Nano-hydrodynamics flow at a solid surface boundary condition E. CHARLAIX University of Lyon, France...

Nano-hydrodynamics

flow at a solid surface boundary condition

E. CHARLAIXUniversity of Lyon, France

The Abdus Salam international center for theoretical physics

INTRODUCTION TO MICROFLUIDICS August 8-26 2005

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

Usual boundary condition : the fluid velocity vanishes at wall

z

VS = 0

Hydrodynamic boundary condition at a solid-liquid interface

v(z)

• 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

z

VS ≠ 0

v(z)

b

VS : slip velocity

S : tangential stress at the solid surface

b : slip length

: liquid-solid friction coefficient

: liquid viscosity

€

b =η

λ

Partial slip and solid-liquid friction

Navier 1823Maxwell 1856

∂V∂z

= : shear rate

Tangentiel stress at interface

€

S =η∂V

∂z= λ VS

€

VS = b∂V

∂z

Interpretation of the slip length

From Lauga & al, Handbook of Experimental Fluid Dynamics, 2005

b

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()

The hydrodynamic bc is linear if the slip length does not depend on the shear rate.

On a mathematically smooth surface, b=∞ (perfect slip).

Some properties of the slip length

No-slip bc (b=0) is associated to very large liquid-solid friction

• weak density = weak momentum transfer between fluid and solid : slip can be large

• simple kinetic theory allows to calculate the friction force at wall

« Ideal » case of gas :

u

h

€

≈α u

h

⎛

⎝ ⎜

⎞

⎠ ⎟2

ρ f mkBT

€

b =η λ

€

F = −λV withV

Bocquet, CRAS 1993

Fluid length scale : b ~ µm

« Ideal » case of gas

Root mean square molecule momentum

Gas nb density

Goldstein 1938

Goldstein S. 1969. Fluid mechanics in the first half of this century. Annu. Rev. Fluid Mech 1:1–28

Batchelor, An introduction to fluid dynamics, 1967

The nature of hydrodynamics bc’s has been widely debated in 19th century

Lauga & al, in Handbook of Experimental Fluid Dynamics, 2005

M. Denn, 2001 Annu. Rev. Fluid Mech. 33:265–87

But wall slippage was strongly suggested in some polymer flows…

Pudjijanto & Denn 1994 J. Rheol. 38:1735

Shark-skin effect in extrusion of polymer melts

And alsoBulkley (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

… and some time suspected in flow on non-wetting surfaces

OUTLINE



What says theory ?



Pressure drop in nanochannelsElektrokinetics effectsMixing

Pressure drop in nanochannels

d

∆P

x

zb

r

Exemple 1

slitd=1 µm

b=20 nm%error on permeability : 12%

Exemple 2

tube

r = 2 nm

b=20 nm

Error factor on permeability : 80

(2 order of magnitude)

B. Lefevre et al, J. Chem. Phys 120 4927 2004 Water in silanized MCM41of various radii (1.5 to 6 nm)

10nm

Forced imbibition of hydrophobic mesoporous medium

The intrusion-extrusion cycle of water in hydrophobic MCM41

mesoporous silica: MCM41

quasi-static cycle, does not depend on frquency up to kHz

L ~ 2-10 µm

Porous grain

Electric fieldelectroosmotic flow

Electrostatic double layernm 1 µm

Electrokinetic phenomena

Electro-osmosis, streaming potential… are determined by interfacialhydrodynamics at the scale of the Debye length

Colloid science, biology, …

- - - - - - - - -+

++

++

+ ++

+ +

E v

z

x

os

- - - - - - - - -+

++

++

+ ++

+ +

E v

z

x

os

Case of a no-slip boundary condition:

no-slip plane

zH

zeta potential

- - - - - - - - -+

++

++

+ ++

+ +

E v

z

x

os

Case of a partial slip boundary condition:

At constant s, the electro

osmotic velocity depends on

Pb for measuring ?

