Indian Journal of Chemistry Vol. 44A, July 2005, pp. 1333-1338

Determination of potential energy function and transport properties of sulphur hexafluoride

Elaheh K Goharshadia.*, Masoud Nahalia & Mahmood Baherololoomb

"Department of Chemistry, Sharif University of Technology, Teheran, Iran

bDepartment of Chemistry, Faculty of Science, Ferdowsi University, Mashhad 91779, Iran Email: gohari@ferdowsi.um.ac.ir

Received 25 April 2004; accepted 31 March 2005

The potential energy function of sulphur hexafluoride has been detennined via the inversion of reduced viscosity collision integrals at zero pressure and fitted to obtain an analytical potential form. A comparison of the potential with the previously determined potential has been included. The interaction potential energy from the inversion procedure reproduces, within experimental error, viscosity, self-diffusion coefficient, thermal conductivity and isotopic thermal diffusion factor of sulphur hexafluoride over a wide temperature range. These properties have also been fitted to very accurate equations.

The forces between molecules are of interest to scientists in a wide range of disciplines as these interactions control the progress of molecular collisions and determine the bulk properties of matter'.

Intermolecular pair potential can be obtained from both experimental observations and theoretical calculations. Two procedures have been used for extracting the intermolecular potential energy function from the experimental data, fitting and inversion methods. In fitting procedure, a model potential is adopted with a certain number of adjustable parameters which are varied until a good fit is obtained to a given set of experimental data. It is now accepted that this procedure usually yields little more than a crude estimate of the intermolecular potential energy function'. The aim of an inversion method is to obtain the potential by considering the experimental data as a functional instead of fitting the data to a constrained potential form having a few parameters I.2. Direct inversion techniques for the potential energy function, derived from gaseous transport coefficients, for example viscosity, have been developed that do not require any explicit assumption about the functional form of the potentiaI 3

-6

. There is a close cOlTespondence between a viscosity-temperature datum and a point on the intermolecular energy-separation curve' . Previous studies have represented the interaction between two SF6 molecules by a spherically symmetric part arising

from the attractive dispersion forces in the short-range while the repulsive contribution has no physical form7

.

The temperature dependence of the zero-density viscosity coefficient is to be recommended for the generation of the intermolecular potential surface of sulphur hexafluoride8

. The primary goal of this work is to deteJTnine the SF6 potential energy function by inversion of the reduced viscosity collision integrals at zero pressure and present the tables of necessary collision integrals at zero pressure which are needed to compute the transport coefficients of SF6 at different temperatures for any pressure.

Tables with reliable thermophysical data of heavy globular gases such as CF4, SiF4, and SF6 in a wide temperature range are requested in contemporary industry. They are user! in different microelectronic and chemical technologies (chemical vapour deposition, thin-film epitaxy, etc). The experimental determination of the thermophysical properties of these toxic and aggressive gases is complicated and expenSive, particularly, at high temperatures9

. The alternative is to calculate their thermophysical properties by means of reliable intermolecular interaction potentials. Thus, the second objective of the present work is to compute the transport properties of SF6 at different temperatures and pressures using the resulting potential. We have computed the viscosity, thermal conductivity, isotopic thermal diffusion factor, and self-diffusion

1334 INDIAN J CHEM, SEC A. JULY 2005

coefficients of SF6 at atmospheric pressure for different temperatures and compared to experiment The viscosity of sulphur hexafluoride has been calculated at zero pressure. The viscosity of SF6 at two constant low densities as a function of temperature has been also computed and compared with experiment.

Theory

Transport properties

Molecular transport processes deal with the transfer or movement of a given property by molecular movement through a system which can be a fluid (gas or liquid) or a solid.

The transport properties of a dilute gas can be expressed in terms of a set of collision integrals dl,s)(T) , characterized by the value of I and s over the intermolecular potential for each of the possible binary encounters in the system lO

. For example, the viscosity and self-diffusion coefficient of a pure gas depend on the collision integrals d 2

.2)(T) and dJ·J)(T) ,

respectively .

ry = j56

( ",:T)''' CY'Q,(;)O (TO) ... (1)

... (2)

where I1l is the mass of the molecule, It number density, k is Boltzmann constant, /" and JD represent the higher-order corrections to the first kinetic theory approximation of viscosity and diffusion coefficients, respectively , and T* is reduced temperature, namely T* = kTlE: (E: is well depth of potential).

