Oscillations of Bubble Shape Cause Anomalous Surfactant Diffusion

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Oscillations of Bubble Shape cause Anomalous Surfactant Diffusion: Experiments, Theory and Simulations. Antonio Raudino, Domenica Raciti, Antonio Grassi, Martina Pannuzzo, and Mario Corti Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02054 • Publication Date (Web): 10 Aug 2016 Downloaded from http://pubs.acs.org on August 16, 2016

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Oscillations of Bubble Shape cause Anomalous Surfactant Diffusion: Experiments, Theory and Simulations. Antonio RAUDINOa*, Domenica RACITIa, Antonio GRASSIb, Martina PANNUZZOc, Mario CORTId,e.

a

Department of Chemical Sciences, University of Catania, Viale A. Doria 6-95125, Catania, Italy

b

Department of Pharmacy, University of Catania, Viale A. Doria 6-95125 Catania, Italy

c

Department of Physics, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, Pennsylvania 15213, USA

d

CNR-IPCF Viale F. Stagno d'Alcontres, 37, 98158 Messina, Italy

e

LITA, University of Milano, Via Fratelli Cervi 93, 20090 Segrate, Milano, Italy

ABSTRACT We investigate, both theoretically and experimentally, the role played by the oscillations of the cell membrane on the capture rate of substances freely diffusing around the cell. In order to obtain quantitative results, we propose and build up a reproducible and tunable biomimetic experimental model system to simulate the phenomenon of oscillation-enhanced (or depressed) capture rate (chemoreception) of a diffusant. The main advantage compared to real biological systems is, that the different oscillation parameters (type of deformation, frequencies and amplitudes) can be finely tuned. The model system we use is an anchored gas drop submitted to a diffusive flow of charged surfactants. When the surfactant meets the surface of the bubble, it is reversibly adsorbed. Bubble oscillations of the order of a few nanometers are selectively excited and surfactant transport is accurately measured The surfactant concentration past the oscillating bubbles was detected by conductivity measurements. The results highlight the role of surface oscillations on the diffusant capture rate. Particularly unexpected is the onset of intense overshoots during the adsorption process. The phenomenon is particularly relevant, when the bubbles are exposed to intense forced oscillations near resonance.

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The capture rate of moving particles by moving (or fluctuating) traps is a fundamental problem in different fields of sciences, ranging from simple bi-molecular chemical reactions to complex preypredator ecological systems. For instance, living cells show endless flickering motion of the membrane enveloping the cells. The nature of the flickering motions associated with thermal and/or metabolic energy exchanges has been pioneered by Brochard and Lennon years ago1 and related to the viscous and elastic properties of the cell, such as cell tension, membrane thickness, membrane-cytoskeleton coupling and so on2-7. For instance, the erythrocyte membrane displays undulation frequencies as high as 103 s-1 and amplitudes ranging from 10 to 100 nm together with much slower motions8.9 (protrusions, blebbing), in the scale of seconds. Oscillations alter the distribution of diffusing particles near the cell surface and then might modify their sorption by the cell (chemoreception). Bongrand et al10. suggested that dynamic deformations of the cell membrane could initiate and modulate signaling events through, e.g., altered binding of messengers. A different, but related class of biological phenomena concerns, e.g., the enhanced catching of diffusing substances by a swimming cell in comparison with a standing cell, as pioneered in a classical paper by Berg and Purcell11. Several properties in different technological devices are strongly affected by the oscillations of the interface. For instance, enhanced exchange of matter (particle desorption) or energy (heat dispersion) has been detected for vibrating solid surfaces in close contact with a fluid phase. The understanding of these dynamic effects would explain the complex mechanisms underlying the ultrasound cleaning techniques12 or the vibration-enhanced heat dissipation13,14. Because of the complexity of the systems described before, in this work we address a key fundamental and specific question: how do interfacial fluctuations affect the diffusive transport of a molecule from the bulk solution to the interior of a biomimetic model of a cell (chemoreception)?. The question is far from being trivial because: a) Different mechanisms couple diffusant transport and interface motions: i) hydrodynamic effects due to the oscillation-induced fluid motion near the interface; ii) direct intermolecular interactions between the oscillating interface and the diffusant; iii) fielddiffusant interactions. Moreover, fields can be either deterministic (e.g., an oscillating electric field), or stochastic (e.g., hydrodynamic fluctuations). b) Both adsorption and desorption rates concur to the total amount of particles at the interface. c) There are different kinds of oscillations: radial oscillations (periodic variations of the cell volume) and shape oscillations (where the cell changes its shape at constant volume). 2 ACS Paragon Plus Environment

