A kinetic model for heterogeneous condensation of vapor on an insoluble spherical particle Xisheng Luo, Yu Fan, Fenghua Qin, Huaqiao Gui, and Jianguo Liu Citation: The Journal of Chemical Physics 140, 024708 (2014); doi: 10.1063/1.4861892 View online: http://dx.doi.org/10.1063/1.4861892 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Development of a molecular-dynamics-based cluster-heat-capacity model for study of homogeneous condensation in supersonic water-vapor expansions J. Chem. Phys. 138, 064302 (2013); 10.1063/1.4790476 Adsorption, desorption, and diffusion of nitrogen in a model nanoporous material. I. Surface limited desorption kinetics in amorphous solid water J. Chem. Phys. 127, 184707 (2007); 10.1063/1.2790432 Kinetics of cluster evaporation and condensation important in homogeneous vapor phase nucleation AIP Conf. Proc. 534, 201 (2000); 10.1063/1.1361846 Condensation of supersaturated water vapor on submicrometer particles of SiO 2 and TiO 2 J. Chem. Phys. 112, 9967 (2000); 10.1063/1.481633 Modeling surface kinetics and morphology during 3C, 2H, 4H, and 6H–SiC (111) step-flow growth J. Vac. Sci. Technol. A 16, 3314 (1998); 10.1116/1.581484
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THE JOURNAL OF CHEMICAL PHYSICS 140, 024708 (2014)
A kinetic model for heterogeneous condensation of vapor on an insoluble spherical particle Xisheng Luo,1 Yu Fan,1 Fenghua Qin,1,a) Huaqiao Gui,2 and Jianguo Liu2 1
Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, China 2 Key Laboratory of Environmental Optics and Technology, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China
(Received 7 October 2013; accepted 30 December 2013; published online 14 January 2014) A kinetic model is developed to describe the heterogeneous condensation of vapor on an insoluble spherical particle. This new model considers two mechanisms of cluster growth: direct addition of water molecules from the vapor and surface diffusion of adsorbed water molecules on the particle. The effect of line tension is also included in the model. For the first time, the exact expression of evaporation coefficient is derived for heterogeneous condensation of vapor on an insoluble spherical particle by using the detailed balance. The obtained expression of evaporation coefficient is proved to be also correct in the homogeneous condensation and the heterogeneous condensation on a planar solid surface. The contributions of the two mechanisms to heterogeneous condensation including the effect of line tension are evaluated and analysed. It is found that the cluster growth via surface diffusion of adsorbed water molecules on the particle is more important than the direct addition from the vapor. As an example of our model applications, the growth rate of the cap shaped droplet on the insoluble spherical particle is derived. Our evaluation shows that the growth rate of droplet in heterogeneous condensation is larger than that in homogeneous condensation. These results indicate that an explicit kinetic model is benefit to the study of heterogeneous condensation on an insoluble spherical particle. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861892] I. INTRODUCTION
Condensation in supersaturated vapor is a very common phenomenon in atmospheric physics,1, 2 gas cleaning technology,3–5 and multiphase flow.6, 7 The condensation can be activated via two ways, homogeneous nucleation8 and heterogeneous nucleation.9 In homogeneous nucleation, a higher nucleation energy barrier should be overcome to create nuclei in the interior of a uniform substance. In heterogeneous nucleation, the foreign surface reduces the nucleation energy barrier and promotes the process of nucleation. As a result, heterogeneous condensation is much common in daily life. The classical condensation theory is based on the kinetic model10–12 where the droplet size distribution can be derived. The framework of the kinetic model for homogeneous condensation is relatively complete so that the kinetic model has been widely used in homogeneous condensation study. Typical examples are to study the steady-state nucleation rate,13 the start-up process of condensation,14 the evolution of the number density of droplets,15 and the influence of vapor depletion.16 The kinetic model is also a useful tool to study the heterogeneous condensation. Some researches9, 17 focused the kinetic study on the heterogeneous condensation occurring on a planar solid surface. As the complicated effects of foreign particle, the kinetic model for heterogeneous condensation on an insoluble spherical particle still has some gaps. In the kinetic model for heterogeneous condensation, there are two mechanisms of clusa) Electronic mail: [email protected]
0021-9606/2014/140(2)/024708/8/$30.00
ter growth:18, 19 direct addition of water molecules from the vapor and surface diffusion of adsorbed water molecules on the particle, where the first one is the way of cluster growth in homogeneous condensation. The integrated consideration of two growth mechanisms and line tension20–22 has not yet appeared in the kinetic model for heterogeneous condensation on an insoluble spherical particle. In addition, the expression for evaporation coefficient is unknown because of the complicated geometrical relation in heterogeneous condensation especially when the effect of line tension is considered. Although the relation between the condensation coefficient and the evaporation coefficient is certain in the kinetic model with detailed balance (the detailed balance will be introduced in Sec. II), the kinetic model only can be used to calculate the steady-state nucleation rate19, 23 for heterogeneous condensation on an insoluble spherical particle, because the exact expression of evaporation coefficient is needed in many other circumstances, such as the growth rate of droplet on the particle, the start-up process of heterogeneous condensation, the evolution of the droplets’ number density, and the influence of vapor depletion. The replacement by the kinetic model for homogeneous condensation or heterogeneous condensation on a planar solid surface is an optional method, but the validity of this approximation is still unknown. In this paper, we shall develop a kinetic model for heterogeneous condensation on an insoluble spherical particle. In our model, the two mechanisms of cluster growth and the effect of line tension are considered, and the exact expression for evaporation coefficient will be derived. We shall also derive the growth rate of droplet on an insoluble spherical
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particle as an example of using our model, which cannot be obtained by the previous kinetic model.19, 23 In Sec. II, the new kinetic model for heterogeneous condensation is described in details. In Sec. III, some discussions of our kinetic model are presented, and the growth rate of droplet on the insoluble spherical particle is derived as an example of our model applications. A brief summary of our results is provided in Sec. IV. II. KINETIC MODEL
In the kinetic model for condensation of vapor, the water cluster is assumed to gain or lose only single water molecules, and cluster-cluster interactions are neglected. We denote a cluster containing g molecules by Bg (note that the subscript “g” denotes the parameters of g-cluster hereafter). The size change of Bg by attaching or detaching of a molecule can be represented by the reversible “reaction:” Bg + B1 Bg+1 ,
g = 1, 2, 3, . . . .
(1)
As shown in Fig. 1, there are two mechanisms for the growth of cluster: (α) direct addition of water molecules from the vapor, and (β) surface diffusion of adsorbed water molecules on the particle. As a result, the cluster Bg Bg can grow by the condensation of molecules from both the vapor and the adsorbed water molecules at rates of Cg (α) and Cg (β), respectively, and shrink by evaporation of molecules to both the vapor and the adsorbed water molecules at rates of Eg (α) and Eg (β), respectively. The number of clusters Bg formed on unit surface of a spherical insoluble particle is denoted by fg . As sketched in Fig. 2, the change rate of fg is dfg = Cg−1 fg−1 − (Cg + Eg )fg + Eg+1 fg+1 , (2) dt where the condensation coefficient Cg is equal to Cg (α) + Cg (β), which denotes the rate where monomers condense on Bg , and the evaporation coefficient Eg is identical to Eg (α)
C g( ) C g( )
E g( )
Bg
g
E g( )
g
rg or R
dg g
oR FIG. 1. A spherical cap shaped cluster Bg forms on the surface of a spherical solid particle. Cg and Eg are the condensation and the evaporation coefficients of water cluster Bg , respectively. (α) and (β) denote the coefficients in direct addition and surface diffusion, respectively. ψ g and φ g are half of the central angles of Bg (radius rg ) and the part of particle (radius R) contacting with Bg , respectively. θ g is the contact angle between Bg and solid. dg is the distance between the centers of cluster Or and solid particle OR .
f g-1
fg
Eg
E g+1
C g-1
f g+1
Cg
FIG. 2. The kinetic process of heterogeneous condensation on an insoluble spherical particle. Cg and Eg are the condensation and the evaporation coefficients of water cluster Bg , respectively. fg denotes the number of clusters Bg formed on unit surface of a spherical insoluble particle.
