Periodic Nodal Surface Research Articles

Inverse Designing Surface Curvatures by Deep Learning

Smooth and curved microstructural topologies found in nature—from soap films to trabecular bone—have inspired several mimetic design spaces for architected metamaterials and bio‐scaffolds. However, the design approaches so far are ad hoc, raising the challenge: how to systematically and efficiently inverse design such artificial microstructures with targeted topological features? Herein, surface curvature is explored as a design modality and a deep learning framework is presented to produce topologies with as‐desired curvature profiles. The inverse design framework can generalize to diverse topological features such as tubular, membranous, and particulate features. Moreover, successful generalization beyond both the design and data space is demonstrated by inverse designing topologies that mimic the curvature profile of trabecular bone, spinodoid topologies, and periodic nodal surfaces for application in bio‐scaffolds and implants. Lastly, curvature and mechanics are bridged by showing how topological curvature can be designed to promote mechanically beneficial stretching‐dominated deformation over bending‐dominated deformation.

Heat transfer and pressure drop characteristics of heat exchangers based on triply periodic minimal and periodic nodal surfaces

The popularity of additive manufacturing has increased interest in the use of triply periodic minimal surfaces (TPMS) in engineering applications due to their potential for superior mechanical, heat and mass transfer properties. Periodic nodal surfaces (PNS) are a class of periodic continuous surfaces that also divide the space into non-intersecting, smooth and continuous domains like TPMS and can potentially have superior mechanical, heat and mass transfer properties. To evaluate the potential for superior performance, in this manuscript we characterize the flow and heat transfer properties of seven TPMS and PNS based structures by numerically computing the friction factors and Nusselt numbers in the laminar flow regime. This is the first study to evaluate the use of PNS as heat exchangers. In addition, it adds to the limited quantitative data available on the flow and thermal properties of TPMS. The friction factors associated with most of the TPMS and PNS based structures in this study are about an order of magnitude higher than that for laminar flow in tubes. These structures also had higher Nusselt numbers compared to tubes, with the enhancement increasing with increase in Reynolds number. Among the TPMS and PNS studied, Schwarz-D had the best heat transfer performance while Schwarz-P was the poorest performing structure. Basic heat exchanger design calculations showed that to remove the same amount of heat and operate under the same pressure conditions, the Schwarz-D based heat exchanger was 3–10 times smaller than a tubular heat exchanger. This shows how TPMS or PNS can be used to design heat exchangers with superior performance, especially in applications where space and weight are at a premium. • First demonstration of the use of non-TPMS Periodic Nodal Surfaces as heat exchanger. • Under laminar flow most TPMS/PNS had friction factors 3–10 times higher than pipes. • Under laminar flow most TPMS/PNS had Nusselt numbers 1–15 times higher than pipes. • TPMS/PNS heat exchangers can be significantly smaller than tubular heat exchangers.

On the continuous transformations between genus-3 minimal surfaces approximants

AbstractA continuous, genus-preserving transformation relating four minimal surfaces –P,G,DandH– was suggested within a common trigonal group-subgroup pair (P-3c1–P321). Periodic nodal surfaces (PNS) were used as approximants of minimal surfaces to illustrate the transition path. ThemeanandGausscurvatures of PNS were calculated in order to quantify their similarity to minimal surfaces. Theinhomogeneityof the Gauss curvature distributions on PNS was used to producetilingsdirectly on the surfaces. Such tilings on theP,GandDsurfaces can be easily related to three-dimensional nets important in structural chemistry (rhr, analcime, sodalite frameworks, respectively). A tiling on theHsurface gave rise to a completely new 3,4-coordinated net.

Periodic nodal surfaces and exponential scale

Abstract Both, Periodic Nodal Surfaces (PNS) and exponential scale surfaces (ES), are suitable tools to investigate and describe the organisation of crystal structures. Therefore the mathematical relationship between ES and PNS is investigated. It is shown that the function cos (x) within the range 0 ≤ x ≤ 2π can be developed into a series in terms of orthogonal functions Φ n (x) = Σ k = 1 n c n k exp (cos ((k–1)x)). An orthonormal set of functions Φ n (x) has been evaluated using Gram-Schmidt orthonormalization. Series representation of cos (x) and sin (x) in terms of exp (sin ((k–1)x)) is impossible.

