File:Hard-sphere phase diagram pressure vs packing fraction.png

Phase diagram of hard sphere system (Solid line - stable branch, dashed line - metastable branch): Reduced pressure ${\displaystyle \beta P\sigma ^{3}}$ as a function of the volume fraction (or packing fraction) ${\displaystyle \eta }$.

With inverse thermal energy ${\displaystyle \beta =(kT)^{-1}}$, sphere diameter ${\displaystyle \sigma =2R}$, packing fraction ${\displaystyle \eta =V_{0}\rho }$, volume of a single sphere ${\displaystyle V_{0}=4\pi R^{3}/3=\pi \sigma ^{3}/6}$, number density ${\displaystyle \rho =N/V}$, system volume ${\displaystyle V}$, total particle number ${\displaystyle N}$.

Vertical grey lines indicate:

• freezing: ${\displaystyle \eta _{\mathrm {f} }\approx 0.494}$
• melting: ${\displaystyle \eta _{\mathrm {m} }\approx 0.545}$
• random close packing: ${\displaystyle \eta _{\mathrm {rcp} }\approx 0.644}$
• close packing: ${\displaystyle \eta _{\mathrm {cp} }=\pi {\sqrt {2}}/6\approx 0.74048}$

Reference:

• A. Mulero: Theory and Simulation of Hard-Sphere Fluids and Related Systems. Springer 2008, ISBN 978-3-540-78766-2