Article pubs.acs.org/Langmuir
Dynamic Equilibrium Model for a Bulk Nanobubble and a Microbubble Partly Covered with Hydrophobic Material Kyuichi Yasui,* Toru Tuziuti, Wataru Kanematsu, and Kazumi Kato National Institute of Advanced Industrial Science and Technology (AIST), 2266-98 Anagahora, Shimoshidami, Moriyama-ku, Nagoya 463-8560, Japan ABSTRACT: The dynamic equilibrium model for a bulk nanobubble partly covered with hydrophobic material in water is theoretically and numerically studied. The gas diffusion into a bubble near the peripheral edge of the hydrophobic material on the bubble surface balances that out of the bubble from the other part of the uncovered bubble surface. In the present model, gas diffusion in quiescent liquid is assumed and there is no liquid flow. The total changes of energy and entropy are both zero as it is a kind of equilibrium state. The main origin of the dynamic equilibrium state is the gradient of chemical potential of gas near the peripheral edge of the hydrophobic material. It is caused by the permanent attractive potential of a hydrophobic material to gas molecules dissolved in liquid water as there is permanent repulsion of a hydrophobic material against liquid water. Thus, the gas supply will not terminate. It is numerically shown that stable nanobubble could be present when the fraction of surface coverage by hydrophobic material is from about 0.5 to 1. The stable size of a nanobubble changes with the liquid temperature as well as the degree of gas saturation of water. In slightly degassed water, not only a nanobubble but also a microbubble could be stable in mass balance when the fraction of surface coverage for a microbubble is on the order of 10−4 or less. For hydrophilic materials, however, a bubble could not be stable unless the fraction of the surface coverage is exactly 1. It is suggested that in many experiments of bulk nanobubbles there could be aggregates of nanobubbles.
I. INTRODUCTION Although a surface nanobubble has been confirmed as a stable gas state with a lens shape on a solid surface both experimentally and theoretically, many researchers in fluid dynamics have strong skepticism on the existence of stable bulk nanobubbles.1 In the typical experiments of bulk nanobubbles, they are first produced as microbubbles using a Venturi tube or a membrane with high pressure gas or ultrasonic irradiation and so on.2−10 Initially the liquid water is milky with a lot of microbubbles as well as even larger bubbles. After a while, the liquid water becomes transparent with a negligible amount of microbubbles. Instead, bulk “nanobubbles” have been experimentally observed with dynamic light scattering, laser diffraction scattering, particle tracking analysis, and so on. Such “nanobubbles” have typical diameter of 100−200 nm and are stable for more than a month after their generation;8 however, solid impurities are not distinguishable from gas nanobubbles by the above methods. Nanobubbles are defined as gas bubbles 10 nm considered in the present study.39,52 Thus, pvapor = 0.61075 × 103 Pa at 0 °C, 2.3376 × 103 Pa at 20 °C, 12.339 × 103 Pa at 50 °C, and 47.364 × 103 Pa at 80 °C. When the total gas influx J is negative, it means outflux. The sign of J (influx or outflux) can be seen from the following quantity (j). j=
KJ = 2(1 − r )(pdis, ∞ − pgas ) 2πRDρH O 2
+ (sin θ )(pdis, ∞ ea / kBT − pgas )
The condition for the mass balance is J = 0 or equivalently j = 0. If a slight increase (decrease) in R from the condition of the mass balance results in further increase (decrease) in R, the mass balance condition is unstable and could not be observed in actual experiments. Actually, observable conditions are stable ones that a slight change of R results in the return of R to the initial equilibrium value. Thus, we need to study the stability of the mass balance condition (J = 0 or j = 0). For this purpose, we calculate the change of j when R is changed by ΔR. When the changed j has the opposite sign to that of ΔR, the initial condition of the mass balance is stable because positive (negative) j results in the increase (decrease) in R. When the mass balance condition (j = 0) is satisfied, the equilibrium bubble radius is denoted by R0. When it is slightly changed as R0 → R0 + ΔR, the following quantities are accordingly changed as r → r′, sin θ → sin θ′, pgas → pgas′, and j → j′. If the surface area of the hydrophobic material covering the bubble surface is kept constant during the change, the following relationship holds.
