## Abstract

Star-shaped molecules with three mutually immiscible arms self-assemble to form a variety of novel structures, with conformations that attempt to minimize interfacial area between the domains composed of the different arms. The geometric frustration caused by the joining of these arms at a common centre limits the size and shape of each domain, encouraging the creation of complex and interesting solutions. Some solutions are tricontinuous, and these solutions (and others) share aspects of bicontinuous structures with amphiphilic assemblies as similar molecular segregation factors are at work. We describe both highly symmetric and balanced structures, as well as unbalanced solutions that take the form of intricately striped amphiphilic membranes. All these patterns can result in chiral assemblies with multiple networks.

## 1. Introduction

This paper is motivated by recent experimental and theoretical studies of novel molecules, namely *three-arm star polyphiles*, with three mutually immiscible moieties linked to a common central core. These star polyphiles are generalizations of conventional amphiphiles, which have two joined immiscible domains—hydrophilic and hydrophobic. Amphiphiles form an array of segregated domains at the mesoscale, thereby pooling hydrophobic and hydrophilic fractions into sheets, rods and globular domains, depending on the chemical composition and the amphiphilic species [1]. Three-arm star polyphiles are analogous to amphiphiles in that they form mesoscaled lyotropic domains in solvents that selectively solubilize specific moieties [2,3]. Owing to their molecular polyfunctionality, the resulting assemblies contain three distinct domain types instead of the amphiphiles' two.

Both amphiphiles and star polyphiles have macromolecular analogues on much larger scales than the star polyphiles discussed here. The macromolecular analogues of amphiphiles are linear diblock copolymers, which also self-assemble in their pure melt phase, or with added selectively compatible polymers or solvents, to give a variety of mesostructured materials [4]. Macromolecular analogues of star polyphiles, the so-called star or mikto-arm copolymers have been widely explored, and shown to also form a rich array of mesostructured self-assemblies [5–7].

The theories described in this paper motivated our choice of experimental star polyphile to study and were developed as a consequence of the resulting studies (which are to date only partially published). With the goal of ‘three mutually immiscible moieties linked to a common central core’ a star polyphile was chosen with one hydrophilic (oligo-ethyleneglycol chain) and two hydrophobic (hydrocarbon and perfluorocarbon chains; figure 1). By design, in addition to the usual hydrophilic–hydrophobic immiscibility common to amphiphiles, the hydrophobic domains are themselves mutually immiscible, giving three distinct domains [3]. The hoped-for highly symmetric tricontinuous structures described in §2 were not found; instead an alternative motif emerged, strongly reminiscent of bicontinuous mesophases in amphiphilic systems. Current data suggest that these hydrophobic membranes are striped with the hydrocarbon (Hc) and fluorocarbon (Fc) domains, and their theoretical study forms the focus of the paper.

As we have this experimental system with precisely the properties we theoretically explore, throughout this paper we use its architecture and properties (e.g. a hydrophobic bilayer composed of striped Hc and Fc surrounding a hydrophilic channel system) as a concrete shorthand for any ABC star-molecular system with comparable molecular segregation and aggregation. It even stands in for the similar geometry but larger scale ABC mikto-arm star copolymers, or indeed any other geometrically similar molecule with appropriate interaction between its arms.

Our journey will repeatedly use hyperbolically curved films, so we first revisit the geometry and topology of the most prevalent bicontinuous mesophases, related to the so-called *Gyroid*, *D* and *P* surfaces,^{1} to understand the role of hyperbolic geometries in molecular-scale self-assembly. We then discuss generalizations of those structures capable of supporting three-arm star polyphile and copolymer self-assemblies. These structures fall into two principal classes that we discuss in turn: tricontinuous patterns, where each chemical species forms a distinct network (§2), and striping patterns, where stripes can form multiple networks (§3). A unifying concept of our discussion is the vital role of two-dimensional surface geometries in directing and sculpting likely assemblies.

## 2. From bicontinuous to tricontinuous

Molecules with ABC star geometries may self-assemble to form tricontinuous patterns. In order to understand these tricontinuous patterns, it is instructive to step back and explain the origin of crystalline (three-periodic) hyperbolic surfaces in bicontinuous cubic phases. The latter typically consist of bilayers that are wrapped onto saddle-shaped surfaces that are related to the three-periodic *Gyroid*, *D* and *P* minimal surfaces (figure 2*a*). The bilayer is ‘reversed’ or ‘normal’, depending on the amphiphilic geometry. Here we assume for simplicity that the phases are ‘type 2’, containing normal bilayers, which are curved towards the polar regions. We shall see later that all three surfaces share a common intrinsic two-dimensional (hyperbolic) symmetry, described by their surface orbifolds (§2.1.1). In all three bicontinuous structures, the bilayer separates two distinct water channels whose midpoints define labyrinth graphs that follow well-known nets. For the *Gyroid*, the two labyrinth graphs are 3-valent and form a pair of enantiomeric **srs** nets. In the *D* phase, the two nets are 4-valent and form a **dia** net, while in the *P* phase, the two nets are 6-valent and form a **pcu** net. (Nets are named according to the RCSR database [9].) We can also label the minimal surfaces according to the nets that define their labyrinth graphs [8], as the surface can be re-generated by inflating the graphs into ever-larger tubes until each tube presses against its neighbours, the interfaces being the location of the original surface. We describe the surface partition and resulting labyrinths for a generic balanced three-periodic symmetric *n*-component pattern by the symbol *n***abc**(*i*) [10], where **abc** is the three-letter RCSR code that describes the net [9] and ‘*i*’ is the number of the space group of the surface according to the International Tables [11]. Figure 2*b* summarizes these names and spacegroups for the *Gyroid*, *D* and *P* surfaces, as encountered for bicontinuous cubic phases: in this case, the two nets both represent water and are exchangable by a symmetry element. For the case that the two nets are not exchangable, the naming scheme is given in figure 2*c*.

In general, these remarkably convoluted ‘bicontinuous’ structures result from a natural tendency for the amphiphilic bilayer (or reversed bilayer) to adopt a warped saddle-shape, in response to a mismatch in molecular cross-sectional area at their antipodal polar and hydrophobic ends. In other words, the complex three-dimensional crystalline forms observed in bicontinuous mesophases—all related to the simplest triply periodic minimal surfaces (or TPMS)—are the global structural expression of rather simple local form expected for the simplest (chemically monodisperse) assemblies built of just a single amphiphile type, that of a uniformly curved saddle [12]. It is most likely, and supported by numerical studies, that these simpler TPMS are the simplest possible ways of forming uniformly curved extended saddles in 3-space, with minimal deformation away from their preferred saddle geometry (or Gaussian curvature; a more detailed introduction to curvature will be given in §3.4.1). A subtle feature of the *Gyroid*, *D* and *P* surfaces is that they have in fact *identical* curvature distributions, and all three patterns are equally deformed away from the preferred local form of constant (negative) Gaussian curvature (equally anticlastic saddles) everywhere. For convenience, we call these curvature variations their *local homogeneity* and say that they share identical local homogeneity. They differ however in their surface-to-volume ratios, so are expected (and do) form at slightly different compositions in the amphiphile–water phase diagrams. We note, without detailed explanation, that these considerations of amphiphilic self-assembly apply also to block copolymer assemblies with two distinct blocks (or related triblocks). The physics of copolymers is somewhat different, governed by the chain entropy of the constituent chains, but it can be mapped onto amphiphilic assembly, with some provisos [13].

In addition to the degree of splay in the molecular form, amphiphiles (and—as a consequence of entropic constraints—copolymers) have certain fixed geometric dimensions, tied to their atomic make-up. Those dimensions impose a preferred *global homogeneity* on the bicontinuous forms. A good measure of their global (in)homogeneity is the variation of effective channel diameter within the convoluted labyrinths of these surfaces. While the *Gyroid*, *D* and *P* patterns have identical curvature and thus equal local homogeneity, their global homogeneity differs. In order of decreasing homogeneity (a crude measure of decreasing stability for monodisperse assemblies), these three TPMS are ranked *Gyroid*, *D* and *P* [14], an ordering consistent with the observed relative abundance of each type of mesophase.

Our goal in this paper is to move beyond amphiphilic or related copolymeric assemblies characterized by their *Gyroid*, *D* and *P* domain interfaces to consider *star-shaped* molecules. We pose the following question: What are the analogues of the simplest bicontinuous forms, the *Gyroid*, *D* and *P* patterns, for three-arm star-shaped molecules?

The introduction of the third radial arm (or ‘block’) to make an ABC star-shaped molecule enforces multiphase segregation at the molecular scale, provided the three chemically distinct blocks are sufficiently mutually immiscible. In the language of block copolymer theory, our assumed mutual immiscibility of blocks ensures that they are in the ‘strong segregation limit’ [15]. In that case, three distinct two-dimensional interfaces separate the three block pairs, namely the AB, BC and CA interfaces. In addition, a new feature emerges in any strongly segregated star assembly: the junctions at the centre of the stars, which are common to all three arms and all three interfaces, assemble to form one-dimensional ABC threefold branch lines or ‘triple lines’, as shown in figure 1. These triple lines necessarily extend without boundary; either they are endless and span the entire material, or they form closed loops, analogous to dislocations in crystals [10].

