Fig. 4.

Out-of-plane structures and electrostatic interactions. a The 1D X-ray scattering profiles extracted along the q_z direction at the peak positions of the GISAXS patterns during stage II at surface pressures of 0.36, 1.28, 2.86, 4.95, 8.52, and 14.14  mN  m⁻¹ (red diamonds). The theoretical simulations are given as black solid lines and the corresponding parameters are listed in Supplementary Table 2. b The immersion depth H and contact angle $\theta$ (inset) plotted as functions of the interparticle distance d. The lines are the polynomial fittings to the data as guides to the eye. c The magnitude of the out-of-plane component of the electrostatic fore plotted as a function of d. The electrostatic forces are estimated from surface tension analyses (red squares) and the numerical simulation results are obtained from FEA (blue line). Inset: schematic illustration of a pair of PNSs at the air/water interface with all the relevant parameters shown. d The electrostatic potential distribution of a pair of PNSs at the air/water interface with the equipotential lines displayed in white. The simulation is carried out using the same set of parameters as given in Supplementary Note 4. The colloidal particles (solid circles) have the radius R = 63.2 nm, the immersion depth H = 91.7 nm, and their separation d = 200 nm