Modified quadrupole Cubic Plus Association Equation of State (mqCPA EoS) for thermodynamic modeling of polymer-supercritical CO2 systems

Accepted Manuscript Modified quadrupole Cubic Plus Association Equation of State (mqCPA EoS) for thermodynamic modeling of polymer-supercritical CO2 systems Ali Haghtalab, Hesam Hasannataj, Hamidreza Soltani Panah PII:

S0378-3812(16)30603-3

DOI:

10.1016/j.fluid.2016.12.007

Reference:

FLUID 11351

To appear in:

Fluid Phase Equilibria

Received Date: 24 July 2016 Revised Date:

4 November 2016

Accepted Date: 5 December 2016

Please cite this article as: A. Haghtalab, H. Hasannataj, H.S. Panah, Modified quadrupole Cubic Plus Association Equation of State (mqCPA EoS) for thermodynamic modeling of polymer-supercritical CO2 systems, Fluid Phase Equilibria (2017), doi: 10.1016/j.fluid.2016.12.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Modified quadrupole Cubic Plus Association Equation of State

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(mqCPA EoS) for thermodynamic modeling of polymersupercritical CO2 systems

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Ali Haghtalab1, Hesam Hasannataj, Hamidreza Soltani Panah

Department of Chemical Engineering, Tarbiat Modares University, P.O. Box 14115-143, Tehran,

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Iran ABSTRACT

In this work, a modified quadrupole Cubic Plus Association Equation of State (mqCPA EoS) is used to model the cloud point pressure of polymers-supercritical carbon dioxide (scCO2)

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systems. The residual Helmholtz free energy is expressed as sum of SRK and Quadrupole terms. We used the experimental data of the cloud point pressure for the different copolymer-scCO2 systems to investigate the capability of the present model through liquid-liquid equilibrium

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calculations. Using mqCPA EoS, the cloud point pressures of the polymer-scCO2 systems for the three types of poly(lactic acid) (PLA), the three types of poly(1-O-(vinyloxy) ethyl-2,3,4,6-

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tetra-O-acetyl-
-d-glucopyranoside) (P(AcGlcVE) and seven types of poly(vinyl acetate-altdibutyl maleate) (PVAc-alt-PDBM) were calculated. These copolymers have different molecular structures, end groups and molar masses. Moreover, we applied both PC-SAFT and PCP-SAFT equations of state for the same polymer-scCO2 systems and their results were compared to those

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Corresponding author; E-mail address: [email protected] (A. Haghtalab) 1

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obtained by mqCPA EoS that showed in overall the both PCP-SAFT and mqCPA EoSs give better results with very good accuracy.

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Keywords: Cubic Plus Association EoS, cloud point pressure, CO2, PLA, P(AcGlcVE), PVAcalt-PDBM

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1. INTRODUCTION

The supercritical fluids such as carbon dioxide, nitrogen, methane, ethane, ethylene, propane and

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propylene are useful materials that have been presented many applications in industry. Among these fluids, CO2 as a known supercritical fluid has a wide application in chemical, petroleum and petrochemical industries. Carbon dioxide has interesting properties such as low critical point (TC= 304.1 K and PC= 73.8 bar), chemically inert, low cost and environmentally benign [1].

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Carbon dioxide has been used in different applications such as enhancement of oil recovery (EOR) [2], formation of microcellular materials, protein extraction and polymer engineering products [1]. However, CO2 as an injection gas has low viscosity (0.03- 0.1 cP) at the reservoir

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temperature and pressure conditions while reservoir oil viscosity is much higher (0.1-50 cP) at reservoir compositional conditions [2]. The big difference in the viscosities of CO2 and oil leads

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to fingering phenomena that cause to escape gas from oil in the reservoir. The soluble polymers such as the polymers based on siloxane, vinyl acetate and fluoro polymers are often used to overcome this problem that leads to thick the supercritical CO2 (scCO2) as close as oil viscosity [2]. The fluorinated acrylic polymers, a group of fluoropolymers, such as poly- (1,ldihydroperfluorooctyl) acrylate are highly soluble in scCO2. Using this fluoropolymer, some

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experimental results showed that the solubility of this polymer in supercritical CO2 leads to enhancement of the viscosity of scCO2 by a factor of 100 [3]. From an application point of view, the two categories of the CO2 systems have been

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investigated so far that in the first group, i.e. CO2-polymer system, the gas is solubilized in polymer melt or polymer solution so that most of the researches in the literatures are belong to this category. A review article in modeling phase equilibria of CO2-polymer systems has been

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given by Hossain and Teja [4]. Also, the solubility of CO2 in polymers and its applications in polymer processing have been given in the literature [5,6]. However, in the other group of works,

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i.e. polymer-scCO2 systems, the polymer is solubilized in scCO2 [7-10] which limited studies have been reported for this type of systems. Determination of the cloud point pressure of the polymer-scCO2 system is a key point for application of such systems in EOR processes. To model polymer-scCO2 systems, there are three general approaches such as lattice, eg. Flory-

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Huggins [11-13], off-lattice and EoS models like perturbed hard chain theories [14]. On the other hand, only a few thermodynamic equations of state have been applied for calculation of the phase behavior of polymer-scCO2 systems [15-20]. As one can expect, despite the wide use of

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the equations of state such as cubic and non-cubic relations in design of industrial plants, only a few of them have focused on polymer-scCO2 systems. Using equations of state, a few works are

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introduced here as follows. Xiong and Kiran [21] applied Sanchez-Lacombe (SL) EoS to model poly(dimethyl siloxane)s (PDMS) in supercritical carbon dioxide. In their model, a temperature and molecular weight dependent binary interaction parameter (kij) was used to correlate the data. Ozkan and Teja [22] used the generalized Associative Reformulation of Thermodynamics (gART-L) model for the associated polymer solutions that is a simple reformulation of the lattice-based Flory–Huggins model. They used a variable as the segment number that is

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calculated by the ratio of the molar volume of CO2 to the segmental polymer molar volume at ambient condition. This “segment number” changes with temperature and pressure of the cloud point pressure of the polymer- scCO2 systems, since the density of CO2 varies dramatically.

