Modified Cell Model for Liquids: Order Defectsand Intermolecular PotentialsBY FRIEDRICH KOHLBR AND J. FISCHERinstitute of Physical Chemistry, University of Vienna, AustriaReceived 16th January 1967On the basis of a cell model, an explicit introduction of coordination defects is possible. In binarymixtures, consideration of coordination defects gives only a small correction to the thermodynamicexcess properties of mixing. The excess free energy is lowered, and its concentration dependenceis influenced in about the same way as by using the quasichemical approximation. Furthermore,the modified cell model has been used to calculate the thermodynamic properties of solid and liquidargon, employing a Kihara pair potential. Comparison is made with previous calculations using aLennard-Jones pair potential.The parameters of the Kihara potential giving the best fit at highdensities differ markedly from those giving the best fit for transport properties and second virialcoefficients. The influence of the exact form of the pair potential on the thermodynamic properties isnot very important.The modified cell model 1 9 2 combines some development of Kirkwood’s varia-tional method 3 with an explicit consideration of order defects. The latter areresponsible for density fluctuations, by analogy with Kirkwood’s 3 undeterminedparameter for multiple occupancy of the cells. Therefore, we have carried out thedevelopment of Kirkwood’s variational method for single occupancy of the cells,and then consider order defects.In developing Kirkwood’s variational method, we have had two aims: (i) toaccount for the distribution of the locations of the neighbouring molecules withintheir cells, (ii) to allow a certain correlation of the motion of the central moleculewith the motion of the neighbouring molecules, at least near the edges of the cells.The first point has been considered by Mayer and Careri.4 As we want to employthe Lennard-Jones 6 : 12 pair potential or the Kihara potential, we have used asquare-well approximation for the probability density in the neighbouring cells ratherthan the Gaussian approximation used by Mayer and Careri.Like them, we haveadjusted the size of the square well consistently to the free volume available in thecentral cell. The second point has been raised by Hirschfelder, Dahler, and Thacher,swho have pointed out that without such a correlation the cell model will always givetoo positive an energy and too small a free volume, especially at low densities.Where-as Kirkwood 3 has based his variational method on the assumption of independentmotions of molecules in their cells, our method shows a certain degree of dependenceupon these motions. Therefore, it is necessary to introduce the correlation functionbefore minimizing the free energy. Our starting equation isN k=N- 1Here P(r1,rz. . . rN)drldr2. . . drN denotes the probability of finding the first particlea the dislocation rl of the centre of the first cell, the second particle at thedislocation r2 of the centre of its cell, etc.+(rl)drl is the probability that a particle is3F . KOHLER AND J . FISCHER 33at a distance r from the centre of its cell, provided that no correlation with the motionof neighbouring molecules is taking place. The effect of the correlation functionx(rik), which is supposed to be a function only of the mutual distance of pairs ofparticles, is to overweigh pairwise configurations which are strongly attractive and toexclude pairwise configurations which are strongly repulsive. The correlation islimited to small mutual distances. For higher distances, the x are set arbitrarilyequal to zero. The limitation is made in such a way that the product n(l + x i k ) canbe developed into various sums, each containing factors with different subscriptsonly, analogous to the treatment of real gases at low densities :n(l + X i k ) = +&ik + C & i k h n + * * (2)i > kInserting eqn.(1) and (2) into the expression for the free energy, and minimizing itwith respect to each 4 and x, these functions can be calculated. We have approxi-mated the probability density 4 by a square well, and the correlation function x by (x = -1) for small pairwise distances, and by (x = 0) for larger distances. Byusing the minimizing procedure a consistent size for the square well is establishedand a value of the pairwise distance where the step should occur is obtained.An order defect is thought of as the removal of a nearest neighbour into the shellof next-nearest neighbours. In this process, the distance between the centres ofneighbouring cells has been stretched from the nearest neighbour distance a to thedistance A = 23a.Let Fa be the free energy for a nearest neighbour distance aaccording to the model above, without order defects. Let FA be the same for anearest neighbour distance A . Then the free energy for an assembly with nearestneighbour distance a, but with a fraction x of stretched contacts, is assumed to begiven byHere [&)IN denotes the number of possible arrangements due to order defects.The fraction x is again determined by a minimization process. For densities lowerthan the critical density, x becomes equal to about 0-5 ; R In g(x) is then the communalentropy per mole of gas. Including a correction for the void of a face-centeredcubic packing, In g(x) has been written1 :In eqn.