Chapter 2 — Case Study 2: Diamond, Graphite, and Graphene — Three Forms of Carbon

How Chapter 2's hybridization concepts explain three of the most important materials on Earth.

1. The puzzle

Carbon exists in nature in several stable forms called allotropes. Three of them — diamond, graphite, and graphene — have very different physical properties despite being made of exactly the same atoms.

• Diamond is the hardest naturally occurring material on Earth (Mohs hardness 10), has extraordinarily high thermal conductivity (2,200 W/m·K, higher than copper), is electrically insulating, and is transparent to visible light.
• Graphite is soft and slippery (feels greasy to the touch), is a reasonable electrical conductor (more than 3,000 S/cm in-plane), and is black.
• Graphene (a single atomic layer of graphite) is the strongest material ever measured (tensile strength of about 130 GPa), has the highest electron mobility ever observed in any material (200,000 cm²/V·s at room temperature), and is essentially transparent to visible light.

All three are pure carbon. How can the same element produce materials this different?

The answer is hybridization. Diamond is $sp^3$. Graphite and graphene are $sp^2$. Every property above can be traced back to that single fact.

2. Diamond: $sp^3$ carbon in three dimensions

In diamond, every carbon atom is $sp^3$-hybridized. Each carbon is bonded to four other carbons in a regular tetrahedral arrangement, with $C-C$ bond lengths of 1.54 Å and bond angles of 109.5°. The network extends in all three dimensions — every carbon is surrounded by four carbons, which are surrounded by four more, and so on, forming a giant covalent crystal.

Three consequences:

1. Hardness. Breaking diamond requires breaking many $C-C$ $\sigma$ bonds simultaneously. The $C-C$ BDE is 83 kcal/mol, and a macroscopic diamond crystal contains an enormous number of such bonds. Scratching, cutting, or crushing diamond requires enough energy to break entire planes of bonds. The $sp^3$ network is the stiffest configuration of covalent bonds known.
2. Electrical insulator. All the valence electrons of diamond are tied up in $\sigma$ bonds, which are localized between specific pairs of atoms. There are no free or delocalized electrons. The energy gap between the filled $\sigma$ bonding orbitals and the empty $\sigma^*$ antibonding orbitals is huge (about 5.5 eV, well above the thermal energy at room temperature). So diamond does not conduct electricity.
3. Transparent. That same 5.5 eV band gap is larger than the photon energies of visible light (which span about 1.6 to 3.1 eV). Visible-light photons cannot be absorbed by diamond, so they pass through unchanged. Diamond is transparent for the same reason window glass is transparent — the band gap is in the ultraviolet.

Synthetic diamond, produced at high pressure and high temperature (HPHT) or by chemical vapor deposition (CVD), is a multi-billion-dollar industry today. Industrial diamond is used for cutting and grinding, for heat sinks in semiconductor electronics, and increasingly as a substrate for quantum sensors and quantum computers.

3. Graphite: $sp^2$ carbon in sheets

In graphite, every carbon atom is $sp^2$-hybridized. Each carbon forms three $\sigma$ bonds to three neighboring carbons, with bond angles of 120°, creating a flat honeycomb (hexagonal) pattern. The unhybridized $2p_z$ orbital on each carbon, perpendicular to the sheet, contributes one electron to a delocalized $\pi$ system that extends across the entire sheet.

The sheets stack on top of each other, held together only by weak van der Waals interactions. Successive sheets can slide over each other easily.

Three consequences:

1. Softness and slipperiness. The in-plane $C-C$ bonds are strong (1.42 Å, shorter than diamond because of the added $\pi$ component). But the between-plane interactions are weak. Graphite shears easily — you can rub a graphite rod against paper and transfer a layer of carbon (this is how a pencil works). Graphite is used industrially as a dry lubricant for the same reason.
2. Electrical conductor. The delocalized $\pi$ system means that electrons can move along the sheet without being stuck between specific atoms. Graphite conducts electricity in-plane comparably to many metals. It does not conduct well between planes, because each plane's $\pi$ system is separate.
3. Black color. Unlike diamond, graphite has no band gap — the $\pi$ and $\pi^*$ orbitals span a range of energies that overlap with the visible spectrum. Graphite absorbs visible-light photons of essentially all energies, which is why it appears black.

