This is a simple, but technical, description of the phenomenon of fluid adsorption or *Wetting.*

**1. INTRODUCTION TO WETTING ( MACROSCOPICS)**

*Wetting* is a phenomenon that occurs on the interfaces (surfaces) of systems. This phenomenon is physical in nature and does not involve any chemical reaction. The prototypical example of wetting is a droplet of liquid that spreads over a solid substrate when the Contact angle \(\,\theta\,\) becomes zero:

The contact angle \(\theta\) satisfies Young’s Law (1804):

\[ \gamma_{SG}\;=\;\gamma_{SL}\;+\;\gamma_{LG}\,\cos{\theta} \]

and is, therefore, determined by the Surface Tensions: \(\,\gamma_{SG}\,\) (solid-gas), \(\,\gamma_{SL}\,\) (solid-liquid) \(\,\gamma_{LG}\,\) (liquid-gas), which represent the energetic cost per unit of area of each of these interfaces. Clearly, these tensions depend on the particular liquid, gas and solid we are considering. Every time that these surface tensions satisfy \(\;\gamma_{SG}=\gamma_{SL}+\gamma_{LG}\,\), consistent with a contact angle \(\,\theta=0^\circ\), the liquid spreads over the solid substrate and is said to *wet* it.

In this example, wetting occurs because the solid substrate much more prefers to be in contact with the liquid rather than with the gas. In fact, wetting is a consequence and a clear indication of the existence of intermolecular interactions. For fluids, these interactions are mainly Dispersion Forces.

If the solid prefers to be in contact with the gas instead of the liquid, the same phenomenon occurs with gas and liquid playing each other’s role:

This phenomenon is called “wetting by gas” or *Drying*, and corresponds to the contact angle of a liquid drop becoming \(\,180^\circ\). If we place drops instead of bubbles on the substrate, we observe this:

Wetting is a phenomenon described predominantly for fluids, but also manifests itself in many other contexts: magnetic systems, superconductors, colloids, and liquid crystals, among others.

**2. A DIFFERENT VIEW OF WETTING ( MICROSCOPICS)**

We describe here an alternative, but somehow more fundamental, view of the wetting transition described above.

**UNDERSTANDING THE BULK**

Imagine we have bulk water. Depending on the temperature and the pressure of our water, it can be in one of three different phases: liquid (water, as such), gas (water vapour) or solid (ice). This is a simplified version of water (in reality, 15 different phases of solid ice have been observed experimentally), but for our purposes is accurate enough. The following plot, called a *bulk phase diagram*, shows schematically which phase of water corresponds to a given temperature \(\,T\,\) and pressure \(\,p\,\). The black lines, called ** coexistence lines**, separate the domains of the different phases.

For example, at room temperature \(\,T=25^\circ C\) and atmospheric pressure \(\,p\approx 10^5 Pa\), water is liquid. This is represented by the point **A** of the diagram above. Suppose now that the pressure is given by our usual atmospheric pressure \(\,p\approx 10^5 Pa\) (our water is in an open container, say) and we can only change the temperature. In this case, we need to restrict to the blue line of the diagram, for which the pressure is the atmospheric pressure. Along that line, the phase of water depends on temperature: water is solid for \(\,T < 0^\circ C\,\) (left part), liquid in the range \(\,0^\circ C < T < 100^\circ C\,\) (central part), and gas at \(\,T > 100^\circ C\,\) (right part). By fixing \(\,T\,\) and \(\,p\,\), our water can be at any point of the diagram. In fact, if we change \(T\) and \(p\), we can move within the diagram. If so, every time we cross a coexistence line in the phase diagram, our water undergoes a *phase transition* and changes phase: it melts/solidifies (freezes), condensates/evaporates (boils), or sublimates/is deposited.

All this very familiar picture is the consequence of the interactions between the molecules, which, depending on the available energy (temperature \(\,T\,\)) and how many of them there are (pressure \(\,p\,\)), force the molecules to: travel erratically far from each other (gas), travel erratically “touching” each other (liquid) or get ordered in a compact manner (solid). Note that if there are many molecules (high pressure – towards the top of the diagram), the molecules are necessarily ordered. However, if there is a lot of energy available (high temperature – towards the right of the diagram), the molecules travel so fast they cannot be ordered and are a gas if a few (lower right part), a liquid if many (middle right part), or a solid if the pressure is really high and there is no other possibility but ordering themselves in order to fit (upper right part).

