Programming Interfacial Porosity and Symmetry with Escherized Colloids

Abstract

We simultaneously designed the porosity and plane symmetry of self-assembling colloidal films by using isohedral tiles to determine the location and shape of enthalpically interacting surface patches on motifs being functionalized. The symmetries of both the tile and motif determine the plane symmetry group of the final assembly. Previous work has either ignored symmetry considerations altogether or accounted for only the tile’s properties, applicable only when the motif is asymmetric; this approach provides a complete account and enables the design of symmetric colloids using this tile-based approach, which are often more practical to manufacture. We present the methodology, based on the type of the tile, and provide computational tools that enable the automatic classification of all tiles for a given motif and the optimization of the tile to fit the motif, sometimes referred to as “Escherization”.

Graphical Abstract

1. INTRODUCTION

Colloidal films play a critical role in technologies ranging from microelectronics to pharmaceutical delivery systems.^1–4 The two-dimensional (2D) pattern of the film and its void fraction affect features such as catalytic activity, mass transfer resistance, optical properties, and hydrophobicity.^5–9 Engineering colloidal self-assembly to achieve specific designs often involves determining the shape of a colloid and where to create enthalpically interacting patches on its surface; however, it is challenging to design systems that can assemble into highly complex patterns.

Here, we develop an approach, based on a technique known as “Escherization,”¹⁰ to design “patchy” colloids that enable a priori control over the self-assembled film’s porosity and symmetry simultaneously. These patches are derived from isohedral (IH) tiles. These tiles have no specific shape requirements and can adopt many complex configurations; the Dutch graphic artist, M. C. Escher, widely employed these tiles in his artwork to produce interlocking designs that illustrate how flexible these tiles can be.¹¹ IH tiles are linked to the plane symmetry group of the pattern, $𝒫$, they belong to, and the formal definition of an IH tile is one for which the pattern’s symmetry group, $𝒮(𝒫)$, contains transformations that map any tile onto all other tiles in the pattern.¹² This property confers a set of features on IH tiles that make them practically useful for designing singlecomponent self-assembling patchy colloidal systems.

First, all IH tiles have the same shape, although some are reflected images of others. They tessellate the plane by connecting with neighboring tiles in an edge-to-edge fashion. The second important property is that this edge-matching pattern is always the same throughout the entire pattern (cf. Figure 1a). Thus, edges are “paired” in a fixed, complementary way, akin to DNA base-pairing. IH tiles may be regarded as closed shapes, whose perimeters are composed of pairs of matching line segments. Our strategy is to use IH tiles to create a set of complementary patches on a colloid in the following way. After a tile is selected, the unfunctionalized colloid is placed within the tile and the tile’s shape is optimized to fit its cargo. Most portions of the tile’s edges will be erased, leaving behind complementary “patches.” All patches attract only their complementary patch (edge), which is sometimes themselves (cf. Figure 1b where pairs are shown with identical colors).

Figure 1.

Self-assembly of Escherized colloids. (a) Examples of tiling using IH4 in which each tile has six neighbors; two are connected by creating translated copies of the central tile while the other four are formed by a ${180}^{\circ }$ rotation. IH4 is associated with the wallpaper group, $S(𝒫)=p2$; the 2-fold rotation centers which define this group are shown with black ellipses on the example at the right. (b) Example of how Escherization proceeds and can sometimes lead to patchy colloids which assemble into an unexpected symmetry group. Mirror lines are denoted in black, and ellipses corresponds to 2-fold rotation centers. (c) Motif encircled by an IH tile (IH30) that does not match perfectly leaving behind some void space. The void fraction of this unit is the same as the overall pattern. (d) Asymmetric motif from (c) is placed in IH30, whose shape and size are modified to fit the motif differently, resulting in different coverage fractions, $f$, when assembled. The genetic codes (e.g., $\mathrm{I}\mathrm{H}4=\mathrm{TCCTCC}$, $\mathrm{IH30}=\overline{{\mathrm{A}}_3}\overline{\mathrm{A}}{\mathrm{C}}_3{\mathrm{C}}_3$) follow from ref 13. For reasons discussed therein, the patches highlighted by red ellipses that are far away from the colloid’s surface for $f={1.5}^{-2}$ and $f={1.6}^{-2}$ may be removed (cf. Section 3).

