1. Velocity and temperature slip at the liquid solid interface Jean-Louis BarratUniversité de Lyon Collaboration : L. Bocquet, C. Barentin, E. Charlaix, C. Cottin-Bizonne, F. Chiaruttini, M. Vladkov

2. Outline Interfacial constitutive equations Slip length Kapitsa length Results for « ideal » surfaces Slip length on structured surfaces Thermal transport in « nanofluids »

3. 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

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

5. INTERFACIAL HEAT TRANSPORT Micro heat pipes Evaporation Layered materials Nanofluids

6. Hydrodynamic description of transport phenomena Bulk constitutive equations 1) Bulk 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

7. Interfacial constitutive equation for flow at an interface surface: the slip length (Navier, Maxwell)  Flow rate is increased  Hydrodynamic 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) Electro- osmosis Poiseuille Electrostatic double layer ~ nm - 1µm  = viscosity = friction Continuity of stress

8. Thermal contact resistance or Kapitsa resistance defined through Kapitsa length: Interfacial constitutive equation for heat transport Kapitsa resistance and Kapitsa length l K = Thickness of material equivalent (thermally) to the interface

9. Important in microelectronics (multilayered materials) solid/solid interface: acoustic mismatch model Phonons are partially reflected at the interface Energy transmission: Z i = acoustic impedance of medium i See Swartz and Pohl, Rev. Mod. Phys. 1989

10. 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 v Evanescnt 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)

11. 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

12. 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 ( 10 8 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)

13. Linear response theory (L. Bocquet, JLB, PRL 1994) Kubo formula: Density at contact. Associated with wetting properties Response function of the fluid Corrugation

14. Note: the slip length is intrinsically a property of the interface. Different from effective boundary conditions used for porous media (Beavers Joseph 1967)

15. Density at contact is controlled by the wetting properties and the applied hydrostatic pressure.  =140°  =90° b/  P/P 0 P 0 ~MPa Slip length as a function of pressure (Lennard- Jones fluid) Density profiles

16. 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)

17. Analogy: hydrodynamic slip length  Kapitsa length Energy  Momentum ; Temperature  Velocity ; Energy current  Stress Kubo formula q(t) = energy flux across interface, S = area

18. Results for Kapitsa length -Lennard Jones fluids - equilibrium and nonequilibrium simulations Wetting properties controlled by c ij

19. Nonequilibrium temperature profile – heat flux from the « thermostats » in each solid. Equilibrium determination of R K Heat flux from the work done by fluid on solid

20. Dependence of l K on wetting properties (very similar to slip length) J-L Barrat, F. Chiaruttini, Molecular Physics 2003

21. 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

22. Time resolved reflectivity experiments

23. Influence of roughness (slip length) Perfect slip locally – rough surface Richardson (1975), Brenner Fluid 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 µm D. Quéré et al

24. 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  = 140°

25. Flow parallel to the grooves (similar results for perpendicular flow ) P/P cap b (nm) imbibited Superhydrophobic rough surface flat surface  * Pressure dependence

26. 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)

27. Complete hydrodynamic description is complex (cf Cottin et al, Eur. Phys. Journal E 2004). Some simple conclusions: Partial slip + complete slip, small scale pattern b 1 >>L Works well at low pressures (additional dissipation associated with meniscus at higher P)

28. Partial slip + complete slip, large scale pattern b 1 << 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)

29. How to design a strongly slippery surface ? For fixed working pressure (0.5bars) size of the posts (a=100nm), and value of b 0 (20nm) Compromise between Large L to obtain large  P cap large enough P<P cap  1/( L) 

30. Contact angle 180 degrees Possible candidate : hydrophobic nanotubes « brush » (C. Journet, S. Purcell, Lyon) P. Joseph et al, Phys. Rev. Lett., 2006, 97, 156104

31. Effective slippage versus pattern: flow without resistance in micron size channel ? (Joseph et al, PRL 2007) Characteristic size L  z (µm) L b eff b eff ~ µm

32. 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

33. 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

34. Simulation of the cooling process Accurate determination of R K : fit to heat transfer equations No influence of Brownian motion on cooling

35. Simulation of heat transfer hot Cold Maxwell Garnett effective medium prediction  = (Kapitsa length / Particle Radius ) M. Vladkov, JLB, Nanoletters 2006

36. 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 0,010,1110 MW CNT Al 2 O 3 CuO TiO 2 Au ??? Cu (²  ) exp / (²  ) 2 phases model Fe Volume fraction Our MD

37. Effective medium calculation for a linear string of particles

38. 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 ?)

39. 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