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ARTICLE https://doi.org/10.1038/s41467-021-22595-1 OPEN Explosive fragmentation of Prince Rupert’s drops leads to well-defined fragment sizes ✉ Stefan Kooij 1 , Gerard van Dalen2, Jean-François Molinari 3 & Daniel Bonn1 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 char- 1234567890():,; acteristic length scale. The origin of this power-law behavior is still unclear, especially why it is such an universal feature. Here we study the explosive fragmentation of glass Prince Rupert’s drops, and uncover a fundamentally different breakup mechanism. The Prince Rupert’s drops explode due to their large internal stresses resulting in an exponential fragment size distribution with a well-defined fragment size. We demonstrate that generically two distinct breakup processes exist, random and hierarchical, that allows us to fully explain why fragment size distributions are power-law in most cases but exponential in others. We show experimentally that one can even break the same material in different ways to obtain either random or hierarchical breakup, giving exponential and power-law distributed fragment sizes respectively. That a random breakup process leads to well-defined fragment sizes is surprising and is potentially useful to control fragmentation of brittle solids. 1 Van der Waals-Zeeman Institute, University of Amsterdam, Amsterdam, The Netherlands. 2 Unilever Research and Development Vlaardingen, Olivier van Noortlaan, Vlaardingen, The Netherlands. 3 Department of Materials Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), ✉ Lausanne, Switzerland. email: [email protected] NATURE COMMUNICATIONS | (2021) 12:2521 | https://doi.org/10.1038/s41467-021-22595-1 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-22595-1 ragmentation is a process that plays a role in a great number CT. This fragment size distribution and all the others throughout of phenomena and applications, from tectonic plate motion the paper are represented by a probability density as a function of F 1–12 to coffee grinding . It covers length scales ranging from the equivalent spherical diameter. Though there are many ways of those of particle physics up to the size of the universe13–15.In presenting fragment size data, this way allows one to directly many applications, control over the fragment size and fragment compare length scales and enables a clear separation between data size distribution is of paramount importance. However, experi- points when changes in the size distribution happen over a short ments have thus far shown that even if an object is broken in a interval such as this distribution for the Prince Rupert’s drop fully controlled way, there is generally a large spread in the sizes (Fig. 1 d). The main conclusions are of course not changed by the of the resulting fragments. This spread in fragment sizes is a clear way the data is presented. characteristic of power-law size distributions that are frequently The distribution of the Prince Rupert’s drop (Fig. 1d), reveals reported in studies of brittle fragmentation16–23, power laws, two distinct exponential regimes, with two associated character- fi = = having only dimensionless t parameters, contain no characteristic length scales (d1 0.31 mm and d2 0.064 mm), suggesting istic length scale, meaning they are scale invariant. This intriguing an inhomogeneous fragmentation. This result seems in conflict behavior has led, among other things, to the formulation of the with a previous study of Prince Rupert’s drops, where the frag- concept of self-organized criticality24, which suggests that it is ment size distribution was fitted with a power law29, but a direct impossible to control the fragment sizes, since they are given by comparison in fact shows the data is comparable (Supplementary the same underlying process for all fragmentation events. Note 4). In that study, a sieving technique was applied. Our more We study here the fragmentation of a remarkably strong, yet precise micro-CT technique, allows each individual fragment to explosively disintegrating piece of glass, the Prince Rupert’s drop be accurately measured and leads to a fundamentally different (also known as “Dutch tears”)25–27. Unlike the studies mentioned conclusion. We measured each fragment size with great accuracy, above, we find that Prince Rupert’s drops undergo a self-sustained i.e., all 20.000+ fragments were measured up to a size of 50 μm. fragmentation process driven by internal stresses only. We use In addition, Silverman et al. mixed fragments from different high-resolution micro-computed X-ray tomography (micro-CT) Prince Rupert’s drops of a range of sizes, while we measure to show that a mm-sized drop explodes into more than 20.000 fragments from individual drops. The mixing of the fragments pieces and measure its fragment size distribution down to 50 μm. from different drops likely obscures the sharp transition between Surprisingly, we find that the size distribution is not power-law, the two exponential regions that we find and broadens the size but has fragments that follow an exponential size distribution. distribution. This together with more noisy data due to the lim- Since an exponential size distribution naturally has a characteristic itations of the sieving technique, likely explains why Silverman length as a fit parameter, this intrinsic length scale shows that the et al. fit an power law to their data, while our data show that for fragments, in this case, are of a well-defined size. We show that an individual drop an exponential fit is more appropriate. The this parameter can be understood theoretically, and is directly difference in glass type between the two studies is not likely to linked to the residual stress within the Prince Rupert’s drop. play any role, as will become clear in the following, our conclu- Comparing the outcomes with fragmentation measurements on sions do not depend on the particular used material type, nor other stressed and unstressed (glass) systems, allows us to show does it depend on the precise shape. 2D shapes of different glass that two different types of breakup processes exist; random and types are found to break in a similar way to the more complicated hierarchic, each with their distinct size distribution. Our results 3D shape of the Prince Rupert’s drop. provide a direct answer to the question of why power-law dis- Analyzing the micro-CT images from a Prince Rupert’s drop tributions are so ubiquitous, while at the same time they explain fractured inside a Carbopol matrix with the typical consistency of why size distributions are exponential in some other cases. hairgel30 so that it has sufficiently high yield stress to prevent the fragments from flying away (Supplementary Note 1), we find that the smallest fragments belonging to the first exponential regime Results and discussion are spread homogeneously throughout the broken droplet, indi- The Prince Rupert’s drops are made by melting the end of a glass cating that these fragments are due to a secondary fragmentation rod with a high thermal expansion coefficient and letting the process. Note that the small fragment regime is referred to in resulting red hot droplets fall into a beaker with cold water. other studies as the “remaining dust”. The micro-CT analysis The rapid cooling causes thermal contraction, and the fact that makes it possible to determine the distribution of these fragments the outside of the drop cools before the inside then leads to very as well. However, the main conclusion is that the 97% of the mass large tensile stresses in the droplet center and compressive of the drop contained in the second regime, exhibits an expo- stresses on the exterior. The very large compressive forces sup- nential fragment size distribution, which points to a random press crack growth, giving rise to the drop’s extreme strength: Poisson process, though theoretical efforts suggest that the pro- they do not break when hit with a hammer; in controlled cess could also be a Matérn Hardcore Point Process due to the experiments, Prince Rupert’s drops can withstand loads of more fact that fragments are not stress free31. As the difference is than 10 kN without breaking27. If, however, a crack is initiated subtle, our work does not allow for this distinction. Interestingly, and it is able to reach the tensile central region, by e.g. breaking the Prince Rupert’s data show resemblance with Mott’s data of the tail of the droplet, it quickly propagates, causing the whole fragmented munition, where the distribution is also exponential, droplet to explosively disintegrate (Fig. 1a). The unique combi- with the argument of the exponential equal to the square root of nation of their extreme strength and unstable nature has made the fragment area32. Though in this research only one type of Prince Rupert’s drops the topic of scientific study since the first Prince Rupert’s drop was used, an extended study could focus on part of 17th-century28. In our experiments, to measure the including data of multiple drops, measuring stress levels before fragment size distribution of a fragmented Prince Rupert’s drop, and after fragmentation as well as using different drop sizes with we break a single drop inside a latex glove so that no secondary different levels of tempering. This can be achieved by using dif- fractures can occur and all fragments can be easily collected. As a ferent glass types with different thermal expansion coefficients generic example, our micro-CT analysis (Fig. 2) shows that a and by changing the flameworking of the glass.

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