1. Nano-hydrodynamics flow at a solid surface boundary condition
E. CHARLAIX
University of Lyon, France
INTRODUCTION TO MICROFLUIDICS August
The Abdus Salam international center for theoretical physics

2. OUTLINE
The no-slip boundary condition (bc):
a long lasting empiricism regularly questionned
Some examples of importance in nanofluidics
What says theory ?
Experiments: a slippery subject
Measurements with dSFA -
Intrinsic slip lengths on smooth surfaces

3. Hydrodynamic boundary condition at a solid-liquid interfacez
v(z)
VS = 0
Usual boundary condition : the fluid velocity vanishes at wall
OK at a macroscopic scale and for simple fluids
But is this boundary condition always valid ?
Phenomenological : derived from experiments on low molecular mass liquids

4. Partial slip and solid-liquid friction
Navier 1823
Maxwell 1856 z
VS ≠ 0
v(z) b
Tangentiel stress at interface
VS : slip velocity
sS : tangential stress at the solid surface
b : slip length
l : liquid-solid friction coefficient
h : liquid viscosity ∂V
= g
: shear rate ∂z

5. Interpretation of the slip lengthb
From Lauga & al, Handbook of Experimental Fluid Dynamics, 2005

6. Some properties of the slip length
No-slip bc (b=0) is associated to very large liquid-solid friction
The bc is an interface property. The slip length has not to be related to an internal scale in the fluid
The hydrodynamic characteristic of the interface is given by b(g)
The hydrodynamic bc is linear if the slip length does not depend on the shear rate.
On a mathematically smooth surface, b=∞ (perfect slip).

7. « Ideal » case of gas « Ideal » case of gas :
weak density = weak momentum transfer between fluid and solid : slip can be large
simple kinetic theory allows to calculate the friction force at wall
with V u h
Root mean square molecule momentum
Gas nb density
Fluid length scale : b ~ µm
Bocquet, CRAS 1993

8. The nature of hydrodynamics bc’s has been widely debated in 19th century
Goldstein S Fluid mechanics in the first half of this century. Annu. Rev. Fluid Mech 1:1–28
Lauga & al, in Handbook of Experimental Fluid Dynamics, 2005
M. Denn, 2001 Annu. Rev. Fluid Mech. 33:265–87
Batchelor, An introduction to fluid dynamics, 1967
Goldstein 1938

9. extrusion of polymer melts
But wall slippage was strongly suggested in some polymer flows…
Shark-skin effect in
extrusion of polymer melts
Pudjijanto & Denn 1994
J. Rheol. 38:1735

10. … and some time suspected in flow on non-wetting surfaces
And also
Bulkley (1931),Chen & Emrich (1963), Debye & Cleland (1958)…
More recently, precise experiments of Churaev & al J Colloids Interf Sci (1984)
Slip length of water in silanized capillaries: b=70nm

11. The no-slip boundary condition (bc):
OUTLINE
The no-slip boundary condition (bc):
a long lasting empiricism regularly questionned
Some examples of importance in nanofluidics
Pressure drop in nanochannels
Elektrokinetics effects
Mixing
What says theory ?
Experiments: a slippery subject
Measurements with dSFA -
Intrinsic slip lengths on smooth surfaces

12. Pressure drop in nanochannelsz b d x ∆P r

13. Exemple 1 d=1 µm b=20 nm slit Exemple 2 r = 2 nm b=20 nm tube
%error on permeability : 12%
slit
Exemple 2
tube
r = 2 nm
b=20 nm
Error factor on permeability : 80
(2 order of magnitude)

14. Forced imbibition of hydrophobic mesoporous medium
mesoporous silica: MCM41
10nm
B. Lefevre et al, J. Chem. Phys Water in silanized MCM41
of various radii (1.5 to 6 nm)
The intrusion-extrusion cycle of water in hydrophobic MCM41
quasi-static cycle, does not depend on frquency up to kHz

15. Porous grain
L ~ 2-10 µm

16. Electrokinetic phenomena
Colloid science, biology, …
Electrostatic double layer
nm µm
Electric field
electroosmotic flow
Electro-osmosis, streaming potential… are determined by interfacial
hydrodynamics at the scale of the Debye length

17. z E v os + + + x + + + + + + +

18. E v - - - - - - - - - zH z os no-slip plane + + + x + + + + + + + zH
Case of a no-slip boundary condition:
zeta potential

