Langmuir 2004, 20, 7323-7332
7323
Assembling and Manipulating Two-Dimensional Colloidal
Crystals with Movable Nanomagnets
L. E. Helseth,*,† H. Z. Wen,† R. W. Hansen,‡ T. H. Johansen,‡ P. Heinig,†,§ and
T. M. Fischer†
Department of Chemistry and Biochemistry, Florida State University,
Tallahassee, Florida 32306, Department of Physics, University of Oslo, Oslo, Norway, and
Laboratoire de Physique des Solides, Université Paris-Sud, Orsay, France
Received April 13, 2004. In Final Form: May 29, 2004
We study crystallization of paramagnetic beads in a magnetic field gradient generated by one-dimensional
nanomagnets. The pressure in such a system depends on both the magnetic forces and the hydrodynamic
flow, and we estimate the flow threshold for disassembling the crystal near the magnetic potential barrier.
A number of different defects have been observed which fluctuate in shape or propagate along the crystal,
and it is found that the defect density increases away from the nanomagnet. We also study the melting
of the crystal/fluid system after removal of the nanomagnet and demonstrate that the bond-oriental order
parameter decreases with time. The nanomagnet can be moved in a controlled manner by a weak external
magnetic field, and at sufficiently large driving velocities we observe self-healing crack formation
characterized by a roughening of the lattice as well as gap formation. Finally, when confined between two
oscillating nanomagnets, the colloidal crystal is shown to break up and form dipolar chains above a certain
oscillation frequency.
1. Introduction
Colloid science is of considerable importance in industry
as well as for the understanding of basic physical
phenomena.1-24 The fact that colloids can be visualized
directly using a microscope has made them ideal for model
studies of structure and crystallization processes. In the
past decade the photonic crystals have boosted the interest
in three-dimensional colloidal crystals growth. However,
†
Florida State University.
University of Oslo.
§ Université Paris-Sud.
‡
(1) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.;
Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183.
(2) Murray, C. A.; Grier, D. G. Annu. Rev. Phys. Chem. 1996, 47, 421.
(3) Ghezzi, F.; Earnshaw, J. C.; Finnis, M.; McCluney, M. J. Colloid
Interface Sci. 2001, 238, 433.
(4) Fudouzi, H.; Xia, Y. Langmuir 2003, 19, 9653.
(5) Fudouzi, H.; Kobayashi, M.; Shinya, N. Langmuir 2002, 18, 7648.
(6) Sun, Y.; Walker, G. C. J. Phys. Chem. B 2002, 106, 2217.
(7) Gong, T. Y.; Marr, D. W. M. Langmuir 2001, 17, 2301.
(8) Gong, T. Y.; Wu, D. T.; Marr, D. W. M. Langmuir 2002, 18, 10064.
(9) Nadal, F.; Argoul, F.; Hanusse, P.; Pouligny, B.; Ajdari, A. Phys.
Rev. E 2002, 65, 061409.
(10) Ristenpart, W. D.; Aksay, I. A.; Saville, D. A. Phys. Rev. Lett.
2003, 90, 128303.
(11) Su, G.; Guo, Q.; Palmer, R. E. Langmuir 2003, 19, 9669.
(12) Auer, S.; Frenkel, D. Phys. Rev. Lett. 2003, 91, 015703.
(13) Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Neumann,
B.; Annesley, J.; Bochway, S.; Parker, A. W.; Ward, A. D.; Burgess, A.
N.; Paunov, V. N. Langmuir 2002, 18, 9587.
(14) Guo, Q.; Arnoux, C.; Palmer, R. E. Langmuir 2001, 17, 7150.
(15) Stancik, E. J.; Widenbrant, M. J. O.; Laschitsch, A. T.; Vermant,
J.; Fuller, G. G. Langmuir 2002, 18, 4372.
(16) Cong, H.; Cao, W. Langmuir 2003, 19, 8177.
(17) Golding, R. K.; Lewis, P. C.; Allard, M.; Sargent, E. H.;
Kumacheva, E. Langmuir 2004, 20, 1414.
(18) Vickreva, O.; Kalinina, O.; Kumacheva, E. Adv. Mater. 2000,
12, 110.
(19) Rugge, A.; Ford, W. T.; Tolbert, S. H. Langmuir 2003, 19, 7852.
(20) Rugge, A.; Tolbert, S. H. Langmuir 2002, 18, 7057.
(21) Trau, M. G.; Aksay, I. A.; Saville, D. A. Langmuir 1997, 13,
6375.
(22) Kumacheva, E.; Garstecki, E.; Wu, H.; Whitesides, G. M. Phys.
Rev. Lett. 2003, 91, 28301.
(23) Helseth, L. E.; Fischer, T. M. Phys. Rev. E 2003, 68, 051403
(24) Helseth, L. E.; Wen, H. Z.; Fischer, T. M.; Johansen, T. H. Phys.