Hans Georg von Schnering 1931–2010

Hans Georg von Schnering 1931–2010

A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint

In this paper, we present an accurate and efficient algorithm to generate constant mean curvature surfaces with volume constraint using a phase-field model. We implement our proposed algorithm using an unconditionally gradient stable nonlinear splitting scheme. Starting from the periodic nodal surface approximation to minimal surfaces, we can generate various constant mean curvature surfaces with given volume fractions. We generate and study the Schwarz primitive (P), Schwarz diamond (D), and Schoen gyroid (G) surfaces with various volume fractions. This technique for generating constant mean curvature surfaces can be used to design biomedical scaffolds with optimal mechanical and biomorphic properties.

Deconstructing Sodalite with Hyperbolic Tilings: A PNS Approach

AbstractHyperbolic tilings on periodic nodal surfaces (PNS) are used to construct scaffoldings of interpenetrating nets, which are employed to deconstruct sodalite into interpenetrating (10,3)‐a Wells nets and to describe MOF frameworks. This is intended as an illustration of a two‐dimensional, non‐Euclidean perspective on the understanding and engineering of structured matter.

Topological Analysis of Mesoporous Solids and Their Ordered Pore Structures by Periodic Nodal Surfaces, PNS

It is shown that Periodic Nodal Surfaces, PNS, give a general and easy to use tool to analyze and characterize mesoporous solids. The comparison of PNS ( Symmetry) with different MCM-48 micrographs indicates two different possible structure types, both closely related to the structure of analcime, and the possibility of a polymorphism for MCM-48.

Hyperbolic Cation Diffusion Paths in α‐RbAg4I5 Type Superionic Conductors

AbstractThe positions of the mobile cations in the superionic conductors of the cubic α‐RbAg4I5 type follow a 3D infinite hyperbolic layer, which is moulded by the symmetry governed Periodic Nodal Surface.

Crystal Structure and Chemical Bonding of Mg3Ru2

Mg 3Ru 2 was prepared by a reaction between the elements in the ideal ratio in a sealed tantalum ampule. Its beta-manganese type crystal structure was refined on the basis of the single-crystal data: space group P4 132, a = 693.52(6) pm, wR2 = 0.024, 168 F (2) values, and 10 parameters. The magnesium (CN = 14) and ruthenium (CN = 12) atoms are completely ordered on the 12d and 8c sites of the crystal structure of beta-manganese. Both environments can be considered as Frank-Kasper related polyhedra. A periodic nodal surface P4 132(110) pi (1) P4 132 separates the magnesium and ruthenium positions in two different labyrinths, suggesting different chemical interactions within different parts of the structural motif. Analysis of the chemical bonding with the electron localizability indicator (ELI-D) reveals covalently interacting three-bonded ruthenium atoms, forming a 3D network. The network interacts with the magnesium substructure by multicenter bonds.

Gerüstverbindungen mit beweglichen LaIII‐Kationen: Synthesen, Kristallstrukturen und Strukturdynamik der Lanthan(III)‐Eisen(II)‐Sulfid‐Halogenide La53Fe12S90X3 (X = Cl, Br, I)

AbstractFramework Compounds with Mobile LaIII Cations: Syntheses, Crystal Structures and Structural Dynamics of the Lanthanum(III) Iron(II) Sulfide Halides La53Fe12S90X3 (X = Cl, Br, I)Black crystals of La53Fe12S90X3 (X = Cl, Br, I) were synthesized from La2S3 and FeS in a reactive LaX3 flux at 1320 K. The structures were determined by single‐crystal X‐ray diffraction. The compounds are isostructural, crystallizing in the rhombohedral space group $R{\bar 3}m$ with Z = 1 (La53Fe12S90Cl3: a = 14.0154(7), c = 21.888(1) Å, V = 3723.5(3) Å3; La53Fe12S90Br3: a = 14.0048(9), c = 22.040(2) Å, V = 3743.6(4) Å3; La53Fe12S90I3: a = 13.9805(8), c = 22.108(2) Å, V = 3742.2(4) Å3). The structure adopted is a stuffed variant of the La52Fe12S90 structure type. [FeII2S9] dimers of face‐sharing octahedra are linked by face‐ and vertex‐sharing bi‐ or tri‐capped [LaIIIS6+n] trigonal prisms, forming a three‐dimensional framework containing cuboctahedral cavities of two sizes. The larger cavities, which remain empty in the structure of La52Fe12S90, are filled by halide ions in La53Fe12S90X3. The smaller cavities accommodate numerous sites for disordered lanthanum cations, modelling a network of diffusion pathways through the structure. An analogous picture is obtained from the calculation of the periodic nodal surface (PNS): The PNS separates a labyrinth containing the framework atoms from a labyrinth containing the mobile lanthanum cations. Molecular dynamic simulations confirm a strong coupling between the motions of the mobile lanthanum ions and the neighbouring sulfide ions.

