Nano-hydrodynamics down to which scale do macroscopic concepts hold ? and how to describe flows beyond ? E. CHARLAIX University of Lyon, France INTRODUCTION TO MICROFLUIDICS August 8-26 2005 The Abdus Salam international center for theoretical physics OUTLINE Why nano-hydrodynamics ? Surface Force Apparatus: a fluid slit of thickness controlled at the Angstrom level First nano-hydrodynamic experiments performed with SFA Experiments with ultra thin liquid films solid or glass transition ? (90’s) a controversy resolved (Becker & Mugele 2003) Nanofluidic devices Microchannels… …nanochannels Miniaturization increases surface to volume ratio: 500 nm 50 nm channels Wang et al, APL 2002 importance of surface phenomena Nanochannels are more specifically designed for : manipulation and analysis of biomolecules . with single molecule resolution ensure specific ion transport Mesoporous materials Large specific surface (1000m2 /cm3 ~ pore radius 2nm) catalysis, energy/liquid storage… 10nm Water in mesoporous silica (B. Lefevre et al, J. Chem. Phys 2004) Water in nanotube Koumoutsakos et al 2003 Electrokinetic phenomena Colloid science, biology, nanofluidic devices… Electric field Electrostatic double layer 3 nm 300 nm electroosmotic flow Electro-osmosis, streaming potential… are determined by nano-hydrodynamics at the scale of the Debye length Tribology : lubrication of solid surfaces Mechanics, biomechanics, MEMS/NEMS friction Nano-rheology of thin liquid films (monomolecular) Controled studies at the nanoscale: Surface force apparatus (SFA) Tabor, Israelaschvili OUTLINE Importance Surface Force Apparatus : a slit of thickness controlled at the Angstrom level First nano-hydrodynamic experiments performed with SFA : Experiments with ultra thin liquid films solid or glass transition ? (90’s) a controversy resolved (Becker & Mugele 2003) Surface Force Apparatus (SFA) Tabor et Winterton, Proc. Royal Soc. London, 1969 Israelachvili, Proc. Nat. Acad. Sci. USA 1972 D Ag mica Ag Optical resonator Franges of equal chromatic order (FECO) Tolanski, Multiple beam Interferometry of surfaces and films, Clarendon Press 1948 Spectrograph Source of white light l Tabor et Winterton, Proc. Royal Soc. London, 1969 Israelachvili, Proc. Nat. Acad. Sci. USA 1972 D=28nm contact Distance between surfaces is obtained within 1 Å l l (nm) r : reflexion coefficient n : mica index a : mica thickness D : distance between surfaces Force measurement In a quasi-static regime (inertia neglected) Piezoelectric displacement Piezoelectric calibration At large D, very low speed Oscillating force in organic liquid films Static force in confined organic liquid films (alkanes, OMCTS…). Oscillations reveal liquid structure in layers parallel to the surfaces The Chan & Horn, J. Chem Phys 1985 Electrostatic and hydration force in water films Horn & al Chem Phys Lett 1989 OUTLINE Importance Surface Force Apparatus : a slit of thickness controlled at the Angstrom level First nano-hydrodynamic experiments performed with SFA : thick liquid films (Chan & Horn 1985) Experiments with very thin liquid films solid or glass transition ? (90’s) a controversy resolved (Becker & Mugele 2003) Drainage of confined liquids : Chan & Horn 1985 Run-and-stop experiments D(t) L(t) D ts Inertia negligible : K ∆(t) = Fstatic (D) + Fhydro (D, D) t Hydrodynamic force When D<<R (cylinders radii) and Reynolds number Re < 1 the hydrodynamic force is essentially dominated by the lubrication flow of liquid drained out of the gap region. R D R D:Å µm R ~ cm D ~ Å/s n ~ 10-6 m2/s Re ≤ 10-9 Re = DD n n : fluid kinematic viscosity Velocity in the drainage flow R Parabolic approximation 2 x z=D+ 2R Crossed cylinders are equivalent to sphere-plane Mass conservation 2pxz U(x) = - p x2 D z(x) D U(x) x U(x) R D √ 2D √ 2RD ~ 10 µm Lubrication flow in the confined film Hypothesis Newtonian fluid u(x,z) z(x) x Quasi-parallel surfaces: dz/dx <<1 Low Re Slow time variation: T >> z2/n Properties Pressure gradient is // Ox Velocity profile is parabolic Average velocity at x: No-slip at solid wall 2 dP z U(x)= 12h dx h: fluid dynamic viscosity Pressure profile R z(x) D U(x) x P(x)-P∞ √ 2RD x Hydrodynamic force between the surfaces Reynolds force: 6 p h R2 D Fhydro = D D<<R Drainage of confined liquids : run-and-stop experiments D(t) ∆(t) L(t) D ts D < 6nm 6p h R2 D K ∆(t) = Fstatic (D) D 6p h R2 D K (D - D) = D 2 D(t) D 6 p h R = ln (t - ts ) + Cte D(t) KD t Chan & Horn 1985 (1) ln D(t) - D D(t) = 6p h R2 (t - t ) + Cte s KD D > 50 nm : excellent agreement with macroscpic hydrodynamics Various values of D : determination of fluid viscosity h excellent agreement with bulk value Chan et Horn, J. Chem. Phys. 83 (10) 5311 (1985) Chan & Horn (2) D ≤ 50nm : drainage too slow Hypothesis: fluid layers of thickness Ds stick onto surfaces Sticking layers 6p h R2 D Fhydro = D - 2Ds Reynolds drainage Excellent agreement for 5 ≤D≤ 50nm OMCTS tetradecane hexadecane Molecular size Ds 7,5Å 4Å 4Å 13Å 7Å 7Å Chan & Horn (3) D ≤ 5 nm: drainage occurs by steps Steps height = molecular size Including static interaction (oscillating force) in dynamic equation yields drainage steps BUT Occurrence of steps is NOT predicted by « sticky » Reynolds + static forces Draining confined liquids with SFA: conclusion Efficient method to study flows at a nanoscale Excellent agreement with macroscopic hydrodynamics down to ~ 5 nm (6-7 molecular size thick film) « Immobile » layer at solid surface, about 1 molecular size Israelachvili JCSI1985 : water on mica George et al JCP 1994 : alcanes on metal Becker & Mugele PRL 2003 : D<5nm Draining confined liquids with SFA: questions In very thin films of a few molecular layers macroscopic picture does not seem to hold anymore What is the liquid dynamics in those very thin films ? How can one describe flows ? Drainage of thin water films Water confined between silica surfaces: Horn & al Chem Phys Lett 162 404, 1989 Static force has no oscillations (no smectic layering) but shows electrostatic effects Drainage of thin water films Water confined between silica surfaces: Horn & al Chem Phys Lett 162 404, 1989 Static force has no oscillations (no smectic layering) Macroscopic hydrodynamics holds down to molecular size with bulk value of viscosity and no-slip boundary condition (no sticking layer) Results for water confined between mica surfaces are similar Israelachvili JCSI1985 Why do ultra-thin films of organic liquids behave differently from water ? OUTLINE Importance Surface Force Apparatus : a slit of thickness controlled at the Angstrom level First nano-hydrodynamic experiments performed with SFA : Experiments with ultra thin liquid films solid or glass transition ? (90’s) a controversy resolved (Becker & Mugele 2003) Shearing ultra-thin films (1) Loaded mica surface flatten and form a film of area A and constant thickness D measured by FECO fringes McGuiggan et Israelachvili, J. Chem Phys 1990 Shearign ultra-thin films (1) « stop-and-go » experiments McGuiggan et Israelachvili, J. Chem Phys 1990 V Solid or liquid behaviour depending on V, V/D, history… very high viscosities long relaxation times ‘continuous’ solid-liquid transition Shearing ultra-thin films (2) Shear force Granick, Science 1991 velocity Dodecane D=2,7nm area thickness Giant increase of viscosity under shear OMCTS D=2,7 nm Shear-thinning behaviour Glass transition induced by confinement hbulk = 0,01 poise Shearing ultra thin films (3) High precision device with both normal and shear force Sensitive in shear up to 6 molecular layers Klein et Kumacheva, J. Chem. Phys. 1996 Shearing ultra thin films (3) Imposed tangential motion Klein et Kumacheva, J. Chem. Phys. 1996 OMCTS, cyclohexane Force response Abrupt and reproducible Solid-liquid transition at n=7 n=6 couches only induced by confinement independant of normal pressure times Shearing ultra thin films (3) Creep viscosity of solid film Klein et Kumacheva, J. Chem. Phys. 1996 Shearing ultra-thin flims has open a research area with controversial effects Same fluids, same technique,different results Increase of viscocisites of ORDER OF MAGNITUDE Shear-thinning, Memory effects, slow relaxation times Glass transition (out of equilibrium) Well defined liquid-solid transition under a critical confinment When D << R (cylinders radii) crossed cylinders geometry is equivalent to a sphere of radius R at distance D from a plane R R D When D<<R and Reynolds < 1, the hydrodynamic force is essentially dominated by the lubrication flow of liquid drained out of the gap region. This is the Reynolds force 2 6 p h R Fhydro = D D D h : fluid kinematic viscosity Lubrication flow in the confined film Hypothesis u(x,z) z(x) x Newtonian fluid Small angle: dz/dx <<1 Low Re Slow time variation: T >> z2/n No-slip at solid wall Lubrication flow in the confined film Properties u(x,z) z(x) Stokes flow: x Pressure gradient is // Ox Velocity profile is parabolic Average velocity at x: 2 dP z U(x)= 12h dx h: fluid dynamic viscosity The hydrodynamic force between two crossed cylinders of radii R is the same as between a sphere of radius R and a plane R R D This is the Reynolds force h : fluid dynamic viscosity 2 6 p h R Fhydro = D D D<<R Drainage de liquides confinés (2) : méthode dynamique Israelachvili, J. Coll. Inter. Sci. 1985 D(t) = D +A cos (w t+f) Amortissement visqueux y(t) = y+Ao cos wt mx¨ +K(x-xo) + Ka D = Fs(D) x = y+D Pour w << wo =√K/m Hydrodynamique macroscopique : a= R2 6ph KD f= ( ∂Fs ∂D D ) Drainage de liquides confinés (2) Israelachvili, J. Coll. Inter. Sci. 1985 Tétradécane confiné entre des surfaces de mica Force statique Inverse de l’amortissement a Détermination indépendante de la viscosité et de la condition limite Pas de couche immobile à 3Å près Drainage de liquides confinés (2) Israelachvili, J. Coll. Inter. Sci. 1985 Eau+NaCl confiné entre des lames de mica TB accord avec l’hydrodynamique macroscopique Pas de couche immobile à la paroi Drainage de liquides confinés (1) Chan et Horn, J. Chem. Phys. 1985 Drainage de films fins de liquides non-polaires : importance de l’humidité