Microfluidic Applications of Magnetic
and Dielectric Fluid Suspensions
Markus Zahn
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
Research Laboratory of Electronics
Laboratory for Electromagnetic and Electronic Systems
High Voltage Research Laboratory
Hsin-Fu Huang
National Taiwan University
Department of Mechanical Engineering
Research Workshop of the Israel Science Foundation on
Electrokinetic Phenomena in Nano-colloids and Nano-Fluidics
Technion-Israel Institute of Technology
Haifa, Israel
19-23 December 2010
1
OUTLINE
1.
2.
3.
4.
5.
Introduction to Ferrohydrodynamics
Applications to Magnetic Resonance Imaging (MRI)
Ferrohydrodynamic Instabilities
Dielectric Analog: Von Quincke Electrorotation
Use of Micro-particle Electrorotation in Enhancing
the Macroscopic Suspension Flow Rate
6. Continuum Analysis of Poiseuille Flow with Internal
Micro-particle Electrorotation
2
1. INTRODUCTION TO FERRODYNAMICS
FERROHYDRODYNAMICS
1 2
(Linear Magnetization)

J

B

H



(Force Density)
F 
2
 J   H   ( M  ) H (Magnetizable Medium)
o
o

(Magnetic Diffusion Time)
   2

Rm 
2
(Skin Depth)


2
/v
  v
(Magnetic Reynolds Number)
ELECTROHYDRODYNAMICS
1 2


E

E 
 f
2
F 
  f E  ( P  ) E

   /
 /
Re 
/v
(Linear Dielectric)
(Force Density)
(Polarizable Dielectric)
(Dielectric Relaxation Time)

v

(Electric Reynolds Number)
3
Ferrofluids
Colloidal dispersion of permanently magnetized particles
undergoing rotational and translational Brownian motion.
Established applications
rotary and exclusion seals, stepper motor
dampers, heat transfer fluids
Emerging applications
MEMS/NEMS – motors, generators,
actuators, pumps
Bio/Medical – cell sorters, biosensors,
magnetocytolysis, targeted drug delivery
vectors, in vivo imaging,
hyperthermia, immunoassays
Separations – magneto-responsive
colloidal extractants
permanently
magnetized core
adsorbed
dispersant
N

Rp
Md
S
solvent molecule
Rp ~ 5nm
4
Ferrofluid Magnetic Properties
Water-based Ferrofluid
0Ms = 203 Gauss
f = 0.036 ; c0 = 0.65, =1.22 g/cc, 7 cp
dmin5.5 nm, dmax11.9 nm
B=2-10 s, N=5 ns-20 ms
Isopar-M Ferrofluid
0Ms = 444 Gauss
f = 0.079 ; c0 = 2.18, =1.18 g/cc, 11 cp
dmin7.7 nm, dmax13.8 nm
B=7-20 s, N=100 ns-200 s
5
Langevin Equation
[
M
1
 coth  ]
Ms

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.
6
Ferrofluid Magnetization Force
A magnetizable liquid is drawn upward around a current-carrying wire.
7
2. Application of Magnetic Media to Magnetic
Resonance Imaging (MRI)
The fundamental elements of MRI are (a) dipole
alignment, (b) RF excitation at Larmor frequency
f   B0 / 2 , (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.
Larmor Frequency f   B0 / 2 = 42.576 MHz/Tesla for Protons
( 
= gyromagnetic ratio,

