2124
Langmuir 1992,8, 2124-2129
Kinetics of Recovery of Hexadecyltrimethylammonium Bromide by Flotation John D. Morgan,**tDonald H. Napper,t Gregory G. Warr,? and Stuart K. Nicolt Department of Physical and Theoretical Chemistry, University of Sydney, New South Wales 2006, Australia, and BHP Research, Newcastle Laboratories, P.O.Box 188, Walkend, New South Wales 2287,Australia Received December 11,1991.In Final Form: April 6,1992 The adsorptionof hexadecyltrimethylammonium bromide at an airlsolutioninterfacehas been reported to be controlled both by limitationson the rate of transport of the surfactantto the interfaceand by energy barriers in the adsorption process. It is desirable to understand whether either of these processes is important in the context of ion flotation in order to predict recovery rates and to characterize the adsorptive surface. A model of adsorption to a rising bubble is proposed, based on diffusion controlled adsorption in the presence of a diffusional boundary layer; this predicta that adsorption equilibrium is likely to be achieved in the order of 0.1 s. Recovery rates were measured experimentally for a range of temperaturesand confiimthat equilibriumis indeed rapidly achieved. Despitethe turbulent environment, the recovery kinetics are well described by adopting the Gibbs equilibrium adsorption isotherm.
Introduction Ion flotation is an adsorptive bubble separation technique first proposed by Sebba’ for the recovery of a dissolved ionic species (the colligend) from a solution of a surfactant. It consistsof passing air through the solution and collectingthe foam that forms. If the surfactant bears a charge opposite to that of the colligend, the foam may be enriched with the colligend. The upper limit to the rate of recovery of the colligend is determined by the rate of flotation of the surfactant, since recovery cannot be faster than that determined by the stoichiometry of the association of surfactant with colligend. Thus it becomes important to study the recovery of surfactant by flotation. Further motivation for studying the recovery is the desirability of characterizingthe bubble interface in terms of, for instance, the surface charge density, or ita relationship to the more familiar equilibrium interfacial structure. The importance of this arises from the need to be able to understand the selectivity of the process for different colligends, for which knowledge of the nature of the selective interface is a prerequisite. It would also be useful to know to what extent existing equilibrium interfacial results apply to the dynamic system. The problem then is that of determining the surface excess of surfactant on a bubble entering the foam. The measurement of surface excess is straightforward in the case of a quiescent interface at equilibrium, and the theoretical underpinnings are well established in the form of the Gibbs equation and a variety of adsorption isotherms. The interface of a rising bubble, however, is not quiescent. Rather, it exists in quite a turbulent environment, so it cannot be simply assumed that the surface excess is necessarily determined by the usual equilibrium relationships. Furthermore, the time of bubble residence within the flotation column may not be sufficiently long for an adsorption equilibrium to become established. In such a case, the final adsorption density may be controlled by transport of surfactant to the bubble surface and/or by the presence of energetic barriers to adsorption. The mathematical description of diffusion controlled adsorption from solution to a surface of limited adsorpt University of Sydney. t BHP Research. (1) Sebba, F. Zon Flotation; American Elsevier: New York, 1962.
