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Anisotropy of Draining Foams V. Carrier* and A. Colin CRPP, Avenue Dr. A. Schweitzer, 33600 Pessac, France Received April 15, 2002. In Final Form: July 3, 2002
We assess the question here whether a foam contains the same amount of water in the horizontal plane as in the vertical plane. We show using different measurements of the liquid fraction that, for foam of fine bubbles, with a Plateau border length lower than 0.4 mm, the liquid is not distributed homogeneously in the foam. We argue that this effect can be explained by a swelling of the films due to drainage and is dependent on the orientation of the films.
1. Introduction Aqueous foams consist of gas bubbles dispersed in a soapy solution. Champagne, shampoo, and ice cream are examples of foams we meet every day. These systems have a complex and disordered structure. The interstitial space between the bubbles can be divided into films, Plateau borders, and nodes. Thin liquid films set between the faces of two touching bubbles. They meet 3-fold in regions called Plateau borders, which join 4-fold in nodes and form an interconnected network of liquid. Usually this structure is assumed to be isotropic, which means that the thickness of the films and the section of the Plateau borders are supposed to be independent of the orientation and that the liquid is distributed homogeneously in the foam. One reason is that the capillary forces are so strong in foams, as compared for example with concentrated emulsions, that they tend to erase any gradient in liquid fraction. We are going to show that this hypothesis is not always true, especially for fine foams by measuring the liquid fraction of the foam. The liquid fraction measures the amount of water present in a foam. It is equal to the volume of liquid in a macroscopic foam region divided by the total volume of the region. We use three different methods to measure it: the weight, which is a measurement of the average liquid fraction in the whole volume of the foam; the conductivity, which is in our experimental setup a horizontal measurement of the liquid fraction; and a velocity front propagation measurement, which is a vertical measurement of the liquid fraction. After presenting our experimental setup, we will recall the precise definition of liquid fraction and describe the three methods of liquid fraction measurements. We comment on the obtained results and finally discuss the film thickness measurements. 2. Experimental Measurements of the Liquid Fraction Fabrication and Characterization of the Foam. Foams of sodium dodecyl benzenesulfonate (SDBS) were made; SDBS was purchased from Aldrich and used as received. This surfactant is particularly stable, and contrary to sodium dodecyl sulfate, it does not undergo hydrolysis reaction in water. The critical micellar concentration of SDBS is equal to 1.2 × 10-3 mol/L or 0.03% w/w. To check the generality of our results, foams made with
Dawn were also studied. Dawn is commercial soap and was used as received. The foaming solutions of SDBS 0.1% w/w and Dawn 1% w/w were prepared with deionized water. The experimental setup is presented in Figure 1. Foam is made by bubbling perfluorohexane saturated nitrogen through a capillary (hole diameter: 1, 0.5, 0.2, and 0.1 mm) or a porous glass disk (porosity: 150-200, 90-150, and 40-90 µm) into the foaming solution, inside a Plexiglas column (2.5 cm × 2.5 cm × 60 cm high). Determination of the foam size is made by image analysis of the channels located on the border of the column. These particular channels are called parietal Plateau borders.1 Their length LPBP is different from the length LPB of the Plateau border located in the foam. The average length of the Plateau border in volume LPB can be calculated from the average length of the parietal Plateau borders LPBP, from the work of Cheng and Lemlich.1 For size dispersions, defined as the ratio of the standard deviation over the average,2 of the order of 30%, like the ones we have, we can deduce from their work that LPB ) LPBP/1.2, where LPBP is the average length of the parietal Plateau borders. Statistics are made over 50 parietal Plateau borders. Bubbling is stopped during experiments. The foams studied are stable: no coalescence occurs during the whole experiment, and Ostwald ripening is stopped by the perfluorohexane,3 which allows to study for a few hours fine foams with parietal Plateau borders LPBP as small as 0.24 mm. All the interfaces are saturated with surfactant. Hence, the amount of surfactant needed to create the interfaces Ninterfaces divided by the amount of disposable surfactant in the solution is Nsolution:
Ninterfaces ΓVfoam 27 ) Nsolution 8x2LPB(1 - 1.5x)2Vliquidc
(1)
where Γ is the surface concentration in surfactant, the liquid fraction of the foam, c the volume concentration in surfactant, Vfoam the volume of the foam, Vliquid the volume of the foaming solution, and LPB the Plateau border length. (1) Cheng, H. C.; Lemlich, R. Ind. Eng. Chem. Fundam. 1983, 42, 105. (2) std ) [(∑n1 xi2 - nxj2)/(n - 1)]1/2, where n is the number of data points, xi a data value, and xj the mean. (3) Gandolfo, F. G.; Rosano, H. L. J. Colloid Interface Sci. 1997, 194, 31.
