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(Mass) transport phenomena in
microfluidic devices(some selected examples...)
Jean-Baptiste SalmonLOF, Pessac, France
www.lof.cnrs.fr
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Basicsmicrofluidics, solute, transport equation,...
Coflowslow mixing, gradient, role of gravity,...
Hydrodynamic dispersionscoflow, Taylor Aris dispersion, sensors
Mixingsmall size, chaotic mixers, droplets,...
Membranesfiltration, pervaporation,...
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Back to basics
w
h
≈ cm
microfluidic scale h≈1—100µm
from simple geometry.... to complex "lab on chip"
Protron MicroTeknik Liu et al. 2003
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∇ pflows come from a competition betweenviscous and pressure forces only
wh ∇
p = η∆v
flows are incompressible∇.v = 0
Physics of microfluidics (see P. Tabeling's lecture)
v̄
analogy with an "electric" circuit
⇒ (ex. for )k ≈ h2/12 h w
≈ "hydrodynamic resistance"
∇ p
wh
∇ p
R
η/kv̄ = − kη∇ p
v̄ v̄
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Microfluidic devices are rarely
used with pure solvents only...
ex. crystallization of proteins
H a n s e n e t a l . ,P N A S 2 0 0 2
ex. chemistry in droplets
I s m a g i l o v e t a l .
L e n
g e t a l .
ex. "soft matter"
transport of solutes ?
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Different kind of solutes...
ions and small molecules (1Å - 1 nm)
micelles, small proteins, nanoparticles (1-10 nm)
colloids, virus,proteins,....(10 nm - 1 µm)
cells, bacteria, ... (1-10 µm)
colloidal limit
brownian motion
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Brownian motion:
(small) solutes move
random motion dueto thermal agitation
Brown (1827)
diffusion coefficient Perrin (1909)
L
∼(Dt)1/2
20 nm fluorescent beads
in water
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Stokes-Einstein law:
η
R
A simple estimate for D
D
≈ kBT 6πηR
"motor" of transport
friction forces
Some values for D
D ≈ 103 µm2/s D ≈ 40 µm2/s
< 1 nm 5 nm 10 µm
D ≈ 0.02 µm2/sD ≈ 0.2 µm2/s
1 µm
D ≈ 2 µm2/s
100 nm
colloidal limit
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Microfluidics: solutes also flow...
LLLL ∼∼∼∼ ((((DDDDtttt))))1111////2222
diffusion
LLLL ∼∼∼∼ V VV V tttt
convection
Péclet number
Pe = diffusion time / convective timeL/vL2/D
Pe = vL/D
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From "dots" to concentration fields and fluxes
⇒
c(x)
concentration =quantity / volume
Browian motion ⇒ Fick's law+d x -d x
+d x -d x
flux =quantity / time & surface
Flow ⇒ convection flux
flux =quantity / time & surfacev(x)
j(x) = −D∂ xc
j(x) = cv
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Conservation equation
concentration
diffusion/convectionflux
c(x,y,z)
dxdy
dz
⇒
j(x) j(x + dx)
dN dt = − [ j(x + dx)− j(x)] dy dz + . . .
∂ tc +
∇.j = 0
⇒ ∂ tc + v.
∇c = D∆c
j = −D∇c + cv
convection diffusion
∂ tc + v.
∇c = D∆c + R
reaction
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(i) Some "classical" cases:
spreading through diffusion
LLLL ∼∼∼∼ ((((DDDDtttt))))
1111////2222
∂ tC = D∆C
C (x, t) = C 0√ 4πDt
exp− x2
4Dt
w ∼√
2Dt c o n c e n t r a t i o n
x x
w
remember some values 1 s ? 1 mm ?- small molecules in water 50 µm 10 min- 200 nm colloids in water 1 µm 5 days
C (x, t = 0) = C 0δ (x)
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...and with a uniform flow
C (x, t = 0) = C 0δ (x)
C (x, t) = 1√ 4πDt exp(−
(x−vt)2
4Dt )
∂ tC + v.∇C = D∆C
v
c o n c e n t r a t i o n
x
⇒ no complex coupling with(uniform) flows
NBalong x : diffusion & convection
but only diffusion along y
x
y
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(ii) Some "classical" cases:connecting two reservoirs
C (x, t = 0) = C 0H (x)
∂ tC = D∆C
c o
n c e n t r a t i o n
x
w
w ∼
√ 2Dt
C (x, t) ∼ 1 + erf
x2√ Dt
x
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Parenthesis: an analogy
jc
An analogy with the Navier Stokes equation ?
