Langmuir 1992,8, 3027-3032
3027
Hole Formation and Sheeting in the Drainage of Thin Liquid Films V. Bergeron, A. I. Jimenez-Laguna,?and C. J. Radke' Earth Sciences Division, Lawrence Berkeley Laboratory and Chemical Engineering Department, University of California,Berkeley, California 94720 Received April 4,1992. In Final Form: October 5, 1992 Foam, emulsion, and pseudoemulsion films stabilized by surfactants above the critical micelle concentration exhibit pronounced stepwise thinning or stratification. Dynamic removal of a layer is initiated by the formation and subsequent sheeting of a hole (or holes) at each step of the stratified thinning. We outline a nonlinear hydrodynamic stability analysiswhich successfully models the dynamics of hole formation and sheeting by accounting for an equilibrium oscillatory structural component in the disjoining pressure isotherm. Initiation of a hole is attributed to the nonlinear growth of a hydrodynamic instabilityduring film thinning under the action of aconstant capillary pressure and an oscillatory disjoining pressure. Subsequent growth or sheeting of the hole is explained by outward fluid flow within the inhomogeneousthin film asradial pressure gradientsdevelop from curvaturevariations and viscous resistance to flow. Qualitative comparison is made between the proposed hydrodynamic theory and experimental observations of thinning events and rates.
Introduction Drainage of liquid films under the influence of molecular thin-film forces (i.e., films of thickness ca. less than about 100 nm) does not always occur uniformly. Early on, Reinold and Riicker,l J ~ h o n n o t t , ~ Hagenbach? *~ and Perrins documented the local, abrupt formation of thin black spots. These spots, which we also designate as holes to emphasize their three-dimensional topology, expanded and coalesced with others formed nearby until the entire film sheeted away to a new thickness corresponding to the bottom of the holes. The entire hole-sheeting process may repeat at several film thicknesses, giving rise to stepwise thinning. Indeed, Johonnott3 and notably Perrin5 observed as many as five black-hole-sheetingtransitions. The fourth and fifth transitions left behind films 0 (12 nm) and 0 (5 nm) in thickness, which are now designated as common and Newton black films, respectively. Decades later, Derjaguid and de Vries7 observed the same phenomena in both horizontal liquid films thinning by capillarity and in vertically held foam lamellae thinning by gravity. Derjaguin6attributed the formation of locally thin spota to instabilitiesin the gaslliquid interfacecaused by random fluctuationsin gas pressure. Conversely, de Vries7imposed a small temperature perturbation to initiate the growth of the holes. de Vries suggested that the rate of expansion of the thin-film spots adjacent to thicker film regions depends on the local competition between attractive London-van der Waals forces and repulsive electrostatic forces and on the viscosityof the fluid that has to be pushed between the interfaces. No quantitative theory was proposed.
* To whom correspondence should be addressed.
t Currently at ESPCI,10 rue Vauquelin Paris, France. (1) Reinold, A. W.; Rocker, A. W. Philos. Mag. 1836, 19, 94. (2) Johonnott, E. S. Thicknessof Black Spota in Liquid Films. Philos. Mag. 1899,47,501-522. (3) Johonnott, E. S. The Black Spota in Thin Liquid Films. Philos. Mag. 1906,11,746-753. (4) Hagenbach, A. Electrical Resistance of Particles of Soap. Arch. Sci. Phys. et Nut. 1913,35,329-339. (5) Perrin, J. La stratification des lames liquid-. Ann. Phys. 1918, 9,160-184.
(6)Derjaguin, B. V.; Titijevekaya, A. S. Foams: Static and Kinetic Stability of Free Films and Froths. Int. Cong. Surf. Act., 2nd 1967,1, 211-219. (7) de Vries, A. J. Foam Stability; Part Iv: Kinetics and Activation Energy of Film Rupture. Recueil 1968, 77,383-389.
