# Collective Effects in Breath Figures

Ambre Bouillant  Physics of Fluids Group, Faculty of Science and Technology,
University of Twente, 7500 AE Enschede, The Netherlands  Laboratoire Matière
et Systèmes Complexes, UMR 7057 CNRS, Université Paris Cité, 10 rue A. Domon
et L. Duquet, 75013, Paris, France  Jacco H Snoeijer  Physics of Fluids Group,
Faculty of Science and Technology, University of Twente, 7500 AE Enschede, The
Netherlands  Bruno Andreotti  Laboratoire de Physique de l’ENS, UMR 8550 CNRS,
Ecole Normale Supérieure, Université Paris Cité – Sorbonne Université, 24 rue
Lhomond, 75005 Paris, France

###### Abstract

Breath figures are the complex patterns that form when water vapor condenses
into liquid droplets on a surface. The primary question concerning breath
figures is how the condensing vapor is allocated between the growth of
existing droplets and the nucleation of new ones. Although numerous
theoretical studies have concentrated on scenarios resulting in highly
polydisperse droplet ensembles, a companion paper [Bouillant et al.,
submitted] demonstrates that nearly monodisperse patterns can be achieved on
defect-free substrates in a diffusion-controlled regime. The objective of this
work is to present a theoretical framework that elucidates the formation and
evolution of nearly-monodisperse patterns in breath figures. We discover that,
following a short nucleation phase, the number of droplets remains constant
over an extensive range of timescales due to collective effects mediated by
the diffusion of vapor. The spatial extent of these diffusive interactions is
identified through asymptotic matching, based on which we provide an accurate
description of breath figures through a mean-field model. The model accounts
for the sub-diffusive growth of droplets as well as for the arrest of
nucleating new droplets, and reveal the scaling laws for the droplet density
observed in experiments. Finally, droplets expand and ultimately coalesce,
which is shown to trigger a scale-free coarsening of the breath figures.

##  I Introduction

“The manner in which aqueous vapor condenses upon ordinarily clean surfaces of
glass or metal is familiar to all” Rayleigh (1911). Yet, the discussion of
breath figures, as initiated by Rayleigh and Atkin in 1911 Rayleigh (1911),
still continues today and holds many unresolved puzzles. Breath figures are
the intricate patterns that form when water vapor condenses into liquid
droplets on a surface Baker (1922). The process of nucleation, in which
nascent structures of a stable phase emerge in a metastable phase, is crucial
to the formation of breath figures Kashchiev (2000). In this context, the
nucleation of nanometric droplets occurs on the solid through a heterogeneous
mechanism that is highly sensitive to the substrate nature Knobler et al.
(1991). On rough, patterned or chemically heterogeneous solids, surface
defects can lower the energy barrier to nucleation, resulting in breath
figures that reflect the substrate heterogeneity, such as the density of
defects. In contrast, on atomically smooth substrates, nucleation is
stochastic and only depends on the energy barrier associated with the
formation of a critical nucleus, reflecting the substrate energy Varanasi et
al. (2009); Sikarwar et al. (2011); Lopez et al. (1993); Enright et al.
(2012). The substrate properties can also affect the droplets’ growth through
the influence of the substrate wettability on the vapor flux absorbed by the
droplet Gelderblom et al. (2011); Popov (2005). At a more advanced stage of
the condensation process, the occurrence of coalescence events between
droplets abruptly changes drop configurations, resulting in intermittent
dynamics and increased polydispersity in the breath figure pattern, ultimately
promoting gravity-driven droplet shedding Bintein et al. (2019); Trosseille et
al. (2019). On rough solids, fusions between neighboring droplets free up the
surface defects, enabling secondary nucleation events and markedly increasing
the pattern polydispersity. In recent decades, experiments performed on clean
surfaces have suggested the emergence of a universal scaling form of the
droplet size distribution evolution Viovy et al. (1988); Blaschke et al.
(2012); Stricker et al. (2022); Haderbache et al. (1998). However, the
underlying mechanisms responsible for the formation and evolution of breath
figures are not yet fully understood.

Condensation is a natural phenomenon that has been harnessed for various
applications, from providing a valuable source of water in deserts for humans
Nikolayev et al. (1996); Liu et al. (2022) and animals Parker and Lawrence
(2001); Munné-Bosch and Alegre (1999); Hill et al. (2015), to heat exchangers
for cooling systems Bortolin et al. (2022), desalination systems Khawaji et
al. (2008), designing nano-emulsions Guha et al. (2017) with innovative
optical properties Goodling et al. (2019), and patterning substrates for
microfabrication Böker et al. (2004); Zhang et al. (2015). In order to
optimize the gravity-driven droplet shedding in these applications, it is
crucial to control the density, size, and polydispersity of dew patterns. This
requires a fine control of the substrate heterogeneity, such as the presence
of defects that promote nucleation, and properties. Strategies for controlling
dew patterns may involve surface patterning Park et al. (2016); Trosseille et
al. (2019); Zhao and Beysens (1995); Varanasi et al. (2009); Bintein et al.
(2019) or electrowetting Baratian et al. (2018). Therefore, understanding the
formation and evolution of breath figures is essential for predicting and
controlling dew patterns in a wide range of applications.

In this study, we investigate the early stages of breath figure formation on
defect-free surfaces and the ensuing pattern coarsening due to drop merging.
The work provides the interpretation-framework for an experimental companion
paper Bouillant et al. (2024), which shows the emergence of nearly
monodisperse breath figures and their subsequent coarsening. Here we develop a
many-droplet theory for nucleation and growth, based on an asymptotic matching
between the vicinity of drops and a far field, which describes droplet
interactions through the quasi-static diffusion of the vapor phase. Our model
reveals two distinct phases of growth. At short times, the droplets grow
diffusively while the the number of drops increases linearly in time. However,
at longer times, we show that, in the absence of turbulent mixing in the vapor
phase, nucleated droplets deplete the humidity in their vicinity. This
depletion creates a boundary layer that grows as the pattern evolves. As
droplets approach each other, interactions mediated through the surrounding
air screen the humidity, resulting in a lower effective humidity experienced
by the droplets. This humidity screening effect leads to a sub-diffusive
droplet growth, and also affects the energy barrier to nucleate new droplets
in a highly sensitive manner – ultimately leading to the arrest of nucleation.
A central question that is addressed in this paper is how the experimentally
observed number of drops (starting around the micron-scale Bouillant et al.
(2024)) is related to the actual nucleation rate (at the nano-scale). Finally,
we address the ultimate regime of the nearly-monodisperse breath figures,
where drop come to contact and exhibit coarsening by coalescence. It is
demonstrated that the coarsening exhibits a scale-free dynamics, explaining
the scaling laws observed in experiment Bouillant et al. (2024).

##  II Drop nucleation

Figure 1: Nucleation and growth of breath figures. (a) A single drop of base
radius $R$ wetting its substrate with a contact angle $\theta$. The arrows
indicate the local vapor flux that drives the droplet growth. The inset shows
a typical vapor density $\rho$ along the substrate, increasing from the
saturated value $\rho_{\rm sat}$ at the droplet’s interface. (b) On the scale
of the cell (height $H$), the drops act as “sinks” of vapor, which modify the
humidity fields in their vicinity. In most of the paper, we focus on the
regime where typical distances between the drops $\lambda$ are large compared
to the typical drop size, i.e. in a regime prior to drop coalescence. This
gives a hierarchy of scales $R\ll\lambda\ll H$ that enables a solution of the
many-droplet diffusion problem by matched asymptotics. In the calculation, we
use periodic boundary conditions in the horizontal direction, so that the cell
size is $L\times L\times H$.

In this study, we employ the Classical Nucleation Theory Kashchiev (2000);
Kelton and Greer (2010) to describe the nucleation of drops on a defect-free
rigid substrate. The drops under consideration are expected to be nanometric
in size, allowing us to disregard the effects of gravity. Consequently, the
liquid assumes the shape of a spherical cap, the geometry of which is
characterized using the horizontal base radius $R$ and a contact angle
$\theta$, as depicted in Fig. 1(a). At equilibrium, the contact angle is
selected by the surface energies of the solid-vapor interface ($\gamma_{sv}$),
the solid-liquid interface ($\gamma_{s\ell}$) and the liquid-vapor interface
($\gamma$) through Young’s law: $\gamma\cos\theta=\gamma_{sv}-\gamma_{s\ell}$.
The droplet volume can be written as $V=\frac{\pi}{3}R^{3}\mathcal{E}(\theta)$
where the function
$\mathcal{E}(\theta)\equiv{(2+\cos\theta)(1-\cos\theta)^{2}}/{\sin^{3}\theta}$
accounts for the influence of the contact angle. For a hemispherical drop one
verifies that $\mathcal{E}(\pi/2)=2$, while at small $\theta$ we find
$\mathcal{E}(\theta)\sim 3\theta/4$. The number of molecules $n$ contained
within the drop is then defined as the ratio between its volume $V$ and the
volume of a single molecule $m\big{/}\rho_{\ell}$, where $\rho_{\ell}$ and $m$
are the liquid mass density and the mass of a single molecule. We thus find

$n=\frac{V}{m/\rho_{\ell}}=\frac{\pi\rho_{\ell}}{3m}R^{3}\mathcal{E}(\theta).$
(1)

