PHOSPHATE IONENRICHMENT IN DROPSFROM BREAKING BUBBLES
2163
Phosphate Ion Enrichment in Drops from Breaking Bubbles by Ferren MacIntyrela and John W. Winchesterlb Dejmrtments of Chemistry and of Geology and Geophysics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received August 6 , 1988)
Bubble-produced aerocolloids from a 22Na-a2P04 solution show a phosphate enrichment (POd*-/Na+)aerocolloid E = (P04a-/Na +) solution - 1 which is positive for all drop sizes from 0.25 to 25 pm and may reach 600 for 8-pm drops. Drops arising from the central jets of small bubbles show enrichments up to 10 which are indepen:ent of drop size. Drops arising from the film caps of Iarger bubbles show a superimposed peak enrichment E which is log-normally centered on a drop diameter of 8 fim. The mechanism postulated for enrichment is selective adsorption of PO4+ as a counterion to surface-active organic molecules (present intentionally or as contaminants in distilled water) followed by ejection into the atmosphere of the enriched surface layer by the surface-microtome behavior of a breaking bubble. Introduction Many workers have observed that certain multiplycharged counterions are enriched a t surfaces by selective attraction to charged surface-active organic molecules. Thus, calcium and magnesium are adsorbed with 50to 200-fold preference over sodium.2sa Van Voorst Vader has shown that this preference is entirely an electrostatic eff Judson, et al., studied adsorption of sulfates by surfactant^.^ Perrin seems to have been the first to separate ions (Hi- from Na+) by foam fractionation.6 Sebba has shown that selective adsorption onto rising bubbles (“ion flotation”) can be used to beneficiate mine liquors by concentrating valuable ions.’ Baylor, et al., have shown that phosphate is concentrated by marine bubbles,* and their findings have been confirmed by MacIntyre.9 Finally, the selective adsorption of calcium into hard-water scum (the insoluble soap calcium stearate) is familiar to all. Anomalies in the ion ratios of precipitation led Kohler and B%th,’O Sugawara,l’ Komabayasi,12 and Bloch, et al.,la to examine ion fractionation between solution and bubble-produced spray. E r i k ~ s o n ’sug~ gested that “Some fractionation process seems to take place at the sea surface, possibly involving organic matter derived from surface films.” The work described here shows that the mechanism of the Eriksson process is selective counterion adsorption combined with the ability of small bubbles to eject surface material into the atmosphere as aerocolloid particles. Experimental Section The ion ratio studied was P04*-/Na+. I n simplest terms, an experimental run consisted of adding 100 pCi of P32as sodium phosphate and 100 pCi of NaZ2as NaCl to 90 ml of solution, creating aerocolloid particles by blowing bubbles in t,he solution, catching the resulting droplets, and classifying them by size in a seven-stage
aerosol impactor,16 and determining the P/Na ratio of the several size-classes by simultaneous P and y oounting. Bubbles smaller than about 1 mm have insufficient cap area to make film drops, but their central jets have enough energy to eject one to four jet drops. Bubbles larger than about 3 mm (equivalent spherical diameter) produce a number of film drops from their thin caps, but their jet drops are large and do not remain airborne. Intermediate bubble sizes produce mixtures of both jet and film drops. The bubbles used in this work were blown less than 1 mm in diameter, at rates of 3-100/sec. Size at rupture, however, is dependent on trace sur(1) (a) Department of Chemistry; to whom inquiries should be addressed a t Scripps Institution of Oceanography, University of California, San Diego, L a Jolla, Calif. 92037. (b) Department of Geology and Geophysics. (2) C. Walling, E. E. Ruff, and J. L. Thornton, Jr., J. Phgs. Chem., 61, 486 (1957). (3) K. Shinoda and K. Ito, ibid., 65, 1499 (1961). (4) F. Van Voorst Vader, Third International Congress on Surface Activity, Vol. 11, 1960, p 276. (5) C. M. Judson, A. A. Lerew, J. K. Dixon, and D. J. Salley, J. Phys. Chem., 57, 916 (1953). (6) J. Perrin, Ann. Phys. (Paris), 10, 160 (1918). (7) F. Sebba, “Ion Flotation,” Elsevier Publishing Co., Amsterdam, 1960. (8) E. H. Baylor, W. H. Sutcliffe, and D. S. Hirschfeld, DeepSea Res., 9, 120 (1962). (9) F. MacIntyre, Thesis, Massachusetts Institute of Technology, 1965. (10) H. Kohler and M. B&h, Nova Acta Regiae Societatis Scientiarum Upsaliensis, Ser. I V , 15, 4 (1952). (11) K. Sugawara, International Oceanographical Congress, Preprints, 1959, p 875. (12) M.Komabayasi, J. Met. Soc. Jap., 40, 25 (1962). (13) iM.R. Bloch, D. Kaplan, V. Kertes, and J. Schnerb, Nature, 209, 802 (1966). (14) E. Eriksson, Tellus, 11, 375 (1959). (15) R. I. Mitchell and J. M. Pilcher, Ind. Eng. Chem., 51, 1039 (1959).
