Homogeneous Crystallization in Cyclically Sheared Frictionless Grains

Many experiments over the past half century have shown that, for a range of protocols, granular materials compact under pressure and repeated small disturbances. A recent experiment on cyclically sheared spherical grains showed significant compaction via homogeneous crystallization (Rietz et al., 2018). Here we present numerical simulations of frictionless, purely repulsive spheres undergoing cyclic simple shear via Newtonian dynamics with linear viscous drag at fixed vertical load. We show that for sufficiently small strain amplitudes, cyclic shear gives rise to homogeneous crystallization at a volume fraction $\varphi =0.646\pm 0.001$. This result indicates that neither friction nor gravity is essential for homogeneous crystallization in driven granular media. Understanding how crystal formation is initiated within a homogeneous disordered state gives key insights into the old open problem of glass formation in fluids.

Figure 1

(a) Cell of dimensions $(L+2d)\times (1.675)L\times L$ used in the simulations of frictionless spheres of diameter $d$ undergoing cyclic shear with $\theta =A\sin (\omega t)$. (b) To inhibit crystallization, the dark colored spheres on the cell boundaries are fixed in disordered positions; the boundary conditions in the $z$ direction are periodic. The light-colored spheres in the interior of the shear cell move in response to interactions with one another, the dark spheres, and gravity $g$. A constant pressure $P$ is applied on the cell bottom, which can move up and down. (c) The volume fraction $\varphi$ as a function of cycle number, from simulations with shear amplitudes $A$. The orange points are from an experiment with $A=0.01$ [9]; at the start of that experiment $\varphi =0.627$, and after 970 shear cycles, $\varphi$ had increased to 0.640, which is the first experimental point shown here. The black circles with a colored interior show for each $A$ the value of $\varphi$ at the onset of rapid growth of crystallites, as discussed in the results.

Figure 2

The probability distribution function (PDF) of (a) the local volume fraction ${\varphi }_{\text{local}}$ and (b) the local bond orientational order parameter $q_6$. Each PDF is time invariant during the range of simulation shear cycles shown, and the experimental data [9] in (a) are also time invariant for a wide range of shear cycles. The black curves in (a) and (b) are Gaussian fits to the PDFs. (c) At this larger global volume fraction, the PDF for ${\varphi }_{\text{local}}$ from both simulations and experiment contains a second peak at ${\varphi }_{\text{local}}=0.74$, which corresponds to crystallites that have formed with HCP and FCC symmetries, as determined from analyses of particle positions. The black curve is a fit of the data to the sum of a Gaussian $g(x)$ and an inverse Gaussian (i.e., a Wald distribution), $f(x)$: $P(\varphi )=ag({\varphi }_{\text{local}})+(1-a)f({\varphi }_c-{\varphi }_{\text{local}})$, where ${\varphi }_c=\pi /3\sqrt{2}\approx 0.7405$.

Figure 3

The percentage of spheres in crystallites as a function of the volume fraction is shown for simulations with $A=0.03$ to 0.10 rad and an experiment with $A=0.01\mathrm{rad}$; the horizontal bars on the data points show the standard error of the mean. The results from the simulation with $A=0.05\mathrm{rad}$ and the experiment with $A=0.01\mathrm{rad}$ are remarkably similar. The inset shows that for sufficiently small $A$, ${\varphi }_{\text{onset}}=0.646\pm 0.001$.

Figure 4

(a) The difference between the probability that a crystallite will grow rather than shrink during cyclic shear, from simulations with amplitude $A=0.05\mathrm{rad}$ (black squares) and from an experiment with $A=0.01\text{rad}$ (orange circles); in both cases crystallites with more than 10 spheres tend to grow while smaller crystallites tend to disappear. (b) The largest crystallite, the next largest crystallite, and the percentage of spheres in crystallites as a function of the global volume fraction for simulations with shear amplitude $A=0.05\mathrm{rad}$; the data were averaged over 10 simulations with different initial conditions.

Figure 5

(a) Global volume fraction as a function of the cycle number during cyclic shear with amplitude $A=0.0833\mathrm{rad}$ for ten simulations with continuous shear (red dashed line) and ten simulations with a periodically paused shear protocol (black solid line) (see Supplemental Material, Sec. B.5 [20]). The light blue bars in (a), (b), and (c) show the standard deviation $\pm 0.001$ in the onset of crystallization at $\varphi =0.646$. (b) The percentage of spheres in crystallites as a function of $\varphi$ for the simulations in (a). (c) The percentage of spheres that are in crystallites as a function of $\varphi$ for simulations with and without gravity at $A=0.05\mathrm{rad}$. The vertical bars on the data points in (b) and (c) show the standard deviation. In each panel, the shaded gray rectangle shows the volume fraction for the onset of crystallization, $\varphi =0.646\pm 0.001$.