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Sedimentation equilibria of ferrofluids: I. Analytical centrifugation in ultrathin glass capillaries
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2012 J. Phys.: Condens. Matter 24 245103
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 24 (2012) 245103 (8pp)
doi:10.1088/0953-8984/24/24/245103
Sedimentation equilibria of ferrofluids: I.
Analytical centrifugation in ultrathin
glass capillaries
Bob Luigjes, Dominique M E Thies-Weesie, Albert P Philipse and
Ben H Erné
Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterials Science,
Utrecht University, Padualaan 8, 3584 CH Utrecht, The Netherlands
E-mail: [email protected]
Received 5 April 2012
Published 23 May 2012
Online at stacks.iop.org/JPhysCM/24/245103
Abstract
Analytical centrifugation is used for the first time to measure sedimentation equilibrium
concentration profiles of a ferrofluid, a concentrated colloidal dispersion of strongly absorbing
magnetic nanoparticles. To keep the optical absorbance from becoming too strong, the optical
path length is restricted to 50 µm by placing the dispersion in a flat glass capillary. The
concentration profile is kept from becoming too steep, despite the relatively high buoyant mass
of the nanoparticles, by making novel use of a low-velocity analytical centrifuge that was not
designed to measure equilibrium profiles. The experimental approach is validated by
comparison with profiles obtained using an analytical ultracentrifuge. At concentrations of a
few hundred grams per liter, the osmotic pressures calculated from the equilibrium profiles are
lower than expected for hard spheres or non-interacting particles, due to magnetic dipolar
interactions. By following the presented experimental approach, it will now also be possible to
characterize the interparticle interactions of other strongly absorbing colloidal particles not
studied before by analytical centrifugation.
(Some figures may appear in colour only in the online journal)
1. Introduction
only of macromolecules but also of larger colloidal particles,
because several key colloidal properties, such as dispersion
stability, are sensitive to the interactions [4]. However,
equilibrium profiles of concentrated colloidal dispersions
have rarely been measured, due to practical bottlenecks.
In this paper I, two of the main technical difficulties are
addressed: the low optical transmission of most concentrated
colloidal dispersions and the relatively high buoyant mass
of most colloidal particles. In the adjoining paper II [5], the
first experimental results obtained for oil-based ferrofluids are
discussed in terms of the equation of state of dipolar hard
spheres.
The extent to which low optical transmission and high
buoyant mass are a problem depends on the colloidal system.
For selected materials such as latex or silica, the refractive
index and/or the mass density can largely be matched
using well-chosen solvent mixtures [6, 7]. Moreover, strong
Analytical centrifugation of macromolecules dispersed in a
liquid is usually performed to measure their sedimentation
velocity, from which the hydrodynamic size or buoyant
mass can be calculated [1, 2]. Centrifugation can also be
continued until the concentration profile no longer changes,
when sedimentation and back-diffusion are in equilibrium.
In the low-concentration limit, an exponential profile is
obtained from which the buoyant mass of the particles can be
calculated, without having to know the solvent viscosity or the
shape of the particles. At higher concentrations, interparticle
interactions affect the profile and, conversely, quantitative
information about the interparticle interactions can be derived,
by using the profile to calculate the osmotic equation of
state [1–3]. It would be very interesting to use analytical
centrifugation to characterize the interparticle interactions not
0953-8984/12/245103+08$33.00
1
c 2012 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 24 (2012) 245103
B Luigjes et al
long-range repulsion can simplify the experiments, since
it causes expanded sedimentation equilibrium profiles, and
deviations from ideality are already observed at relatively
low concentrations. This is the case in the studies by Raşa
et al [8, 9] of charged silica colloids in ethanol, where
the volume fraction remained below 2.5%. Nevertheless,
for many important colloidal particles that strongly absorb
light and have a high buoyant mass, no suitable molecular
solvent mixtures exist to match the refractive index or the
mass density. Examples include a great variety of inorganic
nanoparticles of interest for their magnetic properties (Fe3 O4 ,
Co0 , . . .), quantum confinement effects (PbSe, CdSe, . . .),
or catalytic action (Pt, Au, . . .). The study of their colloidal
interactions by analytical centrifugation would provide
new insight relevant for applications of these particles in
liquid media, for instance biomedical applications [10, 11].
