Theoretical Study of Dissolved Gas at a Hydrophobic Interface - The

Article pubs.acs.org/JPCC

Theoretical Study of Dissolved Gas at a Hydrophobic Interface Di Zhou, Jianguo Mi,* and Chongli Zhong State Key Laboratory of Organic−Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China ABSTRACT: In this work, the classic density functional approach is applied to describe the interfacial structure and properties of the dissolved gas and liquid in the presence of a hydrophobic wall. In the theoretical approach, the modified fundamental measure theory is adopted for the hard-sphere reference term, and the weighted density approximation is applied for the attractive term. The vapor−liquid phase coexistence curve and the interfacial tensions of Lennard-Jones binary mixtures are first calculated. The results are in good agreement with the corresponding simulation data, indicating that the approach is suitable to study the inhomogeneous properties of fluid mixtures. The density profiles of dissolved gas and liquid in the vicinity of the hydrophobic surface are then calculated. It is shown that distinctive gas enrichment and liquid depletion occur in the interfacial region, and the effects can be reinforced as the dispersion potential and molecule diameter of the gas increase. Based on the supersaturation induced by the gas enrichment and liquid depletion, the free energy barriers of bubble nucleation on the solid wall are approximately estimated for different gases to analyze the probability of spontaneous bubble nucleation near the hydrophobic surface.

1. INTRODUCTION Nanobubble has been observed in many experiments during wetting of a hydrophobic surface.1−4 In these experiments, gas effects on measured forces have been observed, and gas-specific depletion of water has been detected, showing that adsorbed gases change the structure of water in the vicinity of the hydrophobic surfaces. Some experiments support the idea that the enrichment of dissolved gas plays an important role in nanobubble formation.5−7 Due to the long-range hydrophobic forces, the gas molecules accumulated at the surface swell the density-depleted water layer and initiate evaporation, which are shown to nucleate bubbles. Study of gas enrichment is of importance not only for understanding surface thermodynamics but also for explaining the nanobubble formation in the presence of a hydrophobic wall. Molecular simulation has been extensively applied to investigate gas enrichment. Dammer and Lohse8 performed molecular dynamics simulations of Lennard-Jones (LJ) gas− liquid mixtures to study the density distributions of dissolved gas near the hydrophilic and hydrophobic walls. They concluded that the dissolved gas significantly modified the liquid structure near the hydrophobic wall. Since this original simulation, Bratko and Luzar9 showed that the remarkable effect of dissolved gas on the liquid−hydrophobic wall interface disappeared when the LJ particles were replaced by the more realistic models for water and gas. Lee and Aluru10 reported that both the solvent-induced potential and the gas−wall potential play key roles in gas enrichment. However, the physical origin and quantitative understanding of gas enrichment and nanobubble nucleation near a solid surface are not well-understood. A profound understanding is still to be © 2012 American Chemical Society

considered. Particularly, there are several important questions to illustrate: how significant gas enrichment is at hydrophobic surfaces, whether it contributes to water depletion in such confinement, and if it can facilitate the liquid-to-gas phase transition under confinement conditions. Compared with experiments and simulations, theoretical investigation of gas enrichment and bubble formation is very limited to date. In the past decades, a few theoretical mechanisms have been proposed to account for the longrange attractions between hydrophobic surfaces, in which electrostatic correlations,11 disturbed water structure,12 separation-induced phase transitions,13,14 and bridging submicroscopic bubbles15,16 are analyzed. Although significant advances were made in developing an understanding of surface nanobubble, given the complexity of the process, the models and correlations could not explain recent observations. Much work is still required. To explore the physics of nanobubble in the vicinity of the hydrophobic wall, further fundamental research is needed, especially about bubble nucleation induced by gas enrichment and liquid depletion. Concerning the bubble formation at the liquid−solid interface, heterogeneous nucleation17−20 is an important issue accompanying the first-order phase transition. A general feature for heterogeneous nucleation is the existence of a threshold supersaturation beyond which the barrierless nucleation can take place.19 The supersaturation can emerge due to the gas enrichment, not the variation of temperature or Received: November 11, 2011 Revised: January 7, 2012 Published: January 9, 2012 3042

