CHEM 548: Materials Chemistry

Lecture 6: Networks, Silicates, and Microporous Solids

Mon 09/14/2026

Network topology, silicates, and microporous solids

Core structural motif: $\mathrm{SiO_4}$ tetrahedra

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 1 §§1.4.3–1.4.4, §§1.5.4–1.5.5; Ch. 14 §14.4

Learning Objectives for Networks

Coordination to topologyunderstand how coordination numbers lead to network topologies in crystal structures

Vertex and linkera vertex is a linking point; a linker, or edge, is the connection between two vertices

Linker kindsrecognize a linker as a bond, an atom (O in a zeolite), or a polyatomic group (CN⁻ in Prussian blue, dicarboxylates in MOFs)

Uninodal, binodal, regularall vertices equivalent; two kinds of vertex; all vertices, edges, and angles equivalent

Binary compounds and frameworksapply network ideas to simple binary compounds and to zeolites and MOFs alike

Read a real structurerelate TiO₂ to its underlying topology as a (6,3) binodal net

Network Vocabulary

Vertexa linking point — generally an atom, though in some cases a cluster of atoms

Linkerconnects vertices: a bond, as in diamond; an atom, such as the oxygen in a zeolite; or a molecule, such as a dicarboxylic acid in a MOF

Uninodal netall vertices are the same

Binodal nettwo types of vertex, written as an (N,M)-connected net — TiO₂ is a (6,3) net

Regular netall vertices, edges, and angles are equivalent by symmetry

See: M. O’Keeffe, M. Eddauodi, H. Li, T. Reinke & O.M. Yaghi, J. Solid State Chem. 152, 3-20 (2000).

Let me lay out the few terms we use to describe a structure as a network.

Vertex = a linking point
Generally an atom; in more complex structures a cluster of atoms

The vertices are only half of the description, so next is what connects them.

[advance: the linker]

An edge, or a linker, is the connection between two vertices.

A bond in a simple compound, as in diamond
An atom: the oxygen bridging two silicons in a zeolite
A polyatomic anion: CN⁻ in Prussian blue
A whole molecule: a dicarboxylic acid in a MOF

With vertices and linkers we describe a structure as a network, and we sort networks by the kinds of vertex they contain.

[advance: uninodal nets]

A network whose vertices are all the same is uninodal.

One kind of vertex: same coordination, same surroundings, everywhere

Next we allow a second kind of vertex.

[advance: binodal nets]

A network with two types of vertex is binodal, and we write it as an (N,M)-connected net.

Two vertex types, each with its own coordination number
TiO₂: Ti is 6-coordinate, O is 3-coordinate, so a (6,3) net

Some networks are far more symmetric than that, and those get their own name.

[advance: regular nets and the reference]

A regular net is one where all the vertices, all the edges, and all the angles are equivalent by symmetry.

Regular: vertices, edges, and angles all equivalent, the rare case
Infinitely many topologies, but crystal chemistry uses a finite, well-studied set
Foundational reference: O'Keeffe and Yaghi (2001), many follow-ups since
Pays off for MOFs and zeolites; here we stay with the basics

With those terms in hand, let's start with networks built from a single kind of vertex, beginning with three connections per vertex.

Uninodal 3-Connected Nets

Two uninodal three-connected nets side by side: the graphite 2D net and the SrSi2 silicon network, each with its space group and site symmetry.

Uninodal 4-Connected Nets

Two uninodal four-connected nets side by side: lonsdaleite and diamond, each with its space group and site symmetry.

Uninodal 6- and 8-Connected Nets

Two ball-and-rod net renders side by side: the six-connected primitive cubic net of polonium and the eight-connected body-centred cubic net of iron.

Regular network	Regular network
Primitive cubic (Po)	Body-centered cubic (Fe)
Space Group = Pm3̅m	Space Group = Im3̅m
Site symmetry = m3̅m (O_h)	Site symmetry = m3̅m (O_h)

Site ordered homeotypes

Four parent nets: lonsdaleite and diamond as four-connected nets, primitive cubic as six-connected, body-centered cubic as eight-connected.

Starting from these important unionodal nets we can develop compounds in a couple of different ways.

One: a site ordering of the vertices — let me illustrate

[advance: show the lonsdaleite to wurtzite site ordering]

Lonsdaleite with the silicons replaced by alternating zinc and sulfur → a site ordered homeotype: wurtzite
We could call this a 4,4 net — the two vertices are now different atoms, but both still tetrahedra

[advance: show the diamond to sphalerite site ordering]

Same site ordering on diamond → zinc blende, or sphalerite — another 4,4 net

[advance: show the primitive cubic to rock salt site ordering]

Primitive cubic net + site ordering — we could get the rock salt structure, sodium chloride — a six net

[advance: show the bcc to CsCl site ordering]

bcc net + site ordering → the CsCl structure
Many familiar compound structures — wurtzite, sphalerite, NaCl, CsCl — can be seen as ordered versions of high-symmetry uninodal nets
Recall: sphalerite (ZnS) = ccp anions, half the tetrahedral holes filled; NaCl = ccp anions, all octahedral holes filled — so these structures can be derived from close-packing or from ordering symmetric uninodal nets
Key point, symmetry reduction: diamond → sphalerite, same cell, m3̅m drops to 4̅3m; lonsdaleite → wurtzite, same cell, 6/mmm drops to 6mm — ordering creates inequivalent sites and eliminates some symmetry operations
pc → NaCl and bcc → CsCl: point symmetry stays m3̅m, translational symmetry changes
pc → NaCl: cell doubles in all three directions, primitive cubic to fcc Bravais lattice; bcc → CsCl: cell size unchanged, body-centered lattice becomes primitive cubic

So far we changed the vertices; the next route keeps the vertices and changes the linker instead.

