Precursor film

When a perfectly wetting liquid spreads relatively slowly on a clean smooth substrate, a very thin precursor film may propagate ahead of the apparent or macroscopic wetting line with the average thickness of the film determined by material property of liquid phase and solid wall (Heslot et al. 1989 and 1990). The first reported observation of an ‘invisible’ film spreading ahead of the edge of a macroscopic drop stems from the pioneering work by (Hardy 1919 and 1936), Hardy was unable to detect its presence directly at that time. Numerous studies have subsequently confirmed the existence of precursor films using ellipsometry Beaglehole (1989); Bascom et al. (1964), interference microscopy patterns Bascom et al. (1964), and polarized reflection microscopy (Ausserre et al. 1986). Simultaneous observations of the moving droplet and of the fringe pattern by the laser ellipsometry (Ueno and Watanabe 2005) reveal the existence of the precursor film, which is ahead the moving contact line and traveled with varying its profile.

If the fluid is perfectly wetting, we have seen that van der Waals forces lead to the formation of a precursor film in front of the contact line (de Gennes 1985). Wang (2003) in his thesis claimed the surface feature, particularly micro-structure or roughness, promotes the formation and development of thin precursor film, they claimed that the precursor film spreads much faster than the movement of the apparent contact line (ACL) on a rough solid surface. Amit Sah (2014) studied precursor film with glass capillaries and cover slips. His experimental results imply that the precursor film moving ahead of the contact line controls the wetting behavior. Yuan and Zhao (2010) using molecular dynamics (MD) simulations to explore the atomic details and the transport properties of the precursor film in dynamic wetting. Their results showed that the molecules which finally formed the precursor film came almost from the surface in the initial state of the droplet. The fast propagation of the precursor film is owing to continuous and very fast diffusing of the surface water molecules that have the highest self-diffusion coefficient to the front of the precursor film allows for fast propagation of the precursor film. Precursor film propagates fast with low energy dissipation.

The typical thickness of the precursor film moving in front of macroscopic body of fluid is in the range of 500–3000 Å (Bascom et al. 1964; Beaglehole 1989). For a fixed volume of water droplet, according to Tanner’s laws (Tanner 1979), the spreading of the macroscopic part of the droplet is rather slow; it would take a very long time to completely spread a macroscopic drop (Voinov 1976; Tanner 1979). The precursor which advances at a seemingly faster rate than the normal contact line is indeed well documented. It has been found that the velocity at which the precursor film advances depends on the materials of the gas/liquid/solid system and can vary in wide range. Bascom et al. 1964 reported the speed of propagation of the precursor film of squalane on stainless steel is about 10⁻⁴ cm/s while indirect measurement for water on glass Marmur & Lelah (1980) suggest that there the precursor film velocity is of the order of 10 cm/s. As a result, the existence of precursor film makes the ACL moving on the film rather than on a real solid surface. In other words, the ACL moves over the “wet” solid surface instead of the “dry” solid surface.

Precursor film on dry surface

As stated by de Gennes (1985) and Joanny and de Gennes (1984), long-range forces, for example, van der Waals forces, should be taken into account in the spreading phenomenon of precursor film. It is found that, at sufficiently short time, when the macroscopic droplet still acts as a reservoir, the behavior of the precursor film is diffusive and the radial extension length (l) on a macroscopic scale follows a universal time dependence (t) of the form:

where D is the diffusion coefficient of the liquid within the precursor film. It is different from the conventional diffusion coefficient describing the random motion of particles in the bulk liquid phase or on solid substrates. As a matter of fact, D also depends on the driving forces which cause the film spreading.

As shown in Fig. 1, the macroscopic wedge of liquid is advancing at a constant velocity U on a solid dry surface into an external air. The dynamic contact angle is θ_D and the wedge is preceded by a precursor film of length l. The dynamic contact angle θ_D, which depends on the advancing velocity U, the liquid/vapor interfacial tension γ and the liquid viscosity η through Tanner’s law (Tanner 1979).

Precursor film on superhydrophilic/hydrophilic surface

Hydrophilic surfaces adsorb water from the environment and the amount of water depositing on the hydrophilic surface depends on the relative humidity. It is generally accepted that under ordinary atmospheric conditions, hydrophilic surfaces adsorb at least a monolayer of water. For example, a clean glass surface is covered with a monolayer of adsorbed water at relative humidity of around 30–50 % at 20 °C (Razouk and Salem 1948). Formation of a water film composed of as many as twenty molecular layers, or more, may occur at the clean surface of high-energy solids, especially at high relative humidities >90–95 % (Zisman 1965).

