Controlled foam generation using cyclic diphasic flows through a constriction

Accepted Manuscript

Controlled foam generation using cyclic diphasic flows through a constriction T. Gaillard, M. Roche, ´ C. Honorez, M. Jumeau, A. Balan, C. Jedrzejczyk, W. Drenckhan PII: DOI: Reference:

S0301-9322(16)30212-9 10.1016/j.ijmultiphaseflow.2017.02.009 IJMF 2549

To appear in:

International Journal of Multiphase Flow

Received date: Revised date: Accepted date:

8 April 2016 20 February 2017 22 February 2017

Please cite this article as: T. Gaillard, M. Roche, ´ C. Honorez, M. Jumeau, A. Balan, C. Jedrzejczyk, W. Drenckhan, Controlled foam generation using cyclic diphasic flows through a constriction, International Journal of Multiphase Flow (2017), doi: 10.1016/j.ijmultiphaseflow.2017.02.009

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Controlled foam generation using cyclic diphasic flows through a constriction

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T.Gaillarda , M.Roch´ea,b , C.Honoreza , M.Jumeaua , A.Balana , C.Jedrzejczyka , W.Drenckhana a

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Laboratoire de Physique des Solides, CNRS UMR 8502 and Universit´e Paris Sud, Bˆ atiment 510, 91405 Orsay Cedex, France b Now at Laboratoire Mati`ere et Syst`emes Complexes, CNRS UMR 7057 and Universit´e Paris Diderot, Bˆ atiment Condorcet, 75205 Paris Cedex 13, France

Abstract

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Numerous industrial and academic applications of liquid foams require a fine control over their bubble size distribution and their liquid content. A particular challenge remains the generation of foams with very small bubbles and low liquid content. A simple technique which fulfils these different criteria, the “double syringe technique”, has been exploited for decades in hospital applications. In this technique, the foaming liquid and gas are pushed repeatedly back and forth through the constriction that connects two syringes. After having motorised the technique we investigate here the influence of the different processing conditions on the obtained foam properties in a quantitative manner. We show that this technique is unique in producing foams with the same characteristic bubble size distributions over a wide range of processing conditions (tubing, fluid velocities,...), making it an ideal tool for controlled foam generation. In contrast to other techniques, the liquid fraction in the double syringe technique can be varied without impacting the bubble size distribution. Using high-speed imaging we show that bubbles are dispersed in the aqueous phase at two different places in the device. Through an analysis based on the estimation of the characteristic dimensionless numbers of the flow we bring some insights on the potential hydrodynamic instabilities at play in the dispersion process. We compare our experimental results with bubble size distributions predicted by hydrodynamic fractionation processes. Keywords: Foam generation, Diphasic Flow, Bubble size distribution, Flow pattern, Hydrodynamic instabilities

Preprint submitted to International Journal of Multiphase Flow

February 22, 2017

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1. Introduction

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Liquid foams consist of closely packed gas bubbles within a continuous liquid phase. Due to their rich properties they are at the core of numerous fundamental research problems [4, 37, 25] and of a wide range of applications [32]. The properties of foams depend on the average bubble size, the bubble size distribution and on the liquid fraction, φ (given by the ratio of the liquid volume over foam volume), and thus depend on the mechanism used to generate the bubbles in the liquid phase. A wide range of foaming techniques are routinely applied in research laboratories or for applications [11], ranging from physical (break-up of bubbles under shear, phase transitions, etc.), via chemical (gas releasing reactions) to biological techniques (yeast). A major issue in the use of these techniques is the fact that bubble size and liquid fraction are not only tightly coupled, but that it is also inherently difficult to obtain control and reproducibility over the foam properties. Last but not least, a key challenge lies in the reliable generation of small bubble foams with low liquid fraction. A promising technique - at least for small foam volumes - has been developed about 40 years ago for foam-based sclerotherapy in hospitals [39]. As shown in Figure 1, the technique consists of two connected syringes containing a well-defined volume of gas VG and liquid VL , which are connected by a short piece of tubing. The foam is then generated by pushing both liquid and gas repeatedly through the connecting tube. Similar techniques are used for the generation of emulsions [30, 31]. Figure 2 shows that foams obtained with the double syringe technique have the visual appearance of a typical white shaving foam: the foam scatters light indicating the presence of numerous small bubbles. When care is taken to avoid the destabilisation of the foam during and just after its generation, we finds the characteristic bubble size to be of the order of 20 microns for the three foams shown in Figure 2. Traditionally, these foams are generated “by hand”. Nevertheless, most users have reported an astounding reproducibility in the foam properties obtained for different experimental runs, different experimentalists and even different syringes and connectors. One main goal of our work was therefore to characterise the foams obtained with the double-syringe technique under various conditions. For this purpose we automated the technique to obtain precise control over the processing parameters. In Section 2 we describe how we performed our experiments using two different devices. The first of these devices applies a sinusoidal velocity profile 2

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Figure 1: Principle of the two-syringes technique (a) Initial state (b) After one push: bubbly Liquid with tiny bubbles (c) Foam after several pushes. The liquid fraction is defined as the ratio between the liquid volume VL and the foam volume VL + VG with VG the gas volume.

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to the fluids while the second one applies a squared velocity profile. The frequency of these profiles is adjustable leading to different pushing velocities U . We used these machines to investigate the influence of the device parameters (Section 3.1) and of the foam formulation (Section 3.2). We confirm the robustness of the bubble size distribution. In Section 4 we report high-speed observations of the foaming process within the device from which we suggest possible bubbling mechanisms. A better understanding of these mechanism may lead to the design of efficient and well-controlled foaming techniques to provide foams of adjustable bubble size and liquid fraction, ideally in a continuous manner to remove the finite foam volume restriction currently imposed by the use of the syringes.

