Regarding the Néel relaxation time constant in magnetorelaxometry
J. Leliaert,1, 2, a) A. Coene,1, b) G. Crevecoeur,1 A. Vansteenkiste,2 D. Eberbeck,3 F. Wiekhorst,3 U. Steinhoff,3
B. Van Waeyenberge,2 and L. Dupré1
1)
Department of Electrical Energy, Systems and Automation, Ghent University, 9000 Gent,
Belgium
2)
Department of Solid State Sciences, Ghent University, 9000 Gent, Belgium
3)
Physikalisch-Technische Bundesanstalt, Berlin 10587, Germany
(Dated: 16 July 2014)
Magnetorelaxometry (MRX) is a sensitive measurement technique frequently employed in biomedical applications for imaging magnetic nanoparticles (MNP). In this letter we employ a first principles model to
investigate the effects of different MNP sample properties on the Néel relaxation time constant τN in magnetorelaxometry. Using this model we determined that dipolar interactions start to have an impact on the MRX
signal from Fe concentrations of 100 mmol/l, and result in a smaller τN . Additionally, the micromagnetic
damping constant, closely related to τN , was found to be between 0.0005 and 0.002 by comparison to an
MRX measurement of iron oxide particles. This is significantly lower compared to the bulk value of 0.07 for
this material.
Magnetorelaxometry (MRX) is an established measurement technique for the quantitative imaging of magnetic nanoparticles (MNP)1–5 which can be used in
biomedical applications such as disease detection6 , drug
targeting7 and thermal ablation8 . Furthermore, binding processes of MNP-based biochemical reactions can
be analyzed9–11 and it is actively utilized to investigate
the physical properties of MNP suspensions in biomedical
applications12–15 .
An MRX measurement starts with the application of
a magnetizing field of typically 1 mT to a superparamagnetic MNP sample. After a certain time (typically 1
second) this field is switched off and the sample relaxes
again towards its demagnetized state16 .
Z
t
m=
m0 exp −
P (V )dV
(1)
τN (V )
V
To obtain quantitative information from these measurements a comparison with a reference sample of known
MNP concentration measured under equal conditions is
needed16 . This reference sample must have the same
physical properties because these have a large impact on
the MRX curve. When assuming that dipolar interactions between the MNP are neglegible, the concentration
of the sample can be obtained by the ratio of the two
MRX measurement amplitudes (reference and investigation), as this scales linearly with the concentration.
Equation 1 describes how the magnetization m of a sample of MNP with size distribution P(V) relaxes from m0
towards 0 with a characteristic time constant τN , called
the Néel relaxation constant17 . Note that the magnetization decays exponentially when all particles have the
same volume. Otherwise the signature is characterized
by the size distribution of the particles. The Néel relaxation constant is given by eq. 2:
KV
τN = τ0 exp
(2)
kB T
In this letter we employ a first principles numerical
model to simulate the MRX signal starting from MNP
material parameters and their size distribution. Using
this model we investigate the Fe concentration above
which dipolar interactions affect the MRX curve. As
reference samples in some cases3,13,19,20 contain MNP
amounts which differ to a great extent with respect to
the sample, it might be possible that the comparison between the reference and sample MRX curves is no longer
valid. Furthermore, it has been shown that MNP samples tend to form aggregates during aging of the reference
sample13 , which locally results in even higher Fe concentrations.
with τ0 having values between 10−8 and 10−12 s16 , K
the uniaxial anisotropy constant, V the magnetic core
volume of the particle, kB the Boltzmann constant and
T the temperature. During the relaxation time of typically 0.5 s, the magnetization can be measured at a certain distance from the sample under study with sensitive
magnetometers such as SQUIDs or Fluxgates.
a) Electronic
b) Electronic
mail: jonathan.leliaert@ugent.be
mail: annelies.coene@ugent.be
There exist different theoretical models that describe
the relation between the physical properties of the MNP
and the MRX signal. Examples of such models are the
moment superposition model (MSM)13,18 and the cluster
moment superposition model (cluster-MSM)13,16 . However, these models are high-level and do not take into
account possible dipolar interactions between the MNP.
