1

Comsol Conference 2008Nonlinear Ferrohydrodynamics

of Magnetic Fluids

Markus ZahnMassachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer ScienceLaboratory for Electromagnetic and Electronic Systems

Presented at the COMSOL Conference 2008 Boston

2

1. Introduction to Ferrohydrodynamics2. Applications to Magnetic Resonance Imaging (MRI)3. Applications to Increasing Power Transformer

Electric Breakdown Strength4. Ferrohydrodynamic Instabilities5. Dielectric Analog: Von Quincke Electrorotation

OUTLINE

3

FERROHYDRODYNAMICS2

o

2

2

12

J ( )

2

/

o

m

J B HF

H M H

R vv

µ

µ µ

τ σµ

δωµσ

σµ σµ

× − ∇= × + •∇

=

=

= =

ELECTROHYDRODYNAMICS21

2( )

///

f

f

e

E EF

E P E

vRv

ρ ε

ρ

τ ε σε σ ε

σ

− ∇= + •∇

=

= =

(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

RpδN

SMd

adsorbeddispersant

permanentlymagnetized core

solvent molecule Rp ~ 5nm

5

Ferrofluid Magnetic Properties

Water-based Ferrofluidµ0Ms = 203 Gauss

φ = 0.036 ; χ0 = 0.65, ρ=1.22 g/cc, η≈7 cp dmin≈5.5 nm, dmax≈11.9 nmτB=2-10 µs, τN=5 ns-20 ms

Isopar-M Ferrofluidµ0Ms = 444 Gauss

φ = 0.079 ; χ0 = 2.18, ρ=1.18 g/cc, η≈11 cp dmin≈7.7 nm, dmax≈13.8 nmτB=7-20 µs, τN=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

MM

αα

= −

7

Characteristic Brownian and Néel Relaxation Times

0

0

3

1 exp

1 1 1

B h

a cN

B N

VkTK V

f kT

ητ

τ

τ τ τ

=

=

= +

610 sτ − (7-8 nm water-based magnetite)

Effect of Radius Size on Ferrofluid Time Constant

8

Ferrofluid Magnetization Force

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

9

Larmor Frequency = 42.576 MHz/Tesla for Protons

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

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

2γπ

γ

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 Bγ π=

1H

0 / 2f Bγ π=

10

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.

11

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, φ. At very high concentrations (φ>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

AF

φ

φ

−

−

= ×

= ×φ

12

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 φ

Particle size fit:

0.8892 0.000625.6

TR nm

ϕ−≈=

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

13

Conservation of Linear and Angular Momentum

( )00 2 2µ ζ= × + ∇× −M H v ω

m

H

c eas g /τ

x

y

M = Nm

N = num. particles per m3

ζ = vortex viscosity

η = kinematic viscosity

( )200 ( ) 2pρ µ η ζ ζ= + ⋅∇ −∇ + + ∇ + ∇×g M H v ω

– Viscous dominated regime– Particle spin-velocity is no

longer given by vorticity– Magnetic torque density in

Con of Ang Mom:

14

Effect of Spin-velocity on Dynamics

( )1) eqvt τ

∂•∇ − × = − −

∂M M M M M+ ( ω

M and H are related by a complex susceptibility– Ω (electrical frequency)

– τ (ferrofluid time constant)

– ω (spin-velocity)

– v (flow velocity)

Increasing frequency Ω > 1/τ

m

H

c eas g /τ

x

y

M. Shliomis, JETP, 34, 1972

15

Comsol Problem Definition

Concentric Cylinders of Water and Ferrofluid

FerrofluidWater

xy

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

θ

B0

Brot

16

Complex Susceptibility

• Relating small-signal magnetic field to small-signal ferrofluid magnetization

2 2

(1 ) (ω )(ω ) (1 )

(1 ) (ω )

z

z

z

jj

j

τ ττ ττ τ

+ Ω − + Ω

+ Ω +

m hχ=

mm

x

y

=

hh

x

y

0

0

MH

( )1) eqvt τ

∂•∇ − × = − −

∂M M M M M+ ( ω xy

17

Exporting Comsol Results to MatlabInstantaneous transverse B field

∗ M and H are approx. collinear, H(t=0) is x-directed

∗ Negligible spin-velocity

z

y

x

Ωτ = 0.001

18

• Streamers are low-density conductive structures that form when dielectric liquids are electrically over-stressed [2].

• Streamer characteristics are polarity dependent.

– Positive Excitation: Filamentary structure, Breakdown velocities greater than 1 km/s, 4 propagation modes.

– Negative Excitation: Bushy structure, Breakdown velocities greater than 100 m/s, 3 propagation modes.

Positive streamer growth in transformer oil. Vappl= 82 kV, Gap d= 1.27 cm. [3]

Negative streamer growth in transformer oil. Vappl= 82 kV, Gap d= 1.27 cm. [3]

Photos taken from:J. C. Devins, et al, “Breakdown and Pre-breakdown Phenomena in Liquids,” J. Appl. Phys., 52 (7), July 1981.

3. Applications to Increasing Power Transformer ElectricBreakdown StrengthElectrical Streamer

19

Observable streamer structures are the result of electrically driven thermal dissipation.

