Rheol. Acta 18, 6 6 7 - 672 (1979) © 1979 Dr. Dietrich Steinkopff Verlag, Darmstadt ISSN 0035-4511 / ASTM-Coden: RHEAAK Polish Academy of Sciences, Institute for Fundamental Technological Researeh, Warsaw Influence of the molecular weight distribution of primary macromolecules on the properties of crosslinked polymer systems W. K l o n o w s k i With 1 table (Received December 23, 1976; in revised form October 26, 1978) 1. Introduction mm In this paper the probabilistic theory of crosslinked potymer systems developed by the author in (1) and (2) is generalized to take into account the intluence of the molecular weight distribution of primary macromolecules undergoing a crosslinking process. The presented method is general and may be applied to any primary distribution. The proper averaging of characteristics for a monodisperse system with respect to the number or weight distribution function gives the possibility of finding topological and physical characteristics of a polydisperse system. We are especially interested in the influence of the distribution on the gel point and the fraction of uncrosslinked material. 2. The model Let us assume that we know the number distribution 4~M of primary macromolecules q~M = N f f N , , [1] and ~, #M = 1, [23 M where NM is the number of macromolecules with molecular weight equal to M, N r is the total number of primary macromolecules and the summation is over all molecular weights present in the system. If the mean linear density of reactive groups p (the number of reactive groups per unit of mass) is identical for all macromolecules, then #M is identical with the distribution 4~,ù of reactive groups 538 ~m - -- Nr --- G , , [3] and Z ~,ù = I, [4] where is the number of macromolecules bearing m reactive groups each and m = pM. [6] As in (1) reacted groups may be divided into three main, disjoint classes G, depending on how many free chain ends (r = 0, 1 or 2) the given group contributes to the junction when forming a crosslink with any other group. The number of groups Gr will be denoted by Gr. But now any class G~ is divided into subclasses G,,ù. The subclass G,~ consists of groups G, which in the same time belong to macromolecules bearing m reactive groups, ,i.e. with a molecular mass M. The number of groups of type Gr,~ is denoted by G,m- Of course G, = Z G,m. [7] //1 Let g,m denote the event that a group taken at random from the system is of type Grm. Analogously 9, is the event that the group is of type G,, i.e. that it contributes r free chain ends irrespective of the molecular weight of a macromolecule to which it belongs. We put the probabilities of such events to be equal to the fractions of groups of given types. Any fraction of macromolecules with a given number of reactive groups (or, equivalently, 43* 668 Rheologica Acta, Vol. 18, No. 5 (1979) due to assumptions [5] and [6], with a given molecular weight) may be now treated in the same way as the whole monodisperse system (1). The summation over all m present in the system under consideration with proper statistical weights 4~m (eq. [3]) will then be taken. Let us define probabilities 7c,ù, as follows; Grm 7G" - Z G,ù, (for any m). [8] r The numbers Grmfor any given m can be found if one knows the distribution fqm(«) of groups involved into junctions between primary macromolecules, f~m(Œ) denotes the probability that in a macromolecule bearing m reactive groups, q groups have reacted to form junctions (~ denotes, as previously (1), the fraction of reacted groups in the whole system, i.e. the number of groups involved into all junctions in the system divided by the total number of reactive groups): fqm(Œ) = Nqm(«), and m (for any m), [10] where, Ne,ù is the n u m b e r of primary macromolecules bearing m reactive groups, q of which have formed junctions. The numbers G,m (r = 0,1,2) of the groups of type G,,ù are given by [9] through formulae analogous to [8] - [10] from (1): G2m 