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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 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 solvent under conditions where the leading effective pair excluded volume of a long dilute polymer vanishes. In the simplest Flory–Huggins solution theory this means $\chi =1/2$. The chain is approximately an ideal chain, while three-segment interactions can still affect the osmotic pressure of a concentrated solution. A particular polymer-solvent pair often meets this condition only near its theta temperature.

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.

The Van der Waals equation is a simple real gas equation of state that can also describe a liquid-gas critical point. With particle count $N$ fixed, write the volume per particle as $v=V/N$:or equivalently $(p+a{N}^2/V^2)(V-Nb)=N{k}_{B}T$. Here $p$ is pressure, $T$ is absolute temperature, and $k_B$ is the Boltzmann constant. In SI, $v$ and $b$ have units ${\mathrm{m}}^3$ per particle, $a$ has units $\mathrm{Pa}{\mathrm{m}}^6=\mathrm{J}{\mathrm{m}}^3$, and $k_B$ has units $\mathrm{J}{\mathrm{K}}^{-1}$; the particle count is dimensionless. The pole at $v=b$ excludes smaller volume values from this model.

The $b$ correction represents excluded volume; the attraction term lowers pressure at fixed temperature and particle number density $1/v$. This is a mean-field theory, replacing detailed intermolecular structure by two parameters. At $v\gg b$ with $a/v^2\ll k_{B}T/v$, it approaches the ideal gas law $pv=k_{B}T$. It supplies a useful qualitative route from a dilute gas to phase coexistence, without a claim of quantitative accuracy for every fluid.^[ref]

Do not mix this particle convention with the molar volume convention elsewhere in this article. With amount of substance $n_{\mathrm{mol}}=N/N_A$, Avogadro constant $N_A$, and molar gas constant $R=N_A{k}_B$, one has $v_m=N_{A}v$, $b_m=N_{A}b$, and $a_m=N_A^{2}a$, so $p=RT/(v_m-b_m)-a_m/v_m^2$. The molar parameters have units ${\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$, respectively.

Johannes Diderik van der Waals introduced this equation of state in 1873, connecting molecular attraction with departures from the dilute ideal gas law. Its historical significance is the unified treatment of gases and liquids, rather than a universally exact fit.^[ref]