On How Surfactant Depletion during Foam Generation Influences

Article pubs.acs.org/Langmuir

On How Surfactant Depletion during Foam Generation Influences Foam Properties Julia Boos,† Wiebke Drenckhan,‡ and Cosima Stubenrauch*,† †

Universität Stuttgart, Institut für Physikalische Chemie, Pfaffenwaldring 55, 70569 Stuttgart, Germany Laboratoire de Physique des Solides, UMR 8502 − Université Paris-Sud, Bâtiment 510, 91405 Orsay Cedex, France

‡

S Supporting Information *

ABSTRACT: Although it is known that foaming a surfactant solution results in a depletion of the surfactant in the bulk phase, this effect is often overlooked and has never been quantified. Therefore, the influence of surfactant depletion on foam properties using solutions of the two nonionic surfactants, n-dodecyl-β-D-maltoside (β-C12G2) and hexaethyleneglycol monododecyl ether (C12E6), were investigated. These investigations were conducted in two steps. First, different foam volumes were generated with the same surfactant solution at a concentration of c = 2 cmc. It was found that the higher the foam volume, the larger the surfactant depletion. Second, two different bulk concentrations (c = 2 and 1.33 cmc) were used for the generation of 50 and 110 mL of foam, respectively. For a foam volume of 50 mL, no differences were observed, whereas generating 110 mL led to different results. The surfactant loss in the bulk solution was measured via surface tension measurements and then compared to the results of purely geometric considerations that take into account the amount of interface created in the foam. Both results were in very good agreement, which means that surfactant depletion can be calculated in the way suggested here. Under conditions where depletion plays a role, our approach can also be used to estimate the bubble size of a foam of known volume by measuring the surfactant concentration in the bulk solution after foaming. m2 containing a solution volume of about 20 mL, then depletion becomes measurable at surfactant concentrations below 10−6 M.5 In the case of foams, surface areas of several square meters can be generated easily, which in turn results in depletion effects at concentrations much larger than 10−6 M. For example, it is not possible to generate stable foams with an initial concentration of c > cmc if after foaming the concentration of the bulk solution cend is too far below the cmc as a result of depletion. As already mentioned above, depletion is not a general problem but depends on the experimental conditions. Depletion needs to be taken into account if large surfaces or interfaces are generated at sufficiently low surfactant concentrations. Because of the fact that emulsion droplets are typically 1 order of magnitude smaller than foam bubbles (i.e., the generated interfaces are 2 orders of magnitude larger), depletion is much more important for emulsions. Indeed, surfactant depletion during emulsion generation has been studied in the past,6−8 but corresponding studies for foam do not exist. However, because of the trend of decreasing surfactant concentrations in various applications, depletion

1. INTRODUCTION Foams are dispersions of gas in a foaming solution that typically contains surfactant as a stabilizing agent. The three main factors that limit the lifetime of foams are drainage, coarsening of bubbles, and bubble coalescence. Extensive studies on foam properties have been carried out to determine characteristic features such as the foamability, foam stability, bubble size, and liquid content, to mention just a few.1−4 Foamability describes the ability of a surfactant solution to produce a certain volume of foam in a given period of time. Once the foam is generated, its stability can be studied by monitoring the change in foam volume Vfoam or liquid fraction ε as a function of time t. Simultaneous measurements of bubble size and liquid fraction complement the picture but still do not answer all of the questions about foam properties. One of those questions relates to the depletion of a surfactant solution during foam generation. During the foaming process, the surfactant adsorbs at the water−air interface, which inevitably leads to a reduction of the bulk concentration. Whether this depletion changes the initial bulk concentration to a measurable extent depends not only on the initial bulk concentration itself but also on the total surface area that is generated. Such depletion effects can even play a role if only a single, large interface is present in a vessel. For example, if one measures the surface tension with the ring method in a vessel with a surface area of around A = 2.5 × 10−3 © 2012 American Chemical Society

