Particle-Particle Interaction in Electromagnetic Fields for
Force-Field Tailoring
S.S. Wanis, T. Rangedera, C. Justin, N. M. Komerath
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150
(404)894-9622, komerath@gatech.edu
Abstract. Tailored electromagnetic force fields offer the possibility of forming structures ranging in size
from micrometers to kilometers from loose particles. In prior work, the analogy between primary radiation
forces in acoustics and electromagnetics was used to develop and validate a scaling relationship in the
Rayleigh regime. This scaling indicated that a 100m x 100m radio-frequency solar-powered resonator could
form radiation-shielded structural modules large enough for human habitation. Force predictions on a single
particle in a microwave field have been experimentally validated. In this paper, secondary forces due to
particle interaction are linked to first principles, and a dipole model is used to facilitate solutions. Secondary
forces are shown to determine the shape of structures formed. The dipole model serves to illustrate
similarities and differences between acoustic and electromagnetic fields. The nature of the dipole interaction
shows that walls form naturally in acoustic fields. In electromagnetic fields, chains can be formed, with walls
or other structures formed from these chains. Finite element simulations are used to derive interesting field
patterns using given combinations of low-order modes. A 2-dimensional time-stepping calculation shows
chains forming in given fields, while a 3-dimensional calculation shows small chains forming into more
complex structures. Thus the force-field tailoring problem is reduced to one represented by well-known
equations and solution techniques for both primary and secondary forces.
Keywords: Tailored-force-fields, acoustic shaping, acoustic radiation pressure, electromagnetic radiation
pressure, optic tweezers, polarization, nonlinear acoustics, dielectric, scattering
PACS: 07.87+v; 41.20.Jb; 43.25+y
INTRODUCTION AND SCOPE
In previous work, we considered the possibility of building large structures in space using mass from extraterrestrial
sources, using tailored electromagnetic force fields (Komerath, Wanis, and Czechowski, 2003; Wanis and
Komerath, 2005). This concept is well outside the realm of concepts previously considered for space-based
construction. The purpose of this and previous STAIF papers by our group is to systematically present the technical
underpinnings. Our approach has been to reduce the prediction and design to steps that are well-grounded in theory,
with well-known solution methods. Such theory and solution methods are found in fields far removed from spacebased construction. Their assimilation and integration into a coherent approach is a major challenge in developing
the technology for space-based construction. Once integrated, success is in demonstrating that the individual
solutions themselves are well-known.
In (Komerath, Wanis, and Czechowski, 2003) we summarized the relevant prior work in optics, acoustics and
ultrasonics, and showed the analogy between the radiation forces exerted by sound and electromagnetic waves, on
small particles in the Rayleigh scattering regime. In this regime, the wavelength is an order of magnitude larger than
the particle diameter, and hence several simplifications can be made about the nature of the scattered field from the
particles. Gradient forces were shown to form stable locations in the field, towards which particles would be driven.
By using resonators, it was shown that the forces could be amplified by several orders of magnitude from those
produced by traveling waves. The similarity between the governing equations for acoustic and electromagnetic force
generation was shown, and used to develop a scaling relationship for the acceleration per unit intensity, experienced
by particles of given materials (Komerath, Wanis, and Czechowski, 2003). Experimental evidence was then used to
compare the forces on millimeter-sized particles in resonant acoustic fields (Wanis et al 1998; Wanis, Sercovich,
and Komerath, 1999; Wanis, Matos and Komerath, 2000) to micron-sized particles in visible-range laser beams
(Ashkin, 2000; Rohrbach and Stelzer, 2001; Grier, 2003; Zhan, 2003). In subsequent work we (Wanis and
Komerath, 2004) explored the influence of the particle material properties, and the separation of wavelengths needed
for formation of walls, versus those for heating and sintering the surfaces. In (Komerath, Wanis, and Czechowski,
2003) the scaling relationships for the forces on particles were shown to be validated by experiments (Watkins,
Jackson, Barmatz, 1992) in microwave fields. This comparison also showed that predictions made using the
Rayleigh regime simplifications held accurately for experiments conducted at the border between Rayleigh and Mie
scattering (where particles are of the same order of magnitude as the wavelength). This closed the loop from
acoustics to electromagnetics, as well as scaling through the size and wavelength regime from nanometers to
centimeters in particle size, and microns to decimeters in wavelength. Space-based construction requires extension
to decimeters in particle size and 100m in wavelength; however, no theoretical obstacles are seen in the remaining
ranges of size and wavelength.
