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Inclusion and Exclusion Approximations of Copolymer Solids Applied to Calculation of Solid-Liquid Transitions Hertanto Adidharma and Maciej Radosz* Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, Wyoming 82071-3295
A model of copolymer solids hypothesizing inclusion of the comonomer units in the crystalline domain is applied to calculation of solid-liquid transitions. Such an inclusion model is compared to an exclusion model hypothesizing exclusion of the comonomer units from the crystalline domain. Both models are used with the copolymer SAFT equation of state1 to represent the solid-liquid transition temperatures of polyethylene and poly(ethylene-co-octene-1) in propane, isooctane, and a mixture of isooctane + ethylene. The inclusion model is found to be more predictive and to have a correct high polymer-concentration limit, which is the melting temperature of pure polymer. Furthermore, the inclusion parameters, the fold energy and the energy penalty, are found to be solvent- and composition-independent, while the exclusion parameter, the crystallizability, is found to be both solvent- and composition-dependent. Introduction Polymeric materials are commonly synthesized and processed in the form of subcritical or supercritical fluid solutions. Such polymer solutions easily separate into one or more phases; they exhibit rich, pressure- and temperature-sensitive phase behavior. The prediction of such phase behavior at all of the stages of synthesis and processing is needed to be able either to avoid undesirable phase separations or to induce desirable phase separations. In this work, we focus on solidliquid-phase separations. An example of an undesirable solid-liquid separation is precipitation of the least soluble polymer fraction which leads to the reactor, pipeline, or heat-exchanger fouling, a serious but poorly understood and poorly recognized technological problem that cuts across virtually all macromolecular processes. Examples of desirable phase separations are those utilized in heterogeneous polymerization, solvent recovery, fractionation, morphology control, and particle formation. While there is room for improvement, and plenty of work in progress, our understanding of fluid-liquid separations is relatively well established. By contrast, our understanding of solid-liquid separations is poor. To understand and predict the solid-liquid separation for partially crystallizable polymer or copolymer in solution, one needs predictive thermodynamic models. Copolymer SAFT, which stands for statistical associating fluid theory, is one example of such a thermodynamic model in the form of an equation of state that explicitly accounts for variable polymer microstructure due to the variability in comonomer incorporation.1 This equation of state has been used to describe the solidliquid behavior of polymer solutions.2-4 In addition to an equation of state, one also needs to characterize the solid structure and the nature of equilibrium between the liquid and solid phases. This is not a trivial task because the polymeric solids exhibit a lamellar morphology, which tends to be very complex and thus poorly defined. The issues associated with the * Corresponding author. E-mail:
[email protected]. Tel: 307-766-2500. Fax: 307-766-6777.
Figure 1. Exclusion model. The solid box represents the solid phase, solid lines represent the ethylene monomer units, solid circles represent the comonomer units, and open circles represent the solvent molecules.
structure of polymeric solids include the temperature history, size-dependent phase stability, and metastable phases. Hence, we need simplifying approximations of the solid structure and its interaction with the solvent. An example of one approximation3-5 is shown in Figure 1: the solid polymer is divided into two discrete domains, crystalline and amorphous. We refer to this approximation as the exclusion approximation. The crystalline domain, shown in a box, contains homopolymer units only; the copolymer units are excluded. The amorphous domain, shown as a collection of flexible chains with solid circles (comonomer units), contains both homopolymer and copolymer units. The balance of Figure 1 is to represent the solution phase containing the molecules of solvent and polymer. An example of another approximation is shown in Figure 2: the solid polymer forms a single crystalline domain, shown in a box, that contains both the homopolymer and copolymer units; the comonomer units are included in the crystalline domain. We refer to this approximation as the inclusion approximation. The comonomer units can be viewed as defects that introduce a degree of disorder within the crystal.6 As in Figure 1, the balance of Figure 2 is to represent the solution phase containing the molecules of solvent and polymer.
