Anti-Fog Coating Design
Fog scatters because condensed water forms droplets. A hydrophilic coating lowers the contact angle so the water spreads into a film, which costs only a weak, flat reflection.
Light transmittance τv into the substrate
99.2 %
Under a water film: 97.5 %; uncoated dry: 95.7 %
Droplet area returning light by TIR
0.0 %
Zero below θ = 48.6° (nwater = 1.333)
Adhesion tension γ cos θ
70.3 mN/m
Work of adhesion γ(1 + cos θ) = 143.0 mN/m; γ = 72.74 mN/m
Fog is condensed water. On a surface it wets poorly it forms droplets, spherical caps that refract light away from the line of sight; above a contact angle of asin(1/nw) = 48.6° part of each droplet sends light back by total internal reflection, a fraction 1 − (1/(nw sin θ))² of its footprint (geometric optics, light crossing the window along the normal, θ ≤ 90°). A hydrophilic anti-fog coating makes θ small, so the water spreads into a continuous film that only adds a weak, flat reflection: the film is taken as thick and uneven (no fringes) and added incoherently. The contact angle sets the wetting, not the optics of the film. γ cos θ (Young: γsv − γsl) is the adhesion tension, not the coating's surface energy; γ is water's surface tension from IAPWS (2014). τv weights T with D65 × V(λ); one face, lossless layers, constant indices (water 1.333).
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Coating and a thick water film added incoherently (transfer matrix, normal incidence, lossless layers, constant indices). Droplets: spherical caps in geometric optics, TIR share of the footprint for θ ≤ 90°.
- Macleod H. A. (2010). Thin-Film Optical Filters, 4th ed., ch. 2 (thick incoherent layers). CRC Press. doi:10.1201/9781420073034
- Hecht E. (2016). Optics, 5th ed., ch. 4 (total internal reflection).
- IAPWS R1-76(2014). Revised Release on Surface Tension of Ordinary Water Substance. iapws.org
- Vargaftik N. B., Volkov B. N., Voljak L. D. (1983). International tables of the surface tension of water. J. Phys. Chem. Ref. Data 12, 817–820. doi:10.1063/1.555688