Rev. E 2003, 68, 011402.
also two-dimensional crystals at the air-water or watersolid interface have been used as model systems for
understanding physical properties and pattern formation
of micrometer-sized colloidal systems. Two-dimensional
self-assembly of colloidal crystals is revealed in thin liquid
films or evaporating drops,1,11,16 and these methods have
been fruitful for well-controlled growth of colloidal crystals.
In general, colloidal systems can be manipulated by electric
and magnetic fields1-10,16-26 which induce dipolar interactions between the particles, thereby creating novel phases
as well as model systems for crystal growth and disassembly. This kind of system may function as a model
system for studying the influence of electric or magnetic
fields on growth of thin organic films. It is well-known
that crystallization in confined polymeric and colloidal
systems can be enhanced by flow.22 Moreover, recently it
has been demonstrated that localized electrostatic fields
can be used to generate colloidal patterns.5
In this work we demonstrate the growth of twodimensional magnetic colloidal crystals by using a strong
magnetic field gradient. That is, we take advantage of the
fact that paramagnetic beads feel a force in magnetic field
gradients and investigate the influence of hydrodynamic
drag and confinement on the crystallization process.
Moreover, we investigate how the crystal melts when the
domain wall is totally removed, as well as its response to
an oscillating magnetic domain wall.
2. Experimental Methods
The magnetic potential well was created experimentally using
a bismuth-substituted ferrite garnet film.24 The garnet film has
very large (1-10 mm) in-plane magnetized domains separated
by domain walls, and these domain walls (of effective width
approximately w ) 50 nm) act as nanomagnets creating a
potential well that attracts paramagnetic beads. A domain wall
is defined as a continuous rotation of the magnetization vector
from one direction to another. Here we use mostly Bloch walls,
where the magnetization vector rotates from the positive
(25) Larsen, A. E.; Grier, D. G. Phys. Rev. Lett. 1996, 76, 3862.
(26) Velev, O. D.; Lenhoff, A. M. Curr. Opin. Colloid In. 2000, 5, 56.
10.1021/la049062j CCC: $27.50 © 2004 American Chemical Society
Published on Web 07/14/2004
7324
Langmuir, Vol. 20, No. 17, 2004
Helseth et al.
that in both cases dislocation lines form where the two lattices
of conflicting orientations (growing perpendicular to the two
magnetic domain walls) meet. In the case of R ) 135° this
dislocation line is not very pronounced, although there are a few
5-fold defects present. However, near the dislocation line seen
in Figure 3d) one observes lots of 5-fold defects. The 5-fold defects
are smaller than the 6-fold symmetric defects. Moreover they
can account for the change of crystal orientation across the
dislocation line, which makes them a main ingredient in the
boundary between the two different lattices in Figure 3d).
3. Magnetic Force on Single Magnetic Beads
Figure 1. Schematic drawing of the experimental setup. The
magnetic moment of the beads is aligned along the field from
the domain wall. The domain wall itself is treated as a thin
long magnet with magnetization vector in the z-direction. The
liquid in which the beads are immersed is not shown here. The
dashed arrow denotes the direction of the flow.
y-direction (when x > 0) to the negative y-direction (when x <
0). The domain wall therefore has a net magnetic moment in the
z-direction; see Figure 1. Moreover, the domains do not produce
any stray field themselves, so that only the field generated by
the domain wall (here treated as a one-dimensional (1D) bar
magnet) is of importance. The typical domain wall coercivity is
about 100 A/m (1 Oe), and the domain wall is displaced from its
original x-position by a distance proportional to the external field
in the y-direction (0.1-1 µm per A/m), depending on the local
stress distribution and the geometric shape of the surrounding
domains. A small open glass ring of about 1 cm2 was put on top
of the magnetic film, and beads immersed in deionized, ultrapure
water at a density of 108 beads/mL were confined within the
walls of this cell. The paramagnetic beads used here had a radius
of a ) 1.4 µm, a susceptibility χ ) 0.17, and were manufactured
by Dynal (Dynabeads M270 coated with a carboxylic acid group).
The beads and domain walls were visualized by a Leica DMPL
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 1, where the domain wall
of width w is located on the y-axis. The domain wall was visualized
by the polar Faraday effect, where the z-component of the
magnetization Mz is directly proportional to the Faraday rotation
θ (θ ) VMz, where V is a constant). In this way we could determine
the exact location of the magnetic domain wall before the
experiment started.
A few minutes after deposition, the beads start to sediment
at the magnetic film interface, where they are attracted to the
closest domain wall. After adsorption of several beads, we could
not see that domain wall any more, although we know it is there
through its interaction with the beads. The process is assisted
by lateral liquid flow, which drags the beads across the surface
at velocities of 0.1-10 µm/s. The lateral flow is driven by the
evaporation from the cell and will disappear when the system
reaches equilibrium (or when the cell is closed). In the vicinity
of the domain wall, the magnetic forces trap the beads despite
the drag from the moving liquid. After a period of between 10
min and 1 h, a colloidal crystal self-assembles on one side of the
magnetic potential well, as can be seen in Figure 2. The crystal
seen here has a width of about 40 layers, but using this
combination of flow and magnetically assisted growth we have
managed to grow up to about 100 layers in width and several
hundred layers in length (depending on the length of the domain
wall). The colloids are well-ordered close to the nanomagnet but
become more disordered further away due to the weakening of
the magnetic attraction. Also fluid flow is important here and
may help ordering the crystal further away from the nanomagnet.
