Conformational Study of Branched Vinylpolymers. Cascade Theory Applied to Chain Transfer BY W. BURCHARD,* B. ULLISCH AND CH. WOLF Institute of Macromolecular Chemistry, University of Freiburg, West Germany Received 7th December, I973 Highly branched materials, obtained by chain transfer between polymers in free radical poIymer- ization, are treated by means of cascade theory which is based on the use of probability generating functions. The necessary link probability generating functions, composed of various probabilities of reaction, have been calculated from the kinetic scheme of the polymerization process. Applica- tion of the cascade theory allows prediction of the critical point of conversion where gelation should occur. Analytical expressions are given for the number and the weight average degree of polymer- ization, the z-average mean square radius of gyration and the particle scattering factor of the mole- cules in the pre- and post-gel state of the system.Furthermore, the mass fraction of linear chains and the sol fraction in the gel are calculated, and finally, the density of branching and the number of elastically effective chains in the gel are evaluated. Evidence of branching may not be apparent from the <S2>, and P,(O) measurements of the unfractionated samples due to the presence of large amount of linear chains, but can be detected when the linear fractions are removed from the sample. No noticeable differences are observed in the properties of molecules in the pre- and post-gelation period. The increase in the number of elastically effective chains after gelation with increasing monomer conversion is compared with the case of a randomly branched trifunctional polycondensate. A much lower elastic modulus is found for the present system than for the trifunctional polycondensate.The theoretical expression for DPw, derived in this paper, shows good agreement with the experimental data of Stein on poly-vinyl acetate. While Stein did not take into consideration gel formation, the present theory now predicts gelation at a monomer conversion of about 67 %. The close agreement between experiment and theory lends not only strong support to the present calculations, but also vindicates the use of cascade theory for the study of chain transfer reactions. Chain transfer in free radical polymerization has aroused much interest in the past because of its abilities to alter properties of polymers.In particular, regulators which are mostly of low molecular weight and bear only one functional group have been widely studied. This paper, however, deals with chain transfer with poly- functional transfer reagents and the aim is to show how such transfer reactions, which necessarily lead to branched material and possibly gel formation at the end, can be given a rigid mathematical framework based on cascade theory. Two main cases can be considered. In the first case, low molecular weight polyfunctional transfer reagent with a well-defined number of active groups is added to the monomer -a process analogous to copolymerization. In the second case, to be considered here, a polymer consisting of repeat units with a functional side group acts as a poly- functional, high molecular weight transfer compound.Branched material is ob- tained in this case by homopolymerization. This type of reaction has been treated theoretically in the past by several authors, 1-5 and the method of Bamford and T ~ m p a , ~ who use a Laplace transform for calculating various moments of a given molecular weight distribution, has proved to be very efficient. Recently, Small et aL6* have applied the technique of probability generating functions to derive similar results. This, however, is only a discrete version of the Laplace transform method. In spite of their undeniable merits, these methods are restricted to the evaluation of the various molecular weight averages and molecular 56W.BURCHARD, B . ULLISCH AND CH. WOLF 57 weight distributions. The interest of the present study is focused in calculating conformational averages, by which it is hoped to get information about the structure of the polymer before and after gel formation. For deriving equations of these con- formational averages, a more powerful mathematical framework is required. Cascade theory, known in demography for a long time, has been recently applied to problems of branching in polymer s ~ i e n c e . ~ ' ' ~ The advantage of this theory is twofold: first, the desired averages, e.g., the number and weight averages of the molecular weight and the z-averages of the mean square radius of gyration and the particle scattering factor, are obtained as special cases of a general path-weighting generating function.Secondly, the branching process can be clearly represented by graphs of rooted trees.l OUTLINE OF THE THEORY In the system concerned, branching can be caused in two ways, i.e., (i) by transfer with an active side group of the polymer, and (ii) by transfer with the same side group of a free monomer ; in the latter case polymers are formed with a terminal double bond whose addition to growing radicals form branches. Statistically, any monomer unit of a polymer has to be selected at random to be planted as a root of a tree. These rooted trees appear organized into various genera- tions with the root as zeroth generation, the units linked to the root being in the first generation etc.