Modeling Copolymer Systems Using the Perturbed-Chain SAFT

with a statistical sequence of the monomers in the backbone. Copolymers with a ... equation of state for copolymers, where the repeat-unit composition...
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Ind. Eng. Chem. Res. 2003, 42, 1266-1274

Modeling Copolymer Systems Using the Perturbed-Chain SAFT Equation of State Joachim Gross,†,§ Oliver Spuhl,†,| Feelly Tumakaka,‡,⊥ and Gabriele Sadowski*,‡ Technische Universita¨ t Berlin, Fachgebiet Thermodynamik und Thermische Verfahrenstechnik, Strasse des 17. Juni 135, TK 7, 10623 Berlin, Germany, and Universita¨ t Dortmund, Lehrstuhl fu¨ r Thermodynamik, Emil-Figge-Strasse 70, 44227 Dortmund, Germany

The perturbed-chain SAFT equation of state is extended to heterosegmented molecules and is applied to copolymers with a well-defined (alternating) repeat-unit sequence as well as to systems with a statistical sequence of the monomers in the backbone. Copolymers with a statistical sequence of the constituting repeat units usually require an assumption on the sequence of neighboring repeat units within the chain. A simple approach for defining such repeat-unit arrangements is proposed. Systems containing polyolefine copolymers (poly(ethylene-co-propylene) and poly(ethylene-co-1-butene)) covering the complete range of copolymer composition (including both of the appropriate homopolymers) were modeled in a mixture with solvents. Good results were found for mixtures of copolymer/solvent systems using constant interaction parameters. Copolymers comprising both nonpolar and polar repeat units, for example, poly(ethylene-co-vinyl acetate) and poly(ethylene-co-methyl acrylate), require an interaction parameter correcting the interactions between repeat units of different types, which depends on the repeat-unit composition. 1. Introduction Most of today’s polymers are not strictly homopolymers, but are in fact copolymers, composed of different types of repeat units.1 Technical LLDPE (linear lowdensity polyethylene), for example, is usually produced with 1-octene or 1-hexene as additional constituents.2 Adding small amounts of different types of repeat units to the chain of these “modified homopolymers” is done to modify certain properties of the appropriate polymer. Another group of copolymers is produced using large or equal fractions of two or more types of monomers, leading to macromolecules with specific properties, independent of the behavior of either of the appropriate homopolymers.2 Depending on client demands, copolymers from a single manufacturer may be subject to variation of the repeat-unit composition. In both of the above-described cases it is desirable to provide an equation of state for copolymers, where the repeat-unit composition is taken into account. Such a model needs to be suitable for the entire range of repeat-unit composition and it should be suitable for extrapolations within. Earlier contributions toward this goal are manifold, for example, Song et al.,3 Gupta and Prausnitz,4 and Hasch et al.5 In a previous study, the authors have proposed the perturbed-chain SAFT (PC-SAFT) equation of state.6 * To whom correspondence should be addressed. Tel: ++49231-755 2635. Fax: ++49-231-755 2572. E-mail: G.Sadowski@ ct.uni-dortmund.de. † Technische Universita¨t Berlin. ‡ Universita¨t Dortmund. § Present address: BASF AG, Conceptual Process Engineering GIC/P-Q 920, 67056 Ludwigshafen, Germany. Fax: ++49621-60 73488. E-mail: [email protected]. | Fax: ++49-30-314 22406. E-mail: O.Spuhl@ vt.tu-berlin.de. ⊥ Tel: ++49-231-755 2635. Fax: ++49-231-755 2572. Email: [email protected].

Figure 1. Molecular model for a copolymer of type poly(R-co-β), comprised of segments R and β.

The three pure-component parameters required for nonassociating molecules were determined for numerous substances. The PC-SAFT model was shown to accurately describe vapor pressures, densities, and caloric properties of pure components. Comparisons to an earlier version of the SAFT equation of state (proposed by Huang and Radosz) revealed improvement for pure-component properties and for vapor-liquid equilibria of mixtures. A subsequent investigation was concerned with applying the PC-SAFT equation of state to polymer systems.7 Good results were obtained for liquid-liquid and vapor-liquid equilibria of binary and ternary polymer mixtures. Furthermore, the PC-SAFT model was applied to associating compounds and mixtures.8 In this work, the perturbed-chain SAFT equation of state will be extended to copolymers, aiming at vaporliquid and liquid-liquid phase behavior of copolymersolvent systems. 2. Molecular Model for Real Copolymers with Statistical Distributions of the Constituting Repeat Units The molecular model underlying the PC-SAFT equation of state assumes regular compounds and homopolymers to be chains of spherical segments of the same type. This molecular model is extended for copolymers by allowing different types of segments within the molecular chain, as depicted in Figure 1.