2
= 42.576 MHz/Tesla for 1H nucleus.)
8
Ferrofluids as Contrast Agents in MRI
MRI Image of the Brain (A) Before and (B) After Gadalinium contrast enhancement
BEFORE
AFTER
Two categories of contrast agents in MRI:
• Ferrofluids called superparamagnetic iron oxide (SPIO) contrast agents (currently smaller market share)
• Gadolinium-based contrast agents
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.
N.S. El Saghir et al., BioMed Central Cancer, 5, 2005
9
Effect of Ferrofluid on Image Contrast
T1 and T2 results for various water based ferrofluid solid volume
fractions, f. At very high concentrations (f>5 x 10-6), the SNR was
not sufficient to allow for the accurate determination of T1 relaxation
times.
A : f  4.7 107
F : f  1.4 10
5
f =Percentage
solid volume
Experiments at the Martinos Center for
Biomedical Imaging (MGH)
Pádraig Cantillon-Murphy, “On the Dynamics of Magnetic
Fluids in Magnetic Resonance Imaging”, Massachusetts Institute
of Technology PhD thesis, May 2008.
10
T2 Time Constant as Function of Ferrofluid
Solid Volume Fraction f
T 2  0.00062 0.889
R  5.6nm
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.
Particle size fit: R  5.6 nm
Pádraig Cantillon-Murphy, “On the Dynamics of Magnetic
Fluids in Magnetic Resonance Imaging”, Massachusetts Institute
of Technology PhD thesis, May 2008.
11
Conservation of Linear and Angular Momentum
0  g Ω0>M1/τ
H  p  (   ) v  2  
Increasing
2
H
y
M = Nm
m
– Viscous dominated regime
– Particle spin-velocity is no
longer given by vorticity
– Magnetic torque density in
Con of Ang Mom:
x
N = num. particles per m3
ζ = vortex viscosity
η = kinematic viscosity
0  0M  H  2  v  2
12
Effect of Spin-velocity on Dynamics
Increasing
Ω > Ω1/τ
Increasing frequency
> 1/τ
H
y
m
x
M. Shliomis, JETP, 34, 1972
 M and H are related by
a complex susceptibility
–
–
–
–
Ω
τ
ω
v
(electrical frequency)
(ferrofluid time constant)
(spin-velocity)
(flow velocity)
M
1
  v  )M    M    M  M eq 
t

13
Comsol Problem Definition
Concentric Cylinders of Water and Ferrofluid
y
8 cm
0.04
0.02
Brot
µ (r > R2) →∞
Ferrofluid
0
Water
-0.02
z

-0.04
-0.06 -0.04 -0.02
B0
x
Water
y
x
0
0.02
0.04
0.06
Dimensions in meters
Ferrofluid
14
Exporting Comsol Results to Matlab
Instantaneous transverse B field
y
 M and H are
approx. collinear,
H(t=0) is x-directed
 Negligible spinvelocity
x
Ωτ = 0.001
z
15
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.
16
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.
17
4
4
2
2
0
0
-2
-2
-4
-4
-4
-2
0
2
4
4
4
2
2
0
0
-2
-2
-4
-4
-4
-2
0
(a)
2
4
-4
-2
0
2
4
-4
-2
0
2
4
(b)
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.
18
3. Ferrohydrodynamic
Instabilities In DC
Magnetic Fields
19
Labyrinthine Instability in Magnetic Fluids
Magnetic fluid in a thin layer
with uniform magnetic field
applied tangential to thin
dimension.
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.
20
Rotating Magnetic Fields
Uniform
Non-uniform
v (r )
bg
v r
y
RO
Surface Current Distribution

K z  Re Ke
j (  f t  )

x
bg
y
z r
z
Ferrofluid
RO
z
x
 z (r)
Surface Current Distribution
Ferrofluid
, , '
, ,'
 
a. One pole pair stator

K z  Re Ke
 
j (  f t  2 )
b. Two pole pair stator
Observed magnetic field distribution in the 3 phase AC stator
21

Ferrofluid
Drops in Rotating
Magnetic Fields
A Gallery of
Instabilities
Ferrohydrodynamic
Drops
22
Ferrofluid Spiral / Phase Transformations
23
4. Dielectric Analog: Von Quincke’s Rotor (Electrorotation)
Von Quincke’s
Rotor
(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.
24
More on Quincke’s Rotor (Electrorotation)
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)
 0
when
 2 1

 2 1
where
i
i 
i
is the charge relaxation time
in each region
Two Competing Forces (Torques):
The electrical torque and the fluid viscous torque exerted on the particle
25
Torques Exerted on a Micro-particle
The fluid viscous torque
The electric torque
Tv  80 R3 
Te  p  E 
Re
1
61 R 3 E02 1   1  2   MW
2
1  21  2 1   2 2 1  1  2 MW

For a small perturbation of rotation to grow, the equation of angular motion for the
particle is re-written as (Jones, 1995):

61R3 E02 1  1  2  MW
d 
3
I

 80 R  
2 2
dt  1  21  2 1   2 2 1  1    MW 


The bracket term should have a value larger than zero for the small perturbation to
grow (Jones, 1995), thus
 2  1
26
Competition of the Viscous and Electric Torques
2

