7276
Langmuir 2005, 21, 7276-7279
Self-Assembly of Colloidal Pyramids in Magnetic Fields
L. E. Helseth
School of Physical and Mathematical Sciences, Division of Physics and Applied Physics,
Nanyang Technological University, Singapore
Received April 28, 2005. In Final Form: May 30, 2005
We study routes toward the construction of 2D colloidal pyramids. We find that magnetic beads may
self-assemble into pyramids near a nonmagnetic 1D boundary as long as the number of beads in the
pyramid does not exceed 10. We have also found that a strong magnetic field gradient could act as a
boundary, thus assisting the self-assembly of magnetic colloids in water, and have observed the formation
of stable microscopic pyramids within a certain magnetic field range. Our results indicate that colloidal
pyramids can be formed in a number of ways by utilizing external fields.
1. Introduction
During the last 4500 years, the pyramids of Egypt and
Mexico have drawn considerable admiration and interest,
ranging from grave robbery to religious worship. The
largest pyramid, built for Pharaoh Khufu around 2530
B.C., was until the early twentieth century the largest
building on our planet. The question of how these
constructions were built has not yet been answered
properly and therefore continues to fascinate mankind.
Our fascination with pyramids also extends to the
microscopic scale as our building skills improve. The
assembly of colloidal structure in one, two, and three
dimensions has been a major task in materials science
the last few decades.1-15 A variety of colloidal structures
can now be routinely reproduced using various selfassembly or lithographic techniques. Colloidal suspensions
show particular promise because it has been demonstrated
that such suspensions may self-assemble into nanostructures that sometimes mimic biological morphologies.2-10
The self-assembled structures exhibit size-dependent
phase separation2 and show novel responses in external
fluid flow and magnetic or electric fields.1-10
Recently, researchers have been interested in fabricating composites using colloidal particles as building blocks.
To this end, pyramidal structures have attracted attention
(1) Pieranski, P. Contemp. Phys. 1983, 24, 25.
(2) Sommer, A. P.; Ben-Moshe, M.; Magdassi, S. J. Phys. Chem. B
2004, 108, 8.
(3) Sommer, A. P.; Franke, R. P. Nano Lett. 2003, 3, 573.
(4) Su, G.; Guo, Q.; Palmer, R. E. Langmuir 2003, 19, 9669.
(5) Masuda, Y.; Itoh, T.; Itoh, M.; Koumoto, K. Langmuir 2004, 20,
5588.
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18, 4155.
(7) Cong, H.; Cao, W. Langmuir 2003, 19, 8177.
(8) Fudouzi, H.; Xia, Y. Langmuir 2003 19, 9653.
(9) Abe, M.; Orita, M.; Yamazaki, H.; Tsukamoto, S.; Teshima, Y.;
Sakai, T.; Ohkubo, T.; Momozawa, N.; Sakai, H. Langmuir 2004, 20,
5046.
(10) Guo, Q.; Arnoux, C.; Palmer, R. E. Langmuir 2001, 17, 7150.
(11) Helseth, L. E.; Wen, H. Z.; Hansen, R. W.; Johansen, T. H.;
Heinig, P.; Fischer, T. M. Langmuir 2004, 20, 7323.
(12) Wen, H. Z.; Helseth, L. E.; Fischer, T. M. J. Phys. Chem. B 2004,
108, 16261.
(13) Shevchenko, E. V.; Talapin, D. V.; Rogach, A. L.; Kornowski, A.;
Haase, M.; Weller, H. J. Am. Chem. Soc. 2002, 124, 11480.
(14) Yin, Y.; Lu, Y.; Gates, B.; Xia, Y. J. Am. Chem. Soc. 2001, 123,
8718.
(15) Yin, Y.; Li, Z. Y.; Xia, Y. Langmuir 2003, 19, 622.
because of their important role in technologies relying on
sharp structures (e.g., force microscopy). The controlled
construction of colloidal microscopic pyramids has until
now relied on prepatterned substrates or the self-assembly
of nanocrystals.13-15 Such techniques have proven very
useful. Still, we would like to gain more knowledge about
colloidal self-assembly in external fields and possibly be
able to provide templates were the colloidal structures
can be controlled in situ. Our previous studies showed
how strong magnetic field gradients can be used to grow
colloidal crystals or to change the conformation of dipolar
chains.11,12 Here we explore theoretically how magnetic
fields could assist the assembly of pyramidal shapes and
offer experimental demonstrations as proof of this principle.
