Langmuir 1989, 5, 442-447
442 I
I
1
I
I
I
W
from the K to the W dpbased orbitals. The peak height change indicates that approximately half of the W sites were occupied with the K atoms, yielding the surface composition of Ko,5WS2,consistent with the results obtained from XPS. Further deposition of K results in formation of a K intercalation compound via atom diffusion into the bulk. The mechanism of this diffusion process is either directly through layers, as was established in the case of Cu on SnS2,11or via surface diffusion to edge sites or defects. The question of whether the K half fills each layer or totally fills alternate layers (staging) also is unresolved with our data.
Conclusion
K
1 1.0
J
1 0.9
I
I I 0.7 0.6 Reduced Ion Energy 0.8
1 0.5
I
0.1
Figure 6. ISS spectra of WS2 before and after K deposition (180 9).
reduced energy scale at 0.68. These results imply that the K sits directly above the W atoms and coordinates to three sulfur atoms. A coordination directly above the W dZz orbital would provide a mechanism for the charge transfer
We have investigated the semiconductor to metal transition induced by K intercalation into WS2 by UHV techniques. This transition is believed to occur via a charge-transfer reaction of an electron from K metal to the WS2 conduction band, which was identified by an increase in the charge density at the Fermi level and accompanied by a shift of the Fermi level into the conduction band. The spectral features may be explained by the rigid band model. Finally, the K appears to sit on the threefold site directly over the W atoms, suggesting that the potassium in the intercalation compound sits in tetrahedral sites in the van der Waals gaps with stoichiometry K,o.sWS2. The cleanliness and control of intercalate dosage offered by the UHV chamber, plus the multitude of spectroscopies available for physical characterization, provide a new method for studying the details of intercalation reactions which has not previously been widely exploited or appreciated. Registry No. K, 7440-09-7; WS2, 12138-09-9.
Some Limitations in the Interpretations of the Time Dependence of Surface Tension Measured by the Maximum Bubble Pressure Method Karol J. Mysels Departments of Chemistry and of Medicine, University of California, S a n Diego, La Jolla, California 92093-031 7 Received August 9, 1988. I n Final Form: December 5, 1988 The maximum bubble pressure method of measuring surface tension can give precise values for bubble intervals from less than 0.1 s to hours. As there is no doubt that longer bubble intervals correspond to greater ages of the bubble surface, this can give an exceptionallyclear survey of the development of adsorption at the air/solution interface over 5 or more decades of time. However, a more quantitative interpretation encounters complications, and some of these are pointed out. In particular, at short times, the so-called “dead time” during the explosive growth of the bubble prior to its detachment, can be an important early stage of the formation of the next bubble. The complex stretching and contracting of the surface during the detachment and the accompanying convection put the first fraction of a second of adsorption at a bubble surface beyond any simple analysis. However, in sufficiently dilute solutions adsorption is sufficiently slow to make these perturbations negligible. At longer times, on the other hand, unless special precautions are taken, slow convection currents in the solution accelerate adsorption.
Intioduction In the maximum bubble pressure method (MBPM)’ of measuring surface tension (ST), one ascertains the maximum pressure, P, inside a bubble growing at the end of (1) For background references, see ref 2.
0743-7463/89/2405-0442$01.50/0
a submerged fine capillary. This pressure is reached, in principle, when the bubble becomes hemispherical. Any increase of pressure beyond this value causes the bubble to become unstable, grow explosively, and finally detach (2) Mysels, K. J. Langmuir 1986, 2, 428.