dx.doi.org/10.1021/jp210860d | J. Phys. Chem. C 2012, 116, 3042−3049

The Journal of Physical Chemistry C

Article

pressure of the system. Meanwhile, since the bubble is formed on the solid wall, the wall-induced density fluctuations of dissolved gas and liquid in the nucleation region also influence the nucleation. The surface wettability, which is defined through the contact angle of a liquid droplet on the wall, plays a decisive role in the morphology of bubble. The mechanism for such an inhomogeneous system has not been extensively researched so far. In this paper, we apply the classic density functional theory (DFT) approach to investigate gas enrichment as well as free energy barrier for bubble nucleation at a hydrophobic interface. In recent years, the theory has been well developed by integration of the modified fundamental measure theory21−23 (MFMT) for short-range repulsion and the weighted density approximation for long-range attraction.24 The MFMT for the repulsive contribution has been demonstrated to perform very well in describing the properties of inhomogeneous fluids on planar solid surfaces.23 The ability of the improved DFT for predicting a large variety of phase transitions of confined fluids has also been well tested.25,26 Very recently, the thermodynamic consistency of the above version has been examined to be reliable despite some approximations involved.27 To investigate the effects of gas enrichment and liquid depletion, the density profiles of different dissolved gases and the liquid in the vicinity of the hydrophobic wall are calculated. Accordingly, the free energy barriers for bubble nucleation at the particular temperature and pressure are approximately estimated to analyze the probability of spontaneous bubble nucleation. Although the systems are LJ mixtures, it will provide a base for the study of real systems in the future.

Ω[ρ1(r), ρ2(r)] 2

= F[ρ1(r), ρ2(r)] +

i=1

where μi denotes the bulk chemical potential for each component and Vi ext(r) indicates the external force affecting component i. For the homogeneous system, Viext(r) = 0. F[ρ1(r), ρ2(r)], which can be split into ideal, hard sphere repulsion and dispersion attraction contributions, stands for the local Helmholtz free energy of the mixture

F[ρ1(r), ρ2(r)]=F id[ρ1(r), ρ2(r)] + F hs[ρ1(r), ρ2(r)] + F att[ρ1(r), ρ2(r)] F id[ρ1(r), ρ2(r)] 2

= kBT ∑

εij =

∫ ρi(r)[ln(ρi(r)Λi3) − 1] dr

i=1

(7)

where kB is the Boltzmann constant and Λi is the de Broglie wavelength of component i. According to MFMT,21 the hard-sphere contribution can be represented by

F hs[ρ1(r), ρ2(r)] = kBT

∫Φ

hs

[nα(r)] dr

(8)

in which Φhs[nα(r)] is the Helmholtz free energy density of the hard-sphere fluid mixture

n n − nV 1nV 2 Φhs[nα(r)] = − n0 ln(1 − n3) + 1 2 1 − n3 1 ⎛ ⎜⎜n3 ln(1 − n3) + 36π ⎝ ⎞ n 3 − 3n n n n32 2 V2 V2 ⎟⎟ 2 + n33 (1 − n3)2 ⎠

(1)

(9)

with 2

(2)

nα(r) =

where unlike interaction parameters are calculated using the Lorentz−Berthelot mixing rules

σij =

(6)

The ideal contribution is exactly known as

where εij and σij are the LJ energy and length parameters. As the wall is invariant in the x and y directions, the fluid−solid interactions only depend on z. The overall potential exerted by the wall can be described by the Steele-type 10−4 potential28

⎡ 2 ⎛ σ ⎞10 ⎛ σ ⎞4 ⎤ V iext(z) = 2πεisσis2 ⎢ ⎜ is ⎟ − ⎜ is ⎟ ⎥ ⎝ z ⎠ ⎥⎦ ⎢⎣ 5 ⎝ z ⎠

(5)

dr

2. THEORY For simplicity, the gas, liquid, and solid wall particles are characterized by the LJ interaction potential

⎡⎛ σij ⎞12 ⎛ σij ⎞6 ⎤ uij(r ) = 4εij⎢⎜ ⎟ − ⎜ ⎟ ⎥ ⎝ r ⎠ ⎥⎦ ⎢⎣⎝ r ⎠

∑ ∫ ρi(r)(Viext(r) − μi)

