Fluids - B
Joel Voldman
Massachusetts Institute of Technology
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 1
Outline
> Review of last time
> Poiseuille flow
> Stokes drag on a sphere
> Squeezed-film damping
> Electrolytes & Electrokinetic separations
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 2
Last time
> Surface Tension
• Force at a liquid-fluid interface
τ =η
∂U x
∂y
> Viscosity
• Constitutive property relation
D ∂
= + U ⋅∇
Dt ∂t
shear stress to shear rate
> Material Derivative
• Time derivative taking into effect
convection
> Mass continuity
> Navier-Stokes Equation
• Fundamental relation for
Dρ m
+ ρ m∇ ⋅ U = 0
Dt
DU
η
2
ρm
= −∇P + η∇ U + ∇(∇ ⋅ U ) + ρ m g
Dt
3
Newtonian fluids
> Reynolds Number
• The MOST IMPORTANT
dimensionless number
ρ m LU
Re =
η
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 3
Outline
> Review of last time
> Poiseuille flow
> Stokes drag on a sphere
> Squeezed-film damping
> Electrolytes & Electrokinetic separations
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 4
Poiseuille Flow
> Pressure-driven flow through a pipe
• In our case, two parallel plates
> Velocity profile is parabolic
> This is the most common flow in microfluidics
• Assumes that h<<W
y
h
high P
τW
low P
τW
Umax
Ux
Image by MIT OpenCourseWare.
Adapted from Figure 13.5 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001, p. 329. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 5
Solution for Poiseuille Flow
> Assume
•
•
•
•
Incompressible
Steady
η
⎛ ∂U
⎞
2
ρm ⎜
+ U ⋅∇U ⎟ = −∇P + η∇ U + ∇ ( ∇ ⋅ U ) + ρ m g
3
⎝ ∂t
⎠
Ux only depends on y
Ignore gravity
> Assume a uniform pressure
gradient along the pipe
> Result is Poisson’s eqn
> Boundary conditions:
dP
= −K
dx
∂ 2U x
K
=−
2
η
∂y
• Relative velocity goes to zero
at the walls
» no-slip boundary
condition
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 6
Solution for Poiseuille Flow
> Integrate twice to get solution
> Maximum velocity is at center
> Can get linear flowrate [m/s] and
>
volumetric flowrate [m3/s]
Can get lumped resistor using the fluidic
convention
> Note STRONG dependence on h
> This relation is more complicated when
1
Ux =
⎡⎣ y ( h − y ) ⎤⎦ K
2η
U max
h2
K
=
8η
Wh3
Q = W ∫ U x dy =
K
12η
0
h
the aspect ratio is not very high…
y
h
Umax
Ux
Image by MIT OpenCourseWare.
Adapted from Figure 13.5 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 329. ISBN: 9780792372462.
ΔP = effort = KL
12ηL
Q
ΔP =
3
Wh
12ηL
⇒ RPois =
Wh 3
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 7
Development Length
> It takes a certain characteristic length, called the
development length, to establish the Poiseuille velocity
profile
> This development length corresponds to a development
time for viscous stresses to diffuse from wall
> Development length is proportional to the characteristic
length scale and to the Reynolds number, both of which
tend to be small in microfluidic devices
L
* ≈ Re
η
U
LD ≈ (time)U ≈ Re L
2
L
time ≈
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 8
A note on vorticity
> A common statement is to say
that laminar flow has no vorticity
> What is meant is that laminar flow
has no turbulence
> Vorticity and turbulence are
different
> Can the pinwheel spin?
• Then there is vorticity
> Demonstrate for Poiseuille flow
ω = ∇×U
Vorticity
1
[ y(h − y )]K
Ux =
2η
∂U x
∂U x
− nz
ω = ny
∂z
∂y
K
(h − 2 y )
ω = −n z
2η
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 9
Outline
> Review of last time
> Stokes drag on a sphere
> Squeezed-film damping
> Electrolytes & Electrokinetic separations
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 10
Stokes Flow
> Steady-state flow in which
inertial effects can be neglected,
ReÆ0
> The result is a vector Poisson
equation
DU
When ρ m
can be neglected
Dt
> Also called “creeping flow”
> Action is instantaneous
• No “mass” in system
• Incompressible: no springs
η∇ 2 U = ∇ P *
> This is a typical approximation
made in microfluidics
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 11
Stokes’ drag on a sphere
> In creeping flow, one can solve for
z
the flow field around a sphere
placed in an initially uniform flow
field
θ
R
r
> This can be used to find the
stresses on the sphere and sum
them to find the total drag
y
φ
x
> This is called the Stokes’ drag
> This is often the predominant
particle force in microfluidic
systems
> See Deen’s text for derivation
Vz = U
Image by MIT OpenCourseWare.
