On saturated bi-layered disk shaped tetrahedral packings Chemical Zeolites Combinatorial . . . Brigitte Servatius — WPI Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Simulating Large-Scale Morphogenesis in Planar Motions Vertex transitive . . . Tissues Home Page DMS2012330 (Wu PI). $200,000, 06/15/2020-05/30/2023. Title Page This project aims to improve tools for modeling a wide range of living tissues that are relatively planar and have been exten- ◭◭ ◮◮ sively studied experimentally. ◭ ◮ Page 2 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Curcumin nanodisks: formulation and characteriza- Holes in Zeolites Motions tion Vertex transitive . . . Home Page Ghosh, M., Singh, A. T., Xu, W., Sulchek, T., Gordon, L. I., and Ryan, R. O. (2011) Title Page Nanomedicine: nanotechnology, biology, and medicine, 7(2), ◭◭ ◮◮ 162167. https://doi.org/10.1016/j.nano.2010.08.002 ◭ ◮ Page 3 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites A process for synthesizing bilayer zeolite mem- Combinatorial . . . Realization branes Finite 2d Zeolites Disk shaped zeolite From the abstract [6] Holes in Zeolites Motions Vertex transitive . . . A silicalite/mordenite bilayered self-supporting membrane with disc-shape was synthesized from a layered silicate, kanemite by Home Page two steps using solid-state transformation. Title Page The mechanical strength (compression strength) of the mem- ◭◭ ◮◮ brane was greater than 10 kg cm 2 . Both sides of the membrane were much different in the morphology and SiO 2 /Al 2 O 3 ratio. ◭ ◮ One side (silicalite side) consisted of the intergrowth of prism- Page 4 of 49 like crystals (ca. 12 µm ), while the other side (mordenite side) was composed of scale-like crystals (ca. > 1 µm ). Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites Combinatorial . . . Realization Gas separation with zeolite membranes Finite 2d Zeolites Disk shaped zeolite In [9] it is described how Zeolite membranes can be used to sep- Holes in Zeolites Motions arate gases. Membrane technology constitutes an increasingly Vertex transitive . . . important, convenient, and versatile way of separating gas mix- Home Page tures. Zeolite membranes are known to have high permeabilities in gas separations. Due to the well-defined pore structures, ze- Title Page olite membranes can also offer high selectivities. In addition, ◭◭ ◮◮ zeolite-based membranes have high chemical, mechanical, and thermal stability, i.e. can potentially be used at both very high ◭ ◮ and very low temperatures, offering a great advantage over poly- Page 5 of 49 meric membranes. Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Mildred Dresselhaus (1930-2017), the queen of carbon science. Her research has been instrumental in the development of the Chemical Zeolites Combinatorial . . . nanotechnology field. Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 49 Go Back Mildred S. Dresselhaus holding a model of a carbon nanotube. Full Screen Credit: Ed Quinn Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites Combinatorial . . . Realization 1. Chemical Zeolites Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites • crystalline solid Motions Vertex transitive . . . • units: Si + 4O Home Page O Title Page Si O ◭◭ ◮◮ O O ◭ ◮ Page 7 of 49 • two covalent bonds per oxygen Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ • naturally occurring Page 8 of 49 • synthesized Go Back • theoretical Full Screen Used as microfilters. Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
2. Combinatorial Zeolites Chemical Zeolites Combinatorial . . . Realization Combinatorial d -Dimensional Zeolite Finite 2d Zeolites Disk shaped zeolite • A connected complex of corner sharing d -dimensional sim- Holes in Zeolites Motions plices Vertex transitive . . . • At each corner there are exactly two distinct simplices Home Page • Two corner sharing simplices intersect in exactly one vertex. Title Page ◭◭ ◮◮ body-pin graph ◭ ◮ Vertices: simplices (silicon) Page 9 of 49 Edges: bonds (oxygen) Go Back There is a one-to-one correspondence between combinatorial d -dimensional zeolites and d -regular body-pin graphs. Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Chemical Zeolites Combinatorial . . . Graph of a Combinatorial Zeolite Realization Finite 2d Zeolites is obtained by replacing each d -dimensional simplex with K d +1 . Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . The graph of the zeolite is the line graph of the Body-Pin graph. Home Page Title Page ◭◭ ◮◮ Whitney ◭ ◮ [8](1932) proved that connected graphs X on at least 5 vertices are strongly reconstructible from their line graphs L ( X ). Page 10 of 49 Moreover, Aut ( X ) ∼ = Aut ( L ( X )). Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
3. Realization Chemical Zeolites A realization of a d -dimensional zeolite Combinatorial . . . Realization Finite 2d Zeolites A placement (embedding) of the vertices of the d -dimensional Disk shaped zeolite complex in R d . Holes in Zeolites Motions Equivalently a placement (embedding) of the vertices of the line Vertex transitive . . . graph of the body-pin graph. Home Page Title Page unit-distance realization ◭◭ ◮◮ A realization where all edges join vertices distance 1 apart in ◭ ◮ R d . Page 11 of 49 non-interpenetrating realization Go Back A realization where simplices are disjoint except at joined ver- Full Screen tices. Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
The Layer Construction Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
The Layer Construction Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
The Layer Construction Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
Finite 2d Zeolites Chemical Zeolites Combinatorial . . . Realization Smallest 2d zeolite is the line graph of K 4 : The graph of the Finite 2d Zeolites Disk shaped zeolite octahedron with four (edge disjoint) faces. Holes in Zeolites For body-pin graphs on more than 4 vertices, the zeolite can be Motions Vertex transitive . . . recovered uniquely from the line-graph. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 15 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
4. Finite 2d Zeolites Chemical Zeolites Combinatorial . . . Body pin graph: K 3 , 3 . Since the body pin graph is not planar, Realization Finite 2d Zeolites the resulting zeolite cannot be planar. Its underlying graph Disk shaped zeolite is generically globally rigid. However, it has a unit distance Holes in Zeolites Motions realization in the plane which is a mechanism. Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 16 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
The typical situation: Not unit distance realizable. Chemical Zeolites Combinatorial . . . Realization Finite 2d Zeolites Disk shaped zeolite Holes in Zeolites Motions Vertex transitive . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 17 of 49 Go Back Full Screen Close Quit • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit
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