Wetting

Low-energy surfaces primarily interact with liquids through dispersive (van der Waals) forces. William Zisman produced several key findings:[11]

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 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.[6] 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.

Figure 3: Coexistence of three fluid phases in mutual contact; here, α, β, and θ each indicate both a phase and its contact angle.

Figure 4: Neumann's triangle relating the surface energies and contact angles of three fluid phases coexisting in static equilibrium, as depicted in Figure 3

Figure 3 shows the line of contact where three phases meet. In equilibrium, the net force per unit length acting along the boundary line between the three phases must be zero. The components of net force in the direction along each of the interfaces are given by:

${\displaystyle {\begin{aligned}\gamma _{\alpha \theta }+\gamma _{\theta \beta }\cos \left(\theta \right)+\gamma _{\alpha \beta }\cos \left(\alpha \right)&=0\\\gamma _{\alpha \theta }\cos \left(\theta \right)+\gamma _{\theta \beta }+\gamma _{\alpha \beta }\cos \left(\beta \right)&=0\\\gamma _{\alpha \theta }\cos \left(\alpha \right)+\gamma _{\theta \beta }\cos \left(\beta \right)+\gamma _{\alpha \beta }&=0\end{aligned}}}$

where α, β, and θ are the angles shown and γ_ij is the surface energy between the two indicated phases. These relations can also be expressed by an analog to a triangle known as Neumann's triangle, shown in Figure 4. Neumann's triangle is consistent with the geometrical restriction that ${\displaystyle \alpha +\beta +\theta =2\pi }$, and applying the law of sines and law of cosines to it produce relations that describe how the interfacial angles depend on the ratios of surface energies.[12]

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.

If the β phase is replaced by a flat rigid surface, as shown in Figure 5, then β = π, and the second net force equation simplifies to the Young equation,[13]

Figure 5: Contact angle of a liquid droplet wetted to a rigid solid surface

${\displaystyle \gamma _{SG}=\gamma _{SL}+\gamma _{LG}\cos \left(\theta \right)}$[14]

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.

Consider the interface as a curve ${\displaystyle y(x)}$ for ${\displaystyle x\in I=[0,L]}$ where ${\displaystyle L}$ is a free parameter. The free energy to be minimized is

${\displaystyle {\cal {F}}[y,L]=\int _{0}^{L}\left(\gamma _{LG}{\sqrt {1+y'^{2}}}+(\gamma _{SL}-\gamma _{SG})\right)dx}$

with the constraints ${\displaystyle y(0)=y(L)=0}$ which we can write as ${\displaystyle \int _{I}y'dx=0}$ and fixed volume ${\displaystyle \int _{I}ydx=A}$.

The modified Lagrangian, taking into account the constraints is therefore

${\displaystyle {\cal {L}}=\gamma _{LG}{\sqrt {1+y'^{2}}}+(\gamma _{SL}-\gamma _{SG})-\lambda _{1}y'-\lambda _{2}y}$

where ${\displaystyle \lambda _{i}}$ are Lagrange multipliers. By definition, the momentum ${\displaystyle p=\partial _{y'}{\cal {L}}}$ and the Hamiltonian ${\displaystyle {\cal {H}}=py'-{\cal {L}}}$ which is computed to be:

${\displaystyle {\cal {H}}=\gamma _{LG}{\frac {1}{\sqrt {1+y'^{2}}}}-(\gamma _{SL}-\gamma _{SG})+\lambda _{2}y}$

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

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

At the boundary ${\displaystyle y(L)=0}$ and ${\displaystyle (1+y'^{2})^{-1/2}=-\cos \theta }$, therefore we recover the Young equation.

