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The influence of shape anisotropy on the microstructure
of magnetic dipolar particles
Cite this: Soft Matter, 2013, 9, 6594
Sofia Kantorovich,*abc Elena Pyanzinab and Francesco Sciortinoa
In the first part of the present contribution we study theoretically the ground states of magnetic rods and
ellipsoids with point dipoles. We consider two different orientations of dipoles: the dipole moment
oriented along the short axis of the anisotropic particle and the dipole moment aligned along the long
axis. We show that depending on the particle asphericity the ground states of ellipsoids and rods might
be a chain and a ring (daisy) in two dimensions, or a carpet and a bracelet in three dimensions. If the
ellipsoids or rods with the dipole moment coaligned with the long axis are elongated enough, the headto-tail configuration of moments becomes less favourable than the antiparallel side-by-side pair of
dipoles. This structural crossover in the ground state drives crucial changes in the microstructure of
Received 17th January 2013
Accepted 26th April 2013
systems of anisotropic particles when the thermal energy starts being comparable to that of the
interparticle interaction. We investigate the latter regime in the second part of our manuscript and
provide a detailed analysis of the interparticle correlations using molecular dynamics simulations. We
extensively study the influence of aspect ratios and different dipolar strengths on the structural
DOI: 10.1039/c3sm50197c
properties, internal pressure and initial susceptibility of anisotropic particles' systems. We conclude that
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the particle shape anisotropy could be effectively used as a control parameter for the system microstructure.
1
Introduction
Magnetic anisotropic particles in the last several years became
an independent fast-emerging branch of dipolar so matter
research. In the long run, these systems will completely fulll
the idea of ne tuning and designing new materials with
controllable properties.
From the theoretical point of view, it is convenient to
distinguish between two possible types of anisotropy according
to the type of steric interactions.
In the rst case the particle retains spherically symmetric
shape but possesses an internal anisotropy which leads to the
change of the magnetic interaction (see, for example, capped
colloids,1 shied dipolar particles,2–6 or magnetic Janus particles7–11). The change or anisotropy of the magnetic interparticle
interaction might be also induced by the presence of an external
magnetic (electric) eld.12,13
In the second case, the particle is characterised by shape
anisotropy, which leads to the orientation-dependent steric
interparticle interaction (see, for example, magnetic rods,14–17
ellipsoids,18–20 cubes21 and systems of dipolar hard22–25 and
so26–29 dumbbells).
a
Dipartimento di Fisica, Università di Roma La Sapienza, Piazzale Aldo Moro 2, 00185
Roma, Italy
b
Ural Federal University, Lenin Avenue 51, 620083, Ekaterinburg, Russia
c
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria.
E-mail: soa.kantorovich@univie.ac.at
6594 | Soft Matter, 2013, 9, 6594–6603
In the present study we focus on the second type of the
anisotropy; the magnetic part of the interaction will be characterised by simple magnetic dipole–dipole interaction (1), with
dipole mi being always in the ith particle centre of mass.
" #
mi $rij mj $rij
mi $mj
m
Ud ðijÞ ¼ 0 3
;
4p
rij 5
rij 3
(1)
rij ¼ rij ¼ ri rj ;
where rij is the displacement vector of the two particles and m0 is
the vacuum permeability. This interaction will keep its functional
form independent of the position of the dipole as long as the
displacement vector has the meaning of the “inter-dipolar”
distance. In the present manuscript, the dipole moment will always
be xed in the particle's centre of mass, and the shape anisotropy
would be modelled by several orientation-dependent steric potentials, the detailed description of which will be provided below.
