A New Analysis of Foam Coalescence - American Chemical Society

Langmuir 2000, 16, 3873-3883

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A New Analysis of Foam Coalescence: From Isolated Films to Three-Dimensional Foams David Monin, Alexandre Espert, and Annie Colin* Centre de Recherche Paul Pascal, Av. Dr. Schweitzer, 33600 Pessac, France Received December 16, 1998. In Final Form: January 3, 2000 The rupture of standing three-dimensional soap foams was examined. A foam of controlled size and water content was produced in a vertical column and the evolution of its liquid fraction under gravitational drainage was followed by electrical conductivity measurements. Using simple models to describe simultaneously foam geometry and liquid drainage (syneresis), we measured coalescence events in a bulk foam. Various surfactants were used, indicating two mechanisms for foam destruction. In one case, the rupture of soap films is induced by the increase of capillary pressure resulting from liquid drainage: foam breaks at the top of the column and its level goes down with a constant velocity. In the other case, films are not influenced by drainage and their breakage is randomly distributed throughout the foam, which coalesces homogeneously in space. These two mechanisms are interpreted with microscopic arguments, based on monolayer elasticity and disjoining pressure isotherms.

1. Introduction Foams are amazing objects and interest in them has increased considerably in the past decade, because of their wide range of applications. This interest is directly related to the existence of one medium dispersed in another one, yielding specific properties (such as foams used for soundproofing, low density foams used to fight oil fires, foams exhibiting a large active surface for use in flotation processes, and foams displaying specific rheological behaviors). Their structures, first addressed by Plateau and Kelvin in the last century, are not yet understood, although the rules to describe foam mechanical equilibrium are known. Knowledge about the structural evolution of foams is very important in the study of foam rheology, a field where we lack experimental studies to support theoretical models. The evolution of foams with time is a major problem, because foams are metastable systems: their destiny is rupture. The understanding of coarsening mechanisms is of great importance, from a scientific point of view, as well as for industrial applications. The stability of cosmetic emulsions or food foams, the destruction of foams used in enhanced oil recovery, and the inhibition of foaming properties in coating processes are some typical challenges accepted by industrial research. Foam coarsening is governed by two well-identified mechanisms. Ostwald ripening results from the diffusion of the dispersed phase through the continuous medium, driven by Laplace-Young pressure gradients. This phenomenon is slow (typically several hours and the size of bubbles increases as t1/2) and relevant for very fine foams (such as shaving creams) where bubbles are very small. For usual soap foams, Ostwald ripening can be neglected. The second mechanism of coarsening is simple rupture, when bubbles coalesce by breaking the barrier that separates them. This mechanism is still under investigation, and we propose a new approach to describe foam coalescence, in the particular case of dry polyhedrally shaped soap foams. A soap foam can be viewed as an assembly of two types of elementary bricks: films and Plateau borders (PBs). * To whom correspondence should be addressed.

Films are thin planar water lamellae, stabilized by surfactant molecules. They separate bubbles and define the structure of the foam. When a film breaks, foam reorganizes itself to relax stresses and reaches a new equilibrium state. According to Plateau’s rules for foam structure, films meet 3-fold and form pipes named PBs. These borders contain the whole liquid of the foam and meet 4-fold, so that a continuous network exists in the foam. This network, and the geometry of PBs, defines transport properties in the foam. Here we study transport properties to follow the evolution of the network, hence the coalescence of the foam. Soap foams made of various surfactants were studied. Monolayers and isolated thin liquid films have been characterized by the usual techniques. An apparatus was developed to study the evolution of soap foams and methods to measure foam coalescence are presented, using a hydrodynamic model for foam drainage and conductivity measurements for the analysis of the network of PBs. Two different rupture mechanisms are identified. Coalescence is either homogeneous and associated with a single probability of rupture, or controlled by a threshold pressure applied on the films, from which they break. These two mechanisms are interpreted from a microscopic point of view, according to disjoining pressure isotherms and viscoelastic properties of thin liquid soap films. 2. Experimental Section Surface Tension Measurements. Measurements were carried out at room temperature (22 ( 1 °C) in a Teflon trough housed in a special Plexiglas box with an opening for the tensiometer. The surface tension was measured with an openframe version of the Wilhelmy plate allowing us to avoid the wetting problems of a classical full plate.1 The rectangular open frame, made from a 0.19 mm diameter platinum wire, was attached to a force transducer (HBM Q11) mounted on a motor allowing it to be drawn away from the surface at a controlled constant rate. The equilibrium surface tension is quickly reached for the more concentrated solutions, whereas a longer time (from 10 min for short surfactant solutions up to several hours for polyelectrolyte systems) is required in the dilute domain. Thin Film Balance (TFB) Device. To study hydrodynamics and equilibrium of isolated films, a modified version of the porous (1) Mann, E. K. Thesis, University of Paris VI, Paris, 1992.

10.1021/la981733o CCC: $19.00 © 2000 American Chemical Society Published on Web 03/24/2000

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Figure 1. Measurement of thickness and disjoining pressure in a TFB apparatus. plate technique first developed by Mysels and Jones2 was used, allowing thin-film forces to be quantified by measuring the disjoining pressure isotherms. A schematic view of the device is presented in Figure 1. This modified version includes some improvements made by several authors.3-5 A thick liquid lens is formed in the center of a small hole drilled in a porous glass disk fused to a capillary tube or in a Sheludko cell.6 The cell is enclosed in a 200 cm3 hermetically sealed and temperature-controlled aluminum box, with the capillary tube exposed to a constant reference pressure. Our experimental setup enables one to measure simultaneously the thickness of the film and its disjoining pressure under a constant pressure applied in the box. The film thickness was determined by an interferometric method developed by Sheludko.6 Under the effect of a constant pressure difference ∆P between the box and the reference, two kinds of measurements were performed: (i) drainage and (ii) disjoining pressure isotherms. (i) By keeping the film radius constant (i.e., ∆P constant), and recording the reflected intensity versus time, we measured the temporal evolution of the film until it broke or reached an equilibrium state. These experiments were carried out in a Sheludko cell. (ii) When a flat horizontal liquid film was formed, its thickness h could be stabilized if the surface force per unit area balanced the external pressure applied. The disjoining pressure, related to the capillary pressure, is determined from the following equation:

γ πd(h) ) ∆P - Fghc + 2 rcap

(1)

where hc is the liquid height in the capillary tube above the film position, rcap is the capillary tube radius, γ is the surface tension, F is the liquid density, and g is the gravity constant. The available pressure ∆P ranges from 50 Pa to 50 kPa. The maximum imposed capillary pressure πdmax must not exceed the entry pressure for the porous disk. Therefore, disks with smaller pores are required for higher capillary pressures. The minimum pressure one has to impose to create a single film, referred to as πdmin, depends on the hole shape (its thickness and its radius) drilled in the porous medium. Some examples of πdmin values (approximately 50 Pa) are available in the literature.4 Foam Column. In this section, we recall the description of a dry foam and we present the experimental device that allows us to measure the liquid fraction. Description of a Dry Foam. Foam may be viewed as a network of pipes, the section of which can vary if solution accumulates in it or drains out of it. The network of these pipes

(2) Mysels, K.; Jones, M. N. Discuss. Faraday Soc. 1966, 42, 42-50. (3) Exerowa, D.; Sheludko, A. Chim.-Phys. 1971, 24, 47-50. (4) Bergeron, V. Thesis, University of California, Berkeley, 1993. (5) Espert, A. Thesis, University of Bordeaux I, Bordeaux, 1998. (6) Sheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391-464.

