Defects and diffusion

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:

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):

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:

J = −D · dC/dx

The diffusion coefficient D depends strongly on temperature:

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.

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.