Vapor-liquid equilibrium of hexadecapolar fluids from a perturbation-based equation of state

Vapor-liquid equilibrium of hexadecapolar fluids from a perturbation-basedequation of state

Francisco Gámez and Santiago LagoDepartamento Sistemas Físicos, Químicos y Naturales, Universidad Pablo de Olavide,Carretra de Utrera Km. 1, Seville 41013, Spain

Fernando del RíoDepartamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55 534,09340 México D.F., Mexico

Ana Laura Benavidesa�

Instituto de Física, Universidad de Guanajuato, Apartado Postal E-143, León, 37150 Guanajuato, Mexico

�Received 30 May 2006; accepted 1 August 2006; published online 11 September 2006�

In this work a numerically tractable expression for the interaction potential between two pointhexadecapoles with octahedral symmetry and a molecular-based equation of state derived byperturbation theory for hexadecapolar fluids are presented. The polar system is modeled bysquare-well particles with a point hexadecapole with octahedral symmetry at their centers. Thisequation of state is analytical in the state variables and in the potential parameters and allows us tostudy the effects of the hexadecapolar moment strength on the thermodynamic properties andliquid-vapor phase diagram. The equation presented here is applied to the thermodynamics of sulfurhexafluoride and gives very good predictions for the saturation pressures and the vapor-liquid phasediagram. © 2006 American Institute of Physics. �DOI: 10.1063/1.2339018�

I. INTRODUCTION

One of the most fruitful approaches in the statistical me-chanics of fluids has been to model thermodynamic proper-ties with the simplest intermolecular potential functions ableto reproduce the effects of interest. Given the importance ofmultipolar fluids—dipolar, quadrupolar, octupolar, andhexadecapolar—in process engineering and other appliedbranches of applied science, various interaction models havebeen developed for molecules with these electrostatic mo-ments in the last few decades. Multipolar square-well�MSW� fluids are among these models and thus have drawnthe attention of researchers. A MSW interaction constitutesone of the simplest models for systems manifesting disper-sion forces together with either a dipolar, quadrupolar, oroctupolar interaction and is therefore of interest by itself.Thermodynamic properties of MSW fluids can be obtaineddirectly from suitable equations of state and by computersimulation. A MSW equation of state �MSW-EOS� based onmechanical-statistical perturbation theory has been devel-oped and used to analyze on a consistent basis the effects ofthe range of the dispersion as well as of polar forces.1–3 Morerecently and taking a step further, this MSW-EOS has beenapplied successfully to model thermodynamic properties ofreal polar substances such as carbon dioxide, nitrogen, meth-ane, carbon tetrafluoride, water, and ammonia.4–6 Thus,MSW fluids can be taken as simple but reliable model sys-tems to represent effectively the properties of multipolar realsubstances. Briefly, the MSW-EOS is made up of separate

terms representing the effects of overlap and dispersionforces on one hand, modeled by a SW term, and of pointmultipolar interactions on the other. In spite of its success,the MSW equation has been directly tested against simula-tion results only in its SW part7,8 and for the dipolar or qua-drupolar SW.9

Following the reasoning for getting the MSW-EOS, inthis work we will consider hexadecapolar interactions whenthese are the first nonvanishing multipolar expansion terms�no dipolar, quadrupolar, or octupolar terms�. These interac-tions are due to the presence of permanent hexadecapole mo-ments in the molecule. The set of hexadecapolar substancesis not very large, but it includes very important substancescorresponding to general chemical formula XF6 �hexafluo-rides�. Among these substances, sulfur hexafluoride, SF6, isthe best-known one. This substance is mentioned in Article3.8 of Kyoto Protocol as one of the compounds responsiblefor the greenhouse effect whose emissions to the atmosphereshould be carefully controlled. Moreover, this substance isnonflammable and often used in industrial applications as agaseous insulator or as semiconductor in dry/plasma etchingdue to its peculiar dielectric properties. The hexafluoridegroup also includes WF6, used for metallization in the semi-conductor industry, substance with the highest molecularweight that remains liquid just below room temperature �nor-mal boiling point: 290 K�, and UF6 which is often used innuclear power plants as heat transmitter that sublimates at320 K at normal pressure. Thus, the transport properties ofthese substances are relatively well known, and specific in-termolecular properties giving account of these propertieswere proposed some time ago.10 This intermolecular poten-tial has recently been checked against molecular simula-

a�Author to whom correspondence should be addressed. Electronic mail:[email protected]

THE JOURNAL OF CHEMICAL PHYSICS 125, 104505 �2006�

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tions11 showing a good agreement for transport propertiesbut much poorer for equilibrium properties particularly forvapor-liquid equilibrium �VLE�. One of the main goals ofthis work is to show how a simple perturbation thermody-namic theory can provide reliable results for the VLE of SF6.These results are of quality enough to allow for thermody-namic predictions necessary to apply the Kyoto Protocol.Furthermore, the theory would allow us to predict thermody-namic properties of UF6 whose complete phase diagram isknown but for which overall experiment results are scarce.The situation is similar for other hexafluorides such as WF6

and MoF6 for which, as far as we are aware, experiment dataare very scarce or inexistent. To our knowledge there is noother theoretical equation of state for liquids currently avail-able that considers explicitly this type of interaction.

