II IIIIIIIIIIIII IIIIIIIIIII IIII IIII ‘ I I 3 1293 010968929 RETURNING MATERIALS: IV1531_] Place in book drop to remove this checkout from 4::::;:EE:_ your record. FINES will be charged if book is returned after the date 3 stamped below. ”Etna-g '9 21‘:- .x 1 33939032000 OCT106 0 100291 EFFECT OF MOLECULAR ASSOCIATION ON OSMOTIC PRESSURE AND DIFFUSION IN DILUTE POLYMER SOLUTIONS By Kit Leung Yam A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1985 ABSTRACT EFFECT OF MOLECULAR ASSOCIATION ON OSMOTIC PRESSURE AND DIFFUSION IN DILUTE POLYMER SOLUTIONS By Kit Leung Yam Polymer molecules in solution often associate with one another via secondary binding forces (such as hydrogen bonds) to form larger polymer molecules. Examples may be found in many biopolymers and synthetic polymers consisting of proton—donating and proton—accepting pairs. However the influence of these associated complexes on solution properties is still not well understood, for at least two reasons. First, existing theories are incapable of describing the behavior of these systems and, second, very few relevant data are available in the literature. In this work, expressions for the osmotic pressure and the diffusion coefficient are derived for open associating systems. The associating systems are treated as pseudo-binary systems--the two components being solvent molecules and polymer molecules with number average molecular weight of the multimer mixture. J; The osmotic pressure n is expressed as Mlfl and the diffusion coefficient D*is expressed as * * D “Do{‘l’2+[2A2 Ml/xpl-(ks+2VpO)\112]p+... where p and M1 are the mass concentration and the unimer molecular weight of polymer, respectively. A2* is the second virial coefficient of the multimer mixture. W1 and 4% (both dimensionless groups) are functions ofla, M1 and the association equilibrium constant K. The partial specific volume of polymer at infinite dilution, vpo’ can be obtained from density experiments, and the friction parameter, ks, can be estimated from the Pyun-Fixman theory. * The model predicts that the initial slope of N /pRT versus p is * 2 (A2 )obs = A2 ' Km1 which differs from its nonassociating counterpart A2 by * K/Mlz. Similarly, the initial slope of D versus p is (k k - K/M1 * d )obs = d which differs from its nonassociating counterpart kd by K/Ml‘ Predictions from these expressions agree well with osmometry data obtained from the literature and with diffusivity data measured in this laboratory, for polyethylene glycol in benzene. The effect of association is most prominent at low concentrations and increases progressively with decreasing molecular weight. Its magnitude is governed by the dimensionless group Kp/Ml, ACKNOWLEDGMENTS The completion of this dissertation would have been impossible without the constructive suggestions of my major professor, Dr. Donald K. Anderson. More than this, I thank him for helping me to develop the skills and confidence to meet the challenge of the future. I am also indebted to my other committee members, Dr. Eric A. Grulke, Dr. Krishnamurthy Jayaraman, and Dr. Frederick Horne for their support and encouragement throughout the course of this work. ii FOREWORD This dissertation consists of two parts. Part One provides a summary for the essential findings of this work. It was prepared in the form for publication. Part Two provides a more detailed description of the work. iii TABLE OF CONTENTS Page FOREWORD O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0 iii LIST OF TABLES O O O O O O I O O O O O O O O O O O O O O O O O O O O O O O O O O O O I O O C O O Vii LIST OF FIGURES 0.000......0..0.00.00.00.000000000000000 ix PART ONE ABSTRACT ......................................... 2 I. INTRODUCTION ..................................... 2 II. MOLECULAR ASSOCIATION ............................ 4 III. MODEL DEVELOPMENT ................................ 8 IV. EXPERIMENTAL ..................................... 12 V. RESULT AND DISCUSSION ............................ 12 VI. CONCLUSION ....................................... 23 VII. REFERENCES 0.00.00...0.00.0.0...OOIOIOOOOOOOOIOOOO 24 PART TWO I. INTRODUCTION 0.0.0.... 0....OOOOOOOOOOOOOOOOOOOOOOOO 27 II 0 MOLECULAR ASSOCIATION O I O O O O O O O O O O O O I O O O O O O O O O O O O O O 32 A. General Background ............................ 32 iv B. Type of Association ........................... 34 1. Open Association ........................... 35 2. Dimerization 0......OOOOOOOOOOOOOOOOOOOOOOOO 38 III. EXPERIMENTAL METHOD .............................. 42 A. Polymer Samples ............................... 42 B. Osmometry Data ................................ 43 C. Diffusivity Data .............................. 45 IV. THERMODYNAMICS OF ASSOCIATING POLYMER SOLUTIONS .. 48 A. Theoretical Background ........................ 48 1. Flory-Huggins Theory ....................... 48 2. Two Parameter Theory ........ ...... ......... 50 3 Second Virial Coefficient .................. 53 B. Expression for Osmotic Pressure for Associating P01ymer SOlutionS OOOOOOOOOOOIOOOOOOOOOCOO0.... 57 C. Presentation of Osmometry Data and Discussions 60 1. Estimation OfK0..OOOOOOOOOOOOOOOOOOOOOOOU 60 2. Estimation Of A:00.0.0000...OOOOOOOOOOOOOO 62 3. Other Discussions ......................... 67 'V. DIFFUSION IN ASSOCIATING POLYMER SOLUTIONS ....... 70 A. Theoretical Background ........................ 70 1. KirkWOOd-Riseman TheorYOOIOOOOOOOOOOOOOOOOOO 71 2. Modified Pyun-Fixman Theory................. 73 B. Expression for Diffusion Coefficient for Associating Polymer Solutions ................. 76 C. Presentation of PEG Diffusivity Data and DiSCUSSionS 0.0......OOOOOOOOOOOIOIOOOO0.00.... 77 1. Estimation Of Do 00.0.00...OOOOOOOOOOOOOOOOO 77 2. Estimation Of ks .OOOOOOOOOOOOOOOCOCOOOOOOOO 79 3. Other Discussions ...... ............... ..... 81 D. Presentation of PTHF Diffusivity Data and Discussions 1. 2. Effect of solvent ......................... Effect of temperature ..................... 3. Molecular Dimensions ...................... 4. "Old" Polymer Samples ..................... VI. CONCLUSIONS AND RECOMMENDATIONS .................. NOMENCLATURE APPENDICES.... A. sample caICUIati-On OO...OOOOOOOOOOOOOOOOOIOOOOO B. Diffusivity Data for PEG in Benzene ........... C. Diffusivity Data for PTHF in Various Solvents . BIBLIOGRAPHY vi 81 84 90 95 96 100 104 106 110 113 116 LIST OF TABLES PART ONE Table Page I. Values of A2* estimated from the osmometry data Of Elias OO......OOOOOOOC......OOOOCOOOOOOOOOOO 13 II. Comparison of Do from the modified Kirkwood—Riseman theory with DC from experiment for PEG in benzene at 25 OC .................... 21 III. Values of A2* used in calculating the concentration dependence of diffusion coefficient in Figure 2 ......... . ...... ... ..... 22 PART TWO Table Page 3-1 Polydispersity of the PEG samples .............. 42 3-2. Comparison of the diffusivity data determined in this work with the data of Gosting and Morris .. 46 5-1. Comparison of the diffusion coefficient predicted from the modified Kirkwood—Riseman equation with the experimental data for PEG in benzene at 25 0C ......OOOOOIOOOOOOOOOO000...... 78 5—2. Characteristics of "fresh" PTHF samples ........ 82 5—3. Characteristics of "old" PTHF samples (before degradation) 00...... ..... ......OIOOOIOOCOO ..... 83 5-4. Characteristics of solvents (MEK, DE, BOH, EA and BB) ........ ...... ..... ..... . ..... .......... 84 vii 5-5. Parameter estimations for the diffusivity data of "fresh" PTHF samples at 25 0C using the form D=GM--d 0.0.000.000.0000......OOOOOOOOOOOOOOOO 88 viii LIST OF FIGURES PART ONE Figure 1. Concentration dependence of H for PEG in benzene at 25 CO.......OOOOOOOOOO......IOOOOOOOOOOOOOO 2. Concentration dependence of diffusion coefficient for PEG in benzene at 25 0C ........ 3. Plot of Do versus M1 for PEG in benzene at 25 oC PART TWO Figure 2-1. ‘I’landq’l' VEISUSP ..... 00.0900. 000000 0000000000 2-2. Distribution of i-mer for open associating systems ............OOOOOOO......OOOOOOOOOOOOOOO 4-1. The excluded volume for two spheres in contact 0.0.0.0.........OOOOOOOOOOOOO0.000...... * 4.2. (A2 )ObS versus l/Ml coo.ooooooooooooooooooooooo 4—3. Molecular weight dependence of the second virial coefficient for PEG in benzene, predicted by the two-parameter theory ..........OOOOOOCOCCOOOOOOO 4-4. Values of A2 (from Table I in Part One) versus M 4-5. Comparision between the association term with the virial term in Equation (15) ............... Page 15 19 20 Page 39 4O 55 61 64 66 68 ks versus M for PEG in benzene at 25 °C, predicted by the Pyun-Fixman theory 000000000000 Dn/T versus M for the "fresh" PTHF samples in MEK, DE, BOH, EA and BB at 25 °C ............... Dn/T versus M for the "fresh" PTHF samples at 25 0C .....IOOOOOOOCCOCOOO......OOOOOOOOOOOOOOO. Comparison of Dn/T versus M for the "fresh" PTHF samples at 25 C and 35 0C Dn/g versus M for the "old" PTHF samples at 25 C......OOOOOOOIOOOOOO......OOOOOOIOOOOOIOOO Sample calculation for the diffusion coefficient of PEG in benzene ...... . .............. . ..... ... 80 87 89 92 95 PART ONE Abstract Expressions for the concentration dependence of osmotic pressure and diffusion coefficient in dilute polymer solutions are derived in this work for open associating systems. Predictions from these expres- sions are in good agreement with the experimental data of polyethylene glycol in benzene obtained from our laboratory and from the liter- ature. The effect of association increases with decreasing molecular weight and is governed by the dimensionless group P - Kp/Ml. 1. INTRODUCTION The objective of this work is to study the effect of inter- molecular association on osmotic pressure and diffusion coefficient of dilute polymer solutions. Polymer molecules in solution often associate with one another to form larger molecules under favorable conditions.192 The resulting associated complexes may be classified, based on the nature of inter- molecular forces, into hydrogen-bonding complexes, polyelectrolyte complexes, stereocomplexes, and charge-transfer complexes.3 The extent of association is affected by the mechanism of association, the chemical structures of polymer and solvent, solute concentration, temperature, and pressure. The behavior of associating polymer solutions is still not well understood, for at least two reasons. First, there is a lack of sufficient and reliable experimental data available in the literature. Second, existing theories, such as the Flory-Huggins and the two- parameter theories," are incapable of describing the behavior of these systems; their applications are valid only for nonassociating, nonelectrolyte systems. Yet there exist many biopolymers and synthetic polymers which form associated complexes in solution,3 and the formation of these complexes can greatly influence the solution properties. For example, osmotic pressure, an important colligative property, may be strongly affected by molecular association. A plot of reduced osmotic pressure versus polymer concentration generally displays a linear relation. In fact this linear relationship is the basis for molecular weight determination.5 However, if the polymer associates, this linear relationship is no longer valid (especially in the very dilute concentration range) and molecular weight determination based on linear extrapolation may be in serious error.6 Other solution prop- erties such as viscosity and diffusion coefficient are also influ- enced. Hence, there is a need for a better understanding of these systems. 