Comparative study of the dynamics of laser and acoustically generated bubbles in viscoelastic media

PHYSICAL REVIEW E

, 043103 (2019)

Chad T. Wilson,

Timothy L. Hall,

Eric Johnsen,

Lauren Mancia,

Mauro Rodriguez,

Jonathan E. Lundt,

Tim Colonius,

David L. Henann,

Christian Franck,

Zhen Xu,

and Jonathan R. Sukovich

†

Department of Biomedical Engineering, University of Michigan, Ann Arbor, Michigan 48105, USA

Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48105, USA

Department of Mechanical Engineering, California Institute of Technology, Pasadena, California, 91125, USA

Department of Mechanical Engineering, Brown University, Providence, Rhode Island 02912, USA

Department of Mechanical Engineering, University of Wisconsin, Madison, Wisconsin 53706, USA

(Received 27 August 2018; revised manuscript received 24 January 2019; published 10 April 2019)

Experimental observations of the growth and collapse of acoustically and laser-nucleated single bubbles in

water and agarose gels of varying stiffness are presented. The maximum radii of generated bubbles decreased

as the stiffness of the media increased for both nucleation modalities, but the maximum radii of laser-nucleated

bubbles decreased more rapidly than acoustically nucleated bubbles as the gel stiffness increased. For water and

low stiffness gels, the collapse times were well predicted by a Rayleigh cavity, but bubbles collapsed faster than

predicted in the higher stiffness gels. The growth and collapse phases occurred symmetrically (in time) about

the maximum radius in water but not in gels, where the duration of the growth phase decreased more than the

collapse phase as gel stiffness increased. Numerical simulations of the bubble dynamics in viscoelastic media

showed varying degrees of success in accurately predicting the observations.

DOI:

10.1103/PhysRevE.99.043103

I. INTRODUCTION

Cavitation is known to generate complex, often violent,

conditions within and surrounding bubbles and has been

widely studied for a range of associated phenomena [

–

]

and applications [

–

]. Recently, there has been a surge

in interest in cavitation in the medical community where, for

example, cavitation activity is suspected of contributing to

traumatic brain injury associated with exposure to blasts and

impacts [

]. Controlled, acoustically generated cavitation

is also being explored for therapeutic

surgical applications

using a technique known as histotripsy. Histotripsy utilizes

short-duration (

s), high-amplitude (

−

15 MPa)

focused acoustic pulses to controllably generate near-vacuum

microbubbles in tissues, which act to mechanically fractionate

and destroy a wide range of tissues [

–

Despite recent medical interests, however, the mechanisms

by which cavitation damages tissue remain poorly understood

[

]. The susceptibility to cavitation-induced damage can

vary significantly depending on the tissues’ mechanical prop-

erties [

]. Modeling efforts that take into account the vis-

coelastic properties of tissues and tissue-mimicking hydrogels

have shown varying degrees of success in replicating obser-

vations [

–

]. The inability of models to fully replicate

observations is due, in part, to the incomplete characterization

of materials’ viscoelastic properties [

–

], especially when

materials are subjected to the high to ultrahigh strain-rates

(10

–10

−

) expected during cavitation. To the best of our

knowledge, experimental techniques to directly measure the

viscoelastic properties of materials subjected to these types

zhenx@umich.edu

†

jsukes@umich.edu

of strain-rates do not exist, but observations suggest that they

may vary significantly from those measured at quasistatic

strain rates [

Experimental investigations of bubble dynamics in aque-

ous and viscoelastic media have typically relied on two

methods, acoustic- and laser-induced cavitation, to nucleate

bubbles. While it may be reasonable to expect that acous-

tically nucleated bubbles would be more representative of

cavitation in biological contexts, e.g., during brain injury, a

number of challenges have impeded the controlled study of

these bubbles in viscoelastic media. In particular, generating

single, spherical bubbles at precisely controlled locations

using transient acoustic pulses [

] is difficult to accomplish

without modifying the media, for example, by injecting gas

microbubbles or inserting point defects in the media [

As such, experimental studies of bubble dynamics in

these media have generally been conducted using laser-based

mechanisms for nucleation as these methods provide pre-

cise temporal and spatial control over cavitation generation

[

–

]. However, the extent to which the dynamics of

laser-generated bubbles represent those of acoustically gen-

erated bubbles is unclear owing to the nucleation mech-

anism, whereby ionization events may locally modify the

material properties of the surrounding media and can lead

to the generation of excess vapor and gas byproducts within

these bubbles which may thus influence collapse outcomes

[

]. Further, due to the complex physics underlying

the formation and growth of laser-nucleated bubbles [

–

and the broader interest in bubble collapse and rebound events

in general, models of the initial growth of bubbles remain

underdeveloped.

