A combined criterion of surface free energy and roughness to predict the wettability of non-ideal low-energy surfaces

The significance of wettability between solid and liquid substances in different fields encourages scientists to develop accurate models to estimate the resultant apparent contact angles. Surface free energy (SFE), which is principally defined for ideal (flat) surfaces, is not applicable to predict the wettability of real (rough) surfaces. This paper introduces a new parameter, namely normalized surface free energy (NSFE) as a combination of SFE and roughness, to predict the contact angle of liquids on non-ideal low-energy surfaces. The remarkable consistency of the predicted and measured contact angles of liquids on some rough surfaces also confirm the validity of the approach.

💡 Research Summary

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The paper addresses a long‑standing limitation in wetting science: conventional surface free energy (SFE) calculations assume an ideal, perfectly flat solid surface, whereas real engineering surfaces are often rough and heterogeneous. To bridge this gap, the authors introduce a new parameter called Normalized Surface Free Energy (NSFE), which explicitly incorporates surface roughness into the energetic description of a solid.

The starting point is the BS EN 828:2013 standard equation for determining SFE from contact‑angle measurements with two probe liquids:

 0.5 γ_lg (1 + cos θ) = √(γ_sg^d γ_lg^d) + √(γ_sg^p γ_lg^p)

In the classic formulation, θ is the Young contact angle on an ideal surface. The authors replace θ with the apparent contact angle θ_α measured on a rough surface, thereby defining “starred” dispersive and polar components (γ_sg^d* and γ_sg^p*) that belong to the NSFE of the real surface. By measuring θ_α for two liquids—one with a dominant dispersive component (e.g., diiodomethane) and one with a dominant polar component (water)—the two unknown starred components can be solved. The NSFE, γ_sg*, is then obtained as the sum of the square‑root terms:

 γ_sg* = √(γ_sg^d*) + √(γ_sg^p*)

Once NSFE is known, the apparent contact angle of any third liquid can be predicted by inserting its known liquid‑gas surface‑tension components (γ_lg^d, γ_lg^p) into the same equation, leaving only θ_α as the unknown.

Two important constraints are highlighted: (1) the derivation assumes zero equilibrium vapor pressure of the liquid on the solid, which is only satisfied for low‑SFE, hydrophobic surfaces; consequently, the method is limited to low‑energy materials. (2) The two reference liquids must be as close as possible to pure dispersive and pure polar liquids, respectively; any deviation introduces systematic error.

To validate the approach, four distinct low‑energy, rough coatings were examined: (I) silica + PFOTS on steel, (II) candle‑soot templated silica on glass, (III) silica nanotubes functionalized with PFDTS, and (IV) silicone nanofilaments modified with PFDTS. Scanning electron microscopy confirmed micron‑ to nanometer‑scale roughness and characteristic microstructures. For each coating, contact angles were measured with three liquids (diiodomethane, water, and a third test liquid). The measured and predicted angles are compiled in Table 2 of the paper.

Results show excellent agreement for the third liquid when its polar component is negligible (errors below 3°), demonstrating that NSFE captures the combined effect of roughness and intrinsic surface energy for low‑energy surfaces. Larger discrepancies appear for highly polar liquids, which the authors attribute to (i) imperfect fulfillment of the zero‑vapor‑pressure assumption, (ii) the presence of more than two phases (e.g., uncovered high‑energy substrate areas), and (iii) the fact that the reference liquids are not perfectly pure dispersive or polar.

The study’s contributions are twofold. First, it provides a straightforward, experimentally accessible method to predict wetting behavior on realistic rough, low‑energy surfaces without resorting to complex topographical analysis or numerical simulations. Second, by systematically analyzing error sources, it outlines pathways for extending the framework to higher‑energy surfaces, multi‑phase coatings, and dynamic wetting scenarios.

In conclusion, the authors successfully demonstrate that the Normalized Surface Free Energy, derived from a simple modification of an existing standard, can reliably predict apparent contact angles on non‑ideal low‑energy surfaces. The method is limited to hydrophobic, low‑SFE materials but offers a valuable tool for coating design, quality control, and fundamental studies of wetting on textured substrates. Future work is suggested to broaden the applicability to high‑energy surfaces, incorporate additional phases, and refine the selection of reference liquids to further reduce prediction error.