(see fig. 1). As the polymers are synthesized by a chain reaction, and propagation always progresses from a point of initiation to a point of termination, the two functionalities to the left and to the right in fig. 1 are not statistically equiva- lent. ' \@/ r Im 1 \@h I r 2 1 0 5 4 3 2 I 0 gene r at i on FIG. 1.-Tree representation of a branched molecule ; 1, r, m denote the type of functionality. The arrows indicate the direction of propagation; -0- denotes a point of chain coupling. The graphs above show the three types of links for units in the first and all further generation. For the sake of brevity, the branch bearing the initiator will be called the left branch and the other one the right branch. The third functionality initiates a branch when the side group is converted to a radical.This branch leads to a termination point (right branch) ; but the side group radical can be also terminated by recombina- tion with a growing chain radical. In this case the branch leads to an initiation point (left branch).58 CONFORMATIONAL S T U D Y OF BRANCHED VINYLPOLYMERS Before use can be made of the cascade theory, it is necessary to set up generating functions for the link probabilities. These link probabilities are related to prob- abilities of reaction, a, t, q, w, y defined below, referred to a specific monomer conversion say p. These reaction probabilities are accessible from the kinetic scheme of polymerization 2-details of these calculations will be published separately. a is the total number of propagation steps with monomer and terminal double bonds divided by all reaction steps which a growing chain radical can take part in. t is the fraction of propagation steps with monomers only.q is the total number of initiation rezctions caused by a macromolecular initiator (a side group of a polymer) divided by the total number of initiations. p is the total number of radical recombinations divided by the sum of recombination and transfer steps. w is the total number of side chain radicals formed during the polymerization divided by the number of monomer units built in the polymers. y is the total number of reactions with monomer of all side group radicals divided by all reaction steps of these radicals. The six possible link probabilities are defined as follows : at, probability of a link between two monomer units ; a(1- t ) , probability of a link between a monomer unit and a terminal double bond ; (1 -a)q, probability of a link between a macromolecular initiator and a monomer ; (1 -alp, probability of coupling of two growing radicals ; WY, probability of a link formed by a side group radical with a monomer ; w(1 -y), probability of coupling of a side group radical with a chain radical.The probability generating functions for the three functionalities of a monomer unit are now readily set up (see, for instance, Feller 16) : where s = (SM, s1, sr). Three auxiliary variables are used in these equations : sM denotes reaction of the functionality with a macromolecular initiator or with a chain terminal double bond, s1 denotes reaction of a functionality with a repeat unit from the left branch, s, denotes reaction with a repeat unit from the right branch.three functionality generating functions of eqn. (1) The link probability generating function (zeroth generation) is the product of the &(s) = F,(s)m)~m(s). (2) In the first and all further generations, three types of generating functions have to be distinguished depending on whether the left, the right or the middle functionality is used for a link to the preceding generation (see fig. 1). In all three cases, two functionalities are available for further reactions with units in the next higher genera- tion. ThusW. BURCHARD, B . ULLISCH AND C H . WOLF 59 or, in a more compact vector, All properties of interest of a branched and eventually gelled material can now be expressed in terms of these generating functions.Analytic expressions are given below for the number average degree of polymerization, DP,, and for the three quantities obtainable by means of light scattering, viz., the