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9,No. Vol. 2, March-April 1976 (6) J. Preston, U S . Patent 3 225 011, assigned to the Monsanto Co. (7) G. Losse and H. Weddige, Justus Liebigs Ann. Chem., 636, 144 (1960). (8) German Patent 832 891; Chem. Abstr., 47,3342 g (1953). (9) H. S. Fry, J . Am. Chem. SOC., 35,1539 (1913). (10) M. T. Dangyan, Izu. Arm. Fil. Akad. Nauk SSSR, 3 (1944); Chem. Abstr, 40,3410 (1946). (11) Molecular Weights of Nonlinear Polymers 199 J.I. Jones, W. Kynaston, and J. L. Hales, J . Chem. SOC., (1957). 614 (12) A. Fry, J . Am. Chem. SOC., 75,2686 (1953). (13) A 56% yield of benzanilide was prepared by the author from phenyl isothiocyanate and benzoic acid in nitrobenzene with pyridine catalyst. Surprisingly, this reaction was not found during a cursory literature survey. A New Derivation of Average Molecular Weights of Nonlinear Polymers Christopher W. Macosko*l* and Douglas R. Millerlb Department of Chemical Engineering a n d Materials Science, Uniuersity of Minnesota, Minneapolis, Minnesota 55455, and the Department of Statistics, University of Missouri, Columbia, Missouri 65201. Received March 25,1975 ABSTRACT: A new method for calculating average molecular weights is presented for nonlinear polymers. In contrast t o the previous methods of Flory and Stockmayer which first calculate the distribution of all species and then use the distributions to calculate average properties, the new method calculates these properties directly. In contrast to the method of Gordon, probability generating functions are not required. Starting with elementary probability and utilizing the recursive nature of network polymers, property relations can be developed more simply. We illustrate the method for calculations of R,, and the gel point for a wide variety of polyfunctional polymerizaR,, tions. Flory2 and Stockmayer3 laid out the basic relations between extent of reaction and resulting structure in nonlinear polymerizations. Starting with the assumptions of equal reactivity of functional groups and no intramolecular reactions, they used combinatorial arguments to derive expressions for the size distribution of the finite molecules as a function of reaction extent. For cases of practical importance these distribution functions become quite complex (e.g., Stockmayer4). At present, experimentally we can only measure average molecular weights of nonlinear polymers. I t is possible, though algebraically very tedious, to calculate these averages from the distribution functions. General treatment of nonidealities such as intramolecular reactions or ring formation using distribution functions appears to be prohibitive. Gordon5 showed that the molecular weight averages could be calculated directly using the theory of stochastic branching processes.6 He and coworkers have used this theory extensively on nonlinear polymer p r o b l e m ~ . ~Gor-l~ don's technique involves abstract mathematics and requires deriving probability generating functions. The method is quite general but rather difficult to use. Our interest in developing mathematical models for network polymer processing motivated us to seek simpler relations which could readily be incorporated into a larger, complex process model. Below we describe a new, simpler method for deriving average properties of nonlinear polymers. We retain Flory's three simplifying assumptions, namely: (1) all functional groups of the same type are equally reactive; (2) all groups react independently of one another; (3) no intramolecular reactions occur in finite species. (Some departures from these assumptions, such as unequal reactivity and substitution effects and some aspects of intramolecular loops, can also be treated with our method.") Our method uses the recursive nature of the branching process .and an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a ran- dom variable, E ( Y ) its expectation (or average value), and E ( Y A ) its conditional expectation given the event A has occurred. Then the law of total probability for expectations is E ( Y ) = E ( Y ] A ) P ( A )+ E ( Y A ) P ( A ) (1) This law is discussed in most texts on probability theory.12 Stepwise Polymerizations Mwfor Homopolymers. Let us first illustrate the method with some simple examples. Consider the simplest case, the one treated by Flory,2 the reaction between similar f functional molecules. An example is the etherification of pentaerythritol: OH I HO-CH,-C-CH?