Abstract
Understanding the behaviour of thermally excited manyparticle systems, composed of a single particle type having a welldefined shape and size, is important in condensed matter, notably protein crystallization. Here we observe and explain the origin of local chiral symmetry breaking in a surprisingly simple system of hard Brownian particles: achiral regular triangles confined to two dimensions. Using enhanced optical video particletracking microscopy, we show that microscale lithographic triangular platelets form two different triatic liquid crystal phases. Above a particle area fraction φ_{A}≈0.55, the simple triatic phase is spatially disordered, yet has molecular orientational characteristics that distinguish it from a hexatic liquid crystal. At higher φ_{A}≥0.61, we find a second triatic phase exhibiting local chiral symmetry breaking; rotational entropy favours laterally offsetting the positions of nearestneighbouring triangles. By contributing to spatial disordering, local chiral symmetry breaking can limit the range of shapes that can be entropically crystallized.
Introduction
The tendency of certain colloidal shapes to form ordered crystals^{1,2,3,4,5,6}, even when subject to thermal excitations, is important in protein crystallization^{1,7} and directed assembly^{8,9,10}. Although some shapes readily crystallize, others do not^{7}; this limits the range of proteins structures that can be deduced by crystallography. Beyond crystallization, the most basic physical ingredients that can induce chiral symmetry breaking^{11}, even locally, remain unclear^{12}.
Entropydriven ordering in dense multiparticle systems can be strongly influenced by particle shape^{9,13,14,15,16,17,18,19,20}. Although initially controversial, hard uniform spheres freeze through a spatial disorder–order transition as their number density is increased, yet well below the density at which the spheres pack and can no longer translate^{3,4,21}. Such spatial ordering, which might seem paradoxical at first, provides greater translational free volume per sphere, thereby maximizing entropy. By contrast, the classic Onsager fluid, consisting of thin and long hard rods, undergoes a different type of phase transition, as the rod density is increased from a disordered isotropic (I) fluid phase, having shortrange spatial and orientational order, to a nematic (N) liquid crystal phase, having shortrange spatial order but longrange molecularorientational (MO) order^{15}. This I–N transition is driven by an increase in entropy: the loss of entropy suffered by the development of MO order is more than compensated by an increase in entropy due to a larger available translational free volume per rod. Thus, significant shape anisotropy can give rise to spatially disordered liquid crystal phases in multiparticle systems at high densities.
The particular type of liquid crystal that forms can depend in a delicate way on a particle's precise shape and also on the dimensionality of space. For instance, in two dimensions (2D), as the density of hard discs is increased, the isotropic phase transforms into a classic hexatic (H) liquid crystal phase, which possesses quasilongrange (QLR) sixfold bondorientational (BO) order, yet only shortrange spatial order^{22,23}. In three dimensions (3D), differences in particle shapes give rise to a variety of liquid crystal phases, such as smectic^{16} and discoidal phases^{17}. Beyond classic liquid crystals^{16}, new forms, such as the cubatic phase^{9,24}, have been predicted but thus far not observed.
Simulations of regular triangular prisms in 3D show a spatially ordered honeycomb crystal that simply melts into an isotropic phase without any intervening liquid crystal phase^{9}; some other complex polyhedra appear to form liquid crystal phases in 3D at high densities^{9}. Among 2D shapes, the equilateral triangle is the simplest of all regular polygons, and its shape is inherently achiral. Nevertheless, the behaviour of dense Brownian systems of hard triangles has not been fully elucidated. An artificial neural network simulation of triangles in 2D (ref. 20) has predicted only a single type of highdensity liquid crystallike phase above a particle area fraction φ_{A}≈0.57, but its nature remains vague and mysterious.
