Volume 12 Number 37 7 October 2016 Pages 7701–7874

Soft Matterwww.softmatter.org

ISSN 1744-683X

PAPER
Eliot Fried, Benjamin B. Yellen et al.
Crystallization kinetics of binary colloidal monolayers

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Crystallization kinetics of binary colloidal
monolayers†
An T. Pham,‡ab Ryohei Seto,‡c Johannes Schönke,c Daniel Y. Joh,dAshutosh Chilkoti,bd Eliot Fried*c and Benjamin B. Yellen*abd
Experiments and simulations are used to study the kinetics of crystal growth in a mixture of magnetic
and nonmagnetic particles suspended in ferrofluid. The growth process is quantified using both a bondorder parameter and a mean domain size parameter. The largest single crystals obtained in experiments
consist of approximately 1000 particles and form if the area fraction is held between 65–70% and thefield strength is kept in the range of 8.5–10.5 Oe. Simulations indicate that much larger single crystals

Received 8th May 2016,
Accepted 22nd July 2016

containing as many as 5000 particles can be obtained under impurity-free conditions within a few hours.

DOI: 10.1039/c6sm01072equantitatively with the experiments. These findings provide an important step toward developing strategies

www.rsc.org/softmatterfor growing single crystals that are large enough to enable follow-on investigations across many
subdisciplines in condensed matter physics.

If our simulations are modified to include impurity concentrations as small as 1–2%, then the results agree

1 Introduction
In the last few decades, there has been great interest in
colloidal self-assembly, not just for its applications in photonic
bandgaps,1,2 biosensors,3,4 and templates for advanced manufacturing,5 but also for its unique ability to mimic diverse
phenomena in condensed matter physics. The experimentally
accessible length, time, and energy scales of colloidal particles(which are respectively B1 mm, B1 s, and BkBT) allow them tobe tracked using an optical microscope with single-particle
resolution and to achieve equilibrium at room temperature.
The ability to tune interparticle interactions makes it possible
to observe and probe collective ensemble behavior analogous toprocesses that occur at the atomic scale. For example, colloidal
suspensions have been used to study condensed matter phenomena such as spinodal decomposition,6 crystal nucleation,7point defect dynamics,8 grain boundary motion,9 and glasstransitions,10,11 along with various liquid–solid and solid–solidtransformations.12,13aDepartment of Mechanical Engineering and Materials Science, Duke University,
Box 90300 Hudson Hall, Durham, NC 27708, USA. E-mail: yellen@duke.edub
NSF Research Triangle Material Research Science and Engineering Center (MRSEC),
Duke University, Box 90217, Durham, NC 27708, USAc
Mathematical Soft Matter Unit, Okinawa Institute of Science andTechnology Graduate University, Onna, Okinawa, Japan 904-0495.
E-mail: eliot.fried@oist.jp; Fax: +81 (0)98 966 1062; Tel: +81 (0)98 966 1318dDepartment of Biomedical Engineering, Duke University, Durham, NC 27708, USA† Electronic supplementary information (ESI) available: Movies of experimentand simulation of rapid quenching. See DOI: 10.1039/c6sm01072e‡ These authors contributed equally to this work.

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The vast majority of prior investigations of colloidal selfassembly have focused on mono-component colloidal suspensions, in which both short-range and long-range interactions
combine to form large, close-packed single crystals.14–16 Morerecently, interest in studying multi-component colloidal crystals
has developed. These systems exhibit a greater diversity of phases
and allow for the study of a more diverse spectrum of phasetransitions and condensed matter phenomena. For example,
recent experimental work on the three-dimensional (3D) selfassembly of binary colloidal suspensions points to richphase behavior and various types of diffusionless phasetransformations.13,17–19 Processes involving the self-assemblyof multi-component suspensions have also been explored in
two-dimensional (2D) settings, which are simpler to study due
to the ease of particle tracking and the ability to simulate largescale phenomena. Moreover, 2D systems are interesting in theirown right since they incorporate physics and scaling laws that differ
from those arising in 3D systems. For example, the long-range
translational and orientational ordering that favors crystal formation in 3D systems is absent in 2D systems, where it hinders thegrowth of large crystals due to the dominance of long wavelength
fluctuations.20,21 Improved understanding of 2D crystallization processes will therefore have an impact in various applications, such asprotein crystallization processes occurring in lipid membranes,22
the formation of self-assembled monolayers (SAMs),23,24 and theconstruction of novel 2D materials.25
To date, there have been very few examples of large single
crystals formed in 2D systems of binary colloidal suspensions.
One example, involving electrostatically repulsive particles,26
exhibits long-range hexagonal ordering in the absence ofSoft Matter, 2016, 12, 7735--7746 | 7735

