Dripping

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Toward ultra-low interfacial tension measurement through jetting/dripping transition. Marie Moiré, Yannick Peysson, Benjamin Herzhaft, Nicolas Pannacci, François Gallaire, Laura Angello, Christine Dalmazzone, Andrea Fani, and Annie Colin Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b00076 • Publication Date (Web): 20 Feb 2017 Downloaded from http://pubs.acs.org on February 22, 2017

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Toward ultra-low interfacial tension measurement through jetting/dripping transition Marie Moiré,† Yannick Peysson,† Benjamin Herzhaft,† Nicolas Pannacci,† François Gallaire,‡ Laura Augello,‡ Christine Dalmazzone,† and Annie Colin¶ †IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex, France ‡EPFL, LFMI, Bâtiment ME A2, Station 9, CH-1015 Lausanne ¶ESPCI, CNRS, SIMM UMR 7615, 11 rue Vauquelin, 75005 Paris, France E-mail:

Abstract In this paper, we present a dynamic microfluidic tensiometer able to perform measurements over more than four decades and which is suitable for high throughput experimentations. This tensiometer is able to withstand hard conditions such as high pressure, high temperature, high salinity and crude oil. It is made of two coaxial capillaries in which two immiscible fluids are injected. Depending on the flow rate of each phase, either droplets or jetting will be obtained. The transition between these two regimes relays on the Rayleigh-Plateau instability. This transition can be theoretically computed thanks to a linear analysis based on the convective and absolute instabilities theory. From this model, the interfacial tension between the two phases can be calculated.

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Introduction Interfacial tension is one of the parameters governing the phase diagrams and the stability of immiscible systems. Many industries, developing formulations, have to measure this parameter for optimization purposes. It is especially the case for the oil industries interested in chemical surfactant Enhanced Oil Recovery (cEOR). In fact, it has been shown that decreasing the interfacial tension (IFT) between the injected water and the crude oil from the typical 5 × 10−2 N/m down to 10−5 or 10−6 N/m, decreases considerably the residual oil saturation (Sor) in the reservoir. 1 In other words, the total crude oil production significantly increases thanks to this strong decrease of IFT obtained through specific surfactant formulations. Achieving ultra-low IFT will strongly depend on the crude oil composition, the salinity of the injected water and the reservoir conditions (temperature, pressure, quantity and nature of dissolved gas). 2–4 Optimizing a formulation for a specific reservoir will therefore necessitate a large number of experimentations in various conditions. This example of EOR formulations illustrates the need for a tensiometer operating in hard conditions (high temperature, high pressure, high salinity and crude oil components) over a wide range of IFT (more than 4 decades) and allowing to perform high throughput experimentations (HTE). There are various experimental set up allowing the measurement of interfacial tensions. 5 One of the most used is the pendant drop tensiometer. The measurement principle of the pendant drop tensiometer 6 is based on the formation of a droplet of a fluid 1 in a fluid 2 (1 and 2 being immiscible). The droplet of fluid 1 is attached at the end of a needle and its profile depends both on the interfacial tension and on the difference of density between fluids 1 and 2. Consequently, using a numerical analysis of the droplet profile and applying the Young-Laplace equation leads to the determination of the interfacial tension. This set up allows the measurement of interfacial tension higher than 1 mN/m and thus does not fill the previous requirements. Microfluidic tensiometers have also been developed. However, most of them 7–9 are not suitable to measure IFT lower than 10−3 N/m. To the best of our knowledge, only Tsai et al. 10 have explored ultra-low IFT observing the trajectory of 2


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magnetic particles crossing the interface between two fluids. One of the drawbacks of this method for an EOR application, is the difficulty to observe the trajectory of the magnetic particles in complex fluids as crude oil. To measure ultra low interfacial tensions, a laser light-scattering technique has been developed. 11 A monochromatic light beam is sent on the interface. It is scattered by thermal capillary waves. The power spectrum of the scattered light depends upon the viscosities, the densities of the both phases and upon the interfacial tension. This technique has been used to measure the interfacial tension between an aqueous solution containing 1-propanol and toluene. In presence of surfactants, the analysis is more complex and involves the bending elasticity of the monolayer. 12 In industrial laboratories, to measure ultra low interfacial tensions, researchers apply the method of Huh. It is an indirect measurement of interfacial tension. It is based upon the phase diagram of microemulsions. There are three types of microemulsions called Winsor I, Winsor II and Winsor III. In Winsor I (respectively Winsor II), an oil-in-water (respectively water-in-oil) microemulsion is in equilibrium with the organic phase (respectively aqueous phase) in excess. In Winsor III, a bicontinuous microemulsion (oil and water) is in equilibrium with both the aqueous and organic phases in excess. For each type of microemulsion, Huh 13 demonstrated that the volume of water (respectively of oil) solubilized in the microemulsion is related to the interfacial tension between the microemulsion and aqueous phase (respectively organic phase) in excess. From this work, empirical equations are used for engineering purposes. They write : γ =

