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Articles Foams As Viewed by Small-Angle Neutron Scattering Monique A. V. Axelos*,† and Franc¸ ois Boue´‡ Laboratoire de Physico-Chimie des Macromole´ cules, INRA, BP 71627, 44316 Nantes Cedex 03, France, and Laboratoire Le´ on Brillouin, UMR 12 CEA/CNRS, CE SACLAY, 91191 Gif-sur-Yvette Cedex, France Received December 13, 2002. In Final Form: April 23, 2003 We report on small-angle neutron scattering measurements obtained in situ on three-dimensional aqueous foams stabilized by sodium dodecyl sulfate. Isotropic as well as anisotropic scattering data have been collected for two kinds of foams: wet foams in the steady-state regime of constant gas bubbling and dry foams under free draining conditions. Reliable scattering spectra were obtained within a few minutes, even for very dry foams. All spectra have a basic I(q) ∼ q-4 behavior on which a foam specific structure is superposed. The q-4 decrease at low q can be interpreted in terms of a Porod law from which the average bubble size is determined for both wet and dry foams. For wet foams, the intensity is modulated at high q by the structural organization of the surfactants in the liquid fraction of the foam (micelles). Remarkably, for dry foams only, the structure also appears at intermediate q, corresponding to the film thickness. This structure can be enhanced by using deuterated sodium dodecyl sulfate to suppress the micelle signal. Anisotropic measurements on dry foams reveal spikes in the 2D scattering data, suggesting an interpretation in terms of the reflection of the incident beam on the film surfaces. The isotropic scattering data may also be interpreted in terms of reflectivity instead of Porod scattering. Our experiments show the feasability of scattering measurements as a means for obtaining rapid information about average, global foam properties from the film thickness to the bubble size scale.
1. Introduction Foams belong to a class of systems governed by the structure and the dynamics of the air/water interface. They combine three different scales: small scales in the direction normal to the interface, typically the thickness of the liquid film and the size of the structure of the surfactants (nm); intermediate scales, typically the cross section of the Plateau borders (µm); and large scales, typically the length of the Plateau borders and the size of the gas bubbles (mm). To investigate these different scales, many techniques are used, from microscopic to macroscopic. Up to now, the vast majority of studies addressed the structural, topological, and dynamical features on the bubble scale.1-3 Microscopically there have been numerous studies on film formation, film stability, and film rupture. However, most of these investigations are not carried out in situ on foams but on planar air/water interfaces such as disjoining pressure experiments on single thin-film4,5 or X-ray reflectivity experiments on vertical soap films.6,7 In this paper we will use small-angle neutron scattering (SANS) techniques which probe the foam on a nanometer * To whom correspondence should be addressed. E-mail:
[email protected];
[email protected]. † INRA. ‡ UMR 12 CEA/CNRS. (1) Hutzler, S.; Weaire, D. The Physics of Foams; Clarendon Press: Oxford, U.K., 1999. (2) Monnereau, C.; Vignes-Adler, M. Phys. Rev. Lett. 1998, 80, 5228. (3) Elias, F.; Flament, C. Philos. Mag. B 1999, 79, 729. (4) Bergeron, V.; Radke, C. J. Langmuir 1992, 8, 3020. (5) Espert, A.; Klitzing, R. V.; Poulin, P.; Colin, A.; Zana, R.; Langevin, D. Langmuir 1998, 14, 4251. (6) Be´lorgey, O.; Benattar, J. J. Phys. Rev. Lett. 1991, 66, 313. (7) Cohen-Addad, S.; Di Meglio, J.-M.; Ober, R. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers 1992, 315, 39.