2. Colloidal Pyramids in a Homogeneous Field
A 2D colloidal pyramid cannot be in a stable equilibrium
configuration in a homogeneous magnetic field without
any boundaries. To this end, consider N identical dipoles
assembled into a 2D pyramid, with their magnetic
moments aligned in the x direction. The colloids considered
here are modeled as dipoles of radius a with dipole moment
mi, and the dipolar interaction energy between two such
dipoles is given by
Eij )
[
]
µ0 3(mi‚ri)(mj‚ri) mi‚mj
4π
ri5
ri3
(1)
where µ0 is the permeability of the surrounding medium
and ri is the distance between the colloids. To obtain the
total energy, one must sum over all such pair potentials.
We will assume that the magnetic moment of the beads
is influenced only by the external magnetic field and that
the field from the other particles does not play a role.
That is, mi ) (4π/3)χa3H, where χ is the susceptibility of
the beads and H ) Hxex is the external magnetic field in
the x direction. This is a good approximation because the
dipole moment of the paramagnetic beads considered here
is rather small and the external magnetic field is relatively
weak. The magnetic field required to saturate the magnetic
moment of the beads is on the order of 100 kA/m, and we
operate here with fields that are orders of magnitude
smaller than this.
10.1021/la051140v CCC: $30.25 © 2005 American Chemical Society
Published on Web 07/08/2005
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Langmuir, Vol. 21, No. 16, 2005 7277
Figure 2. Schematic drawing of the experimental setup where
the contact line of a water droplet is used to assemble the
pyramids.
Figure 1. Two possible configurations of paramagnetic spheres
in an external magnetic field: (a) pyramid and (b) linear chain.
Both configurations have a base consisting of NB beads.
In summing eq 1, it can be shown that the total dipolar
energy associated with the 2D pyramid seen in Figure 1a
is given by
3µ0m2
Ep ) (N - NB)
64πa3
(2)
Here NB is the number of beads in the base of the pyramid:
NB
N)
i)
∑
i)1
NB(NB + 1)
(3)
2
However, if the beads assemble into a pearl chain then
the total energy is given by
El ) -
µ0m2
(N - 1)
16πa3
(4)
It is seen that El < Ep under all circumstances, which
means that pyramid formation is not favorable if the
elements interact only via dipolar forces. One must
therefore conclude that pyramids are prohibited from selfassembling in the absence of boundaries when all of the
magnetic dipole moments are identical.
To overcome this barrier, let us now assume that the
base (x ) 0) of the pyramid or chain consisting of NB dipoles
is fixed and cannot move (Figure 1a). The rest of the beads
(N - NB) may then assemble into a pyramid or a pearl
chain aligned along the y axis. The energy of the pyramid
of Figure 1a is still given by eq 2, whereas the dipolar
energy of the structure of Figure 1b is found to be
El ) -
µ0m2
3
N- N
(
2
16πa
3
B
+
1
2
)
(5)
It can therefore be seen that pyramid formation is
favorable as long as NB < 4. When NB ) 4 (N ) 10), we
find that Ep ) El, thus suggesting that ensembles fulfilling
this requirement may be either chain-like or pyramidallike. For ensembles with NB > 4 (N > 10), we expect the
linear chains to form as long as the dipole energy is larger
than the thermal energy.
Figure 3. Formation of 2D colloidal pyramids in a drop of
water. The gray lines on the left in a and b are the contact lines
of the corresponding droplets. Formation of various shapes in
a magnetic field H ) 800 A/m. See the text for details.
To test this theory experimentally, we investigated the
behavior of magnetic beads in a drop of water. The
paramagnetic beads used here had a radius of a ) 1.4 µm
and a susceptibility of χ ) 0.17 and were manufactured
by Dynal (Dynabeads M270 coated with a carboxylic acid
group). The beads were immersed in deionized, ultrapure
water at a density of 107 beads/mL and deposited in
droplets of 10 µL volume. We visualized the system with
a Leica polarization microscope equipped with a halogen
light source and a Hamamatsu CCD camera. The temperature during the experiments was about T ) 300 K.
A schematic drawing of the experimental setup is shown
in Figure 2.