Another topic of debate is the value of the material
parameter, α, the Gilbert damping constant, which is
not accurately known for MNP but determines the value
of τ0 in eq. 2. There are indications21–23 that this value
can be significantly lower in nanoscale magnets than in
the bulk material . Therefore we compare our numerical
model to an actual MRX experiment to determine α.
The MRX simulations are performed with the in-house
developed simulation tool Vinamax ref vinamax. Vina-
2
1
max numerically solves the Landau-Lifshitz equation24 :.
a)
(3)
In this equation, m denotes the magnetization vector normalized with respect to the saturation magnetization, µ0 = 4π ×10−7 Tm/A, γ0 is the gyromagnetic ratio
(1.7595 ×1011 rad/Ts) and α is the dimensionless Gilbert
damping constant. Heff denotes the effective field working on each nanoparticle, taking into account the demagnetizing, thermal25 , and anisotropy field. This software
allows to simulate the micromagnetic behavior of large
MNP ensembles on the long timescales (seconds) customary in experiments, while still being accurate on the ps
timescale of the magnetic dynamics. It was specifically
developed to increase our understanding of the collective
behavior of magnetic nanoparticles, and could help to
increase the performance of several biomedical applications.
Our aim is to investigate the influence of the dipolar
interactions between MNP in MRX experiments. Therefore we first perform a simulation in which these interactions are not taken into account, representing a sample
with an infinitely diluted concentration. Figure 1 shows
a simulated MRX signal of an ensemble of 30000 magnetite nanoparticles having identical diameters of 11 nm,
a saturation magnetization of 400000 A/m and an uniaxial anisotropy constant of 14400 J/m3 .
Insets a) to d) in Fig. 1 show a reduced MNP ensemble
of 3 particles to explain their magnetic states at the different stages in the MRX simulation. The magnetization
of the MNP is represented by a red arrow and the MNP
have a random uniaxial anisotropy axis that is fixed in
time (black line).
As dipolar interactions equally affect the magnetizing
as the demagnetizing phase of an MRX experiment, we
only simulate the demagnetizing phase for practical purposes. To be able to accurately determine the relaxation
constant τ , we initialize the MNP in a fully magnetized
state along the z-axis. The normalized net magnetic moment along the z-axis of the particles is represented by
< mz > and equals 1 in this stage (Fig. 1 a)).
In the next stage (Fig. 1 b)) the MNP relax (with the
thermal field still switched off) towards one of their energetic groundstates, i.e. their magnetization parallel with
their uniaxial anisotropy axis in the positive z direction,
due to the initialization along the z-axis. The theoretical
prediction of < mz > when having randomly orientated
R π/2
anisotropy axes is 0 sin(x) cos(x)dx = 0.5.
In the subsequent stage (Fig. 1 c)) the thermal field is
switched on corresponding to a temperature of 300 K and
the magnetization slightly vibrates around the anisotropy
axis. Due to these thermal fluctuations the magnetization can switch between the two anisotropy directions at
random intervals. Consequently, < mz > decays towards
zero at a rate given by the Néel relaxation constant τN
(eqs. 1 and 2) as shown in Fig. 1 d). By fitting an exponential curve to the simulated MRX curve, τN can be
0.8
c)
<mz> ()
µ0 γ0
dm
(m × Heff + αm × m × Heff )
=−
dt
1 + α2
0.6
d)
0.4
b)
0.2
0
exponential fit
0
20
40
time (s)
60
80
100
FIG. 1. Simulation of an MRX experiment (red line) resulting from an ensemble of 30000 particles with an initial
magnetization along the z-axis. Insets a) to d) show an ensemble of 3 magnetic nanoparticles in the different stages of
the MRX signal. The uniaxial anisotropy axis is represented
by a black line and the MNP magnetization by a red arrow.
a) The magnetizations lay along the z-axis. b) The magnetizations relax towards their respective anisotropy axis in
the positive z-direction. c) The thermal field is switched on
and the magnetizations randomly move slightly around the
anisotropy axes. d) The thermal field randomly switches the
magnetization towards the opposite anisotropy direction, and
the total magnetization relaxes exponentially towards 0.
determined which equals 60 ns for this simulation.
