Polymer

A polymer is a large molecular chain or network built from many chemically linked repeating units. Statistical chain models idealize its microscopic structure by segment vectors and describe its conformation through distributions, correlations and scattering. A freely jointed chain fixes each segment length while allowing independent orientations; a Gaussian chain replaces the segment or coarse-grained displacement distribution by a Gaussian distribution. These ideal models neglect excluded-volume and other segment interactions.

A carboxylic-acid-bearing polymer whose charge and conformation depend on ionization and solution conditions. Single-chain pulling can distinguish low-force configurational compliance from high-force backbone stretching.

Represent a dilute polymer by two beads connected by a spring obeying Hooke's law, with stiffness $H>0$ and bead drag coefficient $\zeta >0$. Neglect bead inertia and hydrodynamic interactions. For component-first velocity gradient $K_{ij}={\partial }_j{u}_i$, the configuration Fokker-Planck probability current isThe factor two comes from relative motion of both beads. At rest its normalized probability density is ${\psi }_{\mathrm{eq}}=(H/(2\pi k_{B}T){)}^{3/2}{e}^{-H{R}^2/(2k_{B}T)}$. The conformation tensor $C=⟨\mathbf{R}{\mathbf{R}}^T⟩$ has dimensions of length squared; $(H/(k_{B}T))C$ is its dimensionless version. The Hookean dumbbell in planar extension is a particular imposed flow in this model.

For a locally uniform distribution of labelled connectors, a connector crosses a small surface when its midpoint lies in a slab of thickness $|\mathbf{R}\cdot \mathbf{n}|$. This gives the displayed unsigned count per unit area. Omitting the absolute value gives a signed count, or requires a restriction to one orientation. The signed spring force transmitted from the positive side to the negative side is $H\mathbf{R}\mathrm{sgn}(\mathbf{R}\cdot \mathbf{n})$, so multiplication by the unsigned count yields traction $nH⟨\mathbf{R}{\mathbf{R}}^T⟩\mathbf{n}$. Each labelled dumbbell is counted once.

For identical no-slip spheres of radius $a$ linked by a zero-rest-length spring, with $R\gg a$ and $\zeta =6\pi \mu a$, the leading Oseen tensor interaction reduces their relative spring relaxation rate by $3a/(2R)$. The midpoint follows an imposed incompressible linear velocity field. Add the two bead force balance equations to prove midpoint advection; subtract them and use $\mathsf{G}(\mathbf{R})\mathbf{R}=\mathbf{R}/(4\pi \mu R)$ to obtain the displayed equation. This deterministic far-field model omits bead inertia, thermal forcing, connector drag and near-contact lubrication.

For constant number density $n$, constant temperature, a homogeneous incompressible linear velocity field and decaying configuration tails, defineThis free energy is a relative entropy times $n{k}_{B}T$. Differentiate it using the configuration Fokker-Planck probability current. One integration by parts gives $\dot{\mathcal{F}}=n\int \psi (H\mathbf{R}+k_{B}T\nabla \log \psi )\cdot \mathbf{V}d\mathbf{R}$. Substituting the probability current velocity gives the displayed balance: the diffusive affine contribution is $-n{k}_{B}T\mathrm{tr}K=0$. The polymer stress tensor ${\sigma }_p=n(HC-k_{B}TI)$ has mechanical power ${\sigma }_p:K=nHC:K$. Its instantaneous power need not be nonnegative, because it includes changes of stored free energy. The nonnegative quantity is $\mathcal{D}$, and equality of stress power and dissipation requires a stationary free energy.

When the midpoint follows a linear velocity field, the two beads' slip velocities are $\pm (\dot{\mathbf{R}}-K\mathbf{R})/2$. Summing their individual drag powers $\zeta |{\mathbf{v}}_{\mathrm{slip}}{|}^2$ proves the displayed expression. Using a probability current velocity in place of the instantaneous relative velocity defines the corresponding overdamped dissipative functional; adding independent thermal velocities is a different stochastic description.