0.3 σi2

and σi =

Vi Vs

where γ is the interfacial tension

between the phase i and the microemulsion, σi is the solubilization ratio of the phase i, Vi is equal to the volume of phase i solubilized in the microemulsion and Vs is the volume of surfactant present in the system. The volume of surfactant is calculated by assuming that the density of the surfactant is 1 g/cm3 , even if the surfactant is in its solid form. There is no doubt that the 0.3 constant and the definition of the volume of surfactant are questionable. However, this method is used for engineering screening purpose and not to obtain a precise

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measurement of an IFT value. To the best of our knowledge, the single method that corresponds to the conditions required by EOR formulation is the spinning drop technique. The spinning drop can measure a wide range of IFT (10−6 to 5 × 10−2 N/m). Droplets are inserted in an horizontal glass capillary that is rotating about its long axis. When equilibrium is reached (i.e after at least 100 s), pictures of the droplets are taken. The profile of the droplet results from a balance between the centrifugal and the interfacial forces. Using a numerical analysis of the droplet profile and the Young-Laplace equation allows determining the interfacial tension. The insertion of the drop inside the apparatus is a tricky procedure and the determination of a single interfacial tension takes at least a quarter of hour. For interfacial tension lower than 10−5 N/m, the uncertainty of this measurement increases considerably and can be higher than 60 %. 14 After this rapid state of the art, it is clear that adapting an existing tensiometer to respond to the three major challenging specifications previously exposed, is not possible. As a consequence, we decided to choose a different strategy for measuring IFT. The recent studies of Guillot et al., 15 , 16 Utada et al., 17 18 and Gallaire et al. 19 on the dripping to jetting transition occurring in coaxial flow-focusing have demonstrated that this transition depends on the IFT, the hydrodynamics constraints and the confinement. Thanks to the model of Guillot, we demonstrate the possibility to measure IFT from conventional values (10−2 N/m) down to ultra-low values (10−6 N/m). We also investigate the impact of the dynamic character of the method. We will first present the developed apparatus, the observations and the model used for the interfacial tension determination and in a second part the experimental results down to ultra-low IFT.

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Materials and Classical Methods Fluids under scrutiny For validation purposes, the following fluid samples were studied: a mixture of distilled water and bidistilled glycerol (99.5 %) purchased from VWR (79.7/20.3)%wt and a mixture 30.3/69.7 %wt of two silicone oils, 20 cSt and 500 cSt, respectively purchased from Prolabo and VWR. Then, surfactants were introduced to study the impact of kinetics on dynamic measurements. These solutions were made with distilled water, with bidistilled glycerol (99.5 %) from VWR and with either (1-dodecyl)trimethylammonium bromide (DTAB) (99 %) from Alfa Aesar or Triton X100 (Laboratory grade) from Sigma Aldrich. For the DTAB solutions, the distilled water/bidistilled glycerol weight ratio was maintained constant at 0.47, while the concentration of DTAB was varied between 1.8 × 10−3 and 7 × 10−2 mol/L. For the Triton X100, the distilled water/bidistilled glycerol weight ratio was maintained constant at 0.5 which leads to a trouble point temperature higher than the ambient temperature for all the Triton X100 concentration used, comprised between 0.2 × 10−3 and 17.3 × 10−3 mol/L. Finally, microemulsions were used to obtain ultra-low IFT. First, the aqueous phase was formulated and then put in contact with n-decane (99 %) purchased from Alfa Aesar. The aqueous solutions, prepared with distilled water, contained 5.83 %vol of isobutanol (≥ 99 %) from Alfa Aesar, 20 g/L of SDBS (100 %) from Sigma Aldrich and respectively 0, 30, 52 and 100 g/L of sodium chloride (99.9 %) from VWR chemicals.

Interfacial tension measurements For validation purpose, the pendant drop tensiometer (Kr¨ uss model DSA 25) 6 was used to characterize the DTAB solutions, which all have an IFT value higher than 10−3 N/m. Because of the environmental conditions of the laboratory (vibrations due to ventilation requirements for crude oils), all the results are given with an error of 7 %. The spinning drop tensiometer (Dataphysics SVT 20N) 20 was used to measure interfacial tensions lower than 5


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10−3 N/m. For interfacial tensions lower than 10−5 N/m, the uncertainty of this measurement increases considerably and can be higher than 60 %. 14 We studied phase diagrams of microemulsions to measure the interfacial tension using the Huh method. For this purpose, samples of 10 mL (5 mL of aqueous phase and 5 mL of organic phase) were prepared.

