This is one of the six papers assembled by Dr. Andy Kraynik from the IUTAM2011 meeting on the “Mechanics of Liquid and Solid Foams”

Scaling of critical velocity for bubble raft fracture undertension

Chin-Chang Kuo and Michael Dennina)

Department of Physics and Astronomy and Institute for Complex Adaptive Matter,University of California at Irvine, Irvine, California 92697-4575

(Received 9 September 2011; final revision received 9 February 2012;published 27 March 2012)

Synopsis

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 vc 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 vcs=R � L=W, where the fluid

viscosity sets the typical time for bubble rearrangements s. We show that this scaling can be under-

stood 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. VC 2012The Society of Rheology. [http://dx.doi.org/10.1122/1.3695152]

I. INTRODUCTION

The rheological behavior of complex fluids exhibits a rich array of phenomenon due

to their combination of solidlike and fluidlike properties [for example, see Bird et al.(1977); Edwards et al. (1991); Sollich et al. (1997)]. The response of these materials to

applied stresses and strains raises interesting questions in areas ranging from yielding

[Mason et al. (1996); Rouyer et al. (2005); Cohen et al. (2006); Møller et al. (2006,

2008); Xu and O’Hern (2006)] to shear localization [Coussot et al. (2002a, 2002b);

Salmon et al. (2003); Lauridsen et al. (2004); Clancy et al. (2006); Katgert et al. (2008)],

a)Author to whom correspondence should be addressed; electronic mail: mdennin@uci.edu

VC 2012 by The Society of Rheology, Inc.J. Rheol. 56(3), 527-541 May/June (2012) 0148-6055/2012/56(3)/527/15/$30.00 527

to creep flow [Vincent-Bonnieu et al. (2006)], to stick-slip behavior [Coussot et al.(2002a, 2002b); Lauridsen et al. (2002)] and jamming [Cates et al. (1998); Liu and Nagel

(1998, 2001); Trappe et al. (2001)]. The richness of their rheological response is one rea-

son for the wide range of technological applications. Complex fluids include pastes, slur-

ries, foams, granular materials, emulsions, and colloidal suspensions. Their applications

include fire fighting, printing, paints, foods, drugs and drug delivery, cosmetics, and oil

recovery. Therefore, understanding the fundamental elements of their flow behavior and

the transition between different flow regimes is critical.

One of the central issues in the understanding of complex fluids is the transition

between flow dominated fluidlike behavior and solidlike behavior. Recently, the study of

this transition has focused on shear geometries, such as cylindrical and planar Couette

flow, and cone and plate geometries. A major question in these geometries is the funda-

mental mechanisms governing shear localization in which flowing and nonflowing

regions of materials coexist [see Dennin (2008); Schall and van Hecke (2010), and the

references therein]. The coexistence of these two-states raises interesting questions in the

context of jamming in complex fluids, such as the possibility of a dynamic transition

between a “solid phase” and a “liquid phase” as a function of applied stress or strain.

Such a transition may best be described by constitutive relations that explicitly treat each

region of the material as existing in a separate phase. This is in contrast to the use of con-

stitutive relations that are continuous as a function of external parameters, such as a Bing-

ham plastic model. Given the wide range of shear localization that has been observed

experimentally [examples include the work in Kabla and Debregeas (2003); Coussot

et al. (2002a, 2002b); Salmon et al. (2003); Lauridsen et al. (2004); Gilbreth et al.(2006); Clancy et al. (2006); Becu et al. (2006); Katgert et al. (2008); Fall et al. (2009)]

and the success of various theoretical models of shear localization [examples include the

work in Varnik et al. (2003); Barry et al. (2011); Møller et al. (2008); Cox and Wyn

(2008); Cheddadi et al. (2008); Denkov et al. (2009)], understanding this transition

between the fluid and solid behaviors is critical. The wide range of behavior points to

another major question: To what degree is the fundamental physics governing the rheo-

logical properties of individual complex fluids general to all complex fluids. In address-

ing all of these questions, it has proven useful to focus on experimental systems that are

relatively simple and serve as models for general behavior in complex fluids.

Foams have proven to be a very important model complex fluid [Kraynik and Hansen

(1987); Kraynik (1988); Stavans (1993); Prud’homme and Khan (1996); Weaire and Hutzler

(1999)]. One of the main reasons is the relative simplicity of the microscale physics. Foams

are gas bubbles with liquid walls. In general, the main microscale forces are well understood

in terms of surface tension of the fluid and internal pressure of the bubbles. This has pro-

duced a range of successful theoretical models of foam that focus on different aspects of the

underlying physics [see, for example, Kraynik and Hansen (1986); Okuzono et al. (1993);

Durian (1995); Hutzler et al. (1995); Jiang and Glazier (1996); Brakke (1996); Asipauskas

et al. (2003); Cox (2005); Janiaud et al. (2006); Denkov et al. (2009)]. In this paper, we

focus on an experimental model foam system that is quasi two-dimensional: bubble rafts.

The bubble rafts are single layers of bubbles floating on the surface of water. They

have been of interest since Bragg first proposed them as a model for crystalline and amor-

phous systems [Bragg (1942); Bragg and Nye (1947)]. One of the advantages of the bub-

ble raft system is the ability to directly track the motion of the bubbles under applied

strains or stresses. Bubble rafts have been instrumental in studying a number of the phe-

nomena already discussed for complex fluids. In this paper, we focus on a different class

of deformations than the standard shear deformations: failure modes of a foam under

applied tension.

528 C. KUO and P. DENNIN

A significant difference between solids and fluids is their response to applied tension

that results in extension of the material. Typically, fluids exhibit an instability induced

pinch-off. The details of the pinch-off are dependent on the fundamental properties of the

fluid, such as surface tension and viscosity. However, the general behavior consists of the

shrinkage of a region of the fluid until it reaches a critical width and the material breaks

into one or more pieces [Shi et al. (1994); Eggers (1997)]. In contrast, most bulk solids

exhibit some form of fracture under applied tension [Bouchbinder et al. (2010); Broberg

(1999)]. Again, the details depend on the material and can include plastic or brittle frac-

tures. The general behavior involves cracks nucleating in the bulk or at the boundaries

and propagating through the material, breaking it into pieces. The question for foams,

and other complex fluids, is whether or not one or both of these failure mechanisms

occur, and if the dominant mechanism can be predicted.

For bubble rafts, initial studies reported in Arciniaga et al. (2011) established that both

pinch-off and fracture were possible under conditions of a constant pulling speed. A domi-

nant factor in determining which mode occurred was the speed at which the system was

pulled. However, the system size also played a role. In this paper, we focus on the transition

from pinch-off to fracture and demonstrate that this transition is controlled by the pulling

speed, systems size, and time for bubble rearrangements. We present a scaling argument for

the critical speed at which the system makes the transition from pinch-off to fracture.

II. EXPERIMENTAL METHODS

We utilize two experimental geometries in this study. First, the failure of the system

under tension is studied using a rectangular trough in which a pair of opposite walls are

pulled apart at constant speed using stepper-motors. A detailed description of the appara-

tus is provided in Arciniaga et al. (2011). The pulling velocities range from 0.01 to about

200 mm/s. The initial distance between the moving walls determines the initial length Lof the system. In our experiment, we use three initial lengths of the bubble raft: 40, 60,

and 80 mm. The maximum length is limited by the overall size of the trough, the pulling

speed, and the pulling distance required to generate failure. The walls are made of poly-

carbonate. The bubbles are observed to wet the walls. To fix the initial width, small

pieces of polycarbonate of the desired width are attached to the moving walls. This cre-

ates a step edge that the bubbles do not cross. Therefore, a discrete set of initial widths

from 40 to 120 mm with 20 mm steps were studied. The combinations of initial width

and length provide us with 15 distinct sets of initial geometries.

