A rough classification by
topic is shown below
Relevant graphical logos are appended to each paper
Relevant graphical logos are appended to each paper
 |
 |
 |
 |
 |
 |
| Nets Networks Embedded Graphs Tangled Patterns |
Biominerals and biomaterials "Biomorphs": living vs. dead matter |
Cellular structures, surfaces | "Hard"
framework materials: Zeolites Metallo-organic frameworks Novel carbon nanofoam etc. |
Soft
matter: Lyotropic liquid crystals microemulsions block copolymers Membranes in vivo |
Theoretical
crystallography |
PDF files are available for private use only as they are copyrighted
- Stephen T. Hyde and Robert Owen, “Sculpting entanglement”, Australian Physics, to appear, (2015).
- Yang-Chih Lo, Wayne Hsu, Hsiu-Yi He, Stephen T. Hyde, Davide M. Proserpio and Jhy-Der Chen, “Structural
directing roles of isomeric phenylenediacetate ligands in the formation
of coordination networks based on a flexible N,N’-
di(3-pyridyl)suberoamide. A rare 5-fold cds net and a 1D network with
new mode of entanglement”, CrystEngComm, in press, (2014).
Abstract:
Reactions of the flexible N,N′-di(3-pyridyl)suberoamide (L) with
Cu(II) salts in the presence of the isomeric phenylenediacetic acids
under hydrothermal conditions afforded three new coordination networks,
{[Cu(L)(1,2-pda)]·H2O}n (1,2-H2pda = 1,2-phenylenediacetic acid), 1,
{[Cu(L)(1,3-pda)]·2H2O}n (1,3- H2pda = 1,3-phenylenediacetic acid), 2,
and {[Cu(L)(1,4-pda)]·2H2O}n (1,4-H2pda = 1,4-phenylenediacetic acid),
3, which have been structurally characterized by X-ray crystallography.
Complex 1 forms a single 3,5-coordinated 3D net with the
(42.65.83)(42.6)-3,5T1 topology, which can be further simplified as a
6-coordinated (412.63)-pcu topology. Complex 2 is a 5-fold
interpenetrated 3D structure with the (65.8)-cds topology, which
exhibits the maximum number of interpenetration presently known for cds
and complex 3 is the first example of a tangled 1D net that contains
neither knots, links nor ravels, thus exemplifying a new mode of
entanglement. The ligand-isomerism of the phenylenediacetate ligands is
important in determining the structural types of the Cu(II)
coordination networks based on the flexible L ligands.
- Michael G. Fischer, Liliana de Campo, Jacob J.K. Kirkensgaard, Stephen T. Hyde and Gerd E. Schröder-Turk, “The tricontinuous 3ths(5) phase: A new network morphology in copolymer melts”, Macromolecules, in press, (2014).
Abstract:
Self-assembly remains the most efficient route to the formation
of ordered nanostructures, including the double gyroid network phase in
diblock copolymers based on two intergrown network domains. Here we use
self-consistent field theory to show that a tricontinuous structure
with monoclinic symmetry, called 3ths(5), based on the intergrowth of
three distorted ths nets, is an equilibrium phase of triblock
star-copolymer melts when an extended molecular core is introduced. The
introduction of the core enhances the role of chain stretching by
enforcing larger structural length scales, thus destabilizing the
hexagonal columnar phase in favor of morphologies with less packing
frustration. This study further demonstrates that the introduction of
molecular cores is a general concept for tuning the relative importance
of entropic and enthalpic free energy contributions, hence providing a
tool to stabilize an extended repertoire of self-assembled
nanostructured materials.
- Jacob J. K. Kirkensgaard, Martin C. Pedersen and Stephen T. Hyde, “Tiling patterns from ABC star molecules: 3-colored foams?”, Soft Matter, 10, 7182-7194 (2014).
Abstract:
We present coarse-grained simulations of the self-assembly of
3-armed ABC star polyphiles. In systems of star polyphiles with two
arms of equal length the simulations corroborate and expand previous
findings from related miktoarm star terpolymer systems on the formation
of patterns containing columnar domains whose sections are 2D planar
tilings. However, the systematic variation of face topologies as the
length of the third (unequal) arm is varied differs from earlier
findings regarding the compositional dependence. We explore 2D
3-colored foams to establish the optimal patterns based on interfacial
energy alone. A generic construction algorithm is described that
accounts for all observed 2D tiling patterns and suggests other
patterns likely to be found beyond the range of the simulations
reported here. Patterns resulting from this algorithm are relaxed using
Surface Evolver calculations to form 2D foams with minimal interfacial
length as a function of composition. This allows us to estimate the
interfacial enthalpic contributions to the free energy of related star
molecular assemblies assuming strong segregation. We compare the
resulting phase sequence with a number of theoretical results from
particle-based simulations and field theory, allowing us to tease out
relative enthalpic and entropic contributions as a function of the
chain lengths making up the star molecules. Our results indicate that a
richer polymorphism is to be expected in systems not dominated by chain
entropy. Further, analysis of corresponding planar tiling patterns
suggests that related two-periodic columnar structures are unlikely
hypothetical phases in 4-arm star polyphile melts in the absence of
sufficient arm configurational freedom for minor domains to form
lens-shaped di-gons, which require higher molecular weight polymeric
arms. Finally, we discuss the possibility of forming a complex tiling
pattern that is a quasi- crystalline approximant for 3-arm star
polyphiles with unequal arm lengths.
- Stephen T. Hyde, “Aqua Reticulata: topology of liquid water networks”,
in "Aqua Incognita: Why Ice Floats on Water and Galileo 400 years on",
Pierandrea Lo Nostro and Barry Ninham, eds., Connor Court Publishing,
Victoria, Australia, pp. 145-175, (2014).
- Jacob J. K. Kirkensgaard, Myfanwy E. Evans, Liliana de Campo and Stephen T. Hyde, “Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks”, Proc. Natl. Acad. Sci. U.S.A., 111(4), 1271–1276 (2014). doi:10.1073/pnas.1316348111
Abstract:
Numerical simulations reveal a family of hierarchical and chiral
multicontinuous network structures self-assembled from a melt blend of
Y-shaped ABC and ABD three-miktoarm star terpolymers, constrained to
have equal-sized A/B and C/D chains, respectively. The C and D majority
domains within these patterns form a pair of chiral enantiomeric gyroid
labyrinths (srs nets) over a broad range of compositions. The minority
A and B components together define a hyperbolic film whose midsurface
follows the gyroid minimal surface. A second level of assembly is found
within the film, with the minority components also forming labyrinthine
do- mains whose geometry and topology changes systematically as a
function of composition. These smaller labyrinths are well de- scribed
by a family of patterns that tile the hyperbolic plane by regular
degree-three trees mapped onto the gyroid. The labyrinths within the
gyroid film are densely packed and contain either graphitic hcb nets
(chicken wire) or srs nets, forming convoluted intergrowths of multiple
nets. Furthermore, each net is ideally a single chiral enantiomer,
induced by the gyroid architecture. However, the numerical simulations
result in defect-ridden achiral patterns, containing domains of either
hand, due to the achiral terpolymeric starting molecules. These
mesostructures are among the most topologically complex morphologies
identified to date and represent an example of hierarchical ordering
within a hyper- bolic pattern, a unique mode of soft-matter
self-assembly.
