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oss-link functionality, and solvent content. Much is known already about chemically cross-linking Systems. The independent variable of chemical gelation, the extent of cross-linking, p, is defined

as the number of bonds formed divided by the total number of possible bonds. The gel point is reaçhed at a critical extent of reaction, p —» pc,1'2 where the molecular weight of the largest cluster diverges to infinity and the molecular weight distribution is infinitely broad {MJMn —» A0 (6)

It should be emphasized that selfsimilar b e h a v i o r occurs at températures much above the glass transition.

46

1.0-

a m 31.6 rad/s .00 rad/s

0316 rad 0.1-

I

I

I

I

T 10*

10

i (min.) Figure 4. Gel point détermination from tan S data of a physical gel. With the Kramers-Kronig relation, one can show that the common intersect of the curves is a conséquence of power law relaxation behavior at the gel point.

Also, the transition to the glassy state at short times (t < A0) is not considered hère. Vitrification would complicate the dynamic mechanical behavior if it interfered with the gelation. Rheological Détermination of the Gel Point The gel point is essential to know, either for avoiding it to process a polymer before it has gelled or for making materials near the gel point. For molécules with cross-links ofhigh functionality, the gel point occurs early.1,2 A set of data similar to that in Figure 1 gives a good estimate of where the gel point occurs. The problem is that the experiment breaks d o w n as the relaxation time diverges for p —» p c . Determining the gel point from dynamic mechanical data (see F i g u r e 4) is much simpler. The gel point is reached when tan 5 becomes i n d e p e n d e n t of frequency 1 8 (intersect in Figure 4), where S is the phase angle of the dynamical mechanical experiment. It has b e e n r e p e a t e d l y s u g g e s t e d in the l i t e r a t u r e t h a t the gel p o i n t

m a y be r e a c h e d w h e n t h e s t o r a g e modulus equals the loss modulus, G' = G". This proposai, however, violâtes the Kramers-Kronig relation except for the spécial case of n = 0.5. Expectations of a maximum in tan 5 or G" at the gel point are also not in agreement with the observed behavior. Nearly Critical G e l s Interesting polymer gels hâve been prepared by stopping the gelation process s o m e w h e r e n e a r the gel p o i n t . Many possibilities can be envisioned for such a stopping mechanism (thermal quenching, poisoning of catalyst, offsetting of stoichiometry, endcapping of functional groups). It does not cause a problem for the rheological description that none of thèse gels will be s t o p p e d exactly at t h e gel p o i n t . The relaxation time spectrum evolves gradually with extent of reaction. We can use the critical gel as a référence state, a n d the d i s t a n c e from the gel point (p — pc) is the independent variable. Techniques as discussed previously allow determining p c (or the gel

MRS BULLETIN/AUGUST1991

Polymer Gels, Materials That Combine Liquid and Solid Properties