# Physiochemical hydrodynamics of the phase-segregation in an evaporating
binary microdroplet

Yaxing Li1,2<EMAIL_ADDRESS>Pengyu Lv3  Christian Diddens1  Detlef Lohse1,4
<EMAIL_ADDRESS>1Physics of Fluids group, Department of Science and
Technology, Mesa+ Institute, Max Planck Center for Complex Fluid Dynamics and
J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217,
7500 AE Enschede, The Netherlands 2Institute of Fluid Dynamics, Department of
Mechanical and Process Engineering, ETH Zürich, Switzerland 3SKLTCS and
Department of Mechanics and Engineering Science, BIC-ESAT, College of
Engineering, Peking University, Beijing 100871, China 4Max Planck Institute
for Dynamics and Self-Organization, 37077 Göttingen, Germany

###### Abstract

Phase segregation triggered by selective evaporation can emerge in
multicomponent systems, leading to complex physiochemical hydrodynamics.
Recently, Li et al. (Phys. Rev. Lett., vol. 120, 2018, 224501) and Kim & Stone
(J. Fluid Mech., vol. 850, 2018, 769) reported a segregative behavior (i.e.,
demixing) in an evaporating binary droplet. In this work, by means of
experiments and theoretical analysis, we investigate the flow dynamics after
the occurrence of the phase segregation. As example, we take the
1,2-hexanediol-water binary droplet system. First, we experimentally reveal
the overall physiochemical hydrodynamics of the evaporation process, including
the segregative behavior and the resulting flow structure close to the
substrate. By quantifying the evolution of the radial flow, we identify three
successive life stages of the evaporation process. At Stage I, a radially
outward flow is observed. It is driven by the Marangoni effect. At the
transition to Stage II, the radial flow partially reverses, starting from the
contact line. This flow breaks the axial symmetry and remarkably is driven by
the segregation itself. Finally at Stage III, the flow decays as the
evaporation gradually ceases. At this stage the segregation has grown to the
entire droplet, and the flow is again controlled by the Marangoni effect. The
resulting Marangoni flow homogenizes the distribution of the entrapped
volatile water over the whole droplet.

###### keywords:

Binary droplet evaporation, phase segregation, Marangoni flow, physiochemical
hydrodynamics

##  1 Introduction

Evaporation of sessile droplets has been vastly studied in the past decades
(Cazabat & Guéna, 2010; Brutin, 2015; Zang et al., 2019; Lohse & Zhang, 2020;
Wang et al., 2022). Beyond the scope of fundamental research, the interest
also originates from its huge relevance to various technological and
biological applications, e.g. inkjet printing (Park & Moon, 2006; Lohse,
2022), surface patterning (Kuang et al., 2014), disease diagnostics (Brutin et
al., 2011), and microfabrication (Kong et al., 2014), among others. A
comprehensive physical picture of the evaporative lifetime of small airborne
droplets can even be crucial to better understand the spreading of the corona
virus in the current pandemic (Mittal et al., 2020; Chong et al., 2021; Wang
et al., 2021a).

While the evaporation (or analogous dissolution) behavior of pure droplets has
been extensively well understood (Deegan et al., 1997; Popov, 2005; Ristenpart
et al., 2007; Cazabat & Guéna, 2010; Gelderblom et al., 2011; Stauber et al.,
2014; Lohse & Zhang, 2015; Hatte et al., 2019; Zang et al., 2019; Chong et
al., 2020; Shiri et al., 2021), much less research has been done on
multicomponent droplets. However, most droplet systems encountered in nature
and technology are multicomponent, containing either multiple solutes
(Bennacer & Sefiane, 2014; Kim et al., 2016; Diddens et al., 2017; Wang et
al., 2021b; Williams et al., 2021; Wu et al., 2021; Jiang et al., 2021) or
dispersed particulates (Manoharan et al., 2003; Lauga & Brenner, 2004; Marin
et al., 2011; Thayyil Raju et al., 2021). The difficulty stems from the
complex physicochemical hydrodynamics brought by the selective evaporation of
each component in the mixture systems, including the coupling of the mutual
interactions between species (Brenn et al., 2007), contact line dynamics
(Mouat et al., 2020), shape formation (Pahlavan et al., 2021), flow structures
(Christy et al., 2011; Edwards et al., 2018; Li et al., 2019; Diddens et al.,
2021; Moore et al., 2021), and even phase segregation (Tan et al., 2016; Li et
al., 2018; Kim & Stone, 2018; Karpitschka, 2018; Sadafi et al., 2020; Li et
al., 2020a). Over the past 20 years, a growing interest in such complex
evaporative systems appears in literature, which was recently reviewed by
Lohse & Zhang (2020).

In (partially) volatile multicomponent droplets, in general, phase segregation
emerges, as such systems are out of equilibrium. This is due to evaporation
(Tan et al., 2016; Li et al., 2018; Kim & Stone, 2018; Karpitschka, 2018;
Sadafi et al., 2020) or dissolution (Dietrich et al., 2017), or
crystallization (Mailleur et al., 2018; Li et al., 2020b). The emerging
segregative patterns commonly lead to a nonvolatile shielding or an
entrapment, where one liquid on the surface of the droplets covers the others,
hindering its further evaporation. A detailed understanding of the shielding
and the entrapment is crucial for many industrial and technical applications
which require precise control of the droplet drying time, e.g. inkjet-printed
droplets drying on papers. Moreover, liquid-liquid phase segregations are also
essential in many biochemical reactions (Hyman et al., 2014). As a very recent
example, a resemblant arch-shaped liquid-liquid-phase-segregation pattern
triggered by evaporation has been observed in an aqueous mixture droplet of
polyethylene glycol (PEG) and dextran (Guo et al., 2021), which is potentially
a robust method for prebiotic compartmentalization.

