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SPECIAL FEATURE: PERSPECTIVE
Mass and heat transfer between evaporation and
condensation surfaces: Atomistic simulation and
solution of Boltzmann kinetic equation
Vasily V. Zhakhovsky (Василий Жаховский)
a,1
, Alexei P. Kryukov
b
, Vladimir Yu. Levashov
b,c
,
Irina N. Shishkova
b
, and Sergey I. Anisimov
d
Edited by William A. Goddard III, California Institute of Technology, Pasadena, CA, and approved March 21, 2018 (received for review
December 25, 2017)
Boundary conditions required for numerical solution of the Boltzmann kinetic equation (BKE) for mass/heat
transfer between evaporation and condensation surfaces are analyzed by comparison of BKE results with
molecular dynamics (MD) simulations. Lennard–Jones potential with parameters corresponding to solid
argon is used to simulate evaporation from the hot side, nonequilibrium vapor flow with a Knudsen
number of about 0.02, and condensation on the cold side of the condensed phase. The equilibrium density
of vapor obtained in MD simulation of phase coexistence is used in BKE calculations for consistency of BKE
results with MD data. The collision cross-section is also adjusted to provide a thermal flux in vapor identical
to that in MD. Our MD simulations of evaporation toward a nonreflective absorbing boundary show that
the velocity distribution function (VDF) of evaporated atoms has the nearly semi-Maxwellian shape be-
cause the binding energy of atoms evaporated from the interphase layer between bulk phase and vapor is
much smaller than the cohesive energy in the condensed phase. Indeed, the calculated temperature and
density profiles within the interphase layer indicate that the averaged kinetic energy of atoms remains
near-constant with decreasing density almost until the interphase edge. Using consistent BKE and MD
methods, the profiles of gas density, mass velocity, and temperatures together with VDFs in a gap of many
mean free paths between the evaporation and condensation surfaces are obtained and compared. We
demonstrate that the best fit of BKE results with MD simulations can be achieved with the evaporation and
condensation coefficients both close to unity.
evaporation
|
condensation
|
Boltzmann kinetic equation
|
molecular dynamics
Evaporation and condensation can be realized in
different natural phenomena and technologies. A
peculiarity of these processes is the coupled mass
and heat transfer from evaporation to condensation
surface. Nowadays, a correct description of the trans-
port processes across the interfacial surfaces is required
for the development of new advanced technologies
and solution of the known engineering problems.
Among them, the problem of removing heat from
space vehicles, developing technologies based on the
interaction of matter in the form of cryogenic corpus-
cular targets with high-energy beams, evaporation of
droplets on superhydrophobic surfaces using liquid
droplets as the molecular concentrators of ultradilute
solutions, the development of effective vacuum drying
methods—all of these tasks are inextricably linked with
the solution of the evaporation–condensation problem.
The main goal in considering the coupled mass/heat
transfer is an accurate evaluation of the masses of
evaporated or condensed material. Determination of
these quantities is important because a significant heat
can be diverted from the interfacial surface, which, as a
consequence, leads to cooling of the condensed phase.
For example, when considering the evaporation of a
liquid droplet placed in a steam–gas mixture, the inac-
curacies in calculating the evaporation rate can lead to
a
Center for Fundamental and Applied Research,Dukhov Research Instituteof Automatics, Moscow127055, Russia;
b
Departmentof Low Temperatures,
Moscow Power Engineering Institute, Moscow 111250, Russia;
c
Institute of Mechanics, Lomonosov Moscow State University, Moscow 119192,
Russia; and
d
Landau Institute for Theoretical Physics, Russian Academy of Science, Chernogolovka 142432, Russia
Author contributions: S.I.A. designed research; V.V.Z., A.P.K., V.Y.L., and I.N.S. performed research; V.V.Z., A.P.K., V.Y.L., and S.I.A. analyzed data;
and V.V.Z., A.P.K., and V.Y.L. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
Published under the PNAS license.
1
To whom correspondence should be addressed. Email: 6asi1z@gmail.com.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714503115/-/DCSupplemental.
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SPECIAL FEATURE:
PERSPECTIVE
errors in determining the pattern of temperature changes at the in-
terfacial surface, which may result in inaccurate forecasts of the drop-
let temperature and a complete evaporation time.
As noted in ref. 1, despite the simplicity of the formulation of
the evaporation–condensation problem, its solution encounters
certain difficulties in the general case. Traditionally, it is assumed
that the heat coming to the interphase boundary is spent on
evaporation and heating of the particles, and the resulting vapor
is diverted from the evaporation surface by diffusion. It is believed
that the concentration of the evaporated gas near the interphase
boundary is equal to the equilibrium vapor concentration. This
assumption was first used in ref. 2 to consider evaporation of a
spherical drop. However, this is not the case, since the vapor near
the interfacial surface will be saturated only if a diffusion rate of
gas escape is lower than an arrival rate of molecules from the
interphase boundary. Ref. 3 showed the existence of a concentra-
tion jump near the interphase boundary. One of the refinements
of the evaporation–condensation theory can be achieved by in-
voking the kinetics of interaction of vapor molecules with the
surface of the liquid phase and also with each other. In refs. 4
and 5, using the methods of molecular–kinetic theory, a mass flux
of evaporated molecules was evaluated. A disadvantage of the
formula proposed in these papers lies in the fact that it was
obtained for a free molecular flow of the evaporated gas—that
is, for the conditions where emitted particles do not interact with
molecules presented near the surface.