The bubble raft is formed by flowing compressed nitrogen gas through a needle under

a soap solution surface. The bubble size is fixed by the pressure, the needle diameter, and

the needle depth under the solution. We are able to manufacture both monodisperse and

polydisperse bubble rafts by this technique. The monodispersed bubble raft produces a

highly ordered crystalline 2D structure within well-defined domains that are separated by

grain boundaries. The polydispersed bubble raft produces a highly disordered amorphous

2D structure which is created by fixing the nitrogen pressure, the needle diameter, but

oscillating the needle depth under the solution.

The structure and the dynamics of the raft are recorded using a conventional charge-

coupled device (CCD) camera (Pixelink Corporation). Custom developed MATLAB programs

are used to perform the image processing to characterize the individual bubble size and fail-

ure mode. The images of the typical monodisperse and polydisperse bubble rafts are illus-

trated in Figs. 1(a) and 1(c). Figure 1(b) shows a 2D Fourier transform pattern indicative of

the ordered, hexagonal structure of the monodispersed bubble raft, whereas Fig. 1(d) shows

a 2D Fourier transform indicative of the amorphous structure of the polydisperse raft.

529BUBBLE RAFT FRACTURE UNDER TENSION

We define the characteristic bubble radius from the peak wavevector of the angular aver-

aged power Spectral density (PSD) of the 2D Fourier transform pattern (Fig. 2). The charac-

teristic bubble diameter is defined by the peak position of the PSD. The bubble size

distribution is defined by the full width at half maximum of the PSD peak. It is worth noting

that as expected, the PSD peak is broader in the polydisperse raft than in the monodisperse

raft. Upon expansion, we monitor the raft for motion out of the plane of the water surface,

and the bubbles are observed to remain confined to the air–water interface.

Unlike three-dimensional foam or bubbles confined between the plates, the fact that the

bubbles in a bubble raft float on the water surface allows for the existence of “voids” in the

raft. A void is a region of air between the bubbles where bare water surface is present. For

the polydisperse bubble raft, there can be initial voids due to the difference in bubble size.

These represent a negligible fraction of total area. However, given the existence of initial

voids, we use the evolution of the total void area in the bubble raft as a function of time, as

in Arciniaga et al. (2011), to distinguish between pinch-off and fracture. For pinch-off, void

area remains essentially zero throughout the process. Fracture is defined by a monotonic

increase in void area until failure is reached. For some regions of parameter space, there is a

mixed-mode where void area increases in time, but the voids “self-heal” (void area returns to

zero) before failure. For the purposes of this study, we focus on the onset of fracture and

define the critical velocity, vc, to be the velocity at which the void area is observed to

increase with time. The existence of the mixed-mode is accounted for in the error bars

reported for vc, as it does result in some uncertainty in determining the onset of fracture.

FIG. 1. 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.

530 C. KUO and P. DENNIN

The final experimental parameter is the viscosity of the solution used to generate the

bubbles. We focused on two different soap solutions: 5% Miracle Bubble, 15% glycerol,

and 80% deionized water; 5% Miracle Bubble, 32% glycerol, and 63% deionized water

by volume. The 32% glycerol solution has about twice the viscosity of the 15% solution.

One issue for the system is the relative impact of surface tension versus the bulk viscosity

of the fluid, as well as the surface viscosity of the surfactants at the interface. For the pull-

ing speeds considered here, the bubbles remain effectively spherical at all times. There-

fore, it is safe to ignore the surface tension effects. Additionally, the bubble rafts are a

very wet foam, so one expects the bulk viscosity to dominate T1 dynamics rather than

the surface viscosity [Durand and Stone (2006); Rognon and Gray (2009)].

