J. Phys. Chem. 1995,99, 14619-14627
14619
Confined Electron Model for Single-Bubble Sonoluminescence Lawrence S. Bernstein* and Mitchell R. Zakin* Spectral Sciences, Inc., 99 South Bedford Street, Burlington, Massachusetts 01803 Received: June 5, 1 9 9 9
The origin of sonoluminescence, the conversion of acoustic energy into ultraviolethisible radiation in liquids, has remained elusive. We propose that the emission arises from electrons confined to voids in the hot, dense fluid formed during the final stages of bubble collapse. Such electrons are produced by high-temperature ionization of the bubble constituents. A hard-sphere-based model was developed for the fluid structure, thermodynamics, and confined electron emission. The model is consistent with the observed spectral distributions, power output, and time scale associated with emission from single cavitating rare gas bubbles. Effective temperatures during emission in the 200-700 nm spectral window are predicted to range from 20 000 to 60 000 K.
I. Introduction Sonoluminescence (SL), the emission of light from the violent collapse of bubbles formed when resonant acoustic energy is introduced into a liquid, has been a source of fascination and speculation for over six decades.'-30 The recent discovery of repetitive single-bubble sonoluminescence(SBSL) from a single, stable, cavitating gas bubble in H20'1-'4 has enabled SL to be studied in a highly controlled and systematic fashion, both experimentally' -25 and t h e ~ r e t i c a l l y . ~However, ~-~~ a definitive emission mechanism remains elusive. Considerable interest has been generated by the recently developed imploding shock wave mode126-29in which extraordinarily high temperatures, in excess of lo6 K, are predicted and assumed to generate bremsstrahlung radiation. While this model predicts emission intensities and time scales consistent with SBSL observations, it does not account well for the observed spectral distributions or their variability with bubble gas composition. A plausible model has to be consistent with the following key features of SBSL: (1) an upper limit emission time scale in the range 50100 P S , I ~ - ~(2) * peak powers in the range 1-100 mW,23 and (3) high sensitivity of both the emitted power and spectral distribution to bubble gas c o m p o ~ i t i o n . ~ ~ We propose that SBSL originates from discrete transitions of electrons which are confined to voids (Le., empty cavities) in the dense fluid formed during the final stages of bubble collapse. These electrons are produced by ionization of the atomic and molecular bubble constituents, driven by the high temperatures generated within the collapsing bubble. The small spatial dimensions of the voids result in electron energy level spacings corresponding to transitions in the visible and ultraviolet spectral regions. Blurring of the discrete energy levels, primarily due to spatial variability of the void dimensions, gives rise to a continuous emission spectrum. A substantial experimental and theoretical basis for this SBSL mechanism already exists from previous investigations of the behavior of excess electrons in f l ~ i d s . ~ lIt- is~ well ~ established that for a dense fluid in which the electron-atom (or molecule) interactions are dominated by the repulsive part of the interaction potential (e.g. a hard sphere fluid) the electrons prefer to reside in voids and that these confined electrons give rise to a strong continuum absorption ~ p e c t r u m . ~ ' Previous -~~ work has focused on relatively low temperatures, typically room temperature and @Abstractpublished in Advance ACS Absrracts, September 15, 1995.
0022-3654I95l2099-14619$09.00/0
below, in which the absorption spectrum originating from the ground states of the confined electrons is observed. Because of the much higher temperatures produced in a collapsing bubble, the excited states of the confined electrons will become appreciably populated and thus give rise to an observable emission spectrum. At the high temperatures attained in the collapsing bubble, the electron-atom and atom-atom dynamics will be dominated by the repulsive part of the interaction potential. Thus, in the present work, a hard-sphere-based model of the fluid structure, thermodynamics, and confined-electron emission during the bubble collapse process was used to evaluate the proposed SBSL mechanism. Synthetic emission spectra for the confined electrons were generated using a particle-in-a-rectangular-box model for the electron energy levels and radiative decay rates. Similar particle-in-a-spherical-box models32-34,36have been shown to reproduce the salient features of absorption spectra arising from excess electrons in fluids. The synthetic emission spectral contours are controlled by three variables: the mean void size, the void size variance, and temperature. These variables are all directly related to the reduced density of the fluid inside the cavitating bubble. Monte Carlo simulations of the structure of a hard sphere fluid were performed in order to quantify the relationship of both the mean void size and void size variance to the reduced density. The relationship between bubble temperature and reduced density was derived on the basis of an adiabatic description of the collapse process. The adiabatic thermodynamic calculations utilized the Camahan-Starling equation of state3' for hard sphere fluids and also i-ncluded the effect of excited electronic states on the specific heat ratio. The present study focuses on SBSL from pure rare gas (Rg) bubbles, systems which are well described as hard sphere fluids. Model-generated synthetic emission spectra for Xe, Ar, and He are shown to encompass the experimentally observed SBSL thus providing an estimate of the effective emission temperature. As detailed below, the hard sphere fluid/electronin-a-box model is consistent with the key features of SBSL and yields emission temperatures for the Rg bubbles in the 20 OOO60 000 K range. 11. Electron-in-a-Box Spectral Model As will be demonstrated in section 111, the shapes of the individual fluid voids are highly irregular and are more 0 1995 American Chemical Society
Bemstein and Zakin
14620 J. Phys. Chem., Vol. 99, No. 40,1995
reasonably described as rectangular than spherical. The energy levels and radiative decay rates of the quasi-free electrons confined to such voids are therefore approximated by those for a particle in a rectangular box. For simplicity, it is assumed that the height, width, and length of each box are uncorrelated, allowing each void to be expressed as the sum of three onedimensional particle-in-a-box systems. The energy levels E,, and radiative decay rates A,,-,,-1 for a one-dimensional particlein-a-box with infinite walls are given by38
h2 E,, = i n 2 8ma
a VARIATIONS
(1)
and
'.
