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Hydrodynamic simulation of bubble collapse and picosecond sonoluminescence

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Numerical hydrodynamic simulations of the growth and collapse of a 10 [mu]m air bubble in water were performed. Both the air and the water are treated as compressible fluids. The calculations show that the collapse is nearly isentropic until the final 10 ns, after which a strong spherically converging shock wave evolves and creates enormous temperatures and pressures in the inner 0.02 [mu]m of the bubble. The reflection of the shock from the center of the bubble produces a diverging shock wave that quenches the high temperatures ([gt]30 eV) and pressures in less than 10 ps (full width at half maximum). The picosecond pulse widths are due primarily to spherical convergence/divergence and nonlinear stiffening of the air equation of state that occurs at high pressures. The results are consistent with recent measurements of sonoluminescence that had optical pulse widths less than 50 ps and 30 mW peak radiated power in the visible.
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Hydrodynamic simulations of bubble collapse and picosecond
sonoluminescence
William
C.
Moss, Douglas
B.
Clarke, John
W.
White, and David A. Young
Lawrence Livermore National Laboratory,
P.O.
Box 808, Livermore, California 94550
(Received
21
September 1993; accepted 20 May 1994)
Numerical hydrodynamic simulations
of
the growth and collapse
of
a
10
.urn
air bubble in water
were performed. Both the air and the water are treated
as
compressible fluids. The calculations show
that the collapse
is
nearly isentropic until the final 10
ns,
after which a strong spherically converging
shock wave evolves and creates enormous temperatures and pressures in the inner
0.02
.urn
of
the
bubble. The reflection
of
the shock from the center
of
the bubble produces a diverging shock wave
that quenches the high temperatures
(>30
eV) and pressures in less than 10 ps (full width at half
maximum). The picosecond pulse widths are due primarily
to
spherical convergence/divergence and
nonlinear stiffening
of
the air equation
of
state that occurs at high pressures. The results are
consistent with recent measurements
of
sonoluminescence that had optical pulse widths less than 50
ps and 30 m W peak radiated power in the visible.
I.
INTRODUCTION
Sonoluminescence involves the conversion
of
acoustical
energy
to
optical energy.
It
arises from the nucleation,
growth, and collapse
of
gas-filled bubbles in a liquid. Recent
experimental advances by Gaitan
1
,2 have resulted in the
stable synchronous sonoluminescence
of
a single acousti-
cally levitated bubble. A typical
"Gaitan" sonoluminescence
experiment consists
of
a flask with dimensions
of
a few cen-
timeters. The flask contains a degassed liquid, usually water
or a glycerine-water mixture. Acoustic transducers placed
around the outside
of
the flask provide an oscillatory driving
pressure that creates standing waves in the liquid.
An air
bubble inserted into the liquid
is
trapped (levitated) near the
center
of
the flask and oscillates radially. At sufficiently high
transducer amplitudes, the compressed air radiates optically
at the conclusion
of
the collapse phase. Using Gaitan's
method, Barber and Putterman
3
(BP) have reported emis-
sions from a single pulsating bubble that are synchronous
with the driving (oscillatory) acoustic field, and have a mea-
sured pulse width
ps. The emission spectra are consis-
tent with a
;;.,2
eV blackbody radiator.
4
The underlying phys-
ics
of
the short pulse widths and the inferred high
temperatures cannot be explained adequately by any current
theory
of
sonoluminescence.
In this article we show that the
BP
data can be explained
by two physical mechanisms; one that creates the high tem-
peratures rapidly near the center
of
the bubble, and another
that quenches them rapidly.
We
show via numerical simula-
tion
of
the hydrodynamic equations
of
motion that as the
bubble collapses, the spherically convergent flow generates a
strong shock near the center of the bubble. The shock causes
the temperature and pressure
of
the air
to
rise rapidly only in
a very small region
«0.02
f.lm radius), which
is
heated
to
tens
of
eV and radiates. The reflection of the shock from the
center
of
the bubble produces a spherically divergent re-
shock. Hydrodynamic expansion (P
dV
cooling) occurs very
rapidly behind the shock, and quenches the optical emissrons
in a few picoseconds.
