Terminal model of copolymerization ($p_{ij}=k_{ij}{c}_j/{\sum }_{\ell }{k}_{i\ell }{c}_{\ell }$)

The terminal model of copolymerization represents a growing copolymer chain's addition kinetics using only the identity of its final monomeric unit. For two incoming monomers, there are two end types and four chain-growth propagation rate constants $k_{ij}$, where $i$ identifies the existing end and $j$ the incoming monomer. Earlier sequence history is omitted from this kinetic state description.^[ref]

At a fixed reacting-mixture composition, addition rates give a two-state Markov chain. With $f_1+f_2=1$ the monomer mole fractions and $r_1=k_{11}/k_{12}$, $r_2=k_{22}/k_{21}$ the monomer reactivity ratios, its transition matrix isRows label the current terminal unit; columns label the newly added unit. Each row sums to one. This interprets a sequence at fixed feed and neglects initiation, termination and feed drift over the sampled run.

For an original example, take $f_1=f_2=1/2$ and $r_1=r_2=2$. Then $P_{11}=P_{22}=2/3$ and both cross-addition probabilities are $1/3$. The stationary distribution is $(1/2,1/2)$, so the long-chain mean composition is equimolar, while equal consecutive units are favored. A run of type 1 has a geometric distribution with stopping probability $1/3$ and mean length 3. Independent equimolar placement would instead give mean run length 2. Mean composition alone does not determine sequence correlations.

The copolymer composition equation concerns the instantaneous incorporated composition implied by this model. Applying it over a finite conversion interval requires following changes in the reacting mixture. The IUPAC recommendation explicitly bases ratio estimation on conversion and composition observations at different starting feeds.^[ref]

Historically, Frank R. Mayo and Frederick M. Lewis published a 1944 theoretical comparison of monomers with a study of styrene and methyl methacrylate.^[ref] A successful terminal-model fit is evidence about a chosen kinetic representation, not proof that every chemical influence vanishes beyond the last unit.