# The fundamental gap of horoconvex domains in $\mathbb{H}^{n}$

Xuan Hien Nguyen  Iowa State University<EMAIL_ADDRESS>,  Alina Stancu
Concordia University<EMAIL_ADDRESS>and  Guofang Wei  UC Santa
Barbara<EMAIL_ADDRESS>
###### Abstract.

We show that, for horoconvex domains in the hyperbolic space, the product of
their fundamental gap with the square of their diameter has no positive lower
bound. The result follows from the study of the fundamental gap of geodesic
balls as the radius goes to infinity. In the process, we improve the lower
bound for the first eigenvalue of balls in hyperbolic space.

##  1\. Introduction

In this article, the fundamental gap of a domain is the difference between the
first two eigenvalues of the Laplacian with zero Dirichlet boundary
conditions. For convex domains in $\mathbb{R}^{n}$ or $\mathbb{S}^{n}$, $n\geq
2$, it is known from [1, 13, 7, 9] that $\lambda_{2}-\lambda_{1}\geq
3\pi^{2}/D^{2}$, where $D$ is the diameter of the domain.

In hyperbolic space, this quantity behaves very differently from the Euclidean
and spherical cases. Recently, the authors showed [5] that for any fixed
$D>0$, there are convex domains with diameter $D$ in $\mathbb{H}^{n}$, $n\geq
2$, such that $D^{2}(\lambda_{2}-\lambda_{1})$ is arbitrarily small. Since
convexity does not provide a lower bound, one naturally asks if imposing a
stronger notion of convexity, such as horoconvexity, would imply an estimate
for $D^{2}(\lambda_{2}-\lambda_{1})$ from below. Recall that for a domain with
smooth boundary, convexity corresponds to nonnegative principal curvatures of
the boundary, while horoconvexity corresponds to principal curvatures greater
or equal to 1. We show that the quantity $D^{2}(\lambda_{2}-\lambda_{1})$
still tends to zero for all horoconvex domains in hyperbolic space when the
diameter tends to infinity.

######  Theorem 1.1.

For every $n\geq 2$, there exists a constant $C(n)$ such that the Dirichlet
fundamental gap of every horoconvex domain $\Omega$ with diameter $D\geq 4\ln
2$ satisfies

$\lambda_{2}(\Omega)-\lambda_{1}(\Omega)\leq\frac{C(n)}{D^{3}}.$

In particular, as $D\to\infty$, the quantity $(\lambda_{2}-\lambda_{1})D^{2}$
tends to $0$.

We prove this by first obtaining the following estimate for the fundamental
gap for special horoconvex domains, the geodesic balls in hyperbolic space.

######  Theorem 1.2.

Let $B_{R}$ be the geodesic ball of radius $R$ in $\mathbb{H}^{n}$ and
$\lambda_{i}(B_{R})$ be the $i$-th eigenvalue of the Laplace operator
$-\Delta$ in $B_{R}$ with Dirichlet boundary conditions. Then there is a
constant $C(n)$ so that

(1) $\lambda_{2}(B_{R})-\lambda_{1}(B_{R})\leq\frac{C(n)}{R^{3}}.$

In particular, as $R\to\infty$, the quantity $(\lambda_{2}-\lambda_{1})R^{2}$
tends to $0$.

In the authors’ earlier work [5], it was shown that, for any fixed $D>0$, one
can find a domain $\Omega$ for which
$(\lambda_{2}(\Omega)-\lambda_{1}(\Omega))D^{2}$ can be made arbitrarily
small. The domains $\Omega\subset\mathbb{H}^{n}$ in [5] are convex, but not
horoconvex. Their first eigenfunction is not log-concave either. In contrast,
note that the first eigenfunction of $B_{R}$ is log-concave (see [10,
Corollary 1.1] and Lemma 4.3). On the one hand, while the log-concavity of the
first eigenfunction plays a very important role in estimating the fundamental
gap of convex domains in the Euclidean space and sphere, Theorem 1.2 shows
that the log-concavity of the first eigenfunction in the hyperbolic case does
not imply a lower bound estimate for $(\lambda_{2}-\lambda_{1})D^{2}$. On the
other hand, we believe that $D^{2}$ is not the appropriate factor for domains
in the hyperbolic space and we conjecture that, for all horoconvex convex
domains $\Omega\subset\mathbb{H}^{n}$, we have
$\lambda_{2}(\Omega)-\lambda_{1}(\Omega)\geq c(n,D)$ for some function
$c(n,D)$ depending on the dimension and diameter, that can lead to a lower
bound on the fundamental gap appropriately compared with the diameter. This is
true for balls in $\mathbb{H}^{n}$, see (9).

Theorem 1.2 is proved by transforming the eigenvalue equation of balls to the
eigenvalue equation of a Schrödinger operator. As a result, we obtain some
immediate upper and lower bound estimates on the first two eigenvalues of
balls, which improve and simplify earlier estimates on the first eigenvalues
of balls. See Sections 2, 3.

To prove Theorem 1.1, we exploit the fact that all big horoconvex domains
contain a large ball [4], see Theorem 4.1. We then combine Theorem 1.2 with
Benguria and Linde’s [3] comparison result for the fundamental gap to conclude
the proof, see Section 4.

##  2\. Basic Facts on Eigenvalues of Balls in $\mathbb{H}^{n}$

Here we review some basic facts about first two Dirichlet eigenvalues of balls
in the hyperbolic space. By transforming the eigenvalue equation of balls to
its Schrödinger form, we obtain some immediate upper and lower bound estimates
on the first two eigenvalues which improve and simplify earlier estimates.

###  2.1. The first eigenvalue

In this section, let $\lambda_{i}$ be the $i$-th eigenvalue of the Laplacian,
with Dirichlet boundary conditions, of geodesic balls with radius $r$ in
$\mathbb{H}^{n}$.

By [6, 3], the first eigenvalue $\lambda_{1}$ is the first eigenvalue of the
$1$-dimensional problem on $[0,r]$

(2) $u^{\prime\prime}+\frac{n-1}{\tanh t}u^{\prime}+\lambda u=0,\ \ u(r)=0,\
u^{\prime}(0)=0.$

With the change of variable $u(t)=(\sinh t)^{\frac{1-n}{2}}\bar{u}(t)$, we
have the associated Schrödinger equation

(3)
$-\frac{d^{2}}{dt^{2}}\bar{u}+\frac{n-1}{4}\left(n-1+\frac{n-3}{\sinh^{2}t}\right)\bar{u}=\lambda\bar{u}$

with Dirichlet boundary conditions at $0$ and $r$, and $\lambda_{1}$ is the
first eigenvalue of (3). Note that the nonconstant potential term changes sign
at $n=3$. We immediately notice that, when $n=3$,
$\lambda_{1}=1+\frac{\pi^{2}}{r^{2}}$. Since $\sinh^{-2}t\geq\sinh^{-2}r$ on
$(0,r]$, the ODE comparison theorem implies:

######  Lemma 2.1.

For $n>3$,

(4)
$\lambda_{1}>\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{(n-1)(n-3)}{4\sinh^{2}r}.$

For $n=2$,

$\lambda_{1}\leq\frac{1}{4}+\frac{\pi^{2}}{r^{2}}-\frac{1}{4\sinh^{2}r}.$

The lower bound is sharper than the estimate of [2, (1.7)], which followed the
earlier estimate of McKean [11]. It is also an improvement over [12, Theorem
5.6] and an earlier estimate in [8, Theorem 5.2] when $r$ is large and $n>3$.
The upper bound in the case $n=2$ is that found by Gage [8, Theorem 5.2].

The bounds in the other direction do not follow directly from the Schrödinger
equation (3). In [12, Theorem 5.6] the following uniform upper and lower
bounds for the first eigenvalue $\lambda_{1}$ is obtained for all $n\geq 2$:

(5)
$\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}-\frac{4\pi^{2}}{(n-1)r^{3}}\leq\lambda_{1}\leq\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{C}{r^{3}},$

with
$C=\frac{\pi^{2}(n^{2}-1)}{2}\int_{0}^{\infty}\frac{t^{2}}{\sinh^{2}t}dt=\frac{\pi^{4}(n^{2}-1)}{12}$.

We will use this lower bound and improve the upper bound in Section 3.

###  2.2. The second eigenvalue

The second eigenvalue $\lambda_{2}$ is studied in [3, Lemma 3.1], where it is
shown that it is the first eigenvalue of the following equation (see also (16)
with $k=1,l=1$):

(6) $u^{\prime\prime}+\frac{n-1}{\tanh
t}u^{\prime}-\frac{n-1}{\sinh^{2}t}u+\lambda u=0,\ \ \ u(r)=0,\ u(t)\sim t\
\mbox{as}\ t\rightarrow 0.$

Again with the change of variable $u(t)=(\sinh t)^{\frac{1-n}{2}}\bar{u}(t)$,
we have the associated Schrödinger equation

(7)
$-\frac{d^{2}}{dt^{2}}\bar{u}+\frac{n-1}{4}\left(n-1+\frac{n+1}{\sinh^{2}t}\right)\bar{u}=\lambda\bar{u}$

with Dirichlet boundary conditions at $0$ and $r$, where the second eigenvalue
$\lambda_{2}$ is the first eigenvalue of (7). Using once more the ODE
comparison theorem, we obtain

(8)
$\lambda_{2}\geq\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{n^{2}-1}{4\sinh^{2}r}.$

To find an upper bound estimate for $\lambda_{2}$, we will seek in the next
section an upper bound for the first eigenvalue of a more general Schrödinger
equation and, as such, we will simultaneously obtain an upper bound for
$\lambda_{1}$, slightly improve the one in (5).

From (3) and (7) we immediately have the following lower bound on the
fundamental gap of the ball $B_{R}\subset\mathbb{H}^{n}$ for all $n\geq 2$.

(9) $\lambda_{2}-\lambda_{1}\geq\frac{n-1}{\sinh^{2}R}.$

##  3\. First Eigenvalue Upper Bound for Schrödinger Equation

Let $\lambda_{1}^{\alpha}$ be the first eigenvalue of the following equation

(10)
$-\frac{d^{2}}{dt^{2}}u+\frac{n-1}{4}\left(n-1+\frac{\alpha}{\sinh^{2}t}\right)u=\lambda
u$

with Dirichlet boundary conditions at $0$ and $r$.

######  Proposition 3.1.

For $\alpha\geq 0$, we have

(11)
$\lambda_{1}^{\alpha}<\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{(n-1)\alpha}{12r^{3}}\pi^{4}.$

In particular, the first two eigenvalues of the geodesic ball of radius $r$ in
$\mathbb{H}^{n}$ satisfy

(12) $\displaystyle\lambda_{1}$ $\displaystyle<$
$\displaystyle\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{(n-1)(n-3)}{12r^{3}}\pi^{4},\
\mbox{for}\ n\geq 3,$ (13) $\displaystyle\lambda_{2}$ $\displaystyle<$
$\displaystyle\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{(n-1)(n+1)}{12r^{3}}\pi^{4},\
\mbox{for}\ n\geq 2.$

The upper bound (12) improves the upper bound in [12, Theorem 5.6], see (5).

