# Are nucleation bubbles in a liquid all independent?

Joël Puibasset11footnotemark: 1<EMAIL_ADDRESS>
###### Abstract

The spontaneous formation of tiny bubbles in a liquid is at the root of the
nucleation mechanism during the liquid-to-vapor transition of a metastable
liquid. The smaller the bubbles the larger their probability to appear, and
even for moderately metastable liquid, it is frequent to observe several tiny
bubbles close to each other, suggesting that they are not all independent. It
is shown that these spatially correlated bubbles should be seen as belonging
to one single density depression of the liquid due to fluctuations (called LDR
for Low Density Region) and should be counted as one event instead of several.
This has a major impact on the characterization of the bubble density in a
liquid, with consequences (i) for understanding liquid-to-vapor transitions
which proceed through growing and merging of these correlated bubbles, and
(ii) for free energy profile and barrier calculations with molecular
simulation techniques which require to convert the calculated size
distribution of the largest bubble into the size distribution of any bubble.
Remarkably, the average number of LDRs in a given volume simply relates to the
probability of not having bubbles in the liquid.

###### keywords:

nucleation , bubble , cavitation , phase transition , molecular simulation

††journal: Journal of Molecular Liquids

[label1]organization=ICMN, CNRS, Université d’Orléans,addressline=1b rue de la
Férollerie, CS 40059, city=45071 Orléans Cedex 02, country=France

##  1 Introduction

The structure of fluids is an important feature in relation with their
thermodynamic properties, not only in terms of atomic or molecular pair
correlations, but also in terms of voids or cavities between molecules [1, 2,
3, 4, 5, 6]. One can cite for example their importance to understand the
solubility of large molecules which is related to the ability of the fluid to
form cavities of sizes comparable to that of the solute [7, 8, 9].

In the context of phase transitions, in particular the liquid-to-vapor
transition of a metastable superheated or stretched liquid (cavitation) [10,
11, 12, 13, 14, 15, 16, 17, 18, 19], the classical nucleation theory also
relies on the idea that spontaneous cavities can appear in the liquid [20, 21,
22]. As a matter of fact, such cavities are expected to appear spontaneously
in any thermodynamic conditions, in metastable state or at equilibrium, the
probability of occurrence being given by the Boltzmann factor [2, 23]

$p(s)\propto e^{-W(s)/kT}$ (1)

where $W(s)$ is the free energy of formation of a cavity of size $s$, $k$ is
Boltzmann’s constant and $T$ the temperature. This relation is important in
the context of molecular simulations, since, in principle, a simple
measurement of the size of the spontaneous cavities in the liquid gives their
relative free energies through the determination of $p(s)$.

Another important quantity is the number density of cavities in the liquid,
regardless of their size. It appears at least in two important situations:

(i) In the context of nucleation, where the nucleation rate, as given by the
classical nucleation theory, is proportional to the number density of initial
germs [20, 21, 22]. It also raises the subsequent question of the
identification of germs with the budding bubbles and more generally the bubble
growth mechanisms [24, 25, 26, 27, 28, 29, 30, 31].

(ii) In the context of molecular simulations, the distribution of the number
of bubbles $B(n)$ in a given liquid volume $V$ is also an important piece of
information to determine the free energy profile from Eq. (1). The reason is
as follows: since it is impossible to observe spontaneous formation of large
bubbles in molecular simulations due to small system sizes and short
simulation runs, Frenkel and collaborators proposed to use biased Monte Carlo
to sample bubbles up to the critical size [32, 33, 34, 35]. However, the
obtained size distribution is not $p(s)$ but the size distribution of the
_largest_ bubble $p_{L}(s)$ which should not be used in Eq. (1), in particular
for small cavities [36, 37, 38, 39, 40, 41, 29]. But it has been shown very
recently that it is actually possible to transform $p_{L}(s)$ into $p(s)$
thanks to the distribution $B(n)$ [42].

What is the general form for $B(n)$? Thirty years ago, it was proposed that
the number $n$ of bubbles in a given volume follows the geometric law
$\nu^{n}$ [23, 35, 43], while more recently it was suggested that it should
follow the Poisson law $\nu^{n}/n!$ [37]. Given the rapid increase of $n!$ it
is important to propose arguments to decide between the two laws. Let us
denote $\nu$ the average number of bubbles in the volume. In the context of
spontaneous cavity formation in a liquid that is not under extreme stress or
superheat, $\nu$ is small, and therefore it corresponds to the probability to
have one bubble. The probability to have $n$ independent bubbles is then
argued to be proportional to $\nu^{n}$ [23, 35, 43]. However, one has to take
into account the fact that the $n$ bubbles appear in the _same_ volume and not
in $n$ replicas. This is analogous to the decay problem which considers the
probability to have $n$ decays in the _same_ interval of time. The consequence
is the appearance of the factor $1/n!$, giving the Poisson law (see appendix).

