Comparison of different mixing rules for prediction of density and residual internal energy of binary and ternary Lennard Jones mixtures

Fluid Phase Equilibria 178 (2001) 87 95 Comparison of different mixing rules for prediction of density and residual internal energy of binary and ternary Lennard Jones mixtures Jian Chen a,, Jian-Guo Mi a, Kwong-Yu Chan b a Department of Chemical Engineering, Tsinghua University, Beiing 100084, PR China b Department of Chemistry, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China Received 15 November 1999; accepted 27 September 2000 Abstract Mixing rules are necessary when equations of state for pure fluids are used to calculate various thermodynamic properties of fluid mixtures. The well-known van der Waals one-fluid (vdw1) mixing rules are proved to be good ones and widely used in different equations of state. But vdw1 mixing rules are valid only when molecular size differences of components in a mixture are not very large. The vdw1 type density-dependent mixing rule proposed by Chen et al. [1] is superior for the prediction of pressure and vapor liquid equilibria when components in the mixture have very different sizes. The extension of the mixing rule to chain-like molecules and heterosegment molecules was also made with good results. In this paper, the comparison of different mixing rules are carried out further for the prediction of the density and the residual internal energy for binary and ternary Lennard Jones (LJ) mixtures with different molecular sizes and different molecular interaction energy parameters. The results show that the significant improvement for the prediction of densities is achieved with the new mixing rule [1], and that the modification of the mixing rule for the interaction energy parameter is also necessary for better prediction of the residual internal energy. 2001 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Mixing rule; Lennard Jones mixtures 1. Introduction The Lennard Jones (LJ) interaction potential has been widely used in the study of molecular fluid structure and thermodynamic properties of fluids. It is also the main reference fluid to study more complex fluids with perturbation theories [2,3]. For a pure Lennard Jones fluid, the interaction potential u(r) is expressed as [ (σ ) 12 ( σ ) ] 6 u(r) = 4ε (1) r r Corresponding author. Tel.: +86-10-62784540; fax: +86-10-62770304. E-mail address: c-dce@mail.tsinghua.edu.cn (J. Chen). 0378-3812/01/$20.00 2001 Elsevier Science B.V. All rights reserved. PII: S0378-3812(00)00478-7

88 J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 in which there are two parameters, σ and ε, which are parameters for the molecular diameter and the molecular interaction energy, respectively and r is the distance between centers of two molecules. For a mixture of Lennard Jones fluids, parameters between different molecules should be defined. In computer simulation, the following Lorentz Berthelot combining rules are widely used: σ i = 1 2 (σ i + σ ) (2) ε i = ε i ε (3) in which i and refer to different molecules. But in the expression of thermodynamic properties of a mixture with simple equations of state, mixing rules producing effective one component parameters are usually necessary. The mixing rules are very important for better prediction and calculation of the properties of fluid mixtures which are widely encountered in chemical engineering. For LJ mixtures, the van der Waals one-fluid (vdw1) type of mixing rules [4] are still the best and the most widely used mixing rules up to present, as pointed by Harismiadis et al. [5] and Tsang et al. [6] σ 3 = i εσ 3 = i x i x σ 3 i x i x ε i σ 3 i in which x is the mole fraction of a component, and i and refer to kinds of molecules. Different mixing rules have been proposed and widely used also for real fluids. But as a theoretically based mixing rule, the comparison with computer simulation data is very important, because the computer simulation can give exact results for a given set of molecular interaction parameters. The vdw1 mixing rules with Eqs. (2) (5) are exactly valid only when the molecular size parameters of component in a mixture are almost equal to each other. When differences of molecular sizes becomes large, they are not satisfactory over the whole range of density. Chen et al. [1] proposed a new density-dependent mixing rule. With this mixing rule, the pressure of hardsphere mixtures of different sizes, Henry constant and vapor liquid equilibria of Lennard Jones mixtures of different molecular sizes can be predicted with higher accuracy than those with other mixing rules. A special example of the mixing rule has been used in cubic equations of state combined with a g E mixing rule and perfect results have been received for asymmetric systems, such as methane + n-alkane (C 3 C 44 ) and ethane + n-alkane (C 3 C 44 ) [7]. The extension of the new mixing rule to chain-like and heteronuclear polyatomic molecules are also satisfactory [8]. In this paper, different mixing rules are compared further with simulation data of the density and the residual internal energy for binary and ternary Lennard Jones mixtures. 2. Theory In general, when Eqs. (2) (5) are used for mixtures, the following vdw1 mixing rule is obtained for the diameter parameter: σ 3 = ( ) σi + σ 3 x i x (6) 2 i (4) (5)

