Measurements and Modeling of Cloud Point Behavior for Poly

Dec 3, 1999 - Department of Chemical Engineering, Auburn University, Auburn, Alabama ..... As the temperature increases further, the cloud point press...
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Ind. Eng. Chem. Res. 2000, 39, 185-194

185

GENERAL RESEARCH Measurements and Modeling of Cloud Point Behavior for Poly(propylene glycol) in Ethane and in Ethane + Cosolvent Mixtures at High Pressure Todd M. Martin, Ram B. Gupta, and Christopher B. Roberts* Department of Chemical Engineering, Auburn University, Auburn, Alabama 36849

The phase behavior of poly(propylene glycol) (PPG) in ethane and ethane + cosolvent mixtures was studied using a high-pressure, variable-volume view cell. The cosolvents studied were chloroform and carbon tetrachloride. Cloud point pressures for PPG in ethane were measured for three different weight-average molecular weight (Mw) PPG samples (450; 830; 2160) at several different polymer concentrations (ranging from 0.5 to 4 mass %). Cloud point pressures for PPG (Mw ) 830, 4%) in ethane + cosolvent mixtures were measured for several different cosolvent concentrations (ranging from 9 to 42 solvent mass %). The phase behavior of the PPG + ethane system transitioned from lower critical solution temperature (LCST)-like to a merged U-LCSTlike behavior as the polymer concentration or the polymer molecular weight was increased. In general, the addition of the cosolvents reduced the cloud point pressure. The addition of chloroform cosolvent reduced the cloud point pressure more than carbon tetrachloride due to hydrogen bonding between PPG and chloroform. The Lattice Fluid Hydrogen Bonding equation of state was used to model the experimental data. 1. Introduction Supercritical fluids (SCFs) have received much attention as solvents in a variety of polymer processes including extractions and separations,1 fractionations,1-3 particle formation,4 and reactions.5 This attention stems primarily from the ability to dramatically change bulk properties, such as density and solubilities, with small variations in temperature and pressure. However, because SCFs such as carbon dioxide and ethane have a relatively low solvent strength compared with typical organic solvents, nonvolatile substances such as polymers tend to be sparingly soluble even at large pressures. To design SCF-based polymer separation processes, one must first characterize the phase behavior of the solute in the SCF of interest. In this study, our first goal was to characterize the phase behavior of poly(propylene glycol) (PPG) in ethane. The phase behavior of PPG has been studied in several different solvents: propane,6 carbon dioxide,7 and pentane.8 The phase behavior of two similar polymers, poly(ethylene glycol) (PEG)7,9-11 and poly(dimethylsiloxane) (PDMS)12,13 has also been studied. The PEG + carbon dioxide system was studied by several researchers.7,9-11 O’Neill et al.7 also studied the phase behavior of PEG-PPG copolymers in carbon dioxide. The phase behavior of PDMS in lower alkanes (ethane, propane, and butane) was studied by Zeman and co-workers.12 The phase behavior of PDMS in carbon dioxide was studied by Xiong and Kiran.13 The structure of PPG is as follows: By inspection of the structure of PPG, it is apparent that PPG can self-associate through hydrogen bonding. PPG

has two types of hydrogen-bonding acceptors: the oxygens in the terminal hydroxyl groups and the ether oxygens in the polymer repeat units. However, the only source of donor sites that PPG has is the hydrogen in the terminal hydroxyl groups. To examine the selfassociation of PPG, it was necessary to select a solvent that has no specific interaction with PPG. Therefore, we selected ethane as our solvent. It is well-known that the addition of certain cosolvents can significantly increase the solubility of a polymer in a SCF. One manner by which the addition of a cosolvent affects solubility is by increasing the solvent density (which reduces the free volume difference between the polymer and the solvent). The cosolvent can also enhance the solubility if it can interact favorably with the polymer (e.g., by hydrogen bonding).14 The cosolvent effect on polymer + solvent mixtures at high pressures has been studied by several researchers.14-20 As an example, the PEG + carbon dioxide + ethanol system was studied by Mishima et al.20 The addition of ethanol was found to increase the solubility of PEG. In addition, the phase behavior of PDMS in carbon dioxide + cosolvent mixtures was studied by West et al.19 * Author to whom correspondence should be addressed. Telephone: (334) 844-2036. Fax: (334) 844-2063. E-mail: [email protected].

