Glass Transition — Mode Coupling Theory

Mode coupling theory (MCT) predicts a two-step structural relaxation in supercooled liquids: fast β-relaxation (cage rattling) followed by slow α-relaxation (cage escape). The intermediate scattering function F(q,t) shows a plateau at the non-ergodicity parameter f_q, with relaxation time τ_α diverging as (T−Tc)^{−γ}.

F(q,t) current T

β regime

α Kohlrausch fit

Parameters

Temperature T/Tc: 1.20 Non-ergodicity f_q: 0.70 MCT exponent b: 0.60 KWW stretch β_KWW: 0.70

Observables

τ_α / τ₀—

Plateau f_q—

von Schweidler exp b—

Theory

β-regime: F(q,t) ≈ f_q + h_q(t/τ_β)^{−a}
von Schweidler: F(q,t) ≈ f_q − h_q(t/τ_β)^b
α-regime: F(q,t) ≈ f_q·exp(−(t/τ_α)^β)
τ_α ∼ (T/Tc−1)^{−γ}, γ=(½a+½b)/(ab)
MCT critical: Tc ≈ 0.8–0.85Tm for LJ