~~~
w
11
4w
100
T VARIATIONS
oz where h is Planck's constant, m and q are the electron mass and charge, respectively, a is the box size, n is the quantum number (n = 1, 2, 3, ,..), and c is the speed of light. For an electron in such a box, the transition wavelengths A (nm) and radiative decay rates A (s-l) can be expressed as a function of the box size a (A); for the three lowest transitions (2 1, 3 2, and 4 3 ) , these are given by
10
1
- -
-
A,, = 1.76 x 10"/a4;A21 = 1 0 . 9 ~ ~
(3)
= 9.50 x 10"/a4;A32 = 6 . 5 5 ~ ~
(4)
A,, = 2.72 x 10'2/a4;4, =4 . 6 8 ~ ~
(5)
A32
Only these three lowest transitions are required to fully describe the emission spectrum within the 200-700 nm wavelength region covered by the SBSL experiments. Taking a representative box size of a = 5 8, results in A21 = 2.82 x lo8 s-l and 1121 = 545 nm. The transitions are strongly allowed with radiative lifetimes on the order of r = 1/A21 = 3 ns. These short lifetimes enable a significant fraction of the excited-state electrons to radiate during the approximately 10-100 ps time scale for SBSL. To predict the emitted power and its spectral distribution, a prescription is required for the absolute number of confined electrons, as well as for their relative distribution among the particle-in-a-box energy levels. The instantaneous emitted power for a given n m transition and box size can be expressed as
-
Prim = 36hvnflR&qnm
exP(-En/kBT)/Qe(T)
(6)
where E is the ionization efficiency, hv is the photon energy, k~ is the Boltzmann constant, N R is~ the number of Rg atoms in the initially formed bubble, En is the energy of the upper transition level, and Qe = O . S ( J ~ ~ B T / E is I ) "the ~ electron-in-abox partition function. The multiplicative factor of 3 arises because the box is three-dimensional. For a fixed box size, the particle-in-a-box model predicts discrete emission lines. Blurring of these emission lines, due to the distribution of void dimensions, produces a continuous spectrum. The blurring function is represented as
Wa) = H@(a) Q,/kBT)
(7)
where HO is a normalization constant, D(a) is the void size distribution function, and the last two factors account for the
loo 200 300 400 500 600 700 WAVELENGTH (nm)
800
Figure 1. Sensitivity of electron-in-a-box synthetic emission spectra to individual variations in mean box size ao, box size variance a,and
temperature T. relative thermal distribution of electrons among the different size voids. The void size distribution is assumed to be Gaussian,
where DOis a normalization constant, a0 is the mean void size, and a is the void size variance. In section HI, it is shown that a0 and a can be directly determined from Monte Carlo simulations of the structure of a hard sphere fluid and that they depend only on the density of the fluid. The sensitivity of SBSL spectral shape to independent variations in a0, a, and T,as computed by the particle-in-a-box model, is illustrated in Figure 1. The relative spectral shapes do not depend on NR$,A, or 4, although as discussed below, these variables determine the absolute power levels. The variations shown in Figure 1 are relative to a nominal set of parameter values: a0 = 6.9 A, a = 0.9 A, and T = 26 000 K. The top panel indicates that the peak location shifts significantly for relatively modest changes in mean box size. The middle panel demonstrates that the width, and even the appearance, of the spectrum depends strongly on the box size variance. Note that for the smallest value of a in Figure 1 (Le. 0.4 A), the three component transition wavelengths are distinguishable. The bottom panel indicates that the spectral shape depends only mildly on T. It is clear that the present model is capable of representing a broad variety of SBSL spectral features.
III. Void Size Distribution Monte Carlo simulations of the three-dimensional structure of a hard sphere fluid were undertaken to characterize the void size distribution as a function of the reduced density,
e* =ed
(9)
where 8 is the particle number density and 0 is the hard sphere diameter. The Metropolis algorithm39was used in conjunction with 1000 simulated atoms and 1000 moves for each atom.