Fd = 6πηRU
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 12
Stokes’ drag on a sphere
> This is strictly valid ONLY in a uniform
flow
> These are hard to make…
> Instead, we can use Ux(y) of the
parabolic flow profile to calculate a
height-dependent drag force
> This approach fails when the particle is
too big
> Instead, take advantage of published
solutions
• Shear flow: Goldman et al., Chem. Eng.
Sci. 22, 653 (1967).
U ( y)
• Poiseuille flow: Ganatos et al. J. Fluid
Mech. 99, 755 (1980).
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 13
Outline
> Review of last time
> Stokes drag on a sphere
> Squeezed-film damping
> Electrolytes & Electrokinetic separations
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 14
Squeezed-Film Damping
> This is how we will get our b (or R) for the parallel-plate actuator
> The result of motion against a fluid boundary
• If the fluid is incompressible, there can be a large pressure rise, so
•
large back forces result
If the fluid is compressible, it takes finite motion to create a
pressure rise
> In either case, the dissipation due to viscous flow provides a
damping mechanism for the motion
> This is related to “lubrication theory”
F
Moveable
h(t)
Fixed
Image by MIT OpenCourseWare.
Adapted from Figure 13.6 in Senturia, Stephen D. Microsystem Design. Boston,
MA: Kluwer Academic Publishers, 2001, p. 333. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 15
The Reynolds Equation
> Assumptions for compressible
isothermal squeezed-film
damping
• One-dimensional pressure
•
•
•
•
•
•
gradient: P(r,t) = P(y,t) only
» No pressure gradient in z or
along plate (x)
Stokes flow
Poiseuille flow profile in the plane
Ideal gas law
Isothermal (temperature rise due
to compression is small, and heat
flow to the walls is rapid)
No-slip BC’s
Rigid plate: h(r,t) = h(t) only
> The result is a version of the
∂ ( Ph) h3 ⎛ 1 2 2 ⎞
=
⎜ ∇ P ⎟
∂t
12η ⎝ 2
⎠
z
y
x
h
L
W
Image by MIT OpenCourseWare.
Adapted from Figure 13.7 in Senturia, Stephen D. Microsystem Design. Boston,
MA: Kluwer Academic Publishers, 2001, p. 334. ISBN: 9780792372462.
Reynolds equation
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 16
Example: Rigid Plate Damping
> Now, assume small motions
Î Linearize
> The result is (guess what!) the
heat-flow equation
∂ ( Ph) h3 ⎛ 1 2 2 ⎞
=
⎜ ∇ P ⎟
∂t
12η ⎝ 2
⎠
If we linearize:
h = h0 + δ h P = Po + δ P
z
y
x
h
Normalize: ξ =
L
W
Image by MIT OpenCourseWare.
Adapted from Figure 13.7 in Senturia, Stephen D. Microsystem Design. Boston,
MA: Kluwer Academic Publishers, 2001, p. 334. ISBN: 9780792372462.
y
W
pˆ =
δP
Po
h02 Po ∂ 2 pˆ h
∂pˆ
=
−
2
2
ho
∂t 12ηW ∂ξ
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 17
Suddenly Applied Motion
> We already solved a very
similar problem
• Impulse of heat into 1-D
resistor
⎡
⎢ 96η LW 3
F ( s) = ⎢
4 3
⎢ π ho
⎢⎣
2
αn =
> Now we have a velocity
impulse: sudden change of
height
⎤
1 1 ⎥
⎥ sz ( s )
∑
4
s
n odd n
1+ ⎥
α n ⎥⎦
2 2
n π ho Po
= ωn
2
12ηW
n=1
96η LW 3
b=
π 4 ho3
> We get a series of 1st-order
terms, as before
π 2 ho2 Po
ωc =
12ηW 2
> Only need 1st term, which
is an RC circuit
> R for viscous damping
> C for gas compressibility
> Details are in the book
R=b
C=
1
bωc
Image by MIT OpenCourseWare.
