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Simulations of magnetic nanoparticle Brownian motion
Daniel B. Reeves1and John B. Weaver1
Department of physics and astronomy, Dartmouth College, Hanover, NH 03755,
USAa)
Magnetic nanoparticles are useful in many medical applications because they interact with biology on a cellular
level thus allowing microenvironmental investigation. An enhanced understanding of the dynamics of magnetic
particles may lead to advances in imaging directly in magnetic particle imaging (MPI) or through enhanced
MRI contrast and is essential for nanoparticle sensing as in magnetic spectroscopy of Brownian motion (MSB).
Moreover, therapeutic techniques like hyperthermia require information about particle dynamics for effective,
safe, and reliable use in the clinic. To that end, we have developed and validated a stochastic dynamical
model of rotating Brownian nanoparticles from a Langevin equation approach. With no field, the relaxation
time toward equilibrium matches Einstein’s model of Brownian motion. In a static field, the equilibrium
magnetization agrees with the Langevin function. For high frequency or low amplitude driving fields, behavior
characteristic of the linearized Debye approximation is reproduced. In a higher field regime where magnetic
saturation occurs, the magnetization and its harmonics compare well with the effective field model. On
another level, the model has been benchmarked against experimental results, successfully demonstrating that
harmonics of the magnetization carry enough information to infer environmental parameters like viscosity
and temperature.
Keywords: Magnetic nanoparticles, Brownian stochastic simulations, biosensing
I. INTRODUCTION TO BROWNIAN NANOPARTICLES
There is a wide range of possible uses for magnetic
nanoparticles (MNPs) in medical applications. Novel
modalities image particles themselves as in magnetic
particle imaging (MPI)1and particles are also used
as contrast agents in conventional magnetic resonance
imaging2. Magnetic spectroscopy of Brownian mo-
tion (MSB) uses particle dynamics to gather informa-
tion about the microscopic environments the particles
inhabit3. A therapeutic method called hyperthermia
damages unwanted cells (e.g in cancer treatment) by
depositing energy from particles that are unable to ro-
tate in time with oscillating fields due to viscous drag or
anisotropic energy barriers4. In each of these techniques,
it is necessary to understand the rotational dynamics of
the particles in magnetic fields. Furthermore, simula-
tion studies can be used by practitioners to choose ideal
nanoparticle conditions and satisfy their specific needs2.
In this work we examine Brownian rotation where par-
ticles physically rotate and experience viscous drag when
placed in an oscillating field. The physics of Brownian ro-
tation is governed by the average time it takes a particle
to relax to an equilibrium value given its surroundings.
The Brownian time constant τBis based accordingly on
the viscosity ηand temperature Tof the suspending fluid,
the hydrodynamic particle volume V, and Boltzmann’s
constant kB5,6:
τB=3ηV
kBT.(1)
Brownian rotation is the dominant physical process when
the particles are thermally blocked. This means that the
a)Electronic mail: dbr@Dartmouth.edu.
energy barrier for internal magnetic moment relaxation is
greater than the thermal energy of the system. If the par-
ticle is not thermally blocked, the electronic spins within
single magnetic domains can rotate in unison, switching
the magnetization of the particle. This spin rearrange-
ment is referred to as N´eel relaxation, and as described
in Ref.5has a time constant of τN=τ0exp(KVc/kBT)
where Kis the magnetic anisotropy constant, Vcthe par-
ticle core volume, and τ0the N´eel attempt time charac-
teristic of the material. Therefore, if a measurement is
taken on a scale shorter than the relaxation time, no N´eel
spin flips are expected on average.
It is important to note also that both relaxation pro-
cesses occur without an applied field as the probabilities
of random relaxation increase with heightened thermal
activity. In practice, because the particle volume is a
defining element of the blocking energy, the two regimes
are often separated by the size of the particles. There-
fore, when we consider larger 100nm diameter particles
like those used in magnetic spectroscopy we are confident
the dominant mechanism of relaxation is Brownian.