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2. Materials and Methods

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2.1. Materials Most of the experiments were conducted with solutions of the anionic surfactant Sodium Dodecyl Sulfate (SDS, Sigma-Aldrich, used as received). The solutions were prepared by dissolving SDS in ultra-pure water at 23 g/L which corresponds to ten times the critical micellar concentration (CMC). Ultra-pure water with a resistivity greater than 18 M Ω.cm was obtained from a Millipore Simplicity 185 filtering device. For each experiment freshly prepared solutions were used to avoid surfactant hydrolysis. To probe the importance of the foaming liquid we varied the viscosity of the SDS solution by 3

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Figure 2: Typical foams obtained for various liquid fractions φ using the double syringe technique of Figure 1. (a) φ = 30% (b) φ = 10% (c) φ = 3%.

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adding different amounts of glycerol (Fischer Scientific, purity > 99%). Solutions were made of 30 wt% and 50 wt% of glycerol, which leads to solution viscosities of 2.4 mPa.s and 5.8 mPa.s respectively (at room temperature). In order to establish the influence of the foam stabiliser we used different surfactant mixtures. These include the cationic surfactant TTAB (at 10 CMC), the non-ionic sugar-based surfactant β − C12 G2 (at 10 CMC), a surfactant mixture known to make very rigid interfaces and highly stable foams (the “Bulgarian mixture” [18]), and the commercial dishwashing liquid “Fairy” (at 10 wt%). Common names together with abreviations, values of the critical micelle concentrations, and equilibrium surface tension of all these stabilising agents are summarised in Table 1. Nearly all experiments were done using nitrogen gas with traces of C6 F14 (Sigma-Aldrich) in order to reduce gas exchange between bubbles. Finally to compare the influence of the gas on the foaming mechanism as well as on the ageing of the foam during the generation experiments were conducted using pure C2 F6 or CO2 (Air Liquide).

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2.2. Foam generation To generate the foams, the entire body of one syringe (60 mL, Codan Medical) is filled with both the solution (volume VL ) and the gas (volume VG ) in proportions corresponding to the initial liquid fraction, i.e φ = VL /(VL + VG ). In the case of the pure gases the gas is introduced in the syringe by simply connecting it to the gas tank of interest, whereas in the case of C6 F14 , which is liquid at room temperature but extremely volatile, the syringe is 4

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Abbreviation CMC (g/l) SDS 2.3 TTAB 1.3 C12 G2 0.1 Bulgarian mixture Fairy -

Table 1: Stabilising agents.

Surface tension γ (mN/m) 36 38 34 23.8 31

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Surfactant Sodium Dodecyl Sulfate [22] Tetradecyl Trimethyl Amonium Bromide [21] β − C12 G2 [17] SLES+betaine+MAc [18] Fairy dishwashing Liquid [10]

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connected to a bottle containing the liquid C6 F14 and some gas is bubbled slowly through it in to the syringe. It is important to note that the liquid fraction φf of the obtained foam has a lower limit, i.e. if too much gas is placed in the syringe the generated foam will not fill completely the syringe and therefore the final liquid fraction of the generated foam will not correspond to the initially placed liquid fraction in the syringe. In most experiments the gas finally gets entirely included within the liquid, so that φf = φ and we will discuss (in Section 3.1) how this initial liquid fraction φ influences the foaming mechanism. Both syringes are horizontal and this first syringe is then connected to the second one, whose piston is in the fully closed position, as shown in Figure 3a. In our case this connection is ensured by a tubing of variable length and diameter connected to the outlet of the syringes. For most cases we use rigid PVC tubing with an inner diameter of 3.5 mm and a length of 20 cm. The outlets of the syringes, being of a diameter smaller than the tubing used, form constrictions in the flow path (Figure 17). Foaming action occurs upon repeatedly pushing the gas/liquid mixture through the connection. In order to automate this pushing action we built two types of machines. The first one (Figure 3a) is made of two hydraulic pistons going alternatively back and forth at a constant velocity that can be set between 10 and 50 mm.s−1 . The second device, shown in Figure 3b, is made of two rotating arms which perform the pushing action on the pistons. The piston velocity varies therefore in a sinusoidal manner and depends on the frequency f . We use two frequencies, f = 1/3 s−1 and f = 1/6 s−1 . Since the piston path in the syringe is 100 mm long, this gives a pushing velocity which goes up to about 100 mm.s−1 (resp. 50 mm.s−1 ) at the point of maximum velocity and an average velocity of 66 mm.s−1 (resp. 33 mm.s−1 ). 5

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Figure 3: Photographs of the two devices which are used for foam generation: (a) Two hydraulic pistons which perform a pushing action on the syringe pistons at constant, but adjustable, velocity (10 - 50 mm.s−1 ). (b) Crankshaft set-up. Typical rotation frequencies 1/6-1/3 s−1 . The average piston velocities used here are then 33 and 66 mm.s−1 , leading to peak velocities of 50 and 100 mm.s−1 , respectively.