σii + σjj 2

(3)

εiiεjj

(4)

∑ ∫ ρi(r′)wi(α)(r − r′) dr′ i=1

(10)

wi(α)

in which the weight functions = 0, 1, 2, 3, V1, V2 are expressed in terms of the Heaviside step function H(r) and the Dirac delta function δ(r)

wi(2)(r) = πdi2wi(0)(r) = 2πdiwi(1)(r)

The equilibrium structure of binary mixtures at a given temperature can be obtained using the DFT, in which the grand potential of the mixture Ω[ρ1(r), ρ2(r)], can be expressed by a functional of different component densities, ρ1(r) and ρ2(r)

= δ(di/2 − r ) wi(3)(r) = H(di/2 − r ) 3043

(11) (12)



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r δ(di/2 − r ) (13) r 29,30 where di is the Barker−Henderson diameter. Using the weighted density approximation, the attractive part can be expressed as31

2

wi(V 2)(r) = 2πdiwi(V 1)(r) =

ρi̅ (r) =

j=1

ωijatt(r ) =

2

∫ ρi(r){F1[ρi̅ (r)] + F2[ρi̅ (r)]} dr

i=1

F1[ ρi̅ (r)] = − 2πρi̅ (r)β ⎛

∑ ∑ xixjεij[k1, ij⎜⎜G0, ij(z1, ij)e z1,ijR ij ⎝

j

1 + z1, ijR ij ⎞ ⎟ − ⎟ z1,2 ij ⎠ ⎛ − k 2, ij⎜⎜G0, ij(z 2, ij)e z 2, ijR ij ⎝ 1 + z 2, ijR ij ⎞ ⎟] + 8πρ (r)β − i̅ ⎟ z 2,2 ij ⎠

−β

∑ ∑ xixj εijg0, ij(R ij)R ij3Iij ,1 j

F2[ ρi̅ (r)] = − πρi̅ (r)β ∑ ∑ xixj εij [k1, ij(G1, ij(z1, ij)e z1, ijR ij) − k 2, ij(G2, ij(z 2, ij)e z 2, ijR ij)] − 4πρi̅ (r)β ∑ ∑ xixj εijg1, ij(R ij)

⎧ r < R0 ρ ⎪ v, i ρi(r ) = ⎨ ⎪ρ ⎩ nl, i r < R 0

j

R ij3Iij ,1

(16)

⎛ ⎞12 ⎛ ⎞6 1 ⎜ σij ⎟ 1 ⎜ σij ⎟ Iij , ∞ = ⎜ ⎟ − ⎜ ⎟ 9 ⎝ R ij ⎠ 3 ⎝ R ij ⎠ ⎛ ⎞ 1 σij Iij ,1 = ⎜⎜ ⎟⎟ 9 ⎝ R ij ⎠ ξn =

(17)

ρk0,+i 1(z) = f ρik(z) + (1 − f )ρk0, i(z)

3

⎛ ⎞ ⎛ ⎞ 1 σij 2 σij − ⎜⎜ ⎟⎟ + ⎜⎜ ⎟⎟ 3 ⎝ R ij ⎠ 9 ⎝ R ij ⎠

∑ ρmR mn , Δ = 1 − m

6

(22)

(23)

(24)

where ρnl,i is the liquid density of component i around the bubble in given supersaturation state. R0 is the radius of the nucleus. Accordingly, eq 22 can be solved using the Picard iteration

with

12

δρi(r)

⎫ ⎪ ⎬ − βV iext(r)⎪ ⎭

is given, where ρl,i and ρv,i are the equilibrium liquid and vapor densities of component i, respectively, and “0” is the position of the vapor−liquid interface. For the bubble nucleation, the density varies only in the radial direction (r direction), the initial guess can be given by

j

i

δF att[ρ1(r), ρ2(r)]

⎧ ρ z 0

(15)

i

(21)

At the vapor−liquid or fluid−solid interface, the density varies only in the direction perpendicular to the interface (z direction); thereby, the density profile of component i can be written as ρi(z). To solve the above Euler−Lagrange equation, an initial density profile

j

i

∫ cijatt(r ) dr

⎧ δF hs[ρ1(r), ρ2(r)] ⎪ ρi(r) = exp⎨ βμ − β i ⎪ δρi(r) ⎩

∑ ∑ xixjεijR ij3Iij , ∞ − 8πρi̅ (r)β i

cijatt(r )

