Homogeneous vapor nucleation of water in 3 M NaCl solution within a nanopore

International Communications in Heat and Mass Transfer 68 (2015) 252–257

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Homogeneous vapor nucleation of water in 3 M NaCl solution within a nanopore☆ Jaekyoon Oh, Yungpil Yoo, Ho-Young Kwak ⁎ Mechanical Engineering Department, Chung-Ang University, Seoul 156-756, Republic of Korea

a r t i c l e

i n f o

Available online 1 October 2015 Keywords: Vapor nucleation Superheat limit Nano bubble Electrolyte solution

a b s t r a c t Homogeneous vapor nucleation of water in the electrolyte solution within a nanopore at its superheat limit was studied using the bubble nucleation model based on molecular interaction. The wall motion of the bubble that evolved from the evaporated water was obtained using the Keller–Miksis equation and the distribution of temperature inside the bubble was obtained by solving the continuity, momentum and energy equations for the vapor inside the bubble. Heat transfer at the interface was also considered in this study. The nucleation rate of the 3 M NaCl solution at 571 K is estimated to be approximately 0.15 × 1028 clusters/(m3 s). With this value of the nucleation rate, the complete evaporation time of the 50 nm radius of the electrolyte solution is approximately 0.60 ns. The calculated life time of the bubble that evolved from the evaporated solution, or the time duration for the growth and subsequent collapse of the bubble, is approximately 32 ns, which is close agreement with the observed result of 28 ns. The bubble reaches its maximum radius of 301 nm at 13.2 ns after the bubble evolution. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The superheat limit of liquids is the maximum temperature at which liquid evaporates explosively. The measured superheat limits of various hydrocarbons by the droplet explosion technique are approximately 90% of the critical temperature of the liquids [1]. It has been experimentally found that, when the temperature of a butane droplet in ethylene glycol solution at 101.3 kPa reaches its superheat limit of 378.0 K, the droplet vaporizes explosively and the fully evaporated droplet that retains its liquid volume expands to behave as a bubble [2]. In fact, the first order phase transition from liquid to vapor at the superheat limit of liquids occurs with a pressure discontinuity rather than a volume discontinuity. The measured evaporation speed of the butane droplet at its superheat limit is approximately 14.3 m/s so that it takes 50 μs for the 1 mm diameter droplet to evaporate fully [2]. For water, however, the maximum temperature obtained by the droplet explosion technique in benzyl benzoate solution is 552.7 K [3], which is below by 30 K from the 90% of the critical temperature of 647.2 K. Recently, a phase change from liquid to vapor of water in 3 M NaCl solution was observed at 603 K at the center of a nanopore in a thin silicon nitride membrane [4]. The classical bubble nucleation theory which assumes the formation of a critical size bubble in mechanical equilibrium condition with environment was widely accepted because it is quite successful in predicting the superheat limit of hydrocarbons with a nucleation rate approximately 1010–1012 bubbles/(m3 s) [1]. However, the classical ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (H.-Y. Kwak).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.026 0735-1933/© 2015 Elsevier Ltd. All rights reserved.

bubble nucleation theory fails to predict the evaporation process at the superheat limit of liquid, which is clear when one considers how many bubbles form inside the 1 mm diameter droplet within 50 μs with a nucleate rate of 1012 bubbles/(m3 s). On the other hand, the bubble nucleation model based on molecular interactions [5] predicts the superheat limit of various hydrocarbons and the evaporation speed at their superheat limits with a nucleation rate value of 1028 clusters/(m3 s) [6]. In this study, the homogeneous vapor nucleation of water in a 3 M NaCl solution within a nanopore at its superheat limit was studied using the bubble nucleation model based on molecular interaction. The wall motion of the bubble that evolved from the fully evaporated of water molecules in electrolyte solution was obtained using the Keller–Miksis equation [7]. The pressure and temperature inside the bubble were obtained by solving the continuity, momentum and energy equation for the vapor inside the bubble [8]. Heat transfer at the interface was also considered, however, condensation or evaporation at the interface was not considered in this study. 1.1. Joule heating of electrolyte solution in a solid nanopore A schematic view of a nanopore in a silicon nitride membrane [4] is shown in Fig. 1. The membrane separates two 3 M NaCl electrolyte solution chambers connected electrically through the nanopore with a radius of 53.5 nm and a thickness of 71 nm. The two electrodes, whose radius is 10 μm are separated by 2 μm. When a voltage is applied to the two electrodes, electric current flows through the nanopore. A hot spot is generated because Joule heating occurs in the nanopore, where current density increases enormously because of a small cross-