Possibility of very large electro-

osmotic flow by decreasing

b=20nm

= 3nm at 10-2M

Exemple

Churaev et al, Adv. Colloid Interface Sci. 2002L. Joly et al, Phys. Rev. Lett, 2004

Mixing

No-slip

rb

Vs ≠ 0

Partial slip

Dispersion shoud be reduced

OUTLINE



What says theory ?



locally: perfect slip

Far field flow : no-slip

Effect of surface roughness

roughness « kills » slip

Richardson, J Fluid Mech 59 707 (1973), Janson, Phys. Fluid 1988

Fluid mechanics calculation :

Robbins (1990) Barrat, Bocquet (1994)Thomson-Troian (Nature 1997)

Lennard-Jones liquidsmolecular size : corrugation of surface potential : u

u

q = 2

In general very small surface corrugation is enough to suppress slip effects

Slip at a microscopic scale : molecular dynamics

b

Thermodynamic equilibrium determination of b.c.with Molecular Dynamics simulations

Be j(r,t) the fluctuating momentum density at point r

Assume that it obeys Navier-Stokes equation

And assumed boundary condition

Bocquet & Barrat, Phys Rev E 49 3079 (1994)

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 bythermal 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 datameasured in equilibrium Molecular Dynamics simulation

Green-Kubo relation for the hydrodynamic b.c.:

(assumes that momentum fluctuations in fluid obey Navier-Stokesequation + b.c. condition of Navier type)

Slip at a microscopic scale : linear response theory

Barrat, Bocquet Phys Rev E 49 3079 (1994)

Liquid-solid Friction coefficient total force exerted

by the solid on the liquid

canonicalequilibrium

Friction coefficient (i.e. slip length) can be computed at equilibrium fromtime decay of correlation function of momentum tranfer

Barrat, Bocquet (1994)

Lennard-Jones liquidsmolecular size : corrugation of surface potential : u

u

q = 2

u/ b/

00.01>0.03

∞400

very small surface corrugation isenough to suppress slip effects

Slip at a microscopic scale : molecular dynamics

If liquid-solid interactions are strong(liquid wets solid)

Barrat et al Farad. Disc. 112,119 1999Molecular Dynamics simulationsLennard-Jones liquids

Linear b.c. up to 108 s-1

Slip at a microscopic scale: liquid-solid interaction effect

α = {liquid,solid}, : energy scale, : molecular diametercα : wetting control parameter

=140°

130°

=90°

b/

P/P0P0~MPa

Polymer melts Priezjev & Troian, PRL 92, 018302 (2004)

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)

For a sinusoidal wall « corrugation » a exp(q// • R//)

density at wall,depends onwetting properties

fluid structure factor

wall corrugation

molecular size

1

10

100

1000

slip

leng

th (

nm)

150100500

Contact angle (°)

Tretheway et Meinhart (PIV) Pit et al (FRAP) Churaev et al (perte de charge) Craig et al(AFM) Bonaccurso et al (AFM) Vinogradova et Yabukov (AFM) Sun et al (AFM) Chan et Horn (SFA)

Zhu et Granick (SFA) Baudry et al (SFA) Cottin-Bizonne et al (SFA)

Some recent experimental results on smooth surfaces

MD Simulations

Non-linear slip

Brenner, Lauga, Stone 2005

Velocimetry measurements

V(z)

Particule Imaging Velocimetry

V(z)

Fluorescence recoveryin TIR

Fluorescence Double Focus Cross Correlation

O. Vinogradova, PRE 67, 056313 (2003)Pit & Leger, PRL 85, 980 (2000)

Schmadtko & Leger, PRL 94 244501 (2005)

Tretheway & Meinhart Phys Fluid 14, L9, (2002)

Dissipation measurementsPressure drop

Colloidal Probe AFM

Surface Force Apparatus

Churaev, JCSI 97, 574 (1984)Choi & Breuer, Phys Fluid 15, 2897 (2003)

Craig & al, PRL 87, 054504 (2001) Bonnacurso & al, J. Chem. Phys 117, 10311 (2002) Vinogradova, Langmuir 19, 1227 (2003)

Chan & Horn 1985 Israelachvili 1986 Georges 1994Granick PRL 2001Mugele PRL 2003Cottin-Bizone PRL 2005

• Particle Image Velocimetry (PIV)

Measurement of velocity profile

V(z)

Spatial resolution ~ 50-100nm

Fluorescent particules High resolution cameraPair of images

Use for bc : are velocity of tracor and velocity of flow the same ?

Poiseuille flow profile in a capillary

With Micro-PIV (see S. Wereley)

Meinardt & al, Experiments in Fluids (1999)

Meinardt & al (2002)

Effect of tracor-wall interactions

Hydrodynamical lift

O. Vinogradova, PRE 67 056313 (2003)

z

Vsphere ≠ Vflow (zcenter)

because of hydrodynamical sphere-plane interaction

F. Feuillebois, in Multiphase Science and Technology, New York, 1989, Vol. 4, pp. 583–798.

d

0.75 slower than flow at d/R=0.1 ~ 1 µm in 10 -6 M

Colloidal lift

z

d+

+

+

+

+ + + +

+

+

electrostatic force:

depletion layer:

Fsphere ~ R exp (-d)

d ~ 3 -1

Vsphere > Vslip

OUTLINE



What says theory ?



