Giant Supramolecular Liquid Crystal Lattice

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Science  21 Feb 2003:
Vol. 299, Issue 5610, pp. 1208-1211
DOI: 10.1126/science.1078849


Self-organized supramolecular organic nanostructures have potential applications that include molecular electronics, photonics, and precursors for nanoporous catalysts. Accordingly, understanding how self-assembly is controlled by molecular architecture will enable the design of increasingly complex structures. We report a liquid crystal (LC) phase with a tetragonal three-dimensional unit cell containing 30 globular supramolecular dendrimers, each of which is self-assembled from 12 dendron (tree-like) molecules, for the compounds described here. The present structure is one of the most complex LC phases yet discovered. A model explaining how spatial arrangement of self-assembled dendritic aggregates depends on molecular architecture and temperature is proposed.

Molecular self-assembly into a variety of bulk phases with two-dimensional (2D) and 3D nanoscale periodicity, such as cubic, cylindrical, or mesh phases, has been researched intensely in lyotropic (e.g., surfactant-water) LCs (1), block copolymers (2–4), and thermotropic (solvent-free) LCs (5,6). Lyotropics can provide templates for porous inorganic materials with well-defined structures (7), and LC complexes of DNA with cationic and neutral lipids are potential carriers for gene delivery (8). Complex organic nanostructures (9, 10) may serve as scaffolds for photonic materials (11) and other nanoarrays (12) and for surface nanopatterning (13). In the case of bicontinuous cubic phases (1, 2, 5), the same structures have been observed in lyotropics, thermotropics, and block copolymers. The recent inclusion of self-assembling dendrons into the category of nanostructured soft matter (14) has shown their tapered shape to be responsible for creating similar bulk phases (13, 15–18) and enabling “self-processing” of electronics components (19). It has also become apparent that manipulating the size and distribution of “micelles” aggregated from self-assembling dendrons can be used in controlling polymerization (16). Regarding creation of structural diversity, dendron shape can be fine-tuned in ways unavailable in lyotropic LCs or block copolymers (14), and hence have the potential of forming hitherto unobserved phases. Here, we report a highly complex noncubic phase and develop a relationship between chemical structure and the self-assembly mode of tapered dendrons.

We concentrate on compounds I and II, labeled [4-3,4,5-(3,5)2]12G3-X in (20) (Scheme 1), where X is CH2OH and COOH, respectively. However, we have also observed the x-ray signature of the noncubic 3D phase in a number of systems, including the following dendrons: [4-3,4,5-(3,4)2]12G3-CO2CH3, [4-(3,4,5)2]12G2-COOH, (3,4-3,4,5) 12G2-CH2OH, [3,4-(3,5)2]12G3-CH2OH, (4-3,4,5-3,5)12G2-CH2OH (20), polyoxazolines with tapered side groups containing alkyl chains of different lengths (17), as well as in rubidium and cesium salts of 3,4,5-tris-(n-alkoxy)benzoic acid (21).

Wedge-shaped dendrons such as I and II, having alkyl tails on their periphery, have so far been found to form either columnar or cubic phases. In the former, dendrons assemble like flat pizza slices into disks, which stack into columns, which in turn form a hexagonal array. Dendrons with more alkyl chains are cone-shaped and assemble into supramolecular spheres. So far, these spherical aggregates have been known to pack on two cubic lattices, one with Pm3̄n and the other with Im3̄m [body-centered cubic (BCC)] symmetry.

Dendron I displays the presently described phase until the isotropic liquid is obtained at 131°C, whereas II shows the following phase sequences: glass < 110°C < Colh < 140°C CubPm3̄n < 153°C < new phase < 163°C < Iso [Colh = hexagonal columnar, CubPm3̄n = cubic phase of symmetry Pm3̄n (15), and Iso = isotropic liquid]. The fact thatII exhibits the CubPm3̄n phase indicates that above 140°C, the dendron tends to adopt a wedge shape approximated by a cone (15). Between crossed polarizers, the phase appears dark, indicating no birefringence. The small-angle x-ray powder diffractogram (22) shows a series of closely spaced sharp reflections (Fig. 1A), which were indexed on a large tetragonal unit cell with a = 16.93 ± 0.08 nm and c = 8.92 ± 0.05 nm for compound I and with a = 16.74 ± 0.08 nm and c = 8.80 ± 0.05 nm for compoundII (tables S1 and S2). For q = 4πsinθ/λ > 0.5 Å−1 (θ = scattering half-angle, λ = wavelength), only diffuse scattering is observed, confirming the lack of long-range order on subnanometer scale and therefore the LC nature of the phase.

