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Acoustic cavitation series: part five rectified diffusion
Acoustic
cavitation
Rectified
series: part five
diffusion
L.A. CRUM A general review is given of the mechanism of rectified diffusion. The equations that describe the threshold acoustic pressure amplitude as well as the growth rate are presented. Simplified versions of the complicated threshold equation are also obtained for two regions that are of particular interest. Graphical representations of the equations for a variety of physical parameters are given as well as a comparison between the available measurements and the theoretical, predictions. Finally, some suggested areas of future research in this- area are presented. KEYWORDS:
ultrasonics, cavitation, bubble dynamics
contracts, the concentration of gas (moles 1-i) in the interior of the bubble increases, and gas diffuses from the bubble. Similarly, when the bubble expands, the concentration of gas decreases, and gas diffuses into the bubble. Since the diffusion rate is proportional to the area, more gas will enter during expansion than will leave during the contraction of the bubble; therefore, over a complete cycle, there will be a net increase in the amount of gas in the bubble.
Introduction A concept of general interest in the area of acoustic cavitation is that of ‘rectified diffusion’. This process involves the slow growth of a pulsating gas bubble due to an average flow of mass into the bubble as a function of time. This ‘rectification of mass’ is a direct consequence of the applied sound field and can be important whenever a sufficiently intense sound field (acoustic pressure amplitude greater than 0.01 MPa, say) exists in a liquid containing dissolved gas. That is, the liquid may be under-saturated, saturated or oversaturated with gas. If the liquid is not sufficiently oversaturated, a free gas bubble will gradually dissolve due to the so-called Laplace pressure generated in the interior of the bubble by the surface tension. The value of this pressure is given by Pi = 20/R, where Pi is the internal pressure, u is the surface tension and R is the radius of the bubble.
The second effect is the ‘shell’ effect. The diffusion rate of gas in a liquid is proportional to the gradient of the concentration of dissolved gas. Consider a spherical shell of liquid surrounding the bubble. When the bubble contracts, this shell expands, and the concentration of gas near the bubble wall is reduced. Thus, the rate of diffusion of gas away from the bubble is greater than when the bubble is at its equilibrium radius. Conversely, when the bubble expands, the shell contracts, the concentration of gas near the bubble is increased, and the rate of gas diffusion toward the bubble is greater than average. The net effect of this convection is to enhance the rectified diffusion. It has been shown that both the area and the shell effects are necessary for an adequate description of the phenomenon.
It is easy to see that for gas bubbles of the order of 1.0 cm in diameter, this pressure is quite small (less than 0.001 bar for a gas bubble in water); however, for a bubble of the order of 1 pm in diameter, this pressure becomes significant (greater than 1 bar). Thus, for free gas bubbles in liquids that are not oversaturated with gas and which do not rise to the surface, their ultimate fate is slowly to dissolve. (Of course, pockets of free gas may be stabilized against dissolution by a variety of mechanisms, but that is another topic).
Historically, the concept of rectified diffusion was apparently first recognized by Harvey’ et al, who considered the importance of this concept in the formation of bubbles in animals. They were able to recognize that the area effect would lead to growth and called rectified diffusion the ‘principle of incremental enlargement’, recognizing that the growth during one cycle may be quite small. Blake2 was the first to attempt a theoretical analysis of rectified diffusion, but considered only the area effect, and thus his crude approach was in considerable disagreement with the first reported measurements by Strasberg+4. Pode5 and Rosenberg6 attempted to refine Blake’s theory but their results were not much different from that of Blake. The
With the presence of an applied sound field, however, the bubble radius is forced to oscillate about an equilibrium value, and these oscillations can prevent the eventual dissolution of a bubble and even cause it to grow because of the following effects. The first effect is an ‘area’ effect. When the bubble The author is in the Physical Acoustics Research Laboratory, Department of Physics and Astronomy. The University of Mississippi, Mississippi 38677. USA Paper received 24 May 1983.
0041-624X/84/0502 ULTRASONICS.
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1984
15-09/$03.00
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Buttetworth
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importance of convection was demonstrated when Hsieh and Plesset’ correctly included this effect in their solution and Strasberg4 showed that their theory generally agreed with the results of his measurements. Eller and Flynn* extended the analysis to include nonlinear or large amplitude effects by treating the boundary condition of the moving wall in a slightly different way. Safar9 showed that the Hsieh-Plesset results were essentially equivalent to those of Eller and Flynn* when inertial effects were added to the HsiehPlesset approach. The few measurements obtained by Strasberg were extended by EllerloJ’ to include both the threshold and the growth rate. He found that the theory was adequate in predicting the thresholds for growth by rectified diffusion but was unable to account for some large growth rates that he observed. He suggested that acoustic streaming may be the cause of the larger-thanpredicted rates of growth. Gould’* was able to directly observe gas bubble growth by rectified diffusion through a microscope and discovered that growth rates could be greatly enhanced by the onset of surface oscillations of the bubble, which in turn seemed to induce significant acoustic microstreaming. Attempts by Gould’* to apply the acoustic streaming theories of Davidsoni and of Kapustina and Statnikov14 to explain his results were not successful. Additional theoretical treatment was presented by SkinneriS@ and Eller” to account for growth through resonance. Additional measurements of the growth of gas bubbles by rectified diffusion were obtained by Crumi8 who made both threshold and growth rate measurements for a variety of conditions. He extended the theory to include effects associated with the thermodynamic behaviour of the bubble interior and found excellent agreement between theory and experiment for both the rectified diffusion threshold and the growth rate for air bubbles in pure water. Anomalous results were obtained, however, when a small amount of surfactant was added to the water. The rate of growth of bubbles by rectified diffusion increased by a factor of about five when the surface tension was lowered by a factor of about two, with no discernible surface wave activity. Although some increase is predicted, the observed growth rates were much higher. A slight reduction in the threshold with reduced surface tension was also observed. Some explanations offered by Crumi8 for this anomalous behaviour were that there was some rectification of mass due to the surfactant on the surface of the bubble and/or there was microstreaming occurring even in the absence of surface oscillations. To bring our limited review of the literature up to date, it is noted that there have been recent applications of the equations of rectified diffusion to studies of the threshold for growth of microbubbles in biological media,i9 the threshold for cavitation inceptionzO, and as a possible explanation for the appearance of gas bubbles in insonified guinea pigs.*’ Recently, Crum and Hansen** have examined the various theoretical expressions available in the literature that predict the threshold and the growth rate for rectified diffusion and have found that the various equations have limited ranges of applicability. They obtained a set of generalized equations that have a broader range of applicability and are not limited to a specific bubble size or acoustic frequency. It is the purpose
216
of this paper to provide
a general
review of the mechanism of rectified diffusion. The general equations that describe the process will be presented as well as some simplified versions that are easier to apply but have limited ranges of applicability. Several figures will also be presented that graphically demonstrate the dependence of the threshold and the growth rate on the various physical parameters.
Equations
for rectified
diffusion
In this section the appropriate equations for rectified diffusion are given. First, the general equations will be given. Next. simplified versions of these equations will be obtained that are easier to apply and are applicable to the primary regions of interest. The general
equations
A complete mathematical description of the growth of a gas bubble by rectified diffusion would require an equation of motion for the bubble, diffusion equations for the liquid and the interior of the bubble, and heat conduction equations for both the liquid and the bubble. Continuity relations at the bubble wall would be required, and the situation is further complicated by the fact that these equations are coupled. Since this general approach presents a formidable mathematical problem, various simplifications are required to obtain a solution. The approach used by Hsieh and Plesset’ and by Eller and Flynn* is to separate the general problem into an equation for the motion of the bubble wall, and a diffusion equation for the concentration of gas dissolved in the liquid alone. The equation of motion for the gas bubble is the well-known equation, given here by
-PA
coswt wR,w,bk
I= 0
0)
where R and R, are the instantaneous and equilibrium values of the bubble radius respectively, p is the liquid density, n the polytropic exponent of the gas contained within the bubble, PA the acoustic pressure amplitude, o the angular frequency, o. the small-amplitude resonance frequency and b is a damping term applied to the bubble pulsations. Here PO = P, + 2u/R, where P, is the ambient pressure and u is the surface tension of the liquid. Further, o, is given by 00
* = (l/pR,*)
(3rlpo - 2dRo)
(2)
The diffusion equation for the gas in the liquid Fick’s law of mass transfer and is given by
dJ = ac at t v.VC=DV2C,
is
(3)
dt
where C is the concentration of gas in the liquid, v is the velocity of the liquid at a point, and D is the diffusion constant. The coupling between (1) and (3) is through the convective term v * VC and represents a major difficulty in the solution. This term was neglected by Blake2 and resulted in a prediction for the threshold that was an order of magnitude too small. The problem with the moving boundary has been solved in two slightly different ways by Hsieh and
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b, = 4w/a~l3qP,
Plesset’ and by Eller and Flynn8. The details of their solutions can be found in their respective works; the solution of the latter will be used here.
and
It is shown by Eller and Flynn8 that the time rate of change in the number of moles n of gas in a bubble is given by
an at
b, = pRo3w3/3~P, c
(13)
In these equations, p is the viscosity and c is the speed of sound. both in the liquid. The expressions for the damping given here should only be used near bubble resonance.
