Predicting the relationship between surface charge density and electrostatic potential for spherical particles remains a fundamental challenge in colloid science. Because the governing non-linear Poisson–Boltzmann equation lacks a general exact analytical solution, researchers typically rely on numerical calculations or various semi-empirical approximations. In this paper, we overcome these limitations by developing a dual-asymptotic framework that provides explicit, closed-form expressions for this surface charge–potential relationship across the entire spectrum of particle curvature. For weakly curved systems, where the Debye length λ_D is much smaller than the particle radius R (λ_D/R ≪ 1), a formal mathematical derivation rigorously establishes the Ohshima–Healy–White formula as an exact regular perturbation expansion. Conversely, for highly curved spheres (R/λ_D ≪ 1), we employ singular perturbation analysis using the scaled particle radius as the small parameter. This approach provides a first-principles mathematical justification for the spherical Debye–Hückel theory, proving that its leading-order expansion remains asymptotically exact within the full non-linear Poisson–Boltzmann framework due to the geometric deactivation of non-linearity. Crucially, we map the exact limits of this geometric regulation, demonstrating how non-linear screening re-emerges as the particle radius increases, with the breakdown threshold governed by the interplay between curvature and surface charge density.
When the concentration of electrolyte solution varies along the channel, the forces arise that drag the fluid toward the higher or lower concentration region, inducing a flow termed diffusio-osmotic. This paper investigates a flow that emerges in channels with constant density of surface charge sigma and thin compared to their thickness electrostatic diffuse layers. An equation for the fluid flow rate Q is derived and used to describe analytically the flux of ions, and local potentials and concentrations. This equation, which allows us to treat the diffusio-osmotic problems without tedious and time consuming computations, clarifies that the global flow rate is controlled only by the surface charge and concentration drop between the channel ends, and indicates that there exist generally two different values of sigma that correspond to a particular Q. Our theory provides a simple explanation of the directions of the fluid flow rate and ionic flux depending on the surface charge and diffusivity of ions, predicts a non-linear concentration distribution along the channel caused by convection, and relates it to the local potential changes by a compact formula. We also present and interpret the variations of the diffusio-osmotic velocity profiles and the apparent slip velocity along the channel and show that the latter is highly non-uniform and could even become alternating. The relevance of our results for diffusio-osmotic experiments and for some electrochemistry and membrane science issues is discussed briefly.
Flat surfaces with different patterns of hydrophobic spots were employed for experimental investigation of boiling heat transfer. In one case, hydrophobic spots were created on a smooth copper surface and on a surface coated with arrays of micrococoons from silicon oxide nanowires by vapor deposition of a fluoropolymer. In the second case, a hydrophobic coating was deposited on heater surfaces with cavity microstructures formed by laser ablation and chemisorption of fluorinated methoxysilane. Water under saturation conditions at atmospheric pressure was used as the working liquid. The temperature of the heating surface was varied from 100 to 125 °C, and the maximum value of the heat flux was 160 W/cm2. Boiling heat transfer on the test biphilic surfaces was significantly (up to 600%) higher than on non-biphilic surfaces. Surface texture, the shape of hydrophobic regions, and the method of their creation tested in this study did not show a significant effect on heat transfer. The boiling heat transfer rate was found to depend on the size of hydrophobic spots, the distance between them, and hence the number of spots. The highest heat transfer efficiency was detected for the surface with the largest number of hydrophobic spots. After long-term experiments (up to 3 years), the heat transfer coefficient on the obtained surfaces remained higher than on the smooth copper surface. Biphilic surfaces with arrays of cavities formed by laser ablation turned out to be the most stable during prolonged contact with boiling water.