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Thus, the segment number should be recalculated in each step to give good results. Lora et al. [10] used SAFT EoS with binary interaction parameter, kij, to model the phase behavior of the polymer-scCO2 system so that without using this interaction parameter, a high deviation was

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obtained. Shin and Wu [23] used a new form of EoS that is a combination of PR and SAFT equations of state to model the phase behavior of near critical or scCO2 with polymers. In the PR

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term for CO2, the both pure co-volume parameter, b, and attractive parameter, a, are accounted as the adjustable parameters so that “b” was assumed independent of temperature and “a” were fitted by a temperature dependent exponential correlation. Xu et al. [24] added another term to Wu et al. EoS that is called fundamental measure theory (FMT). This modified EoS was

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successful in correlating the experimental data up to 35MPa. Haghtalab and Soltani Panah [15] applied PCP-SAFT EoS [25] to model the phase behavior of polymer -scCO2 systems. By taking into account a quadrupole term for quadrupolar interaction between CO2 molecules and polymer

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segments, the pure parameters of PCP-SAFT EoS and an extra parameter were correlated so that the very good results were obtained; however, for some systems, the better results were obtained

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using an interaction parameter, kij. Although there are some other thermodynamic models that use polar (whether quadrupole or dipole) contribution, but to our best knowledge, none of them focuses on polymer-scCO2 systems [26-31]. In addition to above works, several investigations have applied the different equations of state to model the solubility of CO2 in the various polymers as follows. Dong Woo Cho et al. [32] used PR EoS with temperature-depended binary interaction coefficient for the CO2- Ethylene

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glycol dimethacrylate (EGDMA) and CO2- di-ethylene glycol dimethacrylate (DEGDMA) systems. In their work, the critical properties and acentric factor were estimated using two group contribution methods as the Joback and the Marrero-Gani approaches where Marrero-Gani

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method showed better results in correlating the experimental data. Casimiro et al. [33] applied SL EOS to calculate the liquid equilibrium composition of the CO2 - Krytox 157 FSL binary system through fitting two interaction parameters. Elvassore et al. [34] used the perturbed-hard-

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sphere-chain equation of state (PHSC EOS) for correlation of the absorption data of the CO2 in poly(lactic-co-glycolic acid) (PLGA) with different lactide–glycolide copolymer ratios. Liu and

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Tomasko [35] used SL EoS to correlate the experimental data of CO2 sorption isotherms for the induced polymer (PLGA) swelling systems. Lei et al. [36] used both SL and the groupcontribution lattice-fluid (GCLF) equations of state to correlate the solubility of CO2 in molten state of polypropylene (PP). Both these EOSs could correlate the data to within 5% average

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relative deviation in conjunction with a temperature-dependent interaction parameter. They showed that the solubility of CO2 in the rubbery state of PP first reduced and then enhanced with temperature, while the solubility of CO2 in molten state PP always decreased with temperature.

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Mart´ın et al. [37] used PHSC EoS to correlate the phase behavior of the CO2-Poly ethylene glycol (PEG) systems. This model predicts the apparition of a liquid–liquid immiscibility region

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at moderate temperatures due to the co-solvent effect of CO2 on PEG. PR EoS was used by Coimbra et al. [9] to correlate the solubility of Irgacure® 2959 in scCO2 in a pressure range from 10.0 up to 26.0MPa. They considered solid sublimation pressure is a key property that without this parameter, the two adjustable interaction parameters must be employed to correlate successfully the experimental solubility data. Gregorowicz [38] used PR EoS to model the VLE data of binary system carbon dioxide + L-lactide in the temperature range 275–355K and in the

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pressure range 10–80MPa. For the calculation of critical properties and acentric factor of Llactide, the Joback–Lyndersen group contribution method is used. Chen et al. [39] used original statistical associating fluid theory (CK-SAFT), Perturbed-Chain SAFT (PC-SAFT), SL and

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Sako–Wu–Prausnitz (SWP) EoSs to model the solubility of subcritical and supercritical fluids such as N2, C2H4, C4H10, iso-C4H10, CO2 in molten, thermally softened or semicrystalline polymers. Their results showed the PC-SAFT and SL EoSs with only one temperature-depended

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adjustable parameter could successfully correlate the solubility of the gasses in the molten or thermally softened polymers. Aionicesei et al. [40] used SL and SAFT EoSs to model the phase

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behavior of the poly(ethylene glycol) (PEG)-scCO2 system in the solid (298 K) and molten (323 K) states and in the pressure range of 7-25 MPa. Their results showed that the SAFT model is successful for liquid polymers. They also used SL and PC-SAFT EoSs to model the phase equilibrium of the poly(l-lactide)–CO2 and poly(D,L-lactide-co-glycolide)–CO2 systems [41].

