A Lattice-Fluid Equation of State for Associating CO2 + Polymer

Article pubs.acs.org/IECR

A Lattice-Fluid Equation of State for Associating CO2 + Polymer Systems Mohammad Z. Hossain, Yanhui Yuan, and Amyn S. Teja* School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100, United States S Supporting Information *

ABSTRACT: A new associated lattice-fluid equation of state is derived for CO2 + polymer mixtures by incorporating complex formation in the Guggenheim−Huggins−Miller lattice-fluid partition function. The equation contains mixture parameters that must be obtained from experimental data but are independent of the temperature and pressure. Moreover, one of the parameters may be obtained via in situ attenuated total reflection Fourier transform infrared measurements. The solubilities of CO2 in several polymers were correlated over a wide range of temperatures and pressures using only one adjustable parameter. The extent of swelling of the polymers due to the addition of CO2 could then be predicted without any additional parameters. function for association used in the CLM model.33,34 The association contribution contains physically meaningful parameters that do not depend on the temperature or pressure. Moreover, as demonstrated with the CLM model, in situ attenuated total reflection Fourier transform infrared (FTIR) measurements can be used to obtain one of the parameters.14,35 Application of the new EOS to the correlation of the solubility of CO2 in polymers such as poly(vinyl acetate) (PVAc), poly(lactide) (PLA), poly(lactide-co-glycolide) (PLGA50), poly(methyl methacrylate) (PMMA), poly(ethyleneco-vinyl acetate) (EVA40), poly(butylene succinate) (PBS), poly(butylene succinate-co-adipate) (PBSA), poly(n-butyl methacrylate) (PBMA), and poly(ethylene glycol) (PEG 300) is described below. The volumetric properties (swelling) of PMMA, PVAc, and PBMA due to the addition of CO2 are also estimated using parameters obtained by correlating solubilities.

1. INTRODUCTION The phase behavior of CO2 + polymer systems is of interest in polymer synthesis,1 in flue and natural gas processing,2,3 and in the fabrication of porous scaffolds for tissue engineering.4,5 In addition, the fabrication of micro- and nanoparticles of biodegradable polymers using supercritical CO2 is of interest in controlled drug delivery.6−9 Both theoretical arguments and experimental evidence suggest that CO2 is able to interact with electron-donating carbonyl groups in polymers to form weak Lewis acid−base complexes.10−13 These interactions significantly affect the phase behavior in CO2 + polymer systems.10,14,15 However, few thermodynamic models account for Lewis acid− base interactions explicitly and therefore are limited in their ability to describe the behavior of associating CO2 + polymer solutions. The more successful models for associating systems include versions of the statistical association fluid theory16−19 and several lattice-fluid equations of state such as those of Sanchez and Lacombe,20,21 Shin and Kim,22 and Panayiotou et al.23 Although these equation-of-state models are suitable for both phase and volumetric calculations, they generally require self- and crossassociation parameters that depend on the temperature or molecular weight.16,17,24 Excess Gibbs energy models, such as that of Coleman and co-workers,25−27 add terms to the Flory− Huggins equation to account for specific chemical interactions such as hydrogen bonds. However, they are not suitable for calculation of the volumetric properties. More recently, a compressible lattice model (CLM) for associated systems was proposed by Ozkan and Teja28 and extended by Kasturirangan et al.29,30 and Yuan and Teja.31 These authors were able to successfully correlate phase equilibria in CO2 + polymer and CO2 + cosolvent + polymer mixtures over a range of pressures using one adjustable parameter per binary system. However, the CLM model is also an excess Gibbs energy model and is not directly applicable to calculation of the volumetric properties such as swelling. In the present study, we propose a new associated lattice fluid (ALF) equation of state (EOS) by combining the Guggenheim− Huggins−Miller lattice partition function32 with the partition © XXXX American Chemical Society