Our recent study on an evaporating 1,2-hexanediol-water binary droplet
reported a liquid-liquid phase segregation of the non-volatile (at room
condition) component, i.e., 1,2-hexanediol, during the evaporative process (Li
et al., 2018). We demonstrated that for that particular binary droplet system
the concentration gradient on the droplet surface caused by the preferential
evaporation of water only leads to a small gradient of the surface tension.
The resulting solutal Marangoni flow is then initially insufficient to
homogenize the component distribution within the entire droplet. Therefore the
accumulated solute separates from the mixture, starting with the nucleation of
a series of small 1,2-hexanediol droplets at the contact line. These then
merge to arch-shaped structures. The droplet lifetime is significantly
extended due to the shielding effect of this non-volatile component. The
volatile water remains trapped in the interior of the droplet.

In that study, we also showed that the flow structure drastically changes due
to the emergence of the segregation. An evolution of flow structure was
characterized by the temporal evolution of the vorticity (see Fig. 1):

  1. 1.

At the early stage before the segregation occurs, a radial flow near the
substrate towards the droplet edge is driven by Marangoni forces.

  2. 2.

After the first emergence of a segregated phase of 1,2-hexanediol, the radial
flow evolves to a state of multiple vortices.

  3. 3.

Eventually the vortices decay as the evaporation ceases.

While the flow transitions were characterized by the evolution of vorticity,
to the best of our knowledge, a comprehensive understanding of the complex
flow dynamics induced by the segregation is still missing, especially the one
within the segregative phase. Hence a systematic investigation is required. In
this work, we thus revisit the evaporating droplet system of a 1,2-hexanediol-
water binary solution with multiple experimental means, mainly focusing on the
dynamics after the segregation has emerged, aiming to systematically study the
growth of the liquid-liquid-phase-segregation patterns and the resulting flow
structure. We also offer a theoretical understanding of the local flow
structure within the segregative patterns and of the decay of the flow
intensity at the final stage of the evaporation.

figure 1: Schematics of the three life stages of an evaporating
1,2-hexanediol-water binary droplet.

The paper is organized as follows: In §2, we introduce the employed
experimental techniques. In §3, we show the experimental observations on the
overall physiochemical hydrodynamic of the evaporating process, and identify
three successive life stages (Stage I, II & III) by the combination of the
direct observation of the flow structures and the quantification of the
temporal radial flow velocity. In §4, we briefly show and discuss the flow
structure at Stage I prior to the occurrence of the segregation, summarizing
and confirming the results of Li et al. (2018). In §5, we then focus on Stage
II, in which the segregative patterns emerge and grow. We quantitatively
interpret the growth dynamics of the segregative patterns and the local flow
structures within them. In §6, we quantify and quantitatively explain the flow
in the final Stage III, namely the decay of the flow intensity when the
evaporation gradually ceases. The paper ends with a summary and further
discussions (§7).

##  2 Experimental methods

###  2.1 Solution and substrate preparation

The binary solution we used in our droplet system consisted of 90%/w Milli-Q
water (Reference A+, Merck Millipore, 25∘C) and 10%/w 1,2-hexanediol (Sigma-
Aldrich, $\geq$98%). The experiments were performed on a hydrophobized glass
slide coated by octadecyltrichlorosilane (OTS, Sigma-Aldrich, $>$ 90 %) (Peng
et al., 2014). Before each measurement, the substrates were processed with a
sonication cleaning in 99.8 % ethanol and in Milli-Q water for 15 min and 5
min sequentially, and subsequently dried with compressed N2 gas flow for 30 s.
The droplets were deposited by a glass syringe with a full metal needle
(Hamilton, 10 $\mu$L, Model 701 NWG SYR, Cemented NDL).

###  2.2 Confocal microscopic measurement

Confocal microscopy was employed to conduct a series of measurements on the
evaporation dynamics of the droplet. The observations were carried out by
utilizing an inverted Nikon A1 confocal laser scanning microscope system
(Nikon Corporation, Tokyo, Japan) with a 10x dry objective (Nikon, Plan Fluor
10x/0.30, OFN25, DIC, L/N1).

####  2.2.1 Micro-particle image velocimetry

For flow field visualization, we performed Micro-Particle Image Velocimetry
($\mu$PIV) by adding fluorescent particles [microParticles GmbH; PS-
FluoRed-5.0: Ex/Em 530 nm/607 nm] into the working fluids. The PIV
measurements were also conducted on the confocal microscopy setup. The
particles were excited by a laser at a wave length of 561 nm and the
fluorescent signals were captured at a frame rate of 25 frames per second
(f.p.s.). We measured the flow field near the substrate quantitatively by
adding 520 nm diameter fluorescent particles at a concentration of $2\times
10^{-3}$ vol% into the droplet. For prepossessing the PIV images, we removed
the background noise outside the droplet by applying a contact line detection
algorithm, and subsequently a pixel-wise minimum intensity background was
subtracted from every image to enhance the signal to noise ratio. To obtain
the velocity vector field, we performed iterative two-dimensional cross-
correlations with multiple interrogation window sizes with a 50 % overlap,
which were 64 $\times$ 64 pixels, 32 $\times$ 32 pixels, and 16 $\times$ 16
pixels sequentially. A Gaussian fitting function was used to determine the
subpixel displacement. The velocity vector calculation was performed using an
open-source software, PIVlab (Thielicke & Stamhuis, 2014).

####  2.2.2 Refracted shadowgraphy

We modified a method originally proposed by Chao & Zhang (2001), namely laser
refracted shadowgraphy, and applied it to the confocal microscopy. A white-
light beam was collimated to a vertically parallel beam and then passed
perpendicularly through a test droplet, placed on an OTS glass substrate. The
white-light beam produced a refractive image of the sessile droplet which was
captured by the transmitted light detector at 25 f.p.s (Lv et al., 2017). The
disturbances caused by local concentration variances within the test droplet
refracted light rays, leading to shadows on the refractive images.

####  2.2.3 Fluorescence image acquisition

To visualize the local details within the segregative patterns, the binary
mixture was labeled with two different dyes, i.e., Dextran and Nile Red (Li et
al., 2018). Dextran preferentially dissolves in water and it was excited by a
laser at a wavelength of 488 nm, while Nile Red is a lipophilic dye that
dissolves only in 1,2-hexanediol and was excited by a laser at a wavelength of
561 nm simultaneously. Two-dimensional (2D) images were obtained by scanning
in the focal plane with a depth of field of 0.957 $\mu$m at the bottom of the
droplet $\sim$10 $\mu$m above the substrate. It allows us to monitor the
evolution of both the segregative patterns and the flow structure during the
entire evaporation process. The scan started as soon as the droplet was
deposited on the glass substrate. Operating in resonant mode, the 2D images
were captured with a frame rate of 25 f.p.s.. To increase the contrast of
images for further analysis, we converted the images to be binary (black and
white) in colored.

##  3 Overall phenomenology: Hydrodynamics induced by phase-segregation and
the three identified life stages

To obtain the overall phenomenology of the physiochemical hydrodynamics during
the evaporation process in the 1,2-hexanediol-water binary droplet system, we
first optically monitor the whole evaporation process, by using refracted
shadowgraphy in combination with $\mu$PIV. The multiple channels of the
confocal microscope allow us to perform the two visualization methods
simultaneously, see Fig. 2(a0-a2):

  1. 1.

The transmitted light detector captures the refracted light through the
droplet for shadowgraphy;

  2. 2.

The laser channel captures the fluorescent signal for $\mu$PIV.

The focal plane is located at the bottom of the droplet slightly above the
substrate ($\sim 10~{}\mu$m). Fig. 2(a1) displays the shadowgraph, in which
the strong shadows indicate the segregative patterns, whereas the bright areas
represent water-rich regions. Fig. 2(a2) shows a typical snapshot of
fluorescent particles for $\mu$PIV at the same instant.

###  3.1 Axial symmetric breaking induced by inhomogeneous nucleation of
1,2-hexanediol droplets

Fig. 2(b1-b6) displays a series of refracted shadowgraphs of an evaporating
1,2-hexanediol-water droplet with the emergence of the segregation
($t=t_{s}=50$ s). It starts with a homogeneous appearance of the entire
droplet without the segregation at the beginning of the evaporation [Fig.
2(b1)]. Then 1,2-hexanediol microdroplets nucleating at the rim of the binary
droplet [Fig. 2(b2)]. The emergence of nucleation sites is not homogeneously
distributed along the contact line, but initially local at one certain point
and only then extending to the rest of the rim. These inhomogeneous
nucleations of segregative patterns induce an axial symmetric breaking of flow
structures in the following process.

Without the pinning of the contact line, the contact area features a
continuous shrinkage during the entire evaporation process. During the growth
of the nucleated arch-shaped patterns, multiple water-rich streams emerge,
revealed by the bright paths connecting between the center of the droplet and
the tip of each arch pattern [Fig. 2(b3)]. It resembles the thermocapillary
instability appearing in heated droplets of ethanol (Sefiane et al., 2008;
Sáenz et al., 2017). We note that the surface tension varies both with the
local temperature and with the local solutal concentration, that is,
$\gamma=\gamma_{0}-\partial_{T}\gamma(T-T_{0})-\partial_{c}\gamma(c-c_{0})$;
normally for binary droplets,
$\partial_{c}\gamma(c-c_{0})\gg\partial_{T}\gamma(T-T_{0})$. Therefore the
convective flow can be simplified to a purely soluto-capillary flow,
neglecting the thermal Marangoni effect.

figure 2: Confocal microscopic measurement of an evaporating 1,2-hexanediol-
water droplet. (a) Transmitted light through the droplet and fluorescent
signals of tracer particles simultaneously captured by two channels of the
confocal microscopy. (b1-b6) Morphological evolution of convective structures
with the emergence of phase segregation in an evaporating 1,2-hexanediol-water
droplet. Here (b1) corresponds to Stage I, (b2), (b3), (b4) and (b5) to Stage
II, and (b6) to the final stage III. The three-dimensional sketch illustrates
the flow pattern in (b3). The dark blue arrows represent fast flow streams
(inwards at the droplets surface and outwards close to the substrate) and the
light blue ones slow secondary flows. The white stripes represent the water-
rich paths, which also correspond to slow-flow streams. The tip of each arch
becomes a stagnation point of the slow-flow streams. (c) A refractive image of
the droplet recorded by transmitted light detector (c1) and the corresponding
flow field at the same instant (c2). Here the droplet is in Stage II.

The schematics in Fig. 2(b3) illustrates a fingering flow pattern, namely the
growing arches impede the Marangoni flow towards the edge of the droplet,
causing the slow-flow stream. The tip of each arch becomes a stagnation point
of the slow-flow streams (see sketch belonging to Fig. 2(b3)). The radially
outward flow in the streams continues towards the contact line region along
the boundaries of the arches. During further evaporation, the arches grow and
merge with neighbouring ones, which leads to the formation of new slow-liquid
streams [Fig. 2(b4)]. Due to the breaking of axial symmetry caused by the non-
uniform growth of the segregative patterns, irregular convection cells appear
and evolve [Fig. 2(b5)]. Eventually, the convection cells disappear once the
evaporation ceases [Fig. 2(b6)]. As shown in the previous section, the
expansion of the arches reverses the flow direction in the vicinity of contact
line, namely from originally outward to inward. We qualitatively compare the
shadowgraph with the $\mu$PIV results in Fig. 2(c1-c2): the inward radial flow
(indicated by blue color) in the contact line region shown in the velocity
field [Fig. 2(c2)] corresponds to the patterns within segregative region shown
in the shadowgraph [Fig. 2(c1)].

###  3.2 Life stages identified by transitions in the flow

To further quantify the evolution of flow structure, we analyze the measured
results from the $\mu$PIV. In our previous study (Li et al., 2018), the
different life stages of an evaporating 1,2-hexanediol-water droplet were
identified by the temporal evolution of the spatially-averaged vorticity.
Here, by combining the $\mu$PIV results and the observations from
shadowgraphy, more quantitative details of the flow transitions can be
identified by the evolution of the local radial velocity.

Fig. 3(a) depicts a three-dimensional (3D) map showing the temporal evolution
of the radial velocity along the radial position, which is normalized by the
time-dependent footprint radius $R(t)$ of the droplet. The colour bar displays
the intensity of the radial flow velocity, where positive values indicate
outward direction. Three distinct stages of flow structure are identified:

  1. 1.

Stage I: Before the segregation occurs, the flow is radially outwards and most
intensive in the vicinity of contact line, whereas the flow in the interior is
barely visible;

  2. 2.

Stage II: With the emergence of segregation, the radial flow reverses from the
contact line region, and the flow structure becomes irregular;

  3. 3.

Stage III: The flow intensity starts to decay to nearly zero, until the
evaporation finally ceases.

With the simultaneous observation from shadowgraphy, another two sub-regimes
within Stage II can be distinguished, namely the fingering pattern and the
convective cells shown in Fig. 2(b3-b5). Stage III begins when the convection
cells disappear. In this final stage the flow intensity starts to decay to
nearly zero, until the evaporation finally ceases.

figure 3: Results from micro-particle image velocimetry ($\mu$PIV) in an
evaporating 1,2-hexanediol-water binary droplet. The flow field near the
substrate is revealed. (a) Three-dimensional (3D) map showing the temporal
evolution of the radial velocity averaged in azimuthal angle along the radial
position. (b) The probability distribution function (PDF) of the azimuthal
angle of the flow velocity vs. the radial direction at three different life
stages.

Fig. 3(b) displays three snapshots of flow fields at different life stages.
The first row shows the corresponding flow field at each instant. The second
row shows the probability distribution function (PDF) of the azimuthal angle
$\theta_{ur}$ of the local velocity (i.e., the angle between velocity $u$ &
radial direction) vs. the radial direction.The first column exhibits the flow
structure in Stage I, which is before the occurrence of segregation. Driven by
the Marangoni forces along the liquid-air interface, the flow is mainly radial
towards the edge of the droplet (seen by the accumulation of PDF at
$\theta_{ur}=0$), where the flow is also more intensive than in the interior.
With the emergence of segregation, Stage I undergoes a transition to Stage II.
As seen in the second column, the PDF of the velocity direction in Stage II
becomes more uniform, indicating a more irregular flow pattern, which can be
seen in the snapshot of the velocity field. We attribute this to the existence
of multiple vortices induced by segregation. The last column shows the
velocity vector field at the beginning of Stage III, when the segregative
patterns grow and finally fully occupy of the whole droplet. Then the PDF
shows an opposite distribution as compared to Stage I, indicating that the
inward flow radially towards the interior of the droplet is dominant.

##  4 Stage I: Marangoni-effect-dominant flow before segregation

In our previous work (Li et al., 2018), we have studied the early period of
the evaporation process before the occurrence of the segregation (i.e., Stage
I), revealing the Marangoni-effect-dominant flow structure and the mechanism
leading to the segregation. Due to the preferential evaporation of water, a
relatively high concentration of 1,2-hexanediol is left behind and thus
accumulates in the contact line region. This leads to a surface tension
gradient at the liquid-air interface, driving a Marangoni flow from the
contact line to the apex of the droplet. This interfacial flow also induces
convective flows in the bulk of the droplet. Near the substrate, this
convective flow is directed radially outwards to the rim of the droplet, which
is consistent with our observation here in Fig. 2(b1), and Fig. 3(a). For
details of this early Stage I and a quantitative analysis, we refer to Li et
al. (2018).