II. STOCHASTIC MODELING FROM A BALANCE OF
TORQUES
If we consider a dilute sample of MNPs we can imagine
each as a separate isotropic dipole with a single magne-
tization direction determined by an internal crystalline
properties. In a vacuum, the magnetic torque on such
a dipole is expressed as the cross product between its
magnetic moment and the field, T=m×B. The mag-
netic moment can be found from the material saturation
magnetization multiplied by the magnetic core volume
MsVcore. Nanoparticles are commonly dispersed in a
fluid so we must account for viscous drag during rota-
arXiv:1403.6427v1 [cond-mat.mes-hall] 25 Mar 2014
2
tion. To first approximation, this torque is proportional
to the angular velocity of the particle with magnitude
given by the Stokes-Einstein relation for small Reynold’s
number particles6. Written, the torque is T=−6ηV Ω,
with Ωthe angular velocity, Vthe hydrodynamic volume
of the particle, and ηthe fluid’s viscosity.
At the nanoscale we must also consider the powerful
influence of thermal effects caused by random collisions
of nanoparticles with the minute molecules of the fluid.
If we assume that the time between such interactions is
much shorter than the reaction time of the particles, we
are free to consider the statistical fluctuations Marko-
vian – uncorrelated spatially or temporally. Accordingly
we also implement a white noise force Nin our model7.
White noise is characterized theoretically by a flat power
spectrum in frequency so that in Fourier space the be-
havior is delta autocorrelated in time and space with zero
mean value. Explicitly,
hN(t)i= 0 hNi(t)Nj(t0)i= 2Dδij δ(t−t0).(2)
Thus the expectation value of our random force at any
time is zero, and no previous force affects the subsequent
steps. The Einstein-Smoluchowski diffusion constant D
for spherical rotations depends on both the thermal en-
ergy as well as the viscous drag D= 6kBT ηV . This
may be determined from the fluctuation dissipation the-
orem or the associated Fokker-Planck equation for the
dynamics8. We now have the complete balance of torques
T=m×B−6ηV Ω+√2DN.(3)
The acceleration term proportional to the moment of
inertia can be ignored because we have already specified a
low Reynolds number for the MNPs. This means that the
frictional drag forces are sufficiently intense as to prevent
inertial rotation. With this assumption we can simplify
considerably to a first order differential equation
Ω=1
6ηV m×B+√2DN.(4)
We can further clarify the expression by noting that
the magnetic moment vector will experience a tangential
rate change determined by the perpendicular component
of the angular velocity of the moment viz. dm/dt =
Ω×m9. Using this in Eq. 4 we obtain the full stochas-
tic Langevin equation governing rotational dynamics of
a magnetic Brownian particle suspended in fluid and
placed in a magnetic field10,11:
dm
dt =1
6ηV m×B+√2DN×m.(5)
We chose to make no further assumptions, instead re-
sorting to stochastic numerical analysis. The construc-
tion of the stochastic model is described below. Once
prepared, we embarked on a series of simulations in order
to benchmark against theories in various physically per-
tinent regimes. This determines the strengths and weak-
nesses of each analytical model, from the static Langevin
function derived from Boltzmann statistics12, to the weak
field Debye model13,14, and lastly the low frequency ef-
fective field model15.
III. OTHER APPROACHES TO MODELING MNPS
Currently there is no closed form solution to the
stochastic Langevin equation, yet insight can often be
teased from the rotational diffusion equation, sometimes
also referred to as the Fokker-Planck equation. This ap-
proach introduces a distribution function for nanoparti-
cle magnetization that carries all the associated prob-
ability moments. In contrast, the stochastic method
has a more transparent differential equation, but re-
quires solving for all the moments separately through
repeated trials. As in Felderhof and Jones’ paper15
the orientation of a nanoparticle is presumed to satisfy
the Einstein-Smoluchowski equation for the distribution
function f(m, t), where the azimuthal symmetry of the
dipole potential energy U=−m·Bresults in a simplified
relation that is only a function of the polar angle f(θ, t),
∂f
∂t =D1
sin θ
∂
∂θ sin θ∂f
∂θ +E(t) sin2θf.(6)
The diffusion constant Dis the same as the stochastic
approach above. The variable E(t) = mB sin (ωt)/kBT
is a ratio of magnetic to thermal energy with a perfect
sinusoidal field. Eq. 6 could be used to examine the exact
distribution function over magnetic moment angles but
has not admitted an analytical solution.
