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Ind. Eng. Chem. Res. 1998, 37, 2936-2946
Local Composition Models for Lattice Mixtures Dee-Wen Wu, Yuping Cui,† and Marc D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218
A detailed comparison is made of several local-composition lattice models. The models considered include several popular activity coefficient models: the Wilson equation, Guggenheim’s quasichemical theory (GQC), the nonrandom two-liquid theory (NRTL), and the universal quasichemical (UNIQUAC) theory. Also considered are three recently developed lattice models: the Born-Green-Yvon (BGY) model, the nonrandom surface fraction (NRSF) model, and the Aranovich-Donohue (AD) model. Similarities and differences in the assumptions regarding the local compositions are examined. Detailed comparisons are made for both symmetric and asymmetric monomer mixtures as well as for polymer/solvent mixtures with Monte Carlo simulations. Introduction While the properties of some mixtures can be modeled accurately with mean-field equations like regular solution theory, there are a wide variety of systems that exhibit nonrandom behavior. The Wilson equation (Wilson, 1964), the Guggenheim quasi-chemical (GQC) theory (Guggenheim, 1952), the nonrandom two-liquid (NRTL) model (Renon and Prausnitz, 1968), and the universal quasi-chemical (UNIQUAC) theory (Abrams and Prausnitz, 1968) allow one to account for nonrandomness in the activity coefficients of mixtures. These models are used extensively by plant and process engineers to correlate experimental data. There also have been lattice-based models developed recently, including the Born-Green-Yvon (BGY) model (Lipson, 1991), the nonrandom surface fraction (NRSF) model (Cui and Donohue, 1992), and the Aranovich-Donohue (AD) model (Aranovich and Donohue, 1996; Aranovich et al., Donohue, 1997), to describe nonrandom mixtures. Though it is an approximation, the free energy of a mixture usually is separated into three terms: an ideal solution term, a combinatorial term and a residual term, (Prausnitz et al., 1986). The excess free energy of a mixture therefore contains a combinatorial term and a residual term:
gmixture ) gideal + gcombinatorial + gresidual gE ) gcombinatorial + gresidual
(1)
For condensed phases, where the volume is small, the Helmholtz free energy is approximately equal to Gibbs free energy (Prausnitz et al., 1986) at low pressures; therefore, we have
( ) ( ) aE RT
T,V
≈
gE RT
)
T,P
gcombinatorial gresidual + RT RT
(2)
While the Flory-Huggins entropy expression (Flory, 1953) is used widely to estimate the combinatorial entropy, the Guggenheim random mixing theory (GRM) (Guggenheim, 1952) has been shown to be the most * Author to whom correspondence should be addressed. † Present address: Aspen Technology, Inc., Ten Canal Park, Cambridge, MA 02141.
accurate of the common closed-form expressions for polymer systems (Cui and Donohue, 1992). In the limit of monomer mixtures, the GRM theory reduces to the ideal entropy of mixing. The purpose of this paper is to discuss the similarities and differences among the residual terms of the various models. The residual term is related directly to the internal energy of the mixture by the Gibbs-Helmholtz relation:
u≈h)
( ) ∂(g/T) ∂(1/T)
(3)
V,Ni
Equation 3 has been used to derive expressions for the internal energy from the local composition models discussed above. One of the first theories developed to describe nonrandom behavior in mixtures was the quasi-chemical theory of Guggenheim (1952). While correct in most respects, quasi-chemical theory is not used widely for engineering design calculations because of its mathematical complexity. The first local composition model to be adopted widely for engineering design calculations was the Wilson equation (1964). Wilson proposed an empirical relation between the local concentration and the overall concentration for liquid mixtures that is based on a generalization of the Flory-Huggins expression for the excess entropy. Wilson generalized the Flory-Huggins equation by replacing the volume fraction with a local volume fraction. He obtained an expression for the excess Gibbs energy for a binary mixture with species molar volume Vi and concentration xi of the form
gex ) -x1 ln(x1 + Λ12x2) - x2 ln(x2 + Λ21x1) (4) RT where Λij ) (Vj/Vi) exp[-(λij - λii)/RT], and λij is a measure of the interaction energy between molecules i and j. The Wilson equation is used widely in industrial design calculations because it is simple and because it fits the data for mixtures of water with hydrocarbon compounds better than the NRTL and the UNIQUAC models. Additional parameters are needed to predict
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Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2937
the miscibility gaps found in some liquid mixtures (Walas, 1985). The NRTL model (Renon and Prausnitz, 1968) also uses the local-composition concept and gives the excess Gibbs energy of a binary mixture based on the twoliquid theory of Scott (1956). Renon and Prausnitz proposed
(
)
τ12G12 τ21G21 gex ) x1x2 + RT x1 + G21x2 x2 + G12x1
(5)
where τij and Gij are energy parameters which are defined as
τij )
