T UN1VERSITY OF MIHTI.NT ThDUSTRY PROGRAM -OF THE COLLEGE OF'ENGINEERING SOLUTION AND BULK PROPERTIES OF SOME ARTIFICIALLY.BRANCED., POL.YVIYL ACETAUES C(SVI.Guy -C, Ber A dissertation submitted in partial fl1fillment of the requirements for the degree of Doctor of Philosophy in The TJniversity of Michigan 1960 September, 1960 IP-453

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Doctoral Committee: Professor Brymer Williams, Chairman Assistant Professor Robert Go Craig Associate Professor Samuel Krimm. A-ssociate Professr- Donald Wo McCready Associate Professor Giuseppe Parravano ii

ACKNOWLEfDGMENr S I want to thank Professor Brymer'Williams for the helpful advice and. interest he has extended to me throughout the course ~of this study, I also want to express my thanks to Asslstant ProfEssorI Rober:t G, Cralg for his many helpful discussions on various phases of the work, and to the other members of my committee, Associate Professors Samuel Krimm,. Donald W, Mecready, and. Giuseppe Parravano, fo;r'their helpful adviceo I particularly want to acknowledge imy grat;itude to DrE L. M. Hobbs of the Air Reduction Company under whose direction I'initially beame interested in this study, and who has given me. helpful advice during all phases -of the study, This initial interest began while I was working on the. Michigan Memorial Phoenix Project on Ela.stomerts, made possible by a grant from the Goodyear Tire and Rubber.Company, I also want to thank Dra J, A, Manson and PDr W. W. Graessley of the Air Reduc tion Company and Dr. VT C. Long of The.E, I du Pont de Nemours Company for their useful suggestions and advice on all parts of the study. In addition, I would like to thank the Allied Chemical and Dye Coa.oration for financial assistance -over the past ti.o years~. iii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................. o LIST OF TABLES..........O..........8........................... v LIST OF FIGURES.00O.........................................00 000 I. INTRODUCTIONo88...... O 8O88o8 8............8 O..0........... 1 A.o Scope of the Study..o.................000 0........... 1 Bo Occurrence of Branching in Polymers........ oo....... 2 C. Artificially Branched Polymers...................... 5 D0 The Intrinsic Viscosity8..88888.88008..8800.0 17 E. The Mean Square Radius of Gyration in Good Solvents~o. 28 Fo The Second Virial Coefficiento.,8........o8.......... 45 Go Light Scattering in Good Solvents......o.........o.... 53 H The Melt Viscos lty0o...oo....0.0oo.....o.0 o...0 o. 58 1Io EXPERIMENTAL PROCEDTURES......o..oO........ 66 A. The Synthesis of Artificially Branched Polymero...... 66 1. Polymerization Procedure....o OO o...........oo... 66 20 Isolation of the Branched Polymero............. 68 Bo The Intrinsic Viscosity........... o.........8o.... o.o 72 lo Sample Preparation and the Measurement Procedure. 72 28 Data Treatment. o........................ 73 Co Light Scattering,,.....oo.........0..............., 75 1o Sample Preparation and the Measurement Procedure0 75 20 Data Treatment a 0o o o.80080808............08080 00 76 D. The Melt Viscosity o........ o 0...0 0 80 lo Sample Preparation and the Measurement Procedureo 80 20 Data Treatment 88.000000........................ 82

TABLE OF CONTENTS (Concluded) Page III. RESULTS AND DISCUSSION..................., ooooo 84 A. The Graft Polymerization........................... 84 B. The Solution Propertie s............................... 86 1. Properties of the Linear Polymer................. 86 2, Properties of the Branched Polymer............... 91 C. The Melt Viscosity.,................................... 106 IV. SUMMARY AND CONCLUSIONS.......... o. o.... o..o..........o o 112 RIEFERENCES........ o.... o............................ *o oooee o o115 APPENDIXES I KINETICS OF THE GRAFT POLYMERIZATION................. 122 II DIMENSIONS OF A BRANCHED POLYMER IN AN IDEAL SOLVENT. 127 III THE EFFECT OF GRAFTING ON THE MOLECULAR WEIGHT DISTRIBUTION.....o.................... o............ 130 IV TABLES............ o.o..........4O.O.O..ooO.oOOOO~O~o 133 V FIGLIRES..... ooo.......... a.. a a o a a O O O 0 143 VI NOMENCLATUKE....O.e ooo....................... o a o * a o o a a 162

LIST OF TABLES Table Page I Graft Polymerization Data............................... 134 II Intrinsic Viscosity Data........,.........,.....,...,0.... 135 III Intrinsic Viscosity in 1,2,4-trichlorobenzene............. 136 IV Light Scattering Data for the Linear Polymer.......... 137 V Light Scattering Data for the Graft Polymer................ 138 VI Intrinsic Viscosity Comparisons.....................0..... 139 VII Parameters for the Linear Polymer.............0..0... 140 VIII Parameters for the Branched Polymer...........o......... 141 IX Melt Viscosity Data................o..................... 142

LIST OF FIGURES Figure Page 1 Experimentally observed dependence of the degree of polymerization and the apparent reaction rate constant, (% conv)/(hr)(initiator cone)l/2, on the initial mol ratio of solvent to monomer................................... 144 2 Viscosity plot for fraction S-71-1.................... 145 3 Zimm plot for fraction S-70-1.........................o. 146 4 Number of branches per molecule obtained in a graft polymerization versus the backbone polymer molecular weight multiplied by the overall degree of conversion........... 147 5 Mean square radius of gyration versus molecular weight for branched and linear polymers................... 0......... 148 6 Second virial coefficient versus molecular weight for branched and linear polymers..............,.... 149 7 Ratio A2<M>/[r] versus molecular weight for some linear fractions........... Doo o.. *...................... 150 8 Root means square radius of gyration versus the second virial coefficient for three linear fractions o...O....... 151 9 Expansion factor versus the ratio of the intrinsic viscosities in good and ideal solvents...,................. 152 10 Intrinsic viscosity versus molecular weight for branched and linear polymers.....,..,..........................oo 155 11 Parameter g versus the intrinsic viscosity ratio and second virial coefficient ratio for branched and linear polyme rs O....... O... a... a...o.........a...a.o.....o.... a.154 12 Cube of the ratio of the expansion factors for branched and linear polymers versus the parameter g8 0..,..... 10o55 -vii

LIST OF FIGURES (Concluded) Figure Page 13 Ratio of the mean square radius of gyration for branched and linear polymers versus the parameter g................ 156 14 Ratio of the second virial coefficients for branched and linear polymers versus the ratio of the intrinsic viscosities for branched and linear polymers............... 157 15 Huggins k' versus intrinsic viscosity..............158 16 Huggins k' versus molecular weight...................... 159 17 Melt viscosity for branched and linear polymers versus molecular weight.................... 160 18 Ratio of the melt viscosities of branched and linear polymers versus the molecular weight of the branches on the branched polymer............... 161 -viii

I o INTRODIUCTION A, SCOPE OF THE STUDY The existence of departures from strict linearity in macromolecules has been known for some time and has been of considerable interest. The interest has been generated by the deviations of the properties of nonlinear macromolecules from an expected physical behavior. The study of non-linLear polymers has lagged behind the study of linear materials due to the increased complexity of the theoretical developments (many of which are incomplete for even linear polymers) and. the difficulty of obtaining a non-linear polymer with some a priori knowledge of its structure. The aim of this study has been to develop a method of obtaining a series of non-linear polymers in such a way that their structure could be specifiedo This being accomplished, it would then be possible to study some of the solution and bulk properties of these materials and attempt an interpretation of the observed behavior in light of the present theories. The properties to be examined would be the solution properties of dilute solution viscosity and light scattering and the bulk property of the melt viscosity. The remainder of this introduction will'be devoted to a description of the synthesis scheme used to obtain the non-linear polymer, and a discussion of somue of the pertinent theories for the behavior of macromolecules in solution and in bulko

-2Bo OCCURRENCE OF BRANCHING IN POLYMERS One can think of three important types of departure from linearity in polymers. A linear macromolecule is defined as one in which the mers or structural units which make up the molecule are added one after another in linear fashion such that each mer is attached to at most two other mers, the two end mers in the chain being attached to only one other unit. A non-linear chain might then. be caused by the first and last mer joining together to form a ring molecule. Another type of non-linear structure would result if any of the mers (excepting the end groups) added one or more mers to itselfo Additional mers might then add to this group causing a branch of considerable length to be formed. A mer is said to be bifunctional, trifunctional, etc., depending on whether it is attached to two, three, etc., other mers. The third type of non-linear structure results when a considerable number of linear chains are linked together by interconnecting branches to form an infinite network. This molecule is insoluble in a finite media, and such systems are usually termed gels. The molecules of interest in the present study are of the second type described aboveo This type of branching is known to occur in both natural and synthetic high polymers. A mechaniasm for its occurrence in vinyl polymers will be discussed briefly since this mechanism is important for the synthesis used in this study. Consider the possible reactions of a vinyl monomer during a polymerization. Let I be the initiator, M the monomer Ro any radical, anrd P a polymer molecule, then a reaction scheme could bee

-3Initiation: (l) # I + 2R~ Rate Constant ki Addit ion~ (2) R +M + Ro kp (3) R' + P* + PR kp Transfer (4) R~ + M + P + R* Ktr,m (5) Ri + P2 + PZ + R -Ktr,p Termination (6) R' +R + P kt (7) R. + R + P + P* kt2 #This assumes for simplicity that an initiator radical has the same reactivity as an activated mer. The same assumption is made concerning other activated species in steps (3), (4), and (5). *Designates a macromolecule with a vi:yl group. In this scheme, radicals are generated in step (1), vinyl addition occurs in steps (2) and (3), the radicals are transferred from one molecule to another in steps (4) and (5), and two radicals collide to cause their mutual extinction in steps (6) and (7)o Steps (3), (4), (5), and (7) can lead to the production of a branched polymer. Flory was the first to recognize and explain the importance of transfer reactions such as (4) and (5) in polymerization mechanisms, and. has discussed them in some detail (1). In a typical transfer reaction, a reactive species such as the growing polymer chain, abstracts an atom from a second molecule, leaving this molecule in an activated state capable of further addition to monomer. Thus, the total number of growing sites remains constant during a transfer reaction, and the polymerization rate is unaffectedo Transfer to the monomer, reaction (4), will eventually result in a chain containing a vinyl end group.o This type of end group can also occu when the termination

occurs by disproportionation, step (7)o When such an end. group does undergo reaction, step (3)9 then a trifunctional branch is formed. Transfer to a polymer chain, reaction (5), immediately results in a trifunctional branch as soon as the active site which results on the polymer chain adds any monomero The kinetics of this scheme have been thoroughly investigated by Howard (2). The mechanism described above would lead to a polymer in which chains of some average kinetics chain length are joined together to form a branched structure. The kinetic chain length being defined as the average number of monomers reacting with a given active center from its initiation to its termination, and given by the ratio of the rate of the propagation reaction to the initiation reaction (1). Such a structure is pictured below. It is apparent that all of the branches resulting from this scheme would be trifunctio-nal, and that all of the branching reactions are intermolecular, they involve the reaction of a growing chain with a second molecule. Another type of branching mechanism is sometimes important for vinyl polymerso This involves an intramolecular polymer transfer reaction. In this reaction, one imagines that due to steric effects, the active end of the growing chain is brought into the proximity of the chain and an unstable ring is temporarily formed. If this ring is broken by the abstraction of an atom from the chain and resultant transfer of the active site a few mers back down the chain from its original site, then a short branch is formed.

This type of branching is believed to occur in polyethylene (5), (6), polyvinyl chloride (4), and recently Melville has suggested that it may occur in polyvinyl acetate (3). Intramolecular transfer results in branches which are quite short with respect to the kinetic chain length, and usually in relatively more branches than one observes from intermolecular branching (7). A typical structure might be as pictured below. Although the reactions described above are the most usual cause of branching in vinyl polymers, there are other means of producing non-linearity in vinyl polymerso In addition, it should be noted that non-linear structures will result in the condensation polymers if one introduces a component which is capable of adding more than two other units to itself. C. ARTIFICALLY BRANCHED POLYMERS It is, of course, possible to study branching either in naturally occurring polymers or in synthetic polymers which become branched during the normal course of the polymerizationo It is advantageous, however, to study artifically branched polymers if it is possible to gain information concerning the molecular structure wvhich can be used to interpret the observed properties of the polymer. The reason for investigating the properties of the artificially branched polymer then, is to study the properties these polymers exhibit with the a priori knowledge of the degree and type of branching present in the polymer. This section will be devoted to a discussion of some of the methods previously used to obtain an artifically

branched polymer and the method used in this study with the purpose of pointing out the difficulties in interpreting physical properties that are associated with these methods. It would be an extensive task to include all of the methods which have been used to introduce branching into polymeric structures>o It is sufficient, however, for the purposes of this study to consider the merits of some of the approaches representative of those which have been used to prepare branched molecules. Zimm and Thurmond (9) have contributed one of the first and most satisfactory studies to this date. They polymerized styrene with the addition of a small amount of divinyl benzene, so that the divinyl benzene introduced tetrafunctional branch points into the polymer. This method has two rather severe drawbackso It is difficult to induce many branches into the system before an insoluble network polymer is obtained. Assuming that a soluble non-linear polymer could be produced in this manner, the molecular weight distribution of the resulting polymer would be very broad, making it difficult to obtain. reasonably sharp fractions containing a large amount of polymer (10). In addition, the solubility will be a function of the molecular weight and the degree of branching, making it impossible to obtain a fraction containing as narrow a molecular weight distribution as is possible in the fractionation of a linear polymer (9). The results of the study by Zimm and Thurmond are still important, however, and will be considered in detail later. A method which carries with it problems similar to those discussed #Burnett has given a review of some grafting procedures (8).

above is illustrated by the work of Charlesby and Thomas (11)o They have irradiated samples of polyethylene to induce cross-linking, and thus tetrafunctional branching. Here again, it is difficult to induce branching to the exclusion of gel formation and the resulting polymer has a very broad molecular weight distribution. Similar work has been carried out by Schultz, et al., on polystyrene and polymethyl methacrylate (12). A different approach has been used by Manson and Cragg (13) (14). These authors performed a redox polymerization of styrene in the presence of oxidized polystyrene or oxidized copolymers of styrene and 4-vinyl cyclohexene-1o This method offers several advantages. The oxidized polystyrene can be fractionated before the addition of branches in the redox polymerization, so that the backbone of the resulting branched polymer will at least be uniform. In addition, the formation of branched polymer can be followed by the disappearance of monomer and the decrease in the hydroperoxide content of the oxidized backbone polymer. One still has no independent measure of the length of the branches, and the hydroperoxide analysis is apparently not accurate enough to give the exact extent of branching which has occurredo The results of these studies are useful and enlightening, however, and will1 be considered in a later section. Another process similar in concept to that used'by Manson and Cragg has been used by Jones, et al., (15) and Soboleva, Maktletsova, and Medlvedev (16) among others. In this procedure, a polystyrene sample is first brominated, then the brominated polymer is irradiated with ultraviolet light in the presence of monomero The bromine atoms become excited and

separate from the polymer. When this occurs a site for a branch to grow on the backbone polymer is formed and in addition, the free bromine atom can itself initiate polymerizationo These authors further tried to improve the characterization of the branched polymer by using radioactive monomer to allow a direct estimate to be made for the amount of material introduced as branches. This method has the disadvantage of uncertain degradation of the brominated polymer taking place during the irradiation to form branches, and residual bromine in the branched polymer. Melville, et al., (17) (18) have suggested another means of synthesizing a branched polymer which is of special interest here. These workers have formed a linear fractionated polyvinyl acetate polymer containing about lO% vinyl alcohol units caused by a partial saponification. This material is reacted with a linear fractionated polyvinylacetate polymer which has as a terminal group either a carboxyl or an acetyl chloride group. This scheme offers several advantageso The degree of branching can be followed by the decrease in the alcohol groups of the backbone. It can also be measured by radioactive techniques if either the branches or the backbone are tagged with C14 and if the branches and the backbone material are far enough apart in molescular weight to allow separation of the uncombined branches from the branched polymer. Thus, this method results in a material which is in principle very well characterized. Both the backbone and the branches have a narrow molecular weight distribution and the number of branches can be closely specified. A very clear disadvantage is that the resulting branched polymer contains some residual alcohol groups which will

complicate interpretation of their physical properties. There exists in the literatuire a large body of work devoted to the discussion of graft co-polymerizations. This type of pol ymerizationl can be defined as one in which a monomer is polymerized in the presence of a polymer prepared from an?.other monomer. Alfrey (19) suggested such polymerizations and formulated some equations to predict the efficiency of such an operation in 1949. Carlin and Shakesphere (20) had polymerized p-chlorostyrene in the presence of polynreth.acrylate in 1946, and Smets (21) has been. very active in a systematic stud.y of the polymerization of various monomers in the presence of various polymers. All of this work has supported the concept of a polymer chain transfer reaction. The importance of these considerations to the prese:.n.t work is that it has been definitely established that graft co-polymerization does occur. Infra-red analysis carn be used to adva:ntage ine this type of study to establish the presence and amounLt of the species graft.ed onto the backbone chain and thus the effectiveness of the operation.i It is clear that if transfer reactions occur between. a polymer and a monomer in this tye of polymerization, then it is possible for them to occur between a monomer and its ownn polymer. This method has, in fact, been used in recent years to prepare a branched polymer. Melville, Bevington, and Guzman (22) have prepared branched polyvinyl acetate by the polymerization of vinyl acetate in the presence of polyvinyl acetate, as have Guzman (23) and Okamura and Motoyma (24).

-lO0 Of the works quoted, only Guzman has reported solution properties of the branched material. Gleason, Miller, and Sheats (25) have prepared branched polyacrylamide by this technique, characterizing the polymer by radiotracer studies and by light scattering molecular weights and intrinsic viscosity. These latter authors controlled the length of the branches being formed through the catalyst concentration. The scheme used in this study to produce the branched polymer has been of the preceding type, with some modificationso Vinyl acetate has been polymerized in the presence of linear fractions of polyvinyl acetate (PVOAc), a chain transfer solvent and an initiator. The chain transfer solvent to monomer ratio is adjusted so that the new polymer formed during the reaction will have a specified average molecular weight. This average molecular weight is kept considerably below that of the linear backbone material to facilitate the later removal of the new material which did not graft onto the backbone. The branching mechanism in. the polymerization of vinyl acetate has been studied in some detail since this monomer exhibits a greater tendency to yield branched polymers than most monomers. These studies must be considered here since the reaction steps disclosed will be important in establishing the linearity of the backbone polymer used, and in understanding the grafting reaction itself. Wheeler, Lavin, and Crozier (26) have examined some polyvinyl acetates obtained by slow photopolymerization to different conversions at 700C. The mechanism of the polymerization has been interpreted in terms of the seven reactions given below (numbered ac

-11cording to reactions on page 3). Ratios of some of the pertinent rate constants were obtained in this study. Unfortunately, it was not possible to obtain the ratio of the reaction rates for polymer chain transfer to the a-hydrogen or n-hydrogen position compared to the acetate group. The best number they could suggest for this ratio was between 0.33 and 0.500 Of interest here are their conclusions concerning the ratio of the number of branches attached to the acetate group to the total num0ber of branches. This ratio fell from an initial value of near 0.57 to a value of about 0.31 for 98 percent conversion in the batch reaction. R' + M _- Ro (2) H Ro + M - RH + CH2- (4a) C-O HCH 6 (4b) C=0o HCH H H R~ + P - RH + P-C P (5a) =0 HYH H H H Ro + P - RH + C-Py (5b)

-12H R + -cH2=c + R- CHCH~ (C 0 6=0 C-0 0CH1 H H P P P R1 + CH2y-I P ~ R CH2 &C (3b) (h-A0 ~ C=O 01H3 CH3 The contribution of Howard (2) provides further insight into the polymerization of vinyl acetate. As a part of this study, Howard reported a systematic study of the chain transfer solvents, vinyl acetate, the decamer of vinyl acetate and various esterso He concluded that the transfer to the vinyl acetate monomer probably occurs at the acetate group. Since the vinyl acetate polymer was observed, to be about three times as active as the monomer in the transfer reaction, it can be further concluded that about two thirds of the transfer to the polymer will occur at the a-hydrogern (or P-hydrogen), leading to non-hydrolizable branches. Howard has given a relationship which can be used to estimate the ratio of hydrolizable branches to the total number of branches in a system polymerized in a batch-wise process. Using the estimate that the transfer constant to polymer is three times that to monomer and that about one third of the branches caused by transfer to polymer are hydrolizable, one calculates that the ratio has a value of 0.50 initially and falls to about 0.30 for 98% conversion, in close agreement with Wheeler. The kinetic predictions of Howard concerning the total number of branches per molecule were con

_13a sistent with the solution properties of the polymer he obtained (27). Studies of the graft polymerization of two differenlt monomers onto PVOAc have been consistent with the results quoted above in one case and counter to them in another. Roland and Richards (28) polymerized ethylene in the presence of PVOAc. They concluded that a considerable portion of the ethylene grafted onto the PVOAc was placed at the a or P carbon of the backbone, but that some also occurred at the acetate group. Imoto, Ukida, and Tsugio (29) polymerized vinyl trimethyl acetate in the presence of PVOAc. They were able to investigate the degree of branching by an infra-red analysis of the grafted polymer. A similar analysis of the hydrolized polymer then gave a measure of the branching at the a or P position. These authors concluded that the methyl group of the acetate radical is the principle site for transfer, but that the a or a hydrogens also make a contribution. Unfortunately, the question of the point of attack on the polymer transfer reaction remains unsettled. One can conclude, however, from the work of Howard and Wheeler, et al., that a branched PVOAc polymer would contain about 50% hydrolizable branches at low conversion if any branching existed at all. Melville (3) has recently suggested another complicating factor. He has claimed evidence for an intramolecular transfer reaction in vinyl acetate polymerization. If such a reaction is important, then the polymer produced ought -to be degradable on hydrolysis. The polymer used as the linear backbone in this study was prepared and fractionated by Long (30). The linear samples were prepared accord=

-14ing to conditions suggested by Burnett, George, and Melville (31). Purified monomer (the purification used in this study and described in the experimental section is very close to that used by Long) was photopolymerized slowly at 19~C using azo-bis-isobutyronitrile as the initiator. The fractionation of this polymer has been described in detail by Hobbs, et al. (32). The linearity of the resulting fractions was tested by hydrolizing the polymer to polyvinyl alcohol, followed by reactylation to polyvinyl acetate. Since no decrease was observed in the intrinsic viscosity of the samples after this operation, it was concluded that the material contained no hydrolizable branches and thus probably no branches of any kind. A simplified kinetic analysis of the reactions important in the polymerization of monomer in the presence of its polymer and a chain transfer solvent can be found in Appendix I. It is shown there that the average number of branches per molecule, <K>, grafted onto the backbone is approximately given by <K> = Cp <x>X (1) where Cp is the polymer chain transfer constant given by the ratio (Ktr,p/Kp), <x> is the average degree of polymerization of the backbone material and X is the degree of conversion. The fraction, F, of the new polymer formed that is grafted to the backbone is approximately given by Cp (B/M) F..(2) [s ] Cs [M - + Cm where (B/M) is the ratio of the weight of polymer to the weight of monomer, #See Appendix I for the full expressions derived for these relationships.

