€1.53... “mm“ : .5... .7” [rainq :3 , .r. LU'HON R 80 SAND MAcRojMfOLEcu ”Viscosm V . _. _ . : avg. ééwgmfimg i“ Date/4’7 55/ 0-7 639 a , I 1‘ r!) ') A 3‘11!» “ha. 3‘ MlLAhé... ., .2 Lit: University —_v This is to certify that the thesis entitled VISCOSITY AND THERMODYNAMICS OF MACROMOLECULAR SOLUTIONS presented by Bakulesh Navaranglal Shah has been accepted towards fulfillment of the requirements for Ph-D- #degree in anr. ,4 x, 211(7// ' ékéi1”z/ ._—i Major professor 2 5/775 {fr—fa 0 {p ABSTRACT i‘in VISCOSITY AND THERMODYNAMICS OF MACROMOLECULAR SOLUTIONS By Bakulesh Navaranglal Shah In this work, viscosity and its shear dependence were measured for the solutions of polystyrene (PS) and styrene (ST)- acrylonitrile (ACN) copolymers (SAN) in the solvents benzene, methyl ethyl ketone (MEK), dioxane and dimethylformamide (DMF). The low shear viscosity data indicate that for a polymer, the chemical structure of the solvent has a significant influence on fiscosity in both dilute and moderately concentrated solutions. flfis finding is contrary to the widely held view that the nature of the solvent is unimportant when considering viscosities of nmderately concentrated solutions. In a dilute solution the relative viscosity, “r’ in a poor solvent is lower than that in a good solvent. The reverse is the case at higher concentrations, n in a poor solvent being r several orders of magnitude larger than that in a good solvent. Mots of solution viscosity against concentration in thermo— dynamically good andixxar solvents, therefore, have different Hopes and the curves for different solvents cross over each other Eta particular concentration. As the proportion of ACN in a Bakulesh Navaranglal Shah copolymer increases, the "cross-over" concentration--the concentra- tion at which relative viscosities in good and poor solvents are the same--decreases. In dilute solution a polymer molecule exists as an isolated chain. In more concentrated solutions the polymer molecules over- lap and are entangled. Solution viscosity depends upon the first power of concentration in dilute solutions and upon the fifth power in concentrated solutions. This has led to the concept of a critical value of concentration called the entanglement concen- tration where the slope of a viscosity concentration plot is supposed to change dramatically from one to five. The estimate of onset-of—entanglement concentration, c for PS of 501,000 weight ent’ average molecular weight in only good solvents is correlated by a characteristic value of the product cM. The data of this work indicate that cent of a polymer depends on the thermodynamic inter- action between the polymer and the solvent and the solvent effects cannot be neglected. Thus the concept of a “universal" critical entanglement concentration for a particular polymer is invalid. The value of c is lower in poor solvents than that in good sol- ent vents; e.g., for high molecular weight azeotropic SAN copolymer, c is equal to 6 gm/dl in DMF (good solvent) while in benzene ent (poor solvent) it is equal to 3 gm/dl. Polymer solution viscosities are often correlated with concentration and molecular weight using a power law correlation of viscosity with the product ch. The value of b has often been considered in the past to be a universal value of 0.68. This value Bakulesh Navaranglal Shah was found to be inadequate in this work for correlating the data in poor solvents. The value of b depends on the thermodynamic quality of the solvent and it seems to be related to the Mark—Houwink exponent a in the intrinsic viscosity-molecular weight relation, [n] = KMa, which is known to be dependent on the nature of the solvent; a being 0.5 for O-solvents and 0.8 for good solvents. The Simha correlation was found to unify the same data quite well up to high concentrations. M. C. Williams has developed a thermodynamic-hydrodynamic nwlecular rheological model for prediction of polymer solution vis- cosity in moderately concentrated solutions. It was found that Nflliams' model for predicting low shear viscosity, when used with a modified Frankel and Acrivos friction coefficient, gave a better prediction in good solvents than in poor solvents. This model gave order of magnitude estimates of viscosity of moderately con— centrated polymer solutions but failed at higher concentrations where entanglements of polymer chains are of significant density. In most polymer solutions of the type studied in this investigation, the solution viscosity depends upon shear rate; i.e., the solutions are pseudoplastic. It was found that the slope of the non-Newtonian decrease in viscosity with increasing flmar rate is a function of mechanical formation and break—up of entanglements and the polymer—solvent thermodynamic forces are mfimportant. Of the many suggested forms of the relaxation parameter, To, the Graessley form TO a nOM/CT (l + BCM) Bakulesh Navaranglal Shah for dependence on c was observed to be adequate for the systems considered. For the study of the influence of polymer-solvent thermo- dynamics on the viscosity of polymer solutions, two samples of PS and four samples of SAN copolymers were synthesized by free radical bulk polymerization at 60°C using a-a'-azo-bis-isobutyronitrile initiator. The PS homopolymers, PS-l and PS-Z, were of 185,000 and 50l,000, respectively, weight average molecular weight, Mw' The SAN copolymers were SAN C-l of l5 weight per cent ACN content and of 290,000 Mw’ SAN C-2 of 24 weight per cent ACN content and of 180,000 Mw’ SAN C-2' of 23 weight per cent ACN content and of 666,000 Mw’ and SAN C-3 of 38 weight per cent ACN content and of 332,000 MW. The kinetics of SAN copolymerization could not be described by either the chemical-controlled or the diffusion-controlled termination mechanism. Both the mechanisms appear to be acting simultaneously and a single parameter--¢ in the first and kt(l2) in the second-~kinetic expression appears inadequate to describe the rate of SAN bulk copolymerizations. The stiffness factor, 0, which is a measure of short-range interaction in polymer chains was found to be higher for each of the SAN copolymers than those for the individual homopolymers, PS and polyacrylonitrile. This indicates that in the unperturbed state the copolymers are more extended than the constituent homo- polymers. Bakulesh Navaranglal Shah The viscosity measurements were made by using capillary viscometers and a cone-and—plate viscometer at 30°C. The solvents that were selected covered a range of polymer—solvent thermodynamic interactions. A light scattering photometer was used to measure weight average molecular weights of the polymers. From the above measurements, the expansion factor, a, and the Flory thermodynamic parameter, x], were calculated for each of the polymer—solvent systems. The influence of ACN content on thermodynamic interaction can be clearly seen from the values of intrinsic viscosity, [n], q,second virial coefficient, A2, and X1- The better a solvent, the higher are the values of [n], a, A2 and cent and the lower is the value of X] for a polymer in that solvent. VISCOSITY AND THERMODYNAMICS OF MACROMOLECULAR SOLUTIONS By Bakulesh Navaranglal Shah A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1975 To my parents NAVARANG and JASHVANTI and my brother NIRAD ACKNOWLEDGMENTS I would like to express my deep appreciation to my major professor, Dr. R. F. Blanks, for providing constant guidance, assitance and encouragement throughout the course of this investi— gation, and also for his painstaking review of the manuscript. Gratitude is expressed and thanks extended to Dr. J. B. Kinsinger forlfis generous assistance and guidance during the course of this work. Many hours of fruitful discussion with him have furthered my understanding of macromolecules. Sincere appreciation is extended to Drs. M. H. Chetrick, D. K. Anderson and C. M. Cooper for their review of the manuscript. Thanks are also due to Dr. J. Goddard and Bill Talbot of the University of Michigan for providing unlimited use of the Weissenberg Rheogoniometer, to Dr. N. Deal and Dr. J. Trujillo of Michigan State University for their assistance in the use of the light scattering apparatus, to Mr. Donald Childs of the machine shop and Mr. Robert Rose of the electronic shop for their willing- ness to assist when an apparatus did not cooperate, and to Dow Chemical Company for gel permeation chromatography results. I am indebted to the Division of Engineering Research and Um Petroleum Research Fund administered by the American Chemical Society for providing financial support during the course of my graduate work. iii Sincere gratitude is due to my parents for their constant moral support and encouragement. Special thanks are expressed to my brother Nirad for providing thoughtful guidance, encouragement and genuine wisdom throughout my graduate studies. iv TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES LIST OF APPENDICES Chapter I. INTRODUCTION . A. B C. D Motivation Goals . . Polymers and Solvents Experimental Method POLYMERIZATION AND PREPARATION OF SAMPLE MATERIALS . . . A. B I (7311311 Initiator Purification l. Initiator 2. Monomers Homopolymerization . l. Rate of Reaction . 2. Molecular Weight Copolymerization l. Copolymer— —Monomer Composition Equation. With Conversion 3. Rate of Copolymerization a. Chemical —controlled termination b. Diffusion- controlled termination c. Conversion as a function of time d. Small scale experiments . e. Discussion . Large Scale Polymerization . Chemical Analysis of Copolymers 2. Variation of Copolymer Composition Monomer Reactivity Ratios From Composition of Copolymers . Molecular Weights and Molecular weight Distribution of Polymer Samples V Page viii xii . xviii Chapter III. IV. VI. BACKGROUND AND THEORY . A. Intrinsic Viscosity and Expansion Factor 8. Light Scattering and Second Virial Coefficient . . C. Viscosity Correlation Techniques . 1. Power Law Correlation 2. Simha' 5 Correlation . . . D. Williams Model for Zero Shear Viscosity EXPERIMENTAL APPARATUS AND MEASUREMENTS . A. Viscometry l. Capillary Viscometer a. Practice . b. Calibration . . 2. Cone- and- Plate Viscometer . a. Practice b. Reservoir chamber c. Calibration 8. Light Scattering l. Photometer . a. Practice . g. b. Calibration . 2. Differential Refractometer a. Review . b. Practice c. Calibration d. Measurements . PRESENTATION OF EXPERIMENTAL DATA . A. Intrinsic Viscosities and Huggins Constants B. Cone-and- Plate Viscosities . . . C. Refractive Index Increments . . 0. Molecular Weights and Second Virial Coefficients . E. Discussion of Experimental Accuracy . RESULTS AND DISCUSSION A. Thermodynamics and Configuration of Polymer Chains . l. Intrinsic Viscosity and Expansion Factor . 2. Stiffness Factor . . . 3. Second Virial Coefficient . 4 Flory Thermodynamic Parameter vi Page 112 II8 I30 I32 I32 I32 I37 I40 I47 Chapter Page B. Zero Shear Viscosities . . . . . . . . . I48 I. The Influence of Solvent . . . . . . . I48 2. Entanglement Concentrations . . . . . . l58 C. Correlation Techniques . . . . . . . . . l7l l. Power Law Correlation . . . . . . . . I7I 2. Simha Correlation . . . . . 183 D. Williams Model for Zero Shear Viscosity . . . I90 E. Non- Newtonian Viscosity. . . . . . 222 1. Dependence of Relaxation Time on Concentration . . . . . 223 2. Dependence of Non— Newtonian Viscosity on Thermodynamic Quality of Solvent . . . . 234 VII. CONCLUSIONS . . . . . . . . . . . . . . 240 NOMENCLATURE . . . . . . . . . . . . . . . . 245 BIBLIOGRAPHY . . . . . . . . . . . . . . . . 259 APPENDICES . . . . . . . . . . . . . . . . . 267 Table 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 5.I. 5.2. 5.3. 5.4. 5.5. 5.6. LIST OF TABLES Kinetic Parameters, kp and kt». of ST and ACN at 60°C . . . . Initial Rate of Polymerization, R , at 60°C for Different Initial Ratio of ST: A N in Monomer Mixture . . . Values of o With Different Initiator Concentra- tions for Free Radical SAN Copolymerization at 60°C . . . . . . . . Values of kt( (1;) )for Different Copolymer Compo- sitions for ree Radical SAN Copolymerization at 60°C . . Details of Large Scale Polymerization at 60°C With AIBN Initiator . . . . . . Nitrogen Analysis of Polymers . Molecular Weights and Molecular Weight Distribu- tion of Polymers by GPC . . . . Intrinsic Viscosities, [n], and Huggins Constants, k], of Polymers in Various Solvents at 30°C Refractive Indicgs, no, of Various Solvents at 25°C and 4358 . . . . Refractive Index Increments, dn/dc, of Pglymers in Various Solvents at 25°C and 4358 A . Values of the Constants b and d (Eq. 5.2) for SAN Copolxmers in Various Solvents at 25°C and 4358 . . . . . . . . Second Virial Coefficient, A2, of Polymers in Various Solvents at 25°C.. . . Molecular Weight, MW, of PS-l from GPC and Light Scattering Measurements . . viii Page 32 36 37 39 45 46 48 I06 II3 II4 II6 126 I27 Table 5.7. 5.8. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. A.I. A.2. 8.1. 8.2. Light Scattering and GPC Molecular Weights of Copolymers With Refractive Index Increments at 25°C.. Light Scattering Molecular Weights, Mw , and .Q/Mw. of Copolymers Values of Mark-Houwink Constant, K, of Polymers at e-Condition . . . . . . . Expansion Factor, 6, of Polymers in Various Solvents at 30°C . . . . Stiffness Factor, 0, of Polymers . Flory Thermodynamic Parameter, X]: of Polymers in Various Solvents . . . Estimation of Entanglement Concentration, Cent’ for PS. - - Entanglement Concentrations, Cent, from Eq. 6. 7, for SAN Copolymers . . . . . Shift Factors, y, for Superposition of Viscosity- Concentration Data of SAN C- 2 and SAN C- 2' in Various Solvents . . Flow Parameters of Polymer Solutions at 30°C . Suggested Forms of To Constants of Weissenberg Rheogoniometer Constants of Capillary Viscometers Weissenberg Rheogoniometer 0'? Data for PS-l in Benzene at 30°C . . Weissenberg Rheogoniometer n-y Data for PS-l in MEK at 30°C . . . Weissenberg Rheogoniometer n-I Data for PS-l in Dioxane at 30°C . . . Weissenberg Rheogoniometer n-i Data for PS-2 in Benzene at 30°C . . Weissenberg Rheogoniometer n—i Data for PS-2 in MEK at 30°C . ix Page 128 129 136 138 139 I45 I62 163 184 230 231 269 270 272 273 274 275 276 Table B.6. B.7. B.8. 8.9. B.10. B.Il. B.12. B.13. 3.14. B.15. B.16. 8.18. 3.19. 8.21. B.22. Weissenberg Rheogoniometer in Dioxane at 30°C . Weissenberg Rheogoniometer in Dioxane at 30° We1ssenberg Rheogoniometer in Benzene at 30° C . Weissenberg Rheogoniometer in MEK at 30°C . . Weissenberg Rheogoniometer in Dioxane at 30°C We1ssenberg Rheogoniometer in DMF at 30°C Weissenberg Rheogoniometer in Benzene at 30°C . Weissenberg Rheogoniometer in MEK at 30°C . . Weissenberg Rheogoniometer in Dioxane at 30°C . We1ssenberg Rheogoniometer in DMF at 30° C Weissenberg Rheogoniometer in Benzene at 30° C . Weissenberg Rheogoniometer in MEK at 30°C . Weissenberg Rheogoniometer in Dioxane at 30°C . Weissenberg Rheogoniometer in DMF at 30°C . . . Weissenberg Rheogoniometer in MEK at 30°C . . . Weissenberg Rheogoniometer in DMF at 30°C . . Weissenberg Rheogoniometer in DMF at 30°C . n—i n-i n-1 n-1 n-1 n-1 n-1 n-1 n-1 n-1 n-1 n-1 n-1 n-Y n-1 n-1 n-i Data Data Data Data Data Data Data Data Data Data Data Data Data Data Data Data Data for PS-2 for PS-2 for SAN C-l for SAN C—l for SAN C-l for SAN C-l for SAN C-2 for SAN C—2 for SAN C-2 for SAN C—2 for SAN C—2' for SAN C-2' for SAN C—2' for SAN C-2' for SAN C-3 for SAN C—3 for SAN C—3 Page 277 278 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 Table C.I. 0.2. 0.1. Refractive Index Differences, An, Between Potassium Chloride Solutions and Distilled Water . . . Calibration Constant of Differential Refractometer Constants of Photometer for 436 mu Wavelength xi Page 295 296 301 Figure 1.1. 2.la. 2.1b. 2.2a. 2.2b. 2.3. 2.4. 2.5. 3.1a. 3.1b. 3.2. LIST OF FIGURES Chart Showing Rheological and Thermodynamic Studies in Relation to Process Flow Problems . . . . . Dependence of Instantaneous Copolymer Composi- tion, F1, on Comonomer Feed Composition, f], for SAN in Free Radical Copolymerization at 60°C, Mole Basis . . . . Dependence of Instantaneous Copolymer Composition, F on Comonomer Feed Composition, f1”, for SAN in Free Radical Copolymerization at 60°C, Weight Basis . . Variations in Feed and Copolymer Compositions with Conversion for SAN, Mole Basis Variations in Feed and Copolymer Compositions with Conversion for SAN, Weight Basis Variations in Initial Rate of Copolymerization with Mole Fraction of ST in Feed at Differ- ent Initiator Concentrations at 60°C . Fractional Conversion as a Function of Time for SAN Monomer Mixtures of Different Compo- sitions . . . . . . . . Ross- Fineman Plot for Determination of Monomer Reactivity Ratios, r] and r2, from Nitrogen Analysis of Copolymers . . . A Free-Draining Molecule During Translation Through Solvent . . Translation of a Chain Molecule With Perturba- tion of Solvent Flow Relative to the Molecule . Illustration of Experimentally Observed Viscosity-Shear Rate, n-y , Behavior of a Polymer Solution on a Log-Log Plot . xii Page 22 23 29 3O 38 41 47 56 56 74 Figure 3.3. 4.1. 4.2. 5.1. 5.3. 5.4. 5.5. 5.6. 5.8. 5.9. 5.10. 5J1. Illustration of Position Coordinates Referring to Polymer Molecules and Segments in Derivation of Williams Model . Rheogoniometer (Constant Shear Rate Configuration) . Reservoir Chamber for Rheogoniometer Ratio of Specific Viscosity to Concentration, ns /C, vs. c for PS—l in Benzene, Dioxane and MEK at 30°C . Ratio of Specific Viscosity to Concentration, nS /c, vs. c for PS-Z in Benzene, Dioxane angMEKat30°C. ....... Ratio of Specific Viscosity to Concentration, Dsp /c, vs. c for SAN C- l in Dioxane, Ben- zene, DMF and MEK at 30° C . . . Ratio of Specific Viscosity to Concentration, ns /C, vs. c for SAN C- 2 in DMF, Dioxane, E and Benzene at 30°C Ratio/ of Specific Viscosity to Concentration, ns /C, vs. c for SAN C— 2' in DMF, Dioxane, E and Benzene at 30° C . , Ratio of Specific Viscosity to Concentration, nsp/c, vs. c for SAN C—3 in DMF and MEK at 30° C . . . . . . . . . Viscosity, n, vs. Shear Rate, 1, for PS-l in Dioxane at 30°C . . . . . . . . . Viscosity, n, vs. Shear Rate, 9, for SAN C-2' in Benzene and Dioxane at 30°C . . . . . Viscosity, n, vs. Shear Rate, 1, for SAN C—3 in MEK and DMF at 30°C . . . . . . . Variation in Refractive Index Increment, dn/dc, of Copolymers in MEK, Dioxane, DMF and Ben- zene at 25°C as a Function of ACN Content . Variation in Refractive Index Increment, dn/dc, of PS in Different Solvents at 25° C as a Function of Refractive Index, n0,l of Solvents . . . . xiii Page 77 86 91 100 101 102 103 104 105 108 109 110 115 117 Figure 5.12. 5.13. 5.14. 5.15. 5.16. 5.17. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.'h0. Variation in Refractive Index Increment, dn/dc, of PAN in Different Solvents at 25° C as a Function of Refractive Index, no, of Solvents . . VariatiOn in Refractive Index Increment, dn/dc, of SAN C-l, SAN C-2 and SAN C-3 in Different Solvents at 25°C as a Function of Refractive Index, no, of Solvents Zimm Plot for PS-l in Dioxane at 25°C Zimm Plot for SAN C-l in DMF at 25°C Zimm Plot for SAN C-2 in Benzene at 25°C Zimm Plot for SAN C-3 in MEK at 25°C Second Virial Coefficient, A2, vs. Mole Fraction of ACN in Copolymers Specific Viscosity, nsp, vs. Concentration, c, of SAN C- l in DMF and MEK at 30°C . Specific Viscosity,n p,v .Concentration, c, of SAN C- 2 and SAN -2' in DMF and MEK at 30°C . . . Specific Viscosity,n p, vs. Concentration, c, of SAN C— 2 and SAN ”Sp. 2' in DMF and Benzene at 30°C Specific Viscosity, nRE’ vs. Concentration, c, of SAN C-3 in DMF and K at 30°C . Relative Viscosity, nr, vs. Concentration, c, of PS in Benzene at 30°C . . . . Relative Viscosity, n , vs. Concentration, c, of PS in MEK at 30°E . . . . Relative Viscosity, n , vs. Concentration, c, of SAN C-2 and SAN 2' in DMF at 30°C Relative Viscosity,n , vs. Concentration, c, of SAN C-2 and SAN HE ' in Benzene at 30°C . Relative Viscosity, nr, vs. Concentration, c, of SAN C-3 in DMF at 30°C . . xiv Page 119 120 122 123 124 125 142 150 151 153 154 159 160 164 166 167 Figure 6.11. 6.12. 6.13. 6.14. 6.15. 6.16. 6.17. 6.18. 6.19. 6.20. 6.21. 6.22. 6.23. 6.24. 6.25. 6.26. Relative Viscosity, Dr: vs. 30° C Concentration, c, of SAN C- 3 in MEK at . . . . . . Relative Viscosity, ”r’ vs. cMO'68 for PS in Benzene at 30°C.. . . . . Relative Viscosity, 11],,Ivs.cM0'68 for PS in Dioxane at 30°C . . . . . Relative Viscosity, ”r1 vs. cMO' 68 for PS in MEK at 30°C Relative Viscosity, nr, vs. cMO' 68 for SAN C-2 and SAN C-2' in Dioxane at 30°C . . . Relative Viscosity, Dr, vs. cMO'68 for SAN c—2 and SAN C-2' in MEK at 30°C . Relative Viscos1ty, nr, vs. CM0 68 for SAN C— 2 and SAN C— 2' in Benzene at 30° C . . Relative Viscosity, Dr: vs. ch for SAN C—2 and SAN C-2' in DMF at 30°C . . . . Relative Viscos1ty, 0r, vs. cMO' 5 for SAN C-2 and SAN C- 2' in Benzene at 30°C . . . . Simha Plot for SAN C— 2 and SAN C- 2' in Benzene at 30° C . . . . . Simha Plot for SAN C- 2 and SAN C- 2' in Dioxane at 30° C . . . . . Simha Plot for SAN C- 2 and SAN C- 2' in MEK at 30° C . . . . . Simha Plot for SAN C- 2 and SAN C- 2' in DMF at 30° C . . . . Experimental and Calculated Relative Viscosity, Dr: vs. Concentration, c, of PS-l in Benzene at 30°C Experimental and Calculated Relative Viscosity, Dr: vs. Concentration, c, of PS-l in Dioxane at 30°C . . . . . . . . . Experimental and Calculated Relative Viscosity, nr, vs. Concentration, c, of PS-l in MEK at 30°C . . . . . . . . . XV Page 168 172 173 174 175 176 177 179 182 185 186 187 188 200 201 202 Figure 6.27. 6.28. 6.29. 6.30. 6.31. 6.32. 6.33. 6.34. 6.35. 6.36. 6.37. 6.38. Experimental and Calculated or, vs. Concentration, c, at 30°C . Experimental and Calculated nr, vs. Concentration, c, at 30°C Experimental and Calculated nr , vs. Concentration, c, Benzene at 30°C. Experimental and Calculated nr, vs. Concentration, c, Dioxane at 30°C . . Experimental and Calculated , vs. Concentration, c, DMF at 30°C . . Experimental and Calculated , vs. Concentration, c, MEK at 30°C Experimental and Calculated 0r, vs. Concentration, c, DMF at 30°C Experimental and Calculated 0r: vs. Concentration, c, Dioxane at 30°C . Experimental and Calculated or. vs. Concentration, c, MEK at 30°C . Experimental and Calculated or, vs. Concentration, c, DMF at 30°C . . Experimental and Calculated Dr , vs. Concentration, c, Dioxane at 30°C. Experimental and Calculated n , vs. Concentration, c, MEK at 30°C . . xvi Relative Viscosity, of PS-2 in Dioxane Relative Viscosity, of PS-2 in MEK Relative Viscosity, of SAN C-I in Relative Viscosity, of SAN C-l in Relative Viscosity, of SAN C-l in Relative Viscosity, of SAN C-I in Relative Viscosity, of SAN C-2 in Relative Viscosity, of SAN C-2 in Relative Viscosity, of SAN C-2 in Relative Viscosity, of SAN C-2' in Relative Viscosity, of SAN C-2' in Relative Viscosity, of SAN C—2' in 'Page 204 205 206 207 208 209 210 211 212 214 215 216 Figure 6.39. 6.40. 6.42. 6.43. 6.44. 6.45. 6.46. 6.47. 6.48. 6.49. 6.50. 6.51. Experimental and Calculated Relative Viscosity, Dr: vs. Concentration, c, of SAN C—2' in Benzene at 30°C . . . . . . . . Experimental and Calculated Relative Viscosity, nr, vs. Concentration, c, of SAN C-3 in DMF at 30°C . . . . . . . . . Experimental and Calculated Relative Viscosity, nr, vs. Concentration, c, of SAN C-3 in MEK at 30°C . . . . . . . . . Superposition of Viscosity-Shear Rate, n—v, Curves of SAN C-2 and SAN C-2' of Various Concentrations in Benzene at 30°C . . . Superposition of Viscosity-Shear Rate, n-y, Curves of SAN C—2 and SAN C-2' of Various Concentrations in Dioxane at 30°C . . . Superposition of Viscosity-Shear Rate, n-i, Curves ofSAN C—2 and SAN C—2' of Var1ous Concentrations in DMF at 30°C . . . . Superposition of Viscosity—Shear Rate, n-i, Curves of SAN C—3 of Various Concentra- tions in DMF at 30°C . . . . . . Ratio of Experimental to Rouse Relaxation Time, TO/TR, vs. Concentration, c, of PS—2 and SAN C—2' in Dioxane at 30°C . . . . Ratio of Experimental to Rouse Relaxation Time, To/tR, vs. Concentration, c, of SAN C—3 in DMF at 30°C . . . . . . . Superposition of Viscosity-Shear Rate, n-i. Curves of SAN C-2' in Benzene, DMF, Dioxane and MEK at 7 gm/dl and 30 C Superposition of Viscosity—Shear Rate, n-Y, Curves of SAN C-2' in Dioxane, DMF and Benzene at 10 gm/dl and 30°C - Superposition of Viscosity—Shear Rate, n-Y, Curves of SAN C-2' in Benzene, D1oxane and DMF at 20 gm/dl and 30°C . - Superposition of Viscosity—Shear Rate, n-Y, Curves of SAN C-3 in DMF and MEK at 35 gm/dI and 30°C ., . . . . . . . . . . . xvii Page 217 218 220 225 226 227 228 232 233 235 236 237 238 LIST OF APPENDICES Appendix A. MACHINE CONSTANTS 0F WEISSENBERG RHEOGONIOMETER AND CANNON—UBBELOHDE FOUR—BULB SHEAR DILUTION CAPILLARY VISCOMETERS B. WEISSENBERG RHEOGONIOMETER n—i DATA C. CALIBRATION 0F REFRACTOMETER D. RAYLEIGH RATIO FROM LIGHT SCATTERING DATA AND PHOTOMETER CONSTANTS . . . . xviii Page 268 271 294 297 CHAPTER I INTRODUCTION A. Motivation Engineers must frequently deal with systems forprocessing, handling, transporting, storage and characterizing of liquids. In many cases it has been assumed commonly that the liquids are Newtonian. Today the engineer must deal with an increasing variety of "rheologically complex” liquids. These include process streams in the plastics, chemical, pharmaceutical, paper and pulp, food and fermentation and many other industries. Complex fluids such as polymer solutions and melts, emulsions, suspensions, and col- loids are generally non-Newtonian in their flow behavior. In dealing with flow systems, viscosity is a very important design parameter and the viscosity studies of these materials have resulted in renewed and increased interest in the science of rhe— ology. Rheology is a study of the response of a material to external forces. The variety of viscous behavior observed has led to numerous empirical formulations of rheological equations of state. Unfortunately, these empirical equations, although adequate fm~curve fitting, are often not reliable for extrapolation and prediction. The above problem has led to more fundamental studies of flow phenomena. The ultimate objective is to formulate a O1 relationship between the molecular structure of a fluid and its flow behavior and to apply this to problems of interest, namely, mixing, pumping, extrusion, molding, filtration, viscometry, heat transfer and many other engineering operations. Through increased understanding of the relationship between the physical properties and the flow characteristics of polymeric materials, engineers will be better equipped to deal with these materials and to perform the design of equipment needed for engineering of flow systems. Polymer solutions and melts display the viscous nature of liquids, but also show elastic properties of solids. Thus, for describing the rheology of these systems, Newton's and Hooke's laws are of limited use in their original form. Researchers have developed various molecular and phenomenological models to predict or describe the flow behavior of these materials. These models may be classified as empirical, continuum and molecular. The various viewpoints from different disciplines have been expressed in numer- ous excellent books and articles (F-l, M-l, engineering; F-2, F-3, F-l4, Y-l, M-2, chemistry; L-2, mathematics; and E-l, continuum mechanics). In order to put the rheological and thermodynamic studies in the proper perspective with respect to the process flow problem, a chart is shown in Fig. 1.1. The concept pictured in the chart is the following. The aim of rheological studies and molecular model- ing of polymer solutions is to provide sufficient information with Ivhich to solve process flow problems. This aim is represented by the arrow pointing to the upper box in the chart. The equations of Figure l.l.--Chart Showing Rheological and Thermodynamic Studies in Relation to Process Flow Problems. PROCESS FLOW PROBLEMS e.g. Pumping,Agitation Equations of Density Motion, Energy m—T Experiments 1 Conductivity Statements of Continuum 8. Physical Molecular Principles of 1— Theories of Conservation Viscosity Viscosity J —u function 11 (c,M.T.i’,s,p) Rheological & Thermodynamic Experiments continuity, motion, and energy are required for this purpose. They are obtained by input-out balances which are statements of the laws of conservation of mass, momentum, and energy. The conserva- tion equations are not sufficient in themselves to solve the flow problems. In addition, a knowledge is required of the behavior and properties of the material in question, such as viscosity, density and thermal conductivity. The boxes at the right of the chart represent experimental, empirical and theoretical efforts required to obtain the needed material functions. To be useful, the equation of motion requires a knowledge of the relationship between the stress exerted on a material in a flow field and the resulting rate of strain, or material response. For Newtonian fluids the stress, T, is proportional to the strain rate, 1, and the proportionality constant is the viscosity, u, Eq.1.l: (l.l) For solving many flow problems and when dealing with a large class 0f polymer solutions and melts, a generalization of Eq. l.l has proven useful. This generalization replaces the Newtonian vis- COSIty, u, in Eq. 1.1 with a viscosity function, n, but maintains Hm proportionality between stress and strain rate. (1.2) Hfis equation is adequate for the solution of many engineering flow mpblems, although it does not account for the elastic behavior of polymer solutions (B-2a). In the case of polymer solutions, the viscosity, n, is a function of concentration of polymer, c, molecu— lar weight of polymer, M, temperature of solution, T, shear rate, 1, and chemical nature of the solvent and the polymer, 5 and p, respectively. The symbols 5 and p are used in the sense that intermolecular free energies of interaction between solvent and polymer are variables influencing rheological response and these depend on the particular polymer, p, solvent, 5, pair being studied. A discussion of the nature of the solvent in terms of thermodynamics is given later in the chapter. Two of the least studied parameters are s and p. The parameter 5 describes the thermodynamic interaction between polymer and solvent and the parameter p describes the skeletal structure of the polymer such aslinear or branched. Furthermore, both of these parameters, 5 and p, depend upon the chemical composition of the solvent and that of the polymer, and also the resulting intermolecular forces present in any particular polymer solution. In deriving quantitative representations of the viscosity flmction, n, continuum mechanics provides a mathematical framework, nmlecular modeling attempts to relate observed behavior to molecu- lar structural variables and intermolecular forces, and experimental data is needed to test the validity of the models and for curve 'fitting. As mentioned earlier, these concepts are descriped in Ref.(F-l, F-2, F—3, F—l4, Y—l, L-2, E—l, M—1, and M—2). In this work the adequacy of several molecular models for the viscosity function is to be tested with respect to experimental information with particular reference to the variables 5 and p. 8. Goals The investigation reported here was carried out with the following goals in mind: (I) To investigate the effects of polymer—solvent thermo— dynamic interaction on viscosities of dilute to moderately concen— trrated polymer solutions. (2) To investigate techniques of correlating viscosity of polymer solutions with concentration and molecular weight of polymer and to evaluate these techniques. (3) To test the applicability of a model (W—l) that includes polymer-solvent thermodynamic interaction, 5 and p, for predicting viscosity of moderately concentrated polymer solutions and to ascertain the parameters involved for further investigation. (4) To study the effect of solvent character on the non— Newtonian viscosity function, n. Along with the above goals, the following was also of interest with respect to the copolymers investigated: (5) To investigate the effect of acrylonitrile content on Um configuration of polymer chains in terms of both short—range andlong-range effects (long—range effects in different solvent environments). Configuration refers to the spatial configuration Ufthe molecules in various solvent environments. (6) To investigate the kinetic models for the rate of C0Polymerization and their applicability to styrene-acrylonitrile copolymerization. (Copolymer samples used in this study were pre- pared by laboratory, free radical, polymerizations.) C. Polymers and Solvents It was necessary to choose systems havingai wide range of polymer-solvent thermodynamic interactions in order to study their effect on viscosity. In principle this can be accomplished by using ”good” and “poor“ solvents for a given polymer. A good solvent is one in which polymer segments prefer contacts with the solvent molecules and the polymer expands or swells in solution as opposed to a poor solvent in which the polymer segments prefer contacts with their own kind and thereby the polymer molecule tends to coil-up in solution. The configuration of a polymer molecule in solution depends on its environment, i.e., the quality of the solvent. In a good solvent, where the energy of interaction between a polymer element and a solvent molecule adjacent to it exceeds or is about the same as the mean of the energies of interaction between the polymer- polymer and solvent-solvent pair, the molecule will tend to expand so as to reduce the frequency of contacts between pairs of polymer elements. In a poor solvent, on the other hand, where the energy Ofinteraction between polymer segment and solvent molecule is unfavorable, smaller configurations in which polymer-polymer contacts occur, will be favored. Any theoretical considerations of the solubility of a poly- merin a solvent must necessarily consider the free energy of . -.