(Mass) transport phenomena in
microfluidic devices
(some selected examples...)
Jean-Baptiste Salmon
LOF, Pessac, France
www.lof.cnrs.fr
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 basics
≈ cm
microfluidic scale h≈1—100µm
h
w
from simple geometry.... to complex "lab on chip"
Protron MicroTeknik
Liu et al. 2003
Physics of microfluidics
(see P. Tabeling's lecture)
flows come from a competition between
viscous and pressure forces only
∇p
∇p = η∆v
v̄
h
flows are incompressible
w
⇒
∇.v = 0
v̄ =
k
− η ∇p
η/k ≈ "hydrodynamic resistance"
(ex. k ≈ h2 /12 for h w )
analogy with an "electric" circuit
∇p
∇p
v̄
h
w
R
v̄
Microfluidic devices are rarely
used with pure solvents only...
ex. crystallization of proteins
ex. chemistry in droplets
Hansen et al., PNAS 2002
Ismagilov et al.
ex. "soft matter"
transport of solutes ?
Leng et al.
Different kind of solutes...
ions and small molecules (1Å - 1 nm)
micelles, small proteins, nanoparticles (1-10 nm)
brownian motion
colloids, virus,proteins,....(10 nm - 1 µm)
colloidal limit
cells, bacteria, ... (1-10 µm)
Brownian motion:
(small) solutes move
20 nm fluorescent beads
in water
Brown (1827)
random motion due
to thermal agitation
L ∼ (Dt)1/2
diffusion coefficient
Perrin (1909)
Some values for D
D ≈ 103 µm2/s
D ≈ 40 µm2/s
< 1 nm
D ≈ 2 µm2/s
D ≈ 0.2 µm2/s
100 nm
1 µm
5 nm
D ≈ 0.02 µm2/s
10 µm
colloidal limit
A simple estimate for D
Stokes-Einstein law:
R
η
D≈
kB T
6πηR
"motor" of transport
friction forces
Microfluidics: solutes also flow...
convection
diffusion
L∼ Vt
L ∼ (Dt)1/2
Péclet number Pe = vL/D
Pe = diffusion time / convective time
L2 /D
L/v
From "dots" to concentration fields and fluxes
c(x)
⇒
concentration =
quantity / volume
Browian motion ⇒ Fick's law
-dx
+dx
-dx +dx
j(x) = −D∂x c
flux =
quantity / time & surface
Flow ⇒ convection flux
j(x) = cv
v(x)
flux =
quantity / time & surface
Conservation equation
c(x, y, z) concentration
diffusion/convection
flux
j(x)
j = −D∇c + cv
dN
dt
⇒
dz j(x + dx)
dy
dx
= − [j(x + dx) − j(x)] dy dz + . . .
∂t c + ∇.j = 0
⇒
∂t c + v.∇c = D∆c + R
convection
diffusion
reaction
(i) Some "classical" cases:
spreading through diffusion
∂t C = D∆C
1/ 2
concentration
L ∼ (Dt)
C(x, t = 0) = C0 δ(x)
2
C0
x
C(x, t) = √4πDt
exp − 4Dt
w
w∼
x
x
remember some values
1s?
1 mm ?
- small molecules in water
- 200 nm colloids in water
50 µm
1 µm
10 min
5 days
√
2Dt
v
y
...and with a uniform flow
x
∂t C + v.∇C = D∆C
concentration
C(x, t = 0) = C0 δ(x)
(x−vt)2
1
C(x, t) = √4πDt exp(− 4Dt )
⇒ no complex coupling with
(uniform) flows
x
NB
along x: diffusion & convection
but only diffusion along y
(ii) Some "classical" cases:
connecting two reservoirs
∂t C = D∆C
C(x, t = 0) = C0 H(x)
concentration
C(x, t) ∼ 1 + erf
√x
2 Dt
w
x
x
w∼
√
2Dt
Parenthesis: an analogy
c
j
∂t c + v.∇c = D∆c
convection
diffusion
Pe = convection/diffusion= vL
D
An analogy with the Navier Stokes equation ?