-OH I I CH, + branched polyether + H,O OH First we will only consider stepwise or condensation polymerizations, ignoring the effects of any condensation products for the moment. A later section treats chainwise or addition polymerizations. We can schematically represent the polymerization of Af moles of monomer bearing f groups by 1- 1 200 Macosko, Miller Let the system react until some fraction p of the A's have reacted, where P = (A - &)/A Macromolecules (3) Here and throughout this paper A (or Af) represents the initial moles of A type groups and At equals the moles after some reaction time. Pick an A group a t random, labeled as A' above. What is the weight, WAoUt,attached to A' looking out from its par, ent molecule, in the direction 1 ? Since A' is chosen a t random, W A Jis ~ ~ ~ a random variable. W A ,equals ~ if A' has not reacted. If A' has reacted ~~0 (with A", say) then W A , O equals W A , , ~the weight at~~ ~, into tached to A" looking along L,, A"'s parent molecule. W A N= By equation 1, 1 3 Y 1 if A' does not react if A' does react (with A") 1 0 6 i I (4) 1 5 r4 E ( W A O ~ ~E ( WAO'YA reacts)P(A reacts) + =) E ( WAO~YA does not react)P(A does not react) = E(WA'")p + o(1 - p ) = pE(WAin) ( 5 ) 3 1 5 - 2 I zN 1 1 I I I 1 1 4 E ( WA'"),the expected weight on any A looking into its parent molecule, will be the molecular weight of Af plus the sum of the expected weights on each of the remaining f - 1 arms which is just E ( for each arm. Thus 0 I 2 3 p, EXTENT OF REACTION E ( W A ' ~= MA^ + (f - 1)E(WAO~') ) (6) Figure 1. Calculated average molecular weights for the polyether network formation by the stepwise homopolymerization of penta? erythritol (eq 2). The effect of condensation products on h, and M , is shown (see Appendix, eq A7 and A20). and the repetitive nature of this simple branched molecule leads us back to the starting situation. The molecular weight, W A ~of the entire molecule to , which a randomly chosen Af belongs, will just be the weight attached to one of its arms looking in both directions (in and o u t ) WAf = WAin WAoUt (7) Af moles of f functional A-type monomer reacting with Bz moles of bifunctional B type shown schematically: + and thus the average molecular weight attached to a random Af will be nw = E ( WAf) = E ( Wain) f E ( WAoUt) (8) This is the weight average molecular weight because picking an Af or an A group a t random corresponds to picking a unit of mass and then finding the expected weight of the molecule of which it is a part. Flory,2bp 293, comments on this. Solving eq 5 and 6 and substituting into eq 8 yields - $+ B+B A41 or .. (9) Let the system polymerize until some fraction P A of the A groups and some fraction PB of the B's have reacted. If A + B is the only type of reaction then these are not independent and PAfAfO = PB2BZ0 (11) or (12) which is in agreement with Flory's result derived by the much longer process involving the size distributions.2 I t should be noted above that solutions to eq 5 and 6 exist ) p only when 1 > p(f - 1 . If 1 I ( f - l),then the weight average molecular weight diverges and the system forms a gel or infinite network. The tetrafunctional polyether network of eq 2 is used to illustrate eq 9 in Figure 1. If a condensation product is involved, as in eq 2 above, we need to subtract out M c , the molecular weight of the condensate. In polyetherification M c = 18. The effect bf condensation products is shown in Figure 1 and described in more detail in the Appendix. 