According to these goals, this paper is structured as fol-lows. In Sec. II we present the intermolecular potentialmodel used in this work and the hexadecapolar square-welltheoretical equation of state �Hex-SW-TEOS� derived fromthe statistical mechanic perturbation theory. Special attentionis paid to the derivation of the interaction potential betweentwo point hexadecapoles with octahedral symmetry since wedid not find a numerically tractable expression for it in theliterature. In Sec. III, a theoretical study of the thermody-namic properties and phase diagrams as a function of thehexadecapolar moment strength is presented. The applicationof the Hex-SW-TEOS to calculate the thermodynamics of

SF6 is also developed here using an estimate of its intermo-lecular parameters. Finally, in Sec. IV the main conclusionsand perspectives of this work are given.

II. THE HEXADECAPOLE SQUARE-WELL FLUID ANDITS EQUATION OF STATE

The Hex-SW-TEOS is based on the multipolar square-well perturbation theory,1,2 but now we consider a system ofparticles interacting via a square-well potential of variablerange plus a hexadecapole-hexadecapole interaction u��r�.This potential takes into account overlap, dispersion, andelectrostatic forces.

In a hexadecapolar square-well fluid any pair of particleswith their centers a distance r apart interact with the potential

u�r� = uSW�r� + u��r,�1,�2� , �1�

where �i= ��i ,�i ,�i� are the orientation Euler angles of theith hexadecapole, which is assumed with octahedral symme-try.

The SW potential is described by three parameters, di-ameter �, depth �, and range �, and is given by

uSW�r� = � , r �

− � , � � r ��

0, r � �� .� �2�

The hexadecapole-hexadecapole interactionu��r ,�1 ,�2� for a pair of charge distributions with identicaloctahedrical hexadecapole moments is given by

u��r,�� =6�2

r9 �7 + �70��C�448,4 − 40�16 16

�70�4,1 + �70�4,2 + �70�4,3 + �4,4� +C�448,3 − 30�

64�− 70�3,1 + �70�3,2

+ �70�3,3 + 2�3,4� +C�448,2 − 20�

64�20�2,1 − �70�2,2 − �70�2,3 + 7�2,4� + C�448,1 − 10�−

10

16�1,1

+�70

32�1,2 +

�70

32�1,3 +

14

64�1,4 + �

C�448,000�64

�0,1 +�70

2�0,2 +

�70

2�0,3 +

70

4�0,4� , �3�

where C�l1l2l ,m1−m10� are Clebsch-Gordan coefficients andthe �i,j are the polynomials in Table I, involving si=sin �i,ci=cos �i, and �=�2−�1. In expression �3� � denotes themagnitude of the scalar hexadecapolar moment �see Sec. 4.4of the book by Lucas12�. We stress the fact that the strengthof the hexadecapolar interaction is proportional to �2.

In order to use the high temperature perturbation theorywith this anisotropic potential, it is convenient to rewrite it interms of a reference potential u0 and a perturbation one uP as

u�r� = u0�r� + �uP�r,�1,�2� . �4�

The parameter � varies from 0 to 1, so that the potentialchanges from u0 to u0+up. We have selected the well-knownhard-sphere �HS� potential as reference. The perturbation po-tential is the sum of two contributions: the hexadecapolar

interaction and the attractive part of the square well denotedby SW�, which is

uP�r,�1,�2� = uSW��r� + u��r,�1,�2� . �5�

On the basis of the Barker and Henderson high tempera-ture perturbation expansion, the reduced Helmholtz free en-ergy a=A /NkT for a hexadecapolar square-well fluid of N

particles contained within a volume V and at reduced tem-perature T*=kT /� can be written as

a = a0 +a1

T* +a2

T*2 +a3

T*3 + ¯ , �6�

where a0=AHS /NkT and AHS is the free energy of the hard-sphere system. The first-order term in �6� is simply

104505-2 Gámez et al. J. Chem. Phys. 125, 104505 �2006�


a1 =�

2��8�2�2V� uP�12�g�12�d1d2, �7�

where from perturbation theory g�12� is the HS radial distri-bution function and

� di = �0

�0

� �0

2� �0

2�

drl sin �id�id�id�i. �8�

We can substitute �7� in �5� and, because the polar con-tribution of the potential vanishes after integration of theorientations of the hexadecapolar moments, the only surviv-ing term is

a1 =�*

2�� uSW�

�r12* �g�r12

* �dr12* , �9�

with r12* =r12 /�.