**SOLID SUBSTRATE**

Suppose that we have water vapour or, in other words, \(\,T\,\) and \(\,p\,\) are such that we are in the GAS part of the phase diagram. Now, imagine that we put our vapour in contact with a solid substrate towards which water molecules feel attracted (if the molecules felt *repelled* by the substrate, the discussion would be identical but for the phenomenon called *drying* instead of wetting, as discussed in the introduction). The water molecules that happened to be close to the substrate would feel the attraction and “hang around” in its vicinity for a while. As a consequence, there will be more molecules in the vicinity of the solid than far from it. However, we have seen that molecules prefer to be close or far from each other depending on \(\,T\,\) and \(\,p\,\), and, in the gas phase, they definitely prefer to be apart from each other. In this scenario, molecules have “mixed feelings”: they want to be close to the solid (and therefore close to each other) and be apart from each other (as required by the physical conditions \(\,T\,\) and \(\,p\,\)). The only way out of this dilemma is finding an energetic compromise between both tendencies. In most situations, there can’t be much of a compromise since the physical conditions given by \(\,T\,\) and \(\,p\,\) triumph, and the number of molecules close to the substrate (and, in fact, only very close to it) is just slightly higher than in the bulk (far from the substrate). This is so because the bulk is much larger than the surface and, therefore, its energetic contribution is also much larger.

However, when \(\,T\,\) and \(\,p\,\) are close to the coexistence lines of the phase diagram, things are different (how close and how different depend on how attractive the substrate is). Consider, for example, point **B** in the phase diagram below. Here, our gas is very close to condensate into a liquid and the physical conditions \(\,T\,\) and \(\,p\,\) favour the gas state over the liquid state but only very slightly. In this situation, more molecules of water get close to the substrate, and a thin layer of liquid is formed (*adsorbed*) on the solid substrate. For a given gas and substrate, the thickness \(\,\ell\,\) of this liquid layer depends on \(\,T\,\) and \(\,p\,\), and is generally *nanometric* in size. However, under certain circumstances, it can attain much larger sizes. __The phenomenon of wetting corresponds to this liquid layer becoming macroscopic__.

A standard way of representing this phenomenon is plotting the average thickness \(\,\ell\,\) of the liquid layer adsorbed on the solid as a function of the pressure while keeping a constant temperature. The resulting curves are called ** adsorption isotherms**. Note that, as the gas-liquid coexistence occurs at different values of the pressure for different temperatures, different isotherms finish at different values of the pressure (see the

**orange**and

**green**lines in the diagram above). However, it’s customary to represent the pressure coordinate of each isotherm divided by its corresponding coexisting pressure \(\,p_0(T)\,\), so that, for all temperatures, this rescaled pressure \(\,p/p_0\,\), called

*partial pressure*, ranges from \(\,0\,\) to \(\,1\,\). Below, we show two adsorption isotherms for temperatures \(\,150^\circ C\) and \(\,300^\circ C\) (

**orange**and

**green**lines in the diagram, respectively) for water on a hypothetical substrate.

Note that, for \(\,T=150^\circ C\) the liquid layer is about \(\,10 nm\,\) thick at gas-liquid coexistence \(\,(p/p_0=1)\,\), but for \(\,T=300^\circ C\) the liquid layer increases in size enormously (diverges) when approaching coexistence; that is, the liquid layer becomes *macroscopic*. In other words, we say that the substrate is wet by water at \(\,300^\circ C\) but not at \(\,150^\circ C\).

**WETTING TEMPERATURE**

In the example above, our substrate is wet by water at \(\,300^\circ C\) but not at \(\,150^\circ C\). Thus, there must be an intermediate temperature for which one scenario changes into the other. Indeed, that temperature is the ** wetting temperature** \(\,T_w\) (about \(\,200^\circ C\) in the diagram below; look for the

**wetting point**), which divides the gas-liquid coexistence line into two segments: a

*dry*part (to the left of the wetting point) and a

*wet*part (to the right, in

**red**). The wetting phase transition refers to the process of approaching this “wet” segment.

In the paragraphs above, we’ve always approached this segment *off coexistence*: coming from points outside the coexistence line (recall the adsorption isotherms). However, one can also approach the wetting point *at* coexistence. If one moves *along* the coexistence line increasing the temperature \(\,p=p_0(T),\;T\to T_w\), the thickness of the liquid layer (which, recall, was about \(\,10 nm\,\) at \(\,T=150^\circ C\)) increases and “diverges” at the wetting temperature (**blue** arrow). If the divergence is continuous, it’s called *Second order* wetting; otherwise, *First order* wetting.