1T5, for Copolymers. M o s t stepwise polymerizations involve two reactive groups, for example urethane formation from pentaerythritol and 1,6-hexanediisocyanate. Consider Again let us pick an A at random, A' above, and ask what , ~ is the weight W A , in direction 1 ? Here W A , will ~ ~ ~~~ equal zero if A' has not reacted. If A' has reacted (with B', say) it e uals Wglin,the weight attached to B' looking in direction-9 . if A' does not react W*,OUt = (13) Wgfin if A' does react (with B') As in eq 5 , the law of total probability for expectations (eq 1) now implies E ( WAoUt)= PE( WBin) (14) lo In a similar way as before we can write expected weights following the arrows in eq 10 until the recursive nature of Vol. 9, No. 2, March-April 1976 the structure brings us back to eq 14 Molecular Weights of Nonlinear Polymers 201 E ( WB~*) M B ~ E ( W B ' ~ ~ ) = E ( WBO"~)= r p E ( W A ~ " ) + (15) (16) (17) E ( W*OUt) = E ( WAOUYAdoes not react)P(A does not react) + 1 E ( WAOUYAreacts with B,)P(A reacts with BgJ)= j=1 E(W A ~ ~ ) + (f - 1)E( = MA^ W*OUt) Let War be the total molecular weight of the molecule to be which a randomly chosen Af belongs. Let W B ~ the weight for a randomly chosen B2. Then E ( W A ~ )E ( W A ~+~E( W A " ~,~ ) = ) E ( WBJ = E ( WB~") E(W B O ~ ~ ) (18) where b, = mole fraction of all B's on B,, molecules + (19) In direction 4 there will be a relation for each B,. They will all be of the same form as eq 15. E(WB,'~) = MB, Solving eq 14 through 17 and substituting into eq 18 and 19 gives + (gJ - l)E(WBoUt) (27) The expected weights along 2 and , larly to 25 and 27. T o find the weight average molecular weight, we pick a unit of mass a t random and compute the expected weight of the molecule of which it is a part (another application of eq 1) are derived simi(28) (29) E(WB"'~) = pBZiaftE(WAf,'") E(WAr,ln) = MAf, + (fl - l)E(WAO'~) where af, = mole fraction of all A's on AfL ii-i, where = ~ A , E ( W A+ WB&(WBJ J (22) WAf MAfAf MAfAf+ M e 9 2 b'B2 = 1 - W A f = flAf, af, = .zflAfl and p~ is related to PA as in eq 11 (30) Solving this system of equations we obtain Substituting into eq 22 yields IT&, = (2r/f)(l f rp2)MAf2 (1 + (f - l)rp2))MB,2f ~ ~ P M A P B , + (2rM~,/f M B , ) (- r(f - 1 ) ~ ~ ) + ~ (23) which agrees with S t ~ c k m a y e r Treatment of this system .~ to include a condensation by-product is discussed in the Appendix. If some of the starting species are oligomers, as is often the case, with a distribution of molecular weight, Ziegel, Fogiel, and Pariser have discussed which average molecular weights should be used in eq 23.13 In practice Af is often mixed with A2 to control the chain length between branch points and small amounts of A1 can be present as impurities. I t is not hard to write the general system of equations for Af,'s reacting with B,'s. Following Stockmayer's notation4 this can be represented schematically where (33) (34) If W A ~ , the weight of the molecule to which a random Af, is belongs, similarly for WB,, then from eq 18, 19, 27, and 29 it follows that E(WAf,) = MAftif1E(WAOUt) E ( W B ~= MB, ) (35) (36) + g]E(WBoUt) As before, to find Mw,we take a unit of mass a t random and compute the expected weight of the molecule to which it belongs (again using eq 1): - Mw = ZIWAfcE(WAf,) Z,~B,E(WB,) + (37) BJ3 '94 where A%: ++ ! A 12 The expected weight along f is the same form as before but now we must consider all the possible B,'s with which the A can react. Thus by a generalization12 of eq 1 202 Macosko, Miller Macromolecules Consider the stepwise homopolymerization of Ais, eq 2. For this simple case where is the weight attached to the ith branch. = Thus by eq 44 and Var( WA,ioUt) Var( WAoUt) Var( WAJ = f Var( WAoUt) which by using eq 43 becomes E(WA:) Similarly, WA~ = and Var( W ~ i n ) (f = which, using eq 43, becomes Figure 2. flw extent of reaction for several common stepwise vs. copolymerizations calculated from eq 39, PA = p~ = p . (46) -E(WAf)2 = f[E((WAout)2) -E(WAoUt)2] (47) MA^ + ,E W~,io t f-1 r=l (48) - 1)Var( W~out) (49) E((WAi )2) E(WAin) = (f - ~ ) [ E ( ( W A ~ ) E ) W A ~ ~ ) ~ ] -~( (50) Similar to the development of eq 5 we can use eq 1 for the random variable ( W A ~ ~ ~ ) ~ where ma = %MAf,Af,/zLfrAf,= ZrMA,af,/fr m, = ZrMAf,2Af,/Zrf,Af, ZrMAf, af,/f1 = (40) = E((WAoUt)2) pE((WA ) ) (51) Solving eq 5, 6, 50, and 51 simultaneously yields and mb and mb are analogous for the BgJ s. In eq 39 we have obtained a general relation which covers nearly all nonlinear stepwise polymerization. Equations 9 and 23 are just special cases of eq 39. This result is the same as that which Stockmayer obtained by tortuous combinatorial arguments and manipulation of distribution function^.