The experimental procedure is designed to establish the equilibrium contact angles of water, with and without a solute containing an aromatic ring, on two steel plates of different surface modification. The experiment is performed for plate inclinations from 0 to 20°.

The plates are the same as those investigated by Maggiolo et al. (2019a). The silicon-based treatments of the plates are performed using a liquid precursor applied using a wet chemical process, followed by a thermal curing step. The resulting ceramic coatings are purely inorganic silicon oxides with an average thickness of 400 nm, where one of the plates is more hydrophilic and the other is more hydrophobic.

Two solutions are employed in the experiments: pure distilled water and distilled water with dissolved benzoic acid. We employ a concentration of 200 mg/L (prepared by dissolving 50 mg benzoic acid in 250 mL distilled water). The solubility of benzoic acid in water is 3,400 mg/L at 302 K (PubChem, 2019).

The metal plate is fixed on a clamp with the desired inclination/tilting angle, as illustrated in Figure 1a. Drops are placed on the surface by a pipette. The droplet contact angles are analyzed for every 2° inclination of the metallic plate. At 20°, the droplet starts to slide on the hydrophilic coating. Three repeated trials are conducted for every inclination of the metallic plates to quantify the repeatability. The volume of a single drop is calculated by counting the number of drops it takes to fill a volume of 10 mL. This test was also repeated thrice. 210 drops were required to fill a volume of 10 mL, hence the volume of a single drop is approximately 0.05 mL.

Images of the droplets are taken during the experiment and thereafter converted into grayscale images using Matlab R2020b. The grayscale images are analyzed using ImageJ with the DropSnake plug-in to determine the contact angles from a piecewise polynomial fit (ImageJ software, 2015). More precisely, the DropSnake method is based on B-spline snakes (active contours) to shape the drop (DropSnake software, 2016).

It should be stressed here that we will be referring to the different contact angles on the downhill and uphill sides of the drop when the plate is tilted as advancing and receding contact angles, respectively. As these terms are most often associated with a moving drop (and a moving contact line), a more correct term would perhaps be apparent or static advancing and receding contact angles (Lam et al., 2002; Lam et al., 2001; Tadmor, 2004). However, we shall use the shorter nomenclature to facilitate the presentation, whilst reminding the reader that the contact line remains pinned for all scenarios investigated in the current work.

Computational fluid dynamics (CFD) simulations are performed using the Volume-of-Fluid (VOF) method (Renardy et al., 2001), which is a one-fluid multiphase Direct Numerical Simulation (DNS) methodology based on the Navier-Stokes equations (continuity (Equation 1) and momentum balance (Equation 2)): ∇ · v = 0 (1) ρ ∂ v ∂ t + v · ∂ v ∂ x = − ∇ p + ∇ · μ ∇ v + ∇ v T + ρ g + σ ρ ∇ · α ∇ α 1 2 ρ g + ρ l (2)

Here, v is the velocity (m/s), p is the pressure (Pa), g is the gravitational acceleration vector (m/s²), σ is the surface tension at the gas-liquid interface (N/m), α is the volume fraction of liquid (−), and the one-fluid properties (density, ρ, and viscosity, μ) are obtained using Equations 3, 4: ρ = α ρ l + 1 − α ρ g (3) μ = α μ l + 1 − α μ g (4)

The volume fraction field is described by the following advection equation: ∂ α ∂ t + ∇ · α v = 0 (5)

The computational domain is a box of 10 × 5 × 10 mm³, with periodic boundary conditions in the x and z directions, and no-slip wall boundary conditions on the top and bottom in the y direction. The plate-to-plate distance is 5 mm. A pressure gradient of 0.143 Pa/m, chosen to attain an approximate velocity of 0.02 m/s (as in Maggiolo et al. (2019a)), is applied to drive the flow in the z direction.