- S.T. Hyde, S.J. Ramsden and V. Robins, “Unification and classification of 2D crystalline patterns using orbifolds”, Acta Cryst A, 70, 319–337 (2014). doi:10.1107/S205327331400549X
Abstract:
The concept of an orbifold is particularly suited to
classification and enumeration of crystalline groups in the euclidean
(flat) plane and its elliptic and hyperbolic counterparts. Using
Conway’s orbifold naming scheme, this article explicates conventional
point, frieze and plane groups, and describes the advantages of the
orbifold approach, which relies on simple rules for calculating the
orbifold topology. The article proposes a simple taxonomy of orbifolds
into seven classes, distinguished by their underlying topological
connectedness, boundedness and orientability. Simpler ‘crystallographic
hyperbolic groups’ are listed, namely groups that result from
hyperbolic sponge-like sections through three-dimensional euclidean
space related to all known genus-three triply periodic minimal surfaces
(i.e. the P, D, Gyroid, CLP and H surfaces) as well as the genus-four
I-WP surface.
- Yabing He, Zhiyong Guo, Shengchang
Xiang, Zhangjing Zhang, Wei Zhou, Frank
R. Fronczek, Sean Parkin, Stephen T.
Hyde and Banglin Chen, “Metastable interwoven mesoporous metal-organic frame-works”, Inorg Chem, 52, 11580-11584 (2013).
Abstract:
Three isostructural interwoven (3,4)-connected mesoporous
metal-organic frameworks of pto-a topology (UTSA-28-Cu, UTSA-28-Zn, and
UTSA-28-Mn) were synthesized and structurally characterized. Because of
their meta-stable nature, their gas sorption properties are highly
dependent on the metal ions and activation profiles. The most stable
UTSA-28a-Cu exhibits high gas storage capacities.
- Olaf Delgado-Friedrichs, Stephen T. Hyde, Shin-Won Mun, Michael O’Keeffe and Davide M. Proserpio, “Nets with collisions (unstable nets) and crystal chemistry”, Acta Cryst A, 69, 535-542 (2013). doi: 10.1107/S0108767313020655
Abstract:
Nets in which different vertices have identical barycentric
coordinates (i.e. have collisions) are called unstable. Some such nets
have automorphisms that do not correspond to crystallographic
symmetries and are called non-crystallographic. Examples are given of
nets taken from real crystal structures which have embeddings with
crystallographic symmetry in which colliding nodes either are, or are
not, topological neighbors (linked) and in which some links coincide.
An example is also given of a crystallographic net of exceptional girth
(16), which has collisions in barycentric coordinates but which also
has embeddings without collisions with the same symmetry. In this last
case the collisions are termed unforced.
- Liliana de Campo, Olaf Delgado-Friedrichs, Stephen T. Hyde and Michael O’Keeffe, “Minimal nets and minimal minimal surfaces”, Acta Cryst A, 69, 483-489 (2013). doi:10.1107/S0108767313018370
Abstract:
The 3-periodic nets of genus 3 (‘minimal nets’) are reviewed and
their symmetries re-examined. Although they are all crystallographic,
seven of the 15 only have maximum-symmetry embeddings if some links are
allowed to have zero length. The connection between the minimal nets
and the genus-3 zero- mean-curvature surfaces (‘minimal minimal’
surfaces) is explored by deter- mining the surface associated with a
net that has a self-dual tiling. The fact that there are only five such
surfaces but 15 minimal nets is rationalized by showing that all the
minimal nets can serve as the labyrinth graph of one of the known
minimal minimal surfaces.
- N. Francois, T. Arnoux, L. Garcia, S.T. Hyde, V. Robins, M. Saadatfar, M. Saba, and T.J. Senden, “Experimental investigation of the mechanical stiffness of periodic framework-patterned elastomers”, Phil Trans A, 372, 20120035 (2013). doi: 10.1098/rsta.2012.0035
Abstract:
Recent advances in the cataloguing of three-dimensional nets
mean a systematic search for framework structures with specific
properties is now feasible. Theoretical arguments about the elastic
deformation of frameworks suggest characteristics of mechanically
isotropic networks. We explore these concepts on both isotropic and
anisotropic networks by manufacturing porous elastomers with three
different periodic net geometries. The blocks of patterned elastomers
are subjected to a range of mechanical tests to determine the
dependence of elastic moduli on geometric and topological parameters.
We report results from axial compression experiments, three-dimensional
X-ray computed tomography imaging and image-based finite-element
simulations of elastic properties of framework-patterned elastomers.
- Myfanwy E. Evans, Vanessa Robins and Stephen T. Hyde, “Periodic entanglements II: weavings from hyperbolic fibrations”, Acta Cryst A, 69, 262-275 (2013). doi:10.1107/S0108767313001682
Abstract:
Ordered arrays of cylinders, known as rod packings, are now
widely used in descriptions of crystalline structures. These are
generalized to include crystal- lographic packed arrays of filaments
with circular cross sections, including curvilinear cylinders whose
central axes are generic helices. A suite of the simplest such general
rod packings is constructed by projecting line patterns in the
hyperbolic plane (H^2) onto cubic genus-3 triply periodic minimal
surfaces in Euclidean space (E^3): the primitive, diamond and gyroid
surfaces. The simplest designs correspond to ‘classical’ rod packings
containing conventional cylindrical filaments. More complex packings
contain three-dimensional arrays of mutually entangled filaments that
can be infinitely extended or finite loops forming three-dimensional
weavings. The concept of a canonical ‘ideal’ embedding of these
weavings is introduced, generalized from that of knot embeddings and
found algorithmically by tightening the weaving to minimize the
filament length to volume ratio. The tightening algorithm builds on the
SONO algorithm for finding ideal conformations of knots. Three distinct
classes of weavings are described.
- Myfanwy E. Evans, Vanessa Robins and Stephen T. Hyde, “Periodic entanglements I: networks from hyperbolic reticulations”, Acta Cryst A, 69, 241-261 (2013). doi:10.1107/S0108767313001670
Abstract:
High-symmetry free tilings of the two-dimensional hyperbolic
plane (H^2) can be projected to genus-3 3-periodic minimal surfaces
(TPMSs). The three-dimensional patterns that arise from this
construction typically consist of multiple catenated nets. This paper
presents a construction technique and limited catalogue of such
entangled structures, that emerge from the simplest examples of regular
ribbon tilings of the hyperbolic plane via projection onto four genus-3
TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these
patterns are explored and partially characterized using tools from
TOPOS, GAVROG and a new tightening algorithm.
- Stephen T. Hyde, “D'Arcy Thompson's legacy in contemporary studies of patterns and morphology”, Interdisciplinary Science Reviews, 38, 12-34 (2013).
Abstract:
D’Arcy Thompson’s views on the forms of biomaterials are
assessed in the light of current thinking on biomorphology in selected
areas of biology. It is clear that his guiding concepts — that
biological materials are structured in response to physical forces, and
that the biological and abiotic realms share many common features —
remain valid. Advances in the physical and biological sciences are
discussed, from quantum mechanics and molecular biology to liquid
crystalline materials and macroscopic forms. These reveal Thompson’s
clear-sighted view of the role of physical and mathematical sciences in
biology, as well as his blind-spots.
