Phase equilibria in polydisperse and associating copolymer solutions: Poly(ethene-co-(meth)acrylic acid)–monomer mixtures

Fluid Phase Equilibria 241 (2006) 113–123

Phase equilibria in polydisperse and associating copolymer solutions: Poly(ethene-co-(meth)acrylic acid)–monomer mixtures Matthias Kleiner a , Feelly Tumakaka a , Gabriele Sadowski a,∗ , Henning Latz b , Michael Buback b a b

Lehrstuhl f¨ur Thermodynamik, Universit¨at Dortmund, Emil-Figge-Str. 70, D-44227 Dortmund, Germany Institut f¨ur Physikalische Chemie, Universit¨at G¨ottingen, Tammannstraße 6, D-37077 G¨ottingen, Germany Received 17 October 2005; received in revised form 22 December 2005; accepted 22 December 2005 Available online 30 January 2006

Abstract Phase equilibria of copolymer–monomer mixtures have been modeled on the basis of experimental cloud-point pressure curves for poly(etheneco-acrylic acid) and for poly(ethene-co-methacrylic acid). The monomer system was ethene. In case of poly(ethene-co-acrylic acid), in addition, ethene–acrylic acid mixtures were used as the solvent. The association term of the PC-SAFT equation of state has been extended to account for the polydispersity of the copolymer samples. For this purpose, two pseudo-components were introduced for each copolymer. The amount and the molecular weight of these components were related to the number and weight averages of the copolymer material, Mn and Mw , respectively. For the strongly hydrogen-bonded systems, the influence on cloud-point pressures of copolymer composition and of molecular weight distribution can be adequately described by PC-SAFT modeling with temperature-independent parameters. © 2006 Elsevier B.V. All rights reserved. Keywords: Cloud point; Copolymerization; Ethylene–(meth)acrylic acid copolymers; Association; Polydisperse

1. Introduction Copolymers of ethene with highly polar comonomers, such as poly(ethene-co-acrylic acid) (further referred to as poly(Eco-AA)) and poly(ethene-co-methacrylic acid) (further referred to as poly(E-co-MAA)) have a series of attractive properties. The excellent hot tack strength, low seal initiation temperature, seal-through contamination properties combined with high puncture resistance make them versatile film materials for modern high-speed packaging. Moreover, these copolymers provide very good adhesion to polar substrates like nylon, aluminium foil, paper and glass. Therefore, they are widely used as seal and tie layer in laminated and coextruded structures, in wire and cable insulation and for injection moulded parts. The applica-

Abbreviations: AA, acrylic acid; E, ethylene unit (within a polymer molecule); EAA, poly(ethene-co-acrylic acid); EMAA, poly(ethene-comethacrylic acid); L, single fluid phase; LL, two-phase liquid region; MAA, methacrylic acid; poly(AA), poly(acrylic acid); poly(E-co-AA), poly(ethene-coacrylic acid); poly(E-co-MAA), poly(ethene-co-methacrylic acid); poly(MAA), poly(methacrylic acid) ∗ Corresponding author. Tel.: +49 231 755 2635; fax: +49 231 755 2572. E-mail address: [email protected] (G. Sadowski). 0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.12.027

tion range may be further extended by modifying the copolymer properties through neutralization of the acid groups after polymerization. The production of poly(E-co-AA) and poly(E-co-MAA) is performed at conditions close to the ones of high-pressure ethene polymerization, that is at pressures and temperatures up to 3000 bar and 573 K, respectively. The reaction proceeds at supercritical conditions with respect to the monomers. As a consequence, the reaction system may be continuously tuned in its physical variables, thus allowing for an almost unlimited multitude of process conditions. In order to describe and, more importantly, to optimize polymerization conditions, the kinetics and the phase behavior need to be modeled. Kinetic modeling on the basis of independently measured rate coefficients is already well advanced. The present paper deals with the modeling of the phase behavior for poly(E-co-AA)–ethene and poly(E-coMAA)–ethene systems up to high temperatures and pressures. The polymerizing systems should be homogenous to ensure safe processing conditions and constant product properties. After leaving the polymerization reactor, the reaction mixture has to be subjected to separation and recovery steps. The optimization of these operations also requires detailed information on the phase behavior, which is primarily affected by the type