convection diffusion
Pe = convection/diffusion=
∂ tc + v.∇c = D∆c
η/ρ
ρ(∂ t + v.∇)v = −∇ p + η∆v
vLD
⇒ is the « diffusion » coefficient of the velocity⇒ hydrodynamics = transport of "velocity"
⇒ Re ~ Pe. microfluidics: "diffusion" of velocity is immediate
Re = inertial/viscous effects = ρLvη
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Example: start-up of a flow
x
yLU
δ
Navier-Stokes ρ∂ tv = η∂ 2yv ⇔ diffusion equation with D = η/ρ
⇒ « diffusion of the velocity » δ 2 ∼ ηρ t
developed profile for
Note: microfluidics ?water, L = 10 µm, τ = 0.1 ms
multimedia fluid mechanics
τ ∼ ρL2/η
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The same is true for energy...
∂ tc + v.∇c = D∆c mass transportρ(∂ t + v.∇)v = −∇ p + η∆v hydrodynamics
ρC p(∂ t + v.∇
)T = λ∆T + S thermal transfers
Some (very complete) books:
(...)
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Basics
microfluidics, solute, transport equation,...
Coflowslow mixing, gradient, role of gravity,...
Hydrodynamic dispersionscoflow, Taylor Aris dispersion, sensors
Mixingsmall size, chaotic mixers, droplets,...
Membranesfiltration, pervaporation,...
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Back to microfluidics: mixing in a coflow(a simple view)
flow
flow
w
v
convection L ≈ vt
diffusion
Complete mixing for t
≈w2/D (through diffusion)
Convection during t ≈ w2/D ⇒ L ≈ vw2/D = Pe w
large Péclet ⇒ long channels for efficient mixing
≈ √ Dt
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Numerical applications
Colored dyes with D ≈ 103 µm2/s, w ≈ 100 µm, v ≈ 1 cm/s
⇒ Pe = 1000
⇒ efficient mixing after 10 cm
NB:for colloidal species, D ≈ 1 µm2/s⇒ efficient mixing after 100 m...
⇒ need for mixing strategies...
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Mixing in a coflow: concentration fields
flow
flow
wdiffusion
j = −D∂ yc
⇒
convection diffusionv∂ xc = D∂
2
yc
y
x
convection j = cv
⇔ ∂ tc = D∂ 2yc with change of variable x = vt
⇒ co-flow is a good tool to investigate steady kinetics
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flow
flow
wdiffusion j = −D∂ yc
⇒ v∂ xc = D∂ 2yc
y
x
convection j = cv
position y
c o n c e n t r a
t i o n
c ( x , y
)
δ = Dx/v
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Some applications:
data acquisiton
Y a g e r e t a l . ,2
0 0 1
measurements of D
steady kinetics
kinetics of chemical reactions
v ≈ 10 µm/s - 1 cm/sw = 100 µmτ = v/w = 10 s - 10 ms
Salmon et al., 2005
kinetics of protein folding
P o l l a c k e t a l . ,
2 0 0 2
x = vt
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Fast and long kinetics
on a same chip?
v ≈ 1 cm/s, w = 50 µm, D = 10-9 m2/sLm = v w 2 /D ≈ 2 cm, w 2 /D ≈ 2 s
up to 1 min ? ⇒ 60 cm ?≈ 2 s
R
R
R/2
R R
R/2 R
R
R/2
exponential increase of the residence times:q
q/2 q/4 q/8 q/16
(...)
Cristobal et al. 2005
⇔
R
R
R/2
R
R/2 R/2
R
R/2
⇔
R
R
R/2
R
R
R
R/2
(...)