If the relative bead velocity $\mathbf{v}$ is independent of its separation and has a centered Maxwell-Boltzmann distribution with $⟨v_i{v}_j⟩=(2k_{B}T/m){\delta }_{ij}$, averaging the drag dissipation of a bead-spring pair gives the displayed gross loss. The cross term vanishes by the zero conditional mean. At zero velocity gradient, this remains $3n\zeta k_{B}T/m>0$, although mechanical stress power is zero. A thermal bath replenishes microscopic kinetic energy at equilibrium. Equating this gross loss to polymer stress power omits heat-bath input and cannot follow from the independently specified Maxwellian velocity distribution. The overdamped free-energy balance of a Hookean dumbbell uses a probability current velocity instead.

Multiply the Hookean dumbbell model configuration equation by $R_i{R}_j$ and integrate twice by parts, with sufficient decay at large separation. The linear velocity field contributes $KC+C{K}^T$, the spring drift contributes $-4HC/\zeta$, and diffusion contributes $4k_{B}TI/\zeta$. Thus the dimensionless conformation tensor $A=HC/(k_{B}T)$ obeys $\dot{A}-KA-A{K}^T=-(A-I)/\lambda$, with polymer relaxation time $\lambda =\zeta /(4H)$. Its left side is an upper-convected derivative for this gradient convention.

Let $C=⟨\mathbf{R}{\mathbf{R}}^T⟩$ be the unnormalized conformation tensor, $G_{ij}={\partial }_j{u}_i$ the velocity gradient, and $M/\zeta$ a symmetric positive hydrodynamic mobility matrix independent of the individual connector. The connector Fokker-Planck equation has drift $G\mathbf{R}-(2H/\zeta )M\mathbf{R}$ and diffusion matrix $(2k_{B}T/\zeta )M$. Integrating its second moment twice by parts gives the displayed upper-convected derivative, $C^{\nabla }=D_{t}C-GC-C{G}^T$. If $M$ is a polynomial in $C$, the two matrices commute and the relaxation term simplifies to $-(4H/\zeta )M(C-k_{B}TI/H)$. No Gaussian distribution assumption is required for this moment closure.

Repeated transitions between a broad reservoir and a narrow channel can accumulate length-dependent capture and relaxation delays. Uniform inside-channel drift alone may be length independent and therefore insufficient for separation. A time-dependent drive can retain the length-sensitive delays, while stronger advection raises throughput but can erase selectivity. A quantitative ordering requires the actual barrier, chain dynamics and driving protocol.

If only the inserted length $x$ feels a uniform inward electric field, its charge experiences force $qEx$, where $q$ is charge per contour length. Integrating gives electric energy $-qE{x}^2/2$. Adding a constant entropic confinement force gives the displayed free-energy profile. Its interior maximum is at $x=f/(qE)$ and has height $f^2/(2qE)$; entry waiting and subsequent directed insertion are distinct times.

With friction $\gamma (x)={\zeta }_{\ell }x$, inward force $qEx$ gives constant entry speed after capture. With the field off, constant outward entropic force gives $x\dot{x}=-f/{\zeta }_{\ell }$ and a quadratic recoil time. Exterior drag, adsorption and the short final end region are omitted. These ideal transport times alone do not establish steady size separation through a whole uniform channel.

A synthetic aromatic polyamide with a relatively stiff backbone. Dilute-solution light scattering and viscometry provide evidence for rodlike conformations under the conditions investigated.^[ref]

A polymer near an impenetrable nonadsorbing wall has fewer allowed configurations than a coil in the bulk. This entropic loss suppresses its equilibrium density over a scale comparable with its radius of gyration. Strong confinement, adsorption and strong flow can change the profile; electrical neutrality does not remove configurational entropy.

A polymeric material capable of appreciable electronic conduction, commonly through a conjugated backbone with an appropriate charge state. Conductivity, optical absorption, environmental stability and processing history must be specified; the material category does not guarantee a transparent-electrode specification.

A polymer containing pyrrolidone groups, widely used as a soluble stabilizer or capping agent. In silver nanostructure synthesis its adsorption can affect facet growth and aggregation. Residual polymer can also create electrically resistive nanowire junctions.