10.1021/ie0107791 CCC: $22.00 © 2002 American Chemical Society Published on Web 03/07/2002
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Regardless of the approximation, one needs to consider both types of solid-liquid transitions, crystallization and melting. Crystallization is the transition from a disordered state to an ordered one. Considering the fact that a variety of end states usually occur during crystallization, depending on the transition path, a thermodynamically unique crystalline state is hard to reach and define. Melting, on the other hand, has a common disordered state as the end state and, hence, is easier to model as an equilibrium process. Thus, in this work, we only consider the melting process, and all data presented are for melting. Exclusion Model The exclusion model has been commonly reported in the literature and proven to be suitable for the description of relatively structure-insensitive properties, e.g., density and heat of fusion. It has also been applied to solid-liquid equilibrium calculations of semicrystalline polymer-solvent systems.3-5 Because the solid in such calculations is usually assumed to have two phases, the solid-liquid equilibrium exists among three phases, i.e., the polymeric crystalline phase, the polymeric amorphous phase, and the solvent-rich liquid phase. One of the issues associated with this approach, if the equilibrium calculations are carried out by equating the fugacities, is how to define the fugacities of the crystalline and amorphous parts of the polymer molecule; the same molecule can exist in both the amorphous and crystalline phases. One could assume that the fugacity of the crystalline part of the polymer molecule is equal to the fugacity of the polymer molecule as a whole. If we also assume that there is no solvent in the solid phase, the equilibrium melting point of polymer in solution for the exclusion model can then be estimated from the equality of fugacities
f Sp ) ˆf Lp
(1)
where f Sp is the fugacity of a pure polymer in the solid phase and ˆf Lp is the fugacity of a polymer in the solution phase. As described by Pan and Radosz,3 a thermodynamic cycle leads to the following working equation:
ln
( ) [ φˆ Lp
xLp φLp
)-
(
)
]
∆Hu Tm ∆vP -1 + cu RTm T RT
Figure 2. Inclusion model. The solid box represents the solid phase, solid lines represent the ethylene monomer units, solid circles represent the comonomer units, and open circles represent the solvent molecules.
To obtain Tm, following Flory,7 Pan and Radosz3 assumed that the crystal thickness is infinite. Thus, the melting point of a pure copolymer is given by
1 R 1 ) ln(1 - X) Tm Tm° ∆Hu
where Tm° is the melting temperature of pure polyethylene with infinite crystal thickness (≈415 K) and X is the mole fraction of comonomer units in the copolymer. Inclusion Model The inclusion model accounts for the presence of the comonomer defects by introducing an energy penalty that quantifies the effect of defects on the free energy of fusion. In contrast to the exclusion model, the inclusion model does not pose a conceptual dilemma of having the same molecule present in two different phases. If we assume that the solid phase is pure polymer, the equilibrium melting point of a polymer in solution for this model is obtained from eq 1. It can be shown that the result obtained from the thermodynamic cycle developed for the exclusion model can also be used for the inclusion model. A working equation analogous to eq 2 is as follows:
ln (2)
where φˆ Lp is the fugacity coefficient of a polymer in solution, φLp is the fugacity coefficient of a pure liquid polymer, xLp is the polymer mole fraction in solution, ∆Hu is the enthalpy of melting per mole crystal unit (≈8220 J/mol), Tm is the melting temperature of a pure polymer, T is the melting temperature of a polymer in solution at the specified pressure P, R is the gas constant, ∆v is the difference in molar volume between pure liquid and pure solid polymers at T (≈4.937 cm3/ mol), u is the number of crystallizable units in a polymer molecule (for example, the number of ethylene units in ethylene copolymers), and c is a model parameter quantifying the fraction of the crystallizable units that are actually crystallized, which is fitted to experimental solid-liquid data.
