A THREE-COMPONENT MOLECULAR MODEL WITH BONDING THREE-BODY INTERACTIONS

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1 STATISTICAL PHYSICS, SOLID STATE PHYSICS A THREE-COMPONENT MOLECULAR MODEL WITH BONDING THREE-BODY INTERACTIONS FLORIN D. BUZATU Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele 7715, Romania RADU P. LUNGU Department of Physics, University of Bucharest, Bucharest-Magurele 7715, Romania DALE A. HUCKABY Department of Chemistry, Texas Christian University, Fort Worth, Texas 7619, USA DANIELA BUZATU Physics Department, Politehnica University of Bucharest, Bucharest 776, Romania Received February 1, 5 A model is formulated when every bond of a honeycomb lattice is occupied by a rod-like molecule of type AA, BB, or AB, the molecular ends near a common site having three-body and orientation-dependent interactions. The exact coexistence surfaces for the separation into an AA-rich and a BB-rich phase are obtained, and the asymmetric effect of the three-body interactions is studied. 1. INTRODUCTION For ternary solutions, following the simple model of Wheeler and Widom [1], in which each bond of a regular lattice is covered by a rod-like molecule, Huckaby and Shinmi extended the model in order to include temperature effects, by considering first finite two-body interactions between any two molecular ends near a common site [], and then by considering a more general model with bonding and nonbonding states at the molecular ends [3, 4]. These models were extended in order to include three-body interactions, both in the nonbonding case [5, 6] and also in the bonding case [7]. In Section we generalize the model on a honeycomb lattice to include three-body interactions between molecular ends near a common site, each molecular end having many internal states, one of which is bonding [7]. Equations are given for the calculation of the exact surface of coexisting AA-rich and BB-rich phases. The generalized model requires 8 effective adjustable parameters to specify the coexistence surface, so in Section 3 we study a simplified version of the Rom. Journ. Phys., Vol. 5, Nos. 3 4, P , Bucharest, 5

2 418 Florin D. Buzatu et al. model with 5 effective adjustable parameters, each with a physical meaning, one of them being the reduced temperature. The previously studied models with simple states [, 5, 6] are special cases of the simplified model. In Section 4 we calculate some examples of the two-phase coexistence surface for the simplified model that illustrate the roles of the physical parameters, especially the parameters for the three-body interactions.. THE GENERAL MODEL In order to describe a ternary solution having three-body and orientationally dependent interactions between the molecular ends, and using a generalized Wheeler-Widom molecular model on a honeycomb lattice, we consider the following assumptions: A1. Every link connecting two neighboring sites of the lattice is covered by one rod-like molecule of type AA, BB or AB. A. There are three-body interactions between molecular ends in the triangle near each lattice site. A3. Any molecular end (A or B) can be in any one of q A (respectively q B ) states, only one of which is a bond state that can form a bond with another molecular end near the same site, the other q A 1 (respectively q B 1) states being non-bonding states; the bonds occur between AB pairs of molecular ends, when both A and B are in bond states, but we do not consider the possibility of AA and BB bonds. We consider a finite honeycomb lattice with periodic boundaries G t, having N t sites and the associated 3 1 lattice (called Λ t ), with vertices at the molecular ends and links between ends of the same or neighboring molecules [, 5]; the 3 1 lattice is covered by C 3 graphs (the triangles) and C graphs (the links connecting neighboring triangles). In Table 1 are shown the possible three-body energies and numbers of distinct configurations on C 3 triangles formed by the interacting molecular ends (here X and Y means A or B). The microscopic configuration of the whole molecular system (denoted as ξ) is completely specified by the set of the occupancy numbers {P} on all the sites of the Λ t lattice and by the set of internal state types {α} for all the triangles. The grand-canonical Hamiltonian for the generalized model on Λ t having the configuration ξ is H Λ t ( α) X Y Z XYZ Pi Pj Pk X Y XY Pi Pj XYZ,, { AB, } XY, { AB, } ijk,, C ij, C ( ξ ) = ε µ, 3

3 3 A three-component molecular model 419 Table 1 The energies and the number of distinct configurations on a C 3 triangle site configuration energy number of distinct configurations X XXX () ε XXX () XYX ε XYX XYX, XYX XYX XYX 3 q X q q q + 1 total number X Y X ε ( qx 1) of configurations is () εxyx 1 z = q q XYX X Y where P i is the occupancy number on the i position with X-type molecular ( α) end, ε XYZ is the energy in a C 3 graph having molecular ends of types XYZ and (α =, 1, ) bonds, and µ XY is the chemical potential in a C graph for a molecule XY [7]. The grand-canonical partition function of the molecular system is: β Λ ( ξ) Ξ e t Λ =, t H () where the sum is over all possible configurations ξ. We transform the grand-canonical partition function method [7]: ξ Ξ Λ t using the following (i) first we perform a summation over the internal states, and we obtain an averaged Hamiltonian which is formally the same as for a molecular model without internal states, but expressed in terms of temperature-dependent averaged molecular energies; (ii) then we use a spin representation for the occupation numbers, so that the grand-canonical partition function of the molecular model becomes proportional to the canonical partition function of the Ising model on the 3 1 lattice (with three-spin and two-spin interactions); (iii) finally, we perform two successive spin transformations (the first is an inverse star-triangle transformation and the second is a double decorationiteration transformation), resulting in the partition function of the Ising model on the 3 1 lattice being proportional to the partition function on the honeycomb lattice. As a result of these transformations we get: 1 1 3N / [ ( 1 )] t e β KI ( 3 ) K BLL h Z R R L h β,, Ξ I Λ = ( ) t Λ, ;, = ε Z t G K, H, (3) t [ AL (, h)] Nt where we have introduced the following quantities:

4 4 Florin D. Buzatu et al. 4 K l is a function of temperature and depends parametrically on the characteristic energies of the molecular model, the number of internal states and the chemical potentials, but is independent of the spin configurations and therefore has no contribution to a phase transition; () R 3, R, L, h are the reduced parameters corresponding to the Ising model on the 3 1 lattice (they have well defined expressions in terms of the parameters of the molecular model [7]); (3) L 1, h 1, h are the reduced parameters corresponding to the Ising model on the intermediate lattice (these parameters and also the proportionality constants A and B are related to the previous reduced Ising parameters [7]); (4) the partition function of the simple Ising model on the honeycomb lattice is: (4) ZG ( K, H) = exp K S t i Sj + H Si. {} S i, j C i Gt The mole fractions of the original molecular system (X AA, X BB and X AB ) satisfy a conservation equation, and these quantities can be related to the magnetization of the Ising model on the 3 1 lattice MΛ = Si and to the i Λ spin-spin correlation function of the same Ising model on the C graphs σ = SS (both being defined in the thermodynamic limit t ); Λ i j ij, C therefore, we consider the following equations: XAA + XBB + XAB = 1, (5) XAA XBB = M Λ, (6) X 1 AB = (1 σ Λ ). (7) Using the results of the previous papers [5, 6, 7] we can express σ Λ and M Λ as linear combinations of the similar quantities of the simple Ising model (denoted as σ G and M G ): σ Λ = c( τ, z) + cs( τ, z) σ G( z) + cm( τ, z) MG( z), (8) MΛ = m( τ, z) + ms( τ, z) σ G( z) + mm( τ, z) MG( z). (9) We must observe the following specific properties of the previous quantities: since the Ising model on the honeycomb lattice has a ferromagnetic phase transition (K > ) only at zero external field (H = ), and for sufficient large values of the coupling constant (K > K c ), then the expressions for the spin-spin correlation function and the magnetization are needed only at H =, as

5 5 A three-component molecular model 41 functions of the reduced temperature τ and the exponential form of the coupling constant z e K, the two coexisting phases differing in the sign of M G ; () the coefficients c, c s, c m, m, m s, m m have well defined expressions [7], their dependence on the temperature being realized only in terms of the parameters R3( τ ) and R( τ); (3) the characteristic functions of the Ising model on the simple honeycomb lattice (σ G and M G ) have well defined expressions [7]; (4) in the limiting low value of the z variable (z z m ), we get c 1, cs, and cm, so that σ Λ = 1, and thus X AB =, from Eq. (7); since X AB is non-negative, from the last result it follows that z must have an inferior limit z z m, where z m is a specified function of temperature [7]. 3. THE SIMPLIFIED MODEL In order to calculate the two-phase coexistence surface given by Eqs. (5) (9), it is necessary to calculate R and R 3 as functions of temperature. In the general model, R and R 3 depend on 8 effective adjustable parameters that are combinations of the 1 parameters q A, q B, () BAB, ε AAA, ε BBB, ε ABA, ε ABA, () ε ABA, ε BAB, ε BAB, and ε but one of the 8 effective parameters is absorbed in a reduced temperature. To simplify the model by reducing the number of effective adjustable parameters, we first express the 8 energy parameters as an equivalent set of 8 physical meaningful parameters as follows, where X and Y represent either A or B: XXX 3 XX ε = ε, XYX ε = ε + ε, (11) XX XY XYX XX XY XY ε = ε + ε + ε, () () XYX XX XY XY XY ε = ε + ε + ε + ε. (13) The 8 parameters on the right hand side of Eqs. (13) are three-body interaction parameters that become true two-body interaction parameters in the limiting case of two-body interactions. The parameters ε XX are associated with the interaction energy between two molecular ends of type X, and the parameters ε XY ( ε AB in an ABA triangle and ε BA in an BAB triangle) are associated with the interaction energy between molecular ends of type X and Y. (In the two-body interaction limit, ε = ε.) The quantities ε and ε () ε and ε () for an AB BA XY XY ( AB AB