Figure 1

Small-angle x-ray diffraction of the noncubic 3D phase in I. (A) Powder diffractogram. Monodomain oscillation patterns: (B) x*y* net [beam along the tetragonal (z*) axis] and (C)x*z* net, where x*, y*, andz* are reciprocal axes, parallel in this case to real-space axes x, y, and z(22).

Small-angle x-ray patterns parallel to the [001] and [100] directions of a monodomain of the tetragonal phase are shown, respectively, in Fig. 1, B and C. Figure 1B shows no systematichk0 reflection conditions other than h00:h = 2n, and Fig. 1C shows the conditions 0kl: k+l = 2n. This identifies the Patterson symmetry as P4/mmm and reduces the choice to three space groups: P42nm, P4̄n2, or P42/mnm. As is usual in the case of LCs, the latter (number 136) was selected because it has the highest symmetry.

Electron density maps (22) of the unit cell at different elevations are shown (Fig. 2, A to C). Figure 2D is the top view of the whole cell showing spherical or nearly spherical regions of highest density enclosed by a surface of constant electron density. All are views along the z axis. The unit cell contains 30 nearly spherical regions of high electron density embedded in a continuum of uniform low density (electron density histogram is shown in fig. S1). The high-density regions define the centers of micelle-like aggregates containing carbon-rich aromatic regions of the dendrons. The matrix contains molten dodecyloxy chains that are rich in hydrogen and are thus of lower electron density.Figure 3B shows schematically the 3D view of the unit cell. The cell volume in compound II at 158°C is 2466 nm3, i.e., 3.6 times larger than that of the CubPm3̄n unit cell at 145°C. Because the Pm3̄n cell contains eight “micelles” (Fig. 3A), there is only a 4% reduction in volume of a “micelle” in the transition, consistent with earlier findings of thermal shrinkage in “micellar” cubic supramolecular dendrimers (23). Assuming a mass density of 1 g/cm3 (23), there are, on average, 349 molecules of II per unit cell, or 11.6 per “micelle.”

Figure 2

Electron density maps of the TetP42/mnm unit cell. Shown are 2D maps at elevations (A) z = 0, (B) z = ¼ and ¾, and (C) z = ½. (D) Isoelectron surface at relative electron density of 0.35 (scale from 0 to 1) viewed from the top.

Figure 3

Packing of spheres in a unit cell of (A) CubPm3̄n, (B) TetP42/mnm, and (C) BCC. Rows of close contact (thick lines) are shown. Difference in fill pattern distinguishes spheres at different Wyckoff positions. (D) Stacking of tetrakaidecahedra along [001] in TetP42/mnm and along [100], [010], and [001] in CubPm3̄n, indicating preferential in-plane orientation of alkyl chains. Only 8j tetrakaidecahedra in the TetP42/mnm unit cell provide close contacts, whereas those at 8i′ are less oblate. (E) Equivalent representation of the hexagonal columnar phase.

There are five crystallographically nonequivalent types of “micelles” whose electron density maxima are located at five different Wyckoff positions (24) (Fig. 3B). The coordinates of the five positions are given in table S3. Of the 30 “micelles” in the unit cell, 10 have a coordination number 12 (CN12) and are centered at Wyckoff positions 2b and 8i, 16 have CN14 (8i′ and 8j), and four have CN15 (4g). This compares to two CN12 and six CN14 “micelles” in a unit cell of CubPm3̄nphase (Fig. 3A) (25).

The total volume of the unit cell is divided among the 30 micelles by allocating each a Voronoi (Wigner-Seitz) polyhedron (26). Thus, CN12, CN14, and CN15 micelles occupy, respectively, dodecahedra, tetrakaidecahedra, and pentakaidecahedra (fig. S2). Figure 3D shows three stacked CN14 polyhedra enclosing three 8j micelles.

Interestingly, the tetragonal LC phase is structurally equivalent to the σ phase found in Fe46Cr54, which is responsible for embrittlement of steel, and in a number of other alloys (27), as well as to the high-temperature β phase of uranium (28). Similarly, CubIm3̄m (BCC) phase has its equivalents in pure metals, and CubPm3̄nin alloys (the A15 phase) (29). Unlike CubIm3̄m, both TetP42/mnm (σ phase) and CubPm3̄n are members of the family of tetrahedrally close packed (TCP) structures, or Frank-Kasper phases (30). Their structure is characterized by alternating densely and sparsely populated layers; in TetP42/mnm, the former are located at z = 0 and ½ and the latter at z = ¼ and ¾. The dense nets in TetP42/mnm and CubIm3̄m are compared in fig. S2, E and F.