10.1021/ie020509y CCC: $25.00 © 2003 American Chemical Society Published on Web 02/11/2003

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1267 Table 1. Bonding Fractions Biriβ for a Copolymer i Comprising Segments r and β copolymer

repeat-unit composition

BiRiβ

BiRiR

Biβiβ

random random alternating

zi,β < zi,R zi,β > zi,R zi,β ) zi,R

2[(zi,βmi)/(mi - 1)] 2[(zi,Rmi)/(mi - 1)] 1

1 - BiRiβ - BiβiR 0 0

0 1 - BiRiβ - BiβiR 0

The theoretical framework for extending a SAFT-type equation of state to copolymers was first developed by Amos and Jackson9 for trimers and later for polysegmented molecules by Shukla and Chapman10 and by Banaszak et al.11 These earlier studies were concerned with copolymers, for which the sequence of the constituting repeat units in the copolymer backbone was welldefined. However, the exact sequence of all segments in a copolymer chain is not required in the SAFT framework. It is sufficient to define all pairs of segment-segment bonds, using two quantities: the segment fraction ziR and the bonding fraction BiRiβ. The segment fraction ziR is given as

zi,R )

mi,R mi

(1)

where miR is the segment number of segment-type R in chain i and mi represents the total number of segments constituting molecule i. The bonding fraction BiRiβ gives the fraction of bonds between segment-type R and β within chain i. Many technical copolymers show a statistical repeatunit distribution (also termed random sequence), where the exact arrangement of the repeat units is not known. Assumptions are required to make such copolymers readily accessible for thermodynamic modeling using the PC-SAFT equation of state. For a random copolymer of type “poly(R-co-β)” comprised of repeat units R and β (Figure 1), we assume the following: (I) If the segment fraction of constituent β within the backbone is smaller than that of R, all β-segments are bonded to neighboring R-segments, whereas β-β bonds are not considered. (II) In the case of a copolymer with an equal number of R- and β-segments, a strictly alternating sequence of R- and β-segments is considered. These assumptions represent a simple approach for modeling real copolymers with a statistical distribution of constituting repeat units. Table 1 summarizes the relations for the bonding fractions according to this molecular model. 3. Perturbed-Chain SAFT Equation of State for Copolymers The model development of the PC-SAFT equation of state was described in detail by Gross and Sadowski.6 Given the PC-SAFT equation of state for regular compounds and homopolymers, the extension to copolymers is just a modificationsas the studies of Shukla and Chapman10 and of Banaszak et al.11 for the SAFT model show. A nonassociating copolymer (index i) requires purecomponent parameters of all segment types that it consists of, namely, the segment diameter σiR, the number of segments miR of type R in the chain, and the energy parameter iR/k (the index R runs over all constituents of the copolymer). Because copolymers are comprised of different segment types, a rule analogous

to the mixing rules is needed (even) for a pure copolymer. Here, one-fluid mixing rules are adopted in the dispersion term. The accompanying combining rules are

iRiβ ) xiRiβ(1 - kiRiβ)

(2)

1 σiRiβ ) (σiR + σiβ) 2

(3)