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The solubility of CO2 in the two biodegradable polymers was calculated at 308, 313 and 323 K and in the pressure ranges of 10–30 MPa. Their results presented that both SL and PC-SAFT EoSs are reliable models in describing the phase equilibrium of the PLLA–CO2 and PLGA–CO2

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systems at the given conditions.

In this work, a modified quadrupole cubic plus association equation of state, mqCPA EoS, is

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used to correlate the cloud point pressure of the several copolymer-scCO2 systems. Using this EoS, the calculated cloud point pressures of these systems are compared with those obtained by the PC-SAFT and PCP-SAFT EoSs. 2. THERMODYNAMIC FRAMEWORK 2.1 Modified quadrupole Cubic Plus Association Equation of State (mqCPA-EoS)

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In this work, we used the mqCPA EoS to correlate the phase behavior of the polymer-scCO2 systems. This equation of state consists of two physical parts as Soave–Redlich–Kwong (SRK) term for contribution of dispersion forces and Quadrupole-Quadrupole (QQ) term for

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contribution of the quadrupolar interactions between carbon dioxide and polymer segments. Although the qCPA EoS has been already developed by Bjørner et al. [42,43] that includes an association term, but this association contribution is neglected for the polymer-CO2 systems in

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the present EoS. Moreover, a new definition of segment-molar fraction instead of mole fraction is used both in the mixing rule of SRK and Quadrupole terms that allows one to correlate the

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phase behavior of these polymeric-scCO2 systems. Thus, the mqCPA EoS is written as

 =   +   + 

(1)

where  is the residual Helmholtz free energy and the three terms on the right hand side are Helmholtz free

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dispersion (SRK), association (ASS) and quadrupole (QQ) dimensionless energies, respectively. The details of mqCPA are given in the literature [42-45].

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In the present EoS, to reduce the complexity and provide a simple and robust EoS, one may neglect the association term. It should be noted that treating CO2 as an associating or non-

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associating molecule is still a matter of controversy. Thus, CO2 is often assumed as a quadrupolar, self-associating or solvation molecule [44]. 2.2 The SRK Contribution

The dimensionless Helmholtz free energy of the SRK contribution is expressed as [45]

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  = −1 −  −

 1 

+ 

(2)

where R is universal gas constant, T is absolute temperature and ρ denotes density. In the present

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SRK term the parameters “a” and “b” for mixtures are written in terms of the segment-molar fractions instead mole fraction. Thus, by using the conventional van der Waals one-fluid mixing rule,  and  are calculated as



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 =       

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 =    

(3)

(4)

where  is the segment molar fraction that are defined for a binary polymer-scCO2 system by Eqs. (7,8).  stands for the pure co-volume parameter of component i and  denotes the cross

 =   

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where   is given as

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energy parameter which is calculated using the classical geometric mean rule as

(

  =  !1 + "# $1 − % &'

(5)

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(6)

where  and "# are the adjustable parameters that are often optimized by regression of the vapor pressure and liquid density data of the pure components. However, in this work these two parameters for CO2 are taken from the literature [42].  = (/* ) is defined as reduced temperature. For polymers, we assumed as + → +∞ where  → 0. Hence, we ignored temperature dependency for /0123 , then c# for a polymer set to zero. As explained, the segment -molar fraction was used instead of mole fraction and the association term is neglected 8

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in the present EoS. Thus, the segment molar fractions for a binary polymer-CO2 system are defined as  , 6  , 6 +  ,/ 6/

/ =

 ,/ 6  , 6 +  ,/ 6/

(7)

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 =

(8)

where s and p stand for solvent (CO2) and polymer, respectively and 6 is mole fraction of

7 7+8(

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 ,/ =

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component i and  ,/ is a dimensionless segment number that is defined as

(9)

where M is the average molar mass of polymer and M:;( denotes the molar mass of CO2. Thus, for CO2 n=,> is set to unity. In the range of polymer concentration studied in this work, the glass transition temperature as well as the melting temperature of the polymer does not play a role and

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the polydispersity of polymers was neglected.

In this work, a recent perturbation model of Perturbed Chain Polar - Statistical Associating Fluid

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Theory (PCP-SAFT EoS) [15, 25] is also applied that the dimensionless residual Helmholtz free energy (a@A> ) consists of the three terms as a reference hard-chain contribution [46, 47] (aBC ), a

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perturbation contribution [48] (aDE>F ) and a quadrupole-quadrupole contribution [25] (a== ). Thus, one can write as

 G+GH I = BC + DE>F + ==

(10)

The details of PCP-SAFT EoS are given in the literature [25].

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It could be explained that the polymers have all polar interactions such as dipole-dipole (DD), quadrupole-quadrupole (QQ) and dipole-quadrupole (DQ). On the other hand, CO2 has been mostly accepted to have quadrupole moment. Thus, in mqCPA EoS, DD and QQ interactions are

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assumed to be presented in “disp” and “QQ” terms, respectively, and we neglect DQ interactions to reduce the complexity of mqCPA EoS.

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The mqCPA and PCP-SAFT EoSs are both capable of liquid-liquid equilibrium calculation of the cloud point pressure of the polymer-scCO2 systems. At the cloud point of a polymer-scCO2

should

be

satisfied.