= 4nDRoCo
(4)
where C, is the ‘equilibrium’ or ‘saturation’ concentration of the gas in the liquid in moles (unit volume)-‘, the pointed brackets imply time average, t is the time, and H is defined by H = Ci/Co - <(R/‘R,)4 (PglPm)>l<(RIR,)4>
(5)
Here Ci is the concentration of dissolved gas in the liquid far from the bubble and Pg, the instantaneous pressure of the gas in the bubble, is given by
To obtain a usable expression for the growth-rate equation, the time averages, <(R/R,,r>, and <(R/R,J4 (PdP_)> are required. We find that <(R/RJ4>
= 1 + Ka2(Pp/P,)’
(y
= 1 + (3 + 4K) c~~(Pp/Pm)~,
(15)
and <(R/R,,)4(Pg/P,)>
Pg = P,(RJR)“’
=
(6)
The values of R/R,, to be used in the equations above are obtained by an expansion solution of (1) in the form R/R, = 1 + a(P,/P,)
(16) By assuming an ideal gas behaviour, we can relate the equilibrium radius to the number of moles of gas by the equation
cos (wt + 6)
+ c~~K(P~lPce)~+ . . . where we have determined
PO= 3nk Tl(4xRd) that
(Y-’ = (pRozIP,) [(w’ - w,,~)~+ (wo,~)~]~,
(7)
K -_ (377 + 1 - P*)/4 + (u/4RoP,)(6r7 + 2 - 4/3r7) 1 + (2u/RoP,) (1 - l/37/)
(8)
6 = tan-’
(9)
[ow,bl(o*
- wO’)],
and 0” = pw2R,2/3rjP,
b = b, + 6, + b,
(11)
where b, is given by (21) below and where
where k is the universal gas constant and T is the absolute temperature. This equation should be a good approximation for small acoustic pressure amplitudes, so that the density fluctuations do not become excessive, and for low host-liquid temperatures, so that the ratio of vapour/gas within the bubble remains low. Such conditions would be met for normal experimental situations. By combining (4) (5) and (17) an expression is obtained for the rate of change of the equilibrium bubble radius with time:
where d = k TC,IP,. In (18), the time averages, <(WR,J4> and <(R/R,J4 (Pg/P,)> are given by (14) (15) and (16) respectively. The threshold acoustic pressure growth of a gas bubble is obtained by setting dR,ldi = 0, and results in the equation
(pR,ZOz)* [(l-w*/o~)* ‘A” = (3+4K) (CJCO) -([3(17-l)
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(17)
(10)
It is necessary to know the damping of the bubble pulsations when the bubble is driven near resonance. It has been calculated by EllerJ3 using the expressions of Devin;24 a similar but not identical value for the damping term has also been obtained by Prosperetti?5 For completeness, we give below the expression of Eller,23 which expresses the total damping constant b in terms of the contributions due to thermal, viscous, and radiation effects:
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1984
+ b*(ti*/o,?)] (3~-4)/4]
(1 + 20/RoPw - Ci/Co)
+ (4&3~)K}(l
t 2~/RoPw)
(19)
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The gas in the interior of the bubble is treated in a thermodynamic way by (6) which relates the interior gas pressure to the bubble radius via the polytropic exponent, q. This parameter varies between 1.0, the isothermal case, and y (the ratio of specific heats), which describes the adiabatic case. The polytropic exponent has been calculated by Ellel’ based upon the earlier work of Devin.z4 A more detailed calculation has recently been performed by ProsperettiZ5 using a somewhat different technique. We have shown that these two approaches give approximately the same results for the region of interest.26 Given below is the less complicated expression obtained by Eller.23 7-l
I
?j = y(1 + bg-’
(20)
R, = 30 pm, then X N 3 and accordingly sinh X scosh X 9 (sin III. cos X). It is also possible to write approximately (22)
tl = y/(1 + 3(Y-1)/x)
(23)
Further
6, % (b,, b,) and thus
b = 3(y-1)/X
(24)
Finally, for R, > 30 pm, the surface tension term, 2u/ RJ’, < 1; neglecting this term in the expression for K, and o,,~ results in K = (3~ + 1 - p2)/4,
where b, = 3(y-1)
b, = 3(y-1)/X
X(sinh X + sin X) -2 (cash X - cos X) X2 (cash X - cos X) t 3(y-1)X
and
(sinh X - sin X)
(21)
(25)
(26)
oo2 = 3~P,lpR,~. Thus, a simplified version of the rectified diffusion threshold applicable to region (i) is given by
and X = R,(2w/D,)” In these equations y is the ratio of specific heats and D, = K,/p,C,,, where K, is the thermal conductivity of the gas in the bubble, p, is the density of gas, and C,, is the specific heat at constant pressure for the gas. If the diffusion rate is very rapid, or if there is a considerable influx of vapour, then D, may not remain constant over a cycle. Thus the expression for q may not hold when the diffusion rate or the host-liquid temperature is very high. See Ref. 25 for more details about the meaning of the thermal diffusivity D,. Simplified
equations
The equations for the growth rate, (18) and the threshold, (19) for rectified diffusion are rather complicated mathematical expressions that do not lend themselves very easily to inspection in order to determine the effect of a particular parameter. Further, there are ranges of interest in which the equations can be simplified greatly and still present accurate predictions. Since the threshold for rectified diffusion is normally the desirable quantity, attention will be restricted to that equation alone. The two principal ranges of interest in the study of rectified diffusion are: (i) an area applicable to underwater and physical acoustics where the acoustic frequencies are of the order of a kilohertz and the bubble radii are of the order of tens of pm; (ii) an area applicable to bioacoustics where the frequencies are of the order of a megahertz and the radii of the order of 1 pm. (i) Low frequency case For this case, the expressions for the damping constant and the polytropic exponent can be reduced considerably. For example, note that since the parameter X = R,,(2(LJD,)“, with D, 0.2 cm* s-r, o = 2n (20 kHz) and
218
2 _ (~R$o$)~ [(1-0~/02)~ pA
-
+ b2] [l + 2~/RoPm - Ci/Co] 3
(3 + 4K) (G/G) (27)
where n, b, K and wo2 are given by (23) to (26) respectively. It will be shown in the next section that (27) represents a close approximation to (19) in region (0. (ii) High frequency case A second region of interest is the case applicable to the growth of bubbles exposed to diagnostic or therapeutic ultrasound The frequencies of interest are in the megahertz range and, because resonance size is such an important parameter, the radii of interest are in the pm range. In this region, say w = 27r(l MHz) and R, = 2 pm, the bubbles will pulsate isothermally, due to their small size, and tl 3 1.0. Further, the principal contribution to the damping at this frequency is the viscosity term, so that b, S (b,, bJ and hence b = b,. Finally, for small bubbles, the surface tension term can be quite large and cannot be neglected. With these restrictions, the rectified diffusion threshold applicable to region (ii) is given by pi = (~R’oot)~ 11 - a2 10:)~ + b2] [ 1 + 20/RoPm - Ci/Co] (3>+4K)(Ci/Co)-K(l
+2O/RoPm) (28)
where K _ (1-P” /4) + 5oI(3P=Rcr) 1 t 4u/(3P,R0)
(29)
/3’ = ~o~R,~/~P,
(30)
b = 4w/.4’3P,,
(31)
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1984
and oo2 = (l/pR,2)
(3P,
+ 40/R,)
(32)
Equation (28) is nearly identical to an expression given earlier by Eller.” It will be shown in the next section that for high frequency/low radii situations, (19) and (28) give similar results.
Experimental technique rectified diffusion
for examining
In this section is presented a brief description of the principal experimental technique used to make measurements of the growth of bubbles by rectified diffusion. This technique was first utilized by Strasberg,4 with some modifications by EllerioJ1 and Gould.12 The main feature of the technique is the isolation of a single gas bubble in an acoustic stationary wave, with the ability to determine the bubble radius accurately and quickly. The specific description of the technique used by Crum’* is now given. Refer to Fig. 1 for a diagram of the experimental apparatus. The stationary wave system was constructed by cementing a hollow glass cylinder between two matched hollow cylindrical transducers, fitted with a flexible pressure release diaphragm on one end and open at the other. The composite system was approximately 7.5 cm in diameter by 10 cm in height, with the width of the glass in the middle about 2.5 cm. This system was driven at its (r, 8, z) = (2, 0, 2) mode at a frequency of 22.1 kHz. To obtain amplitude
accurate required
values of the acoustic pressure to levitate a bubble at a specific
position, it is necessary to know the spatial variation of the Ctationary acoustic wave along the axis of the cylinder. This variation was measured with the use of a small calibrated probe hydrophone that was mounted on a micromanipulator. It has been determined previously that stationary wave systems such as the one used here often have distorted acoustic pressure profiles and an accurate measurement of this profile is necessary for accurate acoustic pressure amplitude measurements.27 To reduce effects of the hydrophone itself, the actual acoustic pressure amplitude measurements were made by a small pill transducer mounted externally, as shown in Fig. 1. The acoustic pressure amplitude at the bubble’s position was determined from the voltage output of the pill transducer and the known spatial variation of the sound field. The average radius of the bubble was determined by first measuring its terminal rise velocity through the host liquid and then by iterating the following equation.27 Ro2 = (9vu/2g) (1 + 0.20 R,o.63 + (2.6 x 10-4) R,‘.4)
(33)
where v is the kinematic viscosity of the liquid, u is the terminal velocity, g is the gravitational acceleration, and R, = 2R, u/v is the Reynolds number. Two refinements were made to increase the accuracy of the radius measurements. First, several fiducial lines were introduced into the cathetometer so that rise times could be taken over a variety of distances and thus would not be excessively short or long. Second, the rise-time measurements for an individual bubble were introduced directly into a computer program that corrected for the dissolution of the bubble when the sound field was off and the terminal velocity measure-
Hydrophone
=-l Transducer
A
6
Window
1
i
Fig. 1
bath
]
Microscope
Block diagram of a typical experimental arrangement for obtaining measurements of rectified diffusion
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219
ment was being made. For small bubbles, say 40 pm diameter or less, an interval of a few seconds used tp measure the bubble’s rise velocity could result in errors of 10% or more in the bubble radius due to bubble dissolution. It was also discovered that small variations in the temperature of the host liquid could result in substantial changes in the rectified diffusion threshold and growth rate of the bubble. As will be seen in the following section, a small change in the dissolved gas concentration ratio can significantly affect the rectified diffusion threshold. Accordingly, it was necessary to surround the test cell with a constant temperature bath (see Fig. l), to ensure temperature stability and equilibrium gas concentration. The temperature of the host liquid was constantly monitored to ensure no change in temperature during the measurements or, just as importantly, no change from the temperature at which the liquid was brought to equilibrium gas concentration.
