Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

Abstract

A theory for jammed granular materials is developed with the aid of a nonequilibrium steady-state distribution function. The approximate nonequilibrium steady-state distribution function is explicitly given in the weak dissipation regime by means of the relaxation time. The theory quantitatively agrees with the results of the molecular dynamics simulation on the critical behavior of the viscosity below the jamming point without introducing any fitting parameter. DOI: 10.1103/PhysRevLett.115.098001 PACS numbers: 45.70.-n, 05.20.Jj, 64.70.ps, 83.50.Ax Introduction.-A description of granular rheology has been a long-term challenge for both science and technology. The problem extends to a vast range, from solidlike creep motion, gaslike, to liquidlike phenomena [1]. Similar to solid-liquid transitions, granular materials acquire rigidity when the density exceeds a critical value [2][3][4][5]. This phenomenon, referred to as the jamming transition, is universely observed in disordered materials such as colloidal suspensions [6], emulsions, and foams [7], as well as granular materials. The jamming transition and its relation to the glass transition have attracted much interest in the last two decades, and various aspects have been revealed [8][9][10][11][12]. In particular, characteristics in the vicinity of the jamming point, including the critical scaling behavior, have been extensively investigated by experiments and numerical simulations [2][3][4][13][14][15][16][17][18][19][20][21][22][23][24][25]. It has been shown that the shear stress, the pressure, and the granular temperature can be expressed by scaling functions with exponents near φ ∼ φ J , where φ J is the jamming transition density. The shear viscosity exhibits a form η ∼ ðφ J − φÞ −λ with λ ≈ 2 for non-Browninan suspensions, foams, and emulsions [26][27][28][29], and a recent careful analysis demonstrated that λ is in the range between 1.67 and 2.55 [30]. It seems that the exponent λ for granular flows takes a larger value than that for suspensions [18,19,25,31], although a value in the range mentioned above has been reported as well [17]. However, these studies are based on numerical simulations or phenomenologies without any foundation of a microscopic theory.Even when we focus only on the flow properties below the jamming point φ J , which can be tracked back to Bagnold's work [32], we have not yet obtained a complete set in describing the rheological properties of dense granular flows. One of the remarkable achievements is the extension of the Boltzmann-Enskog kinetic theory to inelastic hard disks and spheres [33][34][35][36][37][38]. However, it has been recognized that the kinetic theory breaks down at densities with volume fraction φ > φ f ¼ 0.49 [39][40][41][42],

Universal Relaxation Dynamics of Sphere Packings below Jamming

“…Even for idealised frictionless spheres [34,37], an exact mean-field prediction for ν is not available, but several distinct predictions exist: ν ≈ 2.83 in [39][40][41] and ν = 2 in [42]. Measured numerical values are spread in the range ν ∈ [2,4] [9,34,37,38,[42][43][44][45][46]. We argue that physical situations where dynamics of unjammed packings is relevant go well beyond steady shear rheology.…”

“…Instead, it was shown that the relaxation dynamics after sudden shear cessation is characterised by a time scale τ that diverges as ϕ J is approached [45], with the observation that η ∝ τ ∝ (∆z) −3.7 in d = 3, where ∆z = 2d − z [46]. Strikingly, ν seems to depend on dimensionality [46], unlike other critical exponent of jamming [14] and theoretical predictions [39][40][41][42]. In Ref.…”

Normal stress differences, their origin and constitutive relations for a sheared granular fluid

“…Appendix E. Scaling of transport coefficients in the dense limit There exist hundreds of recent papers on non-Brownian suspensions and dense granular media (Brady & Morris 1997;Singh & Nott 2003;Boyer et al 2011;Couturier et al 2011;Trulsson et al 2012;Dbouk et al 2013;Denn & Morris 2014;Suzuki & Hayakawa 2015) dealing solely with 'how the shear viscosity diverges near the jamming/maximum-packing density'. Based on numerous simulation and experimental data, there is now a consensus that the shear viscosity follows a power-law scaling,…”

An experimental investigation on the speed of sand flow through a fixed porous bed

“…The widely accepted Beverloo’s law shows that the flow rate of grains through orifices depends on the diameter to a 5/2 power law. Research work have shown that even though Beverloo’s correlation fits very well to the flow rate for large orifices, it fails for small orifices where clogging could take place 13–15 . This has been experimentally validated by Mankoc et al .…”