Phase Behavior of Hyperbranched Polymer Systems: Experiments

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J. Phys. Chem. B 2009, 113, 1022–1029

Phase Behavior of Hyperbranched Polymer Systems: Experiments and Application of the Perturbed-Chain Polar SAFT Equation of State Marta K. Kozłowska,† Bas F. Ju¨rgens, Christian S. Schacht, Joachim Gross,* and Theo W. de Loos Engineering Thermodynamics, Process & Energy Department, Delft UniVersity of Technology, Leeghwaterstraat 44, 2628 CA Delft, The Netherlands ReceiVed: May 20, 2008; ReVised Manuscript ReceiVed: October 20, 2008

Vapor-liquid equilibrium data for systems of hyperbranched polymer (HBP) and carbon dioxide are reported for temperatures of 285-455 K and pressures up to 13 MPa. The bubble-point pressures of (CO2 + hyperbranched polyester) and of (CO2 + hyperbranched polyglycerol + CH3OH) samples with fixed compositions were measured using a Cailletet apparatus. The system (CO2 + polyglycerol + CH3OH) also exhibits a liquid-liquid phase split characterized by lower critical solution temperatures. For this system cloud point curves and vapor-liquid-liquid bubble-point curves were also measured. Moreover, a thermodynamic model has been developed for HBP mixtures in the framework of the perturbed-chain polar statistical association fluid theory (PCP-SAFT) equation of state accounting for branching effects. There is no additional binary interaction parameter introduced along with the branching contributions to the model. Although the miscibility gap in the system (CO2 + polyglycerol + CH3OH) is not predicted by the model, PCP-SAFT including branching effects gives a good representation of the bubble-point curves of this system at temperatures lower than the lower solution temperature (LST). 1. Introduction Hyperbranched polymers (HBPs) have recently attracted much interest due to their physical, thermal, and chemical properties as well as their unique structure (Figure 1). Compared with dendrimers, which are highly uniform and monodisperse polymers,1-3 HBPs are randomly branched, three-dimensional, and polydisperse macromolecules.4,5 They can easily be synthesized via economically favorable one-step reactions6-8 but often show similar chemical, thermal, and rheological properties as dendrimers. Therefore, HBPs may replace dendrimers in many industrial applications.9 Molecular entanglement is much less pronounced in hyperbranched polymers compared to their linear analogues of equal molecular mass. As a result, they possess a remarkable thermal stability5,10,11 and show a lower viscosity in comparison to linear analogues.12,13 Many of HBPs are liquids at room temperature. By controlled functionalization of the large number of functional groups the properties of HBPs can be tailored.8,14,15 Hyperbranched polymers can be used as solvents or additives in separations based on absorption to increase separation performance. In contrast to the conventional solvents, these novel compounds have negligible vapor pressure which results in hardly any emission to the environment and leads to low flammability. Seiler et al.14-16 showed that HBPs develop preferred interaction with one component of a mixture, or in terms of thermodynamics, that the fugacities of components in mixtures are selectively altered by the HBPs. Therefore, HBPs are potential green solvents for large-scale industrial applications, for example, as extraction solvents in liquid-liquid extraction processes or as entrainers in extractive distillation processes.8 They can also be considered promising as absorption * Corresponding author: e-mail [email protected]; Tel +31 15 2786658; Fax +31 15 2786975. † Permanent address: Physical Chemistry Division, Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warsaw, Poland.

liquids, for example, for the separation of carbon dioxide from waste gas streamssa greenhouse gas compound which is emitted in large quantities into the environment. The separation of gases through an absorption process using hyperbranched polymers as absorption liquids appears to be an interesting application of these novel compounds. Arlt et al.17 recently evaluated the use of HBPs for absorption processes, and Fang et al.18 have already used hyperbranched polyimide membranes for gas separation applications. However, the research around hyperbranched polymers in the context of chemical engineering is very young. Thus, the development of new processes using HBPs as process liquids requires a broader experimental database on the phase behavior of such mixtures and suitable thermodynamic models. Different models have been proposed to correlate and predict thermodynamic properties and phase equilibria of mixtures containing hyperbranched or dendritic polymers. Among them is the lattice model of Jang et al.19-21 based on the lattice cluster theory (LCT) and the UNIFAC-FV approach of Seiler et al.22 Kouskoumvekaki et al.23 tested the UNIFAC-FV and Entropic-FV models to predict vapor-liquid equilibria of hyperbranched polymers as well. In previous studies the perturbed-chain statistical association fluid theory (PC-SAFT) equation of state (EOS)24,25 has successfully been applied to asymmetric mixtures and to binary polymer-solvent and ternary gas-polymer-solvent systems as well as to binary copolymer-solvent systems.26,27 There is a need to take into account not only the difference between the molecular sizes but also the branching character of HBPs and the multipolar interactions in these investigated complex systems. The PC-SAFT equation of state will thus be applied with expressions accounting for quadrupolar and dipolar interactions.28-31 The model used in this work has previously been termed perturbed-chain polar statistical association fluid theory (PCP-SAFT) equation of state, and it will here be applied together with a branching term for modeling hyperbranched polymers. The additional term accounting for the

10.1021/jp804459x CCC: $40.75  2009 American Chemical Society Published on Web 01/02/2009

Phase Behavior of Hyperbranched Polymer Systems

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Figure 1. Chemical structure of selected hyperbranched polymers: (a) Boltorn H20 (Perstorp AB), (b) polyglycerol (HyperPolymers GmbH).