tive capacity was first given by Ward and TordaiV2This wadater extended by Baret3to include activation barriers to adsorption. The existenceof such barriers was rejected in a recent review: and several workers have reported barrierlessdiffusion controlled adsorption at the air/water interface for solutions of sodium myristate: various nonionic detergents!*’ and dodecanol.s Others, however, claim to have measured an entropic activation barrier to the adsorption of hexadecyltrimethylammoniumbromide (CTAB),SJOalthough this finding is probably due to the presence of highly surface active impurities. These systems were all studied by means of surface tension relaxation experiments in nonturbulent systems, in contrast to the conditionsencountered in flotation. The reported times required to achieve an adsorption equilibriumvaried dramaticallyin these studies, from less than 1s to several minutes for solutions of about lo4 M surfactant. This cautions against assuming equilibrium coverage after a phase contact time of a few seconds. On the other hand, if it is the case that a steady-state adsorption is achieved on a bubble surface, it may be significantlyperturbed from the equilibriumsurface excess expected from the normal Gibbs equilibrium isotherm. The steady state arises from surfactant encountered at the leading edge of the bubble being adsorbed to the interface and swept to the trailing edge of the bubble surface whereupon it desorbs. Theories of the extent of adsorption of surfactant on bubbles have been developed by Dukhin et al.l1-I4 for a wide range of Peclet numbers and for low to intermediate Reynolds numbers using the Henry adsorption isotherm. The high Reynolds and Pe(2) Ward, A. F.; Tordai, L. J. Chem. Phys. 1946,14,453. (3) Baret, J.-F. J. Chim. Phys. 1968, 65, 895. (4) Kretzschmar, G.; Miller, R. Adu. Colloid Interface Sci. 1991,65, 36. (5) Van den Bogaert, R.; J m , P. J. Phys. Chem. 1979,83,2244. (6) J m , P.; Rherts, E. J. Colloid Interface Sci. 1981, 79,96. (7) Miller, R.; Lunkenheimer, K. Colloid Polym. Sci. 1986,264, 357. (8)Lm,5.-Y.;McKeigue, K.; Maldarelli, C. Langmuir 1991,7,1055. (9) Tsikurina, N. N.; Zadymova,N. M.;Pugachevic,P. P.; Rabinovich, I. I.; Markina, Z. N. Kolloid-2. 1977, 39, 513. (10) Markina, Z. N.; Zadymova, N. M.; Tsikurina, N. N. Kolloid-2. 1978, 40, 876. (11) Dukhin, S. S. Kolloid-Z. 1963,25,418. (12) Dukhin, S. S. Kolloid-Z. 1964, 26, 36. (13) Dukhin, S. S. Kolloid-Z. 1983,45, 22. (14) Derjaguin,B. V.; Dukhin, S. 5.; Lisichenko, V. A. Russ. J. Phys. Chem. 1960,34,248.
0743-7463/92/2408-2124$03.00/00 1992 American Chemical Society
Langmuir, Vol. 8, No. 9,1992 2125
Kinetics of Recovery of Surfactant by Flotation
clet number regime has been studied by Andrews et al.15 using an artificial two-phase isotherm. The hydrodynamic conditions considered in refs 14 and 15 correspond to the experimental conditions encountered in the work to be described below, for which a condition was derived14to determine the relative magnitude of the deviation of the steady-state adsorption density from the Gibbs equilibrium adsorption. The application of these theoretical studies to flotation is problematic as they are concerned with a singlebubble rising in a quiescent solution, whereas a bubble swarm is quite turbulent. Further, the time required for establishment of the steady state is not estimated. The extent of adsorption on a bubble surface has been estimated by electrokinetic measurements. In one study Usui and SasakP used the Dorn potential to derive the bubble {-potential and hence the surface excess, finding that bubbles in M CTAB or sodium dodecyl sulfate had achieved only 5 1 0 % of their equilibrium coverage after about a second. However, this work has been criticized by Hunter17and Laskowski et al.18 for neglecting diffusioelectriccontributions to the {-potential, which casts doubt on the inferred adsorption. In another study, Collinset al.l9obtained the {-potential by measuring the electrophoretic velocity of small bubbles in 5 X 10" M CTAB. The bubbles were found to reach a constant electrophoretic velocity in about 0.1 s, the relaxation time being attributed to the buildup of charge on the bubble as the adsorption equilibrium is aproached. No measurement of the surface excess was attempted. When the assumption was made, however,that the limiting surface excess was that expected from the Gibbs equilibrium isotherm, and further that mass transport occurred in a Stokes flow field around the bubble, a correct estimate for the relaxation time was obtained. This result cannot be directly applied to the present work as the Reynolds number here is large (-300), precluding Stokesian flow. Sviridov et aL20 attempted to determine whether or not adsorption equilibrium was achieved in a flotation experiment. They found that the rate of recovery of cetylpyM ridinium halides at a concentration of 7.5 X increased with increasing bubble rise times from 1.5 up to 6 s, implying that equilibrium had not been achieved in those times. The wide variety of adsorption behaviors reported for these systems allows for a multiplicity of conclusions to be drawn about flotation kinetics: conceivably it may be controlled by diffusion, by convective diffusion, by sorption barriers, or by equilibrium bubble capacities. The present work aims to determine which of these mechanisms is important in the flotation of CTAB. We present a model of the time dependence of the surface excess of surfactant on a rising bubble, developed within the frameworks devised by Ward and Tordai2 and Levich.21 The results suggest how a recovery rate coefficient for flotation may be defined. This rate coefficientis measured for a range of temperatures in flotation experiments, and the results are interpreted in terms of the model.
