Equations of state for monomers and polymers

Equations of state for monomers and polymers

ILIIIE Ill ELSEVIER Fluid Phase Equilibria 117 (1996) I - I 0 Equations of state for monomers and polymers + P. D. Condo and M. Radosz* Corporate R...

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ILIIIE

Ill ELSEVIER

Fluid Phase Equilibria 117 (1996) I - I 0

Equations of state for monomers and polymers + P. D. Condo and M. Radosz* Corporate Research Laboratories, Exxon Research and Engineering Company, Route 22 East, Annandale, NJ 08801, USA ABSTRACT

Of the large number of equations of state available in the literature, few treat small molecules as well as polymers, association, and copolymers. Three equations of state that rigorously account for all of these effects and have been applied to real polymer systems are the lattice-fluid theory (LF), the statistical associating fluid theory (SAFT), and the perturbed hard-sphere chain theory (PHSC). We illustrate the versatility of one of these equations of state (SAF'I') for modelling the phase behavior of polyolefin systems, and the effects due to molecular dissimilarity, compressibility, association, and copolymer composition. Keywords: equation of state, polymer, SAFT, association, copolymer 1.

INTRODUCTION

The equation of state is a very efficient and powerful tool to study the thermodynamic properties and phase behavior exhibited by polymers. Such behavior includes the phase equilibrium of polymers and monomers during polymerization and subsequent separation; [1] association effects in the case of functional polymers [1,2] and aqueous polymer systems; [3] and the effect of copolymer composition on blend miscibility. [4-6] Ideally, the researcher would like to have one equation of state that is applicable to all the thermodynamic behavior of polymer systems. With this ideal in mind, an overview of the versatility of selected equations of state is presented.

+ presented at the 7th International Conference on Fluid Properties and Fluid Equilibria for Chemical Process Design, Snowmass, CO, USA, June, 1995. * corresponding author at Chemical Engineering Dept., Louisiana State University, Baton Rouge, LA 70803-7300, USA 0378-3812/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0378-3812(95)02929-X

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P.D. Condo, M. Radosz/FluidPhase Equilibria 117(1996) 1-10

The selected equations of state can be classified into two categories: lattice and continuum models. [7,8] Lattice equations of state can be subdivided into two classes: cell and hole models. The predecessor of the lattice equations of state is the classic Flory-Huggins theory [9] which is applicable to incompressible liquid systems. Figure 1 illustrates the way in which two lattice concepts, the cell and hole concepts, modify the Flory-Huggins theory to account for compressibility. The lattice-cell approach [10-12] allows the cell volume to change as illustrated in Figure 1 for the case of increasing pressure. The lattice-hole approach [13] allows some of the cells to be empty, and thus the lattice is said to contain holes. Increasing pressure decreases the concentration of holes as illustrated in Figure 1. Continuum models [14-22] do not invoke the artificial lattice, but are derived on the basis of perturbation theory. The selected continuum models define a reference system of athermal hard spheres. Chain molecules, such as polymers, are modelled as tangent hard spheres. By different statistical mechanical techniques, the continuum models yield the same general form of the equation of state. That is, an equation of state for hard spheres plus a remainder, or a perturbation term, that is attributed to chain connectivity or bonding of the hard spheres. Figure 1 shows a schematic representation of the effect of pressure on a continuum solution. The versatility of an equation of state may be judged by asking the following questions: Does the equation of state satisfy the ideal gas law? Can the equation of state correlate polymer melt densities? A positive answer to these two questions signifies that the equation of state is applicable to both small molecules and polymers. Does the equation of state rigorously treat association? Does the equation of state rigorously treat copolymers? Has the equation of state been applied to real systems? Below we subject seven equations of state to the above criteria to determine their versatility. 2.