gij - gii RT
Gij ) e-Rijτij
(6)
with gij analogous to λij of the Wilson equation. In the NRTL model, Rij is treated as a third adjustable parameter. The NRTL model can represent both vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) behavior by adjusting Rij and τij; however, the NRTL model does not take into account size differences between molecules. This precludes its application to mixtures containing polymers. The UNIQUAC theory is a combination of the Wilson equation and the Guggenheim random mixing theory. However, it differs from the Wilson equation in that the residual Helmholtz energy is written in terms of surface fractions. UNIQUAC uses the Guggenheim combinatorial entropy of mixing, which takes into account the effects of molecular size and shape. The UNIQUAC model is
(
)
φ1 q1z θ1 gex + - q1 ln(θ1 + θ2τ21) + ) x1 ln ln RT x1 2 φ1 φ2 q2z θ2 x2 ln ln + - q2 ln(θ2 + θ1τ12) (7) x2 2 φ2
(
)
where τij is the energy parameter and is given by τij ) exp[-(uij - ujj)/RT], and z is the lattice coordination number (i.e., the number of nearest neighbors interacting with a central molecule). qi is the surface area of molecule i, θi is the surface area fraction of component i, and φi is the segment fraction of component i (ri is the chain length of molecule i):
θi )
β∆Gmix φ1 φ2 ) ln φ1 + ln φ2 + Ns r1 r2 z q1φ1 θ1 q2φ2 θ2 ln + ln + 2 r1 φ1 r2 φ2 θ1θ2 q1 q2 z q2 θ1φ1β∆ φ1 + φ2 β2∆2 2 r2 8 r1 r2 θ q θ q 2 z 1 2 1 φ (1 - 2θ1) + φ2(1 - 2θ2) β3∆3 (9) 2 48 r1 1 r2
(
(
(
(
) ) )
)
where is the lattice interaction energy, ∆ ) 212 - 11 - 22, ij is the segmental interaction energy difference between segments i and j, and β is defined as 1/kT. A recent version of the BGY model (Lipson and Brazhnik, 1993) uses the same assumptions and gives a closedform expression for the internal energy of mixtures. The NRSF model also uses the surface fraction concept to take into account nonrandomness in polymer mixtures. It combines a nonrandom expression for the internal energy with the Guggenheim random mixing expression for the combinatorial entropy. It was derived as a closed-form approximation to the series expansion of the lattice cluster theory of Freed and co-workers (Dudowitz et al., 1990). The NRSF model has the following expression for the Gibbs free energy of mixing:
θ1 ∆Gmix z ) n1 ln φ1 + q1n1 ln + RT 2 φ1
xiqi
∑j xjqj xiri
NRTL and the UNIQUAC models have broader applicability than the Wilson equation because they are able to predict liquid-liquid immiscibility, but the Wilson equation usually works better for systems containing water. The BGY model was developed recently by Lipson (1991), using the theory of Born, Green, and Yvon. Lipson’s BGY formalism is an expansion of the original BGY theory for the free energy of mixing. The original Born-Green-Yvon theory evaluates the N and N - 1 particle distribution functions through a set of linked integral differential equations. Lipson applied the Kirkwood superposition approximation to express the triplet distribution function as a product of pair distributions and then applied the independence approximation to the pair distributions. The BGY expression for the total free energy of mixing consists of Guggenheim’s entropy of mixing term and an energetic (thermal) term which contains both energetic and noncombinatorial entropic contributions. The BGY theory is a series expansion in terms of the interchange energy and concentration:
θ2 z n2 ln φ2 + q2n2 ln 2 φ2 (8)
z (q n + q2n2) ln(1 + 2θ1θ2(e-∆/2kT - 1)) (10) 2 1 1
The UNIQUAC model predicts both VLE and LLE for a wide variety of mixtures. Since it includes both enthalpic and entropic terms, intuitively it is more appealing than the Wilson and NRTL equations. However, as each of these models has three or more parameters, it is not clear that UNIQUAC does any better in fitting experimental data. In general, the
The BGY and NRSF models take into account nonrandomness in polymer mixtures and predict LLE in polymer mixtures. They both give close agreement with molecular dynamics simulations for the internal energy of mixing and liquid-liquid equilibria (Cui and Donohue, 1992; Lipson, 1991). The recent AD model is based on a generalization to three dimensions of the Ono-Kondo equations for the density profile near a flat surface (Aranovich and
φi )
∑j xjrj
2938 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Donohue, 1996). This is done by considering the “adsorption” of molecules on a central molecule instead of at a surface. Local densities in a solution then can be obtained from the model. Unlike many previous theories, the AD model has the correct limiting behavior at infinite dilution, at high densities, when the interchange energy goes to zero, and for the behavior of a lattice gas. It is not a Gibbs free energy model but gives the local density and internal energy. Like quasichemical theory, for a binary mixture it is of the form:
z ∆ Umixture ) N x111 + x222 + (x1ζ21 + x2ζ12) 2 2
(
)
(11)