Network Expansion

Three parent nets: lonsdaleite and diamond as four-connected nets and primitive cubic as a six-connected net.

O: the linker Si, Po: parent-net vertices SiO₄, ReO₆ polyhedra: Si or Re at the centre

Another way we can go from an element in a unilateral net to a compound is that we could do something called network expansion.

Keep the same vertices, change the linker

[advance: add oxygen linkers to the lonsdaleite net]

Lonsdaleite with an oxygen atom as the linker between the two silicons → one of the polymorphs of SiO₂: the tridymite structure, hexagonal

[advance: add oxygen linkers to the diamond net]

Diamond with oxygen at the midpoint of each bond → the idealized cristobalite structure — another SiO₂ polymorph

[advance: add oxygen linkers to the primitive cubic net]

Primitive cubic net with oxygens between the vertices → the ReO₃ structure
Add a large cation at the cube center → perovskite, one of the most important ternary structure types — we'll return to this later
Na₃N (though unstable) was eventually synthesized and adopts the ReO₃-type structure — nitrogen at cube corners, each surrounded by six sodiums, highlighting 2-coordinate sodium
Na = the red atoms, N = the blue; 1 atom from all the corners, 3 atoms from all the edges

There is one more modification, where the vertices themselves become groups of atoms.

Decoration: CaTe and CaB₆

The CsCl-type CaTe cell beside CaB6, in which the tellurium vertex is replaced by an octahedral B6 cluster, each with its space group and atom positions.

Interpenetration: Cu₂O

The Cu2O unit cell beside an extended view in which two independent cristobalite-like networks, drawn in false colors, thread the same volume.

False colors show two independent, interpenetrating cristobalite networks.

Some Important Binodal Nets

(N,M)-Net	N Coordination	M Coordination	Net (Example)
4,2	Tetrahedron	Linear (bent)	SiO₂ (cristobalite, tridymite)
4,3	Tetrahedron	Triangle	Si₃N₄
4,4	Tetrahedron	Tetrahedron	ZnS (sphalerite, wurtzite)
4,3	Square	Triangle	Pt₃O₄
4,4	Square	Square	NbO
4,4	Square	Tetrahedron	Cooperite (PtS)
6,2	Octahedron	Linear	ReO₃
6,3	Octahedron	Triangle	TiO₂ (rutile, anatase)
6,4	Octahedron	Tetrahedron	Corundum (Al₂O₃)
6,6	Octahedron	Octahedron	Rock salt (NaCl)
6,6	Octahedron	Trigonal prism	NiAs
8,4	Cube	Tetrahedron	Fluorite (CaF₂)
8,8	Cube	Cube	CsCl

Read the two vertex types

N and M count neighbours. The next two columns describe their geometry.

ZnS: two tetrahedral vertices

(4,4): both Zn and S have four neighbours.

SiO₂: use the formula to find M

1 × 4 = 2 × M → M = 2; tetrahedral Si and two-coordinate O.

Si₃N₄: balance the connections again

3 × 4 = 4 × M → M = 3; tetrahedral Si and triangular N.

Pt₃O₄: the same numbers, a different geometry

(4,3) again, but the four-coordinate Pt vertex is square planar.

NbO: squares at both vertices

(4,4): square-planar Nb and square-planar O.

Cooperite PtS: one square and one tetrahedron

(4,4): square-planar Pt and tetrahedral S.

ReO₃: octahedra joined through linear vertices

1 × 6 = 3 × M → M = 2; a (6,2) net.

TiO₂: octahedral Ti and three-coordinate O

1 × 6 = 2 × M → M = 3; rutile and anatase are (6,3) nets.

Al₂O₃: octahedral Al and four-coordinate O

2 × 6 = 3 × M → M = 4; corundum is a (6,4) net.

NaCl: octahedral coordination at both vertices

(6,6): each Na and each Cl has six unlike neighbours.

NiAs: six neighbours need not form an octahedron

(6,6) again: octahedral Ni and trigonal-prismatic As.

CaF₂: cubic Ca and tetrahedral F

1 × 8 = 2 × M → M = 4; fluorite is an (8,4) net.

CsCl: cubic coordination at both vertices

(8,8): the numbers count neighbours; the geometry columns describe their arrangement.

The same compounds can be described directly as binodal nets.

N and M: neighbours at the first and second vertex
Geometry: a separate piece of information
Formula × coordination number: same connection count at both ends

Start with sphalerite and wurtzite, which we made by ordering tetrahedral nets.

[advance: ZnS: two tetrahedral vertices]

Sphalerite and wurtzite are both four-four nets.

Diamond → sphalerite; lonsdaleite → wurtzite
Zn: 4; S: 4; both tetrahedral
Same vertex types can form different structures

Now use the formula of silica to find the oxygen coordination number.

[advance: SiO₂: use the formula to find M]

Four-coordinate silicon in SiO₂ requires two-coordinate oxygen.

1 Si × 4 = 2 O × M
M = 2: each oxygen joins two silicon atoms
Tetrahedral Si; linear or bent O linker

Keep silicon tetrahedral and change the formula to Si₃N₄.

[advance: Si₃N₄: balance the connections again]

In Si₃N₄, nitrogen must be three-coordinate.

3 Si × 4 = 4 N × M
M = 3; tetrahedron + triangle
Coordination balance constrains the net; does not determine it

Compare this with another four-three net whose four-coordinate vertex is square.

[advance: Pt₃O₄: the same numbers, a different geometry]

Four neighbours can form a square as well as a tetrahedron.