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close packing. The structures that form in this setting resemble
those typically observed in mono-component colloidal systems.14,15
Binary crystals assembled under long-range magnetic repulsion
have previously been studied.27 As a consequence of polydispersity,these systems tend to form small crystals that coexist with theglassy phase. On the other hand, it has been shown that large
single crystals are diﬃcult to form in close-packed binary
colloidal suspensions, which provide more relevant models forcondensed matter physics. Recently, our group has assembled
binary colloidal crystals in a uniform magnetic field;12 however,the crystals we obtained were limited in size due to our limited
understanding of the mechanisms underlying the kinetics ofthe assembly process and, thus, a lack of viable strategies forcontrolling the rate of growth.
Here we study the kinetics of crystal growth in a binary
colloidal suspension of magnetic and nonmagnetic particles
immersed in ferrofluid. In a uniform magnetic field aligned
normal to the 2D suspension, like particles attract and unlike
particles repel in a manner similar to particles bearing equaland opposite charges, which favor the formation of a selfassembled checkerboard lattice. The strength of these interactions can be controlled by the external magnetic field. Since
the eﬀective temperature of this system is inversely proportional
to the square of the strength of that field, the annealing conditions can easily be tuned with the external magnetic field. Due tothe absence of shielding or other external eﬀects, experimental
system can be adequately represented by a model of interacting
point dipoles.
Our goal here is to carefully analyze the time evolution of
crystal growth as a function of eﬀective temperature anddensity, with the objective of determining the ideal range ofconditions to form the largest possible single crystals. Our
idealized simulations suggest that it is possible to form single
crystals containing in excess of several thousand particles;
however, our experiments rarely achieve such sizes due to thepresence of particle impurities in concentrations even as small
as B1–2%. When we include similar impurity concentrations
in simulations, we find excellent agreement with experimental
behavior.
The remainder of the paper is structured as follows. Section 2
provides concise descriptions of the materials used in our experiments, the experimental setup and measurement techniques, andthe methods of data analysis. Theoretical and simulation models
are presented in Section 3. In Section 4, we compare the resultsfrom experiments and simulations, including order parameters,
crystal sizes, and the impact of impurities on the kinetics of
crystal growth. Finally, we briefly provide concluding remarks anddiscuss directions for future work.

2 Experiment methods anddata analysis
The experimental system involves a binary mixture of magnetic
and nonmagnetic spherical particles. Specifically, the magnetic
particles (M-270 Dynabeadss, Life Technologiest) are of mean

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diameter 2.8 mm and the nonmagnetic particles (Fluro-Max
R0300, Thermo Fischert) are of mean diameter 3.1 mm. These
particles are mixed with a ferrofluid (EMG 705, Ferrotect,
Bedford, NH) which is adjusted so that the volume fraction ofmagnetic nanoparticles is fixed at B1% in all experiments. To
reduce nonspecific adhesion of particles to substrates, we growa 50 nm thick poly(oligo(ethylene glycol)methyl ether methacrylate) (POEGMA) polymer brush on the glass slides using asurface-initiated, atom-transfer radical polymerization technique (SI-ATRP).28 Though this polymer brush layer significantly
reduces the percentage of particles that are randomly immobilized on the substrate, we still occasionally observe a few
particles stuck to the substrate (typically around B0.1%). In
addition to random binding to the surface, we occasionallyobserve irreversible binding between nonmagnetic particles,
which represents another type of particle impurity in theexperimental system. We also observe some rare, giant particle
contaminants in the magnetic and nonmagnetic particle stock
concentrations provided by the vendors. Due to the compression forces applied when preparing experimental samples,
these giant particles deform plastically into shapes that resemble oblate spheroids. These constitute a third type of impurity.
Due to our inability to perfectly control the mixing ratio of thetwo particle types, our experimental samples often have a slight
excess of one particle type which impedes the growth of single
crystals in a manner similar to the eﬀect of the other mentioned
impurities.
Fig. 1 provides a schematic of the experimental apparatus.
A 2.7 mL aliquot of particle mixture is confined between theglass surface and a coverslip and sealed using Loctites marine
epoxy. Given the dimensions of a typical coverslip, this leads tofluid film of thickness slightly larger than the diameter of anonmagnetic particle. Although we are not able to accurately
measure that thickness, we can estimate it by evaluating thecenter–center distance of a stacked pair of particles, whichsometimes forms within the suspension.12,29 Based on thismeasured distance and knowledge of the actual diameter ofthe particles, we estimated that the film thickness is in therange of 1.1–1.3 magnetic particle diameters (i.e., 3.1–3.6 mm).
The sample is placed on top of a 3D printed platform stage,
which includes an air-core solenoid that can produce a uniform

Fig. 1 The experimental setup consists of a vertical solenoid with anexperimental sample mounted above. The magnified view on the rightprovides an illustration of the experimental sample with magnetic particles
shown in blue and nonmagnetic particles shown in red. A magnetic field H(arrow) is applied in the vertical direction.

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Papermagnetic field throughout the sample. Field strength is adjusted
by applying current to the solenoid, and is controlled with Labview (National Instrumentst, Version 2014, Austin, Texas). An
inverted Leicat DMI6000B microscope (LEICA, Bannockburn, IL),is used to image the self-assembly process with a 40 objective.
The microscope is capable of automated focusing, and images arecaptured at the rate of two frames per minute with a Qimaging
Micropublishert 5.0 RTV Camera with resolution of 2560  1920
pixels (Qimaging, Surrey, Canada).
A custom code is written in MATLAB (Mathworksr, Version
2014, Natick, MA) for image processing and data analysis. The
circular Hough-transform algorithm30 is used to identify thecoordinates of particle centers. This particle identification algorithm is based on the assumption that particles have a sphericalshape, are of a specific size range, and have a diﬀerent intensityfrom that of the background. Since the magnetic and nonmagnetic particles are respectively darker and lighter than the
ferrofluid, particle types are distinguished by their averageintensity. Based on particle coordinates extracted in eachframe, we then generate Voronoi diagrams to ascertain local
symmetries, from which we identify nearest-neighbor bonds foreach particle.
We use two quantitative measures to characterize the timedependence of the crystallization process, including the average bond order parameter and the mean crystal domain size.
Only particles with nearest neighbor bonds nn greater than
three are included in the bond-order analysis. We choose this
cutoﬀ because it accounts for particles at the edge of a growing
crystallite but excludes particles at the corners or particles not
associated with the crystal. The 4-fold bond order parameter ofthe i-th particle is given by
80;nn o 3;
>
<
n
n
Fi4 
(1)


