3.155J/6.

152J Lecture 20:

Fluids Lab Testing

Prof. Martin A. Schmidt

Massachusetts Institute of Technology

11/23/2005

 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 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 2

 Process Flow - Overview

Unexposed
Si SU-8 (100 µm) Surface treatment &
casting PDMS
photolithography
PDMS
UV light
Si
mask

removing elastomer from

Si master

PDMS
development

seal against glass after plasma

treatment and insert tubing
“master”
Si
tubing Our process
was changed
here

Fall 2005 – M.A. Schmidt 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

„ 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 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 6

 Navier-Stokes

„ The Navier-Stokes equation for

incompressible flow:

„ 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

„ Assume width (w) >> height (h)

„ Neglect entrance effects (L >> h)

L
w

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 8

 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 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 9

 Poiseuille Flow

ILLUSTRATING POISEUILLE FLOW

τw h

High P Low P

τw Ux

Umax

Figure by MIT OCW.

„ ‘No-Slip’ Boundary conditions

„ Ux(y=0) = 0
„ Ux(y=h) = 0
Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 10
 Solution

„ Solution is a quadratic polynomial

„ 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

τw h

High P Low P

τw Ux

Umax
Figure by MIT OCW.

„ Maximum velocity

„ Flow rate

„ Average velocity

Fall 2005 – M.A. Schmidt 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

„ Edge effects

„ Flow rate
„ Particle location in channel
„ Dimensions
„ Merging of channels
„ 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

„ Same problem as diffusion in an epi layer

„ As in the case of the design problem

Dopant
n - epi Concentration

n+ - silicon

Solution in Plummer, Chapter 7, p.382

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 18

 Solution

„ Initial Conditions
C
Initial Profile

„ Identical to Infinite
Diffused
Source Problem: 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…

„ Think of the uniform

concentration as a
sum of dopant Figure removed for copyright reasons.

‘pulses’ Refer to Plummer, J., M. Deal, and P. Griffin. Silicon VLSI Technology:
Fundamentals, Practice, and Modeling. Upper Saddle River, NJ: Prentice

Each ‘pulse’ has a

Hall, 2000. ISBN: 0130850373.
„
Gaussian diffusion
profile
„ Dose = C ∆x
„ Apply superposition
since diffusion is
linear
Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 20
 Solution

„ 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

„ 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

„ 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 cm2/s)
„ Identify relevant non-idealities

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 25