The impact of balls and drops: the same battle

The impact of a drop on a surface has always fascinated mankind: Leonardo da Vinci, in the Leicester codex of 1501, already recorded the delicate digitation figures of splashes, and it was in the ^19th century that Worthington published the first exhaustive study of the shapes adopted by liquid drops falling vertically onto a horizontal plate.

Christian Ligoure, University of Montpellier

The development of ultra-fast cameras capable of recording 10,000 images per second some fifteen years ago has enabled considerable progress to be made in the spatio-temporal description of a wide variety of impact phenomena, invisible to the naked eye because of their short duration (typically a hundredth of a second), and the number of scientific publications in this field has exploded in recent years, driven by the variety of natural or industrial situations where this phenomenon is encountered. Paradoxically, the study of the impact of gel beads or, more generally, of small soft objects or drops of viscoelastic and deformable fluids has generated far less work.

We have recently chosen to revisit the impact of various millimetric objects on a solid surface, and have shown that this phenomenon obeys a very simple unified behavior, whatever the nature of the impacting object.

By covering the impact surface with a very thin layer of liquid nitrogen (very cold), which vaporizes on contact with the object at the moment of impact, forming a small cushion of nitrogen vapour, energy losses due to solid or viscous friction, highly dependent on the object/impact surface pair, can be eliminated. This is known as the reverse Leidenfrost effect. The expansion and retraction of the deformed object can be filmed under ideal conditions of minimum dissipation, at different impact speeds.

In free fall from a height of 1m, the small incompressible millimetre spherical object acquires kinetic energy proportional to its mass and the square of the impact velocity, and crashes at a typical speed of around 4m/s. At the moment of impact, the stopped object deforms violently: it flattens and forms a pancake whose diameter grows to a maximum value that can reach six times the initial diameter. This then retracts to reform the initial ball or drop, which bounces back. The sequence lasts around one hundredth of a second. The shock is elastic and the ball's energy must be conserved: its kinetic energy diminishes and is gradually transformed into elastic deformation energy, which is progressively stored in the expanding pancake until it reaches its maximum size, when conversion is complete. The process is then reversed during retraction.

Elastic energy

But what is the nature of this elastic deformation energy? For simple liquids, for which there are no intermolecular bonds, it's the surface energy, which is proportional to the surface area of the object.

The coefficient of proportionality is called surface tension, and has the dimension of energy per unit area: at the surface of a dense medium, the molecules that make up matter are not in exactly the same state as inside, since they interact with fewer neighbors.

This new local state results in a small increase in the energy of each molecule on the surface. The surface of a medium is therefore associated with an energy per unit area, or surface tension, the origin of which is the cohesive force between identical molecules. This force is responsible, for example, for the spherical shape of a drop at rest (corresponding to the minimum surface area for a fixed volume), or for enabling small insects like gerris to walk on water.

And for solids?

For solids, there is also a volumetric elastic energy of deformation; this arises from the deformation of the molecular lattice bonds that make up the solid, and is characterized by an elastic modulus of deformation, which has the dimension of energy per unit volume (or pressure).

By construction, the ratio between a material's surface tension and its elastic modulus has the dimension of a length called the elastocapillary length, which is characteristic of a material (it is infinite for a simple liquid).

For objects much smaller than their elastocapillary length, it is surface energy that controls object deformation; on the contrary, for objects much larger than this capillary length, it is volume elastic energy that prevails.

Pour des objets de taille comparable à la longueur élastocapillaire ; les deux modes de déformation élastique sont à prendre en compte. Pour la plupart des solides usuels (solides cristallins, métaux., verres) le module élastique est très élevé et varie tandis que la tension superficielle varie peu de sorte que la tension superficielle ne joue un rôle seulement significatif dans la déformation que pour des objets de taille colloïdale (<1 micromètre) ou même nanométrique.

In the case of gels, which are made up of a cross-linked network of polymeric chains immersed in water, the elasticity by volume is of a different nature (referred to as rubbery elasticity), with elastic moduli that can be very low, between 1 and 10,000 ^J/m3, making it possible to formulate ultra-soft solids. If the network bonds are transient: we're dealing with a viscoelastic fluid that behaves like a solid for times shorter than its relaxation time, and like a liquid beyond that. For these objects, the elastocapillary length can vary between 1mm and 10 cm, exactly within the observation range of the impact of drops or beads, and both elastic energies play a role!

The impact dynamics of these objects - gel beads, viscoelastic drops, simple liquid drops - is described in the absence of dissipation by very simple scaling laws as a function of impact velocity, and identical whatever the nature of the object.

It's as if the sheet being deformed were a simple spring to be pulled, whose stiffness constant is simply expressed as a function of physical quantities that characterize the object's mechanical properties: its size, surface tension, elastic modulus of deformation and density.

A new characteristic velocity of generalized elastic deformation associated with the impact of each object appears naturally in this description, a combination of the propagation velocity of transverse sound waves in the medium (this is the solid component) and the Rayleigh velocity associated with the vibration of a spherical liquid drop (this is the liquid component). The impact situations of a single liquid drop or a "hard" elastic ball then appear simply as the limits of a unified behavior.

Christian Ligoure, Professor, soft matter physicist, University of Montpellier

The original version of this article was published on The Conversation.