^^^ Equation 39 should also be obtainable in principle from Gordon s eq 67 in ref 5. Figure 2 shows plotted vs. p from eq 39 for the most commonly encountered stepwise copolymerizations. The diverges is called the gel point. This value of p a t which asymptote is indicated on Figure 2. From eq 39 we see that for the general case becomes infinite when Substituting eq 52,8, and 9 into 47 gives Thus for Af polymerization a, a, a, , Our method can be further extended to problems involving A and B groups on the same molecule. This problem does not appear to have been solved in the literature; however, such systems are rare in practice. Hz. Higher molecular weight averages are not readily measured. In principle it is possible to determine these higher averages from ultracentrifuge data. We recall that is the ratio of the second moment of the weight distribution to the first (Flory,2b p 307) or We can calculate E ( W ) from the variance of W using our recursive method. Recall from probability theory that, for a random variable X, Var(X) = E ( X 2 ) - E(X)2 Also if X and Yare independent random variables then Var(X (43) + Y) = Var(X) + Var( Y) (44) which agrees with the result obtained by Gordon (ref 5 , eq 30). Equation 54 is illustrated in Figure 1 for the polyether network of eq 2. One can use the same approach that leads to eq 54 to treat the general case of Af, s reacting with BgJ s. At this point it may be appropriate to discuss how we are using the concept of randomness in our models and derivations. Actually there are three different instances where this arises. (1) Functional units react with each other at random . By this we mean, for example, that any unreacted A has an equal chance of reacting with any of the unreacted B s. Thus if b, is the proportion of B groups residing on B , molecules, A will have probability b , of reacting with a B , . (2) In computing weight average molecular weight we take a unit of mass at random , i.e., all units of mass have an equal chance of being chosen. Thus if proportion W A of ~ ~ the mass consists of Af, s, we will have probability W A ~ of , picking an Af, when we pick a unit of mass at random . (3) Finally we could pick a molecule at random , i.e., all molecules have an equal chance of selection. If we then look a t the expected weight of the randomly selected molecule, we will be computing the number average molecular weight. We have not been able to use this approach to calculate Mn. However, as Flory and Stockmayer point out, without intramolecular reactions R, can always be calculated from stoichiometry. At extent of reaction P A , is just the total mass, mt, over the number of molecules present, N . N mn. nn Vol. 9, No. 2, March-April 1976 is just the number of molecules present initially, No, less the number of new bonds formed, Nb. Molecular Weights of Nonlinear Polymers 203 an m J(No - N b ) = Considering the general case shown in eq 24 then mt = ZiMA,Af, + ZJMB,B, (55) A A (56) Figure 3. Schematic representation of cross-linking of polymer and chains. which agrees with Flory s result for the gel point (ref 2b, eq 13). The same substitutions can be made into eq 59 to give the number average degree of polymerization xn while Thus = X n d ( 1 - PfnJ (66) where P A was defined in eq 3. Note that in the special case of Af, s reacting only themselves, two A s are involved in each bond and P A above is replaced by p.412. R, vs. p from eq 59 is illustrated in Figure 1 for the tetrafunctional polyether network of eq 2. The effect of condensate is also shown. Note that is finite a t pgei. Equation 64 and 66 are useful, simple results which are applicable to cross-linking of chains of arbitrary initial molecular weight distribution provided they obey eq 60. w, Cross-Linking of Polymer Chains Network polymers are often formed by reacting together side groups on long polymer chains or through unsaturation in the chain backbone. Such cross-linking or vulcanization is shown schematically in Figure 3. As Flory2 points out random cross-linking can be treated with the same equations developed for stepwise reactions. If all the reactive groups are of the same type then we have a polymerization of a mixture of