Later, Kolarov et ale8examined the growth rate of black holes during the transition from common to Newton black films in single soap films. They developed a macroscopic one-dimensional force balance to predict the growth rate of a hole as due to the imbalance in the surface tension force at the contact line between the hole and the precursor film. Ivanov and Dimitrovs review more detailed analyses of mature hole expansion in common black films. Recently, de Gennes,loby drawing analogy to the growth of dry spots on liquid-wet surfaces,ll extended the earlier works to recognize that a rim of liquid must build ahead of the expanding hole. To date, little consideration has been given to how such black holes initially develop. The subject of this work is dynamic stepwise thinning (or stratification, Perrin6) of isolated, horizontal foam lamellae under constant capillary-suctionpressure and at surfactant concentrations above the critical micelle concentration (cmc). Figure 1 displays the thinning kinetics12J3of a plane-parallel foam lamella suspended in a porous frit.14 The film is stabilized by 0.1 M aqueous sodium dodecyl sulfate (SDS), and film thickness is measured via a 50-rm aperture optical probe using Scheludko interferometry.lS Clearly, these data demonstrate that isothermal film thinning under a constant capillary-suction pressure is not monotonic but rather evolves as discrete steps. Four thickness steps or jumps are evident. In each step the film thickness first rises and then falls abruptly by about a 10-nm net change, starting with a film 65 nm thick and ending in a metastable film (8) Kolarov, T.; Scheludko, A.; Exerowa, D. Contact Angle between Black Film and Bulk Liquid. Trans. Faraday SOC.1968, No. 550,64, 2864-2873. (9) Ivanov, I. B. and Dimitrov,D. S.,Thin Film Drainage. Thin Liquid Film 1988, 379-496. (10) de Gennes, P. G.Dynamics of Drying and Film-Thinning. Phys. Amphiphilic Layers, Springer Proc. Phys. 1987,21, 64-71. (11) Brochard-Wyart, F.; di Meglio, J.-M.;QuM, D. @tudedu retrait d'un film liquide non mouillant dBpoa6 sur un plan ou une fibre. C. R. Acad. Sci., Paris 1987, 304 SIrie II(lI), 553-558. (12) Bergeron, V.; Radke, C. J. Equilibrium Meaeurementa of Oscil-
latory Disjoining Pressures for Thin Liquid Foam Films. Langmuir, preceding paper in this issue. (13) Bergeron, V. Forces and Structure in Surfactant-Laden, ThinLiquid Films. Ph.D. thesis, University of California, Berkeley (in preparation, 1992). (14) Exerowa, D.; Kolarov,T.;Khristov, K. H. R. Direct Measurement of Disjoining Pressure in Black Foam Films, I. Films from an Ionic Surfactant. Colloids Surf. 1987,22, 171-185. (15) Scheludko, A.; Platikanov, D. Utersuchung diinner FlWinger Schichten auf Quechiber. Kolloid Z. 1961, 175, 150-158.
0743-7463/92/2408-3027$03.00/0 0 1992 American Chemical Society
Bergeron et
3028 Langmuir, Vol. 8, No.12, 1992 70 I
1
0.1M S.D.S Pc = 65 Pa 24OC
h
E ~
C
v
al E
.-Y0
I \
i I
a
"IY
40
30
I
t 0
100
200
300
Tlme ( 8 )
Figure 1. Stratified thinning of a foam film under a constant capillary-suction pressure. Data are from Bergeron.13 of thickness ca. 20 nm some 250 s later. This last film, which is in equilibrium with the 65-Pa suction pressure, can last routinely for many hours, when carefullyprotected against evaporation,vibration, and pressure disturbances. The thinning process in Figure 1actually occurs by the repeated removal of successive layers. During each stepwise transition one or several nearly circular holes, that appear black to reflective light (corresponding to a portion of thinner film), form and grow in area at the expense of the disappearing thicker film. A sequence of videomicroscopypictures in Figure 2 displaysthese events for the transition from a film thickness of 35 nm to 25 nm, similar to the last step (or the removal of the last layer) in Figure 1. Lighter to darker shades of gray in these micrographs correspond to ever thinner films. Newton rings can be seen at the film Plateau border. Figure 2a shows a 25 nm thickness black spot near the center of a 35 nm thickness film at a time t* = 0.13 s after the spot is first noticeable. The spot or hole radius is RH* = 23 pm. Light gray portions of the film near the top edge reveal the remnants of the previously sheeted away 45 nm thickness film. After 0.96 8 in Figure 2b the black hole spreads to RH* = 57 pm, and a light gray ring visible at the perimeter delineates a thick rim preceding ahead of the hole. Parts c and d of Figure 2 demonstrate that the rim at the circumference of the expanding black spot eventually develops very thick (greater than 100 nm) satellite pockets of liquid. The spreading black hole with its pocketed rim finally consumes the entire 35 nm thickness film, leaving behind a new 25 nm thickness film. This entire thinning process is reiterated at each jump in Figure 1, although usually several holes appear almost simultaneously at different locations across the film. Because the film is symmetric, the hole-sheeting or stratification process in Figure 2 is best visualized as a double-sided, expanding crater or volcano whose growing rim eventually disintegrates. We can now understand the peculiar thinning dynamics in Figure 1. When the optical probe encounters a growing spot, the measured thickness first increases as the rim wall passes under the probe. The film thickness then falls dramatically as the probe detects the flat-bottomed crater. Kralchevsky et al.la have previously observed and proposed an explanationfor the stratification of foam films at surfactant concentrations above the cmc in terms of a (16)Kralchevsky, P. A.; Nikolov, A. D.; Wasan, D. T.; Ivanov, I. B. Formation and Expansion of Dark Spota in Stratifying Foam Films. Langmuir 1990,6,1180-1189.
