In this note, I describe a statistical problem concerning the degree of polymerization of polymers formed through free radical polymerization, a possible model of the reaction mechanism, a simulation and why it fails.
Suggesting ideas on the improvement of the problem solving strategy or the model will be greatly appreciated
Chain Length Problem
How does the distribution of the degree of polymerization (number of monomer units) of polyethene vary with the initial concentration of ethenes and the free radicals?
This is benzoyl peroxide.
When heated, it breaks up into two benzoyl free radicals.
The benzoyl further breaks up into a phenyl radical and carbon dioxide
The phenyl radical attacks an ethene, and forms a phenylethyl radical.
The free radicals keep attacking the ethenes and keeps growing.
The chain reaction terminates when the free radicals die out.
For the purpose of the spherical cow model, we will assume that it is also possible that two phenyl radicals join to form biphenyl.
It is evident that the reaction stops when there are no more free radicals.
Agnishom's Spherical Cow Model of Polyethene Radical Polymerization
The spherical cow model is a simulation friendly model that tries to mimic the process of radical polymerization.
Parameters: The number of free radicals (\(n_1)\)) and the number of ethylene molecules (\(n_2\)) in the system.
The model only distinguishes between three kinds of species:
Free Radicals: Each free radical has a variable called chainLength, which is the number of monomer units attached to it.
Closed Chains: They also have the chainLength variable. However, they are incapable of further change.
Ethenes: They are just simple ethenes. All ethenes are identical and can be potentially attacked by a free radical.
The process of polymerization is modelled here as a process with discrete time steps during each of which two objects, chosen with uniform probability from the system, interact.
The interaction rules are as follows:
|Closed Chain (m)||Closed Chain (n)||Nothing happens|
|Free Radical (m)||Closed Chain (n)||Nothing happens|
|Ethene||Closed Chain (n)||Nothing happens|
|Free Radical (m)||Free Radical (n)||Both radicals are removed. A new Closed Chain of length (m+n) is formed|
|Free Radical (m)||Ethene||The ethene is removed. The size of the free radical increases by 1|