LD5655.V855 1986.B464.Pdf (10.55Mb)

A Comparisonof ThermodynamicModels for the Predictionof Phase Behavior in Aqueous-PolymerTwo-Phase Systems

G. Gregory Benge

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

Chemical Engineering

AP.e.ROVED:

Dr. Henry A. McGee, Jr., Chainnan

Dr. Luther K. Brice Dr. Michael L. King

August, 1986

Blacksburg, Virginia A Comparisonof ThermodynamicModels

for the Predictionof Phase Behavior

in Aqueous-PolymerTwo-Phase Systems

G. Gregory Benge

Dr. Henry A. McGee, Jr., Chairman

Chemical Engineering

(ABSTRACT)

Aqueous-polymer two-phase systems consist of various combinations of water, polymer(s), low molecular weight component(s), and salts. These aqueous-polymer systems are comprised of two phases, each of which contains about 90 percent (by weight) water. Due to some very unique properties, these systems have been applied to separations involving biological molecules for at least a quarter of a century. In particular, these systems are inexpensive, efficient, and provide a mild

(aqueous) and possibly stabilizing environment for fragile biologically-active molecules. These systems may also be designed for a high degree of selectivity. Although much effort has been ex- pended in the area of polymer solution theory, the theory of why these systems exhibit this ex- traordinary two-phase behavior that characterizes them as viable liquid-liquid extraction systems for use with biologically-active molecules is not completely understood. A thermodynamic model which could accurately represent the phase equilibria exhibited by these systems would be useful for the design of systems for use in many different applications.

A potpourri of thermodynamic models and their underlying theoretical structure have been .,, critically studied for their particular application to predicting the phase behavior of aqueous- polymer two-phase systems. In particular, the Flory-Huggins model is reviewed (with discussion of its inadequacies and subsequent modifications); the theory of Ogston; the model by Heil; several local composition models (NRTL, Wilson, and UNIQUAC); and two group-contribution models

(ASOG and UNIFAC) are all discussed. The development of a solvent-electrolyte model (Chen's model) based on local composition theory (in particular the NRTL model) is reviewed, and the subsequent possible modification of this theory for solvent-polymer-electrolyte systems is discussed. The pros and cons of each model are discussed and qualitative results are given. Quantitative comparisons with experimental data are made with several of these models when appropriate data are available.

The main conclusions of this work are:

1. A major limitation to the modeling of these aqueous-polymer two-phase systems is the lack of experimental data. Sufficient, accurate data is needed for the reduction of

meaningful thermodynamic parameters by which thermodynamic models can be tested

for their applicability. There exists a definite need for the generation of accurate,

meaningful thermodynamic data from well characterized systems.

2. The most promising model identified in this work is the theory of Ogston. First, the

model is based on the virial expansion and is thus quite suitable for dilute solutions.

The Ogston model is the simplest theoretically-relevant dilute-solution model. Sec-

ond, it appears to be easily extended to solvent-polymer-electrolyte solutions.

3. The Flory equation of state approach appears to be promising for representing polymer solutions. The free volume dissimilarity effect on which it is based is ex-

tremely important for solvent-polymer solutions. The most important aspect of this theory is its ability to predict lower critical solution temperature (LCST) behavior -- for which the Flory-Huggins theory is totally inadequate. Acknowledgements

I would like to acknowledge all the help which I have received along the way. First, I would like to thank my thesis committee for their help and guidance. In particular, I would like to thank

Dr. Henry A. McGee, Jr., for serving as committee chairman. Also, I would like to pay a special tribute to a dear friend and colleague, Dr. David A. Wallis, whose enthusiasm was always so prevalent. Furthermore, I would like to thank Dr. Michael King from Merck & Co., Inc. for his willingness to serve on my committee as a replacement for Dr. Wallis and for his helpful comments.

I also thank Dr. Luther Brice for his helpful comments and for agreeing to serve on "one more" committee.

I would also like to acknowledge the support I received from the Eastman Kodak Company who furnished a graduate fellowship.

I would also like to thank Dr. Y. A. Liu, Danny Thompson, Kirn Hunter, Tom Totah, and

C. W. Cheah for their continuous encouragement and stimulating conversations. And a special thanks goes to those colleagues in the polymer group who were all so willing to share their expertise and advice. A special note of gratitude is in order for Dr. Tom Ward of the Chemistry Department for his help and advice.

And to my loving wife Suzanne, who has unselfishly given up part of her life for me, I am especially grateful for her love, support, and patience. And to our parents, Cecil & Barbara Benge and Rex & Margaret Springston for their support and encouragement, I also give thanks

Acknowledgements iv Table of Contents

1.0 Introduction ......

2.0 General Thermodynamic Background ...... 8 2.1 Solution Thermodynamics ...... 8 2.1.1 Thermodynamic Formalism: Chemical Potential, Fugacity, and Activity ...... 9 2.1.1.1 The Chemical Potential ...... 9 2.1.1.2 The Fugacity ...... 10 2.1.1.3 The Activity ...... 11 2.1.2 Activity Coefficients from Gibbs Free Energy Expressions ...... 12 2.1.3 The Activity Coefficient and Concentration ...... 14 2.1.4 Thermodynamic Relations for Solvent-Polymer Solutions ...... 15 2.1.5 Normalization of Activity Coefficients ...... 16 2.1.5.1 Symmetric Convention for Normalization ...... 16 2.1.5.2 Unsymmetric Convention for Normalization ...... 16 2.1.6 Liquid-Phase Activity Coefficient Models Via Excess Gibbs Free Energy ...... 17 2.2 Phase Equilibria ...... 18 2.2.1 Criteria for Phase Splitting ...... 18 2.2.2 The Liquid-Liquid Equilibrium Problem ...... 26 2.3 Application of Thermodynamic Models ...... 30 2.3.1 Prediction of Activity Coeflicients ...... 30 2.3.2 Comparison of Experiment and Theory ...... 30 2.4 Summary ...... 32

3.0 Experimental Systems and Data ...... 33 3.1 Binary and Ternary Experimental Systems ...... 33 3.2 Experimental Methods for Collection of Binary Data ...... 34 3.3 Analyses of Equilibrium Phase Compositions ...... 39 3.4 Representation of Experimental Data ...... 40

4.0 Solutions of Small and Similar-Sized Molecules: Local Composition and Group-Contribution i\1odcls ...... 42 4.1 Local Composition Theory: Wilson ( 1964) ...... 43 4.1.1 Wilson Model ...... 43 4.1.2 Criterion for Phase Splitting ...... 46 4.1.3 Non-Random Two-Liquid Model (NRTL) ...... 47 4.1.4 Universal Quasi-Chemical Model (UNIQUAC) ...... 52 4.2 Thermodynamics of Models Based on Group-Contribution Schemes ...... 54 4.2.1 Introduction ...... 54 4.2.2 The Solutions-of-Group Concept ...... 55 4.2.3 UNIFAC ...... 56 4.3 Extensions and Applications to Polymer Solutions ...... 59 4.3.1 Universal Quasi-Chemical Model (UNIQUAC) ...... 59 4.3.2 Group-Contribution Theories ...... 60 4.3.2.1 The UNIFAC Free Volume Model (UNIFAC-FV) ...... 60 4.3.2.2 Empirical Correlation of UNIFAC Activity Coefficients for Large Molecules .. 64 4.3.2.3 A Supercharged UNIFAC: SUPERFAC ...... 64 4.3.2.4 The ASOG-Variable Size Parameter Model (ASOG-VSP) ...... 74 4.4 Summary ...... 75

5.0 Polymer Solutions: The Flory-Huggins Model ...... , . , , , . 76

Table of Contents V 5.1 Liquid Lattice Theory ...... 76 5.2 Theoretical Development ...... 77 5.2. l Athermal Solvent-Polymer Solutions ...... 77 5.2.2 Nonathermal Behavior in Real Polymer Solutions ...... 79 5.2.3 Chemical Potential and Activity Expressions from the Flory-Huggins Model . . . . . 82 5.3 Application of the Classical Flory-Huggins Theory ...... 82 5.3. l Binary Solvent-Polymer Solutions ...... 82 5.3.2 Application of the Classical Flory-Huggins Theory to Water-Polyethylene Glycol Systems ...... 88 5.3.3 Extension of the Flory-Huggins Model to Multicomponent Systems ...... 89 5.3.4 Macromolecular Partitioning Predictions Via the Flory-Huggins Theory ...... 107 5.4 Determination of Flory Interaction Parameters (x's) ...... 111 5.4. l Evaluation of Flory Interaction Parameters from Binary Data ...... 111 5.4.2 Evaluation of Flory Interaction Parameters from Ternary Data ...... 111 5.4.3 Experimental Methods for Determination of Flory Interaction Parameters ...... 116 5.4.4 Estimation of Flory Interaction Parameters ...... 116 5.5 Modifications of the Classical Flory-Huggins Theory ...... 118 5.5.1 Re-examining the Flory Interaction Parameter ...... 119 5.5. l.l Concentration Dependence of the Interaction Parameter ...... 119 5.5. l.2 The Inability of the Flory-Huggins Model For Prediction of Lower Critical Solution Temperature Behavior ...... 123 5.5. l.3 Effect of Molecular Weight on the Interaction Parameter ...... 128 5.5.2 Effect of Polydispersity on the Phase Behavior ...... 128 5.5.3 Dilute Solution Theory ...... 133 5.6 Application of the Modified Flory-Huggins Theory ...... 135 5.6. l Determination of Concentration-Dependent x Parameters ...... 136 5.6.2 Prediction of LLE in Ternary Polymer Solutions Using the Modified Flory-Huggins Theory ...... 152 5.7 Summary: The Flory-Huggins Theory in Perspective ...... 158

6.0 Polymer Solutions: Other Models ...... 159 6.1 Heil's Segment-Interaction Equation for Polymer Solutions ...... 159 6.2 The lbeory of Ogston ...... 162 6.2. l lbe Virial Expansion ...... 168 6.2.2 The Osmotic Pressure Relationship ...... 169 6.2.3 Example of Ogston's Theory ...... 171 6.2.3. l Procedure for Application of the Ogston Theory ...... 171 6.2.3.2 Use of the Ogston Theory for Prediction of the Phase Behavior in a Water-PEG-Dextran System at 20°C...... 173 6.3 Free Volume Considerations and the Flory Equation-of-State ...... 179 6.4 The Solubility Parameter ...... 183 6.5 Summary ...... 183

7.0 Soh·ent-Polymcr-Elcctrolyte Solutions: A Local Composition Approach ...... 184 7.1 Chen's Model for Solvent-Electrolyte Systems ...... 184 7.2 Pitzer-Debye-Hi.ickel (POii) Model for Long-Range Interactions ...... 185 7.3 Local Composition Model (LC) for Short-Range Interactions ...... 186 7.4 Modification of the Chen Model for Application to Polymer Solutions ...... 187 7.5 The Empirical Correlation of Adamcovi ...... 189 7.6 Thermodynamics of Polyelectrolytes ...... 189 7.7 Summary ...... 190

8.0 Conclusions: Looking Back to See Ahead ...... 191 8.1 In Retrospect ...... 191 8.2 Conclusions and Recommendations ...... 193

Table of Contents vi Bibliography • • . • • • • • • • • • • . • • • . • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . • • • . . . . • 196

Appendix A. Ternary LLE Predictions Using a Modified Flory-Huggins Model 205

Appendix A. Ternary LLE Predictions Using a Modified Flory-Huggins Model 206

Appendix B. FORTRAN 77 Code for Fitting Concentration-Dependence of Flory Interaction Parameter . • . • • . . • • • • • • • • • • . • • • • • . • • • • • • • • • • • • • • • • • . • • • • • • • • • . . • • • . • • 219

Appendix C. FORTRAN 77 Code for Determination of Flory Interaction Parameter from Experimental Ternary Data • • • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • . • • • • • 232

Appendix D. FORTRAN 77 Code for Predicting LLE Behavior Based on the Flory-Huggins i\-lodel • • • • • . • • • • • • • • • • • . • • • • . . • • • • • • • • • • . • • • . . • . . • • • . • • . . • • • • • • • • • • • 245

Appendix E. Ternary LLE Predictions Using the Theory of Ogston • . • • • • • • • • • • • • . • . . 254

Appendix F. FORTRAN 77 Code for Predicting LLE Behavior Via the Ogston Theory . • • • 259

Appendix G. FORTRAN 77 Code for Conversion of Weight Fraction Data to Molality Basis 262

Vita • • • • • • • • • • . . • • . • . • • • . . . • • . . . • • • • • • • • • • • • • • • • • • . . . • • . • . • . . . . • • • . • 266

Table of Contents vii List of Illustrations

Figure I. Gibbs free energy of mixing versus mole fraction for binary mixture ...... 20

Figure 2. Isothermal phase diagram for a ternary system displaying the binodal, spinodal, and plait point...... 22

Figure 3. Relative chemical potential of the solvent versus polymer volume fraction for a binary solvent-polymer solution ...... 25

Figure 4. Phase diagram for a binary system projected onto the temperature , composition plane ...... 27

Figure 5. Upper and lower critical solution temperature behavior of a binary system . . . . . 28

Figure 6. Example binodal phase diagram of water-PEG6000-Dextran( D 17) system at T= 20°C ...... 41

Figure 7. UNIFAC and UNIFAC FREE VOLUME predictions for benzene activity in polyisobutylene of MW= 40,000 and at T = 25°C ...... 63

Figure 8. Effect of temperature on water activity in water-PEG(MW = 200) solutions as predicted by UNIFAC-NEW ...... 65

Figure 9. Water activity in water-PEG(MW= 200) solutions at T= 313K as predicted by UNIFAC and UNIFAC-NEW ...... 66

Figure 10. Water activity in water-PEG(MW=300) solutions at T=303K as predicted by UNIFAC and UNIFAC-NEW ...... 67

Figure 11. Water activity in water-PEG(MW = 300) solutions at T= 338K as predicted by UNIFAC and UNIFAC-NEW ...... 68

Figure 12. Water activity in water-PEG(MW = 600) solutions at T = 313K as predicted by UNIFAC and UNIFAC-NEW ...... 69

Figure 13. Water activity in water-PEG(MW = 600) solutions at T = 333K as predicted by UNIFAC and UNIFAC-NEW ...... 70

Figure 14. Water activity in water-PEG(MW= 1500) solutions at T= 313K as predicted by UNIFAC and UNif'AC-NEW ...... 71

Figure 15. Water activity in water-PEG( MW= 6000) solutions at T = 313K as predicted by UNIFAC and UNIFAC-NEW ...... 72

Figure 16. Water activity in water-PEG(MW = 6000) solutions at T = 333K as predicted by UNIFAC and UNIFAC-NEW ...... 73

Figure 17. Effect of Flory interaction parameter, x12 , on the decrease in the relative solvent chemical potential ...... 84

List of Illustrations viii Figure 18. Flory-Huggins predictions of water activity in water-PEG(MW = 200) solutions at temperatures of 293, 313, and 333 K ...... 90

Figure 19. Flory-Huggins predictions of water activity in water-PEG(MW=300) solutions at temperatures of 303 and 338 K ...... 91

Figure 20. Flory-Huggins predictions of water activity in water-PEG(MW = 600) solutions at temperatures of 293,313, and 333 K ...... 92

Figure 21. Flory-Huggins predictions of water activity in water-PEG(MW = 1500) solutions at temperatures of 293, 313, and 333 K ...... 93

Figure 22. Flory-Huggins predictions of water activity in water-PEG(MW = 3000) solutions at temperatures of 328 and 338 K ...... 94

Figure 23. Flory-Huggins predictions of water activity in water-PEG(MW = 5000) solutions at temperatures of 333 and 338 K ...... 95

Figure 24. Flory-Huggins predictions of water activity in water-PEG(MW = 6000) solutions at temperatures of 313 and 333 K ...... 96

Figure 25. Effect of molecular weight on the activity of water in water-PEG solutions at T = 333 K as predicted by UNIFAC-NEW ...... 97 Figure 26. Effect of molecular weight on the activity of water in water-PEG solutions at T = 333 K as predicted by Flory-Huggins ...... 98

Figure 27. Comparison of spinodals predicted from concentration-dependent and independent theories with experiment ...... 124

Figure 28. Temperature dependence of the Xu interaction parameter ...... 126

Figure 29. Effect of temperature on the binodal of ternary systems containing a solvent and two polymers ...... 129

Figure 30. Effect of temperature on the experimental binodals of water-PEG 6000-dextran (068) systems ...... 130

Figure 31. Effect of temperature on the experimental binodals of water-PEG 6000-dextran (048) systems ...... 131

Figure 32. Effect of temperature on the experimental binodals of water- PEG 4000-dextran (048) systems ...... 132

Figure 33. Concentration dependence of the x interaction parameter for water-PEG 1500 solutions at 293, 313 and 333 K ...... 141

Figure 34. Concentration dependence of the x interaction parameter for water-PEG 3000 solutions at 328 and 338 K ...... 142

Figure 35. Concentration dependence of the x interaction parameter for water-PEG 5000 solutions at 333 and 338 K ...... 143

Figure 36. Concentration dependence of the x interaction parameter for water- PEG 6000 solutions at 313 and 333 K ...... 144

List of Illustrations ix Figure 37. Water activity in PEG 1500 solutions at 293, 313 and 333 K as predicted by the modified Flory-Huggins theory ...... 146

Figure 38. Water activity in PEG 3000 solutions at 328 and 338 K as predicted by the modified Flory-Huggins theory ...... 147

Figure 39. Water activity in PEG 5000 solutions at 333 and 338 K as predicted by the modified Flory-Huggins theory ...... 148

Figure 40. Water activity in PEG 6000 solutions at 313 and 333 K as predicted by the modified Flory-Huggins theory ...... 149

Figure 41. Comparison of experimental binodal curve with Flory-Huggins predictions for Water-PEG6000-D48 at 20°c ...... 157

Figure 42. Water activity in polyethylene glycol solutions of MW= 3000 at T = 55°C correlated by the Heil model ...... 164

Figure 43. Water activity in polyethylene glycol solutions of MW= 5000 at T = 60°C correlated by the Heil model ...... 165

Figure 44. Water activity in polyethylene glycol solutions of MW= 5000 at T = 65°C correlated by the Heil model ...... 166

Figure 45. Effect of the molecular weight of dextran on the binodal curves of water-polyethylene glycol-dextran systems...... 176

Figure 46. Comparison of experimental and theoretical binodal curves for the system Watcr(l)-PEG6000(2)-D17(3) at 20°C ...... 178

List of Illustrations X List of Tables

Table l. Experimental Binary Aqueous-Polymer Systems ...... 35

Table 2. Experimental Ternary Aqueous-Polymer Systems from Albertsson ...... 36

Table 3. Characterization of Various Molecular Weight Fractions of Dextran ...... 37

Table 4. Characterization of Various Molecular Weight Fractions of Polyethylene Glycol 37

Table 5. Liquid Densities of Water and Various Molecular Weight Fractions of Polyethylene Glycol and Dextran ...... 38

Table 6. Parameters and Interrelatedness of the Wilson, Heil, and NRTL Models ...... 51

Table 7. Concentration Dependence ofx Interaction Parameter for Water-PEG 1500 Solutions at 293, 313, 333 K ...... 137 Table 8. Concentration Dependence of x Interaction Parameter for Water-PEG 3000 Solutions at 328 and 338 K ...... 138 Table 9. Concentration Dependence of x Interaction Parameter for Water-PEG 5000 Solutions at 333 and 338 K ...... 139 Table 10. Concentration Dependence of x Interaction Parameter for Water-PEG 6000 Solutions at 313 and 333 K ...... 140 Table 11. Experimental x Parameters for the System of Water-PEG6000-Dextran(D48) at 20°c ...... 151

Table 12. Water-PEG Systems Correlated with the Heil Model Using Experimental Data from Malcolm and Rawlinson (81) ...... 163

Table 13. Characterization of Polymers for Experimental and Theoretical Systems 175 Table 14. Comparison of Experimental and Theoretical Binodal Curves for the Water-PEG6000-Dl7 System at 20°C ...... 177

Table 15. Binary Data for Water-PEG6000 at Various Temperatures [57)...... 208

Table 16. Ternary Data for Water-PEG6000-Dextran(D48) at 20 °C (3)...... 209

Table 17. "CHIFIT' Input Data File for Water-PEG6000 Example ...... 210

Table 18. "CI-IIFIT' Output Data File for Water-PEG6000 Example ...... 211

Table 19. "CHI" Input Data File for Water-PEG6000-Dextran(D48) Example 214

Table 20. "CHI" Output Data File for Water-PEG6000-Dextran(D48) Example 215

Table 21. "HSU" Output Data File for Water-PEG6000-Dextran(D48) Example 217

List of Tables xi Table 22. "OGSTOW Output File for Prediction of Binodal Curve of Water-PEG6000-Dextran(Dl7) System at 20°C ...... 256

Table 23. "MOLAL" Input Data File for Conversion of Weight Fraction to Molality . . . . 257 Table 24... MOLAL* Output Data File for Conversion of Weight Fraction to Molality . . . 258

List of Tables XII 1.0 Introduction

And God said, "Let there be a firmament in the midst of the waters, and let it separate the waters from the waters. And God made the finnamcnt and separated the waters which were under the firmament from the waters which were above the firmament. And it was so."

Genesis 1:6-7 (Revised Standard Version)

The chemical and pharmaceutical industries exist for the production of useful chemicals with specific, desirable properties. Often, the key to the success or failure of the processes that produce these chemicals lies in the ability to efficiently and economically separate the desired species from a mixture containing a variety of products. Furthermore, development of new and more powerful separation processes rely upon advances in the separation sciences.

Couple the above importance of separations with the emphasis that our society is currently placing on biotechnology and there exists a definite need for enhanced expertise in separation sciences. Chemical engineers, by their very unique undergraduate training in phase equilibrium thermodynamics, mass transfer, and process design, possess the background required in this area. Phase equilibrium thermodynamics is an essential element in most non-mechanical separation processes, and is an area of undergraduate training almost exclusive to chemical engineers. There is a definite place for chemical engineers in understanding and explaining the theory of many modem laboratory techniques (which are currently used simply because they work) and in the rational development of new methods for biological and biochemical separations. In fact, this area has been cited, and rightly so, as an area offering vast opportunities for service by the chemical engineer in the near future [86). This is not to say that chemical engineers arc the only ones who can contribute in this area, for chemists, biochemists, physicists, etc. have made many contributions.

Chapter I: Introduction As Michaels (86) so adeptly discusses, technological advances in the biological and life sciences have brought about a new concept of biology, which in general is not yet ready for industrial exploitation. This "new biology" can be seen in such areas as molecular and cell biology, genetics, genetic engineering, biochemical engineering, and biochemistry. These new discoveries will, in time, have applications in the chemical and pharmaceutical industries where they will be put into practice for the benefit of mankind (hopefully).

Some example applications which illustrate the broadness and importance of work in these areas include genetically engineered seeds which could alleviate crop failure problems for farmers; production of antibodies by fermentation in the pharmaceutical industry; and the research on and production of anti-cancer (or related) drugs which have been found to prevent the growth of or destroy cancerous cells.

The chief reason for this lag in industrialization stems, in a large part, from the lack of coop- eration between chemical engineers, biologists, and biochemists. The problem as stated by Mi- chaels, is that the life sciences largely developed in isolation from interaction with the engineering and applied science disciplines. Thus, the work of chemical engineers in the life science areas is fairly new, and we do not yet have the experience to adapt this "new biology" to industry in terms of profitable products and processes. We must learn to apply our process technology to biological processes. As Michaels suggests "in general, this new biology is not ready for full exploitation, and this is where there are unparalleled opportunities for the chemical engineer over the next five to ten years" (86].

One underlying key to the success of adapting this "new biology" to industrial practice is the development of separation and purification techniques applicable to biological materi.i!s. These methods must not only be efficient and economic, but they must also be mild in that they do not denature or destroy the fragile biological molecules that are being processed.

These new biotechnological (biochemical, biomedical, cell biological, etc.) processes yield very complex mixtures with components differing in size, shape, form, chemical properties, and especially composition. These biologically-active molecules are usually very fragile, and slight changes in their structure can greatly affect their biological activity. Hence, mild techniques are required for

Chapter I: Introduction 2 their subsequent separation and/or purification. As these are typically high-value, low-volume processes, efficient techniques are required to ensure their economic feasibility. Furthermore, as these molecules may aggregate, dissociate, or change their state with time, multiple-step separations which utilize different properties of the species in subsequent steps are often necessary.

One such separation technique is the aqueous-polymer two-phase system. This method is attracting increased interest in the chemical and biological sciences. Walter, Brooks, and Fisher

[ 141]remark that the field of aqueous phase partitioning is rapidly expanding, and they back up this statement by pointing out that the comprehensive bibliography of Sutherland and Fisher [ 1291 shows that the field contains over 750 publications (as of late 1984)and that about 70 papers currently appear annually. This marked increase in interest lies primarily in the fact that these aqueous-polymer two-phase systems have many of the properties needed for separation of biological molecules. For example, they are inexpensive, efficient, and mild to large, fragile biologically-active molecules. Furthermore, slight changes in a system can yield another system which is more efficient for a subsequent step in a multi-step process. This unique phenomenon has also led to the use of these systems in a wide variety of applications.

The mildness of these systems can be seen in the fact that they do not denature the fragile, biologically-active molecules. This property is a particularly desirable one in that these biological molecules are valuable only if they retain their biological activity throughout the separation. In fact, not only do these systems not denature the complex, biological solute molecule, they quite possibly provide a stabilizing (cushioning) environment for the solute molecules (3,4, 1401. Furthermore, the highly aqueous nature of these systems provides a relatively mild environment for these molecules; indeed, most biological molecules "live" in a highly aqueous environment. As compared with typical solvent extraction systems, the aqueous-polymer systems are milder to these molecules than is the environment associated with most organic solvents. Another important property of these systems is the low interfacial tension (0.0001 • 0.1 dyne/cm) compared with the rather strong interfacial tension associated with conventional liquid-liquid phase systems ( I - 20 dyne/cm) 13).

Strong interfacial tension is another possible threat to the integrity of these fragile molecules. Also, biochemical and biological molecules arc usually much more soluble in water-polymer solutions

Chapter I: Introduction 3 than in water-solvent solutions. In addition, these two-phase systems have no complicated mechanical parts and can easily be scaled up to handle very large quantities of material. An econom- ical advantage of using aqueous-polymer systems over typical solvent extraction techniques is that organic solvents are expensive, they are costly and difficult to recover, but these aqueous-polymer systems can essentially be discarded (versus recovery of the polymer) after use. Also, these systems are characterized by high specificity. For instance, the manipulation of certain key parameters, such as the addition of salts and/or other polymers or the variation of polymer compositions allow for the design of very selective separation systems.

One disadvantage of these aqueous-polymer two-phase systems is the settling times, which can be on the order of hours due to the high viscosities of the polymers in solution. This translates into slow rates of mass transfer between the phases present, i.e. slow rates of phase separation. Long settling times are of minor importance though as centrifugation, for example, can be used to speed up the separation.

These two-phase systems have been used for purification of enzymes, concentration and purification of virus, separation of native from denatured DNA, separation of bacterial spores from vegetable cells, analysis of suspensions of cells and cell organelles such as chloroplasts and mito- chondria, and for studying the binding of one particle to another. For a recent collection of papers regarding the applications of these systems, see Walter, Brooks, and Fisher (141); and for a recent comprehensive bibliography on these systems and their uses, see Sutherland and Fisher ( 129).

These aqueous-polymer two-phase systems occur when two aqueous-polymer solutions are mixed together or when one aqueous-polymer solution is mixed with a low-molecular-weight component (LMWC). Nonionic polymers or polyelectrolytes may be used, with or wiihout the addition of salts. Thus, these systems consist of two immiscible phases in direct contact. The most common of these systems is obtained by mixing together aqueous solutions of polyethylene glycol and dextran (number average molecular weights of 4000-6000 and 20,000-300,000 respectively).

Each phase typically contains 80-95% water (weight per cent) and the remainder polymer.

While these aqueous-polymer two-phase systems have been studied extensively for more than a quarter of a century, very little work has been aimed at understanding the factors governing the

Chapter I: Introduction 4 behavior of these systems. Granted much work has been done in solution thermodynamics and

solubility in the past, even in polymer solution theory, but the application of available theory to these systems has been rare. By applying previously developed theory (and perhaps empirical cor-

relations), models of the behavior of these systems could be formulated which could then be used

as predictive tools (versus the trial-and-error methods employed today) to aid those· interested in

designing and using these systems.

The initial direction of this research was to analyze these systems thermodynamically in order

to select or develop specific models that adequately explain and predict their unique behavior.

These models would aid the experimental investigators by providing them with certain information

pertaining to the range of applicability and partitioning properties of systems of interest. This

should allow for the design and optimization of aqueous-polymer two-phase systems with a mini-

mum amount of experimental verification. Also, in order for the model to be practical, a simple

theory was sought.

The first step in modeling or trying to understand these systems requires the identification of

key manipulative parameters which affect the partitioning properties displayed by these systems.

The following list of Kev Manipulative Parameters Affecting Partition has been identified

(3,74, 136,139):

• types and particular polymers used

• polymer molecular weight and molecular weight distribution • polymer concentration

• types and concentration of any low molecular weight components

• types and concentrations of any salts added

• types and concentration of any contaminating solutes

• temperature

• pH

With these variables in mind, the problem was then to develop a model which would provide

a framework for investigating the relative effect of each of these parameters. This is a tremendous

Chapter I: Introduction 5 task, for polymer solution theory and electrolyte theory for multicomponent solutions are difficult enough alone. When combined, these theories offer at best a very complex model that is not aes- thetically pleasing. In light of the complexities involved in such a comprehensive model, the objective of this research was modified to address nonelectrolyte solvent-polymer solutions. Also, due to the enormity of past and ongoing research in the area of polymer solution theory -- most of the research presented here addresses the identification of applicable existing thermodynamic models for solvent-polymer solutions.

Accordingly, the theme of this thesis is geared more toward a detailed literature review and study of existing models than it is toward the development of a model and its subsequent testing with experimental data. The majority of the discussion is related to the many and various models that are available today: how the models were developed, their ability or inability in modeling these systems, how they may be modified to apply to these systems, and the pros and cons of each.

Models of current interest that were not studied here are referenced and briefly introduced. Several existing theories are compared with available experimental data, and the current state of polymer solution theory is discussed.

The format of this thesis is as follows. Chapter 2 provides a general thermodynamic background. A discussion of solution thermodynamics is given, with the overall theme being the definition of both liquid-phase activity coefficients and the corresponding thermodynamic models, and the relation between these. This is followed by a brief description of the comparison of thermodynamic models with experimental data.

Chapter 3 introduces the experimental systems to be modeled. Information is giv-:.non the characterization of the polymers used in the experimental studies and certain other properties needed to perform calculations with the various models being compared in this thesis. A brief discussion is given on the analysis and representation of the phase equilibrium data being used to test the various theories.

Chapter 4 discusses several of the most common thermodynamic models (i.e. liquid-phase activity coefficient models) for use in applications involving molecules of small and similar sizes.

Chapter I: Introduction 6 The applicability of the UNIFAC and ASOG group-contribution models (and their modifications) is discussed here also.

Chapter 5 discusses polymer solution theory and the classical treatment of Flory and Huggins.

Strengths and weaknesses of this model are discussed, and modifications of the theory are presented which alleviate some of the inadequacies of the Flory-Huggins treatment. Also, dilute solution theory is discussed briefly.

Chapter 6 presents a few alternative approaches to modeling polymer solutions. The semi- empirical model by Heil is discussed along with the Ogston theory. The Flory equation of state theory is also briefly discussed.

Chapter 7 introduces a recently proposed model for solvent-electrolyte solutions and describes an attempt to extend this theory to solvent-polymer-electrolyte solutions.

Chapter 8 serves as the conclusion, with the main purposes of summarizing what was learned in regard to modeling aqueous-polymer systems and suggesting what models hold the most promise for future applications involving these systems.

Chapter I: Introduction 7 2.0 GeneralThermodynamic Background

"Science too has its cathedrals ... "

G. N. Lewis and M. Randall

This chapter seeks to provide the thermodynamic background necessary for the understanding, evaluation, and possible modification of existing models of liquid-phase behavior and their application to predicting the phase behavior of aqueous-polymer systems. This is achieved by first re- viewing the basics of solution thermodynamics and the basis for the development of liquid-phase activity coefficient models. Next, the subject of phase equilibria is discussed, with emphasis placed on the prediction of liquid-liquid equilibria from activity coefficient models. Only a general background is given here; the detailed development of specific thermodynamic models and their utility in modeling aqueous-polymer solutions is then covered in the remaining chapters.

2.1 Solution Thermodynamics

Solution thermodynamics deals with the application of classical thermodynamics to describing the behavior of solutions. For the most part, the thermodynamics discussed here applies to solutions of nonelectrolytes. Once a general format is laid upon which molecular models can be based, the development of a thermodynamic model is outlined.

Chapter 2: Thermodynamic Background 8 2.1.1 Thermodynamic Formalism: Chemical Potential, Fugacity, and Activity

First, the formalism to be used in ensuing discussions must be given. In solution thermodynamics, the following quantities are particularly useful:

• chemicalpotential

• fugacity

• activity

2./.1.1 The ChemicalPotential

J. Williard Gibbs, in his monumental work of 1875 [46), first demonstrated the general applicability of thermodynamics to the analysis of multicomponent chemical systems [108). Hence, it was Gibbs who introduced the concept of the "chemical potential".' The chemical potential of component i, µ1 , is a function defined as

(2.1)

where U is the internal energy, S is the entropy, V is the volume, and n, is the number of moles of component i. However, the chemical potential could just as easily be written as

(2.2)

where H is the enthalpy, A is the Helmholtz free energy, and G is the Gibbs free energy. As the most common pair of independent variables (of S,V,T,P) is temperature and pressure, the last quantity in the expression above has the most practical utility.

1 See discussion in Prausnitz [108), Chapters 1 and 2.

Chapter 2: Thermodynamic Background 9 When the partial derivative of a property such as U, S, V, A, or G is taken with respect to the number of moles of component i, with the temperature, pressure, and moles of all other components constant, the result is a partial molar property. Thus, in the expression above in Equation

(2.2), the chemical potential is seen to be the partial molar Gibbs free energy, and the following relations may be used interchangeably

(2.3)

Walas (138) gives an insightful and straightforward definition of the chemical potential. He says,

"Since pressure is the potential function for work transfer and temperature that for heat transfer, µ

(the chemical potential) may be regarded as the potential function for internal energy transfer ac- companying the transfer of mass across the boundary of the system."

The objective in solution thermodynamics is to determine accurate relations between the chemical potential and physically measureable properties such as temperature, pressure, and composition. An absolute value of the chemical potential cannot be obtained. This is due to the inability to assign an absolute energy, and hence, only differences between the point of interest and a reference point (or standard state) are meaningful. The difference between the chemical potential at the point of interest and the reference point is termed the relative chemical potential. The relative chemical potential and other equivalent thermodynamic expressions are typically used in solution thermodynamics. The choice of a standard state is arbitrary, but a very common preference is that of pure components at the temperature and pressure of the system.

2.1.J.2 The Fugacity

The chemical potential is an abstract concept. It is helpful to define an auxiliary function which will be more meaningful, as well as more easily calculable. The fugacity, defined by G. N.

Lewis, is such a function. The partial molar Gibbs free energy at constant temperature is given by

(constant T) (2.4) or

Chapter 2: ThermodynamicBackground 10 (2.5) and for an ideal gas RT p (2.6)

Substituting and integrating yields

= RT 1n _Lpo (2.7)

This expression, Equation (2.7), is only valid for an ideal gas mixture, but by defining the fu. gacity of component i, fi , of a real gas mixture similarly

f: = RT 1n - 1 (2.8) ft

Hence, G. N. Lewis was able to write a general expression for the chemical potential of any real or ideal, pure or mixed, solid, liquid or gas, and at the same time remove some of the ab-

stractness from the chemical potential (108]. The resulting introduction of the fugacity leads to an easier concept to visualize, as it is seen as a "corrected" pressure that contains all of the nonidealities of the phase.

2./.JJ The Activity

Finally, for a pure ideal gas, the fugacity is equal to the pressure, while for a component i in a mixture of ideal gases, the fugacity is equal to its partial pressure of yiP . For ideal or real systems the behavior approaches ideal at very low pressures, then the definition of the fugacity is t:ampleted by specifying that

as P -+ 0 (2.9)

For a pure substance, this becomes

f -+ I p -+ 0 p as (2.10)

Chapter 2: Thermodynamic Background II and these expressions are given the names of partial fug!1city coefficient, ~; = f~ , and fugacity £ ~ coefficient,

" fi (2.11) a; = [° I

and the activity coefficient, y1 , is defined as

Yi - = (2.12)

Also, defining X; fi° in Equation (2.12) as f;id•••, then

= (2.13) and the activity coefficient has the physical significance of being the ratio of the actual fugacity to the fugacity of an ideal solution at the same conditions. As stated by Prausnitz [ 108[, "the activity of a substance gives an indication of how 'active' a substance is relative to its standard state, since it provides a measure of the difference between the substance's chemical potential at the state of interest and that at its standard state." Values other than fld•••may also be used as the reference state in the definition of the activity and activity coefficient.

2.1.2 Activity Coefficients from Gibbs Free Energy Expressions

Activity coefficients are readily obtained from expressions for the excess Gibbs free energy.

In this Section, equations are derived that relate the excess Gibbs free energy to the Gibbs free energy of mixing and the excess Gibbs free energy to the activity coefficient. First, the excess Gibbs free energy is defined as

E = A A ideal g ugmixing - ugmixing (2.14) and in view of the relation

Chapter 2: Thermodynamic Background 12 (2.15) the former equation now becomes

l = Llgmixing - RT r,(xi 1nxi) (2.16) i In dimensionless form, this becomes

Llgmixing (2.17) L=RT RT

Equation (2.17) is commonly used to convert expressions for the Gibbs free energy of mixing into expressions for the excess Gibbs free energy function. Noting previous results of Equations (2.8) and (2.11),

= RT 1n _!!_ = RT 1n llj (2.18) ft and since

Llgmixing = (2.19) RT then

Llgmixing = (2.20) RT and

(2.21) or

[ X· 1n (2.22) I .5_]Xi or equivalently

(2.23)

From the definition of activity and activity coefficients,

Chapter 2: Thermodynamic Background 13 £ 1n 'Yi = 1n -·f.o - 1n y.., (2.24) I and we know from Equation (2.18) that

" (2.25) . Thus, for an ideal solution, where fidea1= X; fi0 , Equation (2.25) becomes

r,ideal" G·ideal _ Go I I (2.26) 1n -·--ft = RT and inserting Equations (2.25) and (2.26) into (2.24)

Gideal _ Go - -ideal I I Gi-Gi 1n 'Yi = (2.27) RT RT

which by the definition of the excess Gibbs free energy (Gr = G1 - Gjdeal)and the definition of a partial molar property becomes

1n 'Yi = = [ a(ng~/RT) l (2.28) oni T,P,ni

This equation defines the activity coefficient as it is related to the excess Gibbs free energy.

2.1.3 The Activity Coefficient and Concentration

It should be noted that the activity coefficient is not always taken to be the ratio of the activity to the mole fraction, as is the case in Equation (2.12). As stated by Prausnitz (108), "The activity

Chapter 2: Thermodynamic Background 14 coefficient y1 is the ratio of the activity of i to some convenient measure of the concentration of i which is usually taken to be the mole fraction. For electrolyte solutions, it is often more convenient to use molality instead of mole fraction. For polymer solutions, mole fractions are not useful; instead, weight fractions or volume fractions are more appropriate." Likewise, for polymer solutions the chemical potential or the solvent activity is often used.

2.1.4 Thermodynamic Relations for Solvent-Polymer Solutions

In dealing with solvent-polymer solutions, where the mole fraction is not a useful composition variable, it is common to work with activity or relative chemical potential expressions instead of activity coefficient expressions. The relative chemical potential is defined as the difference between the chemical potentialµ and the standard state chemical potential µ0 • From the developments in

Section 2.1.2, the relation between the relative chemical potential of the solvent and the Gibbs free energy of mixing for a binary solvent-polymer solution is found to be:

µs - µ/ = [ aagmixing ] (2.29) ans T,P,np

where subscript s denotes the solvent, subscript p denotes the polymer species, and n is the number moles. Now, since

= 1n as (2.30) the activity of the solvent is found as

(2.31)

where the relative chemical potential of the solvent is found by differentiating the Gibbs free energy of mixing expression via Equation (2.29).

Chapter 2: Thermodynamic Background 15 2.1.5 Normalization of Activity Coefficients

A reference state must always be chosen when working with activity coefficient models. That is to say that the activity coefficients must be normalized to some reference state. In particular, they are usually referenced to an ideal solution (Raoult's law) or to an ideal dilute solution (Henry's law).

Both of these normalizations are discussed below.

2.1.5.1 Symmetric Convention/or Normalization

In the case of symmetric normalization, the ideal solution reference (Raoult's law) is chosen; and the activity coefficients are normalized as follows:

"Yi -+ 1 as xi -+ 1 (2.32)

Th.is holds for both solvent and solute, and hence is called the symmetric convention for normalization.

2./.5.2 Unsymmetric Convention for Normalization

In the case of unsymmetric normalization, the ideal dilute solution (Henry's law) is chosen as the reference; and the activity coefficients are normalized as follows:

Yi ..... l as xi ..... 1 (solvent) (2.33a)

"Yj-+ 1 as xi .....0 (solute) (2.33b)

And since the solvent and solute are normalized in different ways, this is called the unsymmetric convention for normalization. Th.is convention is usually denoted with an asterisk ( •); and hence, the activity coefficient of a component which approaches unity as its mole fraction goes to zero is signified by:

* "Yj ..... 1 as xi .....0 (2.34)

Chapter 2: Thermodynamic Background 16 2.1.6 Liquid-Phase Activity Coefficient Models Via Excess Gibbs Free Energy

Two particular approaches stand out in the development of models for liquid-phase activity coefficients. As is shown in Equation (2.28), the activity coefficient expressions can be found by performing appropriate partial differentiations on the excess Gibbs free energy model, i.e. ge. = f(z.,T,P) . The two approaches are as follows. First, a model may be written in terms of

Agmixing, where Agmixingis composed of an enthalpic (6hmixing)and an entropic (6smixing)contribution by the following familiar thermodynamic relation

Agmixing = Ahmixing - T Asmixing (2.35)

This can also be transformed into an excess Gibbs free energy, gE , expression via Equation (2.16) or (2.17). The activity coefficients are found by using Equation (2.28).

The second approach is to develop the model directly in terms of gE , perhaps by the following formulations:

I. Like the former method only, ge. = hE - Tse. 2. By the relation, gE = g - gideal

3. As was done by Renon and Prausnitz [ 114),

gE = L xi (g(i) - g~~re) I

where g(il's are "residual" Gibbs energies of cells with molecules of type i at

the center.

4. By the use of some expression of gE as a function of all Xi , perhaps empirical,

symmetrical, series expansion (eg. a virial expansion), etc.

Many thermodynamic models have been proposed for expressing the chemical potential, activity, activity coefficient, or Gibbs free energy as a function of temperature, pressure, and composition. A handful of these will be examined in the ensuing Chapters, for their application to the

Chapter 2: Thermodynamic Background 17 aqueous-polymer two-phase systems described in the Introduction (Chapter l). But before they are discussed, the relation of thermodynamic models to phase equilibria will be outlined.

2.2 Phase Equilibria

An appropriate description of phase equilibria is the application of solution thermodynamics to solving (modeling) real problems involving the contact of two or more phases in equilibrium.

A discussion of how solution thermodynamics, or more appropriately thermodynamic models, are applied to problems in phase equilibria is presented below. Since phase equilibria involves two or more phases in contact, and the work presented in this thesis applies to liquid-liquid systems, an understanding of how these liquid-liquid systems exist, from a thermodynamic standpoint, is necessary.

