Non-equilibrium theory of rheology for non-Brownian dense suspensions 2017/03/09 15:00 – 15:30 Koshiro Suzuki (Canon Inc.) in collaboration with Hisao Hayakawa (YITP) 2017/03/09 Non-Gaussian fluctuation and rheology in jammed matter 1

Contents Introduction Outline of the theory MD simulation Discussions Microscopic model Equation of continuity Steady-state averages Pressure, shear stress, normal stress differences MD simulation Discussions Summary 2

Contents Introduction Outline of the theory MD simulation Discussions Microscopic model Equation of continuity Steady-state averages Pressure, shear stress, normal stress differences MD simulation Discussions Summary 3

Introduction – shear viscosity Dilute region Einstein (1905) Batchelor, Green (1972) Dense region Chong et al. (1971), Quemada (1977) volume V suspending liquid (viscosity 　) : empirical 1　　　 4

Introduction – pressure viscosity Morris, Boulay (1999) Zarraga et al. (2000) These models are all empirical (Deboeuf et al., 2009) 2　　　 5

Introduction - recent results Pressure-controlled experiment (Boyer et al., 2011) 3　　　 6

Introduction - theoretical approach Brownian suspensions Thermal noise & diffusion Contribution of Brownian & particle-contact stresses (Brady, 1993) (Smoluchowski eq.) 5　　　 7

Introduction - summary Study of the rheology of suspensions Well studied since Einstein Established by experiments and simulations Theoretical framework for dense suspensions Shear viscosity: only for Brownian suspensions Pressure: absent Aim of this work Construct a unified theoretical framework for the pressure and viscosity of dense non-Brownian suspensions 6　　　 8

Contents Introduction Outline of the theory MD simulation Discussions Microscopic model Equation of continuity Steady-state averages Pressure, shear stress, normal stress differences MD simulation Discussions Summary 9

Microscopic model Equation of motion (Lees-Edwards b.c.) No thermal noises (non-Brownian) No fluid interactions (e.g. lubrication forces) Overdamped approximation mass y m diameter d volume V suspension (viscosity ) x 7　　 10

Equation of continuity Stress tensor 8　　　 11

Equation of continuity Averaged stress 9　　　 12

Steady-state averages Averaged stress Inter-particle force and steady-state distribution is necessary ss ss =0 ss ss 9　　　 13

Steady-state averages Inter-particle force y x 10　　　 14

Steady-state averages Steady-state distribution function Grad’s expansion Kinetic theory of gases Extension to dense suspensions ? (Santos et al., 2004) : peculiar velocity : pressure (ideal gas) : kinetic stress 18　　　 15

Steady-state averages Steady-state distribution function T : temperature of the solvent : contact stress 11　　　 16

Pressure, shear stress, normal stress differences Coupled equations Pressure P Shear stress σxy Normal stresses σxx , σyy 12　　　 17

Pressure, shear stress, normal stress difference Final result 13　　　 18

Contents Introduction Outline of the theory MD simulation Discussions Microscopic model Equation of continuity Steady-state averages Pressure, shear stress, normal stress differences MD simulation Discussions Summary 19

MD simulation Event-driven MD (hard spheres) Uniform shear (Lees-Edwards b.c.) Parameters N = 1000, samples S = 100 Sampling time tm = 1000 collisions Procedure sampling time tm initial configuration start sampling equilibrate sample1 sample2 sample3 sampleS 2015/09/09 14　　　 20

Pressure & shear viscosities Density dependence Both exhibit ~ δφ-2 Slight difference → stress ratio 16　　　 21

Stress ratio Approaches 0.1-0.2 as φ→φJ 17　　　 22

Contents Introduction Outline of the theory MD simulation Discussions Microscopic model Equation of continuity Steady-state averages Pressure, shear stress, normal stress differences MD simulation Discussions Summary 23

Grad’s expansion Original Extension : peculiar velocity (Santos et al., 2004) : peculiar velocity : pressure (ideal gas) : kinetic stress (collisional stress) : contact stress 18　　　 24

Diffusivity Brownian suspensions Non-Brownian suspensions Divergence is not related to the diffusion constant (Brady, 1993) 19　　　 25

Contents Introduction Outline of the theory MD simulation Discussions Microscopic model Equation of continuity Steady-state averages Pressure, shear stress, normal stress differences MD simulation Discussions Summary 26

Summary Theory of rheology for dense non-Brownian suspensions is proposed. It relies on the extension of the Grad’s expansion. It successfully describes the correct divergence for the pressure, shear stress, and the normal stress differences. The distinction between Brownian and non-Brownian suspensions is shown. Quantitative validity of the theory is now under investigation. 24　　　 27