Traditional osmometry may still be a successful alternative
in selected systems, such as aqueous systems for which the
osmotic stress method can be applied [12–15]. However, when
it can be carried out, analytical centrifugation has several
advantages: it requires much less sample—a few microliters
suffice to measure the entire equation of state—, it can
be performed in almost any molecular liquid, and it can
be done reliably down to lower pressures than achievable
via osmometry [8, 9]. This warrants the development
of an experimental approach to determine sedimentation
equilibrium profiles of strongly absorbing colloidal particles.
The problem of a low optical transmission at high
colloidal concentrations can be tackled by using sample
cells with a short optical path length. Commercial analytical
centrifugation cells have an optical path length of at least a
few millimeters. However, better suited homebuilt cells can
be prepared. For instance, Page et al [2] prepared 0.1 mm
thin cells, enabling measurements on a surfactant system up
to concentrations of several weight per cent.
The high buoyant mass of colloidal particles is a problem
because it results in profiles where the concentration is so
strongly height dependent that it is difficult to measure with
sufficient spatial resolution. Piazza et al [16] were able to
measure equilibrium profiles of aqueous latex spheres with
a diameter of 90 nm in normal gravity, but the minimum
acceleration of an analytical centrifuge is usually much higher.
Below a lower limit of the rotation rate, the rotor motion
becomes unstable. A typical analytical ultracentrifuge is not
meant to be used at accelerations below a = 700g. For SiO2
spheres with a diameter of 60 nm in ethanol, as studied by
Raşa et al [8, 9], this would lead to a sedimentation length
kT/(1ma) of only 8 µm, less than the spatial resolution of a
typical analytical ultracentrifuge.
In this paper, we present an approach based on lowvelocity analytical centrifugation of concentrated colloidal
dispersions in ultrathin glass capillaries. The feasibility is
demonstrated with a concentrated dispersion of colloidal
iron oxide nanoparticles (magnetite, Fe3 O4 ), which looks
pitch-black to the naked eye. Magnetic nanoparticles are
expected to exhibit dipolar attraction, as was observed before
in sedimentation-velocity experiments at volume fractions
below 0.0025 [17]. Here, relatively small nanoparticles
will be studied, whose buoyant mass is still low enough
to enable comparison with an analytical ultracentrifuge.
The determination of the osmotic equation of state from
analytical centrifugation is first briefly explained in section 2.
Section 3 describes the colloidal system, the centrifuges,
and the homebuilt glass capillary cells. The data analysis is
presented and evaluated in section 4, starting with the raw
signals and ending with the osmotic equation of state. The
conclusion restricts itself to commenting on the presented
experimental approach. The theoretical interpretation of the
measured equation of state is examined in the accompanying
paper II [5].
2. Theory
The principle of characterizing interparticle interactions from
analytical centrifugation of sedimentation equilibrium profiles
is as follows [1–3]. First, the particle number concentration
ρ as a function of the distance r from the rotor axis is
calculated from the position-dependent optical transmission.
Second, the osmotic pressure 5 at any position r0 is obtained
by integrating ρ(r) starting at low r, where the concentration
is negligible:
Z r0
ρ(r)r dr,
(1)
5 r0 = ω2 1m
rρ→0
with 1m the colloidal particle buoyant mass and ω the
radial frequency. In this way, not only 5 and ρ are known
as a function of r, but also 5 as a function of ρ; this is
called the osmotic equation of state. It is the key equation
that describes the thermodynamic behavior of colloidal
systems, and measuring it has the advantage of enabling a
direct comparison to existing analytical theories. For ideal,
non-interacting point particles, the equation of state is the
Van ’t Hoff equation,
5 = ρkT,
(2)
where kT is the thermal energy. The concentration profile is
then exponential:
ρ(r) = ρ(rb ) exp[ω2 1m(r2 − rb2 )/(2kT)]
(3)
with rb the position of the bottom of the profile. Attractions
between the colloidal particles cause the concentration
profile to shrink, corresponding to pressures lower than
expected from Van ’t Hoff. An example is the interaction
of dipolar spheres, whose effect on osmotic pressure was
recently reexamined theoretically [18]. Conversely, repulsions
cause the concentration profile to expand, corresponding to
pressures higher than expected from Van ’t Hoff. One example
is hard sphere repulsion, described by the Carnahan–Starling
equation of state [19],
1 + φ + φ2 − φ3
,
(4)
5 = ρkT
(1 − φ)3
where φ is the colloidal volume fraction. Another example is
long-range repulsion of charged colloids, leading to osmotic
pressures as have been described by Biben and Hansen [20]
and by Van Roij [21].