We are thus motivated to perform a comparative study

of the dynamics of both acoustically and laser-generated

2470-0045/2019/99(4)/043103(10)

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CHAD T. WILSON

et al.

PHYSICAL REVIEW E

, 043103 (2019)

FIG. 1. Schematic drawing of the experimental setup, top-down

view. Gel samples were lowered into the transducer from above (into

the page).

cavitation bubbles, with the goal of providing insight into how

bubble dynamics are affected by the viscoelastic properties

of the media, and whether and

or how the dynamics are

affected by nucleation mechanism. Experimental observations

of the initial growth and first collapse of both acoustically

and laser-nucleated single bubbles are presented for water

and for tissue-mimicking agarose gels of varying stiffness.

Numerical simulations of the bubble dynamics in viscoelastic

media are then compared with the observations in order to

assess their efficacy in reproducing the experimental results,

and to determine the extent to which laser-generated bubbles

may serve as surrogates for acoustically nucleated ones.

II. METHODS

A. Experimental setup

Experiments were carried out in a custom-built, 10-cm-

diameter, open-topped, spherical acoustic array, populated

with 16, 2-cm-diameter, focused transducer elements with a

center frequency of 1 MHz. Two pairs of 25-mm-diameter

optical windows were placed along the equatorial plane of

the transducer for illumination and imaging, and a 50-mm-

diameter laser access window was included for laser nucle-

ation. A 5

8-cm-diameter opening at the top of the transducer

allowed for gel sample insertion. A schematic drawing of the

experimental setup is shown in Fig.

During experiments single bubbles were generated at the

center of the sphere using two methods, laser and acoustic

nucleation. Laser-nucleated bubbles were generated using a

pulsed Nd:YAG laser (Continuum, Surelite I), frequency dou-

bled to 532 nm with a pulse duration of 6 ns. The beam of

the laser was expanded to a diameter of 40 mm and focused

to the center of the sphere using a 75-mm focal length lens.

Acoustically generated bubbles were nucleated using a 1.5-

cycle acoustic pulse containing only a single, large rarefac-

tional pressure half-cycle. As both nucleation methods rely on

threshold phenomena to generate cavitation, the volume of the

field exposed to super-threshold irradiance

pressure capable

of generating bubbles increases with the intensity of the gen-

erating pulses. Instead of generating larger single bubbles this

preferentially generates multiple bubbles in the focal region.

To minimize the generation of multiple bubbles, the irradi-

TABLE I. Mechanical properties of media used in this study as

given in Ref. [

Young’s modulus

Density

Medium

(kPa)

(kg

)

Water content (%)

Water

–

998

100

0.3% Agarose

1003.0

98.8

1.0% Agarose

1010.0

98.1

2.5% Agarose

242

1025.0

96.7

5.0% Agarose

570

1050.0

94.3

ance and rarefactional pressure thresholds for nucleation were

therefore determined empirically prior to experiments. This

was done by adjusting the respective laser pulse energy and

acoustic focal pressure such that the probability of generating

cavitation during a given attempt was approximately 50%.