« direct » methods : flow velocities measurements« indirect » methods : dissipation measurementsadvantages and artefacts

evanescent wave (TIR) + photobleaching (FRAP)

Writing beam

Reading beam

Evanescent wave ~ nm

v

P.M.

spot L ~ 60 µm

Using molecules as tracors: Near Field Laser Velocimetry

Pit & al Phys Rev Lett 85 980 (2000)

fluorescence recoveryat different shear rates

t(ms)

T. Schmatdko PhD Thesis, 2003

Schmadtko & al PRL 94 244501 (2005)

L

V = z

x = z t°

°

Convection //Ox + Diffusion //Oz Pit-Hervet-Leger (2000)

Model for Near Field Laser Velocimetry

Case of no-slip b.c.

Hexadecane on rough sapphire

z(t)=√ Dmt

z(t)=√ Dmt

L

V = (z+b)

x = t (z+b)°

°

° b

Model for Near Field Laser Velocimetry

Pit-Hervet-Leger (2000)

Partial slip b.c.

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)

First measurement of slippage without flow….

Einstein 1905

L. Bocquet, L. Joly, C. YbertCondmat 0507054

e

F

mobilityDiffusion of a colloidal particle

Measuring tangential diffusion as a function of wall distance gives information on the flow boundary condition.

Diffusion of confined colloids measured byFluorescence Correlation Spectroscopy

Float pyrex

OTS-coated pyrexb=20nm

Rough pyrex

b=100nmDmeasured

Dno-slip

Dissipation measurementsPressure drop

Colloidal Probe AFM

Surface Force Apparatus

Churaev, JCSI 97, 574 (1984)Choi & Breuer, Phys Fluid 15, 2897 (2003)

Craig & al, PRL 87, 054504 (2001) Bonnacurso & al, J. Chem. Phys 117, 10311 (2002) Vinogradova, Langmuir 19, 1227 (2003)

Chan & Horn 1985 Israelachvili 1986 Georges 1994Granick 2001Mugele 2003Cottin-Bizone 2005

Princip of SFA measurements

In a quasi-static regime (inertia neglected)

Distance is measured accurately, Force is deduced from piezoelectric drive

D is measured with FECO fringes (Å resolution, low band-pass)

Tabor et Winterton, Proc. Royal Soc. London, 1969

Princip of colloidal probe measurements

7,5 µm

scanner xyz

piézo

substratecantilever particule

Photodetector

laser

feedback Y

X

z

Ducker 1991

Force is measured directly from cantilever bending Probe-surface distance is deduced from piezoelectric drive

Hydrodynamic force with partial slip b.c.

O. Vinogradova Langmuir 11, 2213 (1995)

D

f *( ) Db

R

Reynolds force

Hypothesis:

Newtonian fluid D<<R Re<1 rigid surfaces uniform constant b (linear b.c.)

Wall shear rate in a drainage flow

z = D + x2

2RMass conservation

2xz U(x) = - x2 D

R

z(x)

x

D

Parabolic approximation

(x)

U(x)

√ 2RD

D√RD3/2

AFM/SFA methods are not adapted for investigating shear-rate dependent b.c.

1

10

100

1000

slip

leng

th (

nm)

150100500

Contact angle (°)

Tretheway et Meinhart (PIV) Pit et al (FRAP) Churaev et al (perte de charge) Craig et al(AFM) Bonaccurso et al (AFM) Vinogradova et Yabukov (AFM) Sun et al (AFM) Chan et Horn (SFA)

Zhu et Granick (SFA) Baudry et al (SFA) Cottin-Bizonne et al (SFA)

Some recent experimental results on smooth surfaces

MD Simulations

Non-linear slip

Brenner, Lauga, Stone 2005

f *( ) Db

Sensitivity to experimental errors

Reynolds force

Determination of b requires very precise measurement of F over a large range in D.

f* varies between 0.25 and 1 and has a log dependence in D/b

D(nm)

10 100

calculated b(nm)

D(nm)

1.0

0.8

0.6

0.4

0.2

0.0

10080604020

OUTLINE



What says theory ?



