Tailoring the frictional properties of granular media

A method of modifying the roughness of soda-lime glass spheres is presented, with the purpose of tuning interparticle friction. The effect of chemical etching on the surface topography and the bulk frictional properties of grains are systematically investigated. The surface roughness of the grains is measured using white-light interferometry and characterized by the lateral and vertical roughness length scales. The underwater angle of repose is measured to characterize the bulk frictional behavior. We observe that the coefficient of friction depends on the vertical roughness length scale.

Figure 1

Scanning electron microscope images of the surfaces of MoSci glass spheres of diameter 146 $\pm$ 19 $\mu$m (small Mo-Sci). Left: Smoothed by NaOH. Center: Untreated. Right: Etched for 4 min. The white bars are 10 $\mu$m long.

Figure 2

Optical micrograph of a glass sphere etched in HF. The white bar is 50 $\mu$m long. Instead of homogeneous surface roughening, large craters are produced.

Figure 3

Principle of operation of a vertical scanning interferometric microscope.

Figure 4

Height map of the cap of a large Mo-Sci grain etched for 4 min.

Figure 5

The method employed to measure roughness takes account of the spherical topology of the grain. The height difference correlation $\rho \left(L\right)$ between pairs of points $r(\theta ,\phi )$ and $r^{\prime }({\theta }^{\prime },{\phi }^{\prime })$, separated by their great-circle distance $L$ is calculated. $R$ is the radius of the grain.

Figure 6

The roughness, $\rho \left(L\right)$, on a double logarithmic scale, yields the vertical and lateral saturation length scales, ${\xi }_{\text{vert}}$ and ${\xi }_{\text{lat}}$, here for a Cataphote grain etched for 30 s. The slope of the fit through the first five data points is the Hurst exponent $H$.

Figure 7

The mean vertical roughness length scale ${\xi }_{\text{vert}}$ as a function of ${\mathrm{NH}}_4\mathrm{FHF}$ etch time. In this and all subsequent plots, the error bars are the standard deviation; for each grain type 10–15 grains were measured. The linear fit through small Mo-Sci points and the means of ${\xi }_{\text{vert}}$ are in the range of 30 s to 4 min.

Figure 8

The mean lateral roughness length scale ${\xi }_{\text{lat}}$ as a function of ${\mathrm{NH}}_4\mathrm{FHF}$ etch time.

Figure 9

The mean Hurst exponent $H$ as a function of ${\mathrm{NH}}_4\mathrm{FHF}$ etch time.

Figure 10

Angle of repose measurement method. (a) A sketch of the setup: A cell [1] 25 cm wide, 30 cm high, and 40 grain diameters deep is filled with water. The reservoir [2] is filled with the granular sample. The funnel rod [3] is raised by a screw thread to open the hole enough to let grains flow to the bottom part of the cell [4]. After an avalanche has occurred, the rod is lowered enough to induce granular arching, thus stopping the flow of grains. (b) CCD image of the pile. Linear fits to the bottom and top edges of the pile give the angle of repose (see text).

Figure 11

Underwater angle of repose as a function of roughening etch time. The smoothing protocol, NaOH, is plotted on the negative $x$ axis. The error bars are the standard deviation. The coefficient of friction is computed by $\mu =\tan \alpha$, where $\alpha$ is the angle of repose [33].

Figure 12

Angle of repose $\alpha$ as a function of correlation length scales (a) ${\xi }_{\text{vert}}$ and (b) ${\xi }_{\text{lat}}$.

Figure 13

When analyzing the roughness of the glass spheres, we need to make an angle-of-sight correction to account for our having measured height in the $z$ coordinate instead of in the radial direction. $R$ is the sphere radius, and $\Delta z(x,y)=h(\theta ,\phi )$ is the height of the grain profile measured vertically from above. The correct radial height of the feature is given by $\Delta r=\frac{\Delta z}{\cos \theta }$.

Figure 14

Radial average and standard deviation of a large Mo-Sci grain etched for one minute. This grain is accepted: both lines remain within the range $[-{\xi }_{\text{vert}}...{\xi }_{\text{vert}}]$ for $L<5{\xi }_{\text{lat}}$.

Figure 15

Radial average and standard deviation of a different large Mo-Sci grain etched for 1 min. This grain is rejected: the standard deviation exceeds the range $[-{\xi }_{\text{vert}}...{\xi }_{\text{vert}}]$ for $L<5{\xi }_{\text{lat}}$.

Figure 16

Maximum volume of cola foam as a function of etch time (small Mo-Sci).

Figure 17

Angle of maximum stability ${\alpha }_{\text{max}}$ as a function of packing fraction for two types of grains: small Mo-Sci etched for 4 min (purple closed symbols) and smoothed in NaOH (green open symbols).