a, b, Experimental AFM frequency-shift (∆f) images obtained at tip heights and oscillation amplitudes of 70 pm and 40 pm (a) and 0 pm and 100 pm (b). c, Δf curves (oscillation amplitude, 40 pm) above a vertical water molecule (vertical), a flat molecule (flat) and the hollow site of hexagonal ice lattice (denoted as background, bkgd) as a function of the tip height. z 1 and z 2 denote the tip heights of the two ∆f images in a and b, respectively. d, e, Simulated ∆f images at different tip heights z (given above each image) obtained with quadrupole (\({d}_{{z}^{2}}\), q = −0.25e; d) and neutral (q = 0; e) tips. f, Top view of the 2D bilayer ice structure (top layer) on the Au(111) substrate. The bottom ice layer is hidden to highlight the structure of the top layer. The green and red dashed parallelograms in d–f denote the sub-lattices of the vertical and flat water molecules, respectively. g, Calculated electrostatic potential map of the bilayer ice on the Au(111) in a plane 7.24 Å above the highest atom in the Au substrate. h, Simulated total potential map of the bilayer ice on Au(111) in a plane, corresponding to the position of the CO-tip apex at a tip height of 12.5 Å. i–k, Vertical force above the flat (F z–f ) and vertical (F z–v ) water molecule as a function of tip height. i, Experimental F z obtained by integrating the experimental ∆f(z) in c according to ref. 42. Before the integration, ∆f(z) was smoothed using a moving average filter with a span of 5. j, k, Simulated F z computed with \({d}_{{z}^{2}}\) (j) and neutral (k) tips. l, Simulated lateral deflection of the quadrupole probe particle in the x direction (X q–d ) as a function of the tip height. X q–d–v and X q–d–f correspond to X q–d above the vertical water molecule and the flat water molecule, respectively. Tip-height references are the same as those in Fig. 2. In g and h, H and O atoms in the top-layer ice are denoted as white and red spheres, respectively. The image sizes in a, b and d–h are 1.25 nm × 1.25 nm. See Methods for details.