In the following sections of this paper, we focus on Stages II & III and the
segregative patterns and the resulting flow structures during these later
stages, and offer theoretical interpretations of them.

##  5 Stage II: Evolution of segregative patterns and induced local flow
structure

At Stage II (after the first segregation has emerged), a radial flow reversal
induced by the growth of the segregation occurs. To study this, a detailed
local observation of the segregative patterns is required. Experimentally, we
employ the fluorescence image technique coupled with confocal microscopy,
i.e., labelling the segregated phase with different dyes, in order to locally
resolve the details of the flow field within the segregated phase. Recently,
Seyfert & Marin (2021) reported an experimental study on the influence of
added dye on Marangoni bursting phenomenon. In our case, the observations by
labelling with fluorescent dyes show the emergence of more homogeneously
distributed nucleation sites along the contact line, rather than the
inhomogeneous nucleation sites there. Note that phenomenologically the two
cases with or without the dyes show similar behaviors with respect to the
growth of the segregation and the resulting flow structure within the
segregative patterns.

###  5.1 Growth dynamics of the liquid-liquid phase segregation

Before the study of the local flow structure within the segregative patterns
of 1,2-hexanediol, we first aim to understand its growth dynamics. The images
in Fig. 4(a1-a5) are taken by confocal microscopy from the bottom view: the
dark regions shaping as arches in the vicinity of the contact line are the
segregated phase of 1,2-hexanediol, whereas the grey counterpart is the
mixture phase. The phase segregation starts with the emergence of various tiny
nuclei evenly distributed along the contact line of the binary droplet [Fig.
4(a1)]. With further evaporation of water, arch-shaped segregation areas grow,
and then merge with their neighbors [Fig. 4(a2-a4)]. Eventually, they fully
occupy the entire droplet [Fig. 4(a5)]. Fig. 4(b) shows the temporal evolution
of the azimuthal undulations W($\theta,t$) of the growing arches, namely the
radial distance between the tip of the arches and the contact line of the
droplet. The total number $N_{s}$ and projected area $A_{s}$ of the arches are
displayed in the inset of Fig. 4(c).

We interpret that the growth of liquid-liquid-phase-segregation patterns is
controlled by the depleted water concentration due to evaporation. The mass
reduction rate of the mixture phase is proportional to the evaporative flux
rate of water, i.e., $\textrm{d}m_{m}/\textrm{d}t\sim-R_{m}D_{w,air}\Delta c$
(Sobac & Brutin, 2014; Li et al., 2020a), where $R_{m}$ the distance between
the center of the droplet to the front of the arches [see Fig. 4(a3)],
$D_{w,air}$ the diffusion coefficient of water vapour in air, and $\Delta c$
the vapour concentration difference between the droplet surface and the
surroundings. The concentration dependence of the mixture liquid density
$\rho_{m}$ is neglected here. With the geometrical relations among the volume
$V_{m}$, the projected area $A_{m}$ and the radius $R_{m}$ of the mixture
phase, i.e., $V_{m}\sim A_{m}^{3/2}$ and $R_{m}\sim A_{m}^{1/2}$, the change
of the droplet mass is

$\frac{\textrm{d}m_{m}}{\textrm{d}t}=\frac{\textrm{d}(\rho_{m}V_{m})}{\textrm{d}t}=\frac{\textrm{d}\rho_{m}}{\textrm{d}t}V_{m}+\rho_{m}\frac{\textrm{d}V_{m}}{\textrm{d}t},$
(1)

in which
$\rho_{m}=(\rho_{w}V_{w}+\rho_{H}V_{H})/(V_{w}+V_{H})\approx\textrm{constant}$,
with the density of water $\rho_{w}=997$ kg/m3 and the density of
1,2-hexanediol $\rho_{H}=951$ kg/m3 at the room temperature. So we obtain

$\frac{\textrm{d}m_{m}}{\textrm{d}t}\approx\rho_{m}\frac{\textrm{d}V_{m}}{\textrm{d}t}\sim\rho_{m}\frac{\textrm{d}(A_{m}^{3/2})}{\textrm{d}t},$
(2)

and the mass depletion by evaporation

$\frac{\textrm{d}m_{m}}{\textrm{d}t}\sim-R_{m}D_{w,air}\Delta c\sim-
A_{m}^{1/2}D_{w,air}\Delta c.$ (3)

From equating Eq. (2) and Eq. (3), we obtain
$\rho_{m}\textrm{d}(A_{m}^{3/2})/\textrm{d}t\sim-A_{m}^{1/2}D_{w,air}\Delta
c$, which yields $\textrm{d}A_{m}\sim-D_{w,air}\Delta
c/\rho_{m}\cdot\textrm{d}t$, i.e., linear shrinkage in time for the projected
area of the mixture phase $A_{m}=A_{t}-D_{w,air}\Delta c/\rho_{m}\cdot t$.
Here $A_{t}$ is the total initial footprint area and $D_{w,air}$ and $\Delta
c$ were assumed to remain nearly constant. The segregated area $A_{s}$ then is
$A_{s}~{}=~{}A_{t}-A_{m}$. Rescaling with the lifetime of the droplet $t_{f}$
and the time until the emergence of the first segregation $t_{s}$, which are
all constant for each case, yields