Figure 2. Schematic view of a plateau border (intersection of three films exerting the same force γL on the border). L is the length of the border and r the radius of the three cylinders that enclose the border. is governed by the Plateau rules for foam stability7 (each pipe is the junction of three films at 120°, and pipes meet 4-fold). Figure 2 defines the geometry of such a PB. One can describe a PB using two characteristic lengths L and r: L is the average length of a PB (L is also the average radius of a polyhedral film), 1/r is its curvature which defines the capillary pressure exerted on the film through the Laplace-Young equation. The cross section of such a PB is A ) C2r2, where C is a geometrical constant. Assuming that the volume of the films is negligible, the volume liquid fraction (the ratio of the liquid in the foam to the total volume of the foam) is defined as

Φliq )

NPBAL ) 2nA HS

(2)

where H is the height of the column, S is its section, NBP is the total number of PBs in the foam, and n is the number of PBs crossing a unit surface. This approximation is valid in our experiments. The liquid fraction is less than 1% which implies a radius of curvature r of 0.15 mm. For a film thickness (hfilm) of 10 nm and a film radius (Rfilms) of 1 mm, the volume of a PB is 100 times larger than the volume of one film:

Vfilms ) πRfilm2hfilm ≈ 3 × 10-14 m3 VPB ) AL ≈ C2r2Rfilm ≈ 4 × 10-12 m3 In the following, measurements of A will be needed. The relation among NPB, A, and L depends strongly upon the structure of the foam. For a monodisperse dry Kelvin’s foam, one obtains8 Kelvin Φliq )

3A A 3A ) 2 ≈ 2.44 2 2 R 2L x2 R x2

(3)

where R is the radius of a bubble that would have the same volume as the Kelvin’s cell. In our experiments, we measure this radius by taking some pictures of the foam on the Plexiglas plate of the cell. (We recall that foams are turbid and it is difficult to image the bulk of such samples.) Measurements of the Liquid Fraction. Figure 3 presents a schematic view of the device we used to measure the liquid fraction.9 The foam is produced by blowing nitrogen through a porous glass or a small capillary tube (1) into the soapy solution, until it fills completely the Plexiglas column (2). This foam can be wetted with the soapy solution by means of a peristaltic pump (7) Plateau J. A. F. Statique experimentale et the´ orique des liquides soumis aux seules forces mole´ culaires; Gauthier-Villars: Paris, 1873. (8) Hutzler, Weaire, D.; Verbist, G.; Peters, E. A. J. F. Adv. Chem. Phys. 1997, 102, 315 (see eq 10.1, p 356). (9) (a) Weaire, D.; Findlay, S.; Verbist, G. J. Phys.: Condens. Matter 1995, 7, L217-L222. (b) Verbist, G.; Weaire, D.; Kraynik, A. M. J. Phys.: Condens. Matter 1996, 8, 3715-3731.

Analysis of Foam Coalescence

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The conservation equation for an incompressible fluid leads to

1 dnAu Fg d dnA γc d )) f nA2 nxA A (4) dt 3 dz 3Kµ dz 2Fg dz

(

Figure 3. Experimental device for conductivity measurements in a bulk foam. (3) (Gilson Minipuls2) that injects liquid at the top of the column. An overflow pipe (4) is used to avoid accumulation of water in the reservoir: the limit between liquid and foam does not move (regardless the variation of hydrostatic pressure of the foam). All along the column are arranged 25 brass electrodes (5) connected to an LCR meter (6) (Hewlett-Packard 4284A) via a switch (7) (Hewlett-Packard 3488A). The whole system is controlled by a PC (8), using HPVEE4 software. The electric conductivity of foams has been studied theoretically by Lemlich,10 yielding a linear relation between the relative conductivity of a dry foam (σfoam/σsolution) and its liquid fraction (Φliq ) (Vliq/Vtot)). Corrections have been predicted and experimentally verified for wet foams (Φliq up to 20%).13 We use these corrected relations to determine Φliq. The conductivity (σsolution) is recorded in the reservoir and the conductivity (σfoam) is measured sequentially on each electrode, every 15 s. Materials. The anionic surfactant sodium 1-octanesulfonic (C8SO3Na) and octanol (C8OH) were purchased from SigmaAldrich and used as received. Miranol is a commercial zwitterionic surfactant supplied by Rhodia. AM/AMPS is an anionic polyelectrolyte made of acrylamide and acrylamidomethylpropanesulfonate, supplied by Floerger. Institut Franc¸ ais du Pe´trole donated some samples. The cationic dodecyltrimethylammonium bromide (DTAB) was obtained from Sigma-Aldrich, and recrystallized three times with 90:10 vol % ethyl acetate/ethanol mixtures. All solutions were prepared with distilled water (Millipore MilliQ ultrapure system).

3. Mode of Operation The problem of drainage of standing foams has been addressed widely.8,10-16 Following the work of Weaire and collaborators, we are going to recall the hydrodynamic model for drainage in a bulk foam. Two simple solutions of this model are obtained and used to define methods to measure foam coalescence. Drainage Model. Derivation of the Drainage Equation. In this description, a foam is viewed as a network of pipes, the cross section of which can vary if solution accumulates in it or drains out of it. The PBs are assumed to be randomly oriented. (10) Lemlich, R. J. Colloid Interface Sci. 1978, 64, 107-110. (11) Leonard R. A.; Lemlich R. AIChE J. 1965, 11(1), 18-25. (12) Gol′dfarb, I. I.; Kann, K. B.; Shreiber, I. R. Izv. Akad. Nauk SSSR 1988, 2, 103. (13) Phelan, R.; Weaire, D.; Peters, E. A. J. F.; Verbist, G. J. Phys.: Condens. Matter 1996, 8, L475-L482. (14) Bhakta, A.; Ruckestein, E. Langmuir 1995, 11, 1486. (15) Narsinham, G.; Ruckenstein, E. Foams: theory, measurements, and applications; Prud’homme, R. K., Khan, S. A., Eds.; Marcel Dekker: New York, 1996; Chapter 2. (16) Koehler, S. A.; Stone, H. A.; Brenner, M. P.; Eggers, J. Phys. Rev. E 1998, 58(2), 2097-2105.

