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1: Debye layer, Zeta potential, Electrokine

2: Electrophoresis, Electroosmosis

3: Dielectrophoresis

4: Inter-Debye layer force, Van-Der Waals

5: Coupled systems, Scaling, Dimensionle

Goals of Part IV:

(1) Understand electrokinetic phenomena

in (natural or artificial) biosystems

(2) Understand various driving forces and

identify dominating forces in coupled s

Key Concepts for section IV (Electrokinet

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The oxide or glass surface

become unprotonated (pK ~ 2)when they are in contact with

water, forming electrical

double layer.

When applied an electric field,

a part of the ion cloud near thesurface can move along the

electric field.

The motion of ions at theboundary of the channel

induces bulk flow by viscousdrag.

Electroosmosis

1 2

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Concentration(c)()

Ficks law of diffusion

E a

Navier-Stokes equation

e, Je : source

Osmosis

(aqueous) medium,

Flow velocity (vm

)

Convection

Electrophoresis

Electroosmosis

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Slip boundary, zeta potential

x

-

+

+

+

+- - - - - - - - - - - - - - - - - - -

+ + + + +

+

+

+

++

++

+

Stern layer

Slip (shear) boundary

(0)

Zeta potential

zE

Stern layer : adsorbed ions, linear potential drop

Gouy-Chapman layer : diffuse-double layer

exponential drop

Shear boundary : vz=0


2 0

0 ( )

e

dvp v E

dt

v incompressible

= + +

=

rr

r

r

New

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Poiseuille flow

parabolic flow profile

( )2 2

( ) ( )4

z zo

electroosmotic flow Poiseuille fl

R r v r r E

=

14442444314424

0, 0 :zP E =

Electroosmotic flow

flat (plug-like) profile0, 0 :zP E =

:

z zEEO EEO

EEO

v E E (outside of the

electroosmotic 'mobili

= =

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Electrophoresis : real picture

+

-

+

+

+-

++-

-

-

-

-

-

Er

count

particle motion

ep epv u E=

rr

ep

is a complex, electromechanically coupled pr

- E field is distorted around the particle.

- Counterions are moving in the opposite

- Fluid slip (friction) is localized within th

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Limiting cases 1 : HuckelLimiting case: R>>1 (particle size >> Debye l

High ionic strength (high buffer concentration) co

Electromechanical coupling (friction) happens wit

Rq

epv

l ep

v v=

electrophoresisFluid at rest

elecpar

Er

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Similarity to electroosmosis

++ + + + + + + +

l epv v=

r =

r =

At small Debye length, surface curvature doesnt matter

Situation similar to electroosmosis at planar surface.

Friction due to the particle motion occurs mostly withinOutside of the Debye layer : no fluid flow gradient (elec

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Sugio, S., Kashima, A.,

Mochizuki, S., Noda, M.,

Kobayashi, K. Protein Eng. 12

pp. 439 (1999)

Brown, T., Le

E. D., Chamb

207 pp. 455 (1

Human Serum Albumin

Proteins : 3D structure with

complex charge distribution

DNA (SDS-p

Linear polym

uniform char

Image removed due to

copyright restrictions.

Image rem

copyright r

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Polyelectrolyte electrophoresis : Free

When driven by an electric field

DNA and counterions aredragged in the opposite

direction

Hydrodynamic interaction

screened

Friction with solvents occurs atevery monomers

friction

When drive

pressure

DNA and s

moving tog

Hydrodyna

the blob mo

Friction with

the surface

friction 6

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DNA Sequencers

From Hitachi Review Vol. 48, No. 3, 107 (1999), Kazumichi Imai, Satoshi Takahashi, Ma

Slab gel sequencer Multiple cap

Courtesy of Hitachi Review. Used with permission.

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Micro Total Analysis System (microTAS)

96~356 samples analyzed in a single chip simult

fluorescence detection of DNA at the center of th

optical head)

Yining Shi et al., Analytical Chemistry, 71, 5

Figure 1 removed due to copyright restrictions

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M. Burns et al., Science, 282, 484 (199

Images removed due to copyright restrictions.

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Technology Need for Advanced Bio

Challenges of Sample Complexity

Blood serum / Urine / Saliva

Highly diverse : more than ~10,000

90% of total serum protein: albumin and globulin (~

biomarkers and cytokines : 10ng/ml or less (up to 1

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Electrophoresis is a complicatedelectrokinetic

(determined by zeta potential, not the net charge of the mole

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Three images removed due to copyright restrictions.Source: Alberts et al., Molecular Biology of the Cell.

Slab Gel electrophoresis (Length-based Separation): se Isoelectric focusing (charge-based separation): see Fig 2D protein separation: see Figure 4-45.

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4: Inter-Debye layer force, Van-Der Waa


Goals of Part IV:





Key Concepts for section IV (Electrokine

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Motion of particles in E and BMotion of (bio) Particles in Electric and Mag

ElectrophoresisMotion of charged particles in an electric field

DielectrophoresisMotion of (neutral) particles in an electric field g

MagnetophoresisMotion of magnetic particles (with magnetic dip

field

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Simple electric dipo

lectric dipole

0, ( )netQ r= =r

+

-

Q

-Q

d p Qdz=r

z

Dipole in uniform electric field

Dipole in non-uniform electric field

-

+-- +

+

- -- +

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Induced dipoleE0

{

00

0

3

00 2

0external field

induced dipole field

3

2

cos2

in

i

o iout

i

E z

E RE z

r

=

+

= +

+ 14444244443

Induced dipole by external field

E0

o > i

E0+

+

+

+++

+

---

---

-

-

-

-

---

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Positive / negative DEP+++

++

--

-

- -

r

pr

Positive DEP

Particle moves toward

the high field region.

++

+

++

Er

Particle m

the high fi

Negativ0( ) >

Motion of (induced) dipo

electric field

++

+++

--

--

-

Er

pr +

+++

Er

DEP force is independent of the direction of the field.

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Introduction : Use of nanoparticles and

modern biotechnology

Nanoparticles : Emerging tools for B

From www.evidenttech.com (Evident

Image removed due to copyright restrictions.

Photo of EviDots (TM) vials - 490nm to 680nm.
http://www.evidenttech.com/http://www.evidenttech.com/

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From www.qdots.com (Quantum Dots Corp



Electron microscope photo of Qdot core-shell nanoc
http://www.qdots.com/http://www.qdots.com/

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Introduction 2 : Cancer targeting using

Gao, Cui, Levenson, Chung and Nie, Nature Biotechnology 22, 96

Courtesy of Leland W. K. Chung. Used with permission.

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The problem of collo

nanoparticle stabili

The problem of colloid (nanoparticle)

M. M. Baksh, M. Jaros, J. T.

Groves, Nature 427, 139(2004)

Image removerestrictions.Figure 4 in A.

Blaaderen. Na(2003)

Courtesy of J. T. Groves. Used with permission.

Source: Figure 2b in Baksh, M. M., M. Jaros, and J. T. Groves."Detection of Molecular Interactions

at Membrane Surfaces through Colloid Phase Transitions."Nature427 (January 8, 2004): 139-141.

Coagulation

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Hydrophobic tail

Polar head

(phosphate for the

lecithin layers

used above)

Interlayer distance in lipid layers, separated by aqueous solution containing as determined by K.J. Palmer and F.O. Schmitt.

