Boundary condition at a gas-liquid interphase
R. Meland1, T. Ytrehus
Department of Applied Mechanics,
Norwegian University of Science and Technology.
Abstract. We have used a molecular dynamics simulation to determine the velocity distributions at
equilibrium for molecules evaporated from the liquid and for molecules incoming from the gas reflected in
the interphase. The evaporated molecules have a larger velocity normal to the interphase than might be
expected, the evaporation distribution resembles a Maxwellian with outward drift-velocity of the order of the
thermal velocity. The distribution for the reflected molecules is similar to a drifting Maxwellian toward the
interphase with a velocity shift of the order of the thermal velocity. Not even for equilibrium can reflection
be interpreted as specular reflection, diffuse reflection or a combination of both, since it would imply in all
cases a Maxwellian with zero drift velocity. We believe significant modifications to the boundary conditions
used in the kinetic theory approach to evaporation and condensation are warranted. We discuss two possible
representations of new boundary conditions.
I
INTRODUCTION
The investigation of the dynamics of a gas during evaporation and condensation is a classical problem of
the kinetic theory of gases [2], there has been a huge number of papers solving the full Boltzmann equation
or an appropriate model equation. With a few exceptions [5], very little has been said about the boundary
conditions at the interphase and their influence on the solution of the Boltzmann equation governing the gas
phase. The usual approach is to assume a Maxwellian distribution for the evaporated molecules and diffuse
or specular reflection, or a combination, for the reflected molecules. Unexpected results from kinetic theory
solutions [1] [3], such as the inverted temperature gradient, have led to some doubts about the validity of the
standard gas-kinetic boundary condition used [4], and attempts have been made to generalize these conditions
[5]Molecular dynamics techniques offer the possibility to study the liquid-gas interphase in detail. In our
simulation, Newton's equations of motion for 1728 argon atoms have been integrated with the Verlet method
for 4 • 10~8s after equilibration. The potential used was Lennard-Jones set to zero beyond 2.5 'molecular
diameters' r$. TQ is the inter-molecular separation where the pair-potential is zero. The dimensions of the box
were 602A x 38A x 38A, periodic boundary conditions were used and the temperature was kept at 85.5 K,
slightly above the triple point for argon. The mean free path AQ in the gas phase, calculated for hard-spheres
with radius equal to TO, was 25 TO, or 82 A. The extent of the gas region was 3 mean free paths, since the system
is in equilibrium the length of the gas phase is not critical. The gas was slightly non-ideal with a compression
factor Z = 0.93 and the mean distance between the atoms in the gas phase was Sg = 1/^/nJ = 4.8 TQ.
Figure 1 shows a plot of the average potential energy near the interphase. A "gas boundary" and a "liquid
boundary" are introduced to divide the system into liquid, interphase and gas regions. The interphase is
approximately 6ro thick. An atom is considered as evaporated if it originated in the liquid region and crosses
the interphase into the gas region, condensed if it originated in the gas region, crosses the interphase and enters
the liquid region and reflected if it originated in the gas region, crosses into the interphase and moves back to
the gas region without having been in the liquid region. Reflecting gas molecules do not penetrate far into the
interphase, hence the position of the liquid boundary is not critical. The gas boundary is more critical though,
the cross-section of the molecules is velocity dependent, molecules with small velocity have larger cross-section
and are more susceptible for collisions and reversal of motion. We have at this stage no good definition of
Roar@Meland.as
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
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u/\ut\
LIQUID
/INTERPHASE
-1
10
20
xlrn
FIGURE 1. Interphase
the gas boundary, we use the point where the graph of U starts to deviate from the equilibrium gas value.
However, qualitatively the results do not change when the gas boundary is varied.
II
SIMULATION RESULTS
The probability distribution functions at the interphase have been found by sampling the velocity of the
molecules in a control volume of thickness ro/4 = Ao/100 at the gas boundary. The probability distributions
can also be found by transforming the differential fluxes, but this was not used due to numerical problems
when dividing by the velocity cx, and besides, to find the number densities and the complete distribution
functions there is no other alternative than using a control volume. In figure 2 the probability distribution F+
for the normal component of velocity for the escaping particles at the gas boundary has been compared with
a Maxwellian at the system temperature, normalized to 1 on cx > 0. The velocity is plotted as speed ratio sx,
F+
0
0
1
s
2
FIGURE 2. Probability distribution for escaping particles
defined as sx = —?==.
