Flory–Stockmayer theory

Gelation occurs when a polymer forms large interconnected polymer molecules through cross-linking.[1] 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.[1] 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.[1]

In creating the Flory-Stockmayer Theory, Flory made three assumptions that affect the accuracy of this model.[1][5] 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.[5]

A general image of a multifunctional branch unit, ${\displaystyle A_{f}}$, reacting with bifunctional monomers with A and B functional groups to form a step-growth polymer.

The Flory-Stockmayer Theory predicts the gel point for the system consisting of three types of monomer units[1][5][6][7]

linear units with two A-groups (concentration ${\displaystyle c_{1}}$),

linear units with two B groups (concentration ${\displaystyle c_{2}}$),

branched A units (concentration ${\displaystyle c_{3}}$).

The following definitions are used to formally define the system[1][5]

${\displaystyle f}$ is the number of reactive functional groups on the branch unit (i.e. the functionality of that branch unit)

${\displaystyle p_{A}}$ is the probability that A has reacted (conversion of A groups)

${\displaystyle p_{B}}$ is the probability that B has reacted (conversion of B groups)

${\displaystyle \rho ={\frac {fc_{3}}{2c_{1}+fc_{3}}}}$ is the ratio of number of A groups in the branch unit to the total number of A groups

${\displaystyle r={\frac {2c_{1}+fc_{3}}{2c_{2}}}={\frac {p_{B}}{p_{A}}}}$ is the ratio between total number of A and B groups. So that ${\displaystyle p_{B}=rp_{A}.}$

The theory states that the gelation occurs only if ${\displaystyle \alpha >\alpha _{c}}$, where

${\displaystyle \alpha _{c}={\frac {1}{f-1}}}$

is the critical value for cross-linking and ${\displaystyle \alpha }$ is presented as a function of ${\displaystyle p_{A}}$,

${\displaystyle \alpha (p_{A})={\frac {rp_{A}^{2}\rho }{1-rp_{A}^{2}(1-\rho )}}}$

or, alternatively, as a function of ${\displaystyle p_{B}}$,

${\displaystyle \alpha (p_{B})={\frac {p_{B}^{2}\rho }{r-p_{B}^{2}(1-\rho )}}}$.

One may now substitute expressions for ${\displaystyle r,\rho }$ into definition of ${\displaystyle \alpha }$ and obtain the critical values of ${\displaystyle p_{A},(p_{B})}$ that admit gelation. Thus gelation occurs if

${\displaystyle p_{A}>{\sqrt {\frac {\alpha _{c}}{r(\alpha _{c}+\rho -\alpha _{c}\rho )}}}.}$

alternatively, the same condition for ${\displaystyle p_{B}}$ reads,

${\displaystyle p_{B}>{\sqrt {\frac {r\alpha _{c}}{\alpha _{c}+\rho -\alpha _{c}\rho }}}}$

The both inequalities are equivalent and one may use the one that is more convenient. For instance, depending on which conversion ${\displaystyle p_{A}}$ or ${\displaystyle p_{B}}$ is resolved analytically.

Trifunctional A monomer with difunctional B monomer

A trifunctional branch unit with A functional group reacting with a bifunctional branch unit with a B functional group, forming a continuous step-growth polymer molecule.

${\displaystyle \alpha _{c}={\frac {1}{f-1}}={\frac {1}{3-1}}={\frac {1}{2}}}$

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

${\displaystyle \alpha ={\frac {\frac {p_{B}^{2}\rho }{r}}{1-{\frac {p_{B}^{2}}{r(1-\rho )}}}}={\frac {p_{B}^{2}}{r}}}$

Therefore, gelation occurs when

${\displaystyle {\frac {p_{B}^{2}}{r}}>\alpha _{c}}$

or when,

${\displaystyle p_{B}>{\sqrt {\frac {r}{2}}}}$

Similarly, gelation occurs when

${\displaystyle p_{A}>{\sqrt {\frac {1}{2r}}}}$