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Sedimentation equilibria of ferrofluids: II. Experimental osmotic equations of state of magnetite
colloids
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2012 J. Phys.: Condens. Matter 24 245104
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 24 (2012) 245104 (7pp)
doi:10.1088/0953-8984/24/24/245104
Sedimentation equilibria of ferrofluids: II.
Experimental osmotic equations of state
of magnetite colloids
Bob Luigjes, Dominique M E Thies-Weesie, Ben H Erné and
Albert P Philipse
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, in final form 2 May 2012
Published 23 May 2012
Online at stacks.iop.org/JPhysCM/24/245104
Abstract
The first experimental osmotic equation of state is reported for well-defined magnetic colloids
that interact via a dipolar hard-sphere potential. The osmotic pressures are determined from
the sedimentation equilibrium concentration profiles in ultrathin capillaries using a
low-velocity analytical centrifuge, which is the subject of the accompanying paper I. The
pressures of the magnetic colloids, measured accurately to values as low as a few pascals,
obey Van ’t Hoff’s law at low concentrations, whereas at increasing colloid densities
non-ideality appears in the form of a negative second virial coefficient. This virial coefficient
corresponds to a dipolar coupling constant that agrees with the coupling constant obtained via
independent magnetization measurements. The coupling constant manifests an attractive
potential of mean force that is significant but yet not quite strong enough to induce dipolar
chain formation. Our results disprove van der Waals-like phase behavior of dipolar particles
for reasons that are explained.
(Some figures may appear in colour only in the online journal)
1. Introduction
The experimental challenge is in the first place to develop
magnetic model colloids that mimic the dipolar hard sphere
(DHS) employed in theory and simulations [6–21]. The
DHS potential only comprises the dipole–dipole interaction,
supplemented with a hard-sphere repulsion, whereas it
disregards all other aspects of interparticle interactions, such
as particle shape details and additional isotropic (van der
Waals) attractions. The experimental instance of a DHS-fluid
is therefore a suspension of monodisperse colloidal spheres,
each with one embedded permanent magnetic moment.
Although conventional ferrofluids contain single-domain
magnetite (Fe3 O4 ) particles, they are generally unsuitable
for comparison with a DHS-fluid: the magnetic moment is
rather weak and difficult to adjust and, in addition, magnetite
particles are often quite polydisperse in size and shape.
Metallic iron colloids [22–25] are significantly more magnetic
but have the major drawback that they easily transform to
an (unfortunately nonmagnetic) iron oxide. However, recent
The osmotic equation of state of a colloidal fluid, i.e. the
osmotic pressure as a function of colloid concentration,
has been extensively studied for colloids with isotropic
interactions, in theory as well as via experiments [1–5].
Also the effect of anisotropic magnetic interactions on
the thermodynamics of colloidal fluids has been frequently
addressed, as witnessed by the theory and simulations over
the past 40 years [6–21]. However, in contrast to the
case of isotropic colloids, relevant experimental data for
magnetic particles are scarce. Nevertheless, experimental
osmotic pressures of magnetic fluids would be of considerable
interest, if only because the phase behavior of magnetic fluids
is still poorly understood. For example, the existence of a
liquid–gas critical point for dipolar spheres is the subject of
a long-standing debate [6–21].
0953-8984/12/245104+07$33.00
1
c 2012 IOP Publishing Ltd Printed in the UK & the USA
J. Phys.: Condens. Matter 24 (2012) 245104
B Luigjes et al
2. Second virial coefficient of dipolar hard spheres
progress in the synthesis of magnetite colloids has improved
the control of their size and magnetic moment, as further
explained in [26–28]. As a result magnetite fluids can be
prepared that form a realistic experimental instance of the
DHS-fluid; fluids of this type are employed in this paper, and
the accompanying paper I.
Having suitable magnetic colloids available, the next
challenge is experimental determination of osmotic pressures.
To verify Van ’t Hoff’s law for ideal solutions and to measure
the first-order deviation from ideality, one has to access
osmotic pressures as low as a few pascal. Such minute
pressures cannot be measured with sufficient accuracy using
classical osmometry employing a semipermeable membrane,
or via osmotic stress measurements [29, 30]. The static light
scattering intensities at zero wavevector are proportional to the
osmotic compressibility [1, 5] but also here very low osmotic
pressures are difficult to access, apart from the fact that the
strong light absorption of magnetite anyhow obstructs light
scattering measurements.
