Fig. 2: Solvent dependence of charge-asymmetric cluster formation.

a Structures and average orientation of solvent molecules at an uncharged O-atom surface (dipole moments — green arrows; surface normal — \({n}_{{{{\rm{z}}}}}\)), as inferred from MD simulations. The normal component of the net interfacial molecular dipole moment, \({\mu }_{{{{\rm{av}}}}}\), points slightly away from the surface for H₂O and D₂O and towards the surface for the other cases. b Digitised experimental images of negatively charged silica SiO₂ particles (blue, \(2R\) = 4.82 or 5.17 μm) and positively charged aminated silica NH₂-SiO₂ particles (pink, \(2R\) = 4.63 μm) suspended in H₂O, D₂O, DMSO, 2-propanol (IPA), 1-hexanol, and ethylene glycol (EG) (Supplementary Fig. 11). Scale bars \(20\) μm. Stable clusters were observed in H₂O and D₂O for SiO₂ particles, and in DMSO, IPA and in primary and secondary alcohols for NH₂ particles (Fig. 3), but not in EG (Supplementary Fig. 17, Supplementary Note 3). Measured zeta potentials, \(\zeta\) (in mV) provided in inset. c Experimental \(g(r)\) for SiO₂ (solid blue lines) and NH₂ particles (solid pink lines), and corresponding simulated \(g(r)\) profiles (dashed grey lines). Ordered clusters yield \(g(r)\) profiles with a periodic peak structure indicating interparticle attraction. d Pair-interaction potentials \(U(x)\) of the form of Eq. (1), inferred from BD simulations, that best reflect the experimental \(g(r)\) profiles for SiO₂ (blue) and NH₂ (pink) particles, and the negligible vdW contribution, \({U}_{{{{\rm{vdW}}}}}\) (grey dashed line) (Supplementary Table 25). Error bars denote estimated uncertainties of ± 100 nm on particle diameter and ± 1.5 \({{{{\rm{k}}}}}_{{{{\rm{B}}}}}T\) in \(w\). e Excess interfacial dipole moment density \({\mu }_{{{{\rm{av}}}}}(\sigma )\) for various solvents at surfaces carrying positive and negative surface charge density \(\sigma\), as obtained from MD simulations. At zero surface charge, \(\sigma=0\), we obtain \({\mu }_{{{{\rm{av}}}}}\left(0\right)=+ 0.85\) D nm⁻² for water and \({\mu }_{{{{\rm{av}}}}}\left(0\right)\, < \, 0\) for other solvents (filled symbols). Parameter values in green quadrants of the \({\mu }_{{{{\rm{av}}}}}\) vs. \(\sigma\) plot support interparticle attraction and cluster formation between electrically like-charged particles (since \(B\, < \, 0\) in Eqs. (1) and (2)). f \({\mu }_{{{{\rm{av}}}}}(0)\) for solvents in contact with a neutral O-atom surface, silica surfaces with varying surface group density (\(\varGamma=0.5\) or \(4.7\) OH nm⁻²), and model amine surfaces with varying surface group density (\(\varGamma=1\) or \(4\) NH₂ nm⁻²).