10.1021/ie990553m CCC: $19.00 © 2000 American Chemical Society Published on Web 12/03/1999

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Ind. Eng. Chem. Res., Vol. 39, No. 1, 2000 Table 1. Properties of the Polymer Samples Used

Figure 1. P-T phase diagram for polymer + solvent systems

The second goal of this work was to examine the cosolvent effect on the phase behavior of PPG in ethane. To isolate the relative effects of solvent density and solvent/solute hydrogen bonding interactions, we selected chloroform (CHCl3) and carbon tetrachloride (CCl4) as our cosolvents. These cosolvents were chosen because they have approximately the same relatively large density (FCHCl3 ) 1.5 g/cm3 and FCCl4 ) 1.6 g/cm3 at 25 °C and 1 atm). In addition CHCl3 has the capacity to hydrogen bond with PPG (but not with itself), whereas CCl4 has no capacity for hydrogen bonding. For polymer + solvent systems, there are several different types of cloud point curves that are observed experimentally: lower critical solution temperature (LCST), upper critical solution temperature (UCST), and merged U-LCST.21 In the absence of specific interactions between the polymer and the solvent, LCST behavior is generally an entropic effect that can be induced by a difference in free volume between polymer and solvent.12 In general, UCST behavior is an enthalpic effect caused by a difference in chemical nature and of intermolecular force fields between polymer and solvent.6 In this study, an isopleth is said to exhibit LCST behavior if the pressure versus temperature (P-T) curve has a positive slope and UCST behavior if the P-T curve has a negative slope regardless of whether the composition is equal to the critical composition (see Figure 1). For systems that exhibit both LCST and UCST behavior, an increase in dissimilarity between the polymer and the solvent causes the LCST curve to shift to the left and the UCST curve to shift to the right (that is both curves shift to higher pressures). If the dissimilarity is increased sufficiently, the LCST and UCST curves merge into a single curve, the U-LCST curve, which does not intersect with the LLV curve21 (see Figure 1). At low temperatures, the U-LCST curve in enthalpically driven, whereas at high temperatures it is entropically driven. In this study, we present new data on the phase behavior of PPG in ethane and in ethane + cosolvent mixtures. Cloud point pressures for PPG in ethane were measured for three different molecular weight (Mw) PPG samples (450; 830; 2160) at several different polymer concentrations (ranging from 0.5 to 4 mass %). Cloud point pressures for PPG (Mw ) 830, 4%) in ethane + cosolvent mixtures were measured for two different cosolvents, CHCl3 and CCl4, for several different cosolvent concentrations (ranging from 9 to 42 solvent mass %). The phase behavior of a multicomponent system can be efficiently characterized by fitting equation of state

molecular weight (Mw)

polydispersity (Mw/Mn)

450 830 2160

1.07 1.06 1.06

parameters to a relatively small amount of experimental data. To model our experimental data, we used the Lattice Fluid Hydrogen Bonding (LFHB) model developed by Panyiotou and Sanchez.22 Comprehensive reviews on the equations of state for hydrogen-bonding mixtures are available in the literature.23,24 Several researchers have used the LFHB model to describe the phase behavior of polymer + solvent systems.22,25,26 Specifically, Missopolinou and Panayiotou27 used the LFHB model for glycol systems. Takishima et al.24 used the LFHB model of Gupta and Johnston28 to model the solubility of PEG, PPG, and PPG-PPG copolymers in carbon dioxide. The Lattice Fluid (LF) model has been used to model the phase behavior of PDMS in carbon dioxide13 and in carbon dioxide + cosolvent mixtures.19 2. Experimental Section 2.1. Materials. The PPG samples used in this study were obtained from Scientific Polymer Products, Inc. The polymer samples were used without further purification. The molecular weights and polydispersities of the samples are given in Table 1. The solvent, ethane (Scott Specialty Gases; CP grade, 99.5%), was used as received. The cosolvents, chloroform (Fisher Scientific, 99.9%) and CCl4 (Aldrich, 99%), were also used as received. 2.2. Apparatus and Procedure. The cloud point measurements were made with a high-pressure, variable-volume view cell. The details of our experimental apparatus and procedure can be found in an earlier publication.29 The maximum operating pressures are 300 bar at 450 K and at least 350 bar at 373 K. 3. Theory 3.1. Lattice Fluid Model. The LF model developed by Sanchez and Lacombe30 is as follows:

F˜ 2 + P ˜ +T ˜ [ln(1 - F˜ ) + (1 - 1/r)F˜ ] ) 0

(1)

where F˜ , P ˜ , and T ˜ are the reduced density, pressure, and temperature, respectively, and r represents the number of lattice sites occupied by the molecule. Similar to Flory theory, the LF theory requires three equation of state parameters for each pure component. Typically, temperature (T*), pressure (P*), and closedpacked mass density (F*) are the three characteristic parameters employed by the Sanchez-Lacombe equation of state to characterize a system. The reduced variables are directly dependent on these characteristic parameters. They are defined as follows:32

F˜ ≡ F/F*

F* ≡

MW ν*

(2)

P P*

P* ≡

 ν*

(3)

P ˜ ≡

Ind. Eng. Chem. Res., Vol. 39, No. 1, 2000 187

T ˜ ≡

T T*

ν˜ ≡

T* )

ν ν*

P*νo R

(4)

1 F˜

(5)

ν˜ )

Table 2. Characteristic Parameters of Pure Components

where MW is the molecular weight, ν* is the hard-core volume of a molecule,  is the interaction energy per mer, νo is the volume of a lattice site, and R is the gas constant. The characteristic parameters for the components in this study are given in Table 2. The parameters for poly(propylene oxide) (PPO) are listed in Table 2 because the parameters for PPG were unavailable in the literature. As will be discussed later, we fit our PPG parameters to our PPG + ethane cloud point data using the PPO parameters as an initial estimate. The LF model is extended to mixtures with the use of the appropriate mixing rules. The mixing rules used in this study are given by Xiong and Kiran.32 3.2. Lattice Fluid Hydrogen Bonding Model. Panayiotou and Sanchez22 modified the LF model to incorporate hydrogen bonding. In the LFHB, model the chain length of the mixture, r, in eq 1 is replaced by a modified chain length, rj, defined by

1 1 ) - νH rj r

m n

∑ ∑νij i)1 j)1

(7)

where νij is the fraction of hydrogen bonds between donor type i and acceptor type j and m and n are the number of donor and acceptor types, respectively. In the LFHB model the chemical potential is modified as follows:

µLFHB ) µLF + µHB

(8)

where the expressions for the LF and LFHB contributions to the chemical potential are given by Xiong and Kiran32 and by Panyiotou and Sanchez,22 respectively. The fraction of hydrogen bonds between donor type i and acceptor type j, νij, is given by22 n

νij ) [νid -

∑k

m

νik][νja -

∑k νkj]F˜ exp(-G0ij/RT)

(9)

where νid and νja are the fraction of donor (of type i) and acceptor sites (of type j), respectively. The physically meaningful solution to eq 9 is given by

rνij )

-bij - xb2ij - 4aijcij 2aij

component

T*, K

P*, bar

F*, g/cm3

ref

1 2 3 3

ethane PPO CHCl3 CCl4

315 524 512 535

3273 4210 4560 3808a

0.64 1.093 1.688 1.788

30 22 30 31

a The value of 8020 atm reported by Sanchez and Lacombe30 appears to be a misprint. Using the other parameters listed in Table 1 of ref 30, one obtains a value of 3808 bar, which is in agreement with the value reported by Sandler.31

Table 3. Expressions for the Hydrogen-Bonding Parameters in Equation 10 for the PPG/Ethane System i j donor acceptor aij 1 1 1 2

