Crystallographic realization of the mathematically predicted densest all-pentagon packing lattice by C5-symmetric "sticky" fluoropentamers

Article abstract of DOI:10.1002/anie.201101553

Regular pentagonal adhesive tiles were obtained by designing a cyclic fluoropentamer with an internal CF...HN H-bonded network and "sticky" edges that can glue the molecules together through intermolecular C_ArH...O=C H bonds. In the crystal they form the mathematically predicted densest all-pentagon 2D lattice with a packing density of 0.921 (see picture). Copyright

Full text of DOI:10.1002/anie.201101553

Communications
DOI: 10.1002/anie.201101553
2D Pentagon Packing
Crystallographic Realization of the Mathematically Predicted Densest
All-Pentagon Packing Lattice by C5-Symmetric “Sticky”
Fluoropentamers**
Changliang Ren, Feng Zhou, Bo Qin, Ruijuan Ye, Sheng Shen, Haibin Su, and Huaqiang Zeng*
Two-dimensional periodic and aperiodic tilings have contin-
uously garnered great interest from ancient[1a,b] to present[1c–i]
times. Fascinated by the aesthetically pleasing fivefold
symmetry and pentagonal shape that are found abundantly
from macroscopic morning glories and star fruits, to micro-
scopic virus capsids,[2a] simple liquids,[2b] and to nanoscopic
fullerenes, and biological pentamers (Figure 1a),[2c] a funda-
mental curiosity that has inspired a long-standing quest[1a–c,2a,3]
concerns two-dimensional (2D) packing by non-overlapping
equal regular pentagons that cannot fully tile a 2D plane
without leaving gaps (Figure 1b).
With a 2D packing density of 0.921, the periodic densest
packing configuration of pentagons in the plane is built of
alternating striped rows of pentagons pointing in opposite
directions. This leads to a congruent arrangement of hexa-
gonally coordinated pentagons that forms an ingenious
solution to overcoming the symmetry mismatch between
pentagons and lattice, and creates a 2D packing lattice with
p2mg symmetry. This densest six-neighbor lattice packing was
first conjectured by Henley in 1986[4] and mathematically
substantiated by Kuperberg and Kuperberg in 1990.[3c]
However, till now experimental observations on penta-
gonal small molecules,[1g,h,5a] proteins,[5b,c] and macroscopic
objects,[1e,5g] as well as computational simulations,[5d–f] all
pointed to a great difficulty for regular pentagons to form a
densely packed crystal lattice in flat 2D space. In the instances
where pentagons appear to be densely packed in some
areas,[1g,5c,g] disorders/defects of varying types (orientational/
translational disorder, lattice dislocation, etc.) exist. On rare
occasions, dense packing of pentagons with the highest
packing density of 0.921 could be found in limited areas by
experiments on macroscopic objects[5g] and by computational
simulations on hard pentagons.[5h]
Mathematically, equal regular pentagons can form
abstract compact 2D crystal lattices (Figure 1b).[1a,b,2a,3a–c]
A few recent studies elegantly explored the use of
fivefold-symmetric corannulene and its derivatives to look
into pentagonal packing diagrams on copper surface. A low
packing density of 0.765 for ordered 2D packing by penta-
gons[1g] and surface-induced trimerization of pentagons with a
packing density of less than 0.725[1h] were obtained. There-
fore, down to the molecular level, whether or not and how
regular pentagons of uniform size can pack ultradensely in the
2D plane are still open questions that have remained elusive
from our current understanding of matter and that have been
hampered by the insufficient diversity of pentagonal mole-
cules,[1h,2c,3b,6,7] and especially by the scarce availability of
neutral molecules having fivefold symmetric planarity.[7a]
In our quest for possible means of experimentally
reproducing the densest mathematically predicted crystals
at the molecular level, we envisioned that the key to better
understanding and possibly solving the periodic packing
problem faced by regular pentagons lies in our ability to
produce fivefold-symmetric, planar molecules that resemble
the overall shape of a hard pentagon. To this end, we set out to
engineer a pentagonal molecule that ideally should have
unique fivefold-symmetric planarity. Following a cue given by
pentagonal molecule 1 (Figure 2), recently reported by us,[2c]
and the proven ability of aromatic fluorine atoms to form
intramolecular H bonds that effectively constrain the aro-
matic backbones into crescent or helical shapes,[8] aromatic
Figure 1. a) Selected examples of the fivefold symmetry or pentagonal
shapes that are found abundantly in the world. b) Three mathemati-
cally constructed periodic 2D packing patterns and packing densities
by non-overlapping equal regular pentagons in the plane. The densest
packing containing hexagonally arranged pentagons has a packing
density of 0.921.
