Magnetic properties and size distributions of
spherical magnetite nanoparticles coated with
poly(acrylic) acid
D. B. Gopman a K. D. Sorge a,∗ E. Belogay b S. Santra c
J. M. Perez c
a Department
b Wilkes
of Physics, Florida Atlantic University, Boca Raton, FL 33431
Honors College, Florida Atlantic University, Jupiter, FL 33458
c Department
of Chemistry, University of Central Florida, Orlando, FL 32816
Abstract
We estimate the size distribution of magnetite nanoparticles from their magnetic
properties. The particles were prepared as a water-based suspension and coated with
poly(acrylic) acid. The physical size of the sample core is found to be roughly 10 nm
by TEM measurements. Particle magnetization was measured using a SQUID magnetometer in magnetic fields up to 25 kOe and temperatures ranging from 5 to 370 K.
The distribution in magnetic moments of particles in the sample was estimated by
fitting to various magnetization models. The distribution in the magnetically active volumes was then calculated from this distribution in magnetic moments. This
active volume was found to be smaller than the volumes observed by TEM.
Key words: Langevin superparamagnetism, Ferrofluids, Size distributions
PACS: 75.30.Cr, 75.40.Mg, 75.50.Bb, 75.50.Mm
Preprint submitted to Elsevier
28 July 2008
1
Introduction
Recent interest in nanoparticles and ferrofluids has been fueled by their application to medicine. For example, small-scale, low-concentration fluids of
superparamagnetic iron oxide nanoparticles (SPIONs) have been investigated
for targeted drug delivery [1,2] where medication is functionalized to magnetic
particles and delivered directly to target areas. Also, ferrofluid technology is
used in localized heating applications [3,4] where a magnetic fluid is heated
by a strong oscillating magnetic field. In addition, monodisperse ferrofluids
composed of nanoparticles are widely used as contrast agents in imaging [5].
Some of the main biological applications of ferrofluids rely upon a monodisperse sample, either in magnetic resonance regimes [6,5], or in Brownian relaxation measurements [7,8]. Such applications are very sensitive to slight
changes in the sizes of suspended particles. In consideration of the sensitive
applications using magnetic fluids, efficient techniques for finding physical and
magnetic distributions of suspended magnetic nanoparticles in ferrofluids are
vital.
The current project examines Fe3 O4 nanoparticles coated with poly(acrylic)
acid (PAA) and suspended as a water-based colloid. These particles were synthesized as potential nanosensors for molecular interactions in MRI measurements, as described previously [5]. A log-normal distribution in the particle
magnetic moment is assumed [7]. By taking a parametric approximation of this
log-normal distribution, magnetization measurement are used to find parameters of the distribution. From this, a distribution in particle size is obtained.
∗ Corresponding Author.
Email address: sorge@physics.fau.edu (K. D. Sorge).
2
2
Experimental
Iron oxide nanoparticles were prepared by mixing an iron salt solution (0.49 g
of FeCl3 , 0.21 g of FeCl2 and 88.7 μL of 12(N) HCl in 15 mL of N2 purged
water) with an ammonia solution (830 μL of 30% NH4 OH solution in 15 mL
of N2 purged water) on a digital vortex mixer, forming a dark suspension
of iron oxide nanoparticles. To the resulting suspension of nanoparticles, a
solution containing the stabilizing agent (polyacrylic acid) was added (1.0 g of
polyacrylic acid in 5 mL of DI water, 0.56 mmol) and then stirred vigorously
for 1 hour at 3000 rpm. The solution was then centrifuged at 4000 rpm for
30 min and the supernatant was washed several times with DI water and
concentrated through an Amicon 8200 cell to get rid of free polyacrylic acid
molecules and other reagents. Two different concentrations of the fluid were
investigated (0.9 and 1.9 mg of Fe3 O4 /mL of H2 O) which will be referred to
as F0.9 and F1.9, respectively. A vacuum-dried powder of the fluid was also
synthesized and analyzed.