Although
it
is accepted generally that sonoluminescence
arises from the heating
of
gas inside the bubble during its
collapse,5.6 the detailed hydrodynamics and thermodynamics
have not been understood well. The hydrodynamics
of
col-
lapsing bubbles was studied first by Rayleigh,7 who obtained
an analytic solution for the collapse
of
a spherical void in an
incompressible fluid. Rayleigh's solution produces collapse
times on the order
of
microseconds for typical bubbles (ini-
tial radii less than 1 mm), which agrees with experimental
data. Noltingk and Neppiras
8
(NN) also considered an in-
compressible fluid, but they replaced the void with an ideal
gas, and included the surface tension
of
the liquid. The NN
analysis assumes the gas
in
the bubble
to
be in thermody-
namic equilibrium, because the transit time
of
sound waves
across the bubble
is
small compared
to
the collapse time
of
the bubble. Others have refined the NN model by including
the effects
of
thermal conduction in the gas
9
or a van der
Waals
lO
gas equation
of
state; however, thermodynamic equi-
librium was still assumed. These models also assumed ex-
plicitly that the gas was heated isentropically. Gas heating
by
nearly isentropic compression
ll
or shock compression,12-14
for various initial conditions has also been suggested, but has
not been believed
to
provide a more realistic description
of
the heating
of
the gas than isentropic compression. This
is
due
to
two reasons: (i) a mechanism that generates a shock
has not been identified, and (ii) the temperatures measured in
the gas
(""'6000 K), prior
to
BP,
have been consistent with
isentropic heating. The
BP data reveals serious problems
with the isentropic model, because muIti-e
V temperatures
cannot be obtained easily by isentropic
compression-an
in-
tense compression, e.g., a shock, is required.
If
a shock
is
present in the gas, then the gas
is
not at a uniform tempera-
ture. A complete solution
of
the hydrodynamic equations
of
motion (conservation
of
mass, momentum, and energy),15
with appropriate initial and boundary conditions, and realis-
tic equations
of
state for the liquid and gas
is
required to
study the effects
of
nonequilibrium thermodynamics in the
gas. The most complete analysis
to
date has been performed
by
Wu
and Roberts.
16
They considered the full hydrodynam-
ics
of
a van der Waals gas, with a constant specific heat. The
motion
of
the bubble radius was obtained from the
Phys. Fluids 6 (9), September 1994
1070-6631/94/6(9)/2979/7/$6.00
© 1994 American Institute
of
Physics 2979
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Rayleigh-Plesset equation/,8,17,18 which models the water as
an almost incompressible fluid. Their analysis showed the
development
of
a spherically convergent shock, as the outer
radius
of
the bubble decreased. The shock heated the gas
near the center
of
the bubble to
=1
keY
(1
eV=
11
605 K). A
constant specific heat
van
der Waals gas equation
of
state is
invalid at these high temperatures. Consequently, we per-
formed a more detailed analysis that includes a realistic air
equation
of
state and the nonlinear hydrodynamics
of
the
water that surrounds the bubble.
II. METHOD AND DISCUSSION
We
assume spherical symmetry and consider the motion
of
an air bubble with initial radius Ro surrounded by a shell
of
water, whose outer radius is R w. The air and water are
initially at atmospheric pressure,
P
o
=l
bar, in thermal equi-
librium, and at rest. The outer radius
of
the water is driven by
an oscillatory pressure
P
0-
P a sin
wt.
We
calculate the bub-
ble's response to only the first half cycle
of
the driving os-
cillatory pressure, that is, only the first
of
the many growth
and collapse cycles that the bubble experiences. A complete
simulation
of
steady state sonoluminescence would require
specifying many quantities, such as, the flask dimensions and
thickness, transducer locations, the driving frequency and
pressure, liquid and gas compositions, viscosities, thermal
conductivities, and surface tensions, etc., which is beyond
the scope
of
this paper. Nevertheless, we assume the physics
that governs the creation
of
anyone
of
the steady state
sonoluminescent flashes can be approximated adequately us-
ing typical values for
R
o
,
R
w
,
w,
and P
a'
because the bubble
collapse is primarily an inertial effect
of
the liquid compress-
ing the gas. Any set
of
parameters that produces a typical
bubble radius, as a function
of
time, should be sufficient to
supply the necessary inertial forces that generate the flash.
The parameters
we
have chosen are typical, but not represen-
tative
of
any particular experiment.
