Copolymer

A polymer containing monomeric units derived from more than one monomer species.

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.

A monomer reactivity ratio compares two propagation rate constants for the same terminal chain type in a binary copolymerization. With $k_{ij}$ denoting addition of monomer $j$ to an end of type $i$,These are homopropagation-to-cross-propagation ratios, as defined in the IUPAC terminology.^[ref] Each is dimensionless when numerator and denominator use matching rate conventions. Neither is simply the ratio of the two monomers' total consumption rates.

A chain-end addition probability depends on available concentrations as well as on the rate constants. For a type-1 end at fixed mixture mole fractions, the terminal model of copolymerization givesFor stipulated $r_1=0.5$, an equimolar mixture gives $p_{11}=1/3$. At $f_1=0.8$ it gives $p_{11}=2/3$, whereas at $f_1=0.2$ it gives $p_{11}=1/9$. Even though $k_{11}<k_{12}$ in each case, sufficient abundance of monomer 1 can make self-addition more frequent. These are illustrative kinetic parameters, not measured constants for a named commercial polymer.

The pair $(r_1,r_2)$ determines both rows of the fixed-feed transition matrix. When $r_1{r}_2=1$, its two rows coincide for nonzero feed fractions, giving independent additions within the idealized fixed-feed model. That does not require the incorporated fraction to equal the feed fraction. For example, $r_1=2$, $r_2=1/2$ and $f_1=1/2$ give probability $2/3$ of adding type 1 from either end.

Experimentally, a ratio is associated with the specified monomer pair, reaction conditions and kinetic model. It is not a universal index belonging to an isolated monomer. The IUPAC recommendation uses several starting compositions together with conversion and incorporated-composition measurements to estimate the two ratios.^[ref] One measured copolymer composition at one feed generally cannot identify two unknown ratios. Joint uncertainty and possible composition drift belong to the interpretation.

The copolymer composition equation, also called the Mayo-Lewis equation, relates the instantaneous monomer-consumption composition to the current reacting mixture under the terminal model of copolymerization. For binary mixture mole fractions $f_1+f_2=1$, let $F_1$ be the fraction of type-1 units among the units being incorporated at that instant. Thenwhere $r_1,r_2$ are monomer reactivity ratios.^[ref] The equation concerns molar incorporation, not the mass fraction or cumulative composition of a finished specimen.

A short derivation identifies the steady-chain-end assumption. Let $c_i$ be monomer concentrations and $E_i$ concentrations of active ends. Cross-additions interconvert the end types; balancing them gives $k_{12}{E}_1{c}_2=k_{21}{E}_2{c}_1$. The two positive consumption rates areSubstitution yields $v_1/v_2=c_1(r_1{c}_1+c_2)/[c_2(c_1+r_2{c}_2)]$. Normalizing $F_1=v_1/(v_1+v_2)$ gives the stated equation. This neglects additional sources of end-type changes and uses a terminal kinetic description.

For an original example with $r_1=r_2=2$ and $f_1=0.20$, the numerator is 0.24 and the denominator is 1.68, so $F_1=1/7\approx 0.143$. The instantaneous polymer is poorer in type 1 than the available mixture. At equimolar feed the same ratios give $F_1=1/2$; this composition equality does not imply independent sequence placement.

For a finite closed batch, conversion bookkeeping for binary copolymers relates the changing monomer pool to cumulative incorporated composition. Treating an initial $f_1$ as constant after substantial selective consumption can therefore misinterpret the final sample. The IUPAC recommendation addresses conversion and integrated composition data explicitly.^[ref] Useful experiments distinguish current feed, conversion and product composition instead of treating all three as the same label.

A statistical copolymer contains more than one kind of monomer unit, with their sequence distribution following a specified statistical rule. A random copolymer is a special case when an appropriate independence model describes the sequence.

Monomer reactivity can produce correlations, composition drift, or blocks longer than a simple random model predicts. Describing only the average monomer fractions therefore does not determine the complete sequence structure or material properties.

An ideal dilution estimate for a random copolymer whose comonomer units are excluded from the crystal. Let $x$ be the crystallizable-unit fraction in the melt and $\Delta h_f$ the molar enthalpy of fusion per such unit. Mixing contributes $RT\ln x$ to its liquid chemical potential. Approximate the unmixed liquid-minus-solid chemical-potential difference by $\Delta h_f(1-T/T_m^0)$. Equating the sum to zero gives the displayed relation. For $x<1$, the predicted melting temperature decreases. Nonideal mixing, finite crystallizable sequences, lamellar surface energy and comonomer inclusion can change the result.

A copolymer containing $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CHCl}-$ and $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{OH})-$ units. It describes the intended chemical composition, not a guarantee that unselective alkaline treatment yields an undegraded polymer.

A copolymer containing $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CHCl}-$ and $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{OAc})-$ units. Hydrolysis changes the acetate side groups without requiring cleavage of the carbon backbone, provided competing degradation is controlled.

Alkaline conditions that hydrolyse acetate side groups can also promote elimination from chlorinated backbone units. Dechlorination, unsaturation, changed solubility and possible subsequent cross-linking complicate isolation of an ideal vinyl chloride-vinyl alcohol copolymer. The competition is condition-dependent: analytical work used staged treatment and precipitation to restrain dehydrochlorination, so difficulty under generic conditions is not a proof that every hydroxyl-containing chloride copolymer is impossible.^[ref]

A copolymer with an ideal sequence in which the probability of a monomeric unit is independent of neighbouring units. A real copolymer made by concurrent radical reactions can have sequence correlations; a feed ratio does not by itself establish incorporated composition or independent-unit statistics.