This term a1 is identical to the first-order contribution inthe high temperature expansion for the pure SW fluid andwill be detailed later on. Now we will show that the higher-order terms in �6� also simplify after the angular integration.

The second-order perturbation terms are13

a21 = −�

4�2�8�2�2V� �uP�12��2g�12�d1d2, �10�

a22 = −�2

2�2�8�2�3V� uP�12�uP�13�g�123�d1d2d3,

�11�

a23 = −�3

8�2�8�2�4V� uP�12�uP�34��g�1234�

− g�12�g�34��d1d2d3d4, �12�

a24 =S0

8�2 �

�� 2

�8�2�2V� up�12�g�12�d1d2�2

, �13�

where S0 is the zero-momentum hard-sphere structure factor,related to the isothermal compressibility �T through S0

=kT�N /V��T. In these equations, g�123� and g�1234� are thethree- and four-particle radial distributions, respectively. Thiscontribution has three types of terms: pure SW�, pure polar,and crossed SW�-� terms.

All the SW� terms survive; after angular integration thecrossed terms are null and due to the symmetry of the poten-tial only one polar term survives,

a2� = a21

� = −�

4�2�8�2�2V� �u��12��2g�12�d1d2, �14�

which gives a contribution of order �4. So, the second-orderterm is simply

a2 = a2SW� + a2

� . �15�

The third-order terms, which involve also the five- andsix-particle radial distributions, g�12345� and g�123456�, are

a31 =�

12�3�8�2�2V� �uP�12��3g�12�d1d2, �16�

a32 =�2

6�3�8�2�2V� uP�12�u�13�uP�23�g�123�d1d2d3,

�17�

a33 =�3

6�3�8�2�3V� �uP�12��2uP�13�g�123�d1d2d3,

�18�

a34 =�3

24�3�8�2�4V� �uP�12��2uP�34��g�1234�

− 3g�12�g�34��d1d2d3d4, �19�

a35 =�3

6�3�8�2�4V� uP�12�uP�13�u�14�

g�1234�d1d2d3d4, �20�

a36 =�4

13�3�8�2�5V� uP�12�uP�13�uP�45��g�12345�

− 3g�123�g�45��d1d2d3d4d5, �21�

a37 =�5

48�3�8�2�6V� uP�12�uP�34�uP�56��g�123456�

− 3g�1234�g�56�

+ 2g�12�g�34�g�56��d1d2d3d4d5d6. �22�

As for the second-order contribution, there are threetypes of terms. All the SW� terms survive; the crossed termsSW�-� are not null but will be commented on later, and theonly polar-type term that survives after angular integration is

a3� = a32

�

=�2

6�3�8�2�2V� u��12�u��13�u��23�g�123�d1d2d3.

�23�

This term is of order �6. So, �6� is rearranged as the sum offour terms,

a = a0 + �i=1

aiSW�

T*i+ �

i=2

ai��

T*i+ �

i=3

aiSW�+��

T*i

= a0 + aSW�+ a�� + a��+SW�

. �24�

It can be noticed that the sum of the first two terms givesprecisely the SW free energy. The exclusively polar contri-bution appears at second order. Since the crossed terms donot appear up to third order, in this theory they will be ne-glected as has been done in the MSW-EOS.1,2

Therefore, the Hex-SW-TEOS requires only the SW andhexadecapolar EOS. Several EOSs are available for the SWfluid, and in this work we have used the analytical SW-EOSproposed by Gil-Villegas et al.,14 obtained by perturbationtheory. This equation is very accurate in the range of 1.25

104505-3 Vapor-liquid equilibrium of hexadecapolar fluids J. Chem. Phys. 125, 104505 �2006�


��2.0, for a wide range of densities and temperaturesexcept near the critical point as has been shown recently.9

See Appendix B for its expression.The polar sum in �24� can be represented by a Padé

approximant,

a� =a2

�

T*21 −a3

�

T*a2�−1

. �25�

The use of a Padé expression was based on the originalworks of Larsen et al.

15 concerning dipolar and quadrupolarHS fluids and on the work for multipolar SW fluids �dipolar,quadrupolar, and octupolar�.1–6 The use of this Padé im-proved the results when compared with simulation data ifonly a second- or a third-order perturbation expansion wasused. Since no simulation data for SW hexadecapolar poten-tials are available for the moment, the use of this approxi-mant can be justified by observing that the behavior of thedifferent contributions �second and third orders and Padé� tothe Hex-SW excess free energy aexc=a−aID, as a function ofdensity at T*=1, is similar here to the dipolar and quadrupo-lar cases �see Fig. 1 in the work of Larsen et al.

