Theory of Gelation: Post-Gelation Behavior

Within the framework of the random distribution assumption of cyclic bonds, the preceding theory of gelation is extended to mixing systems with various functionalities. To examine the validity of the assumption, the theory is applied to experimental data in polyurethane network formation, the result showing the soundness of the theory for the prediction of gel points and gel fraction.


Introduction
There remain some unsolved problems in polymer science. Of those, the determination of the gel point of branched polymer solutions has been one of the most attracting subjects for more than 60 years. It has been shown that the gel point varies widely from systems to systems, often deviating markedly from the ideal values [1]. It was pointed out earlier that the discrepancy between the theoretical point and observed ones can be ascribed to the neglect of cyclization [2]. Based on the findings of the early researchers, the author has pursued so far the general theory of gelation that includes cyclization effects [3].
In this report, the author will extend the preceding theory to mixing systems. The essence of our approach is based on the following four premises (three principles and one assumption): Three Principles (1) (ii) The total ring concentration, [ ], is independent of the initial monomer concentration, C; it is a function of D (the extent of reaction) alone.
(iii) Branched molecules behave ideally at C .

One Assumption
(iv) Assumption I: Cyclic bonds distribute randomly over all bonds.
The premise (i) is a mathematical theorem. The premise (ii) is based on the following formal solution of [ ]: which is common to all model systems, where v R j denotes the velocity of j-ring formation and v v R R j j , and v L that of intermolecular reaction. Experiments have shown that the relative velocity, v v R L j , is a decreasing function of the initial monomer concentration, C, which implies that eq. (2) is of the form: C t dD 1 1 , a function rapidly approaching a constant with increasing t (t is a representation of the concentration of reactants [4], since, as C , the monomer density becomes infinite and the notion of atomic radii vanishes. In contrast to the above three principles, the premise (iv) (Assumption I) is an approximation.
Assumption I, however, greatly simplifies the problem. It reduces otherwise an inherently insoluble problem of gelation to an elementary mathematical exercise, giving the equality D inter D co .
Eq. (1) then becomes with D co being the Flory point, and p R the fraction of cyclic bonds to all possible bonds and can be equated with D ring [3].
In this report, the author generalizes, within the framework of the above three principles and one assumption, the preceding theory to include mixing systems of multifunctionalities.

Gel Point
Suppose that a reaction system is comprised of a mixture, f M Α , that a branching unit leads to the next branching unit to form a network is equal to the product of the fraction of reacted functional units and the fraction of intermolecular bonds. Hence where P Z X P Z X P X | is the conditional probability that a randomly chosen bond is a cyclic bond. Thus the quantity, 1 P Z X | , represents the probability that a chosen bond is an intermolecular bond.
The gelation occurs at Α 1, and eq. (4) gives with f f w i i i Χ . The first term of eq. (6) represents the ideal gel point and the second term the correction term due to cyclization. Our task is then to find the mathematical expression of p R for this mixing system.
As every ring possesses only one excess bond and J 1 branches arise by the merger of J FU's, p R should have the form: C f denoting the FU concentration defined by C fM V f i i i (V is a system volume), while having the formal solution: Experiments have shown that v v R L j is in inverse proportion to the concentration of reactants, so that as C , the relative frequency of cyclization becomes negligible. Hence, at high concentration, eq. (8) can be approximated in the form: This means that at sufficiently high concentration, can be equated with the limiting solution of C : At C , eq. (8) has an analytic solution of the form [5]: f w being the mean weight functionality as defined above.
Because of the relationship (10), we can expect that eq. (11) is a good estimate of the ring concentration in actual branching processes. Unfortunately, eq. (11) breaks down as soon as the ideal gel point, D J f co w 1 1 1 , is passed, so eq. (11) cannot directly be linked with eqs. (6) and (7). To resolve this problem, we make use of a linear approximation of eq. (8), or equivalently of eq. (11) [3]. Recall that the ring concentration has the formal solution at D c : while by eq. (10), at high concentration is a function of D alone, so one can expand eq. (8') with respect to D co which, now, can be applied to the new territory, D D D co c . Eq. (12) has sound mathematical basis because of the relationship of eq. (10). From eqs. (6), (7) and (12), one has where is the reciprocal of the initial FU concentration. For a single component system,