Figure 1. Bulk nanobubble or microbubble partly covered with a piece of hydrophobic material. (a) Whole image. The covered area is bordered by a circumference. A cone is considered with its base bordered by the circumference. The cone point is at the center of a spherical bubble. (b) Enlarged view at around the bubble wall partly covered with hydrophobic material.
pdis = pdis, ∞ ea / kBT
(2)
where pdis,∞ is the dissolved gas pressure in the liquid far from a hydrophobic surface, a is the potential of the hydrophobic attraction (a = 1.7 × 10−20 J38,39,51), kB is the Boltzmann constant, and T is temperature in K. Because of the higher gas concentration at the surface of hydrophobic material, gas diffuses into the bubble near the circumferential border of the hydrophobic material. On the contrary, gas diffuses out of the bubble from the other part of the uncovered bubble surface. With regard to the covered part of the bubble surface, it is assumed that there is no mass (gas and vapor) flow. Then, the total gas influx (J) is expressed by eq 3. J = 4πR2(1 − r )D ρH O (pdis, ∞ e
ρH O (pdis, ∞ − pgas )
a / kBT
2
Kδ
2
KR − pgas )
(5)
4πR 02r = 4π (R 0 + ΔR )2 r′
(6)
Neglecting the second order of (ΔR/R0), the following relationship is obtained.
⎛ 2(ΔR ) ⎞ r′ = ⎜1 − ⎟r R0 ⎠ ⎝
+ 2πR(sin θ )Dδ
(7)
Because sin θ is always positive (or zero)
(3)
sin θ′ =
where R is the bubble radius, 4πR2(1 − r) is the uncovered area of the bubble surface, D is the diffusion coefficient of gas in the liquid, ρH2O is the molar concentration of water (= 5.56 × 104 mol/m3), K is the Henry’s constant, pgas is the gas pressure inside a bubble, 2πR(sin θ) is the length of the circumferential border of the hydrophobic material, and δ is the thickness of the diffusion boundary layer at the circumferential border of the
1 − (cos θ′)2 = 2 r′(1 − r′)
(8)
where eq 1 is used. Inserting eq 7 into eq 8 yields eq 9. (2r − 1)(ΔR ) sin θ′ =1+ sin θ (1 − r )R 0
(9)
Equation 4 yields eq 10. 11103
DOI: 10.1021/acs.langmuir.5b04703 Langmuir 2016, 32, 11101−11110
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Figure 2. Schematic diagram of chemical potentials of gas in the dynamic equilibrium model both inside and outside a bubble partly covered with hydrophobic material.
p′gas = pgas −
2σ(ΔR ) R 02
J = 4πR2(1 − nr1)D
(10)
ρH O (pdis, ∞ ea / kBT
Inserting eqs 7, 9, and 10 into eq 5 yields eq 11. 2g (R 0 , θ ) × (ΔR ) j′ = R0
2
j=
KJ = 2(1 − nr1)(pdis, ∞ − pgas ) 2πRDρH O (16)
g(R0, θ) in eq 12 becomes eq 17 in this case. (12)
g (R 0 , θ ) = (1 − nr1) +
2σ nσ sin θ + 2nr1(pdis, ∞ − pgas ) + R0 R0
n(2r1 − 1) sin θ (pdis, ∞ ea / kBT − pgas ) 2(1 − r1)
(17)
The stable condition g(R0, θ) < 0 is equivalent to the following condition in this case. (13)
R0 > ⎡ σ ⎣⎢2(1 − nr1) − 4nr1 + n sin θ +
where (1 + cos θ ) (1 + cos θ + sin θ) f (θ ) = 2 + sin θ
(15)
2
σf (θ ) (pdis, ∞ ea / kBT + pvapor − p0 )
+ 2πnR(sin θ )Dδ
+ n(sin θ )(pdis, ∞ ea / kBT − pgas )
if g(R0, θ) is negative (positive), then R0 is under a stable (unstable) condition. The condition g(R0, θ) < 0 is equivalent to the following condition. R0 >
KR − pgas )
Accordingly, j is expressed by eq 16.
(11)
2σ σ sin θ g (R 0 , θ ) = (1 − r ) + 2r(pdis, ∞ − pgas ) + R0 R0 (2r − 1) sin θ (pdis, ∞ ea / kBT − pgas ) 2(1 − r )
2
Kδ
where the function g(R0, θ) is defined as follows.