Confined Electron Model for Single-Bubble Sonoluminescence
J. Phys. Chem., Vol. 99, No. 40,1995 14621
p*=o . 9
p*=1.0
40
5
40
d
30
w
W
0
z a
30
y
20
20
a
!-
I-
2 0
Kl U
0
10
0
10
0
10
20
30
0
40
0
Figure 2. Representative two-dimensional void map for hard sphere fluid with reduced density e* = 0.9. The voids are represented by the dark areas.
Cavitation typically produces an ES: 103-fold increase in bubble density. For a Rg bubble initially at 1 atm pressure, this yields a reduced density on the order of e* ES: 1 at maximum compression. Calculations were performed for five reduced densities spanning the e* = 0.8-1.1 range, using u = 4 A; however, as indicated below, the derived void size distribution parameters, a0 and a,are directly proportional to u. To visualize the voids, they were computationally filled with small test beads. A bead diameter of 1 8, was selected in order to filter out the many smaller voids which are too small to confine an electron (i.e. the zero-point-energy penalty is too high; see eq 7). The void size distributions were extracted in the following manner: (1) two-dimensional void maps were constructed by taking 1 8, thick slices (corresponding to the diameter of the test bead) through the simulation volume, (2) histogrammed distributions of the one-dimensional length scales of the voids in these slices were constructed, (3) the mean and variance of the histogrammed distributions were used to define a smooth, analytical distribution via eq 8, and (4)the mean and variance were approximately corrected for the effects of ignoring the third dimension. Examples of the two-dimensional void maps for e* = 0.9 and 1.Oare presented in Figures 2 and 3, respectively; the voids are represented by the dark bead-filled areas. The voids have a highly irregular structure with a broad distribution of length scales. Comparison of Figures 2 and 3 illustrates the important result that an increase in reduced density reduces both the average void size and the spread in void sizes. For simplicity, it is assumed that the three dimensions of the rectangular box are uncorrelated. Thus, the length scale distribution we wish to extract from the two-dimensional slices corresponds to the lengths and widths of equivalent area rectangles drawn to approximately match the shapes of the actual two-dimensional voids. Examples of the histogrammed length-scale distributions for e* = 0.9 and 1.0 are displayed in Figures 4 and 5 , respectively. Lengths smaller than u were not considered since they give rise to emission wavelengths below the 200 nm shortwavelength cutoff of the experimental SBSL spectra. For convenience in performing the synthetic spectral calculations, it is useful to have an analytical representation of the histogrammed distributions. A Gaussian distribution was used, D(a) from eq 8, where a0 and a were determined by requiring that the mean and variance calculated with D(a) from a = u to a = match those for the histogrammed distribution. The Gaussian
-
20
10
DISTANCE ( A )
30
4c
DISTANCE ( A ) Figure 3. Same as Figure 2, except e* = 1.0.
p*=o . 9
= 52 + 3
2
1.0
I+
n W
+
-
0.0 -
-
0.6
-
0.4 -
-
0.2 -
-
a IUJ
-;i - Monte C a r l o __._ Gaussian Fit
a
-1
0.0
I
I
I
,
-.--._ -_ I
BOX S I Z E , a ( A ) Figure 4. Void size distribution for hard sphere fluid with reduced density e* = 0.9, obtained from Monte Carlo simulation. Raw histogrammed distribution and Gaussian fit are denoted by solid and
dashed lines, respectively. representations of the histogrammed distributions are also shown in Figures 4 and 5 . Approximate correction factors for a0 and a, due to neglect of the third dimension in the two-dimensional void size distribution analysis, were derived by considering the case of randomly distributed spherical voids of identical radius R. The two-dimensional void map for such a case would consist of a collection of discs whose radii would be distributed between 0 and R, since a slice through any particular sphere can occur with equal probability anywhere between its equatorial plane and its north or south pole. Thus, the average two-dimensional radius would be less than R,and the variance would be nonzero. For this simple case, exact correction factors for the mean and variance can be derived and lead to the following adjustments to a0 and a:
where the subscript u denotes an uncorrected parameter. Derivation of correction factors for other, more realistic, void
Bernstein and Zakin
14622 J. Phys. Chem., Vol. 99, No. 40, 1995
calculation^^^^^^ b was equated to the excluded volume of the bubble gas in its normal liquid state. This choice was based on the assumption that the liquid density is the maximum achievable density for a highly compressed, high-temperature fluid. However, the maximum density is determined by the close-packing limit and is typically about twice that for the normal liquid. The detailed results reported here are for a base-line SBSL model with a temperature-independent hard sphere diameter. However, it is recognized that for a real atom the effective hard sphere diameter will vary considerably over the large bubble collapse temperature range. An initial evaluation of the effect this exerts on both the bubble collapse thermodynamics and the SBSL emission spectra will be briefly described later. For this evaluation the temperature-dependence of the hard sphere diameter was taken to be4*
p*=1.0 Monte C a r l o Gaussian F i t
c
i
0
BOX S I Z E , a ( A ) Figure 5. Same as Figure 4, except @* = 1.0.
