NONADDITIVITY OF ORIENTATIONAL INTERACTIONS AND PERTURBATION THEORY FOR DIPOLE HARD SPHERES INTRODUCTION. G. B. Litinskii UDC :536.

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1 Journal of Structural Chemistry. Vol. 51, No. 6, pp , 010 Original Russian Text Copyright 010 by G. B. Litinskii NONADDITIVITY OF ORIENTATIONAL INTERACTIONS AND PERTURBATION THEORY FOR DIPOLE HARD SPHERES G. B. Litinskii UDC : The procedure of approximate summation is applied to a series of many-body perturbation theory for the internal energy of a dipolar hard sphere (DHS) fluid to produce an expression similar to the mean field approximation in the Widom Rowlinson penetrable sphere model, which is well consistent with the experiment at moderate or high densities. Similar results are obtained from the hindered rotation model generalized for arbitrary density. The critical parameters, P c and T c of both models are consistent with the data of machine experiments and are close to the parameters of the percolation transition in the DHS system. Keywords: thermodynamic perturbation theory, nonadditivity of interactions, dipolar hard sphere fluid, Widom Rowlinson model. INTRODUCTION There are two approaches to describe the properties of polar fluids: one is based on quasi-chemical models such as the theory of association equilibria (AE) [1, ], and the other, on the strict equations of the theory of fluids the integral equation method and thermodynamic perturbation theory (TPT) [3, 4]. In quasi-chemical models, most of the results are analytical; however, this is achieved by an a priori choice of the structure of the associates representing the polar fluid. Here the underlying model is a chain cluster structure based, for rare systems (polar fluids), on the data of machine experiments [5, 6]. The structure of the associates is determined by a set of chemical equilibrium constants that are, as a rule, unknown, and the associates themselves are considered to be rigid (kinetically independent) because only their translational degrees of freedom are taken into account. These approximations correspond to a strongly-interacting rarefied system (fluid) and not a normal polar fluid, in which the internal (rotational and vibrational) degrees of freedom of clusters cannot be neglected. For dense polar fluids another approach related to AE theory (hindered rotation model (HRM)) was proposed in [7, 8]. In this (cellular) model, the polar fluid is simulated by an infinite chain (percolation cluster) of particles bonded by dipolar forces, and their rotational dynamics is simulated by the conformational dynamics of the chain. The core of the model is the rotational and vibrational degrees of freedom of particles; therefore, it can be applied to dense fluids at sufficiently low temperatures. As regards the strict methods, their main weakness is low accuracy and computational complexity: even a simplest model of the polar system of a DHS fluid can give only numerical results, which hampers their physical interpretation. The only exception is the mean sphere approximation (MSA); however, its accuracy leaves much to be desired. The same refers Kharkiv Polytechnic Institute, National Technical University, Kharkiv, Ukraine; litinskii@yandex.ru. Translated from Zhurnal Strukturnoi Khimii, Vol. 51, No. 6, pp , November-December, 010. Original article submitted January 10, /10/ Springer Science+Business Media, Inc. 1081

2 to thermodynamic perturbation theory (TPT), although an accurate and simple empirical formula (the Pade approximant of Rushbrooke, Stell, and Hoye (RSH)) has been derived here [9, 10], however, it proves to be inapplicable at sufficiently low temperatures and densities. An interesting attempt to generalize TPT by combining it with AE theory is made in [11] in order to include the contribution of branched clusters [1, 13] to the free energy of a DHS fluid. This resulted in a better consistency with the experiment at low temperatures; however, it deprived the theory of its main advantage: the analytical representation of the thermodynamic functions (TFs) of the system. Moreover, the approximations used in [11] and the RSH approach [9] underlying the calculations in [11] are based on some assumptions (the superposition approximation for the triple correlation function, the method of dividing the potential into the basis and associated parts), which makes the theory largely semiempirical. Furthermore, the very idea of the presence of branched clusters in a three-dimensional DHS fluid is currently not generally accepted [14] because they have been observed so far only in two-dimensional systems. This paper considers an alternative [9, 11] method of the TPT generalization based on approximate summation of the many-body perturbation theory series for the internal energy of a DHS fluid. We have previously used this approach to validate the Dieterici equation of state [15]. The obtained analytical formula for the DHS internal energy resembles the respective expression in the Widom Rowlinson model [16] in the mean field approximation. Moreover, we get a simple heuristic generalization of HRM expanding its applicability area to any density. Both models yield similar values for TFs of the DHS system and are well consistent with the available data of machine experiments at medium and high densities. The critical parameters of the models are close to the corresponding values of percolation transitions in the system under study and well agree with the experimental data. TPT AND NONADDITIVITY OF DIPOLE INTERACTIONS A calculation of TFs of fluids within TPT is based on dividing the pair potential into the repulsive (reference) and attractive (perturbation) parts and results in series by powers of the perturbation parameter (inverse temperature). The reference of the DHS fluid is a hard sphere (HS) system, for which there are sufficiently accurate expressions for TFs, and the dipole contribution to the free energy has the form [3, 9] Fd 3 f( ) m f3( ) m (1) N where f m and f 3 m 3 are the second and third order TPT corrections; = 1/kT is the inverse temperature; = N 3 /V and m = d / 3 are the reduced density and the square of the dipole moment; and is the HS diameter; 1 (0) 3 d 4 0 f ( ) u (1) g ( r) d r; () 1 (0) d d d f ( ) [ u (1) u (13) u (3)] g ( r, r ) d rd r; (3) (0) (0) g and g are the two- and three-body correlation functions of the HS basis system; the line above a fragment of the 3 formula stands for averaging over the angular parameters included into the potential of the dipole-dipole interaction u d (ij) of the particles i and j. It is easy to derive the corresponding series for the internal energy of the system from the thermodynamic relation U = m(f/m) U N d 3 3 f ( ) m 3 f ( ) m. (4) 108