6 1989 American Chemical Society
Time Dependence of Bubble Surface Tension
Langmuir, Vol. 5, No.2, 1989 443
Figure 1. As the hemispheriral bubble I A I at the end of the capillary becomes unstable, it prows (B.C) and then detaches itself (D. E)and escapas (F). The outer diameter of the capillary IS 0.5 mm. itself and escape to the surface of the liquid while a new bubble restarts the cycle a t the end of the capillary, as shown in Figure 1. As the radius of the hemispherical bubble is the same as the known radius, r, of the capillary, the surface tension a is obtained from Laplace's equation after correcting for the hydrostatic pressure, p , at the tip of the capillary:
P - p = 2a/r (1) The volume of the escaping bubble can also be used to estimate the surface tensions" by analogy to the drop volume method. In contrast to the MBPM, this 'bubble volume" method is seldom used. The MBPM has been used not only to determine the ST of pure liquids, which do not change with the age of the surface, but also for the study of solutions of surfactants where the ST varies with this age as adsorption proceeds. Such dynamic surface tensions obtained by the MBPM have been reported for times down to the millisecond range? In a recent papep I have described some improvements in the MBPM, in particular the use of a manostat, which permits extension of bubble interval (BI) to much longer times of the order of hours. This is done by setting the pressure in the bubble to a desired value and determining the time taken by the surface of the bubble to reach the corresponding ST (at which moment the bubble escapes). In this way one can obtain an overview of the changes of ST,and hence of the development of adsorption, over some 5 decades of time. In a sufficiently purified surfactant solution it is thus possibles to distinguish between the relatively rapid adsorption of the surfactant itself and the slower adsorption of more surface-active impurities. For sodium dodecyl sulfate (SDS), despite its slow hydrolysis, the solutions could be purified to the point where temporal changes in ST were due overwhelmingly to the adsorption of the surfactant itself and not to that of more surface-active impurities. This permitted the measurement of ST values for concentrations below the critical micellization concentration and the calculation of a partial adsorption isotherm for such purified systems! It also (3) K i p p e n h . C.; TFeler, D. AIChE J. I97O0.J6,314. (4)O h", T.;h k t . T.Rev. Sei. Instrum. 1981.52.590. (5) Fainer". V. B.;Lylyk, S.V. Colloid J . USSR, (Engl. Tmnsl.) 1982.44.538, (6) Mynels. K.J. Longmir 1988,2.421.
susm
INIERVLL. SEC
I
I.
Figure 2. Effectof purification on dynamic surface tension of 6.2 mM SDS at short times. The lowest line is for a solution of SDS recryntalized and extracted with hexane. The next one is for the eame solution passed through a hydrophobic C18 silica column. The third line is after repeating the column treatment. The top line is after an additional foaming step (see ref 6 for details). provided kinetic data for solutions of different degrees of purity, some of which are shown in Figure 2. The solutions represented in Figure 2 all have the Same concentration and are prepared from SDS purified by recrystalization and extraction, but three of them were further purified by adsorption. In addition, one was purified by foaming.s It will be easily seen that any interpretation in terms of adsorption rates of the surfactant must lead to very different conclusions for each curve if the presence of impurities is not taken into account. Miller and Lunkenheimer' have used model calculations to emphasize the misleading effect of impurities on the interpretation of kinetics of adsorption. Figure 2 is an experimental illustration of this point. Conversely, the data for the purest system of Figure 2 should, in principle, be interpretable in terms of the adsorption of the surfactant alone. An obvious difficulty is that adsorption occurs on an expanding surface of the initially flattish bubble gradually changing to a hemisphere as the surface tension decreases with increasing adsorption while the pressure remains substantially constant. This problem can a t least be clearly formulated and should be amenable to analysis. (7) Miller. R; Lunkenheimer, K. Colloid h l y m . Sei. lJsz,2so. 114.9.
444 Langmuir, Vol. 5, No.2, 1989
Figure 3. Adsorption proeeeds during the rapid growth of the bubble; as it detaches, part of its surface is left behind to close off the capillary. Molecules adsorbed on the original hemisphere are greatly diluted by those adsorbed during this proeess.
