Mass Transport in liquids
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 16 - 1
Outline
> Chemical potential
> Species conservation including convection
> H-filter design & eigenfunction expansion
> Taylor dispersion, the microfluidicist’s enemy
> Mixing
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 16 - 2
Chemical potential
> It comes from
thermodynamics
> Chemical potential
gradients are the driving
force for the movement of
molecules
> It is the electron Fermi level
⎛ ∂W ⎞
⎟
∂
N
⎝
⎠ T ,V
μ =⎜
For an ideal solution:
μ i ( x) = μ i0 + k BT ln
ci ( x)
ci0
in semiconductors
> At equilibrium, there are no
gradients in μ
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 16 - 3
Chemical potential
> We can derive Fick’s first law
from the chemical potential
> First, note that there are two
concentration units
> Relate flux to velocity
> Then relate the velocity to a
force f, using a mobility M
> Then the force to a potential
(P ) gradient
[m2/V-s]
U = μn E =
[m/s]
[V/m]
ci = N ACi
⎡ # ⎤ ⎡ # ⎤ ⎡ mol ⎤
⎢ m 3 ⎥ = ⎢ mol ⎥ ⎢ m 3 ⎥
⎦⎣
⎦
⎣ ⎦ ⎣
J i = ciU i = N ACiU i
U i = Mf = − M
∂P
∂x
[s/kg]
μn
qe
( qe E ) = −
μn
qe
( ∇qeφ )
[N]
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 16 - 4
Chemical potential
> Finally, find flux due to a
chemical potential
gradient
> Can relate diffusivity to
mobility
D
k BT =
M
Einstein Relation
μ i ( x) = μ i0 + k BT ln
ci ( x)
ci0
∂μi
∂ ⎛ c ( x) ⎞
= −ci Mk BT ⎜ ln i 0 ⎟
∂x
∂x ⎝
ci ⎠
∂
J i = −ci Mk BT ( ln ci ( x ) − ln ci0 )
∂x
∂
J i = − Mk BTci ( ln ci ( x ) )
∂x
1 ∂ci
J i = − Mk BTci
ci ∂x
J i = −ci M
∂ci
∂ci
= −D
J i = − Mk BT
∂x
∂x
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 16 - 5
Outline
> Chemical potential
> Species conservation including convection
> H-filter design & eigenfunction expansion
> Taylor dispersion, the microfluidicist’s enemy
> Mixing
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 16 - 6
Species conservation equation
> One more conservation
equation…
V
d
bdV = − ∫ F ⋅ ndS + ∫ gdV
∫
dt
> Flux now includes
convection and diffusion
> Incompressible flow
b
g
∂b
= −∇ ⋅ F + g
∂t
n
F
S
Image by MIT OpenCourseWare.
∂ci
= −∇ ⋅ J i + RVi
∂t
convection
J i = − Di ∇ci + ci U i
diffusion
∂ci
= −∇ ⋅ (− Di ∇ci + ci U i ) + RVi
∂t
∂ci
= Di ∇ 2 ci − ci ∇ ⋅ U i − U i ⋅ ∇ci + RVi
∂t
∂ci
+ U i ⋅ ∇ci = Di ∇ 2 ci + RVi
∂t
Dci
= Di ∇ 2 ci + RVi
Dt
Convection-Diffusion 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 16 - 7
Convective term
> We have seen this equation
before
> We can compare the convective
to diffusive flux terms, and get a
Peclet number again
• Now for diffusive vs. convective
mass transport
> For BSA (66 kDA) in microscale
flows, L~100 μm, U~1 mm/s,
D~7x10-11 m2/s
> Convection is important because
times
molecular diffusivity is
slower than heat diffusivity and
105 times slower than momentum
diffusivity
107
∂ci
+ U i ⋅ ∇ci = Di ∇ 2 ci + RVi
∂t
c
convection U i ⋅ ∇ci U L LU
~
~
~
2
c
diffusion
Di ∇ ci D 2
D
L
(
)(
)
LU 10 −4 m 10 −3 m / s
3
Pe =
=
~
10
−11 m 2
D
7 ⋅10
s
Dheat
~10-4 m2/s for water
Dmomentum~10-6 m2/s for water
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 16 - 8
Diffusivities
> How can we get diffusivities for
different objects?
