PHYSICAL REVIEW E 95, 032607 (2017)
Quantifying disorder in colloidal films spin-coated onto patterned substrates
Raheema Aslam,
1
Sergio Ardanza-Trevijano,
1,2
Kristin M. Poduska,
3
Anand Yethiraj,
3,*
and Wenceslao Gonz
´
alez-Vi
˜
nas
1,4,†
1
Universidad de Navarra, Complex Systems Group, Pamplona E-31008, Spain
2
Universidad de Navarra, Topology and FUZZY Logic Group, Pamplona E-31008, Spain
3
Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John’s, Newfoundland A1B 3X7, Canada
4
Universidad de Navarra, PHYSMED Group, Pamplona E-31008, Spain
(Received 11 December 2016; published 17 March 2017)
Polycrystals of thin colloidal deposits, with thickness controlled by spin-coating speed, exhibit axial symmetry
with local 4-fold and 6-fold symmetric structures, termed orientationally correlated polycrystals (OCPs). While
spin-coating is a very facile technique for producing large-area colloidal deposits, the axial symmetry prevents
us from achieving true long-range order. To obtain true long-range order, we break this axial symmetry by
introducing a patterned surface topography and thus eliminate the OCP character. We then examine symmetry-
independent methods to quantify order in these disordered colloidal deposits. We find that all the information
in the bond-orientational order parameters is well captured by persistent homology analysis methods that only
use the centers of the particles as input data. It is expected that these methods will prove useful in characterizing
other disordered structures.
DOI: 10.1103/PhysRevE.95.032607
I. INTRODUCTION
Producing perfect, close-packed colloidal crystals is a
cheap means to fabricate photonic band-gap materials [1–3].
Numerous techniques have been used to obtain well-ordered
dried colloidal deposits [4–17]. However, it is challenging to
obtain defect-free colloidal crystals over large areas. There is a
growing realization [18] that nonequilibrium techniques might
need to be employed to achieve better control of colloidal
self-organization.
Spin-coating has recently been introduced in colloid science
as a far-from-equilibrium modality to fabricate colloidal
crystals [11,19–22]. It is very fast, reproducible, simple, and
needs less material [20]. However, the axial symmetry of
spin-coating makes the resultant colloidal films polycrystalline
in a peculiar way, i.e., the orientationally correlated polycrystal
(OCP) [11].
In spin-coating, the thickness and the uniformity of the
films are important and strongly dependent on several control
parameters, including the spin time and speed, the viscosity
of fluids, the density and the evaporation rate of the fluids, the
concentration of the suspension, as well as the substrate surface
characteristics [20,23,24]. Electric fields have been applied
during spin-coating of colloids to produce translationally
ordered structures in Refs. [25,26]. Thus, if one can overcome
the primary challenge set by the axial symmetry of spin
coating, i.e., the emergence of OCPs, numerous ways to
achieve better order would arise.
To achieve reliable and measurable improvements in order,
it is crucial to characterize the degree of structural order and
disorder systematically. Although there exist several methods
for characterizing colloidal structures in real space [27–30],
these translational- and orientational-order-based methods are
more suited to homogeneous structures of reasonably high
crystallinity. For example, a recent comparison of various
*
ayethiraj@mun.ca
†
wens@unav.es
different methods for making colloidal crystals [30,31]uses
bond-orientational order parameters that can sensitively dis-
tinguish crystals for which 0.8 <
6
< 1.
In this manuscript, we address two goals. We first report a
patterned surface topography technique to generate monolayer
colloidal polycrystals, while at the same time eliminating the
appearance of axial symmetry by spin-coating (i.e., the OCP
character). The effect of surface patterning geometry on the
morphology of spin-coated colloidal deposits is observed, and
the influence of the scale spacing of the patterning topography
on the structural order of colloidal deposits is presented.