-:a--‘ " ’ mixing of the two phases. By a statistical mechanical treatment, Flory (F-3a) derived analytical expressions for the free energy of mixing based on a lattice model of the liquid. This model con- siders that both solvent molecules and polymer segments occupy equivalent sites in the lattice. The formation of a solution is considered to occur in two steps: disorientation of the polymer molecules and mixing of the disoriented polymers with solvent. The latter is more important. The entropy of mixing disoriented POIymer and solvent is given as conf _ ASm - -k(n] ln V15 + n2 ln vp) (l.3) Where Vls and vp are the volume fractions of solvent and solute, respectively; 111 and 112 are the number of solvent and polymer moIlecules, respectively, in a solution; and k is Boltzmann's con— stant. There may also be an entropy change owing to orienting inf“! uences on the components in the solution which differ from thOse existing in the pure component. This entropy change associ- ated with first neighbor interactions must be proportional to the nurnber of pair contacts developed in the solution. Flory (F-3a) Obtains this as ASm = -k[8(x]T)/3T]n]vp (1.4) where x] is a reduced residual chemical potential. It consists of e"thalpy and entropy terms. This is discussed later in the chapter. Since AS$°tal is positive, it is the heat of mixing term that is more important in determining the sign of the free energy change of mixing, AFm.v Two substances will mix whenever AFm is negative. _ _ total AFm — AHm TASm (1.5) The heat of mixing results from replacement of solvent- solvent and polymer—polymer contacts. The magnitude of this con- tribution to the free energy depends upon the degree of interaction of the unlike species in solution. The solvent is good when heat of mixing is exothermic which is the case when polymer-solvent specific interactions are large, i.e., when hydrogen bonding takes place. The total excess entropy term is usually positive and good solvents have AHm < 0. Since there are two different monomer species present in a copolymer, the expression for the interaction energy must be amended toinclude the additional interactions. Stockmayer et a1. (S-4) represented x1 for a copolymer solvent system as X1 = )ZAXA + ’IBXB ‘ iAiBXAB where RA and x8 are mole fractions of monomers A and B in the COpolymer, XA and XB are the interaction parameters for the homo- P01ymers A and B with pure solvent, and XAB is a parameter expressing A-B interactions. The definition of "good” and "poor" solvents may be given thermodynamically. The change in chemical potential of the solvent, mm, may be split into an ideal and an excess term: Au] E Apid + Aqu' (1.7) Flory (F-3a) obtains with complete generality excess (i.e., nonideal) chemical potential of the solvent in terms of partial molar heat of dilution, AH], and partial molar entropy, AS]. According to his derivation, Apex = RT(K - w )v2 (1 8) l 1 1 p ' where K1 and P] are heat and entropy parameters such that _ 2 AH - RTKIVp’ (1.9) _ 2 A31 — Rw1vp. (1.10) Within the limits and validity of his theory and simplifying assumptions, he relates parameters K], W] and X1 by K] - m1 = X1 — l/2. (1.11) e also defines an ”ideal“ temperature 0 as uch that x1 = 1/2 — u1(1 — e/T). (1.13) Hence the excess chemical potential may be written as Ap$x = —RTm1(l — O/T)v§. (1.14) In a poor solvent, K] and W] are generally positive. Accord— ing to the above equation, at the temperature T equal to O, the chemical potential due to segment-solvent interaction is zero. Hence at the O—temperature deviations from ideality vanish. From the above equations it can be seen that in poor solvents W] is nearly equal to K] and X1 is close to 1/2. These quantities or equivalent ones may be experimentally determined by light scattering or intrinsic viscosity measurements and these methods were used in this work to quantify the thermodynamic ”goodness” or ”poorness“ of solvents. In the past most of the rheological work reported in the literature has been done in good solvents. In many of these studies, in spite of.a variety of solvents used for a polymer, the solvents were all good and thus distinct thermodynamic effects were not observed. In this work a different approach was con- sidered. Copolymers from two very different monomers were synthe- sized and studied in different solvents. Each of the selected solvents had a different degree of goodness toward the homopolymers 0f the two monomers. Some of the solvents were very highly c0m- patible with one of the homopolymers while others were non—solvents. “a; .1: were» - Thus the copolymers helped in providing distinctly different polymer-solvent thermodynamic interactions in different solvents. Styrene* and acrylonitrile* monomers were selected to syn— thesize linear polystyrene* homopolymers and styrene-acrylonitrile copolymers.* The two vinyl monomers are very different in charac— ter; styrene has a bulky benzene ring while ACN has a polar CEN group. Styrene is non—polar while ACN is polar. The copolymers that were synthesized had different ACN contents, thereby providing different degrees of localized polarity in different copolymers. Excellent references to ST and ACN polymerization can be found in Ref. (B-l, M—1) and (A—4), respectively, and of polymerization in general in Ref. (O—l). Four solvents were selected for this work: (1) Benzene, (2) Dioxane, (3) Methyl ethyl ketone,* and (4) Dimethylformamide.* Benzene is non—polar and is an excellent solvent for PS while it is a non-solvent for polyacrylonitrile.* Dioxane has two symmetric oxygen atoms and hence its dipole moment is zero but it has local- ized charge separation. Dioxane is a good solvent for PS but is a non-solvent for PAN. The two polar solvents, MEK and DMF, are relatively poor solvents for PS while for PAN, MEK is a non-solvent and DMF is an excellent solvent. Thus this choice of solvents gives a wide range of polymer-solvent interaction in terms of a variety 0f POIymer-solvent intermolecular forces. *Henceforth styrene is referred to as ST, acrylonitrile as ACN, polystyrene as PS, styrene-acrylonitrile copolymers as SAN (mpolymers, methyl ethyl ketoneiasMEK, dimethylformamide as DMF and POlyacrylonitrile as PAN. '1. u. u 14 It may be mentioned here that copolymers are often manufac- tured or tailored for specific physical and chemical properties which are often unobtainable from simple homopolymers. The properties of a particular linear homopolymer are determined pri- marvily by two factors: (1) average molecular weight and (2) molecular weight distribution. In copolymers, along with the above two factors, third and fourth important factors are the average chemical composition and the distribution of composition about this average. The polymers that were synthesized in this work were similar to industrial PS and SAN copolymers with respect to their molecular weights and molecular weight distribution. 0. Experimental Method To achieve the goals of this research, a rather wide range (’f’ eaxperimental work was involved. This consisted of polymeriza- t1.On of monomers, characterization of polymers, viscosity measure- me"ts of dilute and moderately concentrated solutions and experimental determination of pol ymer-solvent thermodynamic inter- acit‘ions. This required the use of the following equipment: 1. Glass reactor with baffles and stirrer for poly- merization (Chemical Engineering Department). 2. Capillary viscometers for dilute solution vis- cometry (Chemical Engineering Department). 3. Cone—and-plate viscometer for moderately concen- trated solution viscometry (Chemical Engineering Department, University of Michigan). 4. Light scattering photometer and differential refractometer for thermodynamic parameters and molecular weights of polymers (Biochemistry Department). 15 The laboratories of Dow Chemical Company in Midland, Michigan, performed the measurements of molecular weights and molecular weight distributions of the polymers by gel permeation chromatography (GPC). Spang Microanalytical Laboratory, Ann Arbor, Michigan, did the nitrogen analysis of the copolymers. CHAPTER II POLYMERIZATION AND PREPARATION OF SAMPLE MATERIALS Polystyrene homopolymers and styrene-acrylonitrile copoly- mers used in this work were synthesized by free radical polymeriza- tion in bulk. The reasons for this were (I) to avoid contamination from solvents used in solution polymerization; (2) to obtain a maximum quantity of each polymer with a restricted size reactor and restricted conversion of monomers to copolymers, low conversion be‘ing necessary to obtain uniform composition of copolymers (this is di Scussed in detail later in the chapter); and (3) because of low Conversions, mixing and heat—transfer would not present difficulties 1" these bulk polymerizations. This chapter presents methods for the selection and purifi- cation of materials, the theory of vinyl homo and copolymerization, all‘Id a description of the sample homo and copolymers which were syn- thesized to use in the viscosity and thermodynamic studies. A. Initiator The rate of vinyl, free radical copolymerization in a binary s.YStem depends not only on the rates of the four propagation steps tWt also on the rates of initiation and termination reactions. To simplify the matter, the rate of initiation may be made independent of the monomer composition by choosing an initiator which releases 16 .. 17 primary radicals that combine efficiently with either monomer. The spontaneous decomposition rate of the initiator should be substan- tially independent of the reaction medium, as otherwise the rate of initiation may vary with composition. The initiator a-a'-Azo-bis- isobutyronitrile* (AIBN) meets these requirements satisfactorily (W-2). Also, AIBN offers an advantage in that, unlike benzoyl per- oxide, it is not susceptible to induced decomposition (F-3b). 8. Purification 1. Initiator The AIBN obtained from Eastman Kodak Co.I was purified by recrystallization from acetone. A large quantity was dissolved in acetone at room temperature till saturation. The solution was fil- tered through a funnel under vacuum. The filtered solution was cooled in an ice-water bath until a crop of crystals precipitated. This procedure was repeated twice and the crystals were dried in a vacuum oven at room temperature. After drying, the purified, crys- ‘talline AIBN was stored in a refrigerator. 2. Monomers + High purity ST and ACN were obtained from Eastman Kodak Co. The containers were stored in a refrigerator and only the approxi- mate amount needed for each run was withdrawn at one time. The 'IIQUid monomers, for an experiment, were withdrawn and separated *Henceforth referred to as AIBN. TEastman Kodak Co., Rochester, New York 14650. 18 from the dissolved inhibitor by passing through columns of activa— ted alumina (B-3, F-4). The monomers were then distilled at reduced pressure. Only the middle fractions were collected and used. C. Homopolymerization The kinetic scheme for homopolymerization in the presence of an initiator may be written as "k p * WMX + M _____> WMx+1 (2.1) where superscript * indicates a radical at the end of a growing chain. 1. Rate of Reaction As shown by Flory (F-3b), the rate of propagation in free adical polymerization is given by = —d m _ 1/2 R — kp (fkd [11 /kt) [m] (22} here kd’ kp and kt are the reaction rate constants for initiator acomposition, chain propagation, and chain termination, respec- vely; f is the fraction of primary radicals available for initi- jon of polymerization (efficiency of initiation); [I] is the itiator concentration; and [M] is the monomer concentration. Equation 2.2 can be integrated and solved for the time of action necessary for the required conversion with a known amount initiator. It was found that for 10 per cent conversion with 19 0.1 per cent initiator concentration, the total reaction time was about four hours at 60°C for ST homopolymerization. Hamielec et al. (H—2) have derived an expression relating conversion with the time of reaction and the amount of initiator. This is given by 2k 1 1 szd[1]1‘/2[ -ktt ] = _ P ____. - x 1 exp [1 kd} ktd‘Ikth exp ( 2 ) l (2.3) where x is the fractional conversion of monomer to polymer, ktd and ktc are the reaction rate constants for termination by dispropor- tionation and termination by combination, respectively. In bulk polymerization of ST, ktd can be neglected (F-3b). The calculated time for 10 per cent conversion with 0.1 per cent initiator concen- tration was also about 4 hours. This should not be very surprising since Eq. 2.3 is an integrated form of the rate expression and the origin of Eqs. 2.2 and 2.3 is the same. 2. Molecular Weight The expected molecular weight can be calculated from the knowledge of the kinetic chain length which is given by v = (k 2/2k )[MJZIR (2.4) P t P where Rp is given by Eq. 2.2. The kinetic chain length, v, WM-IM] (2.6) * k12 * * k2] * WMZ ‘1' M1 ~—> WM2M1 (2.81 k * * 9" 21 where superscript * indicates a radical at the end of a growing chain, subscripts l and 2 denote two types of monomers and kij's are the propagation rate constants. 0n assuming that the reactivity depends only on the terminal unit and making the steady state assumption (F-3d, 0-1) for the free radical species, an expression may be obtained which relates the instantaneous mole fraction, F], of monomer M1 in the copolymer fermed from a binary monomer mixture, to f], the mole fraction of monomer M1 in the monomer mixture. This expression is 2 _ "ifi ” f1‘°2 F1 - 2 2 (2.10) rlf] + 2f1f2 + r2f2 where r1 = k11/k12’ (2.11) r2 ‘ k22/k21’ d[M J _ _ 1 F1 ‘ I ‘ F2 ‘ d[M;]—T—3[M2] ’ (2’12) and [M1] 2] fl-I-fZ-[M]]+[M2]. ('3) For ST and ACN at 60°C, r1 is equal to 0.41 and r2 is equal to 0.04 (B-4). Figure 2.1 shows the dependence of instantaneous copolymer composition, F], on the comonomer feed composition, f1 , for ST-ACN 22 Mole fraction of ST in copolymer, F1 o l l l L l L l L L 0 0.2 0.4 0.6 0.8 1.0 Mole traction of ST in comonomer feed,f1 Figure 2.la.--Dependence of Instantaneous Copolymer Composition, F1, on Comonomer Feed Composition, f], for SAN in Free Radical Copoly- merization at 60°C, Mole Basis. 23 Weight fraction of ST in copolymer, F1w 0111111111 O 042 0.4. O-6) 0.8 1-0 Weight fraction of ST in comonomer feed,f1w Figure 2.lb.--Dependence of Instantaneous Copolymer Composition, F1”, on Comonomer Feed Composition, fiw, for SAN in Free Radical Copoly- merization at 60°C, Weight Basis. W’ {as .-J «a: - 24 monomers in radical copolymerization. Subscripts l and 2 refer to ST and ACN, respectively. Actually Fig. 2.1a is the representation of equilibrium between the copolymer and monomer compositions expressed by Eq. 2.10 in molar units while Fig. 2.lb shows the same equilibrium relationship in weight units. Since, except at the cross-over point (with the 45° line), the instantaneous copolymer composition, F], is different from the composition of the monomer mixture, f], from which it is being formed, a drift in both copolymer and monomer mixture compositions occurs during the course of a batch-type copolymerization. The direction of drift has been indicated by arrows in Fig. 2.la. It is also to be noted that the drift is in opposite directions on either side of the cross—over point. Thus monomer mixtures having composi- tions f1 greater than the cross-overP010tC0mp0$lt10n become depleted inACN with polymerization. Monomer mixtures having compositions fl less than the cross-over point composition become depleted in ST Nth polymerization. This occurs because of the different values f reactivity ratios, r1 and r2, which in turn indicate the differ- nce in the reactivity of the two monomers involved. The composition tthe cross-over point is called the azeotropic composition since at MS composition, the composition of the copolymer formed is the same that of the monomer mixture composition and it remains constant Hipolymerization. At the cross—over point F1 is equal to f], and ice, from Eq. 2.10, F1 = f1=(1- r2)/(2 — r1- r2)- (2-14) ,_A$N.. 25 For SAN, the azeotropic composition from Eq. 2.14 occurs at F1 equal to fI equal to 0.6194 mole fraction or at 0.7615 weight fraction of ST. 2. Variation of CgpoLymer Composi- tion WithTonversion The copolymerization equation (Eq. 2.10) gives the jgstgg: taneous copolymer composition, i.e., the composition of the copoly- mer formed at a particular monomer composition. For all copolymeri- zations except azeotropic, the comonomer and copolymer compositions (copolymer formed out of that comonomer) are different from each other. This results in a variation of copolymer composition with conversion since the feed comonomer composition changes at each instant with copolymerization. In order to determine the instan- taneous copolymer composition as a function of conversion for any given comonomer feed, one must resort to an integrated form of the coDolymerization equation. The most general, useful method is that darived by Skiest (S-l). From a material balance one can obtain M d” M f1 df1 I—M—=1n1=ir——y,_- _f . (2.15) 0 M 0 l 1 M f1 "here M denotes the total moles of the two monomers, and superscript 0 denotes initial values. Equation 2.10 allows the calculation of F] as a function of f1 for a given set of r1 and r2 values which can the" be used in Eq. 2.15 to obtain variations in monomer and copoly- mer compositions with the degree of conversion defined as l - M/MO. 26 Equation 2.15 has been integrated (M-3, M-4) to the useful closed form Bf0_6y a f 1 - f 1 M 1 1 - ——-= 1 - —— M0 f? (2.16) which relates the degree of conversion to changes in monomer compo- sition and where r 1“ _ 2 z 1 “’11—???" 8 117—177’ (2.17) l-rr 1-r - 12 z 2 Y (l-r])(l-r2)’and 6 (2-r1-r2)’ Equation 2.16 was used to calculate the drift in the monomer and COpolymer compositions with conversion. The calculations can be conveniently performed by means of a simple computer program with an appropriate computer. The essential feature of the computer program (in FORTRAN IV for one 6500 computer) is that f] is decreased or increased (depending on whether the initial composition, f], is less than or greater than the azeotropic composition) in step increments 0f 0.005 from f1 to 0 or 1.0, respectively. For each value of f] a the corresponding mole conversion is calculated from Eq. 2.16 and the corresponding instantaneous copolymer composition from Eq. 2.10. With the monomer mixture composition, f1 , and mole conversion (1 ~ M/Mo) known, the cumulative average composition can be easily ca'lculated. The output of these calculations can also be easily 27 converted from molar into weight units. Figure 2.2a shows vari- ations in instantaneous compositions of copolymer, F], and monomer mixture, f], with conversion in molar units, and Fig. 2.2b shows the same in weight units for two particular SAN copolymerizations. As can be seen in these figures (and also in Fig. 2.1), a copolymer fanned from the monomer feed composition greater than the azeotropic composition is less rich in ST than the feed composition, and vice versa for the monomer feed composition less than the azeotropic composition. Again, Figs. 2.2a and 2.2b clearly show the drift in composition with respect to conversion. The straight horizontal lines indicate azeotropic composition. For any feed composition other than azeotropic, as the polymerization proceeds, ultimately at some conversion the feed mixture will become depleted in one of the two monomers depending on the initial feed composition. After this happens, simple homopolymerization takes place and this is shown by horizontal lines at fractions 1 and 0, for PS and PAN, respectively. Figures 2.2a and 2.2b show the instantaneous composition of copoly- mers. The average composition at each fractional conversion can be e35ily calculated and plotted on the same figures but it is not shown for the sake of clarity. The line representing average compo- Sition will be more horizontal. Again, it is amply clear from the figures that at low conversion (less than 10 per cent), the compo- Sition of copolymer formed is practically uniform; i.e., the compo- Sition drift is very small. 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Chemical-controlled termination.——This approach found in standard texts (F-3b, O—l, w-3a) assumes the termination reaction to be chemically controlled. Copolymerization is assumed to consist of four propagation reactions, Eqs. 2.6 to 2.9, and three termination steps as k Wm} WM: .flL (2.18) k Dead NM* + WM" & >___. (2.19) 2 2 Polymer k MM1+ MM: 4515 (2°20) )rresponding to termination between like radicals, Eqs. 2.18 and .19, and cross-termination between unlike radicals, E0- 2-20- The me of copolymerization is then (r [M1 :12 + 21111111121 + rZEMZJ >R 111/2 _ ———————————-"‘2—"" (2.21) R 1/ p {r161[M1]2 +2¢Y¥1P25152[M1][M2] +Y‘252EM2]2} 32 where R. is the rate of initiation of chain radicals of both types, M1 and M2, and it is given by R1 = 2fkd[I] (2.22) and where _ l/2 _ 1/2 52 — (2 ktzzlkzz) (2.23b) and _ 1/2 )1/2 ratios for the The 6 terms are simply the reciprocals of kp/(Zkt homopolymerizations of the individual monomers. The ¢ term repre- sents the ratio of half the cross-termination rate constant to the geometric mean of the rate constants for self-termination of like radicals. A value of ¢ < 1 means that cross-termination is not favored, while o > 1 means that cross-termination is favored (F—3d). Table 2.1 lists kinetic parameters for radical chain copoly- merization at 60°C. TABLE 2.1.--Kinetic Parameters,* k and kt’ of ST and ACN at 60°C. P Monomer kp x 10'3, l/mole/sec kt x 10‘7, l/mole/sec Styrene 0.145 2.9 Acrylonitrile 1.96 78.2 *From Ref. (8-4) and (N-3b). 1 r .2 ..._. ,1. “um-marevnwfis' . rm} 33 For AIBN from Ref. (V-l), 15 l k = 1.58 x 10 exp (-30,800/RT) sec" (2.24) d where R is the gas constant and T is the absolute temperature. The 61 and 52 values in Eqs. 2.23a and 2.23b are obtained from homopoly- merization. Experimental determination of the rate of copolymeriza- tion then allows calculation of ¢ from Eq. 2.21. b. Diffusion—controlled termination.--A kinetic expression for the rate of diffusion-controlled copolymerization was obtained by Atherton and North (A-2) by considering the termination reaction as ‘ 'k 'k + MM; t kt ‘2 Dead (2.25) Polymer * IH * * J where the termination rate constant kt(12) is a function of copoly- mer composition. The expression for the rate of copolymerization, Rp. was found to be (amp2+amnm1+gmg6fi” rZIMz] 1 k22 (2.26) R p 1/2 [r1IM1] kt(12) k11 + rfir, .u—u—a '— 34 This equation was used to calculate the value of kt(12) for each copolymer composition from the knowledge of experimental values of R for two concentrations of initiator. c. Conversion as a function of time.—-For diffusion— controlled termination O'Driscoll and Knorr (0-3) have derived an expression which gives conversion as a function of time. Their expression is f a 1 _ f b f0 _ 6 C ln —1- 1 l = 0 0 f a 1.1 1 ‘ f1 1 ' l/2 -k t f[I d 2(k — Xk ) ———-l——*- exp [———-J - l] (2.27) 21 22 [kdkth [ 2 lere x = (k11 — 1(21)/(1<12 — (<22), (2.28) a = o(l - X) + l, (2.29a) b = 3(1 - x) - x, (2.2%) c = y(l - x), (2.29c) it is the time of reaction in sec. ation 2.27 should permit the calculation of f1 as a function of R The resulting values of f1 may be used in Eq. 2.16 to obtain by version as a function of time and in Eq. 2.10 to obtain F1 as a 35 function of time. Equation 2.27 is therefore a complete descrip- tion of time dependency of free radical c0polymerization. d. Small scale experiments.--In order to determine which of these two kinetic mechanisms is useful for the purpose of pre- dicting rates of SAN copolymerizations, several small scale experiments were performed. Free radical chain polymerization was carried out with pure ST monomer to PS and with SAN comonomers in different propor- tions to SAN copolymers of different compositions. Polymerization was carried out in sealed pyrex ampules after bubbling nitrogen through the monomers to displace oxygen. The nitrogen was first passed through a column of drierite to remove moisture and then into the monomers. Pure ST and four different mixtures of the two monomers in the STzACN ratios of 90:10, 76.2:23.8, 32:68 and 7.3:92.7 by weight were used so as to obtain PS and the SAN copoly- mers in the STzACN ratios of 84.6:15.4, 76.1:23.9, 64.9:35.l and 50:50 by weight, respectively. Five monomer mixtures of each ratio were polymerized for different lengths of time at 60° i 1°C. For PS, concentration of initiator was 0.008 moles/l and for all copolymers, 0.032 and 0.016 moles/l concentrations were used. The overall rate of polymerization in each case was determined from the yield of polymer. The polymers were obtained by precipitation in chilled methanol. The volume of methanol used for each precipita- tion was four times the volume of reaction mixture. The polymers were then redissolved in MEK, filtered and reprecipitated in '10.: 36 methanol. The polymers were then dried in a vacuum oven at 55°C to constant weight. The drying time was about 24 hours. e. Discussion.--For all the polymers, plots of moles of monomers, M, remaining versus time, t, of reaction were made and fit by computer and then extrapolated to zero time, from which the initial rate of polymerization, Rp = -d[M]/dt, could easily be found. Table 2.2 shows these values. TABLE 2.2.--Initia1 Rate of Polymerization, R , at 60°C for Dif— ferent Initial Ratios of STzACN iR Monomer Mixture. Initial Rate of Mg]: Fraction of Mole Fraction of Sgnggfitafigr: Pol ymeri zagion , 1n Monomer ACN 1n Monomer moles/l R x 10 . mole/l/sec 1.0 0.0 0.008 5.6 0.821 0.179 0.032 26.7 0.821 0.179 0.016 18.0 0.620 0.380 0.032 44.9 0.620 0.380 0.016 27.8 0.193 0.807 0.032 63.3 0.193 0.807 0.016 35.5 0.0386 0.9614 0.032 56.9 0.0386 0.9614 0.016 31.3 1.0* 0.0 0.032 13.2 1.0* 0.0 0.016 9.2 *From Ref. (B-S). 37 Figure 2.3 shows plots of Rp versus mole fraction of ST in the initial monomer mixture for both [I] equal to 0.032 moles/l and 0.016 moles/l. Using these experimental rates of copolymerization, Rp, in Eq. 2.21, values of ¢ for both concentrations of initiator were determined by using a computer. The procedure was the same as that followed by Walling (N-Z) in his study of the dependence of the rate of radical copolymerization on the comonomer feed compo- sition for the system styrene-methyl methacrylate at 60°C with AIBN. The value of ¢ is determined by a curve-fitting technique such that for a particular value of ¢ the "best" curve is obtained. Table 2.3 shows these values. TABLE 2.3.--Values of ¢ With Different Initiator Concentrations for Free Radical SAN Copolymerization at 60°C. Concentration of AIBN, [1], ¢ moles/l 0.032 2-09 0.016 4.42 The above result is rather baffling in light of the fact that ¢ is supposed to be a constant for a system regardless of the initiator concentration. Das et al. (D-l) determined the values of ¢ for the SAN system in a slightly different manner. They used differ- ent concentrations of the initiator for one monomer ratio and determined ¢. This was then repeated for different values of monomer ratio. The values of 4 thus obtained were nearly constant. The average value of p was 7.5. o 0 o '0. \‘uu..\l|.r-E o I. . u I 3' I... 'InV f.J 3 2 ache-.0. O.~-o! Dunno... a 38 70 11111 60- 50- moles/litre/sec 5 30% 20- v1 10 Rate of copolymerization, Hp 11 10 0 DJ, I I I I 1.0 0.8 0.6 O O —( 0 .. O - 01:1]: 0-032 moles/l em = 0.016 moles/l — I 44] I l 0.4 0-2 0-0 Mole fraction of S T in feed Figure 2.3.--Variations in Initial Rate of Copoly- merization with Mole Fraction of ST in Feed at Different Initiator Concentrations at 60°C. 39 It was found in this work, however, that the value of 4 in his system varies with copolymer composition and hence the use of q. 2.21 to predict the initial rate of copolymerization with a ingle value of ¢ is of dubious value for this system. North and oworkers (N-l, A-2) have also pointed out the variation of o with opolymer composition in several systems. They pointed out that he termination in radical polymerization could be diffusion— Jntrolled. Thus, the interpretation of o primarily in terms of 1e chemical effects of the radical ends appears questionable. Table 2.4 shows the values of kt(12) obtained from the lffusion-controlled equation, Eq. 2.26. BLE 2.4.—-Values of kt(121 for Different Copolymer Compositions for Free Radica SAN Copolymerization at 60°C. -7 . 1eFractionof kt(12) X 10 mOIGS/llter/sec in Copolymer [I] =0.032 moles/liter [I] =0.016 moles/liter ’42 3.74 4.12 $2 3.4 4.49 68 8.6 13.7 37 25.5 41.9 ‘e 2.4 indicates that kt(12) is not constant for any particular lymer composition irrespective of initiator concentration. rton and North (N-l, A-2) and O'Driscoll et al. (0—2) have i to demonstrate the utility of the above method in a few 1. Although their work was not extensive, none have studied 40 the effect of rate of reaction on the termination rate constant, kt(12)’ as demonstrated here. Thus neither mechanism alone appears to satisfactorily explain instantaneous SAN copolymeriza- tion reaction rates. Figure 2.4 shows experimental and theoretical conversion from Eq. 2.27 as a function of time for three different monomer mixtures. As can be seen, there is no agreement between the cal- culated time-conversion plots and the experimental data even at small conversions where any effects due to drift are small. The value of kt(12) used for each monomer mixture was that found from Eq. 2.26 where it is assumed that kt(12) is totally diffusion- controlled. As discussed above, this may not be true and the complete disregard of chemical-controlled termination is open to question. Secondly, kt(12) was assumed to be time independent. Actually, kt(12) is dependent on the composition of the polymer chain, and this could be dependent on time of polymerization. O'Driscoll and Knorr (0-3) compared experimental and theoretical conversion predicted by Eq. 2.27 for a mixture of methyl meth- acrylate and vinyl acetate and found an agreement only up to about 3 per cent conversion. Equation 2.27 has not been tested exten- sively yet, and the result obtained here indicates that it must be used with caution since the assumptions made during its derivation may not hold for many systems. It may be concluded that the o-factor singly may not be suitable for describing the behavior of systems where 4 varies with composition. At any rate, the value of ¢ > 1 emphasizes the fact O. 41 Fractional conversion ,weight basis 0'13 1 1 1 1 1 1 1 1 1 1 _ Initial weight fraction of _1 Plot ST in monomer mixture _ 1 0.9 .. 0'16 2 0.32 _ 3 0-073 _ [I] = 0.032 moles/l 0.11. +- - 0.12 — .. 0.10 - . — 0 08 2 ' _ 2 — I/ .- e . / /3 /' 0-06 "' // // / d / '/ ,/’ / /’ .— _ / / / . / / / 0.01. - / // / -1 // / // ‘ l/ / - - — — Theoretical "‘ // / . 002 __ , Experimental _ . -/ / _ / _ / 0 L l I I A I I l l l 0 20 40 60 80 100 110 Time .t.minutes Figure 2.4.--Fractional Conversion as a Function of Time for SAN Monomer Mixtures of Different Compositions. ‘1‘»1"VI‘: n 'z‘ 1 09:1: 3" '121 :' '9 I|I n. , 1'1: .. .11 bt‘ ',1 . 1 . 1, 42 observed repeatedly in high polymer chemistry that a free radical has a decided preference for combining with an unlike radical. The rate of termination should differ if the last units are the same or different because of polar repulsion effects. If such a repulsion exists, the rate kt for the reaction \AV/v0M1M; + M1Cnv~orshould be less than that for cnvrc~M2Mg + \xecrer1 and the value of ¢ will change since - 1/2 °‘ ktlZ/(ktllktZZ) ° (2°30) Also, in the copolymerization of ST with ACN, it may be postulated that the interaction between the phenyl rings on adjacent ST units will tend to make the segmental motion slower because of hindered rotation about the chain axis. Barb (B-6) has suggested that the effect of the penultimate unit in a radical chain must be con- sidered. It has been recognized that the increased viscosity during the free radical polymerization of some vinyl monomers causes a decrease in the termination rate constant. This may cause the onset of diffusion-controlled termination. The termination reac- tion in free radical polymerization is at_1ga§t_partially diffusion- controlled even in an environment of low viscosity. Thus it seems that characterization by simple 4 or kt(12) factors is inadequate. Further effort to correlate the rate of copolymerization, R , with the monomer composition was not made since this was not P the main goal of this research, although it can be seen that the .1“ 1 “t. I I ( 1 .