The

unknown

J

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system, the equilibrium criteria as the equality of chemical potentials and pressures in each phase variables

are

the

mole

fraction

of

polymer

(6/  and densities of the phases J , T  that are calculated by an iterative numerical method like Newton-Raphson (NR). The superscripts of U and
denote lean and rich polymer phases,

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respectively. The detail of the parameters fitting has been given in the previous work [15]. In the present study using mqCPA and PCP-SAFT EoSs, the cloud point pressures of polymerscCO2 systems for the three types of poly(lactic acid) (PLA), the three types poly(1-O-(vinyloxy)

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ethyl-2,3,4,6-tetra-O-acetyl-
-d-glucopyranoside) (P(AcGlcVE) [49] and seven types of poly(vinyl acetate-alt-dibutyl maleate) (PVAc-alt-PDBM) [50] were calculated. Moreover, to

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correlate the cloud point pressure of the PVAc-alt-PDBM-scCO2 copolymer systems with the different molecular structures, molar masses and end groups, the PC-SAFT EoS was applied. The structures of these copolymers are given in the previous works [15, 20]. 3. RESULTS AND DISCUSSION The mqCPA EoS with PC-SAFT and PCP-SAFT EoSs were used to correlate the cloud point pressure of the soluble polymers in the supercritical carbon dioxide fluid at various temperatures 10

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and polymer concentrations. The parameters of the mqCPA EoS are b0, a0, c1 and Q. Similarly, the parameters of PC-SAFT are m/M, ε/k, σ and the parameters of PCP-SAFT are presented as m/M, ε/k, σ, and Q. the Q value in PCP-SAFT is not an adjustable parameter, but the

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experimental value of the Quadrupole momentum. The pure parameters of CO2 using the qCPA, PCSAFT and PCP-SAFT equations of state are shown in Table 1. As seen the parameters for PC-SAFT, PCP-SAFT and qCPA were taken from [48, 25, 42], respectively. Table 1 shows the deviations of vapor

gives the better results than PC-SAFT and PCP-SAFT EoSs.

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pressure and liquid density of pure CO2 using these three equations of state. As one can observe, qCPA

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For the polymer- scCO2 systems, the following objective function was applied: Pi exp − Pi calc 100 (11) ∑ P exp Np i where Np is the number of experimental cloud point data and P denotes the cloud point pressure AAD %=

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of the polymer-scCO2 system. The superscripts "exp" and "calc" denote the experimental and calculated values, respectively. The details of the calculation and the algorithm are described in the previous study [15]. In this work, we kept the values of the CO2 parameters unchanged so

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that using these three equations of state we only carried out the LLE calculations to find out the pure parameters of the present copolymers through regression of the cloud point pressure of the

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polymer-scCO2 systems as shown in Tables 2-5. 3.1 The Polymer-scCO2 systems at constant T Figure 1 shows the results of the cloud point pressure against polymer wt% for the three P(AcGlcVE)-scCO2 systems with the different molar masses. As one can observe, for P(AcGlcVE) with the different molar masses, the results of mqCPA are in excellent agreement with the experiment. It is worth to mention the present results are obtained without using kij 11

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(binary interaction parameters). As seen from Table 2 using mqCPA EoS, for the P(AcGlcVE)scCO2 systems with the different molar masses AAD% are 2.6, 0.7 and 2.9 while using PCP-

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SAFT EoS the deviations are 2.4, 0.1 and 0.7, which are given in the previous work [15]. Similarly, Figure 2 shows the experimental and the calculated cloud point pressures for the PLA–scCO2 systems. In addition, this figure shows the PLA with the different molar masses and

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the various end groups that present significant influence in solubility of these polymers in scCO2. As shown in Table 2, the calculated results for the PLA with various end groups show an

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excellent agreement with the experiment by 1.9, 1.3 and 4.1 deviation in AAD%. For this system using PCP-SAFT the deviations were 7.7, 3.4 and 5.3, respectively, that have been given in the previous work [15]. It should be noted that using the PC-SAFT EoS one couldn’t correlate the PLA– and P(AcGlcVE)-scCO2 systems, particularly, at high polymer molecular weight and high

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pressure.

3.2 The Polymer-CO2 systems at Constant Concentration The parameters of the equations of state for the PVAc-PDBM-scCO2 systems at various molar

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masses and end groups are shown in Tables 3-5 using mqCPA, PC-SAFT and PCP-SAFT EoSs,

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respectively. Figure 3 shows the experimental and calculated cloud point pressure versus temperature for the PVAc-PDBM 7000-scCO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs. The AAD% for this system are presented in Table 3-5. As one observes the calculated cloud point pressures are in good agreement with the experiment using both mqCPA and PCPSAFT EoSs with almost the same results. Keep in mind the present EoS has more simple form than the complex mathematical form of PCP-SAFT EoS.

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Similarly, Figures 4 and 5 show the experimental and calculated cloud point pressures for both the PVAc-PDBM-x 2500-CO2 and PVAc-PDBM-x 3200-CO2 systems with xanthine end group in the various temperatures using the PC-SAFT, PCP-SAFT and mqCPA EoSs. As one can

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observe in Table 3-5, both mqCPA and PCP-SAFT EoSs give almost the same results for the PVAc-PDBM 2500+scCO2 system, but for PVAc-PDBM 3200+CO2 system, mqCPA gives

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better results.

