12363
J. Phys. Chem. 1995,99, 12363-12366
Equations of State for Hard-Sphere Chains Richard J. Sadus Computer Simulation and Physical Applications Group, School of Computer Science and Software Engineering, Swinbume University of Technology, PO Box 21 8, Hawthorn, Victoria 3122, Australia Received: March 6, 1995; In Final Form: May 15, 1995@
Recent improvements to thermodynamic perturbation theory equations of state are reformulated to yield a simplified equation of state for hard-sphere chains (STF'T-D). The properties of the hard chain can be obtained solely from the hard-sphere compressibility, the hard-sphere site-site correlation function, and an empirical parameter calculated from molecular simulation data for the second virial coefficient of hard-sphere chains. The STPT-D equation is applied to the prediction of both the compressibility of 4-, 8-, 16-, 51-, and 201-mer hard-chains and the second virial coefficients of up to 128-mer chains. Comparison with molecular simulation data indicates that the STPT-D equation generally predicts both the compressibility and the second virial coefficient more accurately than other equations of state.
Introduction Hard-chain equations of state are potentially an important starting point for developing accurate equations of state for macromolecules. Historically, attention has been focused on using accurate hard-sphere models to represent the repulsive interactions of geometrically simple molecules. For example, the Carnahan-Starling' hard-sphere equation is the basis of many successful equations of state for real fluids? More recently, progress has been achieved in depicting the properties of hard nonspherical bodies. Boublik3 used Kihara's4 concept of a hard convex body (HCB) as the basis of an equation of state for nonspherical hard bodies. Equations of state have been developed for hard dumbbells5and assemblies of fused spheres.6 Flory' s concept of excluded volume has been incorporated7 in a reasonably accurate hard-chain equation of state. Chiew* developed a rigorous theory for representing hard chains which has been r e ~ e n t l yformulated ~.~~ into a useful equation of state for polymers. Wertheim' proposed a thermodynamic perturbation theory (TPT) which accommodates hard-chain molecules. The TPT model is the basis of the successful statistical associating fluid theoryI2 (SAFT) for real molecules. Re~ e n t l y , ' ~the . ' ~accuracy of thermodynamic perturbation theory has been improved by incorporating structural information for the diatomic fluid. For practical applications such as the prediction of phase equilibria, it is important to minimize the complexity of the equation of state to reduce computation time. Many of the proposed hard-chain equations of state are probably too complicated to be the basis of equations of state for real macromolecules. In this work, we propose a simple empirical improvement to recent developments in thermodynamic perturbation theory which reduces the complexity while retaining the theoretical basis of the equation of state. We demonstrate that the resulting simplified equation of state also improves the prediction of both the compressibility and the second virial coefficients of large hard-sphere chains obtained from molecular simulation. Development of a Simplified TPT Hard-Chain Equation of State Chapman et al.I3-l5have generalized Wertheim's TPT model to obtain the following equation of state for the compressibility
(Z)of a hard-chain of m segments:
where gHS(a) is the hard-sphere site-site correlation function at contact, a is the hard-sphere diameter, = nmqa3/6 is the packing fraction, and e is the number density. The compress) be accurately determined from ibility of hard spheres ( P Scan the Camahan-Starling (CS) equation:
For the Camahan-Starling equation, the site-site correlation function is
(3) Recently, Ghonasgi and ChapmanI3and Chang and SandlerI4 modified TPT for the hard-sphere chain by incorporating structural information for the diatomic fluid. The compressibility of a hard chain can be determined from the hard-sphere compressibility and the site-site correlation function at contact of both hard spheres and hard dimers ( g H D ) :
( +7
(0.5m - 1) 1
alngHD(a)) (4)
av
ChiewI6 has obtained the site-site correlation for dimers:
If eqs 2 , 3 , and 5 are substituted into eq 4,@en the resulting equation of state is
@Abstract published in Advance ACS Abstracts, July 15, 1995.
