Note: Descriptions are shown in the official language in which they were submitted.
CA 02238254 1998-OS-22
TITLE OF THE INVENTION
Dielectrophoretic System
NAMES OF INVENTORS
Karan V. I. S. Kaler
Reginald Paul
Lee F. Hartley
FIELD OF THE INVENTION
This invention relates to dielectrophoresis systems.
BACKGROUND OF THE INVENTION
The technique of dielectrophoresis (DEP) characterizes
and separates particles according to specific physical
attributes. The frequency dependent DEP force is sensitive
to particle volume, differences between the particle and
media permittivities and the gradient of the square of the
electric field.
Various electrode configurations are known for
applying electric fields to control particle location. For
example, United States patent no. 4,956,065 of Kaler et al,
issued September 11, 1990, describes a method and apparatus
for three dimensional dynamic electric levitation. Also,
United States patent no. 5,626,734, issued May 6, 1997, of
Docoslis et al, describes a filter that uses
dielectrophoresis to filter particles from a fluid stream.
In recent work illustrated in Fig. 2B, a conical
electrode placed close to a ground plane with the central
axis of the conical electrode oriented perpendicular to the
ground plane has been used to levitate particles between
the conical electrode and ground plane. The position of the
particle above the ground plane may be monitored optically
and a feedback loop used to control the voltage applied to
the conical electrode. In this way, DEP forces balance the
gravitational forces on the particle, and the particle
levitates between the ground plane and conical electrode.
CA 02238254 1998-OS-22
2
The conical electrode also produces a field distribution
that causes the particle to migrate directly under the apex
of the conical electrode. Such migration is termed radial
focussing of the particle. Such a structure functions as a
storage cell or particle memory.
In other recent work, isomotive separation of
particles has been achieved by providing a constant DEP
force on a particle throughout the entire isomotive
structure. The structure shown in Fig. 6 provides a pair of
electrode surfaces labeled +V and -V with a ground place G.
The field near the center of the structure is of low
magnitude, but is highly divergent in all directions. As
one moves to the right along the horizontal axis, the field
intensities increase while becoming highly divergent.
Further still to the right, the field lines become nearly
vertical with only a bEy/bx gradient term remaining.
Throughout the isomotive structure, the inner product of E
with its gradient is constant.
Particles having different particle volumes and
particle permittivities, in the same media and electric
field gradient, will thus experience differential DEP
forces in an isomotive structure and separate into bands of
like particles. Such a structure functions therefore as a
particle shift register.
The confining electrode surfaces of the above
mentioned work are, however, hard to make, and this limits
the commercial feasibility of these designs.
SUN~IARY OF THE INVENTION
Therefore, an object of the invention is to provide
dielectrophoretic apparatus which may control particle
position while being easy to fabricate.
According to an aspect of the invention, there is
provided a dielectrophoretic apparatus, comprising a first
body having a first planar surface, a second body having a
CA 02238254 1998-OS-22
3
second planar surface, the first body being spaced from the
second body with the first and second planar surfaces
facing each other; linear electrodes formed in one of the
first and second planar surfaces, and at least one
electrode formed in the other of the first and second
planar surfaces; and a voltage source operably connected to
the linear electrodes for supplying a voltage distribution
to the electrodes, the voltage distribution being
characterized by, in operation, producing a directional
dielectrophoretic field parallel to the planar surfaces.
For a memory device, the linear electrodes may form a
set of nested loops having a central region and voltage
distribution is characterized by producing a directional
dielectrophoretic field that is radially focussed towards
the central region. The linear electrodes preferably form
a set of concentric circles, with the opposed electrode
forming a ground plane.
In another aspect of the invention, the voltage
distribution is characterized by producing an isomotive
dielectrophoretic field. This may be achieved with the
linear electrodes forming straight lines.
These and other aspects of the invention are described
in the detailed description of the invention and claimed in
the claims that follow.
BRIEF DESCRIPTION OF THE DRAWINGS
There will now be described preferred embodiments of
the invention, with reference to the drawings, by way of
illustration only and not with the intention of limiting
the scope of the invention, in which like numerals denote
like elements and in which:
Fig. 1 is a side view schematic, partly in section,
showing a dielectrophoretic system according to an
embodiment of the invention;
CA 02238254 1998-OS-22
4
Fig. 2A shows equipotential surfaces formed by the
structure of Figs. 1 and 3;
Fig. 2B shows a conducting cone electrode in position
over a ground plane, a structure which is mimicked by the
device shown in Figs. 1 and 3;
Fig. 3 is a top view of the planar electrodes shown in
Fig. 1;
Fig. 4 is a horizontal cross-section showing a second
embodiment of the invention;
Fig. 5 is a top view of the electrodes of Fig. 4; and
Fig. 6 is a horizontal section through a prior art
isomotive structure.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
Isomotive structure as used in this patent document,
means a structure in which a particle is subject to a DEP
constant force throughout the structure or a well defined
region in the structure.
