Introduction

Magma fragmentation is the key mechanism determining whether eruptions are explosive1,2. Explosive eruptions produce volcanic ashes, which affects the environment and human society3,4. The size, distribution, and shape of ashes alter their dispersion and residence time in the atmosphere. Fragmented magmas have various sizes and morphologies depending on their compositions and eruption styles5,6,7,8,9,10,11,12. Analysis of volcanic ashes reveals various shapes; some ashes preserve bubbles that were present before fragmentation, whereas others are bubble-free5,6,8. Ashes generated by ash explosions after Strombolian eruptions and by Vulcanian eruptions are relatively dense and contain few bubbles13,14,15,16,17. Vulcanian eruptions generate relatively finer ashes than other explosive eruptions with similar dispersal areas18,19.

Magma is a viscous fluid but can fragment in a brittle manner when deformation is rapid relative to the relaxation time scale20,21,22. The Deborah number, De \(={\tau }_{{{{{{{{\rm{c}}}}}}}}}\cdot \dot{\gamma }\), characterizes the time scale of deformation relative to the relaxation time, where \(\dot{\gamma }\) is the strain rate, and τc = η0/G∞ is the relaxation time. Here, η0 is the shear viscosity at zero strain rate, and G∞ is the shear modulus at an infinitely high strain rate. A melt elongation experiment shows that De > 0.01 is a threshold for brittle fragmentation23. This threshold is also applicable to the crystal bearing magma24. Later compilation suggests 0.01 < De < 0.04 is transitional and De > 0.04 causes brittle failure22. Interestingly, De = 0.01 is in the regime where the strain rate is slow enough \(\dot{\gamma } \, < \, 1/{\tau }_{{{{{{{{\rm{c}}}}}}}}}\) to cause viscous deformation, but solid-like fragmentation is observed. This threshold is widely used to model magma fragmentation in a conduit25,26,27. Brittle fracture increases porosity and permeability, affecting the eruption style28. The rheology of magma depends on the strain rate23,29. However, the rheology at the time of fragmentation has not yet been directly measured. Rapid deformation allows for the development of stress in the melt surrounding bubbles in magma30. Shock-tube-type rapid decompression experiments show that bubbly magma undergoes brittle fragmentation when the product of the overpressure and volume fraction of bubbles exceeds a critical value, ΔPϕ > σ031. Smaller fragments form at larger ΔPϕ32,33. According to this scaling, fragmentation is caused by the pressurized gas phase in pre-existing bubbles. It is not clear whether dense volcanic ashes generated by Strombolian and Vulcanian eruptions, with few bubbles and not accompanied by pumices/scoriae, can be produced by this mechanism.

Volcanological research has focused on the brittle fragmentation of magma, which is a kind of molten glass, at temperatures above the glass transition temperature Tg. In contrast, the fracture of amorphous material at temperatures below Tg has been explored in material science, focusing on its plastic/ductile/viscous characteristics. Here, we use the term “plastic/ductile/viscous” to indicate dissipative deformation in general. We do not consider the low-viscosity process governed by surface tension or inertia, where a fragmented liquid parcel by instability makes round droplets34,35,36. Macroscopically, glass breaks in a brittle manner37; but the region around the crack tip is expected to undergo both nonlinear dissipative and elastic deformation38. Tensional deformation generates voids, and thus the fracture surface shows a wavy structure, whereas shear deformation forms a relatively flat surface39. Cavitation precedes fracture40, and nonlinearity around the crack tip and rapid crack propagation can cause instability41,42,43,44.

The fracture surface of glasses has a typical morphology with regions denoted as mirror, mist, and hackle morphology45,46,47. The mirror area corresponds to the optically smooth portion of the fracture surface; the mist area is the stippled surface at the boundary of the smooth zone, and the hackle is the area with large undulations. The square root of the mirror region size rm is inversely proportional to the tensile stress σt; that is, high tensile stress reduces the smooth area as \({\sigma }_{{{{{{{{\rm{t}}}}}}}}}{r}_{{{{{{{{\rm{m}}}}}}}}}^{1/2}\)46. In the brittle fracture of silicate glasses, stress corrosion at the crack tip has been suggested48,49. Crack propagation proceeds by the nucleation, growth, and coalescence of nanometric cavities to be a crack, which is similar to the ductile fracture of metals. The rough surface of the mist region originates from such secondary cracks, and the coalescence of the secondary cracks generates the rougher region observed as the hackle morphology47,50,51.

It is not obvious whether these complicated fractures observed on low-temperature glass can occur in high-temperature magma. To investigate this, we conducted tensile experiments using rod-shaped soda glass and haploandesite, Fe-free andesite (for details, see the “Methods” section, Fig. 1, Supplementary Figs. 1 and 2, and Supplementary Table 1). We present in situ observations of fracture at high temperatures, which generates small fragments. The appearance of the fracture surface indicates the origin of small fragments. We also calculate the strain-rate-dependent rheology of these silicate melts from the stress–strain curves and discuss the relation between the deformation type and the occurrence of fragmentation.

Fig. 1: Our experimental apparatus.
figure 1

a, b Photographs and c a schematic diagram. A rod-shaped sample is hooked to the Inconel piston by spheres at both ends and is surrounded by a furnace. The upper piston moves upward to elongate the sample. Two holes at the front of the furnace are used for lighting and photography using a high-speed camera. The location of the sample is adjusted to avoid misalignment by changing the orientation of the slit pointed by the red arrow in b.

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Results

Fragmentation of molten soda glass

First, we explain the fragmentation of the molten soda glass; the behaviour at room temperature has already been investigated45,46,47,48. In these experiments, we varied the operating temperature, strain rate, with or without preheating, and preheating temperature (Table 1).

Table 1 Experimental conditions.
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Figure 2 shows a detailed sequence of the fragmentation of the soda glass at 630 °C and a strain rate of \(\dot{\gamma }=0.0036\) s−1 with preheating at 750 °C. When the soda glass rod was stretched vertically (Fig. 2a, b), the imposed stress increased and then suddenly decreased when the glass rod fractured (Fig. 2c). A time sequence of the rod fracture is shown in Fig. 2a and Supplementary Movie 1. The soda glass rod is located between the red lines (0 s). When a fracture occurs, the crack appears as a black line that propagates the entire width of the rod within one frame (red arrow). Necking, that is, local narrowing of the rod diameter by localized viscous deformation, is not observed, which is consistent with previous observations of the fragmentation of magma23. In subsequent frames, smaller fragments appear on the left side but not on the right side. These characteristics suggest that the rod broke in a solid-like brittle manner.

Fig. 2: Fragmentation of soda glass rod at 630 °C.
figure 2

The imposed strain rate is \(\dot{\gamma }=0.0036\) s−1, and the rod has experienced preheating at 750 °C. a Time-sequence photographs of soda glass rod during fracture. The white numbers indicate the elapsed time in units of 10−5 s. The red lines shown in the first frame indicate slightly outside the glass rod to show there are no flaws at the place of subsequent fracture. The yellow bar indicates the diameter of the glass rod. b Schematic diagram of a. c Measured stress–strain curve (blue dots). The pink curve is the stress–strain curve calculated using Eq. (2) assuming the Maxwell fluid with E = 20 GPa and η = 8.5 GPa s. The stepwise data result from the low time resolution of the displacement data. d Topography of the upper and lower fracture surfaces measured by laser microscopy. The colour of the scale bar indicates the topography height. The vertical length of the scale bar indicates the approximate horizontal scale. e Scanning electron microscopy (SEM) images of the fracture surface. The red rectangle and arrow indicate the approximate location, size, and orientation of the region shown in f. f Magnified view of surface structure. The detailed history of the enlarged area is given in Supplementary Fig. 4. The L and R in d, e correspond to the sides in a. The experimental conditions are listed in Table 1. The corresponding movie (Supplementary Movie 1) is provided in the Supplementary Information. Similar formats of these panels are used in the following Figs. 3–7 unless otherwise noted.

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The time scale of the fracture also suggests brittle fragmentation. From the estimated Young’s modulus of 20 GPa (Supplementary Fig. 3), the shear wave velocity is estimated to be on the order of [(E/3)/ρ]1/2 ~ 1.6 km s−1, where ρ = 2500 kg m−3 is the density of soda glass52, and we assume shear modulus of G ~ E/3. The time scale for crack propagation on the 0.69 mm diameter rod is <10−6 s, which is shorter than the frame interval of 1 × 10−5 s. Crack propagation is sufficiently rapid to be considered brittle.

The fracture surfaces are clearly divided into smooth and rough areas similar to those of glass fractured at room temperature (Fig. 2d, e)45,46,47. The morphology of the roughness differs between the upper and lower surfaces. The rough area is lower than the smooth area in the upper rod, and the fracture surface of the lower rod does not show a notable height difference, suggesting that the small fragments observed in Fig. 2a were produced in this rough region. The wavelength of the roughness increases with distance from the smooth area. The magnified view of the interface between the smooth and rough areas shows smaller (1 μm scale) roughness (Fig. 2f). The magnified view is presented in detail in Supplementary Fig. 4.

The newly created rough surface will be a scattering source of light to be observed as a dark area in Fig. 2a. The roughness height estimated from Fig. 2d is the order of 0.1 mm, which corresponds to a few pixels in Fig. 2a with a resolution of 0.02 mm/pixel. Indeed, the maximum thickness of the black crack is a few pixels in Fig. 2a. We thus consider that the thicker dark line on the left side in Fig. 2a corresponds to the rough surface area in Fig. 2d. Given that the room temperature experiments suggest that the crack propagates from smooth to rough surface area45,46,47, the crack in Fig. 2a should propagate from right to left. Depending on the printed resolution, the right side of the dark crack in Fig. 2a may not be visible because the surface roughness is too small on the smooth area.

A similar result was observed at a higher strain rate of \(\dot{\gamma }=0.027\) s−1 under approximately the same sample temperature of 645 °C and preheating temperature of 750 °C (Fig. 3 and Supplementary Movie 2). Finer fragments were generated in this experiment. The narrower smooth area on the fracture surface is surrounded by radial lines, which branch off and create a structure in a perpendicular direction (Supplementary Fig. 5). In the areas distant from the smooth area, the radial lines are more widely spaced, but small-scale textures are also present.

Fig. 3: Fragmentation of soda glass rod at a higher strain rate.
figure 3

Approximately the same as Fig. 2 but at a rapid strain rate of \(\dot{\gamma }=0.027\) s−1 with preheating at 750 °C. The operating temperature is 645 °C. a Time sequence photographs of soda glass rod during fracture. The upper rod moves downward. In the last photograph, the fine fragments are blown away. b Measured stress–strain curve. The pink curve is calculated using Eq. (2), assuming the Maxwell model with E = 18 GPa and η = 1.2 GPa s. c Fracture surface of the lower rod. d Topography image of the fracture surface of the lower rod obtained by laser microscopy. The corresponding movie is provided in the Supplementary Information (Supplementary Movie 2).

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The fracture pattern after preheating at a higher temperature of 820 °C is different (Fig. 4). In this experiment, the operating temperature of 650 °C and strain rate of \(\dot{\gamma }=0.0044\) s−1 are approximately the same as those shown in Fig. 2. In contrast to that in Fig. 2, the fracture does not generate small fragments (Supplementary Movie 3). The fracture surface is smooth. The stress–strain curve fluctuates before the fracture. During preheating, we elongated the soda glass rod and formed a locally thinner region. After elongation was stopped and the temperature was lowered to 650 °C, we elongated the glass rod again. This locally thinner region is expected to reduce the total force required for fragmentation through the cross-sectional area and increase the strain in the radial direction.

Fig. 4: Fragmentation of soda glass rod with a locally thinner region.
figure 4

Approximately the same as Fig. 2 but for the experiment with preheating at 820 °C. The operating temperature is 650 °C, and the strain rate is \(\dot{\gamma }=0.0044\) s−1. a Time sequence of rod fracture. The yellow bar indicates the diameter of the rod at the place of fragmentation, although it is shown at a different place. b Measured stress–strain curve. The pink curve is calculated using Eq. (2), assuming the Maxwell model with E = 24 GPa and η = 2.4 GPa s. c SEM image of the lower fracture surface. The fractured surface is flat, and no structure by the fracture is observed. Only the peeling and heterogeneously deposited carbon shows textures. The corresponding movie is provided in the Supplementary Information (Supplementary Movie 3).

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Fragmentation of haploandesite melt

In the haploandesite experiments, we varied the strain rate and fixed the operating temperature to 880 °C without preheating. Figure 5a shows the fragmentation of haploandesite at a strain rate of \(\dot{\gamma }=0.039\) s−1. Before this experiment, we elongated the same sample at a lower strain rate of \(\dot{\gamma }=0.0055\) s−1 at the same temperature; it showed viscous deformation, but no necking occurred (Table 1).

Fig. 5: Fragmentation of haploandesite melt.
figure 5

a Time sequence photographs of haploandesite melt fracture at 880 °C and strain rate \(\dot{\gamma }=0.039\) s−1. b Schematic diagram of a. c Stress–strain curve for the experiment shown in a. The pink curve is the stress–strain curve calculated using Eq. (2) assuming the Maxwell model with E = 21 GPa and η = 1.1 GPa s. d SEM images of upper and lower fracture surfaces. The deposited carbon is peeling in some places, which is not the original structure. e Measured topography of lower fracture surface observed by laser microscopy. The corresponding movie (Supplementary Movie 4) is provided in the Supplementary Information.

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No signs of fracture appear in the first frame (0 s), but in the next frame (1 × 10−5 s), the rod has cracked into two parts, as denoted by the red arrow. A small crescent-shaped fragment appears between the two parts of the rod and moves leftward, as indicated by the red circle and shown in the schematic diagram (Fig. 5b). Again, necking is not observed, and the rod breaks in a solid-like brittle manner (Supplementary Movie 4). However, the stress–strain curve suggests that viscous deformation is dominant (Fig. 5c). When deformation begins, the measured stress increases rapidly owing to the elastic response. The slope of the stress becomes flat as the strain increases, which indicates viscous deformation (Supplementary Fig. 2). After the stress reaches the asymptotic value by viscous deformation, fragmentation occurs, and the measured stress suddenly decreases.

The fracture surface has smooth and rough areas (Fig. 5d, e) similar to those observed in soda glass experiments (Fig. 2). The smooth area is surrounded by radial lines that become widely spaced away from the smooth region. The shape of the smooth areas is approximately the same on the upper and lower fracture surfaces, suggesting that this region did not produce additional fragments during the fracture process. The undulation of radial lines appears to be a remnant of the shape obtained by viscous deformation (Fig. 5e). The region with radial lines is lower than the smooth region, suggesting that the crescent-shaped fragment originates from this region.

The fracture surface obtained at a slightly slower strain rate at \(\dot{\gamma }=0.027\) s−1 shows different surface features (Fig. 6). The rim of the lower rod shows a protruding shape that deviates from a circle (Fig. 6c, red arrow). A viscous deformation may occur to make this shape in addition to a brittle fracture. The magnified fracture surface shows that the texture around the boundary between the smooth and rough areas is not small-scale like that observed in soda glass (Supplementary Fig. 6). During more rapid deformation \(\dot{\gamma }=0.053\) s−1, finer fragments are generated; the stress–strain curve becomes linear, and the fracture surface does not show clearly differentiated smooth and rough areas (Fig. 7).

Fig. 6: Fragmentation of haploandesite melt at a slower strain rate.
figure 6

Similar to Fig. 5 but with a strain rate of \(\dot{\gamma }=0.027\) s−1. a Time sequence photographs of haploandesite during fracture. The rod is between the red lines. In the last photograph, the fine fragments are blown away, as denoted by the red circle, which is more visible in Supplementary Movie 5. b Measured stress–strain curve. The pink curve is calculated using Eq. (2), assuming the Maxwell model with E = 23 GPa and η = 1.1 GPa s. c Upper and lower fracture surfaces. The upper surface shows smooth and rough areas similar to those observed in other experiments. The position relative to the lower rod is unknown. d Surface topography of the upper rod observed by laser microscopy.

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Fig. 7: Fragmentation of haploandesite melt at a higher strain rate.
figure 7

Similar to Fig. 5 but with a strain rate of \(\dot{\gamma }=0.053\) s−1. a Time sequence photograph. b Measured stress–strain curve. The pink curve was calculated using Eq. (2), assuming the Maxwell model with E = 21 GPa and η = 1.2 GPa s. c SEM and d topographic images. The corresponding movie is provided in the Supplementary Information (Supplementary Movie 6).

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Relationship between rheology and fragmentation

Next, we consider the conditions under which fragmentation occurs using the stress–strain curves (Fig. 8a, b). The shape of the stress–strain curve indicates the degree of elastic/viscous deformation. In our experiments, we elongated the sample at a constant velocity. Under this condition, a Maxwell fluid initially behaves as an elastic material and then shifts to viscous deformation (Eq. (2), Supplementary Fig. 2a). Thus, when a strain is applied, the stress acting on the sample first increases linearly with the elastic response. Eventually, viscous deformation occurs, and the stress asymptotically approaches the maximum value, \(3\eta \dot{\gamma }\) (Fig. 8c, left inset, blue curve). We here assume that the elongation viscosity is three times the shear viscosity. If fragmentation occurs, the stress immediately becomes zero (Fig. 8c, insets, red lines). If the viscosity (η) or strain rate (\(\dot{\gamma }\)) is large, the stress required for viscous deformation (\(3\eta \dot{\gamma }\)) also becomes large (Fig. 8c, right inset, grey curve). Fragmentation may occur before the onset of viscous deformation. In this case, the measured stress–strain curve becomes relatively linear (Fig. 8c, right inset, blue curve). Our measured stress–strain curves suggest that fragmentation occurred after the onset of viscous deformation in most of the experiments (Fig. 8a, b).

Fig. 8: Summary of stress–strain curves.
figure 8

The colour of the curves indicates the temperature in a, where blue is colder, and the strain rate in b, where blue is rapid. Dotted curves indicate that fragmentation occurs where the rod and sphere meet. In a, the pink and black triangles indicate preheating at ≤750 °C and 820 °C, respectively. c Maximum stress measured during each experiment normalized by the maximum viscous stress (3\(\eta \dot{\gamma }\)) as a function of Deborah number, a product of the relaxation time and strain rate (\({\tau }_{{{{{{{{\rm{c}}}}}}}}}\cdot \dot{\gamma }\)). The insets show the meaning of \({\sigma }_{\max }/(3\eta \dot{\gamma })\). The superimposed symbols indicate the sample type, deformation patterns, and initial conditions, and are shown in Table 1. The symbols for the haploandesite are square-rimmed and green to blue in colour. The colour is the same as b. Soda glass is shown without a rim and red to blue. The colour is the same as a. The crosses and circles indicate fragmentation and deformation, respectively. The asterisks formed by overlapping plus signs indicate that fragmentation occurred where the rod is connected to the sphere (see Fig. 1). The pink and black triangles indicate preheating at ≤750 °C and 820 °C, respectively. The solid black line is a reference at which the maximum stress is consistent with viscous drag \(3\eta \dot{\gamma }\). The dotted black lines are previously reported fragmentation thresholds, \({\tau }_{{{{{{{{\rm{c}}}}}}}}}\dot{\gamma }=0.01,0.04\). d Measured strain at which fragmentation occurs. The symbols are the same as in c.

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The stress and strain at the moment of fragmentation may determine the fragmentation mechanism. One possibility is that fragmentation occurs when the loading stress under rapid strain exceeds the threshold stress. However, the measured maximum stress, which corresponds to the threshold stress at fragmentation, varies among the experiments. Even in experiments in which the stress–strain curves are initially similar, fragmentation sometimes occurs, and sometimes it does not (Fig. 8a, b).

To evaluate whether fragmentation occurs in the viscous or elastic deformation regime, we normalized the measured maximum stress in each experiment, \({\sigma }_{\max }\), by \(3\eta \dot{\gamma }\) (Fig. 8c). When fragmentation occurs after the onset of viscous deformation, \({\sigma }_{\max }/(3\eta \dot{\gamma }) \sim 1\). By contrast, if fragmentation occurs during elastic deformation, \({\sigma }_{\max }/(3\eta \dot{\gamma }) \, < \, 1\). The horizontal axis is the strain rate multiplied by the relaxation time τc. We estimate the relaxation time from the ratio of the measured maximum elongation viscosity to the maximum Young’s modulus, as an approximation of the ratio of the zero strain rate viscosity to Young’s modulus at an infinite strain rate, τc = 3η0/E∞ (see the “Methods” section for details).

The experiments with (crosses) and without (circles) fragmentation are plotted on the line of \({\sigma }_{\max }/(3\eta \dot{\gamma }) \sim 1\) irrespective of the normalized strain rate \({\tau }_{{{{{{{{\rm{c}}}}}}}}}\dot{\gamma }\) (Fig. 8c). That is, fragmentation typically occurs after the onset of viscous deformation, and the strain rate does not affect whether fragmentation occurs.

The experiments with cold glass (≤615 °C, blue asterisks) show lower values of \({\sigma }_{\max }/(3\eta \dot{\gamma }) \, < \, 1\) at a rapid strain rate, \({\tau }_{{{{{{{{\rm{c}}}}}}}}}\dot{\gamma } \, > \, 0.01\). Here, the low-temperature experiments were not conducted in a wide range of strain rates, and the relaxation time was estimated from the measured viscosity for each experiment. The viscosity at zero strain rate can be larger, which can also increase τc, as indicated by arrows (Fig. 8c).

Note that our measured rheology indicates that \(\dot{\gamma } \, > \, 0.01/{\tau }_{{{{{{{{\rm{c}}}}}}}}}\) is in the elastic deformation regime, different from previous models based on the Maxwell model and rheology measurements at small strain amplitudes (Supplementary Fig. 3a, b). We thus consider the prefactor of 0.01 used for the fragmentation threshold can vary with the strain, strain rate, and geometry of deformation (for details, see the “Methods” section). In the previous experiments, fragmentation is not reported in the regime of \(\dot{\gamma } \, < \, 0.01/{\tau }_{{{{{{{{\rm{c}}}}}}}}}\), but it occurs in our experiments (Fig. 8c). This discrepancy may arise because previous experiments in the viscous regime have been conducted under compression/shear deformation22,24 while we elongated the rod samples. The tensional deformation efficiently makes voids to be nucleation sites of cracks, causing fragmentation of viscously deforming melt.