‘Ji' I ’ correlation of Rp with monomer composition would help immensely in predicting the time required for a reaction for the required low onversion using a specified initiator concentration. It is very 'mportant to know the required time for a reaction to be able to iroduce a copolymer reasonably uniform in compositional distribu- ion. In this work it was deemed of vital importance that copoly— ers of uniform composition distribution be produced for the iscometric and thermodynamic studies, and therefore empirical eaction time information generated by the small scale experiments 15 used. E. Large Scale Polymerization It was found that the theoretical rate expressions for polymerization could not be relied upon to determine the time of action for a required conversion. It was therefore decided to )duce large amounts of PS homopolymers and SAN copolymers for cosity and other measurements on the basis of small scale experi- ts with corresponding amounts of initiator, and the reactions e carried out for corresponding lengths of time. Each reaction was carried out in a two—liter, round- :omed flask at 60°C under nitrogen atmosphere. Cold monomer Lure was heated up to 58°C as quickly as possible, dumped in reactor and the AIBN was added. After the completion of the tion, the contents of the flask were poured into chilled anol (four times the volume of the flask contents) in a Waring ier to precipitate the polymer. The precipitated polymer was "f 3115 v01 "'t’H‘ L I‘I'. 1 44 redissolved in MEK, filtered and reprecipitated by addition of methanol. The polymer was dried at 55°C in a vacuum oven. Table 2.5 gives the details of large scale polymerization at 60°C using AIBN. F. Chemical Analysis of Copolymers The copolymers were analyzed for nitrogen content (and hence for acrylonitrile content) by Spang Microanalytical Labora— tory.* Also, a sample of polyacrylonitrile (PAN) was analyzed as a reference. It was synthesized in bulk at 60°C using AIBN. Table 2.6 gives the results of nitrogen analysis. It can be seen that the copolymerization reactions were carried out successfully in obtaining the desired compositions. Again the conversion by weight per cent was small in each case, giving a practically uniform copolymer composition. G. Monomer Reactivity Ratios From Composition of Copolymers Equation 2.10 can be rearranged into the form f(l-2F) f2(F -1) 1 1 =1, 1 1 1 l:111 " f1) 2 F1(1 - r1)2 r1 (2.31) as suggested by Fineman and Ross (F-S). The left side of this equation when plotted against the coefficient of r1 should yield a straight line with slope r1 and intercept r2. This plot is called *P.0. Box 1107, Ann Arbor, Michigan 48106. . M10 ow ma 0m_.m —m_0.o 000,— m m_ 0 mm m z_oa .wEe5 .copmrpecH cowpecpcwuzou . . cmaxyoa cowmsm>cou . . . gasocoz 5m :05p5moaaou to uczoE< :ovpommm we uczoe< copeprCH eo pcsos< wsszwz cmsocoz .LopmeuwcH zmH< new: 0000 pm cowpmecwE>FOQ m—mum mace; we mpwmpwo11.m.m m4m<5 III I tnmtrogen Ani H1 . m, 1 A I' . r0 ‘5'» “131 1" 1.: , r 1' 1 I ",‘;('('1 . . 1' o 1 1"" 101,1 —’ . °'-"-l<".r.r -t .’I-11.( -1'11 "1'11!“ 4 ',‘.r N I .1 ‘v ‘r «121 "" ‘7 ”2' o t ..1 ' a [.8 4 1 n1 .. ’ 1 1" «a .1". . C.- 1'! . 1 , 46 TABLE 2.6.--Nitrogen Analysis of Polymers. Polymer wgrggtNlthgggN Calcu;a§$dAgfiight Analy51s PAN-l 99.7 100 SAN C-1 14.2 15.4 SAN C-2 24 23.9 SAN C-2' 23 23.9 SAN C-3 38 35.1 the Ross-Fineman plot. Figure 2.5 is such a plot where the mole fraction F1 of ST in the copolymers is obtained from the nitrogen analysis. The least square values of r1 and r2 from Fig. 2.5 are 0.463 and 0.0429, respectively. These values compare quite well with the literature values of 0.41 and 0.04 that were used in this work. This good agreement reinforces the confidence in the chemi- cal analysis of the copolymers. H. Molecular Weights and Molecular Weight Distribution of Polymer Samples Samples of all the polymers that were synthesized on a large scale were sent to the analytical laboratories of the Dow Chemical Company* for determination of molecular weights and molecular weight distribution by gel permeation chromatography (GPC). Table 2.7 shows the GPC results. *Midland, Michigan 48640. 47 .mcmsx—oaou we mwmzpmc< :mmoguwz socm .Nc new 5; .mowpmm xuw>wpummm cmaocoz mo cowumcwecmumo so» quQ cosmcwaimmom11.m.m mcamwm _. F P _ (‘1-1)‘;1/(‘az-1)‘1 'tti Lin-Molecular H1 Polymers by “s“ .. , Composition, " 1219111151 100.0 100.0 "1- 85.8 ' ‘1" [(51161) ' "1"1’H “ 9).,1110r1b1 ‘1 (ll '11” . . . , ' ' £51101 I". “Cl-'- 11 I - I‘ r." 1, .1 .1 . C' . . 111,117.11”: ”In .‘fl‘ I {'1‘ 48 TABLE 2.7.--Molecular Weights and Molecular Weight Distribution of Polymers by GPC. P011119?“ 1213111221: R. Mn Mw/Mn 42.31;}:333 PS—l 100.0 185,000 103,000 1.79 191,000 PS—2 100.0 501,000 241,000 2.08 504,000 SAN C-1 85.8 275,000 141,000 1.95 290,000 SAN C-2 76.0 203,000 120,000 1.69 180,000 SAN C—2‘ 77.0 634,000 339,000 1.87 666,000 SAN C—3 62.0 332,000 205,000 1.62 332,000 The GPC result is reliable since the copolymers are practically uniform in composition because of low conversions. Again the 1wlecular weights, Mw’ obtained by GPC compare very well with those mtained by light scattering in this work (Chapters III and V). It 3 interesting to note that the molecular weight distribution, M/Mn, is quite close to the most probable molecular weight distri— Ution of 2, as expected from kinetic models of polymerization for andom polymers (F-3C)- ..__________________ *Details are given in Chapters III and V. LIME 7 this section .1 21' ‘) discussed. "”VW (:21? in Solui .1’1 p.11! 0 1 r..~enc diatdr 1'. 11' 131/21 ”10 ”‘91.".‘11'111" 0' M‘ . ' ..'.-7¢(,IJ¢. 1‘ .1- [.C' ’II'INJ'YI .1. ‘ ., '4 '~ , ~“Dt In. _ . I l v at "('11) .5" I 1 ’I‘fr"‘| .1 1 1‘ H '9’. h ' ' v. 4 “.1 '1 0 II. . 1." 11> ; a (,1 . » 1 e, I" ”r . I f" ‘l' ‘.' .1‘ l L»). '1' H ' J " -I 1'4, I t ‘1 CHAPTER III BACKGROUND AND THEORY A. Intrinsic Viscosity and Expansion Factor In this section, the configuration of a macromolecule in solution is discussed. Parameters describing the effective size of a macromolecule in solution are defined. They are the root-mean- 2>1/2 square end-to-end distance, 1/2. These parameters may be found experimentally of gyration, from the measurements of the intrinsic viscosities of the polymer solutions. The size of a polymer chain in solution is shown to depend upon short-range polymer structural parameters and long- range solvent environment factors. The intrinsic viscosity of a polymer in solution depends upon the molecular weight of the poly- mer. The molecular weight dependence is discussed in this section from hydrodynamic, thermodynamic, and empirical points of view. The results of this section are used to show how the goodness, in a thermodynamic sense, of a solvent for a polymer may be deter- mined from intrinsic viscosity measurements, how the viscosity neasurements may be used to estimate the dimensions of macromole- cules in solution and to develop the relationship between viscosity and polymer molecular weight in dilute solutions. The generation of the structure of a macromolecule through repetition of one or a few elementary units is the basic 49 mmristic of poly" 1121th “many “‘9 m (ully stretched l '1.“ :10)er molecul 1:111‘01..orlnbu1k. t a: ‘engttulse. but. dUI .11‘*1w1110ns which ch. ma 1‘12 ends of the ‘“ ".1 11m configurath ‘159'.111lso often mt: mac density 1 ’1‘ dimension 01 ""-*":e 11'. matial ‘11 1. 1.1,..1’119. L. the 1M” {1, . 1' N0. 118! '0‘ :1: u 4. 4 J ju'otlfm', 71,1 Rimzattle y, 4,. .11 . 3v. 1'31”".3,‘ n I I" 'e "0 2.161'1 1‘ 51. “n ....1 7p! 7 4| "“1 ‘. i131 -. '14": HA 1 ’ '70,!" "‘10 l G‘ A ‘3'..1fl- i. ““5" ' an ' “0'1“. '11 l .‘I . 11151,, .1: I .‘r lietign 50 characteristic of polymeric substances as is implied by the term polymer (i.e., ”many member“). A polymer molecule is a molecule whose fully stretched length is much greater than its diameter. Thus, a polymer molecule may be considered as a long chain. In solution, or in bulk, the chain molecule is in general not stretched out lengthwise, but, due to Brownian motion assumes many spatial configurations which change randomly with time. The distance between the ends of the polymer chain is time dependent. Because of its many configurations, a polymer molecule, in bulk or in solution, is also often considered as a spherical cloud of polymer segments whose density varies radially about a center of gravity. The dimension of a polymer molecule most widely used to maracterize its spatial or configurational character is the end- O-end distance, L, the distance from one chain—end group to the ther. For a long, flexible chain, the number of distinguishable “apes or configurations will obviously be very large. It is iearly impossible to describe such a chain molecule in terms of m individual conformations in which the position of each atom nStituting the chain is specified. A time average value of L is erefore required, the usually appropriate average being the Jt—mean-square end—to—end distance, ]/2. Another important lsure of the effective size of a polymer molecule is the root- n-Square distance of the elements of the chain from its center Gravity. This quantity, designated as <52>1/2, is often called radius of gyration of the molecule. For linear chain polymers Nng Gaussian statistics, Flory (F-38) has shown that IL? I 6'52). lot configurati mm on its environm m szivent. where the 'm'. and: solvent to ‘1 we r. the mean of .» W-nziymer and sol‘ upon: :c- as to redm u: «w elements. II " ““1: 2‘ Interactil ' in"uzratln. gm N" L’J'H’J’, (JCQUT. mutt .‘ ff?pr "I ('l 'o. l‘ '."J't'Ydf.f J 1 J‘ o ~l . i » .9" In,” .y'y' .‘ A 1'. J; to! "f V. ’1')». ". . , J L "3'1" . . .l- cit"? In: _ x I. . _ . b J’ ['29 all, ' t'":l"'1‘. VP! 1 .i. t“" w . ll Ih-n I l J’ -_._ 5l = 5. (3.1) The configuration of the polymer molecule in solution depends on its environment, i.e., the quality of solvent. In a good solvent, where the energy of interaction between a polymer element and a solvent molecule adjacent to it exceeds or is about the same as the mean of the energies of interaction between the polymer-polymer and solvent-solvent pairs, the molecule will tend to expand so as to reduce the frequency of contacts between pairs of polymer elements. In a poor solvent, on the other hand, where the energy of interaction between polymer segment and solvent mole- cule is unfavorable, smaller configurations in which polymer- polymer contacts occur, will be favored. It must be emphasized that the problem of polymer configu- ration is twofold. It depends in the first place on the bond dimensions and angles of the atoms along the chain backbone. These are the short-range effects depending on the characteristics of the units of the chain which are very near one another in sequence. Secondly, the configuration is influenced also by ther- nndynamic interactions between the polymer elements and their environment. The latter is referred to as the longfrange effect. It depends on the polymer molecule and its environment, whereas the first effect depends on the parameters of the polymer mole- cule alone. If the solvent medium is sufficiently poor, i.e., a G-solvent defined later in the chapter, the overall dimensions .‘1 a determined sole ”n :me till prevail ,1, 11M temperature sea-ans ulll reflect mum by environm .: we". :mdition a O-C n named as 'LS'] ‘mrnmlons. -L2- ‘N t"t':'.'. in solven '3 1'! J'W'turbed din "r. t' ‘n .. , ermng in "l l ( 2 ‘ J'LU' l .' l 1'” J ‘ (a! " . ‘ - 'w't‘fl'l 'l 4‘1".” . - . "(I ui'/.'lfijg»0. 'V f .. J l (w re d. 'J' ‘ ‘ J5], "’ 'lw '. 4276561 ‘42-, l'o w ' v'J '0'; ‘I . ’I ‘s ‘ 'Q'l“ , . m) ‘1’: U C ’0. .. ' . ll: ' 52 will be determined solely by polymer unit bond lengths and angles. This state will prevail in a poor solvent for the given polymer at a unique temperature. Physical measurements made under these conditions will reflect the characteristics of the polymer molecule unperturbed by environment. Flory (F-3a) calls this state of solvent condition a G-condition. The unperturbed dimensions are designated as 1/2 or 63>”2 to distinguish them from per- turbed dimensions, U2 or ]/2, arising due to the long- range effects in solvents. The perturbed dimensions will differ from the unperturbed dimensions by the average expansion, a, of the molecule arising from the long—range effects. Then one may write V2 = oL]/2 (3.2) and U2 = a]/2. (3.3) The value of a is often appreciably greater than unity. Staudinger (S-Z) called attention to the utility of vis- cosity measurements on dilute polymer solutions as a means of characterization of polymers. High polymer molecules possess the unique capacity to greatly increase the viscosity of the liquid in which they are dissolved, even when present at concentrations which are quite low. This is the manifestation of the voluminous char- acter of randomly coiled long chain molecules. The higher the molecular weight, the greater is the increase in viscosity produced : new. weight conce _; :53! men divided t M'cve viscosity. nr. l nl rm- .. "presses the Z'Z'IJ'JA The .'.."'.f: i'Siflkr ' 1 have I ../’l r‘ (' ‘tv'.ee ' I ' I i I I. ,'l 'trl'l’ I -l . w"{ I", I v ,r; , J "n I 1 .1 ' . 1‘ H. W". .l—ro a .j I'V- . 1,. ‘1 "in-eat: the I'e'. ‘Hlmie 53 by a given weight concentration of polymer. The viscosity of the solution when divided by the viscosity of the solvent gives the elative viscosity, nr. Also, nr is related to specific viscosity, Sp, by n = n - l (3.4) were nsp expresses the incremental viscosity attributable to the ilymeric solute. The ratio, nsp/c, where c is the concentration dissolved polymer, is a measure of the specific capacity of the lymer to increase the relative viscosity. The limiting value of is ratio at infinite dilution is called the intrinsic viscosity, ]; i.e., [n] = (”Sp/c)c+0 = [(nr - ll/C]C+0- (3-5) concentration, c, is customarily expressed in grams per lOO cc solution, the intrinsic viscosity, [n], then being given in the iprocal of this unit. i.e., in deciliters per gram. Plots of c against c usually are very nearly linear for “r < 2, and it been pointed out (M-5, H—3) that the slopes of these plots for ven polymer—solvent system vary approximately as the square of intrinsic viscosity. Thus the equation proposed by Huggins ‘ to empirically represent data of this type is nsp/c = [n] + k1[n]2c (3.6) m t1 is called the C» q thus the interc ms: t. M the logari N: o' fractionated in. mum]: of their new uttin experimen ~ r. ways may be ex "i ' en: a are empir ' "Win! and the 3 J'. ' {’4' Vvlfl‘kl’ dl -" ' In ‘ » mic be (In .91 “ W’i't’este oi . 5‘". I. ., ,! ”HRH, Li'i‘ " 9, .I‘: I-arl 54 where k1 is called the Huggins constant. The intrinsic viscosity, [n]. is thus the intercept on the ordinate of the plot of nSp/c against c. When the logarithms of the intrinsic viscosities of a series of fractionated linear polymer homologs are plotted against the logarithms of their molecular weights, relationships which are linear within experimental error are usually obtained. The linear relationships may be expressed by a simple equation of the form [n] = KMa (3-7) where K and a are empirical constants determined, respectively, by the intercept and the slope of the plot. Values of K and a vary with both the polymer and the solvent and are dependent on tempera- ture. It should be emphasized that Eq. 3.7 is empirical in origin but its convenience of application has maintained its continued use for correlating intrinsic viscosities and molecular weights. One of the earliest quantitative approaches to the problem of predicting the viscosity of dilute polymer solutions is a hydro- dynamic approach of Debye (D-2). He considered an isolated polymer molecule in a simple shear field and developed the so-called "bead- spring" model. This model is convenient for the purpose of discus- sing the hydrodynamic resistance to the flow of surrounding medium. It consists of a sequence of beads. Each of the beads, connected to one another by springs, offers hydrodynamic resistance to the flow of the surrounding medium. The springs do not offer any resistance to the flow. In the bead-spring model of a polymer r eztle. the total at n '8 l sabgmups. each . i W: special F "145’ mower units St was statistics. T ’7’“: ' is dynamic itttre‘.‘ that all if? ' «:- that the to «"11: a 7 the mono ""4 “lie: on 1|. '7' ‘1»;6‘0‘216'.‘ a} ‘3 hi} .3 Jams-es .‘a PM ' ...‘. ‘s. t'rlf V‘fi r y’Y'p f n" 55 molecule, the total m monomer units in the chain are subdivided into N subgroups, each subgroup containing m/N monomer units (m > N). The special property of the subgroup is that it contains enough monomer units so that at equilibrium its dimensions obey Gaussian statistics. The subgroup is referred to as "bead” or "segment." In dynamic calculations employing the bead-spring model, it is assumed that all the mass of the subgroup is concentrated in a bead such that the total frictional resistance offered by the solvent to all the monomers of the subgroup is accounted for by the frictional forces on the fictitious bead. The segment distribution function is Gaussian also. The beads are assumed to be connected by linear springs. These concepts are discussed in detail by Zimm (Z-l) and Rouse (R-l). Debye assumed the frictional effects to be so small that the motion of the surrounding fluid is only very slightly disturbed by the movement of the polymer molecule relative to the medium. This means that the velocity of the medium everywhere is the same as though the polymer molecule were not present. The solvent streams through the molecule almost unperturbed by it. This is called the free-draining coil model of a polymer molecule. Figure 3.1a illustrates this case. According to the Debye theory, [ ] = N cR2/l00 M (3 3) n AV e ns 0 ' where NAv is the Avogadro's number, nS is the viscosity of solvent, M0 is the molecular weight of monomer, c is the frictional (‘I Figure 3.la.—-A Free—Draining Molecule During Translation Through Solvent.* Figure 3.lb.--Translation of a Chain Molecule with Perturbation f Solvent Flow Relative to the Molecule.* *Arrows indicate flow vectors of the solvent relative to the lymer chain. urn-(tent for a bead arm. a! the polymer 1 li'cih‘ifilt measUre of ' itsas been introc ’T “it mal units. on) are winners. R: is “t “in Eq. 3.8. run m, , ,. e constant “"2 and the ir ol 513" rdl‘grlg of 57 coefficient for a bead of the polymer chain and Re is the effective radius of the polymer chain. The parameter Re is a convenient macroscopic measure of a polymer chain size. The factor l00 in Eq. 3.8 has been introduced in the denominator in order to convert 3, for the intrinsic viscosity. For to the usual units, gm/lOO cm linear polymers, R: is pr0portional to molecular weight M, and hence, from Eq. 3.8, [n] = KmM . (3.9) where Km is a constant peculiar to the monomer, the viscosity of the solvent, and the frictional coefficient. As mentioned before, actual observations of [n] as a function of M in a solvent are described by Eq. 3.7 which is [n] = ma. (3.7) Deviations from Eq. 3.9 are attributed to the fact that a polymer molecule is not freely drained. More sophisticated treatments of intrinsic viscosity take a more detailed view of the flow perturba- tion introduced by the monomer units. In particular, the idea of a "shielding effect" is introduced, whereby peripheral monomer units are imagined to be able to shield interior monomer units from the external flow (F-3f). Debye and Bueche (D-3) and Kirkwood and Riseman (K-l) have carried out analyses with the above model. Their treatments are similar in philosophy but different in mathe- matical technique. Each presents a mathematical treatment of perturbation to the flow field interior and exterior to a polymer molecule. Figure 3.lb illustrates this situation. From their m“. the intrinsic 1...!” varying fron r~m yielding. respe ilory (F'3f)' } 'tf and them“ ‘1: the more 906m“ '. int-cry Willaj -. lar-mare Md'to‘ ' a J'Ilvi’rfidi .1" )7. w_a is provided H u .1“) and l ’0'! A . treat .‘v. twain» ,. . 6' weight .e.tw' . t. prr U‘wvr. ' 58 results, the intrinsic viscosity follows Eq. 3.7 in which a is a parameter varying from 0.5 to l.0 for the case of maximum shielding or no shielding, respectively. Flory (F-3f), in considering hydrodynamic shielding (see Fig. 3.lb) and thermodynamic expansion of the polymer coil, fol- lows the more quantitative treatment of Debye and Bueche (D-3). Flory's theory predicts that the intrinsic viscosity is related to the mean-square end-to-end distance and is given as 2>3/2 [n] = o3/2 separate the quantity 3/2 (see Eq. 3.2). Equation 3.10 may then be written as [n] = ¢(/M)3/2M‘/2a3. (3.11) 2>l/2 For a linear polymer of a given unit structure, ‘l/2 = Chi/2 (3.12) where C isaconstant characteristic of the given chain structure. For a polymer chain composed of identical bonds, C will be propor- tional to the length, l,of the bond. It then follows that /M is independent of M for a linear polymer of a given unit structure. Then, from Eq. 3.11, [n] = KM1/2a3 (3.13) where 7Q I - ¢(/M)3/2, (3.14) According to the preceding hydrodynamic analysis, K is a constant for a polymer independent of both the molecular weight of the poly- mer and the nature of the solvent. Ordinarily, [n] should depend on M not only owing to the 1’2 as in Eq. 3.12 but also on the expansion factors 0- factor n The influence of the expansion resulting from intermolecular inter- actions may be eliminated by suitable choice of the solvent and temperature. Flory (F-3f) calls this choice of solvent and temperature a O-condition or ideal condition. At the O-condition, a is equal to 1 and Eq. 3.13 reduces to [n]e = KMl/z. (3.15) The influence of intermolecular interactions on the configuration can thus be neutralized by this choice of solvent medium. The : reruns at the O~c01 nervous. 131/2 am “CM that "an the measuren ‘ we a I the elven“ .g then a n r‘. ‘1' a poll?“- ' ' M": ‘2? the! DC «‘1'1);.a"li1(:h’.6]iy 1.3,, . I.‘ i- . 2 tirw ( 1 .,. . .‘ Cd 'e-yJV’.’ ' 1 e! ,_ u . e . --..t'lng 12,. ‘ ‘ ' 1 'tl’frh ' 1 ' 1 'r ‘ V. f" (.1 ‘11 .. a. | 1 " ,w a I 1"" N l‘ . . ‘K-r ,_ a .‘H I“ D; 4"] '1' c . "519 k'd "Ia , ‘ “Wt-r 60 dimensions at the e-condition are the same as the unperturbed 2 1/2 l/2 dimensions, and (F-3f). From Eqs. 3.12 and 3.14 it follows that mwmh=oi mnm Thus, from the measurements of intrinsic viscosities in good and G-solvents, the expansion factor, a, can be calculated by using Eq. 3.16; a is then a measure of the "degree of goodness" of the solvent for a polymer. The higher the value of a, the better is the solvent for that polymer. These concepts have been amply demon- strated experimentally and Eq. 3.15 for O-conditions is universally accepted at this time (K-2). B. Light Scattering and Second Virial Coefficient Lord Rayleigh (L-l) first correctly explained the phenome- non of light scattering and expressed the intensity of light as observed at a distance r from a scattering center as R = ierz/IO = 8n4ea2ng(l + coszem4 (3.17) 6 where Re is the reduced intensity (often called Rayleigh ratio), 16 is the intensity of light scattered at an angle 6, I0 is the unpolarized incident intensity, 8 is the number of isotropic scat- tering particles per unit volume having polarizability a, and n0 is the refractive index at wave length A. Following Debye's application of Rayleigh theory of scat- tering to the measurement of scattering from polymer solutions (D-4), k “1.51”. 1. '1th 17"! gr unit of 5‘ . ’1 . . r.§..)ph/(l ‘1 c .‘ scattering 'fl‘tlgt‘i‘u'a rnletul 3M. . bun/£1611”) 1.. hauler) Jr"., ,l u‘ ,, ,MJU‘ 1 , Q )7 F F n e ' . ..rli.d ,.. , . fiv‘yr _ , 1m. '1 “(on i I). ,, 9f. '21 lr ‘11 a e" .. .P" (- 'ft ii ‘ ”x I ,7 'sI-I , . a l 02 61 the turbidity, T, which is the reduction of the incident intensity of light per unit of scattering volume, has been commonly used, thus, nier sin 6 T = 2n I de (3.l8) 0 O or T = (l§E)Re/(l + c0526). (3-19) The Rayleigh scattering from gases and liquids arises from their non-homogeneous molecular structure. Making a solvent even more inhomogeneous by adding a solute increases scattering. From this initial work by Rayleigh and Debye, knowledge of the number, size and structure of solute particles can be obtained from observations on the angular distribution of the scattered light. Considering the non-ideality of the system, the expression for the concentration dependence of the scattering can be given in virial form (S-3) as 2n2n3(dn/dc)2c 1 5.6.. ._+2Ac+33c2+...- (3.20) R90 N >‘4 R Mw 2 AV 90 Ol‘ HC-l neglecting higher virial terms. The concentration, c, is in gm/cm3, Mw is the weight average molecular weight and the optical constant H is given by a I lbr/K h: depends only upon ‘n Mt) of the 501w l '1 1:11? ve index increi 1:1me to solute p1 Ht" 1‘ the incident 3’ 't ‘ 1" .ua! I/PD, inter '0- ' a on . " an unsyhme [of :' .. 2.2: 15ml“ " WW1 Liat’ering ".1 n ”WWI. l — ' 9' spy . ‘1 I. . adzhap I:h 'l 'e l‘ 1' , _ 'l w rrl' N ‘M -a . . 'o ,1! «I: ,1 1' a. ' ~'-€'.'.l-r a .O ' '1 .1 ,. let a," r Y ’ a V . I l 'I: 7 1 (I a .h D" I , O .1 r‘. ’ We. ". {-r‘ ‘1' -l C'l‘: u '- r . I 62 H = 16n/K (3.22) and depends only upon the wave length of radiation, A, the refrac- tive index of the solvent, no, at wave length, A, and the specific refractive index increment, (dn/dc), at A. Equations 3.20 and 3.21 apply only to solute particles which are small compared to the wave length of the incident radiation (less than A/20). For particles larger than A/20, interference occurs in the scattered radiation resulting in an unsymmetrical angular scattering pattern. In this case, Eq. 3.21 is multiplied by a factor P(e) (D-4), which corrects the observed scattering to the value it would have in the absence of the interference. This function has been tabulated for various particle sizes and shapes (D-S, B-7). Equation 3.21 then assumes the form 59. T xlx n + 2A c + - - - - (3.21a) 3 "DA CD N co 2 One of the major problems in light scattering has been that of relating the scattering intensity to the dimensions of a ran— domly coiling chain molecule in solution. Owing to its continuous change in configuration, the dimension obtained is a time-average dimension. Zimm (Z-Z) approached the problem by recognizing that the scattering, which is a function of both angle and concentration for large polymer molecules, could be plotted simultaneously as a function of 6 and c. The limiting value of l/P(e) is expressed as 2 lim l/P(6) = 1 + ‘5"2 <52> sin2 §-+ - - - . (3.23) c+0 3A ,,,.52 is the MEG' I visits. for a ”M 31mm-L2H5t m sdtstituting E 1 general methi as u astc eliminat 1i ' 'v; the correct 1 inf/".1“: by lim (2 'i'm' oi solute ,- '., ‘. The . st. whe H'Ie: 93’. In? 2 stale- 63 where <52> is the mean-square radius of gyration of the polymer molecules. For a random coil configuration is equal to /6 where is the mean—square end—to-end distance of the chain. Substituting Eq. 3.23 into Eq. 3.21a, we have 2 A general method for treating experimental light scattering data so as to eliminate the effects of destructive interference and Jbtaining the correct values for the molecular weight of the solute ms outlined by Zimm (Z-2). In this procedure data are obtained br a number of solute concentrations, c, each at a number of scat- ering angles, 6. The ratio Kc/Re is then plotted as a function f sin2 (6/2) + qc, where q is an arbitrary constant selected to Jitably spread out the data. Experimental points obtained at any iven scattering angle may then be extrapolated to c equal to 0 and ta points obtained for any concentration at different angles trapolated to 6 equal to 0. Thus the limiting slope of the zero gle line in a plot of Kc/Re against sin2 (6/2) + qc yields the 20nd virial coefficient, A2. The ratio of the limiting slope of azero concentration line to the intercept gives the mean-square Fto-end dimension, and the reciprocal of the intercept on the Re axis (where the 6 equal to 0 and c equal to 0 lines when rapolated should meet at a point) gives the weight average ecular weight, MW. In a honopolyr 1" and the only dim I c". heterogeneity. " Him a unifor "51.) dilute soluti mi; only 11 the 1. I ”ll MfgughoUz aaaaaa "" flm'i Droduce ."‘¢e -t drift in ' "Q; 7'» (ODOiym 1",“ ” 1"? “tattlt r ,. r r. , are} J'i'yi;ft' "” tvuatior ”Kile , C 1C “HIT” c": w 64 In a homopolymer all the elements scatter in the same man- ner and the only difference within the sample is due to molecular weight heterogeneity. The preceding treatment applies only to polymers with a uniform composition. In measuring the light scat- tered by dilute solutions of c0polymers, the preceding treatment will apply only if the distribution of the two monomer components is uniform throughout the c0polymer. As mentioned in Chapter II, the copolymers produced could be heterogeneous in composition as a result of the drift in composition at high conversion, except for the azeotropic copolymer. In this work this effect was minimized by carrying the reactions only to low conversions. For copolymers where considerable dispersity in composition nay occur, the equation for Rayleigh scattering has been expressed by Stockmayer and coworkers (S-4) and Bushuk and Benoit (B-8) as 70 I . 2 - K 2 ViciMi (3.26) and . 2 2 4 K 2n no/NAVA (3.27) where Vi’ c1, and Mi are the specific refractive index increment, weight concentration and molecular weight, respectively, of the scattering component i. For such a system, the actual measured quantity is - . 2 Re - K vocMapp (3.28) where v0 is the average refractive index increment and Mapp is the apparent molecular weight obtained from the data. The specific r mx'nlr in a solvent nan 'hrrements of th ivri'. weighted by th 114mm of molecul ‘1 on; teen establ sum an: 'oenoit (Ii-8 11w. a: determined '11.! u at attained 1 i am as: rstabli ‘11"- s r, .t. J 2v” 6 Vll ' 1 '1-4) and - a .1. re Mia”: . .11" A . l cure/,1- r_ 65 The specific refractive index increment, dn/dc, of a copolymer in a solvent is a simple sum of the specific refractive index increments of the two component homopolymers in the same solvent weighted by their weight fractions in the copolymer and independent of molecular weight. The dn/dc of homopolymers has since long been established to be independent of molecular weight. Bushuk and Benoit (8—8) showed that the mole fraction of each monomer, as determined from measurements of dn/dc is within 2 per cent of that obtained through chemical analysis. Kinsinger et al. (K-4) have also established the colligative nature of dn/dc after investigation over a wide range of copolymer composition. Kin— singer et al. (K-4) and Klimisch (K-5) have shown in their study of me colligative nature of this increment that it is possible to alculate the average composition of a copolymer based on the easurements of dn/dc of the whole copolymer and that of the indi- idual parent homopolymers (A and B) in the same solvent. Hence, (dn/dc)copolymer = v0 = XAVA + (1 - xA)VB (3'29) mre xA = cA/(cA + cB) is the weight fraction of species A having ight concentration cA in the copolymer. The refractive index crements, vA and VB, are the dn/dc values for the two homo- lymers A and B in the same solvent. Bushuk and Benoit (B—8) through their derivation obtained Mapp = MW + 21mA — vBl/vol + om), - vB)/v012 (3.30) "'3 " 15 the true 11 fr: i . ‘1',""01 ”emess ', m. weight it arc tile (Ic 66 where Mw is the true molecular weight. The-parameters P and 0 represent the heterogeneity in composition; P relates to the compositional skewness about the average xA and Q to its broadness. . - x) (3.3l) Q = Z YiMi(xi - x)2 (3.32) 1 where Yi is the weight fraction of component i whose molecular weight is "i and the composition is xi, and x, given as x = X y.x., (3.33) is the average composition of the sample in weight fraction of com- ponent l. The osmotic pressure of a polymer solution, n, is defined as the pressure which has to be applied to a solution so as to raise the partial molar free energy of the solvent to the standard state value (F-3a). Thus, -0 - - n - e1 - G1 + (J (361/3P)T,x]dP (3.34) where G? is the partial molar Gibbs free energy of the solvent in the standard state, T is the absolute temperature, x1 is the mole fraction of solvent and P is the pressure. The variation of (aGllaP)T x = V] with pressure may be neglected, so that ’ l G] - G? = RT 1n a = -nV] (3.35) l "a. if is the parti( 1 r'? Raoult's law Ir. a] - ln (1 row 151s the mole ' 5y combining E -xf :;r ssure of a 1’JV 4 the solvent. ' It. 5' in the con ~' 47‘ I 1 dd ;. n . i f,”- I... . .‘ '_'(.l t » . O A 67 where V] is the partial molar volume of the solvent. In the range in which Raoult's law applies, one may write 1n a1 = ln (1 - x2) 2 -x2 (3.36) where x2 is the mole fraction of the solute. By combining Eqs. 3.35 and 3.36 one may write for the osmotic pressure of a dilute solution (for which the partial molar volume of the solvent, V1, is indistinguishable from its molar volume, Vs) in the concentration range satisfying Raoult's law n = (RT/vs)x2. (3.37) Since x2/vS is equal to c/M where the solute concentration, c, is expressed in gm/ml, n = (RT/M)c. (3.38) At higher concentrations, where binary and higher order interactions of solute particles have to be taken into account, 2 3 + A c 3 + . . . .] (3.39) n = RT[(c/m) + Azc where A2, A3, etc. are the second, third, and higher order virial coefficients. The coefficients A2, A3, etc. have the same value as in Eq. 3.21a. Thus A2 is related to Gibbs free energy through n and so it is a thermodynamic quantity. C. Viscosity Correlation Techniques In this section two empirical correlations for correlating viscosities of polymer solutions are described. llaver Lav Correla Is the concen‘ w aslyner “spheres" 1m ’entangled' as 1".1 :1‘ute solutions. . armds Much more ”'9‘: than it doe .