Water appears to have unusual lubricating properties and usually gives wearless friction with no stick slip (Raviv et al. 2004). It is also interesting that a 0.25 nm thick water film between two mica surfaces is sufficient to bring the coefficient of friction down to 0.01−0.02, a value that corresponds to the unusually low friction of ice. The effectiveness of water film only 0.25 nm thick to lower the friction force by more than an order of magnitude is attributed to the “hydrophilicity” of the mica surface and to the existence of a strongly repulsive short-range hydration force between such surfaces in aqueous solutions, which effectively removes the adhesion-controlled contribution to the friction force (Berman et al. 1998). Clearly, a single monolayer of water can be a very good lubricant - much better than most other monomolecular liquid films (Ruths & Israelachvili 2011). It also stresses the inertial nature of the spreading: a viscocapillary motion would have been much quicker on a pre-wetted substrate, because of the lubricating effect of this water layer (Biance et al. 2004).

Villette et al. (1996) focused specifically on the role of water on the spreading of molecular films of non-volatile liquids PDMS, PDMS with hydroxyl ends (PDMS-OH) and TK on oxidized silicon wafers. Their analysis revealed the dependence of D on relative humidity (RH). Overall, within the range of 20–90 % RH, D varies by more than two orders of magnitude, attaining the values observed for mesoscopic precursor films. This agrees well with the observations made by (Valignat et al. 1998 and Vou’e et al.1999). Such a remarkable enhancement of D has been explained by the fact that at this value of RH the patches of water on the substrate start to overlap and form a tortuous connected structure on which the spreading of molecules encounters a very low friction. This implies on superhydrophilic/hydrophilic surface, the length of precursor film is much longer than that of the ordinary surfaces. When viewed at the atomic scale, the precursor film plays an important role in the dynamic wetting process on hydrophilic substrate.

As shown in Fig. 2, before the spreading of liquid on superhydrophilic/hydrophilic surface, the solid surface is already covered with a thin film of water with thickness h due to absorption/condensation on hydrophilic surface. The advancing velocity of the macroscopic wedge of liquid is not the average velocity u, but rather u′ (Joanny and de Gennes 1984). Obviously, if h ≠ 0, u′ > u.

We conclude when the superhydrophilic/hydrophilic was previously wetted by a very thin layer of the inner water, the advancing velocity of the macroscopic wedge of liquid is faster than that of dry solid surface.

Lubrication approximation of thin liquid film

We model the spreading of a thin liquid film on an inclined substrate. We use the lubrication approximation of the Navier–Stokes equations, and we provide appropriate initial and boundary conditions. Within the framework of the lubrication approximation, the velocity of the fluid is depth-averaged over the thickness of the film (Greenspan 1978). Following this approach, one obtains the average fluid velocity,

where ∇ = (∂x, ∂y), h is the fluid thickness, p is the pressure, μ is the viscosity, ρ is the density, g is gravity, and α is the inclination angle of the plane of the substrate. The coordinate frame is chosen so that i points down the incline, and j is the transverse direction in the plane. We note that Eq. 10 assumes no-slip boundary condition at the fluid/solid interface. The pressure includes the hydrostatic component, and the contribution following from the Laplace-Young boundary condition at the fluid-air interface.

Nondimensionalization of thin film equations

We consider a layer of liquid on a plane substrate inclined to the horizontal at an angle α. The fluid is Newtonian and incompressible of density ρ, viscosity μ and surface tension γ. We start with the following 2-D lubrication equation (Diez and Kondic 2002):

Here h is the height of the fluid given as a function of x and y. The (x,y) plane is parallel to the substrate and the x direction is the direction of the slope. Then, g is the acceleration of gravity and i is the unit vector in the x direction There are several possibilities to obtain nondimensional variables.

All the theoretical and computational methods of the spreading drop problem require some regularizing mechanism - either assumption of a precursor film in front of the apparent contact line (Troian et al. 1989; Bertozzi and Brenner 1997), or relaxing the no-slip boundary condition at fluid–solid interface (Greenspan 1978; Hocking and Rivers 1982). However, the computational performance of the precursor film model is shown to be much better than that of various slip models. For this reason, in this work, we also use a precursor film model, which assumes that the solid surface is pre-wetted with the wetting layer thickness h' (scaled by h) as a regularizing method in this work.

Thus, the lubrication approximation reduces the Navier–Stokes equations to this nonlinear fourth order partial differential equations that govern the time evolution of the film thickness h(x, y, t).