where ∆ is the interchange energy defined as ∆ ) 212 - 11 - 22. ζ21 is the probability of finding a molecule of component 2 around a central molecule of component 1, and ζ12 is the probability of finding a molecule of component 1 around a molecule of component 2. The difference between the AD model and quasi-chemical theory is in how the local compositions are calculated. In quasi-chemical theory they are calculated using an equilibrium constant. In the AD model, they are calculated by assuming that the chemical potential is independent of any local composition or density gradient. This assumption leads to the result that
ζ21 )
x2 exp(-x2∆/kT) x1 + x2 exp(-x2∆/kT)
x2
)
x1 exp(x2∆/kT) + x2 (12)
ζ12 )
x1 exp(-x1∆/kT) x2 + x1 exp(-x1∆/kT)
x1
)
x2 exp(x1∆/kT) + x1
An empirical expression that is equivalent to that obtained with eqs 11 and 12 was proposed by Lee et al. (1986). They performed Monte Carlo (MC) simulations for the nonrandom behavior of off-lattice square-well molecules and derived a local composition theory using phenomenological arguments (Lee et al., 1985). They also proposed an empirical modification of the LeeLombardo-Sandler (LLS) equation that showed better agreement with their simulation data. The AD derivation of eqs 11 and 12 provides a theoretical justification for the Lee-Sandler-Patel equation. A recent paper on the AD model (Aranovich et al., 1997) extends their model rigorously to multicomponent systems. The derivation results in an equation of the form:
(∑
z Umixture ) N 2
v
x∞i ii +
i)1
1
v
v
)
∑∑x∞i x∞j ∆ijΨij
2i)1j)1
(13)
where ∆ij ) 2 ij - ii - jj is the interchange energy, x∞i is the bulk density of i-monomers, and the weighting factor Ψij is defined as v
exp[(-ij + Ψij )
v
x∞mim)/kBT] ∑ m)1 v
x∞l exp[(-lj + ∑ x∞mlm)/kBT] ∑ l)1 m)1
(14)
These activity coefficient models and lattice models use similar but somewhat different physical variables. The activity coefficient models use the differences in internal energies between the molecules, while the lattice approaches use differences in molecular or segmental potential energies. Because only differences in interaction energies are considered, these energies must be equivalent physically and should be independent of the reference state defined. Therefore, it is a straightforward matter to translate the parameters in the activity coefficient models to a lattice and make comparisons of physical properties predicted by these models. Similarities and Differences among Local Composition Models Both the activity coefficient approach and the lattice approach provide mixture properties in terms of the intermolecular energy, the molecular size, and the number of nearest neighbors. Because of the similar molecular arguments, these models can be compared rigorously for mixtures of lattice fluids and for the lattice gas. By superimposing a lattice on the binary mixture expressions, we can rewrite the Wilson equation, the NRTL model, and the UNIQUAC model using lattice variables. The energy parameters used in activity coefficient models can be related to the segmental interaction energy in a lattice through the following equation (Abrams and Prausnitz, 1975; Renon and Prausnitz, 1968):
z λij ) gij ) uij ) ij 2
(15)
The volume ratio in the Wilson equation corresponds to the ratio of segment numbers for the two components. Rij in the NRTL model can be related to the lattice coordination number z by (Renon and Prausnitz, 1968)
Rij ) 1/z
(16)
Equations 15 and 16 establish the correspondence between the variables used in the activity coefficient models and those used in theories for lattice mixtures. To compare these models, we examine the internal energies of various mixtures, including monomer mixtures and polymer/solvent mixtures. In a mixture of monomers, the size effect is eliminated and what is left is the nonrandomness caused by the molecular interaction energies. Table 1 presents these models in a simple notation for monomer mixtures. Note that, to derive the internal energy of the mixture from the Wilson equation and the UNIQUAC model, we have used the Gibbs-Helmholtz relation (eq 3). The first thing one notices in Table 1 is the way that the summations are performed in the different models. The first five models, except the NRSF model, use a single summation of concentration multiplied by a ratio of single summations to give the nonrandomness corrections. These models treat the mixture using a twofluid approach, which means that the nonrandomness in each component is counted individually and then summed. The NRSF model, however, uses a ratio of double summations to describe the concentration dependence. This suggests that the NRSF model is a one-
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2939 Table 1. Monomer Mixture Properties for Various Local-Composition Models model
equation
z
Wilson UNIQUAC
Umixture ) N 2
param
∑x P
j ij ij
∑
xi
i
j
∑
Pij ) e-(z/2)(ij/kT)
xjPij
j
NRTL
Gres mixture
z
) N 2
∑x
∑x P
j ij ij
j
i
∑
i
Pij ) e-ij/2kT
xjPij
j
z
BGY
Umixture ) N 2
∑x
∑x P
j ij ij
j
i
i
∑
Pij ) e-ij/kT
xjPij
j
z
NRSF
∑ ∑x x P
i j ij ij
i
Umixture ) N 2
j
Pij ) e-ij/kT
∑ ∑x x P
i j ij
i
ADbinarya
z Umixture ) N 2
∑
j
∑x (
xi
j
)
1 + (ii - jj) Pij 2
ij
i
Pij ) e-xj[(2ij-ii-jj)/kT]
∑x P
i
j ij
j
ADmulti
a
z Umixture ) N 2
(
v
∑x
∞ i ii
i)1
+
1
v
v
∑∑ x 2
∞ i
i)1 j)1
)
x∞j ∆ijΨij
see eq 14b
The AD model for binary mixtures is written in a form similar to that of the other models here for comparison purposes.