10.1021/la020361n CCC: $22.00 © 2002 American Chemical Society Published on Web 08/31/2002
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calculation. The volume V of the Kelvin cell is
V ) 8x2LPB3
(2)
where LPB is the length of a Plateau border in the foam. The volume of water on the faces Vf is Vf ) h/2Af, where Af is the surface covered by the films on the bubble and h the average film thickness. Hilgenfeldt et al.15 gave an expression for Af, depending on the liquid fraction , which leads to
Vf ≈
Figure 1. Foam conductivity apparatus.
Assuming a surface coverage of 40 Å2, considering a volume of the foaming solution equal to 50 mL a foam volume equal to 300 mL, a Plateau border length of 0.2 mm, a solution concentration equal to 0.1% w/w, and a liquid fraction of 1%, the ratio Ninterfaces/Nsolution ≈ 0.10. The amount of surfactant is thus 10 times greater than the minimal amount needed to create all the interfaces. The interfaces are saturated. The foam reposes on the foaming solution and is 50 cm high. The room is thermostated at 21 °C. Usually, under the action of gravity and capillarity, water flows through the foam from the top to the bottom of the foam: the foam is dry at the top and wet at the bottom. This phenomenon is called drainage.4-13 To avoid drainage and prepare the sample with a uniform liquid fraction, we wet the foam from above with the foaming solution using a peristaltic pump at constant rates varying from 0.01 to 1000 µL s-1. In this case, a homogeneous and constant profile is obtained. The local liquid fraction differs from the constant value only in a small region at the bottom of the foam where the capillary rise occurs. In the following, after recalling the definition of the liquid fraction and its links with the structural parameters of the foam, we are going to present three different ways to measure the liquid fraction of a foam. Liquid Fraction Definition. In this section, we are going to define precisely the term liquid fraction of the foam and present its calculation as a function of the size of the Plateau borders in the bulk LPB and of the thickness of the films. The liquid fraction measures the amount of water present in a foam. It is equal to the volume of liquid in a macroscopic foam region divided by the total volume of the region. Considering a monodisperse foam and modeling the bubbles by Kelvin cells,14 one obtains the link between the liquid fraction and the size of the structural parameters of the foam through the following (4) Kraynik, A. M. Sandia National Laboratories Report No. 83-0844, 1983. (5) Leonard, R. A.; Lemlich, R. AIChE J. 1965, 11, 18. (6) Weaire, D.; Pittet, N.; Hutzler, S. Phys. Rev. Lett. 1993, 71, 2670. (7) Ramani, M. V.; Kumar, R.; Gandhi, K. S. Chem. Eng. Sci. 1993, 48, 455. (8) Bhakta, A.; Ruckenstein, E. Langmuir 1995, 11, 1486. (9) Verbist, G.; Weaire, D.; Kraynik, A. M. J. Phys.: Condens. Matter 1996, 8, 3715. (10) Weaire, D.; Hutzler, S.; Verbist, G.; Peters, E. Adv. Chem. Phys. 1997, 102, 315. (11) Stoyanov, S.; Dushkin, C.; Langevin, D.; Weaire, D.; Verbist, G. Langmuir 1998, 14, 4663. (12) Koehler, S. A.; Hilgenfeldt, S.; Stone, H. A. Phys. Rev. Lett. 1999, 82, 4232. (13) Koehler, S. A.; Hilgenfeldt, S.; Stone, H. A. Langmuir 2000, 16, 6327. (14) Princen, H. M. Langmuir 1986, 2, 519.