For laser nucleation this resulted in a pulse energy of 5 mJ

per pulse, and for acoustic nucleation the rarefactional focal

pressure was approximately

−

24 MPa [

During all experiments the transducer was filled with

deionized water, filtered to 2

m and degassed to 4 kPa. Bub-

bles were generated by both nucleation mechanisms in water

and agarose gels with concentrations of 0.3%, 1.0%, 2.5%,

and 5.0% (weight

volume). The gels were prepared following

a modified set of the procedures detailed in Ref. [

], where

in the present study the gels were allowed to solidify at room

temperature (17

◦

C) instead of at 4

◦

C. The impact of

this modification on the resultant stiffness values of the gels in

the present study compared to those in Ref. [

] is expected to

5% [

]. The mechanical properties of the gels as given

in Ref. [

] are shown in Table

. It should be noted that the

properties of the gels in Ref. [

] were measured under quasi-

static loading conditions using a parallel-plate rheometer. Gel

samples were prepared in 7

5 cm long, cylindrical syringes

with diameters of 2

5 cm, following which the samples were

mounted to a positioning system, submerged in the water

through the opening on top of the sphere, and positioned at

the focal spot of the laser

acoustic array for nucleation. To

ensure that the dynamics of the generated bubbles were not

influenced by local defects in the gels, i.e., structural changes

due to previously generated cavitation events, only a single

nucleation event was generated at each focal site within the

gels, and all bubbles were nucleated

(

5mm)from

any previously generated bubbles in the gel sample. All gel

samples had specific acoustic impedances within

5% of that

of water’s and

or were physically large enough to be regarded

as infinite with respect to the nucleated bubbles.

Images of the generated bubbles were captured using a

high-speed camera (Vision Research, Phantom v2012) at a

fixed frame rate of 400 kHz. From the time point of nu-

cleation, all image series extended

100

s. Images were

illuminated using a pulsed, blue LED backlight source with

a minimum flash width of 20 ns. To capture the high-speed

dynamics associated with growth and collapse, a multi-flash-

per-camera-exposure imaging technique was used. During

exposures near the initial growth and first collapse, the LED

backlight was pulsed two to four times per single camera

exposure. A schematic drawing of this imaging technique,

with example images, is shown in Fig.

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...

Exposure 1

Exposure M

Exposure N

Flash Sequence M

...

Flash Sequence 1

Flash Sequence N

Time

FIG. 2. Visualization of the multiflash imaging technique with

timing diagrams of the variable flash sequences used during each

phase of the bubble’s evolution with example images.

Flash timings could be adjusted independently of the cam-

era exposure and spacings between individual flashes within

single image frames ranged from 100 ns to 1

s, depending

on the experiment. This allowed the growth and collapse

events to be acquired with greater temporal resolution than

was possible with the native camera settings. This imaging

technique resulted in nested, concentric bubbles in images

and brightness thresholding and edge detection were used to

differentiate them. The radii of the captured bubbles were cal-

culated by least-squares circle fitting to their detected bound-

aries. Bubbles were rejected for inclusion if the variance in

the radii of the individual points detected along the boundary

exceeded that which would be expected for fluctuations about

the fit radius of more than 5% in any frame of the image

series except those captured within

s of the nucleation

event or collapse point. These frames were excluded from

consideration for rejecting bubbles owing to the small size

of the bubbles at these time points and

or interference from

the laser plasma in images. Using the multiflash imaging

technique it was not always possible to reliably determine

which detected bubble corresponded to which flash in a given

exposure, especially near the collapse point where the bubble

wall velocity changes direction. Radii with indeterminate time

points were thus excluded from analysis.

B. Theoretical modeling

The dynamics of a single isolated, spherical, gas-filled

bubble in both water and in homogeneous viscoelastic media

(representative of the agarose gels) are numerically simulated

using an approach similar to previous studies of histotripsy

cavitation [

–

]. In the present study, only acous-

tically nucleated bubbles are modeled. This is because the

physics of laser nucleation is more complex than acoustic

nucleation, involving plasma formation and recombination as

well as rupture of the gel material adjacent to the nucleation

site–developing a new model for the initial growth of laser-

nucleated bubbles is beyond the scope of the present work.

In the model described in this section, there are two im-

portant unknowns: the bubble’s initial size and the material

TABLE II. Physical constants used in simulations.

Parameter

Value

∞

1497 m

∞

1000 kg

∞

101

325 kPa

∞

◦

072 N

w

001 Pa

115 Pa

−

165

−

1.4

55 W

181

kgK

properties of the surroundings. It is not possible to uniquely

extract both of these unknowns by fitting to the experimental

data. Therefore, for simplicity, we assume that the stiffnesses

of the gels measured under quasistatic conditions (Table

)

are representative of the values at high strain-rates relevant to

bubble dynamics. The reported values of agarose’s viscosity

vary widely over several orders of magnitude [

] and

cannot be measured under conditions relevant to the present

study using currently available techniques [

]. As such, the

viscosity of agarose used in simulations is chosen based on

the criterion that it be on the same order of magnitude as

values reported previously [

], but is otherwise arbitrary.

We acknowledge that these simplifications can contribute to

discrepancies between simulations and experiments. A more

detailed exploration of the simulation parameter space in

relation to the present work will be presented in a future study.

