Unveiling relaxation and crystal nucleation interplay in supercooled germanium liquid

Abstract

A well-designed study of the relationship between crystal nucleation and relaxation in supercooled liquids could clarify some critical issues in condensed matter science. In this article, we dwell on the kinetic spinodal temperature, TKS – where the times of first critical nucleus formation and structural relaxation curves cross over – and the Kauzmann temperature, TK – where the entropy of the supercooled liquid and crystal equalize. Germanium liquid was used as a model system, for which a reliable Stillinger-Weber potential and crystal nucleation kinetics are available. Using molecular dynamics (MD) simulations, we obtained the self-diffusion coefficient, D(T), and the shear viscosity, η(T), down to T=0.7·Tm, where Tm is the equilibrium melting temperature. In this way, three relaxation times were determined in the supercooled liquid: (i) via the shear viscosity, τη; (ii) the incoherent intermediate scattering function, τα; and (iii) the self-diffusion coefficient, τD. We found that τα(T)≈τD(T) and τη(T)<τα(T), corroborating the findings of recent experimental and computational studies conducted with other four substances. Hence, the viscosity only gives a lower bound to the intrinsic structural relaxation times,τα(T). We also compared the MD values of the atomic transport coefficient across the supercooled liquid/critical nucleus interface, D*, with those obtained by the classical nucleation theory (CNT) and found that the activation energies of D(T), η(T), and D*(T) are similar. The birth times of the first critical crystal nucleus at steady-state conditions, τ1(T), obtained in our previous study for the same material, were extrapolated to lower temperatures using the CNT for different sample sizes. We found that τ1(T) is much longer than the nucleation time-lags, confirming the establishment of the steady-state nucleation regime. A relevant finding is that the τα(T) and τ1(T) curves indeed cross over, confirming the existence of a kinetic spinodal for this liquid. Finally, we demonstrate that if the Kauzmann temperature existed for germanium, it would be located well below the TKS; hence, crystallization would intervene in the cooling path before the supercooled germanium liquid could reach TK, thus averting the entropy paradox, posed by Kauzmann in 1948. These findings shed light on several fundamental questions related to supercooled liquids.