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Critical Scaling of Shear Viscosity at the Jamming Transition

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Critical Scaling of Shear Viscosity at the Jamming Transition Critical Scaling of Shear Viscosity at the Jamming Transition Peter Olsson1 and S. Teitel2 1Department of Physics, Ume˚aUniversity, 901 87 Ume˚a, Sweden 2Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 (Dated: September 9, 2007) We carry out numerical simulations to study transport behavior about the jamming transition of a model granular material in two dimensions at zero temperature. Shear viscosity η is computed as a function of particle volume density ρ and applied shear stress σ, for diffusively moving particles with a soft core interaction. We find an excellent scaling collapse of our data as a function of the ∆ scaling variable σ/|ρc − ρ| , where ρc is the critical density at σ = 0 (“point J”), and ∆ is the crossover scaling critical exponent. We define a correlation length ξ from velocity correlations in the driven steady state, and show that it diverges at point J. Our results support the assertion that jamming is a true second order critical phenomenon. PACS numbers: 45.70.-n, 64.60.-i, 83.80.Fg Keywords: In granular materials, or other spatially disordered is, systems such as colloidal glasses, gels, and foams, in 2 which thermal fluctuations are believed to be negligible, ǫ(1 − rij /dij ) /2 for rij < dij V (rij )= (1) a jamming transition has been proposed: upon increas- 0 for rij ≥ dij ing the volume density (or “packing fraction”) of parti- cles ρ above a critical ρc, the sudden appearance of a where rij is the distance between the centers of two par- finite shear stiffness signals a transition between flowing ticles i and j, and dij is the sum of their radii. Particles liquid and rigid (but disordered) solid states [1]. It has are non-interacting when they do not touch, and inter- further been proposed by Liu and Nagel and co-workers act with a harmonic repulsion when they overlap. We [2, 3] that this jamming transition is a special second or- measure length in units such that the smaller diameter der critical point (“point J”) in a wider phase diagram is unity, and energy in units such that ǫ = 1. A system whose axes are volume density ρ, temperature T , and ap- of N disks in an area Lx × Ly thus has a volume density plied shear stress σ (the latter parameter taking one out 2 2 of equilibrium to non-equilibrium driven steady states). ρ = Nπ(0.5 +0.7 )/(2LxLy) . (2) A surface in this three dimensional parameter space then separates jammed from flowing states, and the intersec- To model an applied uniform shear stress, σ, we first tion of this surface with the equilibrium ρ − T plane at use Lees-Edwards boundary conditions [22] to introduce σ = 0 is related to the structural glass transition. a uniform shear strain, γ. Defining particle i’s position as Several numerical [3–10], theoretical [11–14] and ex- ri = (xi +γyi,yi), we apply periodic boundary conditions perimental [5, 15–18] works have investigated the jam- on the coordinates xi and yi in an Lx × Ly system. In ming transition, mostly by considering behavior as the this way, each particle upon mapping back to itself under transition is approached from the jammed side. In this the periodic boundary condition in they ˆ direction, has work we consider the flowing state, computing the shear displaced a distance ∆x = γLy in thex ˆ direction, re- viscosity η under applied uniform shear stress. Previ- sulting in a shear strain ∆x/Ly = γ. When particles do ous works have simulated the flowing response to applied not touch, and hence all mutual forces vanish, xi and yi shear in glassy systems at finite temperature [19–21], and are constant and a time dependent strain γ(t) produces a r in foams [4] and granular systems [10] at T = 0, ρ>ρc. uniform shear flow, d i/dt = yi(dγ/dt)ˆx. When particles Here we consider the ρ − σ plane at T = 0, showing for touch, we assume a diffusive response to the inter-particle the first time that, near point J, η−1(ρ, σ) collapses to a forces, as would be appropriate if the particles were im- ∆ universal scaling function of the variable σ/|ρc − ρ| for mersed in a highly viscous liquid or resting upon a rough both ρ<ρc and ρ>ρc. We further define a correla- surface with high friction. This results