+
ρH O (pdis, ∞ − pgas )
⎡ ⎣⎢2nr1(p0 − pdis, ∞ − pvapor ) +
n(1 − 2r1) sin θ ⎤ ⎦⎥ (1 − r1)
n(1 − 2r1) sin θ (pdis, ∞ ea / kBT 2(1 − r1)
⎤ − p0 + pvapor )⎦⎥
(14)
(18)
and the denominator of the right side of the relation 13 is assumed to be positive. Next, we consider the case that the number of pieces of hydrophobic material covering the bubble surface is n. When each piece has the same surface area as well as the fraction of surface coverage (r1), the total fraction of surface coverage (r) is r = nr1. The total circumferential length of all pieces is 2πnRsin θ. Thus, the total gas influx is expressed as follows in this case.
where the denominator of the right side of the relation 18 is assumed to be positive. Finally, we discuss the chemical potentials of gas both inside and outside a bubble to clarify the mechanism of the dynamic equilibrium state (Figure 2). Here the chemical potential of gas in liquid water far from a bubble is denoted by μ1. The chemical potential of gas (μ2) near the surface of hydrophobic material in liquid water apart from the bubble wall by more than the distance δ (Figure 1b) is given as follows.52 11104
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Langmuir Table 1. Changes of Enthalpy (ΔH) and Entropy (ΔS) at Each Process in Figure 2a ΔH ΔS/Rg a
μ1→μ2
μ2→μ3
μ3→μ4
μ4→μ5
μ5→μ1
total
−a −ln(pdis/pdis,∞)
0 ln(pdis/pdis,∞) − ln(pgas/pdis,∞)
a − |ΔsolH| + |ΔsolS|/Rg
+|ΔsolH| −|ΔsolS|/Rg
0 ln(pgas/pdis,∞)
0 0
Rg is the gas constant.
Figure 3. Bubble radius and j defined by eq 5 as a function of the fraction of surface coverage (r) by a piece of hydrophobic material when the surface area of the hydrophobic material is kept constant. The liquid water is saturated with gas (pdis,∞ = p0 = 1 atm =1.01325 × 105 Pa). The temperature is 20 °C. (a) R0,r=1 = 26 nm. (b) R0,r=1 = 5 nm. (c) R0,r=1 = 50 nm.
μ2 = μ1 − a + R gT ln(pdis /pdis, ∞ ) = μ1
caused by the permanent attractive potential of a hydrophobic material to gas molecules in liquid water (eq 2). The potential is permanent like the gravitational potential of the earth because the attractive potential is due to the permanent repulsive force of a hydrophobic material against liquid water. Thus, the gas supply will not terminate. The chemical potential of gas inside a bubble (μ4) is given as follows.
(19)
where the potential of the hydrophobic attraction a is expressed in J/mol, Rg is the gas constant (8.31 J/mol K), and eq 2 has been used in the last equation. The chemical potential of gas in liquid water adjacent to the bubble wall near the peripheral edge of the hydrophobic material (μ3) (Figure 1b) is given as follows. ⎛ p ⎞ gas ⎟ a holds, as shown in Figure 2. (For O2 gas, |ΔsolG| = 16.4 kJ/mol is larger than a = 10.2 kJ/mol.) One of the origins of the dynamic equilibrium state is the decrease in the chemical potential at the bubble wall near the peripheral edge of hydrophobic material (Figure 2).
When the dissolved gas pressure in liquid water adjacent to the bubble wall (pgas) is lower than the dissolved gas pressure at the surface of the hydrophobic material (pdis) apart from the bubble wall by more than the distance δ, μ3 is lower than μ1. This condition holds for the mass balance condition j = 0 with eqs 5 and 16 for r < 1 and nr1 < 1, respectively because sin θ > 0 and pdis,∞ < pgas hold. (Equations 5 and 16 themselves are valid also for the other conditions.) The gradient of the chemical potential drives the gas flux into a bubble (Jin in Figure 2). The gradient is present due to the fact that pdis > pgas, which is 11105
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Figure 4. Mass balance condition (dotted line, j = 0 in eq 5) and the stability threshold curve (dashed line, the relation 13) as a function of the fraction of surface coverage (r) by a piece of hydrophobic material. Above the stability threshold curve, the mass balance condition is stable. (The relationship 13 holds.) The solid line is the bubble radius when the surface area of the hydrophobic material is kept constant as R0,r=1 = 26 nm. The ambient liquid pressure is p0 = 1 atm. The temperature is 20 °C. (a) pdis,∞ = 1 atm. (b) pdis,∞ = 0.8 atm. (c) pdis,∞ = 2 atm.
