University of Lyon, France
INTRODUCTION TO MICROFLUIDICS August 8-26 2005
The Abdus Salam international center for theoretical physics

OUTLINE
Roughness and wetting : a conspiracy ?
Super- hydrophobic effect
Flow on patterned surfaces

Theory, MD simulations, and a number of experimental result: intrinsic slip lengths on smooth surface are at most of the order of 10-20 nm
Hydrodynamic calculations: roughness kills slip
Why are so large slip length sometimes found ?

Hydrodynamic calculations : roughness decreases slip.
On non-wetting surfaces, can roughness increase slip ?

 Watanabee et al J.F.M.1999
Rough surface with water-repellent coating
Contact angle 150 °
Very large slip effects (200 µm )
Drag reduction in high Re flows
100µm
20µm

Treteway -Meinhart, Phys. of Fluids, 2004
PIV in hydrophilic capillaries :
Poiseuille flow, no-slip
30 µm
300 µm
PIV OTS coated capillaries
Average slip 200 à 900 nm
Heterogeneous
Slip disappears under applied hydrostatic pressure

Bico, Marzolin
& Quéré
Europhys. Lett 47, 220 (1999)
« Fakir » effect or
Super-hydrophobicity

BASICS OF WETTING g
SL g
SV
: solid-liquid surface tension
: solid-liquid surface tension g
LV
: solid-liquid surface tension g
LV g
SV g
SL equilibrium contact angle :
Young Dupré relation g
SV g
SL
= g
LV cos q partially wetting liquid : q < 90° non wetting liquid : q < 90° perfect wetting liquid : q =0°

WETTING OF A ROUGH SURFACE
Young’s law on rough surface:
Wenzel law q o : contact angle on flat chemically same surface
1
-
-1 1 q
-1

2a
WETTING OF A PATTERNED SURFACE h
Wenzel law: q
?
Composite wetting

COMPOSITE WETTING
Bico, Marzolin & Quéré
Europhys. Lett 47, 220 (1999)
Composite wetting is favorable if
-
-1
Super- hydrophobic region
Liquid must be non-wetting
Wenzel law
-1

q
CONTACT ANGLE IN COMPOSITE WETTING
Youngs law for composite wetting
Super- hydrophobic state
-1
Wenzel state
-1
Bico, Marzolin & Quéré
Europhys. Lett 47, 220 (1999)

METASTABILITY EFFECT IN COMPOSITE WETTING
∆P q
System such that prepared in SH state d
State is metastable up to
-1 eq. state
Metastable SH state can be obtained on rough surface as soon as q o
< 90 °.
The smallest the scale the more robust is SH state
-1

METASTABILITY OF WETTING ON
MICROPATTERNED SURFACES
Lafuma & Quéré 2003
Nature Mat. 2, 457
Superhydrophobic state
Compression of a water drop between two identical microtextured hydrophobic surfaces. The contact angle is measured as a function of the imposed pressure.
Wenzel state

Lafuma & Quéré 2003
Nature Mat. 2, 457
Contact angle after separating the plates
Superhydrophobic state
Wenzel state
Maximum pressure applied

METASTABILITY EFFECT IN COMPOSITE WETTING q
-1
Super- hydrophobic region
Need to nucleate a bubble to recover
SH state.
lower pressure in liquid, heat…
The largest the scale the more difficult.
eq. state
-1

Cottin-Bizonne & al 2003 Nature Mat 2, 237
1 µm

Lennard-Jones fluid

= {liquid,solid},

: energy scale

: molecular diameter c
b
: wetting control parameter
Non-wetting situation : c
Ls
= 0,5 : q o
=140 °
N : nb of molecule in the cell

C
b = 0.5 q
= 140 °
N is constant
Volume
Imbibated (Wenzel) state Super-hydrophobic state

Wetting transition under applied pressure
Super-hydrophobic state Imbibated (Wenzel) state
Gibbs energy at applied pressure P (neglecting pressure in vapor phase)
Super-hydrophobic state is stable if
Super-hydrophobic transition at zero pressure
For a given material and texture shape, super-hydrophobic state is favored if scale is small

Imbibated (Wenzel) state
Volume
Super-hydrophobic state

Measure velocity profile as a function of z
V bas
V haut

- on the smooth surface : slip = 22

- on the imbibated rough surface : slip = 2

Roughness decreases slip

- on the smooth surface : slip = 24

- on the super-hydrophobic surface : slip = 57

Roughness increases slip

Barentin et al EPJ E 2005
150
Superhydrophobic state
100 d
P cap
=
-2 g lv cos q d
50
Imbibated state
0
0 1 2 3
P/P cap
The boundary condition depends highly on pressure.
Low friction flow is obtained under a critical pressure, which is the transition
Pressure to obtain the super-hydrophobic state

OUTLINE
Roughness and wetting : a conspiracy ?
Super- hydrophobic effect
Flow on patterned surfaces
Apparent bc on a patterned surface: macroscopic calculation

y
Local slip length : b(x,y)
Independant of shear rate x

Local slip length : b(x,y) Shear applied at z =
Bulk flow : Stokes equations Analytic calculation of apparent slip
Apparent slip length does not depend on shear rate

flow
Effective slip length
 Stripes parallel to the shear
(Philip 1972) analytical calculation
The length scale for slip is the texture scale
Even with parallel stripes of perfect slip, effective slip is weak:
B
//
= L for z
= 0.98

flow
Effective slip length
 Stripes perpendicular to the shear (Stone and Lauga 2003)

LENGTH SCALE FOR THE EFFECTIVE SLIP
Velocity corrugation
Bulk equations
Boundary condition
2D Fourier transform
Velocity corrugations of wavelength 2π/q at solid surface decay exponentially in bulk with length scale L/2π
At z=L/2 the velocity field is smooth

WHAT ABOUT LOCAL PARTIAL SLIP CONDITIONS ?
Wenzel state
L b o
= 0 fraction area z b
1
≠ ∞
The smallest length (b
1
,L) determines the effective slip b
1
/L

WHAT ABOUT LOCAL PARTIAL SLIP CONDITIONS ?
Super-hydrophobic state b o
≠ 0 b
1
=
∞
The largest length (b o
,L) determines the effective slip b o
/L

AN APPROXIMATE MODEL z
Slip length heterogeneties induces flow corrugations which decay as exp(-z/L)
L x
Energy balance in a boundary layer of thickness L
Average shear rate in the layer

2 log B eff
0
-2
-4
-4 b
1
=
∞ b o
= 0
-2 log L 0
For high slip : b
1
=
∞
L large z as small as possible
2
Bimodal distribution b o
, fraction area z b
1
> b o
, fraction area 1z

Comparison of MD slip length with a macroscopic calculation on a flat surface with a periodic pattern of h.b.c.

More dissipation than macroscopic calculation because of the meniscus

AN EXPERIMENTAL REALISATION
Slip length
20µm
Hydrophobic silicon microposts
Weaker work pressure (0.03 bars) allows very large values of B

Roughness + hydrophobicity can induce large slip effects due to composite wetting (gaz/vapor at the liquid-solid interface)
Large slip is obtained in the super-hydrophobic state
Relevant length scale for slip length is texture wavelength
Large slip requires few liquid-solid contact (thin pillars texture)
Pressure effects are important.
Applied pressure can cause transition toward sticky imbibated(Wenzel) state
Patterning surfaces for large slip effects needs texture optimization with competing effects :
Increasing L increases slip but decreases robustness of Super-Hydrophobic state

Nanotube forest