Stabilization of nanobubbles under hydrophobic confinement.

Stabilization of nanobubbles under hydrophobic confinement.

Caroline Desgranges and Jerome Delhommelle Department of Chemistry, University of North Dakota, Grand Forks ND 58202
July 26, 2019

It has been recently shown that nanobubbles exhibit a remarkable and unexpected stability. The lifetime of nanobubbles, formed either within liquids or on hydrophobic surfaces, can exceed by more than 10 orders of magnitude the theoretical expectation, which predicts an almost immediate dissolution due to the very high Laplace internal pressure in such small bubbles. This unexpected property of nanobubbles has made them leading candidates for energy applications, e.g. as high-pressure nanoreactors in fuel cells, and for biological systems, as transport systems for gas delivery to membranes and cells. Here we use molecular simulation to shed light on the molecular mechanisms accounting for the formation and stabilization of nanobubbles under an hydrophobic nanoconfinement. Using an entropic reaction coordinate, we elucidate the nucleation pathway and determine the formation free energy of nanobubbles in water confined in carbon nanotubes. We identify a critical volume for which the existence of nanobubbles is thermodynamically favored, as the free energy profile flattens around this critical volume, and mechanically favored, since the nanoconfined fluid pressure, along the nanotube axis, is positive at this juncture. We also show that the stabilization process is assisted by the hydrophobic nature of the carbon nanotube and by the formation of strong hydrogen bonds at the interface.

The formation of nanobubbles, i.e. of nanoscopic gaseous domains within liquids, has drawn considerable interest in recent years Lohse and Zhang (2015); Maheshwari et al. (2016); Seddon et al. (2012, 2011); Parker et al. (1994); Ishida et al. (2000). One of the most striking and intriguing features of nanobubbles is their unexpected stability, e.g. up to two weeks for a nm nanobubble Ohgaki et al. (2010), despite the theoretical expectation that these domains should be unstable and dissolve as a result of their high internal Laplace pressure. This amazing property of nanobubbles has thus made them emerge as promising candidates for many applications, e.g. as high pressure nanoreactors with enhanced reaction kinetics Svetovoy et al. (2011) that could be employed within fuel cells without the need for expensive catalysts. Other potential applications include their use as ultrasound contrast agents, as transport for gas delivery to membranes and cells Dzubiella (2010) which could have effects on transmembrane proteins and on membrane structures, in turn, modifying cell function and promoting biological functions. Furthermore, nanobubbles can serve as a seed of the growth of larger bubbles, responsible for decompression sickness Craig (1996). Other recent developments in nanotechnology involve the role of nanobubbles to promote the motion of nanomotors by bubble propulsion for nanomedicine applications Wang et al. (2013). Several mechanisms have been proposed to account for the remarkable stability of nanobubbles, either through the assistance of nearby substrates in the case of surface nanobubbles Zhang et al. (2007); Brenner and Lohse (2008); Weijs and Lohse (2013) or through the adsorption of ions on the outside of bulk nanobubbles Ohgaki et al. (2010). However an understanding of nanobubble formation at the molecular level still remains elusive as none of these mechanisms fully account for the stabilization of such objects. From an experimental standpoint, surface bubbles can be generated e.g. through electrolysis Zhang et al. (2006), while nanobubble solutions are induced mechanically by passing a solution through a small space Ohgaki et al. (2010). On the basis of these experiments, several possible explanations have been proposed to account for the stabilization of nanobubbles. For instance, in the case of surface nanobubbles, it was suggested that hydrophobic surfaces play a key role in the stabilization process Zhang et al. (2007); Tan et al. (2018) as they tend to adsorb a vapor phase, which, in turn, provides a mechanism for the stabilization process through a thermodynamic equilibrium Brenner and Lohse (2008); Weijs and Lohse (2013). In the case of bulk nanobubbles, the stabilization process seems to be connected to the formation of a strong hydrogen bond network at the interface, as evidenced by the results from attenuated total reflectance infrared spectroscopy Ohgaki et al. (2010). Here, we use molecular simulation to shed light on the stabilization of nanobubbles within carbon nanotubes filled with water. The simulation allows us to identify a mechanism starting with the initial formation of small surface nanobubbles that coalesce to yield a large gaseous nanodomain stabilized by the adsorption of a vapor phase close to the hydrophobic walls. Our results also provide a rationale at the molecular level, for the prolonged stability of surface nanobubbles. More specifically, our analysis leads to the determination of a critical volume for the nanobubble, which becomes thermodynamically favored, since the free energy profile is remarkably flat at this point, as well as mechanically favored, since the pressure, in the direction parallel to the nanotube axis, reaches a positive value for a nanobubble of this size. This also suggests a new way of preparing surface nanobubbles through the choice of the thermodynamic conditions, instead of using e.g. electrolysis or some mechanical means. This new preparation can also be readily applied to other single component systems as well as to mixtures, which are of particular interest in energy applications and fuel cells.

