CHAPTER 1: Non-Wetting Fundamentals

Wetting is a ubiquitous process that occurs in a huge variety of everyday biological and industrial systems. It is a macroscopic process that is very sensitive to surface properties on the nano or molecular scale. In most wetting situations the solid surface is wet only to some extent, depending on its chemical and physical nature. As is well known, the common quantitative measure of wettability is the contact angle (CA), which in most cases is greater than 0° and much less than 180°. However, the extreme cases of either complete wetting (CA = 0°) or non-wetting (very high CA and additional possible criteria to be discussed below) offer interesting scientific challenges as well as practical applications. Actually, nature has been using non-wetting to solve a variety of important needs, and the main scientific principle has been known for about half a century.¹  However, it is only about two decades ago that it started to become a very popular topic in science and engineering.^2–46  The paper by Neinhuis and Barthlott³  served as an important trigger to the vast interest in non-wetting. It introduced the term “lotus effect” that refers to the self-cleaning of the lotus leaf (and many others), achieved by water drops easily rolling off the surface of the leaf, carrying with them dust and dirt particles.

However, non-wettability is relevant not only for self-cleaning of leaves and not only for drops. For example, some aquatic animals breathe air from an air film on their body even when they are under water. This air film is re-created each time the animal goes back into the water.e.g.^24,36  In addition, while natural systems are predominantly aqueous, the non-wettability of solid surfaces by oils, or organic liquids in general, is also of great practical importance in daily life and in industry.^25,26 

At this point it is important to discuss terminology,^25,41  since there is no standard one and the variety of terms may lead to confusion. A surface that is not wetted by water drops in air, or may sustain an air film under water, is in many cases classified as “water repellent”. This usage is unfortunate, because there is nothing active in this process that repels water. The adjective “non-wettable” (or the noun “non-wetting”), on the other hand, appears to be more true to the facts. Moreover, the so-called “water repellent” surfaces are usually classified as “superhydrophobic”. However, when a surface is not wetted even by liquids of lower surface tension than of water, this term cannot be used, since “hydro” specifically means water. For this purpose, other terms are used, seemingly at random. One term is “superoleophobic”. This is a problematic term, since a surface that is “superoleophobic” is usually also superhydrophobic, so “oleophobic” refers only to a part of the picture. On the other hand, a term such as “omniphobic”, which means “fearing everything”, is far too wide, since, after all, the discussion is about liquids, not about everything. Some time ago I suggested^25,42  using the term “superhygrophobic” to imply non-wetting, because “hygro” in Greek means “liquid”. Thus, the terms “hygrophobic” and “superhygrophobic” exactly express various degrees of non-wetting by liquids in general. In summary, “non-wetting” is a generic term that may be specifically complemented by “superhydrophobic” or “superhygrophobic” when it is important to know what the specific case is.

In order to develop useful non-wettable surfaces, it is important to understand the fundamental theory and apply it in choosing the chemical and physical properties of the surfaces. The objective of this chapter is to present the thermodynamic fundamentals of non-wetting, as they are derived from the general theory of wetting equilibrium. An important aspect that has not been sufficiently noticed and is emphasized here is that of thermodynamic stability. In general, qualitative aspects are stressed in this chapter, with only a few necessary equations, in order to give the general picture rather than the mathematical details.

As is well known, minimizing the energy of a system (internal, Gibbs, or Helmholtz energy, depending on the conditions at the system boundary) leads to a few indicators of equilibrium. First, for all systems, irrespective of the existence of interfaces, the temperature as well as the generalized chemical potential of each species must be uniform throughout the whole system. Then, there are two equations that govern the equilibrium state of an interface: the Young equation and the Young–Laplace equation. The former determines the boundary condition for the shape of the liquid–gas interface, in terms of the local CA that must equal the Young CA, θ_Y. For solid–liquid–gas systems it is given by

here, σ and σ^s are the surface tension of the liquid and of the solid, respectively, and σ^sl is the solid–liquid interfacial tension. This equation is correct for radii of curvature much above the nano scale, for which line tension is negligible e.g.ref. 47.

The Young–Laplace equation determines the shape of the interface, in terms of the local curvature that is determined by the local pressure difference across the interface:

in this equation, P^d and P^c are the local pressure in the drop and in the continuous phase, respectively, and R₁ and R₂ are the local radii of curvature. In the absence of gravity (or other external fields), the pressure difference is constant across the interface. This implies that the average curvature is also constant across the interface. This well-known fact is important for understanding the behaviour of liquids inside roughness grooves, as will be discussed later.