41.
diffusive-osmotic mechanism. During thinning, the film traps micelles in an almost crystalline state. When locked in this ordered structure, intermicellerepulsive forces give rise to a strongly repulsive disjoining pressure.17 Random disturbances then nucleate small vacancieswhich grow to become expanding black by diffusion of micellea from the center of the film to the bulk solution at the Plateau border. A gradient in the chemical potential of micelles at the film edge is said to be the driving force for the stratification process. Attainment of a 'metastable" thin-film state after the removal of a layer is accounted for by a succession of nonequilibrium repulsive disjoining pressures associated with each ordered micellar layer. Unfortunately, the mechanism of micelle diffusion fails to explain the formation, expansion, and subsequent breakup of the liquid rim ahead of the spreading black hole. More importantly, it is not coneistent with the recent experimental measurements of Bergeron and Rgdke,12 clearly demonstrating that equilibrium oscillatory disjoining pressures exist in foam films at surfactant concentrations above the cmc. The goal of this work is to explain the origin of the hole-sheeting or stratification process of Figures 1and 2 in terms of well-acceptedhydrodynamic argumentssimilar to those of de Vriesa7The crucialproperty of the surfactant system that drives stratification is an oscillatorydisjoining pressure isotherm. We outline a simple hydrodynamic stability theory that models the initial formation and subsequent expansion of a hole as triggered by the growth of a random, infinitesimal disturbance at the gadliquid interface. Our analysis predicts the early-timegrowthrates associated with the hole expansion shown in Figure 2; it also explains qualitatively the break-off of satellite fluid pockets during the later stages of hole growth.
Theory Since we assert that hole-sheeting derives from the specific form of the equilibrium disjoining pressure, II*, as a function of film thickness, h*, it is useful to review the shape of that isotherm. The pertinent behavior of lI*(h*),expressed in dimensionlessform, can conveniently be described by AH
II(h) = -- + csch2 (h/2) - B exp(-ah) cos (rBh) (1) h3 The first two terms on the right of eq 1 correspond, respectively, to attractive van der Waals forces and repulsive electrical double-layer forces for two constantcharge interfaces.2l These two terms comprise classical DLVO theory. The third term empirically augments DLVO theory with an exponentially decaying oscillatory structural component.22 In eq 1 AH is a dimensionless Hamaker constant equal to c d A ~ * / 2 * qwhile ~ €3 gauges the ratio of the magnitude of the structural force to the repulsive electrostatic force. h is the dimensionless film thickness scaled by the Debye length, 1 / ~ a; and B are (17)Nikolov, A. D.; Kralchevsky, P. A.; Ivanov, I. B.; Wasan, D. T. Ordered Micelle Structuring in Thin F h Formed From Anionic Surfactant Solutions, 11. Model Development. J. Colloid Interface Sci. 1989,133(11,13-22. (18)Derjaguin,B. V.;Prokhorov,A. V. On the Theory of Rupture of Black Films. J. Colloid Interface Sci. 1981,81,108-115. (19)Prokhorov,A.V.;Derjaguin, B. V. On the GeneralizedTheory of Bilayer Film Rupture. J . Colloid Interface Sci. 1988,126(l),111-121. (20)Kashchiev, D.; Exerowa, D. Nucleation Mechanism of Rupture of Newtonian Black Films. J. Colloid Interface Sei. 1980,77, M)1-611. (21)Verwey, E. J.; Overbeak, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (22)Chan, D. Y.C.; Horn,R. G. The Drainage of Thin Filma between Solid Surfaces. J. Chem. Phys. 1986,83(lo),5311-6324.