h 27LPB2(1 - 1.52x)2 2
(3)
The volume occupied by the Plateau borders and the nodes, according to Phelan et al.,7 is
VPB+n ) 12(0.161r2LPB + 0.241r3)
(4)
where r is the curvature radius of the Plateau border. The liquid fraction is then
)
VPB+n + Vf a ) V L
(
2 PB
x )
1.06 + 3.98
a + LPB2
h (1 - 1.52x)2 (5) 1.19 LPB
where a, the section of the Plateau borders, is linked to r through
π 2 r ≈ 0.161r2 2
(
)
a ) δar2 ) x3 -
(6)
Usually the amount of water contained in the films is neglected for wet foams. One commonly admits that the thickness of the film is smaller than 300 nm. For a typical foam with a Plateau border length of 3 mm and a liquid fraction of 1%, the volume fraction of water captured in the films is less than 0.01%, which is negligible in front of 1%. In the following, we will thus use eq 7 to link the liquid fraction of the foam and its structural parameters:
)
VPB+n + Vf a ) V L
(
2 PB
1.06 + 3.98
x ) a
LPB2
(7)
Weight Measurements. The easiest way to measure the liquid fraction of a foam is to weight the foam. The average liquid fraction j
j )
1 H
∫0H(z) dz
(8)
where z is the vertical coordinate and H the foam height, is simply given by the ratio of the mass of the foam divided by the volume of the cell and by the density of the soapy solution. j corresponds to the local liquid fraction (z) at any height z when the liquid fraction profile is homogeneous. This is the case when the foam is wetted with a steady flow from the top and when the liquid fraction is greater than 4%. Then, the local liquid fraction differs from the average value only in a small region at the bottom of the foam where the capillary rise occurs. To perform the measurement, the following procedure is used. A constant flux is imposed to the column, which has an overflow pipe at its bottom. The level of the water (15) Hilgenfeldt, S.; Koehler, S. A.; Stone, H. A. Phys. Rev. Lett. 2001, 86, 4704.
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stabilizes at a height h1, a bit above the height of the overflow pipe exit, the more so, the higher the flow rate. Then, we make a foam, and we impose the same flux to it. We measure the stationary height h2, h2 < h1 of the water level, on which the foam is sat. The height of water contained in the whole foam is h1 - h2. The liquid fraction of the foam is thus
)
h1 - h2 H
(9)
where H is the foam height. The condition for this measurement to be valid is that there be no sticking of the foam on the container sides. This can be tested by wetting the foam, stopping the wetting and let it drain, and then wet it again at the same flow rate. If the water level recovers its initial value, there is no sticking. Experimentally, for SDBS 0.1%, this is true for liquid fractions above 3%. Let us note that this method is not valid for sticking surfactants such as proteins. Weight measurements allow us to measure liquid fraction greater than 4% with an error bar of (0.3% due to errors on the measurements of h1 - h2 and on the amount of water captured in the capillary rise region. This method probes all the Plateau borders independently of their orientation. Both horizontal and vertical Plateau borders have a weight and are supported by the liquid. An average value of the liquid fraction is thus obtained through these measurements. Conductivity Measurements. The conductivity method has long been used to measure the liquid fraction of foams, and compared to other methods, it has the advantage of measuring accurately liquid fractions as low as 10-5. Lemlich and co-workers16-18 and Peters19 gave extensive validations of the method; Phelan20 in 1996 gave the latest relationship between the conductivity of the foam and its liquid fraction, which we used in our study. In his model, Phelan takes into account the water contained in both the channels and in the nodes, considering the films negligible. According to the author, it is valid for < 7%. This work links the relative conductivity of the foam K ) Z0/Z, Z0 being the resistance of the foaming solution and Z the foam resistance to the liquid fraction of the foam through two equations:
)
(
a 1.06 + 3.98 LPB2
x ) a LPB2
a ) x2[x(3.17K)2 + x2K - 3.17K] L2
(10)
(11)
We recall that a is the area of the Plateau border, LPB the length of the Plateau borders in the foam, the liquid fraction, and K the relative conductivity. The apparatus we use is made of 25 nickel-plated brass electrodes (1.5 cm × 2 cm) with their counter electrodes uniformly distributed along two opposite sides of a column of square section. Each couple of electrodes is connected to a multiplexer which sends the chosen canal to an impedance meter; a PC controls the two apparatus and (16) Lemlich, R. J. Colloid Intertface Sci. 1978, 64, 107. (17) Agnihotri, A. K.; Lemlich, R. J. Colloid Interface Sci. 1981, 84, 42. (18) Datye, A. K.; Lemlich, R. Int. J. Multiphase Flow 1983, 9, 627. (19) Peters, E. Master thesis, Eindhoven University of Technology, 1995. (20) Phelan, R.; Weaire, D.; Peters, E.; Verbist, G. J. Phys.: Condens. Matter 1996, 8, 475.