-1 J
20
With this technique it appears that threshold measurements can be made to within an accuracy of approximately 5%. The radii measurements depend upon the applicability of the drag law, and its specific accuracy, but are probably good to within a few percent also.
Graphical representation and measured data In this section is presented a of the equations as a function parameters that are applicable. thresholds and bubble growth comparison. In these figures, made otherwise, it is assumed
II
1
8-
32-
I I
I
I
I
I.1
1.2
1.3
I .4
Dissolved
gas concentration
rotio ( Ci /Co)
Fig. 2 Stability curves for gas bubbles at various drssolved gas concentration ratios in water. The curves are for three different surface tensions. If a gas bubble lies above the appropriate line, it will grow by diffusion at zero acoustic pressure amplitude; if it lies below, it will dissolve
220
of an air of the
It is seen from (19) that the numerator of the threshold equation can be made to equal zero if the gas concentration ratio Ci/Co = 1 + 2u/R, P,. That is, if the liquid is oversaturated by an amount greater than the surface tension term, gas bubbles will grow with no acoustic field applied. Consider Fig. 2, which shows a plot of bubble radius against the dissolved gas concentration ratio (C;/c,) for various surface tensions such that the ‘critical saturation’ condition is met, that is, Ci/Co = 1 + 2alR, P,. This figure is presented to show the importance of the dissolved gas concentration on gas bubble stability. It can be noted that unless there is some degree of oversaturation all gas bubbles will slowly dissolve (although smaller ones dissolve progressively faster). Further, for very small gas bubbles, high degrees of oversaturation are required before a gas bubble will grow spontaneously. This figure has relevance to the growth of bubbles in carbonated liquids and this particular author has mused over its implications on numerous occasions while making observations of a variety of bubble-filled liquids.
9-
1.0
100
an air bubble in water with the following set of experimental conditions: P, = 1.01 x lo5 dyn cmm2, u = 68 dyn cm-‘, p = 1.0 g cme3, D, = 0.20 cm2 s-l, p = 1.0 X 10m2 g cm-’ s-l, c = 1.48 x lo6 cm s-l, y = 1.4, T = 293 K, D = 2.4 x 10m5cm2 s-l and d = 2.0 x 10-2.
IO -
0 ’
80
Bubble radius Qml
graphical representation of the various physical Some measurements of rates are also given for unless specific mention is that we are considering
-
60
Fig. 3 Variation of the threshold for growth by rectified diffusion bubble in water as a function of the bubble radius for a frequency 22.1 kHz, a surface tension of 68 dyn cm-’ and a dissolved gas concentration of 1 .O. The symbols are experimental measurements; solid curve is calculated from [19); the dashed one from (27)
of the equations
I2 w
40
1.5
Central to the study of rectified diffusion are measurements of the acoustic pressure amplitude required for bubble growth and the agreement between these measurements and the applicable theory. Shown in Fig. 3 are measurements of the threshold for a region (i) case (frequency 22.1 kHz, bubble radii ranging from 20 pm to 70 pm) along with the theoretical calculations. The solid curve is the prediction of (19); the dashed line is the simplified version of the threshold applicable to this region of consideration. (27). It is seen that the two equations give similar predictions until the bubble radius becomes so small that the term 2rr/R, P, becomes significant with respect to unity.
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1984
‘resonance radius’ given by a solution of (2) for R, when w,, is replaced by the driving frequency. In Fig. 5, the driving frequency was 1.0 MHz, the surface tension 68.0 dyn cm-‘, and the ambient pressure 0.1 MPa, thus giving a resonance radius of 3.15 pm. It is seen that the threshold becomes very low near resonance due to the large pulsation amplitude of the bubble there. Near resonance, the amplitude of the bubble’s pulsations are determined mostly by the damping, which is relatively small, and consequently the bubble oscillates rather violently. It is not expected that the equations for bubble growth and for the threshold will be very accurate in this region.
0.08
+j 0.06 C -n 5 0.05
E 2 0.04 t h .; 0.03 a 2 0.02
01 0.6
I
I
I
I
0.7
a0
0.9
1.0
3 II I.1
The variation of the rectified diffusion threshold with frequency for a fixed radius, shown in Fig. 6, is similar to the variation with radius for a fixed frequency, shown in Fig. 5. The familiar reduction in the 1.0
Dissolved gas concentration ratio ( Ci /C,) Fig. 4 Variation of the threshold for growth by rectified diffusion of an air bubble in water as a function of the dissolved gas concentration ratio. The solid line is calculated from (19); the dashed line is from (27); the broken line is a numerical calculation by Eller and Flynne; the symbol is an experimental measurement by Strasberg
The agreement between theory and experiment (for this limited range of bubble radii) is seen to be rather good and within the range of experimental error. This bubble range is particularly important at this frequency because it partially bridges the gap between isothermal behaviour (for R, = 20 pm, tl = 1.01) and adiabatic behaviour (for R, = 80 pm, 1 = 1.23) of the bubble pulsations. The gradual reduction in the threshold as the bubble radius increases is mostly due to the fact that as the bubble grows toward resonance size its pulsation amplitude increases, which results in more rectified mass transfer per cycle. The effect of the dissolved gas concentration in the liquid on the threshold is shown in Fig. 4. These data are for a frequency of 24.5 kHz, a radius of 26 pm and a surface tension of 72 dyn cm-‘. The top two curves are calculated from (19), solid line, and (27), dashed line. It is seen that the simplified and more complicated versions agree so long as the gas concentration ratio is not far from unity. Plotted also on the figure is a measurement due to Strasberg? and a numerical result due to Eller and Flynn.* Since the numerical result is in better agreement, it can be implied that the higher-order terms that are neglected in the analytical approach may be significant if the dissolved gas concentration is considerably less than saturation. It is also noted from the curves that the threshold falls to zero whenever Ci/Co = 1 -t- 2a/R, P,. As discussed earlier, if the amount of dissolved gas present in the liquid is sufficient to overcome the Laplace pressure, the bubble will spontaneously grow. Shown in Fig. 5 is the variation in the rectified diffusion threshold as a function of radius for values of the radius both above and below the resonance values. The curves can be calculated from either (19) or (28), the values being essentially equal. A gas bubble present in a liquid containing a sinusoidally varying pressure field behaves very much like a damped driven harmonic oscillator. It has a linear resonance frequency given by (2). For a system in which the frequency is fixed, and the radius varies, as is typical of studies involving rectified diffusion, one can speak of a
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0.5
0.01
I
I
I
0.2
0.5
I.0
1 0.1
I I 2.0
I 5.0
I IO.0
Bubble radius Cpml Fig. 5 Variation of the rectified diffusion threshold as a function of gas bubble radius for three separate dissolved gas concentration ratios (C/C,). The curves can be calculated by the use of either (19) or (28); the acoustic frequency used was 1 .O MHz and the surface tension was 68 dyn cm-‘; the liquid is assumed to be water
lo-3
1 IO4
I
I
I 10s
I
I
I IO6
Frequency CHzl Fig. 6 Variation of the rectified diffusion threshold as a function of acoustic frequency for two different values of the gas bubble radius. The liquid IS assumed to be water having a dissolved gas concentration ratio of 1 .O and with a surface tension of 68 dyn cm-’
221
60 1.8 rz 17 L!+ 1.6 2 ; 1.5 u % 1.4 m’ 1.3 1.2
IO
I.1 0 0
100
200
300
400
500
600
700
Time Csl
0
IO
20
30
40
50
Time Cmsl
Fig. 7 Dependence of the bubble radius on time for an air bubble undergoing rectified diffusion in water. The symbols are measured values; the curves have been calculated by numerical integration of (18).The bottom two curves (and measurements) are for distilled water having a surface tension of 68 dyn cm-‘; the top curve (and associated set of measurements) is for water containing small amounts of surface-active agents, and having a surface tension of 32 dyn cm-’
Fig. 8 Calculated behaviour of a distribution of the sizes of air bubbles present in water as a function of time and exposed to a continuous sound field of frequency 3.0 MHz and acoustic pressure amplitude of 0.12 MPa. The designated line indicates the free decay of an air bubble. The other lines describe the time history of individual air bubbles and were obtained from a numerical integration of (18)
threshold as the resonance frequency is approached is observed. For the two cases considered in Fig. 6, that is, for radii of 10 pm and 50 pm, the threshold is nearly independent of frequency for frequencies considerably less than resonance, but strongly dependent on the frequency above resonance. This behaviour can be easily seen by inspection of (19). It is observed also that the threshold below resonance is higher for the 10 pm than for the 50 pm bubble. This effect is due to the higher Laplace pressure for the smaller bubble.
present in water when exposed to continuous wave ultrasound of frequency 3 MHz and acoustic pressure amplitude of 0.12 MPa. This figure may be applicable to the growth of microbubbles in tissue by therapeutic ultrasound. Note that if this population of ‘free air bubbles’ were suddenly created, those bubbles with radii less than 1.03 pm would slowly dissolve regardless of the ultrasound. Those bubbles in the range of 1.03 pm to 1.4 pm would grow to an asymptotic limit of about 1.42 pm. Bubbles larger than 1.4 pm would decay to the asymptotic limit.
