THERMODYNAMICS AND DIFFUSION IN POLYMER SOLUTIONS CONTAINING ASSOCIATING SPECIES BY Joe Su—Shien Lin A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1980 ABSTRACT THERMODYNAMICS AND DIFFUSION IN POLYMER SOLUTIONS CONTAINING ASSOCIATING SPECIES BY Joe Su-Shien Lin Thermodynamic and diffusion data of polymer—solvent systems in the literature indicate that agreement between theory and experiment is far from satisfactory. One type of polymer solution which offers interesting insights and for which little has been published is the associating polymer solute in dilute solution. Although association and/or strong intermolecular interactions have been cited by a few researchers as the cause of otherwise unexplain- able experimental results, little effort has been made to study the effects of association of polymer molecules on solution properties. As a result, there exists at present no satisfactory theory which describes accurately the thermodynamic and diffusion properties of polymer solutions containing associating species. An association theory is proposed in this work to describe the thermodynamic and diffusion properties of dilute solutions of an associating polymer solute in an inert solvent. Application of the association theory leads to the prediction that the osmotic pressure of a dilute solution containing associating polymer solute can be expressed as: Joe Su-Shien Lin which is similar to the van der Waals equation not only in form but also in the physical significance of the corres- ponding parameters. The term (K/a)/X2 accounts for effects of intermolecular attraction between polymer molecules and the parameter E represents molar excluded-volume of polymer molecules in solution. The theory also suggests that the experimentally observed osmotic second virial coefficient can be split into two parts: (1) The ”true” second virial coefficient which is directly related to the excluded— volume of polymer molecules and (2) An association term which accounts for the effects of association. Based on the same theory, the concentration dependence of the diffusion coefficient of such a solution can be described by the equation: D = D° m Xasso(l + kdfig + ...) where XasSO is a complex function of polymer concentration that reflects the variation of the average size of diffus- ing species due to intermolecular association. The equa- tion correctly predicts that the diffusion rate first decreases sharply with concentration and then attains an almost constant value or passes through a shallow minimum depending on the magnitudes of the thermodynamic and hydrodynamic interactions between solution components. Joe Su-Shien Lin A Mach—Zehnder diffusiometer was used to measure the diffusion coefficients of eight polymer-solvent mixtures at 34.0°C. Osmotic pressures and osmotic second virial coef— ficients at 34.0°C were determined with a Hallikainen auto- matic membrane osmometer. Monodisperse (MW/MN < 1.05) polytetrahydrofurane (PTHF) with and without OH end-groups were chosen for this work. Two solvents (bromobenzene and methyl-ethyl-ketone) having different capability for block- ing the formation of hydrogen bonds were employed to study the solvent influence upon association behavior. The experimental results indicate that the OH end—groups indeed affect thermodynamic and diffusion prop- erties of dilute polymer solutions significantly. Data from this work as well as from the literature are used to test the validity of the association model with good results. ACKNOWLEDGMENT The author wishes to express his sincere appreciation to Dr. Donald K. Anderson for his guidance and continued support during the course of this work. Grateful thanks are extended to Dr. Robert F. Blanks for suggesting this problem and for his continuing encouragement. The author is indebted to the Division of Engineering Research of the College of Engineering at Michigan State University and to the Amoco Foundation for providing financial support. Special thanks are due to Dr. Hans G. Elias and Dr. Karl Solc of Michigan Molecular Institute whose useful discussions and generous assistance are invaluable to the completion of this work. The understanding, support and encouragement of the author's family, especially his mother, is gratefully appreciated. ii TABLE OF CONTENTS LIST OF TABLES ........................................ LIST OF FIGURES ....................................... I. INTRODUCTION ...................................... II. THEORETICAL BACKGROUND ............................ A. Thermodynamics of Polymer Solutions ........... l. Flory-Huggins Theory ...................... 2. Corresponding States Theory ............... 3. Solubility Parameter Theory ............... 4. Excluded Volume Theory .................... B. Diffusion in Polymer Solutions ................ l. Diffusion in Infinitely Dilute Polymer Solutions ................................. a. The Kirkwood-Riseman Theory ........... b. Flory's Theory ........................ c. Johnston's Theory ..................... d. Fedors' Empirical Relation ............ 2. Diffusion in Dilute Polymer Solutions ..... III.EXPERIMENTAL METHOD FOR MEASURING OSMOTIC PRESSURES IV. OF DILUTE POLYMER SOLUTIONS ................ ....... A. Experimental Aparatus and Principle of Operation B. Membrane Conditioning ................ ....... .. C. Experimental Procedure ....................... . D. Error Analysis . ............................. .. EXPERIMENTAL METHOD FOR MEASURING DIFFUSION COEFFICIENTS OF DILUTE POLYMER SOLUTIONS .......... A. Experimental Apparatus .. ..... . ................ B. Experimental Procedure . ................... .... C. Theory and Calculations ....................... D. Calibration ... ............................ .... E. Error Analysis ............................ .... iii 13 20 23 27 29 29 31 35 37 39 41 41 44 46 49 SO SO 58 62 67 69 V. THERMODYNAMICS OF ASSOCIATING POLYMER SOLUTIONS .. 70 A. Methods for Detecting Association of Polymer Molecules in Dilute Solutions .......... ...... 70 B. Types of Association in Polymer Solutions .... 73 C. Thermodynamics of Solutions Containing Association Polymer Species ....... ......... .. 76 D. Presentation of Osmometry Data and Discussions 87 VI. DIFFUSION IN ASSOCIATING POLYMER SOLUTIONS ....... 101 A. Thermodynamic Basis of Diffusion in Solutions 101 B. Theory of Diffusion in Solutions Containing Associating Polymer Solutes .................. 107 C. Presentation of Interferometry Diffusion Data and Discussions ..... ........... ........ . ..... 118 VII.CONCLUSIONS AND RECOMMENDATIONS .... .............. 132 NOMENCLATURE ......................................... 135 APPENDICES ........................................... 142 BIBLIOGRAPHY ...................................... ... 150 iv II. III. IV. LIST OF TABLES Results of the Calibration Runs on the Interferometer ........ . ............... . ........ Characteristics of PTHF Samples ............... Concentration Dependence of Reciprocal Apparent Number Average Molecular Weights of PTHF-AZ—MEK Solutions at 34.0°C ....... . ........... ........ Concentration Dependence of Reciprocal Apparent Number Average Molecular Weights of PTHF-Bl-MEK Solutions at 34.0°C ............. ....... ..... .. Summary of Diffusion Coefficient Data for PTHF—Solvent Systems at 34.0°C ................ 68 87 88 89 125 Figure 1A 18 10 11 12 LIST OF FIGURES A Free—Draining Molecule During Translation Through Solvent ............................ Translation of a Chain Molecule with Pertur- bation of Solvent Flow Relative to the Molecule ................................... Schematic Flow Diagram of Hallikainen Auto— matic Membrane Osmometer ................... Schematic Diagram of Interferometer Showing Position of Mirrors ........................ Photograph of Mach-Zehnder Interferometer Showing Components ......................... Photograph of Diffusion Cell for Measurement of Diffusion Coefficients .................. Diagram of Diffusion Cell .................. Typical Set of Photographs Taken During Diffusion Run .............................. Fringe Pattern ............................. Concentration Dependence of Reciprocal Apparent Number Average Molecular Weights of PTHF—MEK Solutions at 34.0°C .. ......... .... Molecular Weight Dependence of Observed Osmotic Second Virial Coefficients of PEG SOlutiOnS ...... 0....00000000000 000000000000 Comparison of Osmotic Pressure Data Fittings Based on Various Solution Theories for PTHF-Bl-MEK Solution at 34.0°C ............. vi 33 33 43 51 52 55 56 61 62 63 90 95 98 Figure 13 14 15 16 17 Concentration Dependence of D/Do. Xasso m _ O m and (1+kdpp+ ...). D-D Xassél+kdop+") ..... Concentration Dependence of Diffusion Coefficients of Non-Associating PTHF Solutions at 34.0°C ...... ........ ............ Concentration Dependence of Diffusion Coefficients of Associating PTHF-MEK Solutions at 34.0°C ...... . ..... ... ........... Concentration Dependence of Diffusion Coefficients of Associating PTHF-BB Solutions at 34.0°C .......................... Correlations Between Experimental Do Values of PTHF-Solvent Systems and Fedors' Empirical Relation ........................... vii I. INTRODUCTION The dependence of the rate of diffusion of polymer molecules on concentration has long been of considerable empirical and theoretical interest. However, present under— standing of the diffusion process in polymer-solvent systems is far from satisfactory, and estimates of diffusion coeffi- cients are often unreliable except in some limiting cases such as in infinitely dilute solutions of nonassociating solutes. Although methods are available for predicting the concentration dependence of diffusion coefficients in dilute polymer solutions, they are for the most part valid only for non—electrolyte, non-polar polymer solutions. This situa- tion results in large part from the lack of a usable kinetic theory of polymer solutionsanuifrom the lack (A—B) (II-4) AHm is thus proportional to an interchange energy (Aw), which is in turn related to the energy Eij required to break and form the contacts in equation (II-4): £fih1x;Aw = 1/2(€AA + EBB) - EAB (II—5) By assuming a lattice mode1,* Flory was able to derive the following equation for the entropy of athermal mixing: AS = - k (n1 ln¢1 + n2 ln¢2) (II-6) m where O1 and $2 are the volume fractions of solvent and polymer solute, respectively, and are defined by equations (II—7): n1/(n1 + r n2) (II-7a) ¢1 (1’2 r nz/(n1 + r n2) (II-7b) Here r is the ratio of partial molar volume of polymer * Huggins [2A-4] reached the same conclusion as the result of Flory's without the assumption of fluid lattice. 9 solute to that of solvent and n1 and n2 are the number of molecules of solvent and solute, respectively. The entropy of mixing expressed by equation (II-6) is just the config— urational or combinatorial entropy associated with the large number of ways of arranging the segments of polymer Chain molecules and solvent molecules. It depends only on the concentration of the mixture and does not take into account that effects caused by the difference in structural and chemical nature between polymer and solvent molecules. In AGm it iscxflg/AHnlwhich depends on the nature of the molecules (according to the original Flory—Huggins Theory), which is Characterized by the dimensionless parameter: )’=:ZAw/k T (II-8) Here Z is the lattice coordination number. According to Flory, AHm can be expressed by: AH = k T X n ¢ (II-9) Substituting equation (II—6) and (II-9) into equation (II-3), one gets: AGm = k T(n1 1nd)l + n2 1nd)2 +Xl n1 4 ) (II—10) It was soon found experimentally that X1, as determined from vapor pressure measurements of AGm, was not the same as found from AHm. To account for this, Aw was lO interpreted [2A-5] to be an interchange free energy with enthalpic and entropic contributions, i.e., 1w ‘t AwG = AwH - TAwS and X1 = XH + XS = zin/k T - zAws/k (II-ll) Equation (II-ll) suggests that X1 varies inversely with temperature if both Aw and Aws are independent of tempera- H ture. An important assumption of the theory is that volume Changes taking place during mixing, AVm, are neglected. Thus, not only LS'UuatotalAVm zero, but so are any volume effects on the heat and entropy of mixing. Experimental results [2A-6] also showed that the entropic contribution, X is dominant compared to the enthalpic contribution XH. 5. It becomes clear that the noncombinatorial entropy of mix— ing, which may be caused by the different chemical nature and/or difference of size between polymer and solvent molecules, is extremely significant to polymer mixtures and can be considered as a characteristic property of polymer solutions. It is convenient to study thermodynamics of polymer- solvent systems by measuring the change of chemical poten— tial of the solvent in solution due to the addition of an infinitesimal amount of polymer solute. Differentiation of equation (II—10) for LGm with respect to nl and multiplica- tion of the result by Avagadro's number NA gives: 11 o _ _ _ -1 2 * pl - ul _ R T(ln(l $2) + (l r ) ¢2 + xl¢2) (II 12) where U is the Chemical potential of the solvent in the 1 solution and u; the chemical potential of the pure solvent. By using a series expansion of ln(1 — ¢ ), equation (II—12) 2 can be rewritten: o _ _ l 1 _ 2 '3 ___ _ ul - ul - R T(:2/r + (6 'K) $2 + ¢2/3 + ) (II 13) The sum of the second and higher order terms of $2 in equation (11-13) is defined by Flory as the "excess rela- tive chemical potential" of the solvent: . o _ _ 1 _ ,2 3 ___ _ (ul - u1)E - R T((§ X1) ¢2 + ¢2/3 + ) (II 14) In general, the excess function (u1 - ui)E has two contri— butions, namely enthalpic and entropic contributions. Therefore equation (II—l4) can be generalized (with neglect of o; and higher order terms) as: (ul - “3’12: = R T(k1 —‘i’l) (pg (11-15) Here k1 and W1 are introduced as energy and entropy parameters such that: - 2 . '_—’- 2 _ AH — R T k O , ASl-R W1 ¢2 (II 16) The interaction parameter of the preceding treatment X1 may then be related to these parameters by comparing equations 12 (II-l4) and (II-15), i.e., ) (11—17) A Characteristic parameter of polymer solution can then be defined as: a = kl T/‘gl (II—18) Hence the excess Chemical potential may be rewritten as: o _ .2 - (U1 - “1)E — — R T(l - B/T) ¢2 (II 19) Equation (II-l9) predicts that the excess chemical potential of the solvent vanishes when the solution is at ”theta” condition, i.e., T = 6. Under this special condition, the enthalpic and entropic effects compensate each other, and the interaction parameter X1 has a value of 0.5. Despite the fact that the Flory-Huggins Theory fails to explain many important features of polymer solution thermodynamics [2A-6], the theory has been applied very extensively because of its simplicity. The Flory—Huggins Theory serves as an excellent reference state for polymer solutions just as ideal solution theory is used as a reference state for small molecule solutions. 