-15[S]/[M] is the ratio of the mols of solvent to the mols of monomer, Cs is the solvent chain transfer constant (kt,s/kp) and Cm is the monomer chain transfer constant (ktm/kp)p It is shown in Appendix I that for a typical set of conditions in a graft polymerization of 88~C these expressions give <K> ~ 30X F 15/100. Since the conversion must be kept below about 235, these relationships illustrate that a single reaction will on.ly give about 6 to 8 branches per molecule, and that the branched polymer will'be contaminated'by a considerable amount of new polymer not bound to the backbone materialo This was, in fact, the observed behavior. In order to obtain a number of branches sufficient to be measurable, it was necessary to subject the graft polymer to more than one polymerization step. Thus, after a graft polymerization, the graft polymer was separated from the unattached new polymer by a precipitation fractionationo The isolated graft polymer was then redissolved in monomer.an-d chain transfer solvent an.d a new graft polymerization was done~ This recycle procedure was continued as often as necessary to obtain the desired number of branches~ Equation (1) suggests a method of plotting the data obtained from these polymerizations. Calling ZX the sum of the degrees of conversio.n X obtained for each of the graft polymerizations on a particular linear'backbone polymer, then one can plot the data according to the approximation <w> r Cp <o.>n (3) where <ICE> is the average degree of pol~ymerizatiorl of the linear boackboneO

This expression ignores the change In the degree of polymerization of the backbone substrate, but the approximation is not too serious for the polymerizations considered here. The average number of branches, <K>, can'be computed by the expression <K> = <g M> (4) QMb> where <Mg>, ~l>, Ib> are the molecular weights of the graft, linear backbone, and branch polymer, respectively. The molecular weight of the branches is estimated from measurement of the molecular weight of the unattached Cnew polymer (25) (33) (34). This method of synthesis offers several advantages. The average length of the branches can be controlled'by the chain transfer solvent~ Also, the presence of the chain transfer solvent in the quantities used here precludes difficulties such as Sobova, et al, (16) and Jones,.et al., (15) encountered f)rom the termination of branches on different suabstrate molecules by a coupli.ng mecha:n.ism (reaction 7)n This difficulty is not to be anticipated for vinyl acetate in anxy event, since the monomer itself will act as a cha1inr transfer agen:t L:a the absence of a foreign solvent, and any mutual radical termi:nation that. does occur is apparenatly by disproportiornation (2) Due to the con-trol over the average length of the branches, the information about the kinetic scheme of the reaction, the use of a fractionated linear backbone polymer, and the estimate available for the average number of branches per molecule, this scheme gives a reasonably well characterized branched polymer~ The discussion of the molecu

-17lar weight polydispersity of the branched polymer will be deferred to a later section, but it is shown there that the branched polymer has a comparable distribution to the linear backbone o The problem faced by Zimm and Thurmond (9), and others using similar techniques, of fractionating an extremely broad distribution branched polymer has been avoided. in this scheme. It must be pointed out, however, that only the average length of the branches has been held constant. Do THE INTRINSIC VISCOSITY It is the aim of this, and later sections, on the physical properties of polymers to present and discuss the pertinent theories concerning the behavior of polymeric systems. The discussion will attempt to include only those developments which have a direct bearing on the measurements made in this study. The theoretical calculation of the intrinsic viscosity involves consideration of the motion of a flexible macromolecule in a viscous medium under the influence of an external force and of Brownian motion. The statistical mechanical theory of this irreversible process is of course quite complex-and not yet completely written down in rigorous form even for linear polymers. There is available, however, a model which can be treated with certain simplifying assumptions to give a relationship which has met with considerable success in correlating the flow properties of-linear polymers. The model considered is that adopted by Kirkwood and Riseman (35). The Kirkwood-Riseman theory adopts a method due

-18to Oseen to calculate the hydrodynamic interactions of the macromolecule on the assumption that the actual polymer coil can be replaced by an equivalent cloud of independent beads. Physically, this means that the dissolved polymer molecule and entrapped solvent flow as a single unit. The theory of Kirkwood-Riseman eventually leads to the result N (S5 3/2 [] o X Fo(X) s2 (5) 100 MF() where [n ]o is the intrinsic viscosity in an ideal solvent, N the Avogadro number, X a theoretical parameter, M the molecular weight and (S2)o the meat. square radius of gyration in an ideal solvent. In an ideal solvent the spacial distribution of the units in the macromolecule obey a random statistics. The value of the function of X, F (X), is denoted by Fo(X) for ar, ideal solution. The intrinsic viscosity is defined as [I] = lim (5-io)/io 0 where B and Bo are viscosity coefficients for solution C=O and solvent, respectively. Flory (1) has suggested that since the parameter X Fo(X) approaches an asymptotic limit for large M, that this expression can be simplified to give the relationship [l] = 63/2 ( N (6) where 0 is the asymptotic value of XFo(X) times N 33/2 (100 x 63/2). This reduction implies a rather subtle equivalence of a hydrodynamic radiu.s and. a configurational radius of the polymer chain. One can define Ss as the radius of a hydrodynamically equivalent sphere for the polymer coil. The radius is the one that directly determines the molecular fric

-19tion constant, for the translational diffusion constant, D = kT/f as f = 65To(Ss), The ratio of this radius in a good solvent compared. to that in an ideal solvent is defined as as: as - (Ss)/(Ss)o (7) Thus,, assuming that the intrinsic -viscosity is proportional to the hydrodynamic radius, equation (5) gives, for a good solvent, the expression[T3] = (/ /100/O)X Fo(X) Mo (8) (8) In analogy with the parameter Us, an expansionr factor (a can be defined as the ratio of the actual radius of gyration in a good solvent to that, in an ideal solvent, -e2- (s ~)/()o(9) It now becomes clear that the 0 i~n equation (6) is actually given by the expression 63/2 3 = (s33/2/1OO)X O(X)S (10) Equation (6), which was suggested by Flory (1), assumes that as = a, that is, that the hydrodyramic radius changes as the configurational radius as the solvent-polymer system deviates from an ideal system. If such is the case, then the value of 0 = 2~87 x 10; where ['ql is 4 t-t.ipper.: S in cm, M in units of molecular weight, is obtained using the asymptotic limit for X Fo(X). Flory and Fox (36) have investigated the numerical value of q observed for numerous polymers in variou.s good solvents, sThey observed. that the best value for O seems to be 20l (~ 0,2) x 1021, rathber

-20intercepts) can be seen to differ quite markedly depending on which pair of points are selected for the solution. However, the dotted line is a Ln k l/T Figure 4. Data Smoothing (Fictitious Points). fair representation of the data so that the In k versus l/T plot is seen to be a convenient way of smoothing the data.

)- 3/2 [T.] = (N3/2/100) x Fo(X) M [1 + p(X) -.. ] (13) If one assumes again that the asymptotic value of the functions of X as X becomes very large are appropriate for flexible macromolecules, then there results [11] = 63/26 (1 + 1.55 (14) Using the expression for a given in equation (11), it is seen that equation (14) can be rewritten as 23/2 [r-] = 63/2 ( (15) M The treatment of Stockmayer and Albrecht gives the power on a as 2073 instead of 2.,43, but, indicates that it is still less than 3, which it would be if one assumed that a.s -, It must'be mentioned that all of the expansions in z given above are very slowly conavergento This would seem to limit the u-se of such equations to a solvent very nearly ideal, that is, a system where z is quite small o Thbese con-t.siderations have been restricted to a linear polymer whi.ch is monodisperse with respect to molecular weighto Kuaata, Yamakawa, aind Utiyama (46) have shown how to modify the results for a polydisperse system for which the Schu.lz distribution function for molecular weight can be usedo This fum.ction is given by (47) f(M) = (yh+l/ho)Mh exp (-yM) (16) where f(M) is the weight fractionl of the polymer with molecular weight

-22M in the range dcM, and h and y are adjustable parameters. If a z-average is used for the radius of gyration, and a weight average is used for the molecular weight,/ then equation (15) becomes 3/2 [] 63/2 q(h) <(s 2)> z (17) with m = 2.43 (h+l) (h+2)[r(h+l) ]2 (18) (h+l) r(h+3/2) where r is the usual gamma function. For a normal good fraction with an h of 10, equation (18) gives m = 2e2. The function q(h,e) is to account for the fact that the number average should actually be used to average both the mean sqauare radius and the molecular weight (48), This function has been given by Shultz as calculated by Stockm-ayer (49) a.d is found to be (h,) = (+l)'(h + 1.5 + 3C) Fr-(h+2) 13/2 r(h+l) r(h + 3 + 2) (19) The parameter s is defined by assuming that the radius of gyration in a good solveunt is related to the molecular weight by (S2) proportional to M to the power l+2 o. The data of Krigbazum and Carpentelr (40) and of Schulz and Kirste (37) i.ndicate that the numerical value of m in poor solvents (solvents nearly ideal) is about 2.2, in good agreement with the prediction of equation (18). In addition, Schulz and Kirste have shown that experimentally, the value of m varies from 2.2 to about 2.9 as the value of a goes from o,8 to 20. rDefinitions of the various averages are well know, and can be found in any text on pol yer chemistry (1).

-23In the neighborhood of a = 1.5 to 1.7, which corresponds to most good solvents, m is about 2,4o If, instead of setting m equal to some number other than 5, the FloryFox relationship, equation (6), is used, then there will exist a dependence of 0 on the solvent, molecular weight and temperature. Kurata has suggested that for m equal to 2.43, this dependence should be (46) = (2.87 x 1021) -O. 57 (20) This would give a value for 0 of about 2.4 to 2.2 x 10 1 for a between 1.5 and. 1.7, in agreement with the average value found by Flory and Fox in most good solvents. Meyerhoff (51) has applied the theory of Ptitsyn. and Eizner, which is given in somewhat different form, and has reached similar conclusions. The important point for this discussion is that both theory and experiment ind.icate that, the value of m is less thanr 3 for linear polymers in good solven:rts. Whkereas the theory discussed here is limited to poor solvents because of mathematical dif:ficulties, the theory of Ptitsyn and Eizner, and e-perimental evidence indicate that even in the usual good. solvent (a of 1.5 to 1o7) that the value of m is nearer 2.4 than 3, The calculation. of a for linear polymers f.rom viscosity data is obviously uncertain due to the incompleteness of the prese-nt theoryo The calculation of a for branched polymers from intrinsic viscosity data is even more speculative since there is no comparable developmen.t of the theory to accounlt.for nonideality of the systemo The effect of branching on the viscoelastic properties in ideal so

lutions is not yet completely solved QOne can. define g as the ratio of the mean square radii in an ideal solution of a branched and an unbranched polymer of the same molecular weight (50),. Then, if the factor 0 in the Flory-Fox exression is the same for- linear and branched polymers, it follows that the ratio of the irntrinslic. viscosity for branched and linear molecules of the sazre molecular weight is g/2 o This expression was first used by Zimm and. Thurmond (9) in examining data for'branched polystyrenes (the data were takes. in. a good solvent), but it gave only qualitatively satisfactory results. The first attempt to improve the early treatment was made by Stockmayer and Fixman (27). They considered. the change in the distribution of segments of the macromolecules about the center of mass as the shape of the molec.ule was charnged by branching. The calculation is'based or. a branched molecule which has a cruciform, or star shape, inl which. all of the branches emanate from a single branch.node. The d.evelopmen.t is based or. the p-remise that the intrinsic viscosity, which is depen1de-lt of long range Interactic sn wi1 no-t be as sensitive to anny slight change in the mass distributior. in the molecule as will the radius of gyration itself, A qu.atity h is defined such that h is th.e ratio of the ef fective ra ia.s of a crucif@or:. mo. olecule to that of a ll ear molecule of the same molecular weight, The Kir'kwood-Risema.n theory was then applied to find the friction cons tart for the crucif orm chain. The results give the ratio h as a functioon of the number of branches iTn. the cruciform molecuZLeo A separate caLculation for g is possible (see Section I-E) to give g as a function. of the number of branches in the mcolecule. The as

-25 - sumption is then made that the intrinsic viscosity ratio for branched and linear molecules is given by the ratio of the cube of the hydrodynamic radii, or by h3o Stockmayer and Fixmnan further suggest that even though the relationship between h3 and g has been calculated for a cruciform molecule, it is hopefully a good assumption that the same relationship would be good for other kinds of branching. This calculation gives intrinsic viscosity of the branched polymer as ['g] = 63/2 (S)o h (21) 0 (MY or, in terms of (S2), (s2)3 h3 [lg ]o = 63/2 g ) h3 (22) M g3/2 where () and (Sg) refer to the mean square radius of gyration for linear and branched polymer of moleculular weight M, in an ideal solvent~ The relationship between h and g is given hby h = k/[/2k + 2/ (k-) ] (23) g = (3k-2)/k2e Th:is relationship is plotted in figure 11 This result has been used i:n good solvenrts with satisfactory success by several investigators; see for example references (14) (27) (30). A recent contribution by Zimm. and Kilb (52) has given a slightly' modified form for the intrinsic viscosity ratio. Zimm (53) had previously considered the hydrodynamic problem for linear chains using the approximate form for the ilnteractions give:n by K~irkrwood. and Riseman, but with the

-26introduction of a transformation to a set of normal coordinates given by Rouse (54), to allow a more rigorous solution for the intrinsic viscosity. Zimm and Kilb considered the hydrodynamics of a cruciform chain using the method given by Zimm, to obtain the intrinsic viscosity of the cruciform molecule in an ideal solution. The results show that the ratio of the intrinsic viscosities of branched and linear polymers is approximately given by gl/2 for a cruciform molecule. Then, as in the treatment of Stockmayer and Fixman, it is assumed that the same expression will be a good approximation for the intrinsic viscosity to g relationship for all types of branched. molecules. Actually, for the values of g to be considered in this study, there is little practical difference between the results of the two treatments. Thus, for g greater than about o.6, h3 can be approximated by g3/ It can be seen from the preceding developments that any of the three approaches described give the intrinsic viscosity ratio as some function of g. Calling this function f(g) in a good solvent, or fo(g) in an ideal solvent, equation (21) becomes 3/2 [Ng ] 63/2 O fo(g) ( )/ (24) gog3/2 M It is to be noted that this equation applies explicitly to an ideal solvent The corresponding development for the intrinsic viscosity in a good solvent has not been done. Further discussion of the theoretical considerations involved here will be deferred until the discussion of the cal

-27culation of the radius of gyration in good solvents has been given. What few data that are available suggest that f(g) is slightly larger than fo(g) (2) (9). It should be mentioned that there is another theoretical treatment that has been used to interpret the intrinsic viscosity. This method is due to Debye and A. Bueche (55), and considers the polymer coil to be hydrodynamically replaced by an equivalent porous sphere of uniform density. Such a procedure simplifies calculations, but leaves one with an equation involving a radius of an equivalent sphere explicit in the results. F. c- (56) has recently considered an extension. of this model to cover star shaped branched molecules, and the results are in good agreement with those of Zimm and Kilb. The general method, however, is of limited utility due to the problem of equating the equivalent radius to the actual radius of gyration of the branched molecule other than for a cruciform molecule There has existed a semi-empirical correlation between [r] andd M for many years termed the Mark-Houwink equation~ This relationship gives [1] K= a<> (25) where K and.a are constants for a given polymer-solvent system at a given temperature. In ideal solvents, the value of a is almost always found to be 1/2 (1). Looking ahead to the parameter E which is defined by equation (29), one can combine equation (29) and the Flory-Fox expression, equation (6), to give 1/2 + 3c [r] = KM (26)

-28This expression will be discussed further later. E. THE MEAN SQUARE RADIUS OF GYRATION IN GOOD SOLVENTS This discussion has as its purpose the consideration of the theoretical calculation of the mean square radius of gyration, and the relationship between its value in a thermodynamically good solution and in an ideal solution. Following the procedure of the previous section, the theory relating to linear chains will be presented first, followed by a discussion of branched chains. This procedure is adopted because relatively more is known. about linear polymers, and the developments for branched molecules are usually extensions of the theory for linear chains. The mean square radius of gyration can be defined by the expression (S2) = Zi (si)/n (27) where si is the distance of segment i from the center of mass of a chain of n segments. The statistical calculation of (S2) requires some assumption of a physical model for the configuration of the chain in space, The appropriate starting point for high molecular weight vinyl polymers appears to be the assumption of a random distribution of the chain segments in space (1) (57). The problem then becomes quite similar to one involving the domain swept out by a particle undergoing Brownian motion. The actual dimensions of a macromolecule in solution will of course be different than a calculation based on a purely random distribution of chain segments would suggest, except in certain special cases. The difference can be thought of as arising from two sources, one a short-range effect and

-29the other a long-range effecti By short-range effects are meant such things as hindrance to rotation around individual bonds and bond angles inherent in the chain structure. Long-range effects are those involving thermodynamic interactions between the polymeric segments and their environment. If only short-range effects are considered, then the statistics can be worked out for an equivalent random chain having segments of mean square length b2. This length is in general different from'the length of a structural unit such as a mer. The results give for a random chain of n segments (S2) = (1/6) b2n. (28) It has been recognized for some time that equation (28) does not adequately describe the observed behavior of macromolecules in good solutions. The experimental data can usually be fit by an. expression of the form <(s2> = K M>w (29) where E is a number usually greater than zero, but in any case less than one half (e is only less than zero in unusual cases, it has a value of -2/3 when the polymer coil acts like a sphere). For an ideal solution in which long-range forces are absent, E is equal to zero. The problem of predicting this behavior statistically is the next topic to be considered. It should be mentioned here that the experimental evidence is not conclusive regarding the dependence of (S2) on M. The effect of poly#The statistics of chain configuration taking into account short-range effects is discussed by Flory (1) and Hermans (57) from two different approaches.

-30disperity is an. experimental complication that is not easily eliminated. One of the theories discussed later predicts a linear dependence of (S 2) on M even in the case where long-range effects are important. Koningaveld and Tuynman (58) have recently examined the work of Shultz (49) and Elias and Patat (59) on PVOAc in butanone in order to determine the exact dependence of (S2)i on Mi, where the subscripts are meant to define a molecule of a specific molecular weight. They have concluded that it is impossible to decide whether (S2)i is proportional to Mi or to some other power of Mi on the basis of the experimental data. Flory has suggested the definition of an expansion factor, a, given by equation (9) to account for long-range effects. The calculation of the chain dimensions in a good solvent, or in other words the magnitude of a, and the.nrature of the dependence on molecular weight and thermodynamic variables is exceedingly difficult. It must be admitted that no definitive solution to the problem exists even for linear chains, much less branched molecules. Four different approaches will be considered here, three of which have been extended to cover branched polymerso These are due to Wall and co-workers, Hermans and co-workers, Flory and co-workers and a perturbation theory first explored by Zimm, Stockmayer and Fixnnan and later extended by others. The procedure adopted by Wall, et al., has been to use Monte Carlo methods on a digital computer to generate random walks on a certain type of lattice (60)o The excluded volume, or long-range interaction effect, is simulated by discontinuing the walk when an occupied site is encoun

-31tered. The distarice between the end point and. the starting point is then determined. Using this method, it has been found. that for experiments involving walks of up to 800 steps on a tetrahedral lattice, the mean square end-to-end distance for a chain of n segments of unit lerngth is given by (R2) = 1.40 nL18 (30) This expression indicates that the excluded volume can be expected to cause a greater than linear dependence of (R2) on n, or M. Wall and Erpenbeck (60) have used similar techniques to find the relationship between (R2) and (S2) for a linear chain. They have found a direct proportionality between the two variables' but the proportionality constant is smaller than the value of 1/6 for a purely random flight. Hermans, Ullman, and Klanken (57) (61) have used an analytical approach to account for the excluded volume. Their treatment is based on an approximation that the placement of the (t+l) segment to the tth segment is such that the (t+l)th segment points in the direction where the probability of finding the previously added segments is lowest. For a random flight chain, this probability has the same value in all directions, but for chains in which the segments occupy a finite volume, this probability has a maximum where the local density of segments is lowest. This approach makes the assumption that the interaction of previously placed segments with the envirorments will not modify placement of the successive segments, but that only the volume excluded by the segments is important. With the introduction of the approximation that the probability that the (t+l) segment has a given direction is proportional to the local density

-32of segmenets, Hermans, et alb, obtained a modified diffusion equation known as the Fokker-Planck equation~ A perturbation method was utilized to give the approximate result nb2 (S2) - nb (1 + 1072 v/b3). (31) It is to be noted that this expression predicts (S2) to be proportional to n, or thus to M. The calculations of Hermans, et al., have been criticized by James (62) who has used a similar development, but with different assumptions for the probability of placement of successive segments. The theory of James predicts (S2) to increase faster than n, but is not discussed further here because there is no corresponding development for a branched molecule. Flory (1) has made a different assumption regarding the excluded volume effect on the distribution of segments in space than Hermans, et al., and has come to somewhat different conclusions. Flory has assumed that for statistical calculations, -the true segment distribution of the polymer molecule can'be replaced by one for a cloud of disconnected segments whose distribution a:bout the molecular center of gravity is on the average given'by a Gaussian distribution. The segment distribution arrived at by Hermans was not Gaussian. The Gaussian distribution is assumed to persist even in the presence of intramolecular interactions, but it is assumed that the spacial distribution is uniformly expanded by the factor e defined in equation (9), This treatment results in a closed expression for a which can be given by (64)

533 oP U3= C z (32) where C is a constarnt which Flory gives as 53/2/2, and z is defined by equation (11)o Due to the dependence of z on nl/2, this expression predicts that a will be a slowly increasing function of the molecular weight, increasing without limit evenx when M becomes very large. Since C is an increasing function of M, then (S2) which is equal to U(S2)0 will increase more rapidly than M, in disagreement with the conclusion of Hermans, et al. Although equation (32) is a very useful result, Stockmayer (64) has suggested that its agreement with experiment can be improved by substituting a constant based on Wall's calculations for the theoretical value of C given by Floryn Kurata, Stockmayer and Roig (63) have employed. methods very similar to Flory's in treatirng the excluded voliure effect, but instead of a spherically symmetrical distribution fiunctionQ, these iu.vestigators have considered a uniform distribution of segments within an ellipsoid. Their calculation predicts that the ratio (a5 - U3)/nl/2 will be a function of X rather than a constaat as Flory foundo These results seem to be in better accord. with the calculations of Wall. The last calculation of (S2) for a linear chain to be considered here is the perturbation theory developed by Teramoto (65), Zimm, Stockmnayer, and Fixman (44) and Fixman (45)o This method has been outlined by Kurata, et al. (46), in a recent treatment of the excluded volume effecto Without attempting to give the details, the method involves the calculation on the partition functio.m of the ehain. after assuming pairwise additivity of the

potential of mean force between the segments. The integrand of the partition function is then expanded in a manner analogous to the expansion encountered in the theory of an imperfect gas / This expansion allows an approxnimate solution to be obtained for (R2) or (S2) in the form of slowly convergent power series in z (R2) = nb2(l~iz~~z-~z2 + (s-) nb2 6- + 4z Z- 2 + (33) nb2 1z34 2 (s2) l 1 + 1 cp z + (11) The first term in z involves calculation for only one contact between two chains, the second term accounts for two simultaneous contacts, etc. Zimm, Stockmayer, and Fixman have noted an important result which arises rigorously from their treatment. The equations (33) and (10) are complete only to the approximation of a single contact, that is, consideration of only those configurations in which a single pair of elements are interacting. According to the theory given by Zimm, et alo., if only configurations in which there are three or more simultaneous intramolecular interactions are neglected (a very good approximation.), then the general result will have the form. (S2) = 1/6 nb2 f(z) (34) with f(z) = 1 for z = 0, increasing linearly for small z, but having an unknown asymptotic behavior. Since z is proportional to nl/2p, this implies that even for very large n, (S2) will depend on n to a power greater 7See for instance chapter 4 of the book of Green (66).