2.2.1 Criteria for Phase Splitting

Many liquid systems exhibit partial miscibility. This is generally termed phase splitting. In

polymer systems, this phenomenon is usually referred to as polymer-polymer incompatibility (or

immiscibility). The driving force for phase splitting is a decrease in the Gibbs free energy of mixing or the Gibbs free energy of the mixture. That is, if the Gibbs free energy of the two-phase system is lower than that for the single-phase mixture, then the Gibbs free energy of the system will be minimized by phase separation to form two phases. There is another criterion for phase splitting, and that is that phase splitting occurs when there exists a portion of the .1gmixing versus mole fraction Xi (or the ~ixture versus Xi) curve that is concave downward. Recall from calculus that the curve of a function f is concave downward (i.e. has negative curvature) if f' < 0. Thus, the criteria for phase splitting may be written as

(2.36)

Chapter 2: Thermodynamic Background 18 (2.37)

Figure 1 is a graphical depiction of the relation between the Gibbs free energy of mixing and the phenomenon of phase splitting. There are several key features of Figure 1. First, notice that curve 1 has a positive curvature (f" > 0 , concave upward) everywhere, and so the binary system represented by it is completely miscible. Secondly, notice that there is no common tangent on this curve. Compare this, on the other hand, with curve 2. The first observation is that curve 2 exhibits negative curvature (f" < 0 , concave downward) in a portion of the plot, and the Gibbs free energy within this region is minimized by phase separation. This is represented by points a and b. A mixture of composition x will split into phases of compostion x' and x" because the free energy at point b is lower than the energy at point a. Notice also that curve 2 has a common tangent enclosing the region of the curve exhibiting irnmiscibility.

The points x' and x" represent the nodes (or end points) of a tie-line. A set of these tie-line points yields the binodal or phase boundary curve. Examining curve 2 of Figure l closer, one sees that between binodal points x' and x", the curve passes through an inflection point, a maximum, and another inflection point. These inflection points, x'sp and x"sp , are called spinodal points, and the curve passing through a set of these points is called the spinodal. For a given system at some set of specified conditions, for which a plot such as Figure l ( curve 2) could be constructed, the spinodal and binodal would fall on the same tie-line.

The region between the two spinodal points is an unstable region; a total system composition in this region leads to spontaneous phase separation. This process is sometimes referred to as spinodal decomposition. The region between a binodal and spinodal point of the same phase is a metastable region; a total system composition in this region does not spontaneously separate. A nucleation and growth mechanism has been proposed for phase separation in this region (97).

The binodal curve is the more important curve in that it represents the boundary between the one-phase and the two-phase regions. The spinodal, on the other hand, represents the boundary of spontaneous phase separation.

Chapter 2: Thermodynamic Background 19 0------~

-- I .._ .._ 0\ ---- C: - X E C'

0 x' x;P x~'P x" 1 COMPOSITION

Figure I. Gibbs free energy of mixing versus mole fraction for binary mixture: This figure represents

a binary system at constant temperature an'd pressure. Curve I corresponds to a system

exhibiting complete miscibility. Curve 2 represents partial miscibility or phase splitting.

Chapter 2: Thermodynamic Background 20 For a binary mixture, where the Gibbs free energy of the mixture, ~ixtu"' , and the Gibbs free energy of mixing, Agmixins, may be written in terms of the excess Gibbs free energy function, gE , as

E _ real ideal _ A A ideal g = g - g - gmixing - gmixing (2.38)

(2.39)

(2.40) the criterion for phase splitting becomes:

[ a2gE + RT [-1 + _l ] < 0 (2.41) ;i 2 l Xt X2 uX1 T,P

The way to view this criterion is that when Equation (2.41) is applied to a particular thermodynamic model, written in terms of the excess Gibbs free energy function, there must be some value for the model parameters that satisfies this equation in order for phase separation to occur.

The first three derivatives of the Gibbs free energy of mixing are all important. The significance of these expressions is that the first derivative of the Gibbs free energy predicts the binodal curve which is the phase boundary between the one-phase and two-phase regions of the mixture as is shown by the solid curve in Figure 2 for a ternary mixture. The second derivative predicts the spinodal. The spinodal is the locus of inflection points within the region of instability as denoted by points x' sp and x"sp of Figure 1. The spinodal is shown in Figure 2 by a dashed curve. The third derivative can be used to predict the plait point (or critical point). The plait point is the place at which the tie-lines converge with the binodal curve and is represented in Figure 2 by an open circle.

When the two points of inflection (the spinodal points) merge to yield a single point, this is referred to as the point of incipient instability or incipient phase separation. At the point of incipient phase separation, the second and third derivatives of the Gibbs free energy of mixing with respect to compositions, like Equation (2.37), must be equal to zero.

Chapter 2: ThermodynamicBackground 21 3

...... ', ' ' ' ' \ ' ' A ' \ ' I '\ 1 2

Figure 2. Isothermal phase diagram for a ternary system displaying the binodal, spinodal, anrl plait

point.: The solid curve represents the binodal curve; the dashed curve is the spinodal; and

the open circle is the plait point. Point A represents the overall composition of some mix·

ture, and points B and C correspond to the equilibrium compositions of the two phases

(binodal points) formed upon phase separation of that mixture. Points D and E are spi·

nodal points. Notice that all points A,B,C,D, and E fall on the same Lie-line.

Chapter 2: Thermodynamic Background 22 For a binary system, the use of the derivatives described above for describing the phase behavior can be demonstrated as follows. The first derivative of the Gibbs free energy of mixing as shown earlier is equal to the relative chemical potential

(2.42)

and nodes on the binodal curve are determined from Equation (2.42) satisfying the equilibrium equation

for i = 1,2

The second derivative provides another criterion for phase splitting, which is equivalently the spinodal and the curvature of the Gibbs free energy of mixing, as

(2.43)

For a ternary system, the relation for the spinodal is

= 0 (2.44)

with the derivatives being taken at constant T, P, ni.

The third derivative, of a binary mixture,

(2.45)

is used to find the critical or plait point. Note that the binodal and spinodal both pass through the plait point, since at this point the first three derivatives must all be zero.

Chapter 2: Thermodynamic Background 23 Notice from these expressions, Figure 1 could just as easily have been constructed with the relative chemical potential instead of the Gibbs free energy of mixing as the dependent variable.

Thus, another way to graphically represent the phenomenon of phase splitting in terms of thermodynamic functions is to plot the dimensionless relative chemical potential versus composition.

Figure 3 displays the relation of the relative chemical potential of the solvent as a function of polymer volume fraction for the cases of partial miscibility (phase splitting), critical miscibility (incipient phase separation), and complete miscibility. By the relation of Equation (2.42) and the equilibrium equations following it, it is seen that in order for there to be phase separation, there must be two concentrations at which the relative chemical potential has the same value. This criterion is not satisfied by the curve of Figure 3 representing complete miscibility, thus the solvent and polymer are totally miscible. The curve denoting critical miscibility is the point at which this criterion is just satisfied, and thus represents incipient phase separation. The curve depicting partial miscibility clearly satisfies this criterion, and hence yields a region of phase separation of the solvent and polymer.

Another useful plot is constructed by adding temperature as a variable, and projecting the now three-dimensional free energy plot onto the T,x (temperature, composition) plane. This is equivalent to plotting the equilibrium compositions of the phases versus temperature, as is shown in

Figure 4. The point Tc, x,,is the critical point and is analogous to the plait point of Figure 2. For the case shown in Figure 4, this point represents an upper critical solution temperature (UCST) or upper consolute temperature. This Figure suggests that an incompatible polymer blend may be made compatible (miscible) by raising the temperature above the upper critical solution temperature. However, it has been observed that some solutions become incompatible (immisc~ble)upon raising the temperature high enough. This phenomenon is displayed in Figure 5 and is explained by the fact that there may also exist a lower critical solution temperature. Suppose for example that the temperature of the solution is such that there exists a two-phase mixture. As the temperature is increased, the range of immiscibility decreases, and as the temperature passes through the UCST, the range of immiscibility vanishes, producing a single-phase mixture. If the temperature is raised even higher, through the LCST, another two-phase mixture is formed. This is just one type of

Chapter 2: Thermodynamic Background 24 +------.

0 ------PARTIAL ...... MISCIBILITY 0::: ,.._ CRITICAL MISCIBILITY 0" Vl ~ I V) ~ ---- COMPLETE MISCIBILITY - 0 1 POLYMER VOLUME FRACTION

Figure 3. Relative chemical potential of the solvent versus polymer volume fraction for a binary sol-

vent-polymer solution: The various curves display the functional dependence of the relative

chemical potential of the solvent on the concentration of polymer for the cases of partial

miscibility (phase splitting), critical miscibility (incipient phase separation), and complete

misciblity.

Chapter 2: Thermodynamic Background 25 consolute temperature behavior. The interested reader is referred to Prausnitz et al. [108] for a description of other types of consolute temperature behavior.

Thus, since some polymer solutions have been observed to display both an upper and lower critical solution temperature, an adequate theory for polymer solutions should be able to predict both of them. This will be discussed later in relation to several models and their ability or inability to predict this behavior.

Once an expression for the appropriate thermodynamic function ( Gibbs free energy of mixing, for example) is known, it is a straightforward procedure to determine the spinodal and critical point via the equations derived above. However, in general an analytical expression for the binodal is unobtainable, and one is obligated to use numerical methods to obtain its solution. It should also be noted that the solution of the spinodal, even though much simpler, does not yield the most important relation sought, i.e. the phase boundaries or coexistance curves of the two phases in equilibrium. Flory [37], Scott [119,120], Shultz and Flory (123], and Tompa [131,132]have introduced simplifying assumptions that allow for analytical solutions of the simplified cases of binary and ternary solvent-polymer solution behavior.

2.2.2 The Liquid-LiquidEquilibrium Problem

lbe liquid-liquid equilibrium problem arises due to the partial miscibility observed in certain mixtures of liquids. In fact, this phenomenon of phase splitting has led to the development of many useful liquid-liquid extraction systems, which work primarily due to the differences in distribution of solute molecules between the two phases. Again, the primary goal of this research is to investi- gate the applicability of existing thermodynamic models for the prediction of the phase behavior in the aqueous-polymer liquid-liquid extraction system. The liquid-liquid equilibrium problem is setup as follows. Gibbs (46) showed that for a system to be at equilibrium, the following condition had to be true

(2.46)

Chapter 2: Thermodynamic Background 26 BINODAL ----- SPINODAL

w 0:: :::> t- ~ w a.. ~ w t-

~~ x" COMPOSITION

Figure 4. Phase diagram for a binary system projected onto the temperature , composition plane: This

system is exhibiting an upper critical solution temperature. The solid line represents the

binodal, and the dotted curve represents the spinodal. The subscript c and open circle

correspond to the critical point.

Chapter 2: Thermodynamic Background 27 w 0::: ONE :::> LCST t- PHASE

POLYMER VOLUME FRACTION

Figure 5. Upper and lower critical solution temperature behavior of a binary system: This system exhibits both an upper critical solution temperature cucsnand a lower critical solution temperature (LCST).

Chapter 2: Thermodynamic Background 28 which yields (see Smith and Van Ness (125)) µ{ = µ( for i = 1,2, ... ,N (2.47) where the single prime and double prime represent the two different phases. This equilibrium equation may also be expressed in terms of fugacities,

f( = f."'I for i = 1,2, ... , N (2.48) and referring to Equation (2.12),

for i = 1,2, ... ,N (2.49) then, substituting Equation (2.49) into Equation (2.48) and assuming the same reference state for both liquid phases,

= x(y( for i = 1,2, ... ,N (2.50)

Equation (2.50) is the form of the equilibrium equation used in liquid-liquid equilibrium problems.

The activity coefficients, y1 , are obtained from a thermodynamic model via Equation (2.28). No- tice that the solution of the above N equations of the form of Equation (2.50) requires an iterative scheme since the activity coefficient is a function of the mole fraction, and the mole fraction is what is being sought. The iteration is required to get from some initial estimate of the mole fractions, x ;° , to the solution vector ,l!; 1k, which satisfies the objective function of Equation (2.51) below.

The objective of a liquid-liquid equilibrium problem, then, is to find the mole fractions (or other concentration variable) which yield activity coefficients for which the following objective function is satisfied

OBJECTIVE FUNCTION = x( y( - x( y( for i = 1,2, ... ,N (2.51) to some specified tolerance. As is obvious, if an appropriate thermodynamic model is not employed for estimating the activity coefficients, the phase behavior may be predicted erroneously.

Chapter 2: Thermodynamic Background 29 2.3 Application of Thermodynamic Models

In order to apply thermodynamic models, in a quantitative sense, experimental data must exist from which the model parameters may be obtained. In the following Chapters, a number of existing thermodynamic models will be described. Each of these models will be characterized by a set of parameters which are specific to the systems to which they are applied. The determination of these parameters from the existing experimental data will be detailed. It is appropriate to give here only a general discussion.

2.3.1 Prediction of Activity Coefficients

As discussed earlier, expressions for the activity coefficients are obtained by taking the appropriate partial derivatives of g8 • Then, in order to predict activity coefficients from the model, the model parameters must be determined from experimental data available on the system of interest.

This can be a very tedious process, and it has been studied extensively for many of the available models. In general, this procedure is discussed by Anderson et al. [6,7,8), Magnussen et al. [80),

Prausnitz et al. [106), Prausnitz et al. [108), S0rensen et al. [126,127],and Walas [138).

2.3.2 Comparison of Experiment and Theory

As all of the thermodynamic relations described in the foregoing Sections of this Chapter are related (as has also been previously discussed), then each of these relations is essentially equivalent to the others. Thus, it does not matter whether one chooses to describe solutions by models written in terms ~&nixing, g8 , y, orµ. However, in certain cases it is more appropriate, from a practical point of view, to use a certain type of relation. Sometimes, this is also done in keeping with the general notation. Often times the chemical potential relation provides a more suitable relationship than the corresponding activity coefficient expression. This is the particular case in dealing with polymer solutions because the models arc typically written in terms of volume fractions instead of mole fractions. In other words, in the case of polymer solutions, one is generally interested in the

Chapter 2: Thermodynamic Background 30 activity and not the activity coefficient, and in order to avoid mole fractions (recall from previous discussion that mole fractions are not useful for polymer solutions) the activity is better obtained from the chemical potential expression than the activity coefficient expression.

From another point of view, there are many ways in which to compare the model predictions with experimental data. Two such methods are the comparison of activity coefficients and the comparison of predicted equilibrium concentrations, as is done by comparing phase diagrams or binodal (or binodial) curves in liquid-liquid equilibrium systems. The choice of these is many times determined by the type of experimental data that are available. In the case of the aqueous-polymer systems being studied here, there exist limited data on activity coefficients in binary solutions of, for example, water in polyethylene glycol, so preliminary investigations involved predictions of activity coefficients in these solutions. This should provide an adequate test in that if the model could predict the activity coefficients in one solvent-polymer solution, it would stand a good chance of predicting activity coefficients in other solvent-polymer solutions. In other words, the chief problem in predicting liquid-liquid equilibria is the choice or development of a suitable model, and since activity coefficients are an expression of the model, then the comparison of theoretical and experimental activity coefficients provide an adequate test of the model. Likewise, since the model is written for a particular type of system, for example solvent-polymer solutions, then it should be generally applicable to all solvent-polymer solutions.

A better, or more complete, test of models for liquid-liquid applications is the prediction of the phase behavior, i.e. the prediction of the binodal curve from phase splitting and the prediction of the distribution of the components among the phases present.

Lack of appropriate experimental data on the systems of interest limited the amount of quantitative comparisons of experimental and predicted phase behavi9r. The reason being that the model parameters are generally determined from binary data, which at the present does not exist for many of the components of interest.

Chapter 2: Thermodynamic Background 31 2.4 Summary

In summary, the relations between the fugacity, the chemical potential, the activity, the activity coefficient, the Gibbs free energy of mixing, and the excess free energy have been displayed. Sug- gestions have been given for the formulation of thermodynamic models via Gibbs free energy expressions, and the subsequent determination of the activity coefficients from such models was described. The liquid-liquid equilibrium (LLE) problem was then setup with detail given as to the connection between the thermodynamic model and the LLE problem. Finally, the comparison of experimental data with theoretical predictions was discussed.

Chapter 2: Thermodynamic Background 32 3.0 Experimental Systems and Data

3.1 Bi11aryand Ternary Experimental Systems

One of the major limitations in modeling aqueous-polymer two-phase systems is the lack of experimental data. In order to obtain meaningful parameters for use in thermodynamic analyses, sufficient experimental data must exist. In this study, several thermodynamic models will be compared based upon a limited amount of experimental data which is described in this Chapter.

Perhaps the most appropriate contribution in this area at the present time would be the generation of accurate, meaningful thermodynamic data from well characterized systems. The serious lack of data on these systems and the need for accurate thermodynamic measurements is a realiza- tion reached by many including this author (52,105,107,122).

Another important point is that both binary and ternary data are needed. Most available models lend themselves more readily to reduction of binary experimental data for parameter estimation; however, without the use of ternary data to fine-tune these models, predictions of ternary behavior from these binary parameters can often lead to erroneous results. Also, some models simply do not lend themselves, without modifications or simplifying assumptions, to reducing ternary phase equilibrium data. Therefore, it is usually helpful, if not necessary, to have binary data on the various binary pairs comprising a multicomponent system.

The work by Albertsson (3) and others (see (141)) has addressed many different aqueous-polymer systems. These systems, however, may be classified by the following four categories: systems containing (1) two nonionic polymers and water; (2) a polyelectrolyte, a nonionic polymer, and water; (3) two polyeclectrolytes and water; and (4) a polymer, a low molecular weight component and water. The emphasis of this research is on aqueous-polymer two-phase systems formed by mixing aqueous solutions of two nonionic polymers.

Chapter 3: Experimental Systems and Data 33 The binary and ternary experimental systems being studied here are shown in Tables l and 2.

These Tables contain pertinent information such as the temperature, polymer concentration range, and literature reference. Further information on the characterization of the polymers listed in Ta- bles 1 and 2 is given in Tables 3, 4, and 5. These latter Tables contain information on such properties as molecular weights, polydispersity numbers, intrinsic viscosity, and liquid densities. for detailed information of polymer characterization, see any standard polymer textbook. For this thesis, the following discussion should be sufficient. Polymers are specified by a molecular weight number and distribution. The two most common molecular weights reported are the number average molecular weight Mn and the weight average molecular weight Mw. The number average molecular weight is based on the number of molecules of each size comprising the polymer species.

The weight average molecular weight is based on the weight of each species in the polymer sample.

The particular number reported for Mn and Mw is the central tendency of the polymer molecular weight distribution, which is seldom a very narrow, sharp distribution. Thus, the polymer species needs to be characterized not only by a value of Mn or Mw but also by the breadth and shape of the distribution. The ratio of Mwto Mn is called the polydispersity number and is a measure of the breadth of the molecular weight distribution. This is not the most statistically sound measure of the molecular weight distribution, but it is used here since it was reported or is easily obtained for some of the polymers used. A more correct measure of the breadth of the molecular weight distribution is the standard deviation of the distribution.

3.2 Experimental Metlzods for Collection of Binary Data

Malcolm and Rawlinson [81] measured vapor pressures of solutions of polyethylene glycol in water. They measured the difference between the vapor pressure of the solution and that of pure water at the same temperature. From this data, they calculated the activity of water. Herskowitz and Gottlieb 1571reported activity data on aqueous-polyethylene glycol (PEG) solutions with PEG molecular weights ranging from 200 to 6000 over a wide range of concentrations. They employed an isopiestic apparatus as described by Herskowitz and Gottlieb 155].

Chapter 3: Experimental Systems and Data 34 Table l. Experimental Binary Aqueous-Polymer Systems

Polymer Cone. 3 System 10- M 0 (PEG) Temp., K Range (% w/w) Reference

Water-PEG200 0.19-0.21 293.1 0.15-0.97 (57] 313. l 0.15-0.97 333.1 0.15-0.97

Water-PEG300 0.30 303.1 0.30-0.99 (81] 323.l 0.30-0.99 338.1 0.30-0.99

Water- PEG600 0.57-0.63 293.1 0.48-0.985 (57] 313.1 0.23-0.98 333.1 0.27-0.985

Water- PEG 1500 1.43-1.57 293.l 0.345-0.64 (57) 313.1 0.50-0.91 333.l 0.575-0.98

Water-PEG3000 3.0 328.l 0.50-0.97 (81) 338.l 0.50-0.99

Water-PEG5000 5.0 333.l 0.735-0.95 (81) 338.l 0.50-0.99

Water- PEG6000 6.0-7.5 293.l 0.345-0.64 (57) 313.1 0.50-0.91 333.l 0.575-0.98

Chapter 3: Experimental Systems and Data 35 Table 2. Experimental Ternary Aqueous-Polymer Systems from Albertsson

System Temperatures, K

Water-PEG 6000-D17 293.l 273.1

Water-PEG 6000-D48 293.l 277.1 273.1

Water-PEG 6000-D37 293.1 273.1 Water-PEG 6000-D24 293.1

Water-PEG 6000-D17 293.1 273.1

Water-PEG 6000-DS 293.1

Water-PEG 4000-D48 293.l 273.1

Water-PEG 4000-D17 293.1

Water-PEG 20000-D17 293.l

NOTE: The data for these systems are given by Albertsson (3).

Chapter 3: Experimental Systems and Data 36 Table 3. Characterization of Various Molecular Weight Fractions of Dextran

Intrinsic Number Average Weight Average Viscosity Molecular Weight Molecular_Weight 3 3 Abbreviation (in ml/g) 10- M0 10- Mw Polydispersity

D5 4.5 2.3 3.45 1.5 D 17 16.8 23 30 1.3 D 19 or Dextran 40 19 20 42 2.1 D 24 24 40.5 D 37 37 83 179 2.2 D 48 or Dextran 500 48 180 460 2.6 D 68 68 280 2,200 7.9 D 70 70 73

Table 4. Characterization of Various Molecular Weight Fractions of Polyethylene Glycol

Number Average Molecular_Weight 3 Abbreviation 10- M 0

PEG 20000 15-20 PEG 6000 6-7.5 PEG 4000 3-3.7 PEG 1500 1.3-1.6 PEG 1000 0.95-1.05 PEG 600 0.57-0.63 PEG 400 0.38-0.42 PEG 300 0.285-0.315

Chapter 3: Experimental Systems and Data 37 Table S. Liquid Densities of Water and Various Molecular Weight Fractions of Polyethylene Glycol and Dextran

Component MW fractions Liquid Density (g/cc)

Water 0.99

Polyethylene Glycol PEG 200 I.OS Polyethylene Glycol PEG 300 l.l Polyethylene Glycol PEG 600 1.12 Polyethylene Glycol PEG 1500 1.18 Polyethylene Glycol PEG 3000 1.19 Polyethylene Glycol PEG 5000 1.2 Polyethylene Glycol PEG 6000 1.21

Dextran all 1.2

NOTES: (1) Temperature dependence on densities neglected.

(2) PEG density data obtained from Merck Index, 9th ed. (1976) or interpolated from data found therein.

(3) Dextran density estimated from comparison of aqueous-polymer densities for PEG and Dextran given by Albertsson (3).

Chapter 3: Experimental Systems and Data 38 3.3 Analyses of Equilibrium Phase Compositions

For most of the phase systems studied by Albertsson (4), the compositions of the different phases were calculated from determinations of the water content and the concentration of one of the polymers; the concentration of the other polymer was then obtained by subtraction. The total composition of the system was always known since the phase systems were formed by mixing weighed amounts of water and polymer solutions of known concentrations. As a standard practice,

Albertsson determined all concentrations: of the total system and the top and bottom phases. Then, as a check on experimental accuracy, the tie-line data (i.e. the three sets of compositions as determined above) were plotted; the test being whether or not all three composition points would fall on a straight line (the tie-line).

As just described, the equilibrium compositions needed to construct a phase diagram of the system can be determined by measuring the compositions of each of the phases in the equilibrated system. A series of runs to determine different tie-lines can be made by varying the polymer composition of the overall system. Another method involves performing a turbidity (or cloud-point titration) experiment. This is achieved by beginning with a relatively concentrated solution of one polymer and adding dropwise a stock solution of the second polymer. After each drop is added, the solution is mixed thoroughly. The appearance of turbidity suggests the onset of phase separation. Since the amounts of all species are known, this represents one point on the phase diagram. To determine additional points, a small amount of water may be added to again form a single phase mixture and the second polymer can again be added until turbidity is produced [ IO).

This latter procedure is not an accurate one when dealing with polydisperse polymer solutions

[ l 0,22,69). The reason is that the location of the binodal by turbidity will identify the two-phase region formed by separation of the higher molecular weight members of the two fractions in the polydisperse system, i.e. the phase separation of a polydisperse system is a gradual process with higher molecular weight fractions separating first. The first onset of turbidity will therefore result in a binodal which lies at lower polymer compositions than would be measured by analysis of the equilibrium phases (3,10,18).

Chapter 3: Experimental Systems and Data 39 Hence, the analysis of equilibrium phase compositions is preferred over turbidity measurements. Bamberger et al. [ 1OJ suggest a combination of the following methods for analysis of equilibrium phase compositions: refractive index, optical rotation (polarimetry), and dry weights. They also discuss the use, advantage, and disadvantage of each.

3.4 Representation of Experimental Data

Albertsson chose to represent his ternary phase equilibrium data on a binary plot instead of the common ternary diagram. He probably chose to do so, since for these systems the polymer concentrations are the important parameters, and binary plots are more conveniently usable than are ternary plots. The water composition can always be found by subtraction anyway. An example phase diagram is shown in Figure 6. This diagram is constructed from the equilibrium compositions in weight percent as found from the analyses described above. An example of a set of tie-line data is shown in Figure 6 by points P, Q, and R, where P is the total system composition and Q and R are the compositions of the two coexisting equilibrium phases.

The vertical axis is commonly used for the polymer which is most prevalent in the top phase, which for the water-polyethylene glycol(PEG)-dextran systems is PEG. The curve connecting the equilibrium compositions and running through point C is called the binodal. Point C is called the critical point and is defined as the point at which the tie-line and the binodal converge. This system could equivalently be plotted on a ternary diagram as shown in Figure 2 (Chapter 2).

Chapter 3: Experimental Systems and Data 40 20

0 CJ ~ 0\10 Q) C Q) ~ ...... c Q) ~ 0 5 a.

0 ...... _ ...... ,....,,.,...... 0 5 10 15 20 25 30 Dextran (,; w/w)

Figure 6. Example binodal phase diagram of water-PEG6000-Dextran(D 17) system at T = 20°C: The experimental data are from Albcrtsson (3). The circles represent experimental compositions

of the coexisting phases (points Q and R). The squares represent the total system compo-

sition (point P). The line passing through points P, Q, and R is the tie-line. Point C is the

critical poinL

Chapter 3: Experimental Systems and Data 41 4.0 Solutions of Small and Similar-SizedMolecules: Local

Compositionand Group-ContributionModels

In the next few Chapters, the applicability of several existing thermodynamic models for predicting the phase behavior of aqueous-polymer systems as described by Albertsson [3] will be discussed. However, this will be accomplished in a series of steps. First, thermodynamic models for solutions of small molecules will be described. Then, the discussion will tum to models for solvent-polymer solutions and a detailed discussion of polymer solution theory. Next, an extension of the theory of small molecules to describe solvent-electrolyte solutions will be discussed with a brief description of an attempt to modify this model to handle solvent-polymer-electrolyte systems.

Much emphasis has gone into the development of activity coefficient models with van Laar probably being the first to address this problem. The liquid state lies somewhere between the gas state and the solid state. Molecules in a liquid do not have the freedom that molecules in the gas state do, but they have more freedom of movement than do their fellow molecules in the solid state.

Hence, the liquid state in theory could be modeled either by a dense-gas equation of state or by a liquid "lattice" model. The equation of state approach will only be discussed briefly in this work

(later in Chapter 6). Most of the following discussion, then, will be related to thermodynamic models developed from the liquid lattice model.

The following models for the prescribed applications have been suggested by W alas !138). For correlation of activity coefficients in binary and multicomponent systems, the Wilson equations are recommended. For activity coefficients from group-contribution schemes, the UNIFAC method is recommended over ASOG because it has received much more attention and is thus more developed. However, if data are available for both models, they are effectively equivalent. Both are advantageous over the solubility parameter approach (except possibly for polymer solutions). For liquid-liquid equilibria, the NRTL equation performs well, even with only binary parameters; but

Chapter 4: Solutions of Small Molecules 42 it can perform much better with parameters determined from multicomponent data. Walas also suggests that (I) the NRTL model may be better for aqueous systems; (2) the Wilson equation very poorly represents systems exhibiting hydrogen bonding; and (3) the UNIQUAC model is also useful for liquid-liquid equilibria, especially when the molecules differ greatly in size.

Originally, this research was aimed at using a local composition or other semi-empirical model

.to represent aqueous-polymer systems. Thus, the above discussion serves two purposes: ( l) to

suggest models that may prove worthwhile and (2) to cite the particular nuances of each of these

models as they apply to the aqueous-polymer systems being studied. The models to be reviewed

in this Chapter are by far not the only models available, but they are all intertwined in some way

(as will become apparent) to the work and discussions presented in this thesis.

4.1 Local Composition Theory: Wilson (1964)

4.1.1 Wilson Model

In 1964, Wilson 1143)developed a thermodynamic model in which he introduced the local

composition concept in relation to thermodynamic excess free energy functions. His idea was to

take into account nonrandomness in liquid mixtures. This he did by considering the mixture to

have local fluctuations in the composition so that the composition throughout the mixture was not

equivalent to the overall composition. Wilson suggested that the local mole fraction x11 of type I

molecules and the local mole fraction x21 of type 2 molecules which are in the immediate vicinity

of a type I molecule can be related in terms of the overall mole fractions and two Boltzmann factors

as:

x21 = x2 exp ( -g 21/RT) (4.1) Xi 1 X1 exp ( -g 11/RT)

where the - g21 and the - g11 are energies of interaction between molecules of types 2 and 1, and

two type l molecules, respectively.

Chapter 4: Solutions of Small Molecules 43 The final form of the expression obtained by Wilson for a binary mixture, in a manner analogous to the Flory-Huggins development for athermal polymer solutions (to be discussed in

Chapter 5), is:

= x 1n + x 1n [ ~22 (4.2) LRT I [h]Xi 2 X2 J

In this equation, Wilson has replaced the volume fraction with the local volume fractions ~11 and ~22 which are obtained from the definitions of ~11

(4.3)

and ~22 similarly, which yield

(4.4)

and

(4.5)

where v1 and v2 are molar volumes.

Also, the Wilson equation is probably best recognized in another form. As discussed by

Prausnitz [ I 08), the usual form is obtained as follows. Defining two new parameters, /\ 12 and

/\21, in terms of the previous parameters V1, v2, g11, g12, g21, and g22 and assuming g12= g21

!\ = _2V exp [ _ g 12 -g t t J (4.6) 12 Vi RT and

!\ = _1V exp [ _ g 12 -g 22 J (4.7) 21 V2 RT

Chapter 4: Solutions of Small Molecules 44 for which the Wilson model becomes

(4.8)

The derivation of the Wilson equation has no strict theoretical basis, but it does follow intuitively from the Flory-Huggins equation (to be discussed in Chapter 5) (108].

The Wilson model can readily be generalized to model multicomponent solutions, in which case the model is written as:

(4.9)

where

- 2 exp[ - gij - gii J (4.10) Vj RT

(4.11)

and the activity coefficients are given by

m X·Ak Yk = - X·Ak·] + l - L i i (4.12) 1n 1n[j l:= I J J . I m 1= L X·J\.. j = 1 J IJ

Again, the key features of the Wilson model are ( l) it is intuitively based on the :heoretical athermal Flory-Huggins equation; and as such (2) its form is sort of a combinatorial entropy of mixing with a built in interaction energy dependence. As will be shown in Chapter 5, the Flory-

Huggins equation is based on a statistical mechanical lattice treatment for athermal polymer solutions. This suggests that the Wilson model may not have an adequate enthalpic part, which is very likely responsible for the inability of the Wilson model to predict liquid-liquid phase scpa- ration.

Chapter 4: Solutions of Small Molecules 45 The Wilson equation has proven to be very useful, except for systems exhibiting hydrogen bonding, i.e. strong, specific interactions. This is reasonable from the above discussion, for the

Wilson model is seen to be more of an entropy of mixing model, lacking in a proper enthalpic contribution, and therefore it should not be expected to work well for systems exhibiting strong,

specific interactions.

4.1.2 Criterion for Phase Splitting

The Wilson model has been used successfully to describe a wide variety of liquid mixtures.

However, the Wilson equation cannot predict partial miscibility in liquid mixtures. No values of the Wilson parameters can be be found for which the criterion for phase splitting is satisfied. Recall

from Chapter 2 that for binary mixtures,

a2gE]+ RT[-l + < 0 (4.13) [ 2 Xt _l]X2 aXt T,P

Thus, when Equation (4.13) is applied to the Wilson model (Equations (4.2) through (4.5)), no

values of the parameters (~1 - &,) and (&i - ~i) can be found for which Equation (4.13) is satisfied. Hence, the Wilson model is inadequate to represent our system, which exists by the very nature of

polymer-polymer immiscibility.

The classical Flory-Huggins treatment is based on the assumption that energy interactions do not contribute to the entropy of mixing. The energy of interactions is therefore all absorbed into

the £\h term, and the £\s term is only based on the combinatorial entropy of mixing. On the other hand, the Wilson equation is derived from the athermal Flory-Huggins model

(or the £\s term). The Wilson treatment does not add a separate term for £\h but modifies the £\s

term for local fluctuations in the composition by introducing a Boltzmann-type of interaction re-

lation into the athermal Flory-Huggins expression. Thus, the Wilson model accounts for nonran-

domness in the entropy of mixing. Incidentally, the lack of nonrandom mixing is one of the

inadequacies of the basic relation derived by Flory and Huggins. This brings up two more impor-

tant points. First, if the Wilson equation had a separate £\h term (of the van Laar type used by

Chapter 4: Solutions of Small Molecules 46 Flory and Huggins for nonathermal polymer solutions), then it would be a more correct free energy model and should thus perform better. The inclusion of this ~h term should allow for the prediction of phase splitting and the representation of systems exhibiting strong, specific interactions.

Second, if the Flory-Huggins model was modified by this local composition idea, then a major inadequacy of it could probably be overcome. This inadequacy results from the assumption of a random entropy of mixing as discussed above.

4.1.3 Non-Random Two-Liquid Model (NRTL)

In 1968, Renon and Prausnitz (114) proposed a local composition model based on Scott's two-liquid model (121) and a nonrandomness assumption similar to that of Wilson (1431. At the same time, they gave a critical discussion on the use of local composition models for representing excess Gibbs free energies of liquid mixtures. The development of the NR TL model is based on a two-cell theory where the liquid solution is made up of cells of molecules surrounded by various arrangements of other molecules in the solution. Gibbs energies of interaction between the central molecule of the cell and all nearest neighbors surrounding the central molecule are added to yield the following equation for the excess Gibbs free energy for the mixture

(4.14)

where the local mole fractions x12 and x21 are modified by introducing the "nonrandomness factor", a;; , as

X21 X2exp( - a12g21/RT) = ( 4.15) X1I x1 exp( - a 12g11/RT) and

x12 = x1 exp( - a 12gnfRT) (4.16) X22 x2 exp( - a 12g22/RT)

Chapter 4: Solutions of Small Molecules 47 With the insertion of these relations, reduced by x21 + x11 = l and x12 + x22 = l, into Equation

(4.14), the NRTL model for the excess Gibbs free energy becomes:

L (4.17) RT and in terms of the component activity coefficients, N ~ tji Gji Xj J In Yi= (4.18) where ~g-· t .. = __11 = IJ RT Gii = exp [ - aii tij] aii = constant characteristic of the nonrandomness of the mixture

The NRTL model, unlike the Wilson equation, is applicable to partially as well as completely miscible systems. One drawback to the NRTL model is that it contains 3 parameters. Besides the

2 parameters per binary normally required, it contains a third parameter which must be evaluated.

This parameter, aii, represents the nonrandomness of the mixture. However, reduction of a large amount of experimental data on a variety of systems has shown that a;i varies from 0.2 to 0.47, and a;1 is typically taken to be 0.30 [ l 08, 112,114,138); a more comprehensive examination of Cl;i is given by Walas [138).

The NRTL equation performs best for strongly nonideal mixtures, provided that c2.e is exer- cised in the reduction of data to obtain the model parameters. For moderately nonideal mixtures,

NRTL provides no advantage over the simpler models of van Laar and the 3-suffix Margules equation [ 108). One disadvantage of the NRTL model is that it is more suitable as an enthalpy of mixing model than a Gibbs free energy model. This can best be seen by examining the excess free energy form of the NRTL equation. Recall that the Wilson model was developed in a manner analogous to that of the athermal Flory-Huggins model, which is purely an entropy of mixing

Chapter 4: Solutions of Small Molecules 48 model. No such term corresponding to the Wilson or Flory-Huggins equation is present in the

NRTL model. In other words, since

l RT =

then, the NRTL equation seemingly has no term representing s8 , This is not quite true though.

The NRTL expression is based upon the assumption that the a11parameter adequately describes all of the nonidealities associated with the entropy of the mixture (or nonrandomness of mixing).

In their discussion, Renon and Prausnitz (114) give a generalized expression for the excess

Gibbs free energy which encompasses the Wilson, Heil, and NRTL models. The model by Heil will be discussed in detail in Chapter 6 because it was derived for application to solvent-polymer solutions. For now, suffice it to say that for the case of a binary mixture of small, equal-sized molecules, the Heil equation reduces to:

X· 1n + X 1n L= I [k]Xt 2 [_k]X2 RT ( 4.19) + g21 - gll X ~ + gl2 - g22 X ~ RT 1 "21 RT 2 'o\2

In the generalized form given by Renon and Prausnitz (114) and as shown below in Equations

(4.20) and (4.21), the model by Heil [52,53) is seen to be a combination of the Wilson and NRTL models.

(4.20)

Chapter 4: Solutions of Small Molecules 49 and

= (4.21)

where

The fact that the Heil equation is a combination of the Wilson equation and NRTL model is seen by noting the values of the parameters p, q, P;i , and a;1 given in Table 6. This is noteworthy in assessing the qualitative validity of the Heil model for modeling solvent-polymer solutions. Because it is a combination of the Wilson model (chiefly a combinatorial entropy of mixing model) and the

NRTL model (primarily an enthalpy of mixing model), the Heil equation has a wider range of applicability and is not limited by the inadequacies of either.

Chapter 4: Solutions of Small Molecules 50 Table 6. Parameters and Interrelatedness of the Wilson, Heil, and NRTL Models

Equation p q Pii a1i

Wilson 0 1 vJv1 1

Heil 1 1 vJv1 1

NRTL 1 0 1 ao

Chapter 4: Solutions of Small Molecules 51 4.1.4 Universal Quasi-Chemical Model (UNIQUAC)

UNIQUAC (UNiversal QUAsi Chemical model) is a local composition model based on the quasi-chemical solution theory of Guggenheim (51). As such, UNIQUAC has the advantage of a theoretical basis in statistical mechanics. Another advantage of UNIQUAC is its use of a surface fraction as the primary concentration variable (versus usual mole fractions). The utility of this concept based on surface fractions is its applicability to solutions of both large and small molecules.

The UNIQUAC model as derived by Abrams and Prausnitz [I] is as follows:

[L]Combinatorial + [L]Residual £ = (4.22) RT RT RT where the combinatorial part is given by:

(4.23)

and where the residual part is given by:

(4.24)

In terms of activity coefficients,

In Yi = In yfombinatorial + In yfesidual (4.25) where the combinatorial part of the activity coefficient is given by:

Combinatorial __ _i z 9. N In In ·1 Yi + -2 qi In+. + Ii -- L X· l· (4.26) Xj 'VI Xj j J J and the residual part of the activity coefficient is given by:

Chapter 4: Solutions of Small Molecules 52 Residual In - LN [ J t.. IJ ]] (4.27) Yi . N0· 1 I: ek tki -+-In k

and where the segment fraction (similar to volume fraction), <1>1 ; the area fraction, 01; and the adjustable binary interaction parameter, tiJ, are defined as:

(4.28)

(4.29)

~u ] [ (u .. - u ..) ] tij = exp [ - Ri = exp - iJRT JJ (430)

Also,

z = coordination number = 10

r and q are pure-component parameters

The coordination number is the number of nearest neighbors within the fluid lattice.

Maurer and Prausnitz (84] presented an alternate derivation of UNIQUAC where their method was similar to the two-fluid theory approach of Renon and Prausnitz [l 14] in the derivation of the three-parameter NRTL model. The result is a three-parameter UNIQUAC equation which often yields improved correlations over the two-parameter UNIQUAC equation. However, this three- parameter UNIQUAC cannot be extended to multicomponent solutions without additional assumptions. Another important point made by Maurer is that the introduction of a third parameter

Chapter 4: Solutions of Small Molecules 53 is often not attractive because the quantity and quality of experimental binary data do not justify the use of more than two parameters. It is also appropriate to mention that Maurer and Prausnitz

[84) concluded with a discussion of free volume contributions and the Flory "equation of state contributions" to the excess functions and the importance of these concepts. These ideas will be discussed later.

4.2 Thermodynamics of Models Based on Group-Contribution Schemes

4.2.1 Introduction

The chemical industry relies heavily on the ability to separate product mixtures into the desired components. Since many of the useful separation techniques, such as distillation and extraction, involve considerations of phase equilibrium, this area of thermodynamics is extremely important.

That is to say that in order to design these separation processes, one must be able to quantitatively describe the phase equilibria. This may be achieved in various ways. For example, phase equilibria may be predicted from appropriate thermodynamic models (eg. Wilson, UNIQUAC, NRTL, etc.) using limited experimental data to determine the model parameters. Another example is using empirical correlations (Scatchard-Hildebrand, Chao-Seader, or the DePriester K-charts). On the other hand, experimental data could be obtained for the system of interest, but this is a very tedious undertaking. But what about the case where no experimental data exists, and it is not a timely matter to setup experiments to determine the phase equilibrium data? In this case one can only guess what the equilibrium process is like and most often with a great deal of uncertainty. However, the UNIFAC model provides a rational basis for making this engineering estimate. With UNIF AC, no experimental data are required, and it often yields rather quantitative results when compared with experimental phase equilibrium data.

Chapter 4: Solutions of Small Molecules 54 4.2.2 The Solutions-of-Group Concept

UNIFAC ( UNIQUAC Functional-group Activity Coefficients) is based on a group-contribution extension of the UNIQUAC activity coefficient model previously discussed. The idea behind group-contribution theory is that while there are thousands of industrially important chemicals, there is a much smaller number of functional groups from which these chemicals are composed.

Thus, if a liquid solution can be pictured as an aggregation of functional groups, as opposed to a mixture of molecules, the properties of this solution can perhaps be approximated by summing the contributions of each of the functional groups. This yields a correlation which can describe a large number and variety of mixtures based on a much smaller set of data. The data will be in the form of binary interaction parameters between each pair of functional groups as evaluated from extensive experimental studies on many binary and ternary mixtures.

This group-contribution scheme is necessarily an approximation because the contribution of a functional group in one molecule may be somewhat di.fferent than the contribution of that same group in another molecule. The fundamental assumption in group-contribution methods is addi- tivity. In other words, the contribution of one group is assumed to be independent of all other groups in the mixture. Fredenslund et al. 142]give the following example. The contribution of a carbonyl group in a ketone (say, acetone) would not be expected to be the same as that of a carbonyl group in an organic acid (say, acetic acid). However, it seems very likely, and indeed experience suggests, that the contribution of a carbonyl group in, for example, acetone, is nearly the same as (but not identical to) the contribution of a carbonyl group in another ketone, say, 2-buta- none.

Another point when dealing with a group-contribution scheme is the definition of the functional groups. As the distinction of groups is increased, i.e. groups become more and more specific, the accuracy of the prediction increases. In the limit, however, as more and more distinctions are made, the ultimate group is recovered, that is, the molecule itself. Thus, there is some compromise as to the number of distinct functional groups to include. At the present, there are 75-100 functional groups that are typically used; the actual number varies somewhat between users.

Chapter 4: Solutions of Small Molecules 55 The development of UNIF AC followed in principle the work of Derr and Deal (26) and their

Analytical-Solution-Of-Groups (ASOG) model. The main difference between ASOG and UNI-

F AC is the models upon which they are based. Both ASOG and UNIF AC break the activity into two contributions: one based on molecular size and the other based on molecular interactions. The ways that these are accomplished is what distinguishes the two methods. ASOG uses the athermal

Flory-Huggins model to arbitrarily estimate the contribution due to differences in molecular size, and then applies the Wilson equation to functional groups in order to estimate molecular interactions. UNIF AC, on the other hand, removes the arbitrariness of the first part by combining the solutions-of-group concept with the UNIQUAC model. The advantage here is that the UNIQ-

UAC model contains two parts: one based on the differences in size and shape of the molecules in the mixture and the other based on molecular interactions. When combined with the solutions- of-groups concept, the interaction part has parameters based on interactions between the various pairs of functional groups in the mixture. Also, the contribution due to molecular sizes and shapes are introduced through functional group sizes and interaction surface areas obtained independently from pure-component, molecular structure data.