4 In this paper we formulate theoretical expressions for the concentration dependence of osmotic pressure and diffusion coef- ficient for associating polymer-solvent systems. Since these ex- pressions depend on the type of association, we limit our study to the systems which obey the so-called "open association" model.2 However, the procedures described below may also be applied to other types. First, we derive the concentration dependence of osmotic pressure for open association. This expression describes the behavior of a pseudo-binary system--the two components being the solvent molecules, and the polymer molecules with number average molecular weight of the multimer mixture. Second, the osmotic pressure expression is used to derive an expression for diffusion coefficient. To test the validity of the model we compare theoretical pre- dictions of osmotic pressure and diffusion coefficient with experi- mental data. Polyethylene glycol (PEG) in benzene is used. Mutual diffusion coefficients for this system, with molecular weight ranges from 440 to 12600, were measured in our laboratory. II. MOLECULAR ASSOCIATION When considering colligative properties and diffusion, as- sociating polymer-solvent systems are more difficult to study than nonassociating systems. Unlike nonassociating systems, they cannot always be considered to be binary but must be treated as multicom- ponent systems which consist of unimers, dimers, trimers, etc., with molecular weight distributions that change with concentration. In order to construct a model to describe their behavior, a prior knowledge of the mechanism of association must be assumed and its validity tested with experimental data. For simplicity, we restrict our study to polymer-polymer as- sociation in inert solvents, although association can also occur between polymer-solvent and solvent-solvent molecules. We further restrict it to open association because this model is obeyed by many synthetic polymers and is simple to construct.21 Open association is a consecutive association in which suc- cessively higher multimers (dimers, trimers, etc.) are formed one step at a time: .9 B1 +B1 «- B2 , K--1<2 _) B1+B2 (_ B3 , K-K3 B +B I B K-K 1 1-1 <- 1 ’ i ...) Bl + Bn_1 +- Bn , K - Kn (1) where Bi represents i-mer, Ki represents the association equilibrium constant for the formation of i-mer and n takes all positive integers up to infinity. If we assume Ki is independent of molecular size (for example, in the case of end-group association), and thus K1 I K2 I . . . I K, we obtain i-l i ci - K c1 (2) and KC1 I C2/C1 I C3/C2 I . . . etc. (3) where Ci is the molar concentration of i-mer. KCl is a dimensionless group whose value ranges from O to 1: KCl I 0 corresponds to no association (C2 I C3 I C4 I . . . I O), and KC] I 1 corresponds to the maximum allowable association when all multimers have the same concentration (C1 I C2 I C3 . . ., etc.). With the above description of the model and the relation M1 I 1M1, some useful relations can be derived:6 Y1{p} I MllMa . 1 + 5 (4) Kcl - 1 - 11 (5) c /c - (1 - KC )(KC )1"1 (6) i p l 1 M [M I 1 + KC (7) w n 1 where E I l 1 + 4P (8) P a KD/M1 (9) Ha I apparent polymer molecular weight I Cp/p (10) Mw/Mn I polydispersity of the multimer mixture (11) Cp l true molar concentration of polymer solute - )3 C1 (12) Equations (4) through (7) are expressed in terms of measurable quantities: the polymer mass concentration 0, the association constant K, and the unimer molecular weight M1. The apparent molec- ular weight Ma, is equivalent to the number average molecular weight of multimer mixture and can be readily calculated from Equation (4) once M1, K and p are known. It is noteworthy to point out the similarities between open as- sociation and stepwise polymerization. In fact, Equations (4) through (7) can be obtained from the appropriate expressions for stepwise polymerization, by substituting K01 for the fraction of conversion.7 The molecular distribution can be identified with the Schultz-21mm distribution.8 Since 0 5 KCl S 1 and according to equation (7), the polydispersity is bounded between 1 S Mw/Mn S 2. III. MODEL DEVELOPMENT A. Osmotic Pressure As mentioned in Section II, open associating systems are simply heterogeneous systems with molecular weight distribution governed by Equation (4). In the following, the osmotic pressure expression of heterogeneous systems is tailored for describing these systems. In a dilute solution containing heterogeneous polymer molecules, the osmotic pressure I may be expressed as:4 w 1 pRT M + A20 + . . . (13) and the second virial coefficient A2 is given by A2 I 2 wiwJ A11 (14) where Mn is the number-average molecular weight, "i the weight fraction of polymer molecule 1, and Aij the interaction between the pair of polymer molecules 1 and j at infinite dilution. According to the two-parameter theories, Aij is a function of the excluded volume 2 and the ratio of molecular weights of molecules 1 and 1.9910 To adapt Equation (13) for associating systems, ”a is substituted for Mn, and A2* for A2:296 M I pRT * M p + . . . (15) - “(M + A2 1 is defined for convenience. The superscript * denotes open as- sociation. Y1 is a function of 9. A2* is a function of association. Recently Tanaka and Solc8 have suggested that it may be approximated by the second virial coefficient of a monodisperse polymer, with number-average molecular weight of the multimer mixture. Furthermore, if the Mark-Houwink relationship (A2 I K’ M-“) is assumed,11’12 we can incorporate the molecular weight dependence of A2* into Equation (15). However, this leads to a rather complicated diffusivity expression. Moreover, our calculations show that the osmometry data used in this work are relatively insensitive to the molecular weight dependence of A2*. Hence, an A2* independent of concentration is assumed for each molecular weight sample. For K I 0, Ma and A2* reduce to M1 and A2 (the second virial coefficient of unimer), respectively; and Equation (15) reduces to the expression for nonassociating systems. Differentiating Equation (15) with respect to p and evaluating the result at p I O, we obtain the observed second virial coefficient ) I A - -—— (16) for associating systems. Note that (A2*)obs differs from its nonas- sociating counterpart by K/Mlz. 10 B. Diffusion Coefficient The concentration dependence of the mutual diffusion coefficient for a nonassociating dilute polymer solution can be series expansion:13 D I Do (1 + kd p + . . .) where kd I 2 A2 M - ks - 2 V po and V I V (1 + a p + . . .) P P0 f I f (1 + k p + . . .) o 5 expressed by the (17) (18) (19) (20) Here Do is the diffusion coefficient at infinite dilution, Vp the partial specific volume of polymer, and f the friction coefficient. If associating systems are treated as pseudo-binary systems (i.e. use the apparent molecular weight, M8, to represent molecular weight of the multimer mixture), Equation to obtain the chemical potential of the solvent, p8 and the mutual diffusion coefficient may be derived tion:13'1" D - 1 - V p E. Ops N f M "a so ,P the average (15) may be used .4 (21) from the rela- (22) 11 where N8 is the Avogadro's number; Ms Vs and p8 are the molecular weight, the molar volume and the mass concentration of solvent, respectively. Using Equation (15) and Equations (19) through (22), we obtain the mutual diffusion coefficient D* for associating systems as * * D - Do {T2 + [2 A2 MI/Yl- (k8 + 2 vpo) Y2] p + . . . } (23) where 1 + 5 Y2 I 25 (24) Both Y1 and Y2 are functions of concentration and open association. For X I 0, Equation (23) reduces to Equation (17). Differentiating Equation (23) with respect to p and evaluating the result at pIO, we obtain * (k ) - kd — KIM1 (25) d obs * Note that (kd )obs differs from its nonassociating counterpart by K/Ml. isl Inf] intr 12 IV. EXPERIMENTAL Diffusion coefficients were measured in our laboratory using a Mach-Zahnder interferometer.159l6 Monodisperse polyethylene glycol (PEG) polymer samples (Mg/Mn < 1.06) were purchased from Polymer Laboratories, Inc., Massachusetts; these samples were used without further purification. Solution temperature was controlled at 25.0 t 0.1°C. The accuracy of the interferometer was tested by comparing the diffusion coefficients for several aqueous sucrose solutions with those reported by Gosting and Morris;17 average deviation was found to be less that 1%. However, PEG/benzene system has a smaller refractive index difference compared to the sucrose/water system, and the diffusion coefficients reported in this work (some of them represent the average values of two or three runs) are estimated to be within 4%. V. RESULTS AND DISCUSSION A. Osmotic pressure data The vapor pressure osmometry data of Elias18 for PEG in benzene is chosen here for studying the effect of molecular association. Infrared spectroscopic measurements by Langbein19 had shown that intramolecular hydrogen bonds existed between the hydroxyl end-groups l3 and the ether groups in this system; later, Elias found18920 that intermolecular association also existed and the data could be quan- titatively explained by the open association model. The value of K is estimated first from the osmometry data. According to Equation (16), if the dependence of A2 on H1 is weak, a plot of (A2*)obs versus 1/M12 should yield a straight line, with slope equal to -K. In fact the osmometry data of Elias fit such a straight line quite well, and thus the slope provides a good initial value of K. Next, Equation (15) and the method of least squares are used to obtain the final values of K I 11000 cc/mole (which agrees with the value estimated by Elias using a different approachlg) and A2*. The values of A2* are presented in Table I. Table 1. Values of A2* (mole mllgz) estimated from the osmometry data of Elias.18 For comparison, (A2) is the second virial coefficient calculated from two-parameter theories. * M1 A2 (A2) 208 -0.0095 -0.0027 409 -0.0011 0.0011 594 0.0005 0.0018 1518 0.0020 0.0023 6000 0.0022 0.0019 The performance of the model is shown in Figure l. The effect of association is more prominent at low concentrations and increases pro- gressively with decreasing molecular weight. This behavior is well described by the model. 