Granular materials (Assemblies of particles with dissipation )

Granular materials (Assemblies of particles with dissipation ) Saturn ring Sand mustard seed Ginkaku-ji temple

Sheared granular materials packing fraction : Φ Inhomogeneous flow Gas (Φ = 012) Homogeneous flow (Shear rate γ) Dense liquid (Φ = 08) No flow Amorphous solid (Φ = 085)

Jamming transition Below ΦJ Above ΦJ Homogeneous flow (Shear rate γ) Dense liquid (Φ = 08) Transition point ΦJ Onset of rigidity No flow Amorphous solid (Φ = 085)

Rheology under steady shear frictionless case Shear stress σ T Hatano, M Otsuki, S Sasa, J Phys Soc Jpn, 76, 023001 (2007) non-linear rheological property For Φ < ΦJ, σ γ 2 (liquid) For Φ > ΦJ, σ const (solid) For Φ ΦJ, σ γ y γ Shear rate γ

Rheology under steady shear Shear stress σ T Hatano, M Otsuki, S Sasa, J Phys Soc Jpn, 76, 023001 (2007) σ / Φ - ΦJ β frictionless case T Hatano, J Phys Soc Jpn, 77, 123002, (2008) Shear rate γ non-linear rheological property For Φ < ΦJ, σ γ 2 (liquid) For Φ > ΦJ, σ const (solid) For Φ ΦJ, σ γ y γ scaling plot Critical scaling law γ Φ-ΦJ -α σ(γ, Φ) = Φ - ΦJ y Φ S±(γ Φ-ΦJ -α )

Theory for exponents M Otsuki and H Hayakawa, PRE, 80, 011308, (2009) Three Critical scaling laws T(γ, Φ) = Φ - ΦJ x Φ τ±(γ Φ-ΦJ -α ) σ(γ, Φ) = Φ - ΦJ y Φ S±(γ Φ-ΦJ -α ) P(γ, Φ) = Φ - ΦJ y Φ p±(γ Φ-ΦJ -α ) Four Assumptions S / P is constant P in high density region : P ~ Φ Characteristic time : P-1/2 Low density region : collision frequency T 1/2 Theoretical prediction for critical exponents Coulomb s friction : Hatano (2007) O Hern, et al, (2003) Wyart, et al (2005) Kinetic theory xφ = 3, yφ = 1, yφ = 1, α = 5/2 (for disks) Linear repulsive force

Rheology under steady shear frictionless case σ / ΔΦ T Hatano, J Phys Soc Jpn, 77, 123002, (2008) M Otsuki and H Hayakawa, PRE, 80, 011308, (2009) σ(γ, Φ) = Φ - ΦJ y Φ S±(γ Φ-ΦJ -α ) Theoretical prediction : α = 1, yφ = 2/5 (for disk) linear repulsive force γ ΔΦ -α σ(γ, Φ) = ΔΦ S±(γ / ΔΦ 5/2 ) ΔΦ = Φ - ΦJ

Problem The system under steady shear is not suitable to study the rigidity near the jamming transition In experiments, the steady shear is hard to realize We numerically investigate the rheological properties under oscillatory shear (OS)

Previous study on the system under OS B Tighe, PRL 107,158303 (2011) Complex shear modulus G* G : real part close to the critical point close to the critical point G : imaginary part frequency ω System no mass, fixed contact networks, tangential friction Complex shear modulus exhibits critical scalings

Purpose of this work In the previous work, the attention is restricted to the small shear limit and the change of the contact network is not considered However, the change of the network dominates the rheological property near the jamming transition point We investigate the rheological properties under OS in a wide range of shear amplitude

Fn Fn Fn = k δ - η δ Elastic part Dissipative part

Shear strain : γ(t) = γ0 cos (ωt) Amplitude : γ0, Frequency : ω Shear stress : σ(t) Volume fraction : Φ Shear modulus : G* = G + i G G dt σ(t) cos (ωt) / γ0 Real part Storage modulus G - dt σ(t) sin (ωt) / γ0 Imaginary part Loss modulus G * (γ0, ω, Φ)

σ = σe + σk, σe = E γ, σk = η γ G = E, G = η ω Shear stress σ(t) red ω = 10-4 green ω = 10-3 blue ω = 10-2 ω 0 slope G width G Shear strain γ(t) Complex shear modulus G* G storage modulus G loss modulus G ω frequency ω

G * (γ0, ω, Φ) γ0 1 G * (γ0, ω, Φ) γ0 < 1 G * (γ0, ω, Φ) ω 0

G*(γ0, ω, Φ) γ0 1 Complex shear modulus G* Φ= 067, γ0=1 G loss modulus G storage modulus frequency ω Shear stress σ(t) ω 0 Φ= 067, γ0=1 Shear strain γ(t)

