Polymerization Kinetics

Now that we have a working understanding of the reactions and processes that can happen during free radical polymerization, we can begin to discuss the polymerization kinetics for free radical polymerization. We start from the very beginning and define the rate of polymerization:

R p =− d[ M ] dt
Eq 5.1

As shown in Eq. 5.1, we are defining the rate of polymerization as the rate at which monomer M is consumed. Note the negative sign, which is there because we are defining the rate as the disappearance or consumption of monomer.

You may want to refresh your memory of rate equations before going much further!

Rate of Initiation:

We should be able to write a rate equation for each "step" in free radical polymerization: initiation, propagation, and termination. Let's start with initiation; there are actually two parts to initiation: 1) the formation of the radical from decomposition of the initiator and 2) transfer of the radical to a monomer.

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Figure 5.1:
Source: Lauren Zarzar

The first step of initiation, generation of the radicals, is generally much slower than transfer of the active center to the monomer. Thus, the first step of initiation is the rate determining step, meaning that the rate of initiation is really only a function of the slow step, formation of radicals. Thus, we can represent the rate of initiation as the rate of formation of radicals:

R i = d[ R· ] dt
Eq 5.2

Notice that we don't have anything about monomer concentration in rate of initiation. Because we consume so little monomer during initiation, we neglect any changes in monomer concentration here. But we can make this more specific. Let's say we generate two radicals per initiator molecule, which is common, and that the rate of dissociation of the initiator has a rate constant of kd.

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Figure 5.2:
Source: Lauren Zarzar

We can thus re-write the rate for first step of initiation as:

R i = d[ R· ] dt =2 k d [ I ]
Eq 5.3

However, we also know that not all the radicals that are produced actually get transferred to monomer; only some fraction of them make it through step 2 of initiation. Radicals may combine, react with solvent, etc. To account for the fact that not all radicals produced in step 1 are actually used, we introduce the concept of initiator efficiency, f. For example, If half of our radicals produced actually get used and transferred to monomer, then f =0.5. We can update our description of the rate of initiation:

R i =2f k d [ I ]
Eq 5.4

Rate of Propagation:

Propagation involves the transfer of the radical from one monomer to another monomer, or to polymer. We are using up lots of monomer in this step.

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Figure 5.3:
Source: Lauren Zarzar

The rate of disappearance of monomer (note the negative sign in the rate!) is a function of the rate constant, the monomer concentration, and concentration of active centers (M·)

R prop =− d[ M ] dt  = k p [ M ][ M· ]
Eq 5.5

Note that this is the only place where we directly account for the rate as consumption of monomer (but NOT the radical species).

Rate of Termination:

Recall that we have two different pathways for termination: combination and disproportionation.

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Figure 5.4:
Source: Lauren Zarzar

We write the rate equations for these two termination pathways:

R tc =− d[ M· ] dt  =2 k tc [ M· ][ M· ]
Eq 5.6
R td =− d[ M· ] dt =2 k td [ M· ][ M· ]
Eq 5.7

The coefficient of two results from the fact that two growing polymers are consumed by either pathway.

Add up these two rates (ktc +ktc = kt ) to get an overall rate equation:

R t =− d[ M· ] dt =2 k td [ M· ] 2 +2 k tc [ M· ] 2 =2 k t [ M· ] 2
Eq 5.8