- Gerd E. Schröder-Turk and Stephen Hyde, “Geometry of interfaces: topological complexity in biology and materials”, Interface Focus, 2(5), 529-538 (2012). doi: 10.1098/rsfs.2012.0035
Introduction:
The geometrization of physics, which views physical phenomena
through the prism of geometry and topology, has left a lasting imprint
on many areas of physics, from Einstein’s revolutionary adoption of
Riemannian geome- try to build his theories of relativity, to the
rapidly multiplying zoo of topological phases in quantum physics. It is
therefore not surprising that newer areas of condensed matter research,
particularly synthetic and biological soft liquid crystalline matter
and related materials, are best explored using the tools of low-dimen-
sional geometry and topology. This realization is not new.
Twenty years ago, Elisabeth Dubois-Violette and Brigitte Pansu, both then at the Laboratoire de Physique des Solides at Orsay, organized a seminal meeting in Aussois, ‘Geometry and Interfaces’, which brought together physicists, chemists, biologists and mathemati- cians and resulted in a useful volume that summarized the state of things in 1990 [1]. The Orsay group had an impeccable pedigree in condensed materials research (including, for example, liquid crystal research that led to the award of the Physics Nobel Prize to de Gennes). Their scientific culture recognized the importance of crossing traditional disciplinary borders, and the enrichment of conventional condensed matter physics drawn from studies in other areas, from biology to pure mathematics. That approach now pervades many aspects of contemporary physics research into materials, where it is recognized that biology and materials chemistry offers fertile domains for explora- tion. Another approach to materials research remains however less developed: the exploration of the funda- mental science of biomaterial self-assembly and function using the tools of low-dimensional geometry and topology. Few biologists concern themselves with more complex aspects of geometry, despite the earliest forays by D’Arcy Wentworth Thompson in his seminal book ‘On growth and form’. One notable exception was Yves Bouligand, a biologist whose close links with Orsay and personal interest and knowledge of geometry led to the important recognition of the relevance of the liquid crystalline state to many biological assemblies, such as the cholesterol character of the arrangement of chitin fibres in crab shells (http://people.physics. anu.edu.au/sth110/bouligand_papers.html/). Surely Bouligand is one of the very few biologists who have made significant contributions to the physics of liquid crystals?
In an attempt to redress that imbalance, we orga- nized a successor to the Aussois meeting in October 2011 at Primosˇten, Croatia (http://www.geometry-of- interfaces.org/). The aim was to gauge developments since 1990, and to highlight the continued relevance and importance of geometry and topology to condensed materials, whether hard or soft, synthetic or biological. We were fortunate to have the company of two of the semi- nal figures in the field, Alan Schoen and Ka ̊re Larsson, whose contributions to minimal surface theory and the role of those surfaces in biological membrane folding and liquid crystalline mesophases, respectively, continue to influence research. This theme issue is focused on active research in material structure, with papers from a cross-section of participants.more complex aspects of geometry, despite the earliest forays by D’Arcy Wentworth Thompson in his seminal book ‘On growth and form’ [2]. One notable exception was Yves Bouligand, a biologist whose close links with Orsay and personal interest and knowledge of geometry led to the important recognition of the relevance of the liquid crystalline state to many biological assemblies, such as the cholesterol character of the arrangement of chitin fibres in crab shells (http://people.physics. anu.edu.au/sth110/bouligand_papers.html/). Surely Bouligand is one of the very few biologists who have made significant contributions to the physics of liquid crystals?
In an attempt to redress that imbalance, we orga- nized a successor to the Aussois meeting in October 2011 at Primosˇten, Croatia (http://www.geometry-of- interfaces.org/). The aim was to gauge developments since 1990, and to highlight the continued relevance and importance of geometry and topology to condensed materials, whether hard or soft, synthetic or biological. We were fortunate to have the company of two of the semi- nal figures in the field, Alan Schoen and Ka ̊re Larsson, whose contributions to minimal surface theory and the role of those surfaces in biological membrane folding and liquid crystalline mesophases, respectively, continue to influence research. This theme issue is focused on active research in material structure, with papers from a cross-section of participants.
Twenty years ago, Elisabeth Dubois-Violette and Brigitte Pansu, both then at the Laboratoire de Physique des Solides at Orsay, organized a seminal meeting in Aussois, ‘Geometry and Interfaces’, which brought together physicists, chemists, biologists and mathemati- cians and resulted in a useful volume that summarized the state of things in 1990 [1]. The Orsay group had an impeccable pedigree in condensed materials research (including, for example, liquid crystal research that led to the award of the Physics Nobel Prize to de Gennes). Their scientific culture recognized the importance of crossing traditional disciplinary borders, and the enrichment of conventional condensed matter physics drawn from studies in other areas, from biology to pure mathematics. That approach now pervades many aspects of contemporary physics research into materials, where it is recognized that biology and materials chemistry offers fertile domains for explora- tion. Another approach to materials research remains however less developed: the exploration of the funda- mental science of biomaterial self-assembly and function using the tools of low-dimensional geometry and topology. Few biologists concern themselves with more complex aspects of geometry, despite the earliest forays by D’Arcy Wentworth Thompson in his seminal book ‘On growth and form’. One notable exception was Yves Bouligand, a biologist whose close links with Orsay and personal interest and knowledge of geometry led to the important recognition of the relevance of the liquid crystalline state to many biological assemblies, such as the cholesterol character of the arrangement of chitin fibres in crab shells (http://people.physics. anu.edu.au/sth110/bouligand_papers.html/). Surely Bouligand is one of the very few biologists who have made significant contributions to the physics of liquid crystals?
In an attempt to redress that imbalance, we orga- nized a successor to the Aussois meeting in October 2011 at Primosˇten, Croatia (http://www.geometry-of- interfaces.org/). The aim was to gauge developments since 1990, and to highlight the continued relevance and importance of geometry and topology to condensed materials, whether hard or soft, synthetic or biological. We were fortunate to have the company of two of the semi- nal figures in the field, Alan Schoen and Ka ̊re Larsson, whose contributions to minimal surface theory and the role of those surfaces in biological membrane folding and liquid crystalline mesophases, respectively, continue to influence research. This theme issue is focused on active research in material structure, with papers from a cross-section of participants.more complex aspects of geometry, despite the earliest forays by D’Arcy Wentworth Thompson in his seminal book ‘On growth and form’ [2]. One notable exception was Yves Bouligand, a biologist whose close links with Orsay and personal interest and knowledge of geometry led to the important recognition of the relevance of the liquid crystalline state to many biological assemblies, such as the cholesterol character of the arrangement of chitin fibres in crab shells (http://people.physics. anu.edu.au/sth110/bouligand_papers.html/). Surely Bouligand is one of the very few biologists who have made significant contributions to the physics of liquid crystals?
