J. Phys. Chem. 1985,89, 1520-1523
1520
Lifetimes of Metastable Microbubbles Paul C. Engelking+ Department of Chemistry and Chemical Physics Institute, University of Oregon, Eugene, Oregon 97403 (Received: September 21, 1984)
The lifetime of metastable bubbles in solution is determined under conditions applicable to laser dye circulation systems. When the solution is supersaturated in dissolved gas, the population of microbubbles is truly metastable, with a characteristic long-time population distribution of bubble radii and a characteristic decay rate. If the solution is not saturated in dissolved gas, a maximum lifetime is obtained.
Introduction The realization of an idea in experimental or practical devices often depends upon effects that are quite removed from the intended principles of the device operation. It may be nature’s weakest, composite forces that assist the operation of a device operating upon some of nature’s strongest, fundamental forces. One of these weak, composite forces is surface tension. We depend upon this force literally to keep the gears of our machinery lubricated. But should something so modem as a laser depend upon something so old-fashioned as surface tension? In fact, modern dye lasers depend upon surface tension effects in several critical ways. Laser-pumped dye jet lasers depend upon surface tension to define a liquid/air interface in the dye jet. The length of the flat “window” through which the pump and lasing beams must pass is affected by surface tension; if surface tension were too strong, the jet would too rapidly depart from a plane cross section. Also, surface tension provides a mechanism for surface wave propagation. To control surface waves, the formation and “catching” of the dye jet are critical. In another way surface tension enters the operation of pulsed and C W lasers alike: bubbles present in the dye can serve as optical inhomogeneities,leading to beam attenuation or “dropouts”. These bubbles spectacularly affect C W dye laser operation when they are able to concentrate the pump beam, heat up, grow, and finally explode, interrupting laser action and sometimes scattering dye solution onto laser optics. The importance of bubble control has long been recognized and is often attempted by the use of micron-sized filters in the dye line to remove bubbles and by good circulation techniques attempting to prevent bubble formation. However, theoretical understanding of microbubbles has not kept up with the empirical methods used for their control. The circulating dye laser is similar to the recently shaken carbonated beverage in that both will “remember” having been mistreated. Both form small, metastable microbubbles, potentially the nucleation centers for large bubble growth when the liquid is heated or placed under reduced pressure. This memory is one of the most striking aspects of the phenomenon, and it is wellknown that a bottle of champagne or a soft drink will fizz violently if opened within minutes after vigorous shaking. We examine the theory of metastable bubbles here, as especially applied to the problem of circulating dye lasers. We will also provide answers to the following problems: (1) How long will a liquid remember its agitation in the form of microbubbles? (2) What affects microbubble lifetimes and may be used to control them? As first pointed out by Gibbs,’ the evolution of a bubble in solution depends critically upon the two parameters of the hydrostatic overpressure Po and the pressure P, corresponding to the partial pressure of a gas in equilibrium with the concentration of gas dissolved in the liquid. The prasure P inside of each bubble will be higher than the surrounding hydrostatic pressure Po, and this, combined with the liquid surface tension, gives the bubble its characteristic spherical shape. In addition, if the resulting Alfred P. Sloan Fellow.
0022-365418512089-1520%01 .SO10
internal pressure is higher than P,, the bubble will “dissolve” by entering into solution, but if the internal pressure is less than P,, the bubble will grow as gas enters it from the solution. This leads to distinct cases, determined by the relations of Po to P,. If Po equals or exceeds P,, the internal bubble pressure will exceed P,for all bubble radii, and all bubbles will shrink. Bubble lifetime in this case will be set by the length of time it takes for a bubble either to dissolve or to be transported out of the solution. If transport involves floating up to the surface, large bubbles will be removed more rapidly by transport. Thus, there will be a critical size of bubble having maximum lifetime. We will determine this lifetime for this case. If Po is lesss than P,, for some critical bubble radius a bubble will be metastable to growth (which would occur if it were larger) and to dissolution (which would occur if it were smaller). One wishes to know how long this bubble will be metastable. We will reformulate this question into one that will be more useful: How will an original size distribution of bubbles evolve? For long times, we show that the population assumes a characteristic distribution for which a natural definition of lifetime can be given. Noyes and Smith2 have recently investigated microbubbles in supersaturated liquid solutions of gases in connection with chemical oscillations. Reactions which form a gas in solution can drive the solution far into supersaturation if nucleation centers are absent. Microbubbles can then form spontaneously, nucleating large bubble formation, and sweeping the dissolved gas from solution. After a suitable time interval, the solution again becomes supersaturated, and the foaming repeats. This relaxation oscillation has been extensively analyzed in the form of the “bubblator“ model of Noyes and Smitha2Because of the insight contained in their work, the reader is advised to consult their papers. We will be taking a single aspect of their work, that of bubble size distribution in a metastable situation, and developing that idea here. We would like to point out that while the bubblator oscillations require bubble formation through fluctuations, we will assume that agitation originally produces the bubble distribution, and we then watch the evolution of this distribution in time. The actual bubble distribution will be less interesting to us than the time evolution of the total population capable of subsequent nucleation.
