Investigating the validity of Schrage relationships for water using molecular dynamics simulations

Abstract

Recently, molecular dynamics (MD) simulations were utilized to show that Schrage theory predicts evapora-tion/condensation mass fluxes with good accuracy in the case of monoatomic and non-polar molecular fluids. Here we examine if they are equally accurate for molecular polar fluids, such as water. In particular, using molecular dynamics (MD) simulations we study the steady state evaporation/condensation processes of water in a one-dimensional heat-pipe geometry to ascertain the validity of Schrage relationships. Non-equilibrium mass flow is driven by controlling the temperatures of the source/sink. Equilibrium simulations are utilized to evaluate saturation properties and the mass accommodation coefficients as a function of temperature. Our results indicate that Schrage equations predict evaporation/condensation rates of water with good accuracy. Moreover, we show that molecular velocity distributions in the vapor phase are indeed Maxwellian distributions shifted by the velocity of the macroscopic vapor flow, as assumed in Schrage's theoretical analysis.

Full text

Investigating the validity of Schrage relationships for water using molecular dynamics simulations
Investigating the validity of Schrage relationships for water using
molecular dynamics simulations
Anirban Chandra1and Pawel Keblinski2, a)
1)Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180,
USA
2)Department of Material Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180,
USA
(Dated: 18 August 2020)
Recently, molecular dynamics (MD) simulations were utilized to show that Schrage theory predicts evapora-
tion/condensation mass fluxes with good accuracy in the case of monoatomic and non-polar molecular fluids. Here
we examine if they are equally accurate for molecular polar fluids, such as water. In particular, using molecular dy-
namics (MD) simulations we study the steady state evaporation/condensation processes of water in a one-dimensional
heat-pipe geometry to ascertain the validity of Schrage relationships. Non-equilibrium mass flow is driven by con-
trolling the temperatures of the source/sink. Equilibrium simulations are utilized to evaluate saturation properties and
the mass accommodation coefficients as a function of temperature. Our results indicate that Schrage equations predict
evaporation/condensation rates of water with good accuracy. Moreover, we show that molecular velocity distributions in
the vapor phase are indeed Maxwellian distributions shifted by the velocity of the macroscopic vapor flow, as assumed
in Schrage’s theoretical analysis.
I. INTRODUCTION
Liquid vapor phase change processes are ubiquitous
and used extensively in varied disciplines including
climatology1, agriculture2, astronomy3, medical therapy4,
thermal management5, cooling of microelectronics6, solar-
thermal energy conversion7–9 etc. Powering engineered
systems10,11 and solar steam generation12 are few new age
applications which are being currently researched. Moreover,
understanding transport processes at nano-micro scales are
becoming increasingly important due the emergence of
micro/nano technologies13,14.
Despite such importance of evaporative/condensation pro-
cesses accuracy/validity of quantitative theories and models is
still debated and subject of active research. One of the first
theoretical expressions for describing phase change at liquid
vapor interfaces were obtained by Hertz and Knudsen (HK
relation) using Kinetic theory of gases and certain thermody-
namic consideration15,16. This is still one of the most widely
used expressions for describing evaporation/condensation17.
An inherent assumption in this theory is that the velocity dis-
tribution of vapor molecules near the interface is the equi-
librium Maxwell-Boltzmann (MB) velocity distribution. Nu-
merous experimental and numerical studies have been carried
out to test the validity of HK relation and several inconsisten-
cies have been reported18–20. Schrage16 identified one pos-
sible source of discrepancy by pointing out that if net evap-
oration/condensation occurs, the vapor molecules move nor-
mal to the liquid-vapor interface with a nonzero mean (macro-
scopic) velocity. Therefore, he assumed that vapor molecules
adjacent to the interface would have a MB velocity distribu-
tion shifted by the mean velocity16,21. With this consideration,
Schrage modified the Hertz-Knudsen analysis and derived ex-
a)Electronic mail: keblip@rpi.edu
pressions for net evaporation/condensation rates which are
popularly known as Schrage relationships16.
The debate on accuracy of Schrage relationships, or any
other relationship (such as Hertz-Knudsen) describing inter-
facial phase change rates, is partially due to challenges in the
experimental quantification of evaporation/condensation pro-
cesses. Determination of temperature at the liquid-vapor in-
terfaces, pressure and density in the vapor phase, and mass
accommodation coefficients with high accuracy is crucial for
investigating the validity of these theoretical relationships22.
Experimental quantification of parameters like mass accom-
modation coefficient have been challenging despite long-term
efforts23. In typical experimental studies, mass accommoda-
tion coefficient is often treated as fitting parameters17. One
way to mitigate the aforementioned experimental challenges
is to utilize molecular dynamics (MD) simulations. Advan-
tage of MD simulations lies in the fact that, atomic trajectories
can be analysed to directly obtain all the quantities (interfacial
properties and mass accommodation coefficients) in Schrage
relationships. Furthermore, velocity distribution of evaporat-
ing/condensing molecules, which is necessary for assessing
the of validity of Schrage analysis, can be directly obtained