As we focus on long-scale evolution of liquid films, we scale h by the height h' of the precursor film and we define the scaled in-plane coordinates and time by IEq1_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ \left(x*,y*,t*\right) = \left(\frac{x}{x_c},\frac{y}{\ {y}_c},\frac{t}{t_c}\right) \] \end{document} where

And IEq2_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ a=\sqrt{\frac{\gamma }{\rho g}} \] \end{document} is the capillary length, the velocity scale is chosen naturally as IEq3_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ U=\frac{x_c}{t_c}, \] \end{document} and the capillary number is defined by IEq4_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ =\frac{\mu U}{\gamma }. \] \end{document} Using this dimensionless form, Eq. (12) for IEq5_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ {h}^{*}=\frac{h}{h^{\prime }}, \] \end{document} the lubrication equation given above becomes:

where the single dimensionless parameter IEq6_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ D\left(\alpha \right)={(3Ca)}^{\frac{1}{3}} \cot \left(\alpha \right) \] \end{document} measures the influence of gravity. We note that the lubrication approximation requires the slope of the free surface to be small.

Simulation of water droplet spreading on non-porous flat solid surface

We consider a spherical droplet of volume IEq7_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ V=\frac{4}{3}\pi {r}^3 \] \end{document} deposited on a horizontal wall that enters in contact with a horizontal wettable surface at time t = 0. The initial water droplet radius chosen for this simulation is R = 0.82 mm. Water surface tension is σ = 0.0727 Nm⁻¹; the density of water is equal to ρ =1000 kg/m³. To identify the effects of surface wettability, surfaces with equilibrium contact angles from ~0° to ~117° were studied.

We chose θ_e =1° as a superhydrophilic surface (e.g. existing thin water layer, completely wetting); θ_e =21° as hydrophilic surface (e.g. n-Octyltriethoxysilane surface, partial wetting); θ_e = 117° as hydrophobic surface (e.g. triethoxysilane surface). Based on experimental data of Bird et al. (2008), for 0.1 ≤ t ≤8 ms, we derived the power-law exponent α is about 0.52 and the coefficient C is 1.46 for superhydrophilic surface; for hydrophilic surface, α is about 0.45, the coefficient C is equal to 1.21; for hydrophobic surface, α is about 0.3, the coefficient C is equal to 0.69. By applying Eqs. 15, 16 and 17, we obtained the simulation results as shown in Fig. 4a and b

Figure 4a shows the log-log plot of time t vs spreading radius r of water drops spreading on three surfaces with different wettability. When time is t < 0.1 ms, the wetting follows a power law

r = K t^0.5. For 0.1 ≤ t ≤8 ms, the wetting followed a power law r = K^'t^α. The slope, i.e. α is dependent on the surface wettability and increases from ~0.3 to ~0.5 while θ_eq decreases from ~117° to ~0°, which was consistent with previous studies (Bird et al. 2008). Fig. 4a clearly reveals that the wetting condition has now significant effect on the spreading rate. As the equilibrium contact angle increases, demonstrates a monotonic decrease in the power-law exponent. Drops spread faster on relatively hydrophilic surfaces than on relatively hydrophobic surfaces. At the same time, wetted area on relatively hydrophilic surfaces is larger than that on relatively hydrophobic surfaces.

On partially wettable surfaces (here θ_e =21°) and hydrophobic surface (θ_e =117°), water drops reached equilibrium after the inertial wetting stage, as shown in Fig. 4a and b. In contrast, on completely wetting surfaces, a slower wetting process was observed for t ≥ 8 s. The power law fitting of the data gave a slope of ~0.1, which indicates that spreading was dominated by viscous dissipation (Tanner 1979; Cazabat and Cohen-Stuart 1986).

Simulation of water droplet sliding on non-porous inclined flat solid surface

Hydrophilic surfaces are superior in condensation rate from its early stage whereas dry surfaces directly face humid air (Anna Lee et al. 2012). In this study, we use pre-wetted superhydrophilic surfaces caused by condensation as initial boundary conditions.