13 2. Corresponding States Theory The principle of corresponding states of pure liquids rests on the assumption that the potential energy of attraction of a pair of molecules, C, can be expressed by some universal function w (possibly the often used 6—12 potential function) together with two characteristic factors d*, 6* of the molecular species. C(d) = 6*1b(d/d*) (II—20) Here d is the distance between two interacting molecules and the scale factors €*, d* represent the coordinates of the minimum of C(d). Thus, the properties of a given liquid can in principle be determined (if the universal function is known) by these two parameters. The parameters are conveniently embodied in a Characteristic temperature T* and a characteristic pressure P*. For monomeric liquids they are commonly expressed as: T* E*/k (II-21) p* = €*/d*3 where k is the Boltzmann constant. Prigogine and collabor- ators [2A-7] assumed that the principle of corresponding states was also obeyed by polymeric liquids with a polymer molecule considered as a succession of quasi-spherical seg— ments. For polymeric liquids, it is predicted that the contact between two neighboring segments is also described by equation (II-20), in which C(d) is the potential energy l4 and d the distance between the two interacting segments. However, in order to predict the properties of a polymeric liquid accurately, a third parameter C/q, together with c* and d* is necessary to characterize the random configura- tions of chain—like polymeric molecules in a liquid. The law of corresponding states of polymeric liquid was made possible by Prigogine by introducing the concept of "intermolecular degrees of freedom.” The cohesive energy is proportional to Z6* (Z is the coordination num— ber of the molecule) for a monomeric molecule, but to qZ€* for a polymeric molecule where qZ is the number of external contacts made by the r—segments of a polymer chain. q should be less than r because some of the possible ex- ternal contacts of the segments are used up by covalent bonding within the chain. For a lattice model: qz = r(Z - 2) + 2 (II-22) and q is approximately equal to r if the lattice coordina- tion number is l0 or greater. The expansion of the liquid plays a very important role in the theory. It is caused by the thermal vibrations of the molecules acting against the cohesive energies. However, not all thermal vibrations have an effect on expansion. Only the external degrees of freedom of movement of the molecule as a whole count. If segments within a polymer chain can rotate freely about bonds between segments, the total number of external degrees of freedom of the molecule, expressed as 3c, are: 15 3c = r-+3 (r 2 2) (II-23) In other words, the addition of each successive segment to a dimer increases the external degrees of freedom of the molecule by one. If the polymer chain is completely rigid, then 3C = 5, and the number of external degrees of freedom of the molecule is independent of r. Therefore, c depends on the flexibility Of the polymer chain and can be con- sidered as the ”effective number of segments" if each seg- ment were to have three external degrees of freedom like a monomeric molecule. Formally, what appears in the theory is not C, r, or q alone, but ratios like c/q or c/r, which are the number of external degrees of freedom per unit length or polymer chain (and which are taken as a measure of chain flexibility). At least three molecular parameters now Characterize a polymeric liquid: d* and 6* dealing with individual seg- ments, and the so-called "structural factor" C/q (approxi- mately equal to c/r) which has to do with the whole mole- cule. The structural factor was introduced by Prigogine into the Characteristic temperature T* and pressure P* of polymeric liquids by equation (II—24): (II—24) p* = (r/q)d*3 l6 Prigogine and coworkers went one step further in applying corresponding states theory to polymer mixtures by assuming the same universal potential function (equation (II-20)) to characterize all like-like and like—unlike con- tacts. A polymer mixture can then be treated as a “pseudo— pure” liquid characterized by some kind of average charac- teristic temperature and pressure . One can then predict thermodynamic properties of the mixture providing the corresponding properties of both solute and solvent are known. This can be done by first introducing the "surface fractions” of component 1 (solvent) and component 2 (solute): ql xI q2 X2 X1 = x ; X2 = (II—25) ql 1 I q2 x2 ql x1 + qz x2 The average for the mixture is then defined as: C1 C2 = X — 4- X2— (II-26) 1 ql qz Similarly, the average potential energy and average distance between two neighboring segments for the ”pseudopure" liquid can also be evaluated, and the average characteristic temperature and pressure of the mixture are expressed by: (3*) < *> (5*) k ' P = 3 = (II-27) l7 Thermodynamic properties of the mixture can then be expressed in the same form as those of pure liquids. The corresponding states theory predicts [2A-6, 8] that the interaction parameter X1 in equation (II—10) can be expressed by: X, = - (U/R T) v2 + (CF/2R) T 2 (II-28) Here -U is the energy of vaporization of solvent, CP is the configurational heat capacity of the solvent: : C + R (II-29) P, liq. - CP, gas where R is the gas constant and v and T are defined by: T = l - (T*2/T*1) (II-30) v2 = (32/4 + 9 32) (II-31) 6 and D are cohesive energy and size difference parameters, and they are defined by: E (Cg/C?) - l (II—32) E (dg/df) - l (II-33) Thus the enthalpic and entropic contributions to the x1 parameter are found to be: d C 2 P 2 xH = k1 = ((-U + T Cp)/R T) v - T/2R H‘T’ T (II—34) 18 _I dCP 2 — (CP/R) v2 + < C + T — /2 R) r (II-35) P dT Inspecting these equations one can see that the noncombinatorial excess properties are caused by the fol- lowing two effects: (1) Energetic effect: This effect is due to the cohesive energy and size difference between solvent molecule and polymer segment. This kind of effect is exactly the same as in monomeric mixtures. Structural effect: This effect is found to play a very important role in polymer mixtures and is directly related to the structure and chemical nature of the molecules. This effect is specific to polymeric mixtures and it is caused by the significant difference of size or chain length between polymer and solvent. As pointed out by Prigogine [2A-7], the structural effect depends primarily on the configurational specific heat and its derivative with respect to temperature. This explains clearly why in a rigid lattice approach like that used in the simplified lattice model, the structural effect vanishes. The recognition of the structural effect on thermodynamic properties of polymer mixtures has proved to 19 be the key to the success of the corresponding states theory. Many important observations of polymer solution thermodynamics can be explained very well based on such structural arguments [2A—6fZ8,9]. The corresponding states theory is more rigorous than other theories since it allows volume change during mixing process. The theory also gives the temperature and pressure dependence of the interaction parameter X1 [2A-9], and predicts a concentration depend— ence for X1 which is completely ignored by the Flory-Huggins Theory. The corresponding states theory seems to explain some thermodynamic behavior of polymeric liquids/mixtures very well. However, the application of the theory is somewhat limited to qualitative interpretation of experimental results due to, probably, the difficulty of determining the molecular parameters d*, 6*, and c/q. The theory is found to be most useful in predicting thermodynamic properties of oligomer series when the three molecular parameters are known for a reference liquid [2A-lO]. 2O 3. Solubility Parameter Theory Solubility parameter theory [ZA-lla], like the Flory-Huggins Theory, originated from the concept of regu- lar solutions. According to the theory, the polymer (2) - solvent (1) interaction parameter X1 of equation (II-10) is related to the so-called "solubility parameters," 5, of the two components through: —l (6 _ 5 )2 + 8 (II-36) where Oi is defined as the square root of the “cohesive energy density" of component i: 1 ,.V’§ 5. =[=§—] (II—37) Here LE: and Vi are the molar energy of vaporization and molar volume of component i. The quantity 8 is an empiri- cal constant found necessary for systems in which the dif— ference in size and/or shape between solute and solvent is radical [ZA—llb]. 8 was found to have a value of approxi- mately 0.34 for polymer-solvent systems [2A-12]. Similar to the treatments in the Flory-Huggins Theory, the parameter X1 can be split into enthalpic and entropic contributions, i.e., X and XS , where: H - 5.)2 (II—38) 21 x = B (II-39) The parameter XH is related to the heat of mixing of the polymer with the solvent. The use of equation (II—38) allows only positive values for XH. Thus the theory pre— dicts that a polymer is soluble in a solvent when the solu— bility parameters of the two components have nearly the same value. This approach for selecting good solvents for a particular polymer has proved to be very successful. However, exceptions were observed in some cases. Later refinements of the theory [2A-12, 13] suggested the useful- ness of separating the cohesive energy densities into non— polar, polar and hydrogen bonding parts, i.e. I 5i = 5? + 5? p + 5? (II-4O) 1, hp 1, A more general expression for X” can then be written as: V1 _ 2 XH =R—rr- [(61, np - 62’ np)2+(61,p 62,p)+(61, " (S H (II-41) All three contributions to the solubility parameters for most solvents and commercial polymers can easily be found in the literature [2A-l4]. The parameter XS was originally introduced to correct the supposed inadequacy of the Flory combinatorial entropy approximateion. According to the original derivation [2A-16]: 22 x5 =32- (II-42) where Z is the coordination number of the quasi-lattice model and is expected to have a value between 10 and 12. The difference between 0.34 and l/Z is explained by Guggenhaim [2A-15] to be due to the excluded—volume effect. The difference might also be explained as the result of the structural effects discussed in the corresponding states theory. Although the solubility parameter theory is based on the assumption that the volume Change in mixing process is negligible, Patterson [2A-l6, l7] argued that the theory can also take into account the effects of the volume Change of mixing. According to Patterson and co—workers, the sim- ple solubility parameter theory has great similarity to the more rigorous corresponding states theory, and predicts many of the important observations of polymer solution thermodynamics as well. The comparison given by Patterson between the solubility parameter theory and the correspond— ing states theory provides a strong theoretical basis for the solubility parameter theory. Besides the theoretical background, the solubility parameter theory overcomes the serious defect of the Flory—Huggins Theory that the interaction parameter X1 can— not be predicted or calculated from basic data for the pure components, but must be determined experimentally. It seems likely that solubility parameter theory will 23 continue to be widely applied, particularly in the polymer industry, for the foreseeable future. 