(3)
( ) [ φˆ Lp xLp φLp
)-
(
]
)
∆Hu Tm ∆vP u -1 + RTm T RT
(4)
Note that eq 4 does not need parameter c because, in the inclusion model, all crystallizable units are crystallized. To calculate Tm, the Gibbs energy of fusion of a pure polymer should be corrected to account for the incorporation of comonomer units in the crystal (there is an excess free energy of the defect) and the finite thickness of the crystal.8,9 Following Sanchez-Eby,9 the Gibbs energy of fusion of the copolymer crystal with infinite thickness is given by
[
∆G∞ ) ∆G∞° - RT
(
) ( )]
cXc 1 - Xc + + (1 - Xc) ln RT 1-X Xc Xc ln X
(5)
where Xc is the mole fraction of comonomer units in the crystal, c is the free energy penalty for the defect
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created by incorporating a comonomer unit in the crystalline lattice, and ∆G∞° is the Gibbs energy of fusion of the pure crystal (homopolymer) given by
(
∆G∞° ) ∆Hu 1 -
)
T Tm °
(6)
Because we assume that both monomer and comonomer units can be present in (are not excluded from) the solid phase, the mole fraction of comonomer in the crystal is the same as the overall composition of the comonomer units in the polymer, i.e., Xc ) X. Thus, eq 5 reduces to
∆G∞ ) ∆G∞° - cX
(7)
For a finite thickness of the crystal, the Gibbs energy of fusion is corrected further by introducing the fold free energy, which is the energy contribution that accounts for creating a fold. Thus, the Gibbs energy of fusion of a thin copolymer crystal is given by
σv ∆G ) ∆G∞° - cX - 2 l
( )
(
c 2 σv X ∆Hu l ∆Hu
wt % octene branch density Mw (kg/mol) Mw/Mn EO-0-120K EO-4-100K EO-14-120K EO-14-150K
0 13.9 38.7 38.7
0 4 13.6 14
)
(9)
This equation is similar to the Gibbs-Thomson equation.10 Note on Solvent Solubility As described above, the exclusion and inclusion models assume that the solid phase formed is a pure polymer. This assumption is realistic for the wax precipitation where the solid phase is found to be pure wax.11,12 However, this assumption may not be very realistic for the polymeric solid phases that contain some solvent. To account for the solubility of solvent in the solid polymer, one can use a swelling theory. Most of the polymer-swelling approximations postulate that the equilibrium exists among three phases, i.e., the solvent-free crystalline phase, the solvent-containing amorphous phase, and the solvent-rich liquid phase.13 We obtain the solubility of solvent in the amorphous phase by solving the equality of fugacities in the liquid and amorphous phases. However, the solid-liquid transition temperature is governed only by the formation of the crystalline phase, which requires the equality of the polymer fugacity in the crystalline and liquid phases, as in eq 1. If we assume that the crystalline phase does not contain comonomer units, the solidliquid transition temperature is calculated the same way as in the exclusion model. Thus, the polymer swelling approximations suggest that the solubility of solvent in the solid polymer should not affect the solidliquid transition temperature. Reminiscent of how we approach calculating the fluidphase equilibria, we may assume to a first approximation that the solvent and the polymer form an
119.6 104.3 120.0 154.0
1.19 2.34 1.44 2.04
Table 2. SAFT Parameters for EO + Isooctane and EO + Ethylene + Isooctanea ethylene isooctane EO-0-120K
EO-4-100K EO-14-150K
m
voo
uo/k
source
1.464 4.903 4688.7
18.157 15.394 12.000
212.06 213.44 210.00
19 this work 4
mb
(voo)bb
(voo)br
(uo/k)bb
(uo/k)br
2081.2 3872.2
12.000 12.000
12.475 12.475
210.00 210.00
202.72 202.72
4 4
a bb: backbone. br: branch. b Calculated from the correlation for hydrocarbons.19
ideal solid solution. This assumption leads to
zpf Sp ) ˆf Lp
(8)
where σ is the fold surface free energy, l is the crystal thickness, and v is the molar volume of the crystal. In the state of equilibrium, ∆G is set equal to zero. When eq 6 is substituted into eq 8, the melting temperature of a pure copolymer can be obtained as follows:
Tm ) Tm° 1 -
Table 1. Properties of EO Used
(10)
where zp is the mole fraction of polymer in the solid phase. Using the inclusion model, we easily obtain
ln
( ) [ φˆ Lp xLp φLp
)-
(
)
]
∆Hu Tm ∆vP -1 + u + ln(zp) (11) RTm T RT