ρ(∂t + v.∇)v = −∇p + η∆v
ρLv
Re = inertial/viscous effects =
η
⇒ η/ρ is the « diffusion » coefficient of the velocity
⇒ hydrodynamics = transport of "velocity"
⇒ Re ~ Pe. microfluidics: "diffusion" of velocity is immediate
Example: start-up of a flow
y
U
L
δ
Navier-Stokes
ρ∂t v = η∂y2 v
x
⇔ diffusion equation with D = η/ρ
⇒ « diffusion of the velocity » δ 2 ∼ ηρ t
developed profile for τ ∼ ρL2 /η
Note: microfluidics ?
water, L = 10 µm, τ = 0.1 ms
multimedia fluid mechanics
The same is true for energy...
∂t c + v.∇c = D∆c
mass transport
ρ(∂t + v.∇)v = −∇p + η∆v hydrodynamics
ρCp (∂t + v.∇)T = λ∆T + S thermal transfers
Some (very complete) books:
(...)
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: mixing in a coflow
(a simple view)
v
flow
diffusion
√
≈ Dt
flow
convection
Complete mixing for
Convection during
L ≈ vt
t ≈ w 2 /D
2
w
t ≈ w /D
(through diffusion)
⇒
L ≈ vw 2 /D = Pe w
large Péclet ⇒ long channels for efficient mixing
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...
Mixing in a coflow: concentration fields
convection j = cv
flow
j = −D∂y c
diffusion
w
y
x
flow
⇒
v∂x c =
convection
⇔
⇒
∂t c =
2
D∂y c
diffusion
2
D∂y c with change of variable
x = vt
co-flow is a good tool to investigate steady kinetics
convection j = cv
flow
j = −D∂y c
diffusion
flow
v∂x c = D∂y2 c
concentration c(x,y)
⇒
δ = Dx/v
position y
w
y
x
steady kinetics
v ≈ 10 µm/s - 1 cm/s
w = 100 µm
τ = v/w = 10 s - 10 ms
Some applications:
data acquisiton
x = vt
kinetics of chemical reactions
measurements of D
Yager et al., 2001
Pollack et al., 2002
kinetics of protein folding
Salmon et al., 2005
Fast and long kinetics
on a same chip?
v ≈ 1 cm/s, w = 50 µm, D = 10-9 m2/s
Lm = v w2/D ≈ 2 cm, w2/D ≈ 2 s
≈2s
up to 1 min ? ⇒ 60 cm ?
exponential increase of the residence times:
q
R
R/2
R
R
q/8
R/2
R
R
q/4
R/2
R
R/2
q/2
R
(...)
q/16
R/2
R/2
R
R/2
R
R
R
⇔
R/2
R
R
R
R
⇔
R/2
(...)
Cristobal et al. 2005
diffusive
mixing
δ=
Dx/v
small width of diffusion at high velocities
ex: D = 10-10 m2/s, v = 10 cm/s, δ < µm
⇒ possibility to control complex patterns
Mixing is slow:
some opportunities
"Filters" without membranes
solvent
L
only solute
solute
and "dusts"
Dd D
solute
and "dusts"
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 wd = (Dd τ)1/2 < 3 µm
Yager et al., 2001
Mixing is slow:
generating gradients
negligible hydrodynamic resistance
∇p
0
c/2
c
q
q
q
3q/4
q/4
q/2
q/2
q/4
3q/4
0
3q/4 c/3
3q/4
(...)
2c/3
3q/4
c
3q/4
long serpentines for efficient mixing
L > Pew
Dertinger et al., 2001
Complex design = complex gradients
v
Campbell et al., 2007
w
Dertinger et al., 2001
Note: gradients still disappear for L > vw 2 /D = Pew
Mixing is slow:
generating microstructures
Kenis et al., 1999
Back to colloids
for colloidal species, D ≈ 1 µm2/s
w ≈ 100 µm, v ≈ 1 cm/s
⇒ efficient mixing after 100 m...
⇒ but complex phenomena may happen
Boosting migration of colloids:
diffusiophoresis
high Péclet
v = −Ddp ∇logc
Abécassis et al., 2008
What about gravity ?
thermal or solutal gradients
⇒ density gradients ⇒ flows
⇒ mixing
and at small scale ?
