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Polymer adsorption binds a polymer near an attractive interface through contacts of some of its monomers. The binding energy competes with the loss of conformational entropy. A chain need not lie flat: noncontacting loops and tails can extend into the solvent. In a weak-adsorption blob model, the layer thickness exceeds the Kuhn length and local chain statistics remain approximately those of the free chain below that thickness.

Association of multiple polymer chains into a larger solution object.

The connected main-chain skeleton of a polymer, distinguished from attached side groups. Degradation of the backbone can change molar mass and network integrity.

For a strongly segregated, symmetric long-chain polymer blend, take the square-gradient functional $F/(k_{B}T{\rho }_0)=\int [\chi \varphi (1-\varphi )+a^2({\varphi }^{\prime }{)}^2/(24\varphi (1-\varphi ))]dz$. Its first integral equates the two terms, giving ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$. Thus $\varphi =[1+\tanh (z/\ell )]/2$ with $\ell =a/\sqrt{6\chi }$. The inverse-central-slope width $w=1/{\varphi }^{\prime }(0)=2\ell$ differs by a factor two from the tanh-argument scale. The continuum approximation is marginal when these lengths approach the statistical segment size. An observed width can additionally include roughness and instrument broadening.

Polymer bridging occurs when one polymer chain attaches to more than one particle, producing an effective attraction or network connection. Surface affinity and incomplete coverage are essential to this mechanism. It should be distinguished from both steric stabilization by densely coated particles and the depletion force produced by nonadsorbing chains.

Aggregation caused when an adsorbing polymer chain binds more than one particle. Incomplete surface coverage can favour bridges, whereas a densely covered, well-solvated surface can instead be sterically stabilized.

A layer of sufficiently crowded tethered polymer chains stretched away from their attachment surface. In a good solvent, overlap increases the interaction contribution to free energy; extension relieves this crowding at a cost of chain entropy. The mushroom-to-brush crossover occurs at ${\sigma }_g{R}^2\sim 1$, where $R$ is an isolated coil size. In a simple good-solvent scaling model, the brush height is $H\sim Nb(v{\sigma }_g/b{)}^{1/3}$ for effective segment excluded volume $v$ and statistical segment length $b$; if $v\sim b^3$, this becomes $H\sim Nb({\sigma }_g{b}^2{)}^{1/3}$. The scaling requires strong stretching relative to the isolated coil but extension well below the contour length $Nb$.

The path length along a polymer backbone. For $N$ links of length $a$ it is $Na$, whereas the end-to-end distance of a polymer chain cannot exceed it. A statistical segment count need not equal the chemical degree of 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 IUPAC, Basic Definitions of Terms Relating to Polymers, definition 2.2.

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 ${\mathrm{H}}_2\mathrm{N}(\mathrm{C}{\mathrm{H}}_2{)}_6\mathrm{N}{\mathrm{H}}_2$ with $\mathrm{ClC}(=\mathrm{O})(\mathrm{C}{\mathrm{H}}_2{)}_8\mathrm{C}(=\mathrm{O})\mathrm{Cl}$: 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 Paula Hammond, MIT 10.569 lecture 2: step-growth joining and 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, $P_j^{\ast }+M⟶P_{j+1}^{\ast }$, 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 IUPAC: chain polymerization and 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.

A motivating experiment: drawing nylon from a liquid interface. Feldwinn's UCSB demonstration places $25\mathrm{mL}$ of an aqueous chemical solution of hexamethylenediamine, ${\mathrm{H}}_2\mathrm{N}(\mathrm{C}{\mathrm{H}}_2{)}_6\mathrm{N}{\mathrm{H}}_2$, in a $100\mathrm{mL}$ beaker, then gently adds $25\mathrm{mL}$ of sebacoyl chloride, $\mathrm{ClC}(=\mathrm{O})(\mathrm{C}{\mathrm{H}}_2{)}_8\mathrm{C}(=\mathrm{O})\mathrm{Cl}$, 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 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 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 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 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.

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 $p{N}_0$ 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 thereforeThus $p=0.90$, $0.99$, and $0.999$ give $X_n=10$, $100$, and $1000$. Losing the last fraction of unreacted groups matters enormously. See 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 $A_0\le B_0$ initial reactive-group counts, put $r=A_0/B_0$ and now let $p$ denote the fraction of the limiting $A$ groups consumed. Under the same linear, no-ring assumptions, $N_0=(A_0+B_0)/2$ and $N=N_0-p{A}_0$, givingFor $r=0.98$, even complete consumption of $A$ gives $X_n=(1.98)/(0.02)=99$. Stoichiometry controls chain length as well as yield. See 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.