###### Proof.

The first Dirichlet eigenvalue of a Schrödinger operator
$-u^{\prime\prime}+Vu$ is a minimizer of the Rayleigh quotient

$R[u]=\frac{\int|u^{\prime}|^{2}+Vu^{2}}{\int u^{2}},$

among all non-constant $u$ with $u(0)=u(r)=0$.

The equation (10) with $\alpha=0$ has its first eigenfunction equal to
$v=\sqrt{\frac{2}{r}}\sin(\pi t/r)$. It is normalized so that
$\int_{0}^{r}v^{2}dt=1$. Therefore by inserting $v$ into the Rayleigh quotient
associated to (10), we find

$\displaystyle\lambda_{1}^{\alpha}$
$\displaystyle\leq\frac{(n-1)^{2}}{4}+\int_{0}^{r}\left(\frac{dv}{dt}\right)^{2}dt+\int_{0}^{r}\frac{(n-1)\alpha}{4(\sinh
t)^{2}}v^{2}\,dt$
$\displaystyle=\frac{(n-1)^{2}}{4}+\frac{\pi^{2}}{r^{2}}+\frac{(n-1)\alpha}{4}\int_{0}^{r}\frac{v^{2}}{(\sinh
t)^{2}}\,dt.$

Using $\sin|x|\leq|x|$, we have

$r^{2}\int_{0}^{r}\left(\frac{\sin\left(\pi t/r\right)}{\sinh
t}\right)^{2}dt\leq\pi^{2}\int_{0}^{r}\left(\frac{t}{\sinh
t}\right)^{2}dt<\pi^{2}\int_{0}^{\infty}\left(\frac{t}{\sinh
t}\right)^{2}dt=\frac{\pi^{4}}{6}.$

This gives $\int_{0}^{r}\frac{v^{2}}{(\sinh
t)^{2}}\,dt<\frac{\pi^{4}}{3r^{3}}$, hence (11). ∎

Combining the lower bound in (5) with (13) gives the estimate (1) in Theorem
1.2.

##  4\. Horoconvex domains in $\mathbb{H}^{n}$

A stronger definition of convexity in the hyperbolic space considers
horospheres as natural analogues of Euclidean hyperplanes supporting a convex
domain:

######  Definition.

A set $\Omega\subset\mathbb{H}^{n}$ is called horoconvex if, for every point
$p\in\partial\Omega$, there exists a horosphere ${\mathcal{H}}$ through $p$
such that $\Omega$ lies in the horoball bounded by ${\mathcal{H}}$.

Recall that a _horosphere_ is a sphere with center on the ideal boundary of
$\mathbb{H}^{n}$ and that a _horoball_ is a domain whose boundary is a
horosphere.

When $\Omega$ is a compact domain with smooth boundary in the hyperbolic space
of constant negative curvature $-1$, the domain $\Omega$ is horoconvex if and
only if all principal curvatures of the boundary hypersurface are greater or
equal to one. As a special case, $B_{R}$, the geodesic sphere of radius $R$,
is horoconvex as each of the principal curvatures of its boundary is equal to
$\coth R$, and $\coth R>1$ for all $R>0$.

Finally, for any compact domain, recall that its inradius is the radius of the
largest ball contained in the domain, and that its circumradius is the radius
of the smallest ball containing the domain. Part of a result of Borisenko-
Miquel [4, Theorem 1] states the following:

######  Theorem 4.1.

[4] Let $\Omega$ be a compact horoconvex domain in $\mathbb{H}^{n}$ with
inradius $r$ and circumradius $R$. Denoting $\tau=\tanh\frac{r}{2}$, then

(14) $R-r\leq\ln\frac{(1+\sqrt{\tau})^{2}}{1+\tau}<\ln 2,$

and this bound is sharp.

An immediate consequence of (14) is that the diameter of the domain satisfies
$D\leq 2R\leq 2r+2\ln 2$. We are now ready to prove Theorem 1.1.

###### Proof.

Let $\Omega\subset\mathbb{H}^{n}$ be a horoconvex domain of diameter $D$.
Choose $R_{\Omega}$ such that the ball of radius $R_{\Omega}$ satisfies
$\lambda_{1}(B_{R_{\Omega}})=\lambda_{1}(\Omega)$. Theorem 4.1 implies that
$\Omega$ contains a ball of radius $r$ with $r\geq\frac{D}{2}-\ln 2$. By
domain monotonicity of the first eigenvalue, $R_{\Omega}\geq\frac{D}{2}-\ln
2$, hence

(15) $R_{\Omega}\geq\frac{D}{4},$

when $D\geq 4\ln 2$.

Using [3], Benguria-Linde’s upper bound on the second eigenvalue, we have that

$\lambda_{2}(\Omega)-\lambda_{1}(\Omega)\leq\lambda_{2}(B_{R_{\Omega}})-\lambda_{1}(B_{R_{\Omega}}).$

Applying the estimates (1) and (15) concludes the proof of Theorem 1.1. ∎

## Appendix

Small balls and log-concavity of eigenfunction of geodesics balls in
$\mathbb{M}^{n}_{K}$

To round up the discussion on the fundamental gap of balls in the hyperbolic
space, we thought to include here an observation on the fundamental gap of
balls of small radii, as well as a simple argument proving the log-concavity
of the first eigenfunction of geodesic balls in simply connected Riemannian
manifolds with constant negative sectional curvature.

###  4.1. The gap of small balls in negatively curved manifolds

Let $\mathbb{M}^{n}_{K}$ be the simply connected Riemannian manifold with
constant sectional curvature $K$. Here, we assume that $K$ is negative and
write $K=-k^{2},\,(k>0)$. Denote by $\lambda_{i}(n,k,r)$ the eigenvalues of
the Laplacian for geodesic balls with radius $r$ in $\mathbb{M}^{n}_{K}$ with
Dirichlet boundary condition.

By separation of variables, see [6, 3], the eigenvalues $\lambda_{i}(n,k,r)$
are eigenvalues of

(16)
$u^{\prime\prime}+\frac{(n-1)k}{\tanh(kt)}u^{\prime}-\frac{l(l+n-2)k^{2}}{\sinh^{2}(kt)}u+\lambda
u=0,$

where $l=0,1,2,\cdots$, with boundary condition $u^{\prime}(0)=0$ for $l=0$,
$u(t)\sim t^{l}$ as $t\to 0$ for $l>0$, and $u(r)=0$.

By scaling, this immediately gives [3, Lemma 4.1], for $c>0$,

(17) $\lambda_{i}(n,\frac{1}{c}k,cr)=c^{-2}\lambda_{i}(n,k,r).$

Hence

(18) $\lambda_{i}(n,1,r)=r^{-2}\lambda_{i}(n,r,1).$

Therefore, for small balls in $\mathbb{H}^{n}$, the value
$r^{2}\lambda_{i}(n,1,r)$ is close to the corresponding one in the Euclidean
space, as one would expect. Namely,

######  Lemma 4.2.

$\lim_{r\to
0}r^{2}\lambda_{i}(n,1,r)=\lambda_{i}(n,0,1)=r^{2}\lambda_{i}(n,0,r),$

and

(19) $\lim_{r\to
0}r^{2}\left(\lambda_{2}(n,1,r)-\lambda_{2}(n,1,r)\right)=r^{2}(\lambda_{2}(n,0,r)-\lambda_{1}(n,0,r))=j_{\frac{n}{2},1}^{2}-j_{\frac{n}{2}-1,1}^{2},$

where $j_{p,k}$ is the $k$-th positive zero of the Bessel function $J_{p}(x)$.

###  4.2. The first eigenfunction for balls

The first eigenfunction of balls is purely radial, so it is straightforward to
show that it is log-concave, as in the Euclidean and spherical case.

######  Lemma 4.3.

The first eigenfunction $u_{1}$ of (2) is strictly log-concave.

This is in [10, Corollary 1.1], where more general elliptic equations with
power are considered. For convenience, we give a simple and direct proof here.

###### Proof.

First we show $u_{1}$ is strictly decreasing. Multiplying both sides of (2) by
$\sinh^{n-1}t$, we have

$(u_{1}^{\prime}\sinh^{n-1}t)^{\prime}=-\lambda_{1}u_{1}\sinh^{n-1}t<0.$

Since $u_{1}^{\prime}(0)=0$, we have $u_{1}^{\prime}(t)<0$ for $t\in(0,r)$.

Let $\varphi=(\log u_{1})^{\prime}$. Then $\varphi(0)=0$, $\varphi<0$ on
$(0,r)$, and

$\varphi^{\prime}=\frac{u_{1}^{\prime\prime}}{u_{1}}-\left(\frac{u_{1}^{\prime}}{u_{1}}\right)^{2}=-\frac{n-1}{\tanh
t}\varphi-\lambda_{1}-\varphi^{2}.$

Taking the limit as $t\to 0$ gives
$\varphi^{\prime}(0)=-\lambda_{1}-(n-1)\lim_{t\to 0}\frac{\varphi}{\tanh
t}=-\lambda_{1}-(n-1)\varphi^{\prime}(0)$. Hence, $\varphi^{\prime}(0)<0$.
Now, we claim that $\varphi^{\prime}(t)<0$ on $[0,r)$. Otherwise, there exists
$t_{1}\in(0,r)$ such that $\varphi^{\prime}<0$ on $[0,t_{1})$,
$\varphi^{\prime}(t_{1})=0$ and $\varphi^{\prime\prime}(t_{1})\geq 0$. Note
that $\varphi^{\prime\prime}$ satisfies

$\varphi^{\prime\prime}=\frac{n-1}{\sinh^{2}t}\varphi-\frac{n-1}{\tanh
t}\varphi^{\prime}-2\varphi\varphi^{\prime}.$

Evaluating the two sides of the equation at $t_{1}$ gives

$0\leq\varphi^{\prime\prime}(t_{1})=\frac{n-1}{\sinh^{2}t_{1}}\varphi(t_{1})<0.$

This is a contradiction. ∎

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#  Decay dynamics of a single spherical domain in near-critical phase-
separated conditions

Raphael Saiseau<EMAIL_ADDRESS>Laboratoire Ondes et Matière
d’Aquitaine, UMR 5798, CNRS, Université de Bordeaux, F-33400, Talence, France.
Physics of Fluids, University of Twente, 7522 NB Enschede, The Netherlands.
Henri Truong  Laboratoire Ondes et Matière d’Aquitaine, UMR 5798, CNRS,
Université de Bordeaux, F-33400, Talence, France.  Centre de Recherche Paul
Pascal, UMR 5031, CNRS, Université de Bordeaux, F-33600, Pessac, France.
Thomas Guérin  Laboratoire Ondes et Matière d’Aquitaine, UMR 5798, CNRS,
Université de Bordeaux, F-33400, Talence, France.  Ulysse Delabre  Laboratoire
Ondes et Matière d’Aquitaine, UMR 5798, CNRS, Université de Bordeaux, F-33400,
Talence, France.  Jean-Pierre Delville<EMAIL_ADDRESS>Laboratoire Ondes et Matière d’Aquitaine, UMR 5798, CNRS, Université de
Bordeaux, F-33400, Talence, France.