Volume 73, Number 7 July 1869
FERREN MACINTYRE AND JOHN W. WINCHESTER
2164 Table I : Radiochemical Data for Nuclides Used Nuclide
Na2a Pa2
Half-
Energy,
life
MeV
2.58years 1 4 . 3 days
0.54Pc 1.28~ 1.71p No Y
Production method
aotivity
Purity, 76
Na2a(n,2n)Naes
1 mCi/mg
98f
Saz(n,p)PZt
Carrier free
99
AIR INTAKE
Figure 1. Experimental apparatus; Bee text for details.
factants, as these greatly modify breaking behavior. The classification into jet-drop, film-drop, and mixeddrop runs noted in column 2 of Table I1 is based on visual observation of how bubbles were breaking during a particular run. Film-drop runs resulted from the coalescence of small bubbles into large (sometimes 2 cm) bubbles before final rupture. The apparatus is shown schematically in Figure 1. The entire working area is enclosed in a modified “Micro-Void” dust hood (1) as a first-stage protection against contamination. A blower (9) takes 2 m2/min of air from the room and forces it through a 2.5-cm bed of activated charcoal (lo), a Cambridge “Absolute” filter (11) which retains particles larger than 0.3 pm, and out through the access opening (12) at the front. The working volume (2) encloses the test solution (8). Electrical gear (4) is required for generating gas bubbles ( 5 ) and powering heated jet-drop collectors (6). The impactor (7) is supported close to the test solution. Air-flow through the impactor is monitored by a vacuum gauge (13) and flowmeter (14). The enclosure (2) also provides a nodal point for the bubble source, which is an oscillating horizontal capillary (15)l6bent into a J to dip into the solution. A loudspeaker driver (17) which The Journal of Physical Chemistry
Specific
+
Chemical form
NaCl
NasPO,
vibrates the capillary and a magnetic stirrer (16) complete the apparatus. The physical form of the sample obtained from the impactor is a small deposit (perhaps invisible) of dry salt adhering centrally to a 2.5-cm disk of 0.025-cm thick cellulose acetate. After removal from the impactor the disks are immediately sprayed with “Krylon” lacquer to saturate the salt deposit and bind it firmly to the disk with an 0.2-0.3 mg/cm2 covering. The tracers used are described in Table I. Although P32is a pure /3 emitter, Na22produces both y rays and positrons. Furthermore, in practice the y-scintillation counter interprets bremstrahlung of the high-energy P32 p as y rays, so that in effect both nuclides emit mixed radiation. y rays are counted by a 5-cm well-type NaI/Tl scintillator. Bremstrahlung is reduced by a 1.5-cm thick aluminum sample holder. The entire y-ray spectrum from 0.03 to 0.66 MeV gives better counting statistics than the photopeak itself (0.40-0.66 MeV), and hence a lower error estimate. Simultaneous ,6 counting uses a standard end-window proportional counter shielded by 100 mg/cm2 of aluminum to absorb the NaZ2positrons. The raw data for a run comprise two sets @ and y) each consisting of an even number (2 to 6) of replicate 5-min counts of the seven impactor samples, two standards, a duplicate pair (of known volume) from the bulk solution, a blank, plus separate and combined counts of a “split standard.” Counting requires 6 to 8 hr. The raw data are processed by a computer program which returns an error estimate of its results if provided with an error estimate (counting statistics) bf the input.’’ Background is subtracted to give corrected Pand y-counting rates CBand C,. These form a pair of simultaneous equations Nap Na,
+ Pp = C, + P, = C,
The standards provide the further relationships Na, = a”ap
P, = apPp (16) F. MacIntyre, Rev. Sci. Instrum., 38, 969 (1967). (17) F. Madntyre, M I T Computer Center Memo CC-249, 1966.