fluid model. It uses probabilities for the entire mixture in predicting the local-composition effect. Table 1 also shows that the Wilson equation and the UNIQUAC model are equivalent for a monomer mixture. The BGY model is similar to the Wilson equation and the UNIQUAC model except that the BGY model has a different Boltzmann factor. Also shown in Table 1 is the binary mixture version of the AD equation written in a form similar to that of the other equations. This is shown for comparison purposes. This form of the equation is correct for binary mixtures but not for multicomponent mixtures. The actual multicomponent version of the AD model is more complicated and is given by eqs 13 and 14. It cannot be written in the simple form of the other equations. One sees that the binary version of the AD model is similar in form to the BGY but with different energy and probability parameters. Unlike all the other models, the probability factors (Pij) in the AD equation are written with the interchange energy (∆ ) 212 - 11 22) in the exponent. The rigorous multicomponent AD model has a more complicated weighting function Ψij, with the interaction terms in the exponent. The NRTL model expresses the Gibbs or Helmholtz free energy of the mixture in a form that is similar to the mixture internal energies in the Wilson, UNIQUAC, BGY, and AD equations. Though the NRTL has a mathematical form similar to those of Wilson, UNI-
QUAC, BGY, and AD, its predictions of activity coefficient behavior are quite different, because it is an expression for the Gibbs energy rather than the internal energy. The primary difference among these models is in the Boltzmann factor. The Wilson equation and the UNIQUAC model have the same Boltzmann factor, and they both have the coordination number in the exponential. The BGY and NRSF models also share the same Boltzmann factor, with the coordination number not present in the exponential. In the NRTL model, the energy parameter in the Boltzmann factor is half that of the BGY and NRSF because it is a product of Rij and z/2. In the AD model, the coordination number does not appear in the Boltzmann factor. It has quite different energy terms (ij + (ii - jj)/2) and Boltzmann factor (Pij ) exp[-xj(2ij - ii - jj)/kT]). Thus, in the AD model each component of the summation is not Boltzmannweighted by the interaction energy (ij) but rather by the interchange energy (∆ij). Since it is derived by taking a central molecule as the “absorbent”, the energy parameters are asymmetric, i.e., the i-j term is different from the j-i term. For the multicomponent AD model, the coordination number does not appear in the Boltzmann factor either. Finally, it should be noted that the exponential terms in all the models are functions of energy and temperature. However, in the AD models
2940 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 2. Polymer Mixture Properties for Various Local-Composition Models model
equation
z
Wilson
∑x(v /v ) P j
v
∑ ∑
Umixture ) ( Ni) 2 i)1
param
xi
i
ij ij
i
∑
i
Pij ) e-(z/2)(ij/kT)
xjPij
j
z
UNIQUAC
v
∑
∑θ
Umixture ) ( Niqi) 2 i)1
∑θ P
j ij ij
j
i
∑
i
Pij ) e-(z/2)(ij/kT)
θjPij
j
z
BGY
v
∑
∑θ
Umixture ) ( Niqi) 2 i)1
∑θ P
j ij ij
j
i
∑
i
Pij ) e-ij/kT
θjPij
j
z
NRSF
v
∑
∑∑θ θ P
i j ij ij
Umixture ) ( Niqi) 2 i)1
i
j
∑∑ i
ADbinary
Pij ) e-ij/kT
θiθjPij
j
z Umixture ) (N1q1 + N2q2) 2
∑
∑θ (
θi
j
ij
j
)
1 + (ii - jj) Pij 2
∑θ P
i
Pij ) e-θj[(2ij-ii-jj)/kT]
j ij
j
v
ADmultia
(∑
z Umixture ) ( Niqi) 2 i)1
∑
v
θ∞i ii +
i)1
1
v
v
∑∑ θ 2 i)1 j)1
∞ i
θ∞j ∆ijΨij
)
see eq 14b
a The generalizations of the two AD equations are not rigorous but are written by analogy to the other equations. b The weighting factor Ψij is defined the same as in eq 14 but with surface fractions instead of mole fractions.