J. Oh et al. / International Communications in Heat and Mass Transfer 68 (2015) 252–257

Nomenclature Cb Cp Cv dm dw e EI Fnc h j Jnc Js k kB M m nc N Pb Pbo Pv p∞ Qb qr r Rb Ṙb Rd Rv T Tb Tbl Tbo Tf tl Ts uv Ub Vm vs z Zf 

sound velocity at the bubble wall specific heat at constant pressure specific heat at constant volume average distance between molecules van der Waals' diameter of liquid molecules internal energy ionization potential free energy needed to form a critical cluster enthalpy electric current density nucleation rate of the critical cluster per unit volume nucleation rate of the critical cluster per unit area thermal conductivity of solution Boltzmann constant molecular mass mass of a molecule number of molecules in a critical cluster number density = ρm/m pressure inside bubble pressure at the bubble center vapor pressure ambient pressure heat flow rate from/to the bubble heat flux of radial direction distance from the bubble center bubble radius velocity of bubble wall radius of evaporated liquid volume gas constant of the vapor inside the bubble solution temperature temperature inside the bubble temperature at the bubble wall temperature at the bubble center melting temperature of liquid time lag for nucleation process superheat limit of liquid velocity inside bubble bubble wall velocity molecular volume of liquid evaporation velocity coordination number Zeldovich nonequilibrium factor

Greek letters α polarizability of a liquid molecule αl heat diffusivity of liquid γ specific heat ratio of vapor inside bubble δ thickness of the thermal boundary layer adjacent to the bubble wall ΔHvap enthalpy of evaporation ΔHf enthalpy of fusion εo potential parameter of London dispersion attraction εm energy needed to separate a pair of molecules μ dynamic viscosity of liquid ρ density of solution ρc critical density of liquid ρm density of liquid ρv density of vapor inside the bubble ρ∞ ambient density σ interfacial tension σe electrical conductivity

253

Subscripts c critical state or critical cluster l liquid o center or reference value v vapor

sectional area. The temperature distribution of the electrolyte solution in the nanopore can be obtained by the following energy equation coupled charge conservation and Ohm's law. ρC p

∂T j2 ¼ ∇  ðk∇T Þ þ σe ∂t

ð1Þ

The temperature dependent property values of the electrolyte solution were obtained from various correlations: for density [9], heat capacity [10], thermal conductivity [11] and electrical conductivity [12]. The calculated property values obtained using the existing correlations at ambient temperature of 293.15 K, are a density of 1132 kg/m3 [9], a heat capacity of 4187 J/kg [10], a thermal conductivity of 0.601 W/(mK) [11] and a surface tension of 0.071 N/m [12] for the 3 M NaCl solution. With these temperature dependent property values, the energy equation with a Joule heating term was solved using the COMSOL MultiPhysics program (COMSOL, Inc.). In numerical calculation, the region of right hand side demarked by dotted line in Fig. 1 may be calculated. 1.2. Bubble nucleation model based on molecular interactions A vapor bubble formation model based on the molecular interactions between molecules was proposed by Kwak and Panton [5]. This model assumes that bubble formation is initiated by the clustering of the activated molecules in the metastable state and that the clustering process is driven by the difference in the chemical potentials between the saturated and the metastable liquid molecules. This model also assumes that the energy required to cut across a cluster [13] is the surface energy needed for the liquid cluster to become vapor molecules. Using these assumptions, the condition for a stable critical cluster and the corresponding free energy were obtained, which are as follows. −ðp∞ −P v Þnc 1=3 ¼

zεm 3V m


 F nc z εm ¼ nc 2=3 kB T 6 kB T s

ð2Þ ð3Þ

where z is the coordination number (z = 12 for FCC lattice structure).

Fig. 1. Schematic of silicon nitride nanopore and electrodes (not to scale).