In the cell layer next to the wall boundaries, the contact angle prescribed from experiments is used to set the local curvature of the interface such that the experimentally determined wall adhesion is recreated in the simulation. In our previous work, we employed a density-dependent function to recreate the macroscopically observed equilibrium contact angle in a Lattice-Boltzmann framework (Maggiolo et al., 2019a; Maggiolo et al., 2019b). The herein adopted approach enables us to prescribe the experimentally observed contact angle directly (rather than indirectly via parameter tuning) and to use the physically correct density and viscosity ratios without stability problems (Aidun and Clausen, 2010).

The initial condition is that a spherical drop of 2 mm diameter is placed on top of the bottom wall in a quiescent fluid. This drop size is chosen as it is associated with the onset of the self-cleaning mechanism on the hydrophobically treated plates (Maggiolo et al., 2019a). Thereafter, the solution is advanced in time while monitoring the magnitude of the total gas-liquid interface. The area of the interface is obtained from the solver fields by creating an iso-surface at α = 0.5 through interpolation of node values. At the same time, the area of the plate wetted by a drop is obtained as ∑ i α i A i , where the sum is over all wall faces i on the bottom plate in the computational domain, and A _i is the local face area. The material properties used in the simulations are: ρ _g = 1.225 kg/m³, ρ _l = 998.2 kg/m³, μ _g = 1.7894.10^–5 Pa,s, μ _l = 0.001003 Pa,s, and σ = 0.072 N/m. Note that, as benzoic acid is a weak surfactant and water at neutral pH is used in conjunction with low concentration, the surface tension value of the system is kept constant. The role of benzoic acid addition is thus to alter the effective contact angle at the three-phase contact line, as prescribed by the measurement data. When the areas have stabilized after an initial transient, their values are recorded.

The solver used is the pressure-based Navier-Stokes solver of ANSYS Fluent 2022. R1, which employes the SIMPLE scheme for the pressure-velocity coupling, the PRESTO! scheme for pressure discretization, the second-order upwind scheme for the convective term in Equation 2 and the second-order central differencing scheme for the diffusion term. First-order implicit time stepping is used with a fixed time step that always maintains a Courant number less than 0.25. For Equation 5, the Geo-Reconstruct scheme is used for the spatial discretization and an explicit scheme for the temporal discretization.

The computational mesh contains 500,000 hexahedral cells at a resolution of Δx = 0.1 mm. The domain, the mesh, the initial solution, and a sample snapshot of an instantaneous solution are visualized in Figure 1b.

Examples of experimental photos with identified apparent advancing and receding contact angles are provided in Figure 1c, for both plates and at varying inclination. The main trends can be discerned from the chosen examples: drops on a hydrophilically treated plate exhibit lower contact angles than the same drop on a hydrophobically treated plate; plate inclination induces a difference between advancing and receding contact angles on both plates (increasing the former and decreasing the latter); and addition of benzoic acid to the distilled water tends to decrease contact angles. The contact angle was not measured on uncoated steel plates but is expected to be in the interval 68°–113° (Maggiolo et al., 2019a).

The complete experimental data set is presented in Figures 2a, b for the plate with hydrophilic coating, and in Figures 2c, d for the plate with hydrophobic coating. The error bars indicate the standard deviation from three repeated experiments. It is seen that, at 20° inclination, the drop has a tendency to start to slide on the plate, and this phenomenon is reflected in a sharp increase in the uncertainty estimate.

Overall, the decrease due to the presence of benzoic acid in advancing and receding contact angles appears to be relatively constant when varying the inclination of the hydrophilic plate. It is approximately 4.0° for the advancing and 3.4° for the receding contact angles, respectively. On the hydrophobic plate, the differences are initially larger: for inclinations up to approximately 5°, the difference is 12.3° in contact angle on both sides (advancing and receding). For larger inclinations, between 5 and 10°, the difference decreases and starts to deviate on the advancing and receding sides: 11.2° for the advancing contact angle and 10.4° for the receding contact angle. The difference invoked by the addition of benzoic acid remains relatively constant for inclinations up to 20°, whereas it more or less vanishes for the receding contact angle at the higher inclinations.