19. E v - - - - - - - - - z os + + + x + + + + + + +
Case of a partial slip boundary condition:

20. At constant Ys, the electro osmotic velocity depends on k
Pb for measuring z ?
Possibility of very large electro- osmotic flow by decreasing k-1 ?
b=20nm
k-1 = 3nm at 10-2M
Exemple
Churaev et al, Adv. Colloid Interface Sci. 2002
L. Joly et al, Phys. Rev. Lett, 2004

21. Mixing b
Vs ≠ 0
Partial slip
Dispersion shoud be reduced r
No-slip

22. OUTLINE
The no-slip boundary condition (bc):
a long lasting empiricism regularly questionned
Some examples of importance in nanofluidics
What says theory ?
Experiments: a slippery subject
Measurements with dSFA -
Intrinsic slip lengths on smooth surfaces

23. Effect of surface roughness
Fluid mechanics calculation :
Far field flow : no-slip
locally: perfect slip
Richardson, J Fluid Mech (1973),
Janson, Phys. Fluid 1988
roughness  « kills » slip 

24. Slip at a microscopic scale : molecular dynamics
Robbins (1990) Barrat, Bocquet (1994) Thomson-Troian (Nature 1997) Lennard-Jones liquids molecular size : s corrugation of surface potential : u u
q = 2 p/ s
In general very small surface corrugation
is enough to suppress slip effects

25. Thermodynamic equilibrium determination of b.c.
with Molecular Dynamics simulations
Bocquet & Barrat, Phys Rev E (1994)
Be j(r,t) the fluctuating momentum density at point r
Assume that it obeys Navier-Stokes equation
And assumed boundary condition b

26. Then take its <x,y> average
And auto-correlation function b
C(z,z’,t) obeys a diffusion equation
with boundary condition
and initial value given by
thermal equilibrium
2D density
C(z,z’,t) can be solved analytically and obtained as a function of b
b can be determined by ajusting analytical solution to data
measured in equilibrium Molecular Dynamics simulation

27. b

28. Slip at a microscopic scale : linear response theory
Barrat, Bocquet Phys Rev E (1994)
Green-Kubo relation for the hydrodynamic b.c.:
canonical
equilibrium
Liquid-solid
Friction coefficient
total force exerted
by the solid on the liquid
(assumes that momentum fluctuations in fluid obey Navier-Stokes
equation + b.c. condition of Navier type)
Friction coefficient (i.e. slip length) can be computed at equilibrium from
time decay of correlation function of momentum tranfer

29. Slip at a microscopic scale : molecular dynamics
Barrat, Bocquet (1994) Lennard-Jones liquids molecular size : s corrugation of surface potential : u u
q = 2 p/ s
If liquid-solid interactions are strong
(liquid wets solid)
u/s
b/s
very small surface corrugation is
enough to suppress slip effects
0.01
>0.03 ∞ 40

30. Slip at a microscopic scale: liquid-solid interaction effect
Molecular Dynamics simulations Lennard-Jones liquids
Barrat et al Farad. Disc. 112,
q=140°
130°
q=90°
b/s
P/P0
P0~MPa
 = {liquid,solid},
: energy scale,
 : molecular diameter
cab : wetting control parameter
Linear b.c. up to 108 s-1

31. Slip at a microscopic scale: theory for simple liquids
L. Bocquet, J.L Barrat PRE (1994)
Simple fluids L. Bocquet, J.L Barrat Faraday Disc 112,119 (1999)
Polymer melts Priezjev & Troian, PRL 92, (2004)
For a sinusoidal wall « corrugation » a exp(q// • R//)
molecular size
density at wall,
depends on
wetting properties
wall corrugation
fluid structure
factor

32. Some recent experimental results on smooth surfaces
Tretheway et Meinhart (PIV)
Non-linear slip
Pit et al (FRAP)
Churaev et al (perte de charge)
1000
Craig et al(AFM)
Bonaccurso et al (AFM)
Vinogradova et Yabukov (AFM)
Sun et al (AFM)
100
Chan et Horn (SFA)
Zhu et Granick (SFA)
slip length (nm)
Baudry et al (SFA)
Cottin-Bizonne et al (SFA) 10
MD Simulations 1 50
100
150
Contact angle (°)
Brenner, Lauga, Stone 2005