Referring to Figs. 1 and 3, there is shown a first
embodiment of a dielectrophoretic apparatus according to
the invention. An upper plate 10 or other suitable body has
a planar surface 12 on which are formed a series of linear
electrodes 14, shaped in this instance as concentric rings.
The electrodes 14 may be formed of gold or other suitable
metal by any of various well known micromachining
techniques on an insulating material such as glass or
silica. The electrodes 14 need not be exactly circular,
although that is preferred for isolating the particle at a
single point, but should in this embodiment at least form
loops so as to confine the particles. To provide a voltage
to the electrodes 14, a voltage is applied across the inner
V1 and outer V2 electrodes through connectors 16. A
polysilicon resistor strip 18 lies across all of the
electrode rings 14. Due to the increasing resistance of the
polysilicon resistor with length, a different voltage will
CA 02238254 1998-OS-22
be applied at each ring 14. The voltage is supplied through
lines l9 from conventional voltage source 21. Using a three
layer structure with electrodes 14, polysilicon resistor
strip 18 and connectros 16 allows power to be supplied to
5 each ring, without there being a break in the ring. It
would be understood by a skilled person in the art that
various methods may be used to provide power to the
electrodes.
A lower, preferably grounded, electrode 24 is formed
on a planar surface 22 of a second plate, or body, 20,
which is spaced from the upper plate 10 a suitable distance
for dielectrophoresis. Both plates 10 and 20 are supported
by conventional means within a chamber 26. Plates are
preferred for ease of manufacture for bodies 10 and 20, but
any of various shapes may be used. A particle 25 levitated
between the plates may be optically monitored by optical
system 28 and a signal provided in a control feedback loop
to voltage source 21 to control, in conventional manner,
the field strength and therefore the position of the
particle between the plates. A desired height of the
particle may be input at 29 in conventional manner.
By selecting the voltage distributed to the electrodes
14, a directional electrophoretic force parallel to the
planar surfaces 12 and 22 may be formed between the plates .
When the electrodes form nested loops (one loop inside
another and so on) , the particles are guided towards the
inner loop of the loops. If the loops form concentric
circles, particles will migrate towards a central point,
and thus be radially focussed. The electric field produced
by the structure of Figs. 1 and 3 produces a vertically
intensifying electric field whose effect on a particle may
be selected to balance gravitational forces on the particle
and thus levitate the particle. The appropriate voltage for
the electrodes may be readily calculated. It is convenient
to consider the intersection of (a) the equipotential
CA 02238254 1998-OS-22
6
surfaces formed by a conical electrode and plate structure,
which form a series of broadening cones centered on the
conical electrode, and (b) a horizontal plane perpendicular
to the axis of the conical electrode. This intersection
generates a series of concentric rings, and calculation of
the potentials of these rings yields the appropriate
voltage to be applied to the electrodes. In Fig. 2A, the
equipotential surfaces V1-V7 for the device of Fig. 2B are
shown.
The lower electrode 24 need not be at zero volts, but
this is preferred for safety reasons. In addition, the
lower electrode 24 need not be exactly planar. It may for
example be a fine grid, though this is not preferred, since
too large a grid may interfere with focussing. In addition,
the lower body 20 may be formed in various ways such as a
conductive coating on an insulating plate, or as a solid
metal body.
In a further embodiment shown in Figs. 4 and 5, a
structure may be formed whose voltage distribution produces
an isomotive dielectrophoretic field. In this instance,
linear electrodes 34 forming straight lines may be created
on a planar surface 32 of an upper body 30. Voltage is
supplied to the linear electrodes 34 through lines 33 from
conventional voltage source 41. A lower body 40 having a
planar surface 42 has formed on it, again through
conventional micromaching techniques, a matching lower
series of straight linear electrodes 44. Lower electrodes
44 are preferably grounded. The distribution of voltages
required to generate an isomotive force is again found from
the intersection of the eguipotential surfaces of the
structure of Fig. 6 with the desired discrete electrode
plane. The electric field in area A in Fig. 6 corresponds
to the electric field generated by the structure of Figs.
4 and 5. While the linear electrodes do not have to be
CA 02238254 1998-OS-22
7
exactly straight, any deviation from being straight tends
to hurt the sorting function of the structure.
The devices described here have several advantages
over convention constructions. Planar electrode arrays are
readily fabricated with precision using conventional
microfabrication techniques. The use of planar electrodes
results in a chamber with rectangular cross-section in
which a more uniform laminar flow is readily established.
The planar electrode isometric chamber is capable of higher
frequency operation due to a reduction of the chamber
capacitance. The electrode system and associated
electronics may also be contained on the same substrate.
The theory of dielectrophoretic force is known in the
art. A convenient summary is to be found in Lee Hartley's
thesis available through the University of Calgary,
Alberta, Canada. An excerpt from that thesis describing
electrostatic simulation of the forms appendix A to this
patent document in the following pages.