The hydrophilicity of the silicon-based coating is governed by surface silanol (Si-OH) groups that hydrogen-bond with water, whereas hydrophobic behavior can arise from reduced -OH density or surface chemistry that diminishes water adsorption (Kolman and Abbas, 2019). Upon addition of benzoic acid, the wettability may change either due to surface chemistry effects (i.e., effects at the liquid-solid interface) or surfactant effects (i.e., effects at the gas-liquid interface). If the benzoic acid molecules adsorb on the silica coating, its addition tends to increase the contact angle (Satyarthy et al., 2025; Kolman and Abbas, 2019; Terra et al., 2022; Staniscia et al., 2022). If, on the other hand, the benzoic acid preferentially accumulates at the gas-liquid interface (since the aromatic ring is hydrophobic), modification of the air-liquid surface tension becomes more dominant and the contact angle tends to decrease (Satyarthy et al., 2025; Kolman and Abbas, 2019; Staniscia et al., 2022; Ghzaoui, 1999). The present results, with a decrease in contact angle at the introduction of benzoic acid to the distilled water, imply that the coatings applied offer little chemical affinity for benzoic acid and that the mild surfactant effects are instead dominant (Staniscia et al., 2022).

The contact angle hysteresis (defined as the difference between the advancing and the receding static contact angles) is shown in Figure 2e. The hysteresis increases with increasing the tilting angle of the plate, as is expected for a pinned droplet able to balance its own weight by distorting (Quéré et al., 1998). For the plate with the hydrophilic coating, the addition of benzoic acid has no significant effect. This observation indicates that benzoic acid does not interact strongly with the solid surface, and hence the contact line pinning is due to microscopic surface roughness or chemical heterogeneity (e.g., patches of silanol groups) unaffected by the benzoic acid in the liquid (Li et al., 2014). The contact angle hysteresis is similarly unaffected by the addition of benzoic acid on the hydrophobic plate at low-to-moderate tilting angles. It is noted that the hysteresis is lower on the hydrophobic coating than on the hydrophilic one in this range, which supports the conclusion from Maggiolo et al. (2019a) that drops are more mobile on the hydrophobic plates. At larger tilting angles, the hysteresis is reduced upon addition of benzoic acid on the hydrophobic plate, indicating either adsorption of benzoic acid molecules onto the SiO₂-surface (the carboxyl group could attach to polar -OH groups, exposing the hydrophobic benzene ring and thereby increasing the effective hydrophobicity of the surface) or modest benzoic acid accumulation at the liquid-air interface (Kamieński et al., 1965; Park et al., 2022).

In conclusion, we find that on the hydrophilically modified plate, the drops with pure distilled water and drops with benzoic acid behave identically, apart from a constant offset in the equilibrium advancing and receding contact angles on the order of 3–5°. On the hydrophobically modified plate, the addition of benzoic acid has a more pronounced effect, decreasing the equilibrium advancing and receding contact angles by approximately 11–12° at low plate inclination. As the plate inclination is increased, the effect of the benzoic acid on the receding contact angle more or less disappears, whereas the influence on the advancing one remains relatively constant. Contact angle hysteresis is relatively unaffected by benzoic acid addition on both plates, except at high tilting angles on the hydrophobic plate where hysteresis is reduced.