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of monomer units, by copolymer composition, by the molecular weight distribution and, of course, by the type of solvent. Thermodynamic modeling and prediction of the phase equilibrium in copolymer–monomer mixtures is an important and demanding task. This is particularly true in systems where the comonomers largely differ in polarity, e.g., where one monomer is non-(di)polar and the other is highly polar or is even capable of association via hydrogen bonds, as is the case with ethene–(meth)acrylic acid copolymerizations. Powerful engineering models on the basis of statistical mechanics have been developed for describing such systems. Prausnitz’s group pioneered this field by introducing the Perturbed Hard Chain Theory (PHCT) [1,2], thereby setting a milestone for the development of thermodynamic models and providing the inspiration for the development of a series of equation of state (EOS) approaches, such as the Statistical Associating Fluid Theory (SAFT) [3,4], the Perturbed Hard Sphere Chain (PHSC) approach, which is another model developed by the Prausnitz’s group [5] and the PC-SAFT model [6]. Hasch and McHugh [7] used the SAFT–EOS approach proposed by Huang and Radosz [8] to model the phase behavior of poly(E-co-AA)–solvent systems. These calculations were carried out for monodisperse polymers, that is for the idealized situation of all polymeric species being of identical size. Hasch and McHugh used different binary interaction parameters for each polymer–solvent pair, dependent on the copolymers composition. In the present study, the PC-SAFT–EOS approach is used to model cloud-point curves for the systems poly(Eco-AA)–ethene and poly(E-co-MAA)–ethene. Both copolymer composition and copolymer molecular weight (distribution) were varied. In addition to fluid ethene, also ethene–acrylic acid mixtures were considered as solvents (for poly(E-co-AA)). The focus of our study is directed toward providing an adequate representation of measured cloud-point pressure (CPP) curves on the basis of physically reasonable quantities using a minimum of adjustable parameters, thus strictly following the teaching by Prausnitz. The PC-SAFT–EOS was proposed for the thermodynamic modeling of systems containing long-chain (polymer) molecules. The PC-SAFT model belongs to the SAFT family of equations. It takes a chain fluid as a reference fluid rather than a monomer fluid [6,9–12]. The model expressions were derived by applying the perturbation theory of Barker and Henderson [13] to chain molecules. PC-SAFT has already been successfully applied to phase equilibrium studies for solutions of several homopolymers and to ethylene copolymer systems without hydrogen-bonded interactions [12,14]. To account for copolymer polydispersity, the concept of pseudo-components has been applied [15–19]. As described by Tork et al. [20], these pseudo-components are generated such as to match the moments of the molecular weight distribution. The pseudo-component concept has already been applied in conjunction with considering hard-chain contributions and with treating the dispersion term of the SAFT-type equations of state by Gross [21], Jog and Chapman [18] and Behme et al. [19]. In Section 3.2, two approaches of applying the concept of pseudo-components to the association term within PC-SAFT are described. The so-

called classical approach treats each pseudo-component as being independent. The complexity of the system of equations drastically increases with the number of pseudo-components, which restricts this approach to systems with a very few such components. In case that a large number of pseudo-components needs to be considered, an alternative approach should be used which is similar to the one presented in the earlier work of Behme et al. [19]. This latter approach allows for calculations without increasing computational complexity. The CPP data that serve for fitting the PC-SAFT parameters and for demonstrating the applicability of this type of modeling have been determined in experiments where copolymer materials from steady-state copolymerization in an almost perfect continuously operated stirred tank reactor were used [14]. These copolymers are of high chemical homogeneity, i.e., the chemical compositions of individual copolymer molecules are very close to each other. Such material is perfectly suited for fundamental studies into the phase behavior of copolymer solutions. 2. Experimental With the exception of the CPP data for the poly(E-coMAA)–ethene system at high MAA content (10.3 mol%), which were measured within the present study, the experimental CPP curves considered for the modeling have already been presented in Refs. [14,22,23]. The experimental part thus can be very brief, as details about the CPP measurements are given in Ref. [14]. The poly(E-co-(M)AA) samples were synthesized under steadystate conditions in a continuously operated stirred tank reactor (CSTR) of about 50 mL internal volume at 2000 bar and at temperatures up to 550 K. Di-tert-butylperoxide was used as the initiator. Homogeneity of the reacting system was monitored by visual inspection through a sapphire window of 21.2 mm aperture which is fitted into the CSTR cell body. Copolymer composition was determined by elemental analysis. Polymer molecular weight distributions were measured by size-exclusion chromatography applied to the copolymer samples after complete methyl esterification of the ethylene–(meth)acrylic acid copolymers. Thus, poly(E-co-methyl acrylate) and poly(E-comethyl methacrylate) were actually subjected to SEC analysis rather than poly(E-co-AA) or poly(E-co-MAA). The reason for such indirect SEC analysis is that no suitable SEC columns for the acid group containing copolymers were available. CPP curves were measured on copolymer–ethene mixtures prepared from dried copolymer products in an autoclave with variable internal volume [14]. The design of the cell, which is of 171 mm length and of 22 and 80 mm inner and outer diameters, respectively, closely follows the construction principles used with optical transmission-type cells. The internal volume is separated from the pressurizing medium (n-heptane) with a movable piston sealed with a Kalrez® O-ring. The flat piston surface which faces the sapphire window (of 18 mm outer diameter and 10 mm length) is polished to assist the observation of phase separation in the internal volume. This volume is monitored by an endoscope camera (Optikon). The pictures are permanently displayed on a screen and are taped together with the associated pressure and temperature conditions for more detailed analysis

M. Kleiner et al. / Fluid Phase Equilibria 241 (2006) 113–123

after the experiment. The cell is heated electrically from outside. Temperature is measured within ±0.3 K via a sheathed thermocouple that sits in the fluid mixture under investigation. The copolymer–ethene mixture is stirred by a Teflon® -coated small magnet that is driven (through the non-magnetic cell wall) by a larger outside magnet. The internal volume is flushed several times with ethene after introducing the copolymer together with some inhibitor (2,6-ditert-butyl-4-methylphenol, for synthesis, Merck-Schuchardt). The amount of ethene used for preparing the copolymer–ethene mixture is accurately measured by weighing an auxiliary autoclave containing the ethene supply before and after feeding. The copolymer weight fraction with all CPP experiments was 3 ± 0.2 wt.%. Following the suggestion made by McHugh, the CPP is determined as the pressure at which the homogeneous mixture turns opaque to such an extent that the magnetic stir bar can no longer be seen. Each cloud-point pressure is measured at least three times. The reproducibility was better than ±10 bar in most cases and always better than ±20 bar. Phase-behavior studies into ethene-containing systems at high pressures and temperatures may be affected by ethene polymerization taking place during the thermodynamic experiment. This source of error has been minimized by adding a polymerization inhibitor, by measuring CPPs from low to high temperature and by duplicate CPP measurements during cooling the system after taking the CPP data at the highest temperature of an experimental series. Within such duplicate experiments, CPPs were reproduced within the above-mentioned accuracy of ±20 bar. In addition, copolymer films have been analyzed IRspectroscopically before and after taking CPP data. Comparison of the IR spectra provides no indication of any ethene polymerization during the CPP experiments presented in Ref. [14], which are underlying the present modeling study.

