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quasicrystals. Nature, 428 (6979). pp. 157-160. ISSN 0028-0836
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28. Simon, C. & Irvine, W. Robust long-distance entanglement and a loophole-free Bell test with ions and
photons. Phys. Rev. Lett. 91, 110405 (2003).
29. Cabrillo, C., Cirac, J. I., Garcia-Fernandez, P. & Zoller, P. Creation of entangled states of distant atoms
by interference. Phys. Rev. A 59, 1025–1033 (1999).
30. Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state entanglement and
quantum error correction. Phys. Rev. A 54, 3824–3851 (1996).
Acknowledgements We acknowledge discussions with M. Madsen, P. Haljan, M. Acton and
D. Wineland, and thank R. Miller for assistance in building the trap apparatus. This work was
supported by the National Security Agency, the Advanced Research and Development Activity,
under Army Research Office contract, and the National Science Foundation Information
Technology Research Division.
Competing interests statement The authors declare that they have no competing financial
interests.
Correspondence and requests for materials should be addressed to B.B. (bblinov@umich.edu).
..............................................................
Supramolecular dendritic
liquid quasicrystals
Xiangbing Zeng
1
, Goran Ungar
1
, Yongsong Liu
1
, Virgil Percec
2
,
Andre
´
s E. Dulcey
2
& Jamie K. Hobbs
3
1
Department of Engineering Materials, University of Sheffield, Sheffield
S1 3JD, UK
2
Roy & Diana Vagelos Laboratories, Department of Chemistry, University of
Pennsylvania, Philadelphia, Pennsylvania 19104-6323, USA
3
H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
.............................................................................................................................................................................
A large number of synthetic and natural compounds self-
organize into bulk phases exhibiting periodicities on the 10
28
–
10
26
metre scale
1
as a consequence of their molecular shape,
degree of amphiphilic character and, often, the presence of
additional non-covalent interactions. Such phases are found in
lyotropic systems
2
(for example, lipid–water, soap–water), in a
range of block copolymers
3
and in thermotropic (solvent-free)
liquid crystals
4
. The resulting periodicity can be one-dimensional
(lamellar phases), two-dimensional (columnar phases) or three
dimensional (‘micellar’ or ‘bicontinuous’ phases). All such two-
and three-dimensional structures identified to date obey the
rules of crystallography and their symmetry can be described,
respectively, by one of the 17 plane groups or 230 space
groups. The ‘micellar’ phases have crystallographic counterparts
in transition-metal alloys, where just one metal atom is equiva-
lent to a 10
3
2 10
4
-atom micelle. However, some metal alloys are
known to defy the rules of crystallography and form so-called
quasicrystals, which have rotational symmetry other than the
allowed two-, three-, four- or six-fold symmetry
5
. Here we show
that such quasiperiodic structures can also exist in the scaled-up
micellar phases, representing a new mode of organization in soft
matter.
Research on bulk nanoscale self-assembly of organic matter is
partly motivated by the fact that such complex structures may serve
as scaffolds for photonic materials
6
and other nanoarrays, or as
precursors for mesoporous ceramics or elements for molecular
electronics. Larger biological objects, such as cylinder-like or
sphere-like viruses, also pack on similar macrolattices
7
.
Dendrons and dendrimers (tree-like molecules
8
) are proving
particularly versatile in generating periodic nanostructures
(Fig. 1). Two micellar lattices, with space groups Im3
¯
m (body-
centred cubic, b.c.c.)
9
, and Pm3
¯
n (refs 10, 11), have been estab-
lished. An analogue of the Im3
¯
m phase has also been observed in
block copolymers
12
, and that of the Pm3
¯
n phase in lyotropic liquid
crystals
13
. Recently, a complex three-dimensional (3D) tetragonal
lattice (space group P4
2
/mnm) was found, having 30 self-assembled
micelles in the unit cell (Fig. 1f)
14
.
In many dendron systems, thermal transitions between the
phases in Fig. 1 occur. The master sequence Col
h
! Pm
3n !
P4
2
=mnm ! Im
3m is obeyed with increasing temperature; in only
a handful of cases are all these phases displayed in the same material.
In a number of compounds, however, an additional unidentified
phase has been observed below any other 3D phase but above Col
h
.
