Bursting of Soap Films Part 4.-The Behaviour of Ions on a Crowded Surface BY G. FRENS,* KAROL J. MYSELSt AND B. R. VIJAYENDRAN $ R. J. Reynolds Tobacco Co. Research Dept. Winston-Salem North Carolina 27102 U.S.A. Received 13th April 1970 The growing hole of a bursting soap film is preceded by an aureole of accelerating contracting and thickening film. Measurements of the cross-sectional profile of such aureoles are reported and provide the basis for an interpretation in terms of surface tension changes accompanying the contrac- tion of the a m which is complete in less than a millisecond. The surface tension of sodium dodecyl sulphate solution decreases to below 15 mN/m (dynlcm) under these conditions. As desorption is negligible the data can be interpreted in terms of an extension of surface pressure-area per molecule curves towards high pressures and low areas.This extension shows an unexpected decrease of slope at low areas and indicates a rapid increase in intermolecular attractions as the surfactant ions become crowded on the surface. Once a thin liquid film specifically a soap film develops a tiny perforation the latter will keep growing and the whole film disappears rapidly. In soap films the edges of a hole recede at speeds of the order of lo3 cm s-l. The driving force is the surface tension of the two faces of the film which exerts an uncompensated force on the perimeter of the hole. It was assumed that the growing hole is surrounded by undisturbed film except for a very narrow rim which contains all the receding material. Recently however it has been found 1 * that the expanding hole is preceded by a wide zone (the aureole) of moving film material.The width of the aureole is comparable to the radius of the hole and the expansion of the hole and the aureole are approximately proportional. The thickness 6 of the film increases in the aureole region from the thickness do to approximately 2d0 at the rim of the hole. Flash photographs such as fig. 1 which show some of the complicated features of aureoles have been published.'. A theoretical analysis of the aureole phenomenon was given by Frankel and My~els.~ They showed that the existence of aureoles can be explained if there is a gradient in the surface tension of the film in the vicinity of the hole. Such a gradient is closely related to the thickness variations in the aureole. As a film element increases in thickness it decreases in surface area.If the relaxation of the surface through the desorption of surfactant molecules is slow as compared with the rapid compression of the collapsing film structure then the film surface resembles an insoluble monolayer. The surface tension of a film element becomes smaller as its surface area decreases i.e. as its thickness 6 increases. It is the purpose of the present paper to show that an interesting extension of the classical TJ-A curves (surface pressure against area per molecule) becomes available when this theory is applied to experimental data obtained for the thickness profile of the aureoles. * present address Philips Research Laboratories N.V. Philips' Gloeilampenfabrieken Eind- hoven NetherIands. -f present address Research Dept.Gulf General Atomic Inc. San Diego California 921 12 U.S.A. $ present address Research Dept. Pitney-Bowes Staniford Conn. U.S.A. 12 FIG. 1. Bursting soap film photographed using a 0.5 ys flash. The aureole surrounding the hole is relatively narrow and shows a large frontal shock. [To face page 12. G . FRENS K . J . MYSELS AND B . R. VIJAYENDRAN 13 THEORETICAL In principle it is possible to calculate values for the surface tension as a function of the increase in thickness and of the time during a burst for any arbitrary behaviour of an aureole provided that the history of the evolution of the aureole is sufficiently well known.4 In practice one would need too many and too accurate data to make such an analysis feasible. The interpretation of data is much simpler if the physical situation in a bursting soap film corresponds to a pseudo-equilibrium in which there is no desorption from the surface and where the surface pressure depends only on the area per molecule.In such a system there is no intrinsic (relaxation) timescale and the bursting becomes self-similar i.e. all features of the aureole and of the hole expand from the origin each at its own constant velocity. This assumption means that the surface tension o depends only on the relative shrinkage a or thickening p of the film defined by a = 1/p = a0/6 and not on time so that a single (a,a) curve such as that of fig. 2 characterizes the system. do I / 0 I shrinkage a FIG. 2.-Schematic (a,~) curve. The dashed lines indicate portions corresponding to shock waves in unidimensional bursting.If the burst originates from a straight line-is unidimensional-a complete analysis is then p~ssible.