Model We will find it convenient to develop the distribution of bubbles in number space. N(r,t) will be the number of bubbles having radii less than or equal to r at time t . We will take a mean field approach and assume this distribution is continuous, with the derivative of this function given by p(r,t) = aN(r,t)/ar (1) It is this number density that we will examine. Bubble Growth Rare. A bubble in solution will adjust itself rapidly to have minimum surface area under the constraint of the bubble volume, balancing off surface tension against internal (1) J. W. Gibbs, “The Scientific Papers of J. Willard Gibbs”,Vol. 1. Dover. New York, 1961. (2) K. W. Smith and R. M. Noyes, J . Phys. Chem., 87, 1520 (1983); R. M. Noyes, J . Phys. Chem., 88, 2827 (1984).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
Lifetimes of Metastable Microbubbles pressure. It is easily established that bubbles will have the size1** r = 2 u / ( P - Po) (2)
01
(4)
Thus, the model used by Noyes is appropriate in situations in which the bubble surface is only partially permeable to the gas (surface contamination, or high activation barrier a t the liquid surface) or if each bubble has a boundary layer of thickness a separating it from a well-mixed turbulent bulk liquid, in which case an effective linear transport coefficient is given by eq 4. In the case in which turbulence is low, each bubble will determine its own characteristic length, which can be shown3 to be the bubble radius r. In this case kt; will be a function of bubble radius and we have k,;(r)
= D/r
(5)
which characterizes diffusion-limited transport for slowly changing bubbles. We will develop the theory explicitly for the case of constant ktr,the case of Noyes’s model, but will also state the results for diffusion-limited bubble surface transport and compare these to that of the linear transport model. We will find it useful to state transport in terms of gas pressure in the bubble,rather than in terms of the dissolved-gas concentration. The dissolved-gas concentration is related to an equilibrium gas pressure by a Henry’s law constant K*
c, = KP,
where k, = Kk,,RT. Physical meaning can be given to k, by imagining k, = u,llision(accommodation factor) The total flux into the bubble will be
+ 3P,#)(ar/at)
Equating gives2
VI C
0,
D L
a
n
0
0.6 1.2 radius (lo-’ m) Figure 1. Characteristic bubble size distributions corresponding to eq 15 determined for typical values ro = 1.2 X IO” m, rs = 1.0 X 10” m, and k = 2.5 X m s-l, and various values of X. The corresponding value of metastable bubble radius r* is 0.6 X m. The critical value of decay time constant A* is 0.74 s-l, corresponding to the distribution shown by the dotted line. The nonphysical distributions for X less than this, appearing above this line, diverge at r*. Physically allowed distributions occur for X greater than X* and appear in the figure below the critical dotted line. These physical distributions may have a zero at r*. The exponential divergence for large values of bubble radius is harmless: large bubbles are removed from solution by transport, cutting off the bubble distribution at large radius. All distributions were normalized to unity at r = 1.2 x m.
