# Capillarity

This is a simple, but technical, description of the phenomenon of Capillarity.

1. INTRODUCTION

Capillarity refers to a series of phenomena occurring when liquids are affected by the presence of a solid, usually because the liquid feels very much attracted or repelled by the solid. The most typical example of capillarity is the rise of liquid in a narrow tube (or capillary, hence the name) opposing the gravitational force, or also the rising of (say) milk inside a biscuit above the milk level when this is dunked in a cup of milk.

Capillarity has been studied since the time of Leonardo da Vinci (1452–1519), although the first to provide a mathematical formulation of the phenomenon were Thomas Young (1773–1829), and Pierre-Simon Laplace (1749–1827) in 1805. Shortly after, Carl Friedrich Gauss (1777–1855) unified both descriptions in what is now the standard mathematical approach to the problem.

It is difficult to overestimate the role played by capillarity in science. It has been the focus of attention in many different fields: physics, mathematics, chemistry, biology and, certainly, engineering. As an example, experiments by Francis Hauksbee (1666−1713) (Newton’s demonstrator at the Royal Society) showed that the capillary rise of liquid in a tube did not depend on the thickness of the tube walls. This was a clear indication that the (recently postulated) inter-particle forces extended over insensible distances (were very short-ranged).  This happened about 200 years before the existence of atoms was fully accepted by the scientific community, at the dawn of the 20th century. Illustration of experiments on capillarity in Geminiano Montanari’s 1667 book “Pensieri fisico-matematici sopra alcune esperienze fatte in Bologna” (Physical-mathematical ideas about some experiments done in Bologna)

2. THEORY OF CAPILLARITY

Unlike solids, liquids can accommodate to their surroundings: They change their shape easily and move (flow). For example, when one places a small amount of liquid on a solid object, the (liquid) droplet adopts a certain shape. This shape is not arbitrary. It obeys a principle of minimum energy: The total energy of the system (liquid, solid and gas) for that particular shape is lower than for any other shape. This means that, if we know how to compute the total energy of the system for any shape, we can determine the final shape of the liquid droplet just by finding the shape that has less energy!

The relevant contributions to the total energy (those depending on the shape of the liquid droplet) are the following:

1. The Liquid-Gas contact energy. This contribution is proportional to the area of liquid surface in contact with the surrounding gas. Of course, the liquid can adopt different shapes, yielding a different energetic contribution.
2. The Liquid-Solid contact energy. This contribution is proportional to the area of solid surface in contact with the liquid. Again, the droplet can spread on the solid to different degrees, hence varying this contribution.
3. The gravitational energy. The lower the liquid, the lower the energy.

Of course, these three contributions are not independent. In general, one cannot vary one of them without varying the other two. For example, if a liquid droplet spreads on a flat substrate, the first and second contributions increase, while the third contribution decreases. This interdependence is what makes the problem interesting. Mathematically, the relevant total energy (in appropriate units) can be written as a functional:

$E[\ell]\;=\;A[\ell]\;-\;\cos\theta\;S[\ell]\;+\;\frac{1}{a^2}\,G[\ell]$

Here, $$\,\ell\,$$ symbolises a particular shape of the liquid-gas interface (which embodies the whole droplet shape), and all the quantities that depend on that shape are followed by $$\,[\ell]$$:

$$A\,$$ is the area of liquid surface in contact with the surrounding gas
$$S\,$$ is the area of solid surface in contact with the liquids
$$G\,$$ is the gravitational energy of the liquid droplet

This energy also contains two physical constants, that depend on the choice of liquid and solid but not the drop shape:

$$\theta\,$$ is the Contact Angle
$$a\,$$ is the Capillary Length

By means of the Calculus of Variations , one can obtain the two conditions that must be verified by any droplet shape that minimises the total energy:

1. The angle of the liquid-gas interface with the solid substrate at any point of contact is $$\,\theta$$.
2. The liquid-gas interface satisfies the Young-Laplace equation, which in Cartesian coordinates is

$\displaystyle\nabla\cdot\left(\frac{\nabla\ell}{\sqrt{1+(\nabla\ell)^2}}\right)\;=\;\frac{1}{a^2}\,\ell\;+\;\lambda$

where $\,\lambda\,$ is a Lagrange multiplier, determined by the volume of liquid available.

A complete description of the mathematics involved can be found in Robert Finn’s Equilibrium Capillary Surfaces

Note that capillarity is necessarily related to wetting phenomena, as one of the key ingredients of the theory is the contact angle $$\,\theta$$.

3. IMPORTANCE OF CAPILLARITY

Capillarity is an extremely important phenomenon due to its ubiquity in everyday life: it causes fabrics to absorb liquids (for example, when you dry yourself with a towel), helps sap rise to the top of plants, produces the cohesion of grains in wet sand, allows certain insects to float on water… Also, as capillarity usually occurs on small scales, it’s of great importance in nanotechnology, particularly nanofluidics.

4. OUR WORK

Our main work in this field is the description of the emptying of horizontal capillaries, including the study of the meniscus deformation, and its relation with wetting phenomena and nanotechnology.