2. DEVELOPMENT OF AN EOS FOR ASSOCIATING SYSTEMS The Guggenheim−Huggins−Miller partition function for N1 solvent molecules (each consisting of r1 segments) and N2 polymer molecules (each consisting of r2 segments) arranged randomly on a lattice of coordination number z can be expressed as follows: ΩR =

z /2 ⎛ τ ⎞ N2 (r1N1 + r2N2)! ⎛ (r1N1 + qN2)! ⎞ ⎜ ⎟ ⎟ ⎜ ⎝σ ⎠ N1! N2! ⎝ (r1N1 + r2N2)! ⎠

(2.1)

where σ is a symmetry number, τ is a flexibility parameter, and zq is the number of nonbonded segment pairs around each polymer molecule. zq = (r2 − 2)(z − 2) + 2(z − 1) = r(z − 2) + 2

(2.2)

Received: April 17, 2013 Revised: July 25, 2013 Accepted: August 6, 2013

A

dx.doi.org/10.1021/ie401229j | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

Article

If we randomly distribute r̂N̂ (polymer and solvent) segments and N0 holes on a lattice, then the partition function becomes ΩR =

z /2 ̂ ⎛ τ ⎞ N Nr! ⎛ Nq! ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ σ ⎠ N0! N̂ ! ⎝ Nr! ⎠

(2.3)

where Nq = N0 + qN̂ , Nr = N0 + r̂N̂ , and N = r̂N̂ = r1N1 + r2N2. The total number of nonbonded pairs Nqz/2 is now given by Nqz 2

=

Nrz Nz − N̂ (r ̂ − 1) = r − rN ̂ ̂ + N̂ 2 2

(2.4)

The partition function may be simplified using the Stirling (ln N! = N ln N − N) and Flory (r̃/q → 1) approximations36 to give N0 ̂ ̂ ⎛ rτ̂ ⎞ N ⎛ Nr ⎞ ⎛ Nr ⎞ N ΩR = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ σ ⎠ ⎝ N0 ⎠ ⎝ rN ̂ ̂⎠

Figure 2.1. ALF model.

G E PV 1 ⎛ rτ̂ ⎞ ⎛ f ⎞ = + − ln⎜ ⎟ + ⎜⎜ 0 ⎟⎟ ln f0 NRT NRT NRT r ̂ ⎝ σ ⎠ ⎝ 1 − f0 ⎠

(2.5)

If some solvent segments are constrained to specific sites on the lattice because they are associated with polymer segments, then the contribution to the partition function due to complex formation is given by33,34 Ωcomplex

⎞⎤ ⎛ N − N2⎟ ! ⎥ ⎜ ⎠ ⎥⎦ ⎝ r2(1 + α)

α=

⎛ ϕ − αϕ2 ⎞ + ϕ2 + ⎜ 1 ⎟ r2 ⎝ 1+α ⎠

ϕ2

(2.10)

where f 0 represents the fraction of holes in the lattice and is given by

(2.6)

f0 =

N0 N0 + rN ̂ ̂

=1−

rN ̂ ̂ = 1 − ρ̂ N0 + rN ̂ ̂

(2.11)

Noting that the third term of eq 2.10 does not contribute to the pressure,37 we can write ⎧ ⎡⎛ f ⎞ ⎪ G = rN ̂ ̂ ⎨− ρε̃ * + Pvṽ * + RT ⎢⎜⎜ 0 ⎟⎟ ln f0 ⎪ ⎢⎣⎝ 1 − f0 ⎠ ⎩ ⎤ ⎡ ⎛ ϕ ⎞ 1 + ln(1 − f0 )⎥ + RT ⎢ − ⎜ϕ2 − 2 ⎟ ln(z − 1) ⎥⎦ ⎢⎣ ⎝ r̂ r2 ⎠

(2.7)

where ϕ1 and ϕ2 are volume fractions. The final form of the ALF partition function can then be obtained by combining these contributions according to ⎛ E ⎞ ⎟ Q = Ω R Ωcomplex exp⎜ − ⎝ RT ⎠

r2

−

+ αϕ2 ln α

(1 + K )2 − 4ϕ1ϕ2K (1 + K ) 2ϕ2(1 + K )