4.2.3 UNIF AC

UNIFAC (UNIQUAC Functional-group Activity Coefficients) has a few important characteristics which attract many people to using it for phase equilibrium predictions: (I) it requires no data reduction to determine the binary parameters; (2) it has a wide range of applicability (UNI-

FAC contains a large data bank of binary interaction parameters); and (3) it generally yields good results for vapor-liquid equilibrium (VLE) and liquid-liquid equilibrium (LLE) calculations.

In UNIFAC, Fredenslund et al. (42) directly use the combinatorial part of UNIQUAC,

Equation (4.26),

Combinatorial __ In _i z 0- .N In Yi + -2 qi In _l· + Ii -- 1 L X· l· (4.26) Xj I Xj j J J

Chapter 4: Solutions of Small Molecules 56 Recall that only pure component properties (for the functional groups) enter into this equation.

That is, the parameters in this equation, r1 and q1, are determined from the pure component group volume and surface area parameters, Rk and Ot as given in Table 1 of Fredenslund et al. (42]. These are calculated as:

ri = ~ vk(i)Rk (4.31) k and

qi = ~ vk(i)Qk (4.32) k

where vk<1J , always an integer, is the number of groups of type k in molecule i. The group parameters R" and Qk are obtained from the van der Waals group volume and surface areas

Vwk and Awkas given by Bondi (13):

Ywk (4.33) 15.17 and

Awk (4.34) 2.5 · 109

The normalization factors 15.17 and 2.5 • 109 are those given by Abrams and Prausnitz (1).

In UNIFAC, the residual contribution to the activity coefficient uses the solutions-of-group concept. Hence, Equation (4.27) of the UNIQUAC model is replaced by the following equation:

Residual __ 1n Yi (4.35)

where r" is the group residual activity coefficient, and r"(iJ is the residual activity coefficient of group k in a reference solution containing only molecules of type i, and the sum is over all groups

Chapter 4: Solutions of Small Molecules 57 in the mixture. The ln rt(il term in Equation (4.35) is necessary to attain the normalization that the activity coefficient y1 becomes unity as X; -+ 1 (recall discussion in Section 2.1.5). The group residual activity coefficient, rt, is evaluated from the following equation:

(4.36)

The reference group residual activity coefficient, rt!I>, can also be obtained from this expression. In Equation (4.36), em is the area fraction of group m, 'l'km is the group interaction parameter, and the sums are over all of the functional groups present in the mixture. The area fraction here, em, is calculated similarly to 01 in the UNIQUAC model, Equation (4.29):

(4.37)

The group interaction parameter is calculated as:

'I' mn = exp - [ UmnR-;.Unn] (4.38) = exp - [ a;n J

where Umnis a measure of the energy of interaction between groups m and n. The group-interac-

tion parameters, ~" , (two parameters per binary) are the parameters which must be determined

from experimental phase equilibrium data. UNIF AC actually has two data banks, one based on parameters determined from VLE (UNIFAC-VLE) and the other based on LLE (UNIFAC-LLE).

The current status of the UNIFAC-VLE data bank which contains the defined functional groups and model parameters is given by Gmehling et al. (47), Herskowitz and Gottlieb (54), and Macedo

et al. [78J. Earlier contents of the UNIFAC-VLE data bank were given by Skjold-forgensen et al.

[124) and Fredenslund et al. [40,41,42J. The functional group definitions and model parameters for the UNIFAC-LLE data bank are given by Magnussen et al. [79J. For further information on the

UNIFAC model, including the FORTRAN code, see Fredenslund et al. (41J.

Chapter 4: Solutions of Small Molecules 58 4.3 Extensions and Applications to Polymer Solutions

This Section addresses the use of local composition models (at least some of those previously discussed) for describing polymer solutions. In particular, the applicability of UNIQUAC, UNI-

F AC, ASOG, and extensions of these models are discussed. A brief discussion is also given on a couple of other semi-empirical models.

4.3.1 Universal Quasi-Chemical Model (UNIQUAC)

UNIQUAC as derived by Abrams and Prausnitz (1) has previously been discussed. Here it only needs to be reiterated that UNIQUAC is a local composition model based on the quasi- chemical solution theory of Guggenheim [51), and as such, UNIQUAC has the advantage of a theoretical basis in statistical mechanics. Another advantage of UNIQUAC is its use of a surface fraction as the primary concentration variable (versus usual mole fractions), which renders it applicable to solvent-polymer solutions (i.e. mixtures of large and small molecules). Perhaps the most alluring reason for applying UNIQUAC to solvent-polymer solutions is that it contains the essential features required of a polymer solution theory, i.e. UNIQUAC has both a Flory-Huggins type of combinatorial entropy of mixing term and an enthalpic (residual) term. In fact, in introducing UNIQUAC, Abrams and Prausnitz [l) presented an application of UNIQUAC to representing vapor pressure data in aqueous solutions of polyethylene glycols of molecular weights

M0 = 300 and M0 = 5000 at 65°C. The correlation was very good. However, the use of semi-empirical models for representing solvent-polymer solutio.is has not experienced the same triumphs as in the application to solutions of small molecules. The Berkeley group under Professor Prausnitz has previously looked into several semi-empirical approaches for modeling polymer solutions. Heil and Prausnitz (52,531developed a semi-empirical model which works rather well for solvent-polymer solutions. The Heil model will be discussed in more detail in Chapter 6. Renuncio and Prausnitz [109,115,116)have suggested a local composition "nonrandom" modification of the Flory equation of state, which shows considerable improvement over the

Chapter 4: Solutions of Small Molecules 59 Flory equation of state (to be discussed in Chapter 6). Brandani [15,16)has proposed a two-fluid theory modification of the Flory equation of state. Prausnitz [ 105) suggests that the approach of

Brandani is probably the best semi-empirical treatment available for polar solvent-polymer solutions such as those being studied here.

All of this is of little consequence though, as the above treatments are for concentrated polymer solutions, and the systems being studied here are usually dilute in polymer. Furthermore, as suggested by Heil [52), there is currently not enough experimental data available for assessing the ability or inability of these models for representing phase equilibrium data of polar solvent-polymer systems.

4.3.2 Group-ContributionTheories

Because of the particular nuances of the UNIFAC model (as mentioned earlier), many people have sought to modify or extend the UNIFAC model to make it more widely useful. Several modifications and extensions exist and only the ones applying to polymer solutions will be discussed here. One of these examples is the free volume correction. A couple of other noteworthy modifications and extensions to UNIFAC are also presented. Also, a modification to ASOG for polymer solutions is given.

4.3.2./ The UNIFAC Free Volume Model (UNIFAC-FV)

Oishi and Prausnitz [96) have demonstrated that a free volume term may be added to UNI-

F AC which enables the prediction of solvent activities in polymer solutions. This model is referred to as UNIFAC-FV. Due to the differences in densities between the polymer and solvent (the polymer is much more tightly packed), there are nontrivial changes in the free volumes caused by mixing. This effect was considered by Flory [38) and Patterson [98,99) and will be discussed in more detail in Chapter 6 as the equation of state approach for polymer solutions. UNIFAC requires concentrations to be input in mole fractions, but as mentioned in Chapter 2, the mole fraction is not a useful composition variable for polymer solutions. Oishi and Prausnitz therefore modified

Chapter 4: Solutions of Small Molecules 60 the UNIFAC equations to yield activities instead of the usual activity coefficients. The form of the free volume correction can be added to a thermodynamic model such as UNIF AC as follows:

_ C R lnFV ln~ - ln~ + lnai + ai (4.39) where C, R, and FV represent combinatorial, residual, and free volume contributions, respectively.

The free volume correction as proposed by Oishi and Prausnitz [96) is derived from the Flory equation of state approach, with Flory's parameter X12 equal to zero, see Equation (6.30), for which the resulting equation is precisely the free volume correction.

Oishi and Prausnitz obtained the following free volume correction for the solvent activity

FV (V1 - 1) Vt 1 1na 1 = 3c11n [ ·1,3. - c1 -.-- 1 1 - -.- )-11 (4.40) (v~3 - 1) l [(Vm- )( vJl3 where the reduced volume vis given by

~ = ...!_ (4.41) V • where v1 is the volume of solvent per gram, v; is the hard core volume of solvent per gram, 3c1 is the number of external degrees of freedom per solvent molecule, and subscripts 1 and m refer to the solvent and mixture respectively. The reduced volume of the solvent is then given by

Vt Vi=---- (4.42) 15.17 b r1'

where b is a proportionality constant of the order of unity, and r 1' is calculated from the sum of molar group volumes of the solvent. The reduced volume of the mixture, found by assuming ad- ditivity of volumes in the mixture (i.e. no volume change on mixing) is

(4.43)

Chapter 4: Solutions of Small Molecules 61 where v1 and v2 are the volume per gram of solvent and polymer, respectively, w is the weight fraction, and r;' is calculated from the sum of the molar group volumes. Oishi and Prausnitz suggest the values of b = 1.28 and c1 = 1.1. The proper choice of these parameters, in particular the external degrees of freedom parameter, c1, has been addressed by Oishi and Prausnitz (96), Arai and lwai [9),

Gottlieb and Herskowitz (48) and Prausnitz [104].

Oishi and Prausnitz report that the agreement between UNIFAC-FV and experiment is typically within 10% error. They observed that UNIFAC predicted activities were typically lower than the experimental results and that the free volume correction always raises the solvent activity.

One of the examples given by Oishi and Prausnitz (96) was the effect of this free volume correction on the predicted solvent activity in a benzene-polyisobutylene solution. These calculations were performed and the results are shown in Figure 7, which demonstrates the rather dramatic effect the free volume correction has on the solvent activity, with the final results agreeing very well with experiment.

The UNIF AC model is based on mole fractions and yields activity coefficients. The UNIF AC computer programs published by Fredenslund et al. (41] are written in this manner. Thus, there is a problem when applying UNIFAC, as suggested by Oishi et al. (96], to predicting solvent activities in polymer solutions. A simple procedure for calculating solvent activity coefficients, and thus the Flory x-parameters (the x-parameter is the most commonly used parameter for charac- terizing the thermodynamics of polymer solutions) so that the readily accessible UNIFAC computer code may be applied directly (without any modification) has been proposed by van den Berg

I 135).

Unfortunately however, Herskowitz and Gottlieb (57) have recently found and correctly pointed out that the free volume correction proposed by Oishi et al. (96) does not work for aqueous-polymer solutions, due to the approximate unit reduced volume of water causing the free volume term to be exceedingly large. llerskowitz et al. [57] have therefore suggested the use of a new set of parameters for modeling water-polymer systems, in particular, water-polyglycol systems. The

UNIFAC model based on this new set of parameters utilizes the introduction of a new group,

Chapter 4: Solutions of Small Molecules 62 1. 0 ...._ -...... "i,-...,• ...... '- ...... 0.8 ...... ,, ' ,,. ',' ~ ' ' ' ' > 0.6 ', t- ' ' u ' '

o.0...... -.-...... -.- ...... -.-...... -.- ....._,...... -ro...... ,.-ro ...... -.- ...... - ...... - ...... - ...... 0.6 0.7 0.8 0.9 l. 0 VOLUME FRACTIONOF POLYMER

Figure 7. UNIFAC and UNIFAC FREE VOLUME predictions for benzene activity in polyisobutylene

of MW= 40,000 and at T = 25°C: The experimental data are from Eichinger and Flory

(31]. Points represent experimental data. Solid curves represent UNI F AC predictions. Dashed curves represent UNIFAC-FV predictions based on thefree volume correction proposed by Oishi and Prausnitz (96).

Chapter 4: Solutions of Small Molecules 63 namely CH 2 0H (glycol end-group). The values of these new parameters were estimated by Her- skowitz and Gottlieb (56).

The results of UNIFAC with these new parameters (UNIFAC-NEW) for the prediction of the temperature effect in a low molecular weight (Mn = 200) water-polyethylene glycol (PEG) system are shown in Figure 8. Figures 9 through 16 compare UNIFAC predictions with UNIFAC-NEW

(the introduction of new parameters) for various molecular weights of polyethylene glycol at several temperatures. The experimental data shown in these graphs were obtained in tabular form from

Herskowitz et al. (57). There are several observations obtained from these graphs that are noteworthy. First, at low molecular weights the UNIFAC-NEW predictions are good. This is especially seen in Figure 8. UNIF AC is expected to yield better predictions at low molecular weights, but also UNIFAC-NEW is based on a new set of parameters obtained from polyethylene glycol data. Thus, UNIFAC-NEW predictions are expected to be better than UNIFAC. Secondly,

UNIFAC (without the addition of the parameters performs poorly. The third observation is that as the molecular weight increases, the predictions by UNIFAC-NEW get progressively worse and in fact approach the results of UNIFAC. This is also expected as UNIFAC-NEW employs the newly introduced end group -CH 2 0H, and the effect of the end group decreases with increasing molecular weight.

43.1.2 Empirical Co"elation of UNIFAC Activity Coefficientsfor Large Molecules

Recently, Banerjee [ 11J has observed that water solubilities of organic compounds are typically underestimated by UNIFAC-derived infinite dilution activity coefficients. Banerjee found that these deviations were systematic and could therefore be corrected by some empirical correlation.

The proposed empirical equation of Banerjee successfully correlates data for 50 representative compounds ranging in hydrophobicity from alcohols and amines to polynuclear aromatics and polychlorobiphenyls.

4.3.2.3 A Supercharged UNIFAC: SUPERFAC

One last example regarding modifications and extensions of the UNIF AC group-contribution scheme is a recent attempt reported by Fredenslund et al. (43) to modify and extend the UNIFAC

Chapter 4: Solutions of Small Molecules 64 0.8

~ > 0.6 i= () <( ~ w ~ 0.11 ;:; TEMPERATURE.K 293.15 • 0.2 313.15 • 3J3.15 A ••••••

0.0.,..,...... T""I'",.....,...... ,...... ,'"'"""..,...... __ ...... o.o 0.2 0.11 0.6 0.8 1. 0 POLYMERWEIGHT FRACTION

Figure 8. Effect of temperature on water activity in watcr-PEG(MW = 200) solutions as predicted by

UNIFAC-NEW: The experimental data are from Herskowitz and Gottlieb (57). Points

represent experimental data. Curves represent UNIFAC-NEW predictions using parame·

ters for the new functional group proposed by Herskowitz and Gottlieb (56).

Chapter 4: Solutions of Small Molecules 65 1. 0 ·---...___ ...... ---...... , ...., .., o.a ., ', ~ •',' > ' \, '\ t- ' '\ u '\ <( 0.6 •\ a::: '•,' \ w t

i 0.11 LEGEND EXPTL • UNlf'AC 0.2 UNIF'AC-NEW

o.o.,...... ,i""""...... 'P"T",...... , ...... ,...,...,...... ,...... ,,...... ,..,...... ,...,...,..,...... ,..,,...,1 0.0 0.2 0.11 0.6 0.8 1. 0 POLYMERWEIGHT FRACTION

Figure 9. Water activity in water-PEG(MW=200) solutions at T=313K as predicted by UNIFAC and UNIFAC-NEW: The experimental data are from Herskowitz and Gottlieb [57).

Points represent experimental data. Curves represent UNIFAC predictions using old pa-

rameters and new parameters for the new functional group proposed by Herskowitz and

Gottlieb (56).

Chapter 4: Solutions of Small Molecules 66 1.2~------

1.0 ~------..~- ...... • ...... ', o.e ,, • ' ,, ~ ', > ' '\ 1- ' '\ (.) ' '\ . '\ 0.6 \

0.0,...... - ...... ,r-r- ...... --.-...... -r ...... J 0.2 o." 0.6 0.8 1. 0 POLYMERWEIGHT FRACTION

Figure JO. Water activity in water-PEG(MW=300) solutions at T=303K as predicted by UNIFAC

and UNIFAC-NEW: The experimental data are from Malcolm and Rowlinson (81).

Points represent experimental data. Curves represent UNIFAC predictions using old pa-

rameters and new parameters for the new functional group proposed by Herskowitz and

Gottlieb [56).

Chapter 4: Solutions of Small Molecules 67 1.2

1.0 .------...-...... , ..... 0.8 . ', ..., ~ ' '·\ ', > . \: ''Ii t3<( 0.6 a::: w I- ~ 0.11 LEGEND EXPTL • UNIFAC 0.2 UNIFAC-NEW

o.0...... ----...... --...... -...... --.... 0.2 0.11 0.6 0.8 1. 0 POLYMER WEIGHT FRACTION

Figure II. Water activity in watcr-PEG(MW=300) solutions at T=338K as predicted by UNIFAC

and UNIFAC-NEW: The experimental data are from Malcolm and Rawlinson (81).

Points represent experimental data. Curves represent UNIFAC predictions using old pa-

rameters and new parameters for the new functional group proposed by Herskowitz and

Gottlieb (56).

Chapter 4: Solutions of Small Molecules 68 1.2 r------

1. 0 ------~...... "' . . ', . '\ '\ • '\ 0.8 ' '\ '\ ••• '\ t • \ > • t-u •• <( 0.6 0::: w ~ 3: 0.11 LEGEND EXPTL • UNlf'AC 0.2 UNlf'AC-NEW

o.o'r'"',_,...... -.-...... 'T"""'.-.- ...... --.-.,..., ..... -.- ...... ,....,..,,_,....,...... -.- ...... ,...... ,J 0.2 0.6 0.8 1.0 POLYMER WEIGHT FRACTION

Figure 12, Water activity in water-PEG(MW=600) solutions at T=313K as predicted by UNIFAC

and UNIFAC-NEW: The experimental data arc from Hcrskowitz and Gottlieb (57].

Points represent experimental data. Curves represent UNIFAC predictions using old pa-

rameters and new parameters for the new functional group proposed by Herskowitz and

Gottlieb (56).

Chapter 4: Solutions of Small Molecules 69 1.2.r------

1.0 ------~--~...... • ...... • • ...., • ', ' • ' ' o.e '' \ • \I ~ ' > • t-u <( 0.6 a::: w

i 0.11 LEGEND EXPTL • UNIFAC 0.2 UNIFAC-NEW

0.0.,...... ,...,,.....,...T""""...,..., ...... ,...., ...... ,.....,...... ,.....,...... ~...... --J 0.2 0.11 0.6 0.8 l. 0 POLYMERWEIGHT FRACTION

Figure 13. Water activity in water-PEG(MW=600) solutions at T=333K as predicted by UNIFAC and UNIFAC-NEW: The experimental data are from Herskowitz and Gottlieb (57).

Points represent experimental data. Curves represent UNIFAC predictions using old pa-

rameters and new parameters for the new functional group proposed by Herskowitz and

Gottlieb (56).

Chapter 4: Solutions of Small Molecules 70 1.2r------.....

------~ ...... , "',, 1.0 ' '\ '\ \. ' \ . \ ~ \ \ ... \ > \ I- \ (.) 'Ii <( 0.8 • a:: w i • LEGEND EXPTL • 0.6 • • UNIFAC UNIFAC-NEW

0.4'r- ...... ,....,,.....,...... ~ 0.4 0.6 0.8 1. 0 POLYMER WEIGHT FRACTION

Figure 14, Water activity in water-PEG(MW= 1500) solutions at T=313K as predicted by UNIFAC

and UNIFAC-NEW: The experimental data are from Herskowitz and Gottlieb (57J.

Points represent experimental data. Curves represent UN IF AC predictions using old pa·

rameters and new parameters for the new functional group proposed by Hcrskowitz and

Gottlieb (56].

Chapter 4: Solutions of Small Molecules 71 1. 2

1. 0 • • • ~ > • t- (.) • <( 0.8 a:: w ~ == LEGEND 0.6 EXPTL • UNIFAC UNIFAC-NEW

o.11...... - ...... -.--.-...... -...... -.--.-...... -..-- ...... -.--.-...... 0.2 0.11 0.6 0.8 POLYMERWEIGHT FRACTION

Figure 15. Water activity in water-PEG(MW=6000) solutions at T=JIJK as predicted by UNIFAC and UNIFAC-NEW: The experimental data are from Herskowitz and Gottlieb [57).

Points represent experimental data. Curves represent UNIFAC predictions using old pa·

ramcters and new parameters for the new functional group proposed by Herskowitz and

Gottlieb (56).

Chapter 4: Solutions of Small Molecules 72 1.2

1. 0 • • • • • 0.8 ~ > t- () <( 0.6 ~ w ~ ;: 0.4 • LEGEND • EXPTL • UNIFAC 0.2 UNIFAC-NEW ••••••

o.o.,,__...... -...... ,r--r- ...... , ...... ~...... - ...... ,. 0.2 0.4 0.6 0.8 1.0 POLYMER WEIGHT FRACTION

Figure 16. Water activity in water-PEG(MW=6000) solutions at T=333K as predicted by UNIFAC

and UNIFAC-NEW: The experimental data are from Herskowitz and Gottlieb [57).

Points represent experimental data. Curves represent UNIFAC predictions using old pa·

rameters and new parameters for the new functional group proposed by Hcrskowitz and Gottlieb [56).

Chapter 4: Solutions or Small Molecules 73 model to encompass systems of polymers and electrolytes. Some of the modifications include an improved combinatorial term and more temperature-dependent parameters. This supercharged

UNIFAC system for obvious reasons was called SUPERFAC. Initially, very good results were obtained using SUPERFAC as a correlative model. However, a recent attempt (63] to develop

SUPERFAC into a predictive model has unveiled many problems associated with this tremendous task, and this effort has now been abandoned. The problems will not be discussed here, but are described in detail by Fredenslund et al. (43). Suffice it to say that their model got too complex!

4.3.2.4 The ASOG-Variahle Size Parameter Model (ASOG-VSP)

Recently, Misovich et al. (87,88] have modified the ASOG model for use with polymer solutions. They presented a generalized correlation for solvent activities based upon a free volume correction which is based upon a "variable-size parameter". The model reduces to a closed-form solution containing only one adjustable parameter. The form of the model is a weight fraction solvent activity coefficient which is a function of weight fraction composition and contains one adjustable parameter. The adjustable parameter is an infinite dilution weight fraction solvent activity coefficient which can be obtained from a "single physical measurement of equilibrium solubility of a trace of solvent in pure polymer" (88). Misovich notes that it is also possible to correlate the adjustable parameter with concentration and gives a procedure for doing so.

The ASOG-VSP model is compared with the Flory-Huggins and UNIFAC-FV models. The major difference between the ASOG-VSP and the Flory-Huggins model is in the treatment of free volume differences. The Flory x interaction parameter, as seen by Misovich, is a representation of the contribution of free volume effects upon the solvent activity (or other related thenr..-,dynarnic variables). ASOG-VSP, however, corrects the molecular size ratio for free volume effects based on an infinite dilution measurement. Both of the above free volume corrections are essentially empirical in nature, yet the free volume correction used in the UNIFAC-FV model is based upon the more theoretical approach of the Flory (38) and Patterson (98,99) equation of state theory.

Misovich's results show good agreement between ASOG-VSP and the Flory-Huggins model for athermal systems, but ASOG-VSP was seen to out-perform the Flory-Huggins model for non-

Chapter 4: Solutions of Small Molecules 74 athermal systems. The UNIFAC-FV model performed worse than either the ASOG-VSP or the

Flory-Huggins model for all cases.

Misovich concluded that the ASOG-VSP model shows superiority over the other two models in that (l) its predictions are as good or better; (2) it does not require density data; and (3) it is computationally simpler. However, it does require a data point for determination of the infinite dilution parameter, and therefore ASOG-VSP is a correlative model, whereas UNIFAC-FV is a predictive model requiring no experimental activity coefficient data.

4.4 Summary

Many semi-empirical models have been proposed for solutions of molecules of small and similar sizes, and some of these may even be applied to polymer solutions. The UNIF AC and

ASOG group contribution schemes have been modified to account for free volume differences between the solvent and polymer molecules. The resulting UNIFAC-FV and ASOG-VSP models have been successful for representing a variety of solvent-polymer solutions. The use of semi-empirical formulations for representing solvent-polymer solutions, however, has been for concentrated polymer solutions and is thus not suitable for the typically dilute systems being studied here.

Chapter 4: Solutions of Small Molecules 75 5.0 PolymerSolutions: The Flory-HugginsModel

The purpose of this Chapter is to describe models that are applicable to solutions in which the solvent and solute molecules differ greatly in size. The classical treatment of polymer solutions according to liquid lattice theory as proposed by Flory and Huggins is discussed first. The Flory-

Huggins theory is employed for quantitative predictions of phase behavior in aqueous-polymer solutions. Furthermore, the theory is extended to include a qualitative assessment of the theory for predicting the partition of another macromolecular solute between the two phases present. Due to the inherent inadequacies of the Flory model, many modifications have been proposed, and these are discussed in light of the applications sought herein.

5.1 Liquid Lattice Theory

Ideal solutions are defined as having no enthalpy (or heat) of mixing and an entropy of mixing given by

dsmixing = (5.1)

where k is the Boltzmann constant, n1 is the number of molecules of component i, and Xi is the mole fraction of component i. This expression can be derived assuming a binary solutic,n of molecules of equal size, spatial configuration, and external force field (37].

In these highly idealized mixtures, each of the different molecules is equally replaceable by any other molecule. Due to the increased number of possible arrangements of different molecules over that of a pure solution, the entropy of solution is seen to be greater than that corresponding to the pure components (37]. A key assumption in the liquid lattice treatment is that the molecules are so regularly arranged that their spatial arrangement can be represented by a lattice.

Chapter 5: Polymer Solutions: Flory-Huggins Model 76 Based on the Boltzmann relation,

Smixture = k 1nn (5.2) and the number of arrangements, n, of the n1 identical solvent molecules and the nz identical solute molecules within the lattice containing no =n1 + nz total cells, which is

n = (5.3)

nl = n2! 1 the entropy of mixing is ~Smlxlng = So - s, - sl which becomes

where Stirling's approximation has been used. lbis expression immediately reduces to Equation

(5.1).

5.2 Theoretical Development

5.2.1 Athennal Solvent-Polymer Solutions

Following the liquid lattice model approach, Flory (32,33)and Huggins (60,61)independently obtained an equation for the athermal, configurational entropy of mixing. An atherm~l solution is one in which there is no enthalpy of mixing. The Flory-Huggins model will not be derived here, the interested reader is referred to Flory (37). The expression derived by Flory and Huggins represents the athermal, configurational entropy of mixing for solutions of monomeric solvent molecules and long-chain polymeric solute molecules containing x chain segments, each equal in size to the solvent molecules. The number of segments of the polysegmented solute molecule can be obtained as the ratio of the molar volumes of the solute and solvent. The resulting expression for

Chapter S: Polymer Solutions: Flory-Huggins Model 77 the entropy of mixing, which comes from subtracting the entropy of polymer disorientation from the configurational entropy of mixing is

(5.5)

where n 1 and n 2 are the numbers of molecules of solvent and solute (polymer), respectively, and the volume fractions of solvent and solute, q,1 and q,2 are given by:

xn 2 and <1>2= ---"'-- (5.6) n1 + xn 2

The asterisk is used to denote that this expression is the configurational (or combinatorial) entropy of mixing, and as such, it disregards effects on the entropy of mixing caused by specific internal interactions.

Comparison of Equation (5.5) with the ideal entropy of mixing, Equation (5.l), reveals that the only difference in these two equations is the use of volume fractions versus mole fractions. This is an important point, as the mole fraction composition variable is a special case of the more general volume fraction composition variable. Hence, Equation ( 5.1) is a special case of the more general expression of Equation (5.5); one in which the molecular volumes of the solvent and solute are equal.

The Flory-Huggins theory has received much attention and has been discussed and reviewed by Casassa [21,22), Flory [37), Guggenheim [51), Hildebrand, Prausnitz, and Scott [58), Huggins

[62), Koningsveld [66), Kurata [75), Mark and Tobolsky [82), Morawetz [90.911, Prigogine (111),

Tanford [130), and Tompa [131). Casassa (22) provides one of the best and most comprehensive treatises of polymer solution theory and the Flory-Huggins treatment. Olabisi et al. (97) also gives an excellent account of polymer solution theory. The Flory-Huggins expression was found to work well for athermal polymer solutions, but not so well for polymer solutions exhibiting significant heats of mixing.

ChapterS: Polymer Solutions: Flory-HugginsModel 78 5.2.2 Nonathennal Behavior in Real Polymer Solutions

Dobry (27), as early as the 1930's, demonstrated that some solvent-polymer solutions behaved very nonideally and therefore could not be adequately represented by a purely athermal Gibbs free energy of mixing expression. In other words, the assumption of no heat (enthalpy) of mixing is invalid for most solvent-polymer solutions. Furthermore, partial miscibility in solvent-polymer solutions has been a well-known and observed phenomenon (Dobry (27)). Such phenomena which cannot be predicted from purely athermal assumptions. The prediction of partial miscibility (or phase separation) requires an enthalpic contribution to the model.

In order to extend this theoretically based athermal polymer solution model to real polymer solutions (i.e. solutions with a nonzero enthalpy of mixing), an expression for the enthalpy of mixing is needed. Borrowing from the work of van Laar, the heat (enthalpy) of mixing in any two-component system is given by:

(5.7) where z is the lattice coordination number (or number of cells which are nearest neighbors to a given cell), n1 is the number of solvent molecules,

(5.8)

where w12 is the energy of interaction between 1-2 pairs, w 11 is associated with 1-1 pairs, and w22 is associated with 2-2 pairs. For convenience, Equation (5.7) is written as

(5.9)

where k is the Boltzmann constant, T is the absolute temperature, and x.12 is the Flory solvent- polymer interaction parameter defined as

Chapter S: Polymer Solutions: Flory-Huggins Model 79 X12 = (5.10) which is a dimensionless number characteristic of the interaction energy per solvent molecule. Note that the x1 was introduced in order to generalize the expression for solvent molecules comprised of X1 segments.

Hence, the Flory-Huggins model for a real binary solvent-polymer solution is

_ h C . R ) 6.gmixing - 6. mixing - T(6.smixing + 6.smixing (5.11) where the 6.sc is the configurational (or combinatorial) entropy of mixing, and the 6.sRis the residual entropy of mixing. Assuming that the configurational entropy represents the total entropy of mixing (6,sR = 0) then

(5.12) and

(5.13) or

6.gmixing (5.14) kT

The assumption made in developing this relationship considers specific interaction:- between neighboring molecules to contribute only to the enthalpy of mixing. There is no justification for this a priori dismissal of possible orientational influences between the components which could contribute to the interactions because of the spatial confinement of molecules (37).

In order to correct for the above considerations, Flory proposes the following expressions

(5.15)

ChapterS: Polymer Solutions: Flory-Huggins Model 80 and from Equation (5.13),

(5.16)

Note that under the assumption that Aw12 is independent of T (i.e. Aw12 contains no entropy contribution), X12 =zAw12xi/kT will be inversely proportional to T, and the third term of Equation

(5.15) will be zero, hence Asmixins= As~ixins. Also, -T(ax, 2/oT) = X,2, and the Ahmixinsrelation reduces to Equation (5.9) (multiplied by x1 for generalization) (37).

In light of this refinement, it should be noted that "the most serious of these [assumptionsJ is the assumption of a single lattice to describe the configurational character of both pure components and their solutions as well" [37J. This assumption is tantamount to saying that there is no volume change on mixing.

Another limitation of the theory discussed above is that it only applies to concentrated polymer solutions such that the randomly coiled polymer molecules overlap one another extensively

•• a condition which should be fulfilled at concentrations exceeding several weight per cent, unless the molecular weight is less than 105 , in which case higher concentrations may be required [37J.

It should be noted that Equation (5.7) represents what Guggenheim (511termed the "zeroth approximation" in regular solution theory. To refine this, a statistical weighting factor of the form exp( - ZAw12/kT) could be assigned to each microscopic arrangement of the lattice [22J. In this expression, Z represents the total number of 1-2 contacts in the specific configuration. This extra term would therefore represent X, or the entropy of mixing due to nonrandomness. Unfortunately, the introduction of this higher order approximation into the partition function obtained from the statistical lattice treatment renders the partition function intractable to analytical solution (22J.

Recall from Chapter 4 discussions that the use of such a statistical weighting factor is the approach taken in the development of the local composition theory. The inclusion of this statistical weighting factor in the Flory-Huggins expression is reminiscent of the discussion in Section 4.1.2 on the

Wilson equation.

Chapter5: Polymer Solutions: Flory-HugginsModel 81 5.2.3 Chemical Potential and Activity Expressions from the Flory-Huggins

Model

An expression for the relative chemical potential of the solvent (component 1) may be obtained by differentiating the Gibbs free energy of mixing relation of Equation (5.13) or (5.14) via

Equation (2.29). The resulting expression for a binary solvent-polymer solution is

Combining Equations (2.30) and (5.17) yields the following relation for the solvent activity

(5.18) and from the definition of the activity, the activity coefficient of the solvent is found to be,

(5.19)

5.3 Application of the Classical Flory-Huggins Theory

5.3.1 Binary Solvent-Polymer Solutions

lhe application of the classical Flory-Huggins theory to binary solvent-polymer solutions re-

quires the use of one of the thermodynamic relations derived previously, in particular, t!ie solvent chemical potential equation, Equation (5.17), or the solvent activity expression, Equation (5.18).

In general, given a Gibbs free energy of mixing expression, the proper use of standard thermodynamic relations enables the determination of the chemical potential of the solvent, the solvent ac-

tivity, the osmotic pressure, the critical condition for phase splitting, etc. In order to employ these

expressions, however, the solvent-polymer interaction parameter, X12, must be known or deter-

mined. For now, it is assumed that appropriate values of this parameter exist for the systems of

Chapter5: Polymer Solutions:Flory-Huggins Model 82 interest. Later discussions will focus upon the determination of the Flory interaction parameters (

Xii 's). To get an idea of the effect the Flory interaction parameter has on the relative chemical potential of the solvent, Equation ( 5.17) was used to calculate the solvent chemical potential for various values of Xiz, assuming x= 1000. Figure 17, first calculated by Flory (33), shows the results of such a calculation. This figure leads to a discussion of the application of the phase splitting criterion (as discussed in Section 2.2. l) to the Flory-Huggins model. That is, performing a stability analysis on this model yields the following relation for phase splitting, based on the Flory interaction parameter. Phase splitting occurs if Xiz is greater than the critical interaction parameter Xe which represents the miscibility limit (or critical miscibility, see Figure 3).

> + ]2 (5.20) Xe 1-[12 _l Yi X

The critical interaction parameter Xe can be found by applying the phase splitting criteria of

Equation (2.37) to the Flory-Huggins equation.

Recall from Chapter 2 that the equilibrium equations require that in order for phase splitting to occur, there must be two concentrations at which the chemical potential, µ;, has the same value.

This in tum requires that the curve of chemical potential versus concentration passes through a minimum and a maximum as the concentration varies from zero to one. This curve must also have an inflection point, given mathematically by the second derivative; i.e. the curve has no curvature, so

The inflection point lies between the minimum and maximum; and since the minimum and maximum have a slope of zero

Chapter S: Polymer Solutions: Flory-Huggins Model 83 2.0 X=O

l .S

I() * 0* - l.O X= 0.532 * ~ ...... '0::t O.S I VI ~ I o.o

-o.

o.oo 0.01 0.02 0.03 o.oq o.os VOLUME FRACTIONOF POLYMER

Figure 17, Effect of Flory interaction parameter, Xu, on the decrease in the relative solvent chemical potential: The numbers by the curves represent the value of the Flory interaction pa-

rameter. The curve with X12 = 0.532 represents the miscibility timiL Curves with values greater than 0.532 represent partial miscibility (phase splitting), while those with tower va-

lues denote complete miscibility.

Chapter S: Polymer Solutions: Flory-Huggins Model 84 Thus, at the critical point of incipient phase separation, both of the above conditions must be satisfied. This is the miscibility limit displayed by the curve x12 = 0.532 in Figure 17. For low values of x12 , a monotonic decrease in .1µ1 (and hence a monotonic increase in -.1µ1 as observed in Figure

17) is seen. This leads to complete miscibility. However, as the value of X12 is increased (this is achieved by lowering the temperature or choosing a poorer solvent), the critical point is reached when the curve goes through a minimum, a maximum, and an inflection point. As the X12 is increased even more, the chemical potential curve goes through a minimum, such that the curve is concave downward, and therefore produces two phases since there is a point at which two concentrations yield the same value of the chemical potential.

Flory [34,37) derived an analytical expression from which theoretical binodals may be predicted. Flory derived a relation between the composition

the following relation may be obtained [34,37)

- (y + l)h + [(y + I)2h2 + 4(y - 1)3h] 1' 2 ( 5.21) 2(y - 1)3 where

h = ( 1,~{+(y + l)(lny) - (y- l)J (5.22)

Notice here that the two chemical potential equations have been added in such a way as to eliminate the Flory interaction parameter. The approximate relationship above is due to some simplifications imposed by Flory, but the approximation proves inconsequential for concentrations up

Chapter 5: Polymer Solutions: Flory-Huggins Model 85 to ten times the critical concentration,

which shows that for a binary solvent-polymer solution, phase splitting is predicted at a very small volume fraction of polymer.

The use of the Equations (5.21) and (5.22) for predictions of equilibrium phase compositions is as follows. First, an arbitrary value of y is chosen. Also, a value for x is chosen which represents the molecular weight of the polymer; actually, xis equal to the ratio of the molar volume of the polymer to that of the solvent. Thus, x can be calculated from the molecular weights and densities as

where v is the molar volume, MW is the molecular weight, and p is the density. Then, his calculated from Equation (5.22). With the specified y and x and the calculated h, the concentration of polymer in the dilute phase, q>2', is detennined from Equation (5.21). The concentration of the polymer in the more concentrated phase, q>2 ", is calculated as q>2" = yq>/. This procedure is repeated for different values of y to completely describe the binodal of the system.

Starting with the relation for the chemical potential of the polymer

(5.24)

and inserting the ratio y = q>2"/q>/, a relation for X12, the binary interaction parameter, as a function of y is found to be

Chapter 5: Polymer Solutions: Flory-Huggins Model 86 (y - 1)(1 - + ( ln y)/q>/x = +) (5.25) 2(y - 1) - 2'(l - 1)

Thus, for each calculation of q,1' from y, a corresponding value for X12 may be calculated from

Equation (5.25). Hence, binodals may be plotted as a function of the Xiz interaction parameter or temperature. For the latter case of plotting binodal curves as a function of temperature, Flory

(37,1231uses an empirical calibration to convert the Xiz values into corresponding temperature values. Shultz and Flory ( 1231performed this type of calculation and found that: ( 1) the general features of the theoretically-predicted curves qualitatively matched experimental binodals; (2) observed critical points occur at higher polymer concentrations than those predicted from theory; and (3) the experimental curves are much broader (over the composition range) than the predicted curves.

Graphical comparisons of these results were shown by Flory [37] and Shultz and Flory ( 1231, Based on the above observations, Shultz and Flory concluded that the Flory-Huggins relation for the

Gibbs free energy of mixing (and chemical potential) does not properly take into account the functional dependence on concentration.

A severe limitation of the Flory-Huggihs equation is in the use of volume fractions, such that in order to recalculate the measured properties from the interaction parameters correlated by Flo- ry-Huggins, one must first know the partial specific volumes of the components. Thus, Flory-

Huggins relations have sometimes been based on weight fractions and segment fractions [ 17). This is a serious problem if one desires accurate results, for most of the experimental data is on a weight fraction basis, and the densities of the polymers are not always reported. The liquid densities as well as the molecular weights are needed to convert from weight fractions to volume fractions.

Recall from earlier discussions that the densities are also required for determination of the polymer segment numbers. The density data for the experimental systems being studied were not given, and thus as discussed in Chapter 3, they were estimated. Hence, this is one source of error in the calculations, but the magnitude of the error introduced in inaccurate densities is much smaller than the major contributors to the overall error such as assuming monodisperse polymers, a single lattice model for both pure components and their mixtures, and a constant interaction parameter.

Chapter 5: Polymer Solutions: Flory-Huggins Model 87 5.3.2 Application of the Classical Flory-Huggins Theory to Water-Polyethylene

Glycol Systems

As a basic test of the Flory-Huggins model for binary solvent-polymer solutions, solvent activities as predicted by the theory for the model system of water-polyethylene glycol were compared with experimental data. This particular system was chosen as a model system for the following reasons: (l) the availability of binary data on this system for various molecular weights and temperatures; (2) polyethylene glycol is one of the most common components in the aqueous-polymer systems to which this research is addressed; and (3) as water-polyethylene glycol systems are polar and may exhibit strong, specific interactions, i.e. hydrogen bonding, this binary system was felt to provide a stringent test of the theory.

For various temperatures and molecular weights of polyethylene glycol, water activities were calculated using Equation (5.18) and a value of Xiz = 0.45 at 27°C for water-PEG, as obtained from

the Polymer Handbook [ 17). The results of the Flory-Huggins calculations, using x.12 parameter as obtained above, are shown in Figures 18-24. These plots compare the water activity as predicted

by the Flory-Huggins model with experimental values obtained by Herskowitz and Gottlieb (57)

using an isopiestic method (55). Note that while experimental data for several temperatures are

plotted, there exists only one theoretical curve per figure. This is due to the fact that the X12 parameter obtained from the Polymer Handbook (18) cannot be corrected for its temperature de-

pendence since it was obtained as a constant. The inadequacy of these predictions suggest that the x interaction parameter is not constant; it is a function of one or more of the following variables: temperature, concentration, and molecular weight.

Also, comparison of Figures 18-24 with the analogous predictions via UNIFAC-NEW, i.e.

Figures 8-16, show the much better prediction of UNIFAC-NEW over Flory-Huggins (for the

constant X12 value used) for low molecular weights of PEG. A key feature of Figures 18-24 is the

superiority of Flory-Huggins over UNIFAC-NEW for solutions of moderately-high molecular

weight polyethylene glycol. The difference, however, is not as great as expected. Neither Flory-

Chapter5: Polymer Solutions: Flory-HugginsModel 88 Huggins or UNIFAC-NEW yield very good results at moderately-high polymer molecular weights, but the Flory-Huggins predictions of the molecular weight effect are somewhat better than those of UNIFAC-NEW. The effect of molecular weight on water activities in polyethylene glycol solutions at 333 K as predicted by UNIFAC-NEW is shown in Figure 25; while Figure 26 displays the Flory-Huggins constant-x prediction of the effect of molecular weight on the solvent activity.

These Figures also show the effect of molecular weight as observed experimentally by Herskowitz and Gottlieb [57J. It is suspected that the Flory-Huggins equation with a variable interaction pa-

rameter would yield much better predictions. Possible modifications of the interaction parameter

will be discussed in subsequent Sections.

5.3.3 Extension of the Flory-Huggins Model to Multicomponent Systems

The entropy and heat of mixing as previously derived, Equations (5.5) and (5.9), can readily

be generalized for multicomponent solutions as

asmixing* = - k I,ni 1n

and

ahmixing kT I. ni

where the summations include all pairs of unlike species. For a ternary solvent-polymer-polymer

solution, the Gibbs free energy of mixing is then

agmixing = kT [n1 1n

Chapter 5: Polymer Solutions: Flory-Huggins Model 89 l. 0

0.8

TEMPERATURE.K 293. 15 .• 0.2 313. 15 • I 333. 15 A

o.o....,,..,...,,..,...,,..,...,,..,...,....,..,,..,...,....,..,,..,...,,..,...,....,..,,..,...,,..,....,..,...... ,"'T"'l.....,....., ...... "'T"'I..,...,..,...,..,...,..,...,.,. o.o 0.2 0.11 0.6 0.8 l. 0 POLYMER WEIGHTFRACTION

Figure 18. Flory-Huggins predictions of water activity in water-PEG(MW= 200) solutions at temper-

atures of 293,313, and 333 K: The experimental data arc from Herskowitz and Gottlieb

(57). Points represent experimental data. Curve represents Flory-Huggins prediction with

Xu= 0.45 at 27°C (from Polymer Handbook (17)).

Chapter 5: Polymer Solutions: Flory-Huggins Model 90 1. 0

0.8

•

TEMPERATURE,K 0.2 303.15 • 338.15 • I

0.0 ...... ,...... ,...... _..__,,.....,...... o.o 0.2 0.11 0.6 0.8 l. 0 POLYMERWEIGHT FRACTION

Figure 19. Flory-Huggins predictions of water activity in water-PEG(MW = 300) solutions at temperatures of 303 and 338 K: The experimental data are from Malcolm and Rowlinson [81).

Points represent experimental data. Curve represents Flory-Huggins prediction with

;( 12 =0.45 at 27°C (from Polymer Handbook [171).