3. Theory Within Section 3.1, modeling of monodisperse polymer systems will be addressed. The extension to polydisperse copolymer systems is outlined in Section 3.2. Described in Section 3.3 is the determination of the pure-component parameters for poly(acrylic acid) (further referred to as poly(AA)) and poly(methacrylic acid) (further referred to as poly(MAA)).

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monomer units, the expression is extended to: 


 XAiα 1 aassoc Aiα ln X − xi = + kT 2 2 α i

(2)

Aiα

where XAiα is the fraction of the free monomer repeat units of type α in molecule i that are not bonded at the association site A. This fraction may be estimated via Eq. (3): ⎞−1


= ⎝1 + ρ · xj XBjβ · ∆Aiα Bjβ ⎠ ⎛

XAiα

j

β

(3)

Bj

with   εAiα Bjβ hs 3 ∆Aiα Bjβ = giαjβ (diαjβ ) · κAiα Bjβ · σiαjβ · exp −1 kT (4) hs (d where giαjβ iαjβ ) is the pair distribution function and σ iαjβ is the pair-wise segment diameter as given in the work of Gross et al. [12]. For cross-associating systems, the associating energy εAiα Bjβ and the associating volume κAiα Bjβ may be estimated via simple combination rules, without introducing additional binary parameters as the ones proposed by Wolbach and Sandler [24]:

εAiα Bjβ =

εAiα Biα + εAjβ Bjβ 2

κAiα Bjβ =

√ A B κ iα iα · κAjβ Bjβ ·

(5) 

√

σiα · σjβ (1/2)(σiα + σjβ )

3 (6)

To calculate XAiα , Eq. (3) has to be solved iteratively for each association site. For polymers and copolymers with a large number of association sites, this may lead to numerical problems. To avoid such problems, the so-called multi-sites concept is introduced. The main idea here is to define associationsite types for associating molecules, e.g., the association types A and B for two different hydrogen bonds, and to introduce the association-site numbers that accounts for the total number of each type of association site in the copolymer, e.g., N Aiα = association-site number for association-site type A on monomer unit of type α in copolymer i. Each type of association site should have the same association energy and Eq. (3) may be rewritten as:

3.1. Modeling of monodisperse associating copolymers

⎞−1


= ⎝1 + ρ · xj N Bjβ · XBjβ · ∆Aiα Bjβ ⎠ ⎛

The association contribution to the Helmholtz energy (aassoc /kT) within the SAFT approach is:

XAiα



 1 XAi aassoc Ai ln X − = + xi kT 2 2

It should be noted that the summation Bj has to be performed over all association-site types, and not over all association sites. In most cases, it is sufficient to define two types of association sites for each molecule monomer unit which largely simplifies the calculations. The expression for the association contribution to the Helmholtz free energy for monodisperse associating copolymers

i

(1)

Ai

where xi is the mole fraction of the component i and XAi is the fraction of free molecules i that are not bound to the association site A. For associating monodisperse copolymers with different

j

β

(7)

Bj

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reads: aassoc kT

=




xi N Aiα i

α Aiα

 ln XAiα −

XAiα 2

+

1 2

 (8)

As the association term is not written on a segment basis but on a molecular basis, the association-site number N Aiα in Eqs. (7) and (8) can be directly calculated from the monomer and polymer molecular masses according to: N Aiα =

wiα · Mi Miα

where xpi,j is the mole fraction of pseudo-component j within copolymer i. The summation over all association-site types Aiα is independent of the number of pseudo-components because the association energy parameter is assumed to be the same for all association sites. As a new variable, the mean association¯ i for copolymer i, which is analogous to the mean site number N segment number [19], is introduced:
¯i xpi,j · Npi,j :=N (11) j

(9)

where wiα is the mass fraction of the monomer units of type α contained in the copolymer i, Miα the molecular mass of the monomer unit of type α in copolymer i and Mi is the molecular mass of copolymer i.

