3.3.1 Finding the gel point using the branching coefficient
In this section, we study gelation by heteromolecular condensation reaction in binary mixtures. Typical model systems are condensation of f -functional monomers of the type R{Af} andg-functional monomers of the type R{Bg}. They formbinary mixed networks. The mixtures are indicated by R{Af}/R{Bg}. For simplicity, the reaction is limited to only between the A and B functional groups (Figure3.12).
Before detailed study of the molecular weight distribution function, we consider a simple method to find the gel point by using the branching coefficient [1]. The branching coefficientα is defined by the probability that any one of the functional groups on an arbitrarily chosen branching monomer (functional monomer with functionality more than or equal to 3) reaches the next branching monomer of the same species by a connected path (Figure3.13).
Consider that a reacted path reaches a branching monomer R{Af}. In order for the path to extend to infinity without breakage, at least one of the number(f −1) possible directions of the extension must reach the next branching monomer with probability 1 (Figure3.14). Hence the gel condition is given by
(f −1)α = 1. (3.51)
glycerin dicalboxylic acid
A3 B2
3
Fig. 3.12 Esterification of trifunctional monomers R{A3} and bifunctional monomers R{B2}.
A A
1 2
α
Fig. 3.13 Branching coefficientα of a branching monomer R{Af}. The monomer 1 is connected by the next monomer 2 of the same species by a reacted path.
A A
2
3 4
5 1
Fig. 3.14 Gel point condition as seen fromthe existence of a path that continues to infinity.
(g –1)q
(f –1)p
α = p(g–1)q
A B
A
A A
Fig. 3.15 Branching coefficient of the mixture Af/Bg.
The method for finding the gel point by the branching coefficient is very convenient because it does not require information of the molecular weight distribution.
For the binary mixture R{Af}/R{Bg}, the condition (3.51) gives
(f −1)p(g −1)q = 1, (3.52)
since the probability for a pair of R{Af} monomers to be connected by a reaction path isα = p(g −1)q (Figure3.15), wherep and q are the reactivity of the A and B groups, respecitvely.
Let us study the slightly more complex mixtures of R{Af}/R{A2}/R{B2}. R{A2} and R{B2} are nonbranching monomers. The structure of the branched polymers is shown in Figure3.16. Letρ ≡f Nf/(2N2+f Nf) be the fraction of functional A groups on the branching monomers among all A groups in the system. Summing up all the possible reaction paths from one branching monomer R{Af} to the next one (the bottompart of
i = 3 A
A
A
1
2
1
A A2
B2 B2 A2 B2 A2 B2
2
Fig. 3.16 Branching coefficient of the mixture Af/A2/B2.
Figure (3.16), we find the branching coefficient to be
α =
∞ i=0
p{q(1−ρ)p}iqρ = p q ρ
1−p q(1−ρ). (3.53)
The gel point is found by the condition
[(f −2)ρ +1]p q = 1, (3.54)
from(3.51). Because the average functionality of R{A} monomers isfw= 2(1−ρ)+
f ρ = (f −2)ρ +2, the gel point condition is equivalent to
(fw−1)p q = 1. (3.55)
In particular, for stoichiometric mixtures with the same number of A and B groups, the reactivities are equal,p = q, so that α = p2ρ/[1−p2(1−ρ)].
If there are no R{A2} monomers,ρ = 1 and α = p q hold, so that the gel point is (f −1)p q = 1. If there is no R{Af}, we have randomcopolymerization of R{A2} and R{B2}, for whichα = p2. The polymerization point is the point withαc= 1 where all groups are reacted.
3.3.2 Molecular weight distribution function of the binary mixtures R{Af}/R{Bg}
Consider the binary mixture R{Af}/R{Bg}. LetNAbe the number of R{Af} molecules, andNBbe the number of R{Bg} molecules. The number of functional groups in the
systemisOA= f NAandOB= gNBfor each species. The reaction is assumed to take place only between A and B.
Under the assumption of tree statistics (no intramolecular reaction allowed), Stock-mayer [15] found that the number of clusters consisting of l A monomers and m B monomers is given by
λNl,m= (f l −l)!(gm−m)!
l!m!(f l −l −m+1)!(gm−l −m+1)!xlym, (3.56) whereλ is the equilibriumconstant of the reaction of bond formation
A+B A ·B (3.57)
By putting(l,m) = (1,0), or (0,1), parameters x and y turn out to be
x = λf N10,y = λgN01. (3.58)
These are the numbers of A and B monomers that remain unreacted in the system, multiplied by the equilibrium constantλ.
Becausef N10is the number of A monomers on the unreacted monomers, it must be equal toOA(1−p)f by the definition of the reactivityp. Therefore, x can be written as x = λOA(1−p)f. Similarly,y = λOB(1−q)gholds.
Let us express x and y in terms of the reactivity p and q. Because the number of reacted A groupsOAp is the same as the number of reacted B groups OBq, let us write it asγ . This is also equal to the number of bonds formed by reaction. The equilibrium constantλ can be found by the equilibriumcondition in the reaction (3.57) as
λ = OAp
OA(1−p)OB(1−q)= OBq
OA(1−p)OB(1−q). (3.59) We then have the relationλOA= q/(1−p)(1−q), and hence
x = q
(1−p)(1−q)·(1−p)f=q(1−p)f −1
1−q . (3.60)
Similarly, we have
y = p
(1−p)(1−q)·(1−q)g=p(1−q)g−1
1−p . (3.61)
Next, let us express the number of bondsγ in terms of OA,OB, andλ:
γ = λOAOB(1−p)(1−q). (3.62)
Substituting the relationsp = γ /OA,q = γ /OB, we find
γ = λOAOB(1−γ /OA)(1−γ /OB), (3.63)
which is regarded as the equation to findγ . Solving for γ , we find
(The sign is chosen so that the equation holds in the limit ofλ → 0.) The equilibrium numberγ of bonds can thus be found for the given concentration in the preparation stage.
