= Polymerization
<Polymerization> converts a <monomer>, or a mixture of <monomers>, into a <polymer>: repeated <chemical reactions> establish a connected molecular chain or network. The central question is how connections form, not merely whether a material becomes solid. Joining small <molecules> can produce a thread that can be handled, while the individual <molecules> remain invisible. See https://old.iupac.org/publications/pac/1974/pdf/4003x0477.pdf[IUPAC, Basic Definitions of Terms Relating to Polymers, definition 2.2].
\Image[/polymerization-monomers-interface-units.png]
{title=From two liquid layers to a <Nylon 6,10> thread: <monomers>, a <constitutional repeating unit>, and statistical segments}
{id=polymerization-monomers-interface-units}
{description=Original schematic of the authored UCSB <polymerization> demonstration described below. Chemical formulas identify the actual reactants and interior repeat; colored blocks identify their contributions. The lower coarse-graining sketch deliberately groups three repeats per model segment. Shapes, sizes, and that grouping are illustrative, not measured molecular geometry or a fitted <Kuhn length>.}
In <step-growth polymerization>, complementary reactive groups on <monomers> and on shorter or longer chains can react with one another. A two-unit chain can join a three-unit chain to make a five-unit chain; growth does not require a specially active end to add only free <monomers>. An example is the <Nylon 6,10> reaction of $\mathrm{H_2N(CH_2)_6NH_2}$ with $\mathrm{ClC(=O)(CH_2)_8C(=O)Cl}$: each <amine> end can form an <amide> connection with an acid-chloride end, with formal elimination of $\mathrm{HCl}$. Both reactants have two suitable ends, allowing repeated joining. See https://ocw.mit.edu/courses/10-569-synthesis-of-polymers-fall-2006/5474dc576b0669a8a09e9b70e39b1239_lec02_09082006.pdf[Paula Hammond, MIT 10.569 lecture 2: step-growth joining] and https://ocw.mit.edu/courses/10-569-synthesis-of-polymers-fall-2006/d9f9ac7021b0d0f418d3d3c90dbfb94b_lec05_09152006.pdf[Hammond, lecture 5, p. 4: acid chloride–amine reaction].
In <chain-growth polymerization>, a <monomer> reacts with an active site on a growing chain, and the growth step regenerates an active site. Schematically, $P_j^*+M\longrightarrow P_{j+1}^*$, where $j$ counts incorporated <monomeric units> and the star denotes reactivity rather than a particular <electric charge>. <Ethene> forming <polyethylene> is an example: the double <chemical bond> is converted into single <chemical bonds> as the <carbon> backbone grows. Initiation creates the active chains; subsequent reactions can deactivate them or transfer reactivity to another <molecule>. See https://goldbook.iupac.org/terms/view/C00958/plain[IUPAC: chain polymerization] and https://openstax.org/books/organic-chemistry/pages/31-1-chain-growth-polymers[John McMurry, Organic Chemistry, 31.1: chain growth]. Whether growth is stepwise or carried by an active site is a different question from whether small byproducts are released: <chain-growth polymerization> can also release them.
\b[A motivating experiment: drawing nylon from a liquid interface.] Feldwinn's UCSB demonstration places $25\,\mathrm{mL}$ of an aqueous <chemical solution> of hexamethylenediamine, $\mathrm{H_2N(CH_2)_6NH_2}$, in a $100\,\mathrm{mL}$ beaker, then gently adds $25\,\mathrm{mL}$ of sebacoyl chloride, $\mathrm{ClC(=O)(CH_2)_8C(=O)Cl}$, in an organic <solvent>. The procedure specifies hexane; other parts of that handout list cyclohexane. The layers remain separate. The instructor catches material at their interface with tweezers and winds it slowly onto a glass rod; the handout reports continued spooling of <Nylon 6,10>. It specifies freshly prepared acid-chloride <chemical solution> and washing the thread before touching it. These are observations and conditions of a supervised laboratory demonstration, not measurements of chain lengths. See https://people.chem.ucsb.edu/feldwinn/darby/DemoLibrary/DemoPDFs/Demo044.pdf[Feldwinn, UCSB Demonstration Library, Demo 044, pp. 1–3].
The mechanistic interpretation is that the <monomers> encounter one another near the interface, where <amide> connections produce the film; withdrawing it exposes a region where more material can form. An authored research demonstration independently reports a rapidly formed interfacial film collapsing into a thread on withdrawal, using cyclohexane and an aqueous <chemical solution> containing potassium hydroxide. See https://pubs.acs.org/doi/10.1021/acs.jchemed.6b00767[Polymer Day: Outreach Experiments for High School Students, Journal of Chemical Education (2017), nylon demonstration]. This is the striking observation: a thread much longer than the beaker can emerge because material is continually supplied to the formation region, rather than because a pre-existing thread was hidden in the liquid.
