Ind. Eng. Chem. Res. 1997, 36, 2821-2833
2821
Solubility of Block Copolymer Surfactants in Compressed CO2 Using a Lattice Fluid Hydrogen-Bonding Model Shigeki Takishima,* Mark L. O’Neill, and Keith P. Johnston* Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712
The solubilities of the homopolymers poly(ethylene glycol) (PEG), poly(ethylene glycol) dimethyl ether (PEGDME), and poly(propylene glycol) (PPG) in CO2 were correlated with a lattice fluid hydrogen-bonding (LFHB) model, which was then used to predict solubilities of Pluronic L (PEGPPG-PEG) and Pluronic R (PPG-PEG-PPG) triblock copolymers. Simple averaging rules were developed to evaluate the physical properties of the copolymers without introducing any adjustable parameters. For a given average molecular weight, the predictions of the model were quite reasonable and in some cases perhaps more accurate than the data, due to the large polydispersity of the samples. The model predicts the effects of total molecular weight, PEG/ PPG ratio, terminal functional groups, temperature, and density on solubility. The much higher solubility of PPG versus PEG is due primarily to steric hindrance from the methyl branch, which weakens segment-segment interactions, and to a lesser extent to the stronger hydrogen bond donor strength of a primary (in the case of PEG) versus a secondary (in the case of PPG) alcohol terminal group. Consequently, the predicted solubilities of Pluronic L surfactants, which have stronger hydrogen bond donors on the terminal groups, are not much smaller than those of Pluronic R surfactants for given molecular weights of the blocks. Introduction Supercritical carbon dioxide (CO2) is an environmentally benign alternative to organic solvents in chemical processing (Kiran and Sengers, 1994; McHugh and Krukonis, 1994; Dixon and Johnston, 1995; Hutchenson and Foster, 1995; Johnston and Lemert, 1996). Furthermore it is inexpensive, nonflammable, and essentially nontoxic. In the near-critical and supercritical states, the solvent power of CO2 can be tuned over a wide range by varying the pressure or temperature. The low critical temperature of 31 °C is particularly desirable for processing thermally labile materials. However, CO2 has far weaker van der Waals forces than those of organic solvents; thus, nonvolatile substances tend to be insoluble even at a pressure of 500 bar. Because CO2 does not have a dipole moment and has a low dielectric constant, nonvolatile hydrophilic substances such as proteins, ions, and most catalysts are also insoluble. Both low molecular weight and polymeric surfactants may be used to form microemulsions in a supercritical fluid continuous phase to overcome many of the above solubility restrictions. Recent reviews describe the use of microemulsions in supercritical fluids (Bartscherer et al., 1995; McFann and Johnston, 1996. It is particularly difficult to form microemulsions in CO2 due to its low polarizability/volume (McFann et al., 1994). On the basis of phase behavior studies of surfactants, oligomers, and polymers in CO2, the most promising candidates, in order of increasing interactions with CO2, include siloxanes, fluorocarbons, fluoro ethers, and fluoroacrylates (DeSimone et al., 1992; Hoefling et al., 1993; Newman et al., 1993; Harrison et al., 1994; McHugh and Krukonis, 1994). The term “CO2-philic” is quite appropriate for describing this favorable interaction (DeSimone et al., 1994). Polymeric surfactants may be used to stabilize organic dispersed phases in the form of latexes in CO2. DeSimone et al. (1994) carried out dispersion polymerizations of methyl methacrylate in supercritical CO2 by using poly(1,1-dihydroperfluorooctylacrylate) (poly(FOA)) as a stabilizer. Poly(FOA) is the most CO2soluble polymer known to date; solubilities on the order S0888-5885(96)00702-6 CCC: $14.00
of 10 wt % are due primarily to the “CO2-philic” fluorinated side chains (Hsiao et al., 1995). Block and graft copolymers based on poly(FOA) have been used to form reverse micelles with nonpolar styrene cores and hydrated polar poly(ethylene glycol) cores (Fulton et al., 1995). Harrison et al. (1994) solubilized 32 mol of water per mole of surfactant with an ionic hybrid fluorocarbon/ hydrocarbon surfactant. Furthermore, Johnston et al. (1996) found that the protein bovine serum albumin, with a molecular weight of 67 000, was soluble in a microemulsion formed with an ammonium carboxylate perfluoropolyether surfactant in supercritical CO2. This surfactant with an average molecular weight of 740 contains a CO2-philic fluoroether tail and a hydrophilic ionic headgroup. The use of surfactants for the interface between CO2 and water will offer new opportunities in biochemistry, polymer chemistry, separation and reaction engineering, and environmental and materials science. In the investigation of solubilities of polymeric surfactants in compressed CO2, it is important to understand the interactions of CO2-philic and hydrophilic groups of the surfactants with CO2. These interactions also influence the properties of microemulsions in CO2. Consani and Smith (1990) determined the solubility of over 130 surfactants and related molecules in CO2 at 50 °C and at pressures up to 50 MPa. Lee et al. (1994) measured the solubility of poly(ethylene-co-acrylic acid) in small hydrocarbons at supercritical conditions to investigate the effects of solvent quantity, polymer composition, and molecular weight. O’Neill et al. (1996) measured solubilities of poly(ethylene glycol) (PEG) and poly(propylene glycol) (PPG) homopolymers and triblock copolymer surfactants composed of PEG and PPG. They studied the effects of total molecular weight, PEG/ PPG ratio, and the architecture of the blocks. Because PPG is much more CO2-philic than PEG, these triblock copolymer surfactants are well-suited for investigating solubility phenomena. The analysis of these data with an equation of state (EOS) can offer great insight into the relationship between solubility, intermolecular interactions, and surfactant architecture. © 1997 American Chemical Society
2822 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