Here Cijatt(r) is the direct correlation function (DCF) of the equilibrium interfacial density, i.e., (ρl + ρv)/2, from attractive contribution.27 In eqs 15−19, k1,ij, k2,ij, z1,ij, z2,ij are constants related to the LJ potential parameters, Ri = 2Σjxjdij − ΣiΣixixjdij, g0,ij(Rij) and g1,ij(Rij) are the radial distribution functions (RDFs) of hard-sphere and first-order perturbation terms at contact, and G0,ij(zij)and G1,ij(zij) are the corresponding Laplace transforms.33 The density distribution of each component can be obtained by minimizing the grand potential and solving the Euler− Lagrange equation

(14)

in which the expressions of F1[ρ̅i(r)] and F2[ρ̅i(r)] are given by32

i

(20)

where ωijatt(r) is the weight function

F att[ρ1(r), ρ2(r)] = kBT ∑

∑ ∫ ρj(r′)ωijatt(|r − r′|) dr′

In eqs 14−16, ρ̅i(r) is the weighted density written as

in which f is the mixing parameter and the value is 0−1. The step size of the numerical integration is 0.01σ1, and the iteration is repeated until convergence. Once the equilibrium density profile is obtained, the gas enrichment and liquid depletion can be easily evaluated. As a result, the interfacial tensions for gas−liquid, gas−solid, and liquid−solid can be determined using the following thermodynamic relation

(18)

π ξ 6 3

(19) 31

(25)

and can be 3044



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∞

Ω + PV (26) A where γ denotes interfacial tension and P, V, and A are, respectively, the pressure, volume, and interfacial area. In the vicinity of the hydrophobic wall, a supersaturation or nonequilibrium is achieved by gas enrichment at given temperature and pressure, which fascinates to bubble nucleation. Once the bubble emerges, the contact angle θ can be calculated via Young’s equation γ − γvs cos θ = ls γvl (27)

∫ dr = ∫Rcosθ r 2 dr ∫0

γ=

2π

dφ

∫Rcosθ r 2 dr 2π(1 − cos θ)

(29)

ΔΩ = 2π(1 − cosθ) ∞

2

∫Rcosθ

r 2{ ∑ {f id [ρi(r )] + f att [ρi(r )] i=1

− ρi(r )μi} + Φhs[nα(r )] + P} dr

(30)

where f [ρi(r)] is the Helmholtz free energy density of component i with F[ρ1(r), ρ2(r)] = Σi =2 1∫ dr f [ρi(r)]. As the free energy barrier of the bubble formation reaches a maximum, the size of the bubble is defined as the critical radius Rc. Compared with eq 22 for the quantitative calculation of density profile, eq 30 is a qualitative expression for the free energy barrier, in which the effects of the hydrophobic wall on the density fluctuations inside and outside the bubble are neglected. However, since the average densities of dissolved gas and depleted liquid in the gas enrichment region or the nucleation region are employed in eq 30 to represent the densities inside and outside the bubble respectively, the heterogeneous characteristics are partially considered, and the free energy is still reliable to analyze the bubble nucleation qualitatively.

Since the bubble nucleation occurs at the liquid−solid interface with three-dimensional characteristics, the system is heterogeneous, and the nucleation process is more complicated than that of homogeneous nucleation. Accordingly, the present one-dimensional DFT cannot be applied directly. In this work, we try to explore the relationship between the degree of gas enrichment and the spontaneous bubble nucleation on the hydrophobic wall. In this regard, the free energy barrier and the critical nucleus radius need to be calculated. To simplify the issue, the vapor density inside the bubble, once nucleated, is assumed to be uniform. The assumption is based on the facts that the bubble height is only several times the liquid molecular diameter and the wall-induced density fluctuation inside the bubble is very weak. Therefore, the fluctuation can be neglected. On the other hand, the density of the liquid surrounding the bubble is assumed to be the average value of the local density distribution. During the nucleation process, the density profile of the liquid in the vapor−liquid interface is caused by the interaction of the wall-induced fluctuation and the nucleation-induced phase transformation. Although the wall-induced density fluctuation of the liquid is more obvious than that of the gas, the contribution of the vapor−liquid phase transformation is dominant. As a result, the assumption inevitably causes some deviation, but the deviation is still acceptable. Thereby, the free energy barrier for the heterogeneous nucleation can be approximately expressed with the homogeneous nucleation model34−36