Langmuir, Vol. 8, No. 12, 1992 3029
Hole Formation and Sheeting
a
C
.--tic. --?
b
d
Figure 2. Nucleation and sheeting of a black hole of radius RH* in the stratified thinning of a foam film. The seaquence depicts the transition from a film thickness of 35 to 25 nm. Data are from Bergeron.13 (a, top left) t* = 0.13 s,RH*= 23 pm; (b, bottom left) t* = 0.96 s, RH* = 57 pm; (c, top right) t* = 4 s, RH* = 115 pm; (d,bottom right) t* = 10 s, RH* = 195 pm. An = 2~1d 0.1 M
SDS
A
k
*
o
e: -250
-500 0
10
20
30
40
50
60
h+ (nm)
Figure3. Experimental disjoining pressure isotherm for 0.1 M aqueous SDS foam films (open circles).12J3The solid curve is a fit to eq 1 with the listed parameters.
dimensionless parameters dependent on the ratio of characteristic structure size to the Debye length. Finally, q is the gas/liquid interface charge density, and e is the bulk permittivity of the liquid solution. Figure 3 graphs as open circles direct experimental measurements of the disjoining pressure isotherm, II*(h*),for 0.1 M SDS foam films.12 The solid line in Figure 3 corresponds to eq 1with the listed parameters. Data in this figure arise from equilibrium ordering of bilayers and/ or micelles within the films.12 Using density functional theory, Laso calculates that the outer oscillatory branches
in Figure 3 can readily arise from equilibrium micelle layering within the film.23 A t a fixed capillary-suction pressure, Pc*,a thin liquid film attains metastable equilibrium when Pc* = II*. However, equilibrium is not possible a t all film thicknesses in Figure 3. S ~ h e l u d k oand ~ ~Vrij25use thermodynamic and fluid-mechanical analyses to establish that thin films cannot exist at thicknesses for which dII*/dh* > 0. Accordingly, only thickness branches along which dII*/ ah* < 0 are experimentally accessible, as c o n f i i e d by the experiments in Figure 3 and elsewhere.12-14 We assert that the unstable branches of the disjoining pressure isotherm in Figure 3 drive the stratified film thinning seen in Figures 1and 2. It is now possible to outline qualitatively the origin of hole-sheeting observed in foam films a t surfactant concentrations above the cmc. In our discussion only one oscillation in II*(h*)need be considered. Figure 4a shows in side view the top half of a symmetric liquid film undergoing an axisymmetric hole-sheeting transition. The final film thickness corresponding to equilibrium a t the applied capillary-suction pressure, Pc,is hl,as illustrated in Figure 4b. During thinning from the initial state, say l~ = 6 in Figure 4b, the film remains plane parallel while (23) Laso, M. Density Functional Theory of Micelle Structuring in Thin Micellar Liquid Films. Ph.D. thesis, University of California, Berkeley (in preparation, 1992). (24) Scheludko,A. Sur certaines particulariteades lamesmouB88uB88: 11. Stabilit6 cinbtique,Bpaisseur critique et Bpaisseurd’6quilibre. Roc. Koninkl. Ned. Akad. Wet. 1962, B65, 87-96. (25) Vrij,A. Possible Mechanism for theSpontaneowRuptureofThin, Free Liquid Films. Discuss. Faraday SOC.1966,42,23-33.
