Molecular Boundary Conditions and Temperature Jump
at Liquid-Vapor Interface
Takaharu Tsuruta
Department of Mechanical Engineering, Kyushu Institute of Technology, Kitakyushu 804-8550, Japan
Abstract. This paper presents an important role of new molecular boundary conditions at a liquid-vapor interface for
temperature profiles during condensation and evaporation. This is closely related the problem of the inverted temperature
profiles. The molecular boundary conditions for a kinetic theory have been obtained by conducting molecular dynamics
(MD) simulations. Considering behaviors of evaporating and reflecting molecules at the liquid surface as well as the
condensing probability, the velocity distributions of the evaporating and the reflecting molecules have been expressed
with use of a condensation/evaporation coefficient. Here, the condensation/evaporation coefficient is considered to be a
function of surface-normal component of molecular translation energy. The direct simulation Monte Carlo (DSMC)
method is used to examine the temperature profiles based on the molecular boundary conditions. Also, the ordinary
conditions utilizing the uniform value of the condensation coefficient and the Maxwellian velocity distributions for all
kinds of molecules were used for a comparison. The results show that the dependence of the condensation/evaporation
coefficient on kinetic energy plays an important role in the temperature profiles. For some cases where the ordinary
boundary conditions result in the inverted temperature profiles, the new molecular boundary conditions do not show it.
INTRODUCTION
Interphase mass transfer between liquid and vapor phases has been expressed based on the kinetic theory of gases.
Net evaporation or condensation rate can be obtained from the difference of fluxes between molecules moving outward
from and toward the liquid surface, provided that the velocity distribution functions of vapor molecules in the vicinity
of the liquid surface can be given correctly. Also, macroscopic parameters, condensation coefficient and evaporation
coefficient, play an important role to describe the surface characteristics in the expression on mass transfer rate. A huge
number of studies have been done to obtain the mass transfer rate by solving the Boltzmann equation together with the
coefficients [1]. It is, however, important to remember that several assumptions have been introduced in all of these
studies. That is, the condensation coefficient and evaporation coefficient were assumed to be equal even for nonequilibrium conditions and an uniform value was used for all kinds of molecules irrespective of their kinetic motion.
And the Maxwellian distribution was assumed for the velocity distribution of evaporated and reflected molecules from
the liquid surface. These assumptions need detailed discussions.
Recent progress of molecular dynamic (MD) simulation enables us to examine the validity of these assumptions.
According to our MD studies on the condensation and evaporation at liquid-vapor interface for argon [2] and water [3],
the condensation probability of incident molecule depends on its translation motion as well as the liquid surface temCP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
988
perature. This indicates that the condensation coefficient defined as the averaged condensation probability depends on
the macroscopic vapor velocity, i.e. net mass transfer rate. Therefore, the ordinary method with use of a same value
both for the condensation coefficient and for the evaporation coefficient results in misunderstandings for the mass
transfer rate. Of course, the dependency of the condensing and evaporating characteristics on the spectrum of velocity
distributions should affect the temperature profiles near the interface.
In the present paper, we summarize these molecular-scale informations as boundary conditions for the kinetic theory.
For the new boundary conditions, the condensation flow is analyzed by using the direct simulation Monte Carlo
(DSMC) method and the temperature jump at the interface is going to be discussed.
NEW BOUNDARY CONDITIONS FOR KINETIC THEORY
Condensation Coefficient Based on MD and Transition State Theory
In our MD study [2], the condensation coefficient and the velocity distributions of the evaporating and the reflecting
molecules were measured under the thermal equilibrium conditions. Let me summarize the condensation coefficient
as a first boundary condition at the liquid-vapor interface.
In the ordinary expression of the interphase mass transfer rate, the condensation coefficient has been considered to
have a uniform value irrespective of kinetic motion of vapor molecules. However, we are proposing a new type of
formulation on the condensation coefficient based on the MD results. The equation is a function of the normal component of translation energy of incident molecule and the liquid-surface temperature:

 mvz2  
σ c = α 1 − β exp −

 2 kTc  

(1)
where vz is the molecular velocity component perpendicular to the surface, k the Boltzmann constant, m the mass of
molecule and Tc is the temperature of liquid surface. The parameters α and β are constants describing the characteristics of the surface structure and their values are shown in TABLE 1 for argon as well as water. In this stage, α and β
are empirical values in order to fit the MD data by Equation (1). The equation indicates that the condensation coefficient increases with the translation energy and decreases with increasing surface temperature. The physics is very
simple. The incident molecules with larger translation energy in the direction perpendicular to the surface can push
surface molecules away and they can condense. On the other hand, the molecules with lower energy can be easily
TABLE 1. Parameters (α, β ) obtained by MD simulations and averaged value of condensation coefficient.
T [K]
α
Argon (L-J model)
84
0.971
90
0.935
102
0.923
120
0.750
130
0.685
Water (SPC/E model)
450
0.967
474
0.961
500
0.808
515
0.750
550
0.536
989
β
σc
0.086
0.220
0.299
0.388
0.554
0.929
0.832
0.785
0.605
0.495
0.280
0.337
0.640
0.650
0.932
0.832
0.799
0.549
0.487
0.286
reflected by the surface molecules. The surface molecules raise the reflection probability at higher temperature.
Under an equilibrium condition, the number flux of the incident molecules are given by the Maxwellian velocity
distribution. Then, the averaged value of condensation coefficient is obtained as:
σc =
1
(kTc
1/ 2
2πm)
∫
∞
0
 m 
σ c vz 

 2πkTc 
1/ 2
 mvz2 
 β
exp −
 dv z = α 1 − 2 
 2 kTc 
(2)
The values are included in TABLE 1. We can also found that the condensing molecules have larger mean translation
energy than the Maxwellian. The difference is expressed by:
E0 =
1
σc
∫
∞
0
σc

 4 − β
 mvz2 
1 2 m 
mvz 
− 1kTc
v z exp −
dv z − kTc = 


2
 kTc 
 2(2 − β ) 
 2 kTc 
(3)
This energy difference corresponds to activation energy for condensation in the transition state theory. We have
developed the transition state theory by considering 3-dimensional characteristics of liquid surface and obtained the
averaged value of condensation coefficient [4]:

 E 
Vl 
σ c = 1 − 3 g  exp − 0 


V 
 kTc 

(4)
where V l and V g are specific volumes of liquid and vapor, respectively. A comparison between Eqs. (2) and (4) turns
the empirical parameters α and β into the theoretical expressions. That is, they are determined as follows:
 E 
α = exp − 0  ,
 kTc 
β = 2⋅3
Vl
Vg
(5)
Since we have the specific volumes for most substances, we are able to fix the condensation coefficient of Eq. (1) as the
condensation probability of incident molecules with use of Eqs. (3) and (5).
Velocity Distributions of Evaporating and Reflecting Molecules
In the thermal equilibrium situation, the condensation coefficient should coincide with the evaporation coefficient,
therefore the characteristics of evaporating molecules could be described as a function of translation motion in a
similar way to the case of condensation coefficient. To ensure this, the escaping molecules from the liquid surface
were examined in the equilibrium MD simulations. Since the escaping molecules consist of the evaporating and
reflected molecules, the number of evaporating molecules was estimated by subtracting the reflected molecules. Here,
three components of velocity, v x , v y (tangential) and vz (normal) are considered separately. The distribution density
functions of velocity are shown in Fig. 1 for the normal component vz of argon at 120K. We can compare the velocity
distribution densities for the evaporating, reflecting and all leaving molecules with the Maxwellian. For the tangential
components of velocity, v x and v y , the distribution functions were consistent with the Maxwellian and there was no
marked difference between them. However, we can see a systematic difference in the velocity distribution for the
normal component. That is, although the velocity distribution for total molecules is well expressed by the Maxwellian
distribution, yet the evaporating and reflected molecules show different distributions. For the evaporating molecules,
the density function is larger than the Maxwellian in the large velocity region but it is smaller in the low velocity
region. On the contrary, the reflected molecules show the higher density than the Maxwellian in the smaller velocity
region. In other words, most of leaving molecules with large normal velocity are the evaporating molecules but those
990
0.006
All Leaving Molecules
Evaporated
Reflected
Maxwellian
Eq.(8)
Eq.(9)
Argon
Tc=120K
0.005
Fz
0.004
0.003
0.002
0.001
0
0
100
200
300
400
500
Vz [m/s]
600
700
FIGURE 1. MD informations on velocity distribution of normal component for all leaving molecules from liquid surface together
with those for evaporating and reflecting molecules.
of leaving molecules with smaller velocity are reflected molecules. The large density of evaporating molecules in the
higher velocity region corresponds to the larger condensation probability of incident molecules. The Maxwellian
distribution for all leaving molecules is composed of both the distribution functions for the evaporating and reflecting
molecules. Utilizing the condensation coefficient in the form of Eq. (1) as the evaporation coefficient, we shall obtain
the following distribution functions for the evaporating and the reflecting molecules, respectively:
 m 
f e = σ c ⋅ n

