Jamming Transition in Granular Systems

Recent simulations have predicted that near jamming for collections of spherical particles, there will be a discontinuous increase in the mean contact number $Z$ at a critical volume fraction ${\varphi }_c$. Above ${\varphi }_c$, $Z$ and the pressure $P$ are predicted to increase as power laws in $\varphi -{\varphi }_c$. In experiments using photoelastic disks we corroborate a rapid increase in $Z$ at ${\varphi }_c$ and power-law behavior above ${\varphi }_c$ for $Z$ and $P$. Specifically we find a power-law increase as a function of $\varphi -{\varphi }_c$ for $Z-Z_c$ with an exponent $\beta$ around 0.5, and for $P$ with an exponent $\psi$ around 1.1. These exponents are in good agreement with simulations. We also find reasonable agreement with a recent mean-field theory for frictionless particles.

Figure 1

(a) Schematic cross section of biaxial cell experiment (not to scale). Two walls can be moved independently to obtain a desired sample deformation. (b) Examples of contacts and particles that are either close but not actually in contact, or contacts with very small forces. Circles show true contacts, squares show false apparent contacts. (c) Image of a single disk at the typical resolution of the experiment. (d) Sample image of highly jammed/compressed state and (e) almost unjammed state.

Figure 2

Average contact number and pressure at the jamming transition. Top and bottom panels show $Z-Z_c$ and $P$ vs $\varphi -{\varphi }_c$, respectively, with rattlers included (stars) or excluded (diamonds). Dashed and full curves in the top panel give power-law fits $(\varphi -{\varphi }_c{)}^{\beta }$ with $\beta =0.495$ and 0.561 for the case with and without rattlers, respectively. Full curve in the lower panel gives the fit $(\varphi -{\varphi }_c{)}^{\psi }$ with $\psi =1.1$; dashed line shows a linear law for comparison. Inset: $Z$ vs $\varphi$ for a larger range in $\varphi$.

Figure 3

Results from new computer simulations. For all fits ${\varphi }_c=0.84005$. (a) Average overlap per particle in units of the mean particle radius is linear in $\varphi -{\varphi }_c$. (b) $P$ obtained from the Cauchy stress tensor (circles) and the force on the walls (squares) satisfy a power law $(\varphi -{\varphi }_c{)}^{\Psi }$ with $\psi =1.13$; dashed line shows a linear law for comparison. (c) $Z$ (rattlers included) exhibits a power law $Z-Z_c\propto (\varphi -{\varphi }_c{)}^{\beta }$ with $Z_c=3.94$ and $\beta =0.5015$.

Figure 4

Pressure vs $Z$; experimental data and a fit to the model of Henkes and Chakraborty [7]. In this fit, the constant $C$ defined in the text is treated as an adjustable parameter. The other fitting parameter is $Z_c$.