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J. Oh et al. / International Communications in Heat and Mass Transfer 68 (2015) 252–257

A fluid at the critical point has zero surface tension and may be considered to be in a vaporous-like state. The energy required to separate a pair molecules from a distance dm apart to the distance dc (mean distance between molecules at the critical point) may be considered as the energy for the transition from the liquid to vapor state shown in Eq. (3). Assuming a face-centered cubic lattice model for the liquid structure and the van der Waals interactions [14] between molecules, one may estimate the energy needed to separate the pair of molecules, εm, which is as follows. " εm ¼ −4ε0


2 #"
6
12 # ρ dw dw 1− c − ρm dm dm

ð4Þ

The ε0 parameter in Eq. (4) is the potential energy corresponding to the London dispersion interaction. This parameter is given by [15] ε0 ¼

3 EI α 2 : 16 dw 6

ð5Þ

The average distance between the molecules in a given state dm, appears in Eq. (4), and the effective molecular volume of the liquid, Vm can be obtained from the number density of the liquid, N. It is given by π 3 dm N ¼ V m N ¼ 0:7405 6


 z ε m kB T : 3 kB T s V m

ð7Þ

Following the same argument for the kinetics of the clustering process analogous to the condensation case, one may obtain the nucleation rate per unit volume of the nc-mer cluster for the formation of the vapor bubble [16], which is as follows: 
  z εm nc 2=3 J nc ¼ NDnc Z f exp − 6 kB T s

ð8Þ

where Zf is the Zeldovich nonequilibrium factor [17] which is given by 2 1 Z f ¼ 4− 2πkB T s

2

∂ Fn ∂n2

31=2

!

5 n¼nc


¼

z εm 18π kB T s

1=2

2=3

=nc :

ð9Þ

In Eq. (8), Dnc is the rate of molecules striking the surface of the critical cluster, which is given by [5] Dnc


1=2
   ΔHvap ΔH f N 8kB T s 3V m 2=3 2=3 ¼ 4π nc exp − − : 4 πm 4π Rv T s Rv T f

ð10Þ

dRd ¼ 1:3504V m J s nc dt

ð12Þ

where Js (=Jnc∙1 cm) is the nucleation rate at unit depth. The calculated superheat limit and the evaporation velocity for a butane droplet in ethylene glycol solution with a nucleation rate of 1028 clusters/(m3 s) are approximately 378.0 K and 14.4 m/s, respectively, which are very close to the observed values of 378.2 K and 14.3 m/s [2]. The above vapor nucleation model can be applied to the vapor nucleation of water in electrolyte solution with appropriate property data such as the melting temperature of water and the vapor pressure of water depending on temperature. The number density of water molecules, the enthalpy of evaporation of water and the energy to separate a pair of water molecules should be used for the nucleation process because only water molecules are involved in the nucleation process.

1.3. Bubble dynamics Because the pressure of the fully evaporated water molecules that retain their liquid volume at 571 K, which can be estimated from Eq. (7), is approximately 82.4 MPa, the volume expands rapidly to behave as a bubble. Mass transfer through the bubble wall and evaporation or condensation at the interface, which turns out to be negligibly small during the evolution of the nano bubble [18] were not considered in this analysis. However, the heat transfer which is assumed to occur through the thermal boundary layer of thickness δ(t) was considered. The temperature profile in the thermal boundary layer is assumed to be quadratic [19]. It is given by T−T ∞ ¼ ð1−ςÞ2 T bl −T ∞

ð13Þ

where ς = (r − Rb)/δ and δ is the thickness of the thermal boundary layer. The motion of the bubble may be described by the following Keller– Miksis equation [7] or Rayleigh–Plesset equation [20] which can be obtained from the mass and momentum equations for the liquid adjacent to the bubble wall. The Keller–Miksis equation is given by


 U dU 3 U 1 U 1− b Rb b þ U 2b 1− b ¼ 1 þ b ðP B −p∞ Þ: 2 ρ∞ Cb dt 3C b Cb

ð14Þ

The liquid pressure immediately outside the bubble wall, PB is related to the gas pressure inside the bubble wall, Pb by 

It is noted that the steady state nucleation rate given in Eq. (8) can be applicable after the transient state. The time lag tl, which represents the duration of the transient state after the onset of the nucleation occurs may be written as [17]   t l ¼ 1= 4πDnc Z 2f :

vs ¼

ð6Þ

where 0.7405 is the packing fraction of the FCC lattice structure. If any cluster meets the stability condition given in Eq. (2), the molecules in the cluster vaporizes spontaneously. The stability condition of the cluster may represent the vapor state in the critical cluster. Then, the pressure at the vapor state where the molecular volume of the liquid state, Vm, is retained, may be written as [6] P n ¼ ðP v −p∞ Þnc 1=3 ¼

used for determining nc. Once nc and the molecular properties depending on liquid temperature have been determined, the vapor pressure and the corresponding superheat limit, Ts, can be obtained from Eq. (2). The pressure of the evaporated liquid volume, Pn at the superheat limit can also be obtained from Eq. (7). The evaporation velocity of the liquid layer that reaches the superheat limit may be obtained from a one-dimensional approximation [6], which is given by