We now consider a supersaturated vapor above a solid substrate. The
supersaturation implies a difference in chemical potential between the liquid
and vapor phase $\Delta\mu$, which drives the condensation process. Assuming
that vapor behaves like an ideal gas, $\Delta\mu$ can be written as

$\Delta\mu=\mu_{\ell}-\mu_{v}=-\Lambda k_{B}T,\quad{\rm
with}\quad\Lambda=\ln\left(\rho\big{/}\rho_{\mathrm{sat}}\right).$ (2)

In this expression, $\rho$ is the vapor mass density and $\rho_{\mathrm{sat}}$
its saturated value at a given temperature $T$. The nucleation of a droplet
containing $n$ molecules will lower the Gibbs free energy by $n\Delta\mu$.
However, the creation of new interfaces (liquid-vapor, solid-liquid and solid-
vapor) associated to the creation of the drop comes at an energetic cost. This
energy penalty scales with the surface $\sim R^{2}\sim n^{2/3}$. Working out
the geometric details, one can write the total change in Gibbs free energy as
Kashchiev (2000)

$\displaystyle\frac{\Delta G}{k_{B}T}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\left(2\chi n^{2}\right)^{1/3}-n\Lambda,$ (3)

where we introduced the dimensionless parameter

$\chi\equiv\dfrac{4\pi}{3}\left(\dfrac{\gamma}{k_{B}T}\right)^{3}\left(\dfrac{m}{\rho_{\ell}}\right)^{2}\mathcal{E}\sin^{3}\theta.$
(4)

The free energy thus initially increases with droplet size $n$, with a scaling
$\sim n^{2/3}$ owing to surface effects, while the gain in energy due to the
actual condensation $\sim n$. The competition between these effects gives rise
to an energy barrier for nucleation $\Delta G^{*}$, and a critical nucleus
size $n^{*}$, corresponding to the maximum of $\Delta G(n)$. We thus find:

$n^{*}=\frac{2\chi}{\Lambda^{3}},\quad{\rm and}\quad\frac{\Delta
G^{*}}{k_{B}T}=\frac{\chi}{\Lambda^{2}}.$ (5)

The critical nucleus and the corresponding energy barrier are thus controlled
by the two dimensionless parameters: $\chi$, which isolates the droplet
geometrical influence to the energy barrier, and $\Lambda$, which accounts for
the humidity influence on the energy barrier.

Finally, we assume the nucleation of drops to be an activated process that is
described by Classical Nucleation Theory Kashchiev (2000). We refer to
Appendix. A for details. Within this framework, the number of drops nucleated
per unit surface and unit time takes the form:

$J={\mathcal{J}}\left(\frac{\rho}{\rho_{\rm
sat}}\right)^{2}\exp\left(-\frac{\Delta
G^{*}}{k_{B}T}\right)={\mathcal{J}}\left(\frac{\rho}{\rho_{\rm
sat}}\right)^{2}\exp\left(-\frac{\chi}{\Lambda^{2}}\right).$ (6)

In this expression $\mathcal{J}$ can be interpreted as an attempt frequency
per molecule, while the factor $(\rho/\rho_{\rm sat})^{2}$ reflects the fact
that the overall collision frequency is quadratic in drop density (cf.
Appendix A). Most importantly, however, the nucleation rate involves a
Boltzmann factor that is controlled by the energetic barrier.

##  III Collective effects during growth

The nucleation described above is a stochastic process, which depends on the
local humidity at the substrate. Once drops are formed, however, they grow by
further absorbing vapor. This growth is typically limited by the
(deterministic) diffusive transport of vapor inside the cell towards the
drops. This diffusive transport is described in this section, where we
consistently account for collective effects due to droplet interactions.

###  III.1 Theoretical set-up

We model the growth of droplets by the quasi-steady diffusion of vapor, i.e.
by solving $\nabla^{2}\rho=0$. The complexity of the problem arises from the
intricate, mixed boundary conditions that slowly evolve over time as the
breath figure evolves, and which need to be found self-consistently. At a
liquid-vapor interface, the air is saturated with vapor, so that
$\rho=\rho_{\rm sat}$ is imposed at the surface of the drops. Between the
drops, in the dry regions of the substrate, the normal component of the flux
$\nabla\rho\cdot\mathbf{n}$ must vanish, as molecules cannot penetrate the
substrate. These boundary conditions are illustrated on the level of a single,
isolated drop in Fig. 1a, for which analytical solutions are available (Popov,
2005). The breath figure is driven by imposing a humidity $\rho=\rho_{0}$ at
the top plate at $z=H$ (Fig. 1b), which leads to condensation when
$\rho_{0}>\rho_{\rm sat}$. We introduce the relative humidity
$\mathcal{R}={\rho_{0}}\big{/}{\rho_{\rm sat}}$, which is taken larger than
unity throughout.

Figure 2: Modeling the vapor flux collected into dew droplets with the
electrostatic analogy. A homogeneous surface “charge” $-m\dot{n}_{i}/L^{2}$
imposed at the ceiling ($z=H$) provides a vapor flux that distributes over the
$n_{i}$ droplets (modelled as point sinks), each absorbing a “charge”
$m\dot{n}_{i}$.

###  III.2 Non-interacting drops

The quasi-steady diffusive growth of an isolated sessile drop, spherical cap
of contact angle $\theta$ and base radius $R$ as sketched in Fig. 1(a), is
well understood. Introducing $n_{i}$ as the number of molecules inside a
droplet labelled $i$, the growth law is given by (Popov, 2005)

$m\dot{n}_{i}=2\pi\mathcal{F}(\theta)R_{i}D\left(\rho_{\infty}^{i}-\rho_{\rm
sat}\right),$ (7)

where $D$ is the diffusion coefficient of vapor molecules in air, while

$\displaystyle\mathcal{F}(\theta)$ $\displaystyle=$
$\displaystyle\frac{\sin\theta}{2(1+\cos\theta)}+2\int_{0}^{\infty}\frac{1+\cosh
2\theta\tau}{\sinh 2\pi\tau}\tanh[(\pi-\theta)\tau]d\tau.$

the function $\mathcal{F}(\theta)$ follows from solving Laplace’s equation
$\nabla^{2}\rho=0$, with boundary conditions $\rho=\rho_{\rm sat}$ on the
surface of the spherical cap, a no-flux condition at the dry substrate, while
$\rho_{\infty}^{i}$ is the humidity at infinity (Popov, 2005).
$\mathcal{F}(\theta)$ tends to $2/\pi$ as $\theta$ tends to $0$ and is equal
to $1$ for the hemispherical case ($\theta=90^{\circ}$). The diffusive law has
been validated experimentally (Gelderblom et al., 2011) for an isolated
evaporating drops with $\rho_{\infty}^{i}$ taken constant.
For later reference, we will express the growth law for $\dot{n}_{i}$ in terms
of $\dot{R}_{i}$. Making use of (1), we can write (7) as

$\frac{d(R^{2}_{i})}{dt}=\mathcal{D}\;\left(\frac{\rho_{\infty}^{i}}{\rho_{\rm
sat}}-1\right),$ (8)

where we introduced the effective diffusion coefficient $\mathcal{D}$ as

$\mathcal{D}=\frac{4\mathcal{F}\rho_{\rm sat}}{\mathcal{E}\rho_{\ell}}D.$ (9)

Dropping the label $i$, we see that isolated drops grow as
$R=\sqrt{(\mathcal{R}-1)\mathcal{D}(t-T)}$, where $T$ is the time of creation
of the drop and $\mathcal{R}=\rho_{\infty}/\rho_{\rm sat}$ the relative
humidity at infinity.
The complete solution $\rho(\mathbf{r})$ around a spherical cap is intricate,
but in the limit $|\mathbf{r}-\mathbf{r}_{i}|\gg R_{i}$ it reduces to (Popov,
2005),

$\rho(\mathbf{r}-\mathbf{r}_{i})\simeq\rho_{\infty}^{i}+(\rho_{\rm
sat}-\rho_{\infty}^{i})\mathcal{F}\frac{R_{i}}{|\mathbf{r}-\mathbf{r}_{i}|}.$
(10)

In the far-field, the detailed geometry of the droplet is thus no longer
reflected in the spatial structure of the field: the asymptotic solution
corresponds to an isotropic sink at position $\mathbf{r}_{i}$. The growth law
(7) is indeed recovered from the integration of the diffusive flux
$-D\nabla\rho$ obtained from (10), over the half-spherical surface
$2\pi|\mathbf{r}-\mathbf{r}_{i}|^{2}$. It is instructive to write the far-
field solution directly in terms of $\dot{n}$,

$\rho(\mathbf{r}-\mathbf{r}_{i})\simeq\rho_{\infty}^{i}-\frac{m\dot{n}_{i}}{2\pi
D|\mathbf{r}-\mathbf{r}_{i}|}.$ (11)

This form gives an explicit relation between the strength of the sink and the
growth rate $\dot{n}_{i}$ of the drop.

