Nano-hydrodynamics flow at a solid surface boundary condition E. CHARLAIX University of Lyon, France INTRODUCTION TO MICROFLUIDICS August 8-26 2005 The Abdus Salam international center for theoretical physics

OUTLINE The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Some examples of importance in nanofluidics What says theory ? Experiments: a slippery subject Measurements with dSFA - Intrinsic slip lengths on smooth surfaces

Hydrodynamic boundary condition at a solid-liquid interface z v(z) VS = 0 Usual boundary condition : the fluid velocity vanishes at wall OK at a macroscopic scale and for simple fluids But is this boundary condition always valid ? Phenomenological : derived from experiments on low molecular mass liquids

Partial slip and solid-liquid friction Navier 1823 Maxwell 1856 z VS ≠ 0 v(z) b Tangentiel stress at interface VS : slip velocity sS : tangential stress at the solid surface b : slip length l : liquid-solid friction coefficient h : liquid viscosity ∂V = g : 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 The hydrodynamic characteristic of the interface is given by b(g) The hydrodynamic bc is linear if the slip length does not depend on the shear rate. On a mathematically smooth surface, b=∞ (perfect slip).

« Ideal » case of gas « Ideal » case of gas : weak density = weak momentum transfer between fluid and solid : slip can be large simple kinetic theory allows to calculate the friction force at wall with V u h Root mean square molecule momentum Gas nb density Fluid length scale : b ~ µm Bocquet, CRAS 1993

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

extrusion of polymer melts But wall slippage was strongly suggested in some polymer flows… Shark-skin effect in extrusion of polymer melts Pudjijanto & Denn 1994 J. Rheol. 38:1735

… and some time suspected in flow on non-wetting surfaces And also Bulkley (1931),Chen & Emrich (1963), Debye & Cleland (1958)… More recently, precise experiments of Churaev & al J Colloids Interf Sci 97 574 (1984) Slip length of water in silanized capillaries: b=70nm

The no-slip boundary condition (bc): OUTLINE The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Some examples of importance in nanofluidics Pressure drop in nanochannels Elektrokinetics effects Mixing What says theory ? Experiments: a slippery subject Measurements with dSFA - Intrinsic slip lengths on smooth surfaces

Pressure drop in nanochannels z b d x ∆P r

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

Forced imbibition of hydrophobic mesoporous medium mesoporous silica: MCM41 10nm B. Lefevre et al, J. Chem. Phys 120 4927 2004 Water 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

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 E v os + + + x + + + + + + + - - - - - - - - -

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

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

At constant Ys, the electro osmotic velocity depends on k Pb for measuring z ? Possibility of very large electro- osmotic flow by decreasing k-1 ? b=20nm k-1 = 3nm at 10-2M Exemple Churaev et al, Adv. Colloid Interface Sci. 2002 L. Joly et al, Phys. Rev. Lett, 2004

Mixing b Vs ≠ 0 Partial slip Dispersion shoud be reduced r No-slip

OUTLINE The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Some examples of importance in nanofluidics What says theory ? Experiments: a slippery subject Measurements with dSFA - Intrinsic slip lengths on smooth surfaces

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

Slip at a microscopic scale : molecular dynamics Robbins (1990) Barrat, Bocquet (1994) Thomson-Troian (Nature 1997) Lennard-Jones liquids molecular size : s corrugation of surface potential : u u q = 2 p/ s In general very small surface corrugation is enough to suppress slip effects

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 assumed boundary condition b

Then take its <x,y> 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

b

Slip at a microscopic scale : linear response theory Barrat, Bocquet 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 (1994) Lennard-Jones liquids molecular size : s corrugation of surface potential : u u q = 2 p/ s If liquid-solid interactions are strong (liquid wets solid) u/s b/s very small surface corrugation is enough to suppress slip effects 0.01 >0.03 ∞ 40

Slip at a microscopic scale: liquid-solid interaction effect Molecular Dynamics simulations Lennard-Jones liquids Barrat et al Farad. Disc. 112,119 1999 q=140° 130° q=90° b/s P/P0 P0~MPa  = {liquid,solid}, : energy scale,  : molecular diameter cab : wetting control parameter Linear b.c. up to 108 s-1

