Normal Curvature Material Property for Phase-Field Diffuse Interfaces

[laagesen]

Motivation

For off-normal ion irradiation of solid interfaces, etching rate depends on the local curvatures of the interface in directions parallel and perpendicular to the ions' velocity component tangent to the interface. To model off-normal ion irradiation of solid interfaces using the phase-field method, we will need to calculate the local curvatures as a function of the order parameter that represents the solid-vapor interface.

Design

New object: PhaseField/Materials/InterfaceNormalCurvatures

Local frame construction

At each quadrature point a right-handed orthonormal frame {n̂, t̂₁, t̂₂} is built from the order parameter:

n̂  = ∇η / |∇η|                 (unit interface normal)
t̂₁ = (ẑ × n̂) / |ẑ × n̂|       (in-plane tangent; fallback to x̂ when n̂ ≈ ±ẑ)
t̂₂ = n̂ × t̂₁                    (binormal)

Curvature formula

Differentiating n̂ = ∇η / |∇η| and projecting onto each tangent direction via the shape operator S = −∇n̂ gives, for any unit tangent v̂ ⊥ n̂:

κ(v̂) = − (v̂ · H · v̂) / |∇η|

where H = ∇∇η is the Hessian of η. The cross-term (v̂·n̂)(n̂·H·n̂) vanishes exactly by the tangency condition, yielding a compact and efficient formula. The two scalar outputs are:

κ₁ = − (t̂₁ · H · t̂₁) / |∇η|     (in-plane normal curvature)
κ₂ = − (t̂₂ · H · t̂₂) / |∇η|     (out-of-plane normal curvature)

The mean curvature κ = κ₁ + κ₂ = ∇·n̂ is also declared as a diagnostic property.

Impact

None