14 Figure 1. Concentration dependence of H for PEG in benzene at 25°C. Data taken from Elias et a1.18 ( D ) Ml=6000; ( V) M1=1518; ( Q ) M1=594; ( O ) M1=4O9; ( A) 111=208. predictions from Equation (15). 1.8- 1.6- 1,4- 1 02- 3‘ 15 (V l‘ + P 0.3- 0.6- . .~ 0.4- 4 0.2- o ‘ ' 0.55 E. Us 16 B. Diffusivity data As shown in Section III, the prediction of D(p) in dilute solutions requires the evaluation of A2*, K, Vpo’ Do and ks' The first two terms have already been estimated from the osmometry data, and Vp0 can be obtained from density experiments. Do and ks can be estimated from the two-parameter scheme suggested by Vrentas and Duda21: DC from the modified Kirkwood-Riseman theory and k8 from the Pyun-Fixman theory (version II),22 because they seem to be the best methods available at the present time. The short-range interferences, A, for PEG/benzene system at 25°C is given by11 (R2> 0.5 -9 I 7.9 x 10 cm (26) AI where (R2)o is the unperturbed mean-square end-to-end distance. The long-range interferences are estimated indirectly from the empirical Mark-Houwink intrinsic viscosity relationship:“’21 8 - 0.063 M 0'64 1 1 (27) [ml-K M V where [n] is the intrinsic viscosity. The values of RV and B are obtained from curve fitting the raw data of Rossi and Cuniberti23 "Sing the method of least squares. In addition, to adapt the two— l7 parameter scheme to associating systems, the apparent molecular weight Us is substituted for the unimer molecular weight M1 in calculating the necessary parameters. Several investigators had shown that the Kirkwood-Riseman theory provided reasonable predictions for (Do)e, the translational diffusion coefficient at infinite dilution under theta condition524v 25’ 26’ 27. Furthermore, Vrentas and Duds extended this theory for predicting Do under nontheta conditions27 (D ) D . o 9 o a 8 0.196 k g 5 (28) a n A M ' S 8 where k is the Boltzmann's constant, "s is the viscosity of solvent, and as is the expansion factor relating the perturbed and the unper- turbed mean-square radii of gyration. Mutual diffusion coefficients for four PEG samples in benzene at 25°C were measured using a Mach Zehnder interferometer. The con- centration dependence is shown in Figure 2. The experimental Do is found by extrapolation using the model and the method of least squares. They are found to be in good agreement with the predictions (see Table II). The molecular weight dependence of Do is shown in Figure 3 along with a least squares fit to the relation 4 -0.57 D - G M1 - 2.64 x 10' M (29) Figure 2. 18 Concentration dependence of diffusion coefficient for PEG in benzene at 25 OC. Experimental data: (V) M1=44o; (El) M1=960; (o) M1=4250; (A) M1=126OO. predictions from the association model (Equation (23)); ------- predictions from the nonassociation model (Equation (17)). l9 O l D x 106 (cm23'1) I I I I 0 0.005 0.010 0.015 0.020 P (9 ml") 20 Do (cmzs'1) 10 | I 102 103 104 M1 Figure 3. Plot of Do versus M1 for PEG in benzene at 25 oC. 21 Flory suggested the exponents 7 and B may be related by28 7-(8+1)/3 (30) Thus 7 may be calculated to be 0.55, using 8 I 0.64 from Equation (27), and this value agrees reasonably well with 7 I 0.57 from Equation (29). However, larger differences are observed for some systemsll,29 and more experimental data are needed to determine the validity of this relation. Table II. Comparison of Do (cmZ/sec) from the modified Kirkwood- Riseman theory (Equation (28)) with DC from eXperiment for PEG in benzene at 25°C. M1 Mw/Mn Do(theory) Do(exptl) Do(exptl)/ Do(theory) 440 < 1.09 8.31 x 10'"6 8.58 x 10"6 1.03 960 < 1.06 5.30 x 10-6 5.05 x 10‘6 0.95 4250 < 1.03 2.31 x 10-6 2.20 x 10"6 0.95 12600 < 1.04 1.28 x 10‘6 1.25 x 10'6 0.98 The performance of the association model is shown in Figure 2. K I 11000 cc/mole is used, and the values of A2* are taken from the osmometry data (see Table III). The only adjustable parameter used to fit the data is A of the Pyun-Fixman theory (version II).22 For our data A I 0.88 provides the best fit. (This compares to A I 0.86 used by Vrentas and Duda to fit the diffusivity data of polystyrene in cyclohexane.)22 Agreement between experimental and predicted values is good. by 1:.) at, Kirk Val“ 22 Table III. Values of A2* (mole ml/gz) used in calculating the concen- tration dependence of diffusion coefficient in Figure 2. For com- parison, (A2) is the second virial coefficient calculated from the two parameter theories. * M1 A2 (A2) 440 —0.0010 0.0013 960 0.0012 0.0022 4250 0.0021 0.0020 12600 0.0018 0.0016 As for osmotic pressure, the effect of association is found to be the strongest for low molecular weights. In fact, for the polymer sample M1 I 440, (kd*)ob8 can roughly be equated to -K/M1 (see Equation (25)). As molecular weight increases, the virial term and friction term also become important. For comparison, the predictions using the nonassociating model are also shown in Figure 2. At theta conditions and when the contributions of k8 and Vpo are negligible, Tlreduces to P1 and D*/Do reduces to Y2. Since Y1 decreases more rapidly with increasing concentration than Y2, osmotic pressure is affected more strongly compared to diffusion coefficient by association. (As P approaches infinity, ’1 and Y2 approach 0 and 0.5, respectively.) Some D(p) curves from the literatureé’3o'31 display minima, even at dilute concentrations. The diffusion rate first decreases sharply with increasing concentration and then attains an almost constant value or passes through a minimum. This behavior can be explained at b: 23 least qualitatively by the association model if the polymer molecules associate with one another in a thermodynamically good solvent. Equation (23) suggests that, at very dilute concentrations, asso- ciation dominates and causes the diffusion rate to decrease with increasing concentration; at higher concentrations, the thermodynamic term dominates (assuming that the hydrodynamic and volumetric terms are relative small) and causes the diffusion rate to increase with concentration. VI. CONCLUSION The association model and the two parameter theories provide good predictions for both the osmometry and diffusivity data for PEG in benzene. (A2*)°bs and (kd*)obs differ from their nonassociating counterparts only by K/Ml2 and K/Ml, respectively. The effect of association increases with decreasing molecular weight and is governed by the dimensionless group P I Kp/Ml. 24 REFERENCES 1. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. Morawetz, H., "Macromolecules in Solution," 2nd ed., John Wiley & Sons: New York, (1975). Elias, H.-G., "Light scattering from Polymer Solutions (M.B. Huglin, ed.)," Academic Press: London, (1972). Tsuchida, E. and K. Abe, Adv. in Polym. Sci., (1982), 45, 1. Yamakawa, H., "Modern Theory of Polymer Solutions," Harper and Row: New York, (1971). Billmeyer, F., "Textbook of Polymer Science," 3rd ed., John Wiley & Sons: New York, (1984). Lin, J., Ph.D. Thesis, Michigan State University, (1980). Rodriguez, F., "Principles of Polymer Systems," McGraw-Hill: New York, (1970). Tanaka, G. and K. Solc, Macromolecules, (1982), 15, 791. McMillian, W.G. and J.E. Mayer. J. Chem. Phys. (1945), 13, 276. Zimm, B.H., J. Chem. Phys., (1946), 14, 164. Brandrup, J., E. Immergut, and W. McDowell, "Polymer Handbook," 2nd ed., John Wiley & Sons: New York, (1975). Hilderbrand, J.H. and R.L. Scott, "The Solubility of Non- electrolytes," 3rd ed., Dover Publications: New York, (1964). Vrentas, J.S. and J.L. Duds, J. Polym. Sci., Polym. (Phys. Ed.), (1976), 14, 101. Bearman, R.J., J. Phys. Chem., (1961), 65, 1961. Caldwell, C.S., J.R. Hall and A.L. Babb, Rev. Sci. Instr. (1957), 28, 816. Bidlack, D.L., Ph.D. Thesis, Michigan State University. (1964). Gosting, L.J. and M.S. Morris, J. Am. Chem. Soc., (1949), 71, 1998. Elias, H.-G., "Association of Synthetic Polymers," Midland Macro- molecular Institute, Michigan, (1973). Langbein, G. and 2.2. Kolloid, Polym., (1965), 203, 1. Elisa, H.-G. and H. Lys, Makromol Chem., (1966), 96, 64. 31 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 25 Vrentas, J.S. and J.L. Duda, AIChE Journal, (1979), 25, 1. Vrentas, J.S., H.T. Liu, and J.L. Dude, J. of Polym. Sci. (Phy. Ed.), (1980), 18, 1633. Rossi, C. and C. Cuniberti, J. Polym. Sci., Part B, (1964), 2, 681. King, T.A., A. Knox, W.I. Lee, and J.D.G. McAdam, Polymer, (1973). 14, 151. Raju, K., Ph.D. Thesis, Michigan State University, (1978). Han, C.C., Polymer, (1979), 20, 157. Vrentas, J.S. and J.L. Duda, J. Appl. Polym. Sci., (1976), 20, 1125. Flory, P.J., "Principles of Polymer Chemistry," Cornell Uni- versity Press: New York, (1953). King, T.A., A. Knox and J.D.G. McAdam, Polymer, (1973), 14, 293. Holmes, F.H. and D.I. Smith, Trans. Faraday Soc., (1957), 53, 67. Elias, H.-G., Makromol. Chem., (1958), 27, 261. PART TWO 26 CHAPTER I INTRODUCTION The purpose of this work is to study the effect of association on the concentration dependence of osmotic pressure and diffusion coefficient in dilute polymer solutions. The term "association" is defined here as a rapid equilibrium between unimers (unassociated molecules) and multimers (associated molecules). Many synthetic polymers and biopolymers in solution are capable of associating with one another to form larger molecules via secondary binding forces such as hydrogen bonds. Several studies [B—13, B-18, B-37] have indicated that the extent of association depends on the type of association, the chemical structures of polymer and solvent, molecular weight, polydispersity, concentration, and temperature. An associating system is a multi—component system consisting of polymer unimers, polymer multimers, and solvent molecules. The distribution of unimers and multimers may change drastically with polymer concentration, depending on the type of association. Consequently, 27 L_A_ (h ('7 rev, 28 association can greatly influence those solution properties which are functions of molecular size. Moreover, if the molecular weight of a polymer is to be determined accurately, the possibility of association must be investigated--if the polymer associates, traditional methods for molecular weight determination (such as linear extrapolations of osmotic pressure in dilute solutions) should not be used. In spite of its importance, relatively few studies have been reported for association in polymer solutions. There exists no satisfactory theory for describing the solution properties of these systems. In addition, the lack of pertinent data in the literature makes the situation even more unfavorable. The first systematic investigation of associating macromolecules was conducted by Elias [B—18]. He studied a large number of polymer—solvent pairs (both natural and synthetic polymers) under various conditions and was successful in describing the thermodynamic properties of some of these systems using association models. Only recently has the effect of association on diffusion been studied. Lin [B—35] measured the osmotic Pressures and the diffusion coefficients for two forms of P01ytetrahydrofuran. The two ends of these polymer types had different functional groups attached-—one the methyl D) an 29 group and the other the hydroxyl group. Methylethylketone (MEK) and bromobenzene (BB) were chosen as the solvents. Based on an association model, he derived an expression for the osmotic pressure 1r -——————-—1 +AP+A102+ PRT- 2? 