Extended Data Fig. 11: Correlations between force and energy minima, PA and pK_a, using probe molecules.

a, List of probe molecules with their experimental PA and pK_a values. As our aim is to assign PAs to surface O species, only oxygen-based acids and alcohols were selected. The experimental PAs (for the corresponding base, that is, the X−O⁻ anion) are taken from the National Institute of Standards and Technology (NIST) database⁴¹. The number of listed NIST entries is given in parenthesis. In case that data from several experiments are available, averaged values and uncertainties are given. b, Correlation between force and energy minima. The linear fit to the calculated minima of the E(z) and F(z) curves for our probe molecules (red dashed line) shows an almost perfect linear correlation of the data points. The value of the energy minimum in the E(z) curves (that is, F(z) = 0) is the natural measure for the strength of the hydrogen bond that forms between the OH groups of the tip and the molecules. This linear relationship allows us to focus on establishing a correlation between PA or acidity and the force, and not the energy minimum. This is more convenient, because the force minimum is the natural measure from the AFM experiments. Although the energy minimum can be obtained by integrating the F(z) curves, it is not trivial to eliminate ambiguities stemming from a proper choice for the zero point of energy, which determines the integration constant. Furthermore, the force minimum is encountered at a larger OH–tip distance than the energy minimum. The force minimum is therefore less influenced by other interactions between the tip and side groups of the molecule or neighbouring atoms on the surface. c, Correlation between the PA and acidity constant pK_a of the selected probe molecules. The PA also governs the pK_a in wet, solution-based chemical processes: an OH group with a strongly bound proton (high PA of the O atom) is a weak acid, and an OH group with a weakly bound proton (low PA of the O atom) is a strong acid. A linear fit (red dashed line) to the experimental data in a shows the expected trend, that is, strong acids have a low PA, and weak acids bind their proton more strongly. However, in addition to the PA, the pK_a also includes the Gibbs free energy of solvation of the acid (XOH), the conjugated base (XO⁻) and the H⁺ (see the thermodynamic cycle in Methods). These species have different solubilities, which leads to a large scatter of the data points. d, Correlation between calculated AFM force minima and the experimental acidity constants pK_a of the probe molecules. As expected from the discussion in c, the force minima show a larger scatter around the red dashed regression line than when plotted with respect to the PA (see Fig. 4). This is because the AFM measurements are done in vacuum and do not include information about solvation free energies. Thus, the prediction of absolute pK_a values for our OH groups on the In₂O₃(111) surface based on the AFM measurements alone is not possible. Still, there is a clear trend that strong acids form strong H bonds with the AFM tip (deep force minimum, low PA), whereas weak acids form weak H bonds (shallow force minimum, high PA). If we assume that the solvation energies of the structurally rather similar OH groups do not differ too much, then the deviation from the regression line would be similar for all of them, which would allow us to predict at least relative pK_a changes between the OH groups by using the slope of the linear regression. The AFM-measured difference in the force minima of 81 pN for the O_SH(β) and O_SH(γ) sites (see Fig. 2) then translates to a difference in acidity of 5.5 pK_a units, which is reasonable.