allows for the programming of various measurement processes. The frequency of the signal is automatically adapted as to measure pure conductivity and is chosen to be the lowest frequency at which the phase is within (1°. Typically, it varies between 100 and 1000 Hz. This method has the advantages of being a local investigation of the liquid fraction and of giving accurate measurements. Its drawback is the need for a model, which has to be tested. In our experimental setup, the electrical field is horizontal. We thus probe mainly the horizontal Plateau borders. Hence, if all the Plateau borders were vertical, the foam would not conduct the current. From this measurement, we deduce a value of the liquid fraction that assumes that the amount of water is distributed homogeneously in the foam and that all the Plateau borders behave as if they were horizontal. Front Propagation. When a foam is wetted from the top with a constant flow rate, a homogeneous density profile is obtained. Mass conservation links the mean velocity of liquid in the foam to the liquid fraction . The liquid fraction is equal to the flow rate by unit surface Q divided by the mean velocity of the flow v:
Q/S ) v
(12)
One way to measure the liquid fraction is thus to measure the velocity of the flow in the foam. One usual technique, described by Weaire10 et al. and Koehler et al.,12 is to wet a very dry foam with a constant rate Q. A front propagates then down the foam. Theory of drainage models the flow in the foam.13 In Appendix I, we report the main conclusions obtained by this theory. In the following we recall only the needed results to perform the measurements of the liquid fraction. Theory of drainage12,13 predicts two regimes where the velocity of the fluid is proportional to a power law of the liquid fraction: v ) Bχ-1 and Q/S ) Bχ, where S is the container section and B a constant. The values of B and χ depend on the regimes. At high liquid fractions, the flow is slowed down in the nodes and χ equals 1.5. At low liquid fractions the volume of the nodes get smaller, and the liquid is slowed down in the Plateau borders and then χ varies between 2 and 2.5 depending upon the surface viscosity. χ is thus a function of the liquid fraction. However, from an experimental point of view, it is difficult to measure this cross over, and one may assume that v ) Bχ-1 and Q/S ) Bχ in all the range of our experiments with χ close to 1.8.22 In this case, as shown in Appendix II, the velocity of the front propagation is given by
vf ) B
2χ - 1χ 2 - 1
(13)
where 1 corresponds to the liquid fraction of the dry foam and 2 to the liquid fraction of the wetted foam. If the initial foam is very dry compared to the wetted one, then the front velocity vf is equal to Bχ-1 and thus to the velocity of the flowing fluid v. The liquid fraction is deduced by the ratio
)
Q vS
(14)
This simple technique has two disadvantages: it is limited (21) Weaire, D.; Hutzler, S.; Verbist, G.; Peters, E. Adv. Chem. Phys. 1997, 102, 315. (22) Carrier, V.; Destouesse, S.; Colin, A. Phys. Rev. E 2002, 65, 061404.
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Figure 2. Measured relative conductivity vs liquid fraction measured by front propagation. Foams of variable parietal Plateau border lengths LPBP. SDBS 0.1%. The curve gives the model of Phelan.
in the measurement of low liquid fractions since the liquid fraction of the dry foam must remain negligible to the measured one, and second, there had been until now no proof that a strong liquid fraction gradient had no dissipative effect by itself which would lead to an artifact in the measurements. We preferred then a method involving low gradients. We prepare a foam under constant wetting with a flow rate Q1, and at time zero, we increase suddenly the flow rate at Q2 ) Q1R, where R is a constant close to one; we chose R ) 1.5. A small front then propagates down the foam with a speed vf. To get vf, we measure the conductivity at a known distance ζ from the top of the foam; we visualize a jump in conductivity and report the time tf corresponding to the middle of the jump. Then vf ) ζ/tf. Successive measurements are made keeping the ratio R ) Q2/Q1 constant. v1, the speed for Q1, can be deduced from vf following
vf )
R-1 v1 R1/χ - R
Figure 3. Liquid fraction deduced from conductivity measurements through the Phelan equation vs liquid fraction deduced from front propagation measurements. Foams of variable parietal Plateau border lengths LPBP. SDBS 0.1%. The straight line corresponds to perfect agreement between the two methods.