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2.3. Bubble size measurement After the generation, a small amount of foam is deposited rapidly on a microscope cover slide together with a few drops of the foaming solution. These drops ensure a good separation of the bubbles when the second microscope slide is put on the top to spread the foam. The two microscope slides are separated by plate spacers of height 150 µm as sketched in the inset of Figure 4a. Separating the bubbles in this way is not only helpful for the imaging purpose, but it also reduces foam ageing effects during the capture of the image. To provide good statistics on the bubble size distribution we ensured the measurement of at least a thousand of bubbles, with a typical count of 10000. To do so we use a digital microscope (Keyence VHX Series 2000) that scans a complete layer of thousand of bubbles with micrometric resolution. A small section of a typical image is shown in Figure 4a. To obtain the size of the bubbles, the images are first thresholded and then processed by combining binarisation with a watershed algorithm using the ImageJ software (freely available at http://imagej.nih.gov/ij/). After this procedure the bubbles appear as black circles (Figure 4b), which are then detected by the “Analyze particle” tool of the ImageJ software in order to obtain the corresponding bubble radius distribution. Bubbles which are larger than the spacing between the two microscope cover slides (150 µm) 6

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are flattened into pancake shape and a small correction must be done to obtain their true volume. More details on this correction can be found in a previous paper [14]. The resolution of the image processing is reliable down to bubble of radius of 3 µm. The full processing finally gives bubble size distributions like the on shown in Figure 4c. These distributions are then converted into probability density functions (PDF) in number (PDFN ) or in volume (PDFV ), defined respectively as: N (R < RB < R + ∆R ) , Ntot ∆R

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PDFV =

V (R < RB < R + ∆R ) , Vtot ∆R

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PDFN =

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where N (R < RB < R + ∆R ) is the number of bubbles of radius RB between R and R + ∆R , V (R < RB < R + ∆R ) the total volume of the bubbles of radius RB between R and R + ∆R , Ntot the total number of bubbles, Vtot the total volume of the bubbles, and ∆R the bin size of the histogram. When considering the physical properties of the generated foam it is more relevant to discuss volume distributions rather than number distributions. However as our interest is mostly to understand the mechanisms of generation of the bubbles we will mainly discuss the number distributions (designed simply as PDF from now on). Moreover our imaging technique cannot capture both small (20 µm) and big bubbles (1 mm) so volume distribution and number distribution appear to be quite similar. These will be discussed in Section 3.3. 3. Results

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Using both setups described in Section 2.2 we investigate a set of parameters likely to affect the foam generation. In Section 3.1 using exclusively SDS-stabilised foams we investigate the role of the pushing velocity U and of the number of pushing cycles N , together with the influence of the initial liquid fraction φ incorporated in the syringe. We then explore the influence of both the length L and the diameter D of the tubing connecting the two syringes. In Section 3.2 we investigate the influence of the physical-chemistry of the system, including the viscosity η of the liquid and the nature of the foam stabiliser. All results are complemented in Section 4 by observations obtained

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Figure 4: Bubble size measurement technique: (a) Cropped image obtained using the Keyence microscope (b) Corresponding image after binarisation in combination with a Watershed algorithm. (c) Obtained bubble size distribution.

with high-speed imaging and an analysis of the associated non-dimensional numbers.

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3.1. Dependence of the bubble size distribution on device parameters Initial liquid fraction φ and number of pushing cycles N : In both devices foam is generated already during the first pushing cycle, whatever the pushing conditions. If the initial liquid fraction φ in the syringe is high enough, the entire syringe is filled with foam right after the first cycle. This is shown in Figure 5a for the example of φ = 15%. However, if only a small amount of liquid is present, it takes several cycles N to incorporate all the gas into the liquid. For example N = 15 for φ = φf = 3% as shown on Figure 5a. The precise number of cycles depends not only on the amount of liquid, but also on the pushing speed U . The higher U , the more rapidly the gas is fully integrated into the liquid. It is interesting to analyse the bubble size distribution of the generated foam. After the first cycle, a few large bubbles (about 1 mm) are present in the foam, surrounded by numerous tiny bubbles. Since the few large bubbles are two orders of magnitude larger than the small bubbles, we cannot image both populations simultaneously. This is why they do not appear in the distributions, even though at low pushing speed they may take up a significant volume of the foam during the first cycles.The bubble size distribution of the small bubbles are shown in Figure 5b for both devices, after different num8

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bers of cycles and for φ = 15%. After about 10 - 15 cycles all large bubbles have typically disappeared. When taking into account only the population of small bubble, a general trend can be observed already after the first cycle. A characteristic size distribution appears which shows a nearly exponential decay towards larger bubbles and a steep fall-off towards smaller bubbles defining a peak at a characteristic bubble size of 10 - 20 µm. This peak seems more pronounced for the sinusoidal device. With increasing number of cycles N and for N < 12, the distribution becomes slightly narrower while the peak position remains constant, leading to a small drop in the average bubble radius hRB i. This evolution is shown in Figure 5c for the sinusoidal device for two different liquid fractions (φ = 3% and φ = 15%). Interestingly, the normalised bubble size distributions (Figure 5d) seem fairly invariant during the first 1-12 cycles, even though one can see a difference between the two devices. After about 10 cycles (depending on the device and pushing speed), the distribution remains narrow but the cut-off becomes smoother and the peak shifts to slightly larger bubble sizes. Both peak and average bubble radius increase with increasing number of cycles reaching a constant values around 15 cycles. We think that foam ageing does not play an important role in fixing the initial distribution (after the first cycle), but ultimately it shoulds balance the foam generation mechanisms and have a role in fixing this equilibrium distribution. These foam aggeing effects are discussed in more details in Appendix 6.1. The bubble size distribution after a large number of cycles, which we shall from now on call the “equilibrium distribution”, depends little on the foaming device and the liquid fraction of the foam, as shown in Figure 6. Moreover, when this equilibrium distribution is reached, there are no more big bubbles within the foam.

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Pushing velocity U : Let us now discuss the influence of the piston velocity U on the bubble size distribution obtained after the first push N = 1. This is shown in Figure 7 for both devices for a liquid fraction of φ = 15%. In both of them it seems that the characteristic peak position does not vary significantly with the pushing velocity. The width of the distribution however decreases with increasing pushing velocity in the constant velocity device. While an influence of the pushing velocity may be detected during the first cycles, it has negligible influence on the equilibrium distribution after a large number of cycles.