1 P
>
exp 4iyji ; nn  3;:
nn j¼1
where yji is the angle between the reference axis and thevector directed from the particle i to the particle j, and i is an
imaginary number equal to the square root of negative one.
Since we observe checkerboard lattices, we use an order parameter with 4-fold symmetry instead of the 6-fold symmetry
germane to hexagonally packed crystals.31 The absolute value ofthe bond order parameter of a generic particle i in the field ofview is averaged according to
hF4 i N 
1X
F i 
4
N i¼1

(2)

which quantifies the degree of 4-fold symmetry in the entire
system of N particles. Large regular crystals generate higher orderparameter values, with hF4i = 1 representing an ideal single crystal.
In addition to the bond-order analysis, we measure themean domain size of growing crystals based on a modified
cluster aggregation parameter originally developed by Vicsek
and Family.32 The algorithm first evaluates the real part of thebond order parameter of the nearest unlike particles, i and j.
 j4 exceeds a
If, for given i and j, the real part of the product Fi4FThis journal is © The Royal Society of Chemistry 2016

Fig. 2 Particles are colored according to the absolute value of the bondorder parameter |Fi4|. Disks with and without holes represent the magnetic
and nonmagnetic particles, respectively. Correlation bonds are shown inyellow or red. Bonds that form crystal domains and that satisfy the criteria
described in the text are shown in yellow.

threshold value (0.6 is chosen in this work) and the particle
shares the same two correlating neighbors with another particle,
then all four particles are considered associated with the crystal
domain. This process is repeated so that each particle is assigned
to a specific crystal for the entire field of view. An example
resulting from the crystal identification algorithm is shown in
Fig. 2, where particles connected by yellow line segments areincluded in the crystal but those connected by red line segments
are excluded.
We find that the criterion described above robustly identifies polycrystalline domains similar to those identified by visualinspection of the sample when viewed through the eyepiece ofthe microscope. Based on the number of particles included ineach domain, we then evaluate the mean domain size parameterP 2s nðsÞ
shSi  P
;
(3)
snðsÞ
s

where s is the size of domain and n(s) is the number of domains
having size s. This calculation averages out size fluctuations
between diﬀerent crystals, allowing the determination of a mean
domain size within each field of view.
The correlation functiongðrÞ ¼X  
1d r  ~rij  ;
2prDrrðN  1Þ iaj(4)which represents the ratio of the ensemble average of particles
within the region between r and r + Dr and the average number
density of the system r = N/LxLy can be used to ascertain whether
some kinds of impurities are present. Here, N is the numberof particles in the image size of Lx  Ly. In a perfect single
crystal, the centers of like particles are separated by a distancepﬃﬃﬃ
of 2 particle diameters, which is the diagonal length in asquare lattice. Accordingly, peaks of g(r) at smaller separation

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Soft Matterdistances (i.e., 1.0–1.1 particle diameters) between like particles
indicate the presence of aggregated particles.

3 Simulation model

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3.1

Colloid–colloid interactions

A colloidal particle suspension confined in a thin fluid film, having
a thickness slightly greater than the typical particle diameter, canbe considered as quasi-2D since particle translation is limited toin-plane motion. In our experiments, colloidal particles are suspended in a fluid containing iron oxide nanoparticles, called
ferrofluid, which we model as a uniform continuum with magnetic
permeability mf. We assume that mf is closely approximated interms of the bulk magnetic susceptibility wb and volume fraction
j of the nanoparticles by
mf = m0(1 + jwb),(5)

where m0 is the vacuum permeability. The magnetic moment mi ofa spherical particle i exposed to a weakly inhomogeneous external
magnetic field H can be approximated as a magnetic point dipole
mi E winiH,(6)

where wi is the shape-corrected eﬀective magnetic susceptibilityof the particle measured relative to that of the surrounding
ferrofluid, given bymi  mf
wi ¼ 3;(7)
mi þ 2mf
and ni is the volume of the particle. Thus, a magnetic particle
eﬀectively behaves as paramagnetic when its magnetic permeability
is larger than that of the ferrofluid and has an eﬀective magnetic
susceptibility wi 4 0. A nonmagnetic particle eﬀectively behaves as
diamagnetic, with susceptibility wi o 0, since its magnetic permeability is smaller than that of the ferrofluid. The binary suspensionsof interest in this work involve two kinds of particles: magnetic
particles, with wm and nm, and nonmagnetic particles, with wn and
nn. The volume fraction j of the nanoparticles is controlled to ensurethat the eﬀective magnetic moments of the two particle types areequal and opposite, namely to ensure that wmnm = wnnn is satisfied.
In our prior analyses, we have shown that self-consistent magnetic
moment calculations lead only to small corrections and can beignored for computational eﬃciency.12 Consistent with thesestudies, we do not account for these eﬀects here.
The dipole–dipole interaction energy between two colloidal
particles i and j, at positions Xi and Xj, is written as



m  
ðijÞ
UM ¼  f 3 3 mi  nij mj  nij  mi  mj ;
(8)
4prij
where nij  (Xj  Xi)/rij and rij  |Xj  Xi| respectively represent
the unit vector and distance between the centers of particles iand j. Since the external field H is applied along the z-direction(i.e., perpendicular to the monolayer), (8) is a function only ofrij. This leads to the following magnetic force:
3m
F ijM ¼  f 4 wi wj n i n j H 2 nij :4prij7738 | Soft Matter, 2016, 12, 7735--7746(9)Thus, whereas the magnetic force between like particles(
wiwj 4 0) is repulsive, the magnetic force between unlike
particles (wiwj o 0) is attractive.
Since the colloidal particles are nearly rigid spherical balls,
steric eﬀects prevent them from physically overlapping. To simulate this behavior, we employ a soft-sphere model with suﬃciently
large stiﬀness k, in which the following contact force F (ij)C arises
between overlapping two particles of radii ai and aj:(0;
rij 4 ai þ aj ;(10)
F ijC ¼k rij  ai  aj nij ; rij  ai þ aj :The colloid–colloid interactions encoded in (9) and (10) govern
the basic features of the system, the net interaction energy ofwhich is minimized when particles form a checkerboard lattice.12
However, to investigate the kinetics of crystallization, it is also
necessary to account for the solvent’s influence on particle
dynamics; that influence is discussed next.
3.2