Slip at a microscopic scale: theory for simple liquids L. Bocquet, J.L Barrat PRE 49 3079 (1994) Simple fluids L. Bocquet, J.L Barrat Faraday Disc 112,119 (1999) Polymer melts Priezjev & Troian, PRL 92, 018302 (2004) For a sinusoidal wall « corrugation » a exp(q// • R//) molecular size density at wall, depends on wetting properties wall corrugation fluid structure factor

Some recent experimental results on smooth surfaces 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) slip length (nm) Baudry et al (SFA) Cottin-Bizonne et al (SFA) 10 MD Simulations 1 50 100 150 Contact angle (°) Brenner, Lauga, Stone 2005

Schmadtko & Leger, PRL 94 244501 (2005) Velocimetry measurements V(z) Particule Imaging Velocimetry Tretheway & Meinhart Phys Fluid 14, L9, (2002) Fluorescence Double Focus Cross Correlation V(z) Fluorescence recovery in TIR Pit & Leger, PRL 85, 980 (2000) Schmadtko & Leger, PRL 94 244501 (2005) O. Vinogradova, PRE 67, 056313 (2003)

Dissipation measurements Pressure drop Churaev, JCSI 97, 574 (1984) Choi & Breuer, Phys Fluid 15, 2897 (2003) Surface Force Apparatus Colloidal Probe AFM Chan & Horn 1985 Israelachvili 1986 Georges 1994 Granick PRL 2001 Mugele PRL 2003 Cottin-Bizone PRL 2005 Craig & al, PRL 87, 054504 (2001) Bonnacurso & al, J. Chem. Phys 117, 10311 (2002) Vinogradova, Langmuir 19, 1227 (2003)

Particle Image Velocimetry (PIV) Measurement of velocity profile V(z) Fluorescent particules High resolution camera Pair of images With Micro-PIV (see S. Wereley) Spatial resolution ~ 50-100nm Meinardt & al, Experiments in Fluids (1999) Use for bc : are velocity of tracor and velocity of flow the same ? Meinardt & al (2002) Poiseuille flow profile in a capillary

~ 1 µm in 10 -6 M Effect of tracor-wall interactions O. Vinogradova, PRE 67 056313 (2003) Hydrodynamical lift ~ 1 µm in 10 -6 M Colloidal lift z d + electrostatic force: depletion layer: Fsphere ~ R exp (-kd) d ~ 3 k -1 Vsphere > Vslip z d Vsphere ≠ Vflow (zcenter) because of hydrodynamical sphere-plane interaction 0.75 slower than flow at d/R=0.1 F. Feuillebois, in Multiphase Science and Technology, New York, 1989, Vol. 4, pp. 583–798.

The no-slip boundary condition (bc): OUTLINE The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Some examples of importance in nanofluidics What says theory ? Experiments: a slippery subject « direct » methods : flow velocities measurements « indirect » methods : dissipation measurements advantages and artefacts Measurements with dSFA - Intrinsic slip lengths on smooth surfaces

Using molecules as tracors: Near Field Laser Velocimetry Pit & al Phys Rev Lett 85 980 (2000) Schmadtko & al PRL 94 244501 (2005) evanescent wave (TIR) + photobleaching (FRAP) T. Schmatdko PhD Thesis, 2003 fluorescence recovery at different shear rates t(ms) Writing beam v spot L ~ 60 µm Evanescent wave ~ 80 nm Reading beam P.M.

Model for Near Field Laser Velocimetry Convection //Ox + Diffusion //Oz Pit-Hervet-Leger (2000) Case of no-slip b.c. g ° V = z Hexadecane on rough sapphire z(t)=√ Dmt x = z t g ° L

Model for Near Field Laser Velocimetry Pit-Hervet-Leger (2000) Partial slip b.c. g ° b g ° V = (z+b) z(t)=√ Dmt x = t (z+b) g ° Résolution : 100 nm Velocity averaged on ~ 1 µm depth Needs value of diffusion coefficient Find slip length b=175nm for hexadecane on sapphire (perfect wetting) L

First measurement of slippage without flow…. L. Bocquet, L. Joly, C. Ybert Condmat 0507054 Einstein 1905 Diffusion of a colloidal particle mobility F e Measuring tangential diffusion as a function of wall distance gives information on the flow boundary condition.

