# Equivalence between condensation and boiling in a Lennard Jones fluid

I. Sanchez-Burgos1, P. Montero de Hijes1, P. Rosales-Pelaez1, C. Vega1 and E.
Sanz1 1Departamento de Química Física, Facultad de Ciencias Químicas,
Universidad Complutense de Madrid, 28040 Madrid, Spain

###### Abstract

Condensation and boiling are phase transitions highly relevant to industry,
geology or atmospheric science. These phase transitions are initiated by the
nucleation of a drop in a supersaturated vapor and of a bubble in an
overstretched liquid respectively. The surface tension between both phases,
liquid and vapor, is a key parameter in the development of such nucleation
stage. Whereas the surface tension can be readily measured for a flat
interface, there are technical and conceptual limitations to obtain it for the
curved interface of the nucleus. On the technical side, it is quite difficult
to observe a critical nucleus in experiments. From a conceptual point of view,
the interfacial free energy depends on the choice of the dividing surface,
being the surface of tension the one relevant for nucleation. We bypass the
technical limitation by performing simulations of a Lennard Jones fluid where
we equilibrate critical nuclei (both drops and bubbles). Regarding the
conceptual hurdle, we find the relevant cluster size by searching the radius
that correctly predicts nucleation rates and nucleation free energy barriers
when combined with Classical Nucleation Theory. With such definition of the
cluster size we find the same value of the surface tension for drops and
bubbles of a given radius. Thus, condensation and boiling can be viewed as two
sides of the same coin. Finally, we combine the data coming from drops and
bubbles to obtain, via two different routes, estimates of the Tolman length, a
parameter that allows describing the curvature dependence of the surface
tension in a theoretical framework.

##  I Introduction

Understanding first order phase transitions is of great importance to many
fields, ranging from biology cryopres , to atmospheric science
cantrell2005production , physics debenedettibook , geology icemars2018  or
industry foodcrystallization ; brittain .

In the absence of impurities or external surfaces, first order phase
transitions start with the emergence of a nucleus of the stable phase in the
bulk of the parent metastable phase kelton ; skripov1974metastable . A nucleus
is “critical” if it is big enough so that it has 50 per cent chances to either
grow or redissolve.

Although the emerging phase is more stable, the presence of an unfavourable
interface between the nucleus and the parent phase can delay to a great extent
the phase transition. Thus, for instance, alkane vapors can be saturated
thousands of times over their vapor pressure before condensation takes place
wyslouzil2016overview , alkane liquids can be substantially superheated above
the boiling temperature skripov2010phenomenon ; lipnyagov2018going ;
skripov1992metastable  or liquid water can be supercooled up to $\sim 60$ K
below melting until it freezes manka2012 ; amaya2018ice ; hu2008water ;
bhabheJPCA2013 .

Therefore, the surface tension, $\gamma$, or the free energy per unit area
between both phases, plays a key role in the development of first order phase
transitions. Whereas $\gamma$ can be readily measured for a flat interface at
equilibrium –at least between fluid phases ickespccp2015 – it cannot be
directly probed for curved interfaces, which is the relevant case for
nucleation. Moreover, the fact that critical nuclei are nanoscopic objects
makes it very difficult to observe them in experiments, let alone measuring
their $\gamma$. The usual strategy is to infer $\gamma$ by combining a
theoretical description of nucleation with measurements of the nucleation rate
(the number of nuclei that appear per unit of time and volume) kelton ;
ickespccp2015 ; caupinPRL2016curvature ; granasy:6157 . This approach relies
on the validity of theoretical approximations that are difficult to assess.

Computer simulations do have access to the time and length scales relevant for
the observation of critical nuclei. However, whereas the methodology and
theoretical framework for computing $\gamma$ for flat interfaces is very well
established kirkwood1949statistical ; JCP_2005_123_134703 ; broughton:5759 ;
PhysRevLett.86.5530 ; PNAS_2002_99_12562 ; ubertiPRB2010 ; verrocchioPRL2012 ;
PhysRevLett.94.176105 ; Nature_2001_409_1020 ; espinosaJCP2014_2 , that for
curved interfaces is still under development lau2015surface ;
ceriottitolman2018 ; binder2012beyond . One of the key issues is that $\gamma$
for curved interfaces depends on the location of the interface, that can be
defined in different ways rowlinson2013molecular ; binder2012beyond . The
current situation is that the dependence of $\gamma$ with the curvature of the
interface is contradictory between different groups blokhuis2006thermodynamic
; block2010curvature ; sampayo ; malijevsky2012perspective ; binder2012beyond
; size-dependendentgamma2005 ; ceriottitolman2018 ; JCP_1984_81_00530 ;
lau2015surface ; vrabec2006comprehensive ; wilhelmsen2015tolman ;
joswiak2016energetic ; schmelzer2019entropy ; richard2018crystallization ;
caupinPRL2016curvature

In this work we address fundamental questions regarding the liquid-vapor
interface with computer simulations. It has been shown in different simulation
works that spherical nuclei can be equilibrated at constant volume and
temperature in finite systems block2010curvature ; schrader2009simulation ;
matsumoto2008nano ; troster2012numerical ; macdowell2006nucleation ;
richard2018crystallization ; statt2015finite ; koss2018phase ;
gunawardana2018theoretical ; zierenberg2015exploring ; zierenberg2017canonical
. Recently, we showed with simulations of bubbles seedingNVT  and crystals
montero2020interfacial  that nuclei thus equilibrated are critical, in
agreement with Density Functional Theory (DFT) predictions lutsko2018classical
; lutsko2019crystals . On the other hand, we have extensively developed in the
past years the so-called Seeding method jacs2013 ; knottJACS2012 ; baiJCP2006
; seedingvienes  to study nucleation phenomena. This method consists in
obtaining with simulations the properties of critical clusters and “plug” them
in the Classical Nucleation Theory (CNT) formalism ZPC_1926_119_277_nolotengo
; becker-doring ; gibbsCNT1 ; gibbsCNT2  to obtain predictions of the
nucleaiton rate and of the $\gamma$ curvature dependence. This approach has
been successful for a wide range of systems knottJACS2012 ; jacs2013 ;
seedingvienes ; zaragozaJCP2015 ; espinosaPRL2016 ; espinosaJPCL2017 ;
espinosa2018homogeneous ; seedingNpT ; seedingNVT  and we use it here for the
first time to study condensation. In particular, we apply Seeding at constant
volume both to condensation and to cavitation for a Lennard Jones model.

Since Seeding relies on CNT, it is necessary to validate it by comparing its
predictions with independent calculations that do not rely on such framework.
We do so by computing nucleation rates via Umbrella Sampling (US)
torrie1974monte ; JCP_1992_96_4655  and direct brute force simulations
filion:244115  as well as by testing the consistency of the $\gamma$-curvature
dependence obtained via Seeding with the value for a flat interface.

