Volume 94, number 2 I~LIIUZW 1983

THE INFLUENCE OF THE ATTRACTIVE PART OF THE LENNARDJONES POTENTIAL ON THE VISCOSITY

Marvin BISHOP and J.P.J. MICHELS Yaps tier Waals Laboratory. Universiry of Amsterdam, Amsterdam, The Nerhw~ands (X_WI publication of the van der Weals Fund)

Received 18 August 1982; in final form 23 November 1982

hlolecular dynamics studies of Lennard-Jones repulsive and full potential systems have shown that the sttmctivc por- tion of the potential has L\ small effect on the viscosity in comparison to the cffcct of an nttnctiw square +l supcrim- posed on a hard-sphere potential. This indicates the importance of the prccisc shape of the attractive force on the viscosity.

l_ Introduction 2. Method

Recently, the relative importance of the attractive and repulsive portions of the molecular potential on thermodynamic and transport properties has become an active area of research_ Perturbation theories [ I] have been developed which enable one to calculate the properties of complex systenls by a Suitable per- turbation from a simple reference system_ In the

Weeks-Chandler-Anderson theory [2] the purely repulsive part of the potential is taken as the reference system. By simulating both model systems with the full (repulsions and attractions) potential and systems with only the repulsive potential, the roles of repul- sions and attractions in determining the viscosity can be assessed. in this note we report on our molecular dynamics calculations of the repulsive, RW, and full, FLJ? Lennard-Jones systems. A more detailed treat-

ment of the FLJ system will be published later. We ;l~d~ for the densities and temperatures examined, that tk difference in the viscosity between these two systems decreases with increasing density and is un- observable at the highest densities investigated. More- over, at intermediate densities the results of our cal- culations are, within the scatter of the data, in good agreement with experiments.

The molecular dynamics calculations were done for the RIJ systems via the procedure described by Bishop and Michels (31 and for the FU by the method described by Michels and Trappeniers [4]. All results are given in the following reduced units: lengths are measured in units of CT, velocity in units of c/m and time in units of o(m/e)lP-, where u is the crossing

point of the potential, III is the particle mass and E is the potential well depth. In these units the viscos- ity, 17, is given in terms of (em)l j2/uz and the number density, p. in terms of 1/u3.

Systems with N = 108 particles aud uumber densi- ties of 0.05-0.40 were selected for study. The tern-- perature, T, the pressure. P, and the stress tensor components. J@, were determined concurrrently with the trajectory. These are defined by

T = (3A’A’&-1 c c ui’O_) , i j

and

1 + (3’~Vo)-’ ~~ Ri~)-F,O_)] , (2)

(1)

0 009-X 14/S3/0000-0000/S 03.00 0 1983 North-Holland

volllllls 94. Illllllbcr 2 CHEMICAL PHYSICS LETTERS 14 January 1983

wiierc IV,, is the uumhcr of time steps the data is averaged WC~. q(j). u:(j), N,$j) and Fi(j) are the at11 c~wlpoIlc’Il1 (a = I l 1, 3) of the velocity, the square of tlw velocity, the position vector and the force vector

01‘ tlw itI1 particle at the jth time step, and aide) and *Y&j) iIre the otll and fitI1 component of the relative

I;)r~c vector and posiliou vector. respectively. of the it11 31~1 X-111 particles at time stepj.

‘l’llc viscosity has been obtrtined by a Green-Kubo 1-i 1 rcl;itionship:

m

Simpsons’ rule. J can be decomposed into a kinetic part. corresponding to the momentum flux due to translation of molecules. and a potential part, corre- sponding to the momentum flux due to intermolec-

ular forces. Thus, the total viscosity, q, contains

three terms: kinetic, qlrr potential, qp. and cross, r),.

