Liquid-vapour interface for a simple fluid modelled by the Lennard-Jones potential was treated by the first two mernbers of the Born-Green-Yvon-Bogolyubov hierarchy. The pair of coupled integral equations was solved numerically to get the density profile and the pair correlation function in the inhomogeneous region. It was found that one can get a significant result for the pair correlation function if the monotonous density profile is inserted in the equation determining the pair correlation function. On the other hand, we did not succeed in getting a convergentsolution for the density profile. This means that some further efforts will be necessary to overcome the difficulties connected with the unstable numerical procedures

B. Borštnik

D. Janežič

D. Pumpernik

English

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CROATICA CHEMICA ACTA CCACAA 59 (3) 527-532 (1986)CCA-1665YU ISSN 0011-1643UDe 541Original Scientific PaperOne Particle and Two Particle Distributions in theLiquid-Vapour InterfaceB. Borštnik, D. Pumpernik, and D. JanežičBoris Kidrič Institute of Chemistry, Ljubljana, YugoslaviaReceived September 2, 1985Liquid-vapour interface for a simple fluid modelled by theLennard-Jones potential was treated by the first two mernbersof the Born-Green-Yvon-Bogolyubov hierarchy. The pair of cou-pled integral equations was solved numerically to get the densityprofile and the pair correlation function in the inhomogeneousregion. It was found that one can get a significant result for thepair correlation function if the monotonous density profile isinserted in the equation determining the pair correlation function.On the other hand, we did not succeed in getting a convergentsolution for the density profile. This means that some furtherefforts will be necessary to overcome the difficulties connectedwith the unstable numerical procedures.1. INTRODUCTIONThe question of the behaviour of the liquid-vapour interface is not yetfully settled. Early results obtained either by computer simulation techniquesor by integral equation theories gave oscillatory profiles for simple liquidsinteracting with the Lennard-Jones potentiap,2. Later, the belief in monotonousdensity profil es prevailed. It was found that the oscillatory character ofthe density profiles is an artefact of insufficient averaging" in the computersimulation procedures while the integral equation approach was shown tobe extremely sensitive with respect to the choice of the input quantities,such as pair potential, pair correlation functions and so on", The experimentcannot provide a definite answer, either. The X-ray scattering On the liquidsurface can give an estimate of the range in which the liquid-vapour tran-sition is accomplished but not the exact form of the density profile.In this work, we present the results of a calculation where the equationsfor the single particle and for two partic1e correlation functions were solvedsimultaneously as a system of a pair of coupled equations. The equationsare taken from the Born-Green-Yvon-Bogolyubov hierarchy of equations.In the next section, we develope the formalism which turns the BGYBequations to a form suitable for numerical treatment. In the last section,the results are given and it is stat ed that the results obtained for the paircorrelation function are significant while the numerical procedure fails toreproduce a significant result for the density profile. This means that itwill be necessary to make some further efforts to solve this porblem.528 B. BORŠTNIK ET AL2. THE SYSTEM OF EQUATIONS AND THE NUMERICAL SOLUTIONThe first and the second members of the hierarchy of BGYB equationsare usually solved separately. The solution of the second member for thehomogeneous bulk liquid gives the pair correlation function. If the infor-mation about the single particle distribution function is needed, one shouldsolve the first member of the hierarchy. However, this equation needs thepair correlation function as input. In the inhomogeneous region, where theinhomogeneity is introduced along one direction, the pair correlation functionis a function of three variables. A flat Iiquid-vapour interface representssuch a system. A common practice to model the pair correlation functionfor such a case is to mix the liquid bulk and vapour pair correlationfunctions. There is no unique way of doing this and it was found+ that theresults depend strongly upon the details of this procedure. To get moreconsistent results one can try to solve the equations for the density profileand the pair correlation function simultaneously. The two equations readas follows~ '1 ~ --... -lo-'::" -.::..