When the contact angle is less than 40°, the dependence on θ is almost flat and the expression can be reduced to eq 6.

In the hydrophobic situation (90° < θ < 180°), McHale et al.²⁶ suggested a theoretical model based on a diffusion model to describe the evaporation of a spherical drop under two distinct situations. An evaporation for which the contact angle remains constant for the majority of the evaporation process is characterized by a linear decrease in the spherical radius over time

$$R_s^2 = R_i^2 - \frac{2\lambda \sin^2 \theta}{\pi (1 - \cos \theta)(2 + \cos \theta)} t \quad (8)$$

where $λ = 2πDΔc$ and $R_i$ is a constant of integration. When the evaporation is carried out with a constant contact line radius, the authors proposed to analyze this configuration by plotting the function $H_{PB}(θ)$, which is also linear with respect to time

$$ \begin{aligned} H_{PB}(\theta) &= \frac{e_0}{1 + u} + e_1 \ln(1 + u) + e_2 u + e_3 u^2 \\ &= \frac{-2Dc_v(1 - H)}{\rho R^2} t + H_{PB}(\theta_0) \end{aligned} \quad (9) $$

where $u(t) = \cos \theta(t)$, $H_{PB}(θ_0)$ is a constant of integration representing the function value at $t = 0$, and $e_0$, $e_1$, $e_2$, and $e_3$ are constants given in McHale et al.²⁷

Popov²⁸ generalized the models presented by Deegan et al.²⁹ and Hu and Larson,²⁵ which are valid in the limit of small contact angles ($θ < 90°$), to drops of arbitrary contact angles ($0° < θ < 180°$). Thus, he theoretically obtained, for the same boundary conditions, an exact analytical expression of the rate of mass lost from a drop for any contact angle:

$$ \begin{aligned} \frac{dm}{dt} &= \pi R D c_v (1 - H) f_p(\theta) \\ f_p(\theta) &= \frac{\sin \theta}{1 + \cos \theta} + 4 \int_0^\infty \frac{1 + \cosh 2\theta\tau}{\sinh 2\pi\tau} \tanh((\pi - \theta)\tau) d\tau \end{aligned} \quad (10) $$

Recently, the range of applicability of the quasi-steady, diffusion-controlled evaporation model has been considered. Kelly-Zion et al.³⁰ observed that this type of model largely underpredicts the evaporation rate for large drops (drop radius longer than 3 mm) in a wetting configuration. The effect has been attributed to the natural convection of the vapor, which increases the evaporation rate. An empirical model has been developed for combined transport. According to the size of the drop, the evaporation rate follows a power-law relation with respect to the radius, with the exponent ranging from 1 (small drops) to 2 (expected for large radii).

### III. MATERIALS AND METHODS

The experiment consists of the spontaneous natural evaporation of a sessile drop under atmospheric conditions. The drop is gently laid down on the substrate, and evaporation is carried out on an experimental cell to avoid any perturbations from external flow. The drop is created using a micropipet to control the volume of the drop. The environmental temperature $T_{av}$ pressure $P$, and humidity $H$ are measured using a weather station. The dimensions of the drop are recorded laterally using a digital camera (JAI BM500GE) coupled with a microscope lens (VZM100i) with a spatial resolution of 4 µm. Then, image processing is performed to monitor the evolution of the geometrical parameters related to the drop, such as the contact line radius, the drop height, and the contact angle,

<table><caption>Table 1. Summary of Coating Processes Used in This Study</caption><thead><tr><th>material</th><th>notation</th><th>technique</th><th>thickness<sup>a</sup></th><th>roughness<sup>b</sup></th></tr></thead><tbody><tr><td>perfluorocarbon</td><td>PFC</td><td>PECVD</td><td>13.9 nm</td><td>1.75 µm</td></tr><tr><td>polytetrafluoroethylene</td><td>PTFE</td><td>stuck</td><td>10 µm</td><td>284 nm</td></tr><tr><td>high-carbon PDMS</td><td>SiOC</td><td>PECVD</td><td>13.4 nm</td><td>1.75 µm</td></tr><tr><td>low-carbon PDMS</td><td>SiO<sub>x</sub></td><td>PECVD</td><td>3.4 nm</td><td>1.75 µm</td></tr></tbody></table>

<sup>a</sup>Thickness obtained using profilometry. <sup>b</sup>Root-mean-square roughness obtained using atomic force microscopy.

<table><caption>Table 2. Static Contact Angles of a Droplet of Various Liquids on the Sample Surfaces at Room Temperature<sup>a</sup></caption><thead><tr><th></th><th></th><th>PTFE (deg)</th><th>PFC (deg)</th><th>SiOC (deg)</th><th>SiO<sub>x</sub></th></tr></thead><tbody><tr><td>ethanol</td><td>C<sub>2</sub>H<sub>6</sub>O</td><td>36</td><td>26</td><td>17</td><td><sup>b</sup></td></tr><tr><td>water</td><td>H<sub>2</sub>O</td><td></td><td>135</td><td>115</td><td>63</td></tr><tr><td>methanol</td><td>CH<sub>4</sub>O</td><td></td><td>46</td><td></td><td></td></tr><tr><td>propanol</td><td>C<sub>3</sub>H<sub>8</sub>O</td><td></td><td>39</td><td></td><td></td></tr><tr><td>toluene</td><td>C<sub>7</sub>H<sub>8</sub></td><td>41</td><td></td><td></td><td></td></tr></tbody></table>

<sup>a</sup>Bold values indicate evaporation performed under the pinned mode.

<sup>b</sup>Completely wetting situation.

<table><caption>Table 3. Physical Properties of Fluids Used: Surface Tension, Saturation Pressure, and Vaporization Enthalpy</caption><thead><tr><th></th><th>σ (mN/m)</th><th>P<sub>sat</sub> (hPa)</th><th>L<sub>v</sub> (kJ/kg)</th></tr></thead><tbody><tr><td>methanol</td><td>22.7</td><td>12.977</td><td>1177</td></tr><tr><td>ethanol</td><td>22.1</td><td>5.826</td><td>924</td></tr><tr><td>propanol</td><td>23.7</td><td>4.240</td><td>792</td></tr><tr><td>toluene</td><td>28.4</td><td>5.086</td><td>414</td></tr><tr><td>water</td><td>72.0</td><td>2.329</td><td>2452</td></tr></tbody></table>

using commercial software (Kruss DSA3). This process can be carried out only for contact angles of up to 7°, which explains the end of some curves discussed in section IV. The substrate is a complex assembly of layers mainly composed of an aluminum cylinder (diameter = 10 mm, height = 9 mm). A heat-flux meter is placed overhead (diameter = 10 mm, thickness = 0.8 mm, sensitivity = 0.725 µV·W⁻¹·m²) to determine the power absorbed by the drop on the substrate during evaporation. Finally, different coatings are applied to the substrate surface. Three coatings of nanoscale thickness were achieved by plasma-enhanced chemical vapor deposition: PFC, SiOC, and SiOₓ. A microscale PTFE film was used to coat the substrate by montage (Table 1). The PECVD technique, the preparation, and the chemical composition of these coatings were previously presented by Phan et al.³¹ Estimates of the thermal resistances reveal the negligible influence of these coatings on the thermal properties of the substrates. The substrates are thermally identical; only the physicochemical properties of the surface and the roughness are modified. All surfaces were characterized with respect to morphology using an atomic force microscope (Veeco Explorer) with respect to wettability using a contact angle measurement system (Kruss DSA30). All data are summarized in Tables 1 and Table 2. Further details about the measurement materials are provided in ref 32.

Several liquids were used in this study (methanol, ethanol, propanol, toluene, and water) to investigate a large range of volatility and wettability (Tables 2 and 3). The various combinations of these fluids and coatings allow the study of a large range of wettabilities from 17 to 135°.

The calculation of the Bond number, $B_0 = \rho g R h_0 / \sigma$, which accounts for the balance between the gravitational force and the surface tension force, reveals that the shape of our drops can be considered to be a
<figure><img src="image_81.png" /><figcaption>**Figure 1.** Evaporation of an ethanol drop of volume $V = 4.21 \mu L \pm 1\%$ on three substrates: SiOC (black ●), PFC (red ●), and PTFE (blue ●). (a) Evaporative power evolution. (b) Dimensionless evaporative power evolution. (c) Dimensionless evolution of the contact line radius. Respective initial contact line radius, evaporation time, and cooling of the substrate: $R = (2.7, 2.2, 1.9)$ mm, $t_F = (260, 225, 342)$ s, and $\Delta T_{cool} = (0.6, 0.7, 0.4) \,^\circ C$. Atmospheric conditions: $T_a = 25.8 \pm 0.8 \,^\circ C$, $P = 1$ atm $\pm 1\%$.</figcaption></figure>

hemispherical cap. Here, $\rho$ is the fluid density, $g$ is the gravitational constant, $R$ is the contact line radius, $h_0$ is the initial height of the drop, and $\sigma$ is the air-liquid surface tension. Indeed, the maximum calculated Bond number is 0.49, which indicates that the drops are sufficiently small such that the surface tension is dominant and gravitational effects can be neglected. Thus, a spherical fitting has been used to extract the liquid-gaz interface of the drop.

According to the ratio $c_o(1 - H)/\rho \approx 2 \times 10^{-4}$ for ethanol, $3 \times 10^{-4}$ for methanol, $7 \times 10^{-5}$ for propanol, $2 \times 10^{-4}$ for toluene, and $2 \times 10^{-5}$ for water), the evaporation is considered to be quasi-steady and controlled by diffusion in the vapor phase.

By using a heat-flux meter, we are not directly following the evaporation rate $dm/dt$ but rather the power $P$ absorbed by the drop from the substrate during evaporation, which is called the evaporative power. However, by performing an energy balance, the two quantities are directly linked

$$P = -L_v \frac{dm}{dt} \qquad (11)$$

where $L_v$ is the phase-change enthalpy. A verification of the energy balance between the evaporative power obtained using the heat-flux meter and the evaporation rate obtained using the decrease in the drop volume over time reveals good agreement. Thus, the energy required by the drop to evaporate is provided by the substrate.

## IV. RESULTS AND DISCUSSION

### IV.a. Evaporation of Ethanol Drops with $0^\circ < \theta < 40^\circ$: Influence of the Triple Line.
The drops of ethanol were evaporated from only three surfaces, PTFE, PFC, and SiOC, because ethanol completely wets SiOx. As revealed by Figure 1a, the evolution of the evaporative power $P$ varies with the coating type. The intensity and the dynamics of the evaporative power also vary, as does the time required for evaporation. The evaporation time $t_F$ is defined as the time for which the evaporation power is null. Figure 1b,c highlights the dimensionless evolution of the evaporative power as well as the wetting radius according to the coating type. The dynamics of the triple line change with the surface properties of the substrate, and the dynamics of evaporation are consequently modified:

* The evaporation from the PFC coating occurs with the pinning of the triple line because the wetting radius is constant during the entire evaporation time. The evaporative power

is thus constant throughout the evaporation process. The geometrical parameters evolve according to the model of an evaporating spherical drop$^{21,25}$ with a linear decrease in the contact angle

$$\theta(t) = \theta_i \left(1 - \frac{t}{t_F}\right) \qquad (12)$$

where $\theta_i$ is the initial contact angle, as well as the apex thickness

$$h(t) = \sqrt{\frac{R^2}{\sin^2 \theta} - R/\tan \theta} \qquad (13)$$

* The evaporation from the PTFE coating occurs mainly with the recession of the triple line. This shrinking dynamic is correctly fitted by

$$R(t) \propto (t_F - t)^\alpha \qquad (14)$$

where exponent $\alpha \approx 1/2$.33 The evaporative power decreases over time with a dynamic similar to that of the radius.

* The evaporation from the SiOC coating occurs according to triple-line dynamics that are intermediate between those of the two previous evaporation processes; the triple line is pinned before receding. It is also observed that in this case the evaporative power evolves according to the same dynamics as those of the triple line; the evaporative power is constant when the triple line is pinned and decreases as soon as the triple line recedes.

Substrates with a PTFE or a PFC coating have similar surface energies, but their roughnesses are different. The increase in the roughness from 284 nm to 1.75 $\mu$m leads to the pinning of the triple line. Moreover, because the roughness amplifies the wetting or nonwetting behavior of a surface,$^{13}$ a decrease in the contact angle is observed. Substrates with PFC and SiOC coatings have the same roughness, 1.75 $\mu$m, but different surface energies, which leads to more significant spreading in the case of the SiOC coating. The evaporation of ethanol drops on these substrates occurs under oleophilic conditions with low contact angles of $\theta < 40^\circ$ (Table 2).