The parameters of Q U'!)' and d 2.2

)* are the reduced collision integrals which are defined in general as:

Q (I.s) CT) Q (I")*CT *) = ? ... (3)

no-

where tui is the collision integral for the rigid sphere of diameter (J, and QI.s) is the collision integral.

The collision integrals are determined by the dynamics of binary collisions between molecules and hence depend upon the intermolecular potential. All the information about the intermolecular potential

from transport coefficients is therefore contained in the collision integrals.

The temperature-dependent colli~; ion integral Q (2.2)

is explicitly related to potential energy function, VCr) through the classical mechanical expressions 10:

... (5)

and

X(b,E)=n- 2bf11- b2

2

_ VCr) ylf2 !!~ I;' l r E ) r

... (6)

where E is the relative kinetic energy of a pair of colliding molecules, Q(2)( E) is a tr Esport cross section, b is the impact parameter, X is the scattering angle, and ro is the classical di stance of closest approach in a collision.

The initial density correction to gaseous transport properties The physical properties of moderately dense gases

may be expanded in powers of the density, p by the expressions of the form:

X(P,T) = XarT) [1 + BxCDp+ ... J ... (7)

where XorT) is the zero-density physical property. For example, the viscosity is expanded as:

... (8)

where Bll is second viscosity virial coefficient. The parameter 110 represents the viscosity coeffi cient at zero pressure.

Najafi and co-workers J J have calculated th e second viscosity virial coefficient of some gases based on the R . F ' d h 12 13 . • all1water- nen t eory . USll1g the most accurate potentials available for these gases. According to thi s correlation ", the viscosity of moderate dense gas (up to 2 mol dm·3) at specified temperature can be calculated via the equation :

.. . (9)

where .iVA is Avogadro 's number and B'l' is the reduced second viscosity virial coefficient. The

GOHARSHADI el al. : TRANSPORT PROPERTIES OF SULPHUR HEXAFLUORIDE 1335

parameter. B ,/* is the universal function of (T*t in the form of a sixth order polynomial:

(,

B,/' =L hi CT*r i ... (10) i=1

where hi is the coefficient of the universal correlation function. The values of hi for the gases are given in ref. II.

According to the theory of Rainwater-Friend ' 2. ' 3,

the second transport virial coefficients, BIl (jI can be viscosity (1]) , thermal conductivity, etc.) , consists of three contributions:

B - B (2) + B (3) + B (M -D) ,,- II P P .. . (11)

where B,,(2 ) is the contribution of two free monomers,

Bp(3) represents the effect of three monomer

contributions, and Bp(M-D) is the contribution from

monomer-dimer collisions.

The direct determination of intermolecular forces: Inversion method

The remarkable feature of the inversion procedure based on viscosity is that it identifies a point on the

experimental Q (2 .2)' versus temperature curve with a single point on the V (r) function. There exists a number of mathematical approximations of the various integrals linking the collision integral at a particular temperature to the potential I.

It is customary to use the extended principles of d· 16-18 b ' h . I correspon 1I1g states to 0 tam t e expenmenta

values of collision integrals. It has been proved that these principles are capable of correlating equilibrium and transport properties of the noble gases, some polyatomic gases, and multIcomponent mixtures over a very wide temperature range with an accuracy comparable to the best measurements 14- 16.

In the inversion method based on transport property, it has been assumed that the influence of the initial density dependence of transport properties is small and negligible in comparison with the uncerta1l1tles associated with the experimental method 17

·' 8

. The law of corresponding states l4-16 is

based on experimental values at relatively low densities (atmospheric pressure) since values at zero density are not accessible to direct measurements. In principle, such an assumption cannot be accepted

because the initial density dependence must surely affect the low-density values.