[*] C. L. Ren, Dr. B. Qin, R. J. Ye, S. Shen, Prof. H. Q. Zeng
Department of Chemistry, National University of Singapore
3 Science Drive 3, Singapore 117543 (Singapore)
E-mail: chmzh@nus.edu.sg
F. Zhou, Prof. H. B. Su
Division of Materials Science, Nanyang Technological University
50 Nanyang Avenue, Singapore 639798 (Singapore)
[**] We thank Andrey Yakovenko and Prof. Hong-Cai Zhou from Texas
A & M University for their help in refining the crystal structure of
pentamer 2. This study was supported by the National University of
Singapore AcRF Tier 1 Grants (R-143-000-375-112 and R-143-000-
398-112 to H.Z.), A*STAR BMRC research consortia (R-143-000-388-
305 to H.Z.) and A*STAR SERC grant (M47070020 to H.S.).
Supporting information for this article is available on the WWW
under http://dx.doi.org/10.1002/anie.201101553.
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Angew. Chem. Int. Ed. 2011, 50, 10612 –10615

three months. The determined crystal
structure[10] demonstrates folding of 2
into an almost planar disk arrange-
ment of nearly perfect fivefold sym-
metry in solid state (Figure 3a,b and
Table S5 of Supporting Information).
All five fluorine atoms and amide
protons point inward, and contribute
to formation of a continuous intra-
molecularly H-bonded network
(CF···HN 2.08–2.26 and 1.94–2.27 ꢀ
for S(5) and S(6) intramolecular
H bonds, respectively). This H-bond-
ing network provides the required
entropically favorable preorganiza-
tion force to rigidify the amide link-
Figure 2. a) Chemical structures of 1 and 2. In 1, three methyl groups point up and two point down.
Dotted red lines indicate H bonds. b) Top and side views of crystal structure of 1 and computation-
ally optimized structure of 2 (B3LYP/6-31G* level). Red spheres: O atoms, cyan spheres: F atoms.
“Sticky” edges and vertices in 2 are made up of an exterior array of partially charged aromatic
hydrogen (H-bond donor) and carbonyl oxygen atoms (H-bond acceptor).
fluoropentamer 2 (Figure 2) was finally conceived as a
conceptual pentameric molecular framework to explore its
possible shape, molecular symmetry, and solid-state packing.
Its crystal structure was used to realize the densest planar
pentagons with perfect long-range order at the atomic scale
that was previously thought possible by physicists and
mathematicians.[2a,3a,b]
As a first step toward deriving the topography associated
with 2, we constructed ab initio computational molecular
model using DFT at the B3LYP/6-31G(d,p) level, which has
consistently enabled us to accurately predict 3D top-
ographies of oligomers of varying types that were later
experimentally verified by crystal structures.[2c,9] As
shown in Figure 2b, the geometry of the optimized
structure of 2 shows fivefold-symmetric planarity, and
the extent of backbone bending is insignificantly influ-
enced by covalent macrocyclization constriction.[2c,9a,b]
Instead, its crescent-shaped backbone curvature is
primarily induced and maintained by an inward-point-
ages and instruct an otherwise randomly oriented backbone
to fold into a well-defined, rigid, circularly folded conforma-
tion with pentagonal shape. Five fluorine atoms convergently
align to enclose a star-shaped cavity of about 2.85 ꢀ in radius
from the cavity center to the center of the fluorine nucleus
(Figure 3a,b).
Analysis of the solid-state packing pattern in 2 reveals the
long-awaited densest striped crystalline lattice (Figure 3e)
harmonically built from pure pentagonal molecules that pack
precisely in the same manner as the mathematically verified
À
À
ing, continuous C F···H N H-bonded network. Such an
intrinsic backbone peculiarity requiring five identical
repeating units to form a macrocycle is rarely observed
in other systems.[6] Moreover, the exterior array of
partially charged aromatic hydrogen (H-bond donor)
and carbonyl oxygen atoms (H-bond acceptor) may
serve as “sticky” atoms to glue the molecules together
À
=
through intermolecular C H···O C H bonds (Fig-
ure 2a).