RESULTS AND DISCUSSION

Table 1 shows the versatility of selected equations of state. All the equations of state satisfy the ideal gas law. However, both lattice-cell models, PHCT and FOV require an empirical vibrational contribution to the total partition function to satisfy this limit. In general, lattice-cell models do not satisfy the ideal gas law, but perform well for liquid-like densities, such as polymer melt densities. All the equations of state in Table 1 have been used to correlate polymer melt densities to varying degrees of accuracy.[21,23-25] All the equations of state have been applied to real systems. When considering association and copolymer effects, Table 1 shows that the most common approach is to use group contribution techniques. Only three of the selected equations of state employ a rigorous treatment of association and copolymers. These equations of state are derived from the lattice-fluid theory (LF), the statistical associating fluid theory (SAFT), and the perturbed hard-sphere chain

P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (I 996) I- 10

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LATTICE-CELL CONCEPT: CELL VOLUME DECREASES

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Figure 1. Schematic diagram that illustrates how lattice and continuum concepts treat compressibility effects on a binary polymer-solvent system.

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P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (1996) I - I 0

Table 1 Versatility of selected equations of state £ model

type

ideal

polymer

gas limit

melt density

association

copolymer

applications to real systems

PHCT

I-c

12#

23

GC 27

GC 27

23,27-29

FOV

I-c

11#

24

GC 30

GC 30

24,30,31

LF

I-h

2

24

V2

RCT 4

2,4,24,32-35

G FD

c

15

25

SAFT

c

16

26

TPT116,17

TPT137

1,6,17,26,34,35,37-48

COR

c

22

22

GC 20

GC 20

20,22

PHSC

c

21

21

tBK 49

VT 50

5,51

25,36

£ PHCT: perturbed hard chain theory; FOV: Flory-OrwolI-Vrij model; LF: lattice-fluid theory; GFD: generalized Flory dimer theory; SAFT: statistical associating fluid theory; COR: chain-of-rotators theory; PHSC: perturbed hard-sphere chain theory; I-c: lattice-cell model; I-h: lattice-hole model; c: continuum model; GC: group contribution; V: Veytsman; TPTI: first-order thermodynamic perturbation theory; tBK: ten Brinke-Karasz; RCT: random copolymer theory; vr: variational theory. # ideal gas limit satisfied with an empirical correction.

theory (PHSC). In the remainder of this work, we illustrate the versatility of the SAFT equation of state with which our group has the most experience. The SAFT equation of state may be expressed in generic form as follows:[1618,37]

P = 1 + Z hs -i- Z chain -I- Z disp -i- Z ass°c pkT

(1)

where p, p, k, and T are the pressure, density, Boltzmann's constant, and absolute temperature, respectively. The contributions to the equation of state from different molecular forces (i.e. repulsion, dispersion, bonding, and association) may be expressed in the form of compressibility factors, Z. The second term on the righthand side of eq 1 is the contribution due to repulsive forces or the so-called excluded volume effect. The excluded volume contribution is derived from the Carnahan-Starling equation of state for a mixture of athermal hard sphere segments. [52] The third term on the right-hand side of eq 1 gives the effect of chain connectivity or bonding of the hard sphere reference system to form chain molecules. The chain term is derived from first-order thermodynamic perturbation theory (TPT1). [53] The original version of SAFT is restricted to homonuclear chain

P.D. Condo, M. Radosz / Fluid Phase Equilibria I 17 (1996) I - I 0

5

molecules (e.g. homopolymers).[16,17] Banaszak et al. recently extended SAFT to the rigorous treatment of heteronuclear chain molecules (e.g. copolymers).[37] Dispersion forces or so-called mean field interactions are included in the fourth term on the right-hand side of eq 1. [16,17,37] The contribution due to dispersion forces is determined from a molecular dynamics evaluation of a square-well fluid.[54] In addition to the chain term, TPT1 is also used to model the effect of association for such specific interactions as hydrogen bonding. [16,17,53] The association contribution to SAFT is given by the last term on the right-hand side of eq 1. We refer the reader to the literature cited in Table 1 for a more detailed description of SAFT and its application. In the low density limit, the compressibility factor terms on the right-hand side of eq 1 are zero, therefore, the SAFT equation of state satisfies the ideal gas limit. Figure 2 shows the correlation of polymer PVT behavior using the SAFT equation of state. It has been demonstrated that continuum models such as SAFT are superior to LF and comparable to the best lattice-cell models in correlating polymer PVT behavior.[21,25,26]



01,Pa

.

Figure

behavior of (PEP) correlated with the SAFT equation of state. [26]

M

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300

.

.

.

.

2.

PVT

poly(ethylene-alt-propylene)

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MW=96K

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400

.

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450

.

