Extended Data Fig. 5: Distribution of S_OH in experimental superstructures.

a, The statistics of S_OH in two superstructures in experiments. Although the experimentally obtained superstructures don’t exhibit complete proton ordering due to the presence of residual entropy, we can divide the surface area into units that are equivalent to the unit cell in \(\sqrt{19}\times \sqrt{19}\) phase for statistical analysis. The small S_OHs for superstructures in experiments (S_OH ≤ 5) represent the homogeneous distribution of dangling OH bonds at the surface, with a dominant value of ~3. b, c, Typical surface structures of S_OH = 1/12 + 2/3 + 1 + 1/12 + 1 + 1/6 = 3 (calculated \(\sqrt{19}\times \sqrt{19}\) phase) and S_OH = 1/2 + 1/2 + 1/2 + 1 + 1/2 = 3 (experimental). The contribution value of 1 in S_OH is assigned to both of the two nearest-neighboring dangling OH bonds that are located within the unit cell. For dangling OH bonds located at the boundary or corner, the contribution to S_OH is determined based on the proportion of their portion belonging to the unit cell. Pairs of nearest-neighbor dangling OH bonds are denoted by green circles. The orange and yellow triangles represent tetrahedron structures in different stacking types. H and O atoms of the upper-lying and lower-lying water molecules in the topmost bilayer are denoted as white, red and dark blue spheres, respectively. Bilayers below the surface are shown in light blue.