So far, our analysis in this section has been limited to the rectified diffusion threshold. It is in order also to examine the rate of growth by rectified diffusion when the acoustic pressure amplitude exceeds the threshold. An examination of the growth (or decay) of a bubble is shown in Fig. 7. The symbols are experimental measurements; the curves were obtained by numerical integration of (18). It was observed by Eller’O that his measurements of the growth rate were considerably larger than predicted by theory. Gould’* later confirmed the excessive growth rates but also noticed that surface oscillations, with their associated acoustic streaming, were probably the explanation for the increased growth rates that he observed.
The reason for this limit is that due to the Laplace pressure, there is always a tendency to force gas out of the bubble (note the free decay curve on the figure). If the bubble radius is less than resonance size, and greater than a threshold size, the bubble will grow through resonance to a size sufficiently above resonance where its pulsation amplitude will result in an inward mass transfer due to pulsation that just matches the outward mass transfer due to the Laplace pressure. If the bubble is larger than the asymptotic limit, it will not be able to overcome the outward diffusion due to the Laplace pressure and will dissolve slowly until its pulsation amplitude is large enough to stop further dissolution. Consequently, there is a tendency to force microbubbles present in a liquid to a uniform size that is dependent on the intrinsic and extrinsic physical parameters.
Shown in Fig. 7 are some observations of growth rates made by Crum. I8 He observed an agreement between the measured and predicted rates provided surface oscillations were avoided, and provided that the water was relatively pure and free of surface-active contaminants. When surface-active agents were added to the water, growth rates much higher than predicted were observed. Among the explanations offered for this effect were that (a) the surface-active agents were behaving as a rectifying agent, permitting more diffusion in than out, and (b) acoustic streaming was occurring in the absence of surface oscillations, induced somehow by the surface contamination. This interesting anomaly has not yet been explained. As a final result of this section, consider Fig. 8, which demonstrates what would happen to a population of air bubbles with radii varying from 1 pm to 2 pm
222
Conclusions A set of equations were presented that describe the process of rectified diffusion for a variety of physical conditions. The experimental data available tend to confirm the validity of the existing equations provided the acoustic pressure amplitude is not large, the liquid does not contain surface-active additives, and the dissolved gas concentration ratio is near saturation. Areas that need further
study are as follows:
1. It was seen that the addition of surface active agents to water resulted in measured values of the threshold and growth rate considerably different from those
ULTRASONICS.
SEPTEMBER
1984
predicted by theory. An adequate anomaly should be given.
explanation
of this
2. It is observed that the analytical expressions for the rectified diffusion threshold are not in good agreement with the few data that exist at dissolved gas concentration ratios significantly different from saturation. Numerical studies give much better agreement. An extension of the analytical results to higher order to account for the discrepancy with the dissolved gas concentration ratio is needed. Further. since data at under-saturated levels are sparse, additional data should be obtained. 3. It has been recently observed28 that gas bubbles driven at the pressure amplitudes experienced in rectified diffusion studies will experience considerable fluctuations in their pulsation amplitude as they are driven through their harmonic resonances.29 At these radius positions, the bubble may experience rapid growth that is unaccounted for in the analytical expressions given in this article. 4. It was observed by Gould” that increased growth by rectified diffusion occurred in the presence of acoustic streaming in the vicinity of the bubble. An adequate theory has not been presented to account for the effect of this streaming on rectified diffusion enhancement.
7 8
9
10 11 12
13 14
15 16 17 18 19
20
Acknowledgement The author wishes helpful assistance, and the National financial support discussed here.
21
to thank Gary Hansen for his and the Office of Naval Research Science Foundation for providing for the research work of the author
22 23 24
References 25 Harvey, E.N., Barnes, D.K, McElroy, W.D., Whitely, A.H., Pease, D.C., Cooper, K.W. Bubble Formation in Animals, J Cell Comp. PhysioL 24 (1944) l-40 Blake, F.G., Jr The Onset of Cavitation in Liquids, Tech. Memo. No. 12, Acoust. Res. Lab. Harvard Univ. (1949) Strasberg, M. Rectified Diffusion: Comments on a Paper of Hsieh and Plesset, .I Acoust Sot Am. 33 (1961) 359-360 Strasberg, M. Onset of Ultrasonic Cavitation in Tap Water, J Acoust Sot Am. 31 (1959) 163-176 Pode, J. The Deaeration of Water by a Sound Beam, David Taylor Model Basin Dept. No. 854 (1953) Rosenberg, M.D. Gaseous-type Cavitation in Liquids, Tech. Memo. No. 26, Acoust. Res. Lab. Harvard Univ. (1953)
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1984
26
27 28
29
Hsieh, D.Y., Plesset, M.D. Theory of Rectified Diffusion of Mass into Gas Bubbles, J. Acousf. Sec. Am. 33 (1961) 206-215 Eller, A.I., Flynn, H.G. Rectified Diffusion during Nonlinear Pulsations of Cavitation Bubbles, J. Acoust. Sot. Am. 37 (1965) 493-503 Safar, M.H. Comment on Papers Concerning Rectified Diffusion of Cavitation Bubbles, J. Acoust Sot. Am. 43 (1968) 1188-1189 Ellen AI. Growth of Bubbles by Rectified Diffusion, J. Acoust Sot Am. 46 (1969) 1246-1250 Ellen A.I. Bubble Growth by Diffusion in an 1I-kHz Sound Field, J Acousr. Sot. Am. 52 (1972) 1447-1449 Gould, R.K. Rectified Diffusion in the Presence of, and Absence of, Acoustic Streaming, J Acoust Sot. Am. 56 (1974) 1740-1746 Davison, B.J. Mass Transfer due to Cavitation Microstreaming J. Sound Vib. 17 (1971) 261-270 Kapustina, O.A., Statmikov, Y.G. Influence of Microstreaming on the Mass Transfer in a Gas-BubbleLiquid System, Sov. Phys. Acoust. 13 (1968) 327-329 Skinner, LA. Pressure Threshold for Acoustic Cavitation, J. Acoust. Sot Am. 47 (1970) 327-331 Skinner, L.A. Acoustically Induced Gas Bubble Growth, J. Acoust Sot. Am. 51 (1972) 378-382 Eller, A.I. Effects of Diffusion on Gaseous Cavitation Bubb1es.J. Acousr. Sot Am. 57 (1975) 1374-1378 Crum, LA. Measurements of the Growth of Air Bubbles by Rectified Diffusion, J. Acoust Sot. Am. 68 (1980) 203-211 Lewin, P.A., Bjfimu, L. Acoustic Pressure Amplitude Thresholds for Rectified Diffusion in Gaseous Microbubbles in Biological Tissue, J. Acousf. Sot. Am. 69 (1981) 846-852 Apfel, R.E. Acoustic Cavitation Prediction, J Acousr. Sot. Am. 69 (1981) 1624-1633 Cmm, LA., Hansen, GM. Growth of Air Bubbles in Tissue by Rectified Diffusion, Phys. Med. Biol. 27 (1982) 413-417 Crum, LA., Hansen, G.M. Generalized Equations for Rectified Diffusion J. Acoust Sot. Am. 72 (1982) 1586-1592 Eller, A.E. Damping Constants of Pulsating Bubbles, 1 Acoust Sot. Am. 47 (1970) 1469-1470 Devin, C. Survey of Thermal, Radiation, and Viscous Damping of Pulsating Air Bubbles in Water, J. Acoust Sot, Am. 31 (1959) 1654-1667 Prosperetti, A. Thermal Effects and Damping Mechanisms in the Forced Radial Oscillations of Gas Bubbles in Liquids.1 Acoust. Sot Am 61 (1977) 17-27 Crum, LA. The Polytropic Exponent of Gas Contained Within Air Bubbles Pulsating in a Liquid .I Acoust Sot Am. 73 (1983) 116-120 Crum, L.A., Eller, AI. The Motion of Air Bubbles in a Stationary Sound Field, 1 Acoust. Sot Am 48 (1970) 181-189 Crum, L.A., Prosperetti, A. Nonlinear Oscillations of Gas Bubbles in Liquids: An Interpretation of some Experimental _ Results,1 Aco& Sot Am. 73-(1983) 121-127 Prosperetti, A. Nonlinear Oscillations of Gas Bubbles in Liquids: Steady-State Solutions, J Acoust. Sot Am. 56 (1974) 878-885
223
cavitation
Rectified
series: part five
diffusion
L.A. CRUM A general review is given of the mechanism of rectified diffusion. The equations that describe the threshold acoustic pressure amplitude as well as the growth rate are presented. Simplified versions of the complicated threshold equation are also obtained for two regions that are of particular interest. Graphical representations of the equations for a variety of physical parameters are given as well as a comparison between the available measurements and the theoretical, predictions. Finally, some suggested areas of future research in this- area are presented. KEYWORDS:
ultrasonics, cavitation, bubble dynamics
contracts, the concentration of gas (moles 1-i) in the interior of the bubble increases, and gas diffuses from the bubble. Similarly, when the bubble expands, the concentration of gas decreases, and gas diffuses into the bubble. Since the diffusion rate is proportional to the area, more gas will enter during expansion than will leave during the contraction of the bubble; therefore, over a complete cycle, there will be a net increase in the amount of gas in the bubble.