ϕ2

⎛ αϕ ⎞ 2 ⎟⎟ + ϕ2(1 − α) ln(1 − α) + αϕ2 ln⎜⎜ ⎝ ϕ1 − αϕ2 ⎠

where α is the association ratio. In our work, we have assumed that the polymer and solvent associate to form a chemical complex with a binding ratio of 1:1. The association ratio α is then related to the equilibrium constant K of the complexation reaction via (1 + K ) −

r2

ln

⎛ ⎞ ϕ1 ⎟⎟ × ln(ϕ1 − αϕ2) − ϕ1 ln⎜⎜ ⎝ ϕ1 − αϕ2 ⎠

r2

⎞ ⎛ (r1N1)! ×⎜ ⎟ ⎝ (αr2N2) ! (r1N1 − αr2N2)! ⎠

ϕ2

+

⎞ N2 r2! 1 ⎡ (z − 1)N2(r2 − 1) ⎤⎛ ⎢ ⎥⎜ = ⎟ N2! ⎢⎣ N N2(r2 − 1) ⎥⎦⎝ (r2 − αr2) ! (αr2)! ⎠ × [r2(1 + α)]r2N2 ⎡⎛ ⎞ N × ⎢⎜ ⎟! ⎢⎣⎝ r2(1 + α) ⎠

⎛ ϕ ⎞ 1 ln(1 − f0 ) − ⎜ϕ2 − 2 ⎟ ln(z − 1) r̂ r2 ⎠ ⎝

+

+

ϕ2 r2

ln

ϕ2 r2

−

⎛ ϕ − αϕ2 ⎞ + ϕ2 + ⎜ 1 ⎟ × ln(ϕ1 − αϕ2) r2 ⎝ 1+α ⎠

ϕ2

⎛ ⎞ ⎛ αϕ ⎞ ϕ1 2 ⎟⎟ + αϕ2 ln⎜⎜ ⎟⎟ − ϕ1 ln⎜⎜ ⎝ ϕ1 − αϕ2 ⎠ ⎝ ϕ1 − αϕ2 ⎠ ⎫ ⎤⎪ + ϕ2(1 − α) ln(1 − α) + αϕ2 ln α ⎥⎬ ⎥⎦⎪ ⎭

(2.8)

where E is the energy and V is the volume of the lattice that has N occupied sites and N0 unoccupied sites (holes), as depicted in Figure 2.1. Differentiation of the partition function leads to the Gibbs energy:

(2.12)

in which V = r̂N̂ ṽv*, r̂N̂ = N, and E = −ε*ρ̃. The EOS can be obtained using

⎡ ⎛ ∂ ln Q ⎞ ⎤ ⎥ = −RT ln Q + PV ⎟ G = −RT ⎢ln Q − ⎜ ⎝ ∂V ⎠T , N ⎥⎦ ⎢⎣

⎛ ∂G ⎞ ⎜⎜ ⎟⎟ =0 ⎝ ∂f0 ⎠T , P , N

(2.9)

The final expression for the Gibbs energy is then

i

B

(2.13)



Article

μ1L

and is given by ⎡ ⎛ 1⎞ ⎤ ρ 2̃ + P ̃ + T̃ ⎢ln(1 − ρ ̃) + ⎜1 − ⎟ρ ̃⎥ = 0 ⎝ ⎣ r̂ ⎠ ⎦

RT

=

j≠1

(2.14)

=

with ρ̃ =

ρ ρrv̂ * P Pv* = = P= ρ* ε* M P*

T̃ =

T TR = ε* T*

ṽ =

× (z − 2 − α) − β1(ϕ1 − αϕ2)]χu )ρ ̃ −

Note that this is of the same form as the Sanchez−Lacombe (SL) EOS, but with the average lattice energy per segment given by

Δεex = −

(2.15)

+

ΔHmix N

χa =

(2.24)

(2.17)

with

ΔHa RT

β1 = ∂α /∂N1 = (α(1 + K ) − K )/(1 + K )(2αϕ2 − 1)

(2.18)

ΔHa ⎛ 1 K 1⎞ = ⎜ − ⎟ K0 RT ⎝ T T0 ⎠

We have used eqs 2.23 and 2.24 to calculate the solubility of CO2 (1) in polymers (2) by equating the chemical potential of pure CO2 in the gas phase to the chemical potential of CO2 dissolved in the polymer. The extent of swelling due to CO2 is estimated using