8-
Sampling Syringe
-
(15) Andrews, G.F.;Fike, R.; Wong, S. Chem. Eng. Sei. 1988,43,1467. (16) Usui, S.;Sasaki, H. J . Colloid. Interface Sci 1978, 65, 36. (17) Hunter, R.J. Zeta Potential in Colloid Science; Academic Press: London, 1981; p 118. (18) Laskowski, J. S.;Yordan, J. L.; Yoon, R. H. Langmuir 1989,5, 373. (19) Collins, G.L.; Motarjemi, M.; Jameson, G. J. J. Colloid Interface Sei. 1978, 63,69. (20) Sviridov, V. V.;Balakina, T. D.; Kazantsev, E. I. 2. Fiz. Khim. 1983,57,1744. (21) Levich, V. G. Physicochemical Hydrodynamics, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1962; Chapter 8, Section 72.
Water Jacket
Surfactant Solution
-
Porous Frit Air Inlet
'H
E!Figure 1. Water jacketed flotation column.
Experiments Flotation experimentswere performed in a glass column fitted with a water jacket for temperature control, illustrated in Figure 1. The diameter of the inner glass tube was 5 cm. CTAB (Aldrich, 95% pure) was recrystallized once from ethanol, and solutions were made up in singly distilled water. The column was charged with 900 cm3 of the surfactant solution, giving a liquid depth of 40 cm. Air was passed into the column through the porous frit at its base, the flow rate being controlled by a ball-in-tube type flowmeter. Foam formed at the top of the column and was collected through a side arm. Samples were periodically withdrawn from the bulk solution through a hollow needle. The samples were analyzed for bromide ion concentration by a bromide selective electrode (Radiometer F1022Br). The calibration for the electrode gave a linear plot for the logarithm of electrode potential versus concentration over the range investigated. As the solutions were below the critical micelle concentration (cmc) of CTAB, the concentration of bromide ion was taken to be equal to the CTAB concentration. The initial CTAB concentration was 2 X 10"' M and the gas flow rate used was 200 cm3min-l of nitrogen. These were found to be convenient conditions for investigation of flotation, as a reasonably dry foam was steadily produced for between 1and 2 h. The experiments were performed at 0,15, and 30 "C. The nitrogen being passed through the column was not thermostated prior to its introduction, as the thermostating provided by the water jacket was sufficient to maintain a constant temperature in the flotation column. The volume of liquid in the column varied with time as liquid was lost as foam. The collapsed foam volume at the end of a run varied from 100 cm3 at 30 "C to 300 cm3at 0 "C. The volume of sample removed in the course of an experiment was typically about 25 cm3. Bubble size was determined photographically for the three operating temperatures and the size distribution is shown in Figure 2. We shall use an effective bubble radius (r) derived from these distributions, defined as the radius for which a mon-
2126 Langmuir, Vol. 8, No.9, 1992 2500
Morgan et al.
1
I
1.0
,
1
2000
%. 1500 c
5
i
lo00
@
.