2
J. Phys.: Condens. Matter 24 (2012) 245103
B Luigjes et al
36, 57, and 145g and sedimentation lengths Lg = kT/(1ma)
of 2.1, 1.4, and 0.5 mm, respectively. Oppositely positioned
sample cells were balanced to less than 100 mg difference by
adding pieces of parafilm. All measurements were performed
at 22.0 ◦ C at a light intensity factor of 1.
The LUMiFuge instrument was not designed for
measurements lasting several weeks and had to be restarted
every 99 h. Nevertheless, convection was limited inside the
confined geometry of our capillary cells, and equilibrium
profiles could be attained despite the multiple stops and
restarts. Furthermore, the instrument was not intended for
sensitive absolute measurements of the transmission across
a wide range from nearly full transmission to nearly zero
transmission. Therefore, we averaged 11 × 255 scans of the
profile to reduce the digital noise, and our data analysis
procedure included an internal calibration of the 100%
transmission level, as will be explained in section 4. For our
future measurements, LUM GmbH is currently extending the
maximum measurement time to 100 days and improving the
signal-to-digital noise ratio by one order of magnitude.
Figure 2(a) shows how a thin glass capillary was
positioned inside a metal holder. Glass capillaries with a width
of 1 mm and an internal thickness of 50 µm (from Vitrocom)
were first cut to a length of 20 mm. The capillary was partly
filled with ferrofluid, using capillary action, after which it was
sealed on both sides using transparent two-component epoxy
R
). The sealed capillary was glued via
glue (Bison Kombi Snel
its two extremities into a homebuilt metal holder. The holder
consisted of two 0.15 mm thin metal plates glued together at
the corners, one plate with a 1.2 mm wide slit to position
the slightly more narrow capillary and another plate with a
0.9 mm slit to ensure that light would only pass through the
capillary, rather than going around the sides. Two 0.15 mm
thin glass plates were glued to the metal holder with the
capillary, one on top and the other below. Finally, the capillary
holder was glued into place inside a disposable polycarbonate
rectangular LUMiFuge sample cell with an optical path length
of 2 mm. After use, the cell and the glue could be removed
in dimethylformamide to recycle the metal pieces. Further
practical details can be found in [24].
For comparison with the low-velocity centrifuge results,
measurements were also carried out with two analytical
ultracentrifuges, an XL-I and an XL-A Beckman Coulter
Optima AUC, both using absorbance optics and an An-60 Ti
rotor with 3 samples and 1 counterbalance. They operate at
a tunable optical wavelength, and the recommended rotation
rates range from 3000 to 60 000 rpm, corresponding to
accelerations of 724–289 750g at a radial position of 72 mm.
The selected optical wavelength was 541 nm, chosen as a
compromise, because the Xe lamp intensity becomes too
low at longer wavelengths while the iron oxide absorbance
becomes too high at shorter wavelengths. The temperature
was 20.0 ◦ C, and the rotation rates were 1600, 1400, and
1200 rpm, corresponding to accelerations of 206, 158, and
116g. This is below the lowest recommended rotation rate
of the instrument, where stable rotor motion is no longer
guaranteed; however, it was previously found by us that
lower rotation rates are still feasible [8, 9]; sedimentation
Figure 1. Transmission electron micrograph of the Fe3 O4
nanoparticles (code L: 13.4 nm in diameter, ±11%).
