Biomedical Engineering Reference
In-Depth Information
interactions between atomic moments in metals, most com-
monly, Fe, Ni, and Co. Ferrimagnetism is similar to ferromag-
netism, but results from exchange interactions in ionic solids,
such as metallic oxides. Both ferro- and ferrimagnetic materials
demonstrate strong enough interactions to maintain a magnetic
field in the absence of an applied field, but when a strong external
field is applied, the atomic moments will align in the applied field
direction. Diamagnetism, paramagnetism, and antiferromagnet-
ism are additional forms of magnetic behavior, but will not be dis-
cussed in any detail here. Superparamagnetism is a unique form
of magnetic behavior that arises in nanoscale particles and will be
described in more detail below.
Like many physicochemical properties, a material's mag-
netic behavior can change as its characteristic dimensions
approach the nanoscale, and this affects the loss mechanisms
in an alternating field (Jordan et al. 1993; Hergt et al. 2002;
Lu, Salabas, and Schüth 2007). Heat generation in magnetic
materials under alternating magnetic fields can be generally
considered in three regimes: eddy current heat generation
(bulk materials), hysteresis heating in multidomain structures
(nanoscale and larger), and relaxation losses in single-domain,
superparamagnetic nanoparticles (Hergt et al. 2002). A sum-
mary of size-dependent magnetic behavior and heating mech-
anisms is included in Figure 17.2 and will be discussed in more
detail later. Eddy currents have already been described with
regard to bulk heating in tissue and are a significant source of
heat generation in the use of magnetic seeds for hyperthermia
(Atkinson, Brezovich, and Chakraborty 2007). However, eddy
current effects are insignificant in the heating of nanoparti-
cles, due to their small dimensions and the low conductivity
of the iron oxides commonly used (Lu, Salabas, and Schüth
2007), so the subsequent discussion will focus on hysteresis
and relaxation losses in magnetic nanoparticles, which have
both been shown to produce clinically relevant levels of heat-
ing (Etheridge and Bishcof 2012a).
Typical magnetic materials demonstrate unique domains of
magnetism (parallel magnetic moments), separated by narrow
zones of magnetic, directional transition termed domain walls.
Domains form to minimize the overall magnetostatic energy of
the material, but as dimensions approach the nanoscale,
the energy reduction provided by multiple domains is over-
come by the energy cost of maintaining the domain walls, and
it becomes energetically favorable to form a single magnetic
domain. A number of methods for estimating the critical radius
for single-domain behavior have been proposed (Lu, Salabas,
and Schüth 2007; Gubin 2009), and the results can vary notably
depending on the approach. Some estimated values from litera-
ture have been included in Figure 17.2, with typical diameters on
the order of tens of nanometers.
When an external field is applied to a magnetic material, the
potential energy of the magnetic moments is minimized by align-
ing with the external field, but energy is also required to rotate
the moments. In a multidomain material, the domains that are
aligned with the external field expand at the expense of the sur-
rounding domains. This motion of the domain walls is associated
with thermal energy losses. The strength of the external field
determines the extent of domain wall motion, until the mate-
rial reaches magnetic saturation and is maximally aligned with
the external field. Upon field reversal, the reverse process occurs,
but in moving back through a zero field, the domain walls do not
return all the way to their original position and there is a rem-
nant magnetization (
M
r
). Thus, under an alternating field, the
material's magnetization creates a hysteresis loop. The coerciv-
ity (
H
c
) is the field required to reduce the magnetization back to
zero. Comparisons of several example hysteresis curves adapted
from data in Hergt et al. are included in Figure 17.3 (Hergt et
al. 2002). Loop 1 demonstrates lower saturation magnetization
than Loop 2, but a much higher coercivity. The power loss can
be approximated by integrating within the hysteresis B-H loop
for each cycle, and so higher heating rates would be expected for
Loop 1. Estimating the expected losses requires measurement of
the hysteresis behavior at the fields of interest. Significant losses
can be obtained for materials with high magnetic saturation and
coercivity. However, magnetic saturation generally requires rela-
tively high fields, and hysteresis loops produced under clinically
relevant fields can shrink significantly, as shown in Figure 17.3.
Despite the absence of domain walls, hysteresis behavior can
still occur in single-domain particles, but involves more com-
plicated processes for reversal, such as buckling and fanning.
Classical physical treatments of these reversal losses (Stoner-
Wohlfarth model) have fallen short in explaining experimen-
tally measured losses in this single-domain hysteresis range, but
phenomenological modeling has demonstrated potential as a
predictive tool. Hergt et al. utilized experimental data on various
magnetic particles, ranging in size from 30 to 100 nm, to produce
expressions that closely predicted losses based on the applied field
parameters and particle size distributions (Hergt, Silvio Dutz, and
Röder 2008). The experimental values and theoretical predictions
offered heating rates comparable to those of superparamagnetic
nanoparticles.
At even smaller dimensions, magnetic nanoparticles exhibit
another type of unique behavior, superparamagnetism, in which
thermal motion causes the magnetic moments to randomly flip
directions, eliminating any remnant magnetization. Thus, a nor-
mally ferro- or ferrimagnetic material will only exhibit magne-
tism under an applied field. This behavior arises because below
a critical volume, the anisotropic energy barrier (
K
u
V
m
) of the
magnetic particle is reduced to the point where it can be over-
come by the energy of thermal motion (
k
B
T
). The definition of
superparamagnetism is somewhat arbitrary, in that it relies on
the choice of a measurement time (τ
m
), for which the behavior is
observed and is generally taken to be 100 seconds. The approxi-
mate critical diameter (
d
c
) for superparamagnetic behavior can
be determined by assuming a spherical geometry and modifying
the equation describing the probability of relaxation (O'Handley
2000):
1
3
×=
τ
τ
KV
kT
6
τ
τ
kT
K
m
u
m
m
B
=
exp
d
ln
(17. 2)
c
π
0
B
0
u