A more detailed consideration of the experimental conditions brought out, however, that there are other more serious kinds of difficulties. Hence the purpose of this paper is to point out that quantitative interpretation of measurement of the time dependence of ST by the MBPM is much more complex than assumed by the literature. As a result, kinetics much below about 1s cannot be analyzed at present, and long-term values are subject to different uncertainties. Dead Time In 1967 Austin et aL8 attempted to extend the MBPM to very short surface ages by using a stroboscopic method to determine the BIs. They considered a BI to be divided into two parts: one during which 'the unexpanded interface...is adsorbing wetting agent" and the other "during which the surface expands rapidly and the bubble forms and breaks away". They called the second part the "dead time" and subtracted it from the BI to obtain the age of the surface. Although no detailed argument was given as to why the "dead time" adsorption process should be so treated, this approach was adopted by later workers." Closer consideration indicates, however, that adsorption must proceed during the expansion of the bubble. A t the moment of separation, as in the separation of a drop,'O some of the surface of the bubble (and of its neck) must remain behind and form the surface of the new bubble, as shown schematically in Figure 3 and as may be discemed in Figure 1E. In fact, the remaining surface is likely to undergo an initial contraction, thus further increasing the adsorption density at the new surface. In order to form a stable bubble, this contracted surface must be less than hemispherical. In other words, the 'dead time", far from being on insignificant decay of a bubble, may be a n important formative period of the succeeding one. Some indication of how important this period is may be obtained from the following considerations. The "dead time" was measured with an oscilloscope in my apparatus? which features an inclined capillary and a taut diaphragm electronic manometer, both connected to a small manifold. Oscilloscope traces obtained for both water and a ca. 7 mM SDS solution each at a fast and slow (8) Austin, M.;Bright. B.B.;Simpn, E.A. J. Colloid Interface Sei. 1967. 23. 108. (9) Klaubek, J. J . Colloid Interface Sei. 1972,41. 1. 7. Klaubek, J. Tenside 1968,S. 317. Fainerman. V. B. Colloid J) . USSR . d(Enpl. " ? I979 . -.-,d..l ,79 .-. (IO) G u y , P.-A,. Perrot. F.-L. Arch. Sei. Phys. Not. 1903, IS. 133, l78bis. Adam, N. K. The Physics and Chemistry of Surfaces; Dover: New York. 1968: Figure 61. Hauser. E.A,; Edgerton. H.E.;Holt. B. M.; Cox. J. T..Jr. J . Phys. C h m . 1936,40,973. Adamson. A. W. Physicol Chemistry of Surfmes: Willey: New York. 1982: Figure 11-13.
Mysels
Figure 4. Oscilloscope traces showing the "dead time" in water (left)and an SDS solution (right) a t slow (2 s/hubhle, bottom) and rapid (0.1 s/huhble, top) hubbling. One horizontal division correspands to 20 ms and one vertical division to 0.1 dynfcm. For interpretation see Figure 5. hubbling rate are shown in Figure 4. They all give a value of 65 ms for the "dead time" within an uncertainty of about 5 ms. During the explosive growth of the bubble its Laplace pressure becomes rapidly negligible. The drop of pressure in the manifold is very small, of the order of 0.1%. as may be seen in Figure 4. Its magnitude depends mainly on the relative volumes of the escaping bubble and of the manifold and its connections. Hence the pressure drop under which the gas rushes through the capillary is essentially P - p. the maximum Laplace bubble pressure. For different solutions P - p is proportional to their ST as measured by the MBP a t a particular BI. Since the aerodynamic resistance of the capillary and the viscosity and inertia of the water are constant, a constancy of the "dead time" implies that the volumes of the escaping bubbles are proportional to their ST. This would also be expected according to the zeroth approximation of the bubble volume method of measuring surface tension. Thus we must conclude that the escaping bubble has approximately the same ST as that measured a t the maximum bubble pressure: This is as expected for water, whose ST is time-independent, but suggests that in the SDS solution most of the adsorption is completed during the growth stage, i.e., during the 'dead time". Clearly this is an approximation since the measurement of the dead time is not precise. The smaller size of the escaping bubble in SDS solutions is confirmed by the smaller reduction in manifold pressure, which can be seen in Figure 4. This smaller size