All consistency tests are successfully passed for our Seeding simulations
provided that the nucleus surface is identified with that where the density is
the average between the density of both phases (“equi-density” surface).
Therefore, we identify the equidensity surface with the surface of tension. On
the other hand, we directly compare the condensation of liquid drops in a
supersaturated vapor with the cavitation of vapor bubbles in an overstretched
liquid. We find that, for a given temperature, drops and bubbles of the same
radius have the same $\gamma$ when using the equidensity definition of the
surface of tension. Finally, we estimate the Tolman lenght tolman1949effect ,
a parameter useful to predict the $\gamma$-curvature dependence, via two
different routes, as recently proposed in montero2020interfacial .

##  II Simulation details

The Lennard Jones model potential, as well as the simulation details, are the
same as in our previous work seedingNpT ; seedingNVT . In particular, we study
the truncated and force-shifted Lennard-Jones (TSF-LJ) potential
wang2008homogeneous , a model for which the vapor-liquid transition has been
previously investigated wang2008homogeneous ; tanaka2015simple ;
meadley2012thermodynamics ; seedingNpT :

$U_{TSF-LJ}(r)=U_{LJ}(r)-U_{LJ}(r_{c})-(r-r_{c})U^{\prime}_{LJ}(r_{c}),$ (1)

where $U_{LJ}(r)$ is the standard 12-6 Lennard-Jones potential and
$U^{\prime}_{LJ}(r)$ is its first derivative. The interaction potential is
truncated and shifted at $r_{c}=2.5\sigma$, being $\sigma$ the particle’s
diameter and $\epsilon$ the depth of the un-truncated Lennard-Jones potential.
Unless otherwise specified, all magnitudes in this work are given in Lennard-
Jones reduced units seedingNVT . Thus, the reported temperatures are reduced
by $\epsilon/k_{B}$, distances by $\sigma$, densities by $\sigma^{-3}$,
pressures by $\epsilon/\sigma^{3}$, times by
$\tau=\sqrt{m\sigma^{2}/\epsilon}$ (being $m$ the particle mass), interfacial
free energies by $\epsilon/\sigma^{2}$ and nucleation rates by
$1/(\tau\sigma^{3})$.

We use cubic boxes with periodic boundary condition and the Molecular Dynamics
(MD) LAMMPS package lammps_program  to perform all simulations of this work.
The equations of motion are integrated with a leap-frog algorithm leapfrog .

In the MD Seeding simulations we used a time-step of $0.0012$. The system was
kept at constant temperature using the Nosé-Hover thermostat JCP_1984_81_00511
with a relaxation time of 0.46.

For the MD simulations used within the Umbrella Sampling scheme we set the
time step for the integration of the motion equations to $0.0012$. The
relaxation times for the Nose-Hover thermostat and barostat were 0.46 and 1.6
respectively.

All simulations are carried out at T=0.785. The coexistence pressure at such
temperature for the model is $p_{coex}$=0.0267. We determined this value,
refined with respect to that of 0.026 previously published wang2008homogeneous
, by running long ($4\cdot 10^{5}$ $\tau$) MD NVT simulations with an
elongated box (50 x 17 x 17) where the vapor and the liquid were put at
contact at the temperature of interest. The average pressure normal to the
interface in such simulation corresponds to $p_{coex}$.

##  III Seeding of condensation

This work is based on a recent publication where we demonstrate how to compute
bubble nucleation rates in an overstretched Lennard-Jones fluid by
equilibrating critical bubbles in the NVT ensemble, an approach we call “NVT-
Seeding” seedingNVT . The Seeding method, originally developed to study
crystal nucleation jacs2013 ; knottJACS2012 ; baiJCP2006 ; seedingvienes , and
more recently applied to vapor cavitation seedingNpT ; seedingNVT ;
baidakov2020molecular , consists in combining CNT ZPC_1926_119_277_nolotengo ;
becker-doring ; gibbsCNT1 ; gibbsCNT2  with computer simulations to estimate
nucleation free energy barrier heights, $\Delta G_{c}$, interfacial free
energies, $\gamma$, and, most importantly, nucleation rates, $J$.

According to CNT, the Gibbs free energy barrier for the nucleation of a
spherical liquid drop is given by the following expression:

$\Delta G=\gamma A-V\Delta p$ (2)

Where $V$ and $A$ are the volume and the area of the drop respectively. By
maximizing Eq. 2 assuming a spherical drop shape one obtains both the height
of the nucleation free energy barrier,

$\Delta G_{c}=\frac{2\pi R_{c}^{3}\Delta p}{3},$ (3)

where $R_{c}$ is the critical droplet radius and $\Delta p$ is the pressure
difference between the interior of the drop and the surrounding vapor, and the
number of particles in the critical drop,

$N_{c}=(32\pi\rho_{l}\gamma^{3})/(3\Delta p^{3}),$ (4)

where $\rho_{l}$ is the critical drop number density and $\gamma$ is the
liquid-vapor surface tension. By substituting in the equation above $N_{c}$ by
the droplet volume ($4/3\pi R_{c}^{3}$) times $\rho_{l}$ one recovers the
Laplace equation:

$\Delta p=\frac{2\gamma}{R_{c}}.$ (5)

This derivation shows that the Laplace equation, which is valid when the
droplet surface is located at the the surface of tension, is implicit in CNT.
Consequently, $R_{c}$ should be identified with the radius of tension,
$R_{s}$. This is an important point that we will use later on in the paper.

The CNT prediction for the nucleation rate of drops is given by kelton :

$J=A_{0}\rho_{vap}\exp\left(-\frac{\Delta G_{c}}{k_{B}T}\right),$ (6)

where $k_{B}$ is the Boltzmann constant, $\rho_{vap}$ is the density of the
vapor phase that multiplied by $\exp\left(-\frac{\Delta G_{c}}{k_{B}T}\right)$
gives the number density of critical clusters and $A_{0}$ is a kinetic pre-
factor.

$A_{0}$ is computed as the product of the Zeldovich factor, $Z$, and the rate
of attachment to the critical nucleus, $f^{+}$ becker-doring ; kelton :

$A_{0}=Zf^{+}.$ (7)

$Z$ takes into account the establishment of a steady state and, according to
CNT, is given by kelton ; ZPC_1926_119_277_nolotengo ; becker-doring :

$Z=\sqrt{\frac{|\Delta G(N)^{{}^{\prime\prime}}|_{N_{c}}}{2\pi
k_{B}T}}=\sqrt{\frac{\Delta p}{6\pi k_{B}T\rho_{l}N_{c}}}=\sqrt{\frac{\Delta
p}{8\pi^{2}k_{B}T\rho_{l}^{2}\cdot R_{c}^{3}}}$ (8)

where $N_{c}$ is the number of particles in the drop and $|\Delta
G_{c}(N)^{{}^{\prime\prime}}|_{N_{c}}$ is the curvature of $\Delta G(N)$
evaluated at the barrier top.

