Article pubs.acs.org/Langmuir
Hidden Nanobubbles in Undersaturated Liquids Zhenjiang Guo, Yawei Liu, Qianxiang Xiao, and Xianren Zhang* State Key Laboratory of Organic−Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China ABSTRACT: Here, we propose theoretically the existence of a new type of nanobubble in undersaturated liquids. These nanobubbles have a concave vapor−liquid interface featured with a negative curvature rather than a positive curvature for nanobubbles in supersaturated liquids, so that they often hide inside of the substrate textures and it might not be easy to characterize them through atomic force microscopy (AFM) measurements. However, these hidden nanobubbles are still stabilized by the contact line pinning effect and stay at the thermodynamically metastable state. We further demonstrate that similar to the nanobubbles in supersaturated liquids the contact angle of the hidden nanobubbles is more sensitive to the nanobubble size rather than the substrate chemistry, and their curvature radius is dependent on the chemical potential but independent of the base radius. Finally, we show several potential situations for the appearance of the hidden nanobubbles.
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INTRODUCTION The existence of surface nanobubbles1,2 was first put forward by Parker et al. in 1994 to interpret the long-range attractive force observed between two neighboring hydrophobic substrates immersed in aqueous solution.3 Since 2000, the nanobubbles on various substrates have been observed directly or indirectly by many experimental techniques, particularly using atomic force microscopy (AFM).4−18 The surface nanobubbles observed on the fluid−solid interface as well as the bulk nanobubbles19,20 detected in the bulk solutions have diverse potential applications including flotation,21 interfacial slippage in microfluidics,22,23 ultrasound cavitation,24,25 and biomolecular adsorption.26,27 Thus, the preparation methods28−30 and the properties31−34 of nanobubbles and other related stable gaseous states35 (e.g., micropancakes36−39) have been investigated widely. In particular, the superstability of surface nanobubbles has remained most controversial for a long time. According to the classical Laplace theory, the internal pressure of a bubble in the nanoscale size would become so large that the lifetime of the bubble should not exceed tens of microseconds,40 whereas in experimental studies the bubble can be stable at least for hours.4,5,7−13 The extraordinary stability of nanobubbles has been explained with different models, including the dynamic equilibrium model,41−43 the contamination model,44 the high gas density model45,46 as well as the contact line pinning,47−50 and gas oversaturation model.51,52 Now, the existence of the contact line pinning effect of interface nanobubbles has been confirmed experimentally.48 According to recent molecular dynamics (MD) simulations52 and theoretical analysis,51 supersaturation in the liquid phase is another requirement for the stability of protruding interface nanobubbles. Thus, a fundamental question arises: Is there a possibility that the gaseous state of nanobubbles can exist in the © 2016 American Chemical Society
undersaturated liquid? In this paper, we used constraint lattice density functional theory (constraint LDFT)53,54 to investigate the effect of the saturation degree on the stability of nanobubbles and to find out the possibility for the existence of the thermodynamically stable gaseous state in undersaturated liquids.
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MODEL AND METHODS
In the lattice density functional theory (LDFT),53,55 the grand potential is expressed as Ω = kBT ∑ [ρi ln ρi + (1 − ρi ) ln(1 − ρi )] i
ε − ff ∑ ∑ ρi ρi + a + 2 i a
∑ ρi (φi − μ) i
(1)
where kB is the Boltzmann constant, T is the absolute temperature, ρi is the local density at site i, εff and εsf are the fluid−fluid and the solid−fluid interaction strength, respectively, and φi is the fluid−solid interaction (εsf) summing over the nearest neighbors of site i. At a given chemical potential, the density distribution at equilibrium states (i.e., stable or metastable state) can be obtained by solving ∂Ω/∂ρi = 0. On the basis of the LDFT, the constrained lattice density functional theory (constrained LDFT) 53 was proposed particularly to study unstable but equilibrium states by introducing a functional χi defined as Special Issue: Nanobubbles Received: May 9, 2016 Revised: June 1, 2016 Published: June 2, 2016 11328
DOI: 10.1021/acs.langmuir.6b01766 Langmuir 2016, 32, 11328−11334
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Figure 1. (a) The substrate covered with a ringlike roughness with a radius of R = 20 and a height of h = 10 to provide the contact line pinning force. (b) The phase diagram of the metastable gaseous state in the plane of εsf and μ. In the picture, ○ indicates the existence of stable nanobubbles. c−e show typical morphologies of metastable nanobubbles in liquids at different degrees of saturation: (c) supersaturation, (d) saturation, and (e) undersaturation. (f−h) show the corresponding grand energy profiles for three situations shown in (c−e).