Second, we introduce the use of persistent homology
methods to examine structural heterogeneity in these colloidal
polycrystals. One barrier to attempt at improving crystallinity
by varying one of the many control parameters is the absence
of reliable, quantitative measures of order in relatively poorly
ordered colloidal films. Reciprocal-space methods, e.g., using
small-angle and ultrasmall-angle scattering [32,33], are the
standard for characterizing crystalline structures; however,
spatial resolution is important both for identifying subtle
differences in disordered structures, as well as for following
the kinetics of crystallization. Real-space methods such as
those discussed here can also be used in tandem with new de-
velopments in coherent x-ray diffractive imaging [34]. In this
work, we show that information about orientational order in
spin-coated colloidal deposits, obtained by using orientational-
order based Minkowski structure metric [31,35,36], can be
complemented by an examination of structural heterogeneity
via persistent homology using the first Betti number [37].
II. EXPERIMENTAL METHODS
A. Fabrication of patterned substrates
We designed different photomask geometries to make
triangular arrangements of equally sized regular hexagonal
pillars, as shown in Fig. 1(a). The hexagon sides were 0.55 mm.
We call the distance between adjacent pillars the “scale
spacing,” and this was varied from 0.18 to 1.5 mm, while
2470-0045/2017/95(3)/032607(9) 032607-1 ©2017 American Physical Society
RAHEEMA ASLAM et al. PHYSICAL REVIEW E 95, 032607 (2017)
(b)
mm
864
β
Center of rotation
mm
3
2
4
(a)
(c)
FIG. 1. (a) Optical microscope image of the top surface of a
colloidal film deposited onto a patterned substrate (scale bar is
0.680 mm). (b) Sketch of the AFM scanning regions (rectangles),
at 2.24 mm intervals, where every scanned region makes an angle of
β = arctan(1/2) with radial direction (dash-dotted lines) away from
the center of rotation. (c) AFM image of a colloidal monolayer on the
hexagonal pillar with the detected particle centers indicated as red
marks (scale bar is 2 μm).
keeping the height of the pillars constant at 10 μm. To do
so, SU–8, 2010 (UV-sensitive photoresist, MicroChem [38])
was spin-coated at 3000 rpm for 30 s to achieve a thickness
of 10 μm on top of microscope cover slips (22 mm ×
30 mm). Thereafter, the templates were soft-baked at 95
◦
C
for 3 minutes and exposed to UV-light by using a maskless
patterning system (Intelligent Micro Patterning LLC) for
3–4 minutes. Then, the templates were heated to 95
◦
Cfor
3 minutes as a hard postexposure bake. The templates were
then developed for 2 minutes with ethyl-lactate and a 5%
NaOH aqueous solution. This photolithography step enhanced
the hydrophilic character of the template surfaces. Finally, the
patterned cover-slip templates were glued onto commercially
available microscope glass slides prior to performing the
colloidal spin-coating experiments.
B. Materials
Silica particles of diameter 458 ± 2nmweredried
overnight at 150
◦
C in a convection oven to minimize the
amount of absorbed moisture. After removal from the oven
and cooling to room temperature, a volatile solvent (ethanol
95%) was added to the particles to prepare a suspension with
20% (v/v) concentration. Ultrasonication, typically for four
hours, was then used to obtain a homogeneously dispersed
suspension.
C. Experiments
Spin-coating experiments were performed in an air envi-
ronment at room temperature with a commercial spin-coater
(Laurell technologies, WS-650SZ-6NPP), which was housed
in a fume hood to protect users from solvent vapors. When
the patterned substrate reached a constant angular velocity
(3000 rpm), 60 μL of the colloidal suspension was pipetted
onto it. The spinning was stopped when the dispersion had
completely dried on the patterned substrate. A representative
example of a dried monolayer colloidal film is shown in
Fig. 1.
D. Characterization
The lithographic templates were inspected by optical
microscopy (Nikon Eclipse 80-i upright microscope) prior to
and after spin-coating. A representative optical micrography
of a dried spin-coated film on a template is shown in Fig. 1(a).