Mi i i l
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Mixing is slow:some opportunitiesdiffusive mixing
small width of diffusion at high velocitiesex: D = 10-10 m2/s, v = 10 cm/s, δ < µm
δ =
Dx/v
⇒ possibility to control complex patterns
"Fil " i h b
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"Filters" without membranes
Ex.:small molecules (D = 10-9 m2/s)
dusts (> 100 nm, (Dd < 10-12 m2/s)
v = 1 mm/s and L = 1 cm ⇒ τ = 10 s
w = (D τ )1/2 = 100 µm and w d = (Dd τ)1/2 < 3 µm
L
Dd D
solvent
soluteand "dusts"
only solute
soluteand "dusts"
Yager et al., 2001
Mi i i l
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Mixing is slow:generating gradients
long serpentines for efficient mixing
(...)
∇ pnegligible hydrodynamic resistance
3q/4
3q/4
3q/4
3q/4
L > Pew
c/2
q
q
qc
0
q/2
q/2
q/4
3q/4
3q/4
q/4
0
c/3
c
2c/3
Dertinger et al., 2001
C l d i l di t
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Complex design = complex gradients
Campbell et al., 2007 Dertinger et al ., 2001
w
v
Note: gradients still disappear for L > vw2/D = Pew
Mi ing is slo
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Mixing is slow:generating microstructures
Kenis et al ., 1999
B k t ll id
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Back to colloids
for colloidal species, D ≈ 1 µm2/sw ≈ 100 µm, v ≈ 1 cm/s
⇒ efficient mixing after 100 m...
⇒
but complex phenomena may happen
Boosting migration of colloids:
D ∇l
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Boosting migration of colloids:diffusiophoresis
Abécassis et al., 2008high Péclet
v = −Ddp∇logc
Wh t b t it ?
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What about gravity ?
thermal or solutal gradients⇒ density gradients ⇒ flows⇒ mixing
and at small scale ?
Buoyancy-induced flow
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∇ρ∇φ ∇ p
g ρ(φ)
x
z
h L
⇒ gradient of hydrostatic pressure
e.g. water to 5% of glycerol, L=1 mm, h=100 µm ⇒ U ∼1 µm/snote : for h = 10 µm, U ~ 1 nm/s
⇒ lubrication flowU
∼(h2/η)∂ x p
∂ x p ∼ (δφ/L)∂ φρgh
U ∼ (δφ/L)∂ φρgh3/ηbuoyancy-induced flow ⇒
⇒ ⇒⇒ ⇒ small scale = small effects
Buoyancy-induced flow
in microfluidics ?
Competition with diffusion
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g
Ra = Uh/D
U ∼ (δφ/L)∂ φρgh3/η
h L
⇔ diffusion time over h / convection over h
diffusive mixing Ra << 1
buyoancy-driven flows Ra >> 1vs.g
⇒ diffusive mixing for Ra < 1⇒ buoyancy-driven convection for Ra >1
Quake & Squires, 2005
Competition with diffusion
⇒⇒⇒⇒ important note: these flows always exist !Selva et al., 2012
for confined gradients:
Back to colloids
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w
= 5 0 0 µm
h=110 µm
water & colloidsQ=25 µL/hr
Q=125 µL/hr
Q=125 µL/hr
⇒ no mixing of the colloids (high Péclet number)
Back to colloids
Buoyancy (can) enhances the mixing of the colloids
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water & colloids& glycerol (3%)
water & glycerol
water water
h h
g
Buoyancy (can) enhances the mixing of the colloids
water/glycerol ⇒ negligible influence of buoyancy (Ra<1)
but strong effects on colloids Selva et al., 2012
h=110 µm
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Basics
microfluidics, solute, transport equation,...
Coflow
slow mixing, gradient, role of gravity,...
Hydrodynamic dispersions
coflow, Taylor Aris dispersion, sensors
Mixing
small size, chaotic mixers, droplets,...
Membranes
filtration, pervaporation,...
Life is more complex:top view
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Life is more complex:hydrodynamic dispersion
top viewv
diffusion/reaction w
perspective view
⇒
"no slip" condition
Poiseuille profile
⇒
what about the results presented ?