The hydrodynamic sizes of the polymer-coated nanoparticles were found through
dynamic light scattering (DLS) studies. The transverse (T2) proton relaxation
times for the nanoparticles were measured using a Bruker Minispec mq20 NMR
analyzer operating at a magnetic field of 4.7 kOe and 37◦ C. A histogram of
DLS measured sizes was used to compute the actual median and standard
deviation. PAA-coated nanoparticles had hydrodynamic sizes within the interquartile range of 82—120 nm with a median of 103 nm.
The physical size of the iron oxide cores of the nanoparticles was determined
by high-resolution transmission electron microscopy (HRTEM). A drop of
3
nanoparticle solution was placed on a carbon-coated copper grid and dried
completely in vacuum to be used as a TEM sample. The TEM image was
analyzed using software that counted and estimated the sizes of the particle metal cores. The spherical particles depicted in Fig. 1 exhibited diameters
with a right-skewed distribution, in which the middle 50% of diameters ranged
from 7.8 nm to 12.7 nm, with median 9.2 nm.
As a preparation for magnetic measurements, the fluids were inserted near the
center of a glass capillary tube with a long microliter pipette and the ends of
the glass tube were sealed with a flame torch. Since the amount of magnetite
in the sample is extremely small, each ferrofluid can be assumed as dense as
water. All sample tubes were tested for fitness under vacuum and durability
during thermal cycling. For comparison, a small mass of the powder was mixed
into Duco cement and mounted to a silicon chip. Magnetic measurements
were performed in a superconducting quantum interference device (SQUID)
magnetometer. Data was taken at temperatures between 5 K and 365 K and
fields as high as 25 kOe. Magnetic moment was measured as a function of
applied magnetic field at fixed temperatures.
3
Analysis
Magnetization curves at fixed temperature were measured on each of the samples. The background susceptibility of the mounting materials in the powder
sample was estimated from the high-field susceptibility of the sample and the
susceptibility of water in the fluid samples was computed from the nominal
volume of water. For clarity, the background susceptibility signal was subtracted from the magnetization data in Fig. 2. These magnetization curves
4
pass through the origin at all but the lowest temperatures measured—a simple signature of the expected superparamagnetic behavior.
At 300 K, we find magnetizations of 52 emu/g of Fe3 O4 for fluid samples
and 48 emu/g for the powder. Both of these values fall well below the bulk
magnetization of magnetite—92 emu/g [9]. These diminished magnetizations
have also been observed in similar nanoscale systems [10,11].
For an ideal system of N superparamagnetic particles with magnetic moment
μ, we can describe magnetization as a function of field H at fixed temperature
T by
μH
M (H) = N μ L
kB T
.
(1)
Here kB is the Boltzmann constant and the Langevin function is given by
L(x) = coth(x) − 1/x. Notice that, since L(x) → 1 as x → ∞, the saturation
magnetization is Msat = N μ. We refer to this construct as the Simple Langevin
Model for a monodisperse sample. The single parameter μ would be found by
a curve fit of the data.
We tested our sample against the Simple Langevin model for a uniform ensemble of magnetic moments. Using nonlinear least-squares optimization, we found
the magnetic moment that provides the best fit of the the Simple Langevin
Model to the raw magnetization data. As Fig. 3 shows, the single-size model
provides a good fit for low and high fields, but a poor fit for moderate fields.
Our best fit curve underestimates the background magnetic susceptibility in
order to reduce the overall deviation from our data.
A distribution of magnetic moment sizes is a more realistic assumption than
one of a monodisperse ensemble. The process by which these particles are
5
synthesized often results in a log-normal distribution of particle sizes, which
in turn results in a log-normal distribution of particle magnetic moments
[12,8,7,13]. Therefore, we assume that the magnetic moments of the nanoparticles have a log-normal distribution with a density given by
ln(μ/μ0 )2
f (μ) =
,
exp −
2σ 2
σμ 2π
1
√
(2)
where μ0 is the median moment and σ is the standard deviation of ln(μ/μ0 ).