The Young equation assumes a perfectly flat and rigid surface often referred to as an ideal surface. In many cases, surfaces are far from this ideal situation, and two are considered here: the case of rough surfaces and the case of smooth surfaces that are still real (finitely rigid). Even in a perfectly smooth surface, a drop will assume a wide spectrum of contact angles ranging from the so-called advancing contact angle, ${\displaystyle \theta _{\mathrm {A} }}$, to the so-called receding contact angle, ${\displaystyle \theta _{\mathrm {R} }}$. The equilibrium contact angle (${\displaystyle \theta _{\mathrm {c} }}$) can be calculated from ${\displaystyle \theta _{\mathrm {A} }}$ and ${\displaystyle \theta _{\mathrm {R} }}$ as was shown by Tadmor[15] as,

${\displaystyle \theta _{\mathrm {c} }=\arccos \left({\frac {r_{\mathrm {A} }\cos \left(\theta _{\mathrm {A} }\right)+r_{\mathrm {R} }\cos \left(\theta _{\mathrm {R} }\right)}{r_{\mathrm {A} }+r_{\mathrm {R} }}}\right)}$

where

${\displaystyle r_{\mathrm {A} }=\left({\frac {\sin ^{3}\left(\theta _{\mathrm {A} }\right)}{2-3\cos \left(\theta _{\mathrm {A} }\right)+\cos ^{3}\left(\theta _{\mathrm {A} }\right)}}\right)^{\frac {1}{3}}~;~~r_{\mathrm {R} }=\left({\frac {\sin ^{3}\left(\theta _{\mathrm {R} }\right)}{2-3\cos \left(\theta _{\mathrm {R} }\right)+\cos ^{3}\left(\theta _{\mathrm {R} }\right)}}\right)^{\frac {1}{3}}}$

The Young–Dupré equation (Thomas Young 1805; Anthanase Dupré and Paul Dupré 1869) dictates that neither γ_SG nor γ_SL can be larger than the sum of the other two surface energies.[16][17] The consequence of this restriction is the prediction of complete wetting when γ_SG > γ_SL + γ_LG and zero wetting when γ_SL > γ_SG + γ_LG. The lack of a solution to the Young–Dupré equation is an indicator that there is no equilibrium configuration with a contact angle between 0 and 180° for those situations.[18]

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

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

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:

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

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

With improvements in measuring techniques such as AFM, confocal microscopy and SEM, researchers were able to produce and image droplets at ever smaller scales. With the reduction in droplet size came new experimental observations of wetting. These observations confirmed that the modified Young's equation does not hold at the micro-nano scales.

For a sessile droplet, the free energy of the three phase system can be expressed as:[19]

${\displaystyle \delta w=\gamma _{LV}dA_{LV}+\gamma _{SL}dA_{SL}+\gamma _{SV}dA_{SV}-\kappa dL-PdV-VdP}$

At constant volume in thermodynamic equilibrium, this reduces to:[19][20]

${\displaystyle 0={\frac {dA_{LV}}{dA_{SL}}}+{\frac {(\gamma _{SL}-\gamma _{SV})}{\gamma _{LV}}}-{\frac {\kappa }{\gamma _{LV}}}{\frac {dL}{dA_{SL}}}-{\frac {V}{\gamma _{LV}}}{\frac {dP}{dA_{SL}}}}$

Usually, the VdP term has been neglected for large droplets, however, VdP work becomes significant at small scales. The variation in the pressure at constant volume at the free liquid-vapor boundary is due to Laplace pressure, which is proportional to the mean curvature, and is non zero. Solving the above equation for both convex and concave surfaces yields:[20]

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

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

${\displaystyle A={\frac {(\gamma _{SV}-\gamma _{SL})}{\gamma _{LV}}}}$, ${\displaystyle B={\frac {\kappa }{\gamma _{LV}}}}$and ${\displaystyle C={\frac {\gamma }{3\gamma _{LV}}}}$.

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

${\displaystyle \cos(\theta )={\frac {(\gamma _{SV}-\gamma _{SL})}{\gamma _{LV}}}+{\frac {\kappa }{\gamma _{LV}}}{\frac {1}{a}}-{\frac {\gamma }{3\gamma _{LV}}}(2+\cos(\theta )-2\cos ^{2}(\theta )-\cos ^{3}(\theta ))}$

The first two terms in the above equation 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.[20]

For many surface/adsorbate configurations, surface energy data and experimental observations are unavailable. As wetting interactions are of great importance in various applications, it is often desired to predict and compare the wetting behavior of various material surfaces with particular crystallographic orientations, with relation to water or other adsorbates. This can be done from an atomistic perspective with tools including molecular dynamics and density functional theory.[21][22] In the theoretical prediction of wetting by ab initio approaches such as DFT, ice is commonly substituted for water. This is because DFT calculations are generally conducted assuming conditions of zero thermal movement of atoms, essentially meaning the simulation is conducted at absolute zero. This simplification nevertheless yields results that are relevant for the adsorption of water under realistic conditions and the use of ice for the theoretical simulation of wetting is commonplace.[23]