After some initial random motion during drop settlement, the beads eventually drifted to the contact line
because of weak evaporative fluid flow; see also refs 2 and
3. Thus, in our system the contact line functions as a 1D
boundary to which the beads adhere. After a certain
number of beads have reached the contact line, we turn
on a magnetic field of 800 A/m (10 G) along the x axis, thus
aligning the magnetic moments. The results of two such
experiments are shown in Figure 3a and b. In Figure 3a,
we can see a single standing pyramid as well as a pyramid
with a “tail” (i.e., a linear chain attached to its apex). A
physical explanation for the tail has not yet been revealed,
although it is likely that it is related to the way the colloids
were assembled. In the center of the droplet, only isolated
linear chains were observed (no pyramids), and these were
driven to the contact line by a radial flow gradient. The
way that these chains conform when they enter the contact
line may influence pyramid formation but is outside the
scope of the current study.
Experimentally, the method suggested in this section
is not optimal for assembling single pyramids because of
the fact that only limited sizes can be assembled. In fact,
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Langmuir, Vol. 21, No. 16, 2005
Helseth
throughout numerous experiments we never found larger
pyramids than that of Figure 3b, in agreement with our
simple theory. It is therefore necessary to look for new
solutions, and this is the topic of the next section.
3. Colloidal Pyramids in an Inhomogeneous
Field
Above we considered a system where all of the beads
had identical magnetic moments. Then we could use the
contact line of a droplet as a 1D boundary to assemble
colloidal pyramids. However, if a spatial distribution of
magnetic moments can be generated, then one may hope
to generate pyramids in a more efficient manner. Previously we showed that 1D domain walls can assemble a
variety of colloidal structures.8,9 Here we suggest that such
a 1D structure may indeed function as a virtual boundary
in addition to providing a magnetic field gradient. The
magnetic field from a domain wall acting on a paramagnetic sphere resting on the magnetic film a distance r )
xex + aez from the domain wall can be approximated by
HDW )
M sw r
2π r2
(6)
where Ms is the magnetization of the magnetic film
containing the wall and w is the width of the domain wall.
The force from the domain wall on a single magnetic bead
is found by noting that the interaction can be associated
with an energy E ) -µ0m‚HDW, where µ0 is the permeability of water and m is the magnetic moment of the
bead. Moreover, we assume that only the external
magnetic field H ) Hx aligns the magnetic moments of
the beads, such that m ) (4π/3)a3χHx. Again there are two
possible generic structures that can form near the domain
wall. We first assume that the beads are attracted to the
domain wall forming a pearl chain, and the interaction
energy between the beads and the domain wall is given
by
E⊥1 ) -
2µ0χa2MsHw
N
∑
i)0
3
2i + 1
1 + (2i + 1)2
(7)
where the closest bead is assumed to be located a distance
x ) a from the center of the domain wall. The corresponding
dipolar attraction due to interaction between the beads
themselves is associated with the energy
µ0χ2πa3H2
(N - 1)
E⊥2 ) 9
(8)
The total energy associated with the chain structure is
then given by E⊥ ) E⊥1 + E⊥2.
However, the energy of the beads attracted to the domain
wall forming a pyramidal shape is given by
E∆1 ) -
2µ0χa2MsHw NB
∑
i)0
3
1 + ix3
1 + (1 + ix3)
(NB - i)
(9)
2
where the closest beads are assumed to be located a
distance x ) a from the domain wall. Here NB is the number
of beads in the base of the pyramid:
NB
N)
i)
∑
i)1
NB(NB + 1)
2
(10)
Figure 4. Dipolar energy of pearl chains (blue and red lines)
and pyramids (magenta and green lines) for two different values
of N. See the text for details.
The dipolar energy associated with dipolar interactions
in the pyramid is given by
E∆2 ) -
µ0χ2πa3H2
Nb(Nb - 1)
24
(11)
The total energy associated with the pyramid seen in
Figure 1b can therefore be expressed as E∆ ) E∆1 + E∆2.
It is seen that the E⊥ decreases more rapidly than E∆
as the magnetic field is increased. Moreover, there is a
critical field Hc where E∆ e E⊥, thus suggesting that
pyramids are more stable than chain configurations below
this field as long as the magnetic field is sufficiently strong
such that the dipolar interactions overcome the Brownian
motion (H ≈ (9kT/πχ2µ0a3)1/2 ≈ 300 A/m). Figure 4 shows
E⊥ (blue line) and E∆ (magenta line) as a function of the
magnetic field Hx for N ) 6 (NB ) 3) assuming that Ms ≈
105 A/m and w ) 100 nm. Note that the critical field is Hc
) 28 kA/m. However, when N ) 21 (NB ) 6), we see that
E⊥ (red line) becomes smaller than E∆ (green line) at Hc
) 43 kA/m.