The lead in your pencil is graphite. So is the graphite anode of every lithium-ion battery in your phone, laptop, and electric car. So is the graphite moderator in the Chicago Pile-1 nuclear reactor that Enrico Fermi operated in 1942. Graphite is quietly everywhere.

4. Graphene: a single sheet of $sp^2$ carbon

Graphene is, by definition, one sheet of graphite taken in isolation. A single atomic layer. Two-dimensional carbon.

For decades after the structure of graphite was determined, physicists argued about whether a truly isolated graphene sheet could even exist. Thermodynamic arguments suggested that such a thin sheet should be unstable and would spontaneously curl up or tear. In 2004, Andre Geim and Konstantin Novoselov at the University of Manchester solved the problem with a method so crude it is almost embarrassing: they peeled graphite flakes onto transparent tape, pressed the tape onto a silicon wafer, and repeatedly peeled the tape apart until they had isolated single-layer sheets.

They found that the isolated graphene sheets were, in fact, perfectly stable. Twelve years later, Geim and Novoselov shared the 2010 Nobel Prize in Physics for the discovery.

Graphene's properties are extraordinary:

• The strongest material ever measured, about 200 times stronger than steel by weight.
• Electron mobility approximately 100 times higher than silicon at room temperature. Electrons in graphene behave as if they were massless — they obey the Dirac equation of relativistic quantum mechanics, not the usual Schrödinger equation of ordinary solid-state physics. This is unique among materials.
• Single-atom-thick and essentially transparent (97.7% transmittance for visible light).
• Thermal conductivity of about 5,000 W/m·K, higher than any other known material.

The practical applications of graphene have been slower to arrive than the theoretical excitement predicted — making large, defect-free sheets of graphene in commercially useful quantities remains a challenge. But graphene is already used in specialty electronics, flexible displays, high-strength composites, and biomedical sensors. And the broader family of 2D materials it inaugurated — molybdenum disulfide, hexagonal boron nitride, black phosphorus, and others — is a growth area in both academic and industrial research.

5. The chapter-2 lesson

Diamond, graphite, and graphene are the same element organized differently. The difference is hybridization — $sp^3$ versus $sp^2$ — and whether the unhybridized $p$ orbitals form a localized or delocalized $\pi$ system.

This is a remarkable illustration of a point Chapter 2 has been building toward: the physical and chemical properties of any material are set by how its atoms are electronically organized. You cannot tell by looking at a chemical formula whether you are holding something as hard as diamond or as slippery as graphite. You can tell once you know the hybridization.

Every organic molecule in this book obeys the same logic. An $sp^3$ carbon is a tetrahedral joint in a three-dimensional network. An $sp^2$ carbon is a planar joint in a potentially-delocalized $\pi$ system. An $sp$ carbon is a linear spacer with a pair of perpendicular $\pi$ bonds. These three geometries, and no others for organic carbon, organize the whole subject.

When you meet benzene in Chapter 20, you will recognize it as a closed, aromatic ring of six $sp^2$ carbons with a six-electron delocalized $\pi$ system — a molecular analogue, in a sense, of an isolated ring of graphene. When you meet carbonyls in Chapter 24, you will recognize the $C=O$ as an $sp^2$-carbon + $sp^2$-oxygen unit with a localized $\pi$ bond. When you meet polymers in Chapter 40, you will recognize polyethylene as a chain of $sp^3$ carbons (no $\pi$ system, flexible, insulating) and polyacetylene as a chain of $sp^2$ carbons with a delocalized $\pi$ system (conducting, colored).

The hybridization model, which seemed at first like a bookkeeping exercise to explain why carbon forms four bonds, turns out to be the deepest organizing principle for all of carbon's chemistry. Carry it forward.

Further reading. Novoselov, K. S., Geim, A. K., et al. (2005). Two-dimensional atomic crystals. Proceedings of the National Academy of Sciences, 102(30), 10451–10453. The paper that announced isolated 2D materials. Approachable for a student who has finished Chapter 2 of this book.