3 5 than the first unrless it becomes i:ndependent of J, i eo, of the solverto Existing data do not indicate that (S2) becomes independent of the solvent for large n, indicating a dependency on molecular wTeight that increases faster than n. The quadratic term in the expansion for (S2) has not yet been derived. The term from the expansion for (R2), due to Fixman (45), is sometimes used for lack of a better value~ Kurata9 Yamakawa, and Utiyama (46) have tested equlation (11) with some experimental data, and conclude that the use of the quadratic term from equation (33) for C2 is not justified. an.d that equation (11) is good only for Iz =< 0o15, corresponding to solvents very nearly idealO The data of Schulz and Kriste (37) also indicate that equation (11) is quite unsatisfactory for the usual good solvent where z > 0o5, andd. in fact they report that c2 must be about inversely proportional to M to fit the datao Since no more terms are likely to'be found in the seriesexpansion. in z'by.rigorous means, it does not seem that this approach can. be too fruitful in giving ex-act results. It is perhaps betteer to attemrpt ar. approximate closed form for a as a function of z, such as Flory didg sacrificing some accuracy to obtain an approximate closed express.io.uo Having now dealt with some of the theoretical predictions concerning (S2) for linear chains, it is possible to examine their extension to the branched molecule. The first calculation to'be considered is that of (S)'o, the mean square radius of a branched molecule in an ideal solution, The methods usually used for this purpose are based on a theorem due to Kramers (67) which states that for an.y branched molecule containing n

-36segments, (Sg2) is given by (Sg) = (b/n)2 Z x(n-x) (35) the sum being taken over all possible separations of the molecule into two parts, one with x segmen:ts, the other with n-x segments. Kramers used this relationship to establish the ratio g given by equation (23) for cruciform molecules. Zimm. and Stockmayer (44) have show) how to extend this calculatifon to an ex.tremely important type of branched molecu.le They have calculated g for molecules a ronitaining completely random bratnchingo That is, given branched molecule containing ng segments and k branches, the ng segments may'be considered as distributed at random. among the k branches and k+l chain lengths of the backbone (with the restriction that each of these k + (k+l) "boxes" contains at least one segment)o This type of mod.el is nrot appropriate. however, for the type of molecule considered here. A correct mod.l would be the same as that just d.escribed., with the resthrictio. that a certain number of the seg.ents, say n.1, must be used to fill the (k+l)'backbone chain lengths. A calculation based on this model usir.ng the procedure given by Zimm and Stockmayler was attempted as a part of this studvy with unsatisfactory results~ The solutionl could rlot be pult into a closei form. to make it luseful for calculational purposes, so the model was abandonedo. A simpler model consisting of a branched molecule containi:ng ng total segments, nl segments in the backbone, and #The essential drifficul'ty was evaluation of a d.efirite integral which is to be evaluated between limites 0 and x. In the treatment of Z-S the val ue of x is unity and. this integral becomes the Beta-functiono (The BEetafuncltion of the variables p and q can be expressed as ~(p)P(q)/r(p+q) where F is the gamma func;ltiom ) In the problem here, x K 1, and no closed form was found for'the integralo

k branches containing (ng-nl)/k segments, spaced at intervals of ng/(k+l) segments along the backbone is considered instead. The unsuccessful treatment of the more general model considered above indicated that the substitution of an average value for the length of the branches will not seriously affect the results. The restriction of equally spaced branches might be more serious and will be discussed further. This type of model has been used previously, but no analytic form was given (b14) (68). The calculations based on the simplified model are given in Appendix II, the results give for g the general relationship g(k,r) = 1 [r3(3k-4) k+r2(k+2) k+r 2k+l k+l (36) (kr+l)3 L k+l where r is the ratio of the number of segments in a branch to the number of segments in the backbone, r a= (ng-nl)/knl. For the ratio r less than 1/10, this will reduce to the very usable form g(k,r) = 1 = (37) kr+l ng Equation (37) is applicable to all of the branched molecules considered here insofar as the model is good, that is, the approximations from equation (36) to (37) are valid here. Orofino (69) has derived an expression for g based on the general model described above for random placement of branches of a given length on a backbone of nl segments using a procedure not based on Kramer's theorem. His expression, which is good for k >> 1, is approximated by equation (37) for small r, thus giving more confidence in that expression. It cannot be emphasized too strongly that the calculations of g con

sidered above are limited to ideal solutions. The problem of the excluded volume effect in branched polymers is far from settled theoretically or empirically. Three separate kinds of treatments of the excluded volume effect in branched polymers will be discussed here. It will be seen that these treatments do not arrive at conclusions consistent with each other. Experimental results given in the literature will be considered when the results of this study are discussed. Stockmayer and. Fixema. (27) and Fixman (4h) have calculated the ratio of (Sg-$SI) in a good solvent to g, the value in an ideal solution, using the perturbatio:n theory of Zi.mm, Stoc.maryer, and Fixmnan. discussed above. Their calculation. was based o.n a cruciform model containing four branches f equal length. iThe results give (S\ for a chain of ng segments as gS-) = ngb2/6 (I + 1.12z + o..) (38) Dividing equation (38) by equation (11) to obtain the ratio of the radii l. a good solvent there results (s2) _ I (+ 1.282z + ( This implies that g is greater than (S2)/(S2) since z is always positive when the second virial coefficient (see Section I-F) is greater than zero. In fact, to the first approximation, (s2)/(s = g ( + 6 + 6z + o o(4o)

-39The validity of this result must be questioned in view of the failure of the series in z to adequately account for the behavior of linear chains, aside from its basis on a very special kind of molecule. The calculations of Hermans, et al., were extended by Kreussling and Ullman (70) to account for the type of branching of particular interest here. They considered a chain containing ng segments with k branches of length (ng-nl)/k placed on a backbone of nl segments. An alternative form to equation (27) was used to calculate (S2), namely. 1 n n (S2) = z z (r (41) 2n2 tP p=l t=l where (rtp) is the mean square separation of segments t and p. Using the method previously given by Hermans, et alo., an expression was derived for (rp ) Unfortunately, it was not possible to do the entire sum indicated by equation (41), but the sum had to be confined to those configurations such that both t and p were on the backbone. Thus, only part of the terms co.stributing to (S2) were obtained. The result for this part of (S2), deg noted as (Sg), is given by 2 gS-~) ngb (s2) = 1 + 1.72v/b3) + 0o587kv/b2nl x 9 6 _24n~/1/2 3~3b 3/2 7 7/2\7/ x24t/2 + 1n/2 + 6b 24 Unj1+nb) / -n2 )2 nb/2]} + 3 _/'N XL35 nl 35 n n3'25 (42) where nb = (ng-nl)/k is the number of segments in a branch.'Utilizing equation (37) as an estimate for g for the type of branching obtained here, the following results when equation (42) is divided by equation (31) for a

linear chain of ng segmerrnts (provided the ratio of nb/nl is small) 1/2 V9 A- +'7 bb2(43) (1 + lo72 v/b-) bb2 The result is probably similar to what this theory would give for (Sg)/ (SI) itself as a function of g when the number of segients in the branches is not too large compared to the total number of segnents in the molecule. In any case, the factor on the right would tend to be larger if more terms were included. Since all of the terms in the right hand factor are positive for A2 greater than: zero, this equation indicates that the ratio of the radii in a good solvent will be greater than g, in direct contradiction to the result of Stockmayer and Fixmano The validity of this treatme~nt, Thich does not give an analytically complete relationship between the ratio of the rad.ii and g in:r any event, carn be challenged in view of the predicted dependencce of (S) on n, which is questionable. Although there is still some douibt regarding the correct dependernce of (Se) on n) as discussed above, most exerimental and theoretical treat.merts seem to indicate that (S2) increases faster than no In spite of -these uml.ertainties, equation (43) may serve some usefulness in indicating the direction of the effect of the excluded volume on (S2)/(S2) for the kinds of'branching of interest here. The treatment of Ptitsyn (71) [conclusions reported in English. by Volkenstein (72)] for the excluded volume effect has the advantage of giving a closed form for the answer in terms of co7 It is convenient to first #Both of these invest;igators define a'g' which is the square root of the given here, and used exclusively in most of the literatureo

-41define an expansion factor for a branched chain. in analogy with equation (9) for a linear polymer: Cg = (s2)/(sg)0. (44) The expansion factor for a linear chain will be denoted as C1 wherever this is necessary to avoid confusion. Ptitsyn assumes that in the absence of interaction effects, the distribution of the segments of a polymer chain relative to the center of gravity is approximately Gaussian. Employing a procedure analogous to that of Flory it is assumed that the branched moles cule can be considered to act as a cloud of segments with a Gaussian distribution. of mass about the molecular center of gravity, and that this distribution can be modified by increasing all linear dimensions by some factor, egg, to account for intramolecular interactions. Thus, instead of being modified by 1/02, the argument of tne exponential in the Gaussian distribution is modified by the factor 1/g Ogo Carrying out the calculation on this basis, Ptitsyn comes to the result 9 3' (45) 5 3 g Thus, this calculation predicts that (S2)/(S2) will exceed g if the radii are measured in a good solvent. The attractiveness of this treatment is that a closed form is obtained without recourse to any particular model for the branched molecule (cruciform, etc.). The validity of assuming that the actual segmental distribution in a branched molecule can be approximated in this way is not certain, and it should. be remembered that

-42this procedure gives results which are only partially correct even in the case of a linear molecule. It is, however, a useful prediction of the effect of the excluded volume on the ratio of the radii in good solvents, and the only one amenable to useful calculations. The effect of the excluded volume on the intrinsic viscosity ratio for branched molecules in a good solvent to an ideal solvent must be related to 04g by an equation of the form N / 190 =Cmg (46) where mg is some unknown exponent. This is the analogous expression to equation (17) for linear polymers where the exponent is about 2.5. Thus, on combination of equations (24) and (45) there results [rig = 63/2, fo(g) ( 2 3/2 m g3/2 M g Forcing this equation into the form of the Flory-Fox expression, one obtains [rig] = 63/2 fo (g) (S) (mg-3 48) h I 6 0 I Ce (48) [g] = S3/2 ~g/2 M g The corresponding equation for a linear chain is, of course, given by [11 = 63/2 (SL) ) (49 M Since f(g), the intrinsic viscosity ratio in a good solvent, has been measured here rather than fo(g), it is appropriate to find the relationship f(g) and fo(g). Combining the definitions of f(g) and fo(g), there results

-43[hg]o/[l] fo(g) = _ * [ng ]/ zMf0(g) (50) Equation (48) can now be written as [Tg] = 6~3/2 f(g) ()3/2 am (51) Comparing the proportionality factors between [rI] and S3/M in equations (6) and. (51), one sees that the ratio of the effective value of O, defined as O* = M/(S2)3/2 for branched and linear molecules~ is given by -pf( \r 3 = (x(52) g It is seen from equation (52) that depending on the relative magnitude of the excluded volume effect for linear and branched polymers, the ratio g/Al can be greater or less than unity. In ar ideal solution, one would expect this ratio to be greater than unity based on the value of fO(g) from either the theory of Zimm arid. Kilb or Stockmayer and Fixmarno Equation (52) provides a wray to ch.eck for the excluded volume effect in branched polymer provided there is some a priori kanowledge of go Before leaving this section, some comrment should be mad.e concerning the effect of polydispersity of molecular weight on these variables. It was pointed out in the discussion of the synthesis procedure that the branched polymers of this study are not believed to be much more polydisperse than the starting linear fractions used as the backbone material (see Section III-A) o In some cases, however, the problem of polydispersity is quite seriouso Instances of *this are in the work of Long on

-44 - high conversion'branched PVOAc (30)9 and. Zimm. and Thu.Lrmond (9) and Manson and Cragg (14) on branched polystyrenes to cite a few of many works encountering such difficulties. Kilb (73) has considered the effect of branchlrng on the radius of gyration of a polydisperse randomly branched polymer (the kind of branching considered by Zimm and Stockmayer). He has considered a branched molecule such as would be formed by the synthesis of Zimm and Thurmondo Using a relationship derived by Stockmayer (74) to describe the molecular weight distribution of such a branched polymer, Kilb concludes that at the same value of <>w, <(Sg)>z and <(S2)>z would be identical as a consequence of the distribution functions used in the averaging for the radius of gyration and the molecular weight. This result was suggested as an explanation for the fact that Zimm and Thurmond Lfound (S2) essentially the same for branched and linear materials of the same molecular weight. It should be remembered, however, that these data were taken in good solvents, so that direct application of Kilbts theory, which employs Zimm and Kilb s expression for fo(g) to explain the behavior observed in a good solvent is uncertain. In addition, Kilb 2s theory is not applicable for fractions, but is limited to those cases which have a particular, and broad molecular weight distribution. Thus, while the theory may explain the fact that the ratio of the radii was near unity for the whole polymer of Zimm and Thurmond, it cannot be used as an explanation for the same ratio being near unity for the fractions of that whole polymera

-45 - F. TEE SECOND VIRIAL CO:FF=ICIFNT The second virial coefficientt can be defined from the expression for the osmotic pressure, i, of a polymer of molecular weight M, dissolved in a solution of concentration c (gm/cc): i/RTc = 1/M + A2c + Asc +.. (53) where RT has the usual significance. This expression has been derived rigorously by McMillan and Mayer (75) on general statistical mechanical arguments similar to those used in the development of the theory of fluids (66). The A2, etc., are called the virial coefficients, A2 being the second virial coefficient. The attempts to calculate the virial' coefficients have in general been of two kinds: those based on a particular lattice model for the polymer solution; and those employing more formal theories which apply molecular distribution functions. Excellent reviews of the progress of these theoretical calculations for the virial coefficients have been given by Casassa (76) and Stockmayer (64). The development of a complete theory for A2, or the other virial coefficients, has been very slow because of the extreme complexity of the mathematics involved in chain statistics. Thus, while there ha's been some progress in obtaining the relationship between A2 and the molecular weight and other measurable parameters, there has been relatively little advance in the calculations for A3, and virtually no attempts have been made to derive expressions for the higher virial coefficients. The extension of the theory of A2 to solutions of branched molecules has been even less well studied. There appears to be only one theoretical prediction of the ef

-46feet of branching on Aa given by Stockmlayer and Fixman (27), and discussed'below. Krigbaum and Trementozzi (77) have developed a semi-empirical correlation which shows the effect of'branching on A2, and this too will be considered. The purpose of this sectionr then, is to consider the fundamental meaning of A2 and the methods of calculating A2. Predictions of the effect of branching on A2 based on this fundamental theory will then be discussed. No attempt will be made to include all of the most recent calculations for A2, but some of the predictions on how A2 should correlate with other measurable variables will be mentioned. The theory of A2 based on molecular distribution functions has been pioneered by Zimm (78). A2 is obtained as a pair interaction integral over an integrand which is proportional to the difference between the actual probability for a configuration and the probability of the same configuration if there were no interactions of any kind present (as in an ideal solution. The integral is over all configurations of two polymer molecules a and b of molecular weight M, and is given by A2 = N/2VM2 o- * F1(a)Fi(b) [1 - exp (-Uab/kT)] d(ab) (54) where N is the avagadro'number, V is an arbitrarily large volume, the F1 are distribution functions describing the relative probability of finding a molecule with a set of coordinates (a) or (b), and Uab is the potential of the average force between the two molecules in the solution. Zimm has applied this equation to the calculation fcor a flexible chain molecule, and has been able to obtain an approximate relationship for A2 in the form

-47of a power series in z A2 = Nn2/2M2 (1 - 2.88z + )(55) The first term of this series accounts for configurations involving the interaction of one pair of segments from two molecules, the second for the simultaneous interaction of two pairs, etc. Derivation of further terms have not proved possible, except to give a part of the third term which involves triple contacts. The excluded volume integral, I, is defined as = {l - exp [u(ia,ib)]/rkT 4rtr2 dr (56) where u(iaib) is the mean-potential of the average force between segments ia and ib in molecules a and b respectively. In the notation of Flory (1), B is given by (64) = v *1 (1 - /T) (57) where V1 is the molecular volume of the solvent, 9 the temperature at which the chemical- potential due to segment-solvent interactions is zero, and F1 an entropy parameter. Although, the formal theory of Zimm calculates A2 from considerations of interactions between polymer molecules, the effect of polymer-solvent interactions is explicitly felt in the parameter i through the term u(ia,ib), which will be different for different polymer-solvent systems. Quite analogous to the calculation for the radius of gyration, this formal approach seems to have more value in giving theoretical insight into the meaning of A2o than in the usefulness of its results for any practical calculations. It remained for Flory to develop

-48a theory which resulted in some usable form for comparison of theory to experiment The first calculation of Flory (79) was based on a lattice model that was subsequently modified by Flory and Krigbaum (1)o In terms of the calculation described above, the polymer molecule is represented as a cloud of segments with a spherically symmetrical distribution of segments about the center of mass. Whereas Flory originally considered this distribution to be uniforms Flory and Krigbaum assumed that the segments would have a Gaussian distribution around the molecular center of mass, The potential of the average force, Uab, is then averaged over all configurations of two such molecules having their centers of mass separated by the fixed distance r, to give a potential of the average force between to two molecules, U(r). Equation (54) then becomes (1) (27) A2 N /2M..2 f {1- exp [-u(r)/kTj]} 4r2dr. (58) The results of this integration. can'be given in terms of the nomenclature used here as (76) A2 = Nin /2X2 F[z/(2) / ] (59) where F[z/(S2)3/2] is a definite integral which goes from unity to small values as the argument of the function increases. Numerous attempts have been made to improve these theories by various devices (38) (80) (81), the results of which are discussed by Stockmayer and Casassa in the reviews mentioned above~ The methods based on the smoothed density models for the segment distribution functions have

J49generally led to closed expressions, while the formal theory of Zinm rem sults only in the first few terms of a slowly convergent series in zo The validity of a prime assumption in both of these methods has recently received some attention (64) (76)o In both of these treatments, all interactions are characterized by the parameter i, so the vanishing of segment-segment interactions when A2 goes to zero implies the vanishin.g of all of the virial coefficientsO Such has rnot been the case for all experimental data (82), indicating an inadequacy in the theory. Stockmayer (64) has pointed out that it is extremely unlikely from the statistical theory that As should vanish whenr A2 goes to zero, sicne A3 con.tains contributions from ternary clusters. Thus, the situation where A2 goes to zero, which will happen when T = G, is analogous to the Boyle temperature in the behavior of an imperfect gas where the second virial coefficient vanishes and the gas behaves ideally, to the first approximation. The polymer solution for which T = 9 is not, however,~ strictly an. ideal solution, just as the real gas is:rLot ideal at the Boyle temperature. Thus, the value of (S2) observed for T = 8 may not'be exactly the same as for a strictly random distribution. of segments. Stockmayer and Fixmuan (27) have extended the procedure of the formal theory to consider a cruciform molecule with four equal branches. The result gives for the second virial coefficient of the branched molecule, (A2)g = \n_2/2M (1 4 oOz+ ~)* (60) Again, the slowly converxgent series in z precludes any numerical calcula

tions with equation (6o), but it is seea that, e.uations (60) and (55) imply that (A2) is less than (A2)1 for a given value of M. g Noting that the smoothed den.sity treatment requires no assumption concernlrn the presence or absence of branchingg Stockmayer and Fifnan have shov-n that the density f unctions for branched and linear polymers should be very similar in shape. This suggests that the result for A2 based on a smoothed density ca culaation should be as valid for a branched molecule as for a linear chain. Then, if (S2) is smaller for a branched polJymer than for a linear chain of the sare molecular weight, one would expect (A2)g to be less than (A2): due to the character of the function F' [z/(Sa) 3/] It is to be noted that the value of the radius that is called for in these expressions is the radius in a good solvent. The use of one of the functions F[z/(S2)3/2] derived from theory to predict the behavior of the ratio (A2)g/(A2), will be discussed in a later section, after the experimentally observed efe@ t of the excluded volume for blranched polymers has been: considered In any case, the expe..menrtal evidence suggest rather strongly that braniching does decreas. the value of the secord. virial coefficient (9, 14 16, 30, 83-88), There seems to be only one case where the opposite behavioAp was observed, a1nd this may have been due to the presence of res dual bromine groups iLn a polyst yrene branched by a grafting procedure (16)o Physically, there is no reason to expect the parameter B to be ch.oanged by the presence of branching, since it is o tly depende~ t on the consti tutLon of the segrrents aid the solvent (78 ). As Stockmayer and

