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Magnetic field orientation dependent dynamic susceptibility and Brownian relaxation time of magnetic nanoparticles

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Jing Zhong at Beihang University
Jing Zhong
  • Beihang University
Niklas Lucht
Niklas Lucht
Niklas Lucht
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Birgit Hankiewicz at University of Hamburg
Birgit Hankiewicz
Meinhard Schilling at Technische Universität Braunschweig
Meinhard Schilling
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Abstract

This paper investigates the dynamic ac susceptibility (ACS) and the Brownian relaxation time of magnetic nanoparticles (MNPs) in dc magnetic fields with arbitrary orientations with respect to the ac magnetic field. A CoFe2O4 MNP sample, dominated by Brownian relaxation, is used to perform ACS measurements in an ac magnetic field with a constant amplitude of 0.2 mT (from 2 to 3000 Hz) and a superposed dc magnetic field with amplitudes ranging from 0 to 5 mT. Experimental results indicate that the ACS and Brownian relaxation time are significantly affected not only by the strength but also by the orientation of the dc magnetic field. Moreover, a mathematical model is proposed to analyze the ACS and Brownian relaxation time in dependence of the orientation of the dc magnetic field, which extends the established models parallel or perpendicular to arbitrary-oriented dc magnetic fields. Experimental results indicate that the good fitting between the experimental data (ACS and Brownian relaxation time) and the proposed models demonstrates the feasibility of the proposed model for the description of ACS and Brownian relaxation time in arbitrary-orientated ac and dc magnetic fields.
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Magnetic field orientation dependent dynamic susceptibility and Brownian
relaxation time of magnetic nanoparticles
Jing Zhong1,a), Niklas Lucht2, Birgit Hankiewicz2, Meinhard Schilling1, and Frank Ludwig1
1Institut für Elektrische Messtechnik und Grundlagen der Elektrotechnik, TU Braunschweig D-38106 Braunschweig, Germany
2Institut für Physikalische Chemie, Universität Hamburg D-20146 Hamburg, Germany
a)E-mail: j.zhong@tu-braunschweig.de
This paper investigates the dynamic ac susceptibility (ACS) and the Brownian
relaxation time of magnetic nanoparticles (MNPs) in dc magnetic fields with arbitrary
orientations with respect to the ac magnetic field. A CoFe2O4 MNP sample, dominated
by Brownian relaxation, is used to perform ACS measurements in an ac magnetic field
with constant amplitude of 0.2 mT (from 2 to 3000 Hz) and a superposed dc magnetic
field with amplitudes ranging from 0 to 5 mT. Experimental results indicate that the
ACS and Brownian relaxation time are significantly affected not only by the strength
but also by the orientation of the dc magnetic field. Moreover, a mathematical model is
proposed to analyze the ACS and Brownian relaxation time in dependence of the
orientation of the dc magnetic field, which extends the established models for parallel or
perpendicular to arbitrary-oriented dc magnetic fields. Experimental results indicate that
the good fitting between the experimental data (ACS and Brownian relaxation time) and
the proposed models demonstrates the feasibility of the proposed model for the
description of ACS and Brownian relaxation time in arbitrary-orientated ac and dc
magnetic fields.
Magnetic nanoparticles (MNPs) are of great promise
in biological and biomedical applications, such as
heaters in magnetic hyperthermia,1-4 markers in
biosensors,5 sensors in MNP thermometry6-8 and
contrast agents in magnetic particle imaging (MPI).9-14
In these applications, the dynamic MNP magnetization
and its spectra induced by ac and/or dc magnetic fields
significantly affect their performance. For instance, the
MNP dynamic magnetization is crucial to the heating
efficiency in magnetic hyperthermia15 while its spectra
play significant roles in temperature sensitivity when
using the spectra for thermometry16,17 and image
resolution/quality in MPI.12,18 In a sufficiently
high-frequency ac magnetic field, MNP relaxation may
worsen the systematic error19 in MNP thermometry
while it further blurs the MPI images.12,18 A
comprehensive study of the dynamic ac susceptibility
(ACS) and relaxation of MNPs in ac and/or dc
magnetic field is of great significance and interest to
biomedicine and biology.
Brownian relaxation depends not only on the
hydrodynamic size, viscosity and temperature but also
on external excitation magnetic fields. It is well-known
that an increase in the amplitude of an ac magnetic
field as well as of superposed parallel/perpendicular dc
magnetic fields decrease the Brownian relaxation
time.20-23 Physical and/or phenomenological models
were proposed to describe the dependence of MNP
Brownian relaxation on ac and dc magnetic fields, as
well as the ACS spectra.20-22 However, regarding
superposed dc magnetic fields, only the parallel or
perpendicular orientation with respect to the ac
magnetic field were taken into account. In some
specific applications, e.g. MPI, the ac and dc magnetic
fields have arbitrary orientations in the imaging field of
view. Understanding the MNP spectra and Brownian
relaxation in the arbitrary-orientated ac and dc
magnetic fields, as well as the underlying physical
mechanisms, is of great significance to viscosity and
molecule-binding imaging with MPI. To date, there
have not been any studies or physical models to
describe the MNP relaxation and ACS in
arbitrary-oriented dc magnetic fields.
In this paper, we investigate the ACS and the
Brownian relaxation time of a CoFe2O4 sample in
arbitrary-orientated ac and dc magnetic fields. ACS
spectra of a Brownian relaxation-dominated CoFe2O4
sample are measured whereas the Brownian relaxation
time is extracted from the ACS spectra. Moreover, a
mathematical model is proposed to describe the
experimental results.