Atomic force microscopy (AFM, Asylum Research MFP-3D)
was used to get images that we could use to get particle
positions. Prior to AFM measurements, a thin layer (≈100 nm)
of poly(methyl methacrylate) (PMMA) was spin-coated on
top of the colloids to prevent the AFM tip from intermittently
picking up—and then dropping—colloidal particles. The AFM
was operated in contact mode. A representative AFM image
in Fig. 1(c) shows the microscopic structure of the top surface
of a monolayer colloidal deposit on top of an hexagonal pillar.
We scanned and captured regions of 20 × 20 μm
2
at regular
intervals along the radial direction of each sample, choosing
an angle that was not along any high-symmetry direction. We
achieved this by making a series of 2 mm horizontal offsets,
coupled with 1 mm vertical offsets, so that each successive
image was equally along a radial direction that makes an angle
of β = arctan(1/2) ≈ 26.6
◦
, away from the center of rotation.
A schematic diagram is in Fig. 1(b).
III. STRUCTURAL ANALYSIS METHODS
To characterize colloidal structures in terms of their local
order, we further quantify the AFM images of colloidal deposit
on the hexagonal pillar in depth with different methods. To
do so, in each AFM image, the positional coordinates of
the particles are obtained [a sample of which is shown in
Fig. 1(c) with tiny spots at the centroid of each detected
particle], using home-made routines which are based on the
scikit-image processing library [39]. First, we observed from
the pair correlation function g(r) that there is no long-range
translational order in any of the deposits, and that they
were qualitatively indistinguishable. Then, they were analyzed
with bond-orientational order parameters (characterized by a
Minkowski structure metric) and persistent homology tech-
niques. Finally, we present a comparison of structural order in
colloidal films based on these two kinds of analysis.
032607-2
QUANTIFYING DISORDER IN COLLOIDAL FILMS SPIN- . . . PHYSICAL REVIEW E 95, 032607 (2017)
A. The Minkowski structure metric ψ
msm
s
The Minkowski structure metric (msm) is obtained by
modifying the conventional local bond-orientational order
parameters ψ
s
for a given weight s [40]. ψ
s
calculated for
particle a was expected to serve well for a measure of s-fold
symmetry in highly crystalline samples in the surrounding of
the aforementioned particle. The most common arrangements
in two-dimensional (2D) ordered structures are square (s = 4)
and hexagonal (s = 6). As a matter of fact, some fundamental
difficulties [36] were detected in this method. It assumes the
geometrical arrangement of a set of nearest neighbors N N(a)
around a particle (or vertex of the structure) a; where N N(a)
has greater influence on ψ
s
(a) values specifically in the case
of a square structure [41]. Also, the discrete nature of N N(a)
is not a continuous function of the particle coordinates which
is responsible for the lack of robustness of ψ
s
as structure
metric. As a result, the Minkowski structure metric (msm)
ψ
msm
s
is introduced to overcome these issues [31,35,36,42].
In the msm, the contribution of each nearest neighbor to the
structure metric is weighted by a relative length factor l(λ)/L,
where l(λ) is the length between two neighboring vertices of
the Voronoi cell of particle a that corresponds to a given bond
λ (or edge of the structure), and L =
λ
∈B(a)
l(λ
)isthetotal
perimeter length of the Voronoi cell of particle a, as can be seen
in Fig. 2. B(a) is the set of bonds which link particle a with
its nearest neighbors N N(a). This simple modification leads
to a robust, continuous, and parameter-independent structure
metric ψ
msm
s
(a) which avoids the flaws of the bond order
parameters ψ
s
. The Minkowski structure metric ψ
msm
s
(a)in
2D is defined as
ψ
msm
s
(a) =
λ∈B
(
a
)
l
(
λ
)
λ
∈B
(
a
)
l
(
λ
)
e
isθ
ab
(
λ
)
, (1)
where θ
ab(λ)
is the angle of the bond λ (which links a to b;see
Fig. 2) with a reference axis.