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Because of the many variables, the diffusive transport from the bulk phase to the interior of a fluctuating interface is modified in a complex way. Single physical effects may either enhance or depress the total uptake kinetics, the net effect is a combination of different factors. In this paper, we develop an idealized biomimetic model to study chemoreception by a single fluctuating cell. The cell is mimicked by an oscillating air bubble, its vibrations are selectively excited by a periodic electric field and the resulting surface motions are analysed by a recently developed interferometric technique15-17. The bubble is held fixed inside a stream of diffusants that strongly bind to the bubble surface. The sorption of diffusants at the bubble surface has been simulated by employing a charged surfactant. Surfactants are partially soluble in water because of their polar head, and, at the same time, they prefer to settle at the bubble surface because of the hydrophobic repulsion between apolar tails and bulk water. Trapping of surfactants at the interface at equilibrium is well-known, extension to nonequilibrium conditions has been reported in the literature18,19. The studied system is more complicated because: a) Surfactants are not homogeneously distributed around the bubble but unsteadily diffuse along a spatial coordinate; b) the bubble surface periodically oscillates. The paper is organised as follows. We first briefly describe our interferometric technique to measure the amplitude of selectively excited bubble’s oscillations. Next, we discuss the main results of our diffusion experiments, where a diffusive flux of surfactants crosses an oscillating bubble and sticks on its surface. Then, we report extended coarse-grained Molecular Dynamics (MD) simulations of a surfactant-covered oscillating bubble. Experimental and MD results are interpreted by a simple diffusional model.


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RESULTS AND DISCUSSION Experimental results An oscillating air bubble in water mimics a single fluctuating cell. The bubble, of the order of 0.5 mm in diameter, is formed at the top of a stainless steel tube protruding out from the bottom of a small square cell. Taking advantage of the effective net charge existing at the water-air interface (negative for pure water20), bubble oscillations are excited at controlled amplitudes by a periodic electric field. The resulting surface motions are analysed by a recently developed interferometric technique15-17. When traversed by a Gaussian laser beam, the bubble air-water interfaces act as the mirrors of a confocal Fabry-Perot interferometer. This is because the refractive index mismatch causes light reflection at the gas-liquid interfaces. A set of fringes are formed in the backward direction, which allow very accurate measurements of the optical path variations inside the bubble. This effect enables us to measure extremely small variations in the bubble radius due, e.g., to the bubble deformation oscillations. The root mean square noise-amplitude is evaluated to be of the order of a few tens of a mV, which in terms of radius change corresponds to a few hundredth of a nanometre. The unavoidable prerequisite is that both the bubble (or drop) and the surrounding fluid are transparent to the laser beam. At the surface, the periodic boundary conditions provided by the bubble finiteness allow only a discrete spectrum of stationary oscillation modes. The bubble is set into vibration at the lowest frequency mode, which is in the frequency range of 100 -200 Hz depending on bubble radius and its interfacial properties (surface tension). Details of the interferometric principles applied to a vibrating bubble (or drop) are reported in SI 1. The bubble is held fixed within of a diffusing stream of freely moving molecules (diffusants). Two electrodes, placed horizontally above the bubble (Fig.1), measure the time evolution of conductivity, which yields the concentration build-up of charged diffusants after their injection at the bottom of the vessel.


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Fig. 1- Real (upper figure) and schematic (lower figure) setup of the experimental apparatus. The central sphere mimics the oscillating bubble.

A typical result of our interferometric measurements is given in Fig. 2, where we report different resonances of a naked oscillating bubble. Additional details are given in SI 2.


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Frequency (Hz) Fig. 2- Amplitude (arbitrary units) of the bubble shape deformation against the applied frequency (Hz). The curve shows resonances at certain critical frequencies corresponding to the lower eigenvalues. Data have been obtained for a naked gas bubble in pure water.