PIKISPHATE IONENRICHMENT IN DROPS FROM BREAKING BUBBLES where a is the absorption coefficient for the aluminum absorbers. Combining and solving in terms of /3 counts per minute, we have
I n practice, uN = 30, ap = 0.1, so that as expected the determination of Na22 depends chiefly on annihilation radiation (0.51-MeV y ray) from its positrons, while P32 is seen by its 1.71-MeV b’ rays. Counting rates of the two isotopes are separately corrected to zero time. (Some runs required 48 hr to collect-an appreciable fraction of the 14.3-day half-life of P32J Absorption coefficients are computed for each run from the standards; a check of the over-all accuracy is obtained by counting separate Na and P halves of a “split standard,” then counting the combination and subjecting it to the same data processing as the unknown samples. This check shows that the Na determinat)ion is about 1% low while P is 0.5% high; this variation is almost entirely removed by subsequent normalization against the bulk solution. Another check upon the resolution of the method treats the P half of the split standard as though it were a mixed sample. The data reduction program then reports less than 1 count of Na/min to 12,000 counts of P/min, giving credence to the occasional highly enriched drop sample which shows 1 count of Na/min to 1000 counts of P/min. The error bars on the graphs represent the effects of all counting uncertainties, carried through the computations and drawn as f1 standard deviation. The water was distilled in a Barnstead tin-lined still and polished by a mixed-bed demineralizer. The ionexchange resin presumably leaked trace amounts of surface-active substances, l8 and this contamination is probably responsible for the interesting ability of the distilled water surface to attract phosphate ions. Seawater was obtained from the dock a t Woods Hole Oceanographic Institution and is typical of inshore contaminated water in that it contains trace quantities of nearly any pollutant that man or nature is capable of creating. Three commercial surfactants were used for intentional surface-contamination experiments. Sodium lauryl sulfate has an organic anion, cetyltrimethylammonium bromide an organic cation. Both were used “off-the-shelf” with no further purification. The nonionic surfactant, “Maypon” (Stepan Chemical Co.), is an ill-characterized material made by treating “cocoacid” chloride with hydrolyzed collagen. The peptide moiety appears to be three to five amino acids long. It was used because collagen can bind phosphate in vitro in a number of ways,I9 and Maypon was thought to
2165
approximate in some manner the surface-active nitrogencontaining material of seawater, The gas source is an electrolysis cell containing 0.5 N KzS04. Solution and gas come in contact only with glass, platinum, and Teflon and are presumed free of organic contaminants. The Na count of a given sample is taken as a measure of the total volume of the drops collected. This requires comment on two points. First is the uncertainty inherent in drop spectrometry as to what is measured by “drop-size.” Clearly it is size at time of capture and not size at formation. Change of radius between formation and capture is to be expected because the vapor pressure of small drops is a function of both radius and composition. Fortunately these two effects work in opposite directions, and changes can be kept consistent and small by running the impactor a t constant relative humidity near saturation-conditions which were met in this work. Second, the utility of the Na count as a measure of original drop volume depends on a lack of fractionation between Na+ and water. Such fractionation might arise by negative adsorption of Na+ at the air-water interface. Strictly speaking, the measurements reported here depend on differential adsorption of Na+ and P O P by surfactants; nevertheless, we pause to show that purely inorganic rejection of Na+ a t the interface will cause no detectable fractionation. Ions are rejected from the interface by