Lab on a Chip and Microfluidics

BenoîtCHARLOT

h0p://www.ies.univ-montp2.fr/~charlot/

Part III. Fluid transport (pressure driven microfluidics)

Hydrostatic pressure

Electrokinetics Electro osmose (mouvement de liquides par un champ électrique) Electrophorèse (mouvement de particules sous l’influence d’un champ électrique) Dielectrophorèse Capillary pumps

Lab On Chip fluid transport

Lab On Chip fluid transport Fluid transport : Fluid mechanics at small scale

What are the laws that govern these flows at small scale? Pressure/flow relations?

Navier Stokes :

�

ρ ∂! u ∂t

+ ! u .! ∇ ( )! u ⎛

⎝ ⎞ ⎠ = ρ! g −

! ∇ p + µΔ! u

Pressure driven microfluidics Pressure sources

Controllable (On/Off, amplitude)

Hydraulic

On chip Thermal expansion, evaporation, micropump

Off chip macropump, syringe, finger push, ..

Constant On chip (osmotic) Off chip (pressure reservoir)

Pneumatic

On chip liquid evaporation, ` micropump

Off chip macropump, syringe, finger push, .

Pumps Macroscopic pumps (off chip)

Syringe pushers

Peristaltic

Membrane

Pressure driven microfluidics

Image : blue white industries

Image : KDscientific

Pressure controllers (off chips) Elveflow Fluigent

Pumps Pressure driven microfluidics

Integrated Pumps Pressure driven microfluidics

Valves : actives, passives (requires energy or not) Normally On / Off

Leakage Dead volume Response time Reliability Biocompatibility Resistance

Valves

Valves Examples

Quake valves

Valves Examples

Teflon Valves WilliamH.Grover,MarcioG.vonMuhlen,andSco0R.Manalis,LabonaChip8(6),913-918(2008)

Valves Examples

Folchlab

Lab On Chip fluid transport Fluid transport : Fluid mechanics at small scale

Navier Stokes equation

�

ρ ∂! u ∂t

+ ! u .! ∇ ( )! u ⎛

⎝ ⎞ ⎠ = ρ! g −

! ∇ p + µΔ! u

Fluidic particle

�

! A =

Ax

Ay

Az

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

�

div! A =! ∇ .! A =

∂Ax

∂x+∂Ay

∂y+∂Az

∂z

Avectorialfield

DivergenceofvectorAisthescalarproductofvectorNablabyvectorA

AfuncXonF

�

gr! a dF =

∂F∂x

∂F∂y

∂F∂z

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=! ∇ F

Math Reminders Divergence and gradients

Math Reminders Divergence and gradients

�

ΔF = div(g! r adF) = ∇2F =∂ 2F∂x 2

+∂ 2F∂y 2

+∂ 2F∂z2

Laplacien

VectorLaplacien

�

! Δ ! A =! ∇ 2! A =

ΔAx

ΔAy

ΔAz

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

∂ 2Ax

∂x 2+∂ 2Ax

∂y 2+∂ 2Ax

∂z2

∂ 2Ay

∂x 2+∂ 2Ay

∂y 2+∂ 2Ay

∂z2

∂ 2Az

∂x 2+∂ 2Az

∂y 2+∂ 2Az

∂z2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

Dimensionless numbers

Reynolds Number (1883)

Osborne Reynolds ( August 23, 1842 in Belfast - February 21, 1912 in Watchet (England)) is an engineer and Irish physicist who made important contributions to hydrodynamics and fluid dynamics , most notably the introduction of number Reynolds in 1883 .

Reynolds numbers represented the ratio of momentum forces to viscous forces

It is used to label the nature of the flow: laminar, transient or turbulent

�

Re =ρvL

µ

ρ DensityvspeedµviscosityLcaracterisXclength(diameterofthepipe)

Re < 1 : Stokes regime, reversible flow, laminar, time reversal

Laminar 1< Re < 2000 transient

Re > 2000 turbulence.

Reynolds Number (1883)

Reynolds Number Fluid flow over a cylinder

Re=1 Re=10 Re=20

What is the direction of the flow?

Turbulence

Reynolds Number

Video:projetluteXum

Turbulence

h0ps://blog.espci.fr/lufr/

What about Re in microfluidics?