0.1

20

60

100

140

W (A)

CaCl2

W

(inter-bilayerdistance)Measurement of Win different salt concentration

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Source: Introduction to Colloid and Surface Ch

By Duncan J. Shaw (Butterworth Heinemann)

Schulze-Hardy Rule

Critical coagulation concentrations for hydrophsolutions (millimoles per dm3)

As2S3(-ve sol) Al2OAgI (-ve sol)

LiCl

NaCl

NaCl

CaCl2

AlCl3

MgCl2

MgSO4

K2SO4

K2Cr2O

K2 oxala

KCl

KCl

KNO3

KNO3

K3[Fe(C

K acetate

Al2(SO

4)3

1 2

Al(NO3)3

LiNO3

NaNO3

KNO3

RbNO3

AgNO3

Ca(NO3)2

Mg(NO3)2

Pb(NO3)2

Al(NO3)3

La(NO3

)3

Ce(NO3)3

58

51

49.5

50

110

0.65

0.72

0.81

0.093

0.096

0.095

165

140

136

126

0.01

2.40

2.60

2.43

0.067

0.069

0.69

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Interactions and forces in micro / nanElectrostatic interaction within electrolyte

++

+

+

++

+

+

+

+

+

----

--

-

-----

-- -

- ++

+

+

++

-

---

--

++

+

++

+

+

+

+

+

+

----

--

-

-----

- - -

- ++

+

+

++

+

+

+

+

+

----

--

-

-----

-- -

-

1

h

weak or no interaction

significant repulsive interaction

(inter-Debye layer repulsion)

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Van der Waals Forc

Van der Waals Forces (attractive forces)

London Dispersion Forces (F. London, 193

h


h

Attractive interaction

++ --

(indu

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Electrolyte

Midplane

w w w

m

m o

h/2

h

x

D

Potential distribution resulting from the overlap of double layers from

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Goals of Part IV:






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Introduction : Use of nanoparticles and

modern biotechnology


From www.evidenttech.com (Evident T

Image removed due to copyright restrictions.Photo of EviDots (TM) vials - 490nm to 680nm.
http://www.evidenttech.com/http://www.evidenttech.com/

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The problem of collo

nanoparticle stabili

The problem of colloid (nanoparticle)

M. M. Baksh, M. Jaros, J. T.

Groves, Nature 427, 139(2004)

Image removerestrictions.Figure 4 in A.

Blaaderen. Na(2003)

Coagulation / FloCourtesy of J. T. Groves. Used with permission.

Source: Figure 2b in Baksh, M. M., M. Jaros, and J. T. Groves."Detection of Molecular Interactions

at Membrane Surfaces throughColloid Phase Transitions." Nature427 (January8,2004): 139-141.

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Interactions and forces in micro / nanElectrostatic interaction within electrolyte

++

+

+

++

+

+

+

+

+

----

--

-

-----

-- -

- ++

+

+

++

-

---

--

++

+

++

+

+

+

+

+

+

----

--

-

-----

- - -

- ++

+

+

++

+

+

+

+

+

----

--

-

-----

-- -

-

1

h


significant repulsive interaction

(inter-Debye layer repulsion)

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Van der Waals Forc



h


h


++ --

(indu

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Electrolyte

Midplane

w w w

m

m o

h/2

h

x

D


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Van der Waals Forc



h


h


+- -+

(indu

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Source: Introduction to Colloid and Surface Chemistry

By Duncan J. Shaw (Butterworth Heinemann)

Values of Hamaker Constants

MaterialA11(microscopic)

10-20J

A11(ma

1

Water

Ionic Crystals

Metals

Silica

Quartz

Hydrocarbons

Polystyrene

3.3 - 6.4

15.8 - 41.8

7.6 - 15.9

50

11.0 - 18.6

4.6 - 10

6.2 - 16.8

3.0

5.8

22.1

8.6

8.0

6.3

5.6

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Tokay Gecko (Gekko gecko)

Photo

Photo courtesy of elbisreverri.http://www.flickr.com/photos/elbisreverri/53226345/
http://upload.wikimedia.org/wikipedia/en/c/c9/Tokay_foot.jpg

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Goals of Part IV:






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Electrolyte

Midplane

w w w

m

m o

h/2

h

x

D


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From Introduction to Colloid and Surface

By Duncan J. Shaw (Butterworth Hein

Values of Hamaker Constants

MaterialA11(microscopic)

10-20J

A11(m

1

Water

Ionic Crystals

Metals

Silica

Quartz

Hydrocarbons

Polystyrene

3.3 - 6.4

15.8 - 41.8

7.6 - 15.9

50

11.0 - 18.6

4.6 - 10

6.2 - 16.8

3.

5.

22.

8.

8.

6.

5.

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Tokay Gecko (Gekko gecko)

Photo

Photo courtesy of elbisreverri.http://www.flickr.com/photos/elbisreverri/53226345/
http://upload.wikimedia.org/wikipedia/en/c/c9/Tokay_foot.jpg

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K. Autumn et al., Natu

Tokay gekco (Gekko gecko) has amazing f

A lizard from southeast Asia which..

can generate ~10 N of adhesive force.

can run up to ~ 1m /scan generate sheer stress of ~0.1N mm-2 (~

can walk on ANY surfaces

(hydrophobic/hydrophillic/rough/smooth/ch

What is the mechanism for such an amazing adhes

- micro-suction? No, adhesion works

- friction? No, measured frictio

- micro-interlocking? No, it walks on very

- capillary force? No, it walks on hydr- charge-interaction? No, it walks in ionize

- adhesion by glue? No, there are no skin

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K. Autumn et al., PN

Courtesy of National Academy of Sciences, U.S.A. Used with permission.

Source: Autumn, K., et al. "Evidence forVan der Waals Adhesion in Gecko Setae."PNAS 99,no

National Academy of Sciences, U.S.A.2002,

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K. Autumn et al., PN

Courtesy of National Academy of Sciences, U.S.A. Used with permission.Source: Autumn, K., et al. "Evidence forVan der Waals Adhesion in Gecko Setae."

PNAS

99,no2002, National Academy of Sciences, U.S.A.

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W. R. Hansen and K. Autumn, PNAS, 10

Courtesy of National Academy of Sciences, U.S.A. Used with permission.

Source: Autumn, K., et al. "Evidence for Van der Waals Adhesion in Gecko Setae."PNAS 99,no.National Academy of Sciences, U.S.A.2005,

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Courtesy of National Academy of Sciences, U.S.A. Used with permission. Hansen, W., and K. Autumn. "Evidence for Self-cleaning in Gecko Setae."PNAS

2005, National Academy of Sciences, U.S.A.


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Courtesy of National Academy of Sciences, U.S.A. Used with permission.Source: Autumn, K., et al. "Evidence forVan der Waals Adhesion in Gecko Setae."

PNAS

99,no2005,National Academy of Sciences, U.S.A.

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A. K. Geim et al. Nature Materials, 2, 4

Courtesy of A. K. Geim. Used with permission.

Geim, A. K., et al. "Microfabricated Adhesive Mimicking Gecko Foot-hair."Nature Mate

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VR(2)

VA

VR(1)

VR(3)

V(3)

V(2)

V(1)

0

Well Stabilized

Rapid Flocculation

Distance Between Part

Total Interaction Energy Curves

Curves obtained by summation of an attraction curve with v

repulsion curves (after Shaw 1980).