However, the spontaneous formation of tiny cavities is not so rare since the
free energy cost is low. It is thus possible to have two or more bubbles in a
small volume that could possibly interact if their distance is below the
typical range of interactions in the liquid. Furthermore, the underlying
mechanism of bubble formation may produce correlations. For instance, Neimark
and co-workers have shown that the growth of a bubble in a stretched liquid
approaching its spinodal proceeds via the formation of multiple interstitial
cavities that coalesce into a larger one [28]. Far from spinodal, the
formation of bubbles is a consequence of localized density fluctuations in the
liquid. As sketched in Fig. 1, a bubble corresponds to a region of the liquid
where the local density falls below a given threshold $\rho_{\rm th}$ between
liquid and vapor densities (see for example the region denoted a).

Figure 1:  Schematic representation of the fluid density $\rho$ (upper black
curve) along an arbitrary line passing through three bubbles in this example
(dash-dotted line shown below), showing fluctuations around the average liquid
density $\rho_{\rm L}$ (upper horizontal dashed line in dark-blue). Due to
large localized fluctuations, some regions of space can exhibit a density
lower than a given threshold $\rho_{\rm th}$ (black horizontal dashed line)
defining it as a bubble of gas (e.g. the region denoted a). In some cases,
depending on the choice of the threshold, the low density region (LDR) may
appear as divided in several bubbles (e.g. region b).

Depending on fluctuation correlations in the liquid, it is possible to have an
extended region of low average density consisting of two or more neighboring
bubbles separated by intervals where the density may locally exceed the
threshold $\rho_{\rm th}$ (Fig. 1, region b). Such bubbles are obviously not
independent, and the associated number distribution is not expected to be
Poissonian.

From a fundamental point of view, it is important to distinguish between these
two situations corresponding to isolated versus correlated bubbles. For
instance, in the context of nucleation, independent (isolated) bubbles can be
considered as distinct nuclei, while closely correlated bubbles should be seen
as one single nucleus (the growing bubbles close to each other are expected to
coalesce rapidly) [28]. Therefore, in this case, the low density regions
appear as a better concept to define the nuclei. They will be denoted LDR for
Low Density Region, to distinguish them from the cavities or bubbles. It is
important to note that, although the exact number of bubbles in a given LDR
obviously depends on the threshold $\rho_{\rm th}$ and more generally on the
procedure used to define the gas-like regions, the existence of this LDR is
intrinsic and essentially independent of the procedure.

The paper decomposes as follows. The first section presents the molecular
liquid model used for illustration, the method used to map the volume into
liquid-like and gas-like cells on a discrete grid, and the general procedure
to analyze the correlations between the bubbles. The second section presents
the numerical results, in particular the distribution of the number of bubbles
$B(n)$ for two system sizes and two cluster algorithms to define the bubbles.
Exploiting a correlation criterion between bubbles, the distributions $L(n)$
of the number of LDRs per configuration and $C(n)$ of the number of correlated
bubbles per LDR, are established as a function of this criterion, showing an
optimal choice. Then follows a discussion showing in particular that it is
possible to decompose the distribution $B(n)$ of the number of bubbles in a
given volume into a Poisson distribution of the number of LDRs (compatible
with the hypothesis of independence) and a geometric distribution of the
number of bubbles in a given LDR. It is shown in particular that the parameter
associated with the Poisson distribution of LDRs is essentially independent of
the algorithm used to define the bubbles. This allows in particular to define
an intrinsic average number of LDRs relevant for nucleation. The most striking
result of the analysis is that this average number of LDRs can easily be
deduced from molecular simulations by simply evaluating the probability of not
having bubbles in a given volume of liquid.

##  2 Methods and theory

_Molecular model:_ The numerical model of the liquid is a Grand Canonical
Monte Carlo treatment of generic spherical molecules interacting via the
truncated and quadratically shifted Lennard-Jones potential with a cutoff
radius equal to three atomic diameters [44, 45, 46]. Note that the general
features described in this paper are expected to be quite general and weakly
dependent on the particular choice of interacting potentials. One has however
to keep in mind that specific behaviors could emerge for systems that depart
significantly from these assumptions, in particular those with long-range
interactions (electrostatic), strongly anisotropic molecules, or many-body
interactions (metals). It is also emphasized that the results are expected to
be dependent on the statistical ensemble since it controls the density
fluctuations, but no differences were observed between Monte Carlo and
molecular dynamics runs. All quantities will be expressed in reduced units
built with the Lennard-Jones parameters $\epsilon$ and $\sigma$ associated to
interaction intensity and molecular diameter.

To enhance statistics and favor density fluctuations, we work at the reduced
temperature $kT/\epsilon=1.0$ and at $\ln\left(z\sigma^{3}\right)=-3.20$ where
$z=e^{\mu/kT}/\Lambda^{3}$ is the affinity, $\mu$ is the chemical potential
and $\Lambda$ is the thermal de Broglie wavelength. In these conditions, the
average reduced pressure is $p\sigma^{3}/\epsilon=0.026$, a value below the
reduced saturating pressure $p_{\rm sat}\sigma^{3}/\epsilon=0.055$ at the
working temperature, which places the system in a metastable state
(superheated). Although the formation of tiny bubbles is slightly favored by
the metastability, no transition was observed during the simulation runs: the
metastability is too weak to allow the liquid to overcome the energy barrier.
No bias is used neither: the observed bubbles are all spontaneous. Two cubic
system sizes are considered, of edges $L=9\sigma$ and $18\sigma$ with periodic
boundary conditions, containing an average number of molecules equal to 412
and 3300 respectively (see Table 1).

