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Nonspherical laser-induced cavitation bubbles

Kang Yuan Lim,1Pedro A. Quinto-Su,1Evert Klaseboer,2Boo Cheong Khoo,3Vasan Venugopalan,4and Claus-Dieter Ohl1

1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University,

21 Nanyang Link, Singapore 637371, Singapore

2Institute of High Performance Computing, Fusionopolis, 1 Fusionopolis Way, #16-16 Connexis, Singapore 138632, Singapore

3Department of Mechanical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore

4Department of Chemical Engineering and Materials Science and Laser Microbeam and Medical Program, Beckman Laser Institute,

University of California, Irvine, California 92697-2575, USA

?Received 11 October 2009; published 14 January 2010?

The generation of arbitrarily shaped nonspherical laser-induced cavitation bubbles is demonstrated with a

optical technique. The nonspherical bubbles are formed using laser intensity patterns shaped by a spatial light

modulator using linear absorption inside a liquid gap with a thickness of 40 ?m. In particular we demonstrate

the dynamics of elliptic, toroidal, square, and V-shaped bubbles. The bubble dynamics is recorded with a

high-speed camera at framing rates of up to 300 000 frames per second. The observed bubble evolution is

compared to predictions from an axisymmetric boundary element simulation which provides good qualitative

agreement. Interesting dynamic features that are observed in both the experiment and simulation include the

inversion of the major and minor axis for elliptical bubbles, the rotation of the shape for square bubbles, and

the formation of a unidirectional jet for V-shaped bubbles. Further we demonstrate that specific bubble shapes

can either be formed directly through the intensity distribution of a single laser focus, or indirectly using

secondary bubbles that either confine the central bubble or coalesce with the main bubble. The former approach

provides the ability to generate in principle any complex bubble geometry.

DOI: 10.1103/PhysRevE.81.016308PACS number?s?: 47.55.dp, 47.55.dd, 47.61.Jd

I. INTRODUCTION

Short-lived vapor bubbles created by focusing pulsed-

laser light have demonstrated exciting properties for the ac-

tuation of fast flows in microfluidics and the manipulation of

small objects. These laser-induced cavitation bubbles can be

utilized for the lysis of cells ?1–4?, the creation of pores in

cell membranes ?5?, for microfluidic operations such as

switching ?6?, pumping, and mixing ?7,8?. Due to the very

fast bubble dynamics, fluid actuation on the microsecond

time scale and the creation of very high levels of normal and

shear stresses can be achieved. In contrast conventional mi-

crofluidic operating conditions are typically in the low

Reynolds number regime.

Two physical mechanisms for bubble creation with lasers

can be identified: heating and vaporization of the fluid with

and without stress confinement ?9? due to linear absorption

and optical breakdown ?10–12?. In the case of stress confine-

ment, the laser energy is absorbed in the liquid and the cavi-

tation bubble results from an explosive vaporization. In op-

tical breakdown, a plasma is formed at the focal volume of

the focused laser pulse and drives the early stage of bubble

expansion. Generally, the initial shape of the cavitation

bubble and its induced flow field is determined by the spatial

distribution of the injected energy, i.e., the focal volume of

the laser irradiation. Only at later times do the boundaries of

the fluid domain affect the bubble dynamics.

The generic example of a boundary induced bubble shape

is the single laser-induced bubble at a variable distance from

a rigid boundary ?13?. Depending on the distance of the

bubble to the boundary the bubble may obtain a spherical, an

approximate oblate spheroidal, or a hemispherical shape at

maximum expansion. During shrinkage, the so-called col-

lapse, the bubble develops a jetting flow toward the bound-

ary. Nonspherical, elongated bubbles have also been ob-

served as an effect of weak focusing, in particular for

subnanosecond lasers pulses using low numerical aperture

focusing lenses, leading to a filamentous optical breakdown

structure ?10,14?. In contrast, here we deliberately modify

and control the shape of the focused laser pulse such as to

induce a nonspherical bubble shape from the beginning of

the bubble dynamics. These shaped bubbles then lead to a

nonradial flow with novel properties which may be ex-

ploited. For example microrheology depends on the specific

forcing of the object under test. This may be achieved

through purely extensional or compressional flows. Although

we do not study the properties of each of the new bubble

shapes, the toroidal bubble is a prospective candidate to gen-

erate a purely compressional flow.

We demonstrate here how to extend the current tech-

niques and to create nonradial flows. Our long-term goal is to

create on-demand flow patterns from well controlled bubble

shapes and arrangements for pulsed-laser only controlled

microfluidic actuation.