The viscosity was varied as a method of impacting the T1 dynamics. A T1 event is a top-

ological rearrangement in a foam in which two neighboring bubbles move apart and the

space is filled by two bubbles that were previously next-nearest neighbors. We use oscilla-

tory flow to characterize the T1 dynamics. The bubble raft is formed between two parallel

plates, one of which is held fixed and the other is oscillated sinusoidally at constant ampli-

tude and frequency. For sufficiently large amplitudes and low frequencies, isolated T1

events are generated [see Lundberg et al. (2008) for details of the apparatus and the

dynamics of T1 events under oscillation]. The importance of T1 events in general for flow

is discussed in a number of references. Some examples are Weaire and Hutzler (1999), Den-

nin (2004), Cohen-Addad et al. (2004), Vaz and Cox (2005), Wang et al. (2007), Durand

and Stone (2006), and Cox and Wyn (2008). Our interest is on the typical bubble speed dur-

ing a T1 event. We define this as Vs ¼ Rs, where s is the duration of the T1 event and R is

the characteristic bubble radius. The duration is defined by the time between the two neigh-

bors losing contact and the new neighbors touching. For the data reported in this paper, the

T1 time is measured using an amplitude of 10 mm with an oscillation frequency of 0.05 Hz.

The average T1 time is measured in the amorphous bubble raft. We confirmed that the

typical time for a T1 event was independent of the oscillation amplitude and frequency.

III. SCALING BEHAVIOR

To derive a scaling law for the critical pulling speed, vc, we start with the assumption

that the T1 events control the crossover from plastic flow (pinch-off) to fracture.

FIG. 2. 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, d¼ 0.99 mm, with a distribution in size given by the full width at

half maximum of 0.05 mm and for polydisperse bubbles, d¼ 0.94 mm, with a distribution in size given by the

full width at half maximum of 0.16 mm.

531BUBBLE RAFT FRACTURE UNDER TENSION

The main premise is that the failure of a T1 event to complete generates a void in the raft

that is sufficiently large that the void is able to grow in time. This is based on the experi-

mental observation that the number of bubbles in the system is preserved. Therefore,

fracture is not caused by bubbles popping, and the only other option is bubbles losing

contact with their neighbors, as occurs in a failed T1 event. The argument presented here

is expected to apply to a polydisperse raft, for which grain boundaries are not significant.

The dynamics in monodisperse rafts is expected to be dominated by the motion of grain

boundaries and dislocations. In this case, there are similarities to the T1 dominated situa-

tion, in that the motions can be decomposed into chains of T1 events. However, the

motions are highly correlated, and this is expected to impact any scaling behavior.

Our proposed scaling law derives from three main ideas. First, we assume that it is

reasonable to define a typical velocity for bubbles during a T1 event, vs. If the external

driving force causes the bubbles to move apart faster than this typical speed, the net result

is that bubbles are unable to move into the space created from the bubbles moving apart.

This generates the failed T1 event that serves as the initiation of fracture. This process is

illustrated schematically in Fig. 3. Second, we assume that the typical velocity of a T1

event can be decomposed into two parts based on properties of the system: (1) the radius

of the bubbles (R) and (2) a time scale, s, that depends on the bulk viscosity (g) of the

fluid between the gas bubbles: vs � R=sðgÞ.1 Finally, the fact that the number of bubbles

in the system is preserved, and the bubble volumes are essentially constant implies that

the flow is area-preserving until voids are formed. Therefore, the divergence of the veloc-

ity field is zero, which means that

j @vx

@xj � j @vy

@yj: (1)

Taking the geometry defined in Fig. 3 (lower, left corner), we have the speed in the pull-

ing direction (vx), the speed perpendicular to pulling (vy), the initial length in the pulling

direction (L), and the initial width perpendicular to pulling (W). Therefore, the above

condition for incompressible flow gives the following relationship between vx and vy dur-

ing the initial stages of pulling the bubble raft apart,

vx

L� vy

W: (2)

For our geometry, vx is set by the pulling speed v, and vy is set by this relation. If fracture

occurs, it is in the early stage of pulling, which justifies using this relation to rewrite vy as

vy �vW

L: (3)

As discussed, we assume that voids occur when the externally imposed velocity scale is

greater than the intrinsic velocity scale for T1 events, vy � vs, which sets the critical pull-

ing velocity (vx ¼ vc) from

vy � vs �vcW

L(4)

1There may be weak dependence of s on R as well. However, as we will show, our data are consistent with the

assumption that s depends only on the viscosity g.

