Hypothesis/Assumption in polymer reaction:
Long chain hypothesis (LCH): The consumption of monomer by chain-initiation or transfer events is negligible compared to that by propagation. Thus, the rate of polymerization (disappearance of monomer) can be taken as equal to the rate of propagation ($-R_{monomer}=R_{pol}=R_{prop}$) with the rate of heat generation proportional to the rate of this exothermic reaction.
Quasi-Steady-State Assumption (QSSA): With a continuous source of new radicals in the system, an equilibrium is achieved instantaneously between radical generation and consumption, such that $R_{init}=R_{term}$.
The small radical species $ I^* $, $ M^* $ and $ S^* $ are not consumed by side reactions and do not accumulate in the system, but are converted to polymeric radicals with 100% efficiency. Thus, the total rate of polymer radical formation is given by ($R_{init}=R_{tran}$). The net formation of polymeric radicals is $R_{init}$, since transfer events both consume and create a polymeric radical species.
The rates of propagation, transfer, and termination reactions are independent of n, the length(s) of radical(s) involved.
The rate of propagation is significantly higher than any other reactions in the system. $R_{prop}\ggg R_{init},R_{term},R_{tran}\cdots$
1. Initiation
1.1 Initiator Decomposition
\[I\xrightarrow{k_d}2fI^*\] \[R_{d}=2fk_d[I]\]Where $I$ stands for initiator, $I^*$ is the primary radicals, $f$ is initiation efficiency (typically $0.6<f<0.8$), $k_d$ is the rate coefficient value, and $R$ is the reaction rate ($mol\over{L\cdot S}$).
1.2 Half-Life
Now, let’s assume that there is a batch reactor which has constant volume.
As ${d[I]\over dt}=-k_d[I]$, and assume $[I]\mid _{t=0}=[I]_0$, then:
\[\begin{align} d[I]\over dt&=-k_d[I]\\ d[I]\over [I]&=-k_ddt\\ Integration: \int{1\over [I]}d[I]&=\int{-k_d}dt\\ \ln [I]&=-k_dt+C\\ Exponentiate\ both\ side: [I]&=e^{-k_dt+C}\\ Denote\ e^C\ as\ another\ constant\ value\ A: [I]&=A\cdot e^{-k_dt} \end{align}\]As $[I]\mid _{t=0}=[I]_0$:
\[\begin{align} [I]_0&=A\cdot e^0\\ A&=[I]_0 \end{align}\]Therefore:
\[[I]=[I]_0\cdot e^{-k_dt}\]The decomposition kinetics is often expressed by the half-life of the initiator, the time needed for the concentration to decrease to half of its initial value.
Assume $[I]=1/2[I]_0$ at half-life, then:
\[\begin{align} 1/2[I]_0&=[I]_0e^{-k_dt_{1/2}}\\ \ln 1/2&=-k_dt_{1/2}\\ -\ln 2&=-k_dt_{1/2}\\ t_{1/2}&={\ln 2\over k_d} \end{align}\]1.3 Chain Initiation
Assume primary radicals react with monomer, and we will have:
\[I^*+M\xrightarrow{k_{init}}P_1\]Where $M$ stands for monomer, $P_n$ is the growing radical with $n$ unit long.
The reaction rate for primary radical in the system is the difference between the generation of primary radical (initiator decomposition) and the consumption of primary radical (chain initiation).
\[\begin{align} R_{I^*}&=R_d-k_{init}[I^*][M]\\ &=2fk_d[I]-k_{init}[I^*][M] \end{align}\]As QSSA in the hypothesis, the consumption of free radical equals to the generation of it.
\[R_{I^*}\approx 0\] \[2fk_d[I]=k_{init}[I^*][M]\]According to the reaction of chain initiation,
\[R_{P_1formation}=k_{init}[I^*][M]\]Then,
\[R_{P_1formation}=k_{init}[I^*][M]=2fk_d[I]\]According to the reaction rate $R_{P_1formation}=2fk_d[I]$, we can conclude that:
\[I\xrightarrow{k_d}2fP_1\]2. Propagation
For a chain propagation, such as:
\[P_1+M\rightarrow P_2\Rightarrow P_2+M\rightarrow P_3 \Rightarrow \cdots\]Which can be generalized as:
\[P_n+M\xrightarrow{k_{prop}}P_{n+1},\ n=1,2,3\cdots\]According to the assumption, the reaction in polymer synthesis is independent to the length of radicals involved, and hypothesis from LCH assume all the monomer participates in the chain propagation, we have:
\[-R_{mon}=R_{prop}=k_p[M]\cdot\sum_{n=1}^{\infty}{[P_n]}\]Where $\sum_{n=1}^{\infty}{[P_n]}$ is defined as $[P_{tot}]$.
Therefore,
\[R_{prop}=k_p[M][P_{tot}]\]3. Termination
Define $D_n$ as the product polymer OR a dead polymer chain which loses the free radical and is no longer reactive in the system.
Termination can be classified into 2 types:
\[Combination:\ P_n+P_m\xrightarrow{k_{t,c}}D_{n+m}\] \[Disproportionation:\ P_n+P_m\xrightarrow{k_{t,d}}D_n+D_m\]As the reaction rate independent to the chain length,
\[R^{radical\ loss}_{term}\Rightarrow k_{t,c}+k_{t,d}=k_t\]Since the radicals have the chance to terminate with other radicals of any length, we have the reaction rate for radical loss:
\[\begin{align} R^{radical\ loss}_{term}&=k_t[P_1]([P_1]+[P_2]+[P_3]+\cdots+[P_n])\\ &+k_t[P_2]([P_1]+[P_2]+[P_3]+\cdots+[P_n])\\ &\vdots\\ &+k_t[P_n]([P_1]+[P_2]+[P_3]+\cdots+[P_n]) \end{align}\]Since $[P_{tot}]=[P_1]+[P_2]+[P_3]+\cdots+[P_n]$,
\[R^{radical\ loss}_{term}=k_t[P_{tot}]^2=(k_{t,c}+k_{t,d})[P_{tot}]^2\]As mentioned in the hypothesis, the consumption of radicals is equal to the generation of radicals (QSSA), we can conclude that:
\[\begin{align} R_d&=R^{radical\ loss}_{term}\\ 2fk_d[I]&=k_t[P_{tot}]^2\\ [P_{tot}]&=\sqrt{2fk_d[I]\over k_t} \end{align}\]As $[P_{tot}]$ is proportional to $[I]^{1/2}$, an increase in $[I]$ would result in a higher concentration of radicals in the system, leading to a greater volume of polymers. However, $[M]$ remains a fixed value, indicating that the chain length would decrease with an increasing number of radicals. This can also be explained by experiments, as short-chain polymers are more prone to termination with each other.
4. Transfer
Radicals can react with monomers or solvent transferring their radicals to the other.
\[Monomer:\ P_n+M\xrightarrow{k^{mon}_{tr}}D_n+P_1\] \[Solvent:\ P_n+S\xrightarrow{k^{sol}_{tr}}D_n+P_1\]The total transfer rate is:
\[R_{tr}=k^{mon}_{tr}[M][P_{tot}]+k^{sol}_{tr}[S][P_{tot}]\]