from MD simulations.
One of the first attempts to ascertain validity of Schrage re-
lations using molecular dynamics (MD) simulations was un-
dertaken by Yu and Wang24 where they studied the phase
change of liquid Argon thin films and showed that evapora-
tion rates from the MD simulation agree within 10 −15%
with the Schrage predictions. More rigorous investigations
by Liang et al.25 revealed that, a generalized (exact) version
of the Schrage relation16 predicts mass fluxes with extremely
good accuracy. Recently, Bird et al.26 showed that same re-
lation can predict evaporation/condensation fluxes with good
accuracy in non-polar molecular fluids such as n-dodecane
(diesel fuel)26. However, for water (a polar fluid), accuracy
of these relationships have not been thoroughly investigated.
Furthermore, there is a prevalent opinion that for systems with
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 2
strong intermolecular interactions, such as polar fluids and in
particular water, the assumptions made by Schrage are likely
not satisfied. Therefore, to address this gap in knowledge,
in this work, we will use molecular dynamics simulations to
explore the accuracy of Schrage predictions for water. Re-
mainder of the paper is structured as follows. Section II gives
a brief introduction to Schrage relationships. In Section III
we present our simulation methodology/setup. Subsequently,
we discuss results of our non equilibrium molecular dynamics
simulations in Section IV. Finally, in Section V we present
summary and conclusions.
II. THEORETICAL BACKGROUND
Schrage16 obtained an expression for describing evapora-
tion/condensation rates for pure substances using thermody-
namic variables at the interface along with some kinetic con-
siderations. This relation can be written as,
J=
α
(Tl)rkB
2
π
m
ρ
sat (Tl)√Tl−Γ(vR)
ρ
v√Tv,(1)
where, Jis the mass flux at interfaces (J>0 when net evap-
oration occurs),
α
is the mass accommodation coefficient, m
is mass of the fluid molecule,
ρ
sat (Tl)is the saturated vapor
density at temperature Tl,
ρ
vis vapor density, Tland Tvare
interfacial liquid and vapor temperatures, and kBis the Boltz-
mann constant. Mass accommodation coefficient (
α
) is de-
fined as the fraction of vapor molecules which are ‘adsorbed
(accommodated)’ rather than reflected after collision with a
liquid surface. Γ(vR)is given by,
Γ(vR) = e−v2
R−vR√
π
(1−er f (vR)).(2)
Macroscopic velocity (normal to the interface) of the vapor
molecules leaving the interface is denoted by uv
n.vRis defined
as the ratio of uv
nto the most probable thermal speed of vapor
molecules and can be represented as,
vR=uv
n
p2kBTv/m.(3)
In literature18,22, the Schrage relation popularly used is an ap-
proximate version of Equation 1 wherein vR<< 1. Under this
assumption, Equation 1 reduces to,
J=2
α
(Tl)
2−
α
(Tl)rkB
2
π
m
ρ
sat (Tl)√Tl−
ρ
v√Tv.(4)
Another popular variant of this relation is when tempera-
ture continuity (Tl=Tv) is assumed in Equation 417. Using
molecular dynamics simulations it was shown that Equation
4 predicts mass fluxes with good accuracy when the macro-
scopic velocity of vapor is small compared to the thermal ve-
locity of the atoms25,27; whereas Equation 1 provided accurate
predictions in more generalized conditions25–28. Thus, in this
work, we will explore the validity of the exact Schrage rela-
tion (Equation 1) for water. In Schrage’s analysis16, velocity
distribution of vapor molecules was assumed to be a shifted
Maxwellian. Correctness of this assumption will also be in-
vestigated here.
The exact Schrage relation (Equation 1), as written, is a
function(al) of uv
n,Tl,Tv, and
ρ
v. Since, velocity of the vapor
is not truly an unknown, uv
ncan be expressed as J/
ρ
v, and an
implicit equation for mass fluxes (J) can be obtained in terms
of
ρ
v,Tl, and Tv. Independent variables (
ρ
v,Tl,Tv,
ρ
sat (Tl),
α
(Tl)) of this implicit relation are obtained using equilibrium
and non-equilibrium MD simulations. Subsequently, a solu-
tion for Jis obtained, which is denoted by JSc. In Section IV,
we compare JSc with fluxes obtained from MD simulations
(Jmd) and discuss the accuracy of Schrage predictions.
An explicit evaluation of Jfrom Equation 1 is possible by
substitution of all independent variables (including uv
n). This
reduces the need for a non-linear iterative scheme for obtain-
ing a solution and hence prevents amplification of statistical
errors present in determination of independent variables. We
denote this solution as JSc′and use it to corroborate our dis-
cussions on validity of Schrage relations.
III. COMPUTATIONAL DETAILS
A. General simulation setup
All simulations are carried out in the open source molecu-
lar dynamics code LAMMPS29 using a cuboidial simulation
box of dimensions (6nm, 6nm, 220nm) in the (x,y,z) direc-
tions. Velocity-verlet is used as the time integration scheme
and a timestep of 1fs is utilized. As shown in Figure 1, two
slabs of liquid water are placed near the right and left bound-
aries of this box. These cuboidal slabs have a finite length
along zdirection. The distance between the outermost edges
of these liquid slabs is 216nm. Periodic boundary conditions
are applied in x and y directions. The z and x/y coordinates
are interchangeably referred to as the normal and tangential
directions respectively. A zoomed-in view of the condensing
liquid-vapor interface is shown in Figure 2. Region descrip-
tions at the evaporating interface are done in a similar man-
ner. All water molecules in the region RT(‘Tethered’ region)
are tethered to its initial position using harmonic springs with
k=10N/m. The lattice structure, tetragonal with lattice con-
stants (3Å,6Å) and one unit cell thickness, is chosen such that
no water molecule from region RL(regarded as ‘Liquid’ re-
gion) passes through RTover the duration of our simulations.