We
used
Ro=
10 ).tm,
Rw=5
em, and w=21T(45 kHz).19 The pressure amplitude
P a=0.25 bar was chosen to yield a maximum bubble radius
of
-90
,um,
so the ratio RmaxlRo is consistent with experi-
mental data.2°
In the absence
of
viscosity, surface tension, heat conduc-
tion, mass diffusion, and non mechanical energy loss (radia-
tion), the equations for the conservation
of
mass momentum
and energy for the system are
l5
' ,
Dp
-
+pV·v=O
Dt
'
Dv
P
-=-VP
Dt
'
De
DV
-+p-=o
Dt
Dt
'
(1)
where
p,
v, PCp,e),
e,
and V are the density, velocity, pres-
sure, specific internal energy, and specific volume
[p
I],
and
D/Dt=(a/at)+v·v.
1\vo equations
of
state were used to
describe the water: (i)
P(
7])=K
In(7]),
where
7]=p/po
and
K
-=22.5 kbars,
or
(ii) a polynomial equation
of
state.
21
1\vo
equations
of
state were used to describe the air: (i) an ideal
2980
Phys. Fluids, Vol.
6,
No.9,
September 1994
100
80
.....,
!
60
"
40
20
0
30
---
··
..
------
---_ _ - - -
300
34
.....
_
............
...
....
i""
0
.
38
42
Time
(ps)
I
I
46
-300
-900
50
FIG.
1.
Bubble
radius (solid line) and interface velocity (small dashed line)
as functions
of
time for the coarse zoned air
Step
1 calculation.
The
retarded
driving pressure
(1:tO.25 bar, large dashed line) and link time (dots) for the
Step
2 calculation are
shown
also.
gas, P = p(
y-l)e,
where
y=
1.4,
or
(ii) an analytic model
that includes vibrational excitation, dissociation, ionization,
and a repulsive intermolecular potential. The reasons for the
multiple equations
of
state will be explained later.
The equations
of
motion [Eq. (1)], combined with the
boundary and initial conditions given above comprise a com-
plete set
of
equations that can be solved for the radial and
temporal variation
of
all the field quantities. We used the
hydrocode KDYNA,22,23 which solves numerically the differ-
ential equations using a finite difference method.
The
water
and the air are divided into a mesh consisting
of
many con-
centric shells (zones)
of
fixed mass.
The
mass may vary from
one shell to another. There must be enough shells to ensure
that the finite difference solution converges to the differential
equation solution. The solution is obtained incrementally in
time, where each time increment
is
computed
by
taking the
minimum value
of
ar/c
(shell thickness divided by the local
sound speed in the shell) over the entire mesh. This ensures
that information cannot travel farther than the smallest shell
thickness each time increment.
We
solved the equations in two steps. Step 1 obtained
the hydrodynamic motion
of
the system, during the growth
and the initial collapse. Step 2 studied the details
of
the final
stages
of
the collapse. Step 1 had four equally sized air zones
(2.5 ,urn/zone) and 2000 water zones that increased geo-
metrically in size from
0.3
,urn
at
Ro,
to 154
,urn
at R
w.
The
ideal gas and constant bulk modulus equations
of
state [(i)
above] were used. Figure
1 shows the radius (solid line) and
interface velocity (small dashed line)
of
the bubble, as func-
tions
of
time. The retarded driving pressure (large dashed
line)
PO-P
a
sin
w(t-Rw/c),
where
c=1500
rn/s, is also
shown in the figure. The radius begins to increase when the
decreasing driving pressure reaches the bubble
(t=R
w
/c-33.3
j.ts). The bubble attains a maximum radius
of
90
,um.
The figure shows that the initial collapse reaches a
minimum radius
of
2.4
,um,
after which subsequent
"bounces"
occur. Although Fig. 1 shows many details
of
the
bubble motion, the coarse zoning in the air and limited va-
lidity
of
the equations
of
state provide an inaccurate descrip-
Moss
et
al.
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128.115.190.39 On: Mon, 08 Dec 2014 17:07:52
30
25
15
10
5
o
44.92
f.S
---------r--.----
n
D.'
!
__
.J.
0.0
45.0350
4S.0351
.........
4000
10
..
..
.l-._..L.-....-..l
....
-L..
44.96
45.00
Time
(J.lS)
1000
o
-1000
§
-2000
'-
-3000
45.04
FIG.?
radius (solid line) and interface velocity (dashed line) as
functIOns
of
tIme for the Step 2 calculation. The inset shows the last 600 ps
of
the main figure.
tion
of
the nonequilibrium collapse
of
the bubble.
