Equations of State for Nonspherical Molecules Based on the

11 Equations of State for Nonspherical Molecules Based on the Distribution Function Theories 1

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S. B. Kanchanakpan , L. L. Lee , and Chorng H. Twu 1

School of Chemical Engineering and Materials Science, University of Oklahoma, Norman, OK 73019 Simulation Science, Inc., Fullerton, CA 92633

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Based on the distribution function theories of the liquid state, we are able to derive an expression for the contribution from the attractive energy to the pressure of anisotropic fluids. We have adopted the nonspherical square-well potential with Gaussian overlap to model moderately anisotropic molecules. The repulsive pressure is shown to be given essentially by the underlying hard core repulsion and is represented, for example, by the Nezbeda equation of state for hard convex bodies. It is found that the background correlation function, y(12), plays a major role in the determination of attractive pressures. We exhibit the cluster series for this correlation function and derive therefrom a resummat ion formula for the pressure. To obtain adequate description of real fluid properties, a two-step form of the square-well potential as proposed by Kreglewski is adopted. The final equation consists of three terms, clearly separated into the hard core, repulsive and attractive parts. It is used to correlate the P-v-T and thermal behavior of some 69 substances, including hydrocarbons, ketones, alcohols, amines and polar solvents. For pressures up to 69 MPa, the errors in density and vapor pressure predictions are, with few exceptions, within 1%. Comparison with similar equations of state, such as the Peng-Robinson and Mohanty-Davis equations, shows that the present equation is uniformly superior. The use of hard convex molecules as a reference f l u i d f o r r e a l f l u i d s has received much attention l a t e l y , e s p e c i a l l y i n perturbation theory formulations (1-4) dealing with nonspherical molecules. This i s due to the recognition that i n l i q u i d s the structure i s e s s e n t i a l l y determined by the repulsive part of the molecular i n t e r a c t i o n potent i a l . On the other hand, the a t t r a c t i v e i n t e r a c t i o n makes a major 0097~6156/86/0300-0227$06.75/0 © 1986 American Chemical Society

Chao and Robinson; Equations of State ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

228

EQUATIONS OF STATE: THEORIES AND APPLICATIONS

contribution to the energy of the l i q u i d (Note: the energy derived from the hard convex bodies, HCB, i s zero, and i s therefore inade quate for thermal properties.) In order to achieve a complete de s c r i p t i o n of the thermodynamics of the l i q u i d state, both c o n t r i butions, a t t r a c t i v e as well as repulsive, should be considered. Beret and Prausnitz (5) proposed the perturbed hard chain (PHC) equation of state for long chain hydrocarbons. It contained a modified hard sphere term of repulsion and a square well term for a t t r a c t i v e pressure. I t was applied to hydrocarbons such as methane, ethane, n-decane and eicosane as well as molecules involving polar forces, such as carbon monoxide and water. Later the approach has been extended (Donohue and Prausnitz (6) 1978, Gmehling and Prausnitz (7) 1979) to mixtures and highly polar substances such as methanol, ethanol, acetone and acetic acid by using a chemical theory. The BACK (Boublik-Alder-Chen-Kreglewski) equation proposed by Chen and Kreglewski (8) and Simnick, L i n and Chao (9) was based on the Boublik (10) equation of hard convex bodies. It has been success f u l l y applied to f l u i d s such as methane, neopentane and hydrogen s u l f i d e . In these formulations the a t t r a c t i v e contribution to the pressure was expressed i n a 24-parameter temperature-density double power series o r i g i n a l l y given by Alder, Young and Mark (11). This a t t r a c t i v e term was l a t e r modified to a 21-parameter or a 10parameter series. These series, however, are cumbersome to use and do not reveal the underlying physical basis. We propose here a t h e o r e t i c a l formulation of the a t t r a c t i v e contributions to the equations of state based on the d i s t r i b u t i o n function theories of l i q u i d s and to derive equations that are a p p l i cable to moderately anisotropic f l u i d s . F i r s t we use the Gaussian overlap potential of Berne and Pechukas (12) to represent aniso tropic forces. We note that a continuous potential (see Figure 1) can be approximated by an η-step p o t e n t i a l . In the l i m i t η—> oo , the o r i g i n a l p o t e n t i a l i s recovered. The v a l i d i t y of this repre sentation i s closely associated with the Riemann integration theory i n r e a l analysis. Secondly, we apply the cluster theories of c o r r e l a t i o n functions f o r the derivation of the a t t r a c t i v e pressure. The approach i s therefore d i f f e r e n t from the perturbation approach. These developments w i l l be presented i n Sections 2 and 3. E a r l i e r studies on r i g i d nonspherical molecules were based on the so-called scaled p a r t i c l e theory (13) (SPT). This theory made use of the fact that the chemical potential of an N-body system i s related to the energy required to create a cavity i n the f l u i d i n order to accommodate an additional p a r t i c l e . As developed by Reiss et a l . , (13) SPT contains certain approximations. Thus the results of the theory are not exact. Gibbons (14,15) f i r s t applied this theory to hard convex bodies and obtained an equation of state of the form P/pkT = l / ( l - y ) + 3 a y / ( l - y )