We note that these branch lines are a necessary feature for star assemblies, but not sufficient. Branch lines are also found in the patterns of amphiphilic assemblies. For example, the interfaces between hydrophobic tails of a type 2 amphiphilic surfactant assembly [16] are arranged in a hexagonal columnar array: at the centre of three adjacent columns this interface branches, giving the combined interfaces a pattern resembling a hexagonal honeycomb, with each hexagon surrounding a hydrophilic column. (Of course branching is not required in amphiphilic assemblies: the lamellar and cubic bicontinuous mesophases have unbranched interfaces.)

These branched cellular patterns are very reminiscent of froths and foams, which—at equilibrium—likewise contain arrays of triple lines, along which three interfaces meet [17]. Foams are therefore a useful working model for star molecular assemblies. Many foams have closed cells, such as the celebrated Kelvin foam, with distorted truncated octahedral cells [18]. Closed-cell foams contain threefold branch lines, but those lines are themselves branched at vertices. Other foams, such as the hexagonal honeycomb, contain open cells, and their threefold branch lines are vertex-free. The favoured cellular patterns for three-arm star molecular assemblies are those whose cells are open [10].

The analogy with open-cell foams is instructive, though deficient in one additional aspect. All films in ideal foams are subject to identical surface tension, so that their sections through planes normal to the triple line at any point are geometrically similar: the local section in a neighbourhood of the branch point lying in the triple line is a ‘Y’-shaped figure, and all three arms of the Y subtend local angles of 2*π*/3 with each other [19]. For generic molecular assemblies, even within the strong segregation regime, the surface tensions for AB, BC and CA interfaces are generally different, so this local threefold symmetry is violated. It is simplest, however, to explore first the idealized model for three-arm star-shaped molecules assumed to have equal surface tensions for all three interfaces.

### 2.1. The balanced case: tricontinuous patterns

Beyond our most general assumption of strongly segregated domains then, the simplest scenario is one of *balanced* interfaces,^{2} with equal surface tension for all three interfacial classes (AB, BC and CA). For now, assume further that all three (A, B and C) domains are of equal volume. In that balanced and *symmetric* case, we expect the films between distinct domains to form minimal surfaces, pinned to the unbranched triple lines.

This scenario mimics most closely that of amphiphilic (or diblock copolymeric) self-assemblies where a symmetric bilayer is equally curved towards both sides of the surface, resulting in a minimal surface. In that binary case, two generic patterns emerge: a lamellar structure (the so-called *L*_{α} mesophase of lyotropic liquid crystals), with flat bilayers (figure 3*a*), or saddle-shaped (hyperbolic) bilayers, which wind through space carving out a pair of interwoven labyrinths, forming sponge-like patterns, seen in the ‘balanced sponge’ (*L*_{3}) and bicontinuous cubic (*V* _{1} or *V* _{2}) mesophases (figure 3*b*) [16]. (Sponge and cubic mesophases differ only in their geometric details; sponge phases are geometrically disordered and cubic phases are—as their name implies—translationally ordered and highly symmetric, forming three-dimensional liquid crystals.)

The case of tricontinuous patterns with balanced interfaces and equal volumes has been analysed previously [10], and we summarize the results here. First, consider the simplest three-coloured pattern that satisfies the requirements of a balanced 3-arm (ABC) star molecular self-assembly: the hexagonal honeycomb (figure 3*c*). In terms of balanced interfaces, it can be considered as the branched analogue of the lamellar phase in figure 3*a*.

Analogous tricontinuous patterns to the bicontinuous cubic and sponge mesophases are more complex. Just as the *Gyroid*, *D* and *P* surfaces (which define the bilayer geometry of the bicontinuous phases) cleave space into two interwoven labyrinths, novel *tricontinuous* patterns are possible, where branched minimal surfaces cleave space into three interwoven labyrinths (figure 3*d*).

The relationships between lamellar and three-coloured honeycomb (figure 3*a*,*c*), and bicontinuous cubic and three-coloured tricontinuous (figure 3*b*,*d*) are, however, loose in the sense that in the three-coloured form, the minimal surface corresponds to the monolayer interfaces between distinct chemical species (AB, AC and BC), whereas in the amphiphilic system it is decorated by a bilayer separating water channels (AA).

Like the bicontinuous triply periodic minimal surfaces, there is an infinite number of possible (triply) periodic tricontinuous patterns, distinguished by their topologies and symmetries. However, among the plethora of geometrically realizable bicontinuous forms, only the *Gyroid*, *D* and *P* surfaces are realized in molecular systems. This encourages us to ask: which tricontinuous patterns are the most favoured balanced forms for symmetric three-arm star assemblies?

#### 2.1.1. ‘Locally homogeneous’ balanced tricontinuous patterns

A key feature of the bicontinuous *Gyroid*, *D* and *P* patterns formed by amphiphilic systems is their very high intrinsic (two-dimensional, in-surface) symmetry, characterized by their hyperbolic orbifolds [20]. Among all TPMS (which are themselves more symmetric than other hyperbolic surfaces realizable in three-dimensional space, with smallest area orbifolds), the *Gyroid*, *D* and *P* surfaces are the most symmetric [21]. Thanks to standard formulae for orbifolds [22], we can rank this degree of symmetry in terms of increasing area of the orbifold, as in table 1.

All three simple cubic TPMS: the *Gyroid*, *D* and *P* surfaces, have equivalent two-dimensional symmetries, characterized by the *246 orbifold. Their common in-surface symmetry is no accident: they are in fact identical in a two-dimensional sense, and they differ only in how they are ‘embedded’ in three-dimensional space. Their differences are much like those between a flat sheet of paper and the same paper rolled into a cylinder or a cone. All lengths and angles measured in one shape of the paper are equivalent to equivalent lengths and angles in the other two, and their in-surface geometries are identical. In particular, the *Gyroid*, *D* and *P* surfaces have asymmetric surface patches bounded by geodesic triangles, with vertex angles *π*/2,*π*/4 and *π*/6, shown in figure 4, corresponding to the *2, *4 and *6 sites on the surface (figure 4*a*). Hyperbolic orbifolds offer a useful path to explore the symmetries of the periodic minimal surfaces. There is a direct correspondence between the intrinsic symmetry (the two-dimensional orbifold) of a symmetric pattern on the *Gyroid*, *D* and *P* surfaces and the three-dimensional space group of the pattern [23], as well as sites of special local symmetry in the orbifold, and the point group symmetry of those sites on the surface [24]. In particular, the *6 sites of the *246 orbifold mapped onto the TPMS lie on axes of threefold rotational symmetry on the *Gyroid*, *D* and *P* surfaces and all have crystallographic site symmetry . These sites coincide with special singular ‘flat points’ on the minimal surfaces where the surface is locally flat, in contrast to all other points on the surfaces, which are hyperbolically curved.

Orbifolds describe the symmetries inherent to surfaces, which are everywhere locally smooth. As tricontinuous patterns are *stricto sensu* surface complexes rather than surfaces (inherently unsmooth on their threefold branch lines), extension of the orbifold concept to these patterns is non-trivial. The balanced threefold branch lines enforce local threefold rotational symmetry between surface patches adjacent to the triple line, which can take the form of a threefold rotational axis (e.g. in the 3**etc**(187)), but also screw axis (e.g. a 4_{1} axis in 3**dia**(109) or a 2_{1} axis in **qtz**(145)) among others. In any case, the threefold branch line generates a cellular pattern rather than a single smooth surface. To avoid this difficulty, we can instead analyse the intrinsic hyperbolic (orbifold) symmetry of the simpler case where the threefold branch lines are reduced to twofold rotational axes. This action is equivalent to ‘unfolding’ a honeycomb pattern to a lamellar one, the reverse of the branch lines description above. These new twofold axes then coincide with mirror lines in the smooth surface.

Balanced tricontinuous patterns (with geometrically identical 3-periodic volumes in all three interwoven labyrinths) have been explored previously [10]. As in the bicontinuous case, we rank these most symmetric patterns in order of increasing area, which corresponds to ranking from the most to lesser symmetric examples, in table 2. On the basis of maximal in-surface symmetry—a criterion that is consistent with the formation of the *Gyroid*, *D* and *P* as most favoured bicontinuous forms—the 3**etc**(187) and 3**nbo**–**pcu**–**dia**(160) (figure 5) are the most favourable tricontinuous patterns for balanced 3-arm star assemblies.

#### 2.1.2. Globally homogeneous balanced tricontinuous patterns

A preference for local homogeneity (uniform interfacial curvature, via intrinsic symmetry) in tricontinuous patterns is not the only constraint. As described above, bicontinuous systems *Gyroid*, *D* and *P* have equal local curvatures, yet the *P* is less commonly found in bicontinuous mesophases than the *D* and the *Gyroid* (the most prevalent phase). Clearly, despite their equivalent local homogeneity, the three patterns are not equivalent energetically. This is explained by the fact that these patterns have different global geometries, with distinct three-dimensional embeddings in space. Again we follow the analysis of bicontinuous systems to analyse the ‘global homogeneity’ (measuring the variations in channel dimensions in the labyrinths [24]) of tricontinuous systems, using analogous measures of global homogeneity for tricontinuous patterns [25]. Among a range of candidate tricontinuous patterns, the most globally homogeneous patterns are (in no particular order) 3**dia**(109), 3**srs**(24) and 3**eta**–**qtz**(145) and 3**etc**(187), some of which are illustrated in figure 5. We therefore expect these patterns to be the most prevalent for balanced 3-arm star assemblies, whose formation demands uniform labyrinth dimensions.