A person skilled in the art could make immaterial
modifications to the invention described in this patent
document without departing from the essence of the
invention that is intended to be covered by the scope of
the claims that follow.
CA 02238254 1998-OS-22
ELECTROSTATIC SIMULATION
Electrostatic field simulation encompasses the computer aided engineering task
of
numerically solving Maxwell's equations for arbitrary geometric problems.
Electromagnetic simulators have gained acceptance for their ability to solve
problems too
complex for analytical representation. Several different numerical methods
form the
computational engines of various commercial solvers. While numerical
comparison
between different algorithms will not be provided, fundamental algorithmic
differences
among three popular methods (finite difference, finite element and boundary
element) will
be presented. For numerical simulation, only a boundary element method
software
package was utilized. The simulation program is a 3-D electrostatic design
package
entitled Coulomb; it is one of a set of boundary element method simulation
packages
offered by Integrated Engineering Software of Winnipeg, MB, Canada.
Finite Difference Method
The simplest, and typically therefore the least accurate, numerical method
available
for electrostatic modeling is the finite difference method. A domain-type
solver, meaning
it involves the direct solution of the governing differential potential
equation, the finite
difference method is typically an iterative process whereby a truncated Taylor
series
expansion of the differential operator is applied at each point of a
rectilinear grid over the
problem region. The uniform grid used to discretize the problem space makes
for crude
geometric modeling particularly in regions containing highly non-linear field
topographies.
In open field problems (or unbounded problems), the method involves a large
number of
CA 02238254 1998-OS-22
9
unknowns, and consequent lengthy solution times, as the uniform grid must
extend across
expansive regions.
Finite Element Method
Another domain-type approach, the finite element method will solve Maxwell's
equations in differential form. The problem geometry is entered, and the
solution space is
discretized into triangular (for 2-D FEM) or tetrahedral (for 3-D FEM) finite
elements.
From this discretized interpretation of the problem, a linear system of
equations is
compiled to calculate the electrostatic field values at the nodes of each
element. Iteration
to a final solution is achieved by the algorithmic goal of minimizing a
function proportional
to the energy of the system. The result produced by a finite element solver is
the direct
solution of the governing differential equation for the potential.
For bounded geometry problems (the electric field in the core of a solenoid),
the
generation of finite elements is less problematic than for open geometries
(the electric field
everywhere external to a solenoid). In the latter case, artificial boundaries
must be
imposed on the geometry to allow the finite element mesh to terminate. This
truncation of
the problem space, a field of research in itself, must be carefully performed
to minimize
errors in the final solution.
The discretization of space, or meshing, for finite element analysis is time
consuming,
but it is a critically important step. In 2-D finite element analysis, mesh
generation may be
done manually with the aid of a graphical interface, but automatic mesh
generators are
commonly provided with 2-D FEM tools. For 3-D FEM solvers, problems with
providing
effective representation of a 3D mesh on a 2D computer screen makes automatic
3D
CA 02238254 1998-OS-22
10
meshing a virtual necessity. By binding all the objects in the problem
together via the rigid
mesh, the finite element method may be heavily burdened with parametric
analysis of
different geometries. This is because re-meshing may need to be redone prior
to each
solver cycle.
For many applications, shortcomings of the finite element method include:
~ large problem sizes
derivative discontinuities in the geometric model
localized errors in fields calculated by differentiation and interpolation
algorithmic difficulties associated with error checking
~ the need to artificially truncate unbounded problems.
This is not to say that the finite element method is flawed, or flatly
inferior to other
methods, it is not. The method is relatively simple to implement in software,
and given an
appropriate mesh design, will yield solutions with high levels of both
precision and
accuracy.
Boundary Element Method
The boundary element method is a boundary-type solver that has historically
been
ignored in favor of finite difference and finite element domain-type solvers.
This is due, in
large part, to the simplicity of implementing domain-type solvers in software.
The
boundary element method, utilizing complex integration techniques, is a
complicated
algorithm to program into software, whereas domain-type solvers rely on simple
linear
algebra to arnve at their solutions. It has not been the theory of boundary
element analysis
impeding its commercialization, but rather the translation of its theoretical
complexity into
CA 02238254 1998-OS-22
11
usable software.
The boundary element method, as with any numerical method, must also
discretize
the problem space into elemental units, but boundary-type and domain-type
solvers are
distinguished by the architecture of their elements and the form of Maxwell's
equations
those elements are used to solve for. The boundary element method only
discretizes the
boundaries between neighboring media and then solves Maxwell's equations in
integral
form along those boundaries. The unknowns being solved for are physical
charges and
currents adequate to maintain all boundary conditions prescribed by the
geometry. The
equivalent sources solved for in boundary element analysis, due to their
physical
significance, provide intuitively simple means for direct computation of
global quantities
such as force, torque, stored energy and capacitance. Additionally, scalar and
vector field
solutions emerge from integral operations which minimize errors by smoothing
discretization noise in the boundary sources.
In boundary element analysis, unlike finite element analysis, unbounded
problems do
not require an arbitrary truncation of the problem space to be performed.
