How it Works
This simulation visualizes two crystal lattices meeting at a grain boundary. The left grain has fixed orientation; the right grain is rotated by the misorientation angle θ about the selected crystallographic axis. Coincident lattice sites (atoms that appear in both lattices) are highlighted — the fraction of such sites determines the CSL Σ value.
For FCC metals, the Σ3 boundary (60° about [111]) is the coherent twin boundary with the lowest possible energy. The Read-Shockley model gives the energy of low-angle boundaries. Brandon's criterion defines the angular tolerance ±15°/√Σ for CSL identification.
Low-angle: E = E₀ · θ · (A − ln θ) [Read-Shockley]
Brandon criterion: Δθ_max = 15° / √Σ
Σ3 twin (FCC): 60° rotation about [111]
Frequently Asked Questions
What is crystal twinning?
Crystal twinning occurs when two crystals share a common crystallographic plane (twin plane) or axis (twin axis) such that one crystal is a mirror image or rotation of the other. The boundary between them is the twin boundary.
What is the Coincidence Site Lattice (CSL)?
CSL theory describes grain boundaries by finding coincident lattice points when two crystal lattices are superimposed. The Sigma (Σ) value equals the reciprocal of the fraction of coincident sites. Σ3 means 1/3 of sites coincide.
Why do low-Σ boundaries have lower energy?
Low-Σ boundaries have more coincident lattice sites, meaning atoms at the boundary fit more naturally into both lattice structures. This reduces the structural mismatch and associated strain energy, resulting in lower interfacial energy.
What is the difference between coherent and incoherent twin boundaries?
A coherent twin boundary is parallel to the twin plane and has very low energy (nearly perfect atomic matching). An incoherent twin boundary is at an angle to the twin plane, has higher energy and rougher atomic structure.
How does misorientation angle affect grain boundary energy?
The Read-Shockley model describes how grain boundary energy increases with misorientation angle for low-angle boundaries (below ~15°): E = E0·θ·(A - ln θ). Above 15°, the boundary becomes high-angle with roughly constant energy until special CSL orientations.
What is a Sigma 3 twin boundary?
The Σ3 boundary, with 60° rotation about [111] in FCC metals, is the most common coherent twin boundary. It has extremely low energy and is found abundantly in copper, nickel, and austenitic stainless steels.
What is grain boundary engineering?
Grain boundary engineering manipulates the distribution of grain boundary types through thermomechanical processing. By increasing the fraction of low-Σ boundaries, materials gain improved corrosion resistance, creep strength, and intergranular cracking resistance.
How are twin boundaries observed experimentally?
Twin boundaries are observed using Electron Backscatter Diffraction (EBSD), which measures crystal orientation at each point. Boundaries with 60°/[111] misorientation are classified as Σ3 twins. TEM can resolve atomic structure at the boundary.
What causes mechanical twinning vs growth twinning?
Growth twins form during crystal solidification or thin-film deposition when a stacking fault occurs. Deformation twins form under mechanical loading, especially in HCP metals (Mg, Ti, Zr) and some BCC metals at low temperatures or high strain rates.
What is the Brandon criterion?
The Brandon criterion defines the maximum allowable deviation from exact CSL misorientation for a boundary to still be considered a Σn boundary: Δθ ≤ 15°/√Σ. This angular tolerance decreases as Σ increases.