One member of this list, 3**etc**(187), has been found in amphiphilic self-assemblies and mesoporous inorganic derivatives, namely in silica templated by amphiphiles [26,27] and amphiphile–water assemblies [28]. In those cases, the tricontinuous pattern is realized by linear molecules forming a bilayer on the branched minimal surface, rather than by a 3-arm star. Given that these systems also assemble to give a tricontinuous pattern that is very symmetric (as gauged by its orbifold), we believe that our explanatory and predictive paradigm is a useful one.

On the other hand, our analysis is in some respects simplistic, and ignores critical differences between bicontinuous and tricontinuous assemblies. Star molecules are primarily assemblies along threefold lines; the geometries of the accompanying two-phase walls are dictated by the forms of those lines and the surface tension. For star polyphiles, there is no analogue of the preference for a curved (instead of flat) bilayer present in bicontinuous assemblies. That preference can be traced to a mismatch in steric requirements for the two distinct moieties in an amphiphilic molecule [12]. The homogeneity analysis for tricontinuous patterns neglects the competing demand to reduce the surface energy by minimizing the area of the monolayer film in the assembly, while accommodating the three moieties of the star molecule. *Prima facie* the three-coloured honeycomb pattern—with flat interfaces—incurs smaller surface energy cost than the curved interfaces in the tricontinuous patterns. To disfavour the honeycomb and produce more interesting patterns, we must impose specific physical constraints. One potential constraint is to impose a preferred torsion or twist on neighbouring molecules that pack along the triple branch lines, via careful engineering of the molecule [10]. Alternatively, the drive to minimize interfacial area can be reduced by enhancing the relative importance of the global homogeneity (which governs the chain packing entropy). That can be realized by using polymer star molecules, rather than shorter-chained polyphiles. Detailed self-consistent field theoretic (SCFT) calculations have confirmed that tricontinuous patterns can emerge in these systems, provided the star copolymeric molecular architecture contains an extended core region, effectively enhancing the relative contribution of the global homogeneity over the surface tension [29]. Surprisingly—in the context of our expectation of high symmetry—SCFT simulations predict that a balanced tricontinuous pattern of very low symmetry, the 3**ths**(5) phase shown in figure 6, is a thermodynamically stable morphology in star block copolymer melts [29]. It turns out that this monoclinic pattern is a distortion of the highly symmetric 3**dia**(109) pattern: the lattice is sheared to minimize global homogeneity, including other global homogeneity measures such as distance variation from the triple line, at the cost of increasing local homogeneity. Given the structural similarity and differences in detail between this 3**ths**(5) pattern and the parent 3**dia**(109) pattern, this example serves to reinforce the simple analysis above, but reminds us that the assemblies in specific molecular systems can differ in their structural details.

### 2.2. Unbalanced tricontinuous patterns

Bicontinuous and polycontinuous patterns need not carve space into identical labyrinths. For example, Schoen discovered the *I* − *WP* triply periodic minimal surface [8], which has two distinct labyrinths: one described by the **pcu** net and the other by the **nbo** net [9]. Following Fischer and Koch, we call any such pattern with two or more distinct three-dimensional labyrinths ‘unbalanced’ [30]. Given that unbalanced bicontinuous patterns have, to date, not been observed in materials, their relevance is less clear. We note however, that global homogeneity calculations suggest that the *I* − *WP* structure has a degree of global homogeneity that is comparable with the *P* structure, and the former has been suggested as a potential novel cubic bicontinuous morphology in block copolymer melts [31]. Unbalanced tricontinuous patterns, made of minimal surface patches with threefold branching with three geometrically distinct three-periodic labyrinths are possible. A number of unbalanced tricontinuous forms have been constructed which contain threaded nets of equal genus. We stress here that the stability and range of labyrinth volume ratios (for which the partitions are composed of minimal surface patches) is at present mostly uncertain, and further studies are essential. Clearly, however, they offer alternative geometries for tricontinuous 3-arm star assemblies.

In summary, both the balanced and unbalanced tricontinuous patterns introduced here are analogues of the better-known bicontinuous patterns: the former have three interwoven labyrinths in contrast to the labyrinth pairs in bicontinuous structures. To date they have not been reported in three-arm star polyphile or copolymeric materials.

Recall that highly symmetric balanced patterns require that there is equal interfacial tension between each pair of domains (*Γ*_{AB} = *Γ*_{AC} = *Γ*_{BC}), a chemically difficult feature. Indeed, in cases where one interfacial tension is substantially below the other two (*Γ*_{AB} = *Γ*_{AC}≫*Γ*_{BC}), the B and C domains will form an *entente cordiale* in their mutual repulsion, collectively assembling so as to segregate from the A domain.

The resulting structure will look like a standard star-ABB (amphiphile) assembly, with the addition of weaker B–C phase separation only visible on closer examination. We analyse next the case where the surface tensions are unequal, and B and C form a striped minimal surface wrapping around larger-scale channels of A. This scenario allows a number of intriguing morphological possibilities and complexities, discussed in detail below. The imposition of unequal surface tensions results in striping, but it turns out that striping can extend the resulting assemblies beyond tricontinuous patterns (with three interwoven domains) to ‘polycontinuous’ patterns, with multiple interwoven domains.

## 3. From bicontinuous to polycontinuous striping patterns

### 3.1. Unbalanced striped patterns

The foregoing tricontinuous forms for ABC star-shaped molecules have triple lines pinning A–B, B–C and C–A interfaces of zero mean curvature (minimal surfaces), which meet at mutual angles of 120°. These structures are therefore most relevant to self-assemblies whose surface tensions at A–B, B–C and C–A interfaces are all equal. In practice, it is difficult to tune the inter-molecular interactions to balance these surface tensions, in particular for small molecules. Indeed, the 3-arm star polyphiles composed of oligoethylene glycol (A), hydrocarbon (B) and fluorocarbon (C) domains cannot be balanced (figure 1). When water is present, the surface tension between hydrocarbons and fluorocarbons is much lower than the surface tension of each of these domains with the hydrophilic phase (*Γ*_{AB} = *Γ*_{AC}≫*Γ*_{BC}).

In practice, we find that the phases formed by these star polyphiles in solution resemble closely those formed by more conventional amphiphilic lyotropes. Star polyphilic self-assembly is driven by the hydrophilic/hydrophobic interaction, as with amphiphiles. So it is unsurprising that the suite of phases, for example, in water, is similar to that of normal surfactants in water. So, lamellar, bicontinuous cubic, hexagonal and micellar cubic mesophases can form, depending on the various arm lengths, temperature and water content of the sample. In particular, phases reminiscent of all three bicontinuous cubic phases, the *Gyroid*, *D* and *P* phases, can form. The polyphile architecture results in a bulky hydrophobic moiety compared with the hydrophilic fraction, resulting in ‘type 2’ mesophases, where the hydrophobic domains form a (thick) film wrapped on the minimal surfaces.

On the other hand, this observation is perhaps surprising, given the very different molecular architecture of these star polyphiles. However, more careful investigation of the mesophases—lamellar, hexagonal and cubics—revealed the source of that apparent similarity, as well as differences. Like their amphiphilic analogues, they segregate hydrophilic from hydrophobic domains. In contrast to simpler amphiphilic self-assemblies, however, the star polyphilic liquid crystalline mesophases are hierarchically self-assembled, with an additional level of self-assembly at length scales smaller than those formed by hydrophilic–hydrophobic segregation. This hierarchy is a direct result of the novel nature of these star polyphiles, which contain both fluorophilic and oleophilic hydrophobic moieties. Within the hydrophobic domain, the space is segregated into hydrocarbon and fluorocarbon domains, most probably in the form of alternating hydrocarbon/fluorocarbon stripes, as described in the next section. Such striped patterns are very different from the tricontinuous patterns (recall that by definition the three domains in a (balanced) tricontinuous assembly are spatially equivalent): the striped structures can display some degree of similarity between the hydrocarbon and fluorocarbon domains, but these two hydrophobic domains are very different from the third moiety: the surrounding hydrophilic regions.

We can gain insight into the striping patterns on the minimal surface by first considering a simpler case. Detailed scattering experiments on a related hexagonal mesophase in a star polyphile water mixture as in figure 1 [3] led to the structural model shown in figure 7*a*. The hydrophilic domains consist of a two-periodic lattice of discrete, one-dimensional columnar channels, arranged in a hexagonal array (blue). A cross-section normal to the columns can be idealized as dodecagons, separated by the hydrophobic Hc–Fc matrix. The latter matrix is segregated into columnar Hc (red) and Fc (green) domains, themselves with rectangular and hexagonal cross sections. The cross section is therefore a three-coloured tiling pattern in the plane, with adjacent Hc and Fc tiles sharing common edges. The full three-dimensional pattern contains an arrangement of three-phase ‘branch lines’, formed by extending the tile vertices in the third dimension. Those lines define the locations of the vertices of the star polyphile centres, and their three hydrophilic, Hc and Fc arms point into the 12-, 6- and 4-sided columns, respectively, as shown in figure 7*a*. The Hc and Fc domains therefore form a series of alternating columnar ‘stripes’—the former with hexagonal and the latter with rectangular cross section—around the central hydrophilic cores. The dimensions of the stripes are dependent on the molecular dimensions of the polyphiles, since the width of a single (e.g. Hc) stripe is given by the spacing between a pair of adjacent molecular centres on different triple lines (figure 7*c*). The molecular dimensions therefore strongly favour stripes of constant width, as the drive to reduce interfacial area increases the width everywhere up to the maximum level tolerated by the molecular architecture. Evidently, a flat striping pattern, similar to these stripes ‘curled around’ water columns, is also possible in the polyphile lamellar mesophase (figure 7*b*), where adjacent stripes line the lamellae.

