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An adsorption blob is a subchain of size comparable to the adsorbed layer thickness $\xi$. If the free-chain size law is $R\sim b{g}^{\nu }$, it contains $g\sim (\xi /b{)}^{1/\nu }$ monomers. Confinement costs of order $k_{B}T$ per blob produce $F_{\mathrm{conf}}\sim N{k}_{B}T/g$. A simple contact fraction $b/\xi$ gives $F/(N{k}_{B}T)\sim (b/\xi {)}^{1/\nu }-ϵb/\xi$. This model predicts $\xi /b\sim {ϵ}^{-\nu /(1-\nu )}$ and binding gain per monomer of order $k_{B}T{ϵ}^{1/(1-\nu )}$. It assumes a many-blob layer and a monomer-scale attraction range; it does not replace the full surface-critical theory of polymer adsorption.

In chain-growth polymerization, each growth step joins a monomer to a reactive site on a polymer chain and regenerates that reactive site. This is the precise mechanistic restriction in IUPAC's definition: growth occurs exclusively through these monomer–chain-site reactions. The word chain refers to the succession of site-regenerating chemical reactions. It does not merely mean that the product looks like a long chain. Chain-growth initiation starts this succession; chain-growth propagation continues it; chain-growth termination and chain transfer in chain polymerization can change its fate.^[ref]

The contrast with step-growth polymerization concerns the permitted joining partners. There, suitable reactive groups can join two existing chains, for example a two-unit chain and a three-unit chain. In chain-growth propagation, a free monomer adds to a polymerization-active site. Release of a small byproduct is a separate classification question: IUPAC explicitly allows condensative chain polymerization.^[ref] The useful consequence is that a few growing chains can incorporate many monomer molecules while much of the feed remains unreacted. Their degree of polymerization depends on how those units are shared among chains, as finite chain-growth bookkeeping makes explicit.

Chain growth by successive monomer coordination and migratory insertion at a metal-bound active site.

The degree of polymerization counts monomeric units in a chain. $X_n$ is its number average over a specified collection of chains. State whether unreacted monomers are included and distinguish this count from constitutional repeating units or statistical segments. See IUPAC: degree of polymerization.

Let an adsorbed Gaussian chain gain free energy $\mu >0$ per bound monomer. A pulled segment of $n$ Kuhn length steps spanning height $z$ has elastic free energy $3k_{B}T{z}^2/(2n{b}^2)$, giving $F=-(N-n)\mu +3k_{B}T{z}^2/(2n{b}^2)$. Minimizing over $n$ yields $n=(z/b)\sqrt{3k_{B}T/(2\mu )}$ and $F_{\min }=-N\mu +z\sqrt{6k_{B}T\mu }/b$. Its height derivative is the constant peeling force displayed above. The plateau lasts while $0<n<N$ and the Gaussian chain approximation remains valid. In the slab model $F_{\mathrm{ads}}=N{k}_{B}T[(b/\xi {)}^2-ϵb/\xi ]$, optimization gives $\mu =k_{B}T{ϵ}^2/4$ and $f=\sqrt{3/2}ϵk_{B}T/b$.

Ethene is the two-carbon monomer in the polyethylene example in monomer.

Stipulate a feed of exactly $M_0=40$ monomer molecules, four successfully initiated linear chains, one polymerization-active site per chain, no chain transfer in chain polymerization, no loss of incorporated units, and no further chain-growth initiation. Count only initiated chains in $C$; exclude unreacted monomers. Assume each consumed monomer molecule contributes exactly one monomeric unit. With incorporated feed fraction $q=S/M_0$, the number-average degree of polymerization is the counting identityThis is conditional conservation bookkeeping, not a universal empirical law. Immediately after chain-growth initiation, the four lengths are $(1,1,1,1)$, so $(C,A,S)=(4,4,4)$ and $q=0.1$. Stipulate twelve subsequent chain-growth propagation events, three on each chain: the lengths become $(4,4,4,4)$, giving $(C,A,S)=(4,4,16)$, $q=0.4$ and $X_n=4$. Sixteen units have been incorporated, while twenty-four free monomer molecules remain. The chain-length assignment is deliberately chosen, not a prediction of equal growth in an experiment.