The Intrinsic Shape of the cubic cP124 Clathrate Structure

AbstractIt is shown how the Periodic Nodal Surface PNS D*&Y** separates perfectly the two partial structures of the cubic cP124 type. Condensed metal‐centered [M@E20] pentagonal dodecahedra of covalently bonded E atoms form a 3‐connected net. This partial structure is embedded in a zeolite‐type labyrinth containing chains of the remaining metal atoms. The shape of this ensemble represents the principle of the intrinsic curvature of solids. The relations of the structure to the β‐manganese structure are discussed in detail. Furthermore, some details concerning the electron balance in the covalent E100 net are reported.

CuAl 2 revisited: Composition, crystal structure, chemical bonding, compressibility and Raman spectroscopy

The structure of CuAl 2 is usually described as a framework of base condensed tetragonal antiprisms [CuAl 8/4]. The appropriate symmetry governed periodic nodal surface (PNS) divides the space of the structure into two labyrinths. All atoms are located in one labyrinth, whereas the second labyrinth seems to be ‘empty’. The bonding of the CuAl 2 structure was analyzed by the electron localization function (ELF), crystal orbital Hamiltonian population (COHP) analysis and Raman spectroscopy. From the ELF representation it is seen, that the ‘empty’ labyrinth is in fact the place of important covalent interactions. ELF, COHP in combination with high-pressure X-ray diffraction and Raman spectroscopy show that the CuAl 2 structure is described best as a network built of interpenetrating graphite-like nets of three-bonded aluminum atoms with the copper atoms inside the tetragonal-antiprismatic cavities.

Structure and Bonding of the Mixed‐Valent Platinum Trihalides, PtCl3 and PtBr3

AbstractThe isotypical crystal structures of the mixed valent trihalides PtCl3 and PtBr3 were redetermined by single crystal methods (space group R3¯; trigonal setting; PtCl3: a = 21.213Å, c = 8.600Å, c/a = 0.4054; Z = 36; 1719 hkl; R = 0.035; PtBr3: a = 22.318Å, c = 9.034Å; c/a = 0.4048; Z = 36; 1606 hkl; R = 0.027). A cubic closest packing of X— anions forms the basis of an optimized arrangement of cuboctahedrally [Pt6X12] cluster molecules with PtII and enantiomers of helical chains of edge‐condensed [PtX2X4/2] octahedra with PtIV in cis‐Δ‐ and cis‐Λ‐configuration, respectively. The bond lengths vary with the function of the X— ligands (d¯(PtII—X) = 2.315 and 2.445Å; d¯(PtII—PtII) = 3.336 and 3.492Å; d(PtIV—X) = 2.286 — 2.417Å and 2.437 — 2.563Å). The PtII atoms are shifted outwards the X12 cuboctahedra by 0.045Å and 0.024Å, respectively. The symmetry governed Periodic Nodal Surface, PNS, perfectly separates the regions of different valencies. Quantum chemical calculations exclude the possible additional interactions between PtII and one of the exo‐ligands of PtIV.

Li3[ScN2]: The First Nitridoscandate(III)—Tetrahedral Sc Coordination and Unusual MX2 Framework.

AbstractFor Abstract see ChemInform Abstract in Full Text.

Cover Picture: Li3[ScN2]: The First Nitridoscandate(III)—Tetrahedral Sc Coordination and Unusual MX2 Framework (Chem. Eur. J. 17/2003)

The cover picture shows a section of the crystal structure of the first nitridoscandate Li3[ScN2], in which for the first time Sc is observed in tetrahedral coordination. A the periodic nodal surface (PNS: grey mesh) separates Li+ atoms from the [ScN] network. Two isosurfaces of the electron localization function (ELF) at η=0.85 (light yellow) and η=0.74 (transparent orange) show the structuring of the outer-core localization domain of Sc and the valence localization domain of N, respectively. For more details, see the paper by R. Niewa et al. on p. 4255 ff.

