How the structure of metals, ceramics, polymers and composites sets their properties: crystal structure, defects and diffusion, mechanical, thermal and electrical properties, failure, phase diagrams, heat treatment and materials selection.

Materials Science Fundamentals

## Why it matters

Every engineered object is made of something, and choosing that something well is often the difference between a product that lasts decades and one that cracks, corrodes or costs too much. Materials science explains *why* materials behave as they do, by linking four things that engineers often draw as a chain:

```text
processing  →  structure  →  properties  →  performance
```

How a material is made (cast, rolled, sintered, moulded, heat treated) sets its internal structure (atoms, crystals, grains, defects). Structure sets properties (stiffness, strength, conductivity). Properties decide how the part performs in service. Change one link and the others change too. The rest of this course follows that chain.

## The four main classes

Most engineering materials fall into four groups, largely because of how their atoms are bonded.

**Metals and alloys.** Atoms share a "sea" of free electrons (metallic bonding). That makes metals good conductors of heat and electricity, opaque and shiny, and, importantly, **ductile**: they can deform a lot before breaking, so they bend or dent instead of shattering. Examples: steel, aluminium alloys, copper, titanium. Typical densities range from about 2.7 g/cm³ (aluminium) through 4.5 (titanium) to 7.8 (steel) and 8.9 (copper).

**Ceramics and glasses.** Compounds of metals with non-metals (oxides, nitrides, carbides) held by strong ionic and covalent bonds. They are hard, stiff, resistant to heat and corrosion, and usually electrical insulators, but **brittle**: they break suddenly with little warning. Examples: alumina, silicon carbide, porcelain, window glass, cement and concrete.

**Polymers.** Long chain molecules of carbon and hydrogen (often with oxygen, nitrogen or chlorine), with strong covalent bonds along each chain and weak secondary bonds between chains. They have low density (around 0.9 to 1.5 g/cm³), are easy to shape, resist many chemicals, and insulate well, but have low stiffness and soften at modest temperatures. Three families:

- **thermoplastics** soften on heating and can be remoulded (polyethylene, PVC, nylon, PET);
- **thermosets** cross-link permanently when cured and do not melt (epoxy, phenolics);
- **elastomers** are lightly cross-linked and stretch a lot elastically (natural and synthetic rubbers).

**Composites.** Two or more materials combined so the result outperforms either alone, usually strong fibres in a matrix. Glass-fibre and carbon-fibre reinforced polymers are stiff and light; reinforced concrete puts steel where concrete is weak (in tension); wood is a natural composite of cellulose fibres in a lignin matrix. Composites are often **anisotropic**: much stronger and stiffer along the fibres than across them.

Two more groups are often listed separately: **semiconductors** such as silicon, whose conductivity can be controlled precisely by tiny additions of impurities, and **biomaterials**, chosen to work safely inside the body.

## A rough comparison

| property | metals | ceramics | polymers | composites (CFRP) |
|---|---|---|---|---|
| density | medium to high | medium | low | low |
| stiffness (E) | high | very high | low | high along fibres |
| toughness | high | low | medium | medium |
| max service temperature | medium to high | very high | low | low to medium (set by the matrix) |
| electrical conductivity | high | very low | very low | varies |

These are generalisations. Within each class the range is enormous: a soft lead alloy and a high-strength tool steel are both metals.

## Worked example: picking a class

A designer needs three parts.

1. **A kitchen knife blade.** Needs hardness to hold an edge and enough toughness not to chip when dropped. A hardened steel is the usual answer; ceramic (zirconia) blades are harder and stay sharp longer but chip more easily.
2. **A furnace lining.** Must survive 1,400 °C and insulate. Only ceramics (refractory bricks, alumina fibre) work.
3. **A bicycle frame.** Needs stiffness and strength at low weight. Steel, aluminium, titanium and carbon-fibre composite are all used; the choice depends on cost, ride feel and how damage-tolerant it must be. The last lesson of this course turns this kind of reasoning into numbers.

## Common mistakes

- **Equating strength with hardness or stiffness.** Glass is stiff and hard but weak in tension because it breaks at tiny flaws; rubber is soft but very tough.
- **Assuming every plastic is weak.** Engineering polymers and fibre composites carry serious loads in cars and aircraft.
- **Forgetting processing.** The same steel can be soft and ductile or hard and brittle depending on heat treatment.

## Recap

- Processing sets structure, structure sets properties, properties set performance.
- Metals: ductile and conductive. Ceramics: hard, stiff, heat resistant, brittle. Polymers: light, easy to shape, low stiffness and temperature limit. Composites: tailored combinations, often anisotropic.
- Bonding explains most of the differences between the classes.


Compare metals, ceramics, polymers and composites and link their properties to bonding.

Classes of materials

What is Materials Science and Engineering?

What is materials science?

## Why it matters

Most metals and many ceramics are **crystalline**: their atoms sit in a regular, repeating 3D pattern. The pattern matters. Whether a metal is easily formed or tends to crack in the cold, how densely its atoms pack, and how it responds to heat treatment all depend on its crystal structure. Three structures cover most engineering metals, and learning them is the first step to reading the microstructure of anything metallic.

