hemiwicking. The contact angles at which spreading/imbibition occurs are between 0 and π/2. The Wenzel model is valid between θC and π/2. If the contact angle is less than ΘC, the penetration front spreads beyond the drop and a liquid film forms over the surface. Figure 11 depicts the transition from the Wenzel state to the surface film state. The film smoothes the surface roughness and the Wenzel model no longer applies. In this state, the equilibrium condition and Young's relation yields: By fine-tuning the surface roughness, it is possible to achieve a transition between both superhydrophobic and superhydrophilic regions. Generally, the rougher the surface, the more hydrophobic it is. Spreading dynamics If a drop is placed on a smooth, horizontal surface, it is generally not in the equilibrium state. Hence, it spreads until an equilibrium contact radius is reached (partial wetting). While taking into account capillary, gravitational, and viscous contributions, the drop radius as a function of time can be expressed as For the complete wetting situation, the drop radius at any time during the spreading process is given by where γLG is surface tension of the fluid V is drop volume η is viscosity of the fluid ρ is density of the fluid g is gravitational constant λ is shape factor, 37.1m−1 t0 is experimental delay time re is drop radius in equilibrium Modifying wetting properties Surfactants Many technological processes require control of liquid spreading over solid surfaces. When a drop is placed on a surface, it can completely wet, partially wet, or not wet the surface. By reducing the surface tension with surfactants, a nonwetting material can be made to become partially or completely wetting. The excess free energy (σ) of a drop on a solid surface is: γ is the liquid–vapor interfacial tension γSL is the solid–liquid interfacial tension γSV is the solid–vapor interfacial tension S is the area of liquid–vapor interface P is the excess pressure inside liquid R is the radius of droplet base Based on this equation, the excess free energy is minimized when γ decreases, γSL decreases, or γSV increases. Surfactants are absorbed onto the liquid–vapor, solid–liquid, and solid–vapor interfaces, which modify the wetting behavior of hydrophobic materials to reduce the free energy. When surfactants are absorbed onto a hydrophobic surface, the polar head groups face into the solution with the tail pointing outward. In more hydrophobic surfaces, surfactants may form a