OH OH

OH O

bij

(10)

where the expressions for aij, bij, and cij are given in Tables 3 and 4 for the PPG + ethane and for the PPG + ethane + chloroforms systems, respectively. The

cij

1 -[4x2 + A11 - rν12] 2x2(2x2 - rν12) 1 -[(2 + aether)x2 + aetherx2(2x2 - rν11) A12 - rν11]

hydrogen-bonding fractions in eq 10 must be solved simultaneously with the reduced density (given in eq 1). According to Panayiotou and Sanchez,22 the number of ether oxygen (hydrogen bond acceptor) sites per PPG molecule, aether, is approximately one-half the number of ether oxygens due to the steric hindrance of the methyl group (see structure). The aether value for each of our PPG samples is given in Table 5. The expression, Aij, in Tables 3 and 4 is given by

(6)

where νH is the fraction of hydrogen bonds in the system defined by

νH )

component #

Aij )

( )

G0ij r exp F˜ RT

(11)

where G0ij is the Gibbs free energy change associated with hydrogen bond formation defined by

G0ij ) E0ij + PV0ij - TS0ij

(12)

where E0ij, V0ij, and S0ij are the standard energy, volume, and entropy change upon hydrogen bond formation, respectively. The parameters in eq 12 are given in Table 6. Most of the parameters were obtained from the literature except for the parameters for CHCl3 + alcohol hydrogen bonding. For this case, Eo set equal to that of CHCl3 + ether hydrogen-bonding and So was set equal to one-half that of CHCl3 + ether because this is how the parameters for alcohol + alcohol and alcohol + ether are related. This approximation does not impact the calculations greatly because the concentration of hydroxyl groups is always significantly less than the concentration of ether oxygen groups. Takishima et al.24 obtained hydrogen-bonding parameters for PPG from PPG + carbon dioxide phase equilibria data: Eo ) -2.73 kJ/mol, So ) -2.90 J/molK, and Vo ) -0.316 cm3/mol. These values are significantly different from the parameters used in this study (see Table 6). Possible explanations are that Takishima et al.24 calculated the number of acceptor groups differently, they treated the alcohol and ether oxygens as equivalent acceptor groups, and they simultaneously fit the physical and hydrogenbonding parameters to cloud point data. For PPG in ethane and in ethane + cosolvent mixtures, the hydrogen-bonding contribution to the chemical potential for ethane is given by

µ1H ) r1νH RT

(13)

For PPG in ethane, the hydrogen-bonding contribution

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Table 4. Expressions for the Hydrogen-Bonding Parameters in Equation 10 for the PPG + Ethane + CHCl3 System i

j

donor

acceptor

aij

bij

cij

1 1 2 2

1 2 1 2

OH OH CHCl3 CHCl3

OH O OH O

1 1 1 1

-[4x2 + A11 - rν12 - rν21] -[(2 + aether)x2 + A12 - rν11 - rν22] -[2x2 + x3 + A21 - rν11 - rν22] -[aetherx2 + x3 + A22 - rν12 - rν21]

2x2(2x2 - rν21) + rν12(rν21 - 2x2) 2x2(aetherx2 - rν22) + rν11(rν22 - aetherx2) x3(2x2 - rν11) + rν22(rν11 - 2x2) x3(aetherx2 - rν12) + rν21(rν12 - aetherx2)

Table 5. Number of Ether Oxygen Acceptor Sites per PPG Molecule Mwa

nb

aetherc

424 830 2164

7 14 37

3 6.5 18

nc

a Molecular weights used in calculations (experimental values were adjusted slightly to give whole numbers for n). b Mw ) 58n + 18. c aether ) (n - 1)/2.