Received: March 17, 2012 Revised: May 23, 2012 Published: June 13, 2012 9303

dx.doi.org/10.1021/la301140z | Langmuir 2012, 28, 9303−9310

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during foaming will become more practically relevant and thus deserves quantification. Moreover, in many systems the most stable foams are generated at relatively low concentrations because of the presence of non-negligible surface elasticities and the absence of depletion forces that destabilize foams at higher surfactant concentrations.9 The current study examines the depletion effects for two nonionic surfactants, namely, ndodecyl-β-D-maltoside (β-C12G2) and hexaethyleneglycol monododecyl ether (C12E6), at initial concentrations of c = 2 and 1.33 cmc. The critical micelle concentrations (cmc of βC12G2 = 1.5 × 10−4 M,10 cmc of C12E6 = 0.8 × 10−4 M11) are used throughout as reference concentrations for the foam studies. In all cases, a N2 gas flow rate of 50 mL min−1 was used to produce different foam volumes, namely, 20, 30, 50, 70, 80, 100, and 110 mL with bubble sizes of 200−300 μm. The two surfactants were chosen because foams generated by them have completely different properties, although both surfactants are nonionic and have very similar cmc values. However, depletion should affect both surfactants in the same way, which we wanted to show experimentally. The effect of depletion on the foam properties was examined using the commercially available FoamScan device, which uses image analysis and conductivity measurements to monitor foam properties such as foamability, foam stability, and liquid content. In addition, the bubble sizes and the bubble size distributions can be analyzed with the new cell-size analysis (CSA) function that allows the calculation of the Sauter mean radius r32 = ⟨r3⟩/⟨r2⟩6−8 needed to estimate the degree of depletion of the bulk solution. Our calculations of the depletion are based on purely geometrical considerations that allow an estimation of the total generated foam surface. However, one can easily determine the loss of surfactant in the bulk solution, caused by foam creation, via surface tension measurements of the remaining bulk solution. In our first series of measurements, a constant bulk concentration and a constant volume of the surfactant solution were used to create different foam volumes (i.e., to change the surfactant depletion via the amount of generated foam). In our second set of measurements, two different bulk concentrations were used to generate the same foam volumes, namely, 50 and 110 mL. The results will be discussed, and it will be shown that the measured and calculated values agree very well.

Figure 1. Bubble size evaluation with the original cell size analysis (CSA) picture obtained from the FoamScan device (left) and the cell skeleton calculated via ImageJ (right). (height of the solution Hs = 60 mm in the column) was foamed by sparging N2 at a constant gas flow rate Q of 50 mL min−1 through the surfactant solution. Different foam volumes Vend ranging from 20 to 110 mL were produced with a porous disk (pore sizes = 41−100 μm), leading to polydisperse foams with a narrow bubble size distribution that peaks at typical bubble radii of 200−300 μm. Note that per definition t = 0 s is the time at which the gas input is stopped. The foam volume Vfoam is monitored with the first CCD camera along with the conductivity of the liquid at the first electrode (Helec1 = 20 mm). The conductivity measurements are used to determine the liquid fraction using an empirical equation provided by Feitosa et al.14 (Feitosa et al.14 derived an empirical equation to calculate the liquid fraction ε from conductivities σel, namely, ε = 3σel((1 + 11σ)/(1 + 25σel + 10σel2)), where the relative electrical conductivity σel is defined as the conductivity of the foam Λfoam relative to the conductivity of the bulk liquid Λliquid (i.e., Λfoam/Λliquid.) The conductivity measurement and cell size analysis determination of the bubble size and bubble size distribution are carried out simultaneously at fixed positions along the glass column, namely, at the second electrode and the prism of the second CCD camera at heights of Helec2 = 85 mm (∼25 mm above the liquid level) and Hprism = 105 mm (∼45 mm above the liquid level), respectively. Note that it was not possible to calculate the liquid fraction ε for Vend = 20 mL because the conductivity electrode is not fully covered with foam in that case. In all cases, a surfactant concentration of either c = 2 or 1.33 cmc and a constant electrolyte concentration of 10−2 M NaCl were used. Please note that the liquid content is determined via electrical conductivity and thus a minimum electrolyte concentration of 10−2 M is required as explained in refs 15 and 16. All presented measurements were reproduced at least three times. For the sake of clarity, only one experiment for each Vend is shown. The reproducibility of the FoamScan method is shown in Figure S1 of the Supporting Information. All measurements were performed at room temperature (21 ± 1 °C). 2.2.2. Bubble Size Evaluation. Bubble sizes were calculated with the cell size analysis (CSA) software from TECLIS and freeware program ImageJ. A picture is recorded at the wall of the glass column using the CSA camera. A typical example is shown on the left in Figure 1. These images are treated using ImageJ to obtain a water-free skeleton (right side of Figure 1) by reducing the dark lines on the surface plateau borders to a line of one pixel width. Finally, the ImageJ skeleton picture is analyzed using the CSA software in order to obtain the distribution of cell sizes at the container surface. A long debate as to how the distribution of the cell radii of the 2D surface structure is related to that of the internal foam bubbles was recently settled by Wang and Neethling.17 They show that for sufficiently polydisperse foams with normalized standard deviations (polydispersity index) larger than 0.2 the distributions of the radii of the surface cells are well correlated with that of the 3D bubbles. Because the foams treated here have a polydispersity index larger than 0.23, we therefore rely on surface measurements for the bulk characterization. The skeletonization of the original photographs (Figure 1) is necessary to obtain a proper estimation of the bubble size distribution, which is otherwise obscured by the presence of the thick surface