Domain walls may also divide domains into various geometries,
and here we took advantage of this fact to grow crystals near
wedges of various angles. Panels a and b of Figure 3 show the
beads crystallize in a wedge of opening angle about 135°, whereas
in panels c and d of Figure 3 the angles are about 90°. We note
The strong magnetic field gradient generated by the
domain wall is very important here, since it ensures the
attraction of the beads to the domain wall. The magnetic
field from the 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
(1)
The force from the domain wall on a single magnetic bead
is then found by noting that the dipole can be associated
with an energy E ) -µ0m‚HDW, where µ0 is the permeability of water, and the magnetic moment is given by m
) (4π/3)a3χHDW. The magnetic field required to saturate
the magnetic moment of the beads is of the order 100
kA/m. We operate here with fields that are orders of
magnitude smaller than this, so that the linear relationship given above is a good approximation. Then the force
is found to be (assuming z ) a)
Fx ) A)
Ax
(x2 + a2)2
(2)
2µ0 3
χa (Msw)2
3π
It should be emphasized that HDW is much smaller than
the field required to saturate the magnetic moment of the
beads, and therefore the given expression is a good
approximation. Expanding eq 2 to first order (when x is
small) gives Fx ) -kx, where k ) A/a4.
We have measured k for a single sphere in the potential
well through kinetic measurements (i.e., adsorption of a
single bead to the domain wall) and also with direct
sampling of the probability vs position x converted to an
energy landscape assuming that the probability follows
a Boltzmann distribution. A bead located close to the
domain wall (i.e., x is small) has a probability density p(x)
of being at a position x given by
p(x) ) p(0)e-E(x)/kBT
(3)
where kB is Boltzmann’s constant and T the temperature
(T ) 300 K). Equation 3 is easily transformed into
( )
p(0)
E(x) - E(0)
) ln
kBT
p(x)
(4)
Using the expression for the magnetic energy, we find
that a bead in the well experiences a harmonic potential
energy given by
E(x) - E(0) 1 k 2
)
x
kBT
2 kBT
(5)
To obtain the probability distribution, we performed about
Manipulating Colloidal Crystals
Langmuir, Vol. 20, No. 17, 2004 7325
Figure 2. Image a shows a crystal during self-assembly when the bulk liquid flow is approximately 50 µm/s in the x-direction.
The black scale bar is 25 µm, and the nanomagnet is located at x ) 0. Note that some beads are located on top of the lattice, appearing
as black points. We also see that most of the colloids are located in the area x < 0 due to liquid flow in the positive x-direction.
The curve in (b) shows the force (or pressure) distribution as a function of distance from the nanomagnet. See the text for parameter
values.
Figure 3. Self-assembly of two colloidal crystals with mismatched lattices in the presence of magnetic field gradients and flow.
Images a and b are separated by about 300 s in time, whereas images c and d are separated by approximately 120 s. The white
arrows show the 5-fold defects formed as a result of mismatch in crystal lattice growing perpendicular to the two domain walls
forming the wedge/corner. The bulk flow in (c) and (d) is approximately 50 µm/s in the x-direction (black arrows) and somewhat
lower in (a) and (b). Note that some beads are located on top of the lattice, appearing as black points. The white scale bar is 30
µm.
4000 measurements of the x-coordinate of single beads27
and plotted it in a histogram (see Figure 4a). It can be
seen that the distribution is approximately Gaussian,
which is due to the fact that thermal excitations are
responsible for the small excursions from x ) 0. The
probability distribution can then be converted into an
energy landscape using eq 4, and the result is plotted in
Figure 4b. Fitting eq 5 to Figure 4b gives k ) 10-7 N/m,
which corresponds to an effective domain wall width of w
(27) The displacement accuracy is here about 60 nm, and we have
an optical signal-to-noise ratio of about 100. Note that the displacement
accuracy (change in x between pictures) differs from the accuracy in
absolute position, which is determined by the diffraction limit (about
0.5 µm).
7326
Langmuir, Vol. 20, No. 17, 2004
Helseth et al.
at rest the liquid velocity is the only relevant quantity,
and the hydrodynamic drag is given by ηfavL.
In addition to drag forces, the beads are also exposed
to the magnetic field from the nanomagnet. A simple
estimate of the maximum dipolar interactions between
two spheres in a magnetic field HDW ) Msw/(2πa) gives Ed
) µ0m2/[2π(2a)3] ) 3kBT. The field acting on the magnetic
beads far away from the domain wall is orders of
magnitude smaller than HDW ) Msw/(2πa), and the actual
dipolar energy is therefore comparable to the thermal
energy also after summation over the whole lattice. Since
the beads are coated with a carboxylic acid group, we
estimate the electrostatic interaction between two beads
in close contact to be about 50 kBT, which suggests that
such interactions keep the crystal from being close packed.
