David A. Young

The quotidian equation of state (QEOS) is a general-purpose equation of state model for use in hydrodynamic simulation of high-pressure phenomena. Electronic properties are obtained from a modified Thomas–Fermi statistical model, while ion thermal motion is described by a multiphase equation of state combining Debye, Grüneisen, Lindemann, and fluid-scaling laws. The theory gives smooth and usable predictions for ionization state, pressure, energy, entropy, and Helmholtz free energy. When necessary, the results may be modified by a temperature-dependent pressure multiplier which greatly extends the class of materials that can be treated with reasonable accuracy. In this paper a comprehensive evaluation of the resulting thermodynamic data is given including comparison with other theories and shock-wave data.

A new integral equation in which the hypernetted-chain and Percus-Yevick approximations are "mixed" as a function of interparticle separation is described. An adjustable parameter $\ensuremath{\alpha}$ in the mixing function is used to enforce thermodynamic consistency. For simple $\frac{1}{{r}^{n}}$ potential fluids, $\ensuremath{\alpha}$ is constant for all densities, and the solutions of the integral equations are in very good agreement with Monte Carlo calculations. For the one-component plasma, $\ensuremath{\alpha}$ is a slowly varying function of density, but the agreement between calculated solutions and Monte Carlo is also good. This approach has definite advantages over previous thermodynamically consistent equations.

The equations of state for periodic systems of hard disks and hard spheres in the solid phase have been accurately determined and used to evaluate the coefficients in the expansion of the pressure in powers of the relative free volume, α = (V − V0) / V0, where V0 is the close-packed volume. For disks pV / NkT = 2 / α + 1.90 + 0.67α + O(α2) and for spheres pV / NkT = 3 / α + 2.56 + 0.56α + O(α2). These coefficients are compared to cell models, and those models which include correlations between neighboring particles work best. An equivalent expansion of other thermodynamic properties requires the entropy constant to be evaluated in the close-packed limit. This constant is obtained here by integrating the equation of state over the entire density region. The Lennard-Jones–Devonshire cell-theory estimates of the entropy constant are nearly correct; that is, the cell-theory estimate is too small by 0.06Nk for disks and too large by 0.24Nk for spheres. The pressure difference and hence the entropy difference between the hexagonal and face-centered cubic packings of spheres could not be detected, and thus the relative stability of these two phases remains an open question.

The validity of the augmented van der Waals theory of fluids is demonstrated by a rigorous evaluation of the next three terms in the reciprocal temperature expansion of the Helmholtz free energy for the square-well fluid. Each term in the dense fluid is shown to be an order of magnitude smaller than the preceding one except the fourth, which is comparable to the third. Analysis of molecular dynamics data leads to a peak in the heat capacity at the coexistence curve which is not reproduced by the finite series. This peak in turn leads to realistic values of the critical indices α and α′ while the other indices β, γ, and δ have their classical van der Waals values.

Numerical hydrodynamic simulations of the growth and collapse of a 10 μm air bubble in water were performed. Both the air and the water are treated as compressible fluids. The calculations show that the collapse is nearly isentropic until the final 10 ns, after which a strong spherically converging shock wave evolves and creates enormous temperatures and pressures in the inner 0.02 μm of the bubble. The reflection of the shock from the center of the bubble produces a diverging shock wave that quenches the high temperatures (≳30 eV) and pressures in less than 10 ps (full width at half maximum). The picosecond pulse widths are due primarily to spherical convergence/divergence and nonlinear stiffening of the air equation of state that occurs at high pressures. The results are consistent with recent measurements of sonoluminescence that had optical pulse widths less than 50 ps and 30 mW peak radiated power in the visible.