Due to their highly flexible shapes, IH tiles are instead conventionally classified topologically based on the set of relationships between edges abutting adjacent tiles;¹² e.g., the number of edges, their relative orientation, and which edges are paired with each other. There are a total of 93 topologically different IH tiles in 2D, which represent different “families” of permissible shapes and boundaries. In this work, we represent IH tiles with their canonical designation IHX, where $X$ is a number from 1 to 93; this numbering reflects their historical enumeration and is not ordered according to pattern symmetry. This relatively small number enables exhaustive searching over all possibilities to find the most appropriate tile for a given application. Owing to their definition, each of the 93 IH tiles is linked to a specific plane symmetry group. That is, each tile assembles into one of the 17 different plane symmetry groups in 2D (wallpaper groups). However, this connection is nuanced and is related to the concept of an induced (motif) symmetry group, $S(𝒫\mid M)$, which will be explained in detail later. For the purpose of this work, this group may be thought of as the symmetries shared by the unfunctionalized colloid and the overall pattern.

When $S(𝒫\mid M)$ is trivial so that unfunctionalized colloid and the pattern share no symmetries, only the IH tile determines the wallpaper group of the self-assembled structure. This is strictly true when the colloid is asymmetric but can still hold in other cases. The simplicity of the former case has made it the subject of previous study.^13,14 However, in practice it is often easier to manufacture symmetric colloids, such as spheres, or colloidal “molecules”^15–17 for use in hierarchical self-assembly schemes. Moreover, when the unfunctionalized colloid has some symmetry, e.g., a mirror plane as shown in Figure 1b, it is sometimes possible for the colloid to inadvertently provide symmetries that become part of the global pattern. This is one of several complications that can arise which require careful consideration to prevent. One purpose of this work is to extend our previous design approach^13,14 so that it is applicable for these cases of symmetric colloids.

The other main feature of this work derives from the fact that IH tiles represent intensive units of the self-assembled structure since patterns are created by duplicating this unit. As a result, the fraction of an individual tile’s area covered by the unfunctionalized colloid is equal to the entire pattern’s area fraction, $f$ (cf. Figure 1c). The limit where the colloid’s shape coincides with the tile boundary yields a void fraction, $v=1-f=0$. The search for the IH tile that best satisfies this is called the “Escherization problem.”¹⁰ M. C. Escher famously exploited IH tilings to construct interlocking forms that leave no “gaps” between them.¹¹ This $f=1$ case represents an interesting technological limit as it describes how to minimize manufacturing waste produced after cutting shapes out of nonrecyclable planar materials, such as leather.^18,19 A great deal of work has followed to identify optimal tile shapes for highly complex units using various approaches.^20,21

As originally formulated, though, the Escherization problem ignores symmetry since it does not enforce a constraint on which IH tiles are permissible given the symmetry of the unfunctionalized colloid and seeks only the limit of $f=1$. In terms of colloidal engineering, both symmetry constraints and $f_{\text{target}}<1$ are important material properties to consider. To this end, here, we explore how to extend this approach to create “Escherized” colloids that self-assemble into a structure with a predetermined $f<1$ and wallpaper group, shown schematically in Figure 1d. Systematic exploration of this leads us to three categories of tiles for every possible colloid. We characterize these tiles as “safe,” “dangerous,” or “forbidden.” This classification pertains only to whether or not the wallpaper group desired will be affected during the tile optimization to reach $f_{\text{target}}$, i.e., whether something unexpected (cf. Figure 1b) may arise. If symmetry is of no concern, this designation is unimportant, and any IH tile may be used to functionalize the colloid as in the original Escherization problem.

This article is organized as follows. In Section 2, we describe relevant aspects of tiling and symmetry that lead to this classification schema, and subsequently describe the computational algorithm and optimization tools used to create Escherized colloids in Section 3. We illustrate the creation of these colloids and how different void fractions are achieved using molecular dynamics simulations in Section 4. We conclude in Section 5.

2. BACKGROUND

2.1. Terminology, IH Tiles, and Symmetry.

We begin by defining terminology. Henceforth, we refer to the unfunctionalized colloidal object, e.g., the gray patch or the “Vitruvian Man” in Figure 1 as the motif, $M$. The motif may have some symmetry group, $S(M)$; n-fold rotational symmetries are denoted as $cn$ while dihedral groups containing $cn$ and reflection symmetries from $n$ equally inclined mirror lines are denoted as $dn.$¹² An asymmetric motif’s symmetry is $S(M)=c1$. To functionalize the motif, it is encircled by an IH tile whose perimeter is composed of paired line segments with a unique matching pattern, e.g., indicated by the colors in Figure 1a,b. These lines are partially erased, leaving pairs of matching colors which define the location and shape of the enthalpically attractive patches explained in more detail in Section 3.^13,14 Henceforth, we refer to the combination of a tile and motif as “the colloid.”