4. Excluded Volume Theory Excluded volume theory takes a completely different approach from the other three theories to explain the thermodynamic properties of polymer solutions. The theory says that if one can predict the average dimensions of polymer molecules in an infinitely dilute solution, it should be possible to determine the thermodynamic properties of the solution because both are affected by interactions between solvent and polymer segments in the solution. Consider a polymer molecule in an infinite solvent medium. The dimension of the polymer molecule (either the mean—square end—to-end distance or the mean-square radius of gyration ) is determined by the position of each successive segment in the solution. As one would expect, the direction of a given segment, for example the jth segment, is strongly affected by the direction of its predecessor (i.e., the (j-l)th segment) due to bond angle restrictions. It is also influenced to some extent by the directions of other neighbors (the(j-2)th, etc.) due to hindered rotations. It is reasonable to assume that one bond has no appreciable influence upon the rotation of another bond when they are far apart. Thus, such interac- tions between bonds are referred to as "short-range inter- ference." However, two or more segments remote from one 24 another along the chain cannot occupy the same volume at the same time because of their finite volume. In other words, repulsive forces will act between these segments when close to one another. This repulsive force will be altered by the existence of solvent molecules and be affected by the temperature of the solution. Interactions of this sort are usually referred to as the ”excluded— volume effect” and are of long—range nature. It is quite obvious that two types of excluded volume can be distin- guished: the intramolecular and intermolecular excluded volumes. The polymer chain with only short-range interfer- ences is called the ideal or unperturbed chain, and its molecular dimension is called the unperturbed dimension. The term "unperturbed" indicates there is no excluded volume effect. According to excluded volume theory, the average polymer molecular dimension and thermodynamic properties such as osmotic virial coefficients may be expressed in terms of two basic parameters. One is the unperturbed mean-square end-to-end distance o, and the other is the excluded-volume parameter, usually designated by z. The quantity 2 is proportional to the effective excluded volume for a pair of chain segments at infinite dilution and also to the square root of the number of segments in the chain n, i.e.: 25 _ 3 Vz 2 Z — (m) B n (II—43) 3 = ( 3 ) /2 B n2 2 H o where a is the effective bond length and 8 is the "binary cluster integral" for a pair of segments. It represents the effective volume excluded to one segment by the presence of another. Generally, B will have large positive values in good-solvent systems (where preferential attrac- tions occur between the polymer segment and solvent mole- cule) and small negative values in poor-solvent systems. The conclusion of the theory can be summarized by expressions for the molecular dimension and for the osmotic second virial coefficient, A2. aé = 1 + g z — 2.075 22 + 6.459 22 — --- (II-44) a; = 1 + 1.276 2 - 2.082 22 + ——— (II-45) where, a; and a; are defined by the expressions: afi = / / O is the mean square end—to-end distance of the unperturbed polymer chain. The theory predicts that for the non-free-draining case D0 is inversely proportional to 31 the square root of polymer molecular weight, but is independent of the frictional coefficient (Q). For non— theta solutions, Raju [28-2] has suggested that a correc- tion factor (2-2Y) be multiplied to the right hand side of equation (II-52) for the diffusion coefficient at infinite dilution. D0 = 043—6—53- (2 _ 27) (II—53) 0 é o o where the parameter Y is defined by the relation: [0] = xv Mv (II-54) Here [n] is the intrinsic viscosity of the solution and M; is the viscosity average polymer molecular weight. In gen- eral, Y varies from 0.5 for a theta solvent to 0.8 for a good solvent [2B-3]. Thus (2-2Y) represents a small cor- rection for the so—called "excluded-volume effect" of polymer solutions. Raju‘s D0 data for styrene-acryloni- trile copolymer solutions [28-2] are in agreement with equation (II-53). b. Flory's Theory The principle behind Flory's theory of transport properties of polymer molecules in infinitely dilute solutions [2B—4] is similar to that of the Kirkwood-Riseman Theory in that both assume a relative velocity (i.e., the velocity of the solvent with reference to the molecule) gradient along the 32 radius of the molecule. This is illustrated in Figure l. The ”depth of penetration" of the molecule by the flow should be dependent on the structure of the polymer segment, the size and shape of the polymer molecule in solution, and, according to Flory, can be characterized by the variable C/flc. The parameterc/no can be considered as the "effec- tive size" of one segment since it increases with polymer molecular weight provided ;é remains constant. Thus, the lower the ratio c/no, the closer the solvent flow will penetrate to the molecular center, leading to the free- draining case. Flory argued if the ratio 6/00 is great enough such that the solvent flow is excluded from the inner more densely populated regions of the molecule (which is believed to be the case for polymer solution) a condition of comparative insensitivity of the flow pattern t/no should be approached. Flory then concluded that the frictional coefficient f0 of the molecule as a whole (at fixed no) should be independent of the frictional coeffi- cient C of a segment in the limit of sufficiently large C/no. Based on the above discussion, Flory suggested that the molecular frictional coefficient should depend only on the size lg and not otherwise on the nature of the poly- mer if the molecular weight is sufficiently large. That is: 1 fO/T)o = P’i (II-55) 33 Figure 1A. A Free-Draining Molecule During Translation Through Solvent.* Figure 18. Translation of a Chain Molecule with Perturbation of Solvent Flow Relative to the Molecule.* *Arrows Indicate flow vectors of the solvent relative to polymer chain. 34 where P is a universal constant. A random flight model was used to show that P should have a value of 5.11. Using similar arguments, Flory also proposed that the intrinsic viscosity [fl] depends primarily on the ratio 3/2 /M, consisting of a volume divided by the molecular weight: [n] ="“H/2 /M (II-56) where 9 is another universal constant and has a theoretical value of 3.62 x 1021. However, experimental results show that P is approximately equal to 2.5 X 1021. Combining equations (II-55) and (II—56), one gets: -1 / A 3 £0 = p * ”L/ (11-57) (M[n]) 3 Thus, 1 -1 A 1.0 = 52 = m 97/) (11-58) f0 00(M[n]) 3 According to equation (II-57), the molecular frictional coefficient f0 should vary with M to the 0.5 power in a theta solvent, and slightly greater than 0.5 (but never greater than 0.6) in a good solvent. Flory has suggested that f° is related to the molecular weight by: 1 l £0 = Kf M(3 + a ) (II-59) where Kf is a constant, and a' is given by: 35 a' = (Y - %)/3 (II-60) The parameter Y is defined by equation (II—54). Therefore, the theory predicts that Do should vary inversely with M to a power between 0.5 and 0.6. Equation (II-58) allows one to estimate the value of DO provided the viscosity of the solvent no, the polymer molecular weight M and the intrin- sic viscosity [0] of the system are known. c. Johnston's Theory Recently Johnston and Rudin [28—5] derived a simple expression for the diffusion coefficient of an infinitely dilute polymer solution. By combining the well known Einstein equation for the concentration dependence of solution viscosity [2B—6], with the Mark-Houwink equation (equation (II-54)), the mean radius of pseudo-spherical polymer molecules in solution, Re, can be related to the viscosity average molecular weight M; by the expression: —1+Y _ 3 Kv Mv — (10 H NA/3) Re (II-61) where Re is defined by the relation y=§nR (II-62) 3 e Here, X is the mean volume occupied by a single polymer molecule in solution. Johnston went one step further by assuming that Re is the same as the equivalent hydrodynamic radius of a non-free-draining pseudo-spherical polymer Nd VA E F. ‘46. 36 particle, Re, defined by the equation: 0 _ - f — 6 H no Rf (II 63) The frictional coefficient at infinite dilution f0 can then be rewritten as: 10 H NA )_V3 f° = 6 n n ( 0 3 K ELF}, (II-64) V V and the diffusion coefficient at infinite dilution D0 is expressed by: kT 10 R NA 1/ D0 _ a 11-65 6 H no (3 Kv M;+Y) ( ) Equation (II-65) is in exactly the same form as equation (II—58) of Flory's theory, and the corresponding universal constant P-19V3in equation (II-58) can be expressed as: -1 y, l 10 R NA 1A P 0 - 6 (___§_——) (II—66) According to equation (II-66), the universal constant P-1¢V3has a value of 9.80><106 which is about three times as high as the theoretical value 3.39><106 predicted by the Flory theory. The difference between the two theories might arise from Johnston's assumption that R8 = Rf. However, Johnston showed that D0 values evaluated according to equa- tion (II—65) agree with experimental values within ten per- cent for over 40 polymer-solvent systems. 37 d. Fedors' Empirical Relation Recently, Fedors proposed the following empirical relationship for the estimation of the self-diffusion coefficient of pure liquids [28—7]: 4.5 x 10'9 (V* - V) T no V* 1'./3 D“ = (II-67) where D11 is the self-diffusion coefficient in units of cmZ/seC., V* and V are the molar volumes of the liquid at the critical temperatureenuitemperature T(°K), respectively, and no is the viscosity of the liquid at temperature T expressed in Poises. The correlation constant 4.5 x 10-9 and the exponent 4/3 in equation (II-67) were determined by Fedors by correlating data taken from the literature. The relationship is found to be valid not only for small molecu- lar liquids, but also for high molecular weight polymeric liquids. Fedors suggested that critical molar volumes for polymeric liquids (which are generally not measurable) can be calculated using a group contribution method which he described [2B-8]. He also proposed that the limiting diffusion coeffi- cient of an infinitely dilute solute (D0) can be related to the self-diffusion coefficient of the solvent Q1 by the symmetrical relationship [28—9, 10]. 1 o —o 6 _ _ é _ D (V3 — v2) - D11 (V? V1) (II 68) Here, V3 and V? are the molar volumes at the critical 38 temperatures for the solute and solvent, respectively, V; is the partial molar volume of the solute in infinitely dilute solution, and V1 is the molar volume of the solvent. Fedors mentioned that equation (II—68) is suitable for both liquid-liquid and gas-liquid systems. Combining equation (II-68) with equation (II—67), one gets: 3 4.5 x 10'9 (v; — v,)/2 T D0 = , 1 (II—69) v*/3(v* V°)2 n01 2’2 from which the diffusion coefficient at infinite dilution can be estimated directly, given no, V and V” 1 2. Equation (II-69) was tested by Fedors for some 52 solutes in eight different solvents. The average percent error was found to be 22.3 percent. The most serious errors occur in those systems in which one or both of the components associate to form moltimers. The simple expression of equation (II-69) seems to offer a reasonable correlation between the limiting diffusion coefficient Do and properties of pure solute and solvent. 39 2. Diffusion in Dilute Polymer Solutions Most diffusion data in the literature indicate that the diffusion rate of a polymer in solution varies linearly with concentration in the dilute region. In general, diffusivities are found to increase with increasing polymer concentration, although there are some systems which show the opposite behavior. These later systems are of special interest in this work, and are considered in detail later. Whichever the case, the diffusion coefficient for dilute polymer solutions can conveniently be expressed by a series expansion of the following form: D = D0 (l + kd pp + -——) (II-70) where pp is the polymer mass concentration. Thus the dif- fusion coefficient for dilute solutions can be predicted from a knowledge of the parameters D0 and kd. Recently Vrentas and Duda [ZB-ll] showed that the param- eter kd can be predicted using the equation: (II-71) Here A2 is the osmotic second virial coefficient, Mk is the number average molecular weight of polymer, and V3 is the partial specific volume of the infinitely dilute polymer. The quantities b1 and k5 are defined by series expansions for the concentration dependence of the partial specific volume of the solvent (V1) and the frictional coefficient of the polymer. (l + b pp + ---) (II—72) f = f0 (1 + k pp + —--) (II—73) ks accounts for the effects of intermolecular hydrodynamic interactions on the frictional properties of polymer molecules in solutions. According to equation (II-71), kd includes three contributing effects: (1) the thermodynamic term ZAzfifi; (2) the hydrodynamic term ks; and (3) volumetric effects b1 and 2V;. Volumetric effects generally are insignificant but become important when kd is small. When the thermo— dynamic effect dominates the hydrodynamic effect, kd is positive and the diffusion rate increases with concentra- tion. On the other hand, the diffusion rate decreases with concentration if the hydrodynamic effect becomes dominant. It should be pointed out that equation (II—70) is valid only for non—electrolyte non-associating polymer-solvent systems because the concentration dependence of the dif— fusion coefficient of electrolyte and/or associating poly- mer solutions is, in general, nonlinear even within the dilute region. III. EXPERIMENTAL METHOD FOR MEASURING OSMOTIC PRESSURES OF DILUTE POLYMER SOLUTIONS A. Experimental Apparatus and Principle of Operations Measurement of osmotic pressures in determination of number average molecular weights has not been used exten- sively in the past because of the time-consuming nature of the Classical method and the attendant errors caused by the lengthy observation period. Bruss and Stross [3A—l] have discussed some of the dynamic methods described in the literature [BA-2, 3, 4] with attention to the errors caused by solute permeation of the membrane and the importance of completing the measurement as rapidly as possible. A Hallikainen automatic membrane asmometer (Model 1361 - Code D, designed by Shell Development Co.) was used in this work to measure osmotic pressures of polymer solu- tions. The instrument needs only 10 cc sample solution, which is then isolated in the osmometer cell using the inlet and outlet valves. After a period of about 6 to 20 minutes, depending upon the permeability of the membrane, the osmotic pressure is observed on a Veeder-Root type counter which reads osmotic pressure head of the solvent used. A built-in recorder, the pen of which is directly 41 42 driven by a balancing servo, enables the operator to follow the servo balancing of osmotic pressure and degree of mem— brane permeation, if any. Figure 2 is a schematic diagram of the sample flow scheme. The osmometer cell consists of two cavities sepa- rated by a semipermeable membrane. The sample cavity includes a thin metal diaphragm which is responsive to volume Changes and has suitable valves to admit and isolate a sample. The solvent cavity is connected to a servo— driven plummet in a vertical tube of solvent. The plummet is capable of Changing the solvent head, thus causing sol— vent to flow through the membrane in either direction depending upon the differential pressure. Displacement of the metal diaphragm caused by flow through the membrane is sensed as a capacitance change of an oscillator circuit, causing a servo to null the solvent head for zero osmotic flow. At null balance the servo comes to rest with the solvent meniscus depressed by the amount of the osmotic pressure. The mechanical counter geared to the plummet servo registers this depression and hence the osmotic pressure. The recorder shows the entire balancing cycle and (by the slope of the curve following the maximum indi— cated osmotic pressure) the degree of solute permeation through the membrane. A more detailed description of the experimental procedure is given by Rolfson and Coll [BA-5]. 