Carothers equation
The Carothers equation describes the relationship between degree of polymerization and the degree of conversion at a step growth reaction. It is named after Wallace Hume Carothers.
There are several variants for AB systems, AA / BB- systems and non-linear step-growth reactions.
In linear AB systems are monomers in which both the monomer functional groups carries (e.g. HO-R -COOH).
For linear AA / BB systems are two monomers, each carrying one of the functional groups at both ends ( eg HOOC -Ph- COOH and HO - OH polyethylene terephthalate)
For non-linear A-B systems are cross-linking monomers.
- 2.1 A- B systems
Linear step-growth reactions
A- B systems
If the number of monomers initially present and the number of the time remaining monomers is obtained for the conversion
(1)
P is the probability that one of the groups has reacted. At a conversion of the probability that a group has reacted at 50 %.
The degree of polymerization - the average length of the chains - can be expressed as a fraction of the number of monomers initially present at time t by the remaining molecules:
(2)
Rearranging of Eq. 1
And insert it into Eq. 2, one obtains the Carothers equation for AB systems
A-A/B-B-Systeme
For AA / BB systems you must also note that the system can not be stoichiometric composition, ie different monomer ratio can occur. Therefore, we define a parameter:
The parameter is always defined to be, so the system more BB present as AA.
Is thus obtained as
At the time of molecules of type AA are in sales already been implemented. Therefore applies to, the sum of unreacted AA and BB.
The amount of unreacted monomers is therefore
On the way as above is obtained by inserting the following expression for
Which corresponds to the Carothers equation for AA / BB systems
Non-linear step-growth reactions
A- B systems
Substituting the monomer trifunctional monomers to be, there is a network formation.
In order to calculate the degree of polymerization, defined to have an average functionality of the monomers
Here, the number of functional groups on the molecule and the number of monomer molecules i.
Monomer molecules in total of functional groups are available.
After a time t groups have reacted. Thus molecules have formed because of the need to react for 2 end groups for a bond. The probability of reaction is therefore at
(3)
Forming of Eq. 3 results
And after insertion into Eq. 2 one obtains a Carothers equation for nonlinear systems
Equation 3 can be further reshaping
(4)
When the polymerization goes to infinity, occurs and in Eq gelation. 4 is the expression
Thus applies to the conversion, where the mixture starts to gel:
From this relation it can be seen that can be achieved at much lower yields than in the other cases a high degree of polymerization.
This equation holds only for the case that the mixture is stoichiometric composition (the same as the number of A - B groups).
Graphical representation of turnover and degree of polymerization
The importance of the Carothers equation can be seen that if one applies the revenue against the degree of polymerization:
Only at very high conversions of the polymerization reached appreciably large values. So he is at p = 0.5 just 2, a value of 10,000 is reached only at a degree of conversion of p = 0.9999.
Similarly, r has a significant influence on the degree of polymerization:
Even small deviations from the ideal value of r 1 represents a significantly lower degree of polymerization.
With the addition of cross-linkers, however, sharply increases: