^{1}and Michael Dennin

^{1,a)}

### Abstract

The behavior of materials under tension is a rich area of both fluid and solid mechanics. For simple fluids, the breakup of a liquid as it is pulled apart generally exhibits an instability driven, pinch-off type behavior. In contrast, solid materials typically exhibit various forms of fracture under tension. The interaction of these two distinct failure modes is of particular interest for complex fluids, such as foams, pastes, slurries, etc. The rheological properties of complex fluids are well-known to combine features of solid and fluid behaviors, and it is unclear how this translates to their failure under tension. In this paper, we present experimental results for a model complex fluid, a bubble raft. As expected, the system exhibits both pinch-off and fracture when subjected to elongation under constant velocity. We report on the critical velocity below which pinch-off occurs and above which fracture occurs as a function of initial system width *W*, length *L*, bubble size *R*, and fluid viscosity for both monodisperse and polydisperse systems. Though both exhibit a transition from pinch-off to fracture, the behavior as a function of *L/W* is qualitatively different for the two systems. For the polydisperse systems, the results for the critical velocity are consistent with a simple scaling law , where the fluid viscosity sets the typical time for bubble rearrangements . We show that this scaling can be understood in terms of the dynamics of local bubble rearrangements (T1 events). For the monodisperse systems, we observe a critical value for *L/W* below which the system only exhibits fracture.

The authors acknowledge the support of NSF-DMR-0907212 and Research Corporation. The authors also thank C.O., S.C., P.T., and S.H. for useful discussions.

I. INTRODUCTION

II. EXPERIMENTAL METHODS

III. SCALING BEHAVIOR

IV. RESULTS

V. SUMMARY

### Key Topics

- Geologic fractures
- 22.0
- Viscosity
- 21.0
- Fracture mechanics
- 19.0
- Complex fluids
- 15.0
- Foams
- 12.0

## Figures

Illustration of the spatial organization of a typical monodisperse and a typical polydisperse raft are shown in Figs. 1(a) and 1(c), respectively. Figures 1(b) and 1(d) are the 2D Fourier transform corresponding to Figs. 1(a) and 1(c). The Fourier transform is used to define the characteristic bubble size.

Illustration of the spatial organization of a typical monodisperse and a typical polydisperse raft are shown in Figs. 1(a) and 1(c), respectively. Figures 1(b) and 1(d) are the 2D Fourier transform corresponding to Figs. 1(a) and 1(c). The Fourier transform is used to define the characteristic bubble size.

Plot of the typical data of the angular averaged PSD as a function of the wavelength of the raft structure for a polydisperse raft (red dashed line) and a monodisperse raft (black line). The characteristic diameter of the bubble raft is defined as the peak position of the PSD. The characteristic bubble diameter for these systems shown here are for monodisperse bubbles, *d* = 0.99 mm, with a distribution in size given by the full width at half maximum of 0.05 mm and for polydisperse bubbles, *d* = 0.94 mm, with a distribution in size given by the full width at half maximum of 0.16 mm.

Plot of the typical data of the angular averaged PSD as a function of the wavelength of the raft structure for a polydisperse raft (red dashed line) and a monodisperse raft (black line). The characteristic diameter of the bubble raft is defined as the peak position of the PSD. The characteristic bubble diameter for these systems shown here are for monodisperse bubbles, *d* = 0.99 mm, with a distribution in size given by the full width at half maximum of 0.05 mm and for polydisperse bubbles, *d* = 0.94 mm, with a distribution in size given by the full width at half maximum of 0.16 mm.

Illustration of the role of T1 events in the nucleation of fracture zones. A T1 event occurs when two neighboring bubbles are pulled apart, and two next-nearest neighbors fill the gap and become nearest neighbors. This is the sequence illustrated by the top-right image labeled “No fracture.” In contrast, when the bubbles are moving faster than a critical speed associated with the time for a T1 event to occur, the next-nearest neighbors can not respond in time, and a hole is nucleated in the system. This is illustrated by the lower-right image labeled “fracture.”

Illustration of the role of T1 events in the nucleation of fracture zones. A T1 event occurs when two neighboring bubbles are pulled apart, and two next-nearest neighbors fill the gap and become nearest neighbors. This is the sequence illustrated by the top-right image labeled “No fracture.” In contrast, when the bubbles are moving faster than a critical speed associated with the time for a T1 event to occur, the next-nearest neighbors can not respond in time, and a hole is nucleated in the system. This is illustrated by the lower-right image labeled “fracture.”

