Statistical Theory of Polyfunctional Polymerization
1083
o the Statistical Theory of Polyfunctional Polymerization
. Luby lnsiitute of Chemistry, Slovak Academy of Sciences, 809 33 Brafislava, Czechoslovakia Manuscript Received December 17. 7973)
(Received Ocfober 31. 1972: Revised
Molecular size distribution in multichain polymers formed by polyfunctional homopolymerization of a single kind of repeat unit is calculated theoretically. An expression was derived for the weight fraction distribution of isomers of certain composition of units defined by a vector u. Molecular size distribution equations were derived explicitly as functions of link distribution. A formula was deduced for the link distribution, more generally than it was done for a linear substitution effect.
1. Introduction In the last decade the classical statistical theory1 of the molecular size distribution in polyfunctional polycondensation of a single kind of repeating unit of functionality f has been considerably developed. A variety of theoretical studies2-7 applying viirious conceptions contributed to the state of this knowledge. E.g , Whittle2 shows an interesting method of specification of a polymerization process making use of the theory of stochastic processes. Resides the cyclization effect his conception allows for the neighboring-group effect which means that the reactivity of any functional group depends on how many other groups of the same unit have already reacted. This effect was involved in stochastic graph theory of Matula, ~t a1 , 3 also, who called it the “reorganization heat order.” It was also treated by Gordon’s statistical mechanical approach and he called it the “first shell substitution effect.” Although there is 110 attempt a t a detailed analysis of the above thleortes. special attention is devoted to that of Gordon, et al., which appeared to be the most successful. The latter approach which was verified by experimental evidence5 alloas for the effects of substitution and of intramolecular bonds. The theory of cascade processes based on the so-called probability generating functions (pgf) is a powerful too1 to solving this problem. Various statistical moments, gel point, and sol fraction may be calculated in this way? without the need of pertinent summations. However, a distribution equation expressing directly the fractions of individual xmer species is missing, except for the random ideal ease and a small linear substitution efnstead complicated expansions6 of the pgf and extractions of certain coefficients of the power functions are necessary. An expression evaluating these quantities directly is needed, For this purpose link distribution functions (ldf, see section 2 ) were applied in the present paper without the pgf. As i s obvious the use of ldf enables a clear separation of the combinatorial problem from thermodynamical and kinetical This fact is used in this work also and the dbstribution equations (17, 18, 21) thus derived are applicable both to the thermodynamically and kinetically controlled reaction of homopolymerization of a single kind of repeating unit of functionality f . The conversion degree a given by the fraction of reacted groups in the system is taken LIS an independent variable throughout this paper.
Restriction is, however, made to the case, when the influence of any reacted group is limited to the reactivities of groups of the same unit. This effect will be understood when using the term substitution effect (se) hereafter. The problem of higher shell se (these are by far less efficient) as well as intramolecular condensation in the sol phase is beyond the scope of this paper. Calculations related to molecular distribution are not possible without approximations3 in these cases. 2. Link Distribution Function All units of the reaction mixture may be divided into f 1 types U L ,where i = 0, 1 . . . f denotes the number of functional groups of a monomer unit o f f that have reacted. The number fractions p , of these types of units form the link distribution and are equal to the probability that . Idf in the therrnodya randomly chosen unit is an U L The namically controlled reversible reaction is briefly mentioned in this section. The formula ( 5 ) is derived for the ldf in a kinetically controlled (irreversible) reaction with somewhat higher generality than it was done for a linear se (eq 8). When the reaction is reversible. relation b follows directly from the theory of Gordon, et d 4
+
f
Pi = P I C P L
where