Solvent–colloid interactions

The suspended colloidal particles also interact with the solvent
molecules of the host fluid, and we employ a Brownian dynamics
approach to capture the dominant influences of the solvent. Since
solvent molecules are many orders of magnitude less massive thancolloidal particles, it is reasonable to assume that they equilibrate
instantaneously. Accordingly, we can model their influencewith forces that induce viscous dissipation (i.e., hydrodynamic
forces) and thermal fluctuations (i.e., Brownian forces). With
this approach, solvent molecules are not explicitly tracked forthe sake of computational eﬃciency.
The hydrodynamic forces acting on a system of N suspended
particles can be represented as a 2N-dimensional vector FH 
(F1H,. . .,FN
H) of N two-dimensional hydrodynamic force vectors.
In terms of the 2N-dimensional vector U  (U1,. . .,UN) of twodimensional particle velocity vectors, FH has the form
FH = RU,(11)

where R is a 2N  2N resistance matrix. Note that all force
and velocity vectors in this section are 2N dimensional, which
represent in-plane components of N particles.
Since the Brownian forces FB  (F1B,. . .,F N
B ) are random, theirensemble average must vanish: hFBi = 0. To ensure that the
average kinetic energy of each colloidal particle attains kBT, thecorrelations must satisfy33,34
hFB(0)FB(t)i = 2kBTRd(t),(12)

where d is the delta function.
Since R is generally a dense matrix, it is advantageous to
introduce an approximation. In a dilute suspension, resistanceforces acting on particles are oppositely proportional to theirown velocities. Thus R can be approximated by a diagonal
matrix Rsd:
R E Rsd = zI.

(13)

Here, due to the small diﬀerence in size between the two
species of particles, the suspension is assumed to be monodisperse

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z 0 /z E 1.9 is found at farea = 0.68 for a certain range of fieldstrengths; specifically, the range of the eﬀective (dimensionless)temperature defined later is 0.05 t T̃ t 0.15.
3.3

Equation of motion for colloidal particles

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Since particle inertia is neglected, the particles obey a set ofquasi-static equations of motion, in which the individual particle
trajectories are obtained through a force balance:
FH + FB + FM + FC = 0,

Fig. 3 The mean square displacement (solid line) averaged for 10 particles
in the absence of an external magnetic field is shown as a function of time.
The diﬀusion coeﬃcient D0 E 0.0565 mm2 s1 is obtained through a best
fit to experimental data by hDd2i = 4D0t E 0.226t.

1
N
where FM  (F1M,. . .,FN
M) and FC  (FC,. . .,FC ) respectively represent the 2N-dimensional vectors of the magnetic and contact
forces. At each time step, the magnetic and contact forces,
FM and FC, and the resistance matrix, R, are calculated from theparticle configuration, and the random Brownian forces, FB, satisfying (11) are numerically generated. Eliminating FH between (11)and (14), we obtain

U = R1(FB + FM + FC),
for simplicity; thus, the resistance coeﬃcient z is the same forall particles. Due to the presence of confining glass boundaries
in our system, a better match with experiment requires z to be
determined from experimental measurements of the diﬀusion
coeﬃcient D0  kBT/z of isolated particles. The average value
hD0i E 0.0565 mm2 s1 is evaluated from experimental observations (Fig. 3), leading to an eﬀective Stokes’ drag coeﬃcient z about
2.5 times larger than that expected for water.
In dense suspensions, the hydrodynamic resistance is influenced by the presence of other nearby particles, which leads to
non-negligible oﬀ-diagonal terms in the resistance matrix R.
Strong hydrodynamic resistances, called lubrication forces, arise
from the presence of viscous fluid in the narrow gaps between
nearby particles.35 Under these circumstances, R may be approximated by an expression that includes the lubrication terms,
namely R E Rsd + Rlub, where the particular form of Rlub can befound elsewhere.36 If this resistance matrix is used instead of the
diagonal alternative (13), the computational cost of the simulations is obviously much higher. Fortunately, when we compare
simulations with and without lubrication (Fig. 4), we find that it
aﬀects only the time scale of the considered processes. Therefore, we can rescale the resistance matrix (13) in our simulation
with an adjusted resistance coeﬃcient z 0 . The adjustment factor

(14)

(15)

from which we determine the trajectories X of the colloidal
particles by integrating with respect to time.37
The dependence on external field strength H  |H| is ofessential importance in this work. On writing d = am + an, thecharacteristic force scale FMof the system is given by themagnetic force between magnetic and nonmagnetic particles
in contact under the field:FMw m j
wn jn m n n 2
3mf 
H :
4pd 4

(16)

Each term entering the equation of motion (14) is nondimensionalized based on this force scale: for example, FH becomesF~  F H F  . In the dynamics, the characteristic time scale tM
M

related to the magnetic force scale is given by
tM 

zd
 :
2FM

(17)

With force and time scales (16) and (17), the relation (12)determining the strength of Brownian forces is nondimensionalized in accord withhF̃B(0)F̃B(t̃ )i = 2T̃R̃d(t̃ ),(18)

where t̃  t/tM and R̃  R/z. The dimensionless number
2kB T
8pd 3 kB T
T~ ; ¼ 3m w
wn jn m n n H 2
dFM
f m j

(19)

therefore serves as an eﬀective temperature. From (19), we candecrease T̃ by increasing the strength H of the external field.
3.4

Fig. 4 Simulation without lubrication forces (black) compared to a simulation with lubrication forces (blue) using (a) the average order parameterhF4i and (b) the mean domain size hSi. The simulation neglecting lubrication
forces with an adjusted resistance coeﬃcient z 0 E 1.9z (red) can capture thefeatures of the simulation with lubrication forces.