Figure 4. Liquid fractions measured by conductivity and by front propagation vs liquid fraction measured by weight measurements. LPBP ) 0.33 mm. SDBS 0.1%. The straight line gives the perfect agreement.
(15)
The value of the power law χ is given by the computation of Q1 vs Q1/vf. Finally, the liquid fraction is deduced using relationships 14 and 15. This measurement gives the value of the liquid fraction integrated over the highest part of the foam. The speed of the front must be measured before it reaches the lower region of the foam to get the liquid fraction far from the capillary rise (zf∞). Small front propagation velocity measurements allow us to measure liquid fraction above 0.3% with a precision of 10% on the measured value. The propagation is due to gravity. This measurement clearly probes mainly the amount of water distributed in the vertical Plateau borders. If the channels were all horizontal in the foam, then water could not flow from the top to the bottom. This measurement assumes that the horizontal Plateau borders contain the same amount of water as the vertical ones. 3. Results We obtained the results given in Figure 2. We compute the relative conductivity vs the liquid fraction deduced from front measurements. The curve corresponds to the model of Phelan. The last points for the curves at LPBP ) 2.3 and 0.68 mm are probably false, because the bubbles begin to move.
We see that the Phelan model works well for the big sizes but overestimates the liquid fraction measured by front for the bubbles at LPBP ) 0.33 mm. The transformation of K into liquid fraction (see eqs 10 and 11) gives the graph in Figure 3. The line corresponds to perfect agreement between the two methods. We see again that, within the experimental errors, the liquid fraction measured by conductivity and the one measured by front velocity are the same for sizes above LPBP ) 0.68 mm, while the liquid fraction deduced by conductivity is inferior to the liquid fraction measured by front velocity, for LPBP ) 0.33 mm. A comparison of the conductivity liquid fraction and front measurement liquid fraction with the weight measurements for the small bubbles gives the results shown in Figure 4. The line corresponds to perfect agreement between the two methods. We see that the conductivity liquid fraction is a bit above the weight one but that the liquid fraction deduced from front velocity measurements is strongly above, by a factor of 1.5. The liquid fraction as measured by front velocity measurements is superior to the average liquid fraction as measured by the weight. These experiments are reproducible; we have studied the effects of the finite size of the column and verified that they did not affect the measurements, and there is thus a real effect here. As the three measurements are different,
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Figure 5. Apparatus of film thickness measurements in situ in draining foams.
Figure 7. Pattern of flow inside the films.
Figure 6. Vertical film thickness vs liquid fraction. The liquid fraction is measured by conductivity. SDBS 0.1%. LPBP ) 2.5 and 3 mm. Figure 8. Pattern of film swelling.
foams with small bubbles seem to be anisotropic, and we need to understand why. We are going to show that it can be explained by a swelling of the vertical films due to the flow. We measured the thickness of vertical films in situ in foams during steady drainage using the apparatus schematized in Figure 5. A white light is sent on a chosen approximately vertical film inside the foam, close to the border. The reflected light is collected through a beam splitter and then sent on a grating which will give the spectra of the reflected light. The film thickness d is then given by the Bragg law: 2nd ) (m + 1/2)λ, where n is the refractive index of the water, λ the wavelength, and m the interference order. The interference order is deduced from the successive interference peaks. Unfortunately, our experimental setup does not allow us to probe the horizontal films in the foam. We would need to introduce a mirror inside the foam to measure them. However, we may have an idea of the behavior of the horizontal films by simply observing what happens at the top of the foam with a camera. The results of the measurements on the vertical films are given in Figure 6. These measurements have been made on foams of big bubbles, with LPBP ) 2.5 or 3 mm. We see that the films are swollen to more than 2 µm; at this thickness, the films appear white with the pattern of flow schematized in Figure 7. They are actually pinched on the regions in contact with the Plateau borders and swollen inside as shown in Figure 8. What happens for smaller bubbles? We tried to measure the film thickness in finer foams without getting any signal. An apparatus with better time and space resolutions would be needed to get this information. However, the following can be seen: under 1% in liquid fractions, the films are in the CBF (Common Black Film) state, with thickness around
10-20 nm; around 1%, colors can be seen, and above 1%, the films appear white. They may then be thicker than 2 µm. This swelling is obviously due to the flow. No repulsive forces are able to stabilize films at such thickness. As the flow is governed by gravity, we believe that the horizontal films are thinner than the vertical ones. This hypothesis is strengthened by a simple observation of the horizontal films at the top of the foam. Horizontal films remain black and thin. The liquid content is thus not homogeneously distributed in the foam. More water is present in the vertical direction. To quantify this heterogeneousness, we need to consider the contribution of the films to the liquid fraction. We use eq 5 where h is the average film thickness. For LPBP ) 2.5 mm and ) 1.3%, if we consider the average films thickness being 2 µm, the liquid fraction in the films is 0.07%. The liquid fraction in the films is thus negligible, and one may say that the main part of the water is homogeneously distributed inside the foam. The three different liquid fraction measurements agree and give the same value in the three different directions. For LPBP ) 0.33 mm and ) 1%, if we consider the average film thickness being 2 µm, the liquid fraction in the films is 0.5%. This is no longer negligible, and the foam may be anisotopic. As shown in Figure 4, we can thus expect that, in fine foams, the liquid fraction measured vertically appears higher than the average one and the one measured horizontally. One could also expect the conductivity measurements in the horizontal plane to give the lowest value of the liquid fraction. Figure 4 shows that this is not the case. This point is related to two antagonistic effects. As already pointed out, as the horizontal films are thinner, the liquid fraction measured by conductivity would be smaller.
