Frictional Rigidity Percolation: A New Universality Class and Its Superuniversal Connections through Minimal Rigidity Proliferation

We introduce two new complementary concepts, frictional rigidity percolation and minimal rigidity proliferation, to help identify the nature of the frictional jamming transition as well as significantly broaden the scope of rigidity percolation. To probe frictional rigidity percolation, we construct rigid clusters using a (3,3) pebble game for sliding and frictional contacts first on a honeycomb lattice with next-nearest neighbors, and second on a hierarchical lattice. For both lattices, we find a continuous rigidity transition. Our numerically obtained transition exponents for frictional rigidity percolation on the honeycomb lattice are distinct from those of central-force rigidity percolation. We propose that localized motifs, such as hinges connecting rigid clusters that are allowed only with friction, could give rise to this new frictional universality class. And yet, the distinction between the exponents characterizing the spanning rigid cluster for frictional and central-force rigidity percolation is small, motivating us to look for a limit in which they are identical, i.e., a search for mechanisms of superuniversality. To achieve this goal, we construct a minimally rigid cluster generating algorithm invoking generalized Henneberg moves, dubbed minimal rigidity proliferation. For both frictional and central-force rigidity percolation, these clusters appear to be in the same universality class as connectivity percolation, suggesting superuniversality between all three transitions for such minimally rigid clusters. These combined results allow us, for the first time in rigidity percolation, to directly compare two universality classes on the same lattice and to highlight unifying and distinguishing concepts of rigidity transitions in disordered systems.

Popular Summary

Take a collection of randomly packed particles that collectively act as a rigid solid, remove one particle, and the ensemble may destabilize and become “floppy.” Understanding the nature of this transition—which appears to depend on the type of lattice and the type of force—could help a breadth of applications from understanding grain-silo collapse to designing universal robotic grippers. However, no one has yet done an apples-to-apples comparison between, say, systems comprising the same type of lattice but different types of forces. Here, we introduce two new concepts to help identify the nature of this rigidity transition, and we take a first step toward quantifying how universal it is.

Our new concepts are called “frictional rigidity percolation,” in which a rigid cluster emerges as particles are placed randomly on a lattice, and “minimal rigidity proliferation,” where particles are placed strategically to maintain an initial minimally rigid cluster. To explore these concepts, we simulate a pebble game, which allows us to map out the rigid clusters in the system.

The pebble game lets us identify two types of transitions, one for frictional forces and another for central forces. We propose that a “hinge” connecting rigid clusters in the frictional case may give rise to this distinction. And yet, there is an aspect of both types of transitions that appears to be the same when looking at just minimally rigid clusters using minimal rigidity proliferation. Such clusters are surprisingly related to connectivity clusters, which contain no information about forces at all.

We are currently mapping out rigid clusters in experimental data to predict and test how the rigid clusters merge and ultimately span the system as it is being sheared.

Figure 1

Rigid clusters in simulated frictional packings under slow shear. Four snapshots from the molecular dynamics simulation showing partially rigid systems close to the frictional rigidity transition. In black is the largest rigid cluster, while floppy regions are colored gray. Blue, green, and red disks correspond to three, two, and one leftover pebble, respectively. The range of partial rigidity decreases with increasing friction coefficient $\mu$, or equivalently, increasing $q$. (a) $\mu =0.2$, with a transition at $q=0.78$ and average coordination number $z=3.35$, (b) $\mu =0.3$, with a transition at $q=0.86$ and $z=3.15$, (c) $\mu =0.5$, with a transition at $q=0.95$ and $z=3.0$, and finally (d) $\mu =10$ with a transition at $q=1.0$ and $z=2.8$. The last value is due to the large number of contactless particles (rattlers) in the packing, visible in blue.

Figure 2

Graphical abstract. In conjunction with the Introduction, this serves as a visual guide to the paper.

Figure 3

Schematic frictional rigidity percolation models HC1 and HC2. These random networks are constructed by either adding next-nearest neighbor (NNN) bonds to an occupied honeycomb lattice (HC1) or adding random first- and second-neighbor bonds (HC2), all with probability $p$. Double bonds denote frictional or gearlike bonds, which occur with probability $q$, while single bonds denote sliding bonds.

Figure 4

Small minimally rigid clusters. For a triangular constraint network with all double bonds, this network is minimally rigid via the (3,3) pebble game. For a four-site constraint network, there are 5 possible configurations in which this network is minimally rigid via the (3,3) pebble game, two of which are presented.

Figure 5

Rigid clusters in frictional rigidity percolation. Rigid clusters below, at, and above the rigidity transition for HC1 and HC2 with $q=0.5$. Rigid clusters are colored, with the largest cluster in black, while floppy regions are in gray.

Figure 6

Rigid clusters in central-force rigidity percolation. Rigid clusters below, at, and above the rigidity transition for HC1 and HC2. The black indicates the largest rigid cluster, and floppy regions are gray again.