G*(γ0, ω, Φ) γ0 1 Complex shear modulus G* Φ= 067, γ0=1 G loss modulus G storage modulus frequency ω ω 0 elastic plastic plastic elastic σ = { E γ, (γ < γc) σy, (γc < γ)

G*(γ0, ω, Φ) γ0 1 Storage modulus G close to ΦJ red Φ = 067 green Φ = 066 blue Φ = 065 magenta Φ = 0648 γ0=1 γ0=1 Loss modulus G close to ΦJ red Φ = 067 green Φ = 066 blue Φ = 065 magenta Φ = 0648 frequency ω frequency ω

G*(γ0, ω, Φ) γ0 1 G*(ω, Φ) = ΔΦ g(ω / ΔΦ 5/2 ) ΔΦ = Φ - ΦJ γ0=1 γ0=1 G / Δφ red Φ = 067 green Φ = 066 blue Φ = 065 magenta Φ = 0648 G / Δφ red Φ = 067 green Φ = 066 blue Φ = 065 magenta Φ = 0648 ω / Δφ 5 / 2 ω / Δφ 5 / 2 γ0 σ(γ, Φ) = ΔΦ F±(γ / ΔΦ 5/2 )

G * (γ0, ω, Φ) γ0 > 1 G * (γ0, ω, Φ) γ0 < 1 G * (γ0, ω, Φ) ω 0

G*(γ0, ω, Φ) γ0 1 Complex shear modulus G* 006 γ0=10-3 γ0=10-3, ω=10-4 G : storage modulus close to ΦJ red Φ = 067 green Φ = 066 blue Φ = 065 G loss modulus frequency ω G 004 002 G ~ ΔΦ 1/2 0 064 065 066 067 Volume fraction Φ C O Hern, et al, Phys Rev Lett 88, 075507 (2002)

G * (γ0, ω, Φ) γ0 > 1 G * (γ0, ω, Φ) γ0 < 1 G * (γ0, ω, Φ) ω 0

G*(γ0, ω, Φ) ω 0 G0 (γ0, Φ) lim G (γ0, ω, Φ) ω 0 γc(φ) G0 G 0 = const γ0 < γc(φ) close to ΦJ red Φ = 0650, green Φ = 0652 blue Φ = 0655, magenta Φ = 0660 cyan : Φ = 0670 Shear amplitude : γ0 G 0 decreases as γ0 increases for γ0 > γc(φ) G 0 decreases as Φ approaches ΦJ

G*(γ0, ω, Φ) ω 0 G0 (γ0, Φ) = ΔΦ 1/2 h(γ0 / ΔΦ) lim h(x) x -1/2 x G ~ γ0-1/2 G ~ ΔΦ 1/2, for γ0 0 C O Hern, et al, Phys Rev Lett 88, 075507 (2002) G0 /ΔΦ 1/2 red Φ = 0650, green Φ = 0652 blue Φ = 0655, magenta Φ = 0660 cyan : Φ = 0670 γ0 / ΔΦ The yield strain γc is proportional to the contact length γc(φ) ~ ΔΦ B Tighe, et al, Phys Rev Lett 105, 088303 (2010) G0 is independent of Φ for ΔΦ 0

G*(γ0, ω, Φ) ω 0 G0 (γ0, Φ) = ΔΦ 1/2 h(γ0 / ΔΦ), lim h(x) x -1/2 x lim lim G0 (γ0, Φ) ΔΦ 1/2 ΔΦ 0 006 004 γ0 0 γ0=10-4 C O Hern, et al, Phys Rev Lett 88, 075507 (2002) G ~ ΔΦ 1/2 lim G0 (γ0, Φ) ΔΦ ΔΦ 0 006 004 cf TG Mason et al PRE (1997), experiments of emulsions H Yoshino, analysis of the replica method γ0=10-2 G ~ ΔΦ G0 G0 002 002 0 064 065 066 067 Volume fraction Φ 0 064 065 066 067 Volume fraction Φ

G*(γ0, ω, Φ) ω 0 G0 (γ0, Φ) = ΔΦ 1/2 h(γ0 / ΔΦ), lim h(x) x -1/2 x log G 0 C ( ) = 10-1 = 10-2 = 10-3 = 10-4 = 10-5 lim lim G0 (γ0, Φ) ΔΦ 1/2 γ0 0 ΔΦ 0 log 0 lim G0 (γ0, Φ) ΔΦ ΔΦ 0

G*(γ0, ω, Φ) ω 0 G0 (γ0, Φ) = ΔΦ 1/2 h(γ0 / ΔΦ), lim h(x) x -1/2 x lim lim G0 (γ0, Φ) ΔΦ 1/2 ΔΦ 0 006 004 γ0 0 γ0=10-4 C O Hern, et al, Phys Rev Lett 88, 075507 (2002) G ~ ΔΦ 1/2 lim G0 (γ0, Φ) ΔΦ ΔΦ 0 006 004 cf TG Mason et al PRE (1997), experiments of emulsions H Yoshino, analysis of the replica method γ0=10-2 G ~ ΔΦ G G 002 002 0 064 065 066 067 Volume fraction Φ 0 064 065 066 067 Volume fraction Φ