Adapted from Figure 13.8 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 337. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 18
Outline
> Review of last time
> Stokes drag on a sphere
> Squeezed-film damping
> Electrolytes & Electrokinetic separations
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 19
Electrokinetic Phenomena
> It is a coupled-domain problem in which electrostatic forces
result in fluid flow (and vice versa)
> Start with electrolytes, move into double layer, and finally show
how to manipulate the double layer
> It is the driver behind ALOT of early micro-TAS work
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 20
Electrolytes
> Electrolytes: liquids with
mobile ions
Ion mobility (cm2/V-s)
zi
Ni =
ui Ci E − Di ∇Ci
zi
Flux (cm-2-s-1)
• Examples: water, PBS
> Ions can move via
concentration gradients
(diffusion) or electric fields
(drift)
> Macroscopically, the liquid is
approximately charge-neutral
(called quasineutrality)
Valence
Diffusivity (cm2/s)
Concentration (cm-3)
ρe = ∑ zi qeCi ≈ 0 in the bulk
i
In neutral regions: ∇ 2φ =
− ρe
ε
=0
N + = u+ C+ E − D+ ∇C+
For a binary electrolyte
(e.g., NaCl in water)
N − = −u−C− E − D−∇C−
ρ e = qe (C+ − C− ) ≈ 0
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 21
Electrolytes
> Surfaces with fixed
charge can lead to net
space charge in the
liquid
> Diffusion competes
with drift, and at
equilibrium, Boltzmann
distribution follows
> This leads to the
Poisson-Boltzmann
equation
Glass
OOOOOO-
Glass
OOOOOO-
+ -+ -+
+ + + + + + -+ -+
+ - + +
+ + +
C+ C-
+ +
+ + - + +
+ +
+ +
+ + + + - + + + +
C+
C-
Ni =
zi
ui Ci E − Di ∇Ci = 0 at equilibrium
zi
⇒ Ci ( x ) = Ci ,o e
ρ
Near the wall: ∇ 2φ = − e
ε
−
zi qe (φ ( x ) −φo )
k BT
⇒ ∇ 2φ = −
1
zqC
∑
ε
i e
i ,o
−
e
zi qe (φ −φo )
k BT
i
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 22
Diffuse Double Layer
Express in terms of φˆ = φ − φ0 :
∇ 2 φˆ = −
1
zqC
∑
ε
i e
i ,o
−
e
zi qeφˆ
k BT
i
Expand for small potential variations
φˆ
φw
2
q
⎡
⎤ˆ
2
e
∇ φˆ ≅ − ∑ zi qeCi ,o +
z
C
∑
i
i ,o ⎥ φ
⎢
ε i
ε k BT ⎣ i
⎦
Assuming the reference region is charge neutral
1
∇ 2 φˆ = 2 φˆ
For binary
LD
monovalent
/
z
L
±
electrolyte
φˆ = φ e D
2
LD
1
w
Debye
Length
(e.g., NaCl)
2qe2Ci ,o
qe2 ⎡
1
⎤
2
=
∑ z C = εk T
ε k BT ⎢⎣ i i i ,o ⎥⎦
LD
B
LD=1 nm for
LD ~ 1
Ci ,o 0.1 M NaCl
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 23
Double-layer charge
> We can calculate the total charge in the double layer
> It must balance the charge at the wall
φˆ
φw
Near the wall:
φˆ = φ w e
−
z
LD
T he charge density is
ρe
ε ˆ
ˆ
∇ φ =−
⇒ ρe = − 2 φ
ε
LD
2
Total charge per unit area
LD
in diffuse layer
∞
σ d = ∫ ρ e dz = −
0
DL charge/area
[C/cm2]
εφ w
LD
σ w LD − L
ˆ
e
⇒φ =
ε
z
= −σ w
D
wall charge/area
[C/cm2]
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 24
Actual Double Layers
> The actual situation is MUCH more complicated
> Some ions are tightly or specifically adsorbed, forming the Stern layer
and screening the wall charge
> The rest distribute in a diffuse double-layer: the Gouy-Chapman layer
> This is an active area of research
Stern layer
(up to ~0.2 nm)
Gouy-Chapman layer
(~1-20 nm)
Potential
Shear boundary
Zeta potential
0
Image by MIT OpenCourseWare.
