Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
Supplementary Material
Frustrated Rotator Crystals and Glasses of Brownian Pentagons
Kun Zhao and Thomas G. Mason
Anisotropic Depletion Attractions: Model 2D Systems of Platelets
The depletion attraction between particles can be highly sensitive to the details of
the morphology and roughness on the particle surfaces [1, 2]. Depending on the relative
ratio of the diameter of the depletion agent to the height of asperities on a particle’s
surface, the depletion attraction can be greatly suppressed. For pentagonal platelets that
we have fabricated lithographically, scanning electron microscopy reveals that the flat
surfaces have average asperity heights of about 17 nm, whereas the edges are much
rougher and have average asperity heights that are at least 3 times larger. The diameter of
the depletion agent we have used in these experiments is about 20 nm, significantly
smaller than the average asperity height on the edges. We have also used depletion agents
up to 40 nm in diameter and achieved similar results. As we have previously
demonstrated [1], highly anisotropic interactions that are nearly hard can be created
between rougher surfaces, while strongly attractive interactions can be created between
smoother surfaces if the diameter of the depletion agent is selected to lie between these
two roughness length scales.
In previous experiments using the same system of pentagonal platelets using the
same depletion agent at the same concentration [1], no strong depletion attractions
between adjacent rougher edges of pentagonal prisms were found. Aggregation due to
strong depletion attractions was seen only between the smoother faces of the pentagonal
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
prisms, and we did not observe columns of pentagons in edge-to-edge configurations in
bulk experiments. We have fabricated thicker platelets than those reported in the present
2D experiments, which have at least double the edge surface area, but we still didn’t find
any observable evidence of strong depletion attractions that could cause edge-edge
clustering or aggregation. Thus, our direct observations of the anisotropic roughness on
the surfaces of the platelets as well as their interactions in the presence of the depletion
agent in solution both show that the depletion attraction between edges of the pentagons
is smaller than kBT.
Based on the area of a face of a pentagon and the size of the depletion agent
relative to the average roughness on a face, we estimate that the depletion attraction
between the flat pentagon surface and the glass wall is about 10kBT at the conditions in
the reported 2D experiments. For the depletion attraction between edges, we obtain this
estimate by comparing the depletion attraction between faces of pentagons. Even when
we use polystyrene (PS) spheres (0.04 µm diameter) as a depletion agent at a
concentration of about 0.2%, we find that the pentagons aggregate due to a strong faceto-face attraction to form columns [1]. For this aggregation to occur, the depletion
attraction between aligned faces is about 5kBT to 10kBT. Since the depletion attraction is
proportional to the volume fraction of the depletion agent, when the concentration of PS
spheres is raised to about 6%, the depletion attraction between faces, which is linearly
proportional to the concentration of the depletion agent, will have a magnitude that is
very large: about 100kBT. However, for both of these concentrations, we didn’t observe
any effects of depletion attractions between edges of individual pentagonal platelets (i.e.
no side-to-side aggregation). Considering the area of a pentagon’s edge (about 3.6 µm2)
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
is somewhat smaller than the area of a pentagon’s face (about 5.6 µm2), we estimate that
the depletion attraction between the edges is suppressed by about two orders of
magnitude compared to the depletion attraction between faces and the glass wall. Thus,
for the depletion conditions we have purposefully selected, the anisotropy of surface
roughness between edges and faces provides a means of causing the edge-edge attraction
between neighboring pentagons to be less than about kBT/10, much smaller than face-wall
interactions of about 10kBT that prevent the pentagons from tipping out of the 2D plane,
even when they are subjected to Brownian excitations.
Sample Preparation
An aqueous dispersion of pentagons of an epoxy polymer (SU-8) is made using
photolithography using a stepper [1, 3]. The pentagons are nearly equilateral; at most,
edge lengths differ by about 10% due to discretization of the lithographic mask design.
Also, due to lithographic exposure limitations, the vertices are not perfectly sharp; their
average radius of curvature is about 300 nm.
We prepare a 2D layer of pentagons by mixing a dispersion of pentagons at a
volume fraction of about 5 x 10-4 with an aqueous dispersion of a depletion agent
(polystyrene spheres: diameter ≈ 0.02 µm, concentration ≈ 0.5% w/v, sulfate stabilized).
This mixture is sealed in a rectangular optical cuvette (0.2 mm x 4.0 mm x 20 mm).