Table 1: System size $L/\sigma$, average number of molecules $<N_{\rm Mol}>$, bubble algorithm, average number of bubbles $\nu_{\rm b}$, probability of not having bubbles in the liquid $B(0)$, $\lambda_{\rm 0}=-\ln B(0)$, Poisson distribution parameter $\lambda_{\rm bb}$ and geometric distribution parameter $\kappa_{\rm bb}$ for the optimal bubble-bubble correlation length $l_{\rm bb}=3\sigma$, and $\lambda$ and $\kappa$ values from fits with Eqs 2 and 7 (see text for more details on the definitions).  $L/\sigma$ | $<N_{\rm Mol}>$ | algorithm | $\nu_{\rm b}$ | $B(0)$ | $\lambda_{\rm 0}$ | $\lambda_{\rm bb}$ | $\kappa_{\rm bb}$ | $\lambda_{\rm fit}$ | $\kappa_{\rm fit}$
---|---|---|---|---|---|---|---|---|---
9.0 | 412 | (f) | 1.3 | 0.443 | 0.81 | - | - | 1.37 | 0.44
9.0 | 412 | (fev) | 1.0 | 0.443 | 0.81 | - | - | 0.98 | 0.20
18.0 | 3300 | (f) | 6.8 | 0.0155 | 4.17 | 4.15 | 0.47 | 4.11 | 0.40
18.0 | 3300 | (fev) | 5.3 | 0.0155 | 4.17 | 4.40 | 0.24 | 4.42 | 0.15

_Bubble identification and characterization:_ There are several methods to
detect and characterize the bubbles appearing in the liquid, relying on an
estimation of the local fluid density. For instance one can perform a running
Gaussian convolution on the atomic positions. A less time consuming method
consists in working on a discrete grid. Obviously, different methods will
produce different results, but the main features of the fluid structure (like
large deviations from the liquid density) should be detected by all methods.
In this study we have chosen a four-steps algorithm which has proven to be
very efficient, the _M-method_ [47, 40]:

(i) For each molecule in the system, one counts the number of neighbors closer
than $1.625\sigma$ (first minimum of the radial distribution function of the
liquid in our thermodynamic conditions). If this number is larger or equal to
6 (based on the analysis of liquid and vapor phases in equilibrium), the
molecule is labeled as liquid-like, otherwise it is labeled as vapor-like.

(ii) The volume of the simulation box is divided into cubic cells (voxels) of
edge $l_{\rm cell}=0.5\sigma$.

(iii) The grid cells in the vicinity of a liquid-like molecule (center-to-
center distance $<1.625\sigma$) are marked as liquid, while the others are
marked as vapor.

(iv) We then perform a cluster analysis on the vapor cells to define the
bubbles. Two algorithms are used: the cells are considered to belong to the
same bubble if they are connected through the faces only (f), or through the
faces, edges or vertices (fev). The corresponding bubbles will be denoted (f)-
and (fev)-bubbles.

Figure 2 shows the result given by the previous algorithm at step (iii) for
four molecular configurations.

Figure 2:  3D representation of the result given by the M-method at step (iii)
of the algorithm for four molecular configurations of the superheated Lennard-
Jones liquid at $kT/\epsilon=1.0$ and $\ln z\sigma^{3}=-3.20$ (see text). The
grayed cells correspond to those marked as vapor, while the liquid cells are
not shown.

The grayed cells in the figure correspond to those marked as vapor. A large
variety of situations can be observed in these examples. In the first
configuration (Fig. 2a) one can see two well separated clusters of vapor cells
unambiguously associated to two bubbles. In this case, the two cluster
algorithms considered in step (iv) give the same result since all grayed cells
are connected through their facets.

In the second configuration (Fig. 2b) all vapor cells are localized in the
same region of the simulation box. They form one single cluster if one
includes vertex-sharing in the algorithm at step (iv) (giving one
(fev)-bubble), while they separate into several bubbles if one considers only
face-sharing connectivity (at least five (f)-bubbles can be seen). This is an
example showing that the choice of a particular algorithm in step (iv) may
give several bubbles in a situation where obviously only one nucleation is
occurring since all vapor cells are closely connected.

Figure 2c gives an example where the two algorithms give several bubbles,
because some clusters are separated from the others by at least one liquid
cell. However, these clusters appear closely correlated since they are all
localized in the same region of space compared to the simulation box volume,
and most probably correspond to one single density fluctuation or transient
nucleation.

Finally, Fig. 2d shows an even more complex situation where (i) the number of
clusters depends on the algorithm, and (ii) the vapor cells are not all
localized in the same region, suggesting the existence of several nucleation
spots.

These examples show the influence of the (f)- or (fev)-algorithms used in step
(iv). Once the algorithm is specified, it is straightforward to identify the
corresponding bubbles appearing in the liquid for each molecular
configuration. It is then possible to monitor their number, size and position
defined as that of their center-of-mass. The distribution of the number of
bubbles can be cast into a histogram that can be accumulated over the
configurations and normalized to one. The positions of the bubbles can also be
accumulated over the whole simulation, and one can show that they are
distributed uniformly over the simulation box. Obviously, most of the
quantities deriving from the bubbles are expected to depend on the cluster
algorithm. It is however possible to extract intrinsic information on the
LDRs, to be described now.

