Blip function calculation of the radial distribution functions for a binary liquid mixture

1 June 1975

Volume 33, number 2

Received i8 J~?ly 1972 Rev&e< manuscript received 7 Februxy

It

is

sho\vn

the usz OF the h@

that

hard sphere reference

197.5

iempcrnture

approxhaiion

temperntirre, moderate density, binns_q Iznnzd-Jones

blip hnction tieory together with a tv;o-componeni for the radi! distiibu!ioil iunctions iii a moderate

to

fluid leads to reEsonab!y xcurr?te xadictions

r&:;ure.

Chand!er

the foliowing reietionstips

(31 provides

tween the properties of 2 Recent!y, the blip function expansion of Anderson, Weeks and Chand!er [ 1,2] as extended by Sung 2nd Chandler [3] and Steele and Sandier [4]) has been used tc compute radial and orientational ccrrelaiions in nonspherical liquids. From this starting point i’l has been possible to tinterpret X-ray azd neutron diffractier? data for several diatomic liquids to obtain information on both their intermolecular structure and s-ameters [5] >2nd to construct molecular interaction pLl 2 pertUrb2iiOfl polar core

reference

in the structure

liquid

mixtures,

tension in

thtZ0Iy

and quadrupolar kid

to 2SSeSS

forces

[G] . Since

-he importance

and thermodynamic It seems reasonable

of this work

to mixtures.

is much

of the exfirst step

tiiis effort is ‘: verification

that the blip function expzxsion for liquid mktures of spherica! molecules is reasonably accurate. That is the purpose of this communication.

{u}

= {zIL,

ties of a hard sphere

sity and composition

zcz2, zl12}

reference

fluid

and tie

mo!e-

potenproper-

at the same

der,and with the set of diameters

IdI;

Here 9Z = -PA&/V, where fl= (kT)-‘, Vis the -Jo!ume and AA is the excess Helrnho!tz free energy with xspect to an ideal gas mixture at the same temperatwe, density (pj9 and composition (vi>; dBs has a simiiar meaning for the hard sphere refereric e fluid. In eq. (Z), gii is the pair cOrie!atiOn y;s+;p,_Y[,

ftinction,

{d})=expi+!3z;~(~)]

aLId $;S(r;p,+

{dj)?

where the superscript HS denotes the hard sphere re?erence fluid properties. Eqs. (i > a-td (2) are the result of iowest order blip function theory; their XCUGX~ is of the Ordei E” and g2, respective!y, EhCie

2. Tkory ‘The mixture

tial functions

whose

repu!sive

rig%! interest

properties to consider A necess2qJ

mixture

with tkte set of purely

of di-

in a nonspherical, there

binary

cules interact

be-

b!ip function

theory

of Sung and

Vc!wme 33, numbe:

CHEMICAL

2

if the set {d: is choszn

1 Junz 1975

FHYSICS LETTERS

so that

Q(7)

= z+(7)+

E8

t
=

where

in place

‘ihe

In these equationsf(r) = exp[-@c(r)] - I. Sung and CnancUer [3] found that the blip function theory &ove led to psor predictions for ~7! when the reference system was t&en to be a single component rigid sphere reference fluid (i.e., {dj = {d, d, dj), while See and Levesque [7] found that the Weeks, Chandler and Andersen theory [ 1,2], and the TercusYevick binary miuturo, rigid sphere reference fluid with {d} = {c?,, , @I !, + d7&j2, dzz} would reproduce the total therm0 dynamic properties of a Lennard3ones lldtlli? to within 1%. Since our interest in blip function theory is in tie ESB os’its pGr correletjoi? function predictions as a starting point for the study of ordering in liquid Fixtures and for perturbation theories involving molecular mixtures, this communication is restricted to the study of the pair correlation functions. In particu!~, we test the accuracy of the binary mixture pair correlation functions obtained from blip function the3v when 2 single component rigid sphere fluid, and :I binary rigid sphere mixture with {dl = Cdl 1) (ST,1 i d&2, cl22 I, are used 3s the reference Systems. in this test a binary Lennard-Jones 6-12 mixture Is used as the model system, i.e., U:$ (i-) = &ij[(Uii/r)l”

-

(Ojj/r)6]

~

(4)

wi t:l

El? = (q$,,p i_

and

012 = :tOll+o-ZZ).

as it is the ody moderate or hi& density nonrigid sphere ffuid for which the pair correIation functions are available from a campuicr simulation [S] Since the simple blip fimctioa expansion discussed above is

applicable only to repulsive potential functions, the ti$ tenqpeiaiure 2~protination [2] was made. This zsssumption, which-is applicable to hi@ density and! or hi@ temperature fluids, is to neglect the effect of the attractive forces on‘the Iiquid structure. In the piCSSiti