Requiring
elements only on the physical boundaries between media allows for real limits
at infinity to
remain intact. While this elemental variation delivers the benefit of a
dimensional
reduction in the number of elements required, some of this efficiency is lost
to the
increased computational complexity of the integral operation required at each
element.
All things considered, researchers often find it dif~lcult to directly compare
the
performance of finite element and boundary element simulation packages because
their
only practical similarity is that both are seeking discrete representative
solutions to
CA 02238254 1998-OS-22
12
Maxwell's equations. The methods by which each algorithm pursues that goal are
so
fundamentally different that direct comparison is dif~lcult, but it is
generally recognized
that specific advantages do appear for both techniques under certain
circumstances.
CA 02238254 1998-OS-22
13
BEM SIMULATION RESULTS
Integrated Engineering Software's 3-D electrostatic design package, Coulomb v-
2.6,
was used extensively to model and characterize three dielectrophoretic
structures: a dipole
levitator, a quadrupole levitator and an isomotive separator. A three-stage
process was
generally followed for each structure. First, analog models were simulated and
the results
compared with the analytical solutions. Second, information extracted from the
theoretical models was used to determine voltages appropriate for biasing
discrete
electrodes to approximate the analog model. Lastly, the discrete electrodes
were entered
into Coulomb, biased with the boundary voltages obtained above and the
simulations run
again. This three step process forms the closed loop simulator verification
method:
1. the ability of the simulator to match analytical theory is confirmed
2. optimal values for a discrete implementation are obtained from theory
3. the discrete implementation's performance is compared to theory
Dipole Levitator Simulation
rot feedback controlled particle levitation in an axially symmetric electrode
structure
'the traditional approach has been to machine a conducting tip into a rounded
cone shape approximating one of the equipotential surfaces formed by a semi-
infinite line
charge over a ground plane. Experimentation based on this procedure has been
used
successfully to demonstrate feedback controlled levitation and to extract
dielectric
CA 02238254 1998-OS-22
14
information from levitated particles such as plant protoplasts,
The attention paid here to this subject seeks to devise alternative and more
readily
manufacturable means for generating the axially symmetric fields required for
dipole
levitation. The analysis proceeds in similar fashion to that of the prior art,
except that the
r
semi-infinite line charge of ti~~br ~ is replaced by the finite line charge of
Figure 7. The
cost of this modification is that the analytical voltage expression for the
semi-
infinite line charge becomes much more complex in the case of the finite line
charge. This
in turn leads to increased complexity in the electric field (the negative
gradient of the
potential) and ultimately the theoretical dielectrophoretic force term
containing O E 2 .
Utilizing the method of images, the scalar potential fiznction to the geometry
in Figure
7 can be expressed as the integral sum
- f ~~ _ j ~~ 4g
v h 4~c~ (z-~)2 +PZ -a4~c~ (z+~)z + p2 ( )
Evaluation of this integral yields
V= ~ In d-z+ (d-z)2+p2 (h+z+ (h+z)z+pz 49
4~E ( )
h-z+ (h-z)Z+p2 d+z+ (d+z)2+pz
which simplifies to that for the semi-infinite line charge when the limit as d
-~ ~ is
taken.
Continuing with the voltage expression from (49), and restricting the problem
to the
cylindrical axis, i.e. p = 0 , the z-component of the electric field (EZ) is
determined to be
CA 02238254 1998-OS-22
15
Figure 7 - The Finite Line Charge Over Ground Model
E =_ a (vl = ~__ ''i' (hdz-dh2)+(d-h)zz (SO)
8z P ° 2~E z4 -(hz +d2)zz +dzhz
Defining for convenience
al = hdz -~2
a2 =d-h
a3 =h2 +dz (51)
a4 = dzh2
careful differentiation yields the z-axis gradient of the square of the
electric field as
a EZ - ~,Z a, +a2z2 2a2z - (a, +azz2)(4z3 -Za3z)
VZ 2 27CZEz Z4 -a3ZZ +a4 Z4 -a3Z2 +a4 (Z4 -a3Z2 +a4)z (52)
~2
- 2?t 26z G~rrite (Z)
CA 02238254 1998-OS-22
16
where GF;"ite(z) collects the geometric dependencies as has been done before.
Substitution
i-~,,~ y~u',,tu i~ ~-~,r f ~,.e~ ;,.e~' d;~~ ~ I n-rp.~t~.a~ec.~ns,r~,#1.~-
~,~'~
of this result into and normalizing to r3 yields the theoretical
dielectrophoretic force
for a homogeneous sphere centered at height z on the vertical axis in Figure
7:
FDEP = l~'z
r3 ?IEm Re~K, k7Finite (Z) 53
The result of (53) for a finite line charge over a ground plane takes the
exact same form as
that for the semi-infinite line charge over a ground plane, The two
expressions differ only in the geometric terms GFinite(z) (for the finite line
charge) and G~(z)
(for the semi-infinite line charge). Graph 1 illustrates the degree to which
increasing d in
Figure 7 tends the dielectrophoretic force due to a finite line charge (53) to
that of a semi-
infinite line charge as must be the case in the limit.