80 1
 
Ecrit  1  2 
 2 1  31 2  2   1 
 MW
 E0 
 
 1
 Ecrit 
Steady
Maxwell-Wagner
Relaxation Time
 MW
21   2

2 1   2
Te @0.5Ecrit
E0  Ecrit
Tvis
Te @ Ecrit
Te @ 2 Ecrit
27
5. Flow Rate Enhancement using Electrorotation
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.
E0
E
E 0
From Hsin-Fu Huang PhD Thesis research, supervised by M. Zahn
28
6. Continuum Analysis for Couette & Poiseuille
Flows with Internal Micro-particle Electrorotation
The Couette flow geometry
U0

 eff
 i  T  iy
Stress balance  s  eff
h
 MW z
The Poiseuille flow geometry
2D volume flow rate
Q   u y  z  dz
h
0
The Continuum Governing Equations
EQS & Electro-neutrality
(Haus & Melcher, 1989)
Polarization Relaxation
Equilibrium Polarization
Continuity
 E  0
 D  0
DP  P
1

 v  P    P 
P  Peq
Dt
t
 MW





Peq  Peqy  n, E0 ,  i ,  i  iy  Peqz  n, E0 ,  i ,  i  iz
n

 MW 
21   2
2 1   2
Particle # density
 v  0
Incompressible flow: treating as a single phase continuum
Linear Momentum
(Dahler & Scriven, 1961,
1963; Condiff & Dahler,
1964; Rosensweig, 1997)
Angular Momentum
(Dahler & Scriven, 1961,
1963; Condiff & Dahler, 1964;
Rosensweig, 1997)




Dv
 p  Pt  E  2      v  e 2 v
Dt
2
No-slip boundary conditions  1.5f0 e      ' h 
 0 1  2.5f  Zaitsev & Shliomis, (1969);

I
Rosensweig, (1997)
D
 Pt  E  2  v  2   '       '  2 
Dt




Field conditions: free-to-spin, symmetry, stable rotation
Lobry & Lemaire, 1999; He, 2006; Lemaire et al., 2008
Spin field BCs:


2
v
0   1
Kaloni, 1992; Lukaszewicz, 1999; Rinaldi, 2002; Rinaldi & Zahn, 2002
Polarization Relaxation & Equilibrium Polarization
Shliomis’ First Magnetization Relaxation Equation (1972, 2002)
DM  M
1

 v  M   M 
M  M eq
Dt
t
M






 mH
 coth 
fMd
 kT
M eq
 kT

 mH
Cebers’ Dipole Relaxation Equation (1980)



dP
1
  P 
P  Peq
dt
 MW
   1
  
Peqz  41R3n  2
 2 1  E0
 2 1   2 21   2 

DP  P
1

 v  P   P 
P  Peq
Dt
t
 MW






Peqy  0
DP  P
1

 v  P    P 
P  Peq
Dt
t
 MW



Debye form (Shliomis, 1972,
2002) +
Dahler & Scriven (1963)

Pz 
Py 


Peqz   MW  x Peqy
2
1   MW
 x2
Peqy   MW x Peqz
2
1   MW
x2


We only relax the part of
polarization directly related to
the surface charges on the microparticles, i.e., the retarding part
of the fluid polarization instead of
relaxing the whole total ER fluid
polarization.
Huang, (2010); Huang, Zahn, & Lemaire, (2010a,b)
Polarization Relaxation & Equilibrium Polarization
Electric potential and field solutions to a spherical particle subjected to a uniform DC electric field
rotating at an angular velocity of  .
z
R r 
  r , , f  
    f 
r
 0
2
Laplace’s equation with spherical harmonics
(Jackson, 1999)
BCs (Cebers, 1980; Melcher, 1981; Pannacci,
2006)

r  , E  E0 iz  E0 cos   ir  sin   i
  R  , , f     R  , , f 
n  J f   f v    K f  

R

Kf  fV
y
 2 , 2
x
 f  1 Er  R  , , f    2 Er  R  , , f 
t

f
n  J f   1Er  R  , , f    2 Er  R  , , f 
 f
r
1 ,  1
E †  E0 iz
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):
DP  P
1

 v  P    P 
P  Peq
Dt
t
 MW





  2  1
 2  1 


21   2 
z
3  2 1   2
Peq  41 R n
E0
2
1   MW
2x
Peqy  41 R3n
  2  1
  
 2 1 
 2 1   2 21   2  E
0
2
1   MW
 2x
 MW  x 

Retaining macroscopic fluid spin
Including microscopic particle rotation


  1
    MW



 
Ec  1  2 
 2 1 
2
 E0 

  1,
 Ec 
0,
80 1
31 2  2   1 
E0  Ec
x
E0  Ec
E0
Modeling Results for the Couette Flow Geometry
Schematic diagram for the Couette geometry
Stress balance
U0

