Znd. Eng. Chem. Res. 1995,34, 4553-4561
4553
Equations of State for Mixtures of Square-Well Molecules: Perturbation and Local Composition Theories for Binary Mixtures of Equal-Sized Molecules Costas P. Bokist and Marc D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218-2694
In this paper two classes of equations of state for binary mixtures of equal-sized square-well spherical molecules are compared. The first class is perturbation theory, and the second includes several models based on local composition arguments. It is shown that second-order perturbation theory generally is in better agreement with configurational energy and compressibility factor simulation data than the local composition models. However, perturbation theory is written in a form that requires three different mixing rules. A new, simple equation of state for square well mixtures is proposed. This model is a closed-form version of perturbation theory which has the exact second virial coefficient behavior for both pure components and mixtures. This model also converges to the mean-field limit at the correct rate, and it provides very good agreement with Monte Carlo simulation data over the entire density range.
Introduction The square-well potential, though a highly idealized model, is of considerable interest for the development of molecularly-based equations of state since it contains the essential features of more realistic potential models (i.e., it describes both the repulsive and the attractive forces between molecules). In addition, accurate computer simulations (Rotenberg, 1965; Alder et al., 1972; Rosenfeld and Thieberger, 1975)and perturbation theory calculations (Barker and Henderson, 1967a) are available for this model. The square-well potential is used in several thermodynamic models (Ponce and Renon, 1976; Beret and Prausnitz, 1975; Donohue and Prausnitz, 1978). Further, due to its simplicity, it provides a convenient way to apply statistical mechanical principles to the development of practical equations of state. Much research has been devoted to the development of accurate models to predict the properties of pure square-well systems. However, other than the mixture perturbation theory by Henderson (19741, little has been done until recently to treat mixtures of square-well molecules. Most recent work has been devoted t o the development of thermodynamic models using the concept of local compositions (Lee et al., 1986; Lee and Sandler, 1987; Lee and Chao, 1987, 1988; Guo et al., 1990a, 1990b;Wang and Wang, 1990). Recent computer simulation results for square-well mixtures (Lee and Chao, 1987; Wang and Wang, 1990) allow evaluation of these models. Perturbation theory originally was proposed by Zwanzig (19541, Rowlinson (1964), and McQuarrie and Katz (19661, who performed a perturbation expansion on the canonical partition function of the system and on the configurational integral in particular. The total potential is separated into two parts: the potential of the reference or unperturbed system and the perturbation potential. The reference system usually is taken to be the hard-sphere system. The Helmholtz free energy is obtained as a power series in inverse temperature, with the coefficients being functions of density. Barker and
* Author to whom correspondence should be addressed.
Present address: Aspen Technology, Inc., Ten Canal Park, Cambridge, MA 02141. +
0888-5885/95/2634-4553$09.00/0
Henderson (1967a, 1967b) developed the first generally successfulperturbation theory for liquids. They derived expressions for the first-order term, which involves twobody interactions, and the second-order term, which involves three-body and four-body interactions as well. Donohue and Prausnitz (1978)used perturbation theory to develop the perturbed hard-chain theory (PHCT); PHCT is valid for pure and mixture square-well fluids over a large range of densities, temperatures, and compositions. The main idea in local composition theory (originally used by Scott (1956) and Wilson (1964))is that the local concentration around a central molecule can be different from the average concentration. Therefore, one tries to model this local environment and its dependence on density, temperature, and composition, as well as molecular size and energy. Local composition theory has led to the development of density-dependent localcomposition mixing rules for cubic equations of state (Mollerup, 1981). The purpose of this paper is to compare calculations using perturbation theory and several models based on local composition theory with Monte Carlo simulation data for square-well binary mixtures of equal-sized molecules. Calculations also are presented for a new model developed recently (Bokis and Donohue, 19951, which is simpler than perturbation theory and more accurate than the local composition models.
Perturbation Theory In virial theory, the compressibility factor of a fluid,
2,is written as an infinite series of reciprocal volume
z = 1+-+B,+,+ C v
u
D u
...