###### Abstract

Domain decay is at the heart of the so-called evaporation-condensation
Ostwald-ripening regime of phase ordering kinetics, where the growth of large
domains occurs at the expense of smaller ones, which are expected to
‘evaporate’. We experimentally investigate such decay dynamics at the level of
a single spherical domain picked from one phase in coexistence and brought
into the other phase by an opto-mechanical approach, in a near-critical phase-
separated binary liquid mixture. We observe that the decay dynamics is
generally not compatible with the theoretically expected surface-tension decay
laws for conserved order parameters. Using a mean-field description, we
quantitatively explain this apparent disagreement by the stratification of
solute concentrations induced by gravity close to a critical point. Finally,
we determine the conditions for which buoyancy becomes negligible compared to
capillarity and perform dedicated experiments that retrieve the predicted
surface-tension induced decay exponent. The surface-tension driven decay
dynamics of conserved order parameter systems in the presence and the absence
of gravity, is thus established at the level of a single domain.

Phase ordering kinetics is an ubiquitous and fundamental process [1, 2], that
characterizes the irreversible evolution of a system initially out-of-
equilibrium into well separated coexisting phases. For conserved order
parameter systems, the nucleated domains usually evolve from an intermediate
free-growth regime to the so-called late stage Ostwald ripening [3, 4, 5],
where the larger domains grow at the expense of the smaller ones which decay,
or “evaporate” due to interfacial tension, as nicely demonstrated by the
classical evaporation-condensation LSW theory of Lifshitz, Slyozov [6] and
Wagner [7]. This very general phenomenon is at work in all phase transition
dynamics involving conservation of the order parameter and has consequently
huge implications in out-of-equilibrium liquids, soft matter and material
sciences [8, 9, 10].

The decay of the smallest domains was clearly observed in two dimensional
systems, such as metallic islands [11] or during a liquid-solid transition
triggered in confined geometry [12] in order to illustrate the importance of
correlations between evolving domains. In three dimensions, its investigation
is essentially indirect as most coarsening experiments, performed at the scale
of a large number of interacting droplets, and measured by scattering
techniques [13] or by direct visualization of droplet assemblies [14, 15],
focus instead on the evolution of the droplet distribution and its comparison
to LSW predictions. So, although at the basis of the evaporation-condensation
Oswald-ripening, the surface-tension driven evaporation of small droplets in
conserved order parameter systems remains almost unexplored. Conversely,
standard experiments [16, 17] on the evaporation of single liquid drops report
the classical $R$-squared law, in which the drop radius $R$ evolves with time
$t$ as $R\sim(t_{\mathrm{f}}-t)^{\alpha}$ with $\alpha=1/2$, where
$t_{\mathrm{f}}$ is the final evaporation time. The involved mechanism is
nonetheless not driven by surface tension for conserved systems, for which
theory [2] predicts instead $\alpha=1/3$ . This exponent has been observed in
the different context of the thinning dynamics of liquid necks [18, 19].
However, surprisingly, this domain decay prediction at the scale of a single
spherical drop has never been confronted to experiments, whereas it plays a
key role in evaporation-condensation processes. This is the subject of the
present letter.

Starting from a near-critical phase-separated liquid mixture at equilibrium to
keep the universality of Ising models, we use the optical radiation pressure
of a laser wave to destabilize the meniscus and produce a size-controlled
single drop of one phase immersed into the other, at a chosen altitude. Then,
we turn off the laser and look at the drop evaporation. While we expected the
$R\sim(t_{\mathrm{f}}-t)^{1/3}$ scaling, our observations firmly indicate a
richer dynamics due to gravity [20, 21]. We build a dedicated mean-field model
which incorporates gravity and confront our measurements to predictions.
Finally, we ‘close the loop’ by determining from the modeling the conditions
for which buoyancy becomes negligible compared to capillarity and performing
dedicated experiments that retrieve the expected gravity-free decay exponent
$\alpha=1/3$. We thus characterize the surface-tension driven decay dynamics
of conserved systems, at the level of a single spherical domain.

Figure 1:  (a) Schematic phase diagram of the critical micellar solution, with
$\Phi$ the micellar volume fraction, and ($T_{\textrm{c}},\Phi_{\textrm{c}})$
the coordinates of the critical point. A cartoon of a single water-in-oil
micelle, of typical size 4 nm, is also shown (inset). (b) Schematics of the
optical bending and jetting instability of the meniscus ($T>T_{\textrm{c}}$)
induced by a laser beam focused at the meniscus (green). (c) Typical image
sequence of the optical jetting instability. We show (i, ii) the jet
formation, (iii) the Rayleigh-Plateau instability when the laser is turned off
the first time, (iv) the radiation pressure effect at lower laser power to
force droplet coalescence, and (v) the resulting final evaporating droplet
optically pushed to a given altitude. (d) Decay dynamics for $\Delta T=4$ K,
including snapshots, of the produced droplet when the laser is definitively
turned off. $t_{\mathrm{f}}$ is the first time when the droplet cannot be
detected any longer. Scale bars: $5\leavevmode\nobreak\ \mathrm{\mu m}$.

Experimental system. We consider a water-in-oil micellar solution of critical
composition (water, $9\%$ wt, toluene, $79\%$ wt, SDS, $4\%$ wt, butanol,
$17\%$ wt) [22, 23], see the Supplement Material for details [24]. As a pseudo
binary liquid mixture, it shows an Ising ($d=3,n=1$) low critical point at
$T_{\mathrm{C}}\simeq 38^{\circ}\leavevmode\nobreak\ $C, see Figure 1(a). The
microemulsion is contained inside a tight fused-quartz cell of 2 mm thickness
thermally-controlled with a PID in a homemade brass oven. When $\Delta T\equiv
T-T_{\mathrm{C}}>0$, the solution separates into two phases of respective
micellar volume fractions $\Phi_{1}$ and $\Phi_{2}$ (with
$\Phi_{2}<\Phi_{1}$), separated at equilibrium by an horizontal interface,
called hereafter the meniscus. To produce thermodynamically metastable
droplets, we use the optical radiation pressure of a continuous laser wave
focused at the meniscus to induce a liquid jet [25, 26] of the phase
$\Phi_{2}$ into the phase $\Phi_{1}$, see Figure 1(b). When the laser is
turned off, drops are produced through the Rayleigh-Plateau instability [23].
Then, using again the radiation pressure at lower beam intensity, we force
some of these drops to coalesce into a single one which is further pushed
opto-mechanically to a chosen altitude, see Figure 1(c). Thermodynamically
out-of-equilibrium due to its finite size, the produced droplet immediately
starts to evaporate while buoyancy makes it rising toward the meniscus.

The droplet decay is captured using $\times 20$ or $\times 50$ long working
distance Olympus® microscope objectives and a Phantom® VEO340L camera. A
custom-made ImageJ® code is used to detect the intensity gradient maxima at
the drop edges to extract the droplet radius $R$, and we checked the
calibration with other detection methods, see SM [24]. The altitude
$z_{\mathrm{d}}$ of the center of mass of the droplet is also recorded. An
example of droplet evaporation dynamics is shown on Figure 1(d).

Figure 2: Droplet radius $R$ as a function of $t_{\mathrm{f}}-t$ for (a) for
$\Delta T=4\leavevmode\nobreak\ $K and (b) $\Delta T=0.2\leavevmode\nobreak\
$K. The legend indicates the average altitude $\langle z_{\mathrm{d}}\rangle$.

On Figure 2(a), we present several examples of individual droplet dynamics
$R(t)$ for $\Delta T=4$K. While conservation of the order parameter predicts a
regime $R(t)\propto(t_{\mathrm{f}}-t)^{1/3}$, these measurements clearly
demonstrate that this scaling is not verified. Instead, experimental data show
a regime $R(t)\propto(t_{\mathrm{f}}-t)^{\alpha}$, with $\alpha\simeq
0.5-0.6$. This behavior is confirmed down to $\Delta T=0.2$K [Fig. 2(b)]. As
the decay amplitude varies from drop to drop (Fig. 2) and is correlated to the
mean altitude $\langle z_{\mathrm{d}}\rangle$ (defined as the temporal average
of the spanned $z_{\mathrm{d}}(t)$), a natural candidate for an additional
physical effect impacting the evaporation could be gravity. Indeed, gravity is
well known to affect the statics [27] and dynamics [20, 2] of phase
transitions. We thus develop a theoretical framework to understand the impact
of gravity and surface tension on the evaporation dynamics of single droplets.

Theory. We consider the free-energy functional

$\mathcal{H}[\phi]=\int
d\mathbf{x}\left\\{V(\phi)+\frac{\kappa}{2}(\nabla\phi)^{2}-\rho_{\text{e}}gz\phi\right\\},$
(1)

where the order parameter $\phi$ is the local micellar volume fraction,
$V(\phi)$ is a double well potential, and $\kappa$ gives the energy cost of
local gradients of $\phi$. Also, $z$ is the vertical coordinate (increasing in
the downward direction, with $z=0$ at the meniscus),
$\rho_{\text{e}}=\rho_{\mathrm{micelle}}-\rho_{\mathrm{solvent}}$ is the
effective mass density of the micelles, and $g$ is the earth gravitational
acceleration. Next, we assume that the dynamics satisfies the equations of the
(noiseless) model $H$ [28]:

$\displaystyle\partial_{t}\phi+\mathbf{u}\cdot\nabla\phi=\lambda\nabla^{2}\mu,\
\ \ \ $
$\displaystyle\mu(\mathbf{x})=\delta\mathcal{H}/\delta\phi(\mathbf{x}),$ (2)
$\displaystyle\eta\nabla^{2}\mathbf{u}=\nabla p+\phi\nabla\mu,\ \ \ \ $
$\displaystyle\nabla\cdot\mathbf{u}=0,$ (3)

with $\lambda$ a kinetic coefficient, $\mu$ the chemical potential, $\eta$ the
viscosity and $p$ the pressure. Here, the transport equation (2) for the
conserved order parameter $\phi$ is coupled to Stokes’ equation (3) describing
the flow field $\mathbf{u}$ in the fluid (assumed to be incompressible) with
an additional stress induced by chemical potential gradients. To ensure the
conservation of the number of micelles, we impose that $\mathbf{u}$ and
$\nabla\mu$ vanish at the boundaries of the system.