33. Schmadtko & Leger, PRL 94 244501 (2005)
Velocimetry measurements
V(z)
Particule Imaging Velocimetry
Tretheway & Meinhart Phys Fluid 14, L9, (2002)
Fluorescence Double Focus Cross Correlation
V(z)
Fluorescence recovery
in TIR
Pit & Leger, PRL 85, 980 (2000)
Schmadtko & Leger, PRL (2005)
O. Vinogradova, PRE 67, (2003)

34. Dissipation measurements
Pressure drop
Churaev, JCSI 97, 574 (1984)
Choi & Breuer, Phys Fluid 15, 2897 (2003)
Surface Force Apparatus
Colloidal Probe AFM
Chan & Horn 1985
Israelachvili 1986
Georges 1994
Granick PRL 2001
Mugele PRL 2003
Cottin-Bizone PRL 2005
Craig & al, PRL 87, (2001)
Bonnacurso & al, J. Chem. Phys 117, (2002) Vinogradova, Langmuir 19, 1227 (2003)

35. Particle Image Velocimetry (PIV) Measurement of velocity profile
V(z)
Fluorescent particules
High resolution camera
Pair of images
With Micro-PIV (see S. Wereley)
Spatial resolution ~ nm
Meinardt & al, Experiments in Fluids (1999)
Use for bc : are velocity of tracor and velocity of flow the same ?
Meinardt & al (2002)
Poiseuille flow profile in a capillary

36. ~ 1 µm in 10 -6 M Effect of tracor-wall interactions
O. Vinogradova, PRE (2003)
Hydrodynamical lift
~ 1 µm in M
Colloidal lift z d +
electrostatic force:
depletion layer:
Fsphere ~ R exp (-kd)
d ~ 3 k -1
Vsphere > Vslip z d
Vsphere ≠ Vflow (zcenter)
because of hydrodynamical sphere-plane interaction
0.75 slower than flow at d/R=0.1
F. Feuillebois, in Multiphase Science and Technology, New York, 1989, Vol. 4, pp. 583–798.

37. The no-slip boundary condition (bc):
OUTLINE
The no-slip boundary condition (bc):
a long lasting empiricism regularly questionned
Some examples of importance in nanofluidics
What says theory ?
Experiments: a slippery subject
« direct » methods : flow velocities measurements
« indirect » methods : dissipation measurements
advantages and artefacts
Measurements with dSFA -
Intrinsic slip lengths on smooth surfaces

38. Using molecules as tracors: Near Field Laser Velocimetry
Pit & al Phys Rev Lett (2000)
Schmadtko & al PRL (2005)
evanescent wave (TIR) + photobleaching (FRAP)
T. Schmatdko PhD Thesis, 2003
fluorescence recovery
at different shear rates
t(ms)
Writing beam v
spot L ~ 60 µm
Evanescent wave
~ 80 nm
Reading
beam
P.M.

39. Model for Near Field Laser Velocimetry
Convection //Ox + Diffusion //Oz
Pit-Hervet-Leger (2000)
Case of no-slip b.c. g
V = z
Hexadecane on rough sapphire
z(t)=√ Dmt
x = z t g L

40. Model for Near Field Laser Velocimetry
Pit-Hervet-Leger (2000)
Partial slip b.c. g b g
V = (z+b)
z(t)=√ Dmt
x = t (z+b) g
Résolution : 100 nm
Velocity averaged on ~ 1 µm depth
Needs value of diffusion coefficient
Find slip length b=175nm for hexadecane on sapphire (perfect wetting) L

41. First measurement of slippage without flow….
L. Bocquet, L. Joly, C. Ybert
Condmat
Einstein 1905
Diffusion of a colloidal particle
mobility F e
Measuring tangential diffusion as a function of wall distance gives information on the flow boundary condition.