The nuances involved in obtaining the correct wallpaper group by selecting the right tile for a given motif originate from the fact that the motif’s symmetries may combine with those encoded in the IH tile to produce something unexpected. The pattern’s induced group, $S(𝒫\mid M)$, are the group of symmetries that the pattern and motif share, creating a link between the two.¹² For example, a motif with $d1$ symmetry may lie along a mirror line of the overall pattern in such a way that $S(𝒫\mid M)=S(M)=d1$ (cf. Figure 1b). In all patterns $S(𝒫\mid M)$ is necessarily a subgroup of the motif’s symmetry group, $S(M)$. Historically, IH tiles have been further categorized largely according to differences in $S(𝒫\mid M)$ referred to as “henomeric types.”¹² These types provide a natural description of when such issues can arise. For each target wallpaper group, $S(𝒫)$, there are several henomeric types, each of which includes one or more IH tiles. These henomeric types are the basis of our categorization of IH tiles as “safe,” “dangerous,” or “forbidden” based on the motif being functionalized (defined in Section 2.2). To better understand henomeric types, it is instructive to revisit the connection between IH tiles and wallpaper groups.

Wallpaper groups may be described topologically by orbifolds (orbit manifolds).^22–24 Orbifolds, $O$, are surfaces that are obtained by folding up the fundamental domain of a periodic crystal to match symmetrically equivalent positions.^24,25 46 of the 93 IH tiles are considered “fundamental” and are formed by cutting these orbifolds back open into single, “flat” pieces, recovering all topologically different fundamental domains for each wallpaper group.^26,27 Intuitively, this process of cutting open the orbifold determines the edge-matching pattern for these tiles, i.e., which parts are on the left- vs right-hand side of the scissors (cf. ref 13 for illustration). Careful inspection of this cutting process visually illustrates that all point-preserving symmetries (rotations and reflections) end up on the boundaries of these fundamental tiles;^13,26,27 since there are none running through the interior, there are none to share with the motif placed inside. This is why the induced group is trivial for all fundamental tiles, $S(𝒫\mid M)=c1$, meaning the motif and tile are essentially decoupled in terms of symmetry. The only information controlling the wallpaper group of the pattern comes from the tile itself, and to be consistent, the motif should be asymmetric, $S(M)=c1$ as shown in Figure 1c. It might appear permissible for the motif to have more symmetry; however, as illustrated by Figure 1b this is incorrect and will be discussed further in Section 2.2.

The 47 nonfundamental IH tiles are constructed by “gluing” multiple fundamental tiles together;²⁶ this is done by using the point-preserving symmetries on their boundaries to make adjacent copies, ultimately incorporating the generating symmetry into the center of the new tile. This operation visually reveals the $S(𝒫\mid M)$ associated with each henomeric type. If a tile results from gluing two smaller tiles together across a mirror line, then $S(𝒫\mid M)=d1$; if three tiles are combined around a 3-fold rotation center, $S(𝒫\mid M)=c3$, and so on (cf. the Supporting Information for illustration).

2.2. Classification of Motif and Tile Combinations.

There is one final caveat that determines our classification system. Recall that the induced group is always a subgroup of the motif’s symmetry group $S(𝒫\mid M)<S(M)$; however, it turns out that not all supergroups of $S(𝒫\mid M)$ are permissible in the motif.¹² That is, sometimes there exists a forbidden group of motif symmetries, $ℱ$, such that $S(𝒫\mid M)<ℱ\le S(M)$ which always changes the henomeric type of the pattern. This is nearly always synonymous with changing the wallpaper group of the pattern, though certain rare exceptions do exist where the henomeric type changes, but the wallpaper group does not.¹² Still, avoiding these forbidden cases amounts to a mildly overconservative check which ensures that the wallpaper group achieved is always the one expected. Our classification scheme relies on these previously enumerated henomeric types,¹² and their associated constraints, to identify permissible tile/motif combinations. This is illustrated in Figure 2 and described in more detail in the Supporting Information. Here, we briefly summarize how this results in the designations of “safe,” “dangerous,” and “forbidden”.

Figure 2.

Tile classification system. Two safe examples are given; one for a “waving man” (c1) and one for the Vitruvian Man (d1). Matching colors correspond to matching tile edges (patches). Two examples of forbidden tiles are also shown. One where the motif is a circle ($d_{\mathrm{\infty }}$) whose symmetry group is a supergroup of the minimal forbidden supergroup $(ℱ=c2)$ for the henomeric type IH41 belongs to. Instead of yielding $p1,p2$ results. Other tile shapes or motif placements may produce other groups, but $p1$ is never achievable. The second example shows IH90, which requires a motif with 3-fold rotation symmetry at its center being used to encircle a square (d4); the resulting structure does not have the symmetry group associated with IH90 because the motif is inconsistent with the tile. Dangerous tiles comprise all remaining cases. In the lowest row, elements of each wallpaper group are shown in black: solid lines are mirrors, dashed lines are glide reflections, ellipses denote 2-fold rotation centers, while $n$-gons represent other $n$-fold rotation centers.

