Motion of nanometer sized magnetic particles in a magnetic field gradient
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Motion of nanometer sized magnetic particles in a magnetic field gradient Cite as: J. Appl. Phys. 104, 093918 (2008); https://doi.org/10.1063/1.3009686 Submitted: 06 June 2008 . Accepted: 16 September 2008 . Published Online: 14 November 2008 Vincent Schaller, Ulli Kräling, Cristina Rusu, Karolina Petersson, Jan Wipenmyr, Anatol Krozer, Göran Wahnström, Anke Sanz-Velasco, Peter Enoksson, and Christer Johansson ARTICLES YOU MAY BE INTERESTED IN Interaction between two magnetic dipoles in a uniform magnetic field AIP Advances 6, 025004 (2016); https://doi.org/10.1063/1.4941750 Microelectromagnets for the control of magnetic nanoparticles Applied Physics Letters 79, 3308 (2001); https://doi.org/10.1063/1.1419049 Superparamagnetism Journal of Applied Physics 30, S120 (1959); https://doi.org/10.1063/1.2185850 J. Appl. Phys. 104, 093918 (2008); https://doi.org/10.1063/1.3009686 104, 093918 © 2008 American Institute of Physics.
JOURNAL OF APPLIED PHYSICS 104, 093918 共2008兲
Motion of nanometer sized magnetic particles in a magnetic field gradient
Vincent Schaller,1,a兲 Ulli Kräling,2 Cristina Rusu,2 Karolina Petersson,2 Jan Wipenmyr,2
Anatol Krozer,2 Göran Wahnström,3 Anke Sanz-Velasco,1 Peter Enoksson,1 and
Christer Johansson2
1
Department of Microtechnology and Nanoscience, Chalmers University of Technology,
SE-412 96 Göteborg, Sweden
2
Imego Institute, Arvid Hedvalls Backe 4, SE-400 14 Göteborg, Sweden
3
Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
共Received 6 June 2008; accepted 16 September 2008; published online 14 November 2008兲
Using magnetic particles with sizes in the nanometer range in biomedical magnetic separation has
gained much interest recently due to their higher surface area to particle volume and lower
sedimentation rates. In this paper, we report our both theoretical and experimental investigation of
the motion of magnetic particles in a magnetic field gradient with particle sizes from 425 nm down
to 50 nm. In the experimental measurements, we monitor the absorbance change of the sample
volume as the particle concentration varies over time. We also implement a Brownian dynamics
algorithm to investigate the influence of particle interactions during the separation and compare it to
the experimental results for validation. The simulation agrees well with the measurements for
particle sizes around 425 nm. Some discrepancies remain for smaller particle sizes, which may
indicate that additional factors also influence the separation for the smaller size range. We observe
that the separation process includes the formation of chainlike particle aggregates due to the
magnetic dipole-dipole interactions between particles when subjected to an external magnetic field.
We can also see that the hydrodynamic interaction between these chains contributes to reducing the
separation time. In conclusion, we show that the formation of these particle aggregates, and to a less
extent the hydrodynamic interactions between them contributes to significantly enhancing the
particle separation process. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3009686兴
I. INTRODUCTION directly proportional to the induced magnetic moment of the
particle, becomes weaker, which results in a longer separa-
Magnetic particles 共also called magnetic beads兲 are very
tion time. However, it remains unclear to what extent the
promising for use in biomedical applications such as contrast
particle sizes can be reduced and whether the particles can be
agents for magnetic resonance imaging, magnetic markers
separated in a reasonable amount of time. Previous research
for biosensing, or magnetic carriers in magnetic
separation.1,2 In the latter application, magnetic particles
functionalized with biomolecules 共e.g., antibodies兲 are used
for transport and separation of nonmagnetic molecules.3,4 (a) (b)
A magnetic separation system typically consists of func-
tionalized magnetic particles in a test tube or a sample region
and a permanent magnet generating the magnetic field that
M
magnetizes the particles. The magnet also creates the mag-
netic field gradient that together with the particle magnetiza-
tion exerts magnetic forces on the particles. As a result, the
particles are transported to the magnet 共Fig. 1兲. When the
particle size decreases, the ratio of the particle surface to (c) (d)
particle volume increases, and therefore the particles offer a
larger specific surface for chemical binding for an equivalent
total mass of the particle system. Consequently, there is sig-
nificant interest in decreasing the size of the particles being M
used in biomedical separation.1
However, when the size of the particles decreases, the
induced magnetic moment of the particles decreases as well
due to the reduced amount of magnetic material in the par- FIG. 1. Schematic illustration of a typical separation process 共not drawn to
ticles. Thus the magnetic force exerted on the particle, being scale兲. 共a兲 A cuvette contains a suspension of magnetic particles 共in gray兲 to
be separated from nonmagnetic entities 共in white兲. 共b兲 When the permanent
magnet is placed next to the cuvette, the magnetic particles are attracted to
a兲
Author to whom correspondence should be addressed. Electronic mail: the magnet and accumulate at the cuvette wall. 共c兲 The nonmagnetic entities
vincent.schaller@chalmers.se. Tel.: ⫹46-31 772 18 59. FAX: ⫹46-31 772 remaining in the liquid are washed out. 共d兲 Finally, the magnet is removed
36 22. and the magnetic particles redisperse in the liquid 共for instance by agitation兲.
0021-8979/2008/104共9兲/093918/14/$23.00 104, 093918-1 © 2008 American Institute of Physics093918-2 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
TABLE I. The results of the fitting procedure of the magnetization data at 295 K for the different magnetic particle systems: s is the saturation magnetic
moment per particle and M p is the saturation magnetization of the particle 共given by s divided by the particle volume兲. The mean diameters of the single-
domains are calculated using a typical value of the intrinsic magnetization of small magnetite single-domains 共M s = 350 kA/ m兲 and the number of domains
represents the number of single-domains per particle.
Mp Average diameter of Equivalent number of
D 共nm兲 S共Am2兲 / particle⫻ 10−16 共kA/m兲 d / kBT共T−1兲 single-domain 共nm兲 single-domains
50 0.016 24 72.2 11.8 5
100 0.344 66 82.9 12.3 101
150 0.870 49 91.0 12.7 232
200 0.662 16 75.0 11.9 215
250 5.151 63 84.1 12.4 1489
425 19.35 48 72.1 11.7 6523
related to the motion of magnetic particles in a field gradient to biomolecules during the separation process. The particles
has focused on particles in the micrometer range,5–7 rather are suspended in de-ionized water. Different particle sizes
than on nanometer-sized magnetic particles. with the following hydrodynamic diameters have been stud-
In this paper, we report our theoretical and experimental ied: 50, 100, 150, 200, 250, and 425 nm. These values pro-
investigation of the behavior of magnetic nanoparticles in the vided by the particle manufacturer were confirmed by photon
case of magnetic separation in a magnetic field gradient. The correlation spectroscopy measurements.