Periodic BCs are applied in the z-direction too, but the dis-
tance of outermost edges of RTfrom edges of the simulation
box is greater than the cutoff distance, so these two slabs are
essentially non-interacting. Nosé-Hoover thermostatting30,31
is always applied to atoms in ‘Tethered’ (RT) regions with
Thot and Tcold representing the thermostat temperatures. Ini-
tial thickness of liquid (RL) was chosen to be 2.5nm in all our
simulations.
For evaluating properties in the liquid, RT+RLis divided
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 3
into static bins with a thickness of 8Å (RL1,RL2,RL3, and
RL4– see Figure 2) and interfacial liquid properties are based
on the spatial bin where the density of the fluid is approxi-
mately equal (within 20%) to the density of bulk liquid. In
the vapor region, the bin thickness is set to 50nm. The en-
tire vapor region (see Figure 1) is divided into 4 sub-regions
(RV1,RV2,RV3,RV4) each having a width of 50nm. RV1rep-
resents the vapor region closest to the evaporating interface,
while RV4is nearest to the condensing interface. Values in
these subregions are utilized to determine the spatial variation
of properties in the vapor phase. In Figure 2, line L2(dashed)
marks the beginning of vapor region shown in Figure 1. Line
L1(dash-dotted) represents the imaginary line which we use
to evaluate accommodation coefficients; this is described in
detail in Section III D.
B. Potential model
The interaction between water molecules is modeled us-
ing the SPC/E32 potential with a modification in the evalu-
ation of coulombic terms. Long range electrostatic interac-
tions are evaluated using the Wolf summation technique33,34
with a damping factor of 0.12Å−1and cutoff of 10Å for both
Lennard Jones and Coulombic interactions. Omission of a
reciprocal space evaluation of Coulombic interactions allows
us to perform long simulations (∼10ns) in the NVE ensem-
ble without significant errors in energy conservation within
reasonable computational time. The shake algorithm35 is
used to constrain the bond lengths and angles of the water
molecules. To benchmark our potential model we compare
saturation properties obtained using this model with experi-
mental data and other potentials (see Figures 3a and 3b). This
is discussed in Section III C. Although truncation of Coulomb
interactions leads to slight overestimation of saturated vapor
densities and pressures when compared to typical potentials
like SPC/E (with Ewald summation), our model is closer to
experimentally reported saturation properties in the consid-
ered temperature range. In this paper, we investigate evapora-
tion/condensation characteristics of this specific water model
without focusing on how closely this model describes real wa-
ter.
C. Saturation Properties
Saturated vapor density (
ρ
sat ) as function of temperature
is essential for obtaining estimates of mass fluxes from the
Schrage relations (Equation 1). To evaluate
ρ
sat (T)for our
water model, we consider 5 temperatures in the range 370K
to 430K with increments of 15K. Equilibrium MD simula-
tions are performed to obtain the saturation properties of our
model. Equilibration is performed in two stages. First, ther-
mostats (NVT) are applied on the entire system for ∼8ns.
Subsequently, thermostats are applied only on the ‘Tethered’
regions and the remaining molecules follow Newton’s equa-
tions of motions (NVE). In the 2nd stage of equilibration, data
collection begins after letting the system evolve for ∼5ns .
We collect saturated vapor density (
ρ
sat ) and pressure (P
sat)
data over ∼21ns for systems with target temperatures 370K,
385K, 400K, and 415K. For the 430K case, a smaller data col-
lection duration (about 12ns) leads to statistically significant
observations . Variation of
ρ
sat and P
sat over the data col-
lection interval showed a correlation time which was much
less than 3ns. Therefore, the entire data collection inter-
val is divided into intervals of 3ns and a interval-wise av-
eraged value of
ρ
sat and P
sat is obtained. These averaged
values are then treated as independent observations and used
to evaluate the standard error in our measurements. Figures
3b and 3a show the variation of saturation pressure and sat-
urated vapor density for our water model. Fit to the satu-
rated vapor density data (red dashed line in Figure 3a) has
the form, ln(
ρ
sat ) = −4.279 ×104/8.314/Tl+12.77. Fit-
ting for the saturation pressure data is performed using the
same functional form. Saturation properties of water obtained
using other potentials are presented. In Figures 3b and 3a,
SPC/E data is obtained from36, TIP4P/2005 and TIP4P-Ew
from37. Pressure calculation, unlike temperature, is limited to
the molecules in vapor phase only to avoid the large fluctua-
tions which arise from the incompressibility of liquid water.
Although pressure values are not explicitly used in this work,
it is reported here for future reference. Using the saturation
pressure, density, and temperature values the compressibility
factor at 430K is estimated to be 0.88 and monotonically in-
creases to 0.93 at 370K.
Saturation properties obtained using liquid thickness (RL)
ranging from ∼1.8nm to ∼5nm are comparable, thus nullify-
ing the existence of disjoining pressure effects. In our driven
simulations (see Section III E), data collection in terval is cho-
sen such that the width of liquid remaining at the evaporative
end is always ≥1.8nm. At lower liquid thicknesses, instabili-
ties in the liquid film are observed.
D. Mass accommodation coefficients (
α
)
Prediction of mass fluxes from using the Schrage relations
(Equation 1) not only requires apriori determination of satu-
ration properties, but also mass accommodation coefficients.
Direct experimental evaluation of
α
is challenging39, but in
MD simulations, by analyzing molecular trajectories this co-
efficient can be directly determined. To evaluate