To understand the formation of the metastable states spanning the nanobubble formation pathway Debenedetti (1996), we employ a molecular simulation method that allows for the sampling of rare events. Here, we achieve this by using the recently developed method Desgranges and Delhommelle (2017a), which employs an umbrella sampling (US) potential Torrie and Valleau (1977); Ten Wolde et al. (1995); Desgranges and Delhommelle (2007); Remsing et al. (2015); Punnathanam and Monson (2006); Desgranges and Delhommelle (2011, 2014); Desgranges and Delhommelle (2018) to overcome the free energy barrier associated with the bubble formation and allows us to sample the entire pathway. This US potential is function of an order parameter, or reaction coordinate, that characterizes the structural changes taking place within the nanoconfined fluid, as the formation of the nanobubble proceeds. In this work, we choose an entropic reaction coordinate, calculated from the chemical potential and internal energy of the confined fluid Desgranges and Delhommelle (2017a); Waghe et al. (2012). From a practical standpoint, we perform simulations, with decreasing values for the imposed value for the overall entropy of the system, to sample the entire nucleation pathway. This defines overlapping windows, over which statistics are collected, and the free energy profile for the nucleation process can be calculated using the conventional method for US simulations. We use the force field Berendsen et al. (1987) for water, which models well the thermodynamics of the vapor-liquid equilibrium for water Desgranges and Delhommelle (2017b). The thermodynamic conditions used in simulations ( kJ/kg at  K) are chosen close to the coexistence for bulk water. Prior work has shown that the surface tension for the model Vega and De Miguel (2007) at  K is of  , in reasonable agreement with the experimental value of  . This means that, under such conditions, the Laplace (internal) pressure exceeds the fluid pressure by , which corresponds, for a nanobubble of radius  Å, to an excess pressure ranging from  MPa to  MPa range. Such nanobubbles should dissolve quickly in the absence of any surface-mediated stabilization mechanism Seddon et al. (2012). To elucidate this stabilization mechanism, we simulate the nucleation of nanobubbles in water confined in carbon nanotubes Zambrano et al. (2008); Nosonovsky and Bhushan (2008); Rossi et al. (2004); Tas et al. (2003); Gubbins et al. (2011); Alexiadis and Kassinos (2008); Pascal et al. (2011); Dalla Bernardina et al. (2016) (CNTs) with an armchair configuration , and use the water-CNTs interaction parameters of Goddard et al. Pascal et al. (2011). To determine the impact of the nanoscopic confinement on the nucleation process, we systematically vary the CNT diameter by increasing from to . We perform several different structural analyses to characterize the nanoconfined fluid properties, as well as the features of the developing nanobubble. First, we determine the void fraction within the nanotube by dividing the volume inside the nanotube into small volume elements, with a 0.5 SPC/E interval along the nanotube axis, and a 0.05 SPC/E interval in the radial direction. This allows us to calculate the local density of confined water from the configurations generated during the simulations, with void volume elements identified as containing zero water molecules. Second, we evaluate the number of hydrogen bonds , in which each water molecule is involved, according to the usual geometric criterion Kuffel and Zielkiewicz (2012): (i) the distance between two water molecules must be less than 3.5 Å , (ii) the distance between the O of the first molecule and the H of the second molecule involved in the hydrogen bond is less than 2.5 Å  and (iii) the angle along the hydrogen bond is less than . Finally, we evaluate the value taken by a tetrahedral order parameter Errington and Debenedetti (2001) for each water molecule.

Figure 1: (a) Correlation between the free energy profile (top) associated with the dewetting process, from right to left, and the variation of the parallel pressure () of water in CNT . The first arrow indicates the onset of the nanobubble nucleation process, with a sharp increase in free energy, and a sharp drop in which becomes negative. The second arrow corresponds to a flat part of the free energy profile, associated with a positive value for and, at the molecular level, with the nanobubble shown in (b).