Introduction A concept of general interest in the area of acoustic cavitation is that of ‘rectified diffusion’. This process involves the slow growth of a pulsating gas bubble due to an average flow of mass into the bubble as a function of time. This ‘rectification of mass’ is a direct consequence of the applied sound field and can be important whenever a sufficiently intense sound field (acoustic pressure amplitude greater than 0.01 MPa, say) exists in a liquid containing dissolved gas. That is, the liquid may be under-saturated, saturated or oversaturated with gas. If the liquid is not sufficiently oversaturated, a free gas bubble will gradually dissolve due to the so-called Laplace pressure generated in the interior of the bubble by the surface tension. The value of this pressure is given by Pi = 20/R, where Pi is the internal pressure, u is the surface tension and R is the radius of the bubble.
The second effect is the ‘shell’ effect. The diffusion rate of gas in a liquid is proportional to the gradient of the concentration of dissolved gas. Consider a spherical shell of liquid surrounding the bubble. When the bubble contracts, this shell expands, and the concentration of gas near the bubble wall is reduced. Thus, the rate of diffusion of gas away from the bubble is greater than when the bubble is at its equilibrium radius. Conversely, when the bubble expands, the shell contracts, the concentration of gas near the bubble is increased, and the rate of gas diffusion toward the bubble is greater than average. The net effect of this convection is to enhance the rectified diffusion. It has been shown that both the area and the shell effects are necessary for an adequate description of the phenomenon.
It is easy to see that for gas bubbles of the order of 1.0 cm in diameter, this pressure is quite small (less than 0.001 bar for a gas bubble in water); however, for a bubble of the order of 1 pm in diameter, this pressure becomes significant (greater than 1 bar). Thus, for free gas bubbles in liquids that are not oversaturated with gas and which do not rise to the surface, their ultimate fate is slowly to dissolve. (Of course, pockets of free gas may be stabilized against dissolution by a variety of mechanisms, but that is another topic).
Historically, the concept of rectified diffusion was apparently first recognized by Harvey’ et al, who considered the importance of this concept in the formation of bubbles in animals. They were able to recognize that the area effect would lead to growth and called rectified diffusion the ‘principle of incremental enlargement’, recognizing that the growth during one cycle may be quite small. Blake2 was the first to attempt a theoretical analysis of rectified diffusion, but considered only the area effect, and thus his crude approach was in considerable disagreement with the first reported measurements by Strasberg+4. Pode5 and Rosenberg6 attempted to refine Blake’s theory but their results were not much different from that of Blake. The
With the presence of an applied sound field, however, the bubble radius is forced to oscillate about an equilibrium value, and these oscillations can prevent the eventual dissolution of a bubble and even cause it to grow because of the following effects. The first effect is an ‘area’ effect. When the bubble The author is in the Physical Acoustics Research Laboratory, Department of Physics and Astronomy. The University of Mississippi, Mississippi 38677. USA Paper received 24 May 1983.
0041-624X/84/0502 ULTRASONICS.
SEPTEMBER
1984
15-09/$03.00
0
1984
Buttetworth
8 Co (Publishers)
Ltd 215
importance of convection was demonstrated when Hsieh and Plesset’ correctly included this effect in their solution and Strasberg4 showed that their theory generally agreed with the results of his measurements. Eller and Flynn* extended the analysis to include nonlinear or large amplitude effects by treating the boundary condition of the moving wall in a slightly different way. Safar9 showed that the Hsieh-Plesset results were essentially equivalent to those of Eller and Flynn* when inertial effects were added to the HsiehPlesset approach. The few measurements obtained by Strasberg were extended by EllerloJ’ to include both the threshold and the growth rate. He found that the theory was adequate in predicting the thresholds for growth by rectified diffusion but was unable to account for some large growth rates that he observed. He suggested that acoustic streaming may be the cause of the larger-thanpredicted rates of growth. Gould’* was able to directly observe gas bubble growth by rectified diffusion through a microscope and discovered that growth rates could be greatly enhanced by the onset of surface oscillations of the bubble, which in turn seemed to induce significant acoustic microstreaming. Attempts by Gould’* to apply the acoustic streaming theories of Davidsoni and of Kapustina and Statnikov14 to explain his results were not successful. Additional theoretical treatment was presented by SkinneriS@ and Eller” to account for growth through resonance. Additional measurements of the growth of gas bubbles by rectified diffusion were obtained by Crumi8 who made both threshold and growth rate measurements for a variety of conditions. He extended the theory to include effects associated with the thermodynamic behaviour of the bubble interior and found excellent agreement between theory and experiment for both the rectified diffusion threshold and the growth rate for air bubbles in pure water. Anomalous results were obtained, however, when a small amount of surfactant was added to the water. The rate of growth of bubbles by rectified diffusion increased by a factor of about five when the surface tension was lowered by a factor of about two, with no discernible surface wave activity. Although some increase is predicted, the observed growth rates were much higher. A slight reduction in the threshold with reduced surface tension was also observed. Some explanations offered by Crumi8 for this anomalous behaviour were that there was some rectification of mass due to the surfactant on the surface of the bubble and/or there was microstreaming occurring even in the absence of surface oscillations. To bring our limited review of the literature up to date, it is noted that there have been recent applications of the equations of rectified diffusion to studies of the threshold for growth of microbubbles in biological media,i9 the threshold for cavitation inceptionzO, and as a possible explanation for the appearance of gas bubbles in insonified guinea pigs.*’ Recently, Crum and Hansen** have examined the various theoretical expressions available in the literature that predict the threshold and the growth rate for rectified diffusion and have found that the various equations have limited ranges of applicability. They obtained a set of generalized equations that have a broader range of applicability and are not limited to a specific bubble size or acoustic frequency. It is the purpose
216
of this paper to provide
a general
review of the mechanism of rectified diffusion. The general equations that describe the process will be presented as well as some simplified versions that are easier to apply but have limited ranges of applicability. Several figures will also be presented that graphically demonstrate the dependence of the threshold and the growth rate on the various physical parameters.
Equations
for rectified
diffusion
In this section the appropriate equations for rectified diffusion are given. First, the general equations will be given. Next. simplified versions of these equations will be obtained that are easier to apply and are applicable to the primary regions of interest. The general
equations
A complete mathematical description of the growth of a gas bubble by rectified diffusion would require an equation of motion for the bubble, diffusion equations for the liquid and the interior of the bubble, and heat conduction equations for both the liquid and the bubble. Continuity relations at the bubble wall would be required, and the situation is further complicated by the fact that these equations are coupled. Since this general approach presents a formidable mathematical problem, various simplifications are required to obtain a solution. The approach used by Hsieh and Plesset’ and by Eller and Flynn* is to separate the general problem into an equation for the motion of the bubble wall, and a diffusion equation for the concentration of gas dissolved in the liquid alone. The equation of motion for the gas bubble is the well-known equation, given here by
-PA
coswt wR,w,bk
I= 0
0)
where R and R, are the instantaneous and equilibrium values of the bubble radius respectively, p is the liquid density, n the polytropic exponent of the gas contained within the bubble, PA the acoustic pressure amplitude, o the angular frequency, o. the small-amplitude resonance frequency and b is a damping term applied to the bubble pulsations. Here PO = P, + 2u/R, where P, is the ambient pressure and u is the surface tension of the liquid. Further, o, is given by 00
* = (l/pR,*)
(3rlpo - 2dRo)
(2)
The diffusion equation for the gas in the liquid Fick’s law of mass transfer and is given by
dJ = ac at t v.VC=DV2C,
is
(3)
dt
where C is the concentration of gas in the liquid, v is the velocity of the liquid at a point, and D is the diffusion constant. The coupling between (1) and (3) is through the convective term v * VC and represents a major difficulty in the solution. This term was neglected by Blake2 and resulted in a prediction for the threshold that was an order of magnitude too small. The problem with the moving boundary has been solved in two slightly different ways by Hsieh and
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1984
b, = 4w/a~l3qP,
Plesset’ and by Eller and Flynn8. The details of their solutions can be found in their respective works; the solution of the latter will be used here.
and
It is shown by Eller and Flynn8 that the time rate of change in the number of moles n of gas in a bubble is given by
an at
b, = pRo3w3/3~P, c
(13)
In these equations, p is the viscosity and c is the speed of sound. both in the liquid. The expressions for the damping given here should only be used near bubble resonance.
