UNDERSTANDING THE FLUID PHASE BEHAVIOUR

UNDERSTANDING THE FLUID PHASE BEHAVIOUR OF POLYMER SYSTEMS WITH THE SAFT THEORY

PATRICE PARICAUD

Ph.D. IMPERIAL COLLEGE OF LONDON 2003

UNDERSTANDING THE FLUID PHASE BEHAVIOUR OF POLYMER SYSTEMS WITH THE SAFT THEORY

Patrice Paricaud

January 24, 2003

Department of Chemical Engineering and Chemical Technology Imperial College of Science, Technology, and Medicine London SW7 3BY A thesis submitted for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of Imperial College

To Sandra and my family.

2

Abstract

Fluid phase equilibria in polymer systems are of crucial practical importance for polymer process design. In this work, a theoretical study is first undertaken for model molecules. The features governing the phase equilibria in purely repulsive systems of hard spheres and hard sphere chains are first examined, using a simple description based on the Wertheim TPT1 theory. For hard spheres of the same size as the segments of the chain, the system is fully miscible. When the hard spheres are made larger than the chain segments, the system demixes into low and high-density regions. This phase separation is driven by depletion interactions which corresponds to an effective attraction between the large spherical particles. The mechanism of phase separation in polymer solutions is completely different. For simple systems in which the size and energy parameters of the spherical molecules and the chain segments are all equivalent, a region of liquid-liquid coexistence is found when the chain length is increased, and the system exhibits a lower critical solution temperature (LCST). The nature of LCST behaviour is studied using the Wertheim TPT1 description and a mean-field treatment of the attractions. The effect of chain length polydispersity is also examined using continuous thermodynamics, and the results are compared with those for a discrete ternary mixture. Real systems of alkanes and polyethylene have been also examined, using the SAFT-VR theory. The n-alkanes and polymer molecules are modelled as chains of spherical segments. The segment-segment interactions are characterized by a squarewell potential. The phase equilibria of polyethylene solutions can be understood as an extreme case of a binary mixture of a short and a very long n-alkane. Appropriate comparisons are made with experiments in the discussion and excellent agreement is found for adsorption of gases in polyethylene, and cloud curves.

3

Acknowledgments

The person that I am most grateful to is probably Prof. George Jackson. I would like first to thank him for welcoming me, and helping me so much to set up in London. He has also brought me a lot knowledge in Statistical Mechanics and in many other topics. I thank him for his great generosity, enthusiasm and friendship. I would like also to thank very much Dr Amparo Galindo for her precious help, for all her encouragement and kindness. I thank both George and her for letting me participating to international conferences, and for all the delicious meals at ”La Bouchee” and ”de Mario”. I also thank Dr Szabolcs Varga very much, for his clever advices, efficiency at work, and especially for his friendship and the unforgettable holidays I spent with his family in Hungary. I am very grateful to my colleague Guy Gloor, for his permanent good mood, for having helped me a lot in computing, and above all for having saved my life by repairing my computer which was about to die just before the end of this PhD. I would like to thank Mario Franco and Birju Patel for all the good times spent in the office or around a glass, and my great Mexican flatmates: Alex, Javier, and Carlos, for their joy of life and the great Latin parties. I would like also thank the Modelling Research Group of BP Chemicals based in Lavera for funding a studentship. I would like to end with some French words addressed to my relatives. Je voudrais remercier de tout mon coeur mes parents a ` qui je dois tout, ma grand-mère, ma soeur Isabelle et son mari Jérome, Jean-Louis, et toute la famille Rakoto, pour avoir toujours été l` a, m’avoir soutenu et encouragé dans les moments difficiles. Enfin, je remercie Sandra mon épouse, pour son amour, sa gentillesse, son attention, et pour toute la joie de vivre qu’elle apporte a ` ma vie.

Contents 1 Introduction

25

1.1

Fluid Phase Equilibria in Pure Component Systems . . . . . . . . .

26

1.2

Fluid Phase Equilibria in Binary Mixtures . . . . . . . . . . . . . . .

28

1.2.1

Type I Phase Behaviour . . . . . . . . . . . . . . . . . . . . .

29

1.2.2

Type II Phase Behaviour . . . . . . . . . . . . . . . . . . . .

31

1.2.3

Type V Phase Behaviour . . . . . . . . . . . . . . . . . . . .

33

1.2.4

Type IV Phase Behaviour . . . . . . . . . . . . . . . . . . . .

34

1.2.5

Type III Phase Behaviour . . . . . . . . . . . . . . . . . . . .

36

1.2.6

Type VI Phase Behaviour . . . . . . . . . . . . . . . . . . . .

38

1.2.7

Gibbs Phase Rule and Properties of Mixing . . . . . . . . . .

40

Phase Behaviour in Polymer-Solvent Systems . . . . . . . . . . . . .

42

1.3.1

Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . .

46

1.3.2

Binodal and spinodal curves . . . . . . . . . . . . . . . . . . .

49

1.3.3

The Effect of Polydispersity . . . . . . . . . . . . . . . . . . .

49

Equations of State for Polymer Melts and Polymer Solutions . . . .

53

1.4.1

The Flory-Huggins-Staverman Theory . . . . . . . . . . . . .

54

1.4.2

Compressible Lattice Models . . . . . . . . . . . . . . . . . .

56

1.4.3

Continuous Systems: Tangent Spheres Models

. . . . . . . .

59

1.5

Chain Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

1.6

Remaining Questions and Theme of the Thesis . . . . . . . . . . . .

68

1.3

1.4

2 LCST in Polymer Solutions 2.1

71

Wertheim Association Theory . . . . . . . . . . . . . . . . . . . . . .

72

2.1.1

TPT1 Theory for Chain Formation and Polymers . . . . . . .

76

2.1.2

Chain Contribution Derived from the Cavity Function . . . .

82

2.2

SAFT-HS Theory for Polymer Solution . . . . . . . . . . . . . . . .

86

2.3

Phase Diagrams in Attractive Hard Spheres and Chain Binary Mixtures 91 4

5

CONTENTS 2.4

Properties of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

2.4.1

Ideal Gas Mixture . . . . . . . . . . . . . . . . . . . . . . . .

99

2.4.2

Polymer Solution Modelled with Flory-Huggins Theory . . . 100

2.4.3

Polymer Solution Described by TPT1 Theory . . . . . . . . . 102

2.5

Density Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

2.6

Associating Hard Spheres . . . . . . . . . . . . . . . . . . . . . . . . 112 2.6.1

Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 112

2.6.2

Properties of Mixing . . . . . . . . . . . . . . . . . . . . . . . 113

2.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.8

Appendix A: Algorithm to Solve LLE in Polymer Solution . . . . . . 119

2.9

Appendix B: Mixture of Associating Hard Spheres . . . . . . . . . . 123

3 Demixing in Colloids + Polymer Systems

126

3.1

The Depletion effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

3.2

Wertheim TPT1 Approach for Colloid-Polymer Systems . . . . . . . 134

3.3

Spinodal and Binodal Curves . . . . . . . . . . . . . . . . . . . . . . 138

3.4

Properties of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

3.5

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4 The Effect of Polydispersity In Polymer Systems 4.1

4.2

4.3

4.4

151

Polymer-Solvent System: Discrete Distribution . . . . . . . . . . . . 156 4.1.1

Density Free Energy . . . . . . . . . . . . . . . . . . . . . . . 159

4.1.2

Chemical Potentials . . . . . . . . . . . . . . . . . . . . . . . 163

4.1.3

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4.1.4

Cloud and Shadow Curve Calculation . . . . . . . . . . . . . 165

Polymer-Solvent Systems: Continuous Distributions . . . . . . . . . 168 4.2.1

Density Free Energy . . . . . . . . . . . . . . . . . . . . . . . 169

4.2.2

Chemical Potential and Pressure . . . . . . . . . . . . . . . . 171

4.2.3

Cloud and Shadow Curve Calculation . . . . . . . . . . . . . 172

Moment Method Applied to Polymer-Solvent Systems . . . . . . . . 174 4.3.1

Moment Free Energy . . . . . . . . . . . . . . . . . . . . . . . 174

4.3.2

Moment Chemical Potentials . . . . . . . . . . . . . . . . . . 178

4.3.3

Cloud and Shadow Curve . . . . . . . . . . . . . . . . . . . . 179

Ternary Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.4.1

Cloud and shadow curves . . . . . . . . . . . . . . . . . . . . 182

CONTENTS

6

4.4.2

Ternary Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 185

4.4.3

Cloud and Shadow Curves Obtained with Schulz-Flory Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

4.5

Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

5 Polyethylene + Hydrocarbon Systems 5.1

5.2

5.3

5.4

5.5

5.6

198

SAFT-VR Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.1.1

Molecular Model . . . . . . . . . . . . . . . . . . . . . . . . . 203

5.1.2

Calculation of the Helmholtz Free Energy . . . . . . . . . . . 205

5.1.3

Combining Rules . . . . . . . . . . . . . . . . . . . . . . . . . 210

Modelling of Pure Components . . . . . . . . . . . . . . . . . . . . . 211 5.2.1

n-Alkanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

5.2.2

α-Olefins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.2.3

Other Pure Compounds . . . . . . . . . . . . . . . . . . . . . 224

Modelling n-Alkane + linear Polyethylene Systems . . . . . . . . . . 227 5.3.1

Pentane + Polyethylene . . . . . . . . . . . . . . . . . . . . . 227

5.3.2

Influence of the Polymer Parameters on Cloud Curves . . . . 230

5.3.3

Other alkanes + Polyethylene . . . . . . . . . . . . . . . . . . 234

Solubility of Gases in Amorphous Polyethylene . . . . . . . . . . . . 235 5.4.1

Small Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.4.2

Solubility of Hydrogen in Polyethylene . . . . . . . . . . . . . 239

Crystallinity of Polyethylene . . . . . . . . . . . . . . . . . . . . . . . 243 5.5.1

Experimental Measurements of Crystallinity . . . . . . . . . . 244

5.5.2

Flory’s Theory of the Fusion Behaviour of Copolymers . . . . 246

5.5.3

Modelling of Polyethylene Crystallinity . . . . . . . . . . . . 252

Solubility of Gases in Semi-Crystalline Polyethylene . . . . . . . . . 257 5.6.1

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5.6.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

5.7

Effects of co-absorption . . . . . . . . . . . . . . . . . . . . . . . . . 261

5.8

Conclusion

6 Conclusion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 266

List of Figures 1.1

Schematic representation of a pressure-temperature P T diagram of a pure component. The continuous boundary represents the vapourpressure curve, the long-dashed boundary represents the melting curve, and the dotted boundary represents the sublimation curve. The circle denotes the vapour-liquid critical point and the square denotes the triple point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

27

The six main types of behaviour for binary mixtures shown as pressuretemperature P T projections of the phase diagram. The continuous curves represent the vapour pressure curves of pure components 1 and 2 and the three phase lines. The dashed curves represent critical lines. The triangles are used to represent upper and lower critical end points, and the circles the critical points of the two pure components

1.3

28

Pressure-temperature P T projection of the phase diagram of a type I binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components. The dashed curves represent critical lines. The circles denote the critical points of the pure components. 30

1.4

a) Pressure-composition P x and b) temperature-composition T x slices of the phase diagrams obtained for a type I binary mixture. The temperatures (T1 , T2 , T3 ) and pressures (P1 , P2 ) correspond to slices shown on figure 1.3. The continuous curves represent coexistence boundaries. The circles denote vapour-liquid critical points. The dashed-dotted line represents a tie line.

7

. . . . . . . . . . . . . . . .

30

LIST OF FIGURES 1.5

8

Pressure-temperature P T projections of the phase diagram characteristic of a system exhibiting type II phase behaviour. The continuous curves represent the vapor-pressure curve of the two pure components and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP. . . . . . . . . . . . . . . . . . .

1.6

32

a) Pressure-composition P x (at temperature T1 ) and b) temperaturecomposition T x (at pressure P1 ) slices of the phase diagrams obtained for a type II binary mixture. The regions of vapour-liquid and liquidliquid equilibria are labelled on the figure. The continuous curves represent the coexistence curves. The circles denote critical points (vapour-liquid and UCST) . . . . . . . . . . . . . . . . . . . . . . . .

1.7

32

Pressure-temperature P T projections of the phase diagram of a type V binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEP and LCEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8

33

Pressure-composition P x slices of the phase diagram at a) a temperature T1 , and b) a temperature T2 , corresponding to a type V binary mixture. The continuous curves represent the coexistence curves and three phase lines. The circle denotes a critical point (UCSP), the triangle denotes the UCEP. . . . . . . . . . . . . . . . . . . . . . . .

1.9

34

Pressure-temperature P T projection of the phase diagram of a type IV binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase lines. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEPs and LCEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.10 Pressure-composition P x slices of the phase diagram at constant temperatures a) T1 , and b) T2 of a type IV binary mixture. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points (vapour-liquid and UCSP). The triangle denotes a UCEP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

9

LIST OF FIGURES 1.11 Temperature-composition T x slice of the diagram at a constant pressure P1 of a type IV binary mixture. The continuous curves represent coexistence curves and the three-phase line. The circles denote critical points (vapour-liquid, LCST and UCST). . . . . . . . . . . . . . . .

36

1.12 Pressure-temperature P T projections of the phase diagram of type III. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP. . . . . . .

37

1.13 Temperature-composition T x slice of the phase diagram at a): a pressure P1 , and b) a pressure P2 , corresponding to a type III behaviour. The continuous curves represent the coexistence curves and the threephase line. The circles denote critical points (vapour-liquid, LCST and UCST). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

1.14 Pressure-temperature P T projections of the phase diagram of type VI binary mixtures. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote pure component critical points, and the triangles denote the LCEP and UCEP. 39 1.15 Temperature-composition T x slice of the phase diagrams (at a pressure P1 ) obtained for a type VI binary mixture. The continous curves represent the coexistence curves. The circles denote critical points (vapour-liquid, LCST and UCST). . . . . . . . . . . . . . . . . . . .

40

1.16 Molar Gibbs free energy of mixing for a binary mixture at given T and P , as a function of the mole fraction x of one of the two component. The points A and D represent the binodal points, and B and C the spinodal points. The dotted line corresponds to the common tangent, and the dotted-dashed lines denote the compositions of the two phases α and β in coexistence.

. . . . . . . . . . . . . . . . . . . . . . . . .

42

1.17 Schematic representation of different kinds of polyethylene. a) High Density Polyethylene (HDPE), b) Low Density Polyethylene(LDPE), c) Linear Low Density Polyethylene (LLDPE) . . . . . . . . . . . . .

45

1.18 Atomic representation of some polyolefines (from Lipson et al. [27]). Each black circle represents a carbon atom. . . . . . . . . . . . . . .

46

LIST OF FIGURES

10

1.19 Temperature-composition T x slices of the phase diagrams of a) a type IV binary mixture corresponding to figure 1.9, and b) a type IV monodisperse polymer solution. The symbols X and W refer to mole and weight fractions. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points. The dashed-dotted lines represent the limits of the T x diagram. . . . . .

47

1.20 Pressure-composition P w slice of the phase diagram of a polymer solution in the case of monodisperse polymer. The white circle denotes the critical point (UCSP). The black circles represent the binodal points 50 1.21 Pressure-composition P w phase diagram of a polydisperse polymer solution, where the global composition of the sample W0 is higher than the critical composition. The white circle denotes the critical point. The black circles represent the coexistence points. The thick continuous curve represents the cloud curve. The dashed line is the shadow curve. The thin continuous curves represent the coexistence curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

1.22 Pressure-composition P w phase diagram of a polydisperse polymer solution, where the global composition of the sample W0 is lower than the critical composition. The white circle denotes the critical point. The black circles represent the coexistence points. The thick continuous curve represents the cloud curve. The dashed line is the shadow curve. The thin continuous curves represent the coexistence curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

1.23 Lattice model used in the Flory-Huggins theory to represent a mixture of chain molecules of 10 segments and a monomeric solvent. The white circles denote solvent units. The black circles denote polymer units. 1.24 Polymer molecule and the end-to-end distance RE . . . . . . . . . . 2.1

55 66

Top: Schema of an associating sphere with one site. Bottom: mixture of associating spheres and formation of chains . . . . . . . . . . . . .

80

LIST OF FIGURES 2.2

11

P T pressure-temperature diagrams for binary mixtures of hard spheres + chains of m2 tangent hard spheres of same diameter, and same mean field energy per segment, with a) m2 = 2, m2 = 6, b) m2 = 7, and c) m2 = 100. The continuous lines are the vapour-pressure curves of the pure components and the three phase line. The dashed lines are critical lines. The white circles are the critical points of the pure components. The white triangles are LCEP and UCEP. . . . . . . .

2.3

92

Global phase diagram for the binary mixture of two hard sphere chains of respectively m1 and m2 segments. The solid line delimits regions of the diagram where different types of phase behaviour are encountered. It represents all the type V binary mixtures for which the critical point of the pure shorter chain, LCEP and UCEP are confused. . . . . . .

2.4

93

P T pressure-temperature diagram for a binary mixture of hard spheres + chains of 10 tangent hard spheres of same diameter and mean field energy per segment (system (4)). The continuous lines are vapourpressure curves of the pure components. The dashed lines are critical lines. The white circles are critical points of the pure components. The white triangles are LCEP and UCEP joined by the 3-phase line. The dash-dot lines denote constant temperature or pressure slices. .

2.5

96

T x temperature-composition diagrams, corresponding to the P T diagram of figure 2 obtained for the binary mixture of hard spheres + chains of 10 tangent hard spheres with mean field attractions (see figure 2), where xs,2 is the fraction of polymer segments. The considered pressure slices are at P ∗ = a) P1∗ , b) P2∗ , and c) P3∗ . The continuous lines are coexistence curves. The white circles are critical points (UCST and LCST) of the binary mixture. . . . . . . . . . . .

2.6

97

Enlarged T x temperature-composition diagram corresponding to the T x diagram at pressure P2∗ shown in figure 3 b), obtained for the binary mixture of hard spheres + chains of 10 tangent hard spheres with mean field attractions. The continuous lines are coexistence curves. The white circles are critical points (UCST and LCST) of the binary mixture. The dashed line is the three phase line. The dashdot line denotes a constant temperature slice at T2∗ in the liquid-liquid immiscibility region. . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

12

LIST OF FIGURES 2.7

Reduced Gibbs free energy of mixing and its second derivative obtained for different model systems at constant pressure P2∗ and temperatures T1∗ (a), c)) and T2∗ (b), d)). The thin dash line corresponds to an ideal gas mixture (system(1)). The thick dash line corresponds to a mixture of one monomeric solvent and a polymer of 10 segments, and is obtained from Flory-Huggins theory (system (2)). The thin continuous line corresponds to a binary mixture of hard spheres and a chain of 10 tangent hard spheres modelled with SAFT-HS (system (3)). The thick continuous line corresponds to the same binary mixture as system (3) but with mean field segment-segment attractions (system (4)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.8

Reduced properties of mixing (a) internal energy, b) enthalpy, c) entropy, d) molar volume) obtained for a binary mixture of hard spheres and a chain of 10 tangent hard spheres with mean field attractions (system (4)), at constant pressure P2∗ and at temperature T1∗ in the stable region, and at temperature T2∗ in the liquid-liquid immiscibility region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.9

a) Reduced density obtained for system (4) at constant pressure P 2∗ and temperatures T1∗ and T2∗ . b) Isotherms obtained for the same system at temperature T2∗ in the demixing region, for different chain mole fractions x2 : the thick continuous line , and white triangles, correspond to x2 = 0 (pure solvent), the thin continuous line, and white diamonds, correspond to x2 = 0.01, and the dashed line and white squares correspond to x2 = 0.05. . . . . . . . . . . . . . . . . . 108

2.10 a) repulsive and b) attractive parts of the pressure, at temperature T 2∗ , corresponding to the isotherms shown in figure 7 b) for different chain mole fractions x2 : the thick continuous lines correspond to x2 = 0 (pure solvent), the thin continuous lines correspond to x2 = 0.01 and the dashed line correspond to x2 = 0.05.

. . . . . . . . . . . . . . . 109

2.11 Schematic representation the binary mixture of hard spheres and chains to represent the evolution of the density as a function of the composition, at fixed temperature and pressure. Grey spheres are solvent molecules and black spheres are chain segments. . . . . . . . 111

13

LIST OF FIGURES 2.12 T x temperature-composition diagram at pressure P4∗ for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0, and unlike site-site association. The continuous line is the liquid-liquid coexistence curve as a function of the mole fraction x2 of component 2, the dash line is the total mole fraction XT of bonded molecules in the coexistent phases with respect to temperature. The white circles are UCST and LCST.

. . . . . . . . . . . . . . . . . . 113

2.13 Reduced properties of mixing (a) internal energy, b) enthalpy, c) entropy, d) molar volume) obtained for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0 , and unlike site-site association, at constant pressure P2∗ and temperatures T3∗ , T4∗ and T5∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 2.14 Reduced density obtained for a binary mixture of equal-size hard spheres, with mean field attractions α11 = α22 , α12 = 0, and unlike site-site association, at constant pressure P2∗ and at temperatures T3∗ , T4∗ and T5∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.15 Evolution of the chemical potential of component 1 at fixed temperature T and pressure P , as a function of composition x2 in a demixing binary polymer-solvent system. The circles denote binodal points, and the square denotes a metastable point. . . . . . . . . . . . . . . 123 3.1

Representation of the depletion effect in colloid-polymer system. Colloidal particles are represented by grey spheres, and polymer segments are represented by black spheres.The centers of mass of the polymer coils of diameter 2Rg are excluded from a volume represented by a sphere of diameter 2 (Rg + RC ) (dotted-dashed line), for each colloidal particles of diameter σC = 2RC ( see figure a)). However, when the colloid particles are close to each other (see figure b)), the excluded volumes related to the colloids overlap, the total excluded volume is decreased, and the coils particles can then move in a larger volume. The overlapping regions are represented in black. In the SAFT (TPT1) theory, the polymer chain is modelled as a fully flexible chain of segments of diameter σP 0,

µ

∂P ∂v

¶

= 0, Tc

Ã

∂2P ∂v 2

!

= 0, Tc

Ã

∂3P ∂v 3

!

> 0, Tc

where P is the pressure, Tc the critical temperature, and v the molar volume.

(1.1)

CHAPTER 1. INTRODUCTION

1.2

28

Fluid Phase Equilibria in Binary Mixtures

Because of the great variety of phase behaviour exhibited in practice, it is convenient to classify the different types of phase diagrams in some way. Scott and van Konynenburg [3] have proposed a useful classification in of fluid phase equilibria into six types of behaviour, and showed that the van der Waals equation of state [4] is able to predict five of the six types of behaviour. A general discussion of the types of phase equilibria exhibited by mixtures be found in the following references: [5, 6]. A vast amount of experimental data for binary and ternary systems can, of course, be found in the literature; the reader is directed to the extensive reviews of Dohrn et al. [7, 8]. The phase diagrams presented in this section (from figure 1.2 to figure 1.15) do not represent actual experimental data or calculations, but just schematic representations used to exemplify the different types of behaviour. Similar figures corresponding to the phase behaviour of model systems and real mixtures will be presented in subsequent chapters. The six main types of fluid phase behaviour in binary mixtures are summarised as pressure-temperature P T diagrams in figure 1.2.

Figure 1.2: The six main types of behaviour for binary mixtures shown as pressuretemperature P T projections of the phase diagram. The continuous curves represent the vapour pressure curves of pure components 1 and 2 and the three phase lines. The dashed curves represent critical lines. The triangles are used to represent upper and lower critical end points, and the circles the critical points of the two pure components Note that azeotropic behaviour is not considered in this discussion. In order to explain the progression from one type of phase behaviour to another, the different


29

types of phase behaviour are discussed in the following order: type I, II, V, IV, III, and VI). In the thesis we use the term liquid to denote a dense fluid phase and the term gas to denote a low-density fluid; vapour is used to refer to a gas phase in coexistence with a liquid phase.

1.2.1

Type I Phase Behaviour

In the case of type I fluid phase behaviour, the gas-liquid critical line is seen to be continuous, and there is no liquid-liquid immiscibility. The two components tend to be similar in size, are of the same chemical type, and have similar critical properties in such systems. Some examples of mixtures which exhibit type I phase behaviour include argon + krypton, methane + ethane, methane + nitrogen, and carbon dioxide + oxygen. The components giving rise to type I behaviour in mixtures are usually non-polar. Mixtures of chemically similar substances of the same chemical series, such as the n-alkanes, deviate from type I phase behaviour when the size of the two compoenents become significantly different. In the case of binary mixtures of methane with n-alkanes, type I is observed for methane + ethane, + propane, + n-butane and + n-pentane. A change from type I to type V phase behaviour is first observed for methane + n-hexane [5, 9]. With ethane as the more volatile component, the transition from type I to Type V first occurs in the mixture with n-nonodecane. In the case of propane, the transition occurs in the range n-C 40 and n-C50 but the phase diagrams are complicated with a appearance of solid phases as the freezing point of the heavier component moves to higher temperatures. Type V fluid phase behavior is discussed in greater detail in section 1.2.3. Typical pressure-composition P x and temperature-composition T x slices of the fluid phase equilibria of a type I binary mixture (figure 1.3) are shown in figure 1.4. The continuous curves represent the vapour-liquid coexistence boundaries. In the case of the P x slice at the temperature T3 , component 1 is supercritical and the system exhibits a vapour-liquid critical point. The two components are supercritical at the pressure P2 shown on T x sections and the system exhibits two vapour-liquid critical points. The Gibbs phase rule (see section 1.2.7) requires that in a binary mixture the number of degree of freedom is 2 when two phases coexist. Therefore, vapour-liquid equilibrium is defined when both the temperature and the pressure are fixed.


30

Figure 1.3: Pressure-temperature P T projection of the phase diagram of a type I binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components. The dashed curves represent critical lines. The circles denote the critical points of the pure components.

Figure 1.4: a) Pressure-composition P x and b) temperature-composition T x slices of the phase diagrams obtained for a type I binary mixture. The temperatures (T1 , T2 , T3 ) and pressures (P1 , P2 ) correspond to slices shown on figure 1.3. The continuous curves represent coexistence boundaries. The circles denote vapourliquid critical points. The dashed-dotted line represents a tie line.


31

The compositions of the vapour and the liquid phase in equilibrium are represented by the dew point curve and the bubble point curve, respectively, and the coexistence points on the two curves are joined by the tie lines (figure 1.4 b)).

1.2.2

Type II Phase Behaviour

Type II fluid phase behaviour is differentiated from type I behaviour by an additional region of liquid-liquid immiscibility at low temperature; a three phase line extends from the upper critical end point (UCEP) (figure 1.5) to low temperatures and pressures. The pressure-composition P x diagram at low temperature (figure 1.6 a)), is seen to exhibit a three phase line together with separate regions of liquid-liquid and vapour-liquid equilibria. The three phase line disappears at temperatures and pressures above the UCEP. The region of liquid-liquid coexistence can be seen in the constant pressure temperature-composition T x slice (figure 1.6 b)). This region of immiscibility ends at a liquid-liquid critical point often referred to as the upper critical solution temperature (UCST). It is likely that many type I mixtures would conform to type II behaviour at sufficient low temperature, if the presence of a solid phase did not first intervene. Some examples of mixtures which exhibit type II phase behaviour are carbon dioxide + n-octane, + n-decane, + n-undecane, + 2-octanol, in which the UCSTs increases with applied pressure.


