Flows at a nano-scale:
Where does classical hydrodynamics stop ?
(and how to describe flow beyond ?)
E. CHARLAIX
University of Lyon, France
NANOFLUIDICS SUMMER SCHOOL August 20-24 2007
THE ABDUS SALAM INTERNATIONAL CENTER FOR THEORETICAL PHYSICS
OUTLINE

Why nano-hydrodynamics ?
 Surface Force Apparatus: a fluid slit controlled
at the Angstrom level
 First nano-hydrodynamic experiments performed with SFA
 Experiments with ultra-thin liquid films
solid or glass transition ?
a controversy resolved
Nanofluidic devices
Microchannels…
…nanochannels
Miniaturization increases surface to volume ratio:
500 nm
50 nm channels
Wang et al, APL 2002
importance of surface phenomena
Nanochannels are more specifically designed for :
 manipulation and analysis of biomolecules
.
with single molecule resolution
 specific ion transport
Mesoporous materials
Large specific surface (1000m2 /cm3 ~ pore radius 2nm)
catalysis, energy/liquid storage…
10nm
Water in mesoporous silica
(B. Lefevre et al, J. Chem. Phys 2004)
Water in nanotube
Koumoutsakos et al 2003
Electrokinetic phenomena
Colloid science, biology, nanofluidic devices…
Electric field
Electrostatic double layer
3 nm
300 nm
electroosmotic flow
Electro-osmosis, streaming potential… are determined by
nano-hydrodynamics at the scale of the Debye length
Tribology :
Mechanics, biomechanics, MEMS/NEMS friction
Rheology and mechanics
of ultra-thin liquid films
First measurements at a sub-nanometric scale:
Surface Force Apparatus (SFA)
Bowden & Tabor
The friction and lubrication of solids
Clarendon press 1958
J. N. Israelachvili
Intermolecular and surface forces
Academic press 1985
OUTLINE

Importance
 Surface Force Apparatus : a slit controlled
at the Angstrom level
 First nano-hydrodynamic experiments performed with SFA:
 Experiments with ultra thin liquid films
solid or glass transition ?
a controversy resolved
Surface Force Apparatus (SFA)
Tabor et Winterton, Proc. Royal Soc. London, 1969
Israelachvili, Proc. Nat. Acad. Sci. USA 1972
D
Ag
mica
Ag
Optical resonator
Franges of equal chromatic order (FECO)
Tolanski, Multiple beam Interferometry of
surfaces and films, Clarendon Press 1948
Spectrograph
Source of white light
l
D=28nm
contact
Distance between surfaces
is obtained within 1 Å
l
l (nm)
r : reflexion coefficient
n : mica index
a : mica thickness
D : distance between surfaces
Force measurement
In a quasi-static regime
(inertia neglected)
Piezoelectric displacement
Oscillating force in organic liquid films
Static force in confined
organic liquid films
(alkanes, OMCTS…).
Oscillations reveal
liquid structure in layers
parallel to the surfaces
The
Horn & Israelachvili, J. Chem Phys 1981
Electrostatic and hydration force in water films
Horn & al 1989
Chem Phys Lett
OUTLINE

Importance
Surface Force Apparatus : a slit of thickness controlled
at the Angstrom level
 First nano-hydrodynamic experiments performed with SFA:
thick liquid films (Chan & Horn 1985)
 Experiments with very thin liquid films
solid or glass transition ?
a controversy resolved
Drainage of confined liquids : Chan & Horn 1985
Run-and-stop experiments
D(t)
L(t)
D
ts
Inertia negligible :
K ∆(t) = Fstatic (D) + Fhydro (D, D)
t
Lubrication flow in the confined film
Hypothesis
u(x,z)
z(x)
x
Properties
Pressure gradient is // Ox
Velocity profile is parabolic
Average velocity at x:
Mass conservation
 Reynolds force (D<<R):
Newtonian fluid
Quasi-parallel surfaces: dz/dx <<1
Low Re ( Re ≤ 10-6 )
Slow time variation: T >> z2/n
No-slip at solid wall
2
U(x)= - z dP
12h dx
h: fluid dynamic viscosity
2pxz U(x) = - p x2 D
Drainage of confined liquids : run-and-stop experiments
D(t)
∆(t)
L(t)
D
ts
6p h R2 D
K (D - D) = Fstatic (D) +
D > 6nm
D
ln D(t) - D
D(t)
=
6p h R2 (t - t ) + Cte
s
KD
t
Chan & Horn 1985 (1)
ln D(t) - D
D(t)
=
6p h R2 (t - t ) + Cte
s
KD
D > 50 nm : excellent agreement
with macroscpic hydrodynamics
Various values of D :
determination of fluid viscosity h
excellent agreement with bulk value
Chan et Horn, J. Chem. Phys. 83 (10) 5311 (1985)
Chan & Horn (2)
D ≤ 50nm : drainage too slow
Hypothesis:
fluid layers of thickness Ds
stick onto surfaces
Sticking
layers
6p h R2 D
Fhydro = D - 2Ds
Reynolds
drainage
Excellent agreement
for 5 ≤D≤ 50nm
OMCTS tetradecane hexadecane
Molecular
size
Ds
7,5Å
4Å
4Å
13Å
7Å
7Å
Chan & Horn (3)
D ≤ 5 nm:
drainage occurs by steps
Steps height = molecular size
Including static force
in dynamic equation
yields drainage steps
BUT
Occurrence of steps is NOT predicted
by « sticky » Reynolds + static forces
Draining confined liquids with SFA: conclusion
 Efficient method to study flows at a nanoscale
 Excellent agreement with macroscopic hydrodynamics
down to ~ 5 nm (6-7 molecular size thick film)
 « Immobile » layer at solid surface, about 1 molecular size
Israelachvili JCSI1985 : water on mica
George et al JCP 1994 : alcanes on metal
Becker & Mugele PRL 2003 : D<5nm
 In very thin films of a few molecular layers
macroscopic picture does not seem to hold anymore
OUTLINE