_51J Finman have indicated, the eff ect of branching must be felt in that part of the expression which accounts for the magnitude of the various segmnentsegment interactions. A lower value of the virial coefficient for branched than for linear polymers at a correspon.di ~g molecular weight must imply that the segment-semen.t interactions are somehow increased, which indicates a more densely packed coil for the bra:rched moleculeo The fact that the value of A2 becomes zero when P goes to zero indicates that the theta temperature, G, should be identical for bra:nched. and. linear polymers, as has been observed (9)o The treatment of Krigbaum and Trementozzi (77) is based on the combination of several relationships and an empirical observation~ Calli:ng the series in z of equation (59) simply h(z), combination of equations (9), (59), and the definition for z results i:n A2M2/N(12)3/'2 _ 1 (4t)3/2 zh(z) (61) O 2 Neglecting the departure of h(z) from unity, Krigbaum (89) substituted equation (61) into equation. (11) to obtain the result () 3/ = (2)/ + 143 x 10-25 A2M2 +... (62) Assumption of the Flory-Fox relationship then gives (with 0 = 21 x 1021) [l] = ] o + o0oo0044 A2M +. (. (63) Although the assumpt.iors used. to obtain this result are not valid for good solvents, the equation has been found to hold. for values of A2 which are rather large (89)o Equation (61) can be rewritten using the FloryFox relationship to give

-52A2M/[r] = f(c) (64) where f(c) is some function of a. Occasion will arise to discuss this relationship later. In obtaining their correlation, Krigbaum and Trementozzi have assumed that equation (635) is valid for both branched and linear polymers, although the statistics, and hence at least the constants in the series in z, are explicitly derived for linear chains and are known to be different for branched polymers. In justification for this, some data of Zimm and Thurmond (9) are quoted to show that such behavior has been observed. Dividing equation (63) for a, linear chain into the same equation for a branched polymer, there results (A2)g = [rlg]o1 ri ~g/[i]o- (65) (A2)1 [T1]o Og L ]fI/[1 ]o - 1( It is further argued that in the limit of low branching, that O1 is about equal to Og, and the bracketed term becomes near unity, giving (A2)_g h_ [g9o= ( (A)g- fo(g) (66) (A2)1 [i ]o Avoiding these later assumptions, equation (65) can be cast into different, form utilizing equations (46), (50), and (52). Substituting these expressions into equation. (65), there results (A2)g = g3/2, (67) The usefulness of this result will be discussed latero To summnarize this section, the existing theories concerning A2 for

-53linear polymers, while improving, have not been able to quantitatively predict the experimental behavior. Thus, although the relationship A2 = K M- (68) where c is about 0.1 to 0.4, is usually observed experimentally, no theory predicts such a dependence on M. In addition, the magnitudes of A2 calculated theoretically are lower than the observed values. Regarding branched polymers, the theory does give some reason to expect that the presence of branching will reduce A2, but the most convincing evidence for such behavior is found in the experimental observation of such a reduction for many different kinds of branched polymers. Go LIGHT SCATTERING IN GOOD SOLVENTS The measurement of the Rayleigh scattering from dissolved macromolecules in a good solvent at finite concentrations has been an important experimental device for this study. It has been shown elsewhere that this measurement gives unambiguously the weight average molecular weight, the z-average mean square radius of gyration and the second virial coefficient. Under favorable circumstances, it is also possible to obtain information concerning molecular weight distribution, degree of branching, and other molecular parameters. The purpose of this section is to discuss briefly the theoretical justification of interpretations of the light scattering data to be employed later. Te statistical treatment of Zimm (Ls7) has provided a very useful device for the analysis of light scattering data, A theory of Rayleigh scat

-54 - tering must include a correlation of the configuration of all pairs of scattering elements in the scattering solution. It is to be expected then, from the diff iculties encountered in the corresponding formal treatments of A2 and (S2), that the theory will face considerable mathematical difficulty. The calculation of Zimm can account for only single contacts between polymer molecules, but it is exact to that approximation, and is useful in the interpretation of data extrapolated to infinite dilution where such an approximation may not be too bad. The expression derived by Zimm is given by Kc/R(G) = 1/P(G) + 2A c + o (69) where K is a constant involving the geometric and physical parameters of the system, R(G) is the Rayleigh scattering ratio, P(G) is the particle scattering function, equal to unity for G equal to zero and decreasing as G increases, and G is the angle of scattero The particle scattering f-unction corrects for interference in the scattered light caused by the size of the macromolecules. Albrecht (90) has used Zimm's method to extend the formal derivation one step further to include simultaneous contacts between two pairs segments. These calculations show that the second term on the right of equation (69) should include a function of G, Q(G), which is unity for G equal to zero, and decreases as G increases. Characteristically, the estimation of Q(G) from Albrecht's calculation is limited to small z and small 9 due to a slowly convergent series in these variables~ Flory and Bueche (91) have utilized the smoothed density model of Flory and Krigbaum to obtain an estimation of Q(G) for

-55 - large z and G by machine calculations. The same trends predicted by Albrecht's treatment were observed. Of interest here is the prediction that Q(G) is unity for G equal to zero, so that estimation of A2 from the curve extrapolated to 9 equal to zero should not be affected~ Zimm suggested a useful double extrapolation. method based on his theory in which Kc/R(G) is plotted against sin2(G/2) + kc, where k is an arbitrary constant. By this procedure, the data can be extrapolated simultaneously to zero 9 and c to give the intercept, l/M. The initial slople of the curve [Kc/R(9) ]=0 versus c is then equal to 2A2, and the curve for [Kc/R() ]c=0 versus sin2(G/2) is the reciprocal particle scattering function multiplied by the factor 1/M. Zimm showed that for linear chains obeying random flight statistics, the initial slope of the curve P- (0) versus sin2(9/2) is proportional to (S2). It was shown further that if the polymer exhibited molecular weight polydispersity, that the appropriate averages for these parameters were AM>w and <(S2)>z respectivelyi The asymptotic behavior of the reciprocal scattering curve was also examined by Zimm, and the slope of the asymptote was found to have an intercept of 1/2 <M>n and a slope proportional to <(S2)>w under favorable conditions~ These conditions are that the scattering particles are such that the asymptote can be reached, but not so large that the entire curve for P-1(@) represents the asymptote, with the initial part being compressed into a region over so small a range of 9 that it is not seenm Benoit, The z-average of (S2) is strictly correct only if the scattering solution is ideal. The effect of a non-ideal solution on the average will be pointed out later~

-56Holtzer, and Doty (92) have estimated that in order for both the initial and asymptotic slopes to be observed, the value of <(S2)>z should lie between 105 and 15 x 105 (AO)2e Zimm has observed that the reciprocal scattering curve is only very slowly convergent as G goes to zeroo Thus, he has derived a relationship between the correct initial slope and the observed P- () curve as a function of the parameter h in the Shultz distribution function to allow determination of the correct initial slope. A similar treatment was given for the asymptotic slope. The use of these relationships will be described in the experimental section. Although the calculation of Zimm was limited to a discussion of linear molecules obeying random flight statistics, it has been shown many times that regardless of the shape of the molecule (branched, rod-like, etc. ), the initial slope of the reciprocal scattering curve is proportional to (S2) (93) (94) (95). In fact, it has been shown that even the assumption of random statistics is not necessary for flexible chains, so that the slope will yield (S2) regardless of the volume effect (41) (94) (96) (97) (98). The excluded volume effect does, however, cause the average value of (S2) to deviate slightly from a true z-average in the case of polydwisperse systems (98). This deviation is small in any case, and is of little consequence when comparing the results for fractions having about the same degree of polydispersity, as is the case here. Although the shape of the molecules does not affect the determination of (S2), it is quite important as a factor In the determination of the ex

-57act shape of the P-'1() curve, as is the molecular weight polydispersityo Zimm (47) found that the curve for a monlodlsperse system would have a gentle upward curvature, and that polydispersity would cause an increasing downward curvature as the degree of polydispersity increased. Benoit (95) (98) has examined. the effect of branching in the p-l(G) curve, for both monodisperse and polydisperse systems. He has applied the technique of expanding the expression. for the P-l(G) function to find the initial and asymptotic portions of the expression. Calling the initial and asymptotic slopes to the curve Si and Sa respectively, then for a polydisperse system in a good solvent there results (Sa)g = (k/2) <(ZS)>w [k(G)] (70) (Si)g = (k/3) <(g)> (71) where [ is equal to (1/1 + 2e), E being defined by equation (29). The asymptotic slope, it will be noted, gives the value of (S2) for a linear chai.n of the same molecular weight as the branched molecule being examined.o Combination of equations (70) and (71) give a useful result for analysis of the reciprocal scattering curve (Si/Sa)g = (2/3) >/< > (72) where the factor <(S2)>2e has been dropped as being near unity for small values of e. Using the approximation that the Shultz distribution function for random chains can be employed here, equation (72) becomes (Si/Sa)g = (2/3) [<(Sg)>z/(S)>z] [<>z/<v]r() (73) The corresponding equation for a linear molecule is of course

-58(Si/Sa)a (2/3) (A>z/< 4w )(74) Comparison of equations (73) and (74) reveals that the presence of branching ought to cause upward curvature in the reciprocal scattering curve to the extent that the ratio <(S)> z/<(S)>z is less than unity. Branching will certainly cause upward curvature if the measurements are in an ideal solvent, where this ratio is simply g. It is to be emphasized, however, that in order to use equation (73), there must be independent knowledge of the degree of polydispersity present~ For this reason, the method is of limited value, but it will be shown later that the expected upward curvature does appear in. the data of this study. Equation (73) has been used by other investigators to estimate the degree of long chain branching present in polyethylene (99). H o THE MELT VISCOSITY Before consideration of the theoretical interpretation of the melt viscosity in terms of molecular parameters, and possible predictions concerning the effect of branching, it is appropriate to discuss the justification for the experimental procedure used to measure the melt viscosityo The instrument used inl this study, which is described in the experimental section, consisted essentially of two parallel plates between which an initially cylindrical polymer specimen is caused to flow radially by the exertion of a constant force. Dienes and Klemm (100) have examined this viscous flow under the assumption that it is essentially Newtonian in character~ The argument for this assumption is that the average shear rate in

-59this type of viscometer is quite small, averaging about 0o.1 sec l, so that most polymeric materials might be expected to give essentially Newtonian flow behavior. Then, with the restrictions of a steady-state flow, a diameter to height ratio for the specimen exceeding ten, and a flow pattern such that the radial component of the fluid. velocity becomes zero at the plates and that the axial flow is negligible compared to the radial flow, these calculations give the relationship 1/h4 = 8- F t + const (75) 43V2 or 1/h4 m t + consto where V is the specimen volume in cm3, h is the specimen height in cm, t is the elapsed flow time in minutes, F is the applied force, and r is the viscosity in poise. Expressing the applied load as W kilograms, the viscosity can be calculated from the slope m of equation (75) as 1 8itF = 8021 x 106 (76) m 3V2 mV2 The validity of these assumptions has recently been. examined'by Ito (101) He has fou:nd that in certain cases, the apparen.t viscosity as calculated from equation (76) varied with the initial diameter to height ratio, until this ratio exceeded thirtyo Ito feels that for his system, the assumption of negligible axial flow is not valid before this ratio is obtained, the assumption of an essentially Newtonian:flow still being valid. In this regard, it should be pointed out that a linear plot of l/h4 versus t should not be obtained unless the flow is essentially rewtonian (100)o The poly

-60mer studied here did not show amy depenrderecy of the calculated viscosity on the initial diameter to height ratlio The purpose of the experimental study of the melt viscosity of branched polymers was to c;ortrilbute some - formnation concerningE the effect of branch length and, to a lesser degree, the frequ.eny of branching on the melt viscosity. There has been considerable experimental evidence that branching acts to reduce the melt viscosity when comparisons are made at; a given molecular weight. It appears, however, that these studies all had in common the feature of relatively short branches (30). The recent investigations of Long (30) and Charlesby and Thomas (11) have indicated that the previous observations are not valid when the branched molecule has the characteristic of rather loing branches. The complete description of the molecular flow process that results in the macroscopic property of the melt viscosity is only partially understood, even for linear polymers. Characteristic of the development of theories relative to polymeric systems, the theories for the behavior of linear chains are considerably more advanced than those for branched molecules. Thus. the procedure of this section wil1 be the now familiar one of first discussing the existing developments for linear chain, and then giving the possible extensions of this theory to cover branched molecules. A review of th.e experimental and theoretical work up to 1955 has been given by Fox. GratcLh and Loshaek (102). The flow process of mactromolecules is thought to involve the movemenlt of segments of the polymer chain, much in the way that the flow

-61process in liquids is considered to be the movement of individual molecules from one position to another~ The obvious difference in the process, however, is the connection of the segments in the macromolecule by both intramolecular primary bonds and intermolecular entanglements be-'tween separate chains~ This consideration led Flory (103) to the concept of two separate functions as comprising the melt viscosity: = F(- (w) (77) J (Ti) The segnmental jump frequency, J, expressing the probability that a segment will make a jump from one position to another, is assumed to be a function of temperature and the nature of the arrangement of nearest neighbor segments in the liquid, or of the free volume, 0, associated with this configuration. Both ~, and hence J, are thought to be independent of the chain length for long chains, although J can become dependent on M through 0 for short chains. The statistical factor, F, results from the various interconnections between the flow units which are made up of only segments of the chain, andr it is assumed to be only a function of the molecular weight although it may be affected by the presence of branch.ing. The most recent theoretical interpretations of F and J have been given in treatments by Bueche (104) (105) (106) and by Eyring and coworkers (107). Bueche has given -the most successful treatment to date for the in.terpretation of F, the isothermal part of the viscosity function. Since the free volume will probably be the same for branched and

linear molecules of a high molecular weight amorphous polymer, it is the interpretation of F which is of primary interest here It may be remarked that even a crystalline polymer such as polyethylene exhibits the same density for molten branched and linear polymers, thus eliminating J as being different for the two structures (108) (109) o It is pertinent, then, to restrict the bulk of this discussion to a consideration of the function Fo The treatment of Bueche is based. on a simplified solution viscosity treatment of Debye (110) and others (67) (111), which assumes that the polymer molecule does not affect the flow pattern of the solvento This theory gives the result 1 N (s2) (nf) (78) 6 v M where v is the specific volume of a chain of molecular weight M with n effective segments, and f is the effective segmental friction constanto An essential assumption of Bueche s theory is that a relationship like equation (78) will be applicable to the flow of pure polymer after reinterpretation of the friction factor.fo The basis of this assumption is that in such a case, the tsolvent' is in reality the polymer itself, and thus the flow lines are everywhere equivalent and undistorted by any given molecule~ Bueche has shown that equation (78) must be modified somewhat due to the presence of q entanglements in a chain of n links to give 1 m M~ (nf) (1 + n/8q). (79)

-63The remainder of Bueche s contributio:rn lay in the interpretationr of the factor nfr It is assumed that the factor f is comprised of two parts, fo which is a measure of the friction constant for the motion of a segment past its neighbors, and another factor which accounts for all of the forces due to chain entanglements~ The force F, required to pull a single molecule through the polymeric liquid with velocity u is then given by F = nfu = nfou (1 + +Cs + c2s2 + C3 o3 + (80) The Cn are the number of n order couplings to the primary molecule. A first order coupling involves an entanglement directly to the primary chain, a second order coupling involves entanglements to the first order coupled molecules, etco If these entanglements were perfectly rigid, then the polymeric system would resemble a gel i. its flow properties if there existed an average of more than one entanglement per chain. These en.tanglements are not perfectly rigid., however, and the slippage factor, s, is introduced such that an enttangled molecule moves with a fraction s of the velocity of the chain to which it is coupled. Equation (80) then gives the needed reinterpretation of f in terms of the usual friction factor, foo In order to calculate the crj Bueche has assumed that the number of n-order couplings will be approximately (KM)n times a correction factor to allow for the fact that some of the n-order couplings may be to chains alread- y engaged in lower order couplin.gso This correction factor is estimated by computing the density of order couples assueng a Gaussianf distribution of segments in the solido The result gives nf as a function of

-64~ nfo 9 - and s, where K is a constant characteristic of a particular polymer. For long linear chains shere KM is greater than unity, and the average.number of enta:ag;lemenits per moleiule exceeds unity, there results 2f~o [(S13 / lg)M7/2 Cn e0; 3/ (81) = K f% [(S2)/Mi3/2 (2/32)M7/2 Z jn / (8 n=! where the ratio (S2)/ is about corstant for liear chains, and K is a co:nLstanto For shorter chains, where JE4 is less than unLLity, there results - = K' [(S2)/M] o (82) These equations predict that a plot of log T1 versus log M will exhibit two distinct straight line portions, one with a slope of 305 and the other Vith a slope of 1. The experimental evidence does in fact verify this prediction for a very large number of systems (102), except that the lower molecular weight portion usually has a slope a little greater than unity and varying with Mo Bueche attributes this to a dependency on fo on JJ and herne on M through ( for low molecular weight materials. The value of M at which the slope'abruptly' changes is denoted Mc, and is numerically equal to the molecular weight -when KM is equal to unmityo Physically, this corresponds to that chain length at which there exists on the average one entanglement per molecule, leading to a network formation thr ough chain entanglements o It is immediately possible to modify equation (81) in an. attempt to see the effect of bra;nching (112)o Since the value of the expansion factor in the bulk is approximately unity (1), the appropriate value of the

-65mean square radius in equation (81) is that in an ideal sclution~ To retain the ratio (S2)o/M, which is a constant for any given polymer, the expression for a branched polymer must contain the factor g3/2. Calling the slippage factor for the branched chain sg' one might anticipate that the melt viscosity for a branched chain will be given by an equation analogous to that for the linear chain, as 3/2n g = K fg3/2 [( ~]] 7/2 g (2n-3l)3 (83) n=l The equation predicts that the ratio ng/lj at a given M is equal to g3/2 a number less than unity. This conclusion is not in agreement with the results of Long, Charlesby and Thomas, and this study. A qualitative explanation of this apparent contradiction will be discussed later. It has been assumed implicitly in the above development that the system is monodisperse with respect to molecular weight. Bueche (106) has shown that for polydisperse systems, the appropriate averaging shows that r is a function of the weight average molecular weight for farily narrow fractions, and becomes a function of the z-average molecular weight for broad fractions. This seems to be in accord with the experimental evidence, especially the prediction concerning narrow fractions. Since the polymers of this study are all fairly narrow fractions of about the same degree of polydispersity, it is not believed that polydispersity will cause any ambiguity in the interpretations of the resultso

II. EXPERIMENTAL PROCEDURES A. THE SYNTHESIS OF ARTIFICIALLY BRANCHED POLYMER 1. Polymerization Procedure The general concept of the synthesis scheme has been discussed in the introductory section. Basically, the monomer is polymerized in the presence of a chain transfer solvent and the polymer. The linear fractions of PVGAc used as a backbone polymer have been described fully by Long (30). Vinyl acetate (Niacet vinyl acetate, DPA, Carbide and Carbon Chemicals Co.) was the monomer, toluene the chain transfer solvent and azo-bis-isobutyro nitrile (Eastman Organic Chemicals) the catalyst. The monomer was purified by a fractionation at about 300 mm of mercury under an atmosphere of dry oxygen-free nitrogen in a column 2 cm in diameter, 60 cm in height and packed with glass helices. The nitrogen was purified by bubbling it through Fieser solution for oxygen removal, sulfuric acid for water removal, and then passing it through towers packed with calcium sulfate and phosphorous pentoxideo Middle fractions of the monomer with a boiling range of about O.1'C were collected in Pyrex tubes containing the polymer and a magnetic stirring bar encased in glass. The polymer had previously been evacuated in these tubes at a pressure of about 10- 6 mm of mercury for 60 hrs, Toluene had previously been fractionated in the manner described above for the monomer and collected in a vessel from which the solvent could be withdrawn with a hypodermic syringeo The new air drawn into the vessel to re-66

place the removed solvent passed over a bed of calcuim sulfate, and the vessel and syringe were stored in a desiccator over calcium sulfate. After collection of the monomer, the appropriate amount of toluene was transfered to the polymerization tube with the syringe, and sufficient initiator was added to give a concentration of about 10-4 molar. The total amount of liquid added was adjusted so that the initial polymer concentration was about 5 to 10 gm per 100 cc of liquid, The contents of the tube were then cooled to about -90~C using a slurry of n-propanol partially frozen with liquid air as the coolant. The cooled contents of the tube were then degassed at about 10-4 to l0-5 mm of mercury to remove residual oxygen. Little, if any, of the contents of the reaction vessel were lost in this degassing operation. After a degassing period of one hour, the reaction tubes were sealed off and placed on a roller to cause solution of the polymer in the reaction mixture at room temperature. The tubes were placed in an oil bath at 880C for the polymerization period, the contents of the tube being stirred with the magnetic stirring bar. The polymerization was allowed to proceed to about 20% conversion of the monomer. Since the polymerization rate (the per cent conversion divided by the reaction time) was dependent on (S/M)o, the initial solvent to monomer ratio on a mole basis, the reaction time was judged by an empirical relationship observed between these variables,