Table 6. Hydrogen-Bonding Parameters for Equation 12 i

j

Eij (kJ/mol)

Vij (cm3/mol)

Sij (J/mol)

source

1 1 2 2

1 2 1 2

-25.1 -22 -11.44 -11.44

-5.6 0 0 -0.85

-26.5 -52.0 -4.87 -9.74

ref 22 ref 27 this study ref 22

to the chemical potential for PPG is given by

(

)

(

)

µ2H 2x2 2x2 ) r2νH - 2 ln - 2 ln RT 2x2 - rνH 2x2 - rν11 aetherx2 (14) aether ln aetherx2 - rν12

(

)

For PPG in ethane + CHCl3, the hydrogen-bonding contribution to the chemical potential for PPG is given by

µ2H ) r2νH RT

(

2 ln

)

(

)

2x2 2x2 - 2 ln 2x2 - rν11 - rν12 2x2 - rν11 - rν21 aetherx2 (15) aether ln aetherx2 - rν12 - rν22

(

)

and that for the cosolvent, CHCl3, is given by

(

x3 µ3H ) r3νH - ln RT x3 - rν21 - rν22

)

(16)

For PPG in ethane + CCl4, the hydrogen-bonding equations (10, 13, 15, and 16) are all similar except ν21 and ν22 are set equal to zero. 3.3. Phase Equilibria Calculations. The criterion for equilibrium between two phases at a given temperature and pressure is that the chemical potentials of each component in each phase are equal:32

µLi ) µVi

nc

XLi ) ∑XVi ) 1 ∑ i)1 i)1

(17)

where i ) 1, 2, ..., nc, nc is the number components in the system, and L and V indicate the polymer-rich and polymer-lean phases, respectively. Also, the compositions in each phase must also satisfy the constraint that the sum of the mass fractions in each phase must equal unity:32 where Xi is the composition (in mass fraction) of the ith component.

(18)

The phase equilibria calculations were solved using the CONSTR subroutine, which is part of the optimization toolbox of MATLAB. CONSTR is essentially a constrained nonlinear optimization subroutine. The subroutine allows one to place bounds on the compositions to be solved. For our calculations we treated the polymer as a single component with a molecular weight given by the weight average molecular weight (Mw). This was a reasonable assumption because the polydispersities of the samples were relatively low (∼1.06). 3.3.1. PPG + Ethane Binary System. For a two-phase binary system, the Gibb’s phase rule gives two degrees of freedom to fully specify a system. In our calculations, we chose to specify the temperature and the pressure. The variables that are solved for are the polymer concentration in each phase. A point on a P-T curve was determined by first guessing a value for the pressure (at a given temperature) and then calculating the composition of both phases. The pressure was then iterated for until the composition of the polymer lean phase matched the experimental mixture composition. The pressure was not solved for directly due to convergence difficulties. The pressure was then determined at several different temperatures to generate a P-T curve for a given polymer lean phase composition. The entire procedure was then repeated for each experimental composition. 3.3.2. PPG + Ethane + Cosolvent Ternary System. For a two-phase ternary system, there are three degrees of freedom to fully specify the system. In addition to the temperature and the pressure, we chose to specify the polymer composition of the polymer lean phase. The variables that are solved for are the mass fraction of cosolvent in each phase and the mass fraction of polymer in the polymer-rich phase. The rest of the calculation procedure was the same as that for the PPG + ethane binary system. 4. Results and Discussion 4.1. Experimental. 4.1.1. PPG in Ethane. Cloud point curves for several polymer concentrations for PPG molecular weights of 450, 830, and 2160 are shown in Figures 2, 3, and 4, respectively. In all cases, the cloud point pressure isopleths increased with increasing polymer concentration. For PPG (Mw ) 450) in ethane, the phase behavior switched from LCST-like to U-LCSTlike behavior as the PPG concentration increased. This result is expected in systems exhibiting a merged U-LCST critical locus because at low polymer concentrations, the cloud point curve must approach the solvent vapor pressure curve. In this paper we have used the terms LCST and U-LCST to indicate the shape (i.e., nonmonotonic for U-LCST) or slope (i.e., strictly positive for LCST) of the cloud point curves. Thus, our cloud point curves are not implied to represent critical loci. For brevity, future references to LCST-like and

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Figure 2. Cloud point curves for PPG (Mw ) 450) in ethane. Symbols are experimental isopleths for the following polymer concentrations (in mass %): (() 0.5%, (9) 1%, (2) 2%, (b) 3%, (/) 4%, and (O) vapor pressure of ethane.