Figure 1b,c shows that the trend in the dimensionless dynamics of the evaporative power is the same as that of the
<figure><img src="image_82.png" /><figcaption>**Figure 2.** Evolution of the evaporative power *P*(*t*) during the evaporation of ethanol drops of different sizes (2.9 µL *V* < 6.0 µL and 1.81 mm *R* < 2.76 mm). (a) Pinned triple-line dynamics (PFC coating). (b) Receding triple-line dynamics (PTFE coating). (c) Experimental evaporative power *P*(*t*) compared to the theoretical model of Hu and Larson<sup>25</sup> *P*<sub>th</sub> vs the dimensionless time *t*/*t*<sub>F</sub> whatever the triple-line dynamics. Atmospheric conditions: *T*<sub>a</sub> = 25.8 ± 0.8°C, *P* = 1 atm.</figcaption></figure>

<figure><img src="image_83.png" /><figcaption>**Figure 3.** Evaporative power *P* vs time *t* for drops of several liquids: methanol (red ●), ethanol (yellow ●), propanol (green ●), toluene (blue ●), and water (purple ●). Respective initial volumes and contact line radius: *V* = (3.08, 3.01, 3.53, 3.60, 4.55) µL and *R* = (1.66, 1.88, 1.85, 1.80, 1.60) mm. (Inset) Experimental evaporative power *P* compared to the theoretical model of Hu and Larson<sup>25</sup> *P*<sub>th</sub> vs the dimensionless time *t*/*t*<sub>F</sub> in pinned-mode evaporation. Atmospheric conditions: *T*<sub>a</sub> = 27.3 ± 0.8 °C, *P* = 1 atm, and *H* = 33%.</figcaption></figure>

<caption>**Figure 4.** Static contact angles of a water droplet of volume *V* = 4.23 µL ± 7% on the sample surfaces at room temperature. Atmospheric conditions: *T*<sub>a</sub> = 27.6 ± 0.1 °C, *P* = 1 atm, and *H* = 24 ± 3%.</caption>

dimensionless radius. Thus, the evaporative power is directly
linked to the triple-line behavior. This observation is consistent
with the experimental observation in the literature that a nearly
constant evaporation rate occurs for the evaporation of pinned
drop with a low initial contact angle.²⁵ This result is also in
agreement with the models of quasi-steady, diffusion-controlled
evaporation for a spherical drop cap presented in section II. The
model of Hu and Larson²⁵ (eq 5) is considered here because of
the simplicity of the function of the contact angle $f_{\text{HL}}(\theta)$, which is
valid to up to 90°. In Figure 2, the evolution of the evaporative
power of drops of various volumes is plotted for evaporation
under the pinned triple-line mode (a) and the receding triple-line
mode (b). Figure 2c compares these experimental dynamics of
evaporation *P* with those described by the theoretical model of
Hu and Larson²⁵ *P*<sub>th</sub>. The theory correctly models the evapora-
tive dynamics for any behavior of the triple line because the ratio
*P*(*t*)/*P*<sub>th</sub>(*t*) = 1.0 ± 12%. In the theoretical expression for the
evaporative power *P*<sub>th</sub>, the diffusion coefficient *D* is calculated

using the formula provided in Perry's Handbook³⁵ whereas geo-
metrical evolutions are extracted from the experimental record-
ings. The good quantitative agreement between the experimental
data and the model demonstrates that the evaporation process is
quasi-steady and diffusion-driven.

The model allows for a better understanding of the experi-
mental curves. Indeed, for drops of the same liquid evaporating
under identical atmospheric conditions, the dynamics of the
evaporative power depend only on the geometrical evolution
(i.e., as a function of the contact angle f(θ) and the wetting radius R).
When the contact angle is below 40°, the function f(θ) is almost
flat; therefore, the temporal dynamics of the evaporative power
depend only on the evolution of the wetting radius P(t) ∝ R(t)
(eq 6). Because all experiments featured contact angles below
40°, the dynamics of the evaporative power are the same as those
of the triple line because the evaporative power is directly
proportional to the wetting radius. The experiments using the
heat-flux meter illustrate the role and importance of the triple line
in sessile drop evaporation. Whereas previous work has shown
that the constant evaporative mass flux is proportional to con-
stant wetting radii in the case of pinned evaporating drops,<sup>17,20,29</sup>
this study reveals the proportionality over time and highlights the
validity of the model regardless of the dynamics of the triple line.

According to the properties of the surface of our substrate, the
time of evaporation varies by about 50% for drops of the same
volume. Drops of identical volume have the same energy, which
can be expressed as follows: E = Pt = kR(t)t with k = L<sub>v</sub>c<sub>v</sub>(1 − H)-
Df(θ) = cste when θ < 40°. Thus, from a wettability point of view,
one observes that the more a drop wets a substrate, the more
important the evaporative power becomes (one can experimen-
tally observe this result at the beginning of evaporation, as shown
in Figure 1a) and consequently the faster the evaporation will be.
<figure><img src="image_84.png" /><figcaption>**Figure 5.** Evaporation of a water drop of volume $V = 4.23~\mu\text{L} \pm 7\%$ on three substrates in pinned mode: SiO$_x$ (red ○), SiOC (green ○), and PFC (blue ○). Atmospheric conditions: $T_a = 27.6 \pm 0.1$ °C, P = 1 atm, and H = 24 ± 3%. (a) Mass evolution. (b) Dimensionless mass evolution. (c) Evolution of the function $H_{\text{PB}}(\theta)$ for the two hydrophobic substrates: SiOC [experimental data (green ○); linear fit (−) of $H_{\text{PB}}(\theta) = at + b$ with $a = 8.544 \times 10^{-4}$, and $b = -1.349$] and PFC [experimental data (green ○); linear fit (−) of $H_{\text{PB}}(\theta) = at + b$ with $a = 1.239 \times 10^{-3}$ and $b = -2.296$]. (d) Dimensionless evaporation rate versus the contact angle. Predictions from the Popov model$^{28}$ (−) and the model of Hu and Larson$^{25}$ (—) are shown.</figcaption></figure>

However, the evaporation time depends both on the wettability (value of the triple line) and on the behavior of the triple line. Indeed, in our example, the evaporation is faster for evaporation under the pinned triple-line mode whereas it is not the more strongly wetting mode of evaporation. The cooling is also more efficient in this situation. Thus, the control of the triple-line behavior is an important parameter with respect to drop evaporation. The more wetting and rough (pinned triple line) a substrate, the more efficient and faster the evaporation.

**IV.b. Evaporation of Drops of Various Liquids with 0° < θ < 90°: Influence of the Wettability and Volatility.** To validate the previous quasi-steady, diffusion-controlled model for a larger range of contact angles and for various volatilities, drop evaporation experiments were conducted using several liquids: methanol, ethanol, propanol, toluene, and water. All of the experiments were performed in the pinned triple-line mode, displaying an apparent contact angle of less than 90°. Thus, drop evaporation with a constant contact area was studied, in the absence of any contact line dynamics. Experiments were mainly performed on PFC coatings, except those featuring water and toluene, for which SiO$_x$ and PTFE coatings were used, respectively. Table 2 provides all of the initial contact angles for these liquid/substrate couples. The contact angle ranged from 26° for ethanol on PFC to 63° for water on SiO$_x$. Table 3 provides information about the surface tension and the volatility of fluids.

Figure 3 shows the evaporative power evolution for drops of various liquids. The intensity and the dynamics of the evaporative power are different, depending on the fluid. The evaporative power evolution is constant over time in some cases (ethanol and

propanol) whereas it decreases over time in other cases (methanol, toluene, and water). The model of Hu and Larson$^{25}$ succeeds in describing these evaporation processes. Indeed, the inset, which shows the evolution of the ratio between the experimental evaporative power and the theoretical model versus the dimensionless time $t/t_F$, reveals good agreement for all liquids because the ratio $P/P_{\text{th}} = 1.0 \pm 10\%$. This result supports the conclusion drawn by Hu and Larson:$^{25}$ “the quasi-steady evaporation for the vapor concentration field should be valid even for rapidly evaporating droplets” because the condition for which the vapor phase has a density much smaller than that of the liquid (($c_v$)/($\rho$) $\ll$ 1) is always true except under nearly supercritical conditions. Thus, the model considers fluid volatility. Moreover, the influence of the contact angle is clearly observed in the figure. Indeed, the evaporative power is constant for ethanol and propanol when the initial apparent contact angle is lower than 40° whereas the evaporative power decreases over time for methanol, toluene, and water when the initial apparent contact angle is greater than 40°. This result is obtained because when evaporation occurs with the pinning of the triple line the dynamics of the evaporative power depend on the contact angle: $P(t) \propto f(\theta)$. When the contact angle $\theta$ is less than 40°, the function of the contact angle $f(\theta)$ is flat. The influence of the contact angle is negligible, and the evaporative power dynamics are constant. When the contact angle $\theta$ is above 40°, the influence of the contact angle is no longer negligible. The evaporative power decreases as a function of the contact angle $f(\theta)$.

**IV.c. Evaporation of Water Drops with 0° < θ < 180°: Influence of the Wettability.** The influence of the wettability on
<figure><img src="image_85.png" /><figcaption>**Figure 6.** Evaporation of a drop of water of volume $V = 4.23 \mu L \pm 7\%$ on three substrates in pinned mode: $SiO_x$ (red ○), $SiOC$ (green ○), and PFC (blue ○). Atmospheric conditions: $T_a = 27.6 \pm 0.1$ °C, $P = 1$ atm, and $H = 24 \pm 3\%$. (a) Evaporative power evolution. (b) Evolution of the liquid-air interface. (c) Dimensionless evolution of the contact angle. (d) Dimensionless evolution of the height.</figcaption></figure>

drop evaporation from a hydrophilic to a nearly superhydrophobic surface has been investigated by considering water drops evaporating from the following three coatings: SiOC, $SiO_x$, and PFC. The three surfaces have the same roughness (1.75 µm); thus, only the surface energy is changed, allowing for the observation of the evaporation of drops with initial contact angles ranging from 63 to 135° (Figure 4).

For all of our substrates, the contact line remains pinned throughout almost the entire experiment; hence, evaporation occurs in the constant-contact-area mode. This behavior results from the high contact angle hysteresis of the surfaces, which is related to the “sticky” ability associated with a low value of the receding contact angle.³⁴ The evolutions of the geometrical dimensions of the drops on the three surfaces are compared in Figure 6c,d. The contact angle and the apex thickness decrease linearly in hydrophilic situations, as described by eqs 12 and 13. In hydrophobic situations, nonlinear evolution is observed. Initially, the contact angle decreases slowly over time. This initial decrease is followed by a more rapid decrease. The evolutions of the drop mass, which were determined from the analysis of the side-view images, are shown in Figure 5a. In the hydrophilic case, the evolution of the mass is also linear. The evolution is well described by a model based on the small contact angle approximation.²¹,²⁵,²⁸ In the hydrophobic situation, the evolution of mass is nonlinear. These observations are in agreement with the predictions of Popov,²⁸ who showed theoretically that the large angles are responsible for the nonlinearity of the mass.

Figure 6a shows that an increase in the contact angle leads to a decrease in the intensity of the evaporative power as well as an

<table><caption>**Table 4.** Details Regarding the Evaporation of a Water Drop of Volume $V = 4.23 \mu L \pm 7\%$ on Three Different Substrates (Pinned Mode)</caption><thead><tr><th>surface coating</th><th>θ<sub>i</sub> (deg)</th><th>R (mm)</th><th>t<sub>F</sub> (s)</th><th>S<sub>c</sub><sup>i</sup> (mm<sup>2</sup>)</th><th>S<sub>g</sub><sup>i</sup> (mm<sup>2</sup>)</th></tr></thead><tbody><tr><td>SiO<sub>x</sub></td><td>63</td><td>1.6</td><td>1319</td><td>8.04</td><td>11.09</td></tr><tr><td>SiOC</td><td>115</td><td>0.97</td><td>1614</td><td>3.96</td><td>10.87</td></tr><tr><td>PFC</td><td>135</td><td>0.74</td><td>2092</td><td>1.73</td><td>10.34</td></tr></tbody></table>

increase in the evaporation time. Indeed, an increase in the evaporation time of 58% is observed with the change in wettability from a contact angle of 63 to 135°. One can notice that a decrease in the contact angle is accompanied by an increase in the wetting radius (Table 4). In natural evaporation, it is generally expected that for a nonwetting drop the evaporative flux at the liquid-air interface is uniform, and as this interface becomes larger, evaporation proceeds more quickly. The results provided are not in agreement with this expectation because a drop of water on the SiOC coating evaporates faster than that on the PFC coating whereas the liquid-air interface is smaller over time. The value of the evaporative power is also more important in the first case. We should also consider that the triple line plays an important role in hydrophobic cases. The only differences between these two experiments are the contact angle, the length of the triple line, and consequently the contact area. Indeed, in the case of PFC, the contact area is smaller and the free space around the triple line is smaller. Thus, this situation can have an undesirable effect on evaporation: the concentration of water increases near the triple
line, leading to a decrease in the evaporation mass flux. Numerical simulations must be performed to validate this explanation.

The hydrophobic data set are first compared with the model proposed by McHale et al.²⁷ (eq 9). As shown in Figure 5c, the curves are well fitted to a straight line, revealing the linear time dependence of the experimental $H_{PB}(\theta)$ values. Some deviations are observed after a certain time, and these correspond to the transition to hydrophilic conditions.

Because this model describes only the evaporation dynamics under hydrophobic conditions ($\theta > 90^{\circ}$), we consider the model proposed by Popov²⁸ to describe the mass flow rate dynamics for any contact angle (eq 10). This model has been recently validated for water droplets on carbon nanofiber substrates, which leads to a superhydrophobic surface.³⁶ The authors observed that the drop mass and contact angle evolutions can be described by one respective universal relationship for all drop sizes and initial contact angles. They validated the model by performing experiments for various volumes of drops with a constant initial contact angle of 150°. The same analysis was performed for a quasi-constant initial volume but various initial contact angles. Figure 5a shows the mass evolution of a water drop on our three coatings. In Figure 5b, all curves for the mass evolution collapse onto a universal curve by plotting the dimensionless mass $\hat{m}$ versus the dimensionless time $\hat{t}$ considering the following relations:

$$ \hat{m} = \frac{m}{\rho R^3} = \pi \frac{\cos^3 \theta - 3 \cos \theta + 2}{3 \sin^3 \theta} \quad (15) $$

$$ \hat{t} = \frac{c_v(1-H)}{\rho} \frac{(t-t_F)}{R^2/D} \quad (16) $$

Good agreement is observed between the experimental data and the Popov model²⁸ depicted by the straight black line.

The dependence of the mass loss rate on the contact angle is also in good agreement with the Popov model prediction.²⁸ As shown in Figure 5d, the scaled experimental data collapse onto a single curve described by the following nondimensional equation

$$ \frac{d\hat{m}}{d\hat{t}} = -\pi f_p(\theta) \quad (17) $$

where the expression for $f_p(\theta)$ is provided by eq 10. For comparison, the model proposed by Hu and Larson²⁵ is also depicted. Their approximation works well for contact angles up to 90°. However, for larger contact angles, the Popov analytical model is required to describe the dynamics. The two presented figures validate the Popov model for any initial contact angle. The good quantitative agreement between the experimental data and the model demonstrates that the evaporation process is quasi-steady and diffusion-driven. It should be noted that eqs 15 and 17 do not depend on the size of the drop but only on the contact angle.