particles we have used in this study are so-called multicore The initial particle concentration in the sample was di-
particles, i.e., they consist of several single-domains of mag- luted to 0.1–0.3 mg/ml 共giving a mean distance between the
netite 共Fe3O4兲 embedded in a polymer matrix. This can be particles of more than ten times the particle diameter兲. The
considered a typical magnetic particle system used in biom- initial particle concentration was chosen so that the solutions
agnetic separation applications.1 Our aim is to gain a better containing the particles had relatively low optical absorption.
understanding of the mechanisms involved in the magnetic Lower particle concentrations gave an unstable optical signal
separation process of nanometer-sized particles by combin- when compared to water reference samples. All the optical
ing both measurements and simulations. experiments were conducted with the magnetic particles di-
To simulate the motion of the submicrometer sized mag- luted in distilled water.
netic particles in a magnetic field gradient, a numerical The magnetization curves 共i.e., magnetization versus
model based on a Brownian dynamics algorithm has been field兲 of the particle systems were measured with a vibrating
implemented. This model takes into account the magnetic sample magnetometer from Lake Shore Ltd. The values of
forces exerted on the particles as well as the dipole-dipole the magnetic properties of the particles 共see Table I兲, i.e., the
and hydrodynamic interactions between particles. Simulta- saturation magnetic moment per particle s and the intrinsic
neously an optical measurement setup has been built to magnetic moment of the single-domains in the particle d
evaluate the spatial distribution of the particles in a sample were obtained by fitting the magnetization data to the Lange-
cuvette as a function of time. This method is based on moni- vin function L according to
冉 冊 冋 冉 冊 册
toring the change in intensity of a light beam transmitted
dB dB k BT
through the sample volume as the particle concentration M = M 0L = M 0 coth − , 共1兲
changes with time after placing the permanent magnet next k BT k BT dB
to the sample cuvette. We present experimental results on the
where M is the magnetization, M 0 is the saturation magneti-
separation of the magnetic particles as a function of the ini-
zation of the sample, B is the magnetic field, kB is the Bolt-
tial sample concentration, the distance from the magnet, and
zmann constant, and T is the absolute temperature. The result
the strength of the gradient fields. This optical method for
of the fitting for the magnetic particle system with diameter
quantifying the separation process is a very easy and simple
425 nm can be seen in Fig. 2. From the values of M 0, the
way to study the separation behavior of particle systems with
particle concentration and the particle size the magnetic mo-
different sizes and magnetic properties. The comparison be-
ment per particle M p can be calculated for the different par-
tween the simulations and the measurements has revealed
ticle systems 共Table I兲. The average size of the single-
some of the important mechanisms influencing the separation
domains in the particles can be estimated from the d value
process, for instance the formation of particle aggregates.
and is also given in Table I. The Langevin model does not
take into account magnetic interactions between the single-
II. THE MAGNETIC PARTICLES
domains in the particle or between particles or magnetic an-
The magnetic particles used in this study are fluidMAG- isotropy of the single-domains. The experimental data points
D nanoparticles from Chemicell GmbH. They are composed were fitted to Eq. 共1兲 using only one Langevin function
of a number of single-domains of magnetite 共Fe3O4兲 in a meaning that only one size of the single-domains is consid-
starch coating that consists of a polymer matrix containing ered in the fitting. All of these facts imply that the fitting
terminal hydroxyl groups. The hydrophilic polymer protects procedure to the experimental magnetization curves gives
the particles from aggregation due to foreign ions, and the only approximate values of the size of the single-domains
terminal functional hydroxyl groups can be used for binding and the number of single-domains per particle. Magnetic in-093918-3 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
1000
(a)
800
(c) (e) (c)
(g)
600
M [A/m]
A
(b) (f)
B
400 (d) detection volume
200
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
B [T]
FIG. 2. Magnetization vs field for the magnetic particles of diameter 425 nm
共dots兲. The solid line represents the fit of the experimental data to the Lange-
vin function.
teractions, magnetic anisotropy, and single-domain size dis-
tributions can be introduced into the magnetization model.8,9
However, the results seem reasonable, as seen from transmis-
sion electron microscopy images of the particle systems 共not
shown here兲.
III. EXPERIMENTAL SETUP
We perform the experimental measurements by monitor- detection volume
ing the absorbance change of the sample volume as the par-
ticle concentration varies over time. The experimental setup
for the optical measurements consists of the eight following
main components 关Fig. 3共Top兲兴: a light source, a blazed grat-
ing, lenses and fibers, a cuvette containing the sample, a magnet cuvette magnet
Ø 14 mm 12.5 mm 8 mm
magnet, a photodiode, a source measure unit, and a personal
computer. The light source is a 250 W tungsten-halogen sample
lamp. The blazed grating with 300 lines/mm is set to get a column
10 mm
mean wavelength of 450 nm. This wavelength was found to 2 mm
give the best detection signal. The light is sent through a
fiber to the sample chamber 共HORIBA Jobin Yvon Sam-
pleMAX兲. In this chamber, two lenses focus the light from
the fiber onto the sample and two lenses launch the light
passing through the sample back into a fiber connected to a detection volume
standard photodiode. The current from the photodiode is then (Ø 1.8 mm x 2 mm)
5 mm
measured by the Keithley 237 high voltage source measure
unit. Finally, the data is sent to a computer via the GPIB bus FIG. 3. 共Top兲 Main components of the optical setup for the dynamic mea-
of the source measure unit. surements of the particle separation: 共a兲 light source, 共b兲 grating, 共c兲 lenses,
The sample is contained in the tapered base 共hereafter 共d兲 cuvette, 共e兲 separation magnet 共the direction of the magnetic field is also
indicated in the figure兲, 共f兲 photodiode, 共g兲 source measure unit. 共Middle兲
referred to as sample column兲 of an Eppendorf UVette, a Magnified view of the measurement volume with the light path indicated by
plastic cuvette with four optical windows 共UV and visible arrows. 共Bottom兲 Front and side views of the detection volume 共in gray兲 and
transparent between 220 and 1600 nm兲 and external dimen- the cylindrical NdFeB magnet, drawn to scale.
sions of 12.5⫻ 12.5⫻ 36 mm3. The length and width of the
sample column are 10 and 2 mm, respectively. 3共Middle兲兴. The light beam has a diameter of about 1.8 mm
The detection volume is limited to a small measuring and the optical path length 共i.e., the width of the column兲 is
volume within the sample column defined by the volume of 2 mm. Thus, the detection volume is about 5 l while the
the light beam penetrating through the sample column 关Fig. total sample volume is 40 l. The position of the detection093918-4 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
0.4 2.0
1.8 50 nm
L x
0.35 100 nm
1.6
150 nm
−ln(I / I ) [OD]
B 1.4 200 nm
0.3 M
2R 1.2 250 nm
425 nm
0
0.25 1.0
B [T]
0.8
0.2 0.6
0.4
0.15
0.2
0.1 0.0
0.0 0.1 0.2 0.3
c [mg/ml]
0.05
0 2 4 6 8 10
FIG. 5. Absorbance measurement −ln共I / I0兲 vs particle concentration for the
x [mm] different particle systems used in this study. The particles are distributed
homogeneously in the sample.
