Microfluidic Applications of Magnetic and Dielectric Fluid Suspensions

Markus ZahnMassachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer ScienceResearch Laboratory of ElectronicsLaboratory for Electromagnetic and Electronic SystemsHigh Voltage Research LaboratoryHsin-Fu HuangNational Taiwan UniversityDepartment of Mechanical Engineering

Research Workshop of the Israel Science Foundation onElectrokinetic Phenomena in Nano-colloids and Nano-FluidicsTechnion-Israel Institute of TechnologyHaifa, Israel

19-23 December 2010

2

1. Introduction to Ferrohydrodynamics2. Applications to Magnetic Resonance Imaging (MRI)3. Ferrohydrodynamic Instabilities4. Dielectric Analog: Von Quincke Electrorotation5. Use of Micro-particle Electrorotation in Enhancing

the Macroscopic Suspension Flow Rate6. Continuum Analysis of Poiseuille Flow with Internal

Micro-particle Electrorotation

OUTLINE

3

FERROHYDRODYNAMICS 2

o

2

2

1

2J ( )

2

/

o

m

J B HF

H M H

R vv

ELECTROHYDRODYNAMICS21

2( )

/

/

/

f

f

e

E EF

E P E

vR

v

(Linear Magnetization) (Force Density)

(Magnetizable Medium)

(Magnetic Diffusion Time)

(Skin Depth)

(Magnetic Reynolds Number)

(Linear Dielectric) (Force Density)

(Polarizable Dielectric)

(Dielectric Relaxation Time)

(Electric Reynolds Number)

1. INTRODUCTION TO FERRODYNAMICS

4

FerrofluidsColloidal dispersion of permanently magnetized particles undergoing rotational and translational Brownian motion.

Established applicationsrotary and exclusion seals, stepper motor dampers, heat transfer fluids

Emerging applicationsMEMS/NEMS – motors, generators, actuators, pumpsBio/Medical – cell sorters, biosensors, magnetocytolysis, targeted drug delivery vectors, in vivo imaging, hyperthermia, immunoassaysSeparations – magneto-responsive colloidal extractants

RpN

SMd

adsorbed

dispersant

permanently

magnetized core

solvent molecule Rp ~ 5nm

5

Ferrofluid Magnetic Properties

Water-based Ferrofluid0Ms = 203 Gauss

= 0.036 ; 0 = 0.65, =1.22 g/cc, 7 cp dmin5.5 nm, dmax11.9 nm B=2-10 s, N=5 ns-20 ms

Isopar-M Ferrofluid0Ms = 444 Gauss

= 0.079 ; 0 = 2.18, =1.18 g/cc, 11 cp dmin7.7 nm, dmax13.8 nm

B=7-20 s, N=100 ns-200 s

6

Langevin Equation

Measured magnetization (dots) for four ferrofluids containing magnetite particles (Md = 4.46x105 Ampere/meter or equivalently μoMd = 0.56 Tesla) plotted with the theoretical Langevin curve (solid line). The data consist of Ferrotec Corporation ferrofluids: NF 1634 Isopar M at 25.4o C, 50.2o C, and 100.4o C all with fitted particle size of 11 nm; MSG W11 water-based at 26.3o C and 50.2o C with fitted particle size of 8 nm; NBF 1677 fluorocarbon-based at 50.2o C with fitted particle size of 13 nm; and EFH1 (positive α only) at 27o C with fitted particle size of 11 nm. All data falls on or near the universal Langevin curve indicating superparamagnetic behavior.

1[ coth ]

s

M

M

7

Ferrofluid Magnetization Force

A magnetizable liquid is drawn upward around a current-carrying wire.

[R.E. Rosensweig, Magnetic Fluids, Scientific American, 1982, pp. 136-145,194]

8

Larmor Frequency = 42.576 MHz/Tesla for Protons

( = gyromagnetic ratio, = 42.576 MHz/Tesla for nucleus.)

2. Application of Magnetic Media to Magnetic Resonance Imaging (MRI)

2

The fundamental elements of MRI are (a) dipole alignment, (b) RF excitation at Larmor frequency , (c) slice selection and (d) image reconstruction.

T2 decay is characterized by the exponential decay envelope of the magnetization signal in the {xy} plane after excitation. T1 recovery is characterized by the exponential growth of the magnetization along the z axis as the vector returns to the equilibrium magnetization.