1 Im - G,= = N m ~ 2L=, q=2 E ~rm. Nm r [16] [173 Cm = lm/m. If one assumes that [183 «~ = «, i.e. that the probability for a group to react does not depend on the molecular weight M of the macromolecule to which the group belongs, then l,ù=l. m ,, , mo [19] where mo is the number of reactive groups in the macromolecule with number average molecular weight Mo. Il, moreover, the linear density, p, of reactive groups is constant, then from eq. [6] we get 1m = l . - - M [20] Mo From [8], [ 1 1 ] - [ 1 3 ] and [16] one finds ~2m ~ k m [21] lm 2~fq,ù [12] and [15] Orte could also defines probabilities «ù, that a group belonging to a macromolecule bearing m reactive groups has reacted [11] NmA,ù, = m and analogous to [14] one has [93 Nm Zfqm(«) = I q=0 1 [m =q=O '~~ qfqm - ~mq~=oqNqm, q=2 ]m [22] ' and Gom = Nm ~ (q - 2)f~=. [13] q=2 One introduces as previously the global crosslinking density l equal to the mean number of reacted groups per one primary macromolecule 1 1 I= N,EE (~,m= N---7; (~'' ¢ [143 m Z (q - 2) L ~ rCo~ = q=2 [23] Also, it is easy to check using [15] that Z rC,m = 1 [-24] r ??i Crosslinking densities lm for fractions with given molecular weights may be defined as follows, for any m. Introducing global probabilities, p,, (taking into account eqs. [7] and [14]) as follows 669 Klonowski, Influence of the molecular weight distribution of primary macromolecules m P" = P ( g ~ ) - Z G~ - N,l [25] ' The same results can be achieved in a different way. Let us define instead of the 7~rm (eq. [8]), probabilities Prm as follow~: r one sees from [83, [16] and [3] that m The factor (Im/l) occurs in [26] because the probabilities ~rm are counted for any fraction of macromolecules with a given molecular weight, whereas the pr are counted for all population of macromolecules with any m. From [26], using [ 2 1 ] - [ 2 3 3 , one gets immediately 1_ , [273 P2 = --[-~m m f lm Pl = --~-2 ~bm 2f«m ' [283 (q - 2)f«m . [293 q and Po = -T-Z ~b,ù q Using the definition [15] of l,ù and taking into account [19] (which is a consequence of the assumption [18]), it is easy to check that Z 1,ùq~,ù = l. [303 m L(Œ) _ N« _ ~N«m _ Efqm~m Nt Nt m [31] ' and making the summations in [ 2 7 ] - [29] it is easy to show that the pù having exactly the same form as in a monodisperse case (cf. (1) eqs. [15] - [17]). pl = fl l' 2(1 - fo - A) t [32] ' [33], and Po = P(grm) - EŒär m r 2fo + f ~ + 1 - 2 = 1 "Pi -P2. [34] . r . . . Ed r Nt 1 , [35] m i.e. the p~,ù are counted with reference to all reacted groups in the system whereas the "~~m refer only to the reacted groups in the macromolecules which bear m reactive groups. Of eourse [36] Pr = E Prm, m and the Gùm are still given by [ 1 1 ] - [ 1 3 ] , thus, using [35] and [36], one comes again to [27] to [29]. On the other hand the probabilities zc~mfor any given m are identical with the probabilities p, for the monodisperse case. Under the assumption [18], when [20] i s true, eqs.[2O] mean nothing else but an averaging of the ~z,,ù with the weight distribution function of molecular weights, Wu, to get the pù for the whole system, i.e., [37] Pr = E WmTZ,m, in where (cf. eq. [20]) Wm=WM-- N.____~M- ~ M Nt Mo Defining global fractions f«(«) of primary macromolecules bearing q crosslinks irrespective of their molecular weight, M, PŒ- = [26] re,,ùNm lm l,ù Ntl -- 2 ~mrc"---~" P' = Prm M _ ~ m lm [383 Mo -1-" Without evaluating p, in the described way, it is obvious fromtheir definition that one must weight the monodisperse probabilities rcrm with the distribution function Wm (eq. [38]) and not with ~m (eq. [3]) because the number of reactive groups in a macromolecule is proportional to its molecular weight. 3. Topol0gical andphYSicalcharacteristicS The probability, spa¢e, ~, can be defined as in (1) and the fractions of different types ofjun¢tions, n» (i.e. of junctions having k = 0,1,2,3,4 free chain ends; cf. (1)) are given through p~ by the same formulae as in the monodisperse case (eqs. [22] - [26] from (1) with the normalization constant C = 1): no = Po2 , [39] nl = 2poP2, [40] 670 Rheologica A c t a , Vol. 18, N o . 