Nano-hydrodynamics down to which scale do macroscopic concepts hold ? E. CHARLAIX University of Lyon, France The Abdus Salam international center for theoretical physics INTRODUCTION TO MICROFLUIDICS August and how to describe flows beyond ?

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 Miniaturization increases surface to volume ratio: importance of surface phenomena  manipulation and analysis of biomolecules. with single molecule resolution  ensure specific ion transport 50 nm channels Wang et al, APL nm Nanochannels are more specifically designed for : Microchannels… …nanochannels

Large specific surface (1000m 2 /cm 3 ~ pore radius 2nm) catalysis, energy/liquid storage… Mesoporous materials Water in mesoporous silica (B. Lefevre et al, J. Chem. Phys 2004) Water in nanotube Koumoutsakos et al nm

Electric field electroosmotic flow Electrostatic double layer 3 nm 300 nm Electrokinetic phenomena Electro-osmosis, streaming potential… are determined by nano-hydrodynamics at the scale of the Debye length Colloid science, biology, nanofluidic devices…

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)

Tabor et Winterton, Proc. Royal Soc. London, 1969 Israelachvili, Proc. Nat. Acad. Sci. USA 1972 Surface Force Apparatus (SFA) mica Ag Optical resonator D

Franges of equal chromatic order (FECO) Tolanski, Multiple beam Interferometry of surfaces and films, Clarendon Press 1948 Source of white light Spectrograph

D=28nm contact r : reflexion coefficient n : mica index a : mica thickness D : distance between surfaces Distance between surfaces is obtained within 1 Å  (nm) Tabor et Winterton, Proc. Royal Soc. London, 1969 Israelachvili, Proc. Nat. Acad. Sci. USA 1972

Force measurement In a quasi-static regime (inertia neglected) Piezoelectric displacement At large D, very low speed Piezoelectric calibration

The 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 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)

K ∆(t) = F static (D) + F hydro (D, D) Drainage of confined liquids : Chan & Horn 1985 t tsts D(t) DD L(t) Run-and-stop experiments Inertia negligible :

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 D : Å µm R ~ cm R Re = D D ~ Å/s  m 2 /s Re ≤  : fluid kinematic viscosity

Velocity in the drainage flow z = D + x2x2 2R Crossed cylinders are equivalent to sphere-plane  Mass conservation 2  xz U(x) = -  x 2 D R z(x) x D  Parabolic approximation U(x) √ 2RD 2D R √ D ~ 10 µm

Lubrication flow in the confined film z(x) x u(x,z)  Hypothesis  Properties Pressure gradient is // Ox Average velocity at x: Velocity profile is parabolic Quasi-parallel surfaces: dz/dx <<1 Newtonian fluid Low Re Slow time variation: T >> z 2 / z2z2 12  dPdP dxdx U(x)= -   fluid dynamic viscosity No-slip at solid wall

Pressure profile P(x)-P ∞ x R z(x) x DU(x) √ 2RD

Hydrodynamic force between the surfaces Reynolds force: D<<R D F hydro = - 6  R 2 D

Drainage of confined liquids : run-and-stop experiments K ∆(t) = F static (D) - D 6  R 2 D K (D - D  ) = - D 6  R 2 D t tsts D(t) DD L(t) ∆(t) D < 6nm D(t) - D  D(t) KD  6  R 2 ln = (t - t s ) + Cte

Chan & Horn 1985 (1) D(t) - D  D(t) KD  6  R 2 ln = (t - t s ) + Cte D > 50 nm : excellent agreement with macroscpic hydrodynamics Various values of D  : determination of fluid viscosity  excellent agreement with bulk value Chan et Horn, J. Chem. Phys. 83 (10) 5311 (1985)

Chan & Horn (2) D ≤ 50nm : drainage too slow Reynolds drainage Sticking layers Hypothesis: fluid layers of thickness D s stick onto surfaces D - 2D s D 6  R 2 F hydro = - Excellent agreement for 5 ≤D≤ 50nm OMCTS tetradecane hexadecane Molecular size DsDs 7,5Å 13Å 4Å 7Å 4Å 7Å