43 pmppoumm Leucsoo qub L000: umawfifias< O>me + O>me Cam ccmunem: UwumEOu=< smeamxaflam: wwwauw mo EmLOmHQ 30Hm Ufiumamzom .N Unavam Lew .' umeOLQm = was? *1 __ uwquOEmz _ heumaaaomo .m.m _ umuweoemo _ _ Unfimcmm >uaumamo ¢>Hm> ucm>Hom umduzo . - Damion null.|_ 38> 0036 im OCOLDOOHm cache OaQEmm % EOmLLQmMD m w HO>¢J mosau>m x0030 O>Hm> W I A. .l mumumam Ocfium~=mcH 7 n. L Ocmuneoz o>~m> uwficH uco>Hcm OUCaL>m uco>How 44 B. Membrane Conditioning An important factor in osmotic pressure measurements is the careful conditioning of the membrane to the particu- lar solvent employed. Membrane materials of the greatest interest to the polymer chemist and other research workers Concerned with osmotic pressure measurements are the swellable organo—colloid materials. Membranes made of these materials swell to a different extent in each solvent. Thus, the permeability of a membrane will vary with the solvent. The membranes are generally packed with a solution which prevents bacterial growth. This solution must be removed from the membranes. When aqueous solutions are to be employed, the membranes should be rinsed under running water and then laid for sixty minutes in distilled water before use. If the solvent to be used is different from water the membranes must be conditioned; first for removal of all water, and second for uniform permeability to the type of solvent. The following conditioning procedures were used in this work with successful results. The mem- brane must be handled gently with a soft, smooth instrument or gloved hands. It must not be allowed to dry during or after conditioning. Ethanol drying procedure. (The wash intervals indicated are the minimum recommended. Longer washes may be used without adverse effects.) 45 (1) Wash in distilled water for one hour (2) Wash in 25% ethanol/water for % hour (3) Wash in 50% ethanol/water for % hour (4) Wash in 75% ethanol/water for % hour 1 (5) Wash in pure ethanol for 5 hour (6) Wash in fresh, absolute alcohol for % hour In order to condition a membrane to another solvent, condition in 25% increments of increasing solvent concen- tration. For some solvents, it has been found necessary to condition the membrane through an intermediate solvent (e.g., from ethanol, through toluene, to dimethyl— formamide). To condition from ethanol to a solvent. (1) Wash in 25% solvent/ethanol for % hour (2) Wash in 50% solvent/ethanol for % hour ... (3) Wash in 75% solvent/ethanol for 1 hour (4) Wash in pure solvent for 1 hour After conditioning, the membrane should be stored in fresh, cool solvent until ready for use. The permeability of a membrane is better retained if it is not heated prior to use. For high temperature oper- ation, the membrane is installed on the cell block and the temperature control is set before the main power is turned on. As the cell heats the membrane is conditioned to the new temperature. (1) (2) (4) (6) (7) (8) (9) (10) 46 C. Experimental Procedure Part a: Start-Up Procedure Install the membrane. Set the desired operating temperature by adjusting the SET POINT potentiometer. The minimum operating temper- ature should be at least 10°C above ambient temperature for satisfactory temperature control. Turn on main power and wait about 1% hours for thermal stabilization. Flush both solvent and sample cell with clean solvent. Make sure that any air trapped in the cells is removed. Run the manometer plummet down until the mechanical counter reads zero. Make the solvent meniscus in the manometer tube coincide with the reference position index mark by using a syringe to insert extra (or withdraw excess) solvent through the SOLVENT OUTLET. Adjust the oscillator so that the recorded pen is on the chart reference line. This will be the base line. Fill the sample funnel with the sample polymer solu- tion. Open the OUTLET VALVE to allow the sample to flow into the sample cell, then firmly close. Turn on the heater and wait for thermal equilibrium. Close the sample INLET VALVE and immediately open sample OUTLET VALVE. The pen will move to the bottom (3) (4) (5) 47 of the chart. Wait for a few minutes and then close the sample OUTLET VALVE. The resulting osmotic pres— sure is indicated by the difference between the base line reading and the new steady—state reading. Part b: Recommended Procedure for Molecular Weight Determination. Flush sample and solvent chambers with pure solvent from the same container. All solvent used, both the pure and that mixed with the solute, should be from the same supply. Make two or three blank runs, each of 10 to 15 minutes duration, with pure solvent on both sides of the membrane. These readings will indicate the reproducibility of the base line. Make a sample solution of 0.5 grams per 100 cc of solvent. Flush the sample cell thoroughly with two funnels full of this solution. Fill sample cell with above solution and make a trial run. This will indicate the concentrations to use for the molecular weight determination and indicate the presence of solute components small enough to pass through the membrane. Flush the sample cell, and record another run with pure solvent. Check the final value with that of step (2). If this reading is below the initial base line, then there has been solute permeation into the solventside. (6) (7) (9) (10) 48 Make up several concentrations (using the result of step (4)) that will provide on-scale experimental points. Starting with the most dilute solution, flush the sample cell using two funnels (about 14 cm3). Retain the last 2 cm3 in the cell. Take a 5 to 10 minute osmotic pressure reading. Repeat steps (7) and (8) using sample solutions of increasing concentration until at least five experi- mental points have been determined. Check the base line before, after, and several times during operation as a means of determining reference stability. Where readings are taken over an extended period of time, evaporation of solvent as well as ambient temperature changes may cause a slight shift in the base line. Make a plot of the base line posi— tion as a function of time and use it to correct readings for which the shift has been appreciable. 49 D. Error Analysis The main cause of error results from the permeation of solute molecules through the membrane. This became partic- ularly evident when the solute molecular weight was too low. When solute permeation is detectable, extrapolation of osmotic pressure to zero time is necessary to minimize error. The relatively poor temperature control system of this instrument might also cause observable error. Thisvnns minimized by setting the controller at least 10°C higher than room temperature. The repeatability of osmotic pres- sure measurements in this work is found to be within 1 0.02 cm. IV. EXPERIMENTAL METHOD FOR MEASURING DIFFUSION COEFFICIENTS OF DILUTE POLYMER SOLUTIONS A. Experimental Apparatus A Mach—Zehnder interferometer described by Bidlack [4A—3] was used to study binary diffusion in the polymer solutions. The technique involves forming a sharp inter- face between a more concentrated polymer solution with a less concentrated solution in an optical cell where diffu— sion occurs. The cell is immersed in a constant tempera- ture bath, and the diffusion process is followed by measur- ing the rate of change of refractive index of the solution with the Mach-Zehnder [4A-1] interferometer. The two solutions of slightly different concentrations are care- fully layered, one On top of the other, into the cell and free diffusion allowed to take place after a sharp inter- face is formed between the solutions. The diffusion coef- ficient obtained in this way is assumed to be that of a con— centration which is the average of the two solution concentrations. A diagram and photograph of the interferometer are shown in Figures 3 and 4. The various components of the interferometer are supported by ordinary laboratory bench carriages stationed along a continuous rail composed of 50 51 wuusom Dcaom mama ocflumemHHoo H mouufiz wuouuflz mo :Oauflmom Usazocm umumeoumwpmucH mo Emwomfla Oaumsmcom .m Ousoflm N Louuaz m wouuflz news I mumemu v LOLDHZ wcmam OOOEH - .mpcwcomeou mcflkonm umumeoummwmucH uwpcnmwlcumz mo Lawnmouocm .v wusmfim 53 three optical benches. These in turn are bolted to an I-beam mounted on a concrete block on rubber cushions to dampen outside disturbances and vibrations. Monochromatic light from a Cenco quartz mercury arc lamp source, filtered to isolate the 5461 °A green mercury line, is collimated and then split by a half—silvered mir— ror (1). Half of the beam is reflected to a full reflect- ing mirror (2), and the other half passes through a full reflecting mirror (3). The two beams are then combined at a half-silvered mirror (4). Constructive interference of the two beams occurs when the path lengths 1—2—4 and 1-3-4 are identical or differ by a whole multiple of the wave— length of the incident light. The mirrors are adjusted to give straight, vertical, parallel fringes. The interference beam is arranged so that it can be photographed directly by a camera. The camera consists of a three-foot long aluminum tube (3.5 inches diameter) con— taining a lens (343 mm focal length) set in the end towards the interferometer. The image is focused on a type M, 3% x 4% inch Kodak plate located at the opposite end. A lever mechanism on the plate holder allows fourteen succes— sive exposures per plate. The magnification factor of the camera was found to be 1.929. The diffusion cell is fixed in a water bath maintained at 34.0i0.1°C by a proportional temperature controller. The water bath consists of an 18 X 18 X 18 inch stainless steel tank covered with 3/4 inch plywood and resting on the 54 cement block without touching the interferometer. Two round optical flat windows, 3 inches in diameter, are clamped and sealed into the water bath and aligned to allow passage of the light beams through the bath and cell win— dows. Distilled water is used as the temperature control- ling medium. Figures 5 and 6 show a photograph and a diagram of the diffusion cell. The main body of the cell consists of a % x 3% inch slot cut into a stainless steel plate with optical flat windows clamped over the slot to form a sealed Channel. The channel is situated to allow both light beams to pass through it; thus, a vertical concentra- tion gradient in the solution across one of the beams results in a fringe displacement pattern that is a direct plot of refractive index versus distance. All parts of the cell which are in contact with the liquid solutions are stainless steel or glass. A frame is bolted to the cement block and positioned above the bath so the cell can be hung from the top and immersed in the bath. Two small position pins located on the frame insures that the cell is always placed in the same position. The cell is provided with two inlets (one on the top and the other in the bottom) and two outlets directly across from each other about one-third of the way up the channel sides. The two solutions of slightly different concentrations are slowly flowed simultaneously into the Figure 5. Photograph of Diffusion Cell for Measurement of Diffusion Coefficients. 56 Glass Solution Reservoirs made from 50 cc. syringes q‘——pFilling 7i if i -—F I )' Syringe mi __"_4. Valve 2 Valve 1 Cell Cell Window _,,’d; Body Valv Boundary Sharpening Slits Siphon e 4 _/ Valve 3 8 Valve 5 Figure 6. Diagram of Diffusion Cell. 57 cell, the denser solution through the bottom inlet and the lighter through the top, and out through the two outlets. A sharp boundary is thus formed between the two layered solutions. This boundary is located in the center of the lower beam. All the valves are then Closed and the the solutions allowed to diffuse freely. 58 B. Experimental Procedure The following procedure was found to be the most successful for measuring the binary diffusion coefficient in liquid polymer solutions: (1) (2) (3) All cell valves except valve 2 are Closed and,approxi- mately 30ccB .> wusmam . .1.