Sharp interface limit.-  We consider the sharp-interface limit, in which
$\kappa$ is large enough so that, in each bulk phase $i$, the field $\phi$ is
close to its equilibrium value ($\phi\simeq\Phi_{i}+\overline{\phi_{i}}$, with
small $\overline{\phi_{i}}$), while $\phi$ varies sharply across interfaces.
Here, $\Phi_{1}$ and $\Phi_{2}$ are the equilibrium values of $\phi$ on each
side of the meniscus. In the sharp interface limit, following the arguments of
Ref. [2] and adding gravity, an effective dynamics is obtained for
$\overline{\phi}_{i}$ with effective boundary conditions at the interface
[24]. In the bulk phases, at equilibrium,
$\overline{\phi}_{i}\simeq\rho_{\text{e}}gz/V_{i}^{\prime\prime}$, where
$V_{i}^{\prime\prime}\equiv\partial_{\phi}^{2}V|_{\Phi_{i}}$, so that, due to
gravity, the concentration of micelles varies with the altitude $z$ and
differs from its standard value at co-existence equilibrium [29]. The dynamics
for $\overline{\phi}$ obeys the advection diffusion equation

$\partial_{t}\overline{\phi}_{i}+\mathbf{u}\cdot\nabla\overline{\phi}_{i}=D_{i}\nabla^{2}\overline{\phi}_{i},\
\ \ D_{i}=\lambda V_{i}^{\prime\prime}.$ (4)

Next, the boundary conditions at an interface read

$\overline{\phi}=\frac{-\gamma C}{\Delta\Phi V^{\prime\prime}_{i}},\
\left[\Delta\Phi(\mathbf{v}-\mathbf{u})-D_{2}\nabla\overline{\phi}_{2}+D_{1}\nabla\overline{\phi}_{1}\right]\hat{\mathbf{n}}=0,$
(5)

where $\Delta\Phi=\Phi_{1}-\Phi_{2}$, $\mathbf{v}$ is the interface velocity,
$\hat{\mathbf{n}}$ is the unit vector normal to the interface (pointing
towards phase $1$) and $C$ is the total interface curvature. The first
equation in Eq. (5) is a form of Gibbs-Thomson relation, while the second one
expresses the conservation of the number of micelles.

Further assuming that advective and instationary terms in Eq. (4) are
negligible (so that $\nabla^{2}\overline{\phi}_{i}\simeq 0$), these equations
can be solved by requiring that the equilibrium solution should be recovered
far from the droplet. This leads to

$\dot{R}=\frac{-\lambda}{R\Delta\Phi}\left(\frac{2\gamma}{R\Delta\Phi}+z_{\mathrm{d}}\rho_{\text{e}}g\right),\hskip
8.5359pt\dot{z_{\mathrm{d}}}=-\frac{4\rho_{\text{e}}\Delta\Phi gR^{2}}{15\
\eta},$ (6)

where $\dot{R}\equiv\partial_{t}R$. The latter equation corresponds to
Hadamard’s formula for the sedimentation velocity of a liquid drop. Equations
(32) describe the coupling between the evaporation dynamics and gravity. Two
effects are at work: (i) first, if the droplet is at an altitude
$z_{\mathrm{d}}$, its evaporation dynamics is accelerated due to the increased
micelle concentration at this altitude, and second (ii) a droplet moves and
experiences different micelle concentrations due to Hadamard’s law.

According to Eq. (32), when $R\to 0$, $\dot{z}_{\mathrm{d}}$ vanishes and the
final stage of evaporation occurs at constant $z_{\mathrm{d}}$. In this case,
integrating the equation for $\dot{R}$ leads to

$f\left(\frac{z_{\mathrm{d}}R(t)}{L_{\mathrm{c}}^{2}}\right)=\frac{\lambda\Delta\Phi(t_{\mathrm{f}}-t)(\rho_{\text{e}}gz_{\mathrm{d}})^{3}}{\gamma^{2}},$
(7)

where $f(u)=\frac{(u-4)u}{2}+4\ln\left(\frac{u+2}{2}\right)$ and
$L_{\mathrm{c}}=\sqrt{\gamma/(\rho_{\text{e}}g\Delta\Phi)}$ is the
gravitational capillary length. As limiting cases, we retrieve, for small
radii, the law for surface-tension dominated decay

$R(t)\simeq\left[\frac{6\lambda\gamma}{(\Delta\Phi)^{2}}(t_{\mathrm{f}}-t)\right]^{1/3},$
(8)

whereas when $R\gg L_{\mathrm{c}}^{2}/z_{\mathrm{d}}$, one obtains the
gravity-dominated regime

$R(t)\simeq\left[a(t_{\mathrm{f}}-t)\right]^{1/2},\
a=2\frac{\lambda\rho_{\text{e}}gz_{\mathrm{d}}}{\Delta\Phi}.$ (9)

This behavior looks similar to the classical $R-$squared laws, but here
evaporation is driven by the difference between $\phi$ at the drop altitude
and its value at which phase coexistence is allowed.

Figure 3: (a) $R^{2}$ as a function of $t_{\mathrm{f}}-{t}$ for different
droplet mean altitude $\langle z_{\mathrm{d}}\rangle$ for $\Delta T=2$K. Error
bars are shown for one out of four data points for visibility issues. Dashed
lines correspond to the linear fit $R^{2}=a\left(t_{\mathrm{f}}-t\right)$. (b)
Fitted slope $a$ as a function of $\langle z_{\mathrm{d}}\rangle$, for $\Delta
T=$0.2, 2 and 16K. The horizontal ‘error-bars’ indicate the range of
$z_{\mathrm{d}}$ sampled during the evaporation, while vertical error bars
show the standard deviation on $a$. Dashed lines are linear fits. (c)
$a/\langle z_{\mathrm{d}}\rangle$ as a function of $\Delta T$. Blue continuous
line: best fit with the law $a/\langle
z_{\mathrm{d}}\rangle=v_{\mathrm{0}}\left(\Delta
T/T_{\mathrm{C}}\right)^{-0.325}$ with $v_{\mathrm{0}}=0.25\mathrm{nm/s}$.
Black dashed lines: same law with the coefficients
$v_{\mathrm{0}}=0.17,\leavevmode\nobreak\ 0.33\mathrm{nm/s}$. Square symbol:
$a/\langle z_{\mathrm{d}}\rangle$ obtained from the theoretical analysis at
$\Delta T=20$K (see text).

Experimental evidence of gravity effects.  On Figure 3(a), we show
experimental curves $R^{2}$ as a function of time. These curves are compatible
with a linear relation $R^{2}(t)=a\left(t_{\mathrm{f}}-{t}\right)$. The mean
slope $a$ is then reported as a function of $z_{\mathrm{d}}$ on Figure 3(b),
over a large variation of $\Delta T=0.2-16$K. A linear dependence of $a$ on
$z_{\mathrm{d}}$ is obtained, as predicted in the gravity dominated regime,
see Eq. (9). The experimental values of $a/\langle z_{\mathrm{d}}\rangle$ are
reported on Fig. 3(c) and show a weak temperature dependency. Theoretically,
the diffusion coefficient $D_{i}=\lambda V_{i}^{\prime\prime}$ is usually
believed to scale as $D_{i}\propto V_{i}^{\prime\prime}$, as demonstrated for
spin exchange dynamics [30]. In our case, we conclude that $\lambda$ does not
depend on $T$ near the critical point. Then, we expect
$a/z_{\mathrm{d}}\propto 1/\Delta\Phi\sim 1/(\Delta T)^{\beta}$ with
$\beta\simeq 0.325$. Such scaling is compatible with the experiments, if one
takes into account the dispersion of the data [Fig. 3(c)]. Thus, observations
point towards gravity as being responsible for the acceleration of the
evaporation dynamics from the expected exponent $\alpha=1/3$ to the observed
one $\alpha\simeq 1/2$.

Figure 4: Experimental and theoretical dynamics for $\Delta T=20$K. (a) $R$ as
a function of $t_{\mathrm{f}}-t$, (b) same trajectories in the
$(R,z_{\mathrm{d}})$ plane and (c) $R(t)$ in rescaled coordinates, with
$z_{\mathrm{f}}\equiv z_{\mathrm{d}}(t_{\mathrm{f}})$. Symbols: experimental
values for six different drops, with an additional drop in (b) (red
triangles). Dash-dotted black lines: solutions of Eq. (32). In (c), the
continuous black line represents the scaling form (37), and dash-dotted blue
and red lines indicate respectively the regimes with $\alpha=1/3$ and
$\alpha=1/2$. Parameters of the theory: $L_{\mathrm{c}}=20\mu m$,
$\lambda\gamma/(\Delta\Phi)^{2}=0.07\mu$m3/s, $\rho_{\text{e}}\Delta\Phi
g/\eta=0.19\mu$m-1s-1 [24].

Small drops near the meniscus: surface-tension dominated dynamics -
Nonetheless, the model indicates that the decay exponent $\alpha=1/3$ should
be retrieved when $R\ll L_{\mathrm{c}}^{2}/z_{\mathrm{d}}$, i.e. when
evaporating drops are small, close to the meniscus (where $z_{\mathrm{d}}=0$)
and far from criticality (as $L_{\mathrm{c}}$ vanishes at the critical point).
Since the gradient edge-detection method may be intrinsically limited for
small radii [24], we confort our measurement procedure as follows. Starting
from calibrated silica beads of radii $\mathrm{1.15\mu m}$ and
$\mathrm{2.18\mu m}$ in a water/glycerol mixture to reach refractive index
contrasts close to those of our phase-separated system, a complementary image
analysis method is developed, in agreement with the previously mentioned
method [24]. On Figure 4(a), we show the decay dynamics at $\Delta T=20$K,
relatively far from the critical point. It is observed that some droplets
decay with an exponent $\alpha=1/3$, either over a large temporal window or
only at the end of the dynamics, when $t\to t_{\mathrm{f}}$. Fig. 4(b)
represents the trajectories of the same drops in the $(R,z_{\mathrm{d}})$
plane, where one distinguishes between a region where drops reach the meniscus
with a finite size, and a region where drops fully evaporate before reaching
the meniscus, in agreement with the flow lines of the dynamical system (32).
The comparison with Fig. 4(a) shows that the drops displaying $\alpha=1/3$
during the longest time range are near the meniscus (small $z_{\mathrm{d}}$,
see e.g. blue symbols), as expected theoretically. We also observe that
$\alpha=1/2$ appears when drops are far from the meniscus (large
$z_{\mathrm{d}}$, see e.g. brown symbols). To quantify this effect, we rescale
our data according to Eq. (37) on Figure 4(c). The rescaled data sets fall
onto a master curve, in agreement with the expression (37), with a fitted
parameter $a/z_{\mathrm{d}}$ compatible with other experiments [square on Fig.
3(c)], with a capillary length $L_{\mathrm{c}}=20\mu$m. While within the
correct order of magnitude, this value is 3-4 times smaller than expected.
This underestimation could be attributed to the use of a simplified mean-
field, isothermal and quasi-static model, with an a priori assumed model $H$
dynamics, which could describe real dynamics only at the cost of using
renormalized coefficients. Nonetheless, even if refined theoretical
ingredients may be needed, our model not only predicts the surface-tension
($\alpha=1/3$) and the gravity-driven ($\alpha=1/2$) evaporation regimes and
their crossover, but also enables a full rescaling of the data set and a semi-
quantitative comparison with experiments, as shown in Fig. 4(c).