The Step 2 calculation began at 44.925
f.Ls,
with a bubble
radius
of
27.516
f.Lm,
and an interface velocity of 104.68
f.1.ml
flS. The dots in Fig. 1 show the starting conditions. At
this time, the Step 1 solution
is
still accurate, because the
interface velocity is much less than the sound speed
in
the
gas. The complete Step 1 solution at 44.925
f.LS
was mapped
onto the Step 2 mesh, and the calculation proceeded. Step 2
had 600 equally sized air zones (0.046
f.Lm/zone)
and 1200
water zones that increased geometrically
in
size from
0.00029
f.Lm
at the interface
to
603
f.Lm
at R w. The high
pressure equations
of
state
[(ii)
above] were used. They were
constructed from a combination
of
data and theory and are
valid for the domains
of
density and energy
in
this calcula-
tion. The air equation
of
state
is
_,.
. E
cPo
[(
P )
(n/3)
+ I
P-R
--
-
1-(3/n)
Po
-(:Jl
R'T+
1
](1-m
D
)+m
D
R'T
D
3
+:)
R'T(2mD)(1
mT
I I
i
E
+ c
(n/3)-1
[
(
.
p)
n/3
n (
p'
) ]
-
--
-
+E
Po
3
Po
C'
where
mk=
O.5[tanh[7(T-
O.9T
k
)/T
k
]
+ tanh[0.63
]],
5
Ini
1=1
(2)
(3)
and
R'
=
R/28.
8 is the gas constant for air. The first term in
the pressure equation accounts for the increase in pressure
due
to
the dissociation
of
molecular nitrogen
Phys. Fluids. Vol.
6.
No.9.
September 1994
50
-tl4S.
020
!lSJ
-50
---t,f4S.024
!lSI
--
'"
·····-t,!45.028!lSI
-150
!
-250
-350
-450
-550
0
5
10
15
Distance
(JIm)
FIG.
3.
Spatial profiles
of
velocity at three times before the flash. Time t 1
is
15.3
ns
before the flash. Time
t3
is
7.3
ns
before the flash.
T
D=9.7
eV) and ionization
of
the nitrogen atoms, for the
first five ionization states
of
nitrogen T
1
_
S
=14.5,
29.6,47.4,77.5, and 97.5
eV).I5.24
We
have ignored the den-
sity and temperature dependence
of
these energies, and as-
sumed that the values for nitrogen are representative
of
air.
The functional form
of
mk
in Eq.
(3)
was chosen arbitrarily.
The second term in the pressure equation
is
a simplification
of
the results
of
Monte Carlo calculations
of
intermolecular
potentials in fluids and
soIids.25
Real intermolecular poten-
tials are stiff
(n=12)
typically near the potential minimum,
but soften
(n=6)
as
the density increases. Consequently, we
have chosen n
=9
in our simulations. The density
of
solid air
at 0 K (Po=1.113 glcc) and the binding energy of solid air
(E
c
=2.52X
10
9
ergs/g) are obtained from a weighted average
of
the known parameters for solid nitrogen and oxygen.
26
The first term in the energy equation includes vibrational
(0=3340
K)
(Ref. 15)
to
the energy
of
a rigid
diatomiC molecule
(y=7/5);
y-+(9!7), for
We
neglect
the vibrational zero point energy. The second term
is
the
energy
to
dissociate a molecule, weighted by the dissociation
fraction. The third term
is
the translational energy
of
the
11
:I
.P!
("
l
-;;;-
0
f'rr!
-+---,
i I
--
..
4000
I
I,.
\
-8000
-12000
o
0.5
-ti45.035260!lS1
---t;45.03529S
!lSI
·······t.l45.03530S
·'0"
t,r 45.035315
!lSI
Distance
(JIm)
1.5
4. Spatial profiles
of
velocity at three times before the flash, and one
time after the flash. Time
t4
is
51.8 ps before the flash. Time h is 3.2
ps
after
the flash.
Moss
et
al.
2981
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128.115.190.39 On: Mon, 08 Dec 2014 17:07:52
10'
-_.,,-\
...
a
-j
..
10-'
1,-,
::t
'"
=t
1
(['
------ t:[45.035305
JlSI
--0-t,£45.035315
JlSl
1
__
__
--L
__
o
0.5
1.5
Distance
(pm)
FIG. 5. Spatial profiles
of
pressure at four times (same as Fig. 4). Time
t4
is
51.8 ps before the flash. Time
t7
is 3.2 ps after the flash.
atoms and electrons. The fourth term is the ionization energy.
