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Placement of geometric objects without interior overlap. A packing fraction measures how much area or volume they occupy. Circle packing and sphere packing are examples.

A hard-sphere gas is a classical model of identical impenetrable spheres, each of diameter $\sigma$ and radius $\sigma /2$. Its pair potential is $u(r)=+\infty$ for centre separation $r<\sigma$ and $u(r)=0$ for $r\ge \sigma$; the convention at exact contact has no effect on the integral. It isolates the effect of particle size without an attractive potential energy.

The upper-left panel is a cross-section of a three-dimensional forbidden region: the second centre cannot enter a sphere of radius $\sigma$ around the first centre. The particles themselves have radius $\sigma /2$. Consequently, with particle volume $v_{\mathrm{s}}$ and pair excluded volume $v_{\mathrm{ex}}$,The last equality follows directly from the Mayer f-function: $f=-1$ throughout the forbidden volume and zero outside it. Confusing particle radius with contact separation changes the answer by a factor of eight; omitting the unordered-pair factor changes it by a further factor of two.

Define the packing fraction $\eta =n{v}_{\mathrm{s}}$, where $n$ is number density. Then the virial expansion beginsFor example, at the toy value $\eta =0.01$, the leading correction predicts $Z\approx 1.04$; the omitted terms still matter if accuracy beyond this approximation is needed. The classical second virial coefficient is independent of temperature because the geometry of forbidden configurations does not change when the gas is warmed. The pressure itself still scales with temperature at fixed number density.

This model makes the positive correction particularly clear: eliminating overlapping configurations changes the dependence of the available configuration space on volume, even though the allowed configurations all have zero interaction energy. The figure's other panels add a toy attractive shell and show how its Boltzmann factor competes with that geometry. The hard-sphere gas is a useful reference model, while a low-order virial expansion remains a dilute-gas approximation rather than a dense sphere packing formula.

The stipulated DNA volume is $V_D=20000\mathrm{n}{\mathrm{m}}^3$ and the spherical capsid volume is $V_c=4\pi (20\mathrm{nm}{)}^3/3=33510\mathrm{n}{\mathrm{m}}^3$. ThusThe packed DNA occupies a substantial fraction of the available space: the remaining nominal fraction is $40.3\%$. Compared with the paper's stated $0.71$ reference, it reaches about $84\%$ of that value. This indicates dense confinement, but does not prove crystalline order or specify interstrand spacing. DNA is a connected elongated polymer, rather than a collection of independent hard spheres.

There is also a numerical qualification: the exact densest equal-sphere packing fraction is $\pi /(3\sqrt{2})\simeq 0.74048$, not $0.71$. Using it changes the comparison to about $81\%$ without changing the calculated $0.597$ volume fraction or its physical interpretation.