In 1929, Peter Debye developed a first order approx-
imation for small driving fields that neglects non-linear
behavior, and for our purposes is particularly limited to
cases with no magnetic saturation13. Linear response
theory as used in Debye’s method leads to a susceptibil-
ity that is a combination of real (in phase) and complex
(out of phase) components:
χ=χ0+iχ00 (7)
so that the magnetization parallel to the driving field can
be expressed in terms of the Brownian relaxation time
(Eq. 5), an approximation in and of itself combining mul-
tiple relaxation mechanisms into one time constant. The
Debye magnetization along the direction of the applied
field is thus expressed
M(t) = M0
1+(ωτB)2(cos ωt +ωτBsin ωt).(8)
Both perturbative methods16 and a series expansion in
Legendre polynomials17 have been used to develop a
more complete solution to Eq. 6. A particularly use-
ful result, the so called effective field model, assumes
that the frequency of oscillation is low enough so that
the equilibrium distribution remains Maxwellian feq ∝
exp(−U/kBT). In low frequency regimes the quasi-static
3
formulation does produce accurate nonlinear dynamics
for the MNP magnetization with
dM(t)
dt =−2M(t)
τB1−E(t)
αe(t)(9)
where again τBis the Brownian relaxation time (Eq. 5),
and at any moment in time, αe(t) = mB/kBTis the
argument in the Langevin function,
L(α) = coth α−1
α,(10)
changing in time but not necessarily equal to the per-
fectly sinusoidal variation E(t) from Eq. 6. This slight
difference encodes the non-linearity into the magnetiza-
tion to first approximation. In practice the model is used
to find the magnetization in the direction of the applied
field by iterating the differential equation (Eq. 9) while
at each time step inverting the Langevin function to find
the proper effective field αe.
Aside from the two dynamical models we’ve seen, the
Langevin function itself provides a benchmark for the
magnetization Meq =M0L(α) in the static field case,
provided the system is allowed to reach equilibrium. The
three models presented will be compared to our stochas-
tic model later in the work and the strengths and weak-
nesses of each will be discussed.
IV. NUMERICAL METHODS
To simulate the particle dynamics, we developed a
Monte-Carlo scheme for MNPs in various fields and en-
vironmental conditions. To do so, the numerical recipe
below (Eq. 11), was implemented to solve the stochastic
differential equation (Eq. 5) iteratively. Because at root
we must admit that we are approximating some colored
noise process as a white noise process, we take recourse to
the Stratanovich interpretation of stochastic integration.
The transformation also has practical benefits because
the Stratanovich calculus preserves the rules of ordinary
differential calculus (i.e. the chain rule)18. A fictitious
drift term arises during the transformation from the It¯o
calculus and hence we can employ a stochastic numerical
integration scheme which is transparently analogous to
common methods. The simplest option in the stochas-
tic realm, the Euler-Marayuma scheme (see for example
Ref.19 Ch.7) is sufficient because the slightly increased
accuracy achieved through higher order methods is out-
weighed by longer computation times. We interpret the
white noise force accordingly as a Wiener process dW,
the derivative of a Markov process and also uncorrelated
in varying space or time. It can be approximated numer-
ically as a Gaussian distribution scaled by √∆t19. This
results in the numerical algorithm
m+ ∆m=m
6ηV (m×H×m−2kBTm) ∆t
+s2kBT
6ηV ∆tW×m
where mis the magnetic moment magnitude and Wis a
vector of random numbers with zero mean and unit stan-
dard deviation in each cartesian direction. At this level of
analysis, single particle trajectories are practically mean-
ingless. The quantities of interest are instead statistical
moments, often requiring thousands to millions of sam-
ples. To examine equilibrium magnetization, simulation
lengths were multiples of theoretical time constants. For
simulations using oscillating fields, equilibrium dynamics
were determined by converging similarity between peri-
ods. The standard deviation between a cycle and its pre-
decessor divided by the amplitude of oscillation was used
a metric for convergence. Cycles were repeated until this
value was less than the threshold of 1%. The constant nu-
merical parameters were chosen and varied around those
typical of MSB experiments. The mass magnetization
was taken from a nanoparticle supplier (Micromod) data
sheet as 76 emu/g for 100nm diameter particles. The
particles are actually made up of many crystal domains,
but are treated as though the bulk properties mimic the
component properties and density is assumed constant
so that the total magnetic moment can be written in
its more common units of Joule/Tesla. The viscosity of
the solution is assumed to be ∼1cP, near that of water at
room temperature (293K). We pick realistic properties in
order to see the correct physics in each regime of interest
and this aids in the transition to quantitative comparison
with experiments.