The option we followed in the accompanying paper [31]
was to determine an equation of state from the sedimentation
equilibrium (SE) of colloids in the centrifugal field of an
analytical centrifuge. The underlying principle, discussed
in more detail in paper I and elsewhere [4, 32–34] is
briefly as follows. The SE-concentration profile of colloidal
particles is the equilibrium balance between the osmotic
pressure gradient and the external centrifugal force. Therefore
integration of this concentration profile yields the osmotic
pressure versus the particle concentration [4, 33]. This
method, in principle, is direct and does not make any
assumptions on the colloids or their interactions.
For the case of charged silica spheres it has been
shown [32, 33] that the analytical ultracentrifuge (AUC)
indeed yields accurate osmotic pressures, even for very
low particle concentrations. The AUC, admittedly, is a
demanding technique, with for magnetite fluids at least two
additional complications. First, the already mentioned strong
light absorption of magnetite colloids severely hampers their
optical detection in the AUC. Second, the buoyant mass of
the particles is relatively high which may cause too steep
SE-profiles, even at the lowest rotation rates a conventional
AUC can operate at. Therefore we employed a low-velocity
analytical centrifuge (AC), equipped with homebuilt ultrathin
glass capillaries, which allows one to measure SE-profiles of
magnetic fluids. Our sedimentation method and the analysis
of SE-profiles are described in the accompanying paper [31,
28] where also further details are given about the synthesis
and characterization of the magnetic colloids.
Our principal aim here is to determine the equation of
state (EOS) over a concentration range that is wide enough
to get an accurate experimental value of the second virial
coefficient, from which a dipolar coupling constant can
be extracted [21] for comparison with a coupling constant
obtained via independent magnetization measurements. Next
we wish to assess (in section 4.3) the implications of our
experimental results for the existence of a liquid–gas critical
point for dipolar spheres.
The second virial coefficient B2 appears in the virial expansion
of the osmotic pressure 5 of a fluid with colloid number
density ρ as [1]
5
= 1 + B2 ρ + B3 ρ 2 + · · · ,
ρkT
(1)
where we have also included the third virial coefficient B3 (for
reasons explained in section 4.2). The second virial coefficient
can be calculated from the excluded volume integral as
Z ∞
B2 = 2π
[1 − g(r)]r2 dr.
(2)
0
The required pair distribution function g(r) for a DHS-fluid
is obtained as follows. The starting point is the pair potential
for two uncharged hard spheres with diameter σ , each with an
embedded permanent dipole moment µ,
E at a center-to-center
distance r [35]:
u(r)
= ∞,
0≤r<σ
kT
(3)
u(Er, )
σ 3
f (),
r ≥ σ.
=λ
kT
r
This DHS potential comprises the hard-sphere repulsion, plus
the dipolar interaction at r ≥ σ . The latter has an orientational
part f () = b
µ1 · b
µ2 − 3(b
µ1 ·b
r)(b
µ2 ·b
r), where  denotes the
orientations of both dipoles and carets indicate unit vectors.
The amplitude of the dipolar interaction is set by the coupling
parameter
µ0 µ2
π
,
µ = σ 3 ms
(4)
3
6
4π kTσ
comprising the vacuum permeability µ0 , the magnitude µ
of the dipole moment, and the thermal energy kT; ms is the
volume magnetization of magnetite. The potential of mean
force V(r) for two colloids is the average, reversible work [1]
needed to bring one colloid to a position Er, given that the other
colloid is centered at the origin at r = 0. For dipolar spheres
at low number densities ρ, all that need to be averaged are the
orientations of the two dipole moments, such that V(r) follows
from [35]
λ=
V(r, ρ → 0) = −kT lnhexp[−u(Er, )/kT]i ,
(5)
where the brackets denote an average over all dipole
orientations for two dipoles at a center-to-center distance r =
|Er|. The pair distribution follows from
g(r) = exp[−V(r, ρ → 0)/kT].
(6)
Chan et al [35] found an analytic solution for the orientational
averaging in (5) which leads to the following pair correlation
function:


0,
0≤r<σ



X
∞
x 3x 
n
in
,
(−1) in
(7)
g(r) =
2
2

n=0



σ 3


r ≥ σ, x = λ
,
r
2
J. Phys.: Condens. Matter 24 (2012) 245104
B Luigjes et al
Table 1. Properties of magnetite colloids. (Note: details of the
characterization methods are given in [28, 31].)
