an introduction to microfluidics : lecture n°2

AN INTRODUCTION TO MICROFLUIDICS :

Lecture n°2

Patrick TABELING, [email protected], MMN, 75231 Paris0140795153

1 - History and prospectives of microfluidics2 - Microsystems and macroscopic approach.3 - The spectacular changes of the balances of forces aswe go to the small world.

Outline of Lecture 1

- The fluid mechanics of microfluidics - Digital microfluidics

Outline of Lecture 2

∂ρ∂t

+divρu=0

Navier-Stokes equations

DuDt

=∂u∂t

+(u∇)u=−1ρ

∇P+νΔu

Reynolds numbers are small in microsystems

Re = Ul/ ~ l2

One thus may think in the framework of Stokes equations

Microhydrodynamics

0 =−∂p∂xi

+μ∂2ui∂xj∂xj

Stokes regime : inertial terms are neglected

Acceptable approximation in most case. Exceptions are Micro-heat pipes and drop dispensers

Linéarité : si u(x) et v(x) sont solution des équations de Stokes, alors toute combinaison

linéaire de u et v l 'est aussi. Les conditions aux limites seront également une combinaison

linéaire des conditions aux limites associées à et v.

Réversibilité : changer t en -t ne change pas les équations. De manière équivalente, changer u

en -u sur les frontières de l'écoulement inverse la vitesse partout dans l'écoulement.

Minimum de dissipation : l'écoulement donné par l'équation de Stokes minimise la

dissipation d'énergie cinétique par r apport aux champs cinématiquement admissibles,

compatibles avec les conditions aux limites.

Unicité : La solution des équations de Stokes est unique

Réciprocité : Deux solutions, associées à des conditions aux l imites différentes, sont reliées

par une équation intégrale.

Propriétés des écoulements à très petit nombre de Reynolds

Let us reverse U

-U

If it is a Stokes solution, arrows must be invertedeverywhere

This solution cannot be Stokes

U

Because if we reverse U

-U

We obtain a non plausible streamline pattern

Experiment Performed byO Stern (2001)

Flows in cavities at low Reynolds numbers

Hele Shaw flows

Darcy law governs Hele Shaw cells

V=−b2

12μ∇p

In a Hele Shaw cell, flows are potential

An important notion : the hydrodynamic resistance

ΔP=RQmR=

12νb2LS

~l−3

Increases as the system size decreases

Flows in rectangular ducts

Q=CGwb3

μ

3 10-2

4 10-2

5 10-2

6 10-2

7 10-2

8 10-2

9 10-2

0 2 4 6 8 10 12 14

χ

χ=wb

Another important notion : the hydrodynamic capacity

Qm=CdPdt

Example : Deformable tube :dP= dV/V

Now Qm=dV/dt

Thus Qm=mdP/dt and hence C=m

Volume VPressure P

The bottleneck effect

uc(t)

up(x,t)

U

Question : Show that the time to reach a steady state is given byThe expression

τ=

3πlμD2LEb3w

Response : C=m/E=πD2L/4E et R=12l/b3wwith =RC

Expérience effectuée au MMN (2001)- Matthieu Cécillon

Experiment in a microchannel, 1.4 m deep

Beware of dead volumes

Because to reach a steady state, it takes a time equal to

≈ RC then ≈ mR

One must avoid dead volumes, bubbles, etc..

A PDMS actuator, based on Multi-layer Soft Lithography

A. Unger, H-P. Chou, T. Thorsen, A . Scherer et S. R. Quake, Science, 288, 113, (2000).

Actuation channel

Glass slideWorking channelPDMS

From the electrical point of view, pneumatic actuators arerepresented by a capacitance/non linear resistance system . They are not just diodes

Non linearresistances

J.Goulpeau, A. AjdariP. Tabeling,J. Appl.Phys.May 2005

R=f(VC)

VC

R

R

QuickTime™ et undécompresseur Cinepak

sont requis pour visionner cette image.

No actuation : Large localized gradient

Actuation : Producing different Concentration gradients by changing the actuaction parameters

The same, using mechanicalactuators

Mechanical actuators dedicated to the generation of concentration gradients

Passive concentration gradient generator (1)

(1)Jeon et al, Nature Techn., 20, 826 (2002))

Mixer - ExtractorMicrodoser

MixerGradient concentration generatorMicrodoser

ELECTRICAL REPRESENTATIONS OF ELEMENTARY ACTIVE SYSTEMS

Integrated actuators can be used to make progress in the realization of complex systems : an example is a chip for proteomics

The boundary conditions for liquids

u=±Ls∂u∂z

⎛

⎝ ⎜

⎞

⎠ ⎟

z

The slip length

u

Slip length (or extrapolation length)