At the end of the 20th century, the systems of dipolar particles
with shape anisotropy attracted a lot of attention from theoreticians due to the ability to form various liquid–crystalline phases,
such as nematic, antiferroelectric nematic, smectic, etc.,30–43 and
later revisited in other studies.44–46 The central place in these
investigations was taken by the systems of hard/so ellipsoids or
spherocylinders, with the point dipole moment coaligned with the
main axis (axis of revolution), even though systems with higher
order moments were also investigated.43 The particles with the
dipole moment oriented perpendicular to the axis of revolution
were not addressed that actively,47–52 and the main target of these
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investigations was to understand what kind of new crystalline
phases one can acquire by rotating a dipole. Also some more
complex positioning of the dipoles was studied.53 Additionally, it
was found that the system of hard ellipsoids or spherocylinders,
with the dipole moment coaligned with the main axis, could
undergo the vapour–liquid phase transition,54 in contrast to
dipolar hard-sphere systems, where this transition has never been
proven yet. Only a few investigations were done to elucidate an
isotropic gas phase of anisotropic dipolar particles.32,54,55 The main
ndings for the isotropic phase from the latter studies, such as
computer simulations, various perturbation approaches and
density functional theories, could be summarised as follows: (i) for
high elongation, the antiparallel orientation of the dipoles in the
neighbouring side-by-side particles becomes energetically more
advantageous than the head-to-tail one, inherent to the dipolar
interactions of spherical particles; (ii) most probably the latter
peculiarity of the pair correlation function (domination of the
antiparallel orientation) for highly elongated particles plays a
crucial part in the phase-transition scenario and inuences thermodynamic properties of these systems especially at densities
close to the critical ones. However, the details of the dependence of
thermodynamic and structural properties on the particle elongation at low densities were not thoroughly studied. The recent
“revival” of the subject of magnetic anisotropic particles was
caused by the experimental studies aimed at medical and microuidics applications.14,15 Besides that, numerous and versatile
experiments have been recently done with hematite ellipsoids, the
magnetic moments of which are aligned along the short axis.18,20 In
all these experiments, the systems were analysed at low densities
corresponding to the isotropic phase; the interparticle interactions
in some of these experiments are very strong (see the work reported
in ref. 18, for instance) and lead to aggregation. All this gives the
motivation to thoroughly analyse the inuence of particle shape
anisotropy on the properties of diluted and isotropic phases and
the ground states of such systems to be able to predict theoretically
possible structural transitions and cluster topologies. As a result,
several computer simulation studies appeared, showing the
peculiarities of the magnetic rods' systems in the presence of an
external magnetic eld,56 but no systematic cluster analysis at zero
elds, nor the dependence of the behaviour on the semiaxis ratio
for dipolar rods or ellipsoids has been presented so far.
It is worth mentioning that the orientation of the dipole
moment with respect to the ellipsoid axes would necessarily
inuence the micro-, and as a consequence, macro-properties of
the system. Thus, for example, more than 30 years ago, when the
rst birefringence experiments in ferrouids were done,57 one of
the possible explanations was exactly that of slight ellipsoidal
shape, inherent to the crystalline lattice of the ferroparticles, and in
this case the dipole moment was coaligned with the axis of revolution, in contrast to ellipsoids with silica shell described above.18
Thus, the main aim of this contribution is to (i) thoroughly
analyse the ground states (the most energetically favourable
congurations at 0 K) for the systems of dipolar particles with
variable shape anisotropy (see Section 2 for the two-dimensional case and Section 3 for the three-dimensional one):
ellipsoids and rods, with dipoles oriented perpendicular to
(Subsection 2.1) and along (Subsection 2.2) the main axis; (ii) to
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Soft Matter
verify whether the signature of the ground states is preserved at
nite temperature, when the energetic and entropic contributions to the microstructure start competing, for a simple
example of repulsive Gay–Berne-type58–60 dipolar particles.
2
Ground states in 2D
2.1
Ellipsoids with the dipole moment along the short axis
We start with ellipsoidal particles with semiaxes a and b as shown
in Fig. 1(a). Magnetic moments of the particles are pointing
along the short axis. For a pair of these particles the most probable conguration is a head-to-tail one, therefore the system
under study is similar to that of the magnetic dipolar spheres.
Earlier,61 it was shown that the ground state of such a system is a
chain (in case the total number of particles is less than 4) or an
ideal ring. The term “ideal” determines a ring based on a regular
polygon with the side d and all magnetic moments aligned
tangentially to the circle which an ideal ring is based on. In our
case we can easily generalise the total energy of the chain
(Fig. 1(c)), but for a ring (daisy), shown in Fig. 1(c), the straightforward generalisation of the results for dipolar hard spheres is
not possible due to the fact that the neighbouring ellipsoidal
particles should not overlap in the daisy, thus, the side of a
polygon carrying the centres of the main cross-sections (d(n,X0))
depends on the number of particles n and on the semiaxis ratio
X0 ¼ b/a. In order to nd d(n,X0) we solve the following system:
8
a cos t ¼ xc þ a cos t1 cos f b sin t1 sin f;
>
>
>
>
>
>
>
< b sin t ¼ yc þ a cos t1 sin f þ b sin t1 cos f;
b cos t a sin t1 sin f þ b cos t1 cos f
>
¼
;
>
>
a sin t a sin t1 cos f b cos t1 sin f
>
>
>
>
:
ða cos tÞ2 þ ðb sin tÞ2 ¼ ða cos t1 cos f b sin t1 sin fÞ2
þða cos t1 sin f þ b sin t1 cos fÞ2 :
(2)
Fig. 1 (a) Sketch of an ellipsoidal particle with magnetic moment m coaligned
with the short semiaxis a, the long semiaxis is called b; (b) sketch of a cylindrical
particle: the radius of the base a, and the height 2b; magnetic moment is pointing
along the main axis of the cylinder; parameter X0 defines the shape anisotropy; (c)
ground state candidates for ellipsoids with the dipole moment along the short axis:
a chain (left) and a daisy (right); (d) ground state candidates for cylindrical particles
with dipoles coaligned with the main axis: a carpet (left) and a bracelet (right).
