3.155J/6.152J Lecture 22,
Fluids Lab Testing
Prof,Martin A,Schmidt
Massachusetts Institute of Technology
12/1/2003
Information
Quiz
Wednesday,Dec,3,in-class
Closed Book
Covers materials in lectures from 10/15 through
11/26
Does not include Lab Lectures
A formula sheet will be provided (if needed)
Lecture on Monday,Dec,8
th
Lab tour of Analog Devices MEMS Facility
We will leave from the classroom at 2:35PM SHARP
Course evaluations at the end of this lecture
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 2
Outline
Review of the Process and Testing
Fluidics
Solution of Navier-Stokes Equation
Solution of Diffusion Problem
Lab Report Guidance
References
Senturia,Microsystems Design,Kluwer
6.021 Web Site on Microfluidics Lab
Plummer,Chapter 7,p.382-384
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 3
Process Flow - Overview
Si
Unexposed
SU-8 (100 μm)
Surface treatment &
casting PDMS
photolithography
UV light
Si
mask
Si
PDMS
removing elastomer from
master
development
Si
ing
“master”
here
PDMS
seal against glass after plasma
treatment and insert tub
tubing
Our process
was changed
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 4
The Mixer
Width = 250Pm,500 Pm
Depth = 100 Pm
Inlet Length = 25 mm
Outlet Length = 35 mm
Images,Prof,D,Freeman
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 5
Packaging/Testing
Images,Prof,D,Freeman
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 6
Experiment
Gravity feed of fluids
Requires ‘priming’ of channel
Particles for velocity measurement
We will attempt this
Dye for diffusion experiments
Measurements
Particle velocity
Diffusion
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 7
Navier-Stokes
The Navier-Stokes equation for
incompressible flow,
U = velocity
P
*
= pressure (minus gravity body force)
U
m
= fluid density (10
3
kg/m
3
for water)
K = viscosity (10
-3
Pa-s for water)
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 8
Poiseuille Flow
Assume width (w) >> height (h)
Neglect entrance effects (L >> h)
h
w
L
y
x
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 9
Simplify to our problem
No time dependence
d/dt = 0
Flow is constant in x-
direction (and 0 in z)
U = f(y)
Pressure is only a
function of x
A linear pressure drop
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 10
Poiseuille Flow
‘No-Slip’ Boundary conditions
U
x
(y=0) = 0
U
x
(y=h) = 0
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 11
Solution
Solution is a quadratic polynomial
U
x
= a + by + cy
2
Using boundary conditions and
substitution
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 12
Parabolic Flow Profile
Maximum velocity
Flow rate
Average velocity
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 13
Pressure drop over length
'P = UgH
H = height of water
g = gravity
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 14
Flow Issues
Edge effects
Flow rate
Particle location in channel
Dimensions
Merging of channels
How to model
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 15
The Mixer – Mixing by diffusion
Width = 250Pm,500 Pm
Depth = 100 Pm
Inlet Length = 25 mm
Outlet Length = 35 mm
Images,Prof,D,Freeman
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 16
Diffusion Image Sequence
Think of this axis as length or time
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 17
Imaging System Output
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 18
Diffusion
Same problem as diffusion in an epi layer
As in the case of the design problem
n
+
Dopant
Concentration
- silicon
n - epi
Solution in Plummer,Chapter 7,p.382
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 19
Solution
Initial Conditions
C
Identical to Infinite
Source Problem,
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 20
An ‘Intuitive’ way to look at it…
Think of the uniform
concentration as a
sum of dopant
‘pulses’
Each ‘pulse’ has a
Gaussian diffusion
profile
Dose = C 'x
Apply superposition
since diffusion is
linear
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 21
Solution
Taking the limit of 'x
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 22
Solution
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 23
Error Function Solution
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 24
Diffusion Issues
Fitting ideal curve to measured profiles
Scaling time to position
Choice of velocity
Non-ideal flow profiles
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 25
Fluids Lab Report
Follow the Letters format
Purpose,Characterization of a Liquid
Micromixer
Report Flow Velocity
Compare to calculated
Estimate errors
Extract an effective diffusion coefficient
Utilize ‘best estimate’ for flow velocity
Compare to expected (D ~ 2x10
-6
cm
2
/s)
Identify relevant non-idealities
Fall 2003 – M.A,Schmidt 3.155J/6.152J – Lecture 22 – Slide 26