The advent of precisely controlled synthesis of dispersed colloidal systems, made by microlithography, has opened up new experimental realizations of manybody hardcore systems of precisely controlled shapes, having tailored anisotropic interactions^{25,26,27}. In particular, roughness controlled depletion attractions^{28,29}, a variation of shapeselective depletion attractions^{30}, have yielded a route that enables experimental exploration of dense 2D systems of hard Brownian microscale platelets. These microscopic platelets are small enough that the thermal Brownian excitations at room temperature cause substantial fluctuations in positions and orientations of the particles, yet they are large enough that optical video microscopy can be used to visualize and record images of many particles in a single field of view. Studies of 2D systems of hard uniform pentagons^{14}, which cannot fully tile a plane, have shown crystallization into a hexagonal rotator crystal (RX) phase that undergoes glassylike rotational dynamic arrest and spatial disordering under higher compression. By contrast, 2D systems of hard squares^{13}, which can fully tile a plane, exhibit a firstorder crystal–crystal transition between a hexagonal rotator crystal and a rhombic crystal. The characteristic angle of the rhombic unit cell changes continuously towards a square phase under increasing compression. Although interesting and rich, the observed phases of squares and pentagons do not provide any clear predictive power for the phase behaviour of dense 2D systems of regular triangles.
Using enhanced optical video particletracking microscopy, we study dense diffusing systems of hard regular triangular platelets confined to 2D by roughnesscontrolled depletion attractions^{28}. In striking contrast to systems of squares and pentagons, we show the development of liquid crystal order from the isotropic phase, as φ_{A} of the triangles is increased above φ_{A}≈0.55; corresponding Fourier transform intensities predominantly reveal the equilibrium structure factor. Spatial and orientational correlation functions, translational and rotational mean square displacements, and order parameters reveal the emergence of a simple triatic liquid crystal phase T_{Δ} that reflects the underlying threefold symmetry of the triangular particles. Vertices of triangles in the triatic phase effectively point along six welldefined directions in the plane over extended distances, and, around a given central triangle, the three nearestneighbouring triangles point away from the central triangle. Beyond φ_{A}≈0.61, rotational entropy favours lateral offsetting of nearestneighbouring triangles, yielding a triatic phase T_{Δχ} that exhibits local chiral symmetry breaking (LCSB). Three sets of chiral enantiomeric building blocks, consisting of pairs of + and − offset dimers of triangles, can be equivalently used to analyse the realspace images, and over large distances compared with the triangle size, the T_{Δχ} phase is racemic. Both in the T_{Δ} and T_{Δχ} phases, collective translational sliding motions of triangles along lubricated slip lines in the plane, defined by the orientational order, also have a role in the observed entropic fluctuations. Overall, the direct observation of LCSB in triatic liquid crystals of hard triangles shows that the simple combination of entropy and the geometric shape of the constituent particles is sufficient to induce chirality spontaneously, even if locally. Thus, LCSB can have an important role in limiting the range of shapes of colloidal objects, including proteins and other biological structures, that can be entropically crystallized.
Results
Structure of dense systems of hard Brownian triangles
At φ_{A}=0.52 (Fig. 1a), the rotationally symmetric ring pattern, in the structure factor, is indicative of liquidlike shortrange spatial order. At higher φ_{A}=0.56 (Fig. 1b), the liquid ring remains broadened, but the intensity has developed a sixfold azimuthal modulation in the Fourier transform. This modulation is reminiscent of the sixfold QLRBO order of a hexatic liquid crystal^{31}. We thus associate φ_{A}≈0.55 with a transition from an isotropic phase to a liquid crystal phase, as yet unclassified. At even higher φ_{A}=0.58 (Fig. 1c), the modulations become more pronounced and extend to larger wavenumbers. For φ_{A}≥0.63 (Fig. 1d), sixfold MO order extends across the entire viewing region; triangles far from a chosen triangle have vertices that either point nearly in the same direction or in the opposite direction. The modulations are so pronounced at φ_{A}=0.63 that they appear as six highly diffuse spots. At yet larger φ_{A}=0.69 (Fig. 1e), these six spots around the primary ring are less diffuse and a second ring of six diffuse spots becomes better defined at higher wavenumbers, suggesting a greater degree of local spatial order (Supplementary Fig. S1).