followed by its reversed process. In the present model, gas diffusion in quiescent liquid is assumed, and there is no liquid flow. Thus, the present model satisfies both the first and second laws of thermodynamics. In other words, it is not a perpetual motion machine.
The gas diffusion into a bubble itself is driven by the gradient of chemical potential (from μ1 to μ3), which is the main origin of the dynamic equilibrium state. The chemical potential of gas (μ5) in the liquid at the bubble wall in the absence of hydrophobic material is given as follows. μ5 = μ4 + |Δsol G| = μ1 + R gT ln(pgas /pdis, ∞ )
III. RESULTS AND DISCUSSION j in eq 5 is numerically calculated to see the mass balance condition (J = 0 or j = 0). The bubble radius (R) should be specified to calculate j because the gas pressure (pgas) in eq 5 depends on R through eq 4. We consider that the surface area of the hydrophobic material covering the bubble surface is kept constant as eq 23.
(22)
As pgas > pdis,∞, μ5 is higher than μ1. It results in the gas flux out of a bubble (Jout in Figure 2). The first law of thermodynamics is the law of the energy conservation. The second law of thermodynamics is stated as follows.53 The sum of the entropy changes of a system and its exterior never decrease. To discuss the first and second laws of thermodynamics, we summarize the changes of enthalpy (ΔH) and entropy (ΔS) at each process in Figure 2 in Table 1. The chemical potential (μ) is related to molar enthalpy (H) and molar entropy (S) as μ = H − TS, where T is temperature.52 ΔsolH and ΔsolS in Table 1 are enthalpy and entropy for dissolution of gas into liquid water, respectively (ΔsolG = ΔsolH − TΔsolS). Normally, the enthalpy decreases by the dissolution of gas (endothermic). As already noted, the entropy dramatically decreases by the dissolution of gas. The change of entropy except ΔsolS in Table 1 is due to the entropy of mixing.53 From Table 1, the total changes of entropy and enthalpy are both zero. The total entropy change is zero because it is a kind of equilibrium state.54 The total change of energy is zero because each process of the energy change is
4πR2r = 4πR 0,2 r = 1
(23)
where r is the fraction of surface coverage by a piece of hydrophobic material when the bubble radius is R, 4πR2r is the surface area covered by a piece of hydrophobic material, and R0,r=1 is the bubble radius when the bubble is completely covered with the hydrophobic material of a fixed amount. If the hydrophobic material is rigid, it is impossible to cover a bubble completely. In this case, the relationship eq 23 does not hold for relatively small R. If the hydrophobic material is flexible such as liquid oil, it may be possible that a bubble is completely covered with the hydrophobic material, and eq 23 may approximately hold for the whole range of R. In such a case, 11106
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For liquid water saturated with gas (Figure 4a), the curve of eq 24 with the value of R0,r=1 = 26 nm crosses with the mass balance curve by two points (Strictly speaking, r = 1 is also the mass balance point). One is the stable point (r = 0.95) and the other is the unstable point (r = 0.05). For degassed water of 80% in degree of gas saturation, the curve of eq 24 with the value of R0,r=1 = 26 nm does not cross the mass balance curve (Strictly speaking, r = 1 always satisfies the mass balance condition.) In this case, for the amount of hydrophobic material characterized with the value of R0,r=1 = 26 nm, a bubble shrinks until it is completely covered with the hydrophobic material (r = 1) if it is flexible such as liquid oil. If the hydrophobic material is rigid, a bubble completely dissolves into the liquid in this case. For liquid water supersaturated with gas by 200% (Figure 4c), the curve of eq 24 with the value of R0,r=1 = 26 nm does not cross the mass balance curve. In this case, a bubble indefinitely grows and finally disappears at the liquid surface by moving upward by buoyancy. The range of stable bubble radius is from 22 to 44 nm for gas-saturated liquid (Figure 4a), from 28 to 55 nm in slightly degassed liquid with 80% gas saturation (Figure 4 b), and from 11 to 21 nm for supersaturated liquid with 200% gas saturation (Figure 4c). The stable bubble radius in Figure 5 has been
an arbitrary bubble radius is related to the fraction of surface coverage as follows.
