A simple expression for radial distribution functions of pure fluids and mixtures

Transcription

1 A simple expression for radial distribution funtions of pure fluids and mixtures Enrio Matteoli a) Istituto di Chimia Quantistia ed Energetia Moleolare, CNR, Via Risorgimento, 35, Pisa, Italy G. Ali Mansoori Chemial Engineering Department, University of Illinois, 810 S. Clinton St., Chiago, Illinois Reeived 9 January 1995; aepted 5 June 1995 A simple expression for the radial distribution funtion RDF of pure fluids and mixtures is presented. It satisfies the limiting onditions of zero density and infinite distane imposed by statistial thermodynamis. The equation ontains seven adjustable parameters; they have been fitted to extensive literature data of RDF s for a Lennard-Jones fluid at different values of temperature and density. These in turn have been expressed as funtions of redued temperature and density, thus allowing a omplete parametrization with respet to these variables using 21 parameters altogether with fairly good auray. The values of the redued pressure and internal energy alulated by numerial integration of the ompletely parametrized equation ompare fairly with literature moleular dynamis simulation results. The apability of the expression to fit to RDF s of mixtures has been heked against some of the extensive RDF simulation data of binary mixtures of Lennard-Jones fluids with different diameters available in the literature. Data pertaining to different molar frations as well as to different AA / BB ratios have been onsidered, and the agreement between alulated and simulation urves has resulted satisfatory. The proposed expression an be used to alulate by integration related quantities suh as ompressibility, internal energy, pressure and, using the Kirkwood Buff theory, the hemial potentials and partial molar volumes of the omponents of mixtures for whih RDF data are available Amerian Institute of Physis. INTRODUCTION Radial distribution pair orrelation funtions RDF are the primary linkage between marosopi thermodynami properties and intermoleular interations of fluids and fluid mixtures. In the anonial ensemble of statistial mehanis, where the temperature, T, the volume, V, and the number of partiles of eah omponent in the system, N i, are fixed, the radial pair distribution funtion, g ij (r,,t), is given by g ij r,,t V 2 1 ij /N j e /kt dr 3 dr N /Z T,V,N. In this equation, is the number density, ij is the Kroneker delta, N is the set of number of moleules, N 1,N 2,...,N, is the total potential energy of the system, k is Boltzmann onstant, r i is the position vetor of moleule i, and Z (T,V,N is the onfigurational integral, Z T,V,N e /kt dr 1 dr N. The relations between g ij and the internal energy u and pressure p, assuming pairwise additivity of intermoleular potential energy funtion, are the following: 1 u u ig 1 2 N i 1 a Corresponding author. j 1 x i x j 0 ij r g ij r 4 r 2 dr, p kt 2 /6 i 1 j 1 x i x j r ij r g ij r 4 r 2 dr, 0 where is the derivative of the potential funtion with respet to the distane r. Modern theories of fluid and fluid mixtures have benefited a great deal from the onept of radial distribution funtion. The RDF theories have been quite suessful in desribing the behavior of simple liquids and liquid mixture, 1 7 although have been able to provide RDF analyti expressions only for very simple model fluids suh as the hard spheres and hard rods. 7,8 Moreover, the validity of Eqs. 3 and 4 is limited to spherially symmetri potentials, and therefore their appliation to polar nonspherial fluids is impossible. On the other hand, the Kirkwood Buff theory 9 provides equations valid for any kind of moleular shapes whih allow the alulation of thermodynami properties of mixtures suh as ompressibility, partial molar volumes and hemial potentials only requiring the knowledge of the integrals over the distane of the radial enter to enter distribution funtions Kirkwood Buff integrals, G ij : 4 G ij 0 g ij r 1 4 r 2 dr. 5 Although in reent years several studies have been arried out to alulate the Kirkwood Buff integrals of real mixtures from thermodynami properties by the inverted equation of the theory, 6 no extensive study is available at our knowledge to alulate these quantities diretly from Eq. 5, 4672 J. Chem. Phys. 103 (11), 15 September /95/103(11)/4672/6/$ Amerian Institute of Physis