shapes, such as a cylinder or a rectangle, is much more complex. A qualitative consideration of these cases indicates that the and a corrections derived for a sphere underpredict the true corrections. The final results of the distribution analysis are summarized by (note the scaling with a ) ado= 3.25 - 1.64@*
(12)
d o = 1.16 - 0 . 7 9 5 ~ *
(13)
The estimated statistical errors for these quantities are =*lo% for a0 and w t 2 0 % for a. In addition, these fits apply to values of e* between 0.8 and 1.1 and should be used with caution outside this region.
where TO= 297 K was chosen as the reference temperature, and n is the exponent characterizing the repulsive wall of the interaction potential (Le. V = l/F). For a value of n = 12, appropriate for the Lennard-Jones 6-12 potential, and TdT 0.01 at &in, one computes ah0 = 0.68, a substantial reduction in the hard sphere diameter. The specific heat ratio is defined by
where C, is the constant volume heat capacity. For an atom, the heat capacity is the sum of translational C : and electronic C,' components,
IV. Bubble Collapse Thermodynamics It is generally accepted that the macroscopic thermodynamic where C,'lke = 3/2, and C,' is a temperature-dependentfunction state (i.e. not including shocks) of the bubble during the portion that depends on the energy level distribution of the excited of its collapse between its initial radius, Ro, and its minimum electronic states. There are additional contributions to C, due radius, Rmin, is adequately described as an adiabatic p r o c e s ~ . ~ ~to~ ionization ~ and the presence of e- and Rg+. However, it is For an adiabatic expansion or compression, e* and Tare related shown later that the fractional ionization for Rg bubbles is low, in the following manner? on the order of for Xe. Thus, these extra contributions to C, are of secondary importance and are not included in this analysis. At low temperatures Cue= 0, and one obtains the familiar result for the specific heat ratio of a monatomic gas, y = 1.67. However, at the high temperatures generated by the where A@*)is the fluid equation of state cf = 1 for an ideal bubble collapse, the contribution of C,"to y is substantial. The gas), and y(T) is the specific heat ratio. The Carnahan-Starling derivation of a useful analytical approximation for C,"begins (CS) equation of state is employed in the evaluation of eq 14; with its formal definition$O it is given by
where y =(n/6)~*
(16)
The maximum value of y occurs at the close-packing limit for hard spheres where e* = and y = 0.740. For comparison purposes, we also consider the van der Waals (VDW) equation of state:' which is more approximate than the CS equation of state, but is often sed^^,^^ in bubble collapse thermodynamic calculations. It is expressed as
fi)= 1/(1 - by)
(17)
where b is treated as an adjustable parameter. In previous
where Q is the electronic partition function, and gi is the degeneracy of the ith energy level, Ei. A problem arises in the evaluation of Q, due to the infinite number of finite-energy Rydberg states below the ionization energy, Eion. This yields Q = for any finite temperature and is known as the "hydrogen atom catastrophe". Its resolution lies in the recognition that an infinite number of finite-energy Rydberg states exist only for an atom (or molecule) in a completely noninteracting, unbounded universe.43 When the atom is placed in a finite OQ
Confined Electron Model for Single-Bubble Sonoluminescence enclosure of any size, only a finite number of bound levels exist below E,,,, and thus Q becomes finite at all temperatures. The number of bound levels depends on the size of the enclosure. A simple level cutoff criterion, previously utilized43 in the determination of Q, is to include only those states whose Bohr orbital radius is less than the size of the enclosure. This criterion can be expressed in terms of the maximum energy level Emax included in the Q sum, r0
AE = E,,, - E,,, = -R,,, r where ro = 0.529 8, is the Bohr radius of the ground-state hydrogen atom, r is the size of the enclosure, and R R =~ 13.6 ~ eV is the Rydberg constant. For the current problem, r corresponds to the average distance between nearest neighbors (i.e. the first solvation shell) minus the hard core radius, u/2. It is noted that r varies slowly with reduced density, r = (only e* > 0.2 is important for the C,' contribution to y), and for simplicity is taken to be constant. The sensitivity to the choice of r is addressed through consideration of two bounding choices, r = 5 8, for which A E = 1.44 eV, and r = 2.6 8, for which AE = 2.77 eV. At this point, Q can be evaluated by explicitly summing over all states whose energy is an amount AE below Eion. Comprehensive tabulations of atomic energy levels are available4 and can be used for this evaluation; however, for the rare gases, a simplification arises because the excited states reside in a relatively narrow energy band that is well separated from the ground state, EO = 0. For example, considering Ar and taking A E = 1.44 eV, one finds that the excited electronic levels span the range 11.5-14.2 eV. For the rare gases, the partition function is approximated by taking the energies of all the excited states to be equal to the median E of the bounding excitedstate energies, and the degeneracy g is taken as the sum of the degeneracies of all the contributing levels. This results in the following formulas for Q and C,':
J. Phys. Chem., Vol. 99, No. 40, 1995 14623
c
wa
1.4
0 H
LL
;1 . 2 U
a m
_ - _ _ .Xe I
1.0
I
I
(
I
l
l
io5
io4
io3
TEMPERATURE ( K ) Figure 6. Variation of specific heat ratio y with temperature for He, Ar, and Xe, computed with cutoff energy AE = 1.44 eV. Results illustrate the effect of electronic heat capacity on y .
