How many contacts a jammed packing needs

Unknowns are contact forces, equations are force balance

Remove the rattlers first. That leaves $N'$ load-bearing disks with $N_c$ contacts between them. Without friction each contact has one unknown: the size of its force along the line of centres, $f_c\ge0$ .

Each disk must balance in $x$ and in $y$ , which gives $2N'$ equations. Stack the contact forces into a column $\mathbf f$ and the equations become a matrix times that column equal to zero. Each matrix entry is just the cosine or sine of one contact direction.

\sum_{c\,\ni\, i} f_c\,\hat{\mathbf n}_{c,i}=\mathbf 0\quad(i=1,\dots,N'),\qquad\text{i.e.}\quad \mathsf A\,\mathbf f=\mathbf 0

\hat{\mathbf n}_{c,i}

is the direction in which contact

c

pushes disk

i

. A periodic cell has no external force: the pressure comes entirely from this set of internal forces that balance one another.

Source: C. P. Goodrich

Deriving 2N′ − 1

In a periodic cell the forces on all disks always add to zero: each contact pushes its two disks equally and oppositely, so the pair cancels. Two of the $2N'$ equations are therefore redundant, leaving at most $2N'-2$ independent ones.

Positive pressure needs contact forces that are not all zero. A homogeneous system has a non-zero solution only with at least one more unknown than independent equations, so $N_c\ge2N'-1$ . This is Eq. (3) of Goodrich, Liu and Nagel, who reach the same number by asking for a positive bulk modulus.

A walled box is different. The walls are fixed, so the total force on the disks need not vanish, and all $2N'$ equations are independent. Then $N_c\ge2N'+1$ , counting disk–wall contacts as contacts. The Lab’s “contacts beyond isostatic” is $N_c$ minus this minimum.

N_c^{\min}=2N'-1\ \ (\text{periodic}),\qquad N_c^{\min}=2N'+1\ \ (\text{walls}),\qquad Z=\frac{2N_c}{N'}\;\ge\;4-\frac{2}{N'}

The last bound is for the periodic cell. As

N'

grows it approaches 4, the isostatic contact number

2d

of frictionless disks.

Source: C. P. Goodrich · C. S. O’Hern

Why rattlers come out first

Chapter 1 showed that a disk with fewer than three load-bearing contacts cannot balance a generic load. Once it is removed, a neighbour may be left with only two, so the removal is repeated layer by layer until every remaining disk has at least three contacts. What remains is the load-bearing backbone.

O’Hern and co-workers report that about 5 % of the particles near jamming are rattlers. In our threshold packings in periodic cells, 24 for each of four sizes, the mean rattler fraction of each size lay between 4.0 % and 5.0 %. Counting them would make $N'$ too large and the contacts look too few.

Source: C. S. O’Hern

Worked example: the contact number of 128 disks at jamming

Put 128 bidisperse disks in a periodic cell, find the state where they have only just jammed with the protocol of Chapter 4, and suppose 6 of them are rattlers (about 4.7 %, in line with the fractions above). That leaves $N'=122$ .

At the minimum count, $N_c=2\times122-1=243$ and the mean contact number is $Z=2\times243/122=3.984$ . The factor 2 is there because every contact belongs to two disks.

Compare with the literature. Table I of O’Hern et al. gives $Z_c=3.98\pm0.02$ for 2D bidisperse systems, which matches. Our benchmarks found exactly $2N'-1$ contacts for every seed with 32 and 64 disks, and at most one extra with 128 and 256. The extra comes from the finite step of the threshold search, which keeps the first jammed state just past the threshold.

Source: C. S. O’Hern · C. P. Goodrich

Friction changes the count

With friction each contact gains a tangential force, so there are $2N_c$ unknowns, and each disk gains a torque equation, so there are $3N'$ equations. That gives $Z\ge3$ , or $d+1$ . Shundyak and co-workers point out that this lets frictional packings sit anywhere between 3 and 4 contacts per disk, depending on how they were made.