species, where Aft, f i may be quite large. If the functionalities are uniformly distributed along the chains then MA^^ = fiMc (60) where M , is the weight between cross-linkable sites and each chain end weight is assumed to be M,lz, We also assume that the cross-linking reaction occurs by simple coupling of chains with no weight change. For the polymerization of a mixture of Aft eq 39 becomes Mw=a+ - mr pAMa2 ma ma[l - PA(fe - 1 1 1 (61) Chain Addition Up to this point we have considered networks formed by stepwise polymerizations. With a few modifications the approach described above can readily be applied to networks formed by chain addition. Chainwise polymerizations involve an initiation step followed typically by hundreds of addition or propagation steps ending with termination. Thus a t any time the reaction mixture will consist of unreacted monomer and rather long polymer with essentially no species of intermediate size. To treat chainwise systems we need to define q , the probability that an initiated or growing chain will add one more unit. In kinetic terms R, (67) 9= Rp iRtr + Rtd + Rtc where R, is the propagation rate, Rt, the rate of termination by transfer, Rtd the rate of termination by disproportionation, and Rt, the rate of termination by combination. Once the reaction conditions and the type and concentration of reactants is fixed, q will be fixed and it will be approximately constant, on the order of 0.99 to 0.999, throughout the reaction. Consider the reaction of a vinyl with a divinyl such as methyl methacrylate with ethylene dimethacrylate: HC-C(CH3j Let P A = p be the extent of cross-linking or the fraction of repeat units which are cross-linked. Substituting eq 60 into eq 34 and 40 for Ma, ma,and ma/ gives (62) Combining eq 33 and 60 shows that - 1 COOCH + HC-qCHJ -1 COOCH, COOCH, - I H,C=C(CH,) I -CH2-C(CH3)- I COOCH, I COOCH, CH,-C(CH, COOCH, I j-CH2- (68) where fwo the weight average degree of polymerization of is the initial mixture of long chains.I4 Substituting gives -CH,-C(CH,j-CH,-C(CH,)-CH,COOCH, I I Gordon5 in attempting to derive this relation started with the Af relation (our eq 9) rather than the proper one for a mixture of species, Af,. We see that our result takes the proper limit at p = 0 and will diverge at If the system obeys Flory s simplifying assumptions, then we can represent it schematically as A2 reacting with Af: A-A + A-A ATA A-A 1 - --A-AA-AA-AA-AA- -AA-AA-AA-AA- ATA (69) I 204 Macosko, Miller Macromolecules As with &fw, is based only on the long-chain portion of the reacting system. Thus R, will be the total mass of polymeric material divided by the number of molecules. The number of molecules is half the number of chain ends less the number of cross-links times half the cross-link order minus 1, Le., Af s which have reacted more than once. The number of A s in the polymeric portion equals 2pAz f ( 1 - (1 - p)f/2)Af while the number of activated A s is 2pA2 fpAf. Thus the number of unactivated A s in the polymeric portion is f ( 1 - p - (1 - p)f12)Af.An activated A is a chain end with probability 1 - q f and an unactivated A is a chain end with probability 1. Thus the number of chain ends, Nc,*,in the polymeric portion is Again we let the A groups react to extent p . T o calculate flw consider the weight attached to a randomly chosen we A group looking out, W A O ~ ~ will also be convenient to deIt . the ~ ~ , fine w ~ * ~weight attached to an A group which has been activated, Le., is part of a chain. Thus following our previous approach E ( W A O ~ ~p E ( WA*O J~) (1 - p)O =) and E(WA*OUt) qE(WA*in) (1 - q)E(WT) = a, + (70) (71) + + + Here E(WT) is the expected weight added in the termination step. For termination by transfer a hydrogen atom is typically added; for disproportionation, the average of half of a hydrogen; and for termination by combination, two chains couple. Thus (1- q)E(WT) = Rt, MH where Rtrl = R,,/(R, ly. Since M H = 1 and R terms <0.01, to a good approximation eq 71 becomes + Rtd M~/2+ RtJE(WA*in) ( 7 2 ) + Rt, + Rtd + RtJ; Rtd , Rtc , similar(73) We have derived useful average properties as a function of reaction extent for the stepwise polymerization of a mixture of Af, and B , monomers