In a small-amplitude ac magnetic field Hac without a
superposed dc magnetic field Hdc, the Debye model is
used to describe the dynamic ACS24
B
i

+
=1
0
, (1)
where
0 is the static value calculated from the static
Langevin function,
is the angular frequency and
B is
the Brownian relaxation time. Note that at infinite
frequencies some MNPs with relaxation times below 1
s and/or intra-potential-well contributions cause the
dynamic ACS not drop to zero but have a constant
value.25
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A superposed Hdc significantly affects the ac
susceptibility and the effective Brownian relaxation
time. A physical model for the description of the
dynamic ACS
para (
perp) in a parallel (perpendicular)
Hdc is given by20
( ) ( ) ( )

para
para i
L
+
=1
1
30
, (2a)
( ) ( ) ( )

perp
perp id
dL
+
=1
1
30
, (2b)
where
para (
perp) is the ac susceptibility in a parallel
(perpendicular) bias dc magnetic field.
para (
perp) is the
effective Brownian relaxation time in a superposed Hdc
parallel (perpendicular) to the small-amplitude ac
magnetic field. L(
) is the static Langevin function,
=
0mHdc/kBT, m is the MNP magnetic moment, kB is
Boltzmann constant and T is the absolute temperature.
Herein,
para and
perp are given by19
( ) ( )
( )
Bpara d
Ld
ln
ln
=
, (3a)
( ) ( )
( )
Bperp L
L
−
=2
, (3b)
where
B = 3
Vh/kBT,
is the medium viscosity and Vh
is the hydrodynamic volume of the MNPs.
To investigate the dependence of ac susceptibility
and effective Brownian relaxation time of MNPs on
the magnetic field orientation, a CoFe2O4 MNP sample
suspended in water and glycerin (glycerin weight
percentage: 80%) is used for experiments. The
CoFe2O4 MNPs have a nominal core diameter of 15.5
nm and a hydrodynamic diameter of 38.6 nm.26 The
viscosity of the MNP suspension amounts to about
34.8 mPa· s from theoretical calculation.27 The MNPs
are dominated by Brownian relaxation due to the very
high anisotropy energy barrier of CoFe2O4. A
rotating-magnetic-field (RMF) system with
two-dimension Helmholtz coils and a fluxgate-based
measurement system is used to generate
arbitrary-orientation Hac and Hdc with different
frequencies and measure the ACS spectra. The details
of the RMF system can be found in Ref. [28]. A
2-dimensional Helmholtz coil system is used to
generate the arbitrary-orientated ac and dc magnetic
fields in one plane while a fluxgate based measurement
system measures the MNP response in the direction
perpendicular to the plane of the applied magnetic
fields. The ac magnetic field with constant amplitude
of 0.2 mT is applied with a frequency range from 2 Hz
to 3000 Hz.
Fig. 1 shows the measured ACS spectra (real
’ and
imaginary
’’ parts) at room temperature without a
superposed Hdc.
’’ shows a peak frequency at about
128 Hz, which can be used to determine the effective
Brownian relaxation time. To accurately determine the
Brownian relaxation time from the spectra, the
measured ACS spectra are fitted with the
phenomenological Havriliak-Negami model22,29
( )
( )

−
+
+= 1
inf
1B
amp
i
, (4)
where
inf is the measured susceptibility at a very high
frequency,
amp is the amplitude of the
frequency-dependent ACS,
inf +
amp is the DC
susceptibility.
(
) affects the width (asymmetry) of
the imaginary parts of the ac susceptibility. In this
paper, Eq. (4) is only used to fit the ACS spectra at
different dc fields and angles to accurately obtain the
characteristic Brownian relaxation time. The fits of the
experimental spectra with Eq. (4) are presented as
solid/dashed lines in Fig. 1. The effective Brownian
relaxation time
B (about 1.32 ms for the given case) is
determined from the peak frequency of
’’.
Fig. 1. Ac susceptibility spectra of the CoFe2O4 MNP sample.
Symbols are experimental data whereas solid/dashed lines are fits
with Eq. (4).
Fig. 2. Real and imaginary parts vs. angle
at different frequencies.
Symbols are experimental data whereas solid/dashed lines are guides
to the eye.
Due to the symmetry of the Hac and Hdc orientations
in the four quadrants, the ACS of the MNPs in the first
quadrant should be the same as those in the other three
quadrants. To verify this, the ACS, in dependence of
the angle
between Hac and Hdc, are measured at three
different frequencies in all four quadrants with Hdc = 3
mT, as shown in Fig. 2. It demonstrates the symmetry
of the ACS across the four quadrants. In addition, the
values of the ACS (
’ and
’’) in perpendicular Hdc are
greater than those in parallel Hdc. We have performed
simulations with Eqs. (2) and (3) to calculate the ACS
in parallel and perpendicular Hdc, which shows the
same behaviour (simulation results are not presented
here). Therefore, due to the symmetry in four
quadrants, the experiments on the ACS measurements
in this paper are only performed for
arbitrary-orientation Hac and Hdc in the first quadrant.
The superposed Hdc was varied from 1 mT to 5 mT
with a step of 1 mT while the angle
between Hac and
Hdc varies from 0° to 90° with a step of 10°. Fig. 3
shows the experimental results of parallel and
perpendicular imaginary parts
’’ in different-strength
Hdc. The maximum value of the parallel and
perpendicular
’’ decreases while the peak frequency
shifts to higher frequencies (the Brownian relaxation
time decreases) with increasing Hdc. Qualitatively, the
experimental results agree with Eqs. (2a) and (2b), as
well as Eqs. (3a) and (3b). We have also tried to fit the
ACS spectra in different Hdc with Eqs. (2a) and (2b) by
considering a lognormal size distribution of the core
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and hydrodynamic MNP diameters. However, the
fitting does not work very well, which might be caused
by some clusters in the MNP sample. Previous studies
have already demonstrated that the used MNP sample
contains some trimers, causing the fitting of the ACS
spectra to be more complicated. In addition, the trimers
may lead to a change in the effective magnetic moment
in different applied magnetic fields,26,30 which might
also be responsible for the comparably poor quality of
the fits. Nevertheless, it does not affect the study of the
dependent ACS and Brownian relaxation time.