J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 89 which uses Eq. (2) as in computer simulation. This mixing rule can give good prediction for Lennard Jones mixture over a large size difference range only when the density tends to be 0, i.e. near the density of a gaseous phase. But when the density becomes larger and tends to be near the density of a liquid phase, the prediction accuracy also tend to be worse. For mixtures with a large density of a liquid phase, the following mixing rule is preferred: σ 3 = ( ) σ 3 i + σ 3 x i x (7) 2 i But Eq. (7) is not satisfactory when the density tends to be 0. In the previous paper [1], a density-dependent mixing rule was proposed for the diameter parameter σ 3 = i ( σ 3n i x i x in which n = 1 3 + 2 3 ξ ξ = πρ x i σi 3 6 i + σ 3n 2 ) 1/n (8) ξ is the packing factor and ρ the molecular number density of a mixture. In this paper, the different mixing rules are compared further with the simulation data of Shukla and Haile [9,10] and Fotouh and Shukla [11] for the density and the residual internal energy of binary and ternary Lennard Jones mixtures. This kind of comparisons should be beneficial to better understanding of the effect of the molecular size and the molecular interaction to mixture properties and will lead more suggestions for better expression of mixture properties. For pure Lennard Jones fluids, modified Benedit Webb Rubin (MBWR) equation of state of Nicolas et al. [12] with modified parameters of Johnson et al. [2] is still used here as in the previous paper [1]. This equation shows good agreement with computer simulation data, not only for PVT and the residual internal energy but also for critical data. For clarity, detailed equations of MBWR are not listed here. (9) (10) 3. Results 3.1. Binary Lennard Jones mixtures Shukla and Haile [9] published computer simulation data for the density and the residual internal energy of binary Lennard Jones mixtures with different molecular size parameters (1 σ 2 /σ 1 2) while molecular interaction parameters were kept unchanged for different pairs of molecules. Examples of the prediction results with different mixing rules and the simulation data are listed in Table 1 for mixtures σ 2 /σ 1 = 1 2. In the Table, A.R.D. are the total average relative deviations for all 87 data points of computer simulation by Shukla and Haile [9]. The results show that when the diameter ratio is near 1, the prediction deviations of all three mixing rules are small. But when the diameter ratio tends to be 2, Eq. (6) always overestimates the density

90 J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 Table 1 Examples of prediction results for the density and the residual internal energy with different mixing rules for binary equimolar Lennard Jones mixtures (σ = 3.405 Å, ε/k = 119.8K,ε 22 = ε 11 = ε) σ 11 /σ σ 22 /σ ρσ 3 U res /Nε M.D. a % Eq. (6) % Eq. (7) % Eq. (8) M.D. a % Eq. (6) % Eq. (7) % Eq. (8) kt/ε = 1.0, Pσ 3 /ε = 0.5 1.00 1.25 0.5319 0.88 0.79 0.19 5.38 0.20 0.32 0.25 1.00 1.50 0.3762 2.72 2.32 0.40 5.48 0.82 1.23 0.96 1.00 1.85 0.2400 6.06 4.04 0.73 5.54 3.28 4.19 3.51 1.00 2.00 0.2008 7.56 4.55 0.84 5.58 4.04 5.15 4.29 kt/ε = 2.0, Pσ 3 /ε = 1.2 1.00 1.25 0.4091 0.66 0.57 0.20 3.72 0.19 0.73 0.31 1.00 1.50 0.3142 1.91 2.10 0.15 4.07 0.83 2.17 1.10 1.00 1.75 0.2418 3.47 3.73 0.19 4.34 1.81 3.73 2.17 1.00 2.00 0.1875 5.48 4.83 0.40 4.50 3.88 6.04 4.26 kt/ε = 3.0, Pσ 3 /ε = 2.5 1.00 1.25 0.4133 0.29 0.84 0.19 3.38 0.92 0.37 0.83 1.00 1.50 0.3232 1.60 2.13 0.23 3.68 0.13 1.36 0.32 1.00 1.75 0.2527 3.15 3.64 0.68 3.87 1.51 2.92 1.73 1.00 2.00 0.1992 4.76 5.01 1.40 3.94 3.44 4.31 3.60 A.R.D. b 3.0 2.3 0.9 1.6 2.2 1.7 a From Shukla and Haile [9]. b A.R.D. is the total average relative deviation for all 57 points of Shukla and Haile [9]. and meanwhile Eq. (7) underestimates the density. At the same time, the density-dependent mixing rule Eq. (8) gives quite good prediction results over the whole range of the diameter ratio. For the prediction of the residual internal energy, the new mixing rule Eq. (8) gives almost the same results as Eq. (6) gives, and slightly better than those of Eq. (7). And the results show that the mixing rule Eq. (5) needs a further modification for better expression of the residual internal energy of mixtures with large molecular size differences. Two examples of the prediction for the excess volume are shown in Figs. 1 and 2. Figs. 1 and 2 show that the prediction accuracy with the mixing rule Eq. (8) is obviously better than those with other two mixing rules. At the same time, two examples of the prediction for the excess enthalpy are shown in Figs. 3 and 4. The results show that there is no distinct difference between the results of different mixing rules. The deviations for the excess enthalpy are mainly from the prediction deviations for the residual internal energy. A deviation of 3 5% for the residual internal energy can result in a deviation of 100% for the excess enthalpy and even wrong sign of the excess enthalpy. In Table 2, the comparison is showed for the mixtures of different concentrations with the diameter ratio of 2. For all points, the density-dependent mixing rule Eq. (8) is much better than the other two mixing rules Eqs. (6) and (7) for the prediction of the density. But for the residual internal energy, no obvious improvement is achieved as before.