Typical images for pinch-off and fracture of the bubble raft. (a)–(c) are images of a polydisperse bubble raft with an characteristic bubble radius 0.33 mm. (d)–(f) are images of a monodisperse bubble raft with the characeristic bubble radius 0.33 mm. The scale bar on the top of each image corresponds to a length of 20 mm. The initial length in each case is 60 mm, and the initial width is 80 mm. Images (a) and (d) are the rectangular initial state of the rafts. Images (b) and (e) provide an example of fracture. Images (c) and (f) are examples of pinch-off. The pulling velocities are are 2.57 mm/s for (f), 3.42 mm/s for (c) and (e), and 4.29 mm/s for (b).

Typical images for pinch-off and fracture of the bubble raft. (a)–(c) are images of a polydisperse bubble raft with an characteristic bubble radius 0.33 mm. (d)–(f) are images of a monodisperse bubble raft with the characeristic bubble radius 0.33 mm. The scale bar on the top of each image corresponds to a length of 20 mm. The initial length in each case is 60 mm, and the initial width is 80 mm. Images (a) and (d) are the rectangular initial state of the rafts. Images (b) and (e) provide an example of fracture. Images (c) and (f) are examples of pinch-off. The pulling velocities are are 2.57 mm/s for (f), 3.42 mm/s for (c) and (e), and 4.29 mm/s for (b).

vs *L/W* for different bubble sizes and structures. (a) is for polydisperse rafts and (b) is for monodisperse rafts. The symbols distinguish systems with different characteristic radii: 0.35 (solid squares), 0.47 (solid circles), and 0.68 mm (solid triangles). The initial lengths and widths ranged from 40 to 80 mm and 40 to 120 mm with 20 mm steps. For comparison with previous work, the insets are the same data but with vs *W/L* plotted. The scaled critical velocities for polydisperse and monodisperse rafts collapse onto their respective master curves. However, the behavior for the two systems is fundamentally different. The fitting curves (green lines in (a) and (b)) are a linear fit for polydisperse raft (as proposed), and a polynomial fit with the second order for monodisperse raft, with an apparent onset value for *L/W*.

vs *L/W* for different bubble sizes and structures. (a) is for polydisperse rafts and (b) is for monodisperse rafts. The symbols distinguish systems with different characteristic radii: 0.35 (solid squares), 0.47 (solid circles), and 0.68 mm (solid triangles). The initial lengths and widths ranged from 40 to 80 mm and 40 to 120 mm with 20 mm steps. For comparison with previous work, the insets are the same data but with vs *W/L* plotted. The scaled critical velocities for polydisperse and monodisperse rafts collapse onto their respective master curves. However, the behavior for the two systems is fundamentally different. The fitting curves (green lines in (a) and (b)) are a linear fit for polydisperse raft (as proposed), and a polynomial fit with the second order for monodisperse raft, with an apparent onset value for *L/W*.

A direct comparison for vs *W/L* between polydisperse (black square) and monodisperse (red circle). For all the scaled initial widths, the critical velocity for polydisperse raft is larger than the monodisperse bubble raft.

A direct comparison for vs *W/L* between polydisperse (black square) and monodisperse (red circle). For all the scaled initial widths, the critical velocity for polydisperse raft is larger than the monodisperse bubble raft.

The critical velocity vs the initial bubble raft width with the different volume ratio of the glycerol: 15% glycerol (black square) and 32% glycerol (red circle). The data are for an initial *L* of 60 mm and a bubble radius of 0.47 mm. An increase in viscosity results in a consistently lower critical velocity for fracture.

The critical velocity vs the initial bubble raft width with the different volume ratio of the glycerol: 15% glycerol (black square) and 32% glycerol (red circle). The data are for an initial *L* of 60 mm and a bubble radius of 0.47 mm. An increase in viscosity results in a consistently lower critical velocity for fracture.

The histogram of the time for T1 events to occur in the (a) 15% and (b) 32% glycerol solutions. The T1 event time is measured by counting the time for the single T1 event in the oscillatory geometry. The histograms of 15% and 32% glycerol solution reveal an increase of the characteristic T1 time from 0.36 to 0.47 s as the viscosity is increased.

The histogram of the time for T1 events to occur in the (a) 15% and (b) 32% glycerol solutions. The T1 event time is measured by counting the time for the single T1 event in the oscillatory geometry. The histograms of 15% and 32% glycerol solution reveal an increase of the characteristic T1 time from 0.36 to 0.47 s as the viscosity is increased.

Plot of versus *L/W* for 15% (black squares) and 32% (red circles) solutions. The initial lengths and widths ranged from 40 to 80 mm and 40 to 120 mm with 20 mm steps. The bubble radius is about 0.47 mm for both solutions. The T1 velocity is derived from the T1 time() which is measured by the oscillatory experiment.

Plot of versus *L/W* for 15% (black squares) and 32% (red circles) solutions. The initial lengths and widths ranged from 40 to 80 mm and 40 to 120 mm with 20 mm steps. The bubble radius is about 0.47 mm for both solutions. The T1 velocity is derived from the T1 time() which is measured by the oscillatory experiment.

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