Chapter 5: Polymer Solutions: Flory-Huggins Model 91 • 0.8 • ~ > 0.6 I- ~ ~ w ~ 0.11 3: TEMPERATURE.K 293.15 • 0.2 313.15 • a 3.33.15 • •

o.oT"'l.....,..,.,...,..,."T'"l"T"'l""...... ,...,....,..,.....,...,,.....,..,...... ,..,...,...,..,."T'"I" ...... ,..,...... ,..,...,,.....,,.....,,. o.o 0.2 0.11 0.6 0.8 l. 0 POLYMERWEIGHT FRACTION

Figure 20. Flory-Huggins predictions of water activity in water-PEG(MW= 600) solutions at temper-

atures of 293,313, and 333 K: The experimental data arc from Herskowitz and Gottlieb

[57). Points represent experimental data. Curve represents Flory-Huggins prediction with

'X12 = 0.45 at 27"C (from Polymer Handbook [171).

Chapter 5: Polymer Solutions: Flory-Huggins Model 92 1.or-======------, .. 0.8 .. •• •• •

TEMPERATURE.K 293.15 • 0.2 313.15 • .333.15 •

o.0...... --~ ...... --~ ...... __~ ...... , 0.0 0.2 o.s 0.8 1. 0 POLYMERWEIGHT FRACTION

Figure 21. Flory-Huggins predictions or water activity in water-PEG(MW= 1500) solutions at tem-

peratures or 293, 313, and 333 K: The experimental data are from Herskowitz and

Gottlieb [57). Points represent experimental data. Curve represents Flory-Huggins pre-

diction with Xi 2 = 0.45 at 27°C (from Polymer Handbook [171).

Chapter S: Polymer Solutions: Flory-Huggins Model 93 1.or::===:::::::=--:------,

• • • 0.8 I •• ~ > 0.6 t- ~ I a:: w ~ 0.11 3: ' TEMPERATURE,K 328.15 • 0.2 338.15 • •

0.0-f....,_,...... ,..,..,...,...... ,'T"'"'...... ,..,.., ...... T"'""''"T"l""'P"T'"'l'..,..,...T"""""~ ...... ~~-r-,-,'"T"l""..-o:,' o.o 0.2 0.11 o.6 o.e 1.0 POLYMER WEIGHT FRACTION

Figure 22. Flory-Huggins predictions of water activity in water-PEG(MW= 3000) solutions at tem-

peratures of 328 and 338 K: The experimental data are from Malcolm and Rowlinson [81J. Points represent experimental data. Curve represents Flory-Huggins prediction with

X12 =0.45 at 27°C (from Polymer Handbook [l 7J).

Chapter 5: Polymer Solutions: Flory-Huggins Model 94 • I • 0.8 • I

•

TEMPERATURE.K 333.15 • 0.2 338.15 • •

o.o"'""'..,...... ,...... ,P""l""l'..,.....,...... , ...... "T"'T'.,...,..,...... ,""T"T"T""l'"....,...... ,..,.'T""",...... ,..,..,...... ,. o.o 0.2 0.11 0.6 0.8 I. 0 POLYMERWEIGHT FRACTION

Figure 23. Flory-Huggins predictions of water activity in water-PEG(MW=SOOO) solutions at tem-

peratures of 333 and 338 K: The experimental data are from Malcolm and Rowlinson

[81). Points represent experimental data. Curve represents Flory-Huggins prediction with

x12 =0.45 at 27"C (from Polymer Handbook [171).

Chapter 5: Polymer Solutions: Flory-Huggins Model 95 1. or-=====:::::::;;:::::------i • o.e • • •• •

• • •

TEMPERATURE.K 0.2 313.15 • 333.15 •

0.0 ...... "'l""T" ...... ,..,...,., o.o 0.2 0.4 0.6 0.8 1. 0 POLYMERWEIGHT FRACTION

Figure 24. Flory-Huggins predictions or water activity in watcr-PEG(MW = 6000) solutions at tem· pcratures or 313 and 333 K: The experimental data are from Herskowitz and Gottlieb

[57). Points represent experimental data. Curve represents Flory-Huggins prediction with

X12 = 0.45 at 27°C (from Polymer Handbook (17]).

Chapter 5: Polymer Solutions: Flory-Huggins Model 96 ...... ------.....---·-·....,"'' \ ~ 1.0 ...... ' ' \ * ,, '.\ --....._~ ·,...... \ \ '\ \ ...... \ \ \ , . \ '\ o.e ..... \. ~ . \\ • .~ltr > \ ,\ \ \• (.).= <( 0.6 ~ 0::: .l ,i w \ ,. ~ \\~ 3t ~,. 0.11 \\\t PEGMOLECULAR WEIGHT 200 • 600 • 0.2 1500 -- . r 6000 -·- * \•

0.0....,,..,..,,..,..,,...... ,...... l""T" ...... "'l""T ...... ,.....,...,.., ...... T"P' ...... "I"' o.o 0.2 0.11 0.6 0.8 1. 0 POLYMERWEIGHT FRACTION

Figure 2S. Effect of molecular weight on the activity of water in water-PEG solutions at T = 333 K as predicted by UNIF AC-~EW: The experimental data arc from Herskowitz and Gottlieb

(57). Points represent experimental data. Curves represent UNIFAC-NEW predictions.

Chapter S: Polymer Solutions: Flory-Huggins Model 97 1.2

0.8 ~ > i= (.) <( 0.6 a:::: w • ~ 3: * 0.11 PE;GMOLECULAR WEIGHT * 200 • 600 ----- • 0.2 1500 -- .. • 6000 -·- *

0.0.,.., ...... ,...... ,...... _..... o.o 0.2 0.11 0.6 O.B l. 0 POLYMERWEIGHT FRACTION

Figure 26. Effect of molecular weight on the activity of water in water-PEG solutions at T = 333 K as predicted by Flory-Huggins: The experimental data are from Herskowitz and Gottlieb

(57]. Points represent experimental data. Curves represent Flory·I-luggins prediction with

X12 = 0.45 at 27°C (from Polymer Handbook [171).

Chapter 5: Polymer Solutions: Flory-Huggins Model 98 where subscript l denotes the solvent; subscripts 2 and 3 denote the two polymers; X1 is the number of segments of solvent, x1 = l; x2 and x3 are the numbers of segments of the two polymers. The volume fractions,

<9i = (5.29)

where n is the number of molecules. The Flory interaction parameters are again defined as

(5.30)

The chemical potentials, as found by differentiating Equation (5.28) via Equation (2.29), are:

[ X1 = RT In q,1 + (1 -

(5.32)

(5.33)

Chapter 5: Polymer Solutions: Flory-Huggins Model 99 The above equations contain six parameters, the X;J's, which characterize a given ternary solvent- polymer system. Flory [37) gives the following relation which is used to reduce the six Xii parameters to three independent parameters,

(5.34)

where V; and v1 are molar volumes. Also, x1 is typically taken to be unity.

There is a difficulty in comparing the work of various authors. The notation is seldom exactly the same, but more importantly, there exists different ways of expressing the interaction parameter.

The equations reported here for the chemical potential relations in a ternary solvent-polymer-polymer system, Equations (5.31) through (5.33), are from Flory [37), and on comparison with the expressions reported by Scott [120) and Hsu and Prausnitz [59), they may not seem directly compatible. They are equivalent, however, which is demonstrated below. Writing the chemical potential of the solvent by Equation ( 5.31)

Xt Xt = 1n

Xt + (Xt2

l -

Letting the reference molar volume be that of the solvent, then x1 = 1 and

Chapter 5: Polymer Solutions: Flory-Huggins Model 100 1 1 2 = 1n2+ (1 - xj'")

+ XtJ

The above equation is the same as the expressions used by Scott (120) and Hsu and Prausnitz (59)

if their polymer-polymer interaction parameter µ12 (note: they let 0,1,2 represent the solvent, and two polymers, respectively) is related to the Flory interaction parameter x 23 by

Notice also that X1, Xz, and X3used here are denoted as mo, m1, and mz by Scott and Hsu. Scott

and Hsu reported results for µ12 = 0.004 for the case where X1= l and Xz= X3= x= 1000. In terms

of the XZJ, this would be XZJ= x µ12 = 4. An approach similar to that of Scott [1201and Hsu and Prausnitz (59) was taken by Tseng

[118,133,134). He developed the following alternative set of equations. Tseng's XZJ is the same

parameter as the µ12 parameter of Scott and Hsu. These equations result from utilizing Equation

(5.34) to reduce the six Xii's to three, and then rearranging to yield:

(5.35)

(5.36)

(5.37)

with Xi, Xz, and XJ defined as

(5.38)

(5.39)

Chapter 5: Polymer Solutions: Flory-Huggins Model IOI (5.40)

The use of the above chemical potential equations (either of the two forms) for the prediction of the equilibrium phase behavior of a ternary system is as follows. First, the equilibrium equations are written as

.6µ1' = Aµ{

.6µ2' = Aµ{ .6µ/ = .6µ3" and the above expressions for the chemical potential, eg. Equations (5.31) to (5.33), are substituted in these equilibrium equations to yield a set of three nonlinear equations in four unknowns; the unknowns being, for example, cp1',cp2',cpi",and cpz". The volume fractions cp/and cp/ are not independent of the above four composition variables, as cp/ = 1 - q,1' - cpz' and cp/ = 1 - q,1"' - cpz".The segment numbers x2 and x3 are known (they represent the molecular weights of the polymers and are obtained through the relations of the molar volumes, x2 = :: and

3 x3 = : ). The Flory interaction parameters must be known, determined from experiment, or es- 1 titnated by some other means. lbe determination of the Flory interaction parameters from different methods including experimental binary and ternary data will be discussed in Section 5.4.

From phase rule considerations for a three-component, two-phase system, there are three degrees of freedom, which means that by fixing the temperature, pressure, and one composition, the state of the system is known. Thus, by specifying one of the four volume fractions, wh;it remains is a system of three nonlinear equations in three variables. However, unlike the binary case for which an analytical solution can be obtained, the ternary case cannot be solved explicitly, and numerical methods must be employed.

Flory (37) developed analytical expressions for the case of a binary solvent-polymer system, as was discussed in the previous Section. This method was used by Shultz and Flory ( 123)to study the phase behavior in solvent-polymer systems. Due to the lack of agreement between the Flo-

Chapter 5: Polymer Solutions: Flory-Huggins Model 102 ry-Huggins theory and experiment in their studies, Shultz and Flory concluded that "total faith" in the theory for quantitative predictions is not justified. They cite the sensitivity of phase equilibria predictions to inaccuracies in the theory's representation of concentration dependence as a main problem of the theory. Koningsveld and Staverman [69] state that this critique may be too harsh and somewhat unwarranted, as Shultz and Flory neglected the effect of polydispersity on the predicted phase behavior.

The effect of polydispersity is something that must be at least reckoned with, unless the polymers being used have a very narrow molecular weight distribution. Especially in experimental methods such as cloud-point titrations (or turbidity measurements), the polydispersity has a direct effect on the observed phase behavior. In these cloud-point titrations, there is a polymer fractionation taking place with higher molecular weight fractions separating first. The effect of this is that the cloud-point curve (CPC) is not identical to the binodal curve; these curves are only identical if the polymer is monodisperse. Koningsveld and Staverman realized the importance of taking into account the polydispersity and modified the Flory-Huggins theory to include molecular weight distributions. The approach by Koningsveld and Staverman, discussed in a series of papers

[69,70,71),is essentially to introduce a a continuous expression representing the molecular weight distribution into the theory versus the classical approach of using the single characteristic number of x. The effect of this modification on the phase behavior of solvent-polymer systems will be discussed in Section 5.5.

Scott has studied the case of ternary systems of two ordinary liquids and one polymer [ 119) and the case of a solvent and two polymers [120). In his work on the system of a solvent and two polymers, Scott ( 120] limits his analysis to special cases for which he was able to obtain r.1eaningful qualitative results without resorting to the laborious trial-and-error solution of the general case.

Scott examined, for example, the "symmetric" case, in which the two polymers are equally soluble in the solvent, i.e. X12= x13 , and the two polymers have the same molecular weight, i.e. x2 = x3 • For this case, the binodal could be calculated rather easily. From the resulting expression for the binodal, Scott made the following conclusions: (1) the function of the solvent is solely in dimin- ishing the value of the polymer-polymer interaction parameter, Xz3; (2) the phase behavior is in-

Chapter S: Polymer Solutions: Flory-Huggins Model IOJ dependent of the solvent-polymer interaction parameters, X12= X13; and (3) for high polymers, incompatibility is the rule rather than the exception.

As the critical polymer-polymer interaction parameter is very small, compatibility is exhibited only when the two polymers are so chemically similar that X23 is vanishingly small. From a stability analysis, at the critical point,

and for example, if Xz = X3 = 1000, then (X23) = 0.002 , which means that for X23 > 0.002 phase C splitting will occur.

Tompa (132) has also studied solvent-nonsolvent-polymer systems and solvent-polymer-polymer systems. In regards to the system of a solvent, a nonsolvent, and a polymer, Tompa solved for expressions for the spinodal and critical (plait) point. He applied an iterative technique to solve for the binodal. In his study of the solvent-polymer-polymer case, Tompa assumed there was no interaction between the solvent and one of the polymers, and he assumed that the interaction between a segment of the other polymer and a solvent molecule is the same as that between segments of the two polymers.

Two comments seem worthwhile at this point. First, these early attempts at applying the

Flory-Huggins theory to ternary solvent-polymer-polymer systems were aimed at providing some insight into the experimental findings of Dobry and Boyer-Kawenoki (28) regarding the incompatibility of high polymers and on the effect the molecular weight of the polymer has on this phase separation. Dobry and Boyer-Kawenoki (28) performed a systematic study on polymer-polymer compatibility and concluded the following:

1. The majority of pairs of polymers studied (31 out of 35) showed phase separation, and

hence they concluded incompatibility is the rule rather than the exception.

2. When two high polymers are incompatible in one solvent, they are generally incom-

patible in all other solvents.

Chapter5: PolymerSolutions: Flory-Huggins Model 104 3. The critical point in solvent-polymer-polymer systems depends upon the nature of the

solvent.

4. The molecular weight of the polymers has a significant effect on their compatibility.

The higher the molecular weight, the more incompatible the polymers are toward each

other and the more the critical point is shifted toward smaller concentrations.

5. Theoretical considerations suggest that not only does the molecular weight of the po-

lymers affect compatibility, but also the shape of the polymers influences the com-

patibility.

6. No obvious relationship exists between the compatibility of two polymers and the

chemical nature of their monomers.

The findings and conclusions of Dobry and Boyer-Kawenoki have been confirmed by the experimental work of Kem and Slocombe (64). Secondly, due to the complex equations involved, application of the theory to providing analytical expressions for binodals requires certain a priori assumptions of the phase behavior of the system being modeled. Most early applications of the

Flory-Huggins theory to ternary solvent-polymer systems, such as the work of Scott (119,120)and

Tompa (132), involved such simplifications as the symmetric case of Scott, described above.

Several attempts have been made to solve the general ternary solvent-polymer-polymer problem. Zeman and Patterson [144) were the first to examine the more general case, versus the earlier

Scott-Tompa treatment. They, however, took the easier and less meaningful approach of predicting the spinodal curves (versus the binodal curves) for a number of ternary solvent-polymer-polymer systems. Spinodals can be solved for directly whereas the solution of binodals must be done numerically. Also, spinodals only show trends of phase separation, whereas the binodal yields a quantitative description of the phase behavior. In their study, Zeman and Patterson observed that a small difference in the solvent-polymer interaction can have a marked effect on polymer-polymer compatibility (59,144).

Chapter5: PolymerSolutions: Flory-Huggins Model 105 The main reason that spinodals generally have been studied less in the past than binodals, even with their ease of calculation compared to binodals, is the availability and ease of determination of experimental binodal phase equilibrium data, such as cloud-point curves. However, the use and prediction of spinodals is growing, and this is chiefly due to the development of light scattering methods which permit measurement of the spinodal region (22).

Hsu and Prausnitz (59) presented a computational method for predicting the phase behavior in solvent-polymer solutions based on the Flory-Huggins theory. They demonstrated the effect of interaction parameters and polymer molecular weights on polymer compatibility for asymmetric systems (where the two solvent-polymer interaction parameters are different) of different molecular weight polymers. The numerical method employed by Hsu and Prausnitz was a Powell search method (101)(first applied to liquid-liquid equilibrium calculations by Guffey and Wehe (49)) which searches the solution by minimizing the sum of the squares of the functions f1( cp1) = 0 , which for the liquid-liquid equilibrium problem become

.1µ{ - .1µ( (5.41) RT

The resulting objective function to be minimized is then

2 OBJ = L [.1µ{ ~ .1µ(] (5.42) i=t Rf

The Powell search routine is essentially Gauss' method of iteration with Lagrange multirliers chosen by the Levenberg method (77).

As stated by Olabisi et al. (97), the conclusions arrived at by both Zeman and Patterson (144) and Hsu and Prausnitz (59) are:

1. At low polymer concentrations the difference between the two solvent-polymer inter-

action parameters is directly responsible for the polymer-polymer irnmiscibility.

Chapter S: Polymer Solutions: Flory-Huggins Model 106 2. At high polymer concentration, the state of miscibility is governed by the magnitude

and size of the polymer-polymer interaction parameter.

3. When the interaction between the polymers is low or even negative, a closed misci-

bility gap would result if the two solvent-polymer interaction parameters are different.

These conclusions serve as qualitative guidelines, with which applications of the theory need to agree. Furthermore, these conclusions provide a framework for the application of the Flory-

Huggins theory. For example, if one is working with ternary systems of dilute-polymer solutions, as is the case in solutions of interest here, then special attention should be paid to the difference in the solvent-polymer interaction parameters and not so much to the absolute values of all the interaction parameters. On the other hand, for concentrated polymer solutions, the attention should be focused on the polymer-polymer interaction parameter.

5.3.4 Macromolecular Partitioning Predictions Via the Flory-Huggins Theory

A theoretical framework is provided here for qualitative assessment of the Flory-Huggins theory for predictions of macromolecular partitioning behavior in aqueous-polymer two-phase systems. The equations set forth are compared qualitatively with experimental findings such as those presented by Albertsson (3).

Consider now the situation arising from adding another macromolecular solute, component 4 with molecular weight characterized by X4, to the ternary solvent-polymer-polymer systc:n already described. The term macromolecular solute is used in the sense that the theoretical framework applies to polymers, and it is here being applied to biological macromolecules, which are not expected to have extremely different solution behavior from that of the polymers. As before, the relation for the Gibbs free energy of mixing, Equation (5.28) for the Flory-Huggins theory, is written and differentiated to yield relations for the relative chemical potentials. For a quaternary system, the Gibbs free energy of mixing relation, Equation (5.28), becomes

Chapter S: Polymer Solutions: Flory-Huggins Model 107 .1.gmixing= kT [n1 In

The chemical potential equations are found upon differentiation of the Gibbs free energy of mixing expression given above. Since the application being sought here is a qualitative prediction of the partitioning of a macromolecular solute (component 4) between the two aqueous-polymer phases of the system, then a relation for the relative chemical potential of component 4 is required. For qualitative predictions, it is most useful to assume that all macromolecular solutes (components 2,

3 and 4) are equally soluble in the solvent (component I) (18,37,120), for which case

X12= Xll = X14. The chemical potential of component 4 then becomes [ 18)

I + In 4+ ~ [ -

At equilibrium, or and

1nq>4' + ~[ - 3'(X34- ; 3 ) + X14(4" 2"(X24- - 1-) 2"- /J

In order to simplify this relation further, Brooks et al. (18) neglected all second order volume fractions which necessarily renders the resulting equation a qualitative approximation. following this assumption,

Chapter 5: Polymer Solutions: Flory-Huggins Model 108 exp{ ~[(

where K4 is the partition coefficient of component 4, defined as the ratio of the composition of component 4 in the top phase to the composition of component 4 in the bottom phase. The single prime used here denotes the top phase, while the double prime represents the bottom phase.

A relation such as the one above, Equation (5.46), is very useful for qualitatively assessing the ability of the Flory-Huggins theory for describing the partition behavior of a macromolecular solute in the aqueous-polymer two-phase systems being studied here. The partition behavior derived above does in fact qualitatively describe the partition behavior observed experimentally and expected intuitively as was demonstrated by Brooks et al. [ l 8J in the following arguments:

1. The partition coefficient depends exponentially on the relevant properties of the par-

titioned material and the phase system.

2. The partition coefficient becomes much more one-sided the larger the molecular

weight of the material being distributed.

3. The partition coefficient becomes more one-sided the greater the difference in polymer

concentrations between the two phases, (q,/ - q,/) and (q,3' - q,/).

4. The partition coefficient depends on the balance between the energies of interaction

of component 4 with the phase polymer (X.24and ):34) and the energy of interaction

of component 4 with the solvent (X.14), although this last dependence may be negligible

5. If the molecular weight of one of the phase polymers is decreased, then the partition

of component 4 into the phase in which that polymer predominates increases. For

Chapter 5: Polymer Solutions: Flory-Huggins Model I09 instance, if component 2 is considered to be the polymer enriched in the top phase,

- ) decreasing x2 will increase the term ( ; 2 X24 and, since (

will increase. On the other hand, if x3 is decreased, x34) will increase, but since <+3 -

(

will reduce the magnitude of the exponent and K4 will decrease. That is, partition into

the bottom phase will be enhanced. This general phenomenon, that of partition in-

creasing into the phase enriched in the polymer whose molecular weight is reduced,

has been widely observed (3).

Several of these arguments deserve more discussion. First, because of the exponential dependence of the partition coefficient on parameters such as molecular weights, interaction parameters, and volume fractions, small changes in the differences of these parameters as given in Equation

(5.46) have a large effect on the partition coefficient. Second, the smaller the molecular weight of the solute, the more important is the choice of an appropriate partitioning system, i.e. the more important are the terms in the brackets of Equation (5.46). Third, the molecular weights arc very important and must be maintained as constant as possible or drastic changes in the partitioning properties of a given system may be observed. This also suggests that variations in the molecular weight distribution can have a tremendous effect on the partition coefficient.

In order to gain a more quantitative idea of the behavior discussed above, the equations for the chemical potentials could be written out more exactly and solved numerically -- that is provided the interaction parameters arc known. If the interaction parameters are not known, which is probably the case, then they could be determined experimentally or from existing experimental data.

However, this latter approach is not of much use since there is very little suitable experimental data available on these phase systems, much less is available on the partition of biological molecules in these two-phase systems.

Chapter 5: Polymer Solutions: Flory-Huggins Model 110 5.4 Determination of Flory Interaction Parameters (x.'s)

Before the Flory-Huggins theory can be used for predicting the phase behavior of solvent-po-

lymer systems, the interaction parameters must be available, estimated, determined from existing

data, or determined by some appropriate experimental measurement. Several methods exist for the

determination of solvent-polymer and polymer-polymer interaction parameters. In the following

Sections, methods for calculating interaction parameters from existing binary and ternary data will

be discussed. Following these, a brief discussion of methods for predicting interaction parameters

will be given.

5.4.1 Evaluationof Flory InteractionParameters from Binary Data

First of all, for a binary solvent-polymer solution forming two liquid phases, Equation (5.24)

can be employed to calculate the value of the interaction parameter for each pair of experimental

points on the binodal representing a tie-line.

For the case of solvent activity data as a function of composition, as is available on certain

water-PEG systems, the value of the interaction parameter can be calculated from the experimental

activity and polymer concentration at each data point by use of the following rearrangement of

Equation (5.18)

(lna 1) - ln(l -

Expressions similar to Equation (5.47) can be derived which relate the interaction parameter to such thermodynamic properties as the osmotic pressure or the vapor pressure.

5.4.2 Evaluationof Flory InteractionParameters from TernaryData

For determination of the interaction parameters from existing experimental data on ternary

solvent-polymer-polymer systems, the chemical potential equations given earlier may be combined

Chapter 5: Polymer Solutions: Flory-Huggins Model 111 with the equilibrium equations to give a set of 3 equations in 3 unknowns: the unknowns being the interaction parameters, X12, X13, and X23· The equilibrium phase compositions q>1',q>2',q>3',q>i°',q>/,and q>/ are the known variables. The system of equations and unknowns that remains is a linear system of 3 equations in 3 unknowns. The solution of this matrix problem should yield the values of the interaction parameters; however, due to the interdependence of these three equations via the Gibbs-Duhem relation, a singular matrix is obtained. It should be noted that for linear systems exhibiting singularity, a nontrivial solution cannot be obtained. In the numerical solution of systems of linear equations, a term called the condition number is often used to characterize the degree of singularity of the matrix and hence the uniqueness of the solution.

The problem of matrix singularity arises because of the interrelatedness of two of the three equations, i.e. the system of equations is not linearly independent.

With all of this in mind, alternative means are sought for solving this set of equations to yield the interaction parameters. One method, which is followed in this thesis, uses two of the three equations. This leaves a system of 2 equations in 3 unknowns, and hence something must be specified in order to solve the system. Several choices are available here also. For instance, it could be assumed that both polymers were equally soluble in the solvent, for which case X12= X13, and the system is reduced to 2 equations and 2 unknowns. Recall that this assumption has been used frequently, as in the Scott treatment. It seemed appropriate here to use the binary approach discussed earlier to evaluate one of the three unknowns independently -- since binary data are available for aqueous-PEG solutions. Note that if binary data were available for all the binary pairs of components, Equation (5.47) could be used, and this ternary method would not be required. In this work, the problem of determining Flory interaction parameters from ternary Jata is as follows. First, relations for two of the three chemical potentials are written as Equation (5.31) and

(5.32) and are substituted into each side of the equilibrium equations to yield

(5.48)

Chapter 5: Polymer Solutions: Flory-Huggins Model 112 and

X2)[ ,2 + , , ..2 ,. "] + [ , , + ,2 ,, .. ,,2] X12( Xi 1 1J - 1 - 1J X23 J - 1J - J

+ X13 [ - ~tX2 , (j)3 , + ~tX2 "J ·] -- ( 1n2 ') - ( 1 - 2 ') + t '( Xi X2 ) (5.49) , X2 1n " X2 " X2 + J(-) + ( J(-) ~ x, ~

In order to make this system solvable, one of the unknowns must be specified. This is probably one of the reasons that so many previous workers, in application of the Flory-Huggins theory made such simplifying assumptions as symmetry. Not only does this assumption make the problem less complicated and allow for an analytical solution, but for the ternary case, something must be specified anyway.

In order to decide what to specify as the third equation, or how to eliminate one of the unknowns, the particular situation in which we find ourselves is recapitulated. First, binary data are not available for all three binary pairs of the components (eg. water-PEG, water-dextran, and

PEG-dextran). Second, limited ternary data are available, as is limited binary data on one of the binary pairs (water-PEG). Third, it is preferred to use available data and avoid experimental detennination of phase equilibrium data. Under these circumstances, the binary data are reduced to yield one of the interaction parameters, and its values or an expression as a function of temperature and/or concentration is added as a third equation. This linear system of 3 equations and 3 unknowns can be written in matrix form as

(5.50)

where the coefficient matrix ~ is given as

a11 a,2 a13 a21 a22 a23 1-[a31 a32 a33 l with the coefficients of the matrix A as follows

Chapter 5: Polymer Solutions: Flory-Huggins Model 113 , I + 12 " H ,,2 a12 =

- ln(.!L) + (

and where the solution vector x is given as

.ll. = [:::]

X23

and where a= constant or a= Po+ P1

equations. In LU-factorization, the matrix~ is decomposed into an upper triangular matrix l} and

a lower triangular matrix 1, . "Thesetwo matrices l} and ~ arc used in the back substitution to yield

the solution. Also, the condition number (a measure of the degree of singularity of matrix ~ ) is

Chapter 5: Polymer Solutions: Flory-Huggins Model 114 estimated. If the condition number is extremely large -- towards the machine epsilon -- then for all practical purposes, the matrix is singular. The values of the interaction parameters are returned from the above subroutines based on the solution of the matrix composed of the experimental volume fractions.

Tseng [ 133] studied the phase behavior of ternary systems of polystyrene-polybutadiene- chloroform. He also encountered the problem of singularity as discussed above. Tseng and co- workers [118,133,134] have suggested several other alternatives for solving the above system of equations. For example, Tseng et al. employed the formalism of Allen et al. (5) and showed that subtracting the Aµ1' = Aµ( equation (with the relations for the chemical potentials inserted) from the Aµ/ = Aµ/ equation and rearranging yields

( 5.51)

Likewise, subtraction of the Aµ/ = Aµ/ equation from the Aµ3' = Aµ/ equation and rearranging yields

X2i(/ - 3')- (3.. - 3')+ (1')]= x~ 1n[::: ]- 1n[::: l

Also, the two equations above can be added to yield

*1n[ f. ]-*1n[ f. ]-(X13 - X12)(3'+

Tseng et al. [118,133,134)then discuss the use of the above equations. If values for the solvent-polymer interaction parameters (Xi 2 and Xi 3 ) are known, then this last expression, Equation

(5.53), can be used to directly calculate the polymer-polymer interaction parameter (X2 J) from the other two interaction parameters and the experimental equilibrium phase compositions. If only

Chapter 5: Polymer Solutions: Flory-Huggins Model 115 one of the solvent-polymer interaction parameters is known, then the first two of these expressions,

Equations (5.51) and (5.52), are solved simultaneously to obtain the two unknown interaction pa-

. rameters (X23 and either Xtz or x13 ). Note that this last procedure is essentially the method developed in this thesis.

Another important note is that the application of the above procedures to experimental tie-line

data (equilibrium coexisting-phase compositions) enable the interaction parameters to be deter-

mined as a function of concentration. The importance and use of this approach will be discussed

in Section 5.5. For now, suffice it to say that values of the interaction parameters would be deter-

mined at each experimental data point. The interaction parameters can then be fit to a polynomial

in concentration.

5.4.3 Experimental Methods for Detennination of Flory Interaction Parameters

The following list, from Tseng (133),includes some of the experimental methods available for

determining interaction parameters in solvent-polymer systems: cryoscopy (melting point de-

pression), ebulliometry (boiling point elevation), equilibrium vapor pressure lowering or gravimetric

adsorption, osmotic pressure, light scattering, intrinsic viscosity measurements, inverse gas chro-

matography, and piezoelectric crystal equilibrium sorption. See Tseng [ 133)for a brief discussion

of each of these methods.

Recently Olabisi et al. (97] have reviewed many of the techniques currently being used for de-

termination of polymer-polymer miscibility. These discussions logically include determination and

use of interaction parameters.

5.4.4 Estimation of Flory Interaction Parameters

As previously discussed, the Flory interaction parameters are determined from experimental

data. An attractive method for producing interaction parameters would be from group-contribution

schemes like UNIF AC or ASOG or other semi-empirical models for which polymer solution data

has been correlated. Remember that these group-contribution schemes were proposed as methods

Chapter 5: Polymer Solutions: Flory-Huggins Model 116 for estimating activity coefficients without performing experimentation or reducing data to obtain binary parameters.

Tseng (133] tested the ability of UNIFAC for predicting solvent-polymer interaction parameters in order to avoid their experimental determination. As Oishi and Prausnitz (96] had demonstrated the accuracy of the free volume correction of UNIFAC (UNIFAC-FV) to be within 10% error of observed values, Tseng concluded that this might be a promising method for estimating these solvent-polymer interaction parameters. Tseng points out that UNIF AC has no basis for the effect of molecular weight of the activity. The effect of molecular weight on the solvent activities in aqueous-polyethylene solutions as predicted by UNIFAC and the Flory-Huggins model are shown in Figures 25 and 26, respectively. Recall the inability of UNIFAC to adequately predict the effect of molecular weight. Note however that UNIFAC does predict the correct trend of increasing solvent activity with increasing molecular weight, and it also predicts less effect on the molecular weight with increasing molecular weight. The UNIF AC predictions are very poor though.

Tseng found that the influence of molecular weight on the solvent activity was insignificant, but the molecular weights he used were much larger than the ones represented in Figures 25 and

26. Again, the effect of molecular weight should be negligible at very high molecular weights; this is seen by noting that difference in the curves of Figure 26 becomes smaller and smaller with increasing molecular weights. Tseng also found that the UNIFAC-FV model did predict satisfactory results for the solvent activity but failed to satisfactorily predict interaction parameters. He ascribes this inability to the use of constant interaction parameters within the UNIFAC model.

Van den Berg (135) has presented a procedure that allows for direct calculation of solvent activities (corrected for the free volume differences) from UNIFAC. Van den Berg also gives an

equation for estimating the Flory-Huggins x.interaction parameter from UNIFAC which includes

(I) the activity coefficient (on a mole fraction basis) determined from UNIFAC, for example using

the programs of Fredenslund (41); (2) mole fraction and volume fraction terms; and (3) the free volume correction term ( ln aFv) as proposed by Oishi and Prausnitz (96). Note that the primary

Chapter S: Polymer Solutions: Flory-Huggins Model 117 advantage of van den Berg's procedure is that it allows direct application of the UNIFAC code presented by Fredenslund (41).

Misovich et al. (87,88) have presented a method for predicting solvent-polymer interaction parameters and their concentration dependence by application of ASOG-VSP. Recall that

ASOG-VSP is a one-parameter correlation based on a free volume correction of ASOG which requires the experimental determination of the single adjustable parameter at infinite dilution. Mi- sovich showed that ASOG-VSP predicted the correct dependence of the x interaction parameter on polymer composition for the systems of benzene-poly(isobutylene) at 25°C and benzene- poly(ethylene oxide) at 70°C.

Flory-Huggins x interaction parameters can also be predicted from solubility parameters. For a recent, comprehensive discussion of the use of cohesion (solubility) parameters for polymer solutions, see Barton (12). Olabisi (97) also discusses the Hildebrand solubility parameter approach for polymer solutions.

5.5 Modifications of the Classical Flory-Huggins Theory

The modifications to the classical Flory-Huggins theory to be discussed here basically stem from considering the temperature, composition, and molecular weight effect on the interaction parameter X· Recall that the Flory-Huggins theory assumed the x parameter to be a constant, independent of the above factors. In particular, modifications of the x parameter to include the concentration dependence will be discussed; the inclusion of an entropic contribution to x will also be described; and finally, the effect of polydispersity on the x will be briefly covered. Dilute solution theories will also be highlighted, as many solutions of polymers contain dilute concentrations of polymers. Recall that the classical Flory-Huggins theory only applies to concentrated polymer solutions.

Chapter 5: Polymer Solutions: Flory-Huggins Model 118 5.5.1 Re-examiningthe Flory Interaction Parameter

Two of the greatest fallacies in the classical Flory-Huggins theory stem from assumptions regarding the interaction parameter. First, the interaction parameter is assumed to be independent of concentration and molecular weight. Second, the theory assumes that interactions play no part in the entropy of mixing. Recall that this second assumption leads to a free energy of mixing expression based on a configurational entropy term (spatial) and an enthalpic term (interactional).

It is assumed that there is no entropy term based on interactions. Another problem with this theory is that it assumes all polymer species to be single components (or monodisperse).

5.5.1.l Concentration Dependence of the Interaction Parameter

For some systems, such as rubber solutions, the Flory-Huggins model yields excellent results, yet there are some systems for which representation of data has been very unsatisfactory. Fur- thermore, Shultz and Flory (123) concluded from their studies that the Flory-Huggins equation is somewhat inaccurate in representing the concentration dependence of the chemical potential (or free energy of mixing expression) and that this inadequacy leads to ( l) predicted critical points at lower concentrations than experimentally observed and (2) a much narrower concentration range of the predicted binodals than the experimental ones. In fact, a marked concentration dependence on the x.parameter has been observed for many systems. The Flory-Huggins model may be improved to yield much better results by determining the x.parameters as a function of polymer concentration, temperature, and possibly even as a function of polymer molecular weight.

A slightly different approach to the development of a free energy of mixing expression for polymer solutions was taken by Guggenheim (50) and Maron (83) and advanced by Koningsveld

(65). The free energy expression thus chosen has the concentration-independent enthalpy interaction parameter, X, replaced by a concentration-dependent free-energy interaction parameter, g(T,2 ), which includes both enthalpic and entropic parts, and is given by

~gmixing (5.54) RT

Chapter 5: Polymer Solutions: Flory-Huggins Model 119 Note that this correction is essentially the same as the correction proposed by Flory (37) as seen in

Equations ( 5.15) and ( 5.16).

Upon replacing x12 of Equation (5.14) with the concentration-dependent g parameter and performing differentiation on the free energy expression to obtain the chemical potential, it is found that the Flory interaction parameter, x12 is related to the new interaction parameter, g(T,q>2), as

(5.55)

In other words, the x12 in Equation ( 5.17) should be replaced by the above expression.

Several forms of the new interaction parameter, g(T,q>2 ), have been suggested. These include truncated power series expansions in

g = (5.56) where the temperature dependence is introduced in the first constant of this equation as

&l = &l,1 + &l,2 i + &>,3T + &>,4ln T (5.57)

Tseng (133) essentially followed the approach described above. He chose to write the expression for the solvent activity as

where the last term is added to account for the concentration dependence of the new interaction parameter X· This expression can be rearranged to yield

where comparison with Equation ( 5.18) shows that this new parameter x is related to the old parameter X12 as follows

Chapter 5: Polymer Solutions: Flory-Huggins Model 120 Assuming a power series expansion of X,

and integrating yields [ 133)

co a :E __ n -(1 _ q>~+ I) n=O 1 + n X =

For quadratic dependence of x on

where a 0 , a 1 , and a 2 are constants found by performing a nonlinear regression analysis of x versus

(!)2 data.

For this work, the following expression was used

The use of this expression will be described in Section 5.6.1. This expression is the same as the one employed by Tseng except he wrote his concentration dependence in terms of x instead of X12-

As pointed out by Casassa [22], if the x12 interaction parameter is expanded as a power series of q,2 (polymer volume fraction) as

X12 (5.58) then the solvent activity becomes

Chapter 5: Polymer Solutions: Flory-Huggins Model 121 (5.59)

where X.o,Xi, and x.2 are determined from fitting the experimental concentration dependence of

x.12 • Equation (5.59) is essentially a virial expansion, but as Casassa (221points out, the x.'s chosen to fit concentrated polymer solution data will not necessarily correspond to the correct virial coef-

ficients, which are defined at infinite dilution.

Koningsveld and Kleintjens [671 proposed the following four-constant expression for g

g = a + [ Po + TP1 J( 1 - Y

where the four constants are a, Po,p 1 , and y. They used this expression to fit critical temperature

and composition data of Koningsveld, Kleintjens, and Shultz [68). It should be noted that while

there is some basis for this expression, the constants are primarily empirical parameters used to fit

the data. The foundation for this expression comes out of a more accurate lattice combinatory (22).

The modifications described above do not address the dilute solution problem of polymer

chains surrounded by large expanses of solvent. It is very important that this problem be addressed

as Koningsveld, Stockmayer, Kennedy, and Kleintjens (KSKK) (721have reported that the incor-

rect features of binodals and spinodals predicted by the classical Flory-Huggins theory stem from

the fact that the phase system is not always concentrated in polymer ( 10% w/w). Many times

at least one of the two coexisting phases is dilute in polymer (

by Koningsveld et al. (72) interpolates between the excluded-volume theory for dilute solutions and

the lattice theory for concentrated solutions in such a way as to yield a continuous theory over these

two vastly different domains.

The model obtained by Koningsveld et al. (72) utilizes the following expression for the in~er-

action parameter, g,

g = &:one+ '1'1[ l - *Jc I - h(z)](l + A)- 1 exp( - Aq>2) (5.61)

Chapter 5: Polymer Solutions: Flory-Huggins Model 122 where ~nc is determined from Equation (5.60). The term 'Jf1(1 - *)comes from the excluded- volume theory where 'Vi is an entropy parameter introduced by Flory (37) and 0 is the theta or

"ideal" temperature, at which deviations from ideality vanish. Also, in Equation (5.61) ')..- 1 is the polymer volume fraction at which the sum of all coil domains would equal the volume of the sol-

ution if there were no overlapping, and h(z) is a function of the random flight variable z. See Ca-

sassa [21,221for a discussion of this dilute solution theory which treats the solvent as a continuum

and describes the polymer chains by "random-flight statistics perturbed by a mutual volume of

exclusion between pairs of chain segments in close proximity."

Casassa (221presented a comparison of the KSKK model (of Koningsveld et al. (721), the concentration dependent g representation of Koningsveld and Kleintjens (67), and the classical

Flory-Huggins theory. Figure 27 (from Casassa [221)compares the spinodals predicted from these various models with experimental data determined from light scattering measurements. Indeed it is seen that the Flory-Huggins model predicts (I) spinodals much narrower than experiment and

(2) critical points at concentrations much lower than experimental data. The use of a concentra-

tion-dependent interaction parameter greatly improves the prediction -- but at lower molecular

weights there is still considerable discrepancy between experiment and theory. The use of the

KSKK model, which accounts for concentration dependence of the interaction parameters and di-

lute solution theory, yields predictions superior to both of the other models for the ranges of data

displayed in the graph. Thus, the KSKK model is a very attractive model for solvent-polymer so-

lutions -- theoretically valid over the entire concentration range. However, as discussed earlier, a

simple model is desired. Thus, due to its complexity, the KSKK model was not examined further.

However, the use of concentration-dependent interaction parameters was investigated further, and

the results are forthcoming.

5.5./.2 The Inability of the Flory-Huggins Model For Prediction of Lower Critical Solution

Temperature Behavior

Another disconcerting fact about the Flory-Huggins theory is its inability to predict high

temperature phase splitting or the presence of a lower critical solution temperature (LCST). This

Chapter 5: Polymer Solutions: Flory-Huggins Model 123 25 u 0 "' w 20 Ct: ::, I- <( • Ct: 51X 103 w 15 .. , •, a.. : , , ... I ... . I .. . z I .. w I . I- I ... 10 I I . I • 0 0.05 0.10 0.15 0.20 POLYMER VOLUME FRACTION

Figure 27. Comparison of spinodals predicted from concentration-dependent and independent theories

with experiment: Experimental points from light scattering data on polystyrene and cy-

clohexane (closed circles); critical points. Theoretical predictions: for KSKK [72), g func-

tion with A= O.Sx1' 2 (solid curve); for concentration-dependent, g from Koningsveld and

Kleintjens (67] (dashes); for classical Flory-Huggins, X based on 'Vi and 0 parameters

from Shultz and Flory (123] (dots). (faken from Casassa (22]).

Chapter 5: Polymer Solutions: Flory-Huggins Model 124 is a major problem with the Flory theory, as the existence of upper and lower critical solutions temperatures (UCST and LCST) in solvent-polymer solutions is a well-known phenomenon. The

reason for the inability of the Flory-Huggins model to predict the LCST can be revealed upon close

inspection of the temperature dependence of the interaction parameter. Recall from earlier discussions that Flory was aware of this limitation and offered a formalism for its correction. Flory initially assumed that the entropy of mixing had no internal interaction associated with it. In real-

ity, as he well knew, this is not the case. Thus, the correct representation of the interaction pa-

rameter should be

X12 = Xh + Xs (5.62)

where as seen in Figure 28, even though x.12 is not a very strong function of temperature, the in-

dividual enthalpic, Xh , and entropic, x,, contributions may very well be strong functions of tem-

perature. The reason for the observed differences in the x parameters is as follows. First, Xh is

expected to decrease monotonically with temperature. Second, Xs increases with increasing tem-

perature due the free volume differences between polymer and solvent molecules. The polymer

molecules pack more densely than the solvent molecules; hence when solvent and polymer mole-

cules are mixed, the polymer molecules gain some freedom of motion (vibrational and rotational)

whereas the solvent molecules become more restricted in their movement. But, as the temperature

rises, the solvent molecules gain more freedom of motion, especially in comparison to the polymer

molecules which are essentially compressed. This leads to an even larger difference in free volumes,

and consequently to a greater error irI the Flory-Huggins model, sirlce in the classical Flo~·-Huggins

view, the entropy term (combirlatorial in nature) is not a function of temperature.

Note that in Figure 28, with x = Xh + x,,there are two points at which the x parameter equals

the critical interaction. These points represent low and high temperature miscibility limits. For low

temperature solutions in which x exceeds the critical value, Xe,phase splitting occurs, and the system is said to exhibit UCST behavior. Similarly, for high temperature solutions in which x exceeds

the critical value, phase separation is observed and the system is said to exhibit LCST behavior.

Chapter 5: Polymer Solutions: Flory-Huggins Model 125 0::: L1.J t- w ~ <( 0:: ~ 'Xc------z 0 t- -u - 0:: 0 _J LL

TEMPERATURE

Figure 28. Temperature dependence of the :X:12interaction parameter: Individual contributions to :x::

Xhis the enthalpic contribution; and X is the entropic contribution. X is the critical in- s C teraction parameter and represents the miscibility limit; that is, the X must be lower than

X in order for the components to be miscible. Note that there arc two regions of immis· C cibility; a low temperature region (UCST) and a high temperature region (LCST).