3p ”' 1-1 (Mn)app.e ( ) and an expression for the diffusion cofficient __ o m DObS—D anSO (1+kd Pp+ ...) (1_2) where he defined (M ) as the "apparent number average n aPP.6 molecular weight of polymer molecules in associating solutions under theta condition" and kdm as the "linear concentration dependence constant of the average diffusion coefficient of all diffusing associated and non-associated species". He concluded that the hydroxyl endgroups could cause the polymer molecules to associate strongly with one another. However, Lin's work can be questioned in several respects. First, he used a membrane osmometer to measure osmotic pressures and his polymer samples had molecular weights so low (one of them was determined by him to be as low as 2500) that the use of this kind of osmometer is not recommended. It is very difficult to prevent molecules of molecular weight smaller than 20,000 from leaking through 30 the membrane (see Chapter III). Consequently, faulty data might have been obtained by him. However, if leakage of polymer molecules through the membrane did occur, the actual molecular weights for these polymer samples would be even smaller and the effect of association greater than those predicted by him. Second, Equations (1—1) and (1-2) are oversimplified. It has long been recognized that the second virial coefficient is dependent on molecular weight [B-4, B-7, B-9, B-57]. Justification must be provided for Lin's assumption :1: of constant A2 and kdm. Third, the parameters xasso and kdm in Equation (1-2) are devoid of physical meaning (because they are eXpressed in complicated summation terms) and are obtained almost solely from curve fitting the diffusivity data. Actually, the concentration dependence of diffusion coefficient, kd, is governed by three factors [B-49, B-57]--the thermodynamic, hydrodynamic, and volumetric effects. The volumetric effect is usually relatively insignificant, and the thermodynamic and hydrodynamic factors can be estimated from the two-parameter theory [B-SO]. In this work a different approach is used to study the effect of association on polymer solutions. New eXpressions for the osmotic pressure and the diffusion Coefficient, based on the open association model, are 31 derived. These expressions include the molecular weight dependence of the second virial coefficient, and are expressed in terms of physical quantities which can be obtained easily. Predictions from these expressions are compared with the osmometry and the diffusivity data of polyethylene glycol (PEG) in benzene. CHAPTER II MOLECULAR ASSOCIATION A. General Background Under favorable conditions, polymer molecules in solution can form intermolecular complexes with solvent molecules or with other polymer molecules [B-36, B-46]. Examples for association of polymer molecules with solvent molecules are the binding of iodine to amylose, the binding of counterions to polyions, the association of enzymes with substrates, inhibitors, etc. Examples for association of polymer molecules with other polymer molecules are the nonspecific association of cationic and anionic polymers, the formation of hemoglobin and various enzymes from separate protein subunits, the interaction of antigens with antibodies, etc. Furthermore, intermolecular complexes in polymer solutions may be divided, based on the nature of binding forces, into four classes: polyelectrolyte complexes, hydrogen-bonding complexes, stereocomplexes, and charge transfer complexes. Numerous examples for each class have been compiled by Tsuchida and Abe [B-46]. 32 I'll 33 The formation of intermolecular polymer complexes has been studied under many names [B-18]--association, self-association, aggregation, polymerization, multimerization, complex formation, denaturation, sociation, supersociation, agglomerization, etc. The association between polymer molecules in an inert solvent is treated exclusively in this work. The term "association" or "multimerization" is defined here as a rapid equilibrium between unimers (unassociated molecules) and multimers (associated molecules). Thus an associated solution consists of a mixture of unimers, dimers, trimers, etc. The term "monomer" is avoided here because it is more properly used to designate the molecule from which the polymer is formed. Instead, the term "unimer" is used to represent a polymer molecule which is not associated with another polymer molecule via a secondary binding force. Association is a function of polymer concentration. As polymer concentration increases, the polymer molecules are packed more closely together and, consequently, associate with one another to a greater extent. The size and the number of polymer molecules may change drastically depending on the type of association. Two groups of methods are commonly used to study association: "the group specific methods" and "the molecule Specific methods" [B—18]. Examples for the group specific 34 methods are infrared spectroscopy, nuclear magnetic resonance, ultraviolet, etc. As its name implies, these methods can be used to determine the structure of a "group" and the type of interaction that can occur between this group and another group. On the other hand, examples for the molecule specific methods are osmometry, light scattering, ultracentrifugation, viscometry, gel permeation, I' h chromatography, diffusivity, etc. As its name implies, these methods look at the "molecule" as a whole and can be used to determine the characteristics of polymer molecules such as the molecular weight. In general, the group specific methods should not be used to study the association of a polymer which have only a few associogenic groups (groups that are capable of associating). These associogenic groups can escape being detected because they constitute such a small part of the polymer molecule. In contrast, the molecule specific methods are the prime choices for studying association because the drastic change in apparent molecular weight (due to association) can be easily detected. B. Types of Association When studying association, a physical model is assumed a priori and tested for consistency with experimental data. 35 An important consideration in constructing an association model is to determine whether the number of associogenic groups are dependent on the size of the polymer molecules. "End-to-end association" is the kind in which the number of associogenic groups per molecule is constant, regardless of the length of the polymer. An example is the association of two polymer molecules via associating endgroups. On the other hand, "segment-to—segment association" [B-17, B-18, B~44] is the kind in which the number of associogenic groups increases proportionally with the length of the polymer. End-to-end association is discussed exclusively here. It is assumed that the unimers and the multimers are distinguishable only in size, but not in shape and chemical properties. 1. Open association There are two basic types of end—to-end association-- open association and closed association. "Open association" is one in which all types of multimer are present, and successively higher multimers are formed one step at a time: 36 1 i—l i i —' = . _ B1 + Bn_1 +__ Bn , K Kn (2 1) where Bi represents i-mer. Note that the open association model does not exclude associations such as B + B :2’ B , K = K' (2-2) because the molar concentration 1 (2’3) C = K'C = K C (for K' = K) (2-4) which is identical to Equation (2) in Part One for the open association model. An analogy can be made between open association and stepwise polymerization. In stepwise polymerization, each polymer formed can react further with a monomer to form a larger polymer via a chemical bond, in a manner similar to Equation (2—1). Thus the mathematics for open association and stepwise polymerization are closely related. For example, the mole fraction of the i-mer in the multimer mixture (on a solvent free basis) is c /c = (1 - KC )(KC )1"1 (2-5) i p 1 1 and the polydispersity is Mw/Mn = 1 + KC1 (2-6) 37 for open association. On the other hand, the mole fraction of x-mer (where x is the degree of polymerization) and the polydispersity for stepwise polymerization can be obtained from Equations (2-5) and (2-6) respectively, simply by substituting the fraction of conversion (of stepwise polymerization) for the dimensionless group KC1 [B-41]. Another type is the "closed association" in which only unimers and n-mers are present: nB1 :3 B (2-7) n Of course, combinations of open association and closed association are possible. However, only open association is treated in this work because this type of association is very useful for describing the behavior of many synthetic and natural polymers [B-18]. Some of the characteristics of open association are discussed below. As shown in Part One, the behavior of open association can be expressed in terms of the dimensionless group P = KP/Ml. Both K and M1 can be obtained from osmotic pressure measurements. The dimensionless parameter W1 (see Equation (4) in Part One) is the ratio of the unimer molecular weight to the apparent molecular weight. It is related directly to the extent of association and to the reduction of molecules in solution (due to association). A plot of W1 versus P is shown in Figure 2-1. W1 decreases most rapidly in the 38 region near infinite dilution, indicating that the effect of association is most prominent in this region. It is interesting to see how the i-mers are distributed when the apparent molecular weight is equal to that of unimer, dimer, etc. A plot of molecular weight distribution versus apparent molecular weight is shown in Figure 2-2. Ci/Cp is the mole fraction of i-mer in the multimer mixture (on a solvent free basis). For low apparent molecular weights, the distribution tends towards the small i-mers; for high apparent molecular weights, the i-mers are more evenly distributed. 