V. MODEL BENCHMARKING
In this section, we verify that the model predicts
well known physical results including a) the static field
Langevin function, b) the high-frequency, low-amplitude
Debye model, and c) the effective field model. The first
benchmark is that with no applied field, the relaxation to
equilibrium is on a time scale determined by Einstein’s
formulation of Brownian motion (Eq. 5). If a sample of
particles is prepared to align in some direction but ex-
perience no external force, their average magnetization
quickly decays to zero. By fitting to the normalized data
an exponential decay of the form M=e−t/τ , we obtain
the relaxation time. Increasing the viscosity intuitively
increases the time constant. Fig. 1 shows the stochastic
model agrees with Einstein’s theory with nanosecond pre-
cision. It should be noted however that as the viscosity
is decreased, and particularly at the limit where η→0,
Einstein’s model becomes unphysical and the stochastic
model predicts longer relaxation times.
To test the model’s behavior in a static field we begin
with a sample of particles initially in the x-direction, and
employ a static field in the z-direction. The magnetiza-
tion is allowed to evolve, and the value after five time con-
stants is recorded. This gives a good estimate of the equi-
librium magnetization and agrees with Langevin’s theory
as expected. Raising the viscosity does not change the
outcome of the magnetization value, only delaying or ex-
4
FIG. 1. Two relaxation curves with associated relaxation
times found from exponential fits of the stochastic data. The
expected relaxation times are given by Einstein’s equation for
Brownian relaxation to be 379.42µs and 569.13µs respectively,
within nanoseconds of the simulated data. The simulations
used Eq. 11 at two viscosities. 106particles were averaged
beginning all in the z-direction, with no magnetic field.
pediting equilibrium. Increasing the temperature allows
for more thermal motion, and therefore lessens the total
alignment with the field effectively decreasing the final
magnetization value. The data are depicted in Fig. 2.
Many models of MNPs benchmark against the equilib-
rium and static case, but we have extended ours to os-
cillating fields in order to corroborate analytic approx-
imations. Here we begin by considering an oscillating
field of the form B=B0sin ωt applied with high fre-
quency (ω/2π= 20kHz). In this case, the magnetization
does not saturate and follows the behavior of the Debye
Model (see Fig. 3).
However, in a regime where particles come closer to
saturation, the magnetization obeys different dynamics,
including much more severe hysteretic effects, so that a
simple linearization method like Debye’s can no longer be
applied. Described previously, the effective field model
(Eq. 9) uses an adaptive method to solve for the ef-
fective thermal field from the inverse Langevin function
and includes relaxation effects. Simulations demonstrate
equivalence (Fig. 4) to the effective field model. Yet, the
stochastic model extends past the range of the effective
field model, which is only valid for low frequencies. Fur-
thermore, our model replicates the correct behavior of
solely odd harmonics from symmetric distortion. This
is an important result because magnetization harmonics
are used exclusively in practical imaging and sensing20,21.
FIG. 2. The stochastic model (Eq. 11, data points) is com-
pared to the Langevin function (Eq. 10, solid lines) at various
temperatures. The particles are initially aligned in the x-
direction and the field points in the z-direction. 104particles
were averaged, few enough to demonstrate that as tempera-
ture increases, the disorder in the system becomes much more
prevalent.
FIG. 3. In the low-field, high-frequency regime, the stochastic
model (Eq. 11) agrees with the Debye model (Eq. 8). Shown
is the 20th cycle using 105particles in a 20mT, 2MHz field.
In this simulation, the driving field is a cosine so the phase
lag can be seen. The almost complete out of phase nature of
the curve can be understood by examining the Debye model.
Given a relaxation time of hundreds of microseconds and a
frequency of 20kHz, the out of phase component is an order
of magnitude larger than the in phase signal.
5
FIG. 4. The Monte-Carlo code (Eq. 11) matches the effec-
tive field model magnetization (Eq. 9) and harmonic structure
with a 20mT applied field of frequency 1kHz. The tempera-
ture is 300K and 105averages are used for a smooth curve.
Note the applied field is sinusoidal and the phase-lag is evi-
dent.