Particle code
S
L
Average core diameter σ (nm)
Core polydispersity (%)
Magnetic dipole moment µ (10−19 A m2 )
Coupling parameter λ from magnetization
measurements
11.0
11.7
3.4
0.8
13.4
10.5
6.2
1.8
in the form of a stiff dipolar magnetite chain. For the magnetic
colloids in the present study λ is much smaller, as further
discussed in section 4.3.
3. Experimental equations of state
Figure 1. The second virial coefficient B2 (scaled on the
hard-sphere value) as a function of the dipolar coupling
constant [21]. Numerical integration results (open circles) of
integral (2) for the pair correlation function in (7) are compared
with B2 calculated from (8) using only the λ2 term (dashed line) and
adding the λ4 term (solid line). Including the latter term makes the
calculation accurate up to λ ≈ 2.5.
3.1. Magnetite colloids
For the analytical centrifugation experiments in paper [31]
and [28] we prepared iron oxide (magnetite) particles, coated
with oleic acid, and dispersed in solutions of 15.6 mM
oleic acid in decalin. The seeded growth procedure to adjust
the average particle size is described in detail in [28, 31].
Centrifugation experiments were performed for two different
average particle sizes: magnetite colloids referred to as
S-particles having an average core diameter of 11.0 nm and
somewhat larger L-particles with an average core diameter of
13.4 nm. Particle characterization results are summarized in
table 1 (see also figure 2).
where in is the modified spherical Bessel function. For weakly
interacting dipoles such that x 1, the small-x expansion of
(7) can be substituted in (2) with the result [21]
B2
BHS
2
1
29
1
λ6 .
= 1 − λ2 − λ4 −
3
75
55 125
(8)
3
Here BHS
2 = (2/3)π σ is the second virial coefficient of hard
spheres with diameter σ . For strongly coupled dipoles at
x 1, the excluded volume integral (2) has the asymptotic
solution [21]
B2
BHS
2
∼−
exp[2λ]
,
12λ3
λ → ∞.
3.2. Centrifugation experiments
Centrifugation experiments were performed with a conventional Beckman analytical ultracentrifuge (AUC) and a specially adapted low-velocity LUMifuge analytical centrifuge
(AC). The monitoring of sedimentation equilibrium profiles
and their numerical integration to obtain osmotic pressures are
described in detail in [28, 31].
(9)
The weak-coupling expression (8) predicts that, starting
with nonmagnetic hard spheres, the second virial coefficient
monotonically decreases with increasing dipolar coupling
parameter; an outcome that makes sense because the potential
of mean force in (5) is always attractive, which lowers the
osmotic pressure of the fluid. A comparison with numerical
results (figure 1) shows that (8) is quite accurate up to λ ≈ 2.5.
An interesting result from (9) is the ‘Boyle point’ λ ≈ 1.64,
i.e. the magnitude of the dipolar attraction that compensates
the hard-sphere repulsion such that B2 ≈ 0.
With respect to the strong-coupling result (9) we note that
De Gennes and Pincus [6] also evaluated the second virial
coefficient in the strong-coupling limit, albeit specifically for
dipoles that are fully aligned by a strong magnetic field. Their
result is [6]
B2
BHS
2
∼−
exp[2λ]
,
12λ2
λ → ∞,
4. Results and discussion
4.1. EOS for small particles S
The experimental equation of state (EOS) of the relatively
small S-particles is shown in figure 3. Sedimentation
equilibrium profiles have been obtained for various centrifuge
rotor speeds; note that the eventual osmotic pressures agree
quite well with each other. A striking observation in figure 3
is that also for higher particle concentrations, the osmotic
pressures of S-particles deviate only little from Van ’t Hoff’s
law 5 = ρkT. This is not what one would expect for a
hard-sphere fluid; the EOS of nonmagnetic hard spheres
(at volume fraction φ) is given by the Carnahan–Starling
equation [1]
(10)
which is a factor of λ larger than the zero-field result
in (9). An interesting instance of strongly coupled dipoles
is the single-domain magnetite crystals (λ ≈ 50) found in
magnetotactic bacteria [34], which have an internal backbone
5
1 + φ + φ2 − φ3
=
≈ 1 + 4φ + · · · ,
ρkT
(1 − φ)3
3
(11)
J. Phys.: Condens. Matter 24 (2012) 245104
B Luigjes et al
Figure 2. Representative TEM images [28, 31] of the oleic acid coated magnetite particles in this study: (a) S-particles with an average
diameter of 11.0 nm; (b) L-particles with an average diameter of 13.4 nm.