However, for the moment we neglect the repulsive
electrostatic interactions, so that the force on a layer of
beads at rest a distance x from the nanomagnet can be
given by
Fx ) ηfavL -
Figure 4. Probability distribution (a) and energy (b) for a
magnetic bead in the potential well of a domain wall. In (a) the
number of observations in a certain interval of the x-position
has been plotted into a histogram, whereas in (b) this probability
distribution has been converted into an energy landscape. The
energy can to a reasonable approximation be fitted to the
parabola of eq 5.
≈ 20 nm. We emphasize that several different domain
walls have been used in the current study, and measurements similar to the one above (as well as kinetic
measurements) suggest that their effective widths vary
between 20 and 100 nm. The reason for this variation is
that important properties (e.g., magnetic uniaxial anisotropy) depend on the local environment (e.g., stress
distribution), thereby creating domain walls of different
width. The typical width is about w ) 50 nm, which is the
value we will use in our theoretical estimates.
4. Colloidal Crystallization and Melting
In this section we investigate how the crystal assembles
in the presence of flow and show the different types of
defects observed in it. Finally, we characterize its melting
after the domain wall has been removed.
4.1. Crystal Assembly in Liquid Flow. When enough
beads are in the vicinity of the domain wall, they assemble
into a crystal structure due to the attraction to the domain
wall. It should be emphasized that even though the domain
wall may give rise to a symmetric magnetic field distribution, the liquid flow of velocity vL ensures that the twodimensional colloidal crystal only forms at one side of it.
Although the liquid flow can be as high as vL ) 50 µm/s
in the bulk (estimated by tracking beads in the bulk), the
beads move with velocities vB between 0.1 and 10 µm/s
near the interface due to the increased drag coefficient
there. The hydrodynamic drag is then given by ηfav, where
v ) vL - vB is the relative velocity between the liquid and
the beads. Here η ) 10-3 N s/m2 is the viscosity of water,
and the hydrodynamic drag coefficient has previously been
measured to be f ) 30 through diffusion measurements
using the Stokes-Einstein relationship.24 For a crystal
Ax
(x + a2)2
2
(6)
Here we have assumed that the hydrodynamic drag is in
the positive x-direction. Moreover, we assumed that at
every point in the crystal a constant drag force is acting
on the beads. We will argue here that in the presence of
an external flow this constant force assumption can explain
our results at least qualitatively. First, we observe that
the first few layers are always well arranged due to the
strong magnetic forces here. Second, if the liquid flow is
removed (vL ) 0), one observes that while the rest of crystal
becomes soft (and the number of defects increases) the
first few layers are still well-ordered. Figure 2b shows the
force (eq 6) as a function of x when η ) 10-3 N s/m2, f )
30, vL ) 5 µm/s, Ms ) 105 A/m, w ) 50 nm, and χ ) 0.17.
Figure 2b suggests that the pressure is rather high near
the domain wall but quickly decays so that about 5-10
layers away from the domain wall it is mainly governed
by hydrodynamic drag. We emphasize that only the
average pressure is discussed here, and fluctuations have
not been considered so far.
If the drag force from the moving liquid is larger than
the maximum attractive magnetic force, the beads will be
pushed over to the other side of the nanomagnet and finally
released. The maximum magnetic force is (0.6µχ/3π)(Msw)2, which suggests that the maximum velocity is vLmax
) 0.6µχ/(3πηfa)(Msw)2, thus giving vLmax ) 16 µm/s. This
is not too far from the experimental value, found to be
between 1 and 10 µm/s, depending on the local environment and geometry of the boundary. In the rest of this
study we will only be concerned with crystals which are
exposed to negligible flow (i.e., vL ≈ 0), to probe the
influence of the magnetic field on the crystal in more detail.
4.2. Defects. As can be seen from Figures 2 and 3,
several different defects exist in the colloidal crystals, and
we will discuss them in the following.
The most commonly observed defects in colloidal crystals
have 6-fold symmetry.1,2 These are basically lacking only
a bead and therefore cost minimal amounts of energy. In
addition, we also found 5-fold defects, in particular far
away from the nanomagnets and in regions were mismatched lattices met (see, e.g., Figure 3). The 5- and 6-fold
defects are seen to exhibit shape fluctuations due to the
Brownian motion of the surrounding beads. Two examples
of the possible shapes for a 6-fold defect are shown in
panels a and b of Figure 5, which shows two successive
pictures taken with an interval of 2 s. In (a) the bead on
the left moves a little bit toward the vacancy, whereas in
Manipulating Colloidal Crystals
Langmuir, Vol. 20, No. 17, 2004 7327
Figure 6. The number of 5- and 6-fold defects increases as one
moves away from the domain wall. Here we selected four crystals
of size larger than 200 × 250 µm2, all assembled under similar
liquid flow conditions. We then counted the number of 5- and
6-fold defects as a function of distance from the domain wall
within successive windows of 30 × 200 µm2, averaged the results,
and displayed them in a histogram.