B. Persistent homology
The msm successfully quantifies the degree of s-fold
orientational order in crystalline structures. Nevertheless, the
a
λ
b
l( )λ
FIG. 2. Voronoi diagram, overlaid on part of an AFM image of
a spin-coated deposit. The heavy yellow line l(λ) shows the length
between two neighboring vertices (large and open green circles) of the
Voronoi cell of particle a (small and solid blue dots) that corresponds
to a given bond λ with particle b.
information given by it in the case of disordered systems is
limited, and the order (or disorder) characterization has to be
complemented by other, global, techniques.
Persistent homology is a tool with roots in algebraic
topology that has been applied to extract topological infor-
mation from geometric data. It has been notably successful
in the analysis of particulate systems [37,43–48]. To describe
what persistent homology measures, we first introduce the
Vietoris–Rips complex. Then we give an intuitive idea of
what the homology of a simplicial complex is and, finally, we
describe persistent homology. For more details on homology
and persistent homology we refer to Ref. [49].
Given a set of points (in our case the center of the particles),
a metric (Euclidean distance), and a “filtration” parameter
value δ we construct a two-dimensional simplicial complex,
called the two-dimensional Vietoris–Rips complex (VRC) that
has the set of points as vertices, an edge for each pair of vertices
v
i
and v
j
such that d(v
i
,v
j
) <δ, and a face (a triangle in this
case) for each triplet of vertices with pairwise distance less
than δ. The resulting object with vertices, edges, and faces is
the two-dimensional Vietoris complex that we denote K(δ).
Note that, if δ
1
δ
2
,K(δ
1
) ⊆ K(δ
2
). An example of how the
VRC changes for increasing values of the filtration parameter
can be found in Fig. 3(a). Homology measures the number of
“holes” in different dimensions in simplicial complexes. More
concretely, given a simplicial complex K, homology assigns a
sequence H
i
(K)ofZ
2
-vector spaces, where i is a non-negative
integer. The dimension of the vector space H
0
(K) is called the
zeroth Betti number β
0
and counts the number of connected
components of our simplicial complex, and the dimension of
FIG. 3. Examples of persistence homology analysis on three
compact artificial lattices (see text). (a) Vietoris–Rips complex (VRC)
for square (I), sheared square [II, C in panel (b)] and hexagonal (III)
at the same filtration parameter (δ = 1.2σ ) [open circles in panel
(c)]. (b) VRC for increasing values of filtration parameter δ [stars
on dashed, red line in panel (c)], δ
A
= 1σ , δ
B
= 1.1σ , δ
C
= 1.2σ ,
δ
D
= 1.3σ .(c)β
norm
1
vs δ/σ for the three artificial lattices, where
β
norm
1
is normalized in such a way that the maximum value for a
perfect square lattice is 1. The widest plateau appears for the square
lattice.
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RAHEEMA ASLAM et al. PHYSICAL REVIEW E 95, 032607 (2017)
the vector space H
1
(K), which is called β
1
, counts the number
of loops in our graph that are not covered with triangles (the
number of holes or uncovered loops in the two-dimensional
complex). We can find bases of these vector spaces which
correspond to the loops that we are counting.