⎧ 0 ⎪ χi = ⎨ ⎪ ⎩1
ρi < 0.5
i ∈ vapor
ρi > 0.5
i ∈ liquid
ρi =
(2)
in which κ is the Lagrange multiplier, N0L is the given volume, and NL = ∑iχi. The constrained grand potential is defined as ΩC = Ω + Ω′ = kBT ∑ [ρi ln ρi + (1 − ρi ) ln(1 − ρi )] i
ε − ff ∑ ∑ ρi ρi + a + 2 i a +
κ[NL0
−
∑ ρi (φi − μ) i
∑ χi ]
(3)
i
By solving the equations ∂Ω /∂ρi = 0 and ∂Ω /∂κ = 0, the local density ρi is obtained as C
(
∂χ
1 + exp εff ∑a ρi + a − φi + μ + κ ∂ρi i
)
∀i (4)
Finally, the local density ρi and the Lagrange multiplier κ can be obtained by an iteration algorithm. In our work, we used reduced units with the temperature T* = kBT/εff and the chemical potential μ* = μ/εff. The reduced distance is defined as r* = r/σ, σ being the lattice spacing. Hereafter, the superscript asterisk is omitted to simplify the description. In the LDFT with a simple-cubic lattice, μc = −3.0 corresponds to the vapor−liquid coexistence state, namely, the saturated state in this work. Thus, μ < −3.000 and μ > −3.000 imply the supersaturated and the undersaturated liquids, respectively. In all simulations, the temperature was set to T = 0.8. The simulation box with a size of 70 × 70 × 70 has a substrate placed at the bottom of the box. Periodic boundary conditions were employed in both x- and y-directions, whereas the mirror boundary condition was used in the z-direction. In our simulations, the substrates were decorated with roughness of various geometries to induce the contact line pinning effect.
to distinguish the liquid phase from the vapor phase. The constraint on the volume of droplets/bubbles reads Ω′ = κ[NL0 − NL]
1
C
11329
DOI: 10.1021/acs.langmuir.6b01766 Langmuir 2016, 32, 11328−11334
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RESULTS AND DISCUSSION Existence of Concave Nanobubbles in Undersaturated Liquids. At first, the chemical potential was set to −3.025, −3.000, and −2.975 to study the nanobubbles in the supersaturated, saturated, and undersaturated liquid. According to the definition of the saturation degree, s = exp[(μc − μ)/ kBT], the saturation degrees are 103.2, 100, and 96.9%, and thus, the excess saturation degree ζ = s − 1 corresponds to 3.2, 0, and −3.1%, respectively. Our calculations indicate that in the supersaturated liquid nanobubbles can be stabilized by the contact line pinning effect due to the substrate roughness (Figure 1a) as long as εsf ≤ 0.7. As an example, the variation of the grand energy with the nanobubble volume and the morphology of the nanobubble corresponding to the local free energy minimum at εsf = 0.5 are shown in Figure 1c,f. The local minimum of the grand energy shows that the nanobubble with a convex vapor−liquid interface is in fact thermodynamically metastable (Figure 1f). This observation is in good agreement with the stability mechanism for nanobubbles illustrated by Liu and Zhang52 as well as by Lohse and Zhang,51 namely, the contact line pinning effect and supersaturation mechanism. In the saturated liquid, our results still show a metastable gaseous state of a nanobubble with a flat vapor−liquid interface (corresponding to the zero contact angle, see Figure 1d), similar to our previous observation in the MD simulations.52 The grand energy curve in Figure 1g proves that this nanobubble is also in the metastable state. Unexpectedly, in the undersaturated liquid, we also observed a local minimum in the grand potential as shown in Figure 1h. The density distribution and the snapshot for the system at this point (see Figure 1e) show a nanobubble-like gaseous state but with a concave vapor−liquid interface. This indicates that, in undersaturated liquids, thermodynamically metastable nanobubbles still exist but their vapor−liquid interfaces have negative curvature rather than positive curvature for nanobubbles in supersaturated liquids. As a result, the nanobubbles in undersaturated liquids often hide in the substrate textures; therefore, it might not be easy to characterize them in popular AFM measurements. Strong Contact Line Pinning Effect Required To Stabilize Concave Nanobubbles. The phase diagram in Figure 1b shows the appearance of nanobubbles at different values of chemical potential μ and different substrate hydrophobicities εsf. It is found that in supersaturated liquids stable nanobubbles can exist in a wide range of εsf, whereas in undersaturated liquids, concave nanobubbles can be stabilized only on rough substrates made of highly hydrophobic materials (εsf ≤ 0.2). Thus, decreasing the saturation level requires an enhanced hydrophobicity of the substrate to stabilize nanobubbles; that is, a stronger contact line pinning effect is required for stabilizing concave nanobubbles. This is the reason that in undersaturated liquids concave nanobubbles have not been observed from our previous MD simulations:52 The substrate in that case is unable to provide a sufficient pinning effect to stabilize concave nanobubbles. To confirm this point, we built a series of substrates with a frustum-shaped one-open-end pore as shown in Figure 2a.
