Prince Rupert's Drops: An analysis of fragmentation by thermal stresses and quench granulation of glass and bubbly glass

Abstract

When volcanic eruptions involve interaction with external water (hydrovolcanism), the result is an ash-rich and energetic volcanic plume, as illustrated dramatically by the January 2022 Tonga eruption. The origin of the high explosive energy of these events remains an important question. We investigate this question by studying Prince Rupert's Drops (PRDs)-tadpole-shaped glass beads formed by dripping molten glass into water-which have long fascinated materials scientists because the great strength of the head contrasts with the explosivity of the metastable interior when the tail is broken. We show that the fragment size distribution (FSD) produced by explosive fragmentation changes systematically with PRD fragmentation in air, water, and syrup. Most FSDs are fractal over much of the size range, scaling that can be explained by the repeated fracture bifurcation observed in three-dimensional images from microcomputed tomography. The shapes of constituent fragments are determined by their position within the original PRD, with platey fragments formed from the outer (compressive) shell and blocky fragments formed by fractures perpendicular to interior voids. When molten drops fail to form PRDs, the glass disintegrates by quench granulation, a process that produces fractal FSDs but with a larger median size than explosively generated fragments. Critically, adding bubbles to the molten glass prevents PRD formation and promotes quench granulation, suggesting that granulation is modulated by heterogeneous stress fields formed around the bubbles during sudden cooling and contraction. Together, these observations provide insight into glass fragmentation and potentially, processes operating during hydrovolcanism.

Keywords: Prince Rupert's Drops; fragmentation; glass.

Fig. 1.

From Hooke’s (1665) Micrographia. Note the internal bubbles in the small drop in Left as well as the conical cracks revealed in the cross-section (Fig. Y) and the intersection of these cracks with the PRD surface to form circumferential fractures (Fig. X). Reproduced with permission from the Royal Society.

Fig. 2.

Microscope images of small PRDs. A and B are binocular images of bubble-bearing (A) and bubble-free (B) PRDs. C–F are cross-polarized images of small PRDs obtained by photographing through polaroid sheets. (C) Polarization interference colors in a PRD that lacks an internal void. (D–F) Distortion of the stress field (shown as distortion of the polarization fringes) caused by (D) a single bubble in the head, (E) multiple bubbles in the head, and (F) multiple bubbles in the tail.

Fig. 3.

Axial stresses within a PRD illustrate the large compressive stresses (negative) along the outer surface of the drop and large tensile stresses (positive) in the drop interior. Gray shaded areas show the thickness of compressive layers. Adapted from ref. .

Fig. 4.

PRD FSDs. (A) Large (∼20-g) PRDs. Colors denote the mediums of fragmentation (air, water, syrup) for explosively fragmented particles; each line represents a single fragmented particle. FSDs created by quench granulation of both pure (orange) and bubbly (red) melt are combined fragments for quench granulation experiments. (B) FSDs for small (∼1-g) PRDs (same color scheme as used for large PRDs) and measurements from the µCT hair gel experiment, where fragment volume measurements (V) are converted to effective sieve size (L) assuming either cubes (L = V^1/3) or plates [L = (5 × V)^1/3]. (C and D) Fractal analysis of glass fragments separated into two groups—(C) PRDs explosively broken in air or syrup and (D) PRDs explosively broken in water and fragments produced by quench granulation of bubbly and bubble-free samples. (E) FSDs based on number density distributions of samples explosively fragmented in air and syrup. The curve fit for the most linear syrup FSD yields a slope used to calculate dominant size L_D. (F) FSDs for µCT data plotted as both an exponential distribution and histogram to illustrate bimodality.

Fig. 5.

Cross-sectional images through µCT reconstructed volumes. (A–C) Epoxy-encased PRDs. (A) Longitudinal section of PRD2 with a single internal bubble. (B) Longitudinal section of PRD3 with multiple bubbles. (C) Horizontal section of PRD2 (SI Appendix, Fig. S2). (D–F) Gel-encased PRD4. (D) Longitudinal section. (E) View of the outer surface. (F) Horizontal section. Red arrows indicate the direction of fracture propagation. White arrows highlight the radial perpendicular fractures associated with interior bubbles. Yellow arrows show acute branching. Turquoise arrows show oblique branching. Pink arrows show intersections of fractures with PRD exterior. (G) Rendering of individual fragments within the Carbopol-encased PRD4. Note the abrupt change in shape from the interior blocky fragments around the interior void and the surrounding platey fragments. Smaller fragments can be seen throughout wherever fractures intersect. (H) Quantifying fragment shapes using 2D (near-horizontal) cross-sections of µCT scans. Here, we use the axial ratio (short axis/long axis, where a value of one represents a fully equant form) to measure elongation and solidity (fragment area/area of convex hull, where a value of one represents a fully convex form, such as a circle) to measure roughness; the color scale shows radial distance of the fragment centroid relative to the drop center. Note in particular the low axial ratio for particles with large radial distances (the outer platey fragments).

Fig. 6.

Summary of PRD size and shape data. (A) Variation in median fragment size as a function of PRD size (small or large) and fragmentation conditions. Median grain size increases (fragmentation efficiency decreases) with increasing viscosity of the confining medium; hair gel range shows the effect of different assumptions of fragment shape. Quench granulation of bubble-free molten glass produces the largest fragments; explosive fragmentation in air produces the smallest fragments. Larger PRDs produce larger median fragments than small PRDs. Quench granulation of bubbly molten produces fragments with a limited size range that we hypothesize to be controlled by the interbubble spacing within the melt. (B) Sketch illustrating relation between fragment shape and location. AR, aspect ratio; Sol, solidity; rad dist, radial distance (colors and shape measurements as plotted in Fig. 5H).