Size dependence of second-harmonic scattering from nanoparticles: Disentangling surface and electrostatic...

Polarimetric angle-resolved second-harmonic scattering (AR-SHS) is an all-optical tool enabling the study of unlabeled interfaces of nano-sized particles in an aqueous solution. As the second harmonic signal is modulated by interference between nonlinear contributions originating at the particle's surface and those originating in the bulk electrolyte solution due to the presence of a surface electrostatic field, the AR-SHS patterns give insight into the structure of the electrical double layer. The mathematical framework of AR-SHS has been previously established, in particular regarding changes in probing depth with ionic strength. However, other experimental factors may influence the AR-SHS patterns. Here, we calculate the size dependence of the surface and electrostatic geometric form factors for nonlinear scattering, together with their relative contribution to the AR-SHS patterns. We show that the electrostatic term is stronger in the forward scattering direction for smaller particle sizes, while the ratio of the electrostatic to surface terms decreases with increasing size. Besides this competing effect, the total AR-SHS signal intensity is also weighted by the particle's surface characteristics, given by the surface potential Φ₀ and the second-order surface susceptibility χ_s,2 ². The weighting effect is experimentally demonstrated by comparing SiO₂ particles of different sizes in NaCl and NaOH solutions of varying ionic strengths. For NaOH, the larger χ_s,2 ² values generated by deprotonation of surface silanol groups prevail over the electrostatic screening occurring at high ionic strengths; however, only for larger particle sizes. This study establishes a better connection between the AR-SHS patterns and surface properties and predicts trends for arbitrarily-sized particles.