scale. The main purpose is to interpret the scattering spectra recorded in SANS measurements for a common kind of foam. The samples are foams created by gas bubbling in a solution of water with an anionic surfactant: sodium dodecyl sulfate (SDS). We will discuss the advantages in the use of and the information that can be extracted from scattering techniques: ensemble averaging over the whole sample, rapid data acquisition, and in situ measurements for various geometries, sample environments, and physicochemical and rheological conditions. Emulsions have been analyzed by Sonneville-Aubrun et al. using a similar approach.8 2. Materials and Methods. 2.1. SDS Foam Samples. Solutions were prepared by dissolving SDS (purchased from Merck and used without further purification) in water at three different concentrations: CSDS ) 3, 12, and 25 g/L. The critical concentration threshold above which spherical micelles form is around 2.3 g/L for SDS in water. In one of the 12 g/L solutions, 0.1 M NaCl was added to change the film thickness. To increase or decrease the neutron scattering contrast, the solvent was either a mixture of H2O and D2O or 100% deuterated water (Eurisotop). Deuterated SDS (SDS-d) with an isotopic deuterium enrichment of 98% (Eurisotop) or nondeuterated SDS was used. The time between mixing and foaming was between 1 h and 3 days. To prepare the foam sample, nitrogen gas was pushed through a porous stainless steel disk with 2 µm holes (Mott Industrial) situated at the bottom of a Plexiglas cylinder, with 22 cm height and 30 mm inner diameter, as shown in Figure 1. The cylinder is equipped with two 20 mm diameter quartz windows for the beam trajectory. About 10 cm3 of the surfactant solution was poured into the cylinder above the porous plate. (8) Sonneville-Aubrun, O.; Bergeron, V.; Gulik-Krzywicki, T.; Jo¨nsson, B.; Wennerstro¨m, H.; Lindner, P.; Cabane, B. Langmuir 2000, 16, 1566.
10.1021/la020965r CCC: $25.00 © 2003 American Chemical Society Published on Web 07/04/2003
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Figure 1. Drawing of the 22 cm height, 30 mm inner diameter Plexiglas cylinder used to study foams: (1) nitrogen gas entrance; (2) porous stainless steel disk with 2 µm holes; (3) quartz windows for the beam trajectory. Two kinds of foams were made: (1) One was a steady-state wet foam, where continuous bubbling creates a thin white foam column. The column height was slowly increased, and then the gas rate was adjusted such that the foam height remained stable, the bubbling being in equilibrium with the draining at the top of the column. In such a steady-state regime, the bubble diameter was typically between 0.1 and 0.3 mm. The walls between the bubbles and the Plateau borders at the junctions of these walls are very thick such that the bubbles look spherical. (2) The other foam was a free-draining dry foam, where the bubbling was stopped and the solution was drained continually by gravity. Because of drainage and coarsening, the foam evolved and the bubble diameter reached more than 1 mm within a few minutes. After about 10 min the diameter increased to about 2 mm. Most of the reported spectra were taken at this stage of foam evolution. In some cases specified below, the spectra were recorded after up to 30 min when the bubble diameter reached 3-4 mm. In all these very dry foams the film between the bubbles is of nanometer size, whereas the Plateau borders and the vertexes are of micrometer size, and the sample looks like an array of polyhedra. The diameter of the beam is 7 mm, and the thickness of the sample is 30 mm; therefore, the probed volume is 1150 mm3. For the wet foam, this corresponds to more than 1500 bubbles. For
drained foams, with a bubble diameter of 2 mm, this number reduces to 25 and even below 10 for some very dry foams. In parallel with the foam samples, a fraction of each SDS solution was poured into a Hellma cell in order to measure the bulk scattering intensity. 2.2. SANS Spectrometers. SANS measurements were performed at the Saclay Orphe´e reactor, on two different spectrometers. Most of the measurements were done on the spectrometer called PACE, which is well-suited for isotropic scattering experiments since its detector cells form circular rings. Two different configurations were used to cover a wide range of scattering vectors q, defined by q ) (4π sin θ)/λ, where 2θ is the scattering angle and λ the wavelength. In configuration CE1 (large q) collimating was achieved with a first diaphragm of 22 mm located 2.5 m before the sample and a second diaphragm of 7 mm located just before the sample; the wavelength was 7 Å, and the distance between the sample and the detector was 1 m. This configuration gives access to a q range from 0.0277 to 0.285 Å-1. In configuration CE2 (low q), collimating was achieved with a first diaphragm of 12 mm situated 5 m before the sample and a second diaphragm of 7 mm just before the sample; the wavelength was 10 Å, and the detector was at a distance of 4.5 m from the sample. In this configuration the q range is from 0.004 to 0.0435 Å-1. Additional measurements were done on the spectrometer called PAXE, which