###  III.3 Interacting drops

In the presence of many drops, one expects the same growth law, as long as the
drops are sufficiently separated. Specifically, (11) will be valid at large
distance from the drop yet remaining far from neighbouring drops, i.e. for the
hierarchy of scales

$R_{i}\ll|\mathbf{r}-\mathbf{r}_{i}|\ll\lambda=\psi^{-1/2}.$ (12)

Here we introduced the number of drops per unit area $\psi$. The fact that
drops are small, however, does not imply that drops do not interact. We will
see that $\rho_{\infty}^{i}$ represents the “effective humidity” experienced
by droplet $i$, and that its value results from the presence of the ensemble
of drops. Clearly, this effective humidity will in general be lower than the
humidity $\rho_{0}$ imposed at the top-surface of the cell, since the
neighbouring drops will have a screening effect. By (7), it is clear that the
growth rates $\dot{n}_{i}$ for each of the droplets crucially relies on the
value of $\rho_{\infty}^{i}$.
We now set out to formulate and solve the intricate many-body quasi-steady
diffusion problem, $\nabla^{2}\rho=0$ with appropriate boundary conditions,
which enables us to compute the $\rho_{\infty}^{i}$. As sketched in Fig. 1(b),
we consider a system of height $H$, and length and width $L$ with periodic
boundary conditions in both horizontal directions. A number $\psi L^{2}$ drops
are placed on the substrate ($z=0$), each characterised by a position
$\mathbf{r}_{i}$, base radius $R_{i}$ and number of molecules $n_{i}$. The
condensation is driven by prescribing an average concentration
$\rho_{0}=\int_{z=H}\rho\,d^{2}x$ at the top of the cell ($z=H$). As
previously mentioned, the bottom substrate is subjected to a no-flux condition
$\nabla\rho\cdot\mathbf{n}=0$, while $\rho=\rho_{\rm sat}$ at the surface of
each droplet.

####  III.3.1 Solution by matched asymptotics

We solve for the field $\rho(\mathbf{r})$ and the effective humidities
$\rho_{\infty}^{i}$ using a matched asymptotics analysis that employs the
scale separation $R_{i}\ll\psi^{-1/2}$, i.e. that the typical distance between
drops is large compared to their typical size. The inner problem is determined
by the scale $R_{i}$ and corresponds to an isolated drop [Fig. 1(a)]. The far-
field asymptotics of this inner solution, i.e. for
$R_{i}\ll|\mathbf{r}-\mathbf{r}_{i}|$, is already given by (11). The outer
problem is governed by the scale $\psi^{-1/2}\gg R_{i}$ [Fig. 1(b)]. In the
outer problem, the droplets can be represented by isotropic point sinks with a
singularity of the form $\rho_{i}\simeq-\dot{m}n_{i}/2\pi
D|\mathbf{r}-\mathbf{r}_{i}|$. This makes the problem analogous to point
charges in electrostatics, the charge here being directly related to the
growth rate $\dot{n}_{i}$. This electrostatic analogy is sketched in Fig. 2.
Besides the charges associated to the growing droplets, we impose a
homogeneous surface charge density $-m\dot{n}_{i}/L^{2}$ at $z=H$. The total
charge in the system is thereby zero, which ensures the conservation of mass:
the flux imposed at the ceiling feeds the growth of the drops. Using the
superposition principle, the outer solution can be written as

$\rho(\mathbf{r})=\rho_{0}-\sum_{j=1}^{N}\frac{m\dot{n}_{j}}{2\pi
D}\mathcal{G}(\mathbf{r}-\mathbf{r}_{j}),$ (13)

where the sum runs over each drop. Here, $\mathcal{G}$ is the Green’s function
that captures not only the effect of each sink, but also of the uniform flux
that feeds the drop from the ceiling, the no-flux boundary condition at $z=0$,
and self-interaction through the (horizontal) periodic boundary conditions.
The gauge of the Green’s function is chosen such that it has a vanishing
spatial average at $z=H$. With this gauge, $\rho_{0}$ in (13) represents the
average humidity at the ceiling of the cell, which we will use as a control
parameter for the condensation process. The explicit computation of the
Green’s function is provided in Appendix B.
The asymptotic matching consists of equating the inner expansion (11) far from
the drop, to the outer solution (13) expanded in the limit
$\mathbf{r}\to\mathbf{r}_{i}$. The Green’s function
${\mathcal{G}}(\mathbf{r})$ is decomposed into its $1/|\mathbf{r}|$
singularity and the subdominant rest, $\tilde{\mathcal{G}}(\mathbf{r})$, which
is regular at $\mathbf{r}=\mathbf{0}$. Hence we write
${\mathcal{G}}(\mathbf{r})=\tilde{\mathcal{G}}(\mathbf{r})+1/|\mathbf{r}|$, so
that in the limit $\mathbf{r}\to\mathbf{r}_{i}$,

$\rho(\mathbf{r})=-\frac{\dot{n}_{i}}{2\pi
D|\mathbf{r}-\mathbf{r}_{i}|}+\rho_{0}-\frac{\dot{n}_{i}}{2\pi
D}\tilde{\mathcal{G}}(\mathbf{0})-\sum_{j\neq i}\frac{\dot{n}_{j}}{2\pi
D}\mathcal{G}(\mathbf{r}_{i}-\mathbf{r}_{j})+\mathcal{O}\left(|\mathbf{r}_{i}-\mathbf{r}_{j}|\right).$
(14)

This result needs to be compared to the inner asymptotics (11). One observes
that the leading order $1/|\mathbf{r}-\mathbf{r}_{i}|$ already matches, which
reflects that we have chosen the appropriate strengths for the point sinks in
the outer problem. However, matching the next order
$|\mathbf{r}-\mathbf{r}_{i}|^{0}$, we obtain a nontrivial condition

$\rho_{\infty}^{i}=\rho_{0}-\frac{m\dot{n}_{i}}{2\pi
D}\tilde{\mathcal{G}}(\mathbf{0})-\sum_{j\neq i}\frac{m\dot{n}_{j}}{2\pi
D}\mathcal{G}(\mathbf{r}_{i}-\mathbf{r}_{j}).$ (15)

Hence, we can express the effective humidity $\rho_{\infty}^{i}$ of droplet
$i$ in terms of the growth rates $\dot{n}_{j}$ of the full ensemble of the
drops; this reflects the collective nature of the problem. Having solved for
the effective humidities, we can return to (7) and write an equation that only
contains the growth rates:

$\frac{m\dot{n}_{i}}{2\pi
D}\left(\tilde{\mathcal{G}}(\mathbf{0})+\frac{1}{\mathcal{F}R_{i}}\right)+\sum_{j\neq
i}\frac{m\dot{n}_{j}}{2\pi
D}\mathcal{G}(\mathbf{r}_{i}-\mathbf{r}_{j})=\rho_{0}-\rho_{\rm sat}.$ (16)

This expression is valid for each droplet $i$, and thereby forms a closed set
of linear equations for the $\dot{n}_{i}$.

####  III.3.2 Global mass flux

It is of interest to consider the global mass flux of vapor from the ceiling
($z=H$) to the bottom of the cell ($z=0$), which provides the total rate of
condensed mass. Naturally, this flux must be equal to the total mass collected
by all sinks, which per unit time per unit area reads
$L^{-2}\sum_{j}m\dot{n}_{j}$. We can interpret this result using a horizontal
Fourier decomposition of the diffusion problem, since only the zero-mode
(i.e., the average in the horizontal direction) contributes to the net flux
from the ceiling across the cell towards the bottom. Indeed, the global flux
as described by the zero-mode consists of a uniform vertical gradient from the
average concentration $\rho_{0}$ at the top of the cell to the average at the
substrate that we denote $\rho_{\rm eff}$. This flux $D(\rho_{0}-\rho_{\rm
eff})/H$ must be equal to $L^{-2}\sum_{j}m\dot{n}_{j}$, which readily gives
the (average) substrate humidity effectively felt by droplets:

$\rho_{\rm eff}=\rho_{0}-\sum_{j}\frac{Hm\dot{n}_{j}}{L^{2}D}.$ (17)

An alternative derivation of this result via a Fourier analysis of the Green’s
function (Appendix B) shows that the spatial average of $\mathcal{G}$ at $z=0$
is equal to $2\pi H/L^{2}$. By (13), this gives the same expression for
$\rho_{\rm eff}$.

##  IV Statistical model for breath figures

The above sections provide a many-droplet theory for breath figures (in the
regime prior to coalescence). The theory includes the stochastic nucleation of
new drops (Sec. II) and the deterministic diffusive growth of existing drops
(Sec. III), and consistently accounts for drop interactions via the vapor
phase. We proceed by presenting results obtained from numerical simulations of
the proposed model and show that breath figures evolve via two distinct
regimes. Subsequently, we develop a mean field description that captures the
numerical results without adjustable parameter and which is used for the
interpretation of experiments in the companion paper Bouillant et al. (2024).

Figure 3: Typical numerical simulation (not including coalescence): (a) Inset:
snapshot of a numerical realisation of a breath figure taken at
$\mathcal{D}t/H^{2}=10^{-3}$, with the effective diffusion coefficient
$\mathcal{D}$ defined in (9). Main panel: Number of drops $N=\psi L^{2}$ as a
function of time $t$ rescaled by the diffusive time $H^{2}/\mathcal{D}$ (lower
axis) and by the nucleation rate $J_{0}L^{2}$ (upper axis).The red dashed line
corresponds to $\psi=J_{0}t$. Interaction between drops mediated by vapor
diffusion leads to a small decrease of the number of the drops visible at long
time. (b) Root mean square drop radius $R=\sqrt{\langle R_{i}^{2}\rangle}$
plotted as a function of time $t$ (same rescaling as in panel (b)). The red
dashed line corresponds to $R=\sqrt{\frac{1}{2}(\mathcal{R}-1)\mathcal{D}t}$
(see text) and the blue dotted line to $R\propto t^{1/3}$. Numerical
simulation performed at relative humidity $\mathcal{R}=\rho_{0}/\rho_{\rm
sat}=1.5$, the dimensionless energy gap $\chi=1.5$, horizontal box size $L=H$,
$\mathcal{F}=1$ and $J_{0}H^{4}/\mathcal{D}=2.4\;10^{9}$. The latter parameter
determines the diffusion timescale to nucleation rate.

