The critical point for the appearance of an infinitely large product (a gel) can be found by the condition that the weight average molecular weight of the products is divergent.
Mathematically, this is written as
M w= ∞. (3.2)
This gel point uses the definition of a gel based on the connectivity of the system [10,11,12,1]. The gel point is the point at which the reacting systemis geometri-cally percolated by the connected objects. The appearance of a macroscopic object in the products is called gelation, or sol–gel transition. In chemical gels, the transition is irreversible, while in physical gels it is generally thermoreversible.
For the polycondensation of polyfunctional molecules, we can theoretically find the molecular weight distribution of the products, and hence the average molecular weight, as a function of the reactivity of the systemunder the assumption of equal reactivity.
The principle of equal reactivity states that all functional groups of the same species are equivalent, that is, the reactivity of all functional groups on the polymers is the same irrespective of their molecular weight and structure [1]. In other words, there is an intrinsic reactivity of the polycondensation. We shall derive the molecu-lar weight distribution function of nonlinear polymers under the assumption of equal reactivity.
3.2.1 Random branching
Consider the polycondensation of functional monomers of the type R{ABf −1}. The reaction is assumed to take place between the A group and B group only [1]. Nonlinear polymers with a tree structure are formed by reaction. They may have intramolecular cycles, but to find the exact solution we consider only branched tree-type polymers which have no cycles (Figure3.6). These are sometimes called Cayley trees, named after the mathematician who studied tree-type graphs. The approximation under this assumption of no intramolecular cycles is called the tree approximation.
LetNmbe the number ofm-mers (nonlinear polymers consisting of m monomers), and letp be the reactivity of the A groups and q that of the B groups. Then, we have the stoichiometric relationp = (f −1)q. In what follows, we use q as the independent variable, and write it asq = α. It varies in the range 0 ≤ α ≤ 1/(f −1).
An m-mer has a total number m of A groups and (f − 1)m of B groups, among which m − 1 of A groups and m − 1 of B groups are pairwisely reacted. A total of (f − 1)m − (m − 1) = f m − 2m + 1 B groups remain unreacted. Because each m-mer carries only one unreacted A group (A∗in Figure3.6), the probability for an arbitrarily chosen unreacted A group to belong to anm-mer, i.e., the fraction of unreacted A groups in them-mer among the total of unreacted A groups in the system, is given by the number distribution of them-mers
fm≡ Nm/
j
Nj, (3.3)
whereNmis the number ofm-mers produced by reaction. This is given by
fm= ωm αm−1(1−α)f m−2m+1, (3.4)
A*
Fig. 3.6 Tree structure formed by polyfunctional molecules of type AB3carrying one reactive A group (black circles) and three reactive B group (white circles).
whereωm is the number of different ways to form an m-mer from its constituent m monomers. The numberωmcan be found in the following way.
First, we give a sequence of numbers from 1 tom to the total of m monomers. Over-counting by labeling a sequence for the identical molecules will be corrected later. We then choosem−1 of B groups fromthe total f m−m. The number of different ways of choosing themisf m−mCm−1= (f m−m)!/(m−1)!(f m−2m+1)!. We connect them to the A groups without forming cycles. There are(m−1)! ways to do this. Finally, we correct the overcounting by dividing the result bym!. Thus we find
ωm= (f m−m)!
m!(f m−2m+1)!. (3.5)
The number distribution function of them-mers (clusters) is then
fm=1−α
α ωmβm, (3.6)
whereβ is defined by
β = α(1−α)f −2. (3.7)
The physical meaning of β will be detailed later. The first three moments of the distribution (3.6) are calculated in Appendix 3.A.
Because the number of monomer units is reduced by 1 every time a new bond is formed, the total number of clusters is
M ≡
m≥1
Nm= N −(f −1)αN = N[1−(f −1)α]. (3.8)
This is equal to the number of A groups that remain unreacted in the system. From the first few moments shown in Appendix 3.A, we can find
m n= 1
1−(f −1)α (3.9)
for the number average degree of polymerization, and
m w= 1−(f −1)α2
[1−(f −1)α]2 (3.10)
for the weight average degree of polymerization.