 s  eff
 eff
 iz  T  iy
h
 MW
Modeling Results for the Couette Flow Geometry
High shear rates
Low shear rates
We use the continuum
governing equations with the
rotating coffee cup
polarization relaxation model
in the zero spin viscosity limit
'0
We use the continuum governing equations with or
without the rotating coffee cup polarization relaxation
model in the finite spin viscosity small spin velocity limit
2
'0
 MW
x2 1
Figure from Fig. 7b of Lemaire et al., J. Rheol., pp.769, (2008)
1
Comparison of Couette Effective Viscosity Results
Zero spin viscosity Couette effective viscosity results
compared with experimental/theoretical results found
from the literature (Lemaire et al., J. Rheo. pp.769, 2008)
HT: Huang theory (solid line)
LT: Lemaire theory (dashed line)
LE: Lemaire experiment (dotted line)
 1  1.5 108 S m
f  0.05
Ec  0.83 kV mm
'0
Lemaire theory/experimental results are from
Fig. 7a of Lemaire et al. (2008)
Zero spin viscosity results do not
involve ad hoc fitting!
eff  e 
2 h x
U0
Huang, (2010); Huang, Zahn, & Lemaire, (2010a)
Comparison of Couette Effective Viscosity Results
Zero spin viscosity Couette effective viscosity results
compared with experimental/theoretical results found
from the literature (Lemaire et al., J. Rheo. pp.769, 2008)
HT: Huang theory (solid line)
LT: Lemaire theory (dashed line)
LE: Lemaire experiment (dotted line)
 1  1.5 108 S m
Ec  0.83 kV mm
f  0.1
'0
eff  e 
2 h x
U0
Lemaire theory/experimental results are from
Fig. 7b of Lemaire et al. (2008)
Zero spin viscosity results do not
involve ad hoc fitting!
Huang, (2010); Huang, Zahn, & Lemaire, (2010a)
Modeling Results for the Couette Flow Geometry
High shear rates
Low shear rates
We use the continuum
governing equations with the
rotating coffee cup
polarization relaxation model
in the zero spin viscosity limit
'0
We use the continuum governing equations with or
without the rotating coffee cup polarization relaxation
model in the finite spin viscosity small spin velocity limit
2
'0
 MW
x2 1
Figure from Fig. 7b of Lemaire et al., J. Rheol., pp.769, (2008)
2
Comparison of Couette Effective Viscosity Results
Finite spin viscosity small spin velocity Couette
effective viscosity results compared with
experimental/theoretical results found from the
literature (Lemaire et al., J. Rheo. pp.769, 2008)
HT: Huang theory (solid line)
LT: Lemaire theory (dashed line)
LE: Lemaire experiment (dotted line)
 1  1.5 108 S m
  0 In equilibrium
polarization
Ec  0.83 kV mm
Lemaire theory/experimental results are from
Figs. 7a and 7b of Lemaire et al. (2008)
Finite spin viscosity results do not involve ad hoc
fitting!
 1
 '  h2  h20 1  2.5f 
e du*
 
 dz*

*
Huang, (2010)
z 0
*
*
2  z* 0

*
Right sign, voltage dependence, level out or plateau
Comparison of Couette Effective Viscosity Results
Finite spin viscosity small spin velocity Couette
effective viscosity results compared with
experimental/theoretical results found from the
literature (Lemaire et al., J. Rheo. pp.769, 2008)
 *p  0.1
HT: Huang theory (solid line)
Spin viscosity order of
magnitude range consistent
with Elborai (2006) & He (2006)
LT: Lemaire theory (dashed line)
LE: Lemaire experiment (dotted line)
 1  1.5 108 S m
  0 In equilibrium
polarization
Ec  0.83 kV mm
Lemaire theory/experimental results are from
Figs. 7a and 7b of Lemaire et al. (2008)
Physically legitimate ad hoc fitting range:
d 20 1  2.5f    '  h20 1  2.5f 
103   *p   ' h20 1  2.5f   1
e du*
 
 dz*

*
z * 0
*
2  z* 0

*
Huang, Zahn, Lemaire, (2010b)
 1
Right sign, voltage dependence
 *p  0.2
Comparison of Couette Effective Viscosity Results
Finite spin viscosity small spin velocity Couette
effective viscosity results compared with
experimental/theoretical results found from the
literature (Lemaire et al., J. Rheo. pp.769, 2008)
 *p  0.34
HT: Huang theory (solid line)
LT: Lemaire theory (dashed line)
LE: Lemaire experiment (dotted line)
 1  1.5 108 S m
  0 In equilibrium
polarization
Ec  0.83 kV mm
Lemaire theory/experimental results are from
Figs. 7a and 7b of Lemaire et al. (2008)
Physically legitimate ad hoc fitting range:
d 20 1  2.5f    '  h20 1  2.5f 
103   *p   ' h20 1  2.5f   1
e du*
 
