w  FIG. 2: A snapshot of a premelted grain boundary at an undercooling of 2 K and the corresponding average order parameter versus distance across the grain boundary. Atoms in the top panel are colored based on the order parameter value with lighter color (red in the on-line figure) representing atoms in either of the two grains and darker color (blue in the on-line figure) showing atoms in the grain boundary. The construction to determine the grain-boundary width w is illustrated in the bottom panel and a width of w ≈ 13 . 5 ˚ A is obtained. applications of such sharp-interface models, however, the ambiguity presents no problem in practice provided that each of the bulk and interfacial thermodyamic quantities are defined consistently. In order to determine whether the simulation times employed in this study were sufficient to give adequate statistics for the width histograms, we estimated the correlation time for width fluctuations based on an analysis of the decay of a width autocorrelation function C (Δ t ) defined as follows: In Eq. 6, w ( t ) denotes the instantaneous width of the grain boundary at a time t , and 〈 ... 〉 denote ensemble (time) averages. The time required for the autocorrelation function to decay by a fraction of 1 /e was taken as a measure of the correlation time. The correlation times were found to vary significantly with boundary type and temper- ature. The maximum value of the correlation time was obtained as 100 ps for the Σ9 tilt 11 C (Δ t ) = 〈 w ( t ) w ( t + Δ t ) 〉 − 〈 w 〉 $^{2}$, (6)