The remaining terms are the potential energy.
We
included
dissociation and ionization in our air equation
of
state, be-
. I
I'
16
cause
prevIOUs
ca cu atIons, whIch used a constant specific
heat
("1=7/5) and a simple van der Waals equation
of
state,
produced temperatures near the center
of
the bubble
in
ex-
cess
of
1000 e
V;
which is high enough to remove all seven
electrons from nitrogen. Dissociation and ionization increase
the specific heat
of
the gas, which lowers the temperature, at
a given energy and density. The importance
of
this will be
discussed later.
Figures
10
show the results
of
the Step 2 calculation,
which we will describe first qualitatively. As the bubble col-
lapses, the interface velocity increases, due
to
spherical con-
vergence. Although most
of
the collapse is nearly isentropic,
the waves that move between the interface and the center
of
the bubble increase in amplitude and preheat the center
of
the bubble, during the final
....
10
ns.
These waves are shocks
that are superimposed on the main inward radial
flow,
which
eventually becomes a strong spherically convergent shock
near the center
of
the bubble. The reflection
of
this shock
from the center
of
the bubble produces a spherically diver-
-t.r45.035260JlSI
..........
t.r
45.035295
JlSI
------ t.r45.035305JlSI
--o--t,£45.035315
JlSl
-_.
0.5
1.5
Distallce
(/1m)
FIG.
6.
Spatial profiles
of
density at four times (same as Fig. 4). Time
t4
is
51.8 ps before the flash. Time
t7
is 3.2
ps
after the flash.
2982 Phys. Fluids, Vol.
6,
No.9,
September 1994
50
..
-Air
-----r=
1.4
3
40
1,
'"
--------------:
IOns
---
.....
__
...L.
__
45.00 45.01 45.02
45.03
45.04
Time (/1s)
FIG.
7.
Normalized entropy (28.8 SIR) at the center
of
the bubble. Results
are shown for the dissociation/ionization rEqs. (2) and (3)J (solid line) and
ideal gas (dashed line) air equations
of
state. The implosion is nearly isen-
tropic until the final 10 ns.
gent reshock.
We
propose that this strong shock and its re-
flection, which occur before the bubble has reached its mini-
mum radius, generate the sonoluminescent flash that
is
measured experimentally. The air
is
compressed to a thou-
sand times normal density at this time. The large tempera-
tures and pressures occur only near the center
of
the bubble
and are quenched
in
picoseconds, due to the hydrodynamic
expansion behind the reshock.
-.
Figure 2 shows the radius (solid line) and interface ve-
locity (dashed line)
of
the bubble, as functions
of
time. Fine
details are shown
in
the inset: The minimum radius and ve-
locity are 0.64
/Lm
and
-3060
/Lm/JLS.
We
will use the term
"flash" to refer
to
the calculated time at which the third
of
three shocks reflects from the center
of
the bubble. The ar-
row
in
the inset shows when the flash occurs. Figure 3 shows
spatial profiles
of
the velocity at three times (t
1-3)
before the
flash. The bubble radius is 11.6
/Lm
at t
1
,
which is 15.3
ns
before the flash. There is a shock at 2 /Lm. This shock is the
--==-
0
----·r,[r=
- -
..
.......
180
-.::-
..
...
120
'"
60
t
5
t
t
7
.J
,
7
j
.
45.Q3521
U.(J3U7
0
45.03529
45.03531
45.03533
45.03535
Time
(ps)
FIG.
8.
Temporal histories
of
the pressures in four zones near the center
of
the bubble. The radii
of
the centers
of
the zones. at the time
of
the flash, are
shown in the legend. The inset shows the center zone
of
the Step 2 ideal gas
calculation, which has a
I<WHM
nearly an order
of
magnitude greater than
that produced by the dissociation/ionization air equation
of
state.
Moss
et
al.
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averaged temperature in the air bubble (not shown in the
figures), when it has reached its minimum radius
of
0.64
}Lm,
is
-1
eY,
which is consistent with measurements that lack
the temporal resolution
of
the BP data. The inner 0.016
}Lm
(r<
r
3)
of
the compressed bubble is at least an order
of
magnitUde hotter than the average bubble temperature.