5 (1979) n2 = n(1) + n(z2) = pZ -t- 2pop 2 , 2pip2 , /'13 = [41] [423 and [43] n 4 = p~. It has been shown in (1) that if p, has the form [32] _ [34], the nk fulfil the necessary normalization conditions, 4. Example: Application to the Schuitz distribution To illustrate the influence of molecular weight distribution of prifiaary macromolecules on the characteristics of a crosslinked system, we take as an example the Schultz distribution cb M = 7k+ 1 r(k + 1) M k e -~M , [51] where 4 nk= 1, [44] k=O and 4 4z knk - k=o [45] l where z denotes the fraction of primary macromolecules:bearing at least one crosslink. The only difference is that nowfq and p, are averaged over the molecular weight distribution 4,, of primary macromolecules. As in the monodisperse case one has z 1 T = T pl + P2. [46] One may consider the fractions z m and u,,, i.e. the number fraction of macromolecules bearing at least one crosslink and the weight fraction of primary macromolecules which do not bear any junctions, respectively, (defined for each monodisperse group of primary macromolecules with molecular weight M , i.e. bearing m reactive groups) as follows: z., = 1 - fo,., [47] and u., = fore = 1 - z,.. [48] Then, to get adequate characteristics z and u for the whole system, it is necessary to average [47] - [48], so that z = Z ~.,z,, = 1 - f o , [49] //1 and = Y. Wm",, [50] m where W,, is the weight distribution function [38]. k + 1 = ~ Mo [52] M is the number average molecular weight, AT/., in units of the monomer mass (i,e. more exactly the degree of polymerization). The parameter k ( k ~ ( - 1 , oo)) is a measure of the breadth of the distribution (dispersion), because the ratio of the weight-average molecular weight Mw to ~/, is given by ~ w : ~ , = (k + 2):(k + 1). [53] Thus in the limit of k --, oo one comes again to the monodisperse case. For k = 0 one gets F l o r y ' s distribution. As k ~ ( - 1 ) the distribution becomes broader and broader. For the distributionfq,, (~) we take the P o i s s o n distribution (lm)qe -tin fq,. (~) - q~ [54] Using the Schultz distribution [51] we get from [31] fo and ]'1 (for simplicity the sum is replaced by integral, as the Schultz distribution is a continuous one) and from [ 3 2 ] - [34] we calculate the p,. This then enables to count all the characteristics of the system as in the monodisperse case. We shall discuss the influence of the dispersion of the molecular weight distribution on some of the characteristics, e.g. on the critical value of crosslinking density, l~r, on the numerical fraction of macromolecules bearing at least one crosslink, z, and on the weight fraction of macromolecules bearing no crosslinks, u. The results are compiled in table 1. It is easy to see that one would come to the same results for p, if one averages with respect to the weight distribution function, Wm(eq. [37]), ~rm taken in the form analogous to Pr for a Poisson distribution in the monodisperse case. Klonowski, Influence of the molecular wei9ht distribution of primary macromolecules 671 g + r,i ~ E 0 ,0 r~ I + ~ et t--i g (",1 ee~ 2~ ~ ~ ~ ~ ~ ~ ,~ + ~ + ~ ~ ~ (-4 ;;,., , 0 - + ~ + ~ + ~ + ~ + + T T rq I + I T I I 0 T 0 ~. T , T I , T « + -I- ,N ù.~ + +ù,~ "« + ÷ ù~ ù~ N.,~ ~ I I ]*"v --I-- i l..U , t ù.t'+ + +ù,~ .« ~ + +.« i"l- !~ .:~~g ù - VI t + ~" ~"----. -~~ ~ "-' ù.