Figure 5. Defects in the colloidal crystal. Images a and b,
separated by a 3 s time interval, show shape fluctuations of a
6-fold defect. Images c-f spanning over 60 s show the propagation of a hexagonal defect by cooperation of three 5-fold defects.
Note that the whole crystal is moving slightly due to flow in the
y-direction. Finally, in panels g and h, it is seen how multiple
defects (g) transform into a 5-fold defect with two connecting
dislocation lines (h) when exposed to flow. Note that some beads
are located on top of the lattice, appearing as black points.
(b) the two beads above and below the defect squeeze
together. Both processes are associated with energies of
approximately kBT.
In addition to shape fluctuations, the 6-fold defect may
also move to another location. However, diffusion cannot
support this process in a rigid crystal lattice where the
diffusion coefficient is much smaller than that for a free
particle. On the other hand, we observed that the hexagonal defect could propagate by a transformation into
three 5-fold defects in a two-step collective process as seen
in Figure 5 c-f. In (c) the defect is fluctuating slightly and
a bead partially enters and transforms the 6-fold defect
into a 5-fold defect, creating a new 5-fold defect at its
initial opposition. This initializes the movement of a second
bead toward the second 5-fold defect; thereby creating a
third 5-fold defect; see Figure 5d. When the first bead
totally enters the vacancy associated with the initial defect,
two 5-fold defects remain; see Figure 5e. However, this
configuration is very unstable, and the second bead is
therefore sucked into empty room associated with the
lower 5-fold defect, thereby leaving a more stable 6-fold
symmetric defect in a new position; see Figure 5f. Although
the two beads described here are the only ones actually
moving significantly, it should be pointed out that many
beads are involved in this cooperative phenomenon,
thereby causing larger parts of the lattice to deform
slightly.
Finally, we report a particularly interesting phenomenon in panels g and h of Figure 5, which shows how a
small number of localized defects are turned into a longrange dislocation when hydrodynamic drag acts on the
crystal. In Figure 5h lattice planes several bead diameters
away from the distorted 5-fold defect are displaced a little
bit due to the flow-induced squeezing of the defects. As a
result two dislocation lines emerge from the remaining
5-fold vacancy traveling along the main crystallographic
axes that are separated by an angle of about 60°. This
happens at a flow velocity of the order 1 µm/s.
From the discussion above it should be clear that the
colloidal crystals reported in this study are not defectfree. To get a feeling of the distribution of defects
throughout the crystal, we selected four of our largest
crystals, counted the number of 5- and 6-fold defects as
function of the distance x of the defect from the domain
wall, and then took the average. The result is shown in
Figure 6, where a histogram of the average number of
defects vs distance from the domain wall is displayed. We
observe that the number of 5- and 6-fold defects increases
strongly with x, which can be explained qualitatively by
the decay of the magnetic force holding the crystal together.
4.3. Melting. By applying a very small magnetic field,
we are able to remove the magnetic domain wall. As a
result there is no longer any magnetic force to keep the
crystal together. In absence of liquid flow, the beads start
to diffuse away from each other in both the positive and
negative x-directions, as can be seen in Figure 7. In Figure
7a a relatively long and thin crystal is assembled. In Figure
7b-d the domain wall has been removed with a magnetic
field perpendicular to the magnetic film of about 400 A/m
(5 Oe). Note that in Figure 7d the distribution is more or
less random. In absence of liquid flow the transition from
crystalline to liquid behavior is governed not only by
diffusion resulting from concentration gradients (i.e.,
Fick’s law) but also by weak electrostatic repulsive forces.
However, since these repulsive interactions decay quickly
with increasing distance between the beads, they do not
play a role after a few seconds. A particularly interesting
feature of this melting is the strong asymmetry in the
spatial dimensions. Thus, the relative expansion in the
x-direction (divided by the width of the original crystal)
is much larger than the corresponding relative expansion
in the y-direction, and this makes the system strongly
7328
Langmuir, Vol. 20, No. 17, 2004
Helseth et al.
Figure 7. Time evolution of a system initially in the crystalline phase (a) over a period of 20 s after the 1D nanomagnet has been
removed. Here there is no flow. In image b, which was taken 2 s after removal, one observes that several holes have developed.
Images c and d are taken 10 and 20 s after removal, respectively. Τhe black scale bar is 65 µm.
anisotropic. Figure 8a shows the evolution of the average
width of the colloidal system with time (note that the width
is fluctuating slightly in the y-direction), which suggests
a linear relationship for the time span studied here. The
disassembly of the crystal is accompanied by excitations,
where vacancies are created and some of the beads hop
on top of the others or plow through the lattice in the
x-direction thereby creating elliptical vacancies; see Figure
7b). The reason for this is that some more energetic
particles push extra hard on their surrounding, thus
creating vacancies upon leaving. After 3-4 s the number
of vacancies is so large that the colloidal structure appears
rather homogeneous. To obtain a quantitative measure of
the melting of the crystal, we use the 6-fold bond-oriental
order parameter.25 For a site k whose nearest neighbors
labeled by j are arrayed by angles θkj from a certain
reference direction, this order parameter is given by
ψ6k ) ⟨exp(i6θkj)⟩
(7)
where the angle brackets denote an average over nearest
neighbors and also over all the beads in the small area
considered here (about 25 beads). Due to the finite size of
this crystal, we selected a small area of it in order to
describe melting using this order parameter. Its magnitude is equal to unity when we have a perfect hexagonal
lattice. Figure 8b displays the evolution of the magnitude
|ψ6k| of the order parameter. It is clearly seen that it
decreases with time, which can just as well be spotted in
Figure 7. We notice that the order parameter decreases
most strongly directly after release, and the melting therefore happens within few seconds. It should also be pointed
out that after removal of the domain wall periodicity is
also lost. However, we will not investigate this here.