Although Fig. 9 shows high peak temperatures and nar-
row pulse widths, it shows also high temperature
"tails"
that
persist for at least
40 ps, which are not observed experimen-
tally. The apparent discrepancy between our calculations and
the experimental data occurs because the experimental data
are power measurements, not temperature measurements.
The temperature is deduced from the power measurements,
by fitting the data to a blackbody, or other radiation spec-
trum. Figure 9 shows the temperature is not uniform in the
radiating region, so one should not expect the data to
fit
a
single temperature blackbody spectrum. Furthermore, the op-
tical thickness
of
the bubble varies spatially and temporally:
the hot core and cooler outer regions are optically thin, that
is, the mean free path for radiation absorption is larger than
the radius
of
the bubble, whereas a central annular region is
optically thick. It is possible that experimental detectors
would not see all
of
the optical radiation from the hot core,
so that the spectrum is almost certainly more complicated
than that
of
a single temperature blackbody radiator.
It
is
difficult to determine from our calculations an ac-
curate spectral distribution
of
the optical radiation, because
we have ignored radiation in Eqs. (1). The varying optical
thickness causes additional complications. Nevertheless, we
can approximate grossly the total power emitted by the
bubble,
E(t),
by assuming that the bubble is optically thin (a
volume radiator, rather than a surface radiator) and integrat-
ing the radiated energy per unit volume per unit time over
the volume
of
the bubble.
15
The purpose
of
this calculation
is
to show that the high temperature tails in Fig. 9 make a small
contribution
to
the total radiated power.
We
write
where
R is the bubble radius,
(J
is the Stefan-Boltzmann
constant, and
K is the Planck opacity [mean free path
""(pK)-l].
We
ignore the density and temperature depen-
dence
of
the opacity and choose K=3500 cm
2
/g, which is
consistent with the optically thin approximation. Figure
10
shows
E(t):
There is a well-defined peak with a FWHM less
than
lO
ps. The peak radiated power exceeds 0.4
W.
The
experimental power spectra are measured in the visible to
near
UV
(=200-750
nm), whereas, Fig. 10 shows the total
radiated power, so Fig.
10
should overestimate the measured
(Ref.
4)
30 mW peak visible power.
The addition
of
radiation flow
in
Eqs. (1) would remove
additional energy from the central hot spot. Consequently,
the pulse widths in Figs.
8-10,
which result exclusively from
hydrodynamics, represents an upper bound.
We
conclude that
hydrodynamic flow is sufficient
to
supply the energy for pi-
cosecond sonoluminescence. The details
of
the conversion
of
the mechanical energy to optical radiation are still unre-
solved.
2984 Phys. Fluids,
Vol.
6,
No.9,
September 1994
The numerical simulations shown in Figs.
2-10
depend
strongly on the equation
of
state that is used to describe the
air.
We
repeated the Step 2 calculation twice, first using an
ideal gas
(y=
1.4), then using Eqs.
(2)
and (3) with no ion-
ization or dissociation,
to
examine the differences. The inset
in Fig. 8 shows the temporal history
of
the pressure in the
first zone from the center
of
the bubble, for the ideal gas. The
minimum bubble radius (not shown) is
0.25
}Lm,
which is
less than the
0.64
}Lm
obtained
in
the first Step 2 calculation.
The figure shows two pulses. The symmetry
of
the broad first
pressure pulse suggests that the ideal gas compresses rela-
tively gently until the pressure is sufficient to reverse the
velocity field. The second pulse is a shock caused by the
collision
of
the divergent wave generated by the first pulse
and the interface, which is still moving inward. The large
compressibility
of
the ideal gas, relative to Eqs.
(2)
and (3),
produces a much lower peak pressure, but comparable peak
temperature (not shown). The high ratio
of
peak temperature
to peak pressure occurs because all
of
the internal energy in
the ideal gas appears as temperature. Figure 7 (dashed line)
shows the time dependence
of
the normalized entropy
(S'=2.5
In[PV"/PoV1;]),
for the ideal gas. There are only
two jumps, which arise analogously to the first two jumps
shocks in the original Step 2 calculation. The time
of
the
second jump
in
Fig. 7 is shown by the arrow in the Fig. 8
inset, which confirms that the subsequent broad pulse is isen-
tropic. The ideal gas
is
too compressible to generate a third
jump. The total entropy change is much less in the ideal gas
than in the
"real"
air, which also suggests the ideal gas com-
presses relatively isentropic ally, even at the end
of
the col-
lapse. The inset in Fig. 9 shows the temporal history
of
the
temperature in the first zone from the center
of
the bubble,
in
the absence
of
ionization and dissociation. Although the tim-
ing and pulse widths are similar to the comparable zone
(rt)
in the main figure, the temperature in the inset is much
higher. The dissociation and ionization raise the specific heat,
which absorbs energy that would otherwise raise the tem-
perature. The sensitivities
to
variations in the air equation
of
state can be summarized as follows: narrow pulse widths are
obtained from a strongly repulsive potential. The extremely
high temperatures are mitigated by dissociation and ioniza-
tion, which are energy absorbing mechanisms.