Density: 103 Kg.m-3

Viscosity : 10-3 Pa.s Channel dimension : 100µm Speed: 100µm.s-1

Re=10-2<<1

At low Reynolds number, inertia is negligible, flows are reversibles and perfectly laminar, it is the design only that governs the flow

Reynolds Number

Jean Léonard Marie Poiseuille (April 22 , 1797, December 26, 1869 , Paris ) was a French physicist and physician , graduated from the Ecole Polytechnique

Movement of liquids in tubes with small diameters

Ph.D « Recherches sur la force du cœur aortique », 1828

Poiseuille’s law

Also known for the Hagen–Poiseuille equation

Problem :

- The fluid flow is parallel to the walls -  Friction at walls implies that at macroscopic scales the liquid speed is null (non sliping condition) -  Pressure doesn’t change in the section of the flow - Laminar flow Re < 2000

L

z

Solving of the problem with the Navier Stokes equation for an incompressible fluid

What is the liquid velocity distibution along the section?

pe ps

Poiseuille’s law

�

ρ ∂! u ∂t

+ ! u .! ∇ ( )! u ⎛

⎝ ⎞ ⎠ = ρ! g −

! ∇ p + µΔ! u

Mass acceleration

Analog to

F visquous F pressure weight

Navier Stokes equation

�

! F = m! a ∑

Navier Stokes equation for an uncompressible fluid,

�

ρ ∂! u ∂t

+ ! u .! ∇ ( )! u ⎛

⎝ ⎞ ⎠ = ρ! g −

! ∇ p + µΔ! u

�

ρ! g −! ∇ p + µΔ! u = 0

In steady state, for low Reynolds number :

Stokes equation

�

! ∇ p = µΔ! u

Neglecting weigth (microfluidics)

�

∂p∂x

= µ ∂ 2u(z)∂z2If the section is small compared with length, Pressure P is

only function of x and velocity distribution is function of z Equation becomes :

Poiseuille flow

�

∂p∂x

= µ ∂ 2u(z)∂z2

= cte = α

�

p(x) = αx + β

�

u(z) =αz2

µ+ γz + δ

Boundary conditions

�

p(0) = pep(L) = ps

P(x) =ps − peL

x + pe�

u(r) = 0

�

u(z) =ps − pe4µL

r2 1− zr

⎛ ⎝

⎞ ⎠

2⎛

⎝ ⎜ ⎞

⎠ ⎟

Pressure Velocity

Poiseuille flow

�

u(−r) = 0

�

Δp = 8µLSr2

Q = 8µLπr4

Q

L

�

Δp = RhydroQ

�

U = RelecI

�

Rhydro ∝ r−4Hydro resistance increases when the section decreases (power 4)

�

v(z) = vmax 1−z2

r2⎛ ⎝ ⎜

⎞ ⎠ ⎟

�

vm =vmax2

=R2

8µdpdx

=QS

zPoiseuille flow Maximum velocity

Mean velocity

�

Rhydro ∝ L

Poiseuille flow Hydro resistance for different sections

�

Qe = Qsi∑

i∑

The sum of flow rate is conserved at a junction Equivalent to Kirchhoff's circuit laws « The current entering any junction is equal to the current leaving that junction » 

The sum of all the pressure drop around a loop is equal to zero

�

Pi − Pi−1( )i=2

n

∑ = 0

Microfluidic Network

�

Req = R1 + R2

�

1Req

=1R1

+1R2

Microfluidic Network

Microfluidic Network Electric Circuit Hydraulic Circuit

U Battery P Pump (pressure controller)

I Current generator Q Pump (syringe pusher)

ΔU Potential difference ΔP pressure difference

I Current Q flow rate

i current density v flow speed

R Resistance Rh hydro resistance

C Capacitance Ch compliance

L Inductance Fluid Inertia (negligible)

Power ΔUI power ΔPQ

Microfluidic Network : application Stokes trap for multiplexed particle manipulation and assembly using fluidics

Anish Shenoya, Christopher V. Raob, and Charles M. Schroederb, PNAS, vol. 113 no. 15

Using a six-channel microfluidic device, scientists can alter the flow in the device in such a way that they trap and manipulate two particles at the same time.

Recirculation

Tesla valve

Flow visualization Staining Micro particle image velocimetry Doppler holography

Couette flow

Maurice Marie Alfred Couette, Born january 9, 1858 in Tours, France, and died August 18, 1943, is a French Physicist whose work focused mainly on fluid mechanics and especially on rheology . He defended his thesis for the doctorate of science on friction in liquids in the laboratory of physical research of the Faculty of Sciences of Paris . His name is primarily associated with Couette flow but also for the cylinder viscometer that bears his name .

A film of water on a flat substrate Initial velocity ux(h)=v0

What is the velocity distribution ux(z) ?

�

ρ ∂! u ∂t

+ ! u .! ∇ ( )! u ⎛

⎝ ⎞ ⎠ = ρ! g −

! ∇ p + µΔ! u

�

! u .! ∇ ( )! u = 0

�

ρ! g −! ∇ p + µΔ! u = 0

SteadystateNavierflow

�

ux 0( ) = 0ux h( ) = u0

BoundarycondiXons

�

∂ 2

∂z2ux (z) = 0

�

ux (z) = az + b

�

ux (z) =u0zh

Couette flow

Microfluidic circuits printed on a Compact disc Use of centrifuge force induced by the rotation to move liquids

Centrifuge microfluidics : Lab On a CD