(9)
~
Time Evolution of Bubble Distribution. If we are well into the regime of bubble size large enough to avoid fluxations (free energy much greater than kT), bubbles will evolve deterministically, and we can avoid diffusional terms in what would be the Smolachowski equation in number space. Instead we simply writeZ
Since we are interested in long-time behavior, we look at the time eigenfunctions p(r,t) = pA(r)e+‘
(13)
which, for &/at given by eq 11, has solutions by quadrature = (2/3)rors k,
+
~rse~~r/k*(r~-r,)ir(ro - r,) - rOrs~((r~zr~+2rorrf)A/3(ro-r,)zk8)-l (15)
This distribution function is plotted in Figure 1 for a number of values of A. Presumably this distribution is cut off by transport of bubbles out of solution for some macroscopic value of r. This cutoff makes the divergence due to the exponential term harmless, since it will diverge only for the nonphysical values of r, r = m. More serious is the divergence for negative powers of the term r(ro - r,) - rf,, which occur when
(8)
The growth of the bubble is related to this flux by flux = (4r/3RT)(4ur
x Y .-
(6)
Thus, we may refer all concentrations in solution to their equilibrium gas pressures. P, and C, will refer to the saturation pressure and concentration of gas in solution, which we assume to be constant, while we reserve P to be the actual bubble pressure and Poto be the total static pressure on the system. In terms of gas concentration, the linear transport becomes flux = Kk,,P(area) = Kk,,RTC,,(area) = k,CBas(area) (7)
flux = (4rr2)kg(RT)-’(PS- P)
L
(3)
Others have modeled transport due to diffusion in the liquid. If there is a characteristic length a for the diffusion concentration gradient, in terms of the diffusion constant D, we have k,, = D / a
I
-
where r is the radius, u is the surface tension, and P - PO is the difference between bubble internal gas pressure and the hydrostatic pressure of the solution. Gas transport across the bubble surface can be modeled in two ways. In the approach of Noyes: the surface is modeled as a membrane with linear transport flux = k,,(area)(concentration)
1521
X
< X*
= 3k,(ro - r,)z/(2r02r,
+ ror:)
(16)
or X* = 3k,(P,
- Po)2/(2~)(2P,+ Po)
(17)
This places a divergence on the physical real axis a t the radius of metastability r* = rors/(ro - rs)
(3) L. Liebermann, J . Appl. Phys., 28, 205 (1957).
(18)
whenever ro > r,, or equivalently, Po < P,. Thus, if we wish only to consider physical, nondiverging distributions, we must consider an eigenvalue spectrum of values of X greater than A*. The spectrum of physical values of X is shown in Figure 2. Metastability When Po < P,. We concern ourselves with long-time solution, Le., metastability, and especially the time
Engelking
IS22 The Journal of Physical Chemistry, Vol. 89, No. 8, 1985
A eigenvalues
jim
K physical spectrum
Figure 2. Spectrum of physically allowable values of A. A physically realizable distribution will consist of a superposition of distributions corresponding to values of A greater than A*.
constant associated with this. A solution will be expressible as the form p(r,t)
=
1
e-AI- pA(l‘)CA dX
(19)
We see,that p A ( x ) has nonphysical divergences at values of X A*. Thus, we restrict the integral p(r,t) = x : C A 1 p A ( T ) c A dX
P,. Bubbles all tend to dissolve when Po is greater than P,, and in this case there is no actual metastability. Instead, we find a maximum lifetime bubble radius if bubbles either shrink or are transported out of the solution over the distance 1.
Bubbles float to the surface at the Stokes velocity V = 2ggd/9q
(26)
3
491 2gd
0
from which the maximum lifetime can be obtained. The actual situation is a bit more complicated, since as a bubble dissolves, its Stokes velocity will diminish. Thus, we actually need to find the critical initial value of r for which the time of removal by transport equals that for dissolution. If critical, the bubble will wink out just as it reaches the surface. The critical r is given by solution of -ror2
+ r3 + (2/3)rOZr:+ ( l / 2 ) r o r s r 22(ro - rh2
for which the lifetime is found by substitution of this into eq 28. If the liquid is not quiet, but turbulent, or flowing, so that I cannot be considered fixed, the analysis can be extended to this case by considering bubble removal as dependent upon radiusdependent, first-order kinetics. Since this analysis would be highly idiosyncratic to the particular system and geometry, we do not develop this approach further here.
