Imperfections, Dislocations & Diffusion

The flaws that make materials useful. Learn point defects (vacancies, interstitials, substitutional atoms) and their Arrhenius concentration, solid solutions and Hume-Rothery rules, line defects (edge and screw dislocations, Burgers vector) and why they make metals ductile, planar defects (grain boundaries, twins, stacking faults), grain size, volume defects, and diffusion mechanisms, Fick's first and second laws, the error-function solution, temperature dependence of diffusivity, and industrial diffusion processes (carburising, nitriding, semiconductor doping) — with Python for vacancy fractions and carburising depth.

Perfect crystals don't exist, and that's fortunate: dislocations allow metals to deform plastically instead of shattering, alloying atoms strengthen them, and vacancies allow atoms to diffuse — enabling heat treatment, surface hardening and chip doping.

Most strengthening mechanisms in metals work by making dislocation motion harder.

An analogy

Moving a large carpet across a floor is hard; pushing a ruck (wrinkle) through it is easy. A dislocation is that ruck: it lets crystal planes slip one row of atoms at a time, which is why metals yield at stresses thousands of times lower than the theoretical strength.

Point defects and alloys

Vacancies (missing atoms) exist in equilibrium and increase exponentially with temperature. Self-interstitials are rare in metals. Impurity atoms form solid solutions: substitutional (Cu–Ni, similar size) or interstitial (C, N, H in iron — small atoms in gaps). Hume-Rothery rules for extensive substitutional solubility: atomic radii within 15%, same crystal structure, similar electronegativity and valence.

Dislocations and strengthening

Key ideas:

Planar defects and microstructure

Grain boundaries separate differently oriented crystals — they block dislocations (strengthening), are fast diffusion paths and preferred corrosion sites. Twin boundaries (mirror-image), stacking faults, and phase boundaries also matter. ASTM grain size number relates to grains per area; fine grains give both higher strength and toughness. Volume defects include pores, inclusions and cracks.

Diffusion in practice

Mechanisms: vacancy diffusion (substitutional atoms) and interstitial diffusion (C, N, H — much faster). Short-circuit paths (grain boundaries, surfaces, dislocations) dominate at low temperature. Applications: carburising and nitriding of gears and shafts, decarburisation risks, homogenisation of castings, sintering of powders, dopant diffusion and oxidation in silicon, hydrogen embrittlement, and diffusion bonding. Kirkendall effect shows vacancy-mediated diffusion.

Formulas

Worked example: Diffusion time scaling

Given: 0.5 mm case takes 10 h

Find: Time for 1.0 mm at same T.

Answer: 40 h

Worked example: Fick's first law

Given: D = 3 × 10⁻¹¹ m²/s, ΔC = 1.2 kg/m³ over 5 mm

Find: J.

Answer: 7.2 × 10⁻⁹ kg/m²·s

Worked example: Vacancies per m³

Given: Cu: 8.5 × 10²⁸ atoms/m³, fraction 6.5 × 10⁻⁵ at 1300 K

Find: N_v.

Answer: 5.5 × 10²⁴ vacancies/m³

Vacancies and carburising depth

import math
# Vacancy concentration (Arrhenius) and gas carburising depth with Fick's second law
k = 8.617e-5                                   # eV/K
Qv = 1.08                                      # eV, vacancy formation in Cu
for T in (300, 800, 1300):
    print(f"T = {T} K: vacancy fraction {math.exp(-Qv/(k*T)):.2e}")
# Carburising 1020 steel (0.20% C) with surface held at 1.00% C, 927 C
D0, Qd, R = 2.3e-5, 148000, 8.314              # m2/s, J/mol (C in gamma-Fe)
T = 927 + 273
D = D0 * math.exp(-Qd / (R * T))
Cs, C0, Cx = 1.00, 0.20, 0.60
# (Cs - Cx)/(Cs - C0) = erf(x / (2 sqrt(D t)))  -> solve for x at target Cx
target = (Cs - Cx) / (Cs - C0)                  # = 0.5
lo, hi = 0.0, 3.0
for _ in range(60):
    mid = (lo + hi) / 2
    if math.erf(mid) < target: lo = mid
    else: hi = mid
for hours in (4, 10):
    t = hours * 3600
    x = 2 * lo * math.sqrt(D * t)
    print(f"D = {D:.2e} m2/s; after {hours} h, 0.60% C lies {x*1000:.2f} mm below the surface")
# T = 300 K: vacancy fraction 7.18e-19
# T = 800 K: vacancy fraction 1.57e-07
# T = 1300 K: vacancy fraction 6.50e-05
# D = 8.30e-12 m2/s; after 4 h, 0.60% C lies 0.33 mm below the surface
# D = 8.30e-12 m2/s; after 10 h, 0.60% C lies 0.52 mm below the surface

Vacancy concentration rises by 14 orders of magnitude from room temperature to near the melting point — that's why diffusion needs heat. In carburising, the case depth grows only with √t: 2.5× the time gives just 1.58× the depth. Raising temperature increases D exponentially, so furnaces run hot (900–950 °C) to cut cycle times.

Common mistakes

Applications

Key takeaways

Going further