# Building micro-soccer-balls with evaporating colloidal fakir drops

Álvaro G. Marín<EMAIL_ADDRESS>Physics of Fluids Group, Faculty of Science
and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands.  Arturo Susarrey-Arce  Catalytic Processes and Materials, Faculty
of Science and Technology, University of Twente.  Mesoscale Chemical Systems,
Faculty of Science and Technology
University of Twente  Hanneke Gelderblom  Physics of Fluids Group, Faculty of
Science and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands.  Arie van Houselt  Leon Lefferts  Catalytic Processes and
Materials, Faculty of Science and Technology, University of Twente.  Han
Gardeniers  Mesoscale Chemical Systems, Faculty of Science and Technology
University of Twente  Detlef Lohse  Physics of Fluids Group, Faculty of
Science and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands.  Jacco H. Snoeijer  Physics of Fluids Group, Faculty of Science
and Technology, Mesa+ Institute, University of Twente, Enschede, The
Netherlands.

###### Abstract

Evaporation-driven particle self-assembly can be used to generate three-
dimensional microstructures. We present a new method to create these colloidal
microstructures, in which we can control the amount of particles and their
packing fraction. To this end, we evaporate colloidal dispersion droplets on a
special type of superhydrophobic micro-structured surface, on which the
droplet remains in Cassie-Baxter state during the entire evaporative process.
The remainders of the droplet consist of a massive spherical cluster of the
microspheres, with diameters ranging from a few tens up to several hundreds of
microns. We present scaling arguments to show how the final particle packing
fraction of these balls depends on the dynamics of the droplet evaporation.

###### pacs:

Evaporation-driven particle self-assembly is an ideal mechanism for
constructing micro- and nanostructures at scales where direct manipulation is
impossible. For example, in colloidal dispersion droplets with pinned contact
lines, evaporation gives rise to the so-called coffee-stain effect Deegan  _et
al._ (1997): a capillary flow drags the particles towards the contact line to
form a ring-shaped stain. Such a flow not only aggregates the particles, but
is also able to organize them in crystalline phases Fan and Stebe (2004);
Bigioni  _et al._ (2006); Harris  _et al._ (2007); Marín  _et al._ (2011).
Similar mechanisms such as the “convective assembly” Velikov  _et al._ (2002);
Meng  _et al._ (2006) are currently successfully used to produce two-
dimensional colloidal crystal films. To obtain three-dimensional clusters of
micro-particles, colloidal dispersion droplets which are suspended in
emulsions Velev  _et al._ (2000); Manoharan  _et al._ (2003); Dinsmore  _et
al._ (2002) or kept in Leidenfrost levitation Tsapis  _et al._ (2005) are
used. With these techniques, new colloidal structures arise from the
geometrical constraints during the drying Lauga and Brenner (2004). This
problem of organization of particles into a spherical topography dates back to
the days of the first models of the atom and has been extensively studied by
Bausch et al. Bausch  _et al._ (2003); Lipowsky  _et al._ (2005). The main
drawback of these three-dimensional assembly techniques, however, is the lack
of control on both the amount of particles and the particle arrangement in the
remaining structures.

Figure 1: (a) A droplet of colloidal solution is left to evaporate on a
superhydrophobic surface. As the solvent evaporates, the particle packing
fraction increases. Once all the solvent has completely evaporated the
colloidal particles have aggregated to form a spherical particle conglomerate:
a colloidal supraball. (b) Top view of the resultant compact colloidal
supraball left on the superhydrophobic surface after evaporation. The
micropillars forming the structure are seen as circular objects around the
supraball.

In this work, we devise a new, controlled way of generating on-demand self-
assembled spherical micro-structures via droplet evaporation on a
superhydrophobic surface (see figure 1). We present scaling arguments to
predict the particle arrangement in the microstructures formed, based on the
dynamics of the evaporation process. To generate the microstructures, we
evaporate colloidal dispersion droplets on a special type of superhydrophobic
substrates. In most of the cases, a liquid Cassie-Baxter state drop
evaporating on a superhydrophobic surface will eventually suffer a wetting
transition into a Wenzel state, i.e. it will get impaled into the structure
and loose its spherical shape Reyssat  _et al._ (2008); Tsai  _et al._ (2010).
Here, however, we use a surface that combines overhanging pillared structures
Tuteja  _et al._ (2008) with a hierarchical nano-structure (figure 2c). These
surface properties impose a huge energy barrier for the wetting transition to
occur, and therefore the droplet will remain almost floating over the
structure in a Cassie-Baxter state during its entire life Susarrey-Arce  _et
al._ (2012). A typical result can be observed in figure 1: a water droplet
containing $1\mu m$ polystyrene particles (concentration 0.08% weight and
initial volume $5\mu l$) evaporates on the superhydrophobic surface at room
temperature and 30% humidity. After a typical evaporation time of 45 minutes,
the solvent is completely evaporated and only the colloids are left upon the
substrate. Remarkably, the particles ($\sim 10^{7}$ in this particular case)
are not just lying scattered over the substrate but they have aggregated and
form a spherical macro-cluster resting on top of the micro-pillars, which we
call _colloidal supraball_. We do not observe shell formation and buckling
during the evaporation of the droplets Tsapis  _et al._ (2005): the supraballs
we obtain are solid, and present a high mechanical resistance and stability.
When looking closer at the surface of these particular colloidal supraballs as
shown in figure 2d, one can identify crystalline flat patches which resemble
the pentagonal patches present in a soccer ball.

