3.155J/6.152J Lecture 20: Fluids Lab Testing

Prof. Martin A. SchmidtMassachusetts 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

Si Unexposed SU-8 (100 µm) Surface treatment &

casting PDMS

photolithography

UV light

Si

mask Si

PDMS

removing elastomer from master

development

Si “master”

Our process was changed

here

PDMS

tubing

seal against glass after plasma treatment and insert tubing

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

Images: Prof. D. Freeman

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 4

Courtesy of Dennis Freeman.

Courtesy of Dennis Freeman.

Packaging/Testing

Images: Prof. D. Freeman Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 5

Courtesy of Dennis Freeman.Courtesy of Dennis Freeman.

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)

h

w

L

y

x

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

� ‘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

τw

τw

High P Low P

h

y

Umax

Ux

ILLUSTRATING POISEUILLE FLOW

Figure by MIT OCW.

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

� Maximum velocity

� Flow rate

� Average velocity

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 12

τw

τw

High P Low P

h

y

Umax

Ux

ILLUSTRATING POISEUILLE FLOW

Figure by MIT OCW.

Pressure drop over length

∆P = ρgH

H = height of waterg = gravity

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 13 Courtesy of Dennis Freeman.

Flow Issues

� Edge effects� Flow rate

� Particle location in channel � Dimensions � Merging of channels

� How to model

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 14

Courtesy of Dennis Freeman.

The Mixer – Mixing by diffusion

Width = 250µm, 500 µm Depth = 100 µm Inlet Length = 25 mm Outlet Length = 35 mm

Images: Prof. D. Freeman

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 15

Courtesy of Dennis Freeman.

Courtesy of Dennis Freeman.

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

n+

Dopant Concentration

- silicon

n - epi

Solution in Plummer, Chapter 7, p.382Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 18

Solution

� Initial Conditions

� Identical to Infinite Source Problem:

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 19

C

0x

Initial Profile

Diffused Profile

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.

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

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.

Figure removed for copyright reasons.

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

Fall 2005 – M.A. Schmidt 3.155J/6.152J – Lecture 20 – Slide 23

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.

Figure removed for copyright reasons.

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