2. EXPERIMENTAL SECTION 2.1. Materials and Cleaning Procedure. Nonionic sugar surfactant n-dodecyl-β-D-maltoside (β-C12G2) was purchased from Glycon, and nonionic surfactant hexaethyleneglycol monododecyl ether (C12E6) was purchased from Sigma. Both surfactants were used as received. NaCl (purity ≥99.5%, Merck) was roasted overnight at 500 °C to remove all organic content. All of the glassware was cleaned before use with deconex UNIVERSAL 11 (instead of chromic−sulfuric acid) from Borer Chemie and rinsed several times with distilled water to remove all impurities. The solutions were prepared with doubly distilled water at room temperature (21 ± 1 °C). 2.2. FoamScan Method. 2.2.1. Experimental Setup. This study was carried out with the commercially available FoamScan device (TECLIS, France, http://www.teclis.eu), which uses image analysis and conductivity measurements to monitor foaming properties (foamability, foam stability, and liquid content). In addition, the bubble sizes and bubble size distributions can be analyzed with the new commercially available cell size analysis (CSA) function (TECLIS, France, http://www.teclis.eu), which in turn allows for a visualization of the destabilizing processes (Figure 1). For a detailed description of the FoamScan method, the reader is referred to refs 12 and 13. Briefly, a rectangular glass column (square cross-section 25 × 25 mm2, length of column 298 mm) was used. An initial liquid volume of Vs = 40 mL 9304


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Figure 2. Variation of the foam volume Vfoam (left) and the liquid fraction ε (right) of surfactants β-C12G2 (top) and C12E6 (bottom) as a function of time t. Experimentally, an initial liquid volume of Vs = 40 mL with a concentration of c = 2 cmc was foamed with a constant N2 gas flow rate Q of 50 mL min−1 to different preset foam volumes of Vend = 30, 50, 70, 80, 100, and 110 mL. The time t = 0 corresponds to the time at which foam volume Vend is reached.

mL min−1 was used to produce different preset foam volumes Vend that ranged from 30 to 110 mL. In Figure 2 (left), the foam volume Vfoam is shown as a function of time t for solutions of nonionic surfactants β-C12G2 (top) and C12E6 (bottom) at the same initial concentration of c = 2 cmc. To obtain information about drainage, the time evolution of the liquid fraction ε was also monitored (Figure 2, right) directly after the foam was generated. The time t = 0 corresponds to the time at which the preset volume Vend is reached and where the gas flow is stopped. Thus, data obtained at t < 0 describe the foamability whereas those at t > 0 are a measure of foam stability if one assumes that destabilizing processes do not take place during foam generation. After the foaming process was stopped, the foam evolution was monitored as a function of time. As can be seen in Figure 2 (left), the foamability of β-C12G2 is considerably higher than that of C12E6, although both surfactants are nonionic and have similar cmc values. For example, looking at the data for the preset foam volume of Vend = 50 mL, one sees that the foamability of C12E6 is half that of β-C12G2. Moreover the non-linear increase in the foam volume indicates that the foam formed with C12E6 is not very stable and partially collapses during foam generation. This argument is supported by the fact that at Q = 50 mL min−1 no more than 50 mL of foam could be generated whereas for β-C12G2 a foam volume of Vend ≥ 110 mL can easily be created. Finally, the low stability is

plateau borders. For geometrical reason, the cross-section of a surface plateau border is about 3 times as large as that of a plateau border inside the foam. This surface effect is negligible for liquid fraction measurements across the entire foam, but it plays an important role in the image treatment of surface structures. Because the foams treated here have liquid fractions of only a few percent (ε < 5%), the error we make by using the skeleton is of the same order of magnitude. 2.3. Surface Tension Measurements. The equilibrium surface tension σ of the surfactant solution after foaming was measured by the Du Noüy ring method using an STA-1 tensiometer from Sinterface. At t = 0 (the time at which the preset foam volume is reached), the surfactant solution after foaming Vs,end was quickly sucked out of the FoamScan. This solution was diluted to half of its concentration with doubly distilled water and stirred for 5 min. The dilution of the solution was necessary to reach a concentration below the cmc, which in turn is required to determine concentrations from surface tensions. The surface tension curves of β-C12G2 (cmc = 1.5 × 10−4 M10) and C12E6 (cmc = 0.8 × 10−4 M11), which were used to calculate the concentrations, are shown in Figure S2 in the Supporting Information. The surface tension measurements were performed at room temperature (21 ± 1 °C).