)

where n is the mean number of PBs crossing the unit surface, A is the pipe cross section (A ) C2r2), and u is the average velocity of the liquid in a vertical pipe. The factor 3 results from the average for orientations of the PBs. The z axis is oriented downward. The flow of the liquid in a single PB involves the interplay of gravity, surface tension, and viscous forces. F, g, γ, and µ refer respectively to the density, gravitational acceleration, surface tension, and viscosity. K is a constant imposed by the geometry of the PB. For a vertical pipe, a numerical estimate of K was made by Peters17 (K ) 50). The factor f accounts for the effect of finite surface viscosity. Originally introduced by Leonard,11 it represents the mobility of the liquid-gas interface and is equal to 1 for rigid interfaces. It has been computed by Desai and Kumar and by Kraynik18,19 as a function of the inverse of the dimensionless surface viscosity µ(A1/2)/η5, where ηs is the surface viscosity. In our experiments µ ) 10-3 Pa‚s, ηs ) 10-7 kg‚s, and 4 × 10-10 m2 < A < 4 × 10-9 m2. According to ref 18, f varies between 4 and 8. We will neglect these variations19 and assume f ) 6. This equation has to be solved for the following boundary conditions:

At the top of the column

z)0

nAu/3 ) Q (5)

At the bottom of the column z)H

2nA ) 0.26 (6)

Q is the flux at the top of the foam imposed by the pump. In the following, the situation Q ) 0 will be referred to as free drainage, and Q different from zero will be referred to as forced drainage. The boundary condition at the bottom of the column assumes that the bubbles at the foam-liquid interface are arranged as a closely packed sphere. As shown by Weaire,8 the second-order term (capillary effects) is negligible until the liquid fraction reaches values as low as Φliq ≈ 10-5. In all of our experiments, this term is insignificant; we will neglect it. We will now present two simple solutions of the approximate following equation:

( )

Fgf d Φ2 dΦ )dt 6Kη dz n

(7)

For noncoalescing foams, and in the situation of forced drainage, a stationary and uniform state is reached. It links Φ with the flow Q imposed by the pump:

FgfΦ Q ) S 300ηn

(8)

As described by Findlay,20 the simultaneous measurements of Φ and Q allow the experimentalist to count the number of PBs, i.e., the number of bubbles. (17) Peters, E. A. J. F. Theoretical and experimental contributions to the understanding of foam drainage, M.S. Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1995. (18) Desai, D.; Kumar, R. Chem. Eng. Sci. 1982, 37, 1361-1370. (19) Kraynik, A. M. Sandia Report SAND83-0844. (20) Findlay, S. Thesis.

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For noncoalescing foams, and in the situation of free drainage, the solution of the drainage equation is wellapproximated by Kraynik’s solution:

Φ ) Φo Φ)

for t < td

150nzfη Fgt

for t > td

(9)

For coalescing foam and assuming that n depends only on time, a solution of the drainage equation in the situation of free drainage was proposed by the Trinity College group:21

for t < td

Φ ) Φo and

Φ)

150fη Fg

z

∫0

for t > td

td

du/n(u)

(10)

where td is defined by

Φo )

150fη Fg

z

∫0 du/n(u) td

Measurement Methods. In the following, three types of measurements are presented, which allows us to determine the coalescence events. Method a uses eq 10. A foam is produced by blowing nitrogen through a small capillary tube into the soapy solution, until it fills completely the Plexiglas column. This foam is wetted with the soapy solution by means of a peristaltic pump that injects liquid at the top of the column. A stationary state is reached where the liquid fraction is uniform in the entire column. At t ) 0, the pump is turned off. The foams drains. When the drying front reaches the region where measurements are made (i.e., for t > td), the liquid fraction begins to decrease. Equation 10 allows us to estimate the number of PBs as a function of time.

n(t) ) -

Φ Fg 150fηz d ln Φ dt

(11)

The use of method a is limited by two hypotheses. Equation 11 is valid only at the beginning of the process: f is assumed to be a constant in this analysis. Moreover, n has to be independent of the location in the column. Method b (or Findlay’s method) is the rewetting of the foam after a given time of free drainage. The preparation of the sample is the same as described for method a. At t ) 0, the pump is turned off. After a given time, the pump is turned on and the foam is rewetted. We set the same flux as that used during sample preparation. In this case, eq 8 leads us to

Φinit ∝ xninitQ

and

Φend ∝ xnendQ

so

τ)

( )

Φend n(0) - n(t) )1Φinit n(0)

2

(12)

where τ measures the number of broken films, n(0) is the (21) We thank the referee for bringing to our knowledge this reference.

Figure 4. Surface tension isotherms of C8SO3Na/C8OH solutions: open circles, pure sulfonate; full squares, 0.4 mM/L; open diamonds, 0.8 mM/L; full triangles, 1.65 mM/L; open triangles, 3.2 mM/L; hatched squares, 6.4 mM/L. Solutions represented by small symbols are turbid: concentrated solutions in octanol (c > 3.2 mM/L) correspond to the gray part of the graph.

initial number of PBs by unit surface, and n(t) is the number of PBs by unit surface at the time t. This method does not allow us a continuous measurement of τ. Method c consists of continuously wetting the foam: a forced drainage situation. This is the continuous version of method b, and the derivation of τ is the same. But b and c differ by hydrodynamic conditions: for c, the foam is always wet, whereas for b, the foam can get very dry. Methods b and c assume that the characteristic time of coalescence is small compared to the time needed to reach the stationary state. 4. Results Two systems which show different rupture mechanisms were studied. For each system, the results concerning the properties of monolayers, the behavior of isolated thin liquid films and the stability of bulk foams are presented. Mixtures of C8SO3Na and C8OH. The choice of this system corresponds to the following experimental observation: whereas a solution of pure C8SO3Na produces very unstable foams (10 s lifetime), the same surfactant with a very small amount of C8OH leads to stable foams characterized by a lifetime 2 orders of magnitude longer (around 20 min). These different behaviors may be explained by the evolution of viscoelastic properties of monolayers when surface active “impurities” are added to the surfactant. Such observations were reported in the literature by Jayalakshmi.22 Before we report the behavior of free suspended films and three-dimensional bulk foams, we present the influence of C8OH on surface tension of C8SO3Na solutions. In Figure 4, we report the surface tension isotherms of C8SO3Na/C8OH mixtures, versus the C8SO3Na concentration, for various C8OH concentrations. First, adding C8OH induces a large decrease of surface tension. Its value remains constant as the C8OH concentration reaches a threshold value between 3.1 and 6.2 mM/L. For C8SO3Na concentrations larger than the critical micellar concentration (cmc) of pure surfactant, a slight increase of surface tension is observed. This could be attributed to a surface exchange between C8OH and C8SO3Na, alcohol being solubilized into surfactant micelles. Accordingly, solutions with 6.2 and 12.4 mM/L C8OH are turbid, the amount of surfactant being too small to solubilize all of the alcohol. (22) Jayalakshmi, Y.; Langevin, D. J. Colloid Interface Sci. 1997, 194, 22-30.