32

Figure 1.5: Pressure-temperature P T projections of the phase diagram characteristic of a system exhibiting type II phase behaviour. The continuous curves represent the vapor-pressure curve of the two pure components and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP.

Figure 1.6: a) Pressure-composition P x (at temperature T1 ) and b) temperaturecomposition T x (at pressure P1 ) slices of the phase diagrams obtained for a type II binary mixture. The regions of vapour-liquid and liquid-liquid equilibria are labelled on the figure. The continuous curves represent the coexistence curves. The circles denote critical points (vapour-liquid and UCST)


1.2.3

33

Type V Phase Behaviour

The pressure-temperature P T projection of a type V system is represented schematically in Figure 1.7. This type of behaviour is characterised by discontinuous gasliquid critical lines and a three-phase line, which is bounded by lower and a upper critical end points (LCEP and UCEP). Typical pressure-composition P x diagrams corresponding to type V phase behaviour are shown in figure 1.8. At a temperature T1 which is just at the LCEP, the diagram is identical to that of a type I system (figure 1.8 a),1.6 a)). For temperatures between the LCEP and UCEP (see figure 1.8 b)), a region of liquid-liquid immiscibility appears, ending at a critical point at high pressure. The critical points corresponding to that region of liquid-liquid immiscibility are called upper critical solution pressures (UCSPs) in the case of a constant temperature slice, and lower critical solution temperatures (LCSTs) in the case of a constant pressure slice. Type V phase behaviour is experimentally observed for the systems methane + n-alkanes from n-hexane (methane + n-hexane [10], methane + n-eicosane [11]). There are, however, reasons to believe that some of those systems are in reality characterised by a type IV phase behaviour, as we explain in the next section. The type V behaviour and the thermodynamics of LCSTs is discussed in details in the chapter 2, using a molecular theory.

Figure 1.7: Pressure-temperature P T projections of the phase diagram of a type V binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEP and LCEP.


34

Figure 1.8: Pressure-composition P x slices of the phase diagram at a) a temperature T1 , and b) a temperature T2 , corresponding to a type V binary mixture. The continuous curves represent the coexistence curves and three phase lines. The circle denotes a critical point (UCSP), the triangle denotes the UCEP.

1.2.4

Type IV Phase Behaviour

The difference between type V and type IV fluid phase behaviour (figure 1.9) is the additional presence of a liquid-liquid critical line at low temperature (comparable to that found in type II behaviour). The corresponding P x and T x diagrams are represented in figure 1.10. Two constant temperature pressure-composition P x slices are shown in figures 1.10 a) and b). The separate regions corresponding to vapour-liquid and liquid-liquid equilibrium seen at constant temperature T1 merge into a single region of fluid-fluid equilibrium at a temperature T2 , which is just at the second UCEP. Two regions of liquid-liquid immiscibility can be seen in the T x representation of figure 1.10 c): the liquid-liquid coexistence at low temperature terminates at an upper critical solution temperature (UCST). The liquid-liquid coexistence at higher temperature ends at a lower critical solution temperature (LCST). The critical point seen at higher temperature is the usual liquid-vapour critical point. A number of examples of binary mixtures which exhibit type IV are known. It is commonly thought that some type V systems should in fact be characterised as type IV; the additional low-temperature liquid-liquid critical line seen in type IV systems is thought to always exist and be masked by the appearance of solidification. The system carbon dioxide + n-tridecane [12], is an example of a system exhibiting type IV behaviour.


35

Figure 1.9: Pressure-temperature P T projection of the phase diagram of a type IV binary mixture. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase lines. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangles denote the UCEPs and LCEP.

Figure 1.10: Pressure-composition P x slices of the phase diagram at constant temperatures a) T1 , and b) T2 of a type IV binary mixture. The continuous curves represent coexistence curves and three-phase lines. The circles denote critical points (vapour-liquid and UCSP). The triangle denotes a UCEP.


36

Figure 1.11: Temperature-composition T x slice of the diagram at a constant pressure P1 of a type IV binary mixture. The continuous curves represent coexistence curves and the three-phase line. The circles denote critical points (vapour-liquid, LCST and UCST).

1.2.5

Type III Phase Behaviour

Type III behaviour can be thought of as a type IV behaviour with extreme liquidliquid immiscibility. In type III phase behaviour, the two liquid-liquid critical lines seen in type IV phase behaviour join. The UCEP at low temperature and the LCEP of type IV also merge so that only a UCEP remains. Typical pressure-temperature P T projections and temperature-composition T x slices of the phase diagram of a type III binary mixture are shown in the figures 1.12 and 1.13. Two regions of liquid-liquid immiscibility can be seen in T x slices at high pressure above the critical pressure of the two pure components (figure 1.13 b)). Three different critical points are seen in this slice: one LCST, one UCST and one usual vapour-liquid critical point. At a lower pressure which is still above the critical pressure of the two pure components (figure 1.13 a)), the two regions of liquid-liquid immiscibility merge, to form only one region of fluid-fluid immiscibility which ends at a vapour-liquid critical point; at lower temperatures the coexistence is essentially between two high density liquid phases, and as the temperature is increased (above the critical value of the more volatile component) the density of one of the phases decreased rapidly so that the phase behaviour essentially represents vapour-liquid equilibria. Type III binary mixtures are relatively common. An example of a system which exhibits type III behaviour is the mixture of carbon dioxide and tetradecane [13–15].


37

Figure 1.12: Pressure-temperature P T projections of the phase diagram of type III. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent the critical lines. The circles denote the critical points of the pure components, and the triangle denotes the UCEP. Type III behaviour occurs when the immiscibility of two components is sufficiently large. Mixtures of methane + long-alkanes should behave like mixtures of carbon dioxide + n-alkanes with a change from type IV to type III behaviour. The change of fluid phase behaviour from type II to type IV and the change type IV to type III is continuous. When the two pure components become more and more asymmetric in size, the two regions of liquid-liquid immiscibility become more extensive until the three-phase lines merge. Such a transition in behaviour has not been seen experimentally for the n-alkane series as the critical lines at low temperatures are hidden by the appearance of solid phases. [6]. The binary mixtures water + n-alkane are other examples of type III binary mixtures. Those systems have been recently discussed experimentally by de Loos et al. [16], and modelled with the SAFT theory approach by Galindo et al. [17, 18].


38

Figure 1.13: Temperature-composition T x slice of the phase diagram at a): a pressure P1 , and b) a pressure P2 , corresponding to a type III behaviour. The continuous curves represent the coexistence curves and the three-phase line. The circles denote critical points (vapour-liquid, LCST and UCST).

1.2.6

Type VI Phase Behaviour

Type VI behaviour is quite similar to type II, but in this case the liquid-liquid immiscibility disappears again at low temperatures characterised by a locus of LCSTs terminating on the three phase line at a LCEP (see figure 1.14 a). This re-entrant miscibility (also referred to as closed-loop liquid-liquid immiscibility) is the main characteristic of type VI phase behaviour in the temperature-composition representation (see figure 1.14 b)). This type of phase behaviour was first reported for the nicotine + water system in 1904 [19], and is also seen when aliphatic or aromatic alcohols, amines, ethers or ketones are mixed with water or alcohols [5]. Some common examples of type VI binary mixtures are water + 2-butanone [20], water + 2-butanol [20], or water +2-butoxyethanol [21, 22]. It is clear that a common feature of all of these systems is the possibility of hydrogen bonding interactions between the components. Type VI behaviour is only found in mixtures of chemically complex substances where one or both of the pure components exhibit self hydrogen bonding, and in the mixture, there is a strong hydrogen bonding between the two components. Water is usually one of the two components. The regions of liquid-liquid immiscibility occur at low temperatures, well removed from the regions corresponding to the vapour-liquid critical lines. At low temperatures (below the LCST) the strong directional attractive interactions (hydrogen bonding type) give rise to favorable heats of solution and a miscible liquid


39

state; as the temperature is raised above the LCST the hydrogen bonds break and the liquid becomes unstable leading to demixing. The thermodynamic contributions which give rise to LCSTs in mixtures of low molecular weight hydrogen-bonding molecules are very different to those leading to LCSTs in polymer-solvent systems. All the thermodynamics of the type VI behaviour is described is chapter 2 by using the SAFT approach, and a comparison between LCSTs in type V and VI systems is made. The effect of pressure on UCSTs and LCSTs in type V and VI systems has been studied systematically by Timmermans [23,24]. More recently, Franck and Schneider [25, 26] have made excellent reviews in this area. The phase behaviour in type VI binary mixture is often complicated by the presence of azeotropic lines which we will not discuss here.

Figure 1.14: Pressure-temperature P T projections of the phase diagram of type VI binary mixtures. The continuous curves represent the vapor-pressure curve of the two pure components, and the three-phase line. The dashed curves represent critical lines. The circles denote pure component critical points, and the triangles denote the LCEP and UCEP.


40

Figure 1.15: Temperature-composition T x slice of the phase diagrams (at a pressure P1 ) obtained for a type VI binary mixture. The continous curves represent the coexistence curves. The circles denote critical points (vapour-liquid, LCST and UCST).

1.2.7

Gibbs Phase Rule and Properties of Mixing

A system of n components with ϕ phases in equilibrium is fully described by 2+ϕ(n− 1) variables: the pressure P , the temperature T and the ϕ(n − 1) independent mole fractions xi,k of component i in all phases k = 1, .., ϕ. The equilibrium conditions correspond to the equality of the pressure, the temperature, and the n(ϕ−1) relations for the equality of the chemical potential µi of each component i in all phases 1, .., ϕ, as µαi = µβi = . . . = µϕ i , for i = 1, . . . , n.

(1.2)

The Gibbs phase rule expressed as the number of degrees of freedom F is

F

= 2 + ϕ(n − 1) − n(ϕ − 1)

F

= n + 2 − ϕ.

(1.3)

If we apply the Gibbs phase rule (1.3) to a binary mixture(n = 2), the number of degree of freedom F is 2 if two phases are in coexistence (ϕ = 2). As a result, the equilibrium point is defined by arbitrarily fixing two variables, such as the temperature and the pressure. The phase rule is particularly useful in understanding the phase equilibria of multicomponent systems.

41


The Gibbs energy of mixing is a useful thermodynamic variable to discuss the different contributions which cause phase separation. In a binary mixture of components 1 and 2, the molar Gibbs free energy of mixing ∆gm , at given temperature T and pressure P , is defined as:

∆gm = g − x1 g1 − x2 g2 ,

(1.4)

where g is the molar Gibbs free energy at (T , P ) of the mixture of composition (x 1 , x2 ), and g1 and g2 the molar Gibbs free energies of the pure components at (T , P ). The two necessary requirements to form an homogenous solution in a binary mixture are [5] ∆gm ≤ 0, Ã

∂ 2 ∆gm ∂x2

!

(1.5)

> 0.

(1.6)

T,P

If one of the conditions (1.5) and (1.6) is not satisfied, the liquid is not stable at (T , P , x), and the system demixes into two coexisting liquid phases. The Gibbs free energy of mixing is related to the other properties of mixing via the thermodynamic relation

∆gm = ∆hm − T ∆sm ,

(1.7)

where ∆hm and ∆sm are the molar enthalpy and entropy of mixing. According to the relations (1.5), (1.5) and (1.7), a positive enthalpy of mixing induces demixing, whereas a positive entropy of mixing induces mixing; phase separation is governed by a competition between enthalpic and entropic effects. The evolution of the Gibbs free energy of mixing ∆gm as a function of the composition, for a demixing binary system at fixed (T , P ), is shown schematically in figure 1.16. The coexistence point at (T, P ) between phases α and β is given by the common tangent of the curve ∆gm (x). The common tangent criteria comes from the equality of the chemical potentials of the components in both phases, and the Gibbs-Duhem condition. In figure 1.16, the points A and D are coexistence or so-called binodal points, and the corresponding coexistence compositions are xα and xβ . The points of inflection of the curve, B and C, represent the spinodal points which are determined with the following relation

42


Figure 1.16: Molar Gibbs free energy of mixing for a binary mixture at given T and P , as a function of the mole fraction x of one of the two component. The points A and D represent the binodal points, and B and C the spinodal points. The dotted line corresponds to the common tangent, and the dotted-dashed lines denote the compositions of the two phases α and β in coexistence.

Ã

∂ 2 ∆gm ∂x2

!

= 0.

(1.8)

T,P

The second derivative of ∆gm with respect to composition is negative between points B and C, corresponding to the region of material or diffusion instability. The regions A − B and C − D where the derivative is positive correspond to metastable regions.

1.3

Phase Behaviour in Polymer-Solvent Systems

Polymers are molecule of hight molecular weight, consisted of monomers linked to each other by covalent bounds. There are two kinds of polymers: - The homopolymers are consisted of the repetition of one monomer. The degree of polymerisation x represents the number of monomer units on the macromolecule. - The copolymers are consisted of the repetition of several monomers. Copolymers prepared from bifunctional monomers can be subdivided further into four main categories:

• Statistical copolymers where the distribition of the tow monomers in the chain is essentially random, but influenced by the individual monomer reactivities.

43

CHAPTER 1. INTRODUCTION —ABBAAABBABABA—

• Alternating copolymers with a regular placement along the chain.

—ABABABABABAB—

• Block copolymers consisted of substantial sequences or blocks of monomers.

—AAAAAAABBBBBBBBAAAAAAA—

• Graft copolymers in which blocks of one monomer are grafted on the backbone of the other as branches:

B

B

B

B

B

B

—AAAAAAAAAAAAA— B

B

B

B

B

B

The structure of polymers depends on the functionality of the monomers. There are three different structures : - linear polymers ( for instance the high density polyethylene HDPE) - branched polymers ( for instance the low density polyethylene LDPE) - resins which are consisted of 3 dimension networks. The first are obtained from bi-functional monomers, the others from monomers of functionality > 2. The linear and branched polymers are soluble in organic solvents, thermoplastic, and can be melted. The resins are insoluble and can not be melted.


44

Polyethylene (PE) is the most popular plastic in the world. This is the polymer that makes grocery bags, shampoo bottles, pipes, children’s toys. It has a very simple structure, the simplest of all commercial polymers. A molecule of linear polyethylene is nothing more than a long chain of carbon atoms, with two hydrogen atoms attached to each carbon atom, and can be considered as a very long linear alkane. There are several kinds of polyethylene (figure 1.17). Their properties differ resulting from variations in structure. • High density polyethylene (HDPE) has linear chains with a melting point Tm in the range 130 ˚C < Tm RES , can be defined as hµiRES k = kT

Ã

∂f RES /kT ∂hρm ik

!

.

(4.54)

ρ1 ,hρm ij6=k

From the definition of the density moments (4.44) and using the properties of functionals,

µ

∂hρm ik ∂ρm (m)

¶

= mk .

(4.55)

ρ1

Combining equations (4.53) and (4.55), the residual chemical potential µ RES of a i chain molecule i 6= 1 can be expressed as a

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS172

´ µRES (m) 1 ³ RES . hµi0 + mi hµiRES = 1 kT kT

(4.56)

The residual chemical potential is a linear function of the chain length m as in the discrete case (cf., equation (4.33)). The pressure is given by

P

= ρkT − f RES + ρ1 µ1 +

Z

dmρm (m)µRES (m)

= ρkT − f RES + ρ1 µ1 +

Z

dmρm (m) hµiRES + mhµiRES 0 1

(4.57)

h

i

= ρkT − f RES + ρ1 µ1 + hρm i0 · hµiRES + hρm i1 · hµiRES . 0 1

4.2.3

Cloud and Shadow Curve Calculation

As for a discrete distribution, the phase equilibria for a continuous system is solved by imposing the equality of the pressure of both phases to a fixed pressure Pf ixed (cf. equations (4.37) and (4.38)), and the equality of chemical potentials in the two phases. The following equation must be satisfied for all values of m belonging to (0)

the interval of definition of the parent density distribution ρm equal to the density (Cl)

distribution ρm

of the cloud phase Cl µ(Cl) (m) = µ(Sh) (m),

(4.58)

where µ(Cl) (m) and µ(Sh) (m) are the chemical potentials of the chain molecule m in the cloud phase Cl and shadow phase Sh. Inserting equations (4.52) and (4.56) into (4.58), and rearranging the expression as in the discrete case (cf. equation (4.40), (Sh)

the density distribution ρm

of the shadow phase can be expressed as a function of (Cl)

the density distribution of the cloud phase ρm

ρ(Sh) m (m)

=

RES(Cl)

where (hµi0

³

RES(Cl)

hµi0 (Cl) (m)e ρm

RES(Cl)

, hµi1

RES(Sh)

−hµi0

´

/kT

e

RES(Sh)

), and ( hµi0

(equal to the parent distribution)

³

RES(Cl)

hµi1

RES(Sh)

−hµi1

´

m/kT

, (4.59)

, hµiRES 1,Sh ) are the (zero, first) resid-

ual moment chemical potentials in phases Cl and Sh defined in equation (4.54). The

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS173 (Sh)

density distribution ρm

of the shadow phase has to satisfy the two following con-

straints (4.60), (4.61) corresponding to the definitions of the zero and first moments (Sh)

hρm i0

(Sh) hρm i0

(Sh)

and hρm i0

=

Z

=

Z

= e

(Sh)

hρm i1

³

=

Z

=

Z

= e

³

:

dmρ(Sh) m (m)

dm

Ã

(4.60)

(Cl) ρm (m)e

RES(Cl)

hµi0

³

RES(Cl)

hµi0

RES(Sh)

−hµi0

´

/kT

RES(Sh)

−hµi0

Z

dm

Ã

´

/kT

e

³

RES(Cl)

hµi1

³

RES(Cl)

hµi1 (Cl) ρm (m)e

RES(Sh)

−hµi1

´

RES(Sh)

−hµi1

m/kT

´

m/kT

dmρ(Sh) m (m)m

dm

Ã

³

RES(Sh)

−hµi0

!

(4.61) RES(Cl)

hµi0 (Cl) ρm (mi )mi e

RES(Cl)

hµi0

!

´

/kT

Z

RES(Sh)

−hµi0

´

Ã

/kT

e

(Cl) dm ρm (m)me

³

³

RES(Cl)

hµi1

RES(Cl)

hµi1

RES(Sh)

−hµi1

RES(Sh)

−hµi1

´

´

m/kT

m/kT

As in the discrete case (see section 4.1.4), solving the cloud-point calculation at a fixed pressure Pf ixed consists in solving a non-linear system of five equations and five unknown variables. The equations are the equality of the pressure of both phases to the fixed pressure Pf ixed (equations (4.37) and (4.38)), the equality of the chemical potential of the solvent in both phases, and the two equations (4.60) (Sh)

and (4.61). The variables are the mole fraction xpoly of polymer molecules and the number average chain length hmi(Sh) in the shadow phase, the densities ρ(Cl) n (Sh)

and ρ(Sh) of the two phases, and the saturated temperature T . The variables xpoly , hmi(Sh) , ρ(Cl) and ρ(Sh ) are directly related to the density moments as in the discrete n case (cf. equation (4.10). Since the cloud and shadow curve calculation consists in determining only the density moments of the density distribution for each phase in coexistence, one should try to express the pressure and the free energy only in terms

!

!

.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS174 of those moments. Warren [321], and Sollich et al. [320] have precisely derived such expressions and developed the so-called “moment method”.

4.3

Moment Method Applied to Polymer-Solvent Systems

The moment method represents a powerful alternative tool to calculate phase equilibria in polydisperse systems. The method was developed independently by Sollich et al. [320] and by Warren [321] from two different approaches to end up with the same conclusions. A detailed description of the method is given in reference [322]. The method can only be applied with continuous density distributions. The main feature of the method is to express the ideal part of the free energy f IDEAL as a function of a finite number M of density moments hρm ik defined in equation (4.44). In this way, the total free energy f would be a function of only the M density moments. hρm ik . By considering one pseudo-component k corresponding to each moment density hρm ik appearing in the expression of the total free energy, the phase equilibria can be solved with the usual tangent-plane analysis by imposing the equality of each “moment chemical potential” hµik of pseudo-component k in both phases. The“moment chemical potential” are defined as the derivatives of the density free energy f with respect to the density moment hρm ik (see equation (1.31)). The moment method is very useful to get analytical expressions for the determination of multiple critical points and stability criteria. It gives exact results for cloud and shadow curves as we show later in this section, but only an approximate result in the case of multi-phase flash calculation. However in that case, the moment method becomes more and more accurate as the number of density moments used in the expression of the free energy f is increased (see [322] for further details).

4.3.1

Moment Free Energy

The moment method is built as follows: in the case of our polymer-solvent system described in section 4.2, the ideal density free energy f IDEAL is given by equation (4.47). One can add to the ideal term f IDEAL a term −kT

R

dmρm (m) ln R(m)

where R(m) is a continuous function of m independent of the density distribution ρm . This additional term linear in density does not affect the phase equilibria since its derivative with respect to ρm


R

∂ dmρm (m) ln R(m) kT = kT ln R(m), ∂ρm (m)

(4.62)

does not depend on ρm . We will explain the role of this term later in the discussion. The ideal free energy becomes

f IDEAL = ρ1 [ln ρ1 − 1] + kT

Z

ρm (m) dmρm (m) ln −1 . R(m) ·

¸

(4.63)

As was shown earlier, the total density free energy f is a functional of the density distribution ρm . If the system is “truncable”, the residual free energy f res can be expressed in terms of a finite number of moments (zero and first moment in the case of a polymer solution within a SAFT description). The ideal term f IDEAL depends on the same density moments, but also on an infinite number of higher moments because of the logarithm term in the expression of f IDEAL . At fixed temperature T , volume V , and composition, the total density free energy of the system has to be minimised in order to perform phase equilibria or salability analysis. Therefore, if f is expressed in terms of a finite number M of density moments hρm ik , f has to be a minimum for fixed values of the M moments. The ensemble of moments hρm ik includes at least the moments appearing in the expression of the residual free energy (hρm i0 , and hρm i1 in our case). The problem consists in finding the density distribution function ρm which minimises the total free energy f , with the M constraints given by the definition of the M fixed density moments (equation (4.44)). The minimisation criteria is obtained by using M Lagrange multipliers λ k , as

µ

∂f /kT ∂ρm (m)

¶

ρ1 ,hρm ik

∂ + ∂ρm (m)

Ã

X k

µ

λk hρm ik −

Z

dmρm (m)m

k

¶!

= 0. ρ1 ,hρm ik

(4.64) Applying the rules for functionals, the variation of the ideal free energy is,

Ã

∂f IDEAL /kT ∂ρm (m)

!

ρ1 ,hρm ik

ρm (m) = ln . R(m) µ

¶

(4.65)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS176 Since the residual free energy f res only depends on a finite number of density moments included in the M fixed moments,

Ã

∂f RES /kT ∂ρm (m)

!

= 0.

(4.66)

ρ1 ,hρm ik

Inserting (4.65), (4.66), and (4.55) into (4.64), the minimisation criteria becomes

ln

µ

ρm (m) R(m)

¶

−

X

λk mk = 0.

(4.67)

k

Therefore, the density distribution which minimises the density free energy for fixed values of the density moments hρm ik has the form ρm (m) = R(m)e

P

k

λ k mk

,

(4.68)

and the corresponding density moments hρm ik are given by hρm ii =

Z

dm mi R(m)e

P

k

λ k mk

.

(4.69)

Equation (4.68) represents a family of density distributions, generated by R(m) and (Cl)

by the Lagrange multipliers λk . The exact density distributions ρm

(Sh)

and ρm

(cf.

equation (4.59)) of the cloud and shadow curves must be included in this family. On inserting equations (4.68), (4.69), and (4.44) into equation (4.63), the ideal term of the free energy can be written as

f IDEAL IDEAL = ρ1 [ln ρ1 − 1] − Smom , kT

(4.70)

IDEAL is given by where the “ideal moment entropy” Smom

IDEAL Smom = hρm i0 −

X i

λi hρm ii

(4.71)

The total “moment” free energy fmom is equal to X f RES fmom = ρ1 [ln ρ1 − 1] + . λi hρm ii − hρm i0 + kT kT i

(4.72)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS177 The free energy fmom depends only on the solvent density ρ1 and the moments, as the Lagrange multipliers implicitly depend on the moments by inverting equation (4.69); fmom is also called “projected moment free energy” [320,322] as it corresponds mathematically to a projection of the free energy of the system on a hypersurface of density defined by the fixed density moments hρm ik . The Lagrange multipliers λk are related to the density moments hρm ik by equation (4.69). The zero density moment hρm i0 is a function of the λk for fixed R(m), and can be considered as a differential of the λk as

dhρm i0 =

X µ ∂hρm i ¶ 0

∂λi

i

dλi

(4.73)

λj6=i

From equation (4.69),

µ

∂hρm i0 ∂λi

¶

= λj6=i

Ã R

∂ dm R(m)e ∂λi

P

k

λ k mk

!

λj6=i

= hρm ii .

(4.74)

Equation (4.73) then becomes

dhρm i0 =

X i

hρm ii dλi .

(4.75)

IDEAL can be seen as a On inspecting equation (4.71) the ideal moment entropy Smom

Lagrange transform of the zero density moment hρm i0 and

IDEAL dSmom = dhρm i0 − d

=

X i

= −

Ã

X i

hρm ii dλi −

X i

hρm idλi

X i

!

hρm ii dλi −

(4.76) X i

λi dhρm ii

λi dhρm ii .

As a result,

Ã

IDEAL ∂Smom ∂hρm ik

!

hρm ij6=k

= −λk .

(4.77)


4.3.2

Moment Chemical Potentials

As we saw in the previous sections 4.1.2 and 4.2.2, one can define the “moment chemical potentials” hµii as the derivatives of the moment density free energy fmom with respect to the density moments

hµik = kT

Ã

∂f /kT ∂hρm ik

!

,

(4.78)

ρ1 ,hρm ij6=k

where the residual moment chemical potentials hµiRES are defined in (4.54). The k ideal chemical potentials can be obtained from (4.78), (4.70) and (4.77) as

hµikIDEAL = kT

Ã

∂f IDEAL /kT ∂hρm ik

!

= λk .

(4.79)

ρ1 ,hρm ij6=k

The residual moment chemical potentials hµik are given by (4.54). The pressure P obtained from the moment free energy is given by

P

= −fmom + ρ1 µ1 +

X k

hµik hρm ik

= −kT ρ1 [ln ρ1 − 1] − kT + kT ρ1 ln ρ1 +

X k

X k

(4.80)

λk hρm ik + kT hρm i0 − f RES

³

´

hρm ik kT λk + hµiRES . k

After simplification, and since ρ1 +hρm i0 = ρ, the expression of the pressure becomes P = ρkT − f RES + ρ1 µ1 +

X k

hρm ik hµiRES . k

(4.81)

Expressions (4.72), (4.79), and (4.81) are general for any “truncable” systems. In the case of our continuous polymer-solvent system described in section 4.2, only the zero and first density moments and chemical potentials have to be considered in the sum of equation (4.81). Expression (4.81) then becomes . P = ρkT − f RES + ρ1 µ1 + hρm i0 · hµiRES + hρm i1 · hµiRES 0 1

(4.82)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS179 which is equivalent to the expression (4.57). Hence the pressure derived from the moment free energy and moment chemical potentials gives the exact pressure of the system for fixed temperature, density, and composition (see [322] for further details).