A polymer or oligomer with repeated ether linkages in its main chain.

A fluorinated polymer whose polar crystalline phases can exhibit piezoelectricity and pyroelectricity.

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

A long chain or network maintained by reversible noncovalent association of molecular units.

The distribution of molar masses among the chains in a polymer sample. Number-weighted and mass-weighted descriptions assign different relative weights to long chains: for populations $N_i$ of chains with molar mass $M_i$, the number-average molar mass is $M_n={\sum }_i{N}_i{M}_i/{\sum }_i{N}_i$, while the weight-average molar mass is $M_w={\sum }_i{N}_i{M}_i^2/{\sum }_i{N}_i{M}_i$. A single average does not determine the full distribution.

A polymer containing repeated amide linkages in its main chain.

Consistent orientation of constitutionally unsymmetrical repeat units along a polymer chain.

A polymer with an extended sequence of conjugated bonds along its backbone.

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

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.

A polymer composed of connected blocks with different repeat-unit chemistry. Incompatible blocks can form spatially distinct domains while remaining covalently connected.

A block copolymer with blocks having different solvent affinities, permitting solvent-selective assembly into micelles and other morphologies.

A thin block copolymer film forming alternating sheet-like domains. Domain period, film thickness and the preferences of both boundaries constrain orientation and termination.

A boundary has approximately equal interfacial free energies against the two blocks. One neutral boundary does not remove selectivity at the other boundary.

The two boundaries prefer opposite blocks. In a simple parallel lamellar stack, opposite terminations favour half-integer repeat thicknesses.

The two boundaries prefer the same block. Parallel lamellae then tend to favour matching terminations and integer-period commensurability.

A block copolymer comprising two covalently joined blocks. Its domain morphology depends on block composition, segregation strength and chain conformational properties.

A domain structure can persist because chain relaxation is too slow to reach the thermodynamic equilibrium morphology. A glassy block, rapid solvent loss or strong segregation can impede defect correction. Controlled thermal or solvent annealing and time-dependent structural measurements distinguish arrested patterns from equilibrated phase stability.

The fraction of dry polymer volume assigned to one block, $f_A=V_A/(V_A+V_B)$. Selective solvent swelling changes an effective domain-volume ratio without changing this dry chemical composition.

The product $\chi N$ of the Flory interaction parameter and degree of polymerization, using compatible reference-segment conventions. It compares unlike-contact avoidance with chain entropy; solvent dilution need not change the chemical chain length.

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

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

Cleavage of a polymer backbone into shorter molecular fragments. It lowers chain length before it necessarily removes all material from a specimen.

A polymeric material that can soften and flow reversibly on heating over its processing range without requiring cleavage of a permanent covalent network.

A connected polymer structure in which cross-links couple chains. Network connectivity, density and defects determine macroscopic elasticity and flow.

A permanently cross-linked polymeric material that does not undergo reversible melt flow like an uncrosslinked thermoplastic.

A polymer whose backbone includes elements other than a carbon-only chain. Backbone composition alone does not determine electrical conductivity or hydrolytic stability.

Polymeric sulfur nitride, $(\mathrm{SN}{)}_x$, with a sulfur–nitrogen backbone and metallic electronic conduction in crystalline samples. Real band structure, packing and defects matter beyond a simple repeat-unit electron count.

A sulfur-chain polymer with local S–S sigma bonds and filled nonbonding electron pairs. These features do not supply a partially filled delocalized conduction band.

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.

Tacticity describes the stereochemical ordering of configurational units along a polymer chain. It is distinct from the head-to-tail or head-to-head connectivity described by regioselectivity.

Polypropylene is formed from propene. Head-to-tail insertion gives the repeat connectivity $-\mathrm{C}{\mathrm{H}}_2-\mathrm{CH}(\mathrm{Me})-$. The stereochemical sequence of its substituted carbons determines its tacticity.

lignin is an aromatic plant-wall polymer that contributes to mechanical and chemical resistance. It can obstruct access to cellulose and adsorb enzymes nonproductively.