Two sets of simulations are carried out using the specified

material properties: one where the gel viscosity is taken to

be equal to that of water’s [

w

, and another where

it is taken to be the prescribed viscosity for agarose,

(Table

). With the material properties specified, we then

calibrate the bubble’s unknown initial size by comparing with

the experimental data.

To account for near-field compressibility effects, radial

bubble dynamics are described by the Keller-Miksis equation

[

] extended to include elasticity [

(

−

∞

)

(

−

∞

)

∞

(

∞

)

{

−

[

∞

(

)]

−

}

(1)

where

is the time-dependent bubble radius,

∞

and

∞

are the constant sound speed and density of the surrounding

medium, respectively,

is the internal bubble pressure,

the surface tension, and

is the integral of deviatoric stresses

in the surroundings [Eq. (

)]. All constants correspond to wa-

ter or air at a temperature of 25

◦

C (Table

) unless otherwise

specified.

Acoustic excitation is modeled as a time varying far-field

pressure acting on the bubble. The far-field pressure is the

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CHAD T. WILSON

et al.

PHYSICAL REVIEW E

, 043103 (2019)

sum of the ambient pressure,

∞

, and time-varying acoustic

forcing,

(

)

{

(

cos[

(

−

)]

)

−

(2)

The pressure amplitude,

=−

24 MPa, and frequency,

1MHz (

), correspond to experimental measure-

ments. The time delay,

s, and fitting parameter,

7, were chosen as in previous studies [

]. Here,

it is hypothesized that the largest-amplitude cycle of the

waveform gives rise to the bubble growth.

Agarose is modeled with a finite-deformation Kelvin-Voigt

constitutive equation [

] in which the elastic component

of the material response is given through the Neo-Hookean

model, resulting in the following integral of the deviatoric

contribution of the stresses in the surrounding medium:

=−

−

[

−

(

)

−

(

)

]

(3)

where

is the viscosity and

is the ground-state Young’s

modulus. As previously discussed, simulations use the quasi-

static Young’s modulus reported for each agarose gel con-

centration (Table

) and are carried out using the viscosity

of water,

w

, in all media and additionally with the

prescribed viscosity of agarose,

, in the gels. In Eq. (

is the bubble radius when the surroundings are stress-free,

which, in the present work, we take to represent the initial

bubble radius.

Experimentally inferred cavitation nuclei in water are on

the order of nanometers [

], but direct measurements are not

feasible on this scale. For these simulations, the initial radii of

the bubbles in each medium are empirically determined in an

iterative fashion by adjusting the initial radii in simulations in

order to produce the best agreement between the simulated

maximum radii and those measured experimentally from a

representative set of cavitation events in each medium. The

representative data sets for each medium are defined as those

which have maximum radii closest to the mean maximum

radius of all bubbles measured in the given medium.

Heat transfer effects are incorporated by solving for tem-

perature fields inside and outside of the bubble following

the approaches of Refs. [

–

]. These simulations take

the bubble wall to be impervious to gas and neglect va-

por inside the bubble. While this simplification risks under-

predicting the lifespan of the bubble and neglects subsequent

rebounds [

], this error is expected to be minor when consid-

ering the single cycle of bubble growth and collapse presented

herein. The time derivative of the internal bubble pressure,

, couples the Keller-Miksis equation [Eq. (

)] to the energy

equation for air inside the microbubble:

(

−

∂

∣

−

)

(4)

−

[

∂

(

−

∂

−

)

∂

]

∂

(

∂

)

(5)

In the above equations,

(

) is the temperature field of

air inside the microbubble. Air in the bubble is treated as an

ideal gas with a ratio of specific heats

, and its thermal con-

ductivity is given by

, with empirical constants

and

[

] listed in Table

The energy equation outside the bubble is given by

∂

∇

∞

(

)

(6)

where

(

) is the temperature field in the surrounding

medium. Equation (

) uses the specific heat,

, thermal

diffusivity,

(

∞

), and thermal conductivity,

of water. The final term on the right side of Eq. (

) represents

the dissipation due to viscous stresses.

