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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

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On saturated bi-layered disk shaped tetrahedral packings

Brigitte Servatius — WPI

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Simulating Large-Scale Morphogenesis in Planar Tissues DMS2012330 (Wu PI). $200,000, 06/15/2020-05/30/2023. This project aims to improve tools for modeling a wide range

f living tissues that are relatively planar and have been exten-

sively studied experimentally.

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Curcumin nanodisks: formulation and characteriza- tion Ghosh, M., Singh, A. T., Xu, W., Sulchek, T., Gordon, L. I., and Ryan, R. O. (2011) Nanomedicine: nanotechnology, biology, and medicine, 7(2), 162167. https://doi.org/10.1016/j.nano.2010.08.002

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A process for synthesizing bilayer zeolite mem- branes From the abstract [6] A silicalite/mordenite bilayered self-supporting membrane with disc-shape was synthesized from a layered silicate, kanemite by two steps using solid-state transformation. The mechanical strength (compression strength) of the mem- brane was greater than 10 kg

cm2.

Both sides of the membrane were much different in the morphology and SiO2/Al2O3 ratio. One side (silicalite side) consisted of the intergrowth of prism- like crystals (ca. 12 µm), while the other side (mordenite side) was composed of scale-like crystals (ca. > 1µm).

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Gas separation with zeolite membranes In [9] it is described how Zeolite membranes can be used to sep- arate gases. Membrane technology constitutes an increasingly important, convenient, and versatile way of separating gas mix-

tures. Zeolite membranes are known to have high permeabilities

in gas separations. Due to the well-defined pore structures, ze-

lite 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- meric membranes.

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Mildred Dresselhaus (1930-2017), the queen of carbon science. Her research has been instrumental in the development of the nanotechnology field. Mildred S. Dresselhaus holding a model of a carbon nanotube. Credit: Ed Quinn

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1. Chemical Zeolites

crystalline solid
units: Si + 4O

Si O O O O

two covalent bonds per oxygen

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naturally occurring
synthesized
theoretical

Used as microfilters.

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2. Combinatorial Zeolites

Combinatorial d-Dimensional Zeolite

A connected complex of corner sharing d-dimensional sim-

plices

At each corner there are exactly two distinct simplices
Two corner sharing simplices intersect in exactly one vertex.

body-pin graph Vertices: simplices (silicon) Edges: bonds (oxygen) There is a one-to-one correspondence between combinatorial d-dimensional zeolites and d-regular body-pin graphs.

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Graph of a Combinatorial Zeolite is obtained by replacing each d-dimensional simplex with Kd+1. The graph of the zeolite is the line graph of the Body-Pin graph. Whitney [8](1932) proved that connected graphs X on at least 5 vertices are strongly reconstructible from their line graphs L(X). Moreover, Aut(X) ∼ = Aut(L(X)).

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3. Realization

A realization of a d-dimensional zeolite A placement (embedding) of the vertices of the d-dimensional complex in Rd. Equivalently a placement (embedding) of the vertices of the line graph of the body-pin graph. unit-distance realization A realization where all edges join vertices distance 1 apart in

Rd.

non-interpenetrating realization A realization where simplices are disjoint except at joined ver- tices.

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The Layer Construction

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The Layer Construction

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The Layer Construction

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Finite 2d Zeolites

Smallest 2d zeolite is the line graph of K4: The graph of the

ctahedron with four (edge disjoint) faces.

For body-pin graphs on more than 4 vertices, the zeolite can be recovered uniquely from the line-graph.

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4. Finite 2d Zeolites

Body pin graph: K3,3. Since the body pin graph is not planar, the resulting zeolite cannot be planar. Its underlying graph is generically globally rigid. However, it has a unit distance realization in the plane which is a mechanism.

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The typical situation: Not unit distance realizable.

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Harborth’s Example [4, 3]

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The Layer Construction

Z = (T, C) is a combinatorial zeolite realizable in dimension d.