We have demonstrated that complex bubble patterns can

be generated using a holographic element ?15?. For these

devices the intensity distribution is related to the pattern dis-

played on the spatial light modulator ?SLM?—acting as a

phase object—by a Fourier transform. In previous works

multiple circular foci were generated to create arbitrary ar-

rangements of spherical bubbles. Here, we report on deliber-

ately changing the intensity distribution to create nonspheri-

cal bubbles and study their dynamics using high-speed

photography. Additionally, we compare the bubble dynamics

with the solution from an axisymmetric potential flow solver

?16,17?.

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This work is organized as follows: we begin with a pre-

sentation of the experimental setup to create and observe the

bubbles and a brief description of the numerical method.

Then, we present a series of nonspherical bubbles, e.g., el-

liptic, toroidal, and V-shaped bubbles, created by shaping the

focused intensity distribution of the laser into a line, circle,

and v-shape, respectively. Next, we demonstrate that non-

spherical, e.g., square bubbles, can also be generated through

“secondary bubbles” ?patterns with multiple foci? with and

without bubble coalescence. We discuss several aspects of

the observed dynamics such as eccentricity, jetting, and

potential applications.

II. EXPERIMENTAL SETUP

In our experimental setup the bubbles are created through

linear absorption of pulsed-laser light which leads to a super-

heated region which eventually explodes into a vaporous

cavity ?18?. The laser beam is focused through a microscope

objective ?20?, NA 0.7? at the bottom of a chamber filled

with the absorbing liquid.

To create nonspherical bubbles the light intensity distribu-

tion at the focal plane is shaped with a spatial light modula-

tor ?SLM, model LC-R-2500, Holoeye Photonics, Germany?.

The SLM is a two-dimensional reflective array where each

pixel acts as a variable phase plate which can be individually

addressed by a computer. Illuminating the SLM with a col-

limated beam and focusing the reflected light with a lens

leads to an intensity distribution at the lens’ focal plane

which is related to the SLM pattern through a Fourier trans-

form ?15,19?.

Figure 1 shows a schematic of the experimental setup: a

frequency doubled Nd:YAG laser ?New Wave, Fremont, CA?

emits pulses with a temporal width of 6 ns at a wavelength of

?=532 nm. The polarization of the laser pulse is rotated

with a half wave plate ?indicated a ?/2 in Fig. 1?. A tele-

scope formed by the lenses L1and L2expands the beam to

cover the surface of the SLM. In order to remove the zero-

order undiffracted part of the beam ?15?, a lens phase is

added to the hologram displayed at the SLM. Additionally a

third lens L3images the SLM onto the back aperture of the

microscope objective ?20?, NA 0.7? mounted on an inverted

microscope ?Olympus IX-71?. The pattern is projected at the

bottom of the liquid gap. The imaging is done using the same

objective that is used to deliver the beam into the sample

?15?. The sample is illuminated with the microscope con-

denser and the dynamics of the bubbles are recorded with a

high-speed camera ?Photron SA1, San Diego, CA? at frame

rates of up to 300 000 frames per second.

The liquid film is confined by two No. 1 microscope cov-

erslips separated with spacers ?made of aluminum foil, inset

Fig. 1?. The size of the chamber is 20?20 mm2laterally

and has a height of 40 ?m. We are using yellow inkjet

printer ink ?Maxtec Inc., Hongkong? that absorbs the green

light from the laser and is sufficiently transparent for the

illumination light from the microscope condenser to obtain

high framing speeds. The aqueous ink has a density ?

=1046 kg/m3

anda dynamic

?10−3Pa s at room temperature.

viscosity of

?=2.14

III. SIMULATIONS

To obtain more insight into the fluid flow we compare the

experimental high-speed recordings with simulations using a

boundary element method ?BEM? for potential flow basically

similar to those used in ?16,17,20?. More information on the

use of boundary element methods for bubble dynamics can

be found in ?21–23?. In using this model, we neglect viscous

effects in the fluid, at the rigid boundaries and the bubble

interface. It can be argued that viscous effects from boundary

layers may not be important for sufficiently short times; their

growth is governed by a diffusive time scale, ??y2/?, where

y is a characteristic length and the ? is the kinematic viscos-

ity. If we identify y with half the channel height, i.e., 20 ?m

and the kinematic viscosity ?=?/??2?10−6m2s−1the

diffusive time of boundary layer growth is of the order of 0.2

ms, about ten times longer than the oscillation period of the

bubble.

For potential flow, if the velocity vector, u, anywhere in

the flow and the potential ? are related as u=?? and fur-

thermore the Laplace equation holds:?2?=0 in the fluid do-

main. The boundary element method makes use of the fact

that if the potential ?Dirichlet condition? is given on the

bubble surface S, the solution of the Laplace equation is

unique. That is, the normal derivative of the potential,

??/?n, ?Neumann condition with ?/?n=n·? the normal in-

ward derivative at the boundary? is unique. The vector n is

the normal vector on S and is directed out of the fluid. S is

assumed to be the only surface for this particular problem.