J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 91 Fig. 1. The prediction results for the excess volume with different mixing rules for binary Lennard Jones mixtures (kt/ε = 1.0, Pσ 3 /ε = 0.5 in Table 1). Diamond with error bars: molecular simulation data [9]; dashed line: Eq. (6); dashed-dotted line: Eq. (7); and solid line: Eq. (8). Shukla and Haile [10] published further computer simulation data for binary Lennard Jones mixtures with different molecular interactions and different molecular sizes. Examples of the results are shown in Table 3. The conclusion for the prediction of the density is the same as in Table 1. The results for the residual internal energy show also that the mixing rule Eq. (5) for interaction energy parameter needs Fig. 2. The prediction results for the excess volume with different mixing rules for binary Lennard Jones mixtures (kt/ε = 2.0, Pσ 3 /ε = 1.2 in Table 1). Diamond with error bars: molecular simulation data [9]; dashed line: Eq. (6); dashed-dotted line: Eq. (7); and solid line: Eq. (8).

92 J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 Fig. 3. The prediction results for the excess enthalpy with different mixing rules for binary Lennard Jones mixtures (kt/ε = 1.0, Pσ 3 /ε = 0.5 in Table 1). Diamond with error bars: molecular simulation data [9]; dashed line: Eq. (6); dashed-dotted line: Eq. (7); and solid line: Eq. (8). Fig. 4. The prediction results for the excess enthalpy with different mixing rules for binary Lennard Jones mixtures (kt/ε = 2.0, Pσ 3 /ε = 1.2 in Table 1). Diamond with error bars: molecular simulation data [9]; dashed line: Eq. (6); dashed-dotted line: Eq. (7); and solid line: Eq. (8).