Figure 4 shows the experimental curves of
’’
versus frequency at Hdc = 1 mT (Fig. 4a), 3 mT (Fig.
4b) and 5 mT (Fig. 4c). Fig. 4a indicates that – at Hdc =
1 mT – a decrease in
in the first quadrant shifts the
peak frequency of
’’ to a higher frequency and
decreases the amplitude of
’’. In addition, with
increasing Hdc, the peak frequency shifts to an even
higher frequency and the amplitude further decreases,
which qualitatively fits with Eqs. (2) and (3). Herein,
we propose a mathematical model to describe the
dependent susceptibility:
( )
+=22 sincos, perppara
. (5)
The measured
para and
perp obtained from Fig. 3 are
used to calculate the ACS in arbitrary-oriented Hdc
with Eq. (5), the imaginary parts
’’ of which are
presented as dashed lines in Fig. 4. The good
agreement between the calculated and measured
’’
(coefficient of determination better than 0.997)
demonstrates the feasibility of Eq. (5) for the
description of
dependent
(
,
). The squares of
cos(
) and sin(
) in Eq. (5) serve to make
(
= 0,
)
equal to the Debye model – Eq. (1) with Hdc = 0 mT. In
addition, cos2(
) and sin2(
) can be converted to
cos(2
), meaning that the ACS vs.
curve has two
cycles in a
range from 0° to 360°, which reflects the
180° periodicity. It fits very well with our measured
data as shown in Fig. 2. Note that in Eq. (5) the
parameter
is calculated from the absolute value of the
applied dc magnetic field, i.e., not from the projection
of the dc magnetic field in either parallel or
perpendicular direction.
Fig. 3. Imaginary part versus frequency in parallel (a) and
perpendicular (b) dc magnetic fields. Symbols are experimental data
whereas solid lines are fits with Eq. (4).
To quantitatively investigate the dependence of the
effective Brownian relaxation time
B(
) on
,
’’
are fitted with Eq. (4) while the peak frequency is used
to calculate
B(
). Fig. 5 shows
B(
) versus
in
superposed Hdc with different strengths and
orientations. With
= 0,
para =
B(
= ) decreases
from 1.32 ms to 0.23 ms while
perp =
B(
= °)
decreases from 1.32 ms to 0.38 ms by increasing Hdc
from 0 mT to 5 mT. With Hdc ≠ 0 mT,
B(
)
decreases from
perp to
para with decreasing
in the
first quadrant. Herein, we propose a mathematical
model to describe the
dependent effective Brownian
relaxation time
B(
)
( )
+=22 sincos, perpparaB
. (6)
With the measured
parp and
perp, one can calculate the
B(
) in arbitrary-oriented Hdc with Eq. (6), shown
as dashed lines in Fig. 5. It indicates that the calculated
Brownian relaxation time fits very well (the coefficient
of determination better than 0.984) with the
experimental results, which demonstrates the
feasibility of Eq. (6) for the description of
dependent
B(
).
Fig. 4. Imaginary part vs. frequency in different-orientation ac and
dc magnetic fields with dc magnetic field strengths of 1 mT (a), 3
mT (b) and 5 mT (c). Symbols are experimental results whereas
dashed lines are the calculated values with Eq. (5).
Fig. 5. Effective Brownian relaxation time
B(
) versus
curves
at different Hdc. Symbols are experimental results whereas lines are
fits with Eq. (6).
With the proposed mathematical models for the
description of the
dependent ACS and Brownian
relaxation time – Eqs. (5) and (6) – the physical
models – Eqs. (2) and (3) – can be extended to a more
general case, including Hdc strength and orientation.
Inserting Eqs. (4a) and (4b) into Eq. (6), the effective
Brownian relaxation time
B(
) is given by
( ) ( )
( ) ( )
( )
BB L
L
d
Ld
−
+=22 sin
2
cos
ln
ln
,
. (7)
Without a superposed Hdc (
= 0), Eq. (7) simplifies to
B(0,
) = (cos2
+ sin2
)·
B
=
B. With a parallel
superposed Hdc (
≠ 0), Eq. (3a) is obtained for the
case of a parallel Hdc (
= 0 °) while Eq. (3b) is
obtained from Eq. (7) for a perpendicular Hdc (
=
90°). Furthermore, by inserting Eqs. (2a) and (2b) into
Eq. (5),
(
,
) is given by
( ) ( )
( )
=+
+
=+
=
2
0
2
0
sin
)90,(1
1
3
cos
)0,(1
1
3,


B
B
i
L
i
d
dL
. (8)
Without a superposed Hdc (
= 0), Eq. (7) simplifies to
the Debye model – Eq. (1). With a superposed Hdc (
≠
0), Eq. (2a) is obtained for a parallel Hdc (
= 0 °)
while Eq. (2b) is obtained from Eq. (8) for a
perpendicular Hdc (
= 90 °). Therefore, the proposed
models – Eqs. (7) and (8) – are feasible to describe the
Hdc and
dependent Brownian relaxation time and the
regarding ACS spectra.
Note that Eqs. (7) and (8) are the combination of the
original model Eq. (2) and our proposed models Eqs.
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(5) and (6). The original model Eq. (2) for the
established cases of parallel and perpendicular dc
magnetic fields does not fit the experimental data very
well even if core and hydrodynamic size distributions
with a single set of parameters, e.g. mean and standard
deviation, are included. We speculate that it is caused
by the presence of small MNP cluster (mostly trimers),
leading to causing magnetic field dependent magnetic
moments. This results in the poor agreement of the
experimental data with Eqs. (7) and (8) although the
proposed models Eqs. (5) and (6) fit the experimental
data very well. Thus, the main contributions of this
paper – the proposed mathematical models for the
description of the magnetic field orientation dependent
ac susceptibility (Eq. 5) and Brownian relaxation time
(Eq. 6) – have been validated. We emphasize that the
proposed mathematical models are of great interest for
MPI, as well as corresponding biomedical applications.