A polymer with a repeating Si–O backbone and organic substituents at silicon. Its backbone is flexible but lacks a partly occupied electronic conduction band.

A flexible silicone polymer that can form an elastomer when lightly crosslinked. The network's recoverable deformation provides an example of rubber elasticity.

A polymer with repeat-unit composition ${\mathrm{C}}_5{\mathrm{H}}_8$. Its cis-rich form is the main polymer component of natural rubber.

A biopolymer is a polymer produced by a living system, including proteins, nucleic acids and polysaccharides. Local chemical structure, sequence and interactions can give it stiffness or preferred conformations beyond those of an ideal chain. Biopolymer mechanics uses models such as the worm-like chain and the helix-coil transition model to represent those features.

An elastomer is a polymeric material capable of large substantially recoverable deformation. Flexible chain segments supply configurational entropy, while a network of permanent or sufficiently long-lived cross-links prevents unrestricted macroscopic flow on the observation time scale. Its dynamic modulus depends on whether segmental relaxation can follow the imposed deformation. The rubbery plateau and high-frequency glassy response refer to different accessible relaxation regimes.

A rubbery material based mainly on cis-rich polyisoprene. A lightly crosslinked network provides recoverable deformation through rubber elasticity.

In a flexible crosslinked polymer network, deformation restricts the configurations of chain segments between cross-links. The resulting decrease of entropy produces a restoring entropic force. For an ideal affine Gaussian network, the shear modulus is $G={\nu }_s{k}_{B}T$, where ${\nu }_s$ is the number density of elastically active strands. A nearly incompressible isotropic rubber has Young's modulus $E\simeq 3G$. At fixed network density within the rubbery regime, both increase approximately linearly with absolute temperature. This statement does not describe crossing the glass transition or chemical degradation.

A network of flexible thermally fluctuating strands stores an order-$k_{B}T$ elastic free-energy change per active strand under an order-one deformation. For active-strand number density $n$, its shear modulus is therefore of order $n{k}_{B}T$. If one active element occupies a mesh volume $m^3$, this becomes $k_{B}T/m^3$. The numerical coefficient depends on network connectivity, defects and the deformation model. A mesh size alone does not fix the modulus of a general athermal or bending-dominated network; strand stiffness is then additional input.

A chain of $N$ up/down segments of length $a$ has extension $L=Nax$ and binary configurational entropy. An external force $f(L)=f_{0}L/a$ has potential $-f_0{L}^2/(2a)$, obtained by integrating the force. Its mean-field free-energy minima obey the displayed equation. Expansion gives $F-F(0)=N{k}_B(T-T_c)x^2/2+N{k}_{B}T{x}^4/12+\cdots$, so the ordered extension emerges continuously, $|x|\sim \sqrt{3(T_c-T)/T}$, rather than jumping. The equation matches zero-field mean-field theory of the Ising model; finite systems round the ideal singular behavior.

The ideal rubber elasticity relation infers active strand number density from the shear modulus. If a quoted modulus is instead Young's modulus, nearly incompressible behavior gives ${\nu }_s\simeq E/(3k_{B}T)$. This is not automatically the junction density: for junction functionality $f$ and no dangling ends or loops, counting strand ends gives ${\nu }_s=f{\nu }_j/2$. Network fluctuations and inactive strands modify the modulus, so converting it into a chemical cross-link count requires a network model.

A cross-link connects different segments of a polymer network and restricts their independent motion. Permanent chemical cross-links commonly use covalent bonds; physical associations can instead have finite lifetimes. A connected permanent network can sustain long-time shear stress, whereas an uncrosslinked melt eventually flows. Cross-link density and network structure influence elastomer stiffness.

Polyethylene glycol is a water-soluble polymer with repeating ethylene oxide units. Nonadsorbing dissolved chains can cause a depletion force between larger colloids. Chains attached to a surface can instead give steric stabilization. Which effect occurs depends on surface affinity, chain size and concentration.

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.