Figure 7: Wenzel model

The Wenzel model (Robert N. Wenzel, 1936) describes the homogeneous wetting regime, as seen in Figure 7, and is defined by the following equation for the contact angle on a rough surface:[27]

${\displaystyle \cos \,\left(\theta ^{*}\right)=r\cos \,\left(\theta \right)}$

where ${\displaystyle \theta ^{*}}$ is the apparent contact angle which corresponds to the stable equilibrium state (i.e. minimum free energy state for the system). The roughness ratio, r, is a measure of how surface roughness affects a homogeneous surface. The roughness ratio is defined as the ratio of true area of the solid surface to the apparent area.

θ is the Young contact angle as defined for an ideal surface. Although Wenzel's equation demonstrates the contact angle of a rough surface is different from the intrinsic contact angle, it does not describe contact angle hysteresis.[29]

Figure 8: Cassie–Baxter model

When dealing with a heterogeneous surface, the Wenzel model is not sufficient. A more complex model is needed to measure how the apparent contact angle changes when various materials are involved. This heterogeneous surface, like that seen in Figure 8, is explained using the Cassie–Baxter equation (Cassie's law):[27]

${\displaystyle \cos \,\left(\theta ^{*}\right)=r_{f}\,f\,\cos \,\left(\theta _{\text{Y}}\right)+f-1}$

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. It is important to realize that 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 ${\displaystyle f_{i}}$.

A summation of all ${\displaystyle f_{i}}$ equals 1 or the total surface. Cassie–Baxter can also be recast in the following equation:[30]

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

Here γ is the Cassie–Baxter surface tension between liquid and vapor, γ_i,sv is the solid vapor surface tension of every component, and γ_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:[30]

${\displaystyle \gamma \cos \,\left(\theta ^{*}\right)=f_{1}\left(\gamma _{\text{1,sv}}-\gamma _{\text{1,sl}}\right)-\left(1-f_{1}\right)\gamma }$

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.[31][32]

With the advent of high resolution imaging, researchers have started to obtain experimental data which have led them to question the assumptions of the Cassie–Baxter equation when calculating the apparent contact angle. These groups believe the apparent contact angle is largely dependent on the triple line. The triple line, which is in contact with the heterogeneous surface, cannot rest on the heterogeneous surface like the rest of the drop. In theory, it should follow the surface imperfection. This bending in the triple line is unfavorable and is not seen in real-world situations. A theory that preserves the Cassie–Baxter equation while at the same time explaining the presence of the minimized energy state of the triple line hinges on the idea of a precursor film. This film of submicrometer thickness advances ahead of the motion of the droplet and is found around the triple line. Furthermore, this precursor film allows the triple line to bend and take different conformations that were originally considered unfavorable. This precursor fluid has been observed using environmental scanning electron microscopy (ESEM) in surfaces with pores formed in the bulk. With the introduction of the precursor film concept, the triple line can follow energetically feasible conformations, thereby correctly explaining the Cassie–Baxter model.[33]

Figure 9: "Petal effect" vs. "lotus effect"

The intrinsic hydrophobicity of a surface can be enhanced by being textured with different length scales of roughness. The red rose takes advantage of this by using a hierarchy of micro- and nanostructures on each petal to provide sufficient roughness for superhydrophobicity. More specifically, each rose petal has a collection of micropapillae on the surface and each papilla, in turn, has many nanofolds. The term "petal effect" describes the fact that a water droplet on the surface of a rose petal is spherical in shape, but cannot roll off even if the petal is turned upside down. The water drops maintain their spherical shape due to the superhydrophobicity of the petal (contact angle of about 152.4°), but do not roll off because the petal surface has a high adhesive force with water.[34]

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.[35]

Figure 10: Mushroom state

In the Cassie–Baxter model, the drop sits on top of the textured surface with trapped air underneath. During the wetting transition from the Cassie state to the Wenzel state, the air pockets are no longer thermodynamically stable and liquid begins to nucleate from the middle of the drop, creating a "mushroom state" as seen in Figure 10.[36] The penetration condition is given by:

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

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)

Figure 11: Penetration front spreads beyond drop

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.[37]

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:

${\displaystyle \cos \,\left(\theta ^{*}\right)=\phi \cos \,\left(\theta _{C}\right)+\left(1-\phi \right)}$[36]

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.