electrostatic image forces arising from the change in dielectric constant a t the interface. These forces vanish over a few molecular diameters20 so that negative adsorption surface layer. can at most remove material from a 10-8 Consider a 1-pm cube a t the surface of a solution which is, say, 0.17 X 10+ M in a certain ion containing 1000 ions/pma. Complet? negative adsorption will reject all ions from the 10-A skin, but this is only one ion. If it be removed completely from the 1-pm cube, the concentration has changed by 1 part in 1000, Conversely, positive adsorption is not limited t o removing what is there a t low concentration but will proceed until the surface is covered by a monolayer, or the bulk solution depleted, or some lesser demand of equilibrium is met. Positive adsorption from a 0.17 X M solution can concentrate l O I 4 molecules cm-2 at the surface, adding 106/pm2 for a 1000-fold enrichment over the original composition averaged over the 1-pm cube at the surface. Because the effects of positive adsorption can easily be 106 times as great as the effect of negative adsorption, no significant errors will be introduced in this discussion by considering that Wa+ behaves like water at the inter(18) J. H.Schenkel and J. A. Kitchener, Nature, 182, 131 (1968). (19) M.J. Glimoher and 8. M. Krane, Biochemistry, 3 , 195 (1964). (20) W.0.Harkins and E. C. Gilbert, J . Amer. Chem. Soc., 48, 604 (1926). Volume 78, Number 7 July 1969
2166
FERREN MACINTYRE AND JOHN W. WINCHESTER
face and that the Na+/water ratio is constant until the drop is airborne and evaporation begins.
I O‘OE -
Nomenclature
io9 E S - S e a water
I
I
I
I
I
I
I
-
D Distilled water N - N o n - i o n i c surfactant
-1 E 10‘6 E MONOLAYER
Quantities pertaining to the bulk solution
N o = NaZ2/pl.(counts/min) Po = PS2/pl. (counts/min)
IO8
rs
Quantities pertaining to the material collected on a given stage of the aerosol impactor
N
= NaZ2(counts/min)
P
=
F E V D,
= = = =
D,
=
?z
=
A
=
P32(counts/min) PNo/PoN = fractionation ratio F - 1 = phosphate enrichment N / N o = volume of particles (pl.) volume-average diameter of drops (pm) (see Appendix for derivation) surface-average diameter of drops (pm) (see Appendix for derivation) 10gV/(7rD,3/6) = number of particles of diameter D , 1 0 - * n ~ D ,= ~ total surface area of drops of diameter D, (cm2)
We now define the “excess phosphate” P, as the total amount collected on a given stage minus the expected amount, which is the product of the bulk concentration and the volume. Thus
P,
=
P
=
=
P J A (counts/(min cm2))
in which we attribute all of the phosphate discrepancy to the surface of area A . Although the name is formidable, this is the usual thermodynamic definition of the surface excess.21122 We will find it convenient to consider a different sort of excess called the “drop-volume excess phosphate,” I?,, defined by
r,
=
P,/V = ( P - P O V ) / V = Po(P - 1) = PoE (counts/(minpl))
(1)
where the enrichment E , which may be thought of as a normalized excess distributed throughout the volume, is the added phosphate per phosphate expected. That is, E = 3 means that there are four phosphate counts in the volume where one was expected. All of these quantities are defined in terms of counts per minute. Conversion into actual concentration is uncertain because of the high specific radioactivity and small volume of the tracers. An estimate may be made from the decay constant X of P32(= 3.366 X lom6min-l), the efficiency B of the counter ( 5 10%))and the assumption of carrier-free Pa2. Then the drop-surface excess The Journal of Physical Chemistry
10’
IO6
$10‘0
,375 .75 1.5
3
6
12
24
MEAN DROP DIAMETER D v ,pm
Figure 2. Drop-surface excess phosphate: experimental numbers in counts per minute per square centimeter a t left; estimated surface excess in ions per square centimeter a t right. The dotted line is the slope expected if the excess is proportional to the drop volume, not to the drop surface.
phosphate, I’,,, in terms of surface concentration is r,, = F8/A = 3 x lo6 re(ions/cmz).