(both binary and multicomponent), the exponentials also are functions of composition. Though derived from very different concepts, the AD model and the GQC theory give surprisingly similar results. Both give correct behaviors both for binary mixtures and for the lattice gas (where one of the components becomes a “hole”, or a “void”). They also give the correct behavior when the interchange energy, ∆, goes to zero. In this limit, the energy of mixing must be zero. While the other theories give reasonable results for binary mixtures, they give poor results for the lattice gas. In addition, they all fail to give the correct limit when the interchange energy goes to zero. For polymer mixtures, molecular size also affects mixture nonrandomness. Table 2 shows how each model addresses the size dependence. The NRSF model uses a one-fluid double summation over surface fractions multiplied by the interaction energy and the Boltzmann factor. The other models use a two-fluid summation of the ratio of single sums. The Wilson equation can be written as a summation over mole fractions of a ratio of two volume fraction terms. The UNIQUAC model and the BGY model also use the surface fraction as the concentration variable. The differences in the Boltzmann factors are the same as those in monomer mixtures. A rigorous generalization of the AD model to polymer solutions has not been
derived. We present here an ad hoc generalization of the binary and multicomponent AD equations by substituting surface fractions for mole fractions. Comparison with Simulation Data Monomer Mixtures. Each of the models discussed above has desirable characteristics. Some are easy to use but may not produce very accurate results, while others can produce accurate results but are quite complex. To demonstrate the versatility and limitations of these models, three special cases are discussed. They are binary mixtures, the lattice-gas limit, and the situation when the interchange energy approaches zero. In a binary mixture, each component has a nonzero interaction energy, and the cross interaction energy, 12, usually is approximated by the geometric mean 12 ) x1122. A lattice gas is a binary mixture of molecules and holes (or voids). For a lattice gas 11 is not zero while 12 and 22 both are zero. Finally, when the crossinteraction energy approaches the arithmetic mean, the interchange energy approaches zero. Comparisons of these models are shown in Figures 1-4 for these three different types of systems. Each figure has two plates. One is the ratio of the internal energy to the mean field energy, U/Umf, where Umf ) (z/2)∑i∑jxixjij. The other plate shows the energy of mixing, ∆U/kT. Guggenheim
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2941
Figure 1. Internal energy for a binary monomer mixture. Energy parameters used here are 11/kT ) -0.5, 12/kT ) -0.3, and 22/ kT ) -0.4. For monomers, the Wilson model is the same as the UNQUAC model. For these parameters, the two components are not very different from each other; i.e., the system is fairly ideal. As a result, all models except the UNIQUAC and Wilson models give reasonable predictions of ∆U. However, the BGY and NRSF models show deviations from MC data for x1 > 0.5.
quasi-chemical theory is not in these figures because it is nearly identical with the AD model. Figure 1 shows a comparison of the models for the binary mixture case. In this figure all the models except Wilson’s equation and UNIQUAC equation (which are equal) show reasonable agreement with the Monte Carlo simulation results. However, this apparently good agreement is due to the similarities of the interaction energies selected (11 ) -0.5kT, 12 ) -0.3kT, 22 ) -0.4kT); i.e., the systems are fairly ideal. For systems with quite different interaction energies, not all the models show such good agreement with MC simulation results. This can be seen on Figure 2 where only 11 is strongly attractive (-0.65kT), while 12 and 22 are only slightly attractive (-0.2kT and -0.1kT). The results in Figure 2 are similar to what would be seen in a water-hydrocarbon mixture. Figure 3 shows an even more nonideal system, the lattice gas with 11 ) -0.5kT and 12 ) 22 ) 0. The AD model shows good agreement with the Monte Carlo simulation data, while all the other models show large deviations from the MC data. Figure 4 shows the case where the interchange energy approaches zero, i.e., 212 ) 11 + 22. The system
Figure 2. Internal energy for a binary monomer mixture. Energy parameters used here are 11/kT ) -0.65, 12/kT ) -0.2, and 22/ kT ) -0.1. Note that the BGY and the NRSF models are not as accurate as they are in Figure 1. When the system is more nonideal, all models except the AD model tend to be less accurate.
illustrated is an asymmetric system with 11 ) -0.4kT and 22 ) -0.3kT. The cross term, 12, is -0.36kT. This makes the interchange energy -0.02kT. As discussed above, the behaviors of the BGY and NRSF models are not good for asymmetric mixtures. The AD model still agrees best with the Monte Carlo simulation data. When the interchange energy is exactly zero, the energy of mixing must be zero also. Only the AD model and GQC correctly predict a zero energy of mixing; all the other models give nonzero values for ∆U when the interchange energy is zero. Figures 5 and 6 are for multicomponent monomer mixtures. These figures show the predictions of both the rigorous AD multicomponent model given by eqs 13 and 14 and the ad hoc generalization of the binary AD equation listed in Table 1. Figure 5 is a system with two components and holes. The second component is held at a fixed mole fraction of 0.2. The interaction energies used (11 ) -0.6kT, 12 ) -0.4kT, 22 ) -0.3kT) are moderately asymmetric, but only the multicomponent AD model gives correct predictions of the MC data. The binary AD model tends to overpredict the energy of mixing slightly, while all the other models underpredict the energy considerably. The same situation is observed in Figure 6, which shows calculations and
2942 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 3. Internal energy for a lattice-gas system. Energy parameters used here are 11/kT ) -0.5 and 12/kT ) 22/kT ) 0. When x1 approaches 0, U/Umf should approach exp(-11/kT), which is exp(0.5) ) 1.64. While the BGY and NRSF models give the correct limit when x1 approaches 0, they do not give the correct behavior at intermediate concentrations. The UNQUAC and Wilson equations do not predict correct behavior at all. On the ∆U graph, the AD model gives excellent agreement with the MC data.