$\frac{A_{s}}{A_{t}}\sim\frac{t-t_{s}}{t_{f}-t_{s}},$ (4)

i.e., linear growth of the segregated area. Eq. (4) is verified in Fig. 4(c)
by displaying the temporal evolution of the area of the segregated phase,
compensated by the total footprint area $A_{s}/A_{t}$ in logarithmic scale.
The observed linear growth indeed supports that the growth rate of the
segregated phase patterns is controlled by the evaporation rate of water.

figure 4: Growth dynamics of the segregated 1,2-hexanediol phase. (a) Bottom-
view snapshots, captured via confocal microscope, highlighting the temporal
growth of the phase segregation of 1,2-hexanediol in an evaporating binary
droplet with 10%/90% (w/w) 1,2-hexanediol/water initial concentration. (b)
Temporal evolution of the azimuthal undulations W($\theta$) of the nucleated
droplets along the binary droplet radius during the growth of the segregated
phase shown in sequence in (a). (c) Scaled master curve of the segregated
phase area $A_{s}/A_{t}$ versus time $(t-t_{s})/(t_{f}-t_{s})$. Inset: Raw
data for the temporal evolution of $A_{s}$ and the number of nucleated
droplets $N_{s}$ versus the time $t$.

###  5.2 Local flow structure within segregative patterns

Following the understanding of the growth dynamics of the phase segregation of
1,2-hexanediol, we further conducted a direct measurement of the local flow
structure within the segregative patterns, in order to better understand the
observed radial flow reversal at Stage II.

We utilized the water-soluble dye Dextran as “tracer” to realize the flow
visualization in those regions. The dye is originally dissolved in the mixture
system but then separates from the droplet together with 1,2-hexanediol at the
contact line due to the preferential evaporation of water there. Some of the
separated Dextran attaches on the solid surface, causing pinning of the
contact line. Some other Dextran is transported into the 1,2-hexanediol phase
by the local flow [see Fig. 5(a)]. Because of its insolubility in
1,2-hexanediol, Dextran aggregates as small clusters in the segregated phase,
which then play the role as tracer particles [see Fig. 5(a)]. By extracting
the fluorescent signal of Dextran, we can visualize the trajectories of these
tracers, which are indicative of the flow in the arches. We average the
intensity of 30 consecutive images to qualitatively illustrate the flow
structure within the arches, as shown in Fig. 5(a2-a4).

From that figure we see that the flow structure inside the arch-like patterns
consists of two counter-rotating convective rolls originating from the
periphery of the binary droplet, which is indicated by the green arrows in
Fig. 5(a4). It is reminiscent of the convective rolls inside the arch-shaped
patterns observed in an evaporative ethanol-water binary system confined in a
Hele-Shaw cell, which is driven by a Marangoni instability caused by the
gradient of ethanol at the interface (Linde et al., 1964; Lopez de la Cruz et
al., 2021). However, in our case, the generation of the convective rolls
follows a different route. It is schematically shown in Fig. 5(b1-b2): the
expansion of the arches induces the flow originating from the edge towards the
tip of the arches; then it encounters the outward radial flow outside the
arches (indicated by red and orange arrows)[Fig. 5(b1)], and recirculates
backwards to the edge. Note that the flow differs both from capillary flows
caused by a pinned contact line and from Marangoni flows driven by surface
tension gradients; instead, remarkably, it is driven by the growth of the
segregative patterns.

To quantitatively confirm that the radial flow is mainly driven by the growing
phase segregation, we first performed $\mu$PIV within the segregative patterns
by utilizing the demixing Dextran as $\mu$PIV “tracers”. A typical snapshot of
the flow field within the segregative patterns is shown in Fig. 6(a1-a2). A
proper $\mu$PIV analysis is only possible after the segregation reaches a
certain size, so that the area of each pattern is 3-4 times larger than the
interrogation window size.

figure 5: Flow structure within the segregated phase patterns. (a1) The
fluorescent signal at the focal plane near the substrate. (a2) The fluorescent
signal of Dextran within segregative phase patterns. (a3) Contrast-enhanced
image by averaging the intensity of 30 consecutive frames. (a4) A zoom-in
image of the signal within one of the patterns. (b1-b2) Schematics of the flow
structure within a segregative pattern. figure 6: (a1-a2) Experimental results
of the flow field within the arch-shaped patterns. The colormap displays the
vorticity (a1) and radial velocity (a2). (b1-b2) Analytical prediction of the
flow field within one segregative pattern.

We argue that the flow structure within every individual arch-shaped pattern
is induced by the expansion of the segregation and its interaction with the
surrounding flow. Due to the relative flatness of the droplet ($h/R\ll 1$), we
consider the system in a two-dimensional (2D) planar geometry, namely the 2D
incompressible flow is predominant in our system. To obtain an analytically
solvable model, a few assumptions are taken to simplify the system: (i) Every
individual segregative pattern is simplified as a half circular domain by
neglecting the curvature at the contact line. (ii) For the small timescale in
which the patterns remain considerably the same shape, the mass conservation
holds in the half circular domain which is pushed towards the interior of the
droplet by the growing segregation at the contact line region with the
velocity $U^{sp}_{E}\approx 50~{}\mu$m/s. (iii) The coalescence between
neighbouring patterns is not considered. We then obtain the theoretical flow
velocity $u^{sp}_{x},u^{sp}_{y}$ within the segregative region ($x\leq
R_{sp},|y|\leq R_{sp},x^{2}+y^{2}\leq R_{sp}^{2}$) in a Cartesian coordinates
$(x,y)$ in the laboratory frame of reference as shown in Fig. 5(b2) (see
Supplementary Materials for more details):

$u^{sp}_{x}=(1-\frac{x^{2}+3y^{2}}{R_{sp}^{2}})(U^{sp}_{ma}+U^{sp}_{E})+U^{sp}_{E},$
(5)

$u^{sp}_{y}=\frac{2xy}{R^{2}_{sp}}(U^{sp}_{ma}+U^{sp}_{E}),$ (6)

where $U^{sp}_{ma}\approx 20~{}\mu$m/s is the velocity of the convective flow
induced by Marangoni effect and $R_{sp}$ the radius of the segregative
pattern.