Figure 2. Schematic representation in two-dimensional view of a molecule with two branching points, where eight segments participate in branching tetramers. Segments forming a branching tetramer are represented by solid circles, and linear segments are represented by open circles.

branching effects has been derived by Mu¨ller and Gubbins32 based on Wertheim’s thermodynamic perturbation theory of second order.33 This branching term has been applied earlier to complex molecules by Blas and Vega.34 This study explores and facilitates the applicability of HBPs as absorption liquids to selectively remove carbon dioxide from flue gas streams by providing experimental data of the phase behavior of carbon dioxide-HBP and carbon dioxide-HBP-solvent systems. Moreover, the influence of the polymers’ functionalities, the presence of water and of methanol, and the concentration of carbon dioxide on the phase behavior of selected hyperbranched polymer solutions is elaborated. Furthermore, the PCP-SAFT EOS is applied with a term accounting for the molecular branching of the HBPs. The model is here referred to as bPCP-SAFT for brevity and is used to correlate and predict the CO2 solubility in HBP mixtures over a wide range of conditions. 2. Theory 2.1. PCP-SAFT EOS. The foundation of the SAFT family of equations of state has been established by Wertheim33,35-38 in developing the perturbation theory of first and second order (TPT1 and TPT2) for directional interactions and by Chapman, Jackson, and colleagues,39-42 who have extended this formalism to mixtures and developed a simple description of associating interactions. In the limit of infinite association, a description of bonded spheres was obtained, leading to an expression for segment chains. Several successful variants of SAFT-type equations of state have been developed. We here apply the PCPSAFT equation of state,28-31 where the polar interactions are explicitly accounted for. Considering polar interactions is relevant in this study, since carbon dioxide is a prototype quadrupolar compound and the dipole moment of methanol also

contributes significantly to the overall interactions. Molecular polarizability leads to induced dipoles and thus to higher effective polar interactions. For the PCP-SAFT equation of state, this has been addressed in a previous study,31 but the static polarizability is not considered here. Like other SAFT models, the PCP-SAFT equation of state is based on a perturbation approach. The idea of a perturbation theory is that one starts from a reference fluid with wellestablished properties. The reference fluid is here a chain fluid composed of hard spheres. The Helmholtz energy of the hardchain fluid41 is written as

Aref Aid Ahs Achain ) + + NkT NkT NkT NkT

(1)

where N, k, and T denote the total number of molecules, the Boltzmann constant, and temperature, respectively. Aid, Ahs, and Achain denote the ideal gas contribution, the residual hard-sphere contribution, and the increment of the Helmholtz energy accounting for the formation of bonds between the monomers, respectively. The complete equation of state is given as a sum of the hardchain reference fluid term and the perturbation contributions, as

A Aref Adisp Aassoc Amultipole ) + + + NkT NkT NkT NkT NkT

(2)

where Adisp is the contribution due to dispersive attractions and Aassoc represents the Helmholtz energy due to association. This term accounts for the increment of the Helmholtz energy due to the presence of site-site specific short-range interactions among the segments, like hydrogen bonds. Finally, Amultipole accounts for longrange electrostatic interactions due to partial charges of molecules, and it can be decomposed into several terms, with

Amultipole ADD AQQ ADQ ) + + NkT NkT NkT NkT

(3)

where ADD, AQQ, and ADQ are the Helmholtz energy contributions accounting for dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole interactions, respectively. Higher order multipoles like octapoles are known to contribute only weakly to the overall interactions and are neglected. For substances with no dipolar and quadrupolar moment the equation of state simply reduces to the PC-SAFT equation of state.

1024 J. Phys. Chem. B, Vol. 113, No. 4, 2009

Kozłowska et al. TABLE 1: Experimental Bubble-Point Data for the System [w1CO2 + (100 - w1)Boltorn U3000] for Various Mass Percent of the Carbon Dioxide, w1 5% CO2

Figure 3. Bubble-point curves for the system (CO2 + Boltorn U3000). The symbols represent the experimental data with CO2 concentrations (%) as indicated. Lines are from the correlation by polynomials of the second or third order (15%).

For the detailed expressions in eqs 1-3 we refer to previous studies on the PC-SAFT equation of state25,26 and on the polar contributions to the model.28-30 2.2. PCP-SAFT EOS for Highly Branched Fluids. 2.2.1. Molecular Model. The molecular model underlying the PCP-SAFT EOS has implicitly been introduced in section 2.1, and it is described as follows. Molecules are assumed to be chains consisting of spherical segments. These freely joined segments exhibit dispersive (attractive) forces and may possess association sites, like hydrogen-bonding sites. Multipolar segments carry a dipole moment or a quadruple moment. Molecules such as water or hydrogen sulfide possess a dipole moment. Carbon dioxide is a linear molecule with a charge distribution of (-, + +, -) and is thus a strongly quadrupolar compound. The molecular model is a coarse-grained representation of molecules, where details on the exact location of association sites are not preserved and where angular constraints among segments are not accounted for. The segment number is treated as a continuous parameter, so that noninteger values are allowed. In order to approach highly branched molecules, new types of segments appear, namely articulation segments.32,34 An articulation segment has three or more bonds to neighboring segments, forming e.g. a tetramer unit within the molecule as illustrated with solid circles in Figure 2. We only consider articulation segments with three neighbors here, and the neighboring segments are rigidly bonded, forming an angle of 120° between the sites. The tetramer unit may also have flexible linear connections or arms, as illustrated with open circles in Figure 2. 2.2.2. TPT2 Approach for Branched Fluids. For branched molecules, the reduced Helmholtz energy now reads