3. Experimental details
3.1. Colloidal magnetic nanoparticle system
Monodisperse colloidal magnetic nanoparticles of Fe3 O4
(magnetite) were prepared by thermal decomposition of
iron acetyl acetonate in diphenyl ether, in the presence
of oleic acid and oleylamine. Details of the multistep
seeded growth chemical synthesis have been published
previously [22–24]. The particle diameter was 13.4 nm
(±11%) according to transmission electron microscopy, see
figure 1. The particles are referred to with the code L,
because they are larger than the S-particles with which
their properties are compared in paper II [5]. The magnetic
dipole moment of the particles was determined by fitting
magnetization curves measured at low concentration using a
Micromag 2900 alternating gradient magnetometer (Princeton
Measurements Corporation), assuming a monodisperse dipole
moment [23, 25]. The determined magnetic dipole moment of
6.2 × 10−19 A m2 corresponds to the presence of a single
magnetic domain in each nanoparticle. The nanoparticles
were dispersed in decalin at a concentration of 134 g L−1 ,
with 15.6 mM oleic acid to avoid surfactant desorption [26].
3.2. Analytical centrifugation
Measurements were performed using a low-velocity analytical
centrifuge and homebuilt capillary sample cells. The
R
(LUM
centrifuge was a Stability Analyzer LUMiFuge
GmbH, Berlin), which was developed for the ‘fast and
reproducible characterization of settling behavior and
stability’ of colloidal products in ‘pharmacy, foods, cosmetics,
paints and varnishes’ [27]. It operates with near-infrared light
(880 nm), using a pulsed NIR-LED to illuminate the sample
cell and detecting the transmitted light by a CCD-line of
2048 elements positioned every 14 µm. Possible rotation rates
range from 200 to 4000 rpm, corresponding to accelerations
of 6–2325g at a radial position of 130 mm. Here, rotation rates
of 500, 625, and 1000 rpm were chosen, corresponding to
3
J. Phys.: Condens. Matter 24 (2012) 245103
B Luigjes et al
Figure 2. (a) Assembly of a flat capillary into a metal holder for low-velocity analytical centrifugation. (b) Assembly of a homebuilt
ultrathin centerpiece for analytical ultracentrifugation; after the glass pieces are glued to the metal spacer, the sample compartment is filled
with dispersion and the reference compartment with solvent. The drawings are close to actual size.
equilibrium profiles are still reliable down to 1200 rpm.
For sufficient optical transmission through concentrated
ferrofluid, homebuilt centerpieces were made with an optical
path length of 50–70 µm, depending on the amount of glue,
see figure 2(b). They consisted of a glass bottom, glued to a
50 µm thin metal spacer with two liquid compartments (for
the ferrofluid and for a solvent reference), glued to a glass top
with two filling holes, and sealed with a 0.1 mm thin Teflon
cover with a hole in the optical analysis zone. Centrifugation
was continued until two concentration profiles measured at
an 8 h interval showed less than 1% difference, indicating
that equilibrium had been reached between sedimentation and
diffusion.
To verify the linearity of absorbance with concentration and to examine the wavelength dependence, optical
transmissions were measured using quartz cuvettes with
an optical path length of 2 mm (Varian Cary 1E UV–vis
spectrophotometer) and using glass capillary cells with an
internal thickness of 50 µm, mounted in the same way as for
low-velocity analytical centrifugation (PerkinElmer Lambda
35 UV–vis spectrophotometer). For the AUC experiments, the
UV–vis data were also used to select the optimal wavelength
of 541 nm and to determine the extinction coefficient to
calculate the concentrations.
profiles have been shifted slightly up or down along the y-axis
to set the signal equal to zero at those positions. Transmission
is relatively high through the empty part of the capillary,
to the left of the ferrofluid meniscus. The filled part of the
capillary has a transmission which depends on the ferrofluid
concentration. To the right of the ferrofluid bottom, metal
again results in zero transmission. The data are shown before
centrifugation, at time t0 , and after sedimentation equilibrium
has been attained at two different rotation rates, 500 and
1000 rpm (figure 3(b); to simplify the figure, the 625 rpm data
are not shown here). Before centrifugation the ferrofluid has a
homogeneous concentration and transmission, whereas after
centrifugation the transmission increases near the meniscus
and decreases near the bottom of the liquid.