is well-suited for anisotropic scattering since its detector cells form an XY array of rows and columns. In the configuration XE1, collimating was achieved with a first diaphragm of 12 mm located 5 m before the sample and a second diaphragm of 7 mm just before the sample; the wavelength was 6 Å and the sample to detector distance 5 m (the beam axis being on the side of the detector). Other configurations (XE2, XE3, ...) were used, with the same geometry but with larger wavelengths, e.g., 12, 18, and 25 Å. In this way a very large q range could be investigated, well-exceeding the q range of the PACE spectrometer described above. The measurement time was 10 min for the wet foams. For dry foams the measurement time was adjusted to the rate of evolution of the foam during draining, the shortest time being 3 min. 2.3. Scattering Length Densities and Contrasts. We used different mixing fractions of H2O and D2O and deuterated as well as nondeuterated SDS. In Table 1 the scattering length density F of the molecules used, the density difference ∆F between the molecules and their environment (either water or air), and ∆F2, which we will call the contrast and which is important in SANS, are given. Note the large values of ∆F2 for SDS/D2O and D2O/air and the very low contrast values for SDS-d/D2O and the mixture (H2O(86.4%)-D2O)/air. 2.4. Data Corrections. Standard corrections for sample volume, neutron beam transmission, and incoherent scattering due to protons or deuterons were applied. Good knowledge of these parameters requires an accurate estimate of the amounts of liquid and foam in the beam for each recording. In the absence of a direct macroscopic measurement of the foam density during the scattering experiment, we tried to use the transmission data to determine the thickness of the liquid fraction. However, this parameter was determined to be rather inaccurate for dry foams, for which the transmission is always very close to 1. For wet foams, we met other difficulties, common to any strongly scattering system: a part of the scattered beam is collected in the central cells along with the transmitted beam; this can spoil the transmission measurements. This artifact was observed for measurements in the large-q range, but not for measurements in the small-q range.
Table 1. Scattering Length Densities (G) of the Different Molecules Used and Contrasts (∆G2) between Molecules and Solvents F (cm-2 × 1010) D2O H2O H2O(86.4%)/D2O SDS SDS-d
6.345 -0.561 0.407 0.367 6.73
air
∼0
SDS/D2O SDS-d/D2O SDS/(H2O(86.4%)/D2O) D2O/air SDS/air (H2O(86.4%)/D2O)/air SDS-d/air
∆F (cm-2 × 1010)
∆F2 (cm-4 × 1020)
-5.978 0.385 -0.04 6.345 0.367 0.407 6.726
35.74 0.148 0.0016 40.26 0.135 0.166 45.29
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Figure 2. Scattering curves obtained for bulk solutions of sodium dodecyl sulfate (SDS) in D2O at three different concentrations, (9) 3, (2) 12, and (b) 25 g/L, and SDS-d in D2O at ([) 12 g/L.
Figure 3. log-log plots (where, for example, 1.E+04 represents 1 × 104) of the intensity scattered from foams obtained with a solution of SDS at 25 g/L in D2O: ([) steady-state foam during bubbling, (]) foam during drainage. (4) Bulk SDS solution at the same concentration (shifted upward).
Table 2. Variation with Concentration of the Peak Position qpeak of the Scattering Curves I(q) for Bulk SDS Solutiona bulk SDS solution concn (g/L)
qpeak on I vs q plot (Å)
qpeak1 on q4I vs q plot (Å)
3 12 25 12 in 0.1 M NaCl
0.037 0.055 0.065 0.041
0.156 0.147 0.147 0.138
a In the last column the value of the first maximum q peak1 of the q4I(q) curves are also reported.
As a consequence of the uncertainty in the quantity of material in the beam, the determination of the incoherent scattering contribution was difficult in some cases. The data presented were corrected by the following procedure. The scattering of the empty cylinder was subtracted from the scattering of the cylinder filled with the foam, and this difference was divided by the scattering intensity of 1 mm of water. At low q, for wet and for free-draining foams, the solvent contribution was removed by subtracting a constant which takes into account the D2O contribution and the incoherent scattering due to the protons of the SDS. We have estimated the thickness of the solution in the foam using the low-q transmission data measured both in the foam and in the corresponding 2 mm SDS solution. For wet foams the thickness of the total amount of liquid in the foam was found to be between 3 and 6 mm. This implies that the liquid fraction of the foam is between 0.1 and 0.2. For dry foams the thickness was found to be around 0.5 mm which corresponds to a liquid fraction of about 0.017. For dry foams the uncertainty in determining the transmission is very large such that this value is not very accurate. Below we will present another method using the scattering results to determine the liquid fraction in the dry foams. At large q the same treatment of the data (i.e. using the thickness estimated from transmission in the small-q range) has been applied to the wet foams, whereas for dry foams no incoherent scattering contribution was removed.