ð11Þ

After choosing the nucleation rate level of the critical cluster, which is certainly dependent on the experimental conditions, Jnc, Eq. (8) can be

PB ¼ Pb −

2σ 4μRb − : Rb Rb

ð15Þ

The bubble wall velocity, Ub, is the time derivative of the bubble radius. dRb ¼ Ub dt

ð16Þ

The detailed behavior of water vapor inside the bubble was analyzed by solving the mass, momentum and energy conservation equations for the gas or bubble. The mass, momentum and total energy equations for

J. Oh et al. / International Communications in Heat and Mass Transfer 68 (2015) 252–257

255

the vapor inside the bubble with a spherical shape [21], which is notably true for the nano bubble, are given as

Thus, one can obtain the time rate change of the pressure inside the bubble from Eq. (25). That is

∂ρv 1 ∂  þ 2 ρ uv r 2 ¼ 0; r ∂r v ∂t

ð17Þ

dP b 3γP b dRb 6ðγ−1ÞkT bo ðT bl −T ∞ Þ ¼− − : δRb dt Rb dt

∂P ∂ 1 ∂ ðρ uv Þ þ 2 ρ u2 r 2 þ b ¼ 0; r ∂r v v ∂r ∂t v

ð18Þ


 
  ∂ 1 1 ∂ 1 1 ∂ 2 ρv ev þ ρv u2v þ 2 uv ρv ev þ ρv u2v þ P b r 2 þ 2 r qr ¼ 0: 2 2 r ∂r r ∂r ∂t

ð19Þ The vapor density, velocity and pressure profiles inside the bubble can be obtained from the mass and momentum equations [22,23], which are given as follows: ρv ¼ ρv;o þ ρv;r

ð20Þ



Rb r Rb

uv ¼

ð21Þ


€ 1 1 Rb 2 ρv;o þ ρv;r r ; P b ¼ P b0 − 2 2 Rb

ð22Þ

where ρv,oR3b = const. and ρv,r = ar2/R5b (a is a constant.). The pressure inside the bubble becomes uniform unless the acceleration of the bubble wall exceeds 1012 m/s2, which is true for a sonoluminescing gas bubble in water [22,23]. Noted that a uniform temperature distribution is achieved when the characteristic time of bubble evolution is considerably shorter than the characteristic time of the heat diffusion [24]. Because the solutions given in Eqs. (20), (21) and (22) satisfy the kinetic energy equation for the vapor, the internal energy equation remains to be solved. Assuming that the internal energy for the vapor inside the bubble is a function of vapor temperature only as dev = CvdTb, the internal energy equation is given by ρv C v

DT b 1 ∂ 2 1 ∂  2 ¼− 2 r qr −P b 2 uv r Dt r ∂r r ∂r

ð23Þ

ρv C p;v

DT b DP b 1 ∂ 2 ¼ − 2 r qr : Dt Dt r ∂r

ð24Þ

Eliminating DTb/Dt from Eqs. (23) and (24), one can obtain a heat flow rate equation for the vapor inside the bubble. DP b 1 ∂ 2 1 ∂  2 ¼ −ðγ−1Þ 2 r qr −γP b 2 uv r Dt r ∂r r ∂r

ð25Þ

With a uniform pressure approximation and a constant thermal conductivity for the vapor inside the bubble, the temperature distribution for the vapor inside the bubble can be obtained from Eq. (25) using the Fourier law: #
2 r =ð1 þ ξÞ þ T ∞ T b ðr Þ ¼ ðT bo −T ∞ Þ 1− Rb

"

ð26Þ

where ξ = (kv/Rb)/(k/δ). The temperature at the bubble wall can be derived directly from Eq. (26), which is given by T bl ¼

T ∞ þ ξT bo : 1þξ

The time rate change of the temperature at the bubble center Tbo(t) can be obtained from the ideal gas law which holds at the center using Eq. (28). dT bo 3ðγ−1ÞT bo dRb 6ðγ−1ÞkT bo ðT bl −T ∞ Þ ¼− − δRb P b dt Rb dt

ð27Þ

ð29Þ

The differential equation for the time-dependent thermal boundary layer thickness δ can be obtained from the mass and energy conservation equations for the liquid adjacent to the bubble wall. The temperature T in the liquid layer is governed by
 ∂T dRb ∂T α l ∂ ∂T þ ¼ 2 r2 : dt ∂r r ∂r ∂t ∂r