The quantity ??/?n is also equal to the normal velocity at

the surface. The BEM method exploits this by using a

boundary integral formulation as

FIG. 1. ?Color online? Experimental setup. Cavitation bubbles

are generated with a pulsed laser reflected from a digital spatial

light modulator ?SLM? which is focused after passing through sev-

eral optics into a thin liquid film with a 20? microscope objective.

The liquid film is contained in a chamber of 40 ?m height which is

filled with a light absorbing aqueous yellow ink. The fast bubble

dynamics is studied with a high-speed camera.

LIM et al.

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c?x???x? +?

S

??y??G?y,x?

?n

dS =?

S

G?y,x????y?

?n

dS.

?1?

in which x is a fixed point and y the integration variable both

located on S and c?x? is the solid angle at location x. G

is the Green function or kernel function defined in a

three-dimensional ?or axial symmetric? domain as G?y,x?

=1/?x−y?. Equation ?1? provides a means to calculate the

normal velocity at the bubble surface if the potential is

known. This is done by dividing the bubble surface in ?ini-

tially equally spaced? nodes connected by elements and

transforming Eq. ?1? into a matrix—vector equation, which

can be solved for the unknown velocities. The pressure of the

bubble for a time-dependant problem applicable at the

bubble interface equals ?using the Bernoulli equation?,

p = pref− ?D?

Dt

+1

2??u?2.

?2?

Here ? is the density of the liquid surrounding the bubble, t

represents time, and prefthe reference pressure ?taken to be

atmospheric pressure here?. The material derivative D/Dt

=?/?t+u•? is used, since the potential on the moving

bubble surface is needed for the BEM. An adiabatic uniform

pressure is assumed inside the bubble as p=P0?V0/V??,

where P0and V0are the initial pressure and volume of the

bubble. It is important to note that we model the bubble

interior as an ideal thus noncondensable gas with a constant

ratio of the specific heats of ?=1.25. This value is only an

educated guess as we basically ignore the complex thermo-

dynamic processes of the gas and vapor mixture inside the

bubble. The value of ?=1.25 has been chosen to account for

the heat losses during the late expansion and early collapse

dynamics and the adiabatic bubble wall during the faster

dynamics. This approximation follows also Cole’s model for

an underwater explosion ?24?.

The BEM method now works as follows:

?1? Assume a bubble with the desired initial shape

?square, elliptic, etc.? with initial volume V0and pressure P0.

?2? Calculate the potential at the next time step using a

time discretized version of Eq. ?2?.

?3? Then use Eq. ?1? to get the corresponding normal ve-

locities with which the bubble surface is updated.

?4? Calculate the new volume, pressure and velocity at

each node.

?5? Go to the next time step and repeat steps ?2? to ?4? as

many times as required.

The interested reader is referred to Refs. ?16,17? for the

details on the numerical implementation.

Besides the neglect of viscous effects, a second simplifi-

cation of our model is the axisymmetry formulation, i.e., we

compare the experimental pictures with an appropriate plane

cut through the simulated axisymmetric volume. As demon-

strated in Fig. 2 we compare all nonspherical geometries to

the plane ?=const of the axisymmetric volume ?r,z,?? ex-

cept for the toroidal bubble where both the plane z=0 and

?=const are investigated.

The influence of the confining lower and upper plates

placed at a distance of 40 ?m is not accounted for in the

axisymmetric formulation. Only a fully three-dimensional

simulation would take care of the experimental geometry.

Consider a point like laser focus, which would create a

single laser bubble in an infinite medium. Here, because the

laser is focused just above the lower boundary, see inset of

Fig. 1, a mostly hemispherical bubble will emerge. We ex-

pect that the second boundary is mostly slowing down the

bubble dynamics through the confinement of the liquid. For

example, when the bubble shrinks, less liquid is available to

accelerate the bubble interface and the bubble shrinks slower

as compared to an infinite liquid. The boundaries will also

affect the bubble shape during the last stage of the bubble.

Here, we expect that for close distances between the plates

jetting toward the upper boundary will be induced.

The initial conditions for the simulations are chosen by

adjusting the initial pressure inside the bubble and its initial

volume. Further simplifications of the model are the neglect

of surface tension and gravity as they have only minor ef-

fects on the bubble dynamics for the bubble sizes in the

experiments. The simulations start with an initial pressure of

P0?t=0?=50 bar.