15� as shownin Fig. 1. Taking into account that for ��1, a2

� =O��4� anda3

� =O��6�; the ratio in the denominator of �25� grows as �2.And hence a�=O��2�.

The last expression requires the knowledge of thesecond- and third-order polar perturbation terms. After angu-lar integration and changing the reference system to bipolarcoordinates dr1dr2=4�r12

2 Vdr12, we have

a2� = −

3551

126�17 + 2�70�

�4�

�2 � g�r12�

r1216 dr12. �26�

For the radial distribution function of the HS systemg�r12�, we have used the Verlet-Weiss algorithm as imple-mented by Henderson and Grundke �see Appendix D in Ref.

13�. After numerical integration, a2� can be expressed as a

sixth-order polynomial in �*=��3. In reduced units,

a2� = −

3551�17 + 2�70�126

�*4�*�i=0

6

�i�*i. �27�

Following the same reasoning, the expression for a3� can be

rewritten as

a3� =

91000�259 + 31�70�2187

�2�*6�*2�i=0

6

�l�*i �28�

where Kirkwood’s superposition approximation has beenused for the three-body distribution function g�123�. Thepolynomial coefficients in a2

� and a3� are given in Table II.

Finally, the expression for the Hex-SW-EOS is written as

TABLE I. Polynomial coefficients of the hexadecapole-hexadecapole interaction in Eq. �3�.

�4,1=2s14s2

4 cos 4�

�4,2=s14��1−c2�4 cos�4�−4�2�+ �1+c2�4 cos�4�+4�2��

�4,3=s24��1−c1�4 cos�4�+4�1�+ �1+c1�4 cos�4�−4�1��

�4,4= �1+c1�4�1−c2�4 cos�4�−4�1−4�2�+ �1−c1�4�1+c2�4 cos�4�+4�1+4�2�

�3,1=s13s2

3c1c2 cos 3�

�3,2=s13s2c1��1+c2�3 cos�3�+4�2�+ �1−c2�3 cos�3�−4�2��

�3,3=s1s23c2��1+c1�3 cos�3�−4�1�− �1−c1�3 cos�3�+4�1��

�3,4=s1s2��1+c1�3�1−c2�3 cos�3�−4�1−4�2�+ �1−c1�3�1+c2�3 cos�3�+4�1+4�2��

�2,1=s12s2

2�1−7c12��1−7c2

2�cos 2�

�2,2=s12s2

2�1−7c12��1−c2�2 cos�2�−4�2�+ �1+c2�2 cos�2�−4�2��

�2,3=s12s2

2�1−7c22��1+c1�2 cos�2�−4�1�+ �1−c1�2 cos�2�+4�1��

�2,4=s12s2

2��1+c1�2�1−c2�2 cos�2�−4�1−4�2�+ �1−c1�2�1+c2�2 cos�4�1+4�2��

�1,1=s1s2c1c2�3−7c12��3−7c2

2�cos �

�1,2=s1s23c1�3−7c1

2��1−c2�cos��−4�2�− �1+c2�cos��+4�2��1,3=s2s1

3c2�3−7c22��1−c1�cos��+4�2�− �1+c2�cos��−4�2��

�1,4=s13s2

3��1+c1��1−c2�cos��−4�1−4�2�+ �1−c1��1+c2�cos��+4�1+4�2��

�0,1= �3−30c12+35c1

4��3−30c22+35c2

4��0,2=s2

4�3−30c12+35c1

4�cos 4�2

�0,3=s14�3−30c2

2+35c24�cos 4�1

�0,4=s14s2

2 cos�4�1+4�2�

FIG. 1. Excess reduced Helmholtz free energy as a function of the reduceddensity �* at the reduced temperature T*=1 of a Hex-SW fluid. The approxi-mation up to second order is shown with a dashed line, the approximationup to third order is shown with dashed-dotted line, and the continuous linerepresents the Padé approximation.