Li3[ScN2]: the first nitridoscandate(III)-tetrahedral Sc coordination and unusual MX2 framework.

Li(3)[ScN(2)] was prepared from Li(3)N with Sc or ScN in a nitrogen atmosphere at 1020 K as a light yellow powder with an optical band gap of about 2.9 eV. The crystal structure was refined based on X-ray and neutron powder diffraction data (Ia$\bar 3$, Z=16, X-ray diffraction: R(profile)=0.078, R(Bragg)=0.070; Neutron diffraction: R(profile)=0.077, R(Bragg)=0.074; Rietfeld: a=1003.940(8) pm, Guinier: a=1004.50(3) pm). Li(3)[ScN(2)] is an isotype of Li(3)[AlN(2)] and Li(3)[GaN(2)] and crystallizes in an ordered superstructure of the Li(2)O structure type, leading to a three-dimensional framework of all-vertex-sharing tetrahedra 3[infinity[ScN[4/2][3-]]. Li is displaced from the center of a tetrahedron of N atoms in the direction of one trigonal face. Li(3)[ScN(2)] decomposes above 1050 K to form ScN and Li(3)N. Calculations of the periodic nodal surface (PNS) and of the electron localization function (ELF) support the picture of a covalent Sc-N network separated from isolated Li cations, whereby scandium d orbitals are involved in the chemical bonding.

Chemical, thermal, and theoretical analysis of the α—β phase transition of Ba3Tt4 compounds

Abstract The α to β phase transition of Ba3Tt4 (Tt = Si, Ge; Ba3Ge4 and Ba3Si4 structure type, respectively) has been investigated experimentally by differential thermal analysis (transition temperature 628 K) and variation of the composition Ba3Si x Ge4– x which shows that the transition is clearly dependent on the compositional parameter (0.4 ≤ x ≤ 0.5). The five new Zintl compounds Ba3Si1.89Ge2.11, Ba3Si1.05Ge2.95, Ba3Si0.57Ge3.43 (Ba3Si4 structure type) and Ba3Si0.31Ge3.69, Ba3Si0.16Ge3.84 (Ba3Ge4 structure type) were synthesized and structurally characterized. We have also developed a topological model for the description of the phase transition which is based on the Periodic Nodal Surface (PNS) approach.

The application of structure envelopes in structure determination from powder diffraction data

A structure envelope is a special type of periodic nodal surface that separates regions of high electron density from those of low electron density. Once such a surface has been generated, it can be used in combination with direct-space methods to facilitate structure solution from powder data. To generate an informative structure envelope, the phases of the structure factors of a few strong low-order reflections must be determined; an algorithm has been developed for this purpose. The programSayPermcombines (a) the use of error-correcting codes (e.c.c.'s) to sample phase space efficiently, (b) a pseudo-atom approximation of structure fragments to simulate atomic resolution atca2.5 Å, and (c) phase extension and phase set ranking using the Sayre equation. The effect of using a structure envelope in structure solution was first tested in combination with a subroutine for finding zeolite topologies in the programFOCUS. Then extension to molecular structures in combination with a simulated-annealing program was explored. This resulted in the development of the programSafeand the subsequent determination of the structure of a tri-β-peptide (C32N3O6H53) with 17 variable torsion angles.

On Tilings and Patterns on Hyperbolic Surfaces and Their Relation to Structural Chemistry

Hyperbolic Periodic Nodal Surfaces (PNSs) have been investigated with respect to surface modulations. The resulting patterns and tilings are beautiful examples of texture information on hyperbolic objects. The construction principle, based on short Fourier series, allows for a strict symmetry control of the generated pattern by choosing appropriate sets of structure factors for information coding. Furthermore, a tailor-made design of complexity is achieved based on a proper choice of the structure factor moduli. In some cases, the resulting patterns are instantaneously transferred into hyperbolic framework information just by our visual perception. In most cases, such correlations can be systematically worked out by utilizing crystallographic and chemical expertise, as shown in this contribution. The presented approach is much simpler than most attempts to generate three-dimensional frameworks because it is subject to boundary conditions like pseudo-two-dimensional topology, given base topology, and symmetry control.