This relaxation constant is employed to investigate the
impact of dipolar interactions on the MRX curve. Due to
the prohibitive computational cost of evaluating the dipolar interactions, we prefer simulations in which the relaxation constant is of the order of nano- to microseconds.
To this end we used the MNP ensemble of the previous
simulation as this ensemble had a relaxation constant of
60 ns (without dipolar interactions). The damping constant α equals 0.01 in this simulation (note that the dipolar interaction is independent of the value of α, which we
verified numerically). We repeated the MRX simulation
for increasing MNP concentrations, taking into account
the dipolar interactions by evaluating the dipolar field
working on each particle every 10 ps. The MNP in the
experiments are fixed in space at uniformly distributed
positions. As MNP in biomedical applications are typically iron oxides, the MNP amount is represented by a
Fe concentration. Fig. 2 shows the relaxation constant
for increasing Fe concentrations relative to the one obtained from the simulation without dipolar interactions
(Fig. 1). Starting from an Fe concentration of approximately 100 mmol/l we observe a steady decrease in the
relaxation constant.
A simple theoretical argument that only takes into account the dipolar interaction due to the nearest neighbours shows that the concentration from which on the
dipolar interaction starts to affect the relaxation time is
1 + α2
τ0 =
αγ
s
2πkB T
3 M V
HK
s
100 mmol/l
1
0.8
0.6
0.4
0.2
1e-06
relative relaxation constant τ ()
independent of the size of the particles: the size of the
dipolar field of a nearest neighbour on a particle scales
as V /r3 with V the volume of the particle and r the distance to its nearest neighbour. When considering larger
particles at a fixed concentration,
the distance r between
√
the particles scales as 3 V so that the dipolar field is indepent of the volume of the particles V . Nonetheless,
we have performed additional simulations of MNP with
diameters 12.59, 13.86 and 14.94 nm, corresponding to a
MNP volume of 1.5, 2 and 2.5 times the volume of the
MNP in Fig. 2. The relaxation constants τN without
dipolar effects for these particles are 173, 524 and 1522
ns respectively. These values are then used to obtain
a relative relaxation constant for simulations at different Fe concentrations, as shown in the inset of Fig. 2.
These results imply that, although comparable, the effect
of dipolar interactions is not exactly the same for different MNP sizes and appears to become more important
for larger particles, which means that the nearest neighbour approximation of our theoretical argument was too
crude. We therefore emphasize that during experiments
caution should be taken when using samples with an Fe
concentration above 100 mmol/l in experiments (where
typically MNP with diameters in the 16 to 22 nm range
are used16 ).
A second topic of interest when discussing τN is the
inverse of the attempt frequency: τ0 (eq. 2). Brown25
determined that in the high barrier limit, and in the absence of an external field τ0 is given analytically by eq. 4
with HK = 2K/Ms .
relative relaxation constant τ ()
3
1e-03
1e+00
concentration Fe (µmol/l)
1e+03
1
0.8
0.6
0.4
d = 11.00 nm, τ = 59 ns
d = 12.59 nm, τ = 173 ns
d = 13.86 nm, τ = 524 ns
d = 14.94 nm, τ = 1522 ns
0.2
0
10
100
1000
concentration Fe (µmol/l)
10000
FIG. 2. Change in the relaxation constant for increasing Fe
concentrations. When the Fe concentration of 100 mmol/l is
exceeded a decrease of the relaxation constant is observed.
Inset: This relationship also holds for simulations with MNP
of larger volumes and the dipolar interaction appears to have
a larger effect on larger particles.
