Weight fraction of sol

Once P(F ° ) and P(Fg° ) have been calculated, it is possible to calculate a number of network structure parameters including the weight fraction of sol, wg, and the "effective" crosslink density. A given polymer or crosslinker will be part of the sol only if all of its groups are attached to finite chains. Thus, the weight of the sol is given by... 

From the statistical theory by Flory, the weight fraction of sol is given by ... 

Summation of the weight fractions of all finite species according to Eq. (B-6) gives unity when aac, there is obtained for the weight fraction of sol... 

It should be noted here that Wsoi is also the weight fraction of sol, and Wgei — 1 — Wsoi is the weight fraction of gel. 

The size distribution of the sol fraction for a >0.5 is the same as prevailed for the entire mixture when p = 1 — a. In other words, the distribution functions for a > 0.5 are identical with those at the corresponding lower value 1 — a except that they are reduced by a factor Wa, the total weight fraction of sol present at that value of a. The dispersity of the size distribution in the sol above the gel point thus becomes less until, at q = 1, only monomer is left in a vanishingly small amount. The reduction of XwfXn shown for values of a. increasing above 0.5 is another way of expressing the same phenomenon. 

Problem 5.30 A mixture of equivalent amounts of glycerol and tricarballylic acid is condensed to the extent of 65% conversion. Assuming equal reactivity of functional groups, calculate (a) weight fractions of sol and gel in the mixture, and (b) weight fractions of the monomer and the dimer in the sol fraction. 

It may be noted that the curves in Fig. 5.10 representing the weight fractions of various species calculated from Eq. (5.83) continue through the gel point (or = 0.5) without discontinuity. This would be expected as the derivations of the distribution equations (5.81)-(5.83) place no restriction on or or p except that it be in the physically real range from zero to unity. These equations are vahd only for finite species beyond the gel point. If Eq. (5.83) for Wx is summed over all values of X, the weight fraction of sol (w ) would result. When the units are trifunctional (/ = 3), this yields (Flory, 1953) ... 

Using the complexity distribution approach and assuming that the proportion of -functional units is small ( << 1), the weight fraction of sol when cx, is... 

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