The next stage in the study of evaporation–condensation dates
back to the 1960s, when the dynamics of rarefied gases was de-
veloped rapidly. The needs of technology development led to the
emergence of more rigorous calculation techniques based on the
exact or approximate solution of the Boltzmann kinetic equation
(BKE). A linear theory was being formulated at this time. The
beginning was laid by the work in ref. 6, in which the form of
the velocity distribution function (VDF) near the interphase
boundary was adopted rather than in the derivation of the
Hertz–Knudsen formula (4, 5). The authors suggested that the
VDF for molecules moving to this boundary is the same as for a
negative half-space of velocities at a considerable distance
from the interface of the phases. Then, by writing down the
expression for the mass flow from definition, they obtained a
result that was two times different from the mass flow calcu-
lated by the Hertz–Knudsen formula.
Linearized, or more simply a linear theory of, evaporation and
condensation was developed by Labuntsov and Muratova in refs.
7 and 8. At about the same time, many researchers (9–16) and
Anisimov et al. (17–19) were focused on the solution of nonlinear
evaporation–condensation problems with the use of the kinetic
theory of gases.
With the advancement of computers, the direct numerical
solution of the BKE began to be applied to the evaporation–con-
densation problems (20–24). For solving the BKE, it is necessary to
specify the correct boundary conditions for the VDFs of evapo-
rated and condensed molecules. The shapes of VDFs together
with the evaporation and condensation coefficients, determining
the corresponding fluxes through the evaporation and condensa-
tion surfaces, are involved in those boundary conditions. In the
previously listed works, a semi-Maxwellian VDF with zero trans-
port velocity was taken as such a function. However, as noted in
ref. 25, “no serious theoretical conclusion of such a boundary
condition is known to us.”
The measured evaporation and condensation coefficients may
vary greatly from experiment to experiment. The condensation
coefficient for water ranges from ∼0.01 to 1 as noted in the review
(26). It seems likely that such a wide spread can be explained by
the fact that those coefficients were measured not at the interface
but over a distance of many mean free paths in the vapor. More-
over, even the small differences between the experimental con-
ditions (such as chemical impurities on the interface and variation
of surface temperatures of liquids being investigated) may have a
dramatic effect on the measurement results. The dependence of
evaporation and condensation coefficients from the surface tem-
perature is also reported in simulation works (27, 28). It was found
that those coefficients, which are close to unity at low tempera-
tures, begin to decrease if the surface temperature is increased
well above the triple point.
It should be noted that the molecular–kinetic approach allows
us to correctly describe the change in the macroparameters of the
vapor/gas near the interfacial surface, but it is assumed that the
state of the condensed phase remains unchanged. On the other
hand, the VDF of molecules escaping from the surface can be
affected by processes occurring near this surface, both from the
liquid side and from the vapor side. In this connection, the ap-
proach in which both the condensed and vapor phases are con-
sidered within the framework of a single modeling method is
obvious. As a research method, the molecular dynamics (MD) simu-
lation has been widely used presently. There are several known works
in which the calculation of the VDF of molecules emitted from the
interphase boundary layer is performed by the MD method (25, 29–
31) and the analysis of the simulation results lead to the conclusion
about the proximity of the VDF of vapor molecules “flying”from the
interphase to the Maxwellian distribution.
In refs. 28 and 32, the problem of recondensation through a
small vapor gap with the thickness of about 7 nm, which leads to a
Knudsen flow with Kn ∼1 roughly, is investigated via MD simula-
tion and by the solution of the Enskog–Vlasov equation with the
Direct Simulation Monte Carlo (DSMC) method. The authors came
to the following conclusion: “On the basis of the results of this
study, we constructed the kinetic boundary conditions (KBC) for
the hard-sphere molecules in consideration of the liquid temper-
ature dependence in the course of a steady net evaporation and
condensation; however, the application of the KBC during un-
steady net evaporation-condensation is extremely important.”
Furthermore, the same two-surface method used in refs. 28 and
32 was applied in ref. 33 to study the evaporation–condensation
in an unsteady transient regime on a short MD time scale. The
authors of ref. 34 noted that there is a deviation from the Maxwel-
lian VDF for particles flying in the direction normal to the interfa-
cial surface: “However, the detailed mechanism of the deviation
has not been clarified yet.”
Numerous papers using the MD method indicate that an
important question in determining the KBC is at which position of
the boundary between the liquid and gas phases those KBC
should be determined (27, 32, 34, 35). The authors of ref. 35
discuss the influence of the KBC position on the mass flow of
the evaporating substance.
It seems obvious the way in which a joint (cross-linked) version
of the description is used. That is, the liquid phase and the region
near the interphase boundary layer are described by the MD
method, next the methods of the kinetic theory of gases are used,
and finally the continuum mechanics approaches are applied on
distances larger than the 10 to 20 mean free paths. However, such
attempts are faced with certain problems and difficulties. The
characteristic time scale of the MD processes and the kinetic re-
laxation time differ approximately by a factor of 10
4
. Therefore, more
than 10
5
simulation steps must be taken to trace the behavior of an
atomic system during the average interatomic collision time. The MD
method deals with the coordinates and velocities of the particles, but
for a molecular–kinetic approach, “this description of the motion of
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the gas is unnecessarily complete. Therefore, we must resort to a less
complete statistical description of the behavior of the system”as a
note in ref. 36. Also, the results of molecular–kinetic calculations in the
form of VDF cannot be used to obtain information on the particle
coordinates and velocities necessary for MD simulation. Accordingly,
the mutual exchange of results of calculations obtained by molecular–
kinetic and MD methods becomes problematic.
In this work, a nonequilibrium vapor flow from the evaporation
to condensation boundary layer is considered in detail on an
atomic scale using the MD method, which provides the classical
trajectories of interacting atoms starting from a hot bulk material
and reaching a cold material after traveling through a gas gap. To
simplify the modeling of gas flow, the exact atom positions r
i
and
velocities v
i
can be replaced by a probability density function
fðr,v,tÞto find dN molecules having positions near rand veloci-
ties near vwithin a phase-space volume drdvat time tsuch as
dN ¼fðr,v,tÞdrdv. Upon integrating fðr,v,tÞ, also called a distri-
bution function, over a whole range of velocity, we obtain a local
number density nðr,tÞof molecules. In a volume with spatially
uniform distribution of molecules, a VDF Fðv,tÞcan be obtained
by integrating fðr,v,tÞdrover the volume.
The BKE (36) describing evolution of the fðr,v,tÞat the ab-
sence of external forces is given by
∂f
∂t+v∂f
∂r¼I,[1]
where rðx,y,zÞare Cartesian coordinates, vðυx,υy,υzÞis the
molecule velocity in a laboratory coordinate system at time t,
and Iis a collision integral. The different forms of the collisional
integral are considered in refs. 20 and 21. Here in this work, the
simplest form for the hard sphere collisions is used.
Our prime goal is to make a bridge from atomistic simulation to
a probabilistic approach through a more penetrating insight into
the atomic-scale mechanism of evaporation and condensation,
which determines the VDFs at the corresponding interfaces. We
performed the large-scale MD simulations to find the best
boundary conditions, including their positions, evaporation and
condensation coefficients, and the shapes of VDFs, which can be
used in the solution of BKE providing the best agreement with data
obtained from the MD simulations. For consistency of BKE with MD
results, the saturated vapor density and transport cross-section
were precomputed by MD and then used in the BKE method.
Simulation Techniques
Equilibrium evaporation and condensation processes take place in an
interphase transition layer between the coexisting condensed phase
and its vapor. For MD simulation of nonequilibrium evaporation–
condensation, the two surfaces of condensed phase at different
temperatures are required. This condition can be complied with using
both sides of a single film as illustrated in Fig. 1, where the condensed
phase of argon is placed in the middle of the MD computational
domain. Periodical boundary conditions are imposed on all three di-
mensions of the domain, which has typical dimensions of Lx¼400nm
and Ly¼Lz¼200nm. The total number of atoms was about
29.15 million in our MD simulations of evaporation–condensation.
To establish both evaporation and condensation processes in a
single simulation, a temperature gradient is maintained by the Lan-
gevin thermostat with target temperature TðxÞdepending on atom
position in the film. The thermostat is only applied to atoms moving in
a gray zone nearby the center of film as shown in Fig. 1; thus, the
interphase layers between the condensed phase and vapor are not
acted upon by the thermostat forces. The Langevin forces are given by
mid~
υi=dt ¼~
ξiγ~
υi~
ux,[2]
where the force acting on each atom iis a sum of a Gaussian-
distributed random force~
ξiand a frictional damping term (37).
The last is determined by a friction coefficient γand a thermal
velocity of atom in reference to a target flow velocity ~
ux.To
reproduce the temperature gradient within the film, the dis-
persion hξ2iof random forces and the friction coefficient must
satisfy the condition hξ2iΔt=2γ¼TðxÞ, where Δtis an MD sim-
ulation time step. Then, the dispersion becomes a function of
-200 -100 0 100 200
position x(nm)
0.1
1
10
0.2
0.3
0.5
2
3
5
20
0.05
number density n (atoms/nm3)
65
70
75
80
85
temperatures Tx and Ty (K)
thermostat
evaporationcondensation
200 nm
Th
Tc
n
Tx
Ty
Fig. 1. MD computational domain for evaporation from the right hot
side of liquid film and condensation on the left cold side of the same
film placed in periodical conditions. Profiles of atom number density,
longitudinal T
x
,andtransverseT
y
temperatures are taken from MD
simulation providing Tc¼72.3 for a cold surface and Th¼80.4K for a hot
one. The Langevin thermostat maintains a required temperature gradient
for atoms in a gray zone xi∈[7,7nm] inside the film with a thickness of
32.7nm. The thermostat also keeps the mass center of film at rest by
adjusting the mass flow velocity from the cold to the hot side. Here this
flow velocity is ≈ux¼0.129m=satthex¼0. See details in Fig. S1.
T
-4
-3
-2
-1
0
1
ns3
-1
0
1
2
3
4
P
P
P
ns
Fig. 2. Pressure and atom number density n
s
of saturated vapor in
equilibrium with the condensed phase of argon. Red line fitted to the
vapor density in the liquid–vapor system and the blue line fitted to
the vapor density in the solid–vapor system are used for estimating
vapor density between data points. The experimental data from ref.
38, pp 6-128 and 15-10. The triple point of simulated argon
Tt.p.¼76.7K is lower than the experimental Tt.p.¼83.8058K.
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atom position x
i
because the friction coefficient is set to a
constant in our MD simulation.
To maintain the steady positive atom flux across the MD do-
main and keep the film at rest, the Langevin target velocity ~
uxis
coupled with a displacement of film mass center using a negative
feedback control. After reaching the stationary regime of vapor
flow, such feedback control provides almost a fixed position of the
film with negligible irregular fluctuations of mass center in the
range less than ±0.01nm, which is much smaller than a thickness
of the interphase layer of ∼2nm for liquid argon at T¼80K. For
steady evaporation–condensation, the atom flux in direction xis a
constant everywhere regardless of local density. As a result, the mass
velocity at the center of the film is less than gas flow speed by a factor
equal to a density ratio between condensed phase and vapor. In the
simulation shown in Fig. 1, the vapor mass velocity is about 40m=sin
the middle of the gap between evaporation and condensation sur-
faces, while the liquid flows with 0.129m=s near the center of film,
with the thickness of 32.7nm defined as a distance between positions
of hot and cold surfaces having temperatures T
h
and T
c
, respectively.