‘1‘l'cli‘l r“ '~ ‘1 1'12 content 1'1 We Lolvent dQDQI 131111”th 'oOlutlt 68 1. Power Law Correlation As the concentration of polymer in solution is increased, the polymer "spheres" begin to overlap and finally polymer chains become "entangled" as opposed to existing as isolated chains in very dilute solutions. When this happens, the low-shear viscosity, nr, depends much more strongly on polymer concentration and molecu- lar weight than it does in dilute solution. In dilute solutions, 11', = 1 + cha (3.40) where c is the concentration, M is the molecular weight and a and K are the solvent dependent constants for a polymer-solvent system. In concentrated solutions, a, ~ (crab)f3 (3.41) where b and B are correlation parameters. Experimentally, B is often found to have a value near 5, and b, a value near 0.68, so that correlations of the type ”r ~ c5143.4 are frequently successful (M—la, P-l). The value of the product cM, where Eq. 3.40 ceases to describe the low shear viscosity of a polymer solution and where Eq. 3.41 provides a reasonable fit, is termed the "critical" entanglement point or "critical" entanglement density. Below the "critical“ entanglement point, the size and concentration of the effective polymer spheres dominate the flow phenomena and Eq. 3.40 applies. Above the "critical" entanglement point, the network structure of the solution is usually presumed to dominate the flow We and Eq. 3-4 .zmtniration and mol1 #1 '13. entanglemeflt‘ me in concentn‘ WWW“ parameti 1 e'tanglement poir Hog-log plot e11" :.rve for a PO1 : 1",? weights of t 12111 the ”Jillcal" V '1.‘err;ii-Za) a e 1 O a ' ‘ 1'13 type f1 ”a: .trrt-latiol ‘5 . . 1‘ ,uuiff ‘ 16w 1. ‘flfi'W' luv F OD , .11: UM " ' ‘a‘. ‘ - C'J'V‘fi 1 ll'ie’ - diz‘. r0 In 69 phenomena and Eq. 3.41 is used to describe the dependence of "r on concentration and molecular weight. Since the physical nature of the flow entanglements is usually thought to dominate the flow phenomena in concentrated polymer solutions, polymer-solvent thermodynamic parameters are most often neglected above the "criti- cal" entanglement point (B-9). A log-log plot of nr against cMO'68 usually yields a single master curve for a polymer-solvent system covering a wide range of molecular weights of the polymer. This plot is a straight line 0'58 with a slope of 5. Middleman above the "critical" value of cM (M-la), Ferry (F-Za) and Fox et a1. (F-6) summarize available results of this type for a few polymer solvent systems.* 2. Simha's Correlation The power law correlation is based on determination of a parameter from the experimental observation of the dependence of nr on ch. This correlation has been used for correlating the data as explained above. As an alternative, Simha (S-6) suggested a different way of correlating the data. This correlation is based on Einstein's model for the treatment of viscosity of a suspension of rigid spheres. 5 According to Einstein's (M-Za) original hydrodynamic treat- ment of rigid spheres with interparticle distances very large compared to the particle diameters, *williams Model (w-l) predicts n ~ cM0’625. discussed in Section 0 of this chapter. This model is ”'1‘ ‘i the viscos '11:"1'etl and d 15 11‘: treatment the I‘ 9:11?!“ their f I'ld . 6 :Jr(. ”um: .N “‘40) v11 ‘1" a l " °' trot-r or 3 ‘7 and ' 'J’ New“ ' ""7! ,l‘ W‘Jd '11 T..( 3‘ 1’1- [(1’- I' u. . “ " 'fl'tdn a. ‘ 4-a 1. ‘1'! 91 ‘1 w d. .-e.‘. a], 1 l 1 .r. ’31... , . oa (’3ra l I: H t>¢ '4“ i l. ‘ ,a .a , ‘- L .‘l' .' I It]... '. '21‘ "Il' 70 Tl'l'l _ s = Sp n 5/2 a (3.42) S where n is the viscosity of the suspension, n is the viscosity of s the solvent and p is the volume fraction occupied by the spheres. In this treatment the particles are treated as rigid structures which preserve their shape in the course of flow. In general, n = n - l = at (3.43) where a is a pure number depending on particle shape. The actual viscosity of dilute colloidal suspension often exceeds by an order of magnitude the value predicted from Einstein's relation (Eq. 3.42 and 3.43). This is most probably due to con- centration effects. In extremely dilute suspensions, the total viscosity effect is the sum of the effects caused by each of the individual suspended particles. The perturbations of solvent flow produced by the sus- pended particles are therefore independent of each other. However, as the concentration is increased from infinite dilution, flow perturbations are no longer independent. For moderately dilute solutions the interaction of perturbations of the solvent flow can be classified into those caused by two, three, and higher number of suspended particles. The strength of these effects is directly proportional to the second, third, and higher power of concentration of particles, respectively. This indicates that 115 can be pre- P sented as a polynomial in the concentration of the particles. As "a concentration inc mast to obtain a Fiif‘i'dllm behavio i‘mha (5-5) h metal particles a1 "' Iil'lft'ffellon dew in 1+ mtten ,i.. ““‘r a 5. ',1‘ ‘ f a..{ 'J is. ra')' ‘- ,.. 1.1 'J' ‘ ’ 2" k4 . M v a: 71 the concentration increases, the degree of the polynomial must increase to obtain a reasonable description of the experimental concentration behavior. Simha (S-S) has calculated the coefficient of d2 for spherical particles and obtained nsp = (5/2)o + 12.5 o2 + - - - - (3.45) The concentration dependence of spherical particles to terms p3 and higher may be written as (F-7) 1'1 ——S—2= 2 o o o 0 ¢ [n] + 62¢ + 33¢ + (3°47) or n - - -§E-= [n] + k][n]2 + k2[n13¢2 + ° ' ° - (3.48) where [n] is equal to 2.5 for spheres and k], k2, etc. are pure num- bers independent of dimensions or molecular weight of the suspended particles. Frisch and Simha (F-7) have presented details of the concentration dependence of the viscosity of suspensions of spheri- cal and non-spherical particles. Rearranging Eq. 3.48, a power series in concentration may be written as n -%E-= [n]{l + k][n]c + k2[n]zcz + k3[n]3c3 t ' ’ ‘ '} (3-49) Simha and Utracki (S-6) have proposed to make use of the above equation for correlating viscosity-concentration data for a range of molecular weights of a particular polymer in a particular sl'ver'. The linear upended spheres. l «sentence of the re] :Ma' Different cu "”33“ to produce (.13: asainst c a i "' " ' Given so O - 1""- '-"‘~'1€ plots J" mzt'yfi: (2'6) ti '4 11111; .Ua I t tettlon C than . 0"" 1‘30 (g '\H 7' m i ..> Or :5 Ulla . l, i JJ" "1‘“; 'a . .0. 5.! 2“ [Uht aan‘t.‘ I an»; I“. Ur, 'b’l-e \ ’1 " .. DP' I". . “ ') f‘>” ; "'y l '5 hr,“ . a: 72 solvent. The linear macromolecules in solutions are modeled as suspended spheres. The experimentally observed concentration dependence of the relative viscosity in the very dilute region is linear. Different curves for different molecular weights can be superposed to produce a master curve. Log-log plots of nsp/c[n] against c are made for different molecular weights of a polymer in a given solvent. Then y(M) is chosen as a shift factor such that these plots merge into a single master plot. It has been observed (S-6) that y is preportional to M"E where, usually, 0.5 5.8 5.1.1. 0. Williams Model for Zero Shear Viscosity This section describes a molecular theory of Williams which has been developed to describe the viscosity of moderately concen- trated polymer solutions. Continuum rheological models for the treatment of rhe— ological response contain parameters that frequently lack molecular interpretation and yet they are used because of their simplicity. Middleman (M-lb) has given an excellent review of many of these models that contain two, three, or four parameters. These models contain no reference to the structure of a material, but contain empirical "curve fit" parameters that may or may not have molecular significance. In recent years developments have been made in the treatment of rheological response of a material based on its molecu- lar structure. Ferry (F-2) and Middleman (M-l) deal extensively with molecular models. For polymer solutions these models are of the form in!" "'1‘. 'r and r5 u :11". Live" 0' 10' Sh vr'i. ”Sportive” :41‘ ,1 ‘sra system (a, 0'? n‘tcrldl pd 1:.1'9'31109'10 .fi, :4 i"? one shOM 31'1”“ iullCl. ;".;Ib'.l:d (M' 11” :‘fil' ‘h Fl? a. a a..( D J ”rlglh’} A. a: ‘ V M. .7161] u‘ . . .,e ,l {'(JYV'. Gr .1 "c Tall/(2 :1: '11-: w ”1.). ' .,'.. “(a a t 1- ‘14- n " ‘1 ' 1? 4h ‘ .'p‘c 1' a ”a, e. , ,' .r , 'a 5‘1 0 {7" ..‘( . flag”! on 73 -———§- = W?) (3.50) where n, no and 11S are the viscosity of solution at shear rate i, zero shear or low shear viscosity of solution and the viscosity of solvent, respectively. The parameter A is a characteristic time constant for a system and i is the shear rate. In Eq. 3.50, no and A are material parameters of the system. For many typical poly- mer solutions, log-log viscosity-shear rate (n-i) plots have the shape of the one shown in Fig. 3.2. Several functional forms of continuum models and Eq. 3.50 have been proposed (M-lc) to describe a typical n-i curve such as the one shown in Fig. 3.2. These equations have been quite suc- cessful in describing the experimentally observed 0'? behavior of many systems. In all these equations experimental values of the material parameters are required in order to be able to fit the data of the n-i curve. The horizontal portion of the curve in Fig. 3.2 describes the Newtonian—like, low shear rate behavior. In this region viscosity is constant irrespective of the value of the shear rate. The horizontal curve extended to zero shear rate gives the value of the upper Newtonian viscosity, no, for a system. The shear rate at which non-Newtonian behavior, or the falling cruve begins. is related to the parameter A. Williams (W-l) has suggested a model for predicting the curve and has also suggested a molecular model for predicting the no value of a polymer solution from a knowledge of its molecular weight, 74 .qua moplmop a co cowuzpom cmsxpoa a mo Low>wsmm .. a c .mpmm cmmcmlxpwmoumw> um>gwmno a~_m¢:wswcmaxm mo cowpmcumzp—Hai.m.m mc:m_m $.33. Bonn e: lL'MlsooslA Wmnl’c behavit mate: his effort 1 2111".” solutions. was oi isolated r m solutions and 15! "1e problem la In distinct poly ‘Iim'i. lfitfvanolecul 'LJ'W-ded by an ll .1 ' "'1 991' lhtra l ‘h’; {’3 ailliamg ,‘tb\ ‘ "IO-‘1 F. lri 30h ""1""... . (minor. 1 j '1”‘ln'.. in ',r . . I.’ i 4:, , ,. J . "(1:1 LO'.’ .— ,. 9 l A :‘f’.._)(.a ? i 't’ I, . 1e (1.: r '1 t age-1 V' ‘ — JY'I ,. I" a b .‘t‘ O. .l >1 . a ”hymn"! 4K. . v";nu»vaa w. | 75 thermodynamic behavior, theta dimensions, and friction factor. He directed his effort toward describing the flow of moderately con- centrated solutions. These solutions lie between the conceptual extremes of isolated polymer molecules in a sea of solvent in very dilute solutions and the rubber-like network of polymer chains in a gel. The problem was to find a method of treating the forces between distinct polymer chains in a flowing medium. In dilute systems, jgtramolecular forces are important since a macromolecule is surrounded by an infinite sea of solvent while in concentrated solutions both intra and intermolecular forces are important. According to Williams, the latter interactions (between different macromolecules in solution) are probably more dominant in polymer systems most commonly encountered. He used Fixman's (F-8) descrip- tion of stresses in solutions of linear polymers to develop his relationship for no. The model for a polymer molecule is the pearl necklace chain model (see Section A in this chapter for description I of this model). Each polymeric solute molecule is represented by a series of N identical segments which interact with adjacent seg- ments along the chain through spring forces and with other segments through excluded volume forces and hydrodynamic forces. Each seg- ment interacts with the solvent through frictional resistance. The solvent is represented as a continuum phase creating frictional resistance to segmental motion. Fixman's equation describes the total stress tensor ; as ~a'. 3" .«rV‘lz m .' represents 5 was: :NsWreS' ,l 1'. ‘1‘. e1 )5 the p05 v 7. H.711 will”? a raw '.'.£‘. scant-(1t 1 flatness-r {no mlL‘Ci , :la'm' 150')? {'1‘ . u.a muc- (ifrlt‘. 'r':"a'.1:'. ‘llu' " e a.‘ J’FCI‘ ' C. .l- a!” a". , , .e“" " a f ‘ a '1'. hr ”- 76 N + n X (BiViU> + %-n2 i I. = I (3.51) 0 where ;0 represents stresses due to solvent and externally imposed isotropic pressures, n is the number of polymer molecules in a unit volume, 5} is the position vector of ith segment of the first poly- mer molecule relative to the center of mass of its own molecule, U is the interaction potential (or intersegmental potential energy) between that segment and all other segments, r_is the position vec- tor between two molecular centers and V is the interaction poten- tial between those two molecules and 1 runs over the segments of molecule 1 whose center is fixed at 34. The relation between posi- tion coordinates illustrated in Fig. 3.3 is _ (m) a, - to") + B..- (3.52a) [(m) ' Ev") = £(mn) E: {J (3.52b) th where x, is the position of i segment referring to an arbitrary l origin. The brackets < > are used to signify the averaging over all possible conformations. The term <31ViU> is a function of coordi- nates of the segments of a single molecule. If v is a distribution function in the coordinate space of all S segments, then the term (i.e., the average) is obtained by multiplying Elva” by v and integrating over the complete space (H-4). The term is a function of r_only. If g(r)is a pair correlation function for intermolecular interaction, then the term (i.e., the average) 77 .pmuoz mamwppwz we cowum>wcmo cw mucmsmmm ucm mopsu -mFoz cmaxpoa op mcwccmmmm mmumcwugoou cowgwmoa we cowumgumz__Huu.m.m mcammu o .2320: N 2.50.02 1 tamed by "W" aim 5W (”'4)‘ arm by W mater , a m sewm“ t :1"! "Mr 0‘ m“ it should be 1.3. p, "mm! 5 1m 1M CWCW fr 3, iron the ('13 .,. . gag-«113) (F :u‘e ‘0'! $0M"? ' 1m palm?" "’ r: 'Tv'vZWi'dU‘d 30 ~ 'u :1 meme {’01' I ' .‘I" (63" '1‘ 78 is obtained by multiplying [Yrv by 9(5) and integrating over the complete space (H-4). The pair correlation function, 9(5), is defined by the statement that the number of pairs of molecules which are separated by a distance r'is(N2/2V)g(r) 4nr2dr where N is the number of molecules in volume V (H-Sa). It should be pointed out that Eq. 3.51 does not represent simply an arbitrary series expansion of g in powers of n but is developed directly from the equation of motion of solvent and solute. From the examination of the concentration dependence of the pair potential (F-9, F-lO), Fixman expects the model to be applicable for polymer volume fraction less than one-tenth regard- less of the polymer molecular weight. He assumed that in moder- ately concentrated solution (volume fraction 5.0.l) U must be a pairwise additive potential. In the case of very dilute solutions, Eq. 3.51 is simpli- fied because VrV is unimportant for two reasons: n2 is very small and V(§) becomes negligible as [_increases. Thus the force on the ith segment is not a function of the positions of segments belong- ing to separated molecules. In the case of higher polymer concentrations (volume frac- tion 2.0.l), Eq. 3.5l may be inadequate. The parameter, U, is then no longer pairwise additive. This would necessitate the use of averages taken over three-body interactions, with terms in n3 becoming important. This would correspond to the presence of entanglements and the form of Eq. 3.5l would change considerably. 0n the basis of some experimental investigation, Williams proposed :11 i:. 3.5) would .: ci'ans. he then I m in solution, > . 2 5' MT MM n WU “t"- Wt term wit) 3’ "impolite rm {7' HUM“, V" ‘p .4 . h "n'ldhl'. '1" “' W. 91 W» l _.u. , I “I, e yrfj '4‘... 'l‘. h 'l 51:; rho-'1»: '5” 1 I l P‘(' i. ' y , ' l A I ./ . r 4 1| ' “u. 3.7 | 1 ka. ‘ 79 that Eq. 3.5l would be satisfactory for moderately concentrated solutions. He then argued that with higher concentration of polymer in solution, the intermolecular forces become dominant and the term with n2 would be much greater than the term with n. As a result, the term with n2 would suffice to describe the behavior in some intermediate range of concentration. This reduces Eq. 3.5l to 2 %—n <[Yrv> (3.53a) "F! I llv-i O %"2 I rwwodg (3.53b) For estimating intermolecular potential, V([), Williams made use of Fixman's equilbrium theory (F-8). The segment distri- bution of a single polymer molecule about its center of mass can be described by a probability density, v(3). The presence of another molecule at position §_will lead to a repulsive force on each segment of the first molecule. The net force on this mole- cule is obtained by integrating over the segments at all 3_and is given as V(r) = A I v(3)v(r_+ 3)dB_ (3.54) where 2 _ 2 d e A Vp 537' (3.55) P in which Vp is the molecular volume of polymer in solution, a is the free energy of mixing segments with solvent and vp is the r. a fiction of p ' f. is known as. ' ‘am umssed tl Arr-'57 (‘39) 0011 1‘0 I'leiems USEl 3mm (v.3) ‘Ort "' "'0 stresses in .«w ’J "K”.(Jles. -‘1 in ("I f "'er rd 1 H 5,.“ ’6’ 'r‘4 u ' 7" ' 1‘ b‘. 1 ‘ I M. m. , , ‘ ’x ' o (, V v. H" ,2 ' c "4“: "it ,1 (‘7 -I ' '1'“ . ~ I' ' r ‘ '.." h" “In d‘ 80 volume fraction of polymer in solution. Equation 3.54 can be used if v(3) is known as a function of concentration and shear rate. Williams expressed the function v(B) as a Gaussian distribution function (F-3e) but expressed it in terms of a dimensionless shear rate. Williams used the technique adopted by Kirkwood and coworkers (K-3) for estimating 9(3) which has been used to calcu- late the stresses in single component systems composed of simple spherical molecules. He obtained a steady state equation for g as v- {Vg-gV ln go} = (7%): - g - v9 (3.56) where §_is a shear rate tensor, 6 is the friction coefficient, 90 is the zero shear rate form of 9 usually called the radial distri- bution function, k is the Boltzmann constant and T is the absolute temperature. The shear dependence of g(r) was expressed as -] (3.57a) (.0 II (D O l'“! c—J + 7'? —+l A $9 0 CD 0 (D v '6 A 1 v + go[l + (%%)(sin 6 cos 6 cos ¢)w(r) + 0(52y2)] (3.57b) where y is the magnitude of shear rate in simple shear flow. Wil- liams truncated the series after the linear term in y because of the smallness of 5?. Combination of Eqs. 3.56 and 3.57b yields an equation for w(r) whose solution is necessary in the evaluation of Eq. 3.53b rewritten as “‘1 "Md that t":?", 'J Mre) UH " e ‘.’P" each St " ‘l’W'ital 10! ' T. n 'w'. '.' (”(Ha 81 .1 2 2 - so = 7 n f [:VVJQOd: + %—n2[%¥)f[ryv]gow sin 6 cos 6 cos ¢ dr_ (3.58) illiams argued that at high concentration (for high polymers, one er cent or more) the segments of separate molecules begin to inter- ingle. Then each segment is subject to a nearly random distribu- ion of segmental forces which tend to cancel each other; onsequently, go(r) approaches unity. 90(r) E 1. (3.59) hysically, this means that such a solution contains a uniform den- ity of polymer molecules. This results in a great simplification i the equation for w which is obtained by the combination of Eqs. .56 and 3.57b. The intermolecular potential V(r) was obtained from v(3) 1'"9 Eq. 3.54 where v(B) is shear dependent and Williams expressed in terms of dimensionless shear rate. Finally, to compute vis- Slty, Williams found the symmetrized shear component of g from . 3.58 and the result obtained was n - n s . 9 2.2 . (3.60) *= : _——>\ ' ' ' n0 _ n5 f(>\v) 1 14 Y re A is the unspecified time constant for P01Ymer chain response, rm : ‘1 the concel m". V ‘3 the mob “1.". M7 13th: ""“' X 1' W.- mm H“ 82 cN 2 _ AV B3/2 where c is the concentration of polymer, “AV is the Avogadro's number, M is the molecular weight of polymer, k is the Boltzmann's constant and T is the absolute temperature. In the above equation, = 3/2 (3.62) where is the mean-sequare end-to-end distance of a polymer molecule, 3 53 (3.63) 30m [{[14 cNAV A] and g is the friction coefficient. Several models have been pro- posed to predict g and two of them are by Williams (W-l) and Frankel and Acrivos (F-ll). In this model for no, thermodynamic solvent effects are accounted for by the term A/kT, obtained from the measurement of activity of solvent in solution, and related to the second virial coefficient A2. Equation 3.61 can be tested rigorously since it does not contain any unknown constants and all the parameters involved can be measured or estimated. An interesting feature is 0.625 that the model predicts nr ~ cM A common experimental obser- 0.68 ( vation is that nr ~ cM see Section C of this chapter). [XPEJ "I? "@0109“ *C'I‘I:(0811H "1"?le ’ dbl)” ' i: Iva tonne "n?" ug' WHO, . ‘1‘ ' .‘. 'J’l'p ‘I‘ c; ' 3', Mr. l a" 'v - (Icon-d :1 uni-(er. 1';‘.g.H-rj CHAPTER IV EXPERIMENTAL APPARATUS AND MEASUREMENTS A. Viscometry, The rheological behavior of solutions was characterized by zero shear viscosities as well as by viscosities over a range of shear rates. Capillary viscometers were used for measuring vis- cosities of low concentration solutions. A cone-and-plate viscometer was employed for measuring viscosities of high concen- tration solutions. Viscosities less than 0.l poise were measured by the capillary viscometers. Viscosities between 0.1 and 0.5 poise were measured by both the capillary viscometers and the cone- and-plate viscometer. In this latter case, the experimental viscosities obtained by each of the two methods were within 5 per cent of each other. This is within the error bounds of rheological measurements with a cone-and-plate viscometer. Higher viscosities were measured by the cone-and-plate viscometer. Both types of vis- cometers are extensively described in Ref. (V-Z). All viscosities were measured at 30°C. 1. Capjllary,Viscometer a. Practice.--Cannon-Ubbelohde* suspended level U—tube capillary viscometers with four bulbs (for four different shear *Cannon Instrument Company, State College, Pennsylvania.l680l. 83 ,1“, *7! used for u'nms. The basi .wmter. (2) m We viscometer ‘3. ma 5' the can“ a, 4‘3: in the Cap 6 i : '1 0 gm! imp?“ m I-Z [- lower. ”r 1: mm dilutl' :2 11mm; the (6| '} 1". 1) :"Cv‘ldm’j d r ;'.".r *2 mm “L." o. W \a '1'”? (HM rd '1 I _ .' I or. ing; “61‘ ' ' J. N" 11ml "“1”.“4'14 6' J (1‘6“ W? "he «6' 84 rates) were used for measuring viscosities of low concentration solutions. The basic components for capillary viscometry are (l) viscometer, (2) thermostat, and (3) timer. In the Cannon- Ubbelohde viscometers used here, a side arm provided just below the end of the capillary established the same external pressure on the fluid in the capillary above and below the flowing column. This is a great improvement over the very simple Ostwald viscome- ter (V—2). A lower, larger bulb acts as a solution reservoir in order to make dilution directly in the viscometer. The multiple bulbs topping the capillary are used to provide several shear rates by providing different hydrostatic heads. A complete description is available in the instrument bulletin (V-3). The measurements were made at 30° t 0.0l°C by clamping the viscometer vertically in an insulated water bath equipped with a precision hermostat. The time required for a solution to flow through a ulb was measured at least three times to within 0.l seconds and he average value was taken. The flow times were always greater han 200 seconds so that kinetic energy corrections were found to negligible. Before introducing the solutions into the viscometers, they are clarified by pressure filtration through ultrafine sintered ass filters. The minimum volume necessary for making the asurements was 5 ml. Solutions were diluted in the reservoir lb for measurements of viscosity at different concentrations by ling predetermined quantities of clarified solvent. b. Calibrat Slim um provid PM" it maswing ”'3 7M nmured ”1- noorted in 1 ‘ ~ five: way-33.15“) A . F'filfi‘tj‘cfle‘ 'J" 5" (to: r: ; aft ”(0er D It'. .It-d ‘0' "II 'J". "I? “'DC' n.( 'J' 0'3“”: A l v ' uni C...“ b'e,’ 2’“: .i: t vb ’I"'.'_r ,.1' r\ ‘0 ‘ ' ’ SSZCr *1 in, h“. 6 rd h, (‘ . I.” 1 arm J' .h. i ‘ ‘eror «I O r ,. n] T "‘9‘ 4 , ‘ . can't! ._ ‘ :- “7‘11." :.". 1"...” 1 , f e’» f)- . .”;£$ \ l' '1“, . #Q‘ 85 b. Calibration.--The calibration constants of the vis- cometers were provided by the Cannon Instrument Company. They were checked by measuring viscosities of distilled water and benzene at 30°C. The measured viscosities were within 0.1 per cent of the values reported in literature. Appendix A gives constants of the instruments that were used in this work. 2. Cone-and-Plate Viscometer a. Practice.--The basic components of a cone-and-plate viscometer are (1) variable speed motor, (2) cone and plate, (3) torsion bar (to measure torque), and (4) recorder. A cone- and-plate viscometer called a Weissenberg Rheogoniometer,* model R-l6, was used for measuring viscosities of high concentration solutions. The viscometer was used in the constant shear con- figuration. Figure 4.l shows the main body of the viscometer. The apparatus has been sufficiently documented in Ref. (P-Z). An exhaustive description is available in the instruction manual (I-l). A brief description of the minor modification will be given here. Although a range of sizes are available with the instru- ment, a platen diameter of 7.5 cms, and a cone angle of l°-37' were used. Three different torsion bars were available for different ranges of viscosity. Depending on the expected viscosity, the *Manufactured by Farol Research Engineers, Ltd., Bognor Regis, Sussex, England, and made available for this work by the Department of Chemical Engineering and G. G. Brown Laboratory, University of Michigan, Ann Arbor, Michigan 48l05. U003) Ira-n 86 Legend for Figure 4.1 Torsion Bar Clamp Torsion Bar Shear Stress Quartz Load Cell Output of Load Cell to Charge Amplifier Upper Air Bearing Air Input to Air Bearing Radius Arm Rotor Cone Plate Heating Chamber Heat Transfer Fluid Drive Shaft from Motor and Gear Box Base 87 rn on u\\\\\\V Figure 4.l.--Rheogoniometer (Constant Shear Rate Configuration). mgamding torsil m to record 1h! < Setting the in (Mature of n m‘ts. Theoretic! hir‘fn; the center name this is not ’rszmc at the ape ”WWW-hm err '- hr. Win; the in ”AW"! U the co "4 Wm toHn "I ' . . 01H. w; 1 he .pv . the manual . l7“' I v L5 91% -‘i'u ‘~ in J t r ~ . l. an . a" ’ 'Iv'”, 3! "~ 'tzemn . .‘ ‘ 5*..IYM if»! I I « ‘ vb”; h“ _. a] ‘. l‘: a 1')" ’r- _ \J'i .24 “‘n n*'£7;'b Jhu., 88 corresponding torsion bar was used. An ultraviolet recorder was used to record the output signal. Setting the correct gap between the cone and the plate at the temperature of measurements is extremely important for accurate results. Theoretically, the cone should have a perfect point touching the center of the flat plate during the measurements. In practice this is not possible to achieve. The cone is therefore flattened at the apex (tip) but the viscometer is designed so that the experimental error which results is negligible. The gap between the two during the measurements should be such that an imaginary projection of the cone to a point should touch the center of the plate. Before taking the data, the squareness and concentricity of the platens and the gap were adjusted according to theprocedure given in the manual (I-l). Prior to every run, the constant temperature bath was wrned on, allowing the temperature chamber which enclosed the flatens and reservoir (described later in the chapter) to heat p to the desired temperature of 30°C. The temperature was con— inuously sampled by a thermocouple placed in the thermocouple well ocated in the top platen. The test solutions were kept in a con- tant temperature bath at 30° i 0.l°C for several hours prior to msurements to attain the desired temperature. Before preparing the test solutions, the solvents were ltered through an ultra fine sintered glass filter. The test lutions were not filtered because of high viSCOSltleS- It was t, in fact, necessary to filter the solutions for cone-and-plate uczmy measuremeu Dime-free sample n m: the. the top pl: mm; The mm s “on: shear rates "(Attic 'ange Of 5 " Wm; of m "r‘ "W "Musing n.1,: 5'“ to the .1“. ‘MC V15(°.. ‘ like: 1'4 ,‘On 30 . Blend) «.5. . V0‘Qm10110fl u x" I". .f {6* 1“} y. ‘61' a. _, .’ water‘s. "‘03.... ’94. '68 ' t. l’ -' WA in'qrz" ‘ :‘ 1’»!- 89 viscosity measurements (I-l). With a hypodermic syringe, 3 ml of bubble-free sample were placed on the lower platen in the center and then the top platen was lowered slowly to the required gap setting. The steady state data were taken from the lowest to the highest shear rates attainable. Whenever possible, the widest available range of speeds (to cover as wide a range of shear rates as possible) of the gear box was used, starting with a low speed and then increasing to higher values. Occasionally, readings were repeated back to the lowest value of shear rate starting with the highest. The viscosity curves could be essentially retraced from high speed to low speed. b. Reservoir chamber.--With the solvents used in this study, evaporation was severe. This caused gelling of polymer solutions at the lips of the platens and then skinning around the edge of the platens. This could be immediately observed by a tremendous increase in the output reading for the viscosity. It was tried to minimze the evaporation by creating an atmosphere of a solvent inside the temperature box enclosing the platens by leaving small pools of solvent in the temperature box. Williams (W-4) claims to have avoided evaporation by this method but in this study it was not very helpful. Finally, a reservoir chamber of aluminum was constructed to fit around the platens. Along with the gap between the platens, this chamber was also filled up with the test solution. This helped in preventing gelling and skinning around the gap and the viscosity measurements could easily be made. m 5.2 shows til .. m mietely (_.__C_G_UPL° warm-v. (mainly ‘ v1: :w a series 0‘ ., jgvmp' Instrumel 11""..‘1. atandol we! 0". we“! W‘ -. Wu: :1 the "It'i CAMP” ,... , A"! M 90 Figure 4.2 shows the design of the chamber. The reservoir chamber was also completely surrounded by the temperature box. c. Calibration.--Machine calibrations furnished with the instrument (mainly torsion bar constants) were tested by measure- ments on a series of Newtonian viscOsity standards obtained from the Cannon Instrument Company (V—4). These standards conform to the ASTM oil standard. The viscosities obtained with the supplied calibrations were within the experimental error of 6 per cent. The constants of the instrument are given in Appendix A. B. Light Scattering Many different commercial instruments have been developed for performing light scattering measurements. The complete appa- ratus for molecular weight and other measurements consists of light scattering photometer and differential refractometer. l. Photometer ' a. Practice.--The photometer has four major components: (l) optical source, (2) scattering cell, (3) collection optics, and (4) photomultiplier-electronic recording equipment. All the mea- surements were performed with a Brice-Phoenix* light scattering photometer of the series 2000 located in a low humidity constant temperature (25° i l°C) room. Complete details of the instrument and operation are given in the Brice-Phoenix manual (B-lO). *Phoenix Precision Instrument Co., Vir Tis, Gardiner, New York l2525. 91 b c d a. Plan. c. Plan of one-half of chamber. b. Front elevation. d. Sectional elevation. Figure 4.2.——Reservoir Chamber for Rheogoniometer. no solvent l in glass 3) was i 3.1 using blue ligh‘ n yrisurc filtrat ‘m alums were 1 .imum of 0.5 l 'u at “11m had 1 in mum fillet 1'"; i :m 2129 En ”M Mame hem 3’! mmmnts “-"‘- 'fi :9 each so “’"wn‘ -. Cit/dz, me 1" ‘-'- the ahw 1.713(9 92 No solvent was found to fluoresce appreciably when blue light (4358 A) was used and hence all the measurements were carried out using blue light. The dry, distilled solvents were clarified by pressure filtration through ultra fine sintered glass filters. The solutions were clarified by double filtration through three thicknesses of 0.5 mircon Millipore* filters made up of Teflon. Teflon filters had to be used because some of the solvents used here attacked filters made up of other materials. In Teflon fil- ters, a pore size smaller than 0.5 micron was not available. The minimum volume necessary for making the measurements was 30 ml. All the measurements were made in a cylindrical cell, C-lOl (B-lD). A portion of each solution was set aside for refractive index increment, dn/dc, measurements by a differential refractometer. In addition to the above precautions on the clarity of the solutions, the external surface of the cell was checked for the presence of dust particles, smudges, etc. by shining white light on the surface of the cell. If optically not clear, the surface was most effec- tively cleaned by wiping with an acetone-soaked, lint-free tissue, followed by brushing with an anti-static brush. A brief description of the scattering measurements will be given here but for details, the manual (B-lO) must be consulted. A full scale galvanometer deflection was obtained at 90° with all neutral filters removed. Then the shutter was closed and "dark current" zeroed if necessary. Then the highest galvonometer deflection possible was obtained at 0°, 45°, 50°, and then at l0° *Millipore Corporation, Bedford, Massachusetts. inrruls up to l30 in hm as require W order endin PM?) at each of “WM“ {he re «mm reading a‘ "Tunic of the mm a? hy.:,r° Ongle ~tir: Yhe above *iitfltz. “"9" Wm; l “w 1 ““Nm fl 1 «ti-1. A c .