## V. CONCLUSIONS

The influence of the surface properties of various substrates on the evaporation of sessile drops is studied. By varying the roughness and surface energy of the substrate surfaces using various nano- and microcoatings, various dynamics of the triple line and a large range of wettability (from 17 to 135°) are obtained. The influence of these parameters on the kinetics and dynamics of evaporation is studied in order to highlight the separated and coupled roles of the triple line and the wettability of the drops on the sessile drop evaporation process. When the

contact angle is less than 40°, the influence of the contact angle on the dynamics of evaporation is negligible. Thus, the experiments show that the evolution of the evaporative power is linearly related to the evolution of the wetting radius, regardless of the behavior of the triple line. When the contact angle is greater than 40°, the influence of the contact angle is no longer negligible. The influence of the contact angle on the evaporation is investigated for a pinned triple line. All experimental data are in good quantitative agreement with the models of Hu and Larson²⁵ (valid for $0^{\circ} < \theta < 90^{\circ}$) and Popov²⁸ (valid for $0^{\circ} < \theta < 180^{\circ}$), showing that the evaporation process is quasi-steady and driven by diffusion. Furthermore, the models succeed in describing the evaporation of various liquid drops regardless of the drops' volatility. A universal relationship for the evolution of the drop mass, independent of the drop size and initial contact angle, is obtained, which confirms the recent results of Gelderblom et al.³⁶ Ultimately, the experimental results reveal an increase of 50% of the evaporation time between drop evaporation occurring with a pinned and a receding triple line as well as an increase of 58% when the contact angle increases from 63 to 135°. These data clearly show the importance and the effect of surface properties on sessile drop evaporation.

## AUTHOR INFORMATION

### Corresponding Author

*E-mail: benjamin.sobac@polytech.univ-mrs.fr

## ACKNOWLEDGMENT

We thank the French National Research Agency for their financial support through the ANR NanoSurf (ANR-09-BLAN-0093-03) as well as CNES and Carnot STAR. We are also very grateful to J. Gavillet from CEA LITEN, Grenoble, France, for performing nanocoatings.

## REFERENCES

(1) Jing, J.; Reed, J.; Huang, J.; Hu, X.; Clarke, V.; Edington, J.; Housman, D.; Anantharaman, T. S.; Huff, E. J.; Mishra, B.; Porter, B.; Shenker, A.; Wolfson, E.; Hiort, C.; Kantor, R.; Aston, C.; Schwartz, D. C. *Proc. Natl. Acad. Sci. U.S.A.* **1998**, 95, 8046.

(2) Blossey, R. *Nat. Mater.* **2003**, 2, 301.

(3) Leenaars, A. F. M.; Huethorst, J. A. M.; Van Oekel, J. J. *Langmuir* **1990**, 6, 1701.

(4) Bigioni, T. P.; Lin, X.; Nguyen, T. T.; Corwin, E. I.; Witten, T. A.; Jaeger, H. M. *Nat. Mater.* **2006**, 5, 265.

(5) Di Marzo, M.; Tartarini, P.; Liao, Y.; Evans, D.; Baum, H. *Int. J. Heat Mass Transfer* **1993**, 36, 4133.

(6) Kim, D.; Jeong, S.; Park, B. K.; Moon, J. *Appl. Phys. Lett.* **2006**, 89, 264101.

(7) Park, J.; Moon, J. *Langmuir* **2006**, 22, 3506.

(8) Kawase, T.; Sirringhaus, H.; Friend, R. H.; Shimoda, T. *Adv. Mater.* **2001**, 13, 1601.

(9) Norris, D. J.; Arlinghaus, E. G.; Meng, L.; Heiny, R.; Scriven, L. E. *Adv. Mater.* **2004**, 16, 1393.

(10) Yakhno, T. *J. Colloid Interface Sci.* **2008**, 318, 225.

(11) Brutin, D.; Sobac, B.; Loquet, B.; Sampol, J. *J. Fluid Mech.* **2011**, 667, 85.

(12) Sobac, B.; Brutin, D. *Phys. Rev. E* **2011**, 84, 011603.

(13) Quéré, D. *Annu. Rev. Mater. Res.* **2008**, 38, 71.

(14) Bonn, D.; Eggers, J.; Indekeu, J.; Meunier, J.; Rolley, E. *Rev. Mod. Phys.* **2009**, 81, 739.

(15) Tuteja, A.; Choi, W.; Ma, M.; Mabry, J. M.; Mazzella, S. A.; Rutledge, G. C.; McKinley, G. H.; Cohen, R. E. *Science* **2007**, 318, 1618.
(16) Ristenpart, W. D.; Kim, P. G.; Domingues, C.; Wan, P.; Stone, H. A. *Phys. Rev. Lett.* **2007**, *99*, 23402.

(17) David, S.; Sefiane, K.; Tadrist, L. *Colloids Surf., A* **2007**, *298*, 108.

(18) Dunn, G. J.; Wilson, S. K.; Duffy, B. R.; David, S.; Sefiane, K. *J. Fluid Mech.* **2009**, *623*, 329.

(19) Dunn, G. J.; Wilson, S. K.; Duffy, B. R.; Sefiane, K. *Phys. Fluids* **2009**, *21*, 052101.

(20) Grandas, L.; Reynard, C.; Santini, R.; Tadrist, L. *Int. J. Therm. Sci.* **2005**, *44*, 137.

(21) Shin, D. H.; Lee, S. H.; Jung, J.; Yoo, J. Y. *Microelectron. Eng.* **2009**, *86*, 1350.

(22) Maxwell, J. C. Diffusion, Collected Scientific Papers; *Encyclopaedia Britannica*; Encyclopaedia Brittanica, Cambridge, U.K., 1877.

(23) Langmuir, I. *Phys. Rev.* **1918**, *12*, 368.

(24) Picknett, R. G.; Bexon, R. *J. Colloid Interface Sci.* **1976**, *61*, 336.

(25) Hu, H.; Larson, R. G. *J. Phys. Chem. B* **2002**, *6*, 1334.

(26) McHale, G.; Rowan, S. M.; Newton, M. I.; Banerjee, M. K. *J. Phys. Chem. B* **1998**, *102*, 1964.

(27) McHale, G.; Aqil, S.; Shirtcliffe, N. J.; Newton, M. I.; Erbil, H. Y. *Langmuir* **2005**, *21*, 11053.

(28) Popov, Y. O. *Phys. Rev. E* **2005**, *71*, 036613.

(29) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. *Phys. Rev. E* **2000**, *62*, 756.

(30) Kelly-Zion, P. L.; Pursell, C. J.; Vaidya, S.; Batra, J. *Colloids Surf., A* **2011**, *381*, 31.

(31) Phan, H. T.; Caney, N.; Marty, P.; Colasson, S.; Gavillet, J. *Int. J. Heat Mass Transfer* **2009**, *52*, 5459.

(32) Brutin, D.; Sobac, B.; Rigollet, F.; Le Niliot, C. *Exp. Therm. Fluid Sci.* **2011**, *35*, 521.

(33) Shahidzadeh-Bonn, N.; Rafai, S.; Azouni, A.; Bonn, D. *J. Fluid Mech.* **2006**, *549*, 307.

(34) Kulinich, S. A.; Farzaneh, M. *Appl. Surf. Sci.* **2009**, *255*, 4056.

(35) Perry, R. H.; Green, D. W.; Maloney, J. O. *Perry's Chemical Engineers' Handbook*, 7th ed.; McGraw-Hill: New York, 1999; Chapter 2, p 370.

(36) Gelderblom, H.; Marin, A. G.; Nair, H.; Van Houselt, A.; Lefferts, L.; Snoeijer, J. H.; Lohse, D. *Phys. Rev. E* **2011**, *83*, 026306.

<figure><img src="image_86.png" /></figure>

NOTE ADDED AFTER ASAP PUBLICATION

This article was published ASAP on November 22, 2011. Equation 13 has been modified. The correct version was published on December 13, 2011.
# Annexe B

## Thermal effects of the substrate on water droplet evaporation

Benjamin Sobac & David Brutin

*Physical Review E 86*, 021602 (2012).
Thermal effects of the substrate on water droplet evaporation

B. Sobac* and D. Brutin

Laboratoire IUSTI, UMR 7343 CNRS, Aix-Marseille Université, 5 rue Enrico Fermi, 13453 Marseille cedex 13, France

(Received 19 April 2012; published 20 August 2012)

We experimentally investigate the behavior of a pinned water droplet evaporating into air. The influence of the substrate temperature and substrate thermal properties on the evaporation process are studied in both hydrophilic and hydrophobic conditions. Our objective is to understand the effect of thermal mechanisms on the droplet evaporation process. The experimental results are compared with the quasisteady, diffusion-driven evaporation model, which is implemented under the influence of the temperature; the model assumes the isothermia of the droplet at the substrate temperature. The results highlight a favorable correlation between the model and the experimental data at ambient temperatures for most situations considered here. The model works to qualitatively describe the influence of the substrate temperature on the evaporation process. However, with an increase in the substrate temperature, the role of the thermal-linked mechanisms becomes increasingly important; this experiment highlights the need for more accurate models to account for the buoyant convection in vapor transport and the evaporative cooling and heat conduction between the droplet and the substrate. Finally, the experimental data reveal the modification of contact angle evolution as the temperature increases and the crucial role played by the nature of the substrate in the evaporation of a sessile droplet. The influence of the substrate thermal properties on the global evaporation rate is explained by the parallel thermal effusivity of the liquid and solid phases.

DOI: 10.1103/PhysRevE.86.021602

PACS number(s): 68.03.Fg, 68.60.Dv, 68.08.Bc

I. INTRODUCTION

The study of sessile droplet evaporation has been a subject of increasing interest in recent years due to the important role of this process in many new applications. For instance, one can cite DNA mapping [1], ultraclean surfaces [2], self-assembly technologies [3,4], printing and coating technologies [5,6], surface patterning [7], manufacture of new electronic and optical devices [8–10], or medical tests [11–13]. The extensive investigations lead to vital improvements in the understanding and description of the phenomenon [14–17]. Particular emphasis was placed on the problems of wettability during evaporation and the interaction between the two phenomena. However, sessile droplet evaporation is currently a challenging problem of soft matter physics because of the complexity of the associated fluid dynamics, the physical chemistry of the substrate, and the heat and mass transfer.

Numerous studies have been undertaken to understand the influence of substrate surface properties on the wettability of a sessile droplet [18,19]. The surface-free energy and surface roughness are known to be the two key parameters governing the contact angle of the droplet. The awareness of the complementary roles of these two surface parameters has led to the development of textured surfaces in order to produce superhydrophilic [20], superhydrophobic [21–23], or superoleophobic surfaces [24–26].

Related to these surface properties, several evaporation modes have been explored: the constant-angle mode [14,27], in which the contact area of the droplet on the surface vanishes; the constant contact-area mode [14–16,28,29], in which the contact angle vanishes; and the combination of both modes [14,30,31]. A wide range of wettability has also been investigated, from hydrophilic situations [14–16,27,30,32] to

hydrophobic and, more recently, superhydrophobic situations [17,32–34]. According to the surface properties, the dynamics and kinetics of evaporation are totally different [32,35,36] because the overall evaporation rate is directly related to the contact angle and the triple-line behavior (see the next section about theory and model). The separated and coupled roles of the triple line and the wettability on the sessile droplet evaporation process have been investigated [32]. All of these investigations considered the spontaneous natural evaporation of a droplet and demonstrated good agreement with quasistatic, diffusion-driven evaporation models, suggesting that thermal effects are negligible. Recently, universal relations for the time evolution of the droplet mass and the time evolution of the contact angle, independent of the droplet size and initial contact angle in the situation of pinned droplet evaporation, have been deduced from the Popov model [17] and correlate well with experimental data [32,33].

Paradoxically, only a few investigations were concerned with the thermal effects of the substrate on the droplet evaporation process. Some authors studied the influence of the thermal conductivity of the substrate on natural droplet evaporation. Ristenpart et al. [37] showed that the thermal conductivity is an important parameter that needs to be considered because the ratio of the liquid and solid conductivities can modify the internal flow direction. David et al. [38] experimentally observed that an increase in the conductivity of the substrate causes an increase in the overall evaporation rate. This result contradicts the conclusion that thermal effects do not play a role during natural evaporation. Thus, new mathematical and numerical models to generalize the theoretical model of quasisteady, diffusion-limited evaporation were developed by taking into account the thermal effects associated with evaporative cooling [39] and related to the thermal resistance of the substrate [40]. These improvements are notably due to the attention to the variation of the saturation concentration with temperature, hence coupling the problems for the vapor

<footnote>*benjamin.sobac@polytech.univ-mrs.fr</footnote>
concentration in the atmosphere and the temperature of the liquid and substrate. Under some conditions, the heat diffusion in the substrate can become the limiting parameter of the evaporation rate [41].

Other studies address the effect of the surface temperature on the droplet evaporation process. Experimentally, Crafton et al. [42] and Grandas et al. [35] highlighted that the overall evaporation rate still evolves linearly with the wetting radius. Numerically, these studies focused on the evaporation of pinned water droplets in a wetting situation. Girard et al. [43] investigated the influence of the heating temperature and the relative humidity on the evaporation process and proposed an empirical law for droplet evaporation time as a function of these parameters assuming a perfectly conductive substrate. Saada et al. [44] developed a convection-diffusion model to analyze the effect of buoyant convection in the surrounding air on the heat and mass transfer phenomena during evaporation. These authors observed that the diffusion model always underpredicts the overall evaporation rate and that the deviation increases with the temperature of the substrate. Whereas the previous studies considered a droplet evaporating into air with an imposed temperature at the substrate surface as boundary condition, Sodtke et al. [45], for their part, were interested in the situation of a thin drop evaporating into pure vapor on a surface with an imposed heat flux density. A theoretical model of evaporation taking account of the heat conduction in the liquid and solid phases were developed by the authors and reveals a good agreement with their experimental data.