TABLE II. Coefficients of the Wigner rotation matrices.

n

m

4 3 2 1 0 −1 −2 −3 −4

4 1

16�1+c�4

�2

8s�1+c�3

�7

8s2�1+c�2

�14

8s3�1+c�

�70

16s4

�14

8s3�1−c�

�7

8s2�1−c�2

�2

8s�1−c�3

1

16�1−c�4

3−

�2

8s�1+c�3 −

1

8�1+c�3�3−4c� −

�14

8s�1+c�2�1−2c� −

�7

8s2�1+c��1−4c�

�35

8s3c −

�7

8s2�1+c��1−4c�

�14

8s�1−c�2�1+2c�

1

8�1−c�3�3+4c�

�2

8s�1−c�3

2 �7

8s2�1+c�2

�14

8s�1+c�2�1−2c�

1

4�1+c�2�1−7c+7c2�

�2

8s�1+c��1+7c−14c2� −

�10

8s2�1−7c2� −

�2

8s�1−c��1−7c−14c2�

1

4�1−c�2�1+7c+7c2�

�14

8s�1−c�2�1+2c�

�7

8s2�1−c�2

1−

�14

8s3�1+c� −

�7

8s2�1+c��1−4c� −

�2

8s�1+c��1+7c−14c2�

1

8�1+c��3−6c−21c2+28c3� −

�5

4sc�3−7c2� −

1

8�1−c��3+6c−21c2−28c3� −

�2

8s�1−c��1−7c−14c2�

�7

8s2�1−c��1+4c�

�14

8s3�1−c�

0 �70

16s4 −

�35

8s3c −

�10

8s2�1−7c2�

�5

4sc�3−7c2�

1

8�3−30c2+35c4� −

�5

4sc�3−7c2� −

�10

8s2�1−7c2�

�35

8s3c

�70

16s4

−1−

�14

8s3�1−c�

�7

8s2�2−c��1+4c�

�2

8s�1−c��1−7c−14c2� −

1

8�1−c��3+6c−21c2−28c3�

�5

4sc�3−7c2�

1

8�1+c��3−6c−21c2+28c3�

�2

8s�1+c��1+7c−14c2� −

�7

8s2�1+c��1−4c�

�14

8s3�1+c�

−2 �7

8s2�1−c�2 −

�14

8s�1−c�2�1+2c�

1

4�1−c�2�1+7c+7c2�

�2

8s�1−c��1−7c−14c2� −

�10

8s2�1−7c2� −

�2

8s�1+c��1+7c−14c2�

1

4�1+c�2�1−7c+7c2� −

�14

8s�1+c�2�1−2c�

�7

8s2�1+c�2

−3−

�2

8s�1−c�3

1

8�1−c�3�3+4c� −

�14

8s�1−c�2�1+2c� −

�7

8s2�1+c��1−4c� −

�35

8s3c −

�7

8s2�1+c��1−4c�

�14

8s�1+c�2�1−2c� −

1

8�1+c�3�3−4c�

�2

8s�1+c�3

−4 1

16�1−c�4 −

�2

8s�1−c�3

�7

8s2�1−c�2 −

�14

8s3�1−c�

�70

16s4 −

�14

8s3�1+c�

�7

8s2�1+c�2 −

�2

8s�1+c�3

1

16�1+c�4

104505-5

Vapor-liq

uid

equilib

rium

of

hexadecapola

rfluid

sJ.

Chem

.P

hys.

125

,104505

�2006�


a�T*,�*,�,�*� = aSW�T*,�*,�� + a��T*,�*,�*� . �29�

III. RESULTS

Due to the present lack of computer simulations of par-ticles interacting with the Hex-SW potential, the accuracy ofEq. �30� for this form of interaction can only be inferredfrom previous work with MSW fluids. In view of this, wehave bypassed the direct comparison and opted to test ourtheory against experimental results of fluids in which thehexadecapolar interaction is known to predominate. Weshow below that our theory is really useful in the cases con-sidered.

A. Phase diagram of the Hex-SW-TEOS

From the Helmholtz free energy equation �29� one canobtain all thermodynamic properties using the usual phenom-enological relations. This task is straightforward because ourexpression for this energy is analytical. Two questions areaddressed here: the influence of the hexadecapolar moment�* on the thermodynamic behavior, particularly the vapor-liquid equilibrium, and the applicability of the Hex-SWmodel to a real system.

In Fig. 2, the vapor-liquid equilibrium is shown for dif-ferent values of the moment �*. The main effect of thehexadecapole-hexadecapole interaction is a very noticeablerise on the critical temperature. This increase is qualitativelysimilar to that observed for dipolar and quadrupolar fluids,but we shall see below that there are important quantitativedifferences. In contrast, the critical densities are not verysensitive to a variation in �*. The nonpolar SW case is alsoshown for reference. The change in critical temperatureswith hexadecapolar strength �Tc

*=Tc*��* ,�=1.5�−Tc

*��*

=0,�=1.5� is shown in Fig. 3. This change is reproduced bythe following polynomial in �*:

�Tc* = �

i=0di�

*�. �30�

The coefficients di are given in Table III.A similar but weaker effect was found in dipolar, qua-

drupolar, and octupolar SW fluids. The relative importanceof the hexadecapole can be seen by comparing the �Tc

* of

MSW fluids at equal reduced moments �* �dipolar�, Q* �qua-drupolar�, O* �octupolar�, and �*, included in the same fig-ure. We found previously9 that most multipole values for realmolecules lie below M �2 �M =�* ,Q* ,O*� which is therange where theory and simulation agree particularly well fordipoles or quadrupoles.