Results and Discussion
PoV (counts/min)
Further, we define a “drop-surface excess phosphate” Fa as Pa
cpm/cm‘
Perhaps the most logical hypothesis to test is that the observed phosphate enrichment in drops arises from a surface excess on the drops themselves. This excess should be constant for a given run and independent of drop diameter. That this hypothesis is false can be seen from Figure 2 which plots re against the drop diameter. The slopes are not horizontal as required by the hypothesis. However, the average slope of the several runs closely parallels the dotted line Fa = (constant)D,, and since FS = (I’,/6)DV,we conclude that it is not rewhich is constant, but I’,. That is, the enrichment does not come from a surface excess on the drops themselves but from an excess which is distributed throughout the drop volume. I n Figure 3, E has been split into two components k , where E is the for two of the runs, so that E = E “peak enrichment” and will be seen in later runs to be a log-normal peak centered near 8 pm and which i8 present only when the breaking bubbles have large film caps with an extremely high surface-to-volume ratio. This division of E emphasizes the linearity of E. Several features of the data from Figures 3 and 4 and
+
(21) J. W. Gibbs, “Collected Works,” Yale University, New Haven, Conn., 1964,p 229. (22) G. N. Lewis, M. Randall, K. S. Pitrer, and L. Brewer, “Thermodynamics,” McGraw-Hill Book Co., Inc., New York, N. Y., 1961, Chapter 29.
PHOSPHATE IONENRICHMENT IN DROPS FROM BREAKING BUBBLES
2167
Table I1 : Summary of Data
Run number
Drop typea
10 11 12 13 14 15 16 17
F M F
18 19 21 28 30 32
Surfactantb added (at 10-8 M )
B, linear enrichment
...
9.0 6.5 2.1 0.1
..,
J J J
... ... ... ...
M F F M
SLS SLS SLS .*.
J J J F
...
0.75 0.29 0.2
... 1.9 12.0 0.38 0.95 1.3
CTAB Maypon Maypon *.,
2,peak enrichment ocourring at drop diameter, pm 630 a t 7 8 . 6 at 9 20 at 7 None None None 0.64 at 7 1.45 a t 8 None 0 . 6 at 7 None None None 0.78 a t 3
Fr: fractionation for entire runc
Total volume collected, Yl.
56.0 12.6 4.29
2.5 111 20 75 60 170 95 31 24 108 728 1200
9
l.lld
0. 93d
1.79 1.36 1.26 0 94d 2.89 13.1 1.34 2.05 2.62 I
24
16
SLS = sodium lauryl sulfate (negatively charged surfactant), a F = film drops, J = jet drops, M = mixed film and jet drops. CTAB = cetyltrimethylammonium bromide (positively charged surfactant), Maypon = Stepan Chemical's nonionic surfactant (the reaction product of coco acid chloride with hydrolyzed collagen). ' Differences between E and F z - 1 when & is absent arise from different methods of determination. See text for details.