Before relying on Coulomb to model the dielectrophoretic response of arbitrary
discrete planar electrode geometries, the program's ability to correctly model
analytically
manageable dielectrophoretic phenomenon was to be confirmed. The first such
simulation
studied the dielectrophoretic levitation force profile on a homogeneous sphere
subject to
the electric field of a finite-line charge over a ground plane (Figure 7).
Initially anticipated to be a relatively simple exercise, this 'simulator
verification'
phase of the work was made dramatically more difficult by Coulomb's inability
to
correctly calculate the body force on a neutral object in a non-linear
electric field; the
fundamental precept of dielectrophoresis. Complicating the problem was
Integrated
Engineering Software's persistent unwillingness to acknowledge the defect in
their
CA 02238254 1998-OS-22
17
F~ vs Line Charge Length
1 E+32
1 E+31
- d=250um
1 E+30
L o d=300um
1 E+29 a d=500um
o x d=1000um
1 E+28 ~ Infinite
1 E+27
1 E+26
Graph 1- FDA as Function of Line Charge Length
software. Finally, a simulation was prepared demonstrating Coulomb's blatant
violation of
fundamental energy principles. The technical staff at IES, after reviewing
this submission,
acknowledged the error, and approximately 6 weeks later provided an updated
program to
fix the defect.
To study the finite-line charge dielectrophoretic force profile, the
'universe' in
Coulomb was taken to be free-space, test particles were assigned a relative
permittivity
Ep 4.0 and the solver was run in DC-permittivity mode. The analysis therefore
neglects
particle and media conductivities (taken to be zero) and reduces the Claussius-
Mosotti
polarization factor to a real number.
For the dipole levitation simulations, a finite line-charge model was used in
place of
the analytically simpler semi-infinite model. This was required because
Coulomb does not
0 0.2 0.4 0.6 0.8 1
Normalized Height (zlh)
CA 02238254 1998-OS-22
18
support infinite geometric objects. Additionally, Coulomb does not support
true 'line-
charges', so the finite line charge was represented as two perpendicular
rectangles defined
by ~(8,0, h), (-8,0, h), (-8,0, d )~ and ~(0, 8, h), (0,-8, h), (0,-8, d )~ as
illustrated in Figure
8. These parameters were chosen to be h=200p,m, d=300pm and ~l~m. A total
charge
of 1 Coulomb was distributed over these two surfaces to approximate a 100p,m
line-
charge with charge per unit length ~,=1.OOE+04C/m. To eliminate the need to
geometrically define and element either a 'truncated infinite ground plane' or
an 'image
line-charge', anti-symmetry about the z=0 plane was defined in Coulomb. This
condition
mirrors all geometry across the plane of symmetry and negates boundary
conditions on the
mirrored geometry, i. e. an image line-charge approximation carrying a total
charge of -1 C
extends from z=-h to z=-d. The values h, d and ~, define the physical
parameters required
for analytical tabulation of the normalized dielectrophoretic forces
on particles centered at various axial heights z.
An individual simulation run comprised the introduction to the line-charge
model of a
dielectric sphere of radius r, centered at a height z and filled with a
dielectric material of
relative permittivity 4. To confirm that both the z-position and radial
dependencies on the
dielectrophoretic force are modeled correctly by Coulomb, a batch analysis was
run
whereby three particles of radii 2p.m, 4p,m and 8p,m were translated from
z=40p,m to
z=180p,m in 20p,m increments. At each vertical position, for each particle,
the net body
force (in Newtons) acting on the particle was numerically solved for. The
force values
obtained were normalized by 1/r~ and compared with the analytical prediction
(53).
CA 02238254 1998-OS-22
19
~o,-s - ~,o,d~ ~
d
~s,dl
fs,o,
C/m ~ Charge:
Q=~.*(d-h)
uniformly
distributed
overthese
2 surfaces
Figure 8 - Coulomb Representation of Finite Line Charge
Positional Dependence of Fo~lr3
1.OOE+32
1.OOE+31
x r-2um
1.OOE+30
+ r=4um
a
1.OOE+29 ~c r=Bum
1.OOE+28 - Infinite
p Finite
1.OOE+27
1.OOE+26
0.00 0.20 0.40 0.60 0.80 1.00
Normalized Height (zlh)
Graph 2 - Positional Dependence of FD~/r3 in Finite Line Charge Model
CA 02238254 1998-OS-22
20
113
(FCoulomb~FTheory~
9
8
7
-~- r=2um
E 5 -~ r=4um
~ 4
-~- r=Bum
3
2
1
0
Normalized Height (zlh)
Graph 3 - Particle Radius Extraction from Line Charge Simulation
The results of the above analysis are depicted in Graph 2 and Graph 3. Graph 2
is a
log axis plot of the radially normalized dielectrophoretic force. Shown in the
plot are
normalized results from Coulomb for the 2p,m,4p,m and 8p,m particles, the
analytical force
profile for the finite-line charge model and the analytical force profile for
the semi-infinite
line charge model (for reference). As can be seen, the agreement between the
finite
model's analytical profile and the results obtained numerically from Coulomb
are strong.