The number of molecules is reduced by one every time a new bond is formed. Hence the total number of molecules (clusters) in the system is
lm
Nl,m=OA
f +OB
g −γ . (3.64)
Fromthe information ofNl,m, we can find the weight average molecular weight by M w≡
lm
(MAl +MBm)2Nl,m/
l,m
(MAl +MBm)Nl,m (3.65)
whereMAandMBare the molecular weights of the monomers. By using the distribution (3.56), the sumturns out to be
M w=
fromthe divergence condition of M w, as is expected fromthe branching coefficient (3.52).
3.3.3 Polydisperse binary mixture R{Af}/R{Bg}
Let us generalize the above results to the polydisperse binary mixture R{Af}/R{Bg} in which functional monomers carry various numbers of functional groups. Let NfA (f = 1,2,...) be the number of f -functional monomers, and NgB(g = 1,2,...) be the number ofg-functional monomers. The total number ofAand B groups are then given by OA=
f NfA,OB=
gNgB. Let us introduce the distribution function of the functional groups asρfA≡ f NfA/OAandρgB≡ gNgB/OB. These are the fractions of the functional groups on the monomers of specified functionalities.
Under the assumption of the tree statistics, Stockmayer [15] generalized the monodis-perse mixtures to polydismonodis-perse ones, and found that the number of clusters consisting of
l≡ (l1,l2,...) A monomers and m ≡ (m1,m2,...) B monomers is given by
wherexf andygare generalizations of (3.60) and (3.61), defined by
xf≡ ρfA q(1−p)f −1
1−q , yg≡ ρBg p(1−q)g−1
1−p . (3.69)
The reaction constantλ is given by
λ ≡ p OA
OA(1−p)OB(1−q)= q OB
OA(1−p)OB(1−q) (3.70) By using the average functionalities of the functional monomers
fw≡
f ρfA, gw≡
gρgB, (3.71)
and the weight average molecular weights M A≡
MfρfA, M B≡
MgρgB, (3.72)
the weight average molecular weight of the products can be found by replacingMA/f , MB/g by M A/f , M B/g in (3.66). The gel point condition is
(fw−1)(gw−1)p q = 1. (3.73)
3.3.4Gels with multiple junctions
Let us consider the limit of the complete reaction of functional B groups in the preceding section (Figure3.17). In such a limit, a reactive B group can be regarded as a cross-linker producing a junction of multiplicityg. The multiplicity of the junction is the number of functional groups combined into it (see Figure3.3). The B molecules act as glues to paste the A groups.
Let us indicate the multiplicity byk as in the convention. We then replace the notations as mg→ jk and ρfA→ ρf, ρgB→ pk in the molecular distribution (3.68) while the notationlf is kept as it is. Here,pkgives the probability for a chosen A group to belong to the junction of multiplicity k. The average functionality of B monomers becomes gw=
kpk, and we write this as ¯µw, where ¯µwshows the average multiplicity of the cross-links.
A
A
A A
A
B A
B
Fig. 3.17 Multiple junction as seen fromcross-linking by glue molecules B.
There are two fundamental geometrical relations which hold for clusters of tree type with multiple junctions. For the total number of monomers in the cluster, the relation
lf=
(k −1)jk+1 (3.74)
holds. For the total number of functional groups, the relation
f lf=
kjk (3.75)
holds.
The combinatorial factor in the molecular weight distribution function (3.68) simplifies as the factorp(1−p)q(1−q) disappears, and the limit q → 1 of complete reaction can be taken. As a result, the molecular weight distribution is transformed to
N(j,l) =
f Nf
jk−1
!
lf−1
!
f
(ρf)lf lf!
k
(pk)jk
jk! , (3.76)
which agrees with the tree statistics with multiple junctions derived by Fukui and Yamabe [16] directly by using the statistical-mechanical method.
The gel point turns out to be given by
(fw−1)( ¯µw−1) = 1, (3.77)
after the replacement of the symbols as above. We will derive these results more effi-ciently in Appendix3.Bby using the probability generating function (p.g.f.) of the cascade theory.
The branching coefficient of this systemwhen regarded as an A/B mixture is found to beα = (gw− 1)pq by generalizing (3.52) to polydisperse systems. Fixingq = 1 in
k
Fig. 3.18 Branching coefficient of multiple cross-linking systems.
the gel point condition (3.73), and replacing(gw−1)p by
(k −1)pk, we find that the branching coefficient for the multiple cross-linking is
α = ¯µw−1. (3.78)
Because there arek −1 paths going out of the k junction, the result can be understood in the form(Figure3.18)
α =
k≥1
(k −1)pk= ¯µw−1. (3.79)
Thus, the gel point condition (3.77) is also derived fromthe branching coefficient method.
Most physical gels have multiple cross-links. They are formed by the association of the particular segments on the polymer chains. Therefore, gels with multiple junctions may be understood more profoundly when they are treated by thermodynamic theory rather than reaction theory. In Section 7.1, we shall present some of the equilibrium thermodynamics of physical gels with multiple junctions.