Figure 3. Cloud Point Curves for PPG (Mw ) 830) in ethane. Symbols are experimental isopleths for the following polymer concentrations (in mass %): (() 0.5%, (9) 1%, (2) 2%, (b) 3%, (/) 4%, and (O) vapor pressure of ethane.

U-LCST-like behavior will be referred to as LCST and U-LCST behavior. Zeman and Patterson6 observed U-LCST behavior for PPG (Mw ) 790 and 3900) in propane. This type of behavior was also observed for PDMS in carbon dioxide.13 U-LCST behavior has also been encountered in nonpolar polymer + solvent systems such as poly(ethylene propylene) in ethylene and propylene,21,34,35 polybutadiene in toluene + carbon dioxide,36 and polystyrene in acetone.6 This type of behavior was also observed for the polypropylene + propane + 1-propanol system.17 These results show that U-LCST behavior can also be caused by the addition of a hydrogen-bonding cosolvent to a nonpolar polymer + solvent system. We propose that the U-LCST behavior can be ex-

Figure 4. Cloud point curves for PPG (Mw ) 2160) in ethane. Symbols are experimental isopleths for the following polymer concentrations (in mass %): (() 0.5% and (9) 1%.

plained by a hydrogen-bonding effect at low temperature and a density effect at high temperature. At low temperature, hydrogen bonding between the polymer molecules (self-association) causes the cloud point pressure to increase because ethane is effectively trying to solvate a larger molecule (dimer, trimer, etc.). In other words, hydrogen bonding increases the dissimilarity between PPG and ethane. As the temperature increases, the effect of hydrogen bonding decreases so that the cloud point pressure decreases. As the temperature increases further, the cloud point pressure begins to increase to reduce the difference between the free volume of the polymer and the solvent (the density effect). For PPG (Mw ) 830) in ethane, the behavior was similar to the 450 molecular weight sample, except that (for polymer concentrations 9%, there is a significant difference between the isopleths for CHCl3 and CCl4 at low temperatures. This result can be explained by the fact that at low temperatures, the hydrogen bonding between CHCl3 and PPG is significant. As the temperature increases, hydrogen bonding decreases so that the cloud point isopleth for CHCl3 approaches that of CCl4. We could not measure cloud point pressures for 42% CHCl3 for temperatures 9%, the demixing solvent density for CHCl3 is appreciably lower than it is for CCl4 at low temperatures. This result shows that favorable hydrogen bonding interactions between PPG and CHCl3 reduces the demixing solvent density. At higher temperatures (>60 °C), the difference between the demixing solvent densities is negligible. Thus, the solvent strength of the ethane + cosolvent mixture is mainly a function of density at high temperature because of the diminished capacity for hydrogen bonding. 4.2. Modeling. 4.2.1. PPG in Ethane. Using the literature parameters for PPO (see Table 2), the LFHB model significantly overpredicted the cloud point pressures for PPG in ethane. For example, for PPG (Mw ) 830, 1%) in ethane, the LFHB model predicted a cloud point pressure of 650 bar at 30 °C, whereas the experimental demixing pressure was 117 bar (as shown in Figure 3). To improve the predictions, the characteristic temperature of PPO was adjusted to 420 K. The characteristic temperature was adjusted to match the data for PPG (Mw ) 830, 4%) in ethane. Because this value yielded reasonable predictions for the other polymer concentrations and molecular weights, no further iteration was attempted. The characteristic temperature was adjusted because it has the greatest impact on the phase equilibria predictions. The LFHB predictions for PPG in ethane for PPG molecular weights of 450, 830, and 2160 are shown in Figures 7, 8, and 9, respectively. For all predictions in this study, the binary interaction parameters, δij (defined as in the paper by Xiong and Kiran32), were set equal to zero. For PPG (Mw ) 450) in ethane, the LFHB model correctly predicted U-LCST behavior at 2 and 4% PPG. However, because the model also predicts U-LCST behavior at 0.5% PPG, it overestimates the effect of hydrogen