Larger "width"
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Larger widthclose to the walls
Ismagilov et al., 2000
⇒ a problem to obtainvery thin electrodes
Kenis et al ., 1999
From 3d to 2d
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From 3d to 2d
⇒
for i.e.⇒ homogeneous concentration gradients (no 3d effects)t > h2
/D x > vh2
/D
v
w
ex: h = 10 µm, v = 1 cm/s, D = 10-9
m2
/s vh
2
/D = 1 mmh2/D = 100 ms3d effects
⇒ need to take care of 3d effects for rapid kinetics
3d problem:
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ptransverse hydrodynamic dispersion
v h
y
far from to the wall δ =
Dx/v
close to the wall δ ∼ (x/v)1/3"Leveque" dispersion(diffusion in a shear flow)
Ismagilov et al., 2000Salmon et al., 2007
A problem for chemistry?A Bz
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A problem for chemistry?
eg: 2nd order chemical reaction A+B ⇔ C
A
B
C
x
z
z
z
y
A B
C
but averaged profiles are not so different from the 2d case...
v
Salmon et al., 2005
y
A problem for chemistry?A B
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p yyes for more complex reactions
A B
C
v
growth
aggregation
1 nm10 nm
30 nm
100 nm
sols
nucleation
dimer
nucleus
particule
Iler, 79
gels
long residence times: bigger & bigger particles
(due to a smaller & smaller diffusivity...)
⇒ clogging & leakages !
⇒ need for 3d microfluidics (no walls) or droplets
Another hydrodynamic dispersion:
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y y pdispersion of residence times
h
tracer stripe at t = 0
h t =0
v
v
h t > 0v ?
⇒ ⇒⇒ ⇒ Taylor-Aris dispersion (convection & diffusion)
Taylor-Aris dispersion: a simple view
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y p p
in the frame of the flowt = 0
t h2/D
t ∼ h2/D
t ∼ N h2/D
v
r a n d o m s
t e p
W = vt
W = N 1/2W d ⇒ ⇒⇒ ⇒ effective diffusion
Squires et al. 2005
h
W d = vh2/D
Taylor-Aris dispersion: a simple view
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hv
t
h2/D
« effective » diffusion coefficient
large solutes: efficient dispersion !
⇒ take care of prefactors circular channels:shallow channels:
De = v2h2/(48D)
De = v2h2/(210D)
De = v2h2/D
y p p
W ∼ [(v2
h2
/D)t]1/2
Sensors:
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Sensors:
"making it stick"
Squires et al. 2008
⇒ a priori a very complex problem(c 0 , U, k, D, H, W c , L, W s, bm ...)
⇒ scaling arguments for extreme regimes
(i) No flow and fast reaction:
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( )depletion layer
t H 2/D
t ∼ H 2/D
t H 2/D
L
H
⇒ no steady state⇒ diffusive flux j d = -Dc 0 /δ
⇒ collection rate J d= j
dx Area
δ
δ δ ∼ √ Dt
(i) unitless "diagram"
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δ
J d ~ Dc 0 /δ LW s
J d ~ Dc 0 /δ HW sδ
∼
√ Dt
δ
( ) g
δ ∼√
Dt
(ii) "slow" flow (and still fast reaction)
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J d
~ Dc 0
/δ s
HW c
J c
~ Q c 0
steady depletion layer J c ~ J d ⇒ δ s = DHW c/Q
only valid for , i.e.δ s H
δ s
PeH = U H/D = Q/(DW c)
⇒ full collection rate in this regime
PeH 1
(iii) "fast" flow (and still fast reaction)J c ~ Q c 0
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U = γ̇ztransit along the sensor
diffusion time to the sensor z2
/Dδ s
δ s
L ⇒ depletion layer
L/(γ̇z)
δ s = (DL/γ̇ )1/3
unitless: δ s/L = (1/Pe1/3s ) &
⇒ collection rate⇒ unitless collection rate
J d ~ Dc 0 /δ s LW s
F ~ (Pes )1/3
λ = L/H Pes = 6λ2PeH
z
(iii) validity range
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δ s
δ s/L = (1/Pe1/3s )
Pes compares δs to the width of the sensor PeH compares δs to the width of the channel
validity range: PeH 1 butPes
1
⇒ picture not valid for very small sensors λ 1
(see Squires et al. for the case of small sensors)
λ = L/H
H
L
Pes = 6λ2PeH
⇒ unitless "diagram"
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(fast reaction)
small sensorsλ
< 1
λ > 1
(iv) reaction vs. transport
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dbdt
= koncs(bm − b)− koff b
concentration ofbounded receptors
total concentration of receptors
for non-saturated sensors
reactive flux J r ~ k on c s bm (LW s )
solute concentrationclose to the sensor
b bm
(iv) ex: "fast" flow & reactionc0
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δ s
L
c0
cs reactive flux J r ~ k on c s bm (LW s )diffusive flux J d ~ D(c 0 -c s )/δ s (LW s )
steady regime: Jr ~ Jd ⇒ cs = c01+Da
Da = konbmδ/D
Da (Damkohler) compares reaction rates to transport rate
⇒
Da<1 : reaction limited cs ~ c0⇒ Da>1 : mass transport limited cs ~ 0
(see Squires et al. for a full discussion for the other regimes)
Basics
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Basics
microfluidics, solute, transport equation,...
