525
J. Phys. Chem. 1980, 84,525-528
Sonoluminescence Intensity as a Function of Bulk Solution Temperature C. Sehgal, R. G. Sutherland, and R. E. Verrall" Department of Chemistry and Chemical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0 WO (Received July 16, 1979) Publication costs assisted by the University of Saskatchewan
Sonoluminescence flux I is found to decrease experimentally with the bulk solution temperature T. A direct correlation between I and molar Helmholtz free energy of cavitation (A& has been established and can be expressed mathematically as I = a exp(-KAA(T)) (where a and k are constants that depend upon the dissolved gas and the choice of reference state).
Introduction Sonoluminescence is a weak emission of light that is observed when ultrasound is passed through a liquid containing dissolved gases. It is caused by acoustic cavitation, the magnitude of which depends strongly on the physical properties of the medium and the pressure amplitude of the ultrasonics waves. Jarmanl studied sonoluminescence from several liquids and found that the emission intensity I correlated with a2/p, where u and p v are surface tension and vapor pressure of the liquid, respectively. More recently Golubrichii et a1.2 have studied sonoluminescence from 15 liquids and showed that a stronger correlation appears between I and the free energy of molecular interaction. However, to the present time no quantitative correlation between the physical properties of a liquid and sonoluminescence flux I has been established. The results reported here show that in the presence of inert gases a good correlation exists between I and the Helmholtz free energy A associated with cavity growth in aqueous solution. The purpose of this paper is to establish a mathematical relationship between these two variables, Le., I and AA. Various theories have been proposed for the origin of sonoluminescence. The most recent experimental evidence indicates that it occurs within a cavity and is due to the radiative recornbination and radiative relaxation of free radicals formed during the adiabatic compression of the Dirisolved gases may influence sonoluminescence in several ways: by changing the interaction between molecules; by producing different intracavity temperatures due to different values of the specific heat ratio y;and also by participating directly in the chemical reactions within a cavity. To avoid any contribution from the latter only noble gases were chosen for the present studies. Previous results5 indicate that they do not chemically react, at least within the detection limits used in this study.
this are discussed later). A pressure of 0.7 atm was maintained above the solutions at all times by using the saturating gas. Low-resolution sonoluminescence spectra and total emission flux in a given spectral region were obtained in the temperature range 10-70 "C.
Results a n d Discussion Rapid scans of sonoluminescence spectra of neon-saturated solutions in the region 240-420 nm (Figure 1) show the intensity distribution to be invariant, though attenuated with increasing solution temperature. This suggests the chemical reactions producing sonoluminescence do not depend upon the ambient temperature and that the intensity attenuation is largely due to a decrease in the degree of cavitation. Figure 2 shows that with an increase in temperature the relative sonoluminescence intensity I/Io (where Io is the emission intensity at 30 OC17and has values 10000,4500, and 500 counts per second (cps) for krypton-, argon-, and neon-saturated water, respectively) initially decreases and then levels off. In contradiction to the observations reported by Jarman,l sonoluminescence intensity is found to depend on the previous history of the liquid; for example, whether or not the liquid had been previously subjected to cavitation and the length of time it was left standing, etc. It was also found to be affected appreciably by the presence of small amounts of foreign substance^.^ In contrast, the spectral distribution does not depend on the previous history of water. The intensity behavior is due to the change in the tensile strength of the liquid which apparently affects the degree of cavitation but not the intracavity reactions that produce sonoluminescence. The change in tensile strength of gas-saturated aqueous solutions with time may occur for two reasons depending on the nature of the dissolved substance: (i) the radii of nuclei change until they reach a critical radius corresponding to the thermodynamic equilibrium, and (ii) the poor initial wettability of solid impurities (i.e., with large contact angle 50-60") improves with time due Experimentall Section to a gradual dissolution of layers of gas adsorbed on the Distilled water was degassed and saturated with research grade gases by using techniques described p r e v i o ~ s l y . ~ J ~ surface, thereby causing the contact angle to approach zero. Consequently it was found that conditioning the water The gases were obtained from the following suppliers: before intensity measurements gave better reproducibility argon, Carbonic