Because the Fourier transform of an individual microscope image contains significant speckle, reflecting only a single configuration of triangles in a small viewing region at a particular instant (for example, see insets in Fig. 1), it is desirable to obtain a smoother average from different configurations of triangles, in a certain interrogation region, as the system fluctuates in equilibrium. From recorded movies, taken at different locations in the 2D column and therefore different φ_{A}, we extract images every 3 s over a duration of 4.8 min, and then perform Fourier transforms for each extracted image. At each φ_{A}, we obtain a timeaveraged Fourier transform intensity pattern by summing all of the Fourier transform intensities, and then dividing the sum by the total number of Fourier transform images. Because the timeaveraged Fourier transforms are sixfold symmetric, we also average over successively 60°rotated versions, yielding smoother patterns. The results of these temporally and rotationally averaged Fourier transform intensity patterns for φ_{A}=0.52, 0.58, 0.62 are shown in Fig. 2a–c, respectively. Comparing Fig. 2b with c, we find that features at higher wavenumbers are more intense and sharper for φ_{A}=0.62 than for φ_{A}=0.58. Lightscattering measurements of equilibrium 2D gravitational columns of triangles (Methods) also reveal a circularly symmetric intensity pattern for the lowerdensity isotropic phase and a sixfold symmetric intensity pattern of azimuthally smeared spots for higher φ_{A}, as shown in Fig. 2d,e, respectively.
Spatial and orientational correlation functions
To classify liquid crystal phases precisely, it is necessary to calculate correlation functions: the spatial radial distribution function g(r/D), the threefold MO correlation function g_{3}^{mo}(r/D), and the sixfold MO correlation function g_{6}^{mo}(r/D), where r is the centretocentre distance between two triangles and D is the diameter of a circle circumscribed around a triangle (Methods). Using trajectories of particles obtained by optical video particletracking microscopy (see Methods and Supplementary Fig. S2), we find that g(r/D) rapidly decays to a limiting value of unity at large r beyond a few D (Fig. 3a), so there is no QLR spatial order even at the highest φ_{A} we observe. Next, g_{3}^{mo}(r/D) shows a striking negative shortrange anticorrelation peak of −1 at small r/D (Fig. 3b). Remnants of this peak are discernible even for φ_{A} as low as ≈0.35, indicating local antiparallel orientation of at least some nearestneighbouring triangles. However, for larger r/D, g_{3}^{mo} also rapidly approaches its asymptotic value of zero, so there is no threefold QLR–MO order at any φ_{A} we observe. By contrast, g_{6}^{mo}(r/D), does exhibit sixfold QLR–MO order at larger φ_{A} (Fig. 3c): at φ_{A}=0.42, g_{6}^{mo}(r/D) decays rapidly, at φ_{A}=0.52, the decay in g_{6}^{mo} is slower but still approaches zero towards large r/D, and for higher φ_{A}, g_{6}^{mo} decays relatively little over the range of r/D in a single image. Thus, above φ_{A}≈0.55, we identify a new thermodynamic liquid crystal phase, which has sixfold QLR–MO order, yet local anticorrelation in the threefold MO order, as a simple triatic liquid crystal phase: T_{Δ}.
Rotational and translational dynamics
The identification of different phases is also evident from changes in the rotational and translational dynamics of individual triangles. The ensembleaveraged timedependent mean square angular displacement (MSAD) 〈Δθ^{2}(t)〉 and the 2D mean square displacement (MSD) 〈Δr^{2}(t)〉 for individual triangles are shown for a series of φ_{A} in Fig. 4. The ensemble size used for averaging is ≈250 triangles, and the time duration is about 5 min. At φ_{A}=0.42 and 0.52, the MSADs are subdiffusive at early times and become diffusive at longer times (Fig. 4a). By contrast, for higher φ_{A}≥0.58, the MSADs exhibit plateaus that decrease in magnitude; a longtime plateau indicates a transition from rotational ergodicity to nonergodicity and is correlated with the onset of QLR–MO order. Individual triangles are no longer able to rotate and explore all possible orientations because of cagelike confinement by neighbouring triangles (see, for example, Fig. 1b–e). This caged rotational diffusion of triangles additionally differentiates T_{Δ} from the classic H phase of discs, which exhibit unbounded rotational diffusion since they are not packed into frictional contact.