532 C. KUO and P. DENNIN

or

vc

vs� L

W: (5)

To test our assumption regarding the separation of vs in terms of the bubble radius and

the viscosity, we can rewrite Eq. (5) as

vcsR� L

W: (6)

Therefore, to fully test the elements of the proposed scaling behavior, we use two sepa-

rate experiments. First, we focus on the scaling of the critical velocity with bubble radius.

For these experiments, we use the same fluid solution to form the raft and construct rafts

with different initial geometries and bubble sizes. As we will show in the Results section,

the data are consistent with a scaling of vc

R � LW with a fixed relaxation time s. Second, we

vary the viscosity of the bubble forming solutions. To measure the typical T1 velocity,

we use the oscillatory experiments described in the Methods section. This allows us to

test the full scaling behavior presented in Eq. (5).

IV. RESULTS

Figure 4 shows a typical polydisperse [Figs. 4(a)–4(c)] and monodisperse [Figs. 4(d)–4(f)]

bubble rafts in its initial state [(a) and (d)] and at different states of failure. The initial raft

length and width are 60 and 80 mm, respectively, for both cases. The pulling velocities for

Fig. 4 are 2.57 mm/s for (f), 3.42 mm/s for (c) and (e), and 4.29 mm/s for (b). The images

illustrate a number of the qualitative features of the fracture and pinch-off processes. First, as

FIG. 3. 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.”

533BUBBLE RAFT FRACTURE UNDER TENSION

expected, fracture occurs for higher pulling speed. [This was already reported in Arciniaga

et al. (2011).] In general, the fracture is symmetric, as illustrated in the images, with two

voids growing on either side of the system [Figs. 4(b) and 4(e)]. However, there is some vari-

ation in the initial position of the fracture, and occasionally, we observe fracture in only one

side of the bubble raft. A detailed study of the fracture location will be the focus of future

work and requires significantly more statistics than is currently available. One open question

is whether or not the opening of voids is plastic or brittle fractures. The general shape is sug-

gestive of plastic fracture. But, there is also the question of whether or not it is better

described as a cavitation phenomena, instead of fracture. In this case, one of the key issues is

whether or not the void formation can be connected to a rapid drop in pressure within the

bubble raft. For the speeds studied here, the system remains well below the effective speed of

sound, and our expectation is that fracture is a better model. However, the exact nature of the

void opening will need to be studied in more detail.

The images also illustrate a significant difference between the monodisperse and poly-

disperse rafts, especially during pinch-off. The polydisperse raft experiences relatively

“smooth” plastic flow, due to individual T1 events. The net result is that the boundaries

of the system remain relatively smooth [Fig. 4(c)]. In contrast, the monodisperse systems

“flow” by two mechanisms: discrete motions of slip along grain boundaries and motion

of individual dislocations. The net result are relatively rough boundaries of the system

during pinch-off [Fig. 4(f)].

Figure 5 presents the critical velocity as a function of R, L, and W for bubble rafts gen-

erated from the same solution. Both the results for the monodisperse [Fig. 5(a)] and poly-

disperse [Fig. 5(b)] bubble rafts are presented. Three different characteristic bubble radii

of 0.35, 0.47, and 0.68 mm are used. The discrete initial geometries provide 15 data sets

for each bubble radius. As proposed, we scale the critical velocity by the characteristic

bubble radius, and the system size is presented as L/W. For the polydisperse system, we

FIG. 4. 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).

534 C. KUO and P. DENNIN

observed the proposed linear behavior. Furthermore, the collapse of the data justifies the

separation of the T1 velocity in terms of the bubble radius and a separate viscosity

dependent term. For the monodisperse bubble raft, the behavior is qualitatively different.

We observe a minimum value of L/W below which pinch-off does not occur. Also, the

dependence of vc on L/W is not linear. This suggests that the mechanism for fracture in

the monodisperse case is fundamentally different than the polydisperse case. A compari-

son between the critical velocity of polydisperse and monodisperse rafts is shown in

Fig. 6. In this case, we focus on the critical velocities as a function of the scaled initial

width (W/L) for the same data sets plotted in Fig. 5. The results in Fig. 6 demonstrate that

the critical velocity for the crystalline structure is lower than for the amorphous structure.