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Figure 5: (a) Filling fraction (foam volume / syringe volume) as a function of the number of cycles N for two different liquid fractions for the sinusoidal device (f = 1/3 s−1 ). (b) Influence of the number of cycles N and pushing type (constant velocity or sinusoidal velocity) on the obtained bubble size distributions for φ = 15%. (c) Evolution of the average bubble size < RB > and the peak value with the number of cycles N in the sinusoidal device (f = 1/3 s−1 ) for two different liquid fractions. (d) Normalised bubble size distributions from (b) in semi-log scale for both devices (U = 50 mm.s−1 and f = 1/3 s−1 ).

The influence of the velocity U on a foaming process is often capture via the power input PI of the process. For our devices we may define it as PI = Q∆P = U SP ∆P, 10

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Figure 6: “Equilibrium” distributions for both devices (U = 50 mm.s−1 and f = 1/3 s−1 ) after 15 cycles for different liquid fractions.

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Figure 7: Influence of the pushing type and speed on the bubble size distribution after 1 cycle in the constant-velocity device (top) and in the sinusoidal-velocity device (bottom) for φ = 15 %.

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where ∆P is the pressure difference between the syringes, Q is the average flow rate and SP the cross-section of the syringe. Because of the friction of the pistons on the wall of the syringes the motion of the two piston is not perfectly synchronised and the gas is compressed. We can calculate that ∆P ≈ 1.2 bar. We find that during the first cycle, this value depends little on the pushing velocity. We can therefore estimate the power input to vary from 5 Watt at 10 mm.s−1 to 20 Watt at 50 mm.s−1 . This is a variation by a factor of four, which does not seem to be correlated with the obtained bubble sizes. However the effect of an higher power input might be to generate more bubbles during a cycle which is indeed what we observe since more cycles are needed to produce the foam at lower pushing velocity. Tubing dimensions: In order to investigate the influence of the dimensions of the tubing which connects the syringes, we used tubing of two different lengths L = 20 − 40 mm and of three different inner diameters D = 1.6 − 3.5 − 4.6 mm. The obtained bubble size distributions are shown in Figure 8 for φ = 15 % and for N = 15. Within the experimental error the 12

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equilibrium size distribution seems quite independent of the tubing geometry.

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Figure 8: Influence of the tubing length and diameter at φ = 15% and U = 50 mm.s−1 on the equilibrium bubble size distribution.

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In summary, all our results seem to show that the key features of the bubble size distributions are, to first order, independent of the device parameters - at least in the range we investigated. This means that foam generation with this double-syringe technique is highly reproducible: independently of the number of cycles, the pushing velocity, the initial liquid fraction, or the tubing dimensions, the generated bubbles are very small (peak between 10 and 20 µm) with a polydispersity of 30 to 40 %. Finally a very important feature of this technique is that the bubble size is not a function of the liquid fraction. While bubble size and liquid fraction are correlated in most foaming technique [11, 16], the double syringe technique makes wet and dry foams with the same bubble size distributions.

3.2. Dependency of the bubble size distribution on the foam formulation Viscosity: In order to investigate the influence of the foam formulation on the equilibrium distribution we varied systematically the viscosity of the 13

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foaming liquid by adding glycerol. Figure 9a shows how the bubble size distribution after N = 10 cycles depends on the liquid viscosity (η = 1 mPa.s, 2.4 mPa.s, or 5.8 mPa.s) for two different liquid fractions (15 % and 30 %) in the constant velocity device (U = 50 mm.s−1 ). With increasing viscosity the distribution becomes clearly narrower and the characteristic peak position decreases from 20 µm for η = 1 mPa.s down to 7.5 µm for η = 5.8 mPa.s. Figure 9b shows the evolution of the distribution with liquid fraction for η = 5.8 mPa.s. Contrary to our observation made with a surfactant solution with viscosity η = 1 mPa.s, the bubble size distributions here depends on liquid fraction: the decay towards larger bubbles steepens for lower liquid fractions. Nevertheless, Figures 9c and 9d show that the shape of the normalised distribution of RB / < RB > is roughly independent of the viscosity and of the liquid fraction.

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In the inset of Figure 9b we show the evolution of the average bubble radius with number of cycles for two different viscosities (1 and 5.8 mPa.s) for a liquid fraction of 15% . This data shows that when the viscosity of the solution is increased the mean bubble size varies less with the number of pushing cycle. However, while the final bubble size distribution is obtained after less cycles, the final foam volume is actually obtained with more cycles. i.e with increasing viscosity it takes longer to create the foam. In that sense reducing the initial amount of liquid or increasing the viscosity seems to have a similar effect on the foaming mechanism. More importantly, with increasing viscosity it becomes more difficult to fill the entire syringe with foam, i.e. to incorporate all the gas into the liquid even after many cycles. This leads to a minimum achievable liquid fraction φmin which can be obtained with this technique for a given viscosity. This minimum liquid fraction depends sensitively on the pushing speed: the larger the pushing speed, the larger the minimum liquid fraction becomes (Figure 10). Interestingly, the data scales with the capillary number Ca2/3 = (ηU/γ)2/3 , which may indicate that this final minimum liquid fraction may be fixed by friction between the foam and the wall [9]. Foam stabiliser: An important question arises regarding the influence of the nature of the stabilising agent on the final foam properties. While all the experiments up to now have been done with SDS, we investigate here a range of commonly used foam stabilisers. For this purpose we choose a liquid fraction of 15 % and we compare again the equilibrium distributions 14

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(b) f = 0.34 s−1

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(a) U = 50 mm.s−1

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(c) U = 50 mm.s−1

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Figure 9: (a) Bubble size distributions after 10 cycles at U = 50 mm.s−1 for two different liquid fractions and three different viscosities η. Inset: Images of a layer of bubbles for each viscosity. (b) Bubble size distribution after 10 cycles for sinusoidal pushing velocity (f = 1/3 s−1 ) for different liquid fractions. Inset: Evolution of the mean bubble radius < R > with cycle number N for two viscosities. (b) PDF of RB / < RB > in semi-log of the results shown in (a). (d) PDF of RB / < RB > in semi-log of the results shown in(c).