Figures 6 and 7 demonstrate the cloud point pressures versus the weight fraction of polymer for

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the PVAc-PDBM-x 4200-CO2 and PVAc-PDBM-x 5300-CO2 systems with xanthine end group in the various temperatures, respectively, using these three EoSs. As one can see from Table 3-5, the PCP-SAFT is superior to both PC-SAFT and mqCPA for the PVAc-PDBM 4200-scCO2 system, but for the PVAc-PDBM 5300-scCO2 system with xanthine end group, the mqCPA EoS gives the better results than the other EoSs, but PC-SAFT results high deviation. Moreover,

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Figure 8 shows the calculated cloud point pressure of the PVAc-PDBM-x 6400-scCO2 system versus temperatures. As one can see, the results of mqCPA EoS are superior to those obtained by PCP-SAFT EoS so that the deviation of the PC-SAFT results is high. It should be noted that

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using PCP-SAFT EoS, the calculated cloud point pressure runs through minimum in Figs. 8 and

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9, but not the experimental data. This shows possibility of minima of the isopleths for these two systems that indicates a complex phase behavior with two critical points. However, mqCPA EoS results are in good agreement with the experimental data. 3.3 The Polymer-CO2 systems at various Temperatures and Concentrations Figure 8 shows the calculated and experimental cloud point pressures for solubility of the alternative copolymers of vinyl acetate and dibutyl maleate, PVAc-PDBM 3800, in scCO2 versus

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temperature at the various weight percent [41] using mqCPA, PC-SAFT and PCP-SAFT EoSs with AAD% of 11.1, 17.3 and 11.6, respectively. As shown in Tables 3-5, the results of these three EoSs in comparison to the other systems show less accurate, but both mqCPA and PCP-

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SAFT EoS give the better results than PC-SAFT. However, as one can see using PCP-SAFT at 0.6 wt% of polymer, the calculated cloud point pressure runs through minimum as shown in Figure 9, but not the experimental data. One can conclude that with increasing of polymer

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concentration the calculated results of PCP-SAFT EoS demonstrate a minimum. Thus, more experimental data for the PVAc-PDBM 3800-scCO2 system are needed at the different

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composition and temperature to validate the results of these three EoSs. Therefore, through accounting the QQ interaction among the polymer segments and carbon dioxide allows one to correlate the phase behavior of these polymer–scCO2 systems using mqCPA EoS.

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3.4 The model parameter trends of the three equations of state against molecular weight Table 6 shows the model parameter relations against molecular weight of the polymers for the three EoSs. As one can see for the three polymers, the parameters of the mqCPQ EoS are

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expressed against molecular weight with good trends. However, the parameters of the PC-SAFT and PCP-SAFT EoSs don’t show a good trend. On the other hand, as one can see the pure

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compound parameters of the polymers are regressed by the binary data of each polymer-scCO2 system. However, the like copolymers with the different molar masses have the same repeating segment so that one expects the parameters to be the same. Thus, we carried out comprehensive optimizations simultaneously to find out the equation of state parameters for the same copolymers with various molar masses. As shown in Table 7, the equations of state parameters are calculated for the like copolymers using the mqCPQ, PC-SAFT and PCP-SAFT EoSs. As one can see the overall deviations increase for each EoS in respect to those results were obtained 14

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for each single copolymer with own molecular weight, however mqCPA EoS shows better results. It should be noted that the PC-SAFT EoS may not be used for P(AcGlcVE)-scCO2

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systems. As a result, as one can see from Table 3-5 the overall deviations for the PVAc-PDBM -scCO2 systems are 2.4, 7.7 and 3.3 using mqCPA, PC-SAFT and PCP-SAFT EoSs, respectively. It

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should be noted that the PCP-SAFT EoS has an additional parameter respect to both PC-SAFT and mqCPA EoSs. Thus, one may conclude no surprising by more flexibility of PCP-SAFT the

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better results were obtained. On the other hand, one can see both PC-SAFT and qmCPA have the same number of the parameters, but mqCPA presents more capability to correlate the phase behavior of the soluble polymers in supercritical carbon dioxide fluid with very good accuracy in respect to PC-SAFT. Moreover, mqCPA shows mathematically a more simple form than both

4. CONCLUSIONS

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PC-SAFT and PCP-SAFT EoSs so that it needs less calculation.

The Quadrupole Cubic Plus Association (mqCPA) EoS was used to correlate the cloud point

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pressure of the soluble copolymers in scCO2 fluids through liquid-liquid equilibrium

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computation. This EoS consists of SRK part plus a quadrupole term. The present EoS was applied for the three types of PLA-CO2 systems with different molar masses and end groups, the three types of P(AcGlcVE)–CO2 systems with the different molar masses and seven PVAcPDBM-CO2 systems with the different weight percent and various end chemical groups. Moreover, PC-SAFT and PCP-SAFT EoSs were applied for modeling of the same polymerscCO2 systems. The parameters of these three EoSs for each polymer were obtained through optimization of the experimental data of cloud point pressure of the copolymer-scCO2 systems.

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The results showed that both mqCPA and PCP-EoSs present powerful ability to model the phase behavior of soluble copolymer in supercritical carbon dioxide fluid. Moreover, the results showed the present EoS can correlate the phase behavior of the copolymer-scCO2 systems with

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very good accuracy by deviation less than 2% for some copolymer-CO2 systems. Finally, the overall results of both mqCPAand PCP-SAFT EoSs were almost the same for the PVAc-PDBM-

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scCO2 systems and showed superiority respect to PC-SAFT. REFERENCES

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[3] S. Mawson, K.P. Johnston, J.R. Combes, J.M. DeSimone, Formation of Poly(1,1,2,2tetrahydroperfluorodecyl acrylate) Submicron Fibers and Particles from Supercritical Carbon Dioxide Solutions, Macromolecules, 28 (1995) 3182-3191. [4] Mohammad Z. Hossain, Amyn S. Teja, Modeling Phase Equilibria in CO2 + polymer systems, The Journal of Supercritical Fluids, 96 (2015), 313-323.