2. Dimerization Selecting an appropriate association model is so critical that several promising candidates should be tested with experimental data. Here the behavior of the dimerization model and the open association model are compared. The dimerization model is represented by B +B —’B (2-8) and the apparent molecular weight is \pl'EMl/Ma=0.5+1/(1+§) (2-9) 39 1.0 ___qvl 0.5 .. “'1 o I I I I 1 2 3 4 Figure 2-1. ‘1’1 and \Ill' versus P 4O 0.5- 0 004- o 0.3-1 Q i . 0‘ . 002- G o n 8 001- g o O o n o ' n o . 0 0 I I 1 I I I I I 1 2 3 4 5 6 7 8 i-mer Figure 2-2. Distribution of i-mer for open associating systems. (0) Ma/M,-2; (c1) Ma/Mlc3; 41 where K E ,’1+8P (2-10) Equation (4) in Part One for the open association model and Equation (2—6) for the dimerization model are compared in Figure 2-1. ‘P1 of the open association is a more rapidly decreasing function of P, especially at higher concentrations. This can be explained intuitively by the argument that the open association model allows multimers larger than dimers to exist in the solution. The difference is smaller at low concentrations. In fact, the two plots in Figure 2-1 have the same initial slope -K/M12. However, the PEG/benzene data in this work can be described satisfactory only by the open association model. CHAPTER III EXPERIMENTAL METHOD A. Polymer Sample Polyethylene glycol (PEG) in benzene is chosen for studying association in this work because it has been shown [B-15, B-16, B-34] that the hydroxyl endgroups can form hydrogen bonds either with other hydroxyl endgroups or with the ether groups in this system. Monodisperse PEG samples were purchased from Polymer Laboratories Inc., Massachusetts. The samples were used without further purification. The polydispersity of the samples are listed in Table 3—1. Table 3-1. Polydispersity of the PEG samples. M Mw/Mn 440 <1.09 960 <1.06 4250 <1.03 12600 (1.04 42 43 B. Osmometry Data When the free energy of a solution has been diminished by an addition of solute, it is possible to compensate for this reduction by applying an external pressure to the solution. For example, when solvent molecules diffuse through a semi-permeable membrane from a dilute solution into a more concentrated one through the process of osmosis, it is possible to prevent the diffusion by applying an external pressure (called the osmotic pressure) . Osmotic pressures are frequently used to determine the molecular weight of a polymer and to study the thermodynamics of polymer solutions (such as second virial coefficient, Flory-Huggins interaction parameter, and activity coefficient). A plot of reduced osmotic pressure versus concentration for a dilute polymer solution often yields a straight line, and the number-average molecular weight of the polymer can be determined by extrapolating the data to zero concentration [B-7, B-24, B-25]. However, if a straight line is not obtained, it is necessary to repeat the measurements with several different solvents to prevent faulty extrapolations. The membrane osmometer [B-43, B-52] is commonly used for measuring the osmotic pressures of polymer solutions. The best type is the electrical and automatically recording dynamic osmometer which usually allows equilibrium to be 44 reached within 30 minutes. However, its usefulness is 3 5 limited to molecular weights range from 5 x 10 to 5 x 10 [B-7, B-43, B-52]. The upper limit is determined by the smallest osmotic pressure that can be read, and the lower limit is determined by the permeability of the membrane. In addition, it is often difficult to find a suitable membrane for a particular polymer-solvent pair. The membrane must not be dissolved by the solvent. If the pores of the membrane are too small, the measurements take a long time. On the other hand, if they are too large, polymer molecules may leak through the membrane. Such a membrane is not always available. For molecular weights under 2 x 104 , highly sensitive vapor pressure osmometers should be used [B-43, B-52]. Vapor pressure osmometry is an indirect method for measuring molecular weights and osmotic pressures, and a standard is required for calibration. This kind of osmometer offers the advantages that it requires no membrane, needs only a small amount of sample, and is very easy to operate. The vapor pressure osmometry data of Elias [B-15, B-19] for PEG in benzene at 25°C were used in this work. 45 C. Diffusivity data Diffusion coefficients in this work were measured using a Mach-Zehnder interferometer [B-5, B-lO]. The description of this apparatus and its operation have been given in many places [B-5, B—27, B-35, B-39] and are not repeated in detail here. A complete description can be found in the dissertation of Bidlack [B-5]. Although the Mach-Zehnder interfermometer was developed a few decades ago, it remains one of the most accurate methods for measuring the concentration dependence of diffusion in solution [B-ll]. The interferometry technique involves carefully bringing a more concentrated solution into contact with a less concentrated solution to form a sharp interface in an optical cell where free diffusion is allowed to take place. The optical cell is immersed in a well-controlled temperature bath. The diffusion rate can be followed by measuring the refractive index in the cell as a function of time and position. If refractive index is assumed linear with concentration over small concentration ranges, the diffusion coefficient can be calculated from photographs of this fringe pattern at several times during the diffusion process. The concentration for the measurement is taken to be the arithmetic average of the original solution concentrations. 46 The interferometer was checked by comparing the diffusion coefficients of five aqueous sucrose solutions at 25 0C with those reported by Gosting and Morris [B-22]. The data of Gosting and Morris have been confirmed by several investigators [B-2, B-12] and can be fitted by the method of least squares to the empirical relationship 6 D = 5.226 ( 1 — 0.01480 c ) x 10- ‘:-0.002 (3-1) 3 of where c is the sucrose concentration in grams per 100 cm a water solution diffusing into pure water. A summary of the comparison is presented in Table 3—2. The standard deviation is found to be less that 1%. Table 3-2. Comparision of the diffusivity data determined in this work with the data of Gosting and Morris. 6 D x 10 , cm2/sec Z deviation c, grams/100 cm3 This work Equation (3-1) 0.4 5.220 5.195 +0.48 0.6 5.201 5.180 +0.41 0.8 5.191 5.164 +0.52 1.0 5.158 5.149 +0.17 1.2 5.108 5.133 —O.49 47 The experimental procedure described by Bidlack [B—5] was used in this work. The solutions were prepared and agitated gently for about three hours before using. The concentration differences between the two solutions were chosen to be 0.40 g/dl for all runs. The temperature bath was maintained at 25.0 i 0.1 0C. Each experiment took from 30 minutes to an hour for completion, depending on the molecular weight. Six exposures were usually taken for each run. Because of the small refractive index difference between the PEG/benzene solutions, the number of fringes, J (see reference [B-5]), was usually less than 10. It was vital to measure the value of J (i.e., the total refractive difference between the two solutions) with great accuracy, for a slight error could change the result significantly. Each exposure provides a value of J. Since there were six exposures per run, six values of J were obtained and the average was used to calculate the diffusion coefficient. If the difference between the largest and the smallest values of J exceeded 0.3, the run was discarded. The accuracy of the diffusion coefficients reported in this work were estimated to be within 4%. A sample calculation is presented in Appendix A. CHAPTER IV THERMODYNAMICS 0F ASSOCIATING POLYMER SOLUTIONS A. Theoretical Background The behavior of polymer solutions deviates greatly from Raoult's law except at extreme dilutions. Excess thermodynamic properties are large even for systems of negligible heat of mixing. This is due to the large entropy effect for mixing giant long-chain polymer molecules with small solvent molecules. Below are some theories used frequently to describe the thermodynamics for nonassociating polymer solutions. In this work, an attempt is made to extend the applications of these theories to associating systems by replacing the unimer molecular weight with the apparent molecular weight. 1. Flory-Huggins Theory The Flory-Huggins theory [B—21] has been used extensively due to its simplicity. This theory expresses the change in free energy of mixing as AGm --——- = + + R T “8 ””8 np 1MP ¢S¢pms + ms” (4-1) 48 49 where the subscript s designates the solvent and the subscript p the polymer. 0 and n are the volume fraction and number of moles, respectively. m is the ratio of molar volumes of polymer to solvent. X is called the Flory-Huggins interaction parameter. It is a dimensionless quantity and a function of the interaction energy characteristic of a given solute-solvent pair. It consists of both entropic and enthalpic contributions and can be expressed empirically by [B-41] x=x'+‘-I;—:-(5S-6)2 P (4-2) where x' is the entropy parameter with value between 0.3 and 0.4. 58 and 5p are "solubility parameters" of the solvent and the polymer, respectively. VS is the molar volume of the solvent and R is the gas constant. The thermodynamic quality (good solvent versus bad solvent) can be evaluated in terms of X. Note that the free energy must be negative for the polymer to dissolve in the solvent. Since the first two terms in Equation (4-1) are always negative, the solubility of the polymer is determined solely by the magnitude of X. It can be said that the smaller the value for x, the more negative the value of AGm, and the better the solvent for the polymer. A good solvent is defined as one in which the interaction between polymer 50 and solvent is stronger than that between polymer and polymer. 