In an attempt to redress that imbalance, we orga- nized a successor to the Aussois meeting in October 2011 at Primosˇten, Croatia (http://www.geometry-of- interfaces.org/). The aim was to gauge developments since 1990, and to highlight the continued relevance and importance of geometry and topology to condensed materials, whether hard or soft, synthetic or biological. We were fortunate to have the company of two of the semi- nal figures in the field, Alan Schoen and Ka ̊re Larsson, whose contributions to minimal surface theory and the role of those surfaces in biological membrane folding and liquid crystalline mesophases, respectively, continue to influence research. This theme issue is focused on active research in material structure, with papers from a cross-section of participants.
- Gerd E. Schröder-Turk,
Liliana de Campo, Myfanwy E. Evans, Matthias Saba, Sebastian
Kapfer, Trond Varslot, Karsten Grosse-Brauckmann, Stuart Ramsden, and
Stephen Hyde, “Polycontinuous geometries for inverse lipid phases with more than two aqueous network domains”, Faraday Discuss., 161, 215-247 (2012). doi: 10.1039/C2FD20112G
Abstract:
Inverse bicontinuous cubic phases with two aqueous network
domains separated by a smooth bilayer are firmly established as
equilibrium phases in lipid/water systems. The purpose of this article
is to highlight generalisations of these bi- continuous geometries to
polycontinuous geometries, which could be realised as lipid mesophases
with three or more network-like aqueous domains separated by a branched
bilayer. An analysis of structural homogeneity in terms of bi- layer
width variations reveals that ordered polycontinuous geometries are
likely candidates for lipid mesophase structures, with similar chain
packing charac- teristics to inverse micellar phases (that once were
believed not to exist due to high packing frustration). The average
molecular shape required by global ge- ometry to form these
multi-network phases is quantified by the surfactant shape parameter
v/(al); we find that it adopts values close to those of the known lipid
phases. We specifically analyse the 3etc(187,193) structure of
hexagonal sym- metry P63/mcm with three aqueous domains, the
3dia(24,220) structure of cubic symmetry I43d composed of three
distorted Diamond networks, the cubic chiral 4srs*(24,208) with cubic
symmetry P4232 and the achiral 4srs(5,133) structure of symmetry
P42/nbc, each consisting of four intergrown undistorted copies of the
srs net (the same net as in the QGII Gyroid phase). Structural
homogene- ity is analysed by a medial surface approach assuming that
the head group in- terfaces are constant mean curvature surfaces. To
facilitate future experimen- tal identification, we provide simulated
SAXS scattering patterns that, for the 4srs*(24,208) and 3dia(24,220)
structures, bear remarkable similarity to those of bicontinuous
QGII-Gyroid and QDII-Diamond phases, with comparable lattice parameters
and only a single peak that cannot be indexed to the well-established
structures. While polycontinuous lipid phases have to date not been
reported, the likelihood of their formation is further indicated by the
reported observation of a solid tricontinuous mesoporous silicate
structure termed IBN-9 formed in the presence of surfactants [Han et
al., Nat. Chem., 2009, 1, 123].
- Toen Castle, Myfanwy E. Evans, Stephen T. Hyde, Stuart Ramsden and Vanessa Robins, “Trading spaces: building three-dimensional nets from two-dimensional tilings”, Interface Focus, 2(5), 555-566 (2012). doi: 10.1098/rsfs.2011.0115
Abstract:
We construct some examples of finite and infinite
crystalline three-dimensional nets derived from symmetric reticulations
of homogeneous two-dimensional spaces: elliptic (S^2), Euclidean (E^2)
and hyperbolic (H^2) space. Those reticulations are edges and vertices
of simple spherical, planar and hyperbolic tilings. We show that
various projections of the simplest symmetric tilings of those spaces
into three-dimensional Euclidean space lead to topologically and geo-
metrically complex patterns, including multiple interwoven nets and
tangled nets that are otherwise difficult to generate ab initio in
three dimensions.
Abstract:
- T Castle, M Evans and S T Hyde, “Entanglement of embedded graphs”, Prog. Theor. Phys Supplement No. 191, 235-244, (2011)
Abstract:
We discuss the identification of untangled graph embeddings for finite planar and non-
planar graphs as well as infinite crystallographic nets. Two parallel approaches are discussed:
explicit 3-space embeddings and reticulations of 2-manifolds. 2D and 3D energies are pro-
posed that allow ranking of (un)tangled embedded graphs.
planar graphs as well as infinite crystallographic nets. Two parallel approaches are discussed:
explicit 3-space embeddings and reticulations of 2-manifolds. 2D and 3D energies are pro-
posed that allow ranking of (un)tangled embedded graphs.
- M. Saba, M. Thiel , M.D. Turner, S.T. Hyde, M. Gu, K. Grosse-Brauckmann, D.N. Neshev, K. Mecke, and G.E. Schröder-Turk, “Circular dichroism in biological photonic crystals and cubic chiral nets”, Phys Rev Lett, 106, 103902 (2011).
pdf version
Abstract:
Nature provides impressive examples of chiral photonic crystals, with the notable example of the
cubic srs network structure realized in wing-scales of several butterfly species. By a novel circular
polarization analysis of the band structure of such networks, we demonstrate strong circular dichro-
ism effects: The butterfly srs micro-structure, of cubic I 41 32 symmetry, shows significant circular
dichroism for blue to ultra-violet light, that warrants a search for biological receptors sensitive to
circular polarization. A derived synthetic structure based on four like-handed silicon srs nets ex-
hibits a large circular polarization stop band of width exceeding 30%. These findings offer design
principles for chiral photonic devices.
cubic srs network structure realized in wing-scales of several butterfly species. By a novel circular
polarization analysis of the band structure of such networks, we demonstrate strong circular dichro-
ism effects: The butterfly srs micro-structure, of cubic I 41 32 symmetry, shows significant circular
dichroism for blue to ultra-violet light, that warrants a search for biological receptors sensitive to
circular polarization. A derived synthetic structure based on four like-handed silicon srs nets ex-
hibits a large circular polarization stop band of width exceeding 30%. These findings offer design
principles for chiral photonic devices.
- G.E. Schröder-Turk, S. Wickham, H. Averdunk, F. Brink, J.D. Fitz Gerald, L. Poladian, M.C.J. Large and S.T. Hyde, “The chiral structure of porous chitin within the wing-scales of Callophrys rubi”, J Struct Biol, 174, 290-295 (2011). DOI: 10.1016/j.jsb.2011.01.004
pdf version (1.1 MB)
Abstract:
The structure of the porous
three-dimensional reticulated pattern in the wing scales of the
butterfly C. rubi (the Green Hairstreak)
is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction of a fragment of
this material reveals that the predominantly chitin material is assembled in the wing scale to form a structure whose geometry
bears a remarkable correspondence to the srs net, well-known in solid state chemistry and soft materials science. The porous solid
is bounded to an excellent approximation by a parallel/cmc surface to the Gyroid, a three-periodic minimal surface with cubic
crystallographic symmetry I 41 32, as foreshadowed by Stavenga and Michielson. The scale of the structure is commensurate with
the wavelength of visible light, with an edge of the conventional cubic unit cell of the cmc-Gyroid of approximately 310 nm. The
genesis of this structure is discussed, and we suggest it affords a remarkable example of templating of a chiral material via soft
matter, analogous to the formation of mesoporous silica via surfactant assemblies in solution. In the butterfly, the templating is
achieved by the lipid-protein membranes within the smooth endoplasmic reticulum (while it remains in the chrysalis), that likely
form cubic membranes, folded according to the form of the Gyroid. The subsequent formation of the chiral hard chitin framework
is suggested to be driven by the gradual polymerisation of the chitin precursors, whose inherent chiral assembly in solution (during
growth) promotes the formation of a single enantiomer.