+ Eg (β), which denotes the rate where monomers evaporate from Bg . Based on the capillarity approximation,23 the cluster Bg in heterogeneous condensation is assumed as a spherical cap shaped droplet (with radius rg ) containing g molecules (see in Fig. 1). We denote the vapor phase by subscript “v,” the cluster by “c,” and the solid particle by “s.” Therefore, the relation between cluster radius rg and molecule number g can be obtained by an implicit equation: Vc (rg ) = gVwm ,
(3)
where Vwm denotes the volume of one water molecule, Vc denotes the volume of Bg :24, 25 Vc (rg ) =
1 3 π r (2 − 3 cos ψg + cos3 ψg ) 3 g 1 − π R 3 (2 − 3 cos φg + cos3 φg ), 3
(4)
and cos ψg =
m g R − rg , dg
(5)
cos φg =
R − mg rg , dg
(6)
where ψ g and φ g are half of the central angles of Bg and the part of particle (radius R) contacting with Bg , respectively. mg is the wetting degree of the particle surface, which is equal to the cosine of the contact angle θ g (mg = cos θ g ). The distance between the centers of cluster Or and solid particle OR can be found as dg = R 2 + rg2 − 2Rrg mg . When the water cluster is small, the effect of line tension20–22 τ at the three-phase (cluster, vapor, and solid) contact line should be considered which can have a substantial influence on the nucleation behavior. The microscopic contact angle θ g is different from the macroscopic contact angle θ ∞ , and takes the form21 as a function of the line tension τ : τ cos φg mg = m∞ − , (7) σ rg sin ψg where the macroscopic wetting degree m∞ = cos θ ∞ , σ is the surface tension of water. This result is known as the modified Dupre-Young equation12 which can be solved by an iterative method. The expression of condensation coefficient in the kinetic model for heterogeneous condensation of vapor has an
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explicit expression:11, 12, 16, 19 Cg = Cg (α) + Cg (β),
and Eg (β) are given by (8)
Eg+1 (α) = Cgsat (α) ·
and pv Scv (rg ), Cg (α) = αg √ 2π mwm kB T Gdiff Lcvs (rg ), Cg (β) = δn1 ν exp − kB T
Eg+1 (β) = Cgsat (β) ·
(9)
(10)
where kB is the Boltzmann constant, pv is the pressure of vapor, mwm is the mass of one water molecule, T denotes the temperature of vapor, δ denotes the mean jump distance of one water molecule, ν is the vibration frequency of one water molecule on the surface, Gdiff denotes the surface diffusion energy per water molecule. α g is the sticking coefficient of Bg , which denotes the fraction of monomers hitting the cluster that stick. It lies in the range 0.04–1 for water condensation.26 For the sake of convenience, we assume that α g is unity, which denotes that all hitting monomers stick on the cluster. The surface concentration of water molecules n1 , the surface area between cluster and vapor Scv , and the perimeter of the threephase contact line Lcvs are, respectively, Gdes pv , (11) exp n1 = √ kB T ν 2π mwm kB T Scv (rg ) = 2π rg2 (1 − cos ψg ),
(12)
Lcvs (rg ) = 2π rg sin ψg ,
(13)
where Gdes denotes the desorption energy per water molecule. It should be noted that our model assumes that n1 does not change in time because we do not consider the depletion of vapor around the particle. However, the total number of water molecules on the particle surface would change, as clusters are formed because of the decrease of the heterogeneous nucleation area (the difference between the surface area of particle and the area occupied by clusters). Unfortunately, no explicit expression for the evaporation coefficient Eg is available in literature. We shall derive the expression using a method called “detailed balance.”11 Specifically, the phase equilibrium (the physical vapor-liquid equilibrium in the saturated state) is chosen to find the evaporation coefficient. The detailed balance is sat sat Cgsat (α) · nsat g = Eg+1 (α) · ng+1 ,
(14)
sat sat Cgsat (β) · nsat g = Eg+1 (β) · ng+1 ,
(15)
where the superscript “sat” denotes the saturated state, and ng is the equilibrium number density of Bg . We choose the selfconsistent expression13 for ng : W (rg ) W (r1 ) ng = n1 exp − exp , (16) kB T kB T where W (rg ) denotes the energy to form the water cluster Bg . The evaporation coefficient can be assumed to be supersaturation independent.11 Therefore, for any supersaturation, Eg (α)
nsat g nsat g+1 nsat g nsat g+1
,
(17)
.
(18)
According to the classical heterogeneous nucleation theory,24 the formation energy W of the cap shaped cluster Bg with radius rg in heterogeneous nucleation on the spherical particle with radius R can be expressed by a function of rg : W (rg ) = −
Vc (rg ) μ + σ [Scv (rg ) − m∞ Scs (rg )] Vwm
+ τ Lcvs (rg ) + pδ Vc (rg ),
(19)
and μ = kB T ln S,
(20)
Scs (rg ) = 2πR 2 (1 − cos φg ),
(21)
A0 −3 δ , (22) 6π where μ is the chemical potential difference per molecule between the vapor phase and the liquid phase, S is the saturation ratio, and Scs denotes the surface area between water cluster and solid particle. pδ denotes the disjoining pressure,27, 28 which is defined as the difference between the pressure of the film and the pressure in the bulk phase, A0 is the Hamaker constant,28 and δ is the film thickness. The surface of solid particle is only partial wetting in our context, and therefore, the liquid phase forms as some separate spherical cap shaped droplets with a finite contact angle on the surface. The continuous liquid film covering around the perfect wettable particle is out of the scope of our investigation. As a result, we shall not consider the effect of disjoining pressure hereafter. It should be noted that when the temperature is higher than the wetting temperature,29 the surface can be regarded as perfect wetting and the liquid film will automatically form. As a result, the disjoining pressure should be considered in different material systems when the temperature is higher than the wetting temperature. Here, we denote (Scv − m∞ Scs ) by S , which can be considered as the effective area of cluster in heterogeneous condensation. As a result, the formation energy W of the cap shaped cluster Bg with radius rg in Eq. (19) becomes pδ =
W (rg ) = −
Vc (rg ) μ + σ S (rg ) + τ Lcvs (rg ). Vwm
(23)
Substituting Eq. (23) into Eq. (16), the equilibrium number density of Bg in the saturated state (S = 1) is σ S (rg ) + τ Lcvs (rg ) W (r1 ) sat sat exp , ng = n1 exp − kB T kB T (24) where the chemical potential difference μ = 0 in the saturated state.
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From Eqs. (17) and (18), we find that it is critical to get sat the expression of nsat g /ng+1 in the derivation of the evaporation coefficient, which can be obtained from Eq. (24): nsat σ [S (rg+1 )−S (rg )]+τ [Lcvs (rg+1 )−Lcvs (rg )] g . = exp nsat kB T g+1 (25) sat For a system with a large g (g ≥ 20), nsat /n (after a comg g+1 plicated derivation as provided in the Appendix) can be expressed as nsat g nsat g+1
= exp Keg ,
(26)
where Keg is the g-cluster’s Kelvin number, which is defined as Keg = 2σ Vwm /(kB T rg ). As a result, the exact expression for evaporation coefficient can be obtained by combining Eqs. (9), (10), (17), (18), and (26): Eg = Eg (α) + Eg (β),
(27)
and
Eg (α) = exp Keg · Cgsat (α) pvsat 2σ Vwm Scv (rg ), (28) = exp √ kB T rg 2π mwm kB T
Eg (β) = exp Keg · Cgsat (β) Gdiff 2σ Vwm δnsat Lcvs (rg ). ν exp − = exp 1 kB T rg kB T (29)
When the surface is completely hydrophobic, i.e., the contact angle between foreign surface and cluster θ = π , our expression of evaporation coefficient is pvsat 2σ Vwm Eg |θ= = exp 4π rg2 , (30) √ kB T rg 2π mwm kB T which is identical to the evaporation coefficient in homogeneous condensation.11 When the surface is planar, i.e., the radius of solid particle R → ∞, our expression of evaporation coefficient is (31) Eg |R→∞ = Eg (α) R→∞ + Eg (β) R→∞ , Eg (α)|R→∞ = exp
condensation in vapor and the heterogeneous condensation on a planar solid surface. In brief, the gaps for the kinetic model for heterogeneous condensation on an insoluble spherical particle have been overcome by taking two mechanisms of cluster growth and the effect of line tension into account (Eqs. (8), (27), and (23)), and deriving the exact expression for evaporation coefficient (Eqs. (27)–(29)). In this way, a new kinetic model is established and the kinetic studies on the heterogeneous condensation occurring on a spherical solid surface become feasible.