3. Bulk Solutions Figure 2 gives the scattering curves for the 3, 12, and 25 g/L SDS solutions in D2O with no salt added. The intensity has been divided by the SDS concentration to allow comparison of the data. The three curves show a peak which is more and more pronounced as the concentration increases. This peak is due to the interference of the charged micelles. The position of the peak is related to the intermicelle distance and therefore shifts to larger q values when the SDS concentration CSDS is increased from 3 to 12 and 25 g/L (Table 2). The peak position follows the law qpeak ∼ CSDS0.3, as expected for charged spheres. For comparison the scattering intensity of a 12 g/L SDS-d solution in heavy water is also shown. It is low and flat,
Figure 4. log-log plot (where, for example, 1.E+04 represents 1 × 104) of the scattering intensity for foams stabilized with SDS at 12 (4) and 3 g/L (b) during bubbling (upper curves) and at the end of the drainage (lower curves).
in agreement with the low contrast of SDS-d/D2O (listed in Table 1) such that we can consider this solution to be homogeneous from the point of view of SANS. The addition of 0.1 M NaCl to the 12 g/L SDS solution leads to a shift of the peak toward lower values (listed in Table 2), as expected from the partial screening of the charges. To determine the size of the micelles the same scattering data have been drawn as q4I(q) vs q. A pronounced peak is now visible with a q value (listed in Table 2) that is almost the same for all SDS concentrations and whatever ionic strength. This is shown in Figure 5 for the 12 g/L SDS solution. To determine the radius R of the micelles the experimental data have been fitted to the theoretical formula for the scattering of homogeneous spherical particles. A value of 20 Å is found for the SDS solution at 12 g/L in 0.1 M NaCl. This value is in good agreement with data from other experiments.9 A slightly smaller value, around 18.5 Å, is obtained in the absence of salt. The low value of q4I at the first minimum in the q4I vs q curve indicates that the size distribution of the micelles is very narrow. 4. Isotropic Scattering Data of Foams The results of the isotropic scattering experiments are presented in Figures 3-7. Generally, our curves are composed of four different regions: a q-4 decay, clearly visible at low q on wet and dry foams; a bump at intermediate q, which appears during foam drainage; sometimes another shoulder at large q, visible on wet or dry foams when the scattering contrast between liquid (9) Reiss-Husson, F.; Luzzati, V. J. Phys. Chem. 1964, 68, 3504.
Foams As Viewed by SANS
Figure 5. Plot (where, for example, 3.E-07 represents 3 × 10-7) of q4I vs q for a draining foam obtained with SDS at 25 g/L in D2O (dotted line). The solid line corresponds to the intensity scattered by a 25 g/L bulk solution of SDS. To fit it to the dotted line, the intensity has been divided by 750.
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As the drainage proceeds, the intensity decreases steadily. The scattering is still close to q-4 at the lowest q. At intermediate q, a shoulder becomes more and more visible. The abscissa of this peak barely varies with surfactant concentration. At large q, for CSDS ) 25 g/L, another bump is visible, centered around 0.14 Å-1. This second bump is no longer visible at the two lower surfactant concentrations, Figure 4. 4.2. Estimation of the Average Bubble Size from the Porod Law at Low q. In the field of small-angle scattering, q-4 behavior is known to characterize welldefined surfaces, as for example in a two-phase system; it is called the Porod law.10 A foam can be considered to be a porous medium of water with air pores. More precisely, the foam is a more or less dense packing of bubbles with two types of surfaces: the surface between the air and the Plateau borders and the two parallel surfaces which face each other in the films. Since the size Rbub of the bubbles and the borders is large, we are always in the regime qRbub . π, thus giving a q-4 law. The measured signal arises from all surfaces taken separately, i.e., ignoring correlations between them. We can roughly estimate the absolute scattering intensity (cm-1) from the Porod law applied to a porous two-phase medium, air, and D2O (neglecting the SDS) using the equation
I (cm-1) ) 2πδF2(S/V)(1/q4)
Figure 6. Plot (where, for example, 7.E-08 represents 7 × 10-8)of q4I vs q for a dry foam obtained with SDS at 12 g/L in the presence of 0.1 M NaCl (]) and for a dry foam obtained with SDS at 12 g/L without salt ([).
