Flory–Stockmayer theory is a theory governing the
cross-link
In chemistry and biology a cross-link is a bond or a short sequence of bonds that links one polymer chain to another. These links may take the form of covalent bonds or ionic bonds and the polymers can be either synthetic polymers or natural ...
ing and
gelation
In polymer chemistry, gelation (gel transition) is the formation of a gel from a system with polymers. Branched polymers can form links between the chains, which lead to progressively larger polymers. As the linking continues, larger branched p ...
of step-growth polymers.
[Flory, P.J. (1941). "Molecular Size Distribution in Three Dimensional Polymers I. Gelation". ''J. Am. Chem. Soc.'' 63, 3083] 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.
The theory was initially conceptualized by
Paul Flory in 1941
and then was further developed by
Walter Stockmayer in 1944 to include cross-linking with an arbitrary initial size distribution.
[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]
The Flory-Stockmayer theory was the first theory investigating
percolation processes. Flory–Stockmayer theory is a special case of
random graph theory of gelation.
History
Gelation
In polymer chemistry, gelation (gel transition) is the formation of a gel from a system with polymers. Branched polymers can form links between the chains, which lead to progressively larger polymers. As the linking continues, larger branched p ...
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
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
, interspersed with smaller complex molecules, shifting the polymer from a
liquid to a
network solid or
gel
A gel is a semi-solid that can have properties ranging from soft and weak to hard and tough. Gels are defined as a substantially dilute cross-linked system, which exhibits no flow when in the steady-state, although the liquid phase may still dif ...
phase. The Carothers equation is an effective method for calculating the
degree of polymerization
The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule.
For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given b ...
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.
Flory’s assumptions
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:
# All
functional groups
In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the rest ...
on a branch unit are equally reactive
# All reactions occur between A and B
# There are no
intramolecular reaction Intramolecular in chemistry describes a process or characteristic limited within the structure of a single molecule, a property or phenomenon limited to the extent of a single molecule.
Examples
* intramolecular hydride transfer (transfer of a hy ...
s
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
Steric effects arise from the spatial arrangement of atoms. When atoms come close together there is a rise in the energy of the molecule. Steric effects are nonbonding interactions that influence the shape ( conformation) and reactivity of ions ...
effects prevent each functional group from being equally reactive and intramolecular reactions do occur, the gel forms at slightly higher conversion.
General case
The Flory-Stockmayer Theory predicts the gel point for the system consisting of three types of monomer units
[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]
:linear units with two A-groups (concentration
),
:linear units with two B groups (concentration
),
:branched A units (concentration
).
The following definitions are used to formally define the system
: is the number of reactive functional groups on the branch unit (i.e. the functionality of that branch unit)
: is the probability that A has reacted (conversion of A groups)
: is the probability that B has reacted (conversion of B groups)
: is the ratio of number of A groups in the branch unit to the total number of A groups
: is the ratio between total number of A and B groups. So that
The theory states that the gelation occurs only if
, where
:
is the critical value for cross-linking and
is presented as a function of
,
:
or, alternatively, as a function of
,
:
.
One may now substitute expressions for
into definition of
and obtain
the critical values of
that admit gelation. Thus gelation occurs if
:
alternatively, the same condition for
reads,
:
The both inequalities are equivalent and one may use the one that is more convenient. For instance, depending on which conversion
or
is resolved analytically.
Trifunctional A monomer with difunctional B monomer
:
Since all the A functional groups are from the trifunctional monomer, ρ = 1 and
:
Therefore, gelation occurs when
:
or when,
:
Similarly, gelation occurs when
:
References
{{DEFAULTSORT:Flory-Stockmayer theory
Polymer chemistry