the c e n t r e of the s p h e r e t ha t is o b t a i n e d as L is t a k en to zero. T h u s , t h e H C Y F w h i c h is o b t a i n e d in th i s l imi t has an in termolecular p o t e n t i a l Uncvv(r) g i v e n b y

UHCYF(r) = 0% r < G, ] (2.6)

-- - 4 D ~ e x p [ - z ( r - - a ) ] / r , r > g ,

s ince the i n t e r m o l e c u l a r p o t e n t i a l has f o u r s i t e - s i t e c o n t r i b u t i o n s . T h e f u n c t i o n F(r) r e d u c e s to a de l ta f u n c t i o n at r = 0 in the L---~ 0 l imi t , so tha t the Z P A for the

H D Y F r educes to

h(r) - - 1 , r < cr,~ (2.7)

c(r) = K ~ exp E - z ( r - ~)] /r , r > G, 3 w i t h K = D / k s T as be fore . S i n c e e)(k)---~ 1 as L--~ 0, t he S S O Z e q u a t i o n b e c o m e s

/~(k) = 4~(k) + 4p~(k.)h(k), (2.8)

w h e r e p is n o w the d e n s i t y of the a t o m s f o r m e d as L ~ 0.

854 P . T . C u m m i n g s

12 ' I ' I ' I '

4 �9 .,, ~...._ . . f ' ~ . ~ . . - " - -

"" " ~ * ~ ' ~ ' ~ ~ ~ L

0 , I I I i I-- '-- "-~ . . . . . 0.0 0.2 0 . 4 0 .6 0 .8

pa 3

Figure 2. The parameter /3 as a function of density p6 3 for the H D Y F 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.

Addit ional ly, we need to consider the relation between the a tomic fluid correlation funct ions of the H C Y F and the site-site correlat ion funct ions of the H D Y F in the L--* 0 limit. Deno t ing the a tomic fluid total and direct correlat ion func- t ions by h*(r) and c*(r) respectively, then the usual Orns t e in -Zern ike (OZ) equation for a simple fluid such as the H C Y F is given by

l~*(k) = ~*(k) + p~*(k)l~*(k), (2.9)

where p is the n u m b e r densi ty of atoms. T h u s , by compar ing (2.8) and (2.9), it is clear that h*(r) = h(r) (i.e. the si te-site and atomic total correlat ion funct ions are the same) while c * ( r ) = 4c(r) (i.e. the a tomic direct correlat ion funct ion is four t imes the si te-site correlat ion function). N o w consider the M S A for the H C Y F with interaction potential (2.6): application of the M S A yields the fol lowing condi t ions on the h*(r) and c*(r)

h*(r) = - -1 , r < ~,] (2.10)

c*(r) = K * ~ exp [ - z ( r -- ~)]/r, r > ~,

where K * = 4 D / k B T = 4K. Th i s closure on c*(r) is comple te ly consis tent with the closure (2.7) on c(r) and the relat ionship between c*(r) and c(r). Thus , we find that the Z P A for the H D Y F reduces to the M S A for the H C Y F in the limit L--~ 0. Th i s si tuation is to be contras ted with other, non- l inear approximat ions such as the site-site version of the Percus -Yevick approximat ion discussed in detail by M o n s o n [-21]. One consequence is that the results of the M S A for the H C Y F - - i n particular, the behaviour of the parameter /~ [17, 18, 2 0 ] - - c a n be directly compared with those of the Z P A for the H D Y F presented in this paper, keeping in mind that the parameter K (denoted K* here) used in [17, 18, 20] is four t imes the parameter K of the H D Y F .