The literature reveals that the previously developed theory, including the thermal effects, is almost exclusively based on the experimental results of David et al. [38] which is limited to experiment at ambient temperature. In this paper, we propose to revisit the configuration of David et al. [38] and extend the study to the situation of a heated substrate for a wide range of substrate temperatures. The originality of this experiment is to decouple the effects of wetting properties and thermal properties of the substrate. Thus, whereas previously we investigated the role of wetting properties on evaporation by changing the surface energy and the roughness while maintaining the thermal properties constant thanks to nanoscale coatings on the substrate surface [32], here we investigate the influence of the thermal properties of the substrate while keeping the wetting properties the same by using the coating technic. The bulk of the studies presented in the review of literature are limited to contact angles below 90°C, thus we carry out experiments on a large range of contact angles allowing us to investigate hydrophilic and hydrophobic situations. Experimental data are compared to the quasisteady diffusion-driven evaporation model assuming the isothermia of the drop at the substrate temperature. This comparison permits us to highlight several thermal mechanisms linked to evaporation and their respective contributions in regard of pure mass diffusion mechanism. The range of validity of the classical evaporation model is also discussed. These results are important for a complete modeling of the phenomena.

After a brief overview of the theoretical macroscopic models developed for sessile droplet evaporation in Sec. II and a presentation of the experimental setup and protocol in Sec. III, the results are presented and discussed in Sec. IV. The influence of substrate temperature on geometrical evolution,

evaporation rate, and time of evaporation are investigated and compared with theoretical models and numerical simulations in the case of a hydrophilic situation (Sec. IV A1) as well as in the case of a hydrophobic situation (Sec. IV A2). The influence of temperature on the contact angle evolution is discussed in Sec. IV A3. Finally, the last section concerns the influence of the thermal properties of the substrate in the situation of forced evaporation (Sec. IV B).

## II. THEORY: MODELS OF DROP EVAPORATION

For a small drop with a contact radius less than the capillary length, the fluid adopts a spherical cap shape because gravitational forces are negligible. Hence, the droplet volume $V$ can be expressed as a function of the wetting radius $R$ and the contact angle $\theta$ as follows:

$$V(R, \theta) = \frac{\pi R^3 (1 - \cos\theta)^2 (2 + \cos\theta)}{3 \sin^3\theta} \quad (1)$$

The natural evaporation of a droplet is generally assumed to be controlled by the diffusion of vapor molecules in the gas phase. The evaporative flux from a droplet surface of an evaporating droplet is modeled based on the diffusion of molecules from the liquid-gas interface of the droplet to the surrounding gas phase. Vapor transport by free convection, induced by the density difference between dry and humid air, is assumed to be negligible compared to diffusive transport [46]. The influence of evaporative cooling of the droplet on the evaporative rate is also neglected [39]. Hence, vapor transport occurs mainly by diffusion of water vapor and is characterized by a diffusion time $t_D = R^2/D$, with $D$ as the diffusion coefficient. The diffusion time for water vapor in air is on the order of $10^{-2}$ s. Evaporation occurs in quasisteady fashion because the diffusion time is smaller than the evaporation time $t_D/t_F \approx 10^{-5}$, which means that the vapor concentration adjusts rapidly compared with the time required for evaporation. The quasisteady, diffusion-limited evaporation is then governed by the steady Laplace equation. The concentration field around the droplet is given by

$$\nabla^2 c = 0. \quad (2)$$

At the interface between the liquid and the vapor, the vapor concentration $c$ is assumed equal to the saturation value $c_v$. Far above the droplet, the vapor concentration approaches an ambient value of $c_\infty = Hc_v$ where $H$ is the relative humidity of the ambient air. The difference in vapor concentration $\Delta c = c_v(1 - H)$ drives the evaporation into the air. The diffusive flux is given by Fick's law: $J = -D\nabla c$. This problem has been solved in the limit of small contact angle ($\theta < 90^\circ$) by Deegan [15] and Hu and Larson [16]. An analytical solution for droplets of arbitrary contact angles has been obtained by Popov [17] using the analytical solution to the equivalent problem of finding the electric potential around a charged lens-shaped conductor [47]. The rate of mass loss from a droplet of arbitrary contact angle is given by

$$-\frac{dm}{dt} = \int_{0}^{R} J(r) \sqrt{1 + (\partial_{r}h)^{2}} 2\pi r dr \\ = \pi R D(c_{s} - c_{\infty}) f(\theta), \quad (3)$$
where *m* is the droplet mass, *J* is the diffusive outflux from the droplet surface, *h(r,t)* is the droplet height, *t* is the time, *r* is the radial coordinate, and

$$f(\theta) = \frac{\sin\theta}{1 + \cos\theta} + 4 \int_{0}^{\infty} \frac{1 + \cosh 2\theta\tau}{\sinh 2\pi\tau} \tanh[(\pi - \theta)\tau] d\tau. \quad (4)$$

For a pinned droplet, the volume is a function of $\theta$ by the geometric relation presented in Eq. (1). The contact angle evolution can be determined from an ordinary differential equation obtained from Eqs. (1) and (3):

$$\frac{d\theta}{dt} = -\frac{D(c_s - c_\infty)}{\rho R^2} (1 + \cos\theta)^2 f(\theta). \quad (5)$$

Once the evolution of $\theta$ is obtained by numerical integration, $V(\theta)$ and $dm/dt$ can be derived. Recently, Gelderblom et al. [33], introducing the nondimensional mass $\hat{m} = \frac{m}{\rho R^3}$ and time $\hat{t} = \frac{c_s - c_\infty}{\rho} \frac{t}{R^2/D}$, showed that mass and contact angle evolutions can be described by the respective universal law for all droplet sizes [33] and initial contact angles [32].

In the situation of forced evaporation, the complete analytical resolution of the equations governing a heated droplet has remained, until now, an open problem. However, the previous model can be extended to include the substrate temperature $T_s$. A full analytic solution was obtained for a droplet evaporating in the specific case of an isothermal problem. Using previously developed similar assumptions, it is possible to show that the vapor concentration around the droplet is still governed by the Laplace equation. Indeed, at a temperature of 75°C, the ratio of diffusion time to evaporation time is on the order of $10^{-4}$. Assuming the interface is isothermal at the substrate temperature $T_s$, as the literature has revealed the largest temperature gradients in such systems are smaller than a few degrees [48], the evaporation rate is driven by the difference of vapor concentration $\Delta c = c_v(T_s) - Hc_v(T_a)$ where $T_a$ is the ambient temperature far above the droplet. Hence, Eqs. (3) and (5) are generalized by [43]

$$-\frac{dm}{dt} = \pi D R [c(T_s) - Hc(T_a)] f(\theta), \quad (6)$$

$$\frac{d\theta}{dt} = -\frac{D[c(T_s) - Hc(T_a)]}{\rho R^2} (1 + \cos\theta)^2 f(\theta). \quad (7)$$

In the case of a pinned droplet, the evaporation time is expressed by

$$t_F = \frac{\rho R^2}{D \Delta c} \int_{0}^{\theta_0} \frac{d\theta}{(1 + \cos\theta)^2 f(\theta)}, \quad (8)$$

where $\theta_0$ is the initial contact angle.

For small contact angle $\theta \ll 1$ rad, we can show that the contact angle and the volume decrease linearly with time [17],

$$\theta = \theta_0 \left(1 - \frac{t}{t_F}\right), \quad (9)$$

$$V = \frac{\pi R^3 \theta_0}{4} \left(1 - \frac{t}{t_F}\right), \quad (10)$$

with the evaporation time defined by

$$t_F = \frac{\pi \rho R^2 \theta_0}{16 D \Delta c}. \quad (11)$$

However, as soon as the initial contact angle is above 40° [16,32], the evaporation rate decreases nonlinearly over time due to the evolution of the function of the contact angle $f(\theta)$. Then, the global evaporation rate $\frac{dm}{dt}|_g$, which is equivalent to the medium evaporation rate during evaporation, is defined by

$$\left. \frac{dm}{dt} \right|_g = \frac{1}{t_F} \int_0^{t_F} \frac{dm}{dt} dt = - \frac{m_0}{t_F}. \quad (12)$$

### III. MATERIALS AND METHODS

The experiment essentially involves the evaporation of a water droplet onto a substrate. The droplet is created using a micropipette to control the volume of the droplet, which is gently laid down on the substrate. The droplet evaporates into air inside an experimental cell in PMMA (65 × 135 × 40 mm³) to avoid any perturbations from potential external flow. The surrounding air temperature $T_a$, pressure $P_a$, and humidity $H$ are measured using a weather station (Lufft opus). A schematic diagram illustrating the experimental setup is provided in Fig. 1.

The substrate is an aluminum cylinder (diameter = 10 mm, height = 10 mm) at the top of which is applied a coating of nanoscale thickness by plasma-enhanced chemical vapor

<figure><img src="image_87.png" /><figcaption>FIG. 1. (Color online) Schematic diagram illustrating the experimental setup, including a computer with drop shape analysis (DSA) software (1), a cold light source (2), a computer controlled syringe (3), a charge-coupled device (CCD) camera fitted with a macrolens (4), and an experimental cell (5). As observed in the inset, the experimental cell includes a heated aluminum block regulated in temperature (6), the substrate (7), coated with a nanoscale layer (8), and the water droplet (9).</figcaption></figure>
<table><caption>TABLE I. Summary of nanocoating processes used in this study.</caption><thead><tr><th>Material</th><th>Notation</th><th>Thickness<sup>a</sup></th><th>Roughness<sup>b</sup></th><th>Contact angle</th></tr></thead><tbody><tr><td>Low-carbon PDMS</td><td>SiO<sub>x</sub></td><td>4 nm</td><td>1.75 µm</td><td>67 ± 5°</td></tr><tr><td>Perfluorocarbon</td><td>PFC</td><td>14 nm</td><td>1.75 µm</td><td>133 ± 2°</td></tr></tbody></table>

<sup>a</sup>Thickness obtained using profilometry.

<sup>b</sup>Root mean square roughness obtained using atomic force microscopy.

deposition (PECVD). The two coatings used in this study are PFC and SiOₓ. The PECVD technique, the preparation, and the chemical composition of these coatings were previously presented by Phan *et al.* [49]. These two surfaces were characterized with respect to morphology using an atomic force microscope (Veeco Explorer) and with respect to wettability using a contact angle measurement system (Krusss DSA30). All data are summarized in Tables I and II. These nanocoatings allow for changes in the surface energy of the substrate while maintaining the surface roughness and the thermal properties of the substrate [32]. Estimates of thermal resistances reveal a negligible influence of these coatings on the thermal properties of the substrate. All evaporation on these surfaces was performed with the pinning of the triple line throughout almost the entire duration of the evaporation process due to the relative importance of the surface roughness. The SiOₓ coating allows for the study of evaporation in a hydrophilic situation with an initial contact angle θ₀ = 67 ± 5° whereas the PFC allows for the study of evaporation in a hydrophobic situation θ₀ = 133 ± 2° (see Fig. 2).

To investigate the influence of the substrate temperature Tₛ, the substrate can be heated from the base with a heated aluminum block (40 × 40 × 20 mm³) equipped with two heating cartridges and a platinum probe (Pt 100) on the surface. This latter is fitted with a PID temperature regulator (Omron E5GN) to impose a controlled temperature. The contact is performed via a fine layer of thermal grease (compound silicone Jelt Cm). A range of temperatures from the ambient temperature to 75°C were investigated.

To investigate the influence of thermal properties of the substrate, the cylinder can be exchanged for cylinders made of other materials including copper, brass, bronze, and polyoxymethylene (POM). All these materials were chosen for their widely different thermal conductivities from insulating (POM) to highly conductive substrates (copper). The relevant physical properties of the substrates used are listed in Table II.

The experimental cell was inserted into the center of a Kruss DSA30 drop shape analyzer. This measurement system allows one to optically obtain the evolution of the droplet base radius R, the contact angle θ and the volume V, due to a CCD camera positioned laterally and a backlight. The total evaporation rate is calculated from the volume evolution. The drops have a spherical cap shape because the base radius is always inferior to the capillary length Lₑ = √(σ/ρg) = 2.7 mm where ρ is the fluid density, g is the gravitational constant, and σ is the surface tension.

To ensure the reproducibility of the results, new coatings were used for each experiment, and the surrounding gas phase was renewed at the same time. All of the measurements were performed at least twice to verify the reproducibility of the results. The accuracy of the measurements are 0.5 °C with respect to temperature, 0.1° with respect to the contact angle, 3% with respect to the radius, and 10% with respect to the volume, mass, and global evaporation rate.

## IV. RESULTS AND DISCUSSION

### A. Influence of the heating temperature of the substrate

#### 1. Hydrophilic situation

In this section, we propose to study the evaporation of a water droplet on an aluminum substrate coated with SiOₓ in order to look at the effect of the heating temperature of the substrate Tₛ on the drop evaporation in a hydrophilic situation (θ₀ = 68 ± 1°). For all experiments, the triple line is observed to be pinned throughout almost the entire duration of the evaporation process.