3030 Langmuir, Vol. 8, No. 12,1992 I--RH
Bergeron et al.
film thickness, h(t,r)
(t) I
h2l2
1
t
%
1-1
A
m t I \
1
I
\
/
I
+rh:g+
where
Ef
-
1 = &*K,
A = - *&*
R*’
&*
(“)1/2
(3)
29rq2K
and where the subscripts t and r denote differentiation. The first term in eq 2 represents the time derivative of the h local film thickness. The second term, scaled by A2, Figure 4. (a, top) Two-dimensional schematic of the growth of represents the local rate of thinning due to the overall an instabilityat the gas/liquid interface of a film thinning under driving force induced by Plateau-border suction. A is a constant capillary pressure and (b, bottom) corresponding defined as the ratio of the characteristic wavelength of a oscillatory disjoining pressure. disturbance, A?, to the film radius, R*. Liquid drainage described by this term ceases when Pc = II. The third its thickness diminishes uneventfully according to the term dictates local fluid motion due to gradients in Stefan-Reynolds model.26*27However, near ha the slope disjoining pressure. This term triggers the formation of of the disjoining pressure isotherm changes sign. Any a hole by growing a dimple from an initially corrugated infinitesimal corrugations in the interface shape, which film. Later, this same term produces a flat-bottomed hole were damped for h > ha, now begin to grow. As Figure 4b corresponding to a thinner film in equilibrium with the illustrates,the growingdimple does not necessarily rupture imposed capillary pressure. The fourth term reflects local the film because at the thickness of the dimple apex the surface tension forces having contributions from two slope of the disjoining pressure soon becomes negative. different curvatures. The transverse curvature, proporGrowth of the instability now slows, and eventually the tional to h,, acts to deter rim growth to large heights. equilibrium thickness hl emerges. Conversely, the circumferential curvature, proportional Meanwhile, excess liquid from the burrowing dimple to hJr, induces liquid flow away from the crater and drives must escape the film. The viscous resistance of a planethe rim outward. Note that for a perfectly flat film, eq parallel liquid film varies as h-3and prevents easy escape. 2 reduces to the Stefan-Reynolds thinning equation for Thus, the initiallydisplaced liquid builds a thick rim rather flow between plane-parallel plate^.^^^^' than flow out through the precursor film. This rim does Viscous resistance in the proposed evolution equation not build indefinitely because unfavorable transverse is evaluated from that of a homogeneous Newtonian liquid curvature forces (in the plane of the drawing) become of constant viscosity. For the rather thick films in Figures progressively larger. Finally, a radial pressure gradient 1and 2, L ~ s corroborates o ~ ~ this assumption theoretically arises from circumferential curvature forces (out of the and establishes that the effective film viscosity is well plane of the drawing) and drives the rim radially outward. approximated by that of the bulk micellar solution. For Drainage occurs by rim expansion and by thinning of the the hole-sheeting transition between common and Newton thicker portion of the film outside the rim due to the black films, the assumption of a constant bulk viscosity combined action of surface tension, disjoining, viscous, is more questionable. However, even this step transition and capillary-suction forces. The net result is expansion is likely driven by the positive slope of the disjoining of a black hole and the removal of a layer of fluid. Holepressure isotherm. Thus, hole-sheeting is apparently the sheeting must occur at each oscillation of the disjoining mechanism by which thin liquid films undergo spinodal pressure curve. Thus, from Figure 1 the underlying decomposition. disjoining pressure isotherm must exhibit at least three Boundary conditions for eq 2 include symmetry at the oscillations at thicknesses near 55, 45, and 35 nm, film center (i.e., hr(r=O) = 0), a flat film far away from the respectively. The experimental measurements in Figure center (i.e., hr(r=Rp>10) = 0), constant curvature at the 3 strikingly confirm this assertion. film edge (r = R),defined by Pc, and a local efflux at the Jim6nez-Laguna28 has quantified the hole-sheeting f i i edge due only to the overallcapillary-suctionpressure. process using a long wavelength evolution e q u a t i ~ n . ~ , ~The ~ initial condition is a sinusoidal disturbance applied Drainage of an axisymmetric, cylindrical film is predicted locally from r = 0 to a maximum of 1.5Xfwithan amplitude by the followingnonlinearexpression for the dimensionless not to exceed 0.10h and with a dimensionlesswavelength, Xf, estimated from the fastest growing disturbance in a (26) Stefan,J. Vershche fiber die Scheinbare AdhLion. Sitz. Math.linear stability analysis for a planar film g e o m e t ~ y . ~ ~ , ~ ~ Natur. Akad. Wiss. Wein. 1874,l (69), 713-735. Equation 2 is solved numerically by finite elements with (27) Reynolds, 0. On the Theory of Lubrication. R. SOC.London: Philos. Trans. 1886, 177, 157-234. Hermite cubic basis functions. Time stepping is by Crank(28) Jimhez-Laguna, A. I. Stability of Thin Liquid Films: Theory Nicholson and nonlinearities are iterated using Newton’s and Applicationto Foam Flow in PorousMedia. Ph.D. thesis, University 0
1
2
3
4
5
6
of California, Berkeley, 1991. (29) Atherton, R. W.; Homsy,G. M., On the Derivation of Evolution Equations for Interfacial Waves. Chem. Eng. Commun. 1976,2,55-77. (30)Gauglitz, P. A.; Radke, C. J. An Extended Evolution Equation for Liquid Film Breakup in Cylindrical Capillaries. Chem. Eng. Sci. 1988, 43 (7), 1457-1465.
