Scaling Theory for the Frictionless Unjamming Transition

Abstract

We develop a scaling theory of the unjamming transition of soft frictionless disks in two dimensions by defining local areas, which can be uniquely assigned to each contact. These serve to define local order parameters, whose distribution exhibits divergences as the unjamming transition is approached. We derive scaling forms for these divergences from a mean-field approach that treats the local areas as non interacting entities, and demonstrate that these results agree remarkably well with numerical simulations. We find that the asymptotic behaviour of the scaling functions arises from the geometrical structure of the packing while the overall scaling with the compression energy depends on the force law. We use the scaling forms of the distributions to determine the scaling of the total grain area $A_G$, and the total number of contacts $N_C$.

arXiv comment5 pages, 3 figures, +Supplemental Material

References in corpus (26)

• Jamming at Zero Temperature and Zero Applied Stress: the Epitome of Disorder
• A density-independent glass transition in biological tissues
• Jamming of Soft Particles: Geometry, Mechanics, Scaling and Isostaticity
• Random Packings of Frictionless Particles
• Critical Scaling of Shear Viscosity at the Jamming Transition
• Packing of Compressible Granular Materials
• Active Jamming: Self-propelled soft particles at high density
• Vibrations and diverging length scales near the unjamming transition
• Geometric origin of excess low-frequency vibrational modes in amorphous solids
• Slow dynamics in glassy soft matter
• On the Rigidity of Amorphous Solids
• Unified study of glass and jamming rheology in soft particle systems
• Critical scaling in linear response of frictionless granular packings near jamming
• Finite-Size Scaling at the Jamming Transition
• Statistics of the contact network in frictional and frictionless granular packings
• Microfluidic rheology of soft colloids above and below jamming
• A unified framework for non-Brownian suspension flows and soft amorphous solids
• Stability at Random Close Packing
• A statistical mechanics framework for static granular matter
• Jamming in Systems Composed of Frictionless Ellipse-Shaped Particles
• Entropy and Temperature of a Static Granular Assembly
• Jammed frictionless discs: connecting local and global response
• Geometric partition functions of cellular systems: Explicit calculation of the entropy in two and three dimensions
• Contact Changes near Jamming
• On universal structural characteristics of granular packs
• Disordered Contact Networks in Jammed Packings of Frictionless Disks

Cited by in corpus (9)

• Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming
• Athermal fluctuations in disordered crystals
• Stress Response of Granular Systems
• Disorder Perturbation Expansion for Athermal Crystals
• Timescale divergence at the shear jamming transition
• Athermal fluctuations in three dimensional disordered crystals
• Long-Range Correlations in Elastic Moduli and Local Stresses at the Unjamming Transition
• First contact breaking distributions in strained disordered crystals
• Singularities in Hessian element distributions of amorphous media