Although a large number of equations of state may be considered, two types are particularly applicable to small supercritical solvents, polymers, and mixtures composed of the two, over a wide range of densities. One type includes the perturbed hard chain theory (Beret and Prausnitz, 1975; Donohue and Prausnitz, 1978; Sandler and Lee, 1986) and the perturbed soft chain theory (Morris et al., 1987). Hino et al. (1994, 1995a,b,c; Song et al., 1994) studied phase behavior of copolymer mixtures with a new EOS based on the perturbed hardsphere-chain theory. The second type of equation of state is based on the lattice model. The Flory-Orwell-Vrij EOS (Flory et al., 1964a,b), hole theory by Simha and Somcynsky (1969; Somcynsky and Simha, 1971), lattice fluid theories by Sanchez and Lacombe (1974, 1976, 1978; Lacombe and Sanchez, 1976), Panayiotou and Vera (1982), and Kumar et al. (1987), mean field lattice gas theory by Kleintjens and Koningsveld (1980; Kleintjens, 1983), and the cell theory by Dee and Walsh (1988a,b) are all examples of this type of EOSs. Daneshvar et al. (1990) correlated vapor-liquid equilibrium data quantitatively for mixtures of supercritical CO2 and PEGs with molecular weights of 400, 600, and 1000 by using a lattice fluid model similar to that of Panayiotou and Vera (1982). Their model takes into account the nonrandom distribution of segments and holes based on the quasichemical approach by Guggenheim (1954). In the last several years, there has been substantial progress in the development of molecular theories of association for both types of EOS. Chapman et al. (1990) developed the statistical associating fluid theory (SAFT) extending an integral equation theory to mixtures of associating hard spheres and chain molecules. Huang and Radosz (1990, 1991) obtained excellent results for a wide variety of polar low molecular weight compounds and polymers using the SAFT EOS. The ability of this EOS to calculate phase transitions and phase equilibria was demonstrated for supercritical fluid + polymer systems (Chen et al., 1994; Folie and Radosz, 1995). Phase behavior and aggregation of monohydroxyl and dihydroxyl polyisobutylene in several supercritical fluids were also calculated by the SAFT EOS (Gregg et al., 1994). This theory was extended to chain molecules of Lennard-Jones spheres by Ghonasgi and Chapman (1994) in order to explain the behavior of polymer solutions and polymer blends. A lattice fluid molecular theory of association was proposed by Veytsman (1990) and further developed by Panayiotou and Sanchez (1991; Sanchez and Panayiotou, 1995). For this lattice-fluid hydrogen-bonding (LFHB) model, the derivation is elegantly simple. A beneficial feature of the SAFT and LFHB models is that the existence of associates need not be specified a priori as in chemical theory. Instead, hydrogen bonding between specified proton donor and acceptor sites arises naturally in the theory. Because the physical (van der Waals forces) and chemical (association) interactions are treated separately in the SAFT and the LFHB models, they are useful for understanding the effects of intermolecular interactions on phase equilibria. Lele et al. (1995) used the LFHB model along with an elastic free energy term to calculate swelling behavior of poly(ethylene oxide) gels in chloroform and water. Recently, Gupta and Johnston (1994) proposed a new LFHB model, which utilizes the physical contribution
to the free energy from the SAFT theory and an association term of the LFHB theory of Panayiotou and Sanchez (1991). Another modification was that the association term was described by the local density instead of the bulk density. This modified LFHB model gives significantly improved representations for hard sphere molecules with association sites. The model was used to correlate densities and vapor pressures of pure water and 1-alkanols and phase equilibria of a mixture of benzophenone and water. The LFHB model has been applied to only a few systems containing polar polymers (Sanchez and Panayiotou, 1995). Our objective is to use the LFHB model in a systematic way to understand and predict the solubilities of block copolymers in CO2. First, experimental solubilities of PEGs, PPGs, and poly(ethylene glycol) dimethyl ethers (PEGDMEs) with different molecular weights are correlated by the model to determine the physical and chemical interaction parameters for PEG and PPG. PEGDME is used to determine the physical parameters for PEG groups because no hydrogen-bond donors are present. By modeling these three homopolymers, we wish to explain why PPG is more soluble in CO2 in terms of the physical and chemical interactions. Second, solubilities of PEG-PPG-PEG and PPG-PEG-PPG triblock copolymers will be predicted by the model. Interaction parameters for copolymers are calculated with simple averaging rules developed in this work. No new adjustable parameters are used in the prediction of copolymer solubility. The effect of the polymer structure (total molecular weight, type of endblocks, and PEG/PPG ratio) and operating variables (temperature and pressure or density) on the solubility will be investigated. Model In this work, a lattice fluid hydrogen-bonding (LFHB) model proposed by Gupta and Johnston (1994) was used to correlate and predict solubilities of homopolymers and polymeric surfactants in compressed CO2. The only modifications to the model are described in the section below on copolymers. Detailed descriptions of the model are in the literature (Huang and Radosz, 1990, 1991, 1993; Gupta and Johnston, 1994). The following summary identifies the key concepts and the parameters in the model. The residual Helmholtz free energy, Ares, is expressed as a sum of the three physical terms (hard sphere, chain, and dispersion) of the original SAFT equation of state (Huang and Radosz, 1990, 1991, 1993) and a chemical (association) term proposed by Gupta and Johnston (1994)
Ares(T,Vphy,n) ) A(T,V,n) - Aid(T,Vphy,n) ) Ahs + Adisp + Achain + Aassoc
(1)
where T is the absolute temperature, n is the total number of moles, V is the total volume, and Vphy is the total volume due to physical interactions (excluding the volume change on hydrogen bonding). The hard sphere term is given by a common expression for repulsion in a mixture of hard spheres based on the theoretical result of Mansoori et al. (1971)
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2823
[
3 2 6 (ζ2) + 3ζ1ζ2ζ3 - 3ζ1ζ2(ζ3) Ahs ) nRT πF ζ (1 - ζ )2
{
3
}
3
ζ0 -
(ζ2)3 (ζ3)2
gkl(dkl)hs )
]
∑k xkmk(dkk)n
ζn ) (πNAv/6)F
(3)
where NAv is Avogadro’s number, xk is the mole fraction of component k, mk is the number of segments per molecule, and dkk is the effective, temperature-dependent segment diameter. ζ3 is effectively a reduced density. The dispersion term is expressed by a power series as