The number of association sites of a certain type for the pseudocomponent j may be calculated from the mass fraction of the pseudo-component j in the pure copolymer wpi,j characterized by i and j and from the molecular mass of the pseudo-component Mpi,j according to: wiα · wpi,j · Mpi,j Miα

Npi,j = 3.2. Modeling of polydisperse associating copolymers To additionally account for the polydispersity of an associating polymer, the molecular weight distribution may be represented by pseudo-components. If the number of such pseudocomponents is small, e.g., if there are only two such components, each of them may be treated as an independent component and the classical approach described in the previous section may be used. The polydisperse binary copolymer–solvent system thus is treated as a ternary system containing pseudo-component 1, pseudo-component 2 and the solvent. Thus, three equilibrium conditions have to be fulfilled and a system of three equations with three unknowns has to be numerically solved. The so-called moment approach constitutes an alternative method, which is particularly well suited for systems with a large number of pseudo-components. The concept is similar to the one used by Cotterman and Prausnitz in their studies into the so-called continuous thermodynamics [25]. To apply this concept, we assume that • all pseudo-components have the same parameter value for the segment diameter (σ); • the dispersion energy parameter (ε/k) is the same for all pseudo-components; • the binary interaction parameter, kij , is the same for all pseudocomponent–solvent pairs; • the binary interaction parameter for the pseudo-components is zero; • all association sites on the copolymer molecules have the same association energy parameter, εAi Bj /k; • only one monomer repeat unit of the copolymer is associating. The association term for a polydisperse copolymer i which contains one associating monomeric species and which is described by j pseudo-components reads: 


aassoc XAiα 1 Aiα ln X − xi · xpi,j · Npi,j = + kT 2 2 i j Aiα (10)

(12)

where wiα is the mass fraction of the monomer units of type α within copolymer i. Miα is the molecular mass of the monomer unit α of copolymer i. Combining Eqs. (10) and (11), yields as the equation for the association contribution to the Helmholtz free energy: 

 aassoc XAiα 1 ¯i = ln XAiα − xi · N + (13) kT 2 2 i

with

⎛

XAiα = ⎝1 + ρ ·

Aiα




¯j xj · N

j




⎞−1 XBjβ · ∆Aiα Bjβ ⎠

(14)

Bjβ

The association contribution may thus be estimated on the basis ¯ i of the polydisperse polyof the mean association-site number N mer. The moment concept can be applied to the association term since the fractions of the pseudo-components, xpi,j have been eliminated. Applying the same scheme as used by Tork et al. [20] for a mixture of a polydisperse associating polymer yields the following system of four equations which simultaneously apply under equilibrium conditions (component 1 is the polymer). L1 I ϕ1j x1I
I 0 = 1 − II · x · II x1 j=1 p1,j ϕ1j

(15)

L1 I ϕ1j x1I
I · x · m · p1,j II x1II j=1 p1,j ϕ1j

(16)

L1 I ϕ1j x1I
II I ¯ 0 = N1 − II · x · Np1,j · II x1 j=1 p1,j ϕ1j

(17)

0=m ¯ II 1 −

0=1−

xiI ϕiI · , xiII ϕiII

i = 2, . . . , K

(18)

In Eq. (16), m ¯ II 1 is the average segment number of the pseudocomponents for polymer 1 in the second phase and m1,j is the

M. Kleiner et al. / Fluid Phase Equilibria 241 (2006) 113–123

segment number of the pseudo-components within the polymer. Eqs. (16) and (17) have to be introduced to calculate the mean segment number, m ¯ II 1 , and the mean association-site number, II ¯ N1 , in the second phase. According to this approach, a system of four equations has to be solved for a binary polydisperse copolymer–solvent mixture with two polymeric pseudo-components. In contrast to the classical approach, the number of equations is independent of the number of pseudo-components. For systems with up to three pseudo-components the classical approach is numerically favorable, whereas the moment concept is preferable for higher numbers of pseudo-components. To clarify the proposed approach, we give an example how to apply the method described to calculate the association contribution to the Helmholtz free energy of the system poly(E96.2 -coAA3.8 )–ethene, where the subscriptions indicate the copolymer composition, i.e., 96.2 mol% ethene and 3.8 mol% acrylic acid, corresponds to 9.21 mass%, in the copolymer. For each acrylic acid monomer repeat unit, two association-site types with the same association energy parameter are assumed (Section 3.3). Poly(E96.2 -co-AA3.8 ) is a polydisperse associating copolymer with Mn = 19,090 g/mol and Mw = 63,250 g/mol [14]. For this copolymer, two pseudo-components of equal composition are assumed (Section 4.1). The resulted molar mass of the first pseudo-component is 10,400 g/mol and of the second one is 116,100 g/mol. According to Eq. (12), the association-site number for pseudo-component 1: Np1,1 = (0.0921/72.06) × 0.5 × 10400 = 7 and for pseudocomponent 2: Np1,2 = (0.0921/72.06) × 0.5 × 116100 = 74. Eventually, the association contribution according to Eq. (10) or (13), respectively, can be calculated. 3.3. Pure-component parameters for poly(acrylic acid) and for poly(methacrylic acid) The classical approach with only two pseudo-components to represent poly(E-co-AA) will be used within the copolymer–monomer phase equilibrium calculations. For these estimates, the pure-component parameters for polyethylene (LDPE) as well as for poly(AA) need to be known. As been shown by Tumakaka et al. [26], the pure-component parameters for a homopolymer may be determined by fitting procedures applied to the densities of the polymer and to one set of polymer–solvent binary LLE data. The three pure-component parameters for LDPE have already been reported [10]. For poly(AA), five pure-component parameters need to be determined: the segment diameter σ, the segment number, m, the dispersion energy, ε/k, as well as the association volume, κAi Bi , and the association energy, εAi Bi /k. Poly(AA) densities were taken from Zoller and Walsh [27]. As binary LLE data for poly(AA) were not available for any solvent, parameter estimates were carried out as follows: • The segment diameter, σ, and the segment number, m, were determined by fitting to the densities of poly(AA). • The dispersion energy, ε/k, for poly(AA) was assumed to be identical to the ε/k value for LDPE. As a consequence, the