2. Two-Parameter Theory The two-parameter theory [B-50, B-57] is a group of theories which express the properties (such as second virial coefficients and viscosities) of dilute polymer solutions in terms of two basic parameters. Since polymer molecules in solution are constantly coiling and uncoiling owing to thermal fluctuations, it is possible to characterize their dimensions only by averages. One of these averages is the mean-square end-to—end distance of an unperturbed chain, O. The word "unperturbed" implies that the polymer chain is completely free of outside influences. Unperturbed dimensions are affected only by the so-called "short-range interferences" due to fixed bond angles and hinderances to rotation. The parameter [B-21, B-52] = as <52> (4-4) The expansion factor as measures the extent to which the excluded volume perturbs the polymer molecule from its unperturbed state. In contrast to A, information on as is generally not available and it depends on temperature, polymer molecular weight, and solvent characteristics. The excluded volume effects can be eliminated by a judicious choice of solvent and temperature. The polymer chain contracts in poor solvents because the polymer segments prefer to associate with other polymer segments rather than with solvent molecules. When the contraction due to poor solvent balances the expansion due to the excluded volume, the net excluded volume is zero and the solution is in the unperturbed or the theta state. 52 A common application of the two-parameter scheme [B-50, B-57] is to express the expansion factor as as 018 = as (A, B9 M) (4'5) Yamakawa-Tanaka [B-55] suggest the expression: 652 = 0.541 + 0.459 (1 + 6.04 z)0°46 (4-6) where . - 3.3/2 m 2” A3 (4-7) The parameter A represents the short-range interferences, and the parameter B the long-range interferences. As mentioned earlier, A can be obtained rather easily. B is a function of temperature and solvent. The Preferred method for calculating B involves light-scattering measurements to determine aS at several polymer molecular weights [B—57]. The quantity 2 can then be determined for each value of M from Equation (4-6), and by Equation (4‘7) B can be estimated from the slope of 2 versus Ml/2 plot. The shortcoming of this approach is that very limited 1 . lght~scattering measurements are available in the llterature. An alternative for estimating B is to use the i . ntrlnsic viscosity relation [B-21. 3’57] 53 B [n] = Kv M (4-8) Values of Kv and fi have been tabulated for many polymer-solvent pairs [B-9]. The parameter B can be estimated from the equation [B-SO, B-57] B = KV M845 - 1.05<1>A3 0.27874>M;5 (4-9) which can be used when BMl/Z/A3 < 5.06. Kurata et al. [B—32, B-33] recommend = 2.7 x 1023 for well-fractioned polymer (Mw/Mn < 1.1) and <1>= 2.5 x 1023 for ordinary fractioned polymers (MW/Mn > 1.1). Although the two-parameter theory provides satisfactory predictions for the properties of flexible P01 ymer chains in dilute solution, it breaks down for stiff Chains and for the region far away from the theta state (lzl < 0.15) [B-57]. 3. Second Virial Coefficient As a general rule, polymer solutions seldom behave idea:Lly except at infinite dilution. A convenient way to de$Cribe the nonideal behavior of a polymer solution is to e xpress the chemical potential of the solvent, p , in terms 0f the power series [B-21, B-57]: 54 - 2+...) o “s-“s -RTVSP(1/M+A2p +A3p (4-10) where V8 is the molar volume of the solvent. A2 and A3 are the osmotic second and third virial coefficients, respectively. The virial coefficients represent the binary and higher-order interactions of polymer molecules due to excluded volume effects. In dilute solutions, it is sufficient to consider only the second virial coefficient because the influence of the third and higher order virial coefficients are usually relatively small. The second virial coefficient for a dilute polymer solution has several significant meanings. It is directly related to the chemical potential of the solvent, the excluded volume, and the Flory-Huggins interaction Parameter. It may also be used to measure the goodness of a S°1Vent for a particular polymer--good solvents are commonly dEfined as those having positive A2. The relationship between A2 and the excluded volume depends on the geometry of the polymer molecules. If the polylner molecules behave like rigid spheres, the excluded vollltne per sphere, u, (see Figure 4—1) is [B-24] (4-11) u = 4 (volume of sphere) an . d It can be shown that the second virial coefficient can b e related to u by 55 I I, ‘\ / ‘\ I \ I I l l I l | \ \ \ \\ I, \ / \' I \\ ” ~~ ’ F i~gure 4-1. The excluded volume (dotted volume) for two spheres in contact. Since the n rigic rigi< inde] COIII Wherc Poly: Flor it C and 56 2 A2 = Na u / 2 M (4-12) Since the volume of a spherical molecule is proportional to the molecular weight M, the second virial coefficient for rigid sphere molecules is inversely proportional to M. For rigid rod molecules, the second virial coefficient is ijidependent of the molecular weight [B-57]. In general, the second virial coefficient can also be c<31~related with the molecular weight by the relation [B-9] (4-13) and a are empirical constants which depend on the wileelre K' As mentioned earlier, a=l solvent and temperature. The pol ymer, ft>z‘ rigid spheres and¢x=0 for rigid rod molecules. realation is usually valid only within limited molecular "aiiégllt range, and for most systems the values of a are found to be less than 0.5 [B-9]. Since the osmotic pressure can be obtained from the Flor y_HugginS theory [B-Zl] n l .V’Z MT.” (% -X) ‘_E-—p+ .00 (4-14) PRT VS 1. At (lain been shown easily that the second virial coefficient a. nd X are related by V2 A2=(;s-x1—E—§ M (4—15) Vs 57 When X=1/2, A2=0, the excluded volume vanishes, and the solution is said to be under theta conditions. This situation arises because of the apparent cancellation between the enthalpy of mixing and the excess entropy of mixing. B. Expression for Osmotic Pressure for Associating Polymer Solutions In Part One, an expression for the concentration dependence of osmotic pressure has been derived: _. M11?“ * . a: ‘wh531‘€3 A is assumed to be independent of concentration for 2 ea‘:}1 nnolecular weight. In this section, Equation (4-16) is * extended to include the concentration dependence of A2 , and the Osmotic pressure is expressed in terms of the unimer n1 olecular weight, the mass concentration, the association c. onstant K, and the parameter 0: in Equation (4—13). * 1|: First, an approximation for A2 is made. A2 1. S a complicated term because it includes the interactions be 'tWVeen all multimers (such as unimer-unimer, unimer-dimer, 58 dimer-dimer, etc.). Recently Tanaka and Solc have shown a: that A2 can be approximated by [B-45] an: A 3' A 2 (4-17) where A2,n is the second virial coefficient of a monodisperse polymer with molecular weight equal to the number-average molecular weight of the multimer mixture. (In Tanaka and Solc's paper, open association is identified as a heterogeneous polymer solution having a Schultz-Zimm distribution and a polydispersity less than 2.) For excluded volume parameter 2 < 5, this approximation is accurate within 5%. Since the sizes of the multimers vary with concentration, A2 n also varies with concentration. 9 :1: Next, A2 is expressed in terms of the second virial coefficient of unimer, A The motivation is that A21 can 21' be calculated easily by the two-parameter theory. If equation (4-13) can be applied to the multimers such that A21 = K. Mi-a (4-18) where A2i is the second virial coefficient of i-mer, then A21 = K' 111‘“ (4—19) and A = 11' M “" <4-20) 59 Substituting Equations (4-19) and (4-20) into Equation :1: (4-17), A2 can be expressed as * —01 A2 = A21 (Ml/Ma) = A a (4-21) 21 *1 where \111 has been defined previously (Equation (4) in Part One) as Ml/Ma' Substituting Equation (4-21) into Equation (4-16), the osmotic pressure can be expressed as —(X The observed second virial cofficient obtained from differentiating this equation with respect to p and evaluating the result at p=O is 2 (112*)0b = A21 — K / M1 (4-23) S Which is the same as Equation (16) in Part One. Equation (4-18) is valid for positive A2 only. Because A2 may become negative as molecular weight decreases, application of Equation (4-22) is restricted to polymer molecules of ordinary size (with molecular weight higher than 104). on of 510 fro ben: can 510; calc Squa Afif: Pred: and t 60 C. Presentation of Osmometry Data and Discussions 1. Estimation of K According to Equation (4-23), if the dependence of A 21 on M1 is relatively weak compared to that of K/Mlz, a plot a: of (A2 )obs versus l/Ml2 should yield a straight line with * slope equal to -K. The values for (A2 )obs can be obtained a: from the initial slope of 1r /pRT versus p. The osmometry data of Elias [B-16, B-19] for PEG in benzene are used here to test this equation. Since the data can be well fitted by a straight line (see Figure 4-2), the slope is a good initial guess for K. The final value of K calculated based on the association model and the least squares method is 11000 -_+_ 300 cc/mole. Association of PEG in benzene has also been studied by Afifi-Effat and Hay [B-l]. Their association model pred i cted that * (A2 )obs = A21 ‘ K / M1 (4‘24) and they claimed that their data were in good agreement with this prediction. However, this author found a lack of consistency between their raw data. calculations, and r . eSults. The data of Elias do not agree with Equation (4‘24). (A plot of (A 2|: 2 )Obs versus 1/M1 is not linear.) 61 -001- (mole ml/g) )0!» -0021 ( A; "003'J Figure 4—2. (A 2*)obs versus l/Ml. Experimental data take from Elias et al. [B-16, B-19]. n 62 2. Estimation of A2* The two-parameter theories can be used to estimate the second virial coefficient for linear, flexible chain polymers [B-SO, B-57]: Na B 110(5) A2 = "“E;‘—" (4-25) where E'=.