5. Oscillating Colloidal Crystal
A particular advantage of using magnetic domain walls
to assemble colloidal crystals is that they can be modulated
by an external magnetic field. By applying a weak
magnetic field slightly larger than the domain wall
coercivity (100 A/m) in the y-direction, we may displace
the domain wall in the x-direction. Reversing the sign of
the field also reverses the motion. The required magnetic
field is weak and does not induce significant magnetic
moments in the paramagnetic beads. The only relevant
field is therefore due to the domain wall.
Using a TTI waveform generator (TGA1242, 40 MHz)
and a small electromagnet, we imposed a weak, oscillating
in-plane magnetic field on the domain wall, thereby
causing the average relative position of the layers in the
crystal to oscillate as u
j N(t) ) u
j N0(x) sin(ωt), where u
j N0(x)
is the amplitude of the Nth layer, x is the distance from
the domain wall, and ω ) 2πf is the angular frequency.
The corresponding instantaneous velocity is vj N(t) ) vj N0(x)
cos(ωt), where vj N0(x) ) ωu
j N0(x). Figure 9a shows the
oscillation of different layers of the crystal when u10 )
0.4a and f ) 0.3 Hz. The second layer (N ) 2, open circles)
oscillates fully in phase with the external magnetic field,
whereas the fifth layer (N ) 5, open squares) is slightly
out of phase. We also note that this layer does not show
pure sinusoidal behavior. Moreover, in the ninth layer (N
) 9, open triangles) the forced excitations are of the same
magnitude as the thermal fluctuations of the lattice, and
it is therefore difficult to distinguish the oscillations from
noise.
An important observation from Figure 9a is that the
amplitude of the oscillations decreases as one moves away
from the domain wall. A clearer picture of this is shown
Manipulating Colloidal Crystals
Langmuir, Vol. 20, No. 17, 2004 7329
Figure 9. Oscillating colloidal crystal. Part a shows how the
layers N ) 2 (hollow circles), N ) 5 (hollow squares), and N )
9 (hollow triangles) respond to an oscillating domain wall. In
part b the decay of the amplitude is displayed for two different
frequencies, 0.3 Hz (hollow squares) and 1 Hz (hollow circles).
Figure 8. Time evolution of the system seen in Figure 7. In
(a) the width of the crystal/fluid is shown as a function of time,
whereas in (b) the magnitude of the oriental order parameter
is displayed.
in Figure 9b, where uN0/a is displayed as function of the
layer number N for two different frequencies, f ) 0.3 Hz
(open squares) and f ) 1 Hz (open circles). In the
experiments we also observed thermal fluctuations of the
particles in the crystal lattice, and the position xd where
the magnetic energy equals the thermal energy is given
by
xd ) a
(
µ0χa3(Msw)2
-1
π3kBT
)
1/2
(8)
which we here find to be 19 µm. The fact that the observable
oscillations extend to about 25 µm we may attribute to
either the experimental inaccuracy or the result of
collective effects. One may naively expect from our single
particle model that the amplitude of each layer decays as
uN0 ∝ x-3. However, experimentally we observe that the
decay is nearly linear beyond the two first layers, which
suggests a collective behavior where the particles interact
electrostatically. We also found that the decay is very
similar for different frequencies (and velocities), which
may indicate that hydrodynamic interactions are not
essential as long as the collective motion is periodic. Then,
electrostatic interactions between the electrically charged
magnetic beads are expected to be most important in a
nearly close-packed lattice. The characteristic time τ
required to displace a single isolated bead by δ/2 about
a position x0 is given by τ ≈ ηfdδax03/A. As an example,
assume that δ ) a and x0 ) 1.8a (N ) 2), which gives τ
) 0.3 s. This should be contrasted to the case δ ) 0.5a and
x0 ) 7a (N ) 5), which gives τ ) 9 s, and it is more difficult
for this layer to follow oscillations of frequency 0.3 Hz, in
agreement with our observations. Therefore, only by
considering the single bead dynamics one may infer that
the oscillations of crystal layers far away from the domain
wall may not follow those close to it given that the
displacement δ is sufficiently large.
In the case where the hydrodynamic drag is sufficiently
larger than the magnetic force on the Nth layer, the beads
can no longer follow, and the crystal will break in its
vicinity; see Figure 10. In addition, a video in AVI format
available as Supporting Information shows cracking of a
colloidal crystal in real time.28 Experimentally we found
that the position of the crack depends on both the frequency
and amplitude of the oscillations. As an example, we
consider in Figure 10 the case u10 ) 0.7a and f ) 0.3 Hz.