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 $X_n=q{M}_0/C=qR$ 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 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 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.

The fraction of measured material assigned to polymer under a stated separation or detection criterion. For corrected pellet and supernatant measurements, $f_P=P/(P+S)$; the assignment is valid only if sedimentation separates the intended species.

A polymer melt is a dense fluid of polymer chains with little or no solvent. For a long flexible chain in a homogeneous melt, other chains screen its long-range excluded volume interactions: distant segments of the chosen chain displace statistically similar segments belonging to its neighbours. On scales above local stiffness, equilibrium chain dimensions are consequently approximately those of an ideal chain, even though monomers still cannot physically overlap. Entanglements can strongly affect dynamics without changing this leading large-scale equilibrium size law. Crystallinity, strong orientational order or unusual topological constraints require separate treatment.

A dilute layer of end-tethered polymer coils whose lateral separation greatly exceeds the isolated coil size. Each coil retains approximately its isolated-solvent conformation, modified by the impenetrable tethering surface. Increasing the grafting density drives overlap and the crossover to a polymer brush.

The overlap concentration is the concentration at which distinct dilute polymer coils first substantially interpenetrate. It corresponds to roughly one chain per coil volume. If a chain contains $N$ repeat units, has molar mass $M$ and occupies a coil volume $V_{\mathrm{coil}}$, then its chain number density is approximately $1/V_{\mathrm{coil}}$, its repeat-unit number density is $N/V_{\mathrm{coil}}$, and its mass concentration is $M/(N_A{V}_{\mathrm{coil}})$. A scaling convention uses $V_{\mathrm{coil}}\sim R^3$; a commonly chosen geometric convention uses $V_{\mathrm{coil}}=4\pi R_g^3/3$. These give different order-one prefactors because a coil has no sharp surface. For an ideal chain, $R\sim b{N}^{1/2}$ and a site-volume estimate $v_0\sim b^3$ gives overlap volume fraction ${\varphi }^{\ast }\sim N^{-1/2}$. A good solvent chain with the Flory approximation $R\sim b{N}^{3/5}$ instead gives ${\varphi }^{\ast }\sim N^{-4/5}$.

Polymer physics connects the conformations and motion of long molecules to the properties of a material. A polymer can change shape through many segment rotations while its chemical backbone remains intact. This makes the competition between flow and relaxation important: the same chain can behave as a coil in one experiment and an extended object in another. Soft matter includes both these flexible structures and liquid crystals, whose orientational order produces a different route to strong mechanical and optical response.

A characteristic time for a chosen polymer deformation to relax after forcing stops. Its value depends on the measured quantity and the chain model; one polymer can have multiple relaxation modes.

A smallest constitutional unit whose repetition specifies a regular polymer chain.

An interface changes polymer conformations through segment-surface affinity, solvent selectivity and confinement. Adsorbed chains make bound trains, intervening loops and free tails; end-grafted chains cross from isolated mushrooms to overlapping brushes as coverage rises. Surface binding energy competes with chain conformational entropy and crowding.

For $N$ equal scattering sites on one polymer, define $g(\mathbf{k})=N^{-1}{\sum }_{m,n}⟨e^{i\mathbf{k}\cdot ({\mathbf{R}}_n-{\mathbf{R}}_m)}⟩$. It has $g(0)=N$; its normalized form factor is $g/N$. Under a rotationally invariant joint distribution, each separation vector has a uniform direction at fixed magnitude $r$. The angular average is $\frac{1}{2}{\int }_{-1}^1{e}^{ikr\mu }d\mu =\sin (kr)/(kr)$, with value one at $r=0$. Thus $g$ depends only on $k=|\mathbf{k}|$, and $g(k)=N^{-1}{\sum }_{m,n}⟨\sin (k|{\mathbf{R}}_n-{\mathbf{R}}_m|)/(k|{\mathbf{R}}_n-{\mathbf{R}}_m|)⟩$. The diagonal self terms are essential in a finite chain and give the large-$k$ limit one for a discrete Gaussian chain.

A chemical solution with a polymer as a solute. The polymer conformation and interactions depend on the solvent and concentration.

A thermoplastic polymer commonly used as a positive-tone electron-beam resist.