Buoyancy-induced flow
in microfluidics ?
g ρ(φ)
∇φ
∇ρ
∇p
z
hL
x
∂x p ∼ (δφ/L)∂φ ρgh ⇒ gradient of hydrostatic pressure
U ∼ (h2 /η)∂x p
buoyancy-induced flow ⇒
⇒ lubrication flow
U ∼ (δφ/L)∂φ ρgh3 /η
e.g. water to 5% of glycerol, L=1 mm, h=100 µm ⇒ U∼1 µm/s
note : for h = 10 µm, U ~ 1 nm/s
⇒ small scale = small effects
Competition with diffusion
g
hL
U ∼ (δφ/L)∂φ ρgh3 /η
Ra = U h/D
⇔ diffusion time over h / convection over h
diffusive mixing
g
Ra << 1
vs.
buyoancy-driven flows Ra >> 1
Quake & Squires, 2005
for confined gradients:
⇒ diffusive mixing for Ra < 1
⇒ buoyancy-driven convection for Ra >1
Selva et al., 2012
⇒ important note: these flows always exist !
Back to colloids
Q=125 µL/hr
w = 500 µm
water & colloids
Q=25 µL/hr
Q=125 µL/hr
h=110 µm
⇒ no mixing of the colloids (high Péclet number)
Buoyancy (can) enhances the mixing of the colloids
water & colloids
& glycerol (3%)
h=110 µm
water
h
water & glycerol
water
h
g
water/glycerol ⇒ negligible influence of buoyancy (Ra<1)
but strong effects on colloids
Selva et al., 2012
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,...
top view
Life is more complex:
hydrodynamic dispersion
v
diffusion/reaction w
perspective view
"no slip" condition
⇒
Poiseuille profile
⇒ what about the results presented ?
Larger "width"
close to the walls
⇒ a problem to obtain
very thin electrodes
Kenis et al., 1999
Ismagilov et al., 2000
From 3d to 2d
⇒
2
2
x
>
vh
/D
t
>
h
/D
for
i.e.
⇒ homogeneous concentration gradients (no 3d effects)
ex: h = 10 µm, v = 1 cm/s, D =
10-9
3d effects
m2/s
vh2 /D = 1 mm
h2 /D = 100 ms
v
w
⇒ need to take care of 3d effects for rapid kinetics
3d problem:
transverse hydrodynamic dispersion
v
h
y
far from to the wall δ = Dx/v
close to the wall
1/3
δ ∼ (x/v)
Ismagilov et al., 2000
Salmon et al., 2007
"Leveque" dispersion
(diffusion in a shear flow)
z
A
A problem for chemistry?
B
y
eg: 2nd order chemical reaction A+B ⇔ C
v
A
z
B
z
C
z
C
x
y
but averaged profiles are not so different from the 2d case...
Salmon et al., 2005
A
B
v
A problem for chemistry?
yes for more complex reactions
dimer
nucleation
nucleus
particule
growth
1 nm
10 nm
30 nm
C
100 nm
aggregation
gels
sols
Iler, 79
long residence times: bigger & bigger particles
(due to a smaller & smaller diffusivity...)
⇒ clogging & leakages !
⇒ need for 3d microfluidics (no walls) or droplets
tracer stripe at t = 0
h
Another hydrodynamic dispersion:
dispersion of residence times
v
v
v
?
h
t =0
h
t>0
⇒ Taylor-Aris dispersion (convection & diffusion)
Taylor-Aris dispersion: a simple view
in the frame of the flow
random step
v
h
t=0
t h2 /D
W = vt
t ∼ h2 /D
Wd = vh2 /D
t ∼ N h2 /D
W = N 1/2 Wd
⇒ effective diffusion
Squires et al. 2005
Taylor-Aris dispersion: a simple view
t h2 /D
W ∼ [(v 2 h2 /D)t]1/2
v
h
« effective » diffusion coefficient
De = v2 h2 /D
large solutes: efficient dispersion !
⇒ take care of prefactors
circular channels: De = v2 h2 /(48D)
shallow channels: De = v2 h2 /(210D)
Sensors:
"making it stick"
⇒ a priori a very complex problem
(c0, U, k, D, H, Wc, L, Ws, bm ...)
⇒ scaling arguments for extreme regimes
Squires et al. 2008
(i) No flow and fast reaction:
depletion layer
√
δ ∼ Dt
t H 2 /D
t H 2 /D
δ
t ∼ H 2 /D
δ
H
L
⇒ no steady state
⇒ diffusive flux jd = -Dc0/δ
⇒ collection rate Jd = jd x Area
(i) unitless "diagram"
δ
Jd ~ Dc
/δ HWs
√0
δ∼
δ
Jd ~ Dc
√ 0/δ LWs
δ∼
Dt
Dt
(ii) "slow" flow (and still fast reaction)
Jc ~ Q c0
Jd ~ Dc0/δs HWc
δs
steady depletion layer Jc ~ Jd ⇒ δs = DHWc /Q
⇒ full collection rate in this regime
only valid for
δs H
, i.e.