= 4nDRoCo
(4)
where C, is the ‘equilibrium’ or ‘saturation’ concentration of the gas in the liquid in moles (unit volume)-‘, the pointed brackets imply time average, t is the time, and H is defined by H = Ci/Co - <(R/‘R,)4 (PglPm)>l<(RIR,)4>
(5)
Here Ci is the concentration of dissolved gas in the liquid far from the bubble and Pg, the instantaneous pressure of the gas in the bubble, is given by
To obtain a usable expression for the growth-rate equation, the time averages
= 1 + Ka2(Pp/P,)’
(y
= 1 + (3 + 4K) c~~(Pp/Pm)~,
(15)
and <(R/R,,)4(Pg/P,)>
Pg = P,(RJR)“’
=
(6)
The values of R/R,, to be used in the equations above are obtained by an expansion solution of (1) in the form R/R, = 1 + a(P,/P,)
(16) By assuming an ideal gas behaviour, we can relate the equilibrium radius to the number of moles of gas by the equation
cos (wt + 6)
+ c~~K(P~lPce)~+ . . . where we have determined
PO= 3nk Tl(4xRd) that
(Y-’ = (pRozIP,) [(w’ - w,,~)~+ (wo,~)~]~,
(7)
K -_ (377 + 1 - P*)/4 + (u/4RoP,)(6r7 + 2 - 4/3r7) 1 + (2u/RoP,) (1 - l/37/)
(8)
6 = tan-’
(9)
[ow,bl(o*
- wO’)],
and 0” = pw2R,2/3rjP,
b = b, + 6, + b,
(11)
where b, is given by (21) below and where
where k is the universal gas constant and T is the absolute temperature. This equation should be a good approximation for small acoustic pressure amplitudes, so that the density fluctuations do not become excessive, and for low host-liquid temperatures, so that the ratio of vapour/gas within the bubble remains low. Such conditions would be met for normal experimental situations. By combining (4) (5) and (17) an expression is obtained for the rate of change of the equilibrium bubble radius with time:
where d = k TC,IP,. In (18), the time averages
(pR,ZOz)* [(l-w*/o~)* ‘A” = (3+4K) (CJCO) -([3(17-l)
SEPTEMBER
(17)
(10)
It is necessary to know the damping of the bubble pulsations when the bubble is driven near resonance. It has been calculated by EllerJ3 using the expressions of Devin;24 a similar but not identical value for the damping term has also been obtained by Prosperetti?5 For completeness, we give below the expression of Eller,23 which expresses the total damping constant b in terms of the contributions due to thermal, viscous, and radiation effects:
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(12)
1984
+ b*(ti*/o,?)] (3~-4)/4]
(1 + 20/RoPw - Ci/Co)
+ (4&3~)K}(l
t 2~/RoPw)
(19)
217
The gas in the interior of the bubble is treated in a thermodynamic way by (6) which relates the interior gas pressure to the bubble radius via the polytropic exponent, q. This parameter varies between 1.0, the isothermal case, and y (the ratio of specific heats), which describes the adiabatic case. The polytropic exponent has been calculated by Ellel’ based upon the earlier work of Devin.z4 A more detailed calculation has recently been performed by ProsperettiZ5 using a somewhat different technique. We have shown that these two approaches give approximately the same results for the region of interest.26 Given below is the less complicated expression obtained by Eller.23 7-l
I
?j = y(1 + bg-’
(20)
R, = 30 pm, then X N 3 and accordingly sinh X scosh X 9 (sin III. cos X). It is also possible to write approximately (22)
tl = y/(1 + 3(Y-1)/x)
(23)
Further
6, % (b,, b,) and thus
b = 3(y-1)/X
(24)
Finally, for R, > 30 pm, the surface tension term, 2u/ RJ’, < 1; neglecting this term in the expression for K, and o,,~ results in K = (3~ + 1 - p2)/4,
where b, = 3(y-1)
b, = 3(y-1)/X
X(sinh X + sin X) -2 (cash X - cos X) X2 (cash X - cos X) t 3(y-1)X
and
(sinh X - sin X)
(21)
(25)
(26)
oo2 = 3~P,lpR,~. Thus, a simplified version of the rectified diffusion threshold applicable to region (i) is given by
and X = R,(2w/D,)” In these equations y is the ratio of specific heats and D, = K,/p,C,,, where K, is the thermal conductivity of the gas in the bubble, p, is the density of gas, and C,, is the specific heat at constant pressure for the gas. If the diffusion rate is very rapid, or if there is a considerable influx of vapour, then D, may not remain constant over a cycle. Thus the expression for q may not hold when the diffusion rate or the host-liquid temperature is very high. See Ref. 25 for more details about the meaning of the thermal diffusivity D,. Simplified
equations
The equations for the growth rate, (18) and the threshold, (19) for rectified diffusion are rather complicated mathematical expressions that do not lend themselves very easily to inspection in order to determine the effect of a particular parameter. Further, there are ranges of interest in which the equations can be simplified greatly and still present accurate predictions. Since the threshold for rectified diffusion is normally the desirable quantity, attention will be restricted to that equation alone. The two principal ranges of interest in the study of rectified diffusion are: (i) an area applicable to underwater and physical acoustics where the acoustic frequencies are of the order of a kilohertz and the bubble radii are of the order of tens of pm; (ii) an area applicable to bioacoustics where the frequencies are of the order of a megahertz and the radii of the order of 1 pm. (i) Low frequency case For this case, the expressions for the damping constant and the polytropic exponent can be reduced considerably. For example, note that since the parameter X = R,,(2(LJD,)“, with D, 0.2 cm* s-r, o = 2n (20 kHz) and
218
2 _ (~R$o$)~ [(1-0~/02)~ pA
-
+ b2] [l + 2~/RoPm - Ci/Co] 3
(3 + 4K) (G/G) (27)
where n, b, K and wo2 are given by (23) to (26) respectively. It will be shown in the next section that (27) represents a close approximation to (19) in region (0. (ii) High frequency case A second region of interest is the case applicable to the growth of bubbles exposed to diagnostic or therapeutic ultrasound The frequencies of interest are in the megahertz range and, because resonance size is such an important parameter, the radii of interest are in the pm range. In this region, say w = 27r(l MHz) and R, = 2 pm, the bubbles will pulsate isothermally, due to their small size, and tl 3 1.0. Further, the principal contribution to the damping at this frequency is the viscosity term, so that b, S (b,, bJ and hence b = b,. Finally, for small bubbles, the surface tension term can be quite large and cannot be neglected. With these restrictions, the rectified diffusion threshold applicable to region (ii) is given by pi = (~R’oot)~ 11 - a2 10:)~ + b2] [ 1 + 20/RoPm - Ci/Co] (3>+4K)(Ci/Co)-K(l
+2O/RoPm) (28)
where K _ (1-P” /4) + 5oI(3P=Rcr) 1 t 4u/(3P,R0)
(29)
/3’ = ~o~R,~/~P,
(30)
b = 4w/.4’3P,,
(31)
ULTRASONICS.
SEPTEMBER
1984
and oo2 = (l/pR,2)
(3P,
+ 40/R,)
(32)
Equation (28) is nearly identical to an expression given earlier by Eller.” It will be shown in the next section that for high frequency/low radii situations, (19) and (28) give similar results.
Experimental technique rectified diffusion
for examining
In this section is presented a brief description of the principal experimental technique used to make measurements of the growth of bubbles by rectified diffusion. This technique was first utilized by Strasberg,4 with some modifications by EllerioJ1 and Gould.12 The main feature of the technique is the isolation of a single gas bubble in an acoustic stationary wave, with the ability to determine the bubble radius accurately and quickly. The specific description of the technique used by Crum’* is now given. Refer to Fig. 1 for a diagram of the experimental apparatus. The stationary wave system was constructed by cementing a hollow glass cylinder between two matched hollow cylindrical transducers, fitted with a flexible pressure release diaphragm on one end and open at the other. The composite system was approximately 7.5 cm in diameter by 10 cm in height, with the width of the glass in the middle about 2.5 cm. This system was driven at its (r, 8, z) = (2, 0, 2) mode at a frequency of 22.1 kHz. To obtain amplitude
accurate required
values of the acoustic pressure to levitate a bubble at a specific
position, it is necessary to know the spatial variation of the Ctationary acoustic wave along the axis of the cylinder. This variation was measured with the use of a small calibrated probe hydrophone that was mounted on a micromanipulator. It has been determined previously that stationary wave systems such as the one used here often have distorted acoustic pressure profiles and an accurate measurement of this profile is necessary for accurate acoustic pressure amplitude measurements.27 To reduce effects of the hydrophone itself, the actual acoustic pressure amplitude measurements were made by a small pill transducer mounted externally, as shown in Fig. 1. The acoustic pressure amplitude at the bubble’s position was determined from the voltage output of the pill transducer and the known spatial variation of the sound field. The average radius of the bubble was determined by first measuring its terminal rise velocity through the host liquid and then by iterating the following equation.27 Ro2 = (9vu/2g) (1 + 0.20 R,o.63 + (2.6 x 10-4) R,‘.4)
(33)
where v is the kinematic viscosity of the liquid, u is the terminal velocity, g is the gravitational acceleration, and R, = 2R, u/v is the Reynolds number. Two refinements were made to increase the accuracy of the radius measurements. First, several fiducial lines were introduced into the cathetometer so that rise times could be taken over a variety of distances and thus would not be excessively short or long. Second, the rise-time measurements for an individual bubble were introduced directly into a computer program that corrected for the dissolution of the bubble when the sound field was off and the terminal velocity measure-
Hydrophone
=-l Transducer
A
6
Window
1
i
Fig. 1
bath
]
Microscope
Block diagram of a typical experimental arrangement for obtaining measurements of rectified diffusion
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SEPTEMBER
1984
219
ment was being made. For small bubbles, say 40 pm diameter or less, an interval of a few seconds used tp measure the bubble’s rise velocity could result in errors of 10% or more in the bubble radius due to bubble dissolution. It was also discovered that small variations in the temperature of the host liquid could result in substantial changes in the rectified diffusion threshold and growth rate of the bubble. As will be seen in the following section, a small change in the dissolved gas concentration ratio can significantly affect the rectified diffusion threshold. Accordingly, it was necessary to surround the test cell with a constant temperature bath (see Fig. l), to ensure temperature stability and equilibrium gas concentration. The temperature of the host liquid was constantly monitored to ensure no change in temperature during the measurements or, just as importantly, no change from the temperature at which the liquid was brought to equilibrium gas concentration.