III. SUMMARY
We
have shown via numerical simulation
of
the hydro-
dynamic equations
of
motion that sufficient mechanical en-
ergy can be delivered
in
a short enough time, by shock waves
in the imploding bubble, to account for picosecond sonolu-
minescence. The shocks are generated near the center
of
the
bubble, by the convergence and stagnation
of
a strong mo-
mentum wave that evolves into a series
of
shocks. They oc-
cur at the end
of
a long, slow isentropic compression, and are
facilitated by the sudden stiffening
of
the air equation
of
state. The spherical shocks, which are large only in a very
small region
(=0.02
}Lm
radius) near the center
of
the
bubble, cause the temperature
of
the air to rise rapidly. This
inner region is heated to greater than 10 e
V and radiates. The
subsequent hydrodynamic expansion
(P
dV
cooling) occurs
very rapidly, and quenches the optical emissions on a time
Moss
et
al.
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scale
of
a few picoseconds, in agreement with recently re-
ported experimental observations of picosecond
sonoluminescence.
3
Narrow pulse widths do not occur when
the air
is
described by an ideal gas. Extremely high tempera-
tures occur, unless dissociation and ionization
of
the air are
included in the air equation
of
state. Spherical convergence
of
the boundary driving pressure at the center
of
the bubble
is responsible ultimately for producing the high temperatures
and narrow pulse widths that are characteristic of picosecond
sonoluminescence. Although we have not determined con-
clusively the mechanism(s) that convert the internal energy
into optical radiation, we have proposed plausible mecha-
nisms that supply the energy for sonoluminescence
to
occur.
The implosion is isentropic, except for the final few
nanoseconds, so it is an efficient way
to
compress (small
quantities of) gases
to
very high densities and temperatures.
It
should be possible
to
attain even higher temperatures and
densities, by tailoring the driving pressure. Our calculations
indicate that minimizing dissociation and ionization can in-
crease greatly the peak temperatures in the center
of
the
bubble. Temperatures approaching
1000 eV may be achiev-
able. Consequently, it
is
intriguing to consider using the
heavy isotopes
of
hydrogen. In the case
of
hydrogen,
it
is
of
secondary importance whether or not
it
sonoluminesces.
ACKNOWLEDGMENTS
The authors thank
M.
Moss for bringing this subject
to
our attention.
We
thank J. Levatin and A. Attia for providing
excellent computational support, and
X.
Maruyama,
D.
Sweider,
W.
P.
Crowley,
M.
Moran, and
R.
Day for useful
discussions. The authors also thank
L. Glenn and
T.
Gay for
reviewing the manuscript. This work was performed under
the auspices
of
the U.S. Department
of
Energy
by
Lawrence
Livermore National Laboratory under Contract No. W-7405-
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2985
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... Thermal effects occurring during the collapse of gaseous bubbles [1], including sonoluminescence [2], air dissociation and chemical reactions [3] are now well documented. While the bulk liquid temperature does not change significantly compared to the inner bubble content [4][5][6][7], the latter can reach enormous temperatures during the collapse, of the order of thousands of degrees Kelvin as computational studies for both spherical [8,9] and non-spherical bubble collapse cases [10,11] as well as molecular dynamics [12] suggest. A precise determination of the bubble thermodynamics is important in different areas such as in sonochemistry [13,14] [1,3] and ultrasound therapy such as Highintensity Focused Ultrasound (HIFU) to ensure safety and efficiency [15] [4]. ...
... There are only a few studies where a non-ideal gas EoS is used to model the thermodynamics of the bubble. Moss et al. [8] developed a 1D solver with spherical symmetry assumption and used an analytical EoS for the air bubble that includes vibrational excitation, dissociation, ionization, and a repulsive intermolecular potential. Extremely high temperatures up to 1.74 × 10 6 K and 1.16 × 10 7 K have been reported with and without considering the air dissociation and ionization respectively. ...