Interestingly, the ensembleaveraged translational dynamics are quite different than the rotational dynamics. At low φ_{A}=0.42, the MSD is diffusive at long times, whereas it becomes gradually more and more subdiffusive as the density increases towards φ_{A}=0.64 (Fig. 4b). Even though individual triangles are heavily constrained by a cage of neighbours for larger φ_{A}, this does not necessarily imply that there are no strong positional thermal fluctuations because of the possibility of collective transport. Careful inspection of the videos at higher φ_{A} shows that there indeed exist significant sliding fluctuations of many triangles via collective motion along six different slip directions that are dictated by the sixfold QLR–MO order (Supplementary Movies 1 and 2). Thermal fluctuations can drive collective translational motion of many triangles in concert without significantly disturbing their orientations at essentially no cost in energy, indicating a significant translational sliding degeneracy.
Realspace observation of local chiral symmetry breaking
For φ_{A} well above the I–T_{Δ} transition, we observe that the edges of the three nearestneighbouring triangles are often misaligned with the edges of a given central triangle. We calculate the probability distribution p(d_{sh}/D) of a first nearestneighbouring antiparallel triangle that is shifted by a transverse displacement d_{sh}, relative to a given test triangle (Fig. 5a). A symmetric monomodal peak centred around d_{sh}/D=0 is found for φ_{A}<0.6. In this regime, nearestneighbouring triangles are, on average, aligned with a central test triangle, yet, there are a significant number of nearestneighbouring triangles that have plusandminus lateral offsets (that is, transverse shifts) with equal probabilities; we choose counterclockwise shifting to be positive. However, for φ_{A}≥0.61, we clearly resolve a symmetric double peak in p(d_{sh}/D), indicating a preferred magnitude for a transverse offset, producing two dominant, distinct and equal populations of left and rightshifted dimers of triangles. This quantitatively demonstrates that entropy alone can induce LCSB in a dense system of achiral particles^{11,12}.
The origin of LCSB in dense Brownian systems of hard triangles can be understood by effectively incorporating the effect of rotational entropy into a new particle shape based on a triangle. Consider a rotationally swept triangle that is rotated back and forth symmetrically about its centre over a specific range of angles Δθ, generating a trioid shape (Fig. 5b). This trioid shape effectively captures the range of orientations of triangles that are caged by neighbouring triangles at high φ_{A}. Two equivalent local packing arrangements of a pair of trioids maximize Δθ and thus the rotational entropy: two nearestneighbouring antiparallel trioids are shifted to form either of two chiral isomers (that is, + or −) of the offset dimer motif (Fig. 5c).
To graphically show the racemic mixture of offset dimers at high φ_{A}, we consider only the two average configurations corresponding to the doublepeak in the probability distribution p(d_{sh}/D), which determine two parallelogramshaped isomers that enclose corresponding pairs of dimers (Fig. 5d). Since each of the six different directions in the plane, defined by the QLR–MO order, is associated with a parallelogram, three sets of two parallelograms (that is, pairs of + and − offset dimers) can be equivalently used as a basis for analysis (Fig. 5e). The average sizes of the regions having a similar sense of parallelogram define a correlation length for the LCSB, which is typically about 4–5 particle diameters, yet the entire system remains racemic at large length scales (Fig. 5f–i). Although parallelograms can be used to analyse nearly all triangles in a given image, we also observe some defects, such as dislocations and disclinations. Among these, headtail local configurations create frustration and lie outside the offset dimer analysis scheme.