0022-365419512099-12363$09.00/0 0 1995 American Chemical Society
12364 J. Phys. Chem., Vol. 99, No. 32, 1995
Sadus
Adopting the nomenclature of Chang and Sandler,I4 we will refer to eq 6 as TFT-D1. Chang and Sandler also used a polynomial fit for the dimer site-site correlation function to obtain an altemative to eq 6, which they called TPT-D2. The TPT-D2 equation can be represented as
z" = m('
' - - ") 72
(1
0.5m( 1
VI3
coefficients of hard chains containing up to 128 hard-sphere segments. Comparing these data with the values obtained from eq 12 (c = OS), we find that the average absolute deviation is minimized when a = 0.7666. Consequently, eq 10 becomes
+ (1 - 7 x 2 - 7)
0.76667(2 - m) 1.53327 1 (13)
+
~ ( 3 . 4 9 8- 0.247 - 0 . 4 1 4 ~ ~ ) (1 - 7)(2 - 7)(0.534 0.4147)
+
ComparisonI4 with molecular simulation data indicates that both TPT-D1 and W - D 2 represent the compressibility of 4-, 8-, and 16-mer hard-sphere chains more accurately than either the generalized Flory dimer (GF-D) equation or the original TPT equation of state. The predicted second virial coefficients are also in better agreement with molecular simulation data. In particular, the TFT-D1 equation provides a reasonable representation of the asymptotic behavior of the second virial coefficient for large hard-sphere chains. The observation by Yethiraj and Hall" that the dimer sitesite correlation function can be obtained as a linear function of the hard-sphere site-site correlation function suggests an easy simplification of eq 4. In general, we define
Substituting eq 8 into eq 4, we obtain
(9) Using eq 9, the compressibility of the hard chain can be obtained directly from the hard-sphere properties and two unknown parameters, a and c. If we obtain the hard-sphere properties from eqs 2 and 3, then the equation of state for the hard chain given by eq 9 is
The values of a and c can be obtained by fitting the molecular simulation data for gHS and gHD. By fitting the simulation data for hard diatomic spheres, Yethiraj and HallI7 found that c = 0.534. However, we propose an altemative approach which improves the accuracy of the equation of state. First, the value of c is determined by noting that when 7 = 0, the dimer sitesite correlation function given by eq 5 has a value of 0.5. If we choose c = 0.5, then eq 8 will also yield a limiting value of 0.5. The value of a can be determined from the second vinal coefficient. The second virial coefficient can be obtained from the compressibility by using the relationship
Consequently, the value of a depends on the choice of the hardsphere term. For the compressibility given by eq 10, we obtain
Molecular simulation data are availablet8 for the second vinal
We will refer to this simplified thermodynamic perturbation theory-dimer equation as the STFT-D equation of state.
Comparison with Molecular Simulation Data
.
Compressibility. The compressibility predicted by the STPT-D equation and other equations of state is compared in Table 1 with molecular simulation data for hard chains containing 4, 8, 16, 32,51, and 201 hard-sphere segments. The simulation data for m 5 16 and m = 32 were obtained from Chang and SandlerI4 and Denlinger and Hall,I9 respectively. The calculations for the 51-mer and 201-mer chains were compared with the molecular dynamics data reported by Gao and Weiner.20 The comparison in Table 1 also includes results for the generalized Flory dimer equation7 (GF-D) and an equation of state based on a particle-particle description of chains proposed by Chiewa8 The GF-D equation can be represented by