A Aref Abr-ch Adisp Aassoc Amultipole (4) ) + + + + NkT NkT NkT NkT NkT NkT where Abr-ch is the Helmholtz energy contribution due to the branched structure. For this term we adopt the TPT2 applying an analytic parametrization derived by Mu¨ller and Gubbins,32 who used tabulated values43 of an integral equation approach developed by Attard44 for estimating a triplet correlation function. Blas and Vega34 have with high clarity shown that in treating the tetramer units (Figure 2) with the TPT2 the chain term Achain (which is part of the Helmholtz energy Aref according to eq 1) remains unaltered as compared to the case of linear chains. The

10% CO2

15% CO2

T/K

P/MPa

T/K

P/MPa

T/K

P/MPa

328.49 328.56 333.52 333.56 338.47 338.60 343.53 343.62 348.58 348.60 353.60 358.68 363.61 368.69

2.87 2.86 3.04 3.06 3.21 3.26 3.42 3.43 3.61 3.63 3.83 4.00 4.19 4.39

288.50 293.53 298.50 303.49 308.51 313.52 318.50 323.53 328.55 333.62 338.54 343.51 348.61 353.62 358.65 363.70 368.72 372.27 377.35 377.85 382.26 387.23 392.18 397.37 402.17 407.12 412.05 416.99

2.85 3.13 3.44 3.74 4.11 4.46 4.81 5.13 5.54 5.94 6.29 6.72 7.08 7.48 7.91 8.36 8.77 9.01 9.55 9.56 9.92 10.30 10.68 11.15 11.60 11.97 12.32 12.75

283.44 288.47 293.49 298.40 303.50 308.46 313.51 318.44 318.45 323.46 323.51 328.52 333.56 338.56 343.47 348.59 353.66 358.63 363.72

3.55 3.95 4.36 4.82 5.35 5.84 6.40 7.01 6.99 7.58 7.74 8.34 8.99 9.62 10.27 10.90 11.58 12.26 12.97

Achain term represents the first-order perturbation term due to the formation of linear chains. Therefore, the reference fluid for both nonbranched and branched molecules is a chain fluid composed of hard-sphere segments. Wertheim’s second-order theory (TPT2) describes more accurately how the segments are arranged along the chain because it accounts for the correlation between three consecutive segments. The additional term Abr-ch accounts for the branching units that are here given as rigid tetramer units within a hyperbranched macromolecule, as derived by Phan et al.45 with

Abr-ch )NkT

{

∑ xi ln i

(1 + √1 + 4λ)ni 2ni√1 + 4λ

}

(5)

where xi is the mole fraction of a component i, ni is the number of segments which form a branched tetramer unit within the molecule, and λ is a parameter that contains the triplet correlation function and depends on the packing fraction and the molecular geometry. The TPT2 term of eq 5 is an approximation introduced by Phan et al.,45 which is valid if λ , 1.33,45 For a rigid arrangement of the tetramer unit of segments with a fixed bond angle, ω, the parameter λ is given by32,33,45

λ)

hs hs hs g(3) hs (σ , σ , 2σ sin(ω/2))

[ghs(σ hs)]2

-1

(6)

where ghs is the pair radial distribution function of the hard-sphere fluid at the contact length and σhs is the diameter of the hard-sphere fluid. For our application the effective hard-sphere segment diameter, d, defined for PC-SAFT EOS25 will be considered with

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J. Phys. Chem. B, Vol. 113, No. 4, 2009 1025

TABLE 2: Experimental Bubble-Point (VL), Vapor-Liquid-Liquid Bubble-Point (VLL), and Cloud-Point (LL) Data for {w1CO2 + (100 - w1)[w2PG-2 + w3CH3OH]} at Indicated Mass Percentage of CO2, w1, for w2 ) 25% and w3 ) 75% 2% CO2

5% CO2

10% CO2

15% CO2

T/K

P/MPa

T/K

P/MPa

T/K

P/MPa

T/K

P/MPa

313.77 322.97 332.98 343.00 353.03 363.03 373.05 383.03 393.03 403.03 413.00 422.97 432.97 442.93

0.427 (VL) 0.504 (VL) 0.599 (VL) 0.704 (VL) 0.827 (VL) 0.966 (VL) 1.13 (VL) 1.31 (VL) 1.53 (VL) 1.79 (VL) 2.08 (VL) 2.43 (VL) 2.83 (VL) 3.28 (VL)