117

binary interaction parameter for poly(AA)–ethene is identical to the one for LDPE–ethene. • For each acrylic acid monomer repeat unit, two identical association sites were assumed as suggested by Huang and Radosz [8] for carboxylic acids. • The association volume, κAi Bi , for poly(AA) was assumed to be identical to the one for acrylic acid monomer, κAi Bi . • The association energy, εAi Bi /k, for poly(AA) was fitted to the LLE data of one binary copolymer–solvent system. The cloud-point data for the poly(E96.2 -co-AA3.8 )–ethene were selected for this fitting procedure (see Section 4.1). It is important to note, that no additional binary parameter has been introduced into the association term. Moreover, all parameters are constants, i.e., they are independent of temperature, molecular mass and of copolymer composition. The same procedure as described for poly(AA) was used to determine the pure-component parameters for poly(MAA). The association energy was obtained from fitting the binary LLE data of the poly(E95.2 -co-MAA4.8 )–ethene system. The entire set of pure-component parameters for polyethylene (LDPE), poly(AA) and poly(MAA) is summarized in Table 1. 4. Results 4.1. Poly(E-co-AA)–ethene Copolymer composition, number average molecular weight, Mn , weight average molecular weight, Mw , and polydispersity, Mw /Mn , of the ethylene–acrylic acid copolymers under investigation (and of one LDPE sample) are summarized in Table 2. To model the poly(E-co-AA)–ethene system according to the binary approach described by Gross et al. [12], the following parameters need to be known: • • • •

the pure-component parameters for ethene; the pure-component parameters for LDPE; the pure-component parameters for poly(AA); the binary interaction parameter, kij , for an LDPE segment and ethene; • the binary interaction parameter, kij , for a poly(AA) segment and ethene; • the binary interaction parameter, kij , for an LDPE and a poly(AA) segment.

The pure-component parameters for ethene and LDPE are listed in Table 1. The pure-component parameters for poly(AA) were determined as outlined in the preceding section. The binary interaction parameter, kij , for LDPE and ethene has already been reported by Gross and Sadowski [10]. This kij value is assumed to be identical to the one for poly(AA)–ethene (see preceding section and Table 3). The binary interaction parameter for a poly(AA) segment and an LDPE segment was set to zero. Fig. 1 shows modeling results for the poly(E96.2 -coAA3.8 )–ethene system. Cloud-point pressures were calculated under the assumption that poly(E96.2 -co-AA3.8 ) is monodisperse. The modeling was carried out for polymer molecular

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M. Kleiner et al. / Fluid Phase Equilibria 241 (2006) 113–123

Table 1 Pure-component parameters for LDPE, poly(AA) and poly(MAA) Polymer

Polyethylene (LDPE) Poly(acrylic acid) Poly(methacrylic acid)

m/M (mol/g)

0.0263 0.016 0.024

˚ σ (A)

4.02 4.20 3.70

εAi Bj /k (K)

ε/k (K)

249.5 249.5 249.5

κ Ai Bj

– 2035 2610

– 0.33584 0.07189

AAD% ρ (%)

p range (bar)

1.56 1.70 2.25

1–900 1–2000 1–2000

Table 2 Properties of LDPE and poly(E-co-AA) Polymer/copolymer

Acrylic acid content (mol%)

Mn (g/mol)

Mw (g/mol)

Mw /Mn

Reference

Polyethylene (LDPE) Poly(E96.2 -co-AA3.8 ) Poly(E94.5 -co-AA5.5 ) Poly(E93.3 -co-AA6.7 ) Poly(E97.6 -co-AA2.4 ) Poly(E96.9 -co-AA3.1 ) Poly(E96.3 -co-AA3.7 ) Poly(E95.4 -co-AA4.6 ) Poly(E98.8 -co-AA1.2 ) Poly(E97.6 -co-AA2.4 ) Poly(E97.03 -co-AA2.97 )

0 3.8 5.5 6.7 2.4 3.1 3.7 4.6 1.2 2.4 2.97

21900 19090 12610 9280 22713 19930 23443 23730 37500 30000 25000

103149 63250 42140 31910 258214 235146 227427 205024 183000 150000 126000

4.71 3.3 3.3 3.4 11.4 11.8 9.7 8.6 4.9 5.0 5.0

[14] [14] [14] [22] [22] [22] [22] [23] [23] [23]

weights, Mn with the number of association sites NEAA = 24 (dashed line) and Mw with the number of association sites NEAA = 81 (solid line). As can be seen from Fig. 1, no adequate representation of the cloud-point behavior is obtained. To improve the quality of the simulation, the polydispersity of the copolymer (see Table 2) has been taken into account by considering two pseudo-components per copolymer. The pseudo-components were determined such as to be consistent with the average values Mn , Mw and Mz of the copolymers under investigation (Sadowski [28] and Tork et al. [20]). As only Mn and Mw were known, additional assumptions had to be made to generate these pseudo-components. By defining the first parameter ␣ of the three-parameter Hosemann–Schramek distribution function:   b+1 w(M) = ac(b+1)/a Γ −1 M b exp(−cM a ) (19) a