352

case the assumptioil is to use

0 ofeq. US::

r>f+., (4) in eqs. (2) and (3). of the high temperature approximation

(HTA) introduces a complication into the ardysis in that the computer simulation data of ref. [8] are for a binary i_eMardJones mixture at moderate density and temperature. Thus, some error is invo!ved in using this approximation here. To assess the magnitude of this error we have first studied the accuracy of blip function theory and HTA when applied to a single component Lennard-Jones fluid at a temperature and density comparable to the mixture studied here. Fig. 1 shows ‘he agreement between the simulation results of ref. [S] and our calculations using the Percus-Yevick and exacty(r) funcOons generated

by the computer program of refs. 29, IO]. We see, from the figure, that there is only moderate agreement between theory and simulation, though, surptiSin$y, slightly better agreement is obtair?ed by using the Pzrcus-Yevick rather than exacty functions. Other calculations show that the accuracy of the WA-blip function theory improves at hi&er temperatures and/ or densities [ I,21 . Therefcre, if the agreement between theory and simulation in the case of mixtures is similar to iha; in fig. 1, it would be reasonable to expect improved accuracjr in hi@ temperzture,and ES well. Of course, it is densitjl mixture c&ulaiions possible to improve our theoretical calculations by using the EXP or ORPA methods [I 11. However, such ca!culations are so complicated, especia!ly for mixtures, as to be ofIftt!e WB in iterative calculations, such as those ofief. [5]. Another problem that arises in our c2lculations is that there is not a unique solution for eq. (3) in the case of mixtures as there is for a pure fluid. in particu!!ar there are nlimerous sets of Darameters Cd1 which satisfy eq. (3) and Iesd t o sin&r - values of $, but which lead to poor radial distribution function predictions. This arises because ‘Lhese “solutions” result from a combination of oositive md negative v&es for the individud ljj Such-that the mole iractlon wei@ed sum ofeq. (3) vanishes. In principle, this diZIculty could

be resolved by replacing each integzl in eq. (3) with its absolute value or, equivzlent!y, by so!ving the &ree

Volums

33, iltilllbei

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2

2

I

Fig. 1. The radial distribution fucction fo: n single-component knnard-Jcnes fluici with G = 3.521 A c/k = 143.57 K, T = 115.8 K -and V= 33.31 cm3/mole. The points we from the molecule dynamics simuhtion of ref. [7], and the curves -Endie.SUlt functions, respcctive!y. from the present calculations using the e.KW~ and Percus-Yeticky

equations

.iij = 0,

I',j=1,2

(61

for tf?e three itrdependctz t parameters dl1, a’, 2 and d,,. Presently, computer &ogams are only available for hard sphere y functions for pure fluids and binary mixtures with d12 equal to the arithmetic mean of dII and d,,. With these restrictions cn the set id} it is not possible to satisfy eqs. (6). fierefore, in the spirit of blip function iheory we have chosen the single parameter d in the one-fluid theory and the iwo independent parameters tiI1 and dz2 in the &vo-fluid tieory, so as to minimize one of the two objective functions (74

tions for the pair correlation function of thz minor component in a nonequimolar mixture without significantly affecting the predictions for the major component.

3. Results Fig. 2 gives the ihzee pSr correlation gl1 (r), ,012(r) and g7,2 (i) for 2n eqtiiinok

Jones mixture in v&i~Ir el!/k = 1 I. 9.8 K, E~~IJz = 167.0K,o,,=3.405~,~~~=3.633A,T=1i6K, and VI= 33.31 cm3/mo1e. In this figure tie points wxe obtained from the molecular dyxmics experiments of ref. [8]. and the so!id end dashed lines from the cnlcu!ations

hrrs using eg.

(I)-, (2), (5) si?d (TJ$)

and Percus-Yevick y;)+ functions for the binary mixture rigid sphere referer,cc fluid. The remaining curves are those whic!~ result from the use of the on&zomponent h2rd sphere refereflce fluid. Figs. 3 and 4 give similar results fo’olmixtures of ‘Jle same two ienr?ard-Joizes molecu!ers with “1 = O.! and 0.9 respectively. In these kst ‘rwo 5q;es a~!,

where the bra&e+& denote zbsclute values. For ~41the calcuktions reported here the objective function I, wzs used since it was found to lead to better predx-

functions Lennard-

respeciiu;!y,

the exact

353

Volu:rile 33, nun-~r

2

CHEMCAI, PHYSICS LETTERS

? June 1975

Fig.

2. The radial distribution fuxtions for the cquimolar Lennar-Jones mixture. Here the points are from the molecular dynamics of ref. [7], the curves ---, and --result fro.rn rhc present calculatiocs using the exact one-component Iwd ~pherc reference fluid, and the exact and Percur-Yetick two-component hxd sphere reference fluids, respectively. simtiation

the single component reference fluid results have been omitted since, as a result of the composition independence of the reference fluid :Ind the o’bjective Cu~rction [es. (Tbj], the v-1 _ IX of the s’n$e diarneter d and thus the predicted correlation functions gV(f) are indepenc&t of species mo!e fraction 2nd identical wi’& those