Eqn (1.1) and (1.2) completely determine the equilibrium behaviour of an interface. When the solid surface is ideal (i.e. rigid, smooth, chemically uniform, non-reactive, and insoluble) there is only one solution to these equations, which requires the apparent, namely macroscopically measured CA, to equal the Young CA. However, when the surface is rough or chemically non-uniform, there are many possible solutions. Each solution is characterized by its own apparent CA. Naturally, it is important and interesting to find out (a) which of these solutions has the lowest energy, namely which is the thermodynamically most stable CA, and (b) what are the lowest (receding) and highest (advancing) apparent CAs. The difference between the advancing and receding CAs is called the CA hysteresis range.

When a mathematical function has multiple minima, the only way to identify these minima is to search for them one by one. To find out the global minimum, it is necessary to compare all of them and identify the lowest. There is no general mechanism for this. Luckily, for wetting on rough or chemically heterogeneous surfaces, we have approximate equations for the most stable CA.^1,47  The accuracy of these equations improves as the ratio of the radius of curvature to the heterogeneity scale increases.⁴⁹  For rough but chemically uniform surfaces we have the Wenzel equation,⁴⁸  which assumes the liquid to penetrate completely into the roughness grooves. This state will be referred to as the W state. The apparent CA associated with this global minimum, θ_W, is given by

in this equation, r is the roughness ratio, defined as the ratio between the true area of the solid surface and its projection on a horizontal surface. The above discussion of eqn (1.3) also holds for chemically heterogeneous surfaces. The most stable minimum in energy occurs at the angle that is given by¹ 

here, x₁ and x₂ are the ratios of contact area of the solid with each chemistry to the projection of total area of the solid, and θ_Y1 and θ_Y2 are the Young CAs corresponding to the two chemistries. If the heterogeneous solid surface is flat, then x₁ + x₂ = 1; however it is >1 if the heterogeneous solid surface is also rough. We can easily generalize this equation to a higher number of chemistries, using the principle of linear averaging.

When the surface is rough, there may also be equilibrium positions associated with partial penetration of the liquid into the roughness grooves. This case was first studied by Cassie and Baxter,¹  therefore it is referred to as the CB state. The equation for the apparent CA in this case can be derived from eqn (1.3) and (1.4), assuming the solid surface to be represented by θ_Y1, and air (or an inert gas in general) to be represented by θ_Y2. Because of the perfect hydrophobicity of air, θ_Y2 is taken to be 180°. The solid–liquid area per unit projection area is r_f f, where f is the area fraction of the projection of the wetted part of the solid surface, and r_f is the roughness ratio of the wetted solid. The liquid–gas interface within the roughness is assumed to be flat, therefore its true area fraction is well approximated by its projected area fraction, (1 − f). The apparent flatness of the liquid–gas interface stems from the fact that the pressure inside the liquid is very nearly uniform (if the effect of gravity is small), therefore the radius of curvature around the liquid body must be uniform too. Since this radius of curvature is usually very large compared with the distance between the protrusions of the roughness, the liquid–gas interface inside the grooves appears to be almost flat. This theoretical conclusion^12,25,29  has recently been demonstrated experimentally.⁵⁰  Substituting the above information into eqn (1.4), the CB equation reads

A common problem in publications is the omission of r_f. Assuming r_f = 1 is correct only if the roughness protrusions have flat tops that are parallel to the surface.

The essential characteristic of a non-wettable surface is the ease of removal of a drop from the surface by applying a small force, such as a small fraction of the drop weight. This is usually tested by tilting the surface, similarly to the natural slight tilting of leaves, and measuring the angle at which the drop rolls off. The currently existing quantitative definition, which requires CA > ∼150° and roll-off angle < ∼5°, has only an empirical justification. For fundamental understanding and ability to design successful new non-wettable surfaces, it is essential to study this point in more detail. Because of the prevalence of drop-related non-wetting applications, it makes sense to first reach a full understanding of these cases. However, the definition must be made more general. Easy removal of a drop from a solid surface appears to depend on two main factors: (a) the ability of a weak external force to get the drop out of equilibrium, and (b) high rate of removal from the surface. The factor that may keep a drop in equilibrium under the effect of an external force (say, gravity) is contact angle hysteresis, namely the existence of a range of metastable CAs. This allows the drop to assume a non-axisymmetric equilibrium shape as required by the external force. In contrast, on ideal surfaces the drop must be axisymmetric by definition, so it cannot stay in equilibrium even under the influence of a very small force.