3. THERMODYNAMIC AND STRUCTURAL PROPERTIES OF THE H D Y F

In f igure 3, we p r e sen t the sp inoda l curve of the H D Y F (the locus of po in t s in the d imens ion l e s s t e m p e r a t u r e 1/K and dens i ty p lane a long which a = 0) wi th z = 2 a - 1 and for th ree e longa t ions L = 0, a/3 and a/2. T h e cr i t ical t e m p e r a t u r e and d imens ion l e s s cr i t ica l dens i ty Pc a3 are bo th h i g h e r for sho r t e r e longa t ions . In c o m p a r i n g cr i t ical dens i t ies , it is p e r h a p s m o r e a p p r o p r i a t e to c o m p a r e the cr i t ical v o l u m e f rac t ions given by [21]

fc = (~z/h)Pc d3 = (z~/6)pca3( 1 + 3L*/2 -- L ' 3 / 2 ) , / (3.1)

whe re d is the d i a m e t e r of the sphe re o c c u p y i n g the same v o l u m e as the d u m b e l l and L * = L/0-. F o r L = 0, a/3 and 0-/2, fc = 0-1660, 0-1875 and 0-1943 respec- t ively. W h i l e no t ing tha t the 9 pe r cent d i f ference be tween Pc for L = 0-/3 and L = a/2 has been r e d u c e d to a d i f ference of on ly 3 pe r cent in f r we see that fc for the H C Y F (L = 0) is s igni f icant ly lower than the resu l t for e i ther n o n - z e r o e longa t ion . Desp i t e this , it is c lear tha t cr i t ica l v o l u m e f rac t ion is not a ve ry s t rong func t ion of e longa t ion (over a 50 pe r cent range in e longa t ion , the cr i t ica l v o l u m e f rac t ion changes b y on ly 17 per cent) , sugges t ing tha t m o l e c u l a r vo lume m a y be m o r e useful than the r e d u c e d dens i ty p0-3 as a co r re la t ing p a r a m e t e r for h o m o n u c l e a r d ia tomics .

In cons ide r ing the d e p e n d e n c e of the resul t s for T c on e longat ion , we note tha t our c o m p a r i s o n for d i f ferent values of L m a y not be ve ry mean ing fu l . T h a t is, c o m p a r i n g resul ts in the fo rm used in f igure 3 is t a n t a m o u n t to a s s u m i n g tha t s i t e - s i t e pa i r po ten t ia l s are equal . H o w e v e r , the i n t e r m o l e c u l a r po ten t i a l d e p e n d s no t on ly on the s i t e - s i t e po ten t ia l s , bu t also on the elongation of the molecu le , and

' I ' I ' I '

3

I /K

L - O

2

I L : 1 / 2

0 , I ~ I f I L

o.o 0 .2 0 . 4 0 - 6 0 . 8

po-3

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 H D Y F with Yukawa potential parameter z = 2a-1 at three elongations: L = 0, a/3 and a/2. Notice that at L -- 0, the H D Y F reduces to the HCYF.


to compare different elongation molecules meaningfully probably requires adjust- ing 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 H D Y F type. While we have not identified a microscopic basis for a two-parameter CST, we are able to verify that C S T holds for our class of diatomic fluids by considering the spinodal curves for different elongations plotted as functions of the usual C S T reduced variables: TIT c ( = K J K ) 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 H D Y F along a sub- critical isotherm (K > K~). Specifically, we consider the H D Y F 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 H D F 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 H D F correlation function at the same

I - 2 ' I ' ] '

I-0

Kc/K

0.8

0.6

0 .4 0 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.

H a r d dumbel l f l u id 857

g(r)

0 0

I ' I ' I

/ / /

! !

! /

] P ] ~ _ _ 2 3 4

r /(Y

Figure 5. T h e pair d i s t r ibu t ion func t ion g(r) for the H D Y F wi th e longat ion L = a/2 and Yukawa potent ia l p a r a m e t e r z = 2 a - 1 at d i m e n s i o n l e s s inverse t e m p e r a t u r e K = 0-6 and dens i ty per 3 = 0"08. T h e dashed curve is the H D F resul t at the same r educed densi ty .

g(r)

I ' I ' I I

/ /

/ /

/

, I , I F 2 3 4

r/o-

Figure 6. T h e pair d i s t r ibu t ion func t ion g(r) for the H D Y F wi th e longat ion L = a/2 and Yukawa potent ia l pa r ame te r z ~ 2 a - 1 at d imens ion l e s s inverse t e m p e r a t u r e K = 0-6 an d dens i ty pa 3 = 0'126. T h e dashed curve is the H D F resul t at the s a m e r educed densi ty .