Figure 7. Plot (where, for example, 5.E-08 represents 5 × 10-8)of q4I vs q for a dry foam stabilized with 12 g/L SDS-d in D2O.
and surfactant is large; and finally a q-4 decay at large q, with large statistical fluctuations. 4.1. Wet and Dry Foams in D2O at Different SDS Concentrations. Figures 3 and 4 show I(q) in cm-1 in a log-log plot of data from two configurations with overlapping q ranges. For the wet foams, the q dependence at low q clearly behaves as q-4, whatever the concentration. This is visible in Figure 3 and in the upper two curves of Figure 4. At high q, for CSDS ) 25 g/L, the bump observed in the curve for q values larger than 0.065 Å-1 corresponds to the bump seen in the bulk solution, as shown in Figure 3. A similar result was found for CSDS ) 12 g/L. Conversely, at CSDS ) 3 g/L, no such peak is present.
(1)
where δF2 ) 40.2 × 1020 cm-4 is the contrast, S the total surface, V the total volume, and S/V the specific area. For a wet foam, we can consider that we have small air bubbles dispersed in the solution. We assume a sphere radius Rbub) 0.01 cm for the bubbles. From the comparison of the magnitude of the peak at q ) 0.15, corresponding to the size of the spherical micelles, both in the bulk and in the foam, we estimate the air volume fraction to be φ ) 0.9. From this we can calculate the specific area S/V ) 4πRbub2/[((4/3)πRbub3)/φ] ) 3φ/Rbub ) 270 cm-1 and obtain, for q ) 5 × 10-3 Å-1 (1/q4 ) 1.6 × 10-23 cm-4), a scattering intensity of I ∼ 110 cm-1. For a drained foam, with an air volume fraction φ = 1, all of the space is filled by the bubbles. If, for the sake of simplicity, bubbles are considered to be spheres with Rbub ) 3 mm, we calculate S/V ) 10 cm-1 and obtain, for 5 × 10-3 Å-1, the scattering intensity of I ∼ 4 cm-1. These estimates obtained from the Porod law are fairly close to the measured intensities given in Figure 3 or 4. This is not surprising for wet foams, which can really be considered to be a two-phase porous medium. For dry foams, this is more surprising; we will return to this point below. 4.3. q-4I(q) Plots. Since all the scattering curves approximately follow a q-4 law, it is interesting to use q4I(q) plots in which the deviations from the main q-4 behavior are better visible. In general we observe a single peak, at large q, for wet foams and two peaks of very different magnitude for dry foams (Figure 5). The q position of all the different maxima are given in Table 3. The peak at large q, both for wet and dry foams, has the same q position as the peak observed for bulk SDS solution. The most clear proof that this peak is due to the size of the SDS micelles is its disappearance upon suppressing the contrast between the surfactant and the solvent, e.g., for SDS-d in D2O. As expected, the peak position varies little with ionic strength and concentration. For the wet foams, it is not surprising to observe this feature. It is (10) Porod, G. In Small-Angle X-ray Scattering; Glatter, O., Kratky O., Eds.; Academic Press: London, 1982; Chapter 2.