With the extended model, one can simulate the ACS of
Brownian relaxation dominated MNPs and Brownian
relaxation time, in arbitrary-orientated Hdc superposed
to a small-amplitude Hac. In addition, taking into
account the phenomenological model for the
description of Hac dependent Brownian relaxation
time21,22 and the effective-field approaches,31,32 the
spectra of MNP magnetization induced in
arbitrary-orientated Hdc superposed to a
large-amplitude Hdc can also be simulated, which is
generally interesting to MPI related researches.
In conclusion, we investigated the ac susceptibility
and Brownian relaxation time of MNPs in
different-orientated dc and ac magnetic fields. The ac
susceptibility of a CoFe2O4 MNP sample was
measured in arbitrary-orientated dc and ac magnetic
fields while the Brownian relaxation time was
calculated from the peak frequency of the imaginary
parts. Experimental results indicate that an increase in
the dc magnetic field strength decreases the Brownian
relaxation time while increasing the angle between the
dc and ac magnetic fields in the first quadrant increases
the Brownian relaxation time in the given range of dc
magnetic field strengths. Moreover, mathematical
models were proposed to describe the orientation of the
dc magnetic field dependent ac susceptibility and the
Brownian relaxation time. We envisage that the
proposed mathematical models are of great importance
and interest to MPI related research fields.
The financial support from the German Research
Foundation DFG (Project No.: ZH 782/1-1) and via
SPP1681 (LU 800/4-3 and FI 1235/2-2) is gratefully
acknowledged.
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This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5120609
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5120609
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5120609
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5120609
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5120609
This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/1.5120609
... The dc field's influence on the dynamic response and relaxation characteristics of an ensemble of moving noninteracting particles was investigated [21][22][23]. Zhong et al. [24], based on experimental data, have proposed a formula for describing the Brownian relaxation time of magnetic nanoparticles, that takes into account the relative location of the ac and dc magnetic fields. Single-particle theories of the dynamic response of an ensemble of moving noninteracting particles allow us to qualitatively describe the behavior of a ferrofluid in ac and dc fields. ...
... Experimental data and the results of theoretical studies show that if only the ac magnetic field reacts to the ferrofluid, then an increase in the ac field's amplitude reduces its dynamic magnetic response at low frequencies [20,40,42,49]. It is also known that the addition of a bias dc field directed parallel to the weak ac field reduces the ferrofluid's dynamic susceptibility in the low frequency region [24,43]. On the other hand, dipole-dipole interactions increase the ferrofluid's dynamic response in a weak ac magnetic field [50]. ...
Influence of a bias dc field and an ac field amplitude on the dynamic susceptibility of a moderately concentrated ferrofluid
Article
  • Aug 2023
  • PRE
In this paper, we study the effect of a bias dc field on the dynamic response of a moderately concentrated ferrofluid to an ac magnetic field of arbitrary amplitude. The ferrofluid is modeled by an ensemble of interacting moving magnetic particles; the reaction of particle magnetic moments to ac and dc magnetic fields occurs according to the Brownian mechanism; and the ac and dc magnetic fields are parallel. Based on a numerical solution of the Fokker-Planck equation for the probability density of the orientation of the magnetic moment of a random magnetic particle, dynamic magnetization and susceptibility are determined and analyzed for various values of the ac field amplitude, the dc field strength, and the intensity of dipole-dipole interactions. It is shown that the system's magnetic response is formed under the influence of competing interactions, such as dipole-dipole, dipole-ac field, and dipole-dc field interactions. When the energies of these interactions are comparable, unexpected effects are observed: the system's susceptibility can either increase or decrease with increasing ac field amplitude. This behavior is associated with the formation of nose-to-tail dipolar structures under the action of the dc field, which can hinder or promote the system's dynamic response to the ac field. The obtained results provide a theoretical basis for predicting the dynamic properties of ferrofluids to improve their use in biomedical applications, such as, in magnetic induction hyperthermia.
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... where M (t) [A m −1 ] is the non-equilibrium magnetization and τ [s] is the effective relaxation time of the SPIONs. The core diameter of nanoparticles used in MPI is usually larger than 20 nm, and the effective relaxation time is approximately equal to the Brownian relaxation time [38,39]. If a weak AC field H d = H dp sin (ω d t) [A m −1 ] is applied (KH dp ≪ 1), the differential equation (2) will be transformed into ...
... The relaxation time τ is not only determined by the core diameter, hydrodynamic diameter, solution viscosity, and Kelvin temperature, but is also affected by the bias field [38][39][40]43]: ...
An improved point spread function for complex susceptibility-based magnetic particle imaging
Article
Full-text available
Spatial resolution is a key metric for characterizing magnetic particle imaging (MPI), and magnetic relaxation is a critical factor affecting the spatial resolution. This study investigates the point spread functions (PSFs) of MPI and analyzes the potential of breaking through the spatial resolution limit, which equals the full width at half maximum (FWHM) of the Langevin function derivative. In this work, different PSFs of MPI were built based on the magnitude and real and imaginary parts of complex susceptibility. The imaging performance was evaluated using the FWHM and a self-defined convergence parameter. The results show that image reconstruction can achieve a narrower PSF based on the imaginary part of complex susceptibility, and the heavy-tailed distribution of the derivative of the Langevin curve can be optimized. This suggests that there is scope to improve the spatial resolution and image contrast of MPI.