a n w
a x
F
VOW y ( T ) b - 2 . 7 3
............. CS y=1.67
io4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
REDUCED DENSITY, p*
For AE = 1.44 eV, the {g,E(eV)} values for He, Ar, and Xe, are {52,21.4}, {114,12.9}, and {121,10.1}, respectively. For A E = 2.77 eV, the E's are reduced by 0.67 eV and the g's are reduced by a multiplicative factor of approximately 0.25, relative to their values for AE = 1.44 eV. The effect of Cue on y is illustrated in Figure 6, where a substantial decrease in y is evident over a wide temperature range for each of the rare gases. At sufficiently high temperature, y is seen to approach its lowtemperature limit, which can be understood by noting that Cue = 1 / P 0 for very large T. In general, any contribution to the heat capacity that arises from a finite number of energy levels will lead to a local minimum in y. Thus, in generalizing the approach to diatomic molecules, one must take into account the fact that only a finite number of rotation-vibration states exist for any given electronic state. Using the formulas presented above for the equation of state and y , we proceed with the evaluation of the adiabatic compression relationship in eq 14. In Figure 7, the results of using the CS and VDW equations of state with different assumptions for y(T) are depicted for an Ar bubble. For a constant specific heat ratio of y = 1.67, it is seen that the VDW and CS equations of state yield similar curves. For the temperature-dependent y(T), the CS and VDW curves are similar
-
Figure 7. Temperature-density profiles for adiabatic collapse of an Ar bubble, computed from eq 14. The four curves pertain to combinations of the VDW (with b = 2.73 for liquid Ar) and CS equations of state with either a constant ( y ) or temperature-dependent (y(T); AE = 1.44 eV) heat capacity ratio.
e* a 0.7, at which point the VDW curve diverges (i.e.
m)- = as by - 1 in eq 17). The use of the VDW equation up to
of state introduces an unphysical, sharp cutoff to the bubble collapse thermodynamics. Comparison of the constant and temperature-dependent y results highlights the importance of including the electronic heat capacity contribution in bubble collapse thermodynamic calculations. The sensitivities of the thermodynamic calculations to AE and to the use of a temperature-dependent hard sphere diameter are explored in Figure 8. It is expected that refining the approach to include a e* dependence to AE will lead to a curve intermediate to those shown in Figure 8. Note that in Figure 8 the reduced density for the u(T)curve is defined relative to the varying hard sphere diameter. Since u(T)/a(To) 0.7 at high temperatures, for a given e*,u(T)corresponds to a particle number density 8 which is a factor of (1/0.7)3 % 3 higher than for u(T0). The CS y(T) curves for He, Ar, and Xe are compared in Figure 9, which indicates that the effective emission temperature will be a sensitive function of both the bubble gas composition and the maximum value of e* attained in the collapse.
Bemstein and Zakin
14624 J. Phys. Chem., Vol. 99, No. 40, 1995
A r AE=2.77eV a(T,)
Ar AE-1.44eV
u(TJ ............. Ar A E = i . ~ e vu ( T i
I
\ '
r:
,
Z
/
\
zz
. ,
Y
Y
;; lo-" a
a L
u W a cn i
pR=0.8 T-24. 200 K p*=O.9 T=28, 800 K p " = l . O T-42, 900 K
10-12 300
200
400
600
500
700
WAVELENGTH (NM) Figure 10. Comparison of model synthetic spectra with SBSL dataz3 for a repetitively cavitating Xe bubble.