α
we per-
form two stage equilibrium simulations, similar to ones de-
scribed in Section III C. Data is collected after letting the sys-
tem evolve for ∼5ns in the stage 2 equilibration process. In
one simulation, data is collected over 1ns. Approximately 10
independent equilibrium simulations are performed for each
considered temperature.
For evaluating
α
, we place an imaginary Line L1(see
Figure 2) near the liquid-vapor interface at a distance ∼
3
δ
(∼2.0nm)from the liquid vapor interface.
δ
is the 10-
90 thickness of the interfacial density transition layer. Prior
studies40–43 have suggested that placing the imaginary line be-
tween 2
δ
and 4
δ
away from the interface results does not sig-
nificantly alter the observations. All molecules crossing line
L1with a z-velocity component towards the interface are mon-
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 4
~5nm
Vapor Region (200nm) Evaporation
Condensation
T
cold Thot
~5nm
Vapor flow
z
4Ï8(50nm) 4Ï7(50nm) 4Ï6(50nm) 4Ï5(50nm)
.6.6
FIG. 1: A snapshot of the simulation domain. Thot and Tcold denote the thermostat temperatures (see Figure 2 for a zoomed in
view of the liquid-vapor interface). The entire vapor region is divided into 4 regions – RV1,RV2,RV3,RV4– for obtaining the
spatial variation of properties in the vapor.
RTRL (Liquid) L1L2
T
cold
Vapor
Region
~0.6nm ~2.5nm ~2.0nm
RL1
RL2
RL3
RL4
FIG. 2: A zoomed-in view of a condensing liquid-vapor
interface in our simulation setup. RTrepresents the region
where liquid molecules are tethered with springs and Nosé
Hoover thermostatting is performed. RLdenotes the liquid
where no thermostats are applied. RL1,RL2,RL3, and RL4
signify bins of width 8Å which are used to calculate
properties of the liquid. Line L2(dashed) marks the edge of
the vapor region. Line L1(dash-dot) represents the imaginary
line used to evaluate accommodation coefficients. Region
definitions at the evaporating end are done in a similar way.
itored. Figure 4 shows the time of return (time taken to cross
this line again by the same molecule) when the temperature
is 430K. Probability of reflection is defined as the number
of molecules returning to L1in ∆t(Nref ) normalized by the
total number of molecules crossing L1(Ninc). The time of re-
turn at which the slope of this histogram changes separates the
molecules that were reflected (not accommodated), and those
that first condensed and then evaporated (accommodated). In-
tegral of the reflection probability histogram to such deter-
mined time yields the probability of reflection given by, 1-
α
.
Average normal velocity of molecules crossing L1with z ve-
locity towards the interface is given by pkBTv/(2
π
m), which
is ∼178m/s at 430K. Distance of L1from the liquid interface
is about 20Å, so average time taken for a water molecule to
cross this line and return is ∼22.5ps. This time is compa-
rable to the time at which slope change occurs in Figure 4,
thus, corroborating our accommodation coefficient evaluation
methodology. Detailed description of this technique can be
found in Liang et al.25.
Figure 5 shows the variation of accommodation coeffi-
370 380 390 400 410 420 430
T(K)
0
0.5
1
1.5
2
2.5
sat(kg/m3)
Our Model
SPC/E-Ewald
TIP4P-Ew
TIP4P/2005
Expt.
(a) Saturated vapor density
370 380 390 400 410 420 430
T(K)
0
1
2
3
4
5
Psat(atm)
Our Model
SPC/E-Ewald
TIP4P-Ew
TIP4P/2005
Expt.
(b) Saturation pressure
FIG. 3: Saturation Properties of our model. The red dashed
line represents a fit to our data. SPC/E-Ewald values are
obtained from36, TIP4P/2005 and TIP4P-Ew from37. Black
dash-dot lines denote the experimental data obtained from
standard steam tables for water38.
cients with temperature. The second order polynomial fit (red
dashed curve in Figure 5) to the accommodation data is uti-
lized later for predicting mass fluxes using the Schrage rela-
tion. Fitting function has the form
α
(Tl) = −4.16 ×10−6×
(Tl)2+2.15 ×10−3×Tl+0.73. The number of crossing
events is ∼600 when T=430K, and ∼50 at T=370K. At tem-
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 5
10 20 30 40 50 60 70 80 90
t (ps)
0
0.005
0.01
0.015
0.02
0.025
0.03
Probability of Reflection
0
0.05
0.1
0.15
Running Integral
Probability Histogram
Nref/Ninc
Accommodation Evaporation
Reflection
1 - (430K)
FIG. 4: Probability of Reflection and it’s running integral
against the time of return (∆t). The dotted vertical line
represents the location where the slope of the running
integral enters a constant regime (also represents the location
where the slope of the Probability Histogram stops changing
drastically. In other words, collision events occurring to the
left of this line represents Reflection whereas events to the
right represent, first accommodation and then evaporation.
Dashed horizontal line gives the value o f 1 −
α
at 430K.
300 320 340 360 380 400 420
T(K)
0.88
0.9
0.92
0.94
0.96
0.98
1
FIG. 5: Variation of accommodation coefficient(
α
) with Tl.
The dashed red line represents a quadratic fit to the data.
Circular marker denotes the value of accommodation
coefficient obtained from previous MD studies44.
peratures close to 300K our method of calculation of
α
be-
comes practically inapplicable due to low vapor densities.
Therefore, to enrich our fit, we use
α
=1 at 300Kwhich has
been observed in previous MD studies44.
E. Non-equilibrium (driven) simulations
To determine the validity of Schrage relations, we perform
quasi-steady simulations in a planar heat-pipe geometry. At
the beginning of these non-equilibrium simulations, the num-
ber of molecules in liquid phase (RL) on each end is about
∼3200, with about 400 molecules in the entire vapor region.
Every driven simulation is started from an equilibrium state
at 415K and three combinations of (Thot ,Tcold) are considered
–C1(420K,410K),C2(425K,405K), and C3(430K,400K). Five
independent runs are performed for each combination of (Thot ,