Lifetimes of Metastable Microbubbles
Discussion Mass transfer from bubbles has been studied elsewhere. Liebermann3 obtains the diffusion-controlled solubility (equivalent to eq 5 ) of a vapor bubble in a liquid, but neglects the additional internal pressure resulting from surface tension. Thus, his theory is essentially a large-bubble analysis. He has also modified the theory for the case of surface contamination that reduces transport across the surface itself. Coppock and Meicklejohn4find the mass transfer from a stream of freely rising bubbles to a liquid. Similar treatments have been provided by Forster and Zuber5 and have been investigated experimentally by Pattle.6 Mass transfer under chemical reaction conditions has been studied by Gal-Or and Resnick.' An alternative to the approaches used here assumes that mass transfer is limited not by a characteristic boundary thickness nor by surface transport, but by a characteristic time t* for replacement of solution near the surface. This is the approach of Higbie8 and D a n c k ~ e r t s . ~In this case k,,"
= 2(D/.rrt*)'/'
-
Davies'O has shown k,, D112 for highly agitated water/air systems. Calderbank" found this to hold for large bubbles ( r > 2.5 mm). This appears to be a regime that is not of interest here. During preparation of this manuscript, the work of Ward and LevartI2 appeared, concerning the equilibrium of bubble nuclei on solid surfaces. This work is interesting for the insights it provides. The formulation makes explicit use of the Gibbs free energy of the bubbles and reiterates Ward's ob~ervation'~ that a finite liquid solution (one that would be depleted of dissolved gas as bubbles grow) could have a stable equilibrium for bubbles trapped on a surface. Nevertheless, stability is not expected for free bubbles in the bulk, as our kinetics show: if Po < P,, a (4) P. D. Coppock and G. T. Meicklejohn, Trans. Insr. Chem. Eng., 28, 52 (1950). (5) H. K. Forster and N . Zuber, J . Appl. Phys., 25, 474 (1954). (6) R. E. Pattle, Trans. Insr. Chem. Eng., 28, 27 (1950). (7) B. Gal-Or and W. Resnick, Chem. Eng. Sci., 19, 653 (1964). (8) R. Higbie, Trans. Inst. Chem. Eng., 31, 365 (1935). (9) P. V. Danckwerts, Ind. Eng. Chem., 43, 1460 (1951). (10) J. T. Davies, A. A. Kilner, and G. A. Ratcliff, Chem. Eng. Sci., 19, 583 (1964). (11) P. H. Calderbank, "Mixing", Vol. 2, V. W. Uhl, J. B. Gray, Eds.,
Academic Press, New York. (12) C. A. Ward and E. Levart, J . Appl. Phys., 56, 500 (1984). (13) C. A. Ward, P. Tikuisis, and R. D. Venter, J . Appl. Phys., 53,6076 (1982).
The Journal of Physical Chemistry, Vol. 89, No. 8, 1985 1523 steady-state distribution given by eq 15 for the X = 0 state always diverges nonphysically at r = ras/(ro- r,); if Po > P,, a steady-state ro - r, can be found for which the divergence occurs harmlessly for nonphysical, negative r, but this distribution requires a source for the shrinking bubbles at large, finite r. As Ward's thermodynamic analysis indicates, bubble kinetics on surfaces will be more involved than for bubble evolution in solution. Comparison with experience requires knowledge of the physical parameters used to determine A*. In particular, we need k,. Noyes2 uses k,, = 0.1 s-I for C O at an aqueous interface, with K = 10I2 dyn-I cm-I ( K = 5 X dyn-I cm-' for N 2 at 20 O C in H20),14 giving k , = 2.5 X cm s-l. Artificial lungs are cm s-' for 02/blood.'5 Recently, Noyes16 designed with k, = cm s-l, leading to k , = has revised k,, downward to 2.3 X 6X cm s-l. Direct measurement is often difficult, since for surfaces of low curvature diffusion is often limiting. For an aqueous solution u = 60 dyn cm-' and at atmospheric pressure ro = 1.2 X lo4 cm. If P, = 1.2 atm, X* is 0.74 s-l, for k , = 0.1 cm S-I, corresponding to r* = 6 X lo4 cm. For the more recent value of ke, A* is proportionally smaller. Thus, if a dye circulation system supersaturates the dye by only 20%, significant lifetime of the metastable bubbles will result. This lifetime is made longer, rather than shorter, by reducing the amount of supersaturation! Thus, we see the difficulties involved in eliminating bubbles: the controlling factors may work in ways that are not obvious. It is hoped that the analysis presented here may elucidate how various parameters may interact with the bubble population, leading to a firm basis for the technology of their control.
Conclusions An analysis of the kinetics of bubble evolution in a solution of a gas dissolved in a liquid of large extent demonstrates that a natural lifetime can be obtained either for true bubble metastability (P,> Po) or for the case of all bubbles being soluble (P, < Po). In the first case, the first-order decay rate constant of the metastable distribution X* is given by eq 17 when transport from the bubble is controlled at the bubble surface or by A*' (eq 24) when transport is controlled by bulk diffusion in the liquid. Acknowledgment. This work was supported in part by the Sloan Foundation, and the Murdock Charitable Trust. (14) C. J. J. Fox, Trans. Faraday Soc., 5 , 68 (1909); N . W. Rakestraw and V. M. Emmel, J . Phys. Chem., 42, 1211 (1938). (15) R. Berkow, Ed., 'The Merck Manual", 14th ed., Merck Sharp and Dohme Research Laboratories, Rahway, NJ, 1982. (16) S. M. Kaushik and R. M. Noyes, submitted.