Consequently, we have three different paths:

- Complete Wetting: \(\;p\to p_0,\;T > T_w\) (
**green**arrow) - Partial Wetting: \(\;p\to p_0,\;T < T_w\) (
**orange**arrow) - 1st/2nd order Wetting: \(\;p=p_0(T),\;T\to T_w\) (
**blue**arrow)

While these three paths make up the phenomenology of wetting, *partial* wetting is not a wetting transition as such because, along this path, the liquid layer never attains macroscopic size.

**CONTACT ANGLE**

We have seen two rather different views of wetting: the contact angle of a liquid drop on a solid tending to zero, and a microscopic liquid layer adsorbed on the solid substrate attaining macroscopic thickness. However, both views are, of course, consistent with each other. To see this, note that the microscopic liquid layer must also be present in the drop scenario, but at a very different scale!

Observe that, if the contact angle is zero, the liquid spreads, which is equivalent to having a macroscopic layer of liquid on the substrate. In contrast, if the contact angle is \(\,90^\circ\), gas and liquid are (energetically) indistinguishable to the solid substrate, and no liquid layer is formed. If the contact angle is larger than \(\,90^\circ\), no liquid layer is form. However, if liquid was put into contact with the solid, a microscopic *gas* layer would form, separating the liquid from the solid.

This is quite a general description of wetting and, for the sake of clarity, many details have been skipped. The general picture, however, is thought to be correct.

**3. THEORIES OF WETTING**

Wetting is a surface *Phase Transition* described by *Statistical Mechanics*. Although wetting phenomena have attracted people’s attention for long (Young’s Law is dated 1804, about a century before the existence of atoms was accepted by the scientific community), the real starting point of this field was the simultaneous publication in 1977 of two papers, by **John W. Cahn** (see here) and **Charles A. Ebner** and **William F. Saam** (see here), in which the connection of the phenomenon with a microscopic theory was established for the first time.

In both cases, wetting was described by means of *Density Functional Theory *(DFT), which provides a very complete general picture of the phenomenon for different systems and circumstances. However, DFT becomes mathematically too cumbersome when one wants to look into some specific details like, for example, the effect of *Thermal Fluctuations* (which, in this particular case, are *Capillary Waves*), or the effect of the substrate geometry.

To deal with these aspects of wetting, a *Mesoscopic Theory* called **Effective Hamiltonian Theory** was developed in which the element of study is not the density of particles at each point but just the interfaces separating the different parts of the system (the liquid-gas interface in the example of a drop). So, in general, one uses DFT to describe adsorption on flat (chemically) homogeneous substrates, and Effective Hamiltonian Theory to describe the effect of thermal fluctuations, or adsorption on substrates with certain geometrical shape or (chemical) inhomogeneities.

To give an idea of the difference between these two types of models, we include here a couple of examples of the mathematical expression for the energy in each of them. It is from this function that one is capable of describing the phenomenology of the system of study (or not).

**Density Functional Theory**

In these models, the “working” variable is the *density* of molecules \(\,\rho({\bf r})\,\) at each point in space \(\,{\bf r}=(x,y,z)\). One of the simplest examples is given by the Landau-Ginzburg-Wilson model:

\[ E[\rho({\bf r})]\;=\;\int_V d{\bf r}\;\left\{\,\frac{1}{2}\left(\nabla\rho\right)^2\;+\;\Phi(\rho)\,\right\}\;+\;\int_S d{\bf s}\;\,\phi(\rho) \]

where \(\,V\,\) is the total available volume for the gas and liquid, and \(\,S\,\) is the substrate surface. Here, \(\,\Phi(\rho)\,\) is the energy density of the bulk at density \(\,\rho\,\), and \(\,\phi(\rho)\,\) is the energetic contribution due to having molecules at density \(\,\rho\,\) interacting with the substrate at the point \(\,{\bf s}\).