is the excluded volume of a m-segment molecule calculated from
Ve(m)
* ~e(3+ ) (m - 2)(~e(3)- ~e(2))
Ye(m>
(15)
where Ye(1) = 4 ~ ~ ~~~~1( = 32 9,) ~ ~ ~and ~ 1~ 44 ,3 = ) 9.82605~~. The Camahan-Starling' and Tildesley-Streett5 equations were used for the compressibility of the monomer and dimer, respectively. Chiew's8 expression for an m-segment hard chain can be written as9
and when gHS is calculated from eq 3, we obtain
Equation 17, which will we refer to as Chiew's equation, is of particular interest because it forms the basis of a recentlyg~10 proposed equation of state for polymers. The relative simplicity of Chiew's equation makes it very attractive for computationally intensive applications such as phase equilibria. Song et al?.Io have demonstrated that it can be formulated into a useful equation of state for polymers. However, the data in Table 1 clearly indicate that Chiew's equation is not very accurate. It provides a reasonable prediction of the compressibility only at low densities, and the discrepancy with simulation data increases substantially with chain length. The GF-D equation adequately predicts the compressibility at all densities, but it becomes progressively less accurate as the number of hard-sphere segments in the chain is increased. For m 2 32 it tends to overpredict the compressibility at both low and very high densities, whereas the compressibility at inter-
J. Phys. Chem., Vol. 99, No. 32, 1995 12365
Equations of State for Hard-Sphere Chains
TABLE 1: Comparison of Molecular Simulation Data14J9P with Equation of State Calculations for the Compressibility (2)of m-Hard-Sphere Chains as a Function of Reduced Density @* = Z
e*
simulation Chiew
0.1 0.2 0.3 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
1.49 2.22 3.28 4.84 7.09 8.56 10.26 12.49 15.00 18.26 22.10 27.02 32.49
1.53 2.25 3.25 4.64 6.57 7.81 9.30 11.07 13.20 15.78 18.91 22.74 27.47
0.1 0.2 0.3 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
1.76 2.99 4.91 7.75 11.95 14.83 18.26 22.32 27.14 33.34 40.85 50.64 62.03
1.91 3.18 4.96 7.46 10.98 13.25 15.98 19.25 23.19 27.97 33.80 40.95 49.78
0.1 0.2 0.3 0.4 0.5 0.55 0.6 0.65 0.7 0.75 0.8
2.25 4.47 8.09 13.59 21.96 27.13 34.05 41.90 51.80 65.15 8 1.28
2.67 5.04 8.39 13.11 19.79 24.14 29.34 35.60 43.17 52.36 63.57
0.191 0.382 0.478 0.573
7.08 23.0 37.0 57.6
8.28 22.51 34.15 50.35
0.2 0.3 0.372 0.464 0.590 0.649 0.7 0.8 0.9
11.46 23.04 34.87 56.56 101.04 130.17 160.56 238.12 346.51
13.23 23.36 33.32 50.13 84.41 106.89 131.05 194.00 288.99
36.80 79.44 152.11 256.20 407.16 621.54 927.79 1354.76
48.25 87.52 143.16 224.02 338.00 507.04 752.70 1123.78
0.2 0.3
0.4 0.5 0.6 0.7 0.8 0.9
GF-D TPT-Dl TPT-D2 STF'T-D m=4 1.48 1.50 1.47 1.53 2.21 2.25 2.19 2.30 3.34 3.31 3.40 3.25 4.89 4.97 4.90 4.80 7.17 7.14 7.21 7.02 8.66 8.67 8.61 8.48 10.37 10.46 10.42 10.25 12.62 12.37 12.51 12.52 15.24 14.95 15.06 15.09 18.22 18.42 18.08 18.14 22.31 22.06 21.90 21.92 27.08 26.63 26.78 26.50 32.99 32.62 32.19 32.47 m=8 1.91 1.75 1.83 1.72 3.02 3.13 3.28 2.94 5.00 5.09 5.29 4.83 7.96 7.98 8.20 7.67 12.29 12.20 12.40 11.85 14.99 15.15 15.15 14.62 18.36 18.61 18.47 17.98 22.80 22.46 22.48 22.06 27.44 27.90 27.34 27.02 33.52 34.11 33.26 33.09 41.72 40.98 40.49 40.54 50.18 51.09 49.37 49.74 62.70 61.61 61.15 60.37 m = 16 2.67 2.29 2.48 2.23 4.89 4.45 5.24 4.64 8.60 8.00 9.06 8.39 14.14 14.65 14.09 13.42 22.77 22.32 22.52 2 1.49 28.12 27.75 28.12 26.88 34.35 34.59 34.92 33.44 42.37 43.17 41.43 42.41 52.14 