Graph 3 depicts an alternate analysis of the data whereby the force data from
Coulomb is transformed by first dividing by the radially normalized analytical
value and
then taking the cube root of the result. This transformation should yield the
radius (in
microns) of the test particle, and should therefore be constant for each
particle at each z-
position. Graph 3 displays the results of this analysis and confirms the
validity of the
0.0 0.2 0.4 0.6 0.8 1.0
CA 02238254 1998-OS-22
21
Coulomb solutions.
Having confirmed Coulomb's ability to model the dipolar dielectrophoretic
force
response for the finite line charge over ground, a discrete planar electrode
geometry
capable of emulating the dipolar levitation profile configuration is sought.
Figure 9
depicts the concept behind the discrete planar dipole levitator geometry.
Using the
analytical solution for the voltage due to a semi-infinite line charge with
~, =1.25x10-9 C/m at a height h=S 10~m over a ground plane in free space, the
potential
was computed at 20 points defined by
[ p, z] _ [SO~rrn + n * 100 fan,500 fan] for n = 0..19 (54)
Geometry consisting of twenty quarter circles centered at [0, 0, SOOfan] with
radii matching
vo vx
~a-.
. Y2
Y4...
(al Cone-Plate Equipotential Cross Sections
ve
(b) Planar Ring Model
Figure 9 - Discrete Planar Representation of Cone-Plate Dipole Levitator
CA 02238254 1998-OS-22
22
the p co-ordinates set out in (54) were entered into Coulomb. The thickness,
or line-
width, of each of these quarter circles were SO~.m. The circles were confined
to the
quadrant defined by (x > 0, y > 0) . Voltage boundary conditions were assigned
to each of
the quarter circles such that the boundary voltage on a particular arc was the
voltage
predicted by theory at the center of that node's SOp.m width. Symmetry was
then defined
about the x=0 and y=0 planes, while anti-symmetry was defined about the z=0
plane.
Under these conditions, the overall system effectively becomes twenty
concentric rings
centered at [0, 0) SOO,can] carrying positive voltage boundary conditions and
a set of twenty
image rings located at z=-SOOfon carrying the opposite voltage boundary
conditions.
Figure 10 - Coulomb Model of Discrete Planar Dipole Levitator
CA 02238254 1998-OS-22
23
To test this structure's ability to emulate the dielectrophoretic force
profile of the
semi-infinite analytical model, a test particle was introduced into the
geometry. A lOpm
radius spherical test particle with relative permittivity 6.9 is centered
along the z-axis and
translated from a center height of 30p,m up to 480p.m. At each height, the
solver was run
and the net body force acting on the sphere was computed. It should be noted
that with
the symmetry conditions defined, the spherical particle was represented in the
geometry as
only the portion of the total sphere residing in the quadrant (x>O,y>0).
Consequently, to
get the representative force on the total sphere, the drawn '/a sphere was
selected and
Coulomb was instructed to also include the three image segments located around
the z-
axis in the force calculation. The negative image sphere existing below z=0
was not
included as, by symmetry, the net force on the upper sphere and it's image
would be zero.
The total geometry drawn in Coulomb is illustrated in Figure 10 where the
twenty
concentric'/4 rings are visible converging above the'/4 spherical test
particle.
CA 02238254 1998-OS-22
24
In Graph 4, the results of this parametric analysis are presented whereby the
performance of the discrete planar electrode configuration generally agrees
well with the
analytical model from which it was derived. However, the deviation between the
two
devices does become extreme in the region approaching the surface of the
planar
electrodes. As this is presumably the region where feedback assisted
levitation may be
performed, this defect is of concern and needs to be addressed. A modified
simulation
was run whereby the inner two rings were replaced by 10 rings of 8 p,m line
width spaced
l7.Sp,m apart (center-to-center) as shown in Figure 11. This data, also
portrayed in
Graph 4, shows the effect on the force profile over the region of interest to
now be
Discrete Planar Dipole Levitator Performance
10p,m Particle; 8 p=6.9; s m=1
:.:yyN.:yy.yy'Y.:yy . ..' _
E-0
8
1.0 ....
...:
:.
.
"~.
..>,.<:
....
..~~w
.:..: :
:
'...
"
'
~e
e
T
~
~...:::,.yy
fn'
yn ~< ~~a
ar
e Ch
'te
emi-
In
"~~~:~::
S rY
v>
9
y
t;::'t;::::,'sr:~2'.:a~'.''.~.xza<v
::~ ... ...
~ Simple: 20 Rings '..'y~.y:
r'..'~:.y
_':::y~
~
.