A single water droplet sliding on non-porous inclined and smooth solid surface (e.g. glass surface) was simulated under three different surface conditions represented by different levels of condensation. They are dry glass surface (thin water film thickness = 0 μm) and two pre-wetted nano-TiO₂ superhydrophilic surfaces with condensed 50 and 80 μm thickness thin water film, respectively. To simplify the simulation, only the top part of the glass is chosen. The length of glass is chosen to be 5 cm. The origin of the X-axis (x = 0) is set at the top of the glass. The initial height of the water droplet H₀ is 1.5 mm as shown in Fig. 5a. Simulations of three cases start with the same initial water droplet profile. Simulation results are shown in Fig. 5b. Figure 5b clearly shows that the velocity of the water droplets on superhydrophilic surface is faster than on uncoated glass surface at the same inclined angle. The velocity of water droplet is faster on a superhydrophilic surface with a higher condensation rate.

Simulation of water runoff on non-porous inclined flat solid surface

A water runoff on non-porous inclined smooth glass was simulated under three different surface conditions represented by different levels of condensation aforementioned. The origin of the x-axis (x = 0) is set on the top of the glass, with x₀ = 5 mm and x₁ = 6 mm. The initial water runoff thickness H₀ is equal to 0.25 mm. Before x₀, the water front of the runoff is assumed to have the same thickness as H₀, which can be expressed as a Neumann boundary condition. The length of the glass is assumed to be 25 cm long. Water runoff from the top of the glass is as shown in Fig. 6a. Figure 6b shows the velocity of the water film on superhydrophilic surface was faster than that of an ordinary surface at the same inclined angle. The velocity of the water film is faster on a superhydrophilic surface with a higher condensation rate.

Effect of fluid flow velocity on the detachment of the adherent particles

Aerosol particles cover a size range from 1 nm to 100 μm in diameter. Particles in the range 0.01 to 10 μm are most stable in suspension. Particles smaller than 0.01 μm in diameter are not stable in the atmosphere; they will either react with oxygen or tend to coagulate into larger units, while those larger than 10 μm readily settle out in air. Most particles with dimensions greater than 10 μm require strong air currents to keep them aloft (Sharma 1994). Smaller particles < 1 μm float in the air for days or weeks, during this time, they can be transported over 1000 km before they deposit. While larger particles sediment quite soon and easily onto environmental surfaces which can become contaminated because of their weight (Dagsson-Waldhauserova et al. 2014).

Particles begin to move on the adhered surface when the combined lift and drag forces produced by the fluid flow field applied to particles become large enough to counteract the attractive forces (e.g. the gravity, Van der Waals force) between the particle and the surface that hold the particle in place. In the study of particles, in linear shear flow, Zoeteweij et al. (2009) claimed that the particles are in the boundary layer of the fluid flow; these particles will experience a lift force and a drag force exerted on the particle’s body. Both forces are a function of fluid velocity at the position of the particle body. At a certain flow velocity, the particles start to detach from a plate. Among the different possible particle motions (lift, sliding and rotation), particle rotation turns out to be the responsible mechanism of particle removal. Rotation is related to the moment of surface stresses which is also a function of fluid velocity. Dagaonkar (2012) drew a quite similar conclusion, namely that the lift force is strongly a function of the flow velocity of the fluid. Specifically, at high fluid flow velocities, the contribution of the lift force is expected to be higher resulting in the detachment of the larger sized particles.

Grease, dirt particles and other staining materials are easily attached to a surface of a solar panel. In the ‘photocatalytic’ process of TiO₂ coating, the coating reacts with ultraviolet light to break down the organic dirt on the glass and to reduce the adherence of inorganic dirt. In an earlier study, performed by Chabas et al. (2007), on the behaviors of self-cleaning glass in an urban atmosphere, self-cleaning glass is found to have an evident self-cleaning effect, even when it is not subjected to water. The field study shows that particulate organic matter (POM) was destroyed by a percentage of 44–48 % on the self-cleaning surface.

Therefore, we deduce higher water film spreading velocity on superhydrophilic surface ought to have larger detachment force on particles and a better effect on washing away loose particles.

Reflection and light transmittance of Nano-TiO₂ coating

We recently conducted the spectrometric analysis of Nano-TiO₂ coating on PV panels. As we may notice in Fig. 9, in the wave length between 400 to 1200 nm, the reflection on coated surface seems to be 2 ~ 3 % less than that of uncoated surface. Reflections on solar panels need to be avoided. Less reflective surface can be important in increasing a module’s output power. In contrast to light reflection, in the same range of wave length, coated PVT panels shows 2 ~ 3 % more light transmittance than that of uncoated PVT panels. High light transmittance means more photons are absorbed by the solar cells, and more power is generated. We conclude that Nano-TiO2 coating itself has no side effect on the solar cell performance whatsoever. This was very significant since the solar cells could then be coated to improve its other properties without damaging the initial raw performance. Applying this Nano-TiO2 coating on the surface of a clean PV module will potentially increase the efficiency comparing with other coatings which most likely reduce the transmissivity of sunlight.