~ If it originates from a point which is the case in our experiments only certain features of the unidimensional analysis can be extended to this radial case and others have to be computed numerically. A fundamental velocity in this analysis is Culick’s velocity u = (2aO/p&,)* where go is the surface tension of the film at rest and p its density. Culick showed that this is the velocity of the rim of the hole but in his model there was no aureole. In the presence of an aureole it can be shown that uh is always smaller than u in the unidimensional case and is smaller or slightly larger in the radial case. As indicated in fig. 2 only certain parts of the (o,a) curve which are convex downwards are represented by the visible smooth slopes of the aureole; those that have the 14 BURSTING OF SOAP FILMS opposite curvature are hidden in shock waves the most important of which is the rim itself.4 The rim may be considered as the shock wave due to the zero surface tension within the hole.Generally there is a shock wave at the outer edge of the aureole but there niay be 0thers.l In the unidimensional case the shock waves correspond to chords exactly tangent to the (o,a) curve. In the radial case chords are not exactly tangent.6 A special case is that of a (a,a) curve lying entirely above the diagonal joining the initial state to the origin. In that case the entire aureole is compressed into the shock wave represented by that diagonal and the width of the aureole is reduced to A useful extension of the theory is an expression for the limiting properties of the aureole at its thickest point where it joins the final shock wave i.e.the rim. We denote these by the subscript 1. Conservation of mass and momentum gives for a shock wave moving with a velocity us and separating film elements moving with u and uZ respectively (us - 211)(21 - r l ) p 6 = 2(0l - Q) (1) where o- and o2 are the corresponding surface tensions on the two sides of the wave.4 For the limiting point u2 = us since the material passing through the shock wave remains in the rim o2 = 0 as explained earlier and subscript 1 is replaced by 1. Introducing 21 one obtains finally (u,-iu1)2 = U"cJ&Uo. (2) The value of a. is easily obtained and that of (rl is obtained by integration where p may be defined as to/t where to is the time for a reference feature of the aureole to reach a certain position x or Y in space and t is measured at the same point.The velocities zi arc given by eqn (5.18) of ref. (4) for the radial case and yield where I is an integral giving the amount of film material between the centre and the point considered and uo = r / t o . As there is no material in the hole 1 = 0 and combining with eqii (2) we obtain which is satisfied only at the rim. As this is derived on the assumptions of self- similarity but involves only information concerning the profile of a single aureole it can provide a criterion of self-similarity for each aureole. For the unidiinensional case the similar expression I:(uoi%)2 = Pl(Q1lQd (6) is derived similarly using eqii (5.6) of ref.(4). EXPERIMENTAL Experimentally the velocity of the rim and that of any shock waves is probably best estimated from photographs taken at different time intervals after initiation of the burst by an electric spark.' Details of the profile of an aureole are difficult to obtain precisely in this way and methods ' measuring either the variation of thickness with time at a fixed point, G . FRENS K . J . MYSELS AND B . R . VIJAYENDRAN 15 as the aureole moves by or the time required for a feature to proceed from one point to another are preferred. The results reported here were obtained with an instrument using a laser and measuring simultaneously the thickness of the film at two points along a horizontal radius as indicated in fig. 3 which also shows a typical oscilloscope result and its inter- pretation.TOMULTIPLIERS -ASS 'LATE / OSCILLOSCOPE TRACE T I M E 3 -6 t f / l I INTER P E TAT1 0 N FIG. 3.-Scheniatic diagram of the apparatus used. The laser beam is split and concentrated at two points of the film. The reflected interference pattern changing as the aureole passes these points is recorded through photomultipliers and a two-beam oscilloscope and interpreted as shown on the right. RESULTS A major difficulty discussed in detail earlier,l was that whereas the theory predicts that all velocities should vary with Culick's velocity i.e. with l/dS gross deviations were observed for thin mobile films as well as for rigid films of all thicknesses. The explanation offered earlier,l that even the apparently undisturbed film is slightly compressed was supported by direct evidence for rigid films and may be significant when applied