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Boundary conditions

In the experimental system, particles assemble within a fluid filmof approximately 20 mm (in lateral dimensions). The observed
field of view is distant enough from any boundaries of the sample
to ensure that boundary eﬀects are practically negligible.
In simulations, on the other hand, computational limitations
require that we consider only a small fraction of the system,
which can lead to undesirable boundary eﬀects. To avoid thisSoft Matter, 2016, 12, 7735--7746 | 7739

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potential problem, we use square simulation boxes with periodic
boundary conditions. Additionally, to avoid finite system size
eﬀects, which can lead to correlations between a crystal and its
periodic images from the system boundaries, we restrict oursimulations to systems of greater than 10 000 particles.

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4 Results and discussion
4.1

Order parameter and domain size

To explore the kinetics of crystal growth, we developed anexperimental system that permits dense monolayers of particles
to be studied in a static uniform vertical magnetic field. Fig. 5
shows typical snapshots of the liquid–solid transition obtained
from experiments (a–c) and simulations (d–f). The experimental
field of view is a rectangle with a length and width of 85 and 114
particle diameters, respectively, with each field of view containing approximately 8000 particles. In this figure, the particle
concentration is fixed at farea = 0.66, while the magnetic fieldstrength H is held constant at 7 Oe (corresponding to aneffective temperature of T̃ E 0.081 in (19)). In simulations,
we model a system commensurate in size to the experimental
field of view with N = 10 000 particles (see Section 3.4). Prior to
the quenching process, the Brownian simulation with H = 0(i.e., at T̃ = N) is run for 10 minutes to ensure that the initial

Soft Matter

configuration is a thermal equilibrium state in which magnetic
forces are absent. Simulation conditions are calibrated to approximate experimental conditions (see Table 1, Movies S1 and S2, ESI†).
The time evolution of crystal growth is shown for a situationin which the system is suddenly exposed to a constant verticalmagnetic field at t = 0, which mimics a rapid quenching process
initiated at infinite temperature. Applied magnetic fields induce
the initial liquid phase to rapidly solidify, nucleating small crystals
within the first 20 minutes of experiments. In this early stage of thenucleation process, the dominant eﬀect of interfacial energy
causes nuclei containing 4–30 particles to spontaneously appear
and disappear with time. Over the next phase of crystal growth,which occurs from 20–60 min, larger crystals survive at the expenseof the smaller crystals and the system continues to coarsen until
most particles are incorporated into crystals. At low values of H
(close to the solidification temperature), the liquid phase coexists
with small crystallites that continue to grow until the end of theexperiment. At higher values of H, the system coarsens until the
crystals merge and form domain boundaries, such as those shownin experiment (Fig. 5c) and simulation (Fig. 5f).
Some examples of polycrystalline structures are provided inFig. 6a (experiments) and Fig. 6b (simulations), which depict
typical domain structures after 225 minutes of elapsed time
for diﬀerent values of the eﬀective temperature T̃. For ease ofvisualization, the domains are presented as color coded Voronoi

Fig. 5 Time evolution of crystal growth during the rapid quenching process at an eﬀective temperature of T̃ E 0.081 (H = 7.0 Oe) in experiment (a–c)
and T̃ E 0.076 in simulation (d–f). Magnetic (red) and nonmagnetic (white) particles in the experiment (a–c) are assigned gray and red colors in thesimulation (d–f). Images have a size of 250 mm  250 mm. Scale bars on images show 25 mm.

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Table 1

Simulation parameters

Magnetic particle diameter
Nonmagnetic particle diameter
Magnetic particle’s eﬀective susceptibility
Nonmagnetic particle’s eﬀective susceptibility
Magnetic permeability of ferrofluid
Temperature

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dm
dn
wm
wn
mf
T

2.95 mm2.95 mm0.36
0.36
1.47m0298 K

cells to highlight diﬀerent crystal orientations (i.e., arg Fi4).
Disordered regions are indicated with black. The clear dependence of mean crystal size on T̃, or, equivalently, the appliedmagnetic field strength H, is evident in both experiments andsimulations. In a weak magnetic field of H = 5.5 Oe, corresponding to the solidification temperature T̃ E 0.13, small
crystal domains containing 4–100 particles are surrounded by
the disordered phase. On the other hand, in a strong magnetic
field of H = 13.5 Oe (T̃ E 0.022), the mean crystal size is equallysmall but are caused instead by the slow annealing kinetics,
which impede crystal growth during the experimental timeinterval. Maximum crystal sizes are observed in the medium
range of H = 8.5–10.5 Oe (T̃ E 0.036–0.055), which is slightly
below the solidification temperature. This strong dependenceof crystal size on quenching conditions indicates the existenceof an ideal field strength H for promoting the growth of large
single crystals. The value of H should be high enough tomaintain strong particle association with the crystal but weakenough to allow defects to diffuse out of the crystal interior, as
well as to allow domain boundaries to merge and reform.
We use several order parameters to monitor the crystal
growth process in time. Fig. 7 depicts the average bond orderparameter hF4i as a function of time for various values of T̃ inexperiments (a) and simulations (b). The plot shows that hF4i
rapidly intensifies and reaches almost 90% of the maximum
value within the first 20 minutes in experiments, and within thefirst 30 minutes in simulations. After the first 30 minutes,