“ 2: 62 C. Theory and Calculations Consider a differential volume \. \\ . . . . X) 0 \~ dx ‘\ element in the difoSing section of \\ the cell as shown in Figure 8. By ‘\ + ‘\ > \ \FU - . X20 ______ 0 setting up a material balance on the ~\F \.9 . . . \ \: differential volume element which 6 \ \0 describes the mass transfer in and x< 0 \~ \‘ \ \ out of the element, we obtain \\ ~\ \ \ \ \ N 3 C _ l 3C (IV—l) Fig. 8: Diffusion Cell 23xz I D it Coordinates with the following boundary conditions: Case I (x>0) i) x + m t 2 0 C — c1 ii) t = 0 C = C1 w > x > 0 iii) x = 0 c = (C1 + C1)/2 t 2 0 Case II (x<0) i) x + -w t 3 0 c = c2 ii) t = 0 c = c2 0 > x > -m iii) x = 0 c = (c1 + C2)/2 t 2 0 where C is the concentration and x is defined in Figure 8. In order to solve equation (IV-l), it is necessary to make the following assumptions: (1) the concentration dependence of the diffusion coefficient D is negligible over the small concentration range involved, and (2) the diffusion gradient has the properties of normal distribu- tion curves. 63 Equation (IV-l) may be solved with Laplace transforms or by other means to give the following identical solution for both case I and case II. ) (IV—2) where 9315 the concentration at the zero position in the cell, and as a result of assumption (2) above, is equal to %(C1 + C2). The refractive index, n, may be assumed pro- portional to the concentration, so that: o l erf. ( x n - n / 4Dt ) (IV-3) I NI Essentially the fringe pattern is a plot of refractive index versus distance in the cell so that the refractive index difference may be rep- O-——+ resented by the number of fringes displaced. For the A relationship between the xk method development and the x. fringe pattern, refer to Fig. 9, where J is the x0 total number of fringes crossed from top to bottom; k is the local fringe number in the top half of the cell, Figure 9. Fringe Pattern 64 and j is the local fringe number in the bottom half. Let x. and xk be the measured distance corresponding to fringes J j and k respectively. Thus, where x > 0 and equation (IV-3) becomes x —L = erf—l (35—2—3) m-“ / 4Dt J Similarly, where x < 0 n - nO - J _ 2. n - n - 2J 2 and x. —1 J _ 2. ————l— = erf (——3——l) (IV-5) / 4Dt The exact midpoint of the diffusion zone is difficult to determine; however, the distance, X + x k j' is ea511y determined by difference measurements. Therefore: x. x _ —l . -l I + k — erf (J 3 2!) + erf (2k 3 J) (IV—6) / 4Dt J 4Dt The cell distance is not equal to the distance measured on the photographic plate because of the magnification of the image by a factor, M. Therefore: x. + x x.' + x ' k k 3 = 3 (IV-7) / 4Dt M 74Dt where xj' and xk' are distances on the photographic plate. Hence x.' + x ' 2 j k ] _1J—2J -12k—J (——3———)+ erf (———j——J l [ 4M2 erf Dt (IV—8) For each exposure, value of the function 2 ] x.' + x ' [ A3 k erf-1(g—fi—El)+ erf-1(EEELJE) were determined for several j's and k's and averaged. The averages for several exposures were plotted versus exposure time, and the slope of the resulting line was determined. Thus, D = E1283 (IV-9) 4M2 See Appendix A for details of a sample calculation. This diffusion coefficient is assumed equal to the mutual diffusivity at the average concentration, _ 1 co - 1(C1 + C2). The distances on the photographic plate were measured with an optical comparator made from a Gaertner microscope fitted with a traveling eyepiece. The traveling eyepiece could scan a total distance of 5 centimeters by turning a 66 crank and the distance traveled was indicated on a vernier scale accurate to 0.0001 centimeters. 67 D. Calibration For this study, the accuracy of the interferometric technique was established by measurement of the concentra- tion—dependent diffusion coefficient for the binary system of sucrose-water at 25.0°C. This particular system was chosen because accurate, widely accepted diffusion data are availableftu'comparison. The accuracy of the method used in this work was tested by comparing diffusion coefficients at 25.0°C for four aqueous sucrose solutions with those reported by Gosting and Morris [4D—1]. The data of Gosting and Morris have been carefully checked by several investi- gators [4D—2, 3, 4], and are thought to be accurate to :0.2 percent. Gosting and Morris fit their data to the following empirical relationship using the least square technique: DS=5.226 (1 — 0.0148 co) x 10'6 i 0.002 (IV-10) where DS is the binary diffusion coefficient for sucrose- water system at 25.0°C, and cO is the concentration of sucrose solution in grams of sucrose per 100 cm3 of solu- tion. A comparison of results are summarized in Table I. It is concluded that the precision of the interferometer is no worse than i 2 percent. 68 mmw.o COAumfl>mQ pumpcmum Ham.o.u omH.m mna.m 6.0 v Hmm.o.u woo.m mHH.m v.H m ohv.a.+ vmm.m wva.m O.H N mvm.o+ mam.m voa.m m.o H xpoz mflsfi HHIDmH .wmm cofluma>wo Aomm\meomoa xv Jufiuooa\mmouu:m mo mEmuo .oz cam Oomucwuumm ucmauamwwou CmeSMMHQ mmouusm mo COaumuucmocou HmucwEHquxm .LOOOECLOMLmucH wcu CO mcsm Coflumpnflamu mcu mo muasmwm .H OHQmB 69 E. Error Analysis Some sources of error in measurement of diffusion coefficients include: (1) The accuracy in determining the fraction of a fringe when measuring the total fringes for a particular exposure. The percentage error increases as the total number of fringes decreases. (2) The assumption that the concentration dependence of diffusion coefficient is negligible over the narrow concentration range involved in the experimental run. (3) The accuracy of the magnification factor of the camera. (4) Error in measuring distances between fringes on the photographic plate. (5) The accuracy in determining the slope defined by equation (IV-9). The average percentage error of diffusion coefficients obtained in this work is estimated to be within i 2 percent. V. THERMODYNAMICS OF ASSOCIATING POLYMER SOLUTIONS A. Methods for Detecting Association of Polymer Molecules in Solution [SA—l] The purpose of this section is to review the methods available for a quantitative evaluation of solutions con- taining associating polymer molecules. A quantitative description is needed for correct interpretation of associating solution properties. The phenomenon of association is discussed in the literature under different names: aggregation, self—asso- ciation, multimerization, complex formation, etc. All processes leading to the formation of complexes of higher particle weight from molecules of lower molecular weight via physical bonds can be called sociation processes. They may occur between like molecules or unlike molecules. The sociation process between like molecules is called "multimerization." Solvation is restricted to the socia- tion process between solute and solvent. A non-multimer- ized molecule is called a unimer. Like unimers of molecu- lar weight MI can multimerize to form a particle of weight Mn. The multimerization number n describes the number of unimers in a multimer particle. Homologue series of poly- mer molecules with like constitution are considered as 70 71 like molecules. Basically, two groups of methods can be used to detect and determine multimers: group specific methods and molecule specific methods. Group specific methods look at the structure of a group and its interaction with other groups. Typical examples of group specific methods are spectroscopic methods (such as infrared or ultraviolet) and nuclear magnetic resonance. Molecule specific methods look at the molecule as a whole, e.g., their molecular weight and/or their volume. Typical examples are membrane and vapor pressure osmometry, light scattering, ultracentrifu- gation, viscometry and gel permeation chromatography. A multimerizing polymer molecule may have only few associogenic groups. Because these are only a small frac- tion of the total groups present, they may escape detection by group specific methods which typically become insensi- tive at levels of about 1% "impurities." The particle weight of the multimers may, however, increase drastically. One associogenic group in a polymer with a degree of poly— merization of one thousand represents only 0.1% of the total groups, but the particle weight increases 100% if dimerization is complete. Obviously, molecule specific methods are the prime choice for the detection and determination of multimeriza— tions. It should be remembered, however, that they are influenced by intermolecular interactions only. Informa- tion on intramolecular association and the structure of 72 multimers is difficult to get from molecule specific methods. Therefore, both molecule and group specific methods must be employed in order to elucidate the multi— merization process. 73 B. Types of Association in Polymer Solutions Two basic types of association may be distinguished: open and Closed association. Open association [58-1, 2] is a consecutive association in which all types of multimers appear: A1+ A ——‘ A4 (V-l) A1 + AL4;::: An where A1 represents i-mer and Ki is the association equilibrium constant of the formation of i—mer. Closed association exhibits an all—or—none process in which only unimers and n-mers are present: where KC is the equilibrium constant of the n-merization process. Combinations of these types of equilibria are of course possible. However, discussions throughout this paper will be restricted to open association because open 74 association seems to be the most important type for synthetic polymer molecules in solutions. Moreover, it has been shown [SB—2] that a curve fitting using closed associa- tion models can very often be replaced by using an open association model with only one equilibrium constant. Association may be further subdivided according to the variation of the number of associogenic sites per molecule with the degree of polymerization in polymer homologeous series. The number of groups capable of association may be constant for each molecule, regardless of its length. A simple example is a linear unipolymer Chain with associat- ing endgroups. This type of association can thus be called ”end-to-end” association. This term should not imply that this type of association occurs exclusively via endgroups. It should merely indicate that the number of associogenic groups per polymer molecule is independent of its length. On the other hand, the number of associogenic groups may increase proportionally to the Chain length. Associo- genic groups may be special groups or sequences of consti- tutional or configurational groups. This type of associa- tion can thus be called "segment-to-segment" association. In the case of end-to—end association, the proper choice for describing the polymer concentration should be molar concentratiion, whereas in segment-to-segment asso- ciation the association equilibrium constants have to be based on the mass concentrations. This work deals 75 exclusively with open end-to-end association. For those who are interested in segment-to-segment and/or closed associations, references can be found elsewhere [SB—2, 3, 4]. 76 C. Thermodynamics of Solutions Containing Associating Polymer Species The present section is concerned with strong orienta- tional effects of polymers on the thermodynamics of polymer solutions. Important and typical examples are solutions containing associating polymer molecules with specific functional endgroups. The foundation of most existing polymer solution theories (such as those discussed in Chapter II) are based primarily on the assumption that the polymer molecules are homogeneously distributed throughout the solution. This assumption is expected to break down when strong intermolecular interaction (e.g., hydrogen bonding) occurs between polymer molecules. Since the interaction energy of hydrogen bonding is very large com- pared to other intermolecular energies, methods like the perturbation approach applied in statistical thermodynamics to account for effects of weak intermolecular interactions often become useless. In fact, there exists at present no satisfactory theory of strong orientational effects from which one may deduce the thermodynamic properties and especially the excess functions from intermolecular forces and properties of the pure components. In the history of thermodynamics of solutions, the earliest and probably the most popular approach for describing solution properties of associating solutions is the "chemical approach" first proposed by Dolezalek [SC-l]. Although Dolezalek's association model gained some success 77 in interpreting the negative deviations from Raoult's law in some systems, his oversimplified picture of associating solutions also drew serious criticism. However, it remains as a good basis for the treatment of associating solutions in which specific interactions do exist between molecules. The remaining part of this section is dedicated to presenting an association model developed by the author for dilute solutions of associating polymers in inert solvents. This association model describes favorably the thermodynamic properties of associating polymer solutions, and provides good explanations for some discrepancies that exist between experimental data in the literature and theoretical predictions of existing polymer solution theories. Consider a polymer-solvent mixture in which polymer molecules associate with each other to form larger particles according to the open associating process described by equation (V-l). The solvent molecules of the system are assumed to form no associates with either solvent or poly- mer molecules. It is also assumed that there is no overall volume Change of solution due to intermolecular association and the association constant is independent of molecular size, K = K = - - - = K = K (V—3) All i-mer concentrations can be related by the association constant, K. 0 ll 78 1-1 i (v—4) where C1 is the molar concentration of i—mer. Since the true molar concentration of polymer solute, CP, is the sum of the molar concentrations of all i—mers, Also since the less than that —.I therefore, the -1 C + K C2 + K2 C; + --- + K” C? (V-S) molar concentration of (i+l)—mer is always of i-mer (otherwise Cp will not be finiteL true molar concentration of polymer solute can be expressed as: CP C1 (V-7) 1 - K C1 which can be rewritten: 79 C = ———————— (V—8) It can be seen from equation (V-5) that Cp represents the concentration of ”kinetically independent" polymer molecules per unit volume of solution. The osmotic pressure of a polymer solution can be expressed in terms of a virial expansion according to statistic thermodynamics [SC-2]. H 0 RT 1 2 = _— p p -—— — MN + A2 p + A3 p + (V 9) where A2, A3, etc., are the osmotic virial coefficients, H is the osmotic pressure of the polymer solution at concen- tration pp (in grams of polymer per unit volume of solu- tion), and MN is the number average molecular weight of poly- mer solute. For most non-associating polymer-solvent sys- tems, plots of H/Dp versus pp are found to be linear in'Hue dilute concentration range as predicted by equation (V-9). The slope of the plot is equivalent to the second virial coefficient, and extrapolation of the plot to zero concen- tration gives the reciprocal of the number average molecu— lar weight of the polymer. Equation (V-9) also predicts that for a non-associating theta solution (a theta solu- tion is defined as a solution in which all virial terms in equation (V-9) vanish), H/pP R T is independent of concen- tration and is equivalent to the reciprocal "true" number 80 average molecular weight, TT pp R T ) 1 ( M- (V-lO) e N However, for associating polymer—solvent systems, the true number average molecular weight in equation (V-lO) should be replaced by pp/Cp, —l—) = ——l-—.