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Biological fluid mechanics at the micro‐ and nanoscale Lecture 6: From Liquids to Solids: Rheological Behaviour Anne Tanguy University of Lyon (France)

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From Liquids to Solids: Rheological behaviour I.Elastic Solid II.Plastic Flow III.Visco-elasticity IV.Non-Linear rheology

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Al polycristal (Electron Back Scattering Diffraction) Cu polycristal : cold lamination (70%)/ annealing. Si 3 N 4 SiC dense Dendritic growth in Al: TiO 2 metallic foams, prepared with different aging, and different tensioactif agent:

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1)Two close elements evolve in a similar way. 2)In particular: conservation of proximity. « Field » = physical quantity averaged over a volume element. = continuous function of space. 3)Hypothesis in practice, to be checked. At this scale, forces are short range (surface forces between volume elements) In general, it is valid at scales >> characteristic scale in the microstructure. Examples: crystals d >> interatomic distance (~ Å ) polycrystals d >> grain size (~nm ~ m) regular packing of grains d >> grain size (~ mm) liquids d >> mean free path disordered materials d >> 100 interatomic distances (~10nm) What is a « continuous » medium?

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I. Elastic Moduli

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REMINDER: The Navier-Stokes equation: with for a « Newtonian fluid » Thus:for an incompressible, Newtonian fluid. ( dynamical viscosity) The case of an Elastic Solid: No transport of matter, displacement field u Stress is related to the Strain Hooke’s Law: (anisotropy) 21 Elastic Moduli C ijlk in a 3D solid. Thus:for a Linear Elastic Solid

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(1635-1703) 1678: Robert Hooke develops his “True Theory of Elasticity” Ut tensio, sic vis (ceiii nosstuv) “The power of any spring is in the same proportion with the tension thereof.” Hooke’s Law: τ = G γ or (Stress = G x Strain) where G is the RIGIDITY MODULUS

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Example of an homogeneous and isotropic medium: F E, Young modulus, Poisson ratio Traction: u Simple Shear: , shear modulus P Hydrostatic compression: , compressibility. 2 Elastic Moduli (, )

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Onde longitudinale: Le mouvement des atomes est dans le sens de la propagation Onde transverse: Le mouvement des atomes est perpendiculaire au sens de la propagation Onde longitudinale: Ondes transverses: simple shear Sound waves in an isotropic medium: 2 sound wave velocities c L and c T

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Examples of anisotropic materials (crystals): FCC 3 moduli C 11 C 12 C 44 HCP 5 moduli C 11 C 12 C 13 C 33 C 44 C 66 =(C 11 -C 12 )/2 Ex. cobalt Co: HC FCC T=450°C 3 moduli (3 equivalent axis) 6 (5) moduli (rotational invariance around an axis) The number of Elastic Moduli depends on the Symetry

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Voigt notation: 21 independent Elastic Moduli

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Microscopic expression for the local Elastic Moduli: Simple example of a cubic crystal. On each bond: strain stress Elastic Modulus

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C 1 ~ 2 1 C 2 ~ 2 2 C 3 ~ 2 ( + 2D Lennard-Jones Glass N=216 225 L=483 General case: Local Elastic Moduli at small strain M. Tsamados et al. (2007) Example of an amorphous material Progressive convergence to an isotropic material at large scale Born-Huang

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II. Plastic Flow

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Plastic Flow: u LzLz F In the Linear Elastic Regime: F/S = E.u/L z Elastic modulus Compressive stress Strain S Plastic Flow Elasticité F/S E u/L z Plastic Threshold y Visco-plastic Flow flow . vitreloy Elasticity + Viscoelasticity

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Rheological Description of the Plastic Flow: Rheological law: shear stress at a given P and T, as a function of shear strain, strain rate. Creep experiment: at a constant , what is (t)? Relaxation exp.: at a constant , what is (t)? (here: (t)= if Y / and (t)= Y else) Apparent viscosity: (t, ,d /dt) = (t, ,d /dt) / (d /dt) (here: =∞ if Y / and =0 else) Here, no temporal dependence (≠ viscous flow)

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Example: Flow due to an external force (cf. Poiseuille flow) Binary Lennard-Jones Glass at T=0.2