is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction of a fragment of
this material reveals that the predominantly chitin material is assembled in the wing scale to form a structure whose geometry
bears a remarkable correspondence to the srs net, well-known in solid state chemistry and soft materials science. The porous solid
is bounded to an excellent approximation by a parallel/cmc surface to the Gyroid, a three-periodic minimal surface with cubic
crystallographic symmetry I 41 32, as foreshadowed by Stavenga and Michielson. The scale of the structure is commensurate with
the wavelength of visible light, with an edge of the conventional cubic unit cell of the cmc-Gyroid of approximately 310 nm. The
genesis of this structure is discussed, and we suggest it affords a remarkable example of templating of a chiral material via soft
matter, analogous to the formation of mesoporous silica via surfactant assemblies in solution. In the butterfly, the templating is
achieved by the lipid-protein membranes within the smooth endoplasmic reticulum (while it remains in the chrysalis), that likely
form cubic membranes, folded according to the form of the Gyroid. The subsequent formation of the chiral hard chitin framework
is suggested to be driven by the gradual polymerisation of the chitin precursors, whose inherent chiral assembly in solution (during
growth) promotes the formation of a single enantiomer.
- Stephen T. Hyde and Olaf Delgado Friedrichs, “From untangled nets to tangled materials”, Solid State Sci., 13(4), 676-683 (2011). DOI 10.1016/j.solidstatesciences.2010.10.028
pdf version (1.1 MB)
Abstract:
We suggest constructive definitions for the determination of untangled finite graphs and three-periodic
nets, using barycentric embeddings in two and three dimensions. The possibility of deliberately con-
structing tangled graphs and nets is canvassed, and we conclude that tangled patterns offer a novel class
of nano- and meso-structured materials with useful features, including high internal surface area and
volume and chirality.
nets, using barycentric embeddings in two and three dimensions. The possibility of deliberately con-
structing tangled graphs and nets is canvassed, and we conclude that tangled patterns offer a novel class
of nano- and meso-structured materials with useful features, including high internal surface area and
volume and chirality.
- Liliana de Campo, Trond Varslot, Minoo J. Moghaddam, Jacob J. K. Kirkensgaard, Kell Mortensen and Stephen T. Hyde, “A Novel Lyotropic Liquid Crystal Formed by Triphilic Star-Polyphiles: Hydrophilic/Oleophilic/Fluorophilic Rods Arranged in a 12.6.4. tiling”, Phys Chem Chem Phys, 13, 3139-3152 (2010). doi: 10.1039/C0CP01201G
pdf version (2.4 MB)
Abstract:
Triphilic star-polyphiles are short-chain oligomeric molecules with a radial arrangement of
hydrophilic, hydrocarbon and fluorocarbon chains linked to a common centre. They form a number
of liquid crystalline structures when mixed with water. In this contribution we focus on a
hexagonal liquid crystalline mesophase found in star-polyphiles as compared to the corresponding
double chain surfactant to determine whether the hydrocarbon and fluorocarbon chains are in fact
demixed in these star-polyphile systems, or whether both hydrocarbon and fluorocarbon chains are
miscible, leading to a single hydrophobic domain, making the star-polyphile effectively
amphiphilic. We report SANS contrast variation data that is compatible only with the presence of
three distinct immiscible domains within this hexagonal mesophase, confirming that these star-
polyphile liquid crystals are indeed hydrophilic/oleophilic/fluorophilic 3-phase systems.
Quantitative comparison with scattering simulations shows that the experimental data are in very
good agreement with an underlying 2D columnar (12.6.4) tiling. As in a conventional amphiphilic
hexagonal mesophase, the hexagonally packed water channels (dodecagonal prismatic domains)
are embedded in a hydrophobic matrix, but that matrix is split into oleophilic hexagonal prismatic
domains and fluorophilic quadrangular prismatic domains.
hydrophilic, hydrocarbon and fluorocarbon chains linked to a common centre. They form a number
of liquid crystalline structures when mixed with water. In this contribution we focus on a
hexagonal liquid crystalline mesophase found in star-polyphiles as compared to the corresponding
double chain surfactant to determine whether the hydrocarbon and fluorocarbon chains are in fact
demixed in these star-polyphile systems, or whether both hydrocarbon and fluorocarbon chains are
miscible, leading to a single hydrophobic domain, making the star-polyphile effectively
amphiphilic. We report SANS contrast variation data that is compatible only with the presence of
three distinct immiscible domains within this hexagonal mesophase, confirming that these star-
polyphile liquid crystals are indeed hydrophilic/oleophilic/fluorophilic 3-phase systems.
Quantitative comparison with scattering simulations shows that the experimental data are in very
good agreement with an underlying 2D columnar (12.6.4) tiling. As in a conventional amphiphilic
hexagonal mesophase, the hexagonally packed water channels (dodecagonal prismatic domains)
are embedded in a hydrophobic matrix, but that matrix is split into oleophilic hexagonal prismatic
domains and fluorophilic quadrangular prismatic domains.
- Stephen T. Hyde, “Contemporary geometry for the built design?”, Architecture Theory Review, 15(2), 2-10, (2010).
+erratum
Abstract:
I explore the terrain that lies
between architecture and geometry, from the perspective of a structural
scientist with no professional architectural expertise. The divide
between these disciplines perhaps stems from an ancient dichotomy
between the art vs. engineering schools of architecture, fertilised by
the current dogma that art and science can never meet. Architects stand
to gain much from study of the spectacular advances in geometry in
recent decades, such as the growing understanding of cellular patterns
in space, tiles, nets and curved surfaces. Some examples of those
advances are discussed in detail. I conclude that both architecture and
geometry would benefit from a renewed mutual interest.
- T. Castle, Myfanwy E. Evans and S. T. Hyde, “All toroidal embeddings of polyhedral graphs in 3-space are chiral”, New J. Chem., 2009, 33, 2107 - 2113 (2009).
pdf version (1.3 MB)
Abstract:
We investigate the possibility of forming achiral knottings of polyhedral (3-connected) graphs
whose minimal embeddings lie in the genus-one torus. Various analyses to show that all examples
are chiral. This result suggests a simple route to forming chiral molecules via templating on a
toroidal substrate.
whose minimal embeddings lie in the genus-one torus. Various analyses to show that all examples
are chiral. This result suggests a simple route to forming chiral molecules via templating on a
toroidal substrate.