###  IV.1 Stochastic numerical simulations

The numerical integration of the model is performed by two alternating steps.
First, given the positions $\mathbf{r}_{i}$ and sizes $R_{i}$ of the drops, we
compute the growth-rates $\dot{n}_{i}$ from (16); the drop radii $R_{i}$ are
evolved accordingly. Second, the nucleation of a new drop is stochastically
attempted using a Monte-Carlo step. In this step we randomly select a position
on the substrate and compute the local humidity according to (13). From this,
we accept/reject the nucleation in accordance to the statistical law (6).

Figure 3 shows a typical result for the evolution of a single stochastic
realisation of a breath figure; model parameters are listed in the caption).
The corresponding number of drops inside the cell $N=\psi L^{2}$ is plotted as
a function of time in Fig. 3(a). After starting with a single drop, it can be
seen that the number of drops initially increases linearly in time, suggesting
nucleation to occur at a constant rate, that is with a constant energy
barrier. The dotted line corresponds to $\psi=J_{0}t$, which gives a good
description of the initial phase, where we introduced the initial nucleation
rate

$J_{0}=\mathcal{J}\mathcal{R}^{2}\exp\left(-\frac{\chi}{\ln^{2}\mathcal{R}}\right).$
(18)

The number of drops is observed to reach a saturation point around a well-
defined average value, around $10^{3}$ in the present simulation. This
indicates that nucleation is suppressed during the final stages of the
process, revealing a reduction in supersaturation near the substrate. In fact,
as will be quantified later, the effective humidity near the substrate,
denoted as $\rho_{\rm eff}$, is found to decrease over the course of the
condensation process. This decrease can be attributed to the increased volume
of condensed liquid.

A similar crossover is observed for the evolution of the average drop size
$R$, which is plotted as a function of time in Fig. 3(c). Initially, the drops
grow according to $R\sim t^{1/2}$, as is expected for isolated drops. The
exact solution of the growth of a single isolated drop reads
$R=\sqrt{(\mathcal{R}-1)\mathcal{D}t}$. However, the red dashed line
corresponds to $R=\sqrt{\frac{1}{2}(\mathcal{R}-1)\mathcal{D}t}$, which
provides a better fit: drops that nucleated at a later time $T$ have had less
time to grow, which lowers the average radius of the ensemble. Given that the
number of drops increases linearly in time, the average value of $\langle
T\rangle=t/2$, so that, on average, the time to grow $t-\langle T\rangle=t/2$.
This explains the factor 1/2 once multiple drops have nucleated. At the final
stages of the simulation, the growth is sub-diffusive. Again, this is in line
with a reduction of the supersaturation at the bottom of the cell, which slows
down the growth just like it slowed down the nucleation.

The scenario that breath figures evolve according to two distinct regimes is
very robust, and is seen in numerics for a broad range of parameter values. We
note that the crossover to the second regime is seen to arise when the typical
distance between drops $\lambda=\psi^{-1/2}$ is still much larger than the
drop size $R$, i.e. well before coalescence starts to play a significant role.
In what follows, we develop a mean field model that quantitatively explain
these features.

###  IV.2 Lattice of identical drops

The basis for the mean field description is an exact result that can be
obtained for a regular lattice of identical drops. Treating all drops as
equal, we write $R_{i}=R$, $\dot{n}_{i}=\dot{n}$, and
$\rho_{\infty}^{i}=\rho_{\infty}$. The average concentration at the substrate,
as defined by (17), then becomes

$\rho_{\rm eff}=\rho_{0}-\frac{H\psi m\dot{n}}{D}.$ (19)

We now assume that $\rho_{\infty}\approx\rho_{\rm eff}$, i.e. the humidity
experienced for each drop is approximated by the average humidity on the
substrate. Numerical simulations show that the heterogeneities of the humidity
field remain indeed very small with respect to the average. With this, the
growth equation (7) reads

$m\dot{n}=2\pi\mathcal{F}RD\left(\rho_{\rm eff}-\rho_{\rm sat}\right).$ (20)

Inserting into (19), we find

$\rho_{\rm eff}-\rho_{\rm sat}=\frac{\rho_{0}-\rho_{\rm
sat}}{1+2\pi\mathcal{F}\psi HR},$ (21)

which offers an explicit expression for the supersaturation at the surface.
Equation (21) reveals an essential feature for the evolution of breath
figures: the supersaturation $\rho_{\rm eff}-\rho_{\rm sat}$ will decrease
over time owing to the increase of droplet density $\psi$ and droplet size
$R$. Specifically, we see the emergence of a dimensionless combination,

$\psi HR=\frac{HR}{\lambda^{2}},$ (22)

that quantifies the crossover from a regime of low surface coverage to a
regime high surface coverage. In the low-density limit, $\psi HR\ll 1$, we
find $\rho_{\rm eff}\approx\rho_{0}$ so that the supersaturation at the
substrate is $\rho_{0}-\rho_{\rm sat}$. This is the regime of non-interacting
drops, in line with the early phase in Fig. 3. Conversely, the high-density
limit $\psi HR\gg 1$ describes the second regime for which the bottom of the
cell is no longer supersaturated: nucleation is arrested and droplet growth is
subdiffusive. In line with stochastic numerics, the typical distance between
drops at the moment of crossover $\lambda\sim\sqrt{HR}$, which is much larger
than the typical drop size.

###  IV.3 Mean field model

####  IV.3.1 Formulation

We now formulate a mean field model for breath figures, consisting of
evolution equations for the number density $\psi(t)$ and the average drop size
$R(t)$. The latter is defined as $R=\sqrt{\langle R_{i}^{2}\rangle}$. The
nucleation rate (6) depends on the local humidity at the substrate. The mean
field approximation consists of replacing the local humidity by the effective
humidity $\rho_{\rm eff}$, so that the mean field equation for the number of
drops takes the form,

$\frac{\mathrm{d}\psi}{\mathrm{d}t}=\mathcal{J}\left(\frac{\rho_{\rm
eff}}{\rho_{\mathrm{sat}}}\right)^{2}\exp\left(-\frac{\Delta G^{*}(\rho_{\rm
eff}\big{/}\rho_{\mathrm{sat}})}{k_{B}T}\right)={\mathcal{J}}\left(\frac{\rho_{\rm
eff}}{\rho_{\rm sat}}\right)^{2}\exp\left(-\frac{\chi}{\ln^{2}\left(\rho_{\rm
eff}/\rho_{\rm sat}\right)}\right)$ (23)

For $\rho_{\rm eff}$, we take the result obtained for the periodic lattice of
identical drops (21), which we rewrite as

$\rho_{\rm eff}=\frac{\rho_{0}+2\pi\mathcal{F}\psi HR\rho_{\rm
sat}}{1+2\pi\mathcal{F}\psi HR}.$ (24)

Then, we assume the effective humidity (24) also determines the growth of
individual drops. This amounts to setting $\rho_{\infty}^{i}=\rho_{\rm eff}$
in (8), which fixes the growth of $R$ at constant $\psi$. However, the
nucleation of new droplets also affects the average radius: the continuous
addition of drops of vanishing volume at later times lead to a reduction of
the average radius. We have seen (in the discussion around Fig. 3) that during
the initial phase where $\psi\sim t$, the average time at which a droplet
nucleates is $\langle T\rangle=t/2$. Hence, the average radius will grow as
$R^{2}=\frac{1}{2}(\rho_{\rm eff}/\rho_{0}-1)\mathcal{D}t$, where the factor
$1/2$ accounts for the addition of freshly nucleated drops at later times. We
capture the effect of a variable $\psi$ by proposing the mean field growth
equation:

$\frac{\mathrm{d}\left(\psi R^{2}\right)}{\mathrm{d}t}=\left(\frac{\rho_{\rm
eff}}{\rho_{\rm sat}}-1\right)\mathcal{D}\psi.$ (25)

In comparison to (8), the growth equation explicitly depends on $\psi$.
Indeed, these extra factors are harmless at constant $\psi$, as is observed
during the later stages. However, the incorporation of these factors, which
are initially $\psi=J_{0}t$, ensures the correct early-time result.