We start by commenting on the results obtained in the case of the CNT. The free energy profile for the nanobubble nucleation process is shown in Fig. 1. Since the formation of a bubble in a capillary generally corresponds to the onset of a negative pressure in the system, as shown e.g. in acoustic experiments Caupin et al. (2012) and simulations Giovambattista et al. (2012), we also determine the fluid pressure along the nanotube axis, , through the virial expression, and plot its variation along the nucleation pathway. The nucleation pathway starts from the right hand side of the plot, i.e. from a completely filled CNT, with a high water loading and thus a high total entropy , and a positive value for . At this point, the system is a metastable nanoconfined liquid, associated with the local minimum in free energy reached for around  kJ/kg/K. To confirm the nature of the confined fluid, we carry out a structural analysis and show in Fig. 2 that the void fraction is equal to for this value of and the density of water in the CNT is of  , which is typical of nanoconfined water. As decreases, the nanobubble nucleation process starts to take place with the formation of very small cavities close to the hydrophobic surface of the CNT. This is the first part of the nucleation process, which is characterized by an increase in the free energy of the nanoconfined fluid, and by a pronounced dip in , which becomes negative (see e.g. Fig. 1a for  kJ/kg/K). During this stage, the cavities that form close to the surface are very small and have a very large internal pressure. This in turn results in a system that is both mechanically unstable, with a strongly negative , and thermodynamically unstable as the free energy profile exhibits a significant slope during this stage. The formation of these cavities can also be monitored through the steady, albeit slow, increase in the void fraction, and through the decrease of the water loading seen on the right panel of Fig. 2.

Figure 2: (Left) Void fraction in CNT during the nanobubble nucleation process, and (Right) Density change in the nanoconfined fluid along the nucleation pathway.
Figure 3: Energetic and structural features for the three classes of water molecules during the nanobubble nucleation process (CNT for  kJ/kg/K), with from left to right, the water-water interaction energy ( is given in units reduced with respect to ), the water-CNT interaction energy (, same unit as ), the number of H-bonds per water molecule () and the tetrahedral order parameter (). Class I denotes vapor-like water molecules located between the two vapor-liquid interfaces defining the nanobubble, while Class II are molecules at the nanobubble surface (the rest of the molecules are labeled Class III).
Figure 4: Nanobubble formation in CNT (a) and in CNT (b). Nanobubbles, shown here as a translucent surface, are taken when both favored thermodynamically and mechanically, as indicated in Fig. 1a. Nanobubbles critical volumes are of  Å for CNT and of  Å for CNT .

As further decreases, the void fraction undergoes a more rapid increase, as the cavities start to coalesce. Then, as coalescence further advances, a thorough reorganization takes place within the fluid with the formation of a nanobubble across the nanotube. The snapshot, plotted in Fig. 1b, is obtained when reaches  kJ/kg/K. It shows a typical configuration of the nanoconfined liquid, showing that a nanobubble, surrounded by the nanoconfined liquid has nucleated. At this point, the free energy profile reaches a maximum, indicating that the free energy of nucleation of the nanobubble is of  , and that the critical volume for the nanobubble is of  Å. Furthermore, the plot shows that the free energy profile becomes flat, while becomes positive again. The combination of these two factors results in a stabilization of the nanobubble, both mechanically since the fluid pressure is positive again, and thermodynamically, since the system is on a flat part of the free energy profile. This implies that the nanobubble so obtained can remain metastable over a prolonged period of time, thereby providing insight into the unexpected stabilization of a nanobubble. Then, as further decreases, the free energy profile leaves the plateau and starts to decrease again. This decrease in free energy occurs concomitantly with a decrease in , as well as a continued increase in void fraction and a decrease in the density of nanoconfined water. This indicates that nanobubbles, with a volume exceeding the critical volume, start to spontaneously grow and take over the system. The free energy profile then continues to decrease, while converges towards the pressure of a vapor phase of water adsorbed in the nanotube. At this stage, the free energy profile reaches its minimum, indicating that the system has reached its stable phase, the nanoconfined vapor, which marks the end of the pathway for around  kJ/kg/K. We finally add that the free energy profile obtained in this work is consistent with those found for the dewetting process under nanoconfinement using umbrella sampling simulations Remsing et al. (2015) as well as forward-flux sampling simulations Altabet et al. (2017) for rigid confining plates.