Striping patterns are also possible for star polyphilic bicontinuous cubic phases, such as the example shown in figure 7*d*–*f*, and the rest of this paper is devoted to exploring and analysing them. While striped lamellar and striped hexagonal phases (in the form of two-dimensional tilings) have been experimentally found in mikto-arm copolymers [7], a striped cubic *Gyroid* phase has so far only emerged in simulations of ABC/ABD mikto-arm copolymers [32]. However, our current experimental data suggest that all of these striped phases also exist in the star polyphiles shown in figure 1. Looking at only a single unit cell, such striped cubic phases are interesting because they are ordered on two different length scales (hydrophilic/hydrophobic cubic unit cell and hydrocarbon/fluorocarbon striping). However, their full potential complexity and beauty only becomes apparent when simultaneous periodicity is fulfilled on both these length scales: in such a case, the stripes themselves can form poly-continuous and chiral networks, which will be discussed in detail below.

### 3.2. Optimal striping of curved substrates

Flat surfaces, or more generally surfaces with zero Gaussian curvature, can be covered with stripes of uniform width. However, this is not possible for closed objects or for hyperbolic interfaces, as illustrated in figure 8. Any attempt to stripe closed blob-like objects like a sphere will have defects due to the net positive Gaussian curvature, e.g. striping along lines of latitude of a sphere will leave blobs instead of stripes at the poles. By contrast, striping patterns on periodic minimal surfaces (such as the *Gyroid*, *D* and *P* minimal surfaces) must be branched in order to compensate for the sponge-like topology of these surfaces. These branches can occur anywhere on the surface and be of degree three (threefold branching) or higher, as exemplified in figure 8*c*–*f*. The branching required to stripe a surface can be precisely characterized by the surface topology [32]; see electronic supplementary material, S1.

Topological constraints imply that there must be precisely 16 threefold stripe branchings per conventional unit cell of the *Gyroid* (symmetry ), or a commensurate number of higher order branches, e.g. eight fourfold branches or four sixfold branches. Excess stripe branchings beyond this number must co-exist with *cul-de-sacs* (where a stripe terminates). The *D* and *P* surfaces, in their conventional space group setting and , require four and eight threefold branchings of a defect-free stripe pattern per unit cell, respectively.

These topological calculations specify the total amount of branching, but indicate neither how the branch-points are distributed on the surface, nor the number of stripes that radiate from a particular stripe junction; e.g. they do not distinguish between the possible branching patterns illustrated in figure 8*c*–*f*. Fortunately, geometric arguments prohibit many possibilities and encourage others, limiting the potential patterns.

There are two general possible patterns: ‘branching stripes’ or ‘blobs’. Blobs refers to a set of islands of one moiety immersed in the other which branches around it. This may be possible in certain systems, but has confounding factors: firstly, such a configuration requires significantly more curvature of the triple lines, in contrast with the straighter stripes. That curvature imposes severe crowding on adjacent molecules in some directions, and equally extreme splay in others. Secondly, if the two moieties have similar volume fractions, then the branching phase which encloses the islands will have significant variations in domain width, which is unfavourable for a mixture of equal molecules. As a subtle extension of this idea, covered in detail later in this paper, if the islands are elongated in order to minimize this domain-width variation, then the island on the other side of the bilayer will be distorted to an unfavourable shape by its ‘reflection’ through the negatively curved interface. This leaves branching stripes as the preferred pattern motif, which are well adapted to the task.

Simple calculations show that as the number of stripes meeting at a common junction increases, so does the width of the junction (relative to the stripe width away from the junctions) [32]. In other words, junction regions lead to variable stripe widths. But the star polyphile molecular geometry favours a single preferred stripe width, unable to be realized on bicontinuous films due to the necessity of branching. The resulting inherent frustration can be minimized by reducing the branching order of the stripe junctions (while also ensuring there are no unnecessary junctions leading to ‘terminated’ stripes). As a result, simple ordered patterns of stripes with junctions linking just three radiating stripes are preferred. Each threefold stripe junction imbues the stripe pattern with a quantum of negative Gaussian curvature (or, in topological terms, a contribution to the Euler characteristic of the striped surface of *χ* = −1). The curvature of the minimal surfaces is distributed regularly within the surface, so to avoid crowding and severe distortions in the stripe shape, these junctions should be spread as homogeneously as possible over the surface.

Recall from above that the flat points on the minimal surfaces have threefold rotational site symmetry (in fact inversion symmetry ). These flat points are therefore natural locations for threefold stripe junctions, and they are homogeneously distributed over the surface. Now, in each of the *Gyroid*, *D* and *P* minimal surfaces, the number of these flat points per unit cell exactly matches the number of required threefold stripe junctions, implying that very symmetric patterns of stripes can be located on these surfaces. This is promising for finding interesting patterns, as surface symmetry is taken to be driving the formation of the hydrophobic layer (see above). These symmetric patterns have already been explored theoretically [33,34] and more recently catalogued in some detail [35–37]. We discuss below the relevance of these regular patterns to polyphile liquid crystals.

### 3.3. Periodic striping patterns on three-periodic minimal surfaces

Regular branched stripings of the *Gyroid*, *D* and *P* surfaces (containing e.g. threefold junctions) can be derived in a systematic way via two-dimensional hyperbolic geometry [35]. Here we discuss the construction.

As described earlier, there is a precise equivalence between structures on three-dimensional periodic minimal surfaces and corresponding patterns in the two-dimensional hyperbolic plane [34]. This correspondence means that complex structural features on minimal surfaces can be created, analysed and elucidated far more simply in two dimensions on the hyperbolic plane rather than in three dimensions. For example, a branching pattern in the two-dimensional hyperbolic plane can be projected onto the various three-dimensional minimal surfaces to form networks or arrays of cages, as shown in figure 9. In this paper, we use a representation known as the Poincaré disc representation of the hyperbolic plane. This representation of two-dimensional hyperbolic space is convenient, but contains some significant visual distortions, like maps of the earth on a flat page. The distortion can be gauged by recognizing that all of the blue and white triangles in the Poincaré disc of figure 9 are in fact congruent in hyperbolic space (see also figure 4). Owing to our choice of symmetries, these triangles also map to equivalent triangles on the periodic minimal surfaces. We can think of patterns in the hyperbolic plane (and its Poincaré disc model) as ‘unrolled’ versions of patterned minimal surfaces, much as the infinite graphene net drawn in the flat plane is an unrolling of the carbon network of a tubular fullerene. In other words, while the planar maps reveal in precise detail the local geometry on the surfaces, global features, such as closed loops encircling the circumference of a buckytube or a ‘collar’ of a periodic minimal surface, are invisible in the (hyperbolic or flat) planar map.

As described in the above section, the natural locations for threefold branching of stripes on the cubic minimal surfaces occur at the flat points, which have symmetry and correspond to the *6 locations of the tiling by *246 tiles. We can therefore build symmetric three-branched stripes in the hyperbolic plane by consistently connecting *6 sites, retaining this threefold symmetry. One such branched stripe is drawn on the hyperbolic plane in figure 9. In the hyperbolic plane, it is a ‘tree’ in the mathematical sense: a network without loops. We then wrap the single-striped hyperbolic plane back onto the *Gyroid*, *D* and *P* surfaces (also in figure 9). The tree wraps onto the *Gyroid* and *D* surfaces to build an extended three-branched network in space, and a closed cube-like cage on the *P* surface. Note that the wrapping onto the TPMS destroys the tree-like nature of the original stripe; the tree's branches rejoin each other around the tubular collars of the surfaces, forming loops. Even though the hyperbolic plane is the native space for a branching tree, this wrapping onto a TPMS is an example of features (loops) being induced that are not visible from the flat perspective.

The final configuration of the stripe in three-dimensional space is dependent on the surface and the orientation of the stripe edges relative to the *246 triangulation. This phenomenon is a more complex example of a better-known case: an unbranched stripe drawn on the plane, then wrapped onto a cylinder. In that case, the stripe maps to a helix whose pitch depends on the stripe orientation relative to the cylinder axis: it can form a straight line, an infinite helix, or a closed equatorial loop on the cylinder. The hyperbolic analogue is a branched stripe that forms a branched helix, a branched net (in the case of the *Gyroid* and *D* of figure 9) or a branched cage (in the case of the *P* of figure 9). Since there is an infinite number of pairs of *6 sites that can be joined to form three-branched stripes, the stripe pattern is far from unique. Among those possibilities, we consider only the most symmetric ‘regular’ stripes, formed by repeatedly rotating a single stripe edge by 2*π*/3 about successive endpoints. The resulting pattern is very symmetric, with identical stripe junctions and edges between junctions. We annotate the particular regular stripe pattern by the length of the stripe segment between adjacent junctions, measured in the hyperbolic plane. Since these lengths in hyperbolic space invariably involve the arccosh function, we denote a stripe with edges of length arccosh(*l*) by *l* alone. The simplest stripe, with junctions as close to each other as possible, has *l* = 3 [34], and we can sort forests by ascending *l*, giving the sequence *l* = {3, 5, 15, 53, 99, 195, 675, 725, 1155, … } [35].

