Gelation

According to the Carothers equation number-average degree of polymerization ${\displaystyle DP_{n}}$ is given by

${\displaystyle DP_{n}={\frac {2}{2-p.f_{av}}}}$

where ${\displaystyle p}$ is the extent of the reaction and ${\displaystyle f_{av}}$ is the average functionality of reaction mixture. For the gel ${\displaystyle DP_{n}}$ can be considered to be infinite, thus the critical extent of the reaction at the gel point is found as

${\displaystyle p_{c}={\frac {2}{f_{av}}}}$

If ${\displaystyle p}$ is greater or equal to ${\displaystyle p_{c}}$, gelation occurs.

Flory and Stockmayer used a statistical approach to derive an expression to predict the gel point by calculating when ${\displaystyle DP_{n}}$ approaches infinite size. The statistical approach assumes that (1) the reactivity of the functional groups of the same type is the same and independent of the molecular size and (2) there are no intramolecular reactions between the functional groups on the same molecule.[6][7]

Consider the polymerization of bifunctional molecules ${\displaystyle A-A}$, ${\displaystyle B-B}$ and multifunctional ${\displaystyle A_{f}}$, where ${\displaystyle f}$ is the functionality. The extends of the functional groups are ${\displaystyle p_{A}}$and ${\displaystyle p_{B}}$, respectively. The ratio of all A groups, both reacted and unreacted, that are part of branched units, to the total number of A groups in the mixture is defined as ${\displaystyle \rho }$. This will lead to the following reaction

${\displaystyle A-A+B-B+A_{f}\rightarrow A_{f-1}-(B-BA-A)_{n}B-BA-A_{f-1}}$

The probability of obtaining the product of the reaction above is given by ${\displaystyle p_{A}[p_{B}(1-\rho )p_{A}]^{n}p_{B}\rho }$, since the probability that a B group reach with a branched unit is ${\displaystyle p_{B}\rho }$ and the probability that a B group react with non-branched A is ${\displaystyle p_{B}(1-\rho )}$.

This relation yields to an expression for the extent of reaction of A functional groups at the gel point

${\displaystyle p_{c}={\frac {1}{\{r[1+\rho (f-2)]\}^{1/2}}}}$

where r is the ratio of all A groups to all B groups. If more than one type of multifunctional branch unit is present and average ${\displaystyle f}$ value is used for all monomer molecules with functionality greater than 2.

Note that the relation does not apply for reaction systems containing monofunctional reactants and/or both A and B type of branch units.

Gelation of polymers can be described in the framework of the Erdős–Rényi model or the Lushnikov model, which answers the question when a giant component arises.[8]

The structure of a gel network can be conceptualised as a random graph. This analogy is exploited to calculate the gel point and gel fraction for monomer precursors with arbitrary types of functional groups. Random graphs can be used to derive analytical expressions for simple polymerisation mechanisms, such as step-growth polymerisation, or alternatively, they can be combined with a system of rate equations that are integrated numerically.