to these but we have not been able to support it experimentally for mobile films.On the other hand we have found that frictional resistance of the atmospliere- the windage of the moving aureole and rim-greatly affects the velocity of the rapidly moving rim of thinner films and seems to account for the deviations observed with first black and thicker thin films. Fig. 4 shows rim velocities observed in air in helium and in hydrogen for films of various thicknesses of the same solution. The viscosities of these gases are 185,198 and 89 ,UP and their densities 1.20 0.165 and 0.083 g/l. respectively. The deviations from the 1 / JS behaviour indicated by the straight line become less as the inertia and viscosity of the atmospliere decreases.For thin films the deviations in hydrogen are still significant though much smaller than in air but above some 100 nm they cease to be perceptible. Hence further discussion will be restricted to films having greater thicknesses and bursting in hydrogen. That bursting is self-similar to a first approximation is shown by the constancy of rim velocities as the aureole grows which has been reported earlier for films in air. This has been supported now by experiments in hydrogen. The velocity of the frontal shock is also constant and obeys the 1/,/8 relation closely. A more sensitive criterion is given by the shape of the aureoles which should be unchangirig after correction for growth as a function of time and the corresponding spreading in space.A convenient form is to represent the thickness of the aureole as a function 16 BURSTING OF SOAP FILMS of f f / t where tf is the time for the frontal shock to reach the point of observation and t corresponds to the relative thickening B = S/S, as indicated schematically in fig. 4. The solid line of fig. 5 and the corresponding large points show the results obtained P 1.5 1.0 film thickness nm line shows the slope of ao-4. FIG. 4.-Rim velocities in various atmospheres as a function of original film thickness a0. The - ‘S* A;,& - - - ! I$\ 0 - - 1 in three experiments for the later stages of an aureole. The precision of the measure- ments defines the line quite well. The position where the rim should lie under assump- tions of self-similarity according to eqn (5) is indicated by an arrow and is close to that observed.For younger aureoles the precision gradually decreases as shown by G . FRENS K. J . MYSELS AND B . R. VIJAYENDRAN 17 the smaller points and differences in profile appear as indicated by the dashed line. These differences although systematic are not sufficiently larger than all the un- certainties of the measurement to be accepted as definitely real. Furthermore the criterion of eqn (5) indicates a lack of self-similarity for the dashed line. Fig. 6 shows the (a,@) curves corresponding to the profiles drawn in fig. 5. These were obtained by computer integration using Simpson’s approximation of the I 0 I shrinkage a FIG. 6.40,a) curves derived from the lines of fig. 5. differential expression (5.22) of ref.(4). The difference between these curves is relatively minor so that the uncertainty in a is much less than in the profile itself. This is generally true for (a,a) curves for aureoles differing only in their thicker areas and results from the complex relation Deviations such as those of fig. 5 between young and old aureoles could stem from two sources windage which presumably is greatly reduced by operating in hydrogen or relaxation of the surface by desorption of surfactant molecules under surface pressures much above the equilibrium values leading to higher a values for a given a for older aureoles. The curves of fig. 6 do indicate such an effect upon 0 and if the differences are real they could be used to study such desorption. On the other hand extrapolation to zero age of the aureole could then also be used to correct for any relaxation and serve as a basis for interpretation in terms of the self-similar theory.As shown in fig. 6 the original surface tension of the solution is reduced by about 63 % in the last stages of the aureole. In absolute terms this corresponds to a reduction from 38 to only 14 mN/m considerably below values normally encountered at the air-aqueous solution interface. between the two. DISCUSSION The (o,a) curve is closely related to the surface pressure-area per molecule ( l l A ) curve since II = t ~ ~ ~ ~ - t ~ and a = A/Ao where A . is the area per molecule 18 BURSTING OF SOAP FILMS in the original film provided that there is no desorption. Whereas oo is readily measured the direct determination of A . requires relatively delicate experiments with foaming or with tracers.Indirect determination of A . can be based on highly accurate surface tension and activity measurements through the Gibbs equation. Hence in practice literature data which are often conflicting have to be used. For sodium dodecyl sulphate (NaLS) in water a value of 0.415 nm2 (41.5A2) based on foaming seem most likely to be correct for solutions at and slightly above the c.m.c. 60 50 40 E z E \ 30 I=!! 