Rigid Disks at High Density

A technique is developed for estimating the rigid‐disk partition function near close packing, by evaluating the specific free‐energy contributions from correlated motion of larger and larger sets of contiguous particles. In terms of the reduced density θ=A₀/A (A₀ is the close‐packed value of the closed system's area A), the theory leads to an asymptotic series in the limit θ→1 for the Helmholtz free energy per particle, divided by kT, of the form: $$\text{F/NkT∼2}\ln \text{(λ/a)−2}\ln {\mathrm{(\theta }}^{\text{−1}}{\text{−1)+C+D(θ}}^{\text{−1}}\text{−1)+···,}$$ in which C and D are appropriate numerical constants, λ is the mean thermal de Broglie wavelength, and a the disk diameter. It is shown that lattice defects cannot contribute to the above asymptotic series. The constant C is expressed as an infinite series whose leading term is the single‐particle free area result [— ln (√3/2)], and whose successive terms account for high‐compression cooperative motions involving ascending numbers of particles. An approximate value for C is obtained by computing all contributions for four or fewer correlated disks, with a resulting slight decrease below the free area value. The calculated C may be utilized along with machine‐computed rigid‐disk pressures on both fluid and solid isotherm branches to locate accurately the first‐order phase transition by means of a Maxwell double tangent construction. Some concluding remarks are directed to application of the present method to rigid spheres in three dimensions, where the high density stable crystal structure could be established by suitable extension of the present formalism.