While the primary force field is adequate to explain why particles are driven towards certain preferred, stable
locations in a resonator, simulations showed that this is not adequate to explain why walls or other structures
actually form. So the question remained whether the analogy between acoustics (where microgravity flight tests
provide clear visual proof that walls are formed along predicted nodal surfaces) and electromegnetics, provides any
useful basis for scaling to long-wave radio frequency fields and large particles. This paper studies the interaction
forces between the particles, which are classified as secondary forces (Wanis, 2006). These forces operate at close
range and thus dictate the formation of structures. The similarities and differences between secondary forces in
acoustic and electromagnetic fields, and the resulting differences in assembling particles to form structures are
presented. A simple dipole interaction model is used, and emphasis will be placed on arriving at the interaction
forces in electromagnetic fields where such a model is accurate. For a comprehensive study comparing the particleparticle interaction forces in acoustic and electromagnetic fields the reader is referred to Wanis (2006).
The results in this paper show walls forming naturally in acoustic fields, whereas chains can form naturally in
electromagnetic fields, with the possibility of forming chain elements into more complex structures. Thus, in
electromagnetic fields, there is one higher degree of control available.
All types of wave motion exert some type of unidirectional force on a reflecting and/or absorbing obstacle in their
path (Torr, 1984). The term radiation pressure was first used in electromagnetics by Maxwell which was then
introduced in acoustics by Lord Rayleigh (1905) as he searched for an acoustic analogue. The prior work on
acoustic radiation forces relevant to wall formation is summarized in (Komerath, Wanis, and Czechowski, 2003). In
the field of optics, several experiments have demonstrated the capability of manipulating micro-meter sized particles
using momentum of laser light (Ashkin, 1970). Ashkin (1970) reported using radiation pressure to trap neutral
dielectric particles. He used the radiation pressure of two counter-propagating laser beams. Higher forces were
obtained (Ashkin et al, 1986) by focusing a single laser beam and thus creating an intensity gradient. Ashkin et al.
(1986) distinguish between gradient and scattering forces and a variety of optical traps for neutral dielectric particles
in both the Mie and Rayleigh regimes. They have shown that a laser beam going into or out of a dielectric liquid
(water) will exert an outward radiation pressure on the interface separating the water from air, i.e. away from the
water into the air.
Optic tweezers were used to trap and cool neutral atoms (Chu et al, 1986), manipulate single living cells (Ashkin,
1987), and to measure elastic properties of cells and molecules (Svoboda and Block, 1994). Constable et al (1993)
used a pair of pig-tailed optic fibers to deploy both scattering and gradient forces to trap particles in the gap between
the fiber ends. Most recently, Metzger, Jess and Dholakia (2004) used laser tweezers composed of a pair of pigtailed fiber optics to controllably push and pull trapped particles of various sizes.
PRIMARY ELECTROMAGNETIC FORCES
The interaction of a neutral particle with an electromagnetic field can be characterized by scattering and gradient
forces. Scattering forces are a consequence of the momentum transfer that occurs during reflection, refraction and
absorption of the wave when it meets a particle of different impedance than the background host medium. Gradient
forces are a consequence of the differential Lorentz force on the induced dipole within the dielectric placed in a nonuniform electric field. These forces can be described using the Maxwell stress tensor (Mulser, 1985). A neutral
dielectric particle placed in an alternating electric field will polarize in response to the field. The amount of
polarization is determined by the ratio of the dielectric constant of the particle to the host medium. If the physical
dimensions of the particle are smaller than the wavelength (by about one order of magnitude) then the field radiated
by the polarized particle will resemble that of an electric dipole. Rohrbach and Stelzer (2001) gives a theoretical
derivation of the optical forces onto dielectric matter which traces the origin of both forces to one force expression.
The two forces are obtained as components of one electromagnetic force.
In the following sections, expressions for these forces are traced back to first principles. The modeling of the
secondary force is considered. The objective is to show that the issue of force-field tailoring can be described by
equations derived from first principles. These equations are themselves well-known, and their solutions are
described in textbooks and used in commercially available application software. As such the contribution of the
paper is not the development of solution methods for equations, but the demonstration that a complex and new
capability can be reduced to scientific descriptions that link to well-known equations. Once this is done, solutions
obtained using commercially available solvers are presented.
Gradient Forces
The starting point for the discussion is the total electromagnetic force on a charged particle:
f = q( E + v " B)
(1)
the electromagnetic force density f acting on a small dielectric particle or volume element with dipole moment
density P is given by (Rohrbach 2001)
!
f = P"E +
#P
$B
#t
(2)
Expanding the terms by substituting in the electric field of a dipole, P results in
!
(
)
F = % e$ o # E 2 2 + % e$ o
"
[E ! B]
"t
(3)
Thus the total force F can be split up into its components. The first term Fgrad is characterized by the gradient of
intensity ~ E2, and the second term Fsctr is proportional to the time rate of change of momentum due to scattering.