(1)
where u is the molar volume and B,C, D , ... are the virial coefficients and are functions of temperature and composition but are independent of density. In perturbation theory, the compressibility factor of a fluid is written as an infinite series of reciprocal temperature
0 1995 American Chemical Society
4554 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
or, equivalently, for the Helmholtz free energy
(3) where T is the absolute temperature. In perturbation theory, the first term, A(O) or Z(O),accounts for the molecular repulsions, and the remaining terms correct for molecular attractions. The first attractive term also is called the mean-field term, because it does not take into account the nonrandomness of the fluid. Chen et al. (1969) derived the perturbation expansion for multicomponent mixtures of spherical molecules in terms of integrals which include the hard-sphere radial distribution function, go,and the pair intermolecular potential energy, r. In principle, the thermodynamic properties of multicomponent mixtures can be calculated if these integrals are evaluated. However, the Chen et al. (1969) model contains a number of unknown terms, and, therefore, it is not directly applicable. Henderson (1974) overcame this difficulty by considering mixtures where the molecules have the same size. With this assumption, Henderson obtained for the firstorder perturbation term
1 =- p a x
2
i
E xixj -k Eij
(4)
j
where p is the number density, u is the molecular diameter, xi is the mole fraction of component i, u is the pair intermolecular potential, go(12) is the hard-sphere radial distribution function, and the integral Z is given by
I = su’(12) g0(12)drf2
(5)
where u’ is the reduced perturbation potential (u’= (u - u0)/c)and d r ’ 2 = 4n(r/0)~ d(r/o). While the first-order term depends only on two-body interactions, the second-order term depends on two-, three-, and four-body interactions. Henderson (1974) obtained the following expression for the second-order term
A‘2’
-=
NkT
-
x
1
4
i
j
2 ‘ij
xpj -
k2
The integral J3 is a complicated term involvingg0(1234) and need not be specified (Henderson, 1974). The integrals I and J1 have been evaluated as functions of the density (Barker and Henderson, 1971). However, the integral J Z is more difficult to calculate, since it involves the termgO(123)which generally is not known accurately. Henderson (1974) used the superposition approximation, originally proposed by Kirkwood (19351, which is valid at low densities and gives
(9) Thus, the integral JZ can be calculated. Smith et al. (1970) fitted this integral to a polynomial in density. Donohue and Prausnitz (1978) were among the first to develop an equation of state for real fluid mixtures based on perturbation theory. They derived the composition dependence of the individual terms in the perturbation expansion by combining the mixing rule obtained from lattice theory with Henderson’s (1974) nonrandomness corrections for spherical molecules. While the first-order term in perturbation theory was found to have a simple quadratic composition dependence, this was not the case for the higher-order terms. As shown above, the second-order term in the perturbation expansion requires three mixing rules, even for mixtures of spherical molecules having the same diameter (Henderson, 1974). These additional mixing rules in the second-order term correct the random mixing approximation for clusters of two and three molecules. Donohue and Prausnitz (1978) also made approximations for the third- and fourth-order perturbation terms, including only the corrections for clusters of two molecules (these clusters provide the dominant correction for mixture nonrandomness). The analytic forms of these mixing rules are given by Donohue and Prausnitz (1978). Figure 1 shows calculations for the internal energy of three square-well binary mixtures using first- and second-order perturbation theory. The symbols represent data from the Monte Carlo simulations by Lee and Chao (1987). First-order perturbation theory provides the high-temperature limit and underpredicts the simulation data for all three cases shown in Figure 1. The second-order perturbation terms correct for nonrandomness and give a much better agreement with the simulations. In these calculations, we have omitted the higher-order perturbation terms. This does not introduce any significant error. However, we need to point out that the higher-order terms become important at low densities, especially in the second virial coefficient region. Figure 2 shows calculations for the compressibility factors for the same three square-well binary mixtures. Once again, the symbols represent data from Monte Carlo simulation (Lee and Chao, 1987). One sees that perturbation theory gives a very good agreement with the data. First- and second-orderperturbation theories give nearly identical results at moderate and high densities. However, at low densities the differences become much more pronounced. At these very low densities, perturbation theory converges slowly and, therefore, several terms are required for accurate calculations. The conclusion that can be drawn from these results is that perturbation theory is a rigorous thermodynamic model that can predict the behavior of square-well binary mixtures with very good accuracy, without any adjustable parameters. Grundke et al. (1973) compared
Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995 4555
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-12
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Figure 1. Internal energy calculations using first- and secondorder perturbation theory and comparison with Monte Carlo simulation data (Lee and Chao, 1987): (a) KTIEl1 = 2.0,~ & 1 1 = 2.0,~ 1 2 / ~ 1=1 f i ,X I = 0.204;(b) KTIEii = 2.0,4 ~ 1 =1 2.0,~ 1 2 / ~ 1 1 = 4% XI = 0.796; (c) kTltl1 = 3.0, ~ 2 2 / ~ 1=1 3.0, ~ 1 2 / ~ 1=1 4,x1 = 0.500.