2 E. Matteoli and G. Ali Mansoori: Radial distribution funtions of fluids 4673 and then to use these to obtain thermodynami properties. These alulations are linked to the availability of an analytial expression that ould be used to represent the RDF s of fluids and their mixtures. As we will see later on, due to different reasons, previous RDF expressions are not appropriate to the general purpose of our researh, onsisting in the alulation through the Kirkwood Buff theory of the hemial potentials and the partial molar volumes of the omponents of binary mixtures of Lennard-Jones fluids first and of more omplex substanes afterward. In this work we elaborate on the asymptoti onditions required by statistial thermodynamis and on less rigorous geometrial and spatial onsiderations to provide a parametri equation for RDF of Lennard-Jones fluids and mixtures whih an be used to omply with the first part of our researh. urve to deviate from a strit exponential osillatory deay. The other terms guarantee ontinuity of the funtion at r d. In order that the RDF equation may be apable to inlude temperature and density dependene, all parameter p i h,, m, g(d),,,, are to be expressed as funtion of the redued temperature and density, T* kt/, * 3, being the energy parameter. The following funtional forms are proposed: p i q 1,i q 2,i exp /kt exp q 3,i 3, 7 for p i, g(d),,, and h, with h 1000(h 1); and p i q 1,i q 2,i exp /kt 3 2 q 3,i / 3, 8 for p i m and. They have been hosen to satisfy the limiting ondition of RDF above outlined. RDF OF PURE FLUIDS The radial distribution funtion g(r,,t) of a pure fluid must satisfy the following asymptoti relations. At zero density it must redue to its ideal gas limit of unity, lim g r,,t 1. 0 Similarly, at large intermoleular distanes it must redue to unity, lim g r,,t 1. r As the temperature goes toward infinity, the RDF must beome independent of temperature as is the ase with the hard-sphere fluid RDF, g hs : lim g r,,t g hs r,. T In addition, the equation representing the RDF must be ontinuous over all distanes and the first maximum should our at the ontat distane d, with d h, being the distane parameter of the Lennard-Jones potential funtion, and h an adjustable variable with value very near to unity. Considering the above features regarding the general shape of the pure fluid RDF and the above boundary onditions, we propose the following dimensionless funtional form for the radial distribution funtion: g y 1 y m g d 1 y 1 /y exp y 1 os y 1, m 1, y 1, 6a g y g d exp y 1 2, y 1, 6b where y r/d is the dimensionless intermoleular distane, and h d/, m,,,,, g(d) are adjustable parameters. The terms y m and exp[ (y 1) 2 ] provide for the deay of the first peak, while the fator exp[ (y 1)]os[ (y 1)] has the sope to give an osillatory-deaying shape to the funtion as the RDF must have. Parameter h supports the deviation of d from, and provides some flexibility in the setion in between the first and seond peaks, allowing the RDF OF BINARY MIXTURES Equations 6a and 6b an be slightly modified so as to be applied to binary mixtures. A proper mixture RDF equation should take into aount the variation in the position of the peaks and in their separation distane as the omposition of the mixture hanges. From the observation of the shapes of the RDF s obtained by moleular dynamis simulations of pure ompounds and of binary mixtures, the following two general features as regards position of the peaks and their separation distane an be identified for the three RDF s g AA, g BB, g AB of a mixture of two Lennard-Jones fluids, A and B, with diameters A and B. 7,10 These features are based on spatial onsiderations only; the influene on the RDF shape due to the energy parameters, ij, is taken into aount a posteriori by means of the parametrization Eq. 8. i The ontat distane d, where the first peak appears, is determined by the diameters of the omponents of the mixture independently of the molar frations: d h A B /2, 9 where h (h 1) aounts for deviations of the value of d from the average moleular diameter. ii Eah subsequent maximum is loated at a distane r M from the previous one that depends on the omposition. Simulation experiments indiate that the loation of the seond peak is mainly determined by the size of the partiles whih play the major role in filling the spae in between the pair of moleules under observation, and these are the moleules of that speies whih is the most onentrated and/or the largest in size. 7 So, if A and B have similar diameters, A B, at molar fration X A 0.9, the spae is filled mainly by A moleules, whereas at X A 0.1 it is the B partiles whih oupy most of the spae, and onsequently the seond peak maximum will be loated in the two ases at a distane approximately equal to A and B, respetively, from the 1st maximum. If A B, even at X A 0.5, the moleules of speies A will oupy most of the spae, so that r M A. All this is independent of the pair of moleules under observations, A-A, A-B, orb-b. We introdue this feature in our equation by assuming that r M is given by a weighted arithmeti mean of A and B aording to