> Use mobility due to Stokes drag
> Result is Stokes-Einstein
relation
> Larger particles have smaller
diffusivity
> Often used to get an effective
Ui
k BT
D = Mk BT =
f
Ui
1
=
f = 6πηRU i ⇒
f
6πηR
D = Mk BT =
k BT
6πηR
R=45 nm
Rh=44.8 nm
radius (Rh) for a species
Image removed due to copyright restrictions.
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 16 - 9
Outline
> Chemical potential
> Species conservation including convection
> H-filter design & eigenfunction expansion
> Taylor dispersion, the microfluidicist’s enemy
> Mixing
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 16 - 10
H-filter
> What are the minimum
diffusivity differences that
we can separate?
> How to choose channel
width, length, flowrate
Image removed due to copyright restrictions.
Courtesy of Paul Yager, Thayne Edwards, Elain Fu, Kristen Helton,
Kjell Nelson, Milton R. Tam, and Bernhard H. Weigl.
Used with permission. Please see:
Yager, P., T. Edwards, E. Fu, K. Helton, K. Nelson, M. R. Tam,
and B. H. Weigl. "Microfluidic Diagnostic Technologies for Global
Public Health." Nature 442 (July 27, 2006): 412-418.
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 16 - 11
H-filter
> First, let’s try a quick and
dirty diffusion calculation
> Assume 1-D diffusion
across width of channel
> Ignore convection effects
along length of channel
> No generation terms
> Result suggests that
separation will go as √D
∂ci
+ U i ⋅ ∇ci = Di ∇ 2 ci + RVi
∂t
∂ci
∂ 2 ci
= Di 2
∂t
∂x
ci
ci
~ Di 2
τ
δ
δ ~ Diτ
δ ~ Di
L
U
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 16 - 12
H-filter
> Can we do better?
> Yes, using eigenfunction
analysis
> Assumptions
• Ignore convection
• No generation
• No concentration gradients
along channel height or
length
∂ci
+ U i ⋅ ∇ci = Di ∇ 2 ci + RVi
∂t
∂ci
= Di ∇ 2 ci
∂t
∂c
∂ 2c
=D
∂t
∂x 2
» 1-D diffusion
L
• One dilute component in
solvent
z
h
W
y
x
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JV: 2.372J/6.777J Spring 2007, Lecture 16 - 13
H-filter
> Initial condition
• Solute initially fills part of
channel
L
> Boundary condition
• No solute flux through walls
W
Initial condition:
c0
x
0
h
d
W
⎧c0 for 0 < x < d
c( x,0) = ⎨
⎩0 for d < x < W
Boundary condition:
∂c
∂x
= 0 for all t
x = 0 ,W
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JV: 2.372J/6.777J Spring 2007, Lecture 16 - 14
H-filter
> First, separate variables
> Time response is exponential
> Spatial eigenfunctions are
∂Y
= −α Y ⇒ Y = e −α t
∂t
sinusoids
d 2Cˆ
D 2 = −α Cˆ
dx