(2.19)

Sw =

In the above equation, T0 is some arbitrary reference temperature (298.15 K in our work), so that three parameters are required to apply the EOS derived above: ΔHa, K0, and ε12. We may express ε12 as follows: ε12* = (1 − kij) (ε1*ε2*)

(2.21)

ϕ ϕ2 1 = 1 + r̂ r1 r2(1 + α)

(2.22)

then the chemical potential of pure CO2 and of CO2 (1) dissolved in a polymer (2) can be obtained from the Gibbs energy and written as

RT

=

r1 ⎡ Pv * ⎤ ⎢ −ρ1̃ ε1* + 1 ⎥ RT ⎢⎣ ρ1̃ ⎥⎦ ⎡⎛ 1 − ρ ̃ ⎞ ⎤ 1 1 ⎟⎟ ln(1 − ρ1̃ ) + ln ρ1̃ ⎥ + r1⎢⎜⎜ r̂ ⎢⎣⎝ ρ1̃ ⎠ ⎥⎦

(2.25)

3. RESULTS AND DISCUSSION Before the new EOS can be applied to mixtures, it is necessary to estimate the characteristic constants P*, T*, and ρ* (or εi*, vi*, and ri) for the pure components of interest. Table 3.1 lists these constants for the ALF EOS obtained by fitting PVT data. Also listed are the constants for the SL EOS reported in the literature for many of the same polymers. As can be seen in Table 3.1, the two sets of constants are nearly identical for many polymers. Slight differences in values may be attributed to the different sets of PVT data used in the fitting of the two models or to differences in the molecular weights of the polymers considered. Therefore, it is possible to use the literature values of SL EOS parameters when ALF parameters are not available. In Table 3.1, we also list the characteristic constants for CO2 from the work of Condo et al.38 For mixture calculations, we need binary parameters K0 and ΔHa. As discussed in our previous work, the enthalpy of interaction ΔHa can be determined from FTIR measurements, quantum calculations, or molecular modeling.12,13,35 In Table 3.2, we show the results of our calculations of sorption equilibria for four CO2 + polymer systems using the ALF EOS. In these calculations, ΔHa values were obtained from FTIR

(2.20)

v* = ϕ1v1* + ϕ2v2*

V × 100% V0

where V and V0 are the volume of the CO2 + polymer solution and the dry polymer, respectively.

For simplicity, we have set kij = 0 in the following discussion (reducing the number of experimentally determined parameters to 2, namely, ΔHa and K0). If we express the characteristic volume and segment number using

μ1G,pure

⎞ ⎛ ⎞ ϕ1 − αϕ2 ⎛ 1 − ϕ2β1 ϕ1 ⎜⎜ ⎟⎟ − 1⎟⎟ − ln⎜⎜ 1 + α ⎝ ϕ1 − αϕ2 ⎠ ⎝ ϕ1 − αϕ2 ⎠

⎛β 1 − ϕ2β1 ⎞ α ⎟⎟ + β1ϕ2 ln + αϕ2⎜⎜ 1 − ϕ1 − αϕ2 ⎠ 1−α ⎝α

The enthalpy of association (ΔHa) is related to the equilibrium constant for the association reaction via the van’t Hoff relationship: ln

r2

⎛1 − ϕβ ⎞ ⎛ αϕ ⎞ 2 1 2 ⎟⎟ + β1ϕ2 ln⎜⎜ ⎟⎟ − 1 + ϕ1⎜⎜ ⎝ ϕ1 − αϕ2 ⎠ ⎝ ϕ1 − αϕ2 ⎠

(2.16)

(ε1* + ε2* − 2ε12*)2 RT (z − 2 − α)

r1ϕ2

⎡ 1 β1 ⎤ +⎢ − ⎥ ln(ϕ1 − αϕ2) ⎣1 + α (1 + α)2 ⎦

= −[αϕ2χa + ϕ2(ϕ1 − αϕ2)(z − 2 − α)χu ]NRT

χu =

Pr1 r [−f ν* + ν1*] − r1f0 + 1 f0 + ln(1 − f0 ) RTρ ̃ 0 r̂ 1 − [r1ρ ̃(1 − ρ ̃)ε* + r1ε1*ρ ̃] RT + (β1ϕ2χa + ϕ2[{−β1ϕ2 + ϕ2(1 + α)}