3 4674 E. Matteoli and G. Ali Mansoori: Radial distribution funtions of fluids FIG. 1. Plot of radial distribution funtion, g(r), vs the redued distane, r/., moleular dynamis simulation data, 11, best fitting urves Eqs. 6a, 6b with parameters of Table I. From bottom to top, the figures represent data of the 1st and 5th lines of Table I, respetively. r M h X VA A X VB B, 10 where X VA and X VB are the volume molar frations of the omponents A and B in the mixture, respetively. The hanges to be introdued in Eqs. 6a and 6b for these to beome valid also for mixtures should take into aount the variation of both d and r M aording to the mixture omposition and to the i- j ouple as expressed in Eqs. 9 and 10, and at the same time preserving ontinuity of the funtion at r d and as the omposition hanges gradually from one pure ompound to the mixtures to the other pure ompound. This an be aomplished by taking different funtional forms for y aording to whether r d or r d: r d, y r/d; 11 r d, y r d r M / r M. 12 By introduing these onditions and expressions for y in Eqs. 6a and 6b, we obtain a RDF expression whih an be used for all three g ij funtions of a binary mixture. Similarly to pure fluids, the dependene on temperature and pressure is taken into aount by expressing parameters aording to Eqs. 7 and 8. Moreover, it an be realized that the mixture equations tend to the pure fluid ones when the onentration of one omponent goes to zero; i.e., if X VB 0 then X VA 1, and, looking at Eqs. 9 and 10, we have d r M h A. RESULTS AND DISCUSSION Pure fluids In order to hek the apability of Eqs. 6a and 6b to reprodue the atual behavior of the RDF s, we have onsidered the MD simulation data reported by Verlet. 11 They are quite appropriate for this study beause extend to r values of TABLE I. Parameters of the RDF equation determined by best-fitting of data of Ref. 11. a s.d h m g(d) a Parameters of Eqs. 6a and 6b. The lines orrespond in the order to the 25 RDF urves for a Lennard-Jones fluid of Table IV of Ref. 11. Here s.d. is the standard deviation of the fit.

4 E. Matteoli and G. Ali Mansoori: Radial distribution funtions of fluids 4675 TABLE II. Coeffiients of the parametrization equations. a q 1 q 2 q 3 s.d. EQ. h b m g(d) a Eqs. 7 and 8 of text applied to data of Table I. b In this ase the parametrization equation produes h, defined as h 1000(h 1). 5, refer to relatively wide ranges of temperature and density, and the number of points for eah urve is suffiient to desribe its shape aurately. By means of a nonlinear leastsquare routine, parameters p i of Eqs. 6a and 6b have been adjusted to reprodue all 25 RDF urves there reported. As an example, to illustrate the performane of Eqs. 6a and 6b, in Fig. 1 a omparison between simulation points and alulated urves for a ouple of ases is shown. In Table I the values of the parameters and the standard deviation of the fit are reported. We an observe that the standard deviation of the fit is in most ases less than 5%. In the few instanes where less auray of fitting is ahieved, the shape of the simulation urves in the r d setion presents anomalies suh as disontinuities and sharp edges that our equation, as well as any other funtion whih is ontinuous and has ontinuous derivatives, annot reprodue. The parameters have then been fitted to Eqs. 7 and 8. Due to the very small deviations of parameter h from unity, the quantity h 1000(h 1) instead of h was adjusted to Eq. 7. The set of q i values obtained are reported in Table II, and in Fig. 2 the reprodution by Eqs. 7 and 8 of h and FIG. 3. Comparison between values of redued internal energy, u* (u u ig )/N, and pressure, p* p 3 /, obtained by numerial integration of the RDF equation Eqs. 6 a and b using parameters of Table I full points with those alulated by integration of the fully parametrized RDF equation solid lines, that is, with parameters expressed as funtion of the redued density, * 3, and temperature, T* kt/, aording to Eqs. 7 and 8 and values of Table II. (A, ), * 0.5; (B, ), * 0.65; (C, ), * 0.75; (D, ), * g(d), the most important parameters of our RDF equation, is shown. Some years ago, Goldman 12 proposed an expression for the RDF of pure fluids whih was able to reprodue Verlet s data somewhat more aurately than the present equation; the standard deviation averaged over all 25 RDF urves amounts to for our equation and to for his fitting. However, his model equation requires 108 parameters, and this makes its use quite diffiult; moreover, the possibility of its extension to mixtures does not appear trivial. In ontrast, the simpliity of our model, whih only requires 21 parameters on the whole, and its validity also for mixtures, makes it quite affordable for further appliations and studies. FIG. 2. Plot of the reprodution of parameters h d/ and g(d) by the parametrization Eq. 7. Points are best-fitting parameter values of Table I, the urves desribe the trends of h and g(d) as alulated by Eq. 7 at onstant redued density *, * 3. From top to bottom, * 0.88, 0.85, 0.824, 0.75, 0.65, 0.5, FIG. 4. Comparison between values of redued internal energy, u* (u u ig )/N, and pressure, p* p 3 /, obtained by moleular dynamis simulation experiments in Ref. 14 full points and those alulated by numerial integration of the RDF equation Eqs. 6a and 6b with parameters expressed as funtion of the redued density, * 3, and temperature, T* kt/, aording to Eqs. 7 and 8 and values of Table II solid lines. (A, ), * 0.5; (B, ), * 0.65; (C, ), * 0.75; (D, ), * 0.85.

