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##### Liquid flows on surfaces:

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**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**The no-slip boundary condition (bc):**a long lasting empiricism regularly questionned Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Theory of the h.b.c. for simple liquids Slippage effects in macroscopic flows ?**500nm**Microchannels… …nanochannels 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 ?**Hydrodynamic boundary condition (h.b.c.)**at a solid-liquid interface z v(z) VS = 0 Usual b.c. : the fluid velocity vanishes at wall • Phenomenological : derived from experiments on low molecular mass liquids • OK at a macroscopic scale and for simple fluids**The nature of hydrodynamics bc’s has been widely debated**in 19th century Goldstein S. 1969. Fluid mechanics in the first half of this century. Annu. Rev. Fluid Mech 1:1–28 Lauga & al, in Handbook of Experimental Fluid Dynamics, 2005 M. Denn, 2001 Annu. Rev. Fluid Mech. 33:265–87 Batchelor, An introduction to fluid dynamics, 1967 Goldstein 1938**But wall slippage occurs in polymer flows…**Shark-skin effect in extrusion of polymer melts Pudjijanto & Denn 1994 J. Rheol. 38:1735 … and some time suspected on non-wetting surfaces And also Bulkley (1931),Chen & Emrich (1963), Debye & Cleland (1958)…**Drainage experiments with SFA**Ag mica Ag no-slip flow with liquid monolayer sticking at wall various organic liquids/mica C. Chan and R. Horn J. Chem. Phys. (83) 5311, 1985 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**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**z**v(z) b VS ≠ 0 VS : slip velocity sS : tangential stress at the solid surface h : liquid viscosity l : liquid-solid friction coefficient = g b : slip length Partial slip and solid-liquid friction Navier 1823 Maxwell 1856 Tangentiel stress at interface ∂V : shear rate ∂z**Interpretation of the slip length**b From Lauga & al, Handbook of Experimental Fluid Dynamics, 2005**Some properties of the slip length**No-slip bc (b=0) is associated to very large liquid-solid friction The bc is an interface property. The slip length has not to be related to an internal scale in the fluid On a mathematically smooth surface, b=∞ (perfect slip). The hydrodynamic b.c. is fully characterized by b(g) The hydrodynamic bc is linear if the slip length does not depend on the shear rate.**The no-slip boundary condition (bc):**a long lasting empiricism regularly questionned Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Theory of the h.b.c. for simple liquids Slippage effects in macroscopic flows ?**Tube**r Pressure drop in nanochannels Slit z b d x ∆P**Exemple 2**r = 2 nm b=20 nm Error factor on permeability : 80 tube (2 order of magnitude) Exemple 1 d=1 µm b=20 nm %error on permeability : 12% slit**Exemple 3**Forced imbibition of hydrophobic mesoporous medium mesoporous silica: MCM41 10nm B. Lefevre et al, J. Chem. Phys 120 4927 2004Water in silanized MCM41 of various radii (1.5 to 6 nm) The intrusion-extrusion cycle of water in hydrophobic MCM41 quasi-static cycle, does not depend on frquency up to kHz**Porous grain**L ~ 2-10 µm**Dispersion of transported species - Mixing**d Taylor dispersion t=0 injection time t Without molecular diffusion: Molecular diffusion spreads the solute through the width within Solute motion is analogous to random walk:**b**d t=0 time t With partial slip b.c.**b**d t=0 time t With partial slip b.c. 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**Electrokinetic phenomena**Colloid science, biology, … Electrostatic double layer nm 1 µm Electric field electroosmotic flow Electro-osmosis, streaming potential… are determined by interfacial hydrodynamics at the scale of the Debye length**z**v E os + + + x + + + + + + + - - - - - - - - -**zeta potential**z v E os no-slip plane + + + x + + + + + + + - - - - - - - - - zH Case of a no-slip boundary condition:**z**v E os + + + x + + + + + + + - - - - - - - - - Case of a partial slip boundary condition:**At constant Ys the electroosmotic velocity depends on Debye**length. Pb for measuring surface charges • Possibility of very large el-osmotic flow by decreasing k-1 ? Example: b=20nm k-1 = 3nm at 10-2M nos increased by 800% Churaev et al, Adv. Colloid Interface Sci. 2002 L. Joly et al, Phys. Rev. Lett, 2004**The no-slip boundary condition (bc):**a long lasting empiricism regularly questionned Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Theory of the h.b.c. for simple liquids What about macroscopic flows ?