Because the second term on the right-hand side of eq 11 is very small compared to the other terms, in this approximation, the solubility of solvent in the solid polymer should not have a significant effect on the solid-liquid transition temperature. For the record, there have been some theoretical approaches proposed14,15 to approximating the nonideality of solid solutions, but none of them have been extended to polymer solutions. Even for small molecules, they require Monte Carlo simulations to calculate the partition functions and other thermodynamic quantities. Results and Discussion To demonstrate the exclusion and inclusion models for correlating and predicting the solid-liquid equilibrium data, we select a few representative systems originally presented by Chan et al.,4 i.e., polyethylene and poly(ethylene-co-octene-1) (EO) in propane, and by Luszczyk and Radosz,16 i.e., polyethylene and EO in isooctane and isooctane + ethylene. All of the data were measured in a batch optical cell with a transmitted-light probe described by Chan et al.4 The properties of polymers used in this work are listed in Table 1. We label each polymer as in Chan et al.,4 for example, EO14-120K is EO with a branch density of 14, i.e., 14 branches per 100 ethyl units in the backbone, and a molecular weight of 120K. Both models call for an equation of state to calculate the fugacity coefficients of a pure liquid polymer and a polymer in solution. We use the copolymer SAFT equation of state1 in this work. The SAFT parameters and binary interaction parameters for the EO + propane system are taken from work by Chan et al.,4 whereas those for EO + isooctane and EO + isooctane + ethylene systems are derived in this work and listed in Tables 2 and 3.
Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1777 Table 3. Binary Interaction Parameters kij as Empirical Functions of Temperature ethylene-isooctane backbone-ethylene branch-ethylene backbone-isooctane branch-isooctane b
kij b
source
0 0.064 + 0.0001(T - 403.15) 0.056 + 2.5 × 10-5(T - 393.15) 0.03897 - 6.64677 × 10-5T -0.02
this work 20 20 this work this work
T is the temperature in K.
Table 4. Solvent-Dependent Parameters of Equation 12 solvent
A1
A2
A3
propane ethylene + isooctane
0.66074 0.36778
0.18912 0.42222
0.07376 0.46054
The parameter c, needed for the exclusion model, is fitted to the experimental solid-liquid data. Because the parameter c quantifies the fraction of the crystallizable units that are actually crystallized, c is a function of both the comonomer composition and the type of solvent. As mentioned in work by Chan et al.,4 for a particular solvent, this parameter could be represented by the following function:
c ) A1 + A2e-(wco/A3)
Figure 3. Exclusion model parameter c as a function of polymer composition and solvent type. Solid circles are for EO in propane,4 and open circles are for EO in isooctane + ethylene.
(12)
where A1, A2, and A3 are solvent-dependent parameters and wco is the weight fraction of comonomer units in the polymer. In the comonomer composition range studied, the parameter c decreases as the comonomer composition increases, as expected, but it reaches a certain limiting value at high comonomer compositions. Examples of A1, A2, and A3 for EO in propane and isooctane + ethylene are shown in Table 4. Examples of c as a function of the comonomer weight fraction for these systems are shown in Figure 3. Examples of the predicted solid-liquid equilibrium curves in pressuretemperature coordinates are shown in Figure 4. The inclusion model, as represented by eq 9, has two parameters, i.e., the fold energy σv/l and the energy penalty c, which need to be fitted to experimental data. Because these two parameters represent the pure polymer properties, they should be solvent-independent. To obtain these two parameters, only two well-defined uniform samples are used: the NIST sample EO-0-120K and the cross-fractionated sample EO-14-120K. The fold-energy parameter is obtained by fitting the model to the melting temperature of the EO-0-120K + isooctane system. Because c is set to zero for this system, the fold energy can readily be estimated, and it is found to be 70 J/mol, which has a reasonable order of magnitude. Assuming that the fold-energy parameter does not depend on the system temperature and the comonomer composition of the polymer, the remaining energypenalty parameter is fitted to the melting temperature of EO-14-120K + propane. The energy-penalty parameter for this system is found to be 8000 J/mol, which also has a reasonable order of magnitude. Without further readjustments, we use these two parameters to predict the melting temperature of the other EO systems and solvents. Figure 4 shows these SAFT predictions. For uniform polymers, the inclusion model predicts accurately the melting temperatures in different solvents. We note that the exclusion model seems to be more accurate, but this is due to allowing c
Figure 4. Solid-liquid transition for EO in some solvents. EO data in propane are taken from Condo et al.21 and Chan et al.;4 EO data in isooctane and isooctane + ethylene are taken from Luszczyk and Radosz.16 Values in brackets are the solution composition in weight percent.
