Modelling vesicles under flow
Experiments: T. Podgorski, G. Coupier,
V. Vitkova, S. Peponas, A. Srinivas
Theory: G. Danker, T. Biben, G. Ghigliotti
D. Jamet, K. Kassner, B. Kaoui, A. Farutin,
C. Misbah,
CNRS and Univ. J. Fourier Grenoble I
Birmingham, May 11th 2009
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Biological membranes
 1)
Simple model for viscoelastic cell
properties
 2) Blood rheology
 3) Protein transport (Golgi apparatus,
rafts)
 4) Microfluidic (analysis, diagnostic,
sorting out)
 5) Specific drug delivery by vesicles
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John et al, PRL 2008
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Biomimetic entities: vesicles
 Simple
enough to lend themeselves to
experimental control
 Sound modelling
 Sound cooperation between experiments,
theory, and numerical simulations
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Giant Unilamellar Vesicles (GUV)
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A simple model of cell membrane
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Other roles of membranes
Divet, Danker, Misbah (Phys. REv. E. 2002)
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Motivation
blood circulation,…
technological challenges
Soft lithography
G. Coupier (Grenoble)
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Microfluidics
«
Lab on chip »! (medical diagnostics, cell
sorting out… )
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Fundamental questions: transport of
complex fluids
DYFCOM
(Grenoble
G. Coupier
and T. Podgorski)
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Complex fluid/structure problem>
complex fluid
1) Hydrodnamics, free boundary
2) Micro/macro link
3) Soft walls (blood vessels)
Glycocalyx
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Vesicles under shear
(tank-treading; see dark defect that is bound to
the membrane that shows the tank-treading)
(tumbling occuring
beyond a certain viscosity
Contrast between the
Interior and exterior)
(Rolling and sliding close to a substrate)
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Vacillating-breathing (swinging)
Misbah PRL (2006); (prediction of swinging)
Vlahovska, Garcia PRE (2007)
Noguchi, Gompper PRL (2007) (phase diagram)
Lebedev et al. PRL (2007) (phase diagram)
Danker et al. PRE (2007) (phase diagram)
experiments (Kantsler et Steinberg, PRL 2006; Mader et al. EPJE 2006)
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Rheology of a dilute
suspension
G. Danker, et C. Misbah, Rheology of a dilute suspension of vesicles;
Phys. Rev. Lett. 98, 088104 (2007).
Small dformations (Taylor, Cox, Acrivos-Frenkel , Barthès-Biesel, Rallisson)
Vesicles: Non linear constitutive law and unexpected impact on rhelogy
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Equilibrium shapes
Helfrich, Naturforsch (1974)
E

2

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H
dA


KdA
g


2
V /( 4 /3)
A/(4)
3/ 2
Reduced volume

Human RBC: 0.65

V /(4 /3)
A/(4)
1

 0.7
2
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3/ 2
Vesicle de-swelling
Membrane permeability:
High: small uncharged polar molecules (water)
Low: large uncharged polar molecules (sugars)
Very low: very large molecules (polymers) or ions.
Viscosity contrast: vesicles filled with dextran
Reduced volume n = volume / volume of sphere having
same surface area
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Even a unique vesicle is complex!
 Several
types of motions and dynamics
 deformability change drastically rheology
 link between underlying dynamics and
rheology
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Modelling
 Boundary
integral formulation
 Phase-field
, level set
 Analytical
(expansion on spherical
harmonics)
 Lattice
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Boltzmann method
Natural modeling: sharp interface
 u  p  0


u


p

0
.u0incompressibility
2
Stokes equations:
u u velocit ycontinuity
[ ij n ]  f i  Forcebalanceat themembrane  

j 
E
r
E   / 2 dAH 2   dA (r , t )  pV
p,
Lagrange multipl. Enforcing area and volume
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( I  n  n) : u  0
Determines the still unknown Lagrange multiplier

Integral formulation
 u  p  0
2
G  q   (r  r0 ) ,
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.u  0
.G  0
Integral formulation
u (r , t )   G (r  r0 ) f (r0 ) 
(  1)  u (r0 ).K (r  r0 ).n(r0 )
Gij 
 ij
r

ri r j
r
,
3
r  r0
r
r0
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K ijk 
ri r j rk
r5
  in / out
n  normal
sharp
diffuse

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1
2 2
E[ ]   d  (1   ) 
4

 2
2
  d ( ) 
2

2
1

2 2
2
E[ ]   d (1   )  ( ) 
2
4

Minimum in 1D
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  tanh( r /( 2)
Phase-field modelingof vesicles
1) 2D (out of equilibrium) Biben, Misbah, PRE (2003)
2) 3D Biben, Kassner, Misbah, PRE (2005)
3) Thermodynamically consistent, Jamet, Misbah, PRE 2007 and 2008
4) Du, Liu, Wang, J. Comp. Phys. (2005, 2006)
5) Campelo, and Hernandez-Machado , EPJE (2006)
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2
1




2 2
2
2
E[ ]   d (1   )  ( )  
d H
 d 
4
2

 2
2
2

E Ephys
du
u  p  f   v
dt

 E phys
f 
,
r
.u0
  tanh( r /( 2)
  Dirac - like function
Biben, Misbah 2003, 2005
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2
1

2 2
2
E[ ]   d (1   )  ( ) 
2
4



d H

2
2

2
  d 

2
E Ephys
 E phys
du
f 
, u  p  f   v
dt
r
n   /  ,
H  div (n)
.u0

 E 
 u.   
 H 

t
  
E
2
2 2

  (1   )    