Coflow
slow mixing, gradient, role of gravity,...
Hydrodynamic dispersions
coflow, Taylor Aris dispersion, sensors
Mixing
small size, chaotic mixers, droplets,...
Membranes
filtration, pervaporation,...
Back to microfluidics:i i i l
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mixing is slow...diffusive mixing
δ =
Dx/v
L ≈ vw2/D = Pe wlength for efficient mixing
⇒ some innovative strategies for efficient mixing ??
(1) Diminishing the size(at a finite velocity )Qs (water)
ex. hydrodynamic focusing
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(at a finite velocity...)Qs ( )
wf ≈ 100 nm - 1 µm
τ = w2f /D ≈ 10 µs - 1 msKnight et al., 1998
application fo kinetics of folding of bio-molecules(first data point at 5 ms !)
Russel et al., 2002
This is always a( l ) 3d bl
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(complex) 3d problem...Knight et al., 1998
An easy way for hydrodynamic focusing without walls ?
Kinetics of quenching reaction⇒ no walls⇒
no dispersion
Pabit et al . 2002
(2) Multilaminating the flow
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L ≈ vw2/D = Pe w
?L ≈ v(w/N )2/D = Pe w/N 2
Manz et al. 1999
(2) Multilaminating along the flow
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⇒ need for complex 3d structures
Chen et al. 2004
(3) "Grooving" flows: chaotic mixing
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Stroock et al. 2002
in a confined channel:helical stream lines !
grooves
J = fluid flux that that tends to
align along the « easy axis »
(3) "Grooving" flows: chaotic mixing
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length for efficient mixing
⇒ "chaotic mixing" (note: Re ≈ 0)
L ∼ wlog Pe
Stroock et al. 2002
Chaotic mixers helpto make compact devices
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to make compact devices
Whitesides et al. 2002
(4) Active mixing:the "rotary" mixer
fluidic channels
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the rotary mixer
Unger et al. 2000 a c t u
a t i o
n c
h a n n e l
PDMS is soft⇒
valves & pumps for a massiveintegration
(see P. Joseph's lecture)
Quake et al. 2000
Ex: a complete platform for protein crystallization(see N. Candoni's lecture?)
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Lau et al. 2007
i) creating mixtures with 64 ≠ reactants
ii) mixing in a nanoliter volume
iii) store interesting
mixtures in nL plugs
iv) osmotic control
( )
(4) Active mixing:the "rotary" mixerR
height hτ h2/D
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the rotary mixer R
v
Several regimes of mixing
diffusion limited
Taylor Aris regime
convective stirring regime
Pe 1
τ D = h2/D
τ 2 = (2πR)2/Deff ∼ τ 1/Pe2
2πR/v > τ D
τ 3 ∼ Pe−2/3(h/R)4/3τ 1
Squires et al. 2005
⇒ efficient mixing
τ 1 = (2πR)2/D
(5) The case of droplets:perfect microreactors ?
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perfect microreactors ?
Song et al. 2003
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mixing is difficultLm ~ Pe w and Pe >> 1 hydrodynamic dispersion
dispersion of the residence times
Song et al. 2003
c o f l o w
d r o p l e t s
fast mixing no hydrodynamic dispersion
mixing ≈ 5 ms
Droplets are very well-suited forfast chemical reactions
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7 5 m s
Shestopalov et al. 2004
⇒ multistep synthesis of CdSnanoparticles
Chan et al. 2005
the same at high temperature:
Mixing within droplets ?Streamlines in windings
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Song et al. 2003
L
∼wlog Pechaotic mixing
again Re=0
Evidence for a fast mixing
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for a 10X10 µm2 channel
Song et al. 2003
Always true ?silicone oil
≈ 20 ms
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60 µm
Sarrazin et al. 2008
silicone oil
water
dye
≈ 20 ms
1 mm
⇒ take care of scale, flow rates, geometry, etc.