=£ 1 (v-11) P P P (MN)app,8 Equation (V—ll) also serves as the definition of the apparent number average molecular weight at theta condition. It should be noted that (MN)apg 0 is an increasing function of concentration with its functionality depending on the type of association [SC-3]. In analogy to non-associating systems, we may write [SC-4]: "'%_T = (RN):pp,8 + A; pp + A5 p; + ... (V-12) for associating systems not at theta condition. By doing so, we actually assume that all intermolecular associating forces between polymer molecules can be separated from all other interactions of the components. The superscript star is added to individual virial coefficients because they are different from the corresponding virial coefficients defined in equation (V-9) in that those in equation (V-12) 81 are for the associated complexes (multimers). Since the mass concentration of solute is given by: = 2 C M. = M 2 iC. (v-13) where erepresents the average molecular weight of i-mer. Substitution of equation (V-8) and (V-13) into equation (V-ll) yields: — — P N (M ) = M + _ (V—14) N app,0 N (MN)app,9 Equation (V-l4) is the concentration dependence of apparent number average molecular weight for theta solu- tions with consecutive associations. The result has been previously derived by Meyer and van der Wyk [SC—5], and also by Solc and Elias, using a more rigorous approach [SC-4]. It is convenient to define a, the “degree of association” as: n X 1 Ci — ._ (M ) a = 1-1 = N-app,8 (V-15) “ M Z Ci N i=1 Combining equations (V-14) and(V-15) gives: 4Kp 1 (1: 0.5 [1 + (l + )5] (V-l6) 82 When there is no association (K=0)Ciequals 1.0. It can be seen from equation (V-l6) that the degree of asso- ciation is an increasing function of polymer concentration. Equation (V-l4) can be rearranged to: (M ) = M" — Kpp/(MN)ap“B (v-17) N app, 8 N __ Kpp M (1+- ) N “‘N)apge Realizing that a = (M /MN, equation (V-l7) can be N)aPP.e reduced to a more compact form: Kpp OLM _ -1 -—_1 N M (l + ———) N M a N Equation (V-18) is a more convenient form of equation (V-14), since (M ) is an explicit function of concen- tration in equation (V—18). It describes the variation of the number of kinetically independent molecules as a func— tion of concentration in the absence of all other interac- tions. Substituting equation (V-18) into equation (V-12), one gets: Kop 0M 11 - _1_ * * 2 _ N _ Pp R T — (—N A pp + A3 pp + .. ) Kpp (V 19) MN (1 +—'_-—) 83 or H - (—l- + —-7—A2 MN + ___—A 3 MN + ) - K/OL (V'ZO) R T ‘ y y 13 "' y (y + K/O) where V represents the average volume occupied by one mole of nonassociated polymer molecules (unimers). According to excluded volume theory, the third virial coefficient A3 is Closely related to the second virial coefficient A2 by the relation: A3 a g A? MN (V-22) where g is a constant for a particular polymer-solvent system and is a measure of "goodness” of the solvent. The value of g should range from zero for a theta solvent to an asymtotic limit of 4/3 for a good solvent [SC-6]. If we assume that the relation between A2 and A3 (equation (V-22)) is also valid for A5 and Ag and g = 1, equation (V—20) can be approximated to: 11 _ 1 K/a RT ‘ y — B ’ y (y + K/a) (V'ZB) —2 where B = A5 MN (V—24) In a very dilute solution or when the association is relatively weak, 84 K/a << 2 (V—25) and equation (V-23) can be reduced further to: RIT = §———§ _ —VT (V-26) The assumption of g 1 made to arrive at equations (V—23) and (V-26) introduces minor error in evaluating the osmotic pressure of moderately concentrated solutions. However, the error should be negligible for dilute polymer solutions in which we are most interested. Zimm [SC-7] pointed out the exact formal correspond- ence between the osmotic pressure equation (i.e., equation (V—9)) and the virial equation of state of an imperfect gas. It is interesting to note that a similar formal correspond— ence also exists between equation (V—26) and the well-known van der Waals equation of gas, __ = _____ _ __ (V-27) where v is the molar volume of the gas under pressure P and temperature T. The parameter b accounts for the finite volume of the gas molecules with its value depending on the size and nature of the gas molecules and the term a/v2 is a correction made to account for the attractive forces that exist between molecules. The analogy between the two equations is hardly surprising considering the similarities that exist between the two systems. The osmotic pressure 85 equation derived according to the associating polymer solution model is analogous to the van der Waals equation, not only in form but also in the physical meaning of the corresponding parameters. In equation (V-26), the term (K/OPE2 accounts for effects of intermolecular attraction of polymer molecules via specific (such as hydrogen bond- ing) and/or non-specific (such as van der Waals) interact- ing forces. The meaning of parameter E in the association model is not very clear from its definition (i.e. equation (V-24)). To understand the physical significance of B, we have to first understand the physical meaning of A2 the second virial coefficient. In fact, Zimm [SC-7] has shown that A2 can be directly related to g, the volume excluded to a particular polymer molecule due to the presence of other polymer molecules in solution, by the relation 2 3: IC. (V-28) ZN where NA is the Avagadro constant and MN the number average molecular weight of polymer molecules. The parameter B, therefore, represents molar excluded—volume of polymer molecules in solution since B = A2M§ = NAB/2 = Molar excluded volume (V-29) The difference between V and B can therefore be considered as "molar free volume" of polymer molecules in solution. Taking a linear expansion of equation (V-l4) for 86 -1 for (ENLxm 0 followed by substitution into equation (V-12) yields: II 1 K 2K2 2 =F+ (A*—-_r) p + (A*+-—3—) p + ppRT MN 2 MN P 3 MN P (V—30) _ 1 2 — M; + (A2)obs pp + (A3)Obs pp + ... where (A2)Obs = A3 .. g7 (v—31) N 2 (A3be = A»); + £33?— (v-32) N As can be seen from equations (V-3l) and (V-32), the observed virial coefficients of an associating polymer— solvent system have two contributions: a term identified with association, and a second term which might be identified with all other interactions. 87 D. Presentation of Osmometry Data and Discussions The osmotic pressure of PTHF—MEK solutions were measured using a Hallikainen automatic membrane osometer (Model 1361 - Code D) designed by Shell Development Co. The details of the osmometer and the experimental pro- cedures were discussed in Chapter III. Sartorius regener— ated cellulose (Code SM-11539) with pore size <5 mm was employed throughout this work. Four monodisperse poly- tetrahydrofurane (PTHF) samples, designated as Al, A2, El, and BZ, were purchased from Polymer Laboratories, Ltd. (Church Stretton, Shrewsbury, U.K.), and their character— istics are listed in Table II. The polymer samples were used without further purification. Table II. Characteristics of PTHF Samples. Polymer Code Mn Polydispersity End—Groups Al 281,000 <1.05 -CH3 A2 30,000 <1.05 —CH3 Bl 7,660 <1.05 -OH B2 2,500 <1.05 -OH The results are listed in Tables III and IV and also plotted as Figure 10. As can be seen from Figure 10, the agreement between experimental data and theoretical pre- dictions of the association model is good. The signifi— cant difference in the characteristics of the two plots in 88 Table III. Concentration Dependence of Reciprocal Apparent Number Average Molecular Weights of PTHF-AZ-MEK Solutions at 34.0°C . +“ M/Dp RT X 10 (moles/grams) Concentration Polymer (grams/d1) Ca1culated* Experimental PTHF-A2 0.0521 0.3551 0.3570 0.1190 0.3857 0.3916 .1640 0.4055 .3988 .2553 0.4450 .4414 .3036 0.4657 .4680 .3538 0.4876 .4839 Root Mean Square Deviation: 0.0047 MN = 29,900 grams/mole A2 = 0.0043 mole CC/gram2 *Calculation based on equation (VI-9) using least square technique to fit the data. 89 Table IV. Concentration Dependence of Reciprocal Apparent Number Average Molecular Weights of PTHF-Bl-MEK at 34.0°C. +4 II/pp RT X 10 Polymer Cknfgjgg:vgfyi (moles/gram) Calculated* Ca1cu1ated+Experimenta1 PTHF-Bl 0.0235 1.0719 1.0643 1.0677 0.0336 1.0208 1.0271 1.0259 0.0449 0.9797 0.9900 0.9898 0.0749 0.9196 0.9168 0.9082 0.0802 0.9140 0.9076 0.9197 0.1074 0.8999 0.8781 0.8754 0.1226 0.9008 0.8749 0.8729 0.1990 0.9695 0.9962 0.9966 Root Mean Square 0.0184 0.0059 Deviation *Cglculated using association model (equation (V-26)) (MN: 7,660 grams/mole, A§==0.015 mole cc/gramz, K==l.07x 107 cc/mole) ICalculated using non-association model (equations (V—9) and (V-22) (MN: 8,570 grams/mole, A§=~0.0483nmde cc/gramz, g = 0.997). 90 Figure 10. Concentration Dependence of Reciprocal Apparent Number Average Molecular Weight of PTHF-MEK Solutions at 34.0° C. O PTHF-Bl—MEK A PTHF—AZ-MEK 91 NO (ES/910W) VOL 06 90 Figure 10. Concentration Dependence of Reciprocal Apparent Number Average Molecular Weight of PTHF-MEK Solutions at 34.0° C. O PTHF-Bl-MEK A PTHF—AZ—MEK 91 . mo . A _U\ U V CO58HC®HCOU (D/ezow) VOL X we 92 Figure 10 seems to indicate that PTHF-B1 molecules associate to form complexes via OH end-groups in MEK. NO evidence indicates any detectable association between PTHF-A2 molecules. The second virial coefficient A2 of the PTHF—Bl-MEK solution is found, based on the non—association model, to be negative and much smaller than A2 of PTHF-AZ-MEK solution. This finding conflicts with excluded volume theory [SD—1], which predicts that A2 should decrease very slowly with increasing polymer molecular weight. Similar findings have also been reported by other investi- gators for polyvinyl chloride-cyclohexanone solution [SD-2] and for polyethylene oxide-benzene and -acetone solutions [SD-3, 4, 5]. Although the excluded volume theory allows A2 to have a small negative value when weak intermolecular attraction exists in the system, large negative A2 and change of sign of A2 from positive for higher molecular weight to negative for lower molecular weight polymer are not predicted and cannot be explained by the theory. It is concluded, therefore, that the excluded volume theory should be restricted to non-asso- ciating polymer-solvent systems only. Moreover, A2 is traditionally regarded as a measure of "goodness” of solvents for polymer solutes, and a large negative value of A2 for PTHF-MEK solution at 34.0°C would mean MEK is a non—solvent to PTHF. This conflicts the observation that MEK is a theta solvent to PTHF at 25.0°C (that means MEK 93 should be a better-than-theta solvent to PTHF at the higher temperature). In fact, PTHF dissolves readily in MEK to form a homogeneous solution at 34.0°C. The author suggests that it is A*, instead of (A2) which should be obs' used as a measure of solvation power of solvents for asso- ciating polymer—solvent systems in which polymer molecules associate with each other mainly via their end-groups. The conflicts existing between experimental results and theoretical predictions discussed in the preceeding paragraph can be explained by the association model pre— sented in section C of this chapter. If we accept the prediction by excluded volume theory that the second virial coefficient is insensitive to polymer molecular weight and the assumption made earlier that the association constant K is independent of polymer molecular weight, equation (V-31) predicts that a plot of (A2) vs. Mgzshould be obs linear with intercept equivalent to A3 and slope -K. Equation (V-3l) suggests that the observed second virial coefficient (A2)obs decreases with decreasing polymer molecular weight when the association term K/M; is signifi- cant comparing to the virial term A3. According to the equation, the observed second virial coefficient is also expected to change its sign from positive to negative as the polymer molecular weight continues to decrease such that the association term becomes greater than the virial term A3. In order to verify the validity of these predictions the data of Yamada and co-workers [SD-3] for 94 the second virial coefficient of polyethyleneglycol (PEG) solutions were replotted in a manner as suggested by equation (V-3l). Figure 11 shows that the agreement between the straight—line prediction and experimental results is surprisingly good. In aqueous or methyl alcohol solution, the OH end-groups of PEG are surrounded by solvent molecules, and are most likely to form hydrogen bonds with them. Intermolecular association of PEG mole- cules via OH end-groups is therefore expected to be very weak. Among the four solvents under study, the associa- tion constant of PEG is found to be greatest in acetone. Figure 11 also suggests that water is a better solvent for PEG than the others because PEG has the highest A; value in water. The larger and positive A; of the PTHF-Bl-MEK solu— tion, when compared to the value of A2 for PTHF—AZ—MEK solution, is consistent with excluded volume theory, although the difference between the two values is greater than what would be expected. A possible explanation for the discrepancy is that some of the normally non-associat- ing intermolecular attraction forces actually contribute to the association mechanism when hydrogen bonding occurs. Thus, some of the non-associating effects in A2 are actually shifted to the association term. According to excluded volume theory, the value of g defined by equation (V-22) should vary from zero for a theta solvent to a asymtotic limit of 4/3 for a good 95 Figure 11. Molecular Weight Dependence of Observed Second Virial Coefficient of PEG Solutions (Replotted from reference [SD-3]. 