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III. Visco-elasticity

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Progressive flow of a solid

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(1643-1727) 1687: Isaac Newton addresses liquids and steady simple shearing flow in his “Principia” “The resistance which arises from the lack of slipperiness of the parts of the liquid, other things being equal, is proportional to the velocity with which the parts of the liquid are separated from one another.” Newton’s Law: τ = η dγ/dt where η is the Coefficient of Viscosity Newtonian

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Newtonian Viscous Fluid:Ex. Water, honey.. Kelvin-Voigt Solid:Delayed Elasticity (anelastic behaviour). Maxwell Fluid:Ex. Solid close to T f Instantaneous Elasticity + Viscous Flow General Linear Visco-Elastic Behaviour: Different behaviours:

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Dynamical Rheometers: Oscillatory forcing: Response: G’, Storage (Elastic) Modulus Instantaneous response G’’, Loss (Viscous) Modulus Delay Example of Perfect Elastic Solid: Example of Newtonian Viscous Fluid: Example of Maxwell Fluid:

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Pastes

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Energy Balance: Elastic Energy Stored during T/4 And then given back (per unit volume and unit time). Averaged Dissipated Energy per unit time, during T/4, due to viscous friction >0 Loss Factor (Internal Friction)

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Loss factor Material > 10 0 Polymer or Elastomer (example : Butyl rubber) 10 -1 Natural rubber, PVC with plasticizer, Dry Sand, Asphalte, Cork, Composite material with sandwich structure (example 3 layers metal / polymer / metal)PVC 10 -2 Plexiglas, Wood, Concrete, Felt, Plaster, Brick 10 -3 Steel, Iron, Lead, Copper, Mineral Glass 10 -4 Aluminium, magnésium

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Storage ModulusInternal Friction Viscoelasticity of Polymers: General features amorphous crystalline

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Viscoelasticity of Polymers: Examples

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Viscoelasticity of Mineral Glasses: Examples SiO 2 – Na 2 O Si– Al-O-N Lekki et al.

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Viscoelasticity and crystallization Cristallization: G’ increases, mobility decreases Polymer (PET) Mineral Glass ZrF 4

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Frequency dependent behaviour SiO 2 -Na 2 0-Ca0

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Example of Blood Red Cells: G’ G’’ (t)/ 0

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Macroscopic creep in Metals: Lead Romanian Pipe Creep Metals Ceramics Polymers T > 0,3-0,4 T m 0,4-0,5 T m T g Dislocation creep: b=0 m=4-6 Non-Linear behaviour 0.3 T m

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0,300,50,71 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 Plasticity Theoretical Limit Creep Dislocation Creep Diffusion Elasticity Athermal Elastic Limit Core Volume Grain Boundaries

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IV. Non-Linear Rheology

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Metallic Glass Mineral Glass (SiO2, a-Si) Polymers (PMMA,PC) Pastes Colloids Powders

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F. Varnik (2006) 3D Lenard-Jones Glass From the Liquid to the Amorphous Solid: Non-Linear Rheological Behaviour

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Non-Linear Rheological Behaviours: Shear softening Ex. painting, shampoo Shear thickening Ex. wet sand, polymeric oil, silly-putty Plastic Fluid Ex. amorphous solids, pastes

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Ex. Lennard-Jones Glass Tsamados, 2010 with <1 shear softening Example: in amorphous systems (glasses, colloids..) Ex. Beads made of polyelectric gel

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Simulations of Rheological Behaviour at constant Strain Rate and Temperature in an amorphous glassy material (Lennard-Jones Glass) M. Tsamados 2010 Low strain rate Progressive Diffusion of Local Rearrangements Finite Size Effects Large strain rate Nucleation of Local Rearrangements

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Density of nucleating centers per unit strain Diffusion of plasticity Cooperativity Maximum when L 1 =L 2

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Atomistic Modelling: Classical Molecular Dynamics Simulations for fluid dynamics. I.Description II.The example of Wetting III.The example of Shear Deformation Lecture 7

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Classical Molecular Dynamics Simulations consists in solving the Newton’s equations for an assembly of particles interacting through an empirical potentiaL; In the Microcanonical Ensemble (Isolated system): Total energy E=cst In the Canonical Ensemble: Temperature T=cst with if no external force Different possible thermostats: Rescaling of velocities, Langevin-Andersen, Nosé-Hoover… more or less compatible with ensemble averages of statistical mechanics.