Anisotropy of Draining Foams
However, as the films are no longer negligible, the model of Phelan is no longer valid. Agnihotri and Lemlich17 have shown that, if the films are of importance, the ratio between the liquid fraction and the relative conductivity decreases. Then, the model of Phelan overestimates the real liquid fraction. These two antagonistic effects seem, according to our measurements, to compensate each other. 4. Conclusion We have measured the liquid fractions in foams using three methods: weight measurements, conductivity, and front velocity measurements. The results are coherent for big bubbles, with average parietal Plateau border length LPBP above 0.68 mm. However, for fine foams, with LPBP ) 0.33 mm, the front velocity measurements give a liquid fraction superior to the one measured by weight and conductivity. The amount of water is thus not distributed homogeneously in the foam with small bubbles: this foam is anisotropic. Film thickness measurements in situ, in foams in steady drainage, show that vertical films in foam of LPBP ) 2.5 mm can reach thickness as high as 2 µm. We show that we can expect in fine foams the same swelling of the films. This swelling may be responsible of the anisotropy. Hence, the contribution of the films to the global liquid fraction is no more negligible, and the liquid fraction measured vertically by front velocity appears higher than the average one measured by weight or the horizontal one measured by conductivity. The foam is then different in the vertical direction and in the horizontal one. To proceed further, accurate measurements of the film thickness in fine foams are first needed to be able to understand more precisely afterward why and how do the films swell so much. At this stage, we want to point out that the swelling of the films we present is still misunderstood. It seems quite general since we obtained the same behavior in foams made with Dawn. However, it depends strongly on the physical and chemical characteristics of the surfactant solution. For example, in more concentrated samples of SDBS, the same behavior was not observed. The films remain black and do not swell.22 The presented behavior seems then to be related to dilute solution of small surfactants. These measurements and results have strong implications on the study of foams, especially on the study of drainage and Ostwald ripening.23,24 First, to compare experiments and theory in these processes, precise measurements of the local liquid fraction are needed. We show here that, for fine and dry foams, the weight measurement is the most reliable method as it probes the liquid contained in all the Plateau borders and films, independent of their orientation. We believe that fluorescence, γ-rays, and NMR25-29 should also give the same results. Second, we point out that water swells the films and runs across them during the drainage process. For fine foams, films, usually neglected may thus have an (23) Hilgenfeldt, S.; Koehler, S. A.; Stone, H. A. Phys. Rev. Lett. 2001, 86, 4704. (24) Vera, M. U.; Durian, D. J. Phys. Rev. Lett. 2002, 88, 088304. (25) Barigou, M.; Deshpande, N. S.; Wiggers, F. N. Colloids Surf. A: Phys. Eng. Asp. 2001, 189, 237. (26) Gonatas, C. P.; Leigh, J. S.; Yodh, A. G. Phys. Rev. Lett. 1995, 75, 573. (27) Prause, B. A.; Glazier, J. A.; Gravina, S. J.; Montemagno, C. D. J. Phys.: Condens. Matter 1995, 7, 511. (28) Barigou, M.; Crawshaw, J. P.; Davidson, J. F.; Gladden, L. F.; Hollewand, M. P.; Paterson, W. R.; Scott, D. M. Proc. IchemE Res. Event 1993, 552. (29) MacCarthy, M. J. AIChE J. 1990, 36, 287.