Phase
boundary
(Nernst)
potential
GouyChapman (up to several
potential hundred mV)
Distance from interface
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 25
Electroosmotic Flow
> An axial electric field exerts a
force on the charge in the
diffuse double layer, which
drags the fluid down the pipe
Insulating solid
Electric field
Electrolyte
Net charge in diffuse layer
Insulating solid
Image by MIT OpenCourseWare.
Adapted from Figure 13.11 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers, 2001, p. 343. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 26
Analysis of Electroosmotic Flow
> Assume
•
•
•
•
Creeping flow
One dimensional flow Ux(z)
No pressure drop
Electrical body force
η∇ 2U x = ρe Ex
> Express charge density in
terms of wall surface charge
density
ρe = −
> Set up differential equation
=−
z
−
⎛
d Ux
σ w LD
e
=⎜
2
⎜
dz
⎝ η LD
2
ε ˆ
φ
2
LD
σw
LD
−
e
z
LD
⎞
⎟ Ex
⎟
⎠
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 27
Analysis of Electroosmotic Flow
> This leads to a PLUG-FLOW
profile
~3LD
d 2U x ⎛ σ w − LzD ⎞
=⎜
e ⎟ Ex
2
⎜
⎟
dz
⎝ η LD
⎠
Integrate twice, and use boundary conditions:
Ux ( z = δ ) = 0
Ux(z)
δ
δ
dU x
dz
φˆ(δ ) = ζ
−
σ w LD ⎛ − L
L
Ux = −
⎜e −e
η ⎜⎝
z
D
z
z
σ w LD − L
φˆ ( z ) =
e
ε
D
=0
h/2
⎞
⎟ Ex
⎟
⎠
z
Plug flow
Ux
~3LD
U0
Ux = −
D
ε
ζ − φˆ Ex
η
(
)
Image by MIT OpenCourseWare.
Adapted from Figure 13.12 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 344. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 28
Analysis of Electroosmotic Flow
> This leads to a PLUG-FLOW
profile
> Flow depends on zeta
potential ζ
• The potential at the slip
plane δ
• Which is in a different place
than the wall, the Stern layer,
or LD
• It is what is measured
Ux = −
ε
ζ − φˆ Ex
η
(
)
δ
σ w LD − L
εζ
U0 = −
e Ex = −
E
η
η x
D
for z > 3LD
experimentally
• For h>>LD, one typically
assumes
δ =0
φw = ζ
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 29
Electrophoresis
> This is just electroosmosis
around a solid surface
> Each ionic species has its own
mobility
⎛ εζ
U ep = ⎜⎜
⎝η
⎞
⎟⎟E x = μ epE x
⎠
> Therefore, in an electrolyte in
which there is a net electric field,
ions will drift at various rates
> This is the basis of a separation
technology called electrophoretic
separation
For a
“large”
particle
More
generally
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 30
Electrokinetic separation
> This was THE original driver for micro-TAS
• TAS = total analysis systems
> Create fully integrated microsystems that would
go from “sample to answer”
> The KEY enabler was the integration of nonmechanical valves with the separation column
• This creates extremely narrow sample plugs
> Also important is the ability to multiplex
Image removed due to copyright restrictions.
