For some definitions, a polymer is a large molecule, or macromolecule, composed of many repeated subunits. Polymerization is a process of reacting monomer molecules together in a chemical reaction to form polymer chains or three-dimensional networks.
A condensation reaction is a chemical reaction in which two molecules or moieties, often functional groups, combine to form a larger molecule, together with the loss of a small molecule. And in polymer science, the condensation reaction between monomers are called condensation polymerization. The condensation polymerization comprises 2 kinds of reactions - linear polymerization (LP) and three-dimensional polymerization (TDP). The difference between these two reaction types is that the former is the polymerization between monomers containing 2 or less functional groups, and can lead to the formation of one-dimensional polymers. The latter, however, is the reaction between monomers having more than 2 functional groups, and will produce the crosslinked polymers (Figure 1). It is well-known that crosslinking network formed by TDP possess a larger contact area and a higher mechanical strength. We thus decided to use monomers containing more than 3 functional groups, i.e., the poly-branched monomers Af (f>=3, where A stands for “functional group” or “functionality”).
Fig. 1. Common kinds of polymerization. A: Linear Polymerization, B: Three-dimensional Polymerization. Functional groups a and b can react with each other. M can neither react with a nor b.
Gelation, reflecting a certain reaction extent polymerization would perform, is a common phenomenon of TDP, during which the physical properties like the degree of crosslinking, the viscosity and the “rigidity” of polymers increase a lot.
The prediction of gel point is important in the field of polymer science, for the difference between the state before and after crosslinking is obvious. Internationally famous American chemists Wallace Carothers (1896-1937), Paul J. Flory (1910-1985), Walter H. Stockmayer (1914-2004) and others have already made a lot of efforts to explore it and successfully come up with 2 models (Carothers Model and Flory-Stockmayer Model). In our project this year, we would like to anticipate the gel point specific for the TDP with reaction extent p on the basis of Flory-Stockmayer Theory (View the Supplementary Information for more information). Thus we could avoid gelation and increase the crosslinking extent of our polymer network as much as possible meanwhile to better perform the function of our dissolved crosslinking network.
A branched oligomer becomes a crosslinked polymer when the reactive extent reaches the gel point Pc, and the formula used to calculate the gel point Pc is well described by Flory-Stockmayer Theory. We have therefore derived the formula to calculate the Pc for our own case,
Where the parameters are:
Pc: The gel point, at which the gel forms.
NA: The amount of monomers containing functional groups A.
NB: The amount of monomers containing functional groups B.
fA: Functionality of monomers containing A, e.g. when the system contains equivalent amounts of A-A and A-A-A, then the functionalities are 2 and 3 for each.
fB: Functionality of monomers containing B.
The deduction of formula (2) is contained in Supplementary Information- Calculation of the Gel Point.
Assume that the total number of A and B is constant, then the differences of the propensities towards different kinds of configurations can easily be compared via their respective gel points. Hereby a tetrad SpyTag-SUP and double SpyCatcher reaction is a different configuration from a triple SpyTag-SUP and triple SpyCatcher reaction.
We define θ as the fraction of SpyTag-SUP in the mixture of SpyTag and SpyCatcher,
Then the relation between Pc and θ is,
Which can be described in figure 2,
Fig. 2. Theoretical gel points for a number of monomer configurations. The horizontal axis represents the amount (or molar concentration) of nA-SUP over the amount (or molar concentration) of all monomers, θ, while the vertical axis represents the gel point, Pc.
High Pc means low probability of gelation, thus low degree of crosslinking, during the reaction process. As we have mentioned, this will not be easy to form a polymer network with high degree of crosslinking and lead to poor mechanical strength and a small contact area with solution. We thus chose triple, tetrad, and sextuple SpyTag-SUP (3/4/6A-SUP) and triple SpyCatcher (3B) as basic crosslinking monomers.
For a more direct route from initial experimental parameter configuration to its prediction, the linkage between the initial solution state and final reaction extent (when it reaches equilibrium) may be established. A simple way to do this is to use the concentrations of the functional groups A and B, together with the equilibrium constant K of the condensation reaction between A and B:
It is worth noting that the equilibrium constant K describes both the intramolecular reaction and intermolecular reaction. Intramolecular reactions would lead to the formation of “loop”, which decreases the degree of crosslinking and the strength of polymers. That’s to say, the effective part of crosslinking is the intermolecular reaction of the functional groups on monomers (Figure 3).
Fig. 3. Intramolecular and intermolecular reaction. A: Intramolecular reaction leads to the formation of “loop” and decrease the degree of crosslinking. B: Intermolecular reaction will increase the degree of crosslinking and result in gelation.
We assume that the polymerization meets with Flory’s hypotheses (Described in the Supplementary Information). Then K serves as the equilibrium constant of intermolecular reaction.
Now we further define Pf as the reactive extent of all functional groups,
N as the amount of functional groups consumed in the reaction,
x as the initial amount of all functional groups in the reaction.
A and B each means SpyTag and SpyCatcher.
Then Pf can be described as,
Given that the concentration of small molecule (H2O) is constant in the condensation reaction of SpyTag and SpyCatcher, then the K and the dissociation constant Kd can be described as,
So the relation between Pf and NA, NB is,
We thus find out the relation between the concentration of substrates and the reactive extent Pf. When the reaction reaches its equilibrium, and Pf > Pc, the gel will form.
The equation (7) will be further used to build our software.
References:
[1]. Flory, Paul J. Principle of Polymer Chemistry. Cornell University Press, 1953.
Now we would like to give out our deduction of the formula to calculate the gel point.
According to the Flory-Stockmayer Theory, the gel point is closely related to the “branched-point” and the number of its functional groups. And in the three-dimensional polymerization (TDP), the formation of crosslinking products is the result of containing monomers having more than 2 functional groups in the reaction system. As has been mentioned, the polymerization between monomers is actually the reaction between functional groups. The reaction is shown in the picture below:
Fig.1. The polymerization between a-a-a and b-b-b monomers, which performs the Three-dimensional polymerization (TDP).
Firstly we would like to briefly introduce the Flory-Stockmayer Theory.
Af: Branching unit, monomers containing more than 2 functional groups (f>=3).
α: Branching index, the probability of the linkage between branching units.
αc: Critical branching index, the probability of the linkage between branching units leading to the formation of gel.
NA: The amount of monomers containing functional groups A.
NB: The amount of monomers containing functional groups B.
N: The amount of functional groups consumed in the reaction.
fA: Functionality of monomers containing A, e.g. when the system contains equivalent amounts of A-A and A-A-A, thenfA=2.5.
fB: Functionality of monomers containing B.
PA: The reactive extent of functional group A, i.e., the ratio between “number of the reacted functional group A” to “total number of functional group A”.
PB: The reactive extent of functional group B.
Pf: The reactive extent of all functional groups.
Pc: The gel point, at which the gel forms.
ρ: The ratio of number of A groups in the branching unit to the total number of A groups.
Flory made two assumptions for polymerization only between concerned functional groups that affect the accuracy of the Flory-Stockmayer model. These assumptions were:
(1) All functional groups on a branch unit are equally reactive,
(2) There are no intramolecular reactions.
Since steric hindrance effects prevent each functional group from being equally reactive and intramolecular reactions do occur, a conversion slightly higher than that predicted by the Flory-Stockmayer Theory is commonly needed to actually create a polymer gel.
According to Flory’s calculation for A-A, B-B, A-A-A System, the branching index is,
When the functionality of branching unit is f, then the branching unit having been attached by one branched chain are able to produce α(f-1) more new chains. When α(f-1)>1, the formation of gel is possible. So the critical condition is
According to (1) and (2), we can find out the relationship between f and P, and when
PA=PB=Pf=Pc, we have,
Examples given by Flory can be further viewed at Wikipedia ( Flory-Stockmayer Theory ) and in Flory and Stockmayer’s works.
According to Flory-Stockmayer Theory, we set up our model to describe the relation of Pc and fA, fB. As our reactive system only contains 2 kinds of monomers, i.e., the monomers containing SpyTag and SpyCatcher, nA-FP (n SpyTag-Functional Protein) and mB (m SpyCatcher) (n, m describe the functionality of monomers nA-FP and mB).
Since fA and fB here are unknown, we thus need to find out a new criterion to estimate the critical condition on the basis of Flory’s Theory.
Firstly, a free monomer Af is selected, and it will react with Bf, which has (fB-1) functional groups left after the reaction. Now Bf is able to react with (fB-1) Bfs, which leads to (fA-1) new branches each. Then the number of new branches derived by Af is
when α=αc, and
Then we say that the gel will form.
We now turn to find out the relation of Pc and fA, fB.
Consider the reaction
We have,
Then α can be described as,
When α=αc, the relation between Pc and fA, fB is,
References:
[1]. Flory, P.J. (1941). Molecular Size Distribution in Three Dimensional Polymers I. Gelation. J. Am. Chem. Soc. 63, 3083
[2]. Flory, P.J.(1941). Molecular Size Distribution in Three Dimensional Polymers II. Trifunctional Branching Units. J. Am. Chem. Soc. 63, 3091
[3]. Flory, P.J. (1941). Molecular Size Distribution in Three Dimensional Polymers III. Tetrafunctional Branching Units. J. Am. Chem. Soc. 63, 3096
[4]. 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