Boundary conditions are prescribed for the center of the

bubble and far from the bubble:

∇

0at

0 and

→

∞

→

, where

∞

is the ambient temperature

of the medium and

is the arbitrarily-large outer-

boundary of the domain. Boundary conditions at the bubble-

material interface relate the internal bubble temperature to

the temperature field in the surrounding medium,

(

and

∂

. Finally, sum-

marizing the initial conditions,

(

∞

(

)

∞

, and

(

)

∞

. A more detailed treat-

ment of the derivation and numerical implementation of this

model can be found in Ref. [

III. RESULTS

The multiflash imaging technique allowed for clear differ-

entiation of bubbles at different time points within the same

exposure, effectively allowing higher frame-rate imaging of

the bubbles, particularly in the high velocity regimes asso-

ciated with growth and collapse. The utility of this imaging

technique was more limited in the lower velocity regimes

near the maximum radius, where the change in radius of a

bubble between consecutive flashes is smaller. A series of

images captured using this technique, showing the growth

and first collapse of single bubbles generated via laser and

acoustic nucleation in water, 0.3%, and 2.5% agarose gels,

are shown in Fig.

. Note, the bright regions in the centers of

laser-nucleated bubbles in the first frames are light emissions

generated by the optical breakdown process responsible for

nucleation.

The maximum radii of both acoustically and laser-

nucleated bubbles are observed to be the largest in water and

to decrease monotonically in the gels as a function of increas-

ing stiffness (Fig.

). However, while bubbles nucleated in

water by both mechanisms are larger than those nucleated in

the gels, a clear nucleation-mechanism-dependent difference

in the relative decrease in maximum radius is observed. That

is to say, the maximum radii of laser-nucleated bubbles de-

creases by 15% in going from water to 0.3% gel, compared

to a 3% decrease for bubbles nucleated acoustically. While

more data are required to characterize the governing decay

behaviors, the maximum radii of the bubbles nucleated in gels

appear to decrease in an approximately power-law fashion

with increasing gel stiffness.

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t=0μs

t=2.5μs

t=5μs

t=50μs

t=52.5μs

t=55μs

t=0μs

t=2.5μs

t=5μs

t=15μs

t=25μs

t=27.5μs

t=30μs

250μm

†

†,*

†

(a)

t=0μs

t=2.5μs

t=5μs

t=22.5μs

t=40μs

t=42.5μs

t=45μs

t=0μs

t=2.5μs

t=5μs

t=15μs

t=25μs

t=27.5μs

t=30μs

250μm

††

†

†,*

†

(b)

t=0μs

t=2.5μs

t=5μs

t=7.5μs

t=10μs

t=12.5μs

t=15μs

t=0μs

t=2.5μs

t=5μs

t=7.5μs

t=10μs

t=12.5μs

t=15μs

250μm

†

(c)

FIG. 3. Example images showing single bubbles generated using laser and acoustic nucleation mechanisms in: (a) water, (b) 0.3% agarose

gel (

13 kPa), and (c) 2.5% agarose gel (

242 kPa). Asterisks mark frames outside of the bubbles primary lifetime which were not

included in the analysis of the data. Crosses mark frames containing multiple flashes from the light source. The horizontal stripes observed in

images are camera artifacts.

Although the laser-nucleated bubbles are observed to be

larger than the acoustically nucleated bubbles in all media,

this is not believed to be intrinsically related to the mode

of nucleation. Instead, this is likely a consequence of the

fixed acoustic frequency of the transducer and the fixed focal

profile of the laser, which could not be changed during these

experiments and which imposed restrictions on the amount of

energy that could be delivered to the focus without generating

additional bubbles in the field, as described in Sec.

II A

.As

a result, the maximum radii of the bubbles could not be

controlled or adjusted in this study, however, modifications

to the laser beam profile or acoustic frequency are expected

in general to allow the maximum radii to be controlled. It

should also be pointed out here that the inability to control

the maximum radii of the generated bubbles prevented the

acquisition of meaningful data for the acoustically nucleated

bubbles in the 5% gel which, although they could be gener-

ated, had lifespans shorter than the duration of two camera

frames (

s). Hence, their dynamics and maximum radii

could not be accurately assessed.

While the rebound dynamics of bubbles was not within

the scope of the present study, an important nucleation-

mechanism-dependent difference in the decay lifetimes of

the bubbles nucleated in the gels was observed. That is to

say, although the acoustically nucleated bubbles were gener-

ally observed to rebound following their first collapse, they

were typically seen to dissolve after

3 rebounds (typically

s). In contrast, remnant nuclei of the laser-nucleated

bubbles were observed to persist in the field for up to several

seconds following collapse. Evidence to this effect may be

043103-5