3 The coefficients f i () are determined by the properties of the reference system; the studies [9, 10] have obtained quite simple explanations for these coefficients; however, the calculations of TFs of a DHS fluid from series (1) and (4) have shown them to be invalid. Moreover, these series do not satisfy the Onsager inequality [3], which sets the lower limit for the free energy of the DHS system at m due to which the work [9] has proposed the simplest Pade approximation of series (1) F d / N 4 m, (5) RHS d F m f ( ), (6) N 1 m [ f ( )/ f ( )] 3 satisfying inequality (5) and agreeing with the data of machine experiments. The orientation means of various products of the potentials of dipole-dipole interactions in () and (3) have a simple physical meaning: they represent two- and three-body potentials of the orientation forces [17]. Thus, for example, 60 1 d 6 R1 u ( R ) u (1), (7) 3 0 cc 1 c3 1 3 d d d R1R13R3 (1 3 ) u ( R, R, R ) u (1) u (13) u (3), (8) where kt is the energy parameter characterizing the rotational motion of the molecule as a whole; 0 = m/3 is its orientation polarizability; i are the cosines of the internal angles of the triangle formed by particles 1, and 3; and R ij is the distance between them. These potentials are completely similar to the respective potentials of dispersion interactions [18], whose parameters and have the same meaning yet are connected with the electron subsystem the ionization potential I and the electron polarizability e of the particle. Thus, the polar fluid somehow imitates a non-polar fluid: orientation, not dispersion, forces (pair (7), three-body (8), etc.) are working here with the only difference that these orientation interactions are significantly nonadditive since the orientation polarizability depends on T and can take any values. It would be natural to choose the following relation as the parameter for the extent of nonadditivity of the interactions in the system: where u n are the potentials of n-body forces (n 3) [19]. For physical (dispersion) interactions u n ( /R 3 ) n [18, 0] and u / u, (9) n n 3 n n R ~( / ) 1, (10) since 3 (volume of the molecule); R is the average distance between the particles. Thus, they can be considered to be pair-additive because the contribution of the many-body forces is negligible. An elementary estimation of this parameter for chemical interactions [19] results in an opposite inequality 1. (11) n The parameter n depends on the valence state of the interacting particles (the structure of the associates formed in the system); thus, it can be considered as the parameter of specificity of the interactions: it is the presence of strong (comparable to pair interactions) many-body interactions that determines the unique properties of the chemically reacting systems. For orientation interactions (7) and (8), the parameter n has the form (10), but the orientation polarizability of 0, which is a component of these expressions, depends on T so that inequality (11) holds at sufficiently low temperatures, and the system behaves like an associated liquid. 1083