5 4676 E. Matteoli and G. Ali Mansoori: Radial distribution funtions of fluids TABLE III. Parameters of the RDF mixture equation resulting from best-fitting of seleted simulation data of g ij of Lennard-Jones mixtures. a g ij ij x A * s.d h m g(d) 1 AA AB BB AA AB BB AA AB BB AA AB BB a The g ij simulation data were taken from Ref. 10. For all g ij s, B / A 2. The values of BB / AA were 3, 0.5, 2.0, and 0.5 for the g ij s #1, 2, 3, and 4, respetively. For the g ij s #1 and 3, T* kt/ AA 3.0; for the g ij s #2 and 4, T* 1.0, and 2.0, respetively. Also Li et al. 13 proposed an analytial formula for RDF s of fluids and fluid mixture. Their aim was to provide an expression, derived from the wave motion equation, whih ould reprodue the RDF s of Lennard-Jones fluids, and also predit the exess enthalphy of mixtures. As regards the former point, from the omparison they made with five urves of Verlet we have alulated an average standard deviation by , against a value of that we obtain with our equation for the same urves. This very large deviation is mostly due to the poor reprodution of the r d setion and, to a lesser extent, of the d r 2d setion, and renders this equation unsuitable for its use in the alulation of thermodynami properties through the Kirkwood Buff theory. To investigate how the fitting unertainties reflet on alulated thermodynami properties, we have alulated the redued internal energy, u* (u u ig )/N, and pressure, p* p 3 /, by numerial integration of Eqs. 3 and 4 with parameters given by Eqs. 7 and 8. We have arried out this alulation at several values of the redued temperature T*, T* kt/, for eah of four different values of the redued density *, * 3. In Fig. 3 these alulated urves are ompared with the u* and p* values obtained by numerial integration of the RDF funtion with parameters of Table I. We an see that, exept for the first and the last point at * 0.85, a fairly good reprodution is obtained, and this supports the validity of our parametrization equations. In another paper, Verlet 14 reports the results of simulation experiments of thermodynami properties of Lennard- Jones fluids. The data of u* and p* there obtained have been ompared in Fig. 4 with the urves alulated as desribed above. The figure suggests that good agreement is found only limitedly to the trends; if we look at the single values, the simulation data are all lower than their orresponding urves, and the deviation is larger the higher the temperature. These disrepanies annot be due only to inadequay of our model equation or to inaurate fitting see Fig. 3 ; it may be possible that the simulation tehnique has some inonsisteny in the ways of produing the thermodynami properties and the RDF urves, and this drawbak appears more effetive at high T* and leads to negative values of p* at low T*; however, the maximum error admitted by Verlet 11 of 0.2 in the ompressibility fator, p/ kt, whih at * 0.85 orresponds to an average error of 0.17 on p*, explains only a half of the differene between simulated p* values and alulated urves. The unertainty and dispersion in the simulated data is also evident if we ompare p* data with those obtained in a smaller range of T* by Johnson and Gubbins 15 FIG. 5. Plot of g ij (i,j A,B) vs redued distane for a mixture of A B with: AB ( A B )/2, B / A 2, X A 0.5, BB / AA 0.5, T* kt/ AA 1, and * 3 AA g ij #2 in Table III., moleular dynamis omputer simulation data; 10, present alulations by Eqs. 6, 9 12 with parameters of Table III for g ij #2.