An idealized arrangement of confined DNA as approximately concentric loops around a central hole, with neighboring strands locally close-packed. It separates curvature-dependent DNA bending energy from interstrand interactions. It is a geometrical model, not a universal experimentally established arrangement in every capsid.

The change of an isolated flexible polymer from an expanded coil toward a dense polymer globule as its solvent becomes less favorable. In the Flory approximation, $F/(k_{B}T)\sim R^2/(N{a}^2)+v{N}^2/R^3+u{N}^3/R^6$, where the positive three-segment coefficient $u$ prevents collapse to zero size. Positive $v$ gives swollen-coil scaling, $v=0$ gives the theta temperature condition, and negative $v$ favors a dense state with $R^3\sim uN/|v|$. Finite chain length rounds the change; polymer aggregation is a distinct multi-chain process and must be controlled in an experiment.

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.

A polymer conformation model assigning allowed torsional states and statistical weights to successive backbone bonds. Correlations between torsional states can represent local stiffness beyond the fixed-angle, uniformly rotating freely rotating chain.

The effective volume governing a dissolved macromolecule’s hydrodynamic behavior and pore accessibility.

A dilute-solution bead-spring model including solvent-mediated hydrodynamic interactions. In its scaling regime, a chain of size $R$ has translational diffusion coefficient of order $k_{B}T/({\eta }_{s}R)$ and longest relaxation time of order ${\eta }_s{R}^3/(k_{B}T)$. Intermediate-time segment mean-square displacements scale as $t^{2/3}$. Screening of hydrodynamic interactions in a concentrated solution changes the regime toward Rouse model behaviour; constraints from polymer entanglement require additional physics.

An overdamped bead-spring model of polymer dynamics with local friction and no long-range hydrodynamic interaction. For an ideal unentangled chain of $N$ identical frictional units, the centre-of-mass diffusion coefficient is proportional to $N^{-1}$ and the longest internal relaxation time to $N^2$. A tagged segment has an intermediate-time mean-square displacement proportional to $t^{1/2}$ before ordinary whole-chain diffusion dominates. Entanglement adds constraints beyond this model.

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 polymer chain tethered by an end to a surface. Its surface number density ${\sigma }_g$ sets a typical lateral separation $D\sim {\sigma }_g^{-1/2}$. Comparing $D$ with the isolated coil size determines whether grafted chains behave almost independently or must stretch cooperatively away from the surface.

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.

Formation of a tethered polymer layer by attaching reactive ends of preformed chains to a surface. The chains can be characterized before attachment, but inserting further coils into an existing polymer brush costs conformational entropy and produces a coverage-dependent barrier. Reaction chemistry and surface accessibility also limit the final grafting density.

The number of grafted chains per unit substrate area. For roughly uniform attachment spacing $D$, it scales as $D^{-2}$.

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 $Nb$.

A dense uniform polymer brush with $\varphi \simeq 1$ must have height $H\simeq N{v}_{0}s$ by polymer-brush volume fraction. The contour bound $H\le Na$ therefore requires $s\le a/v_0$. If $v_0=a^3$, the limit is $s{a}^2\le 1$; beyond it the putative incompressible branch demands chains longer than their polymer contour length. This is a failure of that microscopic packing model. A Kuhn length does not alone determine $v_0$.

Conservation of material in a uniform polymer brush gives its mean volume fraction as $N{v}_{0}s/H$, with material volume $v_0$ per counted segment, chain length $N$, grafting density $s$ and height $H$. Identifying $v_0$ with $a^3$ is an extra coarse-graining assumption. The Alexander-de Gennes brush model then gives $\varphi \sim (s{a}^2{)}^{(3\nu -1)/(2\nu )}$, including powers $2/3$ in a good solvent and $1/2$ in a theta solvent.

In the Alexander-de Gennes brush model, a theta solvent gives $D\sim a{g}^{1/2}$. Stacking $N/g$ blobs then gives the displayed height, with mean polymer volume fraction $\varphi \sim (s{a}^2{)}^{1/2}$ for segment volume $a^3$. Cancellation of the two-segment excluded volume does not remove three-segment repulsion or make the crowded layer freely interpenetrable at no cost.