which predicts (see figure 3) that the osmotic pressure should
significantly rise above the Van ’t Hoff value within the
accessed concentration range. The absence of this rise in
figure 3 implies that, even though the S-particles are relatively
small, their dipolar attraction is still sufficient to counteract
the contribution of the hard-sphere repulsion to the osmotic
pressure. Since the pressures in figure 3 deviate only little
from their ideal value, it follows that the dipolar coupling
parameter of S-particles must be fairly close to the Boyle
value λ ≈ 1.64.
Two points need to be further addressed in the
comparison of experimental osmotic pressures with the
Carnahan–Starling EOS. First we need to assess the effect
of the choice of the thickness of the oleic acid layer on the
particles. Changing the layer thickness changes the average
particle volume and weight and, consequently, the number
densities in the experimental EOS, as further explained in [28,
31]. Figure 3, however, shows that the effect of the layer
thickness is modest: it certainly does not account for the
discrepancy between the Carnahan–Starling EOS and the
experiments.
Another potential contribution to this discrepancy is a
van der Waals attraction that, just like the dipolar attraction,
would decrease the osmotic pressures. Using recently reported
values for Hamaker constants of iron oxides in various organic
solvents [36] and Hamaker’s expression for the van der Waals
attraction between two spheres [37], we find that for S- as
well as L-particles, this attraction is of order ∼0.1kT at room
temperature. Thus the effect of Van der Waals forces on the
experimental EOS is negligible.
Figure 3. Osmotic pressures as a function of the number density ρ
of S-particles, obtained from sedimentation equilibrium profiles in a
LUMifuge (low-speed analytical centrifuge, see paper [31]
and [28]), operating at various rotor speeds as indicated in the
figure. The experimental data follow Van ’t Hoff’s ideal osmotic
pressure law (solid line) over almost the entire accessed
concentration range, in contrast to the Carnahan–Starling equation
(11) for hard spheres that predicts a significant pressure increase.
The experimental and calculated equations of state slightly depend
on the choice of the oleic acid layer thickness, 2 nm in (a) and 1 nm
in (b), as further discussed in section 4.1.
4.2. EOS for large particles L
Osmotic pressures for large particles L are shown in figure 4.
If the data indeed manifest a thermodynamic equation of
state, they should be independent of the way in which
the osmotic pressures have been obtained. An important
assessment, therefore, is that both the centrifugation setups
(the LUMiFuge and the Beckman AUC, see paper [31])
yield essentially the same experimental EOS. The reliability
of the osmotic pressure results is further confirmed by the
observation in figure 4 that the EOS is quite insensitive to
colloid starting concentrations as well as the rotor speeds that
maintain the sedimentation equilibrium profile. Note also the
extremely low osmotic pressure values in figure 4 that still can
be accurately measured.
4
J. Phys.: Condens. Matter 24 (2012) 245104
B Luigjes et al
Figure 5. Reduced osmotic pressures (see (12)) versus the volume
fraction of L-particles, showing a very good agreement between the
data from two different centrifuges (the LUMifuge and the AUC).
The second virial coefficient follows from a linear fit to the data
according to (12).
experimental virial coefficient, as further explained below.
Here we first address the question of whether the slopes in
figure 5 indeed manifest the correct second virial coefficient:
it is conceivable that a modest contribution of the third virial
coefficient is present, affecting the value of the measured B2 .
The third virial coefficient could even be the dominating term
in the EOS when the magnetite particles are close to the Boyle
value of the coupling parameter. Therefore we rewrite (1) as
1 5
− 1 = B2 + B3 ρ,
(13)
ρ ρkT
Figure 4. Osmotic pressures versus the number density of
L-particles, derived from sedimentation equilibrium profiles
monitored in (a) the LUMifuge low-speed centrifuge and (b) the
Beckman analytical ultracentrifuge (AUC), operating at rotor speeds
as indicated. The initial weight concentrations (in g L−1 ) of the
homogeneous dispersions at the start of a centrifugation experiment
are also listed. The experimental osmotic pressures initially follow
Van ’t Hoff’s law (black solid line) but fall below it at higher
concentrations, with a large discrepancy with the Carnahan–Starling
pressure (purple dotted line) for hard spheres, due to dipolar
attractions as further discussed in section 4.2. Note the extremely
low values of osmotic pressure that still can accurately be measured
via analytical centrifugation.
such that the right-hand side of (13) should level to
a constant when the contribution of B3 is negligible.