The superposition of the magnetization curves with the log-normal weighting
yields the total ensemble magnetization of N superparamagnetic particles of
various sizes
M (H, T ) = N
∞
0
μH
μL
kB T
f (μ) dμ.
Notice that the saturation magnetization here is Msat = N
(3)
∞
0
μf (μ) dμ =
N μ, where μ is the mean moment. The relation between mean and median
moment for the log-normal distribution is μ = μ0 exp(σ 2 /2).
The ensemble magnetization integral in Eq. (3) has no analytical closed-form
and must be computed numerically for each set of parameter values H and T .
Since L(x) ≈ x/3 for small x and L(x) → 1 as x → ∞, one can approximate
the integral analytically for small and for large values of H [12]. This technique
only uses data from the extremes of the measurement range to get distribution
parameters. In addition, the limited precision of the actual measurements can
lead to unreliable results.
In order to ease computation and achieve higher precision, we transform the
log-normal distribution to a normal distribution with zero mean and unit
standard deviation by the change of variables z = ln(μ/μ0 )/σ. By chosing
a uniform and symmetric partition of 15 nodes along the z-axis, spaced at
6
Δz = 0.5, from z = −3.5 to z = 3.5, we are able to approximate the complete
integral by breaking it into discrete parts, which can be evaluated for any
value of H and T .
Instead of using the node values of the normal probability density function
(pdf), we used the differences of the cumulative distribution function (cdf) at
the nodes as discrete weights against the node values of the Langevin function.
We could do so, because the normal cdf of the new variable z can be computed
numerically with high precision. As a result, the integral in Eq. (3) is now
approximated by the finite weighted sum
7
μn H
M (H, T ) = N
μn L
wn ,
kB T
n=−7
(4)
with weights
wn =
zn +Δz/2
zn −Δz/2
f (z) dz,
(5)
where zn = nΔz, μn = μ0 exp(zn ), and the weights wn represent the area cen√
tered around each node zn of the normal distribution f (z) = exp(−z 2 /2)/ 2π.
This approximation scheme is exact for linear functions in the z variable, so
it provides high precision for the Langevin function, which is almost linear
at low and high fields. In addition, the simple and natural uniform spacing
of nodes in the z variable yields a desirable non-uniform spacing of nodes
in the original μ variable, providing more weight at the steepest part of the
log-normal density.
Finally, we fit this model to data by a non-linear least-squares technique. From
this, we find distribution parameters that minimize the sum of the square
residuals at all measured field values. As Fig. 4 shows, this model provides a
better fit at all field values. In addition, the magnified scale for residuals in
7
Fig. 5 exhibits a nearly five-fold improvement in the quality of fit over that of
the Simple Langevin Model.
We found the optimal magnetic distribution parameters μ0 = 2.17×10−17 emu
and σ = 1.33, by minimizing least-squares error between our experimental
data and the discretely-weighted Langevin function, as shown in Eq. (4). The
rather large computed value of σ is another indication that our distribution
in particle moments is far from monodisperse.
Because the magnetic moment of a particle scales with the volume, finding
the distribution in moments is a powerful way of characterizing the distribution in physical size for the particles. From TEM images (Fig. 1), we assume
that we have spherical particles with volumes given by V = πD3 /6, where
D is the particle diameter. If the volumetric magnetization Mvol is assumed
to be constant, the median diameter will be given by D0 = (6μ0 /πMvol )1/3 .
The standard deviation of this physical size distribution σD is related to the
standard deviation in magnetic moment by σD = σ/3.
By using a magnetization of Mvol = 250 emu/cm3 and the parameters of our
magnetic size distribution, we find that the particle diameter distribution has
parameters D0 = 5.5 nm and σD = 0.44 and with quartile range from 4.1 nm
to 7.4 nm (Fig. 6). When compared to the TEM data for these particles
of D0 = 9.2 nm and σD = 0.33, it implies that not all of the particle is
magnetically active. In contrast, the median magnetic size calculated from
the best-fit monodisperse magnetic moment is D0 = 9.8 nm, which is high in
relation to the sizes measured by TEM (Fig. 6).