 2π ⋅ kTc 
1/ 2
 m v2 
exp −
 ,
 2 kTc 
 m 
fr = (1 − σ c ) ⋅ n

 2π ⋅ kTc 
1/ 2
 m v2 
exp −

 2 kTc 
(6), (7)
where v 2 = v x2 + v y2 + vz2 . The density functions of velocity distribution, Fe and Fr , are obtained by normalizing their
velocity distribution functions. The densities for the normal component shown in Fig.1 are written as:
{
}
Fe,z =
1 − β exp − mvz2 /(2 kTc )  m 
 mvz2 
v z exp −



1− β 2
 kTc 
 2 kTc 
Fr ,z =
1 − α + αβ exp − mvz2 /(2 kTc )  m 
 mvz2 
v z exp −



1 − α (1 − β 2)
 kTc 
 2 kTc 
{
}
(8)
(9)
Figure 1 includes these distribution functions, and the comparison shows that these equations can express the behaviors of evaporation and reflection at the liquid surface.
DSMC ANALYSIS OF CONDENSATION FLOW
Simulation Method
The direct simulation Monte Carlo (DSMC) method [5] was used for the condensation problem in order to obtain
information about the temperature profile close to the liquid surface. The simulation system is depicted in Fig. 2,
where one dimensional condensing flow of argon molecule along z-axis is considered under the non-equilibrium con-
991
Tc
Pc=Psat(Tc)
Tc < Tv
Tv
Pv=Psat(Tv)
10λv
Ne
(1−σc)Nin
σcNin
Uv
Nin
O
Condensing Surface
z
Vapor
FIGURE 2. DSMC simulation system for condensing flow.
dition. The condensing surface and the bulk vapor phase are set at the saturation states at ( Pc , Tc ) and ( Pv , Tv ),
respectively. The calculation region is set within 10 times the mean free path of the bulk vapor λ v from the condensing
surface and is divided into 30 cells. Fresh molecules flow in the calculation region from z=10 λ v , those are sampled
from the Maxwellian distribution at temperature Tv and mean velocity U v due to condensation. The number of sample
molecules in a standard cell is 1000. At the condensing surface, the judgement of condensation is made based on the
condensation coefficient of Eq. (1). For the reflected molecules, new normal velocity vz is given in accordance with
the density function of Eq. (8), while tangential velocities v x and v y are determined by the Maxwellian function. In the
similar way, the velocities of evaporated molecule are given by Eq. (9) and Maxwellian. For the collisions between the
molecules, the rigid sphere model and the maximum collision-number scheme are used for argon. The calculation is
carried out in the dimensionless form using the mean free path λ v , the most probable speed 2RTv and the mean free
time λ v 2 RTv as the characteristic length, velocity and time scales, respectively. The time increment is set to about
10-times smaller than the mean free time and 80,000 steps calculation is performed to obtain the time-averaged results.
Results of DSMC Analysis
Condensation Rate
The kinetic theory gives the condensation mass flux for near-equilibrium condition by the following equation:
ṁ = ζ
Pv − Pc
2πRTv
(10)
where ζ is a function of the condensation coefficient as shown by Schrage [6]. In the present study, the coefficient ζ
was obtained from the numerical results of the condensation rate. Figure 3 shows the coefficient ζ as a function of the
supersaturation S = Pv Pc for some condensation coefficients of Eq. (1). We can find that the coefficient ζ increases
with increasing supersaturation depending on the condensation coefficient. The lines in the figure denote the present
results for the case of β=0. It is interesting to note that the results for the new boundary conditions at Tc = 102 K (i.e.
α=0.923 and β=0.299) are on the line for α=0.85 and β=0. This means that we will get the different value of condensation coefficient if we do not consider the dependency of condensation coefficient on the kinetic energy of incident
