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Supplemental materials to be available online
Optimization of force produced by electromagnet needles
acting on superparamagnetic microparticles
Yu Xiang, Jacob Miller, Vincent Sica and David A. LaVan1
Numerical Method
To avoid singularities, it is often easier to solve a magnetic field using the magnetic
potential A as the dependent variable[33], which is defined as B    A . MaxwellAmpere’s law for magnetostatic fields is   H  J , where H is the magnetic field
intensity and J is the external current density. With the constitutive relationship, B=μ0 (H
+M), the equation can be rewritten as:


   01  A  M  J
(A1)
For axisymmetric problems, (A1) can be written in polar coordinates (er, eφ, ez), as:


 u 


M
2
u




    1  r 
z
1
  ,  r 0     0    
   J

u

M
0

r

z



  
r 
 


 z 


(A2)
where u = Aφ / r.
For this work, equation A2 is solved using the finite element method, in a space that
is much larger (50D x 25L) than the EMN’s characteristic dimensions (D x L), to obtain
the magnetic field and magnetic force generated.
To obtain high accuracy of the magnetic force results, the mesh resolution around
the needle tip and the particle has been adjusted to be of ultra fine quality (see Fig 1-b1).
1
david.lavan@yale.edu, (203) 432-9662, fax (203) 432-7654
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The formula F=m·B can be used to estimate magnetic forces on infinitesimal
particles[1]; for particles with finite size, an alternative and more accurate technique is
preferable, and has been adopted in these simulations. The Maxwell stress tensor is integrated
over the entire particle boundary (on the outer surface). Inside the particle, the Maxwell stress
is balanced by non-electromagnetic stresses to assure div(stress)=0; otherwise there will
be momentum transport inside the particle. Only the Maxwell stress on the vacuum side
contributes to the total net force [34]. The Maxwell stress tensor is given by [35]:
T
 1

BB T  
B  B  M  B  I
0
 2 0

1
(A3)
Most previous closed-form solutions and derivative design philosophies related to
this problem have ignored these nonlinear behaviors and relied on far-field assumptions
that are not valid for particles close to the tip. Liu et al. assumed a constant susceptibility
for the magnetic particles when studying the transport of magnetic beads in a micro
device [4]. Warnke modeled separation of magnetic particles in fluids subjected to a uniform
external magnetic field; they assumed a piecewise linear function: their M-H curve has a positive
slope before the saturation threshold; beyond the threshold the curve is flat [28]. In a simple
analytical model, Driscoll et al. estimated the magnetic force exerted on a
superparamagnetic microsphere in blood flow by using a Langevin function to fit the
non-linear M-H behavior of the microsphere.[5]
In this study, the nonlinear B-H curve (see Fig. S-1a) of the core material is
reconstructed from raw data[30]. The curve is composed of two segments[36]. The
initial part assumed B  (   H ) H where μα=20,000, ν=350,000, for 4-79 Mo
Permalloy[36]. The hysteresis part has the form of
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B
 H  HC
arctan 

 HB
2 BS



(A4)
where BS is the saturation value of the flux density, H is the external field, HC is the coercive
field, and HB is a threshold field that has to be overcome in order to saturate [37]. BS is 0.75 T,
HC is 0.557 A/m. HB can be determined mathematically with the knowledge that B=0.3 T at
maximum relative permeability, μmax = 3.25x105, for the core material [30]. The B-H curve of
ASTM A848 is similarly obtained, and is also shown in Figure S-1a.
Similarly, the reconstructed particle’s M-H curve is represented by
M
2M S

 H
arctan 
 HM

 , where MS = 3.25x104 A/m is the saturation level of magnetization,

and HM =1.59x105 A/m [31]. This M-H curve is shown in Figure S-1b
Geometric constraints for numerical optimization were set to avoid geometry that
could not be produced: Lmin < L < Lmax; Dmin < D < Dmax; 0 < Lt < L; rmin < r < D/2; b >
0; 0 < m < 7 (m is an integer); a + Lt > 0 and (n - 1)Δs + a <L- Lt. The last two
constraints were chosen so that the coils do not extend beyond the far end of the core.
The far end of the core was given a radius, Rf, of D/5 to reduce singularities in the model
that would exist with a perfectly sharp corner on the core. For practical reasons, the
geometry was further constrained such that Lt >Rf and L- Lt > Rf.
The optimization algorithm returns optimal geometry for maximum force and/or maximum
force-per-unit-power, based on the geometric parameters and constraints outlined in the main
paper and above. Other fixed parameters for the optimization calculations were the use of 36
gauge magnet wire (125µm-diameter) for the coils, fixing the coil to be 3 layers thick (3 columns
and 8 rows), and assuming the magnetic particle was placed at various fixed distances from the
tip (h = 10, 100 and 1000 µm). Further, b = bmin = tcoil, and a = amin = 0 such that the centroid of
the coil matrix is nearest to the needle tip.
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Figure S-1. (a) Reconstructed B-H curves of 4-79 permalloy and ASTM A848 pure iron. (b)
Reconstructed M-H curve of a superparamagnetic microparticle.
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