simulations for a system with four different components and holes. It appears that, for monomer mixtures, the AD model gives the best predictions of the MC simulation data. The Wilson and UNIQUAC equations, though used widely, are not very accurate in any of the three cases. BGY and NRSF models both give reasonable results for symmetric binary monomer mixtures, but they do not show the correct behavior for asymmetric systems (i.e., systems where there are large differences in the purecomponent interaction energies). In most cases, the BGY model is better than the NRSF model. Polymer-Solvent Systems. Several of the localcomposition models considered here also take molecular size into account. In athermal lattice mixtures, the effect of molecular size can be described accurately by the Guggenheim random mixing theory (Guggenheim, 1952) which introduced the surface fraction concept (Cui and Donohue, 1992). In mixtures that are not athermal, chain connectivity also contributes to the nonrandom distribution of molecules. In the local-composition models considered here, the main difference in describ-
Figure 4. Internal energy for a binary monomer mixture when the interchange energy, ∆, approaches zero. In this case, energy parameters used are 11/kT ) -0.4, 12/kT ) -0.36, 22/kT ) -0.3, and ∆/kT ) -0.02. The internal energy U is very close to the mean field U. When ∆ is exactly 0, the AD model and GQC theory reduce to the correct mean field limit, but all the other models predict incorrect, nonzero ∆U. The AD model gives excellent predictions of the MC results.
ing the molecular size effect is in the concentration variables. The Wilson equation uses both mole fractions and volume fractions. As discussed above, for monomer mixtures the UNIQUAC model is equivalent to the Wilson equation. In polymer mixtures, the UNIQUAC model puts surface fraction variables into the Wilson equation. As a result, the UNIQUAC model uses mole fractions, volume fractions, and surface fractions as concentration variables. This is an important distinction. By dividing a polymer molecule into segments and introducing the concept of surface area for the segment, one is able to consider the molecular size effectively. By using the segmental properties, group contribution methods can be combined with local-composition models to make predictions of thermophysical properties for polymers. Similarly, the BGY model improves over the NRTL model by including surface fractions and volume fractions in its formulation. Like UNIQUAC, the NRSF model uses both segmental fractions and surface fractions to model local composition. By analogy, both the ADbinary and the ADmulti models can be extended phenomenologically to polymer/solvent systems by substituting mole fractions with molecular
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 2943
Figure 5. Internal energy for a binary mixture with holes. This is a ternary mixture where the third component is a hole. The composition of component 2 is held constant at a mole fraction of 0.2. The energy parameters used here are 11/kT ) -0.6, 12/kT ) -0.4, and 22/kT ) -0.3. The multicomponent AD model behaves better than the ad hoc AD binary model, but the AD binary model still is better than all other models listed in Table 1.
surface area fractions. The modified polymer versions of AD equations are listed in Table 2. In Figures 7-9, we compare all the models with Monte Carlo simulation data. Figure 7 is for a mixture of 5-mers (polymer) and monomers (solvent). To compare the accuracy with their monomer/monomer mixture counterparts, the energy parameters are the same as those used in Figure 1, for a monomer mixture. Therefore, the ADbinary and ADmulti are identical. In these polymer/solvent systems, none of the models show the accuracy of the AD model in Figure 1. However, the AD and BGY models still are better than the other models. At low volume fractions, all models except the UNIQUAC equation predict satisfactory internal energies. When volume fraction increases, the NRSF starts to deviate from the simulation data. Both the AD model and the BGY model behave well on the U/Umf graph, but the AD model is slightly better than the BGY model. Here the mean field internal energy Umf is defined as Umf ) (z/2)∑i∑jθiθjij, following Guggenheim’s definition for polymer mixtures. One interesting point is that both the AD and BGY predictions are least accurate when the polymer volume fraction is between 0.5 and 0.7. This
Figure 6. Internal energy for a multicomponent mixture with holes. This system contains five components where the fifth component is a hole. Mole fractions are held constant for x2 ) 0.05, x3 ) 0.1, and x4 ) 0.05. Energy parameters used here are 11/kT ) -0.60, 12/kT ) -0.50, 13/kT ) -0.40, 14/kT ) -0.45, 22/kT ) -0.30, 23/kT ) -0.35, 24/kT ) -0.25, 33/kT ) -0.45, 34/kT ) -0.40, and 44/kT ) -0.20. Again, the rigorous multicomponent AD model is the best in predicting this multicomponent behavior.