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[5] Kimberly F Webb, Amyn S Teja, Solubility and diffusion of carbon dioxide in polymers, Fluid Phase Equilibria, 158–160 (1999) 1029–1034.

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[6] David L. Tomasko , Hongbo Li , Dehua Liu , Xiangmin Han , Maxwell J. Wingert , L. James Lee , and Kurt W. Koelling , A Review of CO2 Applications in the Processing of Polymers, Ind. Eng. Chem. Res., 42 (25) (2003) 6431–6456. [7] Kamel Khimechea, b, Paolo Alessic, Ireneo Kikicc, Abdallah Dahmani, Solubility of diamines in supercritical carbon dioxide: Experimental determination and correlation, The Journal of Supercritical Fluids, 41, Issue 1 (2007) 10–19. [8] L. Li, K. E. Counts, S. Kurosawa, A. S. Teja, D. M. Collard, Tuning the Electronic Structure and Solubility of Conjugated Polymers with Perfluoroalkyl Substituents: Poly(3perfluorooctylthiophene), the First Supercritical CO2-soluble Conjugated Polymer, Advanced Materials, V.16, Issue 2, January (2004), 180–183.

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[9] Patrícia Coimbra, Daniel Fernandes, Paula Ferreira, Maria H. Gil, Hermínio C. de Sousa, Solubility of Irgacure® 2959 photoinitiator in supercritical carbon dioxide: Experimental determination and correlation, The Journal of Supercritical Fluids, 45, Issue 3 (2008) 272–281.

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[11] P.J. Flory, Thermodynamics of High Polymer Solutions, the Journal of Chemical Physics, 10 (1942) 51-61.

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[16] P.K. Jog, W.G. Chapman, An Algorithm for Calculation of Phase Equilibria in Polydisperse Polymer Solutions Using the SAFT Equation of State, Macromolecules, 35 (2002) 1002-1011.

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thermodynamic modeling of the cloud point pressure for solubility of copolymers of vinyl acetate and dibutyl maleate in supercritical CO2, Fluid Phase Equilib., 425 (2016) 136-142. [21] Y. Xiong, E. Kiran, Miscibility, density and viscosity of poly(dimethylsiloxane) in supercritical carbon dioxide, Polymer, 36 (1995) 4817-4826.

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[24] X. Xu, D.E. Cristancho, S. Costeux, Z.-G. Wang, Density-Functional Theory for Polymer– Carbon Dioxide Mixtures, Industrial & Engineering Chemistry Research, 51 (2012) 3832-3840.

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[30] F. Tumakaka, J. Gross, G. Sadowski, Thermodynamic modeling of complex systems using PC-SAFT, Fluid Phase Equilibria, 228–229 (2005) 89-98. [31] E.K. Karakatsani, T. Spyriouni, I.G. Economou, Extended statistical associating fluid theory (SAFT) equations of state for dipolar fluids, AIChE Journal, 51 (2005) 2328-2342. [32] D.W. Cho, J.H. Lee, J. Shin, W. Bae, H. Kim, M.S. Shin, High-pressure phase behaviour measurement of (CO 2+ ethylene glycol dimethacrylate) and (CO 2+ di-ethylene glycol dimethacrylate) binary mixture systems, The Journal of Chemical Thermodynamics, 43 (2011) 1666-1671.

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[37] A. Martín, F. Mattea, L. Gutiérrez, F. Miguel, M.J. Cocero, Co-precipitation of carotenoids and bio-polymers with the supercritical anti-solvent process, The Journal of Supercritical Fluids, 41 (2007) 138-147. [38] J. Gregorowicz, Phase behaviour of l-lactide in supercritical carbon dioxide at high pressures, The Journal of supercritical fluids, 46 (2008) 105-111.

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[42] M.G. Bjørner, G.M. Kontogeorgis, Modeling derivative properties and binary mixtures with CO2 using the CPA and the quadrupolar CPA equations of state, Fluid Phase Equilibria, 408 (2016) 151-169. [43] M.G. Bjørner, G.M. Kontogeorgis, Corrigendum to “Modeling derivative properties and binary mixtures with CO2 using the CPA and the quadrupolar CPA equations of state”[Fluid Phase Equilib. 408 (2016) 151–169], Fluid Phase Equilibria, 421 (2016) 104. [44] I. Tsivintzelis, G.M. Kontogeorgis, M.L. Michelsen, E.H. Stenby, Modeling phase equilibria for acid gas mixtures using the CPA equation of state. Part II: Binary mixtures with CO2, Fluid Phase Equilibria, 306 (2011) 38-56. 19

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[45] G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chemical Engineering Science, 27 (1972) 1197-1203. [46] W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, New reference equation of state for associating liquids, Industrial & Engineering Chemistry Research, 29 (1990) 1709-1721.

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[48] J. Gross, G. Sadowski, Perturbed-Chain SAFT:  An Equation of State Based on a Perturbation Theory for Chain Molecules, Industrial & Engineering Chemistry Research, 40 (2001) 1244-1260.

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[49] D. Tapriyal, Y. Wang, R.M. Enick, J.K. Johnson, J. Crosthwaite, M.C. Thies, I.H. Paik, A.D. Hamilton, Poly(vinyl acetate), poly((1-O-(vinyloxy) ethyl-2,3,4,6-tetra-O-acetyl-β-dglucopyranoside) and amorphous poly(lactic acid) are the most CO2-soluble oxygenated hydrocarbon-based polymers, The Journal of Supercritical Fluids, 46 (2008) 252-257.