Five independent runs were made for each ternper- nture and density and 25000 equilibrium steps were generated per run for the RLJ system; the FLJ calcu- lations employed 25 independent runs for a length of SO000 steps each.

vno = (P/~W? j- (./*~(t).P~(O)) dr . (41 (I

I Icrc. ( ) represents au enscml~le (or time) average. Avcragcs wrc m3dc over the circular permutation of a,& ix. I?, 22. and I 2. The integral \V;IS evalttatcd by

3. Results

The computer simulation results are presented in tables I and 2. We have checked that the two molec-

. . I ‘71 m “P vc d zc %cs ?c -_-. _.._..~ ._~ . -- . _.- --- _._..~_ ._ ._.. .._. ---_.____.__ . .._._ _~--___~--

1.29 0.2 19 0.152 0.030 0.037 1.0060 1.57 1.57 0.25 1

1.19 0.23$ 0.157 0.0’9 0.052 I .oooo 1.55 1.55 027.5 I .‘I6 0.24s 0.1 so 0.02s 0.040 0.9ss-l 1.54 1.53 0.313

2.95 0.34(1 0.25ll 0.032 0.058 0.9706 1.49 1.49 0.396 -t.O$ 0-15 I 0.348 0.043 0.060 0.9193 1.16 1.45 0.520 ‘).9X 0.716 O.SSll 0.045 0.101 0.9’67 1.40 1.41 0.780

--.--

7‘= 1.3,

T = 5.0

_.__. _.-_ .^_- _-___-____-. P 71t Ok q’p 4c d z, zcs 9r:

-_ ..-_--.-- --

0.05 0.1 b9 0.156 0.001 0.010 1.006 1.1’ 1.11 0.204 0.10 0.215 O.lS4 0.006 0.025 1.006 1.25 1.25 0.208 0.15 0.227 0.1 SO 0.013 0.034 1.006 1.40 1.40 0.219 0.7u 0.2 I9 0.152 11.030 0.037 1.006 1.57 1.57 0.25 1 0.25 0.342 0.148 0.045 0.049 1.006 1.77 1.76 0.184 KZO O.ZFlS 0.132 0.064 0.072 1.006 2.00 1.99 0.313 OAtI 0.365 0.120 0.156 0.090 1.005 2.56 2.56 0.446

0.05 O.-l 1 s 0.399 0.001 0.017 0.9493 1.10 1.10 0.442 0.10 (IA0 1 0.355 0.008 0.036 0.9492 1.20 1.20 0.466 &IS 0.466 0.391 0.018 0.056 0.9498 1.32 1.32 0.486

0.2u 0.15 1 0.3?8 0.043 0.060 0.9493 1.46 1.45 0.520 0.25 0.545 0.376 0.064 0.103 0.9493 1.61 1.60 0.570 0.30 0.546 0.332 0.095 0.119 0.949 1 1.77 1.77 0.631 O.-IO 0.690 0.326 0.1 s9 0.175 0.9498 2.17 2.19 0.791

-- -

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Volmnc 94. numbcx 2 CHEMICAL PHYSICS LETTERS 14 January 1963

ular dynamics methods used here, a “box scheme” for the RW and “neighbor tables” for the FLJ, are equivalent by making a run for the FW system using both techniques. At p = 0.2 and T = 2.00 the FLJ, via the “neighbor table” and with the long time averages mentioned previously, gave T)~ = 0.29 1. qk = 0.194, T+, = 0.037 and pc = 0.061 whereas the FW with the “bos scheme” and shorter time averages gave qt

= 0.284, pk = 0.195, VP = 0.035 and qc = 0.054 for T = 2.04. The agreement is excellent. The box scheme program has also been tested by calculating Hoover et al.‘s [6] N = 1 OS, p = 0.S442, T = 0.728 FLi state.

We find for T = 0.7 14 that qt = 2.82 versus their value of 2.97. This agreement indicates the statistival ac-

curacy of our calculation. Fig. 1 presents the RLJ t) as a function of l/T for

p = 0.2. The data are listed in table l_ At this density the kinetic contribution is the dominant one. As the temperature is decreased the kinetic, as well as the

cross term, decrease. Fig -2 presents 9, for T = 1.3 and 5.0 as a function of density. The point at p = 0.0 has been obtained from the Chapman-Enskog

theory [7]

(5)

The s1 function was calculated for the RLJ potential by integration over intermolecular distances, impact

parameters, and relative velocities [S]. The FLJ 52

o., y

: i

p’ 0.2

I . . TOTAL 0 HINETC Y CRCJSS A POTENTIAL

0.6 0

.

0.L

0 .

0 . . .

0.2 - 0 cl 0

x

* ‘L : ‘I &I-;

x

1 1 1

0 0.1 0.‘ 0.6 0.B

. .

I’$. 1. The viscosities of the KLJ system as a function of l/T

for p = 0.2.