Ind. Eng. Chem. Res., Vol. 39, No. 1, 2000 191

Figure 9. Predicted cloud point pressures for PPG (Mw ) 2160) in ethane. Dotted lines are predictions using the LF model and solid lines are predictions using the LFHB model. Figure 7. Predicted cloud point pressures for PPG (Mw ) 450) in ethane. Dotted lines are predictions using the LF model and solid lines are predictions using the LFHB model.

the experimental data could have been improved if a molecular weight-dependent interaction parameter was used (a method that is employed in the literature13, 34) and if the hydrogen-bonding parameters were fit to the experimental data. We did not make these adjustments because we wanted to make our calculations as a priori as possible to test the predictive power of the LFHB model. The predictions for the LF model are also given in Figures 7, 8, and 9. The LF model correctly predicted the cloud point isopleth for 0.5% PPG for the 450 and 830 molecular weight PPG samples. However, as the concentration increased, the LF model underestimates the cloud point pressures and does not predict a transition to U-LCST behavior. Thus, the LFHB model gives a better representation of the phase behavior of the PPG + ethane system by accounting for the contributions of hydrogen bonding. To quantify the effect of PPG concentration on the hydrogen bonding, we calculated the number of hydrogen bonds per molecule of PPG for several different concentrations of PPG (Mw ) 830). The number of hydrogen bonds per molecule (BPM) is given by

BPM )

Figure 8. Predicted cloud point pressures for PPG (Mw ) 830) in ethane. Dotted lines are predictions using the LF model and solid lines are predictions using the LFHB model.

bonding at this concentration. For PPG (Mw ) 830) in ethane, the model correctly predicts the transition from LCST to U-LCST behavior as the PPG concentration increases. Again the model slightly overpredicts the magnitude of the cloud point curves at the lower polymer concentrations. For PPG (Mw ) 2160) in ethane, the LFHB model underestimates the cloud point pressures. The agreement between the LFHB model and

rνH x2

(19)

As shown in Figure 10, the BPM increased with increasing PPG concentration. This result indicates that the transition from LCST to U-LCST phase behavior with increasing PPG concentration is caused by an increase in hydrogen bonding. Also the model predicts that as the temperature increases, the amount of hydrogen bonding decreases. To quantify the effect of molecular weight on the hydrogen bonding, we calculated the number of hydrogen bonds per PPG segment at 1% PPG. We calculated the number of bonds per segment to account for the fact that the larger PPG molecular weight samples have more acceptor sites per molecule. The number of

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Figure 10. Number of hydrogen bonds per PPG molecule for PPG (Mw ) 830) in ethane. Lines are predicted using the LFHB model for the following PPG concentrations: (‚‚‚) 0.5%, (- - -) 2%, and (s) 4%.

Figure 12. Predicted cloud point pressures for PPG (Mw ) 830, 4%) in ethane + cosolvent mixtures. Symbols are experimental isopleths: (b) cosolvent ) CHCl3, and (O) cosolvent ) CCl4. Solid lines are LFHB predictions for CHCl3 as the cosolvent and dotted lines are LFHB predictions for CCl4 as the cosolvent. Figure 11. Number of hydrogen bonds per PPG segment for PPG (1%) in ethane. Lines are predicted using the LFHB model for the following PPG molecular weights: (‚‚‚) 450, (- - -) 830, and (s) 2160.

Table 7. Number of Hydrogen Bonds per PPG Molecule for PPG (Mw ) 830, 4%) in Ethane + Cosolvent Mixtures Predicted from the LFHB Model

hydrogen bonds per segment (BPS) is given by

T, °C

0.090a 0.259 0.417 0.090 0.259 0.417 0.090 0.259 0.417

30 40 50 60 80 100

0.405 0.315 0.245 0.189 0.114 0.070

BPS )

BPM r2

(20)

As shown in Figure 11, the BPS decreases as the PPG molecular weight increases. This relationship can be attributed to a decrease in the hydroxyl concentration. Earlier we showed that the cloud point curve for PPG (Mw ) 450; 4%) is higher than that for PPG (Mw ) 830; 4%). As shown in Figures 7 and 8, the LFHB model does indeed predict that the cloud point isopleth for Mw ) 450 should be larger than that for Mw ) 830 (for temperatures