858 P . T . C u m m i n g s

g(r)

' I ' I ' I '

r ] i I 2 3

r/o"

Figure 7. T h e pair d is t r ibut ion funct ion g(r) for the H D Y F with elongation L = a/2 and Yukawa potential parameter z = 2 a - 1 at d imensionless inverse tempera ture K = 0.6 and densi ty pa3 = 0.342. T h e dashed curve is the H D F result at the same reduced density.

g(r)

I I ~ [ I I '

l I , I I 2 3

r/o-

Figure 8. T h e pair d is t r ibut ion funct ion g(r) for the H D Y F with elongation L -~ a/2 and Yukawa potential parameter z = 2 a - t at d imensionless inverse t empera ture K = 0-6 and densi ty pa3 = 0'45. T h e dashed curve is the H D F result at the same reduced density.


g(r)

' I ' I ' I '

0 I

o , 1 , I ,

2 r / o " 3 4

Figure 9. The pair distribution function g(r) for the H D Y F 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 H D F result at the same reduced density is indistinguishable from the H D Y F curve on the scale of this figure.

dens i ty . On the l iqu id s ide of the sp inoda l reg ion , the pa i r d i s t r i bu t i on func t ion d i sp lays osc i l la t ions charac te r i s t i c of l iqu id s t ruc tu re . As the dens i ty p is increased , the d i f ference be tween the H D F and the H D Y F d imin i shes as the r epu l s ive in te rac t ion inc reas ing ly d o m i n a t e s the s t ruc tu re of the fluid. A t m o d e r - ate l iqu id dens i t ies , the p r i m a r y effect of the a t t r ac t ion is to increase the va lue of g(r) be tween con tac t (r = a) and the cusp (r = a + L).

T h e s t ruc tu re fac tor for a h o m o n u c l e a r d i a tomic fluid is g iven by [25]

S(k) = 1 + co(k) + 2p/~(k). (3.2)

Resu l t s for S(k) for the H D Y F are d i s p l a y e d in f igures 10 to 13. T h e sol id curve in each f igure c o r r e s p o n d s to the subcr i t i ca l i s o t h e r m K = 0"6 for the H D Y F wi th L = a/2 and z = 2a -1, so tha t these s t ruc tu re factors c o r r e s p o n d to the pa i r d i s t r i b u t i o n func t ions shown in f igures 5 to 9. T h e f igures add i t i ona l ly conta in the s t ruc tu re factors for the H D F at the same dens i ty ( shown by the b r o k e n curve) and the H D Y F wi th e longa t ion L = a/3 at the c o r r e s p o n d i n g s tate (i.e. at the same Kc/K and P/Pc as the H D Y F wi th e longa t ion L = a/2) shown by the d o t t e d curve . T h e c o m p a r i s o n be tween the H D Y F and the H D F (bo th wi th L = a/2) shows tha t S(k) is d o m i n a t e d at large w a v e n u m b e r b y r epu l s ive in t e r - act ions , while , excep t at h igh dens i ty , the a t t r ac t ion s igni f icant ly effects the low w a v e n u m b e r par t excep t at h igh dens i ty . In c o m p a r i n g the H D Y F wi th L = a/2 and a/3 at t h e r m o d y n a m i c a l l y c o r r e s p o n d i n g states, we no te tha t s t ruc tu re factors differ bo th in a m p l i t u d e and phase . T h e la t te r d i f ference no d o u b t ar ises in pa r t due to the d i f ferent l eng th scales p r e sen t on the two fluids, wh ich in t u rn impl i e s tha t a c o r r e s p o n d i n g sca l ing of w a v e n u m b e r k w o u l d be necessa ry to c o m p a r e s t ruc tu re factors m o r e mean ing fu l l y . ( F o r example , r esca l ing the w a v e n u m b e r us ing the fac tor (pr = a/3)/pc(L = 6/2)) 1/3 m i g h t be app rop r i a t e . ) H o w e v e r , we