313.82 323.00 333.00 343.00 353.03 358.03 363.03 372.97 383.04 393.07 403.10 413.09 423.04 433.01 442.96 452.95

1.03 (VL) 1.20 (VL) 1.39 (VL) 1.61 (VL) 1.83 (VL) 1.95 (VL) 2.07 (VL) 2.33 (VL) 2.61 (VL) 2.91 (VL) 3.27 (VL) 3.64 (VL) 4.05 (VL) 4.51 (VL) 5.02 (VL) 5.59 (VL)

313.02 323.00 333.00 343.02 353.05 362.99 363.05 373.05 383.04 393.05 403.03 403.06 413.05 423.05 433.02 442.95 445.43 447.91 447.94 450.43 450.44 452.91 452.95

1.97 (VL) 2.31 (VL) 2.67 (VL) 3.05 (VL) 3.45 (VL) 3.86 (VL) 3.86 (VL) 4.28 (VL) 4.72 (VL) 5.17 (VL) 5.63 (VL) 5.63 (VL) 6.11 (VL) 6.62 (VL) 7.15 (VL) 7.74 (VL) 7.85 (VL) 8.00 (VL) 8.39 (VL) 9.10 (VL) 8.13 (LL) 9.87 (LL) 8.29 (LL)

316.48 323.01 333.03 343.03 353.08 363.09 373.10 382.98 393.09 403.07 413.04 418.04 423.04 418.05 423.03 428.02 418.04 423.04 428.04

3.13 (VL) 3.46 (VL) 4.00 (VL) 4.56 (VL) 5.14 (VL) 5.71 (VL) 6.30 (VL) 6.88 (VL) 7.45 (VL) 8.03 (VL) 8.61 (VL) 9.29 (LL) 10.67 (LL) 8.87 (VLL) 9.14 (VLL) 9.39 (VLL) 8.75 (VL) 8.69 (VL) 8.60 (VL)

TABLE 3: Experimental Bubble-Point (VL), Vapor-Liquid-Liquid Bubble-Point (VLL), and Cloud-Point (LL) Data for {w1CO2 + (100 - w1)[w2PG-2 + w3CH3OH]} at Indicated Mass Percentage of CO2, w1, for w2 ) 50% and w3 ) 50% 2% CO2

5% CO2

10% CO2

15% CO2

T/K

P/MPa

T/K

P/MPa

T/K

P/MPa

T/K

P/MPa

313.56 322.97 332.97 342.99 353.03 363.04 373.08 383.05 393.07 403.08 413.08 423.07 433.03 442.99 452.98

0.639 (VL) 0.749 (VL) 0.881 (VL) 1.02 (VL) 1.18 (VL) 1.36 (VL) 1.56 (VL) 1.78 (VL) 2.03 (VL) 2.32 (VL) 2.65 (VL) 3.02 (VL) 3.45 (VL) 3.94 (VL) 4.50 (VL)

313.09 322.96 332.99 343.01 353.16 363.06 373.11 383.10 393.10 403.11 403.14 413.12 413.13 423.11 433.06 443.03 453.02

1.55 (VL) 1.82 (VL) 2.11 (VL) 2.42 (VL) 2.76 (VL) 3.11 (VL) 3.47 (VL) 3.86 (VL) 4.27 (VL) 4.77 (VL) 4.71 (VL) 5.19 (VL) 5.24 (VL) 5.75 (VL) 6.30 (VL) 6.90 (VL) 7.58 (VL)

312.97 323.01 333.02 343.04 353.08 363.13 373.08 383.06 393.08 403.12 413.08 413.10 423.07 425.97 428.05 433.03 433.05 438.02 442.98 452.93 428.04 430.54 433.04 435.53 438.03

3.07 (VL) 3.61 (VL) 4.19 (VL) 4.80 (VL) 5.43 (VL) 6.06 (VL) 6.75 (VL) 7.41 (VL) 8.10 (VL) 8.80 (VL) 9.54 (VL) 9.51 (VL) 10.27 (VL) 10.40 (VL) 10.57 (VL) 10.84 (VL) 10.82 (VL) 11.06 (VL) 11.31 (VL) 11.78 (VL) 11.06 (LL) 11.67 (LL) 12.51 (LL) 13.08 (LL) 13.90 (LL)

313.14 322.98 332.99 353.04 363.06 373.05 383.04 393.01 403.05 413.08 423.05 433.05 343.00 353.04 363.06 373.05 383.04 393.01 403.05 413.08 423.05 433.05 343.00 353.04 363.06 373.05

4.59 (VL) 5.40 (VL) 6.28 (VL) 7.93 (VL) 8.59 (VL) 9.19 (VL) 9.70 (VL) 10.09 (VL) 10.38 (VL) 10.57 (VL) 10.67 (VL) 10.68 (VL) 7.18 (VLL) 8.08 (VLL) 8.95 (VLL) 9.80 (VLL) 10.60 (VLL) 11.33 (VLL) 12.01 (VLL) 12.64 (VLL) 13.20 (VLL) 13.68 (VLL) 7.24 (LL) 9.60 (LL) 11.98 (LL) 14.42 (LL)