~ + g.:, ., i + 1t + I I ~ « E~ + I I I I I 672 Rheolo9ica Acta, Vol. 18, No. 5 (1979) 5. Concluding remarks From table 1 is seen that the critical value ler of crosslinking density at the gel point decreases when the molecular weight distribution becomes broader. Because nk is related to Pr as in the monodisperse case ( [ 3 9 ] - [ 4 3 ] ) the critical value Lù of the reduced crosslinking density L (see (2)) must be equal just to 2. This means that the network core is formed from the largest macromolecules whereas macromolecules of lower molecular weight remain uncrosslinked or are engaged in crosslinks which are not geleffective (mainly in the sol fraction but also in the gel). Indeed, from [48] and [54] it is seen that u m decreases when m (and hence molecular weight M) increases. Simultaneously, z,~ increases with m. This means that ~the number and the weight fraction of very l o n g macromolecules which bear no crosslink decrease very quickly with molecular weight, i.e. the longer a macromolecule the more probable that it belongs to the network core. If we consider the dependence of P2m on m, where (from [35] and [11]) Pein -- ~ m flm = ~b,ùe-t~ l [55] it is obvious that the contribution of macromolecules with molecular weight M to the total P2 (which, in turn, influence their contribution to the junctions of type J4, J 3 and J(22) which are not gel-effective, cf. [41]-[-43]) depends on the form of ~,ù. If ~bù, is monotonically decreasing (e.g. [51] for - 1 < k <_ 0) then the longer a macromolecule the more probable that it will be engaged in gel-effective junctions. On the other hand, if ~bm has a maximum then. there may exist a value M m a x that P2,ù has a maximum for M = M m a x. Macromolecules with M > M m a x will form mainly gel-effective junctions (i. e. the network core) whereas macromolecules with M < M m a x will probably bear no junctions (i. e. will form the fraction u). For example taking for ~,ù the Schultz distribution [51] one gets from [55] It is easy to check that for any given 1 the weight fraction u of macromolecules bearing no crosslinks is greater when k is smaller ((~u/~k) t < 0), i.e. when the distribution is broader. On the other hand for any given k, u monotonically decreases more slowly with l the smaller k. In contrast, the fraction z of macromolecules bearing at least one crosslink tends to unity when l --, oe. For a Poisson distribution of f qm(c), all the characteristics listed in table 1 are not dependent on the mean molecular weight but for other distributions, e.g., for the binomial distribution, they may depend on M o. Summary The probabilistic theory of crossllnked polymer systems developed by the author is generalized to take into account the molecular weight distribution of primary macromolecules undergoing a crosslinking process. Formulae are given for the calculation of topological and physical characteristics of the system for known distribution functions. As an example, the Schultz distribution is discussed in detail. It is shown that the critical value of the crosslinking density at the gel point decreases as the molecular weight distribution becomes broader, whereas the critical value of the reduced crosslinking density remains equal to 2. Zusammenfassun9 Die vom Verfasser entwickelte probabilistische Theorie der vernetzten Polymersysteme wird in der Weise verallgemeinert, daß die Molekulargewichtsverteilung der am Vernetzungsprozeß beteiligten Primärmoleküle einbezogen wird. Es werden Formeln zur Berechnung der topologischen und physikalischen Kenngrößen des Systems in Abhängigkeit von der Verteilungsfunktion angegeben. Als Beispiel wird die Schultz-Verteilung im einzelnen diskutiert. Es wird gezeigt, daß der kritische Wert der Vernetzungsdichte am Gelpunkt bei breiter werdender Verteilung abnimmt, wohingegen der kritische Wert der reduzierten Vernetzungsdichte unverändert den Wert 2 behält. References 1) Klonowski, W., Rheol. Acta 18, 442 (1979). 2) Klonowski, W., Bull. C1. Sci., Acad. Roy. Belgique, Ser. 5E, t. LXIV, 568 (1978 -9). 3) Dobson, G. R., M. Gordon, J. Chem. Phys. 43, 705 (1965). k Mma x = /+k+l 'M 0 [56] which, of course, could be true only for k > 0, as was stated above. M m a x depends on the crosslinking density I. Author's address: Dr. ge..Klonowski Polish Academy of Sciences Institute for Fundamental Technological Research PL-00-049 Warsaw (Poland)