- ,... 'r: MHMMM " Fuel . ' l J 9' Site VI. '1‘. . L . , 1e , five" 0 :0 ":lLle ‘- n . .(éttfir‘r 93 intervals up to 130° and finally at 135° with neutral filters in the beam as required. The measurements were then repeated in the reverse order ending at 0°. Thus, a total of two galvonometer readings at each of the 12 angles were obtained. For each of the measurements the ratio, Ge/Fe was calculated where G9 is the gals vanometer reading at angle 6 and Fe is the product of the trans- mittance of the neutral filters used for that angle. Finally, for each non-zero angle, an average value of (Ge/Fe)/(GOIFO) was cal- culated. The above procedure was carried out for all the solutions and solvents. When using the cylindrical cell the beam geometry to be used is different from that when the standard rectangular cell, T-l04, is used. A calibration factor for the new geometry was determined according to the procedure given in the manual (B-lO). An equation for obtaining Rayleigh ratio, Re’ from the light scat- tering data is given in Appendix D. In principle the data should be corrected for depolariza- tion of the scattered light induced by the anisotropically polarized molecules. While usually appreciable with small molecules, this correction is considered to be negligible for high polymers and amounts to no more than one per cent for molecular weights greater than 10,000 (0-4). b. Calibration.--Most calibration procedures consist of measuring the amount of light scattered by a known pure liquid. Benzene, as an example, has most frequently been used as a result of the intensive amount of study on the absolute scattering of this am): {P-l. B'llv 'mltsu M Palme mm 0' ”9’“ SC in this war y‘ge‘ully calibri w 1’ mm; benzel . r2 9' {erbide CO‘ were Me validi‘ We i-lC) 3130' Wu": method. 31 r“. av izrucrgtion “yum-191nm ., \. p'ei',. “if" (6 . "‘6”. . t . '3'." N65 1 ‘ . . c w. Mum-g 'IrIJI . ‘J ' ". 4".“11’1’ . |'A .. ' H and ’. " {fr-"u .‘, /J “P‘s! I .. . 6‘. ) ”It 0 ' ‘5 ”J "i (r. I' ‘. a. '0 l .'g‘ ' 'I'IJ‘ v 7" ‘ I. II- A. . . 7,. .N .. ‘1‘ . . 1"" 94 liquid (P-3, B-ll, C-l). Kratohvil et al. (K-6, K-7) and Tomimatsu and Palmer (T-l) have reported extensive work on cali— bration of light scattering instruments. In this work the opal glass method (B-lO) was used to periodically calibrate the instrument and then frequent checks were made by using benzene and fresh solutions of standard PS supplied by Union Carbide Corporation.* Tomimatsu and Palmer (T-l) have reported the validity of this method after a careful study. The manual (B-10) also strongly recommends the opal glass-working standard method. Standard polystyrene samples supplied by Union Carbide Corporation were used in benzene and MEK to measure the molecular weights and in repeated measurements the values obtained were within 8 per cent of the supplied values. The Rayleigh ratio of benzene, R90, was measured periodically using the blue light and the values obtained were between 48.5 x 10'6 to 49.2 x 1076. The reported values compiled in (K-6) are 48.5 x 10'6 by Brice et al. and Trossareli and Saini, and 48.2 x 10'6 by Doty and Steiner at 25°C. Thus, the R90 values obtained in this study are in excellent agreement with the literature values. 2. Differential Refractometer a. Review.--In order to determine reliable values of the molecular weights of polymers by light scattering, it is necessary to accurately measure the specific refractive index increment, dn/dc, since this quantity appears as a squared term in the optical *Union Carbide Corporation, Bound Brook, New Jersey 08805. man! in DebyE'S M: value he as l in: {ya polymer :9 : value. The 51' h; :6 whether the 3'» than that Of 1 Walls. For Di 9 a: wwwiered i‘ *1)!" time this 21w: solut .-- t' "m, (hi'. r "“4"“ the Matt '1! '3 r: (.r hiql .8. m c ,‘ am I i l y . I it. I’ ‘u. ., ,-. ' 4'1“? '- 95 constant in Debye's equation, Eq. 3.20. It is desirable that the dn/dc value be as large as possible since the amount of light scat- tered by a polymer solution is also proportional to the square of this value. The sign of dn/dc may be positive or negative depend- ing on whether the refractive index of the polymer is higher or lower than that of the solvent; however, the sign is unimportant to the results. For polymer solutions, the usual range of values of dn/dc encountered is 0.08 to 0.22 dl/gm, though occasionally values below or above this range may be observed. Since solutions are usually at concentrations of one per cent or less, this means that one is dealing with the difference between the refractive index of a solution and that of a solvent in the third or higher decimal place and this requires a measure- 6 units, ment capable of detecting changes of the order of 5 x 10' which is far less than can be obtained with the instruments designed to measure absolute refractive indices. Consequently, instruments which measure only the difference between the refractive index of a solution and that of a solvent are necessary. These instruments are called differential refractometers. b. Practice.--The five basic components of a differential refractometer are (1) optical source, (2) adjustable slit, (3) cell, (4) microscope, and (5) micrometer. For the measurements in this work, a Brice Phoenix Differential Refractometer Model BP-2000-V was used. When carefully calibrated, this instrument is capable of achieving a limiting sensitivity of about three units in the 'm decimal place we descriptio when Drocedure For ordinar i l hiring a rem ' 1"! me of more '”'~ t mew. to pr '” ‘1. maveriment no {I "w" '11} N: r, ”V a: Mess n, . "* 1"?vtion, O It: ”VI-'JHJ'Y‘flJ .le ”hr‘tt." I, “ff " ”“331“: 1: 96 sixth decimal place of refractive index difference (B-12). A detailed description of the instrument and the calibration and operation procedures are given in the manual (B-13). For ordinary liquids having relatively low vapor pressures, a cell having a removable cover plate may be used with confidence. In the case of more volatile liquids like the ones used here, how- ever, a means to prevent mixing of solvent and solution kept in the two compartments of the cell must be assured. With such liquids, the solvent will readily creep up the edges of the cell by capillary action, and unless an impenetrable barrier separates the solvent from the solution, dilution of the solution will occur. This prob- lem was encountered when using benzene and MEK in the cell with the removable cover plate. Contact between the cover glass and cell-rim produced immediate "wetting" around the entire rim, thereby furnishing direct contact between the solvent and the solution and causing dilution of the latter. Various cells have been devised to eliminate this problem, with most of them having either a mer- cury seal (0-5) or a permanent t0p. One of the latter types with all-fused joints was used. This type of cell has aquarter-inch thick permanently fused top having a tapered hole opening into each of the compartments. The holes can be stoppered with small penny- head round glass stoppers. The capacity of the cells is 2 ml per compartment with l-l.5 ml being sufficient for performing the measurements. The calibration constants are principally dependent upon the relative positions of the various Optical components, I'l'tultrly that t 5'! te'estope. r0, withmld not be ' Wruq,1ftm .“Kt'aufyl ‘. 911.195 r-rW'J u, ztlhdard (W' .061 H 19:14:." 0' J": '> t fl lttfrd . - ‘J. 10“., 97 particularly that of the cell and projection lens with respect to the telescope. For this reason the cell, once calibrated, was not and should not be removed from the cell block unless absolutely necessary. If the cell is removed for any reason, recalibration is necessary. c. Calibration.--The instrument may be calibrated by ref- erence to standard solutions. References most frequently employed are sucrose or sodium chloride or potassium chloride solutions. Extensive tables of refractive index data for these solutions are found in the literature (B-l4, K-B). In this study potassium chloride solutions were used. All calibration measurements were made at 25° i 1°C using the blue (4358 A) light. Tabulated data of potassium chloride solutions given in the manual (B-13) were used. Reagent grade potassium chloride was dried at 100°C for several hours and cooled over magnesium perchlorate in a desic- cator before weighing. Four solutions covering a broad range in concentration and having values of concentration close to those of Kruis (K-B, B-13) were prepared using conductivity water. This was necessary to minimize errors in the interpolation of the graph of concentration versus refractive index difference, An. The calibration constant, k, obtained with 4358 A was 1.0246 x 10"3 for k equal to An/Ad where Ad is the difference in the deviations produced before and after the rotation of the cell and corrected for the deviations produced by the solvent alone. Thus, dn'dc ' k g m 1"! sutlSCrtp1 mtcVUim “pm 'he result! ., ,wmtlon v6] 3.96mezwmhi _ ,,' .l r ., .tvtfdl " vtg'wv"13 '43 {f ." H. ., we". “1'11 : M9530?" '..‘ I” ’ .v. (NYCt ‘5 4"”..{4 UV" l._, . — ' ‘ i ' H.’ I "t r ' rut-{r- I if ‘ (,3 . nc‘r' ( .l"‘ 1‘. ’1 “H. ,1 ' 'v 6 1. ./4( ll- 1‘1 MW . . F... I . J 5' 98 -3 dn/dc = k M= LO—Zfiicll—O—m, - d1) — (d3 - d?)1 (4.1) where the subscripts refer to the positions of the lever on the cell table and the superscripts refer to pure solvent (see manual, Ref. B—l3). The results are also shown in Appendix C. As a check on the calibration value, solutions of two different Union Carbide polystyrene samples in benzene were used for measurements over a period of several months. The average value obtained from the measurements was (dn/dc) equal to 0.116 for 4358 A light showing an agreement with the I.U.P.A.C. value (F-12). d. Measurements.—-The refractometer was kept in the same low humidity, constant temperature (25° i l°C) room as the light scattering photometer. The measurements were made on the solu- tions prepared for light scattering measurements after the scat— tering data for each solution were collected. The solution side of the refractometer cell was rinsed thoroughly with the solution to be used, filled and allowed to reach temperature equilibrium by circulating water at 25° i 0.l°C in the temperature box surrounding the cell. Solvent was kept in the other half ofthe cell. At least hree readings were taken for each position of the cell. From the d value of a solution, An could be calculated from 3 An = 1.0246 x 10‘ Ad. (4.2) inally, from the plots of An versus c, specific refractive index crement, dn/dc, was calculated for each polymer-solvent system. CHAPTER V PRESENTATION OF EXPERIMENTAL DATA A. Intrinsic Viscosities and Huggins Constants Relative viscosity. or, is defined as the ratio of the vis- cosity of the solution to that of the solvent. Relative viscosities at low shear rates were measured for all of the polymers in the selected solvents. Measurements were made for different low con- centrations at 30° i 0.01°C. The data were treated according to the Huggins equation, n /c = [n] + k [n]2c (s 1) Sp 1 . . Figures 5.1 to 5.6 show the Huggins plots for PS-l, PS-2, SAN C-l, SAN C-2, SAN C-2' and SAN C-3, respectively, in the sol- vents used. Intrinsic viscosity, [n], in each case was determined by extrapolation to zero concentration and the Huggins constant, k], from the slope. The results are presented in Table 5.1. The diagrams show that all the data lie on straight lines, as expected at low concentrations. From the values of [n], the trend of the "degree of good- ness" of solvents for different polymers can be seen clearly; the best solvents yielding the highest intrinsic viscosity and vice versa. 99 . ooom um viz 28 9.88.5 652% E 73 to; u.m> .33: mam n6 Spam-1.7m 93m: .cowpmgpcmocoo 3 3.38m; oEu .EEm . o. cozmtcoocou mo; 3 ad 90 to 8.0 m.o «.0 ad N.o S o 5 m3 1 1 to 1 1 m.o 1 U .I. 5 d m T .. 8.4 l D. I 1 IN 6 1 1 to w T 1 1 1 90 Va: 5 a J m.o $223.5 5 4 oconcom E o _ _ b — _ . . _ _ . 101 .Uoom pm gm: use mcmxowo .mconcmm cw mum; Low u.m> .U\amc .cowumgucmucoo op qumoomw> uwmwomam we ovummuu.m.m mgamwu mod 90 5.0 .3 Eu . u .co_.m._.coucou m.o m.o v.0 _ l T T q A x w: E 2.235 E oconcom c_ D 4 O N — 8.0 m... I m... h.— w.— m.— o.~ —.~ 1 c.~ \ h :1 ‘1 .. . _,. '1 102 .u com um gm: tam “=8 .6523 .93on c.” Tu z .33: .cowpfucmucoo 3 3.53m; 3:66am .3 6.537166 SSE mo... 0... .6 Thu . u .co_.m...coucou o.o 5.0 o6 md «.o n6 ~.o to o fl 4_d_______dfi4___§o w6/lp ‘aldsll. xwz 5 min 5 oconcom c_ ocmxoa S _ _ _ _ _ h _ _ _ _ _ r ~ _ _ _ Q.— m.— 40.05 1 \ . . 0.1.8: lkil I... _v\Em.u.co:mtcwucou mo; c; We w.o 5.0 o.o m.o «Au m6 gm: .mcmxowo .dzo :? Nuu z .U\gmc .cowumcpcwocou op xpvmoomw .Uoom gm wcmucwm vcm > occwumam to cwpem--.e.m teamed _ _ . _ _ _ _ _ _ _ _ _ _ _ _ 103 mcmncom xwe4 oCMxQQ mega 0 q u o m.o 0.0 0.0 m.o o; wfi/Ip ‘3/dSLL 104 .060m um acm~cmm 0:6 gm: .mcmxoma .520 :5 N10 z .u\amc .cowumgucaucou op xuwmoum5> owmpumam mo owumm1u.m.m mesmwg .2 E0 . u .cozmzcoucou m00 0.0 5.0 0.0 m0 «.0 m0 m0 .0 0 50 _ _ _ _ _ _ _ 0 _ _ ._ q _ a _ 5d 0; 1 0.0 a; 1 mo 0d 0.— eu ¢_ ~.~ N0 1u as Mw_$ 1y: «N .i. w m~ e. m. w m.~ 0.— 5.N 5... 0.~ o; m.~ m; 0.0 o.~ fin _.N 23530 E o «a xmz c_ a 1 Nu 0:335 c_ a «8 .uzo c_ o 1.m~ mu 0N FR .Q . II. . or 000m 2.... 0E: new ”=8 5 m0 z .oxumc 60520588 3 5.0.538; 35.8QO 50 0.58”. m m PS L .0 N Em . u .cozwtcoucou «#54 0... 0.0 0.0 5.0 0.0 m0 «.0 0.0 N0 5.0 0N.— _ _ _ _ _ _ 0 a _ _ _ _ _ _ .1l.ll.|141l.|l_ll m... T l m.— 0.51 l «.5 9— T 1 m.— m.F r. I. O; m.“ M! 5 O.N r. I 5... .3 m. D. TN 1 I 0.5 I” 6 ~.~ 1 1 S m MIN 1 J o.~ €.N I L —.N m.~ l L N15. w.N 1 I m N xmz E n 5.N 1 . LED 5 o .x.N ¢.N _ _ _ _ _ _ _ b _ _ b h _ _ b _ _ F _ _ _ m.N "1&1 €.l.--lntrir of Pol J ' 1 1 t 11' . er I . 1‘ . ’J'If .. t . ' f “1 - tl I "I I ,'l-v'¢ I)! . a » I. 1 13'4 106 TABLE 5.l.--Intrinsic Viscosities, [n], and Huggins Constants, k], of Polymers in Various Solvents at 30°C. Polymer Solvent [n], dl/gm k] PS-l Benzene 0.782 0.33 Dioxane 0.622 0.47 MEK 0.433 0.34 PS-2 Benzene 1.619 0.38 Dioxane 1.503 0.21 MEK 0.785 0.58 SAN C-l Dioxane 1.007 0.34 Benzene 0.988 0.20 DMF 0.867 0.34 MEK 0.783 0.40 SAN C-2 DMF 0.862 0.34 Dioxane 0.816 0.37 MEK 0.742 0.38 Benzene 0.446 0.86 SAN C-2' DMF 2.110 0.35 Dioxane 1.987 0.40 MEK 1.776 0.42 Benzene 0.725 1.40 SAN C-3 DMF 1.653 0.40 MEK 1.219 0.51 A peculiar but interesting behavior was observed in the case of PS-DMF solutions. When a film of PS-DMF solution of any concentration came in contact with air, it immediately formed a dry, powdery, non-sticky film of PS. This presented problems in the measurements of viscosities. Because of this strange behavior, no viscosity measurements were made with the PS-DMF solutions. B. Cone-and-Plate Viscosities Steady state cone-and-plate viscosity data, n(i), are pre- sented in Appendix B. The viscometric curves, viscosity. n, versus .1111 rate. '1. are ' 11:1. ‘cr several :r1zw1'a11ms. F 1' avg“. polymer .1 nil-"ded very 1 "11130111191111 '11. "11121111,, ‘2'" "11» mt .a- ..,, Iv: "GIL“? '"t'." .21-c 1n » I W111 ‘5 ”1.17 I I fl" (1 "(I “U" Ir 1. Hr). . ' I.- 'A o Lri-r 1' . '~ '0'. 6", ”‘ e '1 . 11 »,.. '11 1 U. a 1 1‘ ’ -'_ .' ‘10-. f . ' 141,” I ~ l {"3 1.. t 'e ."".-’1 .. ,. V _ 0' ,1 e I", 107 shear rate, i, are presented in Figures 5.7 to 5.9, as illustra- tions, for several polymers in different solvents at several concentrations. For many systems, particularly for lower molecu- lar weight polymers and lower concentrations, the data could not be extended very far into the non-Newtonian region either because of the machine limit on available shear rates or because of the flow instability. The flow instability is discussed later in the chapter. The viscosities of standard oils* measured with the viscometer used in this work were found to be within 6 per cent of the supplied values. This means that the viscometric curves should have a precision of about 6 per cent. There was no appre- ciable wobble in the rotating cone at higher speeds. From the measurements on standard oils, it could be concluded that viscous heating was not a problem except for a slight heating dur- ing prolonged operation at high shear rates and for high viscosity solutions. It was easy to avoid, however, by recording the data rapidly, and had no influence on the results. No shear degrada- tion took place over the time of testing and the viscosity curves could be essentially retraced from high speed to low speed. One difficulty was experienced with the measurements on polymer solutions. A flow instability developed at higher shear rates which effectively placed an upper limit on the shear rates that could be attained with a given sample. This problem is apparently a common one for viscoelastic fluids in cone-and-plate *Supplied by Cannon Instrument Company, State College, Pennsylvania 16801. 108 2 105 111111111 1111111 _ 66 _. 56 6509nfldl 10: oZng/dl E g : oZOgmldl : a .. dlng/dl - . — 000000000 '- F' .. ° - 3} ‘5 " Q 9 9 9‘? 9 Q 9 o Q Q '- 3 in E; 1 :3 I: : ¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢Z 0.1 l llllllll2 l llllll 10 10 103 Shear Rate-f5] , sec—1 Figure 5.7.—-Viscosity, n, vs. Shear Rate, i, for PS-l in Dioxane at 30°C. o 1 r4 1 m a m W N 0. bun.— _ h bhbhhhh 1h 2 .5 1,5 5.5555517. ~ 2 o 3 r4 p a 0 1. h . -I.a.a¢ pr D. -1..- p Viscosity, 1'1 , poise Benzene and Dioxane at 30°C. 109 Shear Ra.te,'21),sec'1 In Dioxane o 20 gm/dl 0 10 gm/dl £3 7 gm/dl Figure 5.8.—~Viscosity, a. V5- In Benzene ___________ 9 10 gm/dl *1 20 9m/d1 *3 7 gm/dl Shear Rate, 9, for SAN C-2' ea 1 1o 102 ‘02 1 115” 1 1 1 111111 1 1 1 ’03 0 0000000 _ .153 _. O O '- O O _—o a o o 6 b _— o o 5 6 6 6 6 .— ‘7 10 :» :102 — n 3 1111 1 11111111 1 1 {316111130 2 a 10 1o 10 oz 10 10 so 3 4° 1 1 111111 1 l 11 IHII 1 T‘T‘ 4x10 99999 '- 90 9 9 ._ Q ——1 10 ~ b 86 :3103 : b _ : 0000 00090176.? o I 4‘ 11mm IIIIMUI 1°PJ 2 10 102 511102 in , ’.L'\ I . .‘I 110 Figure 5.9.—-Viscosity, 11, vs. Shear Rate, 1}, for SAN C-3 in MEK and DMF at 30°C. In MEK In DMF o 35 gm/dl :5 50 gm/dl d) 20 gm/dl Q 35 gm/dl <3 20 gm/dl o- 10 gm/dl 0.4 '1' 21102 Av :.:___ 2 o l ...°° LLL mm. .F.>._.ao...a.> Hi Hr A: 555-. u Viscosity, 1], poise 10 111 Shear Rate , '3). sec‘1 20.1. 1 10 20 1. 2X10 IIIIF 1 1 1r11111 281° 0 .9 2 _Obooooo _ I. 810 : ‘gfé :10 F.- I I OI): — c— n. 0 g _ O 0000016?) .1 U 0 .2 1- O ‘ > 00 10 111111 1 1 llllLle 2103 lo 10 10 2x10 0- l 10 80 l. lifillllll 1 TIIIHTI I 11111310 : 111111151515 E Z _ ¢¢¢¢¢p¢¢¢ _ _ °¢ _ E 3:103 2.99999129 I: _ '0'0‘0-0 -8-o-o_o ._ - 00000 _ - 0"O-O'O'O-O'C>'O"(>.o_D_o-¢l *1 2 1 1 1 111111 1 1 1 111111 1 1 11 10 2 2 l 10 10 8x10 firm“ 16'1' f‘=:frflt1c.11y :m 1*. W"? 1°“ v :f'lr". 1'" “131‘ w: 11‘ W 9"” 1:19'1001. 11 “6 Wmtc' "91°" 1 «Q. 131.1 (mm! www :1 a 31161 w ' arm-d U "u 'm, the b1 112 instruments (G-l, H—6, K—9, B-lS). In the instability region the fluid erratically exuded from the gap between the cone and the plate at some locations on the rim and pulled back from the rim at others. The instability was probably deferred to higher shear rates by the presence of the reservoir filled with sample solution. In general, it was possible to penetrate deeper into the non- Newtonian region for higher molecular weight polymers and also for higher concentrations. The flow instability was indicated on the recorder by a sudden and erratic movement of the output signal. Since it seemed that this behavior was related to centrifugal forces at the rim, the behavior of Newtonian oils with the same range of viscosities as that of polymer solutions was examined. It was not possible to cast any of these fluids from the gap even at the high- est rotational speed. The highest rotational speed for the viscometer used in this work corresponded to ? equal to 1674.3 sec-1. This is not a very high shear rate in comparison to the shear rates obtainable in capillary viscometers for concentrated solutions. This implies that for high shear measurements one should use a different type of viscometer where, without encounter- ing such problems, high shear rates are obtainable. A capillary viscometer with pumps is one example. C. Refractive Index Increments Table 5.2 presents the values of refractive indices, no, of the solvents used, at 25°C for the wave length of 4358 A. 113 TABLE 5.2.--Refractive Indices,* no, of Various Solvents at 25°C and 4358 A. Solvent n0 Benzene 1.5l94 Dioxane l.4297 MEK l.3863 DMF 1.4406 Acetic anhydride l.3970 Dimethylsulfoxide+ 1.4903 *From Ref. (R-3) and 110 of DMSO from Ref. (T-2). THenceforth referred to as DMSO. Table 5.3 lists the refractive index increments, dn/dc, of the polymers in the above solvents at 25°C and 4358 K. Copolymer SAN C-3 was not soluble in benzene and dioxane and so acetic anhydride and DMSO were used for the Tight scattering measurements. As indicated in Eq. 3.30 in Chapter III, for obtaining the true value of molecular weight of a copolymer, light scattering measurements must be done in at least three solvents. The values of dn/dc (i.e., VA and v3) must be known for the two homopolymers and for the copolymer (v0) in the same solvents. 0f the solvents used in this work, PAN is soluble only in DMF, DMSO and acetic anhydride and hence the values of vB in benzene, dioxane, and MEK could not be obtained by direct experimentation. These values were " D1114 114 TABLE 5.3.-~Refractive Index Increments, dn/dc of Polymers in Various Solvents at 25°C and 4358 A. . Acetic Polymer Benzene Dioxane MEK DMF Anhydride DMSO PS* 0.114 0.185 0.223 0.176 0.212 0.137 SAN C-1 0.0966 0.174 0.212 0.163 -- -— iAN C-2 0.0856 0.166 0.206 0.154 -_ __ LAN C-2‘ 0.0866 0.167 0.207 0.155 -— __ AN C-3 —- -- 0.196 0.140 0.184 0.0967 AN1L —0.0089 0.104 0.152 0.0809 0.138 0.030 *Values in acetic anhydride and DMSO were obtained from 1- 5.3. 1“Values in benzene, dioxane and MEK were obtained from 1. 5 2. Values in DMF, acetic anhydride and DMSO were calculated 0m v0 values of SAN C-3 and using VA values of PS in Eq. 3.29, ich 15 v0 = xAv + (l - xA)v . Values in DMSO and BM? were obtained experimentally. tained indirectly by an extrapolation procedure. From Eq. 3.29 Chapter III, we can show that V0 = be + d (5.2) re xB = l - xA is the weight fraction of ACN in a copolymer and nd d are constants characteristic of each solvent. Thus, >rding to Eq. 5.2, v values of different copolymers (contain— 0 different weight fractions of ACN) in a particular solvent ' Plotted against xB should give a straight line. Figure 5.10 ’44 115 .ucmucoo zo< 0o comuuczm a mm comm an mcm~cmm use use .mcmxowo .xmz cw meme -a_oaou we .oU\:o .ucmsmcucH xmucm m>wuumcwmm cm comumwcm>uu.op.m mcammu mxfioEboas E zu< *o cozum: 29...? 5.0 0.0 m0 «.0 m0 «.0 —.0 0 . _ _ _ d _ a . q 1 _ . _ _ mo o ocoNcom c_ a w 1 18.0 ..... nr e U 3 II I. I1? “.20 En PM. a. .u w m. 10. 6 x C. 1 1 E: ._ 103m w M 1 w a U ___._________ 36.1 w: w plots «3:: ‘n differen ‘21.’.‘ 1'43“” 10001 .1101.) ”.51 1111 and 1" ‘i‘ , rd: - I 1! g .1) , M. '1'! .N. ’ ‘0 J ‘ 1? '1 "V ' w. ”W . ‘ 1) (I .1 . .° '1 116 shows such plots in different solvents. Values of the constants b and d in different solvents are reported in Table 5.4. TABLE 5.4.--Values of the Constants b and d (Eq. 5.2) for SAN Copolymers in Various Solvents at 25°C and 4358 . Solvent b d Benzene —0.123 0.114 Dioxane ~0.0809 0.185 MEK -0.0707 0.223 DEF -0 0951 0.176 From Eq. 5.2 and Table 5.2, the values of VB in different solvents can be easily calculated for PAN by using xB equal to l. The hypothetical values of 08 in benzene, dioxane and MEK, as well as that in DMF, are reported in Table 5.3. The extrapolated value of VB in DMF is 0.0809. This is in excellent agreement with the reported value of 0.08 (B-4a). The experimental value of VB in DMF in this work was found to be 0.083. Polystyrene is not soluble in acetic anhydride and DMSO. ience, VA values in these two solvents were obtained by interpola— fion. The VA values of P3 in different solvents were plotted gainst the 110 values of these solvents at 25°C and 4358 A. igure 5.11 is such a plot. The equation of the line in Fig. .11 is VA = —0.7996 n0 + 1.3291. ‘ (5.3) XWI . Doualoacott - _Q.CUE.etOud O h p p u 117 .mace>Pem we .o: .xeecw e>wpeegwem we :ewuocem e we womm we mpce>wem ucecewwwo cw we we .ee\:u .uceeece:H xmecH e>wueecwmm cw cewuewce>11.wp.m ecsmwm .choZom we .39... o>5omcwom «m4. mm; om;_ oe4. we; «c; No; cc; mm; @9— . _ _ a _ _ _ _ _ _ _ _ _ _ J _ _ wee l 1 3.0 1. .1 NF.o 1. .1 «_.o l ocoNcom m 1 2.0 T oszm owe exec 1 l 0:385 m 1 o~.o 3:235 232 N 1 $2 _ 1 2e 03203.35 x l 7258:»wa o 1 «ad _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ mud 1w 'op/up / '1uawaJou1 xapul any; 3214.13 1115 "an 1111 equatil 111:1 the no "1”. "'1‘1'51008 in 1. ‘w .' ‘-' acetic l 1'» comeer 3 "'1'. can be 1 1 ‘1‘ W liven r ' . We 5. "1'. 10.1151" 1 4 .. .. ‘ '1'”. ilf‘e' .1" 1.10", t 118 From this equation, ”A value in any other solvent can be calculated once the 110 value of that solvent is known. Table 5.3 also lists the ”A values in acetic anhydride and DMSO. Using the VA values for P5 in acetic anhydride and DMSO and the experimental v0 values of the copolymer SAN C-3 in the same solvents, v8 values in these solvents can be calculated by using Eq. 3.29 in Chapter III. These values are given in Table 5.3. Figure 5.12 shows the v3 values for PAN plotted against the 110 values. It can be seen that the extrapolated values lie on a straight line. This renders confidence in the results. Klimisch (K-S) has shown the validity of this technique for copolymers of vinyl chloride and isobutylene. Some of the solvents that were used by Klimisch for the copolymers dissolved only one of the homo- polymers. In this work also the interpolation technique was successful. Figure 5.13 shows a plot similar to Figs. 5.11 and 5.12 where v0 values of different copolymers are plotted against n0 values. The values plotted are for copolymers SAN C-l, SAN C-2 and SAN C-3. The values for SAN C-2' lie very close to those of SAN C-2, and hence are not plotted for clarity. These are all experimental values and they all lie on straight lines, as they should. 0. Molecular Weights and Second Virial Coefficients The molecular weights and second virial coefficients of the polymers in different solvents were determined from Zimm plots of the light scattering data. The measurements were made at 25°C. 0U E05. Ot.l_)u.o.u u — 0C East 0 —‘—C.E.L.QIU I fi . w . w Ic.‘ 119 .mpcw>_em we .0: .xwecH w>wpemcwm mucm>wom pcmcwwwwo cw zw .c Jcozom .6 xouc. o>:um._wom «m; mm; om; we; ocowcom l omZQ mwzo $3on 1 wvtuzccm use“; v.m.z .c Eel T .358?on NIL. .3 so: US$333 l r111L1111w1111w111101111L1111w1111w1111w1111w1111w1111h1111r111L111101111w1111w1111w11uhg m: .31 N: e: m2 e2 5.1111 3.0. r- N0.0.. onYo No.0 ano 00.0 00.0 N—.0 ¢—.0 00.0 0_.0 1110/1Lu 'DP/ up m .8 832:; a 2.. 00mm 3... peecwmm :w :ewuewce>11.mw.m mcamwm iueweJau1xapulaanoeuea values experlmontai MEK All 1 120 .muce>wem we .0: .xeecH e>wpeecwem we :ewuoczm e we 00mm um muce>wom ucecmwwwo cw M10 zwpeecwem cw :ewpewce>11.mw.m mcemwm .c .Eozom we 30:. o>2umtom «m... «m; om... 0.x; me..— vqé «q... 0.?— 0m._. mm; _ _ _ _ A .44 _ _ _ _ _ _ _ _ _ «ed 1 1 00.0 1. .1 00.0 ”a w I 1070 m D 1. 11 ~—.0 n pl 1 1 .:.e Nu econcom 0 Pw. 1 023 m 12d...” min 0 Ww 1 05335 0 13.0.Wu o _c > .cm u. mu m w 1. e. e .L .. < N .1 0N.0 w xwz _. 0 11 l mo3_m>_mucoc:.oaxo :< m 1 -.0 . _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ w .eomm em megaEaLUCH xaecH a>weeeewam newz meass_oaoe we megawaz Le_=ea_oz eae use mcweaeemem “cmw0--.w.m 00m /‘F (6.1) l/2 where is the root-mean-square end—to-end distance of the polymer chainirithe unperturbed state and f is the hypotheti— cal root-mean-square end—to-end distance of a chain in which the internal rotation about the carbon—carbon bond of the main chain IS completely free. The hindrance to internal rotation is called the short-range interaction. In. the abs smiled "freely- “ ”1’71 (Muted r; r. The freel "'" ' "t 0‘ bond 1 ”WT ' ‘5 1hr nuhfl 11 I'll. I""" B’l'IN‘ ‘ th-ra ““ '1’; .. 1 1 ' Itirgf ‘Vf‘l'pa '5? ’ I: ~ 9' .-c l 1' (l' n 1 4 ' ‘ i 5‘”?! Wit 0;“ . In (H l39 In the absence of any solvent or segment interaction, the so-called "freely-rotating chain” is obtained. Its dimension can be easily computed from the given data of bond lengths and bond angles. The freely rotating dimension of a chain consisting of only one kind of bond is given as (F-3e) f = 1112(1 + cos e)/(i - cos 6) (6.2) where n is the number of bonds, l is the bond length, 6 is the valence angle between successive bonds, and the subscript 0 denotes the lack of long-range interaction while f denotes the freely— rotating state. Assuming l equal to 1.54 A for the carbon-carbon bond and the tetrahedral angle 0 equal l09°-28', the stiffness parameter, a, can be calculated. Table 6.3 lists the values of o for different polymers. For PAN, reported value (K—2) of o is 2.25 while for PS it is 2.22 (B—4a). TABLE 6.3.-—Stiffness Factor, 0, of Polymers. ACN ACN Polymer % Weight % Mole O PS—l 0 0 2.23 SAN C—l 14.2 24.5 2.33 SAN C-2 24 37.l 2.36 SAN C—3 38 54.5 2.40 PAN* l00 l00 2.25 *From Ref. (B-4a). The molar w. 89 ml and th 1mm mm, the i therefor:- the W as hindrance 1w the different '-""~’~’Mrs, 1: is '1” t“ M units 1 1111.1“ 11111 the ”It” If 9“]e 6 “limo. y)! (ah 1 I "if v 1,, It '21:,sr‘ng ‘09 x l I 0 J! v_ ”MI ,1 "'3 o 1. . 0“ 'SM '3 1 . ‘J’l H J. ’1‘ ,, 'D‘ ‘ '- l‘l 'v {1.1 «r. 1 ' ‘1“ 1,” P“! B n «th '. A: '"J'e ’ n “if. ”I. ‘5‘” 1 1,” ' to. . 1; .1" 1 "'1". -._ J O a.» 11 3:. [I ”w. l40 The molar volume of the substituent phenyl groups in PS is about 89 ml and that of the nitrile groups in PAN is about 38 ml. Despite this, the values of o for the two homopolymers are similar and therefore the similarity in a must be caused by other effects such as hindrance to rotation caused by system energetics rather than the different molar volumes of the side groups. For the SAN copolymers, it is statistically calculated (A—3) that the propor- tion of ACN units forming sequences of more than three units increases with the proportion of ACN in copolymer. The values of 0 shown in Table 6.3 for different copolymers are not the same. Therefore, it can be said that the dimensions of the SAN copoly- mers are influenced by the fact that the electrostatic interactions of the neighboring nitrile groups are weakened by the phenyl groups. The value of 0 increases very slowly along with the increase in ACN content in the copolymers. It is presumed that there would be a maximum value of 0 between the two homopolymer values. The data here is insufficient to determine the value of ACN content where 0 would have the maximum value. It seems from Table 6.3 that the maximum would probably occur at about 50 mole per cent ACN content in copolymer when the tendency for alternation is the maximum. The greater value of o for the copolymers shows that in the unper- turbed state the copolymers are more extended than the constituent omopolymers. Second Virial Coefficient The stiffness factor, 0, is a measure of short-range inter— ctions. These interactions influence the dimensions of polymer mum in bulk 0" mzrtxbed. A931 rmwou of 901) ‘. aha! Hnd of .: .1‘:m'.7 in oth w : ‘ricracilons :r'czix’t‘ons but r? 1 We 2m: solven ""J’IC 1"‘el cuei 127.111—0 late N" mum: in t 1 : t'lf‘ "130’: J‘T‘QK’. l4l chains in bulk or in a 0-condition when the polymer molecules are unperturbed. Against this, long-range interactions influence the dimensions of polymer chains in solutions. The interesting question is: What kind of interactions can be observed for copolymers in solutions? In other words, what kind of polymer-solvent thermody- namic interactions can be observed from copolymers of different compositions but made out of the same two monomers and dissolved in the same solvents? These interactions may appear as a change of second virial coefficient, A2, or of Flory thermodynamic parameter, x1 (discussed later in the chapter), from one polymer to another when measured in the same solvent. The second virial coefficient, A2, is thermodynamic in nature and this is discussed in Chapter III. Table 5.5 in Chapter V presents the values of A2. The dependence of A2 on molecular weight is negligibly small for both PS and SAN copolymers in the range of molecular weights considered in this work (B—4a). Therefore, we can examine the variation of A2 with composition of polymers in different sol- vents. The higher the value of A2 for a polymer in a solvent, the better is that solvent for that polymer. As the solvents for a polymer become poorer, the values of A2 in those solvents become lower and finally, at the 0-condition, A2 becomes zero. Figure 6.l shows the plots of A2 of the polymers in different solvents versus ACN content. The points, though not many, are joined to demonstrate the trend in behavior. It is clear that the curve for the solutions of copolymers in MEK has a maximum in the neighborhood of 0.5 mole fraction of ACN. Thus, the intermolecular interaction of the SAN "‘mtir, 142 Figure 6.l.