The strength of the glass is frequently interpreted by the depth and distribution of flaws on the glass surface53,54,55. A glass that experiences a high-temperature condition has deeper flaws56. However, the various stress level for fragmentation observed in Fig. 8a, b lies on the same line of \({\sigma }_{\max }/(3\eta \dot{\gamma }) \sim 1\) in Fig. 8c, indicating that the occurrence of the fragmentation is determined by the onset of viscous deformation rather than the initial condition of the sample, such as the distribution of flaws. This may imply that our glass samples fragment at the stress level originated by viscous deformation, below the strength determined by flaws. We also note that we use the same sample which did not fragment in the previous run and, again, they lie on the same line of \({\sigma }_{\max }/(3\eta \dot{\gamma }) \sim 1\) in Fig. 8c. This fact supports our interpretation that viscous deformation promotes fragmentation irrespective of the deformation history of the sample unless the macroscopic deformation makes the stress concentration locations such as locally thin regions.

Figure 8d summarizes the critical strain at which fragmentation occurs, which depends not on the strain rate but on the material. Haploandesite requires a larger strain for fragmentation than soda glass. The strain at which fragmentation occurs varies by approximately an order of magnitude, even within the same glass sample. One possible interpretation of strain difference for fragmentation is due to the difference of non-bridging oxygen atoms per tetrahedrally coordinated cation (NBO/T ratio). Soda glass has a higher NBO/T ratio corresponds to lower polymerization (Supplementary Table 1). It has been suggested recently that subnanometre-scale chemical heterogeneities in Fe-, Na-, and/or Ca-rich clusters may be bubble nucleation sites57. Such clusters of chemical heterogeneities can also become the nucleation sites of voids. The soda glass with higher NBO/T, less polymerized, may able to make clusters of chemical heterogeneities with a smaller strain. This effect may shorten the strain required for fragmentation. Thus, more polymerized haploandesite can undergo more deformation before fragmentation. Similar NBO/T dependence on critical strain for fragmentation is also observed in the comparison between basaltic, andesitic and rhyolitic melts58. The other possibility is that the conditions of soda glass experiments are closer to the elastic regime, as inferred from the shear thinning viscosity (Supplementary Fig. 3a) than what are estimated using τc in Fig. 8c, d.

In summary, the strain rate does not explicitly determine the fragmentation threshold. Some samples fragment at a lower strain rate than others that deform viscously. Visual observations suggest brittle fragmentation, but the stress–strain curves suggest viscous deformation. Neither constant stress nor strain determines the fragmentation threshold.

Discussion

Finally, we consider how smaller fragments of magma are produced to be fine, dense ashes. In our high-temperature experiments, the fracture surface of glasses shows distinct smooth and rough areas. This characteristic is similar to those observed on glass fracture surfaces at room temperature45,46,47, suggesting that the fracture mechanism at high temperature is the same as that at room temperature. In a low-temperature glass or amorphous material, cracks initiate from flaws such as pores59 and propagate with increasing speed. Beyond a critical velocity at high stress, a single crack becomes unstable because of the nucleation of voids, resulting in branching42,43. The rough fracture surface of the glass may originate from a coalescence of small voids47,51.

Our observation of the fracture surface suggests that fracture occurs at high temperatures by a similar mechanism, and the branching of cracks generates small fragments (Figs. 2, 3, 5–7). This interpretation is consistent with the recent observation of the molecular scale strain obtained by the first sharp diffraction peak in time-resolved X-ray diffraction58. The molecular scale strain becomes larger for the tensile direction than its perpendicular direction, suggesting the anisotropic dilation occurs to make voids to be a nucleation site and cause branching.

In our experiment, high stress is applied to a relatively homogeneous silicate melt, which causes rapid strain, and then a single crack is generated. By contrast, magmas in nature usually contain bubbles and crystals, which can be nucleation sites for cracks, even when vesicularity is low. Branching of the fractures formed by an indirect tensile test with ductility deviates toward crystals and bubbles60. Natural magmas also contain volatiles, which should be another chemical heterogeneity affecting nucleation and crack propagation. Indeed, environmental humidity enhances crack propagation in glass at low temperatures61. Thus, when high stress or a rapid strain rate is applied, multiple cracks may begin propagating from the numerous flaws. Eventually, they branch because of instability, forming smaller fragments than the initial distribution of nucleation sites (Fig. 9, upper left). The fragments, which had contact with the branches, may preserve the void shapes and form ashes with irregular shapes. These shapes are usually interpreted as those of pre-existing bubbles. This mechanism can explain the fine ashes from Vulcanian eruptions18,19.

Fig. 9: Schematic diagram of the fragmentation process.
figure 9

When high stress is applied to less vesicular magma, rapid strain occurs, and cracks appear. Ultimately, cracks branch and generate fine lithic ash particles. Under low stress, the strain rate becomes slow, and viscous deformation occurs. Viscous deformation changes the alignment of bubbles and crystals and may provide the conditions for fracture. If the vesicularity is high, the thin magma film separating bubbles is easily fractured by crack propagation under low stress. This mechanism can form pumices/scoriae.

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Glass fracture experiments at room temperature show that the mirror region, which is a smooth region on the fractured surface, becomes narrow with higher stress46. Rough areas where crack brunching creates smaller fragments are broadened by greater stress, which implies that smaller volcanic ash is created by greater stress.