2

2

2

+ 3a y /U-y)

3

(1)

where y i s the packing f r a c t i o n , y=pb, b i s the volume of a single hard convex molecule, α i s a r a t i o of geometries of the HCB, α = rs/3b


(2)

11.

KANCHANAKPAN ET AL.

229

Nonspherical Molecules

where r i s the mean radius of curvature, and s the surface area of the HCB molecule. For d i f f e r e n t shapes, the mean radius of curva ture i s given by d i f f e r e n t geometric formulas. As an example, for spherocylinders, the geometric factors are 2

3

r = R + L/4, s = 4πϊ1 +2πί&, and b = (4/3)πϊ1

2

-HrR L

(3)

where R i s the radius of the hemispheres and L the bond length be tween the centers of the hemispheres. For spheres, r = R, the radius of the hard sphere (HS). Equation 1 i s accurate at low densities but deteriorates at high densities. Thus i t must be improved. Boublik (10,16) proposed a modified form of (1) f o r prolate spherocylinders that reduces to the Camahan-Starling (17) equation at a=l: P/pkT = l / ( l - y ) + 3 a y / ( l - y )

2

2

2

+ 3a y /(l-y)

Nezbeda (18) upon considering the v i r i a l an alternative resummation formula 2

2

3

2

3

-a y /(l-y)

3

(4)

c o e f f i c i e n t s , proposed

P/pkT = [ l + ( 3 a - 2 ) y + ( a + a - l ) y - a ( 5 a - 4 ) y ] / ( l - y ) 3

3

(5)

which, i n comparison with simulation r e s u l t s for spherocylinders, i s more accurate than Equation 4, especially at high densities. These equations have been developed for the specialized geom etry of spherocylinders. In l a t e r studies (Nezbeda and Boublik (19, 20) on fused diatomic hard spheres, i t was found that Equation 4 r e mains applicable i f one substitutes for the dumb-bell an equivalent spherocylinder that has the "neck" f i l l e d i n . Thus the fused spheres can also be described by the equations developed for hard convex bodies. A recent study [Wojcik and Gubbins (21)] showed that Equation 4 i s also applicable to mixtures of hard dumb-bells. Nez beda and Boublik (22) (see also Nezbeda, Smith and Boublik (23), Nezbeda, Pavlicek and Labik (24), Boublik (25)) c l a s s i f i e d hard con vex bodies into three major types (i) l i n e a r molecules (e.g. prolate spherocylinders, l i n e a r fused hard spheres and diamonds), ( i i ) disk l i k e molecules (e.g. oblate spherocylinders) and ( i i i ) cubes. They found that Equation 4, derived for prolate spherocylinders, i s accurate for l i n e a r molecules, and reasonable for disk molecules, but i s less satisfactory f o r cube-like molecule. We s h a l l adopt the Nezbeda Equation 5 here to describe the harsh repulsive forces i n r e a l molecules not of the cubic shape. Theoretical Developments We formulate our approach i n terms of a square-well (SW) p o t e n t i a l . The SW potential has been extensively studied (for a review, see Luks and Kozak (26) and references contained therein). This i s a simple p o t e n t i a l embodying the essential features of the i n t e r a c t i o n forces i n r e a l molecules, i . e . , the excluded volume and a t t r a c t i v e forces. For nonspherical molecules, the o r i e n t a t i o n a l variations of the pair interaction should also be accounted f o r . One class of angle-dependent potentials that can be generated from simple spheri cal ones i s the so-called Gaussian overlap model of Corner (27) and