The reduced viscosity collision integral at zero pressure, 0 0 (2.2)* , can be obtained by inserting Eq . (I ) into Eq. (8) :

1 1 3 * Q (2 .2)* = Q (2. 2)* [1 + N A (J' B" p]

cs 0

•. . C 12)

where O C5(2.2)* is taken from corresponding states correlation .14 The values of 0 0 (2.2)* can be used to determine the interaction potential by an inversion

d d 'b d b f 4-6 17 18 proce ure as escn e e ore . . .

Results and Discussion The intermolecular potential energy function of SF6

has been calculated using the inversion of reduced viscosity collision integrals obtained at zero pressure and compared with Aziz's potential7 and presented in Fig. l. As Fig. I shows, Aziz potential is evidently unrealistic since the repulsive contribution has no physical form . Hence, it can be claimed that our potential obtained using the inversion of reduced viscosity collision integrals at zero pressure is superior representation of SF6-SF6 interaction potential than that provided by Aziz et 01.7 We have also fitted our potential to get an analytical potential form:

V' Cx) = ., .(13)

where x = ria, and V' = VIc: that a is where potential has zero value and c: is the well depth of the potential. The value of parameters of the potential have been given in Table 1.

Table 1 - Parameters for SF6-SF6 potenti al

Parameter Value

A* 23280.7

a -23281.7

f3 10.0052

a -33.8263

b 192.628

c -365.142

d 301.106

e -94.7594

1336 INDIAN J CHEM, SEC A, JULY 2005

10

w 0.1

> 0

· 1

0.8 0.9 0.9 1.0 1.0 1.1 1.1 1.2 1.2 1.3 1.3 1.4 1.4 1.5 1.5

r /a

Fig. I -- Reduced pair potentials of SF6 obtained by inversion of the reduced viscosity collision integrals at zero pressure (e) compared with Aziz potential (--l

100 ,--------------------------------,

80

20

500 1()()() 1500 2()()() 2500 JOOO 3500

T(K)

Fig. 2 -- Viscosity of SF6

pressure for experimental values (solid line).

against temperature at atmospheric values ( .) 14 and our calculated

••• t .... Q O+-----------~·~rT~------------~ lJi. - ......

·1 ••

·2

·3

·4

·5 +-----r---~-----r----~----~--~ 200 300 400 500 600 700 800

T(K)

Fig. 3 -- Plot of the percent deviation %Dev = (llcalc - llexp!llexp) xlOO of the experimental viscosity values(8) with our calculated values against temperature for SF6 at constant density 7 .25X 10 ·) moIL· 1

•

> • • • • • • Q)

0 .... c • e e ~

·1

., ·3 I

I I

·5 ~ 200 300 400 500 600 ,CO 800

T(K)

Fig. 4 -- Same as Fig. 3 at constant densi ty 28. 18 X 10'.1 1110 1 L . 1.

> ................. ... ~ ... ... CI> ... ... ...

0 0

#- , I ·1

·2 ,

~ ·3 ...

.. ·5

200 400 6(.0 000

T(K)

Fig. 5 -- Plot of the percent deviation %Dev := O"ca,cAcxiAexp) X 100 of the experimental thermal conductivity coefficients I with our calculated values against temperature for SF6 a.t atmospheric pressure.

~

'en N

E o:t 1

o o

500 1 000 1500 2000 2500 3000 3500

T(K)

Fig. 6 -- Self-diffusion coefficients of SF6 against temperature at atmospheric pressure for ' experimental values ( . ) 14 and our calculated values (solid line).

Figure 2 shows the viscosity of SF6 at atmospheric pressure at different temperatures. It is obvious that there exists a very good accordance between the experimental values 19 and our calculated values in the temperature range from 273 K to 3273 K at atmospheric pressure. The curve follow s Eq. (14):

11 (JI. Pa s) = -5.7756 + 0.611101 T O.623~73 ... (14)

GOHARSHADI et of.: TRANSPORT PROPERTIES OF SULPHUR HEXAFLUORIDE 1337

Table 2 - Dimensionless collision integral s d'·s)'=d'·')'lCi and the related ratios" for SF6 at zero pressure