At a contact whose friction force has reached the Coulomb limit, the tangential and normal forces are tied together, which adds a constraint. Chapter 5 uses this rule to discuss shear jamming.

Source: K. Shundyak

Try it in the Lab

Count the spare contacts

Predict first: is the count of contacts beyond isostatic closer to zero just past jamming or deep in the squeeze?
Choose Barely jammed and, at the tightest frame, read the spare contacts and the number of rattlers. Then choose Deep squeeze and read the same two numbers.
Open “More numbers”, read the number of load-bearing disks $N'$ , work out $2N'+1$ yourself and check the Lab’s spare-contact count.

Exercises with explained answers

A walled box has $N'=100$ load-bearing disks, and 20 contacts are on the walls. If the contact count is at its minimum, how many disk–disk contacts are there? With the Lab’s definition $Z=(2N_{\mathrm{pair}}+N_{\mathrm{wall}})/N'$ , what is the mean contact number?
Show the answer
The minimum is $2N'+1=201$ ; removing the 20 wall contacts leaves 181 disk–disk contacts. $Z=(362+20)/100=3.82$ , below 4, because the walls supply part of the constraints.
In a periodic cell, what is the mean contact number at jamming for $N'=30$ and for $N'=1000$ ?
Show the answer
It is $4-2/30=3.933$ and $4-2/1000=3.998$ . Small systems sit visibly below 4: the finite-size correction that Goodrich et al. emphasise.
What is the isostatic mean contact number for frictionless spheres in three dimensions, and for frictional ones?
Show the answer
Frictionless: one unknown per contact and three equations per sphere, so $Z=2d=6$ . Frictional: three unknowns per contact (one normal, two tangential) and six equations per sphere (three forces, three torques), so $3N_c\ge6N$ and $Z=2N_c/N\ge4=d+1$ .

Sources and what we recomputed

C. S. O’Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Jamming at zero temperature and zero applied stress: the epitome of disorder, arXiv:cond-mat/0304421v1; Physical Review E 68, 011306 (2003)
Version 1, 17 April 2003. Eq. (1): the finite-range repulsion between soft spheres. Section II: random infinite-temperature starts, conjugate-gradient minimisation in a periodic cell, and the energy test for an unjammed state. Figs. 5 to 7 and Eq. (3): the fraction of jammed states, the distribution of thresholds and its width, which falls as the number of particles to the power −0.55. Table I: pressure and contact-number exponents and prefactors. Fig. 20: a 2D force network. We recomputed the threshold width, the pressure exponent and the contact-number exponent with our own code; the contact-number exponent did not reproduce at 128 disks.
C. P. Goodrich, A. J. Liu and S. R. Nagel, Finite-size scaling at the jamming transition, arXiv:1204.4915v3; Physical Review Letters 109, 095704 (2012)
Version 3, 6 June 2012. Eq. (2): the contact count needed to remove every zero mode except global translations. Eq. (3): one more contact is needed for a positive bulk modulus, so a periodic packing of N disks in two dimensions needs at least 2N − 1 contacts. Eqs. (4) and (5): the corresponding minimum contact number. The finite-size crossover where the usual power laws break down at low pressure. We checked the 2N − 1 count on our own threshold packings.
K. Shundyak, M. van Hecke and W. van Saarloos, Force mobilization and generalized isostaticity in jammed packings of frictional grains, arXiv:cond-mat/0610205v1; Physical Review E 75, 010301(R) (2007)
Version 1, 7 October 2006. The opening paragraphs: with friction, the isostatic contact number drops from 2d to d + 1. Eq. (1) and Fig. 1: each fully mobilised contact, where the friction force sits at the Coulomb limit, adds one constraint, so the number of such contacts per particle is at most z minus d + 1. Fig. 2: the distribution of mobilisation splits into a peak at the limit and a flat part below it. We did not reproduce their simulations.