≈ 1 min
General note onmixing & mass transport in microfluidics
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⇒ studies often concern water & (very) diluted dyes
Real life is again more complex...
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(?)
solvants & concentrated solutionscomplex fluids
⇒ rheology ? dissolved gas ? wetting properties ? etc...there is always plenty of space for fundamental studies
Some examples...viscosity contrasts
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Cubaut et al. 2006
wetting, degasing...water
ethanol
t = 0
t ~ 1 min
lof, unpublished
viscoelasticity
Lindner et al.
The case of concentrated system
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t = 0
t > 0
t 0
diffusive process ⇒ convection of mass
concentrated systems:convection/diffusion inter-related
Basicsmicrofluidics, solute, transport equation,...
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microfluidics, solute, transport equation,...
Coflowslow mixing, gradient, role of gravity,...
Hydrodynamic dispersionscoflow, Taylor Aris dispersion, sensors
Mixing
small size, chaotic mixers, droplets,...
Membranes
filtration, pervaporation,...
Controlling transport using "membranes"
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(1) dialysis & microfiltration(pore size 1 - 100 nm)
solutes below the pore sizecross the membrane
(2) pervaporation
solubility and evaporation of the solventex: PDMS for water
(1) Microfiltration forconcentration gradients "without flow"
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⇒ two levels system incorporating microfiltration membranes
Morel et al., 2012 (see V. Studer's lecture ?)
top level
Concentration landscapes "without flow"
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top e e
v
v
no flow
diffusion
⇒ possibility to applyconcentration gradients to
cells without shear flow
Morel et al., 2012
And complex (spatio-temporal)lanscapes !
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Morel et al., 2012
Another recent & similar studymicrofluidic chip made of "agarose gel"
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Palacci et al., 2010
p g gdiffusion of solute, no flow
(2) Control by pervaporation
PDMS membrane: permeability to water (and to other solvents too)
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ve ≈ 20 nm/s for e ≈ 10 µm
⇒ low pumping flow rates: 1 nL/hr for 100x100 µm2
but not negligible !
50 µme
≈ 1 cm
an efficient pump ?
µL/hr
Wheeler et al., 2008
Manipulating solutes using pervaporation
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Shim et al. 2007 i) create drops ii) store drops iii) concentrate solutes
The same with a formulation stage(see N. Candoni's lecture for the applications)
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Lau et al. 2007
i) creating mixtures with 64 ≠ reactants
ii) mixing in a nanoliter volume
iii) store interesting
mixtures in nL plugs
iv) osmotic control
Continuous pervaporation:microevaporators
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reservoir
0
pdms
dry air
L0
vev0
eh
conservation of flow rate:
ve ≈ 20 nm/s for e ≈ 10 µmv0 ≈ 1 to 100 µm/s , tunable thanks to geometry
Leng et al. 2006
v0 = L0
h ve
v(x) = v0x/L0 linear profile in the microevaporator
v(x)
solutes Microevaporators fora controlled concentration
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reservoir
L0
v0
convectiondominates
diffusion
dominates
transition at Pe~1 ⇒
Pe = v(x)x/D
x2 ∼ Dτ e
v(x) = v0x/L0 = x/τ e
µm-mm
Tunable concentration rate
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reservoir
L0
v0
conservation of solute
p = √ Dτ e
v0φ0
≈ p dφ
dt
φ0
⇒ continuous and controlled concentration processof solutes in a nanoliter box⇒ useful tool to investigate "soft matter"
(1) microfluidic-assisted growthof colloidal crystals
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cm
50 µm
Merlin et al. 2012
(2) engineeringcomplex materials
plasmonic nanoparticles
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Iazzolino & Angly
2012
(3) phase diagram screening
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Leng et al. 2007
more informations:www.lof.cnrs.fr
j b ti t l t i @ h di
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[email protected]