--Second Virial Coefficient, A2, vs. Mole Fraction of ACN in Copolymers. is». I43 'Zollllllllll 18- ... _. _. _. d _. ---D "‘ co 0 —- N w xx 01 m ‘1 I I I I I I I in Benzene — in Dioxane in DMF _ in MEK obxu Second Virial Coefficient, Azxio", mole cm3 Igm2 J l l I I O 0.2 0.1. 06 0-8 1.0 1.1 Mole fraction of ACN gnxlymtfi In "Ex .11., mm (her: mum “III“: S l' f'rhC. The ma t'LLh 5Hh {hes A.) :zr'If'sI Mg???" -. '11'. possible 1::1'11 3:14.“. not 0 “t“cz'. 0'11'1I'i rum '. nah-pol O 1 1.. (‘VI . ' I? W.- a 51m; 1' t ng“ 't ’ . . I,N¢w , u or , I l'l" . , f ‘I . rev 1 ' .9 ‘C .l '11,. ._ 0 H I" :4" r I‘ I l- l ... 1 ’1. I... d ”I! y. ' '11 "1:..1.‘ "I‘l 10 14""1K ’ [a “'36 l I44 copolymers in MEK shows a maximum at approximately equimolar compo- sition where there exists the greatest tendency of alternation of copolymer units, ST and ACN. The behavior in dioxane shows a simi- lar trend. The maximum in this case is at about 0.25 mole fraction of ACN. Both these solvents are non-solvents for PAN. With the ACN content higher than that in SAN C-3, MEK would be a non-solvent. It is not possibleix>predict this limiting concentration. Also, dioxane does not dissolve the copolymer SAN C-3 which has 0.54 mole fraction ACN content. Benzene is also a non-solvent for PAN. Benzene is non-polar and this makes it much poorer than other sol- vents for copolymers with ACN content higher than 0.24 mole frac- tion. This is evident from the deep plunge in the value of A2 for SAN C-2. At the azeotrOpic composition (SAN C-2 and SAN C-2') the second virial coefficient is negative, indicating that the copolymer is in an extremely poor solvent, and the polymer molecules are tightly coiled. Since DMF is a good solvent for PAN and a compara- tively poor one for PS (this can be seen from the A2 or x1 values; see Table 6.4 for the X1 values), with increasing ACN content in the copolymer, A2 increases continuously. The negative value of A2 for the azeotropic copolymers in benzene is entirely due to the very nature of the copolymer-benzene interaction. In an environment of benzene, the chains prefer polymer-polymer contacts rather than polymer-solvent contacts, since the presence of a large number of ACN units in the chains has a great influence on the polymer-solvent interactions. The tightly coiled nature of the chains is also indicated by the a value in “1&5 I I.--IIor_y l’arIOI WW... ICIymer 'Lh ’_ '1 1 V. o I "-' ' .. .1 5“,” v: 1 1 ' “.1 I 1 .. up '- ‘1 145 TABLE 6.4.--Flory Thermodynamic Parameter, X11 of Polymers in Various Solvents. Polymer Solvent x1 PS-l Benzene 0.426 Dioxane 0.443 MEK 0.481 PS-2 Benzene 0.424 SAN C-l Dioxane 0.418 Benzene 0.433 DMF 0.450 MEK 0.453 SAN C-2 DMF 0.426 Dioxane 0.426 MEK 0.429 Benzene 0.507 SAN C-2' DMF 0.420 Dioxane 0.426 MEK 0.430 Benzene 0.507 SAN C-3 DMF 0.382 MEK 0.428 benzene which is less than unity. It should be mentioned that at the same time benzene is not a theta solvent at 25°C. Also, it should be kept in mind that at the 0-condition A2 is equal to 0. Essentially, A2 is a measure of the tendency of the solvent to interact with the segments of the polymer chain. The lower the value of A2, the lower is the tendency for interaction and the poorer is the solvent until at A2 equal to 0 the chain assumes its unperturbed dimensions governed only by the skeletal effects of the chain. Large values of A2 indicate the tendency of polymer WWII to avoid 11:11:95 1 preie :wn then have 1 II IS Int1 '~’ "1 Hiher in d‘ I I. 1 W1" respecl .1111 'UoIU 'GII‘IQF ( .1! “Hm“ ’0' PE '- m’: that M 12,-”.: “"1 1 f1 -u 1”! a." "‘11 ‘l 1 611,6” I46 segments to avoid one another as a result of "excluded volume" and indicates a preference to interact with the solvent molecules. The chains then have a more expanded structure. It is interesting to note that although PAN does not dis- solve either in dioxane or in MEK, there is a maximum in the value of A2 with respect to ACN content in the copolymers. Intuitively, one would rather expect the maximum at ACN content equal to 0; i.e., the maximum for PS homopolymer, as in the case of benzene solutions. This means that both dioxane and MEK become better solvents for copolymers with increased ACN content but only up to a certain ACN content. This may be due to the polarity of MEK and localized polarity of dioxane. The trend in DMF is normal, showing continu- ously increasing values of A2 with ACN content. Table 5.5 shows a slight decrease in the values of A2 for the higher molecular weight azeotropic copolymer, SAN C-Z', compared to those of the lower molecular weight, SAN C-2. It is difficult to make any conclusion regarding this trend with only two molecular weights available and considering the errors involved in the light scattering measurements. However, according to the literature results (B-4a), A2 values decrease slightly with increas- ing molecular weight of polymer. Values of A2 are rarely available over a range of molecular weights extending over two orders of mag- nitude. In this study, the variation in molecular weight is not large, either for the two PS samples or for the azeotropic copolymer samples, and hence for practical purposes the values of A2 may be considered nearly constant in the range of molecular weights con- sidered. ‘ I 1': “WW Ls 111.111 '. a thcrmdy ;-. women-r 1 "‘r '32 crane "T'J.e".“._.11h1 i "' '«EIVI'h "4“‘1’114 1 ' 1/ 1 1 I l " l 1 I l D ,. 1‘ ‘ ,. 1 I \1 1 1 ' l 1 '1: . .I' 11 '1'”. I47 4. Flory Thermodynamic Parameter As explained in Chapter III, the second virial coefficient, A2, is a thermodynamic quantity. This quantity is related to the Flory parameter X] and hence x] can be calculated from A2 obtained from light scattering data (F-3a). The parameter X1 is a dimension- less quantity which includes the interaction energy characteristic of a given solvent-solute pair. According to the theory of second virial coefficient (F-3a), A2 = (cg/vs)(1/z - x])F(X) (6.3) where 2 3 x x x F(X)=1-——+————-————+ (6.4) x = m2 -1) (6.5) 9p is the specific volume of polymer, vS is the molar volume of solvent and a is the expansion factor for the polymer in the sol- vent used. The series given by Eq. 6.4 is an extremely rapidly con- verging series and hence only the first few terms need to be considered. The a values were obtained at 30°C while the A2 values were obtained at 25°C. Since X] is a weak function of temperature, the difference of 5°C was assumed not to introduce any significant error in the estimate of X1 values. The x1 values are listed in Table 6.4 given on page T45. The parameter X] may be used as a criterion for classifying solvents as good solvents, poor solvents or non-solvents. The m we value c '9: between 0 4'14»: ‘or {hep 1111.15, indicate M‘il'f" ’b-lb). L‘. the pr 11¢" ,. in be -i u“. '1' WI w 1,4, . . .Pl- ". mum 'z'rezw, Ll WNW 1. I'l” ,l' Ur H" ‘ 148 lower the value of x], the better is the solvent. If the value of x] lies between 0.5 and 0.6 then the solvent may be poor or non- solvent for the polymer in question. Values of X] larger than 0.6 probably indicate that the polymer is insoluble in the solvent in question (B-lG). As the proportion of ACN in the capolymer increases, the value of X] in benzene also increases since benzene is a non- solvent for PAN while the value of X] in DMF decreases, signifying that DMF is increasing in its degree of goodness for copolymers with increasing ACN content. The values of x1 for SAN C-2 and SAN C-Z' in benzene are greater than 0.5, indicating that it is a very poor solvent. Thus, the values of [n], a, A2 and X] establish the solvent quality for the polymers. B. Zero Shear Viscosities l. The Influence of Solvent Solvent effects have an important influence on the value of low shear specific viscosity, nS , of polymer solutions in both P dilute and concentrated solutions. The importance of solvent effects has not been recognized widely in concentrated solutions. This influence on low shear viscosity will be demonstrated by con- sidering semi-log plots of specific viscosity, nsp’ versus c in good and poor solvents and in the next section by log-log plots of relative viscosity, nr, versus c. At very low concentrations of polymer in solution, the polymer chains are believed to exist as isolated clouds of polymer matrix which dc mi the very d1 -:1~:'sc1'.defc :r. w ‘n soluti :».1--'.; and the c 1" PW Macon '1‘11P’1'n‘10n T o 3 2'”. 44 £1 149 segments which do not interpenetrate with each other. With this model the very dilute polymer solution appears to flow as a suspen- sion of soft deformable spheres in solvent. As the concentration of polymer in solution is increased, the polymer spheres begin to overlap and the chains become entangled. When this happens the low shear viscosity, ns , depends much more strongly on polymer P concentration. The viscometric data are presented as log n versus SP c in Figures 6.2 to 6.5 for copolymers SAN C—l in DMF and MEK, for SAN C—2 and SAN C-2' in DMF AND MEK and in DMF and benzene and for SAN C-3 in DMF and MEK. To make a proper comparison between viscosities of different solutions, the zero shear viscosity should be normalized by the viscosity of the solvent, ms, and then plotted against c. As the proportion of ACN in the copolymers increases, starting with PS homopolymer, MEK and benzene become poorer sol- vents until the copolymers are no longer soluble in them. Progres— sively, DMF becomes a better solvent since PAN is soluble in DMF while it is not soluble in MEK and benzene. For SAN C-2 and SAN C-2‘ benzene is a very poor solvent as indicated by the expansion factor, a, given in Table 6.2. Copolymer SAN C—3 is not soluble in benzene and MEK. The solvents used in this study are good solvents for PS but their degree of goodness varies for copolymers. The viscosity data make it clear that the solvent character can have a significant influence on viscosity in the whole range of concentrations. This reality differs from the oft—encountered belief that solvent effects are ”neutralized” at high concentrations 150 .o.om an x”: use can as .10 z .amc .>g_moumw> uwcwumam--.m.m menace .2 Eu. u.co:m.:coucou m: 2 m. o. m 080 _ _ T fi fi _ _ 4 fl I. r l n u I IL I 1. S I. l d r 1 u H Ho. M S 3 O m. I 1 .M T J .u S VI 1 d T l U HUN? i E: c. o 1 n “:8 E a u w _ _ _ A _ _ _ _ 1 o. l5l Figure 6.3.-~Specific Viscosity, n , vs. Con- centration, c, of SAN c-2 and SAN c-2' in BHF and MEK at 30°C. bhhfh» b b 152 103 : I 2 10 : : O. — 1 o .11 m — — '1' " ‘ ‘ 3 — — '7: - 1 , _ o 3 — ‘ A in DMF _ '5 1 o in MEK U . E 10 ._ ‘ Open points :SAN C-Z' d 3 I -/ Filled points:$AN c-z : a : / _ 1 l2 __. l . - r: : 0:3, 1 L l l O 5 10 15 20 2,5 Concentration,c, gmldl l52 103 _-_ : 2 10 : j i . : 3 " --4 F' - ‘ d 3: — _ 'fi - ' , _ o f; .. ‘ A in DMF - “s ,1 o in MEK U . E 10 ‘ Open points :SAN C-Z _ u 1: / . . I q o b ° Filled pomts:SAN C-Z .. 0- " -l U) h— / .1 1 II __. ' ‘ - r: _ 023 1 L l 1 0 5 10 15 20 2.5 Concentration , c , gm Idl ; it» 5xl0 r I I I I I I I l l l l L l l 10 lllllll 1111111 I l T l 10 I [Till [1111] ‘ll ll 0 o lTlTlTl Specific Viscosity, 115p ..I.. i JJ [11 r l A in DMF o in Benzene- I Open points :SAN C-2 Filled points :SAN c-2'- r l Tlllll ILILII l 01 I I l J I I I I l O 5 10 12.5 Concentration , c , gmldl Figure 6-4.--Specific Viscosity, n p, vs.Concentration, c, of SAN C-2 and SAN C-2' in DMF and Benzene at 30°C. — 154 3 10 _ l I I I I I I I I - 2 10 : Z O- _ — UI -- -I :- Q — . r: A In DMF 01 8 o in MEK .‘2 10 _—_ _. > — : .2 _ d '8 __ __ a: ‘ .. _ O. U) — 1 : : C I 0.4 I I I I I I I I I 0 5 IO Concentration,c, gmldl Figure 6.5.--Specific Viscosity, nsp,vs. Concentration, C, 01° SAN C—3 In DMF and MEK at 30°C. —-L N 0" team they a" “a “ms 0 l .ng Mil uD 1" in this ‘ "5“"; ACN (WW ,1: mi wiIP' I" ,ijegimg and wwgqmm I'm m 12‘ Iizcosli.‘ w,’ 7: L2 IOI .,.‘.,.,1‘..a‘11,¢ In 1‘. flit" amai'fi. ”‘1' ,.A I vui 912,: W”:’ “a” "f I“! - 155 because they can no longer influence the size of the polymer chains (which depends on the solvent in dilute solution) since the polymer coils fill up the space uniformly. In this work the solvent effect is shown on copolymers of varying ACN content in the solvents, MEK and DMF. In all the cases, ”sp is smaller in poor solvents than in good solvents at low con- centrations and it eventually becomes larger in poor solvents as concentration increases. This is a consequence of stronger depend- ence of viscosity in poor solvents on concentration at high concen- trations. At low concentrations, viscosity increases rapidly with concentration in good solvents because of the larger size of the polymer domains. It can be seen in Figures 6.2 to 6.5 that at high concentrations, viscosity increases more rapidly in poor solvents. This is consistent with the following qualitative thermodynamic argument. As the concentration increases, the density of chain segments per unit volume of solution increases along with the entanglements between the polymer chains. In an environment of a poor solvent, the polymer segments prefer polymer-polymer contacts rather than polymer-solvent contacts. This enhances coiling-up of the polymer chain. The coiling-up of the polymer chains in effect enhances the entanglement or makes the entanglements "tighter." In a good solvent, solvent-polymer contacts are at least equally pre- ferred and this deters direct inter-polymer chain interaction. Thus a polymer molecule in a good solvent finds it much easier to move freely among its neighbors while in a poor solvent freedom would be reduced and could eventually lead to aggregation. In the d I t pow OM VII 31'...“ I “'9" no LOIUIIOH I: 'He (haln 'wumtntion aI "" ""”.'. 0‘ ”I 'r the r“p({ u; t‘H-c', i1 156 In the dilute region, n5 is higher in a good solvent than P in a poor one which merely reflects the fact that polymer chains occupy a larger domain due to expansion in the good solvent. This dilute solution behavior is fully discussed in the literature (M—2, R-Z). The chains are isolated in the dilute region. At higher concentration also the polymer chains expand in good solvents but the effect of this expansion on viscosity of solutions is much less than the effect of the entangled polymer chains enhanced by a coil-up effect in poor solvents. A close examination of the data of this work indicates that the "cross—over" concentration (or the concentration at which nsp values in good and poor solvents are equal) is much lower for the copolymer solutions than that for the PS homopolymer solutions. For the copolymers in this work, the cross—over concentration in DMF (good solvent) and MEK (poor solvent) is about 5 gm/dl for SAN C-l, 4 gm/dl for SAN C-2, 2.5 gm/dl for SAN C-2' and 3.5 gm/dl for SAN C-3. For P5 in these solvents, the cross—over concentration will be much higher than the concentrations considered here. Also, for SAN C—2 and SAN C-2' in DMF and benzene (benzene is a poor solvent for SAN C—2 and SAN C—2', poorer than MEK), these values are 5 and 3 gmldl, respectively. The above result suggests that as the proportion of ACN in COpolymer increases, starting with homopolymer PS, the cross-over concentration decreases. With increasing ACN content, an already poor solvent; MEK, becomes poorer and hence there is more prefer- ence for polymer—polymer contacts rather than polymer-solvent :wmti. This ‘: "'1: u: and s 't '.I 1' a poor ""(UM MN (I '"Wt‘t‘ve to II “W'It‘ com fr "‘6' :qu zol {gm 1”“. W '.’1e'. (_r I N' "in“ '4" 1" M11; 1 p , H I I l l '5 II ' 1. by? .,_ . .i' a I 1 1 l . 1.. I“ Pa 3 1 I 1 I W"‘ 1. 'Vle .. 1' . ; 'I A I I l ‘ I . (It 1 ' 1 157 contacts. This is also the case with benzene. This leads to more coiling up and so tighter entanglements and hence increased vis— cosity in a poor solvent, leading to a lower cross—over point with increased ACN content. These interactions may or may not be insensitive to molecular weight. Further evidence of this phe- nomenon comes from a study of PVC-acetate copolymer systems (J-l) in MEK (good solvent) and cyclohexanone (poor solvent). The cross-over concentration was about 2.5 gm/dl. These observations indicate that cross-over concentration becomes smaller as polymer- polymer interactions become progressively stronger. Further investigation is needed to study the molecular weight effects, and to quantitatively correlate this phenomenon with molecular vari- ables. In this study, the variation in molecular weight between the two samples of azeotropic copolymers was about 3.7 fold. With the data of this work, it is difficult to arrive at a definite conclu- sion regarding the molecular weight dependence of the cross-over point. By eliminating the parameter of molecular weight,* one should be able to observe features which can be attributed directly to polymer—solvent interaction (due to different average polarity of copolymers as a result of different ACN content). The number of chain atoms may be a more proper variable to consider. However, the data of this work are still insufficient to show the influence of chain lengthtn~molecular weightonthe cross-over concentration. M *For PS and different SAN copolymers, if molecular weights are the same, the number of chain atoms will be different because ST and ACN have different molecular weights; ST M. W. =lO4, ACN M. W. = 53 31 must ‘3‘"!1011 (Once . 3'" Whunlrg 1,, ”i'ttulos of 1“ m1 i‘ntin', r '1. "'3 "Kid” .II‘J', 1C. :00 l‘ "1‘. 1 J! ' 4. I'.’ l ‘1’". H ‘n - ' Ir ’ 1‘ "t: ( . 1'."-(' 1 '1 . 'I‘I. 1"!" I' WNW.“ ’e .. I" 1 i p.’ '.r I, (.1 II‘ ’I. . ’7’ H N ' 1 l 1 | I ""Ir ' i I. II 158 It must be emphasized that this discussion pertains to moderately concentrated solutions (up to about 20 per cent). At higher concentrations the polymer—solvent mass exists as a gel and the molecules of the solvent would diffuse within the network of polymer chains rather than the polymer molecules diffusing as in dilute and moderately concentrated solutions. As the concentration increases to 100 per cent polymer, the two curves should be sepa- rated only by the ratio of solvent viscosities. 2. Entanglement Concentrations The features of viscometric data can also be examined from the log-log plots of the viscosity—concentration curves. Figures 6.6 and 6.7 show such plots for PS-l and PS—Z in benzene and MEK, respectively. These two solvents present the extreme cases in degree of goodness for PS among thesolvents chosen in this work. The curves show an abrupt increase in relative viscosity when a certain concentration is reached which corresponds to the concen— tration characteristic of entanglement networks. Various methods have been proposed to calculate the value of the concentration at which such a pronounced increase should be expected (P—l, P-4). The onset of entanglement in polymer solutions can be observed from sharp changes in the slopes of the plots of nr(c) versus c, nr(M) versus M or nr(c,M) versus ch. Plots of zero shear viscosity versus molecular weight for polymer melts show two distinct linear regions with a sharp break point (M-la). The molecular weight of the polymer at the break point is referred to I. I .1 I . €[I1‘I1be~pbh » 11» . :Lbhh a _ a . i.e.ca. . a . ... 3 . 0 PU fa I 0 v up; .1 fl. .0 n -0 v.1 ‘w. 159 Relative Viscosity, 1], z, 10 J” I I I IIIIII I I lllTlL : : 103: : 1— —i ‘ 1 - -i 102: i i— A -1— h A :2-2 '1 - a 10: : C a 1 I I I LIJllfl I I 11111 0.71 10 102 Concentration , c , gm ldl Figure 6.6.--Relative Viscosity, nr, vs. Concentra- tion, c, of PS in Benzene at 30°C. 1 » 160 61 ‘0 -II I IIIIIIII l llllllL l()3 t: j: 3 a P ‘ FL.- _ — 3‘ O? 2 APS'I u 10 : APS-Z: In .. 1-1 5: :: :: o ,_ ._ .2 .. 2.3 _ h .. m — lO :; I: t‘ ._ I— —1 1— ‘ A 1 II I IIIIflII IJLIII 0.71 10 102 Concentration, c , gm ldl Figure 6.7.--Relative Viscosity, ”r’ vs. Concen- tration, c, of P5 in MEK at 30°C. 1.1.191 :ntical .1 w curve «it - 5;» than that '1 l Porter ‘1)' t :Itiymer ma . " MIMIC 161 as the critical molecular weight, M. The melt viscosoty— molecular weight curve with molecular weights higher than MC has a greater slope than that of the curve with molecular weights less than Mc (M- -la). Porter and Johnson (P- 1) recommended that this value of MC for a polymer may be used to calculate entanglement concentration, cent’ in solutions. Accordingly, they recommended that entM = pMc (6.6) where p is the density of bulk polymer, and cent is the entangle— ment concentration in solution for a polymer of molecular weight M. This idea ignores solvent effects and essentially considers the solvent type of no importance to the onset of entanglement. With Porter and Johnson's approach to find the concentration Cent for a polymer, all one needs is a tabulated value of pMC for that polymer. According to Coronet (C—3) from a theory based on packing of polymer coils, 3/2 — 2 (c/M)ent = 2.28 x 10 23/ (6.7) . 2 . where the mean—square end-to-end distance, , 15 evaluated at . 2 . _ the unperturbed (theta) condition. However, s1nce lS prODOF 1 2 tional to molecular weight, the PPOdUCt CM / WOUId be a C°"$ta"t’ and hence, 1/2— _ 1/2 —Constant. (5-8) CentM C M ent Other similar relations have also been proposed (0'5)- 1, a;- lhe prec 1‘1 :1' mied ir ‘. 92,-." "19M 'i' T’It Ion-r UK 1"“! di‘ien .1" (:V'I.» 1'11)“ 162 The predicted values of cent for PS using various relations are presented in Table 6.5. The predicted values for the higher molecular weight, PS—2, are in fair agreement with each other but for the lower molecular weight, PS-l, the values predicted are markedly different from different relations. As shown in Fig. 6.6, the abrupt increase in the slope for PS—2 in benzene is probably at about l0 gm/dl and in MEK (Fig. 6.7), at about 7 gm/dl. These values are in fairly good agreement with the values in Table 6.5.. For PS—l, no sharp increase in slope in either solvent can be observed in the range of concentrations considered, although in MEK the slope is much steeper at higher concentrations (about 20 TABLE 6.5.—-Estimation of Entanglement Concentration, Cent’ for PS. Cent’ gm/dl REIation* M=50l,000 M=l85,000 Reference PS-2 PS-l (cM)ent = 4.41 x 106 8.8 23.8 P—l (cM)ent = 3.75 x 106 7.5 20.2 P-l (cM)ent = 3.03 x 105 6.] 16.4 G—2 (cM1/2)ent =4.65 x 103 6.6 10.8 §g%_6(g_ggg+ (cM1/2)ent:=5,28 x 103 7.5 12.3 P-I *(cM)ent stands for the value of cM beyond which entangle- ment networks prevail. 2 O +Obtained by using Eq. 6.7 with /M = 757 A taken from Ref- (B-4a). T I II "t _ that In 1 1.010119 I II‘VPIy ‘ 1":t’1t, "I l'. l ,4". IC‘. I .' ‘. I"C l! I'. 11, ”" T' ‘I 3' 1 7 ‘ '1 1 l".”'"' .,, ' ' '1le i “ . u‘” Il J.- .l H ’ 4 t" I.“ I ’ 11, I 1 '1. I ’I f .1“ 163 gm/dl) than in benzene. From this work a slight superiority of the 1/2) (cM)ent scheme compared with (CH scheme is suggested for the ent two relatively good solvents involved, although MEK is poorer than benzene, it is not a e-solvent for PS. Figures 6.8 to 6.11 show the plots for the copolymers SAN C-2 and SAN C-2' in DMF and benzene and SAN C-3 in DMF and MEK, respectively. For SAN C-2', the abrupt increase in slope is at about 6 gm/dl in DMF (Fig. 6.8) and at about 3 gm/dl in benzene (Fig. 6.9). For SAN C-2 in DMF (Fig. 6.8), the increase in slope seems to be at 20 gm/dl while in benzene (Fig. 6.9), it is about 5 gm/dl. This is a strong indication of the lower value of the entanelement concentration in poor solvents. For SAN C-3, the values in DMF (Fig. 6.l0) and MEK (Fig. 6.ll) are at about l0 gm/dl and 5 gm/dl, respectively. Thus, the influence of solvent on c t is evident from the data on copolymers. en Unfortunately, no relations for c (as shown in Table 6.5 ent for PS) are available in the literature for SAN copolymers, except for Eq. 6.7. Equation 6.7 was used for prediction of cent for the SAN copolymers. These values are listed in Table 6.6. TABLE 6.6.--Entanglement Concentrations, Cent’ from Eq. 6.7, for SAN Copolymers. Copolymer cent’ gm/dl SAN C-2 ll SAN C-Z' 5.8 SAN C-3 9 164 Figure 6.8.--Relative Viscosity, nr, vs. Concen- tration, c, of SAN C-2 and SAN C-2' in DMF at 30°C. I.-.-r’*" pbhbhh b b h 165 3x104 10" 103 5 .5 .5 2 3 10 .Z’ > 2 . ASAN c-z :- l 3 ‘SANC-Z 0 E! A 10 , 1 III IIII IIIIIII 2 0.7 10 10 Concentrafion,c,gnfldl 11.;1-"‘ .L1)»»»,» 8 h 3 h . ... » SIC» a 1‘ ....) 166 4 . 10 I I III! I l I l I 1111 I I ,.—L .5 '171 3 2 .910 > 0 .2 73 ‘6 tr ASANC-Z _l 10 ASANCZ 1 11111 I IIIIIII 0.4 l 10 102 Concentrafion,c,gm/dl ure 6. 9. —-Relative Viscosity,n n5,C vs. Concentration, Fl'9 C 0f SAN C- 2 and SAN C- 2' in Benzene at 3 2110 70 167 5 ZXICI IIII I I I I IIIII I II III III 5 10 : Z 1— -I -- -I .1... fl 4 10 I IIIIII l I I 11111 I I l 3 I? 10 :: I ~ 1— d 2? r' _ ... I'- -I m — - O U I— —i in S? 1— _ CO .2. ~ ~ .2 0 2 a 10 : :1 - a i— —1 1— 4 10 :; Z L‘ :1 2 III I I_ I II IIIt I I l l 1111 0.6 l 10 102 Concentration, c , gmldl Figure 6.l0.--Relative Viscosity, ”r’ vs,Concentration, c, of SAN C-3 in DMF at 30°C. I “germ“ 168 Figure 6.ll.—-Relative Viscosity, nr, vs. Con- centration, c, of SAN C-3 in MEK at 30°C. - legs- » .9“ u. ) ::_ A 2 C .0 ~C1 ‘5- ~ h K o --- a k s 169 11x10 III I I Illllll I I IIIIIII I lllIIII IJIIIII I J IIIIIII IIIIIII O N Relative Viscosity , Th. I I IIII Illllll 11111” I 2 [LI I I Illllll I UIIIIII 0.5 1 10 102 Concentration. c, gmldl lhe val Attained imn I it at it 900d 5 111.: 9‘1", "1 "it 15¢ ”DEV .[ v0.9. :1 (C "'1‘: ‘.'.P: are ' V ”'1'. .I’ I " V ‘n ., 1‘ 1F: IIWIfic 1"”.11I-r‘1 vi 111'. 1: . ..I .I '11?" ' I I' 14 " 4 11b 1 Q 111.. 1. 1,. I 1.. n I" I 'J' ‘1: 1.. .‘J' t. . I ‘I'I I T .11 w w- w 170 The value of c for the polymers, SAN C-2' and SAN C-3, ent’ obtained from Eq. 6.7 compares reasonably well with the observed value in good solvents for the respective polymers. For PS—l and SAN C—2 the value of cent predicted by Eq. 6.7 seems to be too low since the experimental values are higher, perhaps even higher than the range of concentrations considered in this work. The values predicted are very different from the experimental values in poor solvents. In poor solvents the observed values are much less than predicted since Eq. 6.7 is based on the packing of polymer coils at incipient overlap, and in poor solvents, the polymer chains tend to coil up with tighter entanglements. It is therefore reasonable to expect that Eq. 6.7 may not hold for poor solvent solutions. Equa- tion 6.7 seems to hold for solutions in better solvents where conditions of incipient overlap may exist due to the ”unfolded" nature of the chains as opposed to coiled—up nature. It is to be noted that most relations for predicting cent are based on observations in good solvents. This factor probably did not clearly bring out the influence of the nature of the sol- vent on Cent' The observations in this work point out the necessity to use published relations for ce very carefully. Also, Porter nt and Johnson's approach cannot be applied to all the solvents. Unfortunately, more data must be obtained before the influence of solvent on cent may be theoretically explained in a quantitative manner. a a. --" 13"" Lew ( 'he yo, '- I'I‘f" meI War-141.119 ' 1‘II , ‘ .i' 141511 . .‘. "1"“..r ’I [U ""e'.1n 1,._ 1 '> ”In M'.c1 ,J' .: . 1. In ‘I 3. .1, ) .,. 171 C. Correlation Techniques 1. Power Law Correlation The power law correlation is discussed in Chapter III. It has been commonly believed (F-2a, C-4, F-l3, 0-7) that nr is inde- pendent of the solvent nature for values of cMO'68 higher than the "critical" value. Thus, the equation nr ~ (ch)B (6.9) with be equal to 0.68 or b equal to 0.625 and 8 equal to 5 con- tains no thermodynamic parameters. In Figures 6.l2 to 6.l8, the power law correlation is used to correlate the data collected in this work. All the plots indi- cate that there is no sharp change in slope from 1 to 5 as the concentration increases from that typical of the equation for dilute solutions: nr = l + KcMa (6.l0) to that typical of concentrated solutions: nr ~ (ch)8. (5.11). Therefore, the evaluation of a "critical" entanglement point becomes either very subjective, or meaningless. The curves are smooth, indicative of a gradual change from one type of physical phenomenon to another. At very low concentrations of polymer in solvent, the chains exist as isolated clouds in a sea of solvent. The dependence 3 ..IA 0 r; ELII’ _:-b~_ ~ :Zbae . a . or }..Ip..l I .0.“~n 172 1()3 __ I I I I II III I 1* I I I ll' : 1 _ o PS-l — I A PS-Z _ b- O- 2 .. 10 : 1 :- : I g; :: -+ m __ .— 3 m h' .u ; — a: j: .A ‘5 .. I 10: : a: :3 _. r : - A - -1 1 1 Is 1 LI Ill! 1 I 1 1 ll 103 10‘ 105 “40.68 0.68 Figure 6.l2.--Relative Viscosity, nr, vs. cM for PS in Benzene at 30°C. ‘ 1“! “" l73 13x1 0 RelatI'Ve Viscosity, Tlr 8 N III”! I llllllll 5 5 10 2x10 3 10 l0 CM0.68 . . 0.6 Figure 6.l3.—-Relative Visc051ty, ”r! vs. cM 8 for PS C in Dioxane at 30° . 3 2 f4 0 (V . E~__ _ _ h lEhfplersi :Zac a c . L3. s..9.A..Ir.vl.p U. .wa- 174 in MEK at 30°C. 3 10 .. l I IIIIIII I I lllllll : I o PS-l I _ A PS-2 : -— -d 102: : F I E 2.: : - 3 F U "' —l .2 > -— —1 c: .2 15 -. ._ II —- ._ C : 1 J l 103 10” 105 2x105 cM0£8 . . . . 0.68 Figure 6.l4.--Relat1ve Visc051ty, ”r» vs. cM for PS ...? b..vr.uv.) .D~.'~.z _hnhhh s Pia h 175 4x103 0 SAN C-2 A A SAN c-2' 103 O A a: >; A '5 102 O 3 6 S: 1’ '33 8 15 ‘ I: x 0A 10 o ‘A /o O ,0“ 1 l I lllllll I I ILIHH 3 I. 5 lO 10 10 2x105 CM0.68 nr, vs. CMO°68 for Figure 6.l5.--Relative Viscosity, SAN C-2 and SAN C-2' in Dioxane at 30°C. h ...-Q :»»_~ ~ ICU. ; h I v'.‘2 176 3 4x10 I I I IIIIII I rIIIIIII _ A SAN C-2' _ 1(13 :3 :: E Z a; - _. >2 .- 2 7,; 10 : : oi " - u '- -I .3 ' ‘ > — _ o ._ .2 '- — E _ .' .. (I ‘10 :: :: I Z 1 l J,IJ l I Ill] 1 l l l Ill L1 103 10" 105 2on5 cMO-GB . . . . 0.68 Figure 6.16.—-Relative Visc051ty, nr, vs. cM for SAN C-2 and SAN C-Z' in MEK at 30°C. 177 Figure 6.l7.--Relative Viscosity, ”r’ vs. cM0'68 for SAN C-2 and SAN C-2' in DMF at 30°C. » 3x104 Relative Viscositynh. I. 10 10 d O N 10 178 1 1 F111”? 1 1 I lllllr l A - 0 SAN c-z A SAN c-2' A b _- —. A0 —‘ A E A a — o q — 4 .. A .. O A O : o : — A -I : o - — Q — .. o A l I III“ II I l J 1141]] l 103 10" 0.58 105 3x105 cM PC. _ 01103 b h ~bh>LIL » b IL. 1 ~>>bhb p A 3 2 0 fa A..Up¢a' : b: .C b». » .h~»-~h I W9 3 3x10 I IIIIIIII I IIIHIII 0 SAN c-2 A SAN c-2’ °} 3 b=0.68 ‘}b=0«625 103 Relative Viscosity, n, 5 N l l l l 1 III I L4,4L IL I II Ill 1 3 104 chlb Figure 6.l8.——Relative Viscosity, nr, vs. C-2 and SAN C-2' in Benzene at 30°C. 105 2x105 ch for SAN 'I'gnl'.) 0n "n Ifii’ifflh’ ’a‘ :‘.'¢* ’4'. '. ' ".':|" and ‘7 "first 0 H. I I .- v‘.r0 . l - . . l 'i: '- I . I ,- g.. A 180 of viscosity on concentration is linear and is indicated by Eq. 6.10. As the concentration increases, the polymer chains begin to overlap At still higher concentration, they begin to penetrate This gradual change each other. each other and ultimately they become entangled. from isolated clouds of polymer chains to entangled network is evi- dent from the smooth curve. The curves indicate that the viscosity b >5. dependence increases gradually from (CMa)1 to (CM 0 68 Figures 6.12 to 6.14 show plots of nr against cM ' for PS—l and PS-2 in benzene, dioxane and MEK, respectively. Figures 6.15 to 6.17 show plots for SAN C-2 and SAN C-2‘ in dioxane, MEK and DMF, respectively. The solvents may be labeled as good to The data for the two molecular weights in each case The plots fairly good. are reasonably well correlated by the power law equation. of ”r against cMO'625 are not shown for these polymer—solvent sys- tems; however, the exponent 0.625 also works well. Figure 6.l8 shows plots for SAN C—2 and SAN C-2' in benzene with both values of the exponent, b. Clearly, the correlation with the exponent 0.68 is not as effective as with the exponent 0.625. Two distinct curves are obtained for the two samples of azeotropic Copolymers (SAN C—2 and SAN C—2') when using the 0.68 exponent, thus not correlating the data. Correlation with the power 0.625 Shows a marked improvement in unifying the data for the azeotropic c0P01ymers in benzene. Benzene is a poor solvent for this copolymer and it appears that for poor solvent—polymer systems a lower value of the exponent in this type of correlation produces an improvement. The data for SAN C—2 and SAN C—2' in benzene obtained in this work _——J u.—u-' Mr Import 1 1w when: 1. c-“c'etion of ’4' 3* .190 uh "If "7 row . '.l\ 43"." t“ 'w I 4 f 181 lend support for lower exponents of M in the power correlations for poor solvent systems. As a result it may be suggested that a general correlation of the type a ”r ~ cM (6.l2) may be used where a may be related to the Mark-Houwink constant from intrinsic viscosity correlations (G-6). The exponent a depends upon the nature of the solvent and is usually found to lie between 0.5 and 0.8. For good solvents, a has high values (0.65—0.8) while for poor solvents a has low values (0.5-0.6), and for e-solvents a is equal to 0.5. Figure 6.l9 shows a plot of ”r against cMO'5 for SAN C-2 and SAN C-2' in benzene. The correlation is not as good as the 0.625 correlation. This indicates that for this system, the value of the exponent lies between 0.5 and 0.625, probably closer to 0.625. This is due to the fact that benzene is not a e—solvent for azeotropic SAN copolymers at 30°C, but is quite a poor solvent which may have a Mark-Houwink exponent between 0.5 and 0.625, probably closer to 0.625. The above data show that for one particular polymer in dif- ferent solvents, a power law correlation, using a single value of exponent is not possible. In the past, most data have been obtained with good solvents and hence differences caused by polymer-solvent interactions were not observed. This led to the belief that the data in different solvents could be correlated by a single expo- nent. hh-~h b 3 o h h (4 hhhhk hlhILIlIF Illibhhhb » L A 2 o I1 sf .x..u0un.