without condensation products. Specifically eq 39 gives aw, weight average molecthe ular weight, eq 59 gives Un,the number average, and eq 41 , gives pgel,the extent of reaction a t the gel point. a is computed for the Af homopolymerization. Modifications of these equations to treat nonlinear chainwise addition polymerizations and cross-linking are also described. The Appendix discusses the problem of condensation products. From these relations and the nature of the monomer molecules it may also be possible to derive other properties such as thermal and rheological parameters since these are expected to depend on molecular structure. The advantage of our recursive technique over the combinatorial approach of Flory and Stockmayer should be clear. We derive the molecular weight averages directly rather than the complete size distribution and then the averages. Our derivations are thus much simpler and can be generalized further to include chainwise polymerizations, condensation products and other nonidealities, l and post gel properties such as cross-link density.15 Gordon also calculates average properties directly using branching theory. His approach appears to be more powerful than ours since it can be used to derive az+l higher and averages and can be extended to compute the molecular weight averages of the sol fraction after g e l a t i ~ n . ~ J ~ However, Gordon s use of abstractions, such as vectorial probability generating functions, makes formulation of specific equations for the molecular weight average difficult. Furthermore, the complexity of the resulting matrix equations seems to be unnecessary for many network polymerizations. We believe our recursive method can be readily understood and applied by the typical polymer chemist. E ( W A * O ~ ~q E( W A * ~ ~ ) =) with q = q R t i . Following the approach that led to eq 25-29 we obtain + E ( W A * ~ ~(1 - Uf)E(WA2* ) afE(WApin) (74) =) where E ( W A ~ *is the weight attached to a randomly cho~~) sen A* whose parent molecule is an Af, and + E ( W.t,pin) = MA^ + E ( W A * O ~ ~ ) (75) T o determine E ( W A ~ - we note that one A-A pair will be ~) activated but the other f - 2 are not necessarily. Thus E(W,&,) = MA^ + E(WA*OUt) (f - 2)E(WAoUt)(76) + Combining eq 70 with eq 73-76 we obtain aw usually determined only on the polymeric poris tion of the reacting system; the unreacted monomers are separated out. Thus we consider only the expected weight on A* species rather than all A: Bw w ~ ~ * E ( W+ (1* )W A : ) E ( W A ~ * ) (78) = A~ where W A ~ is the weight fraction of polymeric species, con* sisting of Af s Analogues of eq 35 and 36 give E ( WA2*) = E ( W A ~ * ) Maf = MA^ + 2E(WA*OUt) + 2E(WA*OUt) p(f - 2)E(W,4*Out) + where E ( W A ~ * ) the expected weight attached to an Af is chosen a t random from the polymeric portion. Substituting these into eq 78 gives Gw W A ~ * M (1~ WA,*)MA, = +A+ [2 f WA,*P(f - 2)]E(WA*OUt)(79) by the Union Carbide Corporation, the University of Missouri-Columbia Research Council, and the National Science Foundation. Acknowledgment. This work was partially supported The approach used above can readily be generalized as in the stepwise case to a mixture of Afi s, where f i should be even. Since q = constant, the gel point will be the extent of reaction when eq 77 diverges or (80) Appendix. Polymerizations Involving Condensation Products If a condensation product forms during polymerization, such as water in eq 2, the above analysis must be modified to accommodate this phenomenon. Vol. 9, No. 2, March-April 1976 First consider the homopolymerization of Af. Suppose that a condensate C of molecular weight MC i s a by-product of an AA bond. We shall compute the weight average molecular weight for extent of reaction p . We proceed as with eq 4 and 5 except we must account for the loss of weight from the condensate. Molecular Weights of Nonlinear Polymers 205 Equations 27 and 29 remain unchanged. Solving gives E(WA 1 out = P A ( M b - MC) + P A P B ( g e - 1 ) ( M a .