> Must include DC term in series
⎛
⎞
2
2
ˆ
C ( x) = a0 + ∑ ⎜⎜ An
sin kn x + Bn
cos kn x ⎟⎟
W
W
n =1 ⎝
⎠
∞
∂c
∂ 2c
=D 2
∂t
∂x
c( x, t ) = Cˆ ( x)Y (t )
kn2 =
αn
D
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JV: 2.372J/6.777J Spring 2007, Lecture 16 - 15
H-filter
> Sine does not meet BCs
∂c
∂x
> Cosine does
x =0
dCˆ
=
dx
=0
x =0
⎛
⎞
2
2
0 = ∑ ⎜⎜ An kn
cos kn 0 − Bn kn
sin kn 0 ⎟⎟
W
W
n =1 ⎝
⎠
⇒ An = 0
∞
∞ ⎛
⎞
2
2
ˆ
sin kn x + Bn
cos kn x ⎟⎟
C ( x) = a0 + ∑ ⎜⎜ An
W
W
n =1 ⎝
⎠
∂c
∂x
x =W
∞
dCˆ
=
dx
0 = ∑ − Bn kn
n =1
nπ
⇒ kn =
W
=0
x =W
2
sin knW
W
for n = 1, 2,3,…
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 16 - 16
H-filter
> Finally, use eigenfunction expansion to
meet initial concentration profile
∞
c( x, t ) = a0 + ∑ Bn
n =1
2
cos kn x ⋅ e −α nt
W
t=0
∞
c( x, 0) = a0 + ∑ Bn
n =1
⎧c0 for 0 < x < d
2
cos kn x = ⎨
W
⎩0 for d < x < W
multiply both sides by eigenfctn & integrate
∞
2
2
2
2
=
+
c
x
k
xdx
a
k
xdx
B
k
x
(
,
0)
cos
cos
cos
cos km xdx
∑
0
m
m
n
n
∫0
∫
∫
W
W
W
W
n =1 0
0
W
W
W
extract coefficient
W
Bn = ∫ c( x, 0)
0
2
cos ( kn x ) dx
W
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JV: 2.372J/6.777J Spring 2007, Lecture 16 - 17
H-filter
> Get coefficients and
W
Bn = ∫ c( x, 0)
DC term
0
2
cos ( kn x ) dx
W
d
W
⎤
2 ⎡
Bn =
⎢ ∫ c0 cos ( kn x ) dx + ∫ 0 cos ( kn x ) dx ⎥
W ⎣0
d
⎦
d
2 c0
Bn =
sin ( kn x ) 0
W kn
Bn =
2 c0
sin ( kn d )
W kn
for n = 1, 2,3,...
cd
1
a0 = ∫ c( x, 0)dx = 0
W 0
W
W
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 16 - 18
H-filter
> Can plot time evolution
for d=W/2
> Lowest-order mode
(n=1) is dominant
2c0 ⎛ nπd ⎞ ⎛ nπx ⎞ −α nt c0 d
sin ⎜
⎟ cos⎜
⎟⋅e +
W
⎝ W ⎠ ⎝ W ⎠
n =1 nπ
∞
c ( x, t ) = ∑
2
⎛ nπ ⎞
αn = ⎜ ⎟ D
⎝W ⎠
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 16 - 19
H-filter
> What we’d like to
know is how
separation scales
with D, t, etc.
W
cout
> We can determine the
concentration of
solute in output
channel
> Solve for case of
d=W/2
1
c( x, t )dx
=
∫
W −d d
cout
2
=
W
W
∫
W /2
2c0
⎛ nπ
sin ⎜
⎝ 2
n =1 nπ
∞
c ( x, t ) = ∑
c( x, t )dx
⎞
⎛ nπ x ⎞ −α nt c0
cos
⎟
⎜
⎟⋅e +
2
⎠
⎝ W ⎠
n +1
2c0
⎛ nπ x ⎞ −α nt c0
2
1
cos
−
c ( x, t ) = ∑
( )
⎜
⎟⋅e +
2
⎝ W ⎠
n odd nπ
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 16 - 20
H-filter
> Only focus on 1st mode
• Simplifies math
• Is dominant mode
c ( x, t ) ≈
> First mode has error at
⎛ πx ⎞
cos⎜ ⎟ ⋅ e
π
⎝W ⎠
2c0
2
⎛π ⎞
−⎜ ⎟ Dt
⎝W ⎠
+
c0
2
⎛π ⎞
⎛ 2c
⎞
−⎜ ⎟ Dt
x
c
π
2
⎛
⎞
W
⎜
0
0 ⎟
⎝ ⎠
⋅
+
cout =
e
dx
cos
⎜
⎟
t=0
∫
⎟
⎜
W W /2 π
2
⎝W ⎠
⎝
⎠
• Need other terms to meet
2
I.C.