Chan & Horn (3) D ≤ 5 nm: drainage occurs by steps Steps height = molecular size BUT Occurrence of steps is NOT predicted by « sticky » Reynolds + static forces Including static interaction (oscillating force) in dynamic equation yields drainage steps

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 Static force has no oscillations (no smectic layering) but shows electrostatic effects Water confined between silica surfaces: Horn & al Chem Phys Lett , 1989

Drainage of thin water films Macroscopic hydrodynamics holds down to molecular size with bulk value of viscosity and no-slip boundary condition (no sticking layer) Israelachvili JCSI1985 Static force has no oscillations (no smectic layering) Water confined between silica surfaces: Results for water confined between mica surfaces are similar Horn & al Chem Phys Lett , 1989

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) McGuiggan et Israelachvili, J. Chem Phys 1990 Loaded mica surface flatten and form a film of area A and constant thickness D measured by FECO fringes

Shearign ultra-thin films (1) McGuiggan et Israelachvili, J. Chem Phys 1990 « stop-and-go » experiments V Solid or liquid behaviour depending on V, V/D, history… ‘continuous’ solid-liquid transition very high viscosities long relaxation times

Granick, Science 1991  bulk = 0,01 poise Shear force velocity area thickness Glass transition induced by confinement Dodecane D=2,7nm OMCTS D=2,7 nm Giant increase of viscosity under shear Shear-thinning behaviour Shearing ultra-thin films (2)

Shearing ultra thin films (3) Klein et Kumacheva, J. Chem. Phys High precision device with both normal and shear force Sensitive in shear up to 6 molecular layers

Shearing ultra thin films (3) Klein et Kumacheva, J. Chem. Phys Abrupt and reproducible Solid-liquid transition at n=7 n=6 couches only induced by confinement independant of normal pressure Imposed tangential motion times Force response OMCTS, cyclohexane

Klein et Kumacheva, J. Chem. Phys Creep viscosity of solid film Shearing ultra thin films (3)

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  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  : fluid kinematic viscosity DF hydro = 6  R 2 D R R D D

Lubrication flow in the confined film z(x) x u(x,z)  Hypothesis Small angle: dz/dx <<1 Newtonian fluid Low Re Slow time variation: T >> z 2 / No-slip at solid wall

 Properties Pressure gradient is // Ox Average velocity at x: Velocity profile is parabolic z2z2 12  dPdP dxdx U(x)= -  fluid dynamic viscosity Lubrication flow in the confined film z(x) x u(x,z) Stokes flow:

The hydrodynamic force between two crossed cylinders of radii R is the same as between a sphere of radius R and a plane This is the Reynolds force  : fluid dynamic viscosity DF hydro = 6  R 2 D R R D D<<R

Drainage de liquides confinés (2) y(t) = y+Ao cos  t Israelachvili, J. Coll. Inter. Sci D(t) = D +A cos (  t+  x = y+D mx +K(x-x o ) + K  D = F s (D) ¨ Pour  <<  o = √ K/m f = ∂F s ∂D ( ) D : méthode dynamique Hydrodynamique macroscopique : 6  R 2 KD  Amortissement visqueux

Drainage de liquides confinés (2) Israelachvili, J. Coll. Inter. Sci Tétradécane confiné entre des surfaces de mica Force statique Inverse de l’amortissement  Pas de couche immobile à 3Å près Détermination indépendante de la viscosité et de la condition limite

Drainage de liquides confinés (2) Israelachvili, J. Coll. Inter. Sci Eau+NaCl confiné entre des lames de mica TB accord avec l’hydrodynamique macroscopique Pas de couche immobile à la paroi

Chan et Horn, J. Chem. Phys Drainage de liquides confinés (1) Drainage de films fins de liquides non-polaires : importance de l’humidité