hF4i gradually increases for the remaining time in experiments
and simulations. Though simulation and experiment do notmatch perfectly, they are reasonably consistent. The maximum
average value of hF4i in experiments is 6% smaller than values
found in simulations, which is most likely due to the presenceof particle impurities that hinder crystal growth. We do notcontinue our experiments past 300 minutes, due to the onsetof fluid evaporation and bubble formation, which perturb the
fluid and damage the growing crystals.
The crystal growth process can also be analyzed by plotting
the mean domain size as a function of time on a log–log scale,where the linear slope indicates a power law growth process.
Fig. 8 exhibits the time evolution of the mean domain size during
rapid quenching for different values of the applied magnetic field
H. Since the domain sizes in simulations (B5000 particles) tend
to be significantly larger than in experiments (B600 particles),Fig. 8a and b are plotted on different logarithmic scales. The
discrepancy between experiment and simulation is thought toarise from the influence of impurities which are present inexperiments, but not in simulations. Similar to the resultspresented in Fig. 7, the largest crystals form in the range of
8.5 Oe t H t 10.5 Oe (0.036 t T̃ t 0.055), which corresponds
to the global maximum in the bond order parameter. When thesample is quenched at a low temperature around T̃ E 0.022, themean domain size is limited to around B100 particles. Theseobservations highlight the importance of finding an optimal
condition for obtaining large single crystals.
In most simulations and experiments, we observe several
distinct power laws, which are thought to be associated with
diﬀerent crystal growth regimes ranging from diﬀusion-limited tointerface-limited growth kinetics.38–40 These two growth regimes
are clearly visible at low temperatures (T̃ t 0.034) in bothexperiments and simulations. The power law associated with thefirst growth regime is significantly larger than that of the second
growth regime. To grow larger single crystals, it is importantFig. 6 Polycrystalline domains formed in experiments (a) and simulations (b) at diﬀerent temperatures. Crystal domains are colored according to local
orientations of particles. Black coloration depicts the regions where particles do not form crystalline structures. The scale bars are 50 mm.

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Fig. 7 The average bond order parameter hF4i in experiments (a) and simulations (b) are provided as a function of time for diﬀerent eﬀective
temperatures. In each experiment, the external magnetic field was held constant for 225 min. The average order parameters are calculated by taking theensemble average of local order parameters within a given field of view as a function of time, captured at a rate of two frames per minute.

Fig. 8 The mean domain size hSi depicted as a function of time at diﬀerent eﬀective temperatures in the range 0.022 tT̃ t 0.130 in experiments (a) andthe range 0.034 t T̃ t 0.151 in simulations (b). The system size is displayed as a log–log scale for easier visualization of the power law fitting functions
shown with the dashed lines. Domain sizes obtained in simulations are significantly larger than experiments because they do not include impurities.

both to increase the magnitude of the power law as well as thedwell time in this first growth regime. At low temperatures, thecrossover time between the two growth regimes typically occurs
in the first 10–80 minutes. However, at optimal temperatures in
the range of 0.036 t T̃ t 0.081, we only observe one regime of
growth kinetics. Again, we note that the best fitting power laws
in experiments are always smaller than those arising in simulations due to the presence of impurities.
To check the eﬀect of the cooling rate, we conduct a series ofsimulations with the ramping temperature defined in accord with8
 1
< 1=gt; 0 o t o gT~s ;
~ ¼
TðtÞ
(21)
 1
: ~
Ts ;
t  gT~s ;where T̃s is the setting point of eﬀective temperature chosen to
be T̃s E 0.048, and g is the linear temperature ramp ranging
between 0.18 min1 and 1.8 min1. Surprisingly, the averageorder parameters are indistinguishable from those obtained by
rapid quenching to the same final temperature (Fig. 9b) for all
values of g. The polycrystal domains shown in Fig. 9c also show
negligible diﬀerences, which indicates that the growth kinetics
are not influenced by the cooling rates. Nonetheless, thesesimulations cover only the short time intervals accessible inexperiment and are limited to only a specific temperature setting.

7742 | Soft Matter, 2016, 12, 7735--7746

More exhaustive simulations of diﬀerent cooling rates and final
temperatures may lead to even larger crystals than those observedin our experiments.
4.2

Eﬀect of impurities

In contrast to experiments, simulations allow for the consideration of systems in which all particles are identical in size,
perfectly dispersed, at equal population fractions, and free ofother types of interactions. Under such conditions domains
grow continuously up to several thousand particles on average,
which is much larger than in experiments, where the presence of small particle clumps and occasional giant particles(1.3–1.8 times larger than the normal particles), as well as aslight imbalance between the number ratio of particles restricts
the maximum domain size. As previously discussed, the nonmagnetic particles often have pre-existing clumps that frustrate
crystal growth (Fig. 10a). Fig. 10c shows an abnormal peak in
the pair correlation (4) between nonmagnetic particles (n–n),which is found at a distance of 1.04 particle diameters and isindicative of doublets formed from nonmagnetic particles within
the sample. Since direct contact between like particles isenergetically unfavorable, we use the pair correlation function
g(r) as a qualitative indicator of irreversible binding between
nonmagnetic particles.

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Fig. 9 The temperature ramp rates used in simulations (a) all converge to a similar value at long times (b). Snapshots of the domain structures at the endof the simulated 350 minute time interval are shown for the diﬀerent temperature ramp rates (c). The polycrystals are plotted at t = 350 min with the same
scale and color scheme shown in Fig. 6.