Sorption of surfactants at the bubble surface broadens the vibrational spectrum as shown in Fig. 2B of SI 2. In the following sorption experiments, the bubble is always excited at the lowest resonance,

l=2, to obtain the most intense oscillation amplitude. Let us discuss now the central part of the work: the fate of diffusants near an oscillating biomimetic cell. First, we check the validity of our concentration measurements. Results of the diffusion measurements are shown in Fig. 7 of SI, which compares the time evolution of the conductivity of NaCl and SDS water solutions in absence of bubbles. The Diffusion coefficients extracted from these experiments closely match those reported in the literature (DNaCl ≈ 2 x 10-9 m2

s-1 (Ref. 21) and DSDS (monomeric form) ≈ 5 x 10-10 m2 s-1 (Refs. 22-24) at 25 oC). Once the reliability of the conductivity measurements has been ascertained, we next investigate the effect of the bubble on the surfactant diffusional flow. Results are summarized in Fig.3, where the conductivity data in absence of the bubble (red curve) are compared with those obtained by introducing a bubble with surface saturated by charged surfactants (blue curve). Our results show, that the bubble itself has a small effect on the diffusive flow of surfactants. A different and unexpected behavior is observed, when we introduce an empty bubble (no surfactants at the interface) oscillating at resonance frequency (Fig.3, black curve). In this case, we find an initial sharp increase of the solution conductance (REGION I) followed by a fast decrease of 6 ACS Paragon Plus Environment

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the conductivity (REGION II) and, again, a new steady increase in the number of surfactants reaching the detector (REGION III).

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t (sec) Fig. 3- Conductance of the aqueous solution measured over the bubble against time. Red squares: diffusive flux in the absence of the bubble, Blue triangles: saturated bubble submitted to a flux of surfactants, Black squares: void oscillating bubble submitted to a flux of surfactants.

The behavior described above is even more intriguing, because it is observed only, when the bubble is forced to oscillate and critically depends on the amplitude of oscillations, as determined by the applied electrostatic potential. Typical features are shown in Fig. 4, where we report the black curve of Fig.3 obtained for two different values of the applied potential (3 and 0.3 V, respectively). Other potential differences in the range between 3 and 0.3 V yield similar curves (not reported here) spanning between the red and the black ones of Fig.4.


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Fig. 4- Effect of the oscillation amplitude on the conductance of the aqueous solution. Red curve: oscillating bubble with small amplitude, Black curve: oscillating bubble with high amplitude in the range of 10 nm.

Tentatively, the overall conductivity behavior reported in Figs.3 and 4 could be ascribed to competing effects: A) A conductivity increase due to the faster migration rate of charged surfactants at the waterair interface. A large interfacial mobility arises because the surfactant tails protrude in the air, while head groups are embedded in the fluid. Whence, interfacial surfactants experience a smaller friction than those in the bulk fluid (see, e.g., Refs. 25 and 26). B) Convective motions associated to the bubble oscillations may enhance the interfacial surfactant transport. C) Most surfactants reaching the bubble surface are tightly adsorbed and cannot reach the detector. Therefore, the number of surfactants contributing to conductance decreases during adsorption. While experiments performed by saturated bubbles (the blue and red curves in Fig 3 are almost superimposed) exclude the hypothesis that the enhanced transport is due to the bubble-induced convective motions, the overall behavior still remains rather mysterious. On the basis of the theoretical and simulation results we are going to discuss, we will highlight a more complex and, we believe, interesting scenario.


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Molecular Dynamics (MD) Simulations Theoretical calculations are appealing because of their intrinsic simplicity, but relevant effects are often neglected. For instance, the hydrodynamic solvent displacement around a fluctuating bubble may contribute to the diffusant transport and sorption onto the bubble surface. In principle, the velocity field of a fluid around a vibrating sphere can be calculated and combined with the diffusant transport via a diffusion-advection equation. This task, however, is quite difficult, so we attacked the problem from a computational point of view by MD simulations that, in principle, take into account all many-body effects. Details are described in the section Methods. The studied system is reported in Fig. 5A, showing a bubble inside a diffusional flux of surfactants. The response of the bubble to the external electric field in relation to the quantity of adsorbed SDS molecules has been calculated at different surfactant concentration. Under the effect of the external field, the adsorbed surfactants follow the bubble oscillations and, in turn, the oscillation amplitudes are proportional to the number of adsorbed SDS molecules. The behavior of surfactants near the oscillating bubble surface has been quantitatively calculated in terms of the average diffusion coefficient D of SDS molecules as reported in Fig.5B. Low D means, that most SDS molecules are trapped onto the bubble surface, where they slowly diffuse in a concentration-dependent fashion (crowding), while high D suggests desorption of monomers or fragments into the bulk solutions.