4.3.3

Cloud and Shadow Curve (Cl)

(Sh)

The density distributions ρm

and ρm

of phases Cl and Sh corresponding re-

spectively to the cloud and shadow points, must belong to the family of density distributions generated by (4.68). The mole fraction distribution function X (Cl) of the cloud phase is known and equal to the parent distribution X (0) as seen in section (Cl)

4.1.4, and is related to ρm

(Cl)

by ρm

= ρ(Cl) X (Cl) , where ρ(Cl) is the unknown den-

sity of phase Cl. One can choose any function for the distribution R(m) common for all density distributions of the family defined by (4.68) without changing the phase equilibria, as mentioned in section 4.3.1. To assure that the parent density distri(Cl)(m)

bution belongs to the family (4.68), one should choose R(m) = ρm

common for

all density distributions of the family (4.68) (see [322]). The density distributions (Sh)

(Cl)

and ρm

(Cl)

and λk

ρm λk

(Sh)

are then defined by the corresponding vectors of Lagrange multipliers and

(Cl) (Cl) e = ρm ρm

P

(Cl) ρ(Sh) = ρm e m

P

(Cl)

Equation (4.83) is satisfied if and only if λk (Sh)

the vector λk

(Cl)

k

λk

k

λk

mk

(Sh)

mk

(4.83) .

(4.84)

= 0 for all k. One must only determine

to solve the phase equilibria. The moment free energy at fixed

pressure has to be minimised at equilibrium and it can be shown [322] that the equilibrium conditions are the equality of each moment chemical potential in both phases

(Cl)

hµik (Cl)

kT λk

RES(Cl)

+ hµik

(Sh)

(4.85)

= hµik

(Sh)

= kT λk

RES(Sh)

+ hµik

.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS180 (Cl)

Since λk

= 0, equation (4.85) becomes (Sh)

λk

³

RES(Cl)

= hµik

RES(Sh)

− hµik

´

/kT.

(4.86)

By substituting (4.86) into (4.84), the distribution of the shadow phase at equilibrium is obtained as

(Cl) e = ρm ρ(Sh) m

P ³ k

RES(Cl)

hµik

RES(Sh)

−hµik

´

mk /kT

.

(4.87)

In the case of our polymer solution, equation (4.87) is written only in terms of the zero and first density as

(Cl) ρ(Sh) m (m) = ρm (m)e

³

RES(Cl)

hµi0

RES(Sh)

−hµi0

´

/kT

e

³

RES(Cl)

hµi1

RES(Sh)

−hµi1

´

m/kT

, (4.88)

which is identical to expression (4.59). The two equilibrium conditions (4.60) and (4.61) of the exact solution are satisfied in the moment method via the relations given by equation (4.69). The moment method consists then in expressing the (Sh)

Lagrange multipliers λk

(Cl)

in terms of the density moments hρm ik

(Sh)

and hρm ik

via the relations (4.69), and solving the non-linear system of five equations (equality of the pressure of both phases to the fixed pressure Pf ixed , equality of the chemical potential of the solvent in both phase, and relations (4.85) for the zero and first moment chemical potential). The variables of this system are the same as in the exact case (see section 4.1.4). The moment method is mathematically equivalent to the exact solution of section 4.1.4, and it does not bring any improvement in terms of numerical solving. However, its main advantage is to bring a simplification in the analytical expressions, notably in the determination of stability criteria and critical points. The reader can find in reference [322] a detailed description of the matrices and determinants for stability analysis in polydisperse systems.

4.4

Ternary Systems

Following Koningsveld’s work [302,303], we consider a polymer-solvent system where the solvent (component 1) is represented by a hard sphere (m1 = 1) of diameter σ and the “polydisperse” polymer is a mixture of two chain molecules (components 2

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS181 and 3) composed of m2 and m3 spherical segments of same diameter σ. This is the simplest polydisperse system with a discrete distribution and we can study the effect of polydispersity on the liquid-liquid immiscibility, and explain the shapes of cloud and shadow curves by analysing the corresponding ternary diagrams at constant temperature T and pressure P . The polymer is assumed to have a polydispersity index Ip = 2, and a number average chain length hmin = 100. The chain lengths m2 and m3 , and the parent mole fraction distribution X (0) are related with Ip and hmin via equations (4.5) and (4.7). The chain lengths m2 and m3 must satisfy

X (0) (m2 )m2 + X (0) (m3 )m3 = hmin = 100

(4.89)

X (0) (m2 )m2 2 + X (0) (m3 )m3 2 = Ip hmin = 200.

We consider two different “polydisperse” polymers (1) and (2) composed of the two chain molecules 2 and 3, and satisfying the relations (4.89). The corresponding parent distribution functions X (mole fractions), W (weight fractions) and chain lengths m2 , m3 are given in table 1. The discrete weight fraction distributions of the two polymers are shown in figure 4.2. In the case of polymer (1) there is more of the longer chains, while for polymer (2) the shorter chains predominate. A more asymmetrical distribution has been chosen for polymer (2). Table 3.1: Chain lengths, and mole and weight fraction parent distribution functions for polymer (1) and (2), such that hmi(0) n = 100 and Ip = 2.

m2 m3 X (0) (m2 ) X (0) (m3 ) W (0) (m2 ) W (0) (m3 )

polymer (1)

polymer (2)

66.7 400 0.9 0.1 0.6 0.4

88.8 1000 0.988 0.012 0.878 0.122


Figure 4.2: Discrete parent chain length distribution functions in terms of weight fraction, for polymer (1) (black bars), and polymer (2) (white bars).

4.4.1

Cloud and shadow curves

We first focus on the polymer (1) + solvent system and analyse the effect of polydispersity on the liquid-liquid immiscibility with an LCST. In chapter 2, the reader can find a detailed explanation of the LCST behaviour occurring in polymer systems. We re-define a reduced temperature as T ∗ = kbT /α (not to be confused with that defined in the preceding sections) and a reduced pressure as P ∗ = P b2 /α where b = πσ 3 /6 is the volume of a spherical segment. We calculate the cloud and shadow curves (T x diagram) of four different polymer systems, at the same reduced pressure P ∗ = 0.001. The first system is a binary mixture of a spherical solvent and a monodisperse polymer of chain length m = 100. The second system is a ternary mixture of the same spherical solvent, and two chain molecules of length m2 = 66.7 and m3 = 400. The last two systems are the binary solvent + monodisperse chain of length m2 = 66.7, and the binary solvent + monodisperse chain of length m3 = 400. In the four systems, the segments of the chain molecules all have the same diameter σ which is equal to the diameter of the solvent, and all the sphere-sphere attractive interactions correspond to the same mean-field energy Cl. The curves obtained for the four systems are shown in figure 4.3. The binodal curves of the binary mixtures were determined by using the algorithm described in chapter 2, Anpendix A.


0.090

a)

0.085

b)

T* 0.080

c)

0.075 0.0

0.1

wpoly

0.2

0.3

Figure 4.3: Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400). The dashed dotted curves are binodal curves of the binary systems: a) solvent + chain m = 66.7 , b) solvent + chain m = 100, and c) solvent + chain m = 400. The thick continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST); wpoly = w2 + w3 = 1 − w1 is the total polymer weight fraction where w1 , w2 and w3 are the weight fractions of the solvent, and the chain molecules 2 and 3, respectively. For the ternary mixture, we used the method for discrete system, described at the beginning of this chapter in section 4.1. We recall that in the case of the binary systems, the cloud and shadow curves are identical, and correspond to the binodal curves. For the polydisperse ternary system, the cloud curve is different from the shadow curves, and different from the binodal curves (see chapter 1, section 1.3.3 for further explanation). From an inspection the binodal curves obtained for the three binary systems (solvent + chain m = 100), (solvent + chain m = 66.7, solvent + chain m = 400), depicted in figure 4.3, one can see that increasing the chain length of the polymer molecule extends the region of liquid-liquid immiscibility region: the LCST is found at lower temperatures. As mentioned in chapter 2, the LCST behaviour is due to density changes leading to unfavourable entropic effects, which are enhanced as the chain length of the polymer is increased. The critical point (LCST) is always at the minimum of the binodal curves for the binary systems. Furthermore, as the length of the chain is increased, the binodal curves are more

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS184 skewed and the LCST is shifted to lower weight fractions of polymer. Concerning the ternary system, two different curves can be seen: the cloud curve which gives the first temperature where phase separation occurs in the system at a fixed global composition of the sample, and the shadow curve which gives the composition of the first droplet of the new phase (see chapter 1, section 1.3.3). The polydispersity induces an increase in the extent of liquid-liquid immiscibility when compared with the mono-disperse system (solvent + chain m = 100), and makes the cloud curve more skewed. This result is surprising as the mole fraction of short chain molecules X (0) (m2 ) is much bigger than the mole fraction of long chain molecules X (0) (m3 ) in the parent distribution (see table 3.1). One can conclude that the longer chain molecules dominate the liquid-liquid demixing in the system, and that only a small presence of long chain molecules in a system of short chains + solvent can enhance the liquid-liquid immiscibility significantly. Furthermore, the critical point is no longer observed at the minimum of the cloud curve, but is shifted to higher weight fractions of polymer from the minimum of the curve; the critical point is at the intersection point of the cloud and shadow curves. The point critical point of the ternary system is located however almost at the same weight fraction of polymer as in the monodisperse case. The cloud and shadow curves appear to tend to the binodals of the binary mixture in the high temperature regions of the phase diagram. If the global weight fraction of polymer molecules is below the critical point, the cloud curve corresponds to a polymer-poor phase, while the shadow curve is the polymer-rich phase, and vice et versa. For a global weight fraction wpoly of polymer fixed below the critical point and closed to the pure solvent region, the shadow curve appear to tend to the binodal of the binary mixture (solvent + chain m = 400). This can be explained by the fact that at low temperature, the demixing is dominated by the presence of the long chain molecules (component 3), and the polymer-rich phase (shadow phase) predominatly consists of the longest chains (component 3), so that the number average hmi m (Sh) in the shadow phase is very close to m3 = 400. The shortest chains do not tend to demix with the solvent at that temperature, and remain in the solvent rich phase (cloud phase). However, at higher temperature and for a weight fraction wpoly above the critical point, the cloud curve tends to the binodal of the binary system (solvent + chain

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS185 m = 100), while the corresponding shadow curve appears to follow the binodal of the binary (solvent + chain m = 66.7). At higher temperature, both chain molecules (Cl) 2 and 3 demix with the solvent. The number average chain length hmim of the

polymer-rich cloud phase is 100, so that the cloud curve follows the binodal of the system (solvent + chain m = 100). The long chain molecules 3 are almost absent from the solvent-rich phase at that temperature, and the shadow phase consists mainly the shorter chain molecules 2: the shadow curve then follows the binodal of the system (solvent + chain m = 66.7), and the number average hmim (Sh) is close to m2 = 66.7.

4.4.2

Ternary Diagrams

Figure 4.4: Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400). The think continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST); wpoly = w2 + w3 = 1 − w1 is the total weight fraction of polymer, where w1 , w2 and w3 are the weight fractions of the solvent, and the chain molecules 2 and 3, respectively. The dashed and dotted lines represent constant temperature slices corresponding to the ternary diagrams depicted in figure 4.5. The white squares denote cloud points and the black squares are the corresponding shadow points.


Figure 4.5: Ternary diagrams in weight fractions, at constant pressure P ∗ = 0.001, and temperatures a) T1∗ = 0.0839, b) T2∗ = 0.0794, and c) T3∗ = 0.0787) corresponding to the cloud and shadow curves of the ternary system solvent + polymer (1) (chain m2 = 66.7 + chain m3 = 400) shown in figure 4.4. The thick and continuous curves represent the coexistence curves in the ternary mixture, and the circles denote the critical point (LCST). The thin and continuous lines are tie lines. The dashed and dotted lines represent a fixed ratio of composition between chain 2, and chain 3, corresponding to the parent distribution W (m2 )(0) = 0.6, W (m3 )(0) = 0.4. The white squares denote cloud points and the black squares are the corresponding shadow points.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS187 To explain further details of the nature of the cloud and shadow curves, one can study the corresponding ternary diagrams, at different temperatures and at constant pressure P ∗ = 0.001. We consider there different constant temperature slices (T 1∗ = 0.0839, T2∗ = 0.0794, T3∗ = 0.0787), all at the pressure of P ∗ = 0.001, denoted by the dotted and dashed lines in the temperature composition T w diagram of the ternary mixture solvent + polymer (1) (see figure 4.4). The three ternary diagrams are depicted in figure 4.5. The coexistence curves of the ternary diagrams were determined not by using the algorithm for cloud and shadow curves, but by solving the equality of the chemical potential of all the species (solvent, chain 2, chain 3) at constant temperature and pressure. The temperature slices are chosen such that T1∗ is above the critical point, T2∗ is the temperature of the critical point, and T3∗ is the temperature at the minimum of the cloud and shadow curves ( see figure 4.4). It can be seen in figure 4.4 that there are two different cloud points and two corresponding shadow points at temperature T1∗ : one cloud point corresponds to a polymer-rich phase, and the other to a polymer-poor phase. The corresponding ternary diagram is shown is figure 4.5 a). The dashed-dotted line on this diagram corresponds to a fixed length distribution function of the polymer (1), which is equal to the parent distribution: on that line, the ratio between the weight fractions of the chain molecules 2 and 3 is fixed to W (0) (m2 )/W (0) (m3 ). This line crosses the coexistence curve of the ternary system twice, and the intersection points correspond to the two cloud points of figure 4.4 at the temperature T1∗ . The shadow points are given by the tie lines connecting the corresponding cloud points (see the ternary diagram 4.5 a)). At temperature T2∗ , there are still two cloud and shadow points as can be seen in figure 4.4, however one cloud point is identical to its corresponding shadow point. At that point, the cloud and shadow curves cross, the densities and compositions of both phases become equal. This point is therefore a critical point. In the ternary diagram shown in figure 4.5 b), the dashed-dotted line corresponding to a fixed parent distribution crosses the coexistence curve twice at two cloud points. However, one of the cloud points is also the critical point of the ternary system at that temperature. T3∗ is the lowest temperature where demixing can occur in the ternary system solvent + polymer (1) at pressure P ∗ = 0.001. At that temperature, there is only one cloud point and one corresponding shadow point. These two points are the minimum of the cloud and shadow curves.


0.090

0.085

T* 0.080

0.075 0.00

0.10

wpoly

0.20

0.30

Figure 4.6: Cloud and shadow curves calculated with the SAFT-HS equation for the ternary system solvent + polymer (2) (chain m2 = 88.8 + chain m3 = 1000). The think continuous curve represents the cloud curve of the ternary system, and the dotted curve represents the shadow curve. The circles denote the critical points (LCST). The white square is a three phase point. In the ternary phase diagram at temperature T3∗ (figure 4.5 c)), the dasheddotted is tangent to the coexistence curve since it crosses it only once. The tie line going through the tangent point provides the corresponding shadow point on the coexistence curve. The study of the ternary system solvent + polymer (1) enables us to discuss the effect of polydispersity: the shape of the cloud and shadow curves obtained for the ternary system are very similar to those obtained experimentally (see [313]) when the molecular weight distribution function is not too wide. The polydispersity of chain length has a big effect on the liquid-liquid immiscibility at fixed numberaverage chain length. As the index of polydispersity Ip increases, the liquid-liquid immiscibility region becomes more extensive. In a polydisperse system, the presence of longest chains appears to dictate the shape of the shadow curves at low temperatures below the critical point even if their weight fraction is low in the system, while at temperatures above the critical point, the shortest chains govern the shape of the shadow curves.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS189 As shown in figure 4.7, the shadow phase corresponding to a polymer-rich phase (at temperature below the critical point) mainly consists of the longest chains, while the shadow phase corresponding to a solvent rich phase mainly consists of the shortest chains (at temperature above the critical point).The cloud curve of the ternary system tends to the binodal of the corresponding monodisperse binary system for hight weight fractions of polymer, and the minimum of the cloud curve is shifted to low temperatures from the binodal curve. ˇ Some time ago now Solk [308–310] observed that a very asymmetric distribution can lead to equilibria between three liquid phases. To study the effect of the shape of ˇ the chain length distribution function and confirm Solk results which were obtained with a lattice model very different from SAFT, we consider a second ternary system solvent + polymer (2) (solvent + chain m2 = 88.8 + chain m3 = 1000), where polymer (2) has the same number-average chain length hmin = 100 and same polydispersity index Ip = 2 as polymer (1), but a much more asymmetric distribution function (see the distribution functions of polymers (1) and (2) in figure 4.2.) The temperature composition diagram of the ternary system solvent + polymer (2) is shown in figure 4.6. The cloud curve and especially the shadow curve have a different shape than the cloud and shadow curves obtained for the first ternary system, although polymer (1) and polymer (2) have same hmin and Ip . The cloud curve exhibits a break at a temperature higher than the critical point (see the white square on figure 4.6). At that point, the cloud curve is continuous, but its derivative with respect to the weight fraction of polymer wpoly is clearly discontinuous. The corresponding shadow curve is continuous, but exhibits an inflection point at the same temperature. This behaviour is explain by the occurrence of three-phase equilibria, ˇ as has been mentioned by Solk [308–310], de Loos et al. [313] and Phoenix and Heidemann [316], and the break in the cloud curve corresponds to a three phase point. ˇ Solk performed a full study with the Flory-Huggins theory of the relative position between the critical point and the three phase point, by using continuous distributions. We confirm here his results with a simple ternary mixture using the SAFT-HS equation of state. Three phase equilibria occurs in polymer systems with very asymmetrical distributions, because of the large difference of size between the shortest and the longest chain molecules. According to our relation m3 = 7.2346m1.0955 2 between the chain length of the shorter component and that of the longer component which first gives rise to type V behaviour (see chapter2, figure 2.3), the binary

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS190 mixture chain m2 = 88.8 + chain m3 = 1000 exhibits a type V behaviour: the two chain molecules demix close to the critical point of chain 2. The liquid-liquid immiscibility between both chains is enhanced by the presence of the solvent. In the temperature-composition diagram of the ternary system solvent + polymer (2) (figure 4.6), we can see that the cloud curve consists of two stable liquid-liquid branches intersecting at the three phase point. In figure 4.7, it can be seen that the shadow phase rich in polymer consists mainly of the longest chains, while the shadow phase rich in solvent molecules consists mainly of the shortest chains. In the case of a very asymmetrical distribution function, the evolution of the number-average chain length of the shadow phase is quite complicated.

Figure 4.7: Number average chain length in the shadow phase at fixed pressure P ∗ = 0.001, as a function of the polymer weight fraction in the shadow phase, for the ternary systems solvent + polymer (1), and solvent + polymer (2). The circles denote critical points, and the square denotes a three phase point.


4.4.3

Cloud and Shadow Curves Obtained with Schulz-Flory Distributions

We now consider a polymer-solvent system where the solvent is again represented by a hard sphere of diameter σ, and the polymer by an infinite number of chain molecules of different lengths. In this work, we use a Schulz-Flory function [284,285] for the parent distribution function X (0) of chain length. The Schulz-Flory function depends on two parameters a and b: a is called the degree of coupling between polymer chain and b is related to the number average chain length hmin as b = a/hmin . The index of polydispersity Ip of the Schulz-Flory distribution is related to the parameter a as Ip = (a + 1)/a. The parent Schulz-Flory distribution X (0) in terms of mole fractions, which is equal to the length distribution X (Cl) in the cloud phase, is given by

X (0) (m) = X (Cl) (m) =

ba ma−1 e−bm , Γ(a)

(4.90)

where Γ is the gamma function defined as

Γ(a) =

Z

∞

dx xa−1 e−x .

(4.91)

0

The corresponding parent distribution W (0) in terms of weight fractions is given by

W (0) (m) = W (Cl) (m) =

ba+1 ma e−bm . Γ(a + 1)

(4.92)

Following equation (4.45), the parent density distribution is equal to the density distribution of the cloud phase Cl,

(Cl)

(Cl) ρ(0) m (m) = ρm (m) = hρm i0

ba ma−1 e−bm . Γ(a)

(4.93)

The two distributions are defined in the interval [0, ∞[. They are both normalised since

Z

∞ 0

dm ma−1 e−bm = b−a Γ(a).

(4.94)

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS192 It would be more accurate to use a lower boundary of 1 instead of 0 in the integral, since the minimum number of segments is physically 1. However, using a lower boundary equal to 0 does not make any significant change to the cloud curve calculation, and enables one to simplify the analytical expressions of the integrals. The moment method described in section 4.3 is applied to calculate the cloud and shadow curves of the polydisperse polymer solution with the Schulz-Flory distribution function. The moment free energy of this system can be separated into an ideal part, and a residual part (see equation (4.72)); both parts can be expressed in solely terms of the density of solvent, the temperature, and the zero and first density moments of the density distribution. The expression of the residual moment free energy is given in section 4.1.2, equation (4.29). The ideal part of the moment free energy is given by equation (4.70). One must express the Lagrange multipliers (Sh)

λ0

(Sh)

and λ1

(Sh)

and hρm i1

(Cl)

as a function of the density moments hρm i0

(Cl)

, hρm i1

(Sh)

, hρm i0

(Cl)

by inverting equations (4.69). The Lagrange multipliers λ0

in the

cloud phase are equal to zero, as shown in section 4.3.3. Using equations (4.84) and (Sh)

(4.93), the density distribution ρm

of the shadow phase can be expressed in terms

of the density distribution of the cloud phase as

(Sh)

(Cl) λ0 ρ(Sh) = ρm e m

=

(Cl) hρm i0

(Sh)

+λ1

m

(Sh) λ +m ba ma−1 e 0 Γ(a)

(4.95) ³

(Sh)

λ1

−b

´

.

It is shown in equation (4.95) that the length distribution function of the shadow phase is also a Schulz-Flory distribution. The length distribution in the shadow phase does not have the same number average as the parent distribution because (Cl)

the factor of m in the exponential is different for both distributions ρm

(Sh)

and ρm

(see equations (4.93) and (4.95)). However, the polydispersity index Ip is conserved, as the power of m equal to a − 1 is the same for both distributions. Consequently, the SAFT theory combined with the moment method predicts that phase separation in polymer systems with a Schulz-Flory distribution gives rise to a different average chain length in both phases, but that the width of the distribution is not affected. This result is in agreement with those of Cotterman [282] who found that if the

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS193 distribution of the feed is a gamma function then the distribution of the shadow phase is also a gamma function. Applying equations (4.94), (4.95), and (4.69), the (Sh)

zero and first density moments hρm i0

(Sh)

and hρm i1

of the shadow phase can be

expressed as [322]

(Sh) hρm i0

(Sh) hρm i1

=

(Cl) (Sh) hρm i0 eλ0

Ã

=

(Cl) (Sh) hρm i1 eλ0

Ã

1 (Sh)

1 − bλ1 1

(Sh)

1 − bλ1

!a !a+1

(4.96)

.

By rearranging equation (4.96), the Lagrange multipliers of the shadow phase are given by

(Sh)

λ0

(Sh) λ1

(Sh)

= (a + 1) ln

=

1 b

Ã

hρm i0

(Cl)

hρm i0

(Sh)

1−

hρm i0

(Sh)

hρm i1

(Sh)

− a ln (Cl)

/hρm i0

(Cl)

/hρm i1

hρm i1

(Cl)

hρm i1 !

(4.97)

.

One can then substitute equation (4.97) into (4.79) and (4.85), and determine the cloud and shadow point at a fixed pressure Pf ixed by solving the non-linear system of the following five equations: equalities of the zero and first moment chemical potential in both phases (see equation (4.85)), equality of the pressure (see equation (4.80)) of both phases to the fixed pressure Pf ixed , and equality of the chemical potential of the solvent in both phases). The five variables of the system of equations (Sh)

are the temperature T , the densities ρ(Cl) , ρ(Sh) , the polymer mole fraction xpoly

and the average length hmi(Sh) in the shadow phase. A Newton-Raphson method n can be used, and the variables can be guessed by using the values obtained with the corresponding mono-disperse system and the algorithm described in chapter 2, Appendix A. The cloud and shadow curves have been determined in this way for various indexes of polydispersity (Ip = 2, and Ip = 15). The corresponding parent distribution

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS194 function are shown in figure 4.8. As can be seen in figure 4.9, the liquid-liquid immiscibility region expands as the polydispersity is increased, and the critical point is shifted to lower polymer weight fractions. The number-average chain length hmi (Sh) n of the shadow phase as a function of the composition of the shadow phase are shown in figure 4.10: as in the discrete case, the shadow phase rich in polymer molecules consists mainly of long chain molecules, while the shadow phase rich in solvent molecules consists mainly of short molecules. The evolution of the number average chain length hmi(Sh) in the shadow phase is however very different from the disn crete case: hmi(Sh) tends to infinity as the shadow phase becomes richer in polymer, n while hmi(Sh) tends to the longest possible chain length in the discrete case. We can n also compare the cloud and shadow curves obtained for a Schulz-Flory distributions with the cloud and shadow curves obtained with the ternary system solvent + polymer (1). The polymer in both systems has the same number average chain length hnin = 100, and the same index of polydispersity Ip = 2.

Figure 4.8: Parent Schulz-Flory distributions in terms of weight fractions with a number-average chain length hmi(0) n = 100 and polydispersity indexes Ip = 2, Ip = 15. The corresponding cloud and shadow curves are depicted in figure 4.9.


Figure 4.9: Cloud (continuous) and shadow (dotted) curves for the system solvent + polydisperse polymer, obtained with the Schulz-Flory distributions depicted in figure 4.8 and various polydispersity indexes. The dashed-dotted curve represent the binodal obtained for a mono-disperse polymer-solvent system (Ip = 1). The circles denote critical points.

Figure 4.10: Number average chain length in the shadow phase, as a function of polymer weight fraction in the shadow phase, at fixed pressure P ∗ = 0.001, calculated for the system solvent + polydisperse polymer by using a Schulz-Flory feed distribution with various polydispersity indexes. The circles denote critical points. The number-average chain length of the feed distribution is hmin(Cl) = 100 for both cases, and the two curves correspond to those in figure 4.9 .


0.090

0.085

T* 0.080

0.075 0.0

0.1

wpoly

0.2

0.3

Figure 4.11: Cloud (continuous) and shadow (dotted) curves for the systems solvent + polydisperse polymer by using a Schulz-Flory feed distribution (thick lines), and for the ternary system solvent + polymer (1) (thin curves). The circles denote critical points. In both systems, the number-average chain length of the feed distribution is hmin(Cl) = 100, and the index of polydispersity is Ip = 2. For an inspection of figure 4.11 it can be seen that the cloud curve obtained with a continuous Schulz distribution is very similar to the cloud curve obtained for the discrete ternary system. However, the shadow curve in the continuous case has a smoother evolution. One can however conclude that a simple ternary system is sufficient to predict the cloud curve reasonably accurately. However, the shape of the shadow curves depend much on the shape of the feed distribution function.