Fig. 10 Irreversible binding between non-magnetic particles (a) and thevertical stacking of particles (b), respectively. The pair correlation g(r) ofmagnetic–nonmagnetic (m–n), magnetic–magnetic (m–m), nonmagnetic–
nonmagnetic (n–n) are shown for a typical experiment H = 7.5 Oe (T̃ E 0.07)
after 225 min of rapid quenching (c). A log scale is applied to the vertical axis
to compress the data and better visualize the peaks.

Another source of discrepancy between simulation andexperiment is the possibility that particles may rise out of themonolayer plane (Fig. 10b). Although this occurs in our experiments, it is not permitted in our simulations. Our image
processing algorithms are not optimized to recognize stacked
particles; thus, we do not observe any unusual correlation peakThis journal is © The Royal Society of Chemistry 2016of the magnetic particles. However, we can sometimes see thiseﬀect in experimental images such as those shown in Fig. 10b.
Although we do not develop an algorithm to precisely quantify
the concentration of impurities in our experimental system, weare able to estimate the impurity concentration to be in the rangeof B1–3% of particles, including B0.1% large particles, B0.1%
pinned particles, B1% doublets formed between nonmagnetic
particles, and a typical number ratio of 51 : 49 between themagnetic and nonmagnetic particles, which vary from sample
to sample.
In an attempt to better understand the eﬀect of impurities inexperiments, we include two types of impurities in our simulation: (i) particles pinned to substrates, and (ii) giant particles.
For the sake of simplicity, we ignore particle clumps to avoid
issues with trajectory analysis of rigidly linked bodies. As in our
experiments, we perform simulations in which the particle
concentration is kept constant at farea = 0.68, while thepercentage of impurities is varied in the range of 0–6% and isassumed to be initially randomly distributed in the simulation
box. Giant particles are assumed to have diameters 1.5 times
that of other particles. On the other hand, pinned particles are
assumed to be the same diameter as other particles, but are
spatially fixed. The impact of impurities on the average bondorder parameter and the mean domain size in the case ofT̃ E 0.055 are shown in Fig. 11. As expected, these quantities both
decay significantly as the number of impurities is increased.
The best match with experiment is achieved when simulations
employs a impurity concentration of 2%, which is consistent
with the level observed in experiments.
Fig. 12 shows the average bond order parameter hF4i and themean domain size hSi as a function of the quenching temperature after 225 minutes of elapsed time. Simulation results are
obtained by varying the concentrations of pinned particles in
the range of 0–6%. Similar to what is shown in Fig. 11, the best
match with experimental results is obtained when simulations

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Fig. 11 The average bond order parameter hF4i and mean domain size hSi (insets) are shown as a function of time in simulations that include up to 4%
impurities of either pinned particles (a) or giant particles (b). In all simulations, the particle area fraction is fixed at farea = 0.68 and the temperature is kept
at T̃ E 0.056. The black line shows the experimental data obtained in similar conditions. The best match between experiment and simulations is foundwhen simulations include approximately 2% impurities, regardless of its specific type.

Fig. 12 The average bond order parameter hF4i (a) and mean domain size hSi (b) of the system are presented as a function of temperature after 225 minuteof elapsed time. Simulations include diﬀerent amounts of pinned particles ranging from 0% to 6%, and are compared with experimental data (red) withsimilar growth conditions. In all simulations the particle concentrations are held at farea = 0.68, whereas in experiments the particle concentrations were
kept at farea E 0.66 0.02. The impurity-free simulations are shown in black, whereas the simulations that include impurities are colored from dark to light
blue. The best match with experiment across all temperatures is obtained for simulations that include approximately 2% pinned particles.

include an impurity concentration of approximately 2%. Near theeffective solidification temperature, the slight mismatch between
experiment and simulation is not unexpected due to the influenceof unknown short-range particle interactions (namely surface
charge repulsive forces and other surface forces), which are notincluded in our simulations. These interactions should be negligible when the magnitude H of the applied magnetic field is
sufficiently large, where an improved quantitative agreement
between experiments and simulations is observed.
Matching the experimental curve to simulations leads to abest fitting parameter for the magnetic permeability of theferrofluid determined from (7) with wn E 0.36, which yields
mf E 1.47m0. According to (5), this best fitting estimate of theferrofluid permeability implies that the bulk susceptibility ofthe material constituting the nanoparticles is wb E 47. The
permeability of the magnetic bead can be estimated similarly
from (5) with wm E 0.36, which yields mm E 2.07m0. This value
suggests that the magnetic susceptibility of the magnetic particles in vacuum is wm E 1.07, which is larger than the typicallyreported values of these bead types in water falling in the range of
0.17–0.96.41–43 To facilitate comparisons with our prior paper,12in which a correction factor a = 2.4 0.3 was used, that acted as
multiplier in the particle susceptibilities, we estimate the best

7744 | Soft Matter, 2016, 12, 7735--7746

fitting parameters for the magnetic permeabilities of the magnetic particles and ferrofluid to be mm E 2.56m0 and mf E 1.66m0,
respectively. Thus, the magnetic parameters of the present study
provide a slightly better approximation of the expected values
in vacuum.
Despite our best eﬀort to balance the relative particle concentrations, the numbers Nm and Nn of magnetic and nonmagnetic particles diﬀer in experiments. To examine the influenceof this impurity type, we simulate particle concentration diﬀerences x ranging from 0 to 0.1, where x  (Nm  Nn)/(Nm + Nn). The
eﬀects of the population imbalance on the average bond orderparameter hF4i (Fig. 13a) and the mean domain size hSi (Fig. 13b)
are depicted after a 225 minute elapsed time interval in a constanttemperature of T̃ E 0.048. As expected, both hF4i and hSi areobserved to decrease with increasing population imbalance. Thiseﬀect is not surprising, because an excess of one particle type leads
to a non-zero net dipole moment of the crystal, which induces
stress and hinders crystal growth.44 At the ratios typically observedin experiments, xex = 0.035 0.018, the values of hF4i and hSi areslightly smaller than those obtained under ideal simulation conditions (x = 0). This value of x corresponds to one particle typebeing present at an excess of approximately 3%, which is in asimilar range to the other types of considered impurities.