Soft Matter, 2013, 9, 6594–6603 | 6595
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p m0
m2
nsin3
3
n
4p d ðn; X0 Þ
3
2
pk
cos2
þ1
½ðn1Þ=2
X
modðn
þ
1;
2Þ
n
7
6
þ
4
5:
2
3 pk
k¼1
sin
n
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ER ðn; X0 Þ ¼ Fig. 2 (a) Two touching ellipsoids in a daisy with notations used for calculating
d(n,X0); (b) the dependence of d(n,X0) on n for two different values of X0: pink line
is for X0 ¼ 2, blue line is plotted for X0 ¼ 1.1. Both curves converge to d ¼ 20
which is the diameter of the ellipsoid.
Here, the rst two equations enforce the condition that the
touching point M belongs to both ellipsoids and correspond to
the touching point of their main cross-sections (ellipses) parameterised with t(t1) ¼ 0, ., 2p (Fig. 2(a)); the fact that the
tangent lines to both ellipsoids at the point M must be equal is
reected by the third equation; and the last equation tells that
the distances from M to the centres of both ellipses (the main
cross-section of the ellipsoidal particle) are equal. The solution
of this system for two different values of X0 is plotted in Fig. 2(b).
The centre of the second ellipsoid is in xc, yc and f is the angle
between the main axis of the ellipsoids (see Fig. 2(a)). Then,
using xc, yc, we can calculate the distance between ellipsoids'
centres.
Having the interparticle distances at hand, we can write the
energy of a ring and a chain made out of n ellipsoids trapped in
the plane. The energy of the ring is:
(3)
Here, square brackets stand for the integer part of the ratio in
brackets and the function mod(n + 1, 2) denotes the residue of
the division. The energy of a chain made of n particles has a
simpler form:
ECH ðn; aÞ ¼ m0 m2 ð3Þ
nHn Hnð2Þ :
3
4p 4a
(4)
ECH depends on n and the length of the short semiaxis a only.
ð$Þ
The value Hn is the generalised harmonic number. Now that
we know how to calculate the total energy of the chain and the
ring of n ellipsoids analytically, we can compare the results and
nd the genuine ground state of the system of elongated
magnetic ellipsoids in 2D.
In Fig. 3(a)–(f) the energy per particle in chains and rings for
six different semiaxis ratios X0 as a function of number of
particles n is plotted. This set of gures shows that the ring
eventually becomes the ground state similar to the system of
magnetic spheres, but the number of particles at which the
transition occurs depends on shape anisotropy. For dipolar
spheres this critical number, meaning the number of particles
when the ground state structural transition takes place, is four,
and, for example, for semiaxis ratio equals to X0 ¼ 1.5 (Fig. 3(c))
the ring becomes the ground state for the number of particles
more than 17. And if we investigate the system of more and
Fig. 3 Energy per particle, calculated for magnetite (with a ¼ 10 nm), of the daisy (ring) and a chain versus the number of particles in a cluster n. The absolute
values along the ordinate axes were obtained using the value of material saturation magnetisation 480 kA m1 (as for magnetite) and were multiplied by 1019. (a)
X0 ¼ 1.1, the daisy becomes a ground state at n ¼ 6; (b) X0 ¼ 1.3, the daisy becomes a ground state at n ¼ 11; (c) X0 ¼ 1.5, the daisy becomes a ground state at
n ¼ 17; (d) X0 ¼ 2, the daisy becomes a ground state at n ¼ 36; (e) X0 ¼ 2.5, the daisy becomes a ground state at n ¼ 61; (f) X0 ¼ 3, the daisy becomes a ground state
at n ¼ 90. In all figures the blue solid line is the energy of a daisy and violet dashed one is the energy of a chain. Dash-dotted line serves as a symbolic border
between the chain and daisy domination. Note, that the volume of the particle (i.e., the value of the magnetic moment) changes depending on X0. Log scale is used
along the abscissa axes.
6596 | Soft Matter, 2013, 9, 6594–6603
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Fig. 4 Critical number of particles needed for a ring to become a ground state as a
function of X0. Integer values are marked with symbols, the line is a guide for the eye.
more elongated ellipsoids, this number grows dramatically (see,
for example, Fig. 3(f)).
The dependence of the critical number of particles n*(X0)
needed for a ring to become a ground state is shown in Fig. 4.
With increase of the shape anisotropy the critical number of
particles goes to innity.