The interrelated changes in the measured structural and dynamic characteristics of the hard triangle system are summarized in Fig. 6. The increase in the average antiparallel nature of nearestneighbouring triangles is seen in the threefold local MO average 〈cos(3δθ)〉, where δθ represents the relative angle between a given test triangle and its first nearest neighbour (Fig. 6a). At low φ_{A}<0.35, the MO is isotropic, so 〈cos(3δθ)〉∼0. For φ_{A} >0.35 (near the limit 0.375 of circumscribed triangles packed in 2D, φ_{A}*=φ_{hd}(A_{tri}/A_{disc})=[π/(12)^{1/2}][3^{3/2}/(4π)]=3/8, where φ_{hd} is the area fraction of hexagonally packed discs in a plane, and A_{tri}/A_{disc} is the area fraction of a triangle inside its circumscribed circle (inset, Fig. 6a)), we observe a gradual decrease in 〈cos(3δθ)〉 towards complete antiparallel correlation (that is, −1). Simultaneously, the shortrange local anticorrelation peak in g_{3}^{mo} rises, indicating that local edge–edge configurations of triangles become more prevalent than edge–tip or tip–tip configurations as φ_{A} increases. φ_{A}* would be a lowerbound on the onset area fraction associated with a hypothetically possible RX phase of triangles. However, although there is a gradual increase in the MO anisotropy of nearestneighbouring triangles above φ_{A}*, we do not observe a RX phase for triangles. Instead, triangles have a highenough effective aspect ratio to suppress the RX phase compared with other regular polygons such as squares^{13} and pentagons^{14}, which do have RX phases. Of all regular polygons, triangles have the highest effective aspect ratio, defined as the centretovertex distance divided by the centretonearestedge distance, equal to 2. This aspect ratio is high enough to enable the formation of a molecularly oriented liquid crystal phase.
By contrast, the system's rotational dynamics and orientational order change abruptly at the I–T_{Δ} transition. The power law exponent α, characterizing the rotational diffusion at long times 〈Δθ^{2}(t)〉∼t^{α}, drops dramatically at (Fig. 6b), indicating strongly caged rotational diffusion of triangles above (dashed line, Fig. 6b). Structurally, we find a rapid increase in the sixfold MO order parameter Φ_{6} and in the hexatic sixfold BO order parameter for nextnearestneighbouring triangles Ψ_{6} (Fig. 6c) (see also Methods and Supplementary Fig. S3). The sharp changes in α, Φ_{6}, and Ψ_{6} at indicate that dynamics and structure are closely linked through the I–T_{Δ} transition.
To quantify the onset of LCSB, we calculate Δd_{sh}/D, which describes the total lateral shift between the two peaks in the probability distribution p(d_{sh}/D) (Fig. 6d). This dimensionless parameter is zero below about φ_{A}∼0.61, because there is only one peak in p, and it rises abruptly to a value near Δd_{sh}/D≈0.25, for larger 0.61<φ_{A}≤0.7. Thus, φ_{A}≈0.61 is the transition between simple T_{Δ} having a single peak in p(d_{sh}/D) and the LCSB triatic phase T_{Δχ}, which possesses a clearly resolved double peak in p(d_{sh}/D). The lateral offsets associated with LCSB reduce Ψ_{6}, so it only rises to a value of about 0.5 (Fig. 6c). Additionally, the collective translational sliding degeneracies of the triangles further reduce Ψ_{6} without affecting Φ_{6} to nearly the same degree. The trioid model predicts that Δd_{sh}(φ_{A})/D will decrease towards zero at very high compression.
Discussion
In the T_{Δχ} phase, the chiral symmetry is only broken locally: the system as a whole is a racemic mixture of small chiral domains. The doublepeak in p is associated with the formation of offset dimer trianglepairs having two stereoisomer variants in equal proportions; thus, LCSB effectively results from maximizing the combined rotational and translational entropy. Moreover, the local chiral asymmetry, which is racemic, never becomes longrange of a single enantiomer type, even as thermal fluctuations can continuously drive changes in the signs of local offsets. Collective translational sliding fluctuations along slip lines also aids in the destruction of spatial order, even as orientational fluctuations become strongly quenched.
The observation of entropydriven LCSB, in 2D liquid crystals of hard regular triangles, is remarkable, especially because other regular polygons that have smaller aspect ratios, such as squares and pentagons^{13,14}, do not exhibit LCSB. The combination of LCSB and collective translational sliding in triatic liquid crystals, despite the symmetry and monodispersity of achiral triangular constituents, can thus act to frustrate entropic spatial crystallization. Because LCSB can contribute to spatial disordering, it can limit the range of shapes, even if highly symmetric, that can be entropically crystallized. Indeed, we anticipate that rotational entropy of sufficiently anisotropic hard particles could also lead to LCSB in dense 3D systems. The emergence of two types of triatic liquid crystal phases, including one that exhibits LCSB, indicates that the subtle combination of geometry and entropy may combine to produce yet further surprises for other shapes, whether in two or three dimensions.