pvsat 2σ Vwm 2π rg2 (1 − cos θ ), √ kB T rg 2πmwm kB T (32)
2σ Vwm Gdiff sat 2πrg sin θ, δn1 ν exp − Eg (β)|R→∞ = exp kB T rg kB T (33) which is the same as the evaporation coefficient in heterogeneous condensation on a planar solid surface.17 As a result, the obtained expression of evaporation coefficient is not only appropriate for the heterogeneous condensation on an insoluble spherical particle, but also correct in the homogeneous
III. RESULTS AND DISCUSSIONS
In this section, the physical insight of evaporation coefficient is presented at first. Then, the contributions of two mechanisms to heterogeneous condensation including the effect of line tension are evaluated and analysed. As an example of our model applications, the growth rate of the cap shaped droplet on an insoluble spherical particle in the heterogeneous condensation is derived using our model at last. A. Physical insight of evaporation coefficient
The evaporation coefficients have the same form in homogeneous condensation and heterogeneous condensation: 2σ Vwm · Cgsat . (34) Eg = exp kB T rg This identity can be interpreted from the perspective of eq physics. The vapor pressure in the equilibrium state pv (the superscript “eq” denotes the parameters in the equilibrium state hereafter) for a flat water layer is equal to the saturated vapor pressure pvsat . However, the vapor pressure in the equilibrium state for a spherical droplet (with radius rg ) is higher than the saturated vapor pressure, and it satisfies the Kelvin equation:12 2σ Vwm eq pvsat . pv (rg ) = exp (35) kB T rg When the ambient vapor pressure is identical to the vapor pressure in the equilibrium state for a spherical droplet, the eq eq droplet is in the stable state, i.e., Eg = Cg . As mentioned in Sec. II, the evaporation coefficient is supersaturation independent. Therefore, the evaporation coefficient of the droplet (with radius r and water molecules number g) for any supersaturation is identical to that in the stable state: Eg = Cgeq = Cgeq (α) + Cgeq (β)
2σ Vwm sat Cg (α) + Cgsat (β) = exp kB T rg 2σ Vwm Cgsat . = exp kB T rg
(36)
This expression of evaporation coefficient is identical to that in our model. As a result, the same form of evaporation
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4
4
3
3
the condensation coefficient Cg , i.e., the cluster tends to disappear. When g > g∗ , Eg < Cg , i.e., the cluster tends to grow. When g = g∗ , Eg = Cg , i.e., the cluster is in the metastable state. This tendency is identical to the classical nucleation theory’s prediction. Both the identity of evaporation coefficients and the comparison between the evaporation coefficient and the condensation coefficient indicate that our kinetic model is correct in the physical aspect.
11
-1
Coefficients [10 s ]
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Cg Eg
B. Contributions of two mechanisms to heterogeneous condensation
g=g* 2
50
100
2 200
150
g FIG. 3. The evaporation coefficient Eg (solid line) and the condensation coefficient Cg (dashed line) as functions of the number of molecules g in cluster. R = 1μm, m = 0.5, T = 293.15 K, S = 1.8, g∗ = 90, δ = 0.32 nm, ν = 1013 s−1 , Gdiff = 2.9 × 10−21 J molecule−1 , and Gdes = 2.9 × 10−20 J molecule−1 .
coefficients in heterogeneous condensation and homogeneous condensation can be ascribed to the Kelvin effect. We can also compare the evaporation coefficient with the condensation coefficient in heterogeneous condensation on an insoluble spherical particle. The calculating parameters are listed in Table I. We first use the classical nucleation theory to calculate the critical size r∗ and g∗ : ∂W = 0, (37) ∂r r=r ∗ r∗ =
2σ Vwm , kB T ln S
(38)
Vc (r ∗ ) . Vwm
(39)
g∗ =
In this condition, g∗ = 90. As shown in Fig. 3, when the cluster size g is less than the critical size g∗ , the evaporation coefficient Eg is larger than
Some literatures12, 30, 31 reported that the surface diffusion mechanism is more effective and has a higher contribution to the condensation coefficient than that caused by the direct addition mechanism by a factor between 102 and 103 . However, there are no systematic evaluation and analysis of the contributions of two mechanisms to heterogeneous condensation including the effect of line tension. In order to estimate the relative importance (RI) of two mechanisms of cluster growth, we define a dimensionless number RI as RI =
Cg (β) . Cg (α)
(40)
We evaluate the variation of RI with different numbers of water molecules in cluster g and different macroscopic wetting degrees m∞ . The parameters are set as follows: the number of water molecules in cluster g ranges from 50 to 200, the macroscopic wetting degree m∞ varies from −0.9 to 0.9. For other parameters (the radius of solid particle, the parameters of ambient condition and water molecule), we use the same values as listed in Table I. Figure 4(a) shows that the relative importance of two mechanisms of cluster growth as a function of m∞ and g. The value of RI is always between 102 and 103 , except in some extreme cases (large g and small m∞ ). RI decreases with g, raises with small m∞ (−0.9 ≤ m∞ < 0.5 in our case), and falls with large m∞ (0.5 ≤ m∞ < 0.9 in our case). These tendencies can be ascribed to the geometrical effect as shown in Fig. 4(b).
200
200
RI
80 120
0.4 0.5
160
160
100 140
180
0.6
Lcvs/Scv [109 m-1]
0.7
0.8
0.8 0.9
g
150
g
150
200
1
100
100 220
1.1
240
1.2
260
50 -0.9
-0.6
-0.3
0
0.3
0.6
0.9
50 -0.9
1.3
-0.6
-0.3
0
m∞
m∞
(a)
(b)
0.3
0.6
0.9
FIG. 4. (a) The relative importance (RI) of two mechanisms of cluster growth and (b) the ratio of the perimeter of three-phase contact line to the surface area between cluster and vapor (Lcvs /Scv ) as functions of the macroscopic wetting degree m∞ and the number of water molecules in cluster g. R = 1μm, T = 293.15 K, S = 1.8, δ = 0.32 nm, ν = 1013 s−1 , Gdiff = 2.9 × 10−21 J molecule−1 , and Gdes = 2.9 × 10−20 J molecule−1 .
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According to Eqs. (9) and (10), we find that the condensation coefficients in direct addition Cg (α) and in surface diffusion Cg (β) have positive correlations with the surface area between cluster and vapor Scv and the perimeter of three-phase contact line Lcvs , respectively. The value of Lcvs /Scv has the same orψ der of magnitude to (rg tan 2g )−1 . Figure 4(b) shows that the value of Lcvs /Scv decreases with g, which is because rg−1 falls when g increases. The tendency of Lcvs /Scv with different m∞ in Fig. 4(b) identical to RI in Fig. 4(a) is determined by two competing mechanisms: one is the effect of cluster radius rg , the other is the effect of half of the central angle of cluster ψ g . When m∞ increases, rg raises and ψ g falls. In the regime of small m∞ (−0.9 ≤ m∞ < 0.5 in our case), the variation of ψ g is primary, which results in the raise of Lcvs /Scv . In the regime of large m∞ (0.5 ≤ m∞ < 0.9 in our case), the variation of rg is dominant and leads to the decrease of Lcvs /Scv . These results suggest that the cluster growth via surface diffusion of adsorbed water molecules on the particle is more important than the direct addition from the vapor. These also show that both the macroscopic wetting degree and the number of water molecules in cluster have significant effects on the contributions of the two mechanisms. A larger contribution of surface diffusion can be found at medium macroscopic wetting degree and smaller size cluster.