The experimental evolutions of the contact angle and the volume are provided in Figs. 3(a) and 3(b), respectively, for substrate temperatures Tₛ ranging from ambient temperature, 25 °C, up to 75 °C (colored markers). Because the initial contact angle is above 40°, the contact angle decreases nonlinearly with time. Consequently, the volume evolution is also nonlinear due to the influence of the contact angle function [16,32]. These experimental curves appear to be well fit by a quadratic polynomial function [43] (black dash lines). The experimental evaporation time tF is determined from these extrapolations and is defined as the value of time when the contact angles reach zero degrees. Hence, Fig. 3(c) illustrates the dependence of the evaporation time on the substrate temperature. The variation of the evaporation time tF is fitted by a power law tF = aTₛ⁻ᵇ as proposed by Girard *et al.* [43] in a numerical investigation. Values of a = 1.4089 × 10⁶ and b = 2.8077 are obtained with a correlation

<table><caption>TABLE II. Physical properties of liquid and substrate materials used at Tₑ = 20°C.</caption><thead><tr><th>Parameters</th><th>Symbols</th><th>Units</th><th>Water</th><th>Air</th><th>Copper</th><th>Aluminum</th><th>Brass</th><th>Bronze</th><th>POM</th></tr></thead><tbody><tr><td>Density</td><td>ρ</td><td>kg m<sup>-3</sup></td><td>998</td><td>1.2</td><td>8920</td><td>2700</td><td>8450</td><td>8770</td><td>1420</td></tr><tr><td>Specific heat capacity</td><td>Cp</td><td>J kg<sup>-1</sup> K<sup>-1</sup></td><td>4180</td><td>1004</td><td>380</td><td>900</td><td>376</td><td>377</td><td>1500</td></tr><tr><td>Thermal conductivity</td><td>λ</td><td>W m<sup>-1</sup> K<sup>-1</sup></td><td>0.61</td><td>0.02</td><td>401</td><td>237</td><td>117</td><td>50</td><td>0.31</td></tr><tr><td>Thermal diffusivity</td><td>α</td><td>(×10<sup>-7</sup>) m<sup>2</sup> s<sup>-1</sup></td><td>1.5</td><td>217</td><td>1183</td><td>975</td><td>368</td><td>151</td><td>1.5</td></tr><tr><td>Time of heat diffusion</td><td>t<sub>th</sub></td><td>s</td><td>7</td><td></td><td>1</td><td>1</td><td>3</td><td>7</td><td>687</td></tr><tr><td>Thermal effusivity</td><td>β</td><td>(×10<sup>3</sup>) J K<sup>-1</sup> m<sup>-2</sup> s<sup>-1/2</sup></td><td>1.6</td><td>6</td><td>36.8</td><td>24.0</td><td>19.94</td><td>12.9</td><td>0.8</td></tr><tr><td>Parallel thermal effusivity</td><td>β<sub>||</sub></td><td>J K<sup>-1</sup> m<sup>-2</sup> s<sup>-1/2</sup></td><td></td><td></td><td>1525</td><td>1491</td><td>1470</td><td>1414</td><td>538</td></tr></tbody></table>
<figure><img src="image_88.png" /><figcaption>FIG. 2. (Color online) (a) Side view of a 4.2-µL water droplet on the sample surfaces at the initial moment. (b) Atomic force microscopy image of the surface coated with PFC (root mean squared roughness of 1.75 µm).</figcaption></figure>

coefficient close to 1. Consequently, the global evaporation rate $|dm/dt|_g$ increases as the temperature $|dm/dt|_g = \alpha T_s^\beta$ with $\alpha = 1.3315 \times 10^{-13}$, $\beta = 2.9893$ and a correlation coefficient close to 1.

All these experimental data are compared with the quasisteady, diffusion-driven evaporation model, assuming an isothermal droplet at the temperature of the substrate. Indeed, as previously discussed in Sec. II, the comparison of the diffusion time with the evaporation time reveals that the problem is still quasisteady for the entire range of temperatures considered here. Moreover, numerical simulations in similar configurations reveal that, for water droplets evaporating in a substrate at $T_s = 60$ °C at an angle of $\theta = 80^\circ$, even if a small temperature gradient exists at the interface, the temperature of the droplet is close to the temperature of the substrate [43]. The assumption is even more realistic because for the range of contact angles considered, most of the evaporation is performed at the liquid-air interface close to the triple line where the temperature is close to the substrate temperature. The theoretical evolutions of the contact angle and the volume are plotted in the colored dashed lines in Figs. 3(a) and 3(b), with colors corresponding to the substrate temperature.

As previously mentioned in the literature [32,33], this model is a good predictor at ambient temperature: a good agreement on the evolutions of contact angle and volume is observed [see Figs. 3(a) and 3(b)], and a relative deviation of 3.2% is calculated for the evaporation time where

$\epsilon_{t_F} = \frac{t_{F\text{expt.}} - t_{F\text{ih}}}{t_{F\text{expt.}}} \quad \text{and the global evaporation rate where } \epsilon_{\dot{m}_g} = \frac{|\dot{m}_g|_{\text{expt.}} - |\dot{m}_g|_{\text{ih}}}{|\dot{m}_g|_{\text{expt.}}}$.

The model succeeds in describing qualitatively the influence of substrate temperature on the evaporation process. Indeed, as illustrated in Fig. 3, a good prediction of the dynamics of the contact angle, volume, time of evaporation, and global evaporation rate are achieved by the model, which suggests that the model manages to take the global physics of the problem into account. However, one observes a deviation between the model and the experiments, which becomes increasingly important as the temperature increases. Hence, Fig. 3(d) shows that the model always underestimates the evaporation rate and, with an increase in the temperature, the relative deviation increases until reaching 30% at $T_s = 75$ °C. Consequently, the model overestimates the evaporation time and the relative deviation increases with the temperature until reaching 43% at $T_s = 75$ °C. This global underestimation of the global evaporation rate with an increase in the relative deviation is attributed to the effect of buoyant convection in the surrounding air. Indeed, the buoyant convection contribution increases with the temperature and this effect is not included in the model. Recently, numerical investigations revealed the importance of convection in the vapor phase on the evaporation rate. Saada et al. [44] showed that the diffusion model underestimates the overall evaporation rate by 8.5% for a wall temperature equal to an ambient temperature of 25 °C and by 27.3% for a wall temperature of 70 °C. The same order of magnitude of the relative deviation is observed between the two studies. This section highlights the contribution of convective transport in the vapor phase during sessile drop evaporation.

## 2. Hydrophobic situation

In this section, the influence of the heating temperature of the substrate during the evaporation of a pinned water droplet is investigated in a hydrophobic situation. The substrate is coated with PFC which results in an initial contact angle of $\theta_0 = 133 \pm 2^\circ$.

The results will be presented in the same way as the previous section. The experimental evolutions of the contact angle and the volume are respectively provided in Figs. 4(a) and 4(c), respectively, for the same range of substrate temperature (from the ambient temperature, 25–75 °C) (colored markers). Because the initial contact angle is more important in this case compared with the previous case, the nonlinearity of the contact angle evolutions increases [32]. Consequently, the nonlinearity of the volume evolutions becomes increasingly important due to the influence of the function of the contact angle [16,32]. Hence, these experimental curves now appear correctly fitted by a polynomial functions of higher degree: degree 6 for the contact angle evolutions and degree 4 for the volume evolutions (black dashed lines). The experimental evaporation time $t_F$ is extracted from these extrapolations and is defined as the time when the contact angles reach 0°. The variation of the evaporation time $t_F$ as a function of the temperature is plotted in Fig. 4(c) and is fitted by a power law $t_F = aT_s^{-b}$, with $a = 2.6228 \times 10^6$, $b = 2.2481$, and a correlation coefficient close to 1. The global evaporation rate
<figure><img src="image_89.png" /><figcaption>FIG. 3. (Color online) Evaporation of a pinned water droplet on an aluminum substrate coated with SiOₓ as a function of the substrate temperature Tₛ. (a) Contact angle θ variation versus time t. Inset: Zoom. (b) Volume V variation versus time t. Inset: Zoom. (c) Variation of the evaporation time t<sub>F</sub> versus the substrate temperature T<sub>s</sub>. Inset: Relative deviation of the evaporation time ϵ<sub>t<sub>F</sub></sub> versus the substrate temperature T<sub>s</sub>. (d) Variation of the global evaporation rate |dm/dt|<sub>g</sub> versus the substrate temperature T<sub>s</sub>. Inset: Relative deviation of the global evaporation rate ϵ<sub>ṁ<sub>g</sub></sub> versus the substrate temperature T<sub>s</sub>. Atmospheric conditions: T<sub>a</sub> = 25.4 °C, P = 101.5 kPa, H = 47.5 ± 1%. Drop dimensions: V = 3.64 µL ± 2%, θᵢ = 68 ± 1° and R = 1.44 ± 0.03 mm. Complementary information about the equations of fits: in panel (c), (-) t<sub>F</sub> = 1.4089 × 10<sup>6</sup> T<sub>s</sub><sup>-2.8077</sup>, (- -) t<sub>F</sub> = 7.0624 × 10<sup>6</sup> T<sub>s</sub><sup>-2.5711</sup>; in panel (d), (-) |ṁ<sub>g</sub>| = 1.3315 × 10<sup>-13</sup> T<sub>s</sub><sup>2.9893</sup>, (- -) |ṁ<sub>g</sub>| = 1.9157 × 10<sup>-13</sup> T<sub>s</sub><sup>2.8254</sup>.</figcaption></figure>

|dm/dt|_g also increases with the same nonlinear evolution with an increase of the temperature |dm/dt|_g = α T_s^β, with α = 2.2709 × 10⁻¹³, β = 2.7438, and a correlation coefficient close to 1.

The experimental data are compared with the same qua-
sisteady, diffusion-driven evaporation model. The theoretical
evolutions of the contact angle and the volume are plotted
in colored dashed lines in Figs. 4(a) and 4(b), with colors
corresponding to the substrate temperature.

At ambient temperature, a good agreement is observed
between the experimental data and the model, as shown in
Figs. 4(a) and 4(b) with respect to the evolutions of the contact
angle and volume. Moreover, a relative deviation of 3.8% is
calculated for the evaporation time and the global evaporation
rate.

The model succeeds in qualitatively describing the influ-
ence of the substrate temperature on the evaporation process.
Indeed, as illustrated in Fig. 4, a good prediction of the
dynamics of the contact angle, volume, time of evaporation,
and global evaporation rate are exhibited by the model.

However, one observes a deviation between the model and
the experiments, which becomes increasingly important as
the temperature increases. On the whole, in the hydrophobic
situation, the model overestimates the evaporation rate and the
relative deviation reaches a value of 29.7% at T<sub>s</sub> = 75 °C.

Previously, to completely describe the evaporation process,
the hydrophilic situation revealed the necessity of accounting
for the buoyant convection of transport in the vapor phase
because the diffusion-driven evaporation model globally un-
derestimates the global evaporation rate. In the hydrophobic
situation, the convective transport is still missing in the model,
and yet the model overestimates the global evaporation rate.
Hence, the hydrophobic situation highlights another mecha-
nism acting during droplet evaporation that is not included in
the model. Indeed, the assumption of droplet isothermia is no
longer justifiable in this situation as it was in the hydrophilic
one. The thickness of the drop increases and the temperature
gradient at the interface is no longer negligible. Considering
the local evaporative flux at the liquid-air interface, the
maximum of the evaporative flux occurs along the interface
<figure><img src="image_90.png" /><figcaption>FIG. 4. (Color online) Evaporation of a pinned drop of water on an aluminum substrate coated with PFC as a function of the substrate temperature $T_s$. (a) Contact angle $\theta$ variation versus time $t$. Inset: Zoom. (b) Volume $V$ variation versus time $t$. Inset: Zoom. (c) Variation of the evaporation time $t_F$ versus the substrate temperature $T_s$. Inset: Relative deviation of the evaporation time $\epsilon_{t_F}$ versus the substrate temperature $T_s$. (d) Variation of the global evaporation rate $|dm/dt|_g$ versus the substrate temperature $T_s$. Inset: Relative deviation of the global evaporation rate $\epsilon_{\dot{m}_g}$ versus the substrate temperature $T_s$. Atmospheric conditions: $T_a = 23.3 \pm 0.6$ °C, $P = 101.8 \pm 0.1$ kPa, $H = 36.4 \pm 2\%$. Drop dimensions: $V = 3.79 \ \mu L \pm 3\%$, $\theta_i = 133 \pm 2^\circ$, and $R = 0.72 \pm 0.04$ mm. Complementary information about the equations of fits: the dashed lines are sixth-degree polynomial fits in panel (a) and quadratic polynomial fits in the panel (b); in panel (c), $(-) t_F = 2.6228 \times 10^6 T_s^{-2.2481}$, $(-) t_F = 3.1285 \times 10^6 T_s^{-2.3124}$ and in panel (d), $(-) |\dot{m}_g| = 2.2709 \times 10^{-13} T_s^{2.7438}$, $(-) |\dot{m}_g| = 7.428 \times 10^{-13} T_s^{2.4086}$.</figcaption></figure>

where the temperature is below the substrate temperature. Hence, the prediction, calculated for an interfacial temperature equal to the substrate temperature, overestimates the real contribution of diffusion transport during evaporation. The deviation due to this assumption appears to be even more important than the contribution of convective transport in the vapor phase. This section clearly reveals the importance of the evaporative cooling effect on the evaporation rate.

### 3. Contact angle evolutions

Figure 5 illustrates the normalized evolutions of the contact angle as a function of the temperature of the substrate for the two situations previously presented, i.e., the hydrophilic case (a) and the hydrophobic case (b). The experimental data (colored lines) are compared with the dimensionless evolution of the contact angle predicted by the quasisteady, diffusion-driven evaporation model (black dashed line). The results show good agreement between the model and the experiments at ambient temperature. However, one observes

deviation of the curves with an increase in the temperature of the substrate: the higher the temperature of the substrate, the more deviate the curves. An increase in the deviation is more notable in the hydrophobic case compared with the hydrophilic case. A deviation on the order of 10% is reached between the contact angle at ambient temperature and at 75 °C at $t/t_F = 0.8$ in the hydrophobic situation. Hence, the results reveal the influence of the thermal-linked mechanisms to the contact angle evolution. This influence is not accounted for in the model and could be due to the temperature gradient at the free surface induced by evaporative cooling, which leads to a variation of the saturated vapor concentration along the interface: the temperature gradient is a function of the wettability. This effect may affect the contact angle dynamic as well as the buoyant convection.

### B. Influence of the substrate nature

To investigate the influence of the thermal properties of the substrate on droplet evaporation, experiments in which
<figure><img src="image_91.png" /><figcaption>FIG. 5. (Color online) Normalized contact angle $\theta/\theta_0$ versus the normalized time $t/t_F$ as a function of the temperature of the substrate for the hydrophilic case (a) and the hydrophobic case (b). Insets: zooms. For complementary information see captions in Fig. 3 for the hydrophilic case and in Fig. 4 for the hydrophobic case.</figcaption></figure>

the nature of the substrate was changed were conducted. All substrates were coated with SiO$_x$ to perform these experiments with the same physical chemistry of the surface. All evaporation considered here occurs in the hydrophilic situation $\theta_0 = 67.5 \pm 5^\circ$ with the pinning of the triple line. Figure 6 presents the global evaporation rate as a function of the substrate temperature for various substrates including copper, aluminum, brass, bronze, and POM. The experimental data are compared with the quasisteady, diffusion-driven evaporation model calculated for the median value of each parameter.