_Low density regions (LDR) and their characterization:_ To evidence the
existence of LDRs and the correlations between the bubbles, it is interesting
to calculate the average radial distribution function $g_{\rm bb}(r)$ between
all bubbles in a configuration, where $r$ denotes their mutual distance.
$g_{\rm bb}$ is build exactly as for the well-known $g(r)$ for atomic liquids,
except that the positions are replaced by the center-of-mass of the bubbles,
and the algorithm used to perform the average over the configurations is that
used in the case of systems where the number of particles (bubbles in our
case) varies (the self-averaging quantity is not $g_{\rm bb}$ but the two-
point correlation density $\rho_{\rm bb}({\bf r_{1}},{\bf r_{2}})$ where ${\bf
r_{1}}$ and ${\bf r_{2}}$ denote the positions in space of the two particles).
As for a liquid, isotropy hypothesis is expected to hold despite the
underlying discrete grid, which allows to perform an average over
orientations.

If the bubbles all nucleate independently, one expects a correlation function
$g_{\rm bb}$ equal to 1 (ideal gas), while a peak would reveal correlations in
positions. In that case, one may define a correlation length $l_{\rm bb}$
corresponding to this peak. Integration of $g_{\rm bb}-1$ times the bubble
density over the peak would give the excess number of neighboring bubbles.
This correlation length $l_{\rm bb}$ gives a simple geometric criterion to
define LDRs as clusters of correlated bubbles: two bubbles belong to the same
LDR if they are within $l_{\rm bb}$. Two LDRs are then necessarily separated
by a distance larger than $l_{\rm bb}$.

An important feature of LDRs which will be shown below is that they are
intrinsic, in the sense that their existence and number is independent of the
(f)- or (fev)-algorithm used to define the bubbles. Several questions
regarding the LDRs then arise, necessitating the introduction of new
distributions. For instance, what is the typical number of LDRs per
configuration? And what is the typical number of bubbles per LDR? More
generally, we introduce the distribution $L(n)$ of the number of LDRs per
configuration, and the distribution $C(n)$ of the number of correlated bubbles
per LDR. A general question to be addressed below is how the total
distribution $B(n)$ relates to the distributions $L(n)$ and $C(n)$.

##  3 Results

_Distribution of the number of bubbles $B(n)$:_ Figure 3 shows the
distributions of the number of bubbles for the two system sizes $L=9\sigma$
and $18\sigma$ and the two (f)- and (fev)-algorithms used to define the
bubbles.

Figure 3:  Distributions $B(n)$ of the number of bubbles in the Lennard-Jones
liquid at $kT/\epsilon=1.0$ and $\ln z\sigma^{3}=-3.20$ for two systems sizes
$L=9\sigma$ (black circles) and $L=18\sigma$ (red triangles). The bubbles are
defined and build according to two cluster algorithms with criterion of face
sharing (empty symbols, (f)-algorithm) or face-edge-vertex sharing (filled
symbols, (fev)-algorithm). The dashed (respectively solid) lines are the
Poisson distributions with the parameter equal to the corresponding average
number of bubbles $\nu_{\rm b}$.

For the smallest system, the largest probability is for zero bubbles in the
configuration. The probability then decreases monotonically with essentially
an exponential decay above few bubbles for both algorithms.

For the largest system, the distributions exhibit a maximum: it is more
probable to observe several bubbles instead of 0 or 1. This is expected since
the average number of bubbles $\nu_{\rm b}$ increases with the system size, as
shown in Fig. 4 where the average number of (fev)-bubbles is plotted versus
the system size for three different affinities. Note however that one does not
observe a strict proportionality with the volume. A small offset can be seen
for all affinities, that can be attributed to finite size effects and possible
correlations through the periodic boundary conditions for small systems.

Figure 4:  Symbols with solid lines: average number of (fev)-bubbles $\nu_{\rm
b}$ as a function of the reduced system volume $(L/\sigma)^{3}$ at
$kT/\epsilon=1.0$ and for three affinities $z$ given in the figure. The three
system sizes are $L/\sigma=$ 9, 13.5 and 18. Symbols with the dashed line:
evolution of $\lambda_{\rm 0}$ (see text) for $\ln z\sigma^{3}=-3.20$ and for
two system sizes. Lines are guide to the eye.

The distributions $B(n)$ are also sensitive to the cluster algorithm.
Including edges and vertices produces a distribution which is shifted to the
left (less bubbles) compared to that obtained with faces only, since two
(f)-bubbles connected by a corner or an edge are considered as one single
(fev)-bubble (see for example Fig. 2b).

An important feature is that the first bin $B(0)$ is independent of the bubble
algorithm (Fig 3). It corresponds to the probability that all cells of the
volume are in the liquid state. The numerical values are reported in Table 1.