Figure 6 demonstrates that this is indeed the case and
that, consequently, the deviations from Van ’t Hoff’s law in
figures 4 and 5 are purely from the quadratic density term in
the equation of state.
We now return to the above mentioned ranges in the
experimental B2 : taking them together the two ranges imply
according to (8) that the dipolar coupling constant of the
L-particles is in the fairly narrow range of λ = 2.0–2.4.
This outcome is quite close to the value of λ ≈ 1.8 (see
table 1), determined from the magnetization curves of the
L-particles that are completely independent of sedimentation
experiments. We note here that in [38] a value of λ =
1.77 is found from the magnetization data for oleic acid
coated magnetite particles with an average diameter of 14.2
(±11.3%), comparable to the L-particles in table 1.
The small discrepancy between the coupling constants
obtained from magnetization and sedimentation data may
be due to size polydispersity, even though the distribution
is fairly narrow (see table 1). Possibly the second virial
coefficient samples a higher moment of the distribution than
the magnetization data, due to the contribution of the stronger
attraction between larger particles. This statement, however, is
difficult to quantify further in the absence of any calculation
on the osmotic pressure of polydisperse dipolar spheres.
A coupling constant in the range λ = 2.0–2.4 manifests
a significant dipolar interaction that, nevertheless, still falls
Figure 4 shows that at sufficiently low concentrations the
experimental EOS obeys Van ’t Hoff’s law, just as for the
S-particles in figure 3. However, at higher concentrations the
osmotic pressures exerted by the L-particles clearly fall below
their ideal value. The marked decrease with respect to the
Carnahan–Starling EOS manifests dipolar attractions that are
significantly stronger than those between the S-particles. To
obtain the second virial coefficient we plot the data in figure 5,
for a fit to the virial expansion in the form
5
= 1 + B02 φ.
ρkT
(12)
Here B02 = B2 /v is the second virial coefficient from (1) and
(2) divided by the particle volume v. The fit to (12) yields
for the LUMifuge data a B02 in the range −3.1 to −5.9; for
the AUC data it yields a B02 in the range −2.6 to −4.8. Since
B2 depends quite strongly on the dipolar coupling parameter
λ, the spread in the latter is much smaller than that in the
5
J. Phys.: Condens. Matter 24 (2012) 245104
B Luigjes et al
weak dipolar interactions may exhibit a van der Waals-like
coexistence between an isotropic gas and an isotropic liquid
phase.
A simple prediction for the attraction that is needed to
achieve such phase behavior is provided by Lekkerkerker
and Vliegenthart (LV) who argue that whenever B2 /BHS
2 ≈
−1.5, an isotropic fluid must be close to its gas–liquid
critical point [39]. From (8) and figure 1 we find that the
LV-criterion for B2 corresponds to λ ≈ 2.46, which almost
falls in the experimental range of coupling constants for the
L-colloids. Nevertheless, we have not observed any sign of
phase separation such as a plateau in the measured osmotic
pressure [4]. One could argue that the LV-criterion for B2
is an estimate and that, possibly, the virial coefficient of
the L-particles should have been more negative to observe
liquid–gas criticality. However, from an earlier experimental
study on dipolar structures [27] we know that magnetic
particles with a coupling constant of λ = 2.2 are on the brink
of dipolar chain formation. It is, consequently, very unlikely
that a further increase in λ, and a corresponding further
decrease in B2 , would lead to coexistence between isotropic
fluids and gases of unassociated spheres.
In this work we have studied equilibrium sedimentation
profiles in zero magnetic field; in previous work the
sedimentation velocity of oleic acid magnetite colloids in
decalin was studied in detail [38]. Above a particle diameter
of about 8 nm, that is, above a dipolar coupling constant
of about λ = 0.2 [38], the linear concentration dependence
of the sedimentation velocity was found to be positive. This
association effect is the counterpart of the negative second
virial coefficient which we find here from the experimental
equation of state: the same dipolar attractions that enhance
sedimentation also lower the osmotic pressure. In [38] it
was concluded that orientational pre-averaging of the dipolar
anisotropy in the calculation of hydrodynamic interactions
leads to sedimentation velocities that are much smaller than
the experimental values. Thus it appears that a dipolar
hard-sphere fluid, even for weakly coupled dipoles, cannot
be modeled by an effective isotropic fluid, either in its
thermodynamics or in its transport properties.