(31) Jimenez, A. I.; Radke, C. J. Dynamic Stability of Foam Lamellae Flowing Througha Periodically Constricted Pore. Oil-Field Chemistry, Enhunced Recovery and Production Stimulation;ACS SympoeiumSeries 396; Borchardt, J. K., Yen, T. F., EMS.; American Chemical Society Washington, DC, 1989; Chapter 25 (Appendix A).
Langmuir, VoZ. 8, No. 12, 1992 3031
Hole Formation and Sheeting 4
3.5 h
Y
*‘
z
,
3
I
I I
f+
2.5 2
I II
hn.0,e/, t
Figure 6. Schematic of the growth of an azimuthal instability at the gadliquid interface of a film thinning under a constant capillary pressure. A liquid rim is formed around an expanding black hole and subsequently breaks apart into secondary fluid pockets.
1.5 1
0.5
0 0
1
3
2
4
5
r
Figure 5. Nucleation and early time sheeting of a black hole formed during the stratified thinning of a foam film modeled by eq 2. Parameters for the disjoining pressure isotherm of eq 1 are A H = l,B = 100, (Y = 1, and B = 1.
method. Details on the derivation of eq 2 and on the calculation are available elsewhere.28
Results and Discussion Figure 5 reports an illustrative result for hole-sheeting obtained ntimerically for a surfactant system characterized by a n(h)isotherm similar to that in Figure 3, but with AH= 1,B = 100,and a = = 1. The initial film thickness is ho = 2.3, and a disturbance of dimensionless amplitude €0 = 0.lh and wavelength Xf = 0.58 is applied. The first four time steps reveal the nucleation of a hole and the development of a liquid rim surrounding it, driven by the positive thickness gradient in the oscillatory disjoining pressure. Time steps five and six confirm that the hole is indeed flat bottomed at the equilibrium thickness hl= 1.3 corresponding to the applied capillary pressure of Pc = 15.0. At dimensionless times greater than about 5 the crater rim not only expands and grows but alsodevelops fascinating precursor foothills. During the very early stages of hole growth portrayed in Figure 5, the thickness of the surrounding film, h2, hardly diminishes. Although it is difficult to discern the detailed structure of the expanding rim from our experimental videomicrographs, essentially all the features observed in parts a-c of Figure 2 are reproduced by our calculations. Jim6nez-Laguna28is able to predict the correct trends for the hole radius growth using the experimental disjoining pressure isotherm, but only for early times up to about 0.1 s after the hole is visible (i.e., dimensionless times 5 Beyond this time scale, convergence of the film profile becomes increasinglymore difficult. Further, we have not conducted a thorough experimental study of hole expansion rates for films stabilized by differing surfactants, for films of differing radii, or for films thinning at different applied capillary pressures. Our proposed theory for holesheeting strictly applies to the initiation process. There is a second reason for not pursuing the twodimensional,axisymmetric hole growth calculationat later times. Namely, as noted earlier in parts c and d of Figure 2, the walls of the crater break apart in the circumferential direction into large fluid pockets. Our proposed explanation is highlighted in the three-dimensional perspective of Figure 6. Once a large rim forms around a black spot, it is vulnerable to capillary instabilities in the azimuthal direction. The three-dimensional shape of the crater rim resembles a ring of fluid. This ring, when large enough in diameter, apparently undergoes a breakup process not
unlike that of a cylindrical liquid thread.32 Thus, secondary bodies of fluid appear around the circumference of the hole rim (cf. Figure 2d). Governing forces in this case, however, are not limited only to local surface tension force, but consist of the combined action of disjoiningand curvature forces in addition to the overall thinning action of a capillary-suction pressure and a viscous resistance which escalates the thinner the film becomes. The resulting dynamic configuration is a rim that dramatically falls apart into deep pockets of fluid as sketched in Figure 6. Further, as the hole grows radially, satellib fluid pockets are continually generated at different rim locations. The younger, smaller pockets then translate along the rim toward the larger, older ones and coalesce. Smaller pockets possess relatively lower capillary pressures which drive them toward the larger, thicker ones. This behavior is akin to that observed when wetting particles are placed in proximity at a liquid/gas interface. Thus, our hydrodynamic model for the hole-sheeting thinning process provides a consistent picture both for hole formation and expansion and for rim breakup. We believe this to be an important aspect of any theory for stratified thinning.