Because both averages are divergent atα = α∗≡ 1/(f − 1), this is identified as the gel point. Since the gel point thus found is the point where the reactivity of A groups is 1 (complete reaction), networks are formed in the limiting state where all A groups are reacted. In other words, there is no postgel regime in this system.
3.2.2 Polycondensation
We next consider the condensation reaction of polyfunctional molecules of the type R{Af}. The molecular weight distribution for the special casef = 3 was first studied by Flory [10]. The result was later extended to the general case off by Stockmayer [11]
under the assumption of no intramolecular cycle formation. Their theories are called the classical theory of gelation reaction.
Pregel regime
Letp be the reactivity of A groups, and write it as α (the reason for this will be detailed below). Let N be the total number of monomers in the reacting system. An m-mer contains 2(m−1) reacted groups, and f m−2(m−1) = f m−2m+2 unreacted groups (Figure3.7). The probability for an unreacted A group, which is arbitrarily chosen from f N(1−α) unreacted A groups in the system, to belong to an m-mer is
Pm=[(f −2)m+2]Nm
f N(1−α) . (3.11)
The number of different ways of connecting the remainingm−1 monomers is the same asωmderived in the preceding section, and hence we find
Pm= ωmαm−1(1−α)f m−2m+1. (3.12) Comparing with (3.11), we find
Nm= f N(1−α)2
α ωmβm, (3.13)
where the parameterβ is the same as (3.7). The new number of configurations, ωm≡ (f m−m)!
m!(f m−2m+2)!, (3.14)
has appeared instead ofωm .
A*
Fig. 3.7 Tree structure formed by polyfunctional monomers of the type R{A3}.
Because one monomer is connected every time a new bond is formed in the tree structure, the number of molecules reduces by one. The number of reacted A groups, or equivalently the number of bonds, is (f N)α/2, and the total number of clusters M ≡
m≥1Nmis given by
M = N −(f N)α/2 = N(1−f α/2), (3.15)
fromwhich the number distribution functionfm≡Nm/M of the products takes the form
fm= f (1−α)2
α(1−f α/2)ωmβm. (3.16)
By using the first few moments calculated in Appendix3.A, we find that the number average degree of polymerization is
m n= 1
1−f α/2. (3.17)
Similarly, the weight distribution function wm≡ mNm/
m≥1
mNm, (3.18)
is found to be
wm=f (1−α)2
α mωmβm. (3.19)
Hence the weight average molecular weight is
m w= 1+α
1−(f −1)α. (3.20)
Gel point
Since the weight average molecular weight diverges when the reactivityα reaches
α = 1/(f −1) ≡ α∗, (3.21)
we find that this is the gel point in the tree approximation. The number average remains at a finite value m n= 2(f −1)/(f −2) at the gel point. The solid lines in Figure3.8 belowα∗show these two averages together with thez-average defined by
m z≡
m2wm/
mwm. (3.22)
The explicit formof thez-average is given in Appendix 3.B.
Average MW
Reactivity α m* = 2(f–1)/(f–2)
α* = 1/(f–1)
0 1
(m)z (m)w
(m)n
Fig. 3.8 Number-, weight-, andz-average molecular weights as functions of the reactivity in the gelation reaction of polyfunctional molecules R{Af}.
Postgel regime
After the gel point is passed, the gel part coexists with the sol part in the reacting system.
The reactivityα in each part may in principle be different. Let αSbe the reactivity of the sol part, and letαGbe that of the gel part. The average reactivityα of the entire system should then be given by
α = αS(1−w)+αGw, (3.23)
wherew is the fraction of the A groups that are connected to the gel part, and called the gel fraction. It agrees with the weight fraction of the gel for the monodisperse sys-tem consisting of polyfunctional molecules whose functionality (the number of reactive groups) and molecular weight are uniquely fixed. The sol fraction is 1−w.