Chapter 5: Polymer Solutions: Flory-Huggins Model 126 Recently, Sheu [ 122) has suggested that water-polymer solutions, and in particular water-polyethylene glycol solutions, exhibit LCST behavior and therefore cannot be modeled using the classical Flory-Huggins approach. As evidence of this, Sheu references the data of Malcolm and

Rawlinson (81), which does show LCST behavior for water-polyethylene glycol (PEG 5000) at temperatures above l 25°C.

In regards to the data of Malcolm and Rowlinson (81), Freeman and Rawlinson (44) suggest

the following explanation. The LCST behavior is found only in solutions where both components

are highly polar, and the existence of the LCST is related to the decrease in entropy associated with

the formation of hydrogen bonds at low temperatures. They suggest that the miscibility decreases

upon heating because hydrogen bonds are broken by rotation of the molecules. They also suggest

that the LCSTs are at low temperatures for these polar systems; temperatures usually below the

boiling point of the solvent.

These results yield serious implications in regards to the "blind" use of the Flory-Huggins

equation. For the work conducted here, the following rationalizations were used. First, possibly

the water-PEG systems also exhibit UCST at low temperature (behavior like that shown in Figure

5 of Chapter 2). The systems being studied here are at temperatures well below those reported for

LCST behavior observed by Malcolm and Rawlinson, so that if this was the case, the Flory-Hug-

gins theory would be applicable. However, this seems doubtful since the low-temperature binary

water-PEG data of llerskowitz and Gottlieb (57) does not exhibit two-phase behavior. Assuming

the system exhibits both UCST and LCST behavior as demonstrated in Figure 5 (of Chapter 2),

then, there indeed could be an UCST observed at low temperatures.

Another point is that the ternary water-PEG-dextran systems being studied here Jo indeed

exhibit UCST behavior (that is, provided the data of Albertsson is correct). This is seen by ex-

amining the effect of temperature on the binodal of ternary solvent-polymer-polymer systems ex-

hibiting UCST behavior as shown in Figure 29 and discussed by Koningsvcld and Staverman (69).

It is observed from Figure 29 that as the temperature is increased from T I to T 5 , the two-phase

region decreases (and the solubility range increases). Figures 30-32 show the effect of temperature

on experimental phase diagrams for various molecular weights of dextran and polyethylene glycol,

Chapter 5: Polymer Solutions: Flory-I luggins Model 127 as determined by Albertsson (3). Note that these binary plots are equivalent to the ternary plot of Figure 29, and undoubtedly show that as the temperature is increased, the two-phase region does indeed decrease. Therefore, it is concluded that at low temperatures, these ternary systems exhibit

UCST behavior and thus may be modeled using the Flory-Huggins theory! However, in order to describe these systems, the three interaction parameters X12, Xll, and l23 are needed. Recall that the x12 is being determined from binary water-PEG data using the Flory-Huggins equation which, as described above, appears to be invalid for these systems.

5.5.IJ Effect of Molecular Weight on the Interaction Parameter

The effect of the molecular weight on binodal curves is predicted qualitatively by the Flory-

Huggins theory, as was shown by Scott [119,120),Shultz and Flory [123), and Tompa (1321. The influence of molecular weight on the phase behavior vanishes above very large molecular weights

(eg. 109). Koningsveld and Staverman (70) present a relation of ;c as a function of several variables including the molecular weight. They suggest that the influence of molecular weight and molecular weight distribution is the strongest in regions of low molecular weight, as mentioned above. Also, referring to the work of Dobry and Boyer-Kawenoki (28),the higher the molecular weight, the more incompatible the polymers. This suggests that as molecular weight increases, the x interaction parameter between the two polymers increases.

5.5.2 Effect of Polydispersity on the Phase Behavior

Koningsveld and Staverman (69,70,71)studied liquid-liquid phase separation in multicomponent polymer solutions using a different approach. They studied the effect of the molecular weight

(or chain length) distribution on the phase behavior of polymer solutions in a single solvent. A qualitative description of the effect was given, and a numerical calculation based on the Flory-

Huggins model was suggested for obtaining quantitative predictions of the phase behavior. Appli- cation of the classical Flory-Huggins theory assumes that the polymers being used have been fractionated so effectively that they can be regarded as a single component, or in other words, they are characterized by a very sharp, narrow molecular weight distribution. This brings up two very

Chapter S: Polymer Solutions: Flory-Huggins Model 128 s P,

Figure 29. Effect of temperature on the binodal of ternary systems containing a solvent and two po-

lymers: The polymers P 1 and P 2 are characterized by molecular weights of M 1 and

M2, differing as M2 > M 1• The temperatures in increasing order are

T 1 < T 2 < T 3 < T • < T 5. (Taken from Koningsvcld and Staverman (691).

Chapter 5: Polymer Solutions: Flory-Huggins Model 129 15

T•OC ---·---

Tm20C -•-

0 0 ~ 0\ (1) C (1) ~ 5 ..-J.c (1) ~ 0 0..

0 ,.-.,__ ...... _.,._,....,,...... _.,._,....,,...... ""l""T_,....,,..,...... _.,._,....,""T" ...... _.,. __ ,...... ,. 0 5 10 15 20 Dextran ("- w/w)

Figure 30. Effect or temperature on the experimental binodals or water-PEG 6000~extran (D68) sys-

tems: The experimental data are from Albertsson [3]. Number average molecular weights: PEG, 6000-7500; Dextran, 280,000.

Chapter 5: Polymer Solutions: Flory-Huggins Model 130 15

T = 0 C ---•---

T=20C _.__

0 (,J ~ O'I Q) C Q) ~ 5 +J.c Q) ~ a..0

0 5 10 15· 20 25 Dextran (7.w/w)

Figure 31. Effect of temperature on the experimental binodals of water-PEG 6000-dextran (D48) sys-

tems: The experimental data are from Albertsson (3). Number average molecular weights: PEG, 6000-7500; Dextran, 180,000.

Chapter 5: Polymer Solutions: Flory-Huggins Model 131 15

T a O C -- - 6 -- -

Tc::r20C -•-

:0.. ~ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

Dextran (,; w/w)

Figure 32. Effect of temperature on the experimental binodals of water-PEG 4000-dcxtran (D48) sys-

tems: The experimental data are from Albertsson (3). Number average molecular weights: PEG, 3000-3700; Dextran, 180,000.

Chapter S: Polymer Solutions: Flory-Huggins Model 132 important points. First, strict application of the Flory-Huggins theory to ternary solvent-polymer solutions requires a monodisperse polymer species. Secondly, this reveals the need for proper characterization of the polymers being used.

Koningsveld and Staverman present continuous distribution functions which are used to represent certain molecular weight distributions. Their thermodynamic model is based on the Flory-

Huggins theory with the interaction parameter assumed to be a function of temperature and independent of polymer concentration and molecular weight. Koningsveld and Staverman propose a method whereby the Gibbs free energy of mixing relation is used directly. In the case of a binary mixture, volume fractions of each polymer in the coexisting equilibrium phases are found by means of an iterative, numerical construction of the double tangent to the ~gmixingversus volume fraction curve at different values of the interaction parameter.

For the emphasis of this thesis research, the Koningsveld-Staverman approach, while being theoretically more correct, is too involved, and thus will not be employed.

5.5.3 Dilute Solution Theory

The classical Flory-Huggins theory is not theoretically valid for dilute polymer solutions

(i.e.,

in polymer that the domain of each polymer is surrounded completely by pure solvent.

Several different approaches are available for describing dilute polymer solutions, and a few will be introduced below. First, as pointed out by Flory [37J, even though the lattice model is no longer adequate, the osmotic pressure form of the theory can be quite useful when dealing with dilute polymer solutions.

The osmotic pressure may be related to the solvent activity and chemical potential as

(5.63)

Chapter 5: Polymer Solutions: Flory-Huggins Model 133 where v1 is the molar volume of the solvent, and n is the osmotic pressure. Following the classical Flory-Huggins treatment, the solvent activity is given by Equation

(5.18) for which the osmotic pressure becomes

(5.64)

Expanding the logarithm in

(5.65)

Flory (37) writes the osmotic pressure equation in terms of the concentration c, in grams per milli- liter, where he lets

n = RT + RT[ v~ x12]c + RT[ v1 ]c2 + ... (5.66) C M Vi ][-1-2 3v 1

The first term in this equation is the ideal term, and the higher terms represent deviations from ideality. Note however that the nonideality predicted by these terms will be in error since the model was based on the lattice theory which in not valid for dilute solutions.

Casassa (22] and Staverman (128] discuss the virial coefficients obtained from the above expression from the lattice theory and make two important observations concerning them. First, the second virial coefficient has no molecular weight dependence if Xiz is independent of molecular weight -- which is assumed in the lattice theory. Second, the third and higher virial coefficients show no dependence at all on thermodynamic interactions, which shows that these coefficients are theoretically (from a statistical mechanical point of view) meaningless.

Hence, a new, special theory is needed for dilute polymer solutions. Flory [35,36) developed theories to take into account the discontinuity of the polymer in dilute solutions. Flory's theory is a statistical mechanical treatment of a small number of polymer chains in solution, based on the

Chapter 5: Polymer Solutions: Flory-Huggins Model 134 excluded-volume principle in which it is assumed that a polymer molecule excludes all others from the volume it occupies. Flory's approach leads to a partition function which yields the virial coefficient expansion equation of state

n (5.67) RTc where n is the osmotic pressure, c is weight per volume polymer concentration, M is the molecular weight of the polymer, A2 and A 3 are the second and third virial coefficients.

As discussed earlier, the KSKK theory is based on a different approach for dilute polymer solutions which avoids the use of a lattice. Here, "the solvent is treated as a continuum and polymer chain configurations are described by random-flight statistics modified only by a mutual volume of exclusion between chain segments in close proximity" (21).

5.6 Application of the Modified Flory-Huggins Theory

This Section describes the procedure for using the concentration-dependent interaction parameter form of the Flory-Huggins theory to predict the phase behavior of ternary solvent-polymer-polymer systems. In particular, one of the solvent-polymer interaction parameters, ;c12 , is determined from binary data and these values are fit to an expression quadratic in polymer concentration. Then, with this expression for ;c12 , the system of equations shown in Equation (5.50) is solved to obtain ;c13 and ;c23 Values of these parameters are found at each experimental point, and they each are to be subsequently fit to an expression quadratic in polymer concentration.

However, if the X interaction parameters are constant over the concentrations region of interest, then a constant value for the x parameters can be used -- since this radically simplifies the computational procedure. Once the ;c-inieraction parameters are determined, they are used to predict the phase behavior of water PEG-dextran systems. Each of the above procedures is described in detail below.

Chapter5: PolymerSolutions: Flory-Huggins Model 135 5.6.1 Determination of Concentration-Dependent x Parameters

The detemtlnation of the x parameters needed to represent a ternary system would be trivial if binary data were available on each binary pair. Unfortunately, these data are not available for the systems being studied here. However, it should be noted that binary equilibrium data are often not sufficient to yield parameters that quantitatively predict ternary behavior. Ternary data are available on the systems of interest, but as discussed in Section 5.4.2, in order to apply the chemical potential expressions and equilibrium relations to detemtlnation of x parameters, one of the x parameters must be independently specified. The other two x parameters are found from the specified one by the methods described in Section 5.4.2.

Here, the x12 (water-PEG) interaction parameter was to be specified, but not as a constant as is found in the Polymer Handbook (18). As is seen in Figures 18-24, and 26, the Flory-Huggins model with a constant interaction parameter does not fit the data very well. It is expected that the interaction parameter is a function of concentration, hence Xi 2 was to be specified as a function of polymer concentration. This concentration dependence of Xiz (water-PEG) has also been observed by Heil [52). Binary data were needed on water-PEG systems for detemtlnation of x12 • Data on various water-PEG systems has been presented by Malcolm and Rawlinson [81) and Herskowitz and Gottlieb (57), as was presented in Chapter 3. Thus, the Xiz interaction parameters were calculated from the experimental data for several of the various PEG molecular weight fractions.

These results appear in Tables 7-10 and Figures 33-36. These results reveal a strong dependence of X12 on polymer concentration, <()z, and the methods described in Sections 5.3 to 5.5 were employed to detemtlne the concentration dependence of the solvent-polymer interaction r,arameters in these binary systems.

The procedure for these calculations is as follows. Equation (5.47) from Section 5.4.l was employed to calculate the "experimental" value of x12 (called x!T11 ) from the experimental data.

The experimental data were based on weight fractions, and thus for use of Equation (5.47) these were converted to volume fractions using the liquid densities displayed in Table 5 of Chapter 3.

These X!T11 values were then plotted versus polymer concentration in order to display the concen-

Chapter 5: Polymer Solutions: Flory-Huggins Model 136 Table 7. Concentration Dependence of x Interaction Parameter for Water-PEG 1500 Solutions at 293, 313, 333 K

Wt. Fract. x.rr0 X.fl"d ayxpu af"'d Polymer

Temperature = 293 K

0.6378 0.3948 0.3929 0.8346 0.8340 0.5825 0.3756 0.3777 0.8732 0.8738 0.5437 0.3682 0.3707 0.8964 0.8970 0.4631 0.3686 0.3648 0.9358 0.9352 0.3454 0.3751 0.3762 0.9714 0.9715

Note: Xffd = 0.523 - 0.752

Temperature = 313 K

0.9075 l.0614 l.0511 0.6060 0.6010 0.8930 l.0072 1.0031 0.6395 0.6375 0.8175 0.7664 0.7905 0.7379 0.7491 0.7223 0.5975 0.6055 0.8178 0.8209 0.6259 0.5174 0.5042 0.8823 0.8783 0.6095 0.5045 0.4950 0.8903 0.8876 0.5995 0.5001 0.4904 0.8958 0.8931 0.5035 0.4727 0.4874 0.9392 0.9421

Note: X.ffd= 1.45 - 3.85

Temperature = 333 K

0.9779 1.6350 1.5812 0.3248 0.3086 0.9163 l.1849 1.2539 0.6288 0.6651 0.8506 0.9517 0.9838 0.7522 0.7689 0.7734 0.7835 0.7629 0.8284 0.8191 0.7071 0.6853 0.6501 0.8709 0.8573 0.6481 0.6326 0.6052 0.9037 0.8947 0.5746 0.5811 0.6169 0.9333 0.9427

Note: Xffd = 2.73 - 7.33

Chapter 5: Polymer Solutions: Flory-Huggins Model 137 Table 8. Concentration Dependence of X Interaction Parameter for Water-PEG 3000 Solutions at 328 and 338 K

Wt. Fract. XH"u Xf~ ayxpu af"'d Polymer

Temperature = 328 K

0.9700 1.5615 1.4493 0.4000 0.3604 0.9500 1.4003 1.3862 0.5240 0.5175 0.8440 0.9186 1.0827 0.7590 0.8470 0.7020 0.6583 0.7511 0.8710 0.9071 0.4990 0.5683 0.4045 0.9640 0.9323

Note: Xffd = 0.033 + 0.249q>1 + l.27q>j

Temperature = 338 K

0.9900 1.7832 1.5666 0.1840 0.1490 0.9700 1.5623 1.4924 0.4020 0.3767 0.9500 1.4110 1.4207 0.5310 0.5355 0.9040 l.1515 1.2647 0.6790 0.7421 0.8970 l.1280 1.2421 0.6970 0.7610 0.8460 0.9545 1.0854 0.7740 0.8450 0.7650 0.7809 0.8650 0.8470 0.8856 0.7040 0.7049 0.7205 0.8880 0.8941 0.4990 0.5919 0.3547 0.9690 0.9234

Note: Xffd = 0.056 - 0.065q>2 + l.61q>j

Chapter 5: Polymer Solutions: Flory-Huggins Model 138 Table 9. Concentration Dependence of X Interaction Parameter for Water-PEG 5000 Solutions at 333 and 338 K

Wt. Fract. xt111 lffd afxpu af"'d Polymer

Temperature = 333 K

0.9500 1.4335 1.3820 0.5470 0.5227 0.9020 1.1615 1.2055 0.6990 0.7234 0.8350 0.9340 0.9845 0.7960 0.8225 0.7350 0.7504 0.7054 0.8770 0.8582

Note: Xffd = 0.002 - 0.269q>2 + l.85q>}

Temperature = 338 K

0.9900 1.8066 1.5903 0.1910 0.1547 0.9700 l.6005 l.5135 0.4220 0.3893 0.9510 1.4488 1.4431 0.5470 0.5442 0.9490 l.4244 1.4358 0.5520 0.5576 0.9020 l.1760 1.2719 0.7080 0.7629 0.9000 l.1652 l.2653 0.7110 0.7683 0.8390 0.9644 l.0734 0.8030 0.8624 0.8350 0.9497 1.0615 0.8050 0.8655 0.7640 0.7889 0.8662 0.8570 0.8925 0.7350 0.7655 0.7941 0.8840 0.8962 0.6620 0.6660 0.6309 0.9140 0.9020 0.4960 0.6026 0.3484 0.9740 0.9262

Note: Xffd = 0.064 - 0.103q>2 + l.67q>}

Chapter 5: Polymer Solutions: Flory-Huggins Model 139 Table 10. Concentration Dependence of): Interaction Parameter for Water-PEG 6000 Solutions at 313 and 333 K

Wt. Fract. XH"11 XVfd arxptl ayred Polymer

Temperature = 313 K

0.7047 0.5890 0.5867 0.8464 0.8455 0.6280 0.5133 0.5200 0.8892 0.8912 0.5341 0.4774 0.4700 0.9344 0.9328 0.4408 0.4482 0.4516 0.9624 0.9629 0.3292 0.4664 0.4660 0.9862 0.9862

Note: Xffd = 0.697 - l.292 + l.70q>j

Temperature = 333 K

0.9910 2.5474 1.9624 0.3576 0.2018 0.9761 I. 7914 1.8536 0.4162 0.4414 0.9677 1.6767 1.7944 0.4826 0.5379 0.9375 1.3765 1.5935 0.6170 0.7426 0.8932 l.1352 l.3319 0.7244 0.8412 0.8761 1.0646 1.2411 0.7498 0.8523 0.8095 0.8705 0.9386 0.8207 0.8550 0.7852 0.8227 0.8476 0.8411 0.8529 0.6832 0.6688 0.5696 0.8985 0.8631 0.5800 0.5764 0.4440 0.9376 0.9035 0.5344 0.5553 0.4334 0.9524 0.9257 0.5068 0.5425 0.4394 0.9595 0.9392 0.3248 0.5115 0.6898 0.9905 1.0046

Note: XVfd= l.87 - 5.90q>2 + 6.06j

Chapter 5: Polymer Solutions: Flory-Huggins Model 140 2.0

TEMP,K EXPTLX 1. 8 293 313 1. 6 I 333 I 0::: I w I 1- 1.11 I w I ~ I I ~ I ~ 1.2 I 0... I I I.' >< I l r 1.0 // 0::: / / .. g / .... LL 0.8 /,// .,., .,,,, _,,..I 0.6 -~-' .,..·"' ...... ,..... 0.11

0.2

0.0,._, ...... ,1 o.o 0.2 0.11 0.6 o.e 1. 0 VOLUME FRACTIONOF POLYMER.

Figure 33. Concentration Dependence or the X Interaction Parameter for Water-PEG 1500 solutions

at 293, 313 and 333 K: The curves shown here represent experimental X interaction pa-

rameters.

Chapter 5: Polymer Solutions: Flory-Huggins Model 141 2.0

TEMP.K EXPTLX I. 8 328 ' I' ' 338 I' ---- .' 1. 6 .•' ~ w WI.III- ~

~<( 1.2 a.. >< >-1.0 ~ 0 ...J l.J...0.8

0.6

0.11

0.2

o.o 0.0 0.2 0.11 0.6 0.8 I. 0 VOLUME FRACTIONOF POLYMER

Figure 34. Concentration Dependence of the 'X Interaction Parameter for Water-PEG 3000 solutions

at 328 and 338 K: The curves shown here represent experimental 'X interaction parameters.

Chapter S: Polymer Solutions: Flory-Huggins Model 142 2.0

TEMP,K EXPTL ~ 1. 8 3.33 :.' 3.38 ----- :: 1. 6 ! a::: .: w :' ' Wt- 1.11 ~ < a:::<: 1.2 a.. >< >-1.0 a::: 0_, LL 0.8 .... /. ·" 0.6 ...... ·--·

0.11

0.2

o.o o.o 0.2 0.11 0.6 0.8 1. 0 VOLUME FRACTIONOF POLYMER

Figure 35. Concentration Dependence of the X Interaction Parameter for Water-PEG 5000 solutions

at 333 and 338 K: The curves shown here represent experimental X interaction parame· ters.

Chapter 5: Polymer Solutions: Flory-Huggins Model 143 2.0

TEMP,K EXPTL )( 1. 8 313 :. : 333 .I. 1. 6 .• 0::: . w .•. 1- . w 1.11 :. ~ •. <( . :. 0::: :. <( 1.2 ..• a.. .:. >< / >-1.0 ,•. 0::: ,, g .,·/ LL 0.8 / ,/ ...... 0.6 ...... 0.11

0.2

0.0 ...... o.o 0.2 0.11 0.6 0.8 1. 0 VOLUMEFRACTION OF POLYMER

Figure 36. Concentration Dependence of the X.Interaction Parameter for Water-PEG 6000 solutions

at 313 and 333 K: The curves shown here represent experimental X.interaction parame-

ters.

Chapter 5: Polymer Solutions: Flory-Huggins Model 144 tration dependence. The resulting concentration dependency of Xu for several different temperatures and molecular weights of PEG are shown in Figures 33-36. A nonlinear least-squares routine based on Marquardt's method (obtained from Kuester and Mize (731) was employed to fit the quadratic dependence of the x12 parameter. The procedure for fitting the concentration-dependency of Xu interaction parameter is described in Appendix A, and the FORTRAN code necessary for the calculations is given in Appendix B. The form of concentration dependence of X12 was chosen to be

Notice that even though the results reveal a strong dependence of x.on polymer concentration, cp2 , at high polymer concentration, the curve is fairly constant at moderate to low polymer concentration. This suggests that a constant value for X.12 can be used at low polymer concentrations.

The water activities predicted by the concentration-dependent x12 parameter expressions determined above for several of the binary water-PEG systems are shown in Figures 37-40; the results are shown in Tables 7-10. These expressions necessarily yield better correlations than the constant x parameters. However, if the concentration-dependency of the x.expression is not found with sufficient accuracy, then the resulting solvent activity versus concentration plots may yield unex- pected behavior, such as is demonstrated vividly in Figure 40. The local maximum seen in this graph is just an artifact of the inaccurate fit of the concentration-dependency of Xi2 •

Once expressions for Xu have been obtained, the methods described in Section 5.4.2 can be employed to determine I13 and 'X.23 values for each ternary experimental data point. The resulting

X13 and XZ3 values for different volume fractions can also be fit to an expression quadrai1c in concentration dependence. As described in Section 5.4.2, a 3x3 matrix is solved to determine Xi 3 and

X23 from ternary data and X12. Subroutine ·ocOMP .. was employed to perform LU-factorization on the matrix, and subroutine #SOLVED" was employed to solve the system. The FORTRAN

77 code for these routines is given in Appendix C.

For the case of water-PEG6000-dextran(D48) at 20°C, the x parameters were determined from the experimental data of Albertsson at several different total compositions of the system. The x

Chapter 5: Polymer Solutions: Flory-Huggins Model 145 1.2

1. 0

0::: 0.8 w ~ 3: LL 0 0.6 ~ > I- (.) TEMP,K EXPTL QUAD.~ <( 0.11 293 • 313 • 333 . -·- 0.2

0.0 ...... o.o 0.2 0. II 0.6 0.8 1. 0 VOLUME FRACTIONOF POLYMER

Figure 37. Water activity in PEG 1500 solutions at 293, 313 and 333 K ns predicted by the modified

Flory-Huggins theory: The experimental data are from Hershowitz and Gottlieb (57).

Chapter 5: Polymer Solutions: Flory-Huggins Model 146 1.2

1.0 •...... --···~·1·······Ii o.e • ~ > ut- <( 0.6 ~ w ~ 3: TEMP,K EXPTL QUAD. X 0.11 328 • I I. • .I . .I I 0.2 •

I •I

0.0 ...... 0.0 0.2 0.11 0.6 o.a 1. 0 VOLUME FRACTIONOF POLYMER

Figure 38. Water activity in PEG 3000 solutions at 328 and 338 K as predicted by the modified Flo- ry-Huggins theory: The experimental data are from Malcolm and Rowlinson (81).

Chapter 5: Polymer Solutions: Flory-Huggins Model 147 1.0 • ...... A ....········• I ...... _ ·,. ,...... 0:::o. 8 w ..... ~ ... 3: .. LI.. .I 0 0.6 ~ I > •. •I t; TEMP.K EXPTL OUAD.x <( 0.11 I •I .333 • I. .338 '.: • I. .I •I 0.2 •.

o.o.... ____ _..._..._...... __...... ______...... o.o 0.2 0.11 0.6 0.8 1. 0 VOLUME FRACTIONOF POLYMER

Figure 39. Water activity in PEG 5000 solutions at 333 and 338 K as predicted by the modified Flo-

ry-Huggins theory: The experimental data are from Malcolm and Rowlinson [81).

Chapter 5: Polymer Solutions: Flory-Huggins Model 148 1.2

1.0

······--·--······,•., . •• a::o.e • \ w • .'I. ~ • 'I... := .I .I LL. \ 0 0.6 I .I ~ ~ I. > ~. ti TEMP.K EXPTL QUAD.X . <( 0.11 ..I I 313 • I .I 333 • ---- - •I .I .I .I 0.2 .•

o.01...,...... ,...... ,...... r-- ...... ,. o.o 0.2 0.11 0.6 0.8 1. 0 VOLUME FRACTIONOF POLYMER

Figure 40. Water activity in PEG 6000 solutions at 313 and 333 K as predicted by the modified Flo- ry-Huggins theory: The experimental data are from Herskowitz and Gottlieb (57).

Chapter 5: Polymer Solutions: Flory-Huggins Model 149 interaction parameters (for each system composition) resulting from the procedures described above are shown in Table 11. These parameters were obtained using a constant value of X12 =0.45 since from Figure 36 it is seen that X12 appears to be approximately constant at low polymer concentrations, and the systems being modeled are typically comprised of small concentrations (percent by weight) of polymer. Notice that the x parameters are relatively constant over the region of low polymer concentration. This use of a constant value of the x12 parameter was done to simplify the computational procedure. Notice also that since the concentration range of the ternary data is much smaller than that of the binary data, the correct concentration dependence of Xi 3 and XZJat high polymer concentration could not be predicted from these ternary data. Thus, use of a constant x for the mid-to-low polymer concentration regions is probably more correct anyway. There is another problem. Since the X12 parameters are a strong function of

Supposing that the ternary data were available over a wider concentration range, then the data,

such as presented in Table 11, could also be fit to a polynomial in polymer concentration, like was initially done with Xi 2 • Also the fit of the x parameters to temperature, concentration and molecular weight could be found using a modification of the "CHIFIT" routine (described in Appendices

A and B), given a relation between these variables and the interaction parameter, assuming of course that Xu is first independently determined as such from binary data. For example, for the concentration and temperature dependence, a suitable relation is

where

~o = ~0.1 + ~0.2 ~

Chapter5: Polymer Solutions: Flory-Huggins Model 150 Table 11. Experimental X Parameters for the System of Water-PEG6000-Dextran(D48) at 200C

Vol. Fract. Vol. Fract. X.12 X.13 X.23 PEG Dextran

0.0308 0.0372 0.450 0.514 2.91 0.0296 0.0424 0.450 0.515 3.52 0.0319 0.0439 0.450 0.514 5.68 0.0369 0.0525 0.450 0.522 6.96 0.0419 0.0594 0.450 0.528 8.09 0.0491 0.0716 0.450 0.538 7.28 0.0593 0.0838 0.450 0.548 8.98

Average 0.526 6.20 Standard Deviation 0.013 2.28

Chapter 5: Polymer Solutions: Flory-Huggins Model 151 5.6.2 Prediction of LLE in Ternary Polymer Solutions Using the Modified

Flory-Huggins Theory

This Section discusses the application of the chemical potential expressions given in Equations

(5.31) through (5.33) to the prediction of the liquid-liquid phase behavior of the ternary solvent- polymer-polymer systems discussed earlier. Substitution of these relations into the equilibrium equations

and reduction of the six. Xii to three by use of Equation (5.34), yields a set of 3 equations and 6 unknowns; the unknowns being the volume fractions. Insertion of the mass balance for each phase into these relations simplifies the system to 3 equations and 4 unknowns. Furthermore, phase rule considerations require the specification of one of the remaining 4 volume fractions in order to solve the system. However, as mentioned before, this nonlinear system of equations cannot be solved analytically. Numerical methods must be employed for the solution of this system. The use of a nonlinear equation solver called "HYBRD" in the MINPACK software from Argonne National

Laboratories 145]was chosen to solve the aforementioned system of nonlinear equations. The

*HYBRD" routine is based on a modification of the Powell hybrid method [ 102]. The purpose of the routine is to find a zero of a system of N nonlinear functions in N variables by minimizing the sum of the squares of the functions.

Guffey and Wehe [49] tested six different optimization techniques for the solution of the nonlinear system of equations for the liquid-liquid equilibrium problem based on Black's (modified van

Laar) equation and the NR TL equation. Of those tested, they found that the Powell equation solving routine was by far the quickest and most reliable method in terms of convergence. Guffey and Wehe found a complication in the solution of the equilibrium equation due to the existance of a trivial solution, which like the actual equilibrium solution, satisfies the equilibrium relations.

The problem as set up by Guffey and Wehe is:

Chapter 5: Polymer Solutions: Flory-1 luggins Model 152 a/ = a( or

Ri = (a{ - a() = 0 and

OBJECTIVE FUNCTION = I,(R/ i

The chemical potential or objective function surface profile is seen as flat with two narrow, steep walls; one at the trivial solution and one at the optimum (actual equilibrium solution). Thus, in order to avoid the trivial solution, the search must be started close to the desired solution. This is sort of "cheating" and would prove difficult for predicting unknown equilibrium.

Thus, Guffey and Wehe chose to modify the objective function so that the trivial solution would not represent an optimum solution. In order to do this, they introduced a constraint factor into the objective function by dividing it by a penalty function as follows:

(a{ - a() (x/ - x() 4

Thus, as the trivial solution is approached, the equilibrium function R1 becomes large since x;' is approaching Xi" , and this solution is therefore not an attractive one. They also tried several other penalty functions, but this particular one proved most useful. This penalty function c:ssentially changes the surface profile and erects a high barrier around the trivial solution without actually changing the location of the desired solution. The fourth degree exponent of the penalty function serves to make the objective function very large near the trivial solution because the denominator is approaching zero much faster than the numerator. Hsu and Prausnitz (591suggest using an exponent of 2 or 4. One must be aware that the inclusion of the penalty function can introduce a local minimum in the objective function. Optimization techniques such as the Powell search rou-

Chapter S: Polymer Solutions: Flory-Huggins Model IS3 tine have a problem of "hanging up" on local minimums or maximums. It has been suggested that

"homotopy" methods are more adequate for the typical phase and chemical equilibrium problems often studied in chemical engineering, but the use of these methods was not investigated in this study.

Also, Guffey and Wehe suggest that the introduction of this penalty function alleviates the problem of providing a starting point near the desired solution. Hsu and Prausnitz [59), on the other hand, found that a guess reasonably close to the solution was essential for convergence (or a successful search). Titls is also the case found in the work conducted here. The fact that Guffey and Wehe found otherwise was probably due to the particular form of the activity coefficient equations used, and hence they were lucky.

Recall from earlier discussions (Section 5.3.3) that Hsu and Prausnitz (59) presented a computational procedure for predicting phase equilibrium in ternary solvent-polymer-polymer solutions based on the Flory-Huggins theory. They used the Powell search routine suggested by Guffey and

Wehe. The same procedure was used here with the ..HYBRD" subroutine of the MINPACK software package. Recall that ..HYBRD" is also based on the Powell search routine. The code is not described here, for details see Powell [102,103).

The use of the "HYBRD" code requires a set of N nonlinear functions in N unknowns. The

set of nonlinear functions fi to be solved are the equilibrium equations

(Aµj' - Aµ() RT where the chemical potential expressions [:*]of Equations (5.31) through (5.33) are used.

Due to the computational complication arising from the existence of the trivial solution, the nonlinear functions have been modified by the introduction of the penalty function proposed by

Guffey and Wehe [49) as

(Aµ{ - Aµ()/RT (q>{- (i

ChapterS: Polymer Solutions: Flory-HugginsModel 154 where the denominator is the penalty function and r is a positive, even integer. The value of r must be chosen large enough so that the denominator approaches zero faster than the numerator when q:,;'approaches q:,( near the trivial solution. The value of r should not be too large as this will cause the F1 to become too large. Hsu and Prausnitz (59) have suggested a value of r= 2 or 4.

Hsu and Prausnitz also suggest that a separate constant penalty function should be assigned to the nonlinear function pertaining to the solvent, F5 • The penalty function above is used only for components whose composition in both phases varies during the search process. They chose to use a constant which would make the magnitude of F5 comparable to F1 and Fz. They chose to calculate the solvent penalty function as the average of the penalty functions of F 1 and F z evaluated at the starting point, so it is written as

where the subscript zero represents the starting point (or initial guess).

Hsu and Prausnitz also suggest that since the Powell search, in nature, performs a least-squares regression (minimizing the sum of the squares of the objective function) then all of the Fi should be scaled to the same magnitude. For example, the chemical potential of the polymer in the solution is greater than that of the solvent by a factor of x, the segment number. Hsu and Prausnitz therefore suggest that instead of using the chemical potential of the polymer in the calculations, it is divided by the segment number x and the resulting chemical potential per unit segment is used in the calculations. They also suggest that scaling of the q:,1 is important, but this must be done within the software.

There is another problem which must be corrected. The search routine often yields negative values of q:,1 when the value of q:,1 is close to zero. Hsu and Prausnitz suggested that when this happens, the chemical potential is arbitrarily set to zero and the F1 are multiplied by an additional penalty function, exp( - lOOq:,1), to make q:,1 positive.

The FORTRAN code used to access the MINPACK subroutine 'HYDRO" is given in Ap- pendix D. An example which illustrates the use of the code is given in Appendix A. It should be

Chapter 5: Polymer Solutions: Flory-Huggins Model 155 noted, however, that there are several problems with applying the Powell search routine to the liquid-liquid equilibrium systems being studied. First, there are all of the modifications just described that must be included. Then, the "HYBRD'" parameters must be varied along with the initial guess in order to obtain convergence. The sensitivity of the code to the initial guess and the "HYBRD" parameter called "FACTOR" are the two most noteworthy problems. In general, the convergence tolerance had to be set at 10-7 in order to obtain convergence. For one of the cases, the initial guess was varied and at a tolerance of 10-7 , the code converged to different solutions corresponding to different initial guesses. Toe overall feeling of the author is that the code has too many little intricacies to render itself practically useful for this research.

Toe x parameters given in Table 11 were used with the Flory-Huggins model for predicting the phase behavior of the water-PEG6000-dextran(D48) system at 20°C. The results from this prediction are compared with the experimental phase equilibrium data of Albertsson 13)in Figure

41. Notice in this Figure that the predicted curve appears at lower concentrations than the experimental curve. Recall from previous discussions that this behavior is typical of the Flory-Huggins theory. The actual comparison is good considering all of the inadequacies present in the Flory-

Huggins model. However, as previously mentioned, these solutions were difficult to obtain due to intricacies in the computer code. For example, some of the binodal points plotted in Figure 41 required 20-30 runs before convergence was obtained. The manipulated variables were primarily the initial guess and the "FACTOR* parameter of the "HYBRD" routine. Toe most discouraging fact was that the experimental data were used as the initial guess without yielding convergence.

The initial guess and/or "FACTOR" would then be varied to produce convergence. Many times though this procedure numerous trials. Therefore, it is concluded that for the quality of fit of the data by the Flory-Huggins model, the computer algorithm used is inefficient. Hence, no further comparisons were made with this procedure.

Chapter5: PolymerSolutions: Flory-Huggins Model 156 0. 12

c., W 0.10 a.. u... 0 0.08 i z ' LEGEND 0 \\ 1-- \ 0.06 '\ u \ \ EXPERIMENTAL

Figure 41. Comparison of experimental binodal curve with Flory-Huggins predictions for

Water-PEG6000-D48 at 20°C: The experimental data are from Albertsson [3]. The

Flory-Huggins parameters used are: X 12 = 0.45, X13 = 0.526, and X2 3 = 6.20.

Chapter 5: Polymer Solutions: Flory-Huggins Model 157 5.7 Summary: The Flory-Huggins Theory in Perspective

Perhaps the current state of polymer solution theory and the classic Flory-Huggins treatment is best stated by Casassa 122). The points made by Casassa are as follows. First, the classical lattice treatment of Flory and Huggins has proven to be quite useful for modeling solvent-polymer solutions exhibiting UCST behavior. Furthermore, if UCST systems were the only systems of interest, minor modifications could be made to the classic theory to render it sufficient for modeling these UCST systems. However, discoveries such as the existence of LCST's, coupled with the inadequacies of the Flory-Huggins model (such as concentration dependence on x) observed over the years, have undoubtedly demonstrated the need for fundamental revisions in the theory.

The newer theories, such as the Flory-Prigogine-Patterson equation of state theory, which have addressed the non-combinatorial entropy of mixing (and hence x,), have proven useful for overcoming the inadequacies of the classical Flory-Huggins model.

There is another key point, as made by Flory 138) and Casassa 122). Future progress in polymer solution theory may require a sacrifice in terms of less generality and more specific molecular assumptions regarding the systems being studied. Casassa suggests that current theory may actually be at this "optimum balance".

Chapter 5: Polymer Solutions: Flory-Huggins Model 158 6.0 Polymer Solutions: Other Models

Chapter 5 discussed polymer solution theory and the classical treatment of Flory and Huggins.

This Chapter discusses several alternative models for solvent-polymer solutions. First, the semi-

empirical equation of Heil is discussed. Next, the very promising theory of Ogston is examined.

Then, the Flory equation of state theory is discussed. Also, a brief mention is given to the solubility

parameter approach.

6.1 Heil's Segment-Interaction Equation for Polymer Solutions

In 1966, Heil and Prausnitz [52,53) proposed what they called the "segment-interaction

equation" which makes use of the local composition concept of Wilson (143), and which (like

Wilson's model) contains two adjustable parameters per binary pair. The model was designed for

solvent-polymer solutions and as such it meets the following conditions, as stated by Heil [52,53), required to adequately describe phase equilibria in polymer solutions:

• the model gives adequate variation of chemical potential (or activity) with polymer concentration, • the equation contains only a small number of adjustable parameters -- which are uniquely defined and have an approximate physical significance,

• the expression is able to predict equilibria for polymer-solvent systems with the parameters obtained from binary data, and no new parameters are required for representing multicomponent phase-equilibrium data,

• the expression is thermodynamically consistent and has general applicability for solutions of linear polymers regardless of relative sizes of components or the nature of interactions between them, and

• the model satisfies the criterion for phase splitting.

As proposed by Heil 152,53)the "segment-interaction equation" is:

Chapter 6: Polymer Solutions: Other Models IS9 (6.1)

where the local volume fraction for like molecules is given by:

~ Vj !;,ii= -N------(6.2) L xk vk exp [ - (gik - gii)/RT] k=l and the local volume fraction for unlike binary interactions is given by:

xi vi exp [ - (gii - gii)/RT] (6.3) N L xk vk exp [ -(gik - gii)/RT] k=t

and where s1 is the number of segments per polysegmentcd molecule i.

Talcing the appropriate derivative of Equation ( 6.1) yields the following relation for the relative chemical potential of the solvent

:; = St 1n[X1 :1Ax2] + s1A[ X1 2Ax2] - s2B[ Bx1x~ X2] + (l - St) 1n[ X1Vt ] + (l - S1)[ X2V2 ] + (S2 - l)[ X2Vt ] XtVt + X2V2 X1Vt+ X2V2 XtVt + X2V2 (6.4) s1AG 1x1x2 s2BG2x2 s2B2G2x1x2 + ---'-...a-~;_ + --':;...... ;:;....=....- (x1 + Ax2)2 (Bx1 + X2) (Bx1 + x2)2 where

(6.5)

B = : 1 exp( - [g12 - g22J/RT) (6.6) 2

(6.7)

Chapter 6: Polymer Solutions: Other Models 160 (6.8)

The parameters required for the Heil model are the interaction differences (g;1 - &1) and

(g;1 - ~;) which represent binary interactions between molecules i and j. This model has been tested for systems containing polymers of the following types: polystyrene, polyglycols, polyethers, rubber, polypropylene and cellulose derivatives. The results were in very good agreement with the Heil model, giving a better representation of data than the Flory-

Huggins model (52,53J.

A limitation of this model exists in regard to generalizing the model to include more than one polymer species. As stated by Heil, "In principle, the treatment applies with complete generality to ternary systems composed of one or more polymer species regardless of molecular weight.

However, the analytical form of the relations between liquid compositions in the two phases is sufficiently cumbersome to render impractical the application to such a completely general case"

(52,53J. Again, for the general case of multicomponent systems of more than one polymer, numerical techniques must be employed for solving the phase equilibrium equations; an analytical solution is not always possible.

In their discussion, Renon and Prausnitz (114) give a generalized expression for the excess

Gibbs free energy which encompasses the Wilson, Heil, and NR TL models. In the generalized form, as shown in Chapter 4, the model by Heil is seen to be a combination of the Wilson and NRTL models, where the equation used is a simplification of the Heil model for describing solutions of small and equal-sized molecules. Knowledge of this leads one to believe that the Heil model is quite suitable for polymer solutions, in that it has both an entropic part (sterruning from the Wilson model) and an enthalpic part (stemming from the NRTL part). Even though neither the Wilson or NRTL model alone is applicable, their combination in the form of the Heil model seems rather attractive and useful. The reason for suggesting their inability to represent polymer solutions is that the Wilson model appears to be largely an entropic model (since in fact it is based on the athermal Flory-Huggins model), and this lack of a separate enthalpic contribution may be one factor in its inability to predict phase splitting. Also, the NRTL is largely an enthalpic model,

Chapter 6: Polymer Solutions: Other Models 161 with no representation of the combinatorial entropy;however, it does have an empirical parameter to take into account the nonrandomness of mixing.

Heil (52,531presented interaction parameters for the water-polyethylene glycol systems shown in Table 12. These parameters were used to predict water activities in solutions of PEG molecular weights 3000 (at T= 55°C) and 5000 (at T= 60°C and 65°C). These results are shown in Figures

41-43. Two curves are plotted on these figures; they represent different values for the ~: ratio.

The value cited by Heil (as shown in Table 12) was compared with a value calculated from liquid densities and molecular weights as given in Tables 4 and 5 of Chapter 3. The predictions resulting from both values are shown since they are different and should therefore yield different results. This comparison was necessary since Heil did not reference how he determined these values. From

Figures 41-43, the Heil model is seen to correlate the data quite well, especially using Heil's ~: ratio. Also, the parameters were determined from the experimental data shown in these plots, so that the test being made is a determination of the correlative ability of the Heil model for phase equilibrium data of polar solvent-polymer systems.

6.2 The Theory of Ogston

Another approach to modeling aqueous-polymer two-phase systems was proposed in 1962 by

Ogston (94J. Ogston had observed an interesting phenomenon regarding solutions of large mole-

cules as early as 1937 [93). He observed apparent excesses of osmotic pressures of mixed solutions

of proteins. At the time of this observation, Ogston was curious as to whether the nonideal interactions which these results suggested might be of importance in systems of macromolecules, proteins in particular, in high concentrations. He reasoned that these interactions, under the right conditions, might lead to phase separations -- which might account for phase heterogeneities commonly observed, for instance, in the cytoplasm of cells. This phenomenon, however, was not investigated further at that time. Then in 1961, Ogston and Phelps (95J obtained results on the

nonideal partition of solutes between solutions of hyaluronic acid and buffer solutions; results

which stimulated a new attack on this problem.

Chapter 6: Polymer Solutions: Other Models 162 Table 12, Water-PEG Systems Correlated with the Heil Model Using Experimental Data from Mal- colm and Rowlinson 181)

Polymer V2 V1 Temp. Wt. Fract. [gl2 - gll] [g.2 - g22J PEG MW (OC) Cone. Range (cal/gmol) (cal/gmol)

3000 55 0.50-0.97 167.7 839.9 -342.5 5000 60 0.74-0.95 279.5 837.7 -328.3 5000 65 0.50-0.99 279.4 855.2 -323.5

Chapter 6: Polymer Solutions: Other Models 163 1.0 ,------, • ..... 0.9 ----...... --.., ...... , 0.8 ...... , ... •'\ ~ ' '' \ > o.7 ..... ' \ ' \ \ \ ~ \ \ ~ \ w o.s \ \ '\ ~ \ ~ 1 o.s LEGEND EXPTL • V2/V1 = 137 .9 0.11 V2/V1 =167.7

0.3,._..._...... ,,....,...,...... - ...... r-.--.--.-...-.,-,....,--,.--.,. 0.11 o.6 0.8 l. 0 WEIGHTFRACTION OF POLYMER

Figure 42. Water activity in polyethylene glycol solutions ofMW=3000 at T=SS°C correlated by the

Heil model: Points represent experimental data from Malcolm and Rowlinson (81).

