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Velocity and temperature slip at the Velocity and temperature slip at the liquid solid interfaceliquid solid interface
Jean-Louis Barrat Université de Lyon
Collaboration : L. Bocquet, C. Barentin, E. Charlaix, C. Cottin-Bizonne, F. Chiaruttini, M. Vladkov
Outline
• Interfacial constitutive equations
• Slip length
• Kapitsa length
• Results for « ideal » surfaces
• Slip length on structured surfaces
• Thermal transport in « nanofluids »
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
Hydrodynamics of confined fluids
Bowden et Tabor, The friction and lubrication of solids, Clarendon press 1958
D. Tabor
Micro/Nano fluidics (biomedical analysis,chemical engineering)
Microchannels ….Nanochannels
Lee et al, Nanoletters 2003
Importance of boundary conditions
INTERFACIAL HEAT TRANSPORT
•Micro heat pipes
•Evaporation
•Layered materials
•Nanofluids
Hydrodynamic description of transport phenomena
1) Bulk constitutive equationsBulk constitutive equations : Navier Stokes, Fick, Fourier
2) Boundary conditions: mathematical concept
Physical material property, connected with statistical physics
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)
b z
VbVS
z
VVSS
Flow rate is increasedHydrodynamic dispersion and
velocity gradients are reduced.
micro-channel : S/V~1/L surface effect becomes dominant.
Influence of slip:
Also important for electrokinetic phenomena (Joly, Bocquet, Ybert 2004)
++ +++ + +- - - -
--
electric field
E vElectro-osmosis
Poiseuille
Electrostatic double layer~ nm - 1µm
= viscosity
= friction
Continuity of stress
Thermal contact resistance or Kapitsa resistance defined through
Kapitsa length:
Interfacial constitutive equation for heat transportKapitsa resistance and 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)
AFM with colloidal probe
Optical tweezres
Optical methods:: PIV, fluorescence recovery
Experiments often difficult – must be associated with numerical:theoretical studies
vEvanescnt wave
Drag reduction in capillaries
Churaev, JCSI 97, 574 (1984)
Choi & Breuer, Phys Fluid 15, 2897 (2003)
Craig & al, PRL 87, 054504 (2001)
Bonnacurso & al, J. Chem. Phys 117, 10311 (2002)
Vinogradova, Langmuir 19, 1227 (2003)
1
10
100
1000
150100500
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
Nonlinear
Slip length(nm)
MD simulations
•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
Simulation results (intrinsic length: ideal surfaces, no dust particles, distances perfectly known)
Linear response theory (L. Bocquet, JLB, PRL 1994)
b 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.
=140°
=90°
b/
P/P0P0~MPa
Slip length as a function of pressure (Lennard-Jones fluid)Density profiles
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
-equilibrium and nonequilibrium simulations
Wetting properties controlled by cij
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)
Perfect slip locally – rough surface
Richardson (1975), BrennerFluid mechanics calculation – Roughness suppresses slip
Far flow field – no slip
But: combination of controlled roughness and dewetting effects can increase slîp by creating a « superhydrophobic » situation.
1 µmD. Quéré et al
Simulation of a nonwetting patternRef : C. Cottin-Bizonne, J.L. Barrat, L. Bocquet, E. Charlaix, Nature Materials (2003)
Superhydrophobic state
« imbibited » state
Hydrophobic walls = 140°
Flow parallel to the grooves
(similar results for perpendicular flow)
P/Pcap
b (n
m)
imbibitedimbibitedSuperhydrophobic
rough surfacerough surface
flat surfaceflat surface*
Pressure dependence
Vertical shear rate: Inhomogeneous surface:
Far field flow
is the slip velocity
Macroscopic description
(C. Barentin et al 2004; Lauga et Stone 2003; J.R. Philip 1972)
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
a
L= spatial scale of the pattern
a= lateral size of the posts
s = (a/L)2 Area fraction of no-slip BC
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
Pcap large enough
P<Pcap 1/( L)
Contact angle 180 degrees
Possible candidate : hydrophobic nanotubes « brush » (C. Journet, S. Purcell, Lyon)
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)
Characteristic size L
z (µm)
L beff
-2 0 2 4 6
0
0.2
0.4
v/v 0
-2 0 2 4 6
0
0.2
0.4
v/v 0
-2 0 2 4 6
0
0.2
0.4
z (m)
v/v 0
beff ~ µm
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
• 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
Thermal transport in nanofluids
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
Cold
Maxwell Garnett effective medium prediction
= (Kapitsa length / Particle Radius )
M. Vladkov, JLB, Nanoletters 2006
Effective medium theory seems to work well for a perfectly dispersed system.
Explanation of experimental results ? Clustering, collective effects ?
0,1
1
10
100
1000
0,01 0,1 1 10
MW CNT
Al2O
3
CuO TiO
2
Au
Au ???
Cu
(² )exp
/ (² )2 phases model
Fe
Volume fraction
Our MD
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
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