####  IV.3.2 Dimensionless mean field model

We now present the mean-field equations in their dimensionless forms. The
model predicts the number of drops per unit area $\psi$, the average droplet
radius $R$, the effective humidity at the substrate $\rho_{\rm eff}$, each of
which is evolved over time. In the simulations, we used the size of the cell
$H$ as the length-scale, and $H^{2}/\mathcal{D}$ as the time-scale. Aside from
geometric parameters, the dimensionless parameters include the relative
humidity $\mathcal{R}=\rho_{0}/\rho_{\rm sat}$, the energy gap $\chi$, and the
ratio of timescales $J_{0}H^{4}/\mathcal{D}$. The latter can be scaled out by
choosing appropriate dimensionalization, using the fact that the crossover
between the low-coverage and high-coverage limits occurs when $\mathcal{F}\psi
RH$ is order unity. Specifically, we propose a rescaling where
$\mathcal{F}\psi RH=\tilde{\psi}\tilde{R}$, where tildes denote dimensionless
variables. Imposing that $R^{2}/t$ scales with $\mathcal{D}$, and $\psi/t$
with $J_{0}$, we obtain the following non-dimensionalization:

$\displaystyle R$ $\displaystyle=$
$\displaystyle\left(\frac{\mathcal{D}}{J_{0}\mathcal{F}H}\right)^{1/3}\,\tilde{R},$
(26) $\displaystyle\psi$ $\displaystyle=$
$\displaystyle\left(\frac{J_{0}}{\mathcal{D}\mathcal{F}^{2}H^{2}}\right)^{1/3}\
\tilde{\psi},$ (27) $\displaystyle t$ $\displaystyle=$
$\displaystyle\left(\mathcal{D}J_{0}^{2}\mathcal{F}^{2}H^{2}\right)^{-1/3}\,\tilde{t},$
(28) $\displaystyle\rho_{\rm eff}$ $\displaystyle=$ $\displaystyle\rho_{\rm
sat}\,\tilde{\rho}_{\rm eff},$ (29)

where tildes indicate dimensionless variables. The corresponding dimensionless
mean field equations become:

$\displaystyle\frac{\mathrm{d}\tilde{\psi}}{\mathrm{d}\tilde{t}}$
$\displaystyle=$ $\displaystyle\left(\frac{\tilde{\rho}_{\rm
eff}}{\mathcal{R}}\right)^{2}\exp\left(\frac{\chi}{\ln^{2}\mathcal{R}}-\frac{\chi}{\ln^{2}\tilde{\rho}_{\rm
eff}}\right),$ (30)
$\displaystyle\frac{\mathrm{d}\left(\tilde{\psi}\tilde{R}^{2}\right)}{\mathrm{d}\tilde{t}}$
$\displaystyle=$ $\displaystyle\tilde{\psi}\left(\tilde{\rho}_{\rm
eff}-1\right),$ (31) $\displaystyle\tilde{\rho}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{R}+2\pi\tilde{\psi}\tilde{R}}{1+2\pi\tilde{\psi}\tilde{R}}.$
(32)

The mean field model has thus been reduced to depend only on two dimensionless
parameters: the dimensionless energy gap $\chi$, and the relative humidity
imposed at the top of the cell $\mathcal{R}=\rho_{0}/\rho_{\rm sat}$.

##  V Results

###  V.1 Stochastic numerics versus mean field model

A comparison between different realizations of the stochastic simulations and
the mean-field model (integrated numerically) is presented in figure 4, which
shows the time evolution of the (dimensionless) droplet density
$\tilde{\psi}$, the effective humidity at the substrate $\tilde{\rho}_{\rm
eff}$, and the average radius $\tilde{R}$. In all cases, the average of many
stochastic realizations of breath figures is indistinguishable from the mean-
field prediction. The success of the mean-field model can be rationalized by
examining Fig. 4(d), which displays the corresponding cumulative distribution
function (c.d.f.) of drop sizes at a time within the saturated regime. The
c.d.f. is directly obtained by ordering the $N$ drops by increasing radius and
assigning the value $i/N$ to the $i^{\rm th}$ drop. For the mean-field model,
a list of drops is gradually created in a deterministic way by creating a new
drop when $\psi L^{2}$ crosses an integer value. The radius of all drops is
increased according to the mean-field law:

$\frac{\mathrm{d}\tilde{R}^{2}}{\mathrm{d}\tilde{t}}=\left(\tilde{\rho}_{\rm
eff}-1\right),$ (33)

The c.d.f. is calculated in the same way as for the simulations. The
distributions reveal that the droplets become increasingly monodisperse as
they grow at the same rate without any creation of new drops. This weak
polydispersity aligns with experiments reported in the companion paper
Bouillant et al. (2024), which validates our mean-field model to explain the
monodisperse breath figure patterns obtained in diffusion-controlled
condensation experiments on smooth substrates. In the remainder, we
analytically derive asymptotic predictions from the mean-field model
(30,31,32).

Figure 4: Different realisations of the stochastic model (red) compared to the
mean field model (black) without adjustable parameters. Dimensionless
variables are defined in (26-29), results correspond to $\chi=1.5$ and
$\mathcal{R}=1.5$. (a) Drop density $\tilde{\psi}$ as a function of time
$\tilde{t}$. (b) Effective humidity $\tilde{\rho}_{\rm eff}$ at the substrate
as a function of time $\tilde{t}$. The inset provides the result over a larger
time range, showing the transition from $\tilde{\rho}_{\rm
eff}=\mathcal{R}=1.5$ to the fully saturated regime where $\tilde{\rho}_{\rm
eff}\to 1$. For both panels, the average over many realisations cannot be
distinguished from the mean field prediction. (c) Root mean square drop size
$\tilde{R}$ as a function of time $\tilde{t}$. Dotted lines indicate the
analytical asymptotes of the mean field model (see text). (d) Cumulative
distribution of drop sizes at $\tilde{t}=0.2$. For both panels, the average
over many realisations cannot be distinguished from the mean field prediction.
For this particular time, approximately 80% of the drops fall within 10% of
the largest drop size.

###  V.2 Regimes

The dynamics of $\tilde{\psi}$ and $\tilde{R}$ exhibits two distinct regimes,
the origin of which lies in the expression for effective humidity of the
substrate (32). As is illustrated in the inset of Fig. 4(b), the effective
humidity $\tilde{\rho}_{\rm eff}$ starts at $\mathcal{R}$ (the relative
humidity imposed at the top of the cell), but gradually decreases as the
breath figures evolve. Indeed, the condensation of vapor locally reduces the
average humidity at the substrate, until reaching a fully saturated regime
where $\tilde{\rho}_{\rm eff}\to 1$. We first focus on the initial stages, the
low-density limit, during which $\tilde{\psi}\tilde{R}\ll 1$. In this regime,
(32) reduces to $\tilde{\rho}_{\rm eff}\simeq\mathcal{R}$, and (30,31) reduce
to

$\displaystyle\frac{\mathrm{d}\tilde{\psi}}{\mathrm{d}\tilde{t}}$
$\displaystyle\simeq$ $\displaystyle 1,$ (34)
$\displaystyle\frac{\mathrm{d}\tilde{\psi}\tilde{R}^{2}}{\mathrm{d}\tilde{t}}$
$\displaystyle\simeq$ $\displaystyle\tilde{\psi}\left(\mathcal{R}-1\right).$
(35)

In this regime the nucleation occurs at a constant rate, which allows for an
exact solution

$\displaystyle\tilde{\psi}$ $\displaystyle\simeq$ $\displaystyle t,$ (36)
$\displaystyle\tilde{R}^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}\left(\mathcal{R}-1\right)t.$ (37)

These results captures the early-time asymptotics of the breath figures, as
seen in Fig. 4(a) and 4(c) and already indicated as dashed lines in Fig.
3(a,b).
A second regime arises when $\tilde{\psi}\tilde{R}\gg 1$, for which
$\tilde{\rho}_{\rm eff}\to 1$. In this limit, the right hand side of (30)
vanishes so that no new drops are nucleated and $\tilde{\psi}$ reaches a
plateau value that we indicate by $\tilde{\psi}_{0}$. Note, however, that the
arrest of nucleation already happens long before the humidity reaches the
fully saturated state. In Fig. 4(a), the plateau of $\tilde{\psi}$ is reached
at $\tilde{t}\sim 10^{-1}$, when the effective humidity has only dropped by a
small amount to $\tilde{\rho}_{\rm eff}\approx 1.45$ [cf. Fig. 4(b)]. Indeed,
such a relatively small reduction of humidity is sufficient to dramatically
slow down the nucleation process, owing to the Boltzmann factor that implies
an exponential sensitivity of the nucleation rate. For the dynamics of the
average radius $\tilde{R}$, the late-time asymptotic regime is reached
significantly later, around $\tilde{t}\sim 10^{3}$ in Fig. 4(c), i.e. once the
humidity has dropped to values close to $\tilde{\rho}_{\rm sat}=1$. This
regime corresponds to $\tilde{\psi}\tilde{R}\gg 1$, for which (31) reduces to:

$\displaystyle\frac{\mathrm{d}\tilde{\psi}\tilde{R}^{2}}{\mathrm{d}\tilde{t}}$
$\displaystyle\simeq$ $\displaystyle\frac{\mathcal{R}-1}{2\pi\tilde{R}}.$ (38)

Noting that $\tilde{\psi}=\tilde{\psi}_{0}$ is constant in this regime, this
equation can be integrated to

$\frac{4\pi}{3}\tilde{R}^{3}\simeq(\mathcal{R}-1)\frac{\tilde{t}}{\tilde{\psi}_{0}}.$
(39)

This result implies a linear growth of the average droplet volume,
corresponding to a sub-diffusive asymptotics $\tilde{R}\sim\tilde{t}^{1/3}$,
and is indicated on Fig. 4(c). This growth law has been validated
experimentally in the companion paper Bouillant et al. (2024). Given that the
number of drops in the cell is constant in this regime, the linear growth of
the volume can be interpreted from a constant flux of vapor condensing across
the cell. This argument was already given in previous studies Viovy et al.
(1988); Rogers et al. (1988); Sokuler et al. (2010a). However, the present
analysis explains why both the global flux and the number of drops remains
constant: after a certain time, the humidity at the substrate is no longer
supersaturated, which prevents nucleation and which (at the scale of the cell)
gives rise to a uniform humidity gradient. Our analysis extends to situation
where number of drops in the cell is not constant for instance owing to
coalescence events or coarsening.