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Table 1. Area Per Sulfonate Molecule as a Function of the Bulk Sulfonate Concentration

Table 2. Area per Octanol Molecule for a Given Sulfonate Concentration at the cmc of the Solution

CC8SO3Na

cmc

3 cmc/4

cmc/2

cmc/4

cmc/10

CC8SO3Na

cmc/2

cmc/4

cmc/10

cmc/40

AC8SO3-(Å2/head) Gibbs (mN/m)

58 66

61 58

68 48

82 33

113 17

AC8OH (Å2/head)

38

35

33

27

In the following, we focus only on foams made of (sub)micellar solutions, where alcohol is dilute enough to be considered as an impurity (large symbols in Figure 4). For the pure sulfonate solution, the Gibbs equation23 allows us to write

Table 3. Area per Sulfonate Molecule and Gibbs Elasticity for a Given Octanol Concentration; the Concentration of Sulfonate is cmc/2 CC8OH (mM/L) (Å2/head)

A Gibbs (mN/m) C8SO3-

0

0.4

0.8

1.65

68 48

164 67

210 77

224 126

dγ ) -RTΓC8SO3-d ln FC8SO3-cC8SO3- RTΓNa+d ln FNa+cNa+ (13a) To maintain electrical neutrality at the interface, the two surface coverages ΓC8SO3- and ΓNa+ are equal. Activity coefficients FC8SO3- and FNa+ are calculated using the Debye Hu¨ckel model.24 They vary from 0.94 to 0.96 and we will take them as equal to 1 in the following:

dγ ) -2RTΓC8SO3- d ln cC8SO3-

(13b)

Using a second-order polynomial fit of the evolution of the surface tension as a function of the logarithm of the sulfonate concentration,25 we deduce ΓC8SO3- as a function of the sulfonate concentration. The values are reported in Table 1. The area per octylsulfonate head at the cmc is equal to 58 Å2. This value is close to that of undecylsulfonate 54 Å2 26 or sodium dodecyl sulfate 45 ( 5 Å2.25 From these measurements, we can also estimate the Gibbs elasticity of the monolayer:

Gibbs ) -

dγ d ln ΓC8SO3Na

The values are reported in Table 1. For a mixed monolayer, the Gibbs equation allows us to write

dγ ) -2RTΓC8SO3- d ln cC8SO3- RTΓC8OHd ln cC8OH (14) We calculate the area per surfactant head A ) 1/ΓC8SO3of C8SO3Na by using the following procedure. We fix the bulk concentration of octanol and we measure the evolution of the surface tension as a function of the bulk concentration of sulfonate. Under these experimental conditions, eq 14 becomes

dγ ) -2RTΓC8SO3- d ln cC8SO3as cC8OH ) constant (15) This allows us to estimate ΓC8SO3- for a fixed concentration of octanol. The same procedure is followed to determine the area per surfactant head of octanol ΓC8OH. In this case, the bulk concentration of sulfonate is fixed and we have

dγ ) -RTΓC8OHd ln cC8OH

(16)

Equations 15 and 16 are applied at the cmc of the solutions. This method does not allow us to measure the surface coverages after the cmc where the bulk chemical potential of the components vary. We recall that the most (23) Adamson, A. W. In Physical chemistry of surfaces, 3rd ed.; John Wiley & Sons: New York, 1976; paragraph II.8, pp 68-76. (24) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777-1785. (25) Rehfeld, S. J. J. Phys. Chem. 1967, 71(9), 738-745. (26) Voorst Vader, F. Trans. Faraday. Soc 1959, 56, 1067-1077.

Figure 5. Temporal evolution of the thickness of isolated films of C8SO3Na/C8OH solutions in TFB: open circles, cmc + 2%; full squares, 3/4 cmc + 2%; open diamonds, cmc + 4%; open triangles, cmc + 8%; full triangles, cmc/2 + 2%. Lines are exponential fits.

reliable way to measure areas is by neutron reflection. The measurements are reported in Tables 2 and 3. Two main results are indicated. The area per molecule of C8OH depends slightly on the concentration of surfactant, whereas a large increase in the area per molecule of C8SO3Na is computed when C8OH is added to the solution. This reveals that C8OH molecules replace C8SO3Na molecules in the interface, for a submicellar solution, even if the amount of alcohol in solution is very small. Such behavior has already been reported for SDS/dodecanol mixture.27,28,31 The formalism presented by Joos29 for the elasticity of mixed monolayer allows us to calculate the evolution of the Gibbs elasticity when octanol is added.

Gibbs ) -(ΓC8SO3- + ΓNa+ + ΓC8OH) × dγ (17) d(ΓC8SO3- + ΓNa+ + ΓC8OH) This derivative is estimated by using finite difference. Table 3 reports the estimate of the Gibbs elasticity. The elasticity of a pure monolayer of octanol is equal to 41 mN/m 30 for a concentration of 0.8 mM/L in octanol and that of a pure monolayer of sulfonate is equal to 48 mN/m. We show that the mixed monolayer is much more elastic. As in the case of the SDS/dodecanol mixture, a slight amount of alcohol induces a large increase of the Gibbs elasticity for submicellar solutions. Figure 5 shows the evolution of the film thickness as a function of time for various C8SO3Na/C8OH solutions (27) Staples, E. J.; Thompson, L.; Tucker, I.; Penfold, J. Langmuir 1994, 10, 4136-4141. (28) Petrov, P.; Joos, P. J. Colloid Interface Sci. 1996, 182, 214-219. (29) Petrov, P.; Joos, P. J. Colloid Interface Sci. 1996, 182, 179-189 (30) Lunkenheimer, K.; Serrien, G.; Joos, P. J. Colloid Interface Sci. 1990, 134, 407-411. (31) Baets, P. J. M.; Stein, H. N. J. Colloid Interface Sci. 1994, 162, 402-411.

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Table 4. Distribution of Time Drainage for Isolated Films of Sulfonate/Octanol Mixtures CC8SO3CC8OH 2% 4% 8%

cmc

3/4 cmc

cmc/2

τ ) 9.6 s σ ) 3.3 tlife ) hc τ ) 14.7 s σ ) 5.5 tlife ) hc τ ) 15.2 s σ ) 5.2 tlife ) hc

τ ) 17.7 s σ ) 7.8 tlife ) minb τ ) 13.2 s σ ) 5.6 tlife ) minb τ ) 9.3 s σ ) 1.8 tlife ) hc

τ ) 11.3 s σ ) 4.6 tlife ) seca τ ) 15.7 s σ ) 5.1 tlife ) minb τ ) 12.8 s σ ) 2.7 tlife ) minb

a As soon as the NBF appears, the films breaks; bthe NBF grows and covers the whole surface of the film, which breaks after a few minutes; cwe did not observe spontaneous rupture within 1 h).