ð30Þ

Integrating Eq. (30) from r = Rb to r = Rb + δ yields the differential equation for the boundary layer thickness, which is as follows [25]. "




δ 1þ Rb



"

 #
 # 3 δ 2 dδ 6α l δ 1 δ 2 dRb ¼ − þ þ 10 Rb dt Rb 2 Rb δ dt "

 # δ 3 δ 2 1 dT b : −δ 1 þ þ Rb 10 Rb T bl −T ∞ dt ð31Þ

The instantaneous bubble wall velocity, bubble radius, pressure inside the bubble, the temperature at the bubble center and thickness of the thermal boundary layer can be obtained by calculating Eqs. (14), (16), (28), (29) and (31) simultaneously using the 4th order Runge–Kutta numerical method with the appropriate initial conditions. The heat transfer at the bubble wall can be obtained using the Fourier law. 

where qr is the radial component of heat flux inside the bubble. Using the definition of enthalpy, h = e + p/ρ, the enthalpy representation of the internal energy equation can be written as

ð28Þ

Q b ¼ −4πR2b k


 ∂T ¼ 8πR2b kðT bl −T ∞ Þ=δ ∂r r¼Rb

ð32Þ

The ambient solution properties such as density, surface tension and dynamic viscosity are very important in evaluating the bubble evolution. The properties for the 3 M NaCl solution (See Section 2.) evaluated at 298.2 K were used for the calculation of the bubble motion. 2. Calculation results and discussion Fig. 2 shows the time dependent solution temperature at the center of the nanopore after a voltage of 8.22 V is applied to the electrode. At 10 μs after the voltage is applied, the center temperature reaches 606 K. It is noted that the center temperature in the nanopore depends crucially on the thermal conductivity of the electrolyte solution and the membrane material and the distance between the electrodes. For example, the maximum temperature inside the nanopore is achieved within a microsecond if the distance between the electrodes is shortened to 600 nm. Additionally, the appropriate area of the electrode is needed to make high current density inside the nanopore. However, a diameter greater than 20 μm for the electrode does not affect the center temperature achieved in the nanopore for the 3 M NaCl solution. The temperature distribution of the 3 M NaCl solution in the nanopore at 10 μs is shown in Fig. 3. The average temperature of the heated liquid region inside the nanopore whose radius is approximately 50 nm is 571 K, and may be regarded as the superheat limit of water. In fact, 90% of the critical temperature of water is approximately 583 K. For reference purposes, the center temperatures and the maximum electric

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J. Oh et al. / International Communications in Heat and Mass Transfer 68 (2015) 252–257 Table 1 Center temperatures and the maximum current density in the nanopore depending on the applied voltage to the electrodes. The temperatures in parentheses are calculated by Nagashima et al. [4]. Voltage applied (V) Center temperature in nanopore (K) Current density in nanopore (A/m2)

Fig. 2. Time-dependent temperature at the center of nanopore after 8.22 V is applied to the electrodes.

current density in the nanopore depending on the applied voltage to the electrodes are shown in Table 1. The calculated center temperatures depending on the applied voltage are slightly higher than those by Nagashima et al. [4]. This might be due to the difference in the used solution properties and the difference in the distance between the electrodes. Nagashima et al. reported that they used water property data to calculate the temperature distribution inside the nanopore. However, the size of the Ag/AgCl electrode and the distance between the electrodes were not reported [4]. When the temperature of a region for the electrolyte solution reaches the superheat limit of water, evaporation (nucleation) of the water molecules starts at the hottest point of the solution, the center of the nanopore. The lag time for the steady state nucleation event calculated using Eq. (11) is approximately 1.85 μs. Thereafter, the evaporation continues step by step to spread, eventually covering the whole region. This evaporation process can be considered as the steady state nucleation of the critical cluster which satisfies the condition given in Eq. (2). The evaporation speed calculated from Eq. (9) is approximately 85 m/s, which is comparable to the observed value of 50 m/s [4]. A larger evaporation velocity can be obtained when the evaporation starts at higher temperatures, which is possible for the case of laser-induced cavitation [26]. The time for the evaporation of the superheated region,

Fig. 3. Temperature distribution inside the nanopore at 10 μs after a voltage of 8.22 V is applied to the electrodes.