-.:>.V In e (r) = -- - S V V (I r - t I) g (r, t) e (t) dtkT(1)1 1V In e (r) + V In g (r, s) = - - V V ( I r-s I ) -kT kTS \! V ( I r - ti) g (r, s) g (s, t) e (t) dt (2)The density profile is marked by p (r) and the pair correlation function by9 (rit r2). The superposition approximation was taken into account on theright side of (2) by expressing the three particle correlation function by thetriple product of the two particle correlation functions. In the case of a flatIiquid-vapour interface, the density profile is a function of one variable onlyp (r) = p (z) while the pair correlation function depends upon three variables:the distance of the two particles, position of the gravity centre of the twoparti cles with respect to the Kirkwood-Buff dividing surface" and the orien-tation of the pair with respect to the dividing surface. In our calculationwe shall neglect the orientational dependence so that the pair correlationfunction will be the function of two variables onlyg (rl, Ti) = g (r, z*) (3)with z* being the gravity centre coordinate of the two particles,Neglect of the angular anisotropy of the pair correlation function is arather crude approximation, but it is necessary in order to keep the nume-rical procedure within the reach of the present generation of computers.Having defined the arguments of our unknowns p and g, equations (1) and(2) can be partly integrated to give:(4)LIQUID-VAPOUR INTERFACE 529coP (ZI' Z2) = S V' (r) 9 (r, z*) drI Z2 - ZI I(5)1S v dv 9(rI3, z13*) [g (r23, Z23) -1] e (Z3); v = cos<r. (z,r13)-1(6)Equations (4) and (6) follow directly from (1) and (2). Equation (6) resultsfrom substracting (1) from (2). The meaning of the variables appearing in(4), (5) and (6) is depicted in Figure 1.Z vapourI Z3Z Z, z 2O1Z2 liquidyxFigure 1. Meanings of symbols appearing in equations (4) to (6).Equations (4) and (6) were solved for the Lennard-Jones 6, 12 pairpotential. The density profile p (z) was tabulated in discrete points spacedequidistantly within 0.04 a, Also, the pair correlation function was calculatedon a square lattice with /::,.r = /::,.z = 0.04~. The integrals were calculatedby means of the trapez rule. Since equations (4) and (6) represent a systemof coupled nonlinear equations, we used the iterational procedure whichstarts with an initial guess and afterwards the output of i-th step isinserted into the r. h. s. of the equations in the i + 1 - th step.3. RESULTS AND DISCUSSIONLiquid and vapour can coexist at any temperature in the intervalbetween the triple point temperature and the critical temperature. For theLennard-Jones system the two temperatures are determined by variousmethods, such as molecular dynamics calculations" and by the calculations530 B. BORSTNIK ET ALbased on the integral equations approach", Their approximate values arekTtp = 0.72 I:: and kTc = 1.351::.Experimental measurements on noble gases are also in rough agreementwith the abovementioned values. The calculation should be performed forsome temperature from the central part of the interval between the triplepoint and critical point. Close to the triple point one usually encountersdifficulties due to the existence of the solution for the pair correlationfunction. On the other hand, the surrounding of the critical point is chara-cteristic of the long ranged correlations which cannot be treated numericallywithin equation (6). The study of the vicinity of the critical point is adifficult problem even for the homogeneous case where the pair correlationfunction is a function of one variable. Table I shows some possible pointswhich can be chosen for the calculation of the density profile and paircorrelation function. The corresponding temperatures, densities of liquidphase and vapour phase densities are given. We chose the point atkT = 1.30 I:: where liquid and vapour densities are 0.430--3 and 0.170--3,respectively. It should be noted that this point is not as close to the criticalpoint as one would conclude by comparing this temperature with the above-mentioned critical temperature kTc = 1.351::. The superposition