- M C ) 1 - P A P B k e l ) ( f e - 1) - (A101 E ( W A O U ~ ) = p [E( W A ~- Mc] ~) Equation 6 remains unchanged: (AI) E(WB 1 out = PB(Ma - Mc) + P A P B ( f e - 1 ) ( M b - M C ) 1 - P A P B k e - l ) ( f e - 1) (All) E ( W A ~= MA[+ (f - 1)E(W A O ~ ~ ) ~) Solving we get Again let Af,n represent an Af, unit with n reacted arms and f l - n unreacted. As in the case of Af's reacting with themselves E ( W A ~ ~ ) - nMc = MA[, Now consider an Af unit which has n reacted arms and f n unreacted; denote such a unit as Afn. Let W A equal the ~ weight of the molecule to which a random Afn belongs. Then Similarly E(WBun) + nZ,bgjE(WBglln) (A121 = MB, - nMc + nZ,af,E(WAftLn) (A131 As before Af,n = ( k ) p A n ( l BgJ" E ( W A ~= ) MA^ - nMc 4- nE( WAin) 643) - p,dfr-"Af, - pB)gJ-nBgJ (A14) Now consider the moles of Af" for n = 0, 1,. . . , f at extent of reaction p. If all A's are equally reactive and there are no substitution effects, then the number of reacted A's on an Af is a binomial random variable with parameters f and p. = (g,')pB"(1 W5) Suppose that whenever an AB bond forms, that A contributes McA to the condensate and B contributes McB, McA McB = Mc. Then an Af," weighs MA[,- n M C A and + Afn = (L)pn(l- p ) f - n A f (A41 W A , , ~= (MA[, nMcA)[AfLn1/mt (ME, - n M c B ) [ B g J " l / m t (A161 We can assume that each reacting A contributes mass Mc/2 to the condensate,'thus an Afn unit has mass MA[n(Mc/2). If we define wn to be the proportion of mass (condensate removed) consisting of Af" units then WBgln = (A171 are the proportion of mass (minus condensate) accounted for by Af," and BgJnunits (for mt, see eq A19). As before If a unit of mass is picked at random, it will be an Afn with probability wnr consequently Rw= n=O 5 E(WAp)Wn (A6) Substituting from eq A2-A5 this becomes It is straightforward, but tedious, to make the substitutions into eq A18. The simplest way to use eq A18 is to program eq 27, 29, and A10-18 on a calculator or small computer then evaluate them for the specific network polycondensation of interest. We should also note that is affected by condensation products through the average weight of a monomer unit. For example with Af homopolymerization the total mass of thesystem, mt, will go from M A f A f o a t p = 0 to (MAf - ( f / 2)Mc)AfO t full conversion. In general eq 56 becomes a an mt = r = l n=O C k 6 (MA,- nMcA)Aftn+ j=1 ns0 Note that if Mc = 0 this agrees with eq 9. Figure 1 shows a comparison of eq A7 and eq 9 for the tetrafunctional polyetherification. Stockmayer4 suggested that networks with condensation products could simply be treated by replacing MA[with Maf - fMcI2 in the relations derived without condensation. This approach is better than ignoring Mc entirely but it neglects the unreacted ends of the molecules and can only be strictly valid as p approaches 1. The second term in eq A7 can be viewed as accounting for these unreacted ends. Gordon's approach5 appears to correctly account for condensation. Now consider the general reaction of Af,'s with B,'s as in eq 26. Equations 25 and 28 must be modified C 1 5 (MB, - nMcB)Bgjn (A191 Thus for Af only The effect on of condensation products, eq A20, is illustrated in Figure 1. an References and Notes (1) (a) University of Minnesota; (b) University of Missouri. (2) (a) P. J. Flory, J . Am. Chem. SOC., 63,3083,3097 (1941); (b) P. J. Flory, "Principles of Polymer Chemistry", Cornell University Press, Ithaca, N.Y., 1953, Chapter 9. (3) W. H. Stockrnayer, J. Chem. Phys., 11,45 (1943); 12,125 (1944). (4) W. H. Stockmayer, J . Polym. Sci., 9 , 6 9 (1952); 11,424 (1953). (5) M. Gordon, Proc. R. SOC. London, Ser. A, 268,240 (1962). (6) T. E. Harris, "The Theory of Branching Processes", Springer-Verlag, West Berlin, 1963, Chapter 1. (7) D. S. Butler, G . N. Malcolm, and M. Gordon, Proc. R. SOC. London, Ser. A, 295,29 (1966). 