(for each solution, the concentration of C8SO3Na is given as a fraction of cmc and the concentration of C8OH is given by the ratio of bulk concentrations: [C8OH]/[C8SO3Na]). The thickness decreases exponentially from 200 to 30 nm quickly. Below 30 nm, films undergo a thickness transition leading to a Newton black film (NBF) state. In Table 4, we report the average characteristic time of drainage (τdrainage) and its standard deviation (σ) computed over 2030 measurements for each solution. These results show no clear dependence of the characteristic drainage time with the composition of the film or with the Gibbs elasticity as in previous studies.31 Furthermore, this time does not depend on the film radius (experiments were performed on films with radii ranging from 0.2 to 0.5 mm). Whatever the composition and size of the film, the final thickness is obtained within 1 min. These data (especially the exponential decay) are not expected from the models of drainage of thin liquid films. These theories32 always consider films thinning in an axisymmetric flow and yield power dependencies on time and radius for the film thickness. In our experiments, drainage is obviously nonsymmetric and no model can account for this yet. Nevertheless, new theoretical approaches are proposed to consider these features.33 Table 4 also shows an indicative lifetime of the films in their final state, in the TFB device. Without C8OH, films break immediately, whereas adding C8OH stabilizes the films. The disjoining pressure measurements were performed under a small initial capillary pressure that drives drainage (typically 50 Pa, i.e., the Πdmin value). When the thickness reaches 30 nm (common black film (CBF)), black spots corresponding to thinner domains (10 nm, NBF) appear and grow, leading to a new homogeneous thickness and an equilibrium state. The transition from CBF to NBF is associated with an energetic barrier in the disjoining pressure isotherm. Because we are not able to stabilize CBFs, the height of this barrier does not exceed the applied capillary pressure. Because of limited experimental accuracy, the variation of thickness in the NBF state could not be measured as the capillary pressure increased. In conclusion, disjoining pressure isotherms of C8SO3Na/ C8OH mixtures have a hard-sphere-type form. Using the three methods described in the drainage model, we measure the coalescence of C8SO3Na/C8OH bulk foams. Figure 6 shows the evolution of liquid fraction as a function of time for each method. By rewetting the foam after 10 min, we measure τ with method b (equation 12). (32) Ivanov, I. B.; Dimitrov, D. S. In Thin liquid films, fundamentals and applications; Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988; Chapter 7. (33) Manev, E.; Tsekov, R.; Radoev, B. J. Dispersion Sci. Technol. 1997, 18, 769-788.

Figure 6. Temporal evolution of the liquid fraction of a C8SO3Na/C8OH foam (cmc + 8%) in three experiments illustrating the three methods of measurement: open circles, method a; full squares, method b; open diamonds, method c. The measurements are performed at h ) 16 cm. The initial radius of the bubbles is 0.5 mm.

Figure 7. Compatibility of the three methods used to measure rupture ratio fraction of a C8SO3Na/C8OH foam (cmc + 8%): open circles, method a, full diamond, method b; crosses, method c. The measurements are performed at h ) 16 cm. The initial radius of the bubbles is 0.5 mm.

Figure 8. Rupture ratio of C8SO3Na/C8OH foams versus time, for two systems: open symbols, cmc + 8%; closed inverted triangles, cmc + 2%; and various positions: circles, bottom (h ) 13 cm); diamonds, center (h ) 25 cm); squares, top (h ) 37 cm). The initial radius of the bubbles is 0.5 mm.

The second experiment corresponds to forced drainage (method c), allowing τ to be determined continuously. Finally, because these two analyses show that the number of coalescence events is independent of the position, we also use method a. Figure 7 shows that the three rupture ratios are in good agreement. Figure 8 shows the rupture ratio of the same foam measured in three different positions in the column. These curves are superimposed,


Langmuir, Vol. 16, No. 8, 2000 3879

Figure 9. An unexpected result on a Miranol (8 g/L) foam showing the independence toward hydrodynamics: full squares, method c; open circles, method b twice. The initial radius of bubbles is 0.35 mm. The measurements were performed at h ) 16 cm.

Figure 10. Surface tension isotherms of AM/AMPS/DTAB mixtures (from ref 34) defining the regime studied with bulk foams: open circles, pure DTAB; full circles, DTAB and 746 ppm of AM/AMPS.

demonstrating that rupture occurs homogeneously in the foam, regardless of hydrodynamics. Finally, the rupture ratio was measured for solutions cmc + 8% and cmc + 2%. As shown in Figure 8, for the same position in the column, the higher the C8OH concentration, the more stable the foam, as previously observed for isolated films. Similar results were obtained with a commercial surfactant, Miranol, supplied by Rhodia. For isolated films, drainage is quick and we observe the same hard-spheretype disjoining pressure isotherm. The height of the energetic barrier between CBF and NBF is once again too small to be measured. In bulk foams, the mechanism of coalescence is also homogeneous. In Figure 9 reports two drainage experiments. A Miranol foam is allowed to drain and is rewetted two times (method b). The evolution of its liquid fraction is the same as for continuous forced drainage (method c). This complementary result indicates the independence of coalescence toward hydrodynamics. These two systems indicate a new mechanism for coalescence in soap foams, characterized by a homogeneous destruction, which is not influenced by the drainage of liquid from the cellular material. In both cases, the stability of bulk foams is governed by the behavior of NBFs. Investigations of C8SO3Na/C8OH isolated films show that their stability depends strongly on the quantity of fatty alcohol in solution, demonstrating the importance of viscoelastic parameters of monolayers for foam coalescence. Mixtures of AMPS and DTAB. Solutions of polyelectrolyte and surfactant are of great interest: because of specific interactions (such as electrostatic and steric), colloid properties can be easily modified. Special interest was given by Asnacios and co-workers to solutions of AM/ AMPS and DTAB.34,35 This leads us to analyze the role of surface properties (such as surface forces) on foam stability. AM/AMPS is an anionic polyelectrolyte available in various molecular weights and degrees of charge (we use here two samples: MW ) 2.2 or 0.4 M with a degree of charge 0.25). For a concentration of AM/AMPS less than 36.2 mM/L (in monomer concentration), solutions are not surface active and do not foam. We first present results obtained with the longer polyelectrolyte (MW ) 2.2 M). Figure 10 (from ref 34)