E
IP I
L
I
-i-€-r-+- f-i-
19
0. I
.375 ,751 1.5
3 6 12 24 MEAN DROP DIAMETER D v , p m
Figure 3. Representative jebdrop runs. This is the data of Figure 2 plotted as enrichment E, which is equivalent to the drop-volume excess phosphate rv. Note that a positive surfactant enhances PO2- enrichment, while a negative surfactant depresses it. The parabolas associated with runs 11 and 16 are evidence for an admixture of film drops to the jet drops under discussion. The original data can be reconstructed by adding the parabolas to the horizontal lines.
i -1
Table I1 merit comment. Runs 10-12 were a time sequence using the same 90 ml of solution. The decrease in E with time suggests that some highly surfaceactive component present at low concentration is gradually being depleted. This, however, represents fairly rapid removal and therefore a very low concentration of surfactant, since run 28, from M Maypon lost 1200 p1. or 1.3% of the bulk solution volume with only slight decrease in E. Thus in the presence of adequate amounts of surfactant the enrichment is an equilibrium process which regenerates a surface excess as fast as it is removed. The pair of runs 13 and 14 (not shown graphically) were a continuation in the same solution as 10-12, with the addition of low4M Na2HP04to test the effect of nontrace amounts of P043--. Unexpectedly, the acid phosphate attacked the stainless steel bubble orifice, yielding a flocculent green precipitate of large surface area. The precipitate had no effect on the Pazactivity of the solution, but it completely suppressed enrichment (E for the two runs averaged only 0.02). This observation is consistent with the removal of all of the surface-active organic material from the bulk solution by adsorption onto the large surface of the floc. These were the only runs to yield zero enrichment. The effect of charged surfactants was tested in runs 16-18 with the addition of sodium lauryl sulfate, whose surface-active moiety is negative, and in runs 21 and 22 with positively charged cetyltrimethylammonium bromide. Figure 3 clearly shows the depression of enrichment caused by a surface charge of the same sign as the Pod3- ion and increase of enrichment when the surfactant has the opposite charge. Volume 78, Number 7 July 1969
2168
FERREN MACINTYRE AND JOHNW. WINCHESTER ness t (om) of the surface skimmed from the inside of the bubble. If we write P, and rVin terms of the surface area s (emz) which goes into a single drop, the surface excess on the bulk solution (ions/cm2) and the number of drops n, we have
P, = nsehri
v = 103nst rv = P,/V
= eAri/lO’t
(2)
and since ri =
,375 ,75
1,5
3
6
12
24
is an observable quantity (y being the surface tension, a the activity, R the gas constant, and T the Kelvin temperature), we have, in principle, a direct measure of Y. However, ri could not be measured for any of the solutions studied because the surface concentrations were so low that the surface tension was indistinguishable from “pure” water. We may estimate Pi in the following way. Taking average values of Po (= 100, corresponding t o a M bulk solution) and E (= 3), we have from eq 1 and 2
M E A N DROP DIAMETER D v , ,urn Figure 4. Representative film-drop runs. The parabolas centered a t 8 pm are, in fact, log-normal curves and represent a gaussian distribution of the enrichment .& on a logarithmic drop-size scale. The peak of h’ appears to arise from the high surface-to-volume ratio of the film caps of large bubbles, which are only 1 to 2 N r n thick a t rupture and which produce 8-pm drops &g the initial product of film break-up. The D? mark indicates the presumptive presence of some unknown contaminant in distilled water.
