1. Liquid flows on surfaces: the boundary condition E. CHARLAIX University of Lyon, France NANOFLUIDICS SUMMER SCHOOL August 20-24 2007 THE ABDUS SALAM INTERNATIONAL CENTER FOR THEORETICAL PHYSICS

2. The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Theory of the h.b.c. for simple liquids Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Slippage effects in macroscopic flows ?

3. Miniaturization increases surface to volume ratio: importance of surface phenomena The description of flows requires constitutive equation (bulk property of fluid) + boundary condition (surface property) Yesterday we saw that N.S. equation for simple liquids is very robust constitutive equation down to (some) molecular scale. What about boundary condition ? 500 nm Microchannels… …nanochannels

4. Usual b.c. : the fluid velocity vanishes at wall z V S = 0 Hydrodynamic boundary condition (h.b.c.) at a solid-liquid interface v(z) OK at a macroscopic scale and for simple fluids Phenomenological : derived from experiments on low molecular mass liquids

5. Goldstein 1938 Goldstein S. 1969. Fluid mechanics in the first half of this century. Annu. Rev. Fluid Mech 1:1–28 Batchelor, An introduction to fluid dynamics, 1967 The nature of hydrodynamics bc’s has been widely debated in 19 th century Lauga & al, in Handbook of Experimental Fluid Dynamics, 2005 M. Denn, 2001 Annu. Rev. Fluid Mech. 33:265–87

6. And also Bulkley (1931),Chen & Emrich (1963), Debye & Cleland (1958)… … and some time suspected on non-wetting surfaces But wall slippage occurs in polymer flows… Pudjijanto & Denn 1994 J. Rheol. 38:1735 Shark-skin effect in extrusion of polymer melts

7. C. Chan and R. Horn J. Chem. Phys. (83) 5311, 1985 mica Ag no-slip flow with liquid monolayer sticking at wall various organic liquids/mica Ag J.N. Israelachvili J. Colloid Interf. Sci. (110) 263, 1986 Water on mica: no-slip within 2 Å George et al., J. Chem. Phys. 1994 no-slip flow w. monolayer sticking at wall various organic liquids/ metal surfaces Drainage experiments with SFA

8. N.V. Churaev, V.D; Sobolev and A.NSomov J. Colloid Interf. Sci. (97) 574, 1984 Water slips in hydrophobic capillaries slip length 70 nm

9. z V S ≠ 0 v(z) b V S : slip velocity  S : tangential stress at the solid surface b : slip length : liquid-solid friction coefficient  : liquid viscosity Partial slip and solid-liquid friction Navier 1823 Maxwell 1856 ∂V∂V ∂z∂z =  : shear rate Tangentiel stress at interface

10. Interpretation of the slip length From Lauga & al, Handbook of Experimental Fluid Dynamics, 2005 b

11. The bc is an interface property. The slip length has not to be related to an internal scale in the fluid The hydrodynamic b.c. is fully characterized by b(  ) The hydrodynamic bc is linear if the slip length does not depend on the shear rate. On a mathematically smooth surface, b= ∞ (perfect slip). Some properties of the slip length No-slip bc (b=0) is associated to very large liquid-solid friction

12. The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Theory of the h.b.c. for simple liquids Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Slippage effects in macroscopic flows ?

13. Pressure drop in nanochannels d ∆P x z b Slit r Tube

14. Exemple 1 slit d=1 µm b=20 nm %error on permeability : 12% Exemple 2 tube r = 2 nm b=20 nm Error factor on permeability : 80 (2 order of magnitude)

15. B. Lefevre et al, J. Chem. Phys 120 4927 2004 Water in silanized MCM41 of various radii (1.5 to 6 nm) 10nm Forced imbibition of hydrophobic mesoporous medium The intrusion-extrusion cycle of water in hydrophobic MCM41 mesoporous silica: MCM41 quasi-static cycle, does not depend on frquency up to kHz Exemple 3

16. L ~ 2-10 µm Porous grain

17. Dispersion of transported species - Mixing t=0 injection d time t Taylor dispersion Without molecular diffusion: Molecular diffusion spreads the solute through the width within Solute motion is analogous to random walk:

18. With partial slip b.c. t=0 d time t b

19. With partial slip b.c. t=0 d time t b Same channel, same flow rate Hydrodynamic dispersion is significantly reduced if b ≥ d b = 0.15 d reduction factor 2 b = 1.5 d reduction factor 10

20. Electric field electroosmotic flow Electrostatic double layer nm 1 µm Electrokinetic phenomena Electro-osmosis, streaming potential… are determined by interfacial hydrodynamics at the scale of the Debye length Colloid science, biology, …

21. - - - - - - - - - + + + + + + + + ++ E v z x os

22. - - - - - - - - - + + + + + + + + ++ E v z x os Case of a no-slip boundary condition: no-slip plane zHzH zeta potential

23. - - - - - - - - - + + + + + + + + ++ E v z x os Case of a partial slip boundary condition:

24.  At constant  s the electroosmotic velocity depends on Debye length. Pb for measuring surface charges  Possibility of very large el-osmotic flow by decreasing    b=20nm   = 3nm at 10 -2 M Example: Churaev et al, Adv. Colloid Interface Sci. 2002 L. Joly et al, Phys. Rev. Lett, 2004 os increased by 800%

25. The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Theory of the h.b.c. for simple liquids Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing What about macroscopic flows ?