Is there a general law to describe these distributions? If bubbles are
supposed to be independent (like in Fig. 2a exhibiting two well-separated
bubbles), the distribution of their number is expected to be Poissonian (see
Appendix), with a parameter equal to the average number of bubbles $\nu_{\rm
b}$. Figure 3 shows the normalized Poisson distributions $e^{-\nu_{\rm
b}}\nu_{\rm b}^{n}/n!$ corresponding to the two system sizes and the two
cluster algorithms (see Table 1). As can be seen, a fairly good agreement is
found between Poisson distributions and data for the largest system
$L=18\sigma$ and the (fev)-algorithm. However, noticeable discrepancies can be
noticed: first, for the same (fev)-algorithm but a smaller system size
$L=9\sigma$, the disagreement is noticeable, especially for large numbers of
bubbles. The Poisson law significantly underestimates the probability to
observe more than five bubbles in the system. Furthermore, for the
(f)-algorithm, the disagreement is significant for both system sizes and for
the whole range of the number of bubbles, which is particularly evident for
the largest system. As a general trend one observes that the probability to
observe small or large number of simultaneous bubbles is larger than predicted
by the Poisson law with its parameter equal to the average number of bubbles,
while the probability to observe a number of bubbles close to its typical or
average value (around its maximum) is overestimated by the Poisson law. These
observations clearly indicate that the detected bubbles are most probably not
all independent, as suggested by the qualitative analysis of configurations
(Fig. 2). This is particularly true for (f)-bubbles, but also for
(fev)-bubbles.

_Correlations between bubbles:_ Clearly, the independence assumption is not
fulfilled in general, in particular for the (f)-algorithm, which obviously
overestimates the number of bubbles in the liquid (Fig. 2b). Despite a better
agreement for the (fev)-algorithm, visual inspection of configurations shows
that some bubbles may appear as not so well defined structures, with several
disconnected sub-bubbles in the same region of space (see for instance Figs.
2c and d). For a quantitative evaluation of the correlations between the
bubbles, the radial distribution function $g_{\rm bb}(r)$ has been calculated
and reported in Fig. 5 for the largest system and the two algorithms.

Figure 5:  Radial distribution function $g_{\rm bb}(r)$ of the positions of
the center-of-mass of the bubbles in the Lennard-Jones liquid at
$kT/\epsilon=1.0$ and $\ln z\sigma^{3}=-3.20$ for $L=18\sigma$ and for the
(f)-algorithm (empty symbols) and the (fev)-algorithm (filled symbols).

The rapid oscillations are due to the discretization of the underlying grid
used to map the simulation box and should be ignored. As can be seen, $g_{\rm
bb}(r)$ is zero below the shortest possible distance between unconnected vapor
cells ($l_{\rm cell}\sqrt{2}=0.5\sigma\sqrt{2}$ for (f)-bubbles and $2l_{\rm
cell}=\sigma$ for (fev)-bubbles), and reaches 1 at large distances, as
expected for uncorrelated positions. At intermediate distances, a strong
correlation appears for (f)-bubbles, the height of the peak (roughly 100)
being amplified by the low density of bubbles in the simulation box. The peak
then rapidly decreases (logarithmic scale): $g_{\rm bb}\simeq 10$ at
$r=1.8\sigma$ and 3 at $r=3\sigma$. At $r=6\sigma$, $g_{\rm bb}$ has reached
1, meaning that correlations have disappeared. Interestingly, for
(fev)-bubbles, the correlation function is significantly attenuated, which
could be anticipated from the fact that the associated number distribution
follows more closely a Poisson distribution.

Integration of $g_{\rm bb}-1$ times the density of bubbles gives the total
number of bubbles in excess around any given bubble, the excess being
understood with respect to the situation where the positions are uncorrelated.
This quantity reaches 1.4 for (f)-bubbles and 0.4 for (fev)-bubbles, meaning
that, in practice, only 1.4 (respectively 0.4) bubbles are found in the
vicinity of any given bubble where the correlation function contributes the
most. Here again, the low excess found for the (fev)-bubbles suggests that
they are less correlated.

At odds with the usual radial distribution functions found in dense liquids,
one does not observe a clear end to the correlation peak (associated to a
minimum for instance). It is therefore difficult to give a precise value for
the correlation length except that it is most probably in the interval
$2\sigma\leq l_{\rm bb}\leq 6\sigma$.

_Distribution of the number of LDRs $L(n)$:_ The correlation length $l_{\rm
bb}$ can be used to define the LDRs as “clusters of correlated bubbles”, i.e.
two bubbles belong to the same LDR if their mutual distance is lower than
$l_{\rm bb}$. The configurations have been reanalyzed in terms of LDRs, for
different values of $l_{\rm bb}/\sigma$ between 1 and 5, i.e. in a broad range
above the correlation peak of $g_{\rm bb}$. Two quantities have been
monitored: the distribution $L(n)$ of the number of LDRs in the
configurations, and the distribution $C(n)$ of the number of bubbles per LDR.

The results for $L(n)$ are given in Fig. 6 for $L=18\sigma$ and the (f)- and
(fev)-bubbles.