Just as we form a striping of the flat plane by copying and pasting an original stripe repeatedly, we can stripe the hyperbolic plane by pasting multiple copies of the hyperbolic (branched) stripe. The result is a collection of symmetrically equivalent trees, known as a ‘forest’ (figure 10*a*) [34]. Mapping of those forests back on to the *Gyroid*, *D* and *P* surfaces results (in general) in multiple interpenetrating, highly entangled networks [34,35].

For example, the most symmetric forest with edge-length *l* = 3, namely ‘forest 3’ (introduced above) wraps onto the D minimal surface to form four disconnected, interpenetrating three-periodic chiral nets of equal handedness, as shown in figure 10*b*–*d* (coloured brown, red, orange and yellow). Each of these nets is in fact an **srs** net, which is well known in crystal structures [38], and the same net that forms the labyrinth graph of a *Gyroid* structure. The global spatial pattern of the overall structure is a complex one: it has a pair of labyrinths formed by the *D* surface, whose skeletons are interpenetrating **dia** nets, separated by the hyperbolic striped film decorating the *D* surface. Within the film, the stripes are three-branched trees, which wrap onto the *D* surface to form four distinct labyrinths, each centred by like-handed **srs** net, also mutually threaded (all right- or all left-handed depending on the orientation of the stripe edges relative to the *246 triangulation).

In a self-assembled material, this pattern of four interwoven nets could for example describe four distinct interwoven labyrinths lining the minimal surface, each filled with a distinct and mutually immiscible chemical species. However, 3-arm polyphile assemblies are instead consistent with 2-coloured (hydrophobic) hydrocarbon/fluorocarbon stripings of the minimal surface, while the third (hydrophilic) moiety (oligo-ethyleneglycol, plus water) fills the pair of labyrinths formed by the minimal surface itself. This 2-colouring can be achieved by, for example, colouring all trees equivalently (green) and inserting ribbons (red) between adjacent trees, illustrated on the hyperbolic plane in figure 10*e*. After projection onto the *D* surface (figure 10*f*), the trees and ribbons can be widened to form stripes that tile the minimal surface (figure 10*g*), and then thickened to both sides of the minimal surface to represent the hydrophobic membrane in the polyphilic self-assembly (figure 10*h*). We call such a system a ‘tree–ribbon’ pattern, where one colour forms trees and the other colour forms ribbons. Alternatively, the surface can often, though not always, be striped in two colours by alternately colouring adjacent trees and widening the tree edges to tile the surface, resulting in ‘tree–tree’ patterns. (Occasionally, a specific forest maps on to one of the *Gyroid*, *P* or *D* surfaces to give an odd number of tree edges surrounding collars, so that they cannot be 2-coloured.) Both tree–tree and tree–ribbon patterns are illustrated in the hyperbolic plane and on the *Gyroid* surface in figure 11*a*. This pattern can be extended by the insertion of an arbitrary number of additional ribbons between adjacent trees, and 2-colouring the pattern by choosing alternate colours for adjacent (ribbon- or tree-shaped) tiles. Added ribbons result in thinner stripes, relative to the dimensions of the TPMS. Evidently, a series of patterns of decreasing stripe width can be built: it starts with the simplest tree–tree pattern, then the tree–ribbon pattern (with one ribbon intercalating adjacent trees), then a tree–tree pattern with two parallel ribbons between the trees, etc. (figure 11*a*).

Intercalation of extra ribbons between branching stripes of a given forest, as described above, is one way to produce patterns with decreasing stripe width; another is to choose a different forest. Intercalating multiple ribbon domains to tune the stripe width results in one colour having both ribbon and tree geometries. Given that those colours represent specific chemical domains formed by aggregation of one of the star molecular arms, another application of the principle of global structural homogeneity disfavours intercalated multi-ribbon models: homogeneous domain shapes are surely preferred over multiple geometries. This claim is supported by the observation that the most frequently formed amphiphilic molecular self-assemblies have just one domain geometry for hydrophilic and hydrophobic domains.^{3} By extension, star polyphiles are more likely to adjust to a preferred stripe width by tuning the forest (via *l*) than by intercalating additional ribbons. Recall that forest 3 is the simplest member of a family of related structures; higher order forests are built of edges linking more distant *6 sites. For example, the first three members of a sequence, sorted by increasing *l* are mapped onto the *Gyroid* in figure 11*b*. As the length *l* grows, the stripe width decreases, to accommodate additional stripe sections in the same region. An infinite sequence can be constructed in the hyperbolic plane as explained above. All these forests can be mapped onto the *Gyroid*, *D* and *P* surfaces: the first six members of the sequence are shown in figure 12.

We focus on stripe width here because it is an adjunct to the most interesting feature: the shapes of each striped domain. As *l* varies, so does the direction of the stripe edges on the surface. The directions of the edges combine on a periodic surface to give radically different stripe morphologies in three dimensions, dependent upon the surface the forest is projected on. As an example, figure 13 shows a single tree on the *D* surface for the three simplest forests, forming, respectively, a three-periodic **srs** network (as discussed in figure 10), a closed cage (discussed later, in figure 20) and a two-periodic hexagonal mesh. Detailed studies of the three-dimensional structures that emerge from tree–tree and ribbon–ribbon patterns are given in [35,36], and briefly summarized in the electronic supplementary material, S2.

The particular angles (and lengths) of a forest in the hyperbolic plane have a profound impact on the resultant three-dimensional structure, so we now explore a wider range of forests in the hyperbolic plane and their geometries when wrapped onto the *Gyroid*, *D* and *P* surfaces. The images in figures 11*b* and 12 illustrate that the sequence of striping patterns are related to each other by a rotation about a flat point on the minimal surfaces (coupled to a change in the edge length within the trees). As the forests are very symmetric, we need only explore a radial slice of the hyperbolic plane to extract all possibilities. Such a slice with angle *π*/3 is illustrated in figure 14.

The pattern of forests represented by the edges in figure 14 is symmetric with respect to the horizontal axis in the hyperbolic plane. Note however, that upper (*a*) and lower (*b*) edges map on to the surfaces with distinct orientations, resulting (in general) in left- and right-handed chiral enantiomers on the *D* and *P* surfaces, related through the inversion centre/*6 site (a flat point) on the original minimal surfaces. Projections of the upper and lower pattern onto the *Gyroid* are less symmetric due to the lack of mirrors in this surface. In fact, the same patterns in the hyperbolic plane, but in the *a* or *b* regions (above and below the horizontal axis in figure 14), map to topologically distinct forests on the *Gyroid* [37,42]; see also the electronic supplementary material, S3. An important feature of the maps of these various forests on the *Gyroid*, *D* and *P* surfaces is the distinct space groups of the striped surfaces, depending on whether all trees have the same colour (tree–ribbon patterns), or if the trees alternate in colour (tree–tree patterns). The tree–tree symmetry is always an index (2) subgroup of the tree–ribbon pattern. The space group symmetries for all possible tree–tree and tree–ribbon patterns are collected in table 3.

We will revisit this analysis of the forests, within the hyperbolic plane, in the next section.

### 3.4. Bilayers and the asymptotic requirement

So far, our analyses have been for striping patterns confined to the minimal surfaces, of vanishing thickness. However, self-assembled materials formed from 3-arm star molecules build striped domains that evidently occupy a volume rather than a surface. We build this volume from the surface stripe by thickening the stripe equally on both sides of the minimal surface. The central cores of the molecules then lie along the triple lines that are lifted off the minimal surface.

In order to explore in detail the likely geometry of these stripe domains, we have modelled the striped membrane structures described in the previous section using the freely available software Surface Evolver [43]. The constructions are discussed in detail in the electronic supplementary material, S4. Briefly, we constructed fluid nanostructures for the three-arm (ABC) star polyphile self-assemblies as follows. First, a striped (BC) membrane within a single asymmetric patch of the minimal surface(s) was created, and then extruded to give (B and C) volumes by thickening the membrane to both sides of the surface. The symmetry operations for the relevant space groups were then applied to form a single unit cell, containing two-phase interfaces (A–B, B–C and C–A) and single-phase volumes (A, B and C nanoscale domains) accounting for the hydrophilic (A) and the two (B, C) hydrophobic components in the three-arm star polyphile assemblies. Surface Evolver was then run to relax all the interfaces at fixed volumes in order to minimize interfacial areas and balance competing surface energies subject to imposed (and different) A–B, B–C and C–A surface tensions between the three components, with fixed volume fractions of the A, B and C components. The resulting relaxed structures are therefore useful models for the fluid domains of star polyphilic assemblies.

The forms that emerge from these numerical calculations reveal that not all striping patterns are equally favourable as solutions for bilayer membrane striping. Figure 15*a* (forest 3 on the *P* surface) illustrates the shape of a typical thickened stripe: the stripe width of the (green) branched ribbon increases on one side of the membrane and decreases on the other side, and is therefore ‘unbalanced’. This is energetically problematic, as (for a given lattice size) the stripe widths tend to maximize themselves to the upper bound given by the molecular architecture to reduce the energetic cost of the (B–C) hydrophobic interface. Added to this, it is typically impossible to equalize the stripe widths of one colour on one side of the membrane without further unbalancing the other coloured stripes in between. This generic imbalance grows with increasing membrane thickness (or, equivalently AB membrane volume fraction), as illustrated by the asymmetric cross section of figure 15*a*.