approximationof BGYB equations gives a too high estimation of the critical temperature(approximately kTc = 1.6371::),8 which means that point kTc = 1.3 E is ratherfar from the critical point.TABLE ICoexistent Liquid and Vapour Densities and the Corresponding Temperatures forthe Lennard-Jones Fluid. Densities are Given in Units 0-3 and the Temperaturesin Ejk, Where o and E are Parameters of the Lennard-Jones Potential and k is theBoltzmann ConstantT Ov OL0.8 0.005 0.770.9 0.02 0.7151 0.03 0.661.1 0.05 0.601.2 0.095 0.5231.3 0.17 0.431.35 0.30 0.30, The calculation proceeded as follows. The numerical procedure of solutionof eq. (6) was tested to check whether the limiting behaviour of 9 (r, Z2)agrees with the liquid and vapour bulk pair correlation function calculatedby the numerical procedure for the homogeneous bulk system. We foundthat both limits were satisfactorily reproduced. The calculation was thenstarted with the initial guess for the pair correlation function in the formof the vapour pair correlation function, while the density profile wasexpressed by an analytical function'':O (z) = ov + _1_ (OL- Ov) {tanh [(z - zo) u] + I}2(10)LIQUID- VAPOUR INTERFACE 531Parameter a-1 determines the depth of the liquid-vapour interface and wasset equal to a-1 = 20". Coordinate Zo defines the dividing surface. With theinput as defined above equation (6) was first subjected to the iterative pro-cedure. The density profile was held fixed in the form of (10) while 15iterational steps were performed. The resulting two dimensional array re-presenting the pair correlation function in the inhomogeneous region isshown in Figure 2. We can see that the transition between the two limiting~~-----------'222O~ __'-' ~ __~2 3Figure 2. The pair correlation function of the liquid-vapour interface.pair correlation functions belonging to the liquid and vapour bulk phaseis gradual and monotonous. Of course, this result cannot be treated as thefinal result since the iterations should be continued in order to get a eon-vergent result also for the density profile by running the iterative procedureon equation (5). This was found to be amore difficult task than one canimagine. We found that the iterative procedure based on eq. (1) convergesvery poorly. Detailed examination of equation (4) shows that the numericalprocedure fails to reproduce the correct difference between the two bulkdensities. This property destabi1izes the iterational procedure so that theiterations proceed towards an apparently stable density profile which isdepicted in Figure 3. When the iterational procedure was subjected to amore stringent criterium it was established that if the iterations are continuedthe density profile drifts further and may finally diverge. Thus, we canconclude that some further work will be necessary to get significant results.However, according to the positive experience obtained by studying modelsof the crystal-liquid interface with the integral equation theories we canexpect that also the Iiquid-vapour interface should be manageable usingthis approach.:532 B. BORŠTNIK ET ALO.~ ~[tJ0.30.20.12 t 3Figure 3. Density profile of an apparently converged solution. Dashed line depictsthe input function in the form of (10).Acknowledgement. - The financial support of t.he Research Community ofSlovenia is gratefully acknowledged.REFERENCES1. C. A. Cro x t o n and R. P. Fer rie r, J. Phys. Col (1971) 2447.2. G. M. Na zar i a n, J. Chem. Phys. 56 (1972) 1408..3. S. T o x v a e r d, Mol. Phys. 26 (1973) 91.4. B. B o r š t n i k and A. A ž man, Mol. Phys. 29 (1975) 1165.5. J. G. Kir k w o o d and F. P. B u f f, J. Chem. Phys. 17 (1949) 338.6. L. Ver let, Phys. Rev. 159 (1967) 98.'7. J. A. Bar k e r and D. He n der s o n, J. Chem. Phys. 47 (1967) 4714.8. B. B o r š t n i k, D. P u m per n i k, and D. Jan e ž i Č, to be published.:9. J. P. R. B. W alt o n, D. J. Ti 1des 1e y, and J. S. R o w li n s o n, Mol.Phys. 48 (1983) 1357.SAZETAKJednočestične dvočestične raspodjele na granici faza tekuće-plinovito.B. Borštnik, D. Pumpernik i D. JanežičProučavana je granica faza tekuće-plinovito za jednostavne fluide, modeli--rane Lennard-Jonesovim potencijalom, s pomoću prva dva člana hijerarhije Born--Green- Yvon-Bogolyubova.

One Particle and Two Particle Distributions in the Liquid-Vapour Interface

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