An internal correction parameter kiRiβ may be defined, which corrects the cross-dispersive energy between different segment types, as given in eq 1. The required parameters for a mixture of poly(ethylene-co-propylene) and n-pentane as an example are the pure-component parameters of n-pentane, polyethylene, and those of polypropylene. Furthermore, three binary interaction parameters can be used. Two of these describe the homopolymer-solvent interactions and they can be obtained independent of the copolymer from binary data of homopolymer-solvent mixtures. The third interaction parameter accounts for interactions between the polyethylene and the polypropylene segments within the copolymer according to eq 2. Further details of the extension of PC-SAFT to copolymers is given in Appendix A. 4. Results This chapter presents modeling results for liquidliquid equilibria of copolymer systems obtained with the perturbed-chain SAFT equation of state. Details of purecomponent parameters, the copolymer’s molecular mass and concentration, and binary interaction parameters for the systems considered in this work are given in Tables 2-4. 4.1. Polyolefine Systems. Experimental cloud-point data for the three phase equilibria (LCEP: lower critical end point) of different poly(ethylene-co-propylene) (PEP) copolymers in a mixture with n-hexane was given by Charlet and Delmas.12 They investigated the influence of repeat-unit composition and gave measurements for polyethylene, for PEP copolymers with varying amounts of propylene in the PEP backbone, and for polypropylene. These measurements were used to determine the kiRiβ parameter of polyethylene and polypropylene segments in the PEP copolymer (kiRiβ ) -0.009). The kij parameter of the binary system HDPE/n-pentane was fitted to the VLLE of the respective mixture; kij of PP/ n-pentane was independently determined in a previous study.7 Figure 2 compares some of the data published by Charlet and Delmas with calculation results obtained from the PC-SAFT equation of state. Given the kiRiβ parameter for PEP, the copolymer is fully determined for the whole range of repeat-unit composition (i.e., from polyethylene to polypropylene). Figure 3 gives a comparison between experimental LCEP data of some PEP copolymers in a mixture with cyclopentane and calculation results from the PC-SAFT equation of state. The kij values for HDPE/cyclopentane and PP/cyclopentane were obtained from the respective homopolymer/cyclopentane mixture data. Using the

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Figure 2. Vapor-liquid-liquid cloud points for mixtures of random poly(ethylene-co-propylene) (PEP) and n-pentane with varying repeat-unit compositions of the PEP (wPEP ) 0.05, M ) 109-242 kg/mol, LCEP: lower critical end point). Comparison of LCEP data12 to calculation results of the PC-SAFT equation of state (Table 4).

Figure 3. Vapor-liquid-liquid cloud points for mixtures of random poly(ethylene-co-propylene) (PEP) and cyclopentane with varying repeat-unit compositions of the PEP (wPEP ) 0.05, M ) 109-242 kg/mol, LCEP: lower critical end point). Comparison of LCEP data12 to calculation results from the PC-SAFT equation of state (Table 4).

above-determined kiRiβ parameter for PEP, also the three phase equilibria in cyclohexane are well-described over the complete range of repeat-unit composition. The liquid-liquid equilibrium of PEP with an alternating repeat-unit sequence in a mixture with 1-butene is shown in Figure 4. The cloud-point measurements with varying molecular mass of PEP were conducted by Chen et al.13 The kij parameter for the polyethylene1-butene mixture was independently determined to be kij ) 0.028. Given the internal kiRiβ parameter for PEP (Figure 2), the only remaining binary kij parameter for polypropylene segments and 1-butene was fitted to be kij ) 0.03. The correlation results of the PC-SAFT equation of state are in good agreement with the experimental data for all molecular masses of PEP. Cloud-point measurements of PEP with an alternating repeat-unit sequence in a mixture with propylene, 1-butene, and 1-hexene are displayed in Figure 5. For the mixture of PEP with propylene, a fusion of the LCST (lower critical solution temperature) and the UCST (upper critical solution temperature) demixing areas is calculated from PC-SAFT, whereas the experimental data do not indicate the strong increase in demixing pressure with decreasing temperature.

Figure 4. Liquid-liquid equilibrium for mixtures of alternating poly(ethylene-co-propylene) (PEP) and 1-butene for varying molecular mass of PEP (wPEP ) 0.15). Comparison of experimental cloud points13 to calculation results of the PC-SAFT equation of state.

Figure 5. Cloud-point curves for mixtures of alternating poly(ethylene-co-propylene) (PEP) with varying solvents: propylene, 1-butene, and 1-hexene. (wPEP ) 0.15; M ) 26 kg/mol). Comparison of experimental cloud points13 to calculation results of the PCSAFT equation of state.

