Structural Origin of Enhanced Dynamics at the Surface of a Glassy Alloy

The enhancement of mobility at the surface of an amorphous alloy is studied using a combination of molecular dynamic simulations and normal mode analysis of the nonuniform distribution of Debye-Waller factors. The increased mobility at the surface is found to be associated with the appearance of Arrhenius temperature dependence. We show that the transverse Debye-Waller factor exhibits a peak at the surface. Over the accessible temperature range, we find that the bulk and surface diffusion coefficients obey the same empirical relationship with the respective Debye-Waller factors. Extrapolating this relationship to lower $T$, we argue that the observed decrease in the constraint at the surface is sufficient to account for the experimentally observed surface enhancement of mobility.

Figure 1

The free surface of the amorphous solids at $T=0$. The interface width is $\sim 2$ for all surfaces. The density profiles for the unrelaxed surface and surfaces annealed at $T/T_0=1.2$, 1.6, and 2.4 as explained in the text.

Figure 2

The lateral surface diffusion coefficient $D_{xy}$ (red squares) and the bulk liquid diffusion constant (calculated using the VFT expression [23]) $D_{\text{bulk}}$ (black circles) plotted as a function of $T_0/T$. The dashed line is the lateral surface diffusion constant calculated without the contribution from free particles. Inset (top right): The fraction $C(t)$ of surface particles that have not left the surface after a time $t$ for $T/T_0=2.5$ (black), 2.0 (red dashed), and 1.4 (blue). Arrows indicate the times marking the onset of diffusive motion. Inset (bottom left): $\log (D_{xy})$ plotted against $\log (D_{\text{bulk}})$ with the dashed line representing the suggested [25] relation, $D_{xy}\propto \sqrt{D_{\text{bulk}}}$.

Figure 3

The normalized distribution of normal mode as a function of frequency $\omega$ for the amorphous slab with two free surfaces. The average number of surface modes (defined as those modes whose surface participation fraction $f>0.2$) are also plotted. A small number ($\sim 4$) of surface modes are unstable (i.e., ${\omega }^2<0$) and persist after annealing.

Figure 4

The average individual particle Debye-Waller lateral and normal factors $⟨{\mathrm{\Delta }}_{xy}^2(z)⟩$ and $⟨{\mathrm{\Delta }}_z^2(z)⟩$, respectively, for the unrelaxed surface and for the surface annealed at $T=2.4T_0$. Inset: The distribution of ${\mathrm{\Delta }}^2$ for the bulk (blue) and the surface (transverse) (orange dashed line) for $T=2.4T_0$.

Figure 5

The values of $D_{\text{bulk}}$ as simulated (black circles) or extrapolated using the VFT equation (blue and red circles) and $D_{xy}$ (open triangles) plotted against the plateau ${⟨\mathrm{\Delta }{r}^2⟩}_{\text{plateau}}$ (see text), calculated for the bulk and surface systems, respectively. Upper and lower bounds on the bulk ${⟨\mathrm{\Delta }{r}^2⟩}_{\text{plateau}}$ are obtained at low temperature from as-quenched amorphous (blue circles) and the crystal (red circles), respectively. Dashed curves associated with these bounds are provided as a guide to the eye. The values of the surface diffusion coefficient $D_{xy}$ for individual surfaces (i.e. before averaging) at three temperatures are provided in the Supplementary Information [34].