Curves represent correlation through the Heil model; parameters determined by Heil [S2) from above data.

Chapter 6: Polymer Solutions: Other Models 164 1. 0 •

0.8

~ > 0.6 ut- <( 0:::: w 0.11 <3';

LEGEND EXPTL • o.z V2/V1 =225.9 V2/V1 =279.4

o.o..,_...... __ ...... - ...... -...... - ...... -...... - ...... ,_...... ,...... ,.....,, 0.11 0.6 0.8 1. 0 WEIGHTFRACTION OF POLYMER

Figure 43. Water activity in polyethylene glycol solutions of MW=SOOO at T=600C correlated by the Heil model: Points represent experimental data from Malcolm and Rowlinson [81].

Curves represent correlation through the Heil model; parameters determined by Heil (52]

from above data.

Chapter 6: Polymer Solutions: Other Models 165 •...... , ...... , 0.8 ',,... , , ..,, ', ~ ' ' ', > \ t- (.) ' ' \ <( 7 ' o. \ ~ •' ,. w ' \ ,. \ \ \ ~ \ ~ \ \ \ \ \ LEGEND \ \ 0.6 \ EXPTL \ • I \ I V2/V1 =226.5 \ V2/V1 =279.5 I '~ 1

o.s,,,_.,.....__,.....,...... ,.....-,-...,.....T-'T-..- ...... ,...... ,...... ,...... ,...... ,...... ,....,P"""'f 0.7 0.8 0.9 1. 0 WEIGHTFRACTION OF POLYMER

Figure 44. Water activity in polyethylene glycol solutions of MW= SOOOat T ==6S°C correlated by the

Heil model: Points represent experimental data from Malcolm and Rowlinson [81].

Curves represent correlation through the Heil model; parameters determined by Heil (52) from above data.

Chapter 6: Polymer Solutions: Other Models 166 In his new approach, Ogston thought that the task of predicting phase behavior in ternary solutions of the complex macromolecules could be eased significantly if generalized algebraic expressions for the activity coefficients could be written in forms consistent with required relations as given by classical thermodynamics, that is, relations that satisfy the Gibbs-Duhem equation. In his work of 1962, Ogston (94) showed that such generalized, simple expressions do indeed exist.

Ogston chose to apply the equations to predicting the osmotic pressure. The osmotic pressure is a useful and pragmatic choice for comparison of experiment and theory because the osmotic pressure of a solution is the primary colligative property (a directly measureable property proportional to the relative chemical potential) that is experimentally accessible for most macromolecular solutions [ 117). Also, osmotic pressure measurements were probably the most accurate and reliable measurements available at that time.

Ogston's theory has been applied to various aqueous-polymer two-phase systems by Laurent and Ogston (76); Preston, Davies, and Ogston (110); Nichol, Ogston, and Preston (92); Edmond,

Farquhar, Dunstone, and Ogston (29), Edmond and Ogston [30]; Wells (142]; and Foster, Dunnill, and Lilly (39).

Ogston's method is essentially an equation-of-state approach where the relative chemical potential of component 1 in a binary solution is written in terms of a virial expansion, as follows

- RTV •1 p [ -M1 + B p + C p 2 + ...J (6.10) 2 2 2 2

where µ1 is the chemical potential of the solvent, µ1 ° is the standard state chemical potential of the solvent, T is the system temperature, V1• is the molar volume of the solvent, p is the mass con- 2 centration (mass per unit volume) of component 2, M2 is the molecular weight of the solute

(component 2), B is termed the second virial coefficient, and C is termed the third virial coefficient. The values of B and C depend upon the nature of the solution. However, statistical mechanical treatments suggest that B, the second virial coefficient, is a measure of binary interactions and that C, the third virial coefficient, is a measure of ternary interactions.

Chapter 6: Polymer Solutions: Other Models 167 6.2.1 The Virial Expansion

The virial expansion is a simple expression that can be derived as follows .. First, from arguments presented in Chapter 2, the chemical potential of the solvent in a binary solution ( I = solvent,

2 = solute) can be written as

( 6.11)

Now, recalling that for an ideal dilute solution (see Equation (2.33a) of Chapter 2, Section 2.1.S), the solvent is referenced as follows

* y1 -+ 1 as x1 -+ 1 (6.12)

So that if the solution is dilute in solute (component 2) then the chemical potential of the solvent is given by

(6.13)

and for a pure solvent, x1 = 1, so that

(6.14)

As solute is added, the relative chemical potential is no longer zero. The effect of the solute is to lower the chemical potential of the solvent and it is convenient to express this difference as a power series in the concentration of the solute ( 117). From above, the relative chemical potential of the solvent is RT In x1 , and since (x1 = I - x2) for a binary solution, the relative chemical potential of the solvent is also RT In ( 1 - x2) •

Now, expanding the logarithm term as a power series in x2 yields

2 3 4 X2 X2 X2 -x ------··· (6.1S) 2 2 3 4

Chapter 6: Polymer Solutions: Other Models 168 Finally, noting that for a dilute solution, xl = PlMV1 • , then the virial expansion of the relative l l chemical potential of the solvent in an ideal dilute solution is obtained as

• p (6.16) - RTV 1 ~+[~JP+···] 2 [ 2 2M2 2

Since a real solution will approach ideal solution behavior at high dilutions, they both tend to the same limit, and at this limit the concentration dependence of the chemical potential is the same for both cases. Thus, Equation (6.16) can be generalized to apply to any solution as

-RTV;p +Bp +Cp 2 + ... ] (6.17) 2 [Ml l 2 2 2 where as previously stated B is the second virial coefficient and C is the third virial coefficient.

6.2.2 The Osmotic Pressure Relationship

The osmotic pressure relationship is important in that it relates meaningful thermodynamic parameters to experimentally measureable properties. The setup of an osmotic pressure experiment is very simple; it has two compartments that are separated by a semipermeable membrane which will allow solvent molecules to pass freely through it, but it is impervious to the much larger molecules of the macromolecular solute. The osmotic pressure required to stop the passage of solvent through the membrane and thus establish equilibrium across the membrane is easily measured. The osmotic pressure, n, is thus readily obtained experimentally and can also be pred~.:ted from theory by the following relation

n = = RT p [- 1- + B p + C p2 + ... ] (6.18) 2 M2 2 2

Nichol, Ogston, and Preston (921chose to express the activity coefficient as a virial expansion of the form

Chapter 6: Polymer Solutions: Other Models 169 3 In y = M ( a 1 c + a2 c2 + a3 c + ...) (6.19)

where a 1, a 2,a3, ••• are constant coefficients, Mis the molecular weight, c is the concentration, and y is the activity coefficient. Their expression for the osmotic pressure was

.lL = (6.20) RT

This is not one of the usual forms of the virial expansion of the osmotic pressure (see Tompa ( 1321,

Equations 6.4 to 6.6), but it is closely related as the coefficients a 1, a 2,a3, ••• are proportional to second, third, etc. virial coefficients. Edmond, Farquhar, Dunstone, and Ogston (291use this same relation, but they truncate concentration terms of fourth order and higher. Edmond and Ogston

(301chose relations of the type

(6.21a)

(6.21b)

(6.21c)

where m2 and m3 are molalities of solutes, and c, d, and a are constant coefficients. The last

equation (Equation (6.21c) was determined from the first two (Equations (6.21a) and (6.2lb)) by application of the Gibbs-Duhem equation. These equations are simplified forms of the general ones derived by Ogston (94J. As pointed out by Edmond and Ogston (30), Equation (6.21c) is es-

sentially the osmotic pressure equation (for comparison, see Equation (6.18)). On inspection of

Equation (6.21c), the coefficients c and dare seen to be equivalent to second virial coefficients of

components 2 and 3, and the coefficient a is seen to describe the thermodynamic interaction be-

tween macromolecules 2 and 3 (but on the molal scale of composition). Edmond and Ogston [30)

also discussed the "equivalency" of the expressions of Equations ( 6.2 la-c) to the Flory-Huggins and

Flory "dilute-solution" theories.

Chapter6: Polymer Solutions: Other Models 170 6.2.3 Example of Ogston's Theory

Edmond and Ogston (30) investigated some ternary systems of nonionic aqueous-polymer solutions with the objective of determining the applicability of a simple thermodynamic model (discussed earlier), which is admittedly oversimplified for accurately modeling complex biological systems. The study of these systems aimed to provide a simple framework for which more complex quaternary systems containing polyelectrolytes could be modeled. This method also provides a basis for estimating thermodynamic interaction parameters between polymers which exhibit incompatibility.

An approach of the sort described above is the most useful way to begin a study of complex. systems. That is, a model is proposed which is admittedly oversimplified for representing the system. When this model is applied to the experimental data of the complex system, the inadequacies of the model, and their corresponding effect on representation of the data by the model, can be studied. This should then lead to modification of the theory to better correlate or explain the data.

A good example of this type of an approach is the classical Flory-Huggins treatment of polymer solutions. As stated by Casassa (21), in regards to the work of Paul J. Flory and the approach mentioned above, "Like most germinal scientific developments, the theory is basically very simple.

It suffers, as was recognized from the beginning, from serious inadequacies; yet it embodies ideas that remain central in solution theory."

6.2.3.J Procedure for Application of the Ogston Theory

Edmond and Ogston (30) studied various systems containing water, polyethylene glycol, and dextran. Their use of Ogston's theory (94) for comparison with experimental data, such as that of

Albertsson (3), will now be discussed. As discussed in Chapter 2, comparison of binodal curves from experiment and theory affords the best test of the theory. Thus, this approach will be taken.

In order to apply the Ogston theory to the prediction of binodal curves, the equilibrium equations

Chapter 6: Polymer Solutions: Other Models 171 (6.22)

must be combined with the chemical potential expressions, i.e. Equations (6.2la-c). In the equations above, the single primes denote one phase, while the double primes denote the other phase, and the subscripts 1, 2, and 3 represent the solvent and two macromolecular solutes, respectively. Edmond and Ogston (30) substituted Equations (6.2la-c) in the above equilibrium relations. Then, they combined these expressions in such a way as to eliminate two of the concentration variables ml' , m/ , m/ , m3• , leaving for example, a relationship between m/ and

H, m3 •

_g_+ (m , - m 1 + cQ2 [ ep + 1 + .£.(m , 2 - m • 2) a J J 2a2 eP _ l 2 J J 1 (6.23a)

+ P Q (m3' ep - m31 = 0 e - 1 where

p = ..£..1n[m/ + cd - a2 (m , - m " (6.23b) a m3· l a 3 3 '

(6.23c)

and

(6.23d) a(ep - I)

To use these equations, one chooses a suitable value of, say, m/ , and calculates the value of m3" that satisfies Equations ( 6.23a-c) (30J. This can be achieved by guessing a value of m3" , and using an iterative scheme such as the Newton-Raphson method to converge on the value that satisfies Equations (6.23a-c). Then the corresponding values of m/ and mt can be obtained from

Chapter 6: Polymer Solutions: Other Models 172 Equation (6.23d). This procedure is demonstrated in Appendix E, and the computer code for performing the iterative calculation is contained in Appendix F.

6.2.3.2 Use of the Ogston Theory for Prediction of the Phase Behavior in a Water-PEG-Dextran

System at 20°C.

Since Edmond and Ogston (30) did not present or reference their experimental data, the model parameters (a,c,d) they obtained will be used to predict binodal curves for comparison with similar experimental data of Albertsson (3). 11tls will not be a totally quantitative comparison since

Ogston's parameters were determined from systems that were slightly different than those of Al- bertsson. The assumption being made here is that the coefficients are constant for the slight differences existing between the systems being compared.

The system being used in this example is a solution of water, polyethylene glycol (PEG6000), and dextran (D17 or Dextran 19.7). Comparison of several characteristics of the polymer species involved in the experimental and theoretical systems is shown in Table 13. Of particular interest

is the weight-average molecular weight, Mw ; the number-average molecular weight, M0 ; the polydispersity number, which is the ratio of the weight-average to number-average molecular weights; and the intrinsic viscosity. The polydispersity number is important as it represents the dispersity of the molecular weight distribution. A polydispersity number near one signifies a very narrow molecular weight distribution (monodispersity). Recall from earlier discussions that it is most beneficial to work with polymers having a narrow molecular weight distribution; otherwise, the polymer species need to be modeled as a multicomponent polymer mixture.

From Table 13, it is evident that the two systems being compared are not totally cc,nparable.

It is safe to say that the dextran species are very similar; however, this is not so for the polyethylene glycol species. The result of this may not seem apparently obvious, but examining the effect of molecular weight on phase behavior, such as shown in Figure 44, from the data of Albertsson, the following observations can be made. Starting with a molecular weight of dextran that is similar to that of PEG, and increasing it until it is very large, it is seen that the effect of the molecular weight of dextran tends to become less important at very high molecular weights. A generalization can

Chapter 6: Polymer Solutions: Other Models 173 be made here. At low molecular weights of the polymers, an accurate characterization is essential, but at very high molecular weights, the need for a very precise characterization of the molecular weights is not so great. Thus, it is suspected from the beginning that the difference between the molecular weights of polyethylene glycol as is shown in Table 13 is of substantial importance.

These observations are in qualitative agreement with the theoretical findings of Scott [ 120) and

Tompa ( 131). Another observation is that as the molecular weight of dextran increases, the critical point (or plait point) shifts toward lower concentrations of PEG, and it would be expected that if the dextran molecular weight was held constant and the PEG molecular weight varied, the critical points would shift toward lower dextran concentrations with increasing PEG molecular weight.

Comparison of the two systems described in Table 13, using the coefficients a= 580, c = 320, and d = 520 of the Ogston theory as determined by Edmond and Ogston (30) for the theoretical system described above, yields the results shown in Table 14 and Figure 45. On comparison of these curves, one might note that they appear to be quite similar. Closer inspection reveals that the plait point (critical point) of the theoretical curve is shifted more toward the dextran axis. Since the

Ogston parameters were determined from the critical point (30), this should be the best point on the curve for comparison of the system from which the parameters were determined with the experimental system of Albertsson. The fact that the critical point of the theoretical system of Ogston is shifted toward lower concentrations of PEG suggests that the molecular weight of PEG in the system used for determination of Ogston parameters was greater than that in the experimental system of Albertsson. Indeed this is the case, and the comparison being made can now be better as- sessed. If a slightly lower molecular weight of PEG had been used in the system used to obtain

Ogston parameters, the critical point would shift away from the dextran axis toward higher concentrations of PEG and most likely yield a better correlation. In any respect, the predictive ability of the Ogston theory must be judged impressive.

Chapter 6: Polymer Solutions: Other Models 174 Table 13. Characterization of Polymers for Experimental and Theoretical Systems

Theoretical System of Edmond and Ogston [30J

3 Polymer Species 10- M w Polydispersity Intrinsic Viscosity

Dextran 19.7 26.5 19.7 1.35 16.0 PEG 6000 8.0 20.4

Experimental System of Albertsson (3)

Polymer Species Polydispersity Intrinsic Viscosity

D17 30 23 1.3 16.8 PEG 6000 6-7.5

Chapter 6: Polymer Solutions: Other Models 175 0 ...... - ...... -.-.....-"T"- ...... -.....,,.....,.--,..-,-"T""""T"- ...... ,...... ,.....,...... -.,....,,_,. 0 10 20 30 Dextron (,; w/w)

Figure 45. Effect of the molecular weight of dextran on the binodal curves of water-polyethylene gly-

eol-dextran systems.: The molecular weight of polyethylene glycol is held constant.

PEG6000 is used for all systems. The molecular weight fractions of dextran used and their

corresponding curve numbers are as follows (from Albertsson (31):

l DS

Chapter 6: Polymer Solutions: Other Models 176 Table 14. Comparison of Experimental and Theoretical Binodal Curves for the Water-PEG6000-D17 System at 20°C

Theoretical System of Edmond and Ogston (30)

0.82 19.3 24.4 0.40 1.44 15.2 19.5 0.80 2.01 12.9 16.6 1.20 2.59 11.1 14.5 1.60 3.81 8.48 11.1 2.60 4.87 7.00 9.62 3.20 5.78 5.90 7.63 4.40

Experimental System of Albertsson (3)

lQlm,"' 103m3"' 1Qlm2' lQlm/

1.01 15.7 26.7 0.21 1.56 14.8 25.0 0.28 1.64 12.7 21.5 0.45 2.12 11.5 19.5 0.61 2.68 9.64 16.5 0.99 3.74 7.66 13.4 1.67 5.22 6.09 11.1 2.67

NOTE: These data are in molality, which is defined as moles of substance per kilogram of solvent. Albertsson's data were converted from weight fractions to molalities for this comparison (see Appendices E and G).

Chapter 6: Polymer Solutions: Other Models 177 20

...... 18 ..._.,tr) E 16 LEGEND __..,., ___ 0* EXPERIMENTAL 0 0 Ill OGSTONTHEORY • -. 1"')·12 ~ I.LI ::? 10 ~ 0 a. 8 LL 0 6 ~ ::i II ~ 0 ::? 2

0 0 2 II 6 8 10 12 111 16 18 20 22 211 26 28 MOLALITYOF POLYMER2, 1OOO*m(2)

Figure 46. Comparison of experimental and theoretical binodal curves for the system Water(l}-PEG6000(2}-D17(3) at 20°C: Experimental data are from Albertsson [3].

Theoretical predictions via the Ogston theory with the parameters a= 580, c = 320, d = 520

obtained by Edmond and Ogston [30].

Chapter 6: Polymer Solutions: Other Models 178 63 Free Volume Considerations and the Flory Equation-of-State

The lattice theory as employed by Flory and Huggins has provided a good basis for the de~ scription of solution thermodynamic properties of liquid mixtures. However, it has been observed that the excess thermodynamic properties are not adequately described by this simplified lattice treatment. Recall from earlier discussions that this lattice theory assumes no volume change on mixing and that Flory (37) stated that this represents the severest limitation of the Flory-Huggins theory. Also, the combinational entropy of mixing obtained from the lattice treatment often yields excess entropies that differ markedly from those determined experimentally.

The "free volume" concept has been employed to develop a more appropriate theory. Titls concept is based on the fact that different components of a mixture will in their pure form have different free volumes or different degrees of thermal expansion. The free volume effect commonly discussed is actually a "free volume dissimilarity" effect. When components with different free volumes are mixed, the difference in their free volumes contributes to the excess functions. Prausnitz

[ 108) points out that this is especially important for solutions of molecules differing greatly in size and suggests that for a solvent-polymer solution of chemically similar components, where the interactions are similar, the free volume dissimilarity may be large. Note that excess properties predicted by the lattice theory for this case could be in gross error since the lattice theory has no free volume contribution. This is the reason that Oishi and Prausnitz (961 added a free volume correction term to UNIFAC for thermodynamic predictions of solvent-polymer solutions.

Patterson (98,99) used the "free volume dissimilarity" to explain the existence of LCST behavior. This was discussed in Section 5.5.1.2 where it was suggested that this "free volume dissimilarity" effect is represented by x.. Prausnitz (108) introduces several equation of state theories for solutions of large molecules.

Only the Prigogine[l l l]-Flory(381-Pattcrson[98,99J equation is discussed here. Titls theory has been found to be useful in describing polymer solutions. Several reviews of this theory arc available

(21,22,97,108). The theory developed by Prigogine (1111 is essentially a corresponding-states formalism which is based on molecular parameters for chain length, the number of molecular degrees

Chapter 6: Polymer Solutions: Other Models 179 of freedom contributing to thermal expansion, and the cohesive energy between nonbonded segments (22). Flory (38) proposed a similar development for which he modified the usual combinatorial partition function for an x-mer liquid by multiplying the original partition function by terms referring to liquid-state properties of the segments. Flory (38) defines the free volume V, for one molecule containing r segments as

(6.24) where v' is the characteristic or hard core volume of a segment; v= v/v' is the reduced volume; v = V/(N • r) (N is the number of molecules in the total volume); and tis a numerical factor (which disappears in differentiations in obtaining the equation of state and cancels in taking differences with respect to pure components in deriving excess functions for mixtures) [ 108).

The resulting equation of state for a pure component is

Pv __;_1,_J_ - _l_ = (6.25) T ;113 _ vT where the equation is based on the reduced pressure P , the reduced volume v,and the reduced temperature T. These variables are based on characteristic parameters for pressure P', volume v·, and temperature T'. These are defined by Prausnitz (108) as follows. The characteristic or hard- core volume of a polymer segment v· is related to the reduced volume by

V V/(N • r) V = = (6.26) V • V • where v is the volume available to one segment, V is the total volume, and N is the number of molecules which are divided in r segments. lbe reduced temperature T is

* T = = 2v ckT (6.27) 511

Chapter6: Polymer Solutions: Other Models 180 where c comes from the definition of a parameter 3rc which is the number of effective external degrees of freedom per molecule, k is the Boltzmann constant, T is the temperature, s is the number of contact sites per segment, and -ri/v is the intermolecular energy per contact. The reduced pressure is

.2 p 2Pv p = = (6.28) p* Sfl where P is the pressure. Also, the characteristic parameters P', v', and T' satisfy the equation

(6.29)

Two assumptions are employed to extend the partition function to mixtures 1108J:

( 1) Hard core volumes of the components are additive.

(2) The intermolecular energy depends on the surface areas of contact between molecules

and/or segments.

Prausnitz (1081 shows expressions for the residual entropy and enthalpy resulting from the

Flory equation of state approach. From these, the following expression for the residual activity of the solvent is obtained

13 =~ = r;v;[ 3,j. In ;1 - 1 + <;-1_ ;-1>] RT RT 1 "t/3 1 V - I (6.30) + v;[ X12 Jo~+ _L[a6.Vmixing] RT ; RT aN1 T,P,N 2 where subscript 1 denotes the solvent, no subscript denotes the mixture (eg. v), v; denotes the molar hard core volume of the solvent; X12 is the interaction parameter; 8 is a surface fraction defined as

(6.31)

Chapter 6: Polymer Solutions: Other Models 181 and the last term of Equation (6.30) is negligible at normal pressures. Equation (6.30) can be compared with the residual activity obtained from the Flory-Huggins equation

(6.32) and in comparison of these two equations, it is seen that

X - (6.33)

which shows that x does vary with concentration, as is often observed experimentally.

Also, the interaction parameter, X12, is defined as

(6.34)

Note that this equation of state theory requires the characteristic parameters for the pure components plus the segment surface ratio s2/s1 (used for determining 82) and the X12 interaction parameter. Prausnitz suggests that the segment surface ratio can be estimated from pure component structural data as given by Bondi (13) and as used in UNIQUAC. Prausnitz also suggests that

X 12 be determined from experimental data on enthalpy of mixing (or dilution) or volume of mixing.

These two methods are best suited for determining X12 since they are independent of a combinatorial entropy expression.

Prausnitz concludes his discussion on the Flory equation of state by pointing out that even though this theory is still in an elementary stage, it has been found to overcome the basic inadequacies of the Flory-Huggins theory. As stated before, the Flory equation of state does predict a concentration dependence on the interaction parameter, although the theory overestimates this dependence. Also, the theory can explain negative enthalpies of mixing, which are often observed for

polymer solutions, without involving a negative exchange interaction. The main improvement of

this theory over the Flory-Huggins theory is that it can predict the existence of the LCST in sol-

Chapter 6: Polymer Solutions: Other Models 182 vent-polymer solutions. These improvements have been made without the introduction of arbitrary parameters, except for the usual interaction parameter.

6.4 TJ,e Solubility Parameter

Another simple method for correlating and predicting thermodynamic behavior of polymer

solutions is by the use of solubility parameters or other cohesion parameters. As mentioned in Section 5.4.4, the solubility parameter may be used for estimation of x parameters. For a recent comprehensive discussion of the use of cohesion parameters for polymer solutions, see Barton [ 12).

Also, Olabisi (97) discusses the Hildebrand solubility parameter approach for polymer solutions.

6.5 Summary

Several approaches for modeling polymer solutions were discussed in this Chapter. First, the

semi-empirical model of Heil was discussed and its ability to correlate thermodynamic data for polar

solvent-polymer solutions was demonstrated. However, the Heil equation has been tested only

within the region of high polymer concentrations. Next, the simple theory of Ogston was discussed,

and the impressive correlative ability of it was demonstrated by comparing results predicted with

parameters obtained by Ogston with similar experimental data from Albertsson. Also, the Ogston

theory can be used for dilute solutions since it is based on the virial expansion. Finally, the Flory equation of state approach was introduced. This theory takes into account free volume dissimi-

larities between the components and overcomes many of the inadequacies of the Flory-Huggins

lattice theory. Most importantly, the Flory equation of state can predict LCST behavior, which cannot be predicted by the Flory-Huggins theory.

Chapter 6: Polymer Solutions: Other Models 183 7.0 Solvent-Polymer-ElectrolyteSolutions: A Local Composition

Approach

It is a well-known fact that the presence of salts or other ionic substances can have a marked effect on the phase equilibrium of nonionic mixtures. This Chapter discusses a local composition model that has recently been proposed for modeling solvent-electrolyte systems. This model appeared promising and was examined for the possibility of being modified to model solvent-polymer-electrolyte solutions. This approach was however abandoned early in the research due to the algebraic complexity of the modified equations.

7.1 Chen's Model for Solvent-Electrolyte Systems

A conceptually for solvent-electrolyte systems has recently been proposed by Chen et al. (23).

This model is based on the local composition NRTL model (114) and the extended Debye-Hiickel formulation proposed by Pitzer [ 100). This development is applicable to single solvent, single completely dissociated electrolyte systems. Recently, Chen and Mock (24,25,89) have extended this model to multicomponent systems, including partially dissociated electrolytes which involve disso- ciation equilibria.

However, since the NRTL model does not contain a combinatorial term that is ?resent in models such as Wilson, Heil, and UNIQUAC, it does not apply to polymer solutions where the entropic combinatorial term is important. Chen et al. (23) suggest that in electrolyte systems, the entropy of mixing is negligible in comparison to the enthalpy of mixing, and thus they arc justified in employing the NRTL model. On the other hand, the aqueous-polymer two-phase systems being studied here are quite different. Since these systems contain polymers, the entropy of mixing is probably very important due to the size differences between the polymer molecules, solvent mole-

Chapter 7: Solvent-Polymer-Electrolyte Solutions 184 cules (water), and the ions. In a quantitative analysis, the various contributions to the overall activity would be compared in order to determine the validity of the above assumption.

The model proposed by Chen et al. (23) is composed of two parts: a term based on a modification of the local composition (LC) NRTL model which describes the short-range interactions between various solvent and ion molecules; and a term based on the Pitzer-Debye-Hiickel (PDH) model which describes long-range ion-ion electrostatic interactions. The model by Chen is:

E* [ E* ]PDH [ E* ]LC _g_ = _g_ + _g_ (7.1) RT RT RT where the • signifies an unsymmetric local composition model. Alternatively,

(7.2)

7.2 Pitzer-Debye-Hiicke/ (PDH) Model for Long-Range Interactions

The Pitzer-Debye-Hiickel (PDH) model for long-range interactions, which is normalized to mole fractions of unity for solvent and zero for electrolytes, is as follows:

(7.3)

where

M5 = Molecular weight of solvent Aq, = Debye-Hiickel parameter

2 312 Aq, = j (2rcN0 d/1000) y, (e /DkT) Ix = ionic strength, on a mole fraction basis _ 1 2 l -- "C" Z· X· X 2 t I I where Z = absolute value of ionic charge p = "closest approach"' parameter

Chapter 7: Solvent-Polymer-Electrolyte Solutions 185 and

73 Local Composition Model ( LC) for Short-Range Interactions

The local composition model (LC) for short-range interactions is given by:

(7.5) but the above equation is based on a syrrunetric normalization, hence, it must be renormalized to yield an unsymmetric model. The resulting equation, now normalized to mole fractions of unity for solvent and mole fraction of zero for electrolytes, is:

(7.6) which becomes:

where a stands for anion species, c stands for cation species, and m stands for molecular (solvent) species.

So for the cation species,

LC* In Ye =

Chapter 7: Solvent-Polymer-Electrolyte Solutions 186 and for the anion species,

and for the molecular (solvent) species,

1n LC* 'Ym = (7.10)

where and

7.4 Modification of t/ze C/zen Model for Application to Polymer Solutions

Since many of the industrial applications of aqueous-polymer two-phase systems involve the addition of salts or the use of polyelectrolytes, a model which could easily represent the effect of ions is desired. Furthermore, since the local composition concept has been so useful in the past, even somewhat useful in modeling polymer solutions, the local composition electrolyte approach of Chen seemed attractive. Moreover, based on the discussions in Chapters 2 and 4, it does not seem too difficult to modify the model by Chen. For instance, since the Chen model is based upon an essentially enthalpic local composition model, it seems almost obvious to add to this model an entropy of mixing model such as the Flory-Huggins equation. Another approach is to add the

Wilson model, since it has some entropic character. Since the Heil model can be viewed as a combination of the Wilson and NRTL equations (at least for solutions of small molecules), it could

Chapter7: Solvent-Polymer-ElectrolyteSolutions 187 perhaps serve as a more appropriate local composition model to be used in this modified-Chen approach for solvent-polymer-electrolyte systems.

Thus, this modified-Chen model would look like

(7.11)

where PDH represents the Pitzer-Debye-Hilckel long-range ion-ion electrostatic interactions and

LC represents the local composition model which is generally written as

(7.12)

where the C denotes the combinatorial contribution which describes the spatial configuration of molecules, and R denotes the residual part which represents interactions between all types of molecules (for this case: solvent, polymer, and ions).

Several alternatives are available for describing these two contributions. For instance, assuming the entropy of mixing contribution to be negligible, Chen dropped the combinatorial term in

Equation (7.12) above. The alternatives available for the extension to solvent-polymer-electroly1e are:

1. Use NRTL for the residual part and the athermal Flory-Huggins equation for the

combinatorial part.

2. Use NRTL for the residual part and the Wilson equation for the combinatorial part.

3. Almost equivalent to (2) above, use the Heil model for the LC model since it has both

the combinatorial and residual parts.

4. Use UNIQUAC for the LC model since it has both the residual and combinatorial

parts.

Chapter 7: Solvent-Polymer-Electrolyte Solutions 188 In all of the above models, all molecules (solvent, polymer, and ions) must be treated in each part.

Also, the final equations should have some dependence on polymer molecular weight (or segment number) and the ion charge number. Method (1) above was chosen as an initial test model and was seen to have several problems.

The most misfortunate problem was the algebraic complexity of the model written for a solvent, 2 polymers, a low molecular weight component, anion and cation. Based upon this problem, the

Modified-Chen approach was abandoned early in this research.

7.5 The Empirical Correlation of Adamcovti

Adamcova [2] has studied aqueous-electrolyte-polyethylene glycol systems and has proposed an empirical relation for the activity coefficients. In fact, his activity coefficient model in some re- spects resembles the virial expansion model previously discussed. However, the empirical correlation of Adamcova [21required 10 parameters to adequately represent the experimental data, and the parameters used have no physical significance. This method is mentioned in order to make a very important point. As the familiar saying goes, "We could fit a camel's back if we had enough parameters!" The key is to represent phenomena with a few parameters that have physical significance.

7.6 Thermodynamics of Polyelectrolytes

Staverman [ 128] gives a discussion of phase separation in polyelectrolyte solutions. These systems are necessarily more complex since besides containing more species of molecules, the effect of pH and addition of salts must be dealt with. For many years, Bungenberg de Jong and co- workers [ 14,19,20) have studied phase separation of polyelectrolytes in biological systems. They studied the effect of pH and ionic strength on this phenomenon they called "coacervation". The earliest theories aimed at explaining the results of Bungenberg de Jong were proposed by Voom

[137] and Michaeli, Overbeek, and Voom (85). The interested reader is referred also to Brooks et

Chapter 7: Solvent-Polymer-Electrolyte Solutions 189 al. [ 18) for a discussion of the thermodynamics of molecular partition when one or more of the solutes bears a net charge.

7.7 Summary

The Chen model has worked well for correlating solvent-electrolyte data. However, it is not suitable for polymer solutions in its present form, and its extension to solvent-polymer-electrolyte solutions does not appear attractive. A more logical correlative model may be the SUPERFAC model [43) which was not studied here. In fact, Fredenslund et al. [43) have suggested that SU-

PERF AC works well for correlating data; it is the extension to a predictive model that is unat- tractive. On the other hand, an even more appropriate model would be one that could theoretically handle aqueous-dilute polymer-electrolyte systems. Recall from earlier discussions that local composition and other semi-empirical models for solvent-polymer solutions are not expected to perform well in the range of dilute polymer concentrations.

Chapter 7: Solvent-Polymer-Electrolyte Solutions 190 8.0 Conclusions: Looking Back to See Ahead

It is the purpose of this chapter to summarize the findings of this research and to give recommendations for future work in this area.

8.1 In Retrospect

In this work many different thermodynamic models and theories were studied. All of these have some utility and some shortcomings. Below is a general brief synopsis of what each of the models can and cannot do, as they are applied to modeling aqueous-polymer two-phase systems.

Model General Characteristics

Wilson The Wilson model has three major problems which limit its use for solvent-polymer solutions: (l) it cannot predict liquid-liquid phase separation; (2) its form

does not have the enthalpy of mixing term necessary for modeling polymer sol-

utions (and predicting phase splitting); and (3) it has problems representing polar systems which exhibit strong, specific interactions.

NRTL NRTL is chiefly an enthalpy of mixing model that has an empirical constant introduced to account for nonidealities (nonrandomness of mixing). It therefore

lacks the combinatorial contribution necessary for representing polymer solutions.

UNIQUAC The UNIQUAC model is applicable to polymer solutions since (l) it has both

the combinatorial entropy term and the residual (enthalpy) term and (2) its com-

position variable (surface fractions) enables it to apply to solutions of large and

Chapter 8: Conclusions 191 small molecules. It has been shown to successfully correlate solvent-polymer va-

por pressure data. The application of the UNIQUAC model has been limited to

concentrated polymer solutions; its use for dilute polymer solutions remains to

be determined.

UNIFAC-FV The free volume correction of UNIFAC (UNIFAC-FV) has proven very useful for estimating solvent activities in certain nonaqueous solvent-polymer solutions.

However, the free volume correction as originally presented cannot be used for

aqueous solutions. The UNIFAC-FV model has really only been tested for

concentrated polymer solutions; its applicability to dilute polymer solutions is

questionable.

ASOG-VSP The free volume correction to ASOG (ASOG-VSP) has been shown to correlate solvent-polymer solution data as well as the Flory-Huggins model and better than

UNIFAC-FV. As with the above models, ASOG-VSP has been chiefly applied

to concentrated polymer solutions.

Heil The model by Heil is, conceptually, a combination of the Wilson and NRTL

models, and as such would appear to be applicable to polymer solutions. Indeed,

the Heil model has been shown to correlate polymer solution data quite well. Again, application has been chiefly limited to concentrated polymer solutions.

The Heil model has been applied successfully to polar solvent-polymer solutions.

The Flory-Huggins Lattice Theory

This model is capable of providing a very useful qualitative description of the

phase behavior, without the necessity of determining parameters. Quantitative

predictions have been obtained for many UCST systems with the introduction of various modifications (such as concentration dependence of the x.parameters). The classical Flory-Huggins theory is unable to predict LCST behavior. Still, the

Chapter 8: Conclusions 192 performance of the classical Flory-Huggins model, with its simplistic basis and

equations, must be judged impressive. This theory is also based on concentrated

polymer solutions. The virial expansion (and the osmotic pressure relationship)

form of this model can be used for dilute solutions though. However, a new

theory is desired for dilute polymer solutions. The use of a model based on a li-

near combination of the Flory-Huggins lattice theory for concentrated regions and

the perturbation theory for the dilute region has proven very accurate.

The Ogston Theory

The theory of Ogston is another basically simple theory for which the correlative

ability must be judged impressive. This model which is based on the virial coef-

ficient expansion appears quite useful for dilute aqueous-polymer solutions and

has been used successfully for years by the biochemists.

Flory Equation-of-State

The Flory equation of state approach accounts for free volume dissimilarities be-

tween components, which is extremely important when dealing with solutions of

molecules differing greatly in size. This theory is not hindered by the inadequacies

inherent in the Flory-Huggins lattice theory. Most importantly, the Flory

equation of state can predict LCST behavior -- for which the Flory-Huggins

model is totally inadequate.

8.2 Conclusions and Recommendations

The following list of conclusions and recommendations has come about after many long hours of studying the various models that may, in some way, be applicable to aqueous-polymer two-phase systems. It is the purpose of this list to aid in future research in this area, by providing some ra- tionale as to the course of action that is needed to work toward a more complete understanding of

Chapter 8: Conclusions 193 these complex. systems. It must be noted, however, that any complete theory of any complex. phenomena such as that exhibited by aqueous-polymer two-phase systems will by its very nature also be complicated. On the other hand, a theory should be as simple as possible in order to be practically applicable. Therefore, there is some amount of compromise that must be sought in this very difficult task, as was suggested by Flory (481and Casassa (22).

The conclusions reached in this work are:

1. A major limitation to the modeling of these aqueous-polymer two-phase systems is

the lack of experimental data. Sufficient, accurate data is needed for the reduction of

meaningful thermodynamic parameters by which thermodynamic models are tested for

their applicability. There exists a definite need for the generation of accurate, mean-

ingful thermodynamic data from well characterized systems.

2. The most promising model identified in this work is the theory of Ogston. First, the

model is based on the virial expansion and is thus quite suitable for dilute solutions.

The Ogston model is the simplest theoretically-relevant dilute-solution model. Sec-

ond, it appears to be easily extended to solvent-polymer-electrolyte solutions.

3. The Flory equation of state approach appears to be promising for representing po-

lymer solutions. The free volume dissimilarity effect on which it is based is extremely

important for solvent-polymer solutions. The most important aspect of this theory is

its ability to predict LCST behavior.

4. With modifications, the Flory-Huggins theory has proven useful for modeling sol-

vent-polymer solutions. However, when all the corrections have been introduced, the

theory becomes somewhat "unfriendly" and messy. In this work, the Flory-Huggins

model for ternary solvent-polymer-polymer systems was modified by determining one

of the solvent-polymer interaction parameters (x 12 ) form limited binary experimental data. The other two interaction parameters were then obtained from the first x and

Chapter 8: Conclusions 194 limited ternary data. The functional dependence of all parameters with temperature and concentration could then be found. However, the x parameter appeared constant over the ranges of interest in this study. Thus, constant x parameters were employed. The resulting parameters were used to predict the liquid-liquid equilibria of an aque-

ous-polymer two-phase system with limited success. Note that a dilute solution theory

was not used, but probably would have proven useful. The Flory-Huggins model was

tested regardless of its inadequacy for dilute polymer solutions because of the impres-

sive abilities of such a simple model as the classical Flory-Huggins model.

5. Several semi-empirical models (some of them local composition models) were identi-

fied and tested for their applicability to solvent-polymer solutions; most of them work

rather well for concentrated polymer solutions. Their application is limited at the

present time to concentrated polymer solutions, and thus, they may not be totally

applicable to the systems to which this research is addressed. The testing of these

models for dilute polymer solutions has not been done. Of these models, the models

by Heil and Brandani (briefly introduced) are probably the best semi-empirical models

for polar solvent-polymer solutions.

6. The ASOG and UNIF AC group-contribution models have been modified with a free

volume correction term in order to extend their application to polymer solutions. The models have performed quite well for nonaqueous solvent-polymer solutions. The

ASOG-VSP model has not been tested for aqueous-polymer solutions, but would probably work. The UNIFAC-FV model, however, cannot be applied to aqueous-

polymer solutions because the correction term becomes exceedingly large due to the near unit value of the reduced volume of water.

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2. Adamcova, Z. "Activity Coefficients of Electrolytes in Water-Polyethylene Glycol Mixed Solvent by Isopiestic Method," in Thermodynamic Behavior of Electrolyte Mixed Solvents, (Advances in Chemistry Series: 155), William F. Furter, ed. American Chemical Society, Washington, DC, pp. 361-373 (1976).

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4. Albertsson, P.A. "History of Aqueous Polymer Two-Phase Partition," in Partitioning in Aqueous Two-Phase Systems: Theory, Methods, Uses, and Applications to Biotechnology (II. Walter, D.E. Brooks, and D. Fisher, eds.) Academic Press, Orlando, Florida, pp. 1-10 ( 1985).

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Bibliography 204 Appendix A. Ternary LLE Predictions Using a Modified

Flory-Huggins Model

AppendixA, Flory-Huggins Calculations 205 The use of the Flory-Huggins model and its modifications was discussed in detail in Chapter 5. This Appendix describes this process in more detail and discusses the use of the FORTRAN

77 codes shown in Appendices B, C, and D. The purpose of the procedure described in this Ap- pendix and coded into the computer programs is ( 1) to provide a means of determining

"meaningful" Flory-Huggins interaction parameters for the aqueous-polymer two-phase systems being studied and (2) to examine the ability of these parameters for describing the phase behavior of the aqueous-polymer systems.

The objective is to model the water-PEG6000-dextran(D68) system at 20°C. Tables 15 and

16 contain the available experimental data.

As discussed in Chapter 5, one of the interaction parameters must be specified in order to solve for the other two. The binary data of Table 15 is used with program ·cHIFIT"' to find x12 • "CHIFIT .. also performs a quadratic least-squares regression on the concentration-dependence of

;( 12 • The "CHIFIT" routine is given in Appendix B. Note that "CHIFIT' could easily be modified to fit the concentration-dependence of X13 and ;(23. However, recall from Chapter 5 that constant x parameters were chosen for this work since from Figures 33-36 it was observed that x perhaps tends toward a constant value at medium-to-low polymer concentrations. The ·cHIFIT .. routine

was therefore not used in this example, and ;(12 was chosen to be 0.45 (see Figure 36).

An example for using "CHIFIT"' would be to determine the quadratic concentration-depen-

dency of ;(12 from the binary data presented in Table 15. The input file shown in Table 17 demonstrates the required data (also see code in Appendix B for input list). Temperature, polymer molecular weight, densities, and the water activity versus weight fraction data are the innut neces-

sary to run the program. Table 18 displays the results of the computer program. These are in the

form of (1) coefficients of the quadratic concentration-dependent x expression and (2) a table of calculated ~ata representing the experimental x parameter, the x parameter predicted from the obtained expression, and the subsequent predicted water activity.

AppendixA. Flory-HugginsCalculations 206 Table IS, Binary Data for Water-PEG6000 at Various Temperatures (57).

T=313.1K T=333.1K

Polymer Water Polymer Water Wt. Fract. Activity Wt. Fract. Activity

0.3292 0.9862 0.3248 0.9905 0.4408 0.9624 0.5068 0.9595 0.5341 0.9344 0.5344 0.9524 0.6280 0.8892 0.5800 0.9376 0.7047 0.8464 0.6832 0.8985 0.7852 0.8411 0.8095 0.8207 0.8761 0.7498 0.8932 0.7244 0.9375 0.6170 0.9677 0.4826 0.9761 0.4162 0.9910 0.3576

Appendix A. Flory-Huggins Calculations 207 Table 16. Ternary Data for Water-PEG6000-Dextran(D48) at 20 °C (3).

