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Velocity and temperature slip at the liquid solid interface

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### Velocity and temperature slip at the liquid solid interface

Jean-Louis Barrat Université de Lyon

Collaboration : L. Bocquet, C. Barentin, E. Charlaix, C. Cottin-Bizonne, F. Chiaruttini, M. Vladkov

Outline

- Interfacial constitutive equations
- Slip length
- Kapitsa length
- Results for « ideal » surfaces
- Slip length on structured surfaces
- Thermal transport in « nanofluids »

Hydrodynamics of confined fluids

Original motivation: lubrication of solid surfaces

- Mechanical and biomechanical interest
- Fundamental interest:
- Does liquid stay liquid at small scales?
- Confinement induced phase transitions ?
- Shear melting ?
- Description of interfacial dynamics ?

Controlled studies at the nanoscale: Surface force apparatus (SFA) Tabor, Israelaschvili

Bowden et Tabor, The friction and lubrication of solids, Clarendon press 1958

D. Tabor

(biomedical analysis,chemical engineering)

Lee et al, Nanoletters 2003

Microchannels

….Nanochannels

Importance of boundary conditions

- Micro heat pipes
- Evaporation
- Layered materials
- Nanofluids

Hydrodynamic description of transport phenomena

1) Bulk constitutive equations : Navier Stokes, Fick, Fourier

Physical material property, connected with statistical physics

2) Boundary conditions: mathematical concept

- Replace with notion of interfacial constitutive relations

Physical, material property, connected with statistical physics

Interfacial constitutive equation for flow at an interface surface:

the slip length (Navier, Maxwell)

Continuity of stress

h = viscosity

l= friction

micro-channel : S/V~1/L surface effect becomes dominant.

Influence of slip:

- Flow rate is increased
- Hydrodynamic dispersion and velocity gradients are reduced.

Poiseuille

Electro-osmosis

Also important for electrokinetic phenomena (Joly, Bocquet, Ybert 2004)

Electrostatic double layer

~ nm - 1µm

Interfacial constitutive equation for heat transport

Kapitsa resistance and Kapitsa length

Thermal contact resistance or Kapitsa resistance defined through

Kapitsa length:

lK = Thickness of material equivalent (thermally) to the interface

Important in microelectronics (multilayered materials)

solid/solid interface: acoustic mismatch model

Phonons are partially reflected at the interface

Energy transmission:

Zi = acoustic impedance of medium i

See Swartz and Pohl, Rev. Mod. Phys. 1989

Experimental tools: slip length

Drag reduction in capillaries

SFA (surface force apparatus)

Churaev, JCSI 97, 574 (1984)

Choi & Breuer, Phys Fluid 15, 2897 (2003)

AFM with colloidal probe

Craig & al, PRL 87, 054504 (2001)

Bonnacurso & al, J. Chem. Phys 117, 10311 (2002) Vinogradova, Langmuir 19, 1227 (2003)

v

Evanescnt wave

Optical tweezres

Optical methods:: PIV, fluorescence recovery

Experiments often difficult – must be associated with numerical:theoretical studies

Slip length

(nm)

Tretheway et Meinhart (PIV)

Nonlinear

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 (°)

Simulation results (intrinsic length: ideal surfaces, no dust particles, distances perfectly known)

- Robbins- Thompson 1990 : b at most equal to a few molecular sizes, depending on « commensurability » and liquid solid interaction strength
- Barrat-Bocquet 1994: Linear response formalism for b
- Thompson Troian 1996: boundary condition may become nonlinear at very high shear rates (108 Hz)
- Barrat Bocquet 1997: b can reach 50-100 molecular diameters under « nonwetting » conditions and low hydrostatic pressure
- Cottin-Bizonne et al 2003: b can be increased using small scale « dewetting » effects on rough hydrophobic surfaces

Density at contact. Associated with wetting properties

Response function of the fluid

Corrugation

Linear response theory (L. Bocquet, JLB, PRL 1994)

Kubo formula:

Note: the slip length is intrinsically a property of the interface. Different from effective boundary conditions used for porous media (Beavers Joseph 1967)

q=140°

q=90°

P/P0

P0~MPa

Density at contact is controlled by the wetting properties and the applied hydrostatic pressure.

