589
MECHANISM OF JETFORMATION AT A LIQUIDSURFACE
Bubbles: A Boundary-Layer “Microtome” for Micron-Thick Samples of a Liquid Surface by Ferren MacIntyre’ Scripps Institution of Oceanography, University of California, S a n Diego, L a Jolla, California (Received J u l y IO, 1967)
99097
The vertically ejected jet drops from bubbles smaller than 1 mm in diameter are formed from a thin superficial layer of liquid accelerated inward by surface forces. The drops are easily collected and offer a novel nonmechanical surface microtome.
Small gas lbubbles, breaking at a liquid surface as shown in Figure 1, eject droplets several centimeters into the air.2 These jet droplets are composed of surface material and are easily collected by various means (e.g., a filter, an aerosol impactor,a or simply an inclined plate draining into a collection vessel). The thickness of the surface thus collected is not adequately known: it lies between 0.5 and 20 pm for a 600-pm bubble, with indirect evidence4 suggesting 10 pm. This is many times thinner than the surface collected by McBain and 1HumphreysJ6delicate and complex surface microtome, yet requires no more apparatus than a source of uniform bubbles, which may be generated in large numbers by an oscillating capillary tip.6 Uniformity of bubble size may be necessary if a uniform thickness of surface cut is desired. I n contrast to foam-collection methods,’ which sample the surface by forming stable film caps, the bubble microtome depends upon foam instability and is suited for surface concentrations far lower than those which can be sampled by foam methods. The smaller the thermodynamic surface excess Ti (ions/cmz), the thinner the slice 6’ (cm) required to see it, since
contributes only about 1%of the total energy. Surface tension produces a pressure P given by Laplace’s familiar formula P = u(K1 K z ) , where the K’s are the principal curvatures. Since P is exerted normal to the surface, it might seem that the jet drop could be composed of material from the nadir of the bubble through a simple radial displacement. However, there are several arguments for boundary-layer-like flow down the cavity walls. Blanchard2 has observed that monolayer material from the top surface of the liquid (not from the interior bubble surface) appears to cover 200% of the drop surface (for drops from 3-mm bubbles) -that is, there must be some mechanism acting to collect material from outside the cavity, bring it to the center, and compress it. He further notes complete removal of oleic acid surface films by prolonged bubbling. The spreading velocity of surfactants (which falls to zero when the surface is covered with a monolayer) is at best only 10 cm/secg-far too slow to travel from perimeter to jet unless the surface fluid is itself moving parallel to the surface. The photographs of Kientzler, et u1.,l0 which form the basis for Figure 1, clearly show the peculiar corner ahead of the wave traveling down
ri = (di - ci) 6’
(1) Scripps Institution of Oceanography, La Jolla, Calif. (2) D. C. Blanchard in “Progress in Oceanography,” Vol. 1, M. Sears, Ed.,Pergamon Press Inc., New York, N. Y., 1963,p 71. (3) R. I. Mitchell and J. M. Pilcher, I n d . Eng. Chem., 51, 1039 (1959). (4) F. MacIntyre, Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1965. (5) J. W. McBain and C. W. Humphreys, J . Phys. Chem., 36, 300 (1932). (6) F. MacIntyre, Rev. Sci. Instr., 38, 969 (1967). (7) J. W. McBain and R. DuBois, J . Am. Chem. SOC., 51, 3534 (1929); D. M. Gam and W. D. Harkins, ibid., 52, 2289 (1930); F. van Voorst Vader and M. van den Tempel, 3rd Intern. Congr. Surface Activity, 1,248 (1960);A. Lauwers, P. Joos, and R. Ruyssen, ibid., 3, 195 (1960). (8) F. MacIntyre and J. W. Winchester, unpublished data. (9) J. T. Davies and E. K. Rideal, “Interfacial Phenomena,” Academic Press, New York, N. Y., 1961,p 27. (10) C. F.Kientsler, A. B. Arons, D. C. Blanchard, and A. H. Woodcock, Tellus, 6, 1 (1954).