With additive drag from $N$ identical monomers, ${\zeta }_N=N{\zeta }_0$ and centre-of-mass diffusion is $D_N=k_{B}T/(N{\zeta }_0)$. For monomer Stokes law ${\zeta }_0=6\pi \eta a$, this gives $D_N\propto N^{-1}$. The model neglects long-range hydrodynamic coupling and does not alone describe the long-time spatial diffusion of a strongly entangled chain.

A functional group is an identifiable group of chemical atoms associated with characteristic chemical reactions. In step-growth polymerization, $A$ and $B$ name compatible reactive groups, not necessarily entire monomers or particular chemical elements.

In a balanced bifunctional feed there are initially $N_0$ $A$ groups and $N_0$ $B$ groups. Define the functional-group conversion in step growth by $p=J/N_0$: the reacted fraction of either group type. Each join consumes one of each. Therefore $N=N_0(1-p)$. In the eight-$AB$ example, three joins give $p=3/8$ and four give $p=1/2$. In the latter state only three of eight starting monomer molecules remain isolated: their disappearance fraction is $5/8$, different from $p=1/2$. Neither this group-based $p$ nor disappearance of isolated monomers is the incorporated-feed fraction in the ideal living polymerization count.

With unequal initial counts $A_0\le B_0$, this article always uses $p=J/A_0$, the conversion of the limiting $A$ groups. The reacted fraction of $B$ groups is $J/B_0=rp$, where $r=A_0/B_0$. The fraction of all initial functional groups consumed is $2J/(A_0+B_0)=2rp/(1+r)$. At nonzero conversion these three fractions coincide only at $r=1$. Naming the measured group type prevents an apparently correct equation from using the wrong conversion.

Formation of a tethered polymer layer by growing chains from reactive sites on a surface. Small monomers reach the growth sites without inserting an entire preformed coil, making high grafting density more accessible than in grafting-to. Initiator availability, termination and control of chain-length distribution remain important.

Use an ideal feed in which every starting molecule has exactly two suitable functional groups: either one $A$ and one $B$, or two $A$ groups mixed with two $B$ groups. Only an $A$–$B$ pairing can form a joining chemical bond. Assume no loss of feed-derived material, no splitting of chains, no competing consumption of the counted functional groups, no branching, and no cyclization in step-growth polymerization. Linear here describes connectivity; a chain can bend in space. Small byproducts of polycondensation do not enter the chain-bearing molecule count.

Let $N_0$ count all initial feed molecules, $J$ the number of intermolecular joins, and $N$ all remaining chain-bearing molecules, including unreacted monomers. Each join turns two molecules into one, so $N=N_0-J$ exactly. If chain $i$ contains $j_i$ feed-derived monomeric units, conservation gives ${\sum }_{i=1}^N{j}_i=N_0$. Consequently the number-average degree of polymerization isThis is a count of both feed species' monomeric units, rather than a count of interior constitutional repeating units. For $AA+BB$ chemistry an interior repeat often contains two feed-derived units, as explained in monomer. Equal functional group reactivities are not needed for this bookkeeping at a specified $J$; predicting a time course or a chain-length probability distribution requires additional assumptions.

For the figure's eight $AB$ feed molecules, three joins can give lengths $(2,3,1,1,1)$, so $N=5$ and $X_n=8/5$. Joining the first two chains gives $(5,1,1,1)$: the incorporated total stays eight, $N$ becomes four, and $X_n=2$. A graph makes the bookkeeping visible: each starting monomer molecule is a vertex of a graph, and each intermolecular joining chemical bond connects two previously distinct connected components of a graph. The connected pieces are the counted molecules.