To further characterize our design, pentamer 2 was
synthesized (see Scheme S1 of Supporting Information)
to test the ab initio model and shed new light on the old
mystery of 2D pentagons assembled with long-range
order. Compound 2 was unambiguously characterized by
Figure 3. Crystal structure and 2D molecular packing of macrocyclic fluoropen-
tamer 2. a) Top view. b) CPK model of 2 built on the basis of van der Waals
radii (H gray: 1.20; C green: 1.70; O red: 1.52; N blue: 1.55; F cyan: 1.47 ꢀ).
c) Formation of discrete pairs of isolated H bonds (2.50 ꢀ) between two
pentamers, which stabilize the interplanar stacking. d) Pseudo-hexagonal
arrangement with a side length of 16.7 ꢀ and with each pentagon forming two
edge-to-vertex and two smaller and two larger edge-overlap contacts with its
six close neighbors. Small spheres represent partially charged “sticky” hydro-
1
HRMS and H NMR and 19F NMR spectroscopy (see
Figures S1 and S2 of Supporting Information). Evidence
that short oligomers such as dimer 2c and trimer 3 f
adopt crescent-shaped conformations enforced by intra-
=
À
gen (gray) and oxygen (red) atoms that are involved in C O···H C H bonds,
which allow tight packing of pentagonal molecules. e) Top and side views of
alternating striped lattice packing, consisting of pentagons pointing in
opposite directions in CPK model, which reproduces the densest mathemat-
ical crystal formed by pentagons (Figure 1b). The average interplanar distance
is 3.1 ꢀ. The 2D packing density of 0.921 was calculated by using the van der
À
À
molecular C F···H N H bonds is provided in Table S2
and Figures S3 and S4 of the Supporting Information.
To investigate the symmetry and solid-state assembly
of 2 in the 2D plane, single crystals of 2 were eventually
obtained by slow cooling of a solution containing about
1.1 mgmLÀ1 of 2 in hot DMSO from 110 to 258C over Waals radii specified in (b).
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Communications
six-neighbor densest packing (Figure 1b).[3a,c] While each
lize the 2D packing lattice by 5.33 and 2.50 kcalmolÀ1 per
pentamer, respectively (Table S6 of Supporting Information).
Given its nearly planar pentagonal shape with an exterior
array of “sticky” edges and vertices, macrocyclic pentamer 1
(Figure 2) should also be capable of assembly into a 2D lattice
with long-range order and translational symmetry in the
plane. Re-examining the crystal structure of 1[2c] indeed
reveals a periodic 2D six-neighbor packing pattern (Figure 4),
remarkably similar to the densest pentagonal packing of 2
(Figure 3e). The main driving force for organizing the
interplanar stacking is van der Waals interactions among the
protruding methoxymethyl groups. As a result, the average
interplanar distance increases substantially to 4.3 ꢀ, 0.9 ꢀ
larger than the typical distance of 3.4 ꢀ, and the crystal
density decreases to 1.43 MgmÀ3. In contrast to 2, the 2D
lattice built from 1 consists of slightly distorted hexagonal
units that tile the plane, all of which contain seven water
molecules that each participate in forming one strong
parallelogramic unit cell contains two pentamers, the closest
packing pattern can be better represented by the translation-
ally ordered pseudo-hexagonally coordinated pentamers
(Figure 3d,e). These pentamers having no vertices but seg-
ments of their sides in common are arranged to form
alternating stripes that point head-to-tail in opposite direc-
tions. Each pentamer has five complementary “sticky” edges
and one vertex to attract six close neighbors by means of
H bonding of three types: two edge-to-vertex, two smaller
edge-overlap, and two larger edge-overlap bonding contacts.
All of these specific H-bonding interactions are mediated by
À
=
weak C H···O C H-bonding forces. Because of these specific
H bonds conserved among pentamers that continue across the
2D lattice, minimization of the gaps among “sticky” pentam-
ers in the 2D plane (Figure 3d,e) is as good as in the
mathematically conceived packing lattice (Figure 1b). Thus
our crystal has an experimentally determined 2D packing
density of 0.921, a value theoretically predicted to be the
highest possible.[2a,3a,c]
À
=
intermolecular HO H···O C H bond (2.01 ꢀ) with the
closest pentamer molecule. To accommodate these water
molecules in the lattice, 1) the gaps among pentamers become
larger, reducing the 2D packing density of pentamers to 0.898
with a further contribution to packing density of 0.035 from
seven water molecules (Figure 4b), and 2) each pentamer has
two edge-to-vertex, one smaller and one larger edge-overlap,
and one full edge contacts with its five close neighbors, while
the sixth neighbor interacts with it through two water
molecules.