42. Dmeasured Dno-slip Diffusion of confined colloids measured by
Fluorescence Correlation Spectroscopy
Float pyrex
OTS-coated pyrex
b=20nm
Rough pyrex
b=100nm
Dmeasured
Dno-slip

43. Dissipation measurements
Pressure drop
Churaev, JCSI 97, 574 (1984)
Choi & Breuer, Phys Fluid 15, 2897 (2003)
Surface Force Apparatus
Colloidal Probe AFM
Chan & Horn 1985
Israelachvili 1986
Georges 1994
Granick 2001
Mugele 2003
Cottin-Bizone 2005
Craig & al, PRL 87, (2001)
Bonnacurso & al, J. Chem. Phys 117, (2002) Vinogradova, Langmuir 19, 1227 (2003)

44. Princip of SFA measurements
Tabor et Winterton,
Proc. Royal Soc. London, 1969
D is measured with FECO fringes (Å resolution, low band-pass)
In a quasi-static regime
(inertia neglected)
Distance is measured accurately,
Force is deduced from piezoelectric drive

45. Princip of colloidal probe measurements
Ducker 1991
7,5 µm
scanner xyz piézo
substrate
cantilever
particule
Photodetector
laser
feedback Y X z
Force is measured directly from cantilever bending
Probe-surface distance is deduced from piezoelectric drive

46. Hydrodynamic force with partial slip b.c.
Reynolds force
Hypothesis:
Newtonian fluid
D<<R
Re<1
rigid surfaces
uniform constant b
(linear b.c.)
O. Vinogradova Langmuir 11, 2213 (1995)

47. Wall shear rate in a drainage flow
Parabolic approximation
z = D + x2 2R
z(x) D
U(x) x
Mass conservation
2pxz U(x) = - p x2 D
g (x) √R
D3/2 D
√ 2RD
AFM/SFA methods are not adapted for investigating shear-rate dependent b.c.

48. Some recent experimental results on smooth surfaces
Tretheway et Meinhart (PIV)
Non-linear slip
Pit et al (FRAP)
Churaev et al (perte de charge)
1000
Craig et al(AFM)
Bonaccurso et al (AFM)
Vinogradova et Yabukov (AFM)
Sun et al (AFM)
100
Chan et Horn (SFA)
Zhu et Granick (SFA)
slip length (nm)
Baudry et al (SFA)
Cottin-Bizonne et al (SFA) 10
MD Simulations 1 50
100
150
Contact angle (°)
Brenner, Lauga, Stone 2005

49. D b Sensitivity to experimental errors f *( ) Reynolds force
f* varies between 0.25 and 1 and has a log dependence in D/b
Determination of b requires very precise measurement of F
over a large range in D.

50. D(nm)
calculated
b(nm)
D(nm)

51. OUTLINE
The no-slip boundary condition (bc):
a long lasting empiricism regularly questionned
Some examples of importance in nanofluidics
What says theory ?
Experiments: a slippery subject
Measurements with dSFA -
Intrinsic slip lengths on smooth surfaces

52. Dynamic Surface Force Apparatus
F. Restagno, J. Crassous, E. Charlaix, C.Cottin-Bizonne, Rev.Sci. Inst. 2002
k=7000N/m
Interferometric force sensor
Capacitive displacement sensor
Capacitor plates
Micrometer
Excitation :
0.05 nm < hac < 5 nm
w/2p : [ 5 Hz ; 100 Hz ]
Resolution :
Displacement Force
Static nm nN
Dynamic 5 pm nN
Nomarski
interferometer
Mirors
Piezoelectric elements
Coil
Magnet
Plane

54. Investigating dynamics of confined fluids with
Dynamic Surface Forces Measurements
D µm nm
R ~ mm
F(t)
D(t)
Measure the dynamic force response to an oscillatory motion of small amplitude
In-phase component: stiffness
(derivative of eq. interaction force ; elastic deformations …)
Out-of-phase component: damping (viscous dissipation)
The dynamic force response gives informations on the rheology of the confined liquid and on the flow boundary condition.

55. Newtonian liquid with no-slip b.c.
D µm nm
R ~ mm
F(t)
D(t)
Hypothesis :
The confined liquid remains newtonian
Surfaces are perfectly rigid
No-slip boundary condition
No stiffness
The viscous damping is given by the Reynolds force

56. Simple liquid on a wetting surface
N-dodecane
Smooth surface: float pyrex
Molecular Ø : 4,5 Å
Roughness : 3 Å r.m.s.
Molecular length : 12 Å
Perfectly wetted by dodecane
( = 0°)

57. Inverse of viscous damping
D(nm)
Quasi-static force
Inverse of viscous damping
Bulk hydro. OK for D ≥ 4nm
No-slip : b ≤ 2nm