(4), z denotes the coordination number.F = (1 -x)F,+xF,-RT In g ( x ) (3)In&) = -(2/2)(xln x+(l -x) In (1 -x)+1,88x(1 -x)} (4)BINARY MIXTURESThe model for the order defects may be combined with simpler (and poorer)approximations of the cell model than the one described in the introduction. We nowinvestigate the effect of our coordination defects on the thermodynamic excess pro-perties of mixing of simple liquids. For this, we use the cell model for single occupancyin the approximation given by Prigogine and coworkers6,7. Again, a square well isassumed for the probability density, but the square well is arbitrarily extended tothe distance (a- 0 ) / 2 from the centre of the cell (a being the diameter of the cell, andD being the mutual distance between two molecules where the pair potential passesthrough zero).Furthermore, the energy is calculated as if all molecules would belocated at the centres of their cells.In mixtures, a given molecule A is surrounded by both A and B molecules.Therefore, the energy parameter A,, = ZE& (E& . . . minimum of pair potential betweenA molecules) applicable in the pwe A liquid has to be changed to a parameterAa = EAaa + (1 - X)Aab in the mixture. In the same way, we have for cells occupied by34 MODIFIED CELL MODEL FOR LIQUIDSa B molecule an energy parameter A b = EAab + (1 - x)l\bb. Here E denotes the overallmole fraction of A, and random mixing is implied. For non-random mixing, themole fraction applicable for the construction of A, would differ somewhat from themole fraction to be used in the expression for &.Introducing the concept of stretchedcontacts, we define the following quantities : x, and x b denote, for A-cells and B-cellsrespectively, the fraction of neighbouring molecules which have been removed intothe shell of next-nearest neighbours (i.e., the fractions of stretched contacts).Starting from an A-cell, a fraction y of the normal contracts, and a fraction 7 of thestretched contacts will lead to other A-cells. Starting from a B-cell, the correspondingfractions are called 5 and F. The various quantities are interrelated by the followingrelationships : (a) mass balance equations of the form,y( 1 - xu) + yx, = x, ( 5 )z ( l - x a ) ( l - y ) = ( l - z ) ( l - x b ) t , (6)(b) identity between AB and BA contacts like(c) an assumption about the molecular distribution.We have tried as alternativesthe statistical distribution,and the quasichemical approximation 8m$/(maambb) = 4r7*Here N,b,nm, and N b b denote the numbers of AB, AA, and BB normal contacts,respectively, and q stands for exp(&b - Am- I\bb)/(ZkT).The starting point is the equation for the free energy of the mixture :F = - kT In [Ul,".C' -xa)QNaxaYNb(l -Xb)aNbxba b b 1 + ( N a / 2 ) [ ( 1 -xa)Ea+xaE,l fNa and Nb are the numbers of A and B molecules. Y, and Ea denotes the translationalpartition function within the free volume and the energy of an A molecule in a cell ofdiameter a,, a, and E, stands for the same quantities in a cell of diameter aa2*The double bar means that the number of contacts corresponds to the statisticaldistribution.The free energy is essentially a function of the size of A- and B-cells, and of thefraction of stretched contacts F(a,,ab,xa,xb). In order to determine the parameters,F has to be minimized subject to the condition that the volume, which is a functionof the same parameters, remains constant : V(aa,ab,xa,xb) = const.This leads toequations of the typeaF av -+I-- = 0,aab dab (9)where the Lagrange multiplier A can be evaluated from one of these expressions, e.g.F . KOHLER AND J . FISCHER 35Then each of the partial differentials can be expressed by one of them, say, by a,F/da,,and the equation of state can be written asAs long as P can be assumed to be practically zero, each partial derivative of the freeenergy with respect to the parameters vanishes, as the derivatives of Y with respectto the parameters are all different from zero.Thus we get-by analogy with thetreatment of Prigogine and Bellemans6 -the conditionsd F / W = -Y = 4 = (i3Fjaaa)(t3aa/aV). (11)i?F/i?a, = aF/aa, = dF/dxa = dF/dx, = 0. (12)The computation proceeds as follows. First, estimated values of xu and xb areused to calculate the values of y,y,c and f . Then, with a first guess of ab, values forQ, and xu are determined in small iteration cycles. On the basis of these values, thesame iteration cycles are used for the determination of a b and xb. Now a new set ofthe y and 5 is calculated, and the whole procedure is repeated. The most sensitiveparameters are xu and x b .In our example, about five cycles have been necessaryto arrive at values constant to 0-001 %.n 3 2500E5I1d0lg 2450.4 W'5. wc52 4 0 00.0 0.1 0.3 0 . 