The global evaporation rate is found to be, respectively, higher for substrates with higher thermal conductivity. It is worth noting however that all curves for metallic substrates (copper, aluminum, brass, bronze) are grouped as if there is virtually no difference despite an order of magnitude difference in thermal conductivity. This observation was already performed by David et al. [38] about evaporating droplets at ambient temperatures.

When the substrate is at ambient temperature, the experiences reveal a median global evaporation rate of $|\frac{dm}{dt}|_g = 2.18 \pm 0.14 \times 10^{-9}$ kg s$^{-1}$ for metallic substrates. The deviation is below the uncertainty measurement of the global evaporation rate. A difference of 14% on the median value of the global evaporation rate is noticed between metallic and polymeric substrates. A difference of the same order of magnitude was reported in the investigation of David et al. [38]. The authors explained the decrease in the evaporation rate by an important cooling effect when the substrate is thermally insulating.

<figure><img src="image_92.png" /><figcaption>FIG. 6. (Color online) Global evaporation rate $|\frac{dm}{dt}|_g$ of a pinned water droplet versus the temperature of the substrate $T_s$ for various substrate materials. Atmospheric conditions: $T_a = 25 \pm 1$ °C, $P = 1$ atm $\pm 1\%$, $H = 47.5 \pm 5\%$. Drop dimensions: $V = 3.75 \pm 0.55 \mu$L, $\theta_0 = 67.5 \pm 5^\circ$ and $R = 1.44 \text{ mm} \pm 10\%$. Complementary information about the equations of fits: (brown - -) power law fit of copper points $|\dot{m}_g| = 1.0974 \times 10^{-13} T_s^{3.0355}$, (purple - -) power law fit of bronze points $|\dot{m}_g| = 1.4312 \times 10^{-13} T_s^{2.965}$, (blue - -) power law fit of POM points $|\dot{m}_g| = 4.4651 \times 10^{-13} T_s^{2.6267}$, (black - -) power law fit of the diffusion model $|\dot{m}_g| = 1.9493 \times 10^{-13} T_s^{2.8151}$.</figcaption></figure>

The difference in the global evaporation rate increases with the increase in the substrate temperature and reaches 53.9% at $T_s = 75$ °C between a droplet evaporating onto a copper or a POM substrate. The increase in the evaporation rate due to the increase of the substrate temperature amplifies the effects of the thermal properties influence.

These results highlight the crucial role of the thermal properties of the substrate. Indeed, due to evaporation process (latent heat), energy is consumed by the droplet and a part of this energy is supplied from the surrounding environment. Considering the thermal conductivities of the air and the substrate materials (see Table II), the energy should be mainly brought by heat conduction from the substrate to the droplet. Instead of considering these results through the thermal conductivity of the substrate (as it is usually done in the literature), it is preferable to use the concept of thermal effusivity $\beta = \sqrt{\lambda\rho C_p}$. The thermal effusivity characterizes the ability of a material to change temperature when it absorbs or supplies heat energy. It accounts for variation of the surface temperature due to variation in heat flux density at its surface. The thermal effusivity is a broader concept because it includes the change of temperature by heat storage and heat conduction; hence, it also allows us to take unsteady process into account. Considering the characteristic time scales of thermal conduction in the system (see Table II), the heat transfer cannot always be considered as quasistationary state. Indeed, the POM substrate is characterized by a diffusion time of 687 s which is not negligible compared with the lifetime of the droplets in the experiments. The thermal effusivity
of the several materials is provided in Table II. The same interpretation is performed with the thermal effusivity as the thermal conductivity. Hence, the data reveal that the more effusive a substrate, the more important the evaporation rate. However, there is virtually no difference of behavior for the metallic substrates whereas an important difference is noted compared with the polymer. Indeed, the highly effusive substrates are weakly sensitive to the evaporation on its surface which leads to a surface temperature close to the imposed temperature whereas the lowly effusive substrates are very sensitive to the evaporation on its surface which leads to a lower surface temperature. Consequently, the global evaporation rate is lower in the situation of lowly effusive substrate.

However, the only consideration of the substrate thermal effusivity does not satisfactorily explain the similar values of global evaporation rates for the metallic substrates. The repartition of curves as a function of the substrate thermal properties is correctly explained by the equivalent thermal effusivity, which corresponds to the parallel thermal effusivity of the two materials in contact, i.e., the drop and the substrate. Indeed, previous studies were performed considering the contact between a liquid drop at temperature $T_a$ and a wall at temperature $T_s$ [50]. In the first stage, the temperature of the contact is determined by the theory of heat transfer between two semi-infinite materials, $T_c = \frac{\beta_L T_L + \beta_S T_s}{\beta_L \beta_S}$, and the heat flux received by the drop is $q_L = \frac{\beta_L}{\sqrt{\pi t}}(T_c - T_a) = \frac{\beta_{ll}}{\sqrt{\pi t}}(T_s - T_a)$, with the equivalent thermal effusivity $\beta_{ll} = \frac{\beta_L \beta_S}{\beta_S + \beta_L}$ where the subscripts *S* and *L* indicates the materials considered, respectively, solid and liquid. Considering the parallel thermal effusivity for the materials considered in these study (see Table II), the parallel thermal effusivity for the metallic substrate-water couples are of the same order of magnitude and a ratio of 2.8 is obtained between the copper-water and POM-water couples. Hence, the parallel thermal effusivity describes the equivalent behavior obtained for metallic substrates and the important difference with the polymeric substrate. The ratio of the equivalent thermal effusivities between the copper-water and POM-water couples is of the same order of magnitude but slightly overpredicted the ratio of the global evaporation rate between the same materials. The difference is acceptable considering the uncertainties of measurements and the difference in time scales considered between the theory and the experiments.

All of the experimental results are compared with the quasisteady, isothermal, diffusion model of evaporation. The model describes qualitatively the influence of the substrate temperature on the global evaporation rate. However, the model does not include any of the thermal properties

of the liquid and the substrate. As described previously in the specific case of the aluminum substrate, the model underestimates the global evaporation rate for the highly effusive substrates. The relative deviation increases with the increase of the temperature until reaching 58.9% at $T_s = 75$ °C for the copper substrate. The curve obtained for the lowly effusive substrate is close to the model by chance. It suggests that the contribution of the increase of the evaporation rate by the buoyant convection in the vapor transport is counterbalanced by the decrease in the energy transport on the substrate and the evaporative cooling effect.

## V. CONCLUSION

The influence of the thermal effects of the substrate on the behavior of a pinned sessile water droplet evaporating into air has been investigated in the hydrophilic and hydrophobic cases. Hence, the effect of the substrate temperature on geometrical evolutions, time of evaporation, and the global evaporation rate have been studied, as well as the influence of the thermal properties of the substrate on the global evaporation rate. Experimental results are compared with the quasisteady, diffusion-driven evaporation model, which is implemented with the influence of the temperature and assumes isothermia of the interface at the substrate temperature. The results highlight a favorable agreement between the model and the experimental data at ambient temperatures for most situations considered here. However, with an increase in temperature, the contribution of mechanisms linked with thermal effects becomes increasingly important. Indeed, the global evaporation rate appears clearly influenced by the thermal effusivity of the substrate, and the contact angle evolution is modified with an increase in temperature. The theoretical model succeeds in qualitatively describing the influence of the substrate temperature on the evaporation process. However, comparison with the experimental data highlights the need for a more accurate model that accounts for the buoyant convection in the vapor transport and the evaporative cooling and thermal conduction between the droplet and the substrate.

## ACKNOWLEDGMENTS

We would like to thank J. Gavillet from CEA LITEN, Grenoble, France, for performing nanocoatings, F. Rigollet for fruitful discussions, and M. Hadj for carrying out preliminary experiments. Fundings from Agence Nationale de la Recherche (Project NanoSurf ANR-37509-BLAN-0093-03), Centre National d'Etudes Spatiales (Project "Sessile Drop Evaporation"), and Carnot STAR (Project "Nano Phase Change") are gratefully acknowledged.

[1] J. Jing, J. Reed, J. Huang, X. Hu, V. Clarke, J. Edington, D. Housman, T. S. Anantharaman, E. J. Huff, B. Mishra, B. Porter, A. Shenker, E. Wolfson, C. Hiort, R. Kantor, C. Aston, and D. C. Schwartz, Proc. Natl. Acad. Sci. 95, 8046 (1998).

[2] R. Blossey, Nat. Mater. 2, 301 (2003).

[3] A. F. M. Leenaars, J. A. M. Huethorst, and J. J. Van Oekel, Langmuir 6, 1701 (1990).

[4] T. P. Bigioni, X. Lin, T. T. Nguyen, E. I. Corwin, T. A. Witten, and H. M. Jaeger, Nat. Mater. 5, 265 (2006).

[5] D. Kim, S. Jeong, B. K. Park, and J. Moon, Appl. Phys. Lett. 89, 264101 (2006).

[6] J. Park and J. Moon, Langmuir 22, 3506 (2006).

[7] J. Boneberg, F. Burmeister, C. Schafle, and P. Leiderer, Langmuir 13, 7080 (1997).
[8] T. Ondarcuhu and C. Joachim, Europhys. Lett. 42, 215 (1998).

[9] T. Kawase, H. Sirringhaus, R. H. Friend, and T. Shimoda, Adv. Mater. 13, 1601 (2001).

[10] D. J. Norris, E. G. Arlinghaus, L. Meng, R. Heiny, and L. E. Scriven, Adv. Mater. 16, 1393 (2004).

[11] T. Yakhno, J. Colloid Interface Sci. 318, 225 (2008).

[12] D. Brutin, B. Sobac, B. Loquet, and B. Sampol, J. Fluid Mech. 667, 85 (2011).

[13] B. Sobac and D. Brutin, Phys. Rev. E 84, 011603 (2011).

[14] R. G. Picknett and R. Bexon, J. Colloid Interface Sci. 61, 336 (1977).

[15] R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel and T. A. Witten, Phys. Rev. E 62, 756 (2000).

[16] H. Hu and R. G. Larson, J. Phys. Chem. B 6, 1334 (2002).

[17] Y. O. Popov, Phys. Rev. E 71, 036313 (2005).

[18] D. Quéré, Annu. Rev. Mater. Res. 38, 71 (2008).

[19] D. Bonn, J. Eggers, J. Indekeu, J. Meunier, and E. Rolley, Rev. Mod. Phys. 81, 739 (2009).

[20] J. Yu, X. Zhao, Q. Zhao, and G. Wang, Mater. Chem. Phys. 68, 253 (2001).

[21] T. Onda, S. Shibuichi, N. Satoh, and K. Tsujii, Langmuir 12, 2125 (1996).

[22] L. Feng, S. Li, Y. Li, H. Li, L. Zhang, J. Zhai, Y. Song, B. Liu, L. Jiang, and D. Zhu, Adv. Mater. 14, 1857 (2002).

[23] M. Reyssat, J. M. Yeomans and D. Quéré, Europhys. Lett. 81, 26006 (2008).

[24] A. Tuteja, W. Choi, M. Ma, J. M. Mabry, S. A. Mazzella, G. C. Rutledge, G. H. McKinley, and R. E. Cohen, Science 318, 1618 (2007).

[25] A. Mamur, Langmuir 24, 7573 (2008).

[26] L. Joly and T. Biben, Soft Matter 5, 2549 (2009).

[27] H. Y. Erbil, G. McHale, and M. I. Newton, Langmuir 18, 2636 (2002).

[28] K. S. Birdi, D. T. Vu, and A. Winter, J. Phys. Chem. 93, 3702 (1989).

[29] S. M. Rowan, M. I. Newton and G. McHale, J. Phys. Chem. 99, 13268 (1995).

[30] C. Bourges-Monnier and M. Shanahan, Langmuir 11, 2820 (1995).

[31] M. Cachile, O. Benichou, C. Poulard, and A. M. Cazabat, Langmuir 18, 8070 (2002).

[32] B. Sobac and D. Brutin, Langmuir 27, 14999 (2011).

[33] H. Gelderblom, A. G. Marin, N. Nair, A. van Houselt, L. Lefferts, J. H. Snoeijer, and D. Lohse, Phys. Rev. E 83, 026306 (2011).

[34] S. A. Kulinich and M. Farzaneh, Appl. Surf. Sci. 255, 4056 (2009).

[35] L. Grandas, C. Reynard, R. Santini, and L. Tadrist, Int. J. Therm. Sci. 44, 137 (2005).

[36] D. H. Shin, S. H. Lee, J. Y. Jung, and J. Y. Yoo, Microelec. Eng. 86, 1350 (2009).

[37] W. D. Ristenpart, P. G. Kim, C. Domingues, J. Wan, and H. A. Stone, Phys. Rev. Lett. 99, 234502 (2007).

[38] S. David, K. Sefiane, and L. Tadrist, Colloids Surf. A 198, 108 (2007).

[39] G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and K. Sefiane, J. Fluid Mech. 623, 329 (2009).

[40] K. Sefiane and R. Bennacer, J. Fluid Mech. 667, 260 (2011).

[41] G. J. Dunn, S. K. Wilson, B. R. Duffy, and K. Sefiane, Phys. Fluids 21, 052101 (2009).

[42] E. Crafton and W. Black, Int. J. Heat Mass Transfer 47, 1187 (2004).

[43] F. Girard, M. Antoni, S. Faure, and A. Steinchen, Colloids Surf. A 323, 36 (2008).

[44] M. A. Saada, S. Chikh, and L. Tadrist, Phys. Fluids 22, 112115 (2010).

[45] C. Sodtke, V. S. Ajaev, and P. Stephan, Heat Mass Transfer 43, 649 (2007).