£L. as3 (4-26) ho (g) = 0.547 {1 - (l—+ 3.9035 }-0o4683 2 (4-27) Thus the second virial coefficient A2 of PEG in benzene at 25 0C can be easily estimated if the parameters A and B are known. The parameter A, which represents the short-range interferences, is given by [B-9] A={ ° % _ -9 M I - 7.9 X 10 0‘31 (4’28) The parameter B, which represents the long-range interferences, is estimated indirectly from the empirical 63 Mark-Houwink intrinsic viscosity relationship (see Equations (4-8) and (4-9)): [0] = Kv M6 = 0.063 M0°64 (4-29) where [n] is the intrinsic viscosity in ml/g. The values of RV and 5 in Equation (4-29) are obtained from curve fitting the raw data of Rossi and Cuniberti [B-42] using the method of least squares. It should be pointed out that Rossi and Cuniberti fitted their data with different values of Kv and B: [n] = 0.00129 110'5 (4-30) because they were preoccupied by the idea that a should always be 0.5. However, their raw data clearly agree only with Equation (4-29), and the use of Equation (4-30) is incorrect. Figure 4-3 shows the second virial coefficient as a function of molecular weight for PEG in benzene at 25 oC, predicted by the two-parameter theory. The value of A2 increases very rapidly with molecular weight before it reaches a maximum. (When M<297, B Aoco uumm cw H «Haws Eouwv *~< mo mo=Hm> J - - - .els ousmwm I P I O I P I N r 0 (3'6 [w claw) 80‘ x ‘ZV 67 predictions from the two-parameter theory (compare with Figure 4-3). For high molecular weights, this assumption is valid because the second virial coefficient is a weak function of molecular weight [B-4, B-9, B-21, B-32] and concentration. For low molecular weights, the justification for this assumption is as follows. According to Equation (15) in Part One, Ilconsists of two terms: the association term W1 and the virial term A2*M1P. If the virial term is sufficiently small, the error introduced by this assumption in calculating H is negligible. Figure 4-5 shows the magnitudes of these two terms plotted against concentration for two molecular weights. It is seen that the virial term for the low molecular weight sample (M1=594) is very small. Consequently, this assumption can also be used for low molecular weights. 3. Other Discussion As shown in Figure 4-5, the association term is a decreasing function of p, but the virial term is an increasing function of D. For the low molecular weight sample (Ml=594), the association term dominates; for the high molecular weight sample (M1=6000), the virial term dominates. It is interesting that the effects of these two terms sometimes balance each other, as for the sample 68 1.5' /,Aflf-¢UOH3C) M1=6000 1:594 / LEE—l —————————————— M1=594 0 0305 0710 P (g ml") Figure 4-5. Comparison of the association term G————-) with the virial term ( ----- ) in Equation (15), in Part One, for PEG in benzene at 25 °C. 69 (M1=1518). rIfor this sample (see Figure 1 in Part One) is almost unity regardless of concentration, and thus it behaves like one under theta conditions (see Chapter II). When both terms are large, the model predicts that the association term dominates at low concentrations while the virial term dominates at higher concentrations. In this case, a plot of Ilversus p should first pass through a minimium and then increase with p. This behavior has already been observed in some systems [B-18, B-35]. Although the sample (M1=6000) displays a linear behavior and can be satisfactorily descirbed by the nonassociating model, it is incorrect to assume that association does not occur at higher molecular weights. In fact, Elias has shown that the association constant is independent of molecular weight [B-19]. CHAPTER V DIFFUSION IN ASSOCIATING POLYMER SOLUTIONS A. Theoretical Background Diffusion is movement of a chemical species from a region of higher concentration to a region of lower concentration. The flow of solute molecules per unit time across a unit area perpendicular to the direction of flow (x axis), J, is given by Fick's law L. II I U Q) |.. (5-1) Q) x where D is known as the diffusion coefficient. The diffusion coefficients of many polymer solutions depend strongly on the nature of the solvent and polymer concentration. Most of the studies in this area indicate that for dilute polymer solutions in good solvents, the Value of D generally increases with polymer concentration. On the other hand, the values of D for films or solids in the region near the undiluted polymer generallly increase sharply with increasing diluent concentration. Therefore D(p) in good solvents can be expected to exhibit a maximum 70 71 at an intermediate concentration from pure solvent to pure polymer. In addition to solvent and polymer concentration, the diffusion coefficient may also be affected by molecular association. The larger associated complexes (dimers, trimers, etc.) diffuse much slower than the unimers, leading to lower diffusion coefficients. The concentration dependence of diffusion coefficient in dilute polymer solutions is often expressed as [B—SO, B-57] D = DO ( 1 + kd p +-... ) (5-2) Accordingly, predicting the value of D requires the knowledge of DO and kd. At the present time, Do can best be predicted by the Kirkwood-Riseman theory, and kd by the Pyun-Fixman theory. 1. Kirkwood-Riseman Theory The Kirkwood-Riseman theory [B-29, B—30, B—57] provides a simple method for predicting the diffusion coefficient for linear, flexible polymer chains at infinite dilution under theta conditions. Infinite dilution implies that the polymer molecules are widely dispersed in the solvent and there are no interactions between individual polymer chains. 72 This theory is applicable only under theta conditions because the excluded volume effects are not included in the derivation. Its derivation is based on the assumption (the nonfree draining limit) that there exists a very large hydrodynamic interaction between polymer segments, and the polymer chains behave like rigid molecules. The Kirkwood-Riseman theory is expressed as 0.196 k T (D) = --—-— (5-3) 0 9 USAM’li where k is the Boltzman constant and "s is the viscosity of the solvent. Since A and ”s are usually available, the determination of (DO)e for many polymer-solvent pairs are relatively simple. For solutions under nontheta conditions, Duda et al. have suggested the modified form [B-48, B-SO] (D D0 = 23;" <5-4) where Do is the diffusion coefficient at infinite dilution under nontheta conditions, and as is the expansion factor defined previously. Since as a 1 except for very poor solvents, (DO)E3 2 Do (5—5) 73 and thus the Kirkwood-Riseman theory provides an upper bound for the diffusion coefficient at infinite dilution. Duda et. al. [B-Sl] compared the predictions of this theory with the experimental values of (Do)6 for polystyrene in cyclohexane under theta conditions. The predictions were found to be slightly higher than the experimental values. The average ratio of Do(exptl) to Do(theory) was 0.86. 2. Modified Pyun-Fixman Theory The behavior of dilute polymer solutions changes significantly with polymer concentration. As concentration increases, the polymer molecules interact hydrodynamically with each other even though they may not overlap or entangle. The concentration dependence of diffusion coefficient, kd, can be expressed as [B-49] k = 2 A d M - kS — 2 V (5-6) 2 p0 where A2 is the thermodynamic second virial coefficient, and Vp0 is the partial specific volume of the polymer at infinite dilution. The quantity kS is defined by the series expansion [B-49] f=fo(1+ksp+...) (5-7) 74 and f is the friction coefficient defined by the relation force on a polymer molecule = f ( u - u ) (5-8) where 11S and up are the velocities of solvent and polymer, respectively, with respect to a convenient reference frame. According to Equation (5-6), the value of kd depends on the thermodynamic, hydrodynamic and volumetric effects. The thermodynamic effect, A2, has already been discussed in Chapter IV; the hydrodynamic effect, ks, can be estimated by the Pyun-Fixman theory; and the volumetric effect, vpo’ can be measured from density experiments. The effect of Vpo is usually relatively small, and can be ignored if kd is less than 20 cmB/g. The Pyun-Fixman theory [B-38, B-54, B-57] is based on a spherical model in which the spheres are composed of both polymer and untrapped solvent. It can be expressed as S ks = 2.23 6yfl3.NaM A3/512}- Vp0 (version I) (5-9) or .L i 3 3 ks ={2.23 62 flzNaM A /512%.}- Vp0 (version II) (5-10) under theta conditions. Equation (5-9) is called version I, and Equation (5-10) version II, of the Pyun-Fixman theory 75 [B-51]. They are related by the parameter A which is defined as y H (Do)6,exptl / (Do)6,theory (5-11) To extend its application for nontheta conditions, Duda et al. [B—53] have suggested the following expressions * 41ra03 Na k = _ __ s [7.16 K010 )1 3M vpo (5_12) a ___ 6%135 AMlfias ° 16 (5-13) * _ 4096 2 7201531 (5-14) * '21.n[l+x+(2x+x2);5] K(A ) = 24f{ - 1 x2 0 0 (254+ x2);5 expL4%f(l">02(2'+>O]dx (5-15) which are collectively called the modified Pyun-Fixman theory. Thus if A, B and Vp0 are known for a particular polymer-solvent pair, kS can be calculated in a straight-forward manner. When 2:0 and as=1 (i.e., under theta conditions), Equations (5-12) through (5-15) reduce to 76 Equation (5-9). However, it remains for future work to test the modified Pyun—Fixman theory with experimental data. B. Expression for Diffusion Coefficient for Associating Polymer Solutions In Part One, an expression for the concentration dependence of diffusion coefficient for associating systems has been derived, assuming that A2* is independent of concentration for each molecular weight. Following the same procedure as described in Part One, the diffusion a: coefficient D based on Equation (4-16) can be eXpressed as * M1 D ={l‘(ks+2Vm)I-<—-:} aP {2‘21 MK21(2 J11“ s(1+2))P £—_—(1+z)l+-] (5—16) where \/1 + 4 P (5-17) as defined previously. In deriving this equation, the parameters ks, f and Vp were expanded in terms of a power series with concentration, and only the first-order terms 77 are retained. When a = O (i.e., Aikis independent of molecular weight), this expression reduces to Equation (23) in Part One. Note that Equation (5-16) is considerably more complicated compared with Equation (23). In addition, the parameter a must be estimated. C. Presentation of PEG Diffusivity Data and Discussions 1. Estimation of DO The Stoke-Einstein theory [B-6, B-AO] U ll k T / 61rn r (5-18) is often used to estimate the diffusion coefficient of liquids. k is the Boltzmann's constant, n the solvent viscosity, and r the solute radius. This theory is valid only for large, spherical molecules diffusing in dilute solutions. Moreover its direct application is not always possible because the solute radius r is often not available. However, many authors have used the form Dn/T = f(solute size) as a starting point in developing empirical correlations [B—40]. Polymer molecules in solution, in general, do not behave like large spherical molecules. They are best imagined to be like necklaces consisting of spherical beads 78 connected by strings that have no resistance to flow. Consequently, the Stoke-Einstein equation fails to accurately predict the diffusion coefficients of polymer solutions. On the other hand, the Kirkwood-Riseman theory is more successful because it is based on a more realistic random coil model. Although its application was originally limited to polymer solutions under theta conditions, it has been extended to nontheta conditions (the modified Kirkwood-Riseman equation, see Equation (5-4)). Table 5-1 shows that the predictions of the modified Kirkwood-Riseman theory and the