###  V.3 Selection of the drop density

Figure 5: (a) Different realisations of the stochastic model (red) compared to
the mean field model (black) without adjustable parameters. Rescaling of the
drop density as a function of time for different $J_{0}H^{4}/\mathcal{D}$
varied on a logarithmic scale from $2.4\;10^{6}$ to $2.4\;10^{12}$, for
$L/H=1$, $\chi=1.5$ and $\mathcal{R}=1.5$. (b) Ratio of the selected drop
density $\psi_{0}$ and the asymptotic prediction $\psi_{a}$ given by (44), as
a function of $\chi$. Different curves correspond to different humidities
$\mathcal{R}$.

Finally, we wish to determine how the maximum number of drops
$\tilde{\psi}_{0}$ depends on the two dimensionless parameters: the energy
barrier $\chi$ and the imposed relative humidity $\mathcal{R}$. We have seen
in Fig. 4 that the value of $\tilde{\psi}_{0}$ is already reached at
relatively high humidities, i.e. when $\tilde{\psi}\tilde{R}\ll 1$. We
accordingly expand the humidity and the nucleation equation as

$\displaystyle\tilde{\rho}_{\rm eff}$ $\displaystyle\simeq$
$\displaystyle\mathcal{R}-2\pi\tilde{\psi}\tilde{R}\left(\mathcal{R}-1\right),$
(40) $\displaystyle\frac{\mathrm{d}\tilde{\psi}}{\mathrm{d}\tilde{t}}$
$\displaystyle\simeq$
$\displaystyle\exp\left(-\frac{4\pi\chi(\mathcal{R}-1)}{\mathcal{R}\ln^{3}\mathcal{R}}\tilde{\psi}\tilde{R}\right).$
(41)

(34) was derived assuming that the argument of the exponential factor was
small, so that the nucleation rate is constant. For sufficiently large $\chi$,
however, one can identify an asymptotic region that satisfies the hierarchy

$\tilde{\psi}\tilde{R}\ll
1\ll\frac{4\pi\chi(\mathcal{R}-1)}{\mathcal{R}\ln^{3}\mathcal{R}}\tilde{\psi}\tilde{R}.$
(42)

In this regime the average drop size $\tilde{R}$ is still expected to grow
diffusively, but the exponential factor is much reduced to slow down the
nucleation. Given that the number of drops is approximately constant in this
intermediate regime, the diffusive growth takes the form
$\tilde{R}=\sqrt{(\mathcal{R}-1)\tilde{t}}$ (i.e. without the factor $1/2$).
Plugging this intermediate asymptotics in (41), the integral can be explicitly
carried out and gives:

$\tilde{\psi}(\tilde{t})\simeq\tilde{\psi}_{0}-\frac{2}{c^{2}}\left(1+c\sqrt{\tilde{t}}\right)\exp\left(-c\sqrt{\tilde{t}}\right),\quad\mathrm{with}\quad
c=\frac{4\pi\chi(\mathcal{R}-1)^{3/2}}{\mathcal{R}\ln^{3}\mathcal{R}}\tilde{\psi}_{0},$
(43)

which reveals how $\tilde{\psi}_{0}$ is approached. When $\chi$ is assumed
large, one indeed finds an exponentially fast approach to $\tilde{\psi}_{0}$,
much before the product $\tilde{\psi}\tilde{R}$ has reached order unity. In
this scenario, nucleation is effectively arrested once the exponential factor
has decreased by some fixed factor. Hence, we can estimate the arrest time
$\tilde{\tau}_{a}$ from the condition $c\sqrt{\tilde{\tau}_{a}}\sim 1$, which
also leads to the estimation of the number density at arrest as
$\tilde{\psi}_{0}\sim\tilde{\tau}_{a}$. From this, we define the typical
arrest density

$\tilde{\psi}_{a}\equiv\frac{\mathcal{R}^{2/3}\;\ln^{2}\mathcal{R}}{\chi^{2/3}\left(\mathcal{R}-1\right)},$
(44)

which is therefore expected to capture the scaling law with respect to the
model parameters $\chi$ and $\mathcal{R}$.
These relations are indeed confirmed in Fig. 5. In panel a, we first verify
that the stochastic simulations for different $J_{0}H^{4}/\mathcal{D}$ (but
fixed values $\chi$ and $\mathcal{R}$) collapse when introducing the
dimensionless droplet density $\tilde{\psi}$ and dimensionless time
$\tilde{t}$. Then, Fig. 5(b) reports the arrest density $\tilde{\psi}_{0}$ for
different values of $\chi$. The graph confirms that $\tilde{\psi}_{0}$ follows
the asymptotic prediction $\tilde{\psi}_{a}$ given by (44), for sufficiently
large $\chi$.

##  VI Coarsening by coalescence

###  VI.1 Stochastic simulations including merging

So far, we have considered the non-interactive and vapor-mediated interactive
regimes assuming that the typical distance between drops $\lambda=\psi^{-1/2}$
remains much larger than the drop size $R$, that is before the occurrence of
coalescence events. This is consistent for the description of the nucleation
phase and its arrest. Namely, the stage where the number of drops is constant
and drop sizes grow according to $R\sim t^{1/3}$ is reached when $\psi RH\sim
1$, so that the typical distance between drops $\lambda\sim\sqrt{RH}\gg R$.
Hence, the probability of droplets to touch and merge is negligible during the
phases described above. However, drops will continue to grow so that
ultimately they will touch. To investigate this effect, we extend our
numerical simulations by including coalescence, by assuming an instantaneous
merging when two droplet radii overlap. When an overlap occurs between droplet
$i$ and $j$, we instantaneously create a new droplet of radius
$(R_{i}^{3}+R_{j}^{3})^{1/3}$ (ensuring volume conservation). The new droplet
is placed at the centre of mass position of the two preceding drops (ensuring
momentum conservation).

Figure 6 shows a typical evolution for the number of drops $\psi(t)$, in a
simulation that includes nucleation, growth, and coalscence. As in Fig. 3(a),
we observe a fast nucleation stage, followed by a plateau that marks the
arrest of nucleation. Then, drops continue to grow until they touch and merge,
after which the number of drops start to decay, following approximately the
scaling law $\psi\sim 1/t^{2}$.

Figure 6: The effect of coalescence on the number of drops $N=\psi L^{2}$
versus dimensionless time $\mathcal{D}t/H^{2}$. Black solid line: Stochastic
numerical simulation for the same parameters as Fig. 3, but now including drop
coalescence. Red dotted line: Prediction of the mean field model. Blue dotted
line: Best fit by the coarsening theory (50). Inset: drop configuration
produced by the coarsening phase at $N=100$.

###  VI.2 Mean field model for self-similar coarsening

The observed scaling law for coarsening can be explained by assuming self-
similar coarsening dynamics Fritter et al. (1988); Derrida et al. (1991);
Meakin (1992). To illustrate the concept we first consider the idealised
situation of a square lattice of identical drops. When in contact, drops are
assumed to merge as is sketched in Fig. 7. The result is a new square lattice
with a lattice unit that is increased by a factor 2. This operation
instantaneously reduces the number of drops by a factor 4, while volume
conservation implies that the drop radius after merging is increased by a
factor $4^{1/3}$. Denoting $\psi_{n}$ as the number of drops per unit area
after the $n$-th merging event, we thus have $\psi_{n}=\psi_{0}/4^{n}$. If we
furthermore define $R_{n}$ as the droplet size immediately after coalescence,
one verifies that for a square lattice $\psi_{n}R_{n}^{2}=A$ is constant, with
$A=2^{-8/3}$.

To transform this coarsening into a temporal dynamics for $\psi(t)$, we need
to determine the time in between two merging events. For this, we make use of
the growth law in the saturated regime, given by (39), which we write in
dimensional form,

$\psi\frac{dR^{3}}{dt}=\mathcal{U},\quad\mathrm{with}\quad\mathcal{U}=\frac{3\left(\rho_{0}-\rho_{\rm
sat}\right)\mathcal{D}}{4\pi\mathcal{F}\rho_{\rm sat}H}.$ (45)

Here we introduced the characteristic velocity $\mathcal{U}$ that represents
the volume flux per unit area that dictates the growth. We wish to compute the
time interval between the merging events $n$ and $n+1$. In between the
coalescence times $t_{n}$ and $t_{n+1}$, the droplet size grows from $R_{n}$
to $2^{1/3}R_{n}$. Using that $\psi_{n}$ is constant in between these events,
we obtain from (45)

$t_{n+1}-t_{n}=\frac{\psi_{n}R_{n}^{3}}{\mathcal{U}}.$ (46)

Using $\psi_{n}R_{n}^{2}=A$, we can thus express the time interval as

$t_{n+1}-t_{n}=\frac{A^{3/2}}{\psi_{n}^{1/2}\mathcal{U}}=2^{n}\frac{A^{3/2}}{\psi_{0}^{1/2}\mathcal{U}}.$
(47)

which can be summed to

$\left(\frac{\psi_{0}^{1/2}\mathcal{U}}{A^{3/2}}\right)\left(t_{n}-t_{0}\right)=\sum_{k=0}^{n-1}2^{n}=2^{n}-1.$
(48)

From this we infer the temporal evolution of $\psi_{n}$, as

$\psi_{n}=\frac{\psi_{0}}{4^{n}}=\frac{\psi_{0}}{\left(1+\frac{\psi_{0}^{1/2}\mathcal{U}}{A^{3/2}}(t_{n}-t_{0})\right)^{2}}.$
(49)

Hence, the lattice model exhibits the sought-after scaling $\psi\sim 1/t^{2}$.
In hindsight, this result could have been inferred on dimensional grounds. A
merging event transforms the square lattice into a square lattice of larger
size, which renders the problem self-similar. The resulting dynamics must
display scale-invariance: the model indeed does not exhibit an intrinsic
length scale, nor an intrisic timescale. The coupling of space and time is
established via the growth law (45), which involves a parameter $\mathcal{U}$
that has the dimensions of a velocity. Hence, the only way to construct a
growing lattice size $\psi^{-1/2}$ is via the scaling $\sim\mathcal{U}t$.
Changing the type of lattice (e.g. from square to hexagonal) would only affect
the value of $A$, not the scaling law.