Canada Ltd.; neon and krypton, Matheand less experimental scatter (Figure 2). son. They were used without any further purification. In order to determine a mathematical expression relating Sonoluminescence was measured at 459 f 1 kHz by using a single-photon counting system previously d e ~ c r i b e d . ~ I / I o to temperature, values of In (I/Io)were plotted against 1/T (Figure 3) and a straight line was obtained of the form The insonation cell was designed with a port through which a thermometer could be inserted to measure the bulk soIn ( I / I o ) = m l ( l / T ) In m2 (1) lution temperature T. The latter was regulated to fl "C where m1and m2are the slope and intercept, respectively, by pumping externally thermostated water through an of the straight line described by eq 1. When T = To= 303 inner coil which1 acted as heat exchanger in the insonation K, I = Io, and In m2 = -ml/To eq 1 becomes cell. Gas-saturated water was allowed to stand overnight In (I/Io) = [ m l ( l / T- l / T O ) l before any measurements were carried out (the reasons for (2)
+
0022-3654/80/2084-0525$01 .OO/O
0 1980 American Chemical Society
526
The Journal of Physical Chemistry, Vol. 84, No. 5, 1980
Sehgal, Sutherland, and Verrall 2
600
500
I
400
--
0
0 Q
r(
v
'
300
0
c
200 -I
IO0
1
I
24 0
42 0
-2
X(nrn) Flgure 1. Rapkl scans of low-resolution spectra of neon-saturatedwater (240-420 nm) at various bulk solution temperatures: (a) 25 O C ; (b) 32.5 O C ; (c) 52 O C ; and (d) 69 O C .
>
7t In
\
56 !
i1
--- Ne 0 - - - Ar 0
A
---
Kr
-i
a.
2 t
1 0
,
1
0
10
20
j
\ $
30 T
40
50
60
70
"C
Figure 2. Plot of relative sonoluminescence intensity ( I l l o )vs. bulk solution temperature T . I , is the emission intensity at 30 O C and has a value of 10000, 4500, and 500 cps for krypton, argon, and neon, respectively.
It can be seen from Figure 3 that the slope of the temperature dependence of In ( I / I o )vs. 1/T is virtually the same for all the gases under the experimental conditions used in this study. Free Energy of Cavity Formation and Its Dependence on the Bulk Solution Temperature The free energy o f cavity formation (which, for the sake of brevity, shall be called the free energy) is assumed to be a measure of the intensity of ultrasonic cavitation. It may be defined as the work WA done by ultrasound to form bubbles from the nuclei of dissolved gases. If F is the intermolecular force between the molecules of liquid containing dissolved gas(es), then the free energy required to form 1mol of spherical bubbles of average radius R'from nuclei of average radius Ro is given by the equation AA = -WA =
k
F(R') dR'
(3)
Since F(R9 is a measure of surface_tension u, viscosity q, and vapor pressure pvof a liquid, AA implicity takes into
30
31
32
33 ( I/T)
34 x
35
36
37
1 0 3 OK-'
Figure 3. Plot of In (111,) vs. (11T ) for neon-, argon-, and kryptonsaturated solutions.
account these factors, and therefore would apppear to be a preferable parameter to establish a correlation between sonoluminescence flux and p_ropertiesof a liquid medium. One approach to calculate AA would be to determine F(R9 as a function of R'and integrate eq 3. This would require knowledge of R'(Ro,R) and_also of the exact dependence of F on R'. Alternately, AA can be calculated from macroscopic properties by using classical thermodynamics. The use of the latter to describe some aspects of cavitation may be questioned, but because the formation of a cavity occurs isothermally over a period of time at least an order of magnitude greater than the collapse time, therefore during this intial stage of cavitation equilibrium conditions can be considered to occur to a first approximation and the use of classical thermodynamics is valid. It should be noted that nonequilibrium conditions apply primarily during the cavity implosion. In order to calculate AA it is necessary to known how a cavity grows and what changes occur during this process. A vapor-gas nucleus in a liquid medium exerts a pressure ppo+ pv, where pgo is the pressure due to the gas in a nucleus and pvhas been defined previously. In an ultrasonic field of appropriate pressure amplitude a nucleus grows into a bubble and, during this process, liquid at the interface vaporizes into the c_avity to maintain an equilibrium vapor pressure pv. AA for such a process is zero. Therefore the free energy change associated with this process will be due to the pV work done during bubble expansion. The pressure due to the gas-vapor mixture contained in an isolated spherical bubble of radius R' will be equal to pv p,0(R0/R93. The process is represented by the following equation: (4) G(pV + pg0,fio) G h + P$)
+
+
where G represents the gas inside a cavity with partial molar volume Go and 5, before and after expansion, respectively. The molar free energies of the initial and final states are given by the equations
Ai = A*(?") - RT In (Eo - b) - a / f i O A, = A*(!?') - R T In ( 5 - b ) - a / f i
(5)
(6)
Where A*(!?') is the standard molar free energy, independent of volume, and a and b are the effective van der Waals constants for the gas-vapor mixture of the cavity.
