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Prince Rupert’s Drop — A Curious Material

Brittle materials sometimes break randomly and other times they don’t. Scientists have finally figured out why.

From achieving the perfect coffee grind to understanding the motion of tectonic plates or the formation of meteoric craters, the ability to predict the fragment size distribution of shattered materials has proved crucial in numerous fields. Yet for a long time, it was believed that it is impossible to control the sizes of the pieces resulting from such fragmentation events, as experiments have shown that these sizes follow a power-law distribution.

“Anyone who has ever broken a dish or a glass knows that the resulting fragments range from roughly the size of the object all the way down to indiscernibly small pieces: typical fragment size distributions of broken brittle materials follow a power law, and therefore lack a characteristic length scale,” state the authors of a very recent multi-institutional study (published in the high-impact journal Nature Communications) that revealed for the first time the origins of this universal, yet unusual behavior.

To this end, the Dutch and Swiss researchers performed an experiment using a Prince Rupert’s drop. Named after Prince Rupert of the Rhine, who brought them to England in the 17th century, these droplets are curious materials. They can be easily synthesized by melting the end of a glass cylinder and letting a “blob” drop into room-temperature water. Because the outer layer of the bead cools faster than the inside, a Prince Rupert’s drop is both very tough (can withstand forces up to 10kN) and very fragile (explosively disintegrates upon cutting its tail) due to the large compressive forces that form within the object.

Exploding a Prince Rupert’s drop and measuring the resulting fragments using micro-computed tomography shows that the size distribution is in fact exponential and not power law. This means that the broken pieces have well-defined sizes in connection to the characteristic length of the exponential distribution. Furthermore, this length is related to the dimensions of the object.

The discrepancy between the expected power-law distribution and the exponential result stems from the internal stress characteristic of these beads. In fact, the study revealed that materials can exhibit one of two distinct regimes of breaking: random and hierarchical. Random breaking happens in stressed materials and results in an exponential distribution of the fragments with a characteristic length defined by the residual stress inside the material.

On the other hand, hierarchical breaking occurs in unstressed materials, where the internal stress is non-existent. Such objects will break at ever smaller length scales so that the external elastic energy supplied before the fragmentation dissipates as quickly as possible. This phenomenon leads to the ubiquitous power-law size distribution of the fragments.

To further illustrate how fragmentation can in fact be controlled, the scientists broke the same material in two different ways to obtain either the random or hierarchical regime. To this end, sugar glass disks were fractured slow and fast: one disk was broken by the impact with a solid surface, while the other was attached to a quartz plate and heated slowly. The difference in the thermal expansion coefficients produced a slow fragmentation of the latter sugar glass disk. As expected, slow and fast fragmentation produced random and hierarchical breakup, respectively.

This proves that the fragmentation regime and the fragment size distribution can be controlled and that the breakup process does not depend on the material but on the breakup mechanism. In addition, the reason why the power-law distribution is more often encountered is, as the authors explain in their paper, the strong external energy supplied to break the material:

In normal brittle fragmentation, strain energies exceed the values that are needed for equilibrium fragmentation, so that when cracks appear, the breakup process is hierarchic, with crack branching at ever smaller length scales to dissipate the excess energy. This process is scale-free and results in a power-law distribution.

By equating the stored elastic energy with the surface energy that creates the fragmentation, the authors were able to construct a theoretical model that also allows determining the characteristic length in a controlled, slow fragmentation event and even estimating the number of resulting pieces. This may have tremendous implications in industrial processes where a specific fragment size is desired, such as the technological milling of pharmaceutical powders.

© Gianina Buda, PhD 2021

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