σhs ) d. The function in the numerator is the triplet correlation function of three tangentially hard spheres. Mu¨ller and Gubbins32 defined another triplet correlations function g0, with hs hs hs g(3) hs (σ , σ , 2σ sin(ω/2)) ≡

ghs(σ hs)ghs(σ hs)g0(σ hs, σ hs, 2σ hs sin(ω/2)) (7) which is, as the superposition approximation suggests, a wellbehaved smooth function of density for a given angle ω between

segments. Tabulated values for g0 were given by Attard and Stell43 based on an integral equation approach of Attard.44 In analogy to the radial distribution function of hard spheres proposed by Carnahan and Starling,46 Mu¨ller and Gubbins32 proposed an analytical functional form for g0, with

g0 )

1 + aη + bη2 (1 - η)3

where η is the segment packing fraction24

(8)

1026 J. Phys. Chem. B, Vol. 113, No. 4, 2009

η)

π F 6

∑ xi mi di3

Kozłowska et al.

(9)

i

where F, mi, and di are the total number density of molecules, the number of segments, and the temperature-dependent segment diameter of component i, respectively.47,48 The constants a and b in eq 8 depend on the bond angle. The values for bond angles of 120° are a ) -2.9830 and b ) 2.6251.32 This equation correlates tabulated data for the triplet correlation function provided by Attard and Stell43 with a maximum reduced density of ηmax ) 0.47 and should not be used at higher densities. It is important to note that the effect of molecular branching is, according to the recipe outlined here, considered only in the repulsive interactions. This is a significant simplification, and one may argue that a more important part of this effect should be governed by dispersive interactions, since the (intermolecular) accessibility of segments in the molecular center is reduced compared to linear segment chains, whereas the repulsive interactions still prevail. 3. Experimental Section 3.1. Materials. The carbon dioxide used has been purchased from Hoek Loos with a minimal purity of 99.995%. The methanol was obtained from J.T. Baker and has a purity of 98.5%. Two commercially available hyperbranched polymers of different chemical structure and with different functional groups are studied. Boltorn U3000 is a fatty acid modified j w/M j n ) 1.5) and j w ) 6500 g mol-1, M dendritic polyester (M was supplied by Perstorp AB (Sweden). It is a liquid at room temperature; it has an average of 14 unsaturated fatty ester end groups and possesses lower viscosity than hyperbranched polyglycerol. PG-2 is a highly functional polyglycerol, which is a viscous liquid and possesses an inert polyether scaffold. j w/M jn ) j n ) 2700 g mol-1, M This hyperbranched polymer (M 1.5) was supplied by HyperPolymers GmbH. 3.2. Sample Preparation. Samples have been prepared as follows: a certain amount of the liquid (the pure polymer or the solution of polymer) has been injected into the Cailletet glass tube. The amount of liquid has been determined by weighting using an analytical balance. Next, the liquid has been degassed by the successive freezing of the sample using liquid nitrogen and melting under vacuum. After the sample had been frozen and the tube had been evacuated, carbon dioxide was dosed into a glass bulb with a calibrated volume and sealed with mercury. This amount of carbon dioxide of known volume, temperature, and pressure has been added to the Cailletet tube by the displacement using mercury. The estimated error in the weighing of the polymer and the solvent is (0.1 mg. 3.3. Phase Equilibrium Measurements. Vapor-liquid, vapor-liquid-liquid, or liquid-liquid equilibrium (VLE, VLLE, and LLE, respectively) data have been measured for binary (HBP + CO2) or ternary (CO2 + HBP + CH3OH) mixtures for the purpose of determining the applicability of hyperbranched polymers in the absorption/desorption cycles. The bubble-point pressures of the samples were measured from 285 up to 455 K according to the static-synthetic method using a Cailletet apparatus described in detail previously.49 A sample of the mixture with fixed composition is confined over mercury in a glass tube, placed in a thermostatic bath with circulating water or silicon oil. The mixture is agitated by a small soft-iron stirrer sealed in glass. The stirrer is activated by button magnets which move up and down. Two-phase boundaries have been deter-

Figure 4. Influence of carbon dioxide concentration on the phase equilibrium in the ternary system {CO2 + [PG-2 (25%) + CH3OH (75%)]}. The symbols represent the experimental data with CO2 concentrations (%) as indicated. Solid lines are from the correlation by polynomials of the third order. Dotted lines represent the bPCPSAFT modeling results.

Figure 5. Influence of carbon dioxide concentration on the phase equilibrium in the ternary system {CO2 + [PG-2 (50%) + CH3OH (50%)]}. The symbols represent the experimental data with CO2 concentrations (%) as indicated. Solid lines are from the correlation by polynomials of the third order. Dotted lines represent the bPCPSAFT modeling results.

mined visually by changing the pressure in small increments at constant temperature. During the experiments the temperature is maintained constant to within 0.1 K for water and 0.3 K for silicon oil and is being measured using a platinum resistance thermometer with an accuracy of (0.01 K. The pressure is generated hydraulically with a screw pump and is being measured with a dead-weight gauge within an accuracy of (0.005 MPa. 4. Experimental Results Two different hyperbranched polymers have been used for the carbon dioxide solubility measurements. Figure 3 compares the experimental bubble-point isopleths for the binary system Boltorn U3000 + CO2 for different mass fractions of CO2. The data are fitted to second- or third-order polynomials which exhibit almost linear behavior. The experimental bubble-point data of this system are given in Table 1. Polyglycerol revealed to have too high viscosity in order to measure the bubble-point pressures using the above-mentioned VLE technique. The significance of PG-2 in our study is that it is a well-defined structure, where the performance of the PCPSAFT equation of state in correlating and extrapolating phase behavior can be systematically evaluated. We investigate the influence of compositions in mixtures with PG-2 and of the