- Stephen T. Hyde, Liliana de Campo and Christophe Oguey, “Tricontinuous mesophases of balanced three-arm ‘star polyphiles’ " , Soft Matter, 5, 2782–2794, (2009). DOI: 10.1039/b822814k
pdf version (11.1 MB)
Abstract:
We construct simple models to compare ordered tricontinuous patterns that are topologically
consistent with the constraints imposed by three-arm star polyphile self-assembly, analogous to steric
packing and elastic bending models used to analyse bicontinuous mesophases in amphiphilic
assemblies. We find a number of competing low-energy ordered structures, composed of threading of
three identical labyrinths, with three-fold infinite branch lines, that are likely to be of comparable
energy for polyphile shapes with moderately splayed arms. These patterns are triply-periodic analogues
of the hexagonal honeycomb, which is most favoured for unsplayed three-arm polyphile architectures.
consistent with the constraints imposed by three-arm star polyphile self-assembly, analogous to steric
packing and elastic bending models used to analyse bicontinuous mesophases in amphiphilic
assemblies. We find a number of competing low-energy ordered structures, composed of threading of
three identical labyrinths, with three-fold infinite branch lines, that are likely to be of comparable
energy for polyphile shapes with moderately splayed arms. These patterns are triply-periodic analogues
of the hexagonal honeycomb, which is most favoured for unsplayed three-arm polyphile architectures.
- Jacob Judas Kain Kirkensgaard and Stephen T. Hyde, “Beyond amphiphiles: Coarse-grained simulations of star-polyphile liquid crystalline assemblies”, Phys. Chem. Chem. Phys., 11, 2016 - 2022, (2009).
pdf version (1.5 MB)
Abstract:
We have simulated the self-assembly of a novel class of three-arm molecules,
ABC star-architecture polyphiles, using coarse-grained bead simulations. A number of
topologically complex liquid crystalline mesostructures arise that can be related to the
better-known bicontinuous mesophases of lyotropic amphiphilic systems. The simulations reveal
3D self-assemblies whose structural variations follow those expected assuming a simple steric
molecular packing model as a function of star polyphile splay and relative volumes of each arm
in the polyphile. The splay of each arm, characterised by the 3D wedge-shape emanating from the
core of each molecule to its exterior induces torsion of the interfaces along the triple lines,
whereas differences in the relative volumes of arms induce curvature of the triple lines. Three
distinct mesostructures are described, characterised by their micro-domain topologies, which are
unknown in simpler amphiphilic systems, but resemble in some respects bicontinuous mesophases.
These three- (or more) arm polyphilic systems offer an interesting extension to the better-known
self-assembly of (two-arm) amphiphiles in solution.
ABC star-architecture polyphiles, using coarse-grained bead simulations. A number of
topologically complex liquid crystalline mesostructures arise that can be related to the
better-known bicontinuous mesophases of lyotropic amphiphilic systems. The simulations reveal
3D self-assemblies whose structural variations follow those expected assuming a simple steric
molecular packing model as a function of star polyphile splay and relative volumes of each arm
in the polyphile. The splay of each arm, characterised by the 3D wedge-shape emanating from the
core of each molecule to its exterior induces torsion of the interfaces along the triple lines,
whereas differences in the relative volumes of arms induce curvature of the triple lines. Three
distinct mesostructures are described, characterised by their micro-domain topologies, which are
unknown in simpler amphiphilic systems, but resemble in some respects bicontinuous mesophases.
These three- (or more) arm polyphilic systems offer an interesting extension to the better-known
self-assembly of (two-arm) amphiphiles in solution.
Abstract:
Precipitation of barium or strontium carbonates in alkaline silica-rich
environments leads to crystalline aggregates that have been named
silica/carbonate biomorphs because they resemble the morphology of
primitive organisms. These aggregates are self-assembled materials of
purely inorganic origin, with an amorphous phase of silica intimately
intertwined with a carbonate nanocrystalline phase. We propose a
mechanism that explains all the morphologies described for biomorphs.
Chemically coupled co-precipitation of carbonate and silica leads to
fibrillation of the growing front and to laminar structures that experience
curling on their growing rim. These curls propagate surf-like along the rim
of the laminae. Observed morphologies with smoothly varying curvatures
can be explained by the combined growth of counter-propagating curls
and growing laminae.
Comments on this paper can be found at:
environments leads to crystalline aggregates that have been named
silica/carbonate biomorphs because they resemble the morphology of
primitive organisms. These aggregates are self-assembled materials of
purely inorganic origin, with an amorphous phase of silica intimately
intertwined with a carbonate nanocrystalline phase. We propose a
mechanism that explains all the morphologies described for biomorphs.
Chemically coupled co-precipitation of carbonate and silica leads to
fibrillation of the growing front and to laminar structures that experience
curling on their growing rim. These curls propagate surf-like along the rim
of the laminae. Observed morphologies with smoothly varying curvatures
can be explained by the combined growth of counter-propagating curls
and growing laminae.
Comments on this paper can be found at:
- “Oscillating chemistry explains complex, self-assembled crystal aggregates”,Physics Today, (cover article) March, pp. 17-18 (2009)
- Matthias Kellermeier, Fabian Glaab, Anna M. Carnerup, Markus
Drechsler, Benjamin Gossler, Stephen T. Hyde, and Werner Kunz,
“Additive-induced
morphological tuning of self-assembled silica-barium
carbonate crystal aggregates” , J Cryst Growth, 311, 2530-2541, doi:10....sgro.2009.02.044, (2009).
Abstract:
Crystallisation of barium carbonate from alkaline silica
solutions results in the
formation of extraordinary micron-scale architectures exhibiting
non-crystallographic curved shapes, such as helical filaments and worm-like braids.
These so-called “silica biomorphs” consist of a textured assembly of uniform
elongated witherite nanocrystallites, which is occasionally sheathed by a skin of
amorphous silica. Although great efforts have been devoted to clarifying the
physical origin of these fascinating materials, to date little is known about the
processes underlying the observed self-organisation. Herein, we describe the effect
of two selected additives, a cationic surfactant and a cationic polymer, on the
morphology of the forming crystal aggregates, and relate changes to experiments
conducted in the absence of additives. Minor amounts of both substances are shown
to exert a significant influence on the growth process, leading to the formation of
predominantly flower-like spherulitic aggregates. The observed effects are discussed
in terms of feasible morphogenesis pathways. Based on the assumption of a
template membrane steering biomorph formation, it is proposed that the two
additives are capable of performing specific bridging functions promoting the
aggregation of colloidal silica which constitutes the membrane. Morphological
changes are tentatively ascribed to varying colloid coordination effecting distinct
membrane curvatures.
formation of extraordinary micron-scale architectures exhibiting
non-crystallographic curved shapes, such as helical filaments and worm-like braids.
These so-called “silica biomorphs” consist of a textured assembly of uniform
elongated witherite nanocrystallites, which is occasionally sheathed by a skin of
amorphous silica. Although great efforts have been devoted to clarifying the
physical origin of these fascinating materials, to date little is known about the
processes underlying the observed self-organisation. Herein, we describe the effect
of two selected additives, a cationic surfactant and a cationic polymer, on the
morphology of the forming crystal aggregates, and relate changes to experiments
conducted in the absence of additives. Minor amounts of both substances are shown
to exert a significant influence on the growth process, leading to the formation of
predominantly flower-like spherulitic aggregates. The observed effects are discussed
in terms of feasible morphogenesis pathways. Based on the assumption of a
template membrane steering biomorph formation, it is proposed that the two
additives are capable of performing specific bridging functions promoting the
aggregation of colloidal silica which constitutes the membrane. Morphological
changes are tentatively ascribed to varying colloid coordination effecting distinct
membrane curvatures.