> t)..l.§~.. 182 3 “‘0 _IIIIIIII ITIIIII _'_ 0 SAN c-2 A: k A SAN c-2‘ _ O A 1()3 :: :1 I: .. E _ - ,: .A f; 0 ti 2 A '5 10 : : o I Z .2 .. .. 3. — . ~ 0 '— — m - " .A 10 F- ‘5 o ;: :I o 1: 1'- m — — A0 1 — O 1 IIIIIIII I lllllll 2x102 io3 104 <:N10'5 O 5 Figure 6.19.--Relative Viscosity,n , vs. cM ' for SAN C- 2 and SAN C- 2' in Benzene at 30°C in sumn3ry 1 ufi 10. (The data of Simha and Utraki (S-6) dia not exceed concentrations beyond which Usp/CEUJ > 10.) 2. Different curves are obtained for different polymer- solvent systems. The data cannot be unified for the same polymer in different solvents with a single value of v in each case. The shift factors are Specific for particular polymer-solvent pairs. 3. In the absence of the values of intrinsic viscosity, [n], this correlation is not useful. _— 190 D. Williams Model for Zero Shear Viscosity The correlation techniques described previously are not use- ful in predicting the low shear viscosity, hr, of a polymer solution of known concentration if the appropriate data are not available. A general equation for or in powers of c is hr = l + c[n] + k1[n]2c2 + k2[n]3c3 + k3[n]4c4 + . . . (6.13) This equation is discussed in Chapter III. To predict hr of a polymer solution of known concentration, [n], k1, k2, etc. must be known. Thus far attempts have been made to predict k1 (I-2, P-5). The series can be truncated to the c2 term at low con— centration to give an equation of Huggins nSp/c = [n] + k1[n]c (6.14) where k1 is called Huggins constant. Imai made an exact calculation of k1 from the pearl neck- lace model. He evaluated Eq. 3.51 for the stress tensor, however, the procedure followed is rather tedious because of the mathematical complexity involved in attempting to obtain a rigorous solution that includes solvent effects. The solvent effects were included in terms of the coil expansion factor, a. A discussion on a is given in Chapter III. His result was k1 = kO/oc4 + kz/OI5 (6.15) where k0 is the Huggins constant at O-condition, and k is a con- stant unspecified by Imai. 191 z =(3/211)3/2 A2N1/2/b3 (6.16) where A2 is the segment-solvent second virial coefficient, which is zero at e—condition, N is the number of segments per molecule and b is the segment dimension. Another method of predicting k1 is the one proposed by Peterson and Fixman (P—5). The polymer molecules in this model are considered as Spheres penetrable by other polymer molecules when there is a contact between two polymer molecules, but imperme- able to solvent. The solvent simply flows around the spheres. In this calculation they included hydrodynamic interactions to the perturbations in a flow field caused by different polymer ”spheres." This led to the calculation of the Huggins constant for penetrable spheres. Since polymer molecules can penetrate each other, they were assumed to form temporary doublets and the doublets were assumed to behave like rigid dumbbells. In their calculation sol— vent effects were considered and the result obtained was k1 = 0.69 + 0.16 f(A) (6.17) where 2 A = 3 , (6.18) Bur r is the radius of a single polymer sphere, f(A) is a graphical function presented in (P—5) and f(A) has a maximum which means the predicted k1 has a maximum. Such a maximum has not been experi— mentally observed. Also, another serious objection to this treat— ment is that the polymer spheres are considered impenetrable to solvent instead of treating them as porous spheres. 192 Both of the above methods of calculation of k1 have limited application to engineering or industrial purposes since Imai's parameters kO and k are not defined and Peterson's function, f(A), does not agree with experimental observations. Williams (W—l) proposed a model to apply specifically to moderately concentrated polymer solutions. The details of the derivation of the model are given in Chapter III. The equation for low shear viscosity, nr, is cN _ _61 A 3/2 All the terms are defined in Chapter III. One good feature of this model is that all the parameters involved can be estimated or measured directly. An inquiry of the dependence of no on concen- tration, c, and molecular weight, M, warrants a closer examination of the parameters involved in the above equation. In Eq. 6.19, B is a measure of effective molecular size in terms of the end-to- end distance of the polymer molecules. The distortion of molecular configuration by intermolecular interaction is appreciable in dilute solutions (F-3e), but when the polymer concentration is large, the perturbation in dimensions tends to be less. Then the polymer end- to-end distance may be approximated by its O-value, L0. Hence Williams proposed that B = 3/2 . (6.20) 2 0 193 A discussion on L0 is given in Chapter III. The value of for each polymer can be obtained from the knowledge of the Mark-Houwink constant, K, for each polymer at its O—condition by using K = (/M)3/2 (6.21) in which A is a constant and is equal to 2.5 x 102] dl/[(mole) (cm3)] for broad molecular weight polymers (B—4a). The values of K are listed in Table 6.1. From these K values, and hence B can be easily calculated using Eqs. 6.21 and 6.20, respectively. The key parameter for polymer—solvent intermolecular forces (6.22) UN 0. m G. < UN where Vp is the molecular volume of a polymer molecule, 6 is the Gibbs free energy of mixing polymer segments with solvent and vp is the volume fraction of polymer. Williams (W—l) chose the Flory— Huggins form for e (F—3a), which is c = kT[ns ln (1 — v ) + n ln v + p p p x1van] (6.23) Where nS and 11p are, respectively, the number of solvent and polymer molecules per unit volume of solution and x1 is a dimensionless quantity that characterizes interaction between the solvent and the 901ymer; i.e., it is an enthalpy and entropy of mixing parameter. 194 The quantity, ka1, represents merely the difference in energy of a solvent molecule immersed in the pure polymer compared with one surrounded by molecules of its own kind, i.e., in the pure solvent (F-3a). The parameter x1 is called the Flory thermodynamic parame- ter. As polymer molecular weight becomes very high, np becomes vanishingly small and then 8 = kT[nS ln (1 - vp) + lepns]' (6.24) If V5 is the molecular volume of solvent, then nSVS = (l — vp) (6.25) whereby s = BI—(l — v )[ln (1 ~ v ) + x v ] (6-26) VS p P 1 P and 2 dx d c l l ___ = - 2X + 2(1 - 2V )‘—— p d2x1 6 + — ~————. - vp(l vp) dv2 ( 27) P As a first approximation, only the first two terms on the right- hand side of Eq. 6.27 will be considered, although for a few Systems x1 has been found to be a function of vp (H—7). Combining Eqs. 6.22 and 6.27, 195 2 13.: YR. 1 - 2 (6 28) kT v 1 - v Xi ° ° The volume fraction, vp, was calculated from concentration c by assuming additivity of volumes. From the x1 values listed in Table 6.5 and Eq. 6.28, A/kT could be calculated. The parameter C of Eq. 6.19 is given as a function of B, c, and A/kT as 15 c = l 3-53 . (6.29) sow—2n N c AV A A31" [T FT] J For evaluation of E, Williams (W-l) used Kirkwood's original theory of friction coefficient (K—ll). Kirkwood derived an expres- sion for g in terms of intermolecular potential energy between polymer molecules. This formulation has been used with fair success in calculating the viscosity of simple liquids such as argon (K-3). In a polymer solution, since the solvent is assumed to be present as a continuum, it exerts a frictional resistance to polymer molecules. However, Williams assumed that the frictional forces between over- lapping and entangling polymer molecules are the dominant factor in comparison to the polymer-solvent friction. The evaluation of E was done in terms of A/kT and it was found to be weakly dependent on c. The original model gave values of ”r far too low for polymer-solvent systems considered in this work in comparison with the experimental values. The deficiency was believed to be caused by underestimates 196 of g. The values of g (of the order of 1071]) obtained from the model were believed to be far too low in magnitude. From the values of diffusion coefficients at infinite dilution, an estimate of the order of magnitude of the friction coefficient can be made by using the equation (M-2a), 0 , LfT (6.30) where D is the diffusion coefficient at infinite dilution, f is the friction coefficient at infinite dilution, k is the Boltzmann constant and T is the absolute temperature. The parameter f is found to be of the order of 10'8 (B—4a). The analogy of Williams between simple molecules like that of argon and complex polymer molecules in solution where interac— tions between like and unlike molecules are believed to be strong and complex is deficient in concept. Again in Williams formulation 0f E, the viscosity of solvent, n3, is not involved at all. This is equivalent to the assumption that only interpolymer frictional forces are important and hydrodynamic frictional forces between P01ymer and solvent are unimportant. This deficiency has led to an alternative formulation for friction coefficient based on a model of concentrated suspension of spheres (G-6, F-ll). Frankel and Acrivos (F—ll) used an asymptotic technique to derive the dependence of viscosity on concentration for a suspension of uniform solid spheres. Their result contains no empirical con— stants. The analysis of viscosity of a suspension of arbitrary concentration is an extremely diffTCU1t problem bUt their simple , . 197 approach to an asymptotic model for the viscosity of highly concen- trated suspension of rigid spheres agrees well with the limited available data on suspensions. They assumed that the suspension behaved as a Newtonian continuum on a macroscopic scale. The adopted point of view was that the viscous dissipation of energy in highly concentrated suspensions arises mainly from the flow within the narrow gaps separating the solid spheres from one another. The relative motion of each sphere can be decomposed into two com- ponents, one along the axis joining the centers of the spheres and another normal to it. They indicated that the frictional force due to the motion of the spheres along the axis joining their centers is the dominant force. Then the frictional force from the rate of viscous dissipation is given by 2 _ l" _ I F — 3111150 11 _ 2r — 50 (6.31) where U is the approach velocity of the fluid, E' is the friction coefficient, h is the gap width between two spheres, and F 1'S the radius of the spheres. Thus, 3nnsr2 = h — 2r ' .. (6‘32) Assuming polymer coils to be Spheres of radius of gyration U2 1 we may write 30n5<52> g1 = ___._______—— . (6.33 h _ 2<52>W ) 198 One serious drawback is that Eq. 6.31 is for hard spheres with unde- formable boundaries while the equivalent polymer spheres are not so rigid. This means that when the concentration of polymer in solu- tion is very high, leading to highly entangled chains, 5' would not be applicable. Furthermore, when h is equal to 2U2 or when the spheres are in contact with each other (which may occur at some high concentration before they are entangled), €'+m. To alleviate this problem, Gandhi (G-6) made the following simplifi- cations: g' 3nns /h (6.34) where 3' 1| 1/3 (M/cNAv) . (6.35) Equation 6.32 for 5' represents purely hydrodynamic interactions between polymer and solvent and there is no accounting of polymer- solvent thermodynamic interaction. At sufficiently high concen- trations, V2 can be taken at e-condition (F-3e). The predicted values of E' by Eq. 6.34 were found to be of the correct order of magnitude, 10-8. In the absence of data on friction coefficient at high con- centrations, Eq. 6.34 can be used. It should also be mentioned 2>U2 assumes the polymer sphere to be imperme- that the use of <5 able to solvent. This is not true in actual cases. Hence Eq. 6.34 gives an overestimation of friction. It must be kept in mind that both Williams equation and 5' are applicable only in the absence of significant entanglements. These equations are not adequate 199 descriptions of phenomena observed beyond the onset of entangle- ment mechanisms. Williams' model can be tested for predicting ”r using 5' since all the parameters are known or can be calculated. The equation, after reinstating the linear term in c, may be rewritten as 2 cN n, =1 + [nlc + mi Hi] (11411313/21ca'). (6.36) S This is the only available model for or where hydrodynamic and thermodynamic effects are accounted for with independently measur— able properties. Figures 6.24 to 6.26 show plots of experimental and pre— dicted low shear relative viscosity, or, against concentration, c, for PS-l in benzene, dioxane and MEK, respectively. The plots are made on semi-log papers to accommodate the complete range of ”r values. Figures 6.24 and 6.25 show that in good solvents the Williams equation over—predicts the values of or; that the two curves run parallel up to about c equal to 7.5 gm/dl; and that then the predicted values of nr do not increase as fast as the experimental values. In the case of PS—MEK (Fig. 6.26), this devi- ation is observed at lower concentration. This is completely in agreement with Williams' model. The limitations and assumptions involved in the derivation of the model suggest that high concen_ tration behavior could not be described within its framework. This means viscosity at high concentrations cannot be predicted. 5 Relative Viscosity, 111. 200 2 1:10 I 1 I l I I I I _ —— Theoretical _ —o——o—- Experimental 102 :: : : . j r- ' - 11) :: ' .. Z ' : 1 I I I I I I I I O 5 10 15 20 22.5 Concentration, c , gmldl Figure 6.24.--Experimental and Calculated Relative Vis- cosity, ”r’ vs. Concentration, c, of PS-l in Benzene at 30°C. Relative Viscosity , 111. 201 3 ‘0 1— I l I I I I I 1 I I I :1 - Theorefical : : —o——o— Experimental _ _ _1 C I I: , : I- . fi 1- "‘ 1 l l J, Al 1 l l l l 1 l 0 5 10 15 20 25 30 Concentration, c, gm/dl Figure 6.25.--Experimental and Calculated Relative Viscosity, ”r’ vs. Concentration, c, of PS-l in Dioxane at 30°C. 202 7 102 x _ I I I I 1 I I I I I — “——-——1Theorefical — _ —o—o—— Experimental ‘ r‘ .3 L 2 F 10 E : — : '5 : : O U '_ -—1 .9 > i" .— 0 .2 - ._ ‘16 32 10 E: 33 Z ‘1 F— I— F— — b. — 1 I I I I 4 I I I I I O 5 10 15 20 25 27.5 Concentration, c, gm/dl Figure 6.26.—-Experimental and Calculated Relative Viscosity, 0r, vs. Concentration, c, of PS—l in MEK at 30°C. 203 This is precisely shown in Figs. 6.24 to 6.26. The model is valid only for moderate concentrations. In MEK, the entanglements are enhanced at lower concentrations since it is a poor solvent for PS compared with benzene and dioxane and hence the deviation of pre- dicted values from observed values is at lower concentration. Since Williams‘ model does not take into consideration the entangle- ments or aggregation it should be applicable to higher polymer concentrations in better solvents in comparison with poor solvents. The figures show qualitative success of the model. The difference between the predicted and experimental values differ by a fairly constant factor up to the concentrations when entanglements become important. Figures 6.27 and 6.28 Show plots for PS-2 in dioxane and MEK, respectively. For this high molecular weight PS, entangle- ments become important at lower concentrations than for low molecular weight PS. The predicted values are much closer to experimental values in the good solvent, dioxane, than in the poor solvent, MEK. Figures 6.29 to 6.32 show the same plots for SAN C-l in all the four solvents. Here benzene and dioxane are fairly good solvents while MEK becomes relatively poorer, and DMF becomes a better solvent for this polymer than it was for PS. The effect of this change in degree of goodness is observed on the plots. Figure 6.31 (DMF solution) clearly shows that up to a relatively high concentration, 10 gm/dl, the two curves are parallel. Figures 6.33 to 6.35 Show plots for SAN C-2 in DMF, MEK and dioxane, respectively. For this polymer, DMF is the best solvent and this is obvious from the comparison of the three plots. 204 3 4‘10 I I I I I I I r P O ._ Theoretical .J -o——o— Experimental ‘103 : . : E I - -4 1- -1 2 10 : 1 I I Relative Viscosity, 111. I l l l I O\ 10 llllll I 111111 I l I 1 l l l l l l I L 0 5 10 15 20 22.5 Concentration, c , gm ldl Figure 6.27.—~Experimental and Calculated Relative Vis- cosity, "r, vs. Concentration, c, of PS-2 in Dioxane at 30°C. 205 3 5x10 h. 1 1 1 1 1 I I I — -——'—- Theorefical 2 -O-—<>- Experhnental lO3 :: : ’— —1 I.- F' 2: 102: : '171 Z 2 8 _ -1 .‘2 "‘ _1 > — — o — .2 a — _ o I: 10 :: : 3' 3 i L 1 l I I I I I 0 5 1° ‘5 20 22.5 Concentration, C . gm / dl Figure 6.28.--Experimental and Calculated Relative Vis- COSTtYI nr, vs. Concentration, c, of PS-2 in MEK at 30°C. 206 2x103 | I I I I I I I 103 _ ' ’——-—- Theorefical I —0——<>— Experhnental : ‘— F ‘ 2 3? 10 :3 _ .5 __ _ O h _. U _ _ m __ _ g — — o d .2 — E1 _ _ 0 a: _ 10 :3 j I: I — —l — —l 1' I I I l I I I I 0 5 10 15 20 225 Concentration, c, gnn/dl and Calculated Relative Vis- - . ,_—E erimental Figure 5 29 Xp f SAN c-i in Benzene at 30°C. cosity, nr’ vs. Concentration, c, o 207 10 ‘————— Themeficm —”}“<*' EXPErhnental F“ 2 £5 10 171 0 U .9 > 0 .2 i3 113' c: 10 15 20 22.5 Concentration, C . gm Idl Figure 6.30.-—Experimental and Calculated Relative Vis- cosity, ”r: vs. Concentration, c, of SAN C—l in Dioxane at 30°C. Relative Vicositcy , '7lr 208 10 l I I I l I I I - Theoretical -o——o— Experimental 0 N O 1_L I l l A I I 0 5 10 15 20 22.5 Concentration, C I gm ldl 1 igure 6.31.-—Experimental and Calculated Relative Vis- COSTty, nr, vs. Concentration, c, of SAN C-l in DMF at 30°C. 209 3 - Theorefical : - ‘0—‘0- Experimental _ L102 : : F — I z: I _ '61 __ _ o — — U m — t— ‘S d, — — .2 E g i0 : E l 1 l l l l l 1 1 0 5 10 15 20 22 5 Concentration, C, grnldl ' ' ' Relative Vis— Figure 6.32.—-Experimental and Calculated o cosity, nr, vs. Concentration, c, of SAN C—l in MEK at 30 C. Relative Viscosity ,111. 210 3 10 \ Theoretical H— Experimental 2 IO 10 l l l l l l l l 1 10 15 20 22.5 Concentration, C , gm Idl ' ' Relative Vis- Fi ure 6.33.—~Experimental and Calculated o COSTty, fir? vs. Concentration, c, of SAN C—2 in DMF at 30 C. 211 l()3 I I I I I I I I F I- Theoretical : -o-—o— Experimental 2 I? ‘0 : >. _ 3 r _ U P- _1 .91 > — _ 0 .2 _ _ E 0 a:’ 10 :3 : 1 J J. l 1 4J, 41. l l 0 5 10 15 20 22.5 Concentration, C, gnildl Figure 6.34.-—Experimental and Calculated Relative Vis- cosity, nr, vs. Concentration, c, of SAN C-2 in Dioxane at 30°C. 212 3 2X10 1 1 I 1 1 1 I I 3 ——-———- Theorefical 10 : -o——-o- Experimental 1— l()2 ;: r3: ; 2 3s — .. .171 __ .3 o _ _- U .9. > ’- _— 0 .2 '3 io — : 0.l : —‘ m — I— L.- —l - _l 1 I J L I I I L I 0 5 10 is 20 22 5 Concentration,C, gn1/dl Figure 6.35.--Experimental and Calculated Relative Vis- cosity, ”r, vs. Concentration, c, of SAN C-2 in MEK at 30°C. 213 Figures 6.36 to 6.38 show the plots for SAN C-2' in DMF, dioxane and MEK. Since DMF is the best solvent, the deviation between the predicted and experimental values is the smallest. The experimental and predicted values of the viscosities of the solutions of SAN C-2 in benzene are not shown because of the compu- tation difficulties involved due to the negative value of the thermodynamic parameter A2. This leads to negative values of A/kT (see Eq. 6.28) for the lowest concentrations and small posi- tive values for higher concentrations. This in turn gives negative value or fractional positive values of logarithmic term in C (see Eq. 6.29). For the highest concentrations, the computations could be made but the points were too few to plot them. For SAN C-2' in benzene the same difficulty occurred at the lowest concentrations. Figure 6.39 shows plot for the SAN C-2' copolymer in benzene. This system clearly shows the most discrepancy between the experi- mental and predicted values. Benzene is a very poor solvent for this copolymer and this is reflected by the A2, x1 and a values. The polymer chains are very tightly coiled in benzene because of the unfavorable environment and this leads to tight entanglements even at low concentrations. This shows that Williams' model with the modified friction coefficient is not applicable in poor solvent environments. Figures 6.40 and 6.41 Show plots for SAN C-3 in DMF and MEK, respectively. Here also, the same trend as before is observed for the two curves involved for each solvent, DMF (good) and MEK (poor). 214 3 10 — I l I l I _ E . — ... Theoretical _ 1" -o—<>- Expermental ‘ L. .— 1132 :: I: I? I __ 2‘ — ... "vi 0 ... _- U VI 5; g 10 :: :j 2: .. _ 2 - ._ CO '— .— ‘r '— -1 1 l I J l l 0 5 10 15 Concentration, c , gm Idl Figure 6.36--Experimental and Calculated Relative Viscosity, ”g’ vs. Concentration, c, of SAN C-2' in DMF at 30 C. 215 5 10 I I I ,I 1 I 1 I 1 CI _ Theorefical —— -o-——-o— Experimental 104 : E: . Io3 _ _ 2 I : .5 : I: O U —' .— m —. E; ._ 0 l— _. .2 '0‘ 1__ _- m '3 a: 10 : 3 : -I .. .71 L- — 10 2 : 5 J I I I I I I I _ 0 5 10 15 20 225 Concentration, c, gm/dl FiQUre 6.37.-~Experimental and Calculated Relative Viscosity, n5 , vs. Concentration, c, of SAN C-2' in Dioxane at 30°C. p 216 5x103 1 1 1 1 1 —0—0— Experimental - _ §~ Theoretical '- l llllll Relative Viscosity, «1'11. 3 l l Concentration, C,grn/dl Figure 6.38.--Experimental and Calcula- ted Relative Viscosity, nr, vs. Concentration, C, of SAN C-2' in MEK at 30°C. 217 1(34 '——‘——— Theorefical ‘*}——-<>- Expernnental 103 e 2‘ '6 O U ‘2 > 2 w 10 .2 iii '5 a: 10 2111111111 ‘ O 5 10 ll Concentration, c, gni/dl Figure 6.39 —-Experimental and Calculated Relative Vis- COSlty, or, vs. Concentration, c, of SAN C-2' in Benzene at 30°C. 218 Figure 6.40.-~Experimental and Calculated Relative Viscosity, 0r, vs. Concentration, c, of SAN C-3 in DMF at 30°C. Relative Viscosity , 111. 10 10 O N O 219 Concentration, c, gm ldl 1 1 1 I 1 1 T — b _ : Theoretical ° : " -o-—o- Experimental " “ '1 h- . - — -4 I . I L- . A I ° :51 l L J L J l l O 5 10 15 20 22.5 220 I'IIIII Theoreucal Experhnental RelaHve Viscosity,11r l 1 .EL, l 10 15 20 225 Concentration, c, gm/dl Figure 6. 41. --Experimental and Calculated Relative Viscosity, nr , vs. Concentration, c, of SAN C- 3 in MEK at 30°C. 221 It should be noted that in all the cases the two curves run approximately parallel up to a certain concentration before crossing each other. Then the predicted values are much smaller than the experimental values. The model seems to be applicable for good solvents to higher concentrations than in poor solvents. The two curves can in turn be superposed by moving one onto the other by some constant factor. This demonstrates the need to check the model for the friction coefficient, and if possible obtain experi- mental values of 5' from diffusion or ultracentrifugation data. Unfortunately, there is no data available on the friction coeffi- cient for these systems at moderate concentrations. It is also not known how important a role the polymer-solvent hydrodynamics plays at high concentrations. This is the only model available so far that takes polymer- solvent thermodynamics into consideration when predicting zero shear viscosity of moderately concentrated polymer solutions. It is qualitatively successful for polymer solutions of moderate con- centrations in good solvents, and of dilute solutions in poor sol- vents. At higher concentrations, the predicted values are much less than the experimental values because of the aggregation due to entangled polymer chains. The friction coefficient, 6', is propor- tional to . Since aggregation increases the apparent size of 2 the polymer domains, an effective value of <3 > for these "larger" molecules should increase 5' and thus improve the model. Again with aggregation, enormous increases in friction are anticipated. This is not accounted for by the model and hence, at higher 222 concentrations, there is a large difference between the experimental and predicted values of viscosity. E. Non-Newtonian Viscosity The study of non—Newtonian behavior of macromolecular solu- tions and melts has attained an important status in the field of transport phenomena because of the industrial importance of such materials. These fluids differ from Newtonian fluids in that the viscosity of these fluids is dependent on the velocity gradient or shear rate being applied to them. It has been known for a long time that macromolecular fluids are generally shear—thinning; i.e., the viscosity drops dramatically from the zero shear or Newtonian viscosity as the shear rate increases. This behavior is very important for engineering considerations. Along with this behavior, there are many peculiar but interesting phenomena associated with non-Newtonian fluids which are described in Ref. (B-l7). Up to this point all the discussion in this work has been with regard to the viscosity at sufficiently low shear rate where it is independent of shear rate (Newtonian region of the viscosity— shear rate curve). This section is devoted to the discussion of non-Newtonian behavior. As the shear rate increases, at a certain value of the shear rate, the viscosity begins to decrease from its Newtonian value and continues to do so as the shear rate is increased to still higher levels. It is believed and also observed in some cases (T-3) where extremely high shear rates could be attained, that at some range of higher shear rates Newtonian behavior would again be observed. This is called the upper-Newtonian region. 223 1. Dependence of Relaxation Time on Concentration The low shear Newtonian viscosity and the shear rate where the viscosity begins to decrease, which may be called the critical shear rate region, may change by many orders of magnitude from one system to another depending on the nature of the polymer, its molecular weight, the solvent and the concentration of the solution. The distribution of molecular weight or the degree of polydispersity is also important since the functional form of the viscosity-shear rate curve depends on it (M-lc, G—5). Graessley (G—3) has developed a molecular model of polymer behavior which leads to the concept of a non—Newtonian viscosity. He envisions interaction between polymer molecules which he con— siders to be of an entanglement nature, leading to increased dissipation of energy with shear. This entanglement process is a kinetic phenomenon in which two molecules in a shear field entangle at a finite rate when they are sufficiently close. As the molecules pass each other in a flow field, disentanglement occurs. The detailed kinetics of this process is unknown. In Graessley's picture, two molecules must first be within a certain distance of each other, say, within a sphere of radius R, for entanglement to Then the molecules must remain within this sphere for a The greater the OCCUP. finite time, T, or else no entanglement occurs. shear rate, the more rapidly the two molecules move relative to one another. Hence at high shear rates the entanglement density is reduced, thereby causing a reduction in viscosity. At zero rate 224 of shear the time constant for the formation of chain entanglements is To. Rouse (R—l) has calculated the relaxation time for a bead— spring model and according to him, TR = (6/112)(T)0M/CR1) (6.37) where no is the zero shear viscosity, M is the molecular weight, 0 is the polymer concentration (gm/cm3), R is the gas constant, and T is the absolute temperature. The physical significance of the relaxation time is that an imposed orientation of molecules reverts to random orientation with an exponential time decay proportional to ét/T. Graessley assumes that the two parameters, TR and T0, are related by (6.38) where K is a constant of the order of unity (G-3). The viscosity-shear rate curves for several concentrations of a polymer in a solvent can be superposed to form a single master curve by appropriately shifting the curves horizontally at each concentration after plotting the normalized values of n/nO against 1. It is also possible in turn to superpose the master curves for polymers of different molecular weights, again by appropriately shifting the curves horizontally. Later in this chapter a method of obtaining these shift factors is described. Figures 6.42 to 6.44 show mater curves for azeotropic copolymers (SAN C-2 and SAN C—2') in benzene, dioxane and DMF, respectively, while Fig. 6.45 shows a master curve for SAN C-3 225 .UOOm pm mcm~cmm cw meowuweucmocoo maowem> 00 .N10 zczu .+ I c .mumm LumzmIzuwmoomw> 0o cowuwmogemazmii.~e.o mesmwg .3... m _ ...0 5.0 «00.0 . 2...... . ......4. . I... ..o .282 To z wo.~-u zgzo .>-c .mumm gmmgm - zpwmoumw> mo cowuvmoqgmaamuu.m¢.m mgzmwm o... ..o —n80 Nnxwo 6.33.70 2.5 4 B.Emo~.~-u zgzu .m - c . N .vzcmom ~-uz we cowpwmoa. f x0 ,‘ 04‘ 5&3}qu i 5.3.1 111.23 :1 .ill [I.L L.l #mgpcwocou mzowLm> m0 _Nlu wazm--.¢¢.o mLDm.. 228 .ooom um .zo cw mcowumgpcmucou maowgm> we m-u zgzu .w..c .mpwm .mmcmuzpwmoUmP> mo cowuwmoqgmaam--.m¢.o m.=m.m .3... — _u0 _0.0 mxvogu HIIITI :___‘ __ _ _ d _ _—__ _ _ _ ____ _ q X .25.. 2 .23 mm .256 on a O a m¢;3:sazaxaqsqxa 38.8.0.3 483 4 _ .... _— _ __P_. _ l I III] I l l —.0 m. In. 229 in DMF. These curves are formed by finding appropriate shift fac- tors at each concentration and molecular weight as mentioned before. The major differences in behavior between various solutions are reflected in two parameters. One is the zero shear viscosity, no, and the other is the characteristic shear rate which locates the critical shear rate which in turn is related toro. Thus the slopes in the non-Newtonian region appear relatively insensitive to con— centration and molecular weight. There has been considerable success in correlating viscosity— shear rate relationships for polymer melts and solutions using reduced master plots. These master plots are of the form n/n0 = f(TOv). (5-39) The parameter TO also denotes the shift for each curve along the shear rate axis required to effect superposition on the master plots. The function f(toy) depends on the molecular weight dis— tribution. Graessley (G-S) has predicted f (To?) from a theory based on the shear induced changes in the network of inter- molecular entanglements. In his theory, T0 has a meaning of a characteristic time for formation or disruption of entanglements. The effect of molecular weight distribution on the function, f(TOv), was also predicted by him. He (G-5) has given a table of values of n/n0 and corresponding values of Toy/2 for various molecular weight distributions. Thepolymers synthesized in this work have molecular weight distribution represented by polydis_ Perity values close to 2 and hence theoretical master curve for 230 polydisperse entangling chains with polydispersity of 2 was used fer superposing the experimental data. The experimental curves were shifted parallel to the shear rate axis to achieve the best fit with the theoretical curve. This allowed the determination of To from a direct comparison of the ? axis of the experimental curve with the TO§/2 axis of the theoretical curve. The values of To thus obtained are listed in Table 6.8 along with the Rouse relaxa- tion times, TR. It can be seen that To and TR are always of the same order of magnitude. Many forms of To have been suggested as a result of the attempts to correlate data for solutions and melts. Most of the suggested shift factors are of the form T—nOMa < > T a . 6.40 0 T F(c) TABLE 6.8.--Flow Parameters of Polymer Solutions at 30°C. Mw no, 10x103, tRx103, Polymer Solvent c, gm/dl Poise sec. sec. PS—2 501,000 Dioxane 10 2.14 0.465 0.291 l5 9.56 l.lS 0.77l 50 4493 l36 l09 SAN C-2' 666,000 Dioxane 7 5.83 1.38 1.34 10 25.9 3.85 4.l6 20 806 50 64.8 SAN C-3 332,000 DMF lo 3.38 0.332 0.27 20 52.6 2.33 2.11 35 768 17.4 l7.6 231 Table 6.9 summarizes many of the suggested forms. It is interest- ing to note that Graessley's form (G-4, G-l), n M 0 To “ ciii + BcM) ’ (5-4') includes other forms as special cases. It is not possible to make a complete comparison of the forms of To given in Table 6.9 for the solutions studied since M and T were not varied. The ratio TO/TR is plotted against c in Fig. 6.46 for PS-2 in dioxane and SAN C-Z' in dioxane, and in Fig. 6.47 for SAN C-3 in DMF. According to Eq. 6.40, TO/TR is inversely proportional to F(c)/c. Because of the limited concen- tration range, the exact form of F(c) cannot be decided but the curves in Figs. 6.46 and 6.47 indicate that the data can be described adequately by the Graessley form. TABLE 6.9.--Suggested Forms of To. F (c) a b Reference c l l B-l8, B-l9, B-20 c2 o 0 M-6 c and c2 (depending on 1 D-5 concentration) ' T./'rR r./TR 1.9 1-8 1.7 1.6 1-5 1.4 1.3 1.2 1.1 1.3 1.2 1.1 1.0 0.9 0-8 0.7 0-6 232 0.25 0.275 rlIlllll-FII PS-Z ‘ q IJIIIILIJIL (t1 0.2 ‘0-3 (3.4 (3.5 C). lllflllllll—m SANc-z' c—q q -—l llJlJLllill (J (3435 (L1 0.15 (3.2 c, grn/cfl Figure 6.46.