4 It is the strong nonadditivity of the orientation forces (rather than the anisotropic nature or higher energy of dipole interactions) that underlies the success of the various versions of AE theory and invalidity of the first two orders of TPT in the description of polar fluids. INTERNAL ENERGY OF THE DHS FLUID For an adequate description of the DHS fluid at sufficiently low temperatures it is necessary to consider all the many-body interactions (TPT orders), which is usually made by constructing a Pade approximant either for the free (1) or internal (4) energy of the system [1]. Presentation of TFs as a geometric progression is not based on any physical considerations; it only ensures the fulfillment of Onsager s inequality (5). In this regard, it seems appropriate to search for approaches that consider the properties of specific interactions when summing the TPT series. An example of this approach is the method of Carnahan and Starling [], who have obtained a highly accurate equation of state for a HS fluid by approximating the seven known virial coefficients of this system by a simple algebraic relation, or the method used in [15] to derive the Dieterici equation of state. Let us consider a many-body expression for the internal energy of the system U 1 un (, m), (1) N n! n where u n (, m) is the contribution of n-body interactions. For the DHS fluid, the first two terms in this series are the same as the respective terms of series (4); it is easy to see that a good approximation for them is where z = m is the parameter of the DHS interaction, and 3 3 u (, z) ( kz) / ; u (, z) ( kz) /, (13) k (1 ) f ( ) (14) 9 is the parameter characterizing the interaction with the surrounding particles: at 0, it gives an approximate (accurate to m ) value of the second virial DHS coefficient, and at finite, it approximates the function f () [9]. Let us derive an expression for the internal energy of the DHS fluid by expanding relations (13) to arbitrary n and summing series (1) WR Ud 1kz exp( kz), N which is similar to the corresponding formula of the Widom Rowlinson (WR) penetrable sphere model in the mean field approximation [16] U WR 1exp( ), (16) N where is the depth of the potential well of the penetrable sphere. The distinctive feature of these formulas is that expression (15) contains the interaction parameter z instead of. It is easy to obtain an expression for the free energy of the DHS fluid by integrating (U/m) from (15) with respect to m Fd ln( kz) Ei1 ( kz) kz, (17) N where = is the Euler constant; Ei 1 (kz) is an integral exponential function. Before comparing formula (15) to the available experimental data, note that it (like standard TPT formulas (1), (4), (6)) is a product of some function G(z) depending only on the cell interaction parameter z and the function 1/. In the same (15) 1084

5 way (multiplying by 1/), we get the TF of the attractive interactions in van der Waals theory from the respective functions of the Lennard Jones Devonshire model. Thus, we have a very simple opportunity to expand the cell model of hindered rotation [7, 8] to medium and low densities, and the corresponding fluid equation of the second approximation of HRM for the internal energy of DHS fluid has the form () l Ud / N [ zl ( zl LK I ( x){ z(1 L ) LK}/)]/, (18) where the functions L(z), K(z), and I(x) are given in [7, 8], and the parameter = determines the contribution of the second neighbors in the chain. The calculations of the DHS fluid internal energy by formulas (15) and (18) (curves and 3) are shown in Figs. 1 and, where they are compared to the data of machine experiments [5, 6, 13, 3, 4] (circles) and the internal energy obtained by differentiating Pade approximant (6) (curves 1) RSH RSH Ud Fd 1 1. N N (19) 1 m [ f3( )/ f( )] Fig. 1a-e presents the temperature dependences of the internal energy of DHS fluids with the density = 0.1, 0., 0.4, and 0.8 in a wide temperature range (m < 13). The figure shows that a decrease in the density worsens the consistency of all the models ( RSH (19), WR (15) and HRM (18)) with the experiment, with the maximum deviation being at low temperatures. In this region, the modified HRM (18) proves to be the closest to the experiment (curves 3), and the worst result is observed in TPT-based models (curves 1 and ). Fig. presents the dependences of U d /N on the density for a weakly polar (m = ) and strongly polar (m = 1.5) DHS fluid. In the first case (Fig. ), the best result is provided by the standard perturbation theory (19) (curve 1) and TPT modified in [11] (curve 6) while, in the strongly polar system (Fig. b), none of the models is consistent with the experiment. It is only TPT [11] and HRM (18) that give a qualitative picture of the experimental data, yielding saturated curves (3 and 6), while TPT (19) and the WR model (15) result in monotonously decreasing dependences (curves 1 and ). This is due to the inadequacy of formula (14) at low temperatures. If we assume that, at sufficiently large m stable associates form in the DHS fluid, with the average particle coordination number in the associate being weakly dependent on density, then instead of (14) we should use its maximum value corresponding to 1 k 4 /9, (0) because the idea of the multiplier (1+) in the function k (14) is exactly this average coordination number. This choice of k considerably improves the consistency of WR model (15) with the experiment (curves 4) at low temperatures and 0. (Figs. 1b-d and b): this model is practically a full match of the experiment, being much more accurate than TPT [11] and HRM (18). At the same time, for a weakly polar fluid (Fig a), the choice of k as in (0) is inadequate: curve 4 largely underestimates the internal energy. These questions are discussed in more detail in [5]; here, we will only briefly review the reasons for this inconsistency. As shown above, the parameter n for the orientation forces is set by expression (10); hence, inequality (11), which is typical of chemical (associated) systems, is fulfilled at m 3. (1) Since the density of the fluid is 1, this means that the DHS fluid demonstrates a chemical behavior in a strongly polar system only at m > 3. The chemical behavior means that attractive interactions make a significant contribution to the fluid 0 structure, and using HS (the function g () r in relations (), (3), and (14)) as a reference system in chemical (strongly polar) systems proves to be insufficient. By the way, note that this fact is considered by the authors of [11] in the development of their version of TPT. 1085