Lipids are a chemically diverse family of chemical substances, conventionally associated with biological origin and solubility in nonpolar solvents. They include triglycerides, phospholipids, and cholesterol. The name does not specify one molecular formula, one repeating monomer, or one kind of polymer. Some lipids have both hydrophilic and hydrophobic regions rather than being uniformly nonpolar. See IUPAC Gold Book: lipids.

Structure explains the different jobs. A triglyceride has three fatty acid-derived acyl groups attached to glycerol and serves as an energy store. A typical glycerophospholipid instead combines two such chains with a phosphate-containing head, making it an amphiphile suited to a lipid bilayer. Cholesterol has a different structure and also contributes to cell membrane properties. Thus a lipid is a chemical building material, while a lipid bilayer is an organization of many molecules. See Clark, Douglas and Choi, Biology 2e, 3.3: lipid structures and functions.

Living polymerization is chain-growth polymerization without chain termination or irreversible chain transfer. A linear mean degree of polymerization versus feed conversion additionally needs the fixed-chain assumptions in polymerization; reversibility of monomer addition is not required. See Shipp, IUPAC workshop, slide 50.

A simple route to deuterated polystyrene block copolymers by reverse iodine transfer polymerisation by Trevor Gavin Wright and Harald Pasch (2015, Experimental section, pp. 3237–3238) provides a concrete study of consumption and chain products. The authors used a valve-sealed J Young tube, freeze–pump–thaw preparation and argon. NMR monitoring of chain polymerization compared a $25{}^{\circ }\mathrm{C}$ reference with spectra acquired at $70{}^{\circ }\mathrm{C}$ every fifteen minutes for twenty-four hours. Their formulation contained an azo initiating reagent, molecular iodine and a vinyl monomer. Companion flask syntheses used stirring and a heated silicone-oil bath.

The authors followed initiating and transfer-agent signals; Figure 3 reports a transfer-agent signal rising during induction and then falling as polymerization proceeds. Their size-exclusion characterization of chain polymers used porous columns, refractive-index detection and polystyrene calibration. These distinct consumption and size readouts concern transfer-mediated chemistry; they do not establish strict living polymerization. ^[ref]

Useful proposed controls, rather than reported measurements, are a matched mixture omitting the initiating reagent, an unheated mixture, and a repeat of the complete formulation. Keep monomer amount, solvent, tube preparation and acquisition conditions matched where possible. The unheated comparison needs a readout that does not itself initiate polymerization; placing it in the hot probe would defeat that control. A mixture omitting iodine asks a different question: it changes chain transfer in chain polymerization and induction chemistry, so it is not necessarily a no-polymerization control. Record the pre-reaction baseline and compare crude material with isolated product; isolation can selectively remove shorter chains. These comparisons help separate initiation-dependent monomer consumption from signal drift and unrelated chemical reactions.

The mechanistic inference is deliberately limited. Consumption plus independently characterized larger chain products supports polymerization; evidence of an induction stage helps test a proposed reaction mechanism. Neither readout directly counts stars in $P_j^{\ast }$, proves the absence of chain-growth termination, nor recovers every chain length. A chain-extension test would probe retained growth capability, but even a successful extension would not by itself establish that all original chains extended. No measured molecular distribution is reconstructed here. IUPAC cautions that reversible deactivation can produce living-like behaviour even when some termination occurs.^[ref]

A monomer is a chemical substance whose molecules can undergo polymerization and contribute units to the essential structure of a polymer. A monomer molecule is the individual reactant; its incorporated monomeric unit is the portion identifiable as derived from that reactant. Incorporation can change chemical bonds and release small byproducts, so the unit in the chain need not have the complete molecular formula of the feed molecule. See IUPAC: monomer molecule and IUPAC, A brief guide to polymer terminology: monomeric units.