where di is the concentration of i in the collected drops (ions/cm3) and ci is the concentration in the bulk liquid. The detection limit of (di - ci) is set by the analytic method used, and further sensitivity must come from a reduction in 6’. Using the bubble microtome, Fits from lo9 down to lo7 ions/cm2 (or 10-7 monolayer) have been estimated using radiotracers for assay;* these surface concentrations (although 100-fold higher than thebulk fluid) were toosmall to give measurable surface tension (6)changes to evaluate ri independently as -(&/a In a)/RT. The energy for drop production comes from the surface free energy dG = adA, where dA is the change in area. (See Figure 2.) Gravity (hydrostatic pressure)
+
V o l u m e 79, Number 0 February 1068
590
FERREN MACINTYRE
Figure 1. Time-sequence diagram of a breaking bubble a t 6000 frames/sec. The top jet drop, which is about to break off from the ascending jet, leaves with a velocity of 500 cm/sec for the 1.7-mm bubble shown. The diagram is deduced from published10 and unpublished high-speed photographs by Kientzler, et al., and the theoretical work of Toba [ J . Oceanog. Japan, 16, 1 (1959)]. I n the photographs the upper surface is distorted by a nearby meniscus, so that the drawing is to some extent interpretive. The small ripples ahead of the wave descending the bubble wall appear in all photographs and strongly suggest that although the outer edge of the deformation has the character of a normal capillary wave (with orbital motion of the water), the inner edge is more like a cascade of water running down a wall, with its maximum velocity at the surface.
i
(;I
1
1
1
I
0 1
2
3
4
5
I
1
1
6 7 8 STAGE OF RUPTURE
1
1
1
9 1 0
Figure 2. Liquid volume V and surface area S for the bubble of Figure 1. The initial steep drop in S is caused by the rapid loss of film-cap area, which is a microsecond process. Loss of surface area in the early stages and corresponding conversion of surface free energy into kinetic energy is much more rapid than the rate of cavity filling. The breaks in the curves before stage 1 indicate the uncertainty of time of rupture, which occurred between frames. Equilibrium (flat-surface) values Ve and S, are shown a t right. Datum for liquid volume is horizontal water surface.
the cavity; this strongly suggests that the wave is not produced by radial motion but by tangential motion. Again, photographs of larger cavities by Worthington and Cole," with flow markers injected near the cavity surface, show a boundary-layer flow whose thickness is some 10% of the cavity diameter. When one liquid is dropped into another, the jet drop that returns has The Journal of Physical Chemistry
nearly the same composition as the input drop, showing that the flow pattern during cavity formation and collapse is a highly reversible one. The input drop spreads into a thin lamina lining the cavity at its greatest extent-a cavity resembling stage 5 of Figure 1. This lamina, with a minimum of mixing with the substrate, then regenerates into the ascending drop. A momentum balance requires that there be a second jet directed downward. The vortex ring resulting from this downward jet is shown in Figure 3. If the velocity in the boundary layer decreases linearly with depth, 0.7 of the moving thickness must be directed downward. This reverse jet sweeps the original nadir material down and away, ensuring that the rising jet is composed solely of surface material. Jet formation is not a property of the driving force which collapses the cavity, but simply a result of radial symmetry. When hemispherical cavitation bubbles collapse against a metal plate, high-energy jets erode the metal. The jets appear to arise by amplification of small deformations of the shrinking hemisphere.12 Large and small underwater explosions (depth charges and exploding wires) form cavities which frequently pass through a toroidal figure during their collapse, as a rapid upward jet squirts through the cavity, driven simply by the greater hydrostatic pressure at the bottom of the cavity and focused by the symmetry of the con(11) A. M. Worthington and R. s. Cole, Phil. Trans. Roy. SOC. (London), A189, 137 (1897); A194, 175 (1900). (12) C. F. Naud6 and A. T. Ellis, Trans. A S M E , Ser. D . , J. Basic Eng., 648 (1961).