-1 J
20
With this technique it appears that threshold measurements can be made to within an accuracy of approximately 5%. The radii measurements depend upon the applicability of the drag law, and its specific accuracy, but are probably good to within a few percent also.
Graphical representation and measured data In this section is presented a of the equations as a function parameters that are applicable. thresholds and bubble growth comparison. In these figures, made otherwise, it is assumed
II
1
8-
32-
I I
I
I
I
I.1
1.2
1.3
I .4
Dissolved
gas concentration
rotio ( Ci /Co)
Fig. 2 Stability curves for gas bubbles at various drssolved gas concentration ratios in water. The curves are for three different surface tensions. If a gas bubble lies above the appropriate line, it will grow by diffusion at zero acoustic pressure amplitude; if it lies below, it will dissolve
220
of an air of the
It is seen from (19) that the numerator of the threshold equation can be made to equal zero if the gas concentration ratio Ci/Co = 1 + 2u/R, P,. That is, if the liquid is oversaturated by an amount greater than the surface tension term, gas bubbles will grow with no acoustic field applied. Consider Fig. 2, which shows a plot of bubble radius against the dissolved gas concentration ratio (C;/c,) for various surface tensions such that the ‘critical saturation’ condition is met, that is, Ci/Co = 1 + 2alR, P,. This figure is presented to show the importance of the dissolved gas concentration on gas bubble stability. It can be noted that unless there is some degree of oversaturation all gas bubbles will slowly dissolve (although smaller ones dissolve progressively faster). Further, for very small gas bubbles, high degrees of oversaturation are required before a gas bubble will grow spontaneously. This figure has relevance to the growth of bubbles in carbonated liquids and this particular author has mused over its implications on numerous occasions while making observations of a variety of bubble-filled liquids.
9-
1.0
100
an air bubble in water with the following set of experimental conditions: P, = 1.01 x lo5 dyn cmm2, u = 68 dyn cm-‘, p = 1.0 g cme3, D, = 0.20 cm2 s-l, p = 1.0 X 10m2 g cm-’ s-l, c = 1.48 x lo6 cm s-l, y = 1.4, T = 293 K, D = 2.4 x 10m5cm2 s-l and d = 2.0 x 10-2.
IO -
0 ’
80
Bubble radius Qml
graphical representation of the various physical Some measurements of rates are also given for unless specific mention is that we are considering
-
60
Fig. 3 Variation of the threshold for growth by rectified diffusion bubble in water as a function of the bubble radius for a frequency 22.1 kHz, a surface tension of 68 dyn cm-’ and a dissolved gas concentration of 1 .O. The symbols are experimental measurements; solid curve is calculated from [19); the dashed one from (27)
of the equations
I2 w
40
1.5
Central to the study of rectified diffusion are measurements of the acoustic pressure amplitude required for bubble growth and the agreement between these measurements and the applicable theory. Shown in Fig. 3 are measurements of the threshold for a region (i) case (frequency 22.1 kHz, bubble radii ranging from 20 pm to 70 pm) along with the theoretical calculations. The solid curve is the prediction of (19); the dashed line is the simplified version of the threshold applicable to this region of consideration. (27). It is seen that the two equations give similar predictions until the bubble radius becomes so small that the term 2rr/R, P, becomes significant with respect to unity.
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‘resonance radius’ given by a solution of (2) for R, when w,, is replaced by the driving frequency. In Fig. 5, the driving frequency was 1.0 MHz, the surface tension 68.0 dyn cm-‘, and the ambient pressure 0.1 MPa, thus giving a resonance radius of 3.15 pm. It is seen that the threshold becomes very low near resonance due to the large pulsation amplitude of the bubble there. Near resonance, the amplitude of the bubble’s pulsations are determined mostly by the damping, which is relatively small, and consequently the bubble oscillates rather violently. It is not expected that the equations for bubble growth and for the threshold will be very accurate in this region.
0.08
+j 0.06 C -n 5 0.05
E 2 0.04 t h .; 0.03 a 2 0.02
01 0.6
I
I
I
I
0.7
a0
0.9
1.0
3 II I.1
The variation of the rectified diffusion threshold with frequency for a fixed radius, shown in Fig. 6, is similar to the variation with radius for a fixed frequency, shown in Fig. 5. The familiar reduction in the 1.0
Dissolved gas concentration ratio ( Ci /C,) Fig. 4 Variation of the threshold for growth by rectified diffusion of an air bubble in water as a function of the dissolved gas concentration ratio. The solid line is calculated from (19); the dashed line is from (27); the broken line is a numerical calculation by Eller and Flynne; the symbol is an experimental measurement by Strasberg
The agreement between theory and experiment (for this limited range of bubble radii) is seen to be rather good and within the range of experimental error. This bubble range is particularly important at this frequency because it partially bridges the gap between isothermal behaviour (for R, = 20 pm, tl = 1.01) and adiabatic behaviour (for R, = 80 pm, 1 = 1.23) of the bubble pulsations. The gradual reduction in the threshold as the bubble radius increases is mostly due to the fact that as the bubble grows toward resonance size its pulsation amplitude increases, which results in more rectified mass transfer per cycle. The effect of the dissolved gas concentration in the liquid on the threshold is shown in Fig. 4. These data are for a frequency of 24.5 kHz, a radius of 26 pm and a surface tension of 72 dyn cm-‘. The top two curves are calculated from (19), solid line, and (27), dashed line. It is seen that the simplified and more complicated versions agree so long as the gas concentration ratio is not far from unity. Plotted also on the figure is a measurement due to Strasberg? and a numerical result due to Eller and Flynn.* Since the numerical result is in better agreement, it can be implied that the higher-order terms that are neglected in the analytical approach may be significant if the dissolved gas concentration is considerably less than saturation. It is also noted from the curves that the threshold falls to zero whenever Ci/Co = 1 -t- 2a/R, P,. As discussed earlier, if the amount of dissolved gas present in the liquid is sufficient to overcome the Laplace pressure, the bubble will spontaneously grow. Shown in Fig. 5 is the variation in the rectified diffusion threshold as a function of radius for values of the radius both above and below the resonance values. The curves can be calculated from either (19) or (28), the values being essentially equal. A gas bubble present in a liquid containing a sinusoidally varying pressure field behaves very much like a damped driven harmonic oscillator. It has a linear resonance frequency given by (2). For a system in which the frequency is fixed, and the radius varies, as is typical of studies involving rectified diffusion, one can speak of a
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1984
0.5
0.01
I
I
I
0.2
0.5
I.0
1 0.1
I I 2.0
I 5.0
I IO.0
Bubble radius Cpml Fig. 5 Variation of the rectified diffusion threshold as a function of gas bubble radius for three separate dissolved gas concentration ratios (C/C,). The curves can be calculated by the use of either (19) or (28); the acoustic frequency used was 1 .O MHz and the surface tension was 68 dyn cm-‘; the liquid is assumed to be water
lo-3
1 IO4
I
I
I 10s
I
I
I IO6
Frequency CHzl Fig. 6 Variation of the rectified diffusion threshold as a function of acoustic frequency for two different values of the gas bubble radius. The liquid IS assumed to be water having a dissolved gas concentration ratio of 1 .O and with a surface tension of 68 dyn cm-’
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60 1.8 rz 17 L!+ 1.6 2 ; 1.5 u % 1.4 m’ 1.3 1.2
IO
I.1 0 0
100
200
300
400
500
600
700
Time Csl
0
IO
20
30
40
50
Time Cmsl
Fig. 7 Dependence of the bubble radius on time for an air bubble undergoing rectified diffusion in water. The symbols are measured values; the curves have been calculated by numerical integration of (18).The bottom two curves (and measurements) are for distilled water having a surface tension of 68 dyn cm-‘; the top curve (and associated set of measurements) is for water containing small amounts of surface-active agents, and having a surface tension of 32 dyn cm-’
Fig. 8 Calculated behaviour of a distribution of the sizes of air bubbles present in water as a function of time and exposed to a continuous sound field of frequency 3.0 MHz and acoustic pressure amplitude of 0.12 MPa. The designated line indicates the free decay of an air bubble. The other lines describe the time history of individual air bubbles and were obtained from a numerical integration of (18)
threshold as the resonance frequency is approached is observed. For the two cases considered in Fig. 6, that is, for radii of 10 pm and 50 pm, the threshold is nearly independent of frequency for frequencies considerably less than resonance, but strongly dependent on the frequency above resonance. This behaviour can be easily seen by inspection of (19). It is observed also that the threshold below resonance is higher for the 10 pm than for the 50 pm bubble. This effect is due to the higher Laplace pressure for the smaller bubble.
present in water when exposed to continuous wave ultrasound of frequency 3 MHz and acoustic pressure amplitude of 0.12 MPa. This figure may be applicable to the growth of microbubbles in tissue by therapeutic ultrasound. Note that if this population of ‘free air bubbles’ were suddenly created, those bubbles with radii less than 1.03 pm would slowly dissolve regardless of the ultrasound. Those bubbles in the range of 1.03 pm to 1.4 pm would grow to an asymptotic limit of about 1.42 pm. Bubbles larger than 1.4 pm would decay to the asymptotic limit.