FIG. 4. Calculated magnetic field along the symmetry axis of a cylindrical
NdFeB magnet 共with R = 7 mm, L = 8 mm, and Br = 1 T兲 as a function of
the distance from the magnet pole face. By placing the magnet next to the cuvette 共as illustrated
in Fig. 3兲 containing a sample of magnetic particles with a
volume can be changed by adjusting the position of the light homogenous initial particle concentration, the particles start
beam with respect to the length of the sample column. The moving toward the magnet due to the magnetic force exerted
sample column is aligned to the focus point of the light with on them. The particle concentration measured with the opti-
micrometer screws. This arrangement is set such that the cal method represents the actual magnetic particle concentra-
light beam crosses perpendicular to the symmetry axis of the tion within the detection volume in the sample column. Usu-
magnet. ally the degree of separation is defined as the ratio of
A neodymium 共NdFeB兲 magnet with a radius of 7 mm particles that have reached the magnet to the total amount of
and a thickness of 8 mm creates the magnetic field and mag- particles. In our case, the degree of separation is defined as
netic field gradient. The sample column, the detection vol- the ratio of the actual particle concentration within the detec-
ume, and the cylindrical magnet are illustrated at scale in tion volume to the initial particle concentration. The separa-
Fig. 3共Bottom兲. The small width of the sample column 共2 tion can be evaluated at different positions in the sample by
mm兲 compared to the magnet diameter 共14 mm兲 guarantees adjusting the position of the detection volume with respect to
that the whole sample is close to the symmetry axis of the the length of the sample column. For instance, by positioning
magnet where the magnetic field is mainly directed in the the detection volume closer to the magnet, the separation
direction of the magnet 共x-direction兲. time defined by when all the particles have reached the mag-
We performed a finite-element analysis of the off-axis net 共the more commonly used definition of separation兲 can
components of the magnetic field and magnetic field gradient be determined.
in order to evaluate the longitudinal and transversal compo- A change in particle concentration produces a change in
nents of the magnetic force acting on the particles, Fx and Fy the absorption 共or transmission兲 intensity of the light beam,
respectively. The results show that the ratio Fx / Fy is around and thus in the current measured by the source measurement
20 for the particles located where the field lines diverge the unit. The relation between the particle concentration c and
most in the sample column 共at x = 11.25 mm and y = 1 mm兲 the measured current I is given by
and higher in the rest of the sample column. This means that I = I0e−␣lc , 共2兲
the field perpendicular to the symmetry axis in the sample
column is negligible in this case, which justifies the use of a where I0 is the current of a reference sample with no par-
one-dimensional approximation in the numerical simulation ticles 共i.e., de-ionized water only兲, l is the optical sample
described in Sec. IV. The magnetic field at the surface of the length, and ␣ is the absorption coefficient. The product of the
magnet is about 0.4 T, and its amplitude along the symmetry optical sample length and the absorption coefficient can be
axis of the magnet is plotted in Fig. 4. Along the symmetry determined from a calibration procedure with a known par-
axis, the magnetic field gradient is 52.5 T/m at x ticle concentration and no magnet attached to the cuvette,
= 1.25 mm and down to 7.7 T/m at x = 11.25 mm. The ratio and the particle concentration can then be calculated for ev-
of the magnetic force in the x-direction to the gravity force of ery measured current. In the experiments, we use a low ini-
particles 共taking the buoyancy into account兲 is about 77 at tial particle concentration since in this particle concentration
x = 11.25 and higher in the rest of the sample column. This range the absorbance coefficient is independent of particle
means that the separation process due to the magnetic force concentration, and therefore the relation between absorbance
has a much higher rate than any particle sedimentation of the and particle concentration becomes linear.
particles. In Fig. 5, the absorbance 共−ln关I / I0兴兲 is plotted versus the093918-5 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
−15
particle concentration for the particle systems with different x 10
sizes. It is interesting to notice that the slope of the absor- 1.9
bance curves for the different particle systems increases ac-
cording to the particle size in the following order: 50, 200,
150, 100, 250, and 425 nm. The particle systems are equal 1.8
with respect to the starch coating holding the single-domains
together and single-domain material 共magnetite兲 and size −15
µx [A m2]
1.7 x 10
共about 12 nm diameter兲. According to the Mie theory, the
slope of the absorbance curves is expected to increase with 1.8
µx [A m ]
2
increasing particle size. However, the slopes for the particle
1.6
systems with sizes of 200, 150, and 100 nm are in the oppo- 1.6
site order. We believe that this effect can be explained by the
fact that the number density of single-domains is larger in the 1.5 1.4
100 and 150 nm particle systems than in the 200 nm particle 0 2 4 6 8 10
system. This can bee seen in Table I, where the values of the x [mm]
volume magnetization per particle 共M p兲 for the 100 and 150 1.4
nm particle systems are much higher than the value for the 0 0.1 0.2 0.3 0.4
200 nm particle system. In fact, the 200 nm particle system Bx [T]
has the lowest M p value compared to all the other particle
systems in this study. Since the starch coating is transparent FIG. 6. Induced magnetic moment x of a fluidMAG-D425 nanoparticle as
a function of the external magnetic field B as the particle moves from x
in the visible range, the light is mainly scattered by the mag- = 10 mm to x = 0 mm. Inset: Induced magnetic moment x as a function of
netite single-domains contained in the particles. Therefore, the position x along the axis of the cylindrical NdFeB magnet.