All of the Voronoi polyhedra in TetP42/mnm are distorted to a degree; the distortion of CN14 polyhedra at 8i′ is not the same as that of CN14 polyhedra at 8j. Similarly, CN12 polyhedra at 2b and 8i are different. The fact that, in the same structure, “micelles” of a pure compound can have five different environments, and hence shapes, suggests that the free energy is minimized globally at some expense to that of the individual micelles. An even larger difference in micellar sizes is found in the inverted micellar (“water in oil”) lyotropic cubic phase with Fd3̄m symmetry (31), a TCP phase made up of CN12 and CN16 micelles. It is unlikely that this phase will be found in pure thermotropic LCs, although it may possibly appear in some mixtures where larger molecules might inhabit the larger CN16 micelles. Notably, the equivalent metallurgical Laves phase C15 occurs only in alloys with a large atom size difference.

What makes dendrimers and metals adopt equivalent complex packing patterns? If atoms were hard spheres, all pure metals would adopt either the face-centered cubic (FCC) or the hexagonal close packed (HCP) structure, not a TCP type. However, atomic orbitals are “soft” and d orbitals play an important role in determining TCP structures (32); not surprisingly, these structures are particularly abundant in transition metals (29). Supramolecular dendrimers are also soft spheres. They have been approximated by hard spherical cores and soft aliphatic coronas with a tendency for surface minimization (33). The optimum solution of the packing problem for such a system coincides with the solution of the old Kelvin problem of the minimum energy dry foam (minimum surface area per bubble) (34). To date, the best solution to the Kelvin problem, proposed by Weaire and Phelan (35), is the CubPm3̄n structure. The abundance of CubPm3̄n in supramolecular dendrimers (15) was taken as the experimental vindication of the Weaire-Phelan solution (33).

The present observation of the P42/mnm phase does not challenge this conclusion, because TetP42/mnm occurs at a higher temperature than CubPm3̄n. Area minimization of the aliphatic corona is the driving force at low temperatures, whereas increased conformational disorder at higher temperatures would lead to some lateral expansion of the aliphatic layer. In most supramolecular dendrimers that we studied, where there is an LC phase below the temperature range of CubPm3̄n, that phase is hexagonal columnar (21,23). This suggests that, of the known structures, only a honeycomb foam (a 2D structure) (Fig. 3E) is a better solution than that of Weaire and Phelan.

In order to explain why spherical aggregates of dendrons pack on different 3D lattices, we calculate the ideal average radial distribution of volume dV/dr for supramolecular dendrimers that would fill the unit cell of the different phases perfectly uniformly. Imagine spheres of radius r growing simultaneously from the center of each Voronoi polyhedron. Here,vi (r) is that part of the volume of the Voronoi cell i which is within the sphere. Figure 4A shows the averaged functionsdV/dr, where V = (1/Ni=1 N v1 is the mean v and N is the number of polyhedra in the unit cell. It is worth noting that a similar calculation was carried out to compare simple cubic, BCC, and FCC micellar packings in diblock copolymers (36).

Figure 4

(A) Radial distribution of volume allocated to an average “micelle,” dV/dr, wherer = 0 at the center of a Voronoi polyhedron.dV/drr 2 until a sphere of radiusr = r 0 (the hard sphere radius) first touches a face of its surrounding polyhedron (r 0 is marked with a vertical tick for each structure). The shape of the dV/dr functions to the right of the vertical markers, i.e., beyond the hard sphere radius, is governed by the filling of the interstices and is specific for each structure. Distributions are shown for FCC (CubFm3̄m), BCC (CubIm3̄m), CubPm3̄n, and TetP42/mnm. The curves were calculated numerically. The distributions were normalized to the same total volume of the average “micelle,” i.e., to the same average Voronoi cellEmbedded Image For insets, see text. (B) The r dependence of the rate of decrease in average nonoverlapped solid angle Ω. For a CN14 polyhedron, the nonoverlapped solid angle is subtended by the spherical surface in inset γ. The function -/dr is shown for the same four structures as in (A). The curves are normalized toEmbedded Image

The significance of dV/dr is that it indicates the dendron shape ideally suited for a given crystal structure. A way to relate these functions to the molecular shape is to imagine the conical dendron (inset α) rolled out flat in the direction perpendicular to the cone axis, into a parabolic (r 2) fan as schematically drawn in inset β in Fig. 4A (compare also with the chemical structures ofI and II in Scheme 1). The ideal envelope of the molecular fan for each phase is represented by the correspondingdV/dr, with r = 0 at the acute apex of the dendron. The rectangle in inset β indicates the limited region of dV/dr shown in the diagram.