5 0.7 0.9 -XFIG. l.-GE/[X(l-x)] as calculated for 293-15°K with the parameters given in the text. Upperpair of curves : no coordination defects considered, statistical and strictly regular distribution, respec-tively. Lower pair of curves : the same with coordination defects.The example presented here has been calculated with constants appropriate tothe mixture 1,2,4-trichlorobenzene + n-hexane 99 10 (A, = 6623 1.5 ; Aab = 53 124.7 ;-Abb = 43425.7 J/mole; RZ, (the distance at the minimum of the pair potential of Amolecules) = 6.6466 x 10-8 ; R& = (R&+ R&)/2 ; Rib = 6-6782 x 10-8 cm).Thetemperature variation of Am, caused by the dipole-dipole contribution, has beenneglected. Fig. 1 shows the excess free energy of mixing GE, divided by the productof the mole fractions. The upper pair of curves is calculated without coordinationdefects (Le., xa = x b = O), for statistical and strictly-regular distribution, respectively.The lower pair of curves show the results of the consideration of coordination defects.Their effect on the absolute value of GE is not very marked, which is satisfying fromthe point of view of current theories of mixtures. The main effect of the coordinationdefect is the enhanced cilrvature of the GE/[-F(I -X)] function.For this detail, theconsideration of coordination defects is at least as important as the effect of thequasichemical approximation. As the curvature of the GE/[X( 1 - X2] function isrelzted to the shape of consolute curves,ll one might expect that the consideration o36 MODIFIED CELL MODEL FOR LIQUIDScoordination defects will facilitate the explanation of the consolute curves foundexperimentally. On the other hand, our calculations-which are based on theassumption of central force fields of the molecules-have given no indication of aninversion of the curvature of the GE/[X( 1 - E)] function at high Z values, as has beenfound e~perimentally.~ Fig. 2 gives the concentration dependence of the fractions ofI I0.1 0.3 0.5 0.7 0.9 - XFIG.2 . T h e fraction of stretched contacts for A- and B-cells, respectively, corresponding to thelower pair of curves in fig. 1.stretched contacts, starting from A- and B-cells. Finally, the order of magnitudeof the TSE/[X(l -X)] function for statistical distribution is - 150 J/mole deg without,and + 70 J/mole deg. with, consideration' of coordination defects. Introduction ofthe quasichemical approximation makes this function more negative, the value forX = 0.5 being -20 J/mole deg.COORDINATION I N DENSE SUPERCRITICAL VAPOURSAnother possible application of the model of coordination defects is based on thefact that volume expansion may be caused primarily either by diminishing thecoordination or by extending the average distance to the nearest neighbours.Fig. 3shows the result of calculations for argon at O"C, which demonstrates that up to avolume, about twice that of the liquid at the boiling point, expansion is causedprimarily by diminution of the coordination. The nearest neighbour distanceincreases only about 10 %. One might speculate and correlate this with the findingsof Hensel and Franck l2 on the strong increase in conductance with pressure foundin the dense, supercritical vapour of mercury. The assumption seems to be reasonablethat electron transfer occurs only between atoms which are below a certain minimumdistance. Fig. 3 shows that there might be a large range of volumes, where normalcontacts are less than this minimum distance and stretched contacts are greater.Then the conductance would be determined by the number of interruptions whichoccur in a row of normal contacts.By interruption we mean that the row of normalcontacts has either no continuation or the continuation would result in an opposingpotential. Therefore, the last atom must be the starting point for nine stretchedcontacts at least. The specific resistance would be essentially proportional to thP. KOHLER AND J . FISCHERninth power of the fraction of stretched contacts. In detail,13x9( 1 - x)( 1 + 2x2)'37Here IC is the specific conductivity, which is referred to the nearest neighbour distance13 0 4 0 5 0 6 0 7 0 8 0V(cm3)FIG. 3.