For a given motif, we consider a combination to be safe if the tile belongs to a henomeric type where $S(𝒫\mid M)=S(M)$. In this case, the motif is sharing all of its symmetries with the pattern, so it contains nothing “extra” that could create potential issues. In the case where $S(M)=c1$, the motif has nothing to contribute, which is why the 46 fundamental tiles, for which $S(𝒫\mid M)=S(M)$, can be used to functionalize an asymmetric motif and only the tile controls the pattern’s wallpaper group. Figure 1d illustrates how the fundamental tile IH30 can be used to functionalize an asymmetric motif in different ways, producing a self-assembled film with progressively decreasing surface coverage, $f$. In this safe example, the motif may be placed in any position or orientation and the tile is free to adopt any shape. Safe combinations also occur when $S(P\mid M)=S(M)>c1$ but require that the motif be placed in a fashion consistent with $S(P\mid M)$ (cf. the Supporting Information for implementation details); e.g., Figure 2 illustrates how the Vitruvian Man’s mirror line must be aligned with the tile’s mirror so that it is part of the overall pattern.

Forbidden combinations can occur under two different circumstances. First, the tile’s henomeric type may have $S(𝒫\mid M)$ that is not a proper subgroup of $S(M)$; in other words, the pattern may require a symmetry, like a reflection, that the motif does not have to share, e.g., if it only has rotational symmetries. Second, the motif’s symmetry may contain a forbidden supergroup of $S(𝒫\mid M)$; e.g., the circle motif in Figure 2. In these cases, regardless of how the motif is placed in the tile, or what specific shape the tile adopts, the henomeric type of the pattern changes, which nearly always incurs a change in the wallpaper group, so it should be avoided.¹²

The remaining situations are considered “dangerous.” They occur when $S(𝒫\mid M)<S(M)$ and $ℱ\nless S(M)$; that is, when the motif (unfunctionalized colloid) has more symmetry than the tile’s henomeric type requires and these extra symmetries are not strictly forbidden. In such a case, there are generally ways that the right motif placement or tile shape could accidentally produce a pattern with more global symmetry than what the tile programs for, but it is not guaranteed; there are numerous possibilities, which cannot be explicitly selected or prevented in our computational approach (cf. the Supporting Information). If a dangerous tile is selected, then the pattern should be inspected after assembly to see if this has occurred. Regardless, it is possible to anticipate which alternative wallpaper groups might manifest.

When an unexpected result occurs, the targeted wallpaper group of the pattern $S_{\text{target}}(𝒫)$ is necessarily a motif transitive proper subgroup (MTPS) of the tiling that results, which belongs to the $S_{\text{result}}(𝒫)$ wallpaper group. These MTPSs have also been enumerated for each henomeric type¹² and can be used to identify which alternative wallpaper groups might result from a given dangerous tile/motif combination by collecting all unique $S(𝒫)$ from all the henomeric types where $S_{\text{target}}(𝒫)$ occurs as a MTPS. Some additional consistency checks should be performed, and this lookup operation is further detailed in the Supporting Information and automated in the code provided. For example, $S_{\text{target}}(𝒫)=p2$ is a MTPS of the henomeric type associated with $S_{\text{result}}(𝒫)=p2mg$ and $S(𝒫\mid M)=d1$ as shown in Figure 1. This is why the IH4 tile $(S(𝒫)=p2,S(𝒫\mid M)=c1$ but $ℱ=\varnothing$) can result in $p2mg$ when using a motif with $S(M)=$ $d1>c1$. Regardless of these complications, dangerous tiles are very useful since they expand the possible symmetry groups into which a motif can be assembled; the conditions required to be safe often greatly limit the possibilities.

3. METHODS

The procedure for Escherizing a colloid is as follows. First, a motif is selected to construct the pattern, which has some symmetry group, $S(M)$. This defines which henomeric types are safe, dangerous, or forbidden for all wallpaper groups. Second, a desired wallpaper group for the pattern, $S(𝒫)$, is chosen. From the safe or dangerous types corresponding to this wallpaper group, a tile is then selected to functionalize the motif. The tile shape and orientation relative to the motif are then optimized to achieve the desired $f$; this can be repeated with several tiles to find the one with the most convenient, or realizable, set of patches.