to be adjustable (see Figure 3), while the inclusion parameters are constant. For nonuniform systems, all of the predictions are within the experimental uncertainty of the data. A more demanding test is reported below. Using the inclusion parameters found from the data fitting described above, without further readjustment, we predict from eq 9 and plot in Figure 5 the melting temperature of pure copolymers versus the number of comonomer units per 100 carbons in the backbone. The points represent the experimental data for several copolymers taken from Richardson et al.17 and Kim et al.18 We also plot in Figure 5 the melting temperature of pure copolymers calculated from eq 3, which is used in the exclusion model. We find that the inclusion model represents the experimental data much better than the exclusion model. Figure 5 also suggests that the parameters of the inclusion model can be constant, regardless of the type of branches, which is very attractive for predicting other solid-liquid equilibria in copolymer solutions.
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Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 T ) temperature u ) number of crystallizable units in a polymer molecule v ) molar volume wco ) weight fraction of comonomer units in the polymer x ) mole fraction in the liquid phase X ) mole fraction of comonomer units in the copolymer Xc ) mole fraction of comonomer units in the crystal/solid phase z ) mole fraction in the solid phase Greek Letters c ) free energy penalty φ ) fugacity coefficient of a pure component φˆ ) fugacity coefficient of a component in solution σ ) fold surface free energy Superscripts
Figure 5. Melting temperatures of several EOs as a function of composition: experimental (squares, polypropylene;17 circles, poly(ethylene-co-butene-1);17 diamonds, poly(ethylene-co-pentene-1);17 crosses, EO18) and predicted (dashed line, exclusion model; solid line, inclusion model).
On the other hand, we find that the exclusion model does not predict the correct high-concentration limit, which is the melting temperature of pure copolymer, because of the inaccuracy of eq 3. The melting temperature of pure copolymer predicted using this equation tends to be realistic only for copolymers with low comonomer compositions. This is demonstrated in Figure 5: as the weight fraction of comonomer increases, the accuracy of eq 3 deteriorates. Conclusion The inclusion model, based on the Sanchez-Eby melting point depression theory, is compared to an exclusion model, which is based on the Flory melting point depression theory for calculating solid-liquid transitions in polymer solutions. Both models are used with the copolymer SAFT equation of state to represent the solid-liquid transition temperatures of polyethylene and EO in propane, isooctane, and a mixture of isooctane + ethylene. The inclusion model is found to be more predictive and to have a correct high polymer-concentration limit, which is the melting temperature of a pure polymer. Furthermore, the inclusion parameters, the fold energy and the energy penalty, are found to be solvent- and composition-independent, while the exclusion model parameter, the crystallizability, is found to be solvent- and composition-dependent. The inclusion model parameters also seem to be branch-type-independent, but this observation calls for more data. Acknowledgment This work was supported by NSF Grant CTS9908610. Notation c ) crystallizability (exclusion model parameter) f ) fugacity ˆf ) fugacity of a component in solution G ) Gibbs free energy ∆Hu ) enthalpy of melting per mole of crystal units l ) crystal thickness P ) pressure R ) gas constant