The phase behavior of mixtures of random poly(ethylene-co-1-butene) (PEB) and propane was investigated by Chen et al.14 They have carried out cloud-point measurements for the liquid-liquid equilibrium of this system at high pressures as depicted in Figure 6. The repeat-unit composition of the PEB copolymer ranges from polyethylene to almost pure polybutene (PEB with 97 mass % 1-butene). The comparison of experimental data to calculation results of the PC-SAFT equation of state is given in Figure 6. Measurements of polyethylene (PEB with 0 mass % 1-butene in the backbone) and propane were used to obtain the binary kij parameter for this binary system. The kij parameter for the interactions of polybutene segments and propane was determined from PEB data with 97 mass % 1-butene in the backbone. The remaining internal kiRiβ parameter correcting the interactions between polyethylene segments and polybutene segments of PEB was fitted to be kij ) 0.008. The cloud-point data indicate a significantly improved solubility of PEB in propane with increasing amounts of 1-butene repeat units in the PEB. Over the temperature range of about 200 K and over the entire range of repeat-unit composition the PCSAFT equation of state is in good agreement with the experimental data using constant kij parameters.

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Figure 6. Cloud-point curves for mixtures of poly(ethylene-co-1butene) (PEB) and propane with varying repeat-unit composition (from 0 to 97 mass %) of 1-butene (wPEB ) 0.053, M ) 62-120 kg/mol). Comparison of experimental cloud points14 to calculation results of the PC-SAFT equation of state (Table 4).

Figure 7. High-pressure equilibrium for mixtures of polyethylene (LDPE) with varying solvents: ethylene, ethane, propylene, propane, butane, and 1-butene. (wLDPE ) 0.05, Mw ) 106 kg/mol, Mn ) 20.1 kg/mol). Comparison of experimental cloud-point measurements5 to calculation results of the PC-SAFT equation of state. (LDPE-1-butene: open triangles and dashed line).

4.2. Poly(ethylene-co-methyl acrylate) Systems. The synthesis of the copolymers to be discussed in this section and in section 4.3 (poly(ethylene-co-methyl acrylate) and poly(ethylene-co-vinyl acetate), respectively) follows a radical polymerization reaction. When no comonomer is added to the polymer, this mechanism leads to low-density polyethylene (LDPE). Hence, purecomponent parameters7 of LDPE were used for the polyethylene segments within the copolymers. Figure 7 gives cloud-point measurements for LDPE in a mixture with several solvents in a pressuretemperature diagram. The PC-SAFT equation of state was applied to all of the appropriate mixtures with constant kij parameters. The calculation results are in good agreement with the experimental data for the given temperature range. The phase behavior of poly(ethylene-co-methyl acrylate) (EMA) in different solvents and cosolvents was extensively studied experimentally.5,15,16 The high-pressure equilibria of EMA-propylene, EMA-1-butene, and EMA-butane mixtures were investigated by Hasch et al.5 Cloud-point measurements for random copolymers with different amounts of methylacrylate (MA) repeat units are depicted in Figures 8-10. Compared with the

Figure 8. High-pressure equilibrium for mixtures of poly(ethylene-co-methyl acrylate) (EMA) and propylene for different repeat-unit compositions of the EMA. (wPolymer ) 0.05). Comparison of experimental cloud-point measurements15 to calculation results of the PC-SAFT equation of state. (EMA (0% MA) is equal to LDPE: open diamonds and dashed line).

Figure 9. High-pressure equilibrium for mixtures of poly(ethylene-co-methyl acrylate) (EMA) and 1-butene for different repeat-unit compositions of the EMA. (wPolymer ) 0.05). Comparison of experimental cloud-point measurements5 to calculation results of the PC-SAFT equation of state. (EMA (0% MA) is equal to LDPE: open diamonds and dashed line).

Figure 10. High-pressure equilibrium for mixtures of poly(ethylene-co-methyl acrylate) (EMA) and butane for different repeat-unit compositions of the EMA. (wPolymer ) 0.05). Comparison of experimental cloud-point measurements5 to calculation results of the PC-SAFT equation of state. (EMA (0% MA) is equal to LDPE: open diamonds and dashed line).

phase behavior of LDPE-propylene, the demixing pressure at first declines when MA repeat units are added to the polymer chain (see 25 mass % of MA, Figure 8). An inversion of this behavior is observed for a further increase of MA repeat units, leading to an elevation of the demixing pressure (see 58 and 68 mass % of MA, Figure 8). Because no data of poly(methyl acrylate) (PMA) in either of the solvents were available, the pure-compo-

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Table 2. Pure-Component Parameters of the Perturbed-Chain SAFT Equation of State for Polymers binary system reference

polymer

m/Mb (mol/g)

σ (Å)

/k (K)