C. Growth rate of droplet in heterogeneous condensation
Because the exact expression of the evaporation coefficient has been obtained from our derivation, our kinetic model actually paves the way for further studies of the heterogeneous condensation of vapor on an insoluble spherical particle. As an example of our model applications, the growth rate of the cap shaped droplet on an insoluble spherical particle in the heterogeneous condensation is derived using the new model. The droplet growth depends on an important parameter, the Knudsen number Kn. This dimensionless parameter is defined as the ratio of the mean free path λ of a vapor molecule to the droplet diameter 2r: Kn = λ/2r. For a large Kn, i.e., relatively small droplet, the growth is controlled by vapor kinetic process of vapor molecular collision from the surrounding mixture onto the droplet.6 As a result, the growth rate of single droplet with a large Knudsen number can approximately be given by the difference of the condensation coefficient and the evaporation coefficient.8 For heterogeneous condensation
on an insoluble spherical particle, the growth rate of a droplet is dr dg dr = · dt dg dt het
het
≈ =
het
Vwm − E ) · (C g g het π rg2 (2 − 3 cos ψg + cos3 ψg ) π rg2 (2
Vwm · (S − eKeg ) Cgsat het . 3 − 3 cos ψg + cos ψg ) (41)
It should be noted that clusters are formed by nucleation process and only the cluster whose size is larger than the critical size can grow, which is described by Eq. (2). As a result, Eq. (41) is only an approximation for the case when clusters can be considered as independent. In addition, the effect of line tension can be ignored, because the droplet in this case is large enough. We evaluate the growth rate of droplet dr/dt with different macroscopic wetting degrees m∞ . The parameters are set as follows: the macroscopic wetting degree m∞ varies from −1 to 1, the radius droplet r = 2r∗ (only the droplet larger than the critical size can grow). For other parameters (the radius of solid particle, the parameters of ambient condition and water molecule), we use the same values as listed in Table I. As shown in Fig. 5(a), when m = −1, i.e., the surface of particle is completely hydrophobic, the value of dr/dt in heterogeneous condensation is identical to that in homogeneous condensation, because the heterogeneous condensation can be considered as homogeneous condensation when the surface is completely hydrophobic. When m > −1, the value of dr/dt in heterogeneous condensation is larger than that in homogeneous condensation, because there is an additional contribution to the droplet growth (surface diffusion of adsorbed water molecules on the particle) for heterogeneous condensation compared with the homogeneous counterpart. Figure 5(a) also shows that dr/dt rapidly increases with relatively small m (−1 ≤ m < −0.95), and slightly increases with medium m (−0.95 ≤ m < 0.8), and rapidly increases again with relatively large m (0.8 ≤ m < 1). This tendency is ascribed to the influx of water molecules and the shape of droplet dr/dg. It should be noted that the influx of water molecules is primarily determined by surface diffusion which is dependent on the value of Lcvs , and the radius of droplet with a larger wetting degree increases more quickly when the influx of water
TABLE I. Parameters for the comparison between the evaporation coefficient and the condensation coefficient in heterogeneous condensation on an insoluble spherical particle.
Solid particle Ambient condition Water moleculea
a
Parameter
Symbol
Value
Radius Macroscopic wetting degree Temperature Saturation Mean jump distance Vibration frequency on surface Surface diffusion energy Desorption energy
R (μm) m∞ T (K) S δ (nm) ν (s−1 ) Gdiff (J molecule−1 ) Gdes (J molecule−1 )
1 0.5 293.15 1.8 0.32 1013 2.9 × 10−21 2.9 × 10−20
These data are from Ref. 19.
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Lcvs [nm]
Lcvs dr/dg
10
3
10
2
10
1
10
0
10
-1
-11
10
101
100
-1
-0.5
0
0.5
1
m]
J. Chem. Phys. 140, 024708 (2014)
dr/dg [10
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10-2
m (a)
(b)
FIG. 5. Relations between the macroscopic wetting degree of the particle surface m∞ and (a) the growth rates of droplet dr/dt in heterogeneous condensation (solid line) and in homogeneous condensation (dashed line), (b) the perimeter of three-phase contact line Lcvs (solid line) and the effect of droplet’s shape dr/dg (dashed line). r = 2r∗ , R = 1 μm, T = 293.15 K, S = 1.8, δ = 0.32 nm, ν = 1013 s−1 , Gdiff = 2.9 × 10−21 J molecule−1 , and Gdes = 2.9 × 10−20 J molecule−1 .
molecules is the same (dr/dg increases with wetting degree). As shown in Fig. 5(b), when m is relatively small, the rapid increase of Lcvs improves the surface diffusion of adsorbed water molecules on the particle. When m is medium, Lcvs is almost unchanged, the slight raise of dr/dt is ascribed to the shape of droplet. When m is relatively large, although Lcvs decreases fast, the shape of droplet is much more significant for the droplet growth.
IV. CONCLUSIONS
We have developed a kinetic model to describe the heterogeneous condensation on an insoluble spherical particle. This model considers two mechanisms of cluster growth: direct addition of water molecules from the vapor and surface diffusion of adsorbed water molecules on the particle. Based on the capillarity approximation, we assumed the water cluster as a spherical cap shaped droplet, and obtained the relation between radius and molecule number of the cluster. We also considered the effect of line tension on the heterogeneous condensation. For the first time, we have derived the exact expression of evaporation coefficient for heterogeneous condensation of vapor on an insoluble spherical particle by using the detailed balance. It is confirmed that the expression of evaporation coefficient in our model is also correct in the homogeneous condensation in vapor and the heterogeneous condensation on a planar solid surface. The evaporation coefficients have the same form in homogeneous condensation and heterogeneous condensation, which is ascribed to the Kelvin effect. We also compared the evaporation coefficient with the condensation coefficient to check the coefficient calculated in our model. The results indicate that our kinetic model is correct in physical aspect. The contributions of two mechanisms to heterogeneous condensation including the effect of line tension were evaluated and analysed. It is found that the cluster growth via surface diffusion of adsorbed water molecules on the particle is more
important than the direct addition from the vapor, and both the macroscopic wetting degree and the number of water molecules in cluster have significant effects on the contributions of the two mechanisms. To provide an example of our model applications, we have derived the growth rate of the cap shaped droplet on the insoluble spherical particle in the heterogeneous condensation by our model, which cannot be obtained by previous kinetic model. We also have evaluated the growth rate of droplet with different wetting degrees. It is found that when the surface of particle is completely hydrophobic, the growth rate of droplet in heterogeneous condensation is identical to that in homogeneous condensation. Otherwise, the growth rate of droplet in heterogeneous condensation is larger than that in homogeneous condensation because of an additional contribution to the droplet growth. It is also found that the growth rate of droplet in heterogeneous condensation experiences three regimes (a quick increasing regime, a slight increasing regime, and another rapid increasing regime) when the wetting degree increases. These results indicate that an explicit kinetic model is benefit to the study of heterogeneous condensation on an insoluble spherical particle, and the approximation by using the kinetic model for homogeneous condensation or heterogeneous condensation on a planar solid surface is unsuitable. It is expected that our kinetic model will promote the kinetic studies of heterogeneous condensation on an insoluble spherical particle, such as the start-up process of condensation, the evolution of the number density of droplets, and the influence of vapor depletion, which will also be a part of our future work. In addition, the influences of different material systems on the heterogeneous condensation need to be analysed in further investigation.
ACKNOWLEDGMENTS
This work was supported by the Natural Science Foundation of China (Grant Nos. 11172292 and 10976029), by the
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Strategic Priority Research Program of the Chinese Academy of Sciences (XDB05040400), by the Hefei Physical Science and Technology Center (2012FXCX005), by the CAS Special Grant for Postgraduate Research, and by the USTC Special Grant for Postgraduate Research. APPENDIX: DERIVATION OF THE EXPRESSION OF ngsat / ngsat +1
For a system with a large g (g ≥ 20), Eq. (25) reduces to nsat ∂S 1 ∂r ∂Lcvs g σ . (A1) = exp +τ nsat kB T ∂g ∂r ∂r g+1 First, we calculate the derivatives of the cap shaped cluster’s volume Vc , effective area S , and three-phase contact line’s perimeter Lcvs with respect to r: ∂Vc ∂A = π r 2 (2 − 3A + A3 ) + r 3 (A2 − 1) ∂r ∂r ∂B 3 2 − R (B − 1) , (A2) ∂r ∂A ∂B ∂S = 2π 2r(1 − A) − r 2 + m∞ R 2 , (A3) ∂r ∂r ∂r B ∂B ∂Lcvs = −2πR √ , 2 ∂r 1 − B ∂r
(A4)
A = cos ψg ,
(A5)
B = cos φg .
