Wetting

 ( Fluid Mechanics ) 

• Hydrophobe – molecular definition of hydrophobicity and its energetic basis (DIT)
• Contact angle – classical macroscopic measure of wetting related to hydrophobic/hydrophilic behavior

The other type of solid is weak molecular crystals (e.g., , , etc.) where the molecules are held together essentially by physical forces (e.g., van der Waals forces and ). Since these solids are held together by weak forces, a very low amount of energy is required to break them, thus they are termed "low-energy". Depending on the type of liquid chosen, low-energy surfaces can permit either complete or partial wetting.

Zisman observed that cos θ increases linearly as the surface tension (γ_LV) of the liquid decreased. Thus, he was able to establish a linear function between cos θ and the surface tension (γ_LV) for various organic compound liquids.

A surface is more wettable when γ_LV and θ is low. Zisman termed the intercept of these lines when cos θ = 1 as the critical surface tension (γ_c) of that surface. This critical surface tension is an important parameter because it is a characteristic of only the solid.

Knowing the critical surface tension of a solid, it is possible to predict the wettability of the surface. The wettability of a surface is determined by the outermost chemical groups of the solid. Differences in wettability between surfaces that are similar in structure are due to differences in the packing of the atoms. For instance, if a surface has branched chains, it will have poorer packing than a surface with straight chains. Lower critical surface tension means a less wettable material surface.

 \gamma_{\alpha\theta}                        + \gamma_{\theta\beta}\cos\left(\theta\right) + \gamma_{\alpha\beta}\cos\left(\alpha\right) &= 0 \\
 \gamma_{\alpha\theta}\cos\left(\theta\right) + \gamma_{\theta\beta}                        + \gamma_{\alpha\beta}\cos\left(\beta\right)  &= 0 \\
 \gamma_{\alpha\theta}\cos\left(\alpha\right) + \gamma_{\theta\beta}\cos\left(\beta\right)  + \gamma_{\alpha\beta}                        &= 0
     

Because these three surface energies form the sides of a triangle, they are constrained by the triangle inequalities, γ_ij < γ_jk + γ_ik meaning that not one of the surface tensions can exceed the sum of the other two. If three fluids with surface energies that do not follow these inequalities are brought into contact, no equilibrium configuration consistent with Figure 3 will exist.

$\backslash gamma\_\{SG\}\; =\; \backslash gamma\_\{SL\}\; +\; \backslash gamma\_\{LG\}\backslash cos\backslash left(\backslash theta\backslash right)$

which relates the surface tensions between the three phases: solid, liquid and gas. Subsequently, this predicts the contact angle of a liquid droplet on a solid surface from knowledge of the three surface energies involved. This equation also applies if the "gas" phase is another liquid, immiscible with the droplet of the first "liquid" phase.

$\{\backslash cal\; F\}y,L\; =\; \backslash int\_0^L\; \backslash left(\; \backslash gamma\_\{LG\}\backslash sqrt\{1+y\text{'}^2\}\; +\; (\backslash gamma\_\{SL\}-\backslash gamma\_\{SG\})\; \backslash right)\; dx$

The modified Lagrangian, taking into account the constraints is therefore

$\{\backslash cal\; L\}\; =\; \backslash gamma\_\{LG\}\backslash sqrt\{1+y\text{'}^2\}\; +\; (\backslash gamma\_\{SL\}-\backslash gamma\_\{SG\})\; -\backslash lambda\_1\; y\text{'}\; -\; \backslash lambda\_2\; y$

$\{\backslash cal\; H\}\; =\; -\backslash gamma\_\{LG\}\backslash frac\{1\}\{\backslash sqrt\{1+y\text{'}^2\}\}\; -$

Now, we recall that the boundary is free in the $x$ direction and $L$ is a free parameter. Therefore, we must have:

$\backslash frac\{\backslash partial\{\backslash cal\; F\}\}\{\backslash partial\; L\}\; =\; -\{\backslash cal\; H\}\; =\; 0$

$\backslash theta\_\backslash mathrm\{c\}\; =\; \backslash arccos\backslash left($

