Biological fluid mechanics at themicro and nanoscale‐

Lecture 6:From Liquids to Solids:Rheological Behaviour

Anne TanguyUniversity of Lyon (France)

From Liquids to Solids: Rheological behaviour

I.Elastic SolidII.Plastic Flow

III.Visco-elasticity IV.Non-Linear rheology

Al polycristal (Electron Back Scattering Diffraction)

Cu polycristal : cold lamination (70%)/ annealing.

Si3N4 SiC dense

Dendritic growth in Al:

TiO2 metallic foams, prepared with different aging, and different tensioactif agent:

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?

I. Elastic Moduli

REMINDER:

The Navier-Stokes equation:

with for a « Newtonian fluid »

Thus: for an incompressible, Newtonian fluid.( dynamical viscosity)

v with v.)3

2(2 SeIeIP

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 Cijlk in a 3D solid.

Thus: for a Linear Elastic Solid

dt

dveu with

SS

kllk

ijklijij C .,

0

fzyxCrt

u S

u)):,,(.(2

2

(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 sameproportion with the tension thereof.”

Hooke’s Law: τ = G γ or (Stress = G x Strain)where G is the RIGIDITY MODULUS

Example of an homogeneous and isotropic medium:

u

3/2

1

tr

tr3

P

V

V

1-ility compressib

)u(.)u.(.2u

:motion of equations

.tr..2 components stress

2

2

0

ext

ijijijij

ft

σ

F

Lu

Lv

Lu

ESF

.

.

E, Young modulus

, Poisson ratio

Traction:u

Simple Shear:

Lu

SF

.

, shear modulus

PHydrostatic compression:

E

)21(3

23

3

, compressibility.

2 Elastic Moduli (,)

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

2

,. 22..

LL cc

222.2

.2

. lmnL

cc LLlmn

Onde longitudinale:

LTT ccc ,. 22

..

Ondes transverses: simple shear

Sound waves in an isotropic medium:2 sound wave velocities cL and cT

Examples of anisotropic materials (crystals):

FCC3 moduliC11 C12 C44

HCP5 moduliC11 C12 C13 C33 C44 C66=(C11-C12)/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

Voigt notation:

6)12(

5)31(

4)23(

3)33(

2)22(

1)11(

21 independent Elastic Moduli

)

)

)

E

(CC

(CC

(CC

Microscopic expression for the local Elastic Moduli:

Simple example of a cubic crystal.

On each bond:

....)(2

1).( 02

22

0000 rdr

drrr

dr

drrrr ijijij

ijijijij

EEEE

strain

stress

0

011 r

rrij

20

2

02

0

20

11

).(

4

4'

rdr

rdrr

r

f ijij

ijE

Elastic Modulus

30

0ij

2

0ij2

0111111

EE.

1/'

r

r

dr

rd

rC

C1 ~ 2 1 C2 ~ 2 2 C3 ~ 2 (+2D Lennard-Jones Glass N=216 225 L=483

General case: Local Elastic Moduli at small strain

M. Tsamados et al. (2007)

)(..

....

1)( 4321)(

)

,,,,2

4321

4321 4321

43432121

4321

iiiiinrr

rrrr

rrViC iiii

iiiieq

iieq

ii

eqii

eqii

eqii

eqii

iiiii

E

(

Example of an amorphous material

Progressive convergence to an isotropic material

at large scale

Born-Huang

II. Plastic Flow

Plastic Flow:

u

Lz

F

In the Linear Elastic Regime: F/S = E.u/Lz

Elastic modulus

Compressive stress

Strain

S

Plastic Flow

Elasticité

F/S

E

u/Lz

Plastic Threshold y

Visco-plastic Flow flow.

vitreloy

Elasticity + Viscoelasticity

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)

Example: Flow due to an external force (cf. Poiseuille flow)Binary Lennard-Jones Glass at T=0.2<Tg for tW=104 LJ

F. Varnik (2008)

Not a Poiseuille Flow at small T

(Visco-Plastic)

Poiseuille

≠Poiseuille

P

III. Visco-elasticity

Progressive flow of a solid

(1643-1727)

1687: Isaac Newton addresses liquids and steady simple shearing flow in his “Principia”

“The resistance which arises from the lack ofslipperiness 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/dtwhere η is the Coefficient of Viscosity Newtonian

tdttttdt

d

cstet

.)'(1

)()(.1 0

0

c

.

