liquid flows on surfaces:

Liquid flows on surfaces:the boundary condition

E. CHARLAIXUniversity of Lyon, France

NANOFLUIDICS SUMMER SCHOOL August 20-24 2007THE ABDUS SALAM INTERNATIONAL CENTER FOR THEORETICAL PHYSICS

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 nanochannelsElektrokinetics effectsDispersion & mixing

Slippage effects in macroscopic flows ?

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)

Yesterday we saw that N.S. equation for simple liquids is very robust constitutive equation down to (some) molecular scale.

What about boundary condition ?

500 nm

Microchannels…

…nanochannels

Usual b.c. : the fluid velocity vanishes at wall

z

VS = 0

Hydrodynamic boundary condition (h.b.c.) at a solid-liquid interface

v(z)

OK at a macroscopic scale and for simple fluids

Phenomenological : 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 19th century

Lauga & al, in Handbook of Experimental Fluid Dynamics, 2005

M. Denn, 2001 Annu. Rev. Fluid Mech. 33:265–87

And alsoBulkley (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 with liquid monolayer sticking at wallvarious organic liquids/mica

Ag

J.N. IsraelachviliJ. Colloid Interf. Sci. (110) 263, 1986 Water on mica: no-slip within 2 Å

George et al., J. Chem. Phys. 1994

no-slip flow w. monolayer sticking at wallvarious organic liquids/ metal surfaces

Drainage experiments with SFA

∆P

N.V. Churaev, V.D; Sobolev and A.NSomovJ. Colloid Interf. Sci. (97) 574, 1984

Water slips in hydrophobic capillariesslip length 70 nm

z

VS ≠ 0

v(z)

b

VS : slip velocity

S : tangential stress at the solid surface

b : slip length

: liquid-solid friction coefficient

: liquid viscosity

€

b =η

λ

Partial slip and solid-liquid friction

Navier 1823Maxwell 1856

∂V∂z

= : shear rate

Tangentiel stress at interface

€

S = η∂V

∂z= λ VS

€

VS = b∂V

∂z

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()

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

Pressure drop in nanochannels

d

∆P

x

zb

Slit

r

Tube

QuickTime™ et undécompresseur TIFF (non compressé)

sont requis pour visionner cette image.

Exemple 1

slitd=1 µm

b=20 nm%error on permeability : 12%

Exemple 2

tube

r = 2 nm

b=20 nm

Error factor on permeability : 80

(2 order of magnitude)

B. Lefevre et al, J. Chem. Phys 120 4927 2004 Water in silanized MCM41of 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: QuickTime™ et un

décompresseur TIFF (non compressé)sont requis pour visionner cette image.

QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.Molecular diffusion spreads the solute through the width within

Solute motion is analogous to random walk: QuickTime™ et un






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 2b = 1.5 d reduction factor 10

Electric fieldelectroosmotic flow

Electrostatic double layernm 1 µm

Electrokinetic phenomena

Electro-osmosis, streaming potential… are determined by interfacialhydrodynamics at the scale of the Debye length

Colloid science, biology, …

- - - - - - - - -+

++

++

+ ++

+ +

E v

z

x

os

- - - - - - - - -+

++

++

+ ++

+ +

E v

z

x

os

Case of a no-slip boundary condition:

no-slip plane

zH

zeta potential

- - - - - - - - -+

++

++

+ ++

+ +

E v

z

x

os

Case of a partial slip boundary condition:

At constant s the electroosmotic velocity depends on Debye length.

Pb for measuring surface charges

Possibility of very large el-osmotic

flow by decreasing

b=20nm

= 3nm at 10-2M

Example:

Churaev et al, Adv. Colloid Interface Sci. 2002L. Joly et al, Phys. Rev. Lett, 2004

os increased by 800%





What about macroscopic flows ?

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)

u

q = 2

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 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 bythermal 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 datameasured in equilibrium Molecular Dynamics simulation

Green-Kubo relation for the hydrodynamic b.c.:

(assumes that momentum fluctuations in fluid obey Navier-Stokesequation + 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

canonicalequilibrium

Friction coefficient (i.e. slip length) can be computed at equilibrium fromtime decay of correlation function of momentum tranfer

« soft sphere » liquid interaction potential (r) = ( r)12 molecular size :

u

q = 2

u/ b/

00.01>0.03>0.03

∞400-2

very small surface corrugation isenough 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 z)= sf (1/z9-1/z3)

Strong wall-fluid attraction induces an immobile fluid layer at wall

sf =15

Effect of liquid-solid interaction

D

€

vαβ (r) = 4εσ

r

⎛

⎝ ⎜

⎞

⎠ ⎟

12

− cαβ

σ

r

⎛

⎝ ⎜

⎞

⎠ ⎟6 ⎡

⎣ ⎢

⎤

⎦ ⎥

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

= {fluid,solid}

Simple Lennard-Jones fluid with fluid-fluid and fluid-solid interactions

Barrat et al Farad. Disc. 112,119 1999

c parameter controls wettability

Wettability is characterized by contact angle (c.a.)

cFS=1.0 : =90°

cFS=0.5 : =140°

cFS=0 : =180°

Two types of flow

Here : =140°, P~7 MPaSlip length b=11 is found (both case)

Poiseuille flow

V(z)

z/

F0

b=0

Couette flow

V(z)

z/

U

b=0

Linear b.c. up to ~ 108 s-1

Slip at a microscopic scale: liquid-solid interaction effect

=140°

130°

=90°

b/

P/P0P0~MPa

substantial slips occurs on strongly non-wetting systems

slip length increases with c.a.

essentially no (small) slip in partial wetting systems ( < 90°)

slip length increases stronly as pressure decreases

Po ~ 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 = 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 onwetting properties

fluid struct.factorparallel to wall

wall corrugationa exp(q// • R//)

molecular size

//

Barrat et al Farad. Disc. 112,119 1999

Intrinsic b.c. on smooth surfaces : conclusion

substantial slips occurs in strongly non-wetting systems

slip length increases with c.a.

slip length depends strongly on pressure

at moderate shear rate ( < 108 s-1 ) essentially no slip in wetting systems

slip length amplitude is moderate (~ 5 nm at )

.

slip length is not expected to depend on fluid viscosity (≠ polymers)

non-linear slip develops above a (high) critical shear rate (~ 109 s-1 )

Thomson-Troian Nature 1997

QuickTime™ et undécompresseur sont requis pour visionner cette image.QuickTime™ et undécompresseur


Duez, Ybert, Clanet, Bocquet Nature Physics 3, 180, 2007

MOVING CONTACT LINE INSTABILITY

U

d

ze

d

static contact angle

dynamic ‘ ‘

Tangential stress on L/S surface diverges at c.l.

LV SV

SLQuickTime™ et un


Adds a dynamic force at c.l.:QuickTime™ et un






The dynamic c.a. increases with flow velocity:

Capillary numberAbove threshold Ca > Cac, d = 180°

the c.l. destabilizes, a fluid film is trapped

SV - SL = LV cos e

LANDAU-LEVITCH EFFECT



U

QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.



De Gennes, Brochart et Quéré, Gouttes bulles perles et ondes, 2005

ANTI LANDAU-LEVITCH EFFECT

U

Duez & al Nature Physics 3, 180, 2007



Duez & al Nature Physics 3, 180, 2007





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