Methods
2D osmotic compression and measurement of area fraction
Equilateral triangular platelets (that is, triangles) having edge length L=3.7±0.1 μm and thickness h=2.0±0.1 μm are fabricated from SU8 polymer by photolithography, released into aqueous solution, and stabilized against aggregation by adsorbed sodium dodecyl sulfate (1 mM). We mix a dilute dispersion of triangles (∼5×10^{−4} v/v) with a dispersion of nanoscale polystyrene spheres, which act as a depletion agent, and seal the mixture in a rectangular glass microcapillary (0.2 mm×4.0 mm×20 mm). The size and concentration of polystyrene spheres (polystyrene spheres: diameter ∼0.02 μm, concentration ∼0.5% w/v, sulfatestabilized) are selected to cause desired anisotropic attractions that leads to a good realization of a 2D hardcore system: roughnesscontrolled depletion attractions between the faces of the triangles and the microcapillary surface are strong enough, relative to thermal energy to inhibit Brownian rotational or translational excitations out of the plane^{28}, yet the triangles retain their capacity to diffuse translationally and rotationally in the plane. The triangles are osmotically compressed by gravity in the plane by slightly tilting the microcapillary, between about 2° and 5°, with respect to the horizontal. Because the particles have a higher mass density (ρ_{SU8}≈1.2 g cm^{−3}) than water, the 2D osmotic pressure Π at a given location in the monolayer arises from the effective gravitational mass of particles above it, projected along the tilted plane. The system achieves a steady state after several weeks, when the concentration profile as a function of distance along the microcapillary stops changing, as measured using optical video microscopy (Nikon Eclipse TE2000 inverted microscope, ×40 planapo long working distance objective lens with correction collar, brightfield transmission illumination; Point Grey Flea2 colour CCD camera, 8bit depth per colour channel, 1,024×768 pixels, 30 Hz max acquisition rate). In this steadystate 2D column of particles, Π increases towards the lower regions of the monolayer, so the particle area fraction φ_{A} also increases, yielding a slowly varying gradient in φ_{A} within the monolayer. About 400 particles can be observed in a single field of view (64 μm×64 μm) at high φ_{A}. All φ_{A} are calculated using particle centre locations, determined by tracking software, and the average measured size of the triangles, determined by scanning electron microscopy. Because of edge roughness of the triangles, effective area fractions may be slightly larger (for example, 1–2%) than our reported values. In 2D colloidal systems of microscopic particles, the particle area fraction φ_{A} corresponds directly to the particle density associated with phase transitions, whereas, in condensed molecular systems (which have significant intermolecular attractions, not just hardcore repulsions), altering the temperature is often used to change the molecular density in addition to the degree of thermal fluctuations.
Enhanced particletracking image analysis
We have created enhanced video particletracking microscopy software, improving on existing routines for isotropic spheres and discs (for example, refs 32,33), by detecting not only the centre position but also a vertex of every triangle in each successive video frame, thereby providing both positional and orientational trajectories as a function of time (Supplementary Fig. S2). These measured particle trajectories are then used to calculate ensembleaveraged spatial, orientational and dynamical correlation functions, order parameters, and mean square displacements.
Fourier transforms of video images
For each image, using standard procedures, we crop a 512 pixel×512 pixel portion of the image from the centre of the raw 550 pixel×550 pixel image, convert to grayscale, and use the standard fast Fourier transform command in Interactive data language software to perform the Fourier transform. This command returns a 512 pixel×512 pixel grayscale Fourier transform intensity having 8bit dynamic range. Because the primary features of interest in the Fourier transform are within a 100 pixel×100 pixel region at the centre, we crop the Fourier transform and only show this smaller region. We block the centre of the Fourier transform using a small black square (7 pixel×7 pixel). We use a colour lookup table to colourcode the intensity values over the range from zero to 255.