6 E. Matteoli and G. Ali Mansoori: Radial distribution funtions of fluids 4677 in simulation experiments using the more aurate Niholas equation of state. Binary mixtures Ely and Huber 10 have obtained a large olletion of very detailed simulation data of g ij of binary mixtures of Lennard-Jones fluids. These data refer to B / A 2 and to a variety of molar frations, of AA / BB, of density and temperature. In order to examine the behavior of Eqs. 6a, 6b, 9 12 as to their apability of reproduing g ij of mixtures, among Ely and Huber s data we have hosen some at X A 0.5 and X A 0.95, and for eah molar fration two different sets orresponding to different AA / BB ratios. This hoie allows a wide variation in the spetrum of the RDF shape here examined. To apply Eq. 10, the volume molar frations were alulated through: X VA X A 3 A / X A 3 A X B 3 B 13 whih in turn allows alulation of r M and hene of y in Eq. 12. Table III ollets the values of the parameters of Eqs. 6a and 6b for all 12 g ij s onsidered, obtained by a least squares routine, as well as the standard deviation. The fairly good reprodution of the simulation data is evident from inspetion of Fig. 5. Worth mentioning is also the fat that the values of Table III are not very different from 2, whih is the expeted value if Eq. 10 were stritly obeyed. The flexibility of Eqs. 6a, 6b, 9 12 guaranteed by the adjustable parameters should also allow their appliation to multiomponent mixtures, as long as appropriate onditions similar to Eqs are introdued. Extension of this treatment to all Ely and Huber s data is in progress, in order to get a full parametrization of the mixture RDF equation, and to determine possible relationships between these parameters and those of the single omponents. This subjet will be overed in a future paper, together with an extensive alulation via Eq. 5 of the Kirkwood Buff integrals of Lennard-Jones fluid mixtures, and from these, by means of the Kirkwood Buff theory, of the hemial potential and partial molar volume, to be ompared with those of real mixtures of apolar and polar ompounds. CONCLUSIONS It is shown that the simple parametri equation for RDF of fluids and fluid mixtures here proposed is able to reprodue the shape of the simulated RDF of Lennard-Jones fluids and mixtures at different values of the redued temperature and density with good auray. The expression ontains seven adjustable parameters, eah of whih an be expanded in terms of temperature and density by appropriate funtional forms, eah ontaining three additional parameters. By using a parametrization obtained from literature simulated RDF data of Lennard-Jones fluids, values of redued internal energy and pressure alulated by integration agree fairly with literature data for the same type of fluids. Thanks to its good fitting apability, the expression lends itself to be used as a onvenient tool for the alulation of relevant thermodynami properties one a good reprodution of experimental or simulated RDF data has been obtained. ACKNOWLEDGMENT This researh is supported in part by the NSF Grant No. CTS T. L. Hill, Statistial Mehanis MGraw-Hill, New York, I. R. MDonald, Equilibrium theory of liquid mixture, in Statistial Mehanis, edited by K. Singer The Chemial Soiety, London, UK, D. A. MQuarrie, Statistial Mehanis Harper and Row, New York, Moleular-Based Study of Fluids, edited by J. M. Haile and G. A. Mansoori Amerian Chemial Soiety, Washington D.C., E. Z. Hamad and G. A. Mansoori, Fluid Phase Equilibria 54, Flutuation Theory of Mixtures, Vol. II of Advanes in Thermodynamis, edited by E. Matteoli and G. A. Mansoori Taylor and Franis, New York, A. Ben-Naim, Water and Aqueous Solutions, Introdution to a Moleular Theory Plenum, New York, R. D. Gianotti, J. Wang, I. L. Mlaughlin, and M. Silbert, Mol. Phys. 78, J. G. Kirkwood and F. P. Buff, J. Chem. Phys. 19, M. L. Huber and J. F. Ely, NIST Tehnial Note #1331, L. Verlet, Phys. Rev. 165, S. Goldman, J. Phys. Chem. 83, J-D. Li, Y-G. Li, J-F. Lu, and T. Teng, Fluid Phase Equilibria 55, L. Verlet, Phys. Rev. 159, J. K. Johnson and K. Gubbins, Mol. Phys. 77,