Chemistry:Flory-Stockmayer Theory
Flory-Stockmayer Theory is a theory governing the cross-linking and gelation of step-growth polymers.[1] The Flory-Stockmayer theory represents an advancement from the Carothers equation, allowing for the identification of the gel point for polymer synthesis not at stoichiometric balance.[1] The theory was initially conceptualized by Paul Flory in 1941[1] and then was further developed by Walter Stockmayer in 1944 to include cross-linking with an arbitrary initial size distribution.[2]
History
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]
Flory’s assumptions
In creating the Flory-Stockmayer Theory, Flory made three assumptions that affect the accuracy of this model.[1][3] These assumptions were
- All functional groups on a branch unit are equally reactive
- All reactions occur between A and B
- 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[3]
General case
File:Gelation General Case.tiff The Flory-Stockmayer Theory predicts the gel point for the system consisting of three types of monomer units[1][3][4][5]
- linear units with two A-groups (concentration [math]\displaystyle{ c_1 }[/math]),
- linear units with two B groups (concentration [math]\displaystyle{ c_2 }[/math]),
- branched A units (concentration [math]\displaystyle{ c_3 }[/math]).
The following definitions are used to formally define the system[1][3]
- [math]\displaystyle{ f }[/math] is the number of reactive functional groups on the branch unit (i.e. the functionality of that branch unit)
- [math]\displaystyle{ p_A }[/math] is the probability that A has reacted (conversion of A groups)
- [math]\displaystyle{ p_B }[/math] is the probability that B has reacted (conversion of B groups)
- [math]\displaystyle{ \rho = \frac{ f c_3 }{2 c_1+f c_3} }[/math] is the ratio of number of A groups in the branch unit to the total number of A groups
- [math]\displaystyle{ r=\frac{2 c_1+f c_3}{2 c_2}=\frac{p_B}{p_A} }[/math] is the ratio between total number of A and B groups. So that [math]\displaystyle{ p_B=r p_A. }[/math]
The theory states that the gelation occurs only if [math]\displaystyle{ \alpha \gt \alpha_c }[/math], where
- [math]\displaystyle{ \alpha_c = \frac{1}{f-1} }[/math]
is the critical value for cross-linking and [math]\displaystyle{ \alpha }[/math] is presented as a function of [math]\displaystyle{ p_A }[/math],
- [math]\displaystyle{ \alpha(p_A) = \frac{ r p_A^2 \rho } { 1 - r p_A^2 ( 1 - \rho ) } }[/math]
or, alternatively, as a function of [math]\displaystyle{ p_B }[/math],
- [math]\displaystyle{ \alpha(p_B) = \frac{ p_B^2 \rho } { r - p_B^2 ( 1 - \rho ) } }[/math].
One may now substitute expressions for [math]\displaystyle{ r, \rho }[/math] into definition of [math]\displaystyle{ \alpha }[/math] and obtain the critical values of [math]\displaystyle{ p_A, (p_B) }[/math] that admit gelation. Thus gelation occurs if
- [math]\displaystyle{ p_A\gt \sqrt{\frac{ \alpha_c } { r(\alpha_c + \rho - \alpha_c \rho )}}. }[/math]
alternatively, the same condition for [math]\displaystyle{ p_B }[/math] reads,
- [math]\displaystyle{ p_B\gt \sqrt{\frac{ r \alpha_c } { \alpha_c + \rho - \alpha_c \rho }} }[/math]
The both inequalities are equivalent and one may use the one that is more convenient. For instance, depending on which conversion [math]\displaystyle{ p_A }[/math] or [math]\displaystyle{ p_B }[/math] is resolved analytically.
Trifunctional A monomer with difunctional B monomer
- [math]\displaystyle{ \alpha_c=\frac{1}{f-1}=\frac{1}{3-1}=\frac{1}{2} }[/math]
Since all the A functional groups are from the trifunctional monomer, ρ = 1 and
- [math]\displaystyle{ \alpha=\frac{\frac{p_B^2\rho}{r}}{1-\frac{p_B^2}{r(1-\rho)} }=\frac{p_B^2}{r} }[/math]
Therefore, gelation occurs when
- [math]\displaystyle{ \frac{p_B^2}{r}\gt \alpha_c }[/math]
or when,
- [math]\displaystyle{ p_B\gt \sqrt{\frac{r}{2}} }[/math]
Similarly, gelation occurs when
- [math]\displaystyle{ p_A\gt \sqrt{\frac{1}{2r}} }[/math]
References
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Flory, P.J. (1941). "Molecular Size Distribution in Three Dimensional Polymers I. Gelation". J. Am. Chem. Soc. 63, 3083
- ↑ Stockmayer, Walter H.(1944). "Theory of Molecular Size Distribution and Gel Formation in Branched Polymers II. General Cross Linking". Journal of Chemical Physics. 12,4, 125
- ↑ 3.0 3.1 3.2 3.3 Stauffer, Dietrich, et al.(1982) "Gelation and Critical Phenomena". Advances in Polymer Science 44, 103
- ↑ Flory, P.J.(1941). "Molecular Size Distribution in Three Dimensional Polymers II. Trifunctional Branching Units". J. Am. Chem. Soc. 63, 3091
- ↑ Flory, P.J. (1941). "Molecular Size Distribution in Three Dimensional Polymers III. Tetrafunctional Branching Units". J. Am. Chem. Soc. 63, 3096
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