G*(ω, Φ) = ΔΦ g(ω / ΔΦ 5/2 ) G (Φ - ΦJ) 1/2, G ω G0 (γ0, Φ) = ΔΦ 1/2 h(γ0 / ΔΦ)

Model of granular materials Fn Ft Φ < ΦJ Φ > ΦJ Ft Normal force Fn Tangential force Friction coefficient : μ Fn = k δ Δ - η vn Ft < μ Fn (Coulomb s friction) Elastic part Dissipative part Δ = 1 (Disk) Frictionless : μ = 0 Δ = 3 / 2 (Sphere) Frictional : μ > 0 Important parameters : Δ, μ

Critical property (without shear) Frictionless case, Δ = 1 Φ < ΦJ Φ > ΦJ Shear modulus G ~ (Φ- ΦJ) 1/2 Pressure P ΦJ Viscosity η η ~ (ΦJ- Φ) -3 P ~ (Φ- ΦJ) ΦJ Φ - ΦJ

Critical scalings Shear stress σ Frictionless case, Δ = 1 σ(γ, Φ) / Φ - ΦJ β scaling plot Shear rate γ non-linear transport property For Φ < ΦJ, σ γ 2 (liquid) For Φ > ΦJ, σ const (solid) For Φ ΦJ, σ γ y γ γ Φ-ΦJ -α Hatano, 2008 σ(γ, Φ) = Φ - ΦJ β S±(γ Φ-ΦJ -α ) second order transition

Characteristic features Mean field theory Dimension D = 2, 3, 4 with the same exponents obtained from the theory The critical exponents depend on the type of the contact force Fn = k δ Δ The critical exponents are independent of the dimension M Otsuki and H Hayakawa, Phys Rev E, 80, 011308, (2009)

Effect of Friction σ σ Frictionless (μ = 00) Frictional (μ = 20) Hysteresis loop for frictional case M Otsuki and H Hayakawa, Phys Rev E 83, 051301 (2011)

Effect of friction (pressure in the zero shear limit) Pressure P Pressure P ΔP Frictionless (μ = 00) Continuous transition Frictional (μ = 20) Discontinuous transition

Effect of friction (type of the transition) 0005 ΔP 0004 0003 0002 ΔP 0001 0 0 05 1 15 2 Continuous transition Discontinuous transition Friction coefficient μ

Phase diagram P ~ (Φ- ΦS) Area of the hysteresis loop ΔP ΦC ΦS ΦS ΦC Many critical densities

An amount of the hysteresis loop phase diagram

Scaling relations Solid branch P ~ ( ) Δ, S ~ ( ) Δ,

Scaling relations liquid branch P ~ γ 2 ( ) -4, S ~ γ 2 ( ) -4

Granular rheology Shear stress : S 9 7 5 2 0 low density critical density high density 0 70 Shear rate γ2 Yield stress : SY low density S γ 2 critical density S γ y γ high density S SY Yield stress : SY (Φ - ΦJ) y Φ Bagnold law

Theory for exponents Assumption S / P is constant P in high density region : P ~ Φ Δ Δ-dependent critical exponents PTP, PRE (2009) Coulomb s friction : Hatano (2007) O Hern, et al, (2003) Characteristic time : P-1/2 Wyart, et al (2005) Low density region : collision frequency T 1/2 Kinetic theory normal force cf Hatano 2010, Teigh 2010 (yφ = Δ + 05)

Derivation of exponents γ 0, Φ > 0 Δ average force : Fc(Φ) k δ(φ) compression length δ(φ) Φ Fn=kδ Δ δ C S O Hern, et al (2003) Assumption : S/P is constant Coulomb s law

Two branches Solid branch Liquid branch different contact number contact number > 3 contact number < 2