Conclusion.- With the aim of investigating the surface-tension driven decay at
work in Ostwald-ripening mechanisms, we used an opto-mechanical strategy to
produce the model situation of a single drop of one phase immersed into the
other one. Our observations indicated a decay exponent $\alpha\simeq 1/2$,
which was attributed to gravity-induced stratification of the concentration.
However, focusing on experimental conditions predicted by a theoretical
analysis, we indeed measured the decay exponent $\alpha\simeq 1/3$
characteristic of surface-tension driven decay for conserved systems. Thus,
our work raises experimentally and theoretically new insights on the decay
component of the evaporation-condensation mechanism, so important in material
sciences.

###### Acknowledgements.

The authors thank the machine and electronic shops of LOMA for their technical
contributions. The authors acknowledge financial support from the ANR through
the projects FISICS ANR-15-CE30-0015-01 and ComplexEncounters
ANR-21-CE30-0020. We thank Antoine Aubret for providing calibration beads,
Quentin Moreno and Benjamin Sapaly for preliminary experiments, and Justine
Roux and Gary Croise for their help in control experiments.

SUPPLEMENTAL MATERIAL

Supporting the main text, we provide details on:

  * •

the near-critical properties of the binary liquid used (Section A),

  * •

the experimental setup (Section B),

  * •

the edge detection procedures, data acquisition, calibration and error
calculation (Section C),

  * •

the theoretical analysis (Section D).

##  Appendix A Near-critical micellar phases of microemulsion

The choice of near-critical micellar phases of microemulsion and the
properties of the chosen mixture were described in detail previously, for
instance in the Supplementary Information of a previous publication on near-
critical spreading of droplets [22]. In brief:

1) We used a micellar phase of microemulsion composed of water ($9\%$ wt), oil
(toluene, $70\%$ wt), surfactant (sodium-dodecylsulfate, SDS, $4\%$ wt), and
alcohol (n-butanol-1, $17\%$ wt), of critical composition at
$T_{\mathrm{C}}=38^{\circ}$C. As concentrations of water and surfactant are
much smaller than that of oil + alcohol, the quaternary mixture organizes at
thermodynamic equilibrium as a suspension of surfactant-coated water
nanodroplets, the micelles of size $r\simeq 4$nm, dispersed in a toluene
continuum; miscible in water and oil, alcohol is present in both water and
toluene, and serves as co-surfactant. Consequently, our microemulsion can be
considered as a pseudo binary liquid mixture of micelles in toluene. As for
any binary mixture, it there exists a line of critical points, which separates
the miscible to the phase-separated state, consisting in two micellar phases
of different micellar concentrations and the chosen composition is critical at
$T_{\mathrm{C}}=38^{\circ}$C.

2) In near-critical conditions, observations obtained from one system are
totally transposable to any other belonging to the same Ising universality
class, so that one can choose a particular system which is ‘better adapted’ to
the dedicated experimental situation without losing generality of the
investigation; micelle systems are particularly interesting when looking at
supramolecular liquids. When the ratio between water and oil is small,
micellar phases of microemulsion belong to the Ising ($d=3,n=1$) universality
class, where $d$ and $n$ are respectively the space and the order parameter
dimensions, which is representative of isotropic liquids. Close to the
critical point, the properties vary as power laws of the temperature shift to
the critical temperature with universal exponents related to the Ising class.

3) For the chosen composition (volume fraction $\Phi_{C}$), the liquid-liquid
phase separation (volume fraction $\Phi_{i=1,2}$) occurs by increasing the
temperature above the critical temperature $T_{\mathrm{C}}=38^{\circ}$C (so-
called low critical point) as indicated in the schematic phase diagram shown
in Fig. 1 in the main text.

4) Many critical properties of the microemulsion were already characterized
and of particular interest for the present investigation in the two-phase
region ($T>T_{\mathrm{C}}$), we mention [22]:

  1. 1.

the correlation length of density fluctuations in the phase separated state,
$\xi^{-}=\xi_{0}^{-}[(T-T_{\mathrm{C}})/T_{\mathrm{C}}]^{-\nu}$, with
$\nu\simeq 0.63$ and $\xi_{0}^{-}\simeq 2$nm,

  2. 2.

the micellar volume fraction of the coexisting phases:

$\displaystyle\Phi_{i=1,2}=\Phi_{C}+b\left(\frac{T-T_{\mathrm{C}}}{T_{\mathrm{C}}}\right)\pm\frac{\Delta\Phi_{0}}{2}\left(\frac{T-T_{\mathrm{C}}}{T_{\mathrm{C}}}\right)^{\beta},$
(10)

with $\beta\simeq 0.325$, the concentration at criticality $\Phi_{C}=0.11$,
the asymmetry parameter $b=1.185$ and the coexistence amplitude
$\Delta\Phi_{0}=0.275$,

  3. 3.

the density of the two coexisting phases:
$\rho_{i}=\rho_{\text{micelles}}\Phi_{i}+\rho_{\text{solvent}}(1-\Phi_{i})$,
with $i\in\\{1,2\\}$, with $\rho_{\text{micelles}}=1040.0$kg/m3 and
$\rho_{\text{solvent}}=839.5$kg/m3 the respective densities of the micelles
and of the oil continuum,

  4. 4.

the density contrast between the coexisting phases:
$\Delta\rho=\rho_{1}-\rho_{2}=\Delta\rho_{0}[(T-T_{\mathrm{C}})/T_{\mathrm{C}}]^{\beta}$,
with
$\Delta\rho_{0}=(\rho_{\text{micelles}}-\rho_{\text{solvent}})\Delta\Phi_{0}=55.1$
kg/m3 (we also found $\Delta\rho_{0}=53.5$ kg/m3 using the relation
$\Delta\rho=(\partial\rho/\partial\Phi)\Delta\Phi$),

  5. 5.

the interfacial tension between the coexisting phases:
$\gamma=\gamma_{0}[(T-T_{\mathrm{C}})/T_{\mathrm{C}}]^{2\nu}$, with
$\gamma_{0}=(5.0\pm 0.2)10^{-5}$N/m,

  6. 6.

the shear viscosity of the coexisting phases measured empirically:
$\eta=(1.46-0.014\times(T-273\text{K}))(1+2.5\Phi_{i})10^{-3}$ Pa.s,

  7. 7.

the refractive index contrast between coexisting phases: $\Delta
n=4.62.10^{-2}[(T-T_{\mathrm{C}})/T_{\mathrm{C}}]^{-0.325}$,

  8. 8.

while just pertinent for the production of drops by the optical radiation
pressure, the optical absorption of the microemulsion is $\alpha_{a}=(4.8\pm
0.5)10^{-4}$cm-1, so that laser heating can be neglected at the used beam
powers (smaller and smaller when the critical point is neared).

##  Appendix B Experimental setup

Figure 5: Experimental setup.

The experimental setup was also described several times [31] with regular
improvements [23, 25, 22]. In brief, the schematics on Fig. 5 shows that it is
composed of three main parts.

1) We focus a continuous laser beam ($\lambda=532$nm, TEM00, Coherent Verdi
V5) on the meniscus of a phase-separated liquid system, and use the optical
radiation pressure to locally deform and eventually destabilize this meniscus
to form a liquid jet and droplets by breakup of the phase ($\Phi_{2}$) into
the phase ($\Phi_{1}$) because the index of refraction of $\Phi_{2}$ is larger
than that of $\Phi_{1}$; this contactless approach is particularly suitable
for the manipulation of tight critical systems. The beam power is controlled
using the combination of a half-wave plate and a polarization beam splitter.
The beam size at the meniscus (beam waist) is controlled by varying the
distance between the lens L ($f=80$cm) and the focusing objective using the
movable prism. Even not relevant for the present investigation, where no beam
is present, the choice of beam power and waist is essential to drive the
meniscus instability in a controlled way [25].

2) The temperature of the near-critical phase-separated microemulsion is
controlled using a brass oven, including resistors driven by a PID controller
and a Pt-100 sensor. This oven is also enclosed in a larger box, which
includes a thermally-controlled water circulation in order to eliminate slow
temperature variation of the lab environment. The temperature stability is
locally better than $0.05$K.

3) The illumination part is composed of a high-Intensity Fiber-Coupled
Illuminator (OSL2, Thorlabs) and a critical illumination arrangement to focus
light at the laser beam location; we made this choice to locally increase
lighting, as compared to a Köhler illumination scheme, because the turbidity,
and thus light scattering, increases drastically close to a critical point.
Imaging and frame grabbing are realized using a long working distance
objective (due to the bulk volume of the brass oven, Olympus® LMPLFLN
$\times$20 or LMPL $\times$50) and a CMOS high speed camera with a large 4 Mpx
sensor (Phantom V EO 340L) to be able to easily follow not only the drop
evaporation but also the drop trajectories.

Figure 6: a) Image of a calibrated bead ($R=2.185\pm 0.07$ $\mu m$). The
dashed line indicates the line across the object (here a bead) where the
intensity profile is analyzed. Scale bar represents 100 pixels. b) Intensity
profile of the bead shown in a). The intensity profile is first averaged. In
the first edge detection method (M$1_{\text{max-grad}}$), the diameter of the
object is determined by the position of the maxima of intensity gradients. In
the second edge detection method (M$2_{\mathrm{background}}$), the size of the
object is determined when the intensity is equal to the background intensity.

##  Appendix C Technical details on edge detection procedures, data
acquisition, calibration and considerations on error calculation

###  C.1 Overview

Here, we present the edge detection procedure used in the main article to
measure the diameter of the drops. Two edge detection procedures have been
developed. The first edge detection method - called M$1_{\text{max-grad}}$
method - is based on the position on the maxima of intensity gradients (see in
figure 6) and has been described in details in Ref. [22]. Because this method
is computationally non-ambiguous, it can be easily implemented
algorithmically, and is robust to global intensity variations [32]. However,
as already mentioned in Refs. [32, 22], this method tends to underestimate the
drop radius by a systematic additive error called $d_{\mathrm{cal}}$, which
can be estimated thanks to the analysis of volume conservation (see below).
The radius of the object is then given by : $R^{\mathrm{exp}}=R_{\mathrm{max-
grad}}+d_{\mathrm{cal}}$. However, this systematic error limits intrinsically
the detection of very small drops (when $R\rightarrow 0$).