Total System Bottom Phase Top Phase Poly- Poly- Poly- ethylene ethylene ethylene System Dextran glycol Water Dextran glycol Water Dextran glycol Water % w/w % w/w % w/w % w/w % w/w % w/w % w/w % w/w % w/w A 4.40 3.65 91.95 6.10 2.98 90.92 2.63 4.43 92.94 8 5.00 3.50 91.50 7.34 2.55 90.11 1.80 4.91 93.29 C 5.20 3.80 91.00 9.46 1.85 88.69 1.05 5.70 93.25 D 6.20 4.40 89.40 13.25 1.07 85.68 0.30 7.17 92.53 E 7.00 5.00 88.00 15.89 0.68 83.43 0.14 8.29 91.57 F 8.40 5.80 85.80 19.08 0.52 80.40 0.06 9.93 90.01 G 9.80 7.00 83.20 22.77 0.24 76.99 0.05 12.03 87.92

Appendix A. Flory-Huggins Calculations 208 Table 17. "'CHIFIT"' Input Data File for Water-PEG6000 Example

Watcr(l)-Polyethylene Glycol(2) (MW=6000) at T=333K 6020.0 60.00 0.98324 1.21 13 0.9910 0.3576 0.9761 0.4162 0.9677 0.4826 0.9375 0.6170 0.8932 0.7244 0.8761 0.7498 0.8095 0.8207 0.7852 0.8411 0.6832 0.8985 0.5800 0.9376 0.5344 0.9524 0.5068 0.9595 0.3248 0.9905

Appendix A. Flory-Huggins Calculations 209 Table 18. "CI-IIFIT" Output Data File for Water-PEG6000 Example

Water(l)-Polycthylene Glycol(2) (MW= 6000) at T= 333K

POLYMER MOLECULAR WEIGHT = 6020.00 TEMPERATURE, IN KELVINS = 333.15 DENSITY OF WATER = 0.98324 DENSITY OF POLYMER = l.21000

NUMBER OF POLYMER SEGMENTS = 271.536

POLYMER WEIGHT FRACTIONS WATER ACTIVITIES

0.9910 0.3576 0.9761 0.4162 0.9677 0.4826 0.9375 0.6170 0.8932 0.7244 0.8761 0.7498 0.8095 0.8207 0.7852 0.8411 0.6832 0.8985 0.5800 0.9376 0.5344 0.9524 0.5068 0.9595 0.3248 0.9905

BSOLVE REGRESSION ALGORITHM ICON = 4 PH = 0.55432054D + 01 ITERATION NO. = l ICON = 3 PH = 0.68896259D+ 00 ITERATION NO. = 2 ICON = 3 PH = 0.56765652D+ 00 ITERATION NO. = 3 ICON = 3 PH = 0.56678768D + 00 ITERATION NO. = 4 ICON= 3 PH = 0.56678761D+OO ITERATION NO. = 5 ICON = 0 PH = 0.56678761D+OO ITERATION NO. = 6

SOLUTIONS OF THE EQUATIONS B( I)= 0.18695872D+Ol B( 2) = -0.590 l 6625D + 0 l B( 3) = 0.60625478D + 0 I B( 4) = 0.00000000D + 00

Appendix A. Flory-Huggins Calculations 210 Table 18. "CHIFIT" Output Data File -- Continued

••••••••••••••••••••••••••••••••••••••••••••••••

Water(l)-Polyethylene Glycol(2) (MW= 6000) at T = 333K

THE CONCENTRATION DEPENDENCE OF CI-II IS:

PCHI = (1.8696) + (-5.9017) • SF2 + (6.0625) • SF2••2 + (0.0000) • SF2••3

WF2 SF2 ECHI PCHI EACTl PACTl 0.9910 0.9889 2.5474 1.9624 0.3576 0.2018 0.9761 0.9707 l.7914 l.8536 0.4162 0.4414 0.9677 0.9605 l.6767 l.7944 0.4826 0.5379 0.9375 0.9242 1.3765 l.5935 0.6170 0.7426 0.8932 0.8717 1.1352 1.3319 0.7244 0.8412 0.8761 0.8518 l.0646 1.2411 0.7498 0.8523 0.8095 0.7754 0.8705 0.9386 0.8207 0.8550 0.7852 0.7481 0.8227 0.8476 0.8411 0.8529 0.6832 0.6367 0.6688 0.5696 0.8985 0.8631 0.5800 0.5288 0.5764 0.4440 0.9376 0.9035 0.5344 0.4826 0.5553 0.4334 0.9524 0.9257 0.5068 0.4550 0.5425 0.4394 0.9595 0.9392 0.3248 0.2810 0.5115 0.6898 0.9905 1.0046

Appendix A. Flory-Huggins Calculations 211 The "CHI" routine of Appendix C is used to determine x13 and :(23 from x12 (as specified or found using "CHIFIT') and ternary equilibrium data (as shown in Table 16). Table 19 contains an input data file for use with "CHI·. Again, molecular weights and densities are required as well as x12 and ternary equilibrium data. The output data file generated by "CHI" is displayed in Table 20.

Appendix A, Flory-Huggins Calculations 212 Table 19. "CHI" Input Data File for Water-PEG6000-Dextran(D48) Example

WATER(l)-PEG6000(2)-D48(3) AT 20 C: FIGURE 10.6 SYSTEM B 18.02 6020. 180000. 0.99823 1.21 1.2 0.45 0.9150 0.0350 0.0500 0.9329 0.0491 0.0180 0.9011 0.0255 0.0734

Appendix A, Flory-Huggins Calculations 213 Table 20. "CHI"' Output Data File for Water-PEG6000-Dextran(D48) Example

WATER(l)-PEG6000(2)-D48(3) AT 20 C: FIGURE 10.6 SYSTEM B

THE MOLECULAR WEIGHTS ARE: 18.02 6020.00 180000.00

THE DENSITIES ARE: 0.99823 1.21000 1.20000

THE BET A COEFFICIENTS USED TO FIT X 12 ARE: 0.4500 0.0000 0.0000

THE WEIGHT FRACTIONS IN THE OVERALL MIXTURE AND IN PHASES D AND DD ARE: 0.9150000D + 00 0.3500000D-Ol 0.5000000D-Ol 0.9329000D+OO 0.4910000D-Ol 0.1800000D-Ol 0.9011000D+OO 0.2550000D-Ol 0.7340000D-Ol

THE NUMBER OF SEGMENTS OF EACH COMPONENT IS: 1.0000 275.6049 8309.3507

THE TOTAL AMOUNT OF MASS IN THE MIXTURE, AND THE AMOUNT IN PHASES D AND DD, ARE: 100.0000 41.9352 58.0648

THE MOLES OF EACH COMP., IN PHASES D AND DD ARE: 0.2170997D + 01 0.9426508D-Ol 0.3484544D-Ol 0.2903561D + 0 l 0.6778653D-Ol 0.1967449D + 00

THE VOLUME FRACTIONS OF EACH, IN PHASES D AND DD ARE: 0.9439 0.0410 0.0151 0.9165 0.0214 0.0621

THE OVERALL VOLUME FRACTIONS ARE: . 0.9280 0.0296 0.0424

Appendix A. Flory-I I uggins Calculations 214 Table 20. "CHI" Output Data File -- Continued

••• CONTENTS OF MATRIX A ••• BEFORE LU FACTORIZATION

0.5138782D-03 -0.4335055D-02 0.2568547D-05 0.0000000D + 00 0.0000000D + 00 O.l l 74558D + 02 -0.4624461 D-0 l 0.8293858D-02 -0.2756049D+ 03 0.0000000D + 00 0.0000000D + 00 0.1 OOOOOOD+ 0 l 0.1000000D + 01 0.0000000D + 00 O.OOOOOOOD+ 00 0.0000000D + 00

THE RHS VECTOR, B, IS B=(Bl B2 B3 B4 ...) WIIERE ... X( l)= -0.1990695D-02 X( 2)= 0.6910379D+Ol X( 3) = 0.0000000D + 00 X( 4) = 0.4500000D + 00

••• CONTENTS OF MATRIX A, AFTER L-U FACTORIZATION ••• (MATRIX A CONTAINS U IN UPPER TRIANGLE AND MULTIPLIERS FOR LIN STRICT LOWER TRIANGLE)

-0.2756049D + 03 0.00000000 + 00 0.00000000 + 00 0.10000000 + 0 l 0.00000000 + 00 0.1174558D + 02 -0.4624461 D-0 l 0.8293858D-02 O.l 864547D-05 0.3690797D-03 -0.1449940D-04 0.4925641 D-05 0.3628382D-02 0.00000000 + 00 0.00000000 + 00 0.3628382D-02

••• THE CONDITION NUMBER= 0.1025D+06 •••

THE SOLUTION VECTOR IS X=(Xl X2 X3 X4 ... ) WHERE ... X( I)= 0.4500000D + 00 X( 2)= 0.5146401D+OO X( 3) = 0.352451 SD+ 01 X( 4) = 0.1240222D + 03

Appendix A. Flory-Huggins Calculations 215 The last step is to take the x parameter values and use them to predict the phase behavior of the system using the Flory-Huggins model. This step uses the "'HSU" routine (see Appendix D) which calls subroutine "'HYBRD" of the MINPACK software package. All the pertinent data for this program was entered from within the program. The output is shown in Table 21.

AppendixA. Flory-HugginsCalculations 216 Table 21. "HSU" Output Data File for Water-PEG6000-Dextran(D48) Example

FIGURE 10.6 - SYSTEM 8 - FROM ALBERTSSON

INTERACTION PARAMETERS ARIEH112= 0.4500 CI-1113= 0.5260 CHl23 = 6.2000 SEGMENT NUMBERS ARE: Xl = 1.00 X2= 275.60 X3= 8309.35

THE INITIAL GUESS FOR THE SOLUTION IS: VIP =0.9439 (SPECIFIED) V2P =0.0534 VJP =0.0027 VlDP = 0.9059 V2DP = 0.0001 V3DP = 0.0940

HSU,PRAUSNITZ r PARAMETER (HPR) = 4 SOLVENT CHEM.POT. PENALTY FUNCTION (SPF) = 0.2732456D-04

XTOL= O.lOOOOOOD-06

SPECIFICATION OF "HYBRD" PARAMETERS LDFJAC = 3 LR= 6 MAXFEV = 2000 ML= 2 MU= 2 EPSFCN = O.lOOOD-09 MODE= 1 FACTOR = O.IOOOD+01 NPRINT = 0

Appendix A. Flory-Huggins Calculations 217 Table 21. "HSU" Output Data File -- Continued

••••••• SOLUTION •••••••• FNORM FROM SUBROUTINE HYBRD 0.5833937D+ 03

FINAL L2 NORM OF THE RESIDUALS 0.499271lD-Ol

NUMBER OF FUNCTION EVALUATIONS 140

EXIT PARAMETER 1

THE FINAL APPROXIMATE SOLUTION IS: VIP = 0.9439 (SPECIFIED) V2P =0.0561 V3P =0.0000 VIDP = 0.9034 V2DP = 0.0000 V3DP = 0.0966

Appendix A. Flory-Huggins Calculations 218 Appendix B. FORTRAN 77 Code for Fitting

Concentration-Dependence of Flory Interaction Parameter

Appendix B. "CIIIFIT" FORTRAN 77 Code 219 C******************************************************************** C PROGRAMNAME: CHIFIT (NONLINEARLEAST-SQUARES FIT OF C CHI VS. WT. FRACTION> C******************************************************************** C C G. GREGORYBENGE 17-JUNE-86 C C FIRST, THE RAWDATA (POLYMER WEIGHT FRACTION, WATER ACTIVITY, C POLYMERMW, R, ANDTEMP) FROMHERSKOWITZ IS READIN. C THE DATAWAS OBTAINED BY AN ISOPIESTIC METHODBY C * HERSKOWITZAND GOTTLIEB J. CHEM.ENG. DATA, 30, 233 (1985). C AND C * HERSKOWITZAND GOTTLIEB J. CHEM.ENG. DATA,29, 450 (1984). C C THEN, THE WEIGHTFRACTIONS ARE CONVERTEDTO SEGMENTFRACTIONS. C C THEN, THE FLORYEQUATION CFOR A BINARYSOLVENT-POLYMER SYSTEM) C A IS USEDTO CALCULATEEXPERIMENTAL CHI'S AT EACHCOMPOSITION. C C THESEVALUES IN TURNARE USEDTO FIT THE CONCENTRATIONDEPENDENCE C DEPENDENCEOF THE FLORY-HUGGINSCHI INTERACTIONPARAMETER C EMPLOYINGTHE NONLINEARLEAST-SQUARES METHOD OF MARQUARDT. C THE RESULTINGEQUATION IS USEDTO CALCULATEA CORRECTED C CHI AT EACHCOMPOSITION POINT, FOR COMPARISONWITH THE C EXPERIMENTALCHI CTOSEE HOWGOOD THE FIT IS). C C FINALLY,THE RESULTINGCORRECTED CHI'S ARE USEDTO PREDICTTHE C WATERACTIVITY FOR COMPARISONWITH THE EXPERIMENTALVALUE. C C******************************************************************** C PROGRAMNAME: MARQUARDTNON-LINEAR LEAST SQUARES C******************************************************************** C C WRITTENBY: W. E. BALLCWASH. STATE U.) DATE: TRANSCRIBED5-24-86 C CBYKIM HUNTER) C C PURPOSE:THE PROGRAMSOLVES FOR THE COEFFICIENTSIN A MULTI- C VARIABLENON-LINEAR REGRESSION EQUATION OF THE FORM: C C Y = F CXl, X2, X3, .... ; Al, A2, A3, ..... ) C C WHEREXl ... XK ARE INDEPENDENTVARIABLES AND C Al ... AN ARE THE UNKNOWNCONSTANT COEFFICIENTS C TO BE EVALUATEDBY LEASTSQUARES CURVE FIT TO THE C DATA. N DATAPOINTS ARE SUPPLIEDBY THE USER. C C C MARQUARDT'SMETHOD IS AN EXTENTIONOF THE GAUSSNEWTON C PROCEDURETHAT PERMITS CONVERGENCE WITH RELATIVELY POOR C STARTINGVALUES FOR THE UNKNOWNCOEFFICIENTS AI. C

Appendix B. "CHIFIT" FORTRAN77 Code 220 C THE PROGRAMSEEKS TO OPTIMIZE A LEAST SQUARESOBJECTIVE C FUNCTION.FOR POINTS FAR FROMTHE OPTIMUM,A GRADIENTSEARCH C IS CONDUCTED.NEAR THE OPTIMUMPOINT, THE PROGRAMGRADUALLY C ADJUSTSTO A GAUSSNEWTON SEARCH TO ACCELLERATECONVERGENCE. C C THE MARQUARDTPROGRAM IS COMPOSEDOF 3-4 SUBPROGRAMS: C C SUBROUTINEBSOLVE - PERFORMSPRIMARY CALCULATIONS INCLUDING C NUMERICALEVALUATION OF DERIVATIVES. C SUBROUTINEFUNC - SPECIFIES THE MODELCUSER SUPPLIED>~ C FUNCTIONARCOS - FUNCTIONSUBPROGRAM FOR USE BY BSOLVE. C C C C PARAMETERS: C C NN =NO.DATA POINTS C * NO. EQUATIONSFOR ROOTLOCATION C KK =NO.UNKNOWNCOEFFICIENTS C B = VECTOROF UNKNOWNCOEFFICIENTS C BMIN = VECTOROF MINIMUMALLOWABLE VALUES OF B C BMAX= n n MAXIMUM n n n n C X = VECTOROF INDEPENDENTVARIABLE DATA POINTS C *SET= VECTORB FOR ROOTLOCATION C Y = VECTOROF DEPENDENTVARIABLE C *SET= 0 FOR ROOTLOCATION C PH= LEAST SQUARESOBJECTIVE FUNCTION C Z = COMPUTEDVALUES OF OBJECTIVEFUNCTION C BV = CODEVECTOR - SET= l FOR NUMERICALDERIVATIVES C C C DIMENSIONREQUIREMENTS: C C PCKK* CNN+2) + NN), ACKK,KK+2), ACCKK,KK+2),XCNN), BCKK), C C ZCNN), YCNN), BVCKK), BMINCKK),BMAXCKK), FVCKK), DVCKK) C C C***************************************************************~**** C C CALLINGPROGRAM C C******************************************************************** C C IMPLICIT REAL*8 CA-H,0-Z,M) DIMENSIONPC200), AC25,25), ACC25,25), BC25), BMAXC25), & BMINC25), ZC25), YC25), BVC25), XC25,25), & WF2C20),SF2C20),EACT1C20),PACT1C20),YCHIC20), & ECHIC20),PCHIC20) CHARACTER*80TITLE

Appendix B. "CHIFIT" FORTRAN77 Code 221 EXTERNALFUNC COMMONX C C READINPUT DATA C READC5,500) TITLE 500 FORMATCA80) HRITEC6,500) TITLE READC5,501) MH2,TEMP,DENS1,DENS2 501 FORMATCFl0.2,Fl0.2,Fl0.5,Fl0.5) TEMPK=TEMP+ 273.15DO MHl = 18.01534DO HRITEC6,499) MH2,TEMPK,DENS1,DENS2 499 FORMAT(//TlO,'POLYMERMOLECULAR HEIGHT= ',Fl0.2/ & TlO,'TEMPERATURE,IN KELVINS= ',Fl0.2/ & TlO,'DENSITY OF HATER = ',Fl0.5/ & TlO,'DENSITY OF POLYMER= ',Fl0.5/) C C CALCULATER, THE NUMBEROF POLYMERSEGMENTS, DEFINED AS C THE RATIO OF MOLARVOLUMES OF POLYMERAND SOLVENT C C POLYMERMOLAR VOLUME MH2/DENS2 C R = ------= ------C SOLVENTMOLAR VOLUME MHl/DENSl C R = CMH2/DENS2)/CMHl/DENSl) HRITEC6,495) R 495 FORMAT(//TlO,'NUMBEROF POLYMERSEGMENTS= ',Fl0.3/) C READC5,502) NDATA 502 FORMATCI2) HRITEC6,498) 498 FORMAT(//TlO,'POLYMERHEIGHT FRACTI0NS',T40,'HATER ACTIVITIES'/) DO 600 I=l,NDATA READCS,503) WF2CI),EACT1CI) HRITEC6,497) HF2CI>,EACT1CI) 600 CONTINUE 503 FORMATCFl0.5,Fl0.5) 497 FORMAT(Tl5,Fl0.4,T42,Fl0.4) HRITEC6,496) 496 FORMAT(///) C C CONVERTHEIGHT FRACTION OT SEGMENTFRACTION C DO 601 I=l,NDATA SF2CI) = R*HF2CI)*MH1/ CC1.DO-HF2Cl))*MH2 + R*HF2CI)*MW1) 601 CONTINUE C C CALCULATEEXPERIMENTAL CHI'S CECH!) FROMBINARY FLORY EQUATION C DO 602 I=l,NDATA

Appendix B. ncHIFITn FORTRAN77 Code 222 YCHICI) = C DLOGCEACTlCI))- DLOGC1.DO-SF2CI)) & - Cl.DO - l.DO/R)*SF2CI) ) / SF2CI) ECHICI) = YCHICI) / SF2CI) 602 CONTINUE C NN = NDATA KK = 4 C C ENTEROBSERVED VALUES FOR INDEPENDENTVARIABLES X CN,1,2 .... KK) C C *** SEGMENTFRACTION OF POLYMER==>SF2 *** C DO 603 I=l,NDATA X CI,l) = SF2CI) 603 CONTINUE C C C ENTERTHE OBSERVEDVALUES (DATA POINTS) FOR THE DEPENDENT C VARIABLESYCI). C C EXPERIMENTALFLORY-HUGGINS CHI PARAMETER==> ECHI C DO 604 I=l,NDATA YCI) = ECHICI) 604 CONTINUE C C DATAARE TO BE FIT TO AN EQUATIONOF THE FORM: C C PCHI = BCl) + BC2)*SF2 + BC3)*SF2**2 + BC4)*Cl/T) C C OR IN TERMSOF THE PROGRAMVARIABLES, X=SF2 ANDY=PCHI C C Y = BCl) + BC2)*XC1) + BC3)*XC1)**2 + BC4)*(l/T) C C ENTERINITIAL GUESSESFOR UNKNOWNPARAMETERS BCI). C BCl) = O.SDO BC2) = 0.2DO BC3) = O.lDO BC4) = O.DO C C ENTERLIMITS ON VALUESTHE PARAMETERSCAN HAVE(CONSTRAINTS). C BMINCl) = -1000.DO BMINC2) = -1000.DO BMINC3) = -1000.DO BMINC4) = -1000.DO C BMAX(l) = 1000.DO BMAXC2)= 1000.DO BMAXC3)= 1000.DO

Appendix B. "CHIFIT" FORTRAN77 Code 223 BMAXC4)= 1000.DO C C STARTINGVALUES FOR BSOLVEPARAMETERS. C FNU = O.DO FLA = O.DO TAU = O.DO EPS = O.DO PHMIN= O.DO I = 0 KD = KK FV = O.DO DO 100 J=l,KK BVCJ) = l.DO 100 CONTINUE ICON = KK ITER = 0 C WRITE(6,015) 015 FORMATC25X,'BSOLVE REGRESSION ALGORITHM') C C 200 CALLBSOLVE CKK, B, NN, Z, Y, PH, FNU, FLA, TAU, EPS, PHMIN, & I, ICON, FV, DV, BV, BMIN, BMAX,P, FUNC, DERIV, & KD, A, AC, GAMM> C ITER = ITER + 1 WRITE(6,001) ICON, PH, ITER 001 FORMAT(/,8X,'ICON = ',I3,4X,'PH = ',Dl5.8,4X,'ITERATION NO.=' & , I3) C IF CICON) 10, 300, 200 10 IF (ICON+ 1) 20, 60, 200 20 IF

Appendix B. "CHIFIT" FORTRAN77 Code 224 008 FORMAT(//,2X,'THIS IS NOT POSSIBLE') GO TO 300 300 WRITE (6,002) 002 FORMAT(//,8X,'S0LUTI0NS OF THE EQUATIONS') DO 400 J=l,KK WRITE (6,003) J, BCJ) 003 FORMAT(/,8X,'BC',I2,') = 1 ,Dl6.8) 400 CONTINUE C C USE EQUATIONFOUND ABOVE TO PREDICTCHI'S CPCHI) C TO COMPAREWITH EXPERIMENTALCHI CECH!) C DO 605 I=l,NDATA PCHICI) = BCl) + BC2)*SF2CI) + BC3)*SF2CI)**2 C & + BC4)*SF2CI)**3 C & + BC5)*SF2CI)**4 + BC6)/SF2CI) 605 CONTINUE C C USE PREDICTEDCHI'S CPCHI) TO PREDICT ACTIVITYOF WATERCPACTl) C TO COMPAREWITH EXPERIMENTALWATER ACTIVITY CEACTl) C DO 606 I=l,NDATA PACTlCI) = DEXP C DLOGC1.DO-SF2CI>>+ Cl.DO-l.DO/R)*SF2CI) & 606 CONTINUE C C PRINTOUTRESULTS FOR ABOVEMENTIONEDCOMPARISONS C WRITEC6,700) TITLE 700 FORMAT(///T5,'************************************************' & //Tl0,A80/) WRITEC6,701) BC1),BC2>,BC3),BC4) 701 FORMATC/TlO,'THECONCENTRATION DEPENDENCE OF CHI IS: 1 /// & Tl5,'PCHI = C',Fl0.4, & ') + C',Fl0.4,') * SF2 1 // & T20,' + C',Fl0.4,') * SF2**2 + C',Fl0.4, & ') * SF2**3 1 //) C T20,' + C',Fl0.4, 1 ) * SF2**4 1 // C & T20,'+ C',Fl0.4,')/ SF2'/) WRITEC6,702) 702 FORMATC/T7,'WF2',9X,'SF2',9X,'ECHI',8X,'PCHI',7X,'EACT1', & 7X,'PACTl'/) DO 610 I=l,NDATA WRITEC6,703) WF2CI),SF2CI),ECHICI>,PCHICI),EACT1CI),PACT1CI) 610 CONTINUE 703 FORMAT

Appendix B. "CHIFIT" FORTRAN77 Code 225 C C******************************************************************** C C SUBROUTINEFUNC C C******************************************************************** C C SUBROUTINEFUNC CKK, B, NN, Z, FV) C IMPLICIT REAL*8CA-H,0-Z) DIMENSIONXC25,25), ZC25), BC25) COMMONX C DO 1 I=l,NN ZCI ) = BCl) + BC2>*XCI,l) + BC3)*CXCI,1)**2) C & + BC4)*CXCI,1)**3) 1 CONTINUE C RETURN END C C C******************************************************************** C C SUBROUTINEBSOLVE C C******************************************************************** C C SUBROUTINEBSOLVE CKK, B, NN, Z, Y, PH, FNU, FLA, TAU, EPS, & PHMIN, I, ICON, FV, DV, BV, BMIN, BMAX,P, FUNC, & DERIV, KD, A, AC, GAMM) C IMPLICIT REAL*8 CA-H,0-Z) DIMENSIONPC200), AC25,25), ACC25,25), BC25), BMAXC25), & BMINC25), ZC25), YC25), BVC25), FVC25), DVC25), & XC25,25) C K = KK N = NN KPl = K + 1 KP2 = KPl + 1 KBI1 = K * N KBI2 = KBI1 + K KZI = KBI2 + K IF CFNU.LE. O.DO) FNU = 10.DO IF (FLA . LE. O.DO) FLA = .OlDO IF CTAU. LE. O.DO) TAU = . OOlDO IF CEPS . LE. O.DO> EPS = .00002DO IF (PHMIN .LE. O.DO) PHMIN= O.DO

Appendix B. "CHIFIT" FORTRAN77 Code 226 120 KE= 0 130 DO 160 Il=l,K IF CBVCil) .NE. O.DO) KE= KE+ l 160 CONTINUE IF CKE .GT. O.DO) GO TO 170 162 ICON= -3 163 GO TO 2120 170 IF CN .GE. KE) GO TO 500 180 ICON= -2 190 GO TO 2120 500 Il = l 530 IF CI .GT. 0) GO TO 1530 550 DO 560 Jl=l,K J2 = KBI1 + Jl PCJ2) = BCJU J3 = KBI2 + Jl PCJ3) = DABS CBCJl)) + l.D-02 560 CONTINUE GO TO 1030 590 IF CPHMIN.GT. PH .AND. I .GT. 1) GO TO 625 DO 620 Jl=l,K Nl = CJl-1) 3E N IF CBVCJl)) 601, 620, 605 601 CALL DERIV CK, B, N, Z, PCNl+l), FV, DV, Jl, JTEST) IF CJTEST .NE. C-1)) GO TO 620 BVCJl) = l. DO 605 DO 606 J2=1,K J3 = KBI1 + J2 PCJ3) = BCJ2) 606 CONTINUE J3 = KBI1 + Jl J4 = KBI2 + Jl DEN= .OOlDO 3E DMAX1CPCJ4),DABSCPCJ3))) IF CPCJ3) + DEN .LE. BMAXCJl)) GO TO 55 PCJ3) = PCJ3) - DEN DEN= -DEN GO TO 56 55 PCJ3) = PCJ3) + DEN 56 CALL FUNCCK, PCKBil+l), N, PCNl+l), FV) DO 610 J2=1,N JB = J2 + Nl PCJB) = CPCJB) - ZCJ2)) / DEN 610 CONTINUE 620 CONTINUE C C SET UP CORRECTIONEQUATIONS C 625 DO 725 Jl=l,K Nl = CJl-1) 3E N ACJl,KPl) = O.DO IF CBVCJl)) 630, 692, 630

Appendix B. "CHIFIT" FORTRAN77 Code 227 630 DO 640 J2=1,N N2 = Nl + J2 ACJl,KPl) = A{Jl,KPl) + PCN2) * CYCJ2) - ZCJ2)) 640 CONTINUE 650 DO 680 J2=1,K 660 ACJ1,J2) = O.DO 665 N2 = CJ2-l) * N 670 DO 680 J3=1,N 672 N3 = Nl + J3 674 N4 = N2 + J3 A{Jl,J2) = ACJ1,J2) + PCN3) * PCN4) 680 CONTINUE IF CA{Jl,Jl) .GT. l.D-20) GO TO 725 692 DO 694 J2=1,KP1 ACJ1,J2) = O.DO 694 CONTINUE 695 ACJl,Jl) = l.DO 725 CONTINUE GN = O.DO DO 729 Jl=l,K GN = GN + ACJ1,KP1)**2 729 CONTINUE C C SCALE CORRECTIONEQUATIONS C DO 726 Jl=l,K ACJ1,KP2) = DSQRTCACJl,Jl)) 726 CONTINUE DO 727 Jl=l,K ACJl,KPl) = ACJl,KPl) / ACJ1,KP2) DO 727 J2=1,K ACJ1,J2) = ACJ1,J2) / CACJ1,KP2) * ACJ2,KP2)) 727 CONTINUE 730 FL= FLA/ FNU GO TO 810 800 FL= FNU * FL 810 DO 840 Jl=l,K 820 DO 830 J2=1,KP1 ACCJ1,J2) = ACJ1,J2) 830 CONTINUE ACCJl,Jl) = AC{Jl,Jl) + FL 840 CONTINUE C C SOLVECORRECTION EQUATIONS C DO 9 3 0 Ll =1, K L2 = Ll + 1 DO 910 L3=L2,KP1 ACCL1,L3) = ACCL1,L3) / ACCLl,Ll) 910 CONTINUE DO 930 L3=1,K

Appendix B. "CHIFIT" FORTRAN77 Code 228 IF Cll-l3) 920, 930, 920 920 DO 925 L4=L2,KP1 ACCL3,l4) = ACCL3,l4) - ACCL1,L4) * ACCL3,ll) 925 CONTINUE 930 CONTINUE C DN = O.DO DG = O.DO DO 1028 Jl=l,K ACCJ1,KP2) = ACCJl,KPl) / ACJ1,KP2) J2 = KBI1 + Jl PCJ2) = DMAXlCBMINCJl), DMINlCBMAXCJl),BCJl) + ACCJ1,KP2))) DG = DG + ACCJ1,KP2) * ACJl,KPl) * ACJ1,KP2) DN = DN + ACCJ1,KP2) * ACCJ1,KP2) ACCJ1,KP2) = PCJ2) - BCJl) 1028 CONTINUE COSG = DG / DSQRTCDN*GN) JGAM= 0 IF CCOSG) 1100, 1110, 1110 1100 JGAM= 2 COSG = -COSG 1110 CONTINUE COSG = DMINl CCOSG, l.DO) GAMM=DARCOS CCOSG) * 180.DO / C3.14159265DO) IF CJGAM.GT. 0) GAMM=180.DO - GAMM 1030 CALL FUNC CK, PCKBil+l), N, PCKZI+l), FV) 1500 PHI= O.DO DO 1520 Jl=l,N J2 = KZI + Jl PHI= PHI+ CPCJ2) - YCJ1))**2 1520 CONTINUE IF (PHI .LT. l.D-10) GO TO 3000 IF CI .GT. 0) GO TO 1540 1521 ICON= K GO TO 2110 1540 IF CPHI .GE. PH) GO TO 1530 C C EPSILON TEST C 1200 ICON= 0 DO 1220 Jl=l,K J2 = KBI1 + Jl IF C DABSCACCJ1,KP2))/ CTAU+ DABSCPCJ2))) .GT. EPS) & ICON= ICON+ 1 1220 CONTINUE IF CICON .EQ. 0) GOTO1400 C C GAMMALAMBDA TEST C IF CFL .GT. I.DO .AND. GAMM.GT. 90.DO) ICON= -1

Appendix B. "CHIFIT" FORTRAN77 Code 229 GO TO 2105 C C GAMMAEPSILON TEST C 1400 IF CFL .GT. l.DO .AND. GAMM.LE. 45.DO) ICON= -4 GO TO 2105 C 1530 IF Cil-2) 1531, 1531, 2310 1531 Il = Il + 1 GO TO (530, 590, 800) , 11 2310 IF CFL .LT. l.D+8) GO TO 800 1320 ICON= -1 C 2105 FLA= FL DO 2091 J2=1,K J3 = KBI1 + J2 BCJ2) = PCJ3) 2091 CONTINUE 2110 DO 2050 J2=1,N J3 = KZI + J2 ZCJ2) = PCJ3) 2050 CONTINUE PH= PHI I = I + 1 C 2120 RETURN C 3000 ICON= 0 GO TO 2105 C END C C C******************************************************************** C C FUNCTIONARCOS C C******************************************************************** C C FUNCTIONARCOS CZ) C IMPLICIT REAL*8 CA-H,0-Z) C X = Z KEY = 0 IF ex . LT. C-1.DO)) X = -I.DO IF ex .GT. 1. DO) X = 1. DO IF ex .GE. C-1.DO) .AND. X . LT. O.DO) KEY = 1 IF ex .LT. O.DO) X = DABS CX) IF ex .EQ. O.DO) GO TO 10

Appendix B. "CHIFIT" FORTRAN77 Code 230 ARCOS= DTANCDSQRTCl.DO-X*X) / X) IF CKEY.EQ. 1) ARCOS= 3.14159265DO - ARCOS GO TO 999 10 ARCOS= l.5707963DO C 999 RETURN END

Appendix B. "CHIFIT" FORTRAN77 Code 231 Appendix C. FORTRAN 77 Code for Determination of Flory

Interaction Parameter from Experimental Ternary Data

AppendixC. "CHI" FORTRAN 77 Code 232 C********************************************************************* C PROGRAMNAME: CHI CFLORY-HUGGINS1DETERMINATION OF CHI) C C G. GREGORYBENGE 28-MAY-86 (REVISED: 19-JUNE-86) C C DETERMINATIONOF FLORY-HUGGINSCHI INTERACTIONPARAMETER C FROMEXPERIMENTAL DATA FOR THE TERNARYLIQUID-LIQUID C (2-PHASE> PROBLEM. C C FOR EXAMPLE,SOLVENTCl)-POLYMER PC2)-POLYMER QC3) C C GIVEN THE WEIGHTFRACTIONS OF ALL THREECOMPONENTS C ANDTHEIR DENSITIES, THE CHI ROUTINECALCULATES THE C VOLUMEFRACTIONS AND USES EQUATIONSXIII-12 THRUXIII-14 C OF C FLORY, P.J. "PRINCIPLES OF POLYMERCHEMISTRY". C CORNELLUNIVERSITY PRESS, ITHACA, NEWYORK C P. 549 (1953). C SUBSTITUTEDINTO THE EQUILIBRIUMEQUATIONS TO SETUP C A SET OF 3 LINEAREQUATIONS IN 4 UNKNOWNS. THE C UNKNOWNSARE THE CHI PARAMETERS.TO SOLVETHE SYSTEM C AN ADDITIONALEQUATION MUST BE GIVEN FOR THE EXPRESSION C OF ONE OF THE CHI PARAMETERSAS A FUNCTIONOF CONCENTRATION C CORA CONSTANTVALUE OF CHI). C C THE CHIFIT ROUTINEIS USEDTO FIT THE CONCENTRATION C DEPENDENCEOF THE CHI PARAMETERCHOSEN ABOVE, FROM C AVAILABLEBINARY DATA. C C SUBROUTINEDCOMPD IS CALLEDTO PERFORML-U FACTORIZATION C (MATRIXDECOMPOSITION). ANDSUBROUTINE SOLVED IS CALLED C TO THENSOLVE THE MATRIXPROBLEM, YIELDING THE CHI'S. C C ONCETHE CHI PARAMETERSHAVE BEEN DETERMINED (CHI), C THE CHIFIT ROUTINECAN BE USEDTO FIT THE CONCENTRATION C DEPENDENCECCHIFIT USES A NONLINEARLEAST-SQUARES ROUTINE). C THEN, THE HSU ROUTINEIS EMPLOYEDTO PREDICT C THE LIQUID-LIQUID PHASEBEHAVIOR OF THE SYSTEMOF INTEREST. C C **************************************************************** C C The system to be solved is1 C C A * X = b C = C C where A is the coefficient matrix, and C C T C x = C Xl X2 X3 X4 XS ... ) is initially the C

Appendix C. "CHI" FORTRAN77 Code 233 C vector of unknowns, and it is the solution vector at the C conclusion of the program. C C Procedure1 First, subroutine DCOMPDis employed to perform C LU factorization of the coefficient matrix A and to C estimate the condition number (COND). If the C condition is less than 4000, then subroutine C SOLVEDis called to solve the system. C C Declarations: C C ~ used in subroutine DCOMPDCA,N,COND,IPVT,HKSP) C C C ON ENTRY: C A - doubly subscripted array with dimension CN,N) which C contains the matrix whose factorization is to be computed. C NN - the order of the matrix A and the number of elements in C vectors IPVT (pivot vector) and WKSP (work space vector). C C ON RETURN: C A contains in its upper triangle an upper triangular matrix C U and in its strict lower triangle the multipliers C necessary to construct a matrix L so that A=LU. C IPVT - singly subscripted integer array of dimension N which C contains the pivot information necessary to construct the C permutations in L. Specifically, IPVTCk) is the index of C the k-th pivot row. C COND - an estimate of the condition of matrix A. If C A is singular to working precision, then CONDis set C to 1.0/(unit round-off error). The unit round-off is C found using subroutine MCHEPS. C WKSP - singly subscripted work space vector of dimension N. C C C ~ used in subroutine SOLVED CA,N,B,IPVT) C ------C C ON ENTRY: C A - doubly subscripted array with dimensions CN,N) which C contains the factorization computed by DCOMPD. C It is not changed by SOLVED. C NN order of the matrix A and the number of elements in the C vectors Band IPVT. C !PVT - contains pivot information form SOLVED. C B - singly subscripted array of dimension N which contains the C right hand side b of a system of simultaneous linear C equations Ax=b. C C ON RETURN: C B - contains the solution vector, X = C Xl X2 X3 ... )

Appendix C. "CHI" FORTRAN77 Code 234 C C C********************************************************************* C DOUBLEPRECISION AC4,4),BC4),WKSPC4),COND DOUBLEPRECISION MWC3),DENSC3),BEAT1,BETA2,BETA3,WFDC3),WFDDC3), & WFC3),NDC3),NDDC3),NC3),VFDC3),VFDDC3),VFC3), & SND,SNDD,SNT,Xl,X2,X3,Pl,P2,Ll,L2,LRR INTEGERIPVTC4),NN CHARACTER*80TITLE NN=4 C C READINPUT: C Cl) TITLE C C2) MOLECULARWEIGHTS CMW) AND DENSITIES CDENS) C (3) CHI EQUATIONCONSTANTS FOR CHI12 CBETA) C (4) EXPERIMENTALWEIGHT FRACTIONS C WFCI) - WT.FRAC. IN TOTALMIXTURE C WFDCI) - WT.FRAC. IN PHASED C WFDDCI)-WT.FRAC. IN PHASEDD C READCS,90) TITLE WRITEC6,107) TITLE C READCS,101) CMWCI>,I=l,3) WRITEC6,410) WRITEC6,101) CMWCI>,I=l,3) C READCS,104) CDENSCI),I=l,3) WRITEC6,411) WRITEC6,104) CDENSCI),I=l,3) C C THIS VERSIONUSES A CONSTANTCHI12 CORBETAl), THATIS C BETA2 = BETA3 = O. C C THIS CODESTILL READSAND USES ALL BETA'S IN ORDERTO BE C MOREGENERAL

READCS,100) BETA1,BETA2,BETA3 WRITEC6,412) WRITEC6,100) BETA1,BETA2,BETA3 C READCS,100) CWFCJ),J=l,3) READCS,100) CWFDCJ),J=l,3) READCS,100) CWFODCJ),J=l,3) WRITEC6,415) WRITEC6,499) CWFCI),1=1,3) WRITEC6,499) CWFDCl),1=1,3) WRITEC6,499) CWFDDCI),I=l,3)

Appendix C. "CHI" FORTRAN77 Code 235 C C CALCULATETHE NUMBEROF SEGMENTSIN EACHPOLYMER1 X2,X3 C (ALSO CALLEDR) Xl = l.DO X2 = CMWC2)/DENSC2))/(MWCl)/DENSCl)) X3 = CMWC3)/DENSC3))/(MWCl)/DENSCl)) WRITEC6,420) WRITEC6,100) Xl,X2,X3 C C USING THE LEVERRULE ANDA MASSBALANCE CWITH BASIS OF 100 GRAMS C OF TOTALMIXTURE), THE AMOUNTOF MASSIN EACHPHASE IS C CALCULATEDFROM THE EXPERIMENTALWEIGHT FRACTION DATA C CI.E. WF,WFD,WFDD> C C Pl = MIXTURE; P2 = PHASED ; P3 = PHASEDD C C LI = LENGTHOF LINE P2,Pl C L2 = LENGTHOF LINE Pl,P3 C LI = DSQRTCCWFDC3)-WFC3))**2 + CWFDC2)-WFC2))**2) L2 = DSQRTCCWFC3)-WFDDC3))**2 + CWFC2)-WFDDC2))**2) C C LRR = LEVERRULE RATIO= L2/Ll C LRR = L2/LI C C ASSUMEPl= 100 = P2 + P3 C THEN P2 = Pl - P3 = 100 - P3 C ALSO P2 = LRR * P3 --> 100-P3=LRR*P3 --> P3CLRR+I>=IOO C SO THAT P3 = 100 / CLRR+l) ; AND P2 = LRR*P3 C Pl= 100.DO P3 = Pl/CLRR+l.DO) P2 = LRR*P3 C C ALSO P2 = AMOUNTOF MASSIN PHASED C P3 = AMOUNTOF MASSIN PHASEDD C WRITEC6,701> WRITEC6,100) Pl,P2,P3 C C CONVERTWEIGHT FRACTIONS CWFD,WFDD) TO C VOLUMEFRACTIONS CVFD,VFDD) AND CALCULATE C OVERALLVOLUME FRACTIONS CVF) C C C C FOR CALCULATEMOLES OF SOLVENTCl)= NI C EACH n " " POLYMER(2) SEGMENTS=N2 C PHASE n n n POLYMER(3) SEGMENTS=N3 C

Appendix C. "CHI" FORTRAN77 Code 236 C C MOLESIN PHASESD ANDDD C NDCl) = Xl*WFDCl)*P2/MWCl) NDDCl) = Xl*WFDDCl)*P3/MWCl) NDC2) = X2*WFDC2)*P2/MWC2) NDDC2) = X2*WFDDC2)*P3/MWC2) NDC3) = X3*WFDC3)*P2/MWC3) NDDC3) = X3*WFDDC3)JEP3/MWC3) WRITEC6,SOO) WRITEC6,499) CNDCI),I=l,3) WRITEC6,499) CNDDCI),I=l,3) SND = NDCl) + NDC2) + NDC3) SNDD= NDDCl) + NDDC2)+ NDDC3) C C TOTALMOLES OF EACHIN MIXTURE C SNT = O.DO DO 11 I=l, 3 NCI)= NDCI) + NDDCI) SNT = SNT + NCI) 11 CONTINUE C C DO 10 I=l,3 VFDCI) = NDCI)/SND VFDDCI) = NDDCI)/SNDD VFCI) = NCI)/SNT 10 CONTINUE WRITEC6,520) WRITEC6,100) CVFDCI),I=l,3) WRITEC6,100) CVFDDCI),I=l,3) WRITEC6,530) WRITEC6,100) CVFCI),I=l,3) C C CALCULATECOEFFICIENTS OF MATRIXA C ACl,l) = VFDC2)**2 + VFDC2)*VFDC3) * - VFDDC2)**2 - VFDDC2)*VFDDC3) ACl,2) = VFDC2)*VFDC3)+ VFDC3)**2 JE - VFDDC2)*VFDDC3)- VFDDC3)JEJE2 ACl,3) = - CXl/X2)JEVFDC2)JEVFDC3) JE + CX1/X2)*VF0DC2)*VFDDC3) ACl,4) = O.DO AC2,1) = O.DO AC2,2) = - CX2/Xl)*VFDCl)JEVFDC3) * + CX2/Xl)*VFDD(l)*VFDDC3) AC2,3) = VFDCl)*VFDC3) + VFDC3)**2 * - VFDDCl)*VFDDC3)- VFDDC3)**2 AC2,4) = VFDCl)JEJE2+ VFDCl)JEVFDC3) * - VFDDCl)JEJE2- VFDDCl)*VFDDC3)

Appendix C. "CHI" FORTRAN77 Code 237 AC3,1) = - X2 / Xl AC3,2) = O.DO AC3,3) = O.DO AC3,4> = 1. DO AC4,l> = 1. DO AC4,2> = O.DO AC4,3) = O.DO AC4,4) = O.DO C WRITEC6,300) DO 45 I=l,NN WRITEC6,106) CACI,J),J=l,NN) 45 CONTINUE C C CALCULATECOEFFICIENTS OF VECTORb C BCl) = - C DLOGCVFDCl))+ Cl.DO-VFD(l)) - VFDC2)3EXl/X2 3E - VFDC3)3EXl/X3- DLOGCVFDDCl))- Cl.DO-VFDDCl)) 3E + VFDDC2)3EXl/X2+ VFDDC3)3EXl/X3) BC2) = - C DLOGCVFDC2))+ Cl.DO-VFDC2)) - VFDC1)3EX2/Xl 3E - VFDC3)3EX2/X3- DLOGCVFDDC2))-Cl.DO-VFDDC2)) 3E + VFDD(l)3EX2/Xl + VFDDC3)3EX2/X3) BC3) = O.DO BC4) = BETA! + BETA23EVFC2)+ BETA33EVFC2)3E3E2 C WRITEC6,310) DO 35 I=l,NN WRITEC6,103) I,BCI) 35 CONTINUE C C CALL SUBROUTINEDCOMPD TO DECOMPOSE(PERFORM C L-U FACTORIZATIONON) MATRIXA ANDTO ESTIMATE C THE CONDITIONNUMBER ... C CALL DCOMPDCA,NN,COND,IPVT,WKSP) C C PRINT CONTENTSOF MATRIXA, AFTER L-U FACTORIZATION C WRITEC6,110) DO 50 I=l,NN WRITEC6,106) CACI,J>,J=l,NN) 50 CONTINUE C C CHECKCONDITION NUMBER BEFORE SOLVING ... C WRITEC6,200) COND IF CCOND.GT. l.D20) GOTO400 C C CONTINUETO SOLVESYSTEM ... C