Centrifugation was continued until sedimentation equilibrium had been reached. Initially, the sedimentation front
moved at the sedimentation rate expected for single independent particles. At 500 rpm, for instance, the sedimentation
front moved 8 mm in the first 18 days; this is equal to the
theoretical rate given by the formula for the sedimentation rate
of independent spheres, 2R3 1da/ [9η (R + δ)], where R is the
colloidal core radius of 6.7 nm, 1d is the difference in mass
density between colloidal particle core and solvent (about
4300 kg m−3 ), a is the acceleration (36g), η is the solvent
viscosity (2.2 mPa s), and δ is the surfactant shell thickness
of about 2 nm. This assumes that the surfactant shell had
approximately the same density as the solvent, on the order
of 900 kg m−3 . As the dispersion became more concentrated
at the bottom of the profile, sedimentation slowed down,
as expected due to hydrodynamic interactions [28]. In total,
centrifugation was carried out for 90 days, when changes in
the concentration profile could no longer be detected. The
centrifugation was then continued at 625 and 1000 rpm, each
time, until a new equilibrium had been reached.
4. Results and discussion
Figure 3 illustrates the raw data obtained with the lowvelocity analytical centrifuge for ferrofluid contained inside
a homebuilt capillary cell. The rotor axis is on the left side
beyond the field of view. The detector signal that corresponds
to zero transmission has been determined at positions where
all light is blocked by the metal of the capillary holder, and the
4
J. Phys.: Condens. Matter 24 (2012) 245103
B Luigjes et al
Figure 3. Low-velocity analytical centrifuge measurements on the Fe3 O4 nanoparticles (code L) dispersed in decalin (with initial weight
concentration 134 g L−1 ) contained in a capillary with 50 µm internal thickness. (a) Schematic top view of the capillary positioned in a
metal holder. (b) Corresponding unscaled raw optical transmission signal S before (t0 ) and after centrifugation at 500 and 1000 rpm (36 and
145g). (c) Calibration of the absolute initial transmission T0 of the ferrofluid by taking the ratio between the signal before and after
centrifugation.
the particles compared to the optical wavelength. A beneficial
side-effect of the data treatment in (5) is that it removes
time-independent fluctuations in the signal due to optical
imperfections such as scratches of the polycarbonate cell
(compare figures 3(b) and (c)).
To calculate the osmotic equation of state, the mass
concentration profile was first converted into a number
concentration profile. Therefore, we first determined the
buoyant mass 1m of the nanoparticles from a plot of the
natural logarithm of the absorbance versus the square of the
radial distance, see figure 4(a). According to (3), the slope
of this plot is equal to ω2 1m/(2kT) in the low-concentration
range, where the osmotic pressure is not affected by
interparticle interactions. Somewhat different buoyant masses
were determined at 500, 625, and 1000 rpm, corresponding
to iron oxide core diameters of 13.6, 12.1, and 11.3 nm,
respectively. This is an effect of size fractionation, which
occurs despite the low polydispersity of the nanoparticles
(11%). Second, the effective radius R of the particles was
calculated from the buoyant mass on the basis of the
mass densities dFeOx = 5170 kg m−3 of magnetite and
dliq = 896 kg m−3 of decalin [29], assuming that 1m =
(4π/3)R3 (dFeOx − dliq ). Third, the total mass m of a single
nanoparticle was calculated by assuming an oleic acid surface
layer with thickness δ = 2 nm and a mass density equal to dliq :
m = (4π/3)R3 dFeOx + (4π/3)[(R + δ)3 − R3 ]dliq . Finally, the
mass concentration profiles c(r) were converted into number
concentration profiles ρ(r) by dividing by the mass m per
particle.