The attachment rate, $f^{+}$, can be estimated by multiplying the collision
frequency of the vapor per unit of wall area given by the kinetic theory of
gasses (ktg) by the area of the critical bubble:

$f^{+}_{ktg}=\sqrt{\frac{k_{B}T}{2\pi
m}}\left(\frac{6\sqrt{\pi}N_{c}}{\rho_{l}}\right)^{2/3}$ (9)

where the subscript “$ktg$” stresses the fact that this expression of the
attachment rate is based on the kinetic theory of gasses.

Combining this equation with 8, 4, and 5, the following kinetic pre-factor is
obtained:

$A^{ktg}_{0}=\sqrt{\frac{\Delta pR_{c}}{\pi m}}\frac{\rho_{vap}}{\rho_{l}}.$
(10)

The equations above are quite powerful, because only $R_{c}$, $\Delta p$ and
the density of both phases are required to obtain key nucleation parameters as
free energy barriers, interfacial free energies and ınucleation rates. The
Seeding method consists in performing simulations of a cluster of the stable
phase surrounded by the mestastable phase (a liquid drop surrounded by
supersaturated vapor in our case as shown in Fig. 1) to compute $R_{c}$,
$\Delta p$, $\rho_{l}$ and $\rho_{vap}$ in order to get “cheap” estimates of
$\Delta G_{c}$, $\gamma$ and, most importantly, $J$ through the expressions
above.

The main drawback of Seeding is that the definition of $R_{c}$ is not unique.
Therefore, the resulting free energy barrier depends on the specific
definition of $R_{c}$. This contrasts with rigorous simulation methods like
Umbrella Sampling JCP_2005_122_194501 ; filion:244115  or with theoretical
approaches like DFT shen2001density ; gallo2020nucleation ; lutsko2008density
where the nucleation free energy does not depend on the criterion chosen to
measure the nucleus size, which can be estimated _a posteriori_ via, e. g. the
nucleation theorem JCP_1982_76_05098 ; kashchiev2006forms ; wedekind2007best
(although in the particular case of DFT an approximate functional needs to be
proposed so that the results do also contain approximations). To assess the
suitability of our choice to compute $R_{c}$ we complement Seeding with
Umbrella Sampling simulations.

Figure 1: Snapshot of a critical drop equilibrated in the NVT ensemble at
T=0.785 surrounded by supersaturated vapor. The droplet radius is about 6.8
and the density of the surrounding vapor 0.0550.

###  III.1 $R_{c}$, $\Delta p$, $\rho_{l}$ and $\rho_{vap}$

We use the $NVT$ ensemble to run the simulations of the drops given that in
such ensemble critical nuclei are naturally equilibrated and stabilised for
long times seedingNVT ; montero2020interfacial . We equilibrate drops in 10
different systems. The edge of the cubic simulation box, $L$, and the total
number of particles in each system, $N_{T}$, are reported in table 1. A large
number of particles is used to minimize finite size effects
zierenberg2015exploring ; marchio2018pressure ; seedingNVT . Each system was
simulated for about $10^{3}$ Lennard Jones times of equilibration and $2\cdot
10^{5}$ of production.

To prepare the initial configuration we cut a spherical liquid drop from a
bulk liquid simulation and insert it in a bulk vapor box removing the
overlapping vapor particles. The liquid drop is cut with a certain tentative
radius, but the precise number of particles in each phase is not crucial given
that equilibrium is reached along the course of the $NVT$ simulation.

From a simulation of a drop surrounded by supersaturated vapor one can obtain
an average radial density profile starting from the center of the drop as that
shown in Fig. 2 (to find the drop centre in each configuration we use a
similar strategy to that described in our previous work seedingNVT  consisting
in this case in identifying the maxima of density profiles computed along each
cartesian coordinate). Following Refs. seedingNpT ; seedingNVT , we obtain
$R_{c}$ from such density profile as the distance at which the density is
average between the liquid and the vapor plateaux. This is indicated with a
vertical dashed line in Fig. 2. We refer to this way of obtaining $R_{c}$ as
the “equi-density” criterion. The $R_{c}$’s thus obtained in our NVT-Seeding
simulations are also reported in Table 1. Other definitions of $R_{c}$ are in
principle as valid as the equi-density criterion seedingNpT ;
lutsko2008density ; gallo2020nucleation . We argue later on in the paper that
our $R_{c}$ definition is a good one because it makes Seeding predictions
consistent with independent calculations of $\gamma$, $J$ or $\Delta G_{c}$.

Figure 2: Density profile of a critical drop equilibrated in the NVT ensemble
at T=0.785 surrounded by supersaturated vapor. The droplet radius, indicated
by a red vertical line in the figure, is given by the point at which $\rho(r)$
takes an average value between both plateaux (equi-density criterion). The
density profile corresponds to the system labelled as IV in table 1

.

Figure 3: Box volume versus time in NpT simulations starting from 40
configurations taken from the NVT-Seeding simulation labelled as IX in Table
1. The imposed pressure is the average viral pressure of the NVT-Seeding run.

(a)
(b)
(c)

Figure 4: (a) $\gamma$ vs vapor pressure obtained from NVT-Seeding data of
droplets surrounded by supersaturated vapor. The surface tension at
coexistence (p=0.0267) is included seedingNpT . (b) $\Delta G_{c}$ vs vapor
pressure. NVT-Seeding and US data are compared. Empty black symbols correspond
to Seeding predictions when the Gibbs dividing (equi-molar) –instead of the
equi-density– surface is employed to identify the cluster radius. (c)
Nucleation rate versus vapor pressure as obtained from NVT-Seeding, US and
spontaneous nucleation.

To get $\Delta p=p_{l}-p_{vap}$ we obtain first the vapor density,
$\rho_{vap}$, by counting the number of particles outside a sphere concentric
with the drop but with a larger radius (we use a sphere radius 7$\sigma$
larger than that of the drop, but we have checked for a few selected cases
that any value beyond $\sim 5\sigma$ gives the same result). $\rho_{vap}$ is
given by the number of particles outside the sphere divided by the $L^{3}$
minus the sphere volume. We then use the bulk vapor equation of state to infer
$p_{vap}$ from $\rho_{vap}$. We report $p_{vap}$ and $\rho_{vap}$ in table 1.
We have checked for all studied systems that $p_{vap}$ coincides with the
overall virial pressure of the system. On the other hand, $p_{l}$ is obtained,
as in our previous work seedingNpT ; seedingNVT , by assuming equal chemical
potential between the critical drop and the surrounding vapor:

$\int^{p_{vap}}_{p_{coex}}\frac{1}{\rho_{vap}(p)}dp=\int^{p_{l}}_{p_{coex}}\frac{1}{\rho_{l}(p)}dp$
(11)

where $p_{coex}$ is the coexistence pressure and $\rho_{vap}(p)$ and
$\rho_{l}(p)$ are the bulk vapor and bulk liquid number densities at pressure
$p$. In table 1 we report $p_{l}$ and $\Delta p$ for all studied systems. Once
$p_{l}$ is known, $\rho_{l}$, also reported in the table, can be easily
computed from the bulk liquid equation of state. In all cases, this
computation of $\rho_{l}$, based on the equality of chemical potential between
both phases, is consistent with that obtained from the density profiles. For
instance, for system IV we get $\rho_{l}=0.0680$, which is fully consistent
with the first plateau observed in the density profile shown in Fig. 2. This
means that the mechanical pressure and the thermodynamic pressure inside the
drop coincide, a matter of current debate for solid-liquid nucleation
presiontermod .

