Critical Scaling of Shear 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 in two dimensions at zero temperature. Shear viscosity η is computed as a function of particle volume density ρ and applied 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 , 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 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. 2 we show a log-log plot of inverse shear vis- that they overlap for some range of ρ, before each drops cosity η−1 vs applied shear stress σ, for several different discontinuously into a jammed state. As N increases, the values of volume density ρ. Our results are for systems onset value of ρ for jamming increases to a limiting value large enough that we believe finite size effects are negli- ρc ≃ 0.84 (consistent with the value for random close gible. We use N = 1024 for ρ< 0.844 and N = 2048 for −1 packing [3]) and η vanishes continuously. For finite ρ ≥ 0.844. Again we see that ρc ≃ 0.8415 separates two −1 N, systems jam below ρc because there is always a fi- limits of behavior. For ρ<ρc, log η is convex in log σ, −1 nite probability to find a configuration with a force chain decreasing to a finite value as σ → 0. For ρ>ρc, log η spanning the width of the system, thus causing it to jam; is concave in log σ, decreasing towards zero as σ → 0. and at T = 0, once a system jams, it remains jammed The dashed straight line, separating the two regions of for all further time. As the system evolves dynamically behavior, indicates the power law dependence that is ex- with increasing simulation time, it explores an increasing pected exactly at ρ = ρc (see below). Similar power region of configuration space, and ultimately finds a con- law behavior at ρc was recently found in simulations of a figuration that causes it to jam. The statistical weight three dimensional granular material [23]. 3

5 etai$=0.830 10 1

" etai$=0.834 #c = 0.8415 ! etai$=0.836 4 10-1 etai$=0.838 10 $ = 1.65 etai$=0.840 & = 1.2 $ etai$=0.841 | f (z)

c " $=0.842 3 etai # 10 # < # etai$=0.844 " c

etai$=0.848 #

-2 /|

10 etai$=0.852 1 2

" 10 $=0.856 etai ! $=0.860 etai f+(z) etai$=0.864 101 inverse shear viscosity etai$=0.868 ~ z$/& # > #c 10-3 etai#=0.0012 -5 -4 -3 -2 10 10 10 fit210 100 shear stress # 10-2 10-1 100 & z = %/|# " #c| − FIG. 2: (color online) Plot of inverse shear viscosity η 1 vs −1 applied shear stress σ for several different values of the vol- FIG. 3: (color online) Plot of scaled inverse viscosity η /|ρ− β ∆ ume density ρ. The dashed line represents the power law ρc| vs scaled shear stress z ≡ σ/|ρ − ρc| for the data of dependence expected exactly at ρ = ρc and has a slope Fig. 2. We find an excellent collapse to the scaling form of β/∆ = 1.375. Solid lines are guides to the eye. Points la- Eq. (6) using values ρc = 0.8415, β = 1.65 and ∆ = 1.2. beled σ = 0.0012 correspond to densities ρ = 0.870, 0.872, The dashed line represents the large z asymptotic dependence, ∆ 0.874, 0.876, and 0.878. ∼ zβ/ . Data point symbols correspond to those used in Fig. 2.