Now we are ready to give an idea of what persistent homol-
ogy is. For δ
1
<δ
2
, the inclusion K(δ
1
) ⊆ K(δ
2
) induces a
linear map in each dimension H
i
(K(δ
1
)) → H
i
(K(δ
2
)).Given
a maximum value of the filtration parameter δ
max
, the collection
of all the simplicial complexes K(δ), 0 δ δ
max
is called
a filtered complex. The persistent homology will capture the
information of the homology groups (in our case Z
2
-vector
spaces) for all possible values of the filtration parameter
between 0 and δ
max
giving us a persistence diagram with the
birth and death of each connected component in dimension
0 or the birth and death of each loop in dimension 1. Notice
that a persistence diagram is just a collection of intervals in
each dimension, and hence it is sometimes referred to as a “bar
code.” For simplicity, we proceed to explain the β
0
case. We
order all the vertices. Each represents a connected component
at δ = 0 and hence β
0
(0) is the number of vertices. As the
filtration parameter δ is increased, each component will “die”
at the δ value where it gets connected with a point that appears
earlier in the order. Hence, at δ = 0, K(0) will have as many
connected components as the number of points. The number
of connected components will decrease as δ increases until
everything is connected (provided δ
max
is large enough) and
we have only one component, i.e., β
0
= 1.
We will not work with persistent diagrams, but with a
summary called the Betti profile, a graph that represents, for
each dimension, the corresponding Betti number as a function
of δ. In particular the one-dimensional Betti profile presents the
number of loops [β
1
(δ)] that we have as δ varies from 0 to δ
max
.
A series of free packages are available for computing persistent
homology [50–52]. Here, we use the R package TDA using
“
DIONYSUS” to perform persistent homology calculations [53].
In Fig. 3 we provide a simple example of the above
explanation for persistent homology measurements, using data
given by the centers of particles extracted from three different
artificially generated 2D lattices with low noise. We compare
compact square (I), sheared square (II), and hexagonal (III)
lattices, with a 10% additive noise to each coordinate of
the particle centers and with a nearest-neighbor separation of
σ = 10 pixels. VRC diagrams [Fig. 3(a)] at the same filtration
parameter δ distinguish among these three different structural
orders. The effect of increasing the filtration parameter values
above the diameter of the particles has two possible results,
as shown in Fig. 3(b), using the example of a sheared
square artificial lattice. At moderately high filtration-parameter
values, new connections may lead to the creation of a larger
number of loops (one-dimensional holes), as shown in part B;
this corresponds to higher value of β
1
in Fig. 3(c).Ateven
higher filtration parameter values, the lattice is covered by
triangles, as shown in parts C and D; this corresponds to a
decrease in β
1
in Fig. 3(c).
Profiles of the first Betti number β
1
for artificial lattices
[square, sheared square, and hexagonal; Fig. 3(c)] show that
the lowest β
1
peak height corresponds to the hexagonal
structure, and the widest plateau is associated with the square
structure. For all lattices, the rise in β
1
corresponds to a
filtration parameter δ that equals the nearest-neighbor distance.
For an almost perfect square lattice, the β
1
peak exhibits
a plateau that decreases abruptly beyond the next-nearest
distance (at δ/σ ≈
√
2). For sheared square lattices, which are
intermediate between square and hexagonal (50%), the sudden
decrease of β
1
occurs for δ/σ ≈ (1 +
√
3/4)
1/2
. Furthermore,
its plateau collapses into a peak that is narrower. For sheared
structures that are closer to hexagonal, the β
1
peak value
decreases below 1.
In summary, we use persistent homology analyses to extract
three quantities that describe the structure of colloidal films
(see Sec. IV, especially Figs. 7 and 8 and text): the value of
δ/σ at which β
1
first becomes nonzero, the height of the β
1
peak, and the width of the β
1
peak plateau (or peak).
IV. RESULTS AND DISCUSSIONS
A. Qualitative aspects
A series of experiments were performed in order to examine
the role played by a patterned substrate on the spin-coated
deposit. An unpatterned substrate, acted as the control sample;
it was coated with SU-8 to have comparable wettability
to the patterned substrates. Figure 4(a) shows a colloidal
sediment on an unpatterned substrate, while Figs. 4(b)–4(d)
show sediments on substrates that were patterned with a
lattice of hexagonal pillars with scale spacing of 0.3, 0.55, and
0.82 mm, respectively. The unpatterned substrates [Fig. 4(a)]
produce deposits with large-scale orientationally correlated
polycrystalline (OCP) character that exhibits colored arms
[11], as well as large cracks or big holes. In contrast, the
patterned substrates produce polycrystalline deposits that do
FIG. 4. AFM images of colloidal deposits, obtained at the same distance [(4,2) mm] from the spinning center of each sample. (a) An
unpatterned substrate produces deposits with OCP behavior. (b)–(d) Patterned substrates remove any OCP behavior, at intermediate scale
spacing between hexagonal pillars: (b) 0.3 mm, (c) 0.55 mm, and (d) 0.82 mm. All images are 20 μm × 20 μm.