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Table 3. Position of the First and Second Maxima in the q4I(q) Curves Obtained for Dry Foams Made from Solutions with Different SDS Concentrations SDS concentration (g/L)
qpeak in the low-q range (Å-1)
qpeak in the high-q range (Å-1)
3 12 25 12 in 0.1 M NaCl 12 with SDS-d
0.029 0.024 0.027
no no 0.15 0.13 no
0.026
likely to arise from the bulk solution comprised between the bubbles. In this case the total thickness of the bulk solution crossed by the beam for a sample with a thickness of 30 mm is more than 3 mm, as estimated from the transmission data. For dry foams, this result is at first sight more surprising because the liquid films are very thin, but it can be explained by noting that the Plateau borders and the vertexes are still of micrometric size and may well contain enough liquid to explain the maximum observed in experiments where the foam is not drained too much, i.e., for 25 and for 12 g/L in the presence of salt. In Figure 5 we have plotted q4I(q) vs q for the drained foam obtained with SDS at 25 g/L (black dots). The two peaks are clearly visible. The first peak is at q ) 0.027 Å-1. The second much larger peak is at q ) 0.15 Å-1, at the same q value as the peak for the bulk SDS solution at the same concentration (solid line). The scattering intensity of the bulk SDS solution has been divided by 750 to adjust it to the intensity scattered by the foam. Assuming that the scattering intensity in this q range is due to individual micelles, we can estimate that the total thickness of liquid crossed by the neutron beam is 40 µm in the dry foam. The liquid fraction in this foam sample then is 0.0013. This value is of the right order of magnitude if we consider that most of the scattering comes from the Plateau borders which have a thickness of around 4 µm and a length of the order of the size of the bubbles which is about 3 mm. We also notice, in Figure 5, that the intensity of the q4I(q) curve for the bulk solution is close to zero in the q region where the low-q peak appears in the foam. Therefore the presence of the large-q peak for foams might modify the shape of the right shoulder of the small-q peak, but it does not modify its position. The peak at small q is only observed for drained foams. The peak is at almost the same q values whatever the SDS concentration: 0.027 Å-1 for 25 g/L, 0.029 Å-1 for 12 g/L, and 0.024 Å-1 for 3 g/L. This peak cannot be due to the SDS structure because its q value corresponds to a distance much larger than the interference distance between micelles; it would correspond to concentrations lower than the critical micelle concentration. To explain this peak, we could imagine that there is a lamellar order of the SDS between the walls, but there would again be some difficulties in obtaining the correct q value of the peak. But the most compelling result for interpreting the low-q peak is that the low-q maximum is still observed for SDS-d, which has zero contrast in D2O, and for which the SDS structure therefore cannot be seen. Since, in this case, the bubble film is homogeneous by construction, the observed oscillations must be due to the presence of the two parallel air/liquid interfaces. They depend on the liquid film thickness, similarly to Porod’s law for a flat particle with two surfaces separated by a distance d, where q4I(q) oscillates with a period 2π/d.10 A size distribution of the film thickness usually attenuates the oscillations rapidly. This can explain the presence of only a single peak for SDS/D2O. Following this interpretation we get an average film thickness d between 160 and 180 Å depending on the concentration, between 3 and 25 g/L, in the absence of
salt. In the presence of 0.1 M NaCl the peak moves toward larger q values as shown in Figure 6. This shift of the peak is in agreement with the screening of the electrostatic repulsion between the surfactant layers at the two air/ water interfaces, which leads to an average film thickness around 100 Å. The data are consistent with X-ray reflectivity measurements for vertical soap films made with SDS.6,7 The film thickness in the presence of salt is between the common black film and the Newton black film since 0.1 M ionic strength is not sufficient to lower the thickness down to the 30-50 Å of the Newton black film. In summary, all the measured scattering data are in agreement with the classical Porod description of smallangle scattering, yielding the specific area from the global intensity and the film thickness from the oscillations. This is expected for wet foams, but it is somehow surprising for dry foams. We will now have a closer look at the very dry foam. To better understand the origin of the scattering curves, we have changed the contrast between SDS and water and between liquid and air using the powerful possibilities of neutron scattering. First, we consider the results for SDS (12 g/L) in a 86.4/13.6 H2O/D2O mixture. This mixture matches SDS (i.e., ∆F ) 0), and the contrast with respect to the gas is also very small. The result is a very flat curve (not shown) corresponding to the background due to the hydrogen atoms in the foam. Second, we consider the data for 12 g/L deuterated SDS in pure D2O. The contrast between D2O and SDS-d is very low (from Figure 2 we know that the bulk scattering intensity vanishes and that the contrast between the liquid and the gas is very high, Table 1). The low-q signal obtained from wet and dry foams is very similar to the intensity of SDS/D2O at the same concentration. The bump of the dry foams always lies between 0.010 and 0.032 Å-1. A q-4 decay characterizes the