Scheme 1

The dV/dr distribution for BCC differs from those for CubPm3̄n and TetP42/mnm in that it has a slightly higher peak and shorter tail. This explains why BCC is always seen at a higher temperature than the other two phases (21): it is precisely this change in molecular envelope that is expected from a lateral thermal expansion and longitudinal contraction of alkyl tails.

Figure 4A also suggests the reason that FCC structure has never been observed in supramolecular dendrimers. A high peak and a long tail of the relevant dV/dr are incompatible with the shape of the dendrons studied thus far. We propose that a possible FCC candidate should have a short branch on one of the first carbons of the alkyl tail and should include a small fraction of extra-long tails.

The dV/dr functions for CubPm3̄n and TetP42/mnm are almost indistinguishable in Fig. 4A. It has indeed been observed that in some systems the two phases occur almost interchangeably. However, in equilibrium situations there is a clear pattern of TetP42/mnmoccurring as an intermediate between CubPm3̄n and BCC. In order to understand this, we resort to a more sensitive representation of geometry of each phase. Referring again to a sphere of increasing radius r centered in Voronoi polyhedroni (inset γ in Fig. 4B), let ωi be the solid angle subtended by the nontruncated part of the sphere (round surface in inset γ). Before the sphere first touches a face of the polyhedron ωi = 4π, and after the sphere had completely outgrown the polyhedron ωi = 0. Figure 4B shows the rate of decrease of the mean solid angle, -dΩ/dr, where Ω = (1/Ni=1 Nωi.dΩ/dr is related to dV/dr throughEmbedded Image(1)The differential solid angle function –dΩ/dr for TetP42/mnm is seen to differ markedly from that of CubPm3̄n. For CubPm3̄n, the step increase in –dΩ/dr at r = 0.50 comes from the first contact of black spheres in Fig. 3A along one of the principal directions, i.e., [100], [010], or [001] (solid lines). TetP42/mnm also has close contacts (step nearr = 0.50) but only along [001] (vertical direction inFig. 3B). However, the close contacts in TetP42/mnminvolve only 8 spheres (black in Fig. 3B, Wyckoff positions 8j) of 30 in the unit cell, or 27%, as compared to six of eight, or 75%, in CubPm3̄n. In contrast, BCC structure has no such close contacts (Fig. 3C), and FCC even less so, where the first contact only occurs at r = 0.56. Both in CubPm3̄nand in TetP42/mnm, the closest contacts are established along infinite rows of CN14 polyhedra stacked by their two hexagonal faces (Fig. 3D).

The shape of –dΩ/dr close tor 0 is particularly important because of the high repulsion potential between spherical brushes. This is lowered if the alkyl chains can be moved aside with a low entropic penalty (schematically shown in Fig. 3D). Phases with more close contacts are therefore established preferentially at lower temperatures, or in dendrons with fewer tethered chains, where the penalty is lower. Stacking of the oblate tetrakaidecahedra in supramolecular dendrimers (Fig. 3D) can be regarded as a step toward stacking of disks in the Colh phase (Fig. 3E). Thus, the observed sequence Colh → CubPm3̄n → TetP42/mnm → BCC with ascending temperature (21) is consistent with the increasing avoidance of close contact in that order (Fig. 4B). The apparent paradox of the low- symmetry TetP42/mnmphase occurring above the temperature of the high-symmetry CubPm3̄n phase is, accordingly, explained by the reduction from three to one in the number of principal axes along which close contacts exist.

Based on the principles outlined here, compounds may be designed that form even larger unit cells, containing more than a hundred “micelles,” following the analogy with some other TCP structures in metals. The range of structures may be increased further by employing binary or ternary dendron blends, in analogy with metal alloys and block copolymers.

Supporting Online Material

Materials and Methods

Figs. S1 and S2

Tables S1 to S3

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