may be because, at high volume fraction, the system energy is close to the pure-component energy. In Figure 8, a polymer of length 5 is mixed with holes. All energy parameters are identical with those in Figure 3. In this “lattice-gas” system, the AD model obviously does better than all the other systems. However, in the U/Umf graph, we find that, at low concentration of polymer, even the generalization of the AD model does not give a very good prediction. We believe this is due to an incorrect treatment of intramolecular interactions. This discrepancy is seen in all the two-liquid models. We plan to investigate this behavior in the future. In Figure 9, the polymer/solvent system is the same as in Figure 7, but the chain length of the polymer is 10. The behaviors are similar to those in Figure 8. The AD model and the BGY model are much better than the others, and the AD model is slightly better than the BGY model. In Figure 10, we have a 10-mer lattice gas. In this case, the AD model does a good job in predicting the internal energy of the system. The BGY model shows good agreement also. However, the accuracy of the BGY model is due to the small interaction energy parameter
2944 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 7. Internal energy for a polymer/solvent mixture. Energy parameters used here are 11/kT ) -0.5, 12/kT ) -0.3, and 22/ kT ) -0.4. The length of the polymer is 5. Only the AD and BGY models give reasonable predictions.
used (11 ) -0.1kT). Even so, the NRSF and the UNIQUAC do not give satisfactory predictions. Finally we present calculations for a polymer/polymer/ hole mixture in Figure 11. We mix a 10-mer with a 5-mer. The void fraction of the system remains constant at 0.2 volume fraction. From the results shown, one can see that the AD model still is the best model. For this case, the difference between the binary AD model and the multicomponent AD model is negligible. This is different from the results in Figure 5 and 6 where only the multicomponent AD model gives correct predictions. Monte Carlo Simulations. To verify the validity of the models, we have performed Monte Carlo simulations on a canonical ensemble (NVT ensemble). Simulations were run for monomer mixtures and for polymer/ solvent systems. For monomer mixtures, the system was a cubic box with periodic boundary conditions. The simulations were carried out according to the Metropolis’ algorithm. For different values of interaction energy 11, 22, and 12, the lattice interaction energy ∆ was obtained from ∆ ) 212 - 11 - 22. The coordination number (z) for the cubic lattice system is 6. For the monomer systems considered, all segments were treated as equal-sized spheres on lattices. Only energy parameters and mole fractions were taken into
Figure 8. Internal energy for a “lattice gas like” polymer/solvent system. Energy parameters used here are 11/kT ) -0.5 and 12/ kT ) 22/kT ) 0. The length of the polymer is 5. In this case, only the ADmulti model gives satisfactory energy of mixing. However, none of the models agree with the Monte Carlo simulations when the polymer concentration is very dilute.
account. For each system configuration, the first 4 000 000 steps were discarded for equilibration. Then, 10 000 000 steps were recorded to get thermodynamic properties. For polymer/solvent systems, the polymers were treated as chains of equal-sized spheres in adjacent sites, while solvent molecules were taken as monomers in the rest of the system. We did simulations on chains with chain lengths of 5 and 10. The other constraints are the same as the monomer systems. These Monte Carlo simulation data showed an interesting yet very important result; i.e., that for a binary monomer mixture the energy of mixing (∆U) is symmetric, independent of whether the interaction energies (11 and 22) are symmetric or asymmetric. These simulation results confirm assertions by Madden (1990) that this must be so and dispel contrary arguments (Lipson, 1991). However, the symmetric behavior of the energy of mixing is not present in polymer/solvent mixtures. Conclusions A comparison of local composition models has been made in this paper. The following can be concluded:
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Figure 10. Internal energy for a “lattice-gas-like” polymer/solvent mixture. Energy parameters used here are 11/kT ) -0.1 and 12/ kT ) 22/kT ) 0.0. The length of the polymer is 10. Since the interaction energy is -0.1kT, therefore, it is less “nonideal” than the system in Figure 8. This can be seen by the better predictions of the AD model and the BGY model. However, NRSF and UNIQUAC still fail to give satisfactory predictions.
Figure 9. Internal energy for a polymer/solvent mixture. Energy parameters used here are 11/kT ) -0.5, 12/kT ) -0.3, and 22/ kT ) -0.4. The length of the polymer is 10. The behaviors of the models are similar to those in Figure 10. Again, the AD model gives the best prediction on the internal energy of mixing.