nRT
( )( ) u
∑I ∑J DIJ kT
)m
I
ζ3
J
τ
(4)
where τ ) 0.740 48. DIJ are universal constants fitted to accurate PVT, internal energy, and second virial coefficient data for argon by Chen and Kreglewski (1977). The average number of segments for a mixture, m, is calculated by
m)
∑k xkmk
(5)
The average dispersion energy per segment for a mixture, u, is calculated using an appropriate mixing rule. In the SAFT theory (Huang and Radosz, 1991), a van der Waals one-fluid mixing rule and a volume fraction mixing rule were examined. As the latter mixing rule gave better representations for binary vapor-liquid equilibria near the critical region, it will be used in the present work as follows:
u ) kT
∑k ∑l
fk )
fkfl
() ukl
kT
xkmkvk°
∑l xlmlvl°
ukl ) (1 - kkl)(ukkull)1/2
(6)
(7)
(8)
where fk is the volume fraction of component k, ukl is the (temperature-dependent) dispersion energy of interaction between segments k and l, vk° is the (temperature-dependent) segment volume, and kkl is the binary interaction parameter. The chain term describes covalent bonds between segments to form a chain chain
A
) nRT
2
ln(1 - ζ3) (2)
where ζn (n ) 0-3) is a function of the molar density due to physical interactions, F()n/Vphy). This function, which is applicable to bonded spheres, is expressed as
Adisp
ζ2 3dkkdll 1 + + 1 - ζ3 dkk + dll (1 - ζ )2 3
∑k xk(1 - mk) ln{gkk(dkk)hs}
(9)
where g(d)hs is the pair correlation function for a mixture of hard spheres derived by Mansoori et al. (1971)
(
)
dkkdll 2 (ζ2)2 (10) dkk + dll (1 - ζ )3 3
The effective, temperature-dependent segment diameter, dkk, is related to the temperature-dependent segment volume, vk°, as
v k° )
πNAv (dkk)3 6τ
(11)
Moreover, the temperature dependence of vk° and ukk are expressed as follows
{
(
vk° ) vk°° 1 - C exp
)}
-3ukk° kT
3
(12)
where vk°° is the temperature-independent segment volume, ukk° is the temperature-independent dispersion energy of interaction between segments, and C ) 0.12. The temperature dependence of the dispersion energy is given by
(
ukk ) ukk° 1 +
ek kT
)
(13)
where ek is a constant that expresses the temperature dependence of ukk. In the physical part of the model, four parameters are used for each component: number of segments in a molecule m, temperature-independent segment volume v°°, temperature-independent dispersion energy u°/k, and e/k. The association term was obtained from a partition function that accounts for the number of ways of distributing the hydrogen bonds among donor and acceptor groups in the system. Let dik be the number of proton donor sites of type i in each molecule of type k and ajk the number of proton acceptor sites of type j in each molecule of type k. The total number of donor sites of type i, Ndi, and the total number of acceptor sites of type j, Naj, are given by
Ndi )
∑k dikNk
(14)
Naj )
∑k ajkNk
(15)
where Nk is the number of molecules of type k. When there are Nij hydrogen bonds between a donor site of type i and an acceptor site of type j, the number of donors of type i, Nio, and the number of acceptors of type j, Noj, that are not hydrogen bonded are given by
Nio ) Ndi -
∑j Nij
(16)
Noj ) Naj -
∑i Nij
(17)
The number of ways Ω of distributing the Nij bonds among the functional groups of the system was derived by Panayiotou and Sanchez (1991) by generalizing the argument of Veytsman (1990) to the case of multifunctional molecules
2824 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
Ω)
Ndi!
Naj!
PijNij
∏i N !∏j N !∏i ∏j N ! io
oj
(18)
ij
where Pij is the mean-field probability that a specific acceptor will be proximate to a given donor. This probability was given by
( )
(19)
where N is the total number of molecules and Sij° is the entropy loss associated with the hydrogen bond formation of an i-j donor-acceptor type. The bulk density (ζ3) in the above equation was replaced with the local density (gijhsζ3) using the donor-acceptor pair correlation function by Gupta and Johnston (1994). The Helmholtz free energy due to the association was obtained as follows
) mnRT
[
Aij°
∑i ∑j vij 1 + RT + ln v
]
ln{gij(dij)hsζ3} +
∑i vdi ln
vij iovoj
vio + vdi
∑j vaj ln
voj vaj
(20)
where
vij )
Nij Nio Noj Ndi vio ) voj ) vdi ) mN mN mN mN vaj )
Naj (21) mN
The standard Helmholtz free energy, F°, and standard Gibbs free energy, G°, of hydrogen-bond formation are defined as
Aij° ) Eij° - TSij°
(22)
Gij° ) Eij° - TSij° + PVij°
(23)
In the association term, three association parameters E°, S°, and V° are needed to evaluate number of hydrogen bonds for each type of donor-acceptor pair, where E° is the favorable energy change (excess of any physical interaction energy), S° is the entropy loss, and V° is the volume change upon hydrogen bond formation. Moreover, the numbers of donor and acceptor sites in a molecule are necessary for all components. The total Gibbs free energy of the system, G, is written in terms of residual and ideal gas Helmholtz free energies as follows
G(T,P,n) ) Ares(T,Vphy,n) + Aid(T,Vphy,n) + PV (24) where the system volume V is related to Vphy and Vij° by
V ) Vphy +
∑i ∑j NijVij°
)
T,P,nk,Nij
( ) ∂Ares ∂Vphy
(25)
According to the original LFHB theory, the equation of state, P(T,V,n), the number of hydrogen bonds for each donor-acceptor pair, Nij, and the chemical potential for each component, µk, can be derived by differentiating the Gibbs free energy as
-
T,nk,Nij
( ) ( ) ∂G ∂Nij
µk )
Sij° ζ3 Pij ) gij(dij)hs exp mN R
Aassoc
( ) ∂G ∂Vphy
nRT +P)0 Vphy )0
(26)
(27)
T,P,Vphy,nk,Nst*ij
∂G ∂nk
(28)
T,P,Vphy,nl*k,Nij
From eq 27, the numbers of hydrogen bonds for all donor-acceptor pairs can be obtained by solving a set of simultaneous quadratic equations such as
( )
vij -Gij° ) ζ3gij(dij)hs exp viovoj RT
(29)