List of Figures

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[50] H. Lee, J.W. Pack, W. Wang, K.J. Thurecht, S.M. Howdle, Synthesis and Phase Behavior of CO2-Soluble Hydrocarbon Copolymer: Poly(vinyl acetate-alt-dibutyl maleate), Macromolecules, 43 (2010) 2276-2282.

Figure 1. The experimental [49]and calculated cloud point pressure versus weight percent of copolymer for the different P(AcGlcVE)-CO2 systems at 298 K using mqCPA EoS

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Figure 2. The experimental [49] and calculated cloud point pressure versus weight percent of copolymer for the different PLA-CO2 systems at 298 K using mqCPA EoS

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Figure 3. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM 7000-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs Figure 4. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 2500-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs Figure 5. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 3200-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs Figure 6. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 4200-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs Figure 7. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 5300-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs 20

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Figure 81. The experimental [50] and calculated cloud point pressure vs. temperature for the 0.2 wt% of PVAc-PDBM-x 6400-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

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Figure 9. The experimental [50] and calculated cloud point pressure versus temperature for the different PVAc-PDBM 3800-CO2 systems (0.2, 0.4 and 0.6 wt%) using PC-SAFT, PCP-SAFT and mqCPA EoSs

21

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Table 1. The pure CO2 parameters for qCPA, PC-SAFT and PCP-SAFT EoSs qCPA EoS b0 (mL/mol.)

a0(Pa.m6/mol2)*

c1 (-)

Q (DÅ)

AAD %**

27.93

0.68

0.2982

0.13

4.3

PC-SAFT EoS ε/k (K)

2.0729

σ (Å)

169.2100

-

2.7852

1.5131

σ (Å)

163.33

vl

2.73

2.78

Q (DÅ)

3.1869

Ref.

Psat

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ε/k (K)

[42]

AAD %**

PCP-SAFT EoS m

0.46

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m

ρl

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Psat

Ref.

4.4

[48]

AAD %**

Psat

ρl

0.34

1.27

Ref.

[25]

∗In work of Bjørner and Kontogeorgis [42], the parameter Γ = a0/Rb0 =1284 K is used.   ,  , ∑   ;  ,  

=   ,  ,  

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∗∗AAD% =

Table 2. The parameters of mqCPA EoS for the PLA- and P(AcGlcVE)-scCO2 systems [49] and their AAD% in cloud point pressure Mw

EP

Polymer

b0

a0

Q

Np

AAD%*

1200

21.0469

0.3044

1.5641

3

1.9

PLA (linear end group (R1))

5700

21.8495

0.3628

1.5692

5

1.3

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PLA (linear end group (R2))

PLA (acid end group)

12000

24.6021

0.3628

2.1783

5

4.1

P(AcGlcVE)

6000

24.9385

0.3042

3.2768

6

2.6

P(AcGlcVE)

20000

18.8714

0.2714

2.2125

5

0.7

P(AcGlcVE)

40000

22.9763

0.2913

2.9473

5

2.9

P * AAD %= 100 ∑ i N p

exp − P calc i Pi exp

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Table 3. The parameters of mqCPA EoS for the alternative copolymers of vinyl acetate and dibutyl maleate [50].

PVAc-PDBM

3800

29.0097

PVAc-PDBM

7000

PVAc-PDBM-x

a0

Q (DÅ)

Np

AAD%*

0.3690

8.4180

17

11.1

26.6771

0.9138

6.2018

5

1.7

2500

20.1239

0.8933

3.7439

5

1.4

PVAc-PDBM-x

3200

30.1172

1.2126

5.9854

4

0.6

PVAc-PDBM-x

4200

15.6015

0.3310

5.0917

4

0.9

PVAc-PDBM-x

5300

18.2129

0.3771

PVAc-PDBM-x

6400

17.5010

0.2690

Overall P * AAD %= 100 ∑ i N p

(Pa.m6/mol2)

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b0 (mL/mol.)

SC

Mw

5.7520

5

0.8

5.5594

5

0.2

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exp − P calc i Piexp

2.4

Polymer

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Table 4. The parameters of PC-SAFT EoS for the alternative copolymers of vinyl acetate and dibutyl maleate [50] Mw

m/M

ε/k (K)

σ (Å)

AAD%*

3800

0.0360

205.9420

3.3331

17.3

7000

0.0375

207.6857

3.5118

5.0

PVAc-PDBM-x

2500

0.0445

210.9446

3.4832

2.7

PVAc-PDBM-x

3200

0.0443

219.2516

3.5582

1.6

PVAc-PDBM-x

4200

0.0422

206.4994

3.4029

6.4

PVAc-PDBM-x

5300

0.0442

211.9898

3.4458

9.3

PVAc-PDBM-x

6400

0.0450

211.9259

3.4457

11.9

PVAc-PDBM

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PVAc-PDBM

Overall

P * AAD %= 100 ∑ i N p

exp − P calc i Piexp

7.7

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Table 5. The parameters of PCP-SAFT EoS for the alternative copolymers of vinyl acetate and dibutyl maleate [50] Mw

m/M

ε/k (K)

σ (Å)

Q (DÅ)