A small-angle X-ray powder diffractogram of this phase, recorded
on dendron I (Fig. 1g), is shown in Fig. 2a. The synthesis of I is
described in ref. 15 and Supplementary Information, where
this compound is labelled [3,4,5-(3,5)
2
]12G
3
CH
2
OH. Other
compounds that show the X-ray signature of this phase include
(4-3,4,5-3,5)12G
2
CH
2
OH, [ 4-(3,4,5)
2
]12G
2
COOH, [3,4-(3,5)
2
]
12G
3
COOH, [3,4-(3,5)
2
]12G
3
CH
2
OH, [3,4-(3,4,5)
2
]12G
3
CH
2
OH
(ref. 15), polyoxazolines with tapered side groups containing alkyl
chains of different lengths
16
, as well as certain salts of 3,4,5-tris-(n-
alkoxy)benzoic acid
17
.
On heating, compound I shows the following phase sequence:
room temperature !X ! 718C ! P4
2
=mnm ! 728C! isotropic
liquid, while on cooling phase X forms directly from the liquid
(Supplementary Information). This allowed us to grow mono-
domains of the unknown phase. That phase X is a quasicrystal is
revealed by the distinctive but cr ystallographically forbidden
12-fold symmetry of the small-angle X-ray single-crystal pattern
(Fig. 2b). When the sample is rotated around the 12-fold axis with
the incident beam perpendicular to the axis, the diffraction pattern
repeats every 308. One such pattern is shown in Fig. 2c, where
the Ewald sphere cuts through a pair of strong reflections in Fig. 2b.
The structure of this liquid quasicrystal (LQC) is periodic in the
direction of the 12-fold axis, but quasiperiodi c in the plane
perpendicular to it.
In contrast to normal 3D periodic structures, five instead of three
basis vectors are needed for indexing the diffraction peaks of a
dodecagonal quasicrystal
18
. Four of the vectors, q
1
, q
2
, q
3
and q
4
,
Figure 1 Self-assembly of wedge-shaped molecules. a, Dendrons with fewer tethered
chains adopt a flat slice-like shape (X is a weakly binding group). b, The slices stack up
and form cylindrical columns, which assemble on a two-dimensional hexagonal columnar
(Col
h
) lattice (c). d, Dendrons with more end-chains assume a conical shape. e, The cones
assemble into spheres, which pack on three different 3D lattices (f) with symmetries
Im3
¯
m, Pm3
¯
n and P4
2
/mnm. g, Structure of compound I studied in this work.
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Group
are perpendicular to the 12-fold axis (Fig. 2b). q
5
, along the 12-fold
axis, is indicated in Fig. 2c. Thus each diffraction peak q is indexed
by five integers (n
1
n
2
n
3
n
4
and n
5
) where q ¼ S
5
i¼1
n
i
q
i
. Tentative
indices of some of the diffraction peaks are given in Fig. 2.
The structure of the LQC is closely related to those of the Pm3
¯
n
and P4
2
/mnm phases. Where these phases form from the LQC on
heating, their {002} reflections appear at the same position as the
{00002} reflections of the LQC. Thus the periodicity of the LQC
along the 12-fold axis is the same as that of Pm3
¯
n along any of the
three cubic axes, and P4
2
/mnm along c. This suggests that, like the
above phases, the LQC is also micellar. With increasing temperature,
the observed d-spacings of the LQC decrease proportionally. This
isotropic shrinkage suggests a phase containing isometric objects.
Pm3
¯
n, P4
2
/mnm and Im3
¯
m (b.c.c.) phases that are observed
in dendrimers all have their structural equivalents in transition
metals or alloys (for example, Pm3
¯
n:Cr
3
Si; P4
2
/mnm:Fe
46
Cr
54
and
b
-uranium; b.c.c.:
a
-iron). The dendrimer ‘atoms’ (micelles) have
volumes several thousand times larger than real atoms, and so do
their unit cells. Both Pm3
¯
n and P4
2
/mnm phases belong to the
family of tetrahedrally close packed (t.c.p.) structures of spherical
objects, or Frank–Kasper phases
19
. In a t.c.p. structure, any four
neighbouring spheres pack tetrahedrally, which is locally the densest
packing. However, regular tetrahedral interstices are incompatible
with long-range order. Such frustration leads to the complexity of
Figure 3 Packing of spheres in the LQC and other related t.c.p. structures. All can be
generated from 2D tilings consisting of only squares and equilateral triangles (‘sparse’
nets, elevations z ¼ 1/4 and 3/4, large open circles). These tiles are ‘decorated’ with
spheres at z ¼ 0 (small open circles) and 1/2 (filled circles), forming the ‘dense’ nets.
a, Sphere packing in Pm3
¯
n. b, Sphere packing in P4
2
/mnm. c, Only three different
decorated tiles, one square and two triangular, are used to generate both Pm3
¯
n and
P4
2
/mnm structures. d, The same tiles, when arranged quasiperiodically, will generate
the model of the LQC with 12-fold symmetry. The lattice constant, that is, the length of the
tile edge, as well as the periodicity along the 12-fold axis, is 81.4 A
˚
at room temperature.
e, Two ideal hexagonal antiprisms stacked along the 12 symmetry axis. There is a
distorted hexagonal antiprism at each node of the square-triangular tiling in a, b and d.