4.5

Conclusion

In this chapter, we present three different methods to calculate phase equilibria in polydisperse polymer solutions with SAFT-type equations of states. The first method, developed by Jog and Chapman [318] enables to calculate cloud and shadow curves for discrete chain length distributions. The number of equations to be solved is always of five and does not depend on the number of pseudo-component used to represent the chain length distribution. The second method is an extension of Jog and Chapman method to treat phase equilibria with continuous distributions.

CHAPTER 4. THE EFFECT OF POLYDISPERSITY IN POLYMER SYSTEMS197 The third method, called moment method and developed by Sollich and co workers [320–322], is described and compared with the two first methods. The moment method can be applied only to continuous distributions. We first use the first method to calculate phase equilibria of a simple ternary system where the polymer is represented by two chain molecules. Two ternary systems corresponding to two different polymers (polymer (1) and polymer (2)) have been discussed. Both polymers have the same number average chain length and polydispersity index. By analysing the cloud and shadow curves obtained for these systems, we show that the shape of the distribution can have a big effect on the cloud and shadow curves, and that three phase equilibria may appear if the distribution is very broad. Moreover, we show that the liquid-liquid immiscibility is governed by the presence of the longest chain molecules: only a small presence of long chain molecules in the mixture give rise to an expansion of the liquid-liquid immiscibility region. We have used the moment method to calculate cloud and shadow curves with Schulz-Flory chain length distributions. There phase coexistence is never found for such distributions, confirming previous studies [283, 316]. Furthermore, we compare the results obtained with continuous and discrete distributions (ternary systems), and we show that the simple ternary system (solvent + polymer (1)) exhibits very similar cloud and shadow curves as these obtained with Schulz-Flory distributions. In a near future, we will provide results obtained for polydisperse polymer-colloid systems.

Chapter 5

Polyethylene + Hydrocarbon Systems It is important to know the phase behaviour of polymer systems in order to design and optimise chemical processes. For over fifty years, numerous theories (derived for lattice or tangent-sphere models) have been used to describe the phase equilibria of real polymer systems. A summary of all the various approaches is given in chapter 1. In this last chapter, we mainly deal with the phase equilibria of hydrocarbons and polyethylene (PE) mixtures as polyethylene is the simplest and most wide spread polymer. There are three common types of polyethylene: highdensity polyethylene (HDP E), made from the Ziegler-Natta homopolymerisation of ethylene; low-density polyethylene (LDPE), made by high-pressure free radical homopolymerisation of ethylene; and linear low-density polyethylene (LLDPE), made from the co-polymerisation of ethylene with one α-ofelin (1-butene, 1-hexene, or 1-octene). The thermodynamic experimental studies on polymer systems can be classified into three different classes: Firstly, experiments can be carried out on pure polymer samples to measure their melting points, crystallinities, or specific volumes (PVT data). Excellent introductions to polymer crystallisation can be found in the texts by Mandelkern and by Sharples [323, 324]. Experimental studies on polymer crystallisation are very numerous; the reader can find the most recent experimental studies on high- and low-density polyethylene (HDPE and LDPE) and poly(ethylene-olefins) (or linear low-density polyethylene, LLDPE) crystallisations in various references [325–330]. Theoretical studies are however very rare and most of the approaches are empirical. It is indeed very difficult to predict with accuracy the fusion properties of polymers as they depend on many different factors (co-monomer composition, molecular

198

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS

199

and isomeric structure, branching, polydispersity, lamellar thickness, thermal history of the sample) which are difficult to characterise. Furthermore, the accuracy of the experimental measurements of crystallinity is of order 5 to 10 %. The only available theory to describe the crystallinity of polymers at equilibrium (after an infinite crystallisation time) was derived almost 50 years ago by Flory [331] who proposed an analytical expression for the melting point and the crystallinity of an ideal co-polymer. Flory’s theory provides a good qualitative description of the melting point of polymers as a function of the co-monomers composition, and of the evolution of the crystallinity with temperature. Sanchez and Eby [332] improved Flory’s expression for the melting points of co-polymers by introducing some extra parameters to take into account defects (branching) and chain ends. Kim et al. [333] have recently tested the adequacy of the Flory [331] and the Sanchez and Eby [332] expressions to predict the melting points of LLDPE samples. They conclude that Sanchez and Eby [332] theory can give better results than Flory’s theory if the parameters characterising the defects are fitted on experimental data. However, their study is restricted to poly(ethylene-octene). Crist and Howard [334] tested Flory’s theory on both melting point and crystallinity data of poly(ethylene-butene) samples, and conclude that Flory’s equation overestimates the crystallinity. In this chapter, we propose a semi-empirical model based on Flory’s theory, to predict the melting points and crystallinities of any polyethylenes and poly(ethylene-olefins). The model only requires the crystallinity of the sample at 25◦ C. Another type of experiment carried out in polymer systems are measurements of the solubility of gases in the polymer. It is important to know the solubility of gases in polyethylene, to design low-pressure separation processes, and polymerisation reactors in the gas phase. During the reaction, polymer particles are suspended by a gas flow, and the monomers must sorb into the polymer sample and diffuse through the grain to reach the catalytic sites [335]. The rate of the reaction is directly proportional to the solubility of the monomer at the active site. Furthermore, it is important to know the solubility of gases in polymer in order to eliminate as much as possible of the remaining solvent from the polymer for safety and economic reasons [336]: fires have been experienced in numerous plants involved in polymer production due to the adsorbed flammable gas in the polymer [337, 338]; it is also good practice to recycle as much as of the monomer gas which has not reacted as possible.


200

The experimental studies of absorption of gases in polyethylene are scarce as they are expensive and time consuming. Some researchers have also recently studied these systems by Monte Carlo simulation [177, 178, 339–341], or integral equation theories [342]. It is however necessary to use an accurate, predictive, and fast approach to calculate the solubility of gases in polymer in order to design reactors and separation processes. The solubility is a function of temperature and composition of the gas phase. The term ”adsorption” is often used in an industrial context to denote the quantity of gases present in the polymer sample, as the adsorption experiments are usually carried out at temperatures under the melting point of the polymer (the polymer sample is often a powder). However, the term ”adsorption” is not relevant in this case, as an adsorption process occurs when the gas molecules sorb only at the surface of a solid. The sorption of gases in a polymer is a bulk process as the gas molecules can diffuse through the amorphous parts of the polymer sample, and the terms ”absorption” or ”solubility” are more appropriate. The crystalline parts of the polymer sample, also called ”crystallites”, behave as a barrier against the diffusion of the gas molecules in the sample. As a result, it is usually assumed that the gas can not penetrate the crystallites for steric reasons, and absorb only in the amorphous regions [327, 336, 339, 343]. In order to quantify the total absorption of gases in the polymer, it is hence necessary to know the crystallinity of the polymer sample as a function of temperature. If the temperature of the absorption experiment is above the melting point of the polymer sample, the crystallinity is zero and the gas absorbs in the whole sample. The amorphous regions present a liquid-like structure, and one can model the solubility of gases in polyethylene as a vapour-liquid equilibria between a gas phase where no polymer molecules are present, and a liquid phase consisting of gases molecules absorbing into an amorphous polymer matrix. Two main difficulties can be encountered in the modelling of phase equilibria in gas-polymer systems: firstly, the absorption of gases in a semi-crystalline polymer is basically a three-phase equilibria between a vapour phase, a liquid phase of amorphous polymer, and a solid phase of crystallites. The assumption that gas molecules absorb only in the amorphous regions may be too crude: if the gas molecules are small enough, they can penetrate into the crystalline lattice and the solubility of gases in the crystallites will not be negligible. Furthermore, the presence of gas in the polymer sample decreases the melting point of the pure polymer due to cryoscopic effects [344–346] and in turn the crystallinity decreases. The assumptions that the


201

gas molecules only absorb in the amorphous regions, or that the crystallinity does not change in presence of gas, can both lead to an underestimate of the total absorption. The adequacy of these two assumptions will be addressed in this chapter by modelling polyethylene + gas systems with the SAFT-VR approach. The second problem in polymer-gas absorption is when two or more gases absorb simultaneously in the same polymer sample. It has been observed by some experimentalists (e.g., see [336, 347]) that the solubility of a gas in a polymer sample at a given partial pressure, may or may not be enhanced by the presence of a second gas in the vapour phase, depending on the nature of this second gas. For instance, the solubility of methane in polyethylene, at a fixed partial pressure of methane, is enhanced in the presence of ethylene in the gas phase, while the solubility of methane does not change in the presence of nitrogen [347]. The co-absorption problem is crucial in the case of the copolymerisation of ethylene + α-olefins to make LLDPE: the solubilities of the co-monomers in the polyethylene grains determine the rate of reaction for each monomer, and then the final composition of the co-monomers in the polymer chains. This type of synergy has been confirmed by Monte Carlo simulation [177,339]; however, the effect remains unclear and is sometimes explained in terms of diffusionnal and swelling mechanisms [339]. An explanation of the cosorption synergy is proposed in this chapter by discussing the interactions between molecules. Another more specific problem in the general area of absorption of gases in polymer is the modelling of vapour-liquid equilibria in hydrogen (H2 ) + hydrocarbon (n-alkanes) and hydrogen + polyethylene systems. Industrial processes such as petroleum refining, coal conversion, enhanced oil recovery and supercritical separation have a great demand for phase equilibria data of very asymmetric compounds such as hydrogen and n-alkanes. The VLE of such systems can not be predicted by using simple mixing rules combined with the usual engineering cubic equations of state (PR [289] or SRK [268]). The reason for this is that hydrogen must be treated at the quantum mechanical level in order to take into account the various vibration modes at low temperatures [179]. One approach to model hydrogen systems is to use quantum Monte-Carlo technics [348] and specific intermolecular potentials. However, Monte-Carlo simulations are computational intensive and an equation of state is more useful in the design and optimisation of chemical processes. To calculate phase equilibria of hydrogen systems with an equations of state, some authors have


202

used the pure parameter of hydrogen obtained from the critical properties, and very specific mixing rules [349]. An other way is to use simple one-fluid mixing rules, but a temperature dependent attractive parameter for hydrogen [350]. Here, we are interested in modelling the solubility of hydrogen in polyethylene only at temperature around 400 K, well above the critical point of hydrogen. To model such a system one has to determine the most reliable parameters for hydrogen at these temperatures, and use simple mixing rules to allow an extrapolation from short n-alkanes to very long n-alkanes and polyethylene. One can determine the optimal SAFT-VR parameters of hydrogen at high temperatures directly by fitting to vapour-liquid experimental data of hydrogen+ n-alkane binary mixtures. As is shown in this chapter, a very good accuracy can be obtain with this method in the predictions of both bubble and dew points, without using any cross-interaction parameters k ij . A last type of phase equilibria experiment which is often carried out on polyethylenehydrocarbon systems is the determination of cloud curves and liquid-liquid immiscibility regions. The experimental studies are very numerous in this case, and the reader can find a rich source of experimental data in the review of Kirby and McHugh [29]. One should also mention papers of Hamada et al [351], de Loos and co-workers [61, 312, 313, 352, 353], and Kiran and co-workers [354–361] in this context. It has been seen shown [362] that the region of liquid-liquid immiscibility becomes more extensive with decreasing pressure, decreasing number of carbon atoms of the n-alkane solvent, and increasing molecular weight of the polyethylene. Many authors have already modelled cloud points in polyethylene + n-alkane systems: some authors use binary cross-interaction parameters [207, 316, 363, 364], while others use the simple Lorentz-Berthelot combining rules [70,144,319,359,365]. No new modelling of cloud curves is presented in this thesis, and instead we test the adequacy of the SAFT-VR theory to predict liquid-liquid immiscibility in polyethylene + n-alkane systems, without the use of binary cross-interaction parameters. Cloud-point modelling is indeed a very good test of the theory as the results are very sensitive to the molecular parameters which are used to represent the polymer. Some predictions of of cloud points in polyethylene + n-alkanes systems with the SAFT-VR equation of state are presented in this chapter, and comparisons will be made with the predictions of the BYG theory [27, 70]. The polyethylene molecules are modelled as a very long n-alkanes and the SAFT parameters of the polymer are


203

obtained by using simple linear relationships with molecular weight, derived from the n-alkanes parameters. In a first section, the main expressions of the SAFT-VR theory are presented. The predictions for the vapour-liquid phase equilibria in pure compounds, hydrocarbon binary mixtures, and polyethylene + hydrocarbons are then shown. The cloud curves in polyethylene + n-alkane systems obtained from the SAFT-VR approach are then presented. We then propose a model based on Flory’s theory to predict the crystallinity of polyethylene, and show the prediction of solubilities of gases in semi-crystalline polyethylene. We finally discuss the synergy in the co-absorption of different monomers in polyethylene.

5.1

SAFT-VR Theory

In this section, the focus is on the statistical associating fluid theory for potentials of variable Range (SAFT-VR) theory [99], and a description is given for each of the terms in the equation. As was mentioned in the earlier chapters, the statistical associating fluid theory (SAFT) was developed by Chapman and co workers [366], [171]. The theory is based on a molecular model derived from statistical mechanics, and has been used to predict successfully the properties of a large variety of compounds and mixtures. The main advantage of the SAFT theory is its ability to take into account the non-sphericity of molecules, and to model short-range directional interactions in associating fluids such as water. The mean-field version of SAFT (SAFT-HS) is described in details in chapter 2. The most sophisticated version of SAFT, SAFT-VR, takes into account the structure of the fluid by incorporating the radial distribution function in the attractive term, and an additional non-conformal parameter is used to describe attractive interactions of variable range.

5.1.1

Molecular Model

In the SAFT-VR theory, the molecules are modelled as a flexible chains formed from m tangent spherical segments. Each segment of the chain i has the same diameter σi , but segments belonging to different species can have different diameters, contrary to the prototype model discussed in chapter 2. The dispersive interactions between the segments are modelled with an intermolecular potential (square well, Sutherland, Yukawa or Mie potentials; see figure 5.1) of variable range λ and depth


204

². In this work, the square-well potential has been used to describe the intermolecular interactions of real molecules. Although the shape of this potential is rather unrealistic, it enables one to simplify dramatically the expressions of the free energy, and obtain a very description of the phase equilibria.

Figure 5.1: Potential models which can be used in the SAFT-VR theory to model segment-segment interactions. ² and λ are the depth and the range of the potential, respectively. Anisotropic attractions such as hydrogen bonds can also be modelled in the SAFT-VR approach by incorporating a number of short-range associating sites on the molecule. A typical molecular model used in the SAFT approach is depicted figure 5.2. In this case the molecule is made of 6 segments of diameter σ and 4 different association sites. In figure 5.3, models used to represent some common molecules are also shown. Methane is simply represented by one spherical segment since the methane molecule is almost spherical, while butane is represented by two tangent spherical segments.

Figure 5.2: General molecular model used in the SAFT theory. In this case, the molecules comprised five spherical segments of diameter σ with four association sites.


205

Note that the number of spherical segments is not equal to the number of carbon atoms in the n-alkane series, but represents the effective non-sphericity of the molecule (see section 5.2.1). Water is also considered to be a spherical molecule. The hydrogen bonds are modelled by four sites: two sites represent the two hydrogen atoms, and two represent the two electron pairs of the oxygen atom.

Figure 5.3: Examples of models used for common molecules in the SAFT theory. The n-alkanes are represented by a flexible chain of segments without association sites. Water is represented by a spherical core with 4 sites to represent the hydrogen bonds.

5.1.2

Calculation of the Helmholtz Free Energy

In this section, we summarise the main expressions of the SAFT-VR equation for mixtures with the square-well potential. The reader can find further details in references [99,100]. Here, we assume that the segments belonging to different types of chain molecules can have various diameters and can interact with different potentials. However, the segments belonging to the same molecule have all equivalent. For mixtures of chain molecules composed of hard spherical segments with attractive interactions and associating sites, the expression of the Helmholtz free energy can be written in four separate contributions as

AIDEAL AM ON O. ACHAIN AASSOC A = + + + . N kT N kT N kT N kT N kT

(5.1)

Note that the associating term AASSOC , which is derived from Wertheim’s theory, is not required in the description of alkane mixtures and polyethylene-alkane systems

206


as the n-alkane molecules only interact through van der Waals dispersive forces. The ideal contribution to the free energy is given as a sum over all species i in the mixture:

AIDEAL = N kT

Ã

n X i=1

!

xi ln ρi νi − 1,

(5.2)

where xi = Ni /N is the mole fraction of component i, ρi = Ni /V is the molecular number density, Ni is the number of molecules of type i, Λi is the thermal de Broglie volume (incorporating the translational and rotational kinetic contributions), and V is the total volume of the system. AM ON O. is the contribution due to the interactions between the spherical monomeric segments, which is given by

AM ON O. N kT

= (

xi mi )

X

xi mi )aM ,

i=1

= (

AM Ns kT

X

(5.3)

i=1

where mi is the number of spherical segments in a chain of species i, and Ns is the total number of segments in the system. The monomer free energy per segment of the mixture, aM , is obtained from the high-temperature perturbation theory of Barker and Henderson [46, 172, 173] with a hard-sphere reference system:

aM = aHS +

a2 a1 + + ..., kT (kT )2

(5.4)

The expression of the free energy contribution for a multicomponent mixture of hard spheres, aHS , is described by Boubl´ık [187] and Mansoori et al. [188] as a generalisation of the Carnahan and Starling repulsive term [74]:

a

HS

6 = πρs

"Ã

!

#

ζ23 ζ23 3ζ1 ζ2 + − ζ . ln(1 − ζ ) + 0 3 (1 − ζ3 ) ζ3 (1 − ζ3 )2 ζ32

(5.5)

Here, ρs = Ns /V is the total number density of spherical segments, Ns the total number of segments, V the volume of the system, and ζl are the reduced densities defined by


n πρs X xs,i σiil , 6 i=1

ζl =

207

(5.6)

where σii is the diameter of the spherical segments of chain i, and xs,i is the mole fraction of segments of type i in the mixture. Notice that ζ3 = η. The first term a1 is the mean-attractive energy and is obtained from the sum of the partial terms corresponding to each type of pair attractive interaction:

a1 =

n n X X

xs,i xs,j aij 1,

(5.7)

i=1 j=1

where

aij 1 = −2πρs ²ij

Z

λij σij σij

2 HS rij gij [rij ; ζ3 ] drij ,

(5.8)

HS is the radial pair distribution function for a mixture of hard spheres. where gij

Using the mean-value theorem, the expression for a1 is obtained in terms of the contact value of g HS at an effective packing fraction ζ3ef f : h

i

V DW HS gij σij ; ζ3ef f , a1 = −ρs Σi Σj xs,i xs,j αij

(5.9)

where the van der Waals attractive constant is obtained as 3 V DW = 2πεij σij (λ3ij − 1)/3. αij

(5.10)

The parameters εij and λij are respectively the depth and the range of the square well potential for i−j segment interactions. σij defines the contact distance between spheres i and j. The expression of this potential (square-well) is given by

Φij (r) =

   +∞

−εij   0

if r < σij , if σij ≤ r < λij σij , if r ≥ λij σij .

(5.11)

The van der Waals (one-fluid) mixing rule is used in this work. The radial distribution function of the mixture g HS is obtained as the radial distribution function


208

of a hypothetical pure fluid of packing fraction ζxef f . The expression for a1 is then written as h

i

V DW HS a1 = −ρs Σi Σj xs,i xs,j αij g0 σx ; ζxef f ,

(5.12)

where g0HS is the radial distribution function at contact for a hard-sphere system, and its expression can be obtained from the Carnahan and Starling repulsive term as

1 − ζxef f /2 g0HS = ³ ´3 . 1 − ζxef f

(5.13)

The effective packing fraction ζxef f in the one-fluid description is given by: ζxef f (ζx , λij ) = c1 (λij )ζx + c2 (λij )ζx2 + c3 (λij )ζx3 ,

(5.14)

where

ζx =

πρs 3 σ , 6 x

(5.15)

with σx3 =

XX i

3 xs,i xs,j σij .

(5.16)

j

The parameters c1 , c2 , c3 are given in terms of the following matrices: c1 2.25855  c2  =  −0.669270 c3 10.1576 





−1.50349 0.249434 1 1.40049 −0.827739   λij  . λ2ij −15.0427 5.30827 



(5.17)

The coefficients of this matrix have been obtained by integrating equation (5.8) for a pure fluid with the radial distribution function g HS (r) of the hard sphere system given by the expression of Malijevski and Labik [367]. The second perturbation term a2 is the fluctuation term in the Baker and Henderson perturbation theory. It is evaluated using the local compressibility approximation for mixtures :

a2 =

n n X X

i=1 j=1

xs,i xs,j aij 2,

(5.18)

209

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS where aij 2 corresponds to each of the partial fluctuation terms defined as 1 HS ∂aij 1 aij = K ² ρ . ij s 2 2 ∂ρs

(5.19)

K HS is the hard-sphere isothermal compressibility factor of Percus-Yevick [368] and is given by

K HS =

ζ0 (1 − ζ3

)2

ζ0 (1 − ζ3 )4 . + 6ζ1 ζ2 (1 − ζ3 ) + 9ζ23

(5.20)

Finally, the contribution to the free energy due to chain formation is expressed in terms of the contact value of the background correlation function (or cavity function) of the unbounded monomer fluid y SW as n X ACHAIN xi (mi − 1) ln yiiSW (σii ), =− N kT i=1

(5.21)

where yiiSW (σii ) = giiSW (σii ) exp(−β²ii ) and β = 1/kT . yiiSW (σii ) is obtained from the high-temperature expansion of giiSW (σii ) as giiSW (σii ) = giiHS (σii ) + β²ii g1SW (σii ),

(5.22)

where the term g1SW (σii ) is obtained from a self-consistency between with the pressure equation (Clausius theorem [189]) and the density derivative of the Helmholtz free energy:

giiSW [σii ; ζ3 ] = giiHS [σii ; ζ3 ] + βεii

"

giiHS [σii ; ζ3ef f ]

+

(λ3ii

− 1)

∂giiHS [σii ; ζ3ef f ] ∂ζ3ef f

Ã

∂ζ ef f λii ∂ζ3ef f − ζ3 3 3 ∂λii ∂ζ3

HS [σ ; ζ ef f ] is The contact value of the hard sphere pair distribution function gij ij 3

obtained from the expression of Boub´ık [187] as

HS gij [σij ; ζ3ef f ] =

1 1 − ζ3ef f

+3

Dij ζ3ef f (1 − ζ3ef f )2

+2

³

Dij ζ3ef f

´2

(1 − ζ3ef f )3

,

(5.23)

!#

.


210

where n σii σjj xs,i σii2 Pi=1 Dij = , σii + σjj ni=1 xs,i σii3

P

(5.24)

and where another effective packing fraction ζ3ef f different from ζxef f is defined as ζ3ef f (ζ3 , λij ) = c1 (λij )ζ3 + c2 (λij )ζ32 + c3 (λij )ζ33 ,

(5.25)

The coefficients c1 , c2 , and c3 are the same as those used in equation 5.17.

5.1.3

Combining Rules

In order to evaluate the unlike interaction parameters (also called cross parameters) in the mixture, the standard Lorentz-Berthelot combining rules can be used:

σij =

σii σjj , 2

(5.26)

²ij =

√ ²ii ²jj .

(5.27)

A simple arithmetic combining rule is used for λij :

λij =

λii σii λjj σjj . σii + σjj

(5.28)

The first combining rule (equation (5.26)) is exact since all the segments are hard spherical cores. The second combining rule (equation (5.27)) is an approximation which is reliable when the two compounds interact with only dispersive forces and are chemically similar (such as n-alkanes). The third combining rule (equation (5.28)) is seen to be reliable for n-alkane mixtures. If the two last combining rules do no give (²)

(λ)

satisfactory results, binary adjustable parameters kij and kij can be used, and the relations 5.27 and 5.28 become √ (²) ²ii ²jj (1 − kij )

(5.29)

λii σii λjj σjj (λ) (1 − kij ) σii + σjj

(5.30)

²ij =

λij =

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS (²)

211

(λ)

The parameters kij and kij can be fitted to experimental data of a binary mixture and then transferred to other binary mixtures of similar compounds.

5.2

Modelling of Pure Components

We recall that in the SAFT-VR approach each pure component is characterised by a set of molecular parameters: m is the number of spherical segments which form the chain molecule; each segment belonging to the same chain has the same diameter σ, and interacts with a square well potential of depth ² and range λ. The pure components which only interact with van de Waals dispersive forces like the n-alkanes are usually characterised by these four parameters: m, σ, ², and λ. Some components like water (H2 O) or hydrogen fluoride (HF) exhibit anisotropic associations which can be modelled by a certain number of associating sites, according to the theory of Wertheim (see chapter 2). Each type of bonds is characterised by a bonding volume KHB (or cut-off distance rc , see chapter 2, figure 2.1) and an association energy ²HB . Water is thus characterised by 6 parameters in the SAFT-VR approach, m, σ, ², λ, ²HB , rc .

5.2.1

n-Alkanes

We have modelled the n-alkanes as a flexible chain of m spherical segments of diameter σ [99]. Methane can be considered as the first member of the n-alkane series. The molecule of methane has spherical shape and is represented by only one spherical segment (m = 1). For the longer n-alkanes, we use a simple semi-empirical relationship (equation 5.31 ) between the number of carbon atoms C and the number of segments in the chain m, which was found in previous work [99, 369]:

m = 1 + 1/3 (C − 1)

(5.31)

As a result, butane (C = 4) is modelled by only 2 tangent hard spheres (see chapter 3, figure 5.3). In the SAFT-VR theory, the parameter m does not have to take integer values: for instance, the number of segments representing the molecule of ethane is methane = 1/3 ' 1.333.


212

Determination of the Molecular Parameters Following previous work [99, 369], we have fixed m according to the relation 5.31, and the other molecular parameters σ, ² and λ are determined to provide an optimum description of experimental vapour pressure and saturated liquid densities. Experimental data are available from the triple point up to the critical point for the first members of the n-alkane series. However, such experimental data become increasingly difficult to find for alkanes heavier than decane because hydrocarbons are thermally unstable at a temperature above 650K. Although some data are available in the literature up to hexatriacontane (n-C36 ) [370, 371], they are not very reliable, as the purity of the n-alkanes used was not very high. New techniques are have been used to determine accurately vapour pressures and critical points of the long alkanes [372, 373]. Monte Carlo simulation data on n-alkanes are reliable and can be used as pseudo-experimental data [374–378]. We have determined the molecular parameters of the n-alkanes up to C28 , and the fitting was carried out with the well-known simplex method of Nelder and Mead [379], coupled with an ”annealing” method [209] which introduces some randomness in the parameters searching and enables one to find the global minima of the objective function. The results are presented in various projections slices of the P V T surface: temperature vs saturated molar volume (T V ), temperature vs saturated density (T ρ), vapour pressure vs temperature (P T ), and Clausius Clapeyron diagram ln P vs 1/T . The vapourpressure curves shown in the diagram (ln P vs 1/T ) are almost linear, according to the Clapeyron relation. It is clear from the figures 5.4 a) and b) that the SAFT-VR approach provides a net improvement over the Carnahan and Starling equation of state + mean-field attractions (equivalent to SAFT-HS with m = 1, see chapter 2 for the description of theory), for both saturated liquid densities (figure 5.4 a)) and vapour pressures (figure 5.4 b)). In general, very good agreement with experimental vapour pressure and saturated densities is obtained with the SAFT-VR approach for all of the n-alkanes up to C28 (figure 5.5 to figure 5.7). The critical regions are in general less well represented. The overvaluation of the critical temperature and pressure is a common drawback of most equations of state , and a specific treatment based on normalisation group theory [64, 380, 381] is required to obtain a better agreement in the critical region.