However, specific cases corresponding to specific stripe directions violate this generic feature: instead, the stripes taper symmetrically on both sides of the minimal surface and have a symmetric cross section. For example, stripes derived from the forest 5 pattern on the *P* surface have a ‘balanced’ character on both sides of the minimal surface (figure 15*b*), in stark contrast to the more common unbalanced situation. As the examples in figure 15 show, the degree of balance/unbalance depends on the specific forest projected onto each minimal surface. Further to this, the three-dimensional geometry of each of the *Gyroid*, *D* and *P* surfaces varies despite their common intrinsic curvatures, so the favoured directions—and hence favoured forests—also vary between surfaces.

This geometric observation implies that the various patterns have different self-energies as star polyphilic self-assemblies. Suppose that the self-assemblies are composed of monodisperse star polyphiles. This scenario is analogous to that of conventional monodisperse amphiphilic–water patterns, where the amphiphile dimensions and the concentration impose area : volume constraints on the self-assemblies. The molecular dimensions of the polyphile, and its composition (in, for example, ABC–water mixtures, where A is hydrophilic) impose surface : volume ratios for the two-phase interfaces as well as triple-line length : surface area and length : volume ratios. Whereas the (atypical) balanced forms can be realized by equivalent polyphile packing and matched monolayers on both sides of the bilayer, generic and unbalanced morphologies can only be realized if the membrane bilayer is no longer composed of equivalent monolayers; an unlikely scenario for a system composed of monodisperse molecules, and unknown in analogous amphiphile–water bicontinuous liquid crystalline mesostructures.

We quantify the degree of imbalance in the monolayer geometries after relaxation of the model structures in Surface Evolver with a parameter called ‘stripe-width mismatch’. It is a measure of the mismatch in triple-line lengths on either side of the minimal surface:
where tripL_{1} and tripL_{2} are the measured lengths of the triple-line on each side of the hydrophobic film. The triple-line mismatch is a good approximant for the stripe-width mismatch, although small corrections can be expected due to factors such as the branching of the stripes and the curvature of the membrane. A stripe-width mismatch of, for example, 0.1, means that—compared with the average stripe width on both sides of the membrane—the stripe width is 10% smaller on one side and 10% larger on the other side.

Figure 16*a* shows that the monolayer imbalance for striped patterns on the *Gyroid* depends on both the forest type and the membrane volume fraction. In particular, forest 53a (orange symbols without cross) remains balanced for all membrane volume fractions, in contrast to forest 15b (red symbols with cross), which is the most unbalanced form. The stripe-width mismatch increases linearly with the membrane volume fraction and at large membrane volume fractions (70%), this parameter can be as large as 50%, corresponding to a stripe being three times thicker on one side of the membrane than on the other.

Figure 16*b* illustrates the stripe-width mismatch assuming a membrane volume faction of 30%, plotted as a function of the angle the central edge of the forest subtends on the hyperbolic plane with the tree edge of the simplest forest (forest 3), as shown in figure 14. The plot indicates a smooth variation of the stripe-width mismatch with angle. (Recall also that some intervals of angles are inaccessible for these forests, cf. figure 14.) Clearly, the imbalance depends on the particular minimal surface (the *Gyroid*, *D* or *P* surfaces) onto which the forests are projected. In the case of *Gyroid* patterns (purple curve), the curve implies that balanced monolayers (with zero stripe-width mismatch) form for edges subtending angles of 16.1° and −43.9°, corresponding to the enantiomeric pair of forest 53a. The most unbalanced cases are found at −19.1° and 40.9°, corresponding to the enantiomeric pair of forest 15b, although there are many highly unbalanced forests. Patterns on the *D* surface (orange curve) are balanced when the angle is equal to 0° and 60°, corresponding to the enantiomeric pair of forest 3; the most unbalanced cases form at −30° and 30° (enantiomeric pair of forest 5). These angles and forests correspond to the most unbalanced and balanced patterns on the *P* surface, respectively (with the role of most and least balanced reversed from the *D*). Minimal surfaces striped with forests at these extrema, balanced and most unbalanced, are indicated by blue and red boxes in figure 12.

The variations in the degree of (im)balance between striped monolayers on either side of the minimal surfaces are not present in flat bilayers; it is a direct consequence of the curvature of the minimal surfaces. It can be shown from differential geometry that the degree of imbalance depends on the angle the stripes subtend with two important directions in the surface, namely the principal and asymptotic directions, described in detail below. These directions vary from point to point on a curved surface, and are invisible in the map in the hyperbolic plane, since they are a feature of the way the surface winds in three-dimensional space. The intrinsic (in-surface) geometries of the *Gyroid*, *P* and *D* minimal surfaces are identical, and they cannot be distinguished within the hyperbolic plane. However, their extrinsic geometries are different enough that the principal and asymptotic directions change from surface to surface. To better understand these differences, it is helpful to outline the rudiments of differential geometry on a curved surface.

#### 3.4.1. Brief excursion into differential geometry

Each point on an arbitrarily curved surface has two kinds of curvature, namely ‘mean’ and ‘Gaussian’ curvature [44]. The Gaussian curvature distinguishes a closed object from a flat sheet or a hyperbolic surface, as in figure 8*a*–*c*. The mean curvature describes the difference between a flat sheet of paper and the same paper rolled up into a cylinder, both of which share the same Gaussian curvature (zero). The sign of the mean curvature (positive or negative) distinguishes hills and valleys and determines the change in area of parallel surfaces formed by sweeping the surface by some distance along its normal vectors (growing or shrinking respectively). Thus, for example, the mean curvature of interfaces between hydrophilic and hydrophobic domains in type-1 and type-2 self-assembled amphiphilic systems—which curve towards oil or towards the polar regions, respectively—is respectively positive and negative. Both mean and Gaussian curvatures at a point on the surface depend on the curvature of the most convex and concave plane curves that run through that point along the surface. The directions of those plane curves (that are usually mutually orthogonal, though ill-defined at flat points) define the ‘principal directions’, *p*_{1} and *p*_{2}, at the point Q denoted by a red cross in figure 17*a*. We quantify the curvature of those plane cross-sectional curves in terms of the radius of the largest circle that sits in the surface along each of *p*_{1} and *p*_{2}, namely *R*_{1} and *R*_{2}. At any point on the surface the mean curvature, *M*, is defined by *M* = 1/*R*_{1} + 1/*R*_{2}, while the Gaussian curvature, *K*, is given by *K* = 1/*R*_{1}*R*_{2} [44]. On minimal surfaces such as the *Gyroid*, *D* and *P* surfaces, the principal directions (red cross) are equal in magnitude but have opposite sign (concave and convex) (*R*_{1} = −*R*_{2}), resulting in negative Gaussian curvature and zero mean curvature. The pair of ‘asymptotic directions’ (blue cross, figure 17*a*) lie between these principal directions, and for minimal surfaces they bisect the angle between principal directions. These are the tangential directions along which the sectional curves are uncurved. Along these directions the surface twists rather than bends.

The asymptotic and principal directions form nets on the *Gyroid*, *D* and *P* surfaces, as shown in figure 18. It is evident that these nets are, respectively, straight-edged and maximally curved on the *D* and *P* surfaces; they define the least and most curved edges on the *Gyroid*. We consider next the effect of locating stripes along these alternate directions.

The interplay between bend and twist along trajectories in a curved surface is dependent on the Gaussian curvature of the surface along that trajectory. Every curve on a surface of negative Gaussian curvature must either bend (with curvature *κ* = 1/*R*) or twist (*τ*), or some combination of the two, at every point along its length. The degree of bend and twist depends on the Gaussian curvature, via the Bonnet–Kovalevsky formula: *K* = *κ*^{2} + *τ*^{2} [12]. Stripes running along principal directions are torsion-free and maximally bent, while stripes running along the asymptotic direction run along as straight lines as possible, for example, the (straight) twofold axes in the *P* and *D* surfaces. Such unbent straight stripes necessarily twist in order to compensate for the negative Gaussian curvature.

#### 3.4.2. Differential geometry and optimal striping directions

How does this differential geometry relate to optimal striping directions? Consider a stripe travelling along a principal direction of a minimal surface at a generic (not flat) point in the surface. The stripe bends just like the three joined stripes of figure 15*a*, which together form a bowl shape, as the surface itself bends to follow the red circles of figure 17. The degree of bending is the stripe curvature, whose magnitude is *κ* = 1/*R*_{1}, and whose sign depends on whether the stripe is locally concave or convex. Recall that there is a second principal direction of the surface at right angles to the first (*p*_{2} in figure 17), and as the surface is ‘minimal’ 1/*R*_{1} = −1/*R*_{2} by definition. So, if the long axis of the stripe is convex, the shorter axis—along the width of the stripe—is equally concave. Recall again that we think of the volume of the stripe being generated by thickening the membrane to both sides of the surface. The change in width of the stripe, as it is thickened, depends on the concave or convex nature of the stripe across its width; with a concave stripe getting thinner and a convex stripe getting fatter. This is illustrated in the green stripes of figure 15*a*, with the top layer having convex cross sections and thus flaring out as it moves away from the minimal surface, while the bottom layer is correspondingly concave in this cross-sectional direction, and so tapers in.

This mismatch in stripe widths on either side of the surface must be accommodated by the molecular conformation of the chains that fill the domain volume, from the triple line at the outermost stripe surfaces on the top and undersides of the membrane, to their free dangling ends on the minimal surface. Since the stripe width decreases to one side of the minimal surface, and increases to the other, the corresponding molecular chain conformations must be very different in either monolayer, imposing a significant energy cost (due to attendant chain stretching or compression). On the other hand, if stripes run along a direction in the minimal surface such they have zero curvature along their long axis (figure 15*b*), both monolayer domains are identical, with equal variations of stripe width from the minimal surface to the top and underside. As stated above, this non-bending comes at the cost of torsion, but the energetic cost of torsion in the midsurface of a bilayer is much lower than that for bending for most chemical systems.