3. RESULTS AND DISCUSSION 3.1. Foam Volume and Liquid Fraction. In the first series of measurements, a constant bulk concentration and a constant volume of surfactant solution were used to create different foam volumes (i.e., to change the surfactant depletion via the amount of generated foam). For all experiments, a gas flow rate Q of 50 9305


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Figure 3. Cell size analysis (CSA) pictures taken at time t = 0 (i.e., at the time at which the gas input is stopped because the preset foam volume Vend is reached). Different preset foam volumes Vend were generated with an aqueous solution of β-C12G2 at c = 2 cmc.

its large tendency toward coalescence and thus reflects its low stability. In Figure 4, the relative number of bubbles n/ntotal is plotted as a function of the radius, where n is the number of bubbles in the radius range and ntotal is the total number of bubbles (∼100 bubbles) in an area of A ≈ 20 mm2. For the sake of clarity, the bubble size distributions are shown only for preset foam volumes of Vend = 50, 80, and 110 mL. The results for all other foam volumes can be found in Table 1. As can be seen in Figure 4, bubble radii of 0.05 mm < r < 0.36 mm were observed for all foams. We characterize the polydispersity of the distributions using the Sauter mean radius

reflected in the steep, quick decay of both the foam volume and the liquid fraction whereas the corresponding data of foam formed with β-C12G2 are typical of very stable foams. In the latter case, the foam volume Vfoam is almost time-independent for 0 s < t < 1000 s regardless of the Vend value although the liquid fraction decreases significantly. In other words, the decay in the liquid fraction (drainage) has no effect on the foam volume, which in turn means that coalescence plays no significant role during the measurement time. All of these observations are in line with our previous work,15,16 which serves as a reference for qualitative comparison. With regard to the liquid fraction, one sees in Figure 2 that for foams stabilized by β-C12G2 the initial liquid fraction ε0 is approximately the same (0.041−0.047) for all preset foam volumes Vend (30−110 mL). When comparing this result with that obtained for C12E6, one sees that the ε0 values of foams formed by C12E6 solutions differ considerably with Vend and are much smaller (up to 1 order of magnitude). In addition, the decay of liquid fraction ε is much quicker and complete foam destruction is observed after 40−60 s. As seen in Figure 2 (right), the drainage rate for foams stabilized by β-C12G2 is almost the same for the different preset foam volumes Vend. Thus, one can conclude that foams formed from aqueous solutions of C12E6 are much drier and very unstable compared to those stabilized by β-C12G2. 3.2. Bubble Size Distribution. The bubble size distribution was studied using the new cell size analysis (CSA) function in combination with the ImageJ software as described in Section 2. CSA pictures of β-C12G2 foams (c = 2 cmc) at the wall of the glass column are shown for different foam volumes Vend = 50−110 mL in Figure 3. Note that independent of the amount of generated foam very similar bubble size distributions are obtained for all foams stabilized with β-C12G2, which will be quantified in the following text. For the sake of comparison, a CSA picture of a foam formed from a C12E6 solution is shown in Figure S3 in the Supporting Information. In comparing the pictures for Vend = 50 mL in Figure S3 and Figure 3, one sees that the C12E6 foam has much larger bubbles, which indicates

r32 =

⟨r 3⟩ ⟨r 2⟩

(1)

and the polydispersity index (or normalized standard deviation) PI =

⟨r 2⟩ − ⟨r ⟩2 ⟨r ⟩

(2)

The numerical results for both values are summarized in Table 1 for all preset foam volumes Vend. By looking at the values, one sees that the average bubble size and the bubble size distribution of foams stabilized by β-C12G2 are reasonably similar. For the Sauter mean radius, we find 0.2 < r32 < 0.27, and for the polydispersity index, we find 0.23 < PI < 0.3. However, for foams formed from C12E6 solutions totally different bubble sizes were measured. The bubble radii were between 0.14 and 0.45 mm with a Sauter mean radius of r32 ≈ 0.36 and PI ≈ 0.41. This is due to the presence of significant coalescence during foam generation. With the calculated Sauter mean radii r32 it is now possible to approximate the total surface area generated during the foaming process. This is crucial in determining the depletion of the bulk solution based on purely geometrical considerations as described in the following section. 3.3. Calculation of Surfactant Depletion during Foam Generation. During foam generation, surfactants from the 9306