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An explicit density-based solver of the Euler equations for inviscid and immiscible gas-liquid flow media is coupled with real-fluid thermodynamic equations of state supporting mild dissociation and calibrated with shock tube data up to 5000 K and 28 GPa. The present work expands the original 6-equation disequilibrium method by generalising the numerical approach required for estimating the equilibrium pressure in computational cells where both gas and liquid phases co-exist while enforcing energy conservation for all media. An iterative numerical procedure is suggested for taking into account the properties of the gas content as derived from highly non-linear real gas equations of state and implemented in a tabulated form during the numerical solution. The developed method is subsequently used to investigate gaseous bubble collapse cases considering both spherical and 2D asymmetric arrangements as induced by the presence of a rigid wall. It is demonstrated that the predicted maximum temperatures are strongly influenced by the equations of state used; the real gas model predicts a temperature reduction in the bubble interior up to 41% space-averaged and 50% locally during the collapse phase compared to the predictions obtained with the aid of the widely used ideal gas approximation.
... The reason for the violent collapse is the spherical geometry of a collapsing bubble and the inertia of the inflowing liquid [95]. According to the numerical simulations of the fundamental equations of fluid dynamics inside a collapsing bubble neglecting the effect of thermal conduction, a sharp spherical shock-wave is formed inside a bubble [105][106][107]. It converges at the bubble center, where temperature and pressure dramatically increase. ...
... It was proposed that the convergence and subsequent reflection of a spherical shock-wave is the reason for the extremely short pulse-width of SBSL. However, the shock-wave model resulted in considerably shorter pulse-width than the experimental data [105][106][107][108]. Furthermore, numerical simulations of the fundamental equations of fluid dynamics taking into account the effect of thermal conduction have shown that under many conditions of SBSL, a shock wave is absent inside a bubble because sound velocity increases as the distance from the bubble center decreases due to the increase in temperature caused by thermal conduction to the colder surrounding liquid [109,110]. ...
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In comparison with the first-principles calculations mostly using partial differential equations (PDEs), numerical simulations with modeling by ordinary differential equations (ODEs) are sometimes superior in that they are computationally more economical and that important factors are more easily traced. However, a demerit of ODE modeling is the need of model validation through comparison with experimental data or results of the first-principles calculations. In the present review, examples of ODE modeling are reviewed such as sonochemical reactions inside a cavitation bubble, oriented attachment of nanocrystals, dynamic response of flexoelectric polarization, ultrasound-assisted sintering, and dynamics of a gas parcel in a thermoacoustic engine.
... (1)-(3) are solved numerically by the technique of [13]. This technique was tested in detail in [14], including comparison with the results obtained by the technique of [15]. Figure 1 shows the dependences of the maximum density at the boundary of the small central region of the bubble with r ≤ 0.25 μm on the liquid pressure p 0 , attained during bubble collapse in cold and cool acetone. ...
... Regarding cavitation-induced fusion, light pulses have been noticed during bubble collapse, and extremely high temperatures have been measured [169]. Although the Moss EoS [170] describes such temperature increases and it has been combined with a front-tracking approach [171], it does not include dissociation and ionization of the air. It is necessary to derive a numerical methodology to model such phenomena precisely. ...
Chapter
Cavitation induction is of high interest for a wide range of applications, from hydraulic machines to bioengineering applications. Numerous experimental and numerical studies have aimed to unveil the dynamics of cavitation to enhance the performance and lower the impact of erosion on machinery but also to employ its mechanics in advanced noninvasive medical procedures. The current work provides a comprehensive review of the methodologies that have been developed in the framework of computational fluid dynamics in order to study cavitating flows, highlighting the link of the application with the utilized approach. The methods are presented and assessed according to the class of physical problems addressed, which, in turn, are classified into problems of single-bubble dynamics, bubble cluster dynamics, and cavitating flows at engineering scales.
... At the beginning of the SBSL research, it was believed that a spherical shock wave is formed inside a collapsing bubble [3,4,[29][30][31]. When the spherical shock wave is converged at the bubble center, temperature at the bubble center increases very sharply to about 10 6 K, which was thought to be responsible for the short pulse of SBSL light [29]. ...
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