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!
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