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between depletion and surfactant coverage is much more complex.) The number Nsurf of molecules captured by the gas/ liquid surfaces is given by Nsurf =

S A mol

(3)

where S is the total surface area of the foam and Amol is the area per molecule, which, around and above the cmc, is 40 Å2 (4 × 10−19 m2) for β-C12G2 and 51 Å2 (5 × 10−19 m2) for C12E6 assuming a densely packed surfactant monolayer.18 The total depletion of a solution of volume Vs after foam generation can therefore be calculated as Δc = c − cend =

1 S VsNA A mol

(4)

where NA is Avogadro's number (= 6.022 × 1023 mol−1). This can be rewritten with S = kVg/r32 as Δc = c − cend =

1 Vg A mol NA r32 Vs k

(5)

where Vg is the total volume of the integrated gas and k is a constant that depends on the bubble shape because the surface to volume ratio for a bubble depends on its geometry. In foams, these variations tend to be small. For example, if all bubbles are assumed to be spherical, then k = 3, but if the bubbles are assumed to have the same volume and to be arranged in the Kelvin structure with negligible liquid content, then k ≈ 3.3 .3 Polydisperse foams tend to be well approximated by spheres,19 which is why we use k = 3 in the following analysis. As can be seen in eq 5, one needs to know the volume ratio of the dispersed and continuous phase in order to calculate the depletion. Although this ratio is easily obtained in the case of emulsions, it is more difficult to quantify in the case of foams. Assuming that all liquid is contained in the foam, one can introduce the liquid fraction ε and rewrite eq 5 as Δc = c − cend =

k 1 (1 − ε) ε A mol NA r32

(6)

This, however, is not the case in our measurements, where (unlike in emulsions) − due the large difference in density − the gas bubbles cream rapidly towards the surface of the liquid, creating a well-separated foam on top of a deep liquid pool. The maximum liquid fraction of this foam can be approximated by ε ≈ 0.36 in the case of modest polydispersity, which corresponds to randomly close-packed spheres. In reality, however, most foams drain strongly because of their large bubble sizes and have liquid fractions of ε < 15%, meaning that the liquid volume contained in the foam is negligible in comparison to that contained in the pool below the foam. For this reason, it is reasonable to approximate the gas volume Vg by the foam volume Vfoam. The equation used by us in the following text is therefore

Figure 4. Relative number of bubbles n/ntotal as a function of the bubble radius r for β-C12G2 foams (c = 2 cmc) shown in Figure 3. Calculations were made using the CSA software and the ImageJ analysis of pictures taken at time t = 0 (time at which the preset foam volume Vend is reached). Numerical results are shown in Table 1.

c − cend Δc 1 3 1 Vfoam = = cmc cmc cmc A mol NA r32 Vs

bulk are transported to the water−air interface, which leads to the depletion of the bulk solution. In the following text, the depletion during the foaming process will be calculated. In our approach, we assume that the surfactant has been able to coat the bubble surfaces i.e., that it diffuses rapidly to the newly generated interface. (In systems with slowly diffusing molecules e.g., protein-stabilized foams or with very low bubble residence times in the liquid, there may not be sufficient time for the surfactant to coat the bubbles. In this case, the relationship

(7)

In Figure 5, we use Vfoam = Vend in order to compare the depletion predicted by eq 7 with that measured directly from surface tension measurements of the remaining bulk phase, following the protocol described in Section 2.3. The measured and calculated values for both surfactants show excellent agreement and are summarized in Table 1. A concentration of c 9307


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meas Table 1. Calculated ccalc end and Measured cend Concentration of the Bulk Phase after Foaming Aqueous Solutions of β-C12G2 and C12E6a

Vend/mL r32/mm PI ccalc end/cmc cmeas end /cmc (Nsurf)/ (Nbulk) Vend/mL r32/ mm PI ccalc end/cmc ccalc end/cmc (Nsurf)/ (Nbulk)

20 0.23

30 0.23

∼1.82 ∼1.78 (6.5 × 1017)/ (7.2 × 1018)

