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# Mass and heat transfer between evaporation and condensation surfaces: Atomistic simulation and solution of Boltzmann kinetic equation

Edited by William A. Goddard III, California Institute of Technology, Pasadena, CA, and approved March 21, 2018 (received for review December 25, 2017)

## Abstract

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 because 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 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 transport 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 corpuscular 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 inaccuracies in calculating the evaporation rate can lead to errors in determining the pattern of temperature changes at the interfacial surface, which may result in inaccurate forecasts of the droplet 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 concentration jump near the interphase boundary. One of the refinements of the evaporation–condensation theory can be achieved by invoking 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 developed 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 calculated 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–condensation problems (20⇓⇓⇓–24). For solving the BKE, it is necessary to specify the correct boundary conditions for the VDFs of evaporated and condensed molecules. The shapes of VDFs together with the evaporation and condensation coefficients, determining the corresponding fluxes through the evaporation and condensation surfaces, are involved in those boundary conditions. In the previously listed works, a semi-Maxwellian VDF with zero transport 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. Moreover, even the small differences between the experimental conditions (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 temperature is also reported in simulation works (27, 28). It was found that those coefficients, which are close to unity at low temperatures, 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 approach in which both the condensed and vapor phases are considered within the framework of a single modeling method is obvious. As a research method, the molecular dynamics (MD) simulation 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

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 relaxation 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 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 *dN* molecules having positions near **r** and velocities near **v** within a phase-space volume *d***r**d**v** at time *t* such as

The BKE (36) describing evolution of the *t*, and *I* is 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 dimensions of the domain, which has typical dimensions of

To establish both evaporation and condensation processes in a single simulation, a temperature gradient is maintained by the Langevin thermostat with target temperature *i* is a sum of a Gaussian-distributed random force *γ* and a thermal velocity of atom in reference to a target flow velocity *t* is an MD simulation time step. Then, the dispersion becomes a function of 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 domain and keep the film at rest, the Langevin target velocity *x* is 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 *T*_{h} and *T*_{c}, respectively. Determination of surface positions is discussed in *Evaporation Coefficient from MD Simulation*. The detailed profiles of target temperature 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*r* is an interatomic distance and *a*_{2} and *a*_{3} were chosen to satisfy the conditions

A density of vapor in equilibrium with the condensed phase is required to run the BKE calculation of evaporation and condensation because the density **3** the 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

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 interphase 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. **1** can 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 Δ*t* guarantees that even fastest nodes cannot move more than one spatial step Δ*x*. 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*. The last is an adjustable parameter that must be determined from MD simulation for consistency between BKE and MD methods. The collision cross-section determined by the diameter *d* is adjusted to provide a steady heat flux close to that derived from 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 cold zones in the MD domain with a rigid wall at *x* = 0. Using those MD data, the cross-section was found to equal

For the vapor density

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 *x* positions are calculated as

The steady profiles of density, mass flow velocity, and temperature are obtained by averaging the corresponding values in spatial slabs with the small thickness of *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

## 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 *ε*_{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 overcome the barrier. As a result, the atoms with kinetic energy *ε*_{l} in direction *x* toward the vapor exceeding the binding energy *l* indicates a condensed phase. Taking into account a new kinetic energy of atom in vapor *x*–axis. Hence, assuming the Maxwellian VDF for *υ* is positive (i.e., directed toward the vapor) and *A* is a normalization factor (41). The VDFs *f*_{y} and *f*_{z} do not change during evaporation.

The above-stated simple model ignores the density redistribution 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 required 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 in the interphase until the density drops by one order of magnitude near the interphase edge. Thus, VDF changes gradually with decreasing density inside the interphase from a symmetrical Maxwellian form in the bulk of condensed phase to an almost semi-Maxwellian VDF for evaporated atoms, which has a form described by Eq. **4** with the binding energy

Evaporation is always associated with condensation of evaporated 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 *l*_{gap} of several nanometers from the evaporation surface. In contrast to the large vapor gap

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

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 reasonable to take the position and temperature

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 *T*_{x} drops dramatically, but the averaged kinetic energy of evaporated atoms is little affected, as it is illustrated in Fig. 4 by the longitudinal kinetic energy

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 absorbing boundary controlled by the Maxwell demon. Nevertheless, 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 velocities *y* and *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 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 *j*_{HK} provided by the Hertz–Knudsen formula

Thus, the net evaporation coefficient defined here as

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 different 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 transported 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 *E* decreases about twice in the inner part of the interphase and drops almost to zero at the end of the outer nonequilibrium part of the interface layer. Thus, the atoms releasing 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

The net evaporation coefficient calculated as a function of the vapor gap length is shown in Fig. 8. It is defined as a ratio *α* for a numerical solution of BKE can be obtained as a limit of *l*_{gap} smaller than the interaction cutoff distance of *n*_{s}, which gives the different evaporation coefficient, as an example

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 *α* = 1 for use in the numerical solution of BKE.

## 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 temperatures of which are determined from MD simulation, the boundary conditions have to be imposed on VDFs on these surfaces. Assuming that the hot surface with temperature *T*_{h} is located on the left end of the gap, the positive flux *j*^{–} may be absorbed with probability *β*, which determines a condensation coefficient. Thus, the reflected atoms form the reflected flux **5** results in the positive flux

As we demonstrated in *Evaporation Coefficient from MD Simulation*, the VDF of evaporated atoms is represented by the semi-Maxwellian VDF *T*_{h}, which yields the Hertz–Knudsen flux. The VDF of reflected atoms can also be represented by a similar semi-Maxwellian function *f*_{HK} must be normalized by the *j*_{HK} and multiplied by the

Similarly, the right boundary condition imposed on the cold surface at *T*_{c} is given by*α* is equal to that on the hot surface. As indicated in *Evaporation Coefficient from MD Simulation*, the evaporation coefficient is almost independent of temperature, and *β* is 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 divergence between the longitudinal *T*_{x} and transverse *T*_{y} temperatures 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 interphase 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 coefficient leading to a good agreement between those VDFs. We find numerical solutions of BKE with *SI Appendix*. Fig. 10 shows the VDFs taken at hot and cold surfaces of liquid film for *β* at 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 *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, *β* as a fitting parameter is determined primarily by the VDF at the cold surface. Lower *β* produces 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 density at the cold surface becomes higher than this in MD simulation.

The above comparison of VDFs from MD simulations and solutions of BKE indicates that the best condensation coefficient is *β* = 1 in the considered temperature range. The best agreement between the flow profiles obtained in MD and BKE calculations is also achieved at *β* = 1, as illustrated in Figs. S3 and S4.

*SI Appendix* provides the detailed description of the simulation technique and comparison of MD and BKE temperature profiles. Movies S1 and S2 show snapshots of density and temperature profiles across the interphase layer on the condensation side for

## 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 probabilistic 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 coefficient 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.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: 6asi1z{at}gmail.com.

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.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714503115/-/DCSupplemental.

Published under the PNAS license.

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