213

Figure 5.4: Saturated densities a) and vapour pressures b) for methane: comparison between experimental data (circles) from reference [382] with the SAFT-HS (thin lines) and SAFT-VR (thick lines) theories after fitting of parameters.


214

Figure 5.5: Vapour pressures of the n-alkanes from C1 to C8 , compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) vapour pressure curve. b) Clausius-Clapeyron representation.


215

Figure 5.6: Vapour-liquid coexistence curves of the n-alkanes from C1 to C8 compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes.


216

Figure 5.7: Vapour-liquid coexistence curves of the n-alkanes from C12 to C28 , compared with the SAFT-VR predictions. The circles represent the experimental data [373, 383], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) vapour pressures in Clausius-Clapeyron representation.


217

The parameters of the n-alkanes obtained after optimisation are shown in table 5.1. Table 5.1. SAFT-VR square-well parameters of the n-alkanes obtained after optimisation.

Compound

MW/(g.mol−1 )

m

CH4 C2 H6 C3 H8 C4 H10 C5 H12 C6 H14 C7 H16 C8 H18 C9 H20 C10 H22 C11 H24 C12 H26 C13 H28 C14 H30 C15 H32 C16 H34 C17 H36 C18 H38 C19 H40 C20 H42 C24 H50 C28 H58

16.04 30.07 44.10 58.12 72.15 86.18 100.20 114.23 128.26 142.28 156.31 170.33 184.36 198.39 212.41 226.44 240.47 254.49 268.52 282.55 338.65 394.76

1 1.3333 1.6667 2 2.3333 2.6667 3 3.3333 3.6667 4 4.3333 4.6667 5 5.3333 5.6667 6 6.3333 6.6667 7 7.3333 8.6667 10

λ

σ (˚ A)

²/k (K)

1.4479 1.4233 1.4537 1.4922 1.5060 1.5492 1.5574 1.5751 1.5745 1.5925 1.5854 1.6101 1.6479 1.6023 1.5978 1.6325 1.6091 1.6565 1.6625 1.6637 1.6819 1.6542

3.6847 3.8115 3.8899 3.9332 3.9430 3.9396 3.9567 3.9455 3.9635 3.9675 3.9775 3.9663 3.9583 3.9745 3.9964 3.9810 3.9954 3.9562 3.9713 3.9726 3.9809 3.9896

167.30 249.19 260.91 259.56 264.37 251.66 253.28 249.52 251.53 247.08 252.65 243.03 227.31 249.74 252.87 237.33 249.76 228.81 226.31 227.07 220.00 233.50

Long n-Alkanes Since experimental vapour pressure and saturated densities are not available in the literature for the long n-alkanes and polyethylene, it is necessary to find a method to evaluate the SAFT-VR parameters of those compounds. The polyethylene molecule can be considered as a very long n-alkane and it can be also modelled as a flexible chain of m spherical segments of the same diameter σ. The chain length m is calculated with the same linear relation as that used for the short n-alkanes (equation 5.31. Following the previous work of McCabe et al. [191], we have calculated the SAFT-VR parameters for the long n-alkanes and polyethylene by using simple relations as a function of the molecular weight, obtained for the short alkanes. We


218

can be confident that such an extrapolation will be valid because the parameters are based on a clear molecular description. Very good agreement is obtained between the correlations and the SAFT-VR parameters of the short alkanes obtained after fitting (see figures 5.8). However, we observe that the well depth ² exhibits more fluctuations than the other parameters and the correlation for this parameter is more difficult to determine (figure 5.8 (d)). The analytical expressions of the correlations are:

m = 0.02376M W + 0.6188

(5.32)

mλ = 0.04024M W + 0.6570 mσ 3 = 1.53212M W + 30.753 mε/k = 5.46587M W + 194.263,

where M W is the molecular weight of the alkane. The parameters apart from the chain length m tend to a finite limit when M W tends to ∞, They do not vary much with the molecular weight for very long polymeric chains. The limits of the parameters can be easily determined. For instance, the limit value of λ for very high molecular weight is given by

lim

M W →∞

λ

= =

lim

M W →∞

(mλ) m

(5.33)

0.04024 ' 1.694 0.02376

The limits of the other parameters are obtained as

lim

M W →∞

lim

²/k ' 230.04 K

M W →∞

(5.34)

σ ' 4.010 ˚ A.

Note that the first correlation (5.32) becomes m ' ξMi , where ξ = 0.02376, when the molecular weight of the long n-alkane is very high. The number of segments is


219

then proportional to the molecular weight, as already mentioned in chapter 3, section 4.1. We have tested the SAFT-VR correlations by comparing with the Monte-Carlo simulation data obtained for the pure long-alkanes C16 , C24 , and C48 [384].


220

Figure 5.8: SAFT-VR parameters: a) λ ; b) σ; c) and ² of the n-alkanes as a function of the molecular weight, obtained after fitting. Simple linear correlations have been used to extrapolate the parameters to longer alkanes and polyethylene. The circles denote the values of the parameters fitted on vapour pressures and saturated densities, and the continuous lines represent the correlations (5.32).


221

Figure 5.9: Vapour-liquid coexistence curves obtained for the n-alkanes C 16 , C24 , and C48 . The circles represent experimental data [373, 383]. The crosses represent Monte-Carlo simulation data [384]. The continuous curves correspond to the SAFTVR predictions, and the parameters of the long n-alkanes are calculated with the correlations (5.32). . The coexistence densities are well represented, but the critical point is overestimated as for short n-alkanes. One should note that there might also be some errors in the simulation data around the critical point. The coexistence curves obtained with the correlations for the n-alkanes C16 and C24 (see figure 5.9) are very similar to those obtained with the fitted parameters (see figure 5.7).

5.2.2

α-Olefins

Ethylene and α-olefins (propene, 1-butene, 1-hexene) are the monomers used in the polymerisation of polyethylene (only ethylene for HDPE and LDPE, and ethylene + α-olefins for LLDPE). It is then necessary to calculate the phase equilibria of mixtures of polyethylene + olefins for process design. The SAFT-VR parameters of ethylene and α-olefins have been determined by fitting experimental vapour pressures and coexistence densities of pure compounds. The results are shown in figure 5.10 to 5.10. We use the same relation (5.31) between the number of segments m and the number of carbon atoms of the olefins. Good agreement is obtained with both experimental densities and vapour pressures. The parameters for the olefins are given in table 5.2.


222

Figure 5.10: Vapour pressures of ethylene and α-olefins (propene, 1-butene, 1hexene), compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) vapour pressure curve. b) Clausius Clapeyron representation.


223

Figure 5.11: Vapour-liquid coexistence curves of of ethylene and α-olefins (propene, 1-butene, 1-hexene) compared with the SAFT-VR predictions. The circles represent the experimental data [382], the continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes.


224

Table 5.2. SAFT-VR square-well parameters of the α-olefins obtained after optimisation.

5.2.3

Compound

MW/(g.mol−1 )

m

ethylene propene 1-butene 1-hexene

28.05 42.08 56.11 84.16

1.333 1.667 2.000 2.667

λ

σ (˚ A)

²/k (K)

1.4432 1.4465 1.5564 1.6244

3.6627 3.7839 3.7706 3.82590

222.17 259.80 228.49 217.78

Other Pure Compounds

The SAFT-VR parameters of other pure components have also been optimised as a test of the SAFT-VR theory and the optimisation algorithm. We have modelled nitrogen (N2 ), carbon dioxide (CO2 ), hydrogen fluoride (HF) and water (H2 O), which exhibit very different intermolecular interactions and thermodynamic properties. Nitrogen is modelled as a “dumbbell” (m = 1.3) and only dispersive forces are taken into account via square well potentials. CO2 is treated as an elongated molecule (m = 2) without any association sites. Water is represented as a spherical molecule with four association sites (2 0, 2 H) to model the hydrogen bonds (see figure 5.3): 2 sites model the hydrogen atoms and 2 sites model the electron loan pairs on the oxygen atom. Hydrogen fluoride is also modelled as a spherical molecule with three association sites (2 F, 1 H) which allow for hydrogen fluorine hydrogen bonds. The coexistence curves of the pure components are shown in figures 5.12 and 5.13. Good agreement with the experimental data is also obtained on both coexistence densities and vapour pressure, except in the critical region. One can note in figure 5.13 that the gaseous molar volumes of HF are overestimated by SAFT-VR. This is due to the fact that HF can form rings in the gas phase due to hydrogen bonding; a Wertheim theory taking into account ring formation could be used to improve the prediction of the gaseous volumes and the vapour pressures, as suggested by Galindo et al. [385]. The square-well SAFT-VR parameters obtained after fitting are presented in the table 5.3


225

Figure 5.12: Vapour pressures for CO2 , H2 O, and HF compared with the SAFTVR predictions. The circles represent the experimental data [386], [387], [388]. The continuous curves correspond to the SAFT-VR approach. a) vapour pressure curves. b) Clausius Clapeyron representations.


226

Figure 5.13: Vapour-liquid coexistence curves for CO2 , H2 O, and HF compared with the SAFT-VR predictions. The circles represent the experimental data [386], [387], [388]. The continuous curves correspond to the SAFT-VR approach. a) coexistence densities. b) coexistence volumes.


227

Table 5.3 SAFT-VR square-well parameters obtained after fitting for water, hydrogen fluoride and carbon dioxide. The parameters ²HB and KHB are site-site association energy and bonding volume respectively.

Compound

MW/(g.mol−1 )

m

λ

σ (˚ A)

²/k (K)

²HB /k(K)

KHB (˚ A3 )

N2 CO2 H2 O (4 sites) HF (3 sites)

28.01 44.01 18.02 20.01

1.3 2.0 1.0 1.0

1.534 1.516 1.800 1.483

3.194 2.786 3.036 2.856

84.53 179.27 253.30 168.82

0 0 1365.92 1782.11

0 0 1.020 9.443

5.3

Modelling n-Alkane + linear Polyethylene Systems

A lot of experimental data are available in the literature for polyethylene + short alkanes solutions, though most of the experiments are quite old, as polyethylene was one of the first synthetic polymers. One problem with polyethylene is that it is not well-defined in terms of branching and polydispersity, and one has to be careful with the reliability of the experimental data. As was shown in chapter 3 the polydispersity of the polymer can have a great influence on the fluid phase behaviour, especially the liquid-liquid equilibria. Polydispersity and branching can differ from one polymer to another. Most of the predictions of cloud curves in the literature have been carried out with the use of cross interaction parameters kij fitted to the experimental data, and generally depending on temperature. Good agreement is usually obtained with this kind method [207, 316, 363, 364]. However, the dependence on temperature of the cross parameters is physically suspect as a true intermolecular potential should not depend on temperature, and transferable parameters which are independent of temperature are usually required for truly predictive applications. In our approach, we take the same parameters for low-density (LDPE) and high-density (HDPE) polyethylenes from the correlations (5.32).

5.3.1

Pentane + Polyethylene

The ability of the SAFT-VR approach combined with the relations (5.32) to predict the fluid phase equilibria of polyethylene solutions is first examined by using the


228

combining rules of Lorentz and Berthelot (no cross interaction parameters). We first discuss n-pentane + polyethylene systems for which extensive and reliable experimental data are available [355, 359, 360, 389]. The pressure-temperature projection of the phase diagram of binary mixture of pentane + long n-alkanes and pentane + monodisperse linear polyethylene are calculated first. The results are shown in the figure 5.14.

Figure 5.14: Pressure-temperature P T projection of the phase diagram obtained for the binary mixtures pentane + long alkanes (n-C36 ) and pentane + polyethylene(PE) of molecular weights 1, 10 and 100 kg mol−1 , predicted with SAFT-VR. The white triangles denote calculated upper and lower critical end points. The continuous curves are the vapour pressure curve of the pure compounds. The dashed curves are critical lines. The dashed-dotted lines represent the temperature range where fluid phase are stable: at temperatures below about 400K, crystallisation occurs, and at temperatures above 650K, the alkyl chains decompose. No cross interaction parameter is used (kij = 0). The P T diagram 5.14 of the pentane+PE mixture is similar to those obtained for the prototype polymer solutions discussed in Chapter2 (see figures 2.2 and 2.4). However, the diagram is limited at temperatures below 400K where crystallisation occurs, and above 650K where the alkyl chains decompose. Pentane + polyethylene systems do not exhibit a change from type IV to type III behaviour (see chapter 1 for a description of the various types) contrary to what is found in methane-polyethylene systems. Type IV is always observed, even for very high molecular weight of the polyethylene (500 kg mol−1 ). As the asymmetry of the two compounds is increased, the region of liquid-liquid immiscibility becomes more extensive, but the two critical lines never meet. The liquid-liquid critical line at low temperature is predicted at temperature around 40 K (not shown in figure 5.14).


229

Figure 5.15: a) Vapour-liquid equilibria (bubble point curves) obtained with SAFTVR for the mixture pentane polyethylene (LDPE, M W = 76 kg.mol−1 ), at temperatures T = 150.5◦ C and T = 201◦ C. b) Vapour liquid and liquid-liquid equilibria obtained at T = 201◦ C for the mixture n-pentane + polyethylene (LDPE, M W = 76kg.mol−1). W pentane is pentane weight fraction. No cross interaction parameter is used (kij = 0). The calculated vapour liquid coexistence curve (continuous curves) is compared with experimental data [389] (black circles). The while circle denotes the calculated UCSP. The dashed curves represent the three phase line.


230

The lower critical end points appears to tend to a limiting temperature as the molecular weight of the polyethylene is increased. These results are in agreement with experimental data since UCST behavior is never observed in practice for pentane-polyethylene systems. Following the work of McCabe et al. [192], we have modelled both vapour-liquid and liquid-liquid equilibria for pentane + polyethylene systems. It is assume that there are no polymer molecules in vapour phases. As a result, the polymer chain length polydispersity has no effect on the vapour-liquid coexistence curve. The polymer is assumed to be monodisperse and the experimental number average molecular weight hM W in is used. A pressure composition diagram obtained for the system n-pentane + low density polyethylene (LDPE) of molecular weight 76 kg mol−1 is represented in figure 5.15. The temperatures considered (150.5 and 201◦ C) are above the melting point of polyethylene so that the polymer is completely amorphous. Only the bubble point curves are shown in the diagram as the dew point curve are confused with the x axes. Good agreement with the experimental vapour-liquid experimental data [389] is found all temperatures (see figure 5.15 a)). We observe that at temperature 201 ◦ C K, the vapour pressure is slightly overestimated. In figure 5.15 b), the liquid-liquid coexistence predicted with SAFT-VR is also shown. This curves ends at an upper critical solution pressure (UCSP). We have also compared the predictions with the liquid-liquid equilibria data of Xiong et al [360]. The results are shown in the figures 5.16-5.17. Here again the polyethylene is assumed to be monodisperse as the experimental polydispersity index is close to 1. Good agreement is obtained with the cloud curves: SAFT-VT can be used to accurately predict the influence of pressure and polyethylene molecular weight on the cloud curves and LCSTs. We recall that no cross parameters are used here. Some deviations can however be noted and the correlations (5.32) could be slightly modified to improve the predictions of the LCSTs. To do so, the correlations (5.32) could be determined by fitting simultaneously pure n-alkane data (vapour pressures and saturated densities), P V T data of polyethylene, and cloud curves of PE + n-alkane systems. We plan to follow this type of approach in future work.

5.3.2

Influence of the Polymer Parameters on Cloud Curves

It can be seen in figure 5.18 that a tiny change in the depth of the potential ² 22 for the polymer segments has a big effect on the cloud curves. In figure 5.19 we


231

represent the different segment-segment attractions between solvent-polymer and polymer-polymer molecules. The overall attractive energy of for a solvent-molecule interaction is proportional of m²12 , while the energy for a polymer-polymer interaction is proportional of m2 ²22 . The chain length m is around 1000, a small changes in ²22 and ²12 makes a big change on the total attractive energy and thus on the phase behaviour. This result has been also observed by Lipson et al. [27, 70]


232

Figure 5.16: Cloud curves calculated with SAFT-VR at several pressures for the mixture n-pentane+polyethylene (M W = 108 kg mol−1 ) and compared to experimental data [360] (circles). No cross interaction parameter is used. k ij = 0

Figure 5.17: Cloud curves calculated at pressure P = 5 MPa with SAFT-VR for the mixtures n-pentane + polyethylene(MW= 16.4 kg mol−1 ) and pentane + polyethylene(MW = 2.150 kg mol−1 ), and compared with experimental data [360] (circles). No cross interaction parameter is used (kij = 0).


233

Figure 5.18: Influence of the parameter ²22 of the polymer on the cloud curve calculated with SAFT-VR at pressure P = 10 MPa for the mixtures pentane + polyethylene (M W = 108 kg mol−1 ), and compared with experimental data [360]. (circles). A small change of ²22 give rise to a shift of 5 K for the LCST.

Figure 5.19: Schematic representation of the attractions between solvent-polymer and polymer-polymer segments. The attraction energy for solvent-polymer interactions evolve with a factor m2 ²12 , while the attraction energy for polymer-polymer interactions evolve with a factor m22 ²22 .


5.3.3

234

Other alkanes + Polyethylene

Finally, we have examined other n-alkanes solvents and observed the effect of the molecular weight of the solvent on the LCSTs. As the solvent becomes heavier, it is more compatible with polyethylene and the LCST is increased. We have compared the LCSTs calculated with SAFT-VR and with the BYG theory [70]. One can see from 5.20 that the SAFT-VR approach gives a better description of the LCST than the BYG for short alkanes, however the results are less satisfactory for long alkanes. The deviations can be due to the presence of polydispersity, branching or that our correlations are not adequate enough.

Figure 5.20: Effect of the chain length of the n-alkane on LCST, for solutions of n-alkane + polyethylene (M W = 140 kg.mol−1 ). The predictions of the SAFT-VR theory (black square) and the BYG theory (white circles) are compared with the experimental data [390], [351], [391], [392] (circles). No cross interaction parameter is used (kij = 0).

235


5.4

Solubility of Gases in Amorphous Polyethylene

We focus now on the vapour-liquid equilibria in polyethylene + gas systems. The solubility Sol of a gas in a polymer is defined as the mass of absorbed gas divided by the mass of polyethylene (in percent), and is given by

Sol = 100

massgas wgas = 100 , masspoly wpoly

(5.35)

where wgas and wpoly are the weight fractions of gas and polymer in the liquid phase. The absorption curve (solubility vs Pressure) corresponds to the bubble point curve of the vapour-liquid equilibria (Pressure vs gas weight fraction) for the gas and the polymer. The corresponding absorption curve to the pressure composition diagram of the system pentane + LDPE is shown in figure 5.21. It can be seen in figure 5.22 that the solubility of gas decreases as the molecular weight of polyethylene is increased. This is due to the increase of incompatibility between the gas and the polymer as the difference in size is increased (see chapter 2 for further explanations). It can be also observed that the absorption curve tends to a limiting curve as the molecular weight of polyethylene tends to infinity. The limiting values of solubility are reached for molecular weights higher than about 5 kg mol−1 . When the chain length is large enough, the solubility of the gas is no longer affected, as the gas molecules interact with polymer at the level of the polymer segments.


236

Figure 5.21: Solubility of pentane in amorphous LDPE (M W = 76 kg mol−1 ). SAFT-VR predictions (continuous curves are compared with experimental data [389] (black circles). No cross interaction parameter is used (kij = 0).

Figure 5.22: Solubility of pentane in polyethylene calculated with SAFT-VR at T = 150.5◦ C, for different molecular weights of the polyethylene. No cross interaction parameter is used (kij = 0). The thick line represents the absorption curve of pentane in an infinitely long and linear polyethylene.

237


5.4.1

Small Gases

The solubility of some small gas molecules (ethylene, and nitrogen) in amorphous polyethylene have been calculated with SAFT-VR. To predict the experimental data, one requires some cross interaction parameters. The mixing rules for small molecules deviate from from Lorenz-Berthelot combining rules because the difference between the ² parameters of the small gases and polyethylene is important. However, thye knowledge of a single cross interaction parameter is sufficient for each gas to predict (²)

the solubility over the entire range of temperatures. We have used k12 = 0.075 for (²)

the system ethylene + PE and k12 = 0.15 for the system N2 + polyethylene. The absorption curves are well predicted by SAFT-VR for both gases. It can be seen in figures 5.21, 5.23 and 5.24, that the solubility of the gas is increased if the gas is less volatile (or longer). The solubilities are in this order : SolnC5 > SolC2 = > SolN2 . Usually the solubility of the gas decreases with increasing temperature (see pentane + PE in figure 5.21, and ethylene + PE in 5.23), as the gas becomes more volatile with increasing temperature. Moreover, the absorption curve is more and more concave as the gas becomes less and less volatile. This feature is analogue the phenomenon of condensation which occurs in the adsorption of gas on the surface of solids. The liquid phase is a mixture of pentane and PE. As pentane molecules absorb in this mixture, they encounter more and more pentane molecules. Since the pentane-pentane interactions are attractive and more favorable than the polymer segment - pentane interactions, there is a synergy effect and the absorption curve has an increasing exponential shape. However, the ethylene-ethylene interactions are not as attractive as the pentane-pentane interactions, and the synergy effect is less pronounced: the solubility curve of ethylene is almost a straight line (see figure 5.23) and Henry’s law could be applied. At high temperatures, the nitrogen-nitrogen interactions are almost purely repulsive (low ²N2 ). In this case the synergy is inverse and the curve is convex (see figure 5.24) .


238

Figure 5.23: Solubility of ethylene in amorphous LDPE (M W = 248 kg mol−1 ). Comparison of experimental data [393] (T = 126◦ C, squares, and T = 155◦ C, circles) and SAFT-VR predictions (continuous lines). One cross interaction parameter (²) k12 = 0.075 is used for all temperatures.

Figure 5.24: Solubility of Nitrogen (N2 ) in amorphous HDPE (M W = 111 kg mol−1 ). Comparison of experimental data [394] (T = 160◦ C, squares, and T = 300◦ C, circles) and SAFT-VR predictions (continuous lines).One cross interaction (²) parameter k12 = 0.15 is used for all temperatures.


239

An other particularity of the solubility of nitrogen in polyethylene is observed experimentally and well predicted by SAFT-VR, where the solubility of N2 in PE is seen to decrease with decreasing temperature (see figure 5.24), contrary to the solubility of pentane and ethylene in PE. This result can be explained as follows: at high temperatures, the nitrogen-nitrogen are almost purely repulsive (nitrogen behaves almost as a hard dumbbell) so that the volatility of nitrogen does not increase much as the temperature is increased. Due to the ideal entropy of mixing, the nitrogen molecules will tend to mix more with polyethylene at higher temperatures. Consequently, the solubility of nitrogen increases with increasing temperature. These high temperatures correspond to the right side of the pressure temperature P T diagram of the binary mixture N2 + PE : in this region of the P T diagram, the vapour-liquid critical pressure of the mixture N2 + PE decreases as the temperature is increased, and this is consistent with the evolution of nitrogen solubility. Similar results are obtained for the solubility of hydrogen (H2 ) in n-alkanes and PE (see next section, figures 5.25 to 5.28 ).

5.4.2

Solubility of Hydrogen in Polyethylene

It is necessary to determine the SAFT-VR parameters of pure hydrogen H 2 in order to predict the absorption of hydrogen in polyethylene. Within the SAFT approach one is not able to model hydrogen with only one set of parameters, since for the hydrogen molecule, the quantum effects due to the vibration modes of the H-H bond are not negligible at low temperatures below the critical point of H 2 . As a result, one can not obtain reliable parameters for H2 by fitting vapour pressures and saturated liquid densities. Hoverer, at higher temperatures, the quantum effects can be neglected. One could then optimised the SAFT parameters (m, ², σ, λ) for P V T data of pure hydrogen at high temperatures. However, at these temperatures, hydrogen almost behaves as an ideal gas and many different sets of parameters (m, ², σ, λ) would be consistent with the P V T experimental data. In order to obtain reliable parameters for hydrogen at high temperatures, an other way is to fit the pure hydrogen parameters on vapour-liquid equilibria data of binary mixtures. The binary mixtures H2 + n-alkanes is one of the most convenient mixtures that we can use in this regard, as we would like to predict the solubility of H2 in polyethylene. We have used the VLE experimental data of binary systems (H2 + propane [395], H2 + n-decane [396], H2 + n-hexadecane [397]) and carried out two kinds of optimisation:


240

• The parameters (m, ², σ, λ) of pure hydrogen are optimised for the VLE data of H2 + n − alkane binary mixtures, without any binary cross interaction (²)

(λ)

parameters (see equations (5.29), (5.30), kij = 0, kij = 0). • The parameters (m, ², σ, λ) of pure hydrogen and the cross interaction param(²)

(λ)

eters (kij , and kij ) are both optimised for the VLE data of H2 + n − alkane (²)

(λ)

binary mixture. Only one set of parameters kij and kij is used for all binary mixtures and temperatures considered. The results are shown in figures 5.25, 5.26 and 5.27 and the optimised parameters are given in table 5.4. Good agreement can be obtained on both bubble an dew point curves, for all temperatures and binary mixtures. It can be seen that the use (²)

(λ)

of binary parameters kij and kij does not improve much the results, but the pure hydrogen parameters are quite different in the two cases. As a result, we decide (²)

(λ)

not to use any binary parameter (kij = 0, kij

= 0) in order to have a more

predictive model. Our approach enables us to obtained similar accuracy as other methods (complex mixing rules with binary parameters sometimes depending on temperatures and on the number of carbon atoms of the n-alkane, see [349,350,398]). However, as we do not use any cross interaction parameters, our method is more predictive and can be extrapolated with confidence to predict the absorption of hydrogen in polyethylene. The pure parameters of hydrogen obtained in the first optimisation are very similar to those obtained by Wang et al. [348] by path integral simulation with a potential different from square-well. This result confirms the validity of our approach. We use the parameters of optimisation 1 to predict the solubility of hydrogen in polyethylene (see figure 5.28). It is seen that the solubility of H2 is increased if the temperature is increased, as for the solubility of N2 in HDPE (cf. figure 5.24).


241

Figure 5.25: Pressure composition P x diagram of the binary system H 2 + propane. The symbols denote the VLE experimental data from the DECHEMA series [395]: squares: T= 223 K; crosses: T = 298 K; circles: T = 348K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions of SAFT-VR with the cross inter(²) (λ) action parameters kij = −0.0524, kij = −0.1963 (opt 2).