PotentialEnergy

ofInteraction,

V

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By Duncan J. Shaw (Butterworth Heinema

Critical coagulation concentrations for hydrop

solutions (millimoles per dm3)

As2S3(-ve sol) AAgI (-ve sol)

LiCl

NaCl

NaCl

CaCl2

AlCl3

MgCl2

MgSO4

K2SO

K2Cr2

K2 ox

KCl

KCl

KNO3

KNO

K3[Fe

K acetate

Al2(SO4)31 2

Al(NO3)3

LiNO3

NaNO3

KNO3

RbNO3

AgNO3

Ca(NO3)2

Mg(NO3)2

Pb(NO3)2

Al(NO3)3

La(NO3)3

Ce(NO3)3

58

51

49.5

50

110

0.65

0.72

0.81

0.093

0.096

0.095

165

140

136

126

0.01

2.40

2.60

2.43

0.067

0.069

0.69

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By Duncan J. Shaw (Butterworth Heinema

Graphs removed due to copyright restrictions.Figures 8.3 and 8.4: Influence of electrolyte con

and Stern potential on the total potential energyinteraction of two spherical particles.

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From Introduction to Modern Colloid Sc

By Robert J. Hunter (Oxford Science Publ

Three images removed due to copyright restrict Fig. 9.10: Apparatus to measure long-range f

between sheets of mica immersed in liquid.

Fig. 9.11: Graph of double-layer repulsion in presence of potassium chloride.

Fig. 9.12: Graph of attractive van der Waals dforces between mica surfaces.

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Carries and stores genetic information ofspecies

Chemically stable

Very long

Nucleic Acids

base pairs (kb) length (m)

SV40 5.1 1.7

lambda phage 48.6 17

T2 phage 166 56

Mycoplasma 760 260

E.coli 4,000 1,360

Yeast 13,500 4,600

Drosophila 165,000 56,000

Human 2,900,000 990,000

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Diagrams of hydrophilic amino acids removed dcopyright restrictions. (From Lodish et al)

Chemical structure of adenosine triphosphate (A2.9 in Alberts et al., Molecular Biology of the Ce

Diagrams of aggrecan (proteoglycan) referencby Dick Heinegard (1989) and chondroitin sul

Two slides removed due to copyright restriction

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Courtesy of Prof. Alan Grodzinsky. Used with permission.

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Exception : Migratory Birds (pigeons)


Figure 1 in Mora, Cordula V. "Magnetoreception and itsTrigem

Mediation in theHoming Pigeon." Nature432 (2004): 508-511.

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See B. H. Lapizco-Encinas, B. A. Simmons, E. B. Cummings,

Anal. Chem.2004, 76,1571-1579

Collection of bacteria using Dielectrophoretic

Figures 3 and 4 removed due to copyright restr

See Grey et al. Cells trapped by dielectrophoresis.

Biosensors and Bioelectronics 19 (2004) 17651774

Dielectrophoretic Manipulation of Cells

See Prof. Joel Voldmans group website: http://www.rle.mit.ed

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Design of nanofluidic chann

Pyrex coverslip

Si substrateOxide for insulation

dd

> 1 DNA

direction of

DNA motionDirec

buffer

solution

Cross sectional diagram of the chan

Constriction much smaller than Rg (the radius of gyratio

Open (deep) region where DNA can relax into equilibri

Entropic hindrance for DNA from entering the shallow r

Trapping affects DNA motion driven by an electric field

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Motion of DNA in Channel

channel

Shallow region (90 nm)

Deep region (1

35.7 V/cm

From

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1: Lorentz force law, Field, Maxwells equa

2: Ion Transport, Nernst-Planck equatioin

3: (Quasi)electrostatics, potential function

4: Laplaces equation, Uniqueness

5: Debye layer, electroneutrality

Goals of Part II:

(1) Understand when and why electromag

interaction is relevant (or not relevant)systems.

(2) Be able to analyze quasistatic electric

and 3D.

Key Concepts for this section

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Lorentz Force Law

Electric Force on a charge q : qE

Magnetic Force on a charge q : q v B

(F q E = + r r

Source (charge, current) E and B field

H&M Figure 1.1.1

The field theory of electromagnetic force

Courtesy of Herman Haus and James Melcher. U

Source: http://web.mit.edu/6.013_book/www/

7/25/2019 20330spring 2001

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1 2

2

12

1

48.85 10 ( )

Coulomb

o

o

Q QF

r/m

=

=

Coulombs law

- F

Silk Thread

Two positively charged gl

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Gausss Law : Electric vs Magnetic field

Image source: MIT 8.02 class notes.

Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, and

Professor John W. Belcher. Used with permission.

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Oerstead (1820)


Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, and


7/25/2019 20330spring 2001

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Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin,

Professor John W. Belcher. Used with permission

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Amperes law1

B d s= J d a= I

C S

o


Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, an


7/25/2019 20330spring 2001

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1

oC S S

o

dB d s J d a E d

dt

= +

(Current) (Displaceme

Image source: MIT 8.02 class notes.Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, and


Maxwell displacement c

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Faraday


Courtesy of Dr. Sen-ben Liao, Dr. Peter

Dourmashkin, and Professor John W. Belcher. Used

with permission.

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Maxwells four equatio

and B field

d

J ds= S ( dVV ) Charge Continudt

F q= +( )E v

B Lorentz Force Image sour

Courtesy of Dr. Sen-be

Professor John W.

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2: Ion Transport, Nernst-Planck equation




Goals of Part II:




and 3D.


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+V0

+ + + + + + + + ++ + ++ + ++ + +- - - - - - - - - - - - - - - - -

+

+

+

++

++

+

+

+

-

---

-

-

-

--

-

-

-

Cell

Dielectr

v

diffusion

Chemical reaction

hydrodynamic flow

electroosmosis

Debye layer

Example : BioMEMS system

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Differential form of Maxwells equa

0 eS V

E d s dV =

0sdBS

=

C S

dE d B d s

dt

=

0

1

( )S V

d s A dV =

( )C Sdl A ds =

Gauss t

Stokes

1e o

C S So

dB d s J d a E d adt

= +

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Maxwells equation in source-free space

~ sin( ) cos( )E t k r or t k r

General solution for the Wave equation


Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, and Professor John W.

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Gary Suizdaks tutorial page

(http://masspec.scripps.edu/MSHistory/whati

Related MIT links :

http://web.mit.edu/toxms/www/links2.htm

Mass Spectrometry

Courtesy of Dr. Gary Siuzdak. Used with permission.
http://web.mit.edu/toxms/www/links2.htmhttp://web.mit.edu/toxms/www/links2.htm

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How good is this approxim

2 2 2 2

2 2 2 2~ ~ ~ ( : wavelengterrorE L L LE c T c

Frequency (f) T ~ 1/f

60 Hz 0.167 s

1 MHz 1 s

100 MHz 10 ns

10 GHz 0.1 ns

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EQS approximation

Figure 3.5.1

H&M

urtesy of Herman Haus and James Melcher. Used with permission.

urce: htt ://web.mit.edu/6.013 book/www/

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EM interactions in media - polarization



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EM interactions in media - polarization (lin

0extE =

- + - + - + - +

- + - + - + - +

- + - + - + - +

extE

pol ext media ext E E E E =

r: relative permittivity (dielectric constant ) o

(=>1)

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of various media

Medium r

Water (pure) ~80

0.9% NaCl solution ~60

Ethanol 24

Methanol 34

Acetic acid 15~16

Gases ~1

Glass 3~4

Plastics and rubbers 2~9

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Maxwells equation in a medium (Magnet