0 / 2f B

1H

0 / 2f B

9

Ferrofluids as Contrast Agents in MRI

Two categories of contrast agents in MRI:• Ferrofluids called superparamagnetic iron oxide (SPIO) contrast agents (currently smaller market share)• Gadolinium-based contrast agents

N.S. El Saghir et al., BioMed Central Cancer, 5, 2005

BEFORE AFTER

MRI Image of the Brain (A) Before and (B) After Gadalinium contrast enhancement

T1 weighted MRI of the brain (a) before and (b) after Gadolinium contrast enhancement, showing a metastatic deposit involving the right frontal bone with a large extracranial soft tissue component and meningeal invasion. This figure comes from a case report of suspected dormant micrometatasis in a cancer patient.

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Effect of Ferrofluid on Image Contrast

=Percentage solid volume

Experiments at the Martinos Center for Biomedical Imaging (MGH)

T1 and T2 results for various water based ferrofluid solid volume fractions, . At very high concentrations (>5 x 10-6), the SNR was not sufficient to allow for the accurate determination of T1 relaxation times.

Pádraig Cantillon-Murphy, “On the Dynamics of Magnetic Fluids in Magnetic Resonance Imaging”, Massachusetts Institute of Technology PhD thesis, May 2008.

7

5

: 4.7 10

: 1.4 10

A

F

11

Comparison of the theoretical prediction for T2 at various superparamagnetic particle radii with the experimental results over a range of MSG W11 water-based ferrofluid concentrations of the original 2.75% solution by volume.

5.6R nm

T2 Time Constant as Function of Ferrofluid Solid Volume Fraction

Particle size fit:

0.8892 0.00062

5.6

T

R nm

Pádraig Cantillon-Murphy, “On the Dynamics of Magnetic Fluids in Magnetic Resonance Imaging”, Massachusetts Institute of Technology PhD thesis, May 2008.

– Viscous dominated regime– Particle spin-velocity is no longer

given by vorticity alone– Magnetic torque density in

Angular Momentum equation

12

Conservation of Linear and Angular Momentum

00 2 2 M H v M = Nm

N = num. particles per m3

ζ = vortex viscosity

η = kinematic viscosity

200 ( ) 2p g M H v

1) eqv

t

M

M M M M

Magnetization Relaxation Equation

13

Comsol Problem Definition

Concentric Cylinders of Water and Ferrofluid

Ferrofluid

Waterx

y

z

8 cm

Dimensions in meters

0-0.02-0.04-0.06 0.02 0.04 0.06

0

-0.02

-0.04

0.02

0.04

x

yµ (r > R2) →∞

Ferrofluid

Water

B0

Brot

14

Measured magnetization using a vibrating sample magnetometer (VSM) for (a) Feridex and (b) Magnevist MRI commercial contrast agents. The units of magnetic moment are (a) Am2/kg Fe for Feridex and (b) Am2/kg Gd for Magnevist.

15

The measured phase map for the center coronal slice of a tube of Feridex agent surrounded by water. The phase map only shows the net field component along the x-axis, i.e., Bx. The B0 field is left to right (x-directed) and has a value of 3 T. 

16

Comparing (a) theoretical MRI results and (b) measured B0 maps from the scanner at 3 T for Feridex

and Magnevist contrast agents. The black line indicates the line of maximum field variation.

(a) (b)

4

2

0

- 2

- 4

- 4 - 2 0 2 4

4

2

0

- 2

- 4

- 4 - 2 0 2 4

4

2

0

- 2

- 4

- 4 - 2 0 2 4

4

2

0

- 2

- 4

- 4 - 2 0 2 4

17

3. Ferrohydrodynamic Instabilities In DC

Magnetic Fields

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Labyrinthine Instability in Magnetic Fluids

Stages in magnetic fluid labyrinthine patterns in a vertical cell, 75 mm on a side with 1 mm gap, with magnetic field ramped from zero to 535 Gauss. [R.E. Rosensweig, Magnetic Fluids, Scientific American, 1982, pp. 136-145,194]

Magnetic fluid in a thin layer with uniform magnetic field applied tangential to thin dimension.