Figure 5.26: Pressure composition P x diagram of the binary system H 2 + n-decane. The symbols denote the VLE experimental data from Sebastian et al. series [396]: squares: T = 462.45 K; crosses: T = 503.35 K; circles: T = 583.45 K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent the predictions of SAFT-VR with (²) (λ) the cross interaction parameters kij = −0.0524, kij = −0.1963(opt 2).


242

Figure 5.27: Pressure composition P x diagram of the binary system H 2 + nhexadecane. The symbols denote the VLE experimental data from Lin et al. [397]: squares: T = 461.65 K; crosses: T = 542.25 K; circles: T = 622.85 K; triangles: T = 664.05 K. The continuous curves represent the predictions of SAFT-VR without any cross interaction parameter (opt 1), and the dotted lines represent (²) the predictions of SAFT-VR with the cross interaction parameters k ij = −0.0524, (λ)

kij = −0.1963 (opt 2).

Figure 5.28: Solubility of hydrogen in HDPE (M W = 100 kg.mol−1 ), at T = 120◦ C and T = 180◦ C. The hydrogen parameters of optimisation 1 are used. No cross (²) (λ) interaction parameter is used (kij = 0, kij = 0)


243

Table 5.4. SAFT-VR square-well parameters for pure gas hydrogen and binary cross interaction parameters with n-alkanes (optimisation 2) obtained after optimisation on the VLE data of the binaries (H2 + propane, H2 + n-decane, H2 + n-hexadecane).

Opt 1 Opt 2

5.5

(²)

(λ)

m

λ

σ (˚ A)

²/k (K)

kij

kij

1.0 1.0

1.800 1.594

3.1503 3.1930

37.0182 17.9868

0.0 -0.0524

0.0 -0.1963

Crystallinity of Polyethylene

As for other polymers, polyethylene does not exhibit the crystallisation behaviour of a simple pure component. The introduction into the polymeric chain of units that differ chemically, stereo-chemically, or structurally from the predominant chain repeating units, imposes restrictions on the crystallisation and fusion process. According to the Gibbs phase rule (see chapter 1, section 1.2.7) applied to a pure component, there is only one melting point Tm corresponding to a given pressure P ; if the temperature T is just below Tm , the equilibrium state pure component sample is totally solid, and at temperature T just above the fusion temperature T m the component is totally liquid. A polymer is not a pure component, but a mixture of compounds varying in size, and chemical structures. Moreover, some elements of the polymer chain can not crystallise, such as alkyl branches, as they can not arrange themselves into a solid structure (in the usual timescale of the experiment). One can define a melting point Tm for the polymer sample similar to the melting point of a pure component, such that the polymer sample is totally amorphous at temperatures T > Tm . However, at temperature T < Tm , only a fraction wcyrs of the polymer exists as crystallised units; this is referred to as the crystallinity of the polymer. Under this melting point, the polymer sample is semi-crystalline, as it consists of amorphous regions which exhibit a liquid-like structure, and ordered crystallised regions called crystallites (see figure 5.29). Polyethylene crystallises in lamellar structures with a certain average thickness hζi. The lamellar thickness hζi affects the melting point Tm due to surface effects [323]: the melting temperature decreases as the average lamellar thickness decreases according to the ThompsonGibbs equation.

244


Figure 5.29: Schematic representation of the structure a semi-crystalline polymer. The crystallites have an average thickness hζi.

5.5.1

Experimental Measurements of Crystallinity

A summary of the various techniques to measure the crystallinity of a polymer sample is given in reference [324]. The main experimental techniques include: Dilatometry The most widely used property is the density of the polymer sample, for the reason that it can be measured simply and with great accuracy. The polymer densities can be measured with a dilatometer. The density of the crystallites can be as much as 10% greater than that of the amorphous regions. As the density measurements are very precise, the crystallinity wcrys can be obtained with the relation [399]:

wcrys =

ρ − ρ a ρc , ρc − ρ a ρ

(5.36)

where ρ, ρc , and ρa are the densities in (g cm−3 ) of the polymer sample, the crystallites, and the amorphous regions respectively. The relation (5.36) is exact as it is derived from a mass balance equation; the problem consists in determining accurately the densities ρc and ρa . In the case of polyethylene systems, these two densities may vary from one sample to another, however, it is a good approximation to use the same densities ρc and ρa for all kinds of polyethylene samples (HDPE, LDPE, LLDPE). We propose slightly different values from those of Yiagopoulos [399] for the densities ρc and ρa by fitting ρc and ρa to experimental data involving different types of


245

polyethylene (published data by [327, 328, 330, 336]). The crystallinities were measured by DSC or X-rays (see the next sections for further explanations about these techniques). We obtained ρc = 1.005 g mol−1 and ρa = 0.862 g mol−1 . It can be seen from figure 5.30 that the agreement between the experimental data and equation (5.36) is good. Some significant deviations (max 12%) are however observed for certain polymer samples, which can be due to the assumption that ρc and ρa have the same values for all polyethylenes, or due to some errors in the measurements as the accuracy of the DSC technique is about 10%.

Figure 5.30: Crystallinity of various polyethylene samples (HDPE, LDPE, LLDPE) as a function of their densities. The experimental data are from McKenna [327], Jordens et al. [328], Moore et al. [336], Starck et al. [330], and the continuous line corresponds to the equation (5.36) with ρc = 1.005 g. mol−1 and ρa = 0.862 g mol−1 .

X-ray Diffraction The scattering of X-rays by a polymer sample gives rise to diffraction rings. The amorphous regions give rise to broad diffraction regions, while crystallites give rise to diffraction peaks. If the broad diffraction regions due to amorphous regions is distinguishable from the crystallite diffraction peaks, then the ratio of the two intensities provides a measure of the ratio between amorphous and crystalline regions. This technique is widely used and can measure crystallinities ranging from 0.2 to 0.8.


246

Differential Thermal Analysis Differential Thermal Analysis, also called Dynamic Scanning Calorimetry (DSC), is a versatile technique for a rapid characterisation of several aspects of polymer crystallisation and melting. The DSC method consists in measuring the heat flow received by a sample as the temperature is increased. The curve of the heat flow vs temperature provides several peaks, and the highest peak corresponds to the melting point of the sample. Knowing the enthalpy of fusion ∆Hu of the polymer units (for polyethylene, ∆Hu ' 296 J/g. [330, 333, 334]), the crystallinity can be obtained by calculating the integral of the curve of the heat flow/ temperature. Improved DSC techniques can be used to measure crystallinity as a function of temperature [327], [330].

5.5.2

Flory’s Theory of the Fusion Behaviour of Copolymers

The most robust theory in the literature to predict the crystallinity of polymer samples at equilibrium as a function of temperature was developed fifty year ago by Flory [331]. Although this theory is old, tests of the theory are very scarce in the literature [333, 334], since accurate techniques to measure crystallinities as a function of temperature have only been developed recently [327], [330]. Flory’s idea is based on the following argument: the crystallinity wcrys represents the weight fraction of crystallised polymer in the sample. As the mass of a polymer molecule is proportional to the number of units in the chain (the proportionality factor is the molecular weight of the monomer), the crystallinity also corresponds to the total fraction of crystallised polymer units, or to the probability that a given polymer unit is crystallised. Flory developed then his theory by evaluating this probability as a function of temperature and some properties of the copolymer. In Flory’s approach, one can call a ”copolymer” any polymer containing different types of units along the main chain, such as alkyl branches. For instance, HDPE and LDPE which are made with solely with ethylene, can be treated in this approach as a pseudo-copolymer, since they contain both crystallisable ethylene groups and noncrystallisable units such as alkyl branches along the chain. Following Flory’s development, we consider a model polymer which contains crystallisable units designated as ”A” units, and noncrystallisable units designated as ”B” units. We define the crystallinity wcrys as the mass of crystallised polymer in the sample. wcrys is


247

then equal to the fraction of crystallised polymer units at equilibrium. One should not confuse the fraction of crystallisable units A XA = 1 − XB in the chain, which is the fraction the polymer units susceptible to crystallise, with the fraction of crystallised units equal to the crystallinity wcrys . In the case of polyethylene, A units would represent ethylene groups while B units represent branches along the chain. It may be reasonable to assume that polymer samples containing a high fraction X A of crystallisable A units are likely to have at equilibrium a larger crystallinity wcrys , however this is not always the case. The frequency of A units along the chain is also an important factor.

Figure 5.31: Schematic representation of alternating and block-type copolymers. Thick lines represent crystallisable A units, and thin lines represent noncrystallisable B units. Block-like copolymers is more likely to crystallise than alternating-type copolymers, with the same mole fraction XA of crystallisable A units. As is illustrated in figure 5.31, a block-like copolymer has a higher crystallinity than an alternative-like copolymer, as the probability that a block of consecutive A units is crystallised increases with an increase in the number of consecutive A units in the block. One can define the probability p that a given A unit in the chain is followed by another A unit along the same chain (in any direction). The probability p is used as an average, independently on the direction and the position of the A unit chosen in the chain. This probability p provides a quantitative differentiation between random, alternative, and block-type copolymers (see chapter 1): for random-like copolymers, p is equal to the fraction of A units p = XA ; for block-type copolymers,


248

p greatly exceeds XA , as the A units are more likely to be in blocks; for alternating types of copolymers, p is around zero as each A unit is surrounded by two B units. In Flory’s theory, the longitudinal development of the crystallites is restricted by the occurrence of noncrystallisable B units along the polymer chain. The lateral development is governed by the availability or concentration of sequences of suitable length in the residual melt, and by the decrease in free energy that occurs when a sequence of ζ A units is transferred from the amorphous to the crystalline phase. A quantitative formulation is developed by relating to the melt composition the probability Pζ that a given A unit in the amorphous phase is located within a sequence of at least ζ such A units. Let wζ be the probability that a unit chosen at random from the amorphous regions is an A unit, and also that this A unit belongs to a block of exactly ζ A units. It can be shown [323, 331] that the probabilities P ζ and wζ are related by

wζ = ζ (Pζ − 2Pζ+1 + Pζ+2 ) .

(5.37)

The quantities Pζ and wζ can be related to the constitution of both the initially molten polymer, and to the partially crystalline one. In the completely molten copolymer, prior to any crystallisation, the initial probability wζ0 is given by

wζ0 =

NA,ζ , N

(5.38)

where NA,ζ is the numbers of A units in the molten polymer belonging to blocks (or sequences) containing exactly ζ consecutive A units, and N the total number of units (A + B). Let νζ0 be the number of sequences of ζ A units in the molten polymer. As NA,ζ = ζνζ0 , and as NA = XA N where NA and XA are the total number of mole fraction of A units respectively, wζ0 can be written as

wζ0 =

XA ζνζ0 . NA

(5.39)

By assuming that the probability p of an A unit being succeeded by another A unit is independent of the number of preceding A units in the sequence, one can write equation (5.39) as


wζ0 =

XA ζ (1 − p)2 pζ . p

249

(5.40)

By combining equations (5.37) and (5.40), the probability Pζ0 that in the initial state a given A unit belongs to a sequence of at least ζ units can be written as Pζ0 = XA pζ−1 .

(5.41)

Equation (5.41) means that the probability Pζ0 is equal to the product of the probability XA of choosing a A units beyond the N total units, and the probability that this given A unit is followed by at least ζ − 1 successive A units. When crystallisa-

tion occurs and thermodynamic equilibrium is maintained, the probability Pζeq that a given A unit in the amorphous regions is located in a sequence of at least ζ units is equal to the Boltzmann factor of the change of energy due to the fusion of a block of ζ units (similar to the acceptance probability of a move in an NVT Monte Carlo simulation), and is given by

Pζeq

∆Fζ = exp − RT µ

¶

,

(5.42)

where ∆Fζ is the free energy of fusion per mole of sequence of ζ units from a mole of crystallites of ζ A units. ∆Fζ can be expressed as

∆Fζ = ζ∆Fu − 2σe ,

(5.43)

where ∆Fu = ∆Hu −T ∆Su is the free energy of fusion per mole of units and σe is the surface free energy at the crystallite ends per mole of sequences. For polyethylene, σe /R = 1190.7 K where R is the ideal gas constant (R = 8, 314411 J. mol −1 . K−1 ); 0 where T 0 is the melting point of an ideal ∆Hu /R = 996.34K, and ∆Su = ∆Hu /Tm m

totally crystallisable polyethylene (see references [330, 333, 334]). Equation (5.42) can be expressed as

Pζeq = where

1 exp (−ζθ) , D

(5.44)


∆Hu θ= R

µ

1 1 − 0 , T Tm

250

¶

(5.45)

.

(5.46)

and

2σe D = exp − RT µ

¶

Equations (5.37) and (5.44 ) can then be combined to give wζeq = ζD−1 (1 − exp((−θ))2 exp (−ζθ) ,

(5.47)

which is the fraction of molten sequences of ζ A units. The necessary and sufficient condition that blocks containing at least ζ A units crystallise is Pζ0 > Pζeq .

(5.48)

There is a critical length ζcrit under which the blocks are not crystallised. This critical length corresponds to , Pζ0crit = Pζeq crit

(5.49)

so that ζcrit is expressed as

ζcrit = −

³

³

ln (DXA /p) + 2 ln (1 − p) / 1 − eθ θ + ln p

´´

.

(5.50)

The inequality (5.48) can also be expressed as

XA ζ 1 p > e−θζ . p D

(5.51)

Except for copolymers exhibiting a high tendency to for alternation, 1/D is greater than XA /p. Thus the inequality (5.51) becomes

θ > − ln p, or

(5.52)


1 R 1 − 0 >− ln p, T Tm ∆Hu

251

(5.53)

Equation (5.52) gives the limiting temperature where crystallisation can occur. The melting point Tm is then 1 R 1 − 0 =− ln p, Tm Tm ∆Hu

(5.54)

Since XA is usually closed to p, one can neglect the term XA /p in equation (5.50) (the term ln (DXA /p) does not affect much the final crystallinity, as shown in [323]), and equation (5.50) becomes

ζcrit = −

³

³

ln D + 2 ln (1 − p) / 1 − eθ θ + ln p

´´

.

(5.55)

0 = ∆H /∆S is the melting temperature of the homopolymer. The temperature Tm u u

This temperature can be considered as a reference melting point including all the effects (molecular weight, lamellar thickness [323]) different from the copolymer 0 from the expressions, one can combine equations (5.54), effects. To remove Tm

(5.55) and (5.45) to give

∆Hu θ= R

µ

1 1 − T Tm

¶

− ln p,

(5.56)

and

ζcrit = −

³

³

ln D + 2 ln (1 − p) / 1 − eθ ∆Hu R

³

1 T

−

1 Tm

´

´´

.

(5.57)

It can be seen in equation (5.57) that only very long sequence of A units can crystallise if the temperature T is close to the melting point. Let wζcris be the fraction (concentration) of sequence of ζ A units which are crystallised, then wζcris = wζ0 − wζeq .

(5.58)

252


Applying the properties of the mathematical series, the total fraction of crystallised units wcris (crystallinity) can be obtained as

wcris (T ) =

∞ X

wζcris

(5.59)

ζ=ζcrit

=

5.5.3

"

XA e−θ p 1 1 − (1 − p)2 pζcrit 2 2 + ζcrit 1 − p − 1 − e−θ −θ p (1 − p) (1 − e ) µ

¶#

.

Modelling of Polyethylene Crystallinity

The main difficulty to model the crystallinity of polyethylene if that the exact mole fraction of crystallisable units XA and probability p are can not be measured. More other, additional effects due to defects, lamellar thickness, molecular weight, and 0 [332–334]. One could chain ends must be included via the reference temperature Tm 0 is equal to the melting temperature of neglect all these effects, and assume that Tm

an infinitely long linear polyethylene which is about 145 ◦ C [194,333]. One also could assume that polyethylene is a random copolymer (p = XA ), and that only ethylene group along the main chain crystallise. Thus XA = 1 − Xbr , where Xbr = XB is the fraction of branches characterised by the number of CH3 groups in the chain which can be measured experimentally. Such approach has been tested recently [333, 334] and it has been shown that these assumptions give rise to an overestimation of both crystallinity and melting point. Sanchez and Eby [332] derived an expression of the melting point taking into account defects, chain ends and lamellar thickness effects. The Sanchez-Eby equation enables to get better predictions of the melting point [333], but some parameters have to be fitted on experimental data, and the prediction capability are restraint. This theory also requires the knowledge of the experimental value of the fraction XA (or XB ) of ethylene units which is not always available. Moreover, the assumption that polyethylene is a random-type copolymer (i.e. p = XA ) may be too crude for block-type polymers: it has been seen experimentally that Ziegler-Natta (ZN) catalysts lead to block-type copolymers, while other catalysts like metallocens (Me) give rise to random-like copolymers [326, 329, 330]. In that case, p ≥ XA depending on the catalyst used. We have then developed a model based on Flory’s theory, to predict the melting point and crystallinity of any polyethylene with the minimum of experimental data.

253


As mentioned by experimentalists ( [61]), the density ρ25 of the polyethylene sample measured at 25◦ C, 1 atm, characterises the degree of branching of the polymer. The density ρ25 is about 1 g.cm−3 for a totally crystallised linear polyethylene, and about 0.86 g.cm−3 for a complete amorphous polyethylene. For HDPE, ρ25 is about 0.96 g.cm−3 and for LDPE and LLDPE, ρ25 is about 0.92 g.cm−3 . ρ25 is the related to the crystallinity wcris,25 at 25◦ C via equation (5.36). Hence wcris,25 also characterises the branching of the polyethylene. Following those ideas, we develop some correlations of the melting point and of the parameters XA , p as a function of wcris,25 only.

Two different correla-

tions have been developed according to the type of catalyst used (ZN and Me), for various polyethylenes (HDPE, LDPE, LLDPE made from different olefins). The mole fraction XA can be removed from equation (5.59) such that the relation wcris (T = 25) = wcris,25 is satisfied. Equation (5.59) then becomes

wcris (T )

=

ζcrit

"

ζcrit,25

"

wcrys,25 p Ã

÷ p

p e−θ 1 1 − 2 2 + ζcrit 1 − p − 1 − e−θ −θ (1 − p) (1 − e ) µ

e−θ25 p + ζcrit25 − (1 − p)2 (1 − e−θ25 )2

µ

¶#

1 1 − 1 − p 1 − e−θ25

(5.60) ¶#!

where wcrys,25 is the experimental value of the crystallinity at 25◦ C (which can be evaluated by equation (5.36) if unknown), and where θ25 and ζcrit,25 are given by equations (5.56) and (5.57) respectively at temperature T = 25◦ C . The melting point Tm in ◦ C is given by

for catalyst ZN

:

2 Tm = 13.689wcrys,25 + 5.015wcrys,25 + 124.33

for catalyst Me

:

2 Tm = −81.498wcrys,25 + 163.3wcrys,25 + 63.415

(5.61)

Both correlations go through the point (wcrys,25 = 1., Tm = 145◦ C) corresponding to an ideal infinity long and linear polyethylene. The probability p i given by

for catalyst ZN

:

2 + 0.1397wcrys,25 + 0.9142 p = −0.0538wcrys,25

(5.62)

,

CHAPTER 5. POLYETHYLENE + HYDROCARBON SYSTEMS for catalyst Me

:

254

2 p = −0.0581wcrys,25 + 0.1279wcrys,25 + 0.9303

For both catalysts, the probability p is equal to 1 for an ideal infinitely long and linear polyethylene which totally crystallises. The melting point Tm and probability p that a A unit is followed by an other A unit along the chain are depicted in figures 5.32 a) and b). It can be seen that both melting Tm and probability p are higher for ZieglerNatta polyethylenes, than for metallocen polyethylenes. This result is in agreement with experimental observations [326, 329, 330]: the Ziegler-Natta polyethylene are block-type copolymers (higher p) and crystallises more (higher Tm ) than metallocen polyethylene which are more random-type copolymers (lower p).


255

Figure 5.32: a) Melting point Tm and b) probability p that a crystallisable A unit is followed by another A units, for polyethylene samples, as a function of the experimental crystallinity wcrys,25 at 25◦ C. In figure a), the white circles denote experimental melting points (references [327, 330]) for PE samples made with metallocen (Me) catalysts, and the black circles denote experimental melting points for PE samples made with Ziegler-Natta (ZN) catalysts. In figure b), the circles (white for Me, black for ZN catalysts) correspond to fitted p parameters of equation (5.60) on experimental crystallinity vs temperatures curves shown in figures 5.33 a) and b).


256

Figure 5.33: Crystallinity wcrys as a function of temperature T , for metallocen a) and Ziegler-Natta polyethylenes b). The symbols represent the experimental data from references [327,330] (see the corresponding papers for the meaning of the names of the PE samples). The continuous curves represent the predictions of the modified Flory equation (5.60) where wcrys,25 is the experimental value, and Tm and p are given by the correlations (5.61) and (5.62).

257


As can be seen in figures 5.33 a) and b), our model can predict very well the crystallinity of a large variety of polyethylenes (ZN or Me PE: HDPE, LDPE, or LLDPE made from butene, hexene or octene). Both qualitative and quantitative agreement is obtained with experimental data: at low temperatures, the crystallinity tends to a finite crystallinity, decreases exponentially as the temperature is increased, then tends to zero as the temperature approaches the melting point Tm . The only necessary parameter is the experimental value of the crystallinity of the sample at 25 ◦ C. If that data is unknown, it can be evaluated by equation (5.36) within 10 % of deviation if the density at 25◦ C is known.

5.6

Solubility of Gases in Semi-Crystalline Polyethylene

At temperatures below the melting point of polyethylene (about 130◦ C), the polyethylene sample is semi-crystalline. The phase behaviour then becomes more complex as there is an equilibrium between three phases (vapour, liquid amorphous polymer, and solid crystallites). It has been observed that the solubility of gases is a linear function of crystallinity [327, 336]. Once can assume that the crystallites behave as steric barriers against the diffusion of the gas molecules so that the gas only absorb in the amorphous regions. One also assume that the polymer molecules are either totally amorphous, or totally crystallised, i.e. there are not partial crystallised blocks along the chain. The latter assumption must be accurate as we showed before that the chain length does not affect much the solubility when the molecular weight is high. We also assume that the presence of gas in the polymer sample does not affect crystallinity and melting point (no cryoscopic effects). A schema representing our model of the system gas + semi-crystalline polyethylene is given in figure 5.34.

5.6.1

Model

The solubility is calculated by

(am)

Sol = 100

wgas wgas = 100 (1 − wcrys (T )) (am) , wpoly w

(5.63)

poly

where wgas and wpoly are the global weight fractions of gas and polymer molecules (am)

in the sample, and wgas

(am)

and wpoly are the weight fractions of gas and poly-


258

mer molecules in the amorphous phase at equilibrium, calculated with SAFT-VR; wcrys (T ) is the crystallinity of the pure polymer sample at temperature T , calculated with equation (5.60).

Figure 5.34: Schematic representation of the absorption of gas in semi-crystalline polyethylene. The grey spheres represent gas molecules and black spheres represent polymer segments. The black zones denote crystallites. It is assumed that the gas only absorbs in the amorphous regions, and that the polymer molecules are either completely amorphous or totally crystallised.

5.6.2

Results

We have predicted the recent data of Moore et al. [336] of absorption ofα-olefins in semi-crystalline HDPE and LDPE. The absorption curves of 1-butene and 1-hexene in HDPE and LDPE are displayed in figures 5.35 and 5.36 respectively. In reference [336], the experimental values of crystallinity at 25◦ C are wcrys,25 = 0.702 for the high density polyethylene and wcrys,25 = 0.504 for the low density polyethylene. For HDPE, we have used the correlations for ZN catalyst and for LDPE, we have used the correlations for Me catalysts (equations (5.61) and (5.62)). The crystallinities of both polymer samples have been calculated at the temperatures of the absorption curves with equation (5.60). It can be seen that SAFT-VR combined with our model for crystallinity can predict very well the solubility of olefins in polyethylene for the entire range of temperatures.


259

Figure 5.35: Solubility of a) 1-butene and b) 1-hexene in semi-crystalline HDPE (M W = 11.49 kg.mol−1 , wcrys,25 = 0.702) at different temperatures. Symbols represent experimental data [336] and continuous lines represent SAFT-VR predictions.


260

Figure 5.36: Solubility of a) 1-butene and b) 1-hexene in semi-crystalline LDPE (M W = 22.01 kg.mol−1 , wcrys,25 = 0.504) at different temperatures. Symbols represent experimental data [336] and continuous lines represent SAFT-VR predictions.


261

The solubility of 1-hexene in LDPE is however underestimated at high pressures (figure 5.36 b)). The deviations may be due to cryoscopic effects: the presence of gas in the polymer sample decreases the melting point and crystallinity of the polymer sample, thus the absorption of the gas is increased. In further work, we are planning to take into account these effects by following some recent approaches [345, 346] to calculate solid-liquid equilibria in polyethylene solutions.

5.7

Effects of co-absorption

The co-absorption effects represent an important industrial problems, as the solubilities of the different gas in polyethylene determine the rate of the polymerisation reaction and the final composition of the monomers included in the polymer chain. We consider here the ternary systems 1-butene + nitrogen +HDPE, and 1-butene + 1-hexene + HDPE. The binary system 1-butene + 1-hexene almost behaves as an ideal mixture. Hence there is no need on cross interaction parameters between 1-butene and 1-hexene to predict accurately the experimental VLE data [400] of that system, as shown in figure 5.37. We also assume that no cross interaction parameters are needed for the binary system 1-butene + N2 . The influence on the solubility of 1-butene in of the presence of an other gas (nitrogen or 1-hexene) is shown in figure 5.38. The partial pressure of 1-butene is fixed, and the partial pressure of the second gas is increased. Very different behaviour are observed depending on the nature of the second: the presence of nitrogen hardly affects the solubility of 1-butene in PE (the solubility slightly decreases), while with increasing partial pressure of 1-hexene, the solubility of 1-butene is dramatically increased. Such effects have been observed experimentally (citeLi69) or predicted by MC simulation [177,339], but have not been explained in terms of the intermolecular interactions.


262

Figure 5.37: Pressure composition diagram at temperature T = 100 ◦ C of the binary 1-butene + 1-hexene. The circles denote experimental data [400] and the continuous lines represent SAFT-VR predictions.

Figure 5.38: Solubility of 1-butene (gas 1) in amorphous HDPE (M W = 100 kg. mol−1 ) at temperature T = 150◦ C, at fixed partial pressure of 1-butene PnC4 = = 0.05 MPa, as a function of the partial pressure of a second gas (nitrogen or 1-hexene).


263

Figure 5.39: Ternary diagrams of the system 1-butene + 1-hexene + amorphous HDPE (M W = 100 kg. mol−1 ) calculated with SAFT-VR at temperature T = 460 K, and pressures P = 1 MPa a) and P = 3 MPa b). The continuous line represent coexistence curves. The dashed lines represent tie lines. The squares denote calculated three phase points.


