Phonon Interface Scattering: Kapitza Boundary Resistance

Interface Model

Materials

Sound speed, medium A (hot) 1.60

Sound speed, medium B (cold) 1.00

Impedance ratio Z_B/Z_A 1.00

Simulation speed med

Live Readouts

Normal-incidence T(0°)

Critical angle θ_c

Measured ⟨T⟩ over all hits (n=0)

How it works

A phonon carrying vibrational energy across an interface between two materials is partly transmitted and partly reflected, exactly like a sound or elastic wave hitting a boundary. Each material is characterised by its acoustic impedance Z = ρc (density × sound speed). In the acoustic mismatch model (AMM), the interface behaves like an ideal elastic mirror: the wavevector obeys Snell's law and the transmitted energy fraction follows a Fresnel-like formula:

Snell:  sinθ1 / c1 = sinθ2 / c2
Transmission (AMM):
  T(θ1) = 4 Z1 Z2 cosθ1 cosθ2 / (Z1 cosθ2 + Z2 cosθ1)²
Normal incidence: T(0) = 4 Z1 Z2 / (Z1 + Z2)²

If c2 > c1, there is a critical angle θ_c = arcsin(c1/c2) beyond which sinθ2 would exceed 1 — no real refracted wave exists, so the phonon undergoes total internal reflection, just like light at a fibre-optic core/cladding boundary.

Real interfaces are rarely atomically perfect mirrors. The diffuse mismatch model (DMM) assumes a phonon completely forgets its incidence direction at the interface: transmission becomes angle-independent (here approximated as T ≈ Z2/(Z1+Z2), weighted toward the higher-impedance side) and the outgoing direction is re-randomised over a cosine-weighted hemisphere on whichever side it ends up. Real rough or disordered interfaces sit somewhere between the AMM and DMM limits.

Sound speed sliders — set c1, c2 in each medium; together with the impedance ratio they fix the Snell refraction and critical angle.
Impedance ratio Z_B/Z_A — the acoustic mismatch that drives the Fresnel-like transmission coefficient.
Mode toggle — switch between specular AMM refraction/reflection and diffuse DMM scattering.
The measured transmitted energy flux is inversely related to the interface's Kapitza (thermal boundary) resistance: a well-matched interface (Z1≈Z2) transmits almost every phonon and has low R_K; a strongly mismatched one reflects most phonons and has high R_K — this is the dominant heat-flow bottleneck at nanoscale material interfaces (e.g. metal-on-semiconductor thin films).

Phonon, medium A (hot) Phonon, medium B (cold) Interface plane

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.