Results and Discussion Experimental Data. The homopolymers and triblock copolymers considered in this study are listed in Table 1. All the polymers have a large polydispersity due to the nature of the polymerizations. The PEGCO2 vapor-liquid equilibrium data were measured and modeled by Daneshvar et al. (1990) at temperatures of 313 and 323 K and pressures up to 29 MPa. The polymer samples were obtained from Aldrich Chemical Co. The compositions of vapor and liquid coexisting phases were measured by the analytical method, in which the phases are sampled and analyzed. Daneshvar and Gulari (1992) measured molecular weight distributions for PEG(400) and PEG(600). The numbers of -C2H4O- repeat units ranged from 4 to 16 for PEG(400) and from 4 to 21 for PEG(600). The resulting number-average and weight-average molecular weights were 406 and 433 for PEG(400) and 538 and 588 for PEG(600), respectively. All other data were cloud point measurements of solubilities in CO2 (O’Neill et al., 1996), each for a fixed temperature and polymer concentration. Molecular weight distributions were not characterized for any of these polymers. A serious limitation of this method is that the high molecular weight fractions will precipitate first as the pressure is lowered. As it is further lowered, lower molecular weight fractions will precipitate. The difference between the pressure where turbidity is visible to the observer and the pressure where the majority of the polymer precipitates can range from 5 to 50 bar or more depending upon the molecular weight distribution. Thus, the reported cloud point pressure may be expected to be somewhat high. Conversely, solubilities of more monodisperse surfactants may be expected to be higher than indicated by these cloud point pressures. The assignment of the cloud point for a polydisperse system is somewhat subjective; typically, it was not chosen as the onset of turbidity. Although this technique has these limitations, it is much faster and simpler than the analytical technique. The vast majority of polymersupercritical fluid data have been obtained by this technique (Kiran, 1992; McHugh and Krukonis, 1994). Although molecular weight distributions will strongly affect solubilities in compressed CO2, the present work assumed that each polymer sample was a monodisperse pure component with the molecular weight listed in Table 1. An enormous amount of work would have been required to fractionate and characterize all of the polymers, which was beyond the scope of this study. We are more interested in trends among the various poly-
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2825 Table 1. Molecular Weights (MW) and Numbers of Donor and Acceptor Sites polymer
data sourcea
total MW
MW of PEG groups
MW of PPG groups
no. of donor sites
no. of acceptor sites
homopolymers PEGDME(400) PEGDME(1000) PEG(400) PEG(600) PEG(1000) PPG(400) PPG(1025) PPG(2000)
2 2 1 1, 2 1 2 2 2
400 1000 400 600 1000 400 1025 2000
400 1000 400 600 1000 0 0 0
0 0 0 0 0 400 1025 2000
0 0 2 2 2 2 2 2
0 0 9.7 14.2 23.3 7.6 18.4 35.2
triblock copolymers L31 L61 L62 L81 L92 10R5 17R2 17R4 25R2
2 2 2 2 2 2 2 2 2
1056 2189 2462 2944 3750 2000 2125 2833 3125
106 219 492 294 750 1000 425 1133 625
950 1970 1970 2650 3000 1000 1700 1700 2500
2 2 2 2 2 2 2 2 2
19.5 39.6 45.9 53.1 69.5 40.6 39.6 55.7 57.9
a
1: Daneshvar et al. (1990). 2: O’Neill et al. (1996). L: PEO-b-PPO-b-PEO. R: PPO-b-PEO-b-PPO.
mers. Furthermore, these polymers will be polydisperse in practical applications. The PEG and PPG homopolymers were purchased from Polysciences and Scientific Polymer Products, respectively, and the PEGDME from Aldrich Chemical Co. The Pluronic L (PEO-b-PPO-b-PEO) and Pluronic R (PPO-b-PEO-b-PPO) copolymer samples were donated by BASF. For the Pluronic L series, the first numeral after the L is an indication of the overall molecular weight and the second is the approximate weight fraction of PEO; e.g., 1 means 10 wt %. The same convention is used for the Pluronic R surfactants, except that the R is placed between the first and second numerals. Each propylene oxide repeat unit in the polymer is in the isopropyl form. For each of these polymers the average molecular weight was specified by the supplier. PEG, PPG, and copolymers of the two have two terminal hydroxyl groups, while PEGDME has two terminal methyl ether groups. Hydrogen atoms of the hydroxyl groups will act as proton donor sites to form hydrogen bonds with oxygen atoms. Philippova et al. (1985) studied the hydrogen bond formation of PEGs with molecular weights from 300 to 6000 in dilute carbon tetrachloride solutions. They identified several types of intramolecular and intermolecular hydrogen bonds, where a hydroxyl group is bound to other hydroxyl groups or ether oxygen atoms. These hydrogen bonds may also occur in the compressed CO2 phase as well as in the polymer phase containing dissolved CO2 and can have a significant effect on their solubilities. With FTIR spectroscopy, it has been shown that the strength of the hydrogen bonds increases modestly with a decrease in the dielectric constant of the solvent, for an inert solvent (Kazarian et al., 1993). We have not included this subtle effect in this study, as it would add more parameters. In this work, it was assumed that each hydroxyl group acts as a proton donor and a proton acceptor site and that each ether oxygen atom acts as a proton acceptor site. For PEGDME, no proton donor site is present. Furthermore, PEGs and PEG-PPG-PEG copolymers have two primary alcohol terminal groups (-CH2OH), while PPGs and PPG-PEG-PPG copolymers have secondary alcohol terminal groups (-CH(CH3)OH). The strength of the hydrogen-bonding parameters will be
Table 2. Modified Parameter Values for CO2 and Correlated Results of Saturated Properties ν°° (cm3/mol)
m
u°/k (K)
e/k (K)
21.310
1
246.43
72.642
exptl calcd
Tc (K)
Pc (MPa)
Fc (mol/L)
304.2 306.3
7.38 7.88
10.6 10.7
AAD of satd vapor pressure AAD of satd liquid density [220 K - Tc]
1.42% 9.07%
different for primary and secondary hydroxyl groups. The numbers of the donor and acceptor sites were calculated from the molecular weights of polymers and are listed in Table 1. Each oxygen including the oxygens in the terminal OH groups contains two acceptor sites. Because the average molecular weights do not correspond to integral numbers of polymer repeat units, the numbers of acceptor sites are not integers. Interaction Parameters for CO2. Since CO2 is a nonpolar fluid that does not self-associate, only physical EOS parameters are needed to calculate thermodynamic properties of pure CO2 by the LFHB model. The parameters have already been determined by using measured vapor pressures and saturated liquid densities in a temperature range from 218 to 288 K (Huang and Radosz, 1990). However, these parameter values give a somewhat higher critical temperature and pressure (Tc ) 320.7 K, Pc ) 9.25 MPa) than the measured values (Tc ) 304.2 K, Pc ) 7.38 MPa). Because the solubility data of the polymers have been measured at temperatures close to the critical temperature of CO2, the model used in this work must give a better representation of the critical point of CO2. Thus, the present work redetermined CO2 parameters to fit literature values (Vargaftik, 1975) of saturated vapor pressures and saturated liquid densities as well as critical temperature, critical pressure, and critical density. The parameter values and the correlated results are listed in Table 2. The new parameter set gives considerably better representations for critical properties and saturated vapor pressures than those calculated by the original parameter set. Furthermore, the new parameters can predict densities in single phases with an average absolute deviation of 2.8% over a temperature