To overcome these limitations, we have implemented a second edge detection
method called hereafter M$2_{\mathrm{background}}$ method. This method is
based on the pixel positions where the intensity is equal to the background
image intensity (see figure 6). Importantly, we have validated the size
measurements obtained with this method using calibrated beads at $R=1.15$ $\mu
m$ and $R=2.185$ $\mu m$. Finally, as shown in figure 7, we have checked that
the two detection methods (the M$1_{\text{max-grad}}$ corrected with
$d_{\mathrm{cal}}$ and the M$2_{\mathrm{background}}$ methods) are compatible
with calibrated values (within $20\%$) over a wide range of defocusing
distances. Below we give the full details of these edge detection procedures.

Figure 7: Calibration curves of the measured radius $R^{\mathrm{exp}}$
renormalized by the calibrated radius $R_{0}$ of the bead as a function of the
renormalized defocusing distance $\Delta x/R_{0}$. Open triangles represent
measurements with M$1_{\text{max-grad}}$ method. Filled triangles represent
results with the M$1_{\text{max-grad}}$ with the correction
$d_{\mathrm{cal}}$. Filled circles represent results with the
M$2_{\mathrm{background}}$ method. a) Calibration curve for the 20X objective
and $R_{0}=2.185$ $\mu$m. b) Calibration curve for the 20X objective for
$R_{0}=1.155$ $\mu$m. c) Calibration curve for the 50X objective for
$R_{0}=1.155$ $\mu m$. Error-bands indicate the uncertainty on
$R_{1}^{\text{exp}}$ and $R_{2}^{\text{exp}}$, respectively.

###  C.2  First edge detection method based on maximal intensity gradient:
M$1_{\text{max-grad}}$ method

The first edge detection method used in the main text has been developed by
Bertollini et al. [33] to measure aerosols droplets distributions and has been
described in details in Ref. [22]. It is based on the positions of the maxima
of the intensity gradients in the image.

#### Radius and position determination

Here, we describe the algorithmic procedure to obtain the droplet position and
radius. For each image:

  * •

an image of the background made after the droplet is completely evaporated is
subtracted and a Gaussian filter is applied using the microscope objective
resolution for the kernel size ($\sim 1$ $\mu m$),

  * •

a first rough estimate of $R$ is obtained using an automated circle
localization algorithm from the imageJ plug-in “Analyze Particles”,

  * •

from the circle center, the positions of the intensity gradient maxima are
obtained using a Sobel kernel for $N=25$ directions with uniformly distributed
angles [Fig. 8(b)],

  * •

the drop center coordinates $(y_{\mathrm{d}},z_{\mathrm{d}})=(\langle
y_{i}\rangle_{N},\langle z_{i}\rangle_{N})$ are then recalculated from these
positions and used to follow drop motion. The drop radius is then obtained
from $R_{\mathrm{max-
grad}}=\langle[(y_{i}-y_{\mathrm{d}})^{2}+(z_{i}-z_{\mathrm{d}})^{2}]^{\frac{3}{2}}\rangle_{N}^{1/3}$
.

#### Volume conservation calibration

Using the intensity gradient maxima, the droplet radius is likely to be
underestimated. To account for such systematic error, following the method
developed by van der Bos et al. [32], a calibrated distance $d_{\mathrm{cal}}$
is estimated by insuring volume conservation for strongly elongated liquid
object that relaxes towards a spherical shape, see Fig. 9. Indeed, the
characteristic time for the relaxation dynamics ($\tau\approx 0.1$ $s$) is
much shorter than the evaporation time $(t_{\mathrm{f}}\approx 100$ s), so
that the volume can be considered as constant during the relaxation. In figure
9, we show the evolution of the volume of the liquid ligament during its
relaxation to a sphere, with and without taking into account a calibrated
distance $d_{\mathrm{cal}}=0.44\mu$m, which is optimised to ensure volume
conservation. This calculation of $d_{\mathrm{cal}}$ has been systematized for
various $\Delta T=20$, $10$, $4$ and $2K$ and drop sizes (ranging between
$R=2.5\mu$ m and $R=6.3\mu$ m). The best calibrated distance is found to be
$d_{\mathrm{cal}}=0.44\pm 0.28$ $\mu m$ with no specific dependency on $\Delta
T$ or $R$.

Figure 8: a) Microscopy image of a near-critical microemulsion drop at $\Delta
T=4$ K. In yellow is shown the reconstructed circle with a radius $R=3.36\mu
m$ obtained from the intensity gradient maxima and the calibrated distance. b)
Intensity gradients obtained using a Sobel kernel method of image shown in a).
The $N=25$ intensity gradients maxima obtained for uniformly distributed
angular directions are shown as well as their centroid.

We call $R^{\mathrm{exp}}_{1}$ the measured radius obtained with this first
edge detection method, which is thus given by:

$\displaystyle R^{\mathrm{exp}}_{1}=R_{\mathrm{max-
grad}}+d_{\mathrm{cal}}^{0}\pm e_{R},$ (11)

with $d_{\mathrm{cal}}^{0}=0.44$ $\mu m$. The uncertainty $e_{R}$ is then
given by $e_{R}=[\sigma_{R}^{2}+\delta_{p}^{2}/12+e_{d}^{2}]^{1/2}$, with
$\sigma_{R}$ the standard deviation of the distribution of maxima positions
from the droplet center, $\delta_{p}$ the image resolution, and $e_{d}$ the
error on the calibration distance $d_{\text{cal}}$.

Figure 9: Volume $V$ of a liquid ligament, normalized by its final value,
obtained following the method of Ref. [32], with (orange) and without (blue)
taking into account a calibrated distance $d_{\text{cal}}=0.44$ $\mu m$. Here,
$\Delta T=20$K. Snapshots: evolution of the shape of the liquid filament.

As shown in figure 7, the measured radius by the M$1_{\text{max-grad}}$ method
alone underestimates the size of the bead. However, once the calibrated
distance $d_{\mathrm{cal}}$ is taken into account, the measured radius
$R^{\mathrm{exp}}_{1}$ is compatible with the calibrated values for different
imaging objectives. As already mentioned, this added calibration distance
$d_{\mathrm{cal}}$ is negligible for large drops but it limits intrinsically
the detection of very small drops. To overcome this limitation, we develop a
second image edge detection technique.

###  C.3 Second edge detection technique : M$2_{\text{background}}$ method

We have implemented another edge detection method, based on the image analysis
of calibrated beads. We use two silica calibrated beads (Microparticle GmBH)
with diameters $D_{1}=2.31\pm 0.05$ $\mu m$ and $D_{2}=4.37\pm 0.14$ $\mu m$,
where the uncertainty indicates the standard deviation of the distribution.

#### Preparation of solutions of calibrated beads in water-glycerol mixtures.

We dilute the initial bead solution into water-glycerol mixtures. First, we
prepare water-glycerol solutions from $57\%$ to $85\%$ in glycerol. For each
solution, we measure the refractive index with an Abbe refractometer.
Secondly, we determine the refractive index of calibrated beads by phase
contrast microscopy as described in Ref. [34]. Briefly, we image calibrated
beads into water-glycerol solutions with different refractive indexes. When
the contrast of the bead is lost in phase contrast microscopy, it means that
the refractive index of the solution matches the refractive index of the
object. We find $n_{\text{bead}}=1.428\pm 0.001$ in agreement with the
commercial value $n_{\text{bead}}=1.42\pm 0.01$. To match the optical
experimental conditions, we dilute the calibrated beads into a $57\%$
glycerol-water mixture ($n=1.408$) which gives a refractive index contrast
between the bead and the solution $\Delta n\approx 0.02$, which is close to
the index contrast between separated phases in the micro-emulsion at $\Delta
T=20K$.

#### Image analysis for calibrated beads and validation of the
$\mathrm{M}2_{\mathrm{background}}$ method.

We image the calibrated beads immersed in water-glycerol mixtures by varying
the focus of the imaging objective with a translation stage in order to get
the intensity profile for each defocusing position $\Delta x$ (see figure 5
and 7). In the M$2_{\text{background}}$ method, the diameter of the object is
measured when the intensity is equal to the background intensity (see figure
6). Calibration results are shown in figure 7 for two calibrated beads
$R_{0}=1.155$ $\mu m$ and $R_{0}=2.185$ $\mu m$ and various imaging objectives
in the setup. The experimental radii $R^{\mathrm{exp}}_{2}$ are compatible
with the calibrated radii within $20\%$ over a wide defocusing range
($-3\leq\Delta x/R_{0}\leq 3$). It is also compatible with the first method.
This allows us to use the M$2_{\text{background}}$ method down to $R=1.155$
$\mu m$ unambiguously.

##  Appendix D Theory

###  D.1 Field profile at equilibrium

Here we derive the effective equations of the dynamics in the sharp interface
limit. Our approach consists in adapting the arguments presented in Ref. [2]
to our situation. First, let us describe the profile of $\phi$ inside the
meniscus at equilibrium. The chemical potential is given by

$\displaystyle\mu=\frac{\delta\mathcal{H}}{\delta\phi}=-\kappa\nabla^{2}\phi+V^{\prime}(\phi)-\rho_{\mathrm{e}}gz,$
(12)

where the prime denotes the differentiation with respect to $\phi$. At
equilibrium, $\mu$ is uniform, $\mu(\mathbf{x})=\mu_{e}$. Let us call
$\phi^{*}(z)$ the profile of $\phi$ across the meniscus. Since the meniscus is
located in the vicinity of $z=0$, we neglect the gravity term in the equation
$\mu=\mu_{e}$, which reads

$\displaystyle\kappa\frac{\partial^{2}\phi^{*}}{\partial
z^{2}}-V^{\prime}(\phi^{*})=-\mu_{e}.$ (13)

This equation is the same as the force-balance equation for a classical point-
like particle moving in the effective “potential” $-V+\mu_{e}\phi^{*}$, with
$z$ the “time”, $\kappa$ the “mass” and $\phi^{*}$ the “position”.
Consequently, the fact that Eq. (13) must admit solutions that converge to
constant values at $z\rightarrow\pm\infty$ imposes that this effective
potential has two extrema at $\phi=\Phi_{1}$ and $\Phi_{2}$, and takes the
same value at these extrema, so that

$\displaystyle
V^{\prime}(\Phi_{1})=V^{\prime}(\Phi_{2})=\mu_{e}=\frac{V(\Phi_{1})-V(\Phi_{2})}{\Phi_{1}-\Phi_{2}}.$
(14)

Hence, the equilibrium values $\Phi_{1}$ and $\Phi_{2}$ on each side of the
interface are obtained by using the method of common tangents. We now define
$U(\phi)=V(\phi)-\mu_{e}\phi$, which has two minima of equal depth at
$\Phi_{1}$ and $\Phi_{2}$. The interfacial tension $\gamma$, that is the
excess energy per unit area of the interface can be calculated as

$\displaystyle\gamma=\int_{-\infty}^{\infty}dz\left\\{\frac{\kappa}{2}(\partial_{z}\phi^{*})^{2}+U(\phi^{*}(z))-U(\Phi_{i})\right\\}.$
(15)

Far from the interfaces, the field $\phi$ varies due to gravity. We recall
that $\overline{\phi}_{i}(\mathbf{x})=\phi(\mathbf{x})-\Phi_{i}$. Expanding
the equation $\mu=\mu_{e}$ at leading order in $\overline{\phi}_{i}$, we
obtain

$\displaystyle\overline{\phi}_{i}=\rho_{\mathrm{e}}gz/V^{\prime\prime}_{i},$
(16)

which describes the increase of micelles concentrations due to gravity when
the coordinate $z$ increases.