Pentagons are concentrated in 2D by tipping the cuvette about its long axis at angles
between 1° and 6°, creating a slowly varying spatial gradient in φA.
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
Order Parameters and Correlation Functions
The order parameters and correlation functions that we calculate are based on
standard definitions [4-7]. For each particle, identified by j, we define a complex number,
the local six-fold bond-orientational order parameter ϕ6:
r
" 6 ( rj ) = N j #1
Nj
%e
i 6$ jk
.
(1)
k=1
Here, Nj is the number of nearest neighbors of particle j, θjk is the angle between an
! the line connecting the centers of particles j and k. The positional
arbitrary fixed axis and
order parameter ζ for each particle j is defined as:
r r
r
iG ! r
" (r j ) = e j
,
(2)
r
where G is the reciprocal lattice vector of the appropriate lattice. For the rotator crystal
r
(RX) phase and frustrated rotator crystal (FRX) state, G is the reciprocal hexagonal
r
lattice vector. For an alternating striped crystalline (ASX) phase of pentagons, G would
be the reciprocal ASX lattice vector [4]. For the isotropic phase, there is no periodic
r
lattice structure, so we choose the reciprocal lattice vector G of a corresponding
hexagonal lattice structure at the same area fraction.
After calculating ϕ6 and ζ for all N particles at a fixed particle area fraction, we
determine the global bond-orientational order parameter ψ6:
" 6e
i#
=N
$1
N
r
&% 6 ( rj ) ,
(3)
j=1
where ω represents a global phase. Similarly, the global positional order parameter S is:
!
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
S = N "1
N
r
$# (rj ) .
(4)
j=1
r
In order to calculate S, the reciprocal hexagonal lattice vector G must be first obtained.
! from the lattice spacing and orientation of the platelets. The
This can be determined
lattice spacing can be inferred from the density, and the lattice orientation is the phase ω
from the calculation of global bond orientation order parameter. In practice, we let the
lattice orientation vary in the range between 0 and 2π, calculate a set of possible S, and
then choose the maximum of this set as the order parameter’s actual value. Without
correction, this numerical approach can lead to non-zero values of S even in the isotropic
phase. Since both positional and orientation order parameters approach zero in the dilute
isotropic phase, in Fig. 2(a) of the main paper, we ensure that the order parameters go to
zero by appropriately correcting for a constant offset introduced by this numerical
approach at low φA, by subtracting using an average of first three points.
In addition to order parameters, correlation functions also reveal important details
about the spatial extent of order in 2D thermal systems of platelets. Using standard
conventions, the bond-orientational correlation function is defined as:
g 6 (r ) = Re ! 6" (0)! 6 (r ) ,
(5)
and the spatial correlation function relating to positional order is defined to be:
g s (r ) = Re ! " (0)! (r ) .
(6)
Here, Re represents an operator returning the real part of the value.
Figures S1 and S2 show examples of typical correlation functions of g6(r) and
gs(r) at different area fractions. The undulations in the correlation functions are due to the
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
finite size of the particles, which correspond to the oscillations of the radial distribution
function g(r). The correlations for particles in the isotropic phase decay very quickly.
When the area fraction is increased, the correlation functions decay over a longer spatial
extent. As the area fraction is further raised, the correlation functions decay very slowly.
This is especially true for g6(r), which is almost flat over the range we are able to probe.
This very slow decay signals the RX phase and FRX state. At even higher φA, beyond the
FRX state, the correlation functions decay rapidly again, indicating the onset of an orderdisorder transition resulting from quenching φA into the FRG state.
We can also estimate the correlation lengths for different phases/states by fitting
the envelope tail of the correlation functions using an exponential decay exp(-r/Ld),
where Ld is a correlation length. For the isotropic phase, we find that both the
orientational correlation length Ldo and the positional correlation length Lds are smaller
than a particle’s effective diameter, D. For the RX phase and the FRX state, the typical
correlation length exhibits Ldo > 300D and Lds > 50D, whereas in the quenched FRG
state, Ldo ≈ 30D and Lds ≈ 4D.