the 200 nm particle system has a lower light scattering effect
from the single-domains as compared to the 100 and 150 nm
particle systems due to the lower number density of single- dBx
domains in the 200 nm particles. Fm = x , 共5兲
dx
IV. THEORETICAL MODEL where x is the induced magnetic moment in the particle due
to the magnetic field that the particle is subjected to. The
As explained previously, magnetic particles in a mag- induced magnetic moment in the particle may be approxi-
netic field gradient are subjected to a magnetic force that mated by using the Langevin function 共as discussed in
accelerates the particles toward the separation magnet. The Sec. II兲
magnetic force Fm on a magnetic dipole in a magnetic field
B is given 共in vector notation兲 by10
ជ = ⵜ共
F m ជ · Bជ 兲. 共3兲 x共B兲 = ndM SVdL 冉 M SV dB x
k BT
冊
, 共6兲
ជ as a pointlike magnetic dipole, then
If we consider ជ is not
a function of the space coordinates and therefore 共B ជ · ⵜ兲
ជ
= 0 and 共ⵜ ⫻ ជ 兲 = 0. Furthermore, in a source-free space where M s is the intrinsic magnetization of small single-
domains of magnetite 共a typical value of M s = 350 kA/ m is
Maxwell’s equation gives 共ⵜ ⫻ Bជ 兲 = 0, and Eq. 共3兲 becomes
used in the numerical calculations兲, Vd is the volume of the
single-domain, nd is the number of single-domains in the
Fជ m = 共
ជ · ⵜ兲Bជ . 共4兲 particle, kB is the Boltzmann constant, and T is the absolute
This equation can be considered as a good approxima- temperature. The number of single-domains in the particles
tion for the magnetic particles used in this study, for the depends on the content of iron oxide and on the size of the
following reasons. First, these magnetic particles contain a single-domains. The content of iron oxide is often given for
number of small nanometer-sized Fe3O4 single-domains for commercial magnetic nanoparticle systems. It is important to
which the magnetic dipole approximation holds well. Thus, note that the magnetic moment of the particle should be used
the total force on the magnetic particle is the sum of the instead of only the magnetic susceptibility of the particle
individual forces on each single-domain contained in the par- since the intensity of the magnetic field for a typical separa-
ticle. Second, the variation in the magnetic field due to the tion magnet is around 0.4 T close to the pole face of the
size of the magnetic particle is also assumed to be suffi- magnet. In this magnitude range, the magnetization of the
ciently small to be neglected since the particle diameter is particle is far from linear 共Fig. 6兲, and therefore the suscep-
small 共425 nm and less兲. tibility value would lead to a significant error in the calcula-
Since in our one-dimensional model we are interested in tion of the particle magnetization.
the motion along the x-direction only, the force on the mag- The magnetic field B along the symmetry axis of a cy-
netic particle finally reduces to lindrical permanent magnet is given by11093918-6 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
Bx =
Br
2
冋冑 L+x
共L + x兲2 + R2
−
x
冑x2 + R2 册 , 共7兲
TABLE II. Relaxation time = m / ␥ of the particle motion momentum for
the Chemicell fluidMAG-D nanoparticles.
Chemicell fluidMAG-D Relaxation time
where R and L are the radius and length of the magnet, 共nm兲 共ns兲
respectively, and Br is the remnant magnetic field in the mag-
425 16
net.
250 6.5
This magnet gives both the magnetic field magnetizing 200 4.7
the particle and the field gradient that together create the 150 1.6
magnetic force on the particle. In our calculations, we have 100 0.9
used a value of Br = 1 T typical for the NdFeB magnet, R 50 0.15
= 7 mm, and L = 8 mm. Measured values of the magnetic
field along the symmetry axis of the magnet agree well with
Eq. 共7兲. dx2 dBx dx
As illustrated in Fig. 6, the induced magnetic moment of m = x共B兲 − ␥ + FB . 共11兲
dt2 dx dt
the particle 关calculated from Eqs. 共6兲 and 共7兲兴 is not linear
with the field for particle positions within the sample cuvette, The above particle motion equation has been solved numeri-
meaning that a major part of the magnetization curve must cally varying 共a兲 particle size, 共b兲 single-domain size in the
be considered 共not only the magnetic susceptibility兲 in the particle, and 共c兲 the iron oxide content 共by changing the
particle separation process. number of domains per particle兲. The numerical model has
When moving in a carrier liquid with a specific viscosity, been compared with the data obtained from the optical mea-
the particle experiences a viscous force ជ FV directed opposite surements described in Sec. III.
to the particle velocity vជ according to In order to solve Eq. 共11兲, we have used the following
strategy. In a long-time dynamics, the friction coefficient ␥ is
ជ = − ␥vជ .
F 共8兲 related to the diffusion coefficient D of the particle by the
V
well-known Stokes–Einstein relation13
As a first approximation, we neglect the particle size distri-
bution and describe our particles as identical hard spheres. In k BT
D= . 共12兲
the limit of low Reynolds numbers, the friction coefficient ␥ ␥
of a spherical particle is given by Stokes law
For submicrometer particles, i.e., for low Reynolds numbers,
␥ = 6a, 共9兲 inertia effects can be neglected compared to viscous effects.
In other words, by restricting the numerical time step ⌬t to
where is the dynamic viscosity of the surrounding fluid values that are larger than the relaxation time = m / ␥ of the
and a is the particle radius. When the size of the particle is in particle motion momentum 共so-called overdamped or long-
the submicrometer range, it becomes important to include the time dynamics兲, the acceleration of the particle can be as-
stochastic Brownian force on the particle. Additional forces, sumed to be essentially zero during the time step. In our
such as hydrodynamic, magnetic, and electrostatic interac- simulation, the numerical time step is 1 ms, whereas the
tions between the particles,12 may also be important if the relaxation time of the particles with diameter 425 nm is
particles come close to each other. Including all forces acting about 16 ns 共Table II兲. This noninertia approximation of the
on the particle, the one-dimensional motion equation of the Langevin dynamics model is commonly called Brownian dy-
particle in the x-direction can be expressed as namics, and the displacement ⌬x of the particle can be ex-
pressed in this case as14
dx2 dBx dx
m = x共B兲 − ␥ + FB + FI , 共10兲
dt2 dx dt D
x共t + ⌬t兲 = x共t兲 + Fm⌬t + ⌬xG , 共13兲
k BT
where m is the mass of the particle, x is the induced mag-
netic moment of the particle in the x-direction, Bx is the where ⌬xG is a random variable of Gaussian distribution
magnetic field from the separation magnet in the x-direction, with zero mean and variance 具共⌬xG兲2典 = 2D⌬t. This random
FB is the Brownian stochastic force 关with an ensemble aver- displacement represents the Brownian motion due to the sto-
age 具FB共t兲典 = 0兴, and FI is the sum of all the interaction forces chastic collisions between the molecules of the fluid and the
due to 共a兲 neighboring particle flow 共pressure disturbance in particle. In this approach, the equation of motion is inte-
the liquid兲, 共b兲 dipolar magnetic interactions, and 共c兲 electro- grated over a time interval ⌬t that is sufficiently short so that
static repulsion between the charged particles. the systematic magnetic force Fm remains approximately
constant during this time interval. The numerical model as
described above has been implemented in MATLAB.15 Figure
7 shows calculated trajectories of three identical 100 nm par-
A. No particle interactions
ticles using Eq. 共13兲. The dashed line in this figure shows the
Let us assume first that the interparticle interactions may particle motion without including the Brownian force while
be neglected. The equation of motion can then be expressed the other three curves show typical irregular particle trajec-
by 共Langevin dynamics兲 tories due to the Brownian motion.093918-7 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
5020
5000
4980
4960
x [um]
4940
4920
4900
4880 (a)
4860
4840
0 20 40 60 80 100
t [s]
FIG. 7. Trajectories of three particles 共particle diameter 100 nm, iron con-
tent 10 wt % and single-domain diameter of 7 nm兲. Without Brownian mo-
tion 共dashed line兲 and with Brownian motion 共continuous lines兲.