Dmeasured Dno-slip Diffusion of confined colloids measured by Fluorescence Correlation Spectroscopy Float pyrex OTS-coated pyrex b=20nm Rough pyrex b=100nm Dmeasured Dno-slip

Dissipation measurements Pressure drop Churaev, JCSI 97, 574 (1984) Choi & Breuer, Phys Fluid 15, 2897 (2003) Surface Force Apparatus Colloidal Probe AFM Chan & Horn 1985 Israelachvili 1986 Georges 1994 Granick 2001 Mugele 2003 Cottin-Bizone 2005 Craig & al, PRL 87, 054504 (2001) Bonnacurso & al, J. Chem. Phys 117, 10311 (2002) Vinogradova, Langmuir 19, 1227 (2003)

Princip of SFA measurements Tabor et Winterton, Proc. Royal Soc. London, 1969 D is measured with FECO fringes (Å resolution, low band-pass) In a quasi-static regime (inertia neglected) Distance is measured accurately, Force is deduced from piezoelectric drive

Princip of colloidal probe measurements Ducker 1991 7,5 µm scanner xyz piézo substrate cantilever particule Photodetector laser feedback Y X z Force is measured directly from cantilever bending Probe-surface distance is deduced from piezoelectric drive

Hydrodynamic force with partial slip b.c. Reynolds force Hypothesis: Newtonian fluid D<<R Re<1 rigid surfaces uniform constant b (linear b.c.) O. Vinogradova Langmuir 11, 2213 (1995)

Wall shear rate in a drainage flow Parabolic approximation z = D + x2 2R z(x) D U(x) x Mass conservation 2pxz U(x) = - p x2 D g (x) √R D3/2 D √ 2RD AFM/SFA methods are not adapted for investigating shear-rate dependent b.c.

Some recent experimental results on smooth surfaces 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) slip length (nm) Baudry et al (SFA) Cottin-Bizonne et al (SFA) 10 MD Simulations 1 50 100 150 Contact angle (°) Brenner, Lauga, Stone 2005

D b Sensitivity to experimental errors f *( ) Reynolds force f* varies between 0.25 and 1 and has a log dependence in D/b Determination of b requires very precise measurement of F over a large range in D.

D(nm) 10 100 calculated b(nm) D(nm)

OUTLINE The no-slip boundary condition (bc): a long lasting empiricism regularly questionned Some examples of importance in nanofluidics What says theory ? Experiments: a slippery subject Measurements with dSFA - Intrinsic slip lengths on smooth surfaces

Dynamic Surface Force Apparatus F. Restagno, J. Crassous, E. Charlaix, C.Cottin-Bizonne, Rev.Sci. Inst. 2002 k=7000N/m Interferometric force sensor Capacitive displacement sensor Capacitor plates Micrometer Excitation : 0.05 nm < hac < 5 nm w/2p : [ 5 Hz ; 100 Hz ] Resolution : Displacement Force Static 0.1 nm 600 nN Dynamic 5 pm 40 nN Nomarski interferometer Mirors Piezoelectric elements Coil Magnet Plane

Investigating dynamics of confined fluids with Dynamic Surface Forces Measurements D µm nm R ~ mm F(t) D(t) Measure the dynamic force response to an oscillatory motion of small amplitude In-phase component: stiffness (derivative of eq. interaction force ; elastic deformations …) Out-of-phase component: damping (viscous dissipation) The dynamic force response gives informations on the rheology of the confined liquid and on the flow boundary condition.

Newtonian liquid with no-slip b.c. D µm nm R ~ mm F(t) D(t) Hypothesis : The confined liquid remains newtonian Surfaces are perfectly rigid No-slip boundary condition No stiffness The viscous damping is given by the Reynolds force

Simple liquid on a wetting surface N-dodecane Smooth surface: float pyrex Molecular Ø : 4,5 Å Roughness : 3 Å r.m.s. Molecular length : 12 Å Perfectly wetted by dodecane ( = 0°)

Inverse of viscous damping D(nm) Quasi-static force 0 10 20 30 Inverse of viscous damping Bulk hydro. OK for D ≥ 4nm No-slip : b ≤ 2nm 0 10 20 30