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Figure 10: Minimum liquid fraction φmin as a function of the viscosity η of the foaming solution for different pushing velocities U in the constant velocity device.

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(N = 15) obtained in both devices. One can see in Figure 11 that the foam formulation has a non-negligible influence on the final bubble size distributions. The peak in the obtained bubble sizes can be pushed down below 10 µm. As shown in Figure 12 the obtained bubble size distributions for each stabilising agent are again very reproducible and quite insensitive to the liquid fraction and the device parameters, as discussed in detail for the SDS foams in Section 3.1. Yet there is no direct correlation between the surface tension of the stabiliser and the typical size of the bubble, and it is surprising to find out that foams made from SDS surfactant and the Bulgarian Mix have the closest characteristic bubble size while their surface properties (Bulgarian Mix are known to make rigid interfaces) are very different [18, 36].

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In summary, we have seen that while the device parameters have a negligible influence on the obtained bubble size distributions, the foam formulation (liquid viscosity and stabilising agent) seems to play a role in fixing the bubble sizes.

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Figure 11: Influence of the foam stabiliser on the bubble size distributions after 10 cycles in both devices at φ = 15%.

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3.3. Number VS Volume distribution Our results focused entirely on the bubble size distribution in number as our goal is to obain insight into the generating mechanism. However, for most users, the physical properties of the obtained foams will be of great interest. For this purpose, it is often more appropriate to use the bubble size distributions by volume (Section 2.3). We therefore compare here the two types of distributions (Figures 13) for different liquid fractions and for the different stabilising agents - all obtained in the equilibrium regime. As mentioned earlier no millimetric bubbles remain in the foam at this stage, so that both number and volume distributions are very similar.

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4. Discussion In Section 3 we have evidenced some of the key features of the double syringe device. In particular, we have seen that the foaming mechanism seems to be fairly independent of the device parameters at least in the ranges we tested. Neither the type/amplitude of the pushing velocity nor the liquid fraction nor the tubing dimensions seems to play a major role in fixing the 17

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(b) Fairy

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(a) TTAB

(c) C12 G2

(d) Comparison of various stabilisers

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Figure 12: Bubble size distributions obtained after 10 cycles for different stabilising agents.

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(b) Volume distribution.

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(a) Number distribution.

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(d) Volume distribution.

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(c) Number distribution.

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Figure 13: (a) Number and (b) Volume distributions of SDS-stabilised foams for various liquid fractions after 15 cycles in both devices. (c) Number and (d) Volume distributions of foams stabilised by various surfactant at a liquid fraction φ = 15% after 15 cycles in both devices.

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obtained bubble-size distributions. On the contrary the effect of the foam formulation (viscosity and stabilising agent) is significant. This indicates the presence of a physical foaming mechanism which is controlled rather by the physico-chemical properties of the gas/liquid dispersion, than by the properties of the device. We advance here some hypothesis on the potential mechanisms.

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Before entering into a more detailed discussion, let us note that any discussion on foaming needs to take into account the potential influence of foam ageing effects (bubble ripening or coalescence) on the obtained bubble size distributions. In order not to confuse the reader, this is discussed in the Appendix 6.1. There we provide different experimental investigations and arguments to demonstrate that the main features of the measured bubble size distributions are expected to be the result of the generating mechanism, rather than of foam ageing effects.

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4.1. Description of foaming mechanisms It is tempting to think that the mechanism of foam production in this system is related to bubble break-up under shear. In such a scenario large bubbles are produced during the first cycles and are then broken up repeatedly in the following cycles by the shear arising in both constriction and tubing. The mean bubble size should then decrease with increasing number of cycles. However, even though this mechanism probably happens, it cannot be the major contribution to the bubble size distribution as the characteristic distribution is already obtained after one single push.

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The goal of this section is therefore to describe the bubbling mechanisms observed within this device and to suggest two mechanisms likely to control the final distributions. For this purpose we work with a high-speed camera to provide images of the process (Photron Fastcam SA3). Visualisation experiments are done with the constant-velocity device and during the first push of the process. The set-up was simplified to allow us to observe what happens in the channel. It now consists of only one syringe containing the surfactant solution and the gas which are then pushed through the syringe constriction connected to the tubing. The rigid tubing is replaced by a flexible and transparent one of equal length and diameter. Figure 14 compares the bubble size distribution obtained with this simplified set-up with those obtained with two syringes after one cycle. These distributions being almost 20

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Figure 14: Comparison of bubble size distribution after one push between one syringe and two syringes

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identical we assume that we can use only one syringe and focus on the mechanisms happening within the constriction and the tubing.

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Figure 15 shows images of the flow within the constriction and all along the tube for two different liquids (pure water and surfactant solution) and at two different pushing speeds, for an initial liquid fraction of φ = 50%. We observe a complex foaming behavior which can be divided into two main steps: one foaming mechanism within the constriction and one at the entrance of the tubing. Both are discussed in detail in the following.

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Within the constriction millimetric bubbles are formed in a reproducible way whatever the pushing velocity and the formulation. This phenomenon is shown in Figure 15 and detailed in Figure 16. As it is known from numerous micro- and millifluidic applications [16, 15, 24, 13] a configuration where a millimetric bubble grows inside a channel is not stable indefinitely: the bubble will detach from the injection point due to a capillary-driven in-

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Figure 15: Images of the flow within the constriction (0 mm) and the connecting tube (10, 75,and 150 mm) for pure water and surfactant solutions at two different speeds U = 10 mm.s−1 and U = 50 mm.s−1 .