: free space permeability (410-7 H/m)

0 0 0r r r

E

B J Jt

= + = +

mag ext media ext mag B B B B B = + =

r: relative magnetic permeability of the med



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of various media

rfor water : very close to 1

r(Ni)~600, r(Fe)~5000

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Mobility of various ions in water

Species Mobility

Ui (cm2/v/s)

D

Cations in H2O (25oC)

H+

K+

Na+

Li+

36.3010-4

7.6210-4

5.1910-4

4.0110-4

Anions in H2O (25oC)

OH-

SO42-

Cl-

NO3-

20.5210-4

8.2710-4

7.9110-4

7.4010-4

Electrons in Si at 25oC

Holes in Si at 25oC

1500

600

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Material ni (#/cm3)

DI water ~1017

0.1M NaCl 61019

Copper ~1022 5

Si (intrinsic) n=p~1010

3Si (doped)

Nd=1016

ne=1016

Np=104

Quartz

In silicon (semiconductor), np~1020 (con

In aqueous solutions, [H+][OH-] = 10-14 =

(pH= -log10[H+])

Comparative Number densities and Conductivitie

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Goals of Part II:




and 3D.


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EM interactions in media - polarization (lin

0extE =

- + - + - + - +

- + - + - + - +

- + - + - + - +

extE

pol ext media ext E E E E =

r: relative permittivity (dielectric constant ) o

(=>1)

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of various media

Medium r

Water (pure) ~80

0.9% NaCl solution ~60

Ethanol 24

Methanol 34

Acetic acid 15~16

Gases ~1

Glass 3~4

Plastics and rubbers 2~9

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Maxwells equation in a medium (Magnet

: free space permeability (410-7 H/m)

0 0 0r r r

E

B J Jt

= + = +

mag ext media ext mag B B B B B = + =

r: relative magnetic permeability of the med


Courtesy of Dr. Sen-ben Liao, Dr. Peter Dourmashkin, and Professor John W. B

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of various media

rfor water : very close to 1

r(Ni)~600, r(Fe)~5000

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Mobility of various ions in water

Species Mobility

Ui (cm2/v/s)

D

Cations in H2O (25oC)

H+

K+

Na+

Li+

36.3010-4

7.6210-4

5.1910-4

4.0110-4

Anions in H2O (25oC)

OH-

SO42-

Cl-

NO3-

20.5210-4

8.2710-4

7.9110-4

7.4010-4

Electrons in Si at 25oC

Holes in Si at 25oC

1500

600

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Material ni (#/cm3)

DI water ~1017

0.1M NaCl 61019

Copper ~1022 5

Si (intrinsic) n=p~1010

3Si (doped)

Nd=1016

ne=1016

Np=104

Quartz

In silicon (semiconductor), np~1020 (con

In aqueous solutions, [H+][OH-] = 10-14 =

(pH= -log10[H+])

Comparative Number densities and Conductivitie

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Electrolytes (biological systems) are conductors.

V=VextE

electrode

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Charge Relaxation in electrolyte

+++

+

+

++ +

+

+

+

+

+

+

+

Fixed charges

+

=0 w

=0 w

~10-9 sec

~10-9 sec

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(Quasi) Electroneutrality Appr

It is an approximation.

Valid only after the relaxation time

and outside of the Debye length (

discussed)

Not valid when there is a discon Valid only within a single medium

Boundary of the medium could ca

free charge

No inter-charge interaction in liq

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Electroneutrality

Protein A

++-

+

+

+

Protei

+

-

---

-

~ De

Buffer counterions

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Capillary Electrophoresis (1980s)

HV

105 V/m

Capillary inlet

10 - 200m Detector

Electrolyte bufferElectrolyte buffer

Reservoir

Electroosmotic flow(UV; laser fluorescence

and mass spec; radioisotope)

C

Generic diagram of a capillary electrophoresis system.

+

+

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96~356 samples analyzed in a single chip simult

fluorescence detection of DNA at the center of th

optical head)

Yining Shi et al., Analytical Chemistry, 71, 5

Figure 1 removed due to copyright restrict

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Goals of Part II:




and 3D.


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Charge Relaxation in electrolyte

+++

+

+

++ +

+

+

+

+

+

+

+

Fixed charges

+

=0 w

=0 w

~10-9 sec

~10-9 sec

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(Quasi) Electroneutrality Appr

It is an approximation.

Valid only after the relaxation time

and outside of the Debye length (

discussed)

Not valid when there is a discon Valid only within a single medium

Boundary of the medium could ca

free charge

No inter-charge interaction in liq

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Electroneutrality

Protein A

++-

+

+

+

Protei

+

-

---

-

~ De

Buffer counterions

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0( ) xx e =

0

0

1 e tanh2 4

( ) ln ,

1 e tanh4

x

x

zF

RT RTx

zFzF

RT

+ = =

When 0zF R

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0 0.5 1 1.5 2 2.5 3

0

2

4

6

8

0 0.5 1

0.9

0.95

1

1.05

1.1

1.15

0

( )c x

c

x

0 2zF

RT

=

zF

0

( )c xcc- (counterion)

c+ (co-ion)c+

c-(c

When0 0

( )zF RT ze kT electrical pot

(diffusion dominates.)

When0 0( )zF RT ze kT >> >>

thermal energy

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0.1Msucrose

0.01Msucrose

Membrane permeable

0.1M

KCl

0.01M

KCl

Membrane permeable

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c1=0.1M

KCl

Nernst Equilibrium Potential

c: K+ concentration

Vm+-[ ]

0 0

2

1

2 212 1 2

1 1

0

ln ( ) ( 0)

ln ln

x x

x x

dc dD E u c E

dx dx

dc dD u dx

c dx

dc dD u dxc dx

cD u x x

c

c cD RTu c zF c

= =

= =

+ = =

=

=

= = =

= = =

Nernst Eq

Diffusion of charged particles -> generate

-> stops diffusion of ions

Membrane

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Goals of Part II:




and 3D.


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Electroneutrality

Protein A

++-

+

+

+

Protei

+

-

---

-

~ De

Buffer counterions

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0( ) xx e =

0

0

1 e tanh2 4

( ) ln ,

1 e tanh4

x

x

zF

RT RTx

zFzF

RT

+ = =

When 0zF R

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0 0.5 1 1.5 2 2.5 3

0

2

4

6

8

0 0.5 1

0.9

0.95

1

1.05

1.1

1.15

0

( )c x

c

x

0 2zF

RT

=

zF

0

( )c xcc- (counterion)

c+ (co-ion)c+

c-(c

When0 0

( )zF RT ze kT electrical pot

(diffusion dominates.)

When0 0( )zF RT ze kT >> >>

thermal energy

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0.1Msucrose

0.01Msucrose

Membrane permeable

0.1M

KCl

0.01M

KCl

Membrane permeable

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c1=0.1M

KCl

Nernst Equilibrium Potential

c: K+ concentration

Vm+-[ ]

0 0

2

1

2 212 1 2

1 1

0

ln ( ) ( 0)

ln ln

x x

x x

dc dD E u c E

dx dx

dc dD u dx

c dx

dc dD u dxc dx

cD u x x

c

c cD RTu c zF c

= =

= =

+ = =

=

=

= = =

= = =

Nernst Eq

Diffusion of charged particles -> generate

-> stops diffusion of ions

Membrane

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Quasi-Electrostatics

( ) eE =

0B =

1B J

=

BE

t

=

0

0E =

( )S CE ds =

1

C

2

a

b

2 2

1( ) 1( )a b

E dl E

Electrostatic force :Potential function

(2) (1) =

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2

( ) ( )

( ' )

e

e

E E

Poisson s Equation

= = =

=

2 cct

=

0q =

q k T=

2c =(Ficks second law)

(ste

(Fouriers law for heat conduction)

(conservation law for heat)

However, biomolecules in the system do not generat

are shielded by counterions (electroneutrality).