Conclusions We have modeled the formation and early-time expansion of black spots or holes that form when foam films thin under constant capillary-suction pressure at surfactant concentrations above the cmc. The crucial driving force for hole-sheeting is an equilibriumdisjoining pressure which exhibits unstable thickness regions where dn*/dh* is positive and which is unique to the surfactant system. Nucleation and hole expansion during stratified film thinning are readily quantified by a two-dimensional, nonlinear hydrodynamic stability analysis under an oscillatory disjoiningpressure. Essentiallyall the early-time features of hole-sheeting are captured by the proposed model. Long-time events of rim breakup can similarly be explained but require a full three-dimensional stability analysis. Collapse of the expanding crater rim into satellite pockets of fluid is attributed to the growth of azimuthal capillary instabilities that develop after enough fluid has accumulated in a ring around the spreading black hole. Smaller satellite pockets are driven along the expanding rim to coalesce with larger ones due to gradients in capillary pressure.
Acknowledgment. This work was partially supported by the U.S. Department of Energy under Contract DE(32) Rayleigh, Lord. On the Instabilityof Jets. h o c . London Math. SOC.1879,10,4-13.
Bergeron et al.
3032 Langmuir, Vol. 8, No. 12, 1992
AC03-76SF00098 to the Lawrence Berkeley Laboratory of the University of California
B
Glossary
t
Hamaker constant, J eK3An*/2?rq2,dimensionless Hamaker constant dimensionlessmagnitude of oscillatory structural B force lamella thickness, m h* Kh*, dimensionless lamella thickness h capillary pressure, Pa Pc* Pc*e/2xq2,dimensionless capillary pressure Pc lamella surface charge density, C/m2 4 radial coordinate, m r* r r*/Xf*, dimensionless radial coordinate film radius, m R* R R*/Xf*, dimensionless film radius hole radius, m Rn* dimensionless hole radius Rn dimensionlessdistance away from the position of RP a liquid rim time, s t* t * 2 ~ q ~ / 3 j ~ t ~dimensionless ~xf*~, time t Greek Symbols ratio of characteristic film thickness to characterU istic micelle size AH* AH
EO tf K
x Xf*
Xr
A fi
n* n 0 U
dimensionless wavelength of the structural component to the oscillatory disjoiningpressure given by eq 1 bulk permittivity of the surfactant solution, C2/J m dimensionless disturbance amplitude K-l/Af*, film aspect ratio inverse Debye length, m-l wavelength of a disturbance on a free f i i , m ( u t / 2 ~ q ~ ~ )characteristic '/~, wavelength, m X/ hf*, dimensionless wavelength Xf*IR*, ratio of wavelength to f i i radius Newtonian viscosity of the bulk surfactant solution, Pa.8 disjoining pressure, Pa n*t/2uq2, dimensionless disjoining pressure, defiied in eq 1 azimuthal coordinate equilibrium surface tension of surfactant solution, N/m
Subscript 0 1 2
initial final equilibrium state film state far from instability
Superscript * dimensional.