Fig. 5A- Molecular Dynamics snapshots of a solution with surfactants diffusing (along xy axes, Blue arrow) around an oscillating gas bubble. Left and right drawings refer to early and late states of the diffusion process, respectively. Surfactants are depicted as hydrophobic tails (Blue) connected to a charged head (Yellow). Counterions (Pink) ensure the electroneutrality of the system. An oscillating electric field is applied along z axes (Red arrow).


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Fig. 5B- Molecular Dynamics snapshots of an oscillating gas bubble partially covered by surfactants (hydrophobic tails in cyan and charged head in yellow). From the top to the bottom, systems contain an increased concentration of SDS molecules (9, 21, 36 respectively). Snapshots on the right refer to the same systems reported on the left side, but under the effect of an oscillating Electric field applied along the z axis (red arrow), perpendicularly to the flux direction (along x,y axes). The averaged SDS diffusion coefficients D are reported for each system at the bottom of the relative frame. To speed-up simulation times, we used smaller drops than those reported in Fig.5A. Each system has been simulated for 100 ns. In the inset we report the oscillation of the electric field during the first 200 ps of the whole run.

Two important messages stem from visual inspection of MD data and from the calculated diffusion coefficients: A) In agreement with related experiments27-30, self-aggregation of adsorbed surfactants is more efficient when submitted to an oscillating electric field. This is not a trivial polarization 10 ACS Paragon Plus Environment

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effect of the adsorbed charged molecules. Indeed, as shown in the upper inset of Fig. 5B, the field rapidly oscillates, reversing its orientation many times during the whole MD simulation. B) As shown in this and in our previous paper17, the surfactant desorption rate is greater once their surface concentration has reached a critical threshold to form clusters. Lastly, the increased SDS concentration enhances the oscillation amplitudes because of a stronger coupling between the applied field and the bubble surface charge (see Fig.4 of SI). These results will be used in the next section to develop a realistic model for surfactant adsorption, which turns to be able to explain our experiments on the empty oscillating bubble. Before to end the simulations section, a word of caution is in order. Simulations and experiments cover different length and time scales. MD calculations are unavoidably constrained by the number of particles and the simulation times. Therefore, the simulated bubbles are much smaller than those used in the experiments and a direct comparison between simulations and experiment would be incorrect. The studied system is characterized by processes occurring on different time scales. A long time scale (of order of seconds) involves diffusion-controlled transport of surfactants from the bottom of the vessel to the bubble, while a faster time scale involves phenomena such as the concentration-dependent adsorption/desorption kinetics, the field-induced surfactants clustering, and so on. MD simulations capture these short-time phenomena, while the slow diffusion-controlled variation of the local surfactant concentration near the bubble surface would require extremely time-consuming simulations. Since diffusion modifies the surfactant concentration near the bubble, we can guess the role of the diffusion-controlled transport by comparing simulations performed at different surfactant mean concentrations (see Fig. 5B).

Diffusional model As said in the introduction, at the beginning of the experiment (t = 0) the surfactant is set at the bottom of the vessel (see Fig.1). Because of the concentration gradient, surfactants diffuse toward the bubble, where they are tightly adsorbed onto the air-fluid interface. Different behaviors arise, when the oscillating field, that excites the bubble is either parallel or perpendicular to the surfactant concentration gradient (see Fig. 5 of SI). In the perpendicular configuration, the surfactant motion is little affected by the field. On the contrary, a field along the diffusive flux introduces a coupling with the surfactant charge. The resulting effective diffusion coefficient strongly depends on frequency and amplitude of the oscillating field31. In order to get a transparent picture, we always used electric fields perpendicular to the concentration gradient. 11 ACS Paragon Plus Environment

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The diffusion-sorption behavior was simulated by a diffusion equation in the presence of a trap. The geometry of the studied system is reported in Fig. 6A, while mathematical details are given in SI 3. At low coverage, the sorption of surfactants at the bubble liquid-air interface was enforced through the boundary condition: c Γ = 0 , where c is the surfactant concentration in the fluid phase calculated at the corrugated surface Γ of the bubble. More advanced development valid at high coverage yields more complex boundary conditions, but they do not substantially change the results.

