Local compositions and thermodynamics of polar/non-polar mixtures.

(1)

UvA-DARE is a service provided by the library of the University of Amsterdam (http

s

://dare.uva.nl)

UvA-DARE (Digital Academic Repository)

de Leeuw, S.W.; Williams, C.P.; Smit, B.

DOI

10.1016/0378-3812(89)80196-7

Publication date

1989

Published in

Fluid Phase Equilibria

Link to publication

Citation for published version (APA):

de Leeuw, S. W., Williams, C. P., & Smit, B. (1989). Local compositions and thermodynamics

of polar/non-polar mixtures. Fluid Phase Equilibria, 48, 99.

https://doi.org/10.1016/0378-3812(89)80196-7

General rights

It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)

and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open

content license (like Creative Commons).

Disclaimer/Complaints regulations

If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please

let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material

inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter

to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You

will be contacted as soon as possible.

(2)

Elsevier Science P u b l i s h e r s B.V., A m s t e r d a m - - P r i n t e d in T h e N e t h e r l a n d s

LOCAL COMPOSITIONS AND THERMODYNAMICS OF

POLAR/NON-POLAR MIXTURES

S W. de Leeuw,

Laboralorium voor Fyslsche Chemle, Nieuwe Achtergracht 127, 1018 WS Amsterdam, The Netherlands,

and

C P Wilhams and B Smit,

Konmklqke/Shell-Laboratonum, Amsterdam, (Shell Research B V.), P.O. Box 3003, 1003 AA Amsterdam, The Netherlands

( R e c e i v e d November 28, 1988)

A B S T R A C T

de Leeuw, S W , Williams, C P and Smit, B 1988 Local composdlons and thermodynamics of polar/non-polar mixtures

Results of computer simulations are presented for the thermodynamic properties of mixing and the local compositions for mixtures of Stockmayer and Lennard-Jones fluids with varying concentrations and dipole strengths The results show a strong asymmetry with respect to the concentration which is attributed to the influence of the orientation dependence of the dipole- dipole interaction and explained in terms of a "frustration" effect

1. I N T R O D U C T I O N

The concept of local composihons, which describes the deviahons of concentrations within a region around a specific particle from the overall bulk concentrations, and the importance of such deviations for liquid models, can be traced back as far as the work of Rushbrooke (1938) and Guggenheim (1944) Recently there has been renewed interest in these ideas pri- marily motivated by efforts to extend the apphcabihty of equations of state to include the liquid-phase descriphon of non-ideal mixtures through the use of fundamental mixing rules Methods have been proposed (Mollerup, 1981, Whiting and Prausndz, 1982, Li et al, 1986) by which the local composition mixing rules may be combined with an equation of state

By using computer simulations it has become possible to determine the local composition of well defined model systems and to use these data for further improvement of the local composition models Previously, computer simulations have been concerned mainly with the local,compos~hon of particles with isotroplc potentials The results of these earlier studies suggested that the local compositions and phase behaviour were h~ghly symmetric with respect to the bulk concentrahon This is m direct contrast wrth experimental observations for polar/non-polar m~xtures

Mixtures of Lennard-Jones and Stockrnayer particles prowde a convenient model system which can be used to study the mpcroscoplc behavlour of polar/non-polar mixtures with computer simulation techniques Furthermore the simulations also allow the calculation of the thermodynamic properties of mlxmg for these theorehcal fluid mixtures

2. C O M P U T A T I O N A L DETAILS

The energy of interaction between two Stockmayer molecules with d~pole moments ,~1 and 1~2 separated by a distance r12 is given by

(3)

~ 2 ~s(l~l,/~2, r12) tL ₃D(1,2) + ~LJ(~12) r12 (la) where D(1,2) = 3(~ 1 ~ • q 2 ) 0 , 2 A • q 2 ) ^ - ~1" ~2 (lb)

where^ denotes the untt vector D(1,2) is the angular dependence of the dipolar interaction and ~LJ ts the usual Lennard-Jones interaction

