Microscopic and macroscopic stress with gravitational and rotational forces

Many recent papers have questioned Irving and Kirkwood’s atomistic expression for stress. In Irving and Kirkwood’s approach both interatomic forces and atomic velocities contribute to stress. It is the velocity-dependent part that has been disputed. To help clarify this situation we investigate (i) a fluid in a gravitational field and (ii) a steadily rotating solid. For both problems we choose conditions where the two stress contributions, potential and kinetic, are significant. The analytic force-balance solutions of both these problems agree very well with a smooth-particle interpretation of the atomistic Irving-Kirkwood stress tensor.

Figure 1

Gravitational isothermal equilibrium at unit temperature for $n_x{n}_y=96\times 96=9216$ moving particles above $6\times 96=576$ boundary particles fixed at the bottom of the system. The width of the system is $n_x=96$. The height is unbounded. The field strength $g=4/n_y$ is chosen so that the maximum density matches that of the fixed particles at the bottom: $\rho =N/V=2$ at $y=0$. This snapshot is typical of a long simulation used to calculate the smooth-particle pressure profiles shown in Fig. 2. In all of the figures dimensionless (or “reduced”) units are used. These follow from the definitions of unity for the particle mass, Boltzmann’s constant, and the length and energy scales in the interparticle forces derived from the cusp potential in Sec. 2 and the Hooke’s-law potential in Sec. 4.

Figure 2

Comparison of the observed and analytic pressure profiles for the gravitational problem shown in Fig. 1. From top to bottom the three curves are the total ($T$ or Irving-Kirkwood), kinetic $\left(K\right)$, and potential ($\Phi$ or Zhou) contributions to the pressure profile. These observed pressure contributions are calculated as smooth-particle averages. The points correspond to the analytic expressions from the isothermal equation of state $P_T=P_{\Phi }+P_K=\left({\rho }^2/2\right)+\rho$.

Figure 3

Stationary rotation snapshots of two 2335-particle Hooke’s-law crystals. In the rotationless stress-free case all 6828 nearest-neighbor distances are unity. In the steady rotational situations shown here, both with an angular frequency $\omega =0.01$, the tensile strain offsetting the centrifugal forces is maximized at the center of the rotating solid. The left view is a cold solid. The right view has a temperature $kT=0.01$.

Figure 4

Angular velocity dependence of the cold-crystal maximum tensile stress on rotation rate. The molecular-dynamics data, shown here as points, for nearly circular solids of the type shown in Fig. 3, agree with the linear-elastic result (shown as a straight line in the figure) for disks to four figures as the rotation rate goes to zero. The linear-elastic result is ${\sigma }_{\text{max}}V/N=\left(5N\sqrt{3/4}/12\pi \right){\omega }^2$.

Figure 5

View of a rotating 2335-particle Hooke’s-law crystal at an angular velocity of 0.01 and a temperature $kT=0.02$.

Figure 6

$PV$ in the rotating cold crystal in Fig. 3 with $\omega =0.01$. The theoretical radial and circumferential components are shown as lines based on the expressions derived in the Appendix.

Figure 7

Time-averaged stresses in the warm rotating thermally excited crystal in Fig. 3 with $\omega =0.01$ and $kT=0.01$. The thermal contributions to ${\left(PV\right)}_{rr}$ and ${\left(PV\right)}_{\theta \theta }$ are the points at the top. The theoretical expressions for the stress (based on the cold-crystal elastic constant) shown as lines in the figure agree well with the points representing results from molecular dynamics. The molecular-dynamics results include both the potential and kinetic contributions to the comoving corotating stresses.