Tcold ). After ramping the source and sink temperatures from
415K to their desired values in 10ps, each system is allowed
to evolve for a certain time (7.2ns – C1, 4.8ns – C2, 2.4ns –
C3) before the data collection step begins. Data is collected
for ‘
τ
’ns depending on the driving conditions. Prediction of
mass fluxes using the Schrage relations requires the knowl-
edge of Tl. In all cases, Tlis the temperature of that spatial
bin (closest to the interface) where the density of the fluid is
approximately equal (within 20%) to the density of bulk liq-
uid. Size of bins in liquid phase is 8Å, therefore we can track
interfaces with an accuracy of 8Å. More robust techniques
for interface tracking and interfacial temperature evaluation
can potentially increase the accuracy of our results, but ex-
plorations in such directions are not performed in this work.
Determination of vapor properties close to the interface are
also necessary for making predictions using the Schrage anal-
ysis; averaged density, temperature and velocities obtained in
regions RV1(evaporation) and RV4(condensation) are defined
as interfacial vapor properties.
IV. RESULTS AND DISCUSSION
A. Comparison of mass fluxes from MD simulations with
Schrage predictions
Using the methodology described in Section II, and by
performing non-equilibrium molecular dynamics simulations
(Section III E) we obtain predictions of mass fluxes using the
Schrage relations. Interfacial properties are shown in Tables I
(evaporation) and II (condensation). [∆Tl]iis the difference in
interfacial liquid temperatures; a larger value [∆Tl]iimplies a
larger driving force. Jmd (=
ρ
vuv
n) represents the fluxes calcu-
lated in RV1and RV4from MD simulations. JSc is the solution
obtained by implicitly solving the Schrage relation, while JSc′
denotes explicit evaluation of the same equation (see Section
II for more details). ∆JSc is defined as (JSc −Jmd)/Jmd ×100;
similarly, ∆JSc′is based on JSc′. Three different combinations
of source/sink temperatures are considered: C1(420K,410K)
,C2(425K,405K), and C3(430K,400K). Data collection inter-
val (
τ
) for these three cases are, 6.4ns (C1), 4ns (C2), and 4ns
(C3).
τ
is chosen such that there is at least 1.8nm of liquid at
the evaporating end. In general, increasing
τ
does not signifi-
cantly affect our results (refer to Section IV D). Explicit eval-
uation of Jfrom Equation 1 shows less deviation as compared
to an implicit solution (see Tables I and II) due to less influ-
ence of measurement errors in the former. Variation of mass
fluxes with [∆Tl]iis shown in Figure 6. Although we do not
explicitly calculate fluxes in an equilibrium setting, by defini-
tion it is zero (circular markers in Figure 6). A linear variation
of flux with [∆Tl]i, as evidenced by the dashed line, is ob-
served similar to prior studies on non-polar fluids25,26 . The
observed slope of average fluxes (evaporation and condensa-
tion) is 5.79kg/m2/s/Kis comparable to 6.45kg/m2/s/K, the
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 6
0 5 10 15 20 25 30
[ Tl]i (K)
0
50
100
150
J (kg/m2/s)
Evaporation
0
50
100
150
J (kg/m2/s)
Condensation Jmd JSc
FIG. 6: Variation of JSc and Jmd with different driving
conditions ([∆Tl]i). Circular markers represent the
equilibrium condition when there is no driving force applied.
Dashed lines represent fits to MD data.
prediction made by Equation 16 in Liang et al.25
Deviations of Schrage predictions from MD results are
shown in Figure 7; a good agreement is seen in all cases.
Smallest deviation is observed in the highly driven case (C3),
which is farther away from equilibrium conditions as com-
pared to C1and C2. This indicates that discrepancies observed
might be statistical in nature. Mathematically, the Schrage re-
lation is a combination of two terms JSc =JI−JII , as shown
in Equation 5. With decreasing driving forces, although JSc
decreases significantly (see Tables I and II), JIand JII change
less severely. The ratio, JI I /JSc, at the evaporating interface
for the three considered cases are: C1– 3.3, C2– 1.6, and C3
– 1.0. Similar values are obtained for the condensing inter-
face. Clearly, relative importance of each term in Equation 5
increases appreciably at low driving forces. Thus, more pre-
cise determination of interfacial properties is necessary as the
driving force reduces. This corroborates our claim that the
discrepancies observed are indeed statistical in nature. Other
potential reasons for larger ∆JSc in case C1are discussed in
Section IV D.
J=
α
(Tl)rkB
2
π
m
ρ
sat (Tl)√Tl
|{z }
JI
−
α
(Tl)rkB
2
π
mΓ(vR)
ρ
v√Tv
|{z }
JII
(5)
5 10 15 20 25 30
[ Tl]i (K)
0
5
10
15
20
25
JSc (%)
Evaporation Condensation
FIG. 7: Percentage deviation of JSc w.r.t. Jmd . The horizontal
axis represents the difference between liquid temperatures at
the evaporating and condensing interfaces, [∆Tl]i.
B. Velocity, temperature, and density profiles
Variation of temperature (Tv), density (
ρ
v), magnitude of
normal velocity (uv
n) is shown in Figure 8 when Thot =430K
and Tcold =400K(case C1). Similar trends are observed for
other driving conditions. Velocity and density profiles show
opposing trends, which indicates that mass flux is almost con-
stant in the vapor phase. This is also evidenced by the Jmd
values reported in Tables II and I. Vapor temperature does not
vary significantly along the length of the channel, indicating
that heat transport is predominantly advective in this region.
For non-polar fluids, similar observations have been reported
in the quasi-steady regime25. Temperature jumps at liquid va-
por interfaces undergoing phase-change h ave been reported by
several authors22,45,46, and is still an area of active research47.
In our simulations, congruous with prior observations, such
temperature discontinuities exist (top panel of Figure 8). More