There has been much simulation, theoretical and experimental work devoted to
study the formation of nuclei confined at constant volume vincent2017statics ;
marti2012effect ; macdowell2006nucleation ; finitesizenucreguera ;
reguera2004fusion ; JCP_2003_118_00340 ; binder2012beyond ; bindernuccryst2017
; richard2018crystallization ; yang1985thermodynamical ; gallo2018thermally ;
gallo2020heterogeneous . In Refs. seedingNVT ; montero2020interfacial  we
showed with simulations that nuclei equilibrated in the NVT ensemble are
critical because they have equal chances to grow or shrink when simulated in
the NpT ensemble at the same temperature and at the average pressure along the
NVT run. Based on this result, we opted to study here drop nucleation in the
NVT ensemble, where statistics is better because clusters remain stable for
very long times seedingNVT . Stabilising nuclei to gain time to study their
properties is something quite desirable. An alternative strategy to the use of
constant volume simulations is to pin the nucleus to a heterogeneous solid
substrate xiao2017experiments .

Despite having already shown the equivalence between stable (NVT) and critical
(NpT) nuclei for cavitation seedingNVT , we check here for one of the NVT-
Seeding simulations if the drops equilibrated at constant volume and
temperature do correspond to a Gibbs free energy maximum. In Fig. 3 we show
the evolution of the box volume in NpT simulations started from 40 independent
configurations gathered along the NVT-Seeding trajectory labelled as IX in
Table 1. The imposed pressure is the average virial pressure along the NVT-
Seeding run. Roughly, in 50 per cent of the cases the box expands (the drop
dissolves) and in the other half of the cases the box shrinks (the drop
grows). This result supports the use of NVT to study drop condensation in the
same manner we did for bubble cavitation and cystal nucleation seedingNVT ;
montero2020interfacial . Furthermore, the equivalence between clusters
equilibrated at constant volume and critical nuclei has been recently proven
with DFT theoretical arguments for crystallization (see supplementary material
of Ref. lutsko2019crystals ).

Label | $L$ | $N_{T}$ | $\rho_{l}$ | $\rho_{vap}$ | $p_{l}$ | $p_{vap}$ | $\Delta p$ | $R_{c}$ | $\gamma$ | $\Delta G_{c}/(k_{B}T)$ | $A_{0}^{ktg}$ | $A_{0}^{af}$ | $\log_{10}(J)$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
I | 38.019 | 3774 | 0.6833 | 0.05690 | 0.09279 | 0.03161 | 0.0612 | 5.86 | 0.1792 | 32.8 | 0.028 |  | -17.1
II | 39.160 | 4291 | 0.6811 | 0.05496 | 0.0846 | 0.03092 | 0.0537 | 6.82 | 0.1831 | 45.5 | 0.028 |  | -22.6
III | 39.160 | 4373 | 0.6801 | 0.05414 | 0.08095 | 0.03063 | 0.0503 | 7.15 | 0.1800 | 49.2 | 0.027 |  | -24.2
IV | 39.160 | 4510 | 0.6797 | 0.05384 | 0.07961 | 0.03052 | 0.0491 | 7.48 | 0.1835 | 54.8 | 0.027 | 0.030 | -26.6
V | 39.160 | 4623 | 0.6792 | 0.05342 | 0.07769 | 0.03037 | 0.0473 | 7.79 | 0.1843 | 59.7 | 0.027 |  | -28.8
VI | 39.160 | 4796 | 0.6782 | 0.05277 | 0.07468 | 0.03013 | 0.0446 | 8.20 | 0.1827 | 65.5 | 0.027 | 0.036 | -31.3
VII | 39.160 | 4964 | 0.6783 | 0.05269 | 0.07432 | 0.03010 | 0.0442 | 8.49 | 0.1878 | 72.3 | 0.027 |  | -34.2
VIII | 39.160 | 5163 | 0.6776 | 0.05216 | 0.07181 | 0.02991 | 0.0419 | 8.86 | 0.1856 | 77.8 | 0.027 |  | -36.6
IX | 39.160 | 5435 | 0.6771 | 0.05176 | 0.06989 | 0.02976 | 0.0401 | 9.30 | 0.1865 | 86.0 | 0.026 |  | -40.2
X | 85.264 | 34519 | 0.6761 | 0.05105 | 0.06638 | 0.02949 | 0.0369 | 9.99 | 0.1843 | 98.2 | 0.026 |  | -45.5
Table 1: NVT-Seeding data for the different drops studied in this work at
T=0.785.

Having computed $R_{c}$, $\Delta p$ and $\rho_{vap}$ and $\rho_{l}$ we have
everything needed to obtain $\gamma$, $\Delta G_{c}$ and $J$ according to the
equations presented in section III. We report the values for these variables
in table 1 and plot them in fig. 4(a)-(c) versus the vapor pressure with black
dots. In the following section we comment each of these graphs.

###  III.2 $\gamma$, $\Delta G_{c}$ and $J$

####  III.2.1 $\gamma$

As shown in Fig. 4 (a) the prediction we obtain from Seeding is that $\gamma$
decreases as the vapor supersaturation increases. This trend is in agreement
with previous work block2010curvature ; troster2012numerical . Accordingly,
using the capillarity approximation (i. e., that $\gamma$ is pressure
independent) would be erroneous. The green square in Fig. 4(a) corresponds to
the surface tension at coexistence seedingNpT  obtained through the pressure
tensor walton1983pressure  in an NVT simulation of a liquid and a vapor at
contact. The trend of the Seeding data is fully consistent with the
coexistence value, as shown by the linear fit in the figure. This is a good
consistency test, although the $\gamma$ values provided by Seeding could still
be incorrect despite the fact that they extrapolate correctly to coexistence.
Therefore, a test for Seeding predictions away from coexistence is needed.

####  III.2.2 $\Delta G_{c}$

To further test our Seeding results we compare $\Delta G_{c}$ obtained with
Seeding with that computed via Umbrella Sampling. In Fig. 4(b), where we plot
$\Delta G_{c}$ versus the vapor pressure, black solid dots correspond to
Seeding and red ones to US (details on US calculations are described in
section IV). Whereas Seeding predictions rely on the validity of CNT and on a
proper definition of $R_{c}$, Umbrella Sampling calculations are rigorous and
independent on the criterion to identify the nucleus size filion:244115 . On
the other hand Seeding is much “cheaper” than US from a computational point of
view. As a matter of fact, Seeding has access to much higher nucleation
barriers than US. The accordance between Seeding and US shown in Fig. 4 is
excellent, which gives us great confidence on Seeding predictions. The choice
of the equi-density surface to identify the drop radius has proven correct. If
we use another criterion, such as the Gibbs (equi-molar) dividing surface, the
agreement between Seeding and Umbrella Sampling deteriorates (empty black
symbols in Fig. 4 (b)). To compute $R_{e}$, the radius associated to the Gibbs
dividing surface, we use $N_{T}=N_{l}+N_{vap}$ where $N_{l}=4/3\pi
R_{e}^{3}\rho_{l}$ and $N_{vap}=[V_{T}-(4/3\pi R_{e}^{3})]\rho_{vap}$, where
$V_{T}$ is the volume of the simulation box and the densities $\rho_{l}$ and
$\rho_{vap}$ are obtained as described in Sec. III.