∼1.73 ∼1.70 (9.8 × 1017)/ (7.2 × 1018) 30 0.36 ∼1.75 ∼1.72 (4.9 × 1017)/ (3.8 × 1018)

50 0.20 0.27 ∼1.48 ∼1.48 (1.9 × 1018)/ (7.2 × 1018)

β-C12G2 70 0.22 0.29 ∼1.34 ∼1.38 (2.4 × 1018)/ (7.2 × 1018) C12E6

80 0.23 0.27 ∼1.28 ∼1.27 (2.6 × 1018)/ (7.2 × 1018)

100 0.24 0.30 ∼1.14 ∼1.16 (3.1 × 1018)/ (7.2 × 1018)

110 0.27 0.23 ∼1.15 ∼1.17 (3.1 × 1018)/ (7.2 × 1018)

50 0.36 0.41 ∼1.58 ∼1.52 (8.2 × 1017)/(3.8 1018)

All measurements were carried out with a solution volume of Vs = 40 mL at a constant N2 gas flow rate Q of 50 mL min−1 and a concentration of c = 2 cmc. In all experiments a constant electrolyte concentration of 10−2 M NaCl was used. The preset foam volume Vend was varied from 20 to 110 mL. Nsurf is the calculated number of required surfactant molecules at the foam surface and Nbulk the total number of surfactant molecules in the bulk phase. All calculations were made according to Section 3.3 using the Sauter mean bubble radius r32 and the molecular area Amol (β-C12G2 = 40 Å2, C12E6 = 51 Å2 18) at the bulk water-air interface. a

In this case, a concentration of c = 2 cmc corresponds to 3.8 × 1018 C12E6 molecules in 40 mL of the bulk solution. Here ∼15% is necessary to cover the surface of 20 mL of foam, which leads to a final concentration of ccalc end = 1.7 cmc. As is obvious from Figure 5 and Table 1, measured and calculated depletion values are in excellent agreement for both surfactants (see Table 1 for C12E6), meaning that calculations based on purely geometrical considerations are very accurate despite the approximations made (spherical bubbles and negligible liquid content in the foam). From the very good agreement between the calculated and the measured depletion, one can conclude that the assumed space requirements for βC12G2 (40 Å2) and C12E6 (51 Å2) are reasonable. Because these surface areas correspond to densely packed layers, one can argue that the surfactant layers in the generated foam are also densely packed. Please note that the linear correlation in Figure 5 is not universal. The slope of the line depends on the volume of the surfactant solution Vs (eq 7). 3.4. Influence of the Initial Surfactant Concentration. The comparison of β-C12G2 foams with different volumes Vend might be a problem because of the measurements at different relative heights (i.e., it was always the same prism and the same electrode in the column used). This means that for a tall foam (Vend = 110 mL) the difference between the top of the foam and the used electrode/prism is larger than for the small 50 mL foam. To obtain comparable results, a second series of measurements were carried out, where two different β-C12G2 bulk concentrations (c = 2 and 1.33 cmc) were used for the generation of 50 and 110 mL of foam, respectively. The respective Vfoam(t) and ε(t) curves for concentration c = 1.33 cmc are shown in Figure S4 of the Supporting Information. One can see that foam behavior equal to that for c = 2 cmc was observed. To learn more about the foam structure, the bubble sizes and the bubble size distributions were studied for an initial bulk concentration of c = 1.33 cmc. The results are illustrated in Figure 6. The relative number of bubbles n/ntotal is plotted as a function of the radius r, where n is the number of bubbles in the radius range and ntotal is the total number of bubbles (∼100 bubbles) in an area of A ≈ 20 mm2. Figure 6 (left) shows the

Figure 5. Measured and calculated amount of β-C12G2 surfactant molecules lost from the bulk phase Δc = c − cend during foaming as a function of Vend/Amolr32 cmc, where c = 2 cmc is the concentration of the used β-C12G2 surfactant solution and cend is the concentration of the bulk phase after foaming (eq 4). Numerical results are shown in Table 1.

