Instability of Ultra-Thin Water Films and the Mechanism of Droplet Formation on Hydrophilic Surfaces

This paper presents a new theory of droplet formation during condensation of water on a hydrophilic surface. The theory uses hydration, electrostatic, van der Waals, and elastic strain interactions between a hydrophilic solid surface and a water film, and shows that contributions to the disjoining pressure are dominated by hydration forces for films thinner than 3 nm. The equilibrium film thickness is found to remain almost constant at about 0.5 nm for a wide range of relative humidity, although it increases sharply as the relative humidity approaches unity. The competition between strain energy on one hand, and hydration, van der Waals, and liquid-vapor surface tension on the other, induces instability for films thicker than a critical value. The critical wavelength of instability, Lcr, is also predicted as a function of film thickness. The theory proposes that as the relative humidity increases, nucleation initially occurs in monolayer fashion due to strong hydration forces. Using nucleation thermodynamics it predicts a critical nucleus size, d, and internuclei spacing, l , as a function of subcooling, ∆T, of the solid surface and shows that both length scales decrease with increasing subcooling. Since these monolayer nuclei are formed on the adsorbed water film, it is shown that when the internuclei spacing is larger than the critical wavelength, l > Lcr , instability occurs in the film resulting in droplet formation. The theory predicts that beyond a certain value of subcooling, the interdroplet spacing is “choked” and cannot decrease further. ASME J. Heat Transfer (in press). 2 Nomenclature a elastic strain at liquid-solid interface A Hamaker constant [J] b thickness of a water monolayer [m] B height perturbation in liquid film [m] d radius of a critical water nucleus [m] e charge of an electron, 1.6 x 10 [C] E elastic modulus [N/m] gv volumetric Gibb’s free energy [J/m ] G Gibb’s free energy [J] h film thickness [m] hfg enthalpy of vaporization of water [J/kg] kB Boltzmann constant, 1.38 x 10 -23 [J/K] l internuclei spacing [m] L length [m] P disjoining pressure [N/m] Pv vapor pressure [N/m ] R gas constant for water vapor [J/kg-K] S supersaturation T temperature [K] U free energy parameter in hydration energy [J/m] v valence w free energy per unit area [J/m] W free energy [J] ε dielectric constant εο permittivity of vacuum, 8.85 x 10 [C/N-m] η nucleation density [m] κ reciprocal of Debye length [m] λ length scale parameter in hydration energy [m] ψ relative humidity ρ ion concentration [m] φο surface potential [V] χ surface charge density [C/m] σ surface tension [J/m] υ volume of a water molecule [m] Subscripts cr critical el electrostatic eq equilibrium hyd hydration lv liquid-vapor s surface sat saturation sr strain v volume vdW van der Waal ASME J. Heat Transfer (in press). 3