Interfacial tension measurement through jetting/dripping transition: method and model The flow of two immiscible coaxial fluids has been widely studied in the literature. Recently, this kind of flow has been revisited using microfluidic devices. Guillot et al. have shown that viscosity of both phases and interfacial tension were key parameters that rule the nature of the flow (drops, plugs or jets). In this work, we aim to propose an inverse analysis and to measure the interfacial tension from the flow diagram. In the following, we will describe the experimental set up, the theoretical analysis and our experimental procedure to measure the interfacial tension.

Dripping to jetting experimental set up We developed an experimental set up that allows the observation of the transition from dripping to jetting between two coaxial fluids. This apparatus, described in figure 1, is composed of 5 main parts: syringe-pumps, a mixing unit to prepare formulations (we did not use it in this study), a set up to fix the capillary tubes, a camera and an observation place. Two syringe-pumps are designed to deliver flow rates ranging from 2 µL/h to 336 mL/h. These small values of flow rates are of prime importance to measure ultra-low IFT. For flow rates lower than 10 µL/h, the uncertainty is lower than 20 % and for flow rates higher than 10 µL/h, the uncertainty is lower than 10 %. These experimental errors are due to syringe pump fluctuations. 21 Note that the vibrations of the syringe pump do not create corrugated 6


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of the flow as a function of flow rates. In the following section, we detail our approach which is mainly based upon the work of Guillot et al..

Modeling of the flow diagram Taking advantage of the symmetry of the cylindrical geometry, Guillot et al. 15,16 attempt to model these phenomena analytically. The unperturbed reference state is easily described using the Stokes equation.

Q0e =

Q0i

rio4 ro R2 π(Rc2 − rio2 )2 (1 − o2c − 2ln( i ))(∂z Pio − ∂z Peo ) − ∂z Peo 4ηe ri Rc 8ηe

πrio4 rio πrio4 πrio2 (rio2 − Rc2 ) 0 0 0 = ln( )(∂z Pi − ∂z Pe ) − ∂ z Pi − ∂z Pe0 2ηe Rc 8ηi 4ηe

(1)

(2)

The two pressure gradients are linked through the continuity of the normal stress at the interface: Pio − Peo =

γ ri0

(3)

In these equations 1, 2 and 3, Q0i and Q0e are the inner and outer flow rates in the unperturbed states, ri0 corresponds to the radius of the unperturbed jet, it is not equal to the radius of the inner capillary tube, Rc is the inner radius of the outer capillary tube, ηi and ηe are respectively the inner and outer viscosities and ∂z Pi0 and ∂z Pe0 are the pressure gradients in the z direction respectively in the inner and outer phases, γ is the surface tension. For the unperturbed flow, the radius of the inner jet ri0 is constant and the two fluids bear the same pressure gradient. They perform a linear stability analysis and consider the spatial-temporal response of the system to small z dependent cylindrically symmetric perturbation Qe , Qi , ∂z Pi , ∂z Pe and ri . As the problem is complex even with a linear stability analysis, they proceed with

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approximations. They neglect inertial effects. They make the perturbations proportional to e(ikz+ωt) with k and ω complex numbers. They restrict their analysis to the lubrication approximation, assuming formally that the perturbation wavelength is larger than the radius of the capillary Rc . Guillot et al. 15,16 demonstrate that to respect the hypothesis of lubrication, x must be higher than 0.3 when the inner viscosity ηi is smaller than the outer viscosity ηe , otherwise x must be higher than 0.6. In this framework, the expressions obtained for the unperturbed flow can still be used locally, so that the local perturbations in the flow rates read as: δQe =

∂Qe ∂Qe ∂Qe δ∂z Pe + δ∂z Pi + δri ∂(∂z Pe ) ∂(∂z Pi ) ∂ri

(4)

δQi =

∂Qi ∂Qi ∂Qi δ∂z Pe + δ∂z Pi + δri ∂(∂z Pe ) ∂(∂z Pi ) ∂ri

(5)

The partial derivatives are calculated using the equations 1 and 2. The Laplace equation requires that for the perturbed flow the two pressure gradients are different and linked by:

∂z (Pio − Peo ) = −γ[∂z (

δri + δz2 δri )] 2 Rc

(6)