264

We give here a simple explanation in terms on thermodynamic interactions: in the system 1-butene + nitrogen + PE, nitrogen behaves almost as an ideal gas at T = 460K and the interactions nitrogen - nitrogen and 1-butene - nitrogen are repulsive. Due to the ideal entropy of mixing, N2 absorbs slightly in PE. The presence of N2 in PE then decreases a little the solubility of 1-butene in PE as 1butene - nitrogen interactions are unfavorable. In the case of the system 1-butene + 1-hexene + PE, the interactions 1-butene - 1-hexene are attractive (almost similar to the interactions 1-butene - 1-butene). As a result, the presence of 1-hexene in PE gives rise to an increase of the solubility of butene and the absorption curve is concave (see section 5.4.1 for further explanations about the concavity of the curve). Ternary diagrams of the system 1-butene + 1-hexene + amorphous HDPE have been calculated with SAFT-VR at T = 460 K, and pressures P = 1 MPa (figure 5.39 a)) and P = 3 MPa (figure 5.39 b)). At low pressures, a broad region of vapourliquid region can be seen and the solubility of both olefins in PE are low. However when the pressure is increased, the diagram changes dramatically as liquid-liquid immiscibility occurs. The solubility of the gas 1-hexene in PE is increased a lot. There is liquid-liquid coexistence between 1-butene and PE and the concentration of 1-butene in both liquid is very high. Such diagrams could be used to design separation processes as the weight fractions of a given gas can change dramatically by changing pressure.

5.8

Conclusion

We have used the SAFT-VR theory to model the vapour pressures are saturated densities of a large variety of pure components (Hydrocarbons and associating compounds). By using simple extrapolations with molecular weight, we have obtained the SAFT-VR parameters of polyethylene from those of the n-alkanes. Good agreement with experimental data is obtained for the absorption of gases in amorphous polyethylene. The absorption of the gas hydrogen in polyethylene can also be predicted, by fitting the SAFT parameters of hydrogen on vapour-liquid experimental data of binary mixtures of hydrogen + n-alkanes. The results obtained for the cloud curves in polyethylene + n-alkane systems are also satisfactory. This confirms the capability of the SAFT theory to extrapolate the parameters from one system to an other. This extrapolation can be made


265

with confidence as the SAFT parameters have physical meanings. One could however improve the predictions of the cloud curves by fitting the SAFT parameters of polyethylene directly on experimental cloud points, and change the relation between the number of segments m and the molecular weight of polyethylene. The liquid-liquid region is indeed very sensitive to the parameters, and one could change slightly the correlations (5.32) to get better agreements with the experimental cloud curves for polyethylene + n-alkane systems. Polyethylene is semi-crystalline when the temperature if below its melting point. In this case, the crystallinity of polyethylene has to be taken into account. We developed an accurate model based on Flory’s theory of copolymer crystallinity to predict the crystallinity of any kind of polyethylene as a function of temperature. This model just requires one data of crystallinity at 25◦ C. Applying this model and the SAFT-VR theory, good predictions can be obtained on the absorption of olefins in polyethylene by assuming that the gas molecule only absorb in the amorphous regions of the polyethylene sample. One could bring improvement to the model by taking into account cryoscopic effects [345, 346]. We also have shown that coabsorption may exhibit strong synergy effects which can be explained by a difference of interactions between the absorbed gases.

Chapter 6

Conclusion After having discussed in a first chapter the different types of phase behaviour encountered in binary mixtures, prototypes of polymer + solvent systems have been studied in order to provide a better understanding of the liquid-liquid immiscibility and LCST behaviour if these mixtures. The phase behaviour and the properties of mixing are examined for simple models of polymer solutions in which the solvent is represented as a hard sphere and the polymer as a chain of tangent spherical segments of same diameter. The TPT1 description of the free energy originally developed by Wertheim is used to treat the hard-sphere chain molecules, and the attractive interactions are described at the simple mean-field level of van der Waals. The diameter and attractive energies are taken to be the same for all segments (symmetrical interactions). One of the main findings presented in the thesis is that attractive interactions play a key role in the fluid phase demixing, and the purely athermal system is shown not to exhibit not exhibit any phase separation. A region of liquid-liquid immiscibility region can be seen at temperatures just below the critical temperature of the pure solvent. The enthalpy of mixing is always negative and decreases as the temperature is increased. The entropy of mixing is positive at low temperature, then becomes negative (for mixtures with low compositions of polymer)at high temperature and favour demixing. The liquid-liquid phase separation in these model polymer-solvent systems corresponds to an LCST which is an entropy driven process. The decrease of entropy at temperature closed to the critical temperature of the solvent can be explained in terms of density effects and a negative volume of mixing. The subtle density changes (contraction) which give rise to demixing occur under specific conditions ( see chapter 2) , and are due to a complex balance between attractive and repulsive terms in the equations of state. The main point is that the demixing is governed by the thermodynamics of the interactions 266

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between the solvent and polymer molecules in terms of the segments, and not by the chain dimension. A comparison is also made between the LCSTs seen in polymer solutions and the LCSTs seen in mixtures of associating compounds. In the latter case, the liquid-liquid phase separation is due to unfavorable dispersive attractions between unlike species, and the LCST behaviour is explained by the presence of an associative attraction (of hydrogen bonding type) between unlike species which is dominant at low temperatures. The nature of the fluid phase separation in colloid-polymer systems is examined in a third chapter, using the Wertheim TPT1 description. Although the common approach to treat colloid-polymer systems is to used scaling theories which involves a knowledge of the chain dimension, we show here that it is possible to treat both components at the “microscopic” level. The colloid particles are represented by a large hard sphere and the polymer by a chain of tangent hard-spherical segments of diameter much smaller than the colloid particles. Contrary to the polymer solutions examined in chapter 2, the colloid-polymer system is treated as athermal (no attractions). The main conclusion is that the incorporation of repulsive interactions between polymer-polymer segments leads to significant differences with the theoretical predictions for ideal chains (which is the common Asakura-Oosawa model): the critical density for fluid-phase separation tends to a finite value as the chain length is increased if these interactions are taken into account, while the critical density tends to zero for ideal chains. This result is in agreement with recent studies which require information about the chain dimension. [162,249]. We have also calculated the properties of mixing of this system. As the system is athermal, the term “entropy driven” phase separation is often used in the literature to describe the demixing in the colloid-polymer fluid. One should, however, apply caution in the use of this term as it is found that an unfavourable enthalpy of mixing ∆Hm = P ∆V (due to the difference in excluded volume between the polymer and colloid particles) is responsible for the fluid-phase separation. The fluid phase separation in colloid-polymer systems is thus due to a completely different mechanism than in polymer solutions where attractive interactions are central. In future work it would be interesting to study the effect of incorporating attractive interactions in the description of the fluid-phase behaviour in colloid-polymers systems, and discuss the limits of the various regions of fluid-fluid immiscibility. A proper treatment of the fluid-solid phase transition is also necessary at this stage.


268

Polymer are naturally polydisperse in terms of molecular weight, and polydispersity may affect a lot the phase behaviour. We present three different methods to calculate phase equilibria in polydisperse polymer solutions with a SAFT-type equation of state. The first method is based on Jog and Chapman [318] work and can be applied to discrete chain length distributions. The second method is an extension of the first one to calculate cloud and shadow curves with continuous distributions. The third method, developed by Sollich and co workers [320–322], can be applied to continuous distribution and is compared with the two first methods. We discuss the effect of polydispersity on the liquid-liquid immiscibility. In agreement with previous studies, we found that polydispersity enhances the phase separation, and that the shape of the distribution has a big effect on the phase equilibria. We then compare the results obtained with discrete and continuous distributions, and we show that a simple ternary system (solvent + polymer (1)) exhibits very similar cloud and shadow curves as these obtained with Schulz-Flory distributions. We are planning to calculate fluid phase equilibria in polydisperse polymer-colloid systems. Finally, the vapour-liquid and liquid-liquid fluid phase equilibria of HDPE and LDPE polyethylene as very long n-alkane molecules has been modelled with the more sophisticated (and realistic) SAFT-VR description, without taking into account any branching. The molecular parameters of the pure polymer are extrapolated from those of the first members of the n-alkane series, with some correlations as a function of molecular weight. The SAFT-VR approach can qualitatively predict the effects of the various parameters (polymer contribution, nature and concentration of the solvent, temperature and pressure) on both vapour-liquid and liquid-liquid equilibria very successfully, without any temperature dependent cross-interaction parameters. Good predictions of the liquid-liquid regions (cloud curves) and LCSTs are obtained for polyethylene and n-alkane systems. The SAFT-VR approach can also be used to accurately predict the absorption of gases in amorphous polyethylene, which is essentially treated as vapour-liquid equilibria A model based on Flory’s work to predict the crystallinity of polyethylene is also developed. By combining this model with the SAFT-VR equation of state, one is able to predict the absorption of olefins in semi-crystalline polyethylene samples. In future work we are planning to extend our approach to other polymer systems which involves more complicated molecular interactions, such as polyethylene oxide - water solutions. We also intend to improve the theoretical approach by using the


269

dimer version of the SAFT equation and extend the Wertheim’s association theory to higher order, in order to take into account branching.

Bibliography [1] Asakura, S., and Oosawa, F., 1958, J. Polym. Sci., 33, 183. [2] Vrij, A., 1976, Pure Appl. Chem., 48, 471. [3] van Konynenburg, P. H., and Scott, R. L., 1980, Philos. Trans. R. Soc. London, A298, 495. [4] van der Waals, J. D. Thesis, English Translation and Introductory Essay by J. S. Rowlinson,1988, ’On the Continuity of Gaseous and Liquid States’, Studies in Statistical Mechanics PhD thesis, 1873. [5] Rowlinson, J. S., and Swinton, F. L., Liquids and Liquid Mixtures, Butterworth Scientific, London, 3rd. ed., 1982. [6] de Loos, T. W., and Poot, W., d. S. A. J., 1988, Fluid Phase Equilib., 42, 209. [7] Dohrn, R., and Brunner, G., 1995, Fluid Phase Equilib., 106, 213. [8] Christov, M., and Dohrn, R., 2002, Fluid Phase Equilib., 202, 153. [9] Freeman, P. J., and Rowlinson, J. S., 1960, Polymer, 1, 20. [10] Davenport, A. J., R. J. S., 1963, Trans. Farad. Soc., 59, 78. [11] Kohn, J. P. and, K. Y. J., and Pan, Y. C., 1966, J. Chem. Eng. Data, 11, 33. [12] Enick, R., Holder, G. D., and Morsi, B. I., 1985, Fluid Phase Equilib., 22, 209. [13] Schneider, G. M., 1965, Ber. Bunsenges, Phys. Chem., 70, 10. [14] Aring, H., and Von Weber, U., 1965, J. Prakt. Chem., 30, 295. [15] Hottovy, J. D., Luks, K. D., and Kohn, J. P., 1981, J. Chem. Eng. Data., 26, 256. [16] de Loos, T. W., Wijen, A. J. M., and Diepen, G. A. M., 1980, J. Chem. Thermodyn., 12, 193. [17] Galindo, A., Whitehead, P. J., Jackson, G., and Burgess, A. N., 1996, J. Phys. Chem., 100, 6781. [18] Patel, B. H., Paricaud, P., Galindo, A., and Maitland, G. C., 2003, Ind. Eng. Chem. Res., p. submitted. [19] Hudson, C. S., 1904, Z. Phys. Chem., 47, 113. [20] Timmermans, J., 1923, J. Chem. Phys., 20, 491. [21] Poppe, G., 1935, Soc. Chim. Belg., 44, 640. 270

BIBLIOGRAPHY

271

[22] Schneider, G. M., 1963, Z. Phys. Chem. Frankf. Ausg., 37, 333. [23] Timmermans, J., and Kohnstann, P., 1909, Proc. Sect. Sci. K. Ned. Akad., 12, 234. [24] Timmermans, J., Les Solutions Concentrees, Masson, Paris, 1936. ¨ dheide, K., and Franck, E. U., 1963, Z. phys. Chem. Frankf, Ausg., 37, [25] To 387. ¨ dheide, K., and Franck, E. U., 1967, Chemie-Ungr-Tech., [26] Danneil, A., To 39, 816. [27] Luettmer-Strathmann, J., and Lipson, J. E. G., 1999, Macromolecules, 32, 1093. [28] McHugh, M. A., and Krukonis, T. W., Supercritical Fluid Extraction : Principal and Practice, Butterworth, Boston, 1986. [29] Kirby, C. F., and Mchugh, M. A., 1999, Chem. Rev., 99, 565. [30] Folie, B., and Radosz, M., 1995, Ind. Eng. Chem. Res., 34, 1501. [31] Malcolm, G. N., and Rowlinson, J. S., 1957, Trans. Faraday Soc., 53, 921. [32] Patterson, D., 1969, Macromolecules, 2, 672. [33] Patterson, D., 1982, Poly. Eng. Sc., 22, 64. [34] McMaster, L. P., 1973, Macromolecules, 6, 760. [35] Chapman, W. G., Jackson, G., and Gubbins, K. E., 1988, Mol. Phys., 65, 1057. [36] Jackson, G., Chapman, W. G., and Gubbins, K. E., 1988, Mol. Phys., 65, 1. [37] Keller, A., and Cheng, S. Z. D., 1998, Polymer, 39, 4461. [38] Koningsveld, R., and Staverman, A. J., 1967, J. Polym. Sci. : Part C, 16, 1775. [39] de Loos, T. W., Poot, W., and Lichtenthaler, R. N., 1995, J. Supercrit. Fluids, 8, 282. [40] Sengers, J. V., Kayser, R. F., Peters, C. J., and White Jr., H. J., Equation of State for Fluids and Fluid Mixtures, Vol. 2, Elsevier, Amsterdam, 2000. [41] Prasanna, K. J., and Chapman, W. G., 2001, Macromolecules, 35, 1002. [42] Sollich, P., 2002, J. Phys. Cond. Matter, 14, R79–R117. [43] Curro, J. G., 1974, J. Macromol. Sci., Revs. Macromol. Chem., C11, 321. [44] Rodgers, P. A., 1993, J. Appl. Polym. Sci., 48, 1061. [45] Lambert, S. M., S. Y., and Prausnitz, J. M., ”Equations of State for Polymer Systems”, in Equations of State for Fluids and Fluid Mixtures, Vol. 2, Elsevier, Amsterdam, 2000. [46] Barker, J. A., and Henderson, D., 1976, Rev. Mod. Phys., 48, 587. [47] Taylor, M. P., and Lipson, J. E. G., 1999, J. Chem. Phys., 111, 18.

BIBLIOGRAPHY

272

[48] Flory, P. J., 1941, J. Chem. Phys., 9, 660. [49] Huggins, M. L., 1941, J. Chem. Phys., 9, 440. [50] Flory, P. J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, N. Y., 1953. [51] Staverman, A. J., and Van Santen, J. H., 1941, Rec. Trav. Chim. Pay. B., 60, 76. [52] Qian, C., Mumby, J., S., and Eichinger, B. E., 1991, Polymer Sci.: Polymer Phys. B, 29, 635. [53] Qian, C., Mumby, S. J., and Eichinger, B. E., 1991, Macromolecules, 24, 1655. [54] Solc, K., and Koningsveld, R., 1992, J. Phys. Chem., 96, 4056. [55] Van Opstal, L., and Koningsveld, R., 1992, Polymer, 33, 3433. [56] Bae, Y. C., Shim, J. J., Saone, D. S., and Prausnitz, J. M., 1993, J. Appl. Polymer Sci., 47, 1193. [57] Flory, P. J., Orwoll, R. A., and Vrij, A., 1964, JACS, 86, 3507. [58] Prigogine, I., and Trappeniers, N., M. V., 1953, 15, 93. [59] Beret, S., and Prausnitz, J. M., 1975, AIChE J., 21, 1123. [60] Kim, C.-H., Vimalchand, P., Donohue, M. D., and Sandler, S. I., 1986, AIChE J., 32, 1786. [61] de Loos, T. W., de Graaf, L. J., and de Swaan Arons, J., 1996, Fluid Phase Equilib., 117, 40. [62] Sanchez, I. C., and Lacombe, R. H., 1976, J. Phys. Chem., 80, 2352. [63] Sanchez, I. C., and Lacombe, R. H., 1978, Macromolecules, 11, 1145. [64] Sengers, J. V., Kayser, R. F., Peters, C. J., and White Jr., H. J., Equation of State for Fluids and Fluid Mixtures, Vol. 1, Elsevier, Amsterdam, 2000. [65] Bokis, C. P., Orbey, H., and Chen, C.-C., 1999, Chem. Eng. Prog., 95, 39. [66] Lipson, J., 1991, Macromolecules, 24, 1334. [67] Lipson, J., 1992, J. Chem. Phys., 96, 1418. [68] Lipson, J., and Andrews, S., 1992, J. Chem. Phys., 96, 1426. [69] Lipson, J., and Brazhnik, P., 1993, J. Chem. Phys., 98, 8178. [70] Strathmann, J. L., Scoenhard, J. A., and Lipson, J. E., 1998, Macromolecules, 31, 9231. [71] Luettmer-Strathmann, J., and Lipson, J. E. G., 1998, Fluid Phase Equilib., 151, 649. [72] Taylor, M. P., and Lipson, J. E. G., 1996, J. Chem. Phys., 104, 4835. [73] Dickman, R., and Hall, C. K., 1986, J. Chem. Phys., 85, 4108. [74] Carnahan, N. F., and Starling, K. E., 1969, J. Chem. Phys., 51, 635.

BIBLIOGRAPHY

273

[75] Dickman, R., and Hall, C. K., 1988, J. Chem. Phys., 89, 3168. [76] Honnell, K. G., and Hall, C. K., 1991, J. Chem. Phys., 95, 4481. [77] Tildesley, D. J., and Street, W. B., 1980, Mol. Phys., 41, 85. [78] Denlinger, M. A., and Hall, C. K., 1990, Mol. Phys., 71, 541. [79] Yethiray, A., and Hall, C. K., 1991, J. Chem. Phys., 95, 8494. [80] Hall, C. K., Yethiray, A., and Wichert, J. M., 1993, Fluid Phase Equilib., 83, 313. [81] Gulati, J. H. S., Wichert, M., and Hall, C. K., 1996, J. Chem. Phys., 104, 5220. [82] Gulati, H. S., and Hall, C. K., 1998, J. Chem. Phys., 108, 7478. [83] Wertheim, M. S., 1987, J. Chem. Phys., 87, 7323. [84] Wertheim, M. S., 1984, J. Stat. Phys., 35, 19. [85] Wertheim, M. S., 1984, J. Stat. Phys., 35, 35. [86] Wertheim, M. S., 1986, J. Stat. Phys., 42, 459. [87] Wertheim, M. S., 1986, J. Stat. Phys., 42, 477. [88] Ben-Naim, A., 1971, J. Chem. Phys., 54, 1387. [89] Pratt, L. R., and Chandler, D., 1977, J. Chem. Phys., 67, 3683. [90] Boubl´ık, T., 1986, Mol. Phys., 59, 775. [91] Cummings, P. T., and Stell, G., 1987, Mol. Phys., 60, 1315. [92] Zhou, Y., and Stell, G., 1992, J. Chem. Phys., 96, 1507. [93] Kierlik, E., and Rosinberg, M. L., 1992, J. Chem. Phys., 97, 9222. [94] Ghonasgi, D., Llano-Restrepo, M., and Chapman, W. G., 1993, J. Chem. Phys., 98, 5662. [95] Muller, E. A., and Gubbins, K. E., 1995, Ind. Eng. Chem. Res., 34, 3662. [96] Muller, E. A., and Gubbins, K. E., ”A Review of SAFT and Related Approaches”, in Equations of State for Fluids and Fluid Mixtures, Elsevier, Amsterdam, 2000. [97] Chen, S.-J., and Radosz, M., 1992, Macromolecules, 25, 3089. [98] Paricaud, P., Galindo, A., and Jackson, G., 2002, Fluid Phase Equilib., 194, 87–96. [99] Gil-Villegas, A., Galindo, A., Whitehead, P. J., Mills, S. J., Jackson, G., and Burgess, A. N., 1997, J. Chem. Phys., 106, 4168. [100] Galindo, A., Davies, L. A., Gil-Villegas, A., and Jackson, G., 1998, Mol. Phys., 93, 241. [101] McCabe, C., Galindo, A., Gil-Villegas, A., and Jackson, G., 1998, J. Phys. Chem. B, 102, 4183. [102] Gregorowicz, J., and de Loos, T. W., 2001, Ind. Eng. Chem. Res., 40, 444.

BIBLIOGRAPHY

274

[103] Muller, E. A., and Gubbins, K. E., 1993, Mol. Phys., 80, 957. [104] Phan, S., Kierlik, E., Rosinberg, M. L., Yu, H., and Stell, G., 1993, J. Chem. Phys., 99, 5326. [105] Sear, R. P., and Jackson, G., 1994, Mol. Phys., 81, 801. [106] Shukla, K. P., and Chapman, W. G., 2000, Mol. Phys., 98, 2045. [107] Blas, F. J., and Vega, L. F., 2001, J. Chem. Phys., 115, 4355. [108] Ghonasgi, D., and Chapman, W. G., 1994, J. Chem. Phys., 100, 6633. [109] Tavares, F. W., Chang, J., and Sandler, S. I., 1995, Mol. Phys., 86, 1451. [110] Chang, J., and Sandler, S. I., 1994, Chem. Eng. Science, 49, 2777. [111] Sadus, R. J., 1996, Macromolecules, 99, 7212. [112] Johnson, J. K., 1996, J. chem. Phys., 104, 1729. [113] Kushwaha, K. B., and Khanna, K. N., 1999, Mol. Phys., 97, 907. [114] Sheng, Y.-J., Panagiotopoulos, A. Z., and Kumar, S. K., 1995, J. chem. Phys., 103, 10315. [115] Shukla, K. P., and Chapman, W. G., 1998, Mol. Phys., 93, 287. [116] Srivastava, V., and Khanna, K. N., 2002, Mol. Phys., 100, 311. [117] Boubl´ık, T., 1989, Mol. Phys., 68, 191. ã, M. D., 1990, J. Chem. Phys., 93, 730. [118] Boubl´ık, T., Vega, C., and Pen [119] Walsh, J. M., and Gubbins, K. E., 1990, J. Phys. Chem., 94, 5115. [120] Vega, C., Garzon, B., and Lago, S., 1994, J. Chem. Phys., 100, 2182. [121] Padilla, P., and Vega, C., 1995, Mol. Phys., 84, 435. [122] Vega, C., MacDowell, L. G., and Padilla, P., 1996, J. Chem. Phys., 104, 701. [123] MacDowell, L. G., and Vega, C., 1998, J. Chem. Phys., 109, 5670. [124] Vega, C., Labaig, J. M., MacDowell, L. G., and Sanz, E., 2000, J. Chem. Phys., 113, 10398. [125] Trusler, J. P. M., ”The Virial Equation of State”, in Equations of State for Fluid and Fluid Mixtures, Vol. 1, Elsevier, Amsterdam, 2000. [126] Schweizer, S., and Curro, J. G., 1997, Adv. Chem. Phys., 98, 1. [127] Chandler, D., and Andersen, H. C., 1973, J. Chem. Phys., 57, 1930. [128] Chandler, D., 1973, J. Chem. Phys., 59, 2742. [129] Schweizer, K. S., and Curro, J. G., 1987, Phys. Rev. Lett., 58, 246. [130] Curro, J. G., and Schweizer, K. S., 1987, Macromolecules, 20, 1928. [131] Curro, J. G., and Schweizer, K. S., 1988, Macromolecules, 21, 3070.

BIBLIOGRAPHY

275

[132] Honnell, K. G., Curro, J. G., and Schweizer, K. S., 1990, Macromolecules, 23, 3496. [133] Janssen, R. H. C., Nies, E., and Cifra, P., 1997, Macromolecules, 30, 6339. [134] Chiew, Y. C., 1990, Mol. Phys., 70, 129. [135] Baxter, R. J., 1968, J. Chem. Phys., 49, 2770. [136] Cummings, P. T., Perram, J. W., and Smith, E. R., 1976, Mol. Phys., 31, 535. [137] Chang, J. E., and Sandler, S. I., 1995, J. Chem. Phys., 102, 437. [138] McCabe, C., Kalyuzhnyi, Y. V., Cummings, P. T., and Stell, G., 2002, Mol. Phys., 100, 2499. [139] Song, Y., Lambert, S. M., and Prausnitz, J. M., 1994, Macromolecules, 27, 441. [140] Song, Y., Lambert, S. M., and Prausnitz, J. M., 1994, Ind. Eng. Chem. Res., 33, 1047. [141] Song, Y., Lambert, S. M., and Prausnitz, J. M., 1994, Chem. Eng. Sc., 49, 2765. [142] Sadowski, G., 1998, Fluid Phase Equilib., 149, 75. [143] Gross, J., and Sadowski, G., 2000, Fluid Phase Equilib., 168, 183. [144] Gross, J., and Sadowski, G., 2001, Ind. Eng. Chem. Res., 40, 1244. [145] Yeom, M. S., Chang, J., and Kim, H., 2002, Fluid Phase Equilib., 194, 579. [146] de Gennes, P. G., Scaling Concepts in Polymer Physics, Cornell University Press, New York, 1979. [147] Stockmayer, W. H., 1960, Macromol. Chem. Phys., 35, 54. [148] Nishio, I., Sun, S. T., Swislow, G., and Tanaka, T., 1979, Nature, 281, 208. [149] Swislow, G., Sun, S. T., Nishio, I., and Tanaka, T., 1980, Phys. Rev. Lett., 44, 796. [150] Dijkstra, M., and Frenkel, D., 1994, Phys. Rev. Lett., 72, 298. [151] Dijkstra, M., Frenkel, D., and Hansen, J.-P., 1994, J. Chem. Phys., 101, 3179. [152] Luna-Barcenas, G., Bennett, G. E., Sanchez, I. C., and Johnson, K. P., 1996, J. Chem. Phys., 104, 9971. [153] Escobedo, F. A., and de Pablo, J. J., 1996, Mol. Phys., 89, 1733. [154] Grayce, C. J., 1997, J. Chem. Phys., 106, 5171. [155] Taylor, M. P., and Lipson, J. E. G., 1998, Fluid Phase Equilib., 150, 641. [156] Lekkerkerker, H., Poon, W., Pusey, P., Stroobants, A., and Warren, P., 1992, Europhys. Lett., 20, 559. [157] Meijer, E. J., and Frenkel, D., 1994, J. Chem. Phys., 100, 6873.