Driven by a combination of aromatic p–p stacking
=
À
interactions and numerous interplanar H bonds (C O···H
N 2.50 ꢀ, Figure 3c), the ordered 2D lattices further crystal-
lize into a 3D structure. One pair of isolated interplanar
H bonds creates a discrete dimeric ensemble composed of two
pentamers (Figure 3c). These interplanar H bonds result in
an observed average interplanar distance as short as 3.1 ꢀ and
a high crystal density of 1.58 MgmÀ3. They form as a result of
À
À
comparatively much weaker C F···H N H bonds causing two
amide bonds in each pentamer to twist out of the pentameric
Comparison of 2D molecular packing between 1 and 2
highlights the important fact that, even for rigid pentagonal
molecules, a great degree of adaptive flexibility in crystal
packing in response to external impurities such as water
molecules can be achieved. The fact that such physically
realistic lattices do exist suggests that molecules with fivefold
rotational symmetry are fully compatible with the symmetry
and translational ordering of 2D crystal lattices, and can pack
with high density. In addition to idealized p2mg symmetry
=
À
plane to form stronger intermolecular C O···H N H bonds
(2.50 ꢀ) with the adjacent neighboring pentamer. Such a twist
causes pentamers to deviate from the idealized fivefold-
symmetric planarity of the computed structure of 2 (Fig-
ure 2b), and changes the 2D lattice symmetry from p2 to
pseudo-p2.
To eliminate the above interplanar crystal-packing effect
that gives rise to distorted amide bonds, and thus obtain
not only the perfect packing pattern and packing
parameters for perfect regular pentagons, but also
quantitative information on driving forces responsible
for the formation of the ordered 2D pentagonal lattice,
we continued our theoretical investigation at the B3LYP/
6-31G(d,p) level on the hexagonal unit comprising one
central pentamer and six surrounding pentamers (Fig-
ure 3d and Supporting Information). A key finding of
this theoretical treatment is that the 2D packing lattice
formed by these pentagons is perfectly coplanar (see
Figure S8a of Supporting Information) in the absence of
interplanar layer-to-layer H-bonding interactions as seen
in Figure 3c. Further computational analysis revealed
driving forces accounting for the bonding energy of
À6.82 kcalmolÀ1 per pentamer in forming a 2D pentag-
onal lattice from regular pentagons, that is, the stabilizing
H-bonding interactions between “sticky” hydrogen and
oxygen atoms (À14.05 kcalmolÀ1 per pentamer; Table S6
of Supporting Information) can more than compensate
for the repulsive interactions among hydrogen atoms and
Figure 4. 2D molecular packing of macrocyclic pentamer 1. a) Distorted
hexagonal unit consisting of seven pentamers and seven water molecules
À
=
(CPK models), each of which forms a strong intermolecular HO H···O C
H bond (2.01 ꢀ) with its closest neighbor. b) Top and side views of the
alternating striped lattice packing as CPK model, consisting of water
molecules and pentagons pointing in opposite directions. In contrast to the
packing pattern in Figure 3e, one pentagonal edge of the pseudo-hexagonally
coordinated central pentagon interacts with one of its close neighbors
through two water molecules. The average interplanar distance is 4.3 ꢀ. The
2D packing densities of 0.933 and 0.898 were calculated on the basis of van
among oxygen atoms, which were estimated to destabi- der Waals radii specified in the legend to Figure 3b.
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J. Phys. Condens. Matter 1995, 7, 3421; h) T. Schilling, S. Pronk,
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their packing patterns, the 2D lattices built from pure
pentagonal tiles can have symmetries of p2 for 2D packing
by 2 (Figure 3e), and p1 for 2D packing by 1 (Figure 4b).
Our above design strategy additionally affords covalent
nanoscaffolding for the expedient exchange of a multitude of
modular functional groups that decorate both the exterior and
interior surfaces of the macrocyclic pentamers. This could
lead to a large diversity of freely standing pentagonal
molecules that offer interesting perspectives on other math-
ematically sketched periodic pentagonal tiling patterns (Fig-
ure 1b)[1a,b,2a,3a–c] whose 2D crystalline packings at the molec-
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[10] The obtained values of R1 (0.1495) and wR2 (0.2957) are larger
than expected, which is presumably due to the disorder as a
result of two orientations of the molecule overlapping in one
asymmetric position. The two disordered parts represent differ-
ent directions in which the amide groups go around the
pentamer ring in (anti)clockwise fashion. Since the atoms of
the two disordered parts are largely overlapping, large numbers
of strict restraints were applied to the refinement so that it
became convergent and the structure remained reasonable. In
the refinement, we used the disorder scheme in which all atoms
are involved in the disorder so that the geometry of the
disordered amides is not distorted. This explains the unusually
high values of the final R factors. Without introducing disorder,
the result was even worse. At the same time, we refined the
structure by changing the weight scheme and omitting several
reflections, all of which help to decrease R factors, correct the
GOF, and remove high residual peaks. The ratio of the
disordered parts, as obtained by free variable refinement, is
about 56:44, which is close to the random ratio of 50:50. From
the structural refinement, the existence of a five-membered ring
is clear.
Received: March 3, 2011
Revised: June 15, 2011
Published online: September 16, 2011
Keywords: fivefold symmetry · hydrogen bonds · macrocycles ·
.
pentagonal molecules · supramolecular chemistry
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