OE-09 .
1 ' i . ,vyz~zi'_'zi~R'..'.2:\~iz
~
a Enhanced: 10 + 18 Rin s ...<
9 ~
. ,~.~:v:.l.::.,.::~;";ri.
.
.:y
:,
x
z
::~,k..~:
a<x
.
.
.
.
;;;;;,::v:.~
:
x Onl Inner 10
v~.~:~hw~:~~~~x~~'~'~zx;;;>::,
Y
-
;~~;~xy.:
~,~ ~:y
~:;,r;. ~:
;,
~
.
.-. _
rn .
OE-10 .
1 .
..::..<'::,:,::::
,..,w .:..<: . ~~:;.,::w,;.' :.:~:y,w
;.::.w ':::<:r". .. ~::~.:~:zz:;
~x::i:'ri<"~.'~,;~,.:~y ;..::.:;:;..:.:~zzy::;,...,,y.zzzz;;
'<'~::.
.F.~
:y;
v. ' :
:::.y.:zy.Y;~ : y::~~;~;~:>
: :~x~
:
x
'
',"
'.
''
~
~
. _
~
.
ri~z;~ :
~'z~:y;y::X
:
:~<.
~
,
x
:w:.. '
'~r:~,'::.: .:.~n,~:~y.z~.:z.::
~i.
:
~
S
;
~~
~~
~
k'
'
~
_
:
.
":.:
c .::w:<ty r
: 5:.'~::.#~,.k~i
...~:~:~'~
.~,
"j
.
~
"
a .
:t
:Z2i~'''' ::
2yx ~ y "x' 2isY::~y'y2~ n'iiiz2
~ii2:':t~
Y:.
i
~
x
~~
'~
w
t
~
,
c"
:
z
;nz2
rik" :
~
y '
~':yt 'y i2v.'i'.:.
'~:~ :yy h 2.,22yy: 3X'
:za
~'' :z
~
~;
~'
a
d z
r
11 ..
OE ~ ~:~tzy .az.
1 .
riz>;z~;..,
.;i:
xyv :_ ~i~. : v . ..
" :"
s' :.
s}
.
:~' ;?,\z:::.:. z2
':''
tx. "''
:'
~
\..:'
- ~
Z ~
. ,.
y
.
~:
i.
C
.
~.:
.v~~ ::::' \' : Y~2:~.~':?F:tz>2:zk~~sj
~y2t~:.2;,.:: ~:x2.:
:2;~y :'.2::t~::'y.':,"'t :::2z:y "~ Y
'':Y: :::iy.~yz:~. :z::w:b ~:.':22y2iz222:'a:;::
x
x
' ~zz
'~;' ' '
:#
'~v ~ z
'
'
'
'
'
'
'
zz
G7 z
ri
~
z'
:.
.r..
:
~z?..x
:.#~~
i,:
i;;.
::
zzz~, ' zx'..: <:~:t~zzzz 'zz.:
::zzz z.
~
~:
:z:xz::z.
~vw:~~:;zz~y:
'
'
.
.
~
.
.
~xz.. .::i
. ":" y:.k :.wy:yz:
~:~z: zzzxy zyy ri::,..z.:.. z::~ z ..;,::
;
~ ':
~
'
'
:
O E-12 ';, .
0 1 .
'y:~:yv.\'y:':
is:
i' ' iu~i'~:i:iv:$yiun2'.
x ~~;z'
~
~.' ~
~
. ;
. h
LL .:~.zz
~:>::.~ ~~A~:~=.;t..
: ::~ i~
' ; :
~~: z:zz::yzxy,z,~,zz:; ~x;,z:~;;;
?
~'
~
z
zz
xz:
x ~:zzizyzzz.:.zzz' :zz. ~z
~z
.2 L. >i'. y~:r;::2
.. :z>' z~;2;t.:.z..::.:., z.:y vz;<::'
' :
CAC~'y" '~k
2 k :2Y&
'
'
'
y~
":..: y
.~2.:kt"~'fit~:3o;y2~2kh':
zwt
~-~. ...tt.
~.?"ty'Y:~ ,h.~.::~y
~:'~.
~1:
~
'~
i~'~
0 ,
E-13 .t
1 .
.
:
:
:
.
. :. . . .::;:.: ..
ri
ys VFW
i~yvt'.'t~'~~''t:z':::j;';:wiztC~:.'~
4i
y ''
'~
'y'
'
x ' '
<
'
.
?
.
~
z
kit: v
st.
. ' ,; s ty.::.~::a'i' :,-
i:.'.:'
.
zv:
~.~fic2 ~
s
y
'
'
.n
,,
~ y
g
y<z.'
:i~;,".z:i.':'..'i.
':'~'ii2'z2zzzt
i'
.'y;~v:2x
z~w'~~ '
~ ~i~~:~t.<i::' 2ia;. i:'"'.:::.:.'~:'"~ka
't~y''.Y';':
2
zzv=kvi ~v2v~C'.2i'<2ynz2y~x:,~t_.,",,'.:if
'C:i '2i'\i~ 2z.'v.