fC, denotes a combinatorial number with the parameters f, i. y is a conversion parameter given as an implicit function of a f
a
f
= CiPJCP, 0
(3)
0
iLm is an additional term to the standard free energy of forming a bond between two units which is responsible for the se of m already reacted groups of the unit under consideration. The se is fully determined by f - 1 of these parameters while $ 0 = $-I =- iLf = 0. The problem of an irreversible (kinetically controlled) reaction was described by the same authors4 for the socalled linear se. Their treatment may easily be generalized to the case of an ordinary se when modifying the kinetic rate equations in the following way. The Journal of Physicai Chemistry Vo!. 78. No. 17. 1974
P. Luby
1084
x
- d p , / d t = k [ i : - z)prexp($,/RT) - (f - i 4- l ) p i - , X f
-1
( f - i ) P j exp($,/RT)
:x-:
(4)
0
where formally p - 1 = 0. When dividing (4)by the expression -dpo/dt one obtains f linear differential equations which may be integrated subsequently in order i = 1, 2 etc. up to ,’ = f. General solution of these equations is as follows
where any p, i s defined as
P
=
if -- I.Nem[-($,/RT)li/f
(6)
(analogously p i ?p s , and p m ) . Equation 5 transforms to eq 8 describing a linear se when substitul ing
$) = i R T In N 2 in (6) and therefore
111
(7)
( 5 ) (cf.ref 4,eq 83)
Figure 1. T h e six distinct ordered trees of composition u1 = 3, u2 = 2, u g = 1 , rooted on a terminal vertex. Their number is given n6,,(3,2,1) = 4 ! / 2 ! 2 ! . Two distinct isomorphic classes are
arranged in the upper and the bottom row separately.
?1=O
mit,
N is a constant detwmining the character of a linear se f equations of tne type ( 5 ) together with the trivial relation (9) form a complete system withf + 1variables. f
ip, = .if I1
(9)
3. Derivation ofthe Weight Fraction Distribution Equation The first step in deducing the distribution equation from the known ldf is to find the weight fraction w,(u) of those xmers nhich have the same composition u. A vectorial notation (Clarendon type) is used to designate this composition u = (upu2... u , ) This means that an xmer of the type X u consists of u1,u2 . . uf units of the types U1, U2 . . . U T ,respectively, where L L ~ u2 . uf K > 1. The idea of classification of isomers according to their u vector is not new. It was used by many other authors who solved problems more or less similar to that of ours (e g ref 2 and 8). The value of w,(u) is equal to the probability of finding a randomly chosen unit to be a part of X u . If we specify that the randomly chosen unit must be U , then the corresponding probability lcZl(u)will be
-+
+
).
reacted group is attached to some U,. This value is given by the quotient of reacted groups of all Ut's (zp,) and all reacted groups in the system ( Z p L ) ,viz. eq 9,nx,(u) is the number of all possible configurations, in which the remaining part of a given X u may occur with respect to the Ut under consideration (see Appendix). In the terminology used in graph theory, any U, may be called a vertex with i edges. nx,(u) would be the number of distinct ordered trees rooted on a vertex with I. edges and consisting of ul,u2, . . , uf vertices which have one, two, etc., up tofedges, respectively. It should be emphasized that distinct isomorphic classes of molecular trees may refer to certain u (Figure 1).The number n,,(u) should therefore be taken as a cumulative one. Gordons theorem 1 ( c f . ref 7) proved to be a useful tool for enumeration of this quantity which in our case takes the form
The problem is thus reduced to enumeration of the cumulative number of distinct ordered trees rooted on a terminal vertex ( U l ) . The latter obeys an equation (see Appendix) n,,(u)
(x - 2)!/(u1
- 1)!U2!U3!..~U,!
(15) Since the randomly chosen unit may be of any type, the probability w,(u) is given by the summation =
,=1
Substituting
h,
=
ip,/af
By subsequent substitution from (15) into (14), then in (12) and then in (10) and applying the trivial relation (19) we obtain after rearrangement
we obtain after rearrangement where the expression bU denotes
b“ = bluib2uf ...biu’
(13) Here, any b, denotes the probability that an arbitrary The Journal of Physical Chemistry. Voi. 78. No. 11. 1974
where the expression in brackets denotes a combinatorial number x ! / u l ! . . . u f ! . Equation 17 represents an important relation denoting weight fraction of a special group of isomers defined by the vector u.
Statisfical Theory of Polyfunctional Polymerization
1085
I?Z,
5 [X - 2 -
u,(k - I) k=i+l
where2 < i < f - 1.