experiment data for PEG in benzene are in good agreement. The difference between the predictions and the data is within 5%. Table 5-1. Comparison of diffusion coefficients predicted from the modified Kirkwood-Riseman equatiog with the experimental data for PEG in benzene at 25 C. Do x 106 (cmZ/sec) M Kirkwood-Riseman Experiment Z difference (This work) 960 5.30 5.05 5.0 4250 2.31 2.20 5.0 12600 1.28 1.25 2.4 79 2. Estimation of kS Despite many experimental and theoretical studies, the prediction of k8 still remains a somewhat unsettled problem. Duda et al. [B-Sl] recently evaluated several existing theories by comparing their predictions with experimental results. They concluded that the Pyun-Fixman theory (versions I and II) was the best theory for predicting ks at the present time. Still, one should not expect very accurate predictions from this theory. The predictions may differ from the experimental values by as much as 60%, as is the case for the data of Duda et al. To compensate for the uncertainty of the Pyun-Fixman theory, the parameter A (see Equation (5-10)) is adjusted to fit the diffusivity data. k=0.88 was used to fit the diffusivity data of this work. This compares favorably with A=O.86 used by Duda et al. [B-51] to fit the diffusivity data of polystyrene in cyclohexane. Figure 5-1 is a plot of kS versus M for PEG in benzene at 25 oC, predicted by version I of the Pyun-Fixman theory. Note that the application of this theory is originally limited to nonassociating systems. In this work, its application was extended to associating systems by replacing the unimer molecular weight with the apparent molecular weight. 80 30- 20- , (ms/.1 J ‘0- 0 i W 5000 10000 M Figure 5-1. ks versus M for PEG in benzene at 25 0C, predicted by the two-parameter theory (Equation (5—12)). 81 3. Other Discussions As shown in Figure 2 (in Part One), the diffusivity data are well described by the association model. The data for the low molecular weight samples (M1=440, M1=96O) are strongly dependent on concentration due to association. The data for the higher molecular weight samples (M1=4250, M1=126OO) are less dependent on concentration because the effect of the association term diminishes as molecular weight increases and its effect is also compensated by other terms (A2 and ks). Comparision between predictions from the Pyun-Fixman theory and experimental data was made only for polystyrene in cyclohexane under theta conditions. Thus it is necessary to test this theory and its modified form (the modified Pyun-Fixman theory) more extensively for future work. Accurate knowledge of kS is vital for predicting diffusion coefficients especially for high molecular weight polymers. D. Presentation of PTHF Diffusivity Data and Discussions Diffusivity data for polytetrahydrofuran (PTHF) in five different solvents were measured. They were used to investigate the effects of solvent, molecular weight and 82 temperature on diffusion; to test the Kirkwood-Riseman theory; and to calculate the molecular size of the polymer. Two sets of PTHF samples were used. The first consisted of three "fresh" PTHF samples purchased recently from Polymer Laboratories, Inc., Massachusetts, and their characteristics are listed in Table 5-2. Table 5-2. Characteristics of "fresh" PTHF samples. Molecular weight Endgroups Polydispersity 2850 -CH3 < 1.15 30800 -CH3 < 1.10 290000 -CH3 < 1.15 The second set consisted of the "old" PTHF samples used earlier by Lin [B-35]. The adjective "old" was used because these samples were purchased more than three years ago. They were labeled, according to Lin, as PTHF-A1, PTHF-Bl and PTHF-B2, with characteristics listed in Table 5-3. Note that the molecular weights for PTHF-B1 and PTHF—B2 determined by Lin were lower than those by the manufacturer. Lin pointed out that the manufacturer's values were in error because the manufacturer overlooked the fact that these polymers were capable of associating in solution [B-35]. 83 Because Lin did not store the unused portions of these samples (which were later used by this author) under nitrogen nor at very low temperature (as they should be), the characteristics of these polymer samples might have changed over the interim period of time. Table 5-3. Characteristics of "old" PTHF samples (before degradation). MM and Ml are respectively the molecular weights determined by the manufacturer and by Lin. Polymer Code Endgroups MM Ml PTHF-A1 —CH3 281,000 ---- PTHF-B1 —OH 25,000 7,660 PHTF-B2 -OH 10,200 2,500 Five solvents were used: methylethylketone (MEK), diethylether (DE), n-butanol (BOH), ethylacetate (EA), and bromobenzene (BB). They represent a wide range of solvent power and hydrogen—bonding capability, with characteristics listed in Table 5-4. 84 Table 5-4. Characteristics of solvents. nZSand "34 are the viscosities (centipoise) at 25°C and 34°c, respectively. X:is the Flory-Huggin interaction parameter. Solvent M.W. W25 "30 H-Bonding strength X MEK 72 0.40 0.36 Medium 0.40 DE 74 0.23 ---- Medium 0.64 BOH 74 2.58 2.05 Strong 1.48 EA 88 0.42 ---- Medium 0.38 BB 157 1.06 0.95 Poor 0.61 Diffusion coefficients were measured using the Mach-Zehnder interferometer. A summary of the data is presented in Appendix C. Unless otherwise stated, a polymer solution of 0.30 g/dl was allowed to diffuse into pure solvent at 25°C during each experiment. Thus the average concentration was reported to be 0.15 g/dl. The accuracy of these data was estimated to be within 3%. 1. Effect of solvent As shown in Appendix C, the diffusion coefficients of PTHF in various solvents decrease in the order DB > MEK > EA > BB > BOH. This trend indicates that polymer molecules diffuse faster in solvents with lower molecular weights (see 85 Table 5-4). The slow diffusion rate for the polymer molecules in BOH is attributed to the fact that the BOH molecules are capable of associating with each other to form larger clusters. It is often useful to use the form Dn/T = f(solute size) as a starting point for correlating diffusivity data. If Dn/T is plotted against M on a log-log graph, the result can be fitted by a straight line for each solvent, as shown in Figure 5-2. The straight lines are almost parallel with each other, and the quantity Dn/T decreases in the order MEK > DE > BOH > EA > BB. Note that BOH takes a higher position in the order because the effect of viscosity has been accounted for. The diffusivity data can also be related by the form D = G M-d. The estimated values for the parameters G and d are presented in Table 5-5. Recall that the Kirkwood-Riseman theory, (D ) = 0.196 k T / A n Mo's o 6 (5—19) predicts that the diffusion coefficient should be inversely proportional to the square root of molecular weight. The data in Table 5-5 agree well with this prediction, for the values of d are very close to 0.5. 86 Figure 5-2. Dn/T versus M for the "fresh" PTHF samples in various solvents at 25 °C. (A) MEK; (0) DE; (A) BOH; (1.) EA; (‘7) BB- 87 8‘ 88% now 88m #0.. 82 00.. p _ . — r OPX N on NP- 5 me am... I :30 _. (”oases/B)1/ha 88 Table 5-5. Parameter estimations for the diffusivity data of "fresh" podlymer samples at 250 C using the form D = G Md . Solvent G x 10"4 d Correlation coefficient MEK 2.134 0.502 -1.00 BB 0.296 0.484 -1.00 It is interesting that if Dn/T is plotted against the solvent molecular weight on a log-log graph, the result fits a straight line for each polymer sample (see Figure 5-3). The slopes of the lines suggest that Dn/T is inversely proportional to the solvent molecular weight. Wilke and Chang also made the similar correlation for low molecular weight organic liquids and found that Dn/T was directly proportional to the square root of the solvent molecular weight [B-53]. However, there exists no satisfactory theory for describing the effect of solvent on diffusion in dilute polymer solutions. Further work is needed to determine the correlation between diffusion coefficient and solvent molecular weight. 89 1640-1 -41 00/7 (9/30c3 OK) 10 I 10 100 Ms Figure 5-3. Dn/T versus M5 for the "fresh" PTHF samples at 25 0C. M and M5 are the molecular weights of polymer and solvent, respectivley. 90 2. Effect of temperature Figure 5-4 is a plot Dn/T versus M for PTHF in MEK and BB at 25 oC and 34 OC. The variation of temperature appears to have no effect on diffusion for MEK. However its effect is rather significant for BB. Few researches have been conducted for investigating the effect of temperature on diffusion for dilute polymer solutions. Although some researchers suggest that the diffusion coefficient is a peculiar function of the combination of polymer/solvent/temperature, systematic study in this area remains for future work. 3. Molecular Dimensions Table 5-6. Estimated values of A (cm) for "fresh" polymer samples. Solvent A (25 0C) x 109 A (34 0C) x 109 MEK 9.6 i 0.2 9.4 i0.1 DE 10.9105 ____________ EA 13.1 i.0-3 ———————————— BB 21.7 i 0.5 11.8 i 0.8 When comparing the equation D = G M.d with the Kirkwood-Riseman theory, it can be seen easily that the parameter G is inversely proportional to the parameter A. Figure 5-4. 91 Comparison of Dq/T versus M at 25°C and 34°C. Aand Q are data measured by Lin and Yam, respectively, for MEK at 34 C. V and D are data measured by Lin and Yam, respectively, for BB at 34°C. The solid lines are replotted from Figure 5-2 (at 25°C). The dashed line connects the data of BB at 25°C. 92 mo— I P P I O P .. 2-2 ("o c°“/5) 1/1‘0 93 Consequently values for A can be calculated (see Table 5-6). These estimated values are in reasonably good agreement with those reported in the literature [B-9]. 4. "Old" Polymer Samples As mentioned earlier, the "old" polymer samples might have degraded during the time they were not properly stored. The molecular weights for these degraded polymer samples can be estimated using the data from the "fresh" polymer samples. The lines which correlate the diffusivity data (for the "fresh" polymer samples) in Figure 5-2 are redrawn, and the data for the "old" polymer samples are adjusted to fit these lines for each solvent so that they are consistent with the "fresh" polymer data (see Figure 5-5). Note the that diffusivity data for each polymer falls consistently at a single molecular weight, suggesting that this procedure provides a good measure for the molecular weight of each degraded polymer sample. The molecular weights of PTHF-A1, PTHF-Bl and PTHF-B2 are determined to be 9700, 2200 and 23500, respectively. 94 Figure 5-5. Estimation of the molecular weights for the "old" polymer samples. A. V andUare the data for PTHF-Al, PTHF-Bl and PTHF-B2, respectively. .is the data for BOH. 95 E _ _ x on _ _ M «....2. N ocmmm Joana os- .