Also the distributions for the tangential components of velocity can be shown to be
V SSjFujT
well described by Maxwellians.
Evaporated and reflected molecules have been tagged. In figure 3 the probability distributions for the normal
component of velocity for the evaporated and reflected atoms are compared with the standard Maxwellian.
All distributions are normalized to 1 on cx > 0. The evaporated atoms have a larger velocity than might be
anticipated, the evaporation distribution resembles the cx > 0 part of a drifting Maxwellian. The tangential
components of the velocity of the evaporated atoms can be shown to be the standard Maxwellian. The reflected
atoms have a lower velocity than might be expected. The distribution is similar to a drifting Maxwellian
toward the interphase with a velocity shift of the order of the thermal velocity. The tangential components
of the velocity of the reflected atoms can be shown to be the standard Maxwellian. Number densities for the
evaporated and reflected molecules have been calculated, ne/n+ = 0.68 and nr/n+ = 0.32. n+ = ne +n r , i.e.
the density of the molecules escaping from the interphase. The probability distributions in figure 2 and 3 are
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0
0
1
2
s
X
FIGURE 3. Evaporation and reflection distributions
related by
f+=n+F+ =
If our probability distributions are transformed to differential fluxes, the results are similar to calculations
of the differential fluxes for evaporated and reflected argon atoms made by Tsuruta [6], he also observed the
velocity shifts. Tsuruta used a different definition of evaporation and condensation though.
Ill
DISCUSSION
The deviation from a half-Maxwellian with zero drift- velocity for the distribution function for molecules
reflected in the interphase makes it evident that not even for equilibrium can reflection be interpreted as
specular reflection, diffuse reflection or a combination, since it would imply in all cases a Maxwellian with zero
drift velocity.
As a consequence of our findings, the usual Maxwellian boundary condition
/+ = aensFM 4- (1 - &c)nrFM
(1)
with evaporation and condensation coefficients
2L
(2)
1F
seem inappropriate as a boundary condition for the Boltzmann equation for evaporation and condensation. The
expressions above for ae and ac are consistent with the more basic definitions in terms of velocity distributions
fe = aensFM
fr = (1 - a
M
(3)
Eq. 2 follows from eq. 3 if the usual assumption of constant coefficients is made. ns is the saturation density,
TL is the temperature at the liquid boundary, J~ is the incoming flux from the gas, nr = J~I\l^S Je ls
the evaporation flux, Jc is the condensation flux and FM is a half-Maxwellian with temperature TL and zero
drift-velocity.
It has been shown by Tsuruta [6] that the condensation probability is velocity dependent, it depends upon
the normal component of velocity at the gas boundary, which is also evident from our simulation, see figure 4.
Hence y=- should be interpreted as the average condensation probability ac.
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0.5
0
I
sX
2
FIGURE 4. Condensation probability
IV
NEW BOUNDARY CONDITIONS
We believe the boundary condition for the Boltzmann equation applied to evaporation and reflection should
be modified, both the number densities and the probability distributions should be accurately represented. The
definition in eq. 3 of the coefficients can be retained, i.e. the usual definition with respect to a Maxwellian, but
the requirement that the coefficients are constants must be abandoned. This is the approach used by Tsuruta
[6]. Then the coefficients will be dependent upon cx
= neFe = ae(cx)ns(TL)FM
f = nrFr = [1- ar(cx)} nrFM
(4)
The redefined coefficients take into account the deviation from a Maxwellian with no drift-velocity for the
cx probability distribution and gives the correct density, cy and cz distributions are not affected. Lower-case
indexes indicates reference quantities with the distribution normalized to 1 on -oo < cx < oo, upper-case
physical quantities where the distribution has been normalized to 1 on cx > 0. Note that nr and nr are not
the same quantity. The reference density nr has not been specified. nr ought to be defined from the reflection
flux to ensure that it scales appropriately in nonequilibrium
/
cx[l - or
= Jr =
c x >0
I cx
-
dcx
(5)
c x <0
0"c(|cx|) is the condensation probability in figure 4. nr can not be calculated without knowledge of <Jr(cx)
and vice versa. We have used the term 'reflection' coefficient since it is not immediately apparent that the
reflection coefficient in general is equal to the condensation probability, i.e. condensation coefficient. However,
in equilibrium
= [ve(cx)ns(TL) + (1 - crr(cx)) nr] FM = n s FjM
It is then consistent to set nr = ns, hence cre(cx) = (Jr(cx) in equilibrium. In equilibrium the evaporation and
condensation fluxes are equal.
