Lecture 5: Mixing, mass transfer, adsorption, NP transport

1) Basics of diffusion, dispersion

Microscopic description of diffusion processes

x = li1

n

∑li

x2 = li1

n

∑⎛

⎝⎜

⎞

⎠⎟

2

≈ l2n ≈ Dt

Continuous approach

C(x, t) = δmδV

J = δmδtδA

J = −D∇C

Area δAVolumeδV

Fick’s law

∂C∂t

= DΔC

Diffusion equation

C(x, t) =C04πDt

exp −x2

4Dt# $ %

& ' (

The gaussian distribution

C

x

C

x

t=0 t

STD σ = (2Dt)1/2

€

DCDt

=∂C∂t

+ u∇C = DΔC

Equation of diffusion-advection

The Peclet number

Pe =UlD~ advectiondiffusion

(a) (b) (c)

u∇C = 0u∇C ≠ 0;u uniform

u∇C ≠ 0;u_non_uniform

Basic situations in microfluidics

Slow mixing in side-to-side flows

100 µm

Dispersion in a uniform flow

C(x,t) =C0δ2πσ

exp −(x −Ut)2

2σ 2

%

& ' (

) with σ2=2Dt

x=0

xU

Dispersion in a pure shear flow

C(x, y, t) = Q2πσ x 2Dt

exp −(x −Ut)2

2σ x2

⎛

⎝⎜

⎞

⎠⎟

with σ x2 = 2Dt (1+α

2

12t2 )

hyperdiffusion

U=αy

Dispersion of Taylor-Aris

From Kirby

∂C

∂t= DeffΔC

DC

Dt=∂C

∂t+ (U(z)− V)

∂C

∂x'= D

∂2C

∂x'2

DC

Dt=∂C

∂t+U(z)

∂C

∂x= DΔC

t >> d2/D

Deff = D(1+ αPe2 )

Dispersion of Taylor-Aris

2) Mixing

Perfectly mixed system: when the concentration is homogeneous The mixing process: process that leads to a perfectly mixed system Two mechanisms play a role in any mixing process:

-  Diffusion -  Advection

Notion of mixing

Diffusion based mixing

€

τ =l2

D

Nanofluidics does not need nanomixers

€

τ =l2

D

Mixing by scale reduction

R.Austin et al (1999)

w

Residence time: tR=L/UDiffusion time : tD=w2/DMixing quality: A = tR/tD

Distributed micromixer

L

w’=w/N

Mixing quality : A’ = t’R/tD=N2A >> A

U

Mixing by scale reduction

Manz (2004)

L ~ Pe √Dt

Mixers based on Taylor Aris dispersion

Length of the spot at Pe>>1 :

The circular micromixer

The rotary mixer

Quake, Scherer (2001)

A chip for DNA purification

Chaotic mixers

The most popular chaotic micromixer

A. Strooke et al, Science (2002)

A. Strooke et al, Science (2002)

The most popular chaotic micromixer

Review on micro/mini mixers

Universal diagram of micromixers

3) Mixing in droplets

Mixing in digital microfluidics

Mixing in droplet based microfluidics

Ismagilov group

Application to the measurement of chemical kinetics

4) Mass manipulations in microfluidics

Measurement of diffusion constants

P. Yager (Seattle)

Filtering of particles

P. Yager (Seattle)

Gradient formation

Q

Ismagilov et al, Anal Chem (2001)

Liquid liquid extraction

5) Adsorption phenomena

Adsorption phenomena

Isotherm of Langmuir

Langmuir, I.,. J. Am. Chem. Soc, 1918. 40: p. 1399-1400.

Choc sans adsorption

Choc avec adsorption

Désorption

€

Γ =KaC1+ KaC

Scaling laws indicate that adsorption phenomena are important in microfluidics

€

Qadsorbée ≈ KaCS ~ l2

Qtransportée ≈ CV ~ l3

6) Microfluidic chromatography

Basics of chromatography

N =traveldis tance

width⎛

⎝⎜

⎞

⎠⎟2

N ~ L2

Deff L /U( )~ ULDeff

Basis of chromatography

Number of theoretical plates

U

l ≈ √Deff t

Estimation à la Taylor-Aris :

Deff≈ Pe2D ≈ U2b2/D

U

Estimate of the efficiency of a chromatographic column

€

N ~ ULDeff

~ ULDU 2b2

~ LDUb2

~ µL2DΔPb4

~ L2

Pro/con of miniaturisation of chromatographic columns

Pilot Plant preparation column chromatography Genzyme Pharmaceuticals

Pro/con of miniaturisation of chromatographic columns

Pro - Small samples - Intégration - Parallelism Con Degradation of analytical performances

7) Transport of nanoparticles in microfluidic channels

PFF – Pinched flow fractionation

x

z

Behavior of a particle close to the wall

- Hindered diffusion: D⊥ ≈z− rr

D;D// ≈ D 1− 9r16z

+...⎛

⎝⎜

⎞

⎠⎟

- Van der Waals forces:

- Electrostatic forces:

!!"#! −!

6!(! − !)!

!! = ! !!!/!!