We have provided various computational tools to perform the classification of IH tiles and Escherization of colloids, and they are available online.²⁹ The first step in the design process is to create a motif. The included Create_Motifs.ipynb Jupyter notebook³⁰ may be used to create a motif as a series of points, or it may be user-generated. Subsequently, the Design.ipynb notebook lets users select a motif and check its symmetry. This tool includes a digitized version of Table 5.2.3 from ref 12 that enumerates henomeric types and associated constraints. Given a motif, this tool lists which henomeric types, and all included IH tiles, are safe, dangerous, or forbidden. The user may select from the safe or dangerous tiles and proceed to Escherization; the forbidden tiles should not be used if the wallpaper group of the final pattern matters. It is the user’s responsibility to check this when using a dangerous tile.

Fitting the tile to the motif is achieved using Tactile,²⁸ OptimLib,³¹ and other associated libraries.³² This process is phrased as an optimization problem that, given the cross-sectional area of a motif, searches for a tile area to achieve the desired $f_{\text{target}}$ by minimizing a loss function, $L_{\text{tot}}$ (cf. the Supporting Information). This contains terms that penalize the motif for exiting the tile, penalize the tile for being too oddly shaped that it may be difficult to practically synthesize, and try to keep the interacting patches attached to the surface of the motif.

Importantly, Tactile enables IH tiles to be described by a vertex parametrization such that a set of scalars, $\overrightarrow{\nu }=\left\{{\nu }_0,{\nu }_1,\dots ,{\nu }_n\right\}$, completely determines the vertices of a polygon whose edges may be deformed to make a tile. All parameters are on the order of unity and $n\le 6$ for all tiles. To completely define the colloid’s degrees of freedom, parameters are added to control the location of the motif’s center of mass $\left(C_{\text{motit},x},C_{\text{motifi}y}\right)$, its orientation, ${\theta }_{\text{motif}}$, and relative scale, $S_{\text{tile}}$; in addition, patch location along each edge is described by $\overrightarrow{u_0}$, and edge shape is controlled by the deflection parameter, $d_{\mathrm{f}}$. These parameters are concatenated into a vector, $\overrightarrow{X}$, as illustrated in Figure 3.

$$\overrightarrow{X}=\langle C_{\text{motif},x},C_{\text{motif},y},{\theta }_{\text{motif}},\overrightarrow{v},{\overrightarrow{u}}_0,d_{\text{f}},S_{\text{tile}}\rangle$$ (1)

Figure 3.

Illustration of the parameters optimized during functionalization of a colloid. Discrete points along lines are mapped to edges of a chosen IH tile, which encompasses the motif, using Tactile.²⁸ More details are available in the Supporting Information.

All terms are bounded and may be nondimensionalized, facilitating optimization. Constraints on the motif’s location or orientation are automatically built into the provided code ensuring the motif is always placed consistent with the tile’s symmetry. Values of $\overrightarrow{X}$ are rounded to the nearest allowable values when constrained. This creates “troughs” in the loss function’s landscape that pose problems for gradient-based optimization methods; similarly, rough edges of the landscape occur as the motif exits the IH tile creating troublesome peaks. To handle this, we used particle swarm optimization,³³ a stochastic swarm intelligence algorithm, to perform the optimization. Particle swarm optimization performed well and generally converged quickly to reasonable results.

When estimating the surface coverage fraction, $f$, the motif’s cross-sectional area is assumed to be known, but the tile’s area must be computed. Despite its potentially complex shape, the area of the tile can be computed trivially from the polygon determined by the tile’s vertices before edge deformation occurs; for example, the six-sided polygon resembling a rectangle as shown in Figure 3. Deformations act according to the tile’s symmetry and do not affect its net area.

Once a motif was functionalized to produce a colloid, we simulated its self-assembly using NPT molecular dynamics simulations in LAMMPS.³⁴ Motifs and functionalized patches were discretized into beads that formed a rigid body in two dimensions. Patches were composed of five total beads: four of which interacted enthalpically, and one which was purely repulsive located at one end denoting an upcoming corner of the tile, following ref 13. Typically, the motif’s beads (depicted in black) have a diameter, ${\sigma }_{\mathrm{m}}=1.0$ while the patch beads’ (depicted in color) diameter is set to the distance between the closest pair after this scaling; typically, ${\sigma }_{\mathrm{b}}\approx 0.5{\sigma }_{\mathrm{m}}$. For randomly generated motifs, such as the random patch in Section 4.3, this was reversed. All beads interacted through a pairwise force-shifted Lennard-Jones interaction

$$U_{i,j}^{\text{fs}}(r)=\{\begin{array}{cc}\begin{array}{l}{U}_{i,j}^{\text{LJ}}(r)-U_{i,j}^{\text{LJ}}(r_{i,j}^{\text{cut}})-(r-r_{i,j}^{\text{cut}})\\ {\frac{\text{d}{U}_{i,j}^{\text{LJ}}(r)}{\text{d}r}|}_{r=r_{\text{cut}}}\end{array}& r\le r_{\text{cut}}\\ 0\hfill & r>r_{\text{cut}}\end{array}$$ (2)

where

$$U_{i,j}^{\text{LJ}}(r)=4\epsilon \left[{\left(\frac{{\sigma }_{i,j}}{r}\right)}^{12}-{\left(\frac{{\sigma }_{i,j}}{r}\right)}^6\right]$$ (3)