PVAc-PDBM

3800

0.0301

192.5330

3.5219

5.8895

11.6

PVAc-PDBM

7000

0.0300

186.2110

3.6115

6.5620

1.9

PVAc-PDBM-x

2500

0.0324

184.0937

3.3622

5.3897

1.3

PVAc-PDBM-x

3200

0.0328

199.2224

3.5015

3.8471

1.4

PVAc-PDBM-x

4200

0.0282

177.2434

3.3183

4.9425

0.1

PVAc-PDBM-x

5300

0.0295

179.5954

3.3615

4.1073

2.7

PVAc-PDBM-x

6400

0.0259

182.2536

3.4330

4.5713

3.8

P * AAD %= 100 ∑ i N p

exp − P calc i Piexp

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Overall

AAD%*

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Polymer

3.3

Table 6. The correlation of the parameters of the three equations of state versus molecular weight of the polymers. Polymer

R2

b0

a0

R2

Q

R2

mqCPA EoS

2

1

1×10-10M2-5×10-6M+0.33

1

3×10-9M2-0.0002M+4.1309

1

1

0.0002M-0.4088

1

-0.0006M+11.582

1

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P(AcGlcVE)

2×10 M -

PVAc-PDBM

-0.0007M+32.34

0.0009M+29.793

PVAc-PDBM-x

-7

2

6×10 M -

0.37

0.0082M+44.86

R

5×10-7M+0.0342

1

2

5×10 M -4×10-

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PVAc-PDBM-x

m/M

-10

2

PVAc-PDBM-x

-3×10-8M+0.0302

1

ε/k (K)

-07

2

4×10 M -

2

ε/k (K)

0.8

R

0.0002M+5.0909 -7

2

1×10 M 0.0013M+7.5298

0.78

-1×10 M +0.0014M+1.0861

0.32

R2

σ (Å)

R2

1

6×10-5M+3.1209 -9

2

-5

1

0.07

8×10 M -9×10 M+3.6987

0.33

R2

σ (Å)

R2

1

3×10-5M+3.4155

PCP-SAFT -0.002M+200.04 -7

2

7×10 M 0.0091M+208.33

Q (DÅ)

2

0.0046M+222.62 R

-2×10 M+0.0369

-7

PC-SAFT

0.64

m/M

-6

5×10 M -

0.0005M+203.87

6M+0.0521

PVAc-PDBM

2

0.0008M+2.9533

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PVAc-PDBM

-8

2

1 0.24

0.22

-8

2

1×10 M -0.0001M+3.6242

1 0.1

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M=Molecular Weight; R2= R-squared value

Polymer Type

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Table 7. The comprehensive parameters of the mqCPA, PC-SAFT and PCP-SAFT EoSs for the different copolymers in scCO2 and their AAD% results Mw Range

b0 (mL/mol.)

P(AcGlcVE)

6000-40000

22.2364

0.3015

2.6088

3.2

PVAc-PDBM

4900-8400

24.5949

0.7425

6.4232

14.7

PVAc-PDBM-x

3200-7800

17.4571

0.6307

4.8101

5.0

a0 (Pa.m6/mol2)

Q (DÅ)

Overall

m/M

PVAc-PDBM

4900-8400

0.0367

PVAc-PDBM-x

3200-7800

0.0437

PC-SAFT EoS

Overall Mw Range

m/M

P(AcGlcVE)

6000-40000

0.0203

PVAc-PDBM

4900-8400

PVAc-PDBM-x

3200-7800

p

exp − P calc i Pi exp

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P * AAD %= 100 ∑ i N

σ (Å)

3.4397

16.0

211.9252

3.4459

7.1 11.6

ε/k (K)

σ (Å)

Q (DÅ)

310.039

3.9795

2.8871

4.7

0.0310

189.9971

3.5499

5.5036

18.1

0.0315

198.9783

3.5068

4.0212

7.3

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Overall

7.6

207.5810

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PCP-SAFT EoS

ε/k (K)

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mqCPA EoS

AAD%*

10.0

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Figure 1. The experimental [49]and calculated cloud point pressure versus weight percent of copolymer for the different P(AcGlcVE)-CO2 systems at 298 K using mqCPA EoS

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Figure 2. The experimental [49] and calculated cloud point pressure versus weight percent of copolymer for the different PLA-CO2 systems at 298 K using mqCPA EoS

Figure 3. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM 7000-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

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Figure 4. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 2500-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

Figure 5. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 3200-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

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Figure 6. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 4200-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

Figure 7. The experimental [50] and calculated cloud point pressure versus temperature for the 0.2 wt% of PVAc-PDBM-x 5300-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

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Figure 8. The experimental [50] and calculated cloud point pressure vs. temperature for the 0.2 wt% of PVAc-PDBM-x 6400-CO2 system using PC-SAFT, PCP-SAFT and mqCPA EoSs

Figure 9. The experimental [50] and calculated cloud point pressure versus temperature for the different PVAc-PDBM 3800-CO2 systems (0.2, 0.4 and 0.6 wt%) using PC-SAFT, PCP-SAFT and mqCPA EoSs

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Highlights

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A modified Cubic Plus Quadrupole Equation of State (mqCPQ EoS) was used for polymer-supercritical CO2 systems. mqCPQ EoS consists of two terms as SRK and Quadrupole-Quadrupole. The cloud point pressures were calculated for scCO2 with poly(lactic acid), poly(1-O(vinyloxy) ethyl-2,3,4,6-tetra-O-acetyl-ߚ-d-glucopyranoside) and poly(vinyl acetate-altdibutyl maleate). The pure component parameters of mqCPQ EoS are calculated through LLE calculation. The results of mqCPA are compared with both PC-SAFT and PCP-SAFT EoSs

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