FIG. 5. vc=R vs L/W for different bubble sizes and structures. (a) is for polydisperse rafts and (b) is for monodis-

perse 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 vc=R vs W/Lplotted. The scaled critical velocities for polydisperse and monodisperse rafts collapse onto their respective mas-

ter 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.

FIG. 6. A direct comparison for vc=R 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.

535BUBBLE RAFT FRACTURE UNDER TENSION

It should be noted that scaling proposed here is different from that used in our original

work in Arciniaga et al. (2011). In Arciniaga et al. (2011), the focus was on the depend-

ence of the critical velocity on the system width W and comparisons were made with

other systems that exhibit transitions from fracture to pinch-off as a function of system

size (such as metallic glass systems). Therefore, the focus was the system width in terms

of the number of bubbles in the system (W/R), and not the mechanism for fracture. The

main result was that the critical velocity decreased with system width. The insets in

[Figs. 5(a) and 5(b)] illustrate that we still observe the same dependence of critical veloc-

ity on the initial raft width. Our current work is focused on the proposed T1 mechanism

and the role of vs; hence, the exploration of a different scaling behavior.

To further test the role of vs, we used solutions with different viscosities. A typical

plot of the critical velocity vs the initial width is shown in Fig. 7 with 15% and 32% glyc-

erol. The characteristic bubble radius is about 0.47 mm with a polydisperse bubble raft in

both measurements. The only difference between these two solutions is the viscosity.

To measure vs, we measure the characteristic T1 time as described in Sec. II. The T1

time corresponds to three stages of the processes: the detaching of the connected bubbles,

the intermediate state of an empty space surrounded by four nearby bubbles, and then the

attaching of the next-nearest neighbor bubbles. The whole T1 process can be observed in

the oscillatory shearing bubble raft with a low frequency and a large amplitude setting.

The characteristic bubble radius in this experiment is about 1.2 mm for both 15% and

32% glycerol solutions. We use a larger bubble size than in the previous pulling apart

experiment because this allows a more accurate T1 time measurement, and hence, a bet-

ter measure of vs for a given viscosity. The results for the distribution of T1 times is

shown in Fig. 8(a) with 15% glycerol and in (b) with 32% glycerol solutions. The charac-

teristic T1 time is defined by the average of the T1 time measurements. We find average

T1 times of 0.36 and 0.47 s, respectively, for 15% and 32% glycerol solutions. It should

be noted that this time is not directly proportional to the bulk viscosity. This is most

likely due to the fact that the bubbles are interacting to some degree with the subphase as

well as each other. It is know from shear flow experiments that dissipation from bounda-

ries can be relevant [Wang et al. (2006); Janiaud et al. (2006); Katgert et al. (2008)].

FIG. 7. 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.

536 C. KUO and P. DENNIN

Therefore, the detailed connection between viscosity and T1 velocity is presumably more

complicated than a simple linear relation. Therefore, we can calculate the T1 velocity

from the T1 time and the bubble radius.

Figure 9 shows the fully scaled critical velocity, vc=vs versus L/W, where we take vs

from our oscillatory measurements. In this case, we use a polydisperse bubble raft with

the range of different initial geometries (L and W) used in Fig. 5 and a characteristic

bubble radius of about 0.47 mm for both 15% and 32% solutions. As expected, the data

collapse for these two solutions. This result supports our scaling model which we

derived in Sec. III. At this point, we recognize that the use of only two different viscos-

ities is limited and future work will focus on exploring a wider range of viscosities and

bubble radii.

FIG. 8. 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 vis-

cosity is increased.

FIG. 9. Plot of vc=vs 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 vs is derived from the T1 time(s) which is measured by the oscillatory

experiment.