~Lu(r) = 4 r . { [ ~ - ]12 _ [ ~ ] 6 } ₍₂₎

In the work reported here we have studied flutd mixtures of Stockmayer and Lennard- Jones molecules in which the size parameter, ~ , and the energy parameter, s, of the Stockmayer molecules are identtcal to those of the Lennard-Jones molecules Devtatrons from ideal mixing m these systems are therefore due enhrely to the polardy of the Stockmayer molecules and allows us to study the effect of polardy and composttion on the mtcroscoplc structure and the thermodynamtc properttes of these mixtures

Isothermal-tsochorlc molecular dynamics (MD) caleulahons have been performed for mixtures with various mole trachons, xs, of Slockmayer molecules The temperature was held constant wtth a Nose thermostat (Nose, 1984) using a Nose mass of 100 m reduced umts

The stmulahons were performed at a reduced temperature of T* = kT/s = 1 15 and a reduced density of p* = pr~ 3 - 0822 Both the pure Lennard-Jones and the pure Stockmayer fluids have been studied extensively at thts state point previously (Adams and Adams, 1981, Neumann el al 1984, Petersen et a l , 1988) The reduced dtpole moment,

1 I* = t~/\,'r~.~ T was varied between 0 0 and 2 0, covering a range of physically reahshc values The long-range dtpolar mteractions were handled with standard Ewald summahon techniques usmg "tmfoil' boundary conddtons (de Leeuw et a l , 1986) A total of 309 reciprocal latttce vectors were used for the evaluation of the Fourier part of the Ewald summation The value of ¢. the parameter whtch governs the convergence of the two series in the Ewald sum, was set to 6 58

The simulahons were carried out f o r a l o t a l o f N = 1 0 8 parhcles A few runs were performed with a larger number of particles to study the N-dependence of the results The total energy was found to be almost independent of the number of parttcles and for the pressure a shght dependence (a few percent) was observed

In the simulattons, the system was equthbrated for at least 5000 hme-steps and followed b y a production run of at least 15,000 time-steps A t i m e - s t e p o f 0 0 0 2 5 T w a s used, where : is the reduced unit of hme defined as f = ~/'mr,2/r

(4)

3. RESULTS

3.1 T h e r m o d y n a m i c P r o p e r t i e s

In table I we have collected the values of the potential energy obtained from our simulations for various compositions and dipolar strengths The contributions due to the Lennard- Jones interactions, including the long-tail correctton, have also been included These contributions are seen to increase wdh dipolar strength and mole fraction xs of the Stockmayer molecules This is of course expected due to the tendency of the dipolar interactions to decrease the distance of closest approach which in turn increases the contribution of the repulswe energy in the Lennard-Jones potentqal

Fraction of Stockmayer particles, xs

i ~.2 0 000 0 167 0 333 0 500 0.667 0 833 1 000 0 50 5 526 5 518 5 589 5 599 5 658 5 749 5 825 5 503 5 548 5 503 5 498 5 515 5 510 1 O0 5 526 5 585 5 679 5 831 6 026 6 219 6 451 5 551 5 517 5 523 5 526 5 481 5 469 1.50 5 526 5 630 6 845 6 "125 6 453 6 822 7 223 5 526 5 514 5 501 5 478 5 456 5 426 2 25 5 526 5 756 6 188 6 683 7 259 7 866 8 517 5 528 5 484 5 463 5 413 5 355 5 331 3 O0 5 526 5 986 6 606 7 331 8 "136 9 049 9 920 5 490 5 432 5 383 5 326 5 253 5 194 4.00 5 526 6 274 7 302 8.364 9.510 10 724 "12 039 5 450 5 353 5 278 5 "181 5 070 4 975

TABLE I Potential energy - U/(Ns) for mixtures of Lennard-Jones and Stockmayer fluids for various composdlons x s and reduced dipolar strengths i L*2 (The lower number is the contribubon of L-J interactions )