Figure 7: Coarsening on a square lattice. Drops are assumed to grow according
to (45) until they touch. In this model, 4 drops are assumed to merge
instantaneously while preserving volume, leading to a new square lattice. The
product $\psi R^{2}=1/4$ just before coalescence while $\psi R^{2}=A=2^{-8/3}$
just after coalescence.

Obviously, the lattice model is not intended as a realistic description of the
stochastic numerics: Drops are not equally sized, not equally spaced, and
merge in pairs of two rather than in pairs of four (cf. inset Fig. 6). If,
however, we assume the coarsening patterns to be statistically self-similar,
then the growth law (45) again offers $\mathcal{U}$ as the sole dimensional
parameter in the problem. For the same reason as for the lattice, this
dictates the growth of the pattern length-scale to satisfy
$\psi^{-1/2}\sim\mathcal{U}t$. We thus anticipate (49) to be valid, but with a
non-universal factor $A$. We therefore propose the form

$\psi=\frac{\psi_{0}}{(1+(t-t_{0})/\tau_{c})^{2}},$ (50)

where $\tau_{c}$ should be of the order $\psi_{0}^{-1/2}\mathcal{U}^{-1}$. The
fit of (50) is superimposed as the black solid line in Fig. 6. The proposed
form also provides an excellent fit of the experimental data, presented in the
companion paper Bouillant et al. (2024).

##  VII Discussion

This study presents a theoretical framework to clarify the formation of nearly
monodisperse breath figures on defect-free substrates under a diffusion-
controlled regime, as observed in the experimental companion paper Bouillant
et al. (2024). Specifically, we address the question of how condensing vapor
is distributed between the growth of existing droplets and the nucleation of
new ones. We demonstrate that, following an initial rapid nucleation stage,
dew droplets begin to compete for vapor absorption, leading to the
establishment of a vapor-depleted boundary layer near the substrate. These
collective effects, mediated by vapor diffusion, lead to two distinct regimes:
a low-density nucleation regime ($\psi RH<1$) and a high-density saturation
regime ($\psi RH>1$) where droplet interactions commence. In the low-density
nucleation regime, the effective humidity at the substrate remains close to
the imposed relative humidity at the top of the cell, resulting in diffusive
droplet growth and a consistent nucleation rate. However, as droplets grow and
attain higher densities, the system shifts to the high-density regime. In this
regime, the effective humidity experienced by droplets decreases, impeding
their growth and ultimately halting nucleation due to the exponential
sensitivity of nucleation rates to humidity variations. This transition
signifies the beginning of the constant-drop-number regime, preceding the
coalescence regime. As such, the model explains the experimental measurements
(detecting droplets once they reached the micron size), which indeed exhibit a
constant number of drops (until coalesce kicks in) Bouillant et al. (2024).

Our mean-field model for the formation and evolution of breath figures yields
predictions for the effective humidity field $\rho_{\rm eff}$, the average
droplet radius $R$, and the droplet density $\psi$. Comparison of our theory
with stochastic simulations reveals excellent agreement, particularly in the
nucleation and growth phases, highlighting the robustness of the mean-field
approximation. Specifically, our simulations have corroborated several key
findings: the plateau in droplet number $\psi\rightarrow\psi_{0}$, the sub-
diffusive growth law $R\sim t^{1/3}$ or linear growth of droplet volume $\psi
R^{3}\sim t$ in the interactive regime $\psi RH>1$, and the nearly
monodisperse distribution of droplet sizes, characterized by a cumulative
distribution function (c.d.f.) skewed towards larger droplets. Both our theory
and stochastic simulations are consistent with the dynamics observed in
experimental findings Bouillant et al. (2024). The mean-field model offers
insights into the maximum droplet density, $\psi_{0}$, which is determined by
dimensionless parameters: the energy barrier $\chi$ and the imposed relative
humidity $\mathcal{R}$. Our analysis indicates that $\psi_{0}$ is attained
before the product $\psi RH$ reaches $1$, primarily due to the exponential
suppression of nucleation rates. The derived scaling law for $\psi_{a}$ (44)
is confirmed by stochastic simulations, demonstrating the predictive
capability of our model.

We have also integrated the late-stage evolution of droplet numbers into our
model. By incorporating coalescence events in our simulations, we identified a
coarsening phase where droplet numbers decrease over time as they merge. This
process follows a scaling law $\psi\sim 1/t^{2}$ indicative of self-similar
coarsening dynamics. The mean-field model for self-similar coarsening, based
on the growth law and assuming statistical self-similarity, accurately
described the evolution of droplet density during this phase. The proposed
scaling for $\psi(t)$ (50) provided an excellent fit to both the simulation
and experimental data.

The monodisperse breath figure patterns and growth laws predicted by our
theory are corroborated by experimental observations detailed in our companion
paper Bouillant et al. (2024), and by the observed patterns obtained on smooth
substrates, liquid Zhang et al. (2020); Steyer et al. (1990, 1993);
Nepomnyashchy et al. (2006); Guha et al. (2017), liquid-infused surfaces Anand
et al. (2015); Sharma et al. (2022) or polymeric substrates Briscoe and Galvin
(1991); Ge et al. (2020); Sokuler et al. (2010b), further validating our
approach. However, our model relies on the formation of a diffusive vapor-
depleted boundary layer near the substrate, which may encounter challenges in
configurations where this boundary layer cannot form or is unpredictable.
Depending on the substrate orientation and the humidity injection system (its
flow rate), the flow can exhibit convective or turbulent characteristics,
leading to heterogeneous mixing Villermaux and Innocenti (1999); Villermaux
(2019). To maintain stable stratification and prevent natural convection, the
substrate must be cooled from below, as cold vapor is denser than warm vapor
in the vicinity of the substrate. This condition is always fulfilled in
experiments conducted on liquid layers to prevent liquid drainage or
destabilization. However, it excludes studies where condensation experiments
are conducted on vertical substrates for water harvesting or on downward-
facing horizontal substrates. These factors, along with the possible presence
of surface defects and the eventual occurrence of coalescence, may explain why
in some experimental studies, conducted on smooth Family and Meakin (1989);
Sikarwar et al. (2011), polymeric films or gels Katselas et al. (2022); Rose
(2002); Phadnis and Rykaczewski (2017); Sokuler et al. (2010b); Stricker et
al. (2022); Kolb (1989); Leach et al. (2006); Trosseille et al. (2019) and
liquid-infused surfaces Ge et al. (2020); Anand et al. (2015); Lavielle et al.
(2023), the dew patterns exhibit polydispersity.

In conclusion, our model provides valuable insights into the mechanisms
underlying the formation of quasi-monodisperse dew patterns on defect-free
surfaces. These findings have important implications for controlling dew
pattern formation in various applications, such as designing nano-emulsions
with innovative optical properties or patterning substrates for
microfabrication. By understanding the fundamental physics of breath figure
formation, we can develop new strategies to harness condensation for a wide
range of technological applications.

Acknowledgments. The authors gratefully acknowledge discussions with U. Thiele
and C. Henkel. Final support from NWO Vici (No. 680-47-632), and DFG (Nos.
SN145/1-1 within SPP 2171).