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Figure 16: A millimetric bubble is created in the constriction of the syringe whose rear end breaks into many tiny bubbles just after detachment. The created bubbles are too small to be resolved by the camera but they strongly scatter light which is why they appear as dark zones in the image.

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stability. During this detachment an interesting phenomenon happens: the downstream end of the bubble seems to “explode” into many tiny bubbles during the few milliseconds of the detachment process. The tiny bubbles are too small to be resolved by the camera, but one can see their presence due to their efficient light scattering that creates the black, blurry zones in the images. While this effect is small at low pushing velocities, it becomes more important at higher pushing velocities. The critical pushing velocity at which we begin to observe this phenomenon varies with the presence of surfactant. This effect is reproducible and occurs at the rear of each bubble. The precise mechanism which leads to this break-up phenomenon is not clear to us but preliminary results suggest that the geometry of the constriction has an important role to play. The second foaming mechanism takes place at the exit of the constriction, i.e at the entrance of the tubing. This corresponds to the position z = 10 mm in Figure 15. While diphasic flows and air entrapment into pure water in pipes or microchannels have been extensively studied [1, 38, 35, 33, 27], 23

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we describe here a system where the presence of surfactant influence the flow pattern and the bubble generation. Whereas at U = 10 mm.s−1 with pure water and air an annular flow is formed which then destabilises in a slug flow further in the tubing (z = 75 mm and z = 150 mm), the presence of surfactant stabilises the big bubbles formed in the constriction which then accumulate to form a bubbly flow down the tubing. We note the presence of tiny bubbles. At U = 50 mm.s−1 air and water keep forming annular flow at the outlet of the constriction, and the generation of tiny bubbles is clearly noticable. This annular flow then destabilises into a bubbly flow which generates a large quantity of tiny bubbles turning the tubing black due to strong light scattering. At first sight one may take this observation as a cavitation effect [20, 2]. Using gases of different solubility we show in the Appendix 6.2 that this could be excluded as potential mechanism.

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4.2. Analysis of foaming mechanism Bubbling in two-phase flow is a vast subject [8, 1]. A wide number of flow conditions - and the associated instabilities of the air/liquid interface - are known to produce bubbles with characteristic sizes and size distributions [34, 28, 29]. Due to the complexity of the flow conditions in our set-up, we are not able yet to identify clearly the physical instabilities at the origin of our process. We nevertheless give some first qualitative analysis here. We shall use the following dimensionless numbers

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Capillary number Reynolds number Weber number

Ca = ηhui γ Re = ρhuiL η We =

ηhui2 L , γ

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in order to compare the relative importance of the viscous, inertial and capillary stresses at different locations in our setup. Here γ is the surface tension, ρ the density, hui the mean velocity of the fluid of interest, and L the characteristic length scale over witch the flow occurs. We calculate the non-dimensionless numbers for each of the fluid phases.

The syringes have a large cross-section (Ds = 24 mm) compared to the smallest cross-section of the constriction (Dc = 1.6 mm) and the tubing (Dt = 3.5 mm) (Figure 17a). This leads to high aspect ratios: (Ds /Dc )2 = 225, (Ds /Dt )2 ≈ 50 and (Dc /Dt )2 ≈ 5 - and hence to abrupt and significant 24

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Figure 17: (a) Sketch of the geometry of the constriction of diameter Dc and the tubing of diameter Dt , in which an annular flow occurs. (b) Sectional view of the annular flow. Dg and el are calculated making the approximation that gas and liquid flow have the same cross-section. Di is the diameter of the tube at a given position.

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changes in the average flow velocity < u >. Since we see that the water level in the syringe remains constant during foaming at roughly half the diameter of the syringe, we assume in the following that the flow rate of the liquid and of the gas are roughly equal and constant, i.e. Ql ≈ Qg ≈ const. (considering the gas as incompressible). The cross-section of the gas and of the liquid in the constriction and in the tubing adjust to the flow conditions. Inspired by our experimental observations, we shall assume in the following that the flow is annular and that both cross-sections are roughly equal. As illustrated in Figure 17b, this allows us to fix the characteristic length scale L of the gas flow as the diameter of the thread of the annular flow Dg , and the one of the liquid flow as the thickness el of the layer in contact with the wall . Using these length scales and the average velocity U imposed by the syringe piston, we can calculate the different dimensionless numbers for the gas and the liquid in the constriction and in the tubing. These are listed in Table 2. Because of the high aspect ratio between the cross-section of the syringe and the constriction ((Ds /Dc )2 = 225), gas and liquid are greatly accelerated in the constriction. For example, flow velocities < u > of the order of 10 m.s−1 are reached in the constriction for a pushing velocity

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In the U = 10 mm.s−1 Liquid Gas 0.5 0.5 −2 1.10 2.10−4 2 2.10 8.101 3 2.10−4

tubing U = 50 mm.s−1 Liquid Gas 2.3 2.3 −2 6.10 10−3 3 10 4.102 8.101 4.10−1

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Piston Velocity Phase < u > (m.s−1 ) Capillary number Ca Reynolds number Re Weber number W e

In the constriction U = 10 mm.s−1 U = 50 mm.s−1 Liquid Gas Liquid Gas 2.2 2.2 11.3 11.3 1 −4 2 8.10 9.10 4.10 5.10−3 2 2 3 5.10 2.10 3.10 9.102 3.101 2.10−1 8.102 4

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Table 2: Flow velocity and corresponding dimensionless numbers calculated at the outlet of the construction (Dc = 1.6 mm) and within the tubing (Dt = 3.5 mm).