° ) pure crystal (homopolymer) L ) liquid phase S ) solid phase Subscripts ∞ ) infinite thickness m ) melting p ) polymer
Literature Cited (1) Banaszak, M.; Chen, C. K.; Radosz, M. Copolymer SAFT Equation of State. Thermodynamic Perturbation Theory Extended to Heterobonded Chains. Macromolecules 1996, 29, 6481. (2) Pan, C.; Radosz, M. Copolymer-SAFT Modeling of Phase Behavior in Hydrocarbon-Chain Solutions: Alkane Oligomers, Polyethylene, Poly(ethylene-co-Olefin-1), Polystyrene, and Poly(ethylene-co-Styrene). Ind. Eng. Chem. Res. 1998, 37, 3169. (3) Pan, C.; Radosz, M. Modeling of solid-liquid equilibria in naphthalene, normal-alkane and polyethylene solutions. Fluid Phase Equilib. 1999, 155, 57. (4) Chan, K. C.; Adidharma, H.; Radosz, M. Fluid-Liquid and Fluid-Solid Transitions of Poly(ethylene-co-octene-1) in Sub- and Supercritical Propane Solutions. Ind. Eng. Chem. Res. 2000, 39, 3069. (5) Harismiadis, V. I.; Tassios, D. P. Solid-Liquid-Liquid Equilibria in Polymer Solutions. Ind. Eng. Chem. Res. 1996, 35, 4667. (6) Wunderlich, B. Macromolecular Physics. Volume 1: Crystal Structure, Morphology, Defects; Academic Press: New York, 1973. (7) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (8) Helfand, E.; Lauritzen, J. I. Theory of Copolymer Crystallization. Macromolecules 1973, 6, 631. (9) Sanchez, I. C.; Eby, R. K. Thermodynamics and Crystallization of Random Copolymers. Macromolecules 1975, 8, 638. (10) Wunderlich, B. Macromolecular Physics. Volume 3: Crystal Melting; Academic Press: New York, 1980. (11) Won, K. W. Thermodynamic calculation of cloud point temperatures and wax phase compositions of refined hydrocarbon mixtures. Fluid Phase Equilib. 1989, 53, 377. (12) Coutinho, J. A. P.; Stenby, E. H. Predictive Local Composition Models for Solid/Liquid Equilibrium in n-Alkane Systems: Wilson Equation for Multicomponent Systems. Ind. Eng. Chem. Res. 1996, 35, 918. (13) Michaels, A. S.; Hausslein, R. W. Elastic Factors Controlling Sorption and Transport Properties of Polyethylene. J. Polym. Sci. C 1965, 10, 61. (14) Cottin, X.; Monson, P. A. Solid-fluid phase equilibrium for single component and binary Lennard-Jones systems: A cell theory approach. J. Chem. Phys. 1996, 105, 10022. (15) Hitchcock, M. R.; Hall, C. K. Solid-liquid-phase equilibrium for binary Lennard-Jones mixtures. J. Chem. Phys. 1999, 110, 11433. (16) Luszczyk, M.; Radosz, M. Unpublished data, 2000. (17) Richardson, M. J.; Flory, P. J.; Jackson, J. B. Crystallization and Melting of Copolymers of Polymethylene. Polymer 1963, 4, 221.
Ind. Eng. Chem. Res., Vol. 41, No. 7, 2002 1779 (18) Kim, M.; Phillips, P. J.; Lin, J. S. The Equilibrium Melting Points of Random Ethylene-Octene Copolymers: A Test of the Flory and Sanchez-Eby Theories. J. Polym. Sci. B 2000, 38, 154. (19) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (20) Chan, K. C.; Adidharma, H.; Radosz, M. Fluid-Liquid Transitions of Poly(ethylene-co-octene-1) in Supercritical Ethylene Solutions. Ind. Eng. Chem. Res. 2000, 39, 4370.
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Received for review September 18, 2001 Revised manuscript received January 14, 2002 Accepted January 15, 2002 IE0107791