20 10 The solid line of fig.-7 shows the (II,A) curve thus obtained \ \ I I 1 0.2 0.3 0.4 0’5 area/molecule nm2 FIG. 7.-The solid line is the (rI,A) curve corresponding to the solid ( ~ p ) curve of fig. 6. The dashed lines are some of the availableliterature data for equilibrium adsorbed monolayers of NaLS solutions R from equations of Rehfeld lo ; W data of Weil ; W + E areas of Weil 9 surface tensions of Elworthy and Mysels.from our data and based on this value. It is essentially a mirror image of the (o,cc) curve of fig. 6 and provides an extension of (II,A) curves for ionic monolayers into a new region. If the interpretation of a (o,ct) curve in terms of (n,A) is justified it should give to a first approximation a sniooth extension of the classical (IT,A) curve for higher areas. The interrupted lines of fig. 7 show some of the literature data 9-11 in this region. The values at higher areas are obtained on solutions of NaLS alone of varying concentrations and therefore varying ionic strengths. Our data pertain to a single solution and therefore presumably to a constant ionic strength of the subjacent liquid.This difference could lead to slightly different slopes. Reducing ionic strength at constant A should lead to an increase of interionic repulsions and therefore to an increased surface pressure. Hence the interrupted lines should tend to be steeper than if they had been taken at the constant ionic strength corresponding G . FRENS K . J . MYSELS A N D B. R . VIJAYENDRAN 19 to the c.m.c. As it is the data of Weil seem to give the smoothest fit with ours. Literature data particularly the surface tension values below the c.m.c. show con- siderable disagreement among themselves despite the precautions taken by each author to ensure purity and acc~racy.~-'~ Our data are obtained above the c.m.c. to reduce the effect of surface-active impurities by solubilizing them. Hence any conclusions from the intercomparison of these data must be highly tentative but is indicative of the possibilities of this approach.The striking feature of fig. 7 is that the heretofore known curves for ionic surfactants as exemplified by the interrupted lines are convex to the abscissa whereas most of our curve is concave. If the change of curvature did occur at the point where the two sets of data meet this would be a strange coincidence. In fact however practically all aureoles examined until now (the only exception being a complex mixture of alkylbenzene sulphonates) show a frontal shock. This shock corresponds to a part of the (a,a) curve which is not directly accessible because it includes a curvature opposite to that of the accessible part (fig. 2). Hence the frontal shock wave indicates an extension of the curvature observed by conventional techniques into the region where A<Ao.The surface compression and therefore the change in A during this frontal shock varies in different solutions and can reach 10 %. The aureole which follows after the frontal shock represents still lower values of A . Fig. 2 indicates that the reversal of the curvature in this part of the (I-I,A) curve is intrinsic to the existence of the aureole phen~inenon.~ Hence the reality of the observation that (Tl[,A) plots have concave portions at A < A requires only a qualita- tive validity of our experimental results and their interpretation. It does not depend on the precision of the measurements nor on the detailed outcome of the calculations. The curvature deduced from bursting experiments would be reduced by any relaxation effects since the shrunken area as measured by a would be attributed to the smaller number of molecules remaining in the surface.It does not seem possible however that any small relaxation effects occurring in our experiments could account for the observed result. An equation of state for the surface would describe the (IT,A) curve for A below as well as above Ao. Such a quantitative interpretation of the (II,A) curve in terms of intermolecular forces should become more meaningful now that the aureoles in bursting soap films give access to the metastable states of a surface where the van der Waals attraction between the molecules becomes strong enough to reverse the curvature of the (II,A) plot. Progress along these lines awaits the improvement of the experimental data including those concerning the surface tension and the area per molecule below the c.m.c.W. R. McEntee and K. J. Mysels J. Phys. 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