3. RESULTS AND DISCUSSION Theory provides a simple and flexible way for obtaining phase coexistence envelopes and interfacial tensions of binary fluids, which represent the homogeneous and inhomogenous properties, respectively. Its predictive ability, however, has to be tested in order to find the range of its applicability. When dealing with a specific intermolecular potential, such as the LJ potential, a simple and effective way to test the theory is to compare its calculation results with simulation data, which can in principle represent exactly the behavior of a system. Figure 2 presents the vapor−liquid coexistence envelope of the binary mixture. The intermolecular interaction parameters are σ2/σ1 = 1, ε2/ε1 = 0.5.37 The parameter σ1 is taken as the length unit, and ε1 as the energy unit. The thermodynamic properties of this system are studied at the reduced temperature T* = kBT/ε1 = 1.0. From the figure one can see that good agreement has been achieved between the theory and the simulation37 below the critical point. The surface tension as a function of pressure for the system is further calculated and compared with the simulation data,37 as shown in Figure 3. The reduced surface tension γ* = γσ12/ε1 decreases monotonously with the increasing pressure. The theoretical predictions are found to be a little lower than the corresponding simulation data, but the deviation is within the reasonable range. Figures 2 and 3 suggest that the present theory is capable of describing the phase behavior and interfacial properties of binary mixtures. Before studying the accumulation effects of different dissolved gases at the liquid−wall interface, it is worth discussing the structure of the vapor−liquid interface. In the following calculations, the reduced temperature and pressure of the binary fluid mixtures are fixed at T* = kBT/εl = 0.7 and P* = Pσ13/ε1 = 0.005. Following Dammer and Lohse’s study,8 the

2 i=1

∫0

By integrating eqs 28 and 29, the free energy barrier, i.e., the constrained free energy for a bubble nucleation, can be expressed as

Figure 1. Schematic of a bubble on a hydrophobic wall.

∑ ∫ ρi(r)μi dr + PV

sinθ′ dθ′

∞

=

Here γls, γvs, γvl are the liquid−solid, vapor−solid, and vapor− liquid interfacial tensions. A schematic of the system is shown in Figure 1 to illustrate a bubble with radius R and contact angle θ formed on a hydrophobic wall.

ΔΩ = F[ρ1(r), ρ2(r)] −

θ

(28)

Meanwhile, in the heterogeneous nucleation, the bubble is a half-sphere cap with the specified contact angle θ as shown in Figure 1, and the volume integration in eq 28 can be written as 3045



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Figure 2. Phase diagram of a binary mixture at T* = 1.0 with σ2/σ1 = 1, ε2/ε1 = 0.5.

Figure 4. Density profiles for gas−liquid mixtures A, B, and C at phase interface at the temperature T* = 0.7 and pressure P* = 0.005.

respectively. In contrast, the interfacial tension of the pure liquid is 0.95; in this regard, the dissolved gases decrease the surface tension. The theoretical approach is then extended to the vapor− liquid−solid ternary equilibrium systems, where the three binary mixtures in contact with a hydrophobic wall are investigated. In the following discussions, these three equilibrium systems are denoted as systems A, B, and C. Figure 5 shows the density profiles of the dissolved gases. The

Figure 3. Surface tension as a function of pressure for the mixture as in Figure 2.