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Fig. 13 The average bond order parameter hF4i, and mean domain size
hSi are plotted as a function of the relative particle concentration ratio
in the range of x = 0–0.1. The simulations were averaged over ten
independent runs in order to obtain the mean and standard deviations.
All simulations were conducted for 225 minutes and at a temperature ofT̃ E 0.048. The vertical blue lines show the average population imbalanceof the typical experiments: xex E 0.035. The red dashed curves areprovided as a guide to the eye.

The influence of population imbalance on the growth kinetics
is shown in Fig. 14, which depicts hSi as a function of time. Twocrystal growth regimes can be distinguished in both the perfectly
balanced system and in the system containing an excess of one
particle type with x = 0.08. Comparing these corresponding
growth curves, we find several features demonstrating how theperfectly balanced system attains larger crystalline domain sizes
than the system containing a particle imbalance. First, in theperfectly balanced system the power law of the first growth
regime attains a higher value (1.26) than that found in the system
including an excess (0.81) of one particle type. This more rapid
initial growth process allows the perfectly balanced system toreach larger crystal sizes before entering into the ripening phaseof the second growth regime. Second, in the perfectly balanced
system the cross-over time between the diﬀerent growth regimes
is delayed in comparison to what occurs for the system containing a particle imbalance. Thus, the perfectly balanced system has
more time to grow domains prior to entering the second regime
with slower growth kinetics. For comparison, we also provide one
experimental plot for a system with similar growth conditions as

Fig. 14 The two regimes of growth kinetics are shown for a perfectly
balanced system (x = 0) and for a system that contains an excess of one
particle type corresponding to x = 0.08, with the eﬀective temperature
fixed at T̃ E 0.048. Simulations (black lines) were averaged for ten independent runs, whereas the result from one experiment with growth conditions similar to simulations is shown in orange. The best fitting power laws
for the first and second growth regimes are shown with the blue dotted andred dotted lines, respectively.

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the simulation with an assumed population balance x = 0.08. The
data from this single experiment contains more noise than thedata arising from simulations, which is averaged over 10 individual trials.
Based on our simulations, we conclude that the number
rather than the specific type of impurities is the determining
factor for matching with experiment. Thus, instead of developing a complex model, which incorporates multiple kinds ofimpurities in one simulation, we instead focus on a single
impurity type. In Fig. 11 and 12, we show that including a small
percentage of pinned or giant particles can provide a matchwith experiments. In Fig. 14, we also show that an imbalance inthe particle number ratio can also match experiments at theeﬀective temperature (T̃ E 0.049) and timescale (t = 225 min),with an assumed population imbalance x = 0.08. It is thus clearthat a small percentage of defects can dramatically influence
the domain sizes of 2D crystals.

5 Conclusions
We developed an experimental system for forming two-dimensional,
close-packed, colloidal crystal alloys with more than B1000 particles
in a single domain. Potentially optimal annealing kinetics wereachieved by simultaneously controlling the density and particle
interactions within a 2D monolayer of magnetic and nonmagnetic particles immersed in a ferrofluid. Specifically, thebest crystals were obtained for area fractions of 65–70%, which
are small enough to avoid glass or jamming transitions occurring near the ideal packing fraction of a square lattice (B78%)
while being large enough to prevent voids, which are common
below an area fraction of 60%. We also found that the largest
crystals were obtained for magnetic fields with strengths in therange of 8.5 Oe t H t 10.5 Oe, which are high enough toovercome thermal fluctuations while being low enough to allow
efficient annealing of domain boundaries and to remove
impurities from crystal interiors.
Brownian dynamics simulations based on a point dipole model
were found able to reproduce the main features of the assembly
process; however, they predict much larger crystals than are
obtained in experiments. Results obtained from simulations of
an idealized, impurity-free system agreed qualitatively but deviatequantitatively from experiment, where a small percentage ofparticles that form clumps and adhere to the substrate are
generally present. An imbalance in the particle number ratio also
led to smaller crystals, mimicking the eﬀect of the other experimental impurities. We found that including a similar percentage
of impurities in simulations (1–2%) yields better quantitative
agreement between theory and experiment. Additionally, weachieved a more precise characterization of the magnetic
susceptibility of the magnetic bead wm = 1.07, which is 30%
smaller than in our prior analyses.12
While the mean domain size we were able to experimentallyobtain is on the order of B600 particles, future improvements ofthe experimental conditions through the use of better passivatedsurfaces, the reduction of particle aggregation and imbalance inSoft Matter, 2016, 12, 7735--7746 | 7745

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particle concentration should enable us to approach crystal
sizes of B5000 particles predicted in the idealized simulations.
The ability to obtain suﬃciently large colloidal crystal alloys
will facilitate future scientific investigations on grain boundary
motions,9 dynamics of vacancies and defects,8 jamming transitions,45 and various types of phase transitions that are diﬃcult
or impossible to observe in atomic systems.

Acknowledgements
The authors are thankful for support from the National Science
Foundation Research Triangle Materials Research Science andEngineering Center (DMR-1121107) and the Okinawa Instituteof Science and Technology Graduate University with subsidy
funding from the Cabinet Oﬃce, Government of Japan.

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