This situation differs from the unperturbed flow. The system is closed by writing mass conservation: δQe + δQi = 0

(7)

∂t [Π(ri0 + δri )2 ] = ∂z δQe

(8)

These equations allow them to obtain the dispersion equation: ω16ηe Rc −ikri04 x3 KaE(x)/Rc + F (x)(ri02 k 2 − ri04 k 4 ) = γ D(x)

(9)

E(x, λ) = −4x + (8 − 4λ−1 )x3 + 4(λ−1 − 1)x5

(10)

F (x, λ) = x4 [4 − λ−1 + 4ln(x)] + x6 (−8 + 4λ−1 ) + x8 [4 − 3λ−1 − (4 − 4λ−1 )ln(x)]

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(11)

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D(x, λ) = x9 (1 − λ−1 ) − x5 where λ = Ka =

−δz P 0 Rc γ

ηi , ηe

x =

ri0 Rc

=

(12)

p p α−1 Qi e Qe ), δz P 0 = − πR8η ( λ−1 +α−1 ), α = (1 + λ−1 Q 2 (1−x2 ) and e c

.

E(x, λ) and F (x, λ) are positive functions in cylindrical geometry. Ka is a genuine capillary number as it is the ratio between viscous forces ηe Ve Rc = −δP 0 Rc3 and the capillary forces γRc . We have however used a notation different from the usual Ca to call the reader’s attention to the fact that Ka is a capillary number at the scale of the capillary Rc rather than at the scale of the jet. Obviously in contrast with this approach, studies of unconfined jets focus on capillary numbers defined using either the average jet velocity or the velocity at the surface of the unperturbed jet. Due to surface tension effects and mainly due to the variation of the curvature in the cross section of the jet (i.e the term ri02 k 2 F (x, λ)) the flow is linearly instable. To analyse the nature of the instability and to understand how the instability grows inside the capillary, Guillot et al. 15,16 follow the work of W. Van Saarloos. 23 ∗ ∗ They calculate the velocity of the fronts of a growing distorsion v+ and v− .

∗ v± =

where C1 =

Kax3 E(x, λ) ± C1 F (x, λ) x9 (1 − λ−1 ) − x5

(13)

√ p 5+ 7 ) ( √24 18 7−1

∗ ∗ ∗ As v+ is always positive the nature of the instability is set by the sign of v− . If v− is ∗ positive, the instability is convected and perturbations are propagated downstream. If v− is

negative, the jet is absolutely unstable and perturbations are propagated upstream. ∗ By solving the equation v− = 0, they can deduce Ka and thus, the pressure drop and

both external and internal flow rates, corresponding to the absolute to convective transition (see 15,16 for more details dealing with the calculations). This model relies on numerous hypothesis. To clarify this last point, we choose to compare it with the theoretical transition obtained with the numerical simulations developed by Gallaire and Augello. 24 This model, based on a linear analysis, takes into account inertia and the entire velocity profile. It does

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not assume lubrication approximation. In other words, this model is more accurate than the simplified model of Guillot. 15,16 Figure 3 displays a comparison of the two approaches for a given set of parameters. They are in perfect agreement in the low external flow rate situation. We note that the model of Guillot differs in the high external flow rate situation from the numerical situations, mainly because the lubrication approximation is not filled. This modeling is paving the way for an inverse analysis. To do so, one needs to define the nature of the flow for the droplets, the plugs, the jetting, the jets and the oscillating jets. In their work, Guillot and coworkers 15,16 have associated droplets and plugs to absolute instability and jetting, jets and oscillating jets to convective instability. The main reason for that is that the modulations are convected downstream in the jetting, jets and oscillations situations. Note that the data presented by Guillot are displayed using logarithmic axis ranging over more than 4 decades, which makes difficult a quantitative comparison between the data and the experiments. In the following, we will check this interpretation precisely on a given simple example. This will allow us to define an experimental procedure to measure surface tension.

