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# Wetting

[wet]

Wetting is the contact between a liquid and a solid surface, resulting from intermolecular interactions when the two are brought together. The amount of wetting depends on the energies (or surface tensions) of the interfaces involved such that the total energy is minimized. The degree of wetting is described by the contact angle, the angle at which the liquid-vapor interface meets the solid-liquid interface. If the wetting is very favorable, the contact angle will be low, and the fluid will spread to cover a larger area of the surface. If the wetting is unfavorable, the fluid will form a compact droplet on the surface. Regardless of the amount of wetting, the shape of a drop wetted to a rigid surface is roughly a truncated sphere. Various degrees of wetting are depicted in Figure 1.

A contact angle of 90° or greater generally characterizes a surface as not-wettable, and one less than 90° as wettable. In the context of water, a wettable surface may also be termed hydrophilic and a non-wettable surface hydrophobic. Superhydrophobic surfaces have contact angles greater than 150°, showing almost no contact between the liquid drop and the surface. This is sometimes referred to as the "Lotus effect". Wetting is also important in the bonding or adherence of two materials. Wetting and the surface forces that control wetting are also responsible for other related effects, including so-called capillary effects.

## Minimization of energy, three phases

Consider the line of contact where three phases meet, as shown in Figure 2. 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:

$gamma_\left\{alphatheta\right\}+gamma_\left\{thetabeta\right\}cos\left\{theta\right\}+gamma_\left\{alphabeta\right\}cos\left\{alpha\right\} = 0$
$gamma_\left\{alphatheta\right\}cos\left\{theta\right\}+gamma_\left\{thetabeta\right\}+gamma_\left\{alphabeta\right\}cos\left\{beta\right\} = 0$
$gamma_\left\{alphatheta\right\}cos\left\{alpha\right\}+gamma_\left\{thetabeta\right\}cos\left\{beta\right\}+gamma_\left\{alphabeta\right\} = 0$

where $alpha$, $beta$, and $theta$ are the angles shown and $gamma_\left\{ij\right\}$ 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 3. Neumann’s triangle is consistent with the geometrical restriction that $alpha+beta+theta=2pi$, 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.

Because these three surface energies form the sides of a triangle, they are constrained by the triangle inequalities,

### Simplification to planar geometry, Young's relation

If the $beta$ phase is replaced by a flat rigid surface, as shown in Figure 4, then $beta=pi$, and the second net force equation simplifies to the Young equation,

$gamma_\left\{SG\right\} =gamma_\left\{SL\right\}+gamma_\left\{LG\right\}cos\left\{theta\right\}$

which relates the surface tensions between the three phases solid, liquid and gas, and which 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.

The Young–Dupre equation dictates that neither $gamma_\left\{SG\right\}$ nor $gamma_\left\{SL\right\}$ can be larger than the sum of the other two surface energies. The consequence of this restriction is the prediction of complete wetting when $gamma_\left\{SG\right\} > gamma_\left\{SL\right\}+gamma_\left\{LG\right\}$ and zero wetting when $gamma_\left\{SL\right\} > gamma_\left\{SG\right\}+gamma_\left\{LG\right\}$. The lack of a solution to the Young–Dupre equation is an indicator that there is no equilibrium configuration with a contact angle between 0 and 180 degrees for those situations.

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

$S = gamma_\left\{SG\right\}-\left(gamma_\left\{SL\right\}+gamma_\left\{LG\right\}\right)$

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

Combining the spreading parameter definition with the Young relation, we obtain the Young-Dupre equation:

$S = gamma_\left\{LG\right\}\left(costheta-1\right)$

which only has physical solutions for $theta$ when S < 0.

## Dynamic wetting

The above derivations all apply only to the state in which the interfaces are not moving and the phase boundary line exists in equilibrium. When a phase boundary is in motion, such as in the case of a spreading droplet or advancing contact edge, different mechanics apply. Many aspects of dynamic wetting are not fully understood, and the subject is an area of great interest to many scientists.

When a contact line such as the one in figure 4 is displaced, by either expansion or retraction of the droplet, there is a hysteresis observed in the contact angle. The static contact angle that results after expansion of a droplet is higher than that observed after a contraction. It is also often observed that the contact line does not move smoothly at the microscale. Rather, it is seen to jump abruptly in increments, by an apparent stick-slip mechanism. This has often been attributed to imperfections in the surface causing the contact line to be momentarily pinned, but this description is not complete.

When a contact line advances, covering more of the surface with liquid, the contact angle is increased and generally is related to the velocity of the contact line. A receding interface likewise has a contact angle that is reduced from the static contact angle. The limits of contact angle as velocity approaches zero in the forward and backward directions are not equal, and the range between them defines a range of contact angles that are observed as static contact angles in hysteresis experiments.

If the velocity of a contact line is increased without bound, the contact angle increases, and as it approaches 180° the gas phase will become entrained in a thin layer between the liquid and solid. This is a kinetic non-equilibrium effect which results from the contact line moving at such a high speed that complete wetting cannot occur.

Dynamic wetting is of great importance in industrial processes, where surfaces often must be coated uniformly and quickly with a liquid. Entrainment of air is unacceptable for the quality of products, but the volume demanded necessitates coating at as high a speed as possible.

### Molecular theories

Several molecular theories of dynamic wetting have been proposed. The determination of a theory that describes dynamic wetting observations is complicated by the apparent contradiction with established theories of wetting. For example, in the standard model of viscous flow, there is no slippage of the surface layer of liquid atoms along the surface, but in the immediate vicinity of a progressing contact line, it is necessary to relax this restriction to prevent the prediction of infinite shear.

When a contact line advances, it is seen to be preceded by a thin “precursor film” of submicrometer thickness, that advances ahead of the motion of the droplet. Initially, the precursor film was thought to be an artifact of volatility, but its observation in systems with no vapor presence requires a new theory. Measurements of the precursor film have been made by optical ellipsometry and also by sensitive electrical measurements. These experiments have all suffered from limitations on the liquids used and transient effects in droplet spreading, and have failed to provide any basis for a useful model of precursor film formation.