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Rotating Magnetic Fields

Observed magnetic field distribution in the 3 phase AC statora. One pole pair stator b. Two pole pair stator

Uniform Non-uniform

RO

v rbg

, , 'Ferrofluid

z rbgx

y

z

Surface Current Distribution

( )Re fj t

zK Ke

RO

, , 'Ferrofluid

z (r)x

y

z

( )v r

( 2 )Re fj t

zK Ke

Surface Current Distribution

20

FerrofluidDrops in Rotating Magnetic Fields

FerrohydrodynamicDrops

A Gallery ofInstabilities

21

Ferrofluid Spiral / Phase Transformations

22

(a) Von Quincke’s rotor consists of a highly insulating cylinder that is free to rotate and that is placed in slightly conducting oil between parallel plate electrodes. As DC high voltage is raised, at a critical voltage the cylinder spontaneously rotates in either direction; (b) The motion occurs because the insulating rotor charges like a capacitor with positive surface charge near the positive electrode and negative surface charge near the negative electrode. Any slight rotation of the cylinder in either direction results in an electrical torque in the same direction as the initial displacement.

4. Dielectric Analog: Von Quincke’s Rotor (Electrorotation)

Von Quincke’sRotor

Von Quincke’sRotor

23

Definition of Quincke Rotation: Spontaneous rotation of insulating particles (or cylinders) suspended in a slightly conducting liquid subjected to a DC electric field with the field strength exceeding some critical value (Jones, 1984, 1995)

2 1

2 1

wherei

ii

is the charge relaxation time in each region

0 when

More on Quincke’s Rotor (Electrorotation)

Two Competing Forces (Torques):

The electrical torque and the fluid viscous torque exerted on the particle

24

The electric torque

3 21 0 1 2

2 21 2 2 1

6 1

1 2 1 2 1MW

e

MW

R ET p E

For a small perturbation of rotation to grow, the equation of angular motion for the particle is re-written as (Jones, 1995):

3 21 0 1 2 3

02 21 2 2 1

6 18

1 2 1 2 1MW

MW

R EdI R

dt

The bracket term should have a value larger than zero for the small perturbation to grow (Jones, 1995), thus

2 1

Torques Exerted on a Micro-particle

The fluid viscous torque 308vT R Re 1

25

0 12

1 1 2 2 1

81

2 3critE

2

0 1MWcrit

E

E

0 critE E

@ 0.5e critT E

@e critT E

@ 2e critT E

visT

Competition of the Viscous and Electric Torques

Steady

1 2

1 2

2

2MW

Maxwell-Wagner Relaxation Time

26

E 0E

0E

Experiments have shown that for a given pressure gradient, the Poiseuille flow rate can be increased (Lemaire et al., 2006) by introducing micro-particle electrorotation into the fluid flow via the application of an external direct current (DC) electric field.

5. Flow Rate Enhancement using Electrorotation

From Hsin-Fu Huang PhD Thesis research, supervised by M. Zahn

6. Continuum Analysis for Couette & Poiseuille Flows with Internal Micro-particle Electrorotation

The Couette flow geometry

Stress balance 0s eff eff z y

MW

Ui T i

h

The Poiseuille flow geometry

0

h

yQ u z dz2D volume flow rate

27

22t e

Dvp P E v v

Dt

22 2 ' 't

DI P E v

Dt

1eq

MW

DP Pv P P P P

Dt t

0v

0 0, , , , , ,y zeq eq i i y eq i i zP P n E i P n E i

Polarization Relaxation

Equilibrium Polarization

Continuity

Linear Momentum (Dahler & Scriven, 1961, 1963; Condiff & Dahler, 1964; Rosensweig, 1997)

Angular Momentum (Dahler & Scriven, 1961, 1963; Condiff & Dahler, 1964; Rosensweig, 1997)

No-slip boundary conditions

Field conditions: free-to-spin, symmetry, stable rotation

Incompressible flow: treating as a single phase continuum

n Particle # density

EQS & Electro-neutrality (Haus & Melcher, 1989)

0E 0D

Lobry & Lemaire, 1999; He, 2006; Lemaire et al., 2008

Spin field BCs:

2v

0 1

Kaloni, 1992; Lukaszewicz, 1999; Rinaldi, 2002; Rinaldi & Zahn, 2002

1 2

1 2

2

2MW

01.5

0 1 2.5 e 2' h

Zaitsev & Shliomis, (1969); Rosensweig, (1997)

The Continuum Governing Equations

28

z

y

x

R

,

,

r

1 1

2 2

Electric potential and field solutions to a spherical particle subjected to a uniform DC electric field rotating at an angular velocity of .