It all comes down to solving.. 2 0 ( apla =

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1=0

Electrostatics

2=0

3=0

4=0

5=0

=?

Steady state

c5=0

c1=0

T4=0

Thermal conduction

T3T5=0

T1=0T2=0

T=02 0T =

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Uniqueness of Solution

2

2

;

;

ea a i i

eb b i i

on S

on S

= =

= =

2 0; 0

d a b

d d on S = = = i (satisf

Lets assume two different solutions, a and b

Then define

S1

S5

2

0

d

d

= =0d =

Answer:

for everywhere

0a b =

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Gel Electrophoresis

Gel (, )

Plastic ( =0)

biomolecules

( ) 0J E = =

0 ( )B

E electrostaticst

= =

0eJt

= =

(steady state, no charge accumu

0E =

E=

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0J =

0y yJ Ey

= = =

y

L

W

=V0 wh=0 when x=0

(no charge accumu

0

0y

y=

=

0y W

y=

=

J=0 (insulator)

xJ x=

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Boundary Conditions (For EQS approximati

( ) eE =

0E =

Jt

=

1 1 2 2 (n E E

1 2 1tang

(n E n E E =

1 1 2 2 (n E E

From H&M

Figure 5.3.1 (a) Differential contour intersecting surface supporting surface cha

volume enclosing surface charge on surface having normal n.

Courtesy of Herman Haus and James Melcher. Used with


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2

20 ( )

dx ax b

dx

= = +

2 2

2 20

x y

+ =

1D case:

2D case:( 1,n m

( ,n m

1( , ) ( 1, ) ( , )

2n m n m n m

x

+ = +

1( , ) ( , ) ( 1, )

2n m n m n m

x

=

2

2

1 1( , ) ( , ) ( , )

2 2

( 1, ) ( 1, ) 2 ( , )

n m n m n mx x x

n m n m n m

= +

= + +

2 2

2 2( , ) ( , )

( 1, ) ( 1, ) ( , 1) ( , 1)

n m n m

x yn m n m n m n m

+ =

+ + + + +

( 1, ) ( 1, ) ( ,( , )

4

n m n m n m n m

+ + + + =

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Goals of Part II:




and 3D.


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2

( ) ( )

( ' )

e

e

E E

Poisson s Equation

= = =

=

r r

2 cct

=

0q =r

q k T= r

2c =(Ficks second law)

(ste

(Fouriers law for heat conduction)

(conservation law for heat)

However, biomolecules in the system do not generat

are shielded by counterions (electroneutrality).

It all comes down to solving.. 2 0 ( apla =

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1=0

Electrostatics

2=0

3=0

4=0

5=0

=?

Steady state

c5=0

c1=0

T4=0

Thermal conduction

T3T5=0

T1=0T2=0

T=02 0T =

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Uniqueness of Solution

2

2

;

;

ea a i i

eb b i i

on S

on S

= =

= =

2 0; 0

d a b

d d on S

=

= = i (satisf

Lets assume two different solutions, a and b

Then define

S1

S5

2

0

d

d

=

=0d =

Answer:

for everywhere

0a b =

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Gel Electrophoresis

Gel (, )

Plastic ( =0)

biomolecules

( ) 0J E = =ur ur

0 ( )B

E electrostaticst

= =

r

r

0Jt

= =

ur

(steady state, no charge accumu

0E =r

E= r

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0J =ur

0y yJ Ey

= = =

y

L

W

=V0 wh=0 when x=0

(no charge accumu

0

0y

y=

=

0y W

y=

=

J=0 (insulator)

xJ x=

ur

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E

=

ur

0E =r

Jt

=

ur

1 1 2 2 (n E E

uur uur

1 2 1tang

(n E n E E = uur uur r

1 1 2 2 (n E E

uur uur



Courtesy of Herman Haus and James Melcher. Used with permission.


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Gel or tissue

(,)

G

=V0

=0 =0

=0

C=0

Electrostatics Stea

2 0 =

eJ =

r

J

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Figure 5.5.1 Two of the infinite number of potential functi

(1) that will fit the boundary conditions = 0 aty = 0 a



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Solution

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Known Solutions for Laplace equations

Cylindrical Coordinates

2 2

2 2

1 1( , , ) 0 sin

sin

( , , ) ( ) ( ) ( )

( )

( ) ( (co

( ) (sin ,cos

n

r rr r r r

r R r

R r Spherical Bessel Functio

Legendre Functions P

Trigonometric

= +

=

Spherical Coordinates

2 22

2 2

1 1( , , ) 0

( , , ) ( ) ( ) ( )

( ) ( , ,

( ) (sin,cos,sin

( ) (sin,cos,sinh

n n

z

z R z

R Bessel Functions J N I

Trigonometric

z Trigonometric

= + +

=

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2

20 ( )

dx ax b

dx

= = +

2 2

2 20

x y

+ =

D case:

D case:

( 1,n m

( ,n m 1( , ) ( 1, ) ( , )

2n m n m n m

x

+ = +

1( , ) ( , ) ( 1, )

2n m n m n m

x

=

2

2

1 1( , ) ( , ) ( , ) ( 1, )

2 2n m n m n m n m

x x x

= + = + +

Solving Laplaces Equation (Numerically)

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( 1,n m

( ,n m

2 2

2 2( , ) ( , )

( 1, ) ( 1, ) ( , 1) ( , 1) 4

n m n mx y

n m n m n m n m

+ =

+ + + + +

( 1, ) ( 1, ) ( ,( , )4

n m n m n mn m + + + +

=

Value in the middle = average of surrou

Laplaces equation

In discretized form

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Finite Element Method

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Goals of Part II:




and 3D.


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Gel or tissue

(,)

G

=V0

=0 =0

=0

C=0

Electrostatics Stea

2 0 =

eJ =

r

J

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2 2 22

2 2 20

x y z

= + + =

( , , ) ( ) ( ) ( )x y z x y z = Assume

2 2 22

2 2 2

2 2 2

2 2 2

1 1 10

function functionfunctionof x of z of y

x y z

x y z

= + + =

+ + =

123 123123

2

22

2

1( ) ,

( ) sin( ), cos

0 ( ) ( , :

x xk x k xx

x x

k x e ex

or k x k x

or x ax b a b co

+

= =

= =

= = +

Three possibilities

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=0

2

2 2

2 2

22

2

0, ( , ) ( ) ( )

1 10

1( ) ~ sin( )

sin( ) 0 ( : integer)

: n

x y x y

x y

k X x kxx

kL kL n n

nEigenvalue k

L

= =

+ =

=

= =

=

22

2

expand (x) using Fourier sine series

( ) sin (This satisfies B. C. at x=0, L)