Post-gelation
We confine ourselves to a mixture of monomers with f 2. Let Q be the probability that a chosen branch emanating from a monomer is finite [6]- [8]. Following the aforementioned Assumption I, Q satisfies the recurrence relation: It is easy to show that eq. (14) has two solutions, Q 1 and The assignment of branching units can be made by the binomial expansion: Ω with being the weight fraction of the ith monomer unit and m i the molecular mass. The first term of eq. (16) represents the weight fraction (w s ) of sol, the second term the fraction (w p ) of pendants, and the higher terms than the third the fraction of active network segments. Hence 7 To calculate the post-gelation behavior, we extend the concept of cyclization in sol phase to gel phase; namely, we apply the foregoing linear approximation shown in eq. (12) to the gel phase and for D D co 1. In this expression, it is assumed that all reactions are smooth and continuous beyond the gel point with no reaction anomaly. Substitution of eq. (19) into eq. (17), with the help of eqs. (7) and (15), gives a solution of w g as functions of D, Κ and Γ f .

Gel Point
Let there be a mixing system comprising of two different types of monomer units, g M i A i and f g M j B j , where M A i and M B j are the mole numbers of the A type and the B type monomers, respectively, and g i and f g j are the corresponding functionalities. Let J be the total number of FU's to form a junction point on which the two types of the FU's are arranged alternately. By the nature of the R A R B g f g model, J must be an even integer and J 1 branches arise as a result of the merger of J 2 A type FU's and J 2B type FU's. Thus, the probability, Α, of branching becomes where s P Z X f g D J BB w B 2 1 1 1 | , the subscripts, AA, AB and BB denote cyclic formation via respective bond species, and the subscript, w, inside the bracket L the weight average quantity. For Α 1, eq. (20) gives the gelation condition: with D c representing the extent of reaction of A FU's. Eq. (22) is the R A R B g f g model version of eq. (6). Since p R is the ratio of cyclic bonds to all possible bonds, it may be written in the form: with the reciprocal of the total FU concentration.
p R as a function of D c can be obtained in the same manner as in eq. (12). We write for which eq. (25) has the asymptotic solution of C : With the relationship,

Post-gelation J = 2
We consider the mixture of f 2 monomers. Let Q A be the probability that a branch emanating from an A type monomer unit is finite, and Q B the corresponding probability for a B type unit.
Then Q A and Q B satisfy The respective first terms represent the weight fraction of sol: where Ω A i and Ω B j are the weight fractions of the ith A type and the jth B type monomer units, respectively.

Special Solution I
Let us consider a mixing system of g i 2 (for all i s), f g 1 2, f g 2 3 and J 2.
This special system has an application to the polyurethane network formation of MDI (4,4'diphenylmethane diisocyanate: m A 250 ) and LHT240 (polyoxypropylene triol: m B 708).
LHT240 is considered to be a mixture of diols and triols [9]: There are conflicting views about chemical constituents of LHT240. Adam and coworkers [10] report that LHT240 is purely trifunctional, while Ilavsky and coworkers [9] report that it is a mixture of diols and triols. The difference of the views stems from the difference in the characterization of LHT240, especially in the estimation of OH content (see Table 1). Although which estimation is Substituting into eq. (31), one has the weight fraction of gel: In Figs. (2) and (3), eq. (33) is plotted as a function of Γ f and Κ at D 1, respectively. The parameters employed here are those [3] of MDI-LHT240 (see Table   2). The relative cyclization frequency, j , the only unknown parameter, is evaluated, based on the premise (iii), by the equation: where d is dimension, l is the standard bond length which can be equated with the C-N bond length , Ν is the incomplete Gamma function, and Ν is the proportionality factor defined by     Table 3. Making use of these values, the theoretical line is drawn in Fig. 4 (solid line). Experimental points (×) are Durand and coworkers'. Agreement between the experiment and the theory is satisfactory.

Special Solution II
Consider the homogeneous case of g i 3 (for all i s), f g j 2 (for all j s) and J 2. This system is a particular case of Χ B2 1 of Special Solution I. Using the boundary condition, 0 1 D c , with D co Κ 2 . In order for the gelation to occur, Γ f must be less than the above critical value,

Conclusion
The theory of gelation was applied to the observed data in polyurethane networks. Good agreement was found between the theory and the observations, both in the predictions of gel points and gel fractions.