This plot shows that, if the area available for the ellipsoids is
innite and, X0 / N the transition from a chain of ellipsoids to
a daisy would shi towards larger and larger values of n, and
would disappear at innity. Besides that, the difference between
the energy per particle in a daisy and in a chain vanishes, when
n / N, thus making the ground state almost degenerate.
2.2 Ellipsoids and cylinders with the dipole moment along
the long axis
Since due to the symmetry reasons, the ground states of cylinders and ellipsoids with dipole moment coaligned with the long
axis are qualitatively the same, below we present theoretical
calculations for cylinders. The sketch of such a cylinder is
provided in Fig. 1(b).
As a result of the existence of two minima in the magnetic
dipole–dipole interaction (1), there are two possible congurations for the ground state of a pair of cylinders: the pair with
antiparallel orientation and the one with a head-to-tail orientation of the magnetic moments. The transition between these
two ground state structures will depend on the shape
anisotropy.
In Fig. 5 we present the energy of the two congurations as a
function of anisotropy parameter X0. We see that below the
critical value of X*0 ¼ 21/3 the head-to-tail orientation becomes
the ground state conguration. If we increase the value of X0,
then the antiparallel conguration becomes the ground state.
So, for two rods with a rather small elongation the head-to-tail
minimum of the dipolar interaction becomes more shallow
than the one corresponding to the antiparallel orientation.
Next, we investigated the ground state structure for the system
of n cylinders. Before the critical value of X0 (if cylinders are
short) the system of magnetic cylinders is similar to that of
spheres, therefore, the ground state of the system is a chain or a
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Soft Matter
Fig. 5 The energy of a pair of dipoles as a function of X0 calculated for
magnetite particles (with 2a ¼ 10 nm). The absolute values along the ordinate
axes were obtained using the value of material saturation magnetisation 480 kA
m1 (as for magnetite) and were multiplied by 1019. Blue solid line is the energy of
a head-to-tail configuration, and the violet dashed one is the energy of an antiparallel orientation of magnetic moments. If X0 > 21/3 the antiparallel pair
becomes energetically more advantageous, than the head-to-tail one.
ring, and their energies could be found using the analogous
formulae for spherical particles.61 Note, that in this case, the
number of particles needed for the transition from a chain to a
ring also grows, but remains nite when X0 / X*0. For elongated
cylinders the ground state is the carpet. When the number of
particles goes to innity, the carpet would not tend to close
itself to a daisy, and one might only speculate that when n / N
the ground state degenerates, and the daisy becomes energetically equivalent to the carpet.
As a conclusion of the 2D investigation, we underline that
the ground state of the system of non-spherical particles in two
dimensions strongly depends on the shape anisotropy. Even a
small deviation from the sphere leads to a drastic change of the
ground state congurations. The next question is: does this
inuence remain in 3D?
3
Ground states in 3D
Firstly, it is worth mentioning that if the dipole moment is
coaligned with the short axis, the systems of both ellipsoids and
cylinders are equivalent to that of the spherical dipolar particles,
thus the ground states will be a chain or a ring. So, in this section
we focus our attention on the investigation of the ground states
for the system of elongated magnetic cylinders and ellipsoids
with dipole moments aligned along the long axis. Besides that, in
the limit of small anisotropy (X0 is smaller than the X*0), both
ellipsoids and cylinders would form chains in three dimensions,
which would close themselves in ideal rings, once the necessary
combination of n and X0 is reached (see the previous section).
Finally, when talking about highly diluted systems, the behaviour of strongly elongated ellipsoids and magnetic cylinders
would not show any qualitative difference in the topology of the
ground states, and only critical values might differ slightly. That
is why, below, we present the results for magnetic cylinders only,
and assume that the elongation of them is large enough for the
antiparallel orientation of dipoles in side-by-side conguration
to be more advantageous than the head-to-tail one.
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Under these assumptions, we pinpoint two probable
congurations of particles as ground state candidates: a
“carpet” and a “bracelet”, presented in Fig. 1(d). The dipole
moments of any two neighbouring particles in the carpet and in
the bracelet have antiparallel orientation.