To understand the final structure of these supraballs, it will turn out
crucial to understand the dynamics of the droplet evaporation. The fact that
we do not observe shell formation, suggests that the particles do not
influence the droplet evaporation. To test this hypothesis, we compare the
evaporation dynamics to that of a liquid drop that does not contain any
particles. The evaporative mass loss from such a drop is typically governed by
the diffusion of vapor molecules in the surrounding air Deegan  _et al._
(1997); Popov (2005); Gelderblom  _et al._ (2011). For diffusion-limited
evaporation, the rate of volume change of the drop is given by

$\frac{dV}{dt}\sim D^{\prime}R,$ (1)

where $R$ is the drop radius, and $D^{\prime}=D_{va}\Delta c/\rho$, with
$D_{va}$ the diffusion constant for vapor in air, $\Delta c$ the vapor
concentration difference between drop surface and the surroundings and $\rho$
the liquid density Marín  _et al._ (2011). One might have expected the
evaporation rate from the drop surface to be proportional to the droplet
surface area $\sim R^{2}$. However, the vapor concentration gradient is
proportional to $1/R$, and therefore the total evaporation rate is
proportional to $R$ Eggers and Pismen (2010). If the droplet evaporates with a
constant contact angle, we find that, since $V\sim R^{3}$,

$R(t)\sim[D^{\prime}(t_{f}-t)]^{1/2}.$ (2)

Here $t_{f}$ is the total droplet lifetime in case no particles are present,
for which the drop radius reaches zero. In the present case the drop radius
saturates at a finite radius, $R_{ball}$, at a time
$\hat{t}=t_{f}-R_{ball}^{2}/D^{\prime}$, corresponding to the moment where the
particles become densely packed. In figure 3 we plot the colloidal droplet
radius versus $t_{f}-t$. Our experimental data for different number of
particles are in very good agreement with the $1/2$-power law. This confirms
that the particles do not influence the evaporation process, until the final
radius $R_{ball}$ is reached. The scaling (1) implies that the speed with
which the interface is moving inwards, is given by $dR/dt\sim D^{\prime}/R$.
Hence, the interface speed increases dramatically as the droplet shrinks and
the maximum speed reached in the experiment will be determined by the final
radius $R_{ball}$. As we will show further on, this increase in interface
speed determines the particle packing inside the supraballs.

Figure 2: (a) Tilted view of the supraball in contact with the microstructure.
(b) Detail of the contact area. (c) Magnified view of the micropillars forming
the microstructure. (d) Close-up of the supraball surface. The distribution of
crystalline patches resemble the pentagons in a soccer-ball.

The final size of the ball depends on the number of particles inside the drop.
This can be tuned by manipulating either the initial particle concentration or
the droplet size. In our experiment, the ball size was in the range
$100<R_{ball}/R_{p}<1000$, with $R_{p}$ the particle radius. Clearly, the
exact final size of the ball will not only depend on the amount of particles
in the system but also on their packing fraction. We define the packing
fraction as

$\Phi\equiv N\left(\frac{R_{p}}{R_{ball}}\right)^{3},$ (3)

where $N$ is the total number of particles in the droplet. The final supraball
radius $R_{ball}$ is accurately determined from Scanning Electron Microscope
(SEM) images. If the packing fractions were identical for all supraballs, one
would expect that $R_{ball}/R_{p}\sim N^{1/3}$, as depicted in figure 4a
111All quantitative results shown in this paper have been performed with
colloids of $1\mu m$ diameter (non-surface-modified fluorescent microspheres
supplied by Thermo Scientific), but the same qualitative behavior has been
observed for $0.2$ and $2\mu m$. The colloidal solutions were always prepared
with deionized water.. However, in figure 4b we see that the packing fraction
strongly depends on the number of particles in the system.

