The Impact of Beads and Drops: Same Battle
The impact of a drop on a surface has always fascinated people; as early as 1501, Leonardo da Vinci, in the Codex Leicester, recorded the delicate patterns formed by splashes, and it was inthe 19th century that Worthington published the first comprehensive study of the shapes taken by drops of liquid falling vertically onto a horizontal plate.
Christian Ligoure, University of Montpellier
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The development of high-speed cameras capable of recording 10,000 frames per second, about fifteen years ago, has led to considerable progress in the spatiotemporal characterization of a wide variety of impact phenomena invisible to the naked eye due to their extremely short durations (typically one-hundredth of a second), and the number of scientific publications in this field has exploded in recent years, driven by the variety of natural and industrial situations in which this phenomenon occurs. Paradoxically, the study of the impact of gel beads—or, more generally, small soft objects or droplets of viscoelastic and deformable fluids—has attracted much less research.
We recently decided to reexamine the impact of various millimeter-sized objects on a solid surface and showed that this phenomenon follows a very simple, unified pattern regardless of the nature of the impacting object.
By coating the impact surface with a very thin layer of liquid nitrogen (which is very cold) that vaporizes upon contact with the object at the moment of impact, forming a small cushion of nitrogen vapor, energy losses due to solid or viscous friction—which are highly dependent on the object/impact surface interaction—can be eliminated. This is known as the reverse Leidenfrost effect. This makes it possible to film the expansion and subsequent contraction of the deformed object under ideal conditions of minimal energy dissipation, at various impact speeds.
As it falls in free fall from a height of 1 m, the small, millimeter-sized, incompressible spherical object acquires kinetic energy proportional to its mass and the square of its impact velocity, and crashes at a typical speed of about 4 m/s. At the moment of impact, the object comes to a sudden stop and deforms violently: it flattens out and forms a pancake whose diameter increases to a maximum value that can reach six times the initial diameter. It then retracts to reform the original ball or droplet, which bounces off. The sequence lasts about one-hundredth of a second. The impact is elastic, and the ball’s energy must be conserved: its kinetic energy decreases and is gradually converted into elastic deformation energy, which is stored as the pancake expands until it reaches its maximum size, at which point the conversion is complete. The process then reverses during the retraction.
Elastic Energy
But what is the nature of this elastic deformation energy? For simple liquids, in which there are no intermolecular bonds, it is surface energy, which is proportional to the area of the object.
The proportionality constant 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 the material are not strictly in the same state as those inside, since they interact with fewer neighboring molecules.
This new local state results in a small increase in the energy of each molecule at the surface. The surface of a medium is therefore associated with an energy per unit area—or surface tension—that arises from the cohesive force between identical molecules. This force is responsible, for example, for the spherical shape of a droplet at rest (which corresponds to the minimum area for a given volume), or for allowing small insects, such as water striders, to walk on water.
What about solids?
For solids, there is also a volumetric elastic deformation energy; this arises from the deformation of the bonds in the molecular lattice that constitutes the solid and is characterized by an elastic deformation modulus, which has the dimension of energy per unit volume (or pressure).
By definition, the ratio of a material’s surface tension to its elastic modulus has the dimension of a length called the elastocapillary length, which is a characteristic of the material (it is infinite for a simple liquid).
For objects much smaller than their elastocapillary length, surface energy controls the deformation of the object; conversely, for objects much larger than this capillary length, volumetric elastic energy predominates.
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.
For gels, which consist of a cross-linked network of polymer chains immersed in water, the volumetric elasticity is of a different nature (referred to as rubber-like elasticity), with elastic moduli that can be very low, ranging from 1 to 10,000J/m³; this makes it possible to formulate ultra-soft solids. If the bonds in the network are transient, the material behaves as a viscoelastic fluid that acts like a solid over time scales shorter than its relaxation time and like a liquid beyond that. For these objects, the elastocapillary length can vary between 1 mm and 10 cm—exactly within the range observed for the impact of droplets or beads—and both elastic energies play a role!
The impact dynamics of these objects—gel beads, viscoelastic droplets, and simple liquid droplets—are described, in the absence of dissipation, by very simple scaling laws that depend on the impact velocity and are identical regardless of the nature of the object.
It is as if the deforming film were behaving like a simple spring being pulled, whose spring constant is expressed simply in terms of physical quantities that characterize the object’s mechanical properties: its size, surface tension, elastic modulus, and density.
A new characteristic velocity of generalized elastic deformation associated with the impact of each object naturally emerges from this description—a combination of the propagation velocity of transverse sound waves in the medium (the solid component) and the Rayleigh velocity associated with the vibration of a spherical liquid drop (the liquid component). The cases of impact by a single liquid drop or a “hard” elastic ball then simply appear as the limits of a unified behavior.
Christian Ligoure, Professor of Soft Matter Physics, University of Montpellier
The original version of this article was published on The Conversation.