860 P . T . C u m m i n g s

S ( k )

' I ' I '

,,.~ "o "o, ,o

,'. / / .~ ~176176176 "~176 ./~176176176 %~

O D O ~ ~ ~ ~

I I I [ , ~ k _ , 0 5 I0 1,5 20

k o -

Figure 10. The s t ructure factor S(k) for the H D Y F with elongation L = a/2 and Yukawa potential parameter z = 2 c r -1 at t empera ture K = 0 ' 6 and densi ty p ~ 3 = 0 . 1 2 6 ( ) (Kc/K---0"8673, P/Pc = 0'573). T h e s t ructure factors for the H D F at the same elongation and densi ty ( - - - ) and the H D Y F with elongation L = a/3 and at the same reduced tempera ture and densi ty ( . . . . . ) are shown for comparison.

e - ' "

s c k l -

, ; -

o I , I , I , 0 5 I0 15 20

ko-

Figure 11. T h e s t ructure factor S(k) for the H D Y F with elongation L = a/2 and Yukawa potential parameter z = 2or -1 at t empera ture K = 0"6 and densi ty pa3= 0-342 ( ) (Kc/K = 0-8673, P/Pc = 1-555). T h e dashed and dot ted curves have the same meaning as the cor responding curves in figure 10.

H a r d dumbell f luid 861

S(k)

' I ' I '

t :

. . . , . o ,"~

"--_..._ ....... ;.'" I I i I , i i

0 5 I 0 15 20

F i g u r e 12. T h e s t r u c t u r e fac tor S(k) for t h e H D Y F w i t h e longa t ion L = 0/2 and Yukawa po t en t i a l p a r a m e t e r z -- 2 o - 1 at t e m p e r a t u r e K = 0-6 a n d d e n s i t y pa 3 = 0-45 ( ) (Kr 0"8673, P/Pc = 2-046). T h e d a s h e d a n d d o t t e d cu rves have the s a m e m e a n i n g as t he c o r r e s p o n d i n g cu rves in f igure 10.

S(k)

2

' I ' I ' I

~

o 5

t

, I ~ I I

tO t5 2 0 kcr

F i g u r e 13. T h e s t r u c t u r e fac tor S(k) for t he H D Y F w i t h e longa t ion L = or/2 and Y u k a w a po ten t i a l p a r a m e t e r z --- 2 0 - x at t e m p e r a t u r e K = 0-6 and dens i t y per 3 = 0 '6 ( ) (Kc /K= 0-8673, P/Pc = 2-729). T h e d a s h e d and d o t t e d cu rves have the s a m e m e a n i n g as t he c o r r e s p o n d i n g cu rves in f igure 10.


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 H D Y F compared to the L = a/2 H D Y F 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 13.

4. CONCLUSION

The H D Y F 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 equi- librium 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 H C Y F 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 H D Y F both from the ZPA energy equation of state and from Monte Carlo simulation (since the accuracy of the ZPA for the H D Y F 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, 1103. 1-2] 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, 1930. 1-4] 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 Uni-

versity 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. 307. [8] BAXTER, R. J., 1968, Aust.ff. Phys., 21,563.

Hard dumbell f luid 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, 1299. [13] WAISMAN, E., 1973, Molec. Phys., 25, 45. [14] HOYE, J. S., LEBOWITZ, J. L., and STELE, G., 1974, J. chem. Phys., 61, 3253. [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, 1917. [21] MONSON, P. A., 1984, Molec. Phys. (in the press). [22] STREETT, W. B., and TILDESLEY, D. J., 1978,07. chem. Phys., 68, 1275. [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.

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

Documents