Figure 2 Experimental and simulated X-ray diffraction patterns of the LQC. a, Powder
diffraction pattern of compound I recorded at 70 8C. b, Precession single-crystal pattern
along the 12-fold axis. The intensities of outer diffraction peaks in b and c are scaled up by
100. c, Single-crystal diffraction pattern perpendicular to the 12-fold axis, cutting
through one of the six pairs of strong diffraction spots in b. The pattern repeats itself every
308 when the sample is rotated around the 12-fold axis. All single-crystal patterns were
recorded at room temperature. Simulated diffraction patterns are superimposed, with
the reflections represented by circles whose area is proportional to log(amplitude). The
apparent deviation in position of diffraction spots in the outer region of the diffraction
pattern in c is due to the existence of other domains in the sample (see Supplementary
Information).
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Group
t.c.p. structures. Ultimately, it leads to quasicrystals.
Dodecagonal quasicrystals have been found so far in five tran-
sition-metal systems
20–23
. In all but one case, dodecagonal structures
were studied by electron microscopy or electron diffraction. The
interpretation of electron diffraction is complicated by multiple
scattering. In the one case where quasicrystals sufficiently large for
single-crystal X-ray diffraction were obtained, the stacking along the
12-fold axis was different from that in other dodecagonal phases,
including the present LQC
23
. As X-ray data can be compared with
models more u nequivocall y than electron diffraction data, we
proceed to construct a model of the LQC. We follow the idea that
dodecagonal structures can be generated by appropriate ‘decora-
tion’ of a quasiperiodic square-triangular tiling
18,21,23
.
Pm3
¯
n and P4
2
/mnm structures are characterized by alternating
densely and sparsely populated layers, as shown in Fig. 3a and b,
respectively. The nets generated by connecting the nearest neigh-
bours in the sparsely populated layers (large circles) are equivalent
to tilings that cover an infinite plane using o nly squares and
equilateral triangles. The full 3D crystal structure is then generated
by ‘decoration’, that is, by associating with each tile a column,
periodic along c, containing individual micelles. Only three kinds of
tiles, one square and two triangular (Fig. 3c), are needed to
cons t ru c t the Pm3
¯
n and P4
2
/mnm structures. (In the actual
P4
2
/mnm phase, the position of the micelles is somewhat different
from those in an idealized structure made up of decorated tiles
(Fig. 3c). Similarly, it is expected that in the real quasicrystal, the
micelles would also be somewhat away from the ideal positions
assumed by our starting model (Fig. 3d)).
Using quasiperiodic tiling, the same elements can be used to
construct models of the LQC (Fig. 3d). For ways of generating such
tilings, see refs 24 and 25. Such models readily explain the 12-fold
symmetry, and the equal periodicities in the third dimension for the
LQC, Pm3
¯
n and P4
2
/mnm phases. Simulated diffraction patterns
based on the quasiperiodic square-triangular tiling of ref. 25 are
superimposed on the experimental ones in Fig. 2b and c (for details
of the simulation, see Supplementary Information). Even the
unrefined starting mo del gives a reasonable explanation of the
positions and relative intensities of experimental diffraction spots
(Fig. 2b and c). Nevertheless, discrepancies in intensities remain (for
example, {10102} and {22202} reflections are stronger than
expected). Improvement of the model is progressing, as the
reliability of acquired diffraction data allows it.
Atomic force microscope images of the LQC phase on glass show
that the structure is periodic in one direction but not in the other, in
agreement with a quasicrystalline structure where the 12-fold axis
lies parallel to the g lass surface. Moreover, they suppor t the
‘micellar’ nature and the layered structure of the LQC (Supplemen-
tary Information).
To explain why spherical aggregates of dendrons pack on different
3D lattices, mathematical f unctions defining molecu lar shapes
ideally suited to typical lattices have been calculated
14
. In a different
approach
26
, micelles were approximated by hard cores and soft alkyl
coronas. Assuming that thermal transitions are primarily driven by
the laterally expanding alkyl chains, the first micellar phase to which
a columnar dendrimer would transform on heating is expected to be
the one with the minimal coronal surface area. Such a structure
ought to coincide with the solution of the Kelvin problem of the
ultimate equilibrium dry foam (minimum surface area per bubble).