The calculated distributions of magnetically active diameters are superimposed upon the TEM histogram of core diameters in Fig. 6. The magnetic size
8
distribution is noticeably shifted to lower diameters, relative to the physical
size distribution. It is reasonable to assume that a magnetically “dead” layer
of atoms is present at the surface of the clusters. A descriptive model for such
clusters would consist of a magnetically active core with a concentric “dead”
layer of conducting metal-oxide material. Early studies of magnetic thin films
attributed magnetic “dead layers” to spin depolarization in the spin tunneling
barriers [14]. Spin canting and disorder at the surface of the particles provide
another explanation for this size disparity [15,8].
4
Conclusions
We investigated the magnetic properties of water-based ferrofluids of Fe3 O4
nanoparticles coated in poly(acrylic) acid. Both the small particle size and
the thick polymeric coating contribute to the superparamagnetic behavior of
all samples by precluding agglomeration of magnetic particles in the fluid and
maintaining the single domain state of each nanocluster. In all samples, we
observed superparamagnetism far below the freezing point of water as well as
relatively high saturation magnetization.
Using a discrete weighting to the Langevin function and non-linear leastsquares optimization, we were able to find a log-normal distribution of sizes
that provided an excellent fit to our raw data. The fact that this fit was at
least five times better than the fit based on the single-moment Langevin model
clearly indicates that the nanoparticles in the sample did not have the same
size — their diameters spread over one order of magnitude.
The magnetic sizes calculated from the distribution of magnetic moments
9
were systematically smaller than the physical sizes determined by transmission
electron microscopy. Even the most efficient methods of fabrication of similar
nanoparticle systems may not be able to stabilize the magnetic outer layers
in order to prevent the surface disorder or the existence of a magnetically
“dead” layer whose effect was observed in our measurements. In a sense, this
dead layer is a “signature” of the chemical synthesis and the geometry of
the ensemble. Therefore, determining the distribution of magnetically active
volumes remains a relevant method for magnetic characterization.
5
Acknowledgments
We would like to thank Mary Beth McEnroe of the Jupiter Medical Center
for the donation of the glass capillary tubes and Dr. Christopher Beetle of
Florida Atlantic University for his assistance with Maple calculations.
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11
Figure Captions
Fig. 1 HRTEM image of Fe3 O4 nanoparticles coated with PAA.
Fig. 2 Initial magnetization curves for fluid samples at T = 200 K. Measured
magnetic moment is normalized by the volume of ferrofluid.
Fig. 3 Assuming a monodisperse sample of magnetic clusters, the best-fit
Langevin is successful over low and high field intervals but is deficient over
intervals where data was taken under moderate field strength.
Fig. 4 The best-fit curve obtained by a log-normal weighting of the Langevin
function closely follows the observed data at all field regions.
Fig. 5 Comparison of average error and residuals in the Simple and LogNormal Langevin models. The Log-Normal model is five times better.
Fig. 6 Histogram of core diameters from TEM (diagonal-shaded) with monodisperse (dotted, shaded) and log-normal (smooth) distribution models of magnetically active diameters.
12
20 nm
Fig. 1.
13
F1.9
M (memu / mL H2O)
100
50
0
F0.9
0
5
10
H (kOe)
Fig. 2.
14
15
m (µemu)
325
300
275
Raw Data
Simple
Langevin Model
250
225
0
5
10
15
H (kOe)
Fig. 3.
15
20
25
Raw
Log-Normal
Langevin Moments
m (µemu)
325
300
275
250
225
0
5
10
15
H (kOe)
Fig. 4.
16
20
25
Average Error (µemu)
Residual (µemu)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
20
Simple Langevin Model
0.51 µemu
Log-Normal
Langevin Model
0.092 µemu
1
2
10
0
Simple
Log-Normal
-10
-20
0
5
10
H (kOe)
Fig. 5.
17
15
20
25
Fig. 6.
18