boundary premelting based on MD simulations, phase-field theory and PFC calculations. The exponential form of the disjoining potential for the Σ9 twist boundary presented here and in Ref. [20] is qualitatively consistent with the results of previous investigations where a diverging grain-boundary width has been observed for high-energy boundaries in a variety of different systems (e.g., [25, 26, 28, 34]). The disjoining potential calculated here for the Σ9 tilt boundary features a weak attractive minimum at w$_{m}$ , and is replusive and attractive for smaller and larger values of w , respectively. This form for the disjoining potential corresponds to a grain-boundary whose width increases with T at low temperature, but remains finite at and above T$_{M}$ (up to some maximum superheating). This type of behavior for the temperature dependence of the grain boundary width is qualitatively similar to that found in recent MD studies for one of three twist boundaries in Si [34] and a symmetric tilt boundary in Cu [33]. 

As discussed in the introduction section disjoining potentials with attractive minima, qualitatively similar to that calculated here for the Σ9 tilt boundary, were obtained in PFC calculations by Mellenthin et al.[19] for low-angle tilt boundaries in a model two-dimensional hexagonal system. The boundaries which displayed this type of disjoining potential had highly non-uniform interface structures, containing premelted regions localized on grain- boundary dislocation cores and separated by well ordered regions which we will refer to as solid “bridges”. For these structures, the width corresponding to the minimum in the disjoining potential, which represents an average over the area of the boundary, corresponds to the thickness of an equivalent uniform layer with the same amount of “liquid” as contained in the premelted cores. This area-averaged width can be quite small (e.g., on the order of the atomic spacing) even though the radius of the premelted cores is much larger. 

It is interesting to compare the structures observed in the PFC calculations of Mellenthin et al. [19] with those for the Σ9 tilt boundary considered in the present study. Visual inspection of the numerous snapshots showed that the disorder in this boundary is indeed often highly non-uniform, as illustrated in Fig. 8. 

This figure shows a representative snapshot from a simulation at T = 1708 K, viewed down the tilt axis. In Fig. 8, the structural disorder in the boundary is clearly nonuniform - the region of high structural disorder is highlighted by the grey ellipse, while the red lines connected across the boundary plane highlight the ordered solid bridges. Thus, even though the Σ9 boundary studied in the present work has a misorientation angle (38.9 degrees) 