Importance
 Surface Force Apparatus : a slit of thickness controlled
at the Angstrom level
 First nano-hydrodynamic experiments performed with SFA :
 Experiments with ultra thin liquid films
solid or glass transition ?
a controversy resolved
Shearing ultra-thin films (1)
McGuiggan &Israelachvili,
J. Chem Phys 1990
Strain
gauges
Frictional force
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Velocity
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Flattened mica surfaces
Solid or liquid behaviour depending on V, V/D, history
very high viscosities, long relaxation times
‘Continuous’ solid-liquid transition
Shearing ultra-thin films (2)
Granick, Science 1991
Shear force
area
thickness
hbulk = 0.01 poise
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velocity
Dodecane D=2,7nm
Giant increase of viscosity under
confinement
Shear-thinning behaviour
OMCTS D=2,7 nm
Confinement-induced
liquid-glass transition
Shearing ultra-thin films (3)
High precision device
with both normal and shear force
Klein et Kumacheva,
J. Chem. Phys. 1998
tangential motion
confined organic liquid
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so ntr equ is p our v is io nner cette ima ge.
Shear force response
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Confinement-induced
solid-liquid transition at n=6 layers
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times
Flow in ultra-thin liquid films: questions
In very thin films of a few molecular layers macroscopic
hydrodynamics does not seem to hold anymore
What is the liquid dynamics:
Liquid-glass transition ?
Liquid-solid transition ?
How can one describe flows ?
OUTLINE

Importance
 Surface Force Apparatus : a slit of thickness controlled
at the Angstrom level
 First nano-hydrodynamic experiments performed with SFA :
 Experiments with ultra thin liquid films
solid or glass transition ?
a controversy resolved
Langmuir 99
Nano- particules are present on mica surfaces when cut with platinum
hot-wire
They affect significantly properties of ultra-thin sheared films
(Zhu & Granick 2003, Heuberger 2003, Mugele & Salmeron)
They seem to be removed by water
Methods to cleave mica without particules have been designed
(Franz & Salmeron 98, recleaved mica).
Drainage of ultra-thin films
Direct imaging with SFA
OMCTS molecule
Ø 9-10 Å
recleaved mica
(particle free)
Monochromatic light
Becker & Mugele
Phys. Rev. Lett 2003
Drainage occurs by steps
corresponding to layering transitions
Layering transitions
2 layers
F. Mugele & T. Becker PRL 2003
The heigth between each steps
is the size of a OMCTS molecule
3 layers
Each step is the expulsion
of a single monolayer
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http://pcf.tnw.utwente.nl/people/pcf_fm.doc/
The growth of the N-1 layers region gives information
on the flow in the N-layers film.
Persson & Tossati model for the dynamics of the layer expulsion
No flow
Average velocity V(x)
P=Cte
x
N -1
layers
r(t)
transition
N layers
transition region moves at velocity r(t)
Hypothesis :
Constant pressure Po in the non-flowing N-1 layers region
Lubrication flow in the N-layers region
(Assumes some linear friction law for the flow in the thin film)
Hydrodynamic limit:
 Mass conservation :
d : layer
thickness
Nd : flowing film
thickness
 + lubrication
xo : maximum extend
of the layered region
 Constant pressure in the non-flowing region :
Ao = p xo 2 maximum area of the layered region
A = p r 2 actual area of the N-1 layers region
PT model:
2
1
3
4
Ao measured
2
3
2
1
Po determined from load
Po = Load / Ao
One ajustable parameter for each curve : µ
PT model describes very well the dynamics of a monolayer expulsion
with an ad hoc friction coefficient µ depending on the flowing film thickness
Comparison with macroscopic hydrodynamics
Macroscopic hydrodynamic:
(with no-slip at wall)
N
N
Ad hoc friction model meets hydrodynamic friction at large N
For N≤5 layers, discrepancies with macroscopic hydrodynamic occur.
Effective friction is larger than predicted by hydrodynamic.
Discrete layers flow model
N-1
P=Cte
N
transition
Force balance on one layer of thickness d and length dx
x
F i+1
F i -1
Hydrodynamic limit:
i
i
x+dx
Solving discrete layers flow model
1≤ i ≤N
 Assume two different friction coefficients
liquid-liquid friction
solid-liquid friction
 Solve for 1D flow : mass conservation
m i,i±1 = m ll
m 1,0 = m N,N+1 = m ls
Velocity of transition
region, measured
N+1 equations give Vi and dP/dx as a function of m ll and m ls
 Adjust m ll and m ls so that
Ad hoc friction coefficient
of the PT model
h
=0.3
2
d
N
Discrete model describes very well the thickness variations of µ
Results of Becker & Mugele 2003
 Flow in ultra-thin films is very well described by a lubrication flow with
. ad-hoc friction coefficient depending on the film thickness.
 For N≤5 layers the friction coefficient is slightly larger than
predicted by . macroscopic hydrodynamics with no-slip b.c.
The dependence of the ad-hoc friction with the film thickness is well
. accounted by 2 intrinsic friction coefficients, one for liquid-liquid friction
. and one for liquid-solid friction
Liquid-liquid friction is close to the value of hydrodynamic limit
Liquid-solid friction is about 20 times larger than liquid-liquid friction