molecules. And if we use the averaged value of the condensation coefficient obtained by assuming the Maxwellian
distribution for the incident molecules, σ c = α (1 − β 2) = 0.785 , we will estimate the smaller condensation rate. Therefore, the dependency of condensation coefficient on the incident kinetic energy should be taken into account in the
condensation and evaporation phenomena.
992
2.4
α=1, β= 0
Coefficient
ζ
2.2
α=0.95, β= 0
α=0.9, β= 0
2
1.8
α=0.85, β= 0
1.6
α=0.8, β= 0
1.4
α=0.971,
α=0.935,
α=0.923,
α=0.785,
1.2
1
β=0.086
β=0.220
β=0.299
β= 0
0.8
1
2
3
4
5
6
7
8
Supersaturation, S
FIGURE 3. DSMC results of coefficient ζ for interphase mass transfer rate.
Temperature Jump at the Surface
The typical profiles of the temperature are shown in Fig.4 for the case of surface temperature at 102 K. Here, the
following dimensionless temperature is used:
θ = (T − Tc ) ∆T = (T − Tc ) (Tv − Tc )
(11)
For the new boundary conditions obtained by the molecular dynamics simulation, α=0.923 and β=0.299, the tempera-
Dimensionless Temperature, θ
ture decreases gradually as approaching the condensing surface and the figure shows temperature jumps at the condensing surface. It is found that the temperature jumps are almost the same as around 60 % of the total temperature
difference for all supersaturations. The figure also includes another two results for β=0. In the case of α=1.0, the
result shows a "inverted temperature profile" accompanied by a superheated state in the vicinity of the condensing
surface. In the case of α=0.85 and β=0, where the coefficient ζ is close to that for α=0.923 and β=0.299 as shown in
Fig.3, the larger temperature jump at the condensing surface and the different temperature profile are seen in Fig. 4.
These results indicate that the parameter β describing the effect of kinetic energy of molecules on the condensation/
evaporation coefficient is important to obtain the precise information on the temperature profiles.
1.20
1.00
0.80
0.60
α=0.923, β=0.299
0.40
S=1.155,
S=1.333,
S=1.737,
S=2.236,
S=1.737, ∆T = 8 K
α=0.850, β= 0
α=1.0, β= 0
0.20
∆T = 2 K
∆T = 4 K
∆T = 8 K
∆T = 1 2 K
0.00
0
2
4
6
8
10
Dimensionless Distance from Condensing Surface, z/λv
FIGURE 4. Temperature profiles at the surface .
993
0.7
0.7
Condensing
Surface
Bulk Vapor
0.6
0.5
T c=102K
α =1.0, β=0.0
S=1.737, ∆ T=8K
Condensing
Surface
Bulk Vapor
0.6
0.5
T c=102K
α =0.85, β=0.0
S=1.737, ∆ T=8K
0.4
Fz
Fz
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
-3
-2
-1
0
Vz* = Vz
1
2
-3
3
-2
2 RTv
-1
0
1
Vz* = Vz
2 RTv
(a)
2
3
(b)
0.7
T c =102K
α =0.923, β=0.299
S=1.737, ∆ T=8K
Condensing
Surface
Bulk Vapor
0.6
0.5
Fz
0.4
0.3
0.2
0.1
0
-3
-2
-1
0
1
2
3
Vz* = Vz 2 RTv
(c)
FIGURE 5. Comparison of molecular velocity distributions between artificial cases and practical (MD) case.
By considering the molecular velocity distributions near the interface, the reason for this difference of the temperature jump becomes clear. Figure 5 shows the comparison of molecular velocity distributions between three kinds of
boundary conditions in Fig. 4. The surface subcooling ∆T is the same for all cases but the cases (a) and (b) are for the
artificial condensation coefficients neglecting the dependency of condensation/evaporation coefficient on the translational energy, β=0. In the case (a) incident molecules condense perfectly and there is no reflecting molecule. So we
can see lower density at the small velocity region in the positive velocity side than that for the bulk vapor phase. This
results in the larger temperature jump at the interface. This is due to the lack of reflecting molecules. Even in the case