In Figure 10a the crystal is maximally compressed and
nearly hexagonally close-packed. Figure 10b shows the
same crystal 1 s later, and it is seen that the lattice planes
are no longer as well ordered as that in Figure 10a. In
Figure 10c the crystal is maximally stretched, and in
addition to the disordering in Figure 10b, we observe that
the lattice planes are separated by gaps on several
locations. In Figure 10d, the lattice is again maximally
compressed, and the hexagonal structure has been
restored. It should be emphasized that our crystal is
different from brittle solids which exhibit fast crack
propagation due to breaking of atomic bonds.29 On the
other hand, we do observe both localized coarsening of the
lattice as well as gap formation between the Nth and (N
- 1)th layer, and crack formation in our system therefore
has some similarities with that of a brittle solid. We will
here characterize the crack formation by a roughness
(28) See the video in AVI format, available as Supporting Information,
that shows an oscillating crystal cracking.
(29) Abraham, F. F. Adv. Phys. 2003, 52, 727.
7330
Langmuir, Vol. 20, No. 17, 2004
Helseth et al.
Figure 10. Breaking of a colloidal crystal subject to a modulation. The images show a portion of a colloidal crystal self-assembled
near a domain wall. Images a and b are separated by 1 s, images b and c by 0.5 s, and finally images c and d by 1.5 s. The scale
bar is 7 µm. See also a video in AVI format available as Supporting Information.
parameter ∆N and a measure of the gap ∆uN. The rootmean-square (rms) deviation from the average position
u
j N(t) of the Nth lattice, ∆N(t) ) ⟨(uN(t) - u
j N(t)⟩2)1/2, is taken
to be the roughness parameter. The gap parameter ∆uN
is taken to be the difference between the average position
of lattice N and N - 1, ∆uN(t) ) u
j N(t) - u
j N-1(t). In Figure
11a, ∆N/a is displayed as function of time for N ) 2 (filled
squares) and N ) 3 (filled circles) when f ) 0.3 Hz. The
roughness of layer N ) 2 does not change significantly as
the crystal boundary oscillates, see also Figure 10c, and
the only contribution to ∆2 is Brownian motion. On the
other hand, the ordering of the layer N ) 3 is partially
destroyed, and ∆3 ) 0.2 when the crystal is maximally
expanded. We also find that the roughening of the lattice
is periodic with slightly varying magnitude, and almost
in phase with the oscillation of the domain wall. In Figure
11b we show the distance between the lattices, ∆u2(t) (filled
squares) and ∆u4(t) (filled circles), when f ) 0.3 Hz. Note
that the average distance between N ) 3 and N ) 4
oscillates between 2.4a and 1.8a, thus indicating a gap
formation in addition to the roughening discussed above.
On the other hand, far away from the domain wall the
distance between lattice planes appears to be uncorrelated
with the domain wall motion. One may expect the crack
to occur at a distance xc where the hydrodynamic drag
force is equal to the magnetic force. We thus obtain
xc ≈
(
)
2µ0χa2(Msw)2
3πηfu10ω
1/3
(9)
which gives xc ≈ 2.4a for u10ω ≈ 2 µm/s (experimental
value from Figure 10) and the parameters given above.
We see that this estimate deviates slightly from the
experimental data, which tell us that the roughening and
gap formation start at x ≈ 4a. Further measurements of
∆N/a show that the roughening of the lattice is rather
localized; see Figure 11c, where the circles correspond to
Figure 11. Crack formation is accompanied by a local change
in the roughness of the lattice. In (a) the rms deviation ∆N/a
for N ) 2 (filled squares) and N ) 3 (filled circles) is shown as
function of time when the domain wall oscillates at a frequency
f ) 0.3 Hz. In (b) the gap ∆uN/a is shown as function of time
for N ) 2 (filled squares) and N ) 4 (filled circles). In (c) the
roughness is displayed for two different times (separated by
about 1.5 s) in the periodic cycle. The filled circles correspond
to an expanded lattice, and the filled squares correspond to a
compressed lattice. We note that the roughness of the lattice
may appear to increase away from the domain wall due to
decreasing magnetic attraction.
an expanded crystal, whereas the squares correspond to
maximal compression. The crack is seen to be localized to
layers between N ) 3 and N ) 7. In a normal brittle solid
one would expect cracks to be a major origin of stress
relief, where big cracks may grow at the expense of smaller
ones.29 However, in our system these cracks do not
Manipulating Colloidal Crystals
Langmuir, Vol. 20, No. 17, 2004 7331
Figure 12. Image (a) shows the crystallization between two domain walls. Since the domains outside and inside the domain walls
have opposite magnetization directions (positive or negative y-direction), the domain walls always move in opposite directions when
we apply an oscillating magnetic field in the y-direction, i.e., along the domain walls. When oscillating the two domain walls out
of phase at f ) 1 Hz, we observe chain formation perpendicular to the wall, as can be seen in b. The scale bar is 35 µm.