 \frac{r_\mathrm{A}\cos\left(\theta_\mathrm{A}\right) + r_\mathrm{R}\cos\left(\theta_\mathrm{R}\right)}
      {r_\mathrm{A} + r_\mathrm{R}}
     

where

 r_\mathrm{A} = \left(
   \frac{\sin^3\left(\theta_\mathrm{A}\right)}{2 - 3\cos\left(\theta_\mathrm{A}\right) + \cos^3\left(\theta_\mathrm{A}\right)}
 \right)^\frac{1}{3}
     

 r_\mathrm{R} = \left(
   \frac{\sin^3\left(\theta_\mathrm{R}\right)}{2 - 3\cos\left(\theta_\mathrm{R}\right) + \cos^3\left(\theta_\mathrm{R}\right)}
 \right)^\frac{1}{3}
     

A useful parameter for gauging wetting is the spreading parameter S,

$S\; =\; \backslash gamma\_\{SG\}\; -\; \backslash left(\backslash gamma\_\{SL\}\; +\; \backslash gamma\_\{LG\}\backslash right)$

When S > 0, the liquid wets the surface completely (complete wetting). When S < 0, partial wetting occurs.

Combining the spreading parameter definition with the Young relation yields the Young–Dupré equation:

$S\; =\; \backslash gamma\_\{LG\}\backslash left(\backslash cos\backslash left(\backslash theta\backslash right)\; -\; 1\backslash right)$

which only has physical solutions for θ when S < 0.

For a sessile droplet, the free energy of the three phase system can be expressed as:

$\backslash delta\; w\; =\; \backslash gamma\_\{LV\}dA\_\{LV\}+\backslash gamma\_\{SL\}dA\_\{SL\}+\backslash gamma\_\{SV\}dA\_\{SV\}-\backslash kappa\; dL-PdV-VdP$

At constant volume in thermodynamic equilibrium, this reduces to:

$0=\; \backslash frac\{dA\_\{LG\}\}\{dA\_\{SL\}\}\; +\; \backslash frac\{\backslash gamma\_\{SL\}-\backslash gamma\_\{SG\}\}\{\backslash gamma\_\{LG\}\}\; -\; \backslash frac\{\backslash kappa\}\{\backslash gamma\_\{LG\}\}\backslash frac\{dL\}\{dA\_\{SL\}\}-\backslash frac\{V\}\{\backslash gamma\_\{LG\}\}\; \backslash frac\{dP\}\{dA\_\{SL\}\}$

Usually, the VdP term has been neglected for large droplets, however, VdP work becomes significant at small scales. The variation in pressure at constant volume at the free liquid-vapor boundary is due to the Laplace pressure, which is proportional to the mean curvature of the droplet, and is non zero. Solving the above equation for both convex and concave surfaces yields:

$\backslash cos(\backslash theta\backslash mp\backslash alpha)=A+B\backslash frac\{\backslash cos(\backslash alpha)\}\{a\}\backslash pm\; C\backslash sin(\backslash theta\backslash mp\backslash alpha)(\backslash cos(\backslash theta)+1)^2\backslash biggl(\backslash frac\{\backslash sin(\backslash alpha)(\backslash cos(\backslash alpha)+2)\}\{(\backslash cos(\backslash alpha)+1)^2\}\backslash mp\backslash frac\{\backslash sin(\backslash theta)(\backslash cos(\backslash theta)+2)\}\{(\backslash cos(\backslash theta)+1)^2\}\backslash biggr)$

Where the constant parameters A, B, and C are defined as:

$A\; =\; \backslash frac\{\backslash gamma\_\{SG\}-\backslash gamma\_\{SL\}\}\{\backslash gamma\_\{LG\}\}$, $B\; =\; \backslash frac\{\backslash kappa\}\{\backslash gamma\_\{LG\}\}$ and $C\; =\; \backslash frac\{\backslash gamma\}\{3\backslash gamma\_\{LG\}\}$

This equation relates the contact angle $\backslash theta$, a geometric property of a sessile droplet to the bulk thermodynamics, the energy at the three phase contact boundary, and the curvature of the surface α. For the special case of a sessile droplet on a flat surface (α=0),