Pt: square; O: triangle
Same (4,3) as Si₃N₄
Four-coordinate geometry: square versus tetrahedron

Next, pair a square vertex with another square.

[advance: NbO: squares at both vertices]

NbO has square-planar coordination at both types of atom.

Nb: square; O: square
(4,4) also describes ZnS
Both vertex geometries differ from ZnS

Now keep one square vertex and make the other tetrahedral.

[advance: Cooperite PtS: one square and one tetrahedron]

Cooperite combines square-planar platinum with tetrahedral sulfur.

Pt: square; S: tetrahedron
Same (4,4); third geometry pairing
Next slide compares PtS and NbO

Move down the table to structures with octahedral cations.

[advance: ReO₃: octahedra joined through linear vertices]

An octahedral rhenium vertex paired with a linear oxygen vertex gives ReO₃.

Re: octahedron; O: linear
1 × 6 = 3 × 2
First of the octahedral-cation rows

Give the oxygen vertex a third neighbour to reach the TiO₂ row.

[advance: TiO₂: octahedral Ti and three-coordinate O]

Rutile and anatase both have six-coordinate titanium and three-coordinate oxygen.

Ti: 6; O: 3; 1 × 6 = 2 × 3
Both rutile and anatase
Triangle is the table’s coordination figure; actual geometry differs

Increase the oxygen coordination to four for corundum.

[advance: Al₂O₃: octahedral Al and four-coordinate O]

Corundum pairs six-coordinate aluminum with four-coordinate oxygen.

2 Al × 6 = 3 O × 4
Octahedron + tetrahedron
Coordination figures may be distorted

Now make both vertices six-coordinate.

[advance: NaCl: octahedral coordination at both vertices]

Rock salt has octahedral coordination around both sodium and chlorine.

Na: octahedron; Cl: octahedron
1 × 6 = 1 × 6
Equal coordination numbers; geometry still matters

Keep six neighbours at both vertices, but change one octahedron to a trigonal prism.

[advance: NiAs: six neighbours need not form an octahedron]

NiAs is also a six-six net, but arsenic has trigonal-prismatic coordination.

Ni: octahedron; As: trigonal prism
Same (6,6) as rock salt
Six-coordinate geometry: octahedron versus trigonal prism

The last two rows start with an eight-coordinate cubic vertex.

[advance: CaF₂: cubic Ca and tetrahedral F]

Fluorite pairs eight-coordinate calcium with four-coordinate fluorine.

Ca: cube; F: tetrahedron
1 × 8 = 2 × 4
Coordination balance gives (8,4)

Finally, pair the cubic vertex with another cubic vertex.

[advance: CsCl: cubic coordination at both vertices]

CsCl has eight-coordinate cubic vertices at both species.

Cs: cube; Cl: cube
1 × 8 = 1 × 8
Formula → coordination balance; geometry and connectivity distinguish structures

Let’s look more closely at two rows of that table, starting with the four-four nets built from square-planar vertices.

(4,4) Nets: Cooperite PtS and NbO

Cooperite PtS, with square-planar platinum and tetrahedral sulfur, beside NbO, in which both atoms are square planar, each with its space group and atom positions.

Polymorphs of TiO₂: Rutile and Anatase

The rutile and anatase forms of titanium dioxide drawn as octahedral frameworks, each with its space group, formula units per cell, and site symmetries.

Corundum

The (6,4) network formed by two of the most common environments — octahedron and tetrahedron — does not give a regular environment for either.

Finally, let's take a closer look at corundum (Al₂O₃).

Structure: hexagonal close-packed O²⁻ with Al³⁺ filling two-thirds of the octahedral sites; vacancies ordered so the Al cations shift apart
Coordination: Al octahedral, O tetrahedral — among the two most common coordination environments in inorganic chemistry
But the network is not perfectly symmetric: corundum is rhombohedral, not cubic, so neither polyhedron is ideal
Al octahedra noticeably distorted; O tetrahedra highly distorted
The distortion reflects the lower symmetry of the rhombohedral space group compared to cubic structures, where perfect octahedral or tetrahedral coordination would occur

That completes the network story, so let me summarize it before you practice.

Summary for Networks

A network abstracts a structure into vertices and linkers, and the coordination numbers at those vertices name the net.

Idea	What it says	Example
Vertices and linkers	Linking points, joined by a bond, an atom, or a molecule	Diamond: C vertices, C–C bonds
Uninodal	All vertices identical	Graphite, diamond, primitive cubic, bcc
Binodal	Two vertex types with different coordination, written (N,M)	TiO₂ is a (6,3) net
Regular	All vertices, edges, and angles equivalent by symmetry	SrSi₂, diamond, primitive cubic, bcc, NbO
Site ordering	Order two elements onto one parent net's vertices	Diamond → sphalerite; bcc → CsCl
Expansion	Keep the vertices and insert an atom as the linker	Diamond → cristobalite; primitive cubic → ReO₃
Decoration	Replace a simple vertex with a polyatomic cluster	CaTe → CaB₆
Interpenetration	Two independent nets thread the same volume without connecting	Cu₂O

Binodal nets are the systematic link between empirical formulas, coordination numbers, and real crystal structures.

Let me summarize what the network picture has given us.

A network says how the atoms connect, nothing about where they sit
The coordination numbers at the linking points name the net

Start with the two things every net is made of.

[advance: vertices and linkers]

We keep the linking points and whatever connects them.

Linking points, joined by a bond, an atom, or a molecule
Diamond: C vertices, C–C bonds

Once a structure is a net, we sort nets by the kinds of vertex in them.