Figure 6:  Symbols: distributions $L(n)$ of the number of LDRs per
configuration for the Lennard-Jones liquid at $kT/\epsilon=1.0$, $\ln
z\sigma^{3}=-3.20$, for $L=18\sigma$ and for different correlation lengths
$l_{\rm bb}$ given in the figure. The lines are the corresponding Poisson
distributions with parameters equal to the average number of LDRs
$\lambda_{\rm bb}$. Upper panel: (f)-bubbles; lower panel (fev)-bubbles. The
insets give the evolution with $l_{\rm bb}$ of $\lambda_{\rm bb}$ (for
(f)-bubbles) and the goodness of fit $\chi^{2}$ (for (f)- and (fev)-bubbles).

As expected, the distributions depend on the choice of the correlation length
$l_{\rm bb}$, with a stronger effect for (f)-bubbles. By definition, for
$l_{\rm bb}=0$ the distribution $L(n)$ identifies with that for all bubbles
$B(n)$ given in Fig. 3. Increasing $l_{\rm bb}$ induces a decrease in the
probability to find a large number of LDRs, which mechanically induces an
increase in the probability to find smaller numbers of LDRs. The variations
with $l_{\rm bb}$ are qualitatively the same for (f)- and (fev)-bubbles except
that for (fev)-bubbles $l_{\rm bb}=1\sigma$ gives essentially the same result
as for $l_{\rm bb}=0$, and variations with $l_{\rm bb}$ are less pronounced.
It is remarkable that for large $l_{\rm bb}\geq 4$ the distributions found for
(f)- and (fev)- bubbles tend to become identical. An important feature is that
the first bin $L(0)$ corresponding to the probability to find zero LDR is
independent of $l_{\rm bb}$ since it is given by the probability to find zero
bubble: $L(0)=B(0)$.

For all $l_{\rm bb}$, the corresponding Poisson distributions calculated with
their parameter taken equal to the average number of LDRs (denoted
$\lambda_{\rm bb}$) are shown as lines in Fig. 6. A goodness of fit $\chi^{2}$
is calculated which measures the departure of the data from the Poisson law
(see insets in Fig. 6). As can be seen, an optimal value ($\chi^{2}$
approaching zero) can be found, which corresponds to $l_{\rm bb}\simeq
3\sigma$. The corresponding $\lambda_{\rm bb}$ values are 4.15 and 4.40 for
the (f)- and (fev)-bubbles respectively (see inset giving $\lambda_{\rm bb}$
versus $l_{\rm bb}$ in Fig. 6 and Table 1). It is remarkable that the two (f)-
and (fev)-algorithms finally give essentially the same value for the optimal
correlation length. This is an important argument in favor of the intrinsic
nature of LDRs.

Let us now focus on a second way to determine the optimal $\lambda_{\rm bb}$.
We already noticed that the first bin $L(0)$ has a constant value independent
of $l_{\rm bb}$ ($L(0)=B(0)=0.0155$, for $L=18\sigma$). Since, by definition
of the Poisson law, the first bin is given by $e^{-\lambda_{\rm bb}}$, the
optimal choice for the Poisson parameter is also expected to be $\lambda_{\rm
bb}^{\rm optimal}=\lambda_{\rm 0}=-\ln L(0)=4.17$. This value is reported as a
horizontal line in the inset of Fig. 6. As can be seen, this value is in close
agreement with $\lambda_{\rm bb}=4.15$ found previously for $l_{\rm bb}\simeq
3\sigma$ for (f)-bubbles (4.40 for (fev)-bubbles). The consistency of the
analysis is also supported by the fact that the dependence of $\lambda_{\rm
0}$ with the system size closely follows the linear behavior observed for the
total number of bubbles (see Fig. 4).

_Distribution of the number of bubbles per LDR $C(n)$:_ Figure 7 gives the
distribution of the number of correlated bubbles in each LDR, averaged over
all LDRs and configurations.

Figure 7:  Symbols: distributions $C(n)$ of the number of correlated bubbles
per LDR for the Lennard-Jones liquid at $kT/\epsilon=1.0$, $\ln
z\sigma^{3}=-3.20$, for $L=18\sigma$ and for different correlation lengths
$l_{\rm bb}$ given in the figure. The lines are guides to the eye
corresponding to geometric distributions $\kappa^{n}$. Upper panel:
(f)-bubbles; lower panel (fev)-bubbles. The insets give the evolution of
$\kappa$ with $l_{\rm bb}$.

Note that these distributions are defined only for non-zero values $n\geq 1$.
As for $L(n)$, these distributions depend on the choice of $l_{\rm bb}$ and
the bubble algorithm. Quite remarkably, except for the very first points, the
general trend is a geometric law $\kappa^{n}$, as materialized by the straight
lines in semi-log scale on Fig. 7. The slopes give the $\kappa$ values and its
evolution with $l_{\rm bb}$ is given in the insets. As can be seen, the
distributions for the (fev)-bubbles decrease much faster than those for the
(f)-bubbles as revealed by the lower $\kappa$ values. This is in agreement
with the fact that the (f)-algorithm overestimates the number of bubbles per
LDR. The obtained $\kappa_{\rm bb}$ values for the optimal choice $l_{\rm
bb}=3\sigma$ are reported in Table 1.

