Liquid flows on surfaces:
the boundary condition
Nanoscale Interfacial Phenomena in Complex Fluids

May 19

June 20 2008
The Kavli Institute of Theoretical Physics China
Pressure driving becomes insufficient
L
h
V
P
o
P
o
+
D
P
V = 1 mm/s, L=1cm,
h
= 10

3
Pa.s
h = 0.1
m
m
D
P = 100 bar
Downsizing flow devices
raises new problems
500
nm
Microchannels…
…nanochannels
New solutions are needed
Miniaturization increases
surface to volume
ratio
:
importance of
surface phenomena
The description of flows requires
constitutive equation (bulk property of fluid)
+ boundary condition (
surface property)
We saw that N.S. equation for simple liquids is very robust
constitutive equation down to (some) molecular scale.
What about boundary condition ?
The no

slip boundary condition (bc):
a long lasting empiricism regularly questionned
Theory of the h.b.c. for simple liquids
Some examples of importance of the b.c. in nanofluidics
Pressure drop in nanochannels
Elektrokinetics effects
Dispersion & mixing
Usual b.c. : the fluid velocity vanishes at wall
z
V
S
= 0
Hydrodynamic boundary condition (h.b.c.)
at a solid

liquid interface
v(z)
OK at a
macroscopic
scale and for
simple
fluids
Phenomenological origin:
derived from experiments on low
molecular mass liquids
Goldstein 1938
Goldstein S. 1969. Fluid mechanics in the first half of this century.
Annu. Rev. Fluid Mech
1:1
–
28
Batchelor, An introduction to fluid dynamics, 1967
The nature of hydrodynamics bc’s has been widely debated in 19
th
century
Lauga & al, in Handbook of Experimental Fluid Dynamics, 2005
M. Denn, 2001
Annu. Rev. Fluid Mech
. 33:265
–
87
And also
Bulkley (1931),Chen & Emrich (1963), Debye & Cleland (1958)…
… and some time suspected on non

wetting surfaces
But wall slippage occurs in polymer flows…
Pudjijanto & Denn 1994
J. Rheol. 38:1735
Shark

skin effect in
extrusion of polymer melts
C. Chan and R. Horn
J. Chem. Phys. (83) 5311, 1985
mica
Ag
no

slip flow over a «
trapped
» monolayer
various organic liquids / mica
Ag
J.N. Israelachvili
J. Colloid Interf. Sci. (110) 263, 1986
Water on mica:
no

slip within 2 Å
George et al.,
J. Chem. Phys. 1994
no

slip flow over «
trapped
» monolayer
various organic liquids/ metal surfaces
Drainage experiments with SFA
N.V. Churaev, V.D; Sobolev and A.NSomov
J. Colloid Interf. Sci. (97) 574, 1984
Water slips in hydrophobic capillaries
slip length 70 nm
z
V
S
≠ 0
v(z)
b
V
S
:
slip velocity
s
S
:
tangential stress at the solid surface
b
:
slip length
l
:
liquid

solid friction coefficient
h
:
liquid viscosity
Partial slip and solid

liquid friction
Navier 1823
Maxwell 1856
∂
V
∂
z
=
g
:
shear rate
Tangentiel stress at interface
Interpretation of the slip length
From Lauga & al, Handbook of Experimental Fluid Dynamics, 2005
b
The bc is an interface property.
The slip length has not to be related to an internal scale in the fluid
The hydrodynamic b.c. is fully characterized 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).
Some properties of the slip length
No

slip bc (b=0) is associated to very large liquid

solid friction
The no

slip boundary condition (bc):
a long lasting empiricism regularly questionned
Theory of the h.b.c. for simple liquids
Some examples of importance of the b.c. in nanofluidics
Pressure drop in nanochannels
Elektrokinetics effects
Dispersion & mixing
Pressure drop in nanochannels
d
∆P
x
z
b
Slit
r
Tube
Exemple 1: slit d=1 µm
gain in flow rate : 12%
Change from no

slip to b=20nm
Exemple 2: tube d= 2 nm
Change from no

slip to b=20nm
gain in flow rate : 8000%
(2 order of magnitude)
B. Lefevre et al, J. Chem. Phys 120 4927 2004
Silanized MCM41
of various radii (1.5 to 6 nm)
10nm
Forced imbibition of hydrophobic mesoporous medium
The intrusion

extrusion cycle of water in
hydrophobic MCM41
mesoporous silica:
MCM41
quasi

static cycle does not depend on
frquency up to kHz
Exemple 3
L ~ 2

10 µm
Porous grain
Dispersion of transported species

Mixing
t=0
injection
d
time t
Taylor dispersion
Without molecular diffusion:
Molecular diffusion spreads the solute through the width within
Solute motion is analogous to random walk:
With partial slip b.c.
t=0
d
time t
b
With partial slip b.c.
t=0
d
time t
b
Same channel, same flow rate
Hydrodynamic dispersion is significantly reduced if b ≥ d
b = 0.15 d reduction factor 2
b = 1.5 d reduction factor 10
Electric field
electroosmotic flow
Electrostatic double layer
nm 1 µm
Electrokinetic phenomena
Electro