0 t timesticcharacteri 1.)(

.)(.)(

t

VE

et

dt

dtt

. )()(.1

.1

00

t

ttdt

d

dt

dVE

Newtonian Viscous Fluid: Ex. Water, honey..

Kelvin-Voigt Solid: Delayed Elasticity (anelastic behaviour).

Maxwell Fluid: Ex. Solid close to Tf

Instantaneous Elasticity + Viscous Flow

General Linear Visco-Elastic Behaviour: effect)(memory ')'(.')(.)(0

.

0 dttttftftt

crystalsin ,3

2

Da

kTL

Different behaviours:

friction" internal" '/'')tan(

)('')(')(

GG

iGGG

)sin(0 t

)sin(0 t

)cos(.).('')sin(.).(' 00 tGtG

Dynamical Rheometers:

Oscillatory forcing:

Response:

G’, Storage (Elastic) ModulusInstantaneous response

G’’, Loss (Viscous) ModulusDelay

Example of Perfect Elastic Solid:

Example of Newtonian Viscous Fluid:

Example of Maxwell Fluid:

0''' GG and

.'' and 0' GG

0'' and ' response elastic :

.'' and 0' response viscous:0

..'' and .'222

2

222

22

GG

GG

GG

Pastes

Energy Balance:

)(''.2

.)('.

.)(

4

)(sin).(''..)2sin().('.2

..

20

20

4/

0

220

20

GGdttT

tGtGdt

d

T

P

P

Elastic Energy Stored during T/4And then given back(per unit volume and unit time).

Averaged Dissipated Energyper unit time, during T/4,due to viscous friction >0

energy stored

energy dissipated

'

''tan

G

G

Loss Factor (Internal Friction)

Loss factor Material

> 100 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)

10-2 Plexiglas, Wood, Concrete, Felt, Plaster, Brick

10-3 Steel, Iron, Lead, Copper, Mineral Glass

10-4 Aluminium, magnésium

Storage Modulus Internal Friction

Viscoelasticity of Polymers: General features

amorphous

crystalline

Viscoelasticity of Polymers: Examples

Viscoelasticity of Mineral Glasses: Examples

SiO2 – Na2O Si– Al-O-N

Lekki et al.

Viscoelasticity and crystallization

Cristallization: G’ increases, mobility decreases

Polymer (PET) Mineral Glass

ZrF4

Frequency dependent behaviour

SiO2-Na20-Ca0

Example of Blood Red Cells:

G’

G’’

(t)/0

Macroscopic creep in Metals:

Lead Romanian Pipe

Creep Metals Ceramics Polymers T > 0,3-0,4 Tm 0,4-0,5 Tm Tg

Dislocation creep: b=0 m=4-6 Non-Linear behaviour0.3 Tm<T<0.7 Tm

Nabarro-Herring creep: b=2 m=1 Linear (Newtonian) flow diffusion of defects

T>0.7 Tm

~ 1h

Metling Temperatures, for P=1 atm, Ice: Tm=273°K, Lead: Tm=600°K, Tungsten: Tm=3000°K

0,30 0,5 0,7 1

10-1

10-2

10-3

10-4

10-5

10-6

Plasticity

Theoretical Limit

Creep Dislocation

Creep DiffusionElasticity

Athermal Elastic Limit

Core

Volume

VolumeGrain Boundaries

mT

T

IV. Non-Linear Rheology

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

Pastes

Colloids

Powders

F. Varnik (2006) 3D Lenard-Jones Glass

From the Liquid to the Amorphous Solid:Non-Linear Rheological Behaviour

Non-Linear Rheological Behaviours:

Shear softeningEx. painting, shampoo

(1925) Ostwald 1 with ..

nK

n

Shear thickeningEx. wet sand, polymeric oil,

silly-putty 1 with ..

nK

n

Plastic FluidEx. amorphous solids, pastes

Casson 1,nBulkley -Herschel 1,n Bingham

.

0

.

.

n

CC

C

K

Ex. Lennard-Jones GlassTsamados, 2010

with <1

..

/ xy

shear softening

Example:in amorphous systems (glasses, colloids..)