Useful \i[proposed controls], not results reported in those demonstrations, omit one <monomer> at a time while retaining the same <solvents>, liquid amounts, and pulling procedure. The prediction is loss of a renewable drawable <Nylon 6,10> film; merely finding cloudiness would not establish <polymerization>, since other <chemical reactions>, including acid-chloride <hydrolysis>, can occur. See https://openstax.org/books/organic-chemistry/pages/21-4-chemistry-of-acid-halides[McMurry, 21.4: competing acid-chloride hydrolysis]. Stopping and restarting withdrawal would probe renewal at the interface. Even successful drawing does not measure the <degree of polymerization>, establish a chain-length <probability distribution>, or prove the entire growth mechanism. Chemical identification and a molecular-size measurement would be needed for those stronger conclusions. These controls and limits are inferences from the setup, not invented experimental data.
\b[Why almost complete reaction matters in step growth.] For ideal linear <step-growth polymerization>, assume exactly two reactive groups per starting <molecule>, equal numbers of complementary groups, only joining of different chains, no rings, no branching, and no loss of chain material. Let $N_0$ count starting <monomer molecules>, including both reactant species, and let $p$ be the fraction of either type of reactive group consumed. There are initially $N_0$ groups of each type, so $pN_0$ connections reduce the number of chain-bearing <molecules> to $N=N_0(1-p)$; the count excludes small byproducts but includes unreacted <monomers>. The number-average <degree of polymerization>, measured in <monomeric units>, is therefore
$$
X_n=\frac{N_0}{N}=\frac{1}{1-p}.
$$
Thus $p=0.90$, $0.99$, and $0.999$ give $X_n=10$, $100$, and $1000$. Losing the last fraction of unreacted groups matters enormously. See https://ocw.mit.edu/courses/10-569-synthesis-of-polymers-fall-2006/c7b7a987f7d5b66e0832be42fbf52b7d_lec03_09112006.pdf[Hammond, lecture 3: balanced step-growth conversion]. This counts residues of both feed species; a <Nylon 6,10> interior <constitutional repeating unit> contains two such <monomeric units>.
For an $AA+BB$ feed with $A_0\leq B_0$ initial reactive-group counts, put $r=A_0/B_0$ and now let $p$ denote the fraction of the \i[limiting $A$ groups] consumed. Under the same linear, no-ring assumptions, $N_0=(A_0+B_0)/2$ and $N=N_0-pA_0$, giving
$$
X_n=\frac{1+r}{1+r-2rp}.
$$
For $r=0.98$, even complete consumption of $A$ gives $X_n=(1.98)/(0.02)=99$. <Stoichiometry> controls chain length as well as yield. See https://ocw.mit.edu/courses/10-569-synthesis-of-polymers-fall-2006/5474dc576b0669a8a09e9b70e39b1239_lec02_09082006.pdf[Hammond, lecture 2, pp. 2–3: stoichiometric imbalance]. The liquid-interface demonstration is not a homogeneous, balanced reactor, so its thread length cannot be inserted into this formula as a molecular chain length.
\Image[/polymerization-monomers-conversion-models.png]
{title=Two ideal <degree of polymerization> counts: late growth in <step-growth polymerization> and fixed-chain growth in ideal <living polymerization>}
{id=polymerization-monomers-conversion-models}
{description=Toy analytic curves, not measured data. Left: $X_n=(1+r)/(1+r-2rp)$, with $p$ the limiting-group conversion and $r$ the initial reactive-group ratio. Right: $X_n=qR$, with $q$ the fraction of feed <monomer molecules> incorporated and $R$ their initial number per initiated chain. The two horizontal axes count different entities; the right plot represents a special ideal model, not all <chain-growth polymerization>.}
\b[A contrasting chain count.] In ideal <living polymerization>, assume no chain termination or irreversible chain transfer, all chains initiated before the counted growth, one chain per effective initiator, a fixed chain count $C$, and every consumed <monomer molecule> incorporated into those chains. If $M_0$ is the initial number of feed <monomer molecules> and $q$ their incorporated fraction, conservation gives $X_n=qM_0/C=qR$ in <monomeric units>, excluding initiator fragments. For $R=200$, incorporating half the feed gives $X_n=100$. If initiation consumes one <monomer molecule> per chain, the fixed-chain interval starts at $q=1/R$, rather than $q=0$. See https://ocw.mit.edu/courses/10-569-synthesis-of-polymers-fall-2006/42d2e82b585136f5b6849919a0ee4449_lec21_11012006.pdf[Hammond, lecture 21, p. 1: ideal living-chain count]. The mean does not imply identical chain lengths. <Living polymerization> does not require reversible loss of <monomers>; the defining absence of chain termination and irreversible chain transfer is explained in https://iupac.org/wp-content/uploads/2017/03/IUPAC_PolymEdu_EduWorkshop_PPT_Shipp2.pdf[Devon Shipp, IUPAC Polymer Education workshop, Radical Polymerizations II, slide 50]. Ordinary <chain-growth polymerization> with new chains, termination, or transfer need not follow the straight line.
The useful connection is between local chemistry and a macroscopic material: selecting reactive ends and their proportions controls connectivity, while the resulting <polymer>'s conformation is a separate problem described by models such as the <freely jointed chain> or <Gaussian chain>. Formation of <chemical bonds>, the count of <monomeric units>, and the count of statistical segments must be kept distinct.
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