Determination of surface positions is discussed in Evaporation Co-
efficient from MD Simulation. The detailed profiles of target tem-
perature and flow variables in the film and surrounding vapor are
presented in SI Appendix.
MD simulations were performed with a smoothed Lennard–
Jones (L–J) potential (39) given by
ϕðrÞ¼4"hð=rÞ12 ð=rÞ6i+a2x2a3x3,[3]
where ris an interatomic distance and x¼ðr=Þ2ðr0=Þ2. The
coefficients "¼1.0312kJ=mol, ¼0.33841nm, a2¼1.3647
104kJ=mol, and a3¼2.4614104kJ=mol are fitted to repro-
duce argon fcc crystal at zero temperature, which has the
lattice parameter of 0.524673 nm and the cohesive energy
of 7.74005 kJ/mol. The position of the potential minimum
r0¼21=6is identical to that in the original L–J potential.
The smoothing coefficients a
2
and a
3
were chosen to satisfy
the conditions ϕðrcÞ¼0andϕ′ðrcÞ¼0 at the cutoff radius
rc¼0.8125nm.
A density of vapor in equilibrium with the condensed phase is
required to run the BKE calculation of evaporation and conden-
sation because the density nsðTÞof saturated vapor is used to set a
boundary condition for VDF. Using the smoothed L-J potential Eq.
3the vapor pressure and atom number density are evaluated in
several MD simulations of equilibrium liquid-vapor and solid-
vapor systems. Fig. 2 shows the calculated and experimental
values for argon. The visible difference in vapor pressure indicates
that the smoothed L–J potential, fitted to the experimental
parameters of solid Ar, overestimates the pressure. Using the
phase-coexistence MD method, we obtained all three phases
in equilibrium at the triple point Tt.p.¼76.7K of simulated ar-
gon, which is appreciably lower than the experimental triple
point Tt.p.¼83.8058K (38).
Numerical solution of BKE of the evaporation–condensation
problem is performed only for vapor gaps between two interfaces
defined from MD simulated profiles as the outer boundaries of in-
terphase transition layers from the bulk condensed phase to vapor.
While the results of BKE calculations are almost insensitive to the
delimitation of interfaces, they depend highly on the definition of
interface temperatures. Those temperatures are also evaluated
from MD simulated profiles at some position inside the transition
layer. The interface definitions are introduced in the next section.
With the vapor density function, positions of interfaces, and their
temperatures provided by MD simulations, the BKE Eq. 1can be
solved numerically in the vapor gap. We use a finite-difference
computational method described in refs. 20, 21, and 40, in which
a spherical velocity domain is represented by a discrete 3D mesh of
velocity nodes. The discretized BKE equation for each node is solved
in two steps. First, the spatial displacements are calculated without
collisions, and the Courant condition used for a time step Δtguar-
antees that even fastest nodes cannot move more than one spatial
step Δx. Then the collisions are calculated and taken into account.
The collisional integral is evaluated by the quasi Monte Carlo
method using the Korobov’s pseudorandom sequences (21). For
simplicity, the interatomic collision of L–J atoms is considered as
a collision between hard spheres of diameter d.Thelastisan
adjustable parameter that must be determined from MD simu-
lation for consistency between BKE and MD methods. The
collision cross-section determined by the diameter dis ad-
justed to provide a steady heat flux close to that derived from
-200 -100 0 100 200
position x(nm)
70
80
90
temperatures T
x
and T
y
(K)
0.028
0.032
0.036
density n (atoms/nm
-3
)
n
T
x
T
y
246 kW/m
2
Fig. 3. MD simulation of steady heat flux from the left heating
thermostat with Th¼90K to the right cooling thermostat with
Tc¼70K applied in the gray zones. A transport cross-section for BKE
is fitted to get the same heat flux of 246kW=m2produced by the
temperature gradient of 0.05K=nm imposed at x¼0.
x
0.3
3
0.03
n
3
T
x
T
y
T
x
T
y
n
0
1
2
3
4
5
K
x
T
h
Fig. 4. Number density and temperatures profiles in a hot interphase
layer and a small vapor gap lgap ¼1.5nm bounded by the Maxwell
demon. The left boundary of the nonequilibrium part of the
interphase layer (the left vertical dashed line) is determined by a
point of divergence between longitudinal T
x
and transverse T
y
temperatures. This point is used for definition of the surface
temperature T
h
. The right boundary (the right dashed line)
corresponds to a point where density and T
x
starts to decrease
linearly. The Maxwell demon ensures that all atoms in the gray zone
have υx>0. Numbers along the density profile indicate positions
where velocity data for VDFs are gathered. Atoms pass the
interphase almost without loss of their longitudinal component of
kinetic energy K
x
expressed in the temperature units.
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MD simulation of heat transfer between hot and cold zones of
gas gap shown in Fig. 3. L–J atoms within the left hot and right
coldzonesintheMDdomainwitharigidwallat±200 nm are
subjected to two Langevin thermostats supporting 90K and 70K,
respectively. Initially atoms were placed in the domain at a constant
density of n¼0.032nm3. After reaching a steady regime, the heat
flux, the number density, and temperature gradient were measured
at the center x¼0. Using those MD data, the cross-section was
found to equal d¼0.55nm to reproduce the almost identical heat
flux of 245kW=m2in BKE calculations.
