Emergent Geometry of Inhomogeneous Planar Crystals

Uniform triangular crystals are the ground state for particles that interact isotropically in two dimensions. However, when immersed in an external potential, for example, one arising from an electric field, a flow field, or gravity, the resulting phases are significantly distorted in a way reminiscent of conformal transformations of planar lattices. We study these “conformal crystals” using colloidal experiments and molecular dynamics simulations. By establishing a projection from these self-assembled inhomogeneous crystals to homogeneous crystals on curved surfaces, we are able to both predict the distribution of defects and establish that defects are an almost inevitable part of the ground state. We determine how the inherent geometry emerges from an interplay between the confining potential and the interparticle interactions. Using molecular dynamics simulations, we demonstrate the generic behavior of this emergent geometry and the resulting defect structures throughout a variety of physical systems.

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Particles that repel each other can form intriguing patterns when confined by external fields. Locally, these patterns look like crystals, but globally, they may show great variation in the size and orientation of the unit cell, reminiscent of the distorted grids that many of M. C. Escher’s drawings are based on. Despite the appearance of these patterns in many systems, such as foams or ensembles of electrically charged particles, there is relatively little understanding of their structure and material properties. We show that the properties of such systems come into focus when viewing them as shadows of uniform crystals projected onto curved surfaces.

While repulsive particles would otherwise arrange themselves in perfect hexagonal tiles, pentagonal and heptagonal tiles emerge when the particles are immersed in external potentials. We find that the distribution of these pentagonal and heptagonal tiles—known as topological defects—can be predicted by mapping the coordinates of the particles onto a curved surface using a conformal projection, as is used to project a flat map of the Earth onto a globe. We explore this insight using simulations and experiments on microscopic particles.

Our results show that when repulsive particles are confined by external fields, topological defects are an almost inevitable feature of the ground state. This perspective can be extended to analyze the properties of systems such as colloidal matter, dusty plasmas, and quantum dots.

Figure 1

Inhomogeneous crystals. (a) M. C. Escher’s The Print Gallery. To the right of the artwork is the conformal grid, which bears striking similarity to the distorted grid used by Escher to create his piece (image from Ref. [1]). All M. C. Escher works © 2018 The M. C. Escher Company–the Netherlands. All rights reserved. Used by permission. www.mcescher.com All images may only be used unaltered, unchanged and not manipulated in any way. (b) Perfectly conformal lattice used in Ref. [14] to model repulsive particles in a gravitational field. (c) Inhomogeneous crystal of repulsive colloidal particles. (d) Experimental setup. Micron-sized PMMA colloids are confined at an oil-water interface. We estimate the total integrated curvature in the region where the particles are confined and, thus, any interactions with the substrate curvature are negligible. An external potential is created by applying an alternating voltage to a needle-shaped electrode above the sample. The voltage is grounded on a transparent indium-tin-oxide cover slip below the sample. The particles are fluorescently labeled and imaged from below.

Figure 2

Conformal geometry and defects. (a) After collecting microscope images of the equilibrated system, each particle is identified and the resulting structure is triangulated. Here, fivefold and sevenfold defects are colored green and red, respectively, while black particles are undefected. The shaded gray area represents the boundary region, which is not used for the analysis. In this region, the net defect charge drops to $+6$, as demanded by topological constraints. (b) Conformal transformation between inhomogeneous and homogeneous structures. Above the flat, experimental structure is the corresponding transformed, curved crystal. (c) Average angle between bonds vs orientation of bisect. The angle between adjacent bonds is plotted against the orientation of the bisect of those two bonds with respect to the radial direction. We find that the angle of bonds is the value expected for a triangular lattice, irrespective of the orientation. In other words, adjacent bonds that are oriented in the azimuthal direction and those oriented in the radial direction both are separated by 60 degrees. (d) Average bond length vs orientation of bond. The length of bonds, normalized by the average local lattice spacing at that distance from the structure’s center, is plotted against the orientation with respect to the radial direction. We find that bond lengths are uniform, irrespective of their orientation. (e) Lattice spacing plotted against radial position in the crystal. Two sets of experiments are shown, each with a different number of particles. Each color represents a different experimental run. By changing the confinement strength, the resulting crystal sizes, $R$, are varied. (f) Integrated defect charge vs integrated effective curvature. The effective integrated curvature is plotted against the net defect charge for varying positions within the resulting structures. The colors shown correspond to those in (e). The dotted line represents a line with slope 1.

Figure 3

Structures of simulated confined crystals. (a) Rotationally symmetric potentials. An example low-energy configuration of Coulomb-repulsive particles confined inside the potential $V_{\mathrm{ext}}\propto r^2$ is shown on the left next to the corresponding uniform curved crystal. On the right of the crystal is a plot of the integrated curvature for particles in rotationally symmetric parabolic potentials interacting via various inverse power-law potentials and the result after scaling by the overall system size. (b) Results for translationally symmetric potentials. We show an example low-energy configuration of $1/r^4$ interacting particles inside of the potential $V_{\mathrm{ext}}\propto y^2$ next to the corresponding uniform curved structures. The simulated system is carried out with periodic boundary conditions. Note that the entire crystal is not shown. Shown on the right is the integrated curvature profile for various inverse power-law interactions and the result after scaling by the overall system size. In this case, the integrated curvature scales linearly by the lateral system size, and thus, the integrated curvature must be scaled by the crystal aspect ratio. (c) Result for particles in a constant force field ($V_{\mathrm{ext}}\propto y$) in the presence of a wall at the bottom o the crystal. Again, the simulated system is carried out with periodic boundary conditions, and the entire crystal is not shown.