ular weight, Mz , and higher order moments can be generated. The set of pseudo-components is then determined from these values. Within the present study, for copolymers with polydispersities, Mw /Mn , around 3, α was set to 1, which matches the Schulz–Flory distribution. For polydispersities around 10, α was set to 0.5, i.e., the distribution was assumed to be of Wesslau type. For intermediate polydispersities, at Mw /Mn values around 5, α was assumed to be 0.9. The resulting pseudo-component concentrations and characteristic molecular weights are listed in Tables 4–6 for α values of 0.5, 1.0 and 0.9, respectively. The

where w(M) is the molecular weight distribution, Γ the Gamma function and a, b and c are adjustable parameters of the analytical distribution, one can obtain several two-parameter distribution functions, such as the Wesslau (α = 0.5), the Schulz–Flory (α = 1) or the Gauss function (α = 2) which differ in the width of molecular weight distribution. By using the experimental Mn and Mw values, the two parameters of each of these distribution functions can be determined. Once they are known, the z-average molecTable 3 Binary interaction parameters for the poly(E-co-AA)–ethene systems System

kij

LDPE–ethene Poly(AA) unit–ethene LDPE unit–poly(AA) unit

0.04 0.04 0

Fig. 1. Cloud-point pressures for a mixture of ethene and poly(E-co-AA) with an acrylic acid content of 3.8 mol%. The polymer concentration is 3 wt.%. Experimental data from Ref. [14] (circles) are compared with results from PC-SAFT calculations for monodisperse systems using either Mn (dashed line) or Mw (full line) as the characteristic copolymer molecular mass.

M. Kleiner et al. / Fluid Phase Equilibria 241 (2006) 113–123

119

Table 4 Pseudo-components of the poly(E-co-AA) samples from Table 2 referring to a Schulz–Flory molecular weight distribution (with α = 1) Copolymer

Pseudo-components j

wpi,j a (–)

Mp1,j (g/mol)

Poly(E96.2 -co-AA3.8 )

1 2

0.5 0.5

10400 116100

Poly(E94.5 -co-AA5.5 )

1 2

0.5 0.5

6864 77416

Poly(E93.3 -co-AA6.7 )

1 2

0.5 0.5

5038 58782

a

Pseudo-component concentration in the solvent-free system.

Table 5 Pseudo-components of the poly(E-co-AA) samples from Table 2 referring to a Wesslau molecular weight distribution (with α = 0.5) Copolymer

Pseudo-components j

wpi,j a (–)

Mp1,j (g/mol)

Poly(E97.6 -co-AA2.4 )

1 2

0.673 0.327

15433 757509

Poly(E96.9 -co-AA3.1 )

1 2

0.673 0.327

13554 692543

Poly(E96.3 -co-AA3.7 )

1 2

0.669 0.331

15874 655259

Poly(E95.4 -co-AA4.6 )

1 2

0.666 0.334

16023 582165

a

Pseudo-component concentration in the solvent-free system.

number of association sites of each pseudo-component was estimated via Eq. (12). Fig. 2 shows simulated CPP curves for poly(E-coAA)–ethene systems at AA copolymer contents up to 6.7 mol% (E93.3 AA6.7 ). The copolymer concentration was 3 wt.%. Molecular weights Mw range from 32 to 63 kg/mol. The polydispersity is Mw /Mn ≈ 3.3. The symbols are measured CPP data from Ref. [14]. The simulated data, represented by the lines in Fig. 2, are calculated for the molecular weight range and for the polydispersity of the copolymers subjected to CPP analysis. The cloud-point pressures for the polyethylene (LDPE)–ethene system are included in Fig. 2. The satisfactory agreement of simulated and experimental CPP data demonstrates that, by considering polydispersity within CPP modeling, the phase behavior of copolymer systems with one type of monomer repeat unit being

Fig. 2. Cloud-point pressure curves for mixtures of ethene and poly(E-co-AA) with acrylic acid copolymer contents up to 6.7 mol%. The polymer concentration in the mixture is 3 wt.%. The copolymer molecular weight, Mw , ranges from 32 to 63 kg/mol. The polydispersity indices (Mw /Mn ) are close to 3.3. The dashed line represents the results of PC-SAFT calculations with the parameter values presented in Section 3.3. The experimental data from Ref. [14] (symbols) are compared with the predictions via PC-SAFT (lines).

strongly associating may be adequately predicted for copolymers varying in comonomer composition. Experimental CPP curves for poly(E-co-AA)–ethene systems reported by Beyer and Oellrich [22] are depicted in Fig. 3. Copolymer composition varies from 2.4 mol% (E97.6 AA2.4 ) to 4.6 mol% acrylic acid (E95.4 AA4.6 ). Both copolymer concentration (5 wt.%) and copolymer molecular weight (Mw = 205–258 kg/mol) are above the values of the materials studied in Ref. [14]. The polydispersity of the copolymer samples is Mw /Mn ≈ 10. With the same parameters as used for

Table 6 Pseudo-components of the poly(E-co-AA) samples from Table 2 referring to a Wesslau molecular weight distribution (with α = 0.9) Copolymer

Pseudo-components j

wpi,j a (–)

Mp1,j (g/mol)

Poly(E98.8 -co-AA1.2 )

1 2

0.523 0.477

20658 361462

Poly(E97.6 -co-AA2.4 )

1 2

0.523 0.477

16507 296799

Poly(E97.03 -co-AA2.97 )

1 2

0.523 0.477

13750 249451

a

Pseudo-component concentration in the solvent-free system.