(A6)
and
Then, differentiating Eqs. (5)–(7), with respect to r yields 2 ∂A ∂m A2 − 1 R =B + , ∂r d ∂r d r 2 ∂m r A2 − 1 ∂B =A + , ∂r d ∂r R d τ ∂m ∂B =− (1 − B 2 )−1.5 . ∂r σR ∂r Differentiating Eq. (3) with respect to g yields ∂Vc ∂r = Vwm . ∂r ∂g
(A7) (A8) (A9)
(A10)
Combining Eqs. (A1) and (A10) yields nsat Vwm σ (∂S /∂r) + τ (∂Lcvs /∂r) g . = exp nsat kB T ∂Vc /∂r g+1
(A11)
Finally, after a great amount of labor, the exact expression sat for nsat g /ng+1 can be obtained by combining Eqs. (A2)–(A4), (A7)–(A9), and (A11): nsat 2σ Vwm g . (A12) = exp nsat kB T rg g+1 1 B.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 024708 (2014)
A kinetic model for heterogeneous condensation of vapor on an insoluble spherical particle Xisheng Luo,1 Yu Fan,1 Fenghua Qin,1,a) Huaqiao Gui,2 and Jianguo Liu2 1
Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, China 2 Key Laboratory of Environmental Optics and Technology, Anhui Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Hefei 230031, China
(Received 7 October 2013; accepted 30 December 2013; published online 14 January 2014) A kinetic model is developed to describe the heterogeneous condensation of vapor on an insoluble spherical particle. This new model considers two mechanisms of cluster growth: direct addition of water molecules from the vapor and surface diffusion of adsorbed water molecules on the particle. The effect of line tension is also included in the model. For the first time, the exact expression of evaporation coefficient is derived for heterogeneous condensation of vapor on an insoluble spherical particle by using the detailed balance. The obtained expression of evaporation coefficient is proved to be also correct in the homogeneous condensation and the heterogeneous condensation on a planar solid surface. The contributions of the two mechanisms to heterogeneous condensation including the effect of line tension are evaluated and analysed. It is found that the cluster growth via surface diffusion of adsorbed water molecules on the particle is more important than the direct addition from the vapor. As an example of our model applications, the growth rate of the cap shaped droplet on the insoluble spherical particle is derived. Our evaluation shows that the growth rate of droplet in heterogeneous condensation is larger than that in homogeneous condensation. These results indicate that an explicit kinetic model is benefit to the study of heterogeneous condensation on an insoluble spherical particle. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4861892] I. INTRODUCTION
Condensation in supersaturated vapor is a very common phenomenon in atmospheric physics,1, 2 gas cleaning technology,3–5 and multiphase flow.6, 7 The condensation can be activated via two ways, homogeneous nucleation8 and heterogeneous nucleation.9 In homogeneous nucleation, a higher nucleation energy barrier should be overcome to create nuclei in the interior of a uniform substance. In heterogeneous nucleation, the foreign surface reduces the nucleation energy barrier and promotes the process of nucleation. As a result, heterogeneous condensation is much common in daily life. The classical condensation theory is based on the kinetic model10–12 where the droplet size distribution can be derived. The framework of the kinetic model for homogeneous condensation is relatively complete so that the kinetic model has been widely used in homogeneous condensation study. Typical examples are to study the steady-state nucleation rate,13 the start-up process of condensation,14 the evolution of the number density of droplets,15 and the influence of vapor depletion.16 The kinetic model is also a useful tool to study the heterogeneous condensation. Some researches9, 17 focused the kinetic study on the heterogeneous condensation occurring on a planar solid surface. As the complicated effects of foreign particle, the kinetic model for heterogeneous condensation on an insoluble spherical particle still has some gaps. In the kinetic model for heterogeneous condensation, there are two mechanisms of clusa) Electronic mail: [email protected]
0021-9606/2014/140(2)/024708/8/$30.00
ter growth:18, 19 direct addition of water molecules from the vapor and surface diffusion of adsorbed water molecules on the particle, where the first one is the way of cluster growth in homogeneous condensation. The integrated consideration of two growth mechanisms and line tension20–22 has not yet appeared in the kinetic model for heterogeneous condensation on an insoluble spherical particle. In addition, the expression for evaporation coefficient is unknown because of the complicated geometrical relation in heterogeneous condensation especially when the effect of line tension is considered. Although the relation between the condensation coefficient and the evaporation coefficient is certain in the kinetic model with detailed balance (the detailed balance will be introduced in Sec. II), the kinetic model only can be used to calculate the steady-state nucleation rate19, 23 for heterogeneous condensation on an insoluble spherical particle, because the exact expression of evaporation coefficient is needed in many other circumstances, such as the growth rate of droplet on the particle, the start-up process of heterogeneous condensation, the evolution of the droplets’ number density, and the influence of vapor depletion. The replacement by the kinetic model for homogeneous condensation or heterogeneous condensation on a planar solid surface is an optional method, but the validity of this approximation is still unknown. In this paper, we shall develop a kinetic model for heterogeneous condensation on an insoluble spherical particle. In our model, the two mechanisms of cluster growth and the effect of line tension are considered, and the exact expression for evaporation coefficient will be derived. We shall also derive the growth rate of droplet on an insoluble spherical
140, 024708-1
© 2014 AIP Publishing LLC
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particle as an example of using our model, which cannot be obtained by the previous kinetic model.19, 23 In Sec. II, the new kinetic model for heterogeneous condensation is described in details. In Sec. III, some discussions of our kinetic model are presented, and the growth rate of droplet on the insoluble spherical particle is derived as an example of our model applications. A brief summary of our results is provided in Sec. IV. II. KINETIC MODEL
In the kinetic model for condensation of vapor, the water cluster is assumed to gain or lose only single water molecules, and cluster-cluster interactions are neglected. We denote a cluster containing g molecules by Bg (note that the subscript “g” denotes the parameters of g-cluster hereafter). The size change of Bg by attaching or detaching of a molecule can be represented by the reversible “reaction:” Bg + B1 Bg+1 ,
g = 1, 2, 3, . . . .
(1)
As shown in Fig. 1, there are two mechanisms for the growth of cluster: (α) direct addition of water molecules from the vapor, and (β) surface diffusion of adsorbed water molecules on the particle. As a result, the cluster Bg Bg can grow by the condensation of molecules from both the vapor and the adsorbed water molecules at rates of Cg (α) and Cg (β), respectively, and shrink by evaporation of molecules to both the vapor and the adsorbed water molecules at rates of Eg (α) and Eg (β), respectively. The number of clusters Bg formed on unit surface of a spherical insoluble particle is denoted by fg . As sketched in Fig. 2, the change rate of fg is dfg = Cg−1 fg−1 − (Cg + Eg )fg + Eg+1 fg+1 , (2) dt where the condensation coefficient Cg is equal to Cg (α) + Cg (β), which denotes the rate where monomers condense on Bg , and the evaporation coefficient Eg is identical to Eg (α)
C g( ) C g( )
E g( )
Bg
g
E g( )
g
rg or R
dg g
oR FIG. 1. A spherical cap shaped cluster Bg forms on the surface of a spherical solid particle. Cg and Eg are the condensation and the evaporation coefficients of water cluster Bg , respectively. (α) and (β) denote the coefficients in direct addition and surface diffusion, respectively. ψ g and φ g are half of the central angles of Bg (radius rg ) and the part of particle (radius R) contacting with Bg , respectively. θ g is the contact angle between Bg and solid. dg is the distance between the centers of cluster Or and solid particle OR .
f g-1
fg
Eg
E g+1
C g-1
f g+1
Cg
FIG. 2. The kinetic process of heterogeneous condensation on an insoluble spherical particle. Cg and Eg are the condensation and the evaporation coefficients of water cluster Bg , respectively. fg denotes the number of clusters Bg formed on unit surface of a spherical insoluble particle.