By making separate image files of dots at the centre positions of the triangles and taking the Fourier transform (Supplementary Fig. S1), we have verified that the Fourier transform patterns of the microscope images, shown in the insets of Fig. 1a, dominantly reflect the triangles' positional structure, that is, centretocentre locations of triangles and structure factor S(q), where q is the wavevector, rather than the form factor F(q) of a singletriangle shape. The sixfold intensity modulation of S(q) can be effectively attributed to BO order. In dense systems of triangles, BO order is inherently coupled to MO order. So besides g_{6}^{mo}, the sixfold BO correlation function g_{6}^{bo} for next nearestneighbouring triangles also shows the developing of QLR order as φ_{A} increases (Supplementary Fig. S3).
Lightscattering measurements
A helium–neon laser beam (Melles Griot, wavelength 632.8 nm, beam waist 650 μm, power 15 mW) illuminates the monolayer of triangles. The scattering pattern is formed on a screen located 10 cm beyond the sample and recorded by a monochrome CCD camera (Point Grey: Flea). The lightscattering patterns in Fig. 2d,e are smoother than the averaged Fourier transforms, which is expected when averaging over a much larger number of triangles in the illuminated region (as compared with the microscope images). However, because there is a gradient in φ_{A} in the column, and the illuminating beam waist is much larger than the viewing region in the microscope, these lightscattering patterns represent averages over relatively large ranges of φ_{A}. We have made an attempt to obtain lightscattering patterns at welldefined φ_{A} by reducing the beam waist (for example, to illuminate about the number of particles, as in our microscope images); however, this reduction in illumination area makes the lightscattering intensity patterns significantly less smooth and speckled, so they essentially resemble what we obtain from Fourier transform of individual microscope images. To obtain superior intensity patterns in reciprocal space at a single welldefined φ_{A}, it would be desirable to perform light scattering by illuminating an extended area of a uniform 2D system of triangles, if an alternative method for concentrating the particles, which does not create a gradient in φ_{A}, could be developed.
Order parameters and correlation functions
General definitions of order parameters and correlation functions have been published previously (see ref. 13). In particular, for triangular shapes, the global mfold molecular orientational order parameter Φ_{m} is:
where θ_{j} is the orientation angle of particle j and N is the total number of particles. To remove the degeneracy of molecular orientation due to the threefold symmetry of triangles, we use
Additional information
How to cite this article: Zhao, K. et al. Local chiral symmetry breaking in triatic liquid crystals. Nat. Commun. 3:801 doi: 10.1038/ncomms1803 (2012).
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Acknowledgements
This work has been supported by University of California startup funds (T.G.M.).
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K.Z. designed experiments, synthesized dispersions of triangular platelets, performed experiments, recorded data, wrote video analysis software, and calculated correlation functions and order parameters. R.B. provided theoretical contributions to the trioid model and LCSB interpretation. T.G.M. initiated the project, designed and supervised the experiments, analysed data, and contributed to the trioid model and LCSB interpretation. All authors performed analysis, contributed to discussions and interpretations, and assisted in preparing the manuscript. T.G.M. assembled and finalized the manuscript.
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Supplementary information
Supplementary Figures
Supplementary Figures S1S3 (PDF 368 kb)
Supplementary Movie 1
Entropy driven spatiotemporal dynamics of Brownian triangles in the simple triatic phase T_{Δ} at φ_{A} = 0.58; the movie plays 6 times faster than real time, corresponding to a measurement duration of 4.8 minutes. (MOV 9440 kb)
Supplementary Movie 2
Entropy driven spatiotemporal dynamics of Brownian triangles in the local chiral symmetry broken triatic phase T_{Δχ} at φ_{A} = 0.63; the movie plays 6 times faster than real time, corresponding to a measurement duration of 4.8 minutes. The highlighted region shows that entropic excitations can change the sign of the lateral shift between nearest neighbouring triangles. (MOV 9444 kb)
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Zhao, K., Bruinsma, R. & Mason, T. Local chiral symmetry breaking in triatic liquid crystals. Nat Commun 3, 801 (2012). https://doi.org/10.1038/ncomms1803
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DOI: https://doi.org/10.1038/ncomms1803
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