1 1 V V = = = ρ̃ 1 − f0 V* rNv ̂ ̂ *

ε* = ϕ1ε1* + ϕ2ε2* + Δεex

1 ⎛ ∂G ⎞ ⎟ ⎜ RT ⎝ ∂N1 ⎠T , P , N

(2.23) C



Article

Table 3.1. Characteristic Constants in the ALF and SL Equations for Pure Components polymer

T*/K

P*/MPa

ρ*/kg m−3

PEG300 EV40

728.9 618.9 613.0 592.0 592.0 717.8 717.8 691.7 691.8 641.6 631.0 638.0 649.6 698.6 742.0 624.3 627.0 308.6

544.8 500.5 495.6 504.2 504.2 523.6 523.5 570.1 570.1 547.7 559.2 536.6 572.7 467.1 488.3 409.6 431.0 574

1169.5 1023.7 1020.0 1282.7 1282.0 1263.5 1263 1257.2 1235 1330.1 1333.0 1458.5 1451.6 1265.3 1249.8 1124.8 1125.0 1505.0

PVAc PBS PBSA PLA PLGA50 PMMA PBMA CO2

data ref

X

K0

ΔHa/kJ mol−1

AAD %

data ref

work work

39 40

work

42

work

44

work

44

work

44

PEG300 EVA40 PVAc PBS PBSA(am) PBSA(cr) PLA PLGA50 PMMA PBMA

0.16 0.08 0.10 0.22 0.21 0.17 0.20 0.09 0.16 0.18

−11.5b −9.3a −9.3b −8.8a −8.8a −8.8a −8.8b −8.5b −8b −8a

4.07 3.30 2.90 0.42 0.73 0.69 0.39 1.00 0.34 0.81

49 41 42 45 45 45 29 46 50 48

work

46

work

47

work

47

ref this this 41 this 43 this 45 this 45 this 46 this 46 this 43 this 48 38

Table 3.3. Binary (X + CO2) Parameters for CO2 Sorption in Polymers

a

SL EOS

this work

kij(T)

AAD %

ref

K0

AAD %

PVAc + CO2 PBS + CO2 PBSA(am)a + CO2 PLGA50 + CO2

0.1158 − 0.00055T 0.310 − 0.00104T 0.261 − 0.00885T −0.0273 − 0.00007T

3.60 1.90 2.20 8.03

42 45 45 46

0.10 0.22 0.21 0.09

2.90 0.42 0.73 1.00

a

10

. am = amorphous and cr = crystalline.

values of K0 and ΔHa used in the calculations. ΔHa values were obtained from FTIR spectra reported in the literature35 or were set equal to the reported values of ΔHa for polymers with a similar functional group. Thus, ΔHa for PVAc + CO2 was assumed to be the same as that for EVA40 + CO2 because both PVAc and EVA40 contain the interacting vinyl ester group in their structures. Similarly, the same value of ΔHa was used for systems of CO2 with PLA, PBS, or PBSA, polymers that incorporate the aliphatic ester group in their structures. This was also the case with mixtures of CO2 and polymers such as PMMA and PBMA that incorporate the acrylate group. The following trends may be ascertained from the values of the parameters reported in Table 3.3: (i) For polymers containing the same associating functional group in their structures, higher values of K0 are obtained when there is greater accessibility of the functional group to CO2. Thus, K0 (=0.18) for CO2 + PBMA is larger than that (=0.16) for CO2 + PMMA because the mobility of the PBMA side chain allows its acrylate group to be more accessible to CO2. Parts a and b of Figure 3.1 illustrate this for CO2 + PMMA and CO2 + PBMA, respectively. (ii) Lower values of K0 were obtained for CO2 + copolymer systems than for the corresponding CO2 + polymer systems. For example, K0 = 0.20 in the case of CO2 + PLA, whereas K0 = 0.09 in the case of the CO2 + PLGA system. The associating group of PLA is probably more accessible to CO2 than the same group in PLGA, leading to higher solubility of CO2 in PLA than in PLGA. Sorption equilibria in CO2 + PLA and CO2 + PLGA50 systems are shown in Figure 3.3a,b. (iii) The value of K0 (=0.22) obtained by fitting sorption data for CO2 + PBS is approximately the same as the value (=0.21)