= 2 cmc corresponds to 7.2 × 1018 β-C12G2 molecules in 40 mL of the initial bulk solution. To create a small foam volume of Vend = 20 mL with r32 = 0.23 mm, around 10% of these β-C12G2 molecules are needed to cover the whole surface with a densely packed surfactant layer, whereas the percentage required for a foam volume of Vend = 110 mL is as high as 45%. Therefore, the initial bulk concentration of c = 2 cmc is reduced for different foam volumes to a final concentration ccalc end of between 1.8 and 1.1 cmc. For foams formed from C12E6 solutions, a depletion of the bulk solution due to the foaming process is also observed. 9308


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Figure 6. Comparison of bubble size distributions for β-C12G2 foams at c = 1.33 cmc (foam pictures are shown in Figure S5 in the Supporting Information) and c = 2.0 cmc. Calculations were made using the CSA software and the ImageJ analysis of pictures taken at time t = 0 (time at which the preset foam volumes of Vend = 50 mL (left) and 110 mL (right) are reached). Numerical results are shown in Table 2.

bubble size distributions for a preset foam volume of Vend = 50 mL for the two initial concentrations of c = 2 and 1.33 cmc. One can see that the two bubble size distributions are very similar. The numerical results of the Sauter mean radius r32 and the polydispersity index PI of the distribution are shown in Table 2. On the contrary, the generation of Vend = 110 mL led

surfactant solution and (b) the same amount of foam with surfactant solutions of different concentrations. The depletion has been determined both theoretically and experimentally. In the former case, we calculated the amount of surfactant required to generate the preset foam volume. For these calculations, the Sauter mean bubble radius and the surface area per surfactant are needed: we extracted the bubble size from the analysis of foam images and used surface areas that correspond to densely packed surfactant layers. However, we determined the depletion experimentally via surface tension measurements. A comparison reveals very good agreement between the calculated and measured values. In other words, under conditions where depletion plays a role, the method suggested here allows an estimation of the average bubble size of a foam of known volume by measuring the surfactant concentration in the bulk solution after foaming. Furthermore, our measurements show that for the prediction of surfactant depletion it is essential to use the Sauter mean radius r32 instead of the arithmetic mean radius ⟨r⟩. Even for modest polydispersity indices of around 0.3, we found a significantly larger discrepancy between the calculated and measured depletion values if we used ⟨r⟩ instead of r32 for the calculations (results not shown). Before concluding, we emphasize here that depletion effects in foams should be taken seriously, even when working with surfactant concentrations well above the cmc. Using eq 5, we can provide some general approximations. Taking into account that (a) typical bubble radii in foams range from 10 μm (shaving foam, whipped cream) to 1 mm (dish-washing foam), (b) typical gas to liquid volume ratios Vg/Vs are between 1 and 100, and (c) typical molecule areas Amol are on the order of 50 Å2, one can approximate that depletion is in the range of 10−5− 1 M. Considering that typical cmc values for standard surfactants are of the order of 10−4−10−6 M, one recognizes that the lower limit of the approximation is of the order of the magnitude of the cmc, whereas the upper limit corresponds to at least 104 times the cmc! Hence, as in the case of emulsions,6−8 foam scientists need to take into account the possible influence of depletion effects.

meas Table 2. Calculated ccalc end and Measured cend Concentration of the Bulk Phase after Foaming an Aqueous Solution of βC12G2a

c = 2 cmc Vend/mL r32/mm PI ccalc end/cmc cmeas end /cmc

50 0.20 0.27 ∼1.54 ∼1.47

c = 1.33 cmc 110 0.27 0.23 ∼1.15 ∼1.17

50 0.29 0.44 ∼0.97 ∼0.99

110 0.29 0.47 ∼0.54 ∼0.72

a All measurements were carried out with a solution volume of Vs = 40 mL at a constant N2 gas flow rate Q of 50 mL min−1 and a concentration of c = 2 cmc (right) and c = 1.33 cmc (left). In all experiments a constant electrolyte concentration of 10−2 M NaCl was used. The preset foam volume Vend was varied between 50 mL and 110 mL. All calculations were performed according to Section 3.3 using the Sauter mean radius r32 and the molecular area Amol (β-C12G2 = 40 Å2 18) at the bulk water−air interface.

to significant differences in the bubble size distribution (Figure 6 right). The foam created with the initial concentration of c = 2 cmc shows a narrow distribution whereas the 110 mL foam with the initial concentration of c = 1.33 cmc shows a broader distribution. This increased polydispersity of the foam bubbles and the reproducibility of the bubble size distribution are shown in Figures S5 (right) and S6 in the Supporting Information. However, all calculated and measured (via surface tension) depletions are in good agreement because the Sauter mean radius takes into account the polydispersity of the foam.