-r -
........... "
I-
i
\
' ;1
; ,
Ar
i/
. ,
3
*
300
200 ;
Ar D a t a l H i l l e r e t a 1 . i 2 3 p " = O . 8 T=32. 600 K ~ " ' 0 . 9 T=39, 500 K p " = l . O T-63. 100 K
0
z
;:'
He
10-10
600
500
700
WAVELENGTH (NM)
*'
104 I ' I ! 0.0 0 . 2
400
0.4
0.6
0.8
1.0
1.2
REDUCED DENSITY, p* Figure 9. Temperature-density profiles for adiabatic collapse of He, Ar, and Xe bubbles, computed using the CS equation of state, y(T), AE = 1.44 eV, and constant hard sphere diameter a(T0). V. Comparison to Experimental SBSL Spectra The sensitivity calculations discussed in section I1 indicated that the electron-in-a-box model has the flexibility to represent a broad variety of spectral curves; however, a more important consideration is its consistency with the observed SBSL spectra.23 Synthetic spectra were generated for each Rg at three reduced densities, using a&*) and a(@*)as given by eqs 12 and 13 and Q*) as shown in Figure 9. The data-model comparisons are presented in Figures 10-12. For display purposes, the model calculations were normalized to the peak value of the experimental data. A wavelength-independent value of 2.2 x Wlnm was added to each model spectrum to account for what appears to be either detector dark noise or a low level of background light leakage in the experimental data. This background feature can be seen most clearly in the longwavelength tail of the He data (Figure 12). Inspection of the data-model comparisons reveals that for each Rg system there exists a value of e* for which the synthetic spectrum reasonably reproduces the observed spectrum. Note that the synthetic spectra correspond to the instantaneous emission at a single
Figure 11. Comparison of model synthetic spectra with SBSL dataz3 for a repetitively cavitating Ar bubble.
10-10 1
r: Z
\
z=
Y
w
0 Z
a 10-11
a
a J Q
a
+ 0 w
b
I
10-121
I
200
'
'
I
"
'
300
I
'
400
"
I
'
I
I
'
"
500
600
"
I
'
I
1
700
WAVELENGTH (NM) Figure 12. Comparison of model synthetic spectra with SBSL dataz3 for a repetitively cavitating He bubble.
density e* along the bubble collapse profile, whereas the observed spectra correspond to an integral along the entire profile. Thus, if the actual emission is dominated by a
Confined Electron Model for Single-Bubble Sonoluminescence
TABLE 1: Correlation of SBSL Model Parameter@with Fundamental Rare Gas Atomic Propertiesb e* T(K) ao(A) a & ado BTIEion Xe 0.90 29 000 7.0 1.8 1.8 0.21 Ar He
0.90 0.70
40000 56000
5.8 5.5
1.4 1.6
1.8 2.1
0.22 0.20
a a. (mean box size), a (box size variance), u (hard sphere diameter), and E,,, (ionization energy). Hard sphere diameters (ref 42) for He, Ar, and Xe are 2.6, 3.24, and 3.93 A, respectively, and ionization energies are 24.6, 15.8, and 12.1 eV, respectively.
sufficiently narrow interval of reduced densities, then it is appropriate to compare the experimental integrated emission to the theoretical instantaneous emission. As discussed in section VI, this is indeed the case: it is estimated that the observed spectra correspond to a reduced density interval A@* 0.1. It is expected that integration of the theoretical spectra over this narrow reduced density interval will not degrade the quality of the agreement with the experimental spectra. Table 1 summarizes ao, a, and T derived from the “best” e* in Figures 10-12, respectively, for the three Rg systems. In view of the above discussion, the derived values should be regarded as characteristic of the average state of the bubble during the emission process; for example, T is thus indicative of an effective (not peak) emission temperature. The effective temperatures for the three Rg systems fall within the 20 00060 000 K range, far below the temperatures of > lo6 K required for the imploding shock wavehremsstrahlung SBSL Note that there are strong correlations between the derived variables given in Table 1 and two fundamental atomic properties, the hard sphere diameter u of the rare gases and the Rg ionization energy Eion. For all systems, the mean box size scales directly with the atomic hard sphere diameter, with a proportionality constant of approximately 2. This indicates that in all cases a high degree of compression is achieved during cavitation, and the confined electrons which emit in the 200700 nm spectral window are primarily confined to voids in which one of the three dimensions is ~ 2 u Note, . however, that a fraction of the confined electrons may reside in smaller voids whose diameter may be as small as RU. For the rare gas systems, u 5 4 A, and thus the emission wavelengths for electrons confined to these smaller voids would occur below 175 nm (see eq 3). Such emission is not directly observable due to attenuation by both H2O and air. From Table 1 it is apparent that the cavitation temperature during emission in the 200-700 nm window is highly system-dependent. However, all systems radiate at quite similar reduced temperatures Tr = k$”/EiOn. This indicates directly that the ionization potential is a driving factor for SBSL. The values presented in Table 1 were obtained using a constant hard sphere diameter and a cutoff parameter of AE = 1.44 eV for the excited-state electronic energy levels in the determination of C;. Calculations were also performed using AE = 2.77 eV and resulted in comparably good spectral fits, where the “best fit” @*’s were reduced by ~ 0 . 0 5and the Ts were increased by ~ 2 0 % relative to the values for AE = 1.44 eV. Calculations performed with a temperature-dependent hard sphere diameter resulted in good spectral fits when the e*’s were reduced by ~ 3 3 %and the Ts were reduced by ~ 2 5 % relative to the constant diameter calculations.