96 o.o_ ow % 06 ON oow- _ A N 966569 CAME - 00- .0586 _Efit \‘\§\.\\\\l‘\\|!.. Ow. (36/33 810w)EOL x sqo(3V) 97 solvent. This prediction is again not confirmed by the data of Nord and co-workers for polyvinyl alcohol—acetate copolymer in aqueous solution [SD-6]. Inspection of their data indicates that the g values of their eighteen systems, evaluated according to equation (V—22), varied from 0.32 to 1.41, even though all of these systems have large nega- tive second virial coefficients which suggests that these systems are on the verge of precipitation. Elias and Gerber [SD-7] have argued that the initial slope of the n/pp RT vs. pp curve should not be equated with the second virial coefficient and any interpretation of solu— tion properties based on negative second and positive third coefficients is questionable. The molecular weight of the PTHF-Bl sample determined in this work is found to be much smaller than indicated by the manufacturer, while the agreement for the molecular weight of PTHF-A2 sample is extremely good (see Tables III and IV). Again, this is suggested to be a result of association. Figure 10 demonstrates how one can be misled with a limited number of data, especially when they cover only a rather narrow concentration range, if the solution under study is an associating system. As indicated by the dotted line in Figure 10, extrapolation of data taken within concentration range 0.20 ~ 0.35 g/dl will yield a polymer molecular weight of 25,000 g/mole for the PTHF—Bl sample, which is identical to the manufacturer's value. Figure 12 demonstrates the superiority of the 98 Figure 12. Comparison of Osmotic Pressure Data Fittings Based on Various Solution Theories for PTHF—Bl— MEK Solution at 34.0°C. Association Theory of This Work ----- Flory-Huggins Theory 99 OF. 00. A _E o EBBLEOOCOO (I), O .11 O) O Q (E/QIOUJ) 1701 x F V 100 association model over the Flory—Huggins theory for associating polymer-solvent mixtures. As can be seen from Figure 12, the one-parameter Flory-Huggins equation (equa— tion (II-12)) cannot fit osmotic pressure data for PTHF— Bl-MEK solution. Based on the theory, x115 assigned a value of as high as 2.5 to best fit these data. This is contrary to what the theory predicts that X1should have a maximum value of 0.5. The idea of associating polymer molecules in solution gains more support from Rquopf's osmotic pres— sure data for polyvinylchloride (PVC) solutions [SD-8]. Their results show that the number average molecular weight of their PVC sample, determined by extrapolation of the 11/0p RT vs. 0 plot to zero concentration, varies with P solvent used and with solution temperature. These results clearly indicate that PVC molecules associate in solution and the degree of association is a function of solvent and temperature. VI. DIFFUSION IN ASSOCIATING POLYMER SOLUTIONS A. Thermodynamic Basis of Diffusion in Solution General diffusion theories normally have been developed either on the basis of a physical model of the diffusion process or using irreversible thermodynamic argu- ments. The latter approach is adopted here, both to avoid dependence on a possible unrealistic model and because the results are consistent with available kinetic theory and experimental data [6A-l, 2]. The thermodynamic approach is based on three postulates in addition to conservation and symmetry arguments: (a)near equilibrium behavior, (b) linear relations between fluxes and affinities, and (C) microscopic reversibility [6A-3]. The first step in the development of flux relations is to recognize that diffusion is an irreversible process, and thus results in an overall increase on entropy. A quanti— tative expression for this increase will yield the desired relations between mass fluxes and concentration gradients. For a n+1 component mixture, it can be shown [6A-4] that: + E D. J. (VI-1) 102 where JS is the overall entropy flux relative to the mass average velocity vm, q is the total heat flux and Ji the mass flux of species 1 relative to the mass average velocity. T is the absolute temperature and “i the chemi- cal potential of species 1. In addition: + Z J. V(u) (VI-2) where RS is the volumetric rate of entropy production. Clearly RS vanishes at thermodynamic equilibrium; that is, when the Chemical potential gradients V(ui),r p are zero. ) The 7(“NI p can then be considered as the "driving forces” for entropy production. The preceding arguments can be generalized to any "driving forces” or affinities, Xk, and the corresponding "fluxes," Pk, of irreversible processes: RS = E Kk xk (VI-3) The second law of thermodynamics requires that the entropy production resulting from all the simultaneous irreversible processes be positive. However, it may happen that a system undergoes simultaneous processes such that: Fj xj < 0, while all other Pi xi > 0 (i # j) (VI—4) provided that the sum 103 2 Pk x > 0 (VI—5) These irreversible processes are called ”coupled processes." Thermodynamic coupling allows one or more of the processes to progress in a direction contrary to hat described by its own affinity. For example, in thermal diffusion, the dif— fusion of matter against its concentration gradient is accompanied by a negative entropy production, but this effect is compensated by the positive entropy production due to the flow of heat. This example makes it clear that any particular flux in a system depends upon not only its own affinity, but also all other affinities existing in the system. Thermodynamic considerations alone cannot, howeverugive the form of the dependence of the fluxes on the driving forces. It is quite natural, at least if the system is Close to equilibrium, to assume that each flux is a linear homogeneous function of each driving force: F. = Z W.. X. (VI-6) j 13 3 Linear laws of this kind are often called the "phenomenolo- gical relations.” In addition to being the simplest physically interesting assumption, this relation is consist- ent with other common flux laws (for example, Newton's law, Fourier's law and Fick's law). Like these relations, it is only a first approximation to real behavior. But also like these others, it is normally a good approximation. The 104 coefficients Wij are called the ”phenomenological coeffi- cients," and depend on the state of the system. The coef— ficient Wii may stand for the heat conductivity or diffusiv— ity, while the coefficients Wij (with i#j) describe the ”interaction” of the two irreversible processes i and j. The phenomenological coefficients have been shown to be symmetric by Onsager on the basis of statistical mechanical arguments [6A-5]: that is: These relations are referred to as the “onsager reciprocal relations,” and although their validity is still in some doubt, large deviations form equation (VI—7) are quite unlikely. It should be noted that for the Onsager recipro- cal relations to be valid, the J1 and Xi must both be independent [6A—6] (i.e., for a system of n+1 components, there are only n independent Ji and Xi)° Returning to the mass flux problem due to chemical potential gradients V(ui)T,p, according to equation (VI-6), one can write for n+1 components: C. II I ILMS J1 wij V(Uj)r,p (i = l, "’ n) (VI-8) For practical purposes, Chemical potential gradients are awkward. Equation (VI-8) can be rewritten in terms of mass fraction gradients. Since 113': Uj (6.11, (02, ...wn, T, p) (VI-9) 105 where oi is the mass fraction of component i, therefore n V(U.) = : (8&3) Vwk (VI-10) =1 k (UR, gfk’n T,P Here, component n+1 is chosen as the solvent. Combining equations (VI—10) and (VI-8), one obtains: n n Bu Ji = — 2 W1. [ Z "a-VJ—j) V ] 3:1 3 k—1 k w£,22k,n wk T,P n n 3U. = - z [ z 1' (A j) 1 v6 k=1 3:1 3 "I 002, 2%k,n k T. (VI-11) by: n z w.. -—71) (VI—12) where p is the mass density of the mixture. It can be seen from equation (VI—12) that, in general, Dr; # DEi' If we rewrite equation (VI-10) in terms of mass density gradients: n 3p. = __l - . V9 COCAELOOOC mum3 memum>m uCO>HOmImIBm qcflumfluommmIcoc wo mmonu pom AmmIH> cofipmsqu >uomcu Coflumfluommm may Op ocflpuouum COCHELOump mnm3 mewum>m ucm>HowIm:9m medumfluommm wo menam> so Hmpcmefiumaxm 0:92 www.0A COHumH>OQ mumsqm cmmz poem omam.a wvm.a I wovm.a ommm.A oom.o mmINm moom.o mmm.HMI homo.a moom.o moo.o mmIHm havm.o www.mMI mmmm.o ommm.o III mmIN< 5 ommm.a ooa.mHI mena.o oAmA.o III mmIH< u oAnm.A 606.6 OAA>.6 ooom.m mmv.o xmznmm mmmv.o mao.a VAmm.m oman.m ova.o meIHm mmmo.oI Noo.h I comm.H OABN.A III XMZIN¢ wham.o omm.v vvvv.o mvov.o III meIH< AE@\UUV coflumfi>oo ACMMOMWMMHWMMMWOL.LAmDCOEALmmev AEG\UUV WMMWMMW NIoH Amx Lovpx OOmACOUme Auwm\meuxbaxoo Aumm\~Evaoach :Ioaxzm\x ums>aom .O.o.em um memum>m ucm>AomImmem Low mymo ucwfloflmwmou coflmswwflo mo >umEEDm .> wanmfi 126 method will not provide data at low enough concentration in all cases, the data clearly indicate increased diffusion coefficients at lower concentrations. As it has been shown by Figure 13, the association theory predicts that the effect of association on diffusion rate becomes much less significant at higher concentrations than it is in the infinitely dilute region. For the four associating PTHF-solvent systems, however, the remaining not-so-significant association effect at higher concentra- tion seems to compensate other effects and, as a result, the diffusion rate is found to be independent of polymer concentration within this concentration region (see Figures 14 and 15). Equation (II—70) and (II—71) of the non—asso- ciation theory are not able to describe the complex nature of the diffusional property of associating polymer—solvent mixtures. Scattered evidences on associating systems seemed to indicate that the association of polymer molecules is restricted to very peculiar polymer/solvent/temperature combinations. This might not be true. The association theory projects that there exists a very narrow range of effective association strength K/MN within which the association of polymer molecules is detectable. For low K/MN, the association effect would be insignificant com- pared to other interactions. On the other hand, if K/MN is high, the association process would be complete at such low concentrations that accurate diffusion measurements are 127 not possible. The latter is exactly the case for the four systems under study. In fact, without the osmotic pressure data which provide clear evidence of polymer molecular association, one would likely fit the diffusion data of these four systems with a non-association model. It is therefore concluded that the interferometry method of dif- fusion coefficient measurements is not a good choice for detecting intermolecular association. The finding that the diffusion coefficient increases sharply with decreasing polymer concentration in dilute associating polymer solutions casts doubt on the reliabil- ity of some D0 values reported in the literature, since these values have been obtained by linear extrapolation. This might also explain the lack of agreement between D0 values observed in some laboratories and those predicted by theoretical considerations. Unfortunately, accurate measurements of diffusion coefficients in the infinitely dilute region is difficult (or impossible in some cases) by using traditional apparatus. The association phenomena might therefore be overlooked. In Figure 15 and 16, only limited evidence shows that the D(DP) curves shall increase with decreasing polymer concentration. In order to support the association arguments, therefore, it is worthwhile and also interesting to compare the D0 values of these associating systems determined from the associa- tion model with corresponding D° values predicted by existing theories. Fedors’ empirical relation, presented 128 in Chapter II, part B, was chosen for this purpose because it needs no extra experimental data besides the viscosity of the solvent and is the easiest method to apply. It can be seen from Figure 17 that the Fedors' relation correlates DG data of all eight PTHF-solvent systems satisfactorily (see also Table V) if D0 values of associating systems were determined according to the associating model. It should be mentioned, however, the constant parameter on the right hand side of equation (II-69) was determined to be 3.28 x 10-9 (using a least square technique), instead of 4.5 x 10-9, for the best correlation. More reliable data will be needed to confirm the exact value for the parameter. The distance between open Circles and the 45° line in Figure 17 shows the deviation of D0 values of the four associating systems, if they were determined by linear extrapolation, from Fedors' prediction. This illustration provides good explanation for the observation that the predicted D0 values are higher than they were determined in the laboratory in some systems [6C—l, 2]. It is interesting to point out that the limiting diffusion coefficient D0 of the four associating systems under study, when determined according to the association model, have values about 1.5 to 2.0 times as much as they will be if the non—association model was to be applied. This might explain the finding reported by a number of investigators that a ”unimer—trimer” [6C-l] or a "unimer- Figure 17. 