230

E Q U A T I O N S O F STATE: T H E O R I E S A N D A P P L I C A T I O N S

Berne and Pechukas (12). They b u i l t the molecular anisotropy into the potentials by incorporating angle dependence into the potential parameters. Therefore a Gaussian overlap model f o r the square-well (GOSW) potential i s , (See Figure 2) r Kincaid, S t e l l and Goldmark (30) have shown that with a p o s i t i v e shoulder at λ^, while setting a l l a t t r a c t i v e energies, ε-^=0, the f l u i d can support a c r i t i c a l point (see also Kreglewski (31)). For the purpose of t h i s study, we con sider a two-step SW p o t e n t i a l consisting of a p o s i t i v e shoulder and a negative w e l l (Figure 3). 2

u(12) = oo,

r < dtoy^)

ε (ω ω ), 1

1

(7)

dCcy^ r ^((γ^)

2

-ε (ω ω ), λ ^ ω ^ ω ^ r ^ ( ω ^ ) 2

χ

2

0,

r >

λ

ω

2

ω

^ 1 2^

The v i r i a l equation f o r anisotropic molecules i n s t a t i s t i c a l mechanics i s given by [Hansen and MacDonald (32)].

p/6J*dr

P/pkT = 1 -

4iTr

2

2

12

(8)

where the angular brackets indicate indicate the angle average

12

2

(4π)" / d 9

1

= 1/(4π) s i n 6 άφ 1

2 2

χ

Ιάω Jdo^

1

d6

2

άω d«

(...)

2

s i n 0 àj> 2

2

(9) (...)


11.


K A N C H A N A K P A N ET A L .

231

Figure 1. Approximation of the Lennard-Jones potential by a s i x step square-well p o t e n t i a l . Note that there are three repulsive shoulders and three a t t r a c t i v e wells following thereafter. The repulsive step acts e f f e c t i v e l y as a temperature dependent hard core dimension commonly used i n perturbation theories f o r soft repulsion.

d' d" d"

Or

!

Χ λ" X"

ι -" -€ 1 e

-e'

L _

...!

;û _ ο

r

-

Figure 2 . Schematic drawing of the Gaussin overlap anisotropic square-well p o t e n t i a l , u(r,θχ,Θ2,Φΐ2)» at three values of Θ2 ( 0 , π/6 and π / 2 ) . θ]_ i s kept at 0 . The dependence on the azimuthal angle, i2» i s not shown. The parameters, d, ε, and λ are the Kihara hard core dimension, the a t t r a c t i v e energy and the range of a t t r a c t i v e i n t e r a c t i o n , respectively. The primed quantities are values at d i f f e r e n t angles.