Viscous damping with partial slip: Specificity of the method Two separate sensors with Å resolution : no piezoelectric calibration required More rigid than usual SFA (no glue) or AFM (no torsion allowed) In and out-of-phase measurement allows to check for unwanted elastic deformations (and associated error on distance) Easy check for linearity of the b.c. with shear rate: change amplitude or frequency of excitation at fixed D No background viscous force that cannot be substracted

Viscous damping with a partial slip h.b.c. f *( ) D b O. Vinogradova : Langmuir 11, 2213 (1995) At large distance (D>>b) :

Smooth hydrophilic and hydrophobic surfaces Smooth float pyrex surfaces : 0,3nm r.m.s. OTS silanized pyrex : 0,7nm r.m.s. octadecyltricholorosilane Contact angle Float pyrex OTS pyrex Water Dodecane 0° 0° 110° 30°

Water confined between plain pyrex Environment : clean room bare pyrex plane and sphere : b≤ 3nm Theory Experiment Water on bare pyrex : Bulk hydro OK for D≥ 3nm no-slip D (nm) C. Cottin-Bizonne et al, PRL 94, 056102 (2005)

Water confined between plain and OTS-coated pyrex Environment : clean room Theory Experiment Water on bare pyrex : no-slip b = 17±3 nm silanized plane bare pyrex sphere Linear b.c. up to .shear rate ~ 5.103 s-1 Water on silanized pyrex : partial slip one single slip length bare pyrex plane and sphere : b≤ 3nm D (nm) C. Cottin-Bizonne et al, PRL 94, 056102 (2005)

Summary of results with dSFA: intrinsic slip of simple liquids on smooth surfaces Boundary slip depends on wetting Plain pyrex < 3nm < 3nm water dodecane OTS-pyrex 15-18nm < 3nm DPPC monolayer 8-10nm (fresh) DPPC bilayer < 3nm wetting non-wetting When slip occurs : 1 well-defined slip length fully linear h.b.c. condition (up to 5.103 s-1) amplitude in good agreement with M.D. simulations of LJ liquids

Flow on soft surfaces: lipid monolayers and bilayers Lipid layers are used in a number of bio-materials DPPC bilayers are model for biological cell membrane Supported dense layers are obtained with various wettability DPPC monolayers are hydrophobic (q =95°) DPPC bilayer are hydrophilic air air water water DPPC Langmuir-Blodgett deposition on float pyrex

Dense DPPC bilayers in water Stable Smooth at small scale roughness ~0,3 nm r.m.s Some holes (bilayer thickness 6,5 nm)

DPPC monolayers in water. 200 nm 200 nm 200 nm 200 nm 200 nm 200 nm after 7h after 1h after 1 day roughness : 0,7 nm r.m.s ~ 3 nm pk-pk roughness : 2,2 nm r.m.s 6,5 nm pk-pk contact angle : 95° contact angle : 80°

water on a DPPC monolayer after 1 day hydratation No-slip b= 0 b= 10nm water on a DPPC monolayer after 1 day hydratation No-slip 0 100 D(nm) D (nm) G’’-1(w) nm/µN 0 10 20 30 40 water on a fresh DDPC monolayer : (1-2 hours in water) slip length b=10±3nm b=10 nm b= 0 0 40 80 D (nm) b= 0 water on DPPC bilayer : no-slip within 3 nm

CONCLUSION Hydrodynamic BC for a simple liquid at a solid surface is most of the time a no-slip bc, verified down to the molecular scale On smooth surface significant slip can develop, in strongly non-wetting conditions Theory, molecular dynamic simulations, and a number of experimental result show that intrinsic slip lengths on smooth surface are at most of the order of 10-20 nm On rough surfaces, hydrodynamic calculations show that slip effects shoud be smaller than on the chemically equivalent smooth surface Apart for instrumentation artefacts, why are so large slip length sometimes found ?

Other possible artefacts in boundary slip measurements

with hydrophobic surfaces Contamination D (nm) Experiment run in usual room with hydrophobic surfaces

with hydrophobic surfaces Contamination D (nm) constant slip length Experiment run in usual room with hydrophobic surfaces

Nanobubbles ? Ishida Langmuir 16, 6377 (2000) Nanobubbles on OTS-coated silicon water