-68The value of (S/M)o to use so that the new polymer formed in the polymerization would have the deisred average molecular weight,M n' was similarly chosen on the basis of a semi-empirical relationship. If the ratio of the polymerization rate to the square root of the monomer concentration, %/[M]2, is held constant, one should expect to find that 1/ KMa2 is proportional to (S/M)o at least when the reactions are only carried to low conversions (1). A rather strong curvature was noted for this relationship, however, which may in part be due to the fact that the initiator concentration, and not Rp/[M]2 was constant for all of the polymerizations. The observed dependency of 1/( > ) and K app the apparent reaction rate constant, on (S/M)o is illustrated in Figure 1. (M> v, the viscosity average of the molecular weight is twice the number average molecular weight for the type of molecular weight distribution obtained here). 2. Isolation of the Branched Polymer After the polymerization, the contents of the tube were added dropwise to a large volume of petroleum ether. This isolated the polymer in a sticky mass which was washed with petroleum ether and immediately dissolved in methanol to give a concentration of about 10 to 15 gm of polymer in 500 cc of methanol. After solution of the polymer, a crude precipitation fractionation was performed to isolate as rquch of the graft polymer from the unbound new polymer as possible. This was done by adding the distilled water dropwise until the mixture becamne turbid, and then adding

a large excess of water~ In'this type of process, it was inevitable that some of the unbound polymer would be carried down with the graft polymer, but this was of no consequence at this stage. The amount of the excess was about 30 cc in a total turbid mixture of about 900 cc. The precipitation was carried out at 34.8~C, in a liter screw top bottle with the contents being stirred magnetically. After the excess water was added, the contents of the vessel were heated to a temperature sufficient to cause complete dissolution of the precipitate, usually about 45~C. The contents were then allowed to cool slowly for 4 to 5 hours with stirring to reprecipitate the polymer fraction. When the contents had been cooled to 34.8~C and stirred for about an hour at that temperature, the precipitate was allowed to settle for about 12 hours, and then recovered by siphoning off the solution after this period The unbound polymer in that solution could then be recovered by evaporation to dryness. In several instances, this solution was further treated with up to 50 cc of water before further turbidity developed. There was never very much polymer in this precipitate, and it had an intrinsic viscosity very near that of the unbound new polymer. Thus, the crude fractionation seemed to be accomplishing its purpose of removing as much of the graft polymer as possible. The polymer now could be subjected. to either or both of two different operations. If the entire amount of the graft polymer was to be recycled in another grafting polymerization, the polymer was placed in a reaction vessel and dissolved in benzene, After freezing this solution

-70along the sides of the tube, the benzene was sublimed off at O0C, leaving the polymer in a porous form. The polymer was then degassed at about 10-6 mm of mercury in preparation for a new graft polymerization. If the entire amount of the graft polymer was to be used as a fraction of the branched polymer, with no further reactionthen the precipitate was dissolved in methanol to give about 5 to 8 gmn of polymer in 500 cc of methanol. The precipitation fractionation described above was then repeated with the exception that an effort was made to precipitate only' aboutl to 2 gm fractions of the graft polymer. Successive precipitations were performed until no polymer came down with further addition of water. The unbound polymer remaining in solution was then recovered by evaporation to dryness and added to that previously recovered. In most cases, most of the graft polymer came down in the first fraction. The fractions of graft polymer were dissolved in acetone and then cast into a film for solvent removal. The film was freed from residual solvent by evacuation at 10-6 mm of mercury for 2 to 3 days. Light scattering and visocsity measurements were made on the top fractions, and in some cases where the top fraction was less than 1 gm, the first two fractions were combined and the measurements repeated. The intrinsic viscosity of the unbound polymer was measured after it had been isolated and freed from solvents by the benzene sublimation technique described above. The molecular weight of the unbound polymer, and thus of the branches one y the graft polymer was estimated from the data on the intrinsic viscosity in benzene by the use of equation (25) with a =

-710o-676 and K = 2.15 x 10-4 (30). The data from these polymerizations are given in Table Io The nomenclature used to designate the polymer fractions is as follows: 1) The letters L and S refer to linear and branched polymers respectively. 2) In branched polymers, the first digit indicates the polymerization group of the polymer. Polymers in the same group have a common backbone and about the same average branch length. The second digit refers to the number of recycles a polymer has undergone (see Table I). The third digit refers to the fraction number, mixtures of first and second fractions having an M in place of the third digit. In special cases where the first fraction has been redissolved and reprecipitated, the fourth digit refers to the fraction number of the secondary fraction. 3) In linear polymers, the second and third digits refer to the fractionation scheme and nomenclature described by Long (30). For example, L-3-2 is the third primary fraction, second secondary fraction of the linear polymer. The polymer apparently did not degrade during the graft polymerization. A linear fraction dissolved in toluene in the presence of the catalyst was held at the reaction temperature for 12 hours and did not show significant degradation ([, ] changed from 2.50 to 2.78). The usual reaction time varied from 1 to 6 hours. It should perhaps be mentioned that considerable effort was expended in an attempt to use ca-trichlorotoluene as the solvent chain transfer agent. The solvent was a satisfactory transfer agent (Cs = 0.35 at 60~C), transfer presumably occurring through abstraction of a chlorine at.om. It was hoped that an analysis for the chlorine in the graft polymer would

-72then give arLother method of measuring KkA, but this procedure was abandoned when no suitable analysis could be fAund for the low chlorine contents involved. In addition, the presence of chlorine in the graft polymer was not in itself too desirable. B. THE INTRINSIC VISCOSITY 1o Sample Preparation and the Measurement, Procedure The intrinsic viscosity measurements were made in a modified Ubbelohode suspended-level capillary viscometer described elsewhere (113). The solvents used for the viscosity measurements were benzene, 1, 2, 4-trichlorobenzene and 2-octanone, The trichlorobenzene was distilled in air at about 280 nmn of mercury before u.se. The stock solutions were prepared by putting the polymer and about 25 cc of solvent in a 4 oz. screw top bottle, and then placing the bottle on a roller mill to provide agitation. Both the solvent and solution were passed through a coarse grade sirntered-glass filter before use. The concenrtration of the stock solution added to the viscoameter was determined by evaporating 10 cc aliquotes to dryness in tared weighing bottles. The solution procedure was varied slightly in the case of 2-octanone. These solutions were held at 450C for 24 hours while being stirred, magnetically. Heated filters and pipettes were used in handling the solutions. The flow time of pure solvent,, stock solution, and the stock solution dil ted by one half and one fourth were measured to wicthin O-O5 sec, The concentrations were such that the flow times of the solution relative to

-73the solvent were in the range 1.1 to 2,0. Measurements were made at 34.8~C for benzene (flow time about 100 sec.) and trichlorobenzene (flow time about 180 sec.), and at 29.8~C for 2-octanone (flow time about 180~C). 2. Data Treatment In order to account for the incomplete conversion of the potential energy of the liquid in the flow process, a small kinetic energy correction must be applied to the observed flow time. This correction is conveniently accomplished by subtracting 20/t from the observed flow time, t, for the viscometer used (30). The data have been plotted according to the expressions rsp/cr = ([F]XCs) + ([]x Cs)2 k' cr + ]x k r + (84) In(lrel)/cr = ([L]x Cs) + ( []X CS)2 kt cr +... (85) where trel is the relative viscosity computed as the ratio of the corrected flow times for solution and solvent, nsp is the specific viscosity computed as lrel - 1, CS is the concentration of the stock solution in gm of polymer per 100 cc, cr is the solution concentration relative to Cs, k' and k'T are constants, and [r] is the intrinsic viscosity. Equations (84) and (85) were plotted simultaneously to result in a double plot to give the intercept [i] x cs. The plots were such that the slopes were related by k' - kt' = 0o500 ~ 0.002 after the suggestion of Heller (114) who showed that these constants are related in this way. The intrinsic viscosity was determined by dividing the intercept by cs. The intrinsic

-74.viscosity values have an error of about one to two per cent, the error in k' or k" depending on the magnitude of [i]. For [r] in the range 1 to 5, the error in kt or k" is about ten per cent, below an [r] of about 1 the error can become larger. A typical viscosity plot is given in Figure 2. The majority of the measurements were made in benzene. Sufficient measurements were made in trichlorobenzene to establish the values of K and a in equation (25) for this solvent. These data are given in Table III. Values of [il] in benzene were converted to values in trichlorobenzene according to the relationship a2/al a2/al [1]2 = (K2/K1 ) [n] (86) where 1 and 2 denote benzene and trichlorobenzene respectively. Substitution of the values for the parameters in the two solvents gives the equation [r ]2 = 0.788 [] 0924 (87) This estimate proved to be good for the branched polymer as well as the linear polymer, which should be true if the function f(g) is the same for both solvents. The error in [TA] is probably increased to about five per cent by using equation (8(). The experimental data concerning the intrinsic viscosity are given in Tables II and IIIo

Co LIGHT SCATTERING 1o Sample Preparation and the Measurement Procedure The solutions were prepared in the same manner as that described for the viscosity measurements. The trichlorobenzene and methanol used as the solvents were distilled in air over calcium sulfate at about 280 mm of mercury and 760 mm of mercury respectively. Both solution and solvent were filtered througha coarse grade sintered-glass filter directly into centrifuge tubes. The solution and solvent were then centrifuged (about 25,000 g) in a type SS-lA Servall centrifuge for from four to five hours to centrifuge out dust. A special graduated tube fitted with a syringe stopcock and needle at one end and a glass stopcock and sinteredglass filter at the other end was gently lowered into the centrifuge tube for solvent or solution removal. This tube was connected to a vessel from which water was allowed to drip at a slow, constant rate, thus drawing the liquid from the centrifuge tube into the graduated tube. About 15 cc were removed from each centrifuge tube, the first five solvent tubes being used to rinse the transfer tube and the scattering cell, and the next two solvent tubes being used to obtain the solvent for the light scattering measurement. After a rinse with. about 1 cc of the solution, 15 cc of the solution were then drawn up into the transfer tube and held there for further use. The measurements were made at the ambient instrument temperature, 34 to 380C, with a Brice-Phoenix light scattering photometer, model number 1410 (the instrument is completely described in the Brice-Phoenix manual

-76(115)). The light from a mercury arc was passed through a filter to isolate the 4360 A~ wave length. Measurements were taken at angles of 0, 30, 38, 45, 60, 75, 90, 105, 120, 135 degrees from the exit beam. Before a ran was started, the dissymmetry ratio of the solvent (the ratio of the intensities at 45 and 135 degrees) was determined. It was arbitrarily specified that this ratio should be less than 1.05 for the solvent to be acceptable for a light scattering determination. A perfectly clarified solvent should have a dissymmetry ratio of exactly unity. After the scattering envelope had been determined for the solvent, a portion of the stock solution was added to the scattering cell and the scattering envelope redetermined when thermal equilibriumn had been attainied. This process was repeated for four different solution concentrations, the amount of solution added being recorded each time. 2, Data Treatment The data were treated in the usual way to give a Zimm plot, including the use of a reflection correction (see for instance reference 30). A sample Zimm plot is included in Figure 3. This figure shows the results of two separate determinations plotted together to give an idea of the reproducibility of the method, The data were plotted as cr/Tcit"() versus sin2 (G/2)+kcr, where Tct"() is -the excess scattering ratio of G degrees after the appropriate corrections, and cr is the concentration relative to the stock concentration, cs, The calculation of T0"(G) from the observed intensity at G involves corrections for neutral filters,

cell reflections, solvent scatter, changes in the scattering volume with 0, and an angular effect arising from the use of unpolarized light. All of these corrections are standard and have been considered in detail in reference 30. The intercept for cr and 0 equal to zero gives (l/<M>w)x cs when multiplied by the factor K*, where 32T no (dn/dc) K* = 3N 0o k cn (88) The factor K* accounts for physical constants of the system and an instrument calibration factor k, where ko is the wavelength of the light in vacuum, no is the refractive index of the solvent at %o0 (dn/dc) is the refractive index gradient for the polymer-solvent system and cn is a factor to correct the calibration constant from water to the solvent of interest. The factor cn is given by cn = no no - 57 (89)n 1 + - ~ (89) The photometer was originally calibrated by Manson (14), whose calibration was used by Long. In using this calibration constant to calculate the values of 4>w from measurements of the linear fractions previously examined by Long, it became apparent that the value of k had changed (probably due to some part changes that had been made in the instrument). A new value of k was estimated using the values of c/Tc'"() obtained for the linear fractions in trichlorobenzene, the values of KM>w

-78for the linear fractions from Long's measurements and the value of K* with the appropriate parameters for trichlorobenzene (K* = 0.2755 x 10-5/ k). The value of k calculated in this way was 0.1813 as compared to 0o131 used by Long. Confidence in the new value of k was increased when it gave values of <M>w in methanol for the linear fractions within experimental error of those obtained in trichlorobenzene here, and in butanone by Long, In addition, a sample of the Debye secondary light scattering standard gave an excess turbidity at 90~ of 3537 x 10-2 cm 1 when the data were calculated with this new value of k, in good agreement with the value of Manson, 3.56 x 10-2 cm-1, obtained on this instrument, and with other literature values, 3.21 to 3.56 x 10-2, (116)(117), Using the new value of k, the value of K* was taken as 1.52 x 10-5 in trichlorobenzene and 2.59 x 10-5 in methanol with the other parameters being: Trichlorobenzene; no = 1.587 (Xo = 4360 A~) (dn/dc) = 0,1030# @c = 1.474 Methanol; no = 1.329 (Xo = 4360 A~) (dn/dc) = 0.1314h cn = 0o986 The values of no were measured with an Abbe refractometer and corrected fMany thanks to Dr. Wo W, Graessley of the Air Reduction Company for the measurement of this value on a Brice-Phoenix differential refractometer. ##Many thanks to Dr. Billmeyer of the Polychemnicals Dept., The Eo Io du Pont de Nemours Co. for this value.

-79to a wavelength of 4360 A~ by a dispersion correction (118). The reciprocal particle scattering factor was computed as (cr/To' (0))c =O divided by the intercept for cr and 0 equal to zero. The inir=O tial slope of this curve multiplied by the factor 3(oC /4mo)2 then gave (iS-,>z. This factor was 14-36 x 104 for trichlorobenzene and 20O47 x 104 for methanol. The initial slope of the curve for 0 equal to zero gave the value of A2 when multiplied by the factor (l/2)x5K* (the factor 5 is the arbitrary constant used in the abscissa of the Zimm plot). The determination of the initial slope of the reciprocal particle scattering curve is not easy to locate, as pointed out above. The method given by Zimm (47) was used to obtain the slope more accurately than would have been possible by simple graphical means. Zimm has examined the expression for Ph- (G), the reciprocal scattering factor for a system with a molecular weight distribution characterized by a polydispersity factor h assuming a Shulz distribution. From that analysis, one has for each value of Ph (G) obtained as a function of sin2(G/2), a corresponding value of Ph () LT, the extrapolated initial slope, and Ph- () A, the extrapolated asymptotic slope, as a function of sin2(G/2). In using this procedure one adjusts the value of h until the plots of Ph'1(G)LT and Ph'()A versus sin2(G/2) give straight lines. This type of correction has applied to all light scattering data to obtain the values of (K>z and the ratio of the initial to the asymptotic slope, Si/Sao This ratio gives a check on h for the linear fractions through equation (74) and the

-8Qo relationship M>z h + 2 h+ 1 Equation (90), strictly valid only for a random chain, is obtained by using equation (16) in the appropriate equations for the z-and weight average for the molecular weight. The light scattering data are recorded in Tables IV and VY The errors for the parameters from light scattering are about seven per cent for <Kw, ten per cent for(S~)>z and A2, and about fifteen per cent for the ratio Si/Sa~ A second light scattering determination was made on several of the samnples, and the values obtained may be seen to lie within these limits. The determination of the polydispersity parameter, h, may be in error by ~+ 4 units since the correction to the Ph (@) curve is not too sensitive to the exact value of ho D. THE MELT VISCOSITY 1L Sample Preparation and the Measuremernt Procedure The specimens were prepared frzomn polymer obtained from filtered solutions, the polymer being recovered from solution as a film or by the benzene sublimation techniqueo In either case, the polymer was then evacuated. at a presstwre near 10-6 mn:of mercuary for about a week prior to use, nitrogen being placed over the polymer after the evacuation, The polymer, usually about a gram, was then molded into a pellet 0~57 inch in diam

eter with a plunger type mold. The molding procedure was~ 1) Preheat the mold to 155 C. 2) Place the polymer in the preheated, mold and allow the assembly to stand for 15 minutes, 3) Apply pressure with a hydrau1li press to a total load of 5,000 pounds, releasing and applying the load in cycles to remove entrapped air. 4) Hold the load at 5,000 pounds for 15 minutes at 155~C. 5) Hold the load at 5,000 pounds and cool the mold to room temperature. The melt viscosity measurements were made in a Williams Parallel Plate Plastometer (100) at 180~C. The instrument and specimen were placed in a pyrex bath covered with a glass top and evacuated to about 100 mm of mercury. Dry nitrogen was then passed into the bath to restore atmospheric pressure, so that the measurement could be done in an inert atmosphere. The nitrogen was dried by passage through sulfuric acid and calcuio sulfate before use. A constant flow of nitrogen was maintained during the measurement of the viscosity data. The viscometer essentially consists of two parallel plates, the upper plate being movable and weighing 5.0 kg (30). The specimen is placed between the plates and the vertical motion of the upper plate recorded as a function of time as the specimen undergoes radial flowo The plate separation, h, was then calculated as a function of time by correcting for the thickness of aluminum foil used as a parting agent for the specimen.

-822. Data Treatment The values of the plate separation were plotted as 1/h4 versus time in accord with equation (75) to obtain the slope of the straight line portion of the curve. The melt viscosity was then computed using equation (76). A linear relationship of 1/h4 was always observed, implying the existence of an essentially Newtonian flow. The straight line portion of the curve usually began after an elapsed time of about 100 minutes, the total test time usually being about 350 minutes. The stability of the polymer under the flow conditions was of some concern. It was found that the linear fractions were sufficiently stable at 1800C, or even 205~C, when under a nitrogen atmosphere. This is in accord with Long's (30) observation for his melt viscosity samples, and the depolymerization studies of Grassie (119). Thus, a linear sample that was rerun to give two values of q gave results that agreed within five per cent. The behavior of the graft pol.ymers, however, was not so satisfactory. The specimens were usually a little darker in appearance than the linear samples after the molding operation, and in most cases if a sample was remolded and rerun a lower value of q was obtained. In addition, the value of [I] for a branched polymer was usually decreased after a melt viscosity determination, It was qualitatively observed that the degradation effect increased as the length of the branches increased. The change E[ ] is tabulated. in Table VI. The values of the melt viscosity which were observed for the branched polymer may be somewhat low due to the observed. degradation, but it is difficult to assess the exact effect

of the degradation, and no attempt has been made to do so here. The linear fractions were found to obey the relationship nT o098 x lo-E <KM)w36 5 (91) Figure 17 shows B as a function of <M>w for the branched and linear polymers. The behavior of the branched material was strongly dependent on the molecular weight of the branches. The ratio of the melt viscosity of branched and linear fractions of the same molecular weight, qg/BI,, is showh in Figure 18 as a function of the branch molecular weights, <Mn. The experimental data are recorded in Table IX. Excluding the effects of degradation, the melt viscosity measurements are probably accurate to within five per cent for between 106 and 109 poise, with the error increasing to nearly ten per cent for q near 109 poise.

III. RESULTS AND DISCUSSION A. THE GRAFT POLYMERIZATION The data indicate that short chains of PVOAc were successfully grafted onto a longer backbone substrate of linear PVOAc fractions. The primary evidence in support of this conclusion is the observed increase in the molecular weight of the backbone polymer recovered after a graft polymerization. Secondary evidence includes the changes in the other solution properties and the melt viscosity for the backbone polymer after a graft polymerization. The values of <k) estimated from equation (4) are shown as a function of <M~iX, according to equation (3), in Figure 4. The two straight lines drawn in that figure correspond to values for the polymer chain transfer at 88~C given in the literature (1) (2). It is seen that the data are consistent with these values of Cp. The best value of Cp from these data is about 15 x 10-4 The vertical lines drawn for each point are indicative of the errors estimated for <k> by assuming a seven per cent error in both K<Mw and <MIAw in equation (4). The error is proportionally larger for the graft polymers with the shorter branches due to the small increase in the molecular weight by the grafting process. It should be noted. that it would be more appropriate to use the number average molecular weight for both Mg and M1, but since the molecular weight distribution of the graft polymer is very close to that for the linear -84

polymer, a small and constant factor would be needed to convert both values to the number average. The effect on the molecular weight distribution of placing an average of (k> branches of constant molecular weight Mb on a monodisperse backbone of molecular weight M1 is calculated in Appendix III. It is shown there that the ratio of <M>w/<M>n changes from unity for the linear polymer to approximately KMAw/KMn 1 + Kk>/(Ml/Mb+Kk>) (92) Since equation (92) is based on a monodisperse backbone polymer, it is not strictly applicable here because the actual backbone is instead a narrow fraction with a weight to number average of about 1.1 (30). It is assumed. however, that this calculation indicates the relative change to be expected in Rw/<MK>n when the backbone is not exactly monodisperse. Thus, if the backbone had a weight to number average molecular weight ratio of 1.1, the graft polymer would be expected to have a ratio of (<Mpw/<KIn) x 1.1 where the ratio <KMI w/KM>n is calculated from equation (92). The change in the molecular weight ratio indicated by equation (92)is quite small for all of the samples studied here, ranging from a value of about 1.003 to 1.03. The ratio increases as the molecular weight of the branches increases and as <k) increases. The effect of this change in the distribution function on <k> is well within the experimental accuracy of the estimate of <k), and no attempt has been made to include this effect, An important point to note inTable VIIIs that the recycle operation always caused an increase in the number of branches per molecule, as would be expected..