232

E Q U A T I O N S O F STATE. T H E O R I E S A N D A P P L I C A T I O N S

For nonlinear molecules, one should also integrate over the r o t a t i o n a l angles and normalize by (8π^)2. Note that the gradient of the pair p o t e n t i a l can be written i n terms of the Boltzmann factor, e(12) = exp[-$u(12)], as 33u(12)/9r

= -

(3e(12)/8r)exp[$u(12)]

(10)

However, due to the form of the two-step SW p o t e n t i a l , the derivative 8 e ( 1 2 ) / 3 r i s a sum of three Dirac deltas, one a r i s i n g from the hard core repulsion at dia^a^), the others from the steps at λχ(ωιω2) and λ2(ω^ω2) 9e(12)/8r

= 6 ( r , d ) e x p i - ^ ) + 6(r,X ) [ e x p ( 3 e ) exp(-3e )] 1

(11)

2

1

-6(r,X )[exp(3e )-l] 2

2

Substitution of Equation 10 and Equation 11 i n Equation 8 and change of r-integration with angle averaging gives P/pkT = 1 + (4π/6)ρ sw

3

+ (4π/6)ρ < l ( e λ

3

- (4π/6)ρ < 2 (

3 e 2

-e"

3 e l

)y

s w

inter

(12)

12

(Xl, l, 2)> W

W

1 2

3ε2

-1) (λ2,ω1,ω2)> sw where y i s the background d i s t r i b u t i o n function for the squarewell p o t e n t i a l defined as 6

Υ

± z

y (12) = ( 1 2 ) gw

8

β υ ( 1 2 )

(13)

β

The second term i s the repulsive contribution at the contact of hard cores. We note that for SW, the contact value, y ( d ) , i s not the same as that for hard spheres. In the following we s h a l l ana lyze the simulation data for spherical SW to determine the r e l a t i o n between y g and y^g. The t h i r d term i s due to the repulsive shoulder at λ^ω^α^)· This term actually plays the role of the temperaturedependent hard core dimension often used i n conventional perturba t i o n theories for soft p o t e n t i a l s . The fourth term i s the a t t r a c t i v e contribution. It can be treated as a mean f i e l d value i n case of continuous p o t e n t i a l s . To gain a better understanding, we analyze simulation data on spherical square wells next. g w

W

Contact Values of Correlation Functions for the SW Potential Henderson et a l . (33,34) have carried out computer simulations for the spherical one-step square well p o t e n t i a l with d i f f e r e n t widths, λ* = λ/d = 1.125 - 2.0, over wide ranges of state conditions. The p o t e n t i a l well i s a negative step, - ε, beginning at d and vanish ing at λ. Table 1 shows the contact values of the background corre l a t i o n functions for SW and HS, respectively. The hard spheres are taken to have the same c o l l i s i o n diameter, d, as that of the SW ( i . e . , at 3 ε = 0, the SW reduces to the HS p o t e n t i a l ) . We make the following observations :


11.

KANCHANAKPAN

233


ET A L .

(i) The data c l e a r l y show that the Weeks-Chandler-Andersen zeroth order approximation

i s inadequate. In f a c t , i t i s true only i n the l i m i t of high tem peratures (or low d e n s i t i e s ) . ( i i ) For a given density, the simple empirical formula y ( d ) = y (d)exp(C3e) s w

(15)

HS

i s found to apply with surprising accuracy for wide ranges of well widths, (λ* from 1.125 to 2.0, see Table 1). The index ζ has the values of -1.18 at p* = pd =0.8, -1.16 at p* =0.6 and -1.1 at p* =0.4 (for the well width λ* = 1.5). Since at low densities, both y and y approach 1, ζ must approach zero at zero density. Figure 4 shows the variations of the index, ζ, with density. The contact value of yg^W (Note that the y function i s continuous at d) i s also dependent on the well width, λ. We have displayed the ζ values for cases λ* = 1.375, 1.5, 1.625 i n Figure 4. The three curves are f a i r l y consistent i n their density dependence, ζ i s around -1.0 at medium densities ( i . e . , near p* = 0.5). ( i i i ) The values of ygy(X) at the a t t r a c t i v e wall can be represented by SW ' Υ

Equations of State for Nonspherical Molecules Based on the

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