Fig. 6A- Schematic drawing of the studied system. Geometrical variables are: a the equilibrium bubble radius a and the thickness of the diffusion layer R-a. zo and z1 are the distance of the bubble and of the detector from the bottom of the vessel-containing diffusants, respectively.

Results of our calculations reported in Fig.6B (full and dashed red curves) well reproduce the experimental behaviour shown in the red and blue curves of Fig. 3, namely the diffusion in absence of the bubble and in presence of a standing bubble saturated with surfactants at the interface. They show a lag time followed by a steady growth and confirm the observed small role played by a steady adsorbing trap (the bubble) on the diffusive flow of surfactants. The model calculation reported in Fig.6B were simplified by neglecting the enhanced surfactant diffusivity at the drop surface. Indeed, the enhanced surface diffusivity would have had the effect of


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even reducing the already small difference between the dashed and full red lines of Fig 6B as highlighted in our experiments. Let us now refine the model introducing the effect of surface oscillations on the surfactant binding. The simplest way to consider the effect of surface undulations is to re-normalize the boundary conditions. Recently, Mocenni et al.32 derived an elegant approximation to solve diffusion equations with corrugated interfaces. After averaging, the result is a new sink condition calculated onto a wider, but smooth, surface. The increase of the surface area, however, is modest, because the oscillation amplitudes are small compared with the bubble radius (of the order of 1 − 10 × 10 −9 m, while the bubble radius is about 10 −3 m ). Whence, the calculated values are almost insensitive on surface oscillations and closely match the red curves of Fig. 6B. Yet these calculations are not able to reproduce the observed overshoot in the SDS release, black line in Fig.3, for the case of the empty bubble excited by the oscillating electric field. This effect is significant only in the early stage of the diffusion process, becoming more and more evident on raising the intensity of the applied field (Fig. 4). New physical effects have therefore to be included to model the observed overshoots. Out-of-equilibrium adsorption (release) higher than the value expected in a relaxed state (overshoot) has been sometimes reported for different adsorption/desorption kinetics. For instance, intense overshoots have been found in the adsorption rate of binary fluid mixtures33,34 or during the adsorption of macromolecules with slowly interchanging geometrical arrangements35-37. Here and in a previous paper17, we have proved by MD simulations a collective desorption of SDS molecules from the bubble surface, the desorption rate depending on the lateral interactions among the adsorbed surfactants. Simulations show that the collective desorption involves the detachment of micellar-like fragments once the transition from a gas-like to a liquid-like phase took place at the surface. This observation is key for the understanding and modeling of the observed overshoot. The idea was already presented in the literature. For example, Borwankar and Wasan38 stated that, in the presence of attractive interactions among the surfactant hydrophobic tails, the free energy depends on the amount adsorbed. Hence, at low surface concentrations the energy barrier to desorption is large and desorption slows down. Also, some experimental data reported in the literature support this conjecture. Fainerman et al.39, for instance, showed that protein desorption from the air-water interface can become immeasurably slow at low surface pressures (i.e., low surface concentration). Similar results were obtained in the case of aggregated colloid particles desorbing from the surface of gas bubbles40. 13 ACS Paragon Plus Environment

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Let us now consider another point, which turns out to be important for the understanding of the overshoot: the effect of an oscillating field on self-aggregation. An increasing number of experiments and theories bring convincing evidence of oscillation-induced self-aggregation27-30,41 (a nice exemplum of noise-induced ordering42). At the same time, observations concerning the substrate oscillation-induced desorption are accumulating43,44. Therefore, it is the synergistic effects of the oscillation-enhanced desorption and the favored self-aggregation (an arrangement more prone to desorption17) which gives rise to the gigantic overshoot once a critical surfactant concentration near the gas-like to liquid-like phase transition has been attained at the bubble surface. Notably, when the applied field is lowered, also the intensity of the overshoot decreases (see Fig.4). The above described effects are then taken into account in the model by introducing a coveragedependent desorption, as shown in detail in SI 3. The model reproduces the intense peak due to surfactant desorption beyond a critical concentration threshold. Although the peak reported in Fig.6B (blue line) stems from the assumption of a sudden, coverage-dependent release of adsorbates from the oscillating surface, the width of the peak was calculated exactly in the framework of the model. The calculated shape of the kinetics curves well reproduces the experimental features of Figs.3 and 4. The width of the peak depends on the surfactant diffusion coefficient, desorption rate and detector-bubble distance: the closer the distance, narrower the peak. After overshooting, the concentration of bound SDS rises again. In principle, a second overshoot could be observed at late time. However, once the SDS concentration in the bulk water near the bubble approaches equilibrium (vanishing diffusional flux), also the ratio between free and bound surfactants reaches a constant value. Then, the probability to observe multiple overshoots is extremely tiny on approaching equilibrium.


