Gelation

Quantitative approaches to determine gelation

It is important to be able to predict the onset of gelation, since it is an irreversible process that dramatically changes the properties of the system.

Average functionality approach

According to the Carothers equation number-average degree of polymerization ${\displaystyle D{P}_n}$ is given by

${\displaystyle D{P}_n=\frac{2}{2-p.f_{av}}}$

where ${\displaystyle p}$ is the extent of the reaction and ${\displaystyle f_{av}}$ is the average functionality of reaction mixture. For the gel ${\displaystyle D{P}_n}$ 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 Stockmayer approach

Flory and Stockmayer used a statistical approach to derive an expression to predict the gel point by calculating when ${\displaystyle D{P}_n}$ 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.

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

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

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.

Erdős–Rényi model

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.

Random graph

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.