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Folch et al
1999
E
2
2 2

  (1   )    

2
2
2
2

1





 2  2  
  2  2 H   2
r
r r
s
r
r
s

 E
 u.    2
t
  

  H 

Collapse due
to Laplacian
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Membrane incompressibility
E phys 

d H

2
2

2
  d 

2

 u.  T ( I  n  n) : u , T  tension  like
t
Biben, Misbah PRE (2003), and with Kassner, PRE (2005)
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Asymtotic analysis (Biben et al. PRE 2005)

Singular perturbation
in powers of 
One recovers the sharp boundary equations
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Numerical solution
Spectral method and implicit time integration
reproduces sharp boundary results at
 0
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Thermodynamical consistent formulation
d
A
dt
.u  0
du

 p  .
dt
d ( E  u 2 / 2)
 .q   T : u  
dt
>0
E
D
E








A


0
0
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dE  d   .d   :  
E
   .(  .T )

  (  .T )    T . 
.  E  [   .(  .T )]
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Energy from physical point of view
E  W ( )   / 2( )   2W    (    ) / 2
2
  / 2( H  H 0 ) 2 
 dif
2
   (   )
dW

  2 2   2W H
d
inc  .(n)
curv  
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2

2
( H  H 0 )[  H ( H  H 0 )  4 K ]   s H
Energy from physical point of view
E  W ( )   / 2( )   2W    (    ) / 2
2
  / 2( H  H 0 ) 2 
 dif
2
   (   )
dW

  2 2   2W H
d
inc  .(n)
curv  
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2

2
( H  H 0 )[  H ( H  H 0 )  4 K ]   s H
Thermodynamical consistent formulation
d
   
dt
.u  0
du

 p  (  ) . D
dt
1 d
d
 ( I  n  n) : u  n.
 dt
dt
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Flow inside and outside, nonlinear and nonlocal problem

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=
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+
Vesicles under shear
(tank-treading; see dark defect that is bound to
the membrane that shows the tank-treading)
(tumbling occuring
beyond a certain viscosity
Contrast between the
Interior and exterior)
(Rolling and sliding close to a substrate)
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Viscosity ratio
Diagram under linear shear flow
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(swinging)
Shear rate
Phase-field results (G. Ghigliotti)
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Migration law in Poiseuille flow
unbounded flow
(Kaoui, Ristow, Cantat, Misbah, Zimmermann, PRE 2008)
V. Vitkova, M. Mader & T. Podgorski, Europhys. Lett. (2004)
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Mode coexistence at center line
parachute
bullet
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Slipper : rigid RBC?
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Blood is a complex fluid
 No
constitutive law yet
 Numerical work in progress: treat contact,
put sensible physics in the model (law of
RBC membrane?)
 Analytical: possible in a dilute suspension
(Homogenisation by Danker and Misbah, Phys. Rev. Lett. 2007)
 2D
with G. Ghigliotti: preliminary
results
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
Asymptotics: size
0
Ammari et al. (rigid particles)
Bonnetier et al.(drops)
Vesicles, RBC? Open from math. Point view
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r 1f
f  fn
Series of spherical harmonics
fn f p 1 p 2..... pn
Example:
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n

p
r
1
p
f 2  Fij (t ) xi x j
1
...... p
2
r
n
n1
Constitutive law
DF
 e   ( F : e) F
Dt
e  strain rate
stress   e   ( F : e) F
.  0
  effective viscosity
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Rigid spheres, droplets, vesicles (TT)
Einstein
Taylor
 0
  
0
vesicles
Contraste de viscosité
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
Minimal viscosity at TT-TB point
     0
0
TB
TT

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Hospital Grenoble (Benoît Polack)
Results: red blood cells – intrinsic viscosity
V. Vitkova, M.A. Mader, B. Polack; C. Misbah T. Podgorski, to appear in Biophys. J. Lett.
experimental data for Ht 5%
theoretical data for vesicles with excess area 1,5
4,50
Intrinsic viscosity of RBC
suspensions
4,00
3,50
3,00
2,50
2,00
1,50
1,00
0,50
0,00
0,10
1,00
10,00
Viscosity contrast
RBC
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Vesicle
100,00
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Lattice Boltzmann Method (LBM)


Pseudo-particle moves to 9 possible positions
Mass, momentum conservation, and Gallilean and rotaional
invariances
6
2
5
ci
3
r
0
1
f i (r , t )  f i (r , t )
eq
7
8
i  
4
f i (r  ci , t  1)  f i (r , t )  i  Fi
Collision+ external force
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
Lattice Boltzmann
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
fi  i a1  a2 (ci .u)  a3 (ci .u)  a4 (u.u)
eq
a1  1, a2 
2
2

1
1
,
a

, a4  a2 , i  4 / 9, 1/9, 1/36
3
2
4
cs
2cs
8
Density:
Velocity
   f i (r , t )
i 0
u (r , t ) 
1
8
f ( r , t )c


i 0
i
Pressure p  cs
Navier-Stokes eqs.
i
2
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Kinematic viscosity
n  cs ( 1 / 2)
2
Badr Kaoui (Phd)
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Conclusion
 Phase-field:
useful, easy enough but still
needs numerical efficiency and stability for
quantitative studies
 BIM: provides the best results to date, but
still slow in 3D, and needs linearity
 LBM: versatile, easy to implement, but
needs a refined control of the results
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