The total energy of the carpet can be written as:
n1 X
n
cos pð j 1Þ
m0 m2 X
carpet
Udd ðn; X0 Þ ¼
:
(5)
4p 8a3 i¼1 j¼iþ1
ð j iÞ3
The energy of a bracelet will have the following form:
n1 X
n
cos pð j 1Þ
m m2 X
brac
ðn; X0 Þ ¼ 0 3
sin3 ðp=nÞ:
Udd
4p 8a i¼1 j¼iþ1 sin3 pð j iÞn
(6)
For a large number of particles in the system (n / N) we can
calculate the asymptotes for the two latter energies (eqn (5) and (6)):
m m2 3zð3Þ
1
carpet
1 ;
Udd
ðn/NÞ ¼ 0 3
n
4p 8a 4
(7)
2
m m 3zð3Þ
brac
:
Udd
ðn/NÞ ¼ 0 3
4p 8a 4
Here, z(3) is the z-Riemann function of three. In Fig. 6(a) and
(b), we present the energy per particle in a carpet and a bracelet as
a function of the number of particles. The energy asymptotes are
also plotted here. In Fig. 6(a), one can see the results for X0 ¼ 1.4,
which corresponds to the system of moderately elongated
cylinders. In Fig. 6(b), we plot the energy per particle for highly
elongated cylinders (X0 ¼ 10). The saw-like behaviour of the
energy per particle in a bracelet follows from the fact that if the
number of particles in a carpet was odd and one tried to close it to
form a bracelet, the magnetic moments of the rst and the last
particles would have the same orientation and it corresponds to
the strong repulsion. In other words, the carpet becomes an
energetically favourable conguration for odd number of particles, and the bracelet is the ground state for even number of
particles. From the asymptotes one can see that the difference
between the energies is decreasing but relatively slow (as 1/n).
In order to summarise the ground state structure investigation we provide a cartoon (Fig. 7), containing all possible ground
state structures depending on the dimensionality of the system,
direction of the dipole moment, number of particles and particle
anisotropy. The new topologies found here and in the previous
Fig. 7 Ground state structures found in this manuscript for particles with various
anisotropy in both two and three dimensions.
section are very different from those of spherically symmetrical
dipolar particles. We expect this to inuence the thermodynamic
properties of the systems of dipolar particles with shape
anisotropy. In the next section we compute radial distribution
functions, pressure and initial susceptibility for the systems of
anisotropic particles to elucidate the inuence of the anisotropy.
4 Finite temperature: what remains from
the ground states?
Here, we present the results of molecular dynamics simulations
performed in ESPResSo.62 We used metallic periodic boundary
conditions in all three directions and simulated NVT ensemble
with N ¼ 512 particles initially randomly placed in the box. We
used the dimensionless volume of the simulation box,
measured in the units of a3 (particle's short semiaxis) 2a ¼ 1.
Particles in simulations were interacting via magnetic dipole–
dipole interaction (1) and modied Gay–Berne potential:58–60
43ð$Þ½A12 ð$Þ A6 ð$Þ þ 3ð$Þ; rij # rc
UGB ui ; uj ; rij ¼
(8)
0;
rij . rc ;
where
.
Að$ÞhA ui ; uj ; r^ij ¼ s0 rij s ui ; uj ; r^ij þ s0 ;
"
s ui ; uj ; r^ij ¼ s0 1 þ
Fig. 6 Energy per particle, calculated for magnetite (with 2a ¼ 10 nm), in a
bracelet (violet solid line) and in a carpet (blue solid line) versus n. (a) X0 ¼ 1.4 and
(b) X0 ¼ 10. Dashed lines are the asymptotes provided by eqn (7). The absolute
values along the ordinate axes were obtained using the value of material saturation magnetisation 480 kA m1 (as for magnetite) and were multiplied by 1019.
6598 | Soft Matter, 2013, 9, 6594–6603
cðX0 Þ
2
(
rij $uj
r^ij $ui þ ^
2
1 þ cðX0 Þui $uj
2 )# 1
2
r^ij $ui r^ij $uj
1 cðX0 Þui $uj
;
h
2 i 12
3ð$Þh3 ui ; uj ¼ 30 1 c2 ðX0 Þ ui $uj
:
Here, ^r ¼ rij/rij ¼ (ri rj)/rij is the unit vector, coaligned with
the vector connecting particle centres, ui(j) is the unit vector
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pffiffiffi
along the b-axis of the particle, s0 ¼ 2 2a, 30 denotes the energy
parameter, and c(X0) ¼ [X02 1]/[X02 + 1]. The critical value rc
equals to (21/6 1)s0 + s(ui,uj,^r ij). It is important to underline
that this potential is not central and depends on mutual
orientation of particles through, for example, the effective
interparticle distance s($), which depends on the particle
orientations. Gay–Berne potential58–60 is one of the widely used
steric short range potentials. The original version of this
potential has the functional form of the Lennard-Jones potential.63 Here, we needed only the repulsive part of the potential,
therefore, the expression in eqn (8) is nothing but a Weeks–
Chandler–Andersen64 modication of the Gay–Berne potential.