4. CONCLUSIONS The generation of a foam is accompanied by the generation of a large water−air interface, which in turn depletes the corresponding surfactant solution. Although this effect is well known, it has not been quantified yet. To fill this gap, we generated (a) different amounts of foam with the same 9309


Langmuir

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Article

(18) Patil, S. P.; Buchavzov, N.; Carey, E.; Stubenrauch, C. Binary Mixtures of β-Dodecylmaltoside (β-C12G2) with Cationic and NonIonic Surfactants: Micelle and Surface Compositions. Soft Matter 2008, 4, 840−848. (19) Kraynik, A. M.; Reinelt, D. A.; van Swol, F. Structure of Random Foam. Phys. Rev. Lett. 2004, 20, 208301/1−208301/4.

ASSOCIATED CONTENT

S Supporting Information *

The results of additional measurements that were carried out to support the main ideas. Complementary measurements are reproducability tests, surface tensions, and additional CSA pictures as indicated throughout the article. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +49-(0)711-685-64470. Fax: +49-(0)711-685-64443. Email: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Dr. Enda Carey, Dr. Anniina Salonen, and Dr. Emmanuelle Rio for helpful discussions. REFERENCES

(1) Bikerman, J. J. Foams; Springer-Verlag: Berlin, 1973. (2) Exerowa, D.; Kruglyakov, P. M. Foam and Foam Films − Theory, Experiments, Application; Elsevier: Amsterdam, 1998. (3) Weaire, D.; Hutzler, S. The Physics of Foams; Clarendon Press: Oxford, U.K., 1999. (4) Langevin, D. Aqueous Foams: A Field of Investigation at the Frontier Between Chemistry and Physics. ChemPhysChem 2008, 9, 510−522. (5) Makievski, A. V.; Grigoriev, D. O. Adsorption of Alkyl Dimethyl Phosphine Oxides at the Solution/Air Interface. Colloids Surf., A 1998, 143, 233−242. (6) Tcholakova, S.; Denkov, N. D.; Sidzhakova, D.; Ivanov, I. B.; Campbell, B. Interrelation between Drop Size and Protein Adsorption at Various Emulsification Conditions. Langmuir 2003, 19, 5640−5649. (7) Tcholakova, S.; Denkov, N. D.; Danner, T. Role of Surfactant Type and Concentration for the Mean Drop Size during Emulsification in Turbulent Flow. Langmuir 2004, 20, 7444−7458. (8) Tcholakova, S.; Denkov, N. D.; Lips, A. Comparison of Solid Particles, Globular Proteins and Surfactants as Emulsifiers. Phys. Chem. Chem. Phys. 2008, 12, 1608−1627. (9) Pugh, R. J. Foaming, Foam Films, Antifoaming and Defoaming. Adv. Colloid Interface Sci. 1996, 64, 67−142. (10) Schlarmann, J.; Stubenrauch, C. Stabilization of Foam Films with Non-Ionic Surfactants: Alkyl Polyglycol Ethers. Tenside Surf. Det. 2003, 40, 190−195. (11) Schlarmann, J.; Stubenrauch, C.; Strey, R. Correlation between Film Properties and the Purity of Surfactants. Phys. Chem. Chem. Phys. 2003, 5, 184−191. (12) Carey, E.; Stubenrauch, C. Properties of Aqueous Foams Stabilized by Dodecyltrimetylammonium Bromide. J. Colloid Interface Sci. 2009, 333, 619−627. (13) Carey, E.; Stubenrauch, C. A Disjoining Pressure Study of Foam Films Stabilized by Mixtures of Nonionic (C12DMPO) and an Ionic Surfactant (C12TAB). J. Colloid Interface Sci. 2010, 343, 314−323. (14) Feitosa, K.; Marze, S.; Saint-Jalmes, A.; Durian, D. J. Electrical Conductivity of Dispersions: From Dry Foams to Dilute Suspensions. J. Phys.: Condens. Matter 2005, 17, 6301−6305. (15) Stubenrauch, C.; Shrestha, L. K.; Varade, D.; Johansson, I.; Olanya, G.; Aramaki, K.; Claesson, P. Aqueous Foams Stabilized by NDodecyl-β-D-maltoside, Hexaethyleneglycol Monododecyl Ether. Soft Matter 2009, 5, 3070−3080. (16) Stubenrauch, C.; Claesson, P. M.; Rutland, M.; Manev, E.; Johansson, I.; Pedersen, J. S.; Langevin, D.; Blunk, D.; Bain, C. D. The relationship between the surface and internal structure of dry foam. Adv. Colloid Interface Sci. 2010, 155, 5−18. (17) Wang, Y.; Neethling, S. J. Colloids Surf., A 2009, 339, 73−81. 9310


On How Surfactant Depletion during Foam Generation Influences

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