+ Eg (β), which denotes the rate where monomers evaporate from Bg . Based on the capillarity approximation,23 the cluster Bg in heterogeneous condensation is assumed as a spherical cap shaped droplet (with radius rg ) containing g molecules (see in Fig. 1). We denote the vapor phase by subscript “v,” the cluster by “c,” and the solid particle by “s.” Therefore, the relation between cluster radius rg and molecule number g can be obtained by an implicit equation: Vc (rg ) = gVwm ,
(3)
where Vwm denotes the volume of one water molecule, Vc denotes the volume of Bg :24, 25 Vc (rg ) =
1 3 π r (2 − 3 cos ψg + cos3 ψg ) 3 g 1 − π R 3 (2 − 3 cos φg + cos3 φg ), 3
(4)
and cos ψg =
m g R − rg , dg
(5)
cos φg =
R − mg rg , dg
(6)
where ψ g and φ g are half of the central angles of Bg and the part of particle (radius R) contacting with Bg , respectively. mg is the wetting degree of the particle surface, which is equal to the cosine of the contact angle θ g (mg = cos θ g ). The distance between the centers of cluster Or and solid particle OR can be found as dg = R 2 + rg2 − 2Rrg mg . When the water cluster is small, the effect of line tension20–22 τ at the three-phase (cluster, vapor, and solid) contact line should be considered which can have a substantial influence on the nucleation behavior. The microscopic contact angle θ g is different from the macroscopic contact angle θ ∞ , and takes the form21 as a function of the line tension τ : τ cos φg mg = m∞ − , (7) σ rg sin ψg where the macroscopic wetting degree m∞ = cos θ ∞ , σ is the surface tension of water. This result is known as the modified Dupre-Young equation12 which can be solved by an iterative method. The expression of condensation coefficient in the kinetic model for heterogeneous condensation of vapor has an
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explicit expression:11, 12, 16, 19 Cg = Cg (α) + Cg (β),
and Eg (β) are given by (8)
Eg+1 (α) = Cgsat (α) ·
and pv Scv (rg ), Cg (α) = αg √ 2π mwm kB T Gdiff Lcvs (rg ), Cg (β) = δn1 ν exp − kB T
Eg+1 (β) = Cgsat (β) ·
(9)
(10)
where kB is the Boltzmann constant, pv is the pressure of vapor, mwm is the mass of one water molecule, T denotes the temperature of vapor, δ denotes the mean jump distance of one water molecule, ν is the vibration frequency of one water molecule on the surface, Gdiff denotes the surface diffusion energy per water molecule. α g is the sticking coefficient of Bg , which denotes the fraction of monomers hitting the cluster that stick. It lies in the range 0.04–1 for water condensation.26 For the sake of convenience, we assume that α g is unity, which denotes that all hitting monomers stick on the cluster. The surface concentration of water molecules n1 , the surface area between cluster and vapor Scv , and the perimeter of the threephase contact line Lcvs are, respectively, Gdes pv , (11) exp n1 = √ kB T ν 2π mwm kB T Scv (rg ) = 2π rg2 (1 − cos ψg ),
(12)
Lcvs (rg ) = 2π rg sin ψg ,
(13)
where Gdes denotes the desorption energy per water molecule. It should be noted that our model assumes that n1 does not change in time because we do not consider the depletion of vapor around the particle. However, the total number of water molecules on the particle surface would change, as clusters are formed because of the decrease of the heterogeneous nucleation area (the difference between the surface area of particle and the area occupied by clusters). Unfortunately, no explicit expression for the evaporation coefficient Eg is available in literature. We shall derive the expression using a method called “detailed balance.”11 Specifically, the phase equilibrium (the physical vapor-liquid equilibrium in the saturated state) is chosen to find the evaporation coefficient. The detailed balance is sat sat Cgsat (α) · nsat g = Eg+1 (α) · ng+1 ,
(14)
sat sat Cgsat (β) · nsat g = Eg+1 (β) · ng+1 ,
(15)
where the superscript “sat” denotes the saturated state, and ng is the equilibrium number density of Bg . We choose the selfconsistent expression13 for ng : W (rg ) W (r1 ) ng = n1 exp − exp , (16) kB T kB T where W (rg ) denotes the energy to form the water cluster Bg . The evaporation coefficient can be assumed to be supersaturation independent.11 Therefore, for any supersaturation, Eg (α)
nsat g nsat g+1 nsat g nsat g+1
,
(17)
.
(18)
According to the classical heterogeneous nucleation theory,24 the formation energy W of the cap shaped cluster Bg with radius rg in heterogeneous nucleation on the spherical particle with radius R can be expressed by a function of rg : W (rg ) = −
Vc (rg ) μ + σ [Scv (rg ) − m∞ Scs (rg )] Vwm
+ τ Lcvs (rg ) + pδ Vc (rg ),
(19)
and μ = kB T ln S,
(20)
Scs (rg ) = 2πR 2 (1 − cos φg ),
(21)
A0 −3 δ , (22) 6π where μ is the chemical potential difference per molecule between the vapor phase and the liquid phase, S is the saturation ratio, and Scs denotes the surface area between water cluster and solid particle. pδ denotes the disjoining pressure,27, 28 which is defined as the difference between the pressure of the film and the pressure in the bulk phase, A0 is the Hamaker constant,28 and δ is the film thickness. The surface of solid particle is only partial wetting in our context, and therefore, the liquid phase forms as some separate spherical cap shaped droplets with a finite contact angle on the surface. The continuous liquid film covering around the perfect wettable particle is out of the scope of our investigation. As a result, we shall not consider the effect of disjoining pressure hereafter. It should be noted that when the temperature is higher than the wetting temperature,29 the surface can be regarded as perfect wetting and the liquid film will automatically form. As a result, the disjoining pressure should be considered in different material systems when the temperature is higher than the wetting temperature. Here, we denote (Scv − m∞ Scs ) by S , which can be considered as the effective area of cluster in heterogeneous condensation. As a result, the formation energy W of the cap shaped cluster Bg with radius rg in Eq. (19) becomes pδ =
W (rg ) = −
Vc (rg ) μ + σ S (rg ) + τ Lcvs (rg ). Vwm
(23)
Substituting Eq. (23) into Eq. (16), the equilibrium number density of Bg in the saturated state (S = 1) is σ S (rg ) + τ Lcvs (rg ) W (r1 ) sat sat exp , ng = n1 exp − kB T kB T (24) where the chemical potential difference μ = 0 in the saturated state.
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From Eqs. (17) and (18), we find that it is critical to get sat the expression of nsat g /ng+1 in the derivation of the evaporation coefficient, which can be obtained from Eq. (24): nsat σ [S (rg+1 )−S (rg )]+τ [Lcvs (rg+1 )−Lcvs (rg )] g . = exp nsat kB T g+1 (25) sat For a system with a large g (g ≥ 20), nsat /n (after a comg g+1 plicated derivation as provided in the Appendix) can be expressed as nsat g nsat g+1
= exp Keg ,
(26)
where Keg is the g-cluster’s Kelvin number, which is defined as Keg = 2σ Vwm /(kB T rg ). As a result, the exact expression for evaporation coefficient can be obtained by combining Eqs. (9), (10), (17), (18), and (26): Eg = Eg (α) + Eg (β),
(27)
and
Eg (α) = exp Keg · Cgsat (α) pvsat 2σ Vwm Scv (rg ), (28) = exp √ kB T rg 2π mwm kB T
Eg (β) = exp Keg · Cgsat (β) Gdiff 2σ Vwm δnsat Lcvs (rg ). ν exp − = exp 1 kB T rg kB T (29)
When the surface is completely hydrophobic, i.e., the contact angle between foreign surface and cluster θ = π , our expression of evaporation coefficient is pvsat 2σ Vwm Eg |θ= = exp 4π rg2 , (30) √ kB T rg 2π mwm kB T which is identical to the evaporation coefficient in homogeneous condensation.11 When the surface is planar, i.e., the radius of solid particle R → ∞, our expression of evaporation coefficient is (31) Eg |R→∞ = Eg (α) R→∞ + Eg (β) R→∞ , Eg (α)|R→∞ = exp
condensation in vapor and the heterogeneous condensation on a planar solid surface. In brief, the gaps for the kinetic model for heterogeneous condensation on an insoluble spherical particle have been overcome by taking two mechanisms of cluster growth and the effect of line tension into account (Eqs. (8), (27), and (23)), and deriving the exact expression for evaporation coefficient (Eqs. (27)–(29)). In this way, a new kinetic model is established and the kinetic studies on the heterogeneous condensation occurring on a spherical solid surface become feasible.