-86B& THE SOLUTION PROPERTIES The results of the solution measurements will be considered in a manner analogous to that used in the discussion of the theoretical developments for the various parameters. Thus, the behavior observed for the linear polymers will be presented and discussed first in an effort to lay a foundation for the discussion of the properties of the branched polymer. 1o Properties of the Linear Polymer The values of o>w obtained for the linear fractions in two solvents in this study are in good agreement with those of Long obtained in a third solvent. The linear fractions were found to have an intrinsic viscosity-molecular weight behavior in 1, 2, 4-trichlorobenzene given by the relationship o 0-4) <M> ~' 623 [] = (3530 x 10-4)(M>W 2 (93) Thus, the exponent o4I>w indicates that trichlorobenzene is a slightly poorer solvent than benzene (30) or butanone (49) for PVOAc. The value of a in equation (93) corresponds to a value of e of 0.041, compared to 0.059 for benzene and 0.o070 for butanone. The relationship between )> and <M>w given in Figure 5, shows that (z can be represented as being proportional to <M>w to the power 1.08, in agreement with equation (29), with the proportionality constant equal to 0~062. It should be admitted, however, that the error in graphically determining this power is of the order ~ 0.1, making any

conclusion concerning the exact power somewhat arbitrary. The relationship between a and e given in equation (26) is obtained with the assumption of the Flory-Fox expression. Although this expression is not expected to be valid, the weak dependence of the expansion factor on molecular weight allows the use of this correlation between a and c for a given polymer-solvent system over the molecular weight range considered here. The data should more correctly be plotted as (S2)w versus <M>w to remove the effect of polydispersity, but since all of the fractions have nearly the same degree of polydispersity, this correction would only act as a constant multiplier, not changing the observed dependence on the molecular weight, The relationship between A2 and <KIw, given in Figure 6' shows that A2 is inversely proportional to KM>w to the power 0.230o This is in disagreement with the prediction of the power from equation (64), if the group A2<KMw/[r] is constant for all polymer-good solvent systems. This expression would predict a power of 0.377 based on the intrinsic viscosity relationship to preserve the constancy of the group. This group is not expected to be strictly constant, however, as discussed above, but rather a function of 0c, and hence of the molecular weight. Stockmayer (64) has discussed this functionality and compared the various theories for the function to the existing experimental data. That analysis shows that while the group does tend to show a decreasing dependence on a as a becomes greater than about 1.2 to 1.4, there is a marked dependence on O even for cf as large as 1.6. The group A2KM>w/[[] is plotted against KMAw

in Figure 7, including the results of Long (30) and Shultz (49) in butanone in addition to those of this study in trichlorobenzeneo It can be seen that a slight dependence on canw can be perceived, a line representing a curve with molecular weight to the power 0.147 (or 0.377 - 0.230) is included to show the trend wit;h molecular wFreight expected from the value of c obtained in this study6 This molecular weight dependence is so slight that if the average value of the group is calculated from the data of this study, the average value of 146 obtained is found to be in good agreement with the values of Chinai (in acetone j1201 ), Schultz, and Long, calculated as 142, 139, and 134 respectively. It can be remarked in addition, that while neither Long or Chinai report the exponent of <M>w in the A2molecular weight relationship, the data of both investigations exhibit a lower value of c than would expected if the group A2<M\/B[-i were strictly constant. Thus, it can be concluded that this group is nearly independent of the molecular weight, but that the functionality on a that does exist for the group is reflected in the molecular weight dependence of A2 being somewhat different from that required for strict constancy of the groupo Three of the linear fractions characterized by Lonlg in butanone were examined in methanol as well as trichlorobenzene. It was hoped that these data might be useful in determining the value of (S2) as A went to zero, by extrapolation of a plot of (-2) versus Am to zero A2o Such a plot is shown in Figure 8. It is seen that while the data can be extrapolated to the values of (S2)o at zero A2 as determined experimentally by Shultz (49)

-89in a critical consolute mixture representing a theta-solvent, the values of A2 are too far from zero to allow an accurate extrapolation in themselves. Thus, such a technique is of limited value in determining the solvent effect on (S2), unless it is possible to use solvents very nearly ideal. It is possible, however, to use the data of Shultz for <(S2) Cz as a function of <X>w can be used to estimate the expansion factor for the linear fractions. Shultz gives the relationship <(S12)O = 0102 <M)w, (94) Use of this expression and the observed values of (S z gives al between 1.3 and 1.5 for the linear fractions. The exponent m1 in equation (17) can also be estimated by using this relationship and the values of [n]o as a function of <M>w given by Shultz, Figure 9 shows a plot of log al, which is calculated from (1/2) log <v))z KS ) z1 as a function of log [i]/[r]o~ The plot includes data for three fractions (Mw = 3. 43; 1.87; and 1o46 x 106) in butanone (data of Long) and in methanol and trichlorobenzene. The value of [T ] in methanol has been estimated from data given by Hobbs, et al., (32). It can be seen that the data fall approximately on a line of slope 1/2.5, and that a slope of 1/3 in accordance with the Flory-Fox equation is not appropriate, This gives ml equal to 2,5 compared to the theoretical estimate of 2.2 from equation (18), and the value of 2.4 observed for polymethyl methacrylate in this range of Of by Schulz and Kriste (37).

-90Another indication that ml is less than 3 is the observed value of O*, as calculated from equation (6) modified for a polydisperse polymer in a good solvent to give toD = (1/63/2) q(he)[ <M,> The polydispersity parameter q(h,e) given by equation (19) is calculated to be 0.854 for e = 0.04 and h - 10. All of the samples were characterized by the polydispersity parameter h = 10 due to the insensitivity of determining h by light scattering techniques. Variation of h by ~4 would only change q(h,e) by a few per cent, which would be well within the experimental accuracy of the determination of tO. The average value of O1 was found to be 1.85(~ 002) x 1021. Due to the relatively large inaccuracy in determining O, and the small dependence of a on the molecular weight, it was not possible to detect a trend of $O on the molecular weight. It is possible, however, to estimate the value of ml consistent with the observed value of OD and the theoretical interpretation of the volume effect given by Kurata, et al. Using the average value of to = 1.85 x 1021, and taking the value of cI to be 2.87 x 1021 in an ideal solvent, the exponent m1 can be estimated by an equation like equation (20) with the power on al taken to be ml - 3, Using an average value for al of 1,4, this calculation gives ml equal to 2,0, in fair agreement with the values of 2.2 and 2.5 from theory and. experiment respectively. At any rate, the exponent ml appears to be definitely less than 3 with a value of 2.5 taken as the best estimate. It should be noted that Stockmayer and Albrecht (42)

91have calculated ml equal to 2.73 for a monodisperse system by a method similar in principle to that of Kurata, et al., but using a more rigorous procedure in one place. Substitution of 2.73 for 2.43 in equation (Q18) would lead to a theoretical estimate for ml of 2.5 for a polydisperse system. with h = 10, in good agreement with the value estimated from the observed data. This new value of 2.5 is not, however, in a good agreement with the data of Krigbaum and Carpenter (40) and Schulz and Kriste (37) in poor solvents. Since the calculations are expected to give results limited to poor solvents, the new theoretical estimate for ml cannot be considered to be better than that of Kurata, et al., on the basis of the data of this study. 2. Properties of the Branched Polymer The properties of the branched polymer will be discussed relative to the properties of the linear polymer, in line with the theoretical developments discussed in the Introduction. Comparison will be made to other studies when this seem appropriate. The intrinsic viscosity has been measured in benzene for all of the branched polymers. Figure 10 shows [r] as a function of (M>w for the branched and linear fractions. It is of interest to note that while the ratio of [Tg] to [P1] for fractions of the same molecular weight was less than unity in all cases, the value of [fg] was usually not too different from the intrinsic viscosity of the linear backbone used in the graft polymerization. The ratio. of [ig] to [~], where the (*) designates the linear backbone polymer of the graft polymer in the comparison, was observed to be greater, equal or less than unity. These data are compared in Table VI. Neglecting polydispersity for the

-92moment, the reason for this can be seen by combining equations (4), (25) and the definition of f(g) to give [Tg] = f(g) K (M* + k Mb)a (96) where Mi* is the molecular weight of the backbone polymer. Dividing this expression by the intrinsic viscosity of the backbone polymer, there results [ g Ml* + kMb a =f f(g),, (97) The expression in the brackets is seen to be the value of (l/g) for the type of branched molecules studied here. Thus, the value of the ratio depends on the relationship of f(g) to (1/g)a, where a is 0.676 in benzene (30). The ratio f(g), which is defined as [ g]/[1]] is plotted against g in Figure 11. Also included in this figure is the ratio (A2)g/(A2)1 as a function of g. The data are seen to be near the theoretical expressions for fo(g) given by Zimm and Kilb and by Stockmayer and Fixman, gl/2 and h3 respectively. Thus, f(g) is approximated by g b where b is between 3/4 and 1/2. Depending on the exact value of b relative to a = 0.676, the ratio [g]/[,1*1] will be greater or less than unity. The observed data than, are consistent with the value of b about equal to 0.676.

-93The measurements of Thurmond and Zimm (9) on branched polystyrenes arc are of interest here. They have obtained measurements in a good solvent anc and in a theta-solvent (a critical consolute mixture) for both linear anc and branched fractions, The ratio f(g) was observed to be only slightly la2 larger than fo(g) in most cases. Zimm and Kilb (52) and Thurmond and Zii Zimm have used the value of f(g) from experiment when comparing the theor( oretical expression of fo(g) = g2 to experiment. Some data for [Il] and [TE [rig] in 2-octanone at 29.8~C, which has been reported to be a thetaso] solvent for PVOAc7, are given in Table II for a few fractions of the polymE ymers studied here. The values of f(g) and fo(g) estimated from these meE measurements are recorded in Table VIII. It is seen that f(g) is appreciE ciably larger than fo(g) for the three branched polymers measured. The va] values of [l]o obtained here are about 10 to 20% lower than those reported ed by Shultz, suggesting that this solvent may actually be a little poorer er than a theta-solvent, but this should not appreciably affect the ratio fo( fo(g) so long as the deviation from a theta-solvent is small. These data on [nr] in 2-octanone also indicate that [ig]/[jg]o exceeds [rio [1rl]/[ml]o at the same molecular weight by an appreciable amount. This suE suggests that cg exceeds al, although exact determination of ag from the int intrinsic viscosity ratio is uncertain due to the lack of knowledge concei cerning the exponent mg. The observation that ag might exceed al has been mac made before. both theoretically and experimentally, The data of Thurmond ATr AThis solvent has been suggested by Dr. To Fox, of the Mellon Institute aE as a theta-solvent for PVOAc (121)o

and Zimm (9) on branched polystyrene, Schaefgen and Flory (122) on branched condensation polymers, Long (30) on branched polyvinyl acetate, Wales, et al., (68) on branched polysaccharides, and Tremerntozzi (83) on polyethylene containing long chain branching have all been suggestive of such behavior. These studies were all complicated by the relatively larger amount of polydispersity in the branched fractions compared to the linear fractions due to fractionation difficulties mentioned in Section I-C. The theoretical predictions concerning the ratio g/oa, were considered in the Introductory sectiono It was pointed out there that different theories arrive at opposite conclusions regarding this effect, but that two different theories did predict that the expansion factor should be larger for the branched polymer. The data on (S)> as a function of <M>w obtained here strongly suggest that ag is greater than al. It can be seen in Figure 5 that there is very little difference between S) z and: (>z at the same <M)w for any of the branched samples. Since the degree of polydispersity should be about the same for the branched and. linear samples, this rather strongly suggests that the branched polymers are expanded relatively more in solution than are the linear fractions. Thus, if one accepts that within the experimental accuracy of the measurements, (SA) and Si are identically equal, then it is seen from equations (9) and (44) that ag2 is about equal to aQ2/g, or that 3 0d 3 -7; 3 (98)

95This is a somewhat stronger effect than that predicted by the theoretical equation (45)~ Assuming in equation (50) that mg and mi are about the same and equal to 205, then one finds after substitution of equation (98) into equation (50) that f(g) ru fo( g) Lgj (99) Values of the ratio f(g)/fo(g) and (l/g)/4 observed here are tabulated below. It is seen that there is, in fact, only a fair agreement between the two. This may indicate that ()z is somewhat smaller than ZS19z but close enough to being equal to it that the error in determining (-2) prevents an unambiguous observation of the excluded volume effect for the range of g obtained here. Fraction f(g)/fo(g) (l/g)/4 (l/g) s-60o- 1.4 1.8 1.6 S-70-l 1.1 14 1.3 S-71-1 1.7 1.8 1.6 Equation (52) provides a means of estimating the ratio ag/al by taking advantage of the relatively more more accurate intrinsic viscosity measurement in conjunction with the parameters from light scattering. The ratio (g*/Dl* has been computed by dividing the 0g* by 1.85 x 1021, the average value of 1*'. The ratio if found to be less than unity in all but three cases, which can only be possible if cgg is greater than Oi1 since f(g)/g3/2 is greater thnan unity. Values of the ratio (cxg/o1i)3 calculated

-96from equation (52) are reported in Table VIII, and plotted in Figure 12 as a function of g(a log-log plot). These data are not fit by the result of the Ptitsyn relationship, equation (45), with a, taken as 1.41 nor are they fit by equation (99), which arises when the radius is the same for the branched. and linear polymers. A closer estimate is given by c_3 i (100) ~13 g Using this relationship, the exponent on (l/g) in equation (99) becomes 5/6, giving results in closer agreement with the observed data for f(g)/ fo(g) - Equation (100) suggests that the ratio of the radii in a good solvent are related. to g by the expression S z/) S z. g /3 (101) This would mean that for the range of g obtained here, the ratio of the radii would, be close enough to unity that it could not be taken as different from unity with any degree of confidence without other supporting evidence, The results given by equations (100) or (101) are entirely empirical. Although there is some reason from the theory to expect such behavior, no theoretical treatment has resulted in a prediction of an effect quite this strong. The result of Ptitsyn, which may be seen plotted in Figure 12, does come fairly close to this observed behavior, and appears to be

-97as good a representation for the data as the theory can provide. The results of Stockmayer and Fixman which predict that cg should be slightly less than cl do not fit these data. The results of Kreussling and Ullman are a little more difficult to evaluate since they give only a partially complete expression for (S), and since the final form includes a parameter which is difficult to assess. Their result, given by equation (43) for the type of branched structures of interest here, contains a parameter v, which corresponds to the volume excluded by a polymer segment. If the constant of proportionality in equation (31) is taken as the proportionality constant for the observed Sz - <M>w relationship, and if v is estimated as 4X/3 b3, then equation (43) becomes (Sg) I/(S ) = g(l + 26o <k) nb/n). Using the parameters measured for fraction S-60-1, this expression gives the ratio of the radii in a good solvent as g within three per cent, a much weaker dependence of the ratio of the radii on the excluded volume effect than observed. here, The ratios of <((S)z/((S -`> observed for the branched fractions of this study are tabulated in Table VIII, and plotted as a function of g in Figure 135 It is seen that these values are consistent with equation (101), but that they are not far enough removed from unity to provide a reliable verification of the relationship. It is useful to examine some data of Long ) for branched PVOAc in this regard, since his branch

-98ed polymers would be expected to exhibit values of g as low as 0.2. Long observed that the estimate of the nlmber of branches per molecule based on the ratio of the radii for branched and linear molecules in a good solvent as giving g directly gave results low in comparison to those based on intrinsic viscosity measurements. Since the data were for a randomly branched polymer, the correlation of Zimm and Stockmayer (50) for g as a function of the number of branches per molecule was used. In the interpretation of the intrinsic viscosity measurements Long took the ratio of [ng]/[El] as being h3, and used the theory of Stockmayer and Fixman to relate h3 to g. Since these data are for a randomly branched polymer, the branching results in a considerably lower value of g for a given <k> than is the case for the type of branched polymer studied here. This offers some reason to expect an observable effect of branching on the ratio of the radii at least for the more highly branched samples. The values of the ratio of the radii for some of Long s polymers are recorded below along with the estimate of g from measurements of the intrinsic viscosity ratio, using the theory of Stockmayer and Fixman (gSF and Zimm and Kilb (gZK). The values of g estimated from the ratio of the radii by the use of equation (101) are &lso included (gS)' Fraction ~ gs gZK M- 1-12 o.a8o 0.20 0 12 0.32 M- 2- 3 o.648 O. 27 O. 16 O. 36 M-3-2 0o678 0.31 0138 0.54 M-4-1. 0.681 0.31 0, 32 0,50 M-4-2 o.858 0.63 0.43 0.58 M-5-1 0.870 0,66 0.o 39 0o,55

-99It is seen that the values of g are appreciable smaller than those in this study, as expected. Further, the ratios of the radii are much too large to be taken as giving g itself, in view of the values of gZK or gSFO The values of gs, however, are seen to be in much better agreement with the estimate of g from the viscosity measurements, a very encouraging result since equation (101) from which gS is estimated has been obtained empirically from an entirely different set of data on a different kind of branched polymer. The ratio of the radii for branched and linear polymer at the same molecular weight is plotted as a function of g in Figure 13, including the results of this study and those of Long. The value of g for Long's samples has been taken as the average of gZK and gSFo It is seen that equation (101) provides a reasonable approximation to these data. The ratio of the initial and asymptotic slopes of the reciprocal particle scattering curve are considered next. It was shown in the theoretical discussion that this ratio if given as a function of the ratio of the radii for branched and linear polymers and the polydispersity parameter, h. This offers, in principle at least, another way to determine the ratio of the radii and hence a measure of the extent of branching. If it is assumed that the ratio <M>z/<M>w will be about the same for the graft polymer as it was for the backbone fraction, then combination of equations (73) and (74) gives the result (si/Sa) g/(Si/Sa)i = K(S)>z/K(S)>z ~ (102)

Using the approximation given by equation (101), it is seen that this ratio ought to be gL/3 The experimental data are reported in Table VIIIo It is seen that in all but one case, the ratio of the slopes is smaller for the branched polymer than a comparable linear polymer, which is in the direction suggested by theory (the linear fraction is comparable in polydispersity, not in molecular weight) It is also seen, however, that this ratio is generally very much greater than the value calculated for g, in accord with equation (101)D A comparison of the ratio of the slopes for branched and linear polymers to the ratio of the radii for branched and linear polymers with the molecular weight of the graft polymer, given in Table VIII, shows a general agreement, but an appreciable scatter in the ratio of the slopes. This is to be expected since the technique of determining the slopes to the reciprocal scattering curve is not too precise. It was mentioned above that the data for the Patio [,qig]/[ql ] are in fair agreement with the theoretical calculations for fo(g). Figure 11 shows this agreement in a plot of [rg]/[Ll] as a function of g. In addition, the experimental data in a theta-solvent (2-octanone) indicate that f(g) is appreciably larger than fo(g) for the experimental data, in accord with equation (99) which arises if mg is about equal to mlo These results indicate that fo(g) is more nearly given by f(g) x g5/+6 If f(g) is estimated to be about g5/8, the average result of the Stockmayer-Fixman and Zimm-Kilb tireatments for g greater than about so6, then this indicates that f0(g) is nearly given by g3/2. This is quite surprising since it im

-101plies that in an ideal solvent, as approximated by a theta-solvent, the hydrodynamic radius and the configurational radius of the polymer molecule are identical. This, of course, is in contradiction to the results of the theoretical treatments of fo(g) discussed above. Data for other kinds of polymers with known degrees of branching are needed to clarify this point. Another parameter which is known to be sensitive to branching is the second virial coefficient, The virial coefficients for the branched polymer are plotted against <M>, in Figure 6. It was shown in the Introductory discussion that the ratio of A2 for a branched and linear polymer of the same molecular weight is expected to be less than unity. The ratios of A2 observed in this study are given in Table VTIIIo It is to be noted that due to the weak dependence of Ag_ on the molecular weight, the value of A2 for the graft polymer is not only less than that for a linear chain of the same molecular weight, but it is also less than the value for the backbone material. Equation (67) provides a method of predicting the effect of branching on Ao If it is assumed that. the quantity in the brackets on the right of equation (67) will behave approximately as (ag/ajl) with regard to g, then substitution of equation (101) into equation (67) gives the result )gSince /(he rato of [/(] to [] bserved to be between g / and (10)/ Since the ratio of [fig] to [nrp] was observed to be between g 3/4 and g:L/2,

-4102this expression predicts that the ratio of A2 should be nearly equal to the intrinsic viscosity ratio, A plot of (A2) g/(A2)1 versus the intrinsic viscosity ratio, given in Figure 14, shows that this is approximately true. The ratio of A2 for the branched and linear polymers is also plotted versus g in Figure 11, where it is seen to vary with g as does the intrinsic viscosity ratio. It is of interest- to note that the result of equation (103) arises if one uses a method suggested by Stockmayer and Fixman to estimate A2 for a branched polymer. This method, discussed in the Introductory section, suggests that the results of the calculation of A based on a smoothed function for the segment distribution in the chain should be directly applicable to branched polymers, since no assumption has been made regarding chain structure in this derivation. One of the most recent calculations based on a smoothed density model has been given by Casassa (80). It was found that for large values of the argument of the function F[z/ (S2)/2 in equation (59) (for z/(S /2) greater than about 3.4), the second virial coefficient becomes proportional to (S2) /2/zo Since the calculation of Casassa, like all such treatments to date, ignores terms due to multiple intermolecular contacts, it cannot be expected that this asymptotic expression will be rigorous. It may be, however, that if the Stockmayer- Fixman suggestion regarding the applicability of this tympe of theory to branched polymers is correct, then the ratio of the asymptotic expressions for A2 for branched and linear polymers will not be too bad. The values of the argument for the linear fractions in the good solvent

103used here appear to be about 3 to 4, large enough to expect the asymptotic behavior of A2 to be obtained~ Employing equation (101) for the ratio of the radii in a good solvent, then one sees that equation (103) arises from the expressions for the asymptotic behavior of the branched and linear polymers at the same molecular weight (it is assumed that the parameter p is the same for the branched and linear polymers). The final parameter to be considered here is the Huggins kt coefficient, obtained from the slope of the reduced specific viscosity versus concentration curve, equation (84)~ Simha (123) first predicted theoretically that a cruciform molecule ought to have a larger ki than a linear molecule of the same intrinsic viscosity, There has been a considerable amount of effort expended to find a quantitative relationship between polymer structure and k', most of it experimental in nature (7, 14, 30, 32, 339 86, 124). The net result of this has been the conclusion that while k' may give a qualitative indication of the presence of branching if one can. compare values of kv for branched and linear polymers of the same [r ], it certainly cannot give quantitative information. The value of k' is not too sensitive to branching, and in some cases it appears that considerable branching must be present in high molecular weight materials before any effect is seen (14). To complicate matters more, k' is known to be very sensitive to the presence of certain kinds of impurities (13). Long has suggested that the effect of branching on kt is very closely related to the non-Newtonian behavior of dilAute solutions of macromolecules He found that the branched samples of PVOAc he examined exhibited the samne