6 1086 Fig. 1. Dependence of the internal energy of a DHS fluid on the reduced square of the dipole moment m = d / 3 with the density = N 3 /V = 0.1 (a), 0. (b), 0.4 (c), and 0.8 (d). Curves: (1) RSH (19); () WR model (15), (14); (3) HRM (18); (4) WR model (15), (0); (5) HRM (1). Circles stand for the data of the MC method [5, 6, 13, 3, 4]. At the same time, for a weakly polar (physical) DHS system (Fig. a), the structure is formed by HS, and relation (14) proves to be a good approximation. As concerns HRM (18), which, as it would seem, takes an a priori account of the system chemism, the reason for its inconsistency to the experiment in the strongly polar case (Fig. b) is its quasi-unidimensionality: the fluid is modeled by a linear chain, and the possibility of the formation of branched associates is not considered at all. HRM (18) can be generalized for this branched case, as it is done in [8, 6] for quadrupole hard sphere fluids, i.e., by adding to formula (18) the analogous summand for the transverse chain multiplied by the weight factor of the probability of branching () l () t d Ud Ud U p. () N N N

7 Fig.. Dependence of the internal energy of a DHS fluid with m = d / 3 = on the density = N 3 /V (curves: (1-5) as in Fig. 1, (6) TPT [11]; circles stand for the data of the MC method [3, 4]) (a); m = 1.5 (curves: 1-6 as in Fig. a; circles stand for the data of the MC method [6]) (b). TABLE 1. Critical Parameters of the DHS Fluid Model T Z c Model T Z c HRM (p = 0) RSH (19) HRM (p = 0.5) TPT [11] WR Model MC [13] (15), (0) MC [8] The parameter for () t d / U N is obtained in [7] and expressed through the modified zero and first order Bessel functions I n (z/). In the case of 0.5, the thus obtained dependence (curve 5) is very similar to curve 4 of the WR model (Figs. 1b-d and b). For all the models under consideration, we have calculated the critical point parameters: the density, temperature T c = 1/m c, and compressibility factor Z c = P c V c /NkT c. They are given in Table 1 together with the TPT parameters [11] and experimental data [13, 8]. The table shows that the critical temperature and density in HRM are weakly dependent on the branching parameter p and are close to the values obtained in [8], although the critical compressibility factor appears to be overestimated 1.5- times. Thus, from the standpoint of HRM, the phase transition in the DHS fluid is a transition from linear to branched clusters, as thought in [1, 11-13]. As concerns the WR model (15), its critical density and compressibility factor are closer to the data in the work [13] while T is the same as in HRM. It is noteworthy that the critical density and temperature in the WR model, which considers the DHS system as a simple fluid with isotropic (averaged over orientations) potentials (7), (8), etc. acting between its molecules, are almost the same as the critical parameters of the percolation transition in the orientation averaged DHS system [9]. However, the critical parameters of HRM prove to be similar to the parameters of the analogous transition in the normal DHS system. This supports the conclusion made in [8] that the transition to the DHS system has a percolation nature. 1087