In a recent publication on cavitation (nucleation of bubbles instead of drops)
we compared the performance of different criteria to identify the cluster
radius and found that the equi-density criterion also made Seeding predictions
consistent with other rigorous calculations seedingNpT . Therefore,
identifying the critical drop radius with the equi-density distance seems to
be quite general for condensation-evaporation transitions.

####  III.2.3 J

Once $\Delta G_{c}$ is known computing $J$ via Eq. 6 is quite straight
forward. The kinetic pre-factor $A_{0}$ given by the kinetic theory of gases,
Eq. 10, depends on parameters we already have under control: $\Delta p$,
$R_{c}$ and the density of both phases. The values of $A_{0}$ computed via Eq.
10, $A_{0}^{ktg}$, are reported in Table 1.

These $A_{0}$ values are approximate since they rely on the validity of the
kinetic theory of gasses to estimate the attachment rate (see section III). We
therefore have to check $A_{0}^{ktg}$ by computing the attachment rate with an
alternative approach. Following the work by Auer and Frenkel auerJCP2004 , the
attachment rate can be computed from the diffusion of $N$, the number of
particles in the liquid drop, around the critical drop auerJCP2004 :

$f^{+}_{af}=\frac{\left<(N(t)-N(0))^{2}\right>_{Nc}}{2t},$ (12)

where the average is performed over several trajectories starting from a
critical drop configuration. The $af$ subscript stresses the fact that this
expression of the attachment rate is based on the work by Auer and Frenkel.

To compute $N$ we follow JCP_1998_109_09901 . We count as neighbors all
particles within a 1.625 distance of a tagged particle. Particles with 8 or
more neighbors are labelled as “liquid”. Two liquid particles belong to the
same drop if their mutual distance is less than 1.625. An example of the
calculation of $f^{+}$ according to equation 12 is illustrated in Fig. 5.
Typically, $\left<N(t)-N(0)\right>_{N_{c}}$ is obtained by averaging 20 NpT
runs started from independent configurations of the critical drop, coming
either from NVT-Seeding or from Umbrella Sampling simulations (see section
IV). In these runs, the pressure is fixed to the virial value of the
simulations were the critical clusters were previously equilibrated. According
to Eq. 12, the slope of Fig. 5 divided by 2 gives $f^{+}$. Multiplying such
$f^{+}$ by the Zeldovich factor we get an estimate of the kinetic pre-factor,
$A_{0}^{af}$, that does not rely on the kinetic theory of gases. $A_{0}^{af}$
is reported it in table 1 for a couple of critical clusters generated with
NVT-Seeding (systems IV and VI). $A_{0}^{af}$ is very close to $A_{0}^{ktg}$.
This agreement suggests the validity of the kinetic theory of gases to
estimate the attachment rate and makes the theoretical framework that supports
the Seeding technique quite powerful given that, since $A_{0}^{ktg}$ can be
used, only $R_{c}$, $\Delta p$ and the density of both phases are required to
get accurate estimates of $J$ in a wide range of orders of magnitude. Note in
Fig. 4(c) that Seeding (black dots) has access to $J$ values many orders of
magnitude lower than US (red dots).

Label | $N_{T}$ | $\langle V\rangle$ | $p_{v}$ | $\rho_{v}$ | $\log_{10}(J)$
---|---|---|---|---|---
BF-1 | 4000 | 57145 | 0.035 | 0.0700 | -9.235
BF-2 | 4000 | 53456 | 0.036 | 0.0748 | -8.287
Table 2: Data corresponding to the brute force calculations.

The green dots in Fig. 4(c) correspond to rate estimates obtained in brute
force NPT molecular simulation runs performed at high supersaturations where
condensation occurs spontaneously from an unseeded vapor. In such cases the
nucleation rate can be estimated as $J=1/(t<V>)$, where $<V>$ is the average
volume before nucleation and $t$ is the nucleation time averaged over a number
of independent trajectories (typically 20 in our case). $N_{T}$, $V$, the
vapor pressure and density, and $J$ for the two states where we studied
spontaneous condensation are reported in Table 2. In Fig. 4(c) we show that
$J$ estimates from Seeding and from spontaneous nucleation are consistent with
each other, which further indicates the ability of Seeding to predict
nucleation rates. It is worth mentioning here that NVT-Seeding and spontaneous
nucleation are complementary techniques. On the one hand, the former does not
have access to such high supersaturations given the difficulty to equilibrate
small clusters in the NVT ensemble seedingNVT ; montero2020interfacial . That
said, it would be nonsense using Seeding where nucleation occurs spontaneously
in a straightforward manner. On the other hand, spontaneous nucleation is
limited to a narrow window of nucleation rates (that enabled by computational
time) whereas Seeding has access to extremely low rates.

We would like to end this section by discussing finite size effects, which
could be present if a nucleus sees its replica through periodic boundary
conditions. On the one hand, we made sure that the density of the outer phase
reaches a plateau before L/2 by looking at radial density profiles such as
that shown in Fig. 2. On the other hand, we note that the box side of system X
is more than twice than those of the other systems. By looking at Figs. 4 (a),
(b) and (c) one can see that the results from system X are fully consistent
with those inferred from the other systems, which strongly supports the
absence of noticeable finite size effects in our simulations.

Figure 5: Time dependence of the mean squared deviation of the number of
particles in the critical drop for system VI in table 1. Half the slope of
this plot gives the attachment rate according to Eq. 12

.

Label | $L$ | $N_{T}$ | $\Delta G_{c}/(k_{B}T)$ | $A_{0}^{af}$ | $\log_{10}(J)$
---|---|---|---|---|---
US-1 | 39.112 | 4000 | 17.7 | 0.041 | -10.3
US-2 | 38.501 | 4000 | 40.7 | 0.039 | -20.3
Table 3: Data corresponding to the US calculations.

##  IV Umbrella Sampling

As previously indicated, to validate the Seeding results we used the US
technique. We followed Refs. JCP_1998_109_09901 ; gonzalezPCCP2014  to compute
$\Delta G_{c}$ for two different vapor pressures: p=0.031 and p=0.033. Details
on the simulation box size and number of particles in the systems used to
perform the US calculations are given in table 3.