Figure 7

Spanning rigid cluster probability. Probability of having a spanning rigid cluster as a function of $p$ for lattices of different lengths $L$ for different models, with HC1 in the top row and HC2 in the bottom row and the (3,3) pebble game results (frictional) on the left, while the (2,3) pebble game results (central force) are on the right. Solid lines are fits to the data using an error function as a fitting function. Data points are averaged over 2500 samples for the (3,3) game and over 1000 samples for the (2,3) game.

Figure 8

Finite-size scaling analysis to obtain exponents $\beta$, $\nu$, and $d_f$. (a) For $HC1$ with $q=0.5$, we plot $P_{\infty }$, the fraction of occupied bonds in the largest rigid cluster, as a function of $p$ for different system sizes. We observe $P_{\infty }\sim (p-p_c(q){)}^{\beta }$ just above critical point and tends towards $p$ further away from the transition. (b) Collapse of (a) using $p_c(q)=0.448$, $\nu =1.56$, $\beta =0.18$. (c) $P_{\infty }$ for $HC1$ with the (2,3) pebble game as a function of $p$ for different system sizes. (d) Collapse of (c) using $p_c(q)=0.448$, $\nu =1.54$, $\beta =0.07$. (e) $\mathrm{\Delta }$, as defined in Eq. (4), versus system length $L$ for six different cases of the model. (f) Log-log plot of the number of bonds in the spanning cluster $M$ versus $L$ for $HC1$ with $q=0.5$ and $q=0.7$.

Figure 9

Analysis to obtain exponent $\tau$ and summary table. (a) HC1 with $L=320$ and $q=0.5$. The red squares show the probability for having finite rigid clusters with different sizes very close to the transition point. Blue circles and yellow triangles show the distribution below and above rigidity transition, respectively. (b) Nonspanning cluster size probability distribution at $p_c$ for the six different cases of the model. (c) This table lists the types of rigidity transitions and some critical exponents for the different models defined as follows: $\xi \sim (p-p_{c,q}{)}^{-\nu }$ is the correlation length and diverges at critical point; the nonspanning rigid cluster size obeys a broad distribution, $n_s\sim s^{-\tau }$, at the critical point; $d_f$ is the fractal dimension of spanning rigid cluster at the rigidity transition; $\beta$ is the order parameter exponent; $q$ is the percentage of contacts as double bonds. As a point of reference, exponents from the triangular lattice (TL) using the (2,3) pebble game, as well as ordinary connectivity percolation ($CP$) exponents, are listed at the bottom of the table.

Figure 10

Rigid cluster merging mechanisms for frictional rigidity percolation. (a) Schematic hinge linking two rigid clusters. (b) Schematic double bond linking two rigid clusters. (c) Merging three rigid clusters linked by three floppy double bonds into one rigid structure with the addition of one double bond indicated by the black arrow. (d) Example of a hinge in HC2. (e),(f) Adding exactly one double bond merges and grows five smaller rigid clusters into a new spanning rigid cluster.

Figure 11

Central-force rigidity percolation on a hierarchical lattice. (a) First three generations of hierarchical Berker lattice. (b) Subnetwork counting: dashed bonds are not occupied. Every type of subnetwork is a way to obtain a spanning rigid network between two ends (black dots) and its probability is calculated, as well as the probability for an occupied bond to be in the spanning rigid cluster. (c) First four $p_n$ as function of $p$. $p_n$ tends to converge to a step function at $p_c=0.9446$, which jumps from 0 to 1.

Figure 12

Frictional rigidity percolation on a hierarchical lattice. (a) Allowed rigid subgraphs for the $q=1$ case with the (3,3) pebble game that are not allowed with the (2,3) pebble game. (b) Plot of $d_f\nu$ versus $q$ for frictional RP on the Berker hierarchical lattice.

Figure 13

Henneberg moves. Schematic of type I and type II Henneberg moves for the central-force (2,3) game (top) and for the frictional (3,3) game (middle and bottom).

Figure 14

Schematic of minimal rigidity proliferation (MRP). (a) Existing rigid cluster (blue) surrounded by nearest and next-nearest bonds with their respective weights (purple). (b) Minimal sum of weights from bond pair has been associated to sites; candidate Henneberg move pairs are in red. (c) The move is executed at the site with the lowest total weight. (d) New rigid cluster surrounded by nearest and next-nearest bonds.

Figure 15

Generated spanning clusters. An example of spanning rigid cluster constructed using minimal rigidity proliferation. Two examples of rigid hinges are shown in more detail on the right.

Figure 16

MRP cluster property analysis. (a) Log-log plot between size of spanning cluster at the critical point $M$ and size of whole system $N$. The slope of less than 1 indicates that the spanning cluster at the critical point has a fractal dimension $d_f=1.916\pm 0.010$. (b) As the system size increases, $P_{\infty }$ goes to zero, suggesting a continuous transition. All plots have been averaged from 935 samples.

Figure 17

First four $p_n$ as function of $p$ for the (3,3) game on general hierarchical lattices; $p_n$ needs at least the first two terms to generate a crossing point.