206 Miller, Macosko (8) M. Gordon and T. G. Parker, Proc. R. SOC. Edinburgh, Sect. A, 69, 13 (1970/71). (9) M. Gordon, T. C. Ward, and R. S. Whitney, "Polymer Networks", A. J. Chompf and S. Newman, Ed., Plenum Press, New York, N.Y., 1971. (IO) C. A. L. Peniche-Covas et al., Faraday Discuss. Chem. SOC., 165 57, (1974). (11) C. W. Macosko and D. R. Miller, to be published. (12) L. Breiman, "Probability and Stochastic Processes: with a View Toward Applications", Houghton-Mifflin, Boston, Mass., 1969, pp 138-144. Macromolecules (13) K. D. Ziegel, A. W. Fogiel, and R. Pariser, Macromolecules, 5 , 95 (1972). (14) For unsaturated homopolymers like polyisoprene or polybutadiene x , , will be the usual weight average degree of polymerization. For copolymers like styrene-butadiene , M c will be the average weight between will butadiene groups and thus Two be a degree of copolymerization and less than usual weight average degree of polymerization. (15) D. R. Miller and C. W. Macosko, Macromolecules, following paper in this issue. London, Ser A, 272,54 (1963). (16) I. J. Good, Proc. R. SOC. A New Derivation of Post Gel Properties of Network Polymers Douglas R. Miller'" and Christopher W. Macosko*lb Department of Statistics, University of Missouri, Columbia, Missouri 65201, and the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455. Receiued March 25,1975 ABSTRACT: A simple recursive method is presented which can be used to derive the probability of a finite or dangling chain in a polymer network. Finite chain probabilities are derived for a variety of networks as a function of type and extent of reaction. From these probabilities useful properties such as sol fraction, cross-link density, and the number of elastically effective network chains can be readily developed. Recently we presented a relatively simple recursive method for calculating molecular weight averages up to the gel point in nonlinear polymerization.2a In this paper we show how a similar recursive method can be used beyond the gel point, particularly to write relations for weight fraction solubles, w s , and cross-link density, X. We again retain Flory's ideal network assumptions:2b (1)all functional groups of the same type are equally reactive; (2) all groups react independently of one another; (3) no intramolecular reactions occur in finite species. We will also use an elementary law of conditional probab i l i t ~ Let A be an event and .& its complement, B any .~ other event, and P(BIA) the conditional probability of B given that A has occurred. Then the law of t o t a l probabilitY P(B) = P(BIA)P(A) Let polymerization procede until some fraction p of the A's have reacted. Pick an A group a t random, A' in eq 2. Now we need to know what is the probability that following L (looking out from the molecule) leads to a finite or dangling chain rather than to the infinite network, i.e., to the walls of the container. Let F A O " ~ be the event that -5 is the start of a f i n i t e chain, then from eq 1 it follows that P(FA""~) = P(FA""YA reacts)P(A reacts) + P(FA""YA does not react)P(A does not react) = P(F*'")p + l(1 - p ) = @(FA'") 1- p + (3) where Fain the event that L in eq 2 is the start of a finite is chain. For A" to lead to a finite chain all of the other arms of Af must be finite. Thus P(FAin) + P(BI.&)P(A) = P(FAoUt)f-l (4) (1) Probability of a Finite Chain Stepwise po~ymerization A ~I.t is most useful to deof termine whether a group selected from the polymerization a t random is part of a finite chain. Consider first the simple reaction between similar f functional monomers. We can schematically represent the stepwise homopolymerization of Af by and, as with the weight average,2athe repetitive nature of this simple branched molecule leads us back to the starting situation. Combining eq 3 and 4 we can solve for FA""^) pP(FAo"t)f-l - P(FAout) - p + 1= 0 (5) or p (~~in) .. A + + A' 41 A"J2 A+A 3 I 1, AA- (2) p 1 We desire roots of eq 5 and 6 between 0 and 1. Note that eq 6 can be rewritten as + ( x ) = x where + ( x ) = ( p x + 1 p ) f - l is the probability generating function4 of a Binomial random variable with parameters f - 1 and p . I t can be shown that our situation is exactly that of a branching process with offspring distribution Binomial(f - 1,p) and that our event of a finite chain corresponds to extinction. The probability of extinction is the unique solution of + ( x ) = x in the interval (0,l) if it exists and 1 o t h e r w i ~ e(The anal.~ ysis in the remainder of this section can be justified in a similar manner.) From eq 4 it follows that eq 5 will have a root in (0,l) if and only if eq 6 does; this will happen when D > (f - l)-' = pgel. Physically when P ( F A O ~ ~= 1 the sys)

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