W
⎛π ⎞
⎡
⎤
−
⎜ ⎟ Dt
c
W
x⎞
π
2 2c0W
⎛
cout = ⎢ 2 sin⎜ ⎟ ⋅ e ⎝ W ⎠ + 0 ⎥
W⎢ π
4 ⎥
⎝W ⎠W
2
⎣
⎦
2
W
cout
⎛π ⎞
2 ⎡ −2c0W −⎜⎝ W ⎟⎠
= ⎢
⋅e
2
W⎢ π
⎣
cout
2
Dt
⎛π ⎞
c0 ⎡
8 −⎜⎝ W ⎟⎠
= ⎢1 − 2 ⋅ e
2⎢ π
⎣
c0W ⎤
⎥
+
4 ⎥
⎦
2
Dt
⎤
⎥
⎥⎦
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 16 - 21
H-filter
> Can also look at all modes
at short time
> Result is that increases as
√Dt for short times
2c0
⎛ nπ
sin ⎜
⎝ 2
n =1 nπ
∞
c ( x, t ) = ∑
⎞
⎛ nπ x ⎞ −α nt c0
cos
⎟
⎜
⎟⋅e +
2
⎠
⎝ W ⎠
Take average over
↓
output channel
cout =
c0 4c0
− 2
2 π
1 −α n t
e
∑
2
n odd n
↓?
cout
Dt
≈ 1.1
W2
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 16 - 22
H-filter design implications
> Since cout scales with both D and t, cout,1/cout,2 will be
independent of time at short times
> If D1>>D2, then increasing time and decreasing W
cout
0.3
helps
• Minimum W is set by
0.25
» Pressure drop increases 0.2
» Clogging and bubbles
1st mode
0.15
√Dt approx
0.1
0.05
0
0
Complete solution
0.5
1
time
1.5
2
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JV: 2.372J/6.777J Spring 2007, Lecture 16 - 23
Outline
> Chemical potential
> Species conservation including convection
> H-filter design & eigenfunction expansion
> Taylor dispersion, the microfluidicist’s enemy
> Mixing
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 16 - 24
Taylor dispersion
> Was ignoring convection OK?
• Not really
> One can solve the 1-D
convection-diffusion problem
> This is called Taylor Dispersion
• Axial convection + transverse
diffusion
> The result is that the plug
spreads out faster than from
simple diffusion
> The apparent diffusivity is K
> EOF does NOT suffer from
Taylor dispersion
• Uniform flow field
⎛ Pe 2 ⎞
U 2h2
⎟⎟
= Di ⎜⎜1 +
K i = Di +
210 Di
⎝ 210 ⎠
Parallel-plate flow channel
U 2h2 ⎛ h ⎞
K i = Di +
f⎜ ⎟
210 Di ⎝ W ⎠
Rectangular flow channel
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 16 - 25
Taylor dispersion
> Can determine Ki for
8.5hW
⎛h⎞ W
f⎜ ⎟≈
2
2
⎝ W ⎠ h h + 2.4hW + W
rectangular channels
> As h/WÆ0, f(h/W)Æ ~7.95
NOT 1
• Because of 2-D profile at
10
wall
> But this means a smaller
channel cross-section
and higher U, therefore
possibly more dispersion
Chatwin solution
Approximate model
Two-dimensional result
Isotropic etch profile
Double etch profile
9
8
> This implies that for a
7
f (h/W)
given h, bigger h/W is
better Æ area small
h
W
Approximation
6
5
4
Exact solution
3
2
Parallel plate sol'n
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
h/W
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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 16 - 26
Convection, diffusion, and mixing
> We can use convection for good as well as evil
> At steady state, fluid mixing time turns into distance
> Short distances from inlet, two fluids appear not to mix
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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 16 - 27
Outline
> Chemical potential
> Species conservation including convection
> H-filter design & eigenfunction expansion
> Taylor dispersion, the microfluidicist’s enemy
> Mixing
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 16 - 28
Mixing
> Mixing is driven by diffusion
> Macroscale mixing uses
turbulence (e.g., stirring) to
reduce length for diffusive mixing
> In liquid microfluidics, there is no
turbulence to decrease mixing
lengths
Image removed due to copyright restrictions.