58. Viscous damping with partial slip:
Specificity of the method
Two separate sensors with Å resolution : no piezoelectric calibration required
More rigid than usual SFA (no glue)
or AFM (no torsion allowed)
In and out-of-phase measurement allows to check for unwanted
elastic deformations (and associated error on distance)
Easy check for linearity of the b.c. with shear rate:
change amplitude or frequency of excitation at fixed D
No background viscous force that cannot be substracted

59. Viscous damping with a partial slip h.b.c.
f *( ) D b
O. Vinogradova :
Langmuir 11, 2213 (1995)
At large distance (D>>b) :

60. Smooth hydrophilic and hydrophobic surfaces
Smooth float pyrex surfaces : 0,3nm r.m.s.
OTS silanized pyrex : 0,7nm r.m.s.
octadecyltricholorosilane
Contact angle
Float pyrex
OTS pyrex
Water
Dodecane
0° 0°
110°
30°

61. Water confined between plain pyrex
Environment : clean room
bare pyrex plane
and sphere : b≤ 3nm
Theory
Experiment
Water on bare pyrex :
Bulk hydro OK for D≥ 3nm
no-slip
D (nm)
C. Cottin-Bizonne et al, PRL 94, (2005)

62. Water confined between plain and OTS-coated pyrex
Environment : clean room
Theory
Experiment
Water on bare pyrex :
no-slip
b = 17±3 nm
silanized plane
bare pyrex sphere
Linear b.c. up to .shear rate ~ s-1
Water on silanized pyrex :
partial slip
one single slip length
bare pyrex plane
and sphere : b≤ 3nm
D (nm)
C. Cottin-Bizonne et al, PRL 94, (2005)

63. Summary of results with dSFA:
intrinsic slip of simple liquids on smooth surfaces
Boundary slip depends on wetting
Plain pyrex < 3nm < 3nm
water dodecane
OTS-pyrex nm < 3nm
DPPC monolayer nm
(fresh)
DPPC bilayer < 3nm
wetting
non-wetting
When slip occurs : 1 well-defined slip length
fully linear h.b.c. condition (up to s-1)
amplitude in good agreement with M.D. simulations of LJ liquids

64. Flow on soft surfaces: lipid monolayers and bilayers
Lipid layers are used in a number of bio-materials
DPPC bilayers are model for biological cell membrane
Supported dense layers are obtained with various wettability
DPPC monolayers
are hydrophobic (q =95°)
DPPC bilayer are hydrophilic
air
air
water
water
DPPC Langmuir-Blodgett deposition on float pyrex

65. Dense DPPC bilayers in water
Stable
Smooth at small scale roughness ~0,3 nm r.m.s
Some holes (bilayer thickness 6,5 nm)

66. DPPC monolayers in water.
200 nm
200 nm
200 nm
200 nm
200 nm
200 nm
after 7h
after 1h
after 1 day
roughness : 0,7 nm r.m.s
~ 3 nm pk-pk
roughness : 2,2 nm r.m.s
6,5 nm pk-pk
contact angle : 95°
contact angle : 80°

67. water on a DPPC monolayer after 1 day hydratation No-slip
b= 0
b= 10nm
water on a DPPC monolayer
after 1 day hydratation
No-slip
D(nm)
D (nm)
G’’-1(w) nm/µN
water on a fresh DDPC monolayer :
(1-2 hours in water)
slip length b=10±3nm
b=10 nm
b= 0
D (nm)
b= 0
water on DPPC bilayer :
no-slip within 3 nm

68. CONCLUSION
Hydrodynamic BC for a simple liquid at a solid surface
is most of the time a no-slip bc, verified down to the molecular scale
On smooth surface significant slip can develop, in strongly
non-wetting conditions
Theory, molecular dynamic simulations, and a number of experimental
result show that intrinsic slip lengths on smooth surface are at most of
the order of nm
On rough surfaces, hydrodynamic calculations show that slip effects
shoud be smaller than on the chemically equivalent smooth surface
Apart for instrumentation artefacts, why are so large slip length
sometimes found ?

69. Other possible artefacts in boundary slip measurements

70. with hydrophobic surfaces
Contamination
D (nm)
Experiment run
in usual room
with hydrophobic surfaces

71. with hydrophobic surfaces
Contamination
D (nm)
constant slip length
Experiment run
in usual room
with hydrophobic surfaces

72. Nanobubbles ? Ishida Langmuir 16, 6377 (2000)
Nanobubbles on OTS-coated
silicon water