2826 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 Table 3. Parameter Values for Homopolymers polymer
ν°° (cm3/mol)
m
u°/k (K)
e/k (K)
PEGDME(400) PEGDME(1000)
19.244 19.244
11.56 28.89
315.59 315.59
10 10
PEG(400) PEG(600) PEG(1000)
19.244 19.244 19.244
11.56 17.34 28.89
315.59 315.59 315.59
10 10 10
PPG(400) PPG(1025) PPG(2000)
22.341 22.341 22.341
9.955 25.51 49.77
237.13 237.13 237.13
10 10 10
range from 220 to 400 K and a pressure range from 2 to 50 MPa except for very close to the critical point. Correlation of Solubilities of Homopolymers To Determine Interaction Parameters. EOS parameters of polymers are generally determined using measured PVT behavior, as vapor pressures are typically too low to measure. The PVT behavior for PEGDME(428), PEG(600), and PPG(1025) has been measured by Fang et al. (1994). In the present work, the PVT data were correlated satisfactorily by the LFHB model with average absolute deviations (AAD) in volume of 0.047%, 0.063%, and 0.054% for PEGDME(428), PEG(600), and PPG(1025), respectively. When these correlated parameters were used in the calculation of binary polymerCO2 vapor-liquid equilibria data, however, the calculated polymer solubilities were several orders lower than experimental ones, even when the binary interaction parameter k12 was adjusted. Apparently, the dispersion energies between polymer segments from the PVT correlation were too strong for calculating vapor-liquid equilibria. Therefore, the PVT data were not used; instead, a single set of pure component parameters for the polymers were determined from the binary solubility data in CO2. As previously mentioned, PEG, PPG, and their copolymers may form intramolecular and intermolecular hydrogen bonds of various types both in the liquid (polymer-rich) phase and the vapor (CO2-rich) phase. Strictly speaking, thermodynamic properties of the hydrogen bond formation will be different for each type. In the present work, we assumed that the association parameters (E°, S°, and V°) depend only on the terminating groups of the polymers, namely primary alcohol groups (for PEGs and PEG-PPG-PEG copolymers) and secondary alcohol groups (for PPGs and PPG-PEG-PPG copolymers). This approach was chosen to keep the number of adjustable parameters reasonable. For a given group such as PEG, there exists only one type of hydrogen bond (referred to as type 1-1), and the solution of eq 29 can be written as
1 [(A + d1 + a1) v11 ) 2m 11 {(A11 + d1 + a1)2 - 4d1a1}1/2] (30) A11 )
( )
G11° m exp RT g22(d22)hsζ3
(31)
d1 ≡ mvd1
(32)
a1 ≡ mva1
(33)
where g22(d22)hs is the pair correlation function between polymer segments.
E° (kJ/mol) 0 0 -10.291 -10.291 -10.291 -2.7343 -2.7343 -2.7343
S° (J mol-1 K-1) 0 0
V° (cm3/mol) 0 0
-25.583 -25.583 -25.583 -2.8973 -2.8973 -2.8973
k12(To)
l12 (K)
0.012 91 0.012 91
18.74 18.74
-4.3830 -4.3830 -4.3830
0.012 91 0.012 91 0.012 91
18.74 18.74 18.74
-0.315 93 -0.315 93 -0.315 93
0.077 16 0.077 16 0.077 16
-19.13 -19.13 -19.13
The first step was to calculate the physical properties v°°, u°/k12 for an ethylene oxide segment from the solubility data for all CO2 + PEGDME systems (see Table 3). In this system, no association was present. The same physical parameters were used for an ethylene oxide segment in PEG, and the PEG association parameters E°, S°, and V° were determined by correlating vapor-liquid equilibrium or polymer solubility data for all CO2 + PEG systems. For PPG, we did not have data for the methyl ether-terminated analog. Therefore, the physical and chemical parameters were determined simultaneously by correlating solubility data for all CO2 + PPG systems. A non-linear least-squares method (Marquardt method) was used in the correlations. Values of e/k for all polymers were set equal to 10 according to the original SAFT EOS. We assumed a linear relationship between the number of segments, m2, and the molecular weight, MW2, for all polymers as follows
v2°°m2/MW2 ) v2°°
(34)
where vs°° is the specific segment volume of the polymers and was set to 0.556 cm3/g according to the results of the above PVT correlations. m2 for each polymer was calculated using this equation and is listed in Table 3. This linear relationship with the constant specific segment volume maintains quantitative representations of the PVT relationships for pure polymers; namely, the parameters can predict the PVT data with AADs in volume of 0.50%, 4.4%, and 1.2% for PEGDME(428), PEG(600), and PPG(1025), respectively. It will also offer the correct molecular weight dependence for polymer solubilities. The binary interaction parameter, k12, depended upon temperature as follows
k12(T) ) k12(To) + l12(1/T - 1/To)
(35)
where the reference temperature, To, was set equal to 323.2 K. Figure 1 shows the results of the model for the solubilities of PEGDME(400) and PEGDME(1000) in CO2. The solubilities of these nonassociating polymers were correlated by using only the physical parameters in the EOS, which are listed in Table 3. The correlated temperature dependence, with the parameter l12, and the molecular weight dependence of solubilities are in good agreement with experiment over the pressures measured. No adjustable parameters were required to determine the molecular weight dependence, except the universal value of vs°° for all polymers. The model predicts that the maximum pressure is higher than the experimental value for the 400 MW data. The results of the model for the CO2 + PEG(400) system are shown in Figure 2. The upper and lower
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2827
Figure 1. Comparison of experimental and correlated solubility of PEGDME in CO2. Experimental data are from O’Neill et al. (1996).