Cover of Science 285, no. 5425 (July 2, 1999): 1-156.
THEREFORE,
> Microfluidic mixing is EASY
> Microfluidic mixing is HARD
> Mixing length scales with Pe
W2
L ~U
~ Pe ⋅ W
D
τ ≈ 2.5 s for a 50 μm channel ( D = 10 -5 cm 2 /s)
τ ≈ 40 s for a 200 μm channel ( D = 10-5 cm 2 /s)
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JV: 2.372J/6.777J Spring 2007, Lecture 16 - 29
Mixing
> How does one define mixing?
> No universal definition
> One definition:
• When concentration profile is
Δcmax = c(0, t ) − c(W , t )
=
uniform to within 1% (or 5%)
> For our rectangular channel,
concentration difference is
biggest between x=0 and x=W
2c
⎛ nπd ⎞ ⎛ nπx ⎞ −α nt c0 d
c( x, t ) = ∑ 0 sin ⎜
⎟ cos⎜
⎟⋅e +
π
n
W
W
W
⎝
⎠ ⎝
⎠
n =1
∞
[
]
2c0 ⎛ nπd ⎞
n
−α t
=∑
sin ⎜
⎟ 1 − (− 1) ⋅ e n
⎝ W ⎠
n =1 nπ
∞
4c0 ⎛ nπd ⎞ −α nt
sin ⎜
⎟⋅e
∑
⎝ W ⎠
n odd nπ
⎛ πd ⎞
sin ⎜ ⎟ ⋅ e
π
⎝W ⎠
Δcmax
= 0.01
d
c0
W
⎛ W 2 ⎞ 1 ⎡ 400 W
⎛ πd ⎞⎤
⎟⎟ 2 ln ⎢
= ⎜⎜
sin ⎜ ⎟⎥
⎝ W ⎠⎦
⎣ π d
⎝ D ⎠π
Δcmax ≈
Tmix
2
⎛π ⎞
−⎜ ⎟ Dt
⎝W ⎠
4c0
2
⎛ nπ ⎞
αn = ⎜ ⎟ D
⎝W ⎠
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 16 - 30
Mixing
2
Tmix
relative
to W
mixing
time relative
to h2/D
/D
> Mixing time scales as expected for semi-infinite diffusion
Tmix
⎛ W 2 ⎞ 1 ⎡ 400 W
⎛ πd ⎞⎤
⎟⎟ 2 ln ⎢
= ⎜⎜
sin ⎜ ⎟⎥
⎝ W ⎠⎦
⎣ π d
⎝ D ⎠π
Tmix
⎛W 2 ⎞
⎟⎟
≈ 0.5⎜⎜
⎝ D ⎠
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
d/W
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 16 - 31
Mixing
> At the microscale various
approaches exist for
reducing diffusion lengths
• Depends on how fast you
need to mix
> Approaches trade off
fabrication complexity,
generality, mixing time, etc.
> All find ways to laminate
two fluids
Figure 2 on p. 249 in Miyake, R., T. S. J. Lammerink, M. Elwenspoek, and J. H. J.