pvsat 2σ Vwm 2π rg2 (1 − cos θ ), √ kB T rg 2πmwm kB T (32)
2σ Vwm Gdiff sat 2πrg sin θ, δn1 ν exp − Eg (β)|R→∞ = exp kB T rg kB T (33) which is the same as the evaporation coefficient in heterogeneous condensation on a planar solid surface.17 As a result, the obtained expression of evaporation coefficient is not only appropriate for the heterogeneous condensation on an insoluble spherical particle, but also correct in the homogeneous
III. RESULTS AND DISCUSSIONS
In this section, the physical insight of evaporation coefficient is presented at first. Then, the contributions of two mechanisms to heterogeneous condensation including the effect of line tension are evaluated and analysed. As an example of our model applications, the growth rate of the cap shaped droplet on an insoluble spherical particle in the heterogeneous condensation is derived using our model at last. A. Physical insight of evaporation coefficient
The evaporation coefficients have the same form in homogeneous condensation and heterogeneous condensation: 2σ Vwm · Cgsat . (34) Eg = exp kB T rg This identity can be interpreted from the perspective of eq physics. The vapor pressure in the equilibrium state pv (the superscript “eq” denotes the parameters in the equilibrium state hereafter) for a flat water layer is equal to the saturated vapor pressure pvsat . However, the vapor pressure in the equilibrium state for a spherical droplet (with radius rg ) is higher than the saturated vapor pressure, and it satisfies the Kelvin equation:12 2σ Vwm eq pvsat . pv (rg ) = exp (35) kB T rg When the ambient vapor pressure is identical to the vapor pressure in the equilibrium state for a spherical droplet, the eq eq droplet is in the stable state, i.e., Eg = Cg . As mentioned in Sec. II, the evaporation coefficient is supersaturation independent. Therefore, the evaporation coefficient of the droplet (with radius r and water molecules number g) for any supersaturation is identical to that in the stable state: Eg = Cgeq = Cgeq (α) + Cgeq (β)
2σ Vwm sat Cg (α) + Cgsat (β) = exp kB T rg 2σ Vwm Cgsat . = exp kB T rg
(36)
This expression of evaporation coefficient is identical to that in our model. As a result, the same form of evaporation
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4
4
3
3
the condensation coefficient Cg , i.e., the cluster tends to disappear. When g > g∗ , Eg < Cg , i.e., the cluster tends to grow. When g = g∗ , Eg = Cg , i.e., the cluster is in the metastable state. This tendency is identical to the classical nucleation theory’s prediction. Both the identity of evaporation coefficients and the comparison between the evaporation coefficient and the condensation coefficient indicate that our kinetic model is correct in the physical aspect.
11
-1
Coefficients [10 s ]
024708-5
Cg Eg
B. Contributions of two mechanisms to heterogeneous condensation
g=g* 2
50
100
2 200
150
g FIG. 3. The evaporation coefficient Eg (solid line) and the condensation coefficient Cg (dashed line) as functions of the number of molecules g in cluster. R = 1μm, m = 0.5, T = 293.15 K, S = 1.8, g∗ = 90, δ = 0.32 nm, ν = 1013 s−1 , Gdiff = 2.9 × 10−21 J molecule−1 , and Gdes = 2.9 × 10−20 J molecule−1 .
coefficients in heterogeneous condensation and homogeneous condensation can be ascribed to the Kelvin effect. We can also compare the evaporation coefficient with the condensation coefficient in heterogeneous condensation on an insoluble spherical particle. The calculating parameters are listed in Table I. We first use the classical nucleation theory to calculate the critical size r∗ and g∗ : ∂W = 0, (37) ∂r r=r ∗ r∗ =
2σ Vwm , kB T ln S
(38)
Vc (r ∗ ) . Vwm
(39)
g∗ =
In this condition, g∗ = 90. As shown in Fig. 3, when the cluster size g is less than the critical size g∗ , the evaporation coefficient Eg is larger than
Some literatures12, 30, 31 reported that the surface diffusion mechanism is more effective and has a higher contribution to the condensation coefficient than that caused by the direct addition mechanism by a factor between 102 and 103 . However, there are no systematic evaluation and analysis of the contributions of two mechanisms to heterogeneous condensation including the effect of line tension. In order to estimate the relative importance (RI) of two mechanisms of cluster growth, we define a dimensionless number RI as RI =
Cg (β) . Cg (α)
(40)
We evaluate the variation of RI with different numbers of water molecules in cluster g and different macroscopic wetting degrees m∞ . The parameters are set as follows: the number of water molecules in cluster g ranges from 50 to 200, the macroscopic wetting degree m∞ varies from −0.9 to 0.9. For other parameters (the radius of solid particle, the parameters of ambient condition and water molecule), we use the same values as listed in Table I. Figure 4(a) shows that the relative importance of two mechanisms of cluster growth as a function of m∞ and g. The value of RI is always between 102 and 103 , except in some extreme cases (large g and small m∞ ). RI decreases with g, raises with small m∞ (−0.9 ≤ m∞ < 0.5 in our case), and falls with large m∞ (0.5 ≤ m∞ < 0.9 in our case). These tendencies can be ascribed to the geometrical effect as shown in Fig. 4(b).
200
200
RI
80 120
0.4 0.5
160
160
100 140
180
0.6
Lcvs/Scv [109 m-1]
0.7
0.8
0.8 0.9
g
150
g
150
200
1
100
100 220
1.1
240
1.2
260
50 -0.9
-0.6
-0.3
0
0.3
0.6
0.9
50 -0.9
1.3
-0.6
-0.3
0
m∞
m∞
(a)
(b)
0.3
0.6
0.9
FIG. 4. (a) The relative importance (RI) of two mechanisms of cluster growth and (b) the ratio of the perimeter of three-phase contact line to the surface area between cluster and vapor (Lcvs /Scv ) as functions of the macroscopic wetting degree m∞ and the number of water molecules in cluster g. R = 1μm, T = 293.15 K, S = 1.8, δ = 0.32 nm, ν = 1013 s−1 , Gdiff = 2.9 × 10−21 J molecule−1 , and Gdes = 2.9 × 10−20 J molecule−1 .
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J. Chem. Phys. 140, 024708 (2014)
According to Eqs. (9) and (10), we find that the condensation coefficients in direct addition Cg (α) and in surface diffusion Cg (β) have positive correlations with the surface area between cluster and vapor Scv and the perimeter of three-phase contact line Lcvs , respectively. The value of Lcvs /Scv has the same orψ der of magnitude to (rg tan 2g )−1 . Figure 4(b) shows that the value of Lcvs /Scv decreases with g, which is because rg−1 falls when g increases. The tendency of Lcvs /Scv with different m∞ in Fig. 4(b) identical to RI in Fig. 4(a) is determined by two competing mechanisms: one is the effect of cluster radius rg , the other is the effect of half of the central angle of cluster ψ g . When m∞ increases, rg raises and ψ g falls. In the regime of small m∞ (−0.9 ≤ m∞ < 0.5 in our case), the variation of ψ g is primary, which results in the raise of Lcvs /Scv . In the regime of large m∞ (0.5 ≤ m∞ < 0.9 in our case), the variation of rg is dominant and leads to the decrease of Lcvs /Scv . These results suggest that the cluster growth via surface diffusion of adsorbed water molecules on the particle is more important than the direct addition from the vapor. These also show that both the macroscopic wetting degree and the number of water molecules in cluster have significant effects on the contributions of the two mechanisms. A larger contribution of surface diffusion can be found at medium macroscopic wetting degree and smaller size cluster.