.. ....y.. .. .
yi
%~
'~~x'~~
1.OE-14 . .
.
.
.'.
0 200 400 600
z (wm)
Graph
4
-
Discrete
Planar
Dipole
Levitator
Performance
CA 02238254 1998-OS-22
Isomotive Simulation
Verification of the isomotive structure by simulation proceeded slightly
differently
than the methods used for the cone-plate and quadrupole geometries. In the two
latter
cases, the existence of a free charge analog model allowed for near exact
geometric
representation in Coulomb. The isomotive electrode structure, on the other
hand, stems
from purely mathematical roots whereby the following condition is enforced:
E ~ OE = Constant (55)
An electrostatic potential expression which satisfies the above relationship
is attained
analytically and used to define equipotential surfaces capable of sustaining
the isomotive
relationship. This scalar potential field is infinite in extent, and no simple
finite charge
model exists which would allow precise representation in a bounded geometry
simulator.
Nevertheless, the methodology for deriving the discrete electrode geometry,
and the
means by which it is tested are very similar tb the method already performed
for the ring
CA 02238254 1998-OS-22
26
dipole levitator. The difference being that the simulator is not used to
characterize an
analytical model (because none exists), but rather the discrete isomotive
structure is
compared to an analytical expectation directly.
As an aside, the isomotive field structure would be more simply represented in
Electro
(the 2D analogue to Coulomb), but this package was not made available to our
facility.
Using a 3D simulator to model a 2D problem requires an arbitrary truncation of
the
infinite third dimension and would be a source of edge effect errors at the
peripheries of
the model.
~or~ "tk~, equation which defines the equipotential surfaces in the isomotive
structure, and considering this potential field as a 2D field, the potential
at a discrete set of
N points (pn) defined by
pn = (X, + (n -1)XS, h) (56)
where
1_<n<_N
X, ~ First point
(57)
XS ~ Electrode spacing
h ~ Vertical position
The electric potential at each of these points can be obtained from (45)
through the
transformations
r" =~(Xo +nXs)2 +hz
8" = atan h (58)
Xo +nXS
CA 02238254 1998-OS-22
27
The set of voltages [V"] may be used to emulate the isomotive field when used
to bias a
set of N coplanar electrodes spaced apart by XS vertically mounted by a height
2h over an
identical set of electrodes biased at the set of potentials [-Vn].
The Coulomb geometry used to represent the discrete isomotive structure is
shown in
Figure 12. The designation of anti-symmetry about the z=0 plane simplified the
geometry
in that only the upper set of electrodes needed to be drawn and elemented. The
length of
each electrode was taken to be 3mm long, and all force computations were
computed at
the midline to minimize truncation edge effects. As was the case when modeling
a finite
line charge over a ground plane, anti-symmetry essentially defines, but does
not require
elementing of, the image conditions necessary to induce an infinite ground
plane. To
utilize the structure of Figure 12 as an isomotive separator, the structure is
mounted
vertically, and a particle stream is injected into the structure from the top
at point x; in
Figure 12. A polarizable particle entering the isomotive structure at this
point will be
forced in a direction parallel to the x-axis. To characterize this discrete
isomotive model,
a hemispherical particle is positioned with its circular flat flush with the
z=0 plane as
illustrated in Figure 12. The particle is parametrically translated from left
to right in
Figure 12. At each position, the 3D solver is run and the net force acting on
the
hemisphere (and its image) is computed. This force is equivalent to the net
force that
would be exerted on a spherical particle centered at (x;, 0) along the midline
between a pair
of anti-symmetrically biased discrete electrode sets.
CA 02238254 1998-OS-22
28
Planar Isomotive Electrode Force Profile
3
2
1
0 0
d
_1 0
a
o -2
0
-3
-4
-5
-6
Graph 7 - Normalized Discrete Isomotive Force Spectrum
The results of the parametric coulomb simulation are presented in Graph 7. The
vertical axis is the simulated force normalized by the expected isomotive
force; FISOM~
An exact representation of the isomotive structure would have yielded a flat
line
throughout the structure at 1.0 in Graph 7. As can be seen from Graph 7, the
discrete
z
Electrode
V Cross Section
,..,.,..,.,. ......,.....,.,..,.,.,..,
Hemispherical
Test Particle
X, ' XN X
Hemispherical ~X~
Image Particle
0 0..~ o o-o.o 0 0 ~ 0 0 0:0 0 0 0 a o 0 0 ~ o o.0 0 0 0 0 o a.:::o 0 0 a ~ o
oo a.
Image Electrodes
Figure 12 - The Discrete Planar Isomotive Electrode System
x position (pm)
CA 02238254 1998-OS-22
29
planar structure is isomotive at the prescribed force magnitude throughout the
central
region of the device, but isomotive behavior breaks down at the devices edges
where
fringing fields due to truncation distort the isomotive field topography.
