mass transport phenomena in microfluidic devices

7/23/2019 Mass transport phenomena in microfluidic devices

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(Mass) transport phenomena in

microfluidic devices(some selected examples...)

Jean-Baptiste SalmonLOF, Pessac, France

www.lof.cnrs.fr



Basicsmicrofluidics, solute, transport equation,...

Coflowslow mixing, gradient, role of gravity,...

Hydrodynamic dispersionscoflow, Taylor Aris dispersion, sensors

Mixingsmall size, chaotic mixers, droplets,...

Membranesfiltration, pervaporation,...



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



∇ 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



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 ?



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



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



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



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



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



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



(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)



...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



(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



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



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



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:

(...)



Basics

microfluidics, solute, transport equation,...



Mixingsmall size, chaotic mixers, droplets,...

Membranesfiltration, pervaporation,...



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



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

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



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



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



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



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



"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



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



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



Mixing is slow:generating microstructures

Kenis et al ., 1999

B k t ll id



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



Boosting migration of colloids:diffusiophoresis

Abécassis et al., 2008high Péclet

v = −Ddp∇logc

Wh t b t it ?



What about gravity ?

thermal or solutal gradients⇒ density gradients ⇒ flows⇒ mixing

and at small scale ?

Buoyancy-induced flow



∇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



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



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



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



Basics


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



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"



Larger widthclose to the walls

Ismagilov et al., 2000

⇒ a problem to obtainvery thin electrodes

Kenis et al ., 1999

From 3d to 2d



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:



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



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



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:



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



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



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:



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:



( )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"



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)



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



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



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"



(fast reaction)

small sensorsλ

< 1

λ > 1

(iv) reaction vs. transport



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



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



Basics


Coflow

slow mixing, gradient, role of gravity,...

Hydrodynamic dispersions

coflow, Taylor Aris dispersion, sensors

Mixing


Membranes


Back to microfluidics:i i i l



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



(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



(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



L ≈ vw2/D = Pe w

?L ≈ v(w/N )2/D = Pe w/N 2

Manz et al. 1999

(2) Multilaminating along the flow



⇒ need for complex 3d structures

Chen et al. 2004

(3) "Grooving" flows: chaotic mixing



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



length for efficient mixing

⇒ "chaotic mixing" (note: Re ≈ 0)

L ∼ wlog Pe

Stroock et al. 2002

Chaotic mixers helpto make compact devices



to make compact devices

Whitesides et al. 2002

(4) Active mixing:the "rotary" mixer

fluidic channels



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?)



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



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 ?



perfect microreactors ?

Song et al. 2003



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



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



Song et al. 2003

L

∼wlog Pechaotic mixing

again Re=0

Evidence for a fast mixing



for a 10X10 µm2 channel

Song et al. 2003

Always true ?silicone oil

≈ 20 ms



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



⇒ studies often concern water & (very) diluted dyes

Real life is again more complex...



(?)

solvants & concentrated solutionscomplex fluids

⇒ rheology ? dissolved gas ? wetting properties ? etc...there is always plenty of space for fundamental studies

Some examples...viscosity contrasts



Cubaut et al. 2006

wetting, degasing...water

ethanol

t = 0

t ~ 1 min

lof, unpublished

viscoelasticity

Lindner et al.

The case of concentrated system



t = 0

t > 0

t 0

diffusive process ⇒ convection of mass

concentrated systems:convection/diffusion inter-related

Basicsmicrofluidics, solute, transport equation,...






Mixing


Membranes


Controlling transport using "membranes"



(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"



⇒ two levels system incorporating microfiltration membranes

Morel et al., 2012 (see V. Studer's lecture ?)

top level

Concentration landscapes "without flow"



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 !



Morel et al., 2012

Another recent & similar studymicrofluidic chip made of "agarose gel"



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)



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



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)



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



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



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



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



cm

50 µm

Merlin et al. 2012

(2) engineeringcomplex materials

plasmonic nanoparticles



Iazzolino & Angly

2012

(3) phase diagram screening



Leng et al. 2007

more informations:www.lof.cnrs.fr

j b ti t l t i @ h di



[email protected]

mass transport phenomena in microfluidic devices

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