VI. EXPERIMENTAL COMPARISON AND VALIDATION
Magnetic particle spectroscopy can be used to dis-
cover information about the environment surrounding
nanoparticles. One of the first experiments to do this
employed MNPs as microscopic thermometers22. To do
so, a scaling relationship between the temperature and
the magnetic field in the argument of the Langevin func-
tion was exploited. The method makes use of the low
frequency, adiabatic limit assumption – that the system
is assumed to always be in equilibrium and thus gov-
erned by the Langevin function. Then if a control curve
is developed for a known temperature, the scaling fac-
tor necessary to map subsequent curves onto the control
can be used to quantify temperature. Note that ratios
of higher harmonics are used for experimental reasons;
to avoid dependence on particle number or systematic
receive coil errors. Because the simulations have vali-
dated the Langevin function explicitly, we should not be
too surprised to find that the scaling of field and tem-
perature in the Langevin argument is consistent when
attempted at low frequencies. Yet, that it works for the
harmonic ratios (see Fig. 5) is an important confirma-
tion. In the simulations, we have neglected any temper-
ature dependence on viscosity or magnetic moment, and
the adiabatic approximation fails above a few hundred
cycles per second, at which point the harmonics do not
carry enough information to distinguish between temper-
atures.
FIG. 5. A scaling argument can be made for temperature es-
timation of nanoparticles as in22. By sweeping through field
strengths and calculating the harmonic ratio for different tem-
peratures, we show that scaling the x-values by the ratio of
the control to new temperature shifts the data back on top
of one another. Thus for an unknown temperature curve, a
least squares fit can determine the scaling constant and the
temperature can be recovered. In this figure, a field of 200Hz
was applied at typical experimental values of 7.5-15mT. The
two curves used 104particles at known temperatures of 293K
and 303K respectively so in this case the scaling factor is 0.96.
Results also agree with experiments that determine
nanoparticle relaxation time due to viscosity shifts23.
Taking ratios of harmonics at various frequencies, there
is a characteristic shift in time which compensates for
the increasing relaxation time from an increased viscos-
ity. By scaling the frequency range by the appropriate
amount, the data can be shifted back onto a control
curve. Then by measuring the relaxation time through
the scaling value, obtained with a least-squares regres-
sion, the nanoparticles can be used as viscometer. The
simulations are able to repeat the experimental method
(Fig. 6). When scaled by viscosity, the curves align. Fur-
thermore, by changing the specific values of the nanopar-
ticles size and magnetic moment slightly from the values
given by the manufacturer, the actual experimental data
from Rauwerdink’s paper can be reproduced.
VII. CONCLUSIONS
A Langevin-type stochastic differential equation for
the magnetization of non-interacting isotropic MNPs was
6
FIG. 6. A depiction of three frequency sweeps taking the har-
monic ratio of the fifth to third harmonic as in23. The higher
viscosity appears lower than the original, but can be scaled
horizontally (in frequency) to align with the old curve. This
compensates for viscous affects that are characterized by the
relaxation time. The 1.5cP curve is shown explicitly before
and after shift, while the 2cP is only showed post scaling, to
justify that the three curves all do align. The simulations
used a 20mT, 1kHz field, 300K, and four runs of 104particles
each.
developed and a numerical integration scheme was ap-
plied to model the particle dynamics in various types
of magnetic fields. The simulations successfully mod-
eled the approach to equilibrium (through the classical
Einstein relaxation time) and the eventual state of ther-
mal equilibrium was validated by the Langevin function
at various temperatures. In oscillating fields, our ap-
proach agreed with other standard analytical approxi-
mations, surpassing the ability of both by achieving ac-
curacy through a wide frequency range. The agreement
included the magnetization shape as well as it’s harmonic
content in a distorted regime. Finally, it was shown that
the model reproduces two experimental results from pre-
vious work on MSB sensing: that the relaxation time
shift due to viscosity changes can be compensated for by
a scaling in frequency so that nanoparticle solution vis-
cosity can be measured, and that a similar method can
be used in the adiabatic limit to measure temperature by
scaling field strength. One of the large benefits of using
stochastic analysis is the transparency of the differen-
tial equation. In the future, we will investigate further
physics including the potential of decoupling temperature
and viscosity effects into separate relaxation times.
VIII. ACKNOWLEDGEMENTS
This work was supported by an NIH-NCI grant
1U54CA151662-01.
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