- S.J. Ramsden, V. Robins and Stephen Hyde, "3D euclidean nets from
2D hyperbolic tilings: Kaleidoscopic examples", Acta
Crystallogr. A,
A65, 81-108 (2009) [Lead Article.]
Abstract:
We present a method for geometric construction of periodic
3D Euclidean nets by
projecting 2D hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMS).
Our techniques extend the combinatorial tiling theory of Dress, Huson, and
Delgado-Friedrichs to enumerate simple reticulations of these TPMS.
We include a taxonomy of all networks arising from kaleidoscopic hyperbolic
tilings with up to two distinct tile types (and dually two vertex types), mapped
to three related TPMS, namely Schwarz's Primitive (P) and Diamond (D) surfaces,
and Schoen's Gyroid (G)
projecting 2D hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMS).
Our techniques extend the combinatorial tiling theory of Dress, Huson, and
Delgado-Friedrichs to enumerate simple reticulations of these TPMS.
We include a taxonomy of all networks arising from kaleidoscopic hyperbolic
tilings with up to two distinct tile types (and dually two vertex types), mapped
to three related TPMS, namely Schwarz's Primitive (P) and Diamond (D) surfaces,
and Schoen's Gyroid (G)
- Stephen T. Hyde, Michael O’Keeffe and Davide M. Proserpio, "A short history of an elusive yet
ubiquitous structure in chemistry, materials and mathematics",
Angew. Chemie Int. Ed., 47, 7996-8000 (2008).
http://dx.doi.org/10.1002/anie.200801519
Abstract:
Herein we describe some
properties and the occurrences of a beautiful geometric
figure that is ubiquitous in chemistry and materials science, however, it is not as
well-known as it should be. We call attention to the need for mathematicians to pay
more attention to the richly structured natural world, and for materials scientists to
learn a little more about mathematics. Our account is informal and eschews any
pretence of mathematical rigor, but does start with some necessary mathematics.
figure that is ubiquitous in chemistry and materials science, however, it is not as
well-known as it should be. We call attention to the need for mathematicians to pay
more attention to the richly structured natural world, and for materials scientists to
learn a little more about mathematics. Our account is informal and eschews any
pretence of mathematical rigor, but does start with some necessary mathematics.
- Toen Castle, Myfanwy E. Evans and S.T Hyde, “Ravels: Knot-free but not free.
Novel entanglements of graphs in 3-space”, New J Chem, 32,
1484-1492 (2008).
doi:10.1039/b719665b
Abstract:
Molecular and extended framework
materials, from proteins to catenanes and metal–organic
frameworks, can assume knotted configurations in their bonding networks (the chemical graph).
Indeed, knot theory and structural chemistry have remained closely allied, due to those
connections. Here we introduce a new class of graph entanglement: ‘‘ravels’’. These ravels—often
chiral—tangle a graph without the presence of knots. Just as knots lie within cycles in the graph,
ravels lie in the vicinity of a vertex. We introduce various species of ravels, including fragile
ravels, composite ravels and shelled ravels. The role of ravels is examined in the context of finite
and infinite graphs—analogous to molecular and extended framework nets—related to the
diamond net.
frameworks, can assume knotted configurations in their bonding networks (the chemical graph).
Indeed, knot theory and structural chemistry have remained closely allied, due to those
connections. Here we introduce a new class of graph entanglement: ‘‘ravels’’. These ravels—often
chiral—tangle a graph without the presence of knots. Just as knots lie within cycles in the graph,
ravels lie in the vicinity of a vertex. We introduce various species of ravels, including fragile
ravels, composite ravels and shelled ravels. The role of ravels is examined in the context of finite
and infinite graphs—analogous to molecular and extended framework nets—related to the
diamond net.
-
Zakaria A. Almsherqi, Stephen T. Hyde, Malarmathy Ramachandran, Yuru Deng, “Cubic membranes: a structure-based design for DNA uptake”, J. R. Soc. Interface, 5, 1023-1029 (2008).
doi:10.1098/rsif.2007.1351
Abstract:
Cubic membranes are soft
three-dimensional crystals found within cell organelles in a variety
of living systems, despite the aphorism of Fedorov: ‘crystallization is death’. They consist of
multi-bilayer lipid–protein stacks, folded onto anticlastic surfaces that resemble triply
periodic minimal surfaces, forming highly swollen crystalline sponges. Although cubic
membranes have been observed in numerous cell types and under different pathophysiolo-
gical conditions, knowledge about the formation and potential function(s) of non-lamellar,
cubic structures in biological systems is scarce. We report that mitochondria with this cubic
membrane organization isolated from starved amoeba Chaos carolinense interact sufficiently
with short segments of phosphorothioate oligonucleotides ( PS-ODNs) to give significant
ODNs uptake. ODNs condensed within the convoluted channels of cubic membrane by an
unknown passive targeting mechanism. Moreover, the interaction between ODNs and cubic
membrane is sufficient to retard electrophoretic mobility of the ODN component in the gel
matrix. These ODN–cubic membrane complexes are readily internalized within the
cytoplasm of cultured mammalian cells. Transmission electron microscopic analysis confirms
ODNs uptake by cubic membranes and internalization of ODN–cubic membrane complexes
into the culture cells. Cubic membranes thus may offer a new, potentially benign medium for
gene transfection.
of living systems, despite the aphorism of Fedorov: ‘crystallization is death’. They consist of
multi-bilayer lipid–protein stacks, folded onto anticlastic surfaces that resemble triply
periodic minimal surfaces, forming highly swollen crystalline sponges. Although cubic
membranes have been observed in numerous cell types and under different pathophysiolo-
gical conditions, knowledge about the formation and potential function(s) of non-lamellar,
cubic structures in biological systems is scarce. We report that mitochondria with this cubic
membrane organization isolated from starved amoeba Chaos carolinense interact sufficiently
with short segments of phosphorothioate oligonucleotides ( PS-ODNs) to give significant
ODNs uptake. ODNs condensed within the convoluted channels of cubic membrane by an
unknown passive targeting mechanism. Moreover, the interaction between ODNs and cubic
membrane is sufficient to retard electrophoretic mobility of the ODN component in the gel
matrix. These ODN–cubic membrane complexes are readily internalized within the
cytoplasm of cultured mammalian cells. Transmission electron microscopic analysis confirms
ODNs uptake by cubic membranes and internalization of ODN–cubic membrane complexes
into the culture cells. Cubic membranes thus may offer a new, potentially benign medium for
gene transfection.
- Alina E. Voinescu, Matthias Kellermeier, Anna M. Carnerup, Ann-Kristin Larsson, Didier Touraud, Werner Kunz, Lorenz Kienle, Arno Pfitzner and Stephen T. Hyde and, “Inorganic Self-organized silica aragonite biomorphic composites”, Cryst Growth Design, 8, 1515-1521 (2008).