Fig. 3. Cloud-point pressure curves for mixtures of ethene and poly(E-co-AA) with acrylic acid copolymer contents up to 4.6 mol%. The polymer concentration is 5 wt.%. Copolymer molecular weights, Mw , are ranging from 205 to 258 kg/mol. Polydispersity indices, Mw /Mn , are close to 10. Experimental data from Beyer and Oellrich [22] (symbols) are compared with predictions via PCSAFT (lines).

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Table 7 Pure-component parameters for acrylic acid component

Acrylic acid a

M (g/mol)

72.064

m

2.875

˚ σ (A)

3.054

ε/k (K)

157.5

εAi Bj /k (K)

2999.9

κ Ai Bj

0.33584

AAD%a psat

ρ

3.28

6.38

T range (K)

Data reference

380–550

[29]

Dimensionless absolute average deviations of vapor pressure and liquid densities.

the simulation of the CPP data from Ref. [14] (see Table 2), close agreement between calculated and measured data is also obtained for the experimental CPP data provided by Beyer and Oellrich [22]. The PC-SAFT model in its extension for polydispersity thus appears to be well applicable toward simulation, with the same set of model parameters, of CPP data for various types of poly(E-co-AA)–ethene systems with this data being provided by different groups. This conclusion is further strengthened by Fig. 4 in which CPP curves for poly(E-co-AA)–ethene measured by Wind [23] are presented for copolymers with acrylic acid contents between 1.2 mol% (E98.8 AA1.2 ) and 2.97 mol% (E97.03 AA2.97 ). Copolymer concentrations were close to 10 wt.% and copolymer molecular weights, Mw , were between 126 and 183 kg/mol, thus between the ones of the copolymers used in the experimental studies reported in Refs. [14,22]. The close agreement of measured and simulated CPP data again demonstrates the high predictive potential of PC-SAFT modeling in conjunction with using two pseudo-components for taking copolymer polydispersity into account. We also investigated the influence on the CPP behavior of adding the comonomer acrylic acid to the poly(E98.8 -coAA1.2 )–ethene system. Fig. 5 shows the CPP curves for this ternary system which have been measured by Wind [23]. The AA monomer content has been varied up to 9.5 wt.%. The copolymer

Fig. 5. Cloud-point pressure curves for ternary mixtures of poly(E98.8 -coAA1.2 )–ethene–acrylic acid for acrylic acid weight percentages in the monomer mixture up to 9.5 wt.%. In the binary mixture, at wAA = 0, the polymer concentrations were 8.2 and 13.5 wt.% in the ternary mixtures. Experimental data from Wind [23] (symbols) are compared with predictions via PC-SAFT (solid lines).

material was identical to the one used in the CPP measurements studies on the poly(E98.8 -co-AA1.2 )–ethene system (see Fig. 4). The copolymer concentrations in these ternary mixtures was 13.5 wt.%. As can be seen from the significant lowering in CPP, acrylic acid acts as a cosolvent. For the PC-SAFT modeling of the data in Fig. 5, the pure-component parameters of acrylic acid listed in Table 7 were used. The three binary parameters, kij , of the systems LDPE–acrylic acid, poly(AA)–acrylic acid and ethene–acrylic acid were set to zero. The binary parameters used for the calculations are summarized in Table 8. The curves in Fig. 5 are predictions which exclusively rest on parameters for pure components and binary mixtures. The close agreement of the experimental data from Wind [23] with the simulated curves demonstrates that PC-SAFT modeling is capable of Table 8 Binary interaction parameters for the ternary system poly(E-co-AA)– ethene–acrylic acid

Fig. 4. Cloud-point pressure curves for mixtures of ethene and poly(E-co-AA) with acrylic acid copolymer contents up to 2.97 mol%. The polymer concentration is close to 10 wt.%. Copolymer molecular weights, Mw , are ranging from 126 to 183 kg/mol. Polydispersity indices, Mw /Mn , are close to 5. Experimental data from Wind [23] (symbols) are compared with predictions via PC-SAFT (lines).

System

kij

LDPE–ethane Poly(AA) unit–ethene LDPE unit–poly(AA) unit LDPE unit–acrylic acid Poly(AA) unit–acrylic acid Ethene–acrylic acid

0.04 0.04 0 0 0 0

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121

Table 9 Properties of LDPE and poly(E-co-MAA) Polymer/copolymer

Methacrylic acid content (mol%)

Polyethylene (LDPE) Poly(E99.0 -co-MAA1.0 ) Poly(E96.5 -co-MAA3.5 ) Poly(E95.2 -co-MAA4.8 ) Poly(E94.2 -co-MAA5.8 ) Poly(E83.0 -co-MAA7.0 ) Poly(E89.7 -co-MAA10.3 )

0 1.0 3.5 4.8 5.8 7.0 10.3

a

Mn (g/mol)

Mw (g/mol)

21900 25680 24310 21820 15130 11540 5000a

103149 83200 88530 61230 43870 32210 15000a

Mw /Mn

Reference

4.71 3.3 3.6 2.8 2.9 2.8 3.0a

[14] [14] [14] [14] [14]

Estimated from correlation of molecular weights with the composition of the reacting monomer mixture.

