Schematic representations of two interacting spheres (top), and of the dominant average orientation of water molecules near an uncharged surface (middle). Bottom: Various contributions to the total interaction free energy for positively (red lines) and negatively charged (blue lines) particles: electrostatic repulsion, F el (black dotted line; applies to both signs of charge); interfacial solvation contribution, F int , depends on the sign of charge (dashed lines); total interaction energy, ${U}_{{tot}}={Delta F}_{{el}}+{Delta F}_{mathrm{int}}$ (solid lines); and the vdW contribution, U vdW (dashed-dotted grey line; applies to both cases). For 0 < 0 (for example, in water) negatively charged particles may display a minimum in U tot of depth w at a separation x min (solid blue line), and positively charged particles repel monotonically (red solid line) (Supplementary Information, section 3.1 ).