Figure 2. Comparison of experimental and correlated vaporliquid equilibria for the CO2 + PEG(400) system. Experimental data are from Daneshvar et al. (1990).
parts of the figure show the polymer concentrations in the liquid and vapor (fluid) phases, respectively. Although some modest discrepancies can be seen in the polymer solubilities in the vapor phase at low pressures, the agreement between the experimental data and the model is satisfactory at high pressures. Figure 3 shows the results for the solubility of PEG(600) measured by O’Neill et al. (1996). The horizontal axis denotes the density of pure CO2 calculated by the IUPAC EOS (Angus et al., 1976) as a measure of the solvent power of CO2. For both the data and the model, the log of the solubility is fairly linear in density, as has been observed for many low molecular weight systems (Johnston and Eckert, 1981; Kumar and Johnston, 1988). On the basis of this relationship, it is likely that the set of experiments performed at the lowest solubility may be less accurate than the model. The correlated temperature dependence of the solubility
Figure 3. Comparison of experimental and correlated solubility of PEG(600) vs density of pure CO2. Experimental data are from O’Neill et al. (1996).
Figure 4. Molecular weight dependence of the solubility of PEG in CO2 at 50 °C. Closed and open symbols denote measured data by Daneshvar et al. (1990) and by O’Neill et al. (1996), respectively.
with the parameter l12 (determined from the PEGDME system) is in excellent agreement with the experiments. The molecular weight dependence of the solubility of PEGs is represented in Figure 4. The model predicts successfully a large decrease in solubility with molecular weight. In all cases, two hydrogen-bond donors are present, and far more than two hydrogen-bond acceptors are available. Thus, the model suggests that the decrease in solubility with molecular weight is due more to physical forces than chemical interactions, in particular, the strong segment-segment interactions. The closed and open symbols denote data measured by Daneshvar et al. (1990) and by O’Neill et al. (1996), respectively. The discrepancy between two experimental isotherms for PEG(600) can be considered to be due to the differences in the samples used and in the experimental method. For a given solubility, one might expect the cloud point pressure from O’Neill et al. to be somewhat high as discussed above, yet the opposite is observed. Considering the uncertainties in the molec-
2828 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
Figure 5. Comparison of experimental and correlated solubility of PPG in CO2. Symbols denote experimental data by O’Neill et al. (1996).
ular weights and their distributions, the model gives a fairly good representation for the molecular weight dependence of the polymer solubility. The experimental and modeling results for CO2 + PPG systems are shown in Figure 5. The model correlates the experimental data for PPG(400) and PPG(2000) quantitatively. However, a large discrepancy can be seen for PPG(1025), indicating the possibility that this sample may have a smaller average molecular weight than indicated by the supplier or an unusually large polydispersity. The polymer parameters correlated from the phase equilibria data listed in Table 3 indicate some important results: (1) v°° for PPG is larger than that for PEG; (2) u°/k for PEG is larger than that for PPG; (3) all association parameters for PEG have larger absolute values than those for PPGs. All of these trends in these parameters are physically meaningful and related to the isopropyl structure of PPG. The extra volume of the methyl pendant group is the reason for the larger v°°. Steric hindrance from the pendant methyl group decreases the ability of the segments to interact, lowering u°/k and the strength of the hydrogen bonds. The dipole moment of diethyl ether is 1.3 D, slightly higher than the value of 1.2 D for diisopropyl ether. Absolute values of association parameters for PEGs are lower than those for 1-alkanols (E° ) -25.1 kJ/mol, S° ) -30 J mol-1 K-1, V° ) -5.6 cm3/mol) (Gupta and Johnston, 1994). This may be expected for several reasons. The ether oxygens are less polar than oxygens in alcohols. Also, ether oxygens are somewhat less accessible for steric reasons. Finally, intramolecular hydrogen bonding of PEG is influenced by conformational restraints, even though PEG is a relatively flexible polymer chain. In Figure 6, solubilities of PEG(400), PEGDME(400), and PPG(400) in CO2 are compared at 50 °C for a constant molecular weight. PEG is the least soluble among the three polymers due to the strongest physical and chemical interactions between polymer segments. The PEGDME, which may be assumed to have the same dispersion interactions as PEG, is more soluble than PEG due to the absence of self-association. However, even though PPG can self-associate, it is more soluble than PEGDME because (1) the chemical interactions for
Figure 6. Comparison of solubility of PEG(400), PEGDME(400), and PPG(400) in CO2 at 50 °C. Experimental data for PEG(400) are from Daneshvar et al. (1990), and others are from O’Neill et al. (1996).