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TABLE 4: Pure-Component Parameters for Polar Substances and a Hyperbranched Polymer for bPCP-SAFTa component CH3OH CO2b PG-2

Mi/g mol-1 32.042 44.01 2700c

mi

σi/Å

εi/k/K

Nsite

κAiBi

εAiBi/k/K

2.3112 1.5131 64.0

2.8270 3.1869 3.8

176.376 166.333 280.0

2 0 2 × 37

0.089 249 0 0.004 890

2332.585 0 2544.6

ni

µ/D

Q/D Å

1.70 0.00

0.00 4.40

64.0

The experimental data for adjusting the methanol parameters are taken from the literature. Dipole moments (1 D ) 3.33564 × 10-30 C m) and quadrupolar moments (1 D Å ) 3.33564 × 10-40 C m2) are taken from the literature.58 b Parameters as published in ref 28. c For PG-2 j n, is used. the number-average molecular mass, M a

j n ) 2000 g mol-1) as a function of Figure 6. Density of PG-2 (M pressure at 343.15 K ((), 383.15 K (9), and 423.15 K (2) as given in ref 8. Solid lines represent the bPCP-SAFT results.

polymers’ molecular mass with varying temperature. Methanol serves as a cosolvent here, so that ternary mixtures of PG-2, methanol, and carbon dioxide are studied. Tables 2 and 3 as well as Figures 4 and 5 present the experimental results for {w1CO2 + (100 - w1)[w2PG-2 + w3CH3OH]} for w2 ) 25%, w3 ) 75% and w2 ) 50%, w3 ) 50% systems, respectively, where % denotes mass percentage. Demixing of the liquid phase occurs at higher temperatures and higher CO2 concentrations, resulting in a liquid-liquid (LL) and a vapor-liquid-liquid (VLL) region. The phase behavior of this system shows the characteristics of a type IV or V system according to the classification scheme of Van Konynenburg and Scott.50 On the basis of knowledge of other polymer-solvent-gas systems,51 the (CO2 + PG-2 + CH3OH) system could probably be classified as a type IV system. As expected, the lower critical solution temperature (LCST) phase split occurred at higher temperatures, i.e., close to the critical temperature of the solvent (methanol). At these temperatures the solvent begins to expand rapidly, which leads to an increased free-volume effect and finally to a liquid-liquid phase split.51 A number of authors have shown that the addition of a supercritical fluid to the polymer-solvent system lowers the solvent power of the polymer and shifts the LCST to higher pressures and lower temperatures.51-54 In the system considered in this work the supercritical CO2 acts as an antisolvent as well. An increase in the concentration of the antisolvent gas shifts the lower solution temperature (LST) curve to lower temperatures and higher pressures (see Figures 4 and 5). The liquid phase demixes at lower temperatures in the (gas + polymer + solvent) system with higher CO2 concentrations. Moreover, an increase of the PG-2 concentration also shifts the LST curve to lower values of temperature but to a smaller extent. 5. Application of the PCP-SAFT Equation of State The equation of state including the branching effect was used for correlating the phase behavior in (CO2 + polyglycerol + CH3OH) system. The experimental results of this system are

55-57

suitable for phase equilibrium correlations due to the absence of the chemical absorption within the system. The regular and well-defined structure of polyglycerol (PG-2) makes mixtures with PG-2 suitable test cases for the equation of state (see Figure 1b). However, the polydispersity of this hyperbranched polymer is not taken into account. PG-2 is synthesized from glycerol. Each glycerol molecule contains three -OH groups and can therefore bind to three other glycerol monomers. It has been assumed that each time a homonuclear planar branched Y-type tetratomic structure is formed as defined by Mu¨ller and Gubbins.32 For this reason the angle between the monomers in PG-2 macromolecule is fixed to 120°. 5.1. Pure-Component Parameters for Methanol and PG2. Pure component parameters of volatile substances are identified by adjusting them to vapor pressures and liquid density data. For methanol and carbon dioxide,28 the parameters are given in Table 4. Vapor pressure data are not available for macromolecules, and pure component parameters obtained solely from liquid density data do not lead to satisfying results for the phase behavior of mixtures.59-62 One of the possible methods to derive the polymer pure-component parameters is to simultaneously adjust them to liquid density data of the pure polymer and to binary phase equilibrium data. This approach has been applied to many polymer systems,26,60-62 and it was seen that the parameters are transferable. This approach for determining pure component parameters necessitates a low number of adjustable parameters. An associating hyperbranched polymer is characterized by six pure component parameters, and four of these parameters were estimated independently in order to reduce this number. The pure component parameters are the segment number mi, the segment size parameter σi, the dispersive energy parameter εi/k, the effective association volume κAiBi, the association energy εAiBi/k, and an additional parameter ni which describes the number of segments that are branched. For the segment size parameter σi we have assigned a value of σ ) 3.8 Å, which is a representative value based on previous work.25,26 The OH equivalent of PG-2, provided by the supplier of the polymer, equals 13.7 mmol g-1. Therefore, an average molecule of polyglycerol contains 37 hydroxyl groups. The number of association sites was thus set to Nsite ) 2 × 37. The association parameters εAiBi/k and κAiBi were estimated through analogy: the number of hydroxyl groups per molecular mass of the hyperbranched polyglycerol (2700 g mol-1/37 ) 72.97 g mol-1) is approximately equal to the same ratio for 1-butanol (74.123 g mol-1). The association parameters of 1-butanol were therefore adopted.25 With this assumption, we certainly overestimate the effect of H-bonding for the hyperbranched polyglycerol because the hydroxyl groups are much less approachable and contribute less to intermolecular interactions compared to 1-butanol. The number of segments that participate in a branching-tetramer ni was set equal to the total number of segments. It therefore assumed that for PG-2 all of the segments are part of a branching tetramer, in analogy to the illustration of Figure 2 which implies