[advance: the three kinds of net]

There are three kinds of net to know.

Uninodal: all vertices identical, as in diamond, graphite, primitive cubic, bcc
Binodal: two vertex types, as in 6-coordinate Ti and 3-coordinate O in TiO₂
Regular: vertices, edges, and angles all equivalent, the rare case

From any of these parent nets we build new structures.

[advance: the four transformations]

We transform nets in four ways.

Site ordering: two elements onto one parent net's vertices (NaCl, CsCl, sphalerite, wurtzite)
Expansion: keep the vertices, insert an atom as the linker (ReO₃, perovskite)
Decoration: a polyatomic cluster in place of a simple vertex (CaB₆)
Interpenetration: two independent nets through one volume, unconnected (Cu₂O)

Last is what binodal nets buy us when we write a formula.

[advance: the payoff]

Binodal nets provide a systematic way to connect empirical formulas, coordination numbers, and real crystal structures.

Two coordination numbers fix the (N,M) net and the empirical formula together
Same bookkeeping: the numbers we read off a structure, the formula we write

Let's put that connectivity bookkeeping to work on the ReO₃ framework in problem 1.27.

You Try It Now: Problem 1.27

GivenReO₃ has a 3D network of octahedrally coordinated rhenium

Findthe N,M-connectivity for this binodal network

ReO₃ is a (6,2) Binodal Net

The ReO3 framework of corner-sharing rhenium-centred octahedra, drawn as linked ReO6 polyhedra with the oxygens in red.

ReO₃ (Pm3̅m); Re(1a), O(3d)

Answer: a (6,2)-connected binodal net.

Reat the corners of a simple-cubic framework, octahedrally coordinated by O, so CN = 6

Both ends of every Re–O bond: $\mathrm{CN_A} \times x = \mathrm{CN_B} \times y$

Counted from Re$1 \times 6 = 6$

Counted from O$3 \times \mathrm{CN_O} = 6$$\mathrm{CN_O} = 2$

Olies midway along a Re–Re edge, perovskite without the A cation, so every octahedron corner is shared, no edges or faces

Nettwo node types, Re and O, so it is binodal, with connectivities 6 and 2 → (6,2)

Here is the ReO₃ framework, rhenium-centered octahedra sharing corners in three dimensions.

Count the O around any one Re and you get six, the CN the problem handed us

Start with the answer and with that given six.

[advance: the answer and the given CN of Re]

The answer is a (6,2)-connected binodal net.

Re at the corners of a simple-cubic framework, octahedrally coordinated, so CN = 6
The 2 is the number we earn, and we earn it by counting bonds

Every rhenium–oxygen bond has one end on a rhenium and one end on an oxygen.

[advance: the balance rule and the two bond counts]

That one fact is the whole rule.

For A_xB_y: CN(A) × x = CN(B) × y, both sides count the same bonds
Per formula unit of ReO₃: 1 Re, 3 O
From Re, 1 × 6 = 6; from O, 3 × CN(O) = that same 6, so CN(O) = 2
Same balance as charge last lecture, m × c = n × a

Now picture what a two means.

[advance: what a CN of two means, and the net]

A coordination number of two means each oxygen touches exactly two rheniums.

O midway along a Re–Re edge, perovskite without the A cation
Every corner shared with one neighbor, no edges and no faces
Two node types, so binodal, connectivities 6 and 2, giving (6,2)

With networks in hand, we turn to the second half of the lecture, silicates and microporous solids.

Silicates and Microporous Solids: Learning Objectives

Corner sharingexplain how silicate structures are built from corner-sharing SiO₄ tetrahedra

Niggli formularelate the oxygen-to-silicon ratio to a structural motif

Motifsidentify isolated tetrahedra, chains, rings, sheets, and frameworks from the O:Si ratio

Zeolitesdescribe how they form from Al and Si tetrahedra, why cations create porosity, and what makes a solid microporous (<2 nm), mesoporous (2–50 nm), or macroporous (>50 nm)

Building blocksrecognize the sodalite cage, zeolite A, and faujasite, and their pore sizes

MOFsunderstand what a metal-organic framework is and how MOF-5 is built from a net

Silicates

Abundancethe most common minerals in the Earth’s crust

Structure typesabout one third of all known inorganic crystal structure types

Tetrahedral Sisilicon sits in SiO₄ tetrahedra, and connected tetrahedra share corners

O:Si ratiosets how far the tetrahedra are connected

Niggli Formula: Pyrosilicate

The Sc2Si2O7 structure of corner-sharing SiO4 tetrahedra, annotated with the Niggli formula SiO 3/1 + 1/2: three oxygens belong to one tetrahedron, one oxygen is shared by two.

O:Si Ratio and Niggli Formula

O:Si	Example	Niggli formula	Network connectivity
2	SiO₂	SiO_4/2	3D network of corner connected tetrahedra
2.5	Na₂Si₂O₅	SiO_{1/1 + 3/2}	2D sheets of corner connected tetrahedra
3	Na₂SiO₃	SiO_{2/1 + 2/2}	1D chains or rings of corner connected tetrahedra
3.5	Sc₂Si₂O₇	SiO_{3/1 + 1/2}	dimers of corner connected tetrahedra
4	ZrSiO₄	SiO_4/1	isolated tetrahedra

bridging oxygen · terminal oxygen

Knowing the Nikle formula, we can examine oxygen-to-silicon ratios in silicates.