molecular diameter of the liquid and the energy scale for liquid−liquid interactions are constant, while three different gases are considered, whose energy scale εg and size σg are expressed in terms of εl and σl. The values of (εg, σg) are (0.40, 1.2), (0.50, 1.2), and (0.50, 1.4), respectively. These three sets of gas and liquid combinations are denoted as A, B, and C. The density profiles of A, B, and C at vapor−liquid equilibria are shown in Figure 4, respectively. For the gas component in each system, the gas dissolved in the liquid phase is in an equilibrium state with the gas in the vapor phase. One can immediately see the enrichment of gas in the interfacial region for these three mixtures, before the gas density declines to its value in the bulk liquid. Gas molecules close to the interface go through attraction forces from the liquid phase as well as the vapor phase. Since the density of the liquid phase is much larger than that of the vapor phase, the resulting force is directed to the liquid phase, which results in the nonmonotonous density profiles of gas in the interfacial region. From A to C, the enrichment of gas in the interfacial region increases. In the bulk liquid phase, the density of dissolved gas decreases as σg increases and εg decreases, and the reduced densities are 2.33 × 10−3, 9.96 × 10−3, and 3.47 × 10−3. The corresponding vapor−liquid interfacial tensions are 0.85, 0.76, and 0.68,

Figure 5. Gas density profiles in systems A, B, C in the liquid phase in the vicinity of the hydrophobic wall at the temperature T* = 0.7 and pressure P* = 0.005.

molecular diameter of the wall is set to σw = σl, and the energy scale for the wall−wall interactions εw is 0.056εl, corresponding to εlw/εl = 0.236. The hydrophobicity of the wall is characterized by the ratio εlw/εl with the energy scale εlw for liquid−wall interactions. One can easily see that the gas in the vicinity of the wall has been remarkably enriched. The maximal enrichment densities are, respectively, 0.106, 0.233, and 0.350 for systems A, B, and C, showing that the enrichment becomes more obvious with the increase of εg and σg. Meanwhile, the gas enrichment results in a significant reduction of the liquid density in the vicinity of the hydrophobic wall, as shown in 3046



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Figure 6. For comparison, the density profile of the pure liquid absorbed on the solid wall is also presented in the figure. The

Figure 7. Pressure as a function of the total density in system C with different mole fraction of gas in the liquid phase. The straight line stands for the equilibrium pressure P* = 0.005. Figure 6. Liquid density profiles in systems A, B, C in the vicinity of the hydrophobic wall at the temperature T* = 0.7 and pressure P* = 0.005.

gas-induced depletion becomes increasingly obvious from system A to C. The gas enrichment is imperative for the bubble nucleation. When the gas component accumulates enough at the liquid− solid interface, the concentration of gas is greater than the equilibrium concentration at a particular pressure, and the solution reaches supersaturation. At given temperature and pressure, bubble nucleation can emerge due to the supersaturation of the dissolved gas. The contact angle of the bubble on the wall (see Figure 1) is of vital importance, which can be approximately calculated by the equilibrium surface tensions via Young’s equation. From system A to C, the vapor−liquid and the liquid−solid interfacial intensions decrease with the similar tendency, while the vapor−solid interfacial intension increases. Since the value of the vapor−solid interfacial intension is relatively small, once the bubble emerges, the calculated contact angle θ is almost the same. For above three systems, the value is 50° at T* = 0.7 and P* = 0.005. To investigate bubble nucleation in a simple way, the free energy barriers are calculated by eq 30, in which the densities inside and outside the bubble are assumed to be the average densities of the dissolved gas and depleted liquid. Figure 7 illustrates the equilibrium and supersaturation states of gas− liquid mixtures for the system C at different mole fractions of gas in the liquid phase. The equilibrium gas mole fraction is xg = 0.003, corresponding to ρ*g = 0.003. Other lines represent different nonequilibrium or supersaturation states. At the given pressure, the densities of gas component are 0.020, 0.025, 0.030, and 0.042, respectively, representing different supersaturation ratios 6, 7, 9, 12. Here the supersaturation ratio is defined as the ratio of the enriched density and the gas density in bulk liquid. The densities of the depleted liquid component are 0.774, 0.759, 0.745, and 0.709, respectively. Using these four sets of value as the initial input, we can calculate the constrained free energy curves in the process of bubble nucleation. For the system C, as shown in Figure 8, ΔΩ for each curve has a local maximum at R = Rc, which correlates to a state of unstable equilibrium. When a cluster has a radius R less than Rc, any increase of R corresponds to an increase in ΔΩ,

Figure 8. Constrained free energy curves of the growing bubble in system C with different enriched densities at the temperature T* = 0.7 and pressure P* = 0.005.