1028 J. Phys. Chem. B, Vol. 113, No. 4, 2009

Kozłowska et al.

j n ) 2000 g mol-1) as a Function of Temperature TABLE 5: Comparison of the bPCP-SAFT Results of the Density of PG-2 (M and Pressure with the Literature Data8 343.15 K, F/kg m-3

383.15 K, F/kg m-3

423.15 K, F/kg m-3

P/MPa

bPCP-SAFT

lit. data

bPCP-SAFT

lit. data

bPCP-SAFT

lit. data

5 10 20 35 60

1277.1 1280.6 1287.4 1297.3 1312.9

1263.5 1266.7 1270.5 1274.7 1282.4

1240.9 1244.5 1251.6 1261.8 1277.7

1234.9 1238.8 1244.7 1250.4 1258.8

1206.7 1210.6 1218.3 1229.1 1246.0

1211.8 1216.8 1222.4 1226.2 1235.9

that in our model for hyperbranched PG-2 all of the segments are solid spheres, with no open (linear) segments present. That leaves the segment number mi and the dispersive energy parameter εi/k for adjusting to pure component density data and j n ratio and εi/k were derived by to mixture data. The mi/M simultaneously fitting them to the vapor-liquid equilibria of the ternary system {w1CO2 + (100 - w1)[w2PG-2 + w3CH3OH]} for w1 ) 5%, w2 ) 25%, and w3 ) 75% and to j n ) 2000 g mol-1). Literature liquid density data of pure PG-2 (M j n ) 2000 g data of the liquid density of pure polyglycerol of M -1 mol are available for different temperatures and pressures.8 For identifying these two parameters only experimental values of the density at pressure of 5 MPa have been used (see Figure 6). The resulting values for the segment number and the segment energy parameter are mi ) 64.0 and εi/k ) 280.0 K, respectively. The results of the bPCP-SAFT calculated density data together with the literature data are shown in Table 5 and Figure 6. The pure component parameters for PG-2 are summarized in Table 4.

Figure 7. VLE of {methanol + CO2 (5%)} (() and {methanol + CO2 (10%)} (2) mixtures. Symbols represent interpolated experimental values,62 and lines are predictions of PC-SAFT EOS and PCP-SAFT EOS (with kij ) 0).

5.2. Results and Discussion. The vapor-liquid equilibria of binary mixtures of methanol with 5 and 10% CO2 are modeled using the two equations of state: PC-SAFT EOS, not accounting for the polar nature of the compounds, and PCPSAFT EOS, which takes the multipolar interactions of methanol and carbon dioxide into account. No additional binary interaction parameter has been used for VLE calculations of the methanolcarbon dioxide system (kij ) 0). A comparison of the calculation results of both models with interpolated experimental values63 is given in Figure 7, and good results are found for the PCPSAFT equation of state. The average absolute deviation in pressure (AAD% ) 100/n∑n(|pcalc - pexp|/pexp)) between PCPSAFT and the interpolated experimental values is 4.53%. The bPCP-SAFT equation of state developed in this work has been applied to correlate the experimental data of carbon dioxide-polymer-solvent systems. No binary interaction parameter has been introduced for the correlation of the results presented here (kij ) 0). Figures 4 and 5 present the comparison of the experimental points with the bPCP-SAFT modeling results of (CO2 + PG-2 + CH3OH) systems for different concentrations of carbon dioxide. A good agreement between the experimental bubble-point results and the model is observed for the vapor-liquid equilibrium. The total deviation between the experimental points and the bPCP-SAFT values is 3.16% (AAD). The model does not predict the liquid-liquid and the vapor-liquid-liquid equilibria at the right temperatures. It is e.g. predicted to be at T ) 468 K for the highest carbon dioxide concentration of Figure 5. This is likely to be attributed to the parametrization of the H-bonding interactions. These association parameters were taken as those of 1-butanol, based on some analogy as described in section 5.1. The H-bonding interactions to methanol are likely to be overestimated because the hydroxyl groups in the hyperbranched PG-2 are in average less accessible. The overestimation of the H-bonds results in cross-association between the HBP and methanol that prevent demixing of the two components. Figure 8 presents a comparison of the experimental points with the modeling results for the different solvent (PG-2 + CH3OH) compositions with 5% of CO2. The solid lines are the results of the bPCP-SAFT model containing the branching term, and the dotted lines are the original PCP-SAFT version. The bPCP-SAFT model describes the experimental points in the carbon dioxide-polyglycerol-methanol system better than the PCP-SAFT EOS. This is due to the addition of TPT2 term to the PCP-SAFT model which accounts for the effects of branching in the macromolecule. 6. Conclusions