Navier Boundary Conditions

Flow rate Q

P

Pressure drop with a slip length

ΔP=12μLQwb3

×1

1+6LS /b

Slip length LS

Depth b

Microfluidics and capillarity

Two or three things importantto know in microfluidics

Laplace’s law

R

V

S

Bubble

δE =−pδV+γδS

V=43πR3 ⇒ δV=4πR3δR

S=4πR2 ⇒ δS=8πRδR

At mechanical equilibrium : E=0

p=2γR

Capillary phenomena are important in microsystems

Pressure dropscaused by capillarityare ~ l-1 while those due toviscosity behavelike l0

THE PATTERNS WHICH DEVELOP IN “ORDINARY” TWO PHASE FLOWS

…OFTEN PRODUCE COMPLEX MORPHOLOGIES; THIS IS DUETO HYDRODYNAMIC TURBULENCE

IN MICROFLUIDIC SYSTEMS, WE OBTAIN MUCH SIMPLER MORPHOLOGIES : ESSENTIALLY DROPLETS

Laure MENETRIER, 2004

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Droplets can be made using a “hydrodynamic focusing” geometry

3 cases

Complete wetting

Partial wetting

Desorption

Wetting are exceedingly important in microsytems

Spreading parameter

S=γSG −γSL −γ

When S is non homogeneous, droplets spontaneouslymove on the surfaces

A<B

Poor wetting (S <0)Good wetting (S ≈ 0)

γSO−γSW=γcosθ

oil water

In liquid, one may easily change S by adding surfactants

WaterOil with or without surfactant (Span 80)

Water

Wetting properties of the walls are important in microfluidic multiphase flows

R Dreyfus, P.Tabeling, H Willaime, Phys Rev Lett, 90, 144505 (2003))

QuickTime™ et un décompresseurDV - PAL sont requis pour visualiser

cette image.

NICE DROPS CAN BE PRODUCEDIN MINIATURIZED SYSTEMS IN COMPLETE WETTING CONDITIONS

200

m

When oil fully wets the surface

Water flow rate (L/min)

Oil flow rate(L/min) Stratified regimeIsolated water drops

Coalescence

Pears

Pearl necklace

Pear necklace

Large-pearl necklace

Oil

flo

w-r

ate

(L

/mn)

(R Dreyfus, P.Tabeling, H Willaime, Phys Rev Lett (2003))

Water flow-rate (L/mn)

WHEN THE FLUIDS PARTIALLY WET THE WALLS

Rayleigh instability is the most important instability to be aware of

d

EC =2πRLγ=2πVγR

Surface energy of a column

E =4πR'2Nγ=3γVR'

Surface energy of N spherical droplets

R'>3

2πR UNSTABLE

Applications :Digital microfluidics

- Liquid liquid flows are used in microsystems, in several circumstances.

Producing drops of one liquid into another liquidso as to generate emulsions, or perform screening

Producing bubbles in a microchannel flow so asto increase heat exchange, or simply because the liquid boils.

Digital microfluidics

1 - In air

2 - In a liquid

The drop moves in a liquidin a microchannel

The drop moves in air over a flat surface

Digital microfluidics is interesting for chemical analysis, protein cristallization, elaborating novel emulsions,…

Ismagilov et al(Chicago University)

AN EXAMPLE OF AN INTERESTING PROBLEM : REDUCING THE DROP SIZE OF AN EMULSION BY USING DIGITAL MICROFLUIDICS TECHNOLOGY

Suppose we are willing to reduce the drop size of an emulsion

10m

One possibility is to cut the drops one by onein a microfluidic system

Water drop

10m

U

u

L0

l =L0uU

WHITE = WATER DROP, BLACK (IN THE CHANNEL) = HEXADECANE

DIFFERENT REGIMES, FOR INCREASING SIDE FLOWS

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Lf

Finger

VS

We would like also to control the drop break-up

Lf (m)

VS

(m/s)

BREAKING

NONBREAKING

Curve suggested bythe theory (Navot, 1999)

To break or not to break

0 1 2 3 4 5 x102

Laure MENETRIER, 2004

AN IMPORTANT PART OF DIGITAL MICROFLUIDICS IS BASED ON ELECTROWETTING

Side where the contact angleis smallerA<B

Électrode+V

B=A +1/2CV2

A microfluidic network along which drops are driven

C.J.Kim (2001)

Digital microfluidic devices based on electrowetting

Liquid-liquid flows in microsystems may be used to produce well controlled drops, emulsions,…

… provided the wetting properties of the exposed surfaces, with respect to the working fluids, are appropriately chosen.

an introduction to microfluidics : lecture n°2

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