$\backslash cos(\backslash theta)\; =\; \backslash frac\{\backslash gamma\_\{SG\}-\backslash gamma\_\{SL\}\}\{\backslash gamma\_\{LG\}\}\; +\; \backslash frac\{\backslash kappa\}\{\backslash gamma\_\{LG\}\}\; \backslash frac\{1\}\{a\}\; -\backslash frac\{\backslash gamma\}\{3\backslash gamma\_\{LG\}\}(2+\backslash cos(\backslash theta)-2\backslash cos^2(\backslash theta)-\backslash cos^3(\backslash theta))$

The first two terms are the modified Young's equation, while the third term is due to the Laplace pressure. This nonlinear equation correctly predicts the sign and magnitude of κ, the flattening of the contact angle at very small scales, and contact angle hysteresis.

$\backslash text\{H\}\; =\; \backslash ,\backslash theta\_\{a\}\; -\; \backslash ,\backslash theta\_\{r\}$

When the contact angle is between the advancing and receding cases, the contact line is considered to be pinned and hysteretic behaviour can be observed, namely contact angle hysteresis. When these values are exceeded, the displacement of the contact line, such as the one in Figure 3, will take place by either expansion or retraction of the droplet. Figure 6 depicts the advancing and receding contact angles. The advancing contact angle is the maximum stable angle, whereas the receding contact angle is the minimum stable angle. Contact angle hysteresis occurs because many different thermodynamically stable contact angles are found on a nonideal solid. These varying thermodynamically stable contact angles are known as metastable states.

Such motion of a phase boundary, involving advancing and receding contact angles, is known as dynamic wetting. The difference between dynamic and static wetting angles is proportional to the capillary number, $Ca$, When a contact line advances, covering more of the surface with liquid, the contact angle is increased and is generally related to the velocity of the contact line.

$\backslash cos\backslash ,\backslash left(\backslash theta^*\backslash right)\; =\; r\_f\backslash ,f\; \backslash ,\backslash cos\backslash ,\backslash left(\backslash theta\_\backslash text\{Y\}\backslash right)\; +\; f\; -\; 1$

Here the r_f is the roughness ratio of the wet surface area and f is the fraction of solid surface area wet by the liquid. When f = 1 and r_f = r, the Cassie–Baxter equations becomes the Wenzel equation. On the other hand, when there are many different fractions of surface roughness, each fraction of the total surface area is denoted by $f\_i$.

A summation of all $f\_i$ equals 1 or the total surface. Cassie–Baxter can also be recast in the following equation:

$\backslash gamma\backslash cos\backslash ,\backslash left(\backslash theta^*\backslash right)\; =\; \backslash sum\_\{n=1\}^N\; f\_i\backslash left(\backslash gamma\_\backslash text\{i,sv\}\; -\; \backslash gamma\_\backslash text\{i,sl\}\backslash right)$

Here $\backslash gamma$ is the Cassie–Baxter surface tension between liquid and vapor, $\backslash gamma\_\{i,sv\}$ is the solid vapor surface tension of every component, and $\backslash gamma\_\{i,sl\}$ is the solid liquid surface tension of every component. A case that is worth mentioning is when the liquid drop is placed on the substrate and creates small air pockets underneath it. This case for a two-component system is denoted by:

$\backslash gamma\backslash cos\backslash ,\backslash left(\backslash theta^*\backslash right)\; =\; f\_1\backslash left(\backslash gamma\_\backslash text\{1,sv\}\; -\; \backslash gamma\_\backslash text\{1,sl\}\backslash right)\; -\; \backslash left(1\; -\; f\_1\backslash right)\backslash gamma$

Here the key difference to notice is that there is no surface tension between the solid and the vapor for the second surface tension component. This is because of the assumption that the surface of air that is exposed is under the droplet and is the only other substrate in the system. Subsequently, the equation is then expressed as (1 – f). Therefore, the Cassie equation can be easily derived from the Cassie–Baxter equation. Experimental results regarding the surface properties of Wenzel versus Cassie–Baxter systems showed the effect of pinning for a Young angle of 180 to 90°, a region classified under the Cassie–Baxter model. This liquid/air composite system is largely hydrophobic. After that point, a sharp transition to the Wenzel regime was found where the drop wets the surface, but no further than the edges of the drop. Actually, the Young, Wenzel and Cassie-Baxter equations represent the transversality conditions of the variational problem of wetting.