(b), the density of slower molecule is small. The reason is considered because the reflecting molecules were given by
the Maxwellian distribution function. In the case (c), since the velocity distribution having a peak at slower velocity
region than the Maxwellian was used for the reflecting molecules according to the MD simulation, almost similar
velocity distribution density to the bulk vapor has been obtained as shown in Fig. 5 (c). It is understood from Fig. 5
that the reflecting molecules are compensating the velocity deficit in a low-speed domain. Therefore, in order to have
true information on the temperature jump at the interface, we have to apply the correct molecular boundary conditions.
Especially, in a discussion about the inverted temperature profile it is indispensable to use our new molecular bound-
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ary conditions presented in this paper. Following this consideration we have examined a criterion for the inverted
temperature profile by using the irreversible thermodynamics [7].
CONCLUSIONS
The condensation and evaporation behaviors at the liquid-vapor interface have been studied using the molecular
dynamics method and the DSMC simulation. The effects of translational motion on the condensation and evaporation
coefficients were investigated to obtain the boundary conditions for the interphase mass transfer rate. The following
conclusions have been derived:
1. The condensation coefficient increases with the normal component of translational energy of incident molecule.
The molecules with the large translational energy can condense easily. At elevated temperatures, the condensation
coefficient decreases due to increasing reflection by the surface molecules. These characteristics are able to expressed
theoretically by Eq. (1) together with Eqs. (3) and (5).
2. The velocity distribution of evaporating molecules shows some differences from the Maxwellian. By considering
that the evaporation coefficient is varies with the surface-normal component of the translational energy in the same
way as the condensation coefficient, the velocity distribution density of evaporating molecules is given by Eq. (8) for
their normal component. The tangential component is the same as the Maxwellian.
3. The distribution density of the normal component of reflecting molecules is expressed by Eq. (9). The tangential
component is also as same as the Maxwellian.
4. The DSMC analyses show some important roles of new molecular boundary conditions in the mass transfer rate
and the temperature jump at the interface. It is impossible to neglect the dependency of the condensation/evaporation
coefficient on the normal component of molecular translation energy.
ACKNOWLEDGMENTS
The author wish to express his appreciation to Dr. Gyoko Nagayama and Dr. Hiroyuki Tanaka for their excellent
works on the molecular dynamics simulation and the direct simulation Monte Carlo analysis.
REFERENCES
1. Ytrehus, T., Multiphase Science and Technology 9-3, 205-327 (1997).
2. Tsuruta, T., Tanaka, H., and Masuoka, T., Int. J. Heat & Mass Transf. 42-22, 4107-4116 (1999).
3. Tsuruta, T., and Nagayama, G., Trans. JSME 68-671, in press (2002).
4. Nagayama, G. and Tsuruta, T., Trans. JSME 67-656, 1041-1048 (2001), submitted English translation in J. Chem. Phys.
5. Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994.
6. Schrage, R.W., A Theoretical Study of Interphase Mass Transfer, Columbia Univ. Press, New York, 1953.
7. Kjelstrup, S., Tsuruta, T., and Bedeaux, D., J. Colloid & Interface Sci. in press.
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