PPG are much weaker than those for PEG and (2) the physical segment-segment interactions are weaker for PPG than PEG due to steric effects. Finally, the higher solubility of PPG versus PEG is due to both the weaker segment-segment interactins and the weaker selfassociation. Over the range studied, the maximum in the pressure versus solubility for PEGDME(400) seems unusual. On the basis of the model, and other data throughout this study, it is reasonable to assume that the model is more accurate than the data for the highest solubilities of PEGDME. Not only does the model offer a means to interpret the effects of physical and chemical interactions on solubilities, but it may be used to identify potentially inaccurate data. Prediction of Solubilities of Copolymers. While measured solubilities for copolymers can always be correlated with further adjustable parameters, we chose a different approach. Our goal was to predict the copolymer solubilities with no new parameters. We only used parameters determined for the homopolymers and the supplied information about the molecular weight of each component in the copolymer. In order to do this, simple averaging rules were developed to estimate parameter values for the copolymers. The segment volume for a copolymer, v2°°, was averaged in terms of the segment numbers of constituent blocks
v2°° )
∑k vk°°mk m2
∑k mk
(37)
vs°°MWk vk°°
(38)
m2 )
mk )
(36)
where m2 is the number of segments per molecule of a copolymer and the sum is over all blocks (here the subscript k denotes PEG or PPG blocks). Equation 37
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2829 Table 4. Parameter Values for Copolymers Calculated by Eqs 36-43 ν°° (cm3/mol)
m
u°/k (K)
k12(To)
l12 (K)
L31 L61 L62 L81 L92
21.987 21.987 21.644 21.987 21.644
26.69 55.35 63.26 74.46 96.33
244.47 244.47 251.93 244.47 251.93
0.069 86 0.069 86 0.062 78 0.069 86 0.062 78
-14.83 -14.83 -10.66 -14.83 -10.66
10R5 17R2 17R4 25R2
20.677 21.644 20.990 21.644
53.78 54.59 75.05 80.28
274.96 251.93 267.17 251.93
0.042 74 0.062 78 0.049 23 0.062 78
1.153 -10.66 -2.669 -10.66
polymer
is based on the same idea as eq 5. These three equations conserve the overall segment volume of a polymer. The parameters u22° and k12 were averaged in a similar manner with mixing rules based upon volume fraction
(u22°)1/2 ) (1 - k12)(u22°)1/2 )
fk° )
∑k fk°(ukk°)1/2
(39)
∑k fk°(1 - k1k)(ukk°)1/2
(40)
mkvk°° m2v2°°
(41)
Figure 7. Comparison of experimental and predicted solubility of L31 copolymer in CO2. Experimental data are from O’Neill et al. (1996).
where f° is the volume fraction of a block in a copolymer and was defined using v°° instead of v° in eq 7. The dispersion energy between unlike blocks was approximated by the geometric mean in the derivation of eq 39. Equation 40 provides similar averaging equations for 1 - k12(To) and l12
{1 - k12(To)}(u22°)1/2 )
∑k fk°{1 - k1k(To)}(ukk°)1/2
(42)
l12(u22°)1/2 )
∑k fk°l1k(ukk°)1/2
(43)
These averaging rules are quite simple because the parameter values for a copolymer are dependent only upon the molecular weights of blocks. These rules are physically realistic since they are based on the volume fractions of the various blocks. Table 4 lists the parameter values for copolymers calculated by eqs 36-43. Either the PEG or PPG association parameter set was used depending upon whether the outer block of the copolymer was PEG or PPG. No new adjustable parameters are introduced in this table; each parameter was calculated without any copolymer phase equilibria data from the above averaging rules. Comparisons of predicted and experimental copolymer solubilities are shown in Figures 7 and 8 for Pluronic L31 and 17R2, respectively. The agreement is good considering that none of the copolymer-phase equilibria data were used in the predictions. For a given solubility, the experimental cloud point pressure may be expected to be high for a polydisperse sample. This trend is seen for all of the isotherms. Unlike the data, the model predicts crossover pressures (Chimowitz and Pennisi, 1986; Johnston et al., 1987). Below the crossover pressure, the solubility decreases with temperature at a given pressure. This is due to a loss in density. Above the crossover pressure, temperature effects on density are less significant. Here, temperature weakens
Figure 8. Comparison of experimental and predicted solubility of 17R2 copolymer in CO2. Experimental data are from O’Neill et al. (1996).
the polymer-polymer interactions (polymer volatility) and raises the solubility. It is likely that the data will show crossover points at higher pressures and solubilities than those measured. The comparison of experimental and predicted cloud point (dew point) pressures for the CO2 + PEG-PPGPEG copolymer systems is shown in Figure 9. The predicted results are satisfactory for L31, L61, and L81, which contain 10 wt % of PEG. The cloud point pressures for L31 and L61 are predicted to be further apart than the experimental values, whereas the L61 and L81 are predicted to be closer together than the data. The model predicted much lower cloud point pressures than the data for copolymers containing 20 wt % of PEG, i.e., L62 and L92. Similar results were also obtained for PPG-PEG-PPG copolymers as shown in Figure 10. The order of the experimental cloud point pressures is predicted cor-
2830 Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997
Figure 11. Predicted molecular weight dependence of the solubility of PEG-PPG-PEG copolymers in CO2 at 40 °C, 20 MPa.
Figure 9. Comparison of experimental and predicted cloud point pressures for CO2 + PEG-PPG-PEG copolymer systems. Experimental data are from O’Neill et al. (1996).