R=
R 0, r = 1 r
(24)
Using the relationship 24, j is calculated as a function of r under a given R0,r=1. In Figure 3, the calculated j is shown by dashed line as a function of r for the cases of R0,r=1 = 26 (a), 5 (b), and 50 nm (c). The solid line shows the bubble radius as a function of r given by eq 24. In the case of Figure 3a, there are three points for the mass balance (j = 0): r = 0.05 (R0 = 119 nm), r = 0.95 (R0 = 27 nm), and r = 1 (R0 = 26 nm). For the first point (r = 0.05 and R0 = 119 nm), it is an unstable condition because slightly smaller R than the equilibrium radius R0 (at right side of the equilibrium point) results in negative j, which causes further decrease in R. On the contrary, for the second equilibrium point (r = 0.95 and R0= 27 nm), it is a stable condition. For the third equilibrium point (r = 1 and R0 = 26 nm), it is unstable. Thus, the actually observable point is only the second one. For the case of Figure 3b, the area covered with hydrophobic material is much smaller than that in the case of Figure 3a. In this case, the equilibrium point is only at r = 1 (R0 = 5 nm), and j is negative except at the point (r = 1). The equilibrium point (r = 1 and R0 = 5 nm) is stable. A bubble shrinks until it is completely covered with flexible hydrophobic material, which corresponds to the skin model mentioned in Introduction. If hydrophobic material is rigid, it is impossible to completely cover a bubble. In that case, a bubble completely dissolves into the liquid. For the case of Figure 3c, the area covered with hydrophobic material is much larger than that in the case of Figure 3a. In this case, the equilibrium point is only at r = 1 (R0 = 50 nm), and j is positive except at r = 1. This equilibrium point is unstable. A bubble indefinitely grows as the amount of hydrophobic material attached to a bubble is too large. In an actual case, a bubble moves upward by buoyancy and disappears at the liquid surface. It should be noted that nanobubbles stay in bulk liquid with Brownian motion;55 however, microbubbles move upward by buoyancy. Without numerically calculating j, it is possible to derive the stable equilibrium radius by calculating the mass balance condition (j = 0 in eq 5) and the stable condition (the relationship 13). In Figure 4, the bubble radius at the mass balance condition (j = 0) is shown with dotted line as a function of the fraction of surface coverage (r) by a piece of hydrophobic material. Among them, stable condition (the relationship 13) is above the dashed line in Figure 4. Equation 24 is also shown in Figure 4 with the value of R0,r=1 = 26 nm. The degrees of gas saturation in liquid water are 100% (Figure 4a), 80% (Figure 4 b), and 200% (Figure 4c), with the fixed ambient pressure of p0 = 1 atm. For all of the cases, the bubble radius at the mass balance condition is above the stability threshold curve in the range of the fraction of surface coverage (r) from ∼0.5 to 1. This conclusion does not depend on the assumption of eq 24. The hydrophobic material could be rigid for this result. In conclusion, a stable bulk nanobubble could be present when the fraction of surface coverage by a piece of hydrophobic material is from about 0.5 to 1. It is different from the previous skin model of 100% coverage as well as the “armored” bubble model and the particle crevice model mentioned in the Introduction.
Figure 5. Stable bubble radius as a function of the area covered with a piece of hydrophobic material for various degree of gas saturation of liquid water (80, 100, and 200%, which correspond to the conditions of Figure 4). The ambient liquid pressure is p0 = 1 atm. The temperature is 20 °C.
derived by the direct numerical calculations of j, as performed in Figure 3. The results completely agree with the numerical calculations of the stability condition (the relationship 13). It supports the validity of the stability condition 13. For slightly degassed water (80%), a microbubble with radius >2.1 μm is also stabilized with the fraction of surface coverage