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In the following subsections, SAFT calculations are compared with experimental cloud point pressures (CPP) for model polyolefin-solvent systems. All the systems studied exhibit LCST, UCST, or U-LCST (lower, upper, upper-lower critical solution temperature, respectively) type of phase behavior. All these systems exhibit molecular dissimilarity and are highly compressible because, in most cases, the solvent is a supercritical fluid.

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P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (1996) 1-10

2.1

Compressibility Effects The effect of polymer molecular weight and solvent composition on the CPPs demonstrate the ability of SAFT to treat molecular dissimilarity and compressibility (equation of state) effects. Figure 3a shows the effect of polymer molecular weight on the CPP of poly(ethylene-alt-propylene) (PEP) in propylene. [38] An increase in polymer molecular weight results in an increase in the CPP. This effect can be quantitatively calculated using SAFT. SAFT also captures the transition from LCST to U-LCST type of phase behavior caused by the polymer molecular weight increase. The effect of solvent composition on the CPP of PEP solutions is shown in Figure 3b. [38] As the concentration of antisolvent (i.e. propylene) increases, the CPP increases. SAFT is successful in correlating the solvent composition effect in this ternary system. SAFT captures the LCST to U-LCST type of phase behavior transition (i.e. LL-L), as well as the VLL-LL and VL-L transitions. The influence of other effects on the CPP, such as, the effect of solvent and polymer polarity,[35,3943] and polymer polydispersity [45-48] have also be studied using SAFT.

500

I

I

I

I

I

I



I

I





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~



300

-

~ 200



LL-,L SAFT

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200

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600

-50

0 50 tO0 TEMPERATURE (oC)

(a)

150

200

e

250

-50

0

~

D

50 100 150 TEMPERATURE (°C)

tZ Eth

200

250

(b)

Figure 3. (a) cloud point pressures (CPP) for poly(ethylene-alt-propylene) (PEP) copolymers in propylene as a function of molecular weight. [38,55] (b) cloud point pressures (CPP) for poly(ethylene-alt-propylene) (PEP) of constant molecular weight (26 k) as a function of solvent composition (polymer free basis). [38,55] Solutions in Figures 3a and b contain about 15 wt % polymer. Copyright 1992 American Chemical Society.

P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (1996) I - I 0

7

2.2

Association Effects For associating systems, the CPP increases with increasing self-association of the polymer in a nonassociating solvent as shown in Figure 4a. [39] As temperature decreases, the self-association effect becomes more pronounced leading to larger differences in the CPP for the three solutions. The phase behavior shifts from LCST to U-LCST type of behavior as the polymer is changed from monoto dihydroxy polyisobutylene (PIB). The effect of self-association on the CPP and type of phase behavior (i.e. LCST, U-LCST) is captured by SAFT. In the case where the polymer can both self- and cross-associate, there is a competition between the two effects. Such is the case for the system of dihydroxy PIB in chlorodifluoromethane shown in Figure 4b. [39] In this case, the robustness of SAFT is demonstrated by its ability to simultaneously capture effects due to molecular dissimilarity, compressibility, and both self- and cross-association. 600

400

CHCIF2

C3"s +

~

SAFT

CHCIF2

I\

500

HO-PIB-OH ( l k )

300 400

No a u o ¢

AL. (0

Cross~

300

~200 (Z.

0.

2O0

Intra-associaUon

100

Cro$1~lls$ociatlon

100

0

-50

0

0

50

100

Temperature

(a)

(C)

150

200

0

50

100

150

Temperature

200

250

(C)

(b)

Figure 4. (a) cloud point pressures (CPP) for polyisobutylene (PIB), mono-, and dihydroxy PIBs in propane. [39,56] (b) cloud point pressures (CPP) for polyisobutylene (PIB) and dihydroxy PIB in chlorodifluoromethane. [39,56] Solutions in Figures 4a and b contain about 5 wt % polymer. Polymer molecular weight is 1 k. Copyright 1994 American Chemical Society.