Table 3.2. Adjustable Parameters Used in Fitting Sorption Equilibria in CO2 + Polymer Systems Using the ALF and SL EOSs system

Assumed. bReference

am = amorphous.

spectra reported in the literature,35 and K0 was treated as an adjustable parameter. Average absolute deviations (AADs) between calculated and experimental values are listed. Also listed are K0 values obtained by fitting sorption equilibria using the ALF EOS and the corresponding kij values obtained using the SL EOS. Note that the kij values in the SL EOS calculations are temperature-dependent, whereas a single value of K0 is required in the ALF EOS calculations for these systems. In Table 3.3, we show the results of our calculations of sorption equilibria in several CO2 + polymer systems using the ALF EOS. AADs between the calculated and experimental solubilities of CO2 in each polymer are listed, together with the

Figure 3.1. Solubility of CO2 in (a) PMMA and (b) PBMA. The lines are calculated from the model, and the points are the experimental results.48,50 D



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Figure 3.2. Swelling of (a) PMMA and (b) PBMA due to CO2. The lines are calculated from the model, and the points are the experimental results.50

Figure 3.3. Solubility of CO2 in (a) PLA and (b) PLGA50. The lines are calculated from the model, and the points are the experimental results.29,46

Figure 3.4. Solubility of CO2 in (a) PBS, (b) amorphous PBSA and (c) crystalline PBSA. The lines are calculated from the model, and the points are the experimental results.45

obtained for CO2 + amorphous PBSA. Tg values for the two polymers are also similar (239.5 K for PBS and 236.0 K for PBSA). This leads to the conclusion that the lower value of K0 for

the associating group in PBSA is balanced by the higher mobility of PBSA, resulting in approximately the same solubility of CO2 in the two polymers. Parts a and b of E