For the vapor density n¼0.0963nm3in equilibrium with L–J
liquid at T¼80K, a mean free path in a hard sphere system is
¼1=ffiffiffi
2
pnd2¼7.73 nm, which gives the Knudsen number
Kn ¼=Lgap ¼0.021 for the vapor gap length of Lgap ¼364.3nm
shown in Fig. 1.
To obtain evaporation and condensation coefficients, which
are the basic parameters governing the boundary conditions in
the BKE method, we use the flow profiles and VDFs gained from
atomistic trajectories simulated by the MD method. The most
appropriated coefficients can be found via comparison of VDFs
from numerical solution of BKE with VDFs obtained from the MD
simulation. The VDFs in a steady vapor flow at xpositions are
calculated as Fðx,υx,tÞ¼Rdydz Rdυydυzfðr,v,tÞ. For steady
flow, the distributions are independent of time, which allows us to
accumulate atom position and velocity statistics during MD sim-
ulation. Further integration over υxgives an atom number density
profile nðxÞ¼RdυxFðx,υxÞ.
The steady profiles of density, mass flow velocity, and tem-
perature are obtained by averaging the corresponding values in
spatial slabs with the small thickness of 0.02nm along the x–axis
and during the entire time of productive MD simulation, which is
performed after the attainment of a steady regime. The profile of
flow velocity uxðxÞ¼hυxi, the longitudinal TxðxÞ¼m
kBhðυxuxÞ2i,
and transverse TyðxÞ¼ m
2kBhυ2
y+υ2
zitemperatures are calculated
from the corresponding components of atom velocities υx,υy,
and υz.
Evaporation Coefficient from MD Simulation
The transition of atoms from bulk phase to vapor can be
imagined as a jump over a potential barrier with the height
equal to the atom binding energy "
b
in the condensed phase.
It is assumed in this naive model that the barrier is infinitely
thin and an atom should spend its kinetic energy to over-
come the barrier. As a result, the atoms with kinetic energy "
l
in direction xtoward the vapor exceeding the binding en-
ergy "l>"
bcan go to the vapor phase, where the subscript l
indicates a condensed phase. Taking into account a new ki-
netic energy of atom in vapor "¼"l"bafter passing the
barrier, one can obtain d"¼mυdυ¼d"l¼mυldυl, where veloci-
ties are along the x–axis. Hence, assuming the Maxwellian VDF for
υl, the evaporated atoms obtain a new VDF given by ref. 41:
fxdυ¼Amυdυ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4kBT
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"b+mυ2=2
pexp"b+mυ22
kBT,[4]
where a new atom velocity υis positive (i.e., directed toward
the vapor) and Ais a normalization factor (41). The VDFs f
y
and
f
z
do not change during evaporation.
The above-stated simple model ignores the density re-
distribution and energy transfer inside the interphase boundary
layer having the finite thickness, which makes a real evaporation
process not as easy as it seems. Using large-scale MD simulation
of nonequilibrium evaporation, we demonstrate below that
evaporated atoms are released from the interphase layer almost
without spending their kinetic energies, because they have a near-
zero binding energy at the interphase edge. The main work re-
quired for evaporation is provided by the bulk phase, which
supports via interatomic collisions a relatively slow drift of atoms
through the interphase by a temperature gradient. Because the
characteristic time of interatomic collisions in the condensed
phase is much shorter than the drift time through the interphase
(hundreds of picoseconds), the temperature remains in equilibrium
Fig. 5. Evolution of VDFs within the nonequilibrium part of the
interphase layer. Numbers indicate positions in the layer shown in
Fig. 4. The thin black line shows VDF for transverse velocity υy.The
last VDF at the edge of the interphase layer is marked by 1, while VDF
in vapor is marked by 0. Dashed line shows a model vapor VDF from
Eq. 4 with "b¼0.025kJ=mol. Major changes in VDF take place in the
interphase layer.
Fig. 6. VDFs in front of the absorbing boundary for different T
h
.A
decrease of vapor density with temperature results in a decreasing
rate of interatomic collisions resulting in reduction of a backward flux.
It also leads to lesser smearing of the VDF peak for small υx>0, which
leads to less changes of VDF in the vapor gap for smaller T
h
. The best
fit (red dashed line) of VDF at Th¼65.1K was obtained by Eq. 4 with
"b¼0.002kJ=mol, while a model VDF (black dashed line) normalized
to the same number density was built with "b¼6.65kJ=mol, which
corresponds to the cohesive energy of the atom in the solid argon at
T¼65.2K.
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in the interphase until the density drops by one order of magnitude
near the interphase edge. Thus, VDF changes gradually with de-
creasing density inside the interphase from a symmetrical Max-
wellian form in the bulk of condensed phase to an almost semi-
Maxwellian VDF for evaporated atoms, which has a form described
by Eq. 4with the binding energy "b∼0.
Evaporation is always associated with condensation of evap-
orated atoms gaining a backward velocity due to interatomic
collisions in a vapor gap. Probability of such collisions increases
with the length of gap l
gap
. An additional flux j
ctoward the hot
surface is generated by evaporation from a cold surface facing the
hot surface. To eliminate the flux j
cfrom the opposite cold surface
and minimize the backward flux produced by collisions, we placed
a nonreflective absorbing boundary on the gap l
gap
of several
nanometers from the evaporation surface. In contrast to the large
vapor gap Lgap ¼365nm used in the preceding and next sections,
here the small vapor gaps lgap Lgap are used to achieve Kn 1
to reduce the probability of atom collisions in the gap. Such a
simple approach allows us to estimate an evaporation coefficient
from an intact evaporation flux undistorted by the atoms arriving
to the hot surface, and those may reflect back, without monitoring
of atom trajectories, which is hard to perform in the multimillion
atom simulations.