8.22 605.8 (603) 2.68 × 109

7.0 492.2 (485) 2.42 × 109

6.0 415.4 (409) 1.74 × 109

5.0 364.5 (359) 1.12 × 109

4.0 334.0 (329) 7.06 × 108

whose radius is 50 nm, is approximately 0.6 ns. The pressure of the fully evaporated solution can be calculated from Eq. (7), which is approximately 82.4 MPa for water at the temperature of 571 K. The fully evaporated liquid volume is expected to expand rapidly due to the high pressure of the evaporated liquid and the volume behaves as a bubble. The instantaneous bubble radius and bubble wall velocity, the pressure inside the bubble, the temperature at the bubble center and the thickness of the thermal boundary layer can be calculated by solving Eqs. (14), (16), (28), (29) and (31) simultaneously using the 4th order Runge–Kutta method. The initial conditions of the bubble radius of 50 nm, the temperature of 571 K and the pressure of 82.4 MPa were used in this calculation. In this calculation, the property values for the electrolyte solution were obtained using the correlations discussed in Section 2, and the dynamic viscosity was calculated by Isdale et al.'s correlation [27]. The initial bubble wall velocity was calculated from the following equation, which can be obtained from the potential energy of the bubble.

Ub ≈

sffiffiffiffiffiffiffiffiffi 2P b 3ρ∞

ð33Þ

The radius–time curve of the bubble that evolved from the fully evaporated liquid region of the electrolyte solution is shown in Fig. 4. The bubble grows rapidly with an initial velocity of 220 m/s to reach its maximum radius of 301 nm at 13.2 ns after the evolution and collapses to liquid so that the life time of the nanobubble is approximately 32.2 ns, which is close to the observed value of 28 ns [4]. After the bubble collapse, the vapor inside the bubble may be condensed homogeneously and abruptly [17]. Certainly the bubble radius–time curve cannot be matched to the conductance data obtained during the bubble evolution when the bubble blocks the ionic conductance through the nanopore [4]. However, the time dependence ionic conductance during the bubble evolution may be inversely proportional to the bubble radius

Fig. 4. Bubble radius as a function of time with an initial radius of 50 nm, a pressure of 82.4 MPa and a temperature of 575 K (solid line) and inverted profile of the electric conductivity through the nanopore (empty circles), which is fitted to the bubble radius–time curve.

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conservations for the vapor inside the bubble and the liquid adjacent to the bubble wall. The bubble grows rapidly due to the high pressure of the evaporated water molecules which retain their liquid volume. Heat transfer through the bubble wall during the bubble evolution is negligibly small. However, the viscosity and surface tension of the surrounding fluid induces the collapse of the nano size bubble. The life time of the 50 nm radius bubble that evolved at the superheat limit of 571 K is approximately 32 ns. Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (20120843). References Fig. 5. The temperature at the bubble center ( ). transfer rate (

), and at the bubble wall (

) and heat

as clearly shown in Fig. 4. For a case of ambient temperature of 298.2 K, the maximum radius reached is 250 nm after 10 ns after the bubble evolution and the life time of the bubble is approximately 26.5 ns mainly due to viscosity. The time dependent temperatures inside the bubble are shown in Fig. 5. In this particular calculation, the ambient temperature was taken as 320 K because of the warm environment. Because the bubble wall expands rapidly, the temperature at the bubble center decreases abruptly at the maximum expansion. The center temperature has a minimum value of 309 K at 13.2 ns after the bubble evolution and increases steadily up to 404 K at the collapse point which occurs at 32.2 ns. On the other hand, the temperature at the bubble wall varies mildly around the ambient temperature. The heat transfer rate due to the temperature difference between the temperature at the bubble center and the temperature at the bubble wall becomes positive during the bubble expansion and is negative during the collapsing phase. The magnitude of the heat transfer rate from/to the bubble is a few pW. The ambient temperature of the solution crucially affects the time needed to reach the superheat limit. For example, the time to reach the center temperature of 606 K inside the nanopore with an ambient temperature of 320 K is approximately 150 ns, which is close to the time needed to the subsequent nucleation event under the continuous voltage supply [4]. 3. Conclusions A mechanism for vapor nucleation of water in 3 M NaCl solution in a nanopore was studied using the bubble nucleation model based on molecular interactions. Detailed calculations were performed on the fact that maximum superheat limit for water of 606 K was achieved at the center of a nanopore of silicon nitride. The nucleation rate of water molecules, which was evaluated at the average temperature of 571 K inside the nanopore is approximately 0.15 × 1028 clusters/(m3 s). The calculated evaporation velocity for the water at the average temperature of 571 K inside the nanopore is approximately 85 m/s, which is comparable to the observed value of 50 m/s. The dynamics of the bubble that evolved from the fully evaporated water in the electrolyte solution was analyzed by solving the mass, momentum and energy

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