From these figures it is clear that the single reference fluid bhp fwction theory fo:: misttires is inaccurate even thou& “11 and ~2, dif.Ter by iess than seven percent. The agreement between the git(p) obtained from the blip function theory using ‘he ‘oinary mixture reference fluid 2nd those obtained from molecu!ar dynamics cdculations is quits: gooti. Generally here, is bciter if the as f’or t!x pure fluid, the eseement Fercu-Yevick yms functions are used. !n particuiar hu-._“‘b, 2s for the p!rllre fluid in [ig. 1, the percus-Ye-tick g(r) ftinctions are in vefy good agreement with the

the value of P ai whichgii achieves its maximum value. The difference in the two sets of g functions in the r2gol-l 0.9 fjjmx < r-- < 1.15 A r-qas is appreciable only for the very dilute &nponen~‘in the nonequimolar , mixtures. However, since the minor component contributes little to either the Grerrnodynamic or X-ray and neutron scattering properties of these mixtures, &is difference in probably of litt!e practical importance T3e aseement bet-ween the mixture reference fluid ptir correlation functions obtained here and the simulation results is considerably better than that found using a one-fluid var! der \Vaa!s theory [8]. %e difference between the radial distribution functions c&&ted here and those, obtained from the mclecuizr dynamics simulation are largely the result of usiris the ‘hi& temperature approximation at the temperature and densities considered here. This is eGdent both from the resuits for the pure Lennerd-

mo!eculzr dynamics calculations ol dl componen.ts for rii < 0.9 rFy3X2nd rij > 1.15 p,F,, wheie fry” is

Jones fluid in fig. I 2nd from the generz!Ily poorer agreement obtained for the mixture in Gg. 4 compared

in fig. 3.

Volume 33, numbs

2

CEE!XCAL

PHYSICS LETTERS

0

j

30

3.5

G.0

4.5

5.0

r(A)

Fig 3. The radis distribution functions for the Lennar-Jones component reference fluid cakulntion hns ken omitted (see

3

mixture text).

with s, = 0.1. Szme key as in fig. 2 CTxpt

that the tingle-

Volunx

33, ilumber 2

CHEWCAL

PHYS!CS LETTERS

1 June I975

also for their assistance in tie use of &tis program. Re author also tl12111s Rofessors 5.S. Rowlinson and

to that for mixture in fig. 3 which has ;i 17% larger reduced density p* = p(:rI c: + x20z). There is also a smal: error irl the calculations due to the fact :hzt the r-vail2ble y functions were restricted to the case in which d12 is the arithmetic average of d,, and d,, . This error probably most affects the prediction of

Professor K. Gubbins for a detailed listing of the data which appeared in ref. [g] _

g!2ir)Based on our calculations it is clear that the errors encountered in ‘Lheuse of the high temperature 2p-

References

proximaiion and the blip function expansion for .mixtires are no worse than those for pure fluids at comparable temperatures and densities. Tnerefore, it seems reasonable to conclude that the accuracy of the theoq’ used here will be improved when it is applied to (a) purely repulsive potan::ial functions, (b) high temperature fluids, and/o: (c) hia density fluids. This suggests that the mixture blip innction theory can be used to study the structure, thermodynamic behavior and radiation scattering properties of mixtures ofboth spherical and nonspherical molecules, just as the pure comFonent theory has already been used to study the properties of uure atomic and molecular fluids.

This work was supported, in part, by the National Sciance Foundation (L.S.A.) and the Camille and trenry b)reyfus Foundation. The author thanks Drs. D. Henderson and E. Grundke for providing him with a computer progiiam written by them and Drs. bonard

and Barker, which was necessary for this work, and

350

R. Sargent and Mr. C. Birmin&am for their hospitaliity and assistance during i%s siay at Imperial College, and

[l]

H.C. Andersen, J.D. iYeeks and D. Chandler, AS (1971) 1597. [2] J.D. Weeks, D. Ciiandler and H.C. Andersen.

Fhys. Rec. J. Chem.

l?nys. 54 (1971) 5237; D. Chandkr and J.E. Weeks, Phys. Rev. Letters 25 (1970) 149. [31 S. Sung and D. Chandler, J. Chem. Fnys. 56 (1972) 4989. 141 W.A. Steele 2nd S.I. Sam&r, J. Chem. Phys. 6! (1974) 1315.

[51 A. Dnsgupta, S.I. Sandlei and W.A. Steele. J. Chem. Phys. 6i (!974) 1326; S.i. Sandier, A. Dnsgupta 2nd W.A. Steele, J. Chcm. Fhys., to bl published. !61 S.!. Sand!er. Mol. fiys. 26 (1973) 1297. [71 L.L. ke and D. Levesque, Mol. Wys. 26 (1973) 1351. [El KC. MO, K.E. Gubbins, G. Jacucci and IX McDonald, hfol. Fhys. 27 (2974) 1173. PI P.J. Leonud, D. Henderson and J.A. Barker, Mol. Phys. 21 (1971) 107. [lOI EA. Grundke and D. Henderson, Mol. Phys. 24 (1972) 269. [Ill XC. Andersen and D. Cnzndler, J. Chem. whys..57 (1972) 191s; H.C. Andersn, D. Chandler and J.D. Weeks, J. Chhem. KIYS. 57 (1972) 2626.