Again, for small particles such that we can consider the external electric field to be constant over its dimensions, the
problem is equivalent to that of an induced point dipole. In this regime, the instantaneous Lorentz force acting on
the dipole can be found using the electrostatics polarization modified to the general dielectric case (Harada 1996) in
the electrostatic limit ∇×E=0 :
' ( ) (o $ 1
"" !E 2 (t )
F(t ) = [p(t )+ ! ]E(t ) = 4*( o a 3 %%
& ( + 2( o # 2
To find the gradient force, the force in Equation 4 is time-averaged over one period of a harmonic electric field:
(4)
' ) ( )o $
2
emag
""! E
Fgrad
= F (t ) = *) o a 3 %%
& ) + 2) o #
(5)
Compare this to the scattering force which always points in the direction of the wave, i.e. along the Poynting vector
direction.
Scattering Forces
A beam of light exerts a force on an infinite plane. This force can be described as F=QI/c where Q=1.0 for a
perfectly absorbing surface and Q=2.0 for a perfect reflector. In the Rayleigh regime where the scattering object is
much smaller than the wavelength of the incident radiation, the scattering is isotropic, and the scattering cross
section equals the cross section for the radiation pressure (Kerker 1969). The scattering cross-section from a perfect
conductor in the Rayleigh regime can be expressed as
C sctr = C pr =
10 2
4
! a (k o a )
3
(6)
This can be generalized to a dielectric sphere having dielectric constant ε and host medium εo by multiplying by the
polarization vector,
& ( ' (o
( o $$
% ( + 2( o
#
!!
"
(7)
This term is a result of the difference between the fields inside and outside the sphere, and was obtained by solving
for the potential inside and outside the sphere with the appropriate boundary conditions. For the perfect conductor,
the field inside is exactly equal to the external field so as to achieve the zero boundary condition at the surface. The
modified radiation pressure cross section becomes
10
4 & ' ( 'o #
!!
C pr = )a 2 (k o a ) $$
3
% ' + 2' o "
2
(8)
Intensity I, and electric field intensity E, are related by the relation, I= ½ εocE2 (Griffiths 1989). For a harmonic
electric field spatial distribution, E=Eosin(kox), the force can be expressed as,
Fsctr =
C pr
c
I = ! o C pr E o
2
(9)
The entire force expression is written out below:
2
emag
sctr
F
5
4 & ' ( 'o #
!! E o 2 sin 2 (k o x )
= )'o a 2 (k o a ) $$
3
% ' + 2' o "
(10)
SECONDARY ELECTROMAGNETIC FORCES
In the case of electromagnetics, it is accurate to model the effect of particles on the field, as the effect of dipoles.
The energy U of a single dipole with moment p in an electric field E is
U = "p!E
(11)
The electric field produced by a dipole p1 is
E1 =
1
[3( p1 • rˆ)rˆ $ p1 ]
4"#0r 3
(12)
If we now introduce a second dipole p2 close to p1 and separated by a distance r should have an interaction energy
given by
!
U = " p2 ! E1 =
1
[p1 ! p2 " 3(p1 ! rˆ )(p2 ! rˆ )]
4#$ o r 3
(13)
MODEL SIMULATION RESULTS
The appropriate boundary condition to use in a resonator to study force field tailoring is to use the concept of a
perfect conductor. Figure 1 shows the interaction potential energy computed using the MATLAB commercial
software package, in a 3D polar plot. The height indicates an increase in potential. Positive height thus thus
corresponds to repulsive force regions and negatives correspond to attractive force regions. Note that this potential
plot is what a second particle would experience as it approaches in the plane spanned out by the outer circle of the
plot.
FIGURE 1. Polar plot where height denotes the 2D potential interaction energy between two interacting dipoles with parallel
dipole moments.
From this energy, the force can be computed by taking the negative of its gradient:
Fs =
p1 p2
1" 3cos2 #
4
r
(
(14)
)
The polarizations p1 and p2 contain both electric field E and the cube of particle radius a3. Thus this makes the force
proportional to E2 and a6. The inset in Figure 2 qualitatively shows the expected 3D behavior, obtained by rotating
the above attraction/repulsion zone about
! the dotted axis shown. This yields a space around each induced dipole that
experiences attraction along one axis and repulsion along the other two. Thus, the resulting structures in
electromagnetic resonant fields are chains that can vary in 3D space. Chain orientations depend on the local field. To
achieve complete surfaces in electromagnetics we postulate that a second mode can be excited once chains have
formed and cured, to force the chains to coagulate and then cure them into complete surfaces. This situation
corresponds to that of positive dielectrophoresis (Pohl, 1958), where the formation of chains of particles
perpendicular to a stable charged surface has indeed been observed (IBMM, 2006). We postulate from the results
above that even in the case of neutral particles within a resonator, and in the absence of a charged surface, such
chains will form.