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P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (1996) I - I 0

2,3 Copolymer Effects

Copolymer composition effects on the phase behavior of polymer-solvent systems are illustrated in Figure 5. [37] As the butene content of the poly(ethylenestat-butene) (PEB) copolymer increases the CPP decreases. In addition, the slope of the CPP curve changes sign as the copolymer composition is varied. This sign change of the CPP slope indicates a change from LCST (positive slope) to UCST (negative slope) type of phase behavior. U-LCST type of phase behavior is exhibited by PEB17. In this case, the minimum in the CPP (i.e. U-LCST) is achieved at temperatures above the depressed melting temperature of the copolymer. [57] The change in phase behavior shown in Figure 5 must be due to copolymer composition since the molecular weight of the copolymer is essentially constant and the solvent is unchanged. The rigorous treatment of copolymers in SAFT allows the effect of copolymer composition on the CPP to be modelled quantitatively. [37] 800 PEB0 ,...,600 I,_

CO0~

cu o

@

~ o

-,.0. o o

o

o

O

O

o

o

t_

= I 400 (0

~ --Q PF-B4

O

Figure 5. Cloud point pressures (CPP) for poly(ethylene-stat-butene) (PEB) copolymers in propane from 0 (PEB0) to 94 (PEB94) mole percent butene in the copolymer. [37]

PEB17

0 PEB35

~

PEB79

P G.

200

I~ 0

~ 50

5,wt °/701yTr in C'~ 100

150

200

250

Temperature (°C)

3. CONCLUSIONS Few equations of state rigorously treat molecular dissimilarity, association, and copolymers. These effects have been studied on real systems using the lattice-fluid theory (LF), the statistical associating fluid theory (SAFT), and the perturbed hard-sphere chain theory (PHSC). The versatility of SAFT is demonstrated by modelling the phase behavior of polyolefin systems containing strong effects of molecular dissimilarity, compressibility, association, and copolymer composition.

P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (1996) 1-10

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REFERENCES

1. B. Folie and M. Radosz, Ind. Eng. Chem. Res., 34 (1995) 1501. 2. C. Panayiotou and I.C. Sanchez, J. Phys. Chem., 95 (1991) 10090. 3. C.A. Haynes, F.J. Benitez, H.W. Blanch, and J.M. Prausnitz, AIChE J, 39 (1993) 1539. 4. P.P. Gan and D.R. Paul, Polymer, 35 (1994) 3513. 5. T. Hino, Y. Song, and J.M. Prausnitz, Macromolecules, 27 (1994) 5681. 6. P.D. Condo, M.A. Duran, and M. Radosz, M., manuscript in preparation. 7. C. Wohlfarth, Makromol. Chem., Theory Simul., 2 (1993) 605. 8. S.M. Lambert, Y. Song, and J.M. Prausnitz, unpublished manuscript. 9. P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953. 10. I. Prigogine, N. Trappeniers, and V. Mathot, Discuss. Faraday Soc.,15 (1953) 93. 11. P.J. Flory, R.A. Orwoll, and A. Vrij, JACS, 86 (1964) 3507. 12. S. Beret, S. and J.M. Prausnitz, AIChE J, 21 (1975)1123. 13. I.C. Sanchez and R.H. Lacombe, Macromolecules, 11 (1978) 1145. 14. K.G. Dickman and C.K. Hall, J. Chem. Phys. 85 (1986) 4108. 15. K.G. Honnell and C.K. Hall, J. Chem. Phys. 90 (1989) 1841. 16. S.H. Huang and M. Radosz, Ind. Eng. Chem. Res., 29 (1990) 2284. 17. S.H. Huang and M. Radosz, Ind. Eng. Chem. Res., 30 (1991) 1994. 18. S.H. Huang and M. Radosz, Ind. Eng. Chem. Res., 32 (1993) 762. 19. Y.C. Chiew, Molec. Phys., 70 (1990) 129. 20. U. Finck, C. Wohlfarth, and T. Heuer, Ber. Bunsenges. Phys. Chem., 96 (1992) 179. 21. Y. Song, S.M. Lambert, and J.M. Prausnitz, Ind. Eng. Chem. Res., 33 (1994) 1047. 22. R. Sy-Siong-Kiao, J.M. Caruthers, and K.C. Chao, Ind. Eng. Chem. Res., submitted 3/95. 23. D.D. Liu and J.M. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev., 19 (1980) 205. 24. P.A. Rodgers, J. Appl. Polym. Sci., 48 (1993) 1061. 25. G.R. Brannock and I.C. Sanchez, Macromolecules, 26 (1993) 4970. 26. S.-j. Chen, Y.C. Chiew, J.A. Gardecki, S. Nilsen, and M. Radosz, J. Polym. Sci.: Part B: Polym. Phys., 32 (1994) 1791. 27. M.D. Donohue and P. Vimalchand, Fluid Phase Eq., 40 (1988) 185. 28. M.D. Donohue and J.M. Prausnitz, AIChE J, 24 (1978) 849. 29. D.D. Liu and J.M. Prausnitz, Macromolecules, 24 (1979) 849. 30. G. Bogdanic and A. Fredenslund, Fluid Phase Eq., 40 (1988) 185. 31. G.T. Dee, J. Supercritical Fluids, 4 (1991) 152. 32. C.W. Haschetts and A.D. Shine, Macromolecules, 26 (1993) 5052. 33. B.M., Hasch, S.-H. Lee, M.A. McHugh, J.J. Watkins, and V.J. Krukonis, Polymer, 34 (1993) 2554.