51.91 53.21 51.17 64.11 63.49 65.48 63.1 1 78.82 77.79 77.66 80.53 m = 32 7.90 7.36 6.99 8.62 24.18 24.00 22.68 25.26 36.74 38.46 38.75 39.47 59.94 57.06 59.63 58.95 m = 51 12.64 13.85 11.75 11.09 21.84 23.93 25.53 23.21 33.25 35.67 37.41 35.30 57.96 56.62 53.50 56.27 99.86 100.79 101.72 96.72 129.25 131.90 129.10 125.82 160.00 164.61 160.86 157.46 240.53 250.60 244.60 241.02 362.56 381.44 372.50 368.78 m = 201 45.77 50.72 42.15 39.47 89.66 96.16 86.74 81.15 163.10 155.20 155.98 145.70 263.10 259.67 256.76 245.04 406.69 41 1.48 403.37 390.51 622.41 641.17 625.92 612.13 938.25 979.15 954.73 940.21 1417.13 1493.70 1457.41 1442.30
mediate densities is adequately predicted. Either the TPT-D 1 or TPT-D2 equations provide a good prediction of the compressibility for m I 16. At very high densities, the TPT-D1
TABLE 2: Percentage Average Absolute Deviation (GAD) of the Calculated Compressibility of m-Hard-Sphere Chains ComDared with Molecular Simulation Data AAD (%) m
Chiew
4 8 16 32 51 201
9.02 11.65 13.32 9.84 13.39 16.38
GF-D 1.72 3.58 6.64 10.44 5.37 9.35
TFT-Dl 1.04 1.88 2.63 4.27 2.72 5.89
TPT-D2 0.76 1.78 3.74 5.15 3.26 6.51
S*-D ~
0.85 1.28 1.67 1.08 3.97 3.92
equation tends to slightly under predict the compressibility. The agreement with simulation data for m z 32 is better than can be obtained from the GF-D equation. These equations also over predict the compressibility at the extremes of density. These features are also shared with the STPT-D equation, however, it is more accurate than either the TPT-D1 or TPT-D2 equations at both low and medium densities. To quantify the agreement with simulation data, we have determined the average absolute deviation (AAD) of the compressibility predicted by the various equations, Le.
where N is the number of data. The AAD (%) are summarized in Table 2. The AAD for the 32-mer chain is inconsistent with the rest of the analysis because the comparison was limited to medium densities. It is evident that there is very little difference between the GF-D, TPT-D1, TPT-D2, and STPT-D equations of state for the case of 4-mers. However, the GF-D deteriorates substantially for larger chains compared with only a relatively moderate deterioration in the accuracy of either TPT-D 1 or TPTD2. The STPT-D equation provides best overall agreement between theory and simulation data with an error of 1 2 % for chains with m 5 32. The accuracy of the STPT-D equation of state is also relatively unaffected by chain length for m 5 32. A slight deterioration is observed for larger chains, but the error (3.92%) for the 201-mer is substantially less than can be obtained from the other equations of state (5.89- 16.38%). Second Virial Coefficients. The second virial coefficient is another property that can be used to assess the accuracy of the equation of state. Using eq 11, the second virial coefficient for Chiew's equation can be obtained from
--
(19)
m Yethiraj et aLi8determined the second virial coefficient for the GF-D from the relationship BGF-D 2 --
m2d
1'396 344 m
+ 0.730 458
(20)
Chang and Sandleri4 reported the following relationships for the virial coefficient of the TPT-D1 and TPT-D2 equations of state
* m a 2
2 =
y3'275 +
281 1.112 360) (22) m2d 6 m From eq 12 with a = 0.7666 and c = 0.5, the STPT-D equation
12366 J. Phys. Chem., Vol. 99, No. 32, 1995
Sadus both the TFT-D1and STPT-Dequations of state are reasonably accurate until m FC: 30 and the asymptotic behavior of the virial coefficient for large chains is also more accurately predicted.