The excess energy of mixing Ue, can be r e a d d y o b t a m e d from the data qntable I In figure 1 the variation of Ue. wdh composition is shown for three different values of the reduced dipolar strength /~.2 As expected U,, increases with increasing reduced dipolar strength, however the most interesting feature of the figure is the asymmetric behavlour of U,~ with respect to the composition The energy of mixing has a maximum at mole fractions x s slighly less than 0.5, r e at compositions rich tn the non-polar component This corresponds with what has been observed experimentally in a number of polar/non-polar mixtures such as CH3OH/C6H 6 and CzHsOH/C6H14 (King, 1969, pp 27-35)

The asymmetric behaviour has also been observed m the variation of the excess free energy of mixing fe~, as shown elsewhere (de Leeuw et a l , 1988) For values of #*z > 4 it was shown that this leads to a phase separation into an almost pure Lennard-Jones fluid and a Stockmayer-nch phase Th~s a g r e e s w t t h the results o f M o r n s s and lsbister (1986) w h o s o l v e d the site-site Ornstem-Zermcke equation, using the mean spherical approxtmatlon closure, for mixtures of polar and non-polar hard-dumbbells

(5)

It is instruchve to compare our results with those of Wong and Johnston (1984) who performed Monte Carlo simulations of strongly non-ideal hquld m~xtures The dtpolar interactions were included m the form of an angle-averaged express}on and the behavtour of the excess properties with respect to the composition was found to be htghly symmetric Thin comparison demonstrates that it Is indeed the onentahon dependence of the dtpolar interachons that Is responslble for the asymmetry in the composdlon dependence ofUo~and f~, A s t m d a r a s y m - metric behavtour IS observed for the local composthon as a funchon of the mole frachon of Stockmayer molecules whtch ~s d~scussed Hi the following sechon

0 -

z.

/ . o z = 1 O0

. T " ' " T

/Z% 2.25

• J- ~.3- /~ = 4 00

,5

o.oo

o:25

o:5o

o:75

1.oo

X s

Fig 1 Excess energy of mixing U~dNsversus concentration x s

In table II values for the compresslbddy factor ( Z = P V / N k T ) are given for the vartous composlhons and reduced dtpole moments Again we observe devrahons from tdeahtywhtch behave asymmetrBcally wdh respect to the compos~hon

3.2 L o c a l C o m p o s i t i o n

The local composdton Fn mtxtures of square-well flutds has been thoroughly examined m a series of papers by Lee and Sandier and their co-workers (1984, 1986a, 1986b, 1987) The use of square-well fluids is advantageous In that the cut-off distance in the mtegrahons for the calculahon of the local mole frachon is unambrguously defined Furthermore the properties of such fluid mixtures are clmmed to resemble mixtures of Lennard-Jones molecules

The local composFtton of equlmolar, equal-stzed Lennard-Jones mtxlures have themselves been studied prevlouly (Nakamsht and Toukubo, 1979, Nakantsht and Tanaka, 1983, Nakantsht el al 1983) using molecular dynamtcs and by Wong and Johnston (1984) usmg Monte Carlo stmulahons In both studies a v a r t e t y o f c o r n b m m g rules were used for the energy parameter

(6)

Fraction of Stockmayer particles, x s #*2 OOOO O167 0333 0500 0667 0833 1000 0 5 0 2 2 2 4 21 2 3 2 2 21 21 100 2 2 21 2 2 2 0 18 19 1 7 150 2 2 2 2 2 0 18 1 7 1 4 12 225 2 2 21 19 15 13 1 0 0 5 3 0 0 2 2 2 0 1 7 13 0 9 0 4 -01 4 0 0 2 2 19 15 0 9 0 3 - 0 3 - 1 0

TABLE II Compresstbdity factor PV/NkT for mixtures of Lennard-Jones and Stockmayer fluids for various compositions x s and reduced dipolar strengths iz 'z

%, representattve of assoc~atton and solvahon forces, in adddeon to the usual Lorentz- Berthelot (L-B) geometrtc mean typical of dispersive-only forces