Three different counts answer three different questions. Monomer molecules describe the feed; monomeric units trace that feed's contributions to the product; a constitutional repeating unit describes the smallest chemical pattern repeating along a regular chain. In Nylon 6,10, two different monomers supply one interior constitutional repeating unit,Its molecular formula is ${\mathrm{C}}_{16}{\mathrm{H}}_{30}{\mathrm{N}}_2{\mathrm{O}}_2$. The diamine feed is ${\mathrm{C}}_6{\mathrm{H}}_{16}{\mathrm{N}}_2$, and the diacid-chloride feed is ${\mathrm{C}}_{10}{\mathrm{H}}_{16}\mathrm{C}{\mathrm{l}}_2{\mathrm{O}}_2$; subtracting two $\mathrm{HCl}$ gives the interior-repeat budget. For a finite linear chain made from $n$ of each reactant, there are $2n-1$ joining connections, hence $2n-1$ formal $\mathrm{HCl}$ eliminations, rather than $2n$. One amine end and one acid-chloride end remain in that ideal unmodified chain. This is a stoichiometry check on the depicted reaction, not a claim that the washed experimental thread retains those ends. The reacting end-group chemistry is documented in Hammond, lecture 5, p. 4; the specified Nylon 6,10 reactant pair is reported in Polymer Day, nylon demonstration.

Ethene shows the reverse counting mismatch. One $\mathrm{C}{\mathrm{H}}_2=\mathrm{C}{\mathrm{H}}_2$ monomer molecule contributes a two-carbon $-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-$ monomeric unit to ideal linear polyethylene, without a small byproduct in that growth step. But the smallest constitutional repeating unit of its uniform backbone is just $-\mathrm{C}{\mathrm{H}}_2-$: a chain containing $m$ incorporated ethene monomeric units contains $2m$ such interior one-carbon repeats. End groups are additional bookkeeping. The frequently drawn $[-\mathrm{C}{\mathrm{H}}_2-\mathrm{C}{\mathrm{H}}_2-{]}_m$ is useful for tracing the feed, although it is not the smallest constitutional repeat. The minimum-pattern definition is given in IUPAC: constitutional repeating unit; the ethene growth chemistry is described by ^[ref].

A segment of a freely jointed chain or Gaussian chain instead belongs to a statistical model. It can represent many chemical bonds and several constitutional repeating units; its length and the segment count are chosen to describe chain conformation at the intended scale. The three-repeats-per-segment drawing in Figure "From two liquid layers to a Nylon 6,10 thread: monomers, a constitutional repeating unit, and statistical segments" is a deliberately arbitrary coarse-graining example, not a measured Kuhn length. Always state the counting convention before substituting a segment count for a degree of polymerization. This separation lets chemistry track which chemical atoms entered the material while statistical models track how the material bends and extends.

The number of reactive connections available on a monomer under a specified polymerization chemistry. One-functional units can terminate chains, two-functional units can build chains, and higher functionality can produce branching and networks.

NMR monitoring of chain polymerization uses nuclear magnetic resonance signals to follow monomer and other distinguishable chemical species. In a proton experiment, suitable signal integrals can track changes in their relative amounts. Disappearance of a monomer signal is a consumption readout; it is not a direct count of polymerization-active sites. Overlapping signals, baseline errors and competing chemical reactions can obstruct the inference. A pre-reaction reference and independent product characterization make the consumption interpretation stronger.

Nylon 6,10 has an amide-linked chain whose constitutional repeating unit contains a six-carbon diamine contribution and a ten-carbon diacid-derived contribution. See monomer for its stoichiometry and polymerization for the authored demonstration.

Radical polymerization, a type of chain-growth polymerization. Peroxide bond cleavage initiates reactive chains; propagation adds monomer to a radical end while regenerating a radical end on the longer chain. Termination and chain transfer can then stop or redirect growth. In the propagation step there is no obligatory small-molecule condensate, so this is not a step-growth condensation.