Ex. Beads made of polyelectric gel

Simulations of Rheological Behaviour at constant Strain Rate and Temperature

in an amorphous glassy material (Lennard-Jones Glass)

M. Tsamados 2010

Low strain rateProgressive Diffusion of Local RearrangementsFinite Size Effects

Large strain rateNucleation of Local Rearrangements

Density of nucleating centers per unit strain

Diffusion of plasticity

Cooperativity Maximum when L1=L2

?

Atomistic Modelling:

Classical Molecular Dynamics Simulationsfor fluid dynamics.

I.Description II.The example of Wetting

III.The example of Shear Deformation

Lecture 7

Classical Molecular Dynamics Simulations consists in solving the Newton’s equationsfor 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.

Equations of motion: the example of Verlet’s algorithm.

Adapt the equations of motion, to the chosen Thermostat for cst T.

• Langevin Thermostat:Random force (t)Friction force –.v(t) with <(t).(t’)>=cste.2kBT.(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

0'dt

dH

( )1/2

)(. tFvdt

dvm ii

ii

Thermostats:

Examples of Empirical Interactions:

The Lennard-Jones Potential:

2-body interactions cf. van der Waals

Length scales ij ≈ 10 ÅMasses mi≈10-25 kgEnergy ij≈ 1 eV ≈ 2.10-19J ≈ kBTm

Time scale or

Time step t = 0.01≈ 10-14 s106 MD steps ≈ 10-8 s = 10 ns or 106x10-4=100% shear strain in quasi-static simulations

N=106 particles, Box size L=100 ≈ 0.1 m for a mass density =1.3.N.Nneig≈108 operations at each « time » step.

sm 12

2

10.

s

TD12

8

202

1010

10

)1(

1.0

The Stillinger-Weber Potential:For « Silicon » Si, with 3-body interactionsStillinger-Weber Potential F. Stillinger and T. A. Weber, Phys. Rev. B 31 (1985)

Melting T Vibration modes Structure Factor

The BKS Potential:For Silica SiO2, with long range effective Coulombian InteractionsB.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)

OSijioùr

CeA

r

qqrE ijrB

ijji

BKSij ,),(

4)( 6

0

111 ).().(,,

)(4, ).()..(),...,2,1(

ararijkkji

arjiSW

ikijefeBrANE

2-body interactions(Cauchy Model) 3-body interactions

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

Microscopic determination of different physical quantities:

-Density profile, pair distribution function

-Velocity profile

-Diffusion constant

-Stress tensor (Irwin-Kirkwood, Goldenberg-Goldhirsch)

-Shear viscosity (Kubo)

II. The example of Wetting

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

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

h

h

zz

drruhrr

rfrddzhF

)()(.2

..

21

321

00

)(...2 321

hh

zS rurdrdhhFW

3

.for cos. LVSLSVSLSVLV

III. The example of Shear Deformation

Boundary conditions:

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 .

).10(10..

10/

418lim

12

LJusa

c

a

c

t

scat

Quasi-Static Limit

stressshear xyS

F

ux

Ly

y

xxy L

u

2strain

Rheological behaviour:

Stress-Strain curve in the quasi-static regime

stressshear xyS

F

ux

Ly

y

xxy L

u

2strain

X

y

Local Dynamics:

Global and Fluctuating Motion of Particles

stressshear xyS

F

ux

Ly

y

xxy L

u

2strain

Local Dynamics:

Global and Fluctuating Motion of Particles

Transition from Driven to Diffusive motiondue to Plasticity, at zero temperature.

cage effect (driven motion) Diffusive

y _

max

n ~ xy

p

Tanguy et al. (2006)

Low Temperature Simulations: Athermal Limit

Typical Relative displacement due to the external strain

larger than

Typical vibration of the atom due to thermal activation

ta ...

>>

h

B

k

Tk

Convergence to the quasi-static behaviour, in the athermal limit:At T=10-8 (rescaling of the transverse velocity vy et each step)

M. Tsamados(2010)

cste

.

4.0..

.

.

T= 0.2-0.5 Tg =0.435Rescaling of transverse velocities in parallel layers

Effect of aging

at finite T

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 kBT,

LL

ctd

1

.

. xydt

dQ

.

. xy

BQ

Tkt

.

.

.

.

xy

BdQ

c

LTktt

Visco-Plastic Behaviour:

Flow due to an external force (cf. Poiseuille flow)F. Varnik (2008)

Non uniform T

The relative importance of Driving and of Temperature must be chosen carefully.

See you in Lyon!