In Fig. 3 we replot the data of Fig. 2 in the scaled −1 β ∆ at z0, jamming at finite σ will be like a second order variables η /|ρ−ρc| vs σ/|ρ−ρc| . Using ρc =0.8415, transition; if f+(z) jumps discontinuously to zero at z0, β =1.65 (the same values used in Fig. 1b) and ∆ = 1.2, it will be like a first order transition. Such a first order we find an excellent scaling collapse in agreement with like transition has been reported in simulations [20, 21] the prediction of Eq. (6). As the scaling variable z → 0, −1 of sheared glasses at finite temperature below the glass f−(z) → constant; this gives the vanishing of η ∼ |ρ − β transition, T ρc, showed that a sim- function approach a common curve, f±(z) ∼ z , so −1 β/∆ ilar first order like behavior was a finite size effect that that precisely at ρ = ρc, η ∼ σ as σ → 0 [24]. vanished in the thermodynamic limit. With these obser- This is shown as the dashed line in both Figs. 3 and 2. A vations, we leave the question of criticality at finite σ to similar scaling collapse of η has been found in simulations future work [20] of a sheared Lennard-Jones glass, as a function of The critical scaling found in Fig. 3 strongly suggests temperature and applied shear strain rateγ ˙ , but only that point J is indeed a true second order phase transi- above the , T >T . By comparing the c tion, and thus implies that there ought to be a diverg- goodness of the scaling collapse as parameters are varied, ing correlation length ξ at this point. Measurements we estimate the accuracy of the critical exponents to be of dynamic (time dependent) susceptibilities have been roughly, β =1.7 ± 0.2 and∆=1.2 ± 0.2. used to argue for a divergent length scale in both the That the crossover scaling exponent ∆ > 0, implies thermally driven glass transition [25], and the density that σ is a relevant variable in the renormalization group driven jamming transition [17]. Here we consider the sense, and that critical behavior at finite σ should be equal time transverse velocity correlation function in the in a different universality class from the jamming tran- shear driven steady state, sition at point J (i.e. σ = 0). The of jamming at finite σ > 0 will be determined by the behavior of g(x)= hvy(xi,yi)vy(xi + x, yi)i , (7) the branch of the crossover scaling function f+(z), that describes behavior for ρ>ρc. From Fig. 3 we see that where vy(xi,yi) is the instantaneous velocity in they ˆ f+(z) is a decreasing function of z. If f+(z) vanishes only direction, transverse to the direction of the average shear −1 when z → 0, then Eq. (6) implies that η vanishes for flow, for a particle at position (xi,yi). The average is over ρ>ρc only when σ = 0, and so there will be no jamming particle positions and time. In the inset to Fig. 4 we plot at finite σ > 0. If, however, f+(z) vanishes at some fi- g(x)/g(0) vs x for three different values of ρ at fixed σ = −1 ∆ −4 nite z0, then η will vanish whenever σ/(ρ − ρc) = z0; 10 and number of particles N = 1024. We see that g(x) there will then be a line of jamming transitions emanat- decreases to negative values at a well defined minimum, ing from point J in the ρ − σ plane given by the curve before decaying to zero as x increases. We define ξ to be ∗ 1/∆ ρ (σ) = ρc + (σ/z0) . If f+(z) vanishes continuously the position of this minimum. That g(ξ) < 0, indicates 4

It also agrees with ν = 0.71 ± 0.08 found by O’Hern et #c = 0.8415 al. [3] from a finite size scaling argument. Wyart et al. [14] have proposed a diverging length scale with exponent 1 $ = 0.6 10 ν = 0.5 by considering the vibrational spectrum of soft $ & = 1.2 | h"(z) c

# modes approaching point J from the jammed side, ρ>ρc. # < #c

" 0.4

(0) -4 While our results cannot rule out ν = 0.5, our scaling g N = 1024 % = 10 # )/ x