AAD% F

P range (bar)

solvent

polyethylene (HDPE)a polyethylene (LDPE)a,c polypropylenea polybutenea poly(vinyl acetate) poly(methyl acrylate)

0.0263 0.0263 0.02305 0.014 0.03211 0.03088

4.0217 4.0217 4.1 4.2 3.397 3.5

252.0 249.5 217.0 230.0 204.6 275

1.62 1.14 5.55 ∼30d 5.95 3.27

1-1000 1-1000 1-981 1-1000 1-1000 1-1000

ethylene ethylene n-pentane 1-butene cyclopentane 2-octanone

de Loos et al.21 Mu¨ller22 Martin et al.23 Koak et al.24 Beyer et al.19 Witteman17

a Previously published in ref 7. b The segment number m depends on the molecular mass M of a polymer. It is determined from (m/M) by multiplying the molecular mass M. c The pure-component parameters of LDPE were determined by modifying parameters of HDPE. d A particular emphasis was put on binary data for the regression of the polybutene pure-component parameters.

Table 3. Characteristics of Copolymers Considered in This Work molecular mass copolymer composition of copolymer system Mj (g/mol) mass % of constituent β

segment fraction zβ

134 000 195 000 109 000 236 000 154 000 145 000 242 000

0 26 33 47 57 75 100

of propylene of propylene of propylene of propylene of propylene of propylene of propylene

)HDPE

PEB-propane14

120 000 62 000 96 000 85 000 91 000 90 000

0 8 32 52 88 97

of 1-butene of 1-butene of 1-butene of 1-butene of 1-butene of 1-butene

EMA-propylene15, EMA-1-butene5, and EMA - butane5

106 000 74 800 108 900 110 400

0 25 58 68

EVA-cyclopentane19

106 000 256 000 254 000 285 000 124 800

0 28 50 70 100

PEP-n-hexane12 and PEP-cyclohexane12

bonding fraction BRR BRβ Bββ

0 0.235 0.302 0.437 0.537 0.724 1

1 0.529 0.397 0.125 0 0 0

0 0.471 0.603 0.875 0.925 0.551 0

0 0 0 0 0.075 0.449 1

)HDPE

0 0.044 0.2 0.366 0.796 0.945

1 0.911 0.599 0.268 0 0

0 0.089 0.401 0.732 0.408 0.110

0 0 0 0 0.592 0.890

of methyl acrylate of methyl acrylate of methyl acrylate of methyl acrylate

)LDPE

0 0.281 0.619 0.714

1 0.437 0 0

0 0.563 0.763 0.572

0 0 0.237 0.428

of vinyl acetate of vinyl acetate of vinyl acetate of vinyl acetate of vinyl acetate

)LDPE

0 0.322 0.550 0.740 1

1 0.356 0 0 0

0 0.644 0.901 0.520 0

0 0 0.099 0.480 1

nent parameters of PMA were determined from binary data of PMA-2-octanone and PMA liquid-density data.17 The pure-component parameters are given in Table 2. The binary interaction parameters kij of LDPE/solvent binaries were given in Figure 7. The kij parameters of PMA/solvent systems were fitted to the data displayed in Figures 8-10. The pronounced inversion of demixing pressure observed in Figure 8 is a result of specific molecular interactions. Small amounts of MA repeat units act as a mediator between propylene and the copolymer. This observation may be attributed to multipolar interactions introduced with the MA repeat units.15 However, higher concentrations of MA in the EMA reinforce a phase splitting because the intramolecular interactions of the methyl acrylate groups in the coplymer chain leads to a more coiled chain configuration which results in a poor solubility in the solvent.15 To describe the inversion of the demixing pressure with increasing amounts of MA repeat units quantitatively, an internal kiRiβ interaction parameter is required, which depends on the fraction of MA repeat units in the copolymer chain (0.5269xMA2 + 0.0506xMA + 0.1117). Using this simple function, the model is able to give good qualitative description of the observed nonmonotonic behavior. Details of the polymer are given in Table 3; parameters of the PC-SAFT model are listed in Tables 2 and 4. Jog and Chapman18 proposed an approach for modeling polar polymers, which may improve the results for

