Macromolecules 1985,18, 1162-1166
1162
on polymer networks. Further experiments on wellcharacterized samples are needed to test the generality of the theoretical predictions. In particular, relaxation experiments on networks in which the dangling-end distribution is known would be of significant interest.
5A into eq 4A and 2A leads to the desired result for large N
Acknowledgment. A portion of this work was performed at Sandia National Laboratories and was supported by the US.Department of Energy under Contract Number DE-AC04-76DP00789.
P = Neq = NePc/Pm
Appendix: Distribution of Chain Lengths For a randomly cross-linked system, the probability Qn of having a dangling chain end of exactly n monomers is given b4p Qn
= q(1 - q)n-'
(1A)
For a lightly cross-linked system, q is simply the ratio of the cross-link density to the monomer density. In eq 8 we require the probability PN of having a chain of N = n/N, segments. From eq 1A we can write PN
= Cq(1 - q)Ny-l
(24
where C is a normalization constant to ensure that m
cPN=1 N= 1
From eq 3A we find that
c = [ l - (1 - q)"]/q(l
- qINe-1
(4A)
We now write (1 - q ) N e = exp[N, In (1- q ) ]
z exp(-N,q)
where we have used the fact that q
(54