Figure 8. VLE of ternary systems {CO2 + [PG-2 (25%) + CH3OH (75%)]} and {CO2 + [PG-2 (50%) + CH3OH (50%)]}. The symbols represent the experimental data with 5% of CO2. Solid lines represent the bPCP-SAFT modeling results, and dotted lines are of the PCPSAFT modeling results.

The experimental results on the solubility of carbon dioxide in different hyperbranched polymers show the potential of HBPs as novel and “designable” process solvents for example for the capture of CO2. However, the systematic investigations need to be continued in order to find the proper polymers which meet

Phase Behavior of Hyperbranched Polymer Systems the requirements of a scrubbing liquid such as large-scale availability and low cost, low viscosity, low glass transition or melting temperature and stability against possible reactions (nonreactivity). Additional experimental investigations concerning the modification of the functional end groups and the study on the solubility of CO2 in the modified HBPs are to be performed. An extension of the perturbed chain polar statistical associating fluid theory to describe branched macromolecules has been presented. The model is based on Wertheim’s second-order perturbation theory for chainlike molecules and has been used to model the phase equilibria of gas–hyperbranched polymersolvent systems. Although the bPCP-SAFT equation of state does not predict the liquid-liquid and the vapor-liquid-liquid equilibria,whichwerefoundexperimentallyinthecarbondioxidepolyglycerol-methanol system, it gives good correlation results of the phase equilibria of the considered system. No additional binary interaction parameter has been introduced along with the branching contributions to the bPCP-SAFT model. Acknowledgment. The authors gratefully acknowledge professor Peter J. Jansens for giving an opportunity to perform this work. We also thank Eugene Straver for his help during the experimental work and Perstrorp AB for the free sample of Boltorn U3000. The research was financially supported by the Delft Centre for Industrial Processes (DCSIP). References and Notes (1) Tomalia, D. A.; Naylor, A. M.; Goddard, W. A., III Angew. Chem., Int. Ed. 1990, 29, 138–175. (2) Fre´chet, J. M. J.; Tomalia, D. A. Dendrimers and Other Dendritic Polymers; John Wiley & Sons: West Sussex, 2001. (3) Aulenta, F.; Hayes, W.; Rannard, S. Eur. Polym. J. 2003, 39, 1741– 1771. (4) Voit, B. J. Polym. Sci., Part A: Polym. Chem. 2000, 38, 2505– 2525. (5) Inoue, K. Prog. Polym. Sci. 2000, 25, 453–571. (6) Yates, C. R.; Hayes, W. Eur. Polym. J. 2004, 40, 1257–1281. (7) Gao, C.; Yan, D. Prog. Polym. Sci. 2004, 29, 183–275. (8) Seiler, M. Fluid Phase Equilib. 2006, 241, 155–174. (9) Seiler, M.; Ko¨hler, D.; Arlt, W. Sep. Purif. Technol. 2002, 29, 245– 263. (10) Hult, A.; Johansson, M.; Malmstro¨m, E. AdV. Polym. Sci. 1999, 143, 1–34. (11) Kim, Y. H. J. Polym. Sci., Part A: Polym. Chem. 1998, 36, 1685– 1698. (12) Malmstro¨m, E.; Hult, A. J. Macromol. Sci., Polym. ReV. 1997, 37, 555–579. (13) Jikei, M.; Kakimoto, M.-A. Prog. Polym. Sci. 2001, 26, 1233– 1285. (14) Seiler, M.; Ko¨hler, D.; Arlt, W. Sep. Purif. Technol. 2003, 30, 179– 197. (15) Seiler, M.; Rolker, J.; Arlt, W. Macromolecules 2003, 36, 2085– 2092. (16) Seiler, M.; Jork, C.; Kavarnou, A.; Hirsch, R.; Arlt, W. AIChE J. 2004, 50, 2439–2454. (17) Arlt, W.; Seiler, M.; Rolker, J. DE 102002014707.0. (18) Fang, J.; Kita, H.; Okamoto, K.-I. J. Membr. Sci. 2001, 182, 245– 256. (19) Jang, J. G.; Bae, Y. C. Chem. Phys. 2001, 269, 285–294. (20) Jang, J. G.; Bae, Y. C. J. Chem. Phys. 2001, 114, 5034–5042. (21) Jang, J. G.; Bae, Y. C. J. Chem. Phys. 2002, 116, 3484–3492. (22) Seiler, M.; Rolker, J.; Mokrushina, L. V.; Kautz, H.; Frey, H.; Arlt, W. Fluid Phase Equilib. 2004, 221, 83–96.

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