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128.115.190.39 On: Mon, 08 Dec 2014 17:07:52
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
gasfilled 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 glycerinewater 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,1214
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 muItie
V temperatures
cannot be obtained easily by isentropic
compressionan
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
10706631/94/6(9)/2979/7/$6.00
© 1994 American Institute
of
Physics 2979
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RayleighPlesset 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
.....
_
............
f·
...
J·
....
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(
yl)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)
POP
a
sin
w(tRw/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
/c33.3
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
PR


1(3/n)
Po
(:Jl
R'T+
1
](1m
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
0t,£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
ot,£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
13)
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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128.115.190.39 On: Mon, 08 Dec 2014 17:07:52
t
I 7
\t
'
'''.[g
......
.
Alr[nolonIZltlon1
sao
•
45.0352'
4S.O:JS3S


o
45.03529
45.03531
45.03533
45.03535
Time
(flS)
FIG. 9. Temporal histories
of
the temperatures in the
same
four zones
shown
in Fig. 8.
The
FWHM
of
zones r
1
2 are less than 10 ps. The inset
shows
the center zone
of
the Step 2 calculation, in the absence
of
dissocia
tion and ionization. Although the timing and pulse widths are similar to the
main figure, the temperature is much higher.
first
of
three to reach the center
of
the bubble. The shock
developed smoothly from the linear velocity profile at the
beginning
of
the Step 2 calculation. The bubble radius is 10.0
p.m
at t
2
,
which is 11.3 ns before the flash. The shock
strength has increased due to spherical convergence. The
bubble radius is 8.2
,urn
at
[3'
which is 7.3 ns before the
flash. The shock has reflected from the center
of
the bubble.
The pressure, density, and temperature are 1.7 bars,
0.0011
glcc, and 530 K at the center
of
the bubble. The outgoing
wave is strong enough to reverse the velocity only for radii
less than 1.5
,urn.
Consequently, the front at 1.5
,urn
develops
into another shock, the second
of
the three shocks.
Figures
46
show
spatial profiles
of
velocity, pressure,
and density at four times
(t
47)
after t
3'
The inner 1.5
,urn
are
shown. The bubble radius is
0.95
,urn
and the second shock
has reached
0.4
,urn
at t
4
,
which is 51.8 ps before the flash.
The bubble radius is
0.86
p.,m
at t
s
,
which is 16.8 ps before
the flash.
The
velocity has a double peak, which becomes the
...
.,
l!:
.,.
0.45
• I
lOps
0.30
0.15
0.00
'""
45.03529 45.03533
Time
(flS)
45.03535
FIG. 10. Total radiated
power
as a function
of
time obtained from the
dissociatiOn/ionization equation
of
state and Eq. (4). We have assumed that
the bubble is optically thin to estimate the radiated power.
The
FWHM
is
less than
10 ps.
Phys. Fluids, Vol.
6,
No.9,
September 1994
second and third shocks.
It
is
also barely visible at t
4
•
The
second shock has reached
0.1
,urn.
The velocity, pressure,
density, and temperature at the front are approximately
11000
p.,ml
p.,s,
0.02 Mbar, 0.2 glcc, and 5 e
V.
The bubble
radius is
0.84
,urn
at t
6
,
which is 6.8 ps before the flash. The
positive velocities show that the second shock has reflected
from the center
of
the bubble, causing a large increase (with
respect to
t
5)
of
the pressure and density [also T (not shown)]
in the air. The temperature at the center
of
the bubble is
""'30
eY.
The momentum in the main implosion quickly reverses
this positive velocity, similarly to
t
3
, and between t6 and
t7
the' flash occurs (at 45.035311 8
p.,s).
The bubble radius is
0.81
p.,m
at
(7,
which is
3.2
ps after the flash.
The
tempera
ture at the center
of
the bubble is
=70
eY.
The outward
moving large positive velocity at
t7
quenches the remaining
inward momentum and arrives at the
airwater
interface 39.7
ps after the flash (see Fig. 2 inset), which is when the mini
mum bubble radius occurs. The interaction
of
the wave and
the interface reverses the inward motion
of
the bubble and
produces a weak fourth shock that travels back toward the
center
of
the bubble. The water temperature at the interface
is
""750 K, and decreases to 300 K beyond a radius
of
5
,urn,
at
h.
The
water pressure and density at the interface are 0.16
Mbar and 1.9 glcc, at t
7
,
which supports using a nonlinear
water equation
of
state.
Figure 7 (solid line) shows the time dependence
of
the
normalized entropy Sf
(=28.8SIR)
at the center
of
bubble,
where
R/28.8 is the gas constant for air. The dashed line will
be discussed later. The entropy was obtained from the calcu
lated timedependent thermodynamic values at the center
of
the bubble, by integrating
dS=[ed(ln
e)+PVd(ln
V)]/T.
This representation
of
dS
improves the accuracy
of
the nu
merical integration, because the coefficients
of
the logarith
mic differentials are slowly varying functions. We have cho
sen arbitrarily to set
Sf
=0
at the beginning
of
the Step 2
calculation
(t=44.925
p.,s).
Although the bubble collapse is
violent at late times, it is nearly isentropic until the final
10
ns, during which three jumps in Sf occur. The three jumps
occur when each
of
the three shocks discussed above reaches
the center
of
the bubble; just prior to
[3,
t
6
,
and
[7'
Figures 8 and 9
show
temporal histories
of
P and T in
four zones
(r
1
4)
near the center
of
the bubble. The insets
will
be
discussed later. The centers
of
these four zones at the
beginning
of
Step 2 were
0.023,0.160,0.985,
and 4.833
p.,m.
The centers at the time
of
the sonoluminescent flash are
0.00060, 0.00319, 0.01586, and 0.09285
p.,m.
The earliest
time shown in these figures is
5
ps before t5 (Figs.
46).
Figure 8 (main figure) shows that the wave reaches r 4
slightly after t
5'
This outer zone compresses gradually until
the wave returns from the center
of
the bubble as a diverging
wave. Spherical convergence causes this same process to oc
cur more violently and rapidly in the inner zones. The maxi
mum pressures, temperatures, and densities in the four zones
are
203, 225, 114, and 28 Mbars, 140, 70, 16, and 2
eV,
and
3.6, 9.5, 13.4, and
10.0 glcc.
It
is possible that the air near
the center
of
the bubble is metallic when the flash occurs,
due to the large peak densities?7 The full width at half maxi
mum (FWHM)
of
the pressures
in
all four zones and the
temperatures in zones r
1 _ 2 are less than 10 ps. The mass
Moss
et
al.
2983
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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 StefanBoltzmann
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 welldefined 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
(=200750
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.
810,
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.
210
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. W7405
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