almost quantitatively predicting the influence of the comonomer acrylic acid on the CPP curves at AA concentrations up to 6.8 wt.%. 4.2. Poly(E-co-MAA)–ethene Cloud-point pressure curves in mixtures with ethene were studied for ethene–methacrylic acid copolymers with MAA contents up to 10.3 mol% (see Table 9). To model these binary systems, the following parameters need to be known (in addition to the ones for ethene and LDPE that have already been presented during the analysis of the poly(E-co-AA)–ethene data): • the pure-component parameters for poly(MAA); • the binary interaction parameter kij for a poly(MAA) segment and ethene; • the binary interaction parameter kij for an LDPE and a poly(MAA) segment. To deduce these parameters, the same procedures were applied as described for the poly(AA) systems: the purecomponent parameters for poly(MAA) were determined as detailed in Section 3.3. For the binary interaction parameter, kij , for a poly(MAA) segment and ethene, the same value was adopted as for kij of the LDPE–ethene pair, because the dispersion energy of poly(MAA) is fixed to the same value as used for LDPE. The binary interaction parameter for an LDPE and a poly(MAA) segment was set to zero, as was the associated kij parameter for an LDPE and poly(AA) pair. The binary interaction parameters for the poly(E-co-MAA)–ethene systems are summarized in Table 10. Shown in Fig. 6 are the CPP curves of poly(E-coMAA)–ethene systems with the MAA copolymer contents ranging up to 10.3 mol% (poly(E89.7 -co-MAA10.3 )). Copolymer concentrations were about 5 wt.%. The molecular weights Mw were between 15 and 103 kg/mol. The polydispersity of the poly(ETable 10 Binary interaction parameters for the system poly(E-co-MAA)–ethene System

kij

LDPE–ethene Poly(MAA) unit–ethene LDPE unit–poly(MAA) unit

0.04 0.04 0

Fig. 6. Cloud-point pressure curves for mixtures of ethene and poly(E-co-MAA) for methacrylic acid copolymer contents up to 10.3 mol%. The polymer concentration is close to 5 wt.%. Copolymer molecular weights, Mw , are ranging from 15 to 103 kg/mol. Polydispersity indices, Mw /Mn , are close to 3.3. The dashed line represents the results from PC-SAFT obtained with the parameter values presented in Section 3.3. Experimental data from Ref. [14] (circles) are compared with predictions via PC-SAFT (solid lines).

Table 11 Pseudo-components of the poly(E-co-MAA) samples from Table 9 referring to a Schulz–Flory molecular weight distribution (with α = 1) Copolymer

Pseudo-components j

wpi,j a (–)

Mp1,j (g/mol)

Poly(E99.0 -co-MAA1.0 )

1 2

0.5 0.5

14021 152378

Poly(E96.5 -co-MAA3.5 )

1 2

0.5 0.5

13128 163931

Poly(E95.2 -co-MAA4.8 )

1 2

0.5 0.5

12106 110353

Poly(E94.2 -co-MAA5.8 )

1 2

0.5 0.5

8361 79378

Poly(E83.0 -co-MAA7.0 )

1 2

0.5 0.5

6407 58012

Poly(E89.7 -co-MAA10.3 )

1 2

0.5 0.5

2752 27247

a

Pseudo-component concentration in the solvent-free system.

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co-MAA) samples was taken into account by assuming each copolymer to be composed of two pseudo-components, as with the poly(E-co-AA)–ethene systems. The amounts and properties of these pseudo-components were determined in the same way as described in Section 4.1. The pseudo-component data for the poly(E-co-MAA) samples are summarized in Table 11. With the polydispersities, Mw /Mn , being around 3.3 for all poly(E-coMAA) materials under investigation, α = 1 was used throughout, thus matching the Schulz–Flory distribution. The satisfactory agreement of measured and predicted CPP data, illustrated in Fig. 6, demonstrates that the concept outlined in the present study is well suited for modeling and predicting the phase behavior of polydisperse copolymer–solvent systems with the copolymer being composed of monomer units which largely differ in association strength.

Greek letters α, β segment type ε/k temperature-independent attraction energy fugacity coefficient of component i in the mixture ϕi ˚ 3) ρ total number density of molecules (1/A σ temperature-independent segment diameter Subscripts i, j component Superscripts assoc contribution due to association I, II phase indices A, B association sites Acknowledgements

5. Conclusions An algorithm has been developed for calculating phase equilibria of polydisperse associating copolymer–solvent systems on the basis of the PC-SAFT equation of state. The concept was applied to poly(E-co-AA) and poly(E-co-MAA) systems in mixtures with ethene (and partly in solvent mixtures composed of both associated monomers). The PC-SAFT approach turns out to be capable of adequately modeling and even predicting the phase behavior of the polydisperse polymeric systems by using two pseudo-components for each copolymer, but no additional adjustable parameters. The influence of copolymer composition, of molecular weight as well as of the copolymer concentration on the cloud-point behavior can be adequately described by the PC-SAFT model using constant, i.e., temperature-independent parameters. Within the calculations for polydisperse associating polymers or copolymers consisting of a large number of pseudocomponents, the moment approach was applied to the association term, thereby defining a mean association-site number. The latter quantity may be directly determined from the molecular structure and the molecular weight of the associating pseudocomponents. List of symbols a molar residual Helmholtz energy (J/mol) k Boltzmann constant (J/K) kij binary interaction parameter m number of segments per chain m ¯ mean segment number M molar mass (g/mol) N number of association sites per molecule ¯ N mean association-site number T temperature (K) wi weight fraction of component i wpi,j weight fraction of pseudo-component j in copolymer i xi mole fraction of component i xpi,j mole fraction of pseudo-component j in copolymer i X fraction of monomer units not bonded to the association site

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