Exponents in other works Author yφ yγ = α / yφ yφ xφ α system critical point shear rate Number of particles Olsson & Titel 2007 Hatano 2008 Otsuki, Hayakawa, 2009 Tighe et al 2010 Hatano 2010 Nordstrom et al 2010 Olsson & Titel 2010 12 = Δ+02 (Δ=1) 12 = Δ+02 (Δ=1) 0413 29 foam 063 (Δ=1) 12 = Δ+02 (Δ=1) 25 (Δ=1) 19 (Δ=1) granular Δ 2Δ / (Δ+4) Δ Δ+2 (Δ+4) / 2 granular Δ+05 1/2 foam 15 = Δ+05 (Δ=1) 21 = Δ+06 (Δ=15) 108 = Δ +008 (Δ=1) 06 (Δ=1) 048 (Δ=15) 028 (Δ=1) 15 = Δ+05 (Δ=1) 108 = Δ +008 (Δ=1) 33 (Δ=1) 25 (Δ=1) 41 (Δ=15) 385 (Δ=1) granular 08415 (diameters 1:14) 0646 (diameters 1:14) 0648 (diameters 1:14) 08423 (diameters 1:14) 06473 (diameters 1:14) foam 0635 foam 084347 (diameters 1:14) 1024 10-4 ~ 10 0 1000 5 x 10-7 ~ 5 x 10-5 4000 10-5 ~ 10-1 1210 10-8 ~ 10-2 4000 10-8 ~ 10-6

Correlation length ξ μ

Correlation function Cp(r) r

glass spheres in oil Sheared granular material Eric Brown, et al (2010) Hysteresis loop appears in this system [private communication] Related experiment

Critical scaling γ : scaling plot Hatano, 2008

Derivation of exponents γ 0, Φ > 0 Δ average force : Fc(Φ) k δ(φ) compression length δ(φ) Φ Fn=kδ Δ δ C S O Hern, et al (2003) Assumption : S/P is constant Coulomb s law

Validity of the theory normal force yφ / α

Scaling law high density region( ) + low shear limit(γ 0) P ~ ( ) Δ, S ~ ( ) Δ low density region( ) + low shear limit(γ 0) P ~ γ 2 ( ) -4, S ~ γ 2 ( ) -4 ( ) -4 ( ) -4 P / γ 2 S / γ 2 low shear limit low shear limit

Protocol We sequentially change shear rate Shear rate Shear rate time time

Shear stress μ=00 μ=20 Similar behavior to the frictionless case γ critical density γ Hysteresis loop appears around the critical point

Scaling law frictionless high density region( ) + low shear limit(γ 0) P ~ ( ) Δ, S ~ ( ) Δ, low density region( ) + low shear limit(γ 0) P ~ γ 2 ( ) -4, S ~ γ 2 ( ) -4

Scaling laws frictional Solid branch liquid branch P ~ ( ) Δ, S ~ ( ) Δ, critical densities P ~ γ 2 ( ) -4, S ~ γ 2 ( ) -4 cf Somfai, et al (2007), Silbert (2010) [unsheared case]

Finite-size effect μ=12 μ=20 solid to liquid :N=8000 :N=16000 :N=32000 :N=8000 :N=16000 :N=32000 solid to liquid

Scaling in the liquid branch S ~ γ 2 ( ) -4 P ~ γ 2 ( ) -4, S / (A γ 2 ) P / (A γ 2 )

Critical exponents

Model (frictionless grains) δ Δ=1(Linear model) Δ=3/2(Hertz model) Δ Interaction Force F=kδ Compressed Length δ The exponent for the interaction Δ Dissipative force between the contacting particles

Scaling function Scaling properties of S, T, P, ω jammed 1st eq

Pressure & shear stress For γ 0, Φ > 0 jammed phase Δ average force : Fc(Φ) k δ(φ) compressed length δ(φ) (σ/dφj)φ C S O Hern, et al (2003) Assumption S/P is independent of Φ Coulomb friction T Hatano (2007) 2nd eq 3rd eq

Characteristic frequency For γ 0, Φ > 0 jammed phase fc D(f) D(f) : Density of state f : frequency, fc : cut-oﬀ frequency,, Wyart et al (2005) f Assumption ω is scaled by fc For γ 0, Φ < 0 unjammed phase vc : characteristic velocity l (Φ) : mean free path 4th eq ﬁnal eq 2 vc T/m l (Φ) (σ/dφj) Φ

Critical exponents The exponents depend on Δ

Previous works Our theory : The results are consistent with Previous works : our prediction 2-dimensional elastic particles R Garcia-Rojo, et al, PRE (2006) 3-dimensional elastic particles Singular behavior around Φ = ΦG < ΦJ L Berthier and T A Witten, EPL (2009)

Pair correlation functions Structure factor Pair correlations The ﬁrst peak exceeds the maximum value of this graph D=3, mono-disperse, Δ=1 S(k) does not show any critical behaviors The ﬁrst peak of g(r) changes drastically near ΦJ

Divergence of the first peak small shear magniﬁcation σ0 The ﬁrst peak diverges as the shear rate gets smaller

Scaling of the first peak coordination number : Z Pressure : P The results are consistent with our predictions dilute g(r) have peak g0 around r=σ0 dense D=3, mono-disperse,δ=1

There is no singularity other than point J

ΦJ for static granular packing Stress control simulation φj1 is related with φj Silbert, et al (2001) Ciamarra, et al (2010) φj2 is related with φl Discussion : previous works