accurate estimations of temperature jump can be obtained by
increasing spatial resolution of bins at the interfacial region.
Dashed green vertical lines in the bottom panel of Figure 8 de-
marcates the four vapor regions – RV1,RV2,RV3, and RV4. Ap-
proximate locations of the interface are shown by the dashed
black vertical lines.
C. Velocity distributions
Earlier in this Section, we demonstrated that predictions of
mass fluxes obtained from Schrage relations agree with MD
simulation results reasonably well. Another major assump-
tion in Schrage’s analysis16 was that the normal velocity dis-
tribution of vapor molecules (close to the interface) are stan-
dard Maxwellian distributions but with a bias velocity. In the
tangential directions, velocity distributions were assumed to
be unaltered. Using the data obtained from our simulations
(Thot =430 and Tcold =400), velocity distributions of the va-
por molecules in RV1, the region closest to the evaporating in-
terface, are studied. Non-equilibrium effects are expected to
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 7
[∆Tl]iTlTvuv
n
ρ
vJmd JSc ∆JSc (%) JSc′∆JSc′(%)
9.31 419.31 410.89 34.05 1.44 49.09 59.21 20.60 54.99 12.01
±0.013 ±1.872 ±2.071 ±0.019 ±2.874
18.73 424.11 413.54 69.09 1.45 100.14 121.33 21.16 113.24 13.08
±0.130 ±2.222 ±1.898 ±0.034 ±2.238
28.19 429.20 415.12 107.89 1.53 165.22 174.69 5.73 171.39 3.73
±0.156 ±4.610 ±4.017 ±0.035 ±6.106
TABLE I: Interfacial properties and Schrage predictions at the evaporating interface for three driving conditions:
C1(420K,410K) – top row, C2(425K,405K) – middle row, C3(430K,400K) – bottom row. Interfacial vapor values are obtained
using region MD data in RV1. All values are reported in SI units: temperature in K, velocity in m/s, density in kg/m3, and
fluxes in kg/m2/s.
[∆Tl]iTlTvuv
n
ρ
vJmd JSc ∆JSc (%) JSc′∆JSc′(%)
9.31 409.99 416.01 34.07 1.41 47.98 53.57 11.65 50.90 6.09
±0.098 ±2.852 ±1.972 ±0.026 ±2.453
18.73 405.37 414.67 67.07 1.42 95.21 111.96 17.59 103.30 8.49
±0.125 ±3.413 ±2.497 ±0.012 ±3.204
28.19 401.00 414.39 113.84 1.45 164.97 173.44 5.13 168.73 2.28
±0.169 ±4.086 ±3.585 ±0.035 ±6.430
TABLE II: Interfacial properties and Schrage predictions at the condensing interface for three driving conditions:
C1(420K,410K) – top row, C2(425K,405K) – middle row, C3(430K,400K) – bottom row. Interfacial vapor values are obtained
using region MD data in RV4. All values are reported in SI units: temperature in K, velocity in m/s, density in kg/m3, and
fluxes in kg/m2/s.
400
410
420
430
T(K)
TlTv
100
110
120
uv
n (m/s)
50 100 150 200
x(nm)
1.4
1.5
1.6
v(kg/m3)
FIG. 8: Variation of properties across the vapor region when
Thot =430Kand Tcold =400K. In the bottom panel, dashed
green vertical lines demarcates the four vapor regions – RV1,
RV2,RV3, and RV4. Approximate locations of the interface
are shown by the dashed black vertical lines.
be most prominent in highly driven systems near the evapo-
rating interface as probability of molecular collisions are less
in such cases. Therefore, validity of Schrage’s assumptions
about distributions are explored in RV1of our most driven sys-
tem – where most deviation is expected.
Figure 9 shows the velocity distribution (z/normal direc-
tion) of water molecules and a shifted Maxwellian fit to MD
data. Mean velocity obtained from the fitting procedure is
comparable to the value shown in Table I for case C3. Only
translational velocity of molecules are considered for obtain-
ing distributions; therefore, a slight mismatch between tem-
peratures reported in Figure 9 and Table I (where total KE of
molecules was used to obtain temperature) is expected. Us-
ing MD simulations of Argon, Bird et al.48 showed that, when
[∆Tl]iis significantly higher, velocity distributions of atoms
moving normal to the interface are narrower as compared to
those moving parallel to the interface. This might be another
possible explanation for the slight mismatch in Tvvalues from
the two methods mentioned above (Figure 9 and Table I). Fig-
ure 10 shows that, distribution of the tangential velocities is
Maxwellian and the temperature estimate obtained are in good
agreement with values in Table I. Therefore, both normal and
tangential velocity distributions obtained from MD simula-
tions are consistent with the assumptions made by Schrage.
D. Deviations from Schrage predictions
In Section IV A, it was shown that deviation of Schrage
predictions from MD results were larger for smaller driving
forces. Here we explore some of the potential causes for these
deviations. Effect of data collection time interval (
τ
) for case
C3is shown in Tables III and IV. With increasing data collec-
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 8
-1500 -1000 -500 0 500 1000 1500
u(m/s)
0
0.2
0.4
0.6
0.8
1
(s/m)
10-3
uv
n = 111.1m/s
Tv = 400.6K
Jmd = 169.53kg/m2s
RV1
FIG. 9: Distribution of normal velocity of water molecules in
RV1(closest to evaporating region) when Thot =430Kand
Tcold =400K. The black-dashed line represents a shifted
Maxwellian fit to MD data.
-1500 -1000 -500 0 500 1000 1500
u(m/s)
0
0.2
0.4
0.6
0.8
1
(s/m)
10-3
Tv = 412.4K
RV1
FIG. 10: Distribution of tangential velocity of water
molecules in RV1(closest to evaporating region) when
Thot =430Kand Tcold =400K. The black-dashed line
represents a Maxwellian fit to MD data.
tion interval, errors bars in our measurement and deviation in
Schrage predictions does not change significantly. A similar
trend is observed for other cases. Duration of the data collec-
tion time interval is predominantly limited by the fact that, as
liquid film reduces beyond 1.8nm instabilities tend to develop
(see Section III C). Thus, further exploration in this direction
is not carried out.
Another reason for deviation of Schrage predictions from
MD results for smaller driving forces is that, averaged ve-
locity of vapor molecules in the tangential direction (uv