Fluitman. "Micro-Mixer with Fast Diffusion." In Micro Electro Mechanical Systems,
1993, MEMS '93: An Investigation of Micro Structures, Sensors, Actuators,
Machines and Systems, February 7-10, 1993. New York, NY: Institute of Electrical
and Electronics Engineers, 1993, pp. 248-253. ISBN: 9780780309579. © 1993 IEEE.
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 16 - 32
Mixing
> 3-D split and recombine
lamination
> Complicated fab
> Typical of early designs
that focused on Si
Figure 1 on p. 442 in Branebjerg, J., P. Gravesen, J. P. Krog, and C. R. Nielsen, C.R. "Fast Mixing by Lamination."
In Micro Electro Mechanical Systems, 1996, MEMS '96: An Investigation of Micro Structures, Sensors, Actuators,
Machines and Systems, February 11-15, 1996. New York, NY: Institute of Electrical and Electronics Engineers,
1996, pp. 441-446. ISBN: 9780780329850. © 1996 IEEE.
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 16 - 33
Mixing
> Laminate in one level of
channels by moving complexity
from fab to packaging
Images removed due to copyright restrictions.
Figures 2 and 4 on pp. 266-267 in Jackman, R. J., T. M. Floyd, R. Ghodssi, M. A. Schmidt, and K. F. Jensen.
"Microfluidic Systems with On-line UV Detection Fabricated in Photodefinable Epoxy."
Journal of Micromechanics and Microengineering 11, no. 3 (May 2001): 263-269.
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 16 - 34
Passive chaotic micromixer
> Fairly simple to make
pw
> Uses simple pressure-
A
z
driven flow
∆φm
X
X
pw
∆ φm
h
X
X
h
x
d
c
u
u
c
d
c u
}
}
u c
> Anisotropic boundary
induces anisotropic
flow
> Stroock et al., Science
295(2002):647
z
w
h
y
αh
x
2π/q
B
z
0 Cycles:
1/2 Cycle:
x
x
1 Cycle:
x
∆φm
x
x
100 µm
Image by MIT OpenCourseWare. Adapted from Figure 2 on p. 648 in Stroock, A. D., S. K. W. Dertinger,
A. Ajdari, I. Mezic, H. A. Stone, and G. M. Whitesides. "Chaotic Mixer for Microchannels." Science
New Series, 295, no. 5555 (January 25, 2002): 647-651.
'
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 16 - 35
Passive chaotic micromixer
> Mixing length
A
∆y = 3 cm
scales with ln(Pe)
• Rather than Pe in
pure diffusive
mixing
Cycles 1-5:
0.2 cm
C
1.9
∆y = 3 cm
∆y90 (cm)
E
100 µm
1.7
1.5
1.3
1.1
0.9
0.7
0.5
200 µm
7
8
9
10
11
12
13
Cycle 15:
14
In(Pe)
Images by MIT OpenCourseWare. Adapted from Figure 3 on p. 649 in Stroock, A. D., S. K. W. Dertinger, A. Ajdari, I. Mezic, H. A. Stone, and
G. M. Whitesides. "Chaotic Mixer for Microchannels." Science New Series, 295, no. 5555 (January 25, 2002): 647-651.
,
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 16 - 36
More info
> Microfluidic flow
• Viscous fluid flow, F. White
• Low Reynolds Number Hydrodynamic, Happel & Brenner
• Gravesen et al., “Microfluidics, A Review”, JMME 3(1993) 168
» Includes lumped resistances for turns, constrictions, etc.
• “Life at Low Reynolds Number”, E.M. Purcell
» http://brodylab.eng.uci.edu/%7Ejpbrody/reynolds/lowpurcell.html
• Stone et al., Ann. Rev. Fluid Mech., 36(2004) 381.
> Transport
• Analysis of Transport Phenomena, W. Deen
• Transport Phenomena, Bird, Stewart & Lightfoot
> Taylor Dispersion
• Dutta & Leighton, Anal. Chem., 73(2001), 504
• Chatwin & Sullivan, J. Fluid Mech., 120(1982), 347
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 16 - 37