( )then, ( ) 0 ( ) ~ sinh cosh

( ) sinh since ( ,0) 0 ( , ) sin

n

n

n

n

n xx A

L

y n y n yk y y or

y L L

n y ny x x y A

L

=

=

= = =

s

n

x

( )

n

0

0

0

Determining A : use boundary condition

( , ) sin sinh

2 (1 cos( ))sin

sinh( )

n

n

L

n

n xx L V A n

L

Vm x noperate on both sides A

L n n

= =

=

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2

20 ( )

dx ax b

dx

= = +

2 2

2 20

x y

+ =

D case:

D case:

( 1,n m

( ,n m 1( , ) ( 1, ) ( , )

2n m n m n m

x

+ = +

1( , ) ( , ) ( 1, )

2n m n m n m

x

=

2

2

1 1( , ) ( , ) ( , ) ( 1, )

2 2n m n m n m n m

x x x

= + = + +

Solving Laplaces Equation (Numerically)

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( 1,n m

( ,n m

2 2

2 2( , ) ( , )

( 1, ) ( 1, ) ( , 1) ( , 1) 4

n m n mx y

n m n m n m n m

+ =

+ + + + +

( 1, ) ( 1, ) ( ,( , )4

n m n m n mn m + + + +

=

Value in the middle = average of surrou

Laplaces equation

In discretized form

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Finite Element Method

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Known Solutions for Laplace equations

Cylindrical Coordinates

2 2

2 2

1 1( , , ) 0 sin

sin

( , , ) ( ) ( ) ( )

( )

( ) ( (co

( ) (sin ,cos

n

r rr r r r

r R r

R r Spherical Bessel Functio


Trigonometric

= +

=

Spherical Coordinates

2 22

2 2

1 1( , , ) 0

( , , ) ( ) ( ) ( )

( ) ( , ,

( ) (sin,cos,sin

( ) (sin,cos,sinh

n n

z

z R z

R Bessel Functions J N I

Trigonometric

z Trigonometric

= + +

=

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2( ) ( ) 0 0 (e

E Laplace = = = =r r

i ,

R

Eext

0

Equation to solve :

2 2

2 2

1

1 1( , , ) 0 sin

sin

( , , ) ( ) ( )

separate and solve,

1( )

( ) ( (cos ))

n

n

n

r rr r r r

r R r

R r Ar Br


+

= + +

=

+

Cell in a field

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Guessing the solution

extE E z as r

r

ext extE z E r = =

Eext +

++

++

Pn(cos) ~ cos n

Only n =1 term contributes

(should be dipole field)

2

2

1cos cos (for r R)

1

cos cos (for r R)

0 ( finite at r=0)

( cos when r )

o

i

i

ext o ext

Ar Br

Cr D r

D

A E E r

= +

= +

=

=

Trial Solution:

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( ) eE =ur

0E =r

eJt

=

uur

1 1 2 2 (n E E

uur uur

1 2 1tang

(n E n E E = uur uur r

1 1 2 2 (n E E

uur uur





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Some plots for the solution

< 0

> 0

Cell is less conductive than media Insul

Perfectly conCell is more conductive than media

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20.330 / 6.023 / 2.793 Fields, Forces and Flows in Biol

systems and nan

mucus

Fields/ forces/ flows/ transport in Transport in livibio-microsystems (bioMEMS) systems

Instructors: Jongyoon Jay Han and Scott Manalis

Relevant forces

TOPICS

Introduction to electric fieldsMaxwells equationsIntroduction to fluid flowsTransport phenomena in biological systemsElectro-quasistaticsElectrokinetics

ElectrophoresisVan der Waals and other forces

Photo courte

http://www.flickr.co

Q?Q?Q?Q???

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Textbooks

Truskey, Yuan and Katz Transport PhenomBiological Systems Prentice Hall (REQUIRE

Haus and Melcher Electromagnetic Fields aContent available on the web for free(http://web.mit.edu/6.013_book/www/)

Physicochemical Hydrodynamics, An Introby Ronald F. Probstein. (e-reserve)

Electromechanics of Particles by Thomas

Cambridge University Press (e-reserve)

Other references:

Bird/Stewart/Lightfoot, Transport Phenomena Wiley

Tom Weiss Cellular Biophysics Volume 1. Transport,

AC Electrokinetics: colloids and nanoparticles, by Mor

Research Studies Press.

Principles of Colloid and Surface Chemistry, by HiemeRajagopalan, Marcel Dekker.

Molecular Driving Forces, by Ken Dill and Sarina BromScience
http://%28http//web.mit.edu/6.013_book/www/)http://%28http//web.mit.edu/6.013_book/www/)http://%28http//web.mit.edu/6.013_book/www/)http://%28http//web.mit.edu/6.013_book/www/)

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How precise can a cell mea

concentration of its enviro

E. Coli trajectory

Images removed due to copyright restrictions.See Figs

Berg, Physics Today 2000http
http://www.aip.org/pt/jan00/berg.htmhttp://www.aip.org/pt/jan00/berg.htm

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Measuring binding kinetic

Surface Plasmon Resonance (Biacore)Courtesy of Biacore. Used with permission.

Label-free enables direct readout of Kon and

adsorptiontarget

binon offcapturec

surface time

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Detecting biomolecules on the n

Nat. Biotech. 23 (2005)

Figure removed due to copyright restrictions.

J. Am. Chem. Soc. 128 (2006)

Figure removed due to cop

Nature 445 (2007)

Courtesy of Dr. Charles M. Lieber. Used with permission.

Source: Fig. 1b in Zheng, G., et al. "Multiplexed electrical detection of cancer

markers with nanowire sensor arrays." Nat Biotech23 (2005): 1294-1301.

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64 oligos at 1 femtomolar concentra

-4

x 10

0

2

12

34

-4

0

1

2

3

4

m

x 10 m 0 m

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after 10 seconds

How often do molecules bind to sph

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Proteins : 3D structure with

complex charge distribution

Human Serum Albumin

Figure removed due to copyright restrictions.

Sugio, S., Kashima, A.,Mochizuki, S., Noda, M.,

Kobayashi, K.Protein Eng. 12

pp. 439 (1999)

DNA (SDS-prote

Linear polymer

uniform charge d

DN

Figure removed due to co

Brown, T., Leona

E. D., Chambers,

207pp. 455 (1989

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Migratory birds uses magnets for po


Figure 1 in Mora, Cordula V. "Magnetoreception and its TrigeminalMediation in the Homing Pigeon." Nature432 (2004): 508-511.

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Introduction 2 : Cancer targeting using nan

Gao, Cui, Levenson, Chung and Nie, Nature Biotechnology 22, 969 (20

Courtesy of Leland W. K. Chung. Used with permission.

Courtesy of Lelan

Courtesy of Lelan

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Dielectrophoretic Manipulation of C

Cells trapped by dielectrophoresis, Gray et al.

Biosensors and Bioelectronics 19 (2004) 1765177

Figures removed due to copyright restrictions.

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Electrophoresis / ElectrokineticsJ. Fu et al. Nature Nanotechnology (2007).

urce: Fu, Jianping, and Jongyoon Han,et al. "A Patterned AnisotropicNanofluidic Sieving Structure for Continuou

ature Nanotechnology2 (2007): 121-128.