##  4 Discussion

_A general model for $B(n)$:_ The previous analysis in terms of LDRs can be
considerably simplified by determining a general form for the distribution
$B(n)$ based on the previous results. The optimal choice for $l_{\rm bb}$
gives a Poisson distribution for $L(n)$ compatible with the independence
between the LDRs. We have also seen that $C(n)$ essentially follows a
geometric distribution. Although a slightly more refined model could be used,
it would have the inconvenience to introduce more free parameters.

It is therefore proposed to write $L(n)=e^{-\lambda}\lambda^{n}/n!$ where
$\lambda$ is the first free parameter of the model and corresponds to the
average number of LDRs per configuration. We also write
$C(n)\propto\kappa^{n}$ for $n\geq 1$ with one single parameter. We use the
convention $C(0)=0$ since by definition an LDR necessarily contains at least
one bubble. Normalizing to one imposes $C(n\geq 1)=(1-\kappa)\kappa^{n-1}$.

With this simple model, the distribution $B(n)$ of the number of bubbles per
configuration can be easily deduced. $B(0)$ being the probability to have zero
bubble, one necessarily has zero LDR and thus

$B(0)=L(0)=e^{-\lambda}.$ (2)

To have exactly one bubble, it is necessary to have exactly one LDR containing
one bubble:

$B(1)=L(1)C(1)=e^{-\lambda}\lambda(1-\kappa).$ (3)

For two bubbles, one can either have one LDR containing two bubbles or two
LDRs each containing exactly one bubble:

$B(2)=L(1)C(2)+L(2)C(1)^{2}=e^{-\lambda}\lambda(1-\kappa)\\{\kappa+\lambda(1-\kappa)/2\\}.$
(4)

For three bubbles there are four terms: one can either have one LDR containing
three bubbles, or two LDRs with one bubble in the first and two bubbles in the
second, or the converse, or three LDRs each containing one bubble:

$\displaystyle B(3)=L(1)C(3)+L(2)(C(1)C(2)+C(2)C(1))+L(3)C(1)^{3}$
$\displaystyle=e^{-\lambda}\lambda(1-\kappa)\\{\kappa^{2}+\lambda\kappa(1-\kappa)+\lambda^{2}(1-\kappa)^{2}/3!\\}.$
(5)

The generic expression for $B(n\geq 1)$ decomposes into a sum over the number
$k$ of LDRs, where each term comprises all combinations of $n$ bubbles
distributed into $k$ LDRs:

$B(n\geq 1)=\sum_{k=1}^{n}L(k)\sum_{i_{1},...,i_{k}\in\\{1...n\\}\atop
i_{1}+...+i_{k}=n}C(i_{1})C(i_{2})...C(i_{k}).$ (6)

In the case where $C(i)$ follows a geometric law, all the terms in the second
sum are identical to $(1-\kappa)^{k}\kappa^{n-k}$, and the number of terms
equals the number of ways to place $k-1$ graduations on a discrete (integer)
segment of length $n-1$, which equals $n-1\choose k-1$. One then gets (note
that the sum now runs from 0 to $n-1$):

$B(n\geq
1)=e^{-\lambda}\lambda(1-\kappa)\sum_{i=0}^{n-1}\frac{1}{(i+1)!}{n-1\choose
i}\lambda^{i}(1-\kappa)^{i}\kappa^{n-1-i}.$ (7)

_Application to data analysis:_ The obtained expression is quite simple and
can easily be used along with Eq 2 to fit the data with the two free
parameters $\lambda$ and $\kappa$. The result of the fits are given in Fig. 8
for the two system sizes and the corresponding values of the free parameters
are given in Table 1.

Figure 8:  Same as Fig. 3 except that the solid and dashed lines are now the
best fits obtained with Eqs 2 and 7 and the parameters given in Table 1.

As can be seen, the proposed model gives a remarkable agreement with the data,
for (f)- and (fev)-bubbles as well. For the largest system $L=18\sigma$ the
$\lambda$ parameters are 4.11 and 4.42 for (f)- and (fev)-bubbles
respectively, i.e. identical within 7%, showing that the model gives
essentially the same average number of LDRs for both algorithms. The
difference is larger for the smallest system, most probably due to larger
uncertainties (lower statistics). On the other hand, the $\kappa$ parameters
are significantly different between the two algorithms. It is larger for
(f)-bubbles, reflecting the fact that LDRs are split in a larger number of
bubbles with the (f)-algorithm. It is actually not surprising to observe this
dependence with the algorithm, since we have already seen that the
(f)-algorithm produces more correlated bubbles. These observations (small
difference for $\lambda$ versus a larger one for $\kappa$) suggest that the
proposed model (Eqs 2 and 7) correctly absorbs the differences between the two
algorithms in the $\kappa$ parameter, allowing to extract the intrinsic number
of LDRs $\lambda$.