t) is
non-zero in our simulations. For the highest driven case, uv
t
uv
nis
∼0.1, while in our least driven system it is as high as ∼0.4. To
address this issue, we perform driven MD simulations where
the cross-section area is increased by a factor of two and four.
Effect of changing cross-section area on interfacial properties
and ∆JSc are shown in Tables V (evaporation) and VI (con-
densation).
τ
is 5ns over all three cases. For the largest cross-
section, uv
t
uv
nis ∼0.1. Overall, mismatch between Schrage pre-
dictions and MD decreases with increasing cross-section, but
is not as small as the highest driven case (3rd row of Tables
I and II). This is most likely due to our inability in estimat-
ing the variables from MD simulations with extremely good
accuracy. Error in
ρ
vvalues for the system with largest cross-
section is ∼1% (see Table V). Increasing the value of
ρ
vby
1% changes ∆JSc from 10.60% to 2.5%; clearly indicating that
our current methodology is not very accurate for small driving
forces. Such significant changes in ∆JSc are not observed for
highly driven systems largely due to fact that magnitudes of
fluxes are higher. Systems with larger cross sections (compa-
rable to mean free path of molecules) and more independent
simulations should be considered for obtaining extremely ac-
curate analysis in the smaller driving force regime.
E. Size effects
Interfacial variables do not change (within error bars) with
changing cross-section area, as shown in Tables V and VI.
Doubling the channel length does not affect density, tempera-
ture, or velocity measurements either. Therefore, our simula-
tions do not show any size dependence. However, increasing
the cross-section does increase the accuracy of Schrage pre-
dictions as the error bars in our measurements decrease.
V. SUMMARY AND CONCLUSIONS
In this work, using results of molecular dynamics simula-
tions we discussed the validity of Schrage relationships for
water. Evaporation and condensation rates in the quasi-state
limit were evaluated. Direct determination of all thermo-
dynamic variables at the interface, such as, vapor density,
liquid and vapor temperatures, as well as mass accommo-
dation coefficients are possible from MD simulations. This
allowed us to compare theoretical predictions of evapora-
tion/condensation fluxes from the Schrage relation with those
observed directly in numerical simulations. The predicted
fluxes were up to 20% larger than the observed values for low
driving forces, while highly driven systems showed deviations
of ∼5%. In MD simulations of Argon by Liang et al.25 , de-
viation from Schrage predictions was extremely small (<5%
in all driving conditions). However, molecular non-polar flu-
ids such as n-dodecane26 showed larger deviations for smaller
driving forces. Therefore, in this aspect, our observations
are consistent with prior studies. In experimental literature,
Schrage relations have been reported to deviate by orders of
magnitude17,18,22 mostly due to uncertainties in determination
of mass accommodation coefficients. As compared to these,
our results indicate that the discrepancy between theory and
observed fluxes were relatively small – some of the discrep-
ancy can be attributed to the uncertainties in determination of
various quantities, such as interfacial liquid temperature and
vapor density. We also verified another major assumption in
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Investigating the validity of Schrage relationships for water using molecular dynamics simulations 9
Averaging interval TlTvuv
n
ρ
vJmd JSc %Deviation
2.4ns 429.14 417.01 107.94 1.54 165.97 172.10 3.70
±0.244 ±5.292 ±5.817 ±0.040 ±10.286
4ns 429.20 415.12 107.89 1.53 165.22 174.69 5.73
±0.156 ±4.610 ±4.017 ±0.035 ±6.106
TABLE III: Effect of changing data collection interval (
τ
) on interfacial properties and Schrage predictions for C3(430K,400K)
at the evaporating interface. All values are reported in SI units: temperature in K, velocity in m/s, density in kg/m3, and fluxes
in kg/m2/s.
Averaging interval TlTvuv
n
ρ
vJmd JSc %Deviation
2.4ns 401.30 416.07 113.09 1.45 164.29 172.22 4.83
±0.176 ±4.856 ±5.586 ±0.039 ±9.196
4ns 401.00 414.39 113.84 1.45 164.97 173.44 5.13
±0.169 ±4.086 ±3.585 ±0.035 ±6.430
TABLE IV: Effect of changing data collection interval (
τ
) on interfacial properties and Schrage predictions for C3(430K,400K)
at the condensing interface. All values are reported in SI units: temperature in K, velocity in m/s, density in kg/m3, and fluxes
in kg/m2/s.
the Schrage analysis – the velocity distribution in the vapor
phase near the liquid-vapor interfaces is indeed Maxwellian
with a shifted velocity component normal to the interface.
While we indicate that the most likely source of discrepan-
cies between Schrage predictions and MD mass fluxes is sta-
tistical in their nature, we recognize that there is a possibility
of systematic deviation. The apparent reduction of discrepan-
cies at higher fluxes might be due to cancellation of system-
atic errors – one being inaccuracy of Schrage relations and
the other arising form non-linear effects that can be present in
highly driven systems which are far away from equilibrium.
ACKNOWLEDGMENTS
We are extremely grateful to Dr. Zhi Liang (California State
University, Fresno) for his constructive comments and sug-
gestions. This work was supported by the Office of Naval
Research Thermal Science Program, Award No. N00014-17-
1-2767.
AVAILABILITY OF DATA
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Cross Section Area TlTvuv
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TABLE VI: Effect of changing cross-section on interfacial properties and Schrage predictions for C1(420K,410K) at the
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This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI:10.1063/5.0018726