The Journal of Physical Chemistry, Vol. 84, No. 5, 1980 527
Sonoluminescence Intensity vs. Solution Temperature
I
Subtracting eq 5 from 6, one obtains AA = RT In [ ( E o - b ) / ( E - b ) ] + ( a / 5 0- a / i i ) ( 7 )
I
I
a ----A,
If the compressibility factors of the initial and final states are 2, and 2, respectively, then eq 7 can be shown to take the form AA=RTln[(-)
Pg +. Pv Pgo
f
Pv
zw - b(Pg0 + Pv) ZRT - b(Pg + P") Pgo + Pv
J
----
Kr
I+
Pg + Pv
(8)
At equilibrium, the pressure due to the intracavity contents, p g pv,is related to the ultrasonic pressure field P , sin ut, hyd:rostatic pressure pH,and surface tension as shown in eq 9, where Pa, and w are the pressure am2a pg -k pv = P , sin ut + pH + -' R plitude and frequency of the ultrasonic wave, respectively. For convenience, the expansion of the nucleus is assumed to begin a t the condition P, sin ut = -P,, since the nucleus will expand when its walls are pulled out during the negative phase of the ultrasonic wave. As well, the ultrasonic pressure is assumed to be the same inside and outside of the nucleus since the nucleus diameter is much less than the wavelength of the sound field at frequencies less than 1 MHz. Therefore, at the beginning of the expansion R' = Rc,,p g , = pgo,and P, sin ut = -Pamand eq 9 becomes 2u (10) Pgo + PIT = - P a m + PH + RO As can be seen, the pressure exerted by the nucleus contents can acquire a broad range of values depending upon the magnitudes of Ro,p H , and P., As a consequence, eq 8 can be approximated to varying degrees depending upon the value of the intranucleus pressure. Three possible cases are discussed below. Case I. At high intranucleus pressures, Le., when the hydrostatic pressure is large and the radius of the nucleus is small, the cavity contents behave as a nonideal gas. Typically, for water as the liquid medium, when pH = (100 X lo6),P, = 10 X lo6, u = 80 (all in cgs units), and Ro = lo4 cm, the value of pgo+ pvobtained from eq 10 is equal to 2.5 X lo8 cgs units. For such a case the contributions of the effective van der Waals constants and ccmpressibility coefficients are significant in evaluating AA and the complete form of eq 8 should be used. As well, application of general mixiing rules would be required to estimate values of ,a and b for the gaseous mixture and one would expect AA to change appreciably with the nature of the dissolved gas. Case 11. At moderate intranucleus pressures, i.e., when the radius of the nucleus is of the order of 1 X cm and both the hydrostatic and sound field pressure amplitude are lo7 cgs units, the compressibility coefficients 2 and 20will not differ significantly, i.e., 2 N Zo. To a first approximation eq 8 will reduce to the following form:
+
-
AA = R T I n
[ (--)
Pg + Pv Z&T - N P g o + Pv) + Pv Z&T - HP, + P")
Pgo
I+
For a nucleus that expands isothermally to approximately 10 times its initial size and contains comparable
!
"\o
-2 -2 6
-2.2 -I 8 - 1.4 a~(T)(kcol/rnole)
-I 0
Figure 4. Plot of In (I/Io)vs. A A ( T ) for neon-, argon-, and kryptonsaturated solutions.
-
amounts of gas and vapor, one can write pg = P,,(R~/R)~