Figure 6. A plot of the osmotic pressure data of L-particles
according to (13). The leveling to a plateau at increasing volume
fraction demonstrates that the contribution of the third virial
coefficient to the osmotic pressure is negligible: the deviations from
Van ’t Hoff’s law in figures 4 and 5 are purely from the quadratic,
second virial term in the equation of state. The plateau at φ ≥ 0.07
corresponds to a virial coefficient in the range B02 = −2.6 to − 4.0.
in the weak-coupling regime (see figure 1). From the pair
potential in (3) it follows that the maximal attraction between
two dipolar hard spheres, umax = −2λ, occurs when the
spheres are in contact, with their dipoles in a head-to-tail
configuration. For L-particles with λ = 2.2 we find a fairly
strong maximum of umax ≈ −4.4kT. This value, however,
overestimates the actual contact attraction that follows from
the potential of mean force, which in the weak-coupling limit
has the leading term [6, 21, 35]
V(r)
λ2 σ 6
= − ln g(r) ∼ −
.
(14)
kT
3 r
Thus, the contact attraction from (14) for λ = 2.2 is −1.6kT,
which is considerably smaller than the head-to-tail attraction.
At first sight it is somewhat surprising that a modest
difference of 2.4 nm between the diameters of the L- and
S-particles (see table 1) has such a marked effect on the EOS.
However, according to (4) the coupling constant scales with
the particle diameter as λ ∼ σ 3 . According to this scaling
the range λ = 2.0–2.4 for L-particles with average diameter
13.4 nm corresponds to λ = 1.1–1.3 for S-particles with σ =
11 nm, close to the coupling λ = 0.8 from the magnetization
data (table 1). The range λ = 1.1–1.3 leads to a small positive
virial coefficient of B2 /BHS
2 = 0.40–0.58. Thus the change
in the EOS in going from the S-particles in figure 3 to the
L-particles in figure 4 is indeed the consequence of the strong
size dependence of the dipolar coupling constant.
5. Conclusions
Oleic acid coated magnetite colloids in an organic solvent
with adjustable size are experimental instances of dipolar hard
spheres with adjustable magnetic dipole moments. Analytical
centrifugation is a reliable technique to obtain accurate
osmotic equations of state (EOSs) of a magnetite fluid. The
osmotic pressure can be determined quite precisely down
to values as low as a few pascal. Higher concentrations
of the strongly light-absorbing colloids can be accessed
via sedimentation in specially designed, ultrathin capillaries.
The EOS of the smaller S-particles in this study exhibits
ideal behavior in a broad colloid concentration range, which
manifests a compensation of hard-sphere repulsions by weak
dipolar attractions. The EOS of the larger, more magnetic
L-particles contains a significant, negative second virial
coefficient B2 , which corresponds to dipolar hard spheres
4.3. Van der Waals-like phase behavior?
Pincus and de Gennes noted [6] that in the weak-coupling
regime the interaction potential (the potential of mean force,
to be more precise) decays as 1/r6 , as given by (14).
Since molecular van der Waals attractions have the same
distance dependence, it was conjectured [6] that spheres with
6
J. Phys.: Condens. Matter 24 (2012) 245104
B Luigjes et al
with a coupling constant of λ ≈ 2.2. Despite such a fairly
strong magnetic attraction, no sign of van der Waals-like
liquid–gas criticality is found for our magnetic fluids, even
though B2 is sufficiently negative to satisfy the LV-criterion.
Both sedimentation equilibria and sedimentation velocity
measurements confirm that dipolar fluids cannot be modeled
by an effective isotropic fluid.
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Acknowledgments
Professor A Vrij and Dr R van Roij are acknowledged
for useful discussions on the subject matter of this paper.
Mr B Kuipers is thanked for providing figure 1 and
Mrs M Uit de Bulten-Weerensteijn for her contribution to the
preparation of the manuscript. This work was supported by
the Netherlands Organization for Scientific Research (NWO).
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