Another clear influence of the hexadecapole-hexadecapole interactions can be observed in Fig. 4. AClausius-Clapeyron plot for �=1.5 and different hexadeca-pole moments is shown. In all cases ln p�

* is almost linear in1/T*. In this plot we have considered the reduced pressurep*= p�3 /�. As we can deduce from

d ln p�*

d�1/T*�= −

�Hv

N�= − �H

v

* , �31�

an increment of the vaporization enthalpy with �* is observedas the slope increases with the hexadecapolar moment. Thismeans that the predominantly attractive polar interactionsmake the vaporization of the liquid more difficult, and thiseffect is higher as the polar moment is increased. In the caseof hexafluorides, the larger the bond distance X–F, the largerthe hexadecapole. Thus, we can expect that ��SF6��MoF6��WF6��UF6�, but this effect can be

FIG. 2. Coexistence densities for hexadecapolar SW fluids ��=1.5� for dif-ferent values of the reduced hexadecapolar moment strength �*.

FIG. 3. Theoretical predictions in the change of the critical temperature�Tc

*=Tc*��*�−Tc

*��*=0� as a function of the multipolar strengths. The con-tinuous curve corresponds to the polynomial fit, Eq. �30�, to the results ofthis work. The dash-dot, long-dashed, and dotted curves correspond to theoctupolar, quadrupolar, and dipolar cases, respectively �Refs. 1–6�.

FIG. 4. Clausius-Clapeyron plot for a Hex-SW fluid for different values ofthe reduced hexadecapole moment strength �*: 0, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8,1.0. Lines are shown in decreasing order from left to right for the values of�*.



dimmed on the reduced moment �* by the effect of molecularvolume on its definition.

B. Equation of state for SF6

The Hex-SW model should be applicable to realistic sys-tems with quasispherical overlap and dispersion forces plus apermanent and dominant hexadecapole moment. A typicaland important substance with these characteristics is SF6,which is a quasispherical molecule, with octahedrical shapewhose first nonvanishing permanent multipole is preciselythe hexadecapolar moment. To our knowledge, there are noprevious models including the effect of this electrostatic in-teraction for the liquid phase. The purpose of applying theHex-SW-TEOS to SF6 is to find out if this electrostatic con-tribution has a perceptible effect on the thermodynamics of areal system. The effect on the saturated pressure can be ob-served by making a comparison between experimental re-sults for SF6 �Ref. 16� and the theoretical prediction of theHex-SW-EOS. The best fit values for the intermolecular po-tential parameters were �=1.5, �*=0.35, � /k=214.113 K,and �=4.6850 Å. The corresponding value in the appropri-ate units for the hexadecapole moment is �=60.275 10−42 esu cm4 as the best fit of parameters. Since there is aconsiderably high uncertainty in the prediction of the highermultipolar moments of molecules, no matter whether theystem from quantum mechanical calculations or fromexperiments17 it is hard to say whether this theoretical pre-

diction of the hexadecapole moment is realistic or not. As anexample of this dispersion of values are the Hirono18 valueof 30 10−42 esu cm4 and those reported in the Table D.3 ofGray and Gubbins:16 5.4 10−42 and �7.2–15� 10−42 esu cm4. In Fig. 5, the VLE phase diagram is shown.In this figure we have included the best nonpolar SW and thepolar Hex-SW-EOS theoretical predictions, the experimentaldata of Funke et al.,17 and the molecular dynamics simula-tion for Pawley’s model.11 Far away from the critical regionone can see that the best prediction can be found by using theHex-SW-EOS. For this phase diagram the nonpolar SW-EOSis also very close from the Hex-SW-EOS. As expected froman analytical EOS, the critical point is not accurately esti-mated by any of the equations. Also one can conclude fromthis figure that the Hex-SW-EOS is a better model potentialin the prediction of VLE than the nonpolar SW model andalso than Pawley’s model. In Fig. 6, the saturation pressure isshown; as expected the Hex-SWEOS overestimates the criti-cal point, but below the critical region the agreement is verygood at least if one compares it with the nonpolar SW-EOSor the simulation data of Olivet et al.