The local composihon of Lennard-Jones mtxtures wdh differing component stzes, obeying the L-B mixing rules, has been studaed using molecular dynamics (Glerycz and Nakamshl, 1984, Gierycz et a l , 1984) Hoheisel and Kohler(1984) investtgated the local composition in L-B mtxtures of Lennard-Jones fluids with both equal and differently sized particles Lee et al (1986) studied the effects of size and energy differences m both Lennard-Jones and Kthara mixtures These studies concluded that packing effects, and not the attracttve forces, are the dominant cause of non-randomness in liquid mixtures obeying the L-B combining rules

The number of i-particles around a central j-particle, within a given distance L, is rigor- ously defined m terms of the distribution funchon g,j(r) (Lee et al , "1983) and ~s gtven by

to-

N~j(L) = x,p . dr gp/r) (3a)

and simtlarly for j-particles around a central j-parhcle

r0-

Njj(L) = xjp. drgjj(r) (3b)

For a binary mixture the local mole frachons may then be defined according to the following

N~](L) (4)

X~/L) - (N,j(L) + Njj(L))

w~th the condition that

(7)

Obviously for continuous interaction potentials the actual value of X,~(L) depends on the cboJce of L, the cut-off point for the mtegrahons Thrs is typically chosen to correspond wdh the end of the first-coordmahon shell and, as pointed out by Hohelsel and Kohler (1984), should be independent of the type of contact so as to fulfill the reqmrement that, as L becomes large

X21-~x ~ (6a)

Xt2-~x~ (6b)

fe this paper we are more concerned with the trends m the local mole fractions rather than

the specific values, however for comparison we have chosen Liar = 1 35 in keeping with pre-

vious work (Hohelset and Kohler, 1984)

The local mole fractions X,~, defined by equations (3-5), have been determined for the mmu-

lations d}scussed earher The results for Xss and XL~ ' for two of the Stockmayer/Lennard-Jones

fluid mtxtures studied are shown tn figure 2(a,b), for the same three values of the reduced

dipole moment shown m f~gure 1 The vatues of Xst.(L/cr = I 35) and Xts (L/~ = 1 35) [or all

mixtures and dipole moments are given in table Ill

O66321 05041 I 03205~ 04980 0 64181 Q 4994 1 031511 04917 062741 n4917 03134 01626

J Fraction of Stockmayer Parhcles, xs

,,*' I 0167 / 0333 I 0500 0 6 6 7 083-3- - - - ~ I 0 I673 i 033151""05046 06773 08312

t 08419 / 066321

03393 01661

lOO I

o ooo

,50 I 0t614t

o6611

o8221

I 08o561

03315 01652

2 25 0 16291 0.3055 04716 0 6269 08294 ! 08103[ 06113 t 04707 03132 01670 I 300 J 01392J 02910 04692 06550 07728

!

068931 05860 04717 03319 01550 400 I 014241 02875t 04476 06171 08038 07155 05798' 04541

TABLE III Loca| mole fractions Xst.(upper ) and XLs(lower ) at L/o- = 1 35 for each composition x s and

reduced dipole moment l~ *z

The results are also presented In the form used by Lee and Sandier (Lee and Sandier, 1987), who suggested using the raho

(8)

1.0 L O C A L M O L E F R A C T I O N S X 0.9 0.8 O,7 0.6 0.5 0,4 0.3 0.2 0.1 0,0 0.00

Xss

L E G E N D FZ "8= 1.00 t J, "~= 2.25 ,U. "~= 4.00

XLL

Temperature = 1.15 Density = 0.822 Composition N s = 9 0 N], --'--18 I I I I ', ... I I I I, I I 0,25 0.50 0,75 1.00 1.25 1.50 1.75 2,00 2,25 2.50 2,75 r/ass 1.0 0,9 0.8 0.7 0.6 0.5 X 0.4 0.3 0.2 O.I 0.0

XLL

L E G E N D

~.a=

1.00 \

/ ~ = 2.25 ,, /J.*~= 4.00 \~

X.