Figure 2. (a,b) The schematic diagrams of the special substrate with inverted triangle-shaped pore. The typical substrate texture and corresponding concave nanobubble are shown in (c) and (d), respectively. (e) Shows the phase diagram of metastable state in the plane of εsf and μ. (f) The variation of nanobubble contact angle θ1 as a function of R2 and εsf. The lines in the picture indicate the intrinsic contact angles that depend on the given interaction between fluid and substrate.
These kinds of re-entrant substrates have been proved to have enhanced ability for stabilizing the Cassie state via strengthening the contact line pinning effect.56 To mimic different reentrant substrates, the radius R2 was varied from 20 to 30 while fixing the base radius R1 = 20 and the height h = 10. In the case of R2 = 20, the re-entrant substrate changes to the aforementioned cylindrical pore, which serves as a contrast to the re-entrant substrates with R2 > R1. In our simulations, μ was set to −2.975 to induce concave nanobubbles in undersaturated liquids. A typical substrate texture and the corresponding stable nanobubble are shown in Figure 2c,d. The simulation results at different values of R2 and different values of εsf were summarized and given in the phase diagram as shown in Figure 2e. The figure shows that to stabilize concave nanobubbles one can increase R2 or decrease εsf or do both of them. Note that increasing R2, as demonstrated before,56 would enhance the contact line pinning effect. This clearly confirms that strengthening the contact line pinning effect is favorable to stabilize concave nanobubbles. 11330
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effect. In our simulations, we first changed εsf to search the existence of stable nanobubbles at the radius of the round hole R3 = 20 and μ = −2.975. Figure 3b gives the obtained contact angle θ2 and the depth d for stable concave nanobubbles, showing that the contact angle for a concave nanobubble is independent of the substrate chemistry, which is the same as that found for nanobubbles in supersaturated liquids.50 For the same substrate, we also considered the effect of the chemical potential on the morphology of hidden nanobubbles at R3 = 20 and εsf = 0.5. The simulation results are shown in Figure 3c, which shows that both the contact angle and the nanobubble depth increase with the chemical potential (the saturation degree). In other words, the curvature radius, Rc = R3/sin θ2, of the nanobubbles decreases with the increase of the level of undersaturation. Finally, we consider the effect of the radius of the hole, R3 at εsf = 0.5 and μ = −2.975. Our simulation results are summarized and given in Figure 3d. The figure indicates that the contact angle θ2 increases with R3, and more interestingly, the curvature radius of the nanobubbles, Rc = R3/sin θ2, roughly remains unchanged for different values of R3 (Figure 3d). Thus, in summary, Figure 3b−d demonstrates that the contact angle of hidden nanobubbles in the undersaturated liquid is dependent on the chemical potential but independent of the substrate chemistry, showing similar tendencies that have been found in experiments for nanobubbles in supersaturated liquids.1 For the curvature radius of hidden nanobubbles, it is dependent on the chemical potential but independent of the pinning radius, again similar to the tendencies found for nanobubbles in supersaturated liquids.47 In other words, the nanobubbles in both the undersaturated liquid and supersaturated liquid behave similarly and have the same origin underlying their stability. Diverse Scenarios for Forming Concave Nanobubbles. The rapid development of nanomaterials inspires the appearance of numerous nanoporous materials with various pore morphologies. Here, we list several typical nanoporous media with different shapes that are able to stabilize concave nanobubbles in undersaturated liquids. Because the undersaturated environment is more easily achieved than the supersaturated environment, we believe that concave nanobubbles may have various potential applications, at least as many as the traditional convex nanobubbles. For example, with the development of the nanofluidics,57 there appears a range of newly synthesized nanomaterials having diverse nanopores, and nanobubbles stabilized by those textured materials may show a significant capability in various potential applications including water purification, desalination, and genomic sequencing. The representative nanomaterials having two openings include carbon nanotubes, boron nitride nanotubes, and graphene with holes. Karnik et al. trapped nanobubbles in the pores with hydrophobic inner walls that have a significant prospect to separate water and nonvolatile ions,58,59 as the nanobubbles can act as ionic barriers that allow the passage of water molecules but not nonvolatile ions. To demonstrate the formation of nanobubbles in the porous media that have tubelike nanosized pores, we built a nanotube made of a hydrophobic wall (Figure 4a). Our simulations show that in the undersaturated liquids stable concave nanobubbles can form inside of the tube, for which the entrance provides the contact line pinning effect (Figure 4d). Note that in this simulation the chemical potential μ = −2.975 and the attractive interaction εsf