)PP

)PVA

varying repeat-unit compositions. This will be the subject of future studies. 4.3. Poly(ethylene-co-vinyl acetate) Systems. The phase behavior of poly(ethylene-co-vinyl acetate) (EVA) in a mixture with cyclopentane was experimentally investigated by Beyer et al.19 Cloud-point measurements for this system are depicted in Figure 11. Again, increasing amounts of vinyl acetate (VA) repeat units in the polymer chain initially cause a decrease of demixing pressure, whereas a further increase of VA leads to a significant elevation of demixing pressure. The pure-component parameters for poly(vinyl acetate) (PVA) and the kij of PVA-cyclopentane (kij ) 0.0233) were determined by simultaneously fitting liquid density data of PVA and binary data of PVAcyclopentane. The binary parameter for LDPE-cyclopentane was obtained from experimental cloud points of LDPE-cyclopentane19 (Figure 11). The remaining internal kiRiβ interaction parameter correcting the interaction of LDPE segments and PVA segments was fitted to the copolymer data shown in Figure 11. The PC-SAFT equation of state requires a kiRiβ parameter, which is a linear function of the amounts of VA in the copolymer chain (Table 4). The temperature behavior of this system is well-described from the PC-SAFT equation of state. Conclusion The perturbed-chain SAFT equation of state was applied for modeling the phase equilibria of ethylene-

Ind. Eng. Chem. Res., Vol. 42, No. 6, 2003 1271 Table 4. Binary Interaction Parameters of the PC-SAFT Equation of State for Polymer and Copolymer Systems copolymer system

segment-segment pair

kiRjβ

PEP-n-pentane

HDPE-n-pentane PP-n-pentane ethylene segment-propylene segment

0.0073 0.0137 -0.009

PEP-cyclopentane

HDPE-cyclopentane PP-cyclopentane ethylene segment-propylene segment

-0.023 0.001 -0.009

PEP-propylene

HDPE-propylene PP-propylene ethylene segment-propylene segment

0.029 0.029 -0.009

PEP-1-butene

HDPE-1-butene PP-1-butene ethylene segment-propylene segment

0.001 0.0335 -0.009

PEP-1-hexene

HDPE-1-hexene PP-1-hexene ethylene segment-propylene segment

0.004 0.004 -0.009

PEB-propane

HDPE-propane PB-propane ethylene segment-1-butene segment

0.0206 0.025 0.008

EMA-propylene

LDPE-propylene PMA-propylene ethylene segment-methyl acrylate segment

0.0257 0.078 0.5269xMA2 + 0.0506xMA + 0.1117a

EMA-1-butene

LDPE-1-butene PMA-1-butene ethylene segment-methyl acrylate segment

0.013 0.082 0.5269xMA2 + 0.0506xMA + 0.1117a

EMA-butane

LDPE-butane PMA-butane ethylene segment-methyl acrylate segment

0.011 0.09 0.5269xMA2 + 0.0506xMA + 0.1117a

EVA-cyclopentane

LDPE-cyclopentane PVA-cyclopentane ethylene segment-vinyl acetate segment

homopolymer systems

LDPE-ethane LDPE-propane LDPE-butane LDPE-ethylene LDPE-propylene LDPE-1-butene

-0.016 0.0233 0.1431 - 0.1548wVAb 0.0325 0.02 0.011 0.039 0.0257 0.013

a x MA is the mole fraction of methyl acrylate monomers in the EMA copolymer. Thus, it is a pure-component property and should not be mixed up with the amount of polymer in solution. b wVA is the mass fraction of vinyl acetate monomers in the EVA copolymer. Thus, it is a pure-component property and should not be mixed up with the amount of polymer in solution.

Figure 11. Liquid-liquid equilibrium for mixtures of poly(ethylene-co-vinyl acetate) (EVA) and cyclopentane for different repeat-unit compositions of the EVA. (wPolymer ) 0.06). Comparison of experimental cloud-point measurements19 to calculation results of the PC-SAFT equation of state. (EVA (0% VA) is equal to LDPE: open diamonds and dashed line).