How does it compare with our previous analysis in terms of correlation length
between bubbles? As can be seen in Table 1 the results of the fit are in close
agreement (within 1%) with the $\lambda_{\rm bb}$ values that were extracted
from the analysis consisting in optimizing the Poisson fit by varying $l_{\rm
bb}$. The agreement with $\kappa_{\rm bb}$ is less good, the difference being
15% for (f)-bubbles and reaching 45% for (fev)-bubbles. The main reason is
that the best fit with Eqs 2 and 7 imposes a geometric law for all number of
bubbles $\geq 1$, while the $\kappa$ values deduced from the slopes in the
semi-log representations in Fig 7 ignore the very first points, and are thus
more reliable. Note that it is possible to improve the fits by replacing the
single-parameter distribution $C(n\geq 1)\propto\kappa^{n}$ by the two-
parameter law $C(1)=\alpha$ and $C(n\geq 2)\propto\kappa^{n}$.

##  5 Conclusion

Using molecular simulations of a simple van der Waals fluid, we have shown
that the liquid phase exhibits spontaneous formation of tiny bubbles. It is
emphasized that the precise definition of these tiny bubbles is not unique and
sensitive to the algorithm used. Inspection of the distribution of the number
of bubbles $B(n)$ in a given volume reveals that it does not follow the
Poisson law $e^{-\lambda}\lambda^{n}/n!$ expected for uncorrelated events. We
have shown that the bubbles are actually spatially correlated, with a typical
length scale of order $3\sigma$, i.e. in the range of the intermolecular
interactions. Such correlated bubbles are expected to coalesce in the case of
a growing nucleation event (in a metastable liquid), and should be seen as one
single event instead of several bubbles. These correlated bubbles are
interpreted as belonging to the same fluctuation in a region of space where
the density is locally gas-like (called LDR for Low Density Region). This
refined analysis in terms of LDRs allows to determine their average number in
the simulation box, which appears to be independent of the algorithm used to
define the bubbles, emphasizing their intrinsic nature. Their distribution is
shown to follow the Poisson law, as expected for independent events. On the
other hand, the simulations show that the distribution $C(n)$ of correlated
bubbles in a single LDR is algorithm-dependent. It is however noticed that
$C(n)$ essentially follows a geometric distribution for $n$ larger than few
bubbles.

Conversely, taking these observations as an assumption, it is possible to
obtain a general expression for the total distribution $B(n)$ with only two
free parameters (Eqs 2 and 7) that can be used to fit the data. The parameter
associated to $C(n)$ is shown to be dependent on the algorithm used to define
the bubbles, while the parameter $\lambda$ associated to the Poisson
distribution of the uncorrelated LDRs is independent of the algorithm used to
define the bubbles, which supports the idea that the LDRs are intrinsic. The
obtained values are in good agreement with the previous analysis in terms of
correlations between bubbles.

A remarkable result is that the $\lambda$ parameter associated to the number
of LDRs can easily be extracted from the data without need to determine the
correlation length between bubbles nor to fit the data with (Eqs 7). Thanks to
Eq 2, the relation $\lambda=-\ln B(0)$ immediately gives the desired average
number of LDRs in the liquid. Note that the determination of $B(0)$, which is
the probability of not having bubbles in the liquid, does not even require to
use the (f)- or (fev)-algorithm to define the bubbles, since this probability
is equivalent to having zero gas-like cell in the system, an information that
can be determined at step (iii) of the M-method. This is particularly useful
for the rapid determination of $\lambda$ entering the transformation of the
distribution of the largest bubble given by biased simulations into the
distribution of all bubbles entering Eqs 1 [42].

## Declaration of interests

The author declares to have no known competing financial interests or personal
relationships that could have appeared to influence the work reported in this
paper.

## Acknowledgments

The author acknowledges fruitful discussions with P. Porion, E. Rolley, P.
Spathis and P.E. Wolf, and the financial support of Agence Nationale de la
Recherche through the project CavConf, ANR-17-CE30-0002.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Distribution of the number of independent bubbles

Let us consider a fixed volume $V$ of liquid with a uniform probability of
having a nucleation (i.e. a bubble) at any position. The volume can be divided
in a large number $N$ of small cells of volume $dV=V/N$. We will suppose $dV$
infinitesimal ($dV\rightarrow 0$, $N\rightarrow\infty$). The probability to
have a nucleation in $dV$ is also an infinitesimal, denoted $\tau dV$. The
probability to have two nucleations or more in a cell is of order $dV^{2}$ and
can be neglected. Therefore, the average number of nucleations in $dV$ equals
$\tau dV$, and, the probability being uniform, the average number of
nucleations in $V=NdV$ is given by $\nu=\tau V$. In the following, we suppose
that the cells are independent, i.e. the probability that a nucleation occurs
in a cell is independent of the other cells.

We are now in position to calculate the probability $B(n)$ to have $n$
independent bubbles in the volume $V$. This situation is obtained for any
configuration where nucleation occurs in $n$ cells (probability $(\tau
dV)^{n}$), while the other $N-n$ cells remain liquid (probability $(1-\tau
dV)^{N-n}$), the total number of possible choice for the $n$ cells being
$N\choose n$. Therefore, $B(n)$ is given by the limit:

$B(n)=\lim\limits_{N\rightarrow\infty}{N\choose
n}\left[\tau\frac{V}{N}\right]^{n}\left[1-\tau\frac{V}{N}\right]^{N-n}=e^{-\nu}\frac{\nu^{n}}{n!}$
(8)

which is the Poisson distribution.

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