The simulations are performed in such a way to get a

matching experimental and numerical maximum bubble ra-

dius, Rmax. To compare the simulations with the experiments

we have to deal with the retardation effect induced by the

FIG. 2. Geometry of the axisymmetric bubble simulations and

the resulting planes used to compare with the high-speed photogra-

phy. The axis of symmetry is the z axis. For the elliptic ?a?, the

v-shaped ?b?, and the square bubble ?d? we compare the bubble

dynamics to the projection in the rz plane of an ellipsoidal, hollow

cone, and double cone, respectively. The toroidal bubble ?c? in the

experiments is compared to its projection in the r? plane and the rz

plane.

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narrow gap. Interestingly, the ratio between the experimental

bubble lifetime, Texp, and the simulated dimensional time,

Tsim, is Texp/Tsim?1.7 for all bubbles. The time in the ex-

periments and the simulations was normalized with the

bubble lifetime. In this way the normalized dimensionless

time, t?, lies in the interval ?0,1? and we plot the simulated

frames that correspond to the experimental ones on the nor-

malized time scale.

IV. NONSPHERICAL BUBBLE GENERATION BY

SHAPING THE LASER INTENSITY

DISTRIBUTION

In all figures showing the bubble dynamics we will depict

in the first frame the intensity profile in black and white

which is used to calculate the hologram. This frame reflects

the ideal intensity distribution of the laser radiation in the

focal plane.

A. Elliptic bubble

Figure 3 shows the dynamics of a bubble created from

focusing the light on a line leading to an essentially elliptical

bubble. The high-speed sequence is recorded at 300000 fps.

On the second frame, t=4.1 ?s, the bubble is fragmented,

yet it coalesces quickly into one bubble at 7.4 ?s. The hori-

zontal extent of the bubble, a, is still larger than the vertical

one, b, such that the eccentricity, ?=a/b, is still larger than

1. On the next frame ?10.7 ?s? the bubble has taken on an

almost spherical shape, with an eccentricity very close to 1.0.

At times 20.7 ?s and at 24.1 ?s ?just before collapse? we

can see clearly that the vertical dimension is now larger than

the horizontal one, so that the eccentricity is smaller than 1.

The phenomenon of inversion of the eccentricity was typical

for all elliptic bubbles in this study.

Figure 4 shows the result of the simulations for the evo-

lution of an elliptical bubble that expands against the hydro-

static pressure. The snapshots of the bubble are taken at the

same relative instants as shown in the experiment, Fig. 3

using Texp/Tsim=1.76 and an experimental lifetime of Texp

=25 ?s. We observe the same trend as in the experiment, the

major axis, a, of the initial elliptical bubbles collapses faster

than the minor axis, b. Thus, the eccentricity reverses during

the collapse of the bubble. The simulations also show a ring-

jet formed during the collapse created along the initial major

axis. Even though the curvature in the last frame of Fig. 4 is

relatively high, the physics is still dominated by inertial ef-

fects due to the high velocities occurring at this instant. Sur-

face tension effects are thus still small. In the experiments

we did not observe the jets, likely because the recording was

not fast enough, the jets would appear between the times t

=24.1 ?s and t=27.4 ?s in Fig. 3. Note that the relative

timing from the last two images is 0.96 to 1.00, thus the

collapse takes place very fast ?only 4% of the lifetime of the

bubble?. Although some blurring on the left and right of the

experimental bubble at t=24.1 ?s may hint to a developing

jet. To strengthen the statement of eccentricity inversion we

present a quantitative comparison of the eccentricities as a

function of time in Fig. 5. Experiment and simulations agree

within the experimental uncertainty.

B. V-shaped bubble

More complex bubble dynamics results from two linear

focusing regions connected under some angle, i.e., forming a

V-shaped focus. This shape is shown in the first frame of Fig.

6. The bubble expands and up to t=12.6 ?s with some

cleavages visible. The bubble reaches its maximum size at

t=15.3 ?s. At t=17.9 ?s, the bubble shape starts to trans-

form into a triangular shape pointing in a direction opposite

to the initial orientation of the bubble. This shape remains

FIG. 3. Experimental observation of an elliptical bubble with

eccentricity inversion. The bubble is generated with a longer hori-

zontal shape but during the collapse phase the vertical dimension

becomes larger. First frame represents the image used to shape the

laser focus ?frame rate 300 000 fps, exposure time 1.76 ?s, laser

energy 22 ?J?. The top right corner shows the time after the laser

pulse in microseconds and the black bar denotes 100 ?m and the

time stamps in the upper right are given in microseconds.

FIG. 4. Numerical simulation of an elliptic bubble, with an ini-

tial pressure of P0?t=0?=50 bar. When compared to Fig. 3 similar

phenomena can be observed: an initially elliptic bubble with longer

horizontal axis turns into an almost spherical bubble during its

maximum size and then collapses with a longer vertical axis. The

breakup of the bubble as shown in the last frame could not be

observed in the experiment, likely due to a lack of temporal

resolution.