shows the surface tension as a function of the surfactant concentration, the polyelectrolyte concentration being fixed. For CDTAB below the cmc, two different regimes have been identified: the more dilute DTAB solutions reveal a strong decrease of surface tension, whereas after the critical aggregation concentration (cac), the surface tension remains constant. According to the Gibbs adsorption model, authors of these measurements have demonstrated a synergetic effect on surface tension, induced by the formation of complexes driven by electrostatic and hydrophobic interactions. Below the cac, surface tension and viscosity measurements reveal that surfaces are covered by complexes, but no effect is seen in bulk properties. Above cac, these complexes aggregate and a phase separation occurs.34 Because DTAB foams are very unstable, our interest is mainly focused on AM/AMPS/DTAB solutions (one decade below cac, a very dilute DTAB solution) to examine the effect of the anionic polyelectrolyte on foam stability, when the synergetic effect is relevant and the solution is monophasic. Thin liquid films made with these mixtures have been studied, showing original structuration forces.35 Involving the structure of the bulk solution, the spatial period of the oscillatory forces is linked to the correlation length of semidilute polyelectrolyte solutions. This analysis is clearly emphasized by independent small-angle neutron scattering (SANS) measurements on various systems.36 The solutions used for this study contain only 0.05 mM/L DTAB. This concentration is too small to form stable foams, even in the presence of polyelectrolyte, because the total area of films produced in bulk foam measurement can reach 1 m2. For that reason we used more concentrated solutions. The minimum amount of DTAB required to form foams is 0.25 mM/L. Figure 11 presents one drainage experiment, showing three important points: drainage first follows Kraynik’s solution and the liquid fraction decreases as 1/t (see eq 9). Then the evolution of liquid fraction becomes slower, until the foam breaks. Two different positions in the column are presented. The foam breaks quickly at the top and breaking propagates: a rupture front is observed, and the whole column of foam disappears within 10 min. Conversely, in forced drainage, the foam remains stable for a much longer time (Figure 12) and breaks in a homogeneous way.

(34) Asnacios, A.; Langevin, D.; Argillier, J. F. Macromolecules 1996, 29, 7412-7417. (35) Asnacios, A.; Espert, A.; Colin, A.; Langevin, D. Phys. Rev. Lett. 1997, 78, 4974-4977.

(36) Espert, A.; Klitzing, v. R.; Poulin, P.; Colin, A.; Zana, R.; Langevin, D. Langmuir 1998, 14, 4251-4260.

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Figure 11. Temporal evolution of the liquid fraction of an AM/AMPS/DTAB foam (MW ) 2.2 M, 750 ppm AM/AMPS, 0.25 mM/L DTAB) at two positions in the column (full squares h ) 40 cm, open circles h ) 30 cm).The initial radius of bubbles is 1.65 mm.

Figure 14. Influence of Cpoly on the evolution of the liquid fraction versus time (MW ) 0.4 M, 6 mM/L DTAB): open circles, 2000 ppm AM/AMPS; full squares, 4000 ppm AM/AMPS. The experiments are performed at h ) 30 cm. The initial radius of the bubbles is 1.65 mm.

Figure 12. Forced drainage for an AM/AMPS/DTAB foam (Mw ) 2.2M, 750 ppm AM/AMPS, 0.25 mM/L DTAB) at two positions in the column (full squares h ) 40 cm, open circles h ) 30 cm). The initial radius of bubbles is 1.65 mm.

surfactant concentrations. By varying CDTAB, results similar to those for the longer polyelectrolyte are obtained (the lifetime increases when CDTAB decreases and the fronts of rupture occur at different liquid fraction). Figure 14 shows the liquid fraction versus time for two polyelectrolyte concentrations at CDTAB ) 6 mM/L. Although a rupture front and a slowing of drainage are observed, a new behavior is clearly measured at a shorter time, characterized by a deviation from Kraynik’s solution. Such a deviation reveals homogeneous rupture. Simultaneously, both independent mechanisms of rupture are observed. Foams made of AM/AMPS/DTAB solutions emphasize a new rupture mechanism characterized by a front rupture. Our investigations on the smaller polyelectrolyte show that this mechanism can be superimposed with the homogeneous mechanism described for C8SO3Na/C8OH solutions. This behavior was also observed with simple surfactants, such as SDS. 6. Discussion

Figure 13. Influence of CDTAB on the evolution of liquid fraction versus time (MW ) 2.2 M, 750 ppm AM/AMPS): open circles, 1 mM/L; open triangles, 0.4 mM/L; full circles, 0.25 mM/L. The experiments are performed at h ) 30 cm. The initial radius of the bubbles is 1.65 mm.

Figure 13 presents three experiments with various DTAB concentrations. The behavior of the three curves is very similar, revealing again the points described above. Two small differences are observed: the liquid fraction for which foam breaks varies slightly and the greater CDTAB, the shorter the lifetime of the foam. With the smaller polyelectrolyte (MW ) 0.4 M), we investigated the influence of both polyelectrolyte and

We now discuss several aspects of the results presented in the preceding sections. The two mechanisms of rupture for bulk foams are analyzed and interpreted with the properties of isolated films. Homogeneous Rupture. Experiments on C8SO3Na/ C8OH foams have shown three elements indicating that coalescence is homogeneous (nondependence of rupture toward position in the foam and hydrodynamic conditions). To quantify coalescence events, a basic model previously developed in the study of concentrated emulsions37 is used: the probability of rupture of a film depends linearly on its area. Thus, the number of rupture events during dt is dN ) 1/2πD2Nωdt, where N is the total number of bubbles, D is their average diameter, ω is a probability of rupture per unit time and unit surface. Each film is involved in two bubbles, yielding the coefficient 1/2, and in this crude model, we suppose that the whole surface of bubbles (estimated as if bubbles were spherical) is a film (which means that the foam is polyhedrally shaped and that the “diameter” of PBs is much smaller than the diameter of bubbles). As the volume of foam is kept constant (the height of the column is constant), ND3 ) Cte, so dN/N ) -3[dD/D]. Finally, dD/D3 ) -(π/6)ωdt, yielding 1/D2 ) 1/D02 - (π/3)ωt, where D0 is the initial diameter of bubbles. Coming back to the number of PBs measured (37) Deminie`re, B.; Colin, A.; Leal-Calderon, F.; Bibette, J. In Modern aspects of emulsion science; Binks, B. P., Ed.; The Royal Society of Chemistry: London, 1998; Chapter 8.


