Determination of Errors for type of contact and total number of contacts
The data presented here result from MRI’s with voxels of .5mmx.5mmx.5mm.
Since the edge size of the thetrahedra is 2.43cm (+/-.02cm, about a 1 percent
polydispersity) a voxel is about 2% of a die edge. There are ~ 1000 voxels in
each face and 4 faces. Therefore the center of a tetrahedral should be
determined to .5mm /(4000)^(1/2) or ~.01mm. This would be the accuracy if there
were no noise in the image and if the particles were monodispersed. Altogether
the biggest inaccuracy is fitting the particles to a monodispersed set of perfect
tetrahedral. As such we find interpenetration of fitted particles with an overlap of
up a voxel, hence about a +/-2% accuracy in determination of position and
orientation.
The real problem is trying to find out whether particles touch and the nature of
the contact, face to face, vertex to face etc. We defined a set of criteria. If two
fitted particles share one or more common or contiguous voxels we say the
particles may be touching and we analyze this “cloud of points” for its geometry.
There are four parameters in our analysis:
P_radius – the size of the inscribed radius (relative to the average) of the
tetrahedra used to define the fitted perfect tetrahedra to the images, this allows
for the polydispersity of the particles. The “cloud of points” increases as P_radius
increases.
P_theta – angle between touching faces, small angle implies face to face, larger
angle implies edge to face.
P_S – the size of the “cloud of points” (in voxels), essentially the radius of
gyration of the cloud. Small clouds are considered as point touches, larger clouds
as edge-face, face-face..
P_R – anisotropy as indicated by the ratio of the two primary axes of the “cloud
of points” in the plane of the cloud. More isotropic implies face-face, more
anisotropic implies edge-face.
With these parameters as thresholds we then allowed variations in all four
variables, P_theta = 5,15,25,35,45,55, P_radius =.95,1,1.05, P_S=3,4,5, P_R =
.3,.4,.5,.6,.7,.8,.9. A typical plot of the total number of constraints, <Z>, as a
function of one of these parameters with the others held fixed is shown below
(from Fig. 37 Jaoshvili thesis NYU 2009, [P_R in the figure’s x-axis is 10 x the
anisotropy]).
The average number of each type of contact and the total number of contacts
and constraints and their statistical errors bars obtained by taking 6x3x3x7=378
cases of different parameters is what is plotted in Fig. 4 in the manuscript.
The uncertainty in each measurement is huge as is evident from the error bars in
the figures. However, when directly averaged over the whole range of
parameters considered the computed average number of constraints is 12.1 and
the standard deviation is .95.
We tried to be as unbiased in this evaluation as possible. A similar experiment
and analysis that we did on ~ 1.5 cm plastic ellipsoids of aspect ratio 0.8:1:1.25
produced by 3D lithography gave a contact and constraint number of 8, far from
the frictionless isostatic limit and probably relating to the roughness of the
surfaces (lithographic resolution of 0.2mm). In the original analysis that we
presented at the APS March Meeting in 2007, where we had 1mmX1mmX1mm
voxels, a different sample and a less refined computer analysis we reported a
constraint number of 14.5 +/- 3.
We should mention that we were not too concerned about whether our value for
<Z> was precisely the isostatic one or just approximately isostatic. The point we
wanted to make is that you don’t get even close to isostatic if you use the number
of contacts (~6) rather than the number of constraints (~12).
Since the reader cannot immediately evaluate the uncertainty we have stated:
“Although there are only 6.3 +/- .5 touching neighbors on average, face-face and
edge-face contacts provide enough additional constraints, 12 +/- 1.6 total, to
roughly bring the structure to the isostatic limit for frictionless particles.” The
uncertainty of +/-1.6 is more accessible. There are large error bars on the last
figure in figure 4 corresponding to the fraction of particles with 8 constraints, 9
constraints etc. Simply doing the statistics on these large error bars the reader
will find <Z> = 11.97+/- 1.6.