Since we are using a one-dimensional model of the par-
ticle motion and assuming that the particle concentration
changes only in the direction of the applied field, the three- (b)
dimensional 共3D兲 particle concentration can be directly esti-
mated from the one-dimensional 共1D兲 particle concentration. FIG. 8. Optical microscopy images of the 425 nm magnetic nanoparticles in
suspension in a droplet of de-ionized water on a microscope slide. 共a兲 In
At the initial time, the particles are assumed to be homoge- absence of any external magnetic field. 共b兲 Chain formation observed a few
neously distributed in the sample cuvette. Thus, the average seconds after an external magnetic field is applied to the particles by placing
distance between particles in the x-direction ⌬x̃ is calculated a NdFeB magnet in the vicinity of the microscope slide. The chains are
aligned parallel to the magnetic field.
from the initial bulk concentration by the following relation:
冉 冊 −1/3 B. Magnetic dipole-dipole interaction
c3D,0
⌬x̃ = , 共14兲 The dimensionless parameter used to estimate the
pV p
strength of the magnetic dipole-dipole interaction energy be-
tween magnetic particles relative to the thermal energy is16
where c3D,0 is the initial bulk particle concentration 共i.e., the
total mass of all the particles in the sample divided by the 0 2
sample volume兲, p is the particle density, and V p = 4a3 / 3 is = . 共16兲
4共2a兲3kBT
the particle volume, where a is the particle radius. Since the
particles are assumed to move only in the x-axis direction, For instance, the particle system with diameter 425 nm has a
the bulk concentration as a function of time c3D共t兲 is propor- value between 650 and 1100 depending on the position in
tional to the linear concentration along the x-axis c1D共t兲 ac- the sample cuvette. Particle systems with diameters 100, 150,
cording to 200, and 250 nm are also characterized by values signifi-
cantly larger than unity. This indicates that the magnetic in-
teraction between neighboring particles is significantly larger
c1D共t兲
c3D共t兲 = c3D,0 , 共15兲 than the thermal energy kBT and therefore particle aggrega-
c1D,0 tion is expected to occur when the particles are subjected to
a magnetic field. Optical microscopy observations confirmed
where the subscript 0 denotes the concentration value at the the aggregation of the 425 nm magnetic particles used in this
initial time. study when applying an external magnetic field 共Fig. 8兲. The
When we compare the numerical model presented above particles are contained in a de-ionized water droplet placed
with experimental results regarding the particle motion, we on a microscope slide. In the absence of an external magnetic
find substantially different behaviors. This deviation shows field, the particles are homogeneously distributed 关Fig. 8共a兲兴,
that the model based on noninteracting particles is not suffi- and as a magnetic field is applied the particles tend to aggre-
cient to predict the separation time with enough accuracy, gate into linear chainlike structures aligned with the external
even for very low particle concentrations. Hereafter, we ex- magnetic field 关Fig. 8共b兲兴. The magnetic field was created by
tend the model by including dipolar 共Sec. IV B兲 and hydro- a NdFeB magnet placed next to but slightly above the mi-
dynamic 共Sec. IV C兲 interactions between particles. croscope slide, which explains why the chains appear on the093918-8 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
−15
image with a small out-of-plane tilt. On the other hand, the x 10
50 nm particle system has a value below unity even at the
closest distance from the magnet 共 = 0.46 at x = 1.25兲. This
means that the motion of the 50 nm particles is dominated by 1.69
thermal fluctuations, which prevents the particles from ag-
gregating.
The one-dimensional magnetic dipole-dipole force be-
tween two particles is given by
µ [A m2]
1.68
3 0 i j
Fdd,ij = , i ⫽ j, 共17兲
2 共xi − x j兲4
i
where xi and x j are the positions of particles i and j with
1.67
induced particle magnetic moments i and j, respectively,
calculated from the Langevin function 关Eq. 共6兲兴. Equation
共17兲 will be used later in the final expression of the particle
motion. However, before the particle motion equation can be
solved, we must consider the difference in magnetic and hy- 1.66
1 2 3 4 5 6 7 8 9 10
drodynamic properties of the particle chain compared to
single particles. Particle i
We assume that the particles aggregate into a chainlike FIG. 9. In this simulation, a chain of ten magnetic particles 共with diameter
structure that is represented as an ellipsoid of revolution with of 425 nm兲 is subjected to a magnetic field of 0.1 T. In the figure, the
a total magnetic moment located at the center of the ellip- effective magnetic moments of each particle are given without 共filled
circles兲 and with 共filled squares兲 inclusion of the magnetic dipole interac-
soid. Dipole interactions between neighboring particles in- tions between the particles in the chain.
side the chain increase the total effective magnetic moment
of the chain, which can be calculated by iteratively solving
the individual particles in the chain. It should be noted that
the coupled equations of the effective magnetic field Beff and
the correction factor ␣ M increases rapidly for lower ampli-
magnetic moment eff for all the particles in the chain until
tudes of the magnetic field and for instance can reach a value
convergence is reached 共typically, less than ten iterations
of ␣ M ⬇ 1.3 at B = 5 mT, which cannot be neglected for ap-
were needed for convergence with an error of less than
plications in this field range.
0.1%兲:
Furthermore, the hydrodynamic friction force acting on a
N
0 eff,j chain of particles must be evaluated since it differs from the
Beff,i = B + 3兺 共i ⫽ j兲, 共18兲 case of individual particles. A chain of N particles is repre-
16a j=1 兩i − j兩3
sented as an ellipsoid of revolution moving along the x-axis
where the induced magnetic moment is given by the Lange- and the friction coefficient ␥N of the particle chain can be
vin expression according to approximated by17
eff,i = ndM SVdL 冉 k BT
冊
M SVdBeff,i
. 共19兲 1.35
1.05
We also define a correction factor ␣ M as 1.3 1.04
α [−]
1.03
⌺i=1
N
eff,i 1.25
␣M = 共20兲
M
, 1.02
N
α [−]
1.2 1.01
where N is the number of particles in the chain and is the
M
induced magnetic moment of the particle when magnetic in- 1
1.15 1 3 5 7 9 111315
teractions are neglected 关Eq. 共6兲兴. The effective magnetic
N
moments and correction factor are illustrated in Figs. 9 and
1.1
10 for the particle system with diameter 425 nm. The correc-
tion factor ␣ M is at most 1.05 for the chains located furthest
away from the pole face of the magnet 共where B ⬇ 50 mT, 1.05
see Fig. 4兲 and lower for all the chains closer to the magnet.