Every corner shared, as in SiO₂ polymorphs: formula SiO₄/2, O:Si = 2:1 → a 3D network of corner-connected tetrahedra
Other extreme, O:Si = 4:1: no oxygens shared → isolated tetrahedra, as in zircon (ZrSiO₄)

[advance: add the 2.5 ratio sheet row]

Ratio 2.5: 1 terminal + 3 bridging oxygens → most common motif: a 2-dimensional sheet of corner connected tetrahedra

[advance: add the 3 and 3.5 ratio rows]

Ratio 3: 2 terminal + 2 bridging — a variety of motifs, most notably an infinite one-d chain
Ratio 3.5: 3 terminal + 1 bridging → the dimers of corner connected tetrahedra we saw on the last slide

Let's see what these motifs look like in real silicate structures.

Silicate Structures

Four silicate motifs at their O:Si ratios: a six-membered ring of tetrahedra in beryl, sheets in mica, chains in sodium silicate, and a pyrosilicate dimer in scandium silicate.

Porous Materials

Zeolites“boiling stone”: frameworks of corner connected SiO_4/2 and AlO_4/2 tetrahedra

Zeotypesthe same frameworks with other elements on the tetrahedral sites

MOFsporous coordination polymers: metal ions or clusters joined by multidentate polyatomic ligands

Focusing on 3D networks of corner-connected tetrahedra, a key class is zeolites.

Frameworks of Si and Al tetrahedra
"Zeolite" ("boiling stone"): heating released trapped solvents like water, causing them to fizz — coined 1756 by Swedish mineralogist Axel Fredrik Cronstedt, heating stilbite
Synthetic zeolites from the mid-20th century (Barrer's work in the 1940s) for industrial uses
Crucial applications: ion exchange, catalysis (petroleum cracking to gasoline), detergents, molecular sieves, gas separation
Zeotypes: same framework structures, other elements substituted for Si and Al
MOFs: coordination polymers with open pores — metal ions or clusters linked by multidentate organic ligands

[advance: add the pore-size classification]

We can classify porous materials as microporous, mesoporous, or macroporous by the size of the pores by which molecules can come in or out
Zeolites and MOFs: by and large microporous

With those classes defined, let's compare the most important zeolites.

Important Zeolites

Name	Framework	Ring	Accessible volume*	Channel dimensions
Sodalite	Al₆Si₆O₂₄⁶⁻	6	0	—
Zeolite A	Al₁₂Si₁₂O₄₈¹²⁻	8	21.4%	4.1 Å × 4.1 Å
Faujasite	Al₅₈Si₁₃₄O₃₈₄⁵⁸⁻	12	27.4%	7.4 Å × 7.4 Å

*Percent of volume accessible to a 2.8 Å diameter probe sphere meant to simulate a water molecule.

Construction diagram: a sodalite cage becomes (a) non-porous sodalite, (b) zeolite A, and (c) faujasite, each a different linkage of the same cage.

A few important zeolites illustrate the key concepts.

Sodalite, zeolite A, faujasite: all share the same basic building block, the sodalite cage; sodalite itself is not porous, the others are
Sodalite and zeolite A: Si:Al ratio 1:1, alternating Si and Al tetrahedra
Replacing Si⁴⁺ with Al³⁺ → negative framework charges, balanced by cations (often Na⁺) and their waters of solvation inside the pores
These counterbalancing ions are crucial — without them, the framework would likely collapse into a dense silicate rather than a porous one
Key features: ring sizes, accessible pore volume (about 25% in zeolite A and faujasite for a 2.8 Å probe sphere), and the pore-opening dimensions that control molecular access
Next: the sodalite cage as the fundamental building block

To see where the sodalite cage comes from, let's go back to our four-connected nets.

Uninodal 4-Connected Nets: Lonsdaleite, Diamond, Sodalite

Three uninodal four-connected nets with their space groups and site symmetries: lonsdaleite, diamond, and the sodalite net.

Sodalite, Na₈Al₆Si₆O₂₄Cl₂

Four sodalite cages sharing hexagonal faces in a body centered cubic arrangement, with the P-43n cell parameters and atom positions.

The sodalite network takes its name from the mineral sodalite (Na–Al–Si–O–Cl).

Framework contains cages shaped like truncated octahedra — blue soccer balls
Faces: hexagonal six-rings and square/diamond four-rings — matching the sodalite net topology
In sodalite itself, Cl⁻ ions occupy the cage centers; Na and O are not shown here
The cage is the fundamental building unit of many zeolites, but here the cages are fused — the structure is non-porous, preventing molecular access

[advance: apply site ordering and oxygen bridges to the cage]

Single cage + site ordering → a homeotype: alternating Al–Si–Al–Si around the vertices
Then oxygen bridges: Al and Si do not bond directly but connect through O atoms at the tetrahedral vertices
O is not at the edge midpoint (that would give a linear 180° Al–O–Si bond); the angle is ~145°, maintaining a continuous tetrahedral network of Al and Si

Let's examine that single cage more closely.

Sodalite Cage

Notice the lack of porosity when oxygens are drawn with realistic radii.

From Sodalite to Zeolite A and Faujasite

The same sodalite cage linked three ways: fused into non-porous sodalite, joined through its four-rings into zeolite A with 4.1 angstrom pores, and joined through its six-rings into faujasite with 7.4 angstrom pores.

Hierarchical Faujasite Catalyst

Micropores0.7 nm pores in the zeolite framework hold the active catalytic sites

Mesopores4 nm channels, opened by dealumination and leaching during catalyst preparation, act as diffusion paths

Effectbig feed molecules barely fit, so they get in and stay until they crack; the products are much smaller, diffuse out fast, and are gone before they crack again

Fluid catalytic cracking: a refinery, an electron micrograph of FCC catalyst beads, and a mesostructured zeolite, over two framework drawings marking the 0.7 nm micropore and the 4 nm mesopore that molecules enter and leave through.

