Energy and Force between Two Hydrophobic Interfaces: Based on a Two-State System

The hydrophobic force occurs widely in fields involving hydrophobic surfaces, such as biology, material science, mineral industry, health, environmental science, and technology. The mechanisms of energy evolution and force at hydrophobic interfaces remain unclear. Here, the hydrophobic interaction energy is regarded as the cohesive energy of the bulk minus the interaction energy between the interfacial water and the solid surface. To describe the decay of hydrophobic energy toward the bulk, interfacial water molecules are divided into multiple layers. Based on microstate number calculations, the number of newly generated water molecules with the same energy as bulk in each layer is obtained. It is thought that the outer layer attaches an equal amount of water molecules with the same energy as the bulk in the adjacent inner layer. Then, the hydrophobic energy decay from the solid surface to the bulk is calculated by the amount of water molecules and their energy. Through the Derjaguin approximation, the hydrophobic interaction force for interfaces with different hydrophobicities is obtained, representing the mutual attraction induced by the interfacial liquid. Using hydrogen-bond energy states and Boltzmann distribution, interfacial ligands are classified, and detailed configurations of interfacial water molecules are derived. The effects of hydrophobicity on the interfacial dielectric constant and refractive index are then determined to modify the Derjaguin-Landau-Verwey-Overbeek theory and electrostatic adsorption. Furthermore, the effect of the temperature on hydrophobic interactions is also theoretically derived. The experimental value is more consistent with the theoretical prediction under medium contact angles than in other research studies. At large contact angles, cavitation caused by weak liquid-solid interaction reduces short-range forces, while rough modified surfaces produce long-range hydrophobic force tails. Elevated temperature disrupts bulk cohesion, weakens hydrophobic interactions, and generates more low-coordinated water molecules. By adjusting the parameters, the theory can also be applied to hydrophobic forces between bubbles and particles, providing theoretical significance for hydrophobic interface studies.