~5nm
Vapor Region (200nm) Evaporation
Condensation
T
cold Thot
~5nm
Vapor flow
z
4Ï8(50nm) 4Ï7(50nm) 4Ï6(50nm) 4Ï5(50nm)
.6.6

RTRL (Liquid) L1L2
T
cold
Vapor
Region
~0.6nm ~2.5nm ~2.0nm
RL1
RL2
RL3
RL4

370 380 390 400 410 420 430
T(K)
0
0.5
1
1.5
2
2.5
sat(kg/m3)
Our Model
SPC/E-Ewald
TIP4P-Ew
TIP4P/2005
Expt.

370 380 390 400 410 420 430
T(K)
0
1
2
3
4
5
Psat(atm)
Our Model
SPC/E-Ewald
TIP4P-Ew
TIP4P/2005
Expt.

300 320 340 360 380 400 420
T(K)
0.88
0.9
0.92
0.94
0.96
0.98
1

10 20 30 40 50 60 70 80 90
t (ps)
0
0.005
0.01
0.015
0.02
0.025
0.03
Probability of Reflection
0
0.05
0.1
0.15
Running Integral
Probability Histogram
Nref/Ninc
Accommodation Evaporation
Reflection
1 - (430K)

0 5 10 15 20 25 30
[ Tl]i (K)
0
50
100
150
J (kg/m2/s)
Evaporation
0
50
100
150
J (kg/m2/s)
Condensation Jmd JSc

5 10 15 20 25 30
[ Tl]i (K)
0
5
10
15
20
25
JSc (%)
Evaporation Condensation

400
410
420
430
T(K)
TlTv
100
110
120
uv
n (m/s)
50 100 150 200
x(nm)
1.4
1.5
1.6
v(kg/m3)

-1500 -1000 -500 0 500 1000 1500
u(m/s)
0
0.2
0.4
0.6
0.8
1
(s/m)
10-3
uv
n = 111.1m/s
Tv = 400.6K
Jmd = 169.53kg/m2s
RV1

-1500 -1000 -500 0 500 1000 1500
u(m/s)
0
0.2
0.4
0.6
0.8
1
(s/m)
10-3
Tv = 412.4K
RV1