From theoretical transition to experimental transition Figure 3 displays the obtained data and the two models. The studied system is a glycerol solution in contact with a silicone oils mixture. The inner and outer viscosities are respectively equal to 47.7 mPa.s and 247.3 mPa.s. By injecting the IFT measured with the pendant drop method (26.5 ± 1.9 mN/m) and the viscosity and density of both phases in the model developed by Gallaire and Augello, 24 the transition symbolized by the red crosses is obtained (figure 3). These data show without ambiguity that the drops and the plugs correspond to an absolute instability whereas the flat jets and the jettings are a convective instability. Strikingly, these experiments evidence that the transition between absolute and convective instability corresponds to the transition between oscillating jets and flat jets. That means that in the linear analysis, the oscillating jet corresponds to an absolute instability. At first 12


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Σn (Qi −)(Qi −)

g i gi 1 i=1 Cov(Qi, Qig ) = n−1 std(Qi )std(Qig ) p and std(Qig ) = ( n1 Σni=1 (Qigi − < Qig >)2 ).

where std(Qi) =

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p 1 n ( n Σi=1 (Qii − < Qi >)2 )

Qii corresponds to the experimental internal flow rate at the transition for a given external flow rate Qe, < Qi > to the mean value of the experimental internal flow rate at the transition for a given internal flow rate Qe, Qigi to the theoretical internal flow rate at the transition estimated by using the model of Guillot, < Qig > to the mean value of the theoretical internal flow rate at the transition for a given external flow rate Qe. n is the number of external flow rates tested and i the corresponding index. The experimental internal flow rate at the transition is defined as the mean value of the internal flow rate obtained from the last absolutely unstable point and the first convected one when the internal flow rate is increased for a given external one. Note that to do so, we have considered only the experimental data respecting the hypotheses of the model (laminar flow and lubrication hypotheses). We get Cov =0.998. Using a least square procedure to fit the experimental data with the model of Guillot, we find that the surface tension is equal to 25.8 mN/m (see black line on figure 3) and that Cov=0.9998. In the following, we will use this procedure, i.e we will fit the interfacial tension using a least square procedure and we will calculate the Covariance to quantify the accuracy of the measurement. The difference between the pendant drop value and the microfluidic tensiometer one comes on one hand from the experimental uncertainties (flow rates, viscosities, internal radius of the outer capillary) and on the other hand on the hypotheses of the model (no inertia and lubrication hypothesis). We obtain an error of 6.5 % with the microfluidic tensiometer which is comparable to the pendant drop method error (7 % of error in the conditions of the laboratory).

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High IFT measurements and kinetic aspects The previous section has shown that the microfluidic tensiometer coupled with the model of Guillot 16 is accurate for high IFT and systems of fluids without any surfactant. We choose to study the kinetic aspects while increasing the range of measurements of the microfluidic tensiometer. Therefore, we introduce two types of surfactants having different times of equilibration. First, we introduce different concentrations of DTAB in a water+glycerol solution and measure their interfacial tension with silicone oil thanks to the microfluidic tensiometer (figure 4). To do so, we set an external flow rate. Experimental data were obtained by applying a logarithmic internal flow rate sweep between 10−2 mL/h and 10 mL/h (7 points by decades). Each point was measured during 10 seconds which was enough to ensure flow equilibrium. Images of the flows are captured and recorded with a fast camera (ispeed-211 from IXcameras). We then increase the external flow rate and redo the experiments in order to get a line of transition between oscillating jets and flat jet and between drops and jetting. Image analysis allows us to detect the nature of the flow (i.e drops, jets) as a function of the external and internal flow rates. The transition between oscillating jets and flat jets and between drops and jetting is fitted according to the model of Guillot et al. 15,16 Figure 5 illustrates the evolution of the IFT measured with the pendant drop and the microfluidic tensiometer in fonction of the concentration in DTAB. This graph shows that the microfluidic tensiometer is accurate and that there are no kinetics issues since the IFT measured with the microfluidic tensiometer is not over estimated. In other words, during the IFT measurements with the microfluidic tensiometer, the DTAB molecules have time to diffuse and adsorb themselves onto the interface. As a consequence, in this case, adding a diluter between the syringe pump and the capillaries can be of great interest in order to determine the Critical Micelle Concentration (cmc) of a surfactant thanks to High Throughput Experimentation (HTE). We do the same experiment with aqueous solutions of water/glycerol (50/50 %wt ) con15


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2π 1/2 26 ) , where R is the radius of the droplet or of the jet and Pe is the interface, hc = R( 3P e

Péclet number defined as P e =

RUe . 2Def f

Ue is the characteristic velocity in the external phase

and Def f is the effective diffusion coefficient that takes into account the contribution of −2/3

the micelles to the mass transfer. Following Glawdel and Ren 27 Def f =D(1+β)(1+β Na

)

where β=(c/cmc)-1, c is the surfactant concentration, cmc is the micellar concentration and Na is the aggregation number of micelles. The surface concentrations of surfactants were obtained from steady surface tension values measured by the pendant drop, based on the c . For both Langmuir-Szyszkowski equation. γ − γo = RT Γinf ln(1 + c/a) and Γeq = Γinf a+c

surfactants as a

Dripping

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