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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!(! − !)!
!! = ! !!!/!!