When the vesicularity and stress are low, the magma deforms viscously. Viscous deformation can reduce the cross-sectional area of the connecting melt and change the alignment of bubbles and crystals, changing the required conditions for fragmentation. When the conditions are met, fragmentation occurs (Fig. 9, lower left).

When the vesicularity is higher, the thickness of the connecting melt films/filaments varies. Our experiments with preheating show that viscous deformation reduces the cross-sectional area of the connecting melt, which enables fracture at low stress. These characteristics are consistent with the fact that highly vesicular magma can fragment at low stress62, and vesicularity is another factor affecting fragmentation63. When highly vesicular magma fragments at low stress, the branching of cracks may not occur. However, thin films/filaments will become fine ashes and erupt with scoriae/pumices. Our experiments suggest that a lower NBO/T number allows larger strain before fragmentation. Silicic pumice sometimes shows fully deformed melt films64, which may be attributed to the higher polymerization.

Thus, we infer that fragmentation can occur under any initial conditions. Some explosive eruptions, such as ash explosions after Strombolian eruptions and Vulcanian eruptions, produce mainly ashes rather than scoriae/pumices13,14,15,17,65. Such ashes may be generated by the branching of cracks.

Methods

Experimental method

A rod-shaped sample is attached to a tensile testing machine using spheres at both ends (AND MCT-1150) (Fig. 1). The sample and the tip of the Inconel rods were locally heated in a vertical annular furnace66. The upper piston moves upward at various controlled strain rates (Table 1). We monitored the force F required for deformation using a load cell (Kyowa LUX-B-200N-ID with an eigenfrequency of 14kHz) and a displacement x measured by a laser displacement sensor (Keyence LK-G155A, response time 100–1000 μs). We then calculated the strain x/L and stress F/(π(w/2)2), where L is the sample length, and w is the sample width (diameter), both at the time elongation begins for each experimental run. The original length and width of the sample were measured by a calliper before the experiments.

The furnace has three small holes; one is used to illuminate the sample, and we observed the sample from another. The third hole, which is behind the sample, provides a black background and better contrast with the surroundings. A transparent sample inside a whitish high-temperature furnace is difficult to observe. These holes are not covered and may decrease the temperature inside the furnace. We observed the deformation and fragmentation of the sample using a high-speed camera (Photron FASTCAM SA-Z).

Given that misalignment may cause stress other than tensile components, we carefully adjusted the axes of the sample and pistons to be aligned. By changing the direction of the slit at the end of the piston, we adjusted the location of the sample and prevented stress other than the tensile component imposed on the sample (Fig. 1b). Even with such careful treatment, the possibility of misalignment still remains. Under a misalignment, fracture occurs around the sphere for grip67. In Fig. 8c, d, Table 1, and Supplementary Fig. 3, fractures near the sphere are marked asterisk to distinguish them. We note that fracture near the sphere is not necessarily due to misalignment, because the joints between the spheres and the rod also act as flaws by stress concentration.

Sample preparation

We used soda glass and haploandesite (iron-free andesite) as samples. Both are silicate melt and consist of a network of SiO4 tetrahedra identical to magma. Because of the similarity of the molecular scale structure, the deformation mechanism obtained from our samples is applicable to natural magma. Soda glass is chosen because the fracture at room temperature has been studied, which helps in the interpretation of our high-temperature experiments. Haploandesite has a composition more like real magma but is iron-free to make the sample transparent and prevent crystallization. If iron exists, magnetite forms during the heating, which change the rheology68. The sample compositions are listed in Supplementary Table 1. We note that the haploandesite has lower NBO/T than soda glass. The lower NBO/T indicates a higher degree of polymerization of the structure.

The samples were prepared in the following procedure. Haploandesite was melted at 1500 °C for 7.5 h and then held at 1500 °C for 40 h. The temperature holding time was determined to be long enough to remove air bubbles. Soda glass was prepared similarly but melted at 1200 °C. A quartz glass rod was inserted into the molten samples taken from a furnace and pulled up to form fibres. After that, they were cut into appropriate lengths and spheres were added at both ends with a burner. This method has been previously reported69. The samples were cooled at room temperature, and we did not anneal them. Even if thermal stress remains in the sample, that should be removed during the high-temperature experiments. The relaxation time of our sample is less than 20 s (Table 1), and the thermal diffusion time of the thin rod is 1 s, where the thermal diffusivity of glass is 10−6 m2s−1. The temperature and stress in the rod are expected to be homogeneous. The detailed characteristics of the soda glass and haploandesite are previously reported58,66.

The importance of surface flaws on the strength of the sample has been recognized53,55. We stored the samples in a hard case which might generate surface flaws on the sample. However, we consider such flaws do not affect our experimental results. In our preliminary experiments, we added a scratch at the middle of the rod with a diamond rasp in hopes that fracturing would occur from the scratch within the view field of the high-speed camera. Contrary to our expectation, fracturing occurred other place outside the view field indicating that the artificial scratch did not become a nucleation site of a crack. Recent experiments with glass fibres show that the heated glass loose strength56, suggesting that heating increases flaw depth. Our experiments are conducted at the similar temperature range of the erupting magma. The stresses required for fracture measured in our experiments are the order of 108 Pa (Fig. 8a, b), which is consistent with previously reported for silicate melt23. We thus consider that the surface condition of our sample is reasonably simulating a natural magma.