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P.D. Condo, M. Radosz / Fluid Phase Equilibria 117 (I 996) 1-10

34. Y. Xiong and E. Kiran, J. Appl. Polym. Sci., 55 (1995) 1805. 35. S.J. Suresh, R.M. Enick, and E.J. Beckman, Macromolecules, 27 (1994) 348. 36. C.P. Bokis, M.D. Donohue, and C.K. Hall, Ind. Eng. Chem. Res., 33 (1994) 4866. 37. M. Banaszak, C.-k. Chen, and M. Radosz, manuscript in preparation. 38. S.-j. Chen, I.G. Economou, and M. Radosz, Macromolecules, 25 (1992) 4987. 39. C.J. Gregg, F.P. Stein, and M. Radosz, Macromolecules, 27 (1994) 4972. 40. C.J. Gregg, F.P. Stein, and M. Radosz, Macromolecules, 27 (1994) 4981. 41. C.J. Gregg, F.P. Stein, and M. Radosz, J. Phys. Chem., 98 (1994) 10634. 42. B. Folie, C.J. Gregg, G. Luft, and M. Radosz, Ind. Eng. Chem. Res., submitted 12/94. 43. S.-H. Lee, M.A. LoStracco, and M.A. McHugh, Macromolecules, 27 (1994) 4652. 44. M. Radosz, in Supercritical Fluids, E. Kiran and J.M.H. Levelt Sengers (eds.), Kluwer Academic Publishers, Netherlands, 1994. 45. M. Radosz, A Process for the Supercritical Mixed-Solvent Separation of Polymer Mixtures, EP 489 574 A2 10 (1993). 46. D. Pradhan, C.-k. Chen, and M. Radosz, Ind. Eng. Chem. Res., 33 (1994) 1984. 47. C.-k. Chen, M.A. Duran, and M. Radosz, Ind. Eng. Chem. Res., 32 (1993) 3123. 48. C.-k. Chen, M.A. Duran, and M. Radosz, Ind. Eng. Chem. Res 33 (1994) 306. 49. Y. Song, S.M. Lambert, and J.M. Prausnitz, Chem. Eng. Sci., 49 (1994) 2765. 50. Y. Song, S.M. Lambert, and J.M. Prausnitz, Macromolecules, 27 (1994) 441. 51. S.M. Lambert, Y. Song, and J.M. Prausnitz, Macromolecules, 28 (1994) 4866. 52. G.A. Mansoori, N.F. Carnahan, K.E. Starling, and T.W. Leland, J. Phys. Chem., 54 (1971) 1523. 53. W.G., Chapman, K.E. Gubbins, G. Jackson, and M. Radosz, Fluid Phase Eq., 52 (1989) 31. 54. S.S. Chen and A. Kreglewski Ber. Bunsen-Ges. Phys. Chem., 81 (1977) 1048. 55. reprinted with permission from Macromolecules, 25 (1992) 4987. Copyright 1992 American Chemical Society. 56. reprinted with permission from Macromolecules, 27 (1994) 4972. Copyright 1994 American Chemical Society. 57. P.D. Condo, E.J. Colman, and P. Ehrlich, Macromolecules, 25 (1994) 750.