represent a similar source of stress relief since there are
basically no attractive interactions between the colloids,
and the attraction to the domain wall is the only force
holding the crystal together. In fact, the crack increases
the distance to layers beyond it, and these will always try
to reduce their magnetic energy by moving toward the
domain wall. The cracks may therefore be said to have a
self-healing behavior (which we have also observed directly
by imposing a pulse instead of sinusoidal oscillations), at
least if the domain wall is brought to rest. One may also
wonder why there is a roughness associated with each
crack, since our simple model predicts that the crystal
should be cleaved nicely upon cracking. However, it must
be emphasized that each bead is not identical, and
therefore the hydrodynamic coupling to the underlying
surface also varies. This results in a distribution of
different values for f, and therefore also xc. Thus, one may
say that hydrodynamic coupling to the underlying surface
is not so essential for the collective behavior (although it
does determine some properties of the crystal) when the
lattice oscillates in a relatively periodic way but becomes
important when the crystal breaks.
walls so that the x-component of the magnetic field from
both the walls points in the same direction, a new dynamic
effect occurs.30 As can be seen in Figure 12a, the beads
crystallize near the domain wall. Moreover, the magnetic
moments of the beads are aligned in the x-direction,
thereby creating dipolar chains. A simple estimate using
the field from the domain walls suggests that the dipolar
energy of the chains is of the order kBT, and they are
therefore rather weak; see also Figure 12a. Thus, as long
as the domain walls are at rest, the attraction to the
domain wall is stronger than the dipolar interactions, and
most of the beads prefer to form a hexagonal crystal
structure. By oscillating the domain walls 180° out of phase
with u10 ) a, we found that at an oscillation frequency
about 0.5 Hz a number of the beads are thrown out of the
crystal and start forming dipolar chains.30 We explain
this phenomenon qualitatively using the single bead model
discussed above. This model predicts that a single bead
is released from the domain wall when the velocity of the
domain wall is larger than that allowed by hydrodynamic
drag. Once the beads are “released” from the domain wall
(i.e., the magnetic force is sufficiently weak), they are free
6. Dynamically Induced Transition
(30) This can be obtained either by choosing the magnetization vector
of the domain walls to be pointing in opposite directions (positive or
negative z-direction) or by selecting one domain wall pointing in the
z-direction and the other in the x-direction (Néel wall). Near a single
domain wall (with magnetization vector in the z-direction) one does not
observe chains. The reason one can observe dipolar chains when two
domain walls are present is that the fields from the two domain walls
enhance each other.
So far we have only discussed what happens when a
single domain wall is used as an oscillating interface. It
is also possible to let the beads crystallize between two
magnetic domain walls as seen in Figure 12. We found
that by choosing the magnetization direction in the domain
7332
Langmuir, Vol. 20, No. 17, 2004
Helseth et al.
to form weak dipolar chains; see Figure 12b. The critical
frequency at which the beads at a distance x0 are thrown
out of the crystal is estimated to be
fc ≈
µ0χa2(Msw)2
πηfu10x03
(10)
Assuming u10 ) a gives fc ≈ 1 Hz when x0 ) 1.8a (N ) 2)
and fc ≈ 0.3 Hz when x0 ) 2.7a (N ) 3). This is in qualitative
agreement with what can be observed from Figure 12b,
which tells us that at 1 Hz a significant portion of all
layers N > 3 have been reformed into chains. Note that
this estimate does not take into account dipolar or other
interactions hindering the motion. However, a detailed
study of these effects is most easily done by computer
simulations, which is outside the scope of the current
study.
7. Conclusion
We have studied crystallization in the magnetic field
gradient generated by one-dimensional nanomagnets. It
is found that the pressure in such a system depends on
both the magnetic forces as well as the hydrodynamic
flow, and we attempted to describe the pressure distribution in the system. A number of different defects are
observed which propagate in the crystal or fluctuate in
shape. Upon removal of the nanomagnet, the crystal
quickly disassembles thereby forming a liquid state. The
domain walls can be driven by weak external magnetic
fields that do not alter the dipolar interactions between
the beads, and we have found that this can be used to
oscillate the crystals or probe dynamic phenomena. We
showed that a single particle model can account for most
of the phenomena observed here. To obtain a quantitative
understanding, a theoretical model based on many-body
electrostatic and hydrodynamic interactions must be
developed, a task which is outside the scope of the current
experimental study.
We believe that nanomagnets of reduced dimensionality
can be used to assist directed assembly of colloidal systems.
The method presented here offers a large amount of
flexibility since it is rather easy to move the nanomagnets
in a controllable manner, thus enabling high-resolution
spatial and temporal control of the system. Such colloidal
systems also represent a new class of self-healing materials
which are not irreversibly destroyed upon cracking. In
this context the self-assembled crystals studied here may
improve our fundamental understanding of driven systems
in soft matter physics.
Supporting Information Available: A video in AVI
format showing cracking of a colloidal crystal in real time. This
material is available free of charge via the Internet at
http://pubs.acs.org.
LA049062J