indicating that the cluster would shrink. On the other hand, when a cluster has a radius greater than Rc, any increase in R corresponds to a decrease of ΔΩ, indicating that the bubble would grow up. It can be seen that, as the supersaturation ratio or the enrichment ratio increases, both the free energy barrier and the critical nucleus radius decrease. Figure 8 shows that the critical nucleus radii are 12.6σl, 9.2σl, 6.2σl, and 5.5σl under four different supersaturation conditions. Since the contact angle is known (50°), the bubble heights can be easily calculated, which are 4.5σl, 3.3σl, 2.2σl, and 2.0σl. These values also account for the corresponding nucleation region. According to the density profile of the system C, the average density of the dissolved gas can be approximately determined within the nucleation region. The results are 0.038, 0.051, 0.072, and 0.077, whereas the necessary enriched densities are 0.020, 0.025, 0.030, and 0.042. Obviously, the actual enriched density is higher than the necessary density at each supersaturation ratio, indicating that the bubble can be nucleated. Figure 8 also shows that the free energy barriers are significantly larger than zero at supersaturation ratios 6, 7, and 9. As a result, the spontaneous nucleation cannot occur. Once the energy is supplemented, 3047



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mechanism of nanobubble formation near a hydrophobic surface. Further investigation of heterogeneous nucleation by a three-dimensional DFT approach is now in progress.

however, the bubble will emerge, and the critical nucleus radius increases with the decreasing supersaturation ratio. If the supersaturation ratio increases to 12, the free energy barrier decreases to zero, suggesting that the bubble can be nucleated spontaneously. In contrast, for the systems A and B, at the supersaturation ratio 12, the densities of the necessary gas accumulation are 0.028 and 0.120, respectively. The constrained free energy curves are calculated and shown in Figure 9. If the bubble is

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

ACKNOWLEDGMENTS This work is supported by the Program for New Century Excellent Talents in University, and the financial support of the National Natural Science Foundation of China (Nos. 21076006 and 20876007) is also greatly appreciated.

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REFERENCES

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Figure 9. Constrained free energy curves of the growing bubble for systems A and B under the given supersaturation ratio 12 at the temperature T* = 0.7 and pressure P* = 0.005.

nucleated, the heights of the bubble are 4.90σl in system A and 4.11σl in system B. In the nucleation region, however, the average densities of the enriched gas are 0.011 and 0.013. Under such a supersaturation condition, bubbles cannot emerge in the systems A and B, because the average values of the enriched gas are far from the necessary accumulation level for nucleation. The same results can be obtained at lower supersaturation ratio because the decrease of supersaturation ratio is unfavorable to the nucleation. In summary, gas enrichment is the necessary prerequisite for bubble nucleation. If the enrichment can be reinforced to some extent, the nucleation becomes possible, and the free energy barrier and the critical nucleus radius can be approximately estimated; once the enrichment is high enough, the spontaneous nucleation occurs, and the critical nucleus radius can also be evaluated.

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CONCLUSIONS In this work, the structure and properties of gas−liquid binary mixtures in the vicinity of a hydrophobic wall are investigated using the DFT approach. The density profiles of the liquid and dissolved gas are presented quantitatively. It is shown clearly that, as the gas−liquid and gas−solid interactions increase, or the diameter of gas molecule increases, the effects of gas enrichment and liquid depletion can be reinforced simultaneously. Based on these effects, the free energy barrier and the critical nucleus radius for the bubble nucleation are estimated in a simple way. Once the gas enrichment ratio is high enough, a spontaneous bubble nucleation process occurs. As a result, the present work provides a simple way to explain bubble nucleation, which will be helpful in understanding the 3048


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Article

Mi, J.; Zhong, C.; Li, Y.-G.; Tang, Y. AIChE J. 2006, 52, 342− Li, Z.; Wu, J. Ind. Eng. Chem. Res. 2008, 47, 4988−4995. Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1995, 102, 2156− Talanquer, V.; Oxtoby, D. W. J. Chem. Phys. 1996, 104, 1993− Lopez-Lemus, J.; Alejandre, J. Mol. Phys. 2003, 101, 743−751.

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Theoretical Study of Dissolved Gas at a Hydrophobic Interface - The

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