The number concentration profiles could now be
integrated to yield the osmotic equation of state. A variant of
The measured optical transmission signal S not only
depends on the transmission of the liquid but also on other
effects, including reflections from the polycarbonate cuvette,
the glass plates, and the glass capillary. Therefore, the raw
signal S had to be rescaled in order to obtain the precise value
of the optical transmission through the liquid. The scaling
factor is determined by dividing the initial signal Sinitial by
the final signal Sfinal at radial positions where the liquid
is practically depleted of nanoparticles once sedimentation
equilibrium has been reached. There, the ratio Sinitial /Sfinal
is equal to the initial transmission T0 of the homogeneous
dispersion before centrifugation, see figure 3(c). On this
basis, the initial absorbance is A0 = − log(T0 ), and the final
absorbance A(r) can be calculated at every radial position
r from the ratio of the optical signals before and after
centrifugation, Sfinal (r) and Sinitial (r):
Sfinal (r)
.
(5)
A(r) = − log T0
Sinitial (r)
Since the initial concentration is known, c0 = 134 g L−1 ,
the absorbance profile can be directly converted into a
profile of the mass concentration c, assuming that absorbance
is linear with concentration. The linearity was verified up
to concentrations of 350 g L−1 by UV–vis absorbance
measurements on the same type of glass capillaries and metal
holders as used in the centrifugation experiments. This was
done using slightly larger particles of 17 nm in diameter,
prepared by single-step decomposition of iron oleate [23]. The
linearity confirms that absorbance dominated and scattering
effects were negligible, as expected from the minute size of
5
J. Phys.: Condens. Matter 24 (2012) 245103
B Luigjes et al
Figure 4. Low-velocity analytical centrifuge measurements on the Fe3 O4 nanoparticles (code L) dispersed in decalin (initially at
134 g L−1 ) contained in a capillary with 50 µm in internal thickness. (a) Determination of the buoyant mass 1m from a plot of the
logarithm of the absorbance versus the square of the radial position. (b) Osmotic equation of state obtained by integration of the equilibrium
concentration profiles measured at 36, 57, and 145g.
(1) was used, taking into account a finite numerical integration
step 1r of 14 µm, given by the center-to-center distance of the
optical detector pixels of the CCD of the analytical centrifuge:
5(r) = 5 (r − 1r) + ρ(r)1mω2 r1r.
block-shaped transmission profile with a width of 0.4 mm.
As long as the rotation rate is not too high, so that the
gravitational length remained more than 0.4 mm, optical
broadening is not likely to have a measurable effect on the
measured profiles. This was confirmed by calculations of the
osmotic equation of state obtained by numerical integration
of theoretical concentration profiles that had been convoluted
with a 0.4 mm broadening function [24].
To test further our low-velocity analytical centrifugation
approach, the equation of state was also measured using an
analytical ultracentrifuge, see figure 5. Once the equilibrium
profile had been measured and the zero absorbance level
had been determined in the depleted part of the liquid, the
buoyant mass was again calculated from the slope of a plot
of the logarithm of the absorbance versus the square of
the radial position (figure 5(a)). The profile was integrated
according to (6), taking into account the measured uneven
step sizes 1r of 10–30 µm of the ultracentrifuge, yielding the
equation of state in figure 5(b). Similar to the data obtained
with the low-velocity analytical centrifuge (figure 4), the
calculated osmotic pressures are below Carnahan–Starling
and Van ’t Hoff, indicating the effect of interparticle
attraction.
As a self-consistency test, measurements were done
starting at three different rotation rates (1600, 1400, and
1200 rpm) and different starting concentrations (134, 103, and
36 g L−1 ), yielding the same osmotic equation of state, as it
should. Compared to the measurements with the low-velocity
analytical centrifuge, the calculated pressures deviate more
weakly from ideality, for reasons that are presently unclear.
Nevertheless, the ultracentrifuge results support the validity
of our low-velocity centrifuge approach.
The feasibility of sedimentation equilibrium measurements using a low-velocity analytical centrifuge and thin
capillary cells has been demonstrated. The centrifuge was
not designed to measure such equilibrium profiles, and
technical improvements are currently in progress to extend
the maximum measurement time and to decrease the digital
(6)
Furthermore, to start integration at negligible number
concentrations, where the osmotic pressure is practically zero,
the dilute part of the concentration profile was extrapolated
using (3) to concentrations much lower than the experimental
detection limit. In this way, the osmotic equation of state
started at the origin, and Van ’t Hoff’s law was imposed at
low pressures, in this case below 7 Pa.