PeH 1
PeH = U H/D = Q/(DWc )
(iii) "fast" flow (and still fast reaction)
Jc ~ Q c0
δs
z
U = γ̇z
δs
L
L/(γ̇z)
transit along the sensor
diffusion time to the sensor z2/D
⇒ depletion layer
δs = (DL/γ̇)1/3
1/3
δ
/L
=
(1/Pe
unitless: s
s )
&
Pes = 6λ2 PeH
⇒ collection rate
⇒ unitless collection rate
λ = L/H
Jd ~ Dc0/δs LWs
F ~ (Pes)1/3
(iii) validity range
H
δs
δs /L =
1/3
(1/Pes )
L
Pes compares δs to the width of the sensor
PeH compares δs to the width of the channel
PeH 1
validity range:
Pes 1
2
Pe
=
6λ
PeH
but
s
⇒ picture not valid for very small sensors
(see Squires et al. for the case of small sensors)
λ = L/H
λ1
⇒ unitless "diagram"
(fast reaction)
λ>1
small sensors
λ<1
(iv) reaction vs. transport
total concentration of receptors
db
dt
= kon cs (bm − b) − kof f b
solute concentration
close to the sensor
for non-saturated sensors
concentration of
bounded receptors
b bm
reactive flux Jr ~ kon cs bm (LWs)
c0
cs
δs
L
steady regime: Jr ~ Jd ⇒
(iv) ex: "fast" flow & reaction
reactive flux Jr ~ kon cs bm (LWs)
diffusive flux Jd ~ D(c0-cs)/δs (LWs)
cs =
c0
1+Da
Da = kon bm δ/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
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,...
diffusive
mixing
δ=
Dx/v
Back to microfluidics:
mixing is slow...
length for efficient mixing
L ≈ vw 2 /D = Pe w
⇒ some innovative strategies for efficient mixing ??
ex. hydrodynamic focusing
Qs (water)
wf
(1) Diminishing the size
(at a finite velocity...)
≈ 100 nm - 1 µm
τ = wf2 /D ≈ 10 µs - 1 ms
Knight et al., 1998
application fo kinetics of folding of bio-molecules
(first data point at 5 ms !)
Russel et al., 2002
Knight et al., 1998
This is always a
(complex) 3d problem...
An easy way for hydrodynamic focusing without walls ?
Kinetics of quenching reaction
⇒ no walls
⇒ no dispersion
Pabit et al. 2002
(2) Multilaminating the flow
?
L ≈ vw2 /D = Pe w
Manz et al. 1999
L ≈ v(w/N )2 /D = Pe w/N 2
(2) Multilaminating along the flow
Chen et al. 2004
⇒ need for complex 3d structures
(3) "Grooving" flows: chaotic mixing
grooves
J = fluid flux that that tends to
align along the « easy axis »
in a confined channel:
helical stream lines !
Stroock et al. 2002
(3) "Grooving" flows: chaotic mixing
length for efficient mixing
L ∼ wlog Pe
⇒ "chaotic mixing" (note: Re ≈ 0)
Stroock et al. 2002
Chaotic mixers help
to make compact devices
Whitesides et al. 2002
(4) Active mixing:
the "rotary" mixer
Unger et al. 2000
act
uat
ion
cha
nne
l
fluidic channels
PDMS is soft ⇒
valves & pumps for a massive
integration
(see P. Joseph's lecture)
Quake et al. 2000
Ex: a complete platform for protein crystallization
(see N. Candoni's lecture?)
i) creating mixtures with 64 ≠ reactants
ii) mixing in a nanoliter volume
iv) osmotic control
iii) store interesting
mixtures in nL plugs
Lau et al. 2007
height
R
h
τD = h2 /D
(4) Active mixing:
the "rotary" mixer
Several regimes of mixing
v
diffusion limited Pe 1
τ1 = (2πR)2 /D
Taylor Aris regime
2πR/v > τD
τ2 = (2πR)2 /Deff ∼ τ1 /Pe2
convective stirring regime
τ3 ∼ Pe−2/3 (h/R)4/3 τ1
⇒ efficient mixing
Squires et al. 2005
(5) The case of droplets:
perfect microreactors ?