( b y / b In a)/RT
rv = POE = 300 Estimating t as 10 pm, we find
ri = iowV/EA=
108
x
10-8
x
3oo/
lo-’ X 3 X 10-6 = lo8 ione/cm2 far below monolayer coverage (which is 1014 ions/cm2 and reached by the usual surfactant a t bulk concentrations of M ) . This is concordant with the inability to measure surface tension changes. The mechanism which concentrates surface material into the jet drops is not fully understood. It is being investigated at SI0 by numerical hydrodynamics and high-speed photographs. Photographs of bubbles containing a dye on and adjacent t o their interior surface, with no color in the bulk solution, show that the tips of both the upward jet and the momentum-balancing downward jet into the bulk liquid are brightly colored by the dye. Bubbles breaking through the mobile film of an oleic acid monolayer eject jet drops carrying ten or more layers of oleic acid, indicating strong surface concentrations onto the ascending jet. Blanchard has observed a similar multilayer removal.2s
A comment is in order on run 18, in which the expected high h’from film-drops did not occur. As mentioned earlier, the gas in the bubbles was oxyhydrogen and explosive. I n run 18 the large bubbles were exploded immediately upon reaching the surface by a spark from a Tesla coil. The film caps had not had time to thin by draining, and it is presumed that the high volume-to-surface ratio of the thick film caps is responsible for the lack of enrichment. The data strongly support the hypothesis of surface enrichment but do not allow this enrichment to occur upon the surface of the ejected drops. The alternative surface is the interior of the bubbles, which must be in Conclusions thermodynamic equilibrium with the solution (or nearly so, if sorption kinetics are slow). M a ~ I n t y r e ~ ~ i ~Surface-active ~ substances and their associated counhas suggested that small bubbles behave as surface terions are selectively ejected as jet drops by bubbles breaking a t an enriched interface. The resulting enmicrotomes, ejecting a thin layer from their interior surface into the atmosphere as jet drops. This process richment of the aerocolloid phase in such components is satisfies the requirements that rv,not rs,be the meaningful measure of surface adsorption, as the entire drop (23) F. MacIntyre, J. Phys. Chem., 72, 689 (1968). is made of enriched material. (24) F. MacIntyre, M. Tegner, and K. Isaaos, in preparation. rVcould in principle provide a measure of the thick- (26) D. C. Blanchard, Science, 146, 396 (1967). The Journal of Physical Chemistry
PHOSPHATE IONENRICHMENT IN DROPS FROM BREAKINU BUBBLES independent of drop size. For the case of P04-a (carried by undetermined surfactants) the enrichment is between one- and tenfold. The surface at which the enrichment originally takes place is the interior of the bubble cavity. During jet-drop formation, it appears that this surface flows down the cavity wall in a boundary-layer, surface-free-energy-driven flow and is preferentially incorporated into the jet drops. The thickness of the incorporated surface appears to be less than 10 pm.23,26 Film drops-from rupture of the bubble cap-show a much higher enrichment (up to 600-fold), because of the higher surface-to-volume ratio of the film caps, which are characteristically only 1 to 2 pm thick, The strong drop-sise dependence of the film-drop enrichment E remains a mystery. A tentative suggestion is that the film cap ruptures at a point particularly rich in surfactant and phosphate (perhaps simply a point depleted in water). The first few drops formed from a 1.5-pm film cap are likely to be 8 pm in diameter,gwith, evidently, a log-normal distribution about this mean (Figure 4). Acknowledgments. We thank Dayton Carritt and George Scatchard for their interest in and guidance of this work, Robert Shrock and the ONR for their support of it (Nonr 083-157), and Edward Baylor and Duncan Blanchard for illuminating discussions. Particular thanks go to Carl Garland, who pointed out the obvious -whereafter all obscurities became clear. The MIT Computation Center provided the necessary computer time.
2169
D, are obtained from the volume (Na count) curve. The assumptions made are (1) each impactor stage collects diameters over a range from a to 2a and (2) within this range, a linear approximation is an adequate representation of the volume distribution curve. and 3, the average volume and diameter of a particle on a given stage, are then given by
V
= (n/6)DV3= (n/6)~2uwD3dD/W
S = nD.2 = nlzUwD2dD/W where w is the weighting function and W its integral. From the second assumption W(X)
= V"
+ v'x
where V" = V(a) - vu and v' = [V(2a) - V ( a ) ] / a . Hence, the denominator is just the simple average, and
aW
=
[2aW(z)da
=
CV(2u)
+ V(a)]/2
which leads to
a3[26V(a)
+ 49V(2a)]/20W
a2[llV(a)
+ 17V(2a)]/12W
and
Appendix Particle Diameters from N u Counts. The volumeaverage diameter D, and the surface-average diameter
(26) J. V. Iribarne and B. J. Mason, Trans. Faraday SOC.,63, 2234 (1967).
Volume 79, Number 7 July l06B