could also be seen in photographs such as those of Figure 1. Thus, experimentally the importance of the "dead time" in adsorption kinetics is clearly apparent. Effective Age of New Bubble In another approach, let us define the effective age of a given surface in a given solution as the age of a model surface which has reached the same surface concentration in a reference system. As a reference we can choose a plane surface in a quiescent solution of same composition and an ideal, linear, adsorption isotherm. This is the model most amenable to calculation. We shall consider various factors determining this effective age for a newly formed bubble, although only some of these factors can be evaluated. In my measurements the radius of the capillary and hence of the hemispherical bubble was 70 p n . The volume of a bubble escaping in water was approximately 1.27 mm3 (as measured by direct collection). The volume of the escaping hubble in a 6.2 mM SDS solution is therefore
Time Dependence of Bubble Surface Tension about 0.81 mm3. This is over 1100 times the volume of the unstable hemisphere and corresponds to an 8.3-fold increase in radius and a 137-fold increase in area. Incidentally, I have not seen any “satellite” or “secondary” small bubbles. We may note that in this process the molecules adsorbed on the original hemisphere are spread out more than a 100-fold before a less than hemispherical area is detached to reclose the capillary. Hence, each bubbles does sample a practically fresh surface. For adsorption kinetics involving diffusion, the natural unit of length is a, the depth of the liquid containing as much sorbate as is adsorbed a t equilibrium on the underlying surface. Thus the convenient dimensionless radius of the bubble is R = r / a , and the dimensionless time is T = t D / a 2 ,where D is the diffusion coefficient of the sorbate.”J2 The data of ref 6 give for a 6.2 mM SDS solution a = 0.66 pm, and on the basis of the limiting c~nductivity,’~ D = 7.0 X 10” cm2/s, so T = 1650t. In these units the “dead time” T‘ = 107. Two opposing factors affect the adsorption on an expanding bubble. One is that the surface expands and thus dilutes the surface concentration. This turns out to be the the dominant one. The other is that the liquid surrounding the bubble is stretched and thinned, which brings the surfactant closer to the surface and increases concentration gradients, thus increasing the rate of adsorption. This situation is similar to that in polarography, where the growing mercury droplet is, in principle, a perfect sink. This is equivalent to the early stages of adsorption, far from saturation, when every molecule reaching the surface is adsorbed and stays adsorbed. The problem of diffusion to a spherical sink whose volume is proportional to time has been solved by Ilkovic14and by Mac Gilavry and Rideal15 under the assumption that concentration changes are confined to a layer whose thickness is small compared to the radius of the sink, so that the problem may in part be treated as involving a plane rather than a sphere. The problem involving an adsorbing surface rather than a sink, i.e., the approach to a dynamic equilibrium, has been recently solved by MilleP under the same assumptions. Because of the symmetry of the plane implied in the assumptions, his results apply to diffusion from either side of the growing surface, i.e., to the drop-weight method as well as to our situation. He found that adsorption is always slower than in the reference system by a factor which increases from ’I3 = 2.333 to about 4 as adsorption proceeds. The assumption of quasi-planarity seems to be well satisfied in our system as the difference of adsorption between the planar reference system and a sphere12having the radius of the capillary ( R = 105) is less than 1%even for the relatively long T involved. From Miller’s graph” adsorption for T = 107 (at the end of “dead time”) equals 93.5% of saturation assuming an ideal, linear adsorption isotherm. (It would be about 96.5 70 in the nonexpanding reference system.) Adsorption of SDS is not ideal. To take this into account, we can use as an approximation Hansen’s law,17that the subsurface concentration for a plane becomes inde(11)Miller, R. Colloid Polym. Sci. 1981, 259,375. (12) Frisch, H. L.; Mysels, K. J. J . Phys. Chem. 1983, 87, 398. (13) Mukerjee, P.;Mysels, K. J.; Dulin, C. I. J.Phys. Chem. 1958,62,
1390. (14) Ilkovic, D. J. Chim. Phys. 1938, 35, 129. (15) Mac Gillavry, D.; Rideal, E. K. Red. Trau. Chim. Pays-Bas. 1937, 56, 1013. (16) Miller, R. Colloid Polym. Sci. 1980,258, 179. (17) Hansen, R. S. J. Phys. Chem. 1960,64, 637.