Figure 2. The four distinct- ordered trees of composition u1 = 4, u 2 = 1, u4 = 1 looted on a terminal vertex. Their number fulfills the expression r 1 ~ , , ~ ( 4 , l , G= , 74) ! / 3 ! .
To obtain an explicit relation for the weight fraction of any zmer as a function of link distribution p L which is more useful practically, one requires f - 2-fold summation of ws(u); uiz , the degrees of freedom of variables u is diminished by Ihe trivial relations 19 and 20
where x 2 2, white w1 := po automatically. For example, the size distribution of a trifunctional system (which appears to be the most spread case) obeys the equation
-c
x!3a mJ b l U 1 b 2 u 2 b ~ 3 .x - I,,=, u,!u,! u3! where u1 = u3 -t- 2, ua := x - 2u3 2, and m3 5 0 . 5 ~- 1,and yx =
+
where the b's are found from eq 11. The upper limits of the summations 18 and 21 may be determined by the assumption that a given xmer must contain the highest possible number of units V I . E.g., m f may be found From the condition (22) that the molecule contains only units Uland UT
mi = x uI
- u1
(22)
may be found by a simple stoichiometric consideration. u , = f m f - 2(mi
- 1)
(23)
Substituting (23)into (22) one obtains
mi 5 (z
- 2)/(f
- 1)
(24)
The inequality sign should be used, because m f must always be an integer. Similarly mr-1 may be found, provided that the molecule&contains units L'1, Ut, and Ur-1 only. It holds in general
Appendix Every xmer molecule defined by a vector u may be represented by a molecular tree having x vertices and x - 1 edges (links). These x - 1 edges are constructed pair wise from 2x - 2 reacted functional groups; we can call them shortly "groups" because the unreacted ones are immaterial from the point of ennumeration of n X l ( u ) .This problem requires a combinatorial approach and it is convenient to consider 2 ( x - 1) groups rather than x - 1 edges. These 2x - 2 groups may be classified as those which participate in a link formation with a terminal vertices, U1 (type A), and those which participate with Uz>l(type B). Each x vertices of a given tree (except that of dimer) must have a t least one group of type B. Therefore only x - 2 groups come into combinatorial consideration whether being a B or A . Provided that the tree is constriieted in a special way so that no equivalent vertices (those having the same number of groups) occur beside terminal ones, the number of distinct ordered trees rooted on UI should refer to the variation number (cf. Figure 21 Nevertheless, a given tree may contain some sets of equivalent vertices and n,l(u) given by eq 26 would include identical configurations. By dividing the right side of (26) by the product of permutation numbers u,! referring to the series of equivalent vertices, we obtain a more general expression (cf. Figure 1) In the case of U1 vertices the factorial number is diminished by one, because the referential Ul is taken as the root that does not contribute to the number of identical configurations given by (26). References and Notes (1) P J Flow "Principles of Polymer Chemistry, Cornell Untverslty Press, Ilhaca, N . Y.,'1953. P. Whittle, Proc, Cambridge Phi/. Soc.. 61, 475 (1965); Proc. Roy. Soc.. Ser. A . 285,501 (1965). D. W . Matula, L.. C. D. Groenweghe, and J . R. Van 'Wazer, J. Chem. Phys.. 41, 3105 (1964). M . Gordon and 6. R. Scantiebury, Trans. Faraday soc.. 60, 604 (1964). M . Gordon and G. R. Scantlebury, J. Chem Soc.. 395 (1967); M . Gordon and K. K:ajiwara, Piaste K a u f . . 19, 245 (1972) I. J. Good, Proc. Roy. Soc.. 272, 54 (1963). M , Gordon, T. G. Parker, and W. B. Temple. J. Comb. Theory. 11, 142 (1971). I . J, Good, Proc. Cambridge Phii. Soc.. 5 1 , 240 (1955); W. I+. Stockmayer.J. Chem. Phys , 12, 125 (1944)
The Journal of Physicai Chemisfry. Vol. 78 No. 7 1 1974
