Spotlight #26 — Materials Science

Materials science sits at the intersection of physics, chemistry and engineering. Every material property — hardness, conductivity, ductility, melting point — can be traced to atomic-scale structure and defects. This spotlight tours eight simulations that make those connections tangible: you can rotate crystal lattices, watch atoms diffuse along grain boundaries, bend a virtual steel rod past its yield point, and explore why silicon has a band gap while copper does not.

1. Crystal Structures — BCC, FCC, HCP and Diamond

Every solid metal organises its atoms into one of fourteen Bravais lattices. The three most common are body-centred cubic (BCC: iron at room temperature, tungsten, chromium), face-centred cubic (FCC: aluminium, copper, gold, nickel) and hexagonal close-packed (HCP: titanium, magnesium, zinc). The diamond cubic structure (silicon, germanium, diamond) underlies all modern electronics.

Structure   Atoms/cell   Coord.no.   APF     Example
BCC         2            8           0.680   Fe-α, W, Cr
FCC         4            12          0.740   Al, Cu, Au
HCP         6 (eff. 2)   12          0.740   Ti, Mg, Zn
Diamond     8            4           0.340   Si, Ge, C(diamond)

APF = packing efficiency = volume of atoms / unit cell volume
FCC and HCP are both close-packed (APF 0.740, differs only in stacking ABCABC vs ABABAB)

Lattice parameter a (FCC from atomic radius r): a = 2√2 · r
Interplanar spacing (Miller indices hkl, cubic): d_hkl = a / √(h²+k²+l²)
          

The 3D Crystal Structures viewer lets you rotate all six common structures — including NaCl (ionic) and diamond — with depth-sorted rendering (artist's algorithm) so interior atoms are visible. Toggle the 2×2×2 supercell to count coordination shell distances, and watch the coordination number panel update as you switch between structures.

2. Atomic Diffusion and Fick's Laws

Atoms in a solid are not stationary — they hop between lattice sites via vacancies, interstitial positions, or grain boundaries. The macroscopic description is Fick's second law: ∂C/∂t = D·∇²C. But the diffusivity D itself depends on temperature through an Arrhenius relationship — an exponential activation barrier arising from the energy needed to squeeze past neighbouring atoms.

Fick's First Law (steady-state flux):
  J = −D · ∂C/∂x        [atoms m⁻² s⁻¹]

Fick's Second Law (time-dependent):
  ∂C/∂t = D · ∂²C/∂x²   (1D, constant D)

Arrhenius diffusivity:
  D(T) = D₀ · exp(−Qd / kT)    or equivalently exp(−Qd / RT)
  D₀ = pre-exponential factor [m² s⁻¹]
  Qd = activation energy for diffusion [eV or J/mol]

Carburisation solution (semi-infinite solid, constant surface concentration Cs):
  (Cs − C(x,t)) / (Cs − C₀) = erf( x / (2√(Dt)) )
  where C₀ = initial bulk concentration

Typical activation energies (self-diffusion):
  Fe in Fe-α (BCC):  0.84 eV
  Cu in Cu (FCC):    2.04 eV
  B in Si:           3.46 eV  (slow — important for transistor fabrication)
          

The Crystal Diffusion simulation implements a 2D vacancy-mechanism model on a square lattice, with hopping rates set by the Arrhenius law. Raise the temperature slider and watch diffusion fronts spread through an initially segregated material — the Fick profile emerges spontaneously from millions of individual atomic jumps.

3. Mechanical Behaviour — Stress, Strain and Fracture

Pull a metal rod with increasing force and it first stretches elastically (Hooke's law, reversible), then yields into plastic deformation (permanent, driven by dislocation motion), work-hardens as dislocations tangle, reaches ultimate tensile strength, necks, and finally fractures. The entire curve encodes the alloy's microstructure: a finer grain size raises the yield stress (Hall-Petch law), while precipitate particles pin dislocations (precipitation hardening, the basis of aluminium aerospace alloys).

Elastic region:     σ = E · ε            (Young's modulus E, typically 70–210 GPa for metals)
Poisson's ratio:    ν = −ε_lateral / ε_axial   (ν ≈ 0.3 for most metals)
Yield criterion (von Mises): σ_eff = √½·[(σ₁−σ₂)²+(σ₂−σ₃)²+(σ₃−σ₁)²] = σ_y

Power-law hardening (plastic): σ = σ_y + K · ε_p^n
  K = hardening coefficient, n = strain-hardening exponent (0.1–0.5)

Hall-Petch (grain-size strengthening): σ_y = σ₀ + k / √d
  d = mean grain diameter, k ≈ 0.7 MPa·m^½ (steel)

Griffith fracture criterion:
  σ_f = √(2Eγ / πa)    for a through-crack of half-length a
  Fracture toughness:   K_Ic = σ_f · Y · √(πa)   [MPa·m^½]
          

The Stress-Strain simulation animates a tensile specimen with real-time necking and fracture. Choose from Steel (E = 200 GPa), Aluminium (E = 69 GPa), Rubber (hyper-elastic), Bone (anisotropic), or Polymer (viscoelastic creep). The toughness area — integral of σdε — updates live so you can compare energy absorption before fracture. The Fracture Mechanics simulation adds crack-tip stress concentration: watch the crack propagate as cyclic loading accumulates fatigue damage.

4. Dislocations — Why Metals are So Much Weaker Than Theory Predicts

The theoretical shear stress to slide one crystal plane past another is τ_th = G/(2π) ≈ 10–30 GPa. But real metals yield at 50–500 MPa — 100× weaker. The explanation, found independently by Taylor, Orowan and Polanyi in 1934, is the dislocation: a line defect at which slip has started but not yet completed. Only the atoms near the dislocation core move at any instant; the dislocation glides through the crystal like a wrinkle through a carpet.