based copolymer systems such as poly(ethylene-copropylene) (PEP) and poly(ethylene-co-1-butene) (PEB), as well as for copolymers consisting of polar and

nonpolar repeat units such as poly(ethylene-co-vinyl acetate) (EVA) and poly(ethylene-co-methyl acrylate) (EMA). The PC-SAFT equation of state was extended to copolymers (i.e., to heterosegmented chain molecules) following the work of Banaszak et al.11 and Shukla and Chapman.10 One-fluid mixing rules are adopted for the dispersion contribution to the perturbed-chain SAFT equation of state for copolymer molecules. The copolymer concept of the perturbed-chain SAFT requires purecomponent parameters of the respective homopolymers composing the copolymer chain and one additional parameter correcting the interactions between different segment types in chains. Copolymers with a statistical distribution of the constituting repeat units require an assumption on the pairwise sequence of the repeat units within the chain. A simple approach for defining the segment-segment sequence is proposed. Using this assumption, the PCSAFT equation of state was able to model the temperature and the molecular-weight dependence for various copolymer compositions. For systems containing polyolefine copolymers such as PEP and PEB, the parameter which corrects the interaction between different segment types in the copolymer chains is constant. Good agreement with experimental phase behavior was achieved for PEP systems with cyclopentane, n-hexane,

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propylene, 1-butene, and 1-hexene, as well as PEBpropane binary systems. In the case of copolymers composed of polar and of nonpolar repeat units such as EVA and EMA, the segment-segment interaction parameter showed dependence on the composition of the copolymer constituents. Simple monotonous functions for the segmentsegment interaction parameter were used to describe the observed nonmonotonic dependence of the demixing pressure with varying amounts of polar repeat unit in the copolymer. Acknowledgment

The segment fraction ziR is given by

zi,R )

Appendix A This section provides a summary of equations for calculating the residual Helmholtz energy of copolymer systems using the perturbed-chain SAFT equation of state. The hard-chain reference contribution was developed earlier and discussed by Shukla and Chapman10 and by Banaszak et al.11 To provide a consistent nomenclature, however, we include the hard-chain contribution to the equations presented here. Derivatives of the Helmholtz energy for calculating pressure, fugacity coefficient, and so forth are not given. Equations for such properties were derived in an earlier study,6 and the transfer of the copolymer concept to these relations is straightforward. In the following, a tilde (∼) will be used for reduced quantities; caret symbols (∧) indicate molar quantities. The reduced residual Helmholtz energy is given by

(A.5)

For the case of a copolymer with a statistical distribution of repeat units, the bonding fractions BiRiβ can now be determined from Table 1. Hard-Chain Contribution. The hard-chain reference term is

j a˜ hs a˜ hc ) m

The results of this study were obtained during a time when all authors were members of the Department of Thermodynamics and Thermal Separations at the Technical University of Berlin. We truly thank Prof. Wolfgang Arlt for his dedication in supporting our work. The authors are grateful to the Deutsche Forschungsgemeinschaft for supporting this work with Grants SAD 700/3 and SAD 700/5.

mi,R mi

hs (diRiβ) ∑i xi(mi - 1)∑R ∑β BiRiβ ln giRiβ

(A.6)

where the Helmholtz energy of the hard-sphere fluid is given by

( )

[

]

ζ23 ζ23 1 3ζ1ζ2 + - ζ0 ln(1 - ζ3) + a˜ ) ζ0 (1 - ζ3) ζ (1 - ζ )2 ζ32 3 3 (A.7) hs

and where m j is the mean segment number in the mixture

m j )

∑i ximi∑R ziR ) ∑i ximi

(A.8)

with the radial distribution function hs (diRjβ) ) giRjβ

(

)

diRdjβ 3ζ2 1 + + diR + djβ (1 - ζ )2 (1 - ζ3) 3

(

)

diRdjβ 2 2ζ22 (A.9) diR + djβ (1 - ζ )3 3

and where

a˜ res )

res

A NkT

(A.1)

where a residual property (res) is the actual property minus the ideal gas contribution of that property. The residual Helmholtz energy consists of the hardchain reference contribution, the dispersion contribution, and the association term:

a˜ res ) a˜ hc + a˜ disp + a˜ assoc

miR )

(m/M)iR

∑R mi,R

with n ) (0,1,2,3)

(

(

iR diR ) σiR 1 - 0.12 exp - 3 kT

)

(A.11)

Dispersion Contribution. The dispersion term is given by

(A.3) j ,η) - 2πFI1(m

(A.4)

(A.10)

The temperature-dependent segment diameter diR of segment type R is given by

a˜ disp )

where MCopoly is the total molecular mass of the copolymer and wiR is the mass fraction of the repeat unit R. The (total) molecular segment number mi of copolymer i is the sum of all segments, according to

mi )