MECHAAISM OF JETFORMATION AT A LIQUID SURFACE
591 jet, while a thin layer from the interior surface of the liner is directed into the high-velocity forward jet.'' Blanchard notes that the jet drops carry off l0-20% of the available surface energy. Does viscous dissips, tion in boundary-layer flow leave this much energy available? A rough answer may be given by attempting to compute an energy balance, as has been done15 for the case of an expanding hole in a thin liquid film, where surface energy is equated with kinetic energy, giving 2uA = p6Av2/2
Figure 3A. An air bubble is I A w n dilute suspension of India ink.
iii
:L
with p the density, 6 the film thickness, and u the spreading velocity, some 1700 cm/sec for a 1-rm film of water. A similar energy balance for an interface of diminishing area must take into account both inertial and viscous terms. The inertial force per unit volume is p ( a u / a t ) , the viscous force pv(a2u/ay2), where y is the coordinate normal to the surface and Y = the kinematic viscosity.16 We multiply these by the volume A6 and integrate over the distance dz = udt to get the energy, which is equated to the available surface energy. Then, dropping A from both sides, we have au = pJT(bv/bt
Figure 313. The nir hiilhle is plnced on a water surface. (The dark horiroutnl siirfnac is ink added hcforehnnd tn equalize surface tension.) Nost of the ink i n the hiibble itself remains at t hc huthle intcrfare.
traction.'J The shaped charge of bazooka fame collapses a metal liner (which behaves as a perfect fluid a t the pressures involved). As in the bubble, the bulk of the liner flows into a low-velocity backwarddirected
(1)
where m is the unknown fraction of surface energy which can be thought of as going directly into boundarylayer flow and not directly into bulk motion of the fluid. Presumably a is a function of bubble diameter. The critical time T-from rupture to jet formation (say,stage 7 of Figure 1)-can be gotten from Blanchard for a 1.7-mm bubble as 1.5 see. Since Kientsler, et al., found that rupture time is directly related to diameter, the critical time for a 6OO-pm bubble may be taken as T = 6.10-'sec. The simplifying assumptions made to intepate eq 1 are: (i) acceleration &/at = a is constant. (If Li is the maximum surface velocity, we take a = U / T . With U i 100 cm/sec, a (l/6)lOScm/sec, or 167 g; this is so large that its change with time may be ignored here); (ii) d2u/by2may be takenas @,ignoring a coefficientof order unity.'J We now have au = p
Figure 3C. At a time I/, see after the bubble breaks, ink from the oiiter shell of the iiiterface has been ejected downward in a jet which leaves behind a vortex ring and B trail. The short thick column at the surface is ink which has heen eollected from the bubble surface and concentrated. I t appenrs to be that portion of the upward jet rhieh did not escape ay jet dropr.
+ vb2v/by2) 6udl
soT
(au6
+ vu2/@ dt
(2)
The velocity distribution u(y,t) aero= the boundary layer of the bubble may in ignorance of any better model be approximated by the distribution in twc(13) J. W. Pritehett. USNRDLTR-1044. Sm,Franeiseo. Calif.. Mny 1966; R. H. Cole. "Underwater Explosions. I'rinceton University Pleas. Princeton. N. J.. 1948. (14) G. Birkhoff. D. P. MneDonald. W. M. Pugh, and G. I. Tnylor. 3. Appl. Phua.. 19, 563 (1948). (15) Lord Rnyleiph. Nature. 44,249 (1891): L.Fney. 3. Sd.Melrm.. 3.86 (1951): W.E. Rans. J . A p $ . Phua.. 30, 1950 (1959). (16) SI. Sehliohting, "Boundary Layer Theory." MoGmr-Hill. New York. N . Y.. 1955, pp 13. 65.