The free energy associated to the formation of an $N$ particle cluster drop
can be obtained from:

$\Delta G(N)=-k_{B}T\ln[P(N)],$ (13)

where $P(N)$ is the probability distribution of $N$. Our criterion to compute
$N$ is described in Section III.2.3. It is important to note that even though
different criteria may give different $N$ for a given configuration, the
height of an US free energy barrier does not depend on the criterion to
determine the cluster size filion:244115 . Therefore, contrary to what happens
in Seeding, the US method does not depend on the specific criterion to
determine the nucleus size. This is why it is important to validate the
Seeding method with other tecniques such us US.

With conventional NpT simulations at the selected pressures $P(N)$ can only be
sampled up to $N\sim 40$ while the critical cluster is much larger in this
regime. To sample the rest of the free energy barrier a biasing potential,
$U_{bias}$, is added to the original hamiltonian:

$U_{bias}=\frac{1}{2}k_{bias}\left(N-N_{0}\right)^{2},$ (14)

where $N_{0}$ controls the cluster size around which the sampling will be
centred and $k$ the width of such sampling. Tens of overlapping sampling
“windows” centered at different $N_{0}$ values are required to reconstruct the
whole free energy barrier. The effect of the bias potential on the calculation
of the free energy barrier is removed as follows torrie1974monte :

$\Delta G(N)=-k_{B}T\ln\left<\frac{\chi_{N}}{e^{-U_{bias}/(k_{B}T)}}\right>+C$
(15)

where $\chi_{N}$ is the fraction of clusters with $N$ particles that appear
within a certain window and $C$ is a constant. The constant is obtained by
gluing together the first part of the energy barrier evaluated without the
biasing potential (Eq. 13 ) with the rest of the windows. The result is the
whole free energy barrier.

To compute each window we use the hybrid Molecular Dynamics-Monte Carlo scheme
labelled as HMC(nM-NpT)/US in Ref. gonzalezPCCP2014 . From the starting
configuration, random velocities are assigned to every particle according to a
Maxwell-Boltzmann distribution and a short ($\Delta t$ 19.2 Lennard Jones
times) MD simulation is run for generating a new configuration, which is
accepted with probability $\min[1,\exp[-(U_{bias}(\Delta
t)-U_{bias}(0))/(k_{B}T)]]$. Either in case of acceptance or rejection new
random velocities are assigned at the beginning of each short MD cycle. For
each window, 10000 of such cycles were performed for equlibration and 60000 to
obtain the free energy barrier. We used $k_{bias}=0.04k_{B}T$ in the biasing
potential (Eq. 14), which gives an acceptance rate of $\sim 25\%$.

In figure 6 we plot both free energy barriers, being $\Delta G_{c}=17.7k_{B}T$
for p=0.033 and $\Delta G_{c}=40.7k_{B}T$ for p=0.031 (also reported in table
3). As already discussed, the agreement between US and Seeding is excellent
(see Fig. 4(b)).

Additionally, we compute the kinetic pre-factor $A_{0}^{af}$ (Eq. 7) to obtain
the nucleation rate (Eq. 6). To do that, we launch tens of unbiased
trajectories from independent configurations at the barrier top in order to
compute the attachment rate via Eq. 12. The Zeldovich factor (Eq. 8) can be
obtained by numerically calculating the curvature of $\Delta G(N)$ at the
barrier top. We report $A_{0}^{af}$ thus calculated and the corresponding $J$
in table 3. As previously discussed, $J$ from US is fully consistent with that
coming from Seeding (see Fig. 4 (c)).

In summary, we have compared Seeding, that relies on the theoretical
assumptions by CNT and $ktg$ and depends on the criterion employed to
determine the cluster size, with US, that does not have these limitations. We
have obtained an excellent agreement between both techniques. This is very
good news because Seeding is much more efficient than US and has access to
much lower values of the nucleation rate.

Figure 6: Free energy for two different pressures (p=0.031 and p=0.033) versus the number of particles in the drop as obtained from US calculations. The different colours represent the different windows performed. Label | $L$ | $N_{T}$ | $\rho_{vap}$ | $\rho_{l}$ | $p_{vap}$ | $p_{l}$ | $\Delta p$ | $R_{c}$ | $\gamma$ | $\Delta G_{c}/(k_{B}T)$ | $A_{0}^{BK}$ | $\log_{10}(J)$
---|---|---|---|---|---|---|---|---|---|---|---|---
I | 36.731 | 30795 | 0.03765 | 0.6453 | 0.02365 | -0.02601 | 0.0497 | 7.35 | 0.1826 | 52.7 | 0.341 | -23.5
II | 36.731 | 30342 | 0.03834 | 0.6484 | 0.02398 | -0.01914 | 0.0431 | 8.50 | 0.1832 | 70.6 | 0.342 | -31.3
III | 36.731 | 29760 | 0.03875 | 0.6501 | 0.02419 | -0.01503 | 0.0392 | 9.53 | 0.1869 | 90.6 | 0.345 | -40.0
IV | 36.731 | 29034 | 0.03907 | 0.6514 | 0.02433 | -0.01191 | 0.0362 | 10.52 | 0.1906 | 112.5 | 0.348 | -49.5
V | 36.731 | 28147 | 0.03949 | 0.6530 | 0.02453 | -0.00776 | 0.0323 | 11.50 | 0.1857 | 131.1 | 0.344 | -57.6
VI | 36.731 | 27082 | 0.03972 | 0.6539 | 0.02464 | -0.00558 | 0.0302 | 12.45 | 0.1881 | 155.7 | 0.346 | -68.3
Table 4: NVT-Seeding data for the different bubbles studied in this work at
T=0.785.

(a)  (b)
(c)  (d)

Figure 7: (a) $\Delta p$ vs. $1/R_{c}$, (b) $\gamma$ vs. $1/R_{c}$, (c)
$\Delta G_{c}$ vs. $R_{c}$, and (d) $log_{10}J$ vs. $1/R_{c}$ for droplets
(black symbols) and bubbles (red symbols) as obtained from NVT-Seeding.

##  V Condensation vs boiling

###  V.1 Comparison for a given $R_{c}$

We have studied quite recently the nucleation of bubbles for the same Lennard
Jones model employed here seedingNVT . Since the study was performed at the
same temperature, the question that naturally arises is whether bubbles and
drops with the same radius have the same interfacial properties. To establish
the comparison we have repeated the analysis performed in Ref. seedingNVT
because in such work we used 0.026 as the coexistence pressure instead of
0.0267. We took the 0.026 value from a paper published more than a decade ago
wang2008homogeneous . However, we have recomputed more carefully the
coexistence pressure at $T=0.785$ and obtain $p=0.0267$ instead, which is the
value we use in this work. The difference is subtle, but given that the
pressure inside the nucleating phase is obtained by integrating from the
coexistence pressure (see Eq. 11) it is very important to use an accurate
value for the latter.