The b.c.c. structure proposed by Kelvin
27
was challenged only
recently by a calculation showing that the Pm3
¯
n structure provides
a better solution
28
. The fact that, of all known micellar phases in self-
assembled dendrimers, the Pm3
¯
n was always found at the lowest
temperature was taken as experimental vindication of the new
minimal surface solution
26
. However, now that the updated phase
sequence is Col
h
! LQC ! Pm
3n ! P4
2
=mnm! b.c.c., the possi-
bility is raised of dodecagonal quasicrystals providing a still better
solution of the Kelvin problem.
Outside metal-based system s, non-crystallographic rotational
symmetry has been observed only in thin films of smectic C twist
grain boundary (TGBC) liquid crystals confined between glass
plates with rubbed surface s
29,30
. q-fold symmetr y is created by
helical packing of smectic C gra ins, and lock-in transitions between
different q-values were obser ved with changing temperature. In
contrast, the present LQC is a bulk phase with intrinsic high-level
quasiperiodic order inherent in the molecular architecture.
The tiling of planes normal to the 12-fold axis in the LQC (Fig. 3d)
can be understood intuitively in the following way. There are six
tetrahedrons packed around the vertical line connecting the two
light-grey spheres at z ¼ 1/4 and 3/4 in Fig. 3e, taking two adjacent
spheres at z ¼ 1/2 as the other two tetrahedral apices. However, the
number of perfect tetrahedrons that can fit around a common edge
is 5.1. In different t.c.p. structures, including LQC, the inability to
cover a flat surface with regular pentagons is resolved in different
tilings of distorted hexagons. The problem is reversed in the case of
curved surfaces; these cannot be tiled exclusively by hexagons. Thus
in ‘buckyballs’ and footballs, a cer tain proportion of pentagons is
required and, as the curvature radius decreases relative to tile size,
this proportion increases until the all-pentagonal icosahedron is
reached. Interesting examples are found in hexagonal and penta-
gonal packing modes of proteins in capsids (coats) of cylindrical
and spherical viruses
31
.
Finally, the unusual symmetry of quasicrystals has been exploited
in recent years for fabrication of photonic bandgap arrays. Owing to
their high symmetr y, two-dimensional (2D) quasicrystalline lattices
are able to induce and widen the photonic bandgap
32
, preventing
light within a range of wavelengths from propagating in any
direction. More recently it was shown that light can be slowed
down in a one-dimensional photonic quasicrystal that follows the
Fibonacci sequen ce. To produce photonic ‘quasicrystals’ with
photonic bandgaps in the visible light region, the distance of two
neighbouring objects must be on the scale of several hundred
nanometres. Whereas the results presented here show how the
characteristic length of a self-assembled quasicrystal can be scaled
up from a few a
˚
ngstro
¨
msinmetalalloystonearly10nmin
supramolecular dendrimers, it may be possible to achieve a further
increase of two orders of magnitude by using appropriately
designed self-assembling dendrons, block copolymers or other
soft sphere systems. A
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Acknowledgements We thank A. Gleeson and P. Baker for assistance with X-ray diffraction
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..............................................................
Links between salinity variation in
the Caribbean and North Atlantic
thermohaline circulation
Matthew W. Schmidt
1
, Howard J. Spero
1
& David W. Lea
2
1
Department of Geolog y, University of California, Davis, California 95616, USA
2
Department of Geological Sciences and Marine Science Institute, University of
California, Santa Barbara, California 93106, USA
.............................................................................................................................................................................
Variations in the strength of the North Atlantic Ocean thermoha-
line circulation have been linked to rapid climate changes
1
during
the last glacial cycle through oscillations in North Atlantic Deep
Water for mation and no rthward oceanic heat flux
2–4
.The
strength of the thermohaline circulation depends on the supply
of warm, salty water to the North Atlantic, which, after losing
heat to the atmosphere, produces the dense water masses that
sink to great depths and circulate back south
2
. Here we analyse
two Caribbean Sea sediment cores, combining Mg/Ca palaeo-
thermometr y with measurements of oxygen isotopes in
foraminiferal calcite in order to reconstruct tropical Atlantic
surface salinity
5,6
during the last glacial cycle. We find that
Caribbean salinity oscillated between saltier conditions during
the cold oxygen isotope stages 2, 4 and 6, and lower salinities
during the warm stages 3 and 5, covarying with the strength of
North Atlantic Deep Water formation
7
. At the initiation of the
Bølling/Allerød warm inter val, Caribbean surface salinity
decreased abruptly, suggesting that the advection of salty tropical
waters into the North Atlantic amplified thermohaline circula-
tion and contributed to high-latitude warming.