0.2
t 500
0.0 ! 0.0
.
, 0.5
0 0 . 1 0 . 2 0 . 3 0 . 4 0.5 0.6 0.7 0.8 0 . 9
1
1.1
Bubbia Radiudmm
I
1 .o
.
I
1.5
.
2.0
tlmdmnd
Figure 3. Time dependence of the surface ~ ~ ( c calcdatd ~ 8 8 using Hansen's approximationto Ward and Tordai'e equation
Figure 2. Bubble size distribution as number of bubbles per cma of bubble volume, for 0, 16, and 30 "C.
for 2 X lo-' M CTAB.
odisperse distribution of spheres would have the same surface area to volume ratio as the polydisperse distribution. For a flow rate of 200 cms min-l r was found to be 6.9 X lo-' m at 0 "C, 7.9 X lo-' m at 15 "C, and 8.6 X 1 W m at 30 "C,with an uncertainty of 10%. The rise time of the bubbles in the surfactant solution was measured with a stopwatch to be 2.3 i 0.2 s for the 30 "C experiments,giving a rise velocity uo of 0.17 m 8'. It must be emphasized that while theae values are repmntative for a bubble swarm,individual bubbles traveled faatar or slower depending on their size. Further, the relative phase velocity and the rise velocity uo are not necessarily equal as there was a macroscopic circulation of liquid in the column, and bubbles were moving within this flow. The velocity u, then is an upper estimate of the relative phase velocities. For this reason no attempt was made to measure the bubble rise velocity at the lower temperatures, or to include the effect of the polydispersity of bubble size. This limits the accuracy of the hydrodynamic calculations that follow in which rise velocity and radius are important factors.
made by HansenZ2and Joos et Of these, the long time approximation derived by Hansen for adsorbing species that follow the Langmuir isotherm has been shown by Joos6tobe the most appropriate when it is the latter stages of adsorption that are of special interest. This is the case here, and Hansen's approximation is given by
Models and Adsorption Kinetics In this section we wish to derive an expression for the time dependence of the surface excess of surfactant at the interface of a bubble rising in a surfactant solution. Our starting point will be the model of adsorption proposed by Ward and Tordai2,which assumes that the surface is instantaneously in equilibrium with a region termed the subsurface, which may have a different concentration of surfactant than the bulk solution. This will be wed to determine the time dependence of the surface excess I' of CTAB at a quiescent interface. The theory will then be extended using the boundary layer theory developed by LevichZ1to obtain an expressionfor U t )for a rising bubble. Ward and Tordai2 derived an expression for the time dependenceof the adsorption on a planar surfaceassuming no energy barrier and no limit to the capacityof the surface. Their expression is
where D isthe diffusioncoefficientof the adsorbingspecies, co is ita bulk concentration, and r is ita surface excess as a function of ti, the age of the interface. This was then extended to take account of the back diffusion that is a consequence of the finite capacity of the surface to give their well-known result
where c, is the subsurface concentration. The integral in eq 2 may be calculatedif the adsorptionisotherm is known. However, the full expressionis inconvenient to work with, and several more accessible approximations have been
c,
=
(.