## Lattices and unit cells

A crystal can be described by its **unit cell**: the smallest box of atoms that, repeated in every direction, builds the whole crystal. For cubic cells the box is a cube with edge length a, the **lattice parameter**, usually a few tenths of a nanometre. Treat atoms as hard spheres of radius R touching their nearest neighbours.

Four numbers describe each structure:

- **n**, the number of atoms belonging to one unit cell (corner atoms are shared by 8 cells, face atoms by 2);
- the **coordination number** (CN), how many nearest neighbours each atom touches;
- the **relation between a and R**, from where the atoms touch;
- the **atomic packing factor** (APF), the fraction of the cell volume filled by atoms.

## The three common metal structures

**Body-centred cubic (BCC).** An atom at each corner and one in the centre. Atoms touch along the body diagonal, so 4R = a·√3.

```text
n = 8 × 1/8 + 1 = 2      CN = 8      a = 4R/√3      APF = 0.68
```

Examples: iron at room temperature (ferrite), chromium, tungsten, molybdenum.

**Face-centred cubic (FCC).** An atom at each corner and one in the centre of each face. Atoms touch along the face diagonal, so 4R = a·√2.

```text
n = 8 × 1/8 + 6 × 1/2 = 4      CN = 12      a = 2R·√2      APF = 0.74
```

Examples: aluminium, copper, nickel, gold, silver, and iron above 912 °C (austenite).

**Hexagonal close-packed (HCP).** Layers of close-packed atoms stacked in an ABAB pattern. Like FCC it is as dense as identical spheres can be:

```text
n = 6 per hexagonal cell      CN = 12      APF = 0.74      ideal c/a = 1.633
```

Examples: titanium, magnesium, zinc, cobalt.

FCC and HCP are both close packed; they differ only in the stacking of the close-packed layers (ABCABC for FCC, ABAB for HCP).

## Why structure changes behaviour

Metals deform plastically by planes of atoms slipping past each other. FCC metals have many close-packed slip planes and directions, so they are typically ductile even at low temperature: aluminium and copper can be drawn into fine wire. BCC metals, including ordinary steels, can become brittle when cold (the ductile-to-brittle transition, covered later). HCP metals have fewer easy slip systems and are often less formable at room temperature.

Some materials change crystal structure with temperature. This is **polymorphism** (called allotropy for pure elements). Iron is BCC below 912 °C, FCC from 912 to 1,394 °C, then BCC again up to its melting point. That change is the basis of steel heat treatment.

## Worked example: the density of copper

The theoretical density of a crystal is the mass of the atoms in one unit cell divided by the cell's volume:

```text
ρ = n · A / (Vc · NA)
```

where A is the atomic mass (g/mol), Vc = a³ for a cubic cell, and NA = 6.022 × 10²³ atoms/mol.

Copper is FCC with R = 0.128 nm and A = 63.55 g/mol.

```text
a  = 2 · 0.128 · √2 = 0.3620 nm = 3.620 × 10⁻⁸ cm
Vc = a³ = 4.745 × 10⁻²³ cm³
ρ  = 4 × 63.55 / (4.745 × 10⁻²³ × 6.022 × 10²³) = 254.2 / 28.57 = 8.90 g/cm³
```

The measured density of copper is 8.96 g/cm³: within 1 %. Doing the same for BCC iron (R = 0.124 nm, A = 55.85 g/mol) gives a = 0.2864 nm and ρ ≈ 7.90 g/cm³, against a measured 7.87 g/cm³.

Watch the units: convert nanometres to centimetres (1 nm = 10⁻⁷ cm) so the answer comes out in g/cm³.

## Polycrystals and non-crystalline materials

A real metal part is **polycrystalline**: millions of small crystals (grains), each with the same structure but a different orientation, meeting at grain boundaries. Glasses and many polymers are **amorphous**: their atoms or chains have no long-range order. That is why glass softens gradually instead of melting at one temperature.

## Common mistakes

- **Counting corner atoms as whole atoms.** Each corner atom is shared by eight cells.
- **Using the wrong diagonal.** BCC atoms touch along the body diagonal; FCC atoms touch along the face diagonal.
- **Unit slips** between nm, Å (1 Å = 0.1 nm) and cm in density calculations.

## Recap

- BCC: 2 atoms, CN 8, APF 0.68. FCC: 4 atoms, CN 12, APF 0.74. HCP: CN 12, APF 0.74.
- ρ = n·A / (a³·NA) predicts density within about 1 % for pure metals.
- Iron changes from BCC to FCC on heating, which makes steel heat treatment possible.

## Try it

On the Do tab, finish the density function and compute the densities of copper and iron from their crystal structures.


Identify FCC, BCC and HCP structures and compute a metal's density from its unit cell.

Crystal structure

Unit Cell Chemistry   Simple Cubic, Body Centered Cubic, Face Centered Cubic Crystal Lattice Structu

## Why it matters

No real crystal is perfect. Missing atoms, extra atoms, foreign atoms, misaligned rows and boundaries between grains are present in every metal and ceramic. Far from being a nuisance, these **defects** are what engineers manipulate to control properties. Pure iron is soft; steel is strong mainly because of how carbon atoms and dislocations interact. Doping silicon with a few parts per million of phosphorus turns it into a useful semiconductor. And atoms can only move through a solid, which is how case hardening, sintering and many failures happen, because of defects.