The output energy of solar panels

The output of a solar panel is usually stated in Watt, and the Watt (the amount of electric power) is determined by multiplying the rated voltage V by the rated amperage I. Both voltage and current were adjusted with the temperature compensation. As our data acquisition system obtains data continuously for thirty seconds every twenty minutes, we use the mean value of I and V within these 30s to obtain the power. By integrating the output power over the testing period, we obtain the curve of “Integrated energy output of testing solar panels via time”. As it can be seen from Fig. 10, a high peak on the figure means a higher energy gain (the weather was sunny, warm and dry), while a low peak is related to a low energy gain (the weather was rainy/cloudy and with a relatively high humidity).

Out-door self-cleaning performance comparison

Figure 11a shows in a pre-coating test, that panels to be coated have a slightly better performance (~2.6 % more energy) than uncoated panels. In a post-coating test, as shown in Fig. 11b, coated panels generated ~6.2 % more energy than uncoated panels (net increase ~3.6 %). This is due to coated surfaces that show less light reflection and smaller transmittance than uncoated surfaces resulting in more generated power.

In an extremity test, solid particles from evaporated muddy water were located on the panel surface. They degraded PV performance very much. However, coated panels generate ~8 % more energy than uncoated panels. This is shown in Fig. 11c (net increase ~5.4 %).

Thanks to two major merits of TiO₂ coating, when a compound (either an organic soil or a pollutant) is present on the surface of nano-TiO₂ coating, it can be degraded by redox reactions involving highly reactive transient species in the presence of UV light. Then, the degradation products are either stored in the coating or washed off the surface by rain water. In addition, Nano-TiO₂ coating is also super-hydrophillic and has high water spreading speed; the super hydrophilicity prevents the formation of water droplets. Water molecules will spread flat immediately to form a thin and uniform water film. This means once there is rain, the rain water forms a uniform film on the coated solar panel surface, which accelerates runoff water spreading and flow, that removes soiling and deters the formation of drying marks as well, and finally results in a better energy generation.

Nomenclature

a, Capillary length (m)

IEq8_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ Ca=\frac{\mu U}{\sigma } \] \end{document}, Capillary number of the runoff equation

C, Prefactor for the power-law

D, Diffusion coefficient (m². s^− 1)

D(α), Dimensionless parameter for the runoff equation

δ, film thickness on (m)

g, Gravitational acceleration (m. s^− 2)

h’, Adimensionalion factor for the film thickness (m)

H₀, Initial droplet or runoff film thickness (m)

h, h*, Film thickness (m) and dimensionless film thickness

h_fg, Latent heat of condensation at the bulk air temperature T∞. (J. kg^− 1)

I, Current (A)

i, Unit vector of the x-axis

j, Unit vector of the y-axis

K, K’, Coefficients of the power law

IEq9_TeX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} \[ {K}_a=\frac{\gamma }{\mu U} \] \end{document}, Kapitza number

k_l, Thermal conductivity of condensate (W. m^− 1. K^− 1)

l, radial extension length (m)

L, Adimensionalion factor for the surface support (m)

L_p, Length of plate (m)

∇ = (∂x, ∂y), Nabla operator

R₀, Drop radius (m)

R_e, the Reynolds number

r, Spreading radius (m)

t, t*, Time (s) and dimensionless time

t_c

T_s, Vapor saturation temperature (297 K)

T_w, Wall temperature (K)

u, Average velocity (m. s^− 1)

Velocity component in the x direction (m. s^− 1)

u', Advancing velocity (m. s^− 1)

U, Adimensionalion factor for the velocity (m. s^− 1)

Characteristic flow velocity (m. s^− 1)

V, Voltage (V)

V, Volume (m³)

v  = (u,v), Velocity vector

v, Velocity component in the x direction (m. s^− 1)

x, x*, X coordinate (m) and dimensionless x coordinate tangentially to the surface support

x_c, Adimensionalion factor for x-axis (m)

α, Angle of the inclined plane(rad)

α, Exponent of the power-law

γ, Air-water surface tension (N. m^− 1)

μ, μ_l, Dynamic viscosity of liquid (kg. m^− 1. s^− 1)

ρ, Density of the water (kg. m^− 3)

ρ_a, Density of air (kg. m^− 3)

θ_D, Dynamic contact angle (°)

θ_e, θ_eq, Equilibrium contact angle (°)

τ, Inertial time scale (s)