If we take the limit of infinite molecular weight in the tree approximation, it is natural to assume that the gel network remains in the tree structure. Because the gel can be regarded as them → ∞ of an m-mer, its reactivity is
αG= lim
m→∞2(m−1)/f m = 2/f ≡ α0. (3.24)
Therefore, in the postgel regime where the average reactivity is larger than the critical gel valueα∗, we see that clusters of finite sizes are connected to the gel in the way such that the reactivity of the sol part stays at a constant valueαS= α∗. The relation (3.23) then gives
w =(f −1)α −1 1−α0
(3.25) for the gel fraction. It rises linearly fromα∗= 1/(f −1), and reaches unity at α0= 2/f . All monomers are connected to the gel before the reaction is completed. The sol part stays at the critical conditionα = α∗. Such a theoretical treatment is first proposed by
Gel Fraction w
α* = 1/ (f–1) α0= 2/f ZS
F S 1
0 1
Fig. 3.9 Weight fraction of the gel part plotted against the reactivity. (Stockmayer’s treatment (S), Flory’s treatment (F), and Ziff–Stell’s treatment (ZS).)
Stockmayer, and is called Stockmayer’s treatment of the postgel regime (the solid line S in Figure3.9).
This theoretical treatment is, however, not a unique way of describing the reaction in the postgel regime. Flory postulated that the reactivity of the sol partαSshould be found by the condition
β = α(1−α)f −2= α(1−α)f −2. (3.26) In other words, for the average reactivityα larger than the critical value α∗, the equation
β ≡ α(1−α)f −2 (3.27)
has two roots, andαSshould be the other rootα(the shadow root) which lies below the critical value. Hence,αS= αis assumed.
The molecular weight distribution function of the sol part is therefore given by replac-ingα by αin (3.19). The weight fraction of the sol in the postgel regime is then calculated to be
m≥1
wm= 1−w =(1−α)2α
(1−α)2α, (3.28)
and hence the gel fraction is
w = 1−(1−α)2α
(1−α)2α. (3.29)
Substituting into (3.23) and solving for the reactivityαGof the gel part, we find αG=α +α−2αα
1−αα . (3.30)
ThisαG takes a value larger than α0= 2/f . Therefore, in Flory’s treatment, cycle formation is allowed in the gel network. The number of independent cycles, or the cycle
w
MW distribution in the Sol
0.4 0.6 0.8 1.0
Reactivity α
Gel Fraction
Fig. 3.10 The molecular weight distribution functionwmand the gel fractionw for polycondensation reaction of trifunctional monomers R{A3} plotted against the reactivity.
rank, of the network is given by
ξ =f
2αG−1. (3.31)
All monomers belong to the gel part only in the limit of complete reactionα = 1 (the solid line F in Figure3.9).
Figure3.10shows the molecular weight distribution function and the gel fraction for the polycondensation of trifunctional monomers by Flory’s treatment plotted against the average reactivity. The gel fractionw rises linearly fromthe gel point α = 0.5 and approaches unity in the limit of α → 1. This result leads to the three curves for the number-, weight-, andz-average in the postgel regime in Figure3.8.
The difference in the two approaches was later clarified froma kinetic point of view by Ziff and Stell [13]. It was shown that Stockmayer’s treatment allows reaction neither between sol and gel, nor between gel and gel. The increase of the gel fraction is only by the cascade growth of the sol clusters to infinity, while in Flory’s treatment sol and gel interact, and reaction within the gel is also allowed. Ziff and Stell proposed a new treatment in which intramolecular reaction of the gel is not allowed but reaction between sol and gel is allowed. Their result on the gel fraction is shown in Figure3.9by the broken line (ZS).
In the classical tree statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of intra- and intermolecular reactions in the gel.
f-functional molecule
f f
1
1
4
4 5
5 7 5
3 2
2 3
Fig. 3.11 Nonlinear polymer of the type m= (1,1,1,2,3,0,1) produced in condensation reaction of polydisperse functional monomers.