C. Growth rate of droplet in heterogeneous condensation
Because the exact expression of the evaporation coefficient has been obtained from our derivation, our kinetic model actually paves the way for further studies of the heterogeneous condensation of vapor on an insoluble spherical particle. As an example of our model applications, the growth rate of the cap shaped droplet on an insoluble spherical particle in the heterogeneous condensation is derived using the new model. The droplet growth depends on an important parameter, the Knudsen number Kn. This dimensionless parameter is defined as the ratio of the mean free path λ of a vapor molecule to the droplet diameter 2r: Kn = λ/2r. For a large Kn, i.e., relatively small droplet, the growth is controlled by vapor kinetic process of vapor molecular collision from the surrounding mixture onto the droplet.6 As a result, the growth rate of single droplet with a large Knudsen number can approximately be given by the difference of the condensation coefficient and the evaporation coefficient.8 For heterogeneous condensation
on an insoluble spherical particle, the growth rate of a droplet is dr dg dr = · dt dg dt het
het
≈ =
het
Vwm − E ) · (C g g het π rg2 (2 − 3 cos ψg + cos3 ψg ) π rg2 (2
Vwm · (S − eKeg ) Cgsat het . 3 − 3 cos ψg + cos ψg ) (41)
It should be noted that clusters are formed by nucleation process and only the cluster whose size is larger than the critical size can grow, which is described by Eq. (2). As a result, Eq. (41) is only an approximation for the case when clusters can be considered as independent. In addition, the effect of line tension can be ignored, because the droplet in this case is large enough. We evaluate the growth rate of droplet dr/dt with different macroscopic wetting degrees m∞ . The parameters are set as follows: the macroscopic wetting degree m∞ varies from −1 to 1, the radius droplet r = 2r∗ (only the droplet larger than the critical size can grow). For other parameters (the radius of solid particle, the parameters of ambient condition and water molecule), we use the same values as listed in Table I. As shown in Fig. 5(a), when m = −1, i.e., the surface of particle is completely hydrophobic, the value of dr/dt in heterogeneous condensation is identical to that in homogeneous condensation, because the heterogeneous condensation can be considered as homogeneous condensation when the surface is completely hydrophobic. When m > −1, the value of dr/dt in heterogeneous condensation is larger than that in homogeneous condensation, because there is an additional contribution to the droplet growth (surface diffusion of adsorbed water molecules on the particle) for heterogeneous condensation compared with the homogeneous counterpart. Figure 5(a) also shows that dr/dt rapidly increases with relatively small m (−1 ≤ m < −0.95), and slightly increases with medium m (−0.95 ≤ m < 0.8), and rapidly increases again with relatively large m (0.8 ≤ m < 1). This tendency is ascribed to the influx of water molecules and the shape of droplet dr/dg. It should be noted that the influx of water molecules is primarily determined by surface diffusion which is dependent on the value of Lcvs , and the radius of droplet with a larger wetting degree increases more quickly when the influx of water
TABLE I. Parameters for the comparison between the evaporation coefficient and the condensation coefficient in heterogeneous condensation on an insoluble spherical particle.
Solid particle Ambient condition Water moleculea
a
Parameter
Symbol
Value
Radius Macroscopic wetting degree Temperature Saturation Mean jump distance Vibration frequency on surface Surface diffusion energy Desorption energy
R (μm) m∞ T (K) S δ (nm) ν (s−1 ) Gdiff (J molecule−1 ) Gdes (J molecule−1 )
1 0.5 293.15 1.8 0.32 1013 2.9 × 10−21 2.9 × 10−20
These data are from Ref. 19.
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Luo et al. 2
Lcvs [nm]
Lcvs dr/dg
10
3
10
2
10
1
10
0
10
-1
-11
10
101
100
-1
-0.5
0
0.5
1
m]
J. Chem. Phys. 140, 024708 (2014)
dr/dg [10
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10-2
m (a)
(b)
FIG. 5. Relations between the macroscopic wetting degree of the particle surface m∞ and (a) the growth rates of droplet dr/dt in heterogeneous condensation (solid line) and in homogeneous condensation (dashed line), (b) the perimeter of three-phase contact line Lcvs (solid line) and the effect of droplet’s shape dr/dg (dashed line). r = 2r∗ , R = 1 μm, T = 293.15 K, S = 1.8, δ = 0.32 nm, ν = 1013 s−1 , Gdiff = 2.9 × 10−21 J molecule−1 , and Gdes = 2.9 × 10−20 J molecule−1 .
molecules is the same (dr/dg increases with wetting degree). As shown in Fig. 5(b), when m is relatively small, the rapid increase of Lcvs improves the surface diffusion of adsorbed water molecules on the particle. When m is medium, Lcvs is almost unchanged, the slight raise of dr/dt is ascribed to the shape of droplet. When m is relatively large, although Lcvs decreases fast, the shape of droplet is much more significant for the droplet growth.
IV. CONCLUSIONS
We have developed a kinetic model to describe the heterogeneous condensation on an insoluble spherical particle. This model considers two mechanisms of cluster growth: direct addition of water molecules from the vapor and surface diffusion of adsorbed water molecules on the particle. Based on the capillarity approximation, we assumed the water cluster as a spherical cap shaped droplet, and obtained the relation between radius and molecule number of the cluster. We also considered the effect of line tension on the heterogeneous condensation. For the first time, we have derived the exact expression of evaporation coefficient for heterogeneous condensation of vapor on an insoluble spherical particle by using the detailed balance. It is confirmed that the expression of evaporation coefficient in our model is also correct in the homogeneous condensation in vapor and the heterogeneous condensation on a planar solid surface. The evaporation coefficients have the same form in homogeneous condensation and heterogeneous condensation, which is ascribed to the Kelvin effect. We also compared the evaporation coefficient with the condensation coefficient to check the coefficient calculated in our model. The results indicate that our kinetic model is correct in physical aspect. The contributions of two mechanisms to heterogeneous condensation including the effect of line tension were evaluated and analysed. It is found that the cluster growth via surface diffusion of adsorbed water molecules on the particle is more
important than the direct addition from the vapor, and both the macroscopic wetting degree and the number of water molecules in cluster have significant effects on the contributions of the two mechanisms. To provide an example of our model applications, we have derived the growth rate of the cap shaped droplet on the insoluble spherical particle in the heterogeneous condensation by our model, which cannot be obtained by previous kinetic model. We also have evaluated the growth rate of droplet with different wetting degrees. It is found that when the surface of particle is completely hydrophobic, the growth rate of droplet in heterogeneous condensation is identical to that in homogeneous condensation. Otherwise, the growth rate of droplet in heterogeneous condensation is larger than that in homogeneous condensation because of an additional contribution to the droplet growth. It is also found that the growth rate of droplet in heterogeneous condensation experiences three regimes (a quick increasing regime, a slight increasing regime, and another rapid increasing regime) when the wetting degree increases. These results indicate that an explicit kinetic model is benefit to the study of heterogeneous condensation on an insoluble spherical particle, and the approximation by using the kinetic model for homogeneous condensation or heterogeneous condensation on a planar solid surface is unsuitable. It is expected that our kinetic model will promote the kinetic studies of heterogeneous condensation on an insoluble spherical particle, such as the start-up process of condensation, the evolution of the number density of droplets, and the influence of vapor depletion, which will also be a part of our future work. In addition, the influences of different material systems on the heterogeneous condensation need to be analysed in further investigation.
ACKNOWLEDGMENTS
This work was supported by the Natural Science Foundation of China (Grant Nos. 11172292 and 10976029), by the
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Strategic Priority Research Program of the Chinese Academy of Sciences (XDB05040400), by the Hefei Physical Science and Technology Center (2012FXCX005), by the CAS Special Grant for Postgraduate Research, and by the USTC Special Grant for Postgraduate Research. APPENDIX: DERIVATION OF THE EXPRESSION OF ngsat / ngsat +1
For a system with a large g (g ≥ 20), Eq. (25) reduces to nsat ∂S 1 ∂r ∂Lcvs g σ . (A1) = exp +τ nsat kB T ∂g ∂r ∂r g+1 First, we calculate the derivatives of the cap shaped cluster’s volume Vc , effective area S , and three-phase contact line’s perimeter Lcvs with respect to r: ∂Vc ∂A = π r 2 (2 − 3A + A3 ) + r 3 (A2 − 1) ∂r ∂r ∂B 3 2 − R (B − 1) , (A2) ∂r ∂A ∂B ∂S = 2π 2r(1 − A) − r 2 + m∞ R 2 , (A3) ∂r ∂r ∂r B ∂B ∂Lcvs = −2πR √ , 2 ∂r 1 − B ∂r
(A4)
A = cos ψg ,
(A5)
B = cos φg .
(A6)
and
Then, differentiating Eqs. (5)–(7), with respect to r yields 2 ∂A ∂m A2 − 1 R =B + , ∂r d ∂r d r 2 ∂m r A2 − 1 ∂B =A + , ∂r d ∂r R d τ ∂m ∂B =− (1 − B 2 )−1.5 . ∂r σR ∂r Differentiating Eq. (3) with respect to g yields ∂Vc ∂r = Vwm . ∂r ∂g
(A7) (A8) (A9)
(A10)
Combining Eqs. (A1) and (A10) yields nsat Vwm σ (∂S /∂r) + τ (∂Lcvs /∂r) g . = exp nsat kB T ∂Vc /∂r g+1
(A11)
Finally, after a great amount of labor, the exact expression sat for nsat g /ng+1 can be obtained by combining Eqs. (A2)–(A4), (A7)–(A9), and (A11): nsat 2σ Vwm g . (A12) = exp nsat kB T rg g+1 1 B.
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