Figure 3: The radius of the droplet plotted against $t_{f}-t$ with $t_{f}$ the
lifetime of the droplet and $t$ the actual time. The triangle indicates the
$1/2$-power law, the dots represent the data sets for 7 different experiments,
where the number of particles was varied. For a certain $\hat{t}<t_{f}$ the
final ball size $R_{ball}$ is reached. The final time was extrapolated as
$t_{f}=\hat{t}+R_{ball}^{2}/D^{\prime}$.

As the number of particles increases, the packing fraction approaches that of
a perfect Hexagonal Close Packing configuration, in which case one would find
$\Phi=0.74$ Kepler (1966), hence, we have an _ordered_ particle packing inside
the balls. On the other hand, the supraballs with a smaller amount of
particles show remarkably low packing fractions, even below the Random Close
Packing (RCP) limit ($\Phi=0.64$) Weitz (2004), corresponding to a
_disordered_ particle arrangement. The balls which show packing fractions
below the RCP limit contain several empty cavities. Remarkably, the final
configuration reached, seems to depend on the number of particles in the
system. In figure 4b we indicated the critical number of particles
$N_{c}\approx 3\cdot 10^{6}$ when the packing fraction reaches that of a RCP.
For $N<N_{c}$ we get a loose, _disordered_ particle packing in the supraball,
whereas for $N>N_{c}$ we get a densely packed, _ordered_ supraball.

What causes the transition from ordered to disordered packings, and what
determines the critical number of particles? To explain this, we follow a
similar approach as in Marín et al. Marín  _et al._ (2011): we compare the
time scale on which particles can arrange by diffusion to the hydrodynamic
time scale for the particle transport by convection, given by the inward
motion of the liquid-air interface. If the diffusion time is small compared to
the hydrodynamic time, particles can arrange into an ordered packing. The
diffusive time scale is $t_{d}=R_{p}^{2}/D$, with $R_{p}$ the particle radius
and $D$ the diffusivity of the particles in the liquid Marín  _et al._ (2011).
The hydrodynamic time-scale is $t_{h}=L/\left|\frac{dR}{dt}\right|$. Here
$R(t)$ is the droplet radius, and $L$ is the typical inter-particle distance,
which depends on the particle concentration as $L=N^{-1/3}R$, as long as the
solution is dilute ($L\gg R_{p}$). We define the ratio of both time scales as:

$\mathcal{A}(t)\equiv\frac{t_{d}}{t_{h}}=\left|\frac{dR(t)}{dt}\right|\frac{t_{d}}{L}=\frac{D^{\prime}}{D}N^{1/3}\left(\frac{R_{p}}{R(t)}\right)^{2},$
(4)

where in the last step we used (1) to replace $dR/dt\sim D^{\prime}/R$.

Figure 4:  (a) Supraball to microparticle diameter $R_{ball}/R_{p}$ plotted
against the total amount of particles $N$ in the system. (b) Packing fraction
$\Phi$ strongly depends on $N$. Blue dots represent experimental measurements
and the red solid line corresponds the most efficient particle packing
$\Phi=0.74$ (hexagonal close packed), the dashed line marks $\Phi=0.64$,
random close packed. $N_{c}$ is the critical number of particles, above which
we find an ordered ball structure

From (4) we observe that $\mathcal{A}(t)$ increases as the droplet radius
becomes smaller during the evaporation (see figure 3), until the limit
$R=R_{ball}$ is reached. A cross-over between the time-scales is reached when
the hydrodynamic time becomes equal to the diffusion time, hence when
$\mathcal{A}=1$. If the cross-over is reached when $R\gg R_{ball}$, the amount
of crystalline clusters is still very small. From this point in time onwards
the interface speed is too high for the particles to further arrange in a
crystalline way Marín  _et al._ (2011). Instead, they are pushed together in a
random arrangement, with a low packing fraction. If the cross-over is reached
when $R\leq R_{ball}$, the particle packing is already dense and ordered, and
we find a high packing fraction. For all droplets the evaporative mass loss,
and hence the decrease in radius, is the same (see figure 3), hence, the
moment when the particle packing becomes sufficiently dense for particles to
arrange depends solely on the number of particles in the droplet. If $N$ is
high ($N>N_{c}$), this moment is reached relatively early, i.e. well before
$\mathcal{A}=1$, and we get an ordered particle packing inside the supraballs.
Using that $R_{ball}/R_{p}\sim N^{1/3}$ and considering $\mathcal{A}=1$, we
find from (4) the critical number of particles above which we obtain ordered
supraballs

