Flory Stockmayer Theory

History

Gelation occurs when a polymer forms large interconnected polymer molecules through cross-linking. In other words, polymer chains are cross-linked with other polymer chains to form an infinitely large molecule, interspersed with smaller complex molecules, shifting the polymer from a liquid to a network solid or gel phase. The Carothers equation is an effective method for calculating the degree of polymerization for stoichiometrically balanced reactions. However, the Carothers equation is limited to branched systems, describing the degree of polymerization only at the onset of cross-linking. The Flory-Stockmayer Theory allows for the prediction of when gelation occurs using percent conversion of initial monomer and is not confined to cases of stoichiometric balance. Additionally, the Flory-Stockmayer Theory can be used to predict whether gelation is possible through analyzing the limiting reagent of the step-growth polymerization.

Flory’s Assumptions

In creating the Flory-Stockmayer Theory, Flory made three assumptions that affect the accuracy of this model. These assumptions were

1. All functional groups on a branch unit are equally reactive
2. All reactions occur between A and B
3. There are no intramolecular reactions

As a result of these assumptions, a conversion slightly higher than that predicted by the Flory-Stockmayer Theory is commonly needed to actually create a polymer gel. Since steric hindrance effects prevent each functional group from being equally reactive and intramolecular reactions do occur, the gel forms at slightly higher conversion

General Case

The Flory-Stockmayer Theory predicts the gel point for the system consisting of three types of monomer units

linear units with two A-groups (concentration c 1 ), linear units with two B groups (concentration c 2 ), branched A units (concentration c 3 ).

The following definitions are used to formally define the system

The theory states that the gelation occurs only if α > α c , where

α c = 1 f − 1

is the critical value for cross-linking and α is presented as a function of p A ,

α ( p A ) = r p A 2 ρ 1 − r p A 2 ( 1 − ρ )

or, alternatively, as a function of p B ,

α ( p B ) = p B 2 ρ r − p B 2 ( 1 − ρ ) .

One may now substitute expressions for r , ρ into definition of α and obtain the critical values of p A , ( p B ) that admit gelation. Thus gelation occurs if

p A > α c r ( α c + ρ − α c ρ ) .

alternatively, the same condition for p B reads,

p B > r α c α c + ρ − α c ρ

The both inequalities are equivalent and one may use the one that is more convenient. For instance, depending on which conversion p A or p B is resolved analytically.

Trifunctional A Monomer with Difunctional B Monomer

α c = 1 f − 1 = 1 3 − 1 = 1 2

Since all the A functional groups are from the trifunctional monomer, ρ = 1 and

α = p B 2 ρ r 1 − p B 2 r ( 1 − ρ ) = p B 2 r

Therefore, gelation occurs when

p B 2 r > α c

or when,

p B > r 2

Similarly, gelation occurs when

p A > 1 2 r