Flory–Stockmayer theory

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.

In creating the Flory–Stockmayer Theory, Flory made three assumptions that affect the accuracy of this model.Stauffer, Dietrich, et al.(1982) "Gelation and Critical Phenomena". Advances in Polymer Science 44, 103 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.

Flory postulated that his treatment can also be applied to chain-growth polymerization mechanisms, as the three criteria stated above are satisfied under the assumptions that (1) the probability of chain termination is independent of chain length, and (2) multifunctional co-monomers react randomly with growing polymer chains.

File:Gelation General Case.tiff

The Flory–Stockmayer Theory predicts the gel point for the system consisting of three types of monomer unitsFlory, P.J.(1941). "Molecular Size Distribution in Three Dimensional Polymers II. Trifunctional Branching Units". J. Am. Chem. Soc. 63, 3091Flory, P.J. (1941). "Molecular Size Distribution in Three Dimensional Polymers III. Tetrafunctional Branching Units". J. Am. Chem. Soc. 63, 3096

: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

:$f$ is the number of reactive functional groups on the branch unit (i.e. the functionality of that branch unit)

:$p\_A$ is the probability that A has reacted (conversion of A groups)

:$p\_B$ is the probability that B has reacted (conversion of B groups)

:$\backslash rho\; =\; \backslash frac\{\; f\; c\_3\; \}\{2\; c\_1+f\; c\_3\}$ is the ratio of number of A groups in the branch unit to the total number of A groups

:$r=\backslash frac\{2\; c\_1+f\; c\_3\}\{2\; c\_2\}=\backslash frac\{p\_B\}\{p\_A\}$ is the ratio between total number of A and B groups. So that $p\_B=r\; p\_A.$

The theory states that the gelation occurs only if $\backslash alpha\; >\; \backslash alpha\_c$, where

:$\backslash alpha\_c\; =\; \backslash frac\{1\}\{f-1\}$

is the critical value for cross-linking and $\backslash alpha$ is presented as a function of $p\_A$,

:$\backslash alpha(p\_A)\; =\; \backslash frac\{\; r\; p\_A^2\; \backslash rho\; \}\; \{\; 1\; -\; r\; p\_A^2\; (\; 1\; -\; \backslash rho\; )\; \}$

or, alternatively, as a function of $p\_B$,

:$\backslash alpha(p\_B)\; =\; \backslash frac\{\; p\_B^2\; \backslash rho\; \}\; \{\; r\; -\; p\_B^2\; (\; 1\; -\; \backslash rho\; )\; \}$.

One may now substitute expressions for $r,\; \backslash rho$ into definition of $\backslash alpha$ and obtain

the critical values of $p\_A,\; (p\_B)$ that admit gelation. Thus gelation occurs if

:$p\_A>\backslash sqrt\{\backslash frac\{\; \backslash alpha\_c\; \}\; \{\; r(\backslash alpha\_c\; +\; \backslash rho\; -\; \backslash alpha\_c\; \backslash rho\; )\}\}.$

alternatively, the same condition for $p\_B$ reads,

:$p\_B>\; \backslash sqrt\{\backslash frac\{\; r\; \backslash alpha\_c\; \}\; \{\; \backslash alpha\_c\; +\; \backslash rho\; -\; \backslash alpha\_c\; \backslash rho\; \}\}$

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.

File:GelationExample.tiff

:$\backslash alpha\_c=\backslash frac\{1\}\{f-1\}=\backslash frac\{1\}\{3-1\}=\backslash frac\{1\}\{2\}$

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

:$\backslash alpha=\backslash frac\{\backslash frac\{p\_B^2\backslash rho\}\{r\}\}\{1-\backslash frac\{p\_B^2\; (1-\backslash rho)\}\{r\}\}=\backslash frac\{p\_B^2\}\{r\}$

Therefore, gelation occurs when

:$\backslash frac\{p\_B^2\}\{r\}>\; \backslash alpha\_c$

or when,

:$p\_B>\backslash sqrt\{\backslash frac\{r\}\{2\}\}$

Similarly, gelation occurs when

:$p\_A>\; \backslash sqrt\{\backslash frac\{1\}\{2r\}\}$

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Category:Polymer chemistry