To test our theoretical predictions experimentally, we
investigated the self-assembly of colloidal pyramids near
1D magnetic domain walls. The domain walls were formed
in a bismuth-substituted ferrite garnet film of thickness
4 µm and magnetization Ms ≈ 105 A/m. A glass ring with
a diameter of about 2 cm was placed on top of the garnet
film, and beads immersed in pure water at a density of
∼105 beads/mL were confined within this ring. The
particles used here are paramagnetic beads as described
above, manufactured by Dynal (Dynabeads M270) and
coated with a carboxylic acid (COOH-) group. In the
absence of colloids, the domain wall is easily recognizable
in a polarization microscope. However, the presence of
light scattering from the colloids makes it more difficult
to visualize the domain wall, but the behavior of the
colloids clearly demonstrates its presence. A schematic
drawing of the experimental setup is shown in Figure 5.
The domain wall (lying along the y axis) acts as a 1D
nanomagnet where the beads adsorb. An external magnetic field parallel to the x axis aligns the magnetic
moments of the beads perpendicular to the wall. In weak
magnetic fields between 400 and 1000 A/m, we observe
the formation of pyramids, whereas in stronger fields
>1000 A/m colloidal chains are dominant. Typically, the
pyramids occurred in formation as seen in Figure 6. The
domain wall is located at the base of the pyramids, and
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Langmuir, Vol. 21, No. 16, 2005 7279
Figure 8. Mondisperse system of colloidal pyramids with N
) 3.
Figure 5. Schematic drawing of the experimental setup where
an inhomogeneous magnetic field is used to assemble the
pyramids.
Figure 6. Ensemble of colloidal pyramids with different values
of N.
Figure 7. Colloidal pyramids with N ) 3, 6, and 21.
the centers of the closest beads are positioned a distance
x ≈ a from the domain wall. We also observed many
individual pyramids as seen in Figure 7. Figure 7a shows
two pyramids composed of N ) 3 and 6 beads (located
next to each other), whereas Figure 7b shows a pyramid
with N ) 21. In stronger magnetic fields, linear pearl
chains are most often observed; see ref 12. It should also
be pointed out that in some cases we also observed
pyramids with tails (i.e., with linear chains attached to
the apex), but a physical explanation for this phenomenon
has not yet been revealed. That is, we do not yet
understand why these tails do not conform to increase the
size of the pyramid in order to make it more stable.
However, it should be emphasized that the tails are indeed
of magnetic origin and are not caused by additional flow
in the system.
According to the theory presented, it should be possible
to create larger and more stable pyramids by increasing
the field (as long as pyramid formation is favorable).
However, our experiments do not provide conclusive
evidence that such a phenomenon occurs. Instead, we find
pyramids with different values of N on the same domain
wall in the same experiment. Only in some cases have we
observed many small pyramids with the same N attached
to the same domain wall (Figure 8).
Our experiments suggest that pyramids with N in the
range between 3 and 21 can exist as long as the external
field is less than 1 kA/m. Although the theory presented
here explains the qualitative features rather well, it also
suggests critical fields an order of magnitude larger than
the experimental values. This discrepancy could be related
to the fact that we have neglected the beads’ influence on
each others’ magnetic moments and that we have assumed
that each bead is aligned only with the external magnetic
field, thus neglecting the induced magnetic moment caused
by the domain wall. Our calculations nonetheless correctly
predict the qualitative behavior of the system and could
prove useful in predicting the formation of colloidal
pyramids. It should be noted that the critical field scales
linearly with the strength of the domain wall (Msw), which
could help us in tuning the structural arrangement of
colloids.
4. Conclusions
We visualized the formation of individual colloidal
pyramids in different geometries. By tuning the strength
of the magnetic field, we were able to obtain either
pyramids or chains. Our results show that the formation
of colloidal structures in strong magnetic is feasible.
Moreover, it could be combined with other manipulation
techniques (e.g., optical trapping), thus providing a new
platform on which microscopic crystallites can be studied.
Acknowledgment. I am very grateful to H.Z. Wen
and T.M. Fischer for useful discussions and support.
LA051140V