--Ratio of Experimental to Rouse Relaxation Time, TO/TR, vs. Concentration, c, of PS-2 and SAN C-2' in Dioxane at 30°C. 233 1.5 I 1 I l l l l l l 1.1. - - tr 1'3 " “ ... \ 1 2 b -1 ,3. 1.1 - - 1.0 - - 0.9 1 1 1 1 l J l 1 l 0 0.1 0.2 0.3 0.4 0.5 c ,ggnildl Figure 6.47.--Ratio of Experimental to Rouse Re1axa- tion Time, To/TR, vs. Concentration, c, of SAN C-3 in DMF at 30°C. 234 2. Dependence of Non-Newtonian Viscosity on Thermodynamic Quality of Solvent Graessley argued that a decrease in viscosity with increas- ing shear rate in polymer solutions having concentration high enough to give rise to entanglements of molecules can be explained on the basis of the change in density of the chain entanglements. The ratio TR/TO increases with the increase in the value of the product of concentration and molecular weight (see Figs. 6.46 and 6.47), i.e., a quantity expressing the density of chain entanglements. The interesting question is: What is the effect of the thermody- namic quality of the solvent on the density of chain entanglements which is reflected in the s1ope of the non-Newtonian curve or how are the mechanical formation and break-up of chain entanglements affected by the quality of the solvent? The effect should be evi- dent from the plot of n/nO against TRy/z for one polymer in differ- ent solvents but of the same concentration in all the solvents so that the density of chain entanglements (expressed as cM) is the same in all the solutions. For this purpose, solutions of SAN C-2' in benzene, DMF, dioxane and MEK were used. In each solvent, con- centrations of 7, 10 and 20 gm/dl were considered. Also, SAN C-3 in MEK and DMF at 35 gm/dl was considered. The solvents that were used have different viscosities and also they are of varying degree of goodness for the polymers as indicated by different values of the expansion factors, a (see Table 6.2). The experimental results are plotted in Figs. 6.48 to 6.51. The figures indicate that correlations of n/n0 with TRV/Z in different solvents form master plots for each concentration. It 235 .mumm cmmcmnxpwmoump> mo cowuwmoqcmazmuu.me.o mgzmwu .u.om new .e\sm N am xmz new mcmxowo .mzo .m=m~:mm cm .muu zcsu .w 1 c N\m~ ELK — ...o pod mood d———_— — — ——fi——— q— _ —_———— I: viwe‘ r: D I l 0cmxomo Cm X l l. .mczna c_ d. I. I 0CONCOm cm 0 I. H act-8.80 o H n to...- ‘xoaxfii‘xtx‘xa Q no u b——_ _ _ — __PF—_ _ _ _ b__rrh ..0 'Lt/u. 236 zc=u .w 1 c .mumm cmmcmuxuwmoumw> mo cowuwmoagwazmnu.me.m mesa.» .uoom new .e\Em O. on m=m~=am new .ZQ .meaxo_o e. .N-u NE. a... 0.nv —.0 p0.nv _ _ d — _ — _ _ _ j 1 x o I I. I e I I. j I 0‘ II. .I. r ‘ .l x 1 II o ‘1 QCONCfim Cm X L20 c. < OCMXOmD cm 0 _ p _ _ l— _ _ _ _ m.0 'u/u 237 .u°om new .u\sm om pm use was mcmxowo .m:w~:mm cm .N-u zc=u .+ - : .mumm cmmgmuxgwmoomw> mo cowawmoacwaam--.om.m mcamwm N\M.mr. m — —u0 p040 . :2... . . :1... fl . ...: ..o I: I. Lu l d a .I. I 11 a s.. I. an I 9" O I .II 10.00 0 II II. ‘9‘ '0'. III n 4 end econ-3:008 Evan a ll: — .mean C_ u 2.3on c. 4 .wconcom c. o 1 _, __p_. . . _ _ ____. . _ _ _ .... m 237 .UOOM use Fu\sm ow um use new mcmxowa .m:m~cmm :m .Nuu zczo .w u c .mumm cumsmuxuwmoumm> we cowpwmoacqumuu.om.m «gamma m . ..o 3.0 . .22... . 1...... . :3. I d l l a I 0 l .... I I 0‘ II n . .0. . . n H e ¢a«u4u«o«n0u$uu Swan 0 H “.20 c. ... 2.365 c. 4 ..wconcom c. o I. . :2... . . 2...... . .... —.0 'u/u 238 .uoom ucw _U\Em mm pm xmz 02m 020 2w mIo zc=u .+ I c .mpwm .mmsmuzuwmoumw> we cowpwmoacqum--._m.m mczmwm {some N . ..o S... 80.0 5.... xmz c. 4 I ...... c. o I I I I.. I < I. / l a a l h. l o< ‘ 0 H 0 Exponent in Eq. 2.27, defined by Eq. 2.29a. Mark-Houwink exponent in the empirical rela- tionship between intrinsic viscosity and molecular weight, Eq. 3.7. Parameter depending on particle shape, Eq. 3.43. Activity of solvent. Constants in Eq. 3.47. Thermodynamic constant that determines mag- nitude of intermolecular potential energy between polymers, (gm)(cm5)/sec2, Eq. 3.55. Angstrom unit, 10"8 cm. Coefficient in the virial expansion, (mole) (cm3)/gm2, Eq. 3.20. Third virial coefficient, Eq. 3.39. 246 CA, CB ent 247 Exponent in Eq. 2.27, defined by Eq. 2.29b. Exponent in power law correlations. Dimension of segment of a polymer molecule, Eq. 6.16. Coefficient in the virial expansion, Eq. 3.20. Factor related to molecular size, cm’z, Eq. 3.62. Exponent in Eq. 2.27, defined by Eq. 2.28c. 3 Concentration in gm/cm in Ch. V, sections C and D, and in gm/dl in other chapters. Height concentration of species A and B in copolymer. Entanglement concentration, gm/dl. Weight concentration of component i in copolymer, Eq. 3.26. Parameter in the Williams equation, Eq. 3.61, defined by Eq. 3.63, cm5. Constant characteristic of given chain struc- ture, Eq. 3.12. Differential refractometer readings for solu- tion at cell positions 1 and 2, respectively. 248 Corresponding readings for solvent. Fraction of primary radicals, released by an initiator, which initiate polymer chains, Eq. 2.2. Mole fractions of monomers 1 and 2, respec- tively, in monomer mixture. Weight fraction of monomer 1 in monomer mix- ture. Initial mole fraction of monomer 1 in monomer mixture. Mole fractions of monomer 1 and 2, respec- tively, in copolymer. Product of transmittance of neutral filters of light scattering photometer at angles 0 and 6, respectively. Free energy change on mixing. Weight fraction of monomer 1 in copolymer. Pair correlation function. Radial distribution function. Galvanometer readings in light scattering measurements at angles 0 and a, respectively. Partial molar Gibbs free energy at any and standard states, respectively. 249 h Gap width between spheres, cm, Eq. 6.31. H Optical constant defined by Eq. 3.22. AHm Heat of mixing. AH] Partial molar heat of dilution. ie Intensity of light scattered at angle 6. I0 Incident intensity of light. [1] Concentration of initiator, mole/cm3. k Differential refractomer constant, Eq. 4.1. k Boltzmann's constant. k Unspecified constant in Eq. 6.15. k] Huggins constant. k2, k3 Constants in Eq. 3.49. k], k2 Constants in Eq. 3.48. kg Huggins constant at O-condition. kd’ kp, kt Reaction rate constants for initiator decom- position, chain propagation, and chain termination, respectively. Copolymerization propagation constants for a radical ofthe type indicated by the first sub- script with a monomer indicated by the second. kll’ klZ’ k2]: k22 itc’ 1.td k k tll’ :12, k1:22 250 Reaction rate constants for termination by combination (coupling) and disproportionation, respectively. Termination constants for a radical of the type indicated by the first subscript with a radical of the type indicated by the second. Termination rate constant in diffusion- controlled copolymerization. Optical constants defined by Eqs. 3.20 and 3.27, respectively. Mark-Houwink constant in the theoretical intrinsic viscosity-molecular weight rela- tionship, Eq. 3.14. Empirical constant in Eq. 3.9. Bond length. Mean-square end-to-end distance of a polymer chain in any and unperturbed states, respec- tively. ' Mean-square end-to-end distance of a "freely rotating" polymer chain having no hindrance to internal rotation about carbon-carbon bond. Amount of monomers, moles. Molecular weights of polymer and monomer, respectively. [M] 251 Monomer concentration. Monomer l and 2, respectively, and their concentrations. Critical molecular weight, Eq. 6.6. Molecular weight of component i in a copoly- mer having weight fraction, Yi’ Eq. 3.31. Number and weight average molecular weight, respectively. Apparent molecular weight. Initial amount of monomers. Chain radicals of types 1 and 2, respectively. Radical at the end of a growing chain. Number of links in a polymer chain, Eq. 3.12. Refractive index of solution and solvent, respectively. Number of solvent and polymer molecules, respectively. Number of polymer and solvent molecules, respectively, per volume of solution. Number of segments per polymer molecule, Eq. 6 16. AV P(0) I": '“I’ '"2 252 Avogadro's number. Pressure. Parameter representing heterogeneity in com— position of copolymer, Eq. 3.31. Factor expressing reduction in scattered intensity at angle 6 due to interparticle interference, Eq. 3.21a. Arbitrary constant of Zimm plot. Parameter representing heterogeneity in composition of copolymer, Eq. 3.32. Distance, Eq. 3.17. Radius of sphere, Eqs. 6.18 and 6.31. Position vector between two molecular centers. Monomer reactivity ratios in copolymerization. The gas constant. Effective radius of polymer chain, Eq. 3.8. Rates of initiation and propagation, respec— tively, of polymerization. Rayleigh ratio at the angles 90° and a, respectively. 253 Entropy. Partial molar entropy of dilution. Configurational entropy of mixing. Mean—square radius of gyration of a polymer molecule in any and in unperturbed states, respectively. Time Absolute temperature. Velocity of spheres, Eq. 6.31. Potential of mean force due to presence of all segments, (Gm)(cm2)/sec2. Volume fraction of polymer. Specific volume of polymer. Molar volume of solvent. Intermolecular potential energy, (gm)(cm2)/ sec2, Eq. 3 54. Partial molar volume of solvent. Volume fraction of solvent. Molecular volume of polymer and solvent, respectively. XI: x2 XA’ XB 254 Fractional conversion of monomer to polymer, Eq. 2.3. Average composition of copolymer, Eq. 3.33. Mole fractions of solvent and solute, respectively, Eqs., 3.34 and 3.36. Height fraction of monomers A and B, respec- tively, in copolymer. Mole fraction of monomers A and B, respectively, in copolymer. Composition of component i in a copolymer, Eqs. 3.31 and 3.32. Position of ith segment of a polymer molecule, referred to arbitrary origin, Chapter III, Section D. Parameter in Eq. 3.27, defined by Eq. 2.28. Exponent in Eq. 2.16, defined by Eq. 2.17. Polarizability of scattering particles, Eq. 3.17. Factor expressing the linear deformation of a polymer molecule owing to solvent-polymer interaction. Exponent in Eq. 2.16, defined by Eq. 2.17. <0 255 Power law correlation parameter, Eq. 3.41. Constant in Eq. 6.41. Exponent in Eq. 2.16, defined by Eq. 2.17. Shift factor in Simha correlation, Chapter III, Section C, Part 2. Shear rate, sec-1. Weight fraction of component i in a copolymer, Eqs. 3.31, 3.32 and 3.33. Parameter in Eq. 2.16, defined by Eq. 2.17. Parameters in Eq. 2.21, defined by Eqs. 2.23a and 2.23b. Number of isotropic scattering particles per unit volume having polarizability a, Eq. 3.17. Free energy of mixing polymer segments with solvent, gm/(cm)(sec2), Eq. 3.55. Frictional coefficient for a bead of polymer chain, Eq. 3.8. Viscosity. Zero shear or low shear viscosity. Relative viscosity, n/no. -. nae-r nsp [n]. [n]e ex 256 Viscosity of solvent Specific viscosity, nr - l. Intrinsic viscosity at any and at O-tempera- ture, respectively, in deciliters per gram. Angle between transmitted and scattered beam, Eq. 3.17. ”Ideal” or "Flory” temperature at which poly- mer chains in a solution assume unperturbed dimensions. Parameter expressing the energy, divided by kT, of interaction between a solvent mole- cule and polymer. Time constant for polymer chain response, Eqs. 3.50 and 3.60. Wave length of light, Eq. 3.17. Newtonian viscosity. Chemical potential of solvent. Ideal chemical potential of solvent. Excess chemical potential of solvent. Kinetic chain length. III—l O 257 Refractive index increment, its average value for a copolymer and its values for homopoly- mers A and B, respectively. Friction coefficient between polymer molecules, gm/sec. 3.14159 ..... Osmotic pressure. Density of bulk polymer. Stiffness or steric factor. Turbidity as determined by light scattering measurements, Eq. 3.18. Shear stress. Experimental relaxation time. Rouse relaxation time. Total stress tensor, Eq. 3.51. Stress tensor representing stress due to solvent and externally imposed isotropic pressure, Eq. 3.51. Parameter in chemical controlled termination, Eq. 2.21. Volume fraction of spheres, Eq. 3.42. X1 XAa XB XAB 258 Flory's parameter relating intrinsic vis- cosity to molecular dimension , Eq. 3.10. Flory's thermodynamic parameter expressing interaction between polymer and solvent. Corresponding parameter for homopolymers A and B, respectively, Eq. 1.6. Flory's thermodynamic parameter expressing interactions between homopolymers A and B, Eq. 1.6. r—dependent factor in shear perturbation of go, Eq. 3.57a. Parameter characterizing the entropy of dilu- tion of polymer with solvent. Distribution function in coordinate space of all segments. Il‘l I. '1‘}! BIBLIOGRAPHY 259 A-l. A-3. A-4. B-l. B-2. B-3. 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V—2. V-3. V—4. W-l. W-2. W-3. W—4. Y-1. Z—1. Z-Z. 266 Van Wazer, J. R., J. W. Lyons, K. Y. Kim, and R. E. Colwell, "Viscgsity and Flow Measurement," Interscience, New York (1963 . “Viscometers Bulletin 82," Cannon Instrument Co., State College, Pa. 16801. "Viscosity Standards Bulletin 71,” Cannon Instrument Co., State College, Pa. 16801. Wil1iams, M. C., A. I. Ch. E. J., 12, 1064 (1966), ibid., 16, 534 (1967), ibid., 16, 955 (1967). Walling, C., J., Am. Chem. Soc.,21_,1930 (1949). Walling, C., "Free Radicals in Solution," John Wiley and Sons, Inc., New York (1957), (a) Ch. 4, (b) Chs. 3-5. Williams, M. C., J. Polym. Sci., Part C, 66, 211 (1971). Yamakawa, H., “Modern Theory of Polymer Solutions," Harper and Row, New York (1971). Zimm, B. H., J. Chem. Phys., 24, 269 (1956). Zimm, B. H., J. Chem. Phys., 16, 1093, 1099 (1948). APPENDICES 267 APPENDIX A MACHINE CONSTANTS OF WEISSENBERG RHEOGONIOMETER AND CANNON-UBBELOHDE FOUR-BULB SHEAR DILUTION CAPILLARY VISCOMETERS 268 APPENDIX A MACHINE CONSTANTS 0F WEISSENBERG RHEOGONIOMETER AND CANNON-UBBELOHDE FOUR-BULB SHEAR DILUTION CAPILLARY VISCOMETERS The rheogoniometer was used in the constant shear configura- tion. Platen diameter and cone angle were not varied in any of the viscosity measurements. Three different torsion bars were available for different ranges of viscosity. The constants of the rheogoni- ometer are listed in Table A.1. TABLE A.1.--C0nstants of Weissenberg Rheogoniometer. Torsion Bar Constant, Tor510n Bar dyne cm/micron ST/6 8.603 ST/7 0.091 x 103 ST/8 0.875 x 103 Platen diameter: 7.5 cm Cone angle: l°-37' Two Cannon-Ubbelohde four-bulb shear dilution type capil- lary viscometers were used for measuring viscosities of dilute polymer solutions. The constants of the viscometers are listed in Table A-2. 269 270 TABLE A.2.--Constants 0f Capillary Viscometers. Bulb Constant, Shear Rate centistokes/sec Constant Viscometer 1, Capillary diameter: 0.0364 cm 1 0.001984 672,000 2 0.002028 317,000 3 0.001874 146,000 4 0.001830 68,000 Viscometer 2, Capillary diameter: 0.0417 cm 1 0.003609 411,000 2 0.003697 201,000 3 0.003826 92,000 4 0.003758 40,000 To obtain viscosity in centistokes, efflux time in seconds is multiplied by the viscometer constant. To obtain shear rate at the wall of the capillary in sec-1, shear rate constant is divided by the efflux time in sec- onds. APPENDIX B WEISSENBERG RHEOGONIOMETER n-y DATA 271 272 TABLE B.1.--Weissenberg Rheogoniometer 0‘? Data for PS-l in Benzene at 30°C. Ggar Box Shear Rgte, Viscosjty, n, Poise etting y, sec c — 20 gm/dl 2 4 6.67 1 082 2 3 8.4 1 087 2.2 10.55 1.082 2.1 13.26 1.089 2.0 16.74 1.107 1.9 21.01 1.087 1.8 26.51 1.102 273 TABLE B.2.—-Weissenberg Rheogoniometer n—t Data for PS-l in MEK at 30°C. Gear Box Shear Rate, Viscosity, n, Poise semng Y’ secfi C = 20 gm/dl c = 25 ngl 2.7 3.34 0.61 —— 2.5 5.28 0.61 —— 2.3 8.40 0.63 -- 2.1 13.26 0.62 -— 1 8 26 51 0.63 1 68 l 7 33.39 -— 1 70 1 6 42 02 0.63 1 7O 1 5 52.77 —- 1 69 1 4 66 67 0 63 1 63 1 3 84.03 —— 1 61 1 2 105 54 0 63 1 55 l 1 132.55 —— 1.53 1 0 167.43 0 63 -- 0.8 265.10 0.63 -- 274 TABLE B.3.--Weissenberg Rheogoniometer n-i Data for PS-l in Dioxane at 30°C. Gear Box Shear Rate, VISCOSIty, n, Poise c Setting Y» sec' :12 gm/dl c=20 gm/dl c=25 gm/dl c=50 gm/d1 2.0 16.74 -- -- 3.85 -- 1.9 21.01 -- -- 3.88 51.6 1.8 26.51 0.456 -- 3.90 52.4 1.7 33.39 0.440 -- 3.90 52.0 1.6 42.02 0.456 1.94 3.95 51.3 1.5 52.77 0.461 1.93 3.89 50.6 1.4 66.67 0.448 1.95 3.80 48.7 1.3 84.03 0.451 1.96 3.72 47.7 1.2 105.54 0.456 1.97 3.60 46.0 1.1 132.55 0.455 1.95 3.40 42.0 1.0 167.43 0.461 1.90 -— 38.8 0.9 210.1 0.451 1.89 -- 36.5 0.8 265.1 -- 1.87 -- 33.9 0.7 333.9 -- 1.85 -- -- 0.6 420.2 -- 1.77 -- -- 275 TABLE B.4.--Weissenberg Rheogoniometer n—y Data for PS-2 in Benzene at 30°C Viscosity, n, Poise Gear Box Shear Rate, satting Y’ sec' c = 10 gm/dl c = 20 gm/dl 2.3 8.4 -- 29_7 2.2 10.55 -- 29.5 2.1 13.26 -- 29.0 2.0 16.74 -- 29.1 1.9 21.01 -- 29.4 1.8 26.51 -- 29.0 1.7 33.39 -— 28.6 1.6 42.02 1.84 27.7 1.5 52.77 1.86 25.6 1.4 66.67 1.85 24.6 1.3 84.03 1.85 23.1 1.2 105.54 1.84 20.9 1.1 132.55 1.83 18.3 1.0 167.43 1.82 17.1 0.9 210.1 1.78 15.8 0.8 265.1 1.74 14.0 0.7 333.9 1.70 -- 0.6 420.2 1.61 -— 0.5 527.7 1.50 -- 276 TABLE B.5.--68issenberg Rheogoniometer n-V Data for PS—2 in MEK at °C. Viscosity, n, Poise Gear Box Shear Rate, Setting Y: sec 10 gm/dl c = 15 gm/dl c = 20 gm/d1 0 ll 1.9 21.01 -- -- 10.9 1.8 26.51 —- -- 10.8 1.7 33.39 0.68 2.56 10.8 1.6 42.02 0.67 2.55 10.8 1.5 52.77 0.69 2.54 10.7 1.4 66.67 0.69 2.54 10.9 1.3 84.03 0.69 2.53 10.7 1.2 105.54 0.69 2.51 10.9 1.1 132.55 0.68 2.50 10.7 1.0 167.43 0.67 2.49 10.5 0.9 210.1 0.67 2.44 9.83 0.8 265.1 0.67 2.42 8.95 0.7 333.9 0.67 2.40 8.10 0 6 420.2 -— 2 30 6.96 O 5 527.7 —— 2 18 6 3O 277 TABLE B.6.--Weissenberg Rheogoniometer n-v Data for PS-2 in Dioxane at 30°C. Viscosity, n, Poise Gear Box Shear Rate, Setting 1, sec‘ c =7 gm/dl c=10 gm/dl c=15 gm/dl c=20 gm/dl 2.3 8.4 -- -- 9.56 -- 2.2 10.55 -- -- -- 31.0 2.1 13.26 -- -- -- 31.0 2.0 16.74 -- -- 9.59 30.9 1.9 21.01 0.85 -- -- 29.8 1.8 26.51 0.83 -- -- 29.4 1.7 33.39 0.83 -- 9.52 28.7 1.6 42.02 0.83 -- 9.48 28.1 1.5 52.77 0.83 2.41 9.42 27 5 1.4 66.67 0.85 2.41 9.28 26.4 1.3 84.03 0.83 2.40 9 20 25.4 1.2 105.54 0.84 2.41 8.93 23.9 1.1 132.55 0.85 2.37 8 58 22.4 1.0 167.43 0.84 2.34 8.27 21.0 0.9 210.1 0.85 2.33 7.89 19.4 0.8 265.1 0.85 2.27 -- 17.4 0.7 333.9 0.84 2.15 -- -- 0.6 420.2 -- 2.08 -- -- 0.5 527.7 -- 1.97 -- -- 278 TABLE B.7.-~Weissenberg Rheogoniometer n-i Data for PS-2 in Dioxane at 30°C. Gear Box Shear Rate, Viscosity, n, Poise Setting 9, sec‘1 c = 50 gm/dl 3.6 0.42 4,562 3.5 0.53 4,438 3.4 0.67 4,480 3.3 0.84 4,380 3.2 1.06 4,200 3.1 1.33 4,057 2.9 2.10 3,674 2.8 2.65 3,431 2.7 3.34 3,144 2.6 4.20 2,890 lll‘. 1111.111. 279 TABLE B.8.—-Weissenberg Rheogoniometer n-y Data for SAN C—l in Benzene at 30°C. Viscosity, n, Poise Gear Box Shear Ra e, SEtt'”9 1’ SEC— c = 10 gm/dl c = 20 gm/dl 2.2 10.55 -- 6.80 2.1 13.26 -- 6.85 2.0 16.74 -- 6.80 1.9 21.01 —- 6.60 1.8 26.51 -— 6.30 1.7 33.39 -- 6.05 1.6 42.02 -— 5.84 1.5 52 77 -- 5.65 1.3 84.03 0.49 -— 1.2 105.54 0.49 -- 1.1 132.55 0.50 -- 1.0 167.43 0.50 —- 0.9 210.1 0.50 -- 0.8 265.1 0.50 -- 0.7 333.9 0.49 -- N TABLE B.9.-—Weisse 280 nberg Rheogoniometer n-V Data for SAN C-l in MEK at 30°C Gear Box Shear Réfea Viscosity, n, Poise Setting Y, sec C = 20 gm/dl c = 35 gm/dl 2.7 3.34 -- 21.9 2.6 4.20 -— 21.8 2.5 5.28 —- 21.7 2.4 6.67 -— 21.8 2.3 8.40 -— 21.1 2.2 10.55 —- 20.7 1.6 42.02 2.01 - 1.5 52.77 2.01 - 1.4 66.67 2.04 — 1.3 84.03 1.95 — 1.2 105.54 1.89 - 280 TABLE B.9.-—Weissenberg Rheogoniometer n-t Data for SAN C-l in MEK at 30°C. Gear Box Shear Rgte, Viscosity, n, Poise satt1ng Y’ sec c = 20 gmldl c = 35 gm/dl 2'7 3-34 -- 21.9 2.6 4.20 -- 21.8 2.5 5.28 -- 21.7 2-4 5.57 -- . 21.8 2.3 8.40 -- 21.1 2.2 10.55 -- 20.7 1.6 42.02 2.01 -- 1.5 52.77 2.01 -- 1.4 66.67 2.04 -- 1.3 84.03 1.95 -- 1.2 105.54 1.89 -- 282 TABLE B.11.-—Weissenberg Rheogoniometer n-i Data for SAN C-l in DMF at 30°C. Gear Box Shear Rate, VISCOSIty. n, Poise Setting 9, sec‘ C = 10 gm/dl c = 20 gm/dl C = 50 gm/dl 2.7 3.34 _— __ 235 2.6 4.20 __ __ 233 2.5 5.28 -— -_ 234 2.4 6.67 —— __ 234 2.3 8.4 -- —- 231 2.2 10.55 —— -— 230 2.1 13.26 —- —- 229 2.0 16.74 —- -- 223 1.9 21.01 0.34 -- 220 1.8 26.51 0.34 2.44 216 1.7 33.39 0.34 2.45 212 1.6 42.02 0.34 2.42 208 1.5 52.77 0.34 2.42 201 1.4 66.67 0.34 2.43 192 1.3 84.03 0.33 2.44 —— 1.2 105.54 0.34 2.40 —- 1.1 132.55 0.34 2.39 -— 1 0 167.43 0.35 2.37 -— 0.9 210.1 0.34 2.36 —— 0.8 265.1 0.34 2.30 -- 0.7 333.9 0.34 2.23 —- 6 420.2 -- 2.18 282 TABLE B.11.--Weissenberg Rheogoniometer n-i Data for SAN C-l in DMF at 30°C. Gear Box Shear Ra e, VISCOSItY. n. Poise SEtt'"9 1' sec' c = 10 gm/dl c = 20 gm/dl c = 50 gm/dl 2.7 3.34 -- -- 235 2.6 4.20 -- -- 233 2.5 5.28 -- -- 234 2.4 6.67 -- -- 234 2.3 8.4 -- -- 231 2.2 10.55 -- -- 230 2.1 13.26 -- -- 229 2.0 16.74 -- -- 223 1.9 21.01 0.34 -- 220 1.8 26.51 0.34 2.44 216 1.7 33.39 0.34 2.45 212 1.6 42.02 0.34 2.42 208 1.5 52 77 0.34 2.42 201 1.4 66.67 0.34 2.43 192 1.3 84.03 0.33 2.44 -- 1.2 105.54 0.34 2.40 -- 1.1 132.55 0.34 2.39 -- 1.0 167.43 0.35 2.37 -- 0.9 210.1 0.34 2.36 -- 0.8 265.1 0.34 2.30 -- 0.7 333.9 0.34 2.23 -- 0.6 420.2 -- 2.18 -- 283 TABLE B.12.--Weissenberg Rheogoniometer 0-1 Data for SAN C-2 in Benzene at 30°C. Viscosity, n, Poise Ggar Box Spear Rgte, ett'"9 Y: 59° c = 10 gm/dl c = 20 gm/dl 1.5 52.77 -- 11.9 1.4 66.67 -- 11.9 1.3 84.03 -- 11.8 1.2 105.54 0.716 11.9 1.1 132.55 0.715 11.9 1.0 167.43 0.719 11.8 0.9 210.1 0.711 11.7 0.8 265.1 0.717 11.4 0.7 333.9 0.720 -- 0.6 420.2 0.711 -- 0.5 527.7 0.690 -- 0.4 666.7 0.670 -- 284 TABLE B.13.--Weissenberg Rheogoniometer n-v Data for SAN C-2 in MEK at 30°C. Gear Box Shear Rate, Viscosity, n, Poise Setting 9, sec‘ c = 20 gm/dl 1.7 33.9 5.09 1.6 42.02 5.11 1.5 52.77 5.10 1.4 66.67 5.03 1.3 84.03 5.08 1.2 105.54 5.08 1.1 132.55 4.99 1.0 167.43 4.97 0,9 210 1 4-77 0,3 265.1 4-59 0,7 333.9 4-20 0.6 420.2 3-83 285 TABLE B.14.--Weissenberg Rheogoniometer n-t Data for SAN C-2 in Dioxane at 30°C. Gear Box Shear Rate, V'SCOS'ty’ n, P0159 setting Y’ sec- c = 7 gm/dl c = 10 gm/dl c = 20 gm/dl 1.7 33.39 -- 0.825 8.40 1.6 42.02 -- 0.820 8.40 1.5 52 77 0.331 0.827 8.50 1.4 66.67 0.331 0.820 8.38 1.3 84.03 0.326 0.813 8.33 1.2 105.54 0.323 0.833 8.23 1.1 132 55 0.332 0.835 8.16 1.0 167.43 0 323 0.830 8.05 0.9 210.1 0 331 -- 7.91 0.8 265.1 0.329 -— 7.82 0.7 333 9 0.337 0 835 7.43 0.6 420.2 0.325 -— 7.12 0.5 527.7 0.326 —- 6.80 0 4 666.7 0 329 —- -— 286 TABLE B.15.--Weissenberg Rheogoniometer n-i Data for SAN C-2 in DMF at 30°C. Gear BOX Shear Rate, Viscosity, n, Poise setting Y’ sec- c = 20 gm/dl c = 50 gm/dl 3.2 1.06 -- 110 3.0 1.67 -- 1]] 2.9 2.10 -- 110 2-8 2.65 —- 110 2.7 3.34 -- 1]] 2.6 4.20 -- 1]] 2.5 5.28 -- 1]] 2.4 6.67 -- 110 2-3 8.40 -- 1]] 2.2 10.55 -— 110 2.1 13.26 -- 1]] 2-0 16.74 2.94 111 1.9 21.01 2.95 110 1.8 26.51 2.94 109 1.7 33.39 2.92 109 1.6 42.02 2.95 108 1.5 52.77 2.94 105 1.4 66.67 2.94 103 1.3 84.03 2.93 97,7 1.2 105.54 2.94 92 1.1 132.55 2,91 __ 1.0 167.43 2.93 -- 0.9 210.1 2.95 -- 0.8 265.1 2.85 -- 0.7 333.9 2.76 -- 0.6 420.2 2.61 -- 287 TABLE B.16.-~Weissenberg Rheogoniometer n-i Data for SAN C-2‘ in Benzene at 30°C. Viscosity, n, Poise Gear Box Shear Rate, Setting Y, SEC C = 7 gm/dl C = 10 gm/dl C = 20 gm/dl 3.7 0.33 —- -- 1,580 3.6 0.42 —- -- 1,582 3.5 0.53 -- -- 1,530 3.4 0.67 —- -- 1,502 3.3 0.84 -— —— 1,434 3.2 1.06 -- -- 1,394 3.0 1.67 -- -- 1,323 2.9 2.10 -- —- 1,234 2.8 2.65 -- —— 1,150 2.7 3.34 -- —- 1,080 2.6 4.20 —- —- 981 2.4 6.67 -- —- 798 2.3 8.40 6.69 -— 749 2.2 10.55 6.65 -- 680 2.1 13.26 6.85 -- 609 2.0 16 74 6.69 —- 530 1.9 21.01 -— -- -- 1.8 26.51 6.62 28.47 -— 1.7 33.39 6.54 28.17 -- 1.6 42.02 6.40 26.69 -- 1.5 52.77 6.28 25.34 -— 1.4 66.67 6.10 23.94 -- 1.3 84.03 5.89 22.64 -- 1.2 105.54 5.71 20.89 -- 1.1 132.55 5.55 19 40 -- 1.0 167.43 5.30 -- 0.9 210.1 -- -- -- 0.8 265.1 -- -- '- 0.7 333.9 -- -- ‘- 0.6 420.2 -- -- “- 288 TABLE B.17.--Weissenberg Rheogoniometer n-i Data for SAN C-2' in MEK at 30°C. Gear Box Shear Rate, V‘SCPSIPY9 0’ P0159 59tt'"9 7’ sec c = 7 gm/dl c = 10 gm/dl c = 20 gm/dl 3.8 0.265 -- -- 399 3.7 0.33 -- -- 397 3.6 0.42 -- -- 399 3.5 0.53 -- -- 393 3.4 0.67 -- -- 397 3.3 0.84 -- -- 389 3.2 1.06 -- -- 378 3.1 1.33 -- -- 369 3.0 1.67 -- -- 347 2.9 2.10 -- -- 328 2.3 8.40 -- -- 226 2.2 10.55 -- 8.64 203 2.1 13.26 -- 8.70 188 2.0 16.74 -- 8.62 171 1.9 21.01 -- 8.55 161 1.8 26.51 -- 8.45 152 1.7 33.39 1.78 8.23 140 1.6 42.02 1.73 8.08 132 1.5 52.77 1.76 7.80 122 1.4 66.67 1.77 -- 110 1.3 84.03 1.76 -- -- 1.2 105.54 1.74 -- -- 1.1 132.55 1.76 -- -- 1.0 167.43 1.70 -- -- 0.9 210.1 1.70 -- -- 0.8 265.1 1.70 -- -- 0.7 333.9 1.68 -- ~- 0.6 420.2 1.63 -- -- 0.5 527.7 1.59 -- -- 0.4 666.7 1.45 -- -- 0.3 840.3 1.36 -- -- 0.2 1,055.4 1.31 -- -- 289 TABLE B.18.--Weissenberg Rheogoniometer n-i Data for SAN C-2' in Dioxane at 30°C. Viscosity, n, Poise Gear Box Shear Rate, satt‘"9 Y: see c = 7 gm/dl c = 10 gm/dl c = 20 gm/dl 4.1 0.133 -- -- 806 3.4 0.67 -- -- 802 3.3 0.84 -- -- 810 3.2 1.06 -- -- 802 3.1 1.33 -- -- 805 3.0 1.67 -- -- 792 2.9 2.10 -- -- 780 2.8 2.65 -- -- 755 2.7 3.34 -- -- 733 2.6 4.20 -- -- 705 2.5 5.28 -- 25.8 671 2.4 6.67 5.87 -- 634 2.3 8.40 5.67 26.0 578 2.2 10.55 5.84 26 1 523 2.1 13.26 5.89 25.8 450 2.0 16.74 5.79 24.9 394 1.9 21.01 5.82 24 6 348 1.8 26.51 5.89 24 2 306 1.7 33.39 5.79 23 5 261 1.6 42.02 5.74 23 1 -- 1.5 52.77 5.68 22 4 -- 1.4 66.67 5.54 21 1 -- 1.3 84.03 5.42 20 0 -- 1.2 105.54 5.32 19 1 -- 1.1 132.55 5.12 17.3 -- 1.0 167 43 5.04 -- -- 0.9 210.1 4.80 -- -- 0.8 265.1 4.47 -- -- 0.7 333.9 4.18 -- -- 0.6 420.2 3.79 -- -- 0.5 527.7 3.40 -- -- 290 TABLE B.19.--Weissenberg Rheogoniometer n-i Data for SAN C-2' in DMF at 30°C. Gear Box Shear Rate, Visc051ty, 0’ P0159 sett‘"9 13 sec c = 7 gm/dl c = 10 gm/dl c = 20 gm/dl 3.6 0.42 -- -- 172 3.5 0.53 -- -- 172 3.4 0.67 -- -- 173 3.3 0.84 -- -- 170 3.2 1.06 -- -- 172 3.1 1.33 -- -- 172 3.0 1.67 -- -- 174 2.9 2.10 -- -- 171 2.8 2.65 -- -- 171 2.7 3.34 -- -- 169 2.6 4.20 -- -- 168 2.5 5.28 -- 10.0 169 2.4 6.67 -- -- 164 2.3 8.40 -- 9.86 161 2.2 10.55 -- -- 156 2.1 13.26 -- 10.1 153 2.0 16.74 -- 9.95 144 1.9 21 01 -- -- 139 1.8 26 51 2.68 -- 131 1.7 33 39 2.69 9.79 122 1.6 42 02 2.72 9 73 116 1.5 52 77 2.66 9 50 106 1.4 66 67 2.68 9.40 96 6 1.3 84 03 2.64 9.03 -- 1.2 , 105 54 2.59 -- -- 1.1 132.55 2.55 8.62 _- 1.0 167.43 2.53 8.25 -- 0.9 210.1 2.46 7.82 -- 0.8 265.1 -- 7.40 -- 0.7 333.9 2.32 -- -- 0.6 420.2 2.20 -- -- 0.5 527.7 2.12 -- -- 0.4 666.7 1.96 -- -- 0.3 840.3 1.80 -- -- 291 TABLE B.20.--Weissenberg Rheogoniometer n-i Data for SAN C-3 in MEK at 30°C. Gear Box Shear Réie’ Viscosity, n, Poise 59tt1"9 Y: sec c = 20 gm/dl c = 35 gm/dl 3.4 0.67 -- 3,500 3.3 0.84 -- 3,533 3.2 1.06 -- 3,435 3.1 1.33 -- 3,350 3.0 1.67 -- 3,225 2.9 2.10 -- 3,150 2.8 2.65 -- 2,950 2.7 3.34 -- 2,796 2.6 4.20 -- 2,654 2.5 5.28 103 2,456 2.4 6.67 104 2,071 2.3 8.40 106 1,847 2.2 10.55 105 1,628 2.1 13.26 105 -- 2.0 16 74 106 -- 1.9 21 01 106 -- 1.8 26.51 100 -- 1.7 33.39 94 -- 1.6 42.02 86 -- 292 TABLE B.21.--Weissenberg Rheogoniometer n-§ Data for SAN c-3 in DMF at 30°C. Ggar Box Shear Riie’ Viscosity, n, Poise Ett'"9 7’ sec c = 10 gm/dl c = 20 gm/dl 2.3 8.40 -- 52.5 2.2 10.55 -- 52.5 2.1 13.26 -- 52.8 2.0 16.74 -- 52.4 1.9 21.01 -- 51.9 1.8 26.51 -- 51.1 1.7 33.39 -- 51.0 1.6 42.02 3.38 50.0 1.5 52.77 3.37 47.9 1.4 66.67 3.40 46.9 1.3 84.03 3.38 45.4 1.2 105.54 3.33 44.3 1.1 132.55 3.34 42.3 1.0 167.43 3.33 40.2 0.9 210.1 3.32 38.0 0.8 265.1 3.21 -- 0.7 333.9 3.16 -- 0.6 420.2 3.05 -- 0.5 527.7 2.91 -- 0.4 666.7 2.81 -- 293 TABLE B.22.--Weissenberg Rheogoniometer n-i Data for SAN C-3 in DMF at 30°C. Gear Box Shear Réfea Viscosity, n, Poise SEtt1ng Y’ sec 6 = 35 gmldl c = 50 gm/dl 4.0 0.167 -- 4,990 3.9 0.21 -- 5,010 3.8 0.265 -- 4,975 3.7 0.33 -- 4,978 3.6 0.42 -- 5,003 3'5 0-53 -- 4,937 3.4 0.67 -- 4,851 3.3 0.84 -- 4,811 3.2 1.06 769 4,757 3.1 1.33 779 4,755 3.0 1.67 770 4,555 2.9 2.10 755 4,490 2.8 2.65 758 4,355 2.7 3.34 740 4,144 2.6 4.20 735 3,934 2.5 5.28 723 3,728 2.4 6.67 718 3,409 2.3 8.40 704 3,120 2.2 10.55 535 __ 2.1 13.26 553 __ 2.0 16.74 517 __ 1.9 21.01 588 -_ 1.8 26.51 535 __ APPENDIX C CALIBRATION OF REFRACTOMETER 294 APPENDIX C CALIBRATION 0F REFRACTOMETER The Brice-Phoenix refractometer was calibrated according to the procedure recommended in the manual (B-13). Table C-l lists the refractive index differences, An, between potassium chloride solutions and distilled water (B-13). For all calibration purposes, potassium chloride solutions were used. Table C-l is taken from Ref. (B-13). TABLE C-l. Refractive Index Differences, An, Between Potassium Chloride Solutions and Distilled Water. Concentration in Water 6 0 Solution An x 10 at 25 C (1) (2) and 4358 A gm/lOO ml gm/lOO gm 1 0.0696 0.0699 100 2 0.1067 0.1070 153 3 0.2799 0.2812 399 4 0.5964 0.5994 845 5 1.0794 1.0869 1,521 6 1.4911 1.5037 2.093 7 2.9821 3.0250 4,135 8 3.9969 4.0703 5,500 9 4.4732 4.5647 6,136 10 5.9642 6.1217 8,105 11 6.4680 6.6526 8.763 Concentration: (1) gm of salt/100 ml of distilled water at 25°C. (2) gm of salt/100 gm of distilled water. 295 296 When concentration, c, is plotted against refractive index difference, An, a straight is obtained whose equation is c = 732.4379 An where c is in gm/lOO cm3. From this equation, values of An for other concentrations can be calculated. Table C-2 presents the results of calibration of the dif- ferential refractometer cell using potassium chloride solutions at 25° 5 1°C. and 4358 3. TABLE C.2.--Ca1ibration Constant of Differential Refractometer. 9m7130e25565522r An x 106 Ad k = An/Ad x 103 1.113 1.519.583 1.4828 1.0248 0.5518 753.374 0.7359 1.0237 0.3223 440.037 0.4298 1.0239 0.1004 137.076 0.1341 1.0221 Average k = 1.0236. APPENDIX D RAYLEIGH RATIO FROM LIGHT SCATTERING DATA AND PHOTOMETER CONSTANTS 297 APPENDIX D RAYLEIGH RATIO FROM LIGHT SCATTERING DATA AND PHOTOMETER CONSTANTS For the measurements at different angles, cylindrical cell C-101 was used with narrow diaphragms as described in the manual B-10). The scattering ratios, (Ge/Fe)/(Go/F0)’ at various angles, 9, both for the solutions and the solvents were measured to obtain net scattering due to the presence of polymer. The Rayleigh ratio, Re, can then be calculated from the observed scattering ratios by means of the following equation which is given in the manual: 2 R = 709" (Rw/Rc) [1;] sin e l 1 9 “’49 "h " l+c0520J (l - R)2(1 - 4R2) . 4 [Ge/Fe] [Ge/Fe] G /F " G /F 0 0 solution 0 0 solvent GlBO-B/FlBO-e GO/F [6180-0/F18O-01 ._ 2R G 0/F 0 solution [ 0 ]solvent (0.1) 298 299 where (Ge/Fe) is the scattering ratio, or average observed ratio of galvanometer deflection for the light scattered by the solution at angle 9 to that of the transmitted light at zero angle position; Fe and F0 are the [woducts of transmittances of the neutral fil- ters used in determining the scattering ratio at angles 0 and zero, respectively; a is the constant that relates the working standard to the Opal glass reference standard; T0 is the experimentally determined product of the diffuse transmittance of the opal glass reference standard; h is the width of the diaphragm; n is the refractive index of the solution which for dilute solutions can be replaced by the refractive index of the solvent; Rw/Rc is an experimentally determined correction factor for incomplete compen- sation for reflection effects. The latter correction is not large and does not differ appreciably from instrument to instrument; however, its value does depend on the refractive index of the solvent and cell size. Average values of Rw/Rc for the wave lengths of 436 mu and 546 mu for 40 x 40 mm and 30 x 30 mm cells are given in the manual along with n values of some comnon solvents. Values for other solvents, or for more concentrated solutions with refractive index differing appreciably from that of the solvent, can be estimated with sufficient accuracy from a plot of Rw/Rc against n, or by simple interpolation in the given values of Rw/Rc' In general, for dilute solutions in common solvents, Rw/Rc values are very close to unity. 300 The factor (r/r') is the calibration relating the narrow beam geometry and cylindrical cell to the standard beam geometry and standard cell. It is dependent on the refractive index of the solution and hence must be determined for each solute-solvent system. This correction is quite large in comparison with Rw/Rc° Complete details of its determination are given in the manual. The factor sin 9 corrects for the volume change on view- ing the solution at different angles, (l-tcosze) accounts for the state of polarization of the scattered light, and the factor R is defined as (0.2) SO 11 r—-\ 31 3| I ..a-a H where H is the refractive index of the glass. For A equal to 436 mu, the value of R is equal 'Uo 0.046 for the sinter-fused cells and equal to 0.039 for Pyrex cells. Equation 0.1 takes into account the change of the scatter- ing envelope due to the scattering of the reflected fraction, R, of the primary beam, the attenuation of scattered light at an angle 9 by reflection at the air-glass interface at the point of measurement, and the contribution of reflection, in the -0 direction, of the light scattered in the +(180-6) direction. Table 0.1 lists the calibration constants of the light scattering photometer used in this work. 301 TABLE D.1.--Constants of Photometer fer 436 mp Wavelength. Diffuse transmittance times TD 0.263 diffusor correction factor Width of primary beam, cm ' h 1.20 Working standard constant a 0.0423 Transmittance of neutral filter No. 1 F1 0.477 No. 2 F2 0.219 No. 3 F3 0.109 No. 4 F4 0.0349 A computer program in FORTRAN IV was written to carry out the calculations of Rayleigh ratios and Zimm plots were made by plotting Kc/R6 against qc + sin2 (0/2) where the terms are defined in Chapter III.