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until the bubble collapse. The corresponding simulation is

shown in Fig. 7, where Texp/Tsim=1.70 and an experimental

lifetime of Texp=36 ?s. The bubble shape evolves in a very

similar way as the experimental one, e.g., at time t?=0.86 a

similar reversed triangular shape is found. Interestingly, nu-

merically the collapse scenario is rather complex with the

formation of multiple jets, one from the right pointed tip, and

two counterpropagating jets from below and above.

C. Toroidal bubble

Figure 8 demonstrates the dynamics of a toroidal bubble

created by shaping the beam into the circle shown on the first

frame. At time t=3.7 ?s after the laser pulse arrival we see

a nonsmooth bubble surface, indicating that the torus con-

sists of distinct and rapidly expanding bubbles. The frag-

mented structure already smoothened at 7.7 ?s, likely be-

cause of coalescence and some out of focus movement of

parts of the bubble interface. The toroidal bubble continues

to expand until it reaches its maximum size at 35.7 ?s, and

then starts collapsing. The darker region in the center is

probably due to a remaining liquid core in the center of the

torus, thus the torus retains its shape during the expansion-

collapse cycle.

The simulations in Fig. 9 show both the top view, Fig.

9?a?, and the cross section of the torus, Fig. 9?b?. Interest-

ingly, a toroidal bubble develops an annular jet toward its

00.10.2 0.30.40.5

t/Tosc

0.6 0.70.80.91

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Eccentricity(a/b)

FIG. 5. This graph shows the eccentricity ?a/b?, where a is the

horizontal axis and b the vertical axis of the bubble. The squares

correspond to the experiment shown in Fig. 3 and the solid line is

the BEM simulations.

FIG. 6. Experimental observation of a more complex bubble

dynamics emerging from a V-shaped focus. The recording shows

selected frames taken at 300 000 fps with an exposure time of

2.2 ?s, laser energy of 58 ?J. The dots in the frames are from

3 ?m diameter polystyrene particles used for flow visualization.

The black bar denotes a length of 100 ?m and the time stamps in

the upper right are given in microseconds.

FIG. 7. Simulation of a V-shaped bubble: the initially V-shaped

bubble quickly turns into a more rounded shape. The “hollow” part

of the V shape remains visible for quite some time, but finally

disappears at t??0.4. It then overshoots and becomes a bulge

clearly visible at the right-hand side of the bubble. This bulge de-

velops into a jet from t?=0.94 onward. A ring jet is formed in the

left central section of the bubble. Once this jet impacts the simula-

tions are being stopped. The jet from the right hand side has not yet

fully penetrated the bubble at this instant. The axis of symmetry is

the z axis.

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center. This behavior can be understood with the center-of-

mass moving toward the torus center during the shrinkage.

According to Benjamin and Ellis ?25? any moving and

shrinking bubble develops a jet in the direction of motion.

Such a jet could not be observed in the experiments ?since it

happens out of the viewing point of the camera?, but is very

likely to be present. This example shows the advantages of

the numerical simulations: it can show details that otherwise

would not have been observable. In this simulation

Texp/Tsim=1.64 and the experimental lifetime is Texp=56 ?s.

V. SQUARE BUBBLE GENERATION WITH

MULTIPLE LASER FOCI

Multiple distinct bubbles can be created using multiple

laser foci. We have observed that these bubbles may coalesce

depending on the laser energy attributed to each focus and

their distance. This gives the possibility to create nonspheri-

cal bubbles through bubble-bubble coalescence. Yet, we also

were able to constrain the shape of a central bubble by sur-

rounding it with “helper” bubbles and without coalescence to

occur.

A. Method using bubble coalescence

Here, five bubbles are nucleated and quickly coalesce into

one large square bubble, visible at t=5.9 ?s in Fig. 10. The

square shape is stable during bubble expansion. During the

collapse however, the bubble rounds up visibly at time t

=15.9 ?s and then it develops a diamond shape at t

=18.9 ?s. Shortly afterward jets emerge from all four cor-

ners in which the fast motion resulted in a blurred image at

time t=22.6 ?s in Fig. 10. Some structure inside the bubble

is evident which we attribute to the remains of the liquid

films from the coalescence process.

The simulation of this geometry, Fig. 11, is started with a

diamond shaped bubble resembling the configuration of

bubbles shortly after the generation, with Texp/Tsim=1.68 and

the experimental lifetime of Texp=23 ?s. Thus we ignore the

time before coalescence. The bubble expands to a square

shape quickly and collapses with the rotation of its shape by

45°, resulting again into a diamond shape. The last frame in

the simulation does not compare with the experiment and is

depicting the already re-expanding bubbles. Likely this is

FIG. 8. Toroidal bubble created from a circular focus depicted in

the first frame. The bubble expands initially with a rough surface

likely due to separate bubbles which only coalesce in a later stage.