Langmuir, Vol. 16, No. 8, 2000 3881 Table 6. Theoretical and Experimental Velocities of Rupture Fronts

sample 750 ppm 0.25 mM DTAB 750 ppm 0.25 mM DTAB 750 ppm 1 mM DTAB 750 ppm 1 mM DTAB

Figure 15. Experimental evidence for a rupture front in an AM/AMPS foam (MW ) 2.2 M, 750 ppm AM/AMPS): full circles, bottom (h ) 10 cm); open squares, (h ) 18 cm); full diamonds, (h ) 26 cm); open diamonds, top (h ) 42 cm). The initial radius of bubbles is 1.65 mm. Table 5. Threshold Pressures in Foams Obtained from AM/AMPS/DTAB Solutions CDTAB (mM)

Φthres

R (mm)

σ (mN/m)

Prupt (Pa)

0.25 0.4 1

0.8% 1.2% 1%

1.65 1.15 1.05

51 48 46

201 221 255

experimentally, n ) 31/2/D h 2 ) n0 - (π/31/2)ωt. This model provides for a linear dependence of n with time. Figure 8 obviously demonstrates this point, showing also that the probability of rupture ω of a single film is higher for a lower concentration of alcohol, i.e., a lower surface elasticity. The microscopic and macroscopic behaviors show the same dependence of foam stability on surface viscoelastic parameters. The homogeneous mechanism must be universal, because it is associated with the fundamental metastability of thin liquid films. However, it can be very slow and a quicker mechanism can overcome homogeneous rupture. Last, we have shown that drainage of thin liquid films occurs on a time scale (1 min) much shorter than rupture of bulk foam (20 min), demonstrating the independence of rupture process toward drainage of isolated films. Front Rupture. As described above, a front of rupture is observed in AM/AMPS/DTAB foams. Figure 15 presents the liquid fraction evolution versus time, for various positions in the column. We observe that the rupture of the foam always arises at the same liquid fraction, in the whole column. According to the geometrical model from Figure 2, we deduce the value of the capillary pressure applied on single films:

Prupt )

γ ) r

x

γC 3 x2 RxΦthres

(18)

Because the foam is monodisperse, this capillary pressure appears as a threshold value, at which all films break. Table 4 shows the threshold pressures obtained in foams made with the longer polyelectrolyte (CPoly ) 750 ppm): threshold pressure increases slightly with DTAB concentration. This point emphasizes a great difference for this system between microscopic and macroscopic behavior. Bulk foams of AM/AMPS/DTAB break for relatively small capillary pressures, whereas isolated films can support much higher pressures (up to 5000 Pa) and break under conditions that are really not reproducible (Table 5).

R (mm) η/ηwater Φthres

calculated experimental velocity velocity (mm/s) (mm/s)

1.65

10.8

1.2%

0.56

0.55

1.65

9

0.8%

0.49

0.34

1.65

4

1%

1.26

1.2

1.05

3.5

1.2%

0.69

0.7

To support our assumptions, a forced drainage measurement (method c) is carried out with a high enough continuous flow, so that the capillary pressure remains smaller than the measured threshold pressure (see Figure 12). This experiment demonstrates the existence of a threshold, and also reveals that homogeneous rupture (which was obscured by the quicker front rupture) occurs slowly. If the threshold pressure is a relevant parameter, the time needed to reach it depends strongly of the solution viscosity, according to eq 4, where η governs the time scale of evolution. Because the viscosity is related to electrostatic and hydrophobic interactions between DTAB and polyelectrolyte, an increase of the DTAB concentration with a fixed AM/AMPS concentration leads to a decrease in bulk viscosity.34 Although adding DTAB increases the threshold pressure, this pressure is reached quicker because of the decrease of viscosity. The balance between these two phenomena characterizes the evolution of foam lifetime with DTAB concentration. In our case, the effect of viscosity is more important, so that the more stable foam is made of films breaking from the smaller capillary pressure. The front velocity is given by v ) [Fgf/(50η)][21/2/9]· R2Φthres. This value is exactly the velocity of the liquid in a PB located at the top of the foam: it is the expression of eq 4 where the geometry of the pipe is expressed in terms of threshold pressure and bubble size. Table 6 shows the measurements of v and the calculations for η, R, and Φthres for four foams: two systems were used twice with some variations in the values of parameters. Viscosity was measured in a Ubbelohde viscosimeter before and after the series of experiments, and the decrease may be attributed to dilution and introduction of ionic impurities. An accurate determination of sizes and especially threshold pressures is difficult and these results should be regarded as a crude comparison. This good agreement confirms the existence of a critical liquid fraction, hence a threshold capillary pressure. For these general systems, two rupture mechanisms were identified: a universal mechanism (C8SO3Na/C8OH, Miranol, MW ) 0.4) leads to homogeneous rupture, and a borderline case characterized by a front (MW ) 0.4 and 2.2). According to the fundamental metastability of liquid film, and the role of fluctuations of liquid surfaces, we propose to relate the macroscopic behavior of foams to microscopic properties of films. Microscopic Interpretation of Bulk Foam Rupture. A main step in the description of thin liquid film physics was achieved with the Derjaguin-LandauVerwey-Overbeek (DLVO) theory,38,39 which expresses the equilibrium of two facing rigid flat interfaces in terms (38) Derjaguin, B.; Landau, L. Acta Physiochim. (USSR) 1941, 14, 633. (39) Verwey, E. J. W.; Overbeek, J. T. G. Theory of stability of lyophobic colloids; Elsevier: Amsterdam, 1948.

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Figure 17. Schematic view of density and surface fluctuations in a thin liquid film.

Figure 16. Maxwell construction leading to a threshold pressure. Disjoining pressure isotherm of AM/AMPS/DTAB (MW ) 2.2 M, 750 ppm AM/AMPS, 1 mM/L of DTAB) solution used for making bulk foams: full diamonds, measurements; full line, DLVO fit (Hamaker constant ) 10-20 J, Debye length ) 13.6 nm, P0 ) 4000 Pa) dotted line, steric repulsion.

of molecular interactions. The thickness of the film is then determined by the balance between the external pressure applied on the film and the disjoining pressure. This pressure results from various molecular interactions. van der Waals and electrostatic forces are present in every system, and specific interactions can be superimposed (such as steric repulsion, depletion, or solvation effects).40 A typical DLVO isotherm is shown in Figure 16. This is a thermodynamic description of the film, in which the rupture of the film is not considered. In this respect, we may say that a film is stable. Four regions can be defined. States (1′) and (1) are stable, the thicker one (typically 20-30 nm) is called CBF state, the latter being NBF state (smaller than 10 nm, as a reference to the original work of Newton). Region (2) is a metastable state, defining an energetic barrier that should be overcome to reach the NBF state. Last, region (3) is an unstable state. We will refer to this generic isotherm in the following discussion. An important omission in the DLVO theory is the role of surface fluctuations. For liquid interfaces, an increase in the height of the energetic barrier does not always promote stability:36 the role of fluctuations in thin liquid film rupture may also be significant. Figure 17 represents a schematic view of spatial and density fluctuations in a thin liquid film. Previous theoretical studies considering the hole nucleation process,41-45 the kinetics of spatial corrugations,41 and short-range interactions within the monolayers45 indicate that the influence of fluctuation is mainly (but not only) governed by surface Gibbs elasticity. This influence has been shown experimentally.46 Let us analyze first the rupture of an isolated film stabilized in its thinner stable state (generally an NBF, state (1)). The interaction responsible for this stabilization is a short-range repulsive force. Because this repulsive potential is high, only density fluctuations can induce the formation of domains where this force disappears, leading eventually to the rupture of the film (we do not consider any kinetic effect which could inhibit rupture). (40) Israelachvili, J. In Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992; paragraph 13.4. (41) de Gennes, P. G. Private communication. (42) Kashchiev, D.; Exerowa, D. J. Colloid Interface Sci. 1980, 77, 501-511. (43) De Vries, A. J. Rec. Trav. Chim. 1958, 77, 383-441. (44) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23-33. (45) Exerowa, D.; Kruglyakov, P. M. Foam and foam films; Elsevier: Amsterdam, 1998. (46) Bergeron, V. Langmuir 1997, 13, 3474-3482.