1-
I'm
(a + cO)(rDti)112
(3)
where I', and a are the parameters in the Langmuir isotherm
L)
r = r,( a + c,
(4)
the saturation surfaceexcess and the Langmuir-von Szyszkowski constant. The adsorption isotherm of CTAB was determined by Shchukin et aLZ3for a range of temperature from surface tension data. Their data for 30 "C is well described by the Langmuir isotherm with a = 1.94 X 1W M and r, = 3.9 X 10-6 mol m-2, so we may use Hansen's approximation for CTAB. Substituting Hansen's expressionfor c,, into eq 4, we get r as afunction of ti. Hansen claimed that thia result agreea with Ward and Tordai's equation to within 1% provided C ~ C > O 0.7. The subsurface concentration cannot be measured directly, but since the subsurface is taken to be in equilibrium with the surface, it may be inferred from the instantaneous surface excess if the adsorption isotherm is known. Thus a conditionis imposed on the surfaceexcess: we may regard Hansen's approximation as precise when the instantaneoussurface excess is greater than rw(0.?cO), where rep is the equilibrium surfaceexcess. For our mtial conditions of co = 2 X lW M the previous inequality > 0.42. becomes r/rm We estimated the diffusion coefficient of the hexadecyltrimethylammonium ion to be 5.6 X m2s-l from literature conductivity data,%using the Nernst-Einstein relation. The time dependence of r calculated for these values using Hansen's approximation is shown in Figure 3, and accordingto the above condition the approximation is precise for times greater than half a second. They indicate that adsorption reached about 90% of ita equilibrium value after 1-2 s. This implies that adsorption may still be occurring at a quiescent interfaceafter a phase contact time of 2 s. (22) Hanwn, R. S. J. Phys. Chem. 1964,60,637. (23! Shchukin,E.D.;Markina,Z.N.;Zadymova,N.M.2.Phys. Chem., Leipzrg 1985,266,1005. (24) CRC Handbook of Chemistry and Physics, 69th ed.; CRC Press, Inc.: Boca Raton, FL,1988.
Kinetics of Recouery of Surfactant by Flotation
Langmuir, Vol. 8, No. 9, 1992 2127
The above approach gives a lower limit to the rate of adsorption to a rising bubble since it neglects mass transport by convection. The word "convection" is here used to describe transport by hydrodynamic flow of any nature, rather than being restricted to that driven by thermal instabilities. To allow for such convection, we draw on the boundary layer theory for bubble adsorption developed by Levich.21 The diffusion boundary layer around a bubble is taken to be that region in which mass transport by diffusion is more important than convection. Levich investigated the structure of the surface defined by requiring that the diffusive flux equal the convective flux around the rising bubble. The exact calculations applied only for bubbles with small Reynolds numbers, but in the case of larger Reynolds numbers, such as those we are interested in, he offered an order of magnitude estimate of the boundary layer thickness 6 as21
i n
tlmdsewnds
Figure 4.
r vs time predicted theoreticallyusing eq 9 for 2 x
1V M CTAEL
9.1 In[Br-]/M . 8.98.7 -
a'
9.3
(5) In the experiments discussed in -the next section the effective bubble radius is r = 8.6 X lo4 m at 30 "C. As eq 5 is only an estimate we use this representative bubble radius, rather than considering the full bubble size distribution explicitly. The relative phase velocity u is estimated to be 0.17 m s-l, the bubble rise velocity. This value is representative of a bubble in a swarm moving in a macroscopic circulatory flow, and so no attempt is made to reduce eq 5 by using the theories of bubble rise velocities in quiescent media, such as the Stokes equation. For a bubble of this size and velocity we get 6 1.7 pm. Since convection effects dominate beyond this distance from the bubble, the concentration at and beyond the boundary layer may be taken to be constant and equal to the bulk concentration. Thus we have the concentration gradient near the bubble as
8.5J'
$
+ 5
+ 0
+ x I
Experiment 1 Experiment 2 Experiment 3
I
I
-
TIT 0
15
By equating the change in surface excess with the flux to the interface given by Fick's law from eq 6, we have (7)
Again we presume the interface to be in equilibrium with the subsurface following a Langmuir isotherm, allowing us to substitute for cs from eq 4
The solution of this differential equation is
where we have substituted an expression for a from the equilibrium form of eq 4
The approach to equilibrium for a CTAB concentration of 2 X M given by eq 9 is shown in Figure 4. We find that the time required for equilibrium adsorption on a rising bubble, taking convection into account, is approximately 0.1 s, which is rapid on the timescale of a flotation experiment, and about 20 times faster than the prediction for diffusion alone. The implication of this is that we do
30
104kls-* 3.62 3.59 3.64 3.56 2.43 2.64 2.62 1.40 1.40 1.37
[email protected] 3.60 f 0.04 2.56 i 0.13 1.39 f 0.02
not expect the bubble surfaceexcessto be transport limited beyond this time. For our experimentalconditions,steadystate surface coverage should be rapidly established.