Regarding the rate of removal, it is intuitively appealing to assume that the lower is the solid–liquid contact area, the higher is the rate of removal of the liquid from the solid surface.^12,25,29  If this is true, then the crux of the matter is to find a way to reduce the wetted area as much as possible. The first idea that comes to mind is making the CA as high as possible. However, a simple geometrical calculation indicates that by increasing the Young CA from 90° (considered usually as the lower limit of hydrophobicity) to 120° (the highest available Young CA in practice), the reduction in the area wetted by a drop is only by a factor of about 2. Thus, a different mechanism, capable of much bigger increase in the CA, is required.

Actually, the above two factors that characterize non-wettability can be translated into the following two objectives: (a) achieving a very small hysteresis range (by making the surface as uniform as possible); and (b) making the CA as high as possible. In principle, both objectives can be attained if the surface that is in contact with the liquid consists mostly of a gas, e.g. air trapped in roughness grooves. A gas is the most hydrophobic “surface” we can have, and is also the most uniform. Therefore, a CB state, where a liquid is supported by relatively few solid peaks, certainly answers the need. This statement leads to a possible unified definition of all types of non-wettable surfaces. Qualitatively, this definition may simply state that the wetted area has to be sufficiently small. Some initial calculations¹⁴  showed that the wetted area in the CB state may be orders of magnitude lower than that in the W state, even for the same CA. Further quantitative work is required, but it is clear that non-wettability has to be associated with the CB state, as was qualitatively concluded above and also by Quéré.⁸ 

Whatever the exact definition, from a practical point of view it is clear that in order to be non-wettable the solid surface must be either rough or porous. The grooves of a rough surface are interconnected and open to the atmosphere. In a porous surface, the pores may be either interconnected or isolated. In the latter case it may be much easier to keep the air in the pores in a stable state, but structural constraints may limit the reduction of the wetted area. Therefore, the following discussion is limited only to structures with interconnected grooves or pores.

As previously discussed, roughness of the solid surface is a necessary condition for non-wettability; however, it is not a sufficient condition. As shown below, the geometric characteristics of the roughness may have a major influence. In general, there may be more than one equilibrium position for the liquid–air interface (i.e. minima in the Gibbs energy) within the roughness grooves. The most stable is, of course, the one that has the lowest Gibbs energy. Identifying equilibrium positions is easy: the two equilibrium indicators, namely the Young and the Young–Laplace equations, have to be fulfilled. The latter is fulfilled by the curvature of the liquid–gas interface inside the roughness grooves being the same as that of the outer liquid–air interface, as explained above. This is achieved by assuming that this interface is practically flat. Thus, the only question that needs to be considered is whether the local CA can equal the Young CA at the position that is tested. Then, the identified CB states as well as the W state (that is always a potential equilibrium position) have to be compared to find out the most stable state. In the following we discuss first the case of a drop on a solid, non-wettable surface and then that of a non-wettable surface beneath a liquid.

To make the above analysis clearer it is best to study some examples. For a drop, it is technically easy to compare energies, since the energy varies monotonically with the apparent CA that the drop makes with the solid surface.^12,25  Thus, all that is needed in order to decide which state is more stable is to find out which is associated with a lower apparent CA. One of the simplest forms of roughness is that of straight pillars with a square cross-section. Let us assume that the height of the pillars is h, and that they have flat, horizontal tops of width that cover an area fraction of f (see Figure 1.1(a)). In this case, there are only two possible equilibrium positions. One is the W state, and the other is the CB state with the liquid–gas interface attached to the top of the pillars. This is so, because it is only at the upper corner of the pillar that the liquid–gas interface can locally attain the Young CA when it is >90° (see Figure 1.1(b)). The roughness ratio is given by

Therefore,

The local roughness ratio of the top of the pillar equals 1, therefore

The CB state is more stable if θ_CB < θ_W, namely if cos θ_CB > cos θ_W. When cos θ_Y < 0, this leads to

Thus, for this simple type of roughness, for a given chemistry (cos θ_Y) and surface density of protrusions (f), the only parameter that determines the stability of the non-wetting state is the roughness ratio that depends on the protrusion height, h. The wetting state turns from W to CB when the roughness ratio, namely height of protrusion, is sufficiently high.