To investigate the critical role played by the contact line pinning effect, we also calculated the contact angle θ1 for the stable concave nanobubbles as shown in Figure 2b. Note that we here determined the vapor−liquid interface at which the liquid sites (ρ > 0.5) meet the vapor sites (ρ < 0.5) and then measured the contact angle of the nanobubble through the vapor from the triple-phase contact line. The contact angles of nanobubbles increase with the decrease of R2 (Figure 2f) and are obviously different from the intrinsic contact angle that solely depends on εsf, namely, the materials composed of the substrates. For R2 = 25, for example, the apparent contact angle of nanobubbles is independent of the substrate chemistry, εsf (Figure 2f). For R2 = 30, however, the contact angle seems to be weakly related to the substrate chemistry. For the latter (R2 = 30), detailed inspection of the snapshots indicates a slight slip of the contact line along the side wall of the pore, and the slip may deteriorate the determination of the contact angle in our work based on the geometric relationship. For all of the situations we studied, the contact angle θ1 of the stable concave nanobubbles is always less than the intrinsic contact angle of the substrate (see Figure 2f). Because the pinning force can be quantitatively described by the difference between the intrinsic and apparent contact angles, 47 this observation again demonstrates the essential role of the contact line pinning effect. In other words, a concave nanobubble can stably exist only if its apparent contact angle is smaller than the intrinsic contact angle of the substrate as a result of the contact line pinning effect; otherwise, the contact line may slip along the pore wall and result in the vanishing of the nanobubble. Characteristics of Concave Nanobubbles. To avoid the effect of the slip of the contact line on the determination of the contact angle, we simplified the substrate to a sheet of plane with a round hole. The substrate was placed along the vapor− liquid interface with the liquid being above the substrate (see Figure 3a). Different from the substrates in Figures 1a and 2a in which the contact lines can move along the side wall of the pores, this special structure can effectively prevent the contact line from slipping and thus enhance the contact line pinning
Figure 3. (a) The schematic diagram of the substrate texture and the formed nanobubble due to the pinning of contact line. (b,c) The angle θ2 and height d of the formed are measured as a function of the attractive fluid−substrate interaction and chemical potential. (d) The angle θ2 and curvature radius of formed as a function of R3. The line indicates the averaged curvature radius. 11331
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Figure 4. (a−c) The constructed substrate models and (d−f) the corresponding concave nanobubbles formed.
= 0.2, indicating the occurrence of nanobubbles inside of hydrophobic tubes embedded in the undersaturated liquid. The one-end pores extensively appear on nanomaterials having rough surfaces that show numerous potential applications. For example, nanoporous membranes have been used to explore diodelike effects.60 Here, the one-end pore on a hydrophobic substrate was built (see Figure 4b), and its ability to stabilize concave nanobubbles in the undersaturated liquid was then confirmed by our constrained LDFT simulation (Figure 4e). To fabricate nanochannels or nanofluidics, the nanoslit pore was also a frequently used model.61 Here, we also built a nanosized slit pore (Figure 4c) to show its ability to stabilize concave nanobubbles. During our simulations, we used periodic boundary conditions to ensure that the width of the slit is infinite but with a finite height and depth. From our simulations, we indeed observed stable concave nanobubbles in the undersaturated liquid that was stabilized by the nanopore (Figure 4f).
Finally, we show several potential situations for the appearance of nanobubbles.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (Nos. 21276007 and 91434204). REFERENCES
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CONCLUSIONS To interpret the unexpected stability of the surface nanobubble, the contact line pinning and supersaturation mechanism was proposed, and it shows good agreement with a variety of experiment results. Within the framework of this model, superheating or gas supersaturation is required for the stability of surface nanobubbles. In this work, we try to answer a reverse question, namely, can a nanobubble-like gaseous state exist in the undersaturated liquid? Here, we used constrained LDFT to investigate the effect of the saturation degree on the stability of nanobubbles and to find out the possibility for the existence of a thermodynamically stable gaseous state in undersaturated liquids. Our theoretical calculations predict the existence of a new type of nanobubble in undersaturated liquids that is different from the traditional nanobubbles stabilized in supersaturated liquids. The nanobubbles are featured with the negative curvature of the vapor−liquid interface, so that they may be difficult to be seen with AFM. We demonstrate that similar to traditional nanobubbles the concave nanobubbles show typically the following properties: they are stabilized by the contact line pinning effect and stay at a thermodynamically metastable state; their contact angles are more sensitive to the nanobubble size rather than the substrate chemistry; their curvature radii of hidden nanobubbles are dependent on the chemical potential but independent of the pinning radius. 11332
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DOI: 10.1021/acs.langmuir.6b01766 Langmuir 2016, 32, 11328−11334
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DOI: 10.1021/acs.langmuir.6b01766 Langmuir 2016, 32, 11328−11334