3.2.3 Polydisperse functional monomers
This section studies the gelation reaction of polydisperse functional monomers carrying different numbers of functional groups [11]. Let us consider the condensation system in which the number Nf off -functional monomers is given by R{Af}(f = 1,2,...) (Figure3.11). The total number of monomers is
Nf ≡ N, and the total number of functional group is
f Nf≡O. Let us define the distribution function of the functional groups by1
ρf≡ f Nf/
f Nf. (3.32)
This is defined not by the number of monomers but by the functional groups. The number average functionality of the monomers is
fn≡
f Nf/
Nf=
ρf/f−1
, (3.33)
and the weight average is fw≡
f2Nf/
f Nf=
f ρf. (3.34)
To specify the type of products during reaction, we use the index m= (m1,m2, ...).
It indicates that the cluster consists ofmff -functional monomers (Figure3.11). For example, the label of the cluster in Figure3.11is m= (1,1,1,2,3,0,1).
1 We use the symbolρf for the distribution function of the reactants to avoid confusion with the molecular weight distributionwmof the reacted products.
By repeating the similar counting method as in the previous section, we find that the numberN(m) of clusters specified by the type m is given by
N(m) =
at the reactivityα. This result can be easily found by replacing the factors as f N →
f Nf,m →
mf,f m →
f mf, 1/m! →
(ρf)mf/mf! in the monodisperse system(3.13).
Because the weight average molecular weight is +f mf
,
w= fw(1+α)
1−(fw−1)α, (3.36)
the gel point condition is given by
(fw−1)α = 1. (3.37)
Let us consider the special case of the binary mixture off =2 (unbranching monomers) andf (≥ 3) (multifunctional branching monomers). Types of clusters can be specified by the label(m2,mf). To simplify the notation, write m2= l and mf= m. The number Let ρf ≡ ρ be the fraction of the functional groups which belong to the branching monomers. We then have the relationρ2= 1−ρ, and
Nl,m=
The special caseρ = 0 reduces to the linear polymerization, and ρ = 1 reduces to the condensation of f -functional monomers. The molecular distribution (3.39) connects these extreme cases.
The total number of clusters M ≡
Nl,m decreases by one every time a bond is formed, and hence
holds. Because the weight average functionality of the monomers isfw=f ρ+2(1−ρ)=
(f −2)ρ +2, the gel condition (3.37) turns out to be
[(f −2)ρ +1]α = 1. (3.43)
3.2.4Cross-linking of prepolymers
Let us next consider that the primary molecules are polymers. Mixing cross-linkers, exposure toγ -ray radiation, etc., results in the random cross-linking of monomers on the prepolymers [12,1]. We assume that the prepolymers are monodisperse withn repeat units, and the cross-linking process is independent and random. Each monomer on the chain can be regarded as a functional molecule, so that we can fixf = n in the previous studies. The DP of the polymer is assumed to be sufficiently large that we may take the limit ofn → ∞ under the condition that the number of reacted monomers on a chain
αn ≡ γ (3.44)
is kept constant. We may take the limit ofα → 0 with a constant cross-link index γ . By using the approximations
β ≡ α(1−α)n−2γ e−γ
n 1, (3.45)
ωm= (nm−m)!
m!(nm−2m+2)!(nm)m−2
m! (3.46)
in the molecular weight distribution (3.19) for largen, we find that the molecular weight distribution function takes the limiting form
wmmm−1
γ m! (γ e−γ)m 1
mγ(γ e1−γ)m. (3.47) Hence, the gel point is found to be
γ∗= 1, (3.48)
by the condition thatγ e−γ reaches the maximum value as a function ofγ . It turns out that one cross-link on average per chain is sufficient for gelation.
In the postgel regime, we take the similar limit in the relation
m≥1
wm=f (1−α)2
α S1(α) (3.49)
in Flory’s treatment, and find
1−w = γ/γ , w = 1−γ/γ (3.50)
for the sol and gel fractions, whereα(orγ) is the shadow root of the equation β = α(1−α)f −2γ
ne−γ,
for a givenβ. For this shadow root γ, which is smaller than unity, the relation S1(α) = α
f (1−α)2γ n2 holds.