Burgers vector b: direction and magnitude of lattice distortion
  |b| = a/2 · √(h²+k²+l²) for FCC slip system ⟨110⟩{111}

Edge dislocation stress field:
  σ_xx = − Gb/(2π(1−ν)) · y(3x²+y²)/(x²+y²)²
  σ_yy =   Gb/(2π(1−ν)) · y(x²−y²)/(x²+y²)²
  τ_xy =   Gb/(2π(1−ν)) · x(x²−y²)/(x²+y²)²

Peierls-Nabarro (lattice friction) stress:
  τ_PN ≈ 2G · exp(−2πw/b)  where w = dislocation width

Dislocation strengthening (Taylor hardening):
  Δσ = M · α · G · b · √ρ
  ρ = dislocation density [m⁻²], α ≈ 0.3, M = Taylor factor ≈ 3.06

Frank-Read source multiplication: dislocations bow between two pinning points
  and sweep complete loops when τ ≥ Gb/(2L),  L = pin spacing
          

The Dislocation Slip simulation shows edge and screw dislocations gliding through a 2D crystal under applied shear stress. Raise the stress and watch dislocations nucleate, multiply via Frank-Read sources, pile up at grain boundaries, and eventually cause yielding. The colour overlay shows the local shear stress concentration around each dislocation core.

5. Phase Diagrams — Two-Component Alloys

A binary phase diagram maps which phases (solid, liquid, mixed) are thermodynamically stable as a function of temperature and composition. The eutectic point is the composition-temperature pair at which the liquid freezes directly into two solid phases simultaneously — the lowest melting point in the system. The lever rule gives the fraction of each phase in a two-phase region without solving any equations.

Gibbs Phase Rule:   F = C − P + 2
  F = degrees of freedom (intensive variables freely adjustable)
  C = number of components, P = number of phases

At constant pressure (isobaric): F = C − P + 1

Binary eutectic (C=2):
  One-phase region (liquid or solid): F = 2 (temperature + composition)
  Two-phase region (liquidus + solidus): F = 1 (fix T → both compositions fixed)
  Eutectic point (3 phases, liquid + 2 solids): F = 0 (invariant)

Lever Rule (fraction of α phase at composition X_0 between Xα and Xβ):
  f_α = (X_β − X_0) / (X_β − X_α)
  f_β = (X_0 − X_α) / (X_β − X_α)

Martensitic transformation: diffusionless shear (Fe-C austenite → martensite)
  Start temperature: Ms ≈ 539 − 423(%C) − 30.4(%Mn) − 17.7(%Ni) − 12.1(%Cr) [°C]
          

The Alloy Phase Diagram simulation draws a full Pb-Sn style binary eutectic phase diagram. Click anywhere in the diagram and the lever rule calculation appears automatically — the relative lengths of the tie line segments scale the phase fractions. You can also simulate a cooling path: follow a composition vertically downward and watch the microstructure form in the animated inset.

6. Semiconductor Band Structure

Why does silicon conduct electricity only when doped, while copper conducts freely, and diamond does not conduct at all? The answer lies in the electronic band structure — the allowed energy levels for electrons in a periodic lattice. In metals the Fermi level sits inside a partially filled band. In semiconductors a band gap separates the filled valence band from the empty conduction band; thermal energy or photons (or dopant atoms) can bridge that gap. In insulators the gap is too wide.

Nearly-free-electron band gap at zone boundary k = π/a:
  E_gap = 2|V_G|   (V_G = Fourier component of periodic potential)

Fermi-Dirac distribution (probability state E is occupied):
  f(E) = 1 / (exp((E − E_F) / kT) + 1)

Intrinsic carrier concentration:
  n_i = √(N_c · N_v) · exp(−E_g / 2kT)
  N_c = 2(2πm_e*kT/h²)^(3/2),  N_v similar with m_h*

n-type doping (donor atoms with level E_d near conduction band):
  Freeze-out → saturation → intrinsic regime as T rises
  n ≈ N_d (at moderate T, complete ionisation)

Hall coefficient (carrier sign determination):
  R_H = −1/(ne)  (n-type)    R_H = +1/(pe) (p-type)

Direct vs indirect band gap:
  Direct (GaAs, GaN): momentum-conserved optical transitions → LED, laser
  Indirect (Si, Ge): phonon required for transition → poor emitter
          

The Semiconductor Bands simulation draws the E–k dispersion for a 1D Kronig-Penney model and lets you adjust the potential well depth, spacing, and temperature. The Fermi-Dirac tail fills in as you increase temperature — you can see the crossover from extrinsic (carrier density set by dopants) to intrinsic (carrier density set by band gap and kT) regimes. Separately, the doping panel shows n-type and p-type donor/acceptor levels sitting just inside the band gap.

Why Materials Science Deserves More Simulations

Materials science is one of the most simulation-friendly fields in the physical sciences. Molecular dynamics, finite-element analysis, density functional theory, phase-field models — all of these are routinely used in materials research, and their visual outputs are immediately intuitive. A dislocation moving through a crystal looks like what it is. A phase boundary migrating across a grain looks like what it is.

Yet materials science is often missing from online physics education. Most introductory courses jump straight from atomic orbitals to thermodynamics without visiting the crystalline solid in between. This spotlight attempts to fill that gap with simulations anchored in real material parameters and real equations — not toys.

Coming in future waves: dislocation density evolution during cold working, the Lennard-Jones pair potential connecting interatomic forces to elastic constants, grain growth under annealing (normal and abnormal), and a full p-n junction simulation showing the depletion region and IV curve.