[46] N. Shahidzaneh-Bonn, S. Rafa, A. Azouni, and D. Bonn, J. Fluid Mech. 549, 307 (2006).

[47] N. N. Lebedev, Special Functions and their Applications (Prentice-Hall, Englewood Cliffs, NJ, 1965).

[48] F. Girard, M. Antoni, S. Faure, and A. Steinchen, Langmuir 22, 11085 (2006).

[49] H. T. Phan, N. Caney, P. Marty, S. Colasson, and J. Gavillet, Int. J. Heat Mass Transfer 52, 5459 (2009).

[50] T. Loulou and J. P. Bardon, Rev. Gen. Therm. 36, 682 (1997).
# Annexe C

Infrared visualization of thermal motion
inside a sessile drop deposited onto a
heated surface

David Brutin, Benjamin Sobac, Fabrice Rigollet &
Christophe Le Niliot

*Experimental Thermal and Fluid Science 35*, 521–530 (2011).
<figure><img src="image_93.png" /></figure>

ELSEVIER

<figure><img src="image_94.png" /></figure>

# Infrared visualization of thermal motion inside a sessile drop deposited onto a heated surface

D. Brutin *, B. Sobac, F. Rigollet, C. Le Niliot

Aix-Marseille Université (U1-U2), Laboratoire IUSTI, Technopôle de Château-Gombert, 5 rue Enrico Fermi, 13453 Marseille, France

## ARTICLE INFO

**Article history:**

Received 7 October 2010
Received in revised form 13 December 2010
Accepted 13 December 2010
Available online 19 December 2010

**Keywords:**

Sessile drops
Convection cells
Evaporation
Infrared camera
Thermal motion
Spectroscopy

## ABSTRACT

Drop evaporation is a basic phenomenon but the mechanisms of evaporation are still not entirely clear. A common agreement of the scientific community based on experimental and numerical work is that most of the evaporation occurs at the triple line. However, the rate of evaporation is still predicted empirically due to the lack of knowledge of the governing parameters on the heat transfer mechanisms which develop inside the drop under evaporation. The evaporation of a sessile drop on a heated substrate is a complicated problem due to the coupling by conduction with the heating substrate, the convection/conduction inside the drop and the convection/diffusion in the vapor phase. The coupling of heat transfer in the three phases induces complicated cases to solve even for numerical simulations. We present recent experimental results obtained using an infrared camera coupled with a microscopic lens giving a spatial resolution of 10 µm to observe the evaporation of sessile drops in infrared wavelengths. Three different fluids fully characterized, in the infrared wavelengths of the camera, were investigated: ethanol, methanol and FC-72. These liquids were chosen for their property of semi-transparency in infrared, notably in the range of the camera from 3 to 5 µm. Thus, it is possible to observe the thermal motion inside the drop. This visualization method allows us to underline the general existence of three steps during the evaporating process: first a warm-up phase, second (principal period) evaporation with thermal-convective instabilities, and finally evaporation without thermal patterns. The kind of instabilities observed can be different depending on the fluid. Finally, we focus on the evolution of these instabilities and the link with the temperature difference between the heating substrate and the room temperature.

© 2010 Elsevier Inc. All rights reserved.

## 1. State of the art

Drop evaporation has been widely investigated theoretically [1], experimentally [2,3] and numerically [4]. Generally, the work deals with either drops with small sizes [5,6] or flat drops which can be comparable with films [7]. Table 1 summarizes numerous studies of drop and droplet evaporation performed since 1973; most of these studies discuss physical mechanisms related to the evaporation dynamics. Scientific production has sharply increased since 1995. Indeed, drop and droplet evaporation is involved in many industrial processes and in many heat and mass transfer applications. The literature evidences that the main physical mechanisms driving droplet evaporation are the following:

* the heat transfer by conduction in the substrate,

* the heat transfer by convection [temperature gradients induces superficial tension gradients (Marangoni effects) and normal convection],

* the liquid/substrate molecular interaction at the roughness scale that tends to modify the wettability of the drop for a given drop volume,

* the vapor mass diffusion around the drop.

In spite of the considerable activity attested by the large quantity of publications, the dominant mechanisms acting on the kinetics of drop evaporation are not well understood because of the strong coupling between the various phenomena. In the studies of drop evaporation, two approaches can be distinguished: drop evaporation either with [13,18] or without [15] wall heating. Without wall heating, the drop evaporation dynamic is slower compared to the heating situation. Also, the wall heating enables a higher evaporation flow rate inducing a high temperature gradient inside the drop; this last situation is useful when working with a given infrared resolution.

In an unsaturated atmosphere and without overheating from the substrate, the mass diffusion of the vapor in the gas surrounding the droplet can become the limiting phenomenon of evaporation kinetics. According to the experiments of Shahidzadeh-Bonn et al. [30], such a phenomenon is much more pronounced for organic fluids than for water, this being mainly due to the difference of densities

<footnote>* Corresponding author. Tel.: +33 491106886; fax: +33 491106969. E-mail address: david.brutin@polytech.univ-mrs.fr (D. Brutin).</footnote>
## Nomenclature

### Roman letters

Cp heat capacity (J kg⁻¹ K⁻¹)

d drop base diameter (mm)

g gravitational constant (9.81 m s⁻²)

h drop height (mm)

k absorption coefficient (m⁻¹)

L_c capillary length (mm)

L_ν latent heat of vaporization (kJ kg⁻¹)

L luminance (W m⁻² s r⁻¹ µm⁻¹)

m drop mass (µg)

P heat power (W)

Q heat flux (W m⁻²)

R drop base radius (mm)

S surface (m²)

t time (s)

T temperature (°C)

V drop volume (µL)

### Dimensionless numbers

Bo Static Bond number (–)

Ma Marangoni number (–)

Nu Nusselt number (–)

Pr Prandtl number (–)

Ra Rayleigh number (–)

### Greek symbols

α thermal diffusivity (m² s⁻¹)

β coefficient of thermal expansion (K⁻¹)

Δ difference (–)

ε emissivity (–)

λ heat conductivity (W m⁻¹ K⁻¹)

μ dynamic viscosity (Pa s)

ν kinematic viscosity (m² s⁻¹)

ρ density (kg m⁻³)

σ surface tension (N m⁻¹)

τ transmittivity (–)

θ contact angle (°)

### Subscripts and superscripts

Conv convection

Drop drop

L liquid

S sensor

Sat saturation

Sub substrate

SW short wave (range of 3–5 µm)

V vapor

between vapor and air. In the cases of drops on a heated substrate, the interactions between the liquid and the substrate play a fundamental role in the evaporation kinetics. A significant difference in temperature between the substrate and the liquid is a source of thermo-convective instabilities induced by the Marangoni effect and buoyancy forces. These instabilities depend mainly on the thermophysical properties of the liquid and the substrate, but also on the drop shape which is correlated with interactions at the liquid/substrate interface.

The experimental studies summarized in Table 1 attempt to explain the role of the mechanisms specific to drop evaporation. The parameters of the drop (radius, height and contact angle) are

usually deduced by visualization using a visible camera. Thus the behavior of drops for various liquid/substrate combinations have been analyzed by several authors under various experimental conditions. Two modes of evaporation are observed: one with a pinned diameter and one with constant angle of contact and both appearing in sequences. Crafton and Black [16] observed that for a water drop the basic radius of the drop remains constant during evaporation, whereas for a heptane drop the contact angle remains constant. Other observations on the rate of evaporation of the drops (based on the measurement of volume or weight variation) indicate a linear relation between the rate of evaporation and the wetting diameter of the drops [7,16,18].

<caption>**Table 1**</caption>

Articles related to drop evaporation (several approaches are presented: experiments, numerical simulations, theoretical approaches) (Exp.: experimental work; Num.: numerical simulations; Th.: theoretical work; SS: stainless steel).

<table><thead><tr><th>Authors [Ref.]</th><th>Year</th><th>Approach</th><th>Fluid</th><th>Substrate</th></tr></thead><tbody><tr><td>Boyes [8]</td><td>1973</td><td>Exp.</td><td>Water with saturated vapor</td><td>Cu, PTFE</td></tr><tr><td>Piknett [9]</td><td>1977</td><td>Exp.</td><td>Methyl-acetoacetate</td><td>PTFE</td></tr><tr><td>Birdi [2]</td><td>1989</td><td>Exp.</td><td>Water</td><td>PTFE</td></tr><tr><td>Bourges-Monnier [10]</td><td>1995</td><td>Num. and exp.</td><td>Water, N-decane</td><td>PTFE, SiO<sub>2</sub>, epoxy</td></tr><tr><td>Hegseth [11]</td><td>1996</td><td>Exp.</td><td>Ethanol, polystyrene particles</td><td>–</td></tr><tr><td>Chandra [12]</td><td>1996</td><td>Exp. and Th.</td><td>Water with surfactant</td><td>SS</td></tr><tr><td>Bernardin [13]</td><td>1997</td><td>Exp.</td><td>Water</td><td>Al</td></tr><tr><td>Zhang [14]</td><td>2002</td><td>Exp.</td><td>Silicon oil, R113, ethanol</td><td>Anodized glass</td></tr><tr><td>Hu [5]</td><td>2002</td><td>Num. and exp.</td><td>Water</td><td>Glass</td></tr><tr><td>Erbil [15]</td><td>2002</td><td>Exp.</td><td>Butanol, toluene, nonane, octane</td><td>PTFE</td></tr><tr><td>Crafton [16]</td><td>2004</td><td>Exp.</td><td>Water, N-heptane</td><td>Cu, Al</td></tr><tr><td>Savino [17]</td><td>2004</td><td>Exp. and Num.</td><td>Silicon oil, hydrocarbons, water</td><td>Several metals</td></tr><tr><td>Grandas [18]</td><td>2004</td><td>Exp.</td><td>Water</td><td>PTFE, Al</td></tr><tr><td>Poulard [19]</td><td>2005</td><td>Exp. and Num.</td><td>Water, alkanes</td><td>SiO<sub>2</sub></td></tr><tr><td>Frassy [20]</td><td>2006</td><td>Exp. and Num.</td><td>Water</td><td>SiO<sub>2</sub>, silane, polycarbonate</td></tr><tr><td>Sodtke [21]</td><td>2007</td><td>Exp. and Th.</td><td>Water with saturated vapor</td><td>SS</td></tr><tr><td>Tarozzi [22]</td><td>2007</td><td>Exp.</td><td>Water</td><td>Black paint on IR materials</td></tr><tr><td>Widjaja [23]</td><td>2008</td><td>Num. and exp.</td><td>Water</td><td>–</td></tr><tr><td>Sefiane [24]</td><td>2008</td><td>Exp.</td><td>Ethanol, methanol, FC-72</td><td>PTFE, ceramic, Ti, Cu</td></tr><tr><td>Tartarini [25]</td><td>2009</td><td>Exp. and Num.</td><td>Water</td><td>BaF<sub>2</sub></td></tr><tr><td>Barash [26]</td><td>2009</td><td>Num. and Th.</td><td>Toluene</td><td>–</td></tr><tr><td>Tam [27]</td><td>2009</td><td>Exp. and Th.</td><td>Water</td><td>Superhydrophobic surfaces</td></tr><tr><td>Dhavaleswarapu [28]</td><td>2009</td><td>Exp. and Th.</td><td>Water</td><td>Glass</td></tr><tr><td>Nikolayev [29]</td><td>2010</td><td>Th. and num.</td><td>Water</td><td>SS</td></tr></tbody></table>
Although the observations using visible camera are useful to give an overall picture of the mass transfer of the drops, such an experimental approach is not sufficient to determine the role of the selected pair liquid/substrate in modeling the internal dynamics of the drop. Few experimental studies attempt to give greater comprehension of the phenomenon of drop evaporation using original techniques. Hegseth et al. [11] and Kang et al. [31] used optical techniques of the PIV type (Particle Imaging Velocimetry) to visualize the fluid motion inside the drop using polystyrene particles in an ethanol drop, whereas Zhang and Chao [14] employed a shadowgraphy laser method. Studies dealing with the phenomena of thermogravitational and thermocapillary convection in a drop deposited onto a silicon surface were carried out by using infrared thermography by Savino and Fico [17]. The distribution of the wall temperature below the drop is also given either by using liquid crystals [21] or by infrared thermography through a transparent substrate [22]. Infrared thermography is generally used to take non intrusive measurements [22,17,32,33]. The advantage of these techniques is the absence of contact which thus makes it possible to observe the thermal phenomenon without contact. However, the difficulty is then to correctly interpret the infrared information measured by the camera according to the radiative properties of the fluid observed. Indeed, the liquid sample can be semi-transparent in the spectral range of measurement, making it difficult to convert the measured brightness into surface temperature. Some authors [25,22] choose not to observe the fluid directly but prefer to observe the back face on which the drop evaporates. The substrate in BaF₂ is coated with an opaque emissive painting which makes it possible to measure its surface temperature; it is also the temperature of the drop contact area with the substrate. In other publications, the authors observe the fluid [32,33,17]. The authors then assume the opacity of the various fluids observed (water, ethanol, N-pentane) in the long-wave spectral field of the camera (9–10 µm); they need only the emissivity of the liquid to deduce the surface temperature.

Two processes play a role in the evaporation of a drop: the diffusion of molecules into the air from the drop surface and the internal flow inside the drop to bring the fresh liquid to the liquid-vapour interface [34]. In between these two physical phenomena, the solid-liquid contact angle plays an important role in the evaporation rate [12]. The reduction of the contact angle increases the surface of contact between the droplet and surface, and also reduces the drop thickness, thus increasing the contribution of thermal conduction from the substrate to the liquid-gas interface. These two effects increase the rate of evaporation.