All units reported herein were nondimensionalized by using ${\sigma }_{\mathrm{m}}$ and $\epsilon =1$. Only functional beads of the same color indicating matching points (cf. Figure 3) had attractive interactions while the remaining interactions were truncated to form a purely repulsive potential akin to Weeks–Chandler–Andersen (WCA).³⁵ Points indicating a tile corner, called “stop codons,”¹³ are depicted in white and are also purely repulsive. Purely repulsive interactions used ${r_{i,j}}^{\text{cut}}=2^{1/6}{\sigma }_{i,j}$, while attractive interactions used ${r_{i,j}}^{\text{cut}}=2.5{\sigma }_{i,j}$. All cross interactions were determined using standard Lorentz–Berthelot mixing rules, e.g., ${\sigma }_{i,j}=\left({\sigma }_i+{\sigma }_j\right)/2$.

A large box with $N=200$ total colloids (100 of each chirality) was simulated at $P^{\mathrm{*}}=P{{\sigma }_{\mathrm{m}}}^3/\epsilon ={10}^{-3}$ starting from $T^{\mathrm{*}}=k_{\mathrm{B}}T/\epsilon =1.0$, where $k_{\mathrm{B}}$ is the Boltzmann constant. After initial randomization, the system was cooled to $T^{\mathrm{*}}\approx 0.4$ over the course of $5\times {10}^7$ steps, then held at constant temperature for another $1.5\times {10}^9$ steps, over which annealing occurred and thermodynamic properties were measured. The optimal final temperature to observe self-assembly varied slightly from colloid to colloid and is reported along with final snapshots in the following sections. A time step of $\delta t=0.005$, temperature damping constant $T_{\text{damp}}={10}^2\delta t$, and pressure damping constant of $P_{\text{damp}}={10}^5\delta t$ were used.

4. RESULTS AND DISCUSSION

The patches on motifs, represented by a series of beads, typically span half of the tile edge to which they belong. This is an exaggeration to illustrate our technique, and in practice, the span may be greatly reduced. Furthermore, we have allowed only simple tile edge deformations, as in previous work¹³ (cf. Figure 3), leading to fairly simple shapes as we consider these to be more experimentally feasible when functionalizing a colloid by attaching functional chemical groups.

As alluded to in Figure 1, not all edges (and therefore patches) are necessary for self-assembly. At a minimum, the patches must lead to a fully connected graph connecting all of the colloids in the final assembly. Many times, an IH tile provides more than necessary; an example is provided in the Supporting Information. A shortcut for determining which edges may be removed for the 46 fundamental tiles is discussed in ref 13. Here, we highlight redundant edges with red ellipses when these edges are far away from the motif and would be difficult to attach to the motif in practice. This further simplifies these Escherized colloids and is especially relevant far from the Escherization limit of $f\to 1$.

We have also assumed that, in practice, these colloids will be synthesized in three-dimensional space and then deposited to a 2D surface for self-assembly. Since rotation out of the assembly plane is equivalent to a reflection through a mirror orthogonal to that plane, rotational diffusion is expected to produce a racemic mixture of colloids.¹⁴ Thus, all simulations shown here contain an equal amount of mirror images to emulate this effect.

4.1. Safe.

In Figure 4, we consider the Escherization limit for several motifs using tiles that are considered safe. In this case, the motif’s symmetry is identical to the induced group associated with the tile. The “Pegasus” motif, inspired by Escher’s own sketch¹¹ has no symmetry, $S(M)=c1$. As a result, the 46 fundamental tiles can be used safely, and since the other 47 tiles require some contribution from the motif, $S(𝒫\mid M)>c1$, they are all forbidden. Despite the Pegasus’ unusual shape, it is clear that patches can be placed in reasonable locations, such as on the hooves or along the tail or wings to induce self-assembly into various structures. The wallpaper group $p2$ is a gyration group,¹⁴ which causes the system to phase separate into different domains for each chirality.

Figure 4.

Examples of Escherized colloids that have been functionalized by seeking $f\to 1$. The columns depict the use of various safe tiles for motifs with different symmetries, $S(M)$. The reduced temperature, $T^{\mathrm{*}}$, of each simulation is listed along with the tile and associated wallpaper group. When chiral phase separation occurs¹⁴ the domains of each enantiomorph are circled in red and blue to distinguish them.