LogT' 0(1,1)' 0 (1,2)' 0(2,2)' 0(1,3)* A* B' c·

-1.00

-0.90

-0.80

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

2.00

3.6232

3.3351

3.0714

2.8274

2.5968

2.3736

2.1549

1.9428

1.7426

1.5604

1.4006

1.2651

1.1529

1.061 3

0.9868

0.9257

0.8748

0.8313

0.7931

0.7586

0.7268

0.6972

0.6699

0.6451

0.6235

0.6054

0.5911

0.5804

0.5730

0.5683

0.5657

3.1869

2.9366

2.7053

2.4856

2.2696

2.0539

1.8422

1.6429

1.4644

1.3121

1.1 869

1.0864

1.0063

0.9420

0.8895

0.8454

0.8071

0.7727

0.7408

0.7108

0.6825

0.6560

0.6321

0.6114

0.5946

0.5819

0.5730

0.5674

0.5643

0.5631

0.5632

3.7904

3.4969

3.2320

2.9950

2.7800

2.5752

2.3688

2.1566

1.9435

1.7403

1.5572

1.4004

1.2712

1.1 670

1.0839

1.0176

0.9642

0.9200

0.8821

0.8482

0.8162

0.7846

0.7527

0.7203

0.6884

0.6585

0.6323

0.6106

0.5940

0.5821

0.5742

2.9031

2.6762

2.4621

2.2502

2.0351

1.8206

1.6174

1.4365

1.2843

1.1617

1.0651

0.9891

0.9287

0.8793

0.8376

0.8010

0.7679

0.7369

0.7074

0.6792

0.6526

0.6285

0.6077

0.5910

0.5787

0.5703

0.5653

0.5628

0.5621

0.5626

0.5637

1.0461

1.0485

1.0523

1.0593

1.0706

1.0849

1.0993

1.1100

1.1153 -

1.1153

1.1118

1.1069

1.1025

1.0996

1.0985

1.0994

1.1022

1.1067

1.1123

1.11 8 1

1.1229

1. 1253

1.1235

1.1165

1.1041

1.0878

1.0697

1.0521

1.0367

1.0243

1.0151

1.1929

1.1928

1.1976

1.21 2 1

1.2351

1.2584

1.2722

1.2706

j .2:::;8

1.2266

1.1955

1.1662

1.1420

1.1 239

1.1 I 18

1.1 048

1.101 8

1.1016

1.1029

1.1040

1.1 033

1.0987

1.0889

1.0741

1.0564

1.0382

1.0221

1.0093

1.0003

0.9948

0.9918

0.8796

0.8805

0.8808

0.8791

0.8740

0.8653

0.8549

0.8456

0.840'+

0.8409

0.8474

0.8587

0.8728

0.8876

0.90 1'+

0.9132

0.9226

0.9295

0.93'+1

0.9370

0.9390

0.9409

0.9435

0.9478

0.9538

0.961 3

0.9695

0.9776

0.9849

0.9910

0.9956

a A' = Q 12.2)*, Q ,I.I)* , S* = (5 QII .2)'_ 4 QIU)'], Q I/. /)', and C' = QU.2)', QU'/ )'

Figures 3 and 4 show percent deviation of the experimental viscositl of SF6 with our calculated values against temperature at constant density 7.25x 10-3 and 28.18 x 10-3 mol L-1

, respectively.

The viscosity of SF6 at zero pressure has been calculated from 20.77 to 6500 K and it obeys Eq. (15)

1]" (f.1 Pas) = -3.03586+ 0.431252T 0.663795 ... (15)

Figure 5 shows percent deviation of the experimental thermal conductivity coefficients l 9 of SF6 with our calculated values against temperature at atmospheric pressure. The experimental and our calculated values of thermal conductivity of SF6 from 223 K to 850 K obey the equation:

A{mW m -1 K - 1) =-16.8612+0.691626 T0650829

... ( 16)

1338 INDIAN j CHEM, SEC A, JULY 2005

0.6,-----------------,

• • • • 0.5

0.4

1:)0

0.3

0.2

0.1 +--~-~-~-~-~~-~-----l

o 500 1()()() 1500 2000 2500 3000 3500

T(K) Fig. 7 - Isotopic thermal diffusion coefficients of SF6 against temperature at atmospheric pressure for: experimental values ( . )14 and ollr calculated values (so lid line).