/ cxae(cx)nsFMdcx = I ac(\cx\] \cx\n~F~dcx
c x >0
c x <0
Also nsFM = n~F~, and ac(|Cg.|) = <7e(cx) = &r(cx) = &(cx) in equilibrium.
In nonequilibrium, which is the region of practical interest, the definition of velocity dependent coefficients
is not so straightforward. The problem is the reflection coefficient and the reflection density nr, which are
related by eq. 4 and 5. Tsuruta assumes that the evaporation, condensation and reflection coefficients are
equal, and identical to the equilibrium expression with the same liquid temperature TL . nr is hence implicitly
defined from the flux condition in eq. 5 with ar known. Tsuruta asserts that the probability distributions for
evaporation and reflection are independent of the degree of nonequilibrium, as is the evaporation density ne.
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The only quantity in the distribution functions in eq. 4 influenced by nonequilibrium is nr, nr ^ ns give rise
to a non-Maxwellian distribution function at the interphase :
/+ = [a(cx}(ns -nr} + nr]FM
An other possible representation of the boundary condition is a suggestion of Ytrehus. The evaporation
and reflection distributions Fe and Fr of figure 3 might be used as base-functions, with redefined evaporation
coefficient &fe and a reflection coefficient <J!r.
fe =neFe = afensFe
1
e
=
ne
__
ns
f = nrFr = (1 - a'r) rirFr
'
r
=
rtr
1 _ __
nfr
The reference density nfr to be used for the reflection coefficient is not immediately apparent. We would like
to have a reference value which is proportional to the reflection flux, so afr is more or less independent of the
degree of nonequilibrium and only depends on the liquid temperature T&. A flux condition gives
, _ (1 -a G )c x <o_______
c x >0
nfr can not be calculated without knowledge of afr and vice versa. nfr = J~ / \j^j^ might be used, but it
scales as the incoming flux, and not necessary as the reflection flux, it might be possible that the average
condensation probability depends upon the degree of nonequilibrium. Note that the coefficients afe and afr will
not be equal in equilibrium. If the simplifying assumptions introduced by Tsuruta [6] are correct, cr^, Fe and
Fr shall be independent of the degree of nonequilibrium. From the plots of Fe and Fr in figure 3 it is seen that
the distributions resemble shifted Maxwellians, a 'temperature' and 'drift-velocity' could be defined for both
distributions and used to represent Fe and F**. For instance Fe in the plot can be associated with a positive
drift velocity —?==
« 0.55 and a temperature of approximately 60 K. But it must be noted that a sum of two
v 2fil
shifted Maxwellians can not produce exactly the equilibrium Maxwellian.
v CONCLUSION
Our equilibrium simulation of the distribution function outside the interphase indicates that the boundary
condition traditionally used in kinetic theory modelling of evaporation and condensation is not accurate.
Extensive nonequilibrium simulations for a variety of conditions are necessary before a better model for the
boundary condition can be established.
REFERENCES
1. K. Aoki and C. Cercignani, Evaporation and condensation on two parallel plates at finite Reynolds numbers. Phys.
Fluids, 26, 1163-1168. 1983.
2. T. Ytrehus, Molecular Flow Effects in Evaporation and Condensation at Interfaces, Multiphase Science and Technology, Vol 9, No 3, 205-327. 1997
3. T. Ytrehus, The Inverted Temperature Gradient in Condensation - Revisited, Proceedings of 21. International
Symposium on Rarefied Gas Dynamics, 471-478. 1998.
4. L.D Koffman, M.S Plesset & L. Lees, Theory of Evaporation and Condensation, Phys. Fluids 27, 876-880. 1984.
5. C. Cercignani, W. Fiszdon & A. Frezzotti, The paradox of the inverted temperature profiles between an evaporating
and condensing surface. Phys. Fluids, 28, 3237-3240. 1985.
6. T. Tsuruta, H. Tanaka & T. Masuoka, Condensation/evaporation coefficient and velocity distributions at liquidvapor interface, International Journal of Heat and Mass Trnnsfer, 42, 4107-4116, 1999.
Financial support from the Norwegian Council of Research is gratefully acknowledged.
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