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Propagation
Propagation proceeds via addition of monomer to the active species, i.e. the carbenium ion. The monomer is added to the growing chain in a head-to-tail fashion; in the process, the cationic end group is regenerated to allow for the next round of monomer addition. [textbook]

(include representative diagram of propagation)

Effect of Temperature
The overall activation energy for the polymerization ($$\mathit{E}$$) is based upon the activation energies for the initiation ($$\mathit{E_i}$$), propagation ($$\mathit{E_p}$$), and termination ($$\mathit{E_t}$$) steps:

$$\textstyle E = E_i + E_p - E_t$$

Generally, $$\mathit{E_p}$$ is less than $$\mathit{E_i}$$ and $$\mathit{E_t}$$, meaning the overall activation energy is negative. When this is the case, a decrease in temperature leads to an increase in the rate of propagation. The converse is true when the overall activation energy is positive. [textbook]

Chain length is also affected by temperature. Low reaction temperatures, in the range of 170-190 K, are preferred for producing longer chains. [textbook] This comes as a result of the activation energy for termination and other side reactions being larger than the activation energy for propagation. [textbook, KM book] As the temperature is raised, the energy barrier for the termination reaction is overcome, causing shorter chains to be produced during the polymerization process. [textbook]

Effect of Solvent and Counterion
The solvent and the counterion (the gegen ion) have a significant effect on the rate of propagation. The counterion and the carbenium ion can have different associations, ranging from a covalent bond, tight ion pair (unseparated), solvent-separated ion pair (partially separated), and free ions (completely dissociated). [Odian, textbook]



The association is strongest as a covalent bond and weakest when the pair exists as free ions. [textbook] In cationic polymerization, the ions tend to be in equilibrium between an ion pair (either tight or solvent-separated) and free ions. [Odian] The more polar the solvent used in the reaction, the better the solvation and separation of the ions. Since free ions are more reactive than ion pairs, the rate of propagation is faster in more polar solvents. [textbook, Raave]

The size of the counterion is also a factor. A smaller counterion, with a higher charge density, will have stronger electrostatic interactions with the carbenium ion than will a larger counterion which has a lower charge density. [Odian] Further, a smaller counterion is more easily solvated by a polar solvent than a counterion with low charge density. The result is increased propagation rate with increased solvating capability of the solvent. [textbook]

Termination
Termination generally occurs via a unimolecular rearrangement involving the counterion or a bimolecular transfer with the monomer. [textbook]

Unimolecular Rearrangement
One method of unimolecular rearrangement is hydride abstraction from the active chain end to the counterion. [textbook, Fahlman, Raave] In this process, the growing chain is terminated, but the catalyst-cocatalyst complex is regenerated to initiate more chains. [textbook, KM book]

(Insert representative reaction scheme of hydride abstraction)

A second method is combination of an anionic fragment of the counterion with the propagating chain end. This not only inactivates the growing chain, but it also terminates the kinetic chain by reducing the concentration of the catalyst-cocatalyst complex. Consequently, this is a more effective termination process. [textbook, Odian]

(Insert representative reaction scheme of anionic attachment)

Bimolecular Transfer
Hydrogen transfer can occur from the active chain end to a monomer. This terminates the growing chain and forms a new active carbenium ion-counterion complex which can continue to propagate, thus keeping the kinetic chain intact. [textbook]

(Insert representative reaction scheme of bimolecular transfer)

Kinetics
The rate of propagation and the degree of polymerization can be determined from an analysis of the kinetics of the polymerization. The reaction equations for initiation, propagation, and termination can be written in a general form: [textbook, KM book]

$$\text{I}^+~+~\text{M}\xrightarrow{k_{i}}\text{M}^+$$

$$\text{M}^+~+~\text{M}\xrightarrow{k_{p}}\text{M}^+$$

$$\text{M}^+\xrightarrow{k_{t}}\text{M}$$

In which I+ is the initiator, M is the monomer, M+ is the propagating center, and $$\mathit{k_i}$$, $$\mathit{k_p}$$, and $$\mathit{k_t}$$ are the rate constants for initiation, propagation, and termination, respectively. For simplicity, counterions are not shown in the above reaction equations. The resulting rate equations are as follows, where brackets denote concentrations:

$$\textstyle\text{rate(initiation)} = k_i[\text{M}][\text{I}^+]$$

$$\textstyle\text{rate(propagation)} = k_p[\text{M}][\text{M}^+]$$

$$\textstyle\text{rate(termination)} = k_t[\text{M}^+]$$

Assuming steady-state conditions, i.e. the rate of initiation = rate of termination: [textbook]

$$ [\text{M}^+] = {k_i[\text{M}][\text{I}^+] \over k_t}$$

This equation for [M+] can then be used in the equation for the rate of propagation:

$$\text{rate(propagation)} = {k_p k_i[\text{M}]^2[\text{I}^+] \over k_t}$$

From this equation, it is seen that propagation rate increases with increasing monomer and initiator concentration.

The degree of polymerization, $$\mathit{X_n}$$, can be determined from the rates of propagation and termination: [textbook]

$$Xn = {\text{rate(propagation)} \over \text{rate(termination)}} = {k_p[\text{M}] \over k_t} $$

Here, the termination reaction is considered to be a result of combination with an anionic fragment from the counterion. Hydrogen transfer from the propagating chain end to the counterion or monomer is considered to be chain transfer since active species are regenerated. [textbook] If chain transfer rather than termination is dominant, the equation for $$\mathit{X_n}$$ becomes [textbook]

$$Xn = {\text{rate(propagation)} \over \text{rate(chain transfer)}} $$