Attention: Under construction-- this is a first draft. I have not had a
chance to go through and check for mistakes. Also, I'd like to have a
second person go through this as well.
DIFFERENTIAL EQUATIONS
KINETICS OF POLYMERIZATION
The polymer text will probably use the following format:
- Initiation
- Chemical Equation shown
- Differential Equation shown
- Discussion
- Propagation
- Chemical Equation shown
- Differential Equation shown
- Discussion
- Termination
- Chemical Equation shown
- Differential Equation shown
- Discussion
The format will scrutinize
one idea at a time (initiation, propagation, etc.), instead of using the
order of the steps of the polymerization as the order of the discussion.
This format will chose another way. The reaction steps will be shown,
and then differential equations will be shown. The math expressions
will be presented, and then some of the assumptions will be discussed.
VARIABLES
I initiator
k rate constant
subscript
M monomer
M. M. covers both RM. and MM.
MM. the results of a monomer bonding to a M., and of course
this result is also a M. species.
R signifies the rate of something happening
subscript
R. a free radical, the result of the decompostion of an
initiator.
RM. the results of a monomer reacting with a free radical R.
SUBSCRIPT CLASSIFICATIONS
- d- decomposition
- i- initiation
- p- propogation
- t_- termination (several types of termination are possible:)
- tc- termination by coupling
- td- termination by disproportionation
- tct- termination by chain transfer
REACTION STEPS
|
|
| | Decomposition
of initiator
| | 
| | |
| |
|
| | Initiation
| | | 
| | |
| |
|
| | Propagation
| | | 
| | |
| |
|
| | Termination
by coupling
| | | 
| | |
| |
|
| | Termination
by disproportionation
| | | 
| | |
| |
|
| | Termination
by chain transfer
| | |
| | |
| |
|
|
Note that M. represents many different polymer radicals of varying length,
and includes the smallest possible polymer radical, RM., that you see
in the propagation step.
DIFFERENTIAL EQUATIONS
|
|
| | Decomposition
of initiator
| | | 
| | |
|
|
|
| | Initiation
| | | 
| | |
|
|
|
| | Propagation
| | | 
| | |
|
|
|
| | Termination
by coupling
| | | 
| | |
|
|
|
| | Termination
by disproportionation
| | |
| | |
|
|
|
| | Termination
by chain transfer
| | |
| | |
|
|
|
RM. might better be represented by R - M. and
MM. might be better represented by M -M. as well.
M. is both RM. and MM.
RATE VARIABLES
Each of the above differential equations is the definition for a rate,
R(subscript):
R(d) =
R(i) =
R(p) =
R(tc) =
R(td) = 
R(tct) =
There is a second way to define R(i):
R(i) = 
where f represents the efficiency of the initiator.
Not every initiator molecule is
going to produce two RM. molecules.
To explain this backwards, imagine that the initiator has an efficency
of 75% (f=.75), and that the rate of decomposition of initiator is
100 molecules
per microsecond (I don't know if this is a reasonable rate.) For a
100% efficient system, the 100 molecules of initiator would produce
200 radicals, but since the system is 75% efficient, 150 radicals are
produced. The rate of initiation is 150 radicals per microsecond.
- R(d) = 75, and R(i) = 150
- R(i) = 2R(d)
- We know that R(d) = k(d)[I]
- If all the above is true, then R(i) = 2 k(d) f [I].
ASSUMPTIONS
The disappearance of monomer depends both on initiation and propagation.
The production of polymer radicals is solely from the reaction of
monomer with free radicals produced by the initiator, R., and the
termination of polymer radicals occurs only by coupling (no chain
transfer, disproportionation, etc.)
This is the famous "steady state" assumption. It is assumed that the
number of polymer radicals remains constant during the reaction because
the rate of polymer radical initiation equals the rate of polymer
radical termination.
This assumption holds true throughout most of the reaction.
is equal to zero.
Last Update- September 2, 1995- wld