11 for Pawley’s model.Even more, results of our analytical equation compare favor-ably in the agreement with the experiment with the results ofmolecular dynamics obtained with a model of six-centerLennard-Jones with no multipole.19

IV. CONCLUSIONS

The Hex-SW fluid was introduced as a simple model forhexadecapolar fluids. An analytical equation for the free en-ergy of the Hex-SW fluid has been presented. This free en-

TABLE III. Coefficients used in the expression for Hex-SW equation and in Eq. �30�.

i 0 1 2 3 4 5 6

�0i 0.773 853 −0.157 937 0.499 370 −0.115 220�1i −5.589 61 2.045 30�2i 1.216 473 −2.034 727 1.238 574 −0.425 229�1i 187.141 8 −335.684 5 185.852 8 −34.873 1�2i 1833.196 −5 284.990 5 488.597 −2453.347 402.468�3i −6185.698 16 431.21 −16 084.08 6886.400 −1091.004qi 4.948 76 0.097 245 −12.912 6 7.863 2�i 0.066 695 0.076 053 0.065 674 0.017 173 0.053 89 −0.031 887 0.021 993�i 0.002 9 0.007 2 0.024 −0.024 9 0.090 7 −0.074 4 0.042 2di 1.360 8 0.268 9 −4.852 7 28.579 −37.808 24.048 −5.996 7

FIG. 5. Vapor-liquid phase diagram for SF6. Circles represent the experi-mental data of Funke et al. �Ref. 17� diamonds the simulation data of Olivetet al. �Ref. 11� for the Pawley potential, the dashed curve the nonpolar SW,and the continuous curve the Hex-SW-EOS predictions. FIG. 6. Saturation pressure for SF6. Same symbol convention as in Fig. 5.



ergy equation allows the straightforward calculation of thethermodynamic properties for given values of the potentialparameters. To our knowledge this is the first theory to pro-duce an analytical free energy equation valid for dense fluids,which includes the hexadecapolar interactions. Previous ex-perience with the MSW fluids leads us to expect that thetrends predicted by the Hex-SW equation are reliable. It wasfound that rather small values of the hexadecapole momentproduce a large increment in the critical temperature. Thesechanges are shown to be much stronger than similar changesinduced by dipole, quadrupole, and octupole moments of thesame reduced strength. So, one can expect that hexadecapo-lar substances have higher boiling and critical temperaturesthan other substances with similar molecular mass withoutmultipoles. Indeed, the critical temperature for Xe �M=131.3 Da, Tc=290 K� is notably lower than that of SF6 �M =146 Da, Tc=318.7 K�.

The Hex-SW-EOS was used to model the saturationpressure of SF6, and it is found that it reproduces very wellthe whole curve except the critical point as it was expectedfrom an analytical EOS. This result indicates that the inclu-sion of the hexadecapole in the model gives better predic-tions than a simple nonpolar model.

ACKNOWLEDGMENTS

This work has been partially supported by Grant Nos.41678-F and 2003-C02-43586 of the Consejo Nacional deCiencia y Tecnología �México�, by PROMEP, SEP �México�and CTQ2004 07730 C02 01 of the Spanish Ministerio deEducación y Ciencia. One of us �F.G.� wishes to thank to theOTRI of the Universidad Pablo de Olavide for supporting atravel to Mexico and to IFUG for its hospitality. We areespecially indebted to Professor Dr. Lourdes Vega for help-ing discussions and addressing us to important references.

APPENDIX A: EVALUATION OF THEINTERMOLECULAR INTERACTION BETWEENOCTAHEDRAL HEXADECAPOLAR SHAPES

In order to develop the potential energy between twooctahedral hexadecapoles, consider the general form of thepotential energy between two poles of order l1 and l2 interms of the spherical harmonics expansion16

ul1l2�r,�� = Al1l2 �

n1=−l1

l1

�n2=−l2

l2 Ql1n1Ql2n2

rl1+l2+1

�m1=−l1

l1

C�l1l2l;m1 − m1m�

Dm1n1

l1 ��1�*D−m1n2

l2 ��2�*Y��*, �A1�

where C�l1l2l ;m1m2m� is a Clebsch-Gordan coefficient,Dmn

l ��* is a generalized spherical harmonic, and Y�� is aspherical harmonic, all in the convention of Rose. �i

= ��i ,�i ·�i� is the molecular orientation i and �= �� ,� . � theorientation along the molecular axis. Qlknk

=�tqirilkY lknk

��i� isa component of the multipolar tensor and the constant

Al1l2=

�− 1�l2

2l + 1 �4��3�2l + 1�!

�2l1 + 1�!�2l2 + 1�!�1/2

.