In Supplementary Fig. 1, we compare the viscosities obtained from our experiments, measured in another furnace with reliable temperature, and the model for natural magma70. The glass viscosities at the temperature range of 600–660 °C shows good agreement between our measurements (crosses) and other measurements and the model (curves). The trend of high temperature differs between measurements (solid curve) and models (dotted curve). This may be because the composition of soda glass is different from typical magmas. The measured viscosity for the haploandesite at 880 °C is larger than the two curves. We infer that this difference is arisen from the ambiguity of the temperature estimation. Given that, our furnace has holes for visualization, it is plausible that the sample temperature is lower than the operating temperature monitored closer to the heaters. We confirmed in our previous study that the temperature at the sample position is 5 and 11 °C lower than the operating temperatures of 600 and 800 °C, respectively66. This difference should be enhanced at a higher operating temperature of 880 °C. We also widened the hole for visualization to obtain a better images after we calibrated the temperature, which also might lower the actual temperature at the sample position.

When the sample was elongated without fragmentation, we halted the deformation and continuously used the same sample for deformation at a higher strain rate. We also elongated the sample at a high temperature before causing fragmentation. The later cases are denoted as “preheat” in Table 1. In these experiments, we added the displacement in the previous run to the sample length L for the next run. To obtain the sample width, we monitored the percentage decrease in width due to elongation at the place close to the fragmentation location of the subsequent run and calculated the width for the next run. The width of the samples preheated at 820 °C varied locally by up to 20%, which can cause an error in the estimated stress of <40%. We initially thought that if the centre of the sample was thinner because of deformation, fragmentation would occur in the field of view of a high-speed camera; unfortunately, this was not the case.

We consider that the deformation history of the sample does not affect the stress condition on the subsequent experiments. This is because the stress relaxes shorter than 20 s (Table 1). The time interval until the next experiment is several minutes which is much longer than the relaxation time. We also monitored the value of the load cell which confirms that the stress is fully relieved. In contrast, molecular-scale structure possibly changes by the deformation during the prior experiments. Recent experiments monitoring the first sharp diffraction peak in time-resolved X-ray diffraction indicate that molecular scale anisotropic strain occurs by the tension deformation of glass rods. The anisotropic characteristic relaxes after stopping the deformation, while isotropic permanent strain remains58. This permanent strain may change the distribution of the possible nucleation site of cracks. However, abundant nucleation sites distribute in the glass rod randomly before the deformation. We thus do not consider the permanent strain affects the subsequent experiments inherently. In Fig. 8c, the experiments with various deformation histories lie on the same line, suggesting that the previous structure change does not affect the sample strength.

Observation of fracture surface

The fracture surfaces of the samples were observed using a scanning electron microscope with a Schottky FE-type gun (JSM-7001F, JEOL Ltd) and a laser microscope (LEXT OLS4000, Olympus Co.).

Method of calculating the viscosity and Young’s modulus

A Maxwell fluid is modelled as a serial connection of viscous and elastic components. The total strain is the summation of elastic γ1 and viscous strains γ2, γ = γ1 + γ2, while the imposed stress is uniform \(\sigma =E{\gamma }_{1}=3\eta \dot{{\gamma }_{2}}\), where E is Young’s modulus, η is shear viscosity, and \(\dot{{\gamma }_{2}}\) is the strain rate of viscous component. We here assume that elongation viscosity is three times of shear viscosity, ηelong = 3η23. We also convert Young’s modulus to the shear modulus E = 3G, assuming a Poisson’s ratio of 0.5. Using these relations, the constitutive equation for an elongating Maxwell fluid is

$$\dot{\gamma }=\frac{1}{E}\frac{d\sigma }{dt}+\frac{\sigma }{3\eta },$$
(1)

where \(\dot{\gamma }\) is total strain rate and t is time.

Integrating Eq. (1) for σ = 0 at t = 0, and \(\sigma =3\eta \dot{\gamma }\) at t = ∞, we obtain the stress–strain relation

$$\sigma =3\eta \dot{\gamma }[1-{e}^{-\gamma /\left(\dot{\gamma }\tau \right)}],$$
(2)

where τ = 3η/E, and we assume a constant strain rate of \(\dot{\gamma }=\gamma /t\). By fitting the stress–strain curve with Eq. (2), we obtain the viscosity η and Young’s modulus E. Note that η and E values calculated by using Eq. (2) depend on the strain rate \(\dot{\gamma }\). Supplementary Fig. 2a plots examples of Eq. (2) at various strain rates.

In the above discussion, we assume Poisson’s ratio ν is 0.5. This value is suitable for incompressible fluid23. Under a rapid strain rate, the glass behaves more like a solid, and the Poisson’s ratio can be smaller as 0.258,71. In this case, Young’s modulus is calculated by E = 2G(1 + ν) = 2.4G. It is reasonable to assume a similar ratio of the bulk and shear deformation to calculate viscosity, which results in ηelong = 2.4η. In this case, the shear modulus and shear viscosity could be higher than the estimation assuming ν = 0.5 by up to 20%, but the relaxation time calculated by their ratio is the same.

Rheology measurements

Here we explain the results of viscosity measurements and how we estimated the relaxation time τc. First, we introduce the expected strain rate dependence of the rheology inferred from oscillatory rheology measurements.

Oscillatory measurement with an angular frequency of ω is an informative method for understanding the rheology of complex fluids. For a Maxwell fluid, the storage modulus \({E}^{{\prime} }\) and loss modulus E″ are analytically described as72,

$${E}^{{\prime} }={E}_{{{{{\infty }}}}}\frac{{\omega }^{2}{\tau }_{{{{{{{{\rm{c}}}}}}}}}^{2}}{1+{\omega }^{2}{\tau }_{{{{{{{{\rm{c}}}}}}}}}^{2}},{E}^{{\prime\prime} }={E}_{{{{{\infty }}}}}\frac{\omega {\tau }_{{{{{{{{\rm{c}}}}}}}}}}{1+{\omega }^{2}{\tau }_{{{{{{{{\rm{c}}}}}}}}}^{2}},$$
(3)