perturbation theory calculations for Lennard-Jones mixtures against Monte Carlo simulation and found that the agreement between theory and simulation also was very good. The disadvantages of perturbation theory are that it is a truncated series of an infinite expansion, it requires several different mixing rules, and, therefore, it is rather complicated for practical calculations. In the next section, we will examine a series of models based on the concept of local composition theory. These models are closed-form equations, and generally they are less complicated than perturbation theory.
Local Composition Theory The premise of local composition theory is that the composition in the immediate vicinity of a molecule can be different from the bulk composition of the fluid, due to differences in intermolecular energies and molecular sizes in a mixture. Further, this local composition depends on thermodynamic variables (such as temperature, density, and composition) and structural variables (such as molecular size and shape) (Lee et al., 1986). Wilson (1964) was among the first to derive a thermodynamic model for fluid mixtures based on local composition theory. He considered a binary solution of components 1 and 2, whose molecules occupy cites on a lattice. He then focused attention on a central molecule of type 1and wrote the probability of finding a molecule of type 2, relative to finding a molecule of type 1,about this central molecule. This ratio is expressed in terms
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,
, , ,
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Figure 2. Compressibility factor calculations using first- and second-order perturbation theory and comparison with Monte Carlo simulation data (Lee and Chao, 1987): (a)KTIE11 = 2.0, ~ 2 2 / €11 = 2.0,~ 1 2 / ~ i= i f i ,X I = 0.204;(b) KTIE11 = 2.0,E Z ~ C I=~2.0, ~ i d ~=i A, i XI = 0.796; (c) kTIE11 = 3.0, ~ 2 d ~ 1=1 3.0, ~ 1 2 / ~ 1=1 8, X I = 0.500.
of the overall mole fractions and two Boltzmann factors, i.e.
_ -- x2 exp(-A2,1RT)
3c21
x11
x1 exp(-A,,/RT)
(10)
where x21 and x11 are the local mole fractions, and A21 and A11 are related t o the interaction energies of a 2-1 and 1-1 pair of molecules, respectively. Wilson then replaced the overall volume fractions used in the expression for the excess Gibbs free energy of Flory (1942)and Huggins (1942)by the local volume fractions and, thus, obtained the following expression E
RT = -xl ln(xl + A l g 2 ) - x 2 1n(A21x1+ x2)
(11)
where gE is the excess Gibbs free energy, and the quantities A12 and A21 are related to the molar volumes of the components and the differences A12 - A11 and A12 - 222. Wilson's model was able to give expressions for the activity coefficients as a function of composition and, more importantly, an estimate of the variation of the activity coefficients with temperature. Equation 11 provides a reasonable representation of excess Gibbs free energies for a variety of miscible systems; however, it is not able to predict limited miscibility (Prausnitz et al., 1986). The basic idea in Wilson's derivation follows from the concept of local composition. This concept also was used by Renon and Prausnitz (1968) in their derivation of the NRTL (nonrandom, two-liquid) equation, which
4556 Ind. Eng. Chem. Res., Vol. 34, No. 12, 1995
contains three adjustable parameters. The NRTL model, unlike Wilson’s, can predict partial miscibility as well as complete miscibility. Later, Abrams and Prausnitz (1975) derived an equation which extends the quasichemical theory of Guggenheim (1952) for nonrandom mixtures to solutions containing molecules of different size. Their expression is called UNIQUAC (universal quasichemical theory). In UNIQUAC, the excess Gibbs free energy is written as a sum of two contributions: a combinatorial part and a residual part. The combinatorial term attempts to describe the dominant entropic contribution and is determined only by the composition and by the sizes and shapes of the molecules. The residual term is due primarily to the intermolecular forces that are responsible for the enthalpy of mixing. UNIQUAC contains two adjustable parameters, which appear only in the residual part of the equation. One of the major disadvantages of these activity coefficient models (such as Wilson, NRTL, and UNIQUAC) is that they take into account liquid-phase nonidealities only, ignoring nonideal behavior in the vapor phase. This limitation of the activity coefficient models leads to poor prediction of multicomponent phase equilibria in systems where the vapor phase exhibits strong nonidealities, such as mixtures containing carboxylic acids, alcohols, etc. (Prausnitz et al., 1986). Since the pioneering work of Flory (1942), Huggins (1942), and Guggenheim (1944a, 1944b, 19521, lattice models have been used to describe the properties of polymer solutions, melts, and blends. Generally, it is assumed that the molecular partition function can be separated into an athermal term and a thermal term. The athermal (or repulsive) term corresponds to the combinatorial entropy and the thermal (or attractive) term corresponds t o the energy plus the noncombinatorial entropy. Cui and Donohue (1992) used Guggenheim’s (1944a, 1944b, 1952) random mixing entropy term for the athermal part of the Gibbs free energy. For the thermal (energetic) term, they proposed a closedform expression which assumes that the energy of a mixture is given as the double summation of the product of surface fractions times pair interaction energies, weighted by their Boltzmann factors. To normalize these Boltzmann factors, this was divided by a double summation of the weighting factors. This model is called NRSF (nonrandom, surface fraction) theory. In the limiting case of a mixture of equal-sized monomer molecules, the surface fractions reduce to the mole fractions, and the NRSF model reduces to the following simple expression
where z represents the lattice coordination number. A similar way of normalizing the Boltzmann factors was developed by Lipson (1991), who derived a contemporary version of the integral equation theory of BornGreen-Yvon (BGY, Born and Green, 1946;Yvon, 1935). The BGY expression for the energy of a mixture of monomers is given by
N
2y-1
Yx:E
Equations 12 and 13 represent two similar ways of normalizing the Boltzmann factors with composition. NRSF uses the ratio of a double summation over a double summation, whereas BGY uses a summation of the ratio of two single summations. Both methods yield similar results for the energy of a mixture (Cui and Donohue, 1992). The local composition models described above (Wilson, NRTL, UNIQUAC, NRSF, and BGY) were developed for mixtures whose components occupy sites on a lattice. A different approach that also is based on the concept of local composition is that of an equation of state. Usually, local composition equations of state are expressed in terms of the “coordination number”, i.e., the average number of molecules inside the well of a particular molecule. The coordination number is related to the generalized van der Waals partition function of the fluid; therefore, one can use a model for the coordination number to derive the partition function and other thermodynamic properties (Lee et al., 1985; Guo et al., 1990b; Bokis et al., 1992). The generalized van der Waals partition function for a mixture can be expressed as (Guo et al., 1990b)
Q = ~ Q ~
(14)
i
where Qi is the partition function of component i, given by
Here, Ni is the number of molecules of component i, Vf is the free volume, Ai is the de-Broglie wavelength of component i, qr,qv,and qe are the rotational, vibrational, and electronic contributions to the partition function corresponding to component i, and CP is the mean potential and is related t o the configurational energy EC by (Sandler, 1985) (16) For the mixture, EC is given by the sum of all molecular pairs, i.e.
For the square-well fluid, one can write
where Nu is the coordination number of molecule i around moleculej. In order to apply local composition theory t o the evaluation of thermodynamic properties of fluids, one needs to assume a functional form for the coordination number, calculate the configurational energy using eqs 18 and 17, and then calculate the partition function using eqs 16, 15, and 14. The low-density limit can be calculated from rigorous statistical mechanical analysis and is given by (Lee and Chao, 1987)
where R is the range of the square well (taken here t o be equal to 1.5, since most simulation and previous theoretical work has been done with this value). It has
Ind. Eng. Chem. Res., Vol. 34, No. 12,1995 4557 been shown (Lee and Chao, 1987; Wang and Wang, 1990)that the low-density limit seriously overestimates the coordination numbers at moderate and high densities, since the Boltzmann factor exaggerates the effect of interaction energies. Several models based on local composition theory have been proposed to dampen the exponential dependence on inverse temperature as the density approaches the mean-field limit. Sandler and co-workers (Lee et al., 1986; Lee and Sandler, 1987) recognized that the local composition effects due to attractive forces are most prominent at low densities and insignificant at high densities. They extended their coordination number expression for pure square-well fluids (Lee et al., 1985) to mixtures. The resulting expression (LS) for mixtures of equal-sized square-well molecules has the form
0
I
L.