Rd ⊆ Rd+1

Label each t ∈ T arbitrarily with ±1. For +1, erect a d + 1 dimensional simplex in the upper half space, For −1, erect a d + 1 dimensional simplex in the lower half space, Call the Complex Za and its mirror image Zb. Alternately staking Za and Zb gives a layered Zeolite in Rd+1.

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The general case starting from a finite zeolite. Theorem: There are uncountably many isomorphism classes

f unit distance realizable zeolites in R3.

(actually in any dimension d > 1. [7])

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5. Disk shaped zeolite

Labels all +1 A two layered zeolite.

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A finite 3-D symmetric example:

1 2 3 4 O B C D E F G H I J Model with its two planes of symmetry 1 2 3 4 O A B C D F G H J E

[1]

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This 16 Tetrahedra model of Harborth and M¨

ller can be

thought of as a bi-layer.

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A 3-regular graph

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A 3-regular graph with line graph

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The Harboth-M¨

ller model

The body pin graph of the Harborth-M¨

ller Model.

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Excluding triangular holes Strip 02

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Strip 04 Excluding triangular holes

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Excluding triangular holes Strip 06

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Ring Harborth [2] showed that the number of triangles in a ring graph is at least 3800.

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Ring bilayer

Does it move?

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6. Holes in Zeolites

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7. Motions

Degree of Freedom Each d-dimensional simplex has d(d + 1)/2 degrees of freedom Each of the d + 1 contacts removes d degrees. By a na¨ ıve count, a zeolite is rigid - (overbraced by d(d+1)/2.)

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Generically globally rigid in the plane.

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Generically globally rigid in the plane. A 4-regular vertex transitive graph is globally rigid unless it has a 3-factor consisting of s disjoint copies of K4 with s ≥ 3. [Jackson, S, S – 2004]

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Are there finite generically flexible 2D Zeolites? Yes, line graphs of 3-regular graphs with edge connectivity less than 3.

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Are there finite generically rigid but not globally rigid 2D Zeo- lites? Yes, line graphs of 3-regular graphs with edge connectivity less than 3.

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Are there finite generically rigid but not globally rigid 2D Zeo- lites? Yes, line graphs of 3-regular graphs with edge connectivity less than 3. See [5]

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8. Vertex transitive 3-regular

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Design nano lentils and prove their realization

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Design nano lentils and prove their realization

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Design nano lentils and prove their realization

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Design nano lentils and prove their realization

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References

[1] P. Fazekas, O. R¨

schel, and B. Servatius, The kinematics of a framework presented by H.

Harborth and M. M¨

ller, Beitr¨

age zur Algebra und Geometrie / Contributions to Algebra and Geometry, pp. 1–9. 10.1007/s13366-011-0079-x. [2] H. Harborth, Plane four-regular graphs with vertex-to-vertex unit triangles, Discrete Math., 97 (1991), pp. 219–222. [3] H. Harborth and M. M¨

ller, Complete vertex-to-vertex packings of congruent equilateral

triangles, Geombinatorics, 11 (2002), pp. 115–118. [4] , Vertex-to-vertex packings of congruent triangles, Abh. Braunschw. Wiss. Ges., 51 (2002),

pp. 49–54.

[5] T. Jord´ an, Generically globally rigid zeolites in the plane, Tech. Report TR-2009-08, Egerv´ ary Research Group, Budapest, 2009. www.cs.elte.hu/egres. [6] Y. Kiyozumi, T. Nagase, Y. Hasegawa, and F. Mizukami, A process for synthesizing bilayer zeolite membranes, Materials Letters - MATER LETT, 62 (2008), pp. 436–439. [7] B. Servatius, H. Servatius, and M. F. Thorpe, Zeolites: Geometry and combinatorics, International Journal of Chemical Modeling, 4 (2012), pp. 253–267. [8] H. Whitney, Congruent Graphs and the Connectivity of Graphs, Amer. J. Math., 54 (1932),

pp. 150–168.

[9] X. Zou and G. Zhu, Gas separations with zeolite membranes, in Microporous Materials for Separation Membranes, John Wiley & Sons, Ltd, 2019, ch. 7, pp. 225–254.