A variation on the Vitruvian Man, termed the “waving man,” is similarly asymmetric. Again, self-assembly can be easily programmed by fitting various tiles to the motif. It is clear that some tiles are still more natural than others. In all cases, $f=1$ could not be strictly obtained because the shape of the motif could not be perfectly obtained with the tiles, given the edge shapes as we have allowed. The Vitruvian Man has a mirror plane, $S(M)=d1$, thus, the 46 fundamental tiles that were safe for the previous two cases are now considered dangerous. As illustrated for other tiles that are safe, patches again can be placed reasonably, such as on the shoulders or feet, and self-assembly proceeds well.

Many patches appear more naturally placed on the Vitruvian Man than on the waving man or Pegasus. This is attributed to the fact that the tile’s natural shape is symmetric in the same way that the motif is, which often leads to a visually better “fit” especially when seeking $f\to 1$. Similarly, the pinwheel is functionalized with tiles that have a 3-fold rotation center at the tile’s center of mass. Optimized designs place patches naturally on the arms.

4.2. Dangerous.

Safety requires that $S(M)=S(𝒫\mid M)$ restricting the number of possible tiles, or shapes, that can be fit to the motif and the wallpaper group of the final pattern. For example, there is at least one tile (shape) permissible for each wallpaper group that can be used to safely functionalize the Pegasus, which corresponds to the fundamental domains of these crystals. However, the Vitruvian Man has only 22 different safe tiles spanning 10 wallpaper groups; the pinwheel has only three safe tiles (IH10, IH89, IH90), spanning only three wallpaper groups ($p3,p31m,p6$, respectively). This illustrates the utility of working with dangerous tiles. The 46 tiles that are safe for the Pegasus are dangerous but not forbidden for both the Vitruvian Man and the pinwheel. This provides these motifs with 46 additional tiles to consider, spanning all 17 wallpaper groups.

In this instance, the final pattern needs to have its wallpaper group checked since specific choices of motif position or orientation or tile shape can increase the overall symmetry of the pattern beyond what is desired. Detailed examples are provided in the Supporting Information (cf. Figure S4) along with an explanation of the included computational tool that can automatically identify which other wallpaper groups could result. Section S2 presents many examples of Vitruvian Man being functionalized by dangerous tiles in the Escherization limit. Most produce a pattern with the wallpaper group the tile’s henomeric type is associated with; however, some fail entirely, and some near misses are illustrated. In these cases, it often happens that the shape of the motif biases the tile into adopting a shape that increases the system’s symmetry, akin to Figure 1b.

4.3. Tunable Porosity.

We have thus far illustrated the functionalization of motifs designed to minimize the void space around them; in this case, it is best to have a tile whose shape can be optimally matched to the motif’s surface everywhere since seeking $f\to 1$ essentially “shrinkwraps” the tile around the motif. If we seek $f_{\text{target}}\ll 1$, however, the motif cannot make contact with the tile’s entire perimeter. In this case, only the shape of the tile very near the motif needs to match the motif’s shape, since this is where the patch will be placed. Moreover, there are generally far more possible solutions since packing constraints are reduced; for example, a motif may be rotated inside a large tile, producing many functionalization patterns without affecting $f$.

Figure 5 illustrates the functionalization of a randomly generated shape $(S(M)=c1)$ with several safe tiles. On the left, the tiles have been shrunk to fit the motif; the patch locations match well with the surface of the motif and seem to be plausibly placed. Since the motif cannot be perfectly fit by these, or any other tile, the best $f$ is necessarily less than unity. Still, this produces a structure that is tightly packed. From left to right, the porosity is increasing.

Figure 5.

Molecular dynamics simulations of colloids (bottom left of each panel) formed from a randomly generated convex shape functionalized with different tiles at different $f$ values. Patches from edges that can be ignored or removed are highlighted with red ellipses. (a) IH30 (p31m) was used to functionalize the motif. (b) IH30 was used again but with a weak attraction $\left({\epsilon }_b={10}^{-3}\right.$, cf. the Supporting Information) between the patches and the motif beads. (c) Functionalization using IH7 (p3) over a similar range of $f$ values.

The same tile (IH30, as in Figure 1c) is used in Figure 5a,b; however, in (b) the patch beads are weakly be attracted to the motif’s beads during Escherization (cf. the Supporting Information) which helps place them closer to the motif’s surface. Both are valid designs but clearly result in different patterns despite having the same wallpaper group $(p31m)$. The pore structure, beyond the average amount of void space, plays an important role in determining properties such as selectivity if used for mass separation. It is beyond the scope of this work to study this in detail but clearly many structures are achievable.