032607-4
QUANTIFYING DISORDER IN COLLOIDAL FILMS SPIN- . . . PHYSICAL REVIEW E 95, 032607 (2017)
not show evidence of the orientation correlation that is typical
for spin-coated colloidal crystals [Figs. 4(b)–4(d)].
Preliminary experiments showed that the OCP charac-
ter disappeared only for intermediate scale spacing (0.30–
0.82 mm). At higher and lower scale spacing, the surface
appears to function similar to an unpatterned substrate for the
drying suspension. This dependence of the structural order
on the scale spacing may be related to hydrodynamics or
surface tension. Intermediate-spaced hexagonal pillars behave
as obstacles that prevent the fluid flows that are responsible for
OCP behavior. We note that all subsequent results discussed
here are restricted to the range of scale spacings where the OCP
does not appear. These data are also compared with results
from unpatterned substrates.
In what follows, the positions of colloidal particles, ex-
tracted from AFM images, are analyzed in detail to assess
4-fold and 6-fold structural ordered regions separately using
different methods: Minkowski structure metric analysis and
persistent homology.
B. Minkowski structure metric analysis
The colloid positions derived from AFM images allow us
to obtain the (complex) Minkowski structure metric ψ
msm
s
for
both 4-fold and 6-fold symmetry (s = 4 or 6), and for each
particle independently. Then, its absolute value |ψ
msm
s
| and
argument arg(ψ
msm
s
) provide useful information about the local
s-fold degree of order and local orientation, respectively.
Figure 5 shows a plot of |ψ
msm
4
| vs |ψ
msm
6
| for regions at
different distances from the spinning center, where the average
is over all particles in a given image. For example, the AFM
0.4 0.5 0.6 0.7 0.8
<|ψ
6
msm
|>
0.2
0.3
0.4
0.5
<|ψ
4
msm
|>
(4,2) mm
(6,3) mm
(8,4) mm
0.30 mm
0.55 mm
0.82 mm
unpatterned
FIG. 5. Average Minkowski structure metric values |ψ
msm
6
| and
|ψ
msm
4
| for particle positions extracted from AFM images of spin-
coated deposits. The three different scale spacings, 0.3, 0.55, and 0.82
mm, are represented by circles, squares, and diamonds, respectively,
while AFM images at distances corresponding to vectors (4,2) mm,
(6,3) mm, and (8,4) mm (all at the same polar angle but at the distances
4.47, 6.71, and 8.83 mm from the center) are represented by black,
dark gray (magenta), and light gray (orange) symbols, respectively.
For the scale spacing of 0.55–0.82 mm we have greater values for
|ψ
msm
6
| and smaller for |ψ
msm
4
|. The dashed line shows a linear
fit for data. The results for unpatterned substrates are shown as well
(triangle symbols) but are not considered in the fit.
images shown in Fig. 4, at a distance (4,2) mm from the center,
are represented in this plot by black triangle [Fig. 4(a)], circle
[Fig. 4(b)], square [Fig. 4(c)], and diamond [Fig. 4(d)].
There is no clear systematic dependence of |ψ
msm
s
| on
the distance from the center of spinning: the intermediate
distance (6,3) mm appears to yield the highest value of
|ψ
msm
6
|. In addition, there is no optimal scale spacing, since an
intermediate value (0.55 mm, squares) appears to yield the best
order at two radial distances [dark gray (magenta) and black
squares], but not for the third [light gray (orange) square].