When comparing the "petal effect" to the "lotus effect", it is important to note some striking differences. The surface structure of the lotus leaf and the rose petal, as seen in Figure 9, can be used to explain the two different effects.

The lotus leaf has a randomly rough surface and low contact angle hysteresis, which means the water droplet is not able to wet the microstructure spaces between the spikes. This allows air to remain inside the texture, causing a heterogeneous surface composed of both air and solid. As a result, the adhesive force between the water and the solid surface is extremely low, allowing the water to roll off easily (i.e. "self-cleaning" phenomenon).

The rose petal's micro- and nanostructures are larger in scale than those of the lotus leaf, which allows the liquid film to impregnate the texture. However, as seen in Figure 9, the liquid can enter the larger-scale grooves, but it cannot enter into the smaller grooves. This is known as the Cassie impregnating wetting regime. Since the liquid can wet the larger-scale grooves, the adhesive force between the water and solid is very high. This explains why the water droplet will not fall off even if the petal is tilted at an angle or turned upside down. This effect will fail if the droplet has a volume larger than 10 μL because the balance between weight and surface tension is surpassed.

$\backslash cos\backslash ,\backslash left(\backslash theta\_\backslash text\{C\}\backslash right)=\; \backslash frac\{\backslash phi\; -\; 1\}\{r\; -\; \backslash phi\}$

where

* θ_C is the critical contact angle

* Φ is the fraction of solid/liquid interface where drop is in contact with surface

* r is solid roughness (for flat surface, r = 1)

The penetration front propagates to minimize the surface energy until it reaches the edges of the drop, thus arriving at the Wenzel state. Since the solid can be considered an absorptive material due to its surface roughness, this phenomenon of spreading and imbibition is called hemiwicking. The contact angles at which spreading/imbibition occurs are between 0 and π/2.

The Wenzel model is valid between θ_C and π/2. If the contact angle is less than Θ _C, the penetration front spreads beyond the drop and a liquid film forms over the surface. Figure 11 depicts the transition from the Wenzel state to the surface film state. The film smoothes the surface roughness and the Wenzel model no longer applies. In this state, the equilibrium condition and Young's relation yields:

$\backslash cos\backslash ,\backslash left(\backslash theta^*\backslash right)\; =\; \backslash phi\backslash cos\backslash ,\backslash left(\backslash theta\_C\backslash right)\; +\; \backslash left(1\; -\; \backslash phi\backslash right)$

By fine-tuning the surface roughness, it is possible to achieve a transition between both superhydrophobic and superhydrophilic regions. Generally, the rougher the surface, the more hydrophobic it is.

$r(t)\; =\; r\_e\backslash left[$

   1 - \exp\left(-\left(\frac{2\gamma_{LG}}{r^{12}_e} + \frac{\rho g}{9r^{10}_e}\right)\frac{24\lambda V^4 \left(t + t_0\right)}{\pi^2\eta}\right)
     

For the complete wetting situation, the drop radius at any time during the spreading process is given by

$r(t)\; =\; \backslash left[$

   \left(\gamma_{LG} \frac{96\lambda V^4}{\pi^2 \eta} \left(t + t_0\right)\right)^\frac{1}{2} +
   \left(\frac{\lambda(t + t_0)}{\eta}\right)^\frac{2}{3} \frac{24\rho g V^\frac{8}{3}}{7 \cdot 96^\frac{1}{3} \pi^\frac{4}{3} \gamma_{LG}^\frac{1}{3}}
 \right]^\frac{1}{6},
     

where

* γ_LG is surface tension of the fluid

* V is drop volume

* η is viscosity of the fluid

* ρ is density of the fluid

* g is gravitational constant

* λ is shape factor, 37.1m⁻¹

* t₀ is experimental delay time

* r_e is drop radius in equilibrium