Fig. 6 displays the comparison between the experimental results and the
analytical predictions by Eqs (5&6), which shows a good agreement of vorticity
and velocity fields within the segregative patterns. It indicates that the
inward flow structure (flow reversal) at the contact line region is induced by
the expansion of the segregative patterns, and the two counter-rotating
circulations are caused by the interaction between the expansion of the
segregation and its surrounding outward flow.

##  6 Stage III: Decay of flow intensity

When the segregated 1,2-hexanediol phase evolves to an almost full occupation
of the entire droplet [see Fig. 4(a5)], the evaporation process enters its
final Stage III, which is characterized by the decay of the flow intensity. In
this section, we quantitatively study the decay of the flow intensity by the
spatially-averaged vorticity. A similar decrease of the magnitude of vorticity
has been observed within the flow transition in an evaporating ethanol-water
binary droplet (Christy et al., 2011; Bennacer & Sefiane, 2014; Diddens et
al., 2017). The transition in that flow occurs at the period when ethanol is
almost depleted by evaporation.

In our 1,2-hexanediol-water droplet, we notice that the start of the decay of
the mean vorticity $\left<w(t)\right>$ corresponds to the moment when the
convective cells disappear. After that, the contact area still slightly
shrinks, reflecting that the contact angle recovers from the dynamic receding
angle to the final equilibrium angle (Li et al., 2018). We argue that at this
stage, there is still some water entrapped in the residue of the droplet,
which causes a weak surface tension gradient at the liquid-air interface which
drive some solute Marangoni flow. We estimate the Marangoni number
$Ma=(\Delta\gamma R)/(\mu D_{w,h})\approx 10^{2}$, with the diffusion
coefficient $D_{w,h}\approx 10^{-10}$ m2/s for water in 1,2-hexanediol, the
dynamic viscosity $\mu\approx 80\times 10^{-3}$ kg m-1 s-1, and the surface
tension difference $\Delta\gamma\approx 10~{}\mu$N/m. $\Delta\gamma$ is
estimated from the numerical simulation in our previous work (Li et al.,
2018), which shows that in the late stage of evaporation, the variance of mass
concentration of 1,2-hexanediol along the droplet surface has the order of
$\sim 1\%$. It corresponds to a surface tension variance $\Delta\gamma\sim
10~{}\mu$N/m (Romero et al., 2007). Furthermore, the Rayleigh number in our
case $Ra=(\Delta\rho R^{3}g)/(\mu D_{w,h})\approx 10$, with density difference
$\Delta\rho\sim 1$ kg m-3, and gravitational acceleration $g=9.8$ m s-2.
Therefore, given these estimation of $Ma$ and $Ra$, the Marangoni convection
is stronger than the Rayleigh convection in this life stage (Diddens et al.,
2021), see in the inset of Fig. 7(b).

The out-of-equilibrium state is generated by the non-uniform concentration
distribution. The dynamic mixing can be described by the general advection-
diffusion equation $\partial_{t}C=\nabla\cdot(D\nabla
C)-\nabla\cdot(\vec{u}C)+S$, where $S$ describes the source term. In this late
stage of the evaporation process, we neglect the source term $S$, take the
diffusion coefficient $D$ as constant, and describe the velocity field as an
incompressible flow which has zero divergence (i.e., $\nabla\cdot\vec{u}=0$).
Then the formula simplifies to $\partial_{t}C=\nabla\cdot(D\nabla
C)-\vec{u}\cdot\nabla C$. With the typical velocity $U\approx 10^{-5}$ m/s,
the Péclet number can be estimated by $Pe=UR/D_{w,h}\approx 10$, which
reflects the dominating role of advection as compared to diffusion in this
case. In order to capture the scales of both sides of the equation at leading
order, we further neglect the diffusion term, hence

$\partial_{t}C\sim-\vec{u}\cdot\nabla C.$ (7)

The typical flow velocity $\vec{u}$ is characterized by the velocity at the
liquid-air interface driven by the surface tension gradient. It is expected to
have the magnitude of $(h_{0}\Delta\gamma)/(\mu R)$ (Kim et al., 2016)
throughout the droplet, where $h_{0}$ is the height of the droplet. We then
take the spatial derivative for both sides of Eq. (7),

$\partial_{t}(\nabla
C)\sim-\frac{h_{0}}{\mu}\frac{\Delta\gamma}{R}\cdot\nabla(\nabla C).$ (8)