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important contribution to drainage by increasing the porosity of the foam.22 Acknowledgment. We thank Atofina for financial support. Appendix I. Expression of the Flow Rate as a Function of the Liquid Fraction The theory of drainage models the flow of liquid inside the foam. It assumes that water is mainly contained and flows in the network built by the Plateau borders. In the region where the capillary rise is negligible, it links the flow rate to the liquid fraction through12,13,22
Q)
2x2 3
Fg(1 - )L2S2
( ( x) x )
K 3η 1 - R′ β(a)
a + L2
a I′n L2
(A1)
where the liquid fraction expression of Phelan et al.5 is
)
(
x ) a L2
a 1.06 + 3.98 L2
(A2)
In these expressions, F is the density of the liquid, g the gravity constant, the liquid fraction, a the area of the Plateau borders, L the length of the Plateau borders in the bulk, S the section of the column, R′ is equal to
4 cos
(109°28 2 ) ≈ 5.8
xx3 - π2
and K is a geometrical constant. Leonard and Lemlich,5 and later Peters,20 found by simulations that, in the case of rigid interfaces, with v ) 0 on the sides of the Plateau borders, K ) 49.699. When the surface viscosity ηs of the surfactant monolayer is finite, Leonard and Lemlich5 and Desaı¨ and Kumar30 have shown numerically that the flow is simply accelerated with an acceleration coefficient β(a), which is a function of the ratio ηxa/ηs. η is the bulk viscosity of the liquid and ηs the surface viscosity of the surfactant monolayer. There is no analytical expression of β(a); however, a good estimation of this function may be obtained using the following expression:
β(a) ) 1 + 10.9
()
η η xa + 6.2 ηS ηS
4/3
()
a2/3 - 12.5
η ηS
5/4
a5/8
(A3)
Recently, Cox et al.31 calculated I′n and found that it could vary between 121 in the case of an infinite surface viscosity and 250 ( 25 in the case of a very low one, which is coherent with the original rough estimation of Koehler et al., based upon the flow through a packed bed of rigid spheres13 which had given I′n ≈ 400. Experimentally, the values found by Koehler et al.12 are comprised between 100 and 300. Two extreme regimes are predicted by this theory. At high liquid fractions β(a) is large, the flow is pluglike in the Plateau borders, and the dissipation occurs mainly in the nodes. Then, Q ) FgL2S1.5/3ηI. At small liquid (30) Desai, D.; Kumar, R. Chem. Eng. Sci. 1982, 37, 1361. (31) Cox, S. J.; Bradley, G.; Hutzler, S.; Weaire, D. J. Phys.: Condens. Matter 2001, 13, 4863.
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fractions, the volume of the nodes is small, and the flow is slowed in the Plateau borders Q ) FgL2Sβ(a)2/3ηK. Q varies thus as β(a)e2, which makes it roughly varying between 2 and 2.5. The transition between these two regimes is clearly ruled by surface viscosity. Appendix II. Expression of the Front Speed When a front propagates down a foam, the flow rate can be written
Q ) Q1 + (Q2 - Q1)H(vft - z)
∂ ∂(Q/S) + )0 ∂t ∂z
vf(2 - 1) -
Q2 - Q 1 Q1 Q2/Q1 - 1 ) 0 S vf ) S S1 2/1 - 1 (B4)
In our method we keep the ratio Q2/Q1 ) R constant. Making the approximation Q/S ) Bχ, with B and χ constants, we get the link between the front speed and the fluid velocity v1 at the flow rate Q1:
2χ - 1χ R-1 ) v1 1/χ vf ) B 2 - 1 R -1
(B2)
where 1 is the initial liquid fraction and 2 > 1 is the final liquid fraction.
(B3)
where S is the column section. It leads to
(B1)
where Q1 is the initial wetting flow rate, Q2 > Q1 the final wetting flow rate, z the vertical ordinate, increasing downward and being zero at the top limit of the foam, vf the front speed, and t the time. H is the Heavyside function. As well, the liquid fraction is
) 1 + (2 - 1)H(vft - z)
The mass conservation equation is
LA020361N
(B5)