TABLE 3: Comparison of MC Second Virial CoefAcient (B2/m2d)Datal8 for m-Hard-Sphere Chains with Various Equations of State Predictions B21m2u3
m
MC
Chiew
2 3 4 6 8 12 16 20 24 32 64 128
1.425 1.147 0.989 0.818 0.7 18 0.608 0.545 0.501 0.472 0.426 0.344 0.282
1.440 1.222 1.113 1.004 0.949 0.895 0.867 0.851 0.840 0.826 0.806 0.796
GF-D TPT-Dl 1.429 1.440 1.091 1.196 0.9 16 1.080 0.742 0.963 0.655 0.905 0.847 0.567 0.818 0.524 0.800 0.497 0.480 0.789 0.774 0.458 0.752 0.425 0.409 0.741
TPT-D2 1.440 1.154 1.011 0.868 0.797 0.725 0.690 0.668 0.654 0.636 0.609 0.596
STPT-D 1.440 1.090 0.912 0.736 0.648 0.560 0.5 16 0.490 0.472 0.450 0.417 0.400
1.6
+ i
1.4
I
Conclusion A method has been presented to deduce a hard-sphere chain equation of state solely from the hard-sphere compressibility, the hard-sphere site-site correlation function, and an empirical parameter deduced from the analysis of second virial coefficients. If the Camahan-Starling equation is used for the hardsphere properties, then the resulting STPT-D equation of state improves the accuracy of both the calculated compressibility and the second virial coefficient of hard-sphere chains. Consequently, the STPT-D equation has the potential to form the basis of an accurate equation of state for polymers. The accuracy of the STFT-D equation could potentially improve the prediction of mixture properties. To apply the method to mixtures, eq 9, which is the starting basis for the STFT-Dequation of state, should be specifically reformulated for mixtures. For example, this could be achieved by applying a method similar to that proposed by Huang and Radosz2' for the extension to mixtures of the SAFT equation of state.
Acknowledgment. This work was supported by an Australian Research Council Small Grant. References and Notes
0
STPT-D
0 0.2 0
I
I
I
I
I
I
I
I
16
32
48
64
80
06
112
128
m
Figure 1. Comparison of the second virial coefficient of the hard-
sphere chain calculated from various equations of state (- - -) with Monte Carlo simulation data'* (0).
of state yields
B Y D
+ 0.7334) m 2 d = -6(- 4'0332 m
(23)
The virial coefficients predicted by the above equations are compared with the Monte Carlo (MC) data of Yethiraj et ala'* in Table 3 and Figure 1. The Chiew, GF-D, and TPT-D2 equations yield reasonable agreement with MC data for m < 10. However, these equations predict much larger values of the virial coefficient than MC results for m > 10. In contrast,
(1) Carnahan, N. F.;Starling, K. E. J. Chem. Phys. 1969, 51, 635. (2) Sadus, R. J. High Pressure Phase Behaviour of Multicomponent Fluid Mixtures; Elsevier: Amsterdam, 1992. (3) Boublik, T. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 1038. (4) Kihara, T. Adv. Chem. Phys. 1963, 5, 147. (5) Tildesley, D. J.; Streett, W. B. Mol. Phys. 1980, 41, 85. (6) Boublik, T. Mol. Phys. 1989, 68, 191. (7) Dickman, R.; Hall, C. K. J. Chem. Phys. 1986, 85, 4108. ( 8 ) Chiew, Y. C. Mol. Phys. 1990, 70, 129. (9) Song, Y.; Lambert, S. M.; Prausnitz, J. M. Macromolecules 1994, 27, 441. (10) Song, Y.; Lambert, S. M.; Prausnitz, J. M. Chem. Eng. Sci. 1994, 49, 2765. (11) Wertheim, M. S. J. Chem. Phys. 1987, 87, 7323. (12) Chapman, W. G.;Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 1709. (13) Ghonasgi, D.; Chapman, W. G.J. Chem. Phys. 1994, 100, 6633. (14) Chang, J.; Sandler, S. I. Chem. Eng. Sei. 1994, 49, 2777. (15) Chapman, W. G.;Jackson, G.;Gubbins, K. E. Mol. Phys. 1988, 65, 1057. (16) Chiew, Y. C. Mol. Phys. 1991, 73, 359. (17) Yethiraj, A.; Hall, C. K. J. Chem. Phys. 1990, 93, 5315. (18) Yethiraj, A.; Honnell, K. G.;Hall, C. K. Macromolecules 1992, 25, 3979. (19) Denlinger, M. A.; Hall, C. K. Mol. Phys. 1990, 71, 541. (20) Gao, J.; Weiner, J. H. J. Chem. Phys. 1989, 91, 3168. (21) Huang, S. H.; Radosz, M.Ind. Eng. Chem. Res. 1991, 30, 1994. JP950638A