where τc is the ratio of the elongation viscosity at zero strain rate to the Young’s modulus at an infinite strain rate, τc = 3η0/E∞. For ωτc < 1 and ωτc > 1, a Maxwell fluid is liquid-like and solid-like, respectively, and viscoelastic characteristics are observed around ωτc ~ 1. Using Eq. (3), we can calculate the complex modulus \(| {E}^{* }| ={({E}^{{\prime} 2}+{E}^{{\prime\prime} 2})}^{1/2}\) and complex viscosity ∣η*∣ = ∣E*∣/(3ω), which depend on ω as shown in Supplementary Fig. 2b. For ωτc < 1, the complex Young’s modulus increases with increasing ω, whereas, for ωτc > 1, the complex shear viscosity decreases. As a result, the ratio 3∣η*∣/∣E*∣, which is similar to the time scale of τ ~ 3η/E, also depends on ω as 3∣η*∣/∣E*∣ ∝ ω−1. Polymer melts and solutions often exhibit similar dependence on the strain rate and angular frequency, which is known as the “Cox–Merz ruls.” That is, the complex viscosity as a function of frequency is almost identical to the shear viscosity as a function of strain rate: \(| {\eta }^{* }| (\omega ) \sim \eta (\dot{\gamma })\). According to this analogy, \(| {E}^{* }| (\omega ) \sim E(\dot{\gamma })\).

Supplementary Fig. 3a, b shows the strain rate dependence of η and E, which is similar to that observed in Supplementary Fig. 2b. The haploandesite (the square-rimmed symbols) shows relatively constant viscosity and increases Young’s modulus with increasing strain rate until it reaches an asymptotic value. These characteristics are similar to those observed for the region of ωτc ≤ 1 shown in Supplementary Fig. 2b. By contrast, most of the soda glass experiments showed shear-thinning viscosity and relatively constant Young’s modulus.

The ratio of the strain rate-dependent elongation viscosity to Young’s modulus has a strain rate dependence \(\tau =3\eta /E\propto {\dot{\gamma }}^{-1}\), which is similar to 3∣η*∣/∣E*∣ ∝ ω−1 (Supplementary Fig. 3c). As a result, the product of the strain rate and τ becomes approximately constant, \(\dot{\gamma }\tau \sim 0.01\), irrespective of strain rate (Supplementary Fig. 3d). Note that τ differs from the relaxation time τc = 3η0/E∞ defined by viscosity at zero strain rate and elastic modulus at an infinitely high strain rate. The experiments, in which the sample behaves as a Maxwell fluid, fall around \(3\eta /E\dot{\gamma } \sim 0.01\).

Here, the glass experiments at low temperature (≤615 °C, blue asterisks) and with preheating at 820 °C (black triangles) do not fall on the line \(\dot{\gamma }\tau \sim 0.01\). The reason is that the glass is too cold to behave as a Maxwell fluid. In the experiments with preheating at 820 °C, the glass rod was locally thin where the strain was concentrated. As a result, the strain in the radial direction became dominant, which is not included in our model [Eq. (2)]. We consider that the estimated rheological parameters for these experiments (cold and preheating at 820 °C) are not well constrained (Fig. 4b).

Relaxation time scales

We adopted two methods to estimate relaxation time. One method uses the ratio of the viscosity at zero strain rate to the elastic modulus at an infinitely large strain rate, τc = 3η0/E∞. The critical strain rate for fragmentation is defined by this relaxation time as \(\dot{\gamma } \, > \, 0.01/{\tau }_{{{{{{{{\rm{c}}}}}}}}}\)20. We here discuss using 0.01 for simplicity, although the prefactor may have a range from 0.01 to 0.0422. We estimate τc from Supplementary Fig. 3a, b and summarize the results in Table 1. We assume that the measured maximum values for each material are the same as the infinite values. For most of the soda glasses, η0 ~ 1.1 × 1010 Pa s and E∞ ~ 24 GPa; for the haploandesite, η0 ~ 2.4 × 109 Pa s and E∞ ~ 23 GPa. Consequently, τc ~ 1.3 s and τc ~ 0.31 s, respectively. For the cold glass and glass with preheating at 820 °C, we do not have data showing the strain rate dependence; therefore, we used τ ~ τc.

The other method of estimating the relaxation time is to use strain-rate-dependent rheology. We can define the angular frequency ω for τc2 = 1/ω, as the value at which the viscosity causes shear thinning and the elastic modulus reaches an asymptotic value (Supplementary Fig. 2b).

For the soda glass, we indicate the strain rate of 0.01/τc (grey line), which falls close to the onset of shear thinning (Supplementary Fig. 3a). Similarly, for the haploandesite, the estimated value of 0.01/τc is close to the strain rate at which the Young’s modulus is close to the asymptotic value (Supplementary Fig. 3b). These results suggest that the time scales estimated by the two methods differ by two orders of magnitude, that is, that 0.01τc2 ~ τc.

This difference may be the origin of the prefactor of 0.01 for the fragmentation threshold \(\dot{\gamma } \, > \, 0.01/{\tau }_{{{{{{{{\rm{c}}}}}}}}}\). Note that the τc2 obtained by small amplitude oscillatory measurements is consistent with the ordinary relaxation time, τc2 = τc (Cox-Merz rule). At the small amplitude rheology measurements, the non-Newtonian behaviour becomes apparent at a time scale of 100τc, which has been interpreted as the origin of the prefactor 0.0120,22. However, glass rheology depends on deformation amplitude and strain rate62. The deformation geometry also can affect the rheology of a complex fluid. We thus infer that the “Cox-Merz rule” breaks with large strain at a high strain rate under the extensional geometry. The prefactor of 0.01 can depend on the strain, strain rate, and deformation geometry.