VI. SBSL Peak Power and Time Scale The number of confined electrons in a cavitating bubble is controlled by a variety of electrodion production and destruction chemical reactions. The simplest representation of this chem-
J. Phys. Chem., Vol. 99, No. 40, 1995 14625 istry for the pure Rg systems involves just two reactions: collisional ionization
Rg
three-body recombination
+ Rg - Rg+ + e- + Rg Rg’ + e- + Rg
-
(Rl)
R g + R g (R2) If one assumes that the electron concentration is determined by the equilibrium limit of these reactions, then an upper limit to the fractional ionization is given by Saha’s formula45
-E* - 2{ ( ~ Q ) R ~ + / ( ~ Q{2nmekB ) R ~ I T/h213’2
X
1-5 exp(-Ei,n/kBr)/@ (26) where 4Q is the product of the translational ( 4 ) and electronic (Q) partition functions. The ratio of the translational partition functions, qRg+/qRg, is equal to unity (Le. the masses of Rg+ and Rg are essentially identical), and the Rg electronic partition function is given by eq 24. For Rg+, the excited-state energy levels lie at much higher energies relative to the ground state than for Rg, and for the temperature regime of interest here (Le. T < 100000 K) it is adequate to set Q R ~ equal + to the degeneracy of the Rg+ ground state gRg+ (gRg+ = 2 for He and 6 for Ar and Xe). Equation 26 is strictly valid only for an ideal, low-density gas. The equilibrium constant for a chemical reaction taking place in a dense fluid (such as a cavitating Rg bubble) will in general be much different than for the same reaction occumng in a low-density fluid. The shift in the fractional ionization on going from a dilute to a dense fluid can be estimated by considering the changes in the Rg, Rg+, and electron partition functions that define the equilibrium constant. The fractional ionization in a dense fluid can be formally expressed as46 F2
where the subscripts i and f denote the ideal gas and fluid, respectively, and Ki is the ideal gas fractional ionization equilibrium given by the right-hand side of eq 26. In the limit that both the Rg and Rg+ interactions with the fluid can be reasonably approximated as those for a hard sphere, which is appropriate at the high temperatures of the nearly collapsed bubble, the product of the first two terms in eq 27 will be close to unity. The effect of a dense fluid is to trap the electrons in voids, giving rise to a finite zero-point energy and reducing the volume of the bubble accessible to the electrons. Taking this into account results in the following approximation to eq 27:
where I$ is the fraction of the total void volume that is accessed by the electrons, (1 - e*)/& is the total void volume (defined as the total bubble volume minus the close-packing volume), and El is the one-dimensional zero-point energy. 4 accounts for the degree of localization of the electron in a void. While I$ is in principle ~ a l c u l a b l e ,given ~ ~ , ~details ~ of the fluid structure and the electron-fluid interactions, such a computation is beyond the scope of the present study. We instead establish approximate limits for 4 and show that the observed SBSL power is consistent with these limits. An upper limit is obtained if the electron can readily tunnel between adjacent voids and thereby access all the available free volume; for this case 4 =
Bernstein and Zakin
14626 J. Phys. Chem., Vol. 99, No. 40, 1995
Xe + = l . O
- Model (Wu and
a
.._. R i s e - T i m e W i n d o w ~
............. Rise-Time W i n d o w ( l / e l
a 115
(l/e)
H
$
0.95
, , , ,I
,
,
(
/
r a
0
= REDUCED DENSITY, P' Figure 13. Calculated instantaneous emitted power for a cavitating Xe bubble for two values of the electron localization parameter 4. Also indicated is the reduced density interval corresponding to a factor of e variation in power for 4 = 1.0 x
1. On the other hand, if the electron is largely confined to a single void, then 4 0.8. This dependence can be used to estimate an effective emission time scale for the SBSL model. To compare with experiment, we take the SBSL
4
0.90 0
30
! l , , i , ,I i
i
i
:
60
90
/ ,,
, , ( , ,
120
~,,, 150
RELATIVE TIME (PSEC) Figure 14. Temporal dependence of normalized bubble radius near R,i,, based on the model of Wu and Robertsz6 Also indicated is the time interval corresponding to a factor of e variation in power.