-2 -4
4 '
=-a -10
0
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0 -1 .2 w=
-3
r2 4
5 -5 -6 -7
where p* = p d and ZMis the maximum coordination number and for the square-well fluid with R = 1.5 has the value of 18 (Lee et al., 1985; Bokis et al., 1992).Also, a is given by a = h/(& - p*). Unlike most previous local composition models, the LS model attempted t o describe both the random mixing at the high-density limit and Boltzmann factor nonrandomness at low densities. However, the LS equation does not reduce to the correct low-density limit (given by eq 19). From examination of computer simulation data, Lee and Chao (1987) obtained the following model for the coordination number of square-well mixtures (LC)
x%R3 - 1)(1+ 0.57~")exp(acii/kT) N u. . = - a (21) 1/2 + p*[Q - 11 where
and a is an empirical polynomial function of the reduced density. Unlike the LS model, the LC expression satisfies the low-density limit. However, this model has a complicated summation in the denominator of eq 21 (as shown in eq 221, and it has not yet been possible to integrate the coordination number expression analytically to derive other thermodynamic properties. More recently, Guo et al. (GWL, 1990a) proposed another model for the coordination number of squarewell mixtures. In the case of equal-sized mixtures, the GWL model has the form
4n
NG= xi$R3
- l)p* exp
The major difference between the GWL model (eq 23) and the LS and LC models is the way of normalizing the Boltzmann factors. In the LS and LC models, the denominator accounts for the normalization of the Boltzmann effect. In the GWL expression for the coordination number, this normalization occurs inside the exponential. Without any adjustable parameters, eq 23 satisfies the low-density limit (eq 19) and can be used for the construction of an equation of state.
0
-2 4
#=4
3 ."I0 -12 -14
Figure 3. Internal energy calculations using local composition models and comparison with Monte Carlo simulation data (Lee and Chao, 1987): (a) kT/c11 = 2.0, czdcll = 2.0, c1dc11 = &,' 51 = 0.204; (b) kTIEi1 = 2.0, ~ 2 d ~=i i2.0, ~ i d t i= i A, xi = 0.796 (c) kT/~11= 3.0, ~2d~11 = 3.0, cidcll = A,XI = 0.500.
However, as noted by Wang and Wang (1990), its performance seems t o deteriorate with increasing density. Figure 3 shows a comparison of these local composition models in predicting the internal energy of the three binary square-well mixtures shown previously. The model proposed by Lee and Sandler (LS; 1987)was one of the first attempts to develop a statistical mechanical model for square-well mixtures by proposing a function of the form given by eq 20. Their model was a compromise between low-density and high-density behavior. Unfortunately, the LS model does not give good agreement with the simulation data of Lee and Chao (1987). In addition, as noted previously, it does not give the correct second virial coeflticient. The Lee and Chao model (LC; 1987) seems to be in good agreement with the simulation data; however, the complicated functional form of the LC model has not allowed integration into an equation of state. The model proposed by Guo et al. (GWL; 1990a, 1990b) is in relatively good agreement with the internal energy simulation data, and it contains no adjustable parameters. Wang and Wang (WW; 1990) modified the GWL model by regressing one constant to the binary simulation data; their model slightly improved the high-density U calculations. However, in contrast with the other models considered in this work, the WW model is not a predictive model and, therefore, is not included in these comparisons. In Figure 4 we compare calculations for the compressibility factor of these three square-well mixtures using the local composition models. The Lee-Chao model is not included in this comparison since, as mentioned
4558 Ind. Eng. Chem. Res., Vol. 34,No. 12, 1995 I
6~
7
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*.......-'-'
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Figure 5. Compressibility factor calculations using perturbation theory and local composition models and comparison with Monte Carlo simulation data (Lee and Chao, 1987): k T / m = 2.0, c2dc11 = 2.0, c12/~11= 1.0, XI = 0.204. 7 t
I
No'N
Figure 4. Compressibility factor calculations using local composition models and comparison with Monte Carlo simulation data (Lee and Chao, 1987): (a) k T l w = 2.0, ~ 2 / & = 2.0, e12/t11 = f i ,XI = 0.204; (b) kTIEi1 = 2.0, ~ 2 2 / ~=l l 2.0, ~1z./c11= f i ,X I = 0.796; (c) kTlti1 = 3.0,~ 2 2 / ~ 1=1 3.0,~ 1 2 / t i i= A,XI = 0.500.
