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Register a new userStatistical Physics of the Jamming Transition: The Search for Simple Models
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We investigate universal features of the jamming transition in granular materials, colloids and glasses. We show that the jamming transition in these systems has common features: slowing of response to external perturbation, and the onset of structural heterogeneities.
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We study the glass and jamming transition of finite-dimensional models of simple liquids: hard- spheres, harmonic spheres and more generally bounded pair potentials that modelize frictionless spheres in interaction. At finite temperature, we study their glassy dynamics via field-theoretic methods by resorting to a mapping towards an effective quantum mechanical evolution, and show that such an approach resolves several technical problems encountered with previous attempts. We then study the static, mean-field version of their glass transition via replica theory, and set up an expansion in terms of the corresponding static order parameter. Thanks to this expansion, we are able to make a direct and exact comparison between historical Mode-Coupling results and replica theory. Finally we study these models at zero temperature within the hypotheses of the random-first-order-transition theory, and are able to derive a quantitative mean-field theory of the jamming transition. The theoretic methods of field theory and liquid theory used in this work are presented in a mostly self-contained, and hopefully pedagogical, way. This manuscript is a corrected version of my PhD thesis, defended in June, 29th, under the advisorship of Frederic van Wijland, and also contains the result of collaborations with Ludovic Berthier and Francesco Zamponi. The PhD work was funded by a CFM-JP Aguilar grant, and conducted in the Laboratory MSC at Universite Denis Diderot--Paris 7, France.
We investigate shear thickening and jamming within the framework of a family of spatially homogeneous, scalar rheological models. These are based on the `soft glassy rheology model of Sollich et al. [Phys. Rev. Lett. 78, 2020 (1997)], but with an effective temperature x that is a decreasing function of either the global stress sigma or the local strain l. For appropiate x=x(sigma), it is shown that the flow curves include a region of negative slope, around which the stress exhibits hysteresis under a cyclically varying imposed strain rate gd. A subclass of these x(sigma) have flow curves that touch the gd=0 axis for a finite range of stresses; imposing a stress from this range {em jams} the system, in the sense that the strain gamma creeps only logarithmically with time t, gamma(t)simln t. These same systems may produce a finite asymptotic yield stress under an imposed strain, in a manner that depends on the entire stress history of the sample, a phenomenon we refer to as history--dependent jamming. In contrast, when x=x(l) the flow curves are always monotonic, but we show that some x(l) generate an oscillatory strain response for a range of steady imposed stresses. Similar spontaneous oscillations are observed in a simplified model with fewer degrees of freedom. We discuss this result in relation to the temporal instabilities observed in rheological experiments and stick--slip behaviour found in other contexts, and comment on the possible relationship with `delay differential equations that are known to produce oscillations and chaos.
We determine the dimensional dependence of the percolative exponents of the jamming transition via numerical simulations in four and five spatial dimensions. These novel results complement literature ones, and establish jamming as a mixed first-order percolation transition, with critical exponents $beta =0$, $gamma = 2$, $alpha = 0$ and the finite size scaling exponent $ u^* = 2/d$ for values of the spatial dimension $d geq 2$. We argue that the upper critical dimension is $d_u=2$ and the connectedness length exponent is $ u =1$.
We discuss the physics of embolic stroke using a minimal model of emboli moving through the cerebral arteries. Our model of the blood flow network consists of a bifurcating tree, into which we introduce particles (emboli) that halt flow on reaching a node of similar size. Flow is weighted away from blocked arteries, inducing an effective interaction between emboli. We justify the form of the flow weighting using a steady flow (Poiseuille) analysis and a more complicated nonlinear analysis. We discuss free flowing and heavily congested limits and examine the transition from free flow to congestion using numerics. The correlation time is found to increase significantly at a critical value, and a finite size scaling is carried out. An order parameter for non-equilibrium critical behavior is identified as the overlap of blockages flow shadows. Our work shows embolic stroke to be a feature of the cerebral blood flow network on the verge of a phase transition.
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 to elastic behavior 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.
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