In order to find a solution of the model, simplifications are usually made. For example, if one assumes that the substrate is flat and there are no thermal fluctuations, the density of particles only depends on the distance \(z\) to the wall. With this assumption, the model reduces to the much more manageable:

\[ E[\rho(z)]\;=\;\int_0^\infty \!\!\!dz\;\left\{\,\frac{1}{2}\left(\frac{d\rho}{dz}\right)^2\;+\;\Phi(\rho)\,\right\}\;+\;\phi\left(\rho(0)\right) \]

**Effective Hamiltonian Theory**

These *effective* models can be thought of as a “simplified” version of the Density Functional Models. The “working” variable is not the density at any point in space but the *thickness* \(\,\ell({\bf s})\) of the adsorbed layer at each point \(\,{\bf s}=(x,y)\) above the substrate . Observe that \(\,\ell\,\) depends on \(\,s\,\) (only 2 coordinates), instead of the three coordinates of \(\,r\,\) on which \(\,\rho\,\) depends. This makes things considerably simpler, though the cost of a coarser physical description needs to be paid. An example of an effective Hamiltonian is:

\[ H[\ell]\;=\;\int d{\bf s}\;\left\{\,\frac{\gamma_{LG}}{2}\left(\nabla\ell\right)^2\;+\;W(\ell)\,\right\} \]

where \(\,W(\ell)\,\) is an *effective potential*, the energetic cost of a liquid layer of thickness \(\,\ell\,\). The function \(\,W\,\) is commonly obtained from a constrained minimization of a DFT theory for a planar substrate.

**4. IMPORTANCE OF WETTING**

Nowadays, wetting studies are important for two main reasons:

*1) Technological Applications*

a) Controlling the *wettability* of a substrate (the contact angle of a liquid on that substrate) is important in many contexts. For instance, a high wettability (low contact angles) is crucial when liquids are to be spread on solids, like inks, paints, fertilizers, insecticides… On the other hand, low wettabilities (high contact angles) are necessary for window panes (anti-fog treatments), frying pans (Teflon), (water repellent) fabrics, etc…

Understanding wetting is also essential in order to recover oil efficiently from reservoir rocks. A mere increase of \(\,1\%\) in the efficiency of oil recovery can have dramatic economic consequences.

b) Nanotechnology has transformed wetting studies for two main reasons: firstly, because now we can control the shape and chemical composition of solid substrates at a nanoscopic scale, which allows us to construct surfaces with novel wetting properties. Also, because devices are being progressively miniaturized to the point that the size of the “microscopic” liquid layer that covers the solid substrates is of a similar size to the devices themselves, and therefore its effects cannot be neglected.

*2) Connections with the fundamentals of statistical mechanics*

Wetting is important in statistical mechanics for several reasons:

a) It’s another example of a phase transition.

b) Its is unavoidable in real systems: liquids and gases are *always* in the presence of a substrate.

c) It shows a particular feature which distinguishes it from bulk phase transitions: the fact that the *order parameter* for wetting diverges at the transition.

d) Its presence modifies the bulk phase diagram: phase coexistence *disappears* when wetting occurs.

e) Wetting is a *two-dimensional* phase transition that can be studied experimentally in the real world.

f) It shares some features with bulk phase transitions, like *Universality*, but it also shows striking non-universal features, which makes its study very rich.

**5. OUR WORK
**

Our work in this area goes mainly in two directions:

a) **Theory of Wetting**. In particular, the development of an effective Hamiltonian theory for systems with short-ranged interactions. The motivation for this work is a long-time discrepancy between a number of different theoretical approaches and computer simulation results that has lingered in the subject for over 15 years (see * here *).

b) **Wetting on Structured Substrates**. Particularly, but not exclusively, substrates whose surface is not flat, but has certain geometrical shape; for example, substrates with grooves. In these systems, the geometry of the substrate has a very strong influence and, more often than not, governs the wetting behaviour of the system.

**6. UNANSWERED QUESTIONS**

As the substrate properties can now be altered both chemically and geometrically in unimaginable ways a few years ago, the wetting properties are modified in a way that only now we are beginning to understand:

- How does the wetting properties change if the substrate is not flat, but has a geometrical shape?
- Can we tailor these properties in a controlled manner?
- Can all the new physics be described by the standard methods?
- Is it possible to construct a simple effective Hamiltonian theory that includes all the relevant microscopic effects?

Nowadays, hundreds of research groups in the world study wetting for very different purposes, using theory, experiments, simulations or, in general, a combination of the three.