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(8) Palakodaty, S.; York, P. Phase Behavioral Effects on Particle Formation Processes Using Supercritical Fluids. Pharm. Res. 1999, 16, 976−985. (9) Yeo, S.-D.; Kiran, E. Formation of polymer particles with supercritical fluids: A review. J. Supercrit. Fluids 2005, 34, 287−308. (10) Yuan, Y. Specific Interactions in Polymer + CO2 + Cosolvent Systems: Experiment and Modeling; Georgia Institute of Technology: Atlanta, GA, 2010. (11) Kazarian, S. G.; Vincent, M. F.; Bright, F. V.; Liotta, C. L.; Eckert, C. A. Specific Intermolecular Interaction of Carbon Dioxide with Polymers. J. Am. Chem. Soc. 1996, 118, 1729−1736. (12) Meredith, J. C.; Johnston, K. P.; Seminario, J. M.; Kazarian, S. G.; Eckert, C. A. Quantitative Equilibrium Constants between CO2 and Lewis Bases from FTIR Spectroscopy. J. Phys. Chem. 1996, 100, 10837−10848. (13) Nelson, M. R.; Borkman, R. F. Ab Initio Calculations on CO2 Binding to Carbonyl Groups. J. Phys. Chem. A 1998, 102, 7860−7863. (14) Kasturirangan, A. Specific interactions in CO2 + polymer systems; Georgia Institute of Technology: Atlanta, GA, 2007. (15) Rindfleisch, F.; DiNoia, T. P.; McHugh, M. A. Solubility of Polymers and Copolymers in Supercritical CO2. J. Phys. Chem. 1996, 100, 15581−15587. (16) Huang, S. H.; Radosz, M. Equation of state for small, large, polydisperse, and associating molecules. Ind. Eng. Chem. Res. 1990, 29, 2284−2294. (17) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (18) Karakatsani, E. K.; Economou, I. G.; Kroon, M. C.; Peters, C. J.; Witkamp, G.-J. tPC-PSAFT Modeling of Gas Solubility in Imidazolium-Based Ionic Liquids†. J. Phys. Chem. C 2007, 111, 15487−15492. (19) Ashrafmansouri, S.-S.; Raeissi, S. Modeling gas solubility in ionic liquids with the SAFT-γ group contribution method. J. Supercrit. Fluids 2012, 63, 81−91. (20) Sanchez, I. C.; Lacombe, R. H. An elementary molecular theory of classical fluids. Pure fluids. J. Phys. Chem. 1976, 80, 2352−2362. (21) Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145−1156. (22) Shin, M. S.; Kim, H. A quasi-chemical nonrandom lattice fluid model for phase equilibria of associating systems. Fluid Phase Equilib. 2007, 256, 27−33. (23) Panayiotou, C.; Tsivintzelis, I.; Economou, I. G. Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 2. Multicomponent Mixtures. Ind. Eng. Chem. Res. 2007, 46, 2628−2636. (24) Arce, P.; Aznar, M. Modeling the thermodynamic behavior of poly(lactide-co-glycolide) + supercritical fluid mixtures with equations of state. Fluid Phase Equilib. 2006, 244, 16−25. (25) Coleman, M., Graf, J.; , and Painter, P. Specific Interactions and the Miscibility of Polymer Blends; Technomic Publishing AG: Basel, Switzerland, 1991. (26) Coleman, M. M.; Painter, P. C. Hydrogen bonded polymer blends. Prog. Polym. Sci. 1995, 20, 1−59. (27) Coleman, M. M.; Pehlert, G. J.; Painter, P. C. Functional Group Accessibility in Hydrogen Bonded Polymer Blends. Macromolecules 1996, 29, 6820−6831. (28) Ozkan, I. A.; Teja, A. S. Phase equilibria in systems with specific CO2−polymer interactions. Fluid Phase Equilib. 2005, 228−229, 487− 491. (29) Kasturirangan, A.; Grant, C.; Teja, A. S. Compressible Lattice Model for Phase Equilibria in CO2 + Polymer Systems. Ind. Eng. Chem. Res. 2008, 47, 645−649. (30) Kasturirangan, A.; Teja, A. S. Phase behavior of CO2 + biopolymer and CO2 + fluoropolymer systems. Fluid Phase Equilib. 2007, 261, 64−68. (31) Yuan, Y.; Teja, A. S. Extension of a compressible lattice model to CO2 + cosolvent + polymer systems. J. Supercrit. Fluids 2010, 55, 358− 362.

Figure 3.4 illustrate the results for CO2 + PBS and CO2 + amorphous PBSA. (iv) As expected, K0 (=0.21) obtained by fitting sorption data for CO2 + amorphous PBSA is larger than K0 (=0.17) in the CO2 + crystalline PBSA system because the solubility of CO2 is greater in amorphous PBSA than in crystalline PBSA. Figure 3.4c illustrates this point. Swelling behavior can be predicted using the same binary interaction parameters that we have used to fit the sorption data. Parts a and b of Figure 3.2 represent the predicted swelling behavior in CO2 + PMMA and CO2 + PBMA systems, respectively, using the K0 values obtained by fitting solubility data in these mixtures.

4. CONCLUSIONS An ALF EOS has been derived for CO2 + polymer systems. The EOS contains two mixture parameters: the enthalpy of association ΔHa and the equilibrium constant at a reference temperature K0, which do not depend on the temperature, pressure, or molecular weight. The solubility of CO2 in a number of polymers was correlated over a wide range of temperatures and pressures, using ΔHa values from independent FTIR measurements and K0 values obtained by fitting solubility data. We have shown that the new model is able to correlate solubility data within experimental error (maximum AAD of about 4%). In addition, the extent of swelling of these polymers by CO2 can be predicted without any adjustable parameters. The ALF EOS, therefore, shows considerable promise in calculating both phase equilibria and volumetric data of CO2 + polymer systems.

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ASSOCIATED CONTENT

* Supporting Information S

Detailed calculations for the CO2 solubility in PVAc and PEG. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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A Lattice-Fluid Equation of State for Associating CO2 + Polymer

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