Figure 6 shows our calculated and experimental self-diffusion coefficients of SF6 at atmospheric pressure for different temperatures l4

. We have also derived an equation for calculation of the self­diffusion coefficients of SF6 at atmospheric pressure in the temperature range from 273 .15 to 3273.15 K:

D(l0-411l2 S- I) = -O.()()7) 1286 + 2.92702TI.671

. .. (17)

The isotopic thermal diffusion factor of SF6 at atmospheric pressure from 273 .15 K to 3273.15 K has been computed with (Fig. 7) and compared to the experimene 4

• The curve follows Eq. 18

q, = 0.508688 -

29.9365 529249 8.90724x 106

---+-----T T2 T>

.. . (18)

The most commonly needed collision integrals and their ratios for SF6 at zero pressure are given in Table 2.

To conclude, an iterative procedure has been used which enabled effective and spherical SF6-SF6

potential energy function to be obtained from reduced viscosity collision integrals at zero pressure. To examine the validity of the potential, we have calculated the different transport properties of SF6 and compared with the experiment. The results of these calculations have justified that the iterative method works well. The results of these calculations and also

our previous works on different compounds such as argon l7 , krypton 17, xenon 17

, helium20, methanel~ and

carbon tetrafluoride6 justifies the ability of inversion method to obtain the potential energy funct ion of different compounds and calculate different transport properties of these compounds using the resulting potentials. For the first time, we have calculated the reduced collision integrals of SF6 at zero pressure which are needed for calculation of the transport properties of this compound at any ternperature and pressure.

References Maitland G C, Rigby M, Smith E B & Wakeham W A. Intermolecular Forces: Their o rig ill and Determination. (Clarendon Press. Oxford), 1987.

2 Stone A, The Theory of Intenllolecular Fo rces . (Oxford University Press, Oxford) , 1997.

3 Boushehri A, Viehland L A & Mason E A. Chelll Ph.".\'. 28 ( 1978) 313.

4 Goharshadi E K, Mir-Afzal i Z & Tavangar Z, J Phvs Soc Japall, 67 ( 1998) 4296.

5 Goharshadi E K, lilt J Thermophys.1 9 ( 1998) 227.

6 Goharshadi E K, Abbaspour M &. Morsali A, Illd Ell/-{ Chelll Res, 42(2003) 2256.

7 Aziz R A. Slaman M J, Taylor W L & Hurly J J, J Chem Phys, 94 (1991) 1034.

8 Strehlow T & Vogel E, Physica, 161 ( 1989) 101.

9 Zarkova L & Hohm U, J Phys Cho ll Rl:{ Oaf({, 31 (2002) 183.

10 Chapman S & Cow li ng T G, Marh emarical Theory oj' NOllulliform Gases, (Cambridge Un ive rsJy Press, New York), 1970.

11 Najati B, Ghayeb Y, Rainwater J C. Alavi S & Sinder R F. Physica, 260 A ( 1998) 31 .

12 Rainwater J C, J Chelll Phy.\', 81 (1984) 495.

13 Friend D G & Rainwater] C, Chem Ph)S Lerr. 107 (1984) 590.

14 Boushehri A. Bzowski 1. Kestin J & Mason E A. J Ph,'s Chem Ref Data, 16 (1987) 445.

15 Bzowski j, Kestin j , Mason E A & Uribe F . .I Ph."s Chelll Ref Data, 19(1990) 1179.

16 Najafi B, Mason E A & Kestin J , Physico. Ill) A (1983) 387.

17 Goharshadi E K, Jami Alahmadi M & Najafi B. Call J Chelll , 81 (2003) 8 66.

18 Goharshadi E K & AbbaspourM. Fluid Phase Eqllilib, 2 12 (2003) 53.

19 www.nist.gov

20 Goharshadi E K & Abbaspour M. Indial! .I Chelll . 43A (2004) 1385.