Figure 4

Simulated hard-wall confined crystal with a corresponding negatively curved surface. (a) Circular wall boundary. An example low-energy configuration of Coulomb-repulsive particles confined inside a circular hard-wall boundary is shown next to the corresponding uniform curved crystal. Embedding the entire structure requires covering the surface multiple times (see Appendix pp5). In order to separate the different coverings, the surface is slightly distorted. (b) Integrated curvature, which collapses when rescaled by the system radius.

Figure 5

Simulated hard-wall confined crystal with corresponding negatively curved surface. (a) Translationally symmetric hard-wall boundary. An example low-energy configuration of Coulomb-repulsive particles confined inside a hard-wall boundary is shown next to the corresponding uniform curved crystal. Embedding the entire structure requires covering the surface multiple times (see Appendix pp5). In order to separate the different coverings, the surface is slightly distorted. (b) Integrated curvature, which collapses when rescaled by the system radius.

Figure 6

Inhomogeneous onset of melting. (a) Order parameter variation. Magnitude of orientational order parameter $|{\mathrm{\Psi }}_6|$. We find that the onset of melting occurs further into the bulk as the confinement strength is reduced, an effect absent in the simulated zero-temperature configurations. (b) Total number of defects. Total number of defects are compared to the approximate minimum number needed. Here, confinement strength is characterized by the resulting crystal size. An experimental crystal representing (c) stronger confinement and (d) weaker confinement is shown.

Figure 7

Approximate external potential for particles in colloidal experiments. Using the geometry of the experimental setup, the radial electric field profile is numerically calculated and used to model the external potential felt by the particles $V_{\mathrm{ext}}\propto |E^2(r)|$. Note that the particles in the experiment are confined to a radius $R<500\mu \mathrm{m}$.

Figure 8

PMMA particles at an oil-water interface. In the absence of a spatially varying potential, isotropically repulsive particles form a triangular lattice.

Figure 9

Experimental comparison of net disclination charge and integrated curvature distribution. The net disclination charge enclosed within varying radii is compared to the integrated curvature within that radii. The two quantities are found to be proportional at each radii. Here, each plot corresponds to a different-sized crystal, and each row of plots corresponds to a crystal with a different number of total particles, $N$.

Figure 10

(a)–(h) Typical lattice spacing variation of various systems examined. To the left of the lattice spacing variation is the simulated particle configuration. Theoretical lattice spacing variations with no free parameters are plotted along with the variations measured from simulation data. Lattice spacings are scaled by predicted initial spacings $a_0$, and coordinates are scaled by calculated system sizes $R$ and $H$. The expressions that determine $a_0$, $R$, and $H$ can be found in Appendix pp6. For rotationally symmetric systems, $R$ refers to the radius of the crystal, as shown in (a). For translationally symmetric systems, $\mathrm{\Delta }x$ refers to the distance lateral size of the system, as shown in (c) and (e). Note that configurations (c)–(f) and (h) are cropped at the edges and thus only partially shown, so the actual lateral sizes are larger than what is pictured. For systems with $V_{\mathrm{ext}}=\gamma y$, $H$ refers to the width of the system in the $y$ direction, as shown in (e). For systems with $V_{\mathrm{ext}}=\gamma y^2$, $H$ refers to half the width in the $y$ direction, as shown in (c).

Figure 11

Yukawa interacting particles in a parabolic confining potential $V_{\mathrm{ext}}=\gamma y^2$. (a) Integrated effective curvature predicts final defect distribution as in the case for pure power-law potentials. (b) Example particle configuration. (c, d) Scaling behavior of the integrated curvature profiles. With the additional length scale introduced by the screening length, our data suggest that the integrated curvature profiles no longer collapse when scaling the coordinates by the system size as shown in (d).

Figure 12

Yukawa interacting particles in a parabolic confining potential $V_{\mathrm{ext}}=\gamma r^2$. (a) Example particle configuration. (b) Integrated effective curvature predicts final defect distribution as in the case for pure power-law potentials. (c)–(d) Scaling behavior of the integrated curvature profiles. With the additional length scale introduced by the screening length, our data suggest that the integrated curvature profiles no longer collapse when scaling the coordinates by the system size.

Figure 13

Yukawa interacting particles in a potential of the form $V_{\mathrm{ext}}=\gamma y$. Particles form an inhomogeneous structure in the presence of a barrier. The particle density increases towards the barrier. (a) Example particle configuration. Note that the entire structure is not shown, and periodic boundary conditions are used. (b, c) Scaling behavior of the integrated curvature profiles. With the additional length scale introduced by the screening length, our data suggest that the integrated curvature profiles no longer collapse when scaling the coordinates by the system size as shown in (c). (d)–(f) Structural change as the screening length is varied. In the case of sufficiently large screening length (small $\kappa$), as shown in (d), the structure resembles that of a Coulomb system. In particular, the long-range interactions result in a concentration of integrated curvature near the barrier at $y=0$.