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Equations of motion: the example of Verlet’s algorithm. Adapt the equations of motion, to the chosen Thermostat for cst T.

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Langevin Thermostat: Random force (t) Friction force – .v(t) with =cste.2 k B T. (t-t’) Andersen Thermostat: prob. of collision t, Maxwell-Boltzman velocity distr. Nosé-Hoover Thermostat: Rescaling of velocities: Berendsen Thermostat:with Heat transfer. Coupling to a heat bath. after substracted the Center of Mass velocity, or the Average Velocity along Layers () 1/2 Thermostats:

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Examples of Empirical Interactions: The Lennard-Jones Potential: 2-body interactions cf. van der Waals Length scales ij ≈ 10 Å Masses m i ≈10 -25 kg Energy ij ≈ 1 eV ≈ 2.10 -19 J ≈ k B T m Time scale or Time step t = 0.01 ≈ 10 -14 s 10 6 MD steps ≈ 10 -8 s = 10 ns or 10 6 x10 -4 =100% shear strain in quasi-static simulations N=10 6 particles, Box size L=100 ≈ 0.1 m for a mass density =1. 3.N.N neig ≈10 8 operations at each « time » step.

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The Stillinger-Weber Potential: For « Silicon » Si, with 3-body interactions Stillinger-Weber Potential F. Stillinger and T. A. Weber, Phys. Rev. B 31 (1985) Melting T Vibration modes Structure Factor The BKS Potential: For Silica SiO 2, with long range effective Coulombian Interactions B.W.H. Van Beest, G.J. Kramer and R.A. Van Santen, Phys. Rev. Lett. 64 (1990) Ewald Summation of the long-range interactions, or Additional Screening (Kerrache 2005, Carré 2008) 2-body interactions (Cauchy Model) 3-body interactions

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Example: Melting of a Stillinger-Weber glass, from T=0 to T=2.

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Microscopic determination of different physical quantities: -Density profile, pair distribution function -Velocity profile -Diffusion constant -Stress tensor (Irwin-Kirkwood, Goldenberg-Goldhirsch) -Shear viscosity(Kubo)

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II. The example of Wetting

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Surface Tension: coexistence beween the liquid and the gas at a given V.

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The Molecular Theory of Capillarity: Intermolecular potential energy u(r). Total force of attraction per unit area: Work done to separate the surfaces: (I. Israelachvili, J.S.Rowlinson and B.Widom) Surface Tension: h

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3

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III. The example of Shear Deformation

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Boundary conditions:

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Quasi-static shear at T=0. Fixed walls Or biperiodic boundary conditions (Lees-Edwards) Example: quasi-static deformation of a solid material at T=0°K At each step, apply a small strain ≈ 10 -4 on the boundary, And Relax the system to a local minimum of the Total Potential Energy V({ri}). Dissipation is assumed to be total during . Quasi-Static Limit

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uxux LyLy Rheological behaviour: Stress-Strain curve in the quasi-static regime

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uxux LyLy X y Local Dynamics: Global and Fluctuating Motion of Particles

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uxux LyLy Local Dynamics: Global and Fluctuating Motion of Particles Transition from Driven to Diffusive motion due to Plasticity, at zero temperature. cage effect (driven motion) Diffusive y _ max n ~ xy pp Tanguy et al. (2006)

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Low Temperature Simulations: Athermal Limit Typical Relative displacement due to the external strain larger than Typical vibration of the atom due to thermal activation >>

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Convergence to the quasi-static behaviour, in the athermal limit: At T=10 -8 (rescaling of the transverse velocity v y et each step) M. Tsamados (2010)

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T= 0.2-0.5 Tg =0.435 Rescaling of transverse velocities in parallel layers Effect of aging at finite T

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Non-uniform Temperature Profile at Large Shear Rate Time needed to dissipate heat created by applied shear across the whole system Heat creation rate due to plastic deformation Time needed to generate k B T,

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Visco-Plastic Behaviour: Flow due to an external force (cf. Poiseuille flow) F. Varnik (2008) Non uniform T

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The relative importance of Driving and of Temperature must be chosen carefully. See you in Lyon!

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