Please note that the bubble retains its toroidal shape during the

expansion and collapse cycle. The recording is taken at a frame rate

of 250 000 fps with an exposure time of 1.8 ?s and laser energy of

51.8 ?J. The black bar denotes a length of 100 ?m and the time

stamps in the upper right are given in microseconds.

(b)

(a)

FIG. 9. Two cross sections of the toroidal bubble simulation

results for ?a? the r? plane ?to be compared with Fig. 8?, ?b? the rz

plane.

FIG. 10. Coalescence method to create a square bubble. The

laser is focused onto five distinct but near by spots as depicted in

the first frame. All five bubbles quickly expand and are already

merged into one bubble at t=5.9 ?s. Please note the diamond

shape of the bubble during collapse. The frames are selected from a

recording taken at 300 000 fps with an exposure time of 1.8 ?s.

The laser energy to all five spots is 20 ?J. The black bar denotes a

length of 100 ?m and the time stamps in the upper right are given

in microseconds.

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due to the higher gas pressure in the simulated bubbles,

thereby allowing for lesser compression. Therefore, the

bubble re-expands before it develops the four jets which are

weakly visible in the experiment, Fig. 10.

B. Method without bubble coalescence

In Fig. 12 four bubbles are used to confine a central

bubble into a rotated square shape. The laser energy is ap-

proximately the same as in Fig. 10, yet the distance between

the foci is increased by almost a factor of two. Therefore, the

films between the bubbles do not drain and we see the ex-

pansion and collapse of the five individual bubbles. The in-

terfaces between the outer “helper” bubbles and the central

bubble flatten during expansion which leads to a transforma-

tion of the initially curved interfaces at t=4.1 ?s to flat in-

terfaces at t=10.7 ?s. The flattening of the liquid film be-

tween two close bubbles has been observed before ?26,27?.

Also the center bubble is shielded by the surrounding

bubbles leading to a low pressure region. The center bubble

only collapses after the shielding has ended; it therefore

shows a delayed collapse ?26?. Interestingly the four “helper”

bubbles jet toward the central bubble during their collapse

which is visible at t=14.1 ?s and t=17.4 ?s.

VI. DISCUSSIONS

Nonspherical bubbles, in particular elliptic, V-shaped, and

toroidal bubbles can be created with a single focus on a line,

see Figs. 3, 6, and 8. A closer look of the initial bubble shape

created reveals that the resulting surface is not smooth but

shows crack like structures which we attribute to the inter-

face between explosively expanding bubbles. Only the coa-

lescence of these initial bubbles leads to later desired smooth

shapes.

For the coalescence to occur the separating films have to

be thinned and drained by the expanding bubbles. Although

the surface of the bubble afterward is smooth, some struc-

tures remains visible, e.g., in Fig. 3 at 7.4 ?s. We speculate

that these features may be remnants from the liquid films

separating the bubbles and rupturing just before the coales-

cence. Similar structures are seen for all other experiments

except the noncoalescing bubbles shown in Fig. 12. This

observation suggests that all of the nonspherical bubbles ex-

cept for Fig. 12 are formed through coalescence of smaller

bubbles during their explosive growth. Unfortunately, the

temporal resolution of our experiment does not allow the

study of this coalescence process for Figs. 3, 6, and 8.

The bubbles are created through linear absorption. After

the laser pulse, pressure and tension waves emerge and su-

perimpose over the illuminated region leading to a complex

acoustic diffraction pattern ?28?. Likely these thermoelastic

waves ?29? lead to some instability and growth of distinct

bubbles. Bubble nucleation from a line focus may therefore

be inherently unstable due to these acoustic interactions.

However, for certain configurations as shown in this work at

some later times coalescence is obtained and a single bubble

of the desired shape emerges.

The simulations show surprisingly good agreement with

the experiment although severe simplifications are done to

the experimental geometry. The height of the experimental

liquid gap of 40 ?m is too large that the bubble can be

considered two-dimensional ?18,27? and too small to be a

purely axisymmetric one which we assume in the simula-

tions. Yet, our results suggest that the confinement is basi-

cally slowing down the dynamics in the projected plane and

the bubble shape is in agreement within the spatial and tem-

poral resolution of the experiment. The slowdown of the

bubble oscillation in the experiments is approximately 1.7

for all simulations conducted. We suggest that because less

liquid is available the fluid acceleration is reduced, which

has also been found for a bubble expanding and collapsing

within a droplet ?30?. Although it is very well established

that a single rigid boundary increases the collapse time

?13,31?, an assessment of the importance of two boundaries

FIG. 11. Simulation of the square bubble dynamics without the

bubble-bubble coalescence. The bubble starts with a diamond shape

resembling the shape of the focus, see first frame of Fig. 10. The

last frame shows distinct differences compared to the experiment

which we contribute to residual gas pressure.