Fluid catalytic cracking (FCC) started in the 1940s with simple silica–alumina catalysts, but their activity was limited.

1962: Union Carbide introduced Zeolite Y — strong acid sites, large pores that let bulky oil molecules enter and react
Result: doubled gasoline yields, boosted octane, longer catalyst life; later USY, rare-earth exchange, and ZSM-5 additives made FCC the backbone of refining
Framework: faujasite — AlO₄ and SiO₄ tetrahedra forming spacious ~12 Å cages connected by 12-membered oxygen rings
Al charges balanced by cations (Na⁺), exchangeable for H⁺ → Brønsted acid sites that protonate molecules to form carbocations
At ~500 °C in the FCC riser: β-scission, isomerization, hydride transfer, aromatization → gasoline-range alkanes and alkenes plus lighter gases; coke blocks sites but is burned off in the regenerator
Why it works: pores ~7.4 Å just wide enough for heavy hydrocarbons, surface area ~900 m²/g, shape selectivity limits over-cracking and steers toward high-octane products
Why the pores do that: big feed molecules barely fit, so they get in and stay until they crack; the products are much smaller, diffuse out fast, and are gone before they crack again
Payoff: gasoline yield ~45% to ~70%, higher octane from branched alkanes and olefins, tunable gasoline/propylene/diesel; today FCC with zeolite Y processes about 40% of global crude oil

From these inorganic frameworks we now step to metal-organic frameworks, built from the same network ideas.

MOF-5

Zn₄O(1,4-bdc)₃(DMF)₈(C₆H₅Cl)

Startwith the 6-connected primitive cubic net

Decoratethe vertices with OZn₄ clusters

Expandwith the 1,4-benzenedicarboxylate linker

S. Natarajan and P. Mahata, Chem. Soc. Rev. 2009, 38, 2304–2318.

MOFs (metal–organic frameworks) are a whole field, alongside COFs (covalent organic frameworks).

To illustrate: MOF-5, introduced by Omar Yaghi in a Nature paper in 1999
Exceptionally high permanent porosity, free volume and a very large internal cavity, at very high surface area
Set many of the field’s design principles and moved it from proof of concept to practical relevance, especially gas storage
Built on the simple cubic net, the 6-connected one we started with

Now put something at each vertex.

[advance: decorate the vertices]

At each vertex sits an OZn₄ cluster: an oxygen atom surrounded by four Zn²⁺ cations.

That is the decoration move from the first half: a cluster in place of a single atom

Now stretch the edges.

[advance: expand with the linker]

These clusters are then connected by benzene dicarboxylate linkers, benzene rings with deprotonated carboxylate groups at both ends, which coordinate to Zn.

That is the expansion move, and the Zn end up tetrahedrally coordinated
Enlarge the linker and the pore grows: a whole series of MOFs
Too much void and the frameworks interpenetrate, weaving through each other; avoiding that is a central design challenge
The reference at the bottom relates MOF frameworks to inorganic crystal structures

Let me say more about why that 1999 paper mattered.

Permanent Porosity: Li, Eddaoudi, O’Keeffe, Yaghi, Nature 402, 276 (1999)

The opening page of Li, Eddaoudi, O'Keeffe and Yaghi, Design and synthesis of an exceptionally stable and highly porous metal-organic framework, Nature 402, 276 (1999), with the MOF-5 figure.

The Mof field really go going when Omar Yagi published this paper in 1999.

The 1999 Nature paper on MOF-5 was pivotal because it showed, for the first time, that you could deliberately build a crystalline solid with permanent porosity, very high surface area (~2900 m²/g), and modular designability.

Unlike earlier coordination polymers that collapsed on solvent removal, MOF-5 remained stable
Proof that metal clusters and organic linkers could be stitched into robust, tunable frameworks
Launched reticular chemistry — designing solids from molecular building blocks
Made clear MOFs could rival zeolites and carbons with vastly greater flexibility
Transformed MOFs from a curiosity into a general design platform for gas storage, separations, and catalysis, igniting the field now central to porous materials research

After all that promise, the natural question is whether MOFs have reached any real commercial application.

The 2025 Nobel Prize in Chemistry

Robson, 1989Linked copper ions to a four-armed molecule and got an ordered, diamond-like crystal full of cavities — fragile, and it collapsed easily

Kitagawa, 1997–1998Built a framework crossed by open channels, showed gases flow in and out of it, and predicted that frameworks could be made flexible

Yaghi, 1999–2002Built MOF-5, stable enough to stay open once emptied, then used rational design to grow a whole family of frameworks with larger and larger cavities

Store toxic gases — the named application behind the only commercial MOF product so far.

Royal Swedish Academy of Sciences, 8 Oct 2025: Susumu Kitagawa, Richard Robson, and Omar M. Yaghi, “for the development of metal–organic frameworks”

In October 2025 this whole field got the Nobel Prize in Chemistry.

Three laureates: Susumu Kitagawa, Richard Robson, and Omar Yaghi — for the development of metal-organic frameworks

[advance: show the Robson row]

Robson, 1989: he linked copper ions with a four-armed molecule and got an ordered crystal shot through with cavities, built on the diamond net we drew earlier
The material was fragile — take the solvent out and it collapsed

[advance: show the Kitagawa row]

Kitagawa, 1997: a framework crossed by open channels, with gases flowing in and out of it
1998: he predicted these frameworks could be made flexible, breathing as guests come and go

[advance: show the Yaghi row]

Yaghi, 1999: MOF-5, the structure on the last two slides — stable enough to stay open after the solvent leaves
Then rational design: keep the net, change the linker, and you get a whole family of frameworks with larger and larger cavities
That is reticular chemistry — the same decorate-and-expand moves we used on nets, now done deliberately in the lab

One of the applications the Academy named was storing toxic gases, and that is exactly where the only commercial MOF product so far sits.

