Fig. 9: Graphical summary and comparison of the experimental observations with expectations based on the interfacial solvation model.

Background colour on the plot presents the qualitative theoretical expectation based on the interfacial solvation model given by Eqs. (1) and (2). For negatively charged particles (\(\sigma \, < \, 0\)), attraction (green area) is expected when the average interfacial dipole moment density is substantial and positive, i.e., \({\mu }_{{{{\rm{av}}}}}\, > \,0\), whereas no attraction is expected when \({\mu }_{{{{\rm{av}}}}} \, \lesssim \, 0\) (grey area). Conversely, for positively charged particles (\(\sigma\, > \,0\)), attraction is expected when \({\mu }_{{{{\rm{av}}}}}\, < \, 0\) and no attraction when \({\mu }_{{{{\rm{av}}}}}\, \gtrsim \,0\). Qualitative experimental outcomes are presented as coloured symbols on a binarized abscissa specifying the sign of the charge of the particle – positive or negative, with the corresponding ordinate value denoting values of \({\mu }_{{{{\rm{av}}}}}\) (filled symbols) or \({\mu }_{{{{\rm{av}}}},{{{\rm{w}}}}}\) (open symbols) obtained from MD simulations. Symbol colour denotes the experimental observation: attraction and cluster formation (green symbols), and absence of attraction and cluster formation (grey symbols). The graph presents results describing a total of 37 experimental situations for which \({\mu }_{{{{\rm{av}}}}}\) values were determined (e.g., \({\mu }_{{{{\rm{av}}}}}\) for Tween 20 and solvent mixtures at an interface are not available). All depicted dipole moment density values for silica reflect a group density of 1 OH nm⁻², except values for the amino acids, TMAG, TMAO and glycerol which entail a group density of 4.7 OH nm⁻².