A subroutine in the program could easily be implemented by 1
0 20 40 60 80 100
using the method described above to compute the correction B [mT]
factor at each time step depending on the position 共and thus
on the magnetic field兲 and on the length of the chains. How- FIG. 10. Correction factor ␣ M as a function of the magnetic field B for a
ever, since the correction factor is small 共less than 1.05兲, and chain of N = 15 particles. Inset: Correction factor ␣ M as a function of the
chain length in number of particles N for different magnetic field intensities:
in order to minimize the computation time, we assume as a
B = 0.05 T 共open circles兲, B = 0.1 T 共open squares兲, and B = 0.2 T 共open
first approximation that the total magnetic moment of the diamonds兲. The correction factor reaches a maximum already for short
chain is equal to the summation of the magnetic moments of chains.093918-9 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
3.5 4.5
Relative error [%]
15
12
4
3 9
6
3
3.5
γ* = γN / γ1
2.5 0
2 4 6 8 10 3
αV [−]
N 5.0
2 2.5 4.0
αV [−]
3.0
2
1.5 2.0
1.5 1.0
1 3 5 7 9 111315
1 N
1 2 3 4 5 6 7 8 9 10 1
1 3 5 7 9 11 13 15
N N
FIG. 11. Friction coefficient ␥N of a prolate spheroid 共filled circles兲 as a
FIG. 12. The velocity enhancement factor defined as ␣V
function of its length in terms of number of particles in the chain, normal-
= N␣ M 共N , B兲 / ␥ ⴱ 共N兲 vs the number of particles in the chain. In our case, the
ized by the friction coefficient ␥1 of a single spherical particle. The slender-
correction factor ␣ M is sufficiently small to be neglected and ␣V simplifies to
body approximation 共open squares兲 gives a discrepancy of less than 1%
␣V = N / ␥ ⴱ 共N兲 共filled circles兲. Inset: Velocity enhancement factor ␣V at dif-
compared to the ellipsoid model only for N ⬎ 10 共inset兲.
ferent magnetic fields: B = 0.05 T 共open circles兲, B = 0.1 T 共open squares兲,
and B = 0.2 T 共open diamonds兲.
␥N = 6Reff , 共21兲
tion of the friction coefficient ␥N 共␥N = ␥1, for N = 1兲.
where the hydrodynamic effective radius Reff is given by
In a long-time dynamics, the mean velocity of a chain of
1 V N particles is given by the ratio of the magnetic force Fm,N to
Reff = , 共22兲
Q + a2x nx the friction coefficient ␥N,
where ax is the principal radius of the ellipsoid along the Fm,N
vN = . 共26兲
direction of motion and V = 4axayaz / 3 is the volume of the ␥N
ellipsoid where ay and az are the principal radii along the y
By combining the results from Eqs. 共20兲 and 共21兲, the veloc-
and z directions of the ellipsoid. Q and nx can be evaluated
ity of a chain of particles can then be expressed as
analytically for prolate spheroids 共i.e., 1 ⱖ e2 ⬅ 1 − az2 / a2x ⱖ 0,
and ay = az兲,18 N␣ M Fm,1 N␣ M
共27兲
冉 冊
vN = = ⴱ v1 ,
a2 1+e ␥ ⴱ␥ 1 ␥
Q = z ln , 共23兲
2e 1−e where v1 = Fm,1 / ␥1 is the mean velocity of a single particle
and ␥ ⴱ = ␥N / ␥1 is a normalized friction coefficient for the
nx =
1 − e2
2e3
冋冉 冊 册
ln
1+e
1−e
− 2e . 共24兲
chain of N particles. Clearly, the mean velocity of a particle
chain compared to a single particle is higher. The velocity
enhancement factor ␣V of the particle chain velocity as com-
In the approximation of long chains 共i.e., ax Ⰷ az兲, the spher- pared to the velocity of a single particle depends on the num-
oid becomes equivalent to an elongated rod and Eq. 共22兲 can ber of particles in the chain, on the correction factor for the
be simplified to effective magnetic moment ␣ M , and on the normalized fric-
2 Ne tion coefficient ␥ⴱ. In Fig. 12, ␣V is plotted with ␣ M = 1, and
Reff = az , 共25兲 in insert with ␣ M ⫽ 1, for different magnetic fields. This
3 ln共2N兲 − 0.5
clearly shows that in our case the correction factor ␣ M has
where Ne = ax / az is the equivalent number of particles in the only a marginal effect on the chain velocity, which supports
chain. This result is often referred to as the slender-body our choice of neglecting this parameter for the reasons dis-
theory and was demonstrated to describe the friction coeffi- cussed previously.
cient of micrometer-sized superparamagnetic colloidal beads In this case, the mean velocity of a chain of N particles is
remarkably well, which were aggregated into linear chains enhanced compared to the mean velocity of the N individual
by applications of a homogeneous magnetic field.19,20 How- particles by a factor ⬃N / ␥ⴱ and, for instance, a chain of 12
ever, as shown in Fig. 11, the slender-body approximation particles moves approximately four times faster than the
gives a discrepancy of less than 1% compared to the ellip- same 12 particles moving individually. Therefore, the aggre-
soid model only for N ⬎ 10. Since we are interested in de- gation of the particles into chainlike structures gives a faster
scribing the transition from individual particles 共N = 1兲 to separation process. However, the overall separation time can-
chains 共N ⬎ 1兲, we adopt the ellipsoid model for the calcula- not be simply extrapolated from the velocity enhancement093918-10 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
magnetic field gradient along the x-direction are the same for
all the particles.
The reason for using this pseudo-3D approach instead of
∆y a real 3D implementation is that the computation time would
∆x be drastically increased by the much more intensive calcula-
tion routines required to simulate the 3D displacements and
interactions of the particles.
y ∆y D. Electrostatic interaction
x At the moment, we are not considering the electrostatic
interactions since the optical microscopy observations show
z that the electrostatic repulsive force does not prevent the par-
FIG. 13. Schematic drawing of the particles coordinates, not drawn to scale. ticles from aggregating into chains even for low magnetic
A one-dimensional model is used to describe the displacement of the par- fields. However, the electrostatic interaction may play a more
ticles along parallel trajectories in the x-direction. Aggregation can occur significant role for other particle systems or magnetic field
only between particles located along the same y and z coordinates. The
hydrodynamic interaction is calculated between neighbors at ⫾⌬y and ⫾⌬z.
conditions.
E. Equation of motion including interparticle
factor given above since the chains interact and eventually interactions
aggregate to form larger chains as they all move toward the
magnet. After including particle interactions the motion equation
for particle i 共and neighbor particle j兲 finally becomes22
1
C. Hydrodynamic interaction
xi共t + ⌬t兲 = xi共t兲 + 兺 F 共t兲⌬t
␥ j dd,ij
The numerical model was further improved by consider- Dij共t兲Fm,j共t兲
ing the hydrodynamic interactions between particles 共or +兺 ⌬t + ⌬xG
i , 共30兲
j k BT
chains兲. A method for simulating the Brownian dynamics of
N particles with the inclusion of hydrodynamic interaction where the second and third terms on the right side represent
was described by Ermak and McCammon.21 The effects of the magnetic dipole-dipole interactions and hydrodynamic
hydrodynamic interactions mediated by the host solvent are interactions, respectively.
included through a position-dependent interparticle diffusion
tensor according to22 V. RESULTS AND DISCUSSIONS
k BT
Dii = , In this section, we will present and discuss the particle
␥i separation curve, i.e., the time-dependent particle concentra-
冉 冊
tion, for different particle sizes under varying experimental
k BT rជijrជij conditions. The numerical model described in the previous
Dij = I+ 2 , 共28兲
8rij rij section will be used to interpret the experimental results.