Importantly, as $f$ decreases, it becomes more challenging for the motif to make contact with all edges of its tile. This would suggest that to achieve high-porosity, low-density films the motif must be functionalized with an attractive patch far from its surface. However, not all edges are strictly necessary for selfassembly. For example, IH30 has four edges: two straight edges corresponding to mirrors and two curved edges that are 3-fold rotation images of each other. As illustrated in Supporting Information for Figure 5b at $f=0.4$, one of the mirror planes is superfluous since the tiles in the final assembly can be fully connected by using only the remaining three edges. Extraneous edges, which lead to patches that are not touching the surface of the motif, are indicated with red ellipses. This is a particularly important feature since it enables low $f$ designs while remaining chemically feasible.

Figure 6 illustrates the functionalization of a simple square motif, $S(M)=d4$, with safe, dangerous, and forbidden tiles. IH76 is a safe tile for this motif, and the resulting pattern belongs to the p4mm wallpaper group. In the two blue columns, this functionalization is performed when the functional beads are weakly attracted to the motif (left) and when no such attraction exists (right) during Escherization. The shape of IH76 is a square, the same as the motif; fitting the tile to the motif in the high $f$ limit creates a natural outline. However, as $f$ decreases, the tile tends to expand isotropically when there is no attraction, but the tile undergoes a rotation of ${45}^{\circ }$ when it is present. This enables the patches to attach to the corners of the motif, creating a more feasible design, while also changing the pore topology in the final assembly.

Figure 6.

Molecular dynamics simulations of colloids (bottom left of each panel) formed by functionalizing a square with different tiles at different $f$ values. Patches from edges that can be ignored or removed are highlighted with red ellipses. IH76 is a safe tile to use for this motif. In the blue rectangle, the left column corresponds to when patch beads are weakly attracted to the motif during Escherization; the right column corresponds to when no attraction is used. IH16 is a dangerous tile, but the resulting design successfully produced a p31m pattern; snapshots are shown in green. IH34 is also dangerous, but all designs resulted in p6mm instead of its target; they are shown in red. IH73 is forbidden and never produces the target p4gm symmetry. Some mirror lines of the wallpaper groups that are produced are shown for the different cases of failure.

IH16 and IH34 are dangerous tiles that share only a subgroup of the motif’s symmetries: a mirror line in the former and a 2-fold rotation in the latter. Fortuitously, the designs created for all depicted $f$ values produce the targeted wallpaper group, $p31m$, using the IH16 tile. Conversely, the IH34 tile always produced p6mm patterns in this case. It is possible for IH16 to produce invalid designs and for IH34 to produce valid ones if the motif’s orientation or position is changed. Finally, IH73 is associated with wallpaper group $p4gm$ but the minimal forbidden supergroup, $ℱ=d4$. In this case, a motif with $S(M)=d4$ inevitably promotes the pattern beyond $S_{\text{target}}(𝒫)$ regardless of the motif’s orientation or position.

5. CONCLUSIONS

In this work, we presented a design scheme to create patchy colloids that self-assemble into 2D films with a preprogramed porosity and wallpaper group. This builds upon the technique of “Escherization” and is achieved using IH tiles to create an encompassing boundary, determining the location, shape, and chemical identity of the patches. Since IH tiles produce patterns by repeatedly copying the unit, the fraction of the tile covered by the motif is identical to the coverage fraction of the final assembly. Furthermore, since IH tiles are linked to the wallpaper group of the pattern produced, they provide a way to encode the wallpaper group of the final structure. As a result, it is possible to simultaneously design these two properties.

However, in general, the symmetry of the motif can also affect the wallpaper group of the self-assembled film. Previous work focused on 46 of the 93 IH tiles useful for asymmetric motifs where this programming can be unambiguously encoded in the tile, i.e., only the colloid’s patches. Asymmetric motifs tend to be hard to manufacture, and a methodology to extend this design approach to symmetric motifs is necessary. By using henomeric types, we addressed this gap by providing a method to classify each of the 93 IH tiles as safe, dangerous, or forbidden for a given motif. This classification refers to whether the motif’s symmetries may interfere with the tile’s programming to yield an unexpected wallpaper group. For applications where the film wallpaper group is irrelevant, this classification is unnecessary and any tile may be used to Escherize the colloid.

The fitting of these IH tiles to the motif has been explored previously in the limit of trying to optimally fit the two together. We have expanded this Escherization process to enable the design of low-density films as well. The included computational tools enable a designer to determine which tile(s) to use for a given motif and target wallpaper group, what groups could arise instead if a dangerous tile is selected, and code to fit the tile and patches given a desired film coverage fraction, $f$. While we have focused on only simple edge shapes, future work may focus on expanding this to include much more complex boundaries.