We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

Selective binding of multivalent ligands within a mixture of polyvalent amphiphiles provides, in principle, a mechanism to drive domain formation in self-assemblies. Divalent cations are shown here to crossbridge polyanionic amphiphiles that thereby demix from neutral amphiphiles and form spots or rafts within vesicles as well as stripes within cylindrical micelles. Calcium and copper crossbridged domains of synthetic block copolymers or natural lipid (PIP2, phosphatidylinositol-4,5-bisphosphate) possess tunable sizes, shapes, and/or spacings that can last for years. Lateral segregation in these ‘responsive Janus assemblies’ couples weakly to curvature and proves restricted within phase diagrams to narrow regimes of pH and cation concentration that are centered near the characteristic binding constants for polyacid interactions. Remixing at high pH is surprising, but a theory for Strong Lateral Segregation (SLS) shows that counterion entropy dominates electrostatic crossbridges, thus illustrating the insights gained into ligand induced pattern formation within self-assemblies.

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure p and friction coefficient . The density of vibrational states exhibits a crossover from a plateau at frequencies տ * ͑p , ͒ to a linear growth for Շ * ͑p , ͒. We show that * is proportional to ⌬z, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have ⌬z → 0, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity. DOI: 10.1103/PhysRevE.75.020301 PACS number͑s͒: 45.70.Ϫn, 46.65.ϩg Granular media, such as sand, are conglomerates of dissipative, athermal particles that interact through repulsive and frictional contact forces. When no external energy is supplied, these materials jam into a disordered configuration under the action of even a small confining pressure ͓1͔. In recent years, much new insight has been amassed about the jamming transition of models of deformable, spherical, athermal, frictionless particles in the absence of gravity and shear ͓2͔. The beauty of such systems is that they allow for a precise study of the jamming transition that occurs when the pressure p approaches zero ͑or, geometrically, when the particle deformations vanish͒. At this jamming point J and for large systems, the contact number ͓3͔ equals the so-called isostatic value z iso 0 ͑see below͒, while the packing density J 0 equals random close packing ͓2,4͔. Moreover, for compressed systems away from the jamming point, the pressure p, the excess contact number ⌬z = z͑p͒ − z iso 0 , and the excess density ⌬ = − J 0 are related by power-law scaling relations-any one of the parameters p , ⌬z, and ⌬ is sufficient to characterize the distance to jamming.Isostatic solids are marginal solids-as soon as contacts are broken, extended "floppy modes" come into play ͓5͔. Approaching this marginal limit in frictionless packings as p → 0, the density of vibrational states ͑DOS͒ at low frequencies is strongly enhanced-the DOS has been shown to become essentially constant up to some low-frequency crossover scale * , below which the continuum scaling ϳ d−1 is recovered ͓2,6-11͔. For small pressures, * vanishes ϳ⌬z. This signals the occurrence of a critical length scale, when translated into a length via the speed of sound, below which the material deviates from a bulk solid ͓9͔. The jamming transition for frictionless packings thus resembles a critical transition.In this paper we address the question whether an analogous critical scenario occurs near the jamming transition at p =0 of frictional packings. The Coulomb friction law states that, when two grains are pressed together with a normal force F n , the contact can support any tangential friction force F t with F t Յ F n , where is the friction coefficient. In typical packings, essentially non...

We conduct experiments on two-dimensional packings of colloidal thermosensitive hydrogel particles whose packing fraction can be tuned above the jamming transition by varying the temperature. By measuring displacement correlations between particles, we extract the vibrational properties of a corresponding "shadow" system with the same configuration and interactions, but for which the dynamics of the particles are undamped. The vibrational spectrum and the nature of the modes are very similar to those predicted for zero-temperature idealized sphere models and found in atomic and molecular glasses; there is a boson peak at low frequency that shifts to higher frequency as the system is compressed above the jamming transition.PACS numbers: 63. 63.50 Lm, 82.70 Dd Crystalline solids are all alike in their vibrational properties at low frequencies; every disordered solid is disordered in its own way. Disordered solids nonetheless exhibit common low-frequency vibrational properties that are completely unlike those of crystals, which are dominated by sound modes. Disordered atomic or molecular solids generically exhibit a "boson peak," where many more modes appear than expected for sound. The excess modes of the boson peak are believed to be responsible for the unusual behavior of the heat capacity and thermal conductivity at low-to-intermediate temperatures in disordered solids [2].It has been proposed that a zero-temperature jamming transition may provide a framework for understanding this unexpected commonality [3]. For frictionless, idealized spheres this jamming transition lies at the threshold of mechanical stability, known as the isostatic point [3,4]. As a result of this coincidence, the vibrational behavior of the marginally jammed solid at densities just above the jamming transition is fundamentally different from that of ordinary elastic solids [5][6][7][8]. A new class of lowfrequency vibrational modes arises because the system is at the threshold of mechanical stability [10]; these modes give rise to a divergent boson peak at zero frequency [5]. As the system is compressed beyond the jamming transition, the boson peak shrinks in height and shifts upwards in frequency [5]. Generalizations of the idealized sphere model suggest that the boson peaks of a wide class of disordered solids may arise from proximity to the jamming transition [9][10][11][12]. Moreover, the jamming scenario predicts that systems with larger constituents such as colloids should also have boson peaks.Colloidal glasses offer signal advantages over atomic or molecular disordered solids because colloids can be tracked by video microscopy. Vibrational behavior has been explored in hard-sphere colloids [13] and vibrated granular packings [14], but difficulties with statistics [13] or micro-cracks [14] were encountered. In contrast, we use deformable, thermosensitive hydrogel particles to tune the packing fraction in situ. Our experiments show unambiguously that the commonality in vibrational properties observed in atomic and molecular glas...

We compare the elastic response of spring networks whose contact geometry is derived from real packings of frictionless discs, to networks obtained by randomly cutting bonds in a highly connected network derived from a well-compressed packing. We find that the shear response of packing-derived networks, and both the shear and compression response of randomly cut networks, are all similar: the elastic moduli vanish linearly near jamming, and distributions characterizing the local geometry of the response scale with distance to jamming. Compression of packing-derived networks is exceptional: the elastic modulus remains constant and the geometrical distributions do not exhibit simple scaling. We conclude that the compression response of jammed packings is anomalous, rather than the shear response.