Today, most of the North Atlantic’s s ubtropical gyre water
circulates through the Caribbean Sea before it is transported to
the subpolar regions of the North Atlantic via the Gulf Stream
8
.Net
evaporation exceeds precipitation in the Atlantic, resulting i n
freshwater removal of , 0.35 £ 10
6
m
3
s
21
from the Atlantic
basin
9
. Because of their influence on North Atlantic surface salinity,
the tropical and subtropical Atlantic play an important part in
regulating North Atlantic Deep Water (NADW) formation. How-
ever, unlike the Gulf of Mexico and North Atlantic, Caribbean
salinity is not significantly affected by freshwater runoff and there-
fore surface salinity primarily reflects the evaporation/precipitation
ratio over the western tropical Atlantic. Hence, changes in tropical
Figure 1 Temperature and
d
18
O
SW
variation in the western Caribbean Sea during the
past 136 kyr. a, Colombian basin
d
18
O
C
and b, Mg/Ca–SST records from ODP 999A
(128 45
0
N, 788 44
0
W; 2,827 m; 4 cm kyr
21
sedimentation rate) and VM28-122 (118 34
0
N, 788 25
0
W; 3,623 m; 4 cm kyr
21
sedimentation rate during the Holocene and
LGM, 10–15 cm kyr
21
sedimentation rate during the deglaciation), based on the
planktic foraminifer G. ruber (white). Mg/Ca was converted to SST
13
using
Mg/Ca ¼ 0.38exp0.09[SST–0.61(core depth, in km)]. c, Computed
d
18
O
SW
calculated from the Mg/Ca-derived SST and
d
18
O
C
using T (in 8C) ¼ 16.5 2 4.80
(
d
C
2 (
d
W
2 0.27)) (ref. 14). The continental ice-volume
d
18
O
SW
reconstruction
15
is
shown for comparison. Note that the amplitude of the calculated
d
18
O
SW
change in the
Colombian basin is considerably greater than the global
d
18
O
SW
change due to ice volume
alone.
letters to nature
NATURE | VOL 428 | 11 MARCH 2004 | www.nature.com/nature160
© 2004
Nature
Publishing
Group
![Figure 1 Temperature and d18OSW variation in the western Caribbean Sea during the past 136 kyr. a, Colombian basin d18OC and b, Mg/Ca–SST records from ODP 999A (128 45 0 N, 788 44 0 W; 2,827m; 4 cm kyr21 sedimentation rate) and VM28-122 (118 34 0 N, 788 25 0 W; 3,623m; 4 cm kyr21 sedimentation rate during the Holocene and LGM, 10–15 cm kyr21 sedimentation rate during the deglaciation), based on the planktic foraminifer G. ruber (white). Mg/Ca was converted to SST13 using Mg/Ca ¼ 0.38exp0.09[SST–0.61(core depth, in km)]. c, Computed d18OSW calculated from the Mg/Ca-derived SST and d18OC using T (in 8C) ¼ 16.5 2 4.80 (dC 2 (dW 2 0.27)) (ref. 14). The continental ice-volume d 18OSW reconstruction 15 is shown for comparison. Note that the amplitude of the calculated d18OSW change in the Colombian basin is considerably greater than the global d18OSW change due to ice volume alone.](/figures/figure-1-temperature-and-d18osw-variation-in-the-western-2ggnardg.webp)
![Figure 1 Temperature and d18OSW variation in the western Caribbean Sea during the past 136 kyr. a, Colombian basin d18OC and b, Mg/Ca–SST records from ODP 999A (128 45 0 N, 788 44 0 W; 2,827m; 4 cm kyr21 sedimentation rate) and VM28-122 (118 34 0 N, 788 25 0 W; 3,623m; 4 cm kyr21 sedimentation rate during the Holocene and LGM, 10–15 cm kyr21 sedimentation rate during the deglaciation), based on the planktic foraminifer G. ruber (white). Mg/Ca was converted to SST13 using Mg/Ca ¼ 0.38exp0.09[SST–0.61(core depth, in km)]. c, Computed d18OSW calculated from the Mg/Ca-derived SST and d18OC using T (in 8C) ¼ 16.5 2 4.80 (dC 2 (dW 2 0.27)) (ref. 14). The continental ice-volume d 18OSW reconstruction 15 is shown for comparison. Note that the amplitude of the calculated d18OSW change in the Colombian basin is considerably greater than the global d18OSW change due to ice volume alone.](/figures/figure-1-self-assembly-of-wedge-shaped-molecules-a-dendrons-v8vpajho.webp)