129 Correlations Between Experimental Do Values of PTHF—Solvent Systems and Fedors' Empirical Relation. Non—associating systems ——————— — A Associating systems (determined according to association model) ----------- Associating systems (determined by linear extrapolation) ------------- - C) 130 0. A6 .0. 5 ) -O. a 4 C {I .0. 6m 3 X %-m 0 -m 0 0 O F A _ _ _ _ A O. O. O. O O O O A8m\mEsooAx .... 593 N. -6600 0AA ..Epobimmm- o l3l tetramer" [6C-2] model interpreted their diffusion data reasonably well. VII. CONCLUSIONS AND RECOMMENDATIONS An association model is presented which provides an explanation for some conflicts between experimental obser- vations and existing thermodynamics and diffusion theories of non-associating systems. The osmotic pressure equation derived in Chapter IV based on the association model is found to be analogous to the van der Waals equation, not only in form, but also in the physical meaning of the cor— responding parameters. This result is hardly surprising considering the similarities between the two systems. The diffusion equation based on the association model predicts that for associating polymer-solvent systems, the diffusion coefficient should first decrease with increasing polymer concentration and then increase slightly, remain constant, or decrease slowly with concentration, depending on the sign of the intermolecular interaction parameter k3. This prediction is consistent with experimental observations. It is worthwhile to point out that both osmotic pressures and diffusion data from this work were found to fit the association model very well, and more importantly, the association constant K and the number average molecular weight EN determined from different measurements are in good agreement, indicating that the association model is 132 133 self-consistent. The following recommendations are proposed for further work: (1) Data should be obtained for polymer samples with identical structures and average molecular weights, but with different functional end—groups. The purpose would be to elucidate the effects of association upon thermodynamic and transport properties in polymer solutions. A family of H/oP RT vs. pp curves of such polymer-solvent systems should provide an excellent test of predictions made by the association model. Polymer molecules would be expected to associate to a different extent in the same solvent if they have dif— ferent functional end—groups. Therefore D(Op) curves of such systems should be similar in their shapes, but different in magnitudes. However, these systems should have identical D0. (2) Obtain experimental data for trace-diffusion coefficients of the polymer samples in the same solvents. The advantages of this are: (a) To overcome the difficulty in obtaining accurate diffusion coefficients in the dilute concentra— tion region. Tracer diffusion coefficients must extrapolate to the mutual diffusion coefficients at infinite dilution and tracer coefficients should be measurable at much lower concentrations. ‘31 134 To determine the limiting diffusion coefficient Do with greater accuracy to justify existing theories for the estimation of Do. To test the validity of the association model proposed in this work. Although the model fits the data from this work, as well as from the literature with good results, more systematic data are needed to test the model more extensively. ... '- L - - "A: NOMENCLATURE 135 1 3c c/q c/r (c/q> NOMENCLATURE Species 1 (equations (V-l) and (V-2)) the ith virial coefficient the ith virial coefficient of associating polymer solutions observed ith virial coefficient defined by equations (V-Bl) and (V-32) van der Walls parameter defined by equation(V-27) parameter defined by equation (VI-32) parameter defined by equations (V-59) and (V-60) effective bond length (equation (II—43)) molar excluded volume defined by equation (V—29) van der Waals parameter defined by equation (V—27) parameter defined by equation (II-72) molar concentration of i-mer true molar concentration of polymer defined by equation (V—S) configurational heat capacity of solvent defined by equation (II-29) total number of external degrees of freedom of a polymer molecule structural factor of pure polymer melt structural factor of pure polymer melt average structural factor of a mixture self-diffusion coefficient limiting diffusion coefficient at infinite dilution diffusion coefficient of component i (or i—mer) in a multi-component system 136 obs 137 observed diffusion coefficient of an associating polymer solution diffusion coefficient defined by equations (VI-12) and (VI-12') molar energy of vaporization of component i flux of irreversible process k frictional coefficient at infinite dilution parameter defined by equation (VI-32) free energy of mixing constant defined by equation (V—22) weighting factor of diffusion coefficient of i-mer enthalpy of mixing partial enthalpy of solvent function defined by equation (II—49) entropy flux relative to the mass average velocity mass flux of species 1 (or i-mer) relative to mass average velocity observed mass flux association equilibrium constant defined by equations (V-l) and (V-3) (open association) association equilibrium constant defined by equation (V-Z) (closed association) Boltzmann's constant enthalpic parameter defined by equation (II-l6) parameter defined by equation (II—70) parameter defined by equation (II-73) parameter defined by equation (VI—30) parameter defined by equation (VI-39) M ,1 3| 3| :fl 2 N app 6 3| 3| 138 parameter defined by equation (II-59) parameter defined by equation (II-54) average molecular weight of polymer number average molecular weight of unimers; true number average molecular weight (= M ) N number average molecular weight of polymer apparent number average molecular weight of polymer molecules in associating solutions under theta condition viscosity average molecular weight of polymer number average molecular weight of n-mer in an associating polymer solution Avagadro's number refractive index (equation IV-3)) number of segments in a polymer chain number of moles in component 1 universal constant defined by equation (II-55) characteristic pressure defined by equations (II-21) and (II—24) average characteristic pressure of a mixture heat flux (equation (VI-1)) number of external contacts of a polymer chain universal gas constant mean—square end-to-end distance unperturbed mean-square end-to-end distance radius defined by equation (II—62) radius of a non-free-draining pseudo-spherical polymer molecule volumetric rate of entropy production < |< I: H . <| H-O wij Aw AwH Aws 139 ratio of partial molar volume of polymer to that of solvent mean-square radius of gyration partial entropy of solvent in solution entropy of mixing absolute temperature characteristic temperature defined by equations (II—21) and (II—24) average characteristic temperature of a mixture time energy of vaporization of solvent excluded—volume per polymer molecule mean volume occupied by a polymer molecule in solution defined by equation (V-Zl) molar volume of component 1 partial molar volume of component i at infinite dilution molar volume of component i at its critical temperature volume change of mixing molar volume of van der Waals gas mass average velocity phenomenological coefficient defined by equation (VI-6) interchange free energy defined by equation (II-5) enthalpic interchange energy (equation (II-11)) entropic interchange energy (equation (II-11)) frictional parameter defined by equation (II-51) dimensionless concentration defined by equation (VI-42) .1 CLR 0‘3 140 surface fraction of component i affinity of irreversible process k dimensionless concentration defined by equation (VI—23) coordinate number excluded-volume parameter defined by equation (II-43) degree of association defined by equation (V—lS) expansion factor defined by equation (II-46) expansion factor defined by equation (II-47) binary cluster integral empirical constant defined by equation (II—36) parameter defined by equation (II-54) parameter of cohesive energy difference between solute and solvent molecules defined by equation (II-32) square root of cohesive energy density of compon- ent 1 defined by equation (II-37) non-polar part of 6i defined by equation (II—40) polar part of 6i defined by equation (II—40) hydrogen bonding contribution to 61 defined by equation (II—40) viscosity of solvent intrinsic viscosity of a mixture theta temperature; a characteristic parameter of polymer solution defined by equation (II—18) chemical potential of solvent in solution chemical potential of pure solvent parameter defined by equation (II-31) osmotic pressure 141 parameter defined by equation (II—30) mass density polymer mass concentration parameter defined by equation (II—33) mass concentration of component k volume fraction of component 1 mass fraction of component 1 contact (potential) energy between molecules 1 and j entropic parameter defined by equation (II-l6) distance between two interacting molecules (polymer segments) universal function of potential energy of a pair of molecules interaction parameter defined by equation (II—8) (dimensionless) parameter defined by equation (VI—40) enthalpic interaction parameter (equation (II-11)) entropic interaction parameter (equation (II-11)) frictional coefficient of a polymer segment in solution universal constant defined by equation (II-56) APPENDICES 142 "h APPENDIX A Sample Calculation for the Determination of Diffusion Coefficient Polymer: PTHF—A2 (EN = 30,000) ‘1 Solvent: methyl-ethyl—ketone Solution I (for the upper level of diffusion cell) Concentration = 0.8474 g/dl Solution II (for the lower level of diffusion cell) Concentration = 1.1464 g/dl Photographic Plate: Exposure Number Time(seconds) l 0 2 720 3 1440 4 2160 5 2880 6 3600 Total number of fringes: J = 14.6 143 Exposure 4 5 6 Exposure 4 5 6 Exposure 4 5 6 Exposure 4 5 6 Exposure 4 5 6 Exposure 4 5 6 (x' -x{)(cm) 3 H H+a POO F—‘HO POO POO .9896 .0272 .0625 .9762 .0163 .0539 .9605 .0036 .0429 .9435 .9892 .0316 .9283 .9766 .0231 .9138 .9648 .0136 144 OKOCD OKOCI) (xé-rxi)(cm) ((x$+ x3) (xé-x3))(cm) 1.1257 1.1580 1.1898 1.1235 1.1581 1.1933 1.1183 1.1555 1.1925 1.1122 1.1511 1.1908 1.1065 1.1475 1.1897 1.1010 1.1455 1.1893 0.1361 0.1308 0.1273 0 .1473 1.1418 0.1394 0.1578 0.1519 0.1496 .1687 .1619 .1592 000 .1782 .1709 .1666 000 0.1872 0.1807 0.1757 145 wmaoo.m mwvao.m mmoom.H A omovm meom .o .o mammo.o d 3:- mum m ommwm.o mwmmm.o mmmmo.o fi..xm OH mamma.o Hmcmm.o omvmv.o womba.o homam.o mommv.o 146 Homa.o hmma.o mama.o hvma.o mnaa.o HHHH.O omaa.o _Hmma.o mmoa.o maoa.o momm.o mmoa.o "momum>< "momum>< ”mOmpw>< Nim + <1 ANEoV 2mx + 1x1 hmba.o mowa.o tha.o o musmomxm ©©®H.o mona.o tha.o m mpsmoaxm NmmH.o meH.o nmoa.o v mpswooxm Amx + waxy ommo.o mmmo.o hmmo.o mmmo.o o~mo.o mhho.o wamo.o mmmo.o mmoo.o mvmo.o mmoo.o maho.o "momum>< "wOmum>< ”wmmuw>< wmva.o mHmH.o mhma.o m musmoaxm vmma.o wava.o mnva.o N wusmoaxm mn~a.0 moma.o Hmma.o H musmoaxm x AEUV Amx + .xv (xfl + x')2/(A + B)2(cm2) 3 k .06 147 Slope of the plot = 1.7117 x 10‘5 cmz/sec Diffusivity = Slope/4M2 1.7117 x 10’5/14.884 1.1500 x lO—scmz/sec l l l l l l J l L I l l 12 24 36 48 60 72 Time (Minutes) APPENDIX B Diffusion Data from Interferometry Polymer Solvent Concentration D x 106 (cmZ/sec) (g/dl) PTHF—A1 MEK 0.1031 0.5547 0.2045 0.5450 0.2126 0.6551 0.3892 0.5718 0.6089 0.7346 PTHF-A2 MEK 0.1806 1.1940 0.2902 1.2290 0.4237 1.2320 0.7541 1.1870 0.9969 1.1500 PTHF-Bl MEK 0.1029 1.5830 0.1759 1.4890 0.2069 1.5010 0.3632 1.4560 0.6051 1.3330 0.7286 1.3000 0.9942 1.3590 PTHF-B2 MEK 0.0792 3.023 0.1128 3.098 0.2150 2.508 0.3203 2.503 0.4279 2.273 0.5721 2.563 0.8197 2.613 0.9920 2.548 PTHF-A1 BB 0.135 0.1732 0.202 0.1866 0.239 0.2210 0.303 0.1986 0.593 0.2687 PTHF-A2 BB 0.105 0.415 0.196 0.416 0.400 0.448 0.806 0.502 148 7‘1 149 Polymer Solvent Concentration D x 106 (cmz/sec) (g/dl) PTHF—Bl BB 0.0500 0.597 0.0825 0.502 0.2510 0.5210 0.3875 0.5420 0.6270 0.6020 0.8235 0.5520 PTHF-B2 BB 0.062 1.486 0.113 1.213 0.196 1.183 0.391 1.060 0.574 1.165 0.785 1.268 0.847 1.306 BI BLIOGRAPHY 150 (lA-l) (lA-2) (lA-3) (lA-4) (1A-5) (lA-6) (lA-7) (lA—8) (2A-l) (2A—2) (2A-3) (2A—4) BIBLIOGRAPHY J.S. 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