Dynamic Surface Force Apparatus

Interferometric force sensor

Capacitive displacement sensor

Nomarskiinterferometer Mirors

MagnetCoil

Plane

Piezoelectric elements

Capacitor plates

Micrometer

F. Restagno, J. Crassous, E. Charlaix, C.Cottin-Bizonne, Rev.Sci. Inst. 2002

k=7000N/m

Excitation : 0.05 nm < hac < 5 nm : [ 5 Hz ; 100 Hz ]

Resolution : Displacement Force

Static 0.1 nm 600 nN

Dynamic 5 pm 40 nN

Measure the dynamic force response to an oscillatory motion of small amplitude

Investigating dynamics of confined fluids withDynamic Surface Forces Measurements

D µm nm

R ~ mm

F(t)

D(t)

The dynamic force response gives informations on the rheology of the confined liquid and on the flow boundary condition.

In-phase component: stiffness (derivative of eq. interaction force ; elastic deformations …)

Out-of-phase component: damping (viscous dissipation)

The viscous damping is given by the Reynolds force

No stiffness

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

Smooth surface: float pyrex N-dodecaneRoughness : 3 Å r.m.s.

Molecular Ø : 4,5 Å

Simple liquid on a wetting surface

Molecular length : 12 Å Perfectly wetted by dodecane ( = 0°)

• Quasi-static force

D(nm)

0 10 20 30

0 10 20 30

• Inverse of viscous damping

No-slip : b ≤ 2nm

Bulk hydro. OK for D ≥ 4nm

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.

O. Vinogradova :

Langmuir 11, 2213 (1995)

Df *( )

Db

At large distance (D>>b) :

R

SiSiSiOOSiSiSiOOO(CH2)-18(CH2)-18

Smooth float pyrex surfaces : 0,3nm r.m.s.

OTS silanized pyrex : 0,7nm r.m.s.

Water

Dodecane

Float pyrex OTS pyrex

0°

0°

110°

30°

Contact angle

Smooth hydrophilic and hydrophobic surfaces

octadecyltricholorosilane

Water confined between plain pyrex

D (nm)

Environment : clean room

Water on bare pyrex :Bulk hydro OK for D≥ 3nmno-slip

0

bare pyrex plane and sphere : b≤ 3nm

TheoryExperiment

C. Cottin-Bizonne et al, PRL 94, 056102 (2005)

Water confined between plain and OTS-coated pyrex

bare pyrex plane and sphere : b≤ 3nm

D (nm)

TheoryExperiment

Environment : clean room

Water on bare pyrex :no-slip

b = 17±3 nm

silanized planebare pyrex sphere

Linear b.c. up to .shear rate ~ 5.103 s-1

Water on silanized pyrex :partial slip one single slip lengthb = 17±3 nm

C. Cottin-Bizonne et al, PRL 94, 056102 (2005)

non-wetting

wettingPlain pyrex < 3nm < 3nm

water dodecane

OTS-pyrex 15-18nm < 3nm

DPPC monolayer 8-10nm (fresh)

DPPC bilayer < 3nm

Summary of results with dSFA: intrinsic slip of simple liquids on smooth surfaces

Boundary slip depends on wetting

fully linear h.b.c. condition (up to 5.103 s-1)

When slip occurs : 1 well-defined slip length

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

• DPPC monolayers are hydrophobic 95°)

• DPPC bilayer are hydrophilic

Supported dense layers are obtained with various wettability

DPPC Langmuir-Blodgett deposition on float pyrex

waterwater

air air

Smooth at small scale roughness ~0,3 nm r.m.s

Dense DPPC bilayers in water

Some holes (bilayer thickness 6,5 nm)

Stable

after 1h

DPPC monolayers in water.

200 nm

after 1 day after 7h

200 nm

200 nm

200 nm

roughness : 0,7 nm r.m.s ~ 3 nm pk-pk

200 nm

200 nm

roughness : 2,2 nm r.m.s 6,5 nm pk-pk

contact angle : 95° contact angle : 80°

b= 0

b= 10nm

water on a DPPC monolayer after 1 day hydratationNo-slip

0 100D(nm)

water on DPPC bilayer :no-slip within 3 nm

D (nm)

G’’-1

()

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 80D (nm)

b= 0

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 experimentalresult show that intrinsic slip lengths on smooth surface are at most ofthe 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

Contamination

D (nm)

Experiment run in usual roomwith hydrophobic surfaces

constant slip length

D (nm)

Contamination

Experiment run in usual roomwith hydrophobic surfaces

Ishida Langmuir 16, 6377 (2000)

Nanobubbles on OTS-coated silicon water

Nanobubbles ?

Nano-hydrodynamics flow at a solid surface boundary condition E. CHARLAIX University of Lyon, France...

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