The simulation data for different bubbles equilibrated at T=0.785 in the NVT
ensemble are reported in table 4. The values of $R_{c}$ corresponding to each
system (obtained with the equi-density criterion as discussed in section III.1
and in Ref. seedingNVT ) are very close to those recently reported by
ourselves seedingNVT . However, the values of $\Delta p$ here reported are not
identical to those of Ref.seedingNVT  due to the coexistence pressure issue
discussed above. In Fig.7(a) we plot $\Delta p$ vs $1/R_{c}$ for bubbles and
and drops at T=0.785. Drops and bubbles of the same size have the same $\Delta
p$, which is perhaps the most important result of the paper. Note that, for a
given $R_{c}$, the pressures of the external and the internal phases are not
the same if one compares cavitation and condensation. What is the same is the
pressure difference between the internal and the external phases. For
instance, let’s focus on the case of drop VII and bubble II, both with
$R_{c}\approx 8.5$. In Fig. 8 we compare their radial density profiles. The
density of the liquid inside the drop is different from that of the liquid
outside the bubble. Also, the density of the vapor inside the bubble is
different from that of the vapor outside the drop. The bubble is surrounded by
a liquid of pressure -0.01914 whereas the drop by a vapor of pressure 0.0301:
the pressures of the external phases do not even have the same sign. The
bubble and the drop also have very different pressures: 0.02398 and 0.07432
respectively. Despite the fact that the external and the internal pressures
are very different, $\Delta p$ is not: 0.043 and 0.044 for the bubble and the
drop respectively.

Figure 8: Radial density profiles of drop VII and bubble II, compared. They
have almost identical radius, $R_{c}$.

According to the Laplace equation, that $\Delta p(R_{c})$ is the same for
drops and bubbles, implies that $\gamma$ must be the also same regardless the
identity of the internal and the external phases. In Fig. 7 (b) we plot
$\gamma$ vs $1/R_{c}$ for bubbles and drops and find that, indeed, they have
the same $\gamma$ within our statistical noise. Of course, attending to Eq. 3,
$\Delta G_{c}$, that only depends on $R_{c}$ and $\Delta p$, is also the same
for a given $R_{c}$, as illustrated in Fig. 7(c).

The nucleation rate for bubbles with a given $R_{c}$ is close to the
corresponding drop, but is not exactly the same, given that the kinetic pre-
factor is not identical. In the case of bubble nucleation we have carefully
assessed seedingNpT  that the following expression by Blander and Katz (BK)
provides a good approximation for $A_{0}$:

$A_{0}^{BK}=\sqrt{\frac{\Delta pR_{c}}{\pi m}},$ (16)

which is very similar, but with a missing $(\rho_{vap}/\rho_{l})$ factor with
respect to the $ktg$ expression we use for drop condensation (Eq. 10). The
$A_{0}^{BK}$ values we use in our Seeding predictions of bubble cavitation are
reported in table 4 alongside the resulting values of $J$ obtained as
$J=\rho_{l}A_{0}^{BK}\exp[{-\Delta G_{c}/(k_{B}T)}$]. As it can be seen in
Fig. 7(d), the rate for bubbles and drops for a given $R_{c}$ is quite
similar, although it is systematically lower for the latter due to the
$\rho_{vap}/\rho_{l}$ factor previously mentioned.

Condensation and cavitation have already been compared in the literature
guermeur1985density ; schmelzer2019entropy ; macdowell2006nucleation ;
JCP_2004_120_05293 ; block2010curvature ; binder2012beyond ; caupin2015 .
However, there are only few cases in which $\gamma$ has been compared for a
given temperature as a function of the droplet/bubble size block2010curvature
; binder2012beyond  as we do in this work. Establishing such comparison in
experiments is difficult because it is not possible to detect the critical
nucleus. In simulations the nucleus can be visualized, but computing $\gamma$
is a hard task. It requires either computing the free energy of a system with
the nucleus inside blokhuis2006thermodynamic ; binder2012beyond  or, more
easily, computing the nucleus size and using a theory to infer $\gamma$
baiJCP2006 ; carignano  as we do in this work. In either approach, one has to
deal with the arbitrariness of establishing a location for the interface.

In our case, we found in a recent work by “trial and error” that the equi-
density surface gives good results for cavitation seedingNpT . By “good
results” we mean that Seeding predictions of nucleation are consistent with
those coming from independent methods that do not rely on a precise definition
of the nucleus size. In this work we have demonstrated that the same criterion
to locate the interface is successful in condensation. Therefore, one of our
main findings is that the equi-density surface is the one that provides good
predictions when CNT is used both for cavitation and for condensation. This
means that the equi-density surface can be identified with the surface of
tension, which is the one for which CNT works and the Laplace equation holds
(see section III) kashchievbook ; montero2020interfacial ;
troster2012numerical ; statt2015finite  . We believe that identifying the
surface of tension with the equi-density surface both for cavitation and
condensation is an important finding of our work. This leads to the relevant
conclusion that condensation and cavitation are two sides of the same coin in
the sense that they share the same surface tension.

In Ref. block2010curvature  $\gamma$ was found to be quite different for both
phenomena, but the comparison was not established for the surface of tension
but for the equimolar surface. In Ref. binder2012beyond , however, the
comparison was established for the first time for the surface of tension and,
although $\gamma$ was similar for condensation and cavitation, there were
significant differences that need to be further investigated in order to match
our work with that of Ref. binder2012beyond .

###  V.2 Comparison for a given metastability degree

In Ref. shen2001density  it was proposed in a DFT study that the work of
formation of critical bubbles studied at different temperatures collapse when
plotted against the metastability degree, $X_{m}$, quantified as:

$X_{m}=\frac{\mu_{nuc}-\mu_{coex}}{\mu_{spinodal}-\mu_{coex}}$ (17)

where $\mu_{nuc}$ is the chemical potential of the parent phase at the
conditions where nucleation is studied, $\mu_{coex}$ is the coexistence
chemical potential at the same temperature and at coexistence pressure, and
$\mu_{spinodal}$ is the chemical potential at the same temperature but at the
pressure where spinodal decomposition takes place. To estimate the spinodal
pressure we run NpT simulations of the bulk liquid and vapor phases with 4000
particles. We estimate the spinodal decomposition pressure as that for which
we the system undergoes a phase transition without any induction period, right
after the start of the simulation. Both chemical potential differences in Eq.
17 can be easily obtained by numerically integrating the molar volume along
pressure at constant temperature. The denominator is the maximum possible
metastability whereas the numerator is the actual metastability of the state
where nucleation is studied. Therefore, $X_{m}$ varies from 0 at coexistence,
to 1 at spinodal decomposition. The mestastability degree above described can
be computed for drop as well as for bubble nucleation. Therefore, we have the
chance to compare nucleation free energy barriers for drops and bubbles as a
function of $X_{m}$. The comparison, shown in Fig. 9, reveals the interesting
conclusion that $\Delta G_{c}$ for bubble and drop nucleation is the same for
a given metastbility degree. Therefore, not only nucleation barriers at
different temperatures can be collapsed via the metastability degree as
proposed in Ref. shen2001density , but also bubble and drop nucleation data
match for a given metastability degree.

Figure 9: Nucleation free energy barrier for drops and bubbles (see legend) as
a function of the metastability degree, $X_{m}$, defined in Eq. 17.