The reduced temperature in simulations is T * ¼ kT/30 ¼ 1,
where kT is the thermal energy, and the dimensionless
dipole
pffiffiffiffiffiffi
moment m is measured in the units of 1= T*. Here, we used
the following values of dimensionless magnetic moment: m ¼ 1;
m ¼ 1.414; m ¼ 1.731; m ¼ 2, these values correspond to the
values of the dipolar coupling of l ¼ m2 ¼ 1; 2; 3; 4. Even
though, it might seem to be a narrow window for l, for the
isotropic particles, this covers the range from weakly interacting
systems to strongly chain-forming ones. In order to analyse the
inuence of the volume fraction we used f ¼ 0.01; 0.05; 0.1 to
remain in the range, where the cluster formation in the
isotropic particle systems is still well dened. The anisotropy
ratios were chosen to be X0 ¼ 1.1; 1.5; 1.7; 2.
In order to visualise the interaction potential in a clear way,
one can, for example, x particle orientations and look at the
distance dependence only. In Fig. 8 the total interaction
potential as a function of distance between two particles is
plotted for three congurations, two of which, namely the headto-tail (r ¼ 2b) and the antiparallel one (r ¼ 2a) were observed in
the ground states. The third one (T-conguration) was chosen
due to its entropic advantage,65 while the dipole–dipole interaction in this case is zero. It can be clearly seen that the interaction potential in the latter congurations behaves differently,
depending on X0. In the case of X0 ¼ 1.1, the head-to-tail
conguration wins (the right-most curves), and with growing
value of the magnetic moment the depth of the potential well,
corresponding to this conguration, increases signicantly. For
the antiparallel conguration, the depth of the well also
increases with the growing dipole moment, but never exceeds
that of the head-to-tail conguration. In Fig. 8(b) (X0 ¼ 2) the
Fig. 8 The total interaction potential for three different mutual orientations of
dipoles. Pink lines correspond to head-to-tail configuration, blue ones correspond
to T-configuration and brown lines are antiparallel configuration. Solid lines
describe the interaction potential for the particles with m ¼ 1; dashed lines
correspond to the case of m ¼ 1.732. (a) Results for X0 ¼ 1.1 and (b) for semiaxis
ratio X0 ¼ 2.
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Soft Matter
antiparallel conguration (le-most curves) becomes more
favourable, however the depth of the potential well for this
conguration is not that large, as in the case of lower semiaxis
ratio; for both head-to-tail and antiparallel congurations in
this case the depth of the well decreases only slightly, when the
dipole moment grows. In other words, taking into account the
soness of the potential (8), the points of minima observed here
completely correspond to the ground states for the system of
two cylinders (see Fig. 5).
We start the analysis with structural properties of the
systems. In Fig. 9 the radial distribution functions are plotted
for various values of X0. It can clearly be seen that for X0 ¼ 1.1
(squares) with an increasing value of the magnetic moment, the
rst maximum of the RDF slightly shis to the region of small
distance, increases and becomes more narrow. It is related to
the fact that more head-to-tail pairs form. However, with
growing X0 this effect vanishes, meaning that the head-to-tail
orientation is either replaced by antiparallel orientation (for
large m) or the system becomes less-and-less correlated. With
increasing f for m ¼ 1 the rst peak of the RDF becomes higher
for every value of X0, whereas for m ¼ 2, the rst peaks of the
RDF become lower independent of X0. The reason for this
behaviour is as follows: when the magnetic moment is small,
the rst peak of the RDF represents an almost purely entropic
probability of nding two particles at a close distance as the
dipole–dipole interaction in this case plays minor part. So, with
the increasing density, the probability increases (compare
Fig. 9(a) and (c)). However, if the magnetic dipole–dipole
interaction is strong in the system, for small X0 it ushers in the
chain formation, which leads to the redistribution of particles
between the coordination spheres, and one can observe the
decrease of the rst peak and the appearance of the second peak
and the rst minima of the RDF. As for X0 ¼ 2, the rst peak of
the RDF almost does not change with density, just the initial
shoulder becomes slightly more pronounced. Besides that, for
high anisotropy, the height of the RDF rst peak is much lower
Fig. 9 Radial distribution functions obtained in simulations. Different symbols
correspond to the different values of the semiaxis ratio: pink squares correspond to
X0 ¼ 1.1; blue rhombuses represent data for X0 ¼ 1.5 and brown circles are
obtained for X0 ¼ 2. Concentration of the particles equals to one per cent (f ¼ 0.01)
in (a) and (b) and to ten per cent (f ¼ 0.1) in (c) and (d). Dipole moment is m ¼ 1 in
(a) and (c) and m ¼ 2 in (b) and (d). Lines here are just guides for the eye.
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Soft Matter
(circles and rhombuses), than the one for almost isotropic
particles (squares). It means that the pair formation is strongly
suppressed in the anisotropic particles' system. Another
important observation might be done comparing the initial part
of the RDF r ˛ [1, ., 1.3] for the highest anisotropy (circles) in
Fig. 9(c) and (d): with growing magnetic dipole–dipole interaction there is an emerging shoulder in the RDF, which might be a
signature of the antiparallel pair formation.