First, we will present the separation curves for different
where is the viscosity of the fluid, rជij is the vector from the particle sizes and same initial particle concentration to deter-
center of particle i to the center of particle j 共r2ij = ⌬x2ij mine the separation behavior for each particle system. We
+ ⌬y 2ij + ⌬z2ij兲, and I is the unit tensor. It should be noted that will also show the results for different particle concentrations
Dij is independent of the shape of the particles. This relation of the same particle system to determine whether the sepa-
was derived under the assumption that the particles may be ration process depends on the initial particle concentration.
considered to act as point forces on the fluid, i.e., the inter- Then the influence of the setup alignment on the separation
particle distance is considered to be always very large com- measurement will be demonstrated by changing the position
pared to the effective hydrodynamic radius of a single par- of the light beam crossing the sample volume. Finally, we
ticle. The hydrodynamic interaction on particle i is calculated will show how the separation behavior is influenced by the
from the neighbor particles along the same y and z coordi- distance between the sample and the magnet.
nates as particle i, and from the neighbor particles at ⫾⌬y Figure 14 shows the separation curves for five different
and ⫾⌬z as well 共Fig. 13兲 with periodic boundary condi- particle systems. The concentration is measured in the
tions, but only the x-component of the resulting diffusion middle of the sample column, i.e., 6.25 mm from the pole
tensor is taken into account. The lateral spacing ⌬y and ⌬z face of the magnet and along its symmetry axis. The initial
have fixed values given by particle concentration for all of the particle systems is 0.3
mg/ml. The separation process is faster for larger particles,
⌬y = ⌬z = ⌬x̃. 共29兲
except for the 100 nm particle system, which separates faster
Since ⌬y and ⌬z are typically in the range of a few microme- than the 150 nm particle system. In fact, the 100 nm particles
ters, we can essentially assume that the magnetic field and sample separates almost as fast as the 200 nm particles093918-11 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
0.35 9.05
0.3
9.04
0.25
D200 D150
c [mg/ml]
0.2 D100 9.03
x [mm]
0.15
9.02
0.1
D250
0.05 9.01
D425
0
0 2 4 6 8
9.00
t [min]
FIG. 14. The concentration of magnetic nanoparticles of different diameters 0 0.2 0.4 0.6 0.8 1
共indicated in nanometers in the figure兲 vs time after the magnet has been t [s]
placed next to the cuvette wall. The concentration is monitored in the middle
of the cuvette along the symmetry axis of the magnet. The initial particle FIG. 15. Simulated trajectories of ten particles. At first, the ten particles are
concentration for all of the particle systems is 0.3 mg/ml. randomly distributed in an interval between 9.00 and 9.05 mm along the
x-direction 共i.e., the symmetry axis of the magnet兲 with the magnet pole face
located at x = 0 mm. The fluctuation of the curve is due to the stochastic
sample. We believe that this is due to the different magnetic term ⌬xG representing the Brownian motion of the particles. Collisions be-
properties of the 100 nm particle system compared to the tween particles are indicated by arrows. At t = 1 s, two particles remain
other particle systems. Indeed, the 100 nm particle system isolated, whereas eight particles have aggregated into three chains.
has the highest saturation magnetization when the magnetic
moment per particle is divided by the particle volume, see constituting the chain. As described in Sec. IV, this leads to a
Table I. higher velocity of the chains and to an acceleration of the
The separation curves first exhibit a slow variation of separation process, which explains why the rate of changes
particle concentration with time 共Fig. 14兲. After some time, of the particle concentration increases after some delay 共Fig.
the particle concentration starts to decrease rapidly. It is pos- 14兲. When most of the individual particles have been ag-
sible to observe an inflexion point where the derivative of the glomerated into the particle chains, there are only small
separation curve is at its maximum. We interpret this change changes in the size of the chains which can further accelerate
in the separation behavior as an effect of two different them. The particle concentration at times longer than 10 min
mechanisms that take place in two different time intervals. of the 100, 150, and 200 nm particle systems does not fall to
Initially, the particles move as separate entities. Gradually zero, which would be the case if all the particles have been
particle chains start to form, as discussed previously 共Sec. moved to the magnet. We believe that for these particle sys-
IV兲. Therefore, we believe that the observed acceleration of tems, there are still particles in the sample cuvette that can-
the separation process is due to the formation of these chains. not be moved to the magnet in reasonable separation times.
Our numerical model predicts that from the initial con- This explanation is supported by the visual observation of
figuration 共where particles are randomly distributed along the the sample presenting a slightly brownish color typical of
symmetry axis of the magnet兲, the particles move in the di- iron oxide particles in suspension.
rection of the magnet and collide to form chains when the For the 50 nm particles 共not shown in Fig. 14兲, we do not
trajectories of two or more particles coincide 共Fig. 15兲. The observe any concentration change over time, i.e., the mea-
formation of chainlike aggregates is due to the attractive sured particle concentration remains 0.3 mg/ml after more
magnetic dipole-dipole interaction between the particles after than 30 min. This experimental result agrees well with Eq.
the magnetic particles have been magnetized by the external 共16兲. As discussed previously, the thermal energy is larger
magnetic field. The chain formation when an external mag- than the dipole-dipole interaction energy 共 ⬍ 1兲 for particles
netic field is applied to a particle system has been shown in in this size range. This means that the Brownian motion of
Fig. 8 and in both theoretical23 and other experimental the particles hinders the formation of particle chains, and as
work.24–26 a result the particles cannot be separated in a reasonable
Therefore, the sigmoidal shape of the curves in Fig. 14 amount of time. This illustrates the importance of particle
can be explained as follows. In the beginning of the measure- aggregation into chainlike structures to accelerate the sepa-
ment, the single particles move slowly toward the magnet ration process.
under the influence of the magnetic field gradient. The chains Figure 16 shows the predicted trajectories of 1210 par-
start to form as the particles come close to each other so that ticles corresponding to a particle concentration of 0.1 mg/ml.
the interparticle magnetic interaction becomes stronger. As explained in detail in Sec. IV, the theoretical particle tra-
Larger chains are subject to higher magnetic forces than jectories are calculated by considering only the one-
smaller chains or single particles since the total magnetic dimensional motion of the particles in the sample cuvette as
moments of such chains grow with the number of particles they move toward the magnet.093918-12 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
12.0 Cuvette wall using the numerical model is longer than the measured value.
We believe that one reason for this may be that smaller par-
ticles build elongated aggregates that are thicker than simply
10.0
one particle as in our numerical model. This would result in
a shorter separation time since particles aggregated into these
8.0 thick chainlike structures would experience comparatively
x [mm]
less friction than the same number of particles grouped into
6.0 single-particle thick aggregates. We plan to consider extend-
ing the numerical model in our future work to take this phe-
4.0 nomenon into account and compare the results with the mea-
sured separation times for smaller particles sizes.