## Point defects

- **Vacancy:** a lattice site with no atom. Every crystal above absolute zero contains some, and the number rises steeply with temperature.
- **Self-interstitial:** an extra atom of the host squeezed between normal sites. Rare in metals because it distorts the lattice heavily.
- **Substitutional impurity:** a foreign atom on a normal site (nickel in copper, zinc in brass).
- **Interstitial impurity:** a small foreign atom in the gaps between host atoms (carbon in iron, hydrogen in steel).

The equilibrium fraction of vacant sites follows an Arrhenius law:

```text
Nv / N = exp(−Qv / (k·T))
```

where Qv is the energy to form a vacancy, k = 8.62 × 10⁻⁵ eV/K is Boltzmann's constant and T is in kelvin. For copper (Qv ≈ 0.9 eV) at 1,000 °C (1,273 K):

```text
Nv / N = exp(−0.9 / (8.62 × 10⁻⁵ × 1273)) = exp(−8.20) ≈ 2.7 × 10⁻⁴
```

About one site in 3,700 is empty near the melting point; at room temperature the fraction is vanishingly small.

## Line defects: dislocations

A **dislocation** is a line where the crystal's rows of atoms are mismatched: for an edge dislocation, picture an extra half-plane of atoms pushed into the crystal. Under shear stress, the dislocation can glide one atomic row at a time, like moving a rug by pushing a ruck along it rather than dragging the whole rug. This is why metals yield at stresses hundreds or thousands of times lower than the theoretical strength of a perfect crystal.

It also explains how metals are strengthened. Anything that makes dislocations harder to move raises the yield strength:

- **solid solution strengthening:** impurity atoms distort the lattice and pin dislocations;
- **work (strain) hardening:** plastic deformation multiplies dislocations until they tangle (bend a paper clip back and forth and it gets harder before it breaks);
- **grain size reduction:** grain boundaries block dislocations, so finer grains mean higher strength (the Hall-Petch relation);
- **precipitation hardening:** tiny particles of a second phase obstruct dislocations (heat-treatable aluminium alloys).

## Planar and volume defects

**Grain boundaries** separate crystals of different orientation. **Twin boundaries** are mirror-image boundaries. **Volume defects** include pores, cracks, inclusions (non-metallic particles left from steelmaking) and voids. Volume defects are usually harmful: a pore or inclusion concentrates stress and can start a fatigue crack.

## Diffusion

Atoms in a solid jump from site to site. Substitutional atoms usually need a neighbouring vacancy to move into (vacancy diffusion); small interstitial atoms such as carbon hop between gaps (interstitial diffusion), which is much faster.

**Fick's first law** gives the flux J (atoms or kg passing a unit area per second) under a concentration gradient:

```text
J = −D · dC/dx
```

The **diffusion coefficient** D depends strongly on temperature:

```text
D = D0 · exp(−Qd / (R·T))
```

with Qd the activation energy (J/mol), R = 8.314 J/(mol·K), and T in kelvin.

## Worked example: carburizing steel

Carburizing hardens the surface of a steel gear by diffusing carbon into it at high temperature, while the iron is FCC. For carbon in FCC iron, D0 ≈ 2.3 × 10⁻⁵ m²/s and Qd ≈ 148 kJ/mol.

```text
at 1,000 °C (1,273 K):  D = 2.3 × 10⁻⁵ × exp(−148,000 / (8.314 × 1273)) ≈ 1.9 × 10⁻¹¹ m²/s
at   900 °C (1,173 K):  D ≈ 5.9 × 10⁻¹² m²/s
```

A 100 °C drop cuts D by a factor of about 3.3. A useful rule of thumb is that atoms spread a distance of order √(D·t): in one hour at 1,000 °C, √(1.9 × 10⁻¹¹ × 3,600) ≈ 0.26 mm. That is why case depths of around a millimetre take hours in the furnace, and why diffusion in metals at room temperature is negligible for most purposes.

## Common mistakes

- **Using °C in Arrhenius equations.** Always convert to kelvin.
- **Mixing eV per atom with J per mole.** Use k with eV and R with J/mol.
- **Thinking defects are always bad.** Controlled defects are how most strengthening and doping work.

## Recap

- Point, line, planar and volume defects exist in all real crystals.
- Dislocation motion is how metals deform; blocking it makes them stronger.
- Diffusion follows J = −D·dC/dx, and D rises exponentially with temperature.
- Distance diffused scales roughly with √(D·t).

## Try it

On the Do tab, finish the Arrhenius function for the diffusion coefficient and compare carburizing temperatures.


Explain how vacancies, dislocations and grain boundaries change properties, and how atoms diffuse in solids.

Defects and diffusion

Defects in Crystals

Diffusion: Introduction

Meet the four classes of materials and see how atoms arrange themselves in crystals, and how defects and diffusion in those crystals control properties.