Regarding a bulk foam, where all the films are in the same state (NBF), fluctuations operate identically in the whole foam. In this respect, a homogeneous distribution of rupture events is expected. This situation corresponds to that observed with C8SO3Na/C8OH and Miranol dispersions. In both cases, films quickly reach the NBF state, and the rupture of foam is homogeneous. The evidence of the role of Gibbs elasticity is clearly emphasized by varying the fatty alcohol concentration. We observe that the rupture frequency decreases as C8OH concentration increases, corresponding to an increase of the surface Gibbs elasticity. In conclusion, C8SO3Na/C8OH and Miranol foams show similar behavior of film ruptures in both microscopic (TFB) and macroscopic experiments. The second scenario described by a front in threedimensional foams is characterized by a great difference in the behavior of isolated films and foams: single liquid films can be stabilized at pressures of 1000 Pa, whereas eq 18 leads to an estimation of much smaller threshold pressure. According to previous results,35,36,47 the branch (1′) of the isotherm can be fitted with a classical DLVO potential (Figure 16). The total isotherm is completed by a short-range repulsion related to the existence of an adsorbed complex (this complex was evidenced by ellipsometric measurements, its thickness is roughly estimated to be 2 nm,34 but we never obtained stable films in state (1)). Let us consider a foam made of liquid films stabilized in area (1′). In a three-dimensional foam, due to drainage through PBs, the capillary pressure increases, leading to a thinning of the film. With a classical Maxwell construction,48 we define a limiting pressure (see Figure 16) for which films undergo a phase transition from stable state (1′) to stable state (1). Two possibilities are expected. Either state (1) is stable over a long time (longer than the time needed to reach the limit pressure), leading to the homogeneous rupture mechanism described above. Or state (1) breaks quickly under the effect of fluctuations and the limit pressure defines the threshold pressure. Our results corresponds to the second assumption and are confirmed by the following estimate. With the parameters obtained in the DLVO fit of the disjoining pressure isotherm (Hamaker constant is 10-20 J, the Debye length is 13.6 nm, and the prefactor of the electrostatic term is 4000 Pa), and considering that steric interactions are represented by a hard-wall repulsion at thickness hsteric, we computed the threshold pressure given by the Maxwell construction: 100 Pa for hsteric ) 2.6 nm. This is in good agreement with the estimated thickness of surface aggregates and confirms that a small threshold pressure can be easily obtained in agreement with microscopic properties. However, this tempting description does not enable us to explain the stability of isolated films in TFB experi(47) Klitzing, v. R.; Espert, A.; Asnacios, A.; Hellweg, T.; Colin, A.; Langevin, D. Colloids Surf. 1999, in press. (48) Callen, H. B. In Thermodynamics, 6th ed.; John Wiley & Sons: New York, 1966; pp 146-154.


ments. Two experimental arguments justify this point. Owing to efficient thermostating and use of an antivibrational device, disturbances are dampened thereby inhibiting the fluctuations. In addition, the capillary pressure is increased step by step, controlled, and kept constant during the thickness measurement (the evolution of the system is quasi-static), whereas in a bulk foam, the increase of capillary pressure is imposed by hydrodynamic conditions. The care used to determine disjoining pressure isotherms enabled us to explore metastable states of region (2). In contrast to the systems leading to a homogeneous rupture mechanism, AM/AMPS/DTAB foams show a great difference in microscopic and macroscopic behaviors. 7. Conclusion An experimental study of soap foams coalescence was carried out. Conductivity measurements allow us to estimate continuously the state of the PB network that constitutes the foam. By means of a simple model for drainage of the liquid in one PB, we are able to count the number of bubbles in the foam, hence to follow foam coalescence. Various surfactants were studied and there is evidence for two limit rupture mechanisms. The foam can coalesce homogeneously, if rupture events are equally distributed in the material. In this case, the rupture ratio increases linearly with time. and a mean-field model provides for this result. The peculiar mechanism and kinetic of film rupture are not known yet, but film rupture is induced by fluctuations, whose amplitude and ability to break the film are controlled by viscoelastic parameters. That is why the addition of a small amount of “impurities” (such as dodecanol in SDS or octanol in octanesulfonic acid), known to increase surface elasticity and enhance foam stability. (49) Khristov, K. I.; Exerowa, D. R.; Kruglakov, P. J. Coll. Interface Sci. 1981, 79(2), 584.

Langmuir, Vol. 16, No. 8, 2000 3883

The second mechanism is well-illustrated by the destruction of the foam in a pint of beer. One may observe that the foam level decreases, revealing a rupture front. This behavior is characterized by a threshold. In the experiments presented above, foam films break as soon as the capillary pressure exerted on them reaches a threshold value that does not exceed a few hundred Pascal. For these systems, this point is in contradiction with TFB measurements, where much higher pressures are applied on films, without breaking them. However, external perturbations are very different in both experiments. Isolated films are studied with extreme care, whereas a bulk foam is subjected to mechanical vibrations, acoustic waves (one can easily hear a film when it breaks), and hydrodynamic flows. In this respect, fluctuations are dampened in the first case and “normal” in the latter. Taking into account these fluctuations in the equilibrium film (with disjoining pressure isotherms) leads to the definition of a threshold pressure, in analogy with the well-known liquid-vapor phase transition. To conclude, it is important to note that this front mechanism is a limit mechanism and that foams from many surfactant solutions can survive to very high pressure up to 105 Pa.49 This analysis gives a new insight into foam rupture and suggests new approaches to control foam stability. It emphasizes the complementarity of TFB and bulk foam coalescence measurements and calls for precise determination of interfacial viscoelastic parameters to check the validity of the homogeneous interpretation. Last, the importance of fluctuations is demonstrated, and the detailed mechanism and kinetics of film rupture have to be investigated, to address film size effects, for instance. Acknowledgment. We thank Rhodia Company for financial support, and Vance Bergeron for fruitful discussions. Institut Franc¸ ais du Pe´trole supplied polyelectrolyte samples. LA981733O

A New Analysis of Foam Coalescence - American Chemical Society

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