The absorbing boundary is implemented with the Maxwelldemon,
which watches over atom velocities in a gray zone beyond the
boundary at 1.5 nm from the interphase edge, as shown in Fig. 4.
If an atom gains a negative velocity υx, the Maxwell demon
subroutine replaced it by a small positive value. Atoms passing
the gray zone return back to the cold side of the film like in Fig. 1;
that is, Fig. 4 shows only part of the MD simulation domain to
provide a better spatial resolution for density and temperature
profiles in a thin interphase layer. Thus, a steady regime of evap-
oration, almost entirely uncoupled from condensation, is estab-
lished from the hot side of film.
The interphase layer can be divided into two almost equal
parts. First is the inner part between the bulk phase and a position
where the temperature profile splits into the longitudinal T
x
and
transverse T
y
temperatures, which is denoted by the left dashed
line in Fig. 4. The temperature equilibrium in the inner part of
interphase is well supported because the drift velocity is too small,
and atoms required more than 200ps to pass this part. It is rea-
sonable to take the position and temperature Th¼80.4K at the
point of temperature divergence as the surface/boundary pa-
rameters for the numerical solution of BKE.
With decreasing density by 3.5 times at the end of inner
equilibrium part, the temperature T
x
begins to drop, and the drift
velocity is accelerated to ux>0.005nm=ps, which is recognized as
a beginning of the outer nonequilibrium part of the interphase.
Acceleration of mass flow to about 0.1nm=ps to the edge of this
part shown by a right vertical line on Fig. 4 reduces a drift time of
atoms through the outer part to about 20 ps, which is comparable
with a collision time there. It leads to the large changes in VDF
shape with the approach to the right edge. Fig. 5 shows VDFs
constructed from velocity data accumulated in MD simulation
during about 1.2ns. VDF wings with negative velocities are re-
duced and shrunken much in approach to the right edge of the
interphase, while the width of positive VDF remains almost intact.
As a result of such evolution, the longitudinal T
x
drops dramati-
cally, but the averaged kinetic energy of evaporated atoms is little
affected, as it is illustrated in Fig. 4 by the longitudinal kinetic
energy Kx¼m
kBhυ2
xiexpressed in the temperature units. The trans-
verse Ky¼m
2kBhυ2
y+υ2
zi¼Tyby definition.
The VDF marked by 0 is accumulated in a thin layer with the
thickness of 0.5 nm at the end of vapor gap just before the ab-
sorbing boundary controlled by the Maxwell demon. Neverthe-
less, a negative velocity tail is formed here due to mostly pair
collisions. Such collision conserving the total energy and P
x
,P
y
,
and P
z
momenta may result in the negative velocity of one atom
along x, even though the colliding atoms have both positive ve-
locities υx. To get the negative atom velocity under such condi-
tions, the required kinetic energy is redistributed from the yand z
degrees of freedom of a colliding pair of atoms, which leads to a
decrease of T
y
seen in Fig. 4. Such collisions increase a backward
flux from the absorbing boundary to the interphase edge, but the
total flux remains constant everywhere. Thus, we see a larger
number of atoms having negative velocities in the interphase
edge VDF marked by 1 in Fig. 5.
The backward flux in vapor can be reduced by decreasing
the vapor density, which can be achieved with a decreasing
10 12 14 16 18 20 22 24
position x(nm)
30
40
50
60
70
80
temperatures T
x
and T
y
(K)
-5
-4
-3
-2
-1
0
potential energy E (kJ/mol)
0.1
1
10
0.2
0.5
2
5
20
0.05
density n (atoms/nm
3
)
0
20
40
60
80
100
120
140
flow velocity u
x
(m/s)
E
E
T
y
T
x
Maxwell demon
above 14 nm
u
x
n
Fig. 7. Profiles of number density, temperatures, flow velocity, and
averaged potential energy of atoms for different vapor gap lengths
but the fixed surface temperature Th¼80.4K. The absorbing
boundaries with the Maxwell demons are placed at 14, 16, 18, 20, 22,
and 24 nm in the MD simulation domain, which correspond to
lgap ¼1.5, 3.5, 5.5, 7.5, 9.5, and 11.5 nm, respectively.
lgap
α = j / jm
Fig. 8. Evaporation coefficient ~
α(lgap)¼j=jHK as a function of small
vapor gap length l
gap
. The fluxes are calculated from flow profiles
presented in Fig. 7. Error bars are from uncertainty in the vapor
density calculated from the red-line fit shown in Fig. 2.
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temperature of evaporation surface T
h
. The VDFs presented on
Fig. 6 were built for T
h
in the range of 65.1 to 80.4 K in the systems
similar to one shown on Fig. 4. The condensed phases were in
solid state at 65.1 and 70 K. The Maxwell demon is always placed
beyond the nonreflective absorbing boundary at the position of
14 nm, and the small vapor gaps l
gap
vary between 1.5 and 2 nm.
The collision rate drops for a larger Knudsen number at a lower
density in the vapor gap, resulting in a smaller negative wing of
VDF. It also prevents the large changes of VDF shape in the vapor
flow from the interphase to the absorbing boundary. The lesser
smearing of the VDF peak for small υx>0 for lower temperatures
produces a lesser impact on the almost semi-Maxwellian VDF for
atoms evaporated from the solid argon at Th¼65.1K. Such VDF
illustrates the idea of zero binding energy for evaporated atoms
even better than VDFs obtained for higher temperatures. In these
conditions, the simulated vapor flux j1.02jHK for Th¼65.1K is
very close to the j
HK
provided by the Hertz–Knudsen formula
jHK ¼nsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kBT=2m
p,[5]
which gives the exact flux of atoms in vapor in equilibrium with
a condensed phase, where a Maxwell VDF can be formally
divided by two semi-Maxwellian functions at υx¼0 for posi-
tive and negative fluxes, respectively.