$N_{c}\sim\left(\frac{D^{\prime}}{D}\right)^{3}.$ (5)

This result emphasizes that the transition is governed by two diffusion
processes: the diffusion of vapor, determining the speed of evaporation,
versus the diffusion of particles inside the drop. The ratio of diffusion
constants selects the critical number of particles. In our experiment
$D^{\prime}=3\times 10^{-10}$ m2/s and $D=2\times 10^{-13}$ m2/s, from which
we find that $N_{c}\sim 10^{9}$. This is 2 to 3 orders of magnitudes larger
than the experimentally observed $N_{c}$. However, in the preceding analysis
we have neglected all prefactors, and we emphasize that the result is strongly
(to the third power) dependent on the experimental parameters included in
$D^{\prime}$, i.e. humidity, liquid density, diffusivity of vapor, and
saturated vapor concentration.

To verify whether the final packing fraction indeed depends on the spacing
between the particles the moment the cross-over time is reached, we go back to
our experimental data. We define the time-dependent packing fraction as
$N(R_{p}/R(t))^{3}$. As the droplet evaporates, this packing fraction will
increase until it reaches its final value $\Phi$. At the cross-over, defined
by ${\cal A}=1$, the droplets will have a packing fraction $\Phi^{*}$. If this
$\Phi^{*}$ is low, the amount of crystalline clusters is still very small. On
the other hand, if $\Phi^{*}$ is high, we expect crystalline clusters to have
formed already. After the cross-over time, the interface moves too fast to
allow for further ordering, and it just presses the ordered particle clusters
closer together. In figure 5 we show that droplets with a high $\Phi^{*}$ have
a high $\Phi$: when $\Phi^{*}\gtrapprox 0.1$ we obtain a final packing
fraction above the RCP limit.

Figure 5: Final packing fraction $\Phi_{f}$ versus the packing fraction at the
cross-over time $\Phi^{*}$. Droplets below a certain $\Phi^{*}$ have a too low
packing fraction at the cross-over time to achieve final packing fractions
above the RCP limit. The particle packing can not only be obtained from the
value of the packing fraction, but it can also directly be observed from the
SEM images of the surface of the supraballs.

We cannot predict the critical $\Phi^{*}$ theoretically. However, we can,
retrospectively, use the experimental critical $\Phi^{*}$ to compute $N_{c}$.
Using that $R_{ball}/R_{p}\sim 0.1N^{1/3}$ at the cross-over, we obtain
$N_{c}\sim 10^{7}$, which is in the same order of magnitude as our
experimental results; see figure 4b.

The particle packing in the supraballs can not only be assessed by measuring
the packing fraction, but it can also directly be seen in SEM images from the
surface of the colloidal supraballs, as shown in figure 5. The size of the
soccer-ball-like crystalline patches on the surface of the ordered supraballs
depends on the ball size: bigger balls will show larger patches due to the
reduced curvature at their surfaces. To explain the size of the crystalline
domains, we hypothesize that a crystalline patch will bend radially no more
than a particle size. Then, it follows by simple trigonometry that the size of
a patch $S$ will be related to the ball radius $R_{ball}$ and the particle
size $R_{p}$ via: $S/R_{ball}=\arccos(1-R_{p}/R_{ball})$. This expression
predicts a typical patch size of $\sim 15\mu m$ for a ball with
$R_{ball}=100\mu m$ and $R_{p}=1\mu m$, which is in the right order of
magnitude as one can observe in figure 2d.

In conclusion, in this Letter we devise a simple technique to create spherical
colloidal supraballs relying only on droplet evaporation over a robust
superhydrophobic surface. The supraballs show a highly ordered structure if
the number of particles inside the drop is large enough to trigger early
particle clustering. The critical number of particles required to obtain an
ordered particle packing inside the balls depends on the parameters driving
the droplet evaporation (through $D^{\prime}$) and the diffusivity of the
particles. Hence, by controlling the humidity and ambient temperature the
supraball packing fraction and hence size can be controlled. Massive
fabrication of micro-compact-supraballs could easily be achieved by simply
spraying a colloidal solution over the micro-structure in a controlled
atmosphere. By tuning the wetting properties of the particles one could also
be able to generate the well-known _colloidosomes_ Dinsmore  _et al._ (2002)
using the same proposed technique.

###### Acknowledgements.

We gratefully acknowledge Vicenzo Vitelli, Martin van Hecke, Devaraj van der
Meer, Chao Sun and many others for their useful comments and their support.
The authors also acknowledge financial support by the NWO grant No. 700.10.408
and No. 700.58.041.

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