###  D.2 Dynamics in the sharp interface limit

In the presence of curved interfaces, or interfaces that are not at $z=0$, the
system is no longer stationary. When interfaces are thin, the system can be
divided into domains where $\phi$ is close to its equilibrium value. Writing
again $\overline{\phi}_{i}(\mathbf{x})=\phi(\mathbf{x})-\Phi_{i}$, the
evolution of $\overline{\phi}_{i}$ follows from the expansion of Eq. (2) at
leading order in $\overline{\phi}_{i}$, and is described by

$\displaystyle\partial_{t}\overline{\phi}_{i}+\mathbf{u}\cdot\nabla\overline{\phi}_{i}=D_{i}\nabla^{2}\overline{\phi}_{i}.$
(17)

To obtain effective boundary conditions at the interface, we consider the
chemical potential $\mu$ at the interface located at the altitude $z=z_{s}$:

$\displaystyle\mu-\mu_{e}=$ $\displaystyle
U^{\prime}(\phi)-\kappa\nabla^{2}\phi-\rho_{\mathrm{e}}gz_{s}$
$\displaystyle\simeq$ $\displaystyle U^{\prime}(\phi)-\kappa\
\partial_{s}^{2}\phi-\kappa C\partial_{s}\phi-\rho_{\mathrm{e}}gz_{s},$ (18)

where we have used differential geometry formulas as described in Ref. [1, 2].
Here, $s$ is the normal coordinate along the unit vector $\hat{\mathbf{n}}$
normal to the interface (pointing towards the phase $\Phi_{1}$ by convention),
and $C$ the interface total curvature (when the orientation of the interface
is $\hat{\mathbf{n}}$). Approximating $\phi(s)$ by the equilibrium interface
profile $\phi^{*}$ [defined by Eq. (13)], integrating Eq. (18) over $s$ after
multiplication by $\partial_{s}\phi^{*}$ leads to

$\displaystyle(\mu-\mu_{e}+\rho_{\mathrm{e}}gz_{s})=-\frac{\gamma
C}{\Delta\Phi},$ (19)

where we have used Eq. (15). In the bulk phases, the chemical potential is
approximated by:

$\displaystyle\mu-\mu_{e}\simeq
V^{\prime\prime}_{i}\mathrm{bar}{\phi}-\rho_{\mathrm{e}}gz.$ (20)

Comparing Eqs. (20) and (19), we see that the boundary condition for the field
$\phi$ at a position $\mathbf{r}_{s}$ of the interface is

$\displaystyle
V^{\prime\prime}_{i}\overline{\phi}|_{\mathbf{r}_{s}}=-\frac{\gamma
C}{\Delta\Phi},$ (21)

which is the Gibbs-Thomson equation in our system. Another boundary condition
comes from the conservation of the number of micelles at the interface, which
writes

$\displaystyle\Delta\Phi\
(\mathbf{v}-\mathbf{u})\cdot\hat{\mathbf{n}}=[D_{2}(\nabla\overline{\phi}_{2})_{\mathbf{r}_{s}}-D_{1}(\nabla\overline{\phi}_{1})_{\mathbf{r}_{s}}]\cdot\hat{\mathbf{n}},$
(22)

where $\mathbf{v}$ is the velocity of the interface.

Last, since $\nabla\mu$ is small, Stokes’ equation can be approximated by

$\displaystyle\eta\nabla^{2}\mathbf{u}$ $\displaystyle\simeq\nabla p\
+[\Phi_{1}\theta_{1}(\mathbf{x})+\Phi_{2}\theta_{2}(\mathbf{x})]\nabla\mu,$
(23)

where $\theta_{i}(\mathbf{x})$ is the function that is unity in the phase $i$
and zero elsewhere. This equation can also be written

$\displaystyle\eta\nabla^{2}\mathbf{u}$ $\displaystyle\simeq\nabla\tilde{p}\
+\gamma C\nabla\theta_{1}(\mathbf{x})$
$\displaystyle+[\Phi_{1}\theta_{1}(\mathbf{x})+\Phi_{2}\theta_{2}(\mathbf{x})](-\rho_{\mathrm{e}}g\hat{\mathbf{e}}_{z}),$
(24)

where
$\tilde{p}=p+(\Phi_{1}\theta_{1}+\Phi_{2}\theta_{2})V_{i}^{\prime\prime}\overline{\phi}_{i}$
is a renormalized pressure. Since $\nabla\theta_{1}$ vanishes everywhere
except at the interfaces, the term proportional to $\gamma$ in the above
equation induces a discontinuity of the normal stress of the fluid, while the
last term is the usual gravity acceleration. Equation (24) can thus be
identified as the equation for the flow of a fluid presenting phases of
different densities $\rho_{\mathrm{e}}\Phi_{i}$, with surface tension
$\gamma$.

Let us finally note that the same equations would be found by considering a
more general dynamics:

$\displaystyle\partial_{t}\phi+\mathbf{u}\cdot\nabla\phi=\nabla[\lambda(\phi)\nabla\mu]$
(25)

which includes Dean’s equation ($\lambda(\phi)=\tilde{\lambda}\phi$) [35] or
previous equations used to study gravity
$\lambda(\phi)=\lambda_{0}-\lambda_{1}(\phi-\Phi_{C})^{2}$ [36, 37]. However,
at our level of approximations, where one assumes that $\phi$ is close to
$\Phi_{i}$ in each phase, modifying the dynamics would only lead to a
renormalization of $D_{i}=\lambda(\Phi_{i})V_{i}^{\prime\prime}$.

###  D.3 Dynamics of a spherical domain

Let us now derive the equations of the dynamics for a domain $\Phi_{2}$ with
spherical shape immersed inside the phase $\Phi_{1}$, of radius $R$ and
altitude $z_{\mathrm{d}}$. We assume $z_{\mathrm{d}}\gg R$, which is always
satisfied in the present experiments. We solve the above equations (24), (22),
(21) and (17), requiring that far from the drop (“at infinity”) the
equilibrium solution (16) is retrieved, with a vanishing flow $\mathbf{u}$.

First, the flow is that of a spherical drop immersed in a fluid with different
density. Its expression is well-known [38]. In particular, the radial flow at
the interface reads

$\displaystyle
u_{r}=\mathbf{u}\cdot\hat{\mathbf{e}}_{r}=-\frac{4R^{2}\rho_{\mathrm{e}}g\Delta\Phi}{15\eta}\cos{\theta},$
(26)

where $\hat{\mathbf{e}}_{r}$ is the normal vector pointing to the exterior of
the drop, and $\theta$ is the angle between $\hat{\mathbf{e}}_{r}$ and
$\hat{\mathbf{e}}_{z}$.

For the field $\overline{\phi}$, we assume that the dynamics of $R$ is slow
enough so that a quasi-stationary limit can be reached and the term
$\partial_{t}\overline{\phi}$ can be neglected. We also assume that the Peclet
number is small, so that Eq. (17) becomes $\nabla^{2}\overline{\phi}_{i}=0$.
Its solution, which satisfies the boundary conditions is

$\displaystyle\overline{\phi}_{1}=$ $\displaystyle\frac{-2\gamma}{\Delta\Phi
V^{\prime\prime}_{1}r}+\frac{\rho_{\mathrm{e}}gz_{\mathrm{d}}}{V^{\prime\prime}_{1}}\left(1-\frac{R}{r}\right)+\frac{\rho_{\mathrm{e}}g}{V^{\prime\prime}_{1}}(z-z_{\mathrm{d}})$
$\displaystyle-\frac{\rho_{\mathrm{e}}gR^{3}(z-z_{\mathrm{d}})}{r^{3}V^{\prime\prime}_{1}},$
(27)

with $r$ the distance to the center of the drop. The field
$\overline{\phi}_{2}$ inside the drop is uniform. In particular, the diffusive
flux reads

$\displaystyle
j_{r}=(-D_{1}\nabla\overline{\phi}_{1}\cdot\hat{\mathbf{e}}_{r})_{r=R}=\frac{-2\gamma\lambda}{\Delta\Phi
R^{2}}-\frac{\lambda\rho_{\mathrm{e}}gz_{\mathrm{d}}}{R}-3\lambda\rho_{\mathrm{e}}g\cos\theta.$
(28)

Next, the radial velocity of the interface results from the variation of the
drop’s radius and the translation of its center-of-mass:

$\displaystyle v_{r}=\mathbf{v}\cdot\hat{\mathbf{e}}_{r}=(\dot{R}\
\hat{\mathbf{e}}_{r}+\dot{z}_{\mathrm{d}}\
\hat{\mathbf{e}}_{z})\cdot\hat{\mathbf{e}}_{r}=(\dot{R}+\dot{z}_{\mathrm{d}}\cos\theta).$
(29)

Using the conservation equation (22), we obtain

$\displaystyle\Delta\Phi$
$\displaystyle\left[\dot{R}+\dot{z}_{\mathrm{d}}\cos\theta+\frac{4R^{2}\rho_{\mathrm{e}}g\Delta\Phi}{15\eta}\cos{\theta}\right]=$
$\displaystyle\frac{-2\gamma\lambda}{\Delta\Phi
R^{2}}-\frac{\lambda\rho_{\mathrm{e}}gz_{\mathrm{d}}}{R}-3\lambda\rho_{\mathrm{e}}g\cos\theta.$
(30)

This equation leads to

$\displaystyle\dot{R}=-\frac{\lambda}{R\Delta\Phi}\left(\frac{2\gamma}{R\Delta\Phi}+z_{\mathrm{d}}\rho_{\text{e}}g\right),$
(31) $\displaystyle\dot{z}_{\mathrm{d}}=-\frac{4\rho_{\text{e}}\Delta\Phi
gR^{2}}{15\ \eta}-\frac{3\lambda\rho_{\mathrm{e}}g}{\Delta\Phi}$ (32)

The first term in the equation for $\dot{z}_{\mathrm{d}}$ is Hadamard’s law.
The second term in Eq. (32) comes from a different mechanism of motion, where
phase $\Phi_{1}$ is converted into phase $\Phi_{2}$ above the drop, while
phase $\Phi_{2}$ is converted into phase $\Phi_{1}$ below the drop. This term
turns out to be negligible and will be discarded it in the analysis.