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Fig. 6B- Diffusant concentration (arbitrary units) at the detector as a function of time (sec). Red dashed curve: surfactant diffusion without bubble. Red full curve: surfactant diffusion in the presence of a static bubble acting as an imperfect trap for the approaching surfactant. Blue full curve: diffusion in the presence of a fluctuating bubble in the hypothesis of a sudden release of the adsorbed surfactants once a critical surface coverage has been reached.

Let us end this section by investigating another possible mechanism for the burst of surfactant release from the bubble surface. One could speculate that a role could be played by the Marangoni effect. This effect is related to the inhomogeneous surfactant distribution onto the bubble surface in the early stages of adsorption (see Fig.7). The unbalanced distribution gives rise to a hydrodynamic flux from the regions of lower surface tension (surfactants rich region) to those of higher surface tension (surfactants poor region). When the system approaches equilibrium ( t → ∞ ), the surfactant distribution becomes homogeneous and the hydrodynamic flux disappears. At a first sight, the Marangoni effect could provide an explanation for the burst of surfactant release since the hydrodynamic flux enhances the surfactants transport toward the detector only in the nonequilibrium early stages of the process. However, measurements performed at different oscillation amplitudes of the bubble (see Fig. 4) show that the anomalous SDS release occurs only under the effect of an oscillating field, a result that highlights the key role of the surface oscillations. We may conclude by saying that the classic Marangoni effect alone cannot explain the experimental findings.


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Fig. 7- Schematic drawing of the Marangoni effect arising when a bubble is submitted to a non-steady diffusive flux of surfactants along the Z-axis. Panel A: the inhomogeneous distribution of surfactants onto the bubble surface gives rise to a transient Marangoni flux which boosts the transport of more surfactant molecules toward the detector. When the diffusive flux of surfactant slows down (Panel B), the surfactants distribution onto the bubble surface becomes homogeneous and the Marangoni flux vanishes.

CONCLUSIONS

As shown previously, understanding the complex phenomena involving the diffusant motion near an oscillating trap is a difficult task. Some mechanisms (e.g., the surface increment associated with the bubble periodic shape deformation) enhance the oscillation-modulated uptake kinetics, while other mechanisms (e.g., the decrease of the ‘apparent’ mobility with frequency or the field-induced enhanced desorption from the bubble’s surface) decrease the uptake of diffusants. The combination of different factors (including, for instance, the Marangoni effect) can either enhance or depress the whole kinetics. By combining experiments, theory and MD simulations, we were able to draw a consistent picture and to single-out some of the most relevant factors affecting the adsorption/desorption process at the surface of a vibrating bubble. This goal was reached by selectively tuning the oscillation parameters of the bubble and by carefully detecting the surfactant concentration. A peculiar and unexpected effect was the onset of intense overshoots that become more and more pronounced on 16 ACS Paragon Plus Environment

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increasing the intensity of the oscillating electric fields. Different arguments suggest the central role of the field-assisted surfactant aggregation and desorption in explaining overshoots. Although the detailed molecular mechanisms by which surfactant desorption occurs is not completely elucidated, it is evident, that it involves a collective gas-to-liquid phase transition of the surfactants at the bubble surface. Then, under the effect of an oscillating field, the two-dimensional surfactant aggregates at the bubble surface may easily desorb through micellar-like fragments. The mechanism of energy exchange between the capillary waves at the bubble surface and the adsorbate desorption, as well as the dependence of the desorption rate on the surfactant aggregation state, are left to future investigations. The general problem of unsteady (oscillating) traps crossed by unsteady (diffusional) fluxes is a relevant topic with many potential applications. We mention, for instance, the modelling of drug permeation enhancement induced by oscillating mechanical pressure (sonophoresis) or electric field (iontophoresis). We have shown in this paper that even a quite simple and controlled system (the oscillating bubble) exhibits a very rich behaviour based on coupled elementary chemico-physical effects. We are aware that this work leaves many unanswered questions. In our opinion, however, our interferometric technique together with conductometry provides an extremely sensitive (and quite inexpensive) tool to investigate different and new phenomena in the broad field of the soft matter.