Hallmarks of Observed Phases and Jammed States
For 2D systems of disks, g6(r) and gs(r) are expected to decay to zero
exponentially in isotropic phase (short-range order). By contrast, in the solid/crystal
phase, g6(r) is expected to be a non-zero constant as r increases (i.e. signals long-range
order), while gs(r) is expected to decay to zero algebraically: g s (r ) ~ r ! s , where the decay
exponent lies in the range 0 > ηs > -1/3, corresponding to quasi-long-range order
(QLRO). The KTHNY theory also predicts a hexatic phase, in which g6(r) decays
algebraically as a power law: g 6 (r ) ~ r !6 , where the exponent lies in the range η6 > -1/4,
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
and gs(r) decays exponentially [5, 7]. Thus, in principle, by analyzing the decay behavior
of these correlation functions, especially for large r well beyond the particle scale,
different phases can be detected.
For the 2D system of Brownian pentagons observed in the present study, we find
that, in the I phase, both g6(r) and gs(r) decay exponentially. However, at the area fraction
φA = 0.63, where images might indicate a possible hexatic phase, g6(r) has an algebraic
decay with η6 close to zero (η6 ≈ -0.03), and gs(r) also exhibits an algebraic decay with ηs
≈ -0.4, which is smaller (i.e. more negative) than -1/3. Due to the limited frame size of
our observations, which are made at high magnification in order to track rotational
dynamics of individual pentagons, we cannot conclusively identify a hexatic phase. In the
RX phase and FRX state, g6(r) has an algebraic decay with η6 → 0, and gs(r) has an
algebraic decay with ηs > -1/3. In the FRG state, g6(r) follows an algebraic decay with η6
≈ -1/4, and, due to disorder introduced by frustration of pentagonal packing, gs(r) decays
rapidly and can be most closely fit to an algebraic decay having ηs ≈ -2.
Based on the results of structural and rotational analysis, as φA is raised in the RX
phase, rotational frustration appears to increase continuously until the FRX state is
reached and tip-tip passage events become essentially unobservable over the time scales
of our experiment. There is not much difference between static images taken in the RX
phase and FRX state, and this similarity in positional features is also reflected in the
measurements of order parameters and correlation functions for RX and FRX. In order to
distinguish the fully rotationally jammed FRX state from the unjammed RX phase,
rotational dynamics need to be analyzed. At φA just below FRX, the instantaneous degree
of rotational jamming can vary significantly among pentagons at different spatial
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
locations in the 2D array, and the degree of rotational jamming of a pentagon appears to
be coupled to thermally driven fluctuations of the centers of the neighboring pentagons
about their lattice positions. Such rotational analysis is the key to identifying the onset of
full rotational jamming and in this continuous ergodic-nonergodic rotational transition.
The FRX-FRG “transition” is less well defined, since both FRX and FRG are
jammed metastable states, at least in the sense of being rotationally nonergodic, rather
than strict thermodynamic phases. Especially for FRG state, the degree of spatial disorder
that is introduced can potentially depend on the quenching rate (i.e. rate of 2D osmotic
compression) as φA is raised. If treated within the framework of a transition, it appears to
be more like a continuous transition, since η6 decreases from about 0 to around -1/4. The
translational order decays quickly from quasi-long-range in FRX to short-range in FRG,
mainly because of lattice distortions and numerous grain boundaries that appear between
small, highly defected poly-crystal clusters in the FRG state.
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
References: Supplementary Material
[1]
K. Zhao and T. G. Mason, Phys. Rev. Lett. 99, 268301 (2007).
[2]
K. Zhao and T.G. Mason, Phys. Rev. Lett. 101, 148301 (2008).
[3]
C. J. Hernandez and T. G. Mason, J. Phys. Chem. C 111, 4477 (2007).
[4]
T. Schilling et al., Phys. Rev. E 71, 036138 (2005).
[5]
K. J. Strandburg, Rev. Mod. Phys. 60, 161 (1988).
[6]
C. A. Murray and D. H. V. Winkle, Phys. Rev. Lett. 58, 1200 (1987).
[7]
D. R. Nelson, Defects and Geometry in Condensed Matter (Cambridge Univesity
Press, Cambridge, 2002).
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
Figure S1. Bond-orientational correlation function g6(r/D) for equilateral pentagons at
particle area fractions: φA = 0.60 (○), 0.63 (□), 0.69 (◇), 0.74 (×), 0.80 (+), and 0.88 ().
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Supplementary Material: Brownian Pentagons
K. Zhao and T.G. Mason
Figure S2. Spatial correlation function gs(r/D) for equilateral pentagons at particle area
fractions: φA = 0.60 (○), 0.63 (□), 0.69 (◇), 0.74 (×), 0.80(+), and 0.88 ().
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