time scale to be the time interval required for the instantaneous power to rise from (l/e)Pm, to Pmm, where Pm, is the maximum power (assumed here to occur at Rmin). As indicated by the dotted lines in Figure 13, the power rises by l/e for a reduced density interval of A@* 0.1 centered about e* = 0.9. This is converted to a time interval by relating the change in bubble radius corresponding to A@*to the temporal collapse rate of the bubble. The fractional change in bubble radius is given by ARJR = (1/3)Ag*/@*= 0.037 (derived from R ( l / ~ * ) l ' ~A ). previous theoretical calculation26 of the temporal dependence of the radius of an air bubble, R(t), in the vicinity of Rmin is shown in Figure 14. We have normalized the calculation by Rmin (i.e. R(t)/Rmin) and assume that the resulting relative time dependence is reasonably characteristic of all SBSL systems. For ARIR = 0.037, which corresponds to R/Rmin= 1.037 in Figure 14, it is seen (dotted lines) that the l/e SBSL time scale predicted by the present model is approximately 18 ps. This is consistent with the experimentally based 50-100 ps upper limit range for the SBSL time ~ c a l e . ' ~ - ' ~
VII. Concluding Remarks A simple confined electron model of SBSL has been developed and shown to be consistent with the experimentally determined spectra, time scale, and power levels for cavitating bubbles containing pure He, Ar, and Xe. This model is based on a physical description of SBSL that includes the following fundamental elements: (1) the bubble collapse produces a highdensity, high-temperature fluid, (2) the thermodynamic state of the bubble fluid is consistent with an adiabatic description of the collapse process, (3) voids form in the high-density fluid and are characterized by a broad distribution of sizes, (4) electrons produced by high-temperature ionization of the bubble atoms become trapped in the voids, (5) the confined electrons, reasonably described by a particle-in-a-box spectral model, give rise to a continuum emission spectrum in the visible and ultraviolet spectral regions, and (6) the strong dependence of the total bubble emitted power on the fluid density gives rise to a very short emission time scale. Analysis of experimental SBSL spectra using the hard sphere model indicates an effective bubble temperature for ultraviolethisible emission of 20 00030 000 K for Xe, 30 000-40 000 K for Ar, and 45 000-60 000 K for He. Application of the hard sphere model to both mixed Rg/N2 SBSL spectra and the strong underlying continuum observed
Confined Electron Model for Single-Bubble Sonoluminescence in most multiple-bubble sonoluminescence spectra will be reported elsewhere.
Acknowledgment. The authors gratefully acknowledge Spectral Sciences, Inc., for supporting this effort. We wish to thank D. Chandler for informative discussions on the behavior of excess electrons in fluids, S . Adler-Golden and M. Bemstein for suggestions and review of the manuscript, and S . Richtsmeier for preparation of the artwork. References and Notes (1) Frenzel, H.; Schultes, H. Z . Phys. Chem. 1934, 827, 421. (2) Crum, L. A. Physics Today 1994, 47, 22. (3) Crum, L. A.; Roy, R. A. Science 1994, 266, 233. (4) Walton, A. J.; Reynolds, G. T. Adv. Phys. 1984, 33, 595. (5) Suslick, K. S.; Doctycz, S. J.; Flint, E. B. Ultrasonics 1990, 28, 280. (6) Flint, E. B.; Suslick, K. S. Science 1991, 253, 1397. (7) Flint, E. B.; Suslick, K. S. J. Phys. Chem. 1991, 95, 1484. (8) Flint, E. B.; Suslick, K. S. J. Am. Chem. Soc. 1989, 111, 6987. (9) Sehgal, C.; Sutherland, R. G.; Verrall, R. E. J. Phys. Chem. 1980, 84, 388. (10) Didenko, Y. T.; Pugach, S. P. J. Phys. Chem. 1994, 98, 9742. (11) Gaitan, D. F.; Crum, L. A. In Frontiers of Nonlinear Acoustics, 12th ISNA; Hamilton, M., Blackstock, D. T., Eds.; Elsevier: New York, 1990; pp 459-463. (12) Gaitan, D. F.: Crum. L. A.; Roy, R. A,: Church. C. C. J. Acoust. Soc. Am. 1992, 91, 3166. (13) Crum, L. A. J. Acoust. SOC.Am. 1994, 95, 559. (14) Roy, R. A. Ultrason. Sonochem. 1994, 1, S5. (15) Barber, B. P.; Putterman, S. J. Nature 1991, 352, 318. (16) Barber, B. P.; Hiller, R.; Arisaka, K.; Fetteman, H.; Putterman, S. J. J . Acoust. SOC. Am. 1992, 91, 3061. (17) Lofstedt, R.; Barber, B. P.; Putterman, S. J. Phys. Fluids 1993, AS, 2911. (18) Barber, B. P.; Putteman, S. J. Phys. Rev. Lett. 1992, 69, 3839. (19) Hiller, R.; Putterman, S. J.; Barber, B. P. Phys. Rev. Lett. 1992, 69, 1182. (20) Hiller, R.; Barber, B. P. J. Acoust. SOC.Am. 1993, 94, 1794. (21) Atchley, A. A. In Advances in Nonlinear Acoustics; Hobaek, H., Ed.; World Scientific: Singapore, 1993; pp 36-42.
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JP95 1571U