Kinetics of Recovery We now wish to describe the time dependence of the bulk concentration of surfactant. Typical graphs of the logarithm of the bulk bromide ion concentration versus time are shown in Figure 5. The graphs are approximately linear. A recovery rate coefficient k may therefore be defined as the absolute value of the slope of such a plot. k was determined for 0,15, and 30 "C for at least three runs at each temperature, and the values are reported in Table I. To relate k to the other experimental parameters, consider a volume V , of surfactant solution subjected to an airflow J. The change in concentration of the cell contents is given by (11)
where @A is the amount of area carried through the column in unit time, which is just the product of the flow rate with the surface area to volume ratio of the bubbles. t is the
Morgan et al.
2128 Langmuir, Vol. 8,No. 9,1992 time for which the solution has been subjected to the airflow. rf is the surface excess carried by a bubble as it leaves the solution. Equation 11 may be written
which can be solved if Fdc) is known. We suggested in the previous section that r(c)reflects a steady-state adsorption. Let us suppose that the surface relaxes as the bubble enters the foam, so that W c ) is the surface excess predicted by the equilibrium isotherm. Before undertaking a more complete analysis, it is instructive to consider the solution of the operating equation in the case of the Henry adsorption isotherm
rf= rep = mc
(13)
in conjunction with the assumption of a dry foam, i.e. V, is constant. Under these conditions the solution of eq 12 is
which on identification with our definition of the recovery rate coefficient gives
(15) Under these assumptions k is proportional to the flow rate and surface activity of surfactant and inversely proportional to the bubble radius. Although the operating equation is solved under more realistic conditions below, the structure of the experimental rate coefficient is not as transparent as it is here. The simplicity of eq 14obtains at the price of two rather bold assumptions. The adsorption isotherm, while slowly varying, is not strictly linear; further, liquid is lost as foam, so V, is not constant. A more realistic treatment that considers both of these limitations now follows. The volume of liquid in the cell decreased with time by as much as 300 cm3 in an hour for the experiments performed at 0 OC. We will assume that this loss was linear in time and write
vc=V0-At
(17) If we use this as our definition of A, then its value was 8.3 X 10-8m3s-l in the casejust cited. This expression together with the Langmuir isotherm may be used in eq 12 to get
85
0
10
20 30 tlmelmlnute
40
50
60
Figure 6. Prediction of the time dependence of bulk CTAB concentration in the case of the greatest deviation of conditions from the dry foam approximation.
the form predicted by the dry foam assumption. As the value of a at 0 O C is unknown, we have used the value for 30 O C and note that this is an underestimate as a increases with temperature. The deviation of the data from linearity in the worst case is still smaller than the error in the measurements,so the use of a straight line slope as a single parameter to describe the recovery kinetics introduces no additional error. We have sufficient information for 30 "C to use eq 20 to predict the recovery curve, and in Figure 7 we compare the prediction made using the experimentally measured value A = 2.8 X m3s-l with our experiments at 30 O C . The approachjust developed works very well in this case. The recovery kinetics are therefore consistent with the bubble surface adsorption being that given by the isotherm for unperturbed equilibrium conditions.