weight average degree of polymer- ization DP,, the z-average of the mean square radius of gyration (S2)>, and of the particle scattering factor Pz(8). Furthermore, a condition for gel formation is derived. Finally, the mass fraction of the non-branched material and the sol fraction of the gelled sample are calculated. NUMBER AVERAGE DEGREE OF POLYMERISATION DP,, Applying general stoichiometric considerations of Stockmayer,' Malcolm and Gordon found l 2 where differentiation with respect to s means differentiation with respect to the com- ponents s,, sl, s,.AVERAGES OBTAINABLE FROM LIGHT SCATTERING MEASUREMENTS The three averages DP,, (S2)>, and P,(O) follow as special cases from the path- weighting generating function which is obtained by a cascade substitution : The exponent #,, is a general function of the distance of a unit in the 12-th generation from the root. Differentiation for a given $,, (see table 1) at s = 1 yields various averages of physical significance, such as DP, = l+r(l-P)-'e, ( 6 ) b2 2DPw 1 D P W (S2>, = -r(I-P)-2e, Pz(0) = -[I+~Z(~-PZ)-'~], (7) where e is a unit vector and r = ((I -a)q+2a(1 - t ) , at+(l -cc)p+w(l -y), cct+wy). (9) (10) (1 -a)q+2a(l -t) cct+(l -a)p P = ( (1 -cc)g+cc(l-t) cct+w(l-y) cc(1 - t ) (1 -a)p+w(l - y ) at+wy60 CONFORMATIONAL STUDY OF BRANCHED VINYLPOLYMERS with Z = exp (- b2h2/6), h = (4n/;l)sin(8/2), R being the wavelength in the medium and 8 the scattering angle. TABLE 1.4 n = 1, Ub(1) = DP,, 4 n = (r,,') = b2n, 4 H = rexp (-h2b2/6)1, Ub(1) = 2DP,(S2),, Ub(1) = DP,P,(@). CONDITION FOR GEL FORMATION Gelation occurs when DP, increases beyond all limits. This happens if II-PI = 0. (14) MASS FRACTION OF THE LINEAR CHAINS Because of the mechanism of a chain transfer reaction, there is always a certain amount of linear material present in the system, and it is of interest to know how large this amount is and how it changes with the monomer consumption. The principle for deriving this mass fraction may be demonstrated with the definition of the weight- generating function.The sum over the weight distribution of molecules can be written as b ( x ) = kljn(X)+CWb(x) = m l i n + m b = 1, (1 5 ) where the indices lin and b refer to the linear and branched fractions respectively. Passing to generating functions one finds where and wb(sb) = c O b ( x > s ~ / ~ wb(x) are the normalized weight generating functions for the linear and the branched mole- cules in the system. Hence, subtracting all terms involving &, from the weight generating function W(s) one finds W(slin), and by setting slin = 1 the mass fraction is obtained : m l i n = W(Slinlsll,== 1- (18) It is now necessary to express sl, s,, S, of eqn (1) in terms of slin and sb defined in eqn (15)-(18). In the two functionality generating functions Fl(s) and Ii,(s) of eqn (l), the variable S, denotes the formation of a branch and is identical to s,, whereas w[(l -y)sl +ysr] in the expression for F&) represents branching of probability w and is therefore equal to wsb.Subtracting now the terms of Sb in each generation of the cascade in eqn (5), one obtains W(slln) and hence the mass fraction mlin (from 18). Normalizing the residual generating functions F1(slin) and Fr(slin) for the two functionalities in (la) and (lb), the probability generating functions for the various generations of the linear molecules and a path-weighting generating function can beW. BURCHARD, B . ULLISCH AND CH. WOLF 61 constructed. Following the prescriptions developed in the previous section, calcula- tion leads to the various measurable parameters, viz., where 1 al+P DPwlin = - + -, 1-a, 1-a1 (S2)z1in = at at a, = a, = (1-a)(l-q)+at' l-a+atg is used as an abbreviation.averages for the branched fraction can be calculated as follows : Having derived mlin and the other averages of the linear fraction, the corresponding DENSITY OF BRANCHING The branching density p, defined as the number of branches per reacted monomer, is easily found from the link probability generating function of eqn (2) by collecting all the coefficients of s b . Multiplying this value by DP,, one finds the average number of branches per macro- p = (1 -a)q+2~(1 - t ) + w . (28) molecule The corresponding number of the branched fraction is N = DP,p. where nb is the number fraction of branched molecules, n b = (1 -??Zlin)DPn/DPn,.SOL FRACTION Beyond the gel point only a part of the material is insoluble and the remaining sol fraction can be extracted from the gel. All properties of the molecules in the sol62 CONFORMATIONAL STUDY OF BRANCHED VINYLPOLYMERS fraction are charzcterized by the extinction probability. This probability indicates whether a branch attached to a repeat unit of the polymer is finite in size. Cascade theory shows that the