> The actual sample prep was (and is) usually
ignored
> This is/was the raison d’être of Caliper & Aclara
> Aclara died, unclear if Caliper will succeed
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 31
Electrokinetic separation
> We use a channel-crossing
structure to select a sample
plug
> Then we switch voltages to
drag the plug down a
separation column
> In one approach, EO and EP co-
Ls
μEO
μEP,1
μEP,2
exist, but EO dominates
> Different species travel at
different rates Î separation
Ls = t sep μ EP ,1 − μ EP , 2 E x
Separation time
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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Schematic Illustration
> The key discovery was that
liquid samples could be
controlled with voltages
V1 (+)
VJ
Injection
V3 > VJ
> This allows one to valve
V4 > VJ
V2 = 0
and pump liquids
V1 < VJ
• Create small sample plugs
Injected sample plug
V3 (+)
V4 = 0
U0
V2 < VJ
V1 < VJ
V3 (+)
ORNL movie
Slower component
U0
V2 < VJ
U0
V4 = 0
Faster component
Image by MIT OpenCourseWare.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 15 - 33
Electrokinetic separation
> The “macro” technology is
“conventional” capillary
electrophoresis
• Most notably used in the
human genome project
Ls
> Sample loading is the problem
> To separate two species
(ignoring diffusion), we need
> Smaller starting W means we
Ls ≥ W
W
can use a shorter channel
> And get a faster separation
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 34
Electrokinetic separation
> Conventional capillaries
Richard Mathies (UCB)
have larger W because
injection is not integrated
• Though this is always
getting better
> This thus requires a
longer channel
> Microfab also allows for
integration
Courtesy of Richard A. Mathies. Used with permission.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 35
An aside on electrodes
> Current in the electrolyte is carried by ions
> Current in the wire is carried by electrons
> At the surface, something must happen to transfer
this current
> This is electrochemistry and the typical byproduct are
gases Æ bubbles
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 36
Electrokinetics
> That is just the beginning
> Dielectrophoresis
• Force on dipoles in non-uniform electric fields
• Can use AC fields
• Can hold things in place
> Other phenomena
• Electrohydrodynamics
• Electrowetting
• Induced-charge electrophoresis/electroosmosis
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 37
Comparing EOF & pressure-driven flow
> If you are designing a chip that needs to move liquids
around, which method is best?
> Issues to consider
•
•
•
•
•
Flowrate scaling
Liquid composition
System partitioning
Materials
Species Transport
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 38
Comparing EOF & pressure-driven flow
> Flowrate
• Water in rectangular SiO2 channel
• h varies, W=1000 μm, L=2 cm, ε=80ε0, ζ=50 mV
• Drive with Ex=100 V/cm, ΔP=5 psi
εζ
Ex
η
εζ hW
Q = U 0 hW = −
Ex
η
U0 = −
Wh3
ΔP
Q=
12η L
Q
h2
ΔP
U0 = =
A 12η L
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 39
Comparing EOF & pressure-driven flow
> Scaling
• Pressure-driven flow
2
larger in large channels
10
» Due to cubic
0
10
dependence of flow
resistance
-2
channels
Q (μl/s)
• EO flow larger in small
10
EO
-4
10
> Both scale equivalently
with channel length
• Larger L requires more
voltage and higher
pressure to get same flow
-6
10
Poiseuille
-8
10 -6
10
-5
-4
10
10
-3
10
h (μm)
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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Comparing EOF & pressure-driven flow
> Other issues typically matter more
> Valving
• EOF has “built-in” valving using E-fields
• Poiseuille needs mechanical valves
> Liquid limitations
• Poiseuille flow can pump any liquid
• EOF ionic strength limits
» Debye length depends on 1/√C0
» Increasing ionic strength decreases LD and thus
•
EOF
» Typically use ~10-100 mM salt buffer
EOF pH limits
» pH affects wall charge Æ affects EOF
» Typically use pH ~7 buffers
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 41
Comparing EOF & pressure-driven flow
> Materials issues
• Poiseuille flow can use any material
• EOF requires defined (and stable) surface charge
» Best is silica (or at least glass)
» Polymers are more difficult
» Surface charge can change as molecules
adsorb, etc.
> System partitioning
• Both approaches involve off-chip components
» Electrodes for EOF
» Pumps for Poiseuille flow
• Integrating complete lab-on-a-chip usually BAD idea
» EOF: Easier to use external electronics
» Poiseuille: Pumps hard to make on-chip
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 42
Comparing EOF & pressure-driven flow
> Transport limitations
• EOF
» Will separate molecules as they are convected
downstream
» Species in flow must tolerate E-fields (DNA/proteins
OK, cells not so good)
» Plugs remain plugs
• Poiseuille flow
» Will not separate molecules
» Objects in flow must tolerate shear/pressure
(DNA/proteins OK, cells OK depending on
shear/pressure)
» Will distort plugs
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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JV: 2.372J/6.777J Spring 2007, Lecture 15 - 43
What’s next
> We will discuss the behavior of the stuff in the liquids
> How to manipulate that stuff at the microscale
> And then we head into system-level issues
• Feedback, Noise, etc.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
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