Temperature Dependence of Recovery Further support for the foregoing notion comes from considerationof the temperature dependenceof recovery. Three effects are responsible for the change of k with temperature: the change in the bubble size distribution; the loss of liquid from the cell;the change in bubble surface excess. In order to test the model against the experiment, we need to separate the latter effect from the first two. This can be done using the operating equation. Substituting eq 17 into eq 12 and rearranging we get
Now dc c dlnc -= dt
=-ck
which on integration gives
q r,
c - co
+ a ln
v, (6))=gin (T)
dt
(22)
from our definition of k, from which it follows At
(19)
Rearranging we have
which implicitly describes the time dependenceof the bulk concentration. Superficially, eq 20 appears to be of a quite different mathematical form to the log-linear form of eq 14. To ascertain its deviation from this form, we plotted eq 20 in Figure 6 using the largest measured value of A, which occurred at 0 OC,and the r value for that temperature. These parameters represent the greatest deviation from
vo- At r,(c) = -kc @A
Equation 22 is an approximationthat reflects the inability of the experimental results to discriminate the curvature of the theoretical curve of Figure 7 from a straight line. T h e surface excess calculated from eq 23 is thus an approximationat this level. Values of A and @A from our experimentsare reported in Table 11. From these data we graph the surface excess derived from eq 23 for different temperatures in Figure 8. The surface excess calculated for the 30 "C experiment agree with the published dataF3 within the experimental error of 10%. Figure 8 shows that decreases as the temperature increases, which is consistent with an equilibrium con-
Langmuir, Vol. 8, No. 9, 1992 2129
Kinetics of Recovery of Surfactant by Flotation 9.1
,
1
. I n [ B 1'1
Q
20 timelminute
40
60
Figure 7. In [Br-] vs time curves for three experiments at 30 OC
compared with theory: solid line, eq 20; X, +,.,experiment. 5
+ 4+ e
+
mol m.3 2
+
1
0.0
* ++*e
e
e
." .. .0.2
0.3
[CTAB]/mol
Figure 8. Approximate surface excess of bubbles entering the foam for three temperatures: ).( 30 O C ; ( 0 )15 O C ; (+) 0 O C . Table 11. Interfacial Flux and Liquid Loss Rates 0 15 30
1.5 X 1.3 X 1.2 x 10-2
The approach to adsorption equilibrium of a bubble rising in a surfactant solution was modeled using the Hansen approximation to Ward and Tordai's model, in conjunction with the Levich diffusion boundary layer theory. It was predicted that a steady state would be achieved in about 0.1 s under the conditions employed in the experimental study, thus supporting the notion that the bubble surface excess is not limited by transport of surfactant to the bubble surface. The recovery kinetics could be described by a single experimental rate parameter k, which was measured for several temperatures. The magnitude and form of k were well described by adopting the assumption of adsorption equilibrium. The temperature dependence of the recovery of the surfactant was inconsistent with the hypothesis of energeticcontrol but was in agreement with the assumption of equilibrium adsorption. This establishes a basis for the application of the results of experiments on equilibrium surfaces to the more inaccessible bubble surface.
Acknowledgment. We thank the Broken Hill Proprietary Company Limited and the Department of Industry, Training and Commerce for the award of a PIRA scholarship to J.D.M. and Dr. Kevin Galvin and Mr. Malcolm Engel of BHP Research for helpful discussions.
e
0.1
Conclusions
8.3 x 10-8 5.6 X 10" 2.8 x 10-8
trolled surface excess. This decrease with temperature precludes the possibility that I'f is controlled by an would activation energy barrier, for, were that so, increase with increasing temperature. In contrast, Figure 8 cannot be used to draw any conclusions about whether I?fis transport controlled. An analysis of eq 10 shows a greater initial rate of adsorption as temperature increases, but a lesser final value of res. Accordingly,the predicted value of rfmay either increase or decrease with a rise in temperature, depending on the degree of approach of the surface excessto its equilibrium value.
Glossary Langmuir-von Szyszkowski constant instantaneous bulk concentration of surfactant bulk concentration of surfactant at time zero concentration of surfactant in subsurface diffusion coefficient of surfactant diffusion boundary layer thickness interfacial flux instantaneous surface excess equilibrium surface excess on a quiescent surface saturation surface excess surface excess on a bubble entering the foam gas flow rate recovery rate coefficient rate of loss of liquid from column bubble radius time interface age temperature bubble rise velocity volume of bulk solution