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U = 50 mm.s−1 . This leads to an increasing importance of inertial effects (high Re and W e numbers). Moreover, as one can see in Figure 16, the cross-section of the gas thread in the constriction goes periodically to zero during pinch-off process of the big bubble. Since both the gas and the liquid flow rate remain roughly constant during this process, this means, that the average liquid velocity reduces slightly, while the average gas velocity increases dramatically. Inertial effects may therefore lead to a destabilisation of the gas thread during pinch-off, resulting in the explosive generation of tiny bubbles observed within the constriction (Figure 16). The second foaming mechanism arises when the fluids enter into the tubing. Here the average velocity decreases by a factor of five with a much more significant decrease of the liquid velocity due to the angular flow geometry. The associated capillary number of the liquid decreases significantly from Cal << 1 in the constriction to Cal >> 1 in the tubing. At the same time, the W e number decreases by a factor of 10. Hence, while the flow is clearly dominated by the inertial and viscous stresses within the constriction, capillary stresses become much more important at the entrance of the tubing. In many situations of free interface flows it has been shown that these kind of transitions are likely to lead to air entrainment [26, 7, 6]. Since we cannot visualise the air entrainment process in our setup, it is interesting to analyse more closely the obtained bubble size distributions as these may be revelatory of the underlying mechanism. Two different types of mechanisms are commonly considered in fragmentation-type processes. These are best known in the case of droplet generation, but Eggers and

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fGamma (x, α, β) =

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Villermaux state in their review [12] that “there is no reason to think that the same phenomenology and ideas should not apply to the opposite situation of bubbles forming in a continuous liquid phase”. In the first case, thin ligaments of gas are formed within the liquid which - below a critical thickness - destabilise into small bubbles. This process leads to a Gamma distribution which is given by [40] β α α−1 x exp(−βx), Γ(α)

(4)

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where α and β are the shape and rate parameters, respectively, and R +∞ Γ(x) = 0 tx−1 et dt the Gamma function. In the second case, a cascade of break-ups occurs, in which larger bubbles break into smaller bubbles, which then break again into smaller bubbles etc. Taking into account a certain probability of coalescence events which reunites bubbles, such a process leads to a Lognormal distribution which is given by [19]   (ln(x) − µ)2 1 √ exp − fLognormal (x, µ, σ) = , (5) 2σ 2 xσ 2µ q P P 1 where µ = n n ln(x) and σ = n1 n (ln(n) − µ)2 .

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In Figure 18 we fit these distributions to a selection of experimental distributions obtained in both setups (i.e. constant or sinusoidal pushing velocity) after the first cycle and after ten cycles. After the first cycle, the distributions are captures slightly better by the Lognormal distribution, which may be a signature of a breakup process via cascades. After 10 Cycles, both distributions fit equally well. Moreover, the dependency of bubble size on viscosity is similar to what has been observed in liquid jet fragmentation [12, 23]: a higher liquid viscosity slows down the break-up of the thin gas threads since they remain stable for a longer time, resulting in narrower Gammadistributions. Combining these different observations, it is impossible at this stage to draw any reliable conclusions from bubble size distribution. Moreover, foam ageing and the presence of surfactants may have a non-negligible influence on the obtained distributions. Future work should therefore aim for a direct visualisation of the foaming process in order to elucidate the underlying mechanisms.

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(b) U = 50 mm.s−1 Ncycle = 10

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(a) U = 50 mm.s−1 Ncycle = 1

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(d) f = 1/3 s−1 Ncycle = 10

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Figure 18: Comparison of experimental results with Lognormal (Equation 5) and Gamma distributiona (Equation 4) for both constant and sinusoidal piston velocity.

5. Conclusion and Outlook We have shown here how a simple technique - the “double syringe technique” - can produce small-bubble foams whose bubble sizes are of the order 28

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of 10 − 20 µm. The bubble size distributions have a well-defined peak with a nearly exponential decay towards larger bubbles. These distributions are highly reproducible and, most importantly, independent of the liquid fraction. This is the only foaming technique known to us in which the liquid fraction can be varied independently of the bubble size distribution. The double syringe technique is therefore an ideal tool to produce well-controlled, small-bubble foams.

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We have been able to show that a characteristic bubble size distribution is obtained after the first pushing action, i.e. we are not dealing with a progressive break-up of bubbles under shear in the constriction, as one may naively expect (even though the foam initially contains some millimetric bubbles). We see, however, that the distribution changes slightly with number of cycles, leading to an ”equilibrium distribution” after a certain number of cycles which is again independent of the liquid fraction. We expect foam ageing to play a role in fixing this distribution. However, it is not clear to us why the strong dependence of foam ageing effects on liquid fraction does not show up in the distributions.

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We have systematically tested the influence of a wide range of parameters on the obtained foam properties. We find that the parameters of the device (pushing speed, pushing type, tubing length and diameter, ...) have very little influence on the obtained foam properties. The foam formulation (foam stabiliser and liquid viscosity), on the contrary, influences non-negligibly the obtained bubble size distributions. In particular, the bubble sizes decrease measurably with the viscosity of the liquid. This viscosity also sets a welldefined minimum liquid fraction which can be produced. Since this minimum liquid fraction increases strongly with liquid viscosity, the double syringe technique is not appropriate for foaming high viscosity liquids.

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Using high-speed imaging, we attempted to identify the foaming mechanism and found two different mechanisms in the device, both related to hydrodynamic instabilities. The first mechanism arises in the constriction in which the rear end of millimetric bubbles ”explodes” into numerous tiny bubbles during pinch-off. The second mechanism, which seems the dominating one, is visible at the entrance of the tubing, where a large quantity of tiny bubbles seems to be generated in an annular-type flow. Analysis of the associated non-dimensional numbers indicates that the mechanism in the 29

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constriction may be dominated by inertia, while the mechanism in the tubing should also be sensitive to capillary stresses. Comparison of the bubble size distributions with those known from the literature does not provide conclusive indications about the nature of these mechanisms. Moreover, it is not clear to us how foam ageing and the presence of surfactants may influence the bubble size distributions in comparison to those obtained from purely hydrodynamic effects.