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- - - - - - - - - - - - - - - - -

Example : BioMEMS systems

electroosmosis

+ + + + + + + + ++ + ++ + ++ + +

+

+

+

++

++

+

+

+

-

---

-

-

-

--

-

-

-

Cell

Dielectroph

Elecv

diffusion

hydrodynamic flowDebye layer

Chemical reaction +V0 -

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Ficks law of diffusion

Concentration(c)

()E and

, J : source

Osmosis

(aqueous) medium,

Flow velocity (vm)

Convection

Electrophoresis

S

p

Electroosmosis


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20.330J / 2.793J / 6.023J

Fields, Forces and Flows in Biological Systems, Spring 2007

Department of Biological, Electrical and Mechanical Engineering

Massachusetts Institute of Technology

Problem Set #1Issued: Friday, February 9

Due: Friday, February 16

Questions (10 points each)

Bird / Stewart / Lightfoot Chapter 2, page 62, exercise 11

1.

2.

3.

A metal bead is dropped in a large tank filled with glycerin, and velocity of bead was

measured. Now, the same experiment was repeated, but this time the bead was dropped

near the wall of the tank (within a distance approximately the same as the radius of the

bead) vertically. In this experiment, do you expect the falling velocity of the bead to be

higher, lower, or the same? Briefly explain.

A flow field v(x, y, z) is said to be irrotational if curl of the field is zero. Which of the

following fields are irrotational?

a) vx=y, vy=0, vz=0

b) vx=y, vy=x, vz=0

c) vx=-y, vy=x, vz=0

Choose one irrotational flow field and one rotational flow field from above, and sketch thedirection of the flow near the origin (x=y=z=0).

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Haus and Melcher website

Problem 2.2.1http://web.mit.edu/6.013_book/www/chapter2/2.prob.html

Problems (20 points each)

1.

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What will happen to the two drops? Explain briefly.a)

Calculate the pressure difference between the two drops. The surface tension

for water is 0.0728 N/m.

b)

3. A microfluidic channel has a two openings as shown below, and is filled with

water. Using the pipette, two spherical droplets, with the radius of 1mm and 5mm

each, were put down at the openings.

r=1mmr=5mm

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Biological Engineering Division

Department of Mechanical Engineering

Department of Electrical Engineering and Computer Science

20.330J/2.793J/6.023J/Fields, Forces and Flows in Biological Systems, Spring 2007

Problem Set #4. Issued: March 9th

(Friday)

Due date: March 16th

(Friday)

Problem 1: Electromagnetic Wave

Starting from the differential form of Maxwells equation in free space (no charge and current),

show that E

and B

fields in free space satisfy the following wave equation.22 2 2

2 2 2 2 2

22 2 2

2 2 2 2 2

1( , ,

1( , ,

i

i

ii

E

)

)

E i x y or zx y z v t

BB i x y or z

x y z v t

+ + = =

+ + = =

Use the following vector identity.

( ) ( ) - ( )A A A =

What is the propagation speed v of this electromagnetic wave? Calculate the numerical value.

Problem 2: FT-ICR MS

Fourier Transform Ion-Cyclotron Resonance Mass Spectrometer (FT-ICR-MS) is the currentstate-of-the-art mass spectrometer for analyzing biomolecules. It has a very high mass resolving

power (M/M~10-5

), which is high enough to detect the mass shift by one mass unit (one proton).

Therefore, it is a viable tool for analyzing small changes (post-translation modification ofproteins, for example) in biomolecules. In FT-ICR-MS,

biomolecules are electro-sprayed into a vacuum chamber,

and accelerated to a velocity v by the acceleratingpotential Vac. Then, biomolecules are introduced into the

area where a magnetic field perpendicular to the direction

of motion exists, essentially trapping the chargedbiomolecules into a circular orbit (radius r), as shown in

the figure.

B

v

ze

F

XX XX

XX XX

r (a) Get the cyclotron angular velocity c=v/r as a

function of B, z(charge number), e, and m (mass of the

molecule).

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(b) One of the issue of this type of mass spectrometer is the strength of magnetic field to trap

heavy biomolecules within a reasonable distance, say r~10mm. (Large r requires larger vacuumchambers and larger vacuum pumps, which is very costly.) Using the following typical values,

Vac ~ 1000V

m ~ 10kD (small proteins)z ~ 1

calculate the required magnetic field to trap the biomolecule within the ~20mm size vacuumchamber (r~10mm).

Problem 3: Quasistatic Approximation

For each of the following experimental situations, determine if the quasistatic (QS)

approximation is appropriate or not. Explain your reasoning. Use the typical, approximate size /time scales when necessary.

(a) Wireless telephone in our home uses 2.4GHz frequency for its communication between the

base station and the phone. When we use the phone, electric fields could affect brain tissue,perhaps inducing currents.

(b) In recent work by Prof. Hamad-Schifferli (Biological Engineering, MIT), metallic (gold)nanoparticles (with diameter of ~3nm) are excited by the 1GHz oscillating magnetic field. (The

end result of this is the heating of the particle, which could (locally) denature DNA molecules

that are attached to the particle. See the K. Hamad-Schifferli, J.J. Schwartz, A.T. Santos, S.Zhang, J.M. Jacobson, "Remote electronic control of DNA hybridization through inductive

coupling to an attached metal nanocrystal antenna,"Nature, 2002, 415, 152-155.)

Problem 4: Isoelectric FocusingIn isoelectric focusing (IEF) a pH-gradient is established along the microchannel or a capillary

column by special buffer called carrier ampholytes, as shown in the figure below. When the

protein is in the environment where its pH is above (below) the isoelectric point of the protein

(pI), they have net negative (positive) charges. When an external field is applied, proteins willget focused around the point x=xip, where the mobility (and net charge) of the molecule becomes

zero. Approximately, one can say that the electrical mobility near the pI of the molecule is linear,

as in

( )ipu p x x= (p: positive constant)

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C(xip)=Co

x

( )ip

u p x x=

x=xip

Cathode (high pH)Anode (low pH)

After a certain time, all the proteins will be focused around the x=x ip, reaching a steady state

peak concentration C(xip)=Co. However, the resulting peak will have a finite peak width, due tothe diffusional transport. The diffusion constant of the protein is given as D, and the electric field

in the microchannel/capillary is uniform ( oE E x=

). One can ignore convection in this case.

(a) At steady state, derive the expression for C(x), the concentration of a protein near theisoelectric point.

(b) Estimate the approximate width of the focused protein peak as a function of other parameters.

Figure by MIT OCW.

P

pl

pH

l

v+ v+= 0v-

i

Cathode

pH = 10

Anode

pH = 3 PP

Separation by isoelectric focusing. A sample protein (P) migrates along the linear pH gradient

formed in a capillary until its resulting charge becomes zero, at the position i.

++

+

+

++

+

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Department of Biological Engineering

Department of Mechanical Engineering

Department of Electrical Engineering and Computer Science

20.330J/2.793J/6.023J/Fields, Forces and Flows in Biological Systems, Spring 2007

Problem Set #10. Issued: May 4th

(Friday)

Due date: May 11th

(Friday)

Problem 1: Electrokinetic Pumping (40 points)

Consider a glass capillary, with length L and the inner radius R, connecting the two electrolyte

reservoirs. The cathode, as shown in the figure below, is contained within a closed container,tightly sealed. The anode is immersed within the reservoir facing the atmospheric pressure. The

glass surface of the capillary has negative surface charges that can be characterized by the

(negative) zeta potential . An electric potential is applied between the two electrodes. Debyelength is

-1, and the viscosity and dielectric constant are and , respectively. Ignore

electrolysis at the electrodes, and assume that R>>1.