The activity coefficient models considered in this work all are based on two-liquid theories. The BGY model and the AD model also are two-fluid models for binary mixtures. For the Wilson equation, the UNIQUAC model, the BGY model, and the AD model, the internal energy of a mixture is a summation of a ratio of single summations. The NRTL model also is a two-fluid model. However, it is a model for the free energy instead of the internal energy. The NRSF model is a one-fluid model that uses double summation to describe the local composition for the internal energy of a mixture. For monomer mixtures, the Wilson equation and the UNIQUAC model are identical and the NRTL model and the BGY model are nearly identical. The AD model is similar to the BGY model, but with a different dependence on interaction energy parameters and a different Boltzmann factor. The multicomponent AD model is different from the others, but it is a two-fluid model (more precisely, it is an N-fluid model). The AD model gives the best agreement with simulation data for all systems studied, i.e., binary mixtures, lattice gas, and the situation when the interaction energy goes to zero. The multicomponent AD model predicts the multicomponent simulation data best. For polymer
Figure 11. Internal energy for a binary polymer mixture with holes. The system has two types of polymer chains: the first is of length 10, and the second is of length 5. Energy parameters used here are 11/kT ) -0.3, 12/kT ) -0.5, and 22/kT ) -0.2. The system has a constant volume fraction of holes of 20%. The ADbinary and the ADmulti models are better in predicting the internal energy than the others. Surprisingly, the ADbinary model predicts almost the same energies as the ADmulti model. This is different from the results in Figures 5 and 6.
solutions, the BGY and NRSF models are in close agreement. The BGY model can be considered an extension of the NRTL model to polymer mixtures, and the UNIQUAC is an extension of the Wilson equation. Neither the Wilson equation nor the NRTL model give the proper size dependence for polymer mixtures. The UNIQUAC model consistently overpredicts the energy of mixing. The NRSF model shows an interesting concentration dependence. At the low concentration end, it reduces to the Wilson equation and the UNIQUAC model. At the high concentration end, it reduces to the NRTL and BGY models. The Wilson equation and the UNIQUAC model have a strong coordination number dependence. Lattice cluster theory (LCT) predicts that the coordination
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number dependence is first-order. (The 1/z terms in LCT correct for the fact that LCT is written in volume fractions instead of surface fractions.) The AD model, the NRTL model, the BGY model, and the NRSF model all have the first-order coordination number dependence. Acknowledgment Financial support of this research by the Division of Chemical Sciences of the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract No. DE-FG02-87ER13777 is gratefully acknowledged. Literature Cited Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116. Aranovich, G. L.; Donohue, M. D. A New Model for Lattice Systems. J. Chem. Phys. 1996, 105, 7059. Aranovich, G. L.; Hocker, T.; Wu, D. W.; Donohue, M. D. NonRandom Behavior in Multicomponent Soutions: Effects of Solute Size and Shape. J. Chem. Phys. 1997, 106, 10282. Cui, Y.; Donohue, M. D. Entropy and Energy of Mixing in Polymer Solutions: Simple Expressions that Approximate Lattice Cluster Theory. Macromolecules 1992, 25, 6489. Dudowitz, J.; Freed K. F.; Madden W. G. Role of Molecular Structure on the Thermodynamic Properties of Melts, Blends, and Concentrated Polymer Solutions. Comparison of Monte Carlo Simulations with the Cluster Theory for the Lattice Model. Macromolecules 1990, 23, 4803. Eckert, C. A.; Prausnitz J. M.; Orye R. V.; O’Connel J. P. Calculations of Multi-Component Vapour-Liquid Equilibria. Adv. Sep. Tech., Proc. Symp. 1965, 1, 75. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
Guggenheim, E. A. Mixtures; Oxford University Press: London, 1952. Lee, K. H.; Lombardo, M.; Sandler, S. I. The Generalized Van Der Waals Partition Function. II. Application to The Square-Well Fluid. Fluid Phase Equilib. 1985, 21, 177. Lee, K. H.; Sandler, S. I.; Patel, N. C. The Generalized Van Der Waals Partition Function. II. Local Composition Models For A Mixture of Equal Size Square-Well Molecules. Fluid Phase Equilib. 1986, 25, 31. Lipson, J. E. G. Polymer Blends and Polymer Solutions: A BornGreen-Yvon Integral Equation Treatment. Macromolecules 1991, 24, 1334. Lipson, J. E. G.; Brazhnik P. K. A Born-Green-Yvon Integral Equation Treatment of Compressible Lattice Mixture. Macromolecules 1993, 98, 8178. Madden, W. G. On the Internal Energy at Lattice Polymer Interfaces. J. Chem. Phys. 1990, 92, 2055. Madden, W. G.; Pesci, A. I.; Freed K. F. Phase Equilibria of Lattice Polymer and Solvent: Tests of Theories against Simulations Macromolecules 1990, 23, 1181. Prausnitz, J. M.; Lichtenthaler R. N.; Azevedo E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall Inc.: Englewood Cliffs, NJ, 1986. Renon, H.; Prausnitz J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968, 14, 135. Scott, R. L. Corresponding States Treatment of Nonelectrolyte Solutions. J. Chem. Phys. 1956, 25, 193. Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth Publishers: Kent, U.K., 1985. Wilson, G. M. Vapor-Liquid Equilibrium. XI: A New Expression for the Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86, 127.
Received for review August 11, 1995 Revised manuscript received September 15, 1997 Accepted September 18, 1997 IE950503R