A scaling model of overlapping end-grafted polymers assigns each chain a column of lateral width $D\sim s^{-1/2}$ at grafting density $s$. Each concentration blob contains $g\sim (D/a{)}^{1/\nu }$ statistical segments, and stacking $N/g$ blobs gives polymer brush height $H\sim ND/g$. For assumed segment volume $a^3$, the mean volume fraction is $\varphi \sim N{a}^{3}s/H$. It assumes a long many-segment-blob brush and supplies scaling powers rather than a precise chain-end distribution.

End-grafted chains are approximately isolated when their lateral spacing ${\sigma }_g^{-1/2}$ exceeds their coil size. When the spacing becomes smaller, excluded-volume crowding favors extension normal to the substrate. The crossover criterion is a scaling estimate, not a sharp phase transition.

In a laterally uniform dense polymer brush with approximately fixed polymer volume fraction $\varphi$, conservation of occupied volume gives $\varphi h=N{v}_0\sigma$, with repeat-unit volume $v_0$. At fixed solvent quality this gives $h\propto N\sigma$, unlike the good-solvent brush height. Isolated collapsed polymer globules and laterally segregated grafted layers need not obey this uniform-layer model.

For an overlapping layer of long flexible grafted polymers, concentration blobs of width $D\sim {\sigma }^{-1/2}$ contain $g\sim (D/a{)}^{5/3}$ statistical units under the three-dimensional Flory approximation. A chain of $N/g$ blobs has height $h\sim ND/g$, giving the displayed good solvent scaling. Numerical prefactors are not supplied by this argument; extension must remain below the polymer contour length.

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.

A polymer entanglement is a constraint on a chain's motion caused by the inability of neighbouring chain contours to pass through one another. It is not a permanent chemical cross-link. In a dense polymer melt, these constraints can create long relaxation times and a transient rubbery plateau. Chains eventually release constraints through mechanisms such as reptation and motion of their neighbours. Entanglement therefore strongly affects dynamics even when the equilibrium size of a long chain remains approximately that of an ideal chain because excluded-volume interactions are screened.

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.

The end-to-end vector joins the two ends of a polymer chain. In an isotropic unforced polymer solution, its vector expected value is zero; a useful measure of chain size is its root-mean-square length. For an ideal chain with $N$ independent isotropic steps of length $b$, the cross terms in the square of the step sum have zero expected value, so $R_{\mathrm{rms}}^2=N{b}^2$. For a long Gaussian chain, its radius of gyration obeys $R_g^2=R_{\mathrm{rms}}^2/6$. If $N$ counts chemical repeat units instead of independent Kuhn length segments, the statistical length used in $R_{\mathrm{rms}}^2=N{b}^2$ must be calibrated to that choice of $N$; it is not automatically the physical bond length or the Kuhn length itself.

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.

The end-to-end distance of a polymer chain with $N$ fixed-length links of length $a$ cannot exceed its polymer contour length $Na$. A Gaussian chain spring has no such bound, so it eventually becomes an unphysical extrapolation. For a three-dimensional freely jointed chain under force $f$, the mean extension along the force is $Na[\coth x-1/x]$, where $x=fa/(k_{B}T)$. Its small-$x$ limit is $Nf{a}^2/(3k_{B}T)$, and its large-$x$ limit approaches $Na$ from below.

Reptation is the snake-like motion of a long polymer through the constraints imposed by neighboring chains or a fixed porous matrix. A tube model restricts large transverse motion while allowing curvilinear motion along the chain contour. For a flexible chain with additive segment friction, contour diffusion scales as $N^{-1}$ and the time to renew a contour of length proportional to $N$ scales as $N^3$. With coil size squared proportional to $N$, this gives a centre-of-mass diffusion coefficient proportional to $N^{-2}$. Applying this tube picture to weak-field DNA gel electrophoresis gives an inverse-length mobility over its appropriate regime; field-induced orientation and trapping can invalidate that simple law.