The purpose of the numerical simulations is to assess the role of the fluid flow in shaping the moving drops with a prescribed equilibrium contact angle in the system. The lateral and streamwise periodicity implies that the drop is moving in a system where the distance between adjacent drops is approximately 5 drop diameters, which agrees with the experimental observations of Maggiolo et al. (Maggiolo et al., 2019a) for industrial two-phase plate heat exchangers. Instantaneous visualizations of iso-surfaces representing the gas-liquid interface at quasi-steady conditions are provided in Figure 3 for prescribed contact angles at the wall of 85, 100, and 113°. The 85-degree case reflects one of the cases investigated in Maggiolo et al. (2019a) and is included here as a reference case, whereas the 113-degree case represents pure distilled water and the 100-degree case water with benzoic acid. As the contact angle increases, the drop becomes more spherical. This observation is in good agreement with the analysis in Maggiolo et al. (2019a), which showed that increased hydrophobicity promoted the formation of small, spherical-like, motile droplets under similar flow conditions.

The simulations enable quantification of the droplet surface area and how it changes with changing the drop-plate interaction in the fully resolved, three-dimensional transient flow field. Figure 4a depicts the increase in the plate area wetted by the drop, as compared to a spherical-cap drop of the same radius as the initial drop. The more efficient the wetting of the plate (i.e., the lower the equilibrium contact angle), the larger the wetted area. These quantitative results align well with the qualitative observations from Figure 3. At the same time, Figure 4b displays the decrease of the surface area of the drop exposed to the flow, as compared to the nominal surface area of the initial, completely spherical drop. The deformation of the drop due to the interaction with the plate and the flow leads to a decrease in the gas-liquid interface exposed to the flow, whereas this effect is relatively mild for the investigated parameter range. Finally, in Figure 4c, the variation of the total drop surface area (sum of wall-wetted area and gas-liquid interface exposed to the flow) with the equilibrium contact angle is shown. In all cases, the deformation of the drop from the perfect spherical shape implies an effective increase in the total drop surface area, and this effect is more pronounced for the least hydrophobic plate. These results underline the role of the hydrophobicity of the plate in promoting the formation of close-to-spherical drops moving over the plate surface.

We further extract the increase in exposed gas-liquid interfacial area due to the flow, as obtained from comparisons of simulations with an applied pressure gradient in the streamwise direction to results obtained without any prescribed macroscopic motion (i.e., zero pressure gradient in the streamwise periodic direction). It is found that this increase is 0.065% at 85° contact angle, 0.28% at 100°, and 0.42% at 113°. Evidently, the role of the flow in modulating the gas-liquid interface is not strong at the current conditions. To substantiate this analysis, we also analyze the streamwise velocity profile along a line from plate to plate, through the middle of the drop, and contrast the profile there to that along a line far away from the drop. The outcome of this analysis is shown in Figure 4d. The streamwise velocity far away from the drop exhibits a characteristic parabolic shape. The velocity profile along the line cutting through the drop has a parabolic-like shape above the drop, with the added complexity arising from the internal circulation within the moving drop that gives rise to negative flow in the streamwise direction just inside the gas-liquid interface at the top of the drop. The role of the flow outside the drop is thus to impart motion in the streamwise direction, similar to that of the gas phase, along with inducing internal circulation. The distorting inertial forces are not significant compared to the restoring surface tension forces at the present conditions, and so the deformation of the droplet shape due to the flow is minimal, as expected.

In conclusion, we find that the drop is more spherical at higher plate hydrophobicity, such that the hydrophobically treated plate promotes the formation of almost-spherical droplets. The disturbances induced by the flow on the droplet shape are not significant at the current operating conditions. The role of benzoic acid addition to a pure distilled water drop is to cause a more pronounced spreading of the drop on the plate, however the differences are not large (cf. panels (b) and (c) in Figure 3). The more pronounced wetting of the plate is associated with a decrease in the gas-liquid interface area exposed to the flow. Therefore, the role of a component such as benzoic acid is to decrease the effectiveness of the two-phase heat exchanger in promoting the flow structure needed to maintain self-cleaning functionality. However, it is also clear from the current results that the effect of benzoic acid is relatively minor: as long as the equilibrium contact angles are within the hydrophobic range, the overall characteristics of the drop properties and behaviors are seemingly maintained.