Many technological processes require control of liquid spreading over solid surfaces. When a drop is placed on a surface, it can completely wet, partially wet, or not wet the surface. By reducing the surface tension with surfactants, a nonwetting material can be made to become partially or completely wetting. The excess free energy (σ) of a drop on a solid surface is:[39]

${\displaystyle \sigma =\gamma S+PV+\pi \,R^{2}\left(\gamma _{\text{SL}}-\gamma _{\text{SV}}\right)}$

• γ is the liquid–vapor interfacial tension
• γ_SL is the solid–liquid interfacial tension
• γ_SV is the solid–vapor interfacial tension
• S is the area of liquid–vapor interface
• P is the excess pressure inside liquid
• R is the radius of droplet base

Based on this equation, the excess free energy is minimized when γ decreases, γ_SL decreases, or γ_SV increases. Surfactants are absorbed onto the liquid–vapor, solid–liquid, and solid–vapor interfaces, which modify the wetting behavior of hydrophobic materials to reduce the free energy. When surfactants are absorbed onto a hydrophobic surface, the polar head groups face into the solution with the tail pointing outward. In more hydrophobic surfaces, surfactants may form a bilayer on the solid, causing it to become more hydrophilic. The dynamic drop radius can be characterized as the drop begins to spread. Thus, the contact angle changes based on the following equation:[39]

${\displaystyle \cos \,\left(\theta (t)\right)=\cos \,\left(\theta _{0}\right)+\left(\cos \,\left(\theta _{\infty }\right)-\cos \,\left(\theta _{0}\right)\right)\left(1-\mathrm {e} ^{-{\frac {t}{\tau }}}\right)}$

• θ₀ is initial contact angle
• θ_∞ is final contact angle
• τ is the surfactant transfer time scale

As the surfactants are absorbed, the solid–vapor surface tension increases and the edges of the drop become hydrophilic. As a result, the drop spreads.

Strands of an uncharged ferrocene-substituted polymer are tethered to a hydrophobic silica surface. Oxidation of the ferrocenyl groups produces a hydrophilic surface due to electrostatic attractions between the resulting charges and the polar solvent.[40]

Ferrocene is a redox-active organometallic compound[41] which can be incorporated into various monomers and used to make polymers which can be tethered onto a surface.[40] Vinylferrocene (ferroceneylethene) can be prepared by a Wittig reaction[42] and then polymerized to form polyvinylferrocene (PVFc), an analog of polystyrene. Another polymer which can be formed is poly(2-(methacryloyloxy)ethyl ferrocenecarboxylate), PFcMA. Both PVFc and PFcMA have been tethered onto silica wafers and the wettability measured when the polymer chains are uncharged and when the ferrocene moieties are oxidised to produce positively charged groups, as illustrated at right.[40] The contact angle with water on the PFcMA-coated wafers was 70° smaller following oxidation, while in the case of PVFc the decrease was 30°, and the switching of wettability has been shown to be reversible. In the PFcMA case, the effect of longer chains with more ferrocene groups (and also greater molar mass) has been investigated, and it was found that longer chains produce significantly larger contact angle reductions.[40][43]

Rare earth oxides exhibit intrinsic hydrophobicity, and hence can be used in thermally stable heat exchangers and other applications involving high-temperature hydrophobicity.[44] The presence of oxygen vacancies at surfaces of ceria or other rare earth oxides is instrumental in governing surface wettability. Adsorption of water at oxide surfaces can occur as molecular adsorption, in which H₂O molecules remain intact at the terminated surface, or as dissociative adsorption, in which OH and H are adsorbed separately[45] at solid surfaces. The presence of oxygen vacancies is generally found to enhance hydrophobicity while promoting dissociative adsorption.[46]