For roughness features that are not flat at the top, the situation is more complex and interesting.^12,25  A simple example of two-dimensional roughness with a circular cross-section clearly demonstrates the phenomena that may be observed. For convex roughness features (see Figure 1.2(a)) it is possible to get a stable CB state above a certain roughness ratio, as explained in the following. First, a position of the liquid–air interface for which the CA equals the Young CA has to be identified. This is feasible only if the maximum position angle, α, (see Figure 1.2(a)) is bigger than (180° − θ_Y). Once this condition is fulfilled, we need to check if this position is a minimum in the Gibbs energy. It turns out¹²  that indeed it is a minimum, and that above a certain roughness ratio (determined by the maximum value of α) the CB state is more stable than the W state.

The picture is reversed when the roughness features are concave (Figure 1.2(b)). In this case, the Gibbs energy keeps going down as the liquid penetration into the grooves advances until the W state is reached. Thus, although there exists a position where the CA equals the Young CA (Figure 1.2(b)), the system is unstable and must get to the W state. As concluded from additional studies,^29,43  it turns out that the specific protrusion shape within the group of convex shapes exerts a major effect. Rounded-top protrusions seem to be more effective than flat-topped ones with a sharp edges.^29,43  This theoretical observation may explain why nature prefers rounded-top protrusions.

The role of fractal or multiscale roughness has attracted attention since the early publications on superhydrophobicity.^{2,15,21,23,28,30,33,34,39}  A relatively recent study⁴³  covered a wide range of parameters: three types of roughness geometries with up to four roughness levels (see Figure 1.3). This study showed that the main effect is in reducing the sizes of the roughness protrusions that are necessary for stable superhydrophobicity. Thus, it is not the multiscale nature of the roughness that is responsible for superhydrophobicity; rather, it helps in making the features smaller, therefore more stable from a mechanical point of view.

An interesting extension of the above cases is the one dealing with superhygrophobic surfaces, namely non-wettable surfaces, for which the CAs of the wetting liquid is less than 90°. This case appears at first sight to contradict the common requirement of hydrophobicity for non-wettable surfaces. However, if we look at the CB eqn (1.5), there is no a priori reason that prevents cos θ_CB from being negative, even if θ_Y < 90°. For example, the Young CA may be acute at the equilibrium positions shown in Figure 1.4. However, stability also needs to be checked for this case. The results of this test lead to conclusions that are similar to those of a drop on a hydrophobic rough surface: it may be stable for convex protrusions, and unstable otherwise.²⁵  Nevertheless, it is important to realize that a superhygrophobic state is necessarily metastable, for the following reason. For a hygrophilic surface, the W state is characterized by a CA that is lower than the Young CA. On the other hand, the whole point in making a super-hygrophobic surface is to increase the CA beyond the Young CA. Thus, the W state, by definition, has a lower CA than the CB state, i.e. it is more stable. The special type of roughness that enables superhygrophobicity has been called by several names, such as “multivalued topography” or “re-entrant”.

As mentioned in the introduction, there are important reasons for keeping a stable air film on a solid surface under water. This situation is not explicitly defined by apparent CAs related to the W or CB state. However, the concepts of the W and CB states remain valid in terms of the contact between the liquid and the solid.

The equilibrium criteria for the CB state turn out to be the same as for a drop.¹⁸  The local CA between the liquid and the roughness protrusion must be the Young CA, and the curvature of the liquid–air interfaces must appear to be approximately zero, since it equals the curvature of the outside surface of the liquid. The condition that differentiates unstable equilibrium from metastable or stable ones, in terms of the roughness geometry, turns out to be the same as for a drop.¹⁸  The stable CB state in this system is determined by eqn (1.9), which gives a minimum roughness ratio above which the CB state is stable.¹⁸ 

The following points summarize the opinion of the author regarding the main fundamentals of non-wetting:

• Non-wettability of solid surfaces may be qualitatively defined by stating that the wetted area must be minimal. This implies that the system must be in the CB state. A quantitative definition of non-wettability is yet to be developed.

• Stable CB states can be achieved by roughness geometry that conforms to a certain mathematical condition.^12,18,25  For example, convex protrusions enable it while concave cavities do not.

• Non-wetting in systems with an acute Young CA (superhygrophobicity) is feasible, but it is always metastable.

• Multiscale roughness is not essential for non-wettability; however, it improves the mechanical stability of the surface by lowering the required protrusion size.

• The detailed optimal topography of non-wettable surfaces has yet to be elucidated. Moreover, it is likely that there is more than one solution to the problem, depending on specific constraints.