1 I t I

0 0.1 0.2 0.3 04

I’&_ 7. The total viscosity of KLJ. NJ. esperiments. xxi Enskog 3s 3 function of density for 7‘ = 1.3 and 5.0.

were taken from the literature [S]. The error bars

shown in fig. 2 were determined by performing two independent series of runs. The points labeled esperi-

mental have been obtained from the equation fitted

to rare-gas data by Trappeniers et ai. [9]_ The Enskog results have been calculated from the equation for hard spheres of diameter d:

Be = (5d/24)p(7rz-)“‘(lj5 +o_s + 0.761-I,).

where

y=P/pT- 1 =Z- 1 (7)

and 2 is the virial. To evaluate these equations one

needs to know the hard sphere diameter. (I_ for the RW systems.

This may be obtained from perturbation theory. Verlet and Weis [lo] have shown that the temperature dependence of d, for the systems we are examining. is

given by

d = 0.3S37 + 1.0&3/T

0.4293 -I- l/T - w The data of Weeks et al. [2] indicate that the density dependence is not important for the temperatures of

our study. There is an excelient representation of the equation of state of hard spheres due to Carnahan and Starling [ 111

zCs =(l +y +y’ ->+/(l -v)j _ _I’ =&pd3 _ (9)

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Voli~me 94, number 2 CHEMICAL PHYSICS LEITERS 14 January 1983

Using eq. (S) to calculate (I and eq. (9) to find&,, one obtains excellent agreement with the molec& dynamics Z,. (See tables 1 and 2.) Thus, the RIJ systems behave like hard spheres with d set by eq. (8).

At densities above the critical one @, z 0.3) the Enskog results diverge significantly from the others. This theory takes into accowt excluded volume and collisional transfer effects but completely neglects the effects of correlated molecular motions. As the den- siry increases these effects become Important. Indeed, Alder et al. [ 171 have shown from molecular dynamics calculatiuns on hard spheres that there are substantial dcvialions from rhe Enskog predictions as the density is increllsed.

At intermediate densitites the viscosity of the RIJ system is comparable with the results obtained for hard spheres and. in fact. both systems can be brought into good agreement by choosing a11 appropriate value for 111~ cifecrivc diamcrer. Moreover. the addition of the atttractive force to the RIJ. i.e. applying the full Ll potential_ has no detectable effect at these densities.

This is in contrast to the effect of superimposing a11 attractive square well on hard spheres [ 131, which c&uses the vicosity fo change appreciably_ The con- clusion to be drawn is that the iniluencc of attrac- tions may be very dependent upon the precise form

of llic attractive potential. chosen for the modelsystem.

Acknowledgement

WC thank K. van der Linden for providing his C2

function program_ This investigation is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (F-O-M), supported by the Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.0).

References

[I] J-A. Bllrker and D. Henderson, Rev. hlod. Phys. 48 (1976) 587.

[a] J.D. Weeks. D. Chandler and H.C. Anderson, J. Chem. Phys. 54 (1971) 5237.

[3] bl. Bishop and J-P-J. Michels, Chem. Phys. Letters 88 (1982) 108.

14 1 J.P.J. Michels and N.J. Trappenicrs. Physica 90A (1978)

179. [S] E. Helfand, Phys. Rev. 119 (1960) 1. 161 W.G. Hoover, D-J. Evans, R-B. llickman, A.J.C. Ladd,

\4’_T_ Ashurst and B. hlorrln. Phys. Rev. A22 (1980) 1690.

17 1 S. Chapmsn and T.G. Cowling, The m;lthematical

throry of non-uniform gases (Czmibridge Univ. Press, London. 1951).

[g] J-0. llirschfclder, CF. Curtiss and R. Bird, hlolecuhr

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der Hers and J. van Oosten. Physic 31 (1965) 1661. [ 101 L. Verlct and J.J. Wcis Phys. Rev. A5 (1972) 939. [ 11 J N.1:. Cxn;lhnn tend K.E. Sturling, J. Chem. Phys. 5 1

(1969) 635. [ 121 B.J. Alder. D.M. Gas and TX. Wainwright. J. Chcm.

I’hys. 53 (1970) 3s13_ ( 13 1 J-P-J. Micbels and N.J. Trrappeniers, Cbrm. Phys.

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