FIG. 6: The distribution function P ( w ) vs w from the MD simulations. The function corresponding to the Σ9 tilt grain boundary is in the right of the panel. The function to the left is from the Σ9 twist boundary discussed elsewhere [20]. The lines represent least square fits of the premelting model of Eqs. 1 and 2 for the Σ9 twist grain boundary and Eqs. 1 and 3 for the Σ9 tilt boundary. Figure 5 further illustrates the nature of the disorder present in the Σ9 twist and tilt boundaries. The snapshots were obtained from a 40 ns MD simulation at 1708 K, and represent the largest, smallest and an average value for the interface widths. Results for the Σ9 twist boundary are shown on the left and those for the Σ9 tilt boundary are on the right of Fig. 5. The snapshots illustrate the highly dynamic nature of the boundary structures at 1708 K. The large fluctuations in interface width displayed by these boundaries forms the basis for the histogram analysis underlying calculations of the disjoining potential. The calculated width histograms for both Σ9 boundaries are displayed over a range of interface undercoolings in Fig. 6. The histograms on the left and right in Fig. 6 correspond to the Σ9 twist and tilt boundaries, respectively. For both of the boundaries the histograms become broader as the bulk melting temperature is approached (i.e., with decreasing un- dercooling). For the Σ9 twist boundary the system is observed to melt completely over the course of the 20 ns runs at the bulk melting temperature, and histograms can be obtained for this boundary only for finite values of the undercooling. By contrast, the width of the Σ9 tilt boundary remains finite at T$_{M}$ and the width histogram can be calculated at zero undercooling for this boundary, as shown in Fig. 6. The inset in each of the panels shows the quantity − k$_{B}$T$_{i}$ log [ P ( w,T$_{i}$ )] /A − Δ G$_{f}$w versus w for all of the interface temperatures. The data for the Σ9 tilt boundary shows clear evidence of an attractive minimum in the disjoining potential, as will be discussed further below. 18