3.155J/6.152J Lecture 20: Fluids Lab Testing
Prof. Martin A. Schmidt
Massachusetts Institute of Technology
11/23/2005
Outline
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Review of the Process and Testing
Fluidics
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Solution of Navier-Stokes Equation
Solution of Diffusion Problem
Lab Report Guidance
References
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Senturia, Microsystems Design, Kluwer
6.021 Web Site on Microfluidics Lab
Plummer, Chapter 7, p.382-384
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 2
Process Flow - Overview
Unexposed
SU-8 (100 µm)
Si
Surface treatment &
casting PDMS
photolithography
PDMS
UV light
Si
mask
removing elastomer from
master
Si
PDMS
development
seal against glass after plasma
treatment and insert tubing
Si
“master”
tubing
Fall 2005 – M.A. Schmidt
Our process
was changed
here
3.155J/6.152J – Lecture 20 – Slide 3
The Mixer
Width = 250µm, 500 µm
Depth = 100 µm
Inlet Length = 25 mm
Outlet Length = 35 mm
Courtesy of Dennis Freeman.
Courtesy of Dennis Freeman.
Images: Prof. D. Freeman
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 4
Packaging/Testing
Courtesy of Dennis Freeman.
Courtesy of Dennis Freeman.
Images: Prof. D. Freeman
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 5
Experiment
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Gravity feed of fluids
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Particles for velocity measurement
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Requires ‘priming’ of channel
We will attempt this
Dye for diffusion experiments
Measurements
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Particle velocity
Diffusion
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 6
Navier-Stokes
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The Navier-Stokes equation for
incompressible flow:
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U = velocity
P* = pressure (minus gravity body force)
ρm = fluid density (103 kg/m3 for water)
η = viscosity (10-3 Pa-s for water)
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 7
Poiseuille Flow
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Assume width (w) >> height (h)
Neglect entrance effects (L >> h)
L
w
h
y
x
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 8
Simplify to our problem
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No time dependence
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Flow is constant in xdirection (and 0 in z)
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d/dt = 0
U = f(y)
Pressure is only a
function of x
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A linear pressure drop
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 9
Poiseuille Flow
ILLUSTRATING POISEUILLE FLOW
y
h
τw
High P
Low P
τw
Ux
Umax
Figure by MIT OCW.
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‘No-Slip’ Boundary conditions
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Ux(y=0) = 0
Ux(y=h) = 0
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 10
Solution
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Solution is a quadratic polynomial
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Ux = a + by + cy2
Using boundary conditions and
substitution
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 11
Parabolic Flow Profile
ILLUSTRATING POISEUILLE FLOW
y
h
τw
High P
Low P
τw
Ux
Umax
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Maximum velocity
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Flow rate
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Average velocity
Fall 2005 – M.A. Schmidt
Figure by MIT OCW.
3.155J/6.152J – Lecture 20 – Slide 12
Pressure drop over length
∆P = ρgH
H = height of water
g = gravity
Courtesy of Dennis Freeman.
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 13
Flow Issues
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Edge effects
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Flow rate
Particle location in channel
Dimensions
Merging of channels
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How to model
Courtesy of Dennis Freeman.
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 14
The Mixer – Mixing by diffusion
Width = 250µm, 500 µm
Depth = 100 µm
Inlet Length = 25 mm
Outlet Length = 35 mm
Courtesy of Dennis Freeman.
Courtesy of Dennis Freeman.
Images: Prof. D. Freeman
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 15
Diffusion Image Sequence
Think of this axis as length or time
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 16
Imaging System Output
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 17
Diffusion
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Same problem as diffusion in an epi layer
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As in the case of the design problem
Dopant
Concentration
n - epi
n+ - silicon
Solution in Plummer, Chapter 7, p.382
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 18
Solution
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Initial Conditions
C
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Initial Profile
Identical to Infinite
Source Problem:
Diffused
Profile
x
0
Figure by MIT OCW.
Reference: Plummer, J., M. Deal, and P. Griffin. Silicon VLSI Technology: Fundamentals,
Practice, and Modeling. Upper Saddle River, NJ: Prentice Hall, 2000. ISBN: 0130850373.
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 19
An ‘Intuitive’ way to look at it…
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Think of the uniform
concentration as a
Figure removed for copyright reasons.
sum of dopant
Refer to Plummer, J., M. Deal, and P. Griffin. Silicon VLSI Technology:
‘pulses’
Fundamentals, Practice, and Modeling. Upper Saddle River, NJ: Prentice
Hall, 2000. ISBN: 0130850373.
Each ‘pulse’ has a
Gaussian diffusion
profile
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Dose = C ∆x
Apply superposition
since diffusion is
linear
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 20
Solution
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Taking the limit of ∆x
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 21
Solution
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 22
Error Function Solution
Figure removed for copyright reasons.
Refer to Plummer, J., M. Deal, and P. Griffin. Silicon VLSI Technology: Fundamentals, Practice,
and Modeling. Upper Saddle River, NJ: Prentice Hall, 2000. ISBN: 0130850373.
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 23
Diffusion Issues
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Fitting ideal curve to measured profiles
Scaling time to position
Choice of velocity
Non-ideal flow profiles
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 24
Fluids Lab Report
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Follow the Letters format
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Report Flow Velocity
Compare to calculated
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Estimate errors
Extract an effective diffusion coefficient
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Purpose: Characterization of a Liquid
Micromixer
Utilize ‘best estimate’ for flow velocity
Compare to expected (D ~ 2x10-6 cm2/s)
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Identify relevant non-idealities
Fall 2005 – M.A. Schmidt
3.155J/6.152J – Lecture 20 – Slide 25