in the following tion length ξ from steady state velocity correlations, and equation of motion, which was first proposed as a model show that it diverges at point J. Our results support that for sheared foams [4], jamming is a true second order critical phenomenon. dr dV (r ) dγ Following O’Hern et al. [3], we simulate frictionless soft i = −D ij + y xˆ . (3) dt dr i dt disks in two dimensions (2D) using a bidisperse mixture Xj i with equal numbers of disks of two different radii. The radii ratio is 1.4 and the interaction between the particles The strain γ is then treated as a dynamical variable, 2 obeying the equation of motion, 3.0 101 "5 $ = 10-5 1 $ = 10 (a) (b) " 1 # = 0.8415 " c ! 2.5 N N ! 0 % = 1.65 dγ dV (rij ) 10 eta^-164 eta^-164 = D L L σ − , (4) 2.0 eta^-1128 γ x y eta^-1128 dt dγ eta^-1256 eta^-1256 Xi6=j eta^-1512 eta^-1512 -1 1.5 10 eta^-11024 eta^-11024 beta=1.6 where the applied stress σ acts like an external force on 1.0 -2 γ and the interaction terms V (rij ) depend on γ via the 10 0.5 r inverse shear viscosity particle separations, ij = ([xi −xj]Lx +γ[yi −yj]Ly , [yi − inverse shear viscosity "1 % ! ~ |# " #c| yj ]L ), where by [. .]L we mean that the difference is 0.0 10-3 y µ -3 -2 -1 to be taken, invoking periodic boundary conditions, so 0.78 0.79 0.80 0.81 0.82 0.83 0.84 10 10 10 volume density # volume density #c " # that the result lies in the interval (−Lµ/2,Lµ/2]. The constants D and Dγ are set by the dissipation of the −1 medium in which the particles are embedded; we take FIG. 1: (color online) a) Plot of inverse shear viscosity η vs volume density ρ for several different numbers of particles units of time such that D = D ≡ 1. −5 γ N, at constant small applied shear stress σ = 10 . As N In a flowing state at finite σ > 0, the sum of the inter- increases, one see jamming at a limiting value of the density −1 action terms is of order O(N) so that the right hand side ρc ∼ 0.84. b) Log-log replot of the data of (a) as η vs of Eq. (4) is O(1). The strain γ(t) increases linearly in ρc − ρ, with ρc = 0.8415. The dashed line has slope β = 1.65 −1 time on average, leading to a sheared flow of the particles indicating the continuous algebraic vanishing of η at ρc with with average velocity gradient dvx/dy = hdγ/dti, where a critical exponent β. vx(y) is the average velocity in thex ˆ direction of the par- ticles at height y. We then measure the shear viscosity, defined by, of such jamming configurations decreases, and hence the average time required to jam increases, as one either de- σ σ η ≡ = . (5) creases ρ, or increases N [3]. In the limit N → ∞, we dvx/dy hdγ/dti expect jamming will occur in finite time only for ρ ≥ ρc. −1 In Fig. 1b we show a log-log plot of η vs ρc − ρ, us- −1 We expect η to vanish in a jammed state. ing a value ρc = 0.8415. We see that the data in the We integrate the equations of motion, Eqs. (3)-(4), unjammed state is well approximated by a straight line −1 β starting from an initial random configuration, using the of slope β = 1.65, giving η ∼ |ρ − ρc| in agreement Heuns method. The time step ∆t is varied according to with the expectation that point J is a second order phase system size to ensure our results are independent of ∆t. transition. We consider a fixed number of particles N, in a square If point J is indeed a true critical point, one expects system L ≡ Lx = Ly, and vary the volume density ρ by that its influence will be felt also at finite values of the adjusting the length L according to Eq. (2). We simulate stress σ, with η−1 obeying a typical scaling law, for times ttot such that the total relative displacement per unit length transverse to the direction of motion is typ- −1 β σ η (ρ, σ)= |ρ − ρc| f± ∆ . (6) ically γ(ttot) ∼ 10, with γ(ttot) ranging between 1 and |ρ − ρc| 200 depending on the particular system parameters. −1 ∆ In Fig. 1 we show our results for η using a fixed Here z ≡ σ/|ρ−ρc| is the crossover scaling variable, ∆ is −5 small shear stress, σ = 10 , representative of the σ → 0 the crossover scaling critical exponent, and f−(z), f+(z) limit. Our raw results are shown in Fig. 1a for several are the two branches of the crossover scaling function for different numbers of particles N from 64 to 1024. Com- ρ<ρc and ρ>ρc respectively. paring the curves for different N as ρ increases, we see In Fig.
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