Thus, the net evaporation coefficient defined here as
~
α¼j=jHK 1.02 is even slightly higher than unity, which we at-
tribute to uncertainties in the determination of surface temper-
ature and corresponding equilibrium vapor density estimated
from MD data profiles and the fit of vapor density presented
in Fig. 2.
We show above that the interatomic collisions produce a
backward flux in vapor, which increases with approach to the
evaporation surface. To study the effect of the vapor gap length
on vapor flow, we performed several MD simulations with differ-
ent positions of the absorbing boundary (see Fig. 7). The higher
longitudinal temperatures T
x
observed in vapor for larger gap
lengths are associated with the wider VDFs, which have larger
wings of negative velocities produced by collisions and trans-
ported by the backward flux to an observation point from outer
vapor layers. In contrast to this, the T
y
profiles for the different gap
lengths remain almost identical because there is no transverse flux
and the energy transfer from transverse degrees of freedom to
longitudinal ones depends on a local density, which does not
change much. Decreasing flow velocity u
x
with the gap length is
also explained by the widening of the negative wings of VDFs,
similarly to T
x
.
Fig. 7 demonstrates that the averaged potential energy of
atoms Edecreases about twice in the inner part of the in-
terphase and drops almost to zero at the end of the outer
nonequilibrium part of the interface layer. Thus, the atoms re-
leasing from the edge of interphase have a near-zero binding
energy, which leads to an almost semi-Maxwellian VDF of
evaporated atoms, as already demonstrated by Figs. 5 and 6.
Such semi-Maxwellian VDF with TThgives the flux jjHK,
which corresponds to α1. Thus, the suggested evaporation
mechanism with a near-zero binding energy yields the evapo-
ration coefficient close to unity.
The net evaporation coefficient calculated as a function of the
vapor gap length is shown in Fig. 8. It is defined as a ratio
~
αðlgapÞ¼j=jHK . The evaporation coefficient as a parameter αfor a
numerical solution of BKE can be obtained as a limit of ~
αðlgapÞ→α
at lgap →0, at which the backward flux vanishes. It can be esti-
mated as α0.97 ±0.01. However, this formal procedure may
have no physical meaning for an l
gap
smaller than the interaction
cutoff distance of 0.8125nm or the diameter 0.55nm of collision
cross-section. Also, it is worth noting that another definition of
temperature of evaporation surface can lead to the different
vapor density n
s
, which gives the different evaporation co-
efficient, as an example ~
α0.8 obtained in MD simulation of L–J
evaporation (25).
Because the vapor fluxes calculated for temperatures in the
range of 65 to 80 K are all close to j
HK
, and due to uncertainty in
the definition of the surface temperature and equilibrium vapor
density, we assume that the evaporation coefficient α¼1 for use
in the numerical solution of BKE.
-20 -19 -18 -17 -16 -15
position x(nm)
0.1
1
10
0.2
0.3
0.5
2
3
5
20
30
0.05
number density n (atoms/nm3)
70
72
74
76
78
80
82
84
temperatures Tx and Ty (K)
Tx
Ty
n
Tc
Fig. 9. Number density and temperatures profiles near a
condensation surface on the cold side of liquid film shown in Fig. 1.
The inner boundary of the nonequilibrium part of the interphase layer
(the right dashed line) is determined by a point of divergence
Tc¼72.4 K between longitudinal T
x
and transverse T
y
temperatures.
The outer boundary corresponds to a point where the density profile
becomes flat. Circles indicate positions where velocity data for VDFs
are gathered during 1.5 ns. Evolution of instantaneous (not time-
averaged) profiles is presented in Movie S1.
Fig. 10. VDFs obtained in MD simulation and from BKE solutions with
the different condensation coefficients. VDFs are taken at the hot
and cold sides of liquid film presented in Figs. 1 and 9. While the VDFs
at the hot surface obtained from BKE are almost insensitive to choice
of β, the VDFs at the cold surface depend strongly on this choice.
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Condensation Coefficient from Evaporation–Condensation
Calculated by MD and BKE Methods
For the numerical solution of BKE in a vapor flow between the
evaporation and condensation surfaces, the positions and tem-
peratures of which are determined from MD simulation, the
boundary conditions have to be imposed on VDFs on these sur-
faces. Assuming that the hot surface with temperature T
h
is lo-
cated on the left end of the gap, the positive flux j+
hfrom the
surface is represented by a sum of evaporated and reflected atom
fluxes. It is supposed that an atom coming to the surface with a
negative flux j
–
may be absorbed with probability β, which de-
termines a condensation coefficient. Thus, the reflected atoms
form the reflected flux ð1βÞjjj, which together with the evap-
orated atom flux expressed as αjHK using Eq. 5results in the
positive flux j+
h¼αjHK +ð1βÞjjj. The required boundary VDF
must yield this flux.
As we demonstrated in Evaporation Coefficient from MD
Simulation, the VDF of evaporated atoms is represented by the
semi-Maxwellian VDF fHK ðThÞfor T
h
, which yields the Hertz–
Knudsen flux. The VDF of reflected atoms can also be represented
by a similar semi-Maxwellian function ∼fHK , because the reflection
is assumed to be diffuse with perfect accommodation to the
surface temperature. To produce the reflected flux, the VDF f
HK
must be normalized by the j
HK
and multiplied by the ð1βÞjjj.