J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 93 Table 2 Prediction results for the density and the residual internal energy with different mixing rules for binary Lennard Jones mixtures (σ 11 = 3.405 Å, ε/k = 119.8K,σ 22 /σ 11 = 2.0, ε 22 = ε 11 = ε) kt/ε Pσ 3 11 /ε x 1 ρσ 3 11 U res /Nε M.D. a % Eq. (6) % Eq. (7) % Eq. (8) M.D. a % Eq. (6) % Eq. (7) % Eq. (8) 1.0 0.5 0.25 0.1461 4.25 2.00 0.78 5.82 2.88 3.43 2.97 1.0 0.5 0.50 0.2008 7.56 4.55 0.84 5.58 4.04 5.15 4.29 1.0 0.5 0.75 0.3190 8.85 6.44 0.72 5.37 3.78 5.10 4.22 3.0 2.5 0.25 0.1521 2.63 2.62 0.81 4.01 2.56 2.17 2.53 3.0 2.5 0.50 0.1992 4.76 5.01 1.40 3.94 3.44 4.31 3.60 3.0 2.5 0.75 0.2885 5.07 6.43 1.49 3.70 2.14 5.33 2.73 A.R.D. 6.46 5.28 1.15 3.52 4.81 3.79 a From [9]. Table 3 Examples of prediction results for the density and the residual internal energy with different mixing rules for binary equimolar Lennard Jones mixtures (σ = 3.405 Å, ε/k = 119.8 K) ε 11 /ε ε 22 /ε σ 11 /σ σ 22 /σ ρσ 3 U res /Nε M.D. a % Eq. (6) % Eq. (7) % Eq. (8) M.D. a % Eq. (6) % Eq. (7) % Eq. (8) kt/ε = 2.0, Pσ 3 /ε = 1.2 1.0 1.0 1.000 1.500 0.3147 1.74 2.26 0.01 4.08 0.58 1.92 0.85 1.0 1.3 1.000 1.500 0.3266 2.16 2.13 0.25 5.03 1.43 2.31 1.55 1.0 1.5 1.000 1.500 0.3372 2.69 1.87 0.62 5.99 2.18 2.63 2.15 1.0 1.8 1.000 1.500 0.3475 2.99 1.79 0.77 6.97 2.68 2.76 2.51 1.0 2.0 1.000 1.500 0.3571 3.15 1.81 0.80 7.98 2.74 2.49 2.43 kt/ε = 2.0, Pσ 3 /ε = 2.5 1.0 1.0 1.000 1.000 0.6641 0.00 0.00 0.00 4.11 0.00 0.00 0.00 1.0 1.3 1.000 1.000 0.6862 0.09 0.09 0.09 4.87 0.08 0.08 0.08 1.0 1.5 1.000 1.000 0.7077 0.29 0.29 0.29 5.64 0.08 0.08 0.08 1.0 1.8 1.000 1.000 0.7264 0.27 0.27 0.27 6.41 0.19 0.19 0.19 1.0 2.0 1.000 1.000 0.7428 0.15 0.15 0.15 7.19 0.49 0.49 0.49 1.0 1.0 1.000 1.125 0.5786 0.02 0.36 0.14 4.29 0.02 0.13 0.04 1.0 1.3 1.000 1.125 0.5946 0.13 0.27 0.04 5.11 0.53 0.62 0.54 1.0 1.5 1.000 1.125 0.6099 0.18 0.23 0.00 5.96 0.32 0.39 0.33 1.0 1.8 1.000 1.125 0.6255 0.01 0.42 0.18 6.80 0.32 0.37 0.32 1.0 2.0 1.000 1.125 0.6383 0.10 0.34 0.10 7.63 0.38 0.40 0.37 1.0 1.0 1.000 1.500 0.3770 1.88 2.37 0.39 4.69 1.11 1.93 1.23 1.0 1.3 1.000 1.500 0.3809 2.60 1.82 0.24 5.67 2.24 2.76 2.23 1.0 1.5 1.000 1.500 0.3869 2.68 1.89 0.23 6.68 2.36 2.59 2.23 1.0 1.8 1.000 1.500 0.3917 2.96 1.74 0.44 7.67 2.66 2.63 2.41 1.0 2.0 1.000 1.500 0.3965 3.13 1.69 0.53 8.67 2.69 2.42 2.34 A.R.D. b 1.92 1.37 0.54 2.59 2.74 2.58 a From Shukla and Haile [10]. b A.R.D. is the total average relative deviation for all 88 points of Shukla and Haile [10].