The local concentration gradient $\nabla C$ induce the surface tension
gradient, which results in a local shear stress
$\tau\sim\nabla\gamma\sim\nabla C\cdot\partial_{c}\gamma$. The vorticity
perpendicular to the image plane is induced by the local stress
$\omega_{z}\propto\tau$, which is defined as the curl of the in-plane velocity
$(u_{x},u_{y})$, i.e.,
$\omega_{z}=|\nabla\times\vec{u}|_{z}=|\partial_{x}u_{y}-\partial_{y}u_{x}|$.
Here we only consider the vorticity perpendicular to the image plane
$\omega_{z}$, which is dominant over the other two components due to the low
contact angle of the droplet. We hence argue that a scaling relation
$\omega_{z}\propto\nabla C$ holds (Christy et al., 2011). By replacing $\nabla
C$ by $\omega_{z}$, we then obtain the scaling relation

$\frac{\partial\omega_{z}}{\partial
t}\sim-\frac{h_{0}}{\mu}\frac{\Delta\gamma}{R}\cdot\nabla\omega_{z}\sim-\frac{h_{0}}{\mu}\frac{\Delta\gamma}{R}\frac{\omega_{z}}{R},$
(9)

implying

$\frac{d\omega_{z}}{\omega_{z}}\sim-\frac{h_{0}\Delta\gamma}{\mu R^{2}}dt.$
(10)

By taking the vorticity $\omega_{0}$ at the beginning moment $t_{d}$ of the
decaying as the initial condition, integration in time gives an exponential
decaying of the vorticity,

$\omega\sim\omega_{0}\textrm{exp}\left(-\frac{h_{0}\Delta\gamma}{\mu
R^{2}}t^{*}\right),$ (11)

with the initial vorticity $\omega_{0}$ and $t^{*}=t-t_{d}$, where $t_{d}$ is
the beginning moment of the decay.

Fig. 7 shows good agreement between the analytical prediction and the
measurement of the normalized mean vortex
$\left<\omega\right>/\left<\omega_{0}\right>$ during decaying. Note that
beyond 10 s after the start of the decaying, the vorticity experiences a more
rapid decay than predicted by our model. At this late stage of decaying, the
flow velocity $U$ decreases to less than $10^{-6}$ m/s, which corresponds to
$Pe<1$. Besides the convection, the mutual diffusion also plays a significant
role in the dynamical mixing to homogenize the concentration gradient. Hence
the intensity of the flow decays faster than the prediction by the model which
only considers the convective mixing. The solution indicates that the decaying
of vorticity is mainly due to the mixing effect by the surface tension
gradient, and the viscosity delays the mixing process by slowing down the
convective flow.

figure 7: Temporal evolution of the normalized vorticity within an evaporating
1,2-hexanediol-water droplet at the focal plane near the substrate. The blue
dots represent the experimental measurement, and the red line shows the
analytical prediction from Eq. (11) for the late phase, with $h_{0}\approx
80~{}\mu$m, dynamics viscosity $\mu=80\times 10^{-3}$ kg m-1 s-1. The time
$t^{*}=0$ s indicates the start of the vortex decay. (a) The measured temporal
spatially-averaged vorticity $\left<\omega\right>$ normalized by the value
$\left<\omega_{0}\right>$ at the beginning of the decay ($t^{*}=0$) for the
whole drying process. (b) The same measured dataset in Stage III. The inset
shows the prediction by Diddens et al. (2021). The dark green bullet at
$Ma=10^{2},Ra=10^{1}$ indicates a typical experimental data point of this
work, which clearly lies in the Marangoni-flow-dominant regime, as all data
points of this work.

##  7 Concluding remarks and discussions

In summary, we have studied the physiochemical hydrodynamics of an evaporating
1,2-hexanediol-water binary droplet on a flat substrate, in which the
preferential evaporation of water leads to the segregation of 1,2-hexanediol.
We quantitatively characterize the evolution of convective flow structures and
identify three successive life stages by flow transitions, namely Stage I, II
& III. Stage I is defined as the period before the segregation occurs. The
flow is mainly controlled by the Marangoni effect caused by the surface
tension gradient induced by selective evaporation. At Stage II, a radial flow
reversal originating from the contact line area is observed with the emergence
of the segregation. We show that the growth rate of the segregative patterns
is controlled by the depletion of water due to evaporation. Then we visualize
the flow structure within the segregative arch-shaped patterns and interpret
that the inward radial flow is remarkably induced by the growth of the
segregative patterns. At the late Stage III of the evaporation, i.e., when the
non-volatile segregative phase almost fully occupies the entire droplet, a
vortex decay is observed and the distribution of the entrapped water is
homogenized by the Marangoni effect, which is again caused by the surface
tension gradient. In contrast, the viscosity delays the mixing effect.
Generally, the flow structures undergo transitions from surface-tension-driven
to segregation-driven and again to surface-tension-driven states, which
clearly shows the richness of the hydrodynamics induced by the segregative
behavior.

This work still requires further investigations of the various processes. The
complex flow structure (e.g. the convective cells) in the entire droplet is
three dimensional, but our present observations have been limited to a 2D
focal plane at the bottom of the droplet only. In order to comprehensively
understand the flow behavior, either an experimental or a numerical technique
which is able to reconstruct a complete 3D flow field is required.

###### Acknowledgements.

This work is part of an Industrial Partnership Programme (IPP) of the
Netherlands Organization for Scientific Research (NWO). This research
programme is co-financed by Canon Production Printing Holding B.V., University
of Twente and Eindhoven University of Technology. DL gratefully acknowledges
support by his ERC-Advanced Grant DDD (project number 740479).

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