Velocity and temperature slip at the
liquid solid interface
Jean-Louis Barrat
Université de Lyon
Collaboration : L. Bocquet, C. Barentin, E. Charlaix, C.
Cottin-Bizonne, F. Chiaruttini, M. Vladkov
Outline
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Interfacial constitutive equations
Slip length
Kapitsa length
Results for « ideal » surfaces
Slip length on structured surfaces
Thermal transport in « nanofluids »
Hydrodynamics of confined fluids
Original motivation: lubrication of solid surfaces
Mechanical and biomechanical interest
Fundamental interest:
•Does liquid stay liquid at small scales?
•Confinement induced phase transitions ?
•Shear melting ?
•Description of interfacial dynamics ?
Controlled studies at the nanoscale: Surface force apparatus (SFA)
Tabor, Israelaschvili
Bowden et Tabor, The friction and lubrication of
solids, Clarendon press 1958
D. Tabor
Micro/Nano fluidics
(biomedical analysis,chemical engineering)
Lee et al, Nanoletters 2003
Microchannels
….Nanochannels
Importance of boundary conditions
INTERFACIAL HEAT TRANSPORT
•Micro heat pipes
•Evaporation
•Layered materials
•Nanofluids
Hydrodynamic description of transport phenomena
1) Bulk constitutive equations : Navier Stokes, Fick, Fourier
Physical material property, connected with statistical physics
2) Boundary conditions: mathematical concept
Replace with notion of interfacial constitutive relations
Physical, material property, connected with statistical physics
Interfacial constitutive equation for flow at an interface surface:
the slip length (Navier, Maxwell)
V
 S  VS  
z

b

Continuity of stress
 = viscosity
V
VS  b
z
= friction
micro-channel : S/V~1/L surface effect becomes dominant.
Influence of slip:

Poiseuille
Flow rate is increased
Hydrodynamic dispersion and
velocity gradients are reduced.
E
Also important for electrokinetic
electric field
phenomena (Joly, Bocquet, Ybert 2004) - - - - + + + + +
Electroosmosis
v
+
-+
Electrostatic
double layer
~ nm - 1µm
Interfacial constitutive equation for heat transport
Kapitsa resistance and Kapitsa length
Thermal contact resistance or Kapitsa resistance
defined through
Kapitsa length:
lK = Thickness of material
equivalent (thermally) to
the interface
Important in microelectronics (multilayered materials)
solid/solid interface: acoustic mismatch model
Phonons are partially reflected at
the interface
Energy transmission:
Zi = acoustic impedance of medium i
See Swartz and Pohl, Rev. Mod. Phys. 1989
Experimental tools: slip length
SFA
(surface
force
apparatus)
Drag reduction
in capillaries
Churaev, JCSI 97, 574 (1984)
Choi & Breuer, Phys Fluid 15, 2897 (2003)
AFM with
colloidal probe
Craig & al, PRL 87, 054504 (2001)
Bonnacurso & al, J. Chem. Phys 117, 10311 (2002)
Vinogradova, Langmuir 19, 1227 (2003)
v
Optical tweezres
Optical methods::
PIV, fluorescence
recovery
Evanescnt
wave
Experiments often difficult – must be associated with numerical:theoretical studies
Some experimental results
Slip length
(nm)
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)
Nonlinear
1000
100
10
1
MD simulations
0
50
100
Contact angle (°)
150
Simulation results (intrinsic length: ideal surfaces, no dust
particles, distances perfectly known)
•Robbins- Thompson 1990 : b at most equal to a few
molecular sizes, depending on « commensurability » and
liquid solid interaction strength
•Barrat-Bocquet 1994: Linear response formalism for b
•Thompson Troian 1996: boundary condition may become
nonlinear at very high shear rates (108 Hz)
•Barrat Bocquet 1997: b can reach 50-100 molecular
diameters under « nonwetting » conditions and low
hydrostatic pressure
•Cottin-Bizonne et al 2003: b can be increased using small
scale « dewetting » effects on rough hydrophobic surfaces

b

Linear response theory (L. Bocquet,
JLB, PRL 1994)
Kubo formula:

Density at contact.
Associated with
wetting properties
Response function
of the fluid
Corrugation
Note: the slip length is intrinsically a
property of the interface. Different from
effective boundary conditions used for
porous media (Beavers Joseph 1967)
Density at contact is controlled by the wetting properties
and the applied hydrostatic pressure.
b/
q=140°
q=90°
P/P0
Density profiles
P0~MPa
Slip length as a function
of pressure (LennardJones fluid)
A weak corrugation can also result in a large slip length.
SPC/E water on graphite
Contact angle 75°
Direct integration of Kubo
formula:
b = 18nm
(similar to SFA experiments
under clean room conditions
– Cottin Steinberger Charlaix
PRL 2005)
Analogy: hydrodynamic slip length  Kapitsa length
Energy  Momentum ;
Temperature Velocity ;
Energy current  Stress
Kubo formula
q(t) = energy flux across interface, S = area
Results for Kapitsa length
-Lennard Jones fluids
Wetting properties
controlled by cij
-equilibrium and nonequilibrium
simulations
Nonequilibrium temperature
profile – heat flux from the
« thermostats » in each solid.
Equilibrium determination of
RK
Heat flux from the work done
by fluid on solid
Dependence of lK on
wetting properties
(very similar to slip
length)
J-L Barrat, F. Chiaruttini, Molecular Physics 2003
Experimental tools for Kapitsa length:
Pump-probe, transient absorption experiments for
nanoparticles in a fluid :
-heat particles with a « pump » laser pulse
-monitor cooling using absorption of the « probe » beam
Time resolved reflectivity experiments
Influence of roughness (slip length)
Far flow field – no slip
Richardson (1975), Brenner
Fluid mechanics calculation –
Roughness suppresses slip
Perfect slip locally – rough surface
But: combination of controlled
roughness and dewetting effects
can increase slîp by creating a
« superhydrophobic » situation.
1 µm
D. Quéré et al
Simulation of a nonwetting pattern
Ref : C. Cottin-Bizonne, J.L. Barrat, L. Bocquet, E. Charlaix, Nature Materials
(2003)
Superhydrophobic
state
« imbibited »
state
Hydrophobic walls
q = 140°
Pressure dependence
Flow parallel to the grooves
(similar results for perpendicular flow)
Superhydrophobic
imbibited
b (nm)
 rough surface
* flat surface
P/Pcap
Macroscopic description
(C. Barentin et al 2004; Lauga et Stone 2003; J.R. Philip 1972)
Vertical shear rate:
Inhomogeneous surface:
Far field flow
is the slip velocity
Complete hydrodynamic description is complex (cf Cottin
et al, Eur. Phys. Journal E 2004). Some simple conclusions:
Partial slip + complete slip, small scale pattern b1>>L
Works well at low pressures
(additional dissipation associated
with meniscus at higher P)
Partial slip + complete slip, large scale pattern b1 << L
L= spatial scale of the pattern
a= lateral size of the posts
Fs = (a/L)2 Area fraction of no-slip BC
a
Scaling argument:
Force= (solid area) x viscosity x shear rate
Shear rate = (slip velocity)/a
Force= (total area) x (slip velocity) x (effective friction)
How to design a strongly slippery surface ?
For fixed working pressure (0.5bars) size of the posts
(a=100nm), and value of b0 (20nm)
Compromise between
Large L to obtain large z
Pcap large enough
P<Pcap  1/( L)

Possible candidate : hydrophobic
nanotubes « brush » (C. Journet,
S. Purcell, Lyon)
Contact angle 180 degrees
P. Joseph et al,
Phys. Rev. Lett., 2006, 97, 156104
Effective slippage versus pattern: flow without resistance in
micron size channel ? (Joseph et al, PRL 2007)
v/v0
0.4
0.2
0
-2
0
2
4
6
v/v0
0.4
0.2
beff ~ µm
0
-2
0
2
4
6
z (µm)
v/v0
L
beff
0.4
Characteristic size
0.2 L 
0
-2
0
2
z (m)
4
6
Very large (microns) slip lengths observed in some
experiments could be associated with:
•Experimental artefacts ?
•Impurities or small particles ?
•Local dewetting effects, « nanobubbles » ?
Tyrell and Attard, PRL 2001 AFM
image of silanized glass in water
Thermal transport in nanofluids
• Large thermal conductivity increase
reported in some suspensions of
nanoparticles (compared to effective
medium theory)
• No clear explanation
• Interfacial effects could be important into
account in such suspensions
Simulation of the cooling process
•Accurate determination of RK: fit to heat transfer equations
•No influence of Brownian motion on cooling
Simulation of heat transfer
hot
Maxwell Garnett effective medium
prediction
a = (Kapitsa length / Particle Radius )
Cold
M. Vladkov, JLB,
Nanoletters 2006
Effective medium theory seems to work well for a
perfectly dispersed system.
Explanation of experimental results ? Clustering,
collective effects ?
(² /)
exp
/ (² /)
2 phases model
1000
Au
???
Cu
MW CNT
100
10
1
0,1
0,01
Al2O3
CuO
TiO2
Fe
Our MD
Au
0,1
1
10
Volume
fraction
Effective medium calculation for a linear string of particles
Similar idea for fractal clusters (Keblinski 2006)
Still other interpretation: high coupling between fluid and
particle (Yip PRL 2007) and percolation of high conductivity
layers (negative Kapitsa length ?)
Conclusion
• Rich phenomenology of interfacial
transport phenomena: also electrokinetic
effects, diffusio-osmosis (L. Joly, L.
Bocquet)
• Need to combine modeling at different
scales, simulations and experiments