8 CONCLUSIONS In conclusion, let us emphasize two main results of this work. First, the similarity of the two above models (WR (15) and HRM (1)) shows that, in principle, it is possible to obtain (and validate) the equations of quasi-chemical models from the firstprinciples, i.e., by considering all the many-body potentials of the mean force acting between the particles of the fluid. Thus, these models could be included into the strict statistical theory of condensed matter. Second, the relationship between the equations of many body TPT DHS system (15) and Widom Rowlinson penetrable sphere model (16) appears to be interesting and promising because, as far as we know, this model has so far had no real applications. However, this relationship calls for further investigation due to the aforementioned difference: the WR model of DHS fluid (15) involves the interaction parameter z rather than the density. Of special interest is the question which system is, in the case of the modified WR model (15), the analog of a two-component mix of nonadditive hard spheres [30], which is shown in [16] to be equivalent to the WR model. The answers to these questions would allow one, inter alia, to understand the nature of the phase transition in dipole systems, and we hope to discuss them in our further papers. REFERENCES 1. P. I. C. Teixeira, J. M. Tavares, and M. M. Telo da Gama, J. Phys.: Cond. Mat., 1, R411 (000).. A. Yu. Zubarev and L. Yu. Iskakova, Zh. Éksp. Teor. Fiz., 107, No. 5, 1534 (1995). 3. I. R. Yukhnovskii and M. F. Golovko, Statistical Theory of Classical Equilibrium Systems [in Russian], Naukova dumka, Kiev (1980). 4. N. A. Smirnova, Molecular Theories of Solutions [in Russian], Khimiya, Leningrad (1987). 5. J.-M. Caillol, J. Chem. Phys., 98, No. 1, 9835 (1993). 6. D. Levesque and J. J. Weis, Phys. Rev., E49, No. 6, 5131 (1994). 7. G. B. Litinskii, J. Struct. Chem., 45, No. 1, 83 (004). 8. G. B. Litinskii, Teplofiz. Vys. Temp., 48, No. 1, 3 (010). 9. G. S. Rushbrooke, G. Stell, and J. S. Hoye, Mol. Phys., 6, No. 5, 1199 (1973). 10. B. Larsen, J. C. Rasaiah, and G. Stell, Mol. Phys., 33, No. 4, 987 (1977). 11. Y. V. Kalyuzhnyi, I. A. Protsykevytch, and P. T. Cummings, Euro. Phys. Lett., 80, No. 5, 5600 (007). 1. T. Tlusty and S. A. Safran, Science, 90, No. 11, 138 (000). 13. P. J. amp, Shelley, and G. N. Patey, Phys. Rev. Lett., 84, No. 1, 115 (000). 14. A. Yu. Zubarev and L. Yu. Iskakova, Zh. Éksp. Teor. Fiz., 13, No. 5, 1160 (007). 15. G. B. Litinskii, Teplofiz. Vys. Temp., 3, No. 5, 867 (1985). 16. B. Widom and J. S. Rowlinson, J. Chem. Phys., 5, No. 4, 1670 (1970). 17. Yu. V. Gurikov, Zh. Fiz. Khim., 64, No. 3, 63 (1990). 18. I. G. Kaplan, Introduction to the Theory of Intermolecular Interactions [in Russian], Nauka, Moscow (198). 19. G. B. Litinskii, Theoretical Chemistry. II. Quantum Chemistry [in Russian], KhITV, Kharkiv (005). 0. G. B. Litinskii, Zh. Fiz. Khim., 70, No. 3, 39 (1996). 1. G. N. Patey and J. P. Valleau, J. Chem. Phys., 61, No., 534 (1974).. N. F. Carnahan and K. E. Starling, J. Chem. Phys., 51, No., 635 (1969). 3. K.-C. Ng, J. P. Valleau, G. M. Torrie, and G. N. Patey, Mol. Phys., 38, No. 3, 781 (1979). 4. D. Levesqu, G. N. Patey, and J. J. Weis, Mol. Phys., 34, No. 4, 1077 (1977). 1088

9 5. G. B. Litinskii, Vestn. Khark. Univ. Khim., No. 895, issue 18, 6 (010). 6. G. B. Litinskii, J. Struct. Chem., 47, No (006). 7. G. B. Litinskii, Vestn. Khark. Univ. Khim., No. 437, issue 3, 59 (1999). 8. A. F. Pshenichnikov and V. V. Mekhonoshin, Pis ma Zh. Éksp. Teor. Fiz., 7, No. 4, 61 (000). 9. D. Laria and F. Vericat, Phys. Rev. A, 43, No. 4, 193 (1991). 30. J. L. Lebowitz and D. Zomick, J. Chem. Phys., 54, No. 8, 3335 (1971). 1089

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Phase transitions of quadrupolar fluids Seamus F. O Shea Department of Chemistry, University of Lethbridge, Lethbridge, Alberta, Canada, T1K 3M4 Girija S. Dubey Brookhaven National Laboratory, Upton, New