This now prompts the question of identifying the exact structure and organization, at the molecular level, accounting for this phenomenon. Experiments, as well as theoretical approaches, have suggested that a specific mechanism and microscopic organization takes place at the vapor-liquid interface to account for the unexpected stability of the nanobubbles Lohse and Zhang (2015); Seddon et al. (2012); Weijs and Lohse (2013). We therefore carry out an analysis of the interaction energy between water molecules (), between molecules and the CNT (), as well of the hydrogen bond network of water, determining the number of hydrogen bonds per molecule Kuffel and Zielkiewicz (2012) () and of the value of the tetrahedral order parameter for each molecule Errington and Debenedetti (2001) (). This analysis allows us to identify three different types of water molecules, as shown in Fig. 3. The first type, Class I, are molecules that interact with few other water molecules (smaller values), are located close to the CNT surface (large values), and are involved in fewer hydrogen bonds than the other types of molecules. Class I are therefore vapor-like water molecules, close to the hydrophobic surface of the CNT, that assist the formation and stability of the nanobubble. This confirms the general hypothesis for surface nanobubble stability Lohse and Zhang (2015). Here, the CNT provides a surfaces that repels liquid water and adsorbs vapor molecules, which, in turn, reduces the surface area through which outfluxing takes place Seddon et al. (2012). Turning to Class II molecules, which are located at the vapor-liquid interface on both sides of the nanobubble, we find dramatically different energetic and structural features than for Class I. From an energetic standpoint, Class II molecules experience more strongly attractive water-water interactions, weaker interactions with the CNT, as well as twice as many hydrogen bonds and an increased amount of tetrahedral order than Class I molecules. This is expected, since roughly half of the environment of Class II molecules is filled with water molecules of Classes II and III. Similarly, the reduced amount of tetrahedral order, when going from Class III to Class II, is correlated with the decrease in the number of neighboring water molecules for surface (Class II) molecules. What is unexpected, however, is the closeness, both in terms of and , found between molecules of Classes II and III. Given the reduced number of neighboring water molecules for Class II, this means that Class II molecules are arranged, on the surface of the nanobubble, in a way that maximizes the attraction between water molecules as well as the hydrogen bond network. This finding is consistent with the experimental observations of Ohgaki et al. Ohgaki et al. (2010), who carried out attenuated total reflectance infrared spectroscopy to show that strong hydrogen bonds formed on the surface of nanobubbles. The simulation results presented here therefore shed light on two molecular processes that assist the formation and stability of nanobubbles, through the adsorption of vapor-like molecules close to the hydrophobic surface of the CNT and through the existence of strong hydrogen bonds between water molecules located at the vapor-liquid interface defining the nanobubble.

How does the extent of the nanoscopic confinement impact the nanobubble nucleation process? To address this question, we compare the free energy profiles obtained for the CNTs under the same conditions of chemical potential and temperature. The snapshots shown in 4 provide a direct comparison between the nanobubbles obtained at the top of the free energy barrier for CNT and CNT . The free energy profiles reveal that the free energy barrier of nucleation are of the same order, i.e. of   for CNT and of   for CNT , with a similar mechanistic pathway followed for the formation of the nanobubbles. Both nanobubbles spread across the entirety of the nanotube, with a critical volume that is shown to exhibit more than a two-fold increase as the CNT diameter increases from about  Å  for CNT to  Å  for CNT . The nanobubbles share the same qualitative features, i.e. are located on a flat part of the free energy profile, are associated with a positive value for , confirming the crucial role played by these two favorable factors and are stabilized by the presence of vapor-like water molecules close to the hydrophobic surface and by the existence of strong hydrogen bonds at the vapor-liquid interface of the nanobubble.

To shed light on the unexpected and mysterious stability of nanobubbles, we unravel in this work the nucleation pathway corresponding to the formation of nanobubbles in water confined in a carbon nanotube. We achieve this by using a molecular simulation method, function of an entropic reaction coordinate, to analyze the ordering processes taking place within the nanoconfined liquid and resulting in the formation of a nanobubble of a critical size. Several key factors accounting for the stabilization of such nanobubbles are identified. First, from a thermodynamic standpoint, configurations containing a nanobubble of a critical size are located on a flat part of the free energy profile. This leads to a prolonged stabilization of the nanobubble, since the absence of a strong free energy gradient will result in a very slow dissolution of the nanobubble. Second, the formation of this nanobubble occurs with a sign change in the fluid pressure, which becomes positive again, leading to a mechanical stability for these nanobubbles. Third, from a structural standpoint, the stabilization of the nanobubble is assisted by the adsorption of vapor-like molecules close to the hydrophobic surface and by the onset of strong hydrogen bonds at the vapor-liquid interface, thereby confirming recent experimental and theoretical findings Ohgaki et al. (2010); Zhang et al. (2007); Brenner and Lohse (2008); Weijs and Lohse (2013). Our results also suggest how the stabilization of nanobubbles can be controlled through the choice of thermodynamic conditions in the absence of any additives, e.g. ions to stabilize the vapor-liquid interface. This is key for many applications of nanobubbles, for energy production applications since nanobubbles can serve as high-pressure nanoreactors for fuell cell systems, and for biological systems, as nanobubbles provide a means for gas transport to membranes and cells.

Acknowledgements Partial funding for this research was provided by the National Science Foundation (NSF) through CAREER award DMR-1052808.


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