ION-X: the status quo problem

GasesIon-implant tools in chip fabs need toxic dopants: arsine (AsH₃), phosphine (PH₃), BF₃

Storage todayHigh-pressure steel cylinders, often above 100 psig

Leak riskOne bottle holds enough toxic gas to endanger workers and trigger a fab-wide shutdown

Regulatory burdenRedundant ventilation, scrubbing, and emergency protocols raise cost and insurance

Cylinder heelUnused residual gas: waste that still poses disposal risks

What the MOF changesThe pores adsorb and store the gas, so the cylinder holds the same usable inventory at sub-atmospheric pressure (near vacuum to ~1 atm) and desorbs it to feed the tool

ION-X: why fabs pay for it

SafetyBreach the cylinder and no pressurized toxic gas rushes out, which cuts both the odds and the severity of a catastrophic event

Regulatory reliefA lower safety classification than high-pressure bottles: lower facility cost, easier permitting, fewer insurance penalties

ContinuityNo shutdowns from leaks or alarms, which cost more than $1M a day in lost production

WasteLower heel volumes mean less toxic waste to manage and dispose of

CompatibilityION-X cylinders plug straight into existing ion-implant tools, with no redesign

They convert a toxic-gas supply chain from a high-pressure hazard into a sub-atmospheric, fail-safe consumable, saving fabs money on risk, compliance, and downtime.

Why do fabs pay for this?

First, massively reduced safety risk: even if the cylinder is breached, no pressurized toxic gas rushes out — that alone cuts the probability and severity of a catastrophic event
Second, regulatory relief: a lower safety classification than high-pressure bottles — lower facility cost, easier permitting, fewer insurance penalties
Third, operational continuity: prevents fab shutdowns due to leaks or alarms — more than a million dollars a day in lost production
Fourth, waste reduction: lower heel volumes, less toxic waste to manage and dispose of
Fifth, compatibility: ION-X cylinders plug directly into existing ion-implant tools, no redesign needed
Value in one line: a toxic-gas supply chain converted from a high-pressure hazard into a sub-atmospheric, fail-safe consumable — saving fabs money on risk, compliance, and downtime

Here is the framework chemistry that makes that work.

Open Cu sites in a MOF channel

Four panels of a copper MOF: the as-made framework with DMF bound in the pores, the same framework after solvent removal with open Cu sites and channels 12.00 by 9.05 angstroms across, the view along [010], and a zoom on the channel walls with 4.29, 2.98, and 4.08 angstrom spacings.

ION-X performance at Axcelis

Left, Fig. 2: SDS 1, 2 and 3 cylinders with beaded zeolite, beaded activated carbon and monolithic activated carbon at 500, 1200 and 1500 square metres per gram, and the ION-X cylinder with a MOF below 8000. Right, Fig. 3: volumetric dopant gas delivery for arsine, phosphine and boron trifluoride, ION-X against two carbon products.

J. Arnó and co-workers, NuMat Technologies, Skokie, IL — Figs. 2 and 3

ION-X (MOF) vs SDS (Carbon) Cylinders

Metric	ION-X (MOF)	SDS (Carbon)
Gas delivered	Higher usable capacity (20–50% more for some gases)	Lower deliverable, larger residual heel
Flow stability	Stable delivery down to very low pressures (few Torr)	Delivery falls off sooner; larger heel
Residual gas (heel)	Very low; most gas is recoverable	Higher heel, more waste
Purity / contamination	Low metals, stable across cycles	Acceptable, but batch variability possible
Reproducibility	Highly consistent (crystalline pores)	More variability (heterogeneous carbon)
Safety	Sub-atmospheric storage	Also sub-atmospheric storage (same safety principle)

This table compares ION-X MOF cylinders with the incumbent SDS carbon cylinders, metric by metric.

Gas delivered: MOF gives higher usable capacity, 20 to 50 percent more for some gases; carbon delivers less, larger residual heel
Flow stability: MOF holds stable delivery down to a few torr; carbon's delivery falls off sooner
Residual gas, the heel: very low for the MOF, most gas recoverable; carbon leaves a higher heel and more waste
Purity and contamination: MOF shows low metals, stable across cycles; carbon acceptable but can vary batch to batch
Reproducibility: highly consistent for the MOF's crystalline pores; more variable for heterogeneous carbon
Safety: both are sub-atmospheric storage — the same safety principle

That brings us to the end of the new material, so let me summarize.

Summary

One tetrahedron, linked more and more openly: the O:Si ratio sets the motif, and the motif sets the size of the pore.

Silicates	Dominate Earth’s crust; their tetrahedral networks are defined by O:Si ratios
Motifs	Isolated tetrahedra, 1D chains, 2D sheets, and 3D frameworks
Zeolites	Al/Si tetrahedral frameworks; charge-balancing cations generate accessible pores
Key zeolites	Sodalite, zeolite A, and faujasite differ in connectivity and pore size, 4 to 7.4 Å
Pore classes	Microporous below 2 nm, mesoporous 2 to 50 nm, macroporous above 50 nm
MOFs	Metal–oxide clusters linked by organic ligands: the same network ideas, made highly porous and tunable

MOF-5 closes the loop: decorate a cubic net, expand the linker, and the pore grows to order.