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... The magnetic field and nanoparticle concentration dependence of the evaluated results are in agreement with the results recently measured (Ota et al. 2016;Morales et al. 2021;Deissler et al. 2014;Neveu-Prin et al. 1993;Dieckhoff et al. 2016Dieckhoff et al. , 2011Goodwill et al. 2011;Rauwerdink and Weaver 2010;Noguchi et al. 2022;Zhong et al. 2019). ...
A study of Brownian relaxation time in magnetic nanofluids: a semi-analytical model
Article
Full-text available
  • Jun 2023
Magnetic nanofluids find application in various fields, including magnetic hyperthermia, which holds significant potential for non-invasive cancer treatment. The dynamics of magnetic nanoparticle systems under the influence of a magnetic field plays a crucial role in all applications, particularly in magnetic hyperthermia, and has been the subject of recent intensive research. Numerous studies investigating magnetic hyperthermia assume that the Brownian relaxation time is independent of the magnetic field and nanoparticle concentration. However, this modeling assumption can introduce errors in estimating certain parameters of interest. Consequently, these errors propagate in determining the effective relaxation time, which holds great importance in estimating quantities such as the Specific Absorption Rate and Intrinsic Loss Power Values for magnetic hyperthermia. This scientific paper presents a study that addresses these errors using a semi-analytical model. The experimental results obtained in our study contribute to the understanding of magnetic relaxation mechanisms in nanofluids under various conditions. Furthermore, these findings can aid in the development of accurate numerical evaluation models for practical applications.
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... The magnetic system's relaxation time under investigation, τ, dictates the system's frequency response. Therefore, AC susceptibility measurements lead to the inference of the relaxation time of a magnetic material and have been extensively used in the characterization of high-criticaltemperature superconductors, 3,4 spin-glasses, paramagnetic salts, 5,6 magnetic nano-particles, [7][8][9] and conventional and low-dimensional ferromagnets. 10,11 In conventional AC susceptometry, the sample under study is excited using a primary coil. ...
Mapping AC susceptibility with quantum diamond microscope
Article
  • May 2023
  • REV SCI INSTRUM
We present a technique for determining the micro-scale AC susceptibility of magnetic materials. We use the magnetic field sensing properties of nitrogen-vacancy (NV-) centers in diamond to gather quantitative data about the magnetic state of the magnetic material under investigation. A quantum diamond microscope with an integrated lock-in camera is used to perform pixel-by-pixel, lock-in detection of NV- photo-luminescence for high-speed magnetic field imaging. In addition, a secondary sensor is employed to isolate the effect of the excitation field from fields arising from magnetic structures on NV- centers. We demonstrate our experimental technique by measuring the AC susceptibility of soft permalloy micro-magnets at excitation frequencies of up to 20 Hz with a spatial resolution of 1.2 µm and a field of view of 100 µm. Our work paves the way for microscopic measurement of AC susceptibilities of magnetic materials relevant to physical, biological, and material sciences.
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... In MH, the imaginary part of the AC magnetic susceptibility (also called as loss component) is responsible for power dissipation (heat generation) during the magnetization reversal in MNPs [43] . It has been shown [44][45][46] that the reduction in the imaginary part of the AC susceptibility in the presence of a DC bias magnetic field is more pronounced when the DC field is applied in the parallel direction com-pared to the perpendicular direction. In our results, also, a similar trend was observed in the SLP distribution outside FFP, i.e. ...
A prediction model for magnetic particle imaging–based magnetic hyperthermia applied to a brain tumor model
Article
  • Apr 2023
  • COMPUT METH PROG BIO
Background and objective: Brain tumor is a global health concern at the moment. Thus far, the only treatments available are radiotherapy and chemotherapy, which have several drawbacks such as low survival rates and low treatment efficacy due to obstruction of the blood-brain barrier. Magnetic hyperthermia (MH) using magnetic nanoparticles (MNPs) is a promising non-invasive approach that has the potential for tumor treatment in deep tissues. Due to the limitations of the current drug-targeting systems, only a small proportion of the injected MNPs can be delivered to the desired area and the rest are distributed throughout the body. Thus, the application of conventional MH can lead to damage to healthy tissues. Methods: Magnetic particle imaging (MPI)-guided treatment platform for MH is an emerging approach that can be used for spatial localization of MH to arbitrarily selected regions by using the MPI magnetic field gradient. Although the feasibility of this method has been demonstrated experimentally, a multidimensional prediction model, which is of crucial importance for treatment planning, has not yet been developed. Hence, in this study, the time dependent magnetization equation derived by Martsenyuk, Raikher, and Shliomis (which is a macroscopic equation of motion derived from the Fokker-Planck equation for particles with Brownian relaxation mechanism) and the bio-heat equations have been used to develop and investigate a three-dimensional model that predicts specific loss power (SLP), its spatio-thermal resolution (temperature distribution), and the fraction of damage in brain tumors. Results: Based on the simulation results, the spatio-thermal resolution in focused heating depends, in a complex manner, on several parameters ranging from MNPs properties to magnetic fields characteristics, and coils configuration. However, to achieve a high performance in focused heating, the direction and the relative amplitude of the AC magnetic heating field with respect to the magnetic field gradient are among the most important parameters that need to be optimized. The temperature distribution and fraction of the damage in a simple brain model bearing a tumor were also obtained. Conclusions: The complexity in the relationship between the MNPs properties and fields parameter imposes a trade-off between the heating efficiency of MNPs and the accuracy (resolution) of the focused heating. Therefore, the system configuration and field parameters should be chosen carefully for each specific treatment scenario. In future, the results of the model are expected to lead to the development of an MPI-guided MH treatment platform for brain tumor therapy. However, for more accurate quantitative results in such a platform, a magnetization dynamics model that takes into account coupled Néel-Brownian relaxation mechanism in the MNPs should be developed.