Langmuir, Vol. 5, No. 2, 1989 445 pendent of the isotherm for large enough T , to estimate that adsorption is increased to 96.5% of equilibrium because of the nonlinearity of the adsorption isotherm? The corresponding effective time increases to about 300 because of the slowness of approach to equilibrium in the reference system. The analogy between bubbles and the better studied drops suggests that in addition to the phenomena which we have evaluated above there are processes which must be highly important and cannot a t present be analyzed more closely. One is the stretching of the neck of the bubbles as it detaches from the capillary.’O This is very rapid and may lead to some lowering of the surface concentration. After the extended neck breaks, the part attached to the capillary retracts even more rapidly,’O and this must lead to some increase in surface concentration. Finally, these processes must be accompanied by considerable convection in the surrounding liquid, which accelerates the approach to surface/solution equilibrium. Thus the above estimate of 96.5% complete adsorption should be a low estimate. It may be noted, however, that this 96.5% estimate for the end of the dead time corresponds to a ST of 47.2 dyn/cm.6 This is well within the range of Figure 2 and thus is not reached in reality until much later. Hence the above considerations, useful as they may be in pointing out the importance of the “dead time”, must neglect some important factor hindering adsorption such as perhaps an energy barrier to the transfer from subsurface to surface. Altogether, what happens during the “dead time” of the growth and detachment of the bubble is complicated and can correspond to significant adsorption and surface tension lowering leading to an ill-defined effective age of the surface of the remaining bubble. This makes it difficult if not impossible to interpret data such as those of Figure 2. It is only when the solution is dilute enough and the a parameter large enough for little adsorption to occur during this time that the still ill-defined effective age becomes small enough to be neglected. This is the case for highly surface-active substances, which lower the equilibrium ST markedly when present in micromolar concentrations but show no significant lowering in the first few seconds at these concentrations.ls
Maximum Bubble Pressure at Short Times The ST of pure water does not change with time. Hence one would expect the maximum bubble pressure to remain constant for bubble intervals decreasing right up to the “dead time” since the increase in pressure within the bubble, the destabilization of the bubble as it becomes hemispherical, and the pressure at this point should all be independent of what happens during the explosive growth and detachment that precedes and follows it. In other words, as the rate of flow of gas increases, the rate of growth of the bubble should increase and the pressure in the manifold should drop less, as shown schematically in Figure 5. But the hemispherical shape should be reached at the same pressure, and the detachment should then be initiated. Experimentally, however, there is a definite rise in the maximum pressure as the bubble interval approaches the dead time as shown in Figure 6. The following is a very tentative explanation. As the bubble grows explosively, it pushes the water away from the capillary. When it escapes, which is a very rapid process, the water rushes back and presses against the face ~~
~
(18)Stafford, R.; Dennis, E., submitted for publication.
446 Langmuir, Vol. 5, No. 2, 1989
IT - 0.1
Mysels
0,o
Figure 5. Schematic of expected pressure changes in the manifold
at increasing rates of air flow. The right-sided arrows indicate the effect of gas escaping with the old bubble. The left-sided arrow indicate the effect of the incoming gas flow. The horizontal dotted line shows the resultant of the two. Note that the pressure changes are of the order of 0.1% and thus do not affect the outflow or the growth and detachment of the bubble and thus the "dead time", as indicated by the vertical dotted line. Only when the inflow exceeds the output (top part) are the initial conditions not restored by the escape of a bubble.