In a nutshell, the differential geometry above allows us to predict that if the energetic cost of ‘unbalanced’ stripes exceeds the cost of stripe torsion (as is the case in many systems), then we can expect the direction of stripes to be aligned with asymptotic directions of the surface. We note in passing that this preference for alignment of domains along asymptotic directions applies regardless of the degree of order, or the shape of the domains. Less regular striping patterns and even ‘blob-shaped’ inclusions within the membrane will tend to run along the asymptotic directions.

We visualize the correspondence between the principal and asymptotic directions and their striping patterns in two ways. Firstly, these two sets of directions have been calculated for the *Gyroid*, *D* and *P* minimal surfaces for a number points using Mathematica, described in detail in the electronic supplementary material, S6. One set of the asymptotic directions on each minimal surfaces are illustrated in white in figure 19*a*, together with an overlay of the appropriate chiral striping pattern with the least stripe-width mismatch. Figure 19*b* includes one set of principal directions in black, again overlaid by the appropriate chiral striping pattern with the greatest stripe-width mismatch. The enantiomeric striping patterns travel along the second set of asymptotic and principal directions (not shown). Clearly, these best and worst stripings are very well aligned with the asymptotic and principal directions.

Secondly, the same striping patterns are traced on the *Gyroid*, *D* and *P* minimal surfaces in figure 20, zoomed out to display structural distinctions. Very different global geometries emerge from differently oriented forests. Aligning the tree edges along an asymptotic direction results in interwoven extended periodic **srs** nets, namely 54, four and eight distinct nets on the *Gyroid*, *D* and *P* surfaces respectively [35]. Those nets are in fact the same as the pair of nets defining the polar labyrinths generated by the *Gyroid* surface, although with (sometimes dramatically) larger lattice spacings. We see in figure 20 that the asymptotic stripings of the minimal surfaces result in unbent, but very twisted labyrinths, entirely consistent with theoretical expectations for optimal 3-arm star polyphile membrane stripings. By contrast, the stripings along principal directions in the surfaces result in untwisted, but highly bent cages, also consistent with our geometric analysis, and far from optimal.

These extended patterns are extraordinarily complex, containing multiple interpenetrating networks at multiple length scales. Thus, for example, the 54 **srs** nets for the ideal forest 53a (tree–ribbon) on the *Gyroid* imply that such a self-assembled material would contain two **srs** networks of one species (here water), separated by a striped membrane that contains 54 networks of branched stripes filled with a second species and intercalated ribbons of a third species. In our terminology, these 56 interlocking networks, all maximally interpenetrating, make this a truly polycontinuous assembly. For the case of the forest 53a, the intercalated ribbons form a *Γ* rod packing composed of triple helical rods [36]. Recall from figure 13 that generic forests can form multiple interpenetrating two- or three-periodic networks, or cages (along the principle directions), as well as layers. A comprehensive list of the three-dimensional structures that form by embeddings of the various forests, as well as intercalating ribbons on the *Gyroid*, *D* and *P* surfaces has been published elsewhere [35,36], and briefly discussed in the electronic supplementary material, S2.

The **srs** nets that lie on asymptotes of the TPMS have two interesting chirality features on different scales. Firstly, while the pair of **srs** nets in the *Gyroid* labyrinths are mutually enantiomeric, the suite of multiple **srs** nets that emerge as asymptotic surface stripings are all the same enantiomer (right- or left-handed). Secondly, the torsion accompanying these asymptotic stripes induces a twisting along the edges of the striping lattice. This twisting is in a consistent direction, so each edge of the net also has matching chirality as illustrated in figure 20*a*.

The discussion above has been confined to the most symmetric three-branched striping patterns. However, the same arguments apply to more disordered striped configurations within a membrane, where preferred alignment of the stripes along the asymptotic directions can ensure that the stripe has the same size and shape on both sides of the membrane. This disorder can cover a spectrum from apparently random branching (and perhaps stripe termination) to almost symmetric solutions. These non-ideal structures can be expected to form in systems due to kinetic trapping. On the other hand, they could be the most favoured solutions for specific compositions, which demand stripe widths incompatible with any ‘defect’-free forest/intercalated stripe patterns for the imposed lattice dimension. A number of more complex quasi-symmetric patterns can be constructed, which are less regular than the ideal symmetric patterns analysed above, but nevertheless optimal. This is perhaps visible in the simulations of [32], depending on how generously one interprets the structures. These quasi-symmetric patterns are characterized by domains of symmetric striping, merging into other symmetric domains, with overall stripe directions close to the asymptotic directions of a ‘perfect’ solution.

More generally, however, we note that the restriction to perfectly balanced stripes is unlikely to be absolute, since small imbalances are likely to be accommodated in a real system, depending on the ease with which molecules can self-assemble to form these patterns. Now, a precise measure of allowed degree of width imbalance is difficult to estimate, and depends on both entropic and enthalpic interactions. For example, both contributions will disfavour imbalance, and depend on the specific polyphile–water chemistry, as well as the polydispersity in chains in the three arms of the constituent polyphile. However, we consider it unlikely for a system of monodisperse molecules to have stripes that are more than 20% larger on one side than on the other, i.e. the value for stripe-width mismatch should not be larger than 0.1.

### 3.5. Which solutions can potentially be achieved in a real system?

With this geometric exploration of various tree–tree and tree–ribbon stripe patterns on the *Gyroid*, *D* and *P* minimal surfaces in place, it is now possible to search for these patterns in star polyphile and mikto-arm copolymeric self-assemblies. Note that these highly symmetric patterns require simultaneous periodicity on two length scales: periodicity of the stripes and periodicity of the underlying surface. Thus, the stripe width must be commensurate with the lattice parameter of the underlying bicontinuous cubic phase or, in other words, only distinct values for the stripe width per unit cell dimension can result in these highly periodic solutions. All other solutions must have defects. We call the stripe width divided by the unit cell edge length the ‘effective stripe width’ here to distinguish it from the actual, physical stripe width in an experimental system.

The effective stripe widths, as well as the stripe width mismatch for all the available tree–tree and tree–ribbon patterns discussed above, were extracted from the Evolver data, electronic supplementary material, S5, and are summarized in figure 21. The plotted stripe-width mismatch was estimated for a membrane volume fraction of 30%.

This image gives an overview over all available idealized symmetric 3-branched striping patterns on the *Gyroid*, *D* and *P*. It can be seen that there are numerous structural solutions with small effective stripe widths (stripe widths in unit cell <0.1), but solutions with large effective stripe widths are rare, and are separated by large gaps. For example, the two patterns with the largest effective stripe widths are the tree–tree patterns produced by the shortest forest edge lengths, namely forest 3 (empty black square) and forest 5 (empty brown squares), where the stripe width of the latter is reduced by 30%. (Note that on the *D* surface, the tree–tree pattern for forest 3 does not exist.)

As the patterns with small stripe-width mismatch, i.e. the more balanced patterns, are preferred, we focus on the lower regions of the graphs in figure 21. It is clear that there are few accessible structures. For example, if we demand an imbalance in the stripe widths on either side of the minimal mid-surface of at most *ca* 10%—corresponding to a stripe-width mismatch of 0.05—there are just five distinct forests on the *Gyroid* at 30% membrane volume fraction (colours refer to horizontal lines in figure 21): forest 15a, red; forest 53a, orange; forest 195a, aqua; forest 725a, purple; and forest 1155a, pink (all with or without extra ribbons). The next most nearly balanced forest (forest 99a in green) has an imbalance in stripe widths of 20%. Among the forests decorating the *D* and *P* surfaces, only two cases have a stripe mismatch below 0.05 at 30% membrane volume fraction (again illustrated in figure 21): forest 3 (black line) for the *D*, and forest 5 (maroon line) for the *P*. In general, however, the type 2 bicontinuous *D* and *P* mesophases typically form at lower membrane volume fractions than their *Gyroid* analogue. If we therefore include lower membrane volume fractions, which reduces the effective value for the stripe-width mismatch (see figures 15*a* and 16*a*), forests 195 and 725 on the *D* (aqua and purple lines in figure 21), and forests 99 and 675 on the *P* (green and blue lines in figure 21), may also give accessible patterns.

Which of these striping patterns are accessible to star polyphile self-assemblies in water, of the type shown in figure 1? In those assemblies, the preferred dimensions of hydrocarbon and fluorocarbon domains, and therefore the optimal stripe width, are determined by the molecular dimensions of the centre and the three arms (figure 7*c*). On the other hand, the lattice parameter of the underlying bicontinuous cubic phase can be adjusted by changing the water content, and thus the effective stripe width can be tuned widely.