As explained in the previous section, our theoretical
2.0
model includes the magnetic interactions between the par-
Cuvette wall ticles, the hydrodynamic interactions both between the indi-
0 vidual particles and the chainlike aggregates, and the Brown-
0 30 60 90 120 150 180
t [s] ian motion of the particles. All of these factors are needed to
obtain good agreement between the model and the experi-
FIG. 16. Trajectories of 1210 particles along the symmetry axis of the mental results.
magnet vs time, predicted by the numerical model. The cuvette has an inner Since the mean distance between the single particles de-
width Win = 10 mm and outer dimensions Wext = 12.5 mm 共the cuvette walls
are indicated by dashed lines兲. The magnet pole face is located at x pends directly on the particle concentration, we repeated the
= 0 mm. measurements with different initial particle concentrations to
evaluate the influence of this parameter on the separation
process. As shown in Figs. 18共a兲 and 18共b兲 for the 150 and
The separation curve can then be calculated for a spe-
425 nm particles, respectively, we found that the separation
cific measurement position in the sample cuvette by counting
process is approximately the same for samples with an initial
the number of particles at a specific distance from the magnet
particle concentration of 0.1, 0.2, and 0.3 mg/ml. Thus from
over time. A very good fit exists between the experimental
these results, we can conclude that the separation process is
and numerical separation curves for the 425 nm particles
not strongly dependent on the initial particle concentration in
共Fig. 17兲. The experimental and the predicted results differ
this specific particle concentration range.
only in the initial stage of the separation process. This dif-
The fact that we observed the chain building of particles
ference is probably due to the fact that in our one-
in the presence of a magnetic field and a magnetic field gra-
dimensional model, the particles are already aligned along
dient means that even at low particle concentrations, when
the x-direction, and therefore they build chains more rapidly
the distance between the particles is much larger than their
than if they were homogeneously distributed in the sample
diameter, particle chain formation takes place. The small dif-
volume.
ferences between the separation curves for different concen-
For smaller particle sizes, the separation time predicted
trations 共Fig. 18兲 could mean that at higher particle concen-
trations the chains are somewhat longer than for low
0.1 concentrations. Longer particle chains speed up the separa-
tion process due to higher magnetic forces on the longer
chains compared to shorter chains.
0.08
The apparent separation time may also depend on the
position of the detection volume, i.e., on the position of the
light beam across the sample with respect to the magnet. If
c [mg/ml]
0.06
the detection volume is close to the magnet, almost all par-
ticles have to cross the light beam and the separation is fin-
0.04 ished when all the particles have reached the magnet. On the
other hand, when the detection volume is at the opposite side
of the cuvette, the concentration change will only depend on
0.02 the particles leaving the detection volume. Thus, the mea-
sured concentration is expected to decrease faster in the sec-
ond case than in the first one. To verify this dependence,
0 measurements were conducted with the 425 nm particle sys-
0 30 60 90 120 150 180
t [s] tem at an initial concentration of 0.1 mg/ml and for two
different positions of the light beam, 4 and 6 mm from the
FIG. 17. Concentration of the magnetic particles with diameter 425 nm vs magnet. The results are shown in Fig. 19. The differences in
time after the magnet has been placed next to the cuvette wall, from the the separation curves and the slightly longer separation time
measurements 共continuous line兲 and from the simulation model based on
observed close to the magnet can be explained by the fact
noninteracting particles 共open squares兲 and interacting particles 共i.e., mag-
netic dipole-dipole and hydrodynamic interactions兲 共open circles兲. Initial that when the measurement volume is close to the magnet
concentration is 0.1 mg/ml. more particles have to pass through the detection volume,093918-13 Schaller et al. J. Appl. Phys. 104, 093918 共2008兲
1.2 0.35
ca. 4 mm
0.3 ca. 6 mm
1
0.25
0.8
c [mg/ml]
c / c [−]
0.2
0
0.6
0.15
0.4
c0 = 0.1 mg/ml 0.1
c0 = 0.2 mg/ml
0.2 0.05
c = 0.3 mg/ml
0
0 0
2 4 6 8 10 0 1 2 3 4 5
t [min] t [min]
FIG. 19. Particle concentration vs time for the magnetic particles with di-
(a) ameters of 425 nm at an initial concentration of 0.3 mg/ml measured at 4
and 6 mm from the magnet in the cuvette. The separation process is depen-
dent on the position of the light beam crossing the cuvette because of the
1.2 different amounts of particles that have to cross the detection area. The
c0 = 0.1 mg/ml fall-off in concentration occurs when the magnet is placed next to the cu-
vette at t = 30 s.
1 c0 = 0.2 mg/ml
c0 = 0.3 mg/ml
ameter is excluded by measuring only one particle system at
0.8 one concentration. As expected, when the distance from the
magnet to the cuvette increases, the magnetic forces on the
c / c [−]
particles become weaker, and thus the separation time be-
0
0.6
comes longer. Figure 20 shows the separation curves of the
425 nm particle system at an initial concentration of 0.3
0.4 mg/ml for three different distances between the magnet and
the measurement volume 共9, 15, and 20 mm兲. It is interesting
to note that the shape of the separation curves is similar for
0.2
the different magnet-to-sample distances, i.e., the different
curves are only stretched out in time. In this figure, each
0
2 4 6 8 10
0.35
t [min] 9 mm
0.3 15 mm (scaled in time)
(b) 20 mm (scaled in time)
FIG. 18. Normalized particle concentration vs time for magnetic particles of 0.25
diameters of 150 nm 共a兲 and 425 nm 共b兲 at particle concentrations of 0.1,
c [mg/ml]
0.2, and 0.3 mg/ml. As can be seen from the figure, there is no strong
relation between the particle concentration and the separation time.
0.2
0.15
and therefore there is a longer decay of the light absorbance
despite the fact that in this region there are strong magnetic 0.1
forces on the particles.
The separation time depends not only on the size of the 0.05
particles and the viscosity of the carrier liquid the particles
are placed in, but also on the external magnetic field that
0
magnetizes the particle and the magnetic field gradient that 0 0.5 1 1.5 2
together with the magnetization of the particles determine t [min]
the magnitude of the magnetic force on the particles. The
magnetic forces on the particles have been varied by chang- FIG. 20. Particle concentration vs time for magnetic particles of diameter
425 nm at different distances between the magnet and the detection volume.
ing the distance between the permanent magnet and the cu-
The separation curve for 9 mm distance to magnet is as measured while the
vette. In this case, both the magnetic field and the magnetic separation curves for 15 and 20 mm are compressed in time by 4.5 and 10.5,
field gradient are changed. The influence of the particle di- respectively.You can also read