So far, our analysis in this section has been limited to the rectified diffusion threshold. It is in order also to examine the rate of growth by rectified diffusion when the acoustic pressure amplitude exceeds the threshold. An examination of the growth (or decay) of a bubble is shown in Fig. 7. The symbols are experimental measurements; the curves were obtained by numerical integration of (18). It was observed by Eller’O that his measurements of the growth rate were considerably larger than predicted by theory. Gould’* later confirmed the excessive growth rates but also noticed that surface oscillations, with their associated acoustic streaming, were probably the explanation for the increased growth rates that he observed.
The reason for this limit is that due to the Laplace pressure, there is always a tendency to force gas out of the bubble (note the free decay curve on the figure). If the bubble radius is less than resonance size, and greater than a threshold size, the bubble will grow through resonance to a size sufficiently above resonance where its pulsation amplitude will result in an inward mass transfer due to pulsation that just matches the outward mass transfer due to the Laplace pressure. If the bubble is larger than the asymptotic limit, it will not be able to overcome the outward diffusion due to the Laplace pressure and will dissolve slowly until its pulsation amplitude is large enough to stop further dissolution. Consequently, there is a tendency to force microbubbles present in a liquid to a uniform size that is dependent on the intrinsic and extrinsic physical parameters.
Shown in Fig. 7 are some observations of growth rates made by Crum. I8 He observed an agreement between the measured and predicted rates provided surface oscillations were avoided, and provided that the water was relatively pure and free of surface-active contaminants. When surface-active agents were added to the water, growth rates much higher than predicted were observed. Among the explanations offered for this effect were that (a) the surface-active agents were behaving as a rectifying agent, permitting more diffusion in than out, and (b) acoustic streaming was occurring in the absence of surface oscillations, induced somehow by the surface contamination. This interesting anomaly has not yet been explained. As a final result of this section, consider Fig. 8, which demonstrates what would happen to a population of air bubbles with radii varying from 1 pm to 2 pm
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Conclusions A set of equations were presented that describe the process of rectified diffusion for a variety of physical conditions. The experimental data available tend to confirm the validity of the existing equations provided the acoustic pressure amplitude is not large, the liquid does not contain surface-active additives, and the dissolved gas concentration ratio is near saturation. Areas that need further
study are as follows:
1. It was seen that the addition of surface active agents to water resulted in measured values of the threshold and growth rate considerably different from those
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1984
predicted by theory. An adequate anomaly should be given.
explanation
of this
2. It is observed that the analytical expressions for the rectified diffusion threshold are not in good agreement with the few data that exist at dissolved gas concentration ratios significantly different from saturation. Numerical studies give much better agreement. An extension of the analytical results to higher order to account for the discrepancy with the dissolved gas concentration ratio is needed. Further. since data at under-saturated levels are sparse, additional data should be obtained. 3. It has been recently observed28 that gas bubbles driven at the pressure amplitudes experienced in rectified diffusion studies will experience considerable fluctuations in their pulsation amplitude as they are driven through their harmonic resonances.29 At these radius positions, the bubble may experience rapid growth that is unaccounted for in the analytical expressions given in this article. 4. It was observed by Gould” that increased growth by rectified diffusion occurred in the presence of acoustic streaming in the vicinity of the bubble. An adequate theory has not been presented to account for the effect of this streaming on rectified diffusion enhancement.
7 8
9
10 11 12
13 14
15 16 17 18 19
20
Acknowledgement The author wishes helpful assistance, and the National financial support discussed here.
21
to thank Gary Hansen for his and the Office of Naval Research Science Foundation for providing for the research work of the author
22 23 24
References 25 Harvey, E.N., Barnes, D.K, McElroy, W.D., Whitely, A.H., Pease, D.C., Cooper, K.W. Bubble Formation in Animals, J Cell Comp. PhysioL 24 (1944) l-40 Blake, F.G., Jr The Onset of Cavitation in Liquids, Tech. Memo. No. 12, Acoust. Res. Lab. Harvard Univ. (1949) Strasberg, M. Rectified Diffusion: Comments on a Paper of Hsieh and Plesset, .I Acoust Sot Am. 33 (1961) 359-360 Strasberg, M. Onset of Ultrasonic Cavitation in Tap Water, J Acoust Sot Am. 31 (1959) 163-176 Pode, J. The Deaeration of Water by a Sound Beam, David Taylor Model Basin Dept. No. 854 (1953) Rosenberg, M.D. Gaseous-type Cavitation in Liquids, Tech. Memo. No. 26, Acoust. Res. Lab. Harvard Univ. (1953)
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27 28
29
Hsieh, D.Y., Plesset, M.D. Theory of Rectified Diffusion of Mass into Gas Bubbles, J. Acousf. Sec. Am. 33 (1961) 206-215 Eller, A.I., Flynn, H.G. Rectified Diffusion during Nonlinear Pulsations of Cavitation Bubbles, J. Acoust. Sot. Am. 37 (1965) 493-503 Safar, M.H. Comment on Papers Concerning Rectified Diffusion of Cavitation Bubbles, J. Acoust Sot. Am. 43 (1968) 1188-1189 Ellen AI. Growth of Bubbles by Rectified Diffusion, J. Acoust Sot Am. 46 (1969) 1246-1250 Ellen A.I. Bubble Growth by Diffusion in an 1I-kHz Sound Field, J Acousr. Sot. Am. 52 (1972) 1447-1449 Gould, R.K. Rectified Diffusion in the Presence of, and Absence of, Acoustic Streaming, J Acoust Sot. Am. 56 (1974) 1740-1746 Davison, B.J. Mass Transfer due to Cavitation Microstreaming J. Sound Vib. 17 (1971) 261-270 Kapustina, O.A., Statmikov, Y.G. Influence of Microstreaming on the Mass Transfer in a Gas-BubbleLiquid System, Sov. Phys. Acoust. 13 (1968) 327-329 Skinner, LA. Pressure Threshold for Acoustic Cavitation, J. Acoust. Sot Am. 47 (1970) 327-331 Skinner, L.A. Acoustically Induced Gas Bubble Growth, J. Acoust Sot. Am. 51 (1972) 378-382 Eller, A.I. Effects of Diffusion on Gaseous Cavitation Bubb1es.J. Acousr. Sot Am. 57 (1975) 1374-1378 Crum, LA. Measurements of the Growth of Air Bubbles by Rectified Diffusion, J. Acoust Sot. Am. 68 (1980) 203-211 Lewin, P.A., Bjfimu, L. Acoustic Pressure Amplitude Thresholds for Rectified Diffusion in Gaseous Microbubbles in Biological Tissue, J. Acousf. Sot. Am. 69 (1981) 846-852 Apfel, R.E. Acoustic Cavitation Prediction, J Acousr. Sot. Am. 69 (1981) 1624-1633 Cmm, LA., Hansen, GM. Growth of Air Bubbles in Tissue by Rectified Diffusion, Phys. Med. Biol. 27 (1982) 413-417 Crum, LA., Hansen, G.M. Generalized Equations for Rectified Diffusion J. Acoust Sot. Am. 72 (1982) 1586-1592 Eller, A.E. Damping Constants of Pulsating Bubbles, 1 Acoust Sot. Am. 47 (1970) 1469-1470 Devin, C. Survey of Thermal, Radiation, and Viscous Damping of Pulsating Air Bubbles in Water, J. Acoust Sot, Am. 31 (1959) 1654-1667 Prosperetti, A. Thermal Effects and Damping Mechanisms in the Forced Radial Oscillations of Gas Bubbles in Liquids.1 Acoust. Sot Am 61 (1977) 17-27 Crum, LA. The Polytropic Exponent of Gas Contained Within Air Bubbles Pulsating in a Liquid .I Acoust Sot Am. 73 (1983) 116-120 Crum, L.A., Eller, AI. The Motion of Air Bubbles in a Stationary Sound Field, 1 Acoust. Sot Am 48 (1970) 181-189 Crum, L.A., Prosperetti, A. Nonlinear Oscillations of Gas Bubbles in Liquids: An Interpretation of some Experimental _ Results,1 Aco& Sot Am. 73-(1983) 121-127 Prosperetti, A. Nonlinear Oscillations of Gas Bubbles in Liquids: Steady-State Solutions, J Acoust. Sot Am. 56 (1974) 878-885
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