BIBLIOGRAPHY

276

[158] Hagen, M. H. J., and Frenkel, D., 1994, J. Chem. Phys., 101, 4093. [159] Suen, J. K., Escobedo, F. A., and de Pablo, J. J., 1997, J. Chem. Phys., 106, 1288. [160] Dijkstra, M., Brader, J. M., and Evans, R., 1999, J. Phys.: Conds. Matter, 11, 10079. [161] Sear, R. P., 2001, Phys. Rev. Lett., 86, 4696. [162] Bolhuis, P. G., Louis, A. A., and Hansen, J.-P., 2002, Physical. Rev. lett., 89, 128302. [163] Aarts, D. A. L., Tuinier, R., and Lekkerkerker, H. N. W., 2002, J. Phys. Condens. Matter, 14, 7551. [164] Varga, S., Galindo, A., and Jackson, G., 2002, J. Chem. Phys., 117, 7207. [165] Bruns, W., and Bansal, R., 1981, J. Chem. Phys., 74, 2064. [166] Bruns, W., and Bansal, R., 1981, J. Chem. Phys., 74, 5149. [167] Toxvaerd, S., 1987, J. Chem. Phys., 86, 3667. [168] Smit, B., Van der Put, A., Peters, C. J., de Swaan Arons, J., and Michels, J. P. J., 1988, Chem. Phys. Lett., 144, 555. [169] Luna-Barcenas, G., Gromov, D. G., Meredith, J. C., Sanchez, I. C., de Pablo, J. J., and Johnston, K. P., 1997, Chem. Phys. Lett., 278, 302. [170] Polson, J. M., and Zuckerman, M. J., 2002, J. Chem. Phys., 116, 7244. [171] Chapman, W. G., Gubbins, K. E., Jackson, G., and Radosz, M., 1990, Ind. Eng. Chem. Res., 29, 1709. [172] Barker, J. A., and Henderson, D., 1967, J. Chem. Phys., 47, 2856. [173] Barker, J. A., and Henderson, D., 1967, J. Chem. Phys., 47, 4714. [174] Heuer, T., Peuschel, G., Ratzsch, M., and Wohlfarth, C., 1989, Acta Polym., 40, 272. [175] Wohlfarth, C., Funck, U., Schultz, R., and Heuer, T., 1992, Die Ang. Makromolekulare Chemie, 198, 91. [176] Nath, S. K., Escobedo, F. A., de Pablo, J. J., and Patramai, I., 1998, Ind. Eng. Chem. Res., 37, 3195. [177] Nath, S. K., Banaszak, B. J., and de Pablo, J. J., 2001, Macromolecules, 34, 7841. [178] Shetty, R., and Escobedo, F. A., 2002, J. Chem. Phys., 116, 7957. [179] Hill, T. L., An Introduction to Statistical Mechanics, Addison-Wesley, New York, 1960. [180] Walsh, J. M., and Gubbins, K. E., 1993, Mol. Phys., 80, 65. [181] Chapman, W. G. PhD thesis, Cornell University, 1988. [182] Wertheim, M. S., 1986, J. Chem. Phys., 85, 2929. [183] Sear, R. P., and Jackson, G., 1994, Phys. Rev. E, 50, 386.

BIBLIOGRAPHY

277

[184] Sear, R. P., and Jackson, G., 1996, Mol. Phys., 87, 517. [185] Ghonasgi, D., Perez, V., and Chapman, W. G., 1994, J. Chem. Phys., 101, 6880. [186] Ghonasgi, D., and Chapman, W. G., 1995, J. Chem. Phys., 102, 2585. [187] Boubl´ık, T., 1970, J. Chem. Phys., 53, 471. [188] Mansoori, G. A., Carnahan, N. F., Starling, K. E., and Leland, T. W., 1971, J. Chem. Phys., 54, 1523. [189] Hansen, J. P., and McDonald, I. R., Theory of Simple Liquids, Academic Press, London, 1986. [190] Jackson, G., and Gubbins, K. E., 1989, Pure Appl. Chem., 61, 1021. [191] McCabe, C., and Jackson, G., 1999, Phys. Chem. Chem. Phys., 1, 2057. [192] McCabe, C., Galindo, A., Garcia-Lisbona, M. N., and Jackson, G., 2001, Ind. Eng. Chem. Res., 40, 3835. [193] McCabe, C., Gil-Villegas, A., and Jackson, G., 1998, J. Phys. Chem. B, 102, 4183. [194] Flory, P. J., Principles of Polymer Chemistry, Vol. Chapters 12 and 13, Cornell University Press, Ithaca, N. Y., 1953. [195] Asakura, S., and Oosawa, F., 1954, J. Chem. Phys., 22, 1255. [196] Paricaud, P., Varga, S., and Jackson, G., 2003, J. Chem. Phys., in press. [197] Blas, F. J., 2000, J. Phys. Chem., 104, 9239. [198] Arnaud, J. F., Ungerer, P., Behar, E., Moracchini, G., and Sanchez, J., 1996, Fluid Phase Equilib., 124, 177. [199] Dolgolenko, W., 1908, Z. Phys. Chem., 62, 499. [200] Jackson, G., 1991, Mol. Phys., 72, 1365. [201] Davies, L. A., Jackson, G., and Rull, L. F., 1999, Phys. Rev. Let., 82, 5285. [202] Davies, L. A., Jackson, G., and Rull, L. F., 2000, 61, 2245. [203] Michelsen, M. L., 1982, Fluid Phase Equilib., 9, 1. [204] Michelsen, M. L., 1982, Fluid Phase Equilib., 9, 21. [205] Michelsen, M. L., 1984, Fluid Phase Equilib., 16, 57. [206] Chen, C.-K., Duran, M. A., and Radosz, M., 1993, Ind. Eng. Chem. Res., 32, 3213. [207] Koak, N., and Heidemann, R. A., 1996, Ind. Eng. Chem., 35, 4301–4309. [208] Xu, G., Brennecke, J. F., and Stadtherr, M. A., 2001. [209] Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannary Guevara, B. P., Numerical Recipes in Fortran, Cambridge university press, London, 1986. [210] Frenkel, D., 1999, Physica A., 263, 26.

BIBLIOGRAPHY

278

[211] Frenkel, D., 2002, Physica A., 313, 1. [212] Biben, T., and Hansen, J., 1991, Phys. Rev. Lett., 66, 2215. [213] Lekkerkerker, H. N. W., and Stroobants, A., 1993, Physica A, 195, 387. [214] Caccamo, C., and Pellicane, G., 1997, Physica A, 235, 149. [215] Dijkstra, M., Van Roij, R., and Evans, R., 1998, Phys. Rev. Lett., 81, 2268. [216] Dijkstra, M., Van Roij, R., and Evans, R., 1999, Phys. Rev. E., 59, 5744. [217] Warren, P. B., 1999, Europhys. Let., 46, 295. [218] Cuesta, J. A., 1999, Europhys. Lett., 46, 197. [219] Sear, R. P., 1998, Europhys. Lett., 44, 531. [220] Poon, W., and Pusey, P. N., Observation, Prediction and Simulation of Phase Transitions in Complex Fluids, Baus, M., and Rull, L., and J. Ryckaert, Kluwer, Dordrecht, 1995. [221] Li-In-On, F. K. R., Vincent, B., and Waite, A., 1975, ACS Symp. Ser.,, 9, 165. [222] Calderon, F. L., Bibette, J., and Bais, J., 1993, Europhys. Lett., 23, 653. [223] Pusey, P. N., Pirie, A. D., and Poon, W. C. K., 1993, Physica A., 201, 322. [224] Zimmerman, S. B., and Minton, A. P., 1993, Annu. Rev. Biophys. Biomol. Struct., 22, 27. [225] Pusey, P. N., Poon, W. C. K., Ilett, S. M., and Bartlett, P., 1994, J. Phys.: Condens. Matter, 6, A29. [226] Ilett, S., Orrock, A., Poon, W., and Pusey, P., 1995, Phys. Rev. E., 51, 1344. [227] Poon, W. C. K., Pirie, A. D., and Pusey, P. N., 1995, Faraday Discuss., 101, 65. [228] Dinsmore, A., Yodh, A., and Pine, D., 1995, Phys. Rev. E., 52, 4045. [229] Verhaegh, N. A. M., Van Duijneveldt, J. S., Dhont, J. K. G., and Lekkerkerker, H. N. W., 1996, Physica A., 230, 409. [230] Bodnar, I., and Oosterbaan, W. D., 1997, J. Chem. Phys., 106, 7777. [231] Verma, R., Crocker, J. C., Lubensky, T. C., and Yodh, A. G., 1998, Phys. Rev. Lett., 81, 4004. [232] Moussa¨ıd, A., Poon, W. C. K., Pusey, P. N., and Soliva, M. F., 1999, Phys. Rev. Lett., 82, 225. [233] Renth, F., Poon, W. C. K., and Evans, R. M. L., 2001, Phys. Rev. E, 64, 31402. [234] Ramakrishnan, S., Fuchs, M., Schweizer, K. S., and Zukoski, C. F., 2002, J. Chem. Phys., 116, 2201. [235] Joanny, J. F., Leibler, L., and De Gennes, P. G., 1979, J. Polym. Sci.: Polym. Phys., 17, 1073.

BIBLIOGRAPHY

279

[236] Hanke, A., Eisenriegler, E., and Dietrich, S., 1999, Phys. Rev. E., 59, 6853. [237] Gast, A., Hall, C., and Russel, W., 1983, J. Colloid Interface Sci., 96, 251. [238] Vincent, B., 1987, Colloids Surf., 24, 269. [239] Vincent, B., Edwards, J., Emmett, S., and Croot, R., 1988, Colloids Surf., 31, 267. [240] Pusey, P. N., Colloidal Suspensions. Les Houches: Liquides, Cristallisation et Transition (Vitreuse/Liquids), Freezing and Glass Transition, Elsevier, Amsterdam, 1989. [241] Bolhuis, P., and Frenkel, D., 1994, J. Chem. Phys., 101, 9869. [242] Lekkerkerker, H. N. W., and Widom, B., 2000, Physica A., 285, 483. [243] Oversteegen, S. M., and Lekkerkerker, H. N. W., 2002, Physica A, 310, 181. [244] Brader, J. M., and Evans, R., 2002, Physica A, 306, 287. [245] Schmidt, M., and Fuchs, M., 2002, J. Chem. Phys., 117, 6308. [246] Fuchs, M., and Schweizer, K. S., 2000, Europhys. Lett., 51, 621. [247] Warren, P. B., Ilett, S. M., and Poon, C. K., 1995, Phys. Rev. E., 52, 5205. [248] Fuchs, M., and Schweizer, K. S., 2001, Phys. Rev. E, 64, 021514. [249] Sear, R. P., 2002, Phys. Rev. E, 66, 051401. [250] Meijer, E. J., Louis, A. A., and Bolhuis, P., 2003, to be published. [251] Louis, A. A., Bolhuis, P. G., Hansen, J.-P., and Meijer, E. J., 2000, Phys. Rev. Lett., 85, 2522. [252] Bolhuis, P. G., Louis, A. A., Hansen, J.-P., and Meijer, E. J., 2001, J. Chem. Phys., 114, 4296. [253] Bolhuis, P., Louis, A. A., and Hansen, J.-P., 2001, Phys. Rev. E, 64, 021801. [254] Bolhuis, P. G., and Louis, A. A., 2002, Macromolecules, 35, 1860. [255] Escobedo, F. A., and de Pablo, J. J., 1996, Mol. Phys., 87, 347. [256] Paricaud, P., Galindo, A., and Jackson, G., 2003, Mol. Phys., p. submitted. [257] Moro, M. G., Kern, N., and Frenkel, G., 1999, Europhys. Lett., 48, 332. [258] Jackson, G., Rowlinson, J. S., and Van Swol, F., 1987, J. Phys. Chem., 91, 4907. [259] Williamson, D. C., and Jackson, G., 1995, Mol. Phys., 86, 819. [260] Vega, C., and MacDowell, L. G., 2001, J. Chem.Phys., 114, 10411. [261] Vega, C., 1998, J. Chem. Phys., 108, 3074.

BIBLIOGRAPHY

280

[262] Cotterman, R. L., and Prausnitz, J. M., Kinetic and Thermidynamic Lumping of Multicomponent Mixtures, Elsevier, Amsterdam, 1991. [263] Browarzik, D., and Kehlen, H., ”Polydisperse Fluids”, in Equations of State for Fluids and Fluid Mixtures, Vol. 2, Elsevier, Amsterdam, 2000. [264] Pedersen, K. S., 1983, Fluid Phase Equilib., 14, 209. [265] Pedersen, K. S., Thomassen, P., and Fredenslund, A., 1984, Ind. Eng. Chem. Proc. Des. Dev., 23, 566. [266] Pedersen, K. S., Rasmussen, P., and Fredenslund, A., 1985, Ind. Eng. Chem. Proc. Des. Dev., 24, 948. [267] Pedersen, K. S., Blilie, A. L., and Meisingset, K. K., 1992, Ind. Eng. Chem. Res., 31, 1378. [268] Soave, G., 1972, Chem. Eng. Sci., 27, 1197. [269] Jaubert, J.-N., and Neau, E., 1995, Ind. Eng. Chem. Res., 34, 640. [270] Avaulle, L., Neau, L., and Jaubert, J.-N., 1999, Fluid Phase Equilib., 139, 171. [271] Pedersen, K. S., Thomasen, P., and Fredenslund, A., 1988, Chem. Eng. Sci., 43, 269. [272] Sportisse, M., Barreau, A., and Ungerer, P., 1999, Fluid Phase Equilib., 139, 255. [273] Jaubert, J.-N., Solimando, R., and Paricaud, P., 2000, Ind. Eng. Chem. Res., 39, 5029. [274] Katz, D. L., and Brown, G. G., 1933, Ind. Eng. Chem., 41, 1373. [275] Bowman, J. R., 1949, Ind. Eng. Chem. Res., 41, 2004. [276] Edminster, W. C., 1955, Ind. Eng. Chem. Res., 47, 1685. [277] Hoffman, E. J., 1968, Chem. Eng. Sci., 23, 957. [278] Aris, R., and Gavalas, G. R., 1966, Phil. Trans. Roy. Soc., 260, 351. [279] Roth, H., 1968, Chem. Techn. (Leipzig), 20, 7720. [280] Roth, H., 1972, Chem. Techn. (Leipzig), 24, 293. [281] Dickson, E., 1980, J. Chem. Soc., Faraday Trans. II, 76, 1458. [282] Cotterman, R. L., Bender, R., and Prausnitz, J. M., 1984, Ind. Eng. Chem. Proc. Des. Dev., 24, 194. ¨ Tzsch, M. T., 1983, Z. Phys. Chem. (Leipzig), 264, [283] Kehlen, H., and Ra 1153. [284] Schulz, G. V., 1935, Z. Phys. Chem., 30, 379. [285] Flory, P. J., 1936, J. Am. Chem. Soc., 58, 1877. [286] Stockmayer, W. H., 1945, J. Chem. Phys., 13, 199. [287] Cotterman, R. L., and Prausnitz, J. M., 1985, Ind. Eng. Chem. Proc. Des. Dev., 24, 194. [288] Du, P. C., and Mansoori, G. A., 1986, Fluid Phase Equilib., 30, 57.

BIBLIOGRAPHY

281

[289] Peng, D. Y., and Robinson, D. B., 1976, Ind. Eng. Chem. Fundam, 15, 59. [290] Mansoori, G. A., Du, D. C., and Antoniades, E., 1989, Int. J. Thermophys., 10, 1181. [291] Cotterman, R. L., and Prausnitz, J. M., 1985, Ind. Eng. Chem. Des. Dev., 24, 434. [292] Chou, G. F., and Prausnitz, J. M., 1986, Fluid Phase Equilib., 30, 75. [293] Shibata, S. K., Sandler, S. I., and Behrens, R. A., 1987, Chem. Eng. Sci., 42, 1977. ¨ Tzsch, M. T., 1990, J. Marcromol. [294] Browarzik, D., Kehlen, H., and Ra Sc.i-Chem. A, 27, 549. [295] Browarzik, D., and Kehlen, H., 1996, Fluid Phase Equilib., 123, 17. [296] HU, Y., and Prausnizt, J. M., 1997, Fluid Phase Equilib., 130, 1. [297] Rochocz, G. L., Castier, M., and Sandler, S. I., 1997, Fluid Phase Equilib., 139, 137. [298] Heidemann, A., and Khalil, A. M., 1980, AICHe J., 26, 769. [299] Browarzik, D., and Kowalewski, N., 1999, Fluid Phase Equilib., 163, 43. [300] Sako, T., Wu, A. H., and Prausnitz, J. M., 1989, J. Appl. Polym. Sci., 38, 1839. [301] Schulz, G. V., 1939, Z. Phys. Chem., 43, 25. [302] Koningsveld, R., 1961, Chem. Weekblad, 57, 129. [303] Koningsveld, R. PhD thesis, University of Leiden, 1967. [304] Okamoto, M., and Sekikawa, K., 1961, J. Polym. Sci., 55, 597. [305] Tung, L. H., 1962, J. Polym. Sci., 61, 449. [306] Huggins, M. L., 1967, J. Polym. Sci. A, 2, 1221. [307] Baysal, B. M., and Stockmayer, W. H., 1967, J. Polym. Sci. A, 2, 315. ˇ [308] SOlk, K., 1969, Chem. Commun., 34, 992. ˇ [309] SOlk, K., 1970, Macromolecules, 3, 665. ˇ [310] SOlk, K., 1975, Macromolecules, 8, 819. [311] Kamide, K., Sugamiya, K., Ogawa, T., Nakayama, C., and Baba, N., 1970, Makromol. Chem., 135, 23. [312] de Loos, T. W., Poot, W., and Diepen, G. A., 1983, Macromolecules, 16, 111. [313] de Loos, T. W., Lichtenthaler, R. N., and Diepen, G. A., 1983, Macromolecules, 16, 117. [314] Koak, N., and Heidemann, R. A., 1996, Ind. Eng. Chem. Res., 35, 4301. [315] Koak, N. PhD thesis, University of Calgary, Alberta, 1997. [316] Phoenix, A. V., and Heidemann, R. A., 1999, Fluid Phase Equilib., 158, 643.

BIBLIOGRAPHY

282

[317] Choi, J. J., and Bae, Y. C., 1998, Europ. Polym. J., 35, 1703. [318] Jog, P. K., and Chapman, W. G., 2002, Macromolecules, 35, 1002. [319] Gross, J., and Sadowski, G., 2002, Ind. Eng. Chem. Res., 41, 1084. [320] Sollcih, P., and Cates, M. E., 1998, Phys. Rev. Let., 80, 1365. [321] Warren, P. B., 1998, Phys. Rev. Lett., 80, 1369. [322] Sollich, P., Warren, P. B., and Cates, M. E., 2001, Adv. Chem. Phys., 116, 265. [323] Mandelkern, L., Crystallisation of Polymers, Advanced Chemistry, McGraw-Hill, New York, 1964. [324] Sharples, A., Introduction to Polymer Crystallisation, Edwards Arnold, London, 1966. [325] Yoon, J., Yoo, H.-S., and Kang, K.-S., 1996, Eur. Polym. J., 32, 1333. [326] Marigo, A., Marega, C., Zannetti, R., and SGARZI, P., 1998, Eur. Polym. J., 34, 597. [327] McKenna, T. F., 1998, Eur. Polym. J., 34, 1255. [328] Jordens, K., Wilkes, G. L., Janzen, J., Rohlfing, D. C., and Welch, M. B., 2000, Polymer, 41, 7175. [329] Moreira, S., C., de Fatima, M., and Marques, F. V., 2001, Eur. Polym. J., 37, 2123. [330] Starck, P., Rajanen, K., and Loefgren, B., 2002, Thermochimica Acta, 7040, 1. [331] Flory, P. J., 1955, Trans. Faraday Soc., 51, 848. [332] Sanchez, I. C., and Eby, R. K., 1975, Macromolecules, 8, 639. [333] Kim, M.-H., Phillips, P. J., and Lin, J. S., 2000, J. Polymer. Sc. Part B, 38, 154. [334] Crist, B., and Howard, P. R., 1999, Macromolecules, 32, 3057. [335] Hutchinson, R. A., and Ray, W. H., 1990, J. Polym. Sc. Part B, 41, 51. [336] Moore, S. J., and Wanke, S. E., 2001, Chem. Eng. Sc., 56, 4121. [337] Beret, S., Muhle, M. E., and Villamil, I. A., 1977, Chem. Eng. Progress, 73, 44. [338] Beret, S., and Hager, S. L., 1979, J. Appl. Polym. Sc., 24, 1787. [339] Nath, S. K., and de Pablo, J. J., 1999, J. Phys. Chem. B, 103, 3539. [340] Fukuda, M., 2000, J. Chem. Phys., 112, 478. [341] Zervopoulou, E., Mavrantzas, V. G., and Theodorou, D. N., 2001, J. Chem. Phys., 115, 2860. [342] Budzien, J. L., McCoy, J. D., Curro, J. G., LaViolette, R. A., and Peterson, E. S., 1998, Macromolecules, 31, 6669. [343] Flaconneche, B., Martin, J., and Klopffer, M. H., 2001, Oil. Gas. Sc. Tech.-Rev. IFP., 56(3), 261.

BIBLIOGRAPHY

283

[344] Pan, C., and Radosz, M., 1999, Ind. Eng. Chem. Res., 38, 2842. [345] Ghosh, A., and Chapman, W. G., 2002, Ind. Eng. Chem. Res., 41, 5529. [346] Adidharma, H., and Radosz, M., 2002, Ind. Eng. Chem. Res., 41, 1774. [347] Li, N. M., and Long, R. B., 1969, AIChE J., 15, 73. [348] Wang, Q., Johnson, J. K., and Broughton, J. Q., 1996, Mol. Phys., 89, 1105. [349] Huang, H., Sandler, S. I., and Orbey, H., 1994, Fluid Phase Equilib., 96, 143. [350] Twu, C. H., Coon, J. E., Harvey, A. H., and Cunningham, J. R., 1996, Ind. Eng. Chem. Res., 35, 905. [351] Hamada, F., Fujisawa, K., and Nakajima, A., 1973, Polym. J., 4, 316. [352] Kennis, H. A. J., de Loos, T. W., de Swaan Arons, J., Van Jer Haegen, R., and Kleintjens, L. A., 1990, Chem. Eng. Sci., 45, 1875. [353] Koak, N., Visser, R. M., and de Loos, T. W., 1999, Fluid Phase Equilib., 158, 835. [354] Kiran, E., and Zuang, W., 1992, Polymer, 33, 5259. [355] Kiran, E., Xiong, Y., and Zhuang, W., 1993, J. Supercritical Fluids, 6, 193. [356] Kiran, E., and Zuang, W., 1994, J. Supercritical. Fluids, 7, 7. [357] Kiran, E., Xiong, Y., and Zhuang, W., 1994, J. Supercrit. Fluids, 7, 283. [358] Xiong, Y., and Kiran, E., 1995, J. Appl. Polym. Sc., 55, 1805. [359] Xiong, Y., and Kiran, E., 1994, Polymer, 35, 4408. [360] Xiong, Y., and Kiran, E., 1994, J. Appl. Polym. Sc., 53, 1179. [361] Zhuang, W., and Kiran, E., 1998, Polymer, 39, 2903. [362] Kleintjens, L. A., and Koningsveld, R., 1980, Coll. Polymer Sc., 258, 711. [363] Hino, T., and Prausnitz, J. M., 1997, Fluid Phase Equilib., 138, 105. [364] Gauter, K., and Heidemann, R. A., 2001, Fluid Phase Equilib., 183, 87. [365] Tumakaka, F., Gross, J., and Sadowski, G., 2002, Fluid Phase Equilib., 194, 541. [366] Chapman, W. G., Gubbins, K. E., Jackson, G., and Radosz, M., 1989, Fluid Phase Equilib., 52, 31. [367] Malijevsky, A., and Labik, S., 1987, Mol. Phys., 60, 663. [368] Percus, J. K., and Yevick, G. J., 1958, Phys. Rev., 1, 110. [369] Jackson, G., and Gubbins, K. E., 1989, Pure Appl. Chem., 61, 1021. [370] Stull, D. R., 1947, Ind. Eng. Chem., 39, 517. [371] Morecroft, D. W., 1964, J. Chem. Eng. Data, 9, 488.

BIBLIOGRAPHY

284

[372] Hust, J. G., and Schramm, R. E., 1976, J. Chem. Eng. data, 21, 7. [373] Nikitin, E. D., Pavlov, P. A., and Skripov, P. V., 1996, Int. J. Thermophys., 17, 455. [374] Jorgensen, W. L., Madura, J. D., and Swenson, C. J., 1984, J. Am. Chem. Soc., 106, 813. [375] Smit, B., Karaorni, S., and Siepmann, J. I., 1995, J. Chem. Phys., 102, 2126. [376] Martin, M. G., and Siepmann, J. I., 1998, J. Phys. Chem. B, 102, 2569. [377] Nath, S. A., Escobedo, F. A., and de Pablo, J. J., 1998, J. Chem. Phys., 108, 9905. [378] Ungerer, P., Beauvais, C., Delhommel, J., Boutin, A., Rousseau, B., and Fuchs, A. H., 2000, J. Chem. Phys., 112, 5499. [379] Nelder, J. A., and Mead, R., 1965, Computer Journal, 7, 308–313. [380] Lue, L., and Prausnitz, J. M., 1998, J. Chem. Phys., 108, 5529. [381] Senger, J. M. H. L., 1999, Fluid Phase Equilib., 158, 3. [382] Smith, B. D., and Srivastava, R., Thermodynamic Data for Pure Compounds, Elsevier, Amsterdam, 1986. [383] Chirico, R. D., Nguyen, A., Steele, W. V., and Strube, M. M., 1989, J. Chem. Eng. Data, 34, 149. [384] Smit, B., Karaborni, S., and Siepmann, J., 1995, J. Chem. Phys., 102, 2126. [385] Galindo, A., Burton, S. J., Jackson, G., Visco JR, D. P., and Kofke, D. A., 2002, Mol. Phys., 100, 2241. [386] Handbook of Chemistry and Physics, CRC Press, Boca Raton, 60th ed., 1981. [387] Frank, E. U., and Spalthoff, W., 1957, Elektrochem, 61, 348. [388] Vanderzee, C. E., and Rodenburg, W., 1970, J. Chem. Thermodyn., 2, 461. [389] Surana, R. K., Danner, R. P., de Haan, A. B., and Beckers, N., 1997, Fluid Phase Equilib., 139, 361. [390] Flory, P. J., Orwoll, R. A., and Vrij, J. A., 1964, J. Am. Soc., 86, 3515. [391] Kodama, Y., and Swinton, F. L., 1978, Brit. Polym. J., 10, 191. [392] Charles, G., and Delmas, G., 1981, Polymer, 22, 1181. [393] Hao, W., Elbro, H. S., and Alessi, P., Polymer Solution Data Collection, Vol. 14, part 1 of DECHEMA Chemistry Data, DECHEMA, Frankfurt, 1992. [394] Sato, Y., Fujiwara, K., Takikawa, T., Sumarno, Takishima, S., and Mauoka, H., 1999, Fluid Phase Equilib., 162, 261. [395] Knapp, H., Doring, R., Plocker, L., and Prausnitz, J. M., DECHEMA, Vol. 6, Frankfurt. [396] Sebastian, H. M., Stmnick, J. J., Lin, H.-M., and Chao, K.-C., 1980, J. Chem. Eng. Data, 25, 68.

BIBLIOGRAPHY

285

[397] Lin, H.-M., Sebastian, H. M., and Chao, K.-C., 1980, J. Chem. Eng. Data, 25, 252. [398] Ioannidis, S., and Knox, D. E., 1999, Fluid Phase Equilib., 165, 23. [399] Yiagopoulos, A., Yiannoulakis, H., Dimos, V., and Kiparissides, C., 2001, Chem. Eng. Sc., 56, 3979. [400] Laugier, S., and Richon, D., 1996, J. Chem. Eng. Data, 41, 282.

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