METHODS Experimental setup. A small bubble (typically of 0.5 mm radius) is built-up by injecting air through a stainless steel tube (internal diameter 0.3 mm). Once the bubble is formed, a small dead volume valve, placed just below the cell, firmly closes the gas inlet in order to control the bubble’s internal pressure. The bubble stays attached to the tube. The poly-methylmetacrylate cell, 8 x 8 x 10 mm3 (figure 1), is filled with about 400 microliter of pure water. The stainless steel tube (0.67 mm outer diameter) stems 2 mm out of the cell bottom. It can be used as an electrode. Two more electrodes are placed at a height of 4 mm. Taking advantage of the effective net charge existing at the water-air interface (negative for pure water20) the bubble’s motion is excited by a modulated oscillating potential applied across the electrodes 2 and 3 (or 1 and 2, see Fig.1). The amplitudes of the different oscillation modes of the bubble are measured by an extremely sensitive (sensitivity is of the order of 0.1 nm) differential interferometric technique15-17. The excitation set shown in Fig.1 enables us to excite the low 17 ACS Paragon Plus Environment

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deformation modes. For an isolated free bubble, the lowest mode (the quadrupolar one) being a sphere-to-ellipse mode. For a bound bubble the vibrational spectrum is a bit more complex, but the general features are retained45,46. As an example, Fig. 3 of SI gives the spectrum of a 0.64 mm radius bubble in pure water excited by a 3 Volt sine wave between electrodes 1 and 2. The full line is a Lorenzian fit. The same spectrum, scaled to a smaller amplitude, is obtained also with the potential applied between electrodes 2 and 3. In this case, the excitation is provided by the field gradients. This type of excitation is used along with the diffusion measurements. At time t = 0, a controlled small amount of a charged surfactant solution is gently injected on the bottom of the vessel containing about 400 microliter of pure water (see Fig.1). A Hamilton syringe provides 1 microliter of a 2.5 % Sodium Dodecyl Sulphate (SDS) solution. Surfactant molecules freely diffuse across the aqueous medium because of the concentration gradient. When the surfactant molecules hit the bubble, they are strongly adsorbed onto the bubble surface. The SDS final concentration, under the assumption of a homogeneous distribution over the whole volume of the cell, is about 10-4 M. The instantaneous surfactant concentration at a given distance from the vessel-water surface, namely at the height of the electrodes 2 and 3, has been followed by measuring the current through these two electrodes at a given potential. A double channel signal analyzer (Stanford SR785) supplies a sinusoidal voltage to the electrodes, measures the interferometric signal in one channel and the electric current through the cell (proportional to the conductance) in the second channel. The sinusoidal signal, swept through the resonance step by step in frequency, is synchronously averaged over many cycles of the sine wave in order to ensure good noise rejection. Sensitivity of our current measurements is of the order of 100 picoampere. After careful calibration with a SDS solution of known concentration, current measurements give precise access to SDS concentration in a selected position of the bulk solution. We estimate an overall uncertainty of about one micromole. The interferometric signal gives access to the bubble amplitude oscillation as a function of frequency. It is convenient to measure oscillation amplitudes and SDS concentrations with the same two electrodes with the excitation provided by the signal analyzer. The performance of the experimental apparatus with the delicate injection procedure has been checked by measuring diffusion against gravity in absence of the bubble, either in the case of NaCl and of SDS. As an example, Fig. 7 of SI shows the time evolution of conductivity measured at the electrodes 2 and 3 (i.e., perpendicular configuration between the surfactant flux and the electric field, see Fig.5 of SI ) for two separate injections of SDS at two quite different voltages, 3 Volt and 0.3 Volt. The two curves are practically identical. This experiment excludes any significant effect of the electric field perpendicularly applied to the diffusive motion of the charged surfactant. On the contrary, 18 ACS Paragon Plus Environment