Appendix C. "CHI" FORTRAN77 Code 238 CALL SOLVEDCA,NN,B,IPVT> C C PRINT OUT THE RESULTSOF VECTORx, FOR Ax=b C WRITEC6,102) DO 40 I=l,NN WRITEC6,103) I,BCI) 40 CONTINUE STOP C C ERRORHANDLING ROUTINE C 400 WRITEC6,601) COND 601 FORMAT(15X,'****** SYSTEMNOT SOLVED ******'/ * 15X,'*** CONDITIONNUMBER IS LARGECCOND=',Dll.4,')', * ***'///) C C READ/WRITEFORMATS C 90 FORMAT(A80) 100 FORMAT(8Fl0.4) 101 FORMATC3Fl0.2) 102 FORMAT(///lOX,'THE SOLUTIONVECTOR IS X=CXl X2 X3 X4 ... ) 1 / &30X,'WHERE... '/) 103 FORMATC20X,'XC',I2,')=',Dl5.7/) 104 FORMAT(3Fl0.5) 106 FORMAT(6CD15.7,3X)) 107 FORMAT(5X,A80///) 110 FORMAT(/////lOX,'*** CONTENTSOF MATRIXA, AFTERL-U',lX, &'FACTORIZATION*** 1 //l5X,'CMATRIX A CONTAINSU IN UPPER TRIANGLE' &/l5X,'AND MULTIPLIERSFOR LIN STRICT LOWERTRIANGLEJ 1 //) 200 FORMATC1Hl///l5X,'*** THE CONDITIONNUMBER =',Dll.4,2X,'***'//) 300 FORMAT(//////115,'*** CONTENTSOF MATRIXA ***'/ * 118,'BEFORE LU FACTORIZATION'/) 310 FORMAT(///lOX,'THE RHS VECTOR,B, IS B=CB1B2 B3 B4 ... ) 1 / &lOX,'WHERE... '/) 410 FORMAT(//15,'THE MOLECULARWEIGHTS ARE1 1 /) 411 FORMAT(//15,'THE DENSITIES ARE:'/) 412 FORMAT(//15,'THE BETACOEFFICIENTS USED TO FIT Xl2 ARE:'/) 415 FORMAT(//15,'THE WEIGHTFRACTIONS IN THE OVERALLMIXTURE'/ & Tl5,'AND IN PHASESD ANDDD ARE11 /) 420 FORMAT(//T5,'THE NUMBEROF SEGMENTSOF EACHCOMPONENT IS1 1 /) 499 FORMAT(4CD15.7,5X)) 500 FORMAT(//TS,'THE MOLESOF EACHCOMP., IN PHASESD ANDDD ARE:1 /) 520 FORMAT(//TS,'THE VOLUMEFRACTIONS OF EACH, IN PHASESD',lX, * 'AND DD ARE11 /) 530 FORMAT(//TS,'THE OVERALLVOLUME FRACTIONS ARE: 1 /) 701 FORMAT(//T5,'THE TOTALAMOUNT OF MASSIN THE MIXTURE,'/ & TlS,'AND THE AMOUNTIN PHASESD ANDDD, ARE:'/) END

Appendix C. "CHI" FORTRAN77 Code 239 SUBROUTINEDCOMPDCA,N,COND,IPVT,WKSP) REAL*8WKSP,A,ANORM,ATHOLD,COND,EPS,ONE,STORE,SUMCOL,YNORM, & ZERO,ZNORM INTEGERIPVT,I,ICOND,J,K,KK,KMINUS,KPLUSl,M,N,NMINUS DIMENSIONACN,N),WKSPCN),IPVTCN) DATAONE,ZERO /l.DO,O.DO/ C******************************************************************** C C DECOMPOSESA REAL MATRIX BY GAUSSIANELIMINATION C ANDESTIMATES THE CONDITIONNUMBER FOR THE MATRIX C ANDIN DOINGSO IT CALLSMCHEPS TO DETERMINETHE C MACHINEROUND-OFF ERROR C C USERSHOULD CALL SOLVED TO COMPUTESOLUTIONS TO C LINEARSYSTEMS ONCE THE MATRIXHAS BEENFACTORED. C C INPUT..... C C A= THE MATRIXTO BE TRIANGULARIZED C N = ORDEROF THE MATRIX C C OUTPUT.... C C A CONTAINSAN UPPERTRIANGULAR MATRIX U ANDA C PERMUTEDVERSION OF A LOWERTRIANGULAR MATRIX SO THAT C (PERMUTATIONMATRIX)*A = L*U C C COND= AN ESTIMATEOF THE CONDITIONOF A. IF C A IS SINGULARTO WORKINGPRECISION THEN C CONDIS SET TO 1.0/UNIT ROUND-OFFERROR C C !PVT= THE PIVOT VECTOR C IPVTCK) = THE INDEXOF THE K-TH ROW C IPVTCN) = C-l)**CNUMBEROF INTERCHANGES C C WORK=WORK SPACE VECTOR OF LENGTHN C C THE DETERMINANTOF A CANBE OBTAINEDON OUTPUTBY C DETCA)= IPVT*AC1,l)*AC2,2)* ... *ACN,N) C C******************************************************************** IPVTCN) = l IFCN.EQ.l) GO TO 140 NMINUS= N-1 C C CALCULATETHE 1-NORMOF A C ANORM= ZERO DO 20 J = l,N SUMCOL= ZERO DO 10 I= l,N

Appendix C. "CHI" FORTRAN77 Code 240 SUMCOL= SUMCOL+ DABSCACI,J)) 10 CONTINUE IFCSUMCOL.GT.ANORM)ANORM= SUMCOL 20 CONTINUE C C GAUSSIANELIMINATION WITH PARTIALPIVOTING C DO 70 K = l,NMINUS KPLUSl = K+l C C FIND PIVOT C M = K DO 30 I= KPLUSl,N IFCDABSCACI,K)).GT.DABSCACM,K)))M = I 30 CONTINUE IPVTCK) = M IFCM.NE.K) IPVTCN) = -IPVTCN) STORE= ACM,K) ACM,K) = ACK,K) ACK,K) = STORE C C SKIP IF PIVOT IS ZERO C IF CSTORE.EQ.ZERO)GO TO 70 C C COMPUTEMULTIPLIERS C DO 40 I= KPLUSl,N ACI,K) = -ACI,K)/STORE 40 CONTINUE C C INTERCHANGEAND ELIMINATE BY COLUMNS C DO 60 J = KPLUSl,N STORE= ACM,J) ACM,J) = ACK,J) ACK,J) = STORE IF CSTORE.EQ.ZERO)GO TO 60 DO 50 I= KPLUSl,N ACI,J) = ACI,J) + ACI,K)*STORE 50 CONTINUE 60 CONTINUE 70 CONTINUE C C ESTIMATETHE CONDITIONNUMBER C C SOLVECA-TRANSPOSE)*Y = E C DO 90 K = l,N STORE= ZERO

Appendix C. "CHI" FORTRAN77 Code 241 IFCK.EQ.l) GO TO 85 KMINUS= K-1 DO 80 I= l,KMINUS STORE= STORE+ ACI,K>*HKSPCI> 80 CONTINUE 85 ATHOLD= ONE IFCSTORE.LT.ZERO>ATHOLD = ZERO - ONE IF CACK,K>.EQ.ZERO)GO TO 150 HKSPCK)= -CATHOLD+ STORE)/ACK,K) 90 CONTINUE DO 110 KK = l,NMINUS K = N - KK STORE= ZERO KPLUSl = K + 1 DO 100 I= KPLUSl,N STORE= STORE+ ACl,K)*HKSPCK) 100 CONTINUE HKSPCK>= STORE M = IPVTCK) IFCM.EQ.K) GO TO 110 STORE= WKSPCM) HKSPCM>= HKSPCK) HKSPCK)= STORE 110 CONTINUE YNORM= ZERO DO 120 I = l,N YNORM= YNORM+ DABSCHKSPCI)) 120 CONTINUE C C SOLVEA*Z = Y **NOTE**---> THIS IS NOT SOLVINGTHE SYSTEM C IT IS ONLYCALLING SOLVED TO C AID IN CONDESTIMATION. C CALL SOLVEDCA,N,HKSP,IPVT) ZNORM= ZERO DO 130 I= l,N ZNORM= ZNORM+ DABSCHKSPCI>> 130 CONTINUE C C ESTIMATECONDITION NUMBER C COND= ANORM*ZNORM/YNORM IF CCOND.LT.ONE>COND = ONE RETURN C C 1 X 1 C 140 COND= ONE IF CACl,l).NE.ZERO) RETURN C

Appendix C. "CHI" FORTRAN77 Code 242 C EXACTSINGULARITY C 150 CALLMCHEPSCEPS) COND= ONE/EPS RETURN END C C********************************************************************** C SUBROUTINEMCHEPSCEPS) C C CALCULATETHE MACHINEUNIT ROUND-OFFERROR C REAL*8EPS,EPS1 EPS = 1. 10 EPS = EPS/2. EPS1 = 1. + EPS IFCEPS1.GT.l.) GO TO 10 EPS = EPS*2. RETURN END SUBROUTINESOLVEDCA,N,R,IPVT) REAL*8 A,R,STORE INTEGER IPVT,I,K,KK,KMINUS,KPLUSl,M,N,NMINUS DIMENSIONACN,Nl,RCN>,IPVTCN) C******************************************************************** C C SOLUTIONOF LINEARSYSTEM A*X = R C DO NOTUSE IF DCOMPDHAS DETECTEDSINGULARITY C C INPUT...... C C A= TRIANGULARIZEDMATRIX OBTAINED FROM DCOMPD C N = ORDEROF SYSTEM C R = RIGHTHAND SIDE VECTOR C IPVT = PIVOT FROMDCOMPD C C OUTPUT...... C C R = SOLUTION C C******************************************************************** IFCN.EQ.l) GO TO 50 C C FORWARDELIMINATION C NMINUS= N - 1 DO 20 K = 1,NMINUS KPLUSl = K + 1 M = IPVTCK) STORE= RCM)

Appendix C. "CHI" FORTRAN77 Code 243 RCM)= RCK) RCK) = STORE DO 10 I= KPLUSl,N RCI) = RCI) + ACI,K)*STORE 10 CONTINUE 20 CONTINUE C C BACKSUBSTITUTION C DO 40 KK = l,NMINUS KMINUS= N - KK K = KMINUS+ l RCK) = RCK)/ACK,K) STORE= -RCK> DO 30 I= l,KMINUS RCI) = RCI) + ACI,Kl*STORE 30 CONTINUE 40 CONTINUE 50 RCl) = RCl)/ACl,l) RETURN END

Appendix C. "CHI" FORTRAN77 Code 244 Appendix D. FORTRAN 77 Code for Predicting LLE Behavior

Based on the Flory-Huggins Model

Appendix D. "HSU" FORTRAN 77 Code 245 C********************************************************************* C PROGRAMNAME1 HSU CMINPACKSOLUTION OF NONLINEARSYSTEM C OF PHASEEQUILIBRIUM EQUATIONS USING C TERNARYFLORY-HUGGINS MODEL WITH C CONSTANTCHI PARAMETERS) C********************************************************************* C THIS IS A REAL RUN FOR SYSTEMB OF FIGURE 10.6 ALBERTSSON C********************************************************************* C C G. GREGORYBENGE 19-AUGUST-86 C C THE HSU ROUTINEUSES THE MINPACKSOFTWARE TO SOLVETHE C SYSTEMOF NONLINEAREQUILIBRIUM EQUATIONS GENERATED BY THE C FLORY-HUGGINSMODEL, EQUATIONS XIII-12 THRUXIII-14 C OF C FLORY, P.J. "PRINCIPLES OF POLYMERCHEMISTRY". C CORNELLUNIVERSITY PRESS, ITHACA, NEWYORK C P. 549 (1953). C C MINPACKSUBROUTINE HYBRID IS EMPLOYED. HYBRIDFINDS THE ZERO C OF A NONLINEARSYSTEM OF EQUATIONS. HYBRDIS A MODIFICATION C OF THE POWELLHYBRID METHOD. HYBRDSEARCHES FOR A ZERO OF THE C SYSTEMSBY MINIMIZINGTHE SUMOF SQUARESOF THE FUNCTIONS. C TWOOF THE CHARACTERISTICSOF HYBRDARE: Cl) THE CHOICEOF THE C CORRECTIONAS A CONVEXCOMBINATION OF THE NEWTONAND SCALED C GRADIENTDIRECTIONS, AND C2) THE UPDATINGOF THE JACOBIANBY C THE RANK-1METHOD OF BROYDEN. THE POWELLSEARCH ROUTINE WITH C LAGRANGEMULTIPLIERS CHOSEN BY THE METHODSUGGESTED BY LEVENBERG C HAS FIRST APPLIED TO TERNARYPOLYMER SOLUTIONS BY HSU AND C PRAUSNITZ(1974). C HSU, C.C.; ANDPRAUSNITZ, J.M. "THERMODYNAMICSOF POLYMER C COMPATIBILITYIN TERNARYSYSTEMS," MACROMOLECULES, C VOL. 7, NO. 3, PP. 320-324 (1974). C*********************************************************************** C MAINVARIABLES - WE SEEK SOLUTIONTO FVEC EQUATIONSIN SUBROUTINE C FCN FOR THE FOLLOWINGVARIABLES1 C XSP = VIP = VOL. FRAC. l IN PHASED (SPECIFIED) C XCl) = V2P = VOL. FRAC. 2 IN PHASED C V3P = VOL. FRAC. 3 IN PHASED C XC2) = VlDP = VOL. FRAC. 1 IN PHASEDD C XC3) = V2DP = VOL. FRAC. 2 IN PHASEDD C V3DP = VOL. FRAC. 3 IN PHASEDD C C V3P ANDV3DP HERE ELIMINATEDFROM THE EQUIL. EQNSUSING C THE FOLLOWINGTWO MASS BALANCES C V3P = l - VlP - V2P C V3DP = l - VlDP - V2DP C

Appendix D. "HSU" FORTRAN77 Code 246 C THENTHE 3 VARIABLESXC1),XC2),XC3) CORV2P,VlDP,V2DP) ARE FOUND C FROMTHE SOLUTIONOF THE 3 NONLINEAREQUIL. EQNS. CFVECCl-3)) C FVECCl) = DMUCl)- DMUC2) WHEREDMUCl)=DMU 1 IN PHASED C AND DMUC2)=DMU1 IN PHASEDD C FVECC2)= DMUC3)- DMUC4) C FVECC3)= DMUCS)- DMUC6) C C*********************************************************************** INTEGERJ,N,MAXFEV,ML,MU,MODE,NPRINT,INFO,NFEV,LDFJAC,LR,HPR DOUBLEPRECISION XTOL,EPSFCN,FACTOR,FNORM,ENORM,REALL2,DPMPAR DOUBLEPRECISION XC3),FVECC3),DIAGC3),FJACC3,3),RC6),QTFC3), & WA1C3),WA2C3),WA3C3),WA4C3) DOUBLEPRECISION DENSC3),MWC3),Xl,X2,X3,CHI12,CHI13,CHI23,XSP, & RG,TEMP,SPF,DMUC6),VlP,V2P,V3P,VlDP,V2DP,V3DP EXTERNALFCN LOGICALFLAG COMMONXl,X2,X3,CHI12,CHI13,CHI23,XSP,RG,TEMP,SPF,HPR,FLAG CHARACTER*80TITLE C C SPECIFY SIZE OF SYSTEM,I.E. ORDEROF MATRIX C N = 3 FLAG= .FALSE. C C INPUT1 CHI'S, DENS, MWCLAST TWO FOR CALCULATIONOF SEGMENTNO.) C CHI12=0.45DO CHI13=0.514DO CHI23=2.9DO DENSC1)=0.99823DO DENSC2)=1.21DO DENSC3)=1.2DO MWC1)=18.02DO MWC2)=6020.DO MWC3)=180.D3 C C CALCULATETHE NUMBEROF SEGMENTSIN EACHPOLYMER, X2,X3 C CALSOCALLED R> Xl=l.DO X2=CMWC2)/DENSC2))/CMWCl)/DENSCl)) X3=CMWC3)/DENSC3))/CMWCl)/DENSCl)) C WRITEC6,199) 199 FORMAT(//TS,'FIGURE10.6 - SYSTEMB - FROMALBERTSSON 1 //) WRITEC6,200) CHI12,CHI13,CHI23,Xl,X2,X3 200 FORMATC//T5,1 INTERACTIONPARAMETERS ARE: 1 ,T40, 1 CHI12=',F7.4/ & T40, 1 CHI13=1 ,F7.4/T40,'CHI23=',F7.4/ & /TS,'SEGMENT NUMBERSARE:',T35,'Xl= 1 ,F7.2/T35,'X2=',F7.2/ & T35,'X3= 1 ,F7.2//)

Appendix D. "HSU" FORTRAN77 Code 247 C C SPECIFY ONE OF THE SIX VOLUMEFRACTIONS==> XSP C ANDPROVIDE AN INITIAL GUESSOF SOLUTIONVECTOR X C C XSP=0.9409DO C XCl)=0. 0370DO EXPERIMENTALRESULTS FROM C XC2)=0.9235DO ALBERTSSON C XC3)=0.0250DO C XSP=0.9409DO XCl )=0. 0370DO XC2)=0.9235DO XC3)=0.0250DO C C WRITINGFLORY MODEL VARIABLES IN TERMSOF SOLUTIONVECTOR C ANDINTRODUCING TWO MASS BALANCES TO RELATEV3P TO VlP ANDV2P C VlP = XSP V2P = XCU V3P = l.DO - XSP - XCU VlDP = XC2) V2DP = XC3) V3DP = l.DO - XC2) - XC3) C WRITEC6,201) VlP,V2P,V3P,VlDP,V2DP,V3DP 201 FORMATC/TS,'THEINITIAL GUESS FOR THE SOLUTIONIS: 1 / & T20,'VlP =',F6.4,SX,'CSPECIFIED) 1 /T20,'V2P =',F6.4/ & T20,'V3P =',F6.4/T20,'VlDP =',F6.4/ & T20,'V2DP =1 ,F6.4/T20,'V3DP =1 ,F6.4//) C C CALCULATEPENALTY FUNCTION FACTOR FOR SOLVENTCHEM. POT. EQN. C CSPF) BASEDON AN AVERAGEOF PENALTYFUNCTIONS OF COMPS2 & 3 C TAKENAT INITIAL CONDITIONS C ALSO C HPR = HSU ANDPRAUSNITZ "R" VALUEFOR POSITIVE, EVEN C INTEGRALNUMBER FOR EXPONENTOF PENALTYFUNCTION C HPR = 4 SPF= CO.SDO*CDABSCV2P-V2DP)+DABSCV3P-V3DP) ))**HPR C WRITEC6,30) HPR,SPF 30 FORMAT(//TS,'HSU,PRAUSNITZr PARAMETERCHPR) =',I2/ & TS,'SOLVENT CHEM.POT. PENALTYFUNCTION (SPF) =',Dl4.7//) C C C SPECIFICATIONSFOR SUBR. HYBRD C C C SET XTOLTO THE SQUAREROOT OF THE MACHINEPRECISION. C UNLESSHIGH PRECISION SOLUTIONSARE REQUIRED, C THIS IS THE RECOMMENDEDSETTING.

Appendix D. "HSU" FORTRAN77 Code 248 C C XTOL = DSQRTC DPMPARCl)) C XTOL=O.lD-06 WRITEC6,202) XTOL 202 FORMATC//T5,'XTOL=',Dl4.7//) C C LDFJAC= 3 LR = 6 MAXFEV= 2000 ML= 2 MU= 2 EPSFCN= l.OD-10 MODE= 1 C DO 20 J = 1, N C DIAGCJ) = IO.DO C 20 CONTINUE C DIAGC3) = I.DO FACTOR= 1. OD+Ol NPRINT = 0 WRITEC6,203) LDFJAC,LR,MAXFEV,ML,MU,EPSFCN,MODE,FACTOR,NPRINT 203 FORMATC/T5,'SPECIFICATI0NOF "HYBRD"PARAMETERS'/Tl5,'LDFJAC =', & I2/Tl5,'LR =',I2/Tl5,'MAXFEV =',I6/Tl5,'ML =',I2/Tl5,'MU =', & I2/Tl5,'EPSFCN =1 ,Dl0.4/Tl5,'MODE =',Il/Tl5,'FACTOR =',Dl0.4/ & T15,'NPRINT =1 ,I2//) C C C C CALL SUBR. HYBRDTO SOLVESYSTEM C C C CALL HYBRDCFCN,N,X,FVEC,XTOL,MAXFEV,Ml,MU,EPSFCN,DIAG, * MODE,FACTOR,NPRINT,INFO,NFEV,FJAC,LDFJAC, * R,LR,QTF,HAl,HA2,HA3,HA4) FNORM= ENORMCN,FVEC) C C CONVERTSOLUTION VECTOR TO FLORYMODEL VARIABLES C VIP = XSP V2P = XCI) V3P = I.DO - XSP - XCI) VlDP = XC2) V2DP = XC3) V3DP = 1. DO - XC2) - XC3) C DMUCl) = C DLOGCVlP)+ Cl.DO-VIP) - V2P*CXl/X2) & - V3P*CXl/X2) + CCHI12*V2P+CHI13*V3P)*CV2P+V3P) & + CHI23*CX1/X2)*V2P*V3P) / Xl DMUC2)= C DLOGCV2P)+ Cl.DO-V2P) - VlP*CX2/Xl) - V3P*CX2/X3)

Appendix D. "HSU" FORTRAN77 Code 249 & + CCHI12*CX2/Xl)*V1P+CHl23*V3P)*CV1P+V3P) & + CHI13*CX2/Xl)*V1P*V3P) / X2 DMUC3)= C DLOGCV3P)+ Cl.DO-V3P) - VlP*CX3/Xl) - V2P*CX3/X2) & + CCHI13*CX3/Xl)*V1P+CHl23*CX3/X2)*V2Pl*CV1P+V2P) & + CHI12*CX3/Xl)*VlP*V2P) / X3 DMUC4)= C DLOGCVlDP)+ Cl.DO-VlDP) - V2DP*CXl/X2) & - V3DP*CX1/X2) + CCHI12*V2DP+CHI13*V3DP)*CV2DP+V3DP) & + CHI23*CXl/X2)*V2DP*V3DP) / Xl DMUCS)= C DLOGCV2DP)+ Cl.DO-V2DP) - VlDP*CX2/Xl) & - V3DP*CX2/X3) & + CCHI12*CX2/Xl)*V1DP+CHI23*V3DP)*CV1DP+V3DP) & + CHI13*CX2/Xl)*V1DP*V3DP) / X2 DMUC6)= C DLOGCV3DP)+ Cl.DO-V3DP) - V1DP*CX3/Xl) & - V2DP*CX3/X2) & + CCHl13*CX3/Xl)*V1DP+CHl23*CX3/X2)*V2DP)*CV1DP+V2DP) & + CHI12*CX3/Xl)*V1DP*V2DP) / X3 C C CALCULATE"REAL" L2 NORMOF THE RESIDUALSOF THE C EQUILIBRIUMFUNCTIONS C REALL2= DSQRTCCDMUC1)-DMUC4))**2 + CDMUC2)-DMUC5))**2 & + CDMUC3)-DMUC6))**2) C C C PRINTOUTFINAL APPROXIMATESOLUTION AND RELATED INFO C WRITE (6,1000) FNORM,REALL2,NFEV,INFO WRITE (6,1001) VlP,V2P,V3P,VlDP,V2DP,V3DP STOP 1000 FORMATClX,28H ******* SOLUTION ********, // & 5X,28H FNORMFROM SUBROUTINE HYBRD,D15.7 // & 5X,31H FINAL LZ NORMOF THE RESIDUALS,D15.7 // & 5X,31H NUMBEROF FUNCTIONEVALUATIONS,110 // & 5X,15H EXIT PARAMETER,16X,I10//) 1001 FORMATC/TS,'THEFINAL APPROXIMATESOLUTION IS:'/ & TZO,'VlP =',F6.4,5X,'CSPECIFIED)'/T20,'V2P =1 ,F6.4/ & T20,'V3P =1 ,F6.4/T20,'VlDP =1 ,F6.4/ & T20,'V2DP =1 ,F6.4/T20, 1 V3DP =',F6.4//) C C LAST CARDOF MAINPROGRAM -HSU C END SUBROUTINEFCN CN,X,FVEC,IFLAG) C********************************************************************** C SUBROUTINEFCN C********************************************************************** C DOUBLEPRECISION XCN),FVECCN) DOUBLEPRECISION XSP,CHI12,CHI13,CHI23,Xl,X2,X3,RG,TEMP,SPF, & DMUC6),APF,NXI,V1P,V2P,V3P,V1DP,V2DP,V3DP INTEGERN,IFLAG,HPR

Appendix D. "HSU" FORTRAN77 Code 250 LOGICALFLAG COMMONXl,X2,X3,CHl12,CHI13,CHl23,XSP,RG,TEMP,SPF,HPR,FLAG c------c APF & NXI FOR SCALING, C APF I ADDITIONALPENALTY FUNCTION C NXI : IS SET EQUALTO THE XCI) WHICHIS NEGATIVE c------c VIP = XSP V2P = XCl) V3P = I.DO - XSP - X(l) VlDP = XC2) V2DP = XC3) V3DP = I.DO - XC2) - XC3) C IF CIFLAG .NE. 0) GOTOS c------c INSERT PRINT STATEMENTSHERE WHEN NPRINT IS POSITIVE C C WRITEC6,8) RG,TEMP,Xl,X2,X3,CHI12,CH1l3,CHI23,XSP,SPF C 8 FORMAT(//TS,'DATAECHO FROM SUB. FCN:', C & T40,'RG=',F6.3/T40,'TEMP CINK) =',F7.2/ C & T40,'Xl=',F5.4/T40,'X2=',F6.2/T40,'X3=',Fl0.2/ C & T40, 1 CHI12=',F5.2/T40,'CHI13=',F5.2/T40,'CHI23=',F5.2/ C & T40,'XSP SPECIFIED=',FS.2/T40,'SPF=',Dl4.7/ C & T40,'HPR=',I2//) c------WRITEC6,9) CI,DMUCl),1=1,6) 9 FORMATC/T5,'RELATIVECHEM. POT.1',6CT30,'DMUC',I1, & ')=',Dl4.7/)//) WRITEC6,10) CI,FVECCl),1=1,3) 10 FORMATC/T5,'FUNCTI0NVECTOR EVALS: 1 ,3CT30,'FVECC',I1, & ')=',Dl4.7/)//) WRITEC6,12) VlP,V2P,V3P,VlDP,V2DP,V3DP 12 FORMATC/T5,1 SOLUTI0NVECTOR X: 1 / & T20,'VlP =',F6.4,5X,'CSPECIFIED)'/T20,'V2P =',F6.4/ & T20,'V3P =',F6.4/T20,'VlDP =',F6.4/ & T20,'V2DP =1 ,F6.4/T20,'V3DP =',F6.4//) RETURN c------5 CONTINUE C C WRITEC6,32) FLAG DMUCl) = C DLOGCVlP)+ Cl.DO-VIP) - V2P*CXl/X2) & - V3P*CXl/X2) + CCH1l2*V2P+CHI13*V3P)*CV2P+V3P) & + CHI23*CX1/X2)*V2P*V3P) / Xl C C WRITEC6,32) FLAG IF CV2P .LT. O.DO) THEN DMUC2)=0.DO FLAG= .TRUE.

Appendix D. "HSU" FORTRAN77 Code 251 NXI = VZP ELSE DMUCZ)= C DLOGCVZP)+ Cl.DO-VZP) - VlP*CXZ/Xl) - V3P*CX2/X3) & + CCHII2*CX2/Xl)*V1P+CHI23*V3P)*CV1P+V3P) & + CHII3*CX2/Xl)*V1P*V3P) / X2 ENDIF C C WRITEC6,32) FLAG IF CV3P .LT. O.DO) THEN DMUC3)=0.DO FLAG= .TRUE. NXI = V3P ELSE DMUC3)= C DLOGCV3P)+ Cl.DO-V3P) - VlP*CX3/Xl) - V2P*CX3/X2) & + CCHI13*CX3/Xl)*VlP+CHI23*CX3/XZ)*VZP)*CVlP+V2P) & + CHII2*CX3/Xl)*V1P*V2P) / X3 ENDIF C C WRITEC6,32) FLAG IF CVlDP .LT. O.DO) THEN DMUC4)=0.DO FLAG= .TRUE. NXI = VlDP ELSE DMUC4)= C DLOGCVlDP)+ Cl.DO-VlDP) - V2DP*CXl/X2) & - V3DP*CXl/X2) + CCHI12*V2DP+CHII3*V3DP)*CV2DP+V3DP) & + CHI23*CXl/XZ)*V2DP*V3DP) / Xl ENDIF C C WRITEC6,32) FLAG IF CV2DP .LT. O.DO) THEN DMUC5)=0.DO FLAG= .TRUE. NXI = V2DP ELSE DMUC5)= C DLOGCVZDP)+ Cl.DO-V2DP) - VlDP*CX2/Xl) & - V3DP*CX2/X3) & + CCHII2*CX2/Xl)*V1DP+CHI23*V3DP)*CV1DP+V3DP) & + CHII3*CX2/Xl)*V1DP*V3DP) / X2 ENDIF C C WRITEC6,32) FLAG IF CV3DP .LT. O.DO) THEN DMUC6)=0.DO FLAG= .TRUE. NXI = V3DP ELSE DMUC6)= C DLOGCV3DP)+ Cl.DO-V3DP) - VlDP*CX3/Xl) & - V2DP*CX3/X2) & + CCHI13*CX3/Xl)*VlDP+CHI23*CX3/X2)*V2DP)*CVlDP+V2DP) & + CHII2*CX3/Xl)*V1DP*V2DP) / X3

Appendix D. "HSU" FORTRAN77 Code 252 ENDIF C C WRITEC6,32) FLAG IF CFLAG) THEN APF = DEXPC-l.D2*NXI) C WRITEC6,31) FLAG,NXI,APF 31 FORMATCIIT5,'FLAGIS ',LSITS,'NXI = ',Dl4.71 & T5,'APF = ',D14.71) ELSE APF=l.DO ENDIF C C WRITEC6,32) FLAG FVECCl) = APF*C DMUCl)- DMUC4)) / (SPF) FVECC2>= APF*C DMUC2)- DMUCS)) / CCV2P-V2DP>**HPR) FVECC3>= APF*C DMUC3)- DMUC6)> / CCV3P-V3DP>**HPR) C C RESET FLAG& SPECIAL PARAMETERS C FLAG= .FALSE. NXI=O.DO APF=l.DO C C WRITEC6,32) FLAG 32 FORMATCIT5,'NEGATIVEVOLUME FRACTION FLAG IS ',LS) C RETURN C C LAST CARDOF SUBROUTINEFCN. C END

Appendix D. "HSU" FORTRAN77 Code 253 Appendix E. Ternary LLE Predictions Using the Theory of

Ogston

Appendix E. Ogston Theory Calculations 254 The use of the Ogston theory was discussed in detail in Section 6.2.3. This Appendix. describes the process used for obtaining the data shown in Table 14 and Figure 46. The system being modeled is water-PEG6000-dex.tran(D17) at 20°C. This procedure is a two-step process. First, the

"OGSTON" routine of Appendix. F is used with appropriate values of the parameters (a, c, and d) to predict the binodal curve. The "OGSTON'" computer code is based on molality, so therefore it is necessary to convert the experimental weight fraction data of Albertsson (3) to molality for easy comparison of experiment and theory. The "MOLAL"' code in Appendix G was written for this purpose.

The data required for the phase equilibrium calculation of "OGSTON" is input within the computer code (see Appendix r), thus an input file is not necessary. The output file generated by

"OGSTON" is shown in Table 22. The essence of the output is that for the specified molality of component 3 in phase P, the initial guess for the molality of component 3 in phase DP, and the tolerance, the equilibrium molalities of the polymers as predicted by the Ogston theory are given.

The "MOLAL" routine requires input data as shown in Table 23. The molecular weights and weight fraction data (of total system, phase P, and phase DP) are needed. The output from

"MOLAL" is displayed in Table 24. Results of intermediate calculations are given as well as the molalities of all the components.

Appendix E. Ogston Theory Calculations 255 Table 22. "OGSTON" Output File for Prediction of Binodal Curve of Water-PEG6000-Dextran(D17) System at 20°C

FOR COEFFICIENTS OF a = 580.000 C = 320.000 d = 520.000 AND AN INITIAL GUESS FOR M3DP OF O.IOOOD-01

AND FOR A TOLERANCE OF O.lOOOOD-05 NE\VfON-RAPHSON CONVERGED AFTER 53 ITERATIONS TO YIELD:

1------M2P ------1------M2DP ------1------M3P ------1------M3DP ------1 0.1293D-01 0.3077D-02 0.2000D-02 0.9910D-02

Appendix E. Ogston Theory Calculations 256 Table 23. "'MOLAL"' Input Data File for Conversion of Weight Fraction to Molality

SYSTEM A (FIG. 10.12): WATER(l)-PEG6000(2)-D17(3) AT 20 C 18.02 6750. 23000. 0.8669 0.0477 0.0854 0.8509 0.0300 0.1191 0.8800 0.0659 0.0541

Appendix E. Ogston Theory Calculations 257 Table 24. "MOLAL" Output Data File for Conversion of Weight Fraction to Molality

SYSTEM A (FIG. 10.12): WATER(l)-PEG6000(2)-Dl7(3) AT 20 C

THE MOLECULAR WEIGHTS ARE:

18.02 6750.00 23000.00

THE WEIGHT FRACTIONS IN THE OVERALL MIXTURE AND IN PHASES D AND DD ARE:

0.86690000 + 00 0.47700000-01 0.85400000-0 l 0.85090000 + 00 0.30000000-01 0.11910000+ 00 0.88000000 + 00 0.65900000-0 l 0.54100000-01

THE TOT AL AMOUNT OF MASS IN THE MIXTURE, AND THE AMOUNT IN PHASES D AND DD, ARE:

100.0000 48.7487 51.2513

THE MASS OF SOLVENT (kg) IN PHASES D AND DD IS:

0.0415 0.0451

THE MOLES OF EACH COMP., IN PHASES D AND DD ARE:

0.23019020 + 0 l 0.21666090-03 0.25243360-03 0.25028370 + 0 l 0.50036420-03 0.12055190-03

THE MOLALITIES OF EACH COMP., IN PHASES D AND DD ARE: 55.4938957 0.0052232 0.0060856 55.4938957 0.0110943 0.0026729

Appendix E. Ogston Theory Calculations 258 Appendix F. FORTRAN 77 Code for Predicting LLE Behavior

Via the Ogston Theory

Appendix F. "OGSTON" FORTRAN 77 Code 259 C*********************************************************************** C PROGRAM: OGSTON C C G. Gregory Benge, 10-JUNE-1986 C C DESCRIPTION: This program uses the Ogston theory to predict binodal C curves for systems of macromolecules. The equations and values C for a, c, and d come from: C Edmond, E.; and Ogston, A.G. "An Approach to the Study of C Phase Separation in Ternary Aqueous Systems," Biochem. J. C Vol. 109, pp. 569-576 (1968). C C METHOD:This program solves the above nonlinear equation by using C the Newton-Raphson iteration. C C INPUT: For the system of interest, the values of a, c, and d must C be entered. Also, the weight fractions of m3' CM3P) and an C initial guess for m3'' CM3DPI) must be entered. C C OUTPUT: The results of the tie-line points of the binodal curve C are output in molality for easy comparison with the data of C Edmond and Ogston or Albertsson. C C C*********************************************************************** REAL*8 A,C,D,M3P,M3DP,M3DPI,M2P,M2DP,W3P,W3DP,P,Q,F,DF, * DQ,DP,TERM1,TERM2,M3DPN,TOL INTEGERITER C C INPUT DATA C A= 580.DO C = 320.DO D = 520.DO M3P = 0.002DO M3DPI = O.OlDO TOL = l.D-06 C C ITER = 0 M3DP = M3DPI C C BEGIN NEWTON-RAPHSONITERATION C 10 ITER = ITER + 1 P = C/A * DLOGCM3P/M3DP)+ CCC*D-A**2)/A)*CM3P-M3DP) Q = - C DLOGCM3P/M3DP)+ D*CM3P-M3DP)) F = Q/A + CM3P-M3DP)+ CCC*Q**2)/(2.DO*A**2)*CCDEXPCP)+l.D0)/ * CDEXPCP)-1.DO))) + CD/2.DO>*CM3P**2-M3DP**2)+ * CQ/CDEXPCP)-l.DO))*CM3P*DEXPCP)-M3DP) DQ = Cl.DO/M3DP) - D

Appendix F. "OGSTON"FORTRAN 77 Code 260 DP= - C/CA*M3DP)- CC*D - A**2)/A TERMl = C CDEXPCP)+l.DO)/CDEXPCP)-1.DO) )*2.DO*Q*DQ * - C2.DO*CQ**2)*DEXPCP)*DP)/( CDEXPCP)-l.D0l**2) TERM2= M3P*CCCDEXPCP)-1.DO)*DEXPCP)*DQ - Q*DEXPCP)*DP * - CDEXPCP)-l.DO)*CM3DP*DQ+Q)- Q*M3DP*DEXPCP)*DP) * / ((DEXPCP)-1.DO)**Z) ) DF = (l.DO/A)*DQ - l.DO + (C/(2.DO*A**2))*TERM1 - D*M3DP+ TERM2 C M3DPN= M3DP- F/DF IF ( DABSCM3DPN-M3DP).LE. TOL ) GOTO20 C C IF ABOVECONDITION IS NOT SATISFIED, SET M3DP=M3DPN C ANDRETURN TO LINE 10 ANDCONTINUE ITERATION C M3DP= M3DPN GOTO10 C 20 CONTINUE C C IF CONDITIONWAS SATISFIED, ITERATIONIS COMPLETE, C CONTINUETO CALCULATEREMAINING RESULTS AND PRINT C M2DP = Q/(A*CDEXPCP)-1.DO)) M2P = M2DP*DEXPCP) C C TO PRINT RESULTS C WRITEC6,70) A,C,D 70 FORMAT(///TlO,'FOR COEFFICIENTSOF'/Tl5,'a = ',F8.3/Tl5, * 'c = ',F8.3/Tl5,'d = ',F8.3/) WRITEC6,80) M3DPI 80 FORMAT(TlO,'AND AN INITIAL GUESS FOR M3DPOF 1 ,Dll.4/) WRITEC6,90) TOL 90 FORMAT(TlO,'AND FOR A TOLERANCEOF ',Dll.5/) WRITEC6,95) ITER 95 FORMAT(TlO,'NEWTON-RAPHSON CONVERGED AFTER',16,lX,'ITERATIONS' * ,lX,'TO YIELD:1 /) WRITEC6,100) 100 FORMAT(/TS, 1 1----- M2P -----,----- M2DP-----', * ':----- M3P -----:----- M3DP-----1 1 /) WRITEC6,110) M2P,M2DP,M3P,M3DP 110 FORMAT(T7,Dll.4,T23,Dll.4,T40,Dll.4,T57,Dll.4/) END

Appendix F. "OGSTON"FORTRAN 77 Code 261 Appendix G. FORTRAN 77 Code for Conversion of Weight

Fraction Data to Molality Basis

Appendix G. "MOLAL" FORTRAN 77 Code 262 C*********************************************************************** C PROGRAM: MOLAL C*********************************************************************** C C G. GREGORYBENGE 21-JUNE-86 C C DESCRIPTION: PERFORMSCONVERSION FROM WEIGHT FRACTIONS C TO MOLALITIES. THIS PROGRAMIS USEDTO CONVERT C EXPERIMENTALCWT. FRAC.) DATAOF ALBERTSSONINTO C MOLALITIESTO COMPAREWITH THEORETICALPREDICTIONS C FROMTHE OGSTONTHEORY CAND ROUTINE 'OGSTON'). C C C INPUT C ------C Cl) TITLE, CA80) C (2) MWCI) , C3F10.2) C C3) EXPTL. WEIGHTFRACTIONS C WFCI) , C3F10.4) C WFDCI) , C3Fl0.4) C WFDDCI), C3Fl0.4) C C C*********************************************************************** C DOUBLEPRECISION MWC3),WFC3),WFDC3),WFDDC3), & NDC3),NDDC3),NC3),MLDC3),MLDDC3),MSD,MSDD, & Pl,P2,Ll,L2,LRR INTEGERIPVTC4),NN CHARACTER*80TITLE C C READINPUT: C Cl) TITLE C C2) MOLECULARWEIGHTS CMW) C (3) EXPERIMENTALWEIGHT FRACTIONS C WFCI) - WT.FRAC. IN TOTALMIXTURE C WFDCI) - WT.FRAC. IN PHASED C WFDDCI>-WT.FRAC. IN PHASEDD

READCS,90) TITLE WRITEC6,107) TITLE C READCS,101) CMWCI),I=l,3) WRITEC6,410) WRITEC6,101) CMWCI),I=l,3) C READCS,100) CWFCJ),J=l,3) READCS,100) CWFDCJ),J=l,3) READCS,100) CWFDDCJ),J=l,3) WRITEC6,415) WRITEC6,499) CWFCI),I=l,3)

Appendix G. nMOLALnFORTRAN 77 Code 263 WRITEC6,499) CWFDCI),I=l,3) WRITEC6,499) CWFDDCI),I~l,3) C C USING THE LEVERRULE AND A MASSBALANCE l00-P3=LRR*P3 --> P3CLRR+l)=l00 C SO THAT P3 = 100 / CLRR+l) ; AND P2 = LRR*P3 C Pl= 100.DO P3 = Pl/CLRR+l.DO) P2 = LRR*P3 C C ALSO P2 = AMOUNTOF MASSIN PHASED C P3 = AMOUNTOF MASSIN PHASEDD C WRITEC6,701) WRITEC6,100) Pl,P2,P3 C C CONVERTWEIGHT FRACTIONS CWFD,WFDD> TO C MOLALITIESCMLD,MLDD) C C MOLALITYCI)= MOLESOF Cl) C ------C kg OF SOLVENT C C C C CALCULATEMASS Ckg) OF SOLVENT= MSD,MSDD C FOR CALCULATEMOLES OF SOLVENTCl) = NDC1),NDC2) C EACH " " " POLYMER(2) SEGMENTS=NDC2),NDDC2) C PHASE n n " POLYMER(3) SEGMENTS=NDC3),NDDC3) C C

Appendix G. "MOLAL"FORTRAN 77 Code 264 C MASSOF SOLVENTIN PHASESD ANDDD, IN KILOGRAMS C MSD= P2*WFDCl)/l.D3 MSDD= P3*WFDD(l)/l.D3 WRITEC6,420) WRITEC6,l00) MSD,MSDD C C MOLESIN PHASESD ANDDD C NDCl) = WFDCl)*P2/MWCl) NDD(l) = WFDDCl)*P3/MWCl) NDC2) = WFDC2)*P2/MWC2) NDDC2) = WFDDC2)*P3/MWC2) NDC3) = WFDC3)*P2/MWC3) NDDC3) = WFDDC3)*P3/MWC3) C WRITEC6,500) WRITEC6,499) CNDCI),I=l,3) WRITEC6,499) CNDDCI),I=l,3) C C CALCULATEMOLALITIES C DO 10 I=l,3 MLDCI) = NDCI)/MSD MLDDCI)= NDDCI)/MSDD 10 CONTINUE WRITEC6,520) WRITEC6,108) CMLDCI),I=l,3) WRITEC6,108) CMLDDCI),I=l,3) C C READ/WRITEFORMATS C 90 FORMATC A80) 100 FORMAT<8Fl0. 4) 101 FORMATC.3Fl0. 2) 106 FORMAT(6CD15.7,3X)) 107 FORMAT(5X,A80///) 108 FORMAT<3Fl5. 7) 410 FORMAT(//T5,'THE MOLECULARWEIGHTS ARE: 1 /) 415 FORMAT(//T5,'THE WEIGHTFRACTIONS IN THE OVERALLMIXTURE'/ & Tl5,'AND IN PHASESD ANDDD ARE:1 /) 420 FORMAT

Appendix G. "MOLAL"FORTRAN 77 Code 265 The vita has been removed from the scanned document