###  D.4 Regimes of the dynamics

Defining $\tilde{R}=R/L_{R}$, $\tilde{z}_{\mathrm{d}}=z_{\mathrm{d}}/L_{Z}$
and $\tilde{t}=(t_{f}-t)/\tau$, we obtain the set of equations

$\displaystyle\partial_{\tilde{t}}\tilde{R}=\frac{2}{\tilde{R}^{2}}\left(1+\frac{\tilde{z}_{\mathrm{d}}\tilde{R}}{2}\right),$
(33)
$\displaystyle\partial_{\tilde{t}}\tilde{z}_{\mathrm{d}}=\frac{4}{15}\tilde{R}^{2},$
(34)

where

$\displaystyle\tau=\frac{1}{\rho_{\mathrm{e}}g}\sqrt{\frac{\eta}{\lambda}},\
L_{R}=\left[\frac{\gamma\sqrt{\eta\lambda}}{(\Delta\Phi)^{2}g\rho_{\mathrm{e}}}\right]^{1/3},\
L_{Z}=\frac{L_{\mathrm{c}}^{2}}{L_{R}}.$ (35)

With this choice of rescaled variables, the dynamics does not depend on any
parameter. Note that we have inverted the time, so that $\tilde{t}=0$
corresponds to $\tilde{R}=0$, and the spherical domain grows when $\tilde{t}$
increases; $\tilde{z}_{\mathrm{d}}(0)$ is the dimensionless final altitude of
the drop. Different regimes can be obtained as follows. First, at the late
stage of the dynamics ($\tilde{t}\to 0$), we may consider that the dynamics
occurs at constant
$\tilde{z}_{\mathrm{d}}=\tilde{z}_{\mathrm{d}}(0)\equiv\tilde{z}_{\mathrm{f}}$,
so that Eq. (33) leads to

$\displaystyle f\left(\tilde{z}_{\mathrm{f}}\tilde{R}\right)=\tilde{t}\
\tilde{z}_{\mathrm{f}}^{3},$ (36) $\displaystyle f(u)=\int_{0}^{u}dy\
\frac{y^{2}}{(2+y)}=\frac{(u-4)u}{2}+4\ln\frac{u+2}{2}.$ (37)

Using this formula and the asymptotics of $f$ for large and small $u$, we find
that when $\tilde{R}\tilde{z}_{\mathrm{f}}\ll 1$,
$\tilde{R}=(6\tilde{t})^{1/3}$, whereas when
$\tilde{R}\tilde{z}_{\mathrm{f}}\gg 1$, we find a $R$-squared laws
$\tilde{R}=\sqrt{2\tilde{z}_{\mathrm{f}}\tilde{t}}$.

Next, for very large $\tilde{t}$, drops are so big that one can neglect the
term which does not contain $\tilde{z}_{\mathrm{d}}$ in Eq. (33), leading to

$\displaystyle\partial_{\tilde{t}}\tilde{R}=\frac{\tilde{z}_{\mathrm{d}}}{\tilde{R}},$
$\displaystyle\partial_{\tilde{t}}\tilde{z}_{\mathrm{d}}=\frac{4}{15}\tilde{R}^{2},$
(38)

which predicts exponential solutions

$\displaystyle\tilde{R}\propto e^{\tilde{t}\sqrt{2/15}}.$ (39)

To summarize, for $\tilde{t}\gg 1$ one obtains exponential solutions (39),
while for $\tilde{t}\ll 1$, the law (37) holds, which exhibits an $\alpha=1/3$
regime, and possibly an $\alpha=1/2$ regime (if $\tilde{z}_{\mathrm{f}}$ is
large enough so that the $R-$squared law is reached for $\tilde{t}\ll 1$,
before one switches to the exponential regime.

###  D.5 Fit to experiments

For $\Delta T=20$K, the choice $\tau\simeq 174$ s, $L_{R}\simeq 2.3\ \mu$m and
$L_{Z}=174\ \mu$m gives a good agreement between the experimental and
theoretical trajectories in the $({z}_{\mathrm{d}},R)$ and the $(R,t)$
representations [Fig. 4 in the main text]. Of note, with these parameters, the
coefficient in Hadamard’s law reads

$\displaystyle\frac{|\dot{z}_{\mathrm{d}}|}{R^{2}}=\frac{4L_{Z}}{15\tau
L_{R}^{2}}=0.050\ \mu\text{m}^{-1}\ \text{s}^{-1},$ (40)

whereas this coefficient can also be estimated as

$\displaystyle\frac{|\dot{z}_{\mathrm{d}}|}{R^{2}}=\frac{4\rho_{\text{e}}\Delta\Phi
g}{15\ \eta}\simeq 0.060\ \mu\text{m}^{-1}\ \text{s}^{-1}$ (41)

where we have used the values given in Section A. The difference between the
value (40) and its expected value (41) is thus of the order of $20\%$. On the
other hand, as mentioned in the main text, $L_{C}=\sqrt{L_{R}L_{Z}}=20\mu$m,
whereas we expect $L_{C}\simeq 80\mu$m. This discrepancy could come from (i)
the use of an a priori assumed isothermal model $H$ dynamics, without noise,
and/or, (ii) possible large scale weak amplitude thermal gradients within the
cell containing the microemulsion, due to the presence of four tiny openings
within the brass oven, two for the laser beam crossing and two perpendicular
ones for lighting and imaging the drop dynamics.

###  D.6 Discussion of the approximations

To discuss the validity of the approximations, we will consider the specific
case of a $\phi^{4}$ potential,
$U(\phi)=-\frac{r}{2}(\phi-\phi_{0})^{2}+\frac{u}{4}(\phi-\phi_{0})^{4}$ for
which the interface profile can be calculated:

$\displaystyle\phi^{*}(z)=\phi_{0}-\frac{\Delta\Phi}{2}\tanh\left(z/\xi\right)$
(42)

where $\xi=\sqrt{\kappa/V_{i}^{\prime\prime}}=\sqrt{\kappa/(2r)}$ is the
correlation length and $\Delta\Phi=2\sqrt{r/u}$. We can evaluate (15) and
obtain

$\displaystyle
V_{1}^{\prime\prime}=V_{2}^{\prime\prime}=\frac{6\gamma}{(\Delta\Phi)^{2}\xi}.$
(43)

#### How small is $\overline{\phi}_{i}$?

Let us first evaluate the increase of concentration at equilibrium at altitude
$z$:

$\displaystyle\frac{\overline{\phi}_{1}}{\Delta\Phi}=\frac{\rho_{\mathrm{e}}gz}{V^{\prime\prime}_{1}\Delta\Phi}=\frac{\xi\rho_{\mathrm{e}}gz\Delta\Phi)^{2}}{6\gamma\Delta\Phi}=\frac{\xi\
z}{6L_{\mathrm{c}}^{2}}.$ (44)

For $\Delta T=20K$, with $\xi\sim 10$nm and $z=1$mm,
$L_{\mathrm{c}}=20-80\mu$m, the approximation
$\overline{\phi}_{1}\ll\Delta\Phi$ is justified. Next, the local decrease of
micellar concentration due to surface tension at the drop surface is

$\displaystyle\frac{|\overline{\phi}_{\mathbf{r}_{s}}|}{\Delta\Phi}\sim\frac{\gamma}{V^{\prime\prime}_{1}(\Delta\Phi)^{2}R}\sim\frac{\xi}{R}.$
(45)

Since $\xi\ll R$ is always satisfied we conclude that we can safely assume
that $\overline{\phi}_{i}\ll\Delta\Phi$.

#### Discussion of the quasi-static approximation

The quasi-static approximation will be valid if $D$ is large compared to
$\partial_{t}R^{2}$. We can estimate $D$ as

$\displaystyle D=\lambda
V_{i}^{\prime\prime}=\frac{6\lambda\gamma}{(\Delta\Phi)^{2}\xi}=6K_{0}/\xi,$
(46)

where $K_{0}$ is such that $R^{3}=6K_{0}(t_{\mathrm{f}}-t)$ in the surface-
tension dominated regime. Let us evaluate the validity of the quasi-stationary
approximation in this regime:

$\displaystyle\frac{R\dot{R}}{D}\sim\frac{K_{0}^{2/3}\xi}{(t_{f}-t)^{1/3}K_{0}}\sim\frac{\xi}{R},$
(47)

meaning that, as long as $\xi\ll R$, the quasi-stationary approximation is
well justified. In the gravity-dominated evaporation regime, we have

$\displaystyle\frac{\partial_{t}R^{2}}{D}=\frac{a}{D}\sim\frac{\rho_{\mathrm{e}}gz_{\mathrm{d}}}{\Delta\Phi
V^{\prime\prime}_{1}}\sim\frac{\xi\ z_{\mathrm{d}}}{6L_{\mathrm{c}}^{2}},$
(48)

which is small compared to $1$, so that in this regime the quasi-static
hypothesis is justified.

Last, in our experiments it is clear that $|\dot{z}_{\mathrm{d}}|<\dot{R}$, so
that the above estimates imply that the Peclet number
$\text{Pe}=\dot{z}_{\mathrm{d}}R/D$ is also very small compared to unity,
justifying the omission of advection terms in the equation for
$\overline{\phi}_{i}$.

#### Effect of quasi-incompressibilty

We now discuss the incompressibility approximation $\nabla\cdot\mathbf{u}=0$
which is not exact for binary liquid mixtures [39], in particular at the
interface since the two phases $\Phi_{1}$ and $\Phi_{2}$ have different
densities. Indeed, mass conservation reads

$\displaystyle\partial_{t}\rho+\nabla\cdot[\rho\ \mathbf{u}]=0,$ (49)

where $\rho$ is the mass density,
$\rho=\phi\rho_{\text{micelles}}+(1-\phi)\rho_{\text{solvent}}$. As already
estimated, the variations of $\phi$ inside one phase are negligible compared
to $\Delta\Phi$, so that one can consider that the fluid is incompressible
inside each phase. At the interface, (49) leads to the equation

$\displaystyle(\Phi_{1}\rho_{\text{e}}+\rho_{\text{solvent}})[u_{1}^{\perp}-v_{\perp}]=$
$\displaystyle\phi\rho_{\text{micelles}}+(\Phi_{2}\rho_{\text{e}}+\rho_{\text{solvent}})[u_{2}^{\perp}-v_{\perp}],$
(50)

where $u_{i}^{\perp}$ is the perpendicular component of the flow in the phase
$i$. For example, at the surface of a drop, in the regime where
$z_{\mathrm{d}}$ does not vary any longer, the flow induced by evaporation is
spherically symmetric, vanishing at the interior ($u_{2}^{\perp}=0$), whereas
the flow at the exterior of the drop reads

$\displaystyle
u_{1}^{\perp}=\dot{R}\frac{\Delta\Phi\rho_{\text{e}}}{\Phi_{1}\rho_{\text{e}}+\rho_{\text{solvent}}}.$
(51)

With typical experimental values at $\Delta T=20$K, we find that
$|u_{1}^{\perp}/\dot{R}|<0.03$. We thus conclude that taking quasi-
incompressibility into account in Eq. (22) would only slightly modify the
dynamics of $R(t)$.

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