###  V.3 Tolman Legth

Since bubbles and drops of the same radius have the same interfacial
properties, we can use the data coming from both systems altogether in order
to compute the Tolmann Lenght, $\delta_{T}$, which is defined as
tolman1949effect ; blokhuis2006thermodynamic :

$\delta_{Tolman}=\lim_{R_{s}\rightarrow\infty}(R_{e}-R_{s})$ (18)

where $R_{e}$ is the Gibbs equi-molar radius and $R_{s}$ is the radius of the
surface of tension. We identify $R_{s}$ with $R_{c}$ (the equi-density radius)
given that (i) we obtain good predictions of nucleation when we use $R_{c}$
and (ii) $R_{s}$ is the radius that enters CNT kashchievbook ;
montero2020interfacial ; troster2012numerical ; statt2015finite . To underline
the fact that we identify $R_{c}$ with $R_{s}$ we label $R_{c}$ as $R_{s=c}$
in the following figures. $R_{e}$ can be easily computed from the radial
density profiles seedingNpT ; montero2020interfacial . In Fig. 10(a) we show
$R_{e}-R_{s=c}$ versus 1/$R_{s=c}$ for all data (either bubbles or drops)
coming from this work. The extrapolation to $1/R_{s=c}=0$ provides an estimate
of $\delta_{Tolman}$, indicated with an empty blue dot in the figure. We
obtain $\delta_{Tolman}=0.15\pm 0.02$. We showed in a recent paper, in which
we analysed spherical hard sphere crystals in equilibrium with the fluid, that
$\delta_{Tolman}$ can be also estimated by fitting $\gamma$ to the following
expression:

$\gamma=\gamma_{0}\left(1-2\frac{\delta_{T}}{R_{s}}\right),$ (19)

where $\gamma_{0}$ is the value of $\gamma$ at coexistence at the temperature
of interest and $\delta_{T}$ is the fitting parameter that serves as an
estimate for $\delta_{Tolman}$ montero2020interfacial . This approach is
similar in spirit to those that include $\gamma$ given by Eq. 19 in CNT to fit
free energy barriers obtained by rare event methods gallo2020nucleation ;
menzl2016molecular . Again, we identify here $R_{s}$ with $R_{c}$.
Consequently, we use the $\gamma$ data coming from such radius (that reported
in tables 1 and 4) to obtain an estimate of $\delta_{T}$ with the expression
above. The data of $\gamma$ vs $1/R_{s=c}$ are shown in green in Fig. 10(b).
The solid line is a linear fit of $\gamma$ vs $1/R_{s=c}$ which includes
$\gamma_{0}$ (the green square in the figure). The $\delta_{T}$ value coming
from such fit, $\delta_{T}=0.21\pm 0.03$, is shown with a red dot in Fig.
10(a). Both values, $\delta_{Tolman}$ obtained via Eq. 18 (blue dot in Fig.
10(a)) and $\delta_{T}$ coming from Eq. 19 (red dot the same figure), are
consistent with each other within the statistical uncertainty of our
estimates. This corroborates the idea, recently checked for the first time for
hard sphere crystals montero2020interfacial , that the Tolman lenght can be
obtained either from Eq. 19 or from Eq. 18. Hence, this idea seems to be a
general one pertaining not only to the crystal-fuid equilibrium but also to
the liquid-vapor one.

This study may shed some light in the intense literature debate about the
magnitude and sign of the Tolman length blokhuis2006thermodynamic ;
block2010curvature ; sampayo ; malijevsky2012perspective ; binder2012beyond ;
size-dependendentgamma2005 ; ceriottitolman2018 ; JCP_1984_81_00530 ;
lau2015surface ; vrabec2006comprehensive ; wilhelmsen2015tolman ;
joswiak2016energetic ; schmelzer2019entropy ; richard2018crystallization ;
caupinPRL2016curvature . We obtain a Tolman length of about twenty per cent
the particle diameter. Its sign is positive, which means that $\gamma$
decreases when one moves away from coexistence at constant temperature.

(a)
(b)

Figure 10: (a) $R_{e}-R_{s=c}$ and (b) $\gamma$ vs. $1/R_{s=c}$ for drops and
bubbles together.

##  VI Conclusions

The main conclusions we draw from our work are the following:

  1. 1.

We have used NVT-Seeding to investigate droplet nucleation in a supersaturated
Lennard-Jones vapor. The results obtained from this technique are consistent
with: (i) independent calculations of the nucleation free energy barrier
performed with Umbrella Sampling (ii) the surface tension of a flat interface
obtained from the pressure tensor in a vapor-liquid coexistence simulation
(iii) the drop nucleation rate obtained both with US and in brute force
spontaneous nucleation simulations.

  2. 2.

NVT-Seeding requires defining the radius of a droplet equilibrated in the NVT
ensemble. The radius definition that passes the consistency tests mentioned in
the previous paragraph is that given by the surface where the density is
average between that of the interior and that of the exterior phases. Such
radius definition was also successful in our earlier studies of bubble
nucleation seedingNpT ; seedingNVT . Therefore, we identify this “equi-
density” radius with the radius of tension, $R_{s}$.

  3. 3.

The good performance of Seeding strongly supports the use of CNT to describe
nucleation. However, the capillarity approximation (that $\gamma$ is curvature
independent) does not provide good results. A $\gamma$ dependent on the
curvature of the critical nucleus must be plugged into the the theory.
Therefore, the theory, although powerful, requires the involvement of
simulations given that the $\gamma$-curvature dependence is obtained by
computing the size of the critical cluster at different pressures.

  4. 4.

The kinetic theory of gases provides very good estimates of the kinetic pre-
factor of the condensation nucleation rate. This makes the theoretical
framework very powerful given that only the size of the critical cluster, the
density of the external phase and the bulk phases equations of state are
needed to estimate nucleation rates.

  5. 5.

We compare NVT-Seeding results of droplets with those obtained for bubbles and
find that, for a given temperature, bubbles and droplets of the same radius
have, within the accuracy of our method, the same pressure difference with the
surrounding medium. Therefore, bubbles and droplets of the same size have the
same surface tension and the same nucleation free energy barrier. In this
respect, condensation and boiling can be seen as two sides of the same coin.
Such duality is only verified if the size of the critical nucleus (either a
bubble or a drop) is determined with the equi-density radius (our empirical
definition of the surface of tension).

  6. 6.

We estimate the Tolman length, $\delta_{T}$, by extrapolating to infinite-size
drops/bubbles the difference between the equimolar radius, $R_{e}$, and
$R_{s}$. Such $\delta_{T}$ is consistent with that obtained by linearly
fitting $\gamma(1/R_{c})$, in accordance with our recent work of hard sphere
crystals montero2020interfacial .

##  VII Acknowledgments

This work was funded by grants FIS2016/78117-P and PID2019-105898GB-C21 of the
MEC. The authors acknowledge the computer resources and technical assistance
provided by the RES. P. R. thanks a doctoral grant from UCM. P. M. H.
acknowledges financial support from the FPI grant no. BES- 671 2017-080074. We
also thank the reviewers of this work for their suggestions.

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