--V
-0 -0 -A
--• 1 i S T I R R I N G R A T E , RPSI'"
Figure 7. Effect of stirring on the dynamic surface tension of
d
-- c
0.130 mM 1-ddecyl-sn-glycerolphosphatidylethanolamine (data of ref 21). For any given maximum bubble pressure and therefore given surface tension, extrapolation to zero stirring rate consistently indicates a longer time, and therefore less convection, than in the absence of stirring (indicated by solid symbols).
h
''
BUBBLE INTERVAL
SEC
"
Figure 6. ST of pure water appears to increase significantly as the bubble interval approaches the "dead time". In view of the argument of Figure 5 this seems to be due to phenomena at the
end of the capillary.
of the capillary and of the remaining bubble, thus increasing the gas pressure necessary for it to reach instability. This hydrodynamic pressure dissipates soon, and the maximum pressure than returns to its normal value. Figure 1F tends to support this explanation, especially when compared with Figure 1A. It shows the remaining bubble protruding less from the capillary than during its normal growth shown in the first photograph. The phenomenon was also regularly observed visually under stroboscopic illumination. I t is presumably due to the extra pressure produced by the onrushing water after the escape of the large bubble. The recession of the bubble can be seen also in surfactant solutions, but the pressure increase cannot be studied here because surface tension changes rapidly with time (see Figure 2). Hence, it is not clear how large a correction should be applied to the pressure measurement when the surface tension is lower, and this introduces an additional uncertainty into measurements at bubble intervals of less than about two "dead times". Longer Term Measurements Estimates of effective surface age a t longer times encounter another difficulty because of the presence of
ubiquitous slow convection currents, which can accelerate adsorption in an unsuspected way. Such convection has plagued the study of gravitational settling of suspensions until McDowell and Usherlg showed that by sufficient precautions such currents could be effectively suppressed in small vessels. Sutherlandmhas already pointed out the importance of this factor in adsorption kinetics. Evidence that convection plays a significant role in experiments such as mine, which utilize a M-100-cm3 volume buried in a thermostated jacket,2 comes from stirring experiments reported elsewhere.21 Figure 7 shows the time required for the surface of a bubble to reach a certain surface tension when the solution is stirred a t different rates. The data are plotted against the square root of the rotation rate of the magnetic stirrer used, which seems to give a straight line relation for the three or four slowest stirring rates used (1or 0.5-0.1 rps) and thus permits an extrapolation to zero stirring rate. Times required to reach the same ST without any stirring are also shown. It is striking that in all cases except one the linear extrapolation to zero stirring gives a significantly longer time than found in the absence of stirring. This indicates very strongly that the unstirred system is not truly quiescent but subject to irregular convection currents equivalent to very slow (of the order of a few revolutions per minute) stirring. Thus, unless special precautions are taken, considerable caution should be exercised in applying model calculations based on diffusion in quiescent solution to experimental data obtained by the MBPM or, for that matter, any (19) McDowell, C.M.; Usher, F. L. h o c . R.SOC.London. 1932, A138, 133. (20) Sutherland, K.L. A u t . J. Sci. Res., Ser. A 1952,5,683. (21) Mysels, K.J.; Stafford, R. Colloids Surf. 1989, in press.
Langmuir 1989, 5, 447-451 method involving a significant volume of solution.
Experimental Section The MBPM apparatus and the cell have been described previously.2 The stroboscope used was a simple Pasco Scientific (Hayward,CA) Model SF9211. Photographs were taken without synchronization by using a 35-mm reflex camera and a reversed 55-mm lens on a 37-cm extension tube. For photographing, the cell was a spectrophotometer cuvette having two round windows separated by 1cm. It was placed close to an optical window sealed into the protecting bell jar.z Despite this protection the bubbling was not sufficiently regular to permit a complete stopping of the motion. The photographs of Figure 1represent a selection from about 100 slides taken over several days at various bubbling rates. Conclusion The conventional maximum bubble pressure method of measuring surface tension provides accurate surface tension data and can show the evolution of bubbles as the
447
bubble ages over a wide span of time. The processes involved in that aging are, however, complicated, and a quantitative interpretation in terms of an equivalent age of the surface, or the like, is complicated by convection for both short and long times. At very short times (