6 4 Florin D. Buzatu et al. 6 ABA triangle and ε BA and ε BA () for an BAB triangle) are associated respectively with the bond energy and with the bond-bond interaction energy in an XYX triangle. We construct a simplified version of the general model that contains the following simplifications: S. 1: qa = qb q, (14) S. : S. 3: AB BA ε = ε ε, (15) () () AB BA 1 ε = ε ε. (16) These three simplifications reduce from 8 to 5 the number of effective adjustable parameters needed to specify R 3 and R. We define the following effective reduced parameters in terms of the energy the reduced temperature a parameter for single-bonding ε εab + εba εaa εbb ; τ 4, βε κ 1 1 ε ε, a three-body bond-bond interaction parameter δ ε ε, and an asymmetric three-body interaction parameter 3( εab εba). ε (17) (18) (19) () In terms of q and the reduced parameters κ 1, δ, and, the dependence of R 3 and R on τ is given by R 3 =, τ R 1 1 ln c c e c e τ 4 where κ κ 1+δ and 4κ/τ 4κ/τ 1 = + 1 +, ()

7 7 A three-component molecular model 43 c = 1 + 1, c =, c 3 =. q q q q q3 If κ 1 = and δ =, the simplified model reduces to the previously studied three-body interaction model without bonding [5, 6], and it further reduces to the two-body interaction model if κ 1 =, δ =, and =. 4. EXACT COEXISTENCE SURFACES The exact surface of coexisting AA-rich and BB-rich phases can be calculated in terms of the mole fraction variables X AB and X AA X BB, and the reduced temperature τ. The parametric form of this surface, in terms of τ and z, is given by Eqs. (5) (9). The surface consists of two branches, one corresponding to the AA-rich phase and the other to the BB-rich phase, the two branches coinciding on the critical line. Fig. 1. Curves on the coexistence surface for q = 6, =, κ 1 = 75,. κ = 13.. In our previous paper [7] we have discussed many interesting properties of the possible types of coexistence surfaces. In the following we shall present an interesting effect, due to the asymmetry of the coexistence surface, in the case of strong bonding, when the system has phase separation only at intermediate temperatures. In order to see this effect we consider two cases having the same values for q = 6, κ 1 = 75,. and κ = 13,. but one with = (when the surface is symmetric) and the other with = 1 (when there is asymmetry); for both cases we have determined the locus of pairs of coexisting phases on each isothermal section, using tie lines and the level rule. For our model, with the coexistence surface parametrized by τ and z, pairs of points representing coexisting phases

8 44 Florin D. Buzatu et al. 8 (one on the AA-rich branch and the other on the BB-rich branch) have the same values of τ and z. We have plotted, in Fig. 1 for the symmetric case and in Fig. for the asymmetric case, the following curves on the coexistence surface: the critical curve (which has a maximum at a double critical point), an isotherm at the temperature of the double critical point, the closed loop corresponding to phase equilibrium of the two-component system (when X AB = ), a second constant X AB loop, and the coexistence curve obtained by heating the phase at the lower critical point of this constant X AB loop. For the symmetric case, by beginning with the single phase at a lower critical point (when XAA = XBB ) and then heating the system, maintaining the same overall mole fractions for the three components, we generate a closed loop of coexisting phases (as is illustrated in Fig. 1) that ends in a single phase at an upper critical point. Because of symmetry, X AB is constant in all the phases on the loop. Fig.. Curves on the coexistence surface for q = 6, = 1, κ 1 = 75,. κ = 13.. In the asymmetric case, we begin again with the initial phase at a lower critical point and heat the system, maintaining constant overall mole fractions. Then, the overall composition of the system evolves along the straight line parallel to the τ axis that contains the lower critical point. Now, using tie lines for each temperature, we find the pairs of points representing coexisting phases. The critical curve in Fig. is located at positive values of XAA XBB, and each phase on the AA-rich branch of a coexistence curve has a lower X AB than the coexisting phase on the BB-rich branch. Because the line parallel to the τ axis, that contains the initial phase at the lower critical point, intersects the AA-rich

9 9 A three-component molecular model 45 branch, the process will end in a single phase at this point on the AA-rich branch, the coexisting phase on the BB-rich branch vanishing. Thus the locus of coexisting phases obtained by heating the critical phase results in an open curve, as is shown in Fig.. Since X AB is unequal in a pair of coexisting phases in the asymmetric case, the closed curves of the coexistence surface with constant X AB (the dotted loop in Fig. ) are not closed-loop diagrams of coexisting phases, as was true in the symmetric case. Acknowledgement. This research was supported by the Robert A. Welch Foundation, grant P-446, and the research project CERES C REFERENCES 1. J. C. Wheeler and B. Widom, J. Am. Chem. Soc. 9, 364 (1968).. D. A. Huckaby and M. Shinmi, J. Stat. Phys. 45, 135 (1986). 3. D. A. Huckaby and M. Shinmi, J. Chem. Phys. 9, 5675 (1989). 4. D. A. Huckaby and M. Shinmi, J. Stat. Phys. 6, 347 (199). 5. D. L. Strout, D. A. Huckaby and F. Y. Wu, Physica A 173, 6 (1991). 6. F. D. Buzatu and D. A. Huckaby, Physica A 99, F. D. Buzatu, R. P. Lungu, and D. A. Huckaby, J. Chem. Phys. 11, 6195 (4).