large-q behavior, both for wet and dry foams. As seen in Figure 7 the q-4 behavior is observable over the entire q range with a first peak at q ) 0.026 Å-1, at the same q value as for all other drained foams (Table 3). A particularity of this SDS-d/D2O mixture is that in addition to the first oscillation two smaller oscillations at 0.061 and 0.1 Å-1 are visible in the q4I(q) plot. We will discuss this in more detail in section 5. For q > 0.1 Å-1 the experimental points are scattered too strongly to extract further information. As expected, the peak at 0.15 Å-1 corresponding to the size of the micelles is absent because the signal of SDS-d/D2O is not sensitive to any feature of the internal structure of the liquid film. 5. Anisotropic Scattering Data for Foams To see whether reflectivity is important, we performed anisotropic measurement, using a two-dimensional X-Y detector. Three different types of behavior are observed: one for the bulk solution, one for wet foams, and one for dry foams. In bulk solutions scattering is centrosymmetric: it displays a halo, which after integration over angles leads to a peak in I(q). For wet foams scattering is still centrosymmetric, but with a different profile. After radial averaging it displays a q-4 law with a shoulder as shown in Figure 3. The peak in I(q) from the liquid in the Plateau borders is seen as a halo in the XY patterns (not shown here). The most interesting behavior can be seen for drained foams: as the drainage proceeds, we observe the progressive appearance of anisotropy in the scattering data, in terms of what we call “spikes”. Spikes correspond to angular sectors where the scattering intensity is
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Qref ) (4π sin θref)/λ [do not confuse the vector Qref ) ki - kf, with ki and kf being the incident and the reflected wavevectors, and the vector qref used in the field of reflectivity, qref ) Qref/2], where θref is the incident as well as the reflecting angle. If the direct beam is reflected by a surface at a angle θref, it will be detected at an angle 2θref ) θSANS, following the definition of θ in SANS, with qSANS ) (4π sin(θSANS/2))/λ ) (4π sin θref)/λ ) Qref. We recall the conditions of total reflectivity: the Descartes-Snell laws impose the absence of a refracted beam below a certain angle θcrit (except the evanescent wave; hence, all the wave is reflected). The range between 0 < θ < θcrit is called the “total reflectivity plateau”, where
sin θcrit ) xNb/π λ
Figure 8. XY patterns of the intensity map on a twodimensional detector: scattering from a very dry foam during the final evolution. Neat “spikes” are distinctly visible.
particularly high. This is illustrated in Figure 8. In the early stages of drainage this is difficult to observe since good statistics requires sufficiently long measuring times. Thus, the foam may evolve during the measurement. For very drained foams, good statistics can be achieved and the geometry of the spikes remains stable over a long time. Frequent visual inspection of such highly drained foams confirms that they are made of a small number of reflecting surfaces on a few bubbles. Moreover, we changed the orientation of the column and observed a corresponding change in the orientation of the spikes. We conclude that we observe a small number of reflections of the beam on some bubble films. The spike shape deserves some attention. A priori, one would expect that a single reflection on a given wall would yield an image of the incident beam, which is a circular spot, centered at a given cell of the detector. However, the primary beam is not completely parallel: it is made of rays with slightly different incident angles. Also, for large bubbles, the reflecting surface is partly curved (due to the Plateau borders, in particular). All rays, which satisfy the condition of total reflection (θ/2 less than the critical angle θC), will produce a spike. If the incident angles are larger than θC, reflectivity will still occur, but will not be total: it should follow the reflectivity laws that are discussed below. The intensity is expected to decrease with increasing θ. At present, a more complete explanation of the spikes is lacking because it involves the curved nature of the bubble walls. Accounting for the different orientations could possibly be done by numerical calculations. For dried foams, the above results strongly support the interpretation in terms of reflectivity. For wet foams the large number of bubbles and their curved shape due to the importance of the Plateau borders yield a large specific surface covering a continuum of angles. This gives a continuum of reflected beams. To each angle θ/2 between an element of the surface and the incident beam there is a reflected beam at angle θ, with a reflectivity rate which follows again the reflectivity laws, as discussed above for the discrete case which leads to spikes. A connection can be established between reflectivity and SANS. For reflectivity the vector Q is defined as
(2)
with Nb ) F as defined above, but in units Å-2 ) 1016 cm-2. This gives for D2O, Nb ) 6.4 × 10-6 Å-2. Hence, sin θcrit ) (6.4 × 10-6/π)1/2λ(Å) ∼ 1.43 × 10-3λ(Å). We can evaluate Qcrit ) (4π sin θcrit)/λ ∼ 1.8 × 10-2 Å-1 . This is a very large value! It suggests that, for q < 1.8 × 10-2 Å-1, most of the scattering corresponds to total reflectivity and therefore should not depend on q (assuming that the angular distribution of the reflecting bubble walls is isotropic). Moreover, for a thin layer of thickness e, the reflectivity laws predict Kiessig fringes with intensity maxima for
Q ) Qcrit + 2mπ/e, m ) 1, 2, 3, ...