For all samples that were spin-coated onto patterned sub-
strates, there is, however, a strong correlation between |ψ
msm
6
|
and |ψ
msm
4
| which is shown by the dashed line in Fig. 5
(linear fit with absolute value of correlation coefficient 0.995).
We do not understand the origin of the linear relation between
|ψ
msm
6
|and |ψ
msm
4
|in this specific kind of experiment, but it
is worth noting that the relationship is statistically significant.
This correlation implies a trend that needs to be explored
further.
Although the points from the unpatterned substrate (trian-
gles) are not in the confidence interval of the linear model
(confidence >0.999), they lie very close to those of the
patterned substrates with highest |ψ
msm
4
|and lowest |ψ
msm
6
|
(dashed rectangle in Fig. 5). At this point, it is not clear if we
can robustly distinguish the differences in disorder [refer to
Figs. 4(a) and 4(b)]byusingthemsm method, despite the fact
that the structures are very different; namely, OCP vs not OCP.
Thus, it is more useful to classify these structures through the
distribution of the cells’ orientation with respect to the radial
direction, rather than by means of the orientational degree of
order.
Hence, we show histograms of the phase of ψ
msm
6
and
ψ
msm
4
for AFM images of samples from experiments using
unpatterned substrates which lead to OCP behavior [Figs. 6(a)
-200 -100 0 100 200
arg(ψ
r
msm.
) (deg)
0
20
40
60
80
100
s = 4
s = 6
-200 -100 0 100 200
arg(ψ
r
msm.
) (deg)
0
20
40
60
80
100
-200 -100 0 100 200
arg(ψ
r
msm.
) (deg)
0
50
100
150
200
-200 -100 0 100 200
arg(ψ
r
msm.
) (deg)
0
50
100
150
200
(a)
(c)
(b)
(d)
FIG. 6. Histograms of the local phases of ψ
msm
6
and ψ
msm
4
that are
obtained in AFM images taken at different distances from the rotation
center [(4,2) mm for panels (a) and (b); and (6,3) mm for panels (c) and
(d)]. Gray thin (red) bins correspond to 6-fold symmetries and black
thick bins to 4-fold symmetries. (a), (c) Using unpatterned substrates
where crystalline domains are aligned along the radial direction, i.e.,
with OCP character. Here the histograms are narrower. (b), (d) Using
a patterned substrate with 0.82 mm spacing between hexagon pillars
without OCP character. Here the histograms are broader.
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![FIG. 3. Examples of persistence homology analysis on three compact artificial lattices (see text). (a) Vietoris–Rips complex (VRC) for square (I), sheared square [II, C in panel (b)] and hexagonal (III) at the same filtration parameter (δ = 1.2σ ) [open circles in panel (c)]. (b) VRC for increasing values of filtration parameter δ [stars on dashed, red line in panel (c)], δA = 1σ , δB = 1.1σ , δC = 1.2σ , δD = 1.3σ . (c) βnorm1 vs δ/σ for the three artificial lattices, where βnorm1 is normalized in such a way that the maximum value for a perfect square lattice is 1. The widest plateau appears for the square lattice.](/figures/fig-3-examples-of-persistence-homology-analysis-on-three-3gmfvmju.webp)
![FIG. 4. AFM images of colloidal deposits, obtained at the same distance [(4,2) mm] from the spinning center of each sample. (a) An unpatterned substrate produces deposits with OCP behavior. (b)–(d) Patterned substrates remove any OCP behavior, at intermediate scale spacing between hexagonal pillars: (b) 0.3 mm, (c) 0.55 mm, and (d) 0.82 mm. All images are 20 μm × 20 μm.](/figures/fig-4-afm-images-of-colloidal-deposits-obtained-at-the-same-3scal27y.webp)