FIG. 12. Creation of a rotated square bubble by confining the

expansion of a central bubble with four helper bubbles. The larger

distance between the laser foci ?first frame? suppressed the coales-

cence although as similar laser energy of 26 ?J is used. Selected

frames taken at 300 000 fps with an exposure time of 1.8 ?s. The

black bar denotes a length of 100 ?m.

NONSPHERICAL LASER-INDUCED CAVITATION BUBBLESPHYSICAL REVIEW E 81, 016308 ?2010?

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demands a fully three-dimensional simulation.

The flow fields of nonspherical bubbles have interesting

properties. For example the torodial bubble possesses a stag-

nation point at its center. An object placed there would pri-

marily experience a pressure field which may be of interest

in the study of pressure-only effects on biological cells.

Chahine and Genoux ?32? have developed a Rayleigh-

Plesset type model for the oscillation of a toroidal bubble.

We have checked their ordinary differential equation against

the boundary element method and obtained very good agree-

ment for the overall dynamics and sufficiently large aspect

ratios. Interestingly, our simulations reveal the formation of

an annular jet during the collapse caused by the inward mov-

ing bubble ring. This feature of toroidal bubbles may have

not been reported before. Yet, it should be mentioned that

toroidal bubbles have been produced with a static hologram

and pulsed-laser-illumination before ?33?.

Except for this jetting the toroidal shape is stable during

expansion and collapse; in contrast to the elliptic bubble

which demonstrates shape instability during the collapse

leading to a rotation of its major axis by 90°. These dynam-

ics are nicely verified in the simulations. We are only aware

of one report demonstrating this rotation of the axis for el-

lipsoidal bubbles ?14?. There the ellipsoidal shape is created

due to imperfection of the laser focus from the femtosecond

optical breakdown process. It is reported that the bubble

starting as an ellipsoid obtains an almost spherical shape

during maximum expansion, i.e., an aspect ratio of approx.

1.0, and collapses with a 90° rotated major axis. It was ar-

gued by Vogel et al. ?13? that higher curved parts of the

bubble shorten the collapse time locally. Therefore, more

curved parts of the bubble, i.e., closer to the major axis,

shrink faster leading to its rotation. This phenomenon is not

based on surface tension but inertia only.

The V-shaped bubble accelerates a directed flow from the

right to left during in the last stage of collapse, see Figs. 6

and 7; thus a unidirectional jet is obtained from a single

bubble, in total absence of boundaries or imposed pressure

gradients. This property may be of interest in cavitation as-

sisted microfluidics for the transport of small objects. Also

fast and localized jets may be suited to probe elastic proper-

ties of microscopic object, e.g., nanotubes and nanowires

?34?, lipid vesicles, or biological cells.

VII. CONCLUSIONS

In summary, we have demonstrated the creation and dy-

namics of nonspherical bubbles in a liquid gap using a

pulsed-laser source in combination with a digital hologram.

To our knowledge this is the first time that the bubble shape

has been altered on-demand. The flows from these bubbles

extend the range of flow fields from the well known

spherical/hemispherical bubbles in various geometries. They

allow to achieve specific tasks, for example unidirectional

jetting from the V-shaped bubble and flow with a fixed stag-

nation point at the center of a toroidal bubble leading to a

compressive only flow. On the long term we envision a digi-

tally controlled microfluidic platform actuated with a pulsed

laser which allows creating arbitrary flow patterns, manipu-

late microscopic objects, and measure their elastic properties.

ACKNOWLEDGMENTS

Funding by the Ministry of Education Singapore ?Grant

No. T208A1238? and the Nanyang Technological University

through Grant No. RG39/07 is greatly acknowledged.

?1? K. R. Rau, P. A. Quinto-Su, A. N. Hellman, and V. Venugo-

palan, Biophys. J. 91, 317 ?2006?.

?2? R. Dijkink, S. Le Gac, E. Nijhuis, A. van den Berg, I. Vermes,

A. Poot, and C. D. Ohl, Phys. Med. Biol. 53, 375 ?2008?.

?3? P. A. Quinto-Su, H. H. Lai, H. H. Yoon, C. E. Sims, N. L.

Allbritton, and V. Venugopalan, Lab Chip 8, 408 ?2008?.

?4? H.-H. Lai, P. A. Quinto-Su, C. E. Sims, M. Bachman, G. P. Li,

V. Venugopalan, and N. L. Allbritton, J. R. Soc., Interface 5,

S113 ?2008?.

?5? S. Le Gac, E. Zwaan, A. van den Berg, and C. D. Ohl, Lab

Chip 7, 1666 ?2007?.