extinction probability is given by ** 11* 12* l9 where v = (uM, ul, vr). Eqn (31) is not easily tractable analytically but has been solved numerically. v = FlW, (31) The sol fraction is sol = F,(v). (32) Also for the sol fraction, the molecular and conformational averages can be calculated. Again, one has to start with the generating functions for the three functionalities of a repeat unit, eqn (1).Multiplying the components of s with the corresponding com- ponents of v, one selects those monomer units which belong to the sol fraction. Normalizing leads to the following : where and vs = (uMsM, ~ 1 ~ 1 , cryr), F,<s> = P,(s)Pr(s)Ern(s>, (34) @1(s) = (P,(s)pr(s), P ~ ( s ) P ~ ( s > , Fr(s)prn(s))- (35) These and other generating functions to be calculated by cascade substitution as outlined earlier, can be used to determine the required averages. The equations for DP,,, <S2)>, and Pz(8) are almost identical to eqn (6)-(8) with the exception that P has to be replaced by PV and r by rV, where V is a diagonal matrix whose elements are theextinction probabilities. The symbol * indicates properties of the sol fraction.A A A ELASTICALLY EFFECTIVE CHAINS The elastic modulus of rubber-like network is given by 20* 21 E = kTNeldlVA/Mo = kTv,l (36) where Nel is the number of elastically effective chains per monomer unit, d is the density of the polymer, Mo the molecular weight of the monomer unit and NA Arogaolrd's number; vel is the number of elastically effective chains per volume. As was shown by Malcolm and Gordon,12 this number can again be evaluated from the probability generating function. By multiplying each link probability by the components of the probability (1 -v), where v is the set of extinction probabilities as defined in eqn (31), one obtains the probability generating function for an infinitely extended branch. where T~(Y,s) = (1 -YI +YA)(~ -Yr +VrSr)(l -yrn+YmSm), (37) yI = [(I - a)q + a( 1 - t)]( 1 - om) +at( I - P ~ ) , yr = a(l -t)(l - u , ) + ( l -a)p(l -co,)+at(l -L$, Y , = w(l -y)( 1 - UJ + wy(1 - ur).W.BURCHARD, B. ULLISCH AND CH. WOLF The number Nel of the active network chains is then given by NeI = O.S[Tb(s),= 1 - Tb(s),=o- T3S)s=0]- 63 (39) RESULTS AND DISCUSSION Stein experimentally investigated branching of poly(viny1 acetate) due to chain transfer and gave a corresponding theoretical treatment for the derivation if DP, and DP, based on the Bamford-Tompa Using the data of Stein for the transfer constant Ct, = 1.8 x loe4 and the reduced reactivity of 0.8 for the terminal double bond, we checked our theory against Stein's experimental results. The agreement is satisfactory.3 2 I 2 2 n X 1 I \ 0 0.5 1.0 B FIG. 2.-DPw as function of monomer conversion for polyvinylacetate. 0 measurements of Stein," curve (a) calculated from eqn (6) with p =+ 0, (1 - t ) =+= 0 and Ct, = 1.8 x and reduced reactivity for terminal double bonds ; curve (b) calculated with p = 0, (1 - t ) =+= 0 ; curve (c) with p =k 0, A (1 - t ) = 0 ; curve (d) with p = (1 - t ) = 0. Dotted line shows DPw of the sol fraction. Fig. 2 shows DP, versus p, calculated from our probability theory. The full circles represent the measurements of Stein, who carried out his experiments up to a monomer conversion of about 60 %. However, our theory now predicts gel forma- tion at about 67 % monomer conversion. The gel point is essentially determined by three parameters : the chain transfer constant, the amount of incorporated chains with a terminal double bond and by the chain coupling.The influence of these factors is clearly expressed in the link probability generating function by the three probabilities w, a(1- t ) andp. Fig. 2 shows also curves of DP, versus for the special cases where64 CONFORMATLONAL STUDY OP BRANCHED VINYLPOLYMERS (i) no recombination of radicals occurs, p = 0, 1 - t # 0 (curve b) ; (ii) no polymer with terminal double bond is incorporated, 1 - t = 0,p # 0, (curve c) ; (iii) 1 - t = 0, p = 0 simultaneously (curve d). One realizes that recombination has only little influence in this system. This results from the fact that p is small, evidently because of the high amount of transfer reactions. On the contrary, the gel point is drastically shifted to higher conversion, if no chains with terminal double bond are incorporated in the polymer.Finally, gelation does not occur at any conversion, if there is no termination due to radical recombination and if no chains with terminal double bonds are added to the growing chain. In the latter case, only branching and no crosslinking takes place. The broken line in fig. 2 exhibits the decrease of DP, of the sol fraction as gelation pro- gresses. A 3 \ \ \ \ 0 0.5 I. 0 B FIG. 3.