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... Then, the decrement of the with the increment of the applied external magnetic field is more emphasised for the nanoparticles with a larger size (dm = 22 nm), compared to those with a smaller size (dm = 10 nm). The magnetic field and nanoparticle concentration dependence of the evaluated results are in agreement with the results recently measured (Ota et al. 2016;Morales et al. 2021;Deissler et al. 2014;Neveu-Prin et al. 1993;Dieckhoff et al. 2016;Goodwill et al. 2011;Rauwerdink and Weaver 2010;Noguchi et al. 2022;Zhong et al. 2019;Dieckhoff et al. 2011). ...
Understanding the Effect of Magnetic Field and Nanoparticle Concentration on Brownian Relaxation Time in Magnetic Nanofluids: A Semi-Analytical Model
Preprint
Full-text available
  • Feb 2023
Magnetic nanofluids are used in many types of applications. Therefore, the dynamics of magnetic nanoparticle systems under the action of magnetic field were intensively studied, lately. Many studies related to biomedical applications consider the Brownian relaxation time independent from the magnetic field and nanoparticle concentration. This modelling assumption can lead to certain errors in the estimation of some parameters of interest. Thus, these errors also propagate in the determination of the effective relaxation time, which is of great importance in the estimation of some quantities of interest such as SAR (Specific Absorption Rate) or ILP (Intrinsic Loss Power Values) for magnetic hyperthermia. This paper presents a study of these errors starting from a semi-analytical model. Our experimental results can be useful to understand the mechanisms of magnetic relaxation of a nanofluid in various conditions and, above all, to create suitable numerical evaluation models.
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... (1) might break down in nonequilibrium situations [14]. Of particular relevance for this study is the finding that the effective relaxation times are strongly dependent on an external magnetic field [15,16]. In particular, the Néel relaxation time can decrease dramatically and therefore can become comparable to Brownian relaxation even when Néel processes would be negligible in the field-free case [17]. ...
Longest relaxation time versus maximum loss peak in the field-dependent longitudinal dynamics of suspended magnetic nanoparticles
Article
Full-text available
  • Oct 2022
Magnetic nanoparticles in suspensions provide fascinating model systems to study field-induced effects. Their response to external fields also opens up promising new applications, e.g., in hyperthermia. Despite significant research efforts, several basic questions regarding the influence of external fields on the magnetization dynamics are still open. Here we revisit the classical model of a suspended magnetic nanoparticle with combined internal and Brownian dynamics in the presence of an external field and discuss the field-dependent longitudinal relaxation. While internal and Brownian dynamics are independent in the field-free case, the coupling of both processes when an external field is present leads to richer and more complicated behavior. Using a highly efficient and accurate solver to the underlying Fokker-Planck equation allows us to study a broad parameter range. We identify different dynamical regimes and study their respective properties. In particular, we discuss corrections to the popular rigid-dipole approximation which are captured in terms of a simplified diffusion-jump model in the Brownian-dominated regime with rare Néel relaxation events. In addition, we discover a regime with surprising mode-coupling effects for magnetically soft nanoparticles. We explain our findings with the help of a perturbation theory, showing that in this regime the magnetization relaxation at late times is slaved by the slow Brownian motion of the nanoparticle. We discuss consequences of these findings such as the discrepancy of the longest relaxation time and the inverse frequency of the loss peak of the magnetic susceptibility.
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Mapping AC Susceptibility with Quantum Diamond Microscope
Preprint
  • Sep 2022
We present a novel technique for determining the microscale AC susceptibility of magnetic materials. We use magnetic field sensing properties of nitrogen-vacancy (\ce{NV-}) centers in diamond to gather quantitative data about the magnetic state of the magnetic material under investigation. In order to achieve the requisite speed in imaging, a lock-in camera is used to perform pixel-by-pixel lock-in detection of \ce{NV-} photo-luminescence. In addition, a secondary sensor is employed to isolate the effect of the excitation field from fields arising from magnetic structures on \ce{NV-} centers. We demonstrate our experimental technique by measuring the AC susceptibility of soft permalloy micro-magnets at excitation frequencies of up to \SI{20}{\hertz} with a spatial resolution of \SI{1.2}{\micro \meter} and a field of view of \SI{100}{\um}. Our work paves the way for microscopic measurement of AC susceptibilities of magnetic materials relevant to physical, biological, and material sciences.
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AC Magnetic Susceptibility of Magnetic Nanoparticles Measured Under DC Bias Magnetic Field
Article
  • Mar 2022
The magnetization dynamics of magnetic nanoparticles are essential for the medical applications of hyperthermia and magnetic particle imaging. In this paper, we clarified the AC magnetic susceptibility of superparamagnetic nanoparticles and their frequency dependence. We measured the AC magnetic susceptibility under an applied DC bias magnetic field. The frequency dependence of AC magnetic susceptibility was clearly observed. The experimentally obtained characteristics of the samples with different effective core diameters and dependence on the intensity of the DC bias magnetic field agreed with the theory assuming a simple particle model. The sample comprising effective core diameters with a wide distribution exhibited different characteristics than the samples comprising magnetically fractionated nanoparticles with a narrower core diameter distribution.
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Investigation of Commercial Iron Oxide Nanoparticles: Structural and Magnetic Property Characterization
Article
Full-text available
  • Feb 2021
Magnetic nanoparticles (MNPs) have been extensively used as tiny heating sources in magnetic hyperthermia therapy, contrast agents in magnetic resonance imaging, tracers in magnetic particle imaging, carriers for drug/gene delivery, etc. There have emerged many MNP/microbead suppliers since the past decade, such as Ocean NanoTech, Nanoprobes, US Research Nanomaterials, Miltenyi Biotec, micromod Partikeltechnologie GmbH, nanoComposix, and so forth. In this paper, we report the physical and magnetic characterizations on iron oxide nanoparticle products from Ocean NanoTech. Standard characterization tools such as vibrating-sample magnetometry, X-ray diffraction, dynamic light scattering, transmission electron microscopy, and zeta potential analysis are used to provide MNP customers and researchers with an overview of these iron oxide nanoparticle products. In addition, the dynamic magnetic responses of these iron oxide nanoparticles in aqueous solutions are investigated under low- and high-frequency alternating magnetic fields, giving a standardized operating procedure for characterizing the MNPs from Ocean NanoTech, thereby yielding the best of MNPs for different applications.