Abstract:
The precipitation of calcium
carbonate in alkaline silica solutions results in the formation of
complex curvilinear
forms if aragonite formation is encouraged by growth at an elevated temperature (80 °C). The resulting coralline self-assembled
silica-calcium carbonate particles are “biomorphs”, bearing a striking resemblance to natural coral forms. These materials, comprised
of calcium carbonate nanocrystals and an amorphous silica matrix, have a complex ultrastructure, made of clusters of gathered
sheets of variable curvatures formed by successive curling. The nanocrystals within these “ruled surfaces” are thin, elongated,
densely packed needles of aragonite. These clusters are outgrowths from central saddlelike cores that resemble developable petaloid
surfaces. The size, shape, crystallography, and chemical composition of the resulting biomorphs were examined by optical microscopy,
field emission scanning electron microscopy (FE-SEM), powder X-ray diffractometry (XRD), Fourier transform infrared spectroscopy
(FT-IR), transmission electron microscopy (TEM and HRTEM), and energy dispersive X-ray analysis (EDX).
forms if aragonite formation is encouraged by growth at an elevated temperature (80 °C). The resulting coralline self-assembled
silica-calcium carbonate particles are “biomorphs”, bearing a striking resemblance to natural coral forms. These materials, comprised
of calcium carbonate nanocrystals and an amorphous silica matrix, have a complex ultrastructure, made of clusters of gathered
sheets of variable curvatures formed by successive curling. The nanocrystals within these “ruled surfaces” are thin, elongated,
densely packed needles of aragonite. These clusters are outgrowths from central saddlelike cores that resemble developable petaloid
surfaces. The size, shape, crystallography, and chemical composition of the resulting biomorphs were examined by optical microscopy,
field emission scanning electron microscopy (FE-SEM), powder X-ray diffractometry (XRD), Fourier transform infrared spectroscopy
(FT-IR), transmission electron microscopy (TEM and HRTEM), and energy dispersive X-ray analysis (EDX).
- G.E. Schröder-Turk, A. Fogden and S.T.Hyde, ” Local v/a variations as a measure of structural packing frustration in bicontinuous mesophases, and geometric arguments for an alternating Im3m (I-WP) phase in block-copolymers with polydispersity”, Eur. Phys. J., B59, 115-126 (2007).
Abstract:
This article explores global
geometric features of bicontinuous space-partitions and their rele-
vance to self-assembly of block-copolymers. Using a robust definition of ‘local channel radius’, based on the
concept of a medial surface [1], we relate radius variations of the space-partition to polymolecular chain
stretching in bicontinuous diblock- and terblock copolymer assemblies. We associate local surface patches
with corresponding cellular volume elements, to define local volume-to-surface ratios. The distribution of
these v/a ratios and of the channel radii are used to quantify the degree of packing frustration of molecular
chains as a function of the specific bicontinuous geometry, modelled by triply-periodic minimal surfaces
and related parallel interfaces. The Gyroid geometry emerges as the most nearly homogeneous bicontinu-
ous form, with the smallest heterogeneity of channel radii, compared to the cubic Primitive and Diamond
surfaces. We clarify a geometric feature of the Gyroid geometry: the three-coordinated nodes of the graph
are not the widest points of the labyrinths; the widest points are at the midpoints of the edges. We also ex-
plore a more complex cubic triply-periodic surface, the I-WP surface, containing two geometrically distinct
channel subdomains. One of the two channel systems is nearly as homogeneous in local channel diameters
as the Gyroid, the other is more heterogeneous than the Primitive surface. Its hybrid nature suggests the
possibility of an “alternating I-WP” phase in polydisperse linear ABC-terpolymer blends, with monodis-
perse molecular weight distributions (MWD) in the A and B blocks and a more polydisperse C block.
vance to self-assembly of block-copolymers. Using a robust definition of ‘local channel radius’, based on the
concept of a medial surface [1], we relate radius variations of the space-partition to polymolecular chain
stretching in bicontinuous diblock- and terblock copolymer assemblies. We associate local surface patches
with corresponding cellular volume elements, to define local volume-to-surface ratios. The distribution of
these v/a ratios and of the channel radii are used to quantify the degree of packing frustration of molecular
chains as a function of the specific bicontinuous geometry, modelled by triply-periodic minimal surfaces
and related parallel interfaces. The Gyroid geometry emerges as the most nearly homogeneous bicontinu-
ous form, with the smallest heterogeneity of channel radii, compared to the cubic Primitive and Diamond
surfaces. We clarify a geometric feature of the Gyroid geometry: the three-coordinated nodes of the graph
are not the widest points of the labyrinths; the widest points are at the midpoints of the edges. We also ex-
plore a more complex cubic triply-periodic surface, the I-WP surface, containing two geometrically distinct
channel subdomains. One of the two channel systems is nearly as homogeneous in local channel diameters
as the Gyroid, the other is more heterogeneous than the Primitive surface. Its hybrid nature suggests the
possibility of an “alternating I-WP” phase in polydisperse linear ABC-terpolymer blends, with monodis-
perse molecular weight distributions (MWD) in the A and B blocks and a more polydisperse C block.
- Alina E. Voinescu, Matthias Kellermeier, Anna M. Carnerup,
Ann-Kristin Larsson, Didier Touraud, Stephen T. Hyde and W. Kunz,
“Co-precipitation of silica
and alkaline-earth carbonates using TEOS as silica source”, J
Cryst Growth, 306 152-158 (2007).
Abstract:
We explore the use of
tetraethoxysilane (TEOS) as a silica source for the formation of
carbonate-silica composite materials known as ‘biomorphs’. The basic hydrolysis of
TEOS furnishes silica in a controllable fashion, allowing a significantly higher reproducibility
of the obtained silica–barium and silica–strontium carbonate co-precipitates compared to
commercial water glass silica used so far. We further discuss the influence of ethanol used
as a co-solvent on the morphologies of biomorphs, which are examined by optical microscopy,
field emission scanning electron microscopy (FESEM) and energy dispersive X-ray analysis (EDX).
carbonate-silica composite materials known as ‘biomorphs’. The basic hydrolysis of
TEOS furnishes silica in a controllable fashion, allowing a significantly higher reproducibility
of the obtained silica–barium and silica–strontium carbonate co-precipitates compared to
commercial water glass silica used so far. We further discuss the influence of ethanol used
as a co-solvent on the morphologies of biomorphs, which are examined by optical microscopy,
field emission scanning electron microscopy (FESEM) and energy dispersive X-ray analysis (EDX).
- S.T. Hyde and G.E. Schröder-Turk, “Tangled (up in) Cubes”, Acta
Cryst A63 186-197 (2007).
Abstract:
The ‘simplest’ entanglements of
the graph of edges of the cube are enumerated,
forming two-cell {6, 3} (hexagonal mesh) complexes on the genus-one two-
dimensional torus. Five chiral pairs of knotted graphs are found. The examples
contain non-trivial knotted and/or linked subgraphs [(2, 2), (2, 4) torus links and
(3, 2), (4, 3) torus knots].
forming two-cell {6, 3} (hexagonal mesh) complexes on the genus-one two-
dimensional torus. Five chiral pairs of knotted graphs are found. The examples
contain non-trivial knotted and/or linked subgraphs [(2, 2), (2, 4) torus links and
(3, 2), (4, 3) torus knots].