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consistent variations were observed when the applied electric field was parallel to the diffusion flux (electrodes 1 and 2). For this reason, we always excited the bubble oscillation by applying a voltage at the electrodes 2 and 3. The involved time scales are consistent with the known NaCl and SDS diffusion coefficients21-24. When the differential potential between the electrodes (1-2 or 2-3, see Fig.1) is zero, the bubble undergoes only a random small perturbation due to the temperature or pressure fluctuations in the thermal bath. By contrast, on rising the oscillation potential at the electrodes, the bubble is excited in a frequency-dependent fashion. Specifically, the amplitudes of the surface deformation motions show a maximum at the bubble resonances, decaying according to a Lorentian function in the neighborhood of the resonance frequencies as shown in Fig. 1 of SI. Intensity and width of the band bring a lot of information on surface charge and local viscosity of the bubble interfacial region. Chemicals.

High grade commercial sodium dodecyl sulphate (SDS) was purified by three

recrystallizations from 95% ethanol and its purity with regard to dodecanol was checked with the method described in reference16. SDS was stored in anidrous atmosphere and solutions were freshly prepared prior to each set of measurements. Molecular Dynamics (MD) Methods. Following our previous papers16,17,47, we studied the system described above by means of extended MD simulations in the framework of coarse-grained methods. Calculations have been performed by GROMACS simulation package48 using a MARTINI Coarse grained force-field parametrization49. For the full topologies, including bonded terms, see http://cgmartini.nl, where topology files in Gromacs format are readily available. Initially, Sodium Dodecyl Sulphate (SDS) molecules were randomly dispersed in a water solution in the presence of counter ions (Na+). The solution was electroneutral. In this work, we used the literature parameters for water, sodium counter-ions and SDS molecules49. The box was not completely filled up with water. This allowed, during the following fast equilibration of 10 ns in a NVT ensemble, spontaneous formation of a bubble having ∼10 nm diameter confined by surfactant islands distributed at the interface. Energy minimization was performed using the method of steepest descents. Equilibration was performed by using the Berendsen weak coupling thermostat algorithm50 in a NVT ensemble with constant of 1.0 ps. After equilibration, the temperature was kept constant at 300 K using the vrescale algorithm51 with a time constant of 1.0 ps. The equation of motion was integrated using a leap-frog algorithm and a timestep of 20 fs. Following standard protocols associated with the 19 ACS Paragon Plus Environment

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Martini force field, the LJ and Coulomb potentials were smoothly shifted to zero between 0 and 1.2 and between 0.9 and 1.2 nm, respectively, using the Gromacs shifting function. The neighbor list was updated every 10 steps with a cutoff of 1.4 nm. Electrostatic interactions were screened implicitly (ε = 15). The centre of mass motion was removed every step. Periodic boundary conditions were applied in all directions. We first have carried out an intensive preliminary study, where the amplitude and frequency of the external electric field and size of the system have been systematically varied. In order to reduce the exceedingly high number of particles and drastically shorten the simulation times, we opted for smaller bubbles than those used in the experiments (10 nm Ø). Such a choice ensures a synergistic reduction of both simulation time and number of particles because the diffusive motion occurs over a much shorter space. Every system composed of approximately 30000 coarse-grained particles has been simulated then for 100 ns. For each system (either in presence or in the absence of an electric field, and at different SDS concentrations) five independent simulations have been performed.

SUPPORTING INFORMATION The Supporting Information is available free of charge on the ACS Publications website at DOI: ….. Section SI 1 briefly describes the optical interferometer, SI 2 accounts for the bubble resonant response to a periodic external excitation. Lastly, in SI 3 we develop an approximate analytical model to rationalize the diffusion experiments.

AUTHOR INFORMATION Corresponding author: *E-mail: [email protected]


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Acknowledgments. Work financially supported by a FIR (University of Catania, Italy) and a PRIN 2010/2011, Cod. prog.: 2010L96H3K. This work was in part supported by the Deutsche Forschungsgemeinschaft (DFG) Research Training Group 1962 Dynamic Interactions at Biological Membranes. REFERENCES

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TOC FIGURE


Oscillations of Bubble Shape Cause Anomalous Surfactant Diffusion

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