Given the close relationship between striped *Gyroid*, *D* and *P* structures and the bicontinuous cubic mesophases in conventional amphiphile–water systems, we can apply swelling laws for a fixed membrane thickness [45]. In systems like our polyphile system from figure 1, the hydrocarbon and fluorocarbon domain sizes are also rather fixed. So we can combine these two parameters to different ratios of stripe-width/membrane thickness and plot the dependence of the effective stripe widths on the hydrophilic channel volume fraction. This dependence is traced out by the blue arcs in figure 22. As the channels swell with water dilution, the effective stripe width decreases. In addition, the figure shows the effective stripe widths of the symmetric striping patterns analysed previously (from figure 21), extended over a range in channel volume fractions corresponding to a maximal stripe-width mismatch of 0.1 (20% difference between the monolayers), beyond which the forest is considered to be an unlikely solution. Recall that the stripe-width mismatch linearly increases with increasing membrane volume fraction, so linearly decreases with increasing channel volume fraction. (We have neglected to add the data for the *b* forests in the *Gyroid*, as their stripe-width mismatch is always significantly greater than that for the *a* versions.)

These data allow us to determine in quantitative detail the expected occurrences of these three-branched symmetric forests on dilution of the polyphile with water. Consider, for example, a polyphile–water system whose polyphilic molecular dimensions impose a stripe aspect ratio of unity (i.e. stripe-width/membrane thickness = 1). Further suppose that the polyphile–water system adopts the *Gyroid* type 2 membrane geometry. From figure 22*a*, we see the following progression as water is mixed into the system (increasing the channel volume fraction). For low *Φ* at large effective stripe widths, accessible symmetric striping patterns are forest 15a (tree–tree, open red squares) at *Φ* = 0.53, forest 53a (tree–tree, open orange squares) at *Φ* = 0.65, possibly forest 99a (tree–tree) at *Φ* = 0.72 (although this is close to the 20% mismatch), and so forth. As the effective stripe widths become thinner, a large number of accessible patterns are found. For molecules in figure 1, we can estimate a membrane thickness of 2.6 nm and a stripe width of 1.5 nm, giving a stripe-width/bilayer thickness ratio of about 0.6. Looking at figure 22, this should give us access to forest 53a (tree–tree), forest 195a (tree–tree), forest 15a (tree–ribbon) and many more forests at high water content. We note, however, that those at such high water contents are difficult to access experimentally, since *Gyroid* type 2 bicontinuous phases with such large water contents are rare. We also note that forest 53a (tree–tree) is also not accessible for these star polyphiles, because the small channel volume required is below the limit required for cubic phase formation.

## 4. Conclusion and summary

We began this paper with a simple query: what are the most likely 3-arm star molecular assemblies that are analogous to the ubiquitous bicontinuous mesophases? The analysis has required an excursion into the fields of differential geometry and surface topology, which we have described in qualitative terms.

In short, most bicontinuous mesophases found in amphiphile–water systems correspond to precisely those hyperbolic patterns in three-dimensional Euclidean space whose (Gaussian) curvature variations and global channel dimensions are as uniform as possible. In a nutshell, the mesophases whose membranes wrap onto the *Gyroid*, *D* and *P* minimal surfaces are (in order of increasing variations) the least frustrated solutions to the twin requirements of hyperbolic, balanced bilayers with preferred (negative, Gaussian) curvature. These correspond to the most symmetric hyperbolic surfaces in three-dimensional space.

In tricontinuous systems with equal surface tension between each pair of moieties, we have presented a set of branched minimal surfaces composed of standard hyperbolic patches spanning the interfaces between the branched triple-lines. We identify the most likely tricontinuous structures, assuming a preference for uniform negative Gaussian curvature. That analysis is analogous to the correlation between local homogeneity of the interface and its intrinsic (orbifold) symmetry found in amphiphilic bicontinuous forms. Amphiphiles assemble into bilayers, which are often spontaneously hyperbolic, due to the steric mismatch between the inner and outer surface areas making up the bilayer. By contrast, there is no analogue in the star polyphile assemblies, since the interfaces between pairs of immiscible domains are not preferentially hyperbolic. As a result, in the absence of further constraints on the molecular packing of the star polyphiles, tricontinuous systems have no corresponding drive to form negatively curved surfaces. The suite of hyperbolic tricontinuous patterns discussed here must compete with the relatively stable, and more prosaic three-coloured honeycomb pattern, shown in figure 3*c*, which has lower interfacial area [29]. Like the tricontinuous patterns, it too has branched interfaces, but flat instead of curved. Nevertheless, our shortlist of tricontinuous structures with curved, branched interfaces are in fact realized in simpler amphiphile systems. This implies that tricontinuous patterns may be realizable too in star assemblies, if one can impose additional constraints. For example, in contrast to the hexagonal honeycomb, the interfaces of the tricontinuous patterns twist along their triple lines. Therefore, we can favour these over three-coloured honeycombs by judicious design of the 3-arm polyphiles, forcing them to stack with a preferred twist between adjacent molecules. One such geometry involves propellor-like forms at the core of the star polyphiles, achievable by judicious chemical design of the polycyclic aromatic cores.

The situation for 3-arm star polyphile assemblies with unbalanced surface tensions, as in figure 1, is more complex. To lowest order, we can view these systems too as hydrophilic–hydrophobic assemblies, and so the formation of hydrophobic bilayers giving the *Gyroid*, *D* and *P* bicontinuous cubic mesophases is expected (and found). However, the star molecules exhibit a second level of assembly within the membrane, which can result in a striping pattern. Again, frustration and symmetry determine the resulting mesostructures. We expect that star polyphiles form symmetric (two-coloured) stripe patterns (one for each of the two hydrophobic moieties of the 3-arm stars). Those patterns are most likely to be branched three-ways, on the same minimal surfaces, building three-dimensional domains that can have extraordinary complexity.

Regardless of the detailed motifs, balanced (unfrustrated) patterns on both sides of the minimal surfaces impose a very strong constraint, that preferred patterns are those whose stripes (whether trees or ribbons) align along the least curved (and maximally twisted) asymptotic directions of the *Gyroid*, *D* and *P* surfaces. In fact, the same argument that predicts that highly symmetric stripe patterns should run along asymptotic directions is valid for less ordered striping patterns and even the boundaries of membrane inclusions: in general, any triple line on a minimal surface will have a tendency to run along asymptotic directions.

Earlier work has explored the specific compositions for which bicontinuous amphiphile–water systems can self-assemble, assuming the assemblies are constituted of amphiphiles with rigid dimensions. Those compositions are dependent on the specific steric details of the amphiphiles: their interfacial area per molecule and membrane thickness [12]. Here we extend that approach, and (introducing a new parameter: the ratio of stripe width to membrane thickness, or effective stripe width) deduce the likely compositions for formation of striped bicontinuous patterns with outstandingly complex morphology.

Consider, for example, the extreme topological complexity possible in even the simplest symmetric striping patterns, such as the tree–ribbon pattern of forest 5 [36]. Let us describe the resulting structure, assuming it is realized in a type 2 bicontinuous system folded onto the *P* surface, forming a hyperbolic hydrophobic bilayer, comprised of fluorophilic trees intercalated by oleophilic ribbons. The resulting assembly, which is essentially nothing more than a decorated bicontinuous structure, nevertheless has (i) periodic features on three different length scales, (ii) chirality on two levels, (iii) in total 10 mutually interpenetrating networks, and (iv) also contains a rod packing of domains! To elaborate: (i) on the smallest scale, the surface is striped, see figure 23*a*; at the intermediate scale we find a the pair of three-periodic **pcu** water labyrinths (figure 23*b*) separated by a hydrophobic membrane; at a larger length scale, the pattern of branching stripes on the surface formed by the fluorophilic domains traces out eight mutually threaded like-handed **srs** networks in space, each with a lattice parameter twice that of the underlying *P*-surface (figure 23*d*). (ii) There are two levels of chirality in the pattern. First, the stripes themselves are twisted, in a consistent direction, see figure 23*d*,*f*. In addition, the **srs** nets themselves are chiral, as their fourfold helices all have the same handedness. (iii) We note also that each of the two **pcu** water networks and **srs** branching ribbon networks are continuous and thread each other. (iv) The remaining complementary volumes, occupied by the oleophile, are intercalated between these sets of three-periodic labyrinths to form an array of interlocking columnar domains, mutually inclined along the four cubic [111] directions, known as the Ω^{+} rod packing [36]. Each one of these self-assembled components is of interest in its own right. For example, the water and fluorophile labyrinths in isolation offer matrices for controlled release of pharmaceuticals of comparable interest to current intensely studied cubosomes [46]. In concert, their novel chemical affinities, combining three quite distinct chemical species that are spontaneously structured over a range of (meso-) length scales typical of lyotropic liquid crystals, present a fascinating target for realization in the laboratory. Work is in progress, and data collected to date suggest that these patterns afford idealized defect-free models, generally realized in the laboratory with defects. The quantitative data presented here will, we hope, guide us to the most favourable compositions and three-arm molecular species for spontaneous formation of these liquid crystalline patterns, that rival the most complex catenated molecular crystals, such as metal–organic frameworks [47].

## Authors' contributions

S.T.H. conceived the project; L.d.C., T.C. and S.T.H. performed the research and wrote the paper.

## Competing interests

We declare we have no competing interests.

## Funding

This research was funded by the Australian Research Council.

## Footnotes

One contribution of 17 to a theme issue ‘Growth and function of complex forms in biological tissue and synthetic self-assembly’.

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3773300.

↵1 We follow the naming convention of Alan Schoen: ‘

*D*’ is short for Diamond, ‘*P*’ is for Primitive, while ‘Gyroid’ is spelled out [8].↵2 The word ‘balanced’ is used in different contexts in this paper, following the different uses of the word in different literatures. We endeavour to make clear by context which meaning we refer to.

↵3 With the exception of the discrete cubic micellar mesophases (of symmetry [39,40] and [41]).

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.