Microscopic theory of the zero-temperature jamming transition in frictional, polydisperse granular packs: structural signatures, force-chain buckling, and protocol-dependent length scales

Cheng's swelling-particle experiment (Physical Review E 81, 031301 (2010)) exposed three unexplained facets of the athermal jamming transition in a real granular system — a frictional, polydisperse pack of soft particles: the maxima of the pair-correlation peak heights $g_1(\phi)$ and $g_2(\phi)$ at the jamming point, with no geometric theory for the second-peak maximum; multi-step, direction-dependent jamming, conjectured to proceed by successive force-chain buckling, with no microscopic theory connecting friction and initial density heterogeneity to the onset sequence; and a cluster static length scale growing to the system size, of unknown universality. The problem is to construct and validate a quantitative microscopic theory of the zero-temperature jamming transition for frictional, polydisperse packs that accounts for all three pillars — taking friction coefficients, stiffness, size distribution, and preparation protocol as inputs — or, within the same framework, to quantitatively refute any of the pillars as a protocol or measurement artifact.

Background

Jamming is the abrupt transition by which a disordered assembly of athermal particles — sand grains, emulsion droplets, foams — develops mechanical rigidity as the packing fraction $\phi$ (the fraction of space covered by particles) increases past a critical value. In the 'zero-temperature' (athermal) limit, thermal fluctuations are absent and the transition is a pure geometry-plus-mechanics problem. Simulations of frictionless soft spheres compressed from random configurations (O'Hern et al., Physical Review E 68, 011306 (2003)) established a sharp onset: below $\phi_c$ no particles touch, and just above it the contact network is isostatic, meaning the number of contact constraints exactly matches the mechanical degrees of freedom, which gives an average coordination $z_c = 2d$ in $d$ dimensions for frictionless spheres ($z_c = d + 1$ when Coulomb friction is sufficiently large, since a frictional contact then transmits $d$ independent force components — one normal and $d-1$ tangential — rather than only the normal one, so fewer contacts are needed to balance the forces and torques on each particle).

The pair correlation function $g(r)$ is the probability density of finding two particle centers a distance $r$ apart, normalized as for an ideal gas; $g_1(\phi)$ and $g_2(\phi)$ denote the heights of its first (nearest-neighbor) and second peaks. Silbert, Liu, and Nagel (Physical Review E 73, 041304 (2006)) showed in simulations of frictionless monodisperse disks and spheres that as jamming is approached, $g_1$ develops a diverging delta-function contribution from particles exactly in contact with a power-law tail, and the second peak splits into two sub-peaks with divergent slopes — structural signatures of the transition's singularity. They noted, however, that no geometric picture existed even there for why the second peak should be singular.

Xiang Cheng (Physical Review E 81, 031301 (2010)) tested these ideas in a real system: a quasi-two-dimensional layer of roughly ten thousand tapioca pearls (8% diameter polydispersity, dry diameter 3.3 mm, Young's modulus about 0.06 GPa when swollen) in a water-filled cell, in which the particles slowly swell over about 24 hours so that $\phi$ rises uniformly and quasi-statically without boundary driving — an athermal 'swelling' protocol. He found: (i) $g_1(\phi)$ and $g_2(\phi)$ both show a maximum at the jamming point $\phi_c \approx 0.84$, located independently by the onset of boundary force and the vanishing of particle displacement, with amplitudes that do not diverge — attributed to polydispersity and particle softness; (ii) the pack self-organizes into clusters whose characteristic static length scale grows to the system size as $\phi \to \phi_c^-$; (iii) in initially heterogeneous packs, friction causes the system to jam in steps, becoming rigid along some directions before others, and Cheng speculated that this multi-step jamming proceeds through successive buckling of force chains — the quasi-linear chains of contacts that carry most of the stress in a granular packing — since small vibrations, which release frictional contacts, eliminate the multi-step phenomenology.

On the theory side, frictionless systems are now understood quantitatively only in the near-contact regime: mean-field calculations, exact in the limit of infinite spatial dimension and based on the glass-theory technique of full replica symmetry breaking (a hierarchical solution of the disordered free-energy landscape), together with marginal mechanical stability arguments (the statement that a jammed packing sits exactly at the boundary of linear stability, with an abundance of soft vibrational modes and weak contacts), give exact exponents for the distribution of small inter-particle gaps and weak contact forces (Charbonneau, Corwin, Parisi, and Zamponi, Physical Review Letters 109, 205501 (2012)), with the associated localized buckling excitations analyzed in Physical Review Letters 114, 125504 (2015). For frictional, polydisperse packs — the experimentally generic case — no equivalent theory exists: friction shifts the isostatic coordination, makes contact networks history-dependent through Coulomb thresholds, and, as a photoelastic-disk experiment showed (Wang et al., PNAS 119, e2204879119 (2022); photoelastic disks are transparent disks whose internal stress is visualized through polarized light), changes the measured marginal-criticality exponents under shear. Whether the structural signatures, the diverging static length scale, and the multi-step phenomenology of Cheng's experiment are universal features of the athermal jamming transition or consequences of a particular protocol and frictional contact mechanics remains unsettled.

Problem Statement

Construct and validate a quantitative, microscopic theory of the athermal (zero-temperature) jamming transition in packs of frictional, polydisperse, soft particles — the material class of Cheng's swelling-particle experiment (Physical Review E 81, 031301 (2010)) — in which the only inputs are microscopic and preparation parameters: inter-particle and particle-boundary friction coefficients, particle stiffness, the particle size distribution, and the preparation protocol including the initial density heterogeneity. The theory must predict, without refitting per observable:

(1) Structural signatures: the curves $g_1(\phi)$ and $g_2(\phi)$ — the heights of the first and second peaks of the pair correlation function — through the transition, including the existence, positions, and heights of their maxima at jamming, together with the geometric or mechanical mechanism responsible for the second-peak maximum in polydisperse frictional packs (why it is a rounded maximum rather than the peak splitting with divergent slopes seen in frictionless monodisperse simulations), and how the signature sharpens or disappears as polydispersity, softness, or plasticity (irreversible particle deformation) are varied.

(2) Multi-step jamming mechanics: the evolution of the average contact coordination number through the transition and the sequence of directional (local) and global jamming onsets in initially heterogeneous packs, including the statistics and mechanics of the successive force-chain buckling events conjectured to mediate them (buckling thresholds, spatial statistics of buckling events, and their connection to localized particle rearrangements), as an explicit function of the friction coefficients, the particle stiffness, and the initial density field.

(3) Static length scales: the cluster static length scale $l$ (defined, as in Cheng's experiment, as the peak position of the positive region of the correlation $D_r(r)$ of the projection of relative particle-pair displacements onto their separation direction) and its growth near jamming, and whether its divergence to the system size is universal or contingent on the athermal swelling protocol — that is, whether comparable static length scales arise under alternative preparation protocols such as quenched or boundary-driven compression.

The theory must account for all three phenomenological pillars within the same framework, quantitatively explaining each or quantitatively refuting it as a protocol or measurement artifact, and must be validated against Cheng's experiment or an equivalent frictional, polydisperse, soft-particle system. The three pillars are not independent and must be connected within a single framework.

References

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