osmosis, streaming potential… are determined by interfacial
hydrodynamics at the scale of the Debye length
Colloid science,
biology,
…
The no

slip boundary condition (bc):
a long lasting empiricism regularly questionned
Theory of the h.b.c. for simple liquids
Some examples of importance of the b.c. in nanofluidics
Pressure drop in nanochannels
Elektrokinetics effects
Dispersion & mixing
locally: perfect slip
Far field flow : no

slip
Effect of surface roughness
roughness
«
kills
» slip
Richardson,
J Fluid Mech 59 707 (1973),
Janson,
Phys. Fluid 1988
Fluid mechanics calculation :
Robbins (1990)
Barrat, Bocquet (1994, 1999)
Thomson

Troian (Nature 1997)
Slip at a microscopic scale :
molecular dynamics on simple liquids
b
Thermodynamic equilibrium
determination of b.c.
with Molecular Dynamics simulations
Be
j
(
r
,t) the fluctuating momentum density at point
r
Assume that it obeys Navier

Stokes equation
And assume Navier boundary condition
Bocquet & Barrat, Phys Rev E 49 3079 (1994)
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
b
Green

Kubo relation for the hydrodynamic b.c.:
(assumes that momentum fluctuations in fluid obey Navier

Stokes
equation + b.c. condition of Navier type)
Slip at a microscopic scale : linear response theory
Liquid

solid
Friction coefficient
total force exerted
by the solid on the liquid
canonical
equilibrium
Friction coefficient (i.e. slip length) can be computed at equilibrium from
time decay of correlation function of momentum tranfer
Bocquet & Barrat, Phys Rev E 49 3079 (1994)
«
soft sphere
» liquid
interaction potential
n
(r) =
e
(
s/
r)
12
molecular size :
s
u
q = 2
p/ s
u/
s
b/
s
0
0.01
>0.03
>0.03
∞
40
0

2
very small surface corrugation is
enough to suppress slip effects
Slip at a microscopic scale : molecular dynamics
Barrat, Bocquet, PRE (1994)
hard wall
corrugation z=u cos qx
attractive wall
interaction potential
f (
z)=
e
sf
(1/z
9

1/z
3
)
Strong wall

fluid attraction induces
an immobile fluid layer at wall
e
sf
/
e
=15
Effect of liquid

solid interaction
D
a, b
= {fluid,solid}
Simple
Lennard

Jones
fluid
with fluid

fluid and fluid

solid interactions
Barrat et al
Farad. Disc. 112,119 1999
c
ab
parameter controls wettability
Wettability is characterized by contact angle (c.a.)
c
FS
=1.0 :
q
=90
°
c
FS
=0.5 :
q
=140
°
c
FS
=0 :
q
=180
°
Two types of flow
Here :
q
=140
°
, P~7 MPa
Slip length b=11
s
is found (both case)
Poiseuille flow
V(z)
z/
s
F
0
b=0
Couette flow
V(z)
z/
s
U
b=0
Linear b.c. up to ~ 10
8
s

1
Slip at a microscopic scale: liquid

solid interaction effect
q
=140
°
130
°
q
=90
°
b/
s
P/P
0
P
0
~MPa
substantial slips occurs on
strongly non

wetting systems
slip length increases with c.a.
essentially no (small) slip in
partial wetting systems (
q
< 90
°
)
slip length increases stronly as
pressure decreases
P
o
~ MPa
Slip increases with reduced fluid density at wall.
However slippage does not reduce to «
air cushion
» at wall.
fluid density profile across the cell
Soft spheres on hard repulsive wall
Lennard

Jones fluid
q
= 137
°
Slip at a microscopic scale: theory for simple liquids
Analytical expression for slip length
Depends only on structural parameters, no dynamic parameter
density at wall,
depends on
wetting properties
fluid struct.factor
parallel to wall
wall corrugation
a exp(q
//
• R
//
)
molecular size
//
Barrat et al
Farad. Disc. 112,119 1999
Theory for intrinsic b.c. on smooth surfaces : summary
substantial slips in strongly non

wetting systems
slip length increases with c.a.
slip length decreases with increasing pressure
no

slip in wetting systems
(except very high shear rate
g
<
10
8
s

1
)
slip length is moderate
(~ 5 nm at
q ~ 120
)
.
slip length does not depend on fluid viscosity
(≠ polymers)
non

linear slip develops at high shear rate
(~
10
9
s

1
)
.
(obtained with LJ liquids, some with water)
1
10
100
1000
slip length (nm
)
150
100
50
0
Contact angle (
°
)
Tretheway et Meinhart (PIV)
Pit et al (FRAP)
Churaev et al (perte de charge)
Craig et al(AFM)
Bonaccurso et al (AFM)
Vinogradova et Yabukov (AFM)
Sun et al (AFM)
Chan et Horn (SFA)
Zhu et Granick (SFA)
Baudry et al (SFA)
Cottin

Bizonne et al (SFA)
Some experimental results….
MD Simulations
Non

linear slip
Brenner, Lauga, Stone 2005
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