Song et al. 2003
coflow
hydrodynamic dispersion
dispersion of the residence times
droplets
mixing is difficult
Lm ~ Pe w and Pe >> 1
fast mixing
no hydrodynamic dispersion
Song et al. 2003
75 ms
mixing ≈ 5 ms
Droplets are very well-suited for
fast chemical reactions
⇒ multistep synthesis of CdS
nanoparticles
Shestopalov et al. 2004
the same at high temperature:
Chan et al. 2005
Streamlines in windings
chaotic mixing
again Re=0
Mixing within droplets ?
L ∼ wlog Pe
Song et al. 2003
Evidence for a fast mixing
for a 10X10 µm2 channel
Song et al. 2003
Always true ?
silicone oil
≈ 20 ms
water
60 µm
dye
Sarrazin et al. 2008
silicone oil
≈ 1 min
1 mm
⇒ take care of scale, flow rates, geometry, etc.
General note on
mixing & mass transport in microfluidics
⇒ studies often concern water & (very) diluted dyes
Real life is again more complex...
(?)
solvants & concentrated solutions
complex fluids
⇒ rheology ? dissolved gas ? wetting properties ? etc...
there is always plenty of space for fundamental studies
Some examples...
viscosity contrasts
wetting, degasing...
water
t=0
Cubaut et al. 2006
viscoelasticity
ethanol
t ~ 1 min
lof, unpublished
Lindner et al.
The case of concentrated system
t=0
t>0
t0
diffusive process ⇒ convection of mass
concentrated systems:
convection/diffusion inter-related
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,...
Controlling transport using "membranes"
(1) dialysis & microfiltration
(pore size 1 - 100 nm)
solutes below the pore size
cross the membrane
(2) pervaporation
solubility and evaporation of the solvent
ex: PDMS for water
(1) Microfiltration for
concentration gradients "without flow"
⇒ two levels system incorporating microfiltration membranes
Morel et al., 2012
(see V. Studer's lecture ?)
Concentration landscapes "without flow"
top level
v
no flow
v
diffusion
⇒ possibility to apply
concentration gradients to
cells without shear flow
Morel et al., 2012
And complex (spatio-temporal)
lanscapes !
Morel et al., 2012
Another recent & similar study
microfluidic chip made of "agarose gel"
diffusion of solute, no flow
Palacci et al., 2010
(2) Control by pervaporation
PDMS membrane: permeability to water (and to other solvents too)
ve ≈ 20 nm/s for e ≈ 10 µm
⇒ low pumping flow rates: 1 nL/hr for 100x100 µm2
but not negligible !
e
an efficient pump ?
50 µm
µL/hr
≈ 1 cm
Wheeler et al., 2008
Manipulating solutes using pervaporation
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)
i) creating mixtures with 64 ≠ reactants
ii) mixing in a nanoliter volume
iv) osmotic control
iii) store interesting
mixtures in nL plugs
Lau et al. 2007
Continuous pervaporation:
microevaporators
pdms
e
ve
dry air
reservoir
v0
h
v(x)
conservation of flow rate: v0 =
L0
0
L0
h
ve
ve ≈ 20 nm/s for e ≈ 10 µm
v0 ≈ 1 to 100 µm/s , tunable thanks to geometry
v(x) = v0 x/L0 linear profile in the microevaporator
Leng et al. 2006
solutes
Microevaporators for
a controlled concentration
reservoir
v0
diffusion
dominates
convection
dominates
v(x) = v0 x/L0 = x/τe
transition at Pe~1 ⇒ x2 ∼ Dτe
µm-mm
L0
Pe = v(x)x/D
Tunable concentration rate
reservoir
v0 φ0
√
p = Dτe
conservation of solute
v 0 φ0 ≈
L0
dφ
p dt
⇒ continuous and controlled concentration process
of solutes in a nanoliter box
⇒ useful tool to investigate "soft matter"
(1) microfluidic-assisted growth
of colloidal crystals
cm
50 µm
Merlin et al. 2012
plasmonic nanoparticles
(2) engineering
complex materials
Iazzolino & Angly
2012
(3) phase diagram screening
Leng et al. 2007
more informations:
www.lof.cnrs.fr
jean-baptiste.salmon-exterieur@eu.rhodia.com