**Far field flow : no-slip**Effect of surface roughness Fluid mechanics calculation : locally: perfect slip Richardson, J Fluid Mech 59 707 (1973), Janson, Phys. Fluid 1988 roughness « kills » slip**u**Slip at a microscopic scale : molecular dynamics on simple liquids Robbins (1990) Barrat, Bocquet (1994, 1999)Thomson-Troian (Nature 1997) q = 2 p/ s**b**Thermodynamic equilibrium determination of b.c. with Molecular Dynamics simulations Bocquet & Barrat, Phys Rev E 49 3079 (1994) Be j(r,t) the fluctuating momentum density at point r Assume that it obeys Navier-Stokes equation And assume boundary condition**C(z,z’,t) obeys a diffusion equation**with boundary condition b 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 Then take its <x,y> average And auto-correlation function**Slip at a microscopic scale : linear response theory**Green-Kubo relation for the hydrodynamic b.c.: canonical equilibrium Liquid-solid Friction coefficient total force exerted by the solid on the liquid (assumes that momentum fluctuations in fluid obey Navier-Stokes equation + b.c. condition of Navier type) Friction coefficient (i.e. slip length) can be computed at equilibrium from time decay of correlation function of momentum tranfer**Slip at a microscopic scale : molecular dynamics**Barrat, Bocquet, PRE (1994) • « soft sphere » liquid interaction potential n (r) = e(s/ r)12molecular size : s • hard wall corrugation z=u cos qx u q = 2 p/ s • attractive wall interaction potential f (z)= esf (1/z9-1/z3) u/s b/s • very small surface corrugation is • enough to suppress slip effects 0 0.01 >0.03 >0.03 ∞ 40 0 -2 • Strong wall-fluid attraction induces an immobile fluid layer at wall esf /e =15**Effect of liquid-solid interaction**Barrat et al Farad. Disc. 112,119 1999 Simple Lennard-Jones fluid with fluid-fluid and fluid-solid interactions D a, b = {fluid,solid} cab parameter controls wettability Wettability is characterized by contact angle (c.a.) cFS=1.0 : q=90° cFS=0.5 : q=140° cFS=0 : q=180°**Couette flow**Poiseuille flow F0 V(z) U V(z) b=0 b=0 z/s z/s Two types of flow Here : q=140°, P~7 MPa Slip length b=11 s is found (both case)**q=140°**b/s 130° q=90° P0~MPa P/P0 Slip at a microscopic scale: liquid-solid interaction effect • essentially no (small) slip in partial wetting systems (q < 90°) • substantial slips occurs on strongly non-wetting systems • slip length increases with c.a. • slip length increases stronly as pressure decreases Po ~ MPa • Linear b.c. up to ~ 108 s-1**fluid density profile across the cell**Lennard-Jones fluid q = 137° Soft spheres on hard repulsive wall Slip increases with reduced fluid density at wall. However slippage does not reduce to « air cushion » at wall.**Slip at a microscopic scale: theory for simple liquids**Analytical expression for slip length Barrat et al Farad. Disc. 112,119 1999 molecular size // density at wall, depends on wetting properties wall corrugation a exp(q// • R//) fluid struct.factor parallel to wall Depends only on structural parameters, no dynamic parameter**Intrinsic b.c. on smooth surfaces : conclusion**. • at moderate shear rate (g < 108 s-1 ) essentially no slip in wetting systems • substantial slips occurs in strongly non-wetting systems slip length increases with c.a. slip length depends strongly on pressure • slip length amplitude is moderate (~ 5 nm at q ~ 120° ) • slip length is not expected to depend on fluid viscosity (≠ polymers) • non-linear slip develops above a (high) critical shear rate (~ 109 s-1 ) Thomson-Troian Nature 1997**The no-slip boundary condition (bc):**a long lasting empiricism regularly questionned Some examples of importance of the b.c. in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Dispersion & mixing Theory of the h.b.c. for simple liquids Slippage effects in macroscopic flows ?**Duez, Ybert, Clanet, Bocquet**Nature Physics 3, 180, 2007**MOVING CONTACT LINE INSTABILITY**U qe static contact angle gSV - gSL =gLV cos qe z gLV gSV qd qd dynamic ‘ ‘ gSL Tangential stress on L/S surface diverges at c.l. Adds a dynamic force at c.l.: The dynamic c.a. increases with flow velocity: Above threshold Ca > Cac, qd= 180° the c.l. destabilizes, a fluid film is trapped Capillary number**U**LANDAU-LEVITCH EFFECT De Gennes, Brochart et Quéré, Gouttes bulles perles et ondes, 2005**ANTI LANDAU-LEVITCH EFFECT**U Duez & al Nature Physics 3, 180, 2007