Therefore, the left boundary condition applied for a positive wing
of VDF at the surface is given by
f+
h¼ðα+ð1βÞjjjjHK ÞfHK ðThÞ¼~
αfHK ðThÞ.[6]
The negative wing βfis eliminated on the left boundary.
Similarly, the right boundary condition imposed on the cold
surface at T
c
is given by
f
c¼ðα+ð1βÞjj+jjHK ÞfHK ðTcÞ,[7]
where we assume that the αis equal to that on the hot surface.
As indicated in Evaporation Coefficient from MD Simulation,
the evaporation coefficient is almost independent of temperature,
and α1inMDsimulationofevaporation;hence,α¼1isap-
plied on both boundaries in BKE calculations. The unknown con-
densation coefficient βis also taken to be independent of
surface temperature.
Definition of position and temperature of the cold surface is
similar to that for the hot surface introduced in the discussion of
Fig. 4. The density and temperature profiles with a point of di-
vergence between the longitudinal T
x
and transverse T
y
temper-
atures on the cold side of the film presented in Fig. 1 in the vicinity
of the interphase layer are shown in Fig. 9. Thus, the position and
temperature of the right boundary used in BKE calculations are
determined at the end of the equilibrium part of the cold in-
terphase layer formed in MD simulation of steady evaporation–
condensation.
By varying the β, governing both boundary conditions in Eqs. 6
and 7, the VDFs obtained by MD and BKE methods can be
compared with the aim to find an optimal condensation co-
efficient leading to a good agreement between those VDFs. We
find numerical solutions of BKE with β¼0.8; 0.9; 1 for two evap-
oration–condensation systems with ThjTc¼80.4j72.4K in the va-
por gap Lgap ¼364.3nm and ThjTc¼79.4j60.5K in the vapor gap
Lgap ¼366.7nm. See details of simulations in SI Appendix. Fig. 10
shows the VDFs taken at hot and cold surfaces of liquid film for
β¼0.9; 1. The VDFs for β¼0.8 are not shown because they give
the largest deviation from the VDFs obtained in MD. One can see
that the VDFs from BKE are weakly dependent on βat the hot
surface, while BKE solution at the cold surface is very sensitive to
the variation of β.
Similar VDFs from BKE solutions of evaporation–condensation
between the liquid and solid sides of the same film at
ThjTc¼79.4j60.5K, respectively, are shown in Fig. 11. The cor-
responding flow profiles are presented in Fig. S2. The steady re-
gime of vapor flow in the corresponding MD simulation is
illustrated in Movies S1 and S2, where the instantaneous (not
time-averaged) profiles of density and temperatures across the
cold and hot interphase layers are shown, respectively. The den-
sity profile is almost static, and only temperatures in the low-dense
vapor experience the natural fluctuations in those videos. It is
readily seen that the point of temperature divergence appears at
very low number density within the interphase layer, which implies
the fast energy exchange between atoms there, as discussed
in Evaporation Coefficient from MD Simulation. Fig. 11 shows
again that the VDF on the hot surface is almost insensitive to the
choice of β.
Thus, βas a fitting parameter is determined primarily by the
VDF at the cold surface. Lower βproduces a larger vapor flux from
the condensation surface, which must be compensated by a
positive flux to the surface since the total flux in the steady
evaporation–condensation is fixed. As a result, the number den-
sity at the cold surface becomes higher than this in MD simulation.
The above comparison of VDFs from MD simulations and so-
lutions of BKE indicates that the best condensation coefficient is
β¼1 in the considered temperature range. The best agreement
between the flow profiles obtained in MD and BKE calculations is
also achieved at β¼1, as illustrated in Figs. S3 and S4.
SI Appendix provides the detailed description of the simula-
tion technique and comparison of MD and BKE temperature
profiles. Movies S1 and S2 show snapshots of density and tem-
perature profiles across the interphase layer on the condensation
side for TcjTh¼72.4j80.4K and on the evaporation side for
ThjTc¼79.4j60.5K, respectively.
Fig. 11. VDFs obtained in MD simulation and from BKE solutions with
the different condensation coefficients in evaporation–condensation
between the liquid and solid sides of the same film at ThjTc¼
79.4j60.5K, respectively. Lower βresults in larger negative flux from
the cold surface, which leads to larger deviation from VDF provided
by MD simulation.
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Conclusions
Consistent application of MD and BKE methods to the problem
of evaporation–condensation bridges the gap between the
atomistic representation of complex atom motion and probabi-
listic evolution of velocity distributions between two surfaces of
the condensed material. Using multimillion atom simulation, we
demonstrate that, contrary to intuition, atoms are released from a
condensed phase without the use of their kinetic energy but with
the support from collisions with other atoms in the interphase
layer. This effectively means that the evaporated atoms have a
near-zero binding energy, which leads to the virtually semi-
Maxwellian VDF of evaporated atoms and evaporation coeffi-
cient close to unity.
We also find that the best agreement between the steady flow
profile and VDFs obtained by MD and BKE methods is achieved if
both evaporation and condensation coefficients are close to unity
in the considered conditions.
We think that the evaporation–condensation of liquid metals,
water, and polyatomic molecular liquids should be studied next to
verify the applicability of our results to more complex materials.
Such studies may provide more appropriate boundary conditions
for use in continuum mechanics approaches to the evaporation–
condensation problem.
Acknowledgments
This work was supported by Russian Foundation for Basic Research Grant 17-08-
00805. S.I.A. was supported by Russian Science Foundation Grant 14-19-01599.
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