26. locally: perfect slip Far field flow : no-slip Effect of surface roughness roughness « kills » slip Richardson, J Fluid Mech 59 707 (1973), Janson, Phys. Fluid 1988 Fluid mechanics calculation :

27. Robbins (1990) Barrat, Bocquet (1994, 1999) Thomson-Troian (Nature 1997) u q = 2  Slip at a microscopic scale : molecular dynamics on simple liquids

28. b Thermodynamic equilibrium determination of b.c. with Molecular Dynamics simulations Be j(r,t) the fluctuating momentum density at point r Assume that it obeys Navier-Stokes equation And assume boundary condition Bocquet & Barrat, Phys Rev E 49 3079 (1994)

29. Then take its average And auto-correlation function b C(z,z’,t) obeys a diffusion equation with boundary condition and initial value given by thermal equilibrium 2D density C(z,z’,t) can be solved analytically and obtained as a function of b b can be determined by ajusting analytical solution to data measured in equilibrium Molecular Dynamics simulation

30. b

31. Green-Kubo relation for the hydrodynamic b.c.: (assumes that momentum fluctuations in fluid obey Navier-Stokes equation + b.c. condition of Navier type) Slip at a microscopic scale : linear response theory Liquid-solid Friction coefficient total force exerted by the solid on the liquid canonical equilibrium Friction coefficient (i.e. slip length) can be computed at equilibrium from time decay of correlation function of momentum tranfer

32.  « soft sphere » liquid interaction potential (r) =  (  r) 12 molecular size :  u q = 2  u/  b/  0 0.01 >0.03 ∞ 40 0 -2  very small surface corrugation is enough to suppress slip effects Slip at a microscopic scale : molecular dynamics Barrat, Bocquet, PRE (1994)  hard wall corrugation z=u cos qx  attractive wall interaction potential  z)=  sf (1/z 9 -1/z 3 )  Strong wall-fluid attraction induces an immobile fluid layer at wall  sf  =15

33. Effect of liquid-solid interaction D  = {fluid,solid} Simple Lennard-Jones fluid with fluid-fluid and fluid-solid interactions Barrat et al Farad. Disc. 112,119 1999 c  parameter controls wettability Wettability is characterized by contact angle (c.a.) c FS =1.0 :  =90° c FS =0.5 :  =140° c FS =0 :  =180°

34. Two types of flow Here :  =140°, P~7 MPa Slip length b=11  is found (both case) Poiseuille flow V(z) z/  F0F0 b=0 Couette flow V(z) z/  U b=0

35.  Linear b.c. up to ~ 10 8 s -1 Slip at a microscopic scale: liquid-solid interaction effect  =140° 130°  =90° b/  P/P 0 P 0 ~MPa  substantial slips occurs on strongly non-wetting systems  slip length increases with c.a.  essentially no (small) slip in partial wetting systems (  < 90°)  slip length increases stronly as pressure decreases P o ~ MPa

36. Slip increases with reduced fluid density at wall. However slippage does not reduce to « air cushion » at wall. fluid density profile across the cell Soft spheres on hard repulsive wall Lennard-Jones fluid  = 137°

37. Slip at a microscopic scale: theory for simple liquids Analytical expression for slip length Depends only on structural parameters, no dynamic parameter density at wall, depends on wetting properties fluid struct.factor parallel to wall wall corrugation a exp(q // R // ) molecular size // Barrat et al Farad. Disc. 112,119 1999

38. Intrinsic b.c. on smooth surfaces : conclusion  substantial slips occurs in strongly non-wetting systems slip length increases with c.a. slip length depends strongly on pressure  at moderate shear rate (  < 10 8 s -1 ) essentially no slip in wetting systems  slip length amplitude is moderate (~ 5 nm at  ).  slip length is not expected to depend on fluid viscosity (≠ polymers)  non-linear slip develops above a (high) critical shear rate (~ 10 9 s -1 ) Thomson-Troian Nature 1997

39. The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Theory of the h.b.c. for simple liquids Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Slippage effects in macroscopic flows ?

40. Duez, Ybert, Clanet, Bocquet Nature Physics 3, 180, 2007

41. MOVING CONTACT LINE INSTABILITY U dd z ee dd static contact angle dynamic ‘ ‘ Tangential stress on L/S surface diverges at c.l.  LV  SV  SL Adds a dynamic force at c.l.: The dynamic c.a. increases with flow velocity: Capillary number Above threshold Ca > Ca c,  d = 180° the c.l. destabilizes, a fluid film is trapped  SV -  SL =  LV cos  e

42. LANDAU-LEVITCH EFFECT U De Gennes, Brochart et Quéré, Gouttes bulles perles et ondes, 2005

43. ANTI LANDAU-LEVITCH EFFECT U Duez & al Nature Physics 3, 180, 2007