## Appendix

##  Appendix A Classical nucleation theory

We recall here the main results of the classical nucleation theory, necessary
to understand the model hypotheses. Drops of type $A_{n}$ are clusters of $n$
molecules deposited on the substrate, which can grow, merge or break up. Their
surface density is noted $\psi_{n}$. A single water molecule is a monomer
$A_{1}^{v}$ in the vapor phase. We consider only binary processes that involve
a monomer, i.e. the aggregation of one vapor molecule at the surface of a drop
containing $n$ water molecules. We first only focus on reversible growth, one
molecule at a time, represented by the chemical balance equations

$\displaystyle{A_{1}}^{v}$
$\displaystyle\stackrel{{\scriptstyle\beta_{1}}}{{\underset{\alpha_{1}}{\rightleftharpoons}}}A_{1}$
$\displaystyle A_{1}+{A_{1}}^{v}$
$\displaystyle\stackrel{{\scriptstyle\beta_{2}}}{{\underset{\alpha_{2}}{\rightleftharpoons}}}A_{2}$
$\displaystyle\,\vdots$ $\displaystyle A_{n-1}+{A_{1}}^{v}$
$\displaystyle\stackrel{{\scriptstyle\beta_{n}}}{{\underset{\alpha_{n}}{\rightleftharpoons}}}A_{n}$
(51) $\displaystyle\,\vdots$

$\beta_{n}$ is the step-forward reaction rate from size $n-1$ to size $n$
while $\alpha_{n}$ the step-backward reaction rate. The number of particles
follows as:

$\frac{\mathrm{d}\psi_{n}}{\mathrm{d}t}=J_{n}-J_{n+1}\quad{\rm with}\quad
J_{n}=\beta_{n}\psi_{n-1}-\alpha_{n}\psi_{n}$ (52)

The vapor deposition rate $\beta_{n}$ onto a drop containing $n$ molecules has
a contribution from the collisions from molecule in the surrounding air $\sim
n^{2/3}$ and a contribution from the diffusion of molecules along the surface,
proportional to $\sim n^{1/3}$. Introducing the cross-over value $n_{c}$
between these two regimes, we write the law under the generic form:
$\beta_{n}=(n^{2/3}+n_{c}^{1/3}n^{1/3})\mathcal{B}$, where $\mathcal{B}$ is
proportional to $\rho$. The adsorption stage can be written under the same
form as eq. (52), introducing a fictive density $\psi_{0}$ proportional to
$\rho$. The equilibrium concentration obey Boltzmann law:

$\psi_{n}^{0}=\psi_{0}\exp{\frac{-\Delta G_{n}}{k_{B}T}}$ (53)

which allows one, using detailed balance, to express $\alpha_{n}$ as:

$\alpha_{n}=\beta_{n}\frac{\psi_{n-1}^{0}}{\psi_{n}^{0}}=\beta_{n}\exp{\frac{\Delta
G_{n}-\Delta G_{n-1}}{k_{B}T}}$ (54)

We therefore get the equation governing the flux of creation of nano-droplets
containing $n$ molecules:

$J_{n}=\beta_{n}\psi_{n-1}^{0}\left(\frac{\psi_{n-1}}{\psi_{n-1}^{0}}-\frac{\psi_{n}}{\psi_{n}^{0}}\right)$
(55)

In the steady state, the flux obeys $J_{n}=J$ for all $n$, where $J$ is the
number of drops formed per unit surface and unit time. Dividing equation (52)
by $\beta_{n}\psi_{n-1}^{0}$, and summing from $n=1$ to $n=\infty$, we get:

$J=\left(\sum_{n=1}^{\infty}\frac{1}{\beta_{n}\psi_{n-1}^{0}}\right)^{-1}={\mathcal{B}}\psi_{0}\left(\sum_{n=1}^{\infty}\frac{\exp{\frac{\Delta
G_{n-1}}{k_{B}T}}}{n^{2/3}+n_{c}^{1/3}n^{1/3}}\right)^{-1}$ (56)

In the limit $\Delta G^{*}\gg k_{B}T$, the sum tends to its first term:

$J={\mathcal{B}}\psi_{0}(1+n_{c}^{1/3})$ (57)

In the activated regime $\Delta G^{*}\ll k_{B}T$, the sum can been
approximated by an integral, leading to:

$J\simeq\beta_{n^{*}}\psi_{0}Z_{*}\exp\left(-\frac{\Delta
G_{*}}{k_{B}T}\right)$ (58)

where $Z_{*}$ is called the Zeldovich factor, which reads:

$Z_{*}=\sqrt{\frac{-1}{2\pi k_{B}T}\left.\frac{\mathrm{d}^{2}\Delta
G}{\mathrm{d}n^{2}}\right|_{n}}=\frac{\Lambda^{2}}{\sqrt{3\pi\chi}}$ (59)

In the following, the calculations will be performed for $n_{c}=0$, for which
the flux on nanodrops takes the final form, parametrized by $\chi$ and the
multiplicative factor ${\mathcal{J}}$:

$J\simeq{\mathcal{J}}\left(\frac{\rho}{\rho_{\rm
sat}}\right)^{2}\exp\left(-\frac{\chi}{\Lambda^{2}}\right)$ (60)

##  Appendix B Electrostatic analogy

The far field drop problem is analogous to point charges in electrostatics,
the charge here being proportional to the growth rate $\dot{n}_{i}$. In terms
of electrostatics, the charge at the bottom ($z=0$) $\dot{n}_{i}$ (Fig 8) is
then balanced by a homogeneous surface charge density $-\dot{n}_{i}/L^{2}$ on
the ceiling ($z=H$). In order to impose the boundary condition on the solid
located at $z=0$, we introduce mirror charges. We decompose the problem as
schematized in figure 8 into the contribution of the sinking drops in $z=0$,
counterbalanced by an uniform background charge $-2\dot{n}/L^{2}$ in $z=0$ and
the contribution of a double capacitor for which the plates carry an opposite
charge $\pm\dot{n}/L^{2}$. We write formally:

$\mathcal{G}(\mathbf{r})=\sum_{i,j}\frac{1}{|\vec{r}-iL\vec{e}_{x}-jL\vec{e}_{y}|}+\mathcal{G}_{g}$
(61)

where $i$ and $j$ label the periodic copies of the drops. The gauge value
$\mathcal{G}_{g}$ is found using the condition: $\int\mathcal{G}d^{2}x=0$ over
the ceiling. For later reference it is useful to split of the $1/r$
singularity according to:
$\tilde{\mathcal{G}}(\vec{r})=\mathcal{G}(\vec{r})-|\vec{r}|^{-1}$ where
$\tilde{\mathcal{G}}$ is regular, with a well-defined finite value
$\tilde{\mathcal{G}}(\vec{0})$. The value is determined numerically in real
space.

Figure 8: Analogy with the fields created by a 2D periodic array of charge
$\dot{n}$. The neutrality condition requires that the array produces an
uniform backgound charge in $z=0$, compensating for the discrete charges
carried by the drops. The non-accumulation condition requires that this
uniform background charge in $z=0$ is compensated by the injection in $z=H$.

The Green’s function $\mathcal{G}$ captures not only the effect of each
condensed drop, but also the uniform flux that feeds the drop from the ceiling
and the boundary condition on the floor. The Green function $\mathcal{G}$
vanishes on the ceiling $z=H$. It is normalised such that a single drop gives
a singularity $\mathcal{G}=1/|\vec{r}|$ and obey:

$\nabla^{2}\mathcal{G}=-2\pi\sigma.$ (62)

with

$\sigma(\vec{r})=\sum_{p,q}2\delta(x-pL)\delta(y-qL)\delta(z)-\frac{1}{L^{2}}\left[\delta(z-H)+\delta(z+H)\right].$
(63)

Here we used the convention that a single drop represents two charge units:
the true drop, associated with a sink $\dot{n}$ and the mirror drop with
another sing $\dot{n}$, the latter of which of course does not contribute to
the growth of the drop. We split the charge according to
$\sigma=\sigma_{1}+\sigma_{2}$, with

$\sigma_{1}(\vec{r})=\sum_{p,q}2\delta(x-pL)\delta(y-qL)\delta(z)-\frac{2}{L^{2}}\delta(z),$
(64)

where $(p,q)$ are integers used to scan the bi-periodic array in the
$(x,y)$-plane. The residual charge reads:

$\sigma_{2}(\vec{r})=\frac{1}{L^{2}}\left[2\delta(z)-\delta(z-H)-\delta(z+H)\right].$
(65)

The latter has a simple solution $\mathcal{G}_{2}=2\pi(H-|z|)/L^{2}$. The
Fourier transform of the Poisson equation (eq.62) gives
$|\mathbf{K}|^{2}\hat{\mathcal{G}}_{1}(\mathbf{K})=-2\pi\hat{\sigma}_{1}(\mathbf{K})$
introducing the 3D-wavevector $\mathbf{K}=(k_{x},k_{y},k_{z})$. The inverse
Fourier-transform in $z$ can be performed, which gives:

$\hat{\mathcal{G}}_{1}(\mathbf{k},z)=\pi\frac{\hat{\sigma}_{1}(\mathbf{k})}{|\mathbf{k}|}e^{-|\mathbf{k}||z|}$
(66)

where $\mathbf{k}=\left(k_{x},k_{y}\right)$ is the 2D-wavevector. For the
discrete charges, $\dot{n}$ distributed along a periodic array in the
$(x,y)$-plane, the wavevectors along the $(x,y)$ directions are quantified
$k_{x}=2\pi p/L$, $k_{y}=2\pi q/L$ and the norm of the 2D-wavenumber is
$|\mathbf{k}|=\dfrac{2\pi}{L}\sqrt{p^{2}+q^{2}}$. We thus get:

$\mathcal{G}_{1}=\frac{1}{L}\sum_{p^{2}+q^{2}\neq
0}\frac{1}{\sqrt{p^{2}+q^{2}}}\cos\left(2\pi\left(\frac{px+qy}{L}\right)\right)\exp{\left(-2\pi\sqrt{p^{2}+q^{2}}\frac{z}{L}\right)}$
(67)

The total Green’s function thus reads

$\displaystyle{\mathcal{G}}(\vec{r})=\frac{2\pi}{L^{2}}(H-|z|)+\sum_{p^{2}+q^{2}\neq
0}\frac{1}{L\sqrt{p^{2}+q^{2}}}\cos\left(2\pi\left(\frac{px+qy}{L}\right)\right)\exp{\left(-2\pi\sqrt{p^{2}+q^{2}}\frac{z}{L}\right)}$
(68)

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