?6? T. H. Wu, L. Gao, Y. Chen, K. Wei, and P.-Yu. Chiou, Appl.

Phys. Lett. 93, 144102 ?2008?.

?7? A. N. Hellman, K. R. Rau, H. H. Yoon, S. Bae, J. F. Palmer, K.

S. Phillips, N. L. Allbritton, and V. Venugopalan, Anal. Chem.

79, 4484 ?2007?.

?8? R. Dijkink and C. D. Ohl, Lab Chip 8, 1676 ?2008?.

?9? L. V. Zhigilei and B. J. Garrison, J. Appl. Phys. 88, 1281

?2000?.

?10? A. Vogel, J. Noack, K. Nahen, D. Theisen, S. Busch, U. Parl-

itz, D. X. Hammer, G. D. Noojin, B. A. Rockwell, and R.

Birngruber, Appl. Phys. B: Lasers Opt. 68, 271 ?1999?.

?11? A. Vogel, Phys. Med. Biol. 42, 895 ?1997?.

?12? C. DeMichelis, IEEE J. Quantum Electron. 5, 188 ?1969?.

?13? A. Vogel, W. Lauterborn, and R. Timm, J. Fluid Mech. 206,

299 ?1989?.

?14? T. Kurz, D. Kröninger, R. Geisler, and W. Lauterborn, Phys.

Rev. E 74, 066307 ?2006?.

?15? P. A. Quinto-Su, V. Venugopalan, and C. D. Ohl, Opt. Express

16, 18964 ?2008?.

?16? Q. X. Wang, K. S. Yeo, B. C. Khoo, and K. Y. Lam, Theor.

Comput. Fluid Dyn. 8, 73 ?1996?.

?17? S. Rungsiyaphornrat, E. Klaseboer, B. C. Khoo, and K. S. Yeo,

Comput. Fluids 32, 1049 ?2003?.

?18? E. Zwaan, S. Le Gac, K. Tsuji, and C. D. Ohl, Phys. Rev. Lett.

98, 254501 ?2007?.

?19? W. Hentschel and W. Lauterborn, in Proceedings of the First

International Conference on Cavitation and Inhomogeneities in

UnderwaterAcoustics, edited by W. Lauterborn ?Springer, Ber-

lin, 1980?, p. 47.

?20? B. B. Taib, Ph.D. thesis, University of Wollongong, 1985.

?21? G. L. Chahine, in Proceedings IUTAM Symposium, edited by

J. R. Blake, J. M. Boulton-Stone, N.H. Thomas ?Kluwer Aca-

demic Publishers, Birmingham, UK, 1994?, p. 195

LIM et al.

PHYSICAL REVIEW E 81, 016308 ?2010?

016308-8

Page 9

?22? J. R. Blake, B. B. Taib, and G. Doherty, J. Fluid Mech. 170,

479 ?1986?.

?23? E. Klaseboer, S. W. Fong, C. K. Turangan, B. C. Khoo, A. J.

Szeri, M. L. Calvisi, G. N. Sankin, and P. Zhong, J. Fluid

Mech. 593, 33-56 ?2007?.

?24? H. Cole, Underwater Explosions ?Princeton University Press,

Princeton, NJ, 1948?.

?25? T. B. Benjamin and A. T. Ellis, Philos. Trans. R. Soc. London,

Ser. A 260, 221 ?1966?.

?26? N. Bremond, M. Arora, C. D. Ohl, and D. Lohse, Phys. Rev.

Lett. 96, 224501 ?2006?.

?27? P. A. Quinto-Su and C. D. Ohl, J. Fluid Mech. 633, 425

?2009?.

?28? M. Frenz, G. Paltauf, and H. Schmidt-Kloiber, Phys. Rev. Lett.

76, 3546 ?1996?.

?29? G. Paltauf and P. E. Dyer, Chem. Rev. 103, 487 ?2003?.

?30? D. Obreschkow, P. Kobel, N. Dorsaz, A. de Bosset, C. Nicol-

lier, and M. Farhat, Phys. Rev. Lett. 97, 094502 ?2006?.

?31? J. R. Krieger and G. L. Chahine, J. Acoust. Soc. Am. 118,

2961 ?2005?.

?32? G. L. Chahine and P. F. Genoux, J. Fluids Eng. Trans. ASME

105, 400 ?1983?.

?33? Paul Prentice, Dundee University ?private communication?.

?34? P. A. Quinto-Su, X. H. Huang, R. Gonzalez-Avila, T. Wu, and

C. D. Ohl, Phys. Rev. Lett. 104, 014501 ?2010?.

NONSPHERICAL LASER-INDUCED CAVITATION BUBBLESPHYSICAL REVIEW E 81, 016308 ?2010?

016308-9