In order to check the latter hypothesis, we performed a
cluster analysis of the data. We considered two particles to
belong to a cluster of an arbitrary topology, if the distance was
smaller than a certain value (rc), and the dipole–dipole interaction between these two particles was negative. This approach
was shown to be robust for systems with various cluster
topologies.24,25,66,67 In Fig. 10 we present simulation snapshots
for two different values of rc and X0 for the same density f ¼ 0.1
and m ¼ 2. In order to make snapshots less crowded we show
only clusters and, additionally, we represent particles by cylinders and not by Gaussian ellipsoids in order to make the
orientations of the dipoles clearer. One can see that for high
anisotropy the number of particles in clusters is much lower
(compare, Fig. 10(a) and (b) or (c) and (d)). The dominant
structures on the le are chains, whereas on the right are
mainly antiparallel pairs. Increasing the value of rc leads to the
higher amount of structures and increases their size. Thus,
longer chains can be seen in Fig. 10(c), than in Fig. 10(a);
signicantly more pairs might be found in Fig. 10(d). Note, that
on the one hand the densities here are low enough, for the
systems to be in a gas phase, and on the other hand, the values
of dipole moment are not high enough for the systems to
exhibit any percolation even for small X0.
In the next gure (Fig. 11), we analyse the particle-related part
of the pressure as a function of the dipole moment m for different
values of the anisotropy parameter X0. For magnetic uids one
Fig. 10 Simulation snapshots of particle clusters, obtained via an energy criterion for f ¼ 0.1 and m ¼ 2. Arrows depict the particle dipole moments, cylinder
shape is used for the clarity of visualisation. In (a) and (b) the value of rc ¼ 1.1; in (c)
and (d) rc ¼ 1.2. The anisotropy parameter is X0 ¼ 1.1 in (a) and (c) and X0 ¼ 2 in
(b) and (d). One can see chains on the left and antiparallel pairs on the right. The
amount of clusters and their size increase from the top to the bottom. The cluster
topology does not depend on rc.
6600 | Soft Matter, 2013, 9, 6594–6603
Emerging Area
Fig. 11 Internal pressure normalised by the value of the pressure for m ¼ 1 as a
function of the dimensionless dipole moment m. (a) f ¼ 0.01 and (b) f ¼ 0.1.
Brown circles are for X0 ¼ 2; blue rhombuses are for X0 ¼ 1.5; pink squares
describe the data for X0 ¼ 1.1. Lines here are just guides for the eye.
could determine the osmotic pressure related to the colloidal
particles removing the contribution from the carrier liquid.
Here, we consider both contributions to the osmotic pressure,
namely the parts related to the kinetic and the potential energy of
the anisotropic particles. It is known that for isotropic dipolar
particles, with growing strength of dipolar interactions, one
observes a decrease of the particle-related part of pressure.68,69
Thus, aer observing the decrease of the RDF rst peak height
with growing anisotropy, we assume this to affect also the pressure. In fact, in Fig. 11 we observe a strong decrease of pressure
for X0 ¼ 1.1 (up to 50 per cent, see squares) and, only a change on
the order of 10 per cent for X0 ¼ 2 (circles). This result conrms
the hypothesis that the magnetic correlations in the system
become signicantly weaker, once the particle's anisotropy
becomes large enough and agrees well with the ndings of earlier
work.30,32,33,55 Here, however, we can easily quantify the effect, and
pinpoint the values of X0, for which the dipolar correlations start
decaying, being still in the gas phase.
As the next step, we calculate the initial magnetic susceptibility. Initial magnetic susceptibility (cin) is one of the most
important characteristics of magnetic so materials, and is
equal to the slope of the magnetisation curve at zero eld. It
shows how strongly the dipole moments in the system are
correlated and how responsive the system is to the application
of an external innitesimal magnetic eld. In simulations, one
usually employs the uctuation–dissipation theorem to calculate cin:
1
(9)
cin ¼
M 2 hMi2 ;
3V m0 T*
where M is the dimensionless total dipole moment of the system,
and the averaging h.i is done over all statistically independent
congurations, and V is a dimensionless volume of the system
measured in the units of a. The results of eqn (9) are plotted in
Fig. 12. It is well known that in systems of isotropic magnetic
particles, the formation of chains and the effective eld, arising
from the dipolar interactions, lead to a rapid growth of the initial
magnetic susceptibility.67,70–72 One sees a similar behaviour also
for slightly anisotropic particles (X0 ¼ 1.1, le most points in
both plots). However, once the anisotropy starts growing, not
only the absolute value of the susceptibility decreases, but also
the relative change of it becomes smaller. The distance between
the right most points, meaning the values of the initial
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Fig. 12 Initial susceptibility as a function of X0. (a) f ¼ 0.01 and (b) f ¼ 0.1.