-CathodeAnode

(a) Find the expression for (steady-state) pressure difference between the cathodic and anodicreservoir (P=Pc-Pa) at the steady state.

(b) Determine the flow velocity profile vz(r) within the capillary at the steady state. Sketch the

flow streamline along the capillary (over the entire length of the capillary). Use the

cylindrical coordinate system with the center line of the capillary being r=0 (z axis).

1

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(c) Calculate P for R=10m, 0.1m, and 0.01m. Use =-100mV, V=1000V, L=5cm, =1-

1m

-1and use the values for water for dielectric constant and viscosity.

(d) Four different experimental situations are compared in the following. Identify the case that

will generate the highest pressure, and explain why.

A: One capillary (with the radius R and length L) bridging the reservoirsB: One capillary (with the radius R and length 2L) bridging the reservoirs

C: Four capillaries (with the radius 0.5R and length L) bridging the reservoirs

D: One capillary (with the radius 0.4R and length 2L) bridging the reservoirs

Problem 2. Time scale for the onset of electroosmotic flow. (15 points)

In the lecture, we all learned that the electroosmotic flow is generated by the motion of surface-bound Debye layer charges. Initially only the surface fluid layer moves (as shown in figure 1

above), but then its momentum is transferred to the entire fluid column, yielding a flat flow

profile as shown in 4 above. Assuming the small Debye length limit (R>>1) and assuming that

the buffer solution in the capillary is water, estimate the approximate time scalefor the transitionfrom 1 to 4 (a scaling result will suffice). What are the values of these time scales when

R=10m, 100nm and 1nm? (Hint: This is indeed a fluid dynamics problem. Write down theNavier-Stokes equation and think about scaling arguments.)

2

1 2 3 4

Figure by MIT OCW.

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Problem 3: Capillary On-line Preconcentration of proteins (45 points)

V1 V2ground

Reservoir 1 Reservoir 2

Capillary 1 Capillary 2

L1 L2

Consider the above capillary system, where two different capillaries with the same radius R but

different lengths L1and L2(blue) are joined by a porous membrane (green) which conducts

electrical (ionic) currents freely but has negligible water permeability. Then a different potential

(V1and V2) are applied to the two reservoirs, which contain positively charged proteins (reddots) with the electrophoretic mobility uep. The capillary surface has (negative) zeta potential of

at the buffer condition used in this experiment, and the Debye length thickness -1

is much

smaller than any other size parameters of the system (R >> 1).

(a) Show that a hydraulic press P (above atmospheric pressure of reservoir 1 and 2) will bedeveloped within the capillary junction, at the steady state. Get the expression for P interms of other parameters. Explain why there is a pressure developed at the junction.

P=PATM+P

Impermeable to fluid flow

(b) Determine the stead-state flow velocity (averaged over the cross section of the capillary)of the system, as a function of other parameters.

(c) Assume V1 > V2.For a given values of L1, L2, uep, and V1, determine the conditions forV2, which will allow the proteins to be continuously focused and concentrated at the

capillary junction. This could be a concentration scheme for protein analysis in capillary

electrophoresis.

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20.330J Fields, Forces and Flows in Biological Systems

Prof. Scott Manalis and Prof. Jongyoon Han

Review: Vector Calculus

Vector Product

v n = vxnx + vyny + vznz

=

v n cos()

Gradient (on a scalar funct ion )

p = ixp

x+ iy

p

y+ iz

p

z

Divergence (operated on vector)

v =vx

x+

vy

y+

vz

z => scalar

Curl (operated on vector)

v =

ix iy iz

x

y

zvx vy vz

=> vector

In 1D integrat ion

f(x2) f(x1) =

x1

x2 dx

...similarly, we have two different integral theorems for vector calculus.

n (normal vector)

v

x y zi i i

x y z

= + +

x1 x2

f(x)

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(1) Gauss theorem (Divergence theorem)For any vector field ,v

vs

n da = (

v)dvv

velocity area

total outgoingvolume flow rate

surface S

volume expansion

vn

Proof: consider infinitesimal cube.

(x,y,z) xyz21

From surfaces 1and 2:

(

vs

n) da (Vx x+x Vx x )yz

12

Similarly, from other surfaces,

(

vs

n) da = (Vx x+x Vx x )yz

+(Vy y+y Vy y )xz

+(Vzz+z Vzz)xy

Divide each terms with , ,y respectively,

= Vxx

+ Vyy

+ Vzz

xyz

= (

VV

)dV

20.330 Fields, Forces and Flows in Biological Systems Vector Calculus ReviewProf. Scott Manalis and Prof. Jongyoon Han Page 2 of 4

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Meaning of

V volume expansion net outgoing flux

for incompressible flow,

V = 0 (no fluid source/sink)

V = 0

V > 0

V < 0 divergence free

(2) Stokes theorem (curl theorem)

For a given vector field v ,

VC

d

s = (

VS

) n da

Surface S

Contour C

n

d

s

Proof: think about the rectangle in the xy plane.

V

C d

s

= (Vx y Vx y +y )x

+(Vy x+x Vy x)y

= Vx y+y Vx y

y

+Vy x+x

Vy x

x

xy

= Vy

x

Vx

y

xy =

V( )zxy

Similar for curves in other planes

Contour C

(x,y)x

1

2

4

3y1

3 y

x


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Meaning of

V Represents circulation of the flow.

V = 0

V 0Laminar flow Turbulent flow

References H&M website: Chapter 2 Appendix of TY & K


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FEMLABtutorial by Y.S. 3/31/07

Weve learned how to solve the problem below by using separation of variables. Now

we can solve the same problem using the finite element model in FEMLAB.

Gel or tissue

(,)

=V0

=0 =0

=0

2 0 =

eJ = r

For the analytical solution, please see lecture notes.

1

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Click on COMSOL Multiphysics 3.3on your desktop

In Model Navigator, under New,

- choose either 2D or 3D space dimension

-

under Electromagnetics, choose either Electrostaticsor Conductive MediaDC

Draw->Specify Objects->Square-> specify the size and position of the square you want

to draw

(If you want to create a composite object , i.e. a square + a circle overlapping: go to

Draw->create composite object, then select all the objects you want to be in the

composite (by holding Ctrl), and click on Union, also uncheck Keep interiorboundaries, then click OK.)

Physics->Subdomain Settings:-Select Subdomains(since you only have a square in this case, its the subdomain1)

-Click on (isotropic), then enter a value for electrical conductivity in the

Value/Expressionbox.

-Click OK.

Physics->Boundary Settings:

-For each boundary (i.e. 1, 2, 3, 4), select the appropriate Boundary condition(i.e. current flow, inward current flow, distributed resistance, electric insulation,

electric potential, ground).-also fill in Value/Expressionif applicable.

-Click OK.

Mesh->Initialize Mesh

Solve->Solve Problem

Postprocessing->Plot Parameters

-Surface: check Surface plot; at Predefined quantities, choose Electric

potential.-Streamline: check Streamline plot; at Predefined quantities, choose Electric

field; you can also change the number of streamlines by specifying the Number

of start points-Arrow: check Arrow plot; at Predefined quantities, choose Electric field; you

can make the arrows bigger or smaller by unchecking Auto(under Arrowparameters) and enter a scale factor.-Click OK

2

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You should get plots similar to the ones shown here.

20330spring 2001

Documents

Transcript of 20330spring 2001