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## Liquid flows on surfaces:

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**Liquid flows on surfaces:**the boundary condition Nanoscale Interfacial Phenomena in Complex Fluids - May 19 - June 20 2008 The Kavli Institute of Theoretical Physics China**500nm**Microchannels… …nanochannels Downsizing flow devices raises new problems Pressure driving becomes insufficient h V Po Po +DP L V = 1 mm/s, L=1cm, h = 10-3 Pa.s h = 0.1 mm DP = 100 bar New solutions are needed**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) We saw that N.S. equation for simple liquids is very robust constitutive equation down to (some) molecular scale. What about boundary condition ?**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**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 origin: 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 C. Chan and R. Horn J. Chem. Phys. (83) 5311, 1985 no-slip flow over a « trapped » monolayer various organic liquids / mica 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 over « trapped » monolayer 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**Tube**r Pressure drop in nanochannels Slit z b d x ∆P**Exemple 2: tube d= 2 nm**Change from no-slip to b=20nm gain in flow rate : 8000% (2 order of magnitude) Exemple 1: slit d=1 µm Change from no-slip to b=20nm gain in flow rate : 12%**Exemple 3**Forced imbibition of hydrophobic mesoporous medium mesoporous silica: MCM41 10nm B. Lefevre et al, J. Chem. Phys 120 4927 2004Silanized 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**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**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**Slip at a microscopic scale :**molecular dynamics on simple liquids Robbins (1990) Barrat, Bocquet (1994, 1999)Thomson-Troian (Nature 1997)**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 Navier 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**Bocquet & Barrat, Phys Rev E 49 3079 (1994) 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**Theory for intrinsic b.c. on smooth surfaces : summary**(obtained with LJ liquids, some with water) . . • no-slip in wetting systems (except very high shear rate g < 108 s-1 ) • substantial slips in strongly non-wetting systems slip length increases with c.a. slip length decreases with increasing pressure • slip length is moderate (~ 5 nm at q ~ 120° ) • slip length does not depend on fluid viscosity (≠ polymers) • non-linear slip develops at high shear rate (~ 109 s-1 )**Some experimental results….**slip length (nm) Tretheway et Meinhart (PIV) Non-linear slip Pit et al (FRAP) 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) 10 MD Simulations 1 0 50 100 150 Contact angle (°) Brenner, Lauga, Stone 2005