537BUBBLE RAFT FRACTURE UNDER TENSION

V. SUMMARY

By varying the initial size of the bubble rafts (length L and width W), the bubble radius

R, and the effective time for T1 events s, we have shown that for polydisperse rafts the

critical velocity vc for the transition from fracture to pinch-off is consistent with the rela-

tion: vc=vs � L=W, where it can be useful to separate vs into two parts and write

vcs=R � L=W. The derivation of this relation assumes that the T1 events control the

nucleation of voids, and the experimental agreement strongly suggests that T1 events

play this central role. Of particular note is the fact that the experimental results for the

polydisperse system are not only linear in L/W, but also extrapolate to vc ¼ 0 at L/W¼ 0,

which is not the case for the monodisperse system.

For monodisperse systems, the collapse of the data when vc is plotted versus L/W sug-

gests that the aspect ratio is the correct physical parameter for determining the crossover

from fracture to pinch-off. However, the detailed mechanism is clearly different from

that for the polydisperse case. Two interesting facts present themselves: (1) a minimum

value of L/W below which pinch-off is not observed and (2) consistently lower values for

vc compared to the polydisperse system. Our current assumption is that grain boundaries

are the source of crack nucleation in the monodisperse case based on our qualitative ob-

servation of the process. The existence of an onset for pinch-off as a function of L/W sug-

gests that for sufficiently wide systems, grain boundaries always open and nucleate

cracks before the system is able to narrow significantly and pinch-off. The lower vc rela-

tive to polydisperse systems suggests that grain boundaries present a lower nucleation

energy for crack formation relative to T1 events. Finally, there is the nonlinear scaling of

vc with L/W that requires explanation. Focusing on these issues in more detail will be the

subject of future work and will be an important step forward in our understanding of the

differences between crystalline and amorphous systems.

The success of our proposed scaling law for explaining the behavior in the polydisperse

case has implications for generic amorphous materials, such as molecular plastics. For mo-

lecular plastic materials, there is evidence that the failure under tension is controlled by the

dynamics of shear-transformation zones (STZs) [Spaepen (1977); Falk and Langer (1998);

Falk (1999); Falk and Langer (2011)], which are regions of nonaffine molecular rearrange-

ments. It is worth exploring if the our arguments regarding the connection between fracture

and T1 events carries over to a connection between failure modes and STZs in molecular

systems. Because a T1 event is a specific, highly local event involving four bubbles and the

films between them, and an STZ is a region (though relatively local) of nonaffine motion,

the exact correspondence between the two types of rearrangements is still an open question.

There is evidence from experiments in bubble rafts under shear that the T1 events represent

the core of STZs [Dennin (2004); Twardos and Dennin (2005); Lundberg et al. (2008)].

The combination of the results presented here and the previous results for shear geometries

strongly suggests that a better understanding of the connection between STZs and T1 dy-

namics will be important in understanding the failure of complex fluids under tension, and

possible extensions to molecular systems.

Another issue that requires additional study is the scaling with the typical T1 velocity,

vs. At this point, we have only tested two different values of this parameter, so further

work is needed in this direction. Especially of interest is the simplification of treating vs

in terms of R and a radius independent time scale s associated with T1 events. It is con-

ceivable that the scaling is based on other typical velocities formed by the ratio of R and

another time scale. Uniquely, distinguishing between the time scale for T1 events and

other time scales requires additional work. However, at this point, the results are certainly

consistent with the T1 velocity as the relevant scaling parameter.

538 C. KUO and P. DENNIN

Finally, it is worth noting that the failure of foams under over-pressure has also been

studied [Arif et al. (2010)] with a focus on crack propagation. In this geometry, one also

observes a velocity dependent transition between different failure modes. In this case, a

transition from plastic deformation and brittle fracture is reported in Arif et al. (2010).

Though there are important difference in the geometry and underlying microscopic

mechanisms (such as film rupture) between the results reported here and in Arif et al.(2010), it is interesting to speculate on the potential connections between the different

failure modes and the implications for complex fluids in general.

ACKNOWLEDGMENTS

The authors acknowledge the support of NSF-DMR-0907212 and Research Corpora-

tion. The authors also thank C.O., S.C., P.T., and S.H. for useful discussions.

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