Contactless rheology of finite-size air-water interfaces
Vincent Bertin, Zaicheng Zhang, Rodolphe Boisgard, Christine Grauby-Heywang, Elie Raphael, Thomas Salez, Abdelhamid Maali
NNon-contact rheology of finite-size air-water interfaces
Vincent Bertin,
1, 2, ∗ Zaicheng Zhang, ∗ Rodolphe Boisgard, ChristineGrauby-Heywang, Elie Rapha¨el, Thomas Salez,
1, 3, † and Abdelhamid Maali ‡ Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33405 Talence, France. UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 75005 Paris, France. Global Station for Soft Matter, Global Institution for Collaborative Research and Education,Hokkaido University, Sapporo, Hokkaido 060-0808, Japan. (Dated: November 11, 2020)We present non-contact atomic-force microscopy measurements of the hydrodynamic interactionsbetween a rigid sphere and an air bubble in water at the micro-scale. The size of the bubble is foundto have a significant effect on the response due to the long-range capillary deformation of the air-liquid interface. To rationalize the experimental data, we develop a viscocapillary lubrication modelaccounting for the finite-size effect. The comparison between experiments and theory allows us tomeasure the air-liquid surface tension, without contact, paving the way towards robust non-contacttensiometry of polluted air-liquid interfaces.
The interface between two media has an energy costper unit surface, called surface tension, resulting fromthe microscopic interactions of the constitutive moleculesat the interface [1, 2]. Surface tension is an importantparameter in soft condensed matter and at small scaleswhere capillary phenomena usually dominate. Examplesinclude wetting properties [3, 4], thin-film dynamics [5,6], multiphase flows...Surface active molecules – i.e. surfactants – are widelyused to stabilize capillary interfaces on purpose, e.g. inemulsions or foams, but are also inevitable due to pol-lution. These contaminants, which are usually adsorbedat the interface between two immiscible fluids, lower thesurface tension and are responsible for specific rheolog-ical properties of the interface [7]. To understand thedynamics of soft materials, the interaction between ob-jects such as droplets and bubbles, or to quantify theamount of interfacial contamination, capillary interfa-cial rheology is essential. Specifically, surface tension ismeasured by a large variety of techniques: pendant-dropmethod [8], spinning-drop method, Wilhelmy plates ordu No¨uy rings [9], for instance. Moreover, the inter-facial rheology is usually measured with the Langmuirtrough [10] or through oscillating-disk devices [11].A complementary device to measure material prop-erties is atomic-force microscope (AFM), which has re-cently been used to study capillary phenomena such asthe interaction between bubbles [12, 13] or droplets [14–16], the hydrodynamic boundary condition at a water-airinterface [17, 18], and dynamical wetting [19–24]. Re-cently, the AFM has also been employed in a dynamicalmode, and appears to be a remarkable tool to quantifyproperties – with the advantage of providing non-contactmeasurements [25–29].In this Letter, we study the force exerted on a water-immersed sphere attached to an AFM cantilever, that isdriven to oscillate near the apex of an air bubble. Thedeformation of the bubble and the force exerted on thespherical probe are coupled, and result from the hydro- r
01 N / m of thecantilever (with the sphere attached to it) is determinedfrom the drainage method [34]. The bulk resonance fre-quency ω / (2 π ) = 1240 ± Q = 3 . ± . C are in the 0 . − . R b are inthe 0 . − . θ (seedefinition in Fig. 1) are in the 40 − ◦ range, with theexact value depending on C . A multi-axis piezo stage(NanoT series, Mad City Labs) is used to control the dis-tance between the sphere and the bubble, by imposing adisplacement to the substrate at very low velocity. Theamplitude A and phase ϕ of the cantilever’s deflectionsignal are measured by a lock-in amplifier (Model 7280,Signal Recovery), and are recorded versus the piezo dis-placement. Additionally, the DC component of the can-tilever’s deflection is also recorded and used to determinethe average gap distance D .The real and imaginary parts of the measured mechan-ical impedance G ∗ = G (cid:48) + iG (cid:48)(cid:48) are plotted in Fig. 2 asfunctions of the average sphere-bubble distance D , fortwo frequencies. Best fits to the model [33] are alsoshown, in good agreement with the data, with the air-
10 100 1000 10
10 100 1000 10 G
D/D c = 10, for the three bubble radii of Fig. 3, as obtainedfrom the model [33]. The insets display zooms near the sym-metry axis. In panel (b), Eq. (S16) [33] is plotted (dashedline) for comparison. Similarly, the real and imaginary partsof the amplitude of the dimensionless deformation field areplotted in panels (c) and (d), respectively. surface tension as a function of the SDS concentrationin water are shown in Fig. 5. We observe that the sur-face tension globally decreases with increasing surfactantconcentration, as expected. At surfactant concentrationssmaller than ∼ . ∼ ± e.g. the curvature radius ofthe undeformed interface, which would modify slightlythe fitted values.Finally, we discuss the capacity of our method to beused as a robust tensiometer. To do so, we performed in-dependent tensiometry experiments on similar air-water-SDS interfaces using the Wilhelmy-plate method [9]. Theresults are shown in Fig. 5, and agree well with the onesobtained with our method. Possible systematic devia-tions at the highest concentrations may result from asurfactant-induced depinning of the contact line of thebubble on the substrate [39]. In such a scenario, the hy-drodynamic pressure would not only trigger a local cap-illary deformation (see Eq. (5)), but would also induce aspreading-dewetting cycle of the bubble on the substrate.In addition, the bubble resonance frequency being lower Surface tension versus concentration.
Concentration (mmol/L) 0.2 0.4 1 2 4 6 8 10 40Surface tension fitted (mN/m) 63 72 58 47 41 44 29 24 29 .
01 0 . Wilhelmy plateAFM [ m N / m ]
ACKNOWLEDGEMENTS
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