The role of four charge generation mechanisms have been explored.– Fowler-Nordheim electron injection (Unimportant)

Injection of electrons from a highly stressed negative electrode.– Electric field assisted ionic dissociation (Unimportant)

Generation of positive and negative ions due to dissociation of weakly bonded neutral ion-pairs at high electric fields.

– Impact ionization (Unimportant) Generation of positive ions and electrons due to inelastic collisions

of free electrons that are accelerated in the high electric field levels.– Electric field dependent molecular ionization, Field ionization

Generation of positive ions and electrons due to the direct ionization of neutral molecules at high electric field levels.

Charge Generation Mechanisms

(Most Important)

20

Nanoparticle Charging Dynamics As a nanoparticle captures charge the electric field distribution in the vicinity of

the particle changes with a decreasing charge capture window

z

Rw(t=0+)=RwMAX

0 zE E i=

_

__

_

_

_

__

__

_

_ _

_

_

_

_

_

_

0t += 0t +> t →∞

Rw(t>0+) < Rw(t=0+)

zz

Rw( ) = 0t →∞

ε2,σ2

ε1,σ1

2

00( ) , 12 ,41

ss

QQ t Q E Rtεπε τρ µτ

= = =+

21

The trapping of electrons by nanoparticles in a nanofluid can be modeled by adding an attachment term to both the electron and negative ion continuity equations

Heaviside function H(ρNPsat-ρ-) is used to account for the charge saturation of the nanoparticles Consider a 1 Gauss ferrofluid manufactured using 10 (nm) diameter particles:

n0 = 3.4×1020 (1/m3)

/

/ (1 ( ))

(1 ( ))

( ) where

(| |)

(| |)

e

e eI

e e e ee

e

NP

e

NP

Ia

e

a

H NPsat

H NPsat

V E V

R RJ G Et e e

RJ G E

t e

RJt e

ρ ρ

ρ ρ

ε ρ ρ ρ

ρ ρρ ρ ρ

ρ ρ ρ ρ ρτ

ρρ ρ ρτ

τ

ρτ

+ −

+ ++ + − ±+

+ +

− + − ±−

− − −

− − −

−∇ ⋅ ∇ = + + = −∇

∂+∇ ⋅ = + +

∂

∂+∇ ⋅ = − − − −

∂

∂+∇ ⋅ = − +

∂

20 30

18

30

3.4 10 (1/m )

1.76 10 (C)11

600 (C/m )

S

NPsat S

n

Qelectrons

n Qρ

−

= ×

≈ − ×

≈

= ≈ −

Electrodynamic Model for Nanofluid

2 2 * 20

2(| |) | |exp| |

fI f f f

B e n a m aG E A E A Bh ehE

π ∆= − ⇒ = =

22

Simulations using nanoparticle attachment time constant values (τNP) of 2, 5 and 50 (ns)

Pure Oil

NF:τNP=50nsNF:τNP=5ns

NF:τNP=2ns

The electric field wave propagation in the nanofluid is slower than in the pure oil

The model predictions correspond exceptionally well with experimental results

Nanoparticles must be conductive!!

Oil NF

Experiment 1.59 km/s 1.02 km/s

Model 1.65 km/s 1.05 km/s

Electric Field Dynamics in Nanofluids

23

Results: Surface Electric FieldVappl = 300 kV

t = 100 ns t = 300 ns t = 500 ns

24

Field ionization results in the development of a moving dissipative source that “sweeps” through the oil.

Major Result: Field ionization results in thermal enhancement values exceeding 200K which is needed overcome the latent heat of vaporization of transformer oil.

2

ρ∂ ⋅

+ ⋅∇ = ∇ +∂

Tl v

T E Jv T K Tt c

Temperature enhancement can be estimated using the thermal diffusion equation

25

4. FerrohydrodynamicInstabilities In DCMagnetic Fields

26

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.

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

27

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

µ →∞

RO

v rθbg

η ζ η, , 'Ferrofluid

ω z rbgx

y

z

Surface Current Distribution

( )Re fj tzK Ke θΩ −=

µ →∞

RO

η ζ η, , 'Ferrofluid

ω z (r)x

y

z

( )v rθ

( 2 )Re fj tzK Ke θΩ −=

Surface Current Distribution

28

FerrofluidDrops in Rotating Magnetic Fields

FerrohydrodynamicDrops

A Gallery ofInstabilities

Traveling WavePumping

29

Ferrofluid Spiral / Phase Transformations

30

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

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

Von Quincke’sRotor

31

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.

Electrorotation—Introduction

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

32

The electric torque ( )( )( )( )

3 21 0 1 2

2 21 2 2 1

6 11 2 1 2 1

MWe

MW

R ET p E

πε τ τ τε ε σ σ τ

− Ω= × =

+ + +Ω

For a small perturbation of rotation to grow, the torque balance on the particle is re-written as (Jones, 1995):

( )( )( )( )

3 21 0 1 2 3

12 21 2 2 1

6 18

1 2 1 2 1MW

MW

R EdI Rdt

πε τ τ τπη

ε ε σ σ τ

− ΩΩ = − Ω

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

2 1τ τ>

Electrorotation—Quincke rotation

33

( )2 1 1

1 1 2 2 1

812 3critE σ η σσ ε σ τ τ

= + −

2

0 1MWcrit

EE

τ

Ω = ± −

0 critE E>

@ 0.5e critT E

@e critT E

@ 2e critT E

visT

Electrorotation—Quincke rotation