(4)
In this equation every quantity except α is known for
MNP. Thus we are able to determine α by comparing to
an MRX experiment. Typically, τ0 is described as a time
constant that can take values between 10−8 and 10−12
s16 . This large range is inconvenient, and an estimate
of α together with eq. 4 could lead to a more accurate
estimate of τ0 . For iron oxide MNP there is no accurate
data available for α. There is however a typical value of
0.07 determined for the bulk material26 . For thin films
there is a clear trend visible in which α decreases with
decreasing dimensions22 , also seen in other materials21,23
, and in Ref.22 an upper limit for α in iron oxide thin
films of 0.0365 is found. We argue that in MNP this
value is expected to be significantly smaller, because of
the nanoscale dimensions in every direction.
We compare our numerical model to an MRX experiment performed at the Physikalisch-Technische Bundesanstalt in Berlin15 to determine the Gilbert damping
constant α. The MNP under investigation are large single magnetite core SHP-20 particles (Ocean Nanotech,
USA) coated by oleic acid combined with an amphiphilic
polymer. The size distribution was obtained by TEM15
and showed a lognormal distribution with a mean diameter of 19 nm and a standard deviation of 0.11 nm. The
saturation magnetization was determined to be 250000
A/m and K = 14500 J/m3 . The MNP are freeze dried
so only Néel relaxation occurs allowing to simulate the
MRX signal using our numerical model. The Fe concentration is 4.85 mmol/l, which is sufficiently low to neglect
dipolar interactions (see Fig. 2). The magnetizing field
was applied for 1 second and had an amplitude of 1600
A/m. In contrast to the numerical method described
earlier in this letter (Fig. 1), we now consider an experiment in which the particles are not fully magnetized.
Consequently, the magnetizing stage of the experiment
also has to be simulated. After this stage, the MRX signal is generated as the sample is allowed to relax for 0.4
seconds.
Figure 3 shows the result of MRX simulations for a
range of different α values. The smooth full blue line
shows the measurement data from the MRX experiment.
The green line shows a simulation of the experiment for
α = 0.07, for which we observe a large discrepancy between the simulated and measured MRX curve. The gray
region shows the simulation results for α ranging from
0.0005 to 0.002, in close correspondence to the MRX
measurement data. Therefore we suggest α = 0.001 for
these type of particles, which is indeed significantly lower
4
magnetic moment (a.u.)
0.12
Experiment
α=0.0005 to 0.002
α=0.07
Exponential fit
0.1
0.08
0.06
0.04
0.02
0
0.001 time (s)
0.01
0.1
FIG. 3. Comparison of simulations of MRX experiments for a
range of different α with an MRX measurement (smooth full
blue line). The gray region, with α between 0.0005 and 0.002
is in close correspondence with the measurement. The simulation for the bulk α = 0.07 (green line) differs significantly
from this measurement. The signature shape of the MRX
curve, which closely matches the experimental one, proves
that the size distribution of the particles is simulated correctly. For reference purposes a purely exponential function
is fit to the data, corresponding to a sample of MNP with
equal diameters.
compared to the bulk value of 0.07. The signature shape
of the MRX measurement curve corresponds well to our
simulations. As a reference, a purely exponential fit to
the measurement data is also shown (red dashed line),
which could only result from a sample consisting of particles with equal diameters. As τ0 depends on the volume
of the MNP and the experiment consisted of √
MNP of different diameters we report τ0 as 4.64e-20 / V s which
leads to a value of 8.65 ns for the average diameter of
19 nm in this experiment, well within the previously reported range.
In conclusion, we employed a numerical model starting
from first principles to simulate MRX experiments taking into account anisotropy, thermal effects and dipolar
interactions. We determined that for Fe concentrations
higher than 100 mmol/l the dipolar interactions affect
the MRX curve and decrease the relaxation time constant. Furthermore, an estimate of the Gilbert damping
parameter α was obtained by comparison to an MRX
measurement of iron oxide MNP. We determined α to
be between 0.0005 and 0.002 and suggest α = 0.001 for
SHP-20 magnetite nanoparticles, which is considerably
lower than the bulk value of 0.07. This leads to an
inverse attempt frequency τ0√of 8.65 ns for 19 nm diameter MNPs and scales as 1/ V for MNP of other sizes.
This work is supported by the Flanders Research Foundation (G.C and A.V)
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