(3)
This equation agrees with SANS only if Qcrit ) 0. Nevertheless, there may be nocontradiction between the two descriptions because the Snell-Descartes laws assume a thick layer. For thin layers, it is known that there is no reflectivity plateau. In other words, one can assume qSANS,crit ) Qcrit ) 0. This could be verified by numerical simulations of the reflectivity of a thin layer. The observed q-4 law can also be explained by reflectivity. Reflection for Q > Qcrit can be approximated by a (Q/Qcrit)-4 variation, which is called the Fresnel law. The above equations reconcile the reflectivity interpretation with Porod’s interpretation. In both cases, for q . qcrit, the Born approximation is used, which assumes that each scatterer is subjected to the same planar wave. Using a two-dimensional detector for the measurements of a wet foam shows the validity of Porod’s law. The evaluation of the specific surface is the same for both descriptions. This supports the more general idea that Porod scattering and the Fresnel law are two descriptions of the same phenomenon.11 6. Conclusion Foams made of bubbles that are large (mm), compared to the sizes probed by SANS, give reliable SANS spectra in a few minutes. A q-4 decay of the scattering intensity dominates all the recorded spectra. Superimposed on this behavior are one or two maxima in the scattering plots. For wet foam a maximum appears at the same large-q value as in bulk solutions, which can therefore safely be attributed to scattering from the liquid in the walls and mostly in the Plateau borders. For dry foams, an additional maximum appears at low q; it cannot be attributed to anything else than the profile normal to the wall. The peak stays visible upon contrast matching between water and SDS, thus demonstrating that it is related to the wall thickness. This suggests an interpretation in terms of (11) Sinha, S. K.; Sirota, E. B.; Garoff, S.; Stanley, H. B. Phys. Rev., B: Condens. Matter Mater. Phys. 1988, 38, 2297.
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neutron interference between the scattering from the two surfaces of the soap film of the bubbles. The peak would then correspond to the film thickness. The q-4 scattering data can be interpreted as a SANS Porod scattering from the surfaces of the foam considered to be a porous medium. Although the specific area is very low here, in particular for drained foams, Porod’s law gives the right order of magnitude of the scattering intensity and allows determination of the average bubble size. Experiments on a two-dimensional XY detector show the progressive rise of a highly anisotropic signal (spikes) when the number of films in the beam is getting lower, i.e., during draining. This suggests and supports another origin of the scattering: Fresnel’s reflectivity of the incident beam by the bubble walls. Such a law also predicts a q-4 decay for q larger than the critical Qcrit, below which reflectivity is total. Reflectivity also predicts a series of maxima at low q. A closer look at the scattering by contrast matched films and using a q-4I(q) representation lets us detect not only one but three maxima. Both the Porod/ SANS and Fresnel/reflectivity interpretations of these peaks and of the q-4 variation agree well with each other if we assume Qcrit ) 0. This is predicted for very thin films. To go beyond the SANS versus reflectivity interpretation, we take a more factual and pragmatical point of view.
Axelos and Boue´
Isotropic SANS experiments on foams can give quick information on two different scales. One obtains the size of the bubbles from their specific area and at the same time one probes the internal structure of the Plateau border and the film thickness with the small-angle data. Anisotropic SANS measurements give additional information about the complicated interfacial wall profiles observed for SDS. It might be a way to characterize the orientations under mechanical deformation. Real-time kinetics may be performed directly on foams and appear very promising as a tool to follow the changes in film thickness and structure. Such a global observation is complementary to single film observation, since a foam is more than a simple collection of individual films. An interesting extension will be the addition of biopolymers such as polysaccharides or proteins to stabilized surfactant foams. Acknowledgment. We are grateful for the valuable discussions with Y. Popineau at INRA Nantes and A. Menelle and F. Cousin at LLB. We thank P. Papineau and A. Sire for their assistance in the conception and technical realization of the foam columns. LA020965R