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Due to the coupling of hydrodynamics, physical chemistry and foam ageing, future work should therefore aim for direct visualisation of the process, potentially with simplified model experiments. A major step would then be to export the acquired understanding in order to exploit this mechanism in a continuous process, rather than a periodic cycling between two syringes which can only provide a small foam volume at a time. Moreover, it would be desirable to find a device parameter which provides explicit control over the obtained size distributions.

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However, even without this fine understanding of the foaming mechanism, the double seryinge technique can already be used reliably in order to generate well-controlled, small-bubble foams with variable liquid fraction for research purposes or applications. Acknowledgment

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We thank Florence Elias and the members of the ANR SAMOUSSE in whose context this work was inspired and regularly discussed. We thank Emmanuelle Rio, Anniina Salonen, Dominique Langevin, Benjamin Dollet, Arnaud Saint-Jalmes, Jens Eggers, Christophe Josseran, St´ephane Popinet, Thomas S´eon, Fr´ed´eric Moisy, and Marc Rabaud for fruitful discussions. We thank Marc Bottinot, Vincent Klein, J´er´emy Sanchez and R´emi Brauge for building the foaming devices. We also thank Christophe Poulard for letting us use excessively his Keyence Microscope, and Alain Cagna and S´everine Besson from TECLIS for additional stimulation of this work from an applied perspective. Funding is greatfully acknowledged from the European Research Counsil (ERC) in form of an ERC Starting Grant (agreement 307280-POMCAPS) and from the French Agence Nationale de la Recherche (project SAMOUSSE, ANR-11-BS09-001).

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6. Appendix

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6.1. Foam ageing To discuss the potential foaming mechanism one needs to quantify the influence of the foam stability on the obtained bubble size distributions. Foams are known to be inherently unstable: bubbles can dissolve in the surrounding liquid (“dissolution”), bubbles can exchange gas due to the difference in internal pressures (“coarsening”) and thin films between bubbles can break (“coalescence”). These different ageing mechanisms are particularly rapid for small-bubble foams [3] and can lead to non-negligible changes in the bubble size distributions. In our case we need to worry about this in two different situations: foam ageing may arise in the syringes during foam generation and/or after foam generation during the time it takes to take the image of the bubbles after their deposition in a monolayer (Section 2.3).

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In order to verify the influence of the second process, we investigated the evolution of the bubble size distribution of an SDS foam containing nitrogen bubbles with with traces of C6 F14 after deposition, taking images of the same foam after t = 300 s, 2400 s, and 4500 s. t = 0 s corresponds to the moment when the bubbles are deposited in the monolayer, as shown in Fig 4.a. The results are shown in Figure 19. Since bubbles cannot coalesce in this configuration, the ageing is essentially due to coarsening. One can see that it takes about one hour for the peak to double its maximum position. This slow evolution is due to the presence of the C6 F14 . Since it takes us about 3 minutes to take the first image we can therefore assume that the bubble size has not evolved more than a few micrometers between foam generation and imaging.

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The analysis of the influence of foam ageing effects during foam generation is more involved. Since bubbles are in close contact within the foam, all ageing mechanism are accelerated with respect to the evolution shown in Figure 19. Hence, it is certain that foam ageing has an influence on the final size distribution - the question being to what extent. A selection of observations tells us that the key features of the distributions we observe should be a result of the generating mechanism - rather than of the foam stability. A first indication is the fact that the distributions depend very little on the liquid fraction. Foam ageing effects increase drastically with decreasing liquid fraction - hence we should observe much larger bubbles for lower liquid 31

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Figure 19: Ageing of a bubble monolayer deposited between two glass plates, made from a SDS solution at a concentration of 10CMC and with air having C6 F14 traces

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fractions, which is not the case. It shows even the opposite behaviour for more viscous foaming liquids: the bubble size is more peaked for low liquid fraction. Equally, the type of foam stabiliser plays an important role in controlling foam stability. If foam ageing was a key parameter in fixing the bubble size distributions, more stable foams would have narrower size distributions. Looking at Figures 11 and 12, this is not at all the case. “Fairy” and the “Bulgarian Mix” are expected to make the most stable foams but do not have at all the narrowest bubble size distribution.

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To sum up: even though we expect foam stability to have an influence on the size distributions, we believe that the controlling factor must be in the generation mechanism. At this stage we are not able to quantify the influence of foam ageing. However, we expect it to be small enough so that our key observations can be assigned to the foaming mechanism which we discuss in the following.

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6.2. Cavitation If cavitation was playing a role in the foam formation, the foaming efficiency should depend on the solubility of the gas. We therefore repeated the experiment under the same conditions using either pure C2 F6 (poorly soluble in water) or pure CO2 (highly soluble in water). Figures 20b and 20c show that the bubbling mechanism with these two gases is undistinguishable from the one with air+C6 F14 , which is why we consider the origin of this observation to be of hydrodynamic nature [5]. Unfortunately we cannot observe directly what happens due to the strong light scattering by the tiny bubbles present on the walls of the tube. The fact that this mechanism is much less efficient with pure water - which has a much higher surface tension (0.072 N/m) than the SDS solution (0.034 N/m) (Figure 15, U = 50 mm.s−1 ) suggests that surface tension- and therefore capillarity effects - may play an important role in the foaming mechanism.

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(a) Air and C6 F14 traces

(c) CO2

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(b) C2 F6

Figure 20: Bubble generation in pure water at the entrance of the tubing for three gases with very different solubilities in water to test a potential influence of cavitation.

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