You Try It Now: Problem 1.22

GivenCrO₃ is built from chains of corner-sharing chromium-centered tetrahedra

Findthe Niggli formula and the simple crystal-chemical formula for this structure

Niggli formula: counting the corners

StructureCorner-sharing tetrahedra in 1‑D chains → CrO₄ tetrahedra

Niggli formulaone cation-centered polyhedron: each oxygen as how many over how many share it

Bridging O2 corners join the next tetrahedron2/2

Terminal Othe other 2 are this Cr’s alone2/1

CrO_{2/2 + 2/1}

2/2 is 1 and 2/1 is 2, so we recover CrO₃, and the numerators 2 + 2 give CN(Cr) = 4.

The SiO4 4- silicate tetrahedron with its oxygen and silicon spheres labelled, above the 1. Single Chain drawing of corner-sharing tetrahedra.

Silicate tetrahedron SiO₄⁴⁻: oxygen and silicon labelled, above “1. Single Chain”. Cr sits where Si does.

Here is one tetrahedron and the chain it sits in.

Four oxygens at the corners, same SiO₄ picture with chromium at the center
The Niggli formula is per one polyhedron: each oxygen as how many of them, over how many polyhedra share one

So split the four corners into the shared ones and the unshared ones.

[advance: the corner count]

A single chain shares a corner at each end of the tetrahedron.

Two bridging: one corner to the tetrahedron ahead, one to the tetrahedron behind, each shared by 2 → 2/2
Two terminal: nothing on the other side, so they belong to this chromium alone → 2/1
Same 2 terminal + 2 bridging as the O:Si = 3 row, the chain row of the table

Write those two pieces side by side and you have the formula.

[advance: the Niggli formula]

The Niggli formula is CrO 2/2 plus 2/1.

Cation first, then one term per kind of oxygen
Nothing here is memorized: it is the corner count, written down

Check it by adding the terms up.

[advance: the check back to CrO3]

Note that 2/2 is 1 and 2/1 is 2, so we recover CrO₃.

Add the numerators, 2 + 2 = 4: the coordination number of Cr, tetrahedral
Add the values, 1 + 2 = 3: the three oxygens per chromium in CrO₃

The next slide gives you a general algorithm for writing these crystal-chemical formulas.

Crystal-Chemical Formula Algorithm

1. List atomsWrite down each unique cation and anion site

2. Tag cationsAdd the coordination number, how many neighbors, and the geometry: t tetrahedral, o octahedral, and so on

3. Tag anionsMark how many cations each anion connects to: O[1] terminal, O[2] bridging, O[3] 3-connected

4. Check balancesBonds from cations = bonds accepted by anions; the anion count = the formula stoichiometry; the charges must cancel

5. Keep it simpleOnly the distinct sites and connectivities needed to show the bonding

Here is a general algorithm for writing a crystal-chemical formula.

Step one, list atoms: every unique cation site and every unique anion site
Unique means crystallographically distinct, not one entry per atom

Then tag each cation.

[advance: step two, tag cations]

Step two is the cations.

Coordination number: how many neighbors
Geometry letter: t tetrahedral, o octahedral, and so on
Read as one tag: Cr[4t] is four oxygens in a tetrahedron

The anions get the same treatment.

[advance: step three, tag anions]

Step three is the anions.

The number in brackets is how many cations it touches
O[1] terminal, O[2] bridging, O[3] 3-connected

Then check the counts against each other.

[advance: step four, check balances]

Step four is the balances, and there are three of them.

Bonds: what the cations send out = what the anions accept
Anion count: matches the stoichiometry in the formula
Charges: cancel

Last, keep the formula short.

[advance: step five, keep it simple]

Step five: write only the distinct sites and connectivities you need to show the bonding.

A formula that lists every atom in the cell is a formula nobody reads

Let's apply it to the chromium oxide chain from problem 1.22.

Worked solution: CrO₃

GivenCr is tetrahedral; tetrahedra link by corners to form 1D chains

SitesCr, bridging O (O_br), terminal O (O_t)

CoordinationCr[4t]; O_br[2n] (bent Cr–O–Cr); O_t[1]

Balances, per one Cr

Stoichiometry$b + t = 3$

Connectivity$4 = 2b + t$

Solvesubtract$b = 1,\ t = 2$

${}^{1}_{\infty}[\mathrm{Cr}[4t]\,\mathrm{O}[2n]\,\mathrm{O}_{2}[1]]$

One crystallographically distinct infinite chain; each Cr tetrahedral; one bridging O per formula (nonlinear) and two terminal O.

Chromium is tetrahedral, and the tetrahedra link by corners into one-dimensional chains.

That is everything the problem gives us; the rest we count

Start by naming the sites and their coordination.

[advance: the sites and their coordination]

There are three sites: chromium, a bridging oxygen, and a terminal oxygen.

Cr[4t]: four oxygens around it, tetrahedral
O[2n]: two chromiums, bent Cr–O–Cr, which is what the n records
O[1]: one chromium and nothing else

Now count how many of each oxygen there are per chromium.

[advance: the two balances]

Call the bridging count b and the terminal count t, both per one chromium.

Stoichiometry: b + t = 3, the three oxygens per chromium in CrO₃
Connectivity: 2b + t = 4, chromium’s four bonds counted from the oxygen end
Same bonds, counted from both ends, as in the ReO₃ problem

Subtract one from the other.

[advance: the solve and the formula]

Subtracting gives b equals one, and then t equals two.

Superscript 1, subscript infinity: one crystallographically distinct infinite chain
Inside the brackets, each chromium tetrahedral, one bridging O and two terminal per formula unit

Last, where to practice this.

Practice questions

Work the Lecture 6 practice questions before the next class.

Open the Lecture 6 practice questions