Volume 7% Number d Febnrorv 1968
FERREN MACINTYRE
592
dimensional Cartesian flow, where, with constant acceleration, it is1’ v(y,t) = a t [ ( l
+ 2Yz)erfc~-(2/v‘i)
ye-”]
with Y = y/(2.\/2). This function drops to 1% of its surface value at Y = 1.45, for a velocity boundarylayer thickness of 60 = 2.91/Jt, or 72.5 pm a t time T for the 600-pm bubble, in good agreement with Worthington and Cole’s 10 % value. (The corresponding thicknesses at 10 and 50% of the surface velocity are 45 and 15 pm, respectively.) For the energy calcule tion, however, we are interested in the energy boundarylayer thickness 6, which may be taken as 6O/4 = 18 pm (to the approximation that v is linear in y). Thus we may use 6 = k v % with k = 2.9/4 = 0.725 to integrate eq 2. We further take linear average velocities of the velocity over the moving thickness, so that if v(surface) = at, ij = v/2, and 0 2 = v2/3. Finally, replacing UT with U yields CZ(T
=
(‘/z
+ ‘/3k2)2p6U2/5
(3)
where the terms in parentheses are the inertial and viscous portions of the energy consumed, 44 and 56%, respectively. Thus, nearly half of the energy going into boundary layer flow is used mechanically and may be carried off by the jet drops. We note in passing that eq 3 gives an estimate of the maximum velocity
u=
[a/(si.5
10-3)]~/~
or 280 4;cm/sec for the 600-pm bubble, satisfactorily higher than monolayer spreading velocities and lower than the ultimate jet-drop ejection velocity of 2000 cm/sec which is compounded by the radial convergence of the boundary layer. We can put a lower limit on the thickness ejected by assuming that the jet drop is made from a uniform thickness 6* of some fraction, f, of the bubble surface. The jet drop has a diameter 0.1D for bubble diameters, D,between 100 and 1000 pm; a mass balance shows that 6* = ( D / 6 f ) lo-*
(4)
or O.l/f pm for a 600-pm bubble. For this case, f can
The Journal of Phgsical C h e d s t r y
hardly be larger than l / 4 , under the assumptions regarding fluid motions above, so that 6* = 0.5 pm is a lower limit for the thickness of surface collected. The best upper limit which can presently be put on 6’ is to note from the momentum balance that it cannot exceed 0.3 6O, or some 20 pm for the 600-pm bubble. Concordance of surface concentration measurements from foam and from jet drops4 suggest that 6’ lies near 10 pm. Although incontrovertible chemical evidence for the bubble microtome is not presently available, the interpretation is supported by dye-distribution experiments as shown in Figure 3. In these, no dye is present in the bulk solution, as the bubble is blown separately (Figure 3A) and set onto the clear solution, where it takes up its equilibrium position below the surface with negligible diffusion of dye into the bulk solution (Figure 3B). The distribution of dye after bubble rupture is axial (Figure 3C) ; the original bubble surface has been swept into the center and ejected either up or down. The material closest to the surface goes up, and the jet drops captured from these bubbles were nearly as dark as the original dye. The coupling (by computer) of realistic free-surface boundary conditions to the Navier-Stokes equations is underway and should lead to a quantitative understanding of the flow pattern which delivers surface material to the jet drops and to reliable estimates of 6’ as a function of bubble diameter. The preliminary note seemed justified by the novelty of the mechanism and the possibilities which it opens for exploration of extremely low surface concentrations.18
Acknowledgment. This work was supported by grants from the ONR (NR 083-157, to MIT) and the ACS (PRF 995-G2). I thank Charles S. Cox for helpful discussions and Ellen Rice for a literacy check. (17) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” Oxford University Press, London, 1947, p 243. (18) NOTEADDEDI N PROOF.J. V. Iribarne and B. J. Mason, Trans. Faradag SOC.,63,2234 (1967), have examined the electrical charge separation during jet-drop formation and conclude that the data are satisfied if the ejected surface thickness is approximated as 6‘ & 0/600. This expression is based on a mass balance similar to eq 4 above and leads to an estimate of a 1-pm cut for a 600-pm bubble, in good agreement with the lower limit 6* computed above.