2 0 , ,

R rr

r

Laplace’s equation with spherical harmonics (Jackson, 1999)

0 0, cos sinz rr E E i E i i

, , , ,R R f

f f fn J v Kt

BCs (Cebers, 1980; Melcher, 1981; Pannacci, 2006) ( ) (sin cos cos )f f f x r fK V i Ri R i i

1 2, , , ,f r rn J E R E R 1 2, , , ,f r rE R E R

†0 zE E i

The proposed “rotating coffee cup model” for the retarding polarization relaxation equations with its accompanying (quasi-static) equilibrium retarding polarization (Huang, 2010; Huang, Zahn, & Lemaire, 2010a, b):

x

0E

1eq

MW

DP Pv P P P P

Dt t

2 1 2 1

1 2 1 231 02 2

2 24

1z

eqMW x

P R n E

2 1 2 1

1 2 1 231 02 2

2 24

1

MW xy

eqMW x

P R n E

0

0

2

0 ,

0,

11 c

c

cMW

E E

E E

EE

0 12

1 1 2 2 1

81

2 3cE

Retaining macroscopic fluid spin

Including microscopic particle rotation

Polarization Relaxation & Equilibrium Polarization

29

,

,

Schematic diagram for the Poiseuille geometry

0

h

yQ u z dz2D volume flow rate

Modeling Results for the Poiseuille Flow Geometry

30

Zero spin viscosity Poiseuille flow velocity profiles compared with experimental results found from the literature (Peters et al., 2010)

81 5.4 10 S m

Lemaire experimental results are from Fig. 9 of Peters et al., J. Rheol., pp.311, (2010)

Cusp structure for zero spin viscosities

1.8cE kV mm

Zero electric field solution of Poiseuille parabolic profile

' 0

5974.6p Pa

L m

The zero spin viscosity solutions of our present continuum mechanical field equations over predicts the value of the spin velocity profile and has a cusp in the mid-channel position, which is not consistent with experimental measurements done by Peters et al. (2010). However, the order of magnitude is correct.

Comparison of Poiseuille Velocity Profile Results

Huang, (2010)

0.05

31

Finite spin viscosity small spin velocity Poiseuille flow rate results compared with experimental/theoretical results found from the literature (Lemaire et al., 2006) Lemaire theory/experimental results are from Figs. 5 and 6 of

Lemaire et al., J. Electrostat., pp. 586, (2006)

HT: Huang theory (solid line)LT: Lemaire theory (dashed line)LE: Lemaire experiment (dotted line)

81 4 10 S m

1.3cE kV mm

1

2 20' 1 2.5h h

Finite spin viscosity results do not involve ad hoc fitting!

Comparison of Poiseuille Flow Rate Results

0.05 0.1

Huang, (2010)32

Note: if we use particle diameter for spin viscosity,

Three orders of magnitude less than best fit value. Likely supports ER fluid parcel physical picture

Finite spin viscosity small spin velocity Poiseuille flow velocity profiles compared with experimental results found from the literature (Peters et al., 2010)

81 5.4 10 S m 1 2 10' 0.012 2.96 10h N s

61.8483 10 1.8cE V m kV mm 1.8cE kV mmround and substitute to analytic solution

Lemaire experimental results are from Fig. 9 of Peters et al., J. Rheol., pp.311, (2010)

Note: At this pressure gradient, MAX spin velocity is not necessarily small. We are kind of pushing the limit of small spin velocities

Zero electric field solution of Poiseuille parabolic profile

Agreement achieved for the all voltages considered! (Rounding of critical electric field strength is only within 3%)

2 13' 6.7 10d N s

Ultrasound velocity profile measurements from Prof. Lemaire’s group likely support our finite spin viscosity theory combined with our new rotating coffee cup polarization model.

Huang Analytic Solutions V.S. Lemaire Numeric Solutions

Best fit, within spin viscosity values calculated by He (2006) and Elborai (2006) for ferrofluids

Huang, 2010

Comparison of Poiseuille Velocity Profile Results

5974.6p Pa

L m

0.05

33

Potential applications of the negative electrorheology phenomenon

1. Micro or nano fluidic pumping

2. Electrically responsive smart materials

3. Electrically variable damping devices

4. Clutch or torque transmission devices

Applications of the theory presented herein

1. Negative electrorheological fluid flow induced by internal micro-particle electrorotation

2. Offers complementary insights towards the ferrofluid spin-up problem in magnetorheology

3. Electrically or magnetically controlled suspensions in bio systems

Some Applications of Phenomenon and Theory

34

Hsin-Fu Huang is financially supported by the National Science Council (Taipei, Taiwan) through the Taiwan Merit Scholarships (contract No. TMS-094-2-A-029).

This work is partially supported by the United States-Israel Binational Science Foundation (BSF) through grant No. 2004081.

35

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