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Influence of signal bandwidth and phase delay on complex susceptibility based magnetic particle imaging
Article
Full-text available
In this paper, the influences of signal bandwidth and phase delay on the spatial resolution of complex susceptibility based magnetic particle imaging (csMPI) are investigated by modeling the transfer function of the measurement circuit. Based on the simulation and experimental results, it is found that the signal bandwidth and the phase delay will significantly impact on the spatial resolution of csMPI. To avoid the influence of narrow signal bandwidth and phase delay of measurement circuit on csMPI, a resonance measurement circuit with −3 dB bandwidth of 2.3 kHz (much larger than the signal bandwidth of the desired csMPI signal) and resonance frequency of 12.8 kHz is adopted to measure the weak nonlinear magnetization of the magnetic nanoparticles; a phase compensation is used when calculating the real and the imaginary components of ac magnetic susceptibility.
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Magnetic nanoparticle temperature imaging with a scanning magnetic particle spectrometer
Article
Full-text available
Non-invasive and in vivo temperature imaging plays a significant role in disease diagnostics and therapy. This paper proposes a method of magnetic nanoparticle (MNP) temperature imaging with a scanning magnetic particle spectrometer (SMPS), which has great potential for non-invasive and in vivo temperature imaging beneath the surface of an object. The SMPS system, consisting of a mechanical scanner and a magnetic particle spectrometer (MPS), is built to measure the spatial distributions of the fundamental f 0 and 3f 0 harmonics of a MNP sample. A reconstruction method based on the point spread function, defined by the sensitivity profile of the pickup coil, is investigated to visualize the spatial distributions of MNP concentration and temperature. Experiments on a phantom with different temperature profiles are performed, which demonstrate the feasibility of the proposed method for MNP temperature imaging with a spatial resolution of about 3.5 mm. The SMPS is a low-cost and versatile setup for MNP concentration and temperature imaging, which has great potential for disease diagnostics and therapy.
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Real-time and quantitative isotropic spatial resolution susceptibility imaging for magnetic nanoparticles
Article
Full-text available
The magnetic transparency of biological tissue allows the magnetic nanoparticle (MNP) to be a promising functional sensor and contrast agent. The complex susceptibility of MNPs, strongly influenced by particle concentration, excitation magnetic field and their surrounding microenvironment, provides significant implications for biomedical applications. Therefore, magnetic susceptibility imaging of high spatial resolution will give more detailed information during the process of MNP-aided diagnosis and therapy. In this study, we present a novel spatial magnetic susceptibility extraction method for MNPs under a gradient magnetic field, a low-frequency drive magnetic field, and a weak strength high-frequency magnetic field. Based on this novel method, a magnetic particle susceptibility imaging (MPSI) of millimeter-level spatial resolution (<3 mm) was achieved using our homemade imaging system. Corroborated by the experimental results, the MPSI shows real-time (1 s per frame acquisition) and quantitative abilities, and isotropic high resolution.
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Magnetic nanoparticle thermometry independent of Brownian relaxation
Article
Full-text available
An improved method of magnetic nanoparticle (MNP) thermometry is proposed. The phase lag φ of the fundamental f 0 harmonic is measured to eliminate the influence of Brownian relaxation on the ratio of 3f 0 to f 0 harmonic amplitudes applying a phenomenological model, thus allowing measurements in high-frequency ac magnetic fields. The model is verified by simulations of the Fokker-Planck equation. An MNP spectrometer is calibrated for the measurements of the phase lag φ and the amplitudes of 3f 0 and f 0 harmonics. Calibration curves of the harmonic ratio and tanφ are measured by varying the frequency (from 10 Hz to 1840 Hz) of ac magnetic fields with different amplitudes (from 3.60 mT to 4.00 mT) at a known temperature. A phenomenological model is employed to fit the calibration curves. Afterwards, the improved method is proposed to iteratively compensate the measured harmonic ratio with tanφ, and consequently calculate temperature applying the static Langevin function. Experimental results on SHP-25 MNPs show that the proposed method significantly improves the systematic error to 2 K at maximum with a relative accuracy of about 0.63%. This demonstrates the feasibility of the proposed method for MNP thermometry with SHP-25 MNPs even if the MNP signal is affected by Brownian relaxation.
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Magnetic particle imaging: From proof of principle to preclinical applications
Article
Full-text available
Tomographic imaging has become a mandatory tool for the diagnosis of a majority of diseases in clinical routine. Since each method has its pros and cons, a variety of methods is regularly used in the clinics to satisfy all application needs. Magnetic particle imaging (MPI) is a relatively new tomographic imaging technique that images magnetic nanoparticles with high spatio-temporal resolution in a quantitative way and in turn is highly suited for vascular and targeted imaging. MPI has been introduced in 2005 and now enters the preclinical research phase, where medical researcher get access to this new technology and exploit the potential under physiological conditions. Within this paper we review the development of MPI since its introduction in 2005. Beside an in depth description of the basic principle, we provide detailed discussions on imaging sequences, reconstruction algorithms, scanner instrumentation, and potential medical applications.