Violet triangles correspond to m ¼ 2; pink circles are plotted for m ¼ 1.732; blue
rhombuses are for m ¼ 1.414; brown squares describe the susceptibility for m ¼ 1.
Lines here are just guides for the eye.
susceptibility for different dipolar strengths for X0 ¼ 2, becomes
approximately twice as small as that for X0 ¼ 1.1 (the le-most
points), which agrees with the results of the dielectric
constant,33,55 unfortunately not much statistics was available at
that time, and only qualitative predictions were provided. Here,
in contrast, we can easily quantify the inuence of the particle
anisotropy on the initial susceptibility of the particle systems.
The overall decrease of interparticle dipolar correlations/
magnetic response observed in this work might be explained by
the change of the ground state congurations. Once, the antiparallel pair starts being energetically more advantageous, than
the head-to-tail orientation, the two main consequences are the
following: (i) at room temperature, the entropy of the antiparallel
pair is very low, and thus, no stable pairs are formed even if the
dipole moment is large; (ii) the total dipole moment of the
antiparallel pairs is a vanishing quantity, and the susceptibility
of such an object is much lower than that of the head-to-tail
dimer. In other words, the particles' shape anisotropy can
signicantly change the microstructure of the system of
magnetic particles at nite temperature.
5
Conclusion
We analysed in detail the microstructure of the systems of
ellipsoids/rods with the central magnetic dipole at low densities, where these systems are isotropic.
As the rst step, the ground state structures for dipolar
ellipsoids and rods in two and three dimensions were found
analytically. For ellipsoids with the dipole moment along the
short axis we showed, both for two and three dimensions, that a
chain or a “daisy” (a ring of ellipsoids) are the ground state
topologies. The number of particles for which the transition
between a chain and a daisy occurs depends not only on the
number of particles in the structure (as for the system of
magnetic spheres), but also on the particle shape anisotropy.
The more elongated the ellipsoids are, the more particles are
needed for a daisy to become the ground state. For ellipsoids or
cylinders, whose point dipole is coaligned with the long axis, in
two dimensional case, the ground state might be a chain or a
ring if the elongation of the particles is small, or a carpet made
of side-by-side ellipsoids (cylinders) with neighbouring dipoles
oriented antiparallelly. In 3D, similar to 2D, the ground state of
strongly elongated particles with dipole moment along the
This journal is ª The Royal Society of Chemistry 2013
Soft Matter
main axis is either a carpet or a bracelet of side-by-side particles
with antiparallel orientation of dipoles.
The antiparallel pair of dipoles in the ground state has a zero
total dipole moment. For nite temperature, when the interparticle interaction energy starts being comparable to kBT, in
the case of the formation of the antiparallel pair, its dipole
moment will be still a vanishing quantity. In order to analyse
the inuence of the anisotropy at nite temperature, we investigated in detail the microscopic structure of the system of
“Gay–Berne-type” ellipsoids with the dipole moment coaligned
with the main axis of the particle using molecular dynamics
simulations. We found that with growing elongation the dipolar
interparticle correlations and their inuence on the system
microstructure become weaker. We analysed the radial distribution functions, internal pressure and initial susceptibility at
nite temperature for various densities, dipolar strengths and
semiaxis ratios. We discovered that for a broad range of dipolar
strength and densities, the cluster formation becomes less
pronounced if the semiaxis ratio grows, the internal pressure
increases and the initial susceptibility decreases.
Finally, using the particle anisotropy as a control parameter
one can change the system from the one full of chains (nearly
spherical particles) to a basically spatially homogeneous system
(for elongated particles), without changing the value of the
saturation magnetisation. This might be very important in
various medical and industrial applications, where the strong
magnetic response of the particles should be combined with the
absence of strong cluster formation. For that, we are planning to
investigate in detail magnetisation, viscosity and diffusion in
systems of anisotropic dipolar particles under the inuence of
an external magnetic eld.
Acknowledgements
The research has been nancially supported by FP7-IDEAS-ERC
Grant PATCHYCOLLOIDS; S.K. and E.P. are grateful to RFBR
grants mol-a 1202-31-374 and mol-a-ved 12-02-33106 and have
been supported by Ministery of Science and Education of RF
2.609.2011. S.K. was supported by Austrian Science Fund (FWF):
START-Project Y 627-N27. The authors are grateful to Dr M. Sega
for fruitful discussions. The research was carried out in terms of
Ural Federal University development program with the nancial
support of young scientists.
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