 dz*

*
z * 0
*
2  z* 0

Spin viscosity order of magnitude
range consistent with Elborai (2006)
& He (2006)
*
Huang, Zahn, Lemaire, (2010b)
 1
Right sign, voltage
dependence, plateau
 *p  0.6
Modeling Results for the Poiseuille Flow Geometry
Schematic diagram for the Poiseuille geometry
2D volume flow rate
Q   u y  z  dz
h
0
Modeling Results for the Poiseuille Flow Geometry
High pressure
gradients
1
Experimental Ec used in
modeling
We use the continuum
governing equations with
the rotating coffee cup
polarization relaxation
model in the zero spin
viscosity limit
'0
Low pressure
gradients
We use the continuum governing equations with the rotating coffee cup polarization relaxation model in
2
the finite spin viscosity small spin velocity limit
 '  0  MW
x2 1
Figure from Fig. 6 of Lemaire et al., J. Electrostat., pp.586 (2006)
Comparison of Poiseuille Flow Rate Results
'0
Zero spin viscosity Poiseuille flow rate results compared with experimental/theoretical results found from the
Lemaire theory/experimental results are from Figs. 5 and 6 of Lemaire et al., J. Electrostat.,
literature (Lemaire et al., 2006)
HT: Huang theory (solid line)
LT: Lemaire theory (dashed line)
LE: Lemaire experiment (dotted line)
f  0.05
pp.586, (2006)
 1  4 108 S m
Ec  1.3 kV mm
f  0.1
Note: Exp. value for Ec was used in
LT modeling (38% from theoretical)
Zero spin viscosity results do not
involve ad hoc fitting!
Huang, (2010); Huang, Zahn, & Lemaire, (2010a)
Comparison of Poiseuille Velocity Profile Results
Zero spin viscosity Poiseuille flow velocity profiles compared with experimental results found from the
literature (Peters et al., 2010) Lemaire experimental results are from Fig. 9 of Peters et al., J. Rheol., pp.311, (2010)
 1  5.4 108 S m Ec  1.8 kV mm
'0
f  0.05
Cusp structure for zero spin viscosities
p
Pa
 5974.6
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.
Huang, (2010)
Zero electric field solution of
Poiseuille parabolic profile
Modeling Results for the Poiseuille Flow Geometry
High pressure
gradients
Experimental Ec used in
modeling
We use the continuum
governing equations with
the rotating coffee cup
polarization relaxation
model in the zero spin
viscosity limit
'0
Low pressure
gradients
2
We use the continuum governing equations with the rotating coffee cup polarization relaxation model in
2
the finite spin viscosity small spin velocity limit
 '  0  MW
x2 1
Figure from Fig. 6 of Lemaire et al., J. Electrostat., pp.586 (2006)
Comparison of Poiseuille Flow Rate Results
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)
 1  4 108 S m
Ec  1.3 kV mm
f  0.05
f  0.1
Finite spin viscosity results do not
involve ad hoc fitting!
 1
 '  h2  h20 1  2.5f 
Huang, (2010)
Comparison of Poiseuille Velocity Profile Results
Finite spin viscosity small spin velocity Poiseuille flow velocity profiles compared with experimental
results found from the literature (Peters et al., 2010)
Lemaire experimental results are from Fig. 9 of
 1  5.4 10 S m   1  '  0.012h   2.96 10
8
2
10
N s
Best fit, within spin viscosity values calculated
by He (2006) and Elborai (2006) for ferrofluids
Ec  1.8483 106 V m  1.8 kV mm
Ec  1.8 kV mm
round and substitute to
analytic solution
Peters et al., J. Rheol., pp.311, (2010)
p
Pa
 5974.6
L
m
f  0.05
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
Huang Analytic Solutions V.S.
Lemaire Numeric Solutions
Agreement achieved for the all voltages
considered! (Rounding of critical electric field
strength is only within 3%)
Note: if we use particle diameter for spin viscosity,
 '  d 2  6.7 1013 N  s
Three orders of magnitude less than best fit value.
Likely supports ER fluid parcel physical picture
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, 2010
Some Applications of Phenomenon and Theory
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 microparticle electrorotation
2. Offers complementary insights towards the ferrofluid spin-up
problem in magnetorheology
3. Electrically or magnetically controlled suspensions in bio systems
This work is partially supported by the United States-Israel Binational Science
Foundation (BSF) through grant No. 2004081.
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).
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