Liquid flows on surfaces:

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Oct 24, 2013 (4 years and 16 days ago)

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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