The experimental osmotic equations of state are shown
in figure 4(b). The same equation of state is found at the
three different rotation rates, which shows that the results
are self-consistent. At different accelerations, the measured
profiles correspond to different gravitational lengths, but
nevertheless, the calculated equation of state is unique. Only
at high concentrations, where transmission is too low for
reliable measurements, do the curves depend on the rotation
rate. At low concentrations, the Van ’t Hoff equation of
state (2) is obtained, not only in the 0–7 Pa range where
it was imposed, but also up to about 30 Pa, validating
our extrapolation approach. At higher concentrations, the
osmotic pressures deviate from ideality. The pressures are
below those expected from the Carnahan–Starling equation
for hard spheres (4) or from the Van ’t Hoff equation for
non-interacting point particles (2), revealing the presence
of attractions between the particles. In paper II [5], it will
be argued that the deviation from ideality is in quantitative
agreement with the magnetic dipole moment of the studied
nanoparticles.
One technical conclusion supported by finding the same
osmotic equation of state at different accelerations is that the
spatial resolution of the analytical centrifuge was sufficient.
Independent experiments were also done to determine the
optical broadening function. A metal slit with a width of
100 µm (from CVI Melles Griot) led to a more or less
6
J. Phys.: Condens. Matter 24 (2012) 245103
B Luigjes et al
Figure 5. Analytical ultracentrifuge measurements on the Fe3 O4 nanoparticles (code L) dispersed in decalin contained in homebuilt
centerpieces with an optical path length of 60 µm (see figure 2(b)). (a) Determination of the buoyant mass 1m from a plot of the logarithm
of the absorbance versus the square of the radial position at 206g. (b) Osmotic equation of state obtained by integration of the equilibrium
concentration profiles measured at 206, 158, and 116g.
particle synthesis, Peter Horsman and Gert de Jong for
constructing the metal capillary holders, Hans van der
Kraan for cutting the glass plates and capillaries, and
Henkjan Siekman, Peter de Graaf and Wim Nieuwenhuis
for constructing the ultracentrifuge centerpieces. This work
was supported by the Netherlands Organization for Scientific
Research (NWO).
noise. The main strength of the low-velocity analytical
centrifuge is its minimum acceleration of merely 6g. This
implies that a gravitational length of 0.5 mm could be
obtained for particles with a buoyant mass of 85 kDa,
for instance 40 nm iron oxide spheres or 70 nm porous
Stöber silica spheres. This opens the way to study colloidal
particles whose sedimentation equilibria have not yet been
studied because of the strong optical absorption or high
buoyant mass of the particles. In the future, our low-velocity
centrifuge approach will be applied to other magnetic
nanoparticles, whose independently known magnetic dipole
moment enables the verification of the strength of interparticle
interactions determined from analytical centrifugation [5].
Other prospective experiments involve the analysis of the
interactions between semiconducting nanoparticles, which
possibly have an electrical dipole moment [30].
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5. Conclusion
In conclusion, our results demonstrate that sedimentation
equilibrium profiles of strongly absorbing colloidal nanoparticle dispersions can be measured by analytical centrifugation
in ultrathin glass capillaries. High concentrations, up to
hundreds of grams per liter, can thus be measured, sufficiently
high that interparticle interactions have a clear effect on
the profile and the corresponding osmotic equation of state.
Using a low-velocity analytical centrifuge, strongly absorbing
nanoparticles with a relatively high buoyant mass become
accessible, particles for which the interparticle interactions
have not yet been studied by analytical centrifugation. Even
very low osmotic pressures can be measured accurately,
which allows a quantitative assessment of dipolar attraction in
ferrofluids, as demonstrated in the accompanying paper II [5].
Acknowledgments
Claire van Lare is thanked for initial centrifugation
experiments, Rick de Groot and Suzanne Woudenberg for
7
J. Phys.: Condens. Matter 24 (2012) 245103
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