∑i ximi ∑R ziRdiRn

(A.2)

The segment number miR of segment type R is obtained from the pure-component parameter (m/M)iR as

wiRMCopoly

π ζn ) F 6

∑i ∑j

(

xixjmimj

∑R ∑β

∂Zhc πFm j 1 + Zhc + F ∂F

ziRzjβ

) ( )

∑i ∑j xixjmimj∑R ∑β ziRzjβ

-1

( ) iR jβ kT

σiRjβ3 -

I2(m j ,η)

iR jβ

2

σiR jβ3

kT with η ) ζ3 (A.12)

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and

(

) (

contributions). Nj,β may be related to the copolymer molecular mass analogous to eq A.19. The definition of Nj,β also allows rewriting of eq A.18, as

-1

∂Zhc 1 + Zhc + F ∂F

8η - 2η2 ) 1+m j + (1 - η)4 20η - 27η2 + 12η3 - 2η4 (1 - m j) [(1 - η)(2 - η)]2

)

(

-1

(A.13)

a˜

assoc

)

Ni,A ln X ∑i xi∑R ∑ A iR

AiR

-

)

1 - XAiR 2

(A.22)

where The association strength in eqs A.20 and A.21 is

6

j)) I1(η,m

ai(m j )ηi ∑ i)0

(A.14)

( ( ) )

∆AiRBjβ ) giRjβ(diRjβ)κAiRBjβσiRjβ3 exp

AiRBjβ -1 kT

(A.23)

6

j)) I2(η,m

∑ bi(mj )ηi

(A.15)

and the combining rules are

i)0

1 AiRBjβ ) (AiRBiR + AjβBjβ) 2

and

j ) ) a0i + ai(m j ) ) b0i + bi(m

j -2 m j -1 m j -1m a1i + a2i m j m j m j

(A.16)

j -2 m j -1 m j -1m b1i + b2i m j m j m j

(A.17)

The model constants a0i, a1i, ... are given in ref 6. Association Contribution. The association term is

a˜ assoc )

[(

∑i ∑R ∑ A xi

ln XAiR -

iR

) ]

XAiR 2

+

WiR 2

(A.18)

where WiR is the number of bonding sites of segmenttype R. WiR can in principle be a fixed value for copolymers, for example, if the bonding sites are located at both ends of the polymer chain only. When in contrast the association sites of a polymer are situated along the chain length (along the backbone of the polymer), it is reasonable to define the number of bonding sites as a ratio of sites to molecular mass. Analogous to the segment number, the appropriate pure-component parameter is (W/M)iR. For a copolymer of molecular mass MCopoly, the number of bonding sites is in this case determined from

WiR ) (W/M)iRziRMCopoly

(A.19)

The “monomer fraction” XAiR of association-site A located at segment-type R is obtained by solving the implicit equation

XB ∆A ∑j xj∑β ∑ B

XAiR ) (1 + F

jβ

)-1

iRBjβ

(A.20)

where the last summation in (A.20) runs over all association-sites B of segment-type β. It should be noted that the numerical problem is often drastically simplified when a variable Nj,β is introduced according to Kraska,20 representing the number of association sites of a certain type so that

Nj,βXB ∆A ∑j xj∑β ∑ B

XAiR ) (1 + F

jβ

)-1 (A.21)

iRBjβ

If a segment type of the chain possesses, for example, two association sites and it is present N times in the polymer chain, then eq A.21 has to be solved twice for this segment type rather than 2N times (and the summation over index B also reduces to the sum of two

κ

AiRBjβ

) xκA

iRBiR

(

xσiRiRσjβjβ

AjβBjβ

κ

1

(A.24)

)

/2(σiRiR + σjβjβ)

3

(A.25)

Abbreviations EMA ) poly(ethylene-co-methyl acrylate) EVA ) poly(ethylene-co-vinyl acetate) HDPE ) high-density polyethylene LDPE ) low-density polyethylene MA ) methyl acrylate PEB ) poly(ethylene-co-1-butene) PEP ) poly(ethylene-co-propylene) PB ) polybutene PMA ) poly(methyl acrylate) PP ) polypropylene PVA ) poly(vinyl acetate) VA ) vinyl acetate

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Received for review July 11, 2002 Revised manuscript received December 9, 2002 Accepted December 20, 2002 IE020509Y