-104o dependence on shear rate as a linear material of the same molecular weight. Long observed that k5 has about the same value for branched and linear polymers if the camparison is made or. a nmolecular weight basis rather than on an intrinsic viscosity basis as is the usual practice, suggesting that k' is effected at least in part by the non-NewtonianL behavior of the systemo The results for the kT measurements made in this study may be found in Table II. Figures 15 and 16 shows k' as a function of [n] and <M>w, respectivel~y, for the branched polymers and some linear fractions of PVOAc (30) It is seen that k' is at most only slightly larger for the branched material, and in some cases the branched material has the same k' as the linear polymer. These data are similar to those observed by Long on PVOAc, Billmeyer (7) on polyethylene, and by Manson and Cragg (14) on polystyrene in so far as the magnitude of the difference between k' for branched and linear polymers is concerned. A study with quite different results is found in the work of Melville, et alO, (17)(18)o They found that their branched polymers had a very much higher k? than a linear material of the same Eq] the synthesis of these branched polymers is discussed in Section I-0C) It was pointed out above that these branched polymers contain residual hydroxyl groups. Examination of the data of Melville, et alo, also reveals tJhat their polymers have a higher frequency of branching (more branches per backbone structual unit) than do the branched samples of this study. Either one, or both of these faetors could conceivably result in tha high kt values reported for the branched polymer~ It should

-105be remarked that in the above study, Melville, et al, report ratios of [rg]/[lh] which exceed unity, The difficulty seems to lie in the manner of estimating [q1]o Melville has estimated [nl] from the computed values of <M;w, using an empirical intrinsic viscosity-molecular weight relationship due to Wagner (125). The values of <Mg>w were estimated from radioactive tracer measurements to give the number of branches per backbone molecule, and. from knowledge of the molecular weight of the backbone and branch polymers. The relationship of Wagner appears to be invalid for Melville's molecular weight range. If the relationship found by Long for benzene is used, the intrinsic viscosity ratio will become less than unity in all cases for Melville's data~ In addition, the ratio is found to be given by the same function of g as observed for the samples of the study [the values of g were calculated for Melville's branched polymers from equation (37) o The data.of this study regarding kt can only support the notion that kT appears to be larger for a branched. molecule than a linear chain of the same [II provided the degree of branching exceeds some lower limit. In addition, the difference may be explained by shear effects as Long has suggested since the increase in k' for the branched polymers has been accompanied by an increase in the molecular weight. Figure 16 shows, however, that contrary to the suggestion of Long, the increase in molecular weight is not sufficient to account for the increase in k' in all cases0 Thus, although shear effects are probably important in causing high k' coefficients for branched polymers, it appears that a universal dependence

-1o6of k' on the molecular weight through the dependence of the shear effect on the molecular weight is not supported by these data. It can be noted that in most cases, the highest values of kl are associated with the branched polymers with the longer branch lengths. The degree of branching, however, does not seem too important, at least not in the range studied here. Thus, these data support the qualitative conclusion that branching may cause an increase in k' if the branches are long enough and above some lower limit in number. There does not, however, appear to be a useful quantitative relationship between k' and the degree of branching, or some parameter of the branching such as g. Co THE MELT VISCOSITY The dependence of the melt viscosity on molecular parameters has been discussed. above in terms of the model proposed by Bueche (104) for linear polymerso This model has been quite successful in the interpretation of data for amorphous high molecular weight materials (102). Fox and Nakayasu (126) have studied the melt viscosity of PVOAc and its diethyl phthlate solutions. Of interest here is the value obtained for the critical molecular weight, that is, the molecular weight where the Bueche model suggests there is one entanglement per chain on the average. They have reported a value of 26,000 for the weight average critical molecular weight, MwO. It is seen in Table IX that the values of KMb>w for the graft polymers of this study vary from 36,000 to 212,000, from slightly above KM4w to about ten times <MW. Following a suggestion of Long (jo30),

-107the ratio of the melt viscosity for branched. and linear polymers of the same molecular weight are plotted as a function of <M2>w in Figure 18. A satisfactory correlation is seen to exist between the ratio of the melt viscosities, Tig/fl, and <KM >w The data are not sufficiently precise to allow the effect of the number of branches per molecule to be determined, however. The value of <Kb>w when g/n1 is unity is of interest. It is qe n that a value of <Mbw = Mc>w when the melt viscosity ratio is unity is quite consistent with the data, although the error in determining the exact value of K<Mw for which the viscosity ratio is unity prohibits direct confirmation of such a relationship. The data of Long have been included in Figure 18, and. it is seen that the results of the two studies are in fair agreement, The experimental observation that jg/l, can exceed unity is in apparent contradiction with equation (83). Certainly if one assumes that the effective slippage factor, s, is the same for branched and linear molecules, then the ratio ng/iql is always less than unity since it is given by g2.5o The difficulty may be in the interpretation of the slippage factor for the branched. molecule. It is known that the length of side groups present in polymers like poly n-alkyl methacrylate. can affect the melt viscosity, and it is presummed that such an effect is in part caused through variation of the slippage factor (127). In the case of branched polymers of the type studied here, one might imagine a mixture comprised of a linear polymer with a molecular weight equal to the backbone polymer and a linear polymer with a much smaller molecular weight equal to the

branch molecule. If the nmliber of the low molecular weight species is smaEL, then the weight average molecular weight of the mixture, and thus its melt virlscosity, wil2 be approximateLy Qqaaa -to th.at of the backbone polymer. The melt viscosity of the milxtuLe, however,, wou.ld. reasonably be qu-ite different- from t;:hat of the equivialernt branekhed polymer in which the low molecular weight material is attached to the backbone. In terms of the entanglement, model, the branched molecules have certain'built in entanglements' in the form of the branch nodes which have a slippage factor of unity. This would indicate that the approach of Bueche, which assigns the same slippage factor to all entanglements., could only be qualitatively applicable for the branched polymer~ The experimental observation -that Tg/r l (at the same molecular weight) can exceed unity indicates that a numnber of chains placed along the backbone will increase the viscosity' more than if they were all attached to the backbone in such a manner as to give a linear po-lymer increased in molecular weight. This is surprising since the branched molecule in the solid state is presumably more compact than a linear chain, and thus less likely to engage in interchain entanglements~ The qualitative discussion given above suggests that while this mayl be so, th.is effect is countered by the btranches acting as a rigidlyr entengled. molecule with no slippage. The observed dependence of the melt viscosity ratio on KbW> is in qualitative agreement with this interpretatio4n since one would expect the effe t of the branches to be direct;:ly related to the probability that they will engage in entanglements, or thus bje related to the molecular

-109weight of the branches. Several attempts have been made to find. a relationship which fit the observed data better than the simple dependence of'lg/i1 on the molecular weight of the branches. The viscosity ratio was plotted against <Mb>w/ KMSw and <Mb3w/<Myw. In addition, the viscosity ratio divided by g2.5 as suggested by equation (83) was plotted against the last two variables. The viscosity ratio as a function of g showed complete scatter of the data. The other relationships did. tend. to bring the data together somewhat, the best fit being obtained with the ratio as a function of <M>/ <Mgw. The data of Long, however, did not fit in at all with the data observed. here when plotted. in this manner, Thus, a relationship with the viscosity ratio given as a direct function of the branch length appears reasonable since it fits the results of two separate studies on different types of branched. polymer with fair success, The behavior of the melt viscosity ratio for the sample with the longest branch length is seen to be much lower than would. have been expected.o This may be due to either one or both of two effects. The branched material had a very high viscosity due to its high molecular weight, even without the effect of branching. Thus, since the melt viscometer used, measures the change in plate separation as the specimen flows, the data for this sample were subject to a relatively large experimental error because the plates underwent very little motion. Compounding this effect is the unknown effect of degradation. This sample had about the same change in the intrinsic viscosity as the other polymers with rela

-110tively long branches, but since its molecular weight was so high, any change in the modlecular weight would have been magnified due to the 3.5 power on M, and thus have been more noticable. In. any event, the drop off observed in the melt viscosity ratio as a function of the molecular weight of the branches cannot be held significant from these data. It was mentioned above that the treatment of Bueche predicts that the melt viscosity should depend on the weight average of the molecular weight in polydisperse systems with a relatively narrow molecular weight distribution (106). It was indicated that the z-average appears to be more appropriate for fractions with a relatively broad molecular weight distribution. This could be an important point if the branched polymers were more polydisperse than the linear materials to which they were compared, but it has been shown above that both linear and branched polymers have about the same degree of polydispersity. Even if the z-average were appropriate for some other, unknown reason, the polydispersity is not sufficient to explain the entire difference in the melt viscosity of the branched and linear polymers, Thus, the ratios of ng/Yl observed would require <M>z/<MK)w to be as large as 2, whereas the largest value of this ratio for any of the fractions is about 11ol to 1,2, To summarize the results of the study of the melt viscosity of branched polyvinyl acetate, it can be stated that these results show that the ratio of the melt viscosities of branched and linear polymers of the same molecular weight carn exceed unity. This effect has been observed before, but the samples here were perhaps better characterized and of narrower

-.111molecular weight distribution due to the method of their synthesis. The interpretation of the melt viscosity given by Bueche, which has been suggested to be valid for both linear and. branched. polymers, does not give an a priori prediction of such a behavior. It is possible, however, to utilize the entanglement modrel of Bueche in qualitatively discussing such behavior.

IV. SUAIA AYND C OONCLUSIONS A graft polymerization has been utll'zed to obtain a branched polymer of pol -viny-l acetate, This grafting procedure allowed the struacture of thoe resultant branched polymers to be closely controlled and characterized. The method has the additional advanltage of giving a branched polymer with a nilecullar weight distribution very close ton thiat of a good linear fract ic making the comparison of the properties of the branched polymer to tho se of a linear polymer more mean ngfull. The solution properties of the branched. polymer and of some linear polymers of comparable molecular weight have been measured in a thermodynami-ally good ssolvent, by viscosity and S light scattering methods. The purpose of these measurements was to complete the characterization of the structure of the branched. polymer and to obtain soame information concerning the radiius of gyration, the secon.d virial coefficient and the intrinsIo v1iscsdty of a branched polymer in a good solvent. ft has been fond that; the branched. polymers studied here were relati.vely mor.9e expanded in solution than linear polymners of the same molecualar weight. In particular, if- the ratio of the mean square radii of branched arad linear polymners of the same molecular weight is defined by the parameter g in a thermodynamically ideal solvent, then this same ratio would be about gl/3 inl a good solvent. ll2

-l13The solvent effect can be conveniently described, by considering an expansion coefficient defined as the ratio of the radii of a polymer in a good. solvent and an ideal solvent~ The cube of the ratio of this coefficient for branched. and linear polymers of the same molecular weight was found. to vary inversely as the parameter go The expansion effect is reflected in the behavior of the intrinsic viscosity and the second. virial coefficient of the branched polymers. The ratio of the intrinsic viscosities of branched and linear polymers of the same molecular weight has been determined, in a good solvent. It if found that the observed. ratio in a good solvent is closely predicted by expressions derived, elsewhere for the viscosity ratio in an ideal solvent. This is rather surprising since the expansion effect implies that the viscosity ratio should. be larger in a good solvent than it is in an ideal solvent. The data of this study indicate that in an ideal solvent, the intrinsic viscosity ratio is approximately given by g3/2, a relationship that arises theoretically when the effective hydrodynamic radius of the polymer in flow (in dilute solution) is the same as the configurational radius of the chain. The ratio of the second virial coefficients for branched. and. linear polymers of the same molecular weight was found. to vary in a manner pred.ictable by theory if the expansion effect is accounted for, The ratio is given by gl/2 to a good approximation, for the samples studied here, and. it is believed. that this relationship should hold. for other types of branched. molecules as well.

It was found that the type of branching obtained. here did not cause a marked increase in the Huggins k' when the value for a branched. polymer is compared to that for a linear polymer of the same intrinsic viscosity. The values of kt for the branched, polymer were found to be slightly larger or about the same as. the values for a comparable linear polymer. Thus, k' can only be used as a qualitative indication of branching and the presence of a branched. structure cannot be excluded on the basis of a'normal' value for k'. The melt viscosity of the branched polymers was found to be strongly dependent on the length of the branches. The ratio of the melt viscosities of branched and linear polymers of the same molecular weight appears to vary directly with the molecular weight of the branches. The ratio is unity for a branched molecule where the branches have some critical molecular weight which is characteristic of the particular mers that make up the polymer. This critical molecular weight corresponds to the length of a polymer chain that would be engaged. on the average in an entanglement with one other molecule in the melt, Thus, the effect of the branches on the melt viscosity appears to be related. to the ability of the branches to engage in intermnnolecular entanglements~ If they can do this, then they seem to increase the amount of coordination between the various molecules in flow, and, by thus increasing the resistance to flow they increase the melt viscosity. If they are not long enough to engage in entanglements, then they appear to act as internal plasticizers, thus reducing the melt viscosity of the branched polymer.

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APPENDIX I KINETICS OF THE GRAFT POLYMERIZATION

The purpose of the following calculation is to obtain an expression for the number of branches grafted onto a backbone substrate in a graft polymerization as a function of the reaction parameters. The pertinent reactions are given below. All concentrations are in mol/liter. (B) is defined as the mol/l of sites available for branching, and is assumed to be given by x(P), where x is the degree of polymerization of the backbone polymer. (F) is the mol/l of the newly formed polymer, which may be bound to the substrate or unbound, (I), (M), (S) and (R') are the mol/l of initiator, monomer, solvent and polymeric radicals respectively. Initiations I -— 2R k Addition: Ro + M ) R k2 R. + R* R- k2 Transfer: Ro + B3 --- F + B. k3 R. + F -- F + F. k3 R. + S F + S' k4 R. + M - F + M. k5 Following transfer: B. + M - R4 k6 S- + M -- R. k M' + M --- R. k8 F, + M --- R' k6 Termination: R +B RB - R + R* kg -123

-124The number of branches per backbone molecule after time t of the grafting reaction is given by k = (i/(P))If ( - d(B) dt (Al) d(B)/dt = k(R ) (B dt where -d(B)/dt = k3(R-) (B) At steady state conditions, d(R.)/dt = 0. 2kl(I) - 2ks(R.)2 (A2) or (R-) = (kl(I)/ks)2 (A2) Similarly, (R-) = k3(B) (R-)/ks(M) (A3) (S-) 2 k4(S) (R-)/k7(M) (A4) (M.) = k5(R-)/k8 (A5) (F.) = k3(F) (R-)/k6(M) (A6) The fraction of the initial monomer, (M)o, converted to polymer at time t is given by t X - (1/(M)o) f (-d(M)/dt) dt (A7) where -d(M)/dt = k2(M)(R.) + ks(M)(R-) + k6(M)(B-) (A8) + k7(M)(S- ) + k8(M)(M-) + k6(M)(F ) Using the relationships for (B-), (So), (M-), and (F.), this becomes -d(M)/dt = (R ) [k2(M) + 2k5(M) + k3(B) + k4(S) + ks(F)] (A9)

-125For low conversions, the integrands in the above definite integrals becomes constant, or nearly so. This approximation is not strictly correct, but it gives results which are in good agreement with a more rigorous method for conversions below twenty percent (2). Using this approximation, the above integrals give x = (RB) [k2 + 2k5 + k3(B)o/(M)o + k4(S)o/(M)o + k3(F)o/(M)o] x t (A1o) k = (l/(P)o)k3(R~)(B)o x t (All) Eliminating (Ro), there results k = CpxX/[l + 2Cm + Cp [(B)o/(M)o + (R)0/(M)o] + CS(S)o/(M) Defining F as the ratio of the newly formed polymer bound to the backbone to the total new polymer after time t, then F = -d(B)/dt / d(F)/dt (A13) where d(F)/dt = k3(B)(R) + k4(S)(Ro) + k5(M)(R') + 2ks(Ro)2 (A14) Thus, F is given by F = k3(B)/ [K3(B) + k4(S) + k5(M) + (4klks(I) )2] (Al5) (B)o (B)o + kg + Or F = C C + C ( + CC + () N (M)0 L P (N)0 ()0(o K0 2 ~~~~~~~~~~~~~(HAo6o

-126Taking the rate constants at 88~ a' k, = 12 6 x 10 ks -= 4 7 x 167 k2 = 7~ 3 x l0 -4 m - 24 x 1o Cs = 50 x 10'-4 Cp = i5 x 10 then with (S)o/(M)o = 0oO8, (I) 0o., M and (nd B)o io the initial weight of polymer to the weight of mo:rnoere takren as 0,08. a typi2cal value for F would.'be F = 120/ 120 + 4o0 + 24o + 4i 2 s /1o00 Equation (2) JIs seen to be a good approximtion -(for. the. L e]u pitUrLIi since the l as term in the denominator is regigigblet Si mlarly, the expression for k) is seen to be app rb Xi 2m, ox ted )by eqwationr (1) since the terms other than unity in th-.e C.&eom irator* cf.-5the full expressio n are negligibly sm-ll compared. to uniIt'yo Tk.;., x l. = x 10e a typical value of Kk> is given by k>. 3ox

APPENDIX II DIMENSIONS OF A BRANCHED POLYMER IN AN IDEAL SOLVENT

The purpose of this section is to show the derivation of g, the ratio of the mean square radii of branched and linear polymers, for the simplified model described in Section I-E. This model has k branches of equal length, t, spaced at equal intervals, s, on a monodisperse backbone such that the total branched polymer has n segments. Kramerst rule gives (see equations (35) and (28)) g = (6/n3') Z x(n - x) (A17) where the sum is over all possible pairs of x and n-x. The model is represented schematically as The indicated sum becomes s t k-l s+p(r+t+l) g - (6/n3) 2 Z x(n - x) + k Z x(n - x) + Z Zx(n - x) (A18) x=l x=l p=l xx=p(r+t+l) Performing the indicated summations, there results g(k,r) = (l/(kr + 1)3) [r(3(k - 4)k + x2 (k +2)k + k + 1 (A19) where r = t/s(k+l). For t = s, this reduces to g(k, l/k +1) = (1/(2k + 1)3)(4k3 +12k2 + 4k + I) (A20) -128

-129This type of branched structure has been considered by Zimm and Stockmayer (50). The values of gt=s computed from this equation agree exactly with those given by Zimm and Stockmayer. For k large, the general expression for g(k,r) reduces to g(k,r) = (l/(kr + 1)3) (3r3k2 + r2k2 + 2rk + 1) (A21) Then for r << 1, this expression gives equation (37) g(k,r) = l/(kr + 1) (37) Equation (37 is found to give results which agree very well with thos calculated from the full expression for r between 0.01 and 0.03 and k between 0 and 100. All of the branched structures of interest here are within these limits. The results of Orofinio (69) on the more general model where the branches of equal length are placed at random on a monodisperse backbone given equation (37) modified by the addition of one term. This term, given by 3kr2, is negligibly small for all of the structures of interest here. Thus, a typical value of g from equation (37))would be 0o8, while the additional term would typically amount to 0.01. Thus, the additional term was ignored for the purposes of this study.

APPENDIX III THE EFFECT OF GRAFTING ON THE MOLECULAR WEIGHT DISTRIBUTION

This calculation is intended to show the effect on the molecular weight distribution of placing k branches of average molecular weight Mb on a monodisperse backbone polymer with n sites for branching and of molecular weight M1. If p is the probability that a site on the backbone contains a branch, then the number fraction of chains with i branches, fi, is given fi = |i() pi(l _ p)n-i (A22) where fi is normalized by n Z f = 1 (A23) i=0 (f is simply the binomial distribution for the probability of i successes in n Bernoulli trials with the probabilities p for success and (1 - p) for failure. See for instance reference (128), page 137. Calling the average molecular weight of the branched polymer <Pa, the number average of Mg is calculated as n <n> n Z Mg fi (A24) i=O n <Mg>n = Z fi(M1 + iMb) (A25) <Mg = M1 + npMb (A26) But p is simply k/n for n very large, so that Mg>n = M1 + Mb (A27) -131

-132The weight average value of Mg becomes n n <Mw = Z Mg2 fi/ Z Mg fi (A28) i-O.. i=O N <Mg> n4w = Z fi(M12 + 2iMlMb + i2:Mb2) (A29) i=O <Mn<Mw = (M1 + kMb)2 + kMb2(1 - 1/n) (A30) Thus, the new value for the ratio of the weight to number average molecular weight is seen to be <M>w\4~ n = 1 + k(l - k/n)/(k + ml/Mb)2 (A31) Equation (92) gives this expression for k/n much less than unity~

APPENDIX IV TABLES

-134TABLE I GRAFT POLYMERIZATION DATA x <Mb> v <M> Polymer Conversion Unbound Unb ound Backone + Unbund Backbone <M~)w Z X KM~v Polme Conversion Polymer Polymer Po lymer S-20 0.365 0.520 113.0 x 10s 0.218 0~563 127o0 2,27 x 106 132 x 104 118 x 103 S-30 0.272 o0.459 99.0 o0.289 0.373 75.0 o,365 0.399 69.0o 1.29 119 79.2 s-40 0o.163 0.213 39.6 0,145 0.219 39.6 0.o106 0o.185 33.0 0.127 O,189 34.0 1.50 81 37.0 S-41 0.159 0,207 37.5 0,116 0.174 31.2 0.171 0.193 34~6 1.50 148 36,0 8-50 0.153 0,219 40.0 0,233 0o267 50.0 0.157 0.270 52.0 0.209 0,263 47.0 2.02 152 47~4 S-51 0o.198 0,267 50,0 0o.185 0,261 49.0 0.148 0.285 54.0 2,02 260 48.8 s-60 0,348 0o821 212 3.76 131 212 S-70 0.323 0,o667 158 0.167 0.582 133 216 106 149,5 S-71 0.316 o0,468 99 0.274 0.605 139 216 230 134.. s-8o 0.257 0.544 122 0,228 0.554 123 2.27 110 122 S-81 0.281 0,501 108 0.176 O0.703 170 2,27 214 127 S-90 0o,226 0.418 86 0,226 0o479 102 2.54 115 94.0 S-91 0.268 0,381 76.5 0.163 0.522 112 2.54 236 87.4 +The average molecular weight of the branches after a series of graft polymerizations is calculated as hbev = m> a omV/ZX where the sums are over all polymerizations in a grating cycle.