above, the NG expression has not been integrated for this model. From Figure 4 it is apparent that the LS and GWL models are not in very good agreement with the computer simulation data for the compressibility factor. Also, their performance seems to deteriorate with increasing density. This is because, as density increases, the molecules of the fluid are pulled equally by other molecules; this leads t o a cancellation of the Boltzmann factors, and, hence, the behavior of the fluid is described quite well by first-order perturbation theory. Similar results were obtained when we calculated the compressibility factors of mixtures where the cross interaction energy is not given by the geometric mean of the pure-component interaction energies. This is shown in Figure 5 , where we plot compressibility factors for a binary square-well mixture (kTIE11= 2, ~ 2 2 / ~ 1= 1 2, and ~ 1 2 / ~ 1=1 11, using local composition theories and perturbation theory, respectively (again, the simulation data are from Lee and Chao, 1987). It is clear that the Donohue and Prausnitz (1978) version of Henderson's (1974) perturbation theory predicts the compressibility factors of this system more accurately than the local composition models. The reason for this is that the repulsive forces determine the structure of liquids t o a great extent, and, therefore, the attractive forces have little influence on local structure or composition.
Development of the New Model The calculations and results described in the previous two sections suggest that perturbation theory is very
successful in describing the thermodynamic behavior of mixtures of square-well molecules. However, it has the disadvantage that it is very complicated. For example, the fourth-order expansion of Donohue and Prausnitz (1978) requires five different mixing rules, even for equal-sized molecules. Recently, we (Bokis, 1995; Bokis and Donohue, 1995) developed a new equation of state for the square-well fluid. This equation has the exact second virial coefficient limit, the correct dense-liquid (mean-field)limit, and it interpolates between these two limits with remarkable accuracy. A comparison with Monte Carlo simulation data for pure-component square-well molecules (Bokis and Donohue, 1995) showed that this model is more accurate than the fourth-order perturbation expansion of Alder et al. (1972) in the dilute-gas region. In terms of the configurational internal energy, U , this equation of state has the form (24) where U1 and UZare the first- and second-order terms in the perturbation expansion, and they are calculated by
Ul = -9.500057 - 1 3 . 2 6 5 5 ~-~3 . 0 8 3 6 ~+~
+
+
2 0 . 2 3 1 8 ~ ~3 0 . 5 4 5 ~ ~1 0 . 2 1 2 6 ~(25) ~ and
-u2 _ - 1 - 5.416517 + 8 . 4 1 5 8 2 ~+~6 . 6 7 2 9 8 ~-~
u,
+
+
1 9 . 3 9 ~ ~2 . 0 4 4 5 4 ~-~9 . 9 2 9 5 6 ~ ~11.9467' (26) where 7 = zp*/6. For a mixture of equal-sized molecules, eq 24 can be written in a form analogous to that of eq 18 T
mixture
(27) where
2 &)
uij= U1EUexp(
Ind. Eng. Chem. Res., Vol. 34, No. 12,1995 4559 0 -1 .2
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Figure 6. Internal energy calculations using perturbation theory, local composition models, and the proposed model (eq 29) and comparison with Monte Carlo simulation data (Lee and Chao, 1987): (a) kTk11 = 2.0, c2dc11 = 2.0, E I ~ E I I= 1.0, xi = 0.500; (b) kTIg11 = 2.0, ~2d~11 = 2.0, ~ i d c i= i XI = 0.796; (c) kTIEi1 = 2.0, E2dcii = 2.0, ~id