-<S2>, as function of /?, for symbols see fig. 2. Fig. 3 shows corresponding behaviour for (S2>, versus conversion. The mass fraction of the sol and that of the linear chains in the system are plotted against monomer consumption in fig. 4. This fraction of linear chains remains appreciable throughout the reaction and amounts still more than 20 % at the gel point.Information on the structure of the material in the pre-gel and post-gel state is obtained from the mean square radius of gyration, from the particle scattering factor and from the number of elastically effective chains. The dependence of (S'), on DP, is shown in fig. 5 for unfractionated samples (curve a) and for the branchedW. BURCHARD, B . ULLISCH AND CH. WOLF 1.0 C 0.5 0 0 I LO I Q5 '2 0 65 FIG. 4.-Variation of the mass fractions of linear molecules and of the sol fraction with monomer conversion. ~ ~ ~ 1 1 0 5 FIG. 5.-Dependence of the mean square radius of gyration on the degree of polymerization ; (a) for unfractionated samples, and (b) for the branched fraction.The dotted line represents behaviour of linear chains with most probable distribution. 57--c66 CONFORMATIONAL STUDY OF BRANCHED VINYLPOLYMERS fraction (curve b). The dotted line represents behaviour of linear chains with most probable distribution in length. The striking feature of the small difference in ( S 2 ) , between the linear and the unfractionated samples can be explained by the presence of 15 - to z s m 5 0 I 0 I0 20 30 40 51 h2<s2>z FIG. &-Plots of Pz(8)-1 versus hz<S2>, for (a) linear chains, and (a’) the branched fraction of the present system. Curves (b’) and (6) show behaviour of star molecules with five homodisperse and polydisperse branches respectively. large amount of linear chains in the system.Comparison of (S2), of the branched fraction with that of the linear chains of same molecular weight shows the familiar decrease, frequently expressed as the g-factor, The g-factor varies slightly between 0.89 at the beginning of the reaction and 0.766 near the gel point. The g-factors for star-molecules with branches of most probable distribution are g = 0.75 for a 3-star molecule, and g = 0.53 for a star with 5.3 branches on the average.24* 2 5 These functionalities correspond respectively to the mean number of branches per molecule in the branched fraction at the beginning and near the gel point. The g-values show similar dependence on branching, although the effect is less marked for the chain transfer system. It appears therefore a little unexpected at the first sight that the particle scattering factors show such vastly different behaviour (see fig.6). However, this can be explained as follows. The regular star-molecules are characterized by a pronounced up-turn of the I/P,(O) against h2(S2>, plot, which is considered to be typical for branching. Recently, one of us showed that the up-turn is flattened out by a molecular weight distribution sample.25* 13* l4 Since our system has a much broader molecular weight distribution (Mw/Mn z 50) compared to the star-molecules, this flattening effect is much more pronounced, although a slight up-turn still exists. Concerning the properties of gel, the change in the number of elastically effective chains with conversion is also of great interest. This behaviour for our system is compared with that of a trifunctional randomly branched polycondensate l1 and fig.7 shows a much slower increase in Nel for the present system. This is the consequenceW. BURCHARD, B . ULLISCH AND CH. WOLF 67 of the lower reactivity of the transfer functionality, compared to the other two. A much softer gel results from this effect. We are aware of the limitation of the present theory. Certainly, some complica- tions will occur at large monomer conversion. Some reactions may become diffusion controlled like the recombination of two chain radicals (Trommsdorf effect), or the 0.1 5 8.1 I 2 0.05 0 I. Q 1.25 1.5 BlBc FIG. 7.-Increase of the number of elastically effective chains with reduced conversion fl/& (a) for present system, and (a') for trifunctional randomly branched polycondensates." addition of chain terminal double bonds to growing chain radicals.It will be interesting to study the deviation from this idealized behaviour in an actual system, and this treatment may help to get more insight into the various processes at high conversions. P. J. Flory, J. Amer. Chern. SOC., 1937, 59, 241 ; 1947, 69, 2593. T. G. Fox and S. Gratch, Ann. N. Y. Acad. Sci., 1953, 57, 367. C. H. Bamford and H. Tompa, Trans. Faraday SOC., 1954,50, 1097. D. J. Stein, Makromol. Chem., 1964, 76, 157 ; 1964, 76, 169. W. W. Graessley, H. Mittelhauser and R. Maramba, Makromol. Chem., 1965, 86, 129. P. A. Small, Polynzet-, 1972, 13, 536. R. A. Jackson, P. A. 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