-The nearest neighbour distance a and the mean coordination number 2, as calculated forargon at 273.15"K.T4 12 2 0 28PIP,FIG.4 . T h e r.h.s. of eqn. (13) for argon at 273.15"K.a, and the subscript 1.d. stands for "low density", where transferable electrons areabsent. The quantity on the r.h.s. of eqn. (13) depends on the pressure accordingto fig. 4, where the calculated results of the 0°C-isotherm of argon are used. It isquestionable to transfer the results from argon to mercury, although it demonstrateshow important the distinction can be between volume expansion by coordinationdefects and volume expansion by increasing the nearest neighbour distance38 MODIFIED CELL MODEL FOR LIQUIDSPAIR POTENTIAL AND THERMODYNAMIC PROPERTIESIn the previous paper,2 the modified cell model has been applied to argon at higherdensities, using a Lennard-Jones 6 : 12 pair potcntial with the minimum coordinatesEO = 128 k, R" = 3-83 A.In the present paper, a Kihara potential&(r) = &0[(-)"-2(->3 R"-s R"-sr - s r - shas been employed with the parameters of Barker, Fock, and Smith 13 (i.e., E' =The free energy of the solid is calculated without order defects, but by consider-ing coupling of the vibrations (e.g., by employing a Debye approximation).Previously,2 the calculated melting point was too low (55°K in comparison with theexperimental value of 84°K). We thought that possibly the Lennard-Jones potentialwas responsible for the free energy of the liquid being too negative, as the Lennard-Jones potential overemphasizes the attraction at distances larger than R".142.9 k, R" = 3.7338 A, s = 0-90069 R").1.0 1.2 1.4 1-6PFIG.5.-The difference of the Gibbs free energy between solid and liquid, as calculated with aLennard-Jones potential (L.J.) and with a Kihara potential (K.).FIG. 6.-The gas-liquid coexistence curve of argon at high densities.The present calculation shows that the effect of the form of the pair potential onthe thermodynamic properties is small in comparison to the effect of the choice of theminimum coordinates, and in comparison to statistical approximations. Thedifference between the free energy curves of liquid and solid is not greatly changed byusing the Kihara potential, and the small change is in the wrong direction. Fig. 5gives the difference of the Gibbs' free energy between solid and liquid according to thepotential used.Entropy and volume difference between solid and liquid phase areabout correct at the temperature of the experimental melting point. Furthermore,fig. 6 shows part of the coexistence curve between liquid and gas at high densities.The parameters of the Lennard-Jones potential used correspond to a ratio E"/R",which is rather high compared to the values most frequently given. The parametersof the Kihara potential are supposed to fit the measurements of gas viscosities andsecond virial coefficients. Nevertheless the use of the Kihara potential gives a smallervolume and a larger energy of solid and liquid than the use of the Lennard-JonespotentialF . KOHLER AND J . FISCHER 39Fig. 7 shows the entropy of the solid as function of the temperature.This functionseems to be most seriously influenced by the square-well approximation for the proba-bility density, which leads to characteristic temperatures rising with temperature.If a constant characteristic temperature 0" is used (taken from the curvature of25h8 -23\ 3 eFIG. 7.-The entropy of solid argon; (L.J.) and (K.) calculated with a square-well approximationfor the probability density in the cells, using a Lennard-Jones and Kihara potential, respectively;0" calculated with a characteristic temperature evaluated directly from the hnnard-Jones potential.FIG. 8.--The volume of solid argon ; descriptions of curves as in fig. 7.the Lennard-Jones potential around the potential minimum), then too high an entropyis obtained. In the square-well approximation, the Kihara curve should differ fromthe Lennard-Jones potential around the potential minimum), then too great an entropyand this is the case. Finally fig. 8 shows the volume of the solid as function of thetemperature. Qualitatively, the situation is the same as for the entropy, but thevolume according to the Kihara potential is much too small.1 F. Kohler, Ber. Bunsenges., 1966, 70, 1068.2 F. Kohler and F. Weissenback, submitted to J. Chem. Physics.3 J. G. Kirkwood, J. Chem. Physics, 1950,18, 380.4 J. E. Mayer and G. Careri, J. Chem. Physics, 1952, 20, 1001.5 J. S. Dahler, J. 0. Hirschfelder and H. C. Thacher, J. Chem. Physics, 1956, 25, 249.6 e.g., I. Prigogine and A. Bellemans, Disc. Fcruday Soc., 1953, 15, 80 ; I. Prigogine, MolecularTheory of Solutions (North Holland 1957).7 see also, F. Kohler, Monatsh, 1957 88, 857.8 E. A. Guggenheim, Mixtures (Oxford, 1952).9 R. J. Munn and F. Kohler, Monarsh, 1960,91, 381 ; A. Neckel and F. Kohler, Monatsh, 1956,10 F. Kohler, Chem. Technik, 1966,18,272.11 F. Kohler, J. Chem. Physics, 1955, 23, 1398.12 F. Hensel and E. U. Franck, Ber. Bunsenges, 1966, 70, 11 54 ; we are indebted to the authors13 cited from J. S. Rowlinson, Disc. Furaduy Soc., 1965,40, 19.87, 176.for communicating their results prior to publication