The term related to intermolecular orientation Y lm��can be simplified if the reference system is taken in orderthat the OZ axis is centered in a molecule cross along theother one, so �=0 and the � dependence will be null, so m

=0. Taking into account that20

Y l0�0,�� = 2l + 1

4�1/2

and the following property:20

Dmnl ��,�,�� = exp�im��dmn

l ��exp�in�� ,

Eq. �A1� can be rewritten using the coefficients of theWigner rotation matrices dmn

l given in Table II in the follow-ing way:20

ul1l2�r,��

= 2l + 1

4�1/2

Al1l2 �n1=−l1

l1

�n2=−l2

l2 Ql1n1Ql2n2

rl1+l2+1

�m1=−l1

l1

C�l1l2l;m1 − m10�

dm1n1

l1 ��1�d−m1n2

l2 ��2�

exp�i�− m1� + n1�1 + n2�2�� , �A2�

where �=�1−�2 has been defined.For the hexadecapolar order �the fourth order in multi-

polar expansion� we have

l1 = l2 = 4,

l = l1 + l2 = 8.

And only three terms of the multipolar expansion sur-vive, which are16

Emult�l1l2l;n1n2;r� = Al1l2

Ql1n1Ql2n2

rl1+l2+1 ,

Emult�448;00;r� = 6�2

r91430�

171/2

,

Emult�448;04;r� = Emult�448;40;r� = 6�2

r93575�

1191/2

.

Substituting these values in �A2� yields

ul1l2�r,�� =

6�2

r9 �715

2+ 5�143

7 f�� ,

where



f�� = �n1=0,4

�n2=0,4

�m1=0

4

�C�l1l2l,m1,m1,0�

�dm1n1

4 ��1�d−m2n2

4 ��2�cos�m1� − n1�1 − n2�2�

+ d−m1n1

4 ��1�dm2n2

4 ��2�cos�m1� + n1�1 + n2�2��

is the real part of the angular term. Notice that in the m1

=0 term, the last term does not exist.Expanding this, we get Eq. �3�.

APPENDIX B: SW CONTRIBUTION FOR AHEXADECAPOLAR FLUID

The reduced Helmholtz free energy of the Hex-SW fluidcan be expressed as

aM�T*,�*,�,�*� = aSW�T*,�*,�� + a��T*,�*,�,�*� , �B1�

with the square-well term aSW given as12

aSW = aID + aHS +a1

T* +a2

T*2 + aR. �B2�

The first term of Eq. �B2�, aID, corresponds to the idealgas contribution to the Helmholtz free energy.

The second term, aHS, is the Carnahan-Starling reducedHelmholtz free energy for a hard-sphere reference fluid,

aHS =��4 − 3��1 − ��2 , �B3�

where �= �� /6��* is the reduced packing fraction. The thirdterm of �A2� represents the first-order square-well perturba-tion term which is given by

a1 = − 4��3 − 1�exp��0 + �1� + �2�2 + �3�3� , �B4�

where the �i and � functions are expressed as follows:

�0 = − ln�1 − �� +42� − 39�2 + 9�3 − 2�4

6�1 − ��3 , �B5�

�1 =�4 + 6�2 − 12�

2�1 − ��3 , �B6�

�2 = −3�2

8�1 − ��2 , �B7�

�3 =− �4 + 3�2 + 3�

6�1 − ��3 , �B8�

and

� = �i=0

2

�i�i, �B9�

�0 = �i=0

3

�0i�i, �B10�

�1 =6

��2 − ��e�10+�11�, �B11�

�2 = �i=0

3

�2i�i, �B12�

The coefficients �ni in Eqs. �B9�–�B12� are given inTable II. The fourth term of �B2� is the second-order square-well perturbation term which is expressed as

a2 = − ��3 − 1�2KHS2 − n��

�1 − ��3 exp��1� + �2�2 + �3�3� , �B13�

with

KHS =�1 − ��4

1 + 8� − 2�2 �B14�

and

�� =− 5�5 − 5�4 + 85�3 − 75�2 − 111� − 111

2��2 + � + 1�. �B15�

In Eq. �B13�, �n are polynomials in � whose coefficients�ni are also given in Table II. The last term of �B2� is theresidual square-well term

aR = − ��3 − 1� 4�1 −3

2s��KHS

2 w��

+ q��KHS3 �t3 − �3�� , �B16�

with �= �1/T*� and the functions s��, q��, t, and w�� are

s�� =− 49�5 − 49�4 + 293�3 + 5�2 − 211� − 211

6��2 + � + 1�,

�B17�

q�� = �i=0

3

qi�i, �B18�

t = e� − 1, �B19�

w�� = t − � −1

2�2. �B20�

The coefficients qi of Eq. �B18� are given in Table III.

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Vapor-liquid equilibrium of hexadecapolar fluids from a perturbation-based equation of state

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