Slip length as a function of pressure (Lennard-Jones fluid)

Density profiles

A weak corrugation can also result in a large slip length. SPC/E water on graphite

Contact angle 75°

Direct integration of Kubo formula:

b = 18nm

(similar to SFA experiments under clean room conditions – Cottin Steinberger Charlaix PRL 2005)

Analogy: hydrodynamic slip length Kapitsa length

Energy Momentum ;

Temperature Velocity ;

Energy current Stress

Kubo formula

q(t) = energy flux across interface, S = area

-Lennard Jones fluids

Wetting properties controlled by cij

-equilibrium and nonequilibrium simulations

Nonequilibrium temperature profile – heat flux from the « thermostats » in each solid.

Equilibrium determination of RK

Heat flux from the work done by fluid on solid

Dependence of lK on wetting properties (very similar to slip length)

J-L Barrat, F. Chiaruttini, Molecular Physics 2003

Experimental tools for Kapitsa length:

Pump-probe, transient absorption experiments for nanoparticles in a fluid :

-heat particles with a « pump » laser pulse

-monitor cooling using absorption of the « probe » beam

Influence of roughness (slip length)

Far flow field – no slip

Richardson (1975), BrennerFluid mechanics calculation – Roughness suppresses slip

Perfect slip locally – rough surface

But: combination of controlled roughness and dewetting effects can increase slîp by creating a « superhydrophobic » situation.

D. Quéré et al

1 µm

Simulation of a nonwetting pattern

Ref : C. Cottin-Bizonne, J.L. Barrat, L. Bocquet, E. Charlaix, Nature Materials (2003)

Superhydrophobic state

« imbibited » state

Hydrophobic walls

q= 140°

Flow parallel to the grooves

(similar results for perpendicular flow)

Superhydrophobic

imbibited

rough surface

flat surface

*

b (nm)

P/Pcap

(C. Barentin et al 2004; Lauga et Stone 2003; J.R. Philip 1972)

Vertical shear rate:

Inhomogeneous surface:

Far field flow

is the slip velocity

Complete hydrodynamicdescription is complex (cf Cottin et al, Eur. Phys. Journal E 2004). Some simple conclusions:

Partial slip + complete slip, small scale pattern b1>>L

Works well at low pressures (additional dissipation associated with meniscus at higher P)

Partial slip + complete slip, large scale pattern b1 << L

L= spatial scale of the pattern

a= lateral size of the posts

Fs = (a/L)2 Area fraction of no-slip BC

a

Scaling argument:

Force= (solid area) x viscosity x shear rate

Shear rate = (slip velocity)/a

Force= (total area) x (slip velocity) x (effective friction)

How to design a strongly slippery surface ?

For fixed working pressure (0.5bars) size of the posts (a=100nm), and value of b0 (20nm)

Compromisebetween

- Large L to obtain large z
- Pcap large enough

P<Pcap 1/( L)

Possible candidate : hydrophobic nanotubes « brush » (C. Journet, S. Purcell, Lyon)

P. Joseph et al, Phys. Rev. Lett., 2006, 97, 156104

Effective slippage versus pattern: flow without resistance in micron size channel ? (Joseph et al, PRL 2007)

beff ~ µm

z (µm)

beff

L

Characteristic size L

Very large (microns) slip lengths observed in some experiments could be associated with:

- Experimental artefacts ?
- Impurities or small particles ?
- Local dewetting effects, « nanobubbles » ?

Tyrell and Attard, PRL 2001 AFM image of silanized glass in water

Thermal transport in nanofluids

- Large thermal conductivity increase reported in some suspensions of nanoparticles (compared to effective medium theory)
- No clear explanation
- Interfacial effects could be important into account in such suspensions

Simulation of the cooling process

- Accurate determination of RK: fit to heat transfer equations
- No influence of Brownian motion on cooling

Maxwell Garnett effective medium prediction

hot

a = (Kapitsa length / Particle Radius )

Cold

M. Vladkov, JLB, Nanoletters 2006

l/l

(²

)

/ (²

)

exp

2 phases model

1000

Au

Cu

MW CNT

???

100

Al

O

2

3

CuO

TiO

2

Fe

10

1

Au

0,1

0,01

0,1

1

10

Effective medium theory seems to work well for a perfectly dispersed system.

Explanation of experimental results ? Clustering, collective effects ?

Our MD

Volume fraction

Similar idea for fractal clusters (Keblinski 2006)

Still other interpretation: high coupling between fluid and particle (Yip PRL 2007) and percolation of high conductivity layers (negative Kapitsa length ?)

Conclusion

- Rich phenomenology of interfacial transport phenomena: also electrokinetic effects, diffusio-osmosis (L. Joly, L. Bocquet)
- Need to combine modeling at different scales, simulations and experiments

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