---_. ~ - - - _ Temperature = 1.15 Density = 0 . 8 2 2 position N e = 1 8 N L ---90 I I ] I I I i ] ] [ 0.00 0 25 0.50 0.75 LO0 1.25 1.50 1.75 2.00 2.25 2.50 2.75

r/~ss

FIG 2 a,b Locat m o l e fractions

Xss

and XLL for two compositions x s (0 167 and 0 833) and t h r e e reduced dipole m o m e n t s #.2 (t.00, 2 25. 4 00)

(9)

NI/x/ (7)

Nj/Xl

wh~ch, if equal to unity would indlcate a completely randomsoluhon These results are shown in figure 3(a,b) and gwen m table IV (again at L/,7 = 1 35 ) for all mtxtures considered

The most notable feature o f these results is the asymmetry wtth respect to the compost- tlon, as seen prewously, with the greatest deviation from a random solutron being found for mixtures wtth low concentrations of Stockmayer particles As discussed above, this is m agreement with our earher work (de Leeuw et al , "1988)

Fraction of Stockmayer Particles, x s #*~ 0 167 0 333 0 500 0 667 0 833 050 `1004f 09916 `10185 10496 09847 1 0653 0 984~ 1 0164 1 0270 0 9961 1 O0 0 9984 0 9433 0 9920 0 9732 0 9662 1 0230 0 8958 0 9975 0 9900 0 99`11 1 50 0 9623 0 9199 0 9674 0 9754 0 9243 08286 08418 09674 09918 09895 2 25 0 9727 0 8799 0 892~ 0 8401 0 9724 08543 07863 08893 09119 10023 300 0 8085 0 8208 0 8839 0 9491 0 6801 O4437 07077 08927 09934 09171 4 O0 0 8300 0 8071 0 8104 0 8057 0 8195 0 5030 0 6898 0 8317 0 913( 0 9707

TABLE IV Local composition ratios NsLXL/NLLXs(Upper ) and NLsXs/Nssx~(Iower ) at L/cr = 1,35 for each composition x s and reduced dipole moment t/.2

Wong and Johnston (1984) also reported values for lhe local mole fractions for non-

p o r a r / ' p o l a r - h k e " mixtures Similar to their observations concerning the thermodynamic properties of mixing discussed prevlouly, they found that the local composdlons of the systems were symmetric with respect to the bulk concentration Thrs again reinforces our sug- geshon that the onentahon-dependence of the dipole-dipole mterachon plays an important role in determining the behawour of this type of system

4. DISCUSSION

It is interesting to speculate about the origin of the asymmetric behaviour shown rn the results given above If one considers two Stockmayer partlclesw~thm close proximity of each other, in a background of Lennard-Jones particles, then the two polar particles are free to orientate themselves in the most energetically favourable onentatron For dipolar particles this occurs when the two are ahgned"nose-to4all" However, l n c r e a s l n g t h e o v e r a l l c o n c e n t r a t r o n of Stockmayer parhcles increases the probabrhty of a third polar parbcle also being m the vi- cinity One can then rmaglne that the particles are frustrated m t h e l r a t t e m p t s t o a c h l e v e t h t s optimal orientation for each patr-mterachon Therefore some compromise must be achfeved

(10)

1.2 L O C A L C O M P O S I T I O N R A T I O S

x"

<

1,0 0.8 0.6 0,4 0.2 0.0 1.2

(~.~DO~

/:"

/

J ... I .... 1 ~ I 0.50 0.75 L O O 1,25 1,50 1.75

r/~,,

Compo,|tton N s = 9 0 N L = [ 8 N s L x u / N ~ x . NL, x , / N , , x z

p."=

1.o0 /~"= 1.oo M, "8= 4.00 ,U, "~= 4.00 ... I ~ I ... I,, 2.00 2,25 2.50 2.75 3.00

LO

( ~ O ~

0,8 O.B 0.4 0.2 0.0 '- ~ ... 0.50 0.75

:

/

/ -

/

f

/

J Compo,itlon. ~s ----18 N~ = 9 0 N , ~ x j N , ~ x s N , a x j N s s x L

p,"~= t.o0 ,u,'~= 1.oo p,-,8= 2.25 /~'~= 2.25 ,u, "8= 4.00 ~'~= 4.00

I I I ... I ... I L ... I l

I O0 1.25 t.50 1.75 2.00 2 25 2 50 2 75 3 O0

r/~s

s

FIG 3 a,b Local composlhan rahos Ns~xL/NcLxs and N~s/Nssx L for two compositions x s (0 167 and

(11)

which will then result ;n a decrease tn the interactPon between each parr This arguement rs then the basis for our explanahon of the observed asymmetry of the results, wtth respect to the concentration, in terms of our so-called"frustrahon effect" Th~s suggestion ~s further re- inforced by the orientation correlahon funchons for these mixtures which will appear elsewhere shortly

Acknowledgements

We would h k e t o t h a n k D Frenkeland J Drexhage for helpful dtscussions concermngthe interpretahon of the results S d L acknowledges SARA for the allocation ofcompuhngtrme on the IBM-3090-180 at the University of Amsterdam

REFERENCES

Adams, DJ and Adams, E M , 1981 Molec Phys,42, 907

Glerycz, P,Tanaka, H and Nakamsht, K, 1984 Fluid Phase Equilibria, 16, 241 Grerycz, P , a n d Nakamshl, K, 1984 Fluid PhaseEquihbria, 16, 255

Guggenheim, E A . 1944 Proc Roy Soc (London) Ser A, 183, 203 Hohelsel, C and Kohler, F, 1984 Fluid Phase Equllrbria, 16, 13

King, M B , 1969 Phase EqulhbNum in Lrqutd M~xtures Pergamon press, Oxford, 27-35 Lee, K-H and Sandler, S I , 1987 Fluid Phase Equdlbna, 34, 113

Lee, K - H , Lombardo. M and Sandler, S I , 1985 Fluid PhaseEqudfbrla, 17, 217 Lee, K-H Sandier, SI and Monson, P A , 1986a Int J Thermophys, 7, 367 Lee, K-H Sandler, SI and Patel, N C , 1986b Flurd PhaseEquihbria, 25,31 Lee, L L Chung, FTH and Starling, K E , 1983 Flurd Phase Equrhbna, 12, 105 Lee, L L Chung, FTH and Landis LH 1986 Fluid Phase EqudFbrra, 31, 253 de Leeuw, S W , Perram, JW and Smith, ER, 1986 Ann Rev Phys Chem, 37, 245 de Leeuw, SW, Wdhams, CP and SmJt, B, 1988 Molec Phys, 65, 1269

Lf, M H , Chung, F T H , S o , C - K , Lee, LL and Starhng K E , 1986 ACSSymposium Series No 300, KC Chaoand IRL Roblnson, Ed, Amerpcan Chemrcal Society, Washington D C, pp 250-280 Mollerup, J, 1981 Fluid Phase Equdlbria, 7, 121

Morrlss, G P and Isbrster, D J, 1986 Molec Phys, 59, 911 Nakanlshl, K and Toukubo, K, 1979 J Chem Phys, 70, 5848 Nakamshr, K. andTanaka, H, 1983 Fluid PhaseEqudrbna, 13, 371

Nakamshi, K, Okazakf, S, IkarJ, K H~guch~,T and Tanaka, H, 1983 J Chem Phys, 76, 629 Neumann. M, SteJnhauser, O and Pawley, G S , 1984 Molec Phys, 52, 97

(12)

Petersen, H G, de Leeuw, S W and Pertain, J W, '1988 Molec Phys (in press) Rushbrooke, G S., 1938 Proc Roy Soc (London) Ser A, 166, 296

Whiting, WB and Prausmtz, J M , 1982 Flusd Phase Equilibria, 9, 119 Wong, J M and Johnston, K P , 1984,1nd Eng Chem Fundam, 23, 320.