M. van Hecke

Jamming of soft particles: geometry, mechanics, scaling and isostaticity

Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. At the jamming point, these systems are marginally stable (isostatic) in the sense of constraint counting, and many geometric and mechanical properties scale with distance to this jamming point. Finally, we discuss current explorations of jamming of frictional and non-spherical (ellipsoidal) particles. Both friction and asphericity tune the contact number at jamming away from the isostatic limit, but in opposite directions. This allows one to disentangle the distance to jamming and the distance to isostaticity. The picture that emerges is that most quantities are governed by the contact number and scale with the distance to isostaticity, while the contact number itself scales with the distance to jamming.Amorphous materials as diverse as foams, emulsions, colloidal suspensions and granular media can jam into a rigid, disordered state where they withstand finite shear stresses before yielding. Here we review the current understanding of the transition to jamming and the nature of the jammed state for disordered packings of particles that act through repulsive contact interactions and are at zero temperature and zero shear stress. We first discuss the breakdown of affine assumptions that underlies the rich mechanics near jamming. We then extensively discuss jamming of frictionless soft spheres. At the jamming point, these systems are marginally stable (isostatic) in the sense of constraint counting, and many geometric and mechanical properties scale with distance to this jamming point. Finally, we discuss current explorations of jamming of frictional and non-spherical (ellipsoidal) particles. Both friction and asphericity tune the contact number at jamming away from the isostatic limit, but in opposite directions. This allows one to disentangle the distance to jamming and the distance to isostaticity. The picture that emerges is that most quantities are governed by the contact number and scale with the distance to isostaticity, while the contact number itself scales with the distance to jamming.

Improved piercing of microneedle arrays in dermatomed human skin by an impact insertion method

An electrical applicator was designed, which can pierce short microneedles into the skin with a predefined velocity. Three different shapes of microneedles were used, namely 300 mum assembled hollow metal microneedle arrays, 300 mum solid metal microneedle arrays and 245 mum hollow silicon microneedle arrays. The latter are available as 4x4, 6x6 and 9x9 arrays. When using a velocity of 1 or 3 m/s reproducible piercing of dermatomed and full thickness human skin was evident from the appearance of blue spots on the dermal side of the skin after Trypan Blue treatment and the presence of fluorescently labeled particles in dermatomed skin. Manual piercing did not result in the appearance of blue spots. Transport studies revealed that i) piercing of microneedles with a predefined velocity into human skin resulted in a drastic enhancement of the Cascade Blue (CB, Mw 538) transport, ii) A higher piercing velocity resulted in a higher CB transport rate, iii) The CB transport rate was also dependent on the shape of the microneedles and iv) no difference in transport rate was observed between 4x4, 6x6 and 9x9 hollow silicon microneedle arrays.

Non-affine response: Jammed packings vs. spring networks

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.

Couette flow of two-dimensional foams

We experimentally investigate flow of quasi–two-dimensional disordered foams in Couette geometries, both for foams squeezed below a top plate and for freely floating foams (bubble rafts). With the top plate, the flows are strongly localized and rate dependent. For the bubble rafts the flow profiles become essentially rate independent, the local and global rheology do not match, and in particular the foam flows in regions where the stress is below the global yield stress. We attribute this to nonlocal effects and show that the fluidity model recently introduced by Goyon et al. (Nature, 454 (2008) 84) captures the essential features of flow both with and without a top plate.

Jamming and geometry of two-dimensional foams

We experimentally probe the vicinity of the jamming point J, located at a density corresponding to random close packing (rcp=0.842), in two dimensional, bidisperse packings of foam bubbles. We vary the density of the foam layer and extract geometrical measures by image analysis. We confirm the predicted scaling of the average contact number Z with and compare the distribution of local contact numbers to a simple model. We further establish that the distribution of areas p(A) strongly depends on . Finally, we find that the distribution of contact forces p(f) systematically varies with density.

Ensemble theory for force networks in hyperstatic granular matter

An ensemble approach for force networks in static granular packings is developed. The framework is based on the separation of packing and force scales, together with an a priori flat measure in the force phase space under the constraints that the contact forces are repulsive and balance on every particle. In this paper we will give a general formulation of this force network ensemble, and derive the general expression for the force distribution P(f). For small regular packings these probability densities are obtained in closed form, while for larger packings we present a systematic numerical analysis. Since technically the problem can be written as a noninvertible matrix problem (where the matrix is determined by the contact geometry), we study what happens if we perturb the packing matrix or replace it by a random matrix. The resulting P(f) s differ significantly from those of normal packings, which touches upon the deep question of how network statistics is related to the underlying network structure. Overall, the ensemble formulation opens up a different perspective on force networks that is analytically accessible, and which may find applications beyond granular matter.

Critical jamming of frictional grains in the generalized isostaticity picture

While frictionless spheres at jamming are isostatic, frictional spheres at jamming are not. As a result, frictional spheres near jamming do not necessarily exhibit an excess of soft modes. However, a generalized form of isostaticity can be introduced if fully mobilized contacts at the Coulomb friction threshold are considered as slipping contacts. We show here that, in this framework, the vibrational density of states (DOS) of frictional discs exhibits a plateau when the generalized isostaticity line is approached. The crossover frequency ω* scales linearly with the distance from this line. Moreover, we show that the frictionless limit, which appears singular when fully mobilized contacts are treated elastically, becomes smooth when fully mobilized contacts are allowed to slip. Finally, we elucidate the nature of the vibrational modes, both for slipping and for non-slipping fully mobilized contacts.

MR Imaging of Reynolds Dilatancy in the Bulk of Smooth Granular Flows

Dense granular matter has to expand in order to flow, a phenomenon known as dilatancy. Here we perform, by means of Magnetic Resonance Imaging (MRI), direct measurements of the evolution of the local packing density of a slow and smooth granular shear flow generated in a split-bottomed geometry. The degree of dilatancy is found to be surprisingly strong. For flows without appreciable transient, the dilated zone follows the region of large strain rate, while for flows with a strong transient, the dilated zone extends also into the region where transient flow took place. In all cases, the dilated zone slowly spreads as a function of time. These findings suggest that the local packing density is governed by the total amount of local strain experienced since the start of the experiment.

Jamming in finite systems: Stability, anisotropy, fluctuations, and scaling

Athermal packings of soft repulsive spheres exhibit a sharp jamming transition in the thermodynamic limit. Upon further compression, various structural and mechanical properties display clean power-law behavior over many decades in pressure. As with any phase transition, the rounding of such behavior in finite systems close to the transition plays an important role in understanding the nature of the transition itself. The situation for jamming is surprisingly rich: the assumption that jammed packings are isotropic is only strictly true in the large-size limit, and finite-size has a profound effect on the very meaning of jamming. Here, we provide a comprehensive numerical study of finite-size effects in sphere packings above the jamming transition, focusing on stability as well as the scaling of the contact number and the elastic response.

Relaxation and flow in linearly sheared two-dimensional foams

We probe the relation between rheology and shear-induced relaxation in experiments on two-dimensional foams at steady shear. The characteristic relaxation time tr, which we extract from the non-affine part of the bubble displacements, scales non-linearly with the local strain rate . In particular, the relative strength of the non-affine part grows when —hence the foam flow is not quasistatic down to the lowest experimentally accessible shear rate. Furthermore, we establish a direct connection between the relaxation time scaling and the macroscopic rheology.