The possibilities of electromagnetic shaping were explored along two approaches, and results from initial studies are
given in this paper. One line of exploration was to efficiently generate solutions for different combinations of
modes, resonator geometries and wall boundary conditions. This was performed using the COMSOL commercial
software, which can be used to model both acoustic and electromagnetic fields. This is a finite element solver for the
describing equations. The boundary conditions used were perfect conductors for the electromagnetic field
boundaries, and hard walls for the acoustic field boundaries. Unforced solutions were obtained for different
geometries. These solutions are adequate to calculate fields including multiple static particles. To compute the
dynamics of the particles to understand wall or chain formation, a separate computation is required. The particle
interaction dynamics were computed using a separate FORTRAN-based time-stepping simulation.
attractive
α=54.73o
External Field
Eejω t
repulsive
repulsive
P
1
Fs
attractive
FIGURE 2. Two dimensional schematic showing the regions of attraction and repulsion between two induced dipoles P1 and P2.
The inset shows a schematic of force directions in 3 dimensions.
Mode shapes
Evidently, several interesting geometries can be obtained by suitable choice of modes and container shapes – a result
with significance to the development of flexible manufacturing facilities, for instance those to be located in Space.
In the case of electromagnetics, the mode shapes do not immediately yield contours of the structure shapes that can
be built – this process requires the time-stepping computation discussed below. Figures 3 and 4 show the intensity
contours corresponding to a TEM modes 110 and 400 in a cylindrical resonator, computed for perfectly conducting
walls using the COMSOL electromagnetics computation module.
FIGURE 3. Intensity contours for the TEM(11) 2D mode in a
cylindrical resonator.
FIGURE 4. Intensity contours for the TEM(40) mode in a
cylindrical resonator.
Particle Interaction Simulations
A time-stepping algorithm was developed to simulate particle interaction dynamics, where particles of specified
shape and material properties start at specified locations in a field corresponding to a given combination of modes in
FIGURE 5. 2-Dimensional (110) mode wth 75 particles, 10
seconds and 3000 time steps.
FIGURE 6. 3-dimensional behavior showing small chains of
particles forming and then drifting along other directions.
a given container geometry. Forces experienced by the particles were used to compute their motion, and the
locations were updated at every time step. A “soft collision” model was developed in order to enforce the condition
that particles could not overlap, in the time-stepping computation. The soft collision model uses a force given by
FMD = " ln( a " r ) " ln(r) " Fatt
(15)
where a is particle diameter, r is distance between particle centers, and Fatt is the attraction force between the
particles. Figure 5 shows results in a 2-D computation with 75 spherical particles in a resonator operating in the 110
mode. At the end of 3000 time steps,
! the particles have accumulated, forming not only cross-shaped chains of
particles, but also a more complex grouping in a surface. In Figure 6, the three-dimensional behavior is studied,
albeit in an incomplete computation, at the end of 3000 time steps. Now we can see that small chains of particles
themselves accumulate along other chains, validating the idea that complex shapes can be formed from the chains
once they are formed.
CONCLUSIONS
The grand objective of tailoring force fields to build structures of many sizes, is seen to be reducible to processes
that are described by the well-known equations of acoustics and electromagnetics. Primary forces on particles in
electromagnetic resonators are generated from the electric field which is governed by the specific resonator mode
being excited. Primary forces are responsible for collecting particles into regions dictated by the minimum potential
of the external field. Secondary forces are induced from the scattered electric fields of particles at close range and
thus dictate the final form of the structure produced. A dipole model is successful in predicting particle interactions.
In transverse electromagnetic modes, particles approaching a stable location are attracted towards particles already
there, along the axis connecting the particles. Weak repulsion acts in the orthogonal directions. Thus it is predicted
that chains are produced in electromagnetic resonators containing neutral dielectric particles. Simulations show that
these chains can be varied in three dimensions as predicted by the primary force field governed by the external
electric field in the cavity.
NOMENCLATURE
All dimensions are in SI units.
a
α
Β
c
C
ε0
ε
E
f
F
I
k
P
Q
q
r
t
U
v
= particle diameter
= polarizability
= magnetic field vector
= speed of l ight in v acuum
= scattering cross section
= dielectric constant of the medium
= dielectric constant of the particle material
= electric field
= electromagnetic force density
= force
= intensity
= wave number
= electric field of a dipole
= reflectance factor (1.0 for perfectly absorbing and 2.0 for perfectly reflecting).
= charge on a particle
= distance from particle center
= time
= energy
= velocity
ACKNOWLEDGMENTS
The first author’s PhD thesis work, which forms the basis of much of this work, was supported through the NASA
Graduate Student Researcher Program. Dr. Sheila Thibeault served as the NASA Mentor. The other authors’ effort
was supported by an NIAC Phase 2 grant from the Universities Space Research Association through NASA Institute
of Advanced Concepts. The technical monitor is Dr. Robert Cassanova.
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