Velocity and temperature slip at the liquid solid interface

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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
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Outline

• Interfacial constitutive equations
• Slip length
• Kapitsa length
• Results for  ideal  surfaces
• Slip length on structured surfaces
• Thermal transport in  nanofluids 

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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
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Micro/Nano fluidics (biomedical
analysis,chemical engineering)
Lee et al, Nanoletters 2003
Microchannels
.Nanochannels
Importance of boundary conditions
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INTERFACIAL HEAT TRANSPORT

• Micro heat pipes
• Evaporation
• Layered materials
• Nanofluids

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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

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Interfacial constitutive equation for flow at an
interface surface the slip length (Navier,
Maxwell)
Continuity of stress
h viscosity l friction
micro-channel S/V1/L ?surface effect becomes
dominant. Influence of slip

• Flow rate is increased
• Hydrodynamic dispersion and velocity gradients
  are reduced.

Poiseuille
Electro-osmosis
Also important for electrokinetic phenomena
(Joly, Bocquet, Ybert 2004)
Electrostatic double layer nm - 1µm
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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
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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
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Experimental tools slip length
Drag reduction in capillaries
SFA (surface force apparatus)
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
Evanescnt wave
Optical tweezres
Optical methods PIV, fluorescence recovery
Experiments often difficult must be associated
with numericaltheoretical studies
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Some experimental results
Slip length (nm)
Tretheway et Meinhart (PIV)
Pit et al (FRAP)
Nonlinear
Churaev et al (perte de charge)
1000
Craig et al(AFM)
Bonaccurso et al (AFM)
Vinogradova et Yabukov (AFM)
Sun et al (AFM)
100
Chan et Horn (SFA)
Zhu et Granick (SFA)
Baudry et al (SFA)
Cottin-Bizonne et al (SFA)
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MD simulations
1
150
100
50
0
Contact angle ()
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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

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Linear response theory (L. Bocquet, JLB, PRL 1994)
Kubo formula
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• Note the slip length is intrinsically a
  property of the interface. Different from
  effective boundary conditions used for porous
  media (Beavers Joseph 1967)

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Density at contact is controlled by the wetting
properties and the applied hydrostatic pressure.
Slip length as a function of pressure
(Lennard-Jones fluid)
Density profiles
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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)
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Analogy hydrodynamic slip length ? Kapitsa length
Energy ? Momentum Temperature? Velocity
Energy current ? Stress
Kubo formula
q(t) energy flux across interface, S area
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Results for Kapitsa length
-Lennard Jones fluids
Wetting properties controlled by cij
-equilibrium and nonequilibrium simulations
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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
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Dependence of lK on wetting properties (very
similar to slip length)
J-L Barrat, F. Chiaruttini, Molecular Physics 2003
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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
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Time resolved reflectivity experiments
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Influence of roughness (slip length)
Far flow field no slip
Richardson (1975), BrennerFluid 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.
D. Quéré et al
1 µm
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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
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Pressure dependence
Flow parallel to the grooves
(similar results for perpendicular flow)
Superhydrophobic
imbibited
?
rough surface
flat surface

b (nm)
P/Pcap
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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
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Complete hydrodynamic description is complex (cf
Cottin et al, Eur. Phys. Journal E 2004). Some
simple conclusions
Partial slip complete slip, small scale
pattern b1gtgtL
Works well at low pressures (additional
dissipation associated with meniscus at higher P)
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Partial slip complete slip, large scale
pattern b1 ltlt 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)
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How to design a strongly slippery surface ?
For fixed working pressure (0.5bars) size of the
posts (a100nm), and value of b0 (20nm)

• Compromise between
• Large L to obtain large z
• Pcap large enough
• PltPcap ? 1/( L)

?
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Possible candidate hydrophobic nanotubes
 brush  (C. Journet, S. Purcell, Lyon)
P. Joseph et al, Phys. Rev. Lett., 2006, 97,
156104
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Effective slippage versus pattern flow without
resistance in micron size channel ? (Joseph et
al, PRL 2007)
beff µm
z (µm)
beff
L
Characteristic size L ?
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• 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
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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

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Simulation of the cooling process

• Accurate determination of RK fit to heat
  transfer equations
• No influence of Brownian motion on cooling

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Simulation of heat transfer
Maxwell Garnett effective medium prediction
hot
a (Kapitsa length / Particle Radius )
Cold
M. Vladkov, JLB, Nanoletters 2006
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Effective medium theory seems to work well for a
perfectly dispersed system. Explanation of
experimental results ? Clustering, collective
effects ?
Our MD
Volume fraction
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Effective medium calculation for a linear string
of particles
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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 ?)
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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