94 J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 Table 4 Examples of prediction results for the density and the residual internal energy with different mixing rules for ternary equimolar Lennard Jones mixtures ε 22 /ε 11 ε 33 /ε 11 σ 22 /σ 11 σ 33 /σ 11 ρσ 3 11 U res /Nε 11 M.D. a % Eq. (6) % Eq. (7) % Eq. (8) M.D. a % Eq. (6) % Eq. (7) % Eq. (8) kt/ε 11 = 3, ρσ11 3 \ε 11 = 2.5 1 1 1 1 0.5150 0.28 0.28 0.28 2.941 0.28 0.28 0.28 1 1 1.05 1.125 0.4664 0.27 0.06 0.19 3.145 0.32 0.20 0.30 1 1 1.125 1.25 0.4144 0.39 0.37 0.08 3.357 0.42 0.05 0.36 1 1 1.25 1.5 0.3295 0.96 1.58 0.26 3.682 0.37 0.49 0.24 1 1 1.5 1.75 0.2458 2.13 2.36 0.35 3.929 0.78 1.64 0.90 1 1 1.75 2 0.1867 3.24 3.17 0.74 4.011 2.34 2.68 2.40 1.75 2 1 1 0.5987 0.28 0.28 0.28 5.863 0.62 0.62 0.62 1.75 2 1.05 1.125 0.5322 0.02 0.28 0.12 6.299 0.02 0.05 0.01 1.75 2 1.125 1.25 0.4628 0.44 0.49 0.05 6.738 0.56 0.69 0.55 1.75 2 1.25 1.5 0.3547 1.67 1.32 0.23 7.439 1.67 1.85 1.55 1.75 2 1.5 1.75 0.2558 3.02 2.08 0.22 8.078 3.02 2.64 2.51 1.75 2 1.75 2 0.1890 4.44 2.67 0.12 8.467 4.77 3.57 3.79 A.R.D. b 2.3 1.8 0.4 3.4 3.3 3.2 a From Fotouh and Shukla [11]. b A.R.D. is the total average relative deviation for all 48 points of Fotouh and Shukla [11]. further modifications for better expression of the residual internal energy for mixtures where size ratio deviates from 1.0 significantly. This modification should also take into account the effect of the size asymmetry. 3.2. Ternary Lennard Jones mixtures Fotouh and Shukla [11] published the computer simulation data for the density and the residual internal energy of ternary Lennard Jones mixtures. Examples of the prediction results with different mixing rules are shown in Table 4. The results also show that for ternary Lennard Jones mixtures, the mixing rule for the molecular size pays a very more important role for the prediction of the density. The old mixing rules Eqs. (6) and (7) are only valid in the two limit regions of the density. Meanwhile, the new one Eq. (8) is valid for the whole density range. From the prediction results for the residual internal energy, it is also concluded that the modification of Eq. (5) is necessary for better expression of the residual internal energy for mixtures with different molecular sizes. 4. Conclusions The vdw1 type of mixing rules are compared with computer simulation data in literature for the density and the residual internal energy of binary and ternary Lennard Jones mixtures. The results show that the mixing rule proposed by Chen et al. [1] for the mixture size parameter is a good improvement for the prediction of the mixture density. The results for the residual internal energy show that the mixing rule

J. Chen et al. / Fluid Phase Equilibria 178 (2001) 87 95 95 for the interaction parameter needs a further modification, which should pay attention to the effect of molecular size asymmetry. List of symbols k Boltzmann factor (J K 1 ) n power parameter in the mixing rule of Chen et al. [1] r distance between centers of two molecules (Å) u molecular interaction potential (J) U internal energy (J mol 1 ) x mole factions of components Greek letters ε molecular interaction energy parameter (J) ρ molecular number density (Å 3 ) σ molecular interaction diameter parameter (Å) ξ packing factor Superscript res residual properties Subscripts i, kind of molecules References [1] J. Chen, J.F. Lu, Y.G. Li, Fluid Phase Equilib. 132 (1997) 169 186. [2] J.K. Johnson, J.A. Zollweg, K.E. Gubbins, Mol. Phys. 78 (1993) 591 618. [3] J.K. Johnson, E.A. Muller, K.E. Gubbins, J. Phys. Chem. 98 (1994) 6413 6419. [4] T.W. Leland, J.S. Rowlinson, G.A. Sather, Trans. Faraday Soc. 64 (1968) 1447 1460. [5] V.I. Harismiadis, N.K. Koutras, D.P. Tassios, A.Z. Panagiotopoulos, Fluid Phase Equilib. 65 (1991) 1 18. [6] P.C. Tsang, O.N. White, B.Y. Perigard, L.F. Vega, A.Z. Panagiotopoulos, Fluid Phase Equilib. 107 (1995) 31 43. [7] J. Chen, Z.C. Li, Tsinghua Sci. Technol. 1 (1996) 410 415. [8] J. Chen, J.F. Lu, Y.G. Li, Fluid Phase Equilib. 140 (1998) 37 51. [9] K.P. Shukla, J.M. Haile, Mol. Phys. 62 (1987) 617 636. [10] K.P. Shukla, J.M. Haile, Mol. Phys. 64 (1988) 1041 1059. [11] K. Fotouh, K. Shukla, Chem. Eng. Sci. 52 (1997) 2369 2382. [12] J.J. Nicolas, K.E. Gubbins, W.B. Streett, D.J. Tildesley, Mol. Phys. 37 (1979) 1429 1454.