| 0.3 (

/ collapse in Fig. 4 does seem somewhat better when using g 1 rho=0.83

" # = 0.830 0.2

! the larger value 0.6. rho=0.834# = 0.834 0.1 rho=0.838# = 0.838 100 This work was supported by Department of Energy $/& h+(z) 0.0 ~ z grant DE-FG02-06ER46298 and by the resources of the -0.1 # > #c velocity correlation 0 5 10x 15 20 Swedish High Performance Computing Center North 10-2 10-1 100 101 (HPC2N). We thank J. P. Sethna, L. Berthier, M. Wyart, & z = %/|# " #c| J. M. Schwarz, N. Xu, D. J. Durian, A. J. Liu and S. R. Nagel for helpful discussion. FIG. 4: (color online) Inset: Normalized transverse velocity correlation function g(x)/g(0) vs longitudinal position x for − N = 1024 particles, applied shear stress σ = 10 4, and vol- ume densities ρ = 0.830, 0.834 and 0.838. The position of the minimum determines the correlation length ξ. Main fig- [1] Jamming and Rheology, edited by A. J. Liu and −1 ν ure: Plot of scaled inverse correlation length ξ /|ρ − ρc| vs S. R. Nagel (Taylor & Francis, New York, 2001). ∆ scaled shear stress z ≡ σ/|ρ − ρc| for the data of Fig. 2. We [2] A. J. Liu and S. R. Nagel, Nature 396, 21 (1998). find a good scaling collapse using values ρc = 0.8415, ∆ = 1.2 [3] C. S. O’Hern et al. Phys. Rev. E 68, 011306 (2003). (the same as in Fig. 3) and ν = 0.6. Data point symbols [4] D. J. Durian, Phys. Rev. Lett. 75, 4780 (1995) and Phys. correspond to those used in Fig. 2. Rev. E 55, 1739 (1997). [5] H. A. Makse, D. L. Johnson and L. M. Schwartz, Phys. Rev. Lett. 84, 4160 (2000). that regions separated by a distance ξ are anti-correlated. [6] C. S. O’Hern et al., Phys. Rev. Lett. 86, 000111 (2001) We can thus interpret the sheared flow in the unjammed and 88, 075507 (2002). 95 state as due to the rotation of correlated regions of length [7] J. A. Drocco et al., Phys. Rev. Lett. , 088001 (2005). [8] L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. ξ. Similar behavior, leading to a similar definition of 95, 098301 (2005) and Phys. Rev. E 73, 041304 (2006). ξ, has previously been found [26] in correlations of the [9] W. G. Ellenbroek et al., Phys. Rev. Lett. 97, 258001 nonaffine displacements of particles in a Lennard-Jones (2006). glass, in response to small elastic distortions. [10] N. Xu and C. S. O’Hern, Phys. Rev. E 73, 061303 (2006). As with viscosity, we expect the correlation length [11] J. M. Schwarz, A. J. Liu and L. Q. Chayes, Europhys. ξ(ρ, σ) to obey a scaling equation similar to Eq. (6). We Lett. 73, 560 (2006). −1 [12] C. Toninelli, G. Biroli and D. S. Fisher, Phys. Rev. Lett. consider here the inverse correlation length ξ , which 96 −1 , 035702 (2006). like η should vanish at the jamming transition, obey- [13] S. Henkes and B. Chakraborty, Phys. Rev. Lett. 95, ing the scaling equation, 198002 (2005). [14] M. Wyart, S. R. Nagel, T. A. Witten, Europhys. Lett. −1 ν σ 72, 486-492 (2005); M. Wyart et al., Phys. Rev. E 72, ξ (ρ, σ)= |ρ − ρc| h± ∆ . (8)  |ρ − ρc|  051306 (2005); C. Brito and M. Wyart, Europhys. Lett. 76, 149 (2006). The correlation length critical exponent is ν, but the [15] V. Trappe et al., Nature (London) 411, 772 (2001). crossover exponent ∆ remains the same as for the vis- [16] T. S. Majmudar et al., Phys. Rev. Lett. 98, 058001 cosity. (2007). [17] A. .S. Keys et al., Nature physics 3, 260 (2007). In Fig. 4 we plot the scaled inverse correlation length, 78 −1 ν ∆ [18] M. Schr¨oter et al., Europhys Lett. , 44004 (2007). ξ /|ρ−ρc| vs the scaled stress, σ/|ρ−ρc| . Using ρc = 58 −1 [19] R. Yamamoto and A. Onuki, Phys. Rev. E , 3515 0.8415 and ∆ = 1.2, as was found for the scaling of η , (1998). −1 we now find a good scaling collapse for ξ by taking the [20] L. Berthier and J.-L. Barat, J. Chem. Phys. 116, 6228 value ν =0.6. By comparing the goodness of the collapse (2002). as ν is varied, we estimate ν =0.6±0.1. From the scaling [21] F. Varnik, L. Bocquet and J.-L.Barrat, J. Chem. Phys. equation Eq. (8) we expect both branches of the scaling 120, 2788, (2004). ν/∆ [22] D. J. Evans and G. P. Morriss, Statistical Mechanics of function to approach the power law h±(z) ∼ z as −1 ν/∆ Non-equilibrium Liquids (Academic, London, 1990). z → ∞, so that ξ ∼ σ as σ → 0 at ρ = ρc [24]. [23] T. Hatano, M. Otsuki and S. Sasa, condmat/0607511. This is shown as the dashed line in Fig. 4. Our result [24] In general, one should consider nonlinear scaling vari- is consistent with the conclusion “ν is between 0.6 and ables. In our case, the most important correction would 2 0.7” of Drocco et al. [7] for the flowing phase, ρ<ρc. be to replace ρ − ρc in Eq. (6) by gρ(ρ,σ) ≡ ρ − ρc + cσ ; 5 this could lead to noticeable corrections to our scaling (1983). equation near ρ = ρc. However, since we find ∆ > 0.5, [25] L. Berthier et al., Science 310, 1797 (2005). −1 β/∆ our conclusion that η ∼ σ at ρ = ρc remains valid. [26] A. Tanguy et al., Phys. Rev. B 66, 174205 (2002). See, A. Aharony and M. E. Fisher, Phys. Rev. B 27, 4394