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Multiparametric Magnetic Particle Spectroscopy of CoFe 2 O 4 Nanoparticles in Viscous Media
Article
  • Feb 2019
The detailed signal generation of the magnetization response of magnetic nanoparticles (MNPs) as a result of externally applied magnetic fields with flux densities of several milliTesla is of high interest for biomedical applications like Magnetic Resonance Imaging (MRI) or Magnetic Particle Imaging (MPI). Although, MNPs are already frequently used as contrast agents or tracer materials, experimental data is rarely compared to model predictions due to distinct deviations. In this contribution, we use a customized Brownian dominated CoFe2O4 particle system to compare experimental Magnetic Particle Spectroscopy (MPS) data with Fokker-Planck simulations considering Brownian relaxation. The influences of viscosity, excitation frequency, field-dependency and size distribution are studied. We show that the effective magnetic moment and cluster sizes can be determined using a sample viscosity series. As introduced, such particle systems can serve as model systems to evaluate mathematical expressions and to study dependencies on physical influencing factors. Investigations of defined MNP systems and detailed characterizations enable a wide field of improved diagnosis and therapy applications, e.g. mobility MPI and magnetic hyperthermia.
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In-Field Orientation and -Dynamics of Ferrofluids Studied by Mössbauer Spectroscopy
Article
  • Dec 2018
By studying the response behavior of ferrofluids of 6 – 22 nm maghemite nanoparticles (IONPs) in glycerol solution exposed to external magnetic fields, we demonstrate the ability of Mössbauer spectroscopy to access a variety of particle dynamics and static magnetic particle characteristics at the same time, offering an extensive characterization of ferrofluids for in-field applications: Field-dependent particle alignment as well as particle mobility in terms of Brownian motion have been extracted simultaneously from a series of Mössbauer spectra for single core particles as well as for particle agglomerates. Additionally, information on Néel superspin relaxation and surface spin frustration could be directly inferred from this analysis. Parameters regarding Brownian particle dynamics, as well as Néel-type relaxation behavior, obtained via Mössbauer spectroscopy, have been verified by complementary AC-susceptometry (ACS) experiments, modulating the AC-field amplitude and using an extended frequency range of 10⁻¹ – 10⁶ Hz, while field-dependent particle alignment has been crosschecked via magnetometry.
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Excitation frequency dependence of temperature resolution in magnetic nanoparticle temperature imaging with a scanning magnetic particle spectrometer
Article
  • Sep 2018
  • J MAGN MAGN MATER
This paper investigates the dependence of temperature resolution in magnetic nanoparticle (MNP) temperature imaging on the frequency of an applied ac magnetic field with a custom-built scanning magnetic particle spectrometer (SMPS). The fundamental f0 and 3f0 harmonics are measured with the SMPS while the amplitude ratio of the 3f0 to f0 harmonics is used to determine the spatial distribution of temperatures. Experiments on a three-line phantom filled with a MNP sample are performed in different-frequency ac magnetic fields with a constant strength of 8 mT. The standard deviation of measured temperatures is used to characterize the temperature resolution. Experimental results show that the temperature resolution improves from 0.9 K to 0.2 K with increasing the excitation frequency from 600 Hz to 5 kHz, which is mainly caused by a higher signal-to-noise ratio due to Faraday's law and by a higher temperature sensitivity of the 3f0 to f0 harmonic ratio.
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Analysis of AC Susceptibility Spectra for the Characterization of Magnetic Nanoparticles
Article
  • Apr 2017
Measurements of the ac susceptibility as a function of frequency have been widely applied for the determination of structure parameters of magnetic nanoparticles. The analysis of spectra of real and imaginary part measured on suspensions of magnetic nanoparticles is generally based on the Debye model, extended by distributions of size parameters. Here we compare different modifications of the Debye model with experimental data recorded on suspensions of single-core and multi-core iron-oxide nanoparticles. The applied models also depend on whether the nanoparticle’s magnetic moments are thermally blocked and whether both Brownian and Neel relaxation have to be taken into account. The obtained core and hydrodynamic size parameters are compared with those from TEM and dynamic light scattering. Whereas structure parameters can be reliably determined for single-core nanoparticles, the interpretation of ac susceptibility spectra measured on multi-core nanoparticles is more complicated, especially regarding the contribution of particles relaxing via the Neel mechanism. Depending on the packing density and thus the interaction between cores in a particle, the effective core parameters derived from the spectrum must be interpreted with care.
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Influence of static magnetic field strength on the temperature resolution of a magnetic nanoparticle thermometer
Article
  • Oct 2016
This paper investigates the influence of dc magnetic field strength on the resolution of a magnetic nanoparticle (MNP) thermometer, which employs the fundamental f 0 and 2f 0 harmonics of the MNP magnetization induced by ac and superimposed dc magnetic fields. In ac and parallel dc magnetic fields, the strength of dc magnetic field modulates the harmonics of the MNP magnetization, which affects their temperature sensitivities and measurement signal-to-noise ratios (SNRs). A temperature-adjustable fluxgate-based magnetic particle spectrometer was used to measure the spectra of the MNP magnetization at different temperatures. To determine the temperature, the amplitudes of the measured f 0 and 2f 0 harmonics were modeled based on the static Langevin function. AC susceptibility measurements on a MNP sample demonstrate the applicability of the static Langevin function for the description of the MNP magnetization spectra at a low frequency ac magnetic field without taking into account the MNP dynamics. Our simulations and experiments show that with increasing dc magnetic field from 0.2 mT to 2.0 mT, both the amplitude of the 2f 0 harmonic and the temperature sensitivity of the amplitude ratio of the 2f 0 to f 0 harmonics increase by a factor of about 10 in an ac magnetic field with a frequency of 70 Hz and an amplitude of 1 mT. Concomitantly, the SNR of the 2f 0 harmonic significantly increases by about 20 dB. Consequently, the temperature resolution of the MNP thermometer is improved from 1.97 K to 0.26 K.
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