Figure 10. Comparison of experimental and predicted cloud point pressures for CO2 + PPG-PEG-PPG copolymer systems. Experimental data are from O’Neill et al. (1996).
rectly, but the predictions become less accurate as the cloud point pressures increase. The deviations between the model and experiment may be due to subtleties in hydrogen-bonding interactions missed by the model. The model does not fully consider the locations of the hydrogen-bonding sites on the chain and does not distinguish between intermolecular and intramolecular hydrogen bonding. It does distinguish between terminal primary and secondary alcohol functionalities in a simple manner with the parameters in Table 3. For PEG homopolymers, the physical and chemical parameters were correlated separately, by using PEGDME. This strategy was not used for PPG, as PPGDME data were not available. Major reasons for the deviations between the model and experiment are the uncertainty in the average molecular weight and the variations in polydispersity for each block. In nearly all cases, the model predicted
a lower cloud point pressure than the data. As stated above, this is expected since the highest molecular weight fractions precipitate first. The deviations from the data were larger for copolymers with higher molecular weights, which have wider molecular weight distributions. The increase in cloud point pressure with molecular weight is much larger for PEG than PPG. Thus errors due to polydispersity may be expected to be larger for copolymers with 20 wt % PEG versus 10 wt % PEG, as was seen in Figure 9. Given that the predictions of the model are in reasonable agreement with experimental data, we now present further predictions to understand the molecular weight dependence of copolymer solubilities. The predictions are shown in Figure 11 for PEG-PPG-PEG triblock copolymers at 40 °C and 20 MPa. When the total molecular weight of the two PEG outside blocks of a PEG-PPG-PEG copolymer increases from 0 to 3000, while keeping the molecular weight of PPG constant, the solubility decreases several orders of magnitude. On the other hand, for a given PEG molecular weight, the solubility first increases and then slightly decreases with an increase in the molecular weight of the PPG middle block. This initial increase in the solubility is due to the sharp decrease in the average strength of dispersion interactions between polymer segments. In essence, PPG is more CO2-philic than PEG. Eventually, with futher increases in PPG molecular weight, the overall molecular weight becomes too large and solubility decreases gradually. This type of behavior has been reported on the basis of cloud point measurements for diblock copolymers (Hoefling et al., 1993; Newman et al., 1993). Figure 12 shows the same effects of molecular weight for PPG-PEG-PPG copolymers. This figure almost coincides with Figure 11 when the axes are aligned. For a given total PPG and PEG molecular weight, the solubilities are slightly higher when the PEG is in the middle versus in the end blocks. This result is consistent with the stronger hydrogen bonding for primary alcohol groups of PEG relative to secondary alcohol groups of PPG. As shown in the data for two block copolymers with similar block lengths, Pluronic L62 has a higher cloud point pressure than Pluronic 17R2. The fact that the choice of end blocks (PPG or PEG) has less influence on the solubility than the MW of the PEG block(s) suggests that the physical interactions are somewhat more important than the hydrogen-bonding interactions, as was indicated above. The results shown
Ind. Eng. Chem. Res., Vol. 36, No. 7, 1997 2831
Figure 12. Predicted molecular weight dependence of the solubility of PPG-PEG-PPG copolymers in CO2 at 40 °C, 20 MPa.
in these two figures indicate important guidelines for surfactant design. For example, a variety of surfactant architectures and molecular weights are available to achieve a given solubility. Conclusions The LFHB model sheds great insight into the effects of physical and chemical interactions on solubilities. The solubilities of copolymers may be predicted with simple averaging rules from interaction parameters determined from homopolymers without adding any adjustable parameters. The model predicts the effects of total molecular weight, PEG/PPG ratio, terminal functional groups, temperature, and density on solubility. The much higher solubility of PPG versus PEG is due primarily to steric hindrance from the methyl branch, which weakens segment-segment interactions. A secondary factor is that the donor-acceptor association parameters are significantly stronger for a primary (in the case of PEG) versus a secondary (in the case of PPG) alcohol terminal group. For triblock copolymers, the fact that the choice of end blocks (PPG or PEG) has less influence on the solubility than the MW of the PEG block(s) suggests that the physical interactions are somewhat more important than the hydrogen-bonding interactions for these polymers. The success of the predictions indicates that these types of models may be used to aid the design of CO2-soluble copolymer surfactants. Nomenclature A ) Helmholtz free energy, J a ) number of acceptor sites in a molecule, 1/molecule C ) constant in eq 12 ) 0.12 DIJ ) universal constants in dispersion term d ) effective, temperature-dependent segment diameter, m, or number of donor sites in a molecule, 1/molecule E° ) molar association energy, J/mol e/k ) parameter representing temperature dependency of u/k, K F° ) standard Helmholtz free energy of hydrogen bond formation, J/mol f ) volume fraction defined by eq 7 f° ) volume fraction defined by eq 41 G ) Gibbs free energy, J G° ) standard Gibbs free energy of hydrogen bond formation, J/mol g ) pair correlation function
k ) Boltzmann’s constant ≈ 1.381 × 10-23 J/K, or binary interaction parameter l ) parameter representing temperature dependency of binary interaction parameter, K MW ) molecular weight m ) number of segments in a molecule, 1/molecule N ) total number of molecules NAv ) Avogadro’s number ≈ 6.02 × 1023 molecules/mol Naj ) total number of acceptor sites of type j Ndi ) total number of donor sites of type i Nij ) total number of hydrogen bonds of i-j type Nio ) total number of donor sites of type i not hydrogen bonded Noj ) total number of acceptor sites of type j not hydrogen bonded n ) amount of substances, mol P ) pressure, Pa, or mean field probability of hydrogen bond formation R ) universal gas constant ≈ 8.314 J mol-1 K-1 S° ) molar entropy loss of hydrogen bond formation, J mol-1 K-1 T ) absolute temperature, K To ) reference temperature ) 323.2 K u/k ) temperature-dependent dispersion energy between segments, K u°/k ) temperature-independent dispersion energy between segments, K V ) total volume, m3 V° ) molar volume change of hydrogen bond formation, m3/mol v° ) temperature-dependent segment volume, m3/mol of segments v°° ) temperature-independent segment volume, m3/mol of segments vs°° ) temperature-independent specific segment volume of polymer, cm3/g x ) mole fraction µ ) chemical potential, J/mol F ) molar density due to physical interactions, mol/m3 τ ) constant ≈ 0.740 48 ζ ) function defined by eq 3 Superscripts assoc ) association chain ) chain formation disp ) dispersion hs ) hard sphere id ) ideal gas phy ) values due to physical interactions res ) residual Subscripts c ) critical point i ) type of donor site j ) type of acceptor site k ) type of component (1 ) CO2, 2 ) polymer) or type of polymer block l ) type of component
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Received for review November 4, 1996 Revised manuscript received April 16, 1997 Accepted April 18, 1997X IE960702Q X Abstract published in Advance ACS Abstracts, June 15, 1997.