-135TABLE II INTRINSIC VISCOSITY DATA Fraction Solvent [1 ] kI k + S-20-111 Benzene 5.20 0.401 0.501 S-20-M Benzene 4.18 0.347 0. 498 S-30-11 Benzene 2.96 0.366 0,502 S-30-M Benzene 2.77 0.367 0.502 S-40-11 Benzene 3.18 0. 368 0o498 S-41-11 Benzene 3,64 0.344 0.498 S-41-M Benzene 3 34 0.347 0.498 S-50-11 Benzene 4.52 0.338 0.502 S-51-11 Benzene 4.48 0.338 0.500 S-51-M Benzene 4.15 0 366 0.501 S 60-1 Benzene 5,76 0.356 0.500 S-70-1 Benzene 4.00 0.366 0.500 S-71-1 Benzene 4.32 0.355 0.500 s-80-1 Benzene 4.27 0 369 0.502 S-81-1 Benzene 4.04 0.359 0.501 S-90-1 Benzene 5.01 e0.38 0,502 S-91-1 Benzene 4.19 0.381 0.502 L-2-2 2-octanone 1.35 0.534 0.501 L-5-3 2-octanone 0.940 0.887 0.500 S-60-1 2-octanone 0. 950 1208 0.500 S-70-1 2-octanone 0.953 O.475 0,500 S-71-1 2-octanone 0,616 0.815 0 500

TABLE III INTRINSIC VISCOSITY IN 1, 2 4-TRICHLOROBENZENE Fraction [] <M>v s-307o 03o50 o.o69 x 106 T-277 0~792 0~,275 L-6-1 2.06 1o34 L-5-2-1 280 lo 96 L-2-3 3,22 2o54 s-60o- 4.18 5o51 7This is the unbound polymer from the third recycle of polymerization S-30, unfractionated, 77An unfractionated polymer prepared with monomer in the presence of toluene, but no backbone polymer.

-1 6(TABLE IV LIGHT SCATTERING DATA FOR THE LINEAR POLYMER Fraction Solvent w (S ) A2 h L-2-1 1,2,4-tri- 3.76 x 106 73.2 x 104 1.75 x 10-4 19 chlorohenzene L-2-2 1,2,4-tri- 5.43 69.5 1.94 9 chlorobenzene L-2-2 Methanol 3.54 66.0 1.33 L-2-3 1,2, 49.3 1.90 18 chlorobenzene L-2-3 1,2,4-tri- 2.54 48. -- 18 chlorobenzene L-4 —tri- 2.27 40.8 2.09 19 chlorobenzene L-4-1 1,2,-tri- 2.16 42.5 2.05 7 chlorobenzene L-5-2 1,2,-tri- 1.87 36.4 1.99 19 chlorobenzene L-5-2 Methanol 1.90 30.9 1.30 L-3-3 1,2,6-tri- 1.81 37.5 1.98 19 chlorobenzene L-5-3 1,2,4tri 1.50 28.0 1.90 19 chlorobenzene L-5-3 21,2,4-tr- 1.53 27.7 2.32 chlorobenzene L-5-3 Methanol 1.62 24.4 1.14 L-6-1 1,2,4-tri- 1.9 26.5 2.39 19 chlorobenzene L-6-1 1,2,4-tri- 1. 8 28.5 2.29 19 chlorobenzene L-6-3 1,2,6 1.05 17.2 2.35 19 chlorobenzene

-138TABLE V LIGHT SCATTERING DATA FOR THE GRAFT POLYMER F ract ion Solvent <M <()>z A2 Si/Sa S-20-111 1,2,4-tri- 3.81 x 106 -- - - S-20-M 1,2,4-tri- 2.80 55.4 x 104 1.68 x 10-4 0.615 chlorobenzene S-30-11 1,2,4-tri- 2.33 chlorobenzene S-30-M 1,2,4-tri- 1.68 chlorobenzene S-30-M 1,2,.-tri- 1.61 -- 1.69 chlorobenzene S-40-11 1,2,4-tri- 2.00 37.5 1.78 0.613 chlorobenzene S-41-11 1,2,4-tri- 2.24 44.2 1.69 0.686 chlorobenzene S-41-M 1,2,4-tri- 2.09 37.2 1.69 0.685 chlorobenzene S-50-11 1,2,4-tri- 2.71 55.1 1.78 o.685 chlorobenzene S-51-11 1,2,4-tri- 3.67 68.7 1.47 0.631 chlorobenzene S-51-M 1,2,4-tri- 3.21 54.3 1.62 0.515 chlorobenzene S-60-1 1,2,4-tri- 5.60 111.5 1.56 --- chlorobenzene S-60-1 1,2,4-tri- 5.43 115.8 1.18 0.575 chlorobenzene S-60-1 Methanol 6.10 99.6 0.99 --- S-70-17 1,2,4-tri- 2.65 52.0 1.75 0.623 chlorobenzene S-70-1 Methanol 2.92 54.8 1.01 S-71-1 1,2,4-tri- 3.43 60.1 1.29 0.627 chlorobenzene S-80-1l 1,2,4-tri- 3.29 61.3 1.52 0.730 chlorobenzene S-80-1 Methanol 3.55 59.7 1.06 --- S-81-1 1,2,4-tri- 4.11 71.8 1.18 0.615 chlorobenzene S-90-1 1,2,4-tri- 3.45 64.1 1.48 0.715 chlorobenzene -91-1 1,2,4-tri- 3.96 74.1 1.29. 624 chlorobenzene /These data were obtained from a plot of the data from two separate light scattering determiinations, see Figure 3 for instance.

-139TABLE VI INTRINSIC VISCOSITY COMPARISONS (All viscosities in benzene) Fraction [rlg] [11]7 [[] g]l Ing I.I..., S-20-111 5.20 6.17 4.29 -- S-20-M 4.18 5.10 4.29 3.88 S-30-11 2.96 4.50 2.92 -- S-30-M 2.77 3.58 2.92 s-4o-11 35.18 4. 05 3.32 S-41-11 3.64 4.36 3.32 S-41-M 3.34 4.15 3.32 -- S-50-11 4.52 5.03 4 o6 3.77 S-51-11 4.48 6.08 4,06 -- S-51-M 4.15 5.62 4.06 -- S-60-1 5.76 8.07 5.60 5.24 S-70-1 4.00 4.97 4.57 3.57 S-71-1 4.32 5.85 4.57 4.08 S-80-1 4.27 5.72 4.29 4.07 S-81-1 4.04 6.37 4.29 S-90-1 5.01 5.90 4.37 4.60 S-91-1 4.19 6.25 4.37 jThe intrinsic viscosity of a linear polymer of the same molecular weight as the graft polymer. 7The intrinsic viscosity of the backbone linear polymer of the graft polymer. m//The intrinsic viscosity of the graft polymer after a melt viscosity determination.

-140TABLE VII PARAMETERS FOR THE LINEAR POLYMER Fraction A2 <M>/ [l ] D L-2-1 155 1o86 x 102 L-2-2 176 1o87 L-2-3 157 1o 78 L-2-3 --- 1o84 L-4-2 156 2.20 L-4-1 134 2.00 L-5-2 142 1o79 L-3-3 126 1.79 L-5-3 --- 1.82 L-5-3 132 1o 71 L-6-1 145 1o62 L-6-1 150 1.5 4 L-6-3 134 2016 Avg 146 + 25 1.85 (+ 0.2) x 1021

-141TABLE VIII PARAMETERS FOR THE BRANCHED POLYMER Fraction <k) gX x 1e21 g [r>] ( (A2)1 (S7/Sa )i cgj S-20-11]1 26 -- 0.616 0.842 --- 0.836 S-20-M 9 1.60 0.821 0.821 0.966 0o858 0o880 1.29 S-30-11 26 -- 0.556 0659 --- 0o.614 S-30-M 10 -- 0.755 0.773 --- 0.769 0.734 S-40-11 27 1.60 0.743 0.786 0.949 0o844 0.876 1.41 s-41-11 41 1.92 0.670 0.836 0.986 0.804 O 686 1.46 S-41-M 33 1,77 0,712 0.804 0,890 0.807 0.978 1o40 S-50.-11 28 1.68 0.744 0.897 10o1 O 0908 0,980 1,53 S-51-11 67 1.o 61 0.524 0.738 o0,896 0.799 0,902 2.10 S-51-M 48 1.86 0.640 0,738 0o823 0.860 0.734 1.43 S-60-1 17 1.49 0o600 0.703 0.924 --- ---,86 s-60-1 16 1.38 0o.600 0.703 -- 0.709 0.794 2,02 S-70-1 8 1,24 0.764 0o.804 0.969 0o.884 0.886 1.79 S-71-1 19 1o77 0.634 0.737 0.853 0 694 0o883 1,52 S-80-1 17 1.64 o.963 o.748. 904. 809 --- 1. 46 s-81-1 23 1.54 0.617 0.666 0o.834 0o701 0o878 1o66 S-90-1 29 1.88 0.736 0o848 0o898 0,797 1.02 1.32 S-91-1 33 1.46 O 641 0o671 0.902 0o714 0.896 1.66 S-60-1 0,507* S-70-1 0.752* S-71-1 0,431* *These values are for the data obtained in 2-octanone at 29,80~C

-142TABLE IX THE MELT VISCOSITY DATA (Viscosities in poise) KM i>w Fraction rlg/r 1 <Mov <r L-7-2 0.012 x 108 -- 0.622 x 106 L-7-1 0.024 --- - 0.71 L-6-2 0.223 ----- 1.14 L-T. 651 ----- 1o 57 L-T (0.610)+ - --- 1.57 L-3-3 1.37 -- 1,81 L-3-3 (1o03) - --- 1.81 L-2-3 2.05 -- 2. 54 S-20-M 66.8++ 5.2 118 x 10 2.80 S 20-M (9o51) (2.9) 118 2,80 S-40-11 0. 506 0. 53 370 2,00 S-40-11 (0.624) (0.66) 37.0 2.24 S-41-M 1,88 1.63 36.o 2.09 s-50-11 1309 4,74 47.4 2o71 S-50-11 (7.87) (2.53) 37.4 2.71 S-51-M 16.4 2.26 48.8 3.21 S-60-1 84.o 2.48 212 5.52 S 70-1 24.7 10,8 149.6 2.65 S-71-1 55.9 7.70 134 3.41 S-80-1 314 5o 29 122 3 29 S-81-1 84.4 6.32 127 4,11 S-90-1 40.7 5.76 94 3~45 S-91-1 40.2 3.47 87.4 3~96 +The data in parentheses are from a second, separate determination on a single sample. This measurement made at 1520C. This value corresponds to r = 17 x 108 poise at 183~0C.

APPENDIX V F IGUIES

-144I Y( XOn Xb~n ij,~/ _X c 2.0 (S/M)0 Figure 1. Experimentally observed dependence of the degree of polymerization and the apparent reaction rate constant, (% conv)/(hr)(initiator conc)l/2, on the initial mol ratio of solvent to monomer.

1.2Cs 0.1988 gm / 100 Cc. 0.9 0(. 0 0.25 0.50 0.75 1.0 Cr Figure 2. Viscosity plot for fraction S-71-1.

50 40 30 90 3~~~~~~~0~ ~60 I- 20 - 120 ) — Cs= 0.4003 gm/100 cc 0 0.5 1.0 1.5 2.0 2.5 1.0 3.5 Si.n2 (8/2) +5Cr Figure 3. Zimm plot t'or fraction S-70-1.

60 50 40 I 30 20 I 0 00 0 0.5 1.0 1.5 2:0 2.5 3.0 3.5 -6 (<M.> X x 10 Figure 4. Number of branches per molecule obtained in a graft polymerization versus the backbone polymer molecular weight multiplied by the overall degree of conversion.

-1486.1 O LINEAR 6.0 X E BRANCHED 5.9 El 5.8 VIr~~,Cn ~ ~ ~ ~ 0; 5.6 5.5 5.4 5.3 5.2 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.6 Log. <M)w Figure 5. Mean square radius of gyration versus molecular weight for branched and linear polymers.

0.4 0.3 0'C o o < 0.2 013 0'.I l —0t LINEAR E BRANCHED 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Log. (M)w Figure 6. Second virial coefficient versus molecular weight for branched and linear polymers.

180 0 THIS STUDY A SHULTZ (49) I I I I I I o LONG (30) ~, 160 140 El 1 20 I 00 5.6 5.7 5.8 5.9 6.0 6-i 6.2 6.3 6.4 6.5 Log. <M.W Figure 7. Ratio <'M>w1/[n] versus molecular weight for some linear fractions.

-151I0 0 THIS STUDY o LONG (30) A SHULTZ (49) 8 / 0 7 6 5 0 1 2 3 4 A2 x I10 Figure 8. Root mean square radius of gyration versus the second virial coefficient for three linear fractions.

0.2 CL0 8' 0.1 0 04, r.' 1, 0 0.1 0.2 0.3 0.4 0.5 0.6 Log [7/] / o Figure 9. Expansion factor versus the ratio of the intrinsic viscosities in good and ideal solvents.

0.8 0 LINEAR (REF 30). 0.7 E1 BRANCHED l l 1 I - l3 0.6 El o 0.5 0.4 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 Log. (M) Figure 10. Intrinsic viscosity versus molecular weight for branched and linear polymers. ~~ZI. and linear polymers.

-1541.0 o (A2)g/(Aa)I 0.9 0.8 0.4 2 CA)/ 0.6 0.7 0.8 0.9 1.0 [7g]/[ 7~R (A2O/(A) Figure 11. Parameter g versus the intrinsic viscosity ratio and second virial coefficient ratio for branched and linear polymers. Curves A, B, and C give [fig]/[rll] equal to g3/2, h3, and gJ/2, respectively.

0.4 0~ 04~~~~~~ 0 V1 0 0.10.72 0.75 0.80 0.85 0.90 Log (g x 10) Figure 12. Cube of the ratio of the expansion factors for branched and linear polymers versus the parameter g. [All curves go through (ag/al) = 1 for g = 1 ],

1.0 0 0 o THIS STUDY O LONG (30) 0 i~~~z". 0.9~~~~0 x EP 0~~~~~~~~~~~~0 ~2~ 0.9 N 0. 03 0. 06. 08. I (;n ~ ~~~~~~~~og(gx 0 130. -I 0.7 0.2 0.3 OAQ 0.5 0.6 0.7 0.8 0.9 1.0 Log (g xIO0) Figure 15. Ratio of the mean square radius of gyration for branched and linear polymers versus the parameter g.

-1571.0 0.9 0.8 - -~ 0.7 0.6 0.5 0.5 0.6 0.7 0.8 0.9 1.0 Figure 14. Ratio of the second virial coefficients for branched and linear polymers versus the ratio of the intrinsic viscosities for branched and linear polymers.

I _ _ 0 LINEAR (REF 30) 0.40 0 BRANCHED 0.38 13 IEl El~ El El ] El 1 0.36 - - 0 0 0 00 0 50 E E G 0 0.0 0l 0 0..~~~~~ [.~ 3 ra I 0.34 - - - - 0 0.32 2 3 4 5 6 [i] Figure 15. Hu ggins kt versus intrinsic viscosity.

0.40 O LINEAR (REF 30) 10E BRANCHED 0.38 El 13 0.36 - ~~0 0 0 0 E 0 _ - 0.:34. —------. - - 0.34 e. zp -- -- 0.32 -.................. 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Log (M/w Figure 16. Huggins k' versus molecular weight.

-160 - 0 LINEAR 10 El BRANCHED 0 E 9o8 7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Log (M)W Figure 17. Melt viscosity for branched and linear polymers versus molecular weight.

-1610 O 10 THIS STUDY A LONG (30) 8 -- 6 _ 4 I 3 A 00~ ~ ~ 0 0 *W0 0.20 (Mb) X 1 Figure 18. Ratio of the melt viscosities of branched and linear polymrs versus the molecular weight of the branches on the branched polymer.

APPENDIX VI NOMENCLATURE

a Exponent in the empirical relationship between the intrinsic viscosity and the molecular weight. A2 9 A3 The second and third virial coefficients, respectively. b Length of an equivalent segment in a polymer chain. B,(B) Reactive sites for a grafting reaction and its concentration, mol/l. c Exponent in the empirical relationship between the second virial coefficient and the molecular weight. c,cs Concentration and concentration of a stock solution, gm/cc, cn Factor in the light scattering calibration constant, equation (89). cr Relative concentration, c/cs. Cm,CpCs Chain transfer constant (ktr/kp) for monomer, polymer, and solvent, respectively. Cn Number of nth order couplings in a system of entangled polymer chains. dn/dc Refractive index gradientg cc/gmO D Translational diffusion constant, f Friction constant; fi Number of fraction of polymer chains with i branches, f(g) The ratio of the intrinsic viscosities of a branched and linear polymer. f(M) Weight fraction of polymer with molecular weight M in the range dMo F Ratio of polymer bound to backbone substrate to the total new polymer formed in a graft polymerization, F The statistical factor in the melt viscosity relationship. -163

-164F, (F) Newly formed polymer in a graft polymerization and its concentration, mol/1, F,W Load applied to a parallel plate viscometero F1 (i) Distribution function describing the relative probability of finding a polymer chain with a set of coordinates (i). g Ratio of the mean square radii for a branched and linear polymer of the same molecular weight, in an ideal solvent, h Separation of the plates in a parallel plate viscometer, cmo h Parameter in the molecular weight distribution function. I,(I) Initiator and its concentration, mol/l. J Segmental jump frequency in the melt viscosity relationshipo k Instrument calibration factor in the light scattering calculation. k Boltzmann's constant. k,<k> Average number of branches per molecule. k' Huggins k' coefficient, see equation (84). k" Associated with the k' coefficient, see equation (85)o k,kl Reaction rate constant for initiator decomposition. kpyk2 Reaction rate constant for chain propagation. kt k9" Reaction rate constant for chain termination. kt rm9k5 Reaction rate constant for monomer chain transfer. ktr,p9k4 Reaction rate constant for polymer chain transfer. ktr,s9k3 Reaction rate constant for solvent chain transfer. K Generally used as a proportionality constant, Kapp Apparent reaction rate constant, (Rapp)/[I]1/2 m Exponent i.n the relationship between the expansion factor and the ratio of the intrinsic viscosity in a good and ideal solvent.

-165M The molecular weight of a polymer, M,(M) The monomer and its concentration, mol/l. Mc Critical molecular weight in the melt viscosity relationship. n Number of equivalent segments in a polymer chain. nno Refractive index of solution and solvent, respectively. N Avagadro's numbero P, (P) Polymer and its concentration, mol/l. P(G) Particle scattering factor, q Number of entanglements in a chain of n segments in a polymeric me it. Q(G) Function to account for bimolecular contacts in the light scattering virial equationo r The ratio of Mb to MNo r Distance between the centers of mass of two polymer chains. R Gas constant. Ro,(R~) Chain radical and its concentration, mol/l, (R2) Mean square separation of the ends of a polymer chain, averaged over all chain configurations, cm2 or (A~)2o R(G) Rayleigh scattering ratio. Rp Rate of polymerization reaction, mol/(l)(sec). Rapp Apparent rate of polymerization, calculated as the yield per volume per time. s Slippage factor, S, (S) Solvent and its concentration, mol/l, (S 2) Mean square radius of gyration of a polymer chain, averaged over all chain configurations, cm2 or (A0)2.

-166Sa Asymptotic slope of the reciprocal scattering curve. Si Initial slope of the reciprocal scattering curve. Ss Radius of a sphere hydrodynamically equivalent to the polymer coil. t Time, min or sec. T Temperature, oK. T"c () Excess scattering ratio at G, corrected for reflection. u The mean potential of the average force between two segments in two separate polymer chains. Uab Potential of the average force between two polymer chains in solution. U(r) Average potential between two molecules whose centers of mass are separated by a distance ro v Specific volume. v Volume excluded by an equivalent segment. V Volume, which may be arbitrarily large. V1 Molecular volume of the solvent. x The number of structural units in a polymer chain, not necessarily the same as n. X Degree of conversion in a polymerization, based on the monomer. X A parameter in viscosity theory. ZX Sum of the separate degrees of conversion In a series of graft polymerizations on one backbone substrate. y Parameter in the molecular weight distribution function. z A thermodynamic parameter. OG Expansion factor, defined as the square root of (S2)/(S2)o. 8 Excluded volume integral.

-167r The gamma functiono e Part of the exponent in the empirical relationship between the radius of gyration and molecular weighto Melt viscosity, poise.,q Ao Viscosity coefficient of polymer solution and solvent, respectivelyo Tlrel Relative viscosity, T/ioo asp Specific viscosity, 1rel-lo [] Intrinsic viscosity, decilieters/gm. G Angle of scatter measured from the transmitted beam, g Temperature at which the chemical potential of a polymer solution due to segment-solvent interactions is zero. o0 Wavelength of light in a vacuum, A~. r Osmotic pressure. Z Summation sign. Free volume associated with the configurational nature of nearest neighbors in a polymeric melto (,D Proportionality constant between the intrinsic voscisity, radius of gyration, and molecular weight in an ideal and a good solvent, respectively0 Entropy parameter. < > Average of the parameter in the brackets with respect to molecular weight.

-168Subscripts b unbound polymer, or polymer bound to backbone in graft polymer. g Any branched polymer. 1 Any linear polymer. o An ideal solution, or the initial value in the case of concentrations, n Number average. w Weight average. z Z-average. U NIVRSTY OF MICHIGANI 3 9015 02223 2154