A recent study has been performed by Sefiane et al. [24] on the use of thermography to observe a droplet under evaporation. However, the paper does not take into account the fluid radiative properties. Also the use of water for infrared properties is very sensitive since water reflectivity is high and transparency in the infrared wavelength of the FLIR SC3000 is almost zero for fluid thicknesses above 0.2 mm (see Table 3). Furthermore, the temperature scales presented in the figures are not related to the drop shape factor or to the fluid spectral emissivity in the infrared wavelength range of camera. The conclusion on the absence of flow motion inside the water droplet cannot be evidenced since water for this drop thickness (more than 1 mm) is totally opaque. The heat flux received by the infrared camera is a surface radiative heat flux which provides only information from a thin layer below the interface of less than 0.2 mm. Furthermore, all experiments are performed in ambient temperature whereas an investigation of the influence of the substrate temperature on the instabilities is carried out in this study. Finally, we characterize the drop evaporation dynamics using a complementary tool which is the heat fluxmeter to carefully analyse the heat transfer to the drop.

An axisymmetric numerical model has been developed by Ruiz and Black [6] to simulate the evaporation of deposited water drops which takes into account evaporation at the surface and convection of the liquid inside the drop due to Marangoni forces. The circulation of the liquid implies a greater rate of evaporation than the previous models and is in accordance with a linear change of volume according to time. Simulations which include convection in the liquid indicate that the flow has a substantial effect on the distribution of the temperature in the drop. Assuming the existence of internal convection in the drop, the distribution of the temperature is very different from the results predicted by a model based on pure conduction in the drop. Ruiz and Black [6] considered the existence of thermocapillary movements in the drop, which gives a distribution of isotherms completely different from that obtained with only conductive heat transfer inside the droplet. Hu and Larson [5] show with a model that the kinetics of evaporation is almost constant during all the phenomena for drops posed with an initial contact angle of $2\pi/9$. In the three cases [5,6,16], the authors studied the phenomena of evaporation for small drops, where the effect of gravity is negligible compared to the effects of surface tension. The shape of these droplets is comparable to a portion of a sphere. In practice, this assumption is valid only for drops below a critical size.

The whole experimental literature (except [24]) do not actually realize that a complicated and non-axisymmetric flow is possible inside the drop due to thermo-capillary instabilities. The strength of this paper is to visualize and investigate this complicated and non-axisymmetric flow inside the drop under evaporation.

The thermal aspects (heat and mass transfer) of the phenomena have never really been approached in a systematic way and the influence of thermal motion on the evaporation process is very largely unknown. This is why we studied the phenomenon of the evaporation of sessile drops using infrared tools in the case of a broad range of sizes, temperature gradients and liquids.

## 2. Experimental setup

### 2.1. *Drop evaporation apparatus and methods*

An experimental device was set up in order to characterize a drop during its evaporation and to analyze the kinetics as well as the heat transfer associated with this phenomenon. This device includes a substrate where the drop is deposited and an instrumentation dedicated to thermal measurements (temperature, heat flux density). The environmental conditions (temperature, pressure, humidity) are measured continuously.

The heater is an aluminum parallelepipedic block ($50 \times 50 \times 20 \text{ mm}^3$) instrumented inside by two heating cartridges 6.35 mm in diameter. A 1-mm-diameter thermocouple of the K type and a platinum probe (Pt 100) are fixed onto the surface. The latter is coupled with a PID regulator in order to maintain the temperature imposed at the surface of the element (between 25 °C and 100 °C) constant. The thermocouple signal is recorded at 1 Hz. A thin foil flexible heat flux sensor (CAPTEC) consists of a thermoelectric panel laminated between flexible heterogeneous very thin plastic layers and is located between the heating aluminum block and the PTFE substrate. The heat flux signal is very sensitive ($0.7 \mu\text{V W}^{-1} \text{m}^{-2}$). The total thickness of the fluxmeter is 0.6 mm with a diameter of 10 mm. A T-type thermocouple is located inside the heat fluxmeter and will be used in the next sections to follow the substrate temperature. A final PTFE (Polytetrafluoroethylene) sheet with a thickness of 200 µm is stuck on the fluxmeter; this surface will receive the sessile drop. The PTFE sheet
<figure><img src="image_95.png" /><figcaption>**Fig. 1.** Experimental setup used for infrared thermography of sessile drop evaporation.</figcaption></figure>

is use for its hydrophobic properties and is changed regularly to avoid the particular problem of cleaning and to guarantee the reproducibility of the surface quality (Fig. 1).

Visible video acquisition is performed using a Pike 800 × 600 pixels camera with 30 images/second coupled to a microscope with a VZM100i lens. This visible imaging allows a visualization of the drop shape from the side. This is used to determine the geometrical parameters of the drop. The sampling rate is 30 images/second because of the low interface displacement velocity. After having selected an area of interest on the image (definition of a base line), an analysis is carried out in order to determine the contact angles on the left and right, the height, the diameter of the wetted area, the external surface and the volume of the drop. The contact angles are given by carrying out a polynomial regression on the drop free interface. Volume is calculated by carrying out an integration with the axisymmetric assumption of drop.

One tool used to analyse flow motion inside a drop measuring a few millimeters in diameter is an infrared camera. Compared to a layer evaporation problem, the first difficulty due to the interface curvature which induces a form factor. This form factor is required to access an accuracy of ±0.1 °C. For drops, a second difficulty rose, since it is necessary to record the interface with a side view camera over a period of time and then model it in relation to the evaporation of the drop. Then each pixel of the film needs to be computerized with the right total emissivity of the fluid (see Section 2.2.2). This procedure is under implementation. Consequently, the infrared images presented in this work were obtained at half the initial drop thickness. The temperature scale has an uncertainty of ±0.5 °C. This is a first approach in the analysis of the infrared data. The infrared diagnosis is carried out using a FLIR SC6000 camera (640 × 512 pixels) coupled to a microscope making it possible to reach a resolution of 10 µm, so, a field of view of 6.4 mm × 5.12 mm. The infrared camera is assembled vertically to the top of the drop. Before doing the experiment, it is checked that the substrate temperature homogeneity is stable at ±0.5 °C.

## 2.2. Properties of liquids used

### 2.2.1. Physical properties

Table 2 provides the properties of the fluids investigated in this study and include water properties for comparison. All fluids used have a low surface tension creating drops with a contact angle ranging from 10° to 40° (mainly because of the surface tension: 22.7 mN m⁻¹ for methanol, 12.0 mN m⁻¹ for FC-72 and 22.0 mN m⁻¹ for ethanol). The three fluids have quite different Prandtl numbers [Pr = (μCp)/λ]. Their heat of vaporization is also very different which makes it possible to observe evaporation over different time ranges and different temperature gradients. The heat of vaporization of FC-72 is very low at 88 kJ kg⁻¹, while for ethanol the heat of vaporization is almost 10 times higher at 841 kJ kg⁻¹; finally the heat of vaporization of methanol is the highest at 1165 kJ kg⁻¹. Ethanol and methanol have almost the same capillary lengths [L<sub>c</sub> = √σ/(g · Δq)] since their surface tension and fluid and vapor surface tension are similar.

### 2.2.2. Global emissivity determination

Here is considered a semi-transparent liquid between x = 0 and x = L. This liquid is homogeneous and is supposed isothermal at temperature T. Furthermore, there is no particle inside that could diffuse radiation. The monochromatic luminance of the entering heat flux (at x = 0) and sorting heat flux (at x = 1) are respectively noted L<sub>z</sub>(0) and L<sub>z</sub>(L). The monochromatic transmittivity of the liquid of thickness L is defined by:

$$ \tau_{\lambda}(L, T) = \frac{L_{\lambda}(L)}{L_{\lambda}(0)} = e^{-k_{\lambda}(T)L} \quad (1) $$

where k<sub>λ</sub> (µm⁻¹) is the monochromatic absorption coefficient. The length 1/k<sub>λ</sub> (m) is then the characteristic length of penetration of a wavelength λ radiation. The dimensionless product k<sub>λ</sub> * L is the monochromatic optical thickness of the liquid. The transmittivity τ<sub>λ</sub>(L) is measured by a spectrophotometer and plotted for wavelengths between 3 and 5 µm for three different fluids and for particular thicknesses (Fig. 2).

Due to the hypothesis of non scattering liquid, radiation that is not transmitted through the liquid is only absorbed inside it (τ<sub>λ</sub>(L) + α<sub>λ</sub>(L) = 1), the monochromatic absorptivity being noted α<sub>λ</sub>(L), with ε<sub>λ</sub>(L) = α<sub>λ</sub>(L) (Kirchhoff's law). The monochromatic emissivity of the liquid of thickness (L) at temperature (T) is then defined by

$$ \epsilon_{\lambda}(L, T) = 1 - e^{-k_{\lambda}(T)L} = 1 - \tau_{\lambda}(L, T) \quad (2) $$

and is then obtained using the measured value of τ<sub>λ</sub>(L, T). If the liquid is transmitting and emitting radiations, the monochromatic luminance of sorting heat flux at x = L is given by

$$ L_{\lambda}(L, T) = L_{\lambda}(0)e^{-k_{\lambda}(T)L} + L_{\lambda}^{0}(T)[1 - e^{-k_{\lambda}(T)L}] \quad (3) $$

$$ \begin{aligned} L_{\lambda}(L, T) = \tau_{\lambda}(L, T)L_{\lambda}(0) &+ \epsilon_{\lambda}(L, T)L_{\lambda}^{0}(T) \\ & = [1 - \epsilon_{\lambda}(L, T)]L_{\lambda}^{0}(T) + \epsilon_{\lambda}(L, T)L_{\lambda}^{0}(T) \end{aligned} \quad (4) $$

After integration on the spectral bandwidth of the infrared camera that is working in the Short Wave range (SW: 3–5 µm), we obtain

<table><caption>**Table 2**

Physical properties and dimensionless numbers of fluids at 25 °C and 1 atm.</caption><thead><tr><th></th><th>ρ<sub>L</sub> (kg m<sup>-3</sup>)</th><th>ρ<sub>V</sub> (kg m<sup>-3</sup>)</th><th>Cp (J kg<sup>-1</sup> K<sup>-1</sup>)</th><th>L<sub>V</sub> (kJ kg<sup>-1</sup>)</th><th>λ (W m<sup>2</sup> K<sup>-1</sup>)</th><th>μ (mPa s)</th><th>σ (mN m<sup>-1</sup>)</th><th>T<sub>sat</sub> (°C)</th><th>L<sub>c</sub> (mm)</th><th>Pr (–)</th></tr></thead><tbody><tr><td>Water</td><td>997</td><td>0.59</td><td>4180</td><td>2449</td><td>0.606</td><td>0.890</td><td>72.7</td><td>100</td><td>2.73</td><td>6.14</td></tr><tr><td>FC-72</td><td>1680</td><td>4.35</td><td>1100</td><td>88.0</td><td>0.057</td><td>0.638</td><td>12.0</td><td>56.0</td><td>0.82</td><td>12.3</td></tr><tr><td>Methanol</td><td>791</td><td>0.21</td><td>2531</td><td>1165</td><td>0.203</td><td>0.560</td><td>22.7</td><td>64.7</td><td>1.71</td><td>6.98</td></tr><tr><td>Ethanol</td><td>789</td><td>1.50</td><td>2845</td><td>923</td><td>0.140</td><td>1.095</td><td>22.0</td><td>78.0</td><td>1.69</td><td>22.3</td></tr></tbody></table>
<figure><img src="image_96.png" /><figcaption>**Fig. 2.** Global emissivity of FC-72, methanol and ethanol as a function of the optical thickness of the fluid in a wavelength of 3–5 µm ($T_{room}$ = 25 °C, fluid optical thickness uncertainty is ±40 µm).</figcaption></figure>

$$L_{SW}(L, T) = [1 - \epsilon_{SW}(L, T)]L_{SW}(0) + \epsilon_{SW}(L, T)L_{SW}^{0}(T) \quad (5)$$

with the 'global emissivity' of the liquid layer in the SW range noted $\epsilon_{SW}(L, T)$:

$$ \begin{aligned} \epsilon_{SW}(L, T) &= \frac{\int_{\lambda=3}^{\lambda=5} \epsilon_{\lambda}(L, T)L_{\lambda}^{0}(T)d\lambda}{\int_{\lambda=3}^{\lambda=5} L_{\lambda}^{0}(T)d\lambda} = \epsilon_{SW}(L, T) \\ &= \frac{\int_{\lambda=3}^{\lambda=5} \epsilon_{\lambda}(L, T)L_{\lambda}^{0}(T)d\lambda}{L_{SW}^{0}(T)} \end{aligned} \quad (6) $$

and assuming

$$ \frac{\int_{\lambda=3}^{\lambda=5} \epsilon_{\lambda}(L, T)L_{\lambda}(0)d\lambda}{\int_{\lambda=3}^{\lambda=5} L_{\lambda}(0)d\lambda} = \frac{\int_{\lambda=3}^{\lambda=5} \epsilon_{\lambda}(L, T)L_{\lambda}(0)d\lambda}{L_{SW}(T)} \sim \epsilon_{SW}(L, T) \quad (7) $$

Eq. (7) above can describe the situation of a drop (assumed isothermal) deposited onto the heated surface, $L_{SW}(0)$ being the spectral luminance of heat flux leaving the PTFE surface at imposed temperature and $L_{SW}(0)$ being the spectral luminance of heat flux leaving the drop surface toward the infrared camera. The two extreme situations are the following. On one hand, if the liquid layer is completely transparent in the SW bandwith, $\epsilon_{SW}(L, T) = 0$ and the camera measures directly the heat flux coming from the PTFE heater. On the other hand, if the liquid layer is opaque in the SW bandwith, $\epsilon_{SW}(L, T) = 1$ and the camera measures only the heat flux coming from the drop surface. The effect of wave reflexions due to change of refractive index at the liquid-gas interface is neglected here. The value of $\epsilon_{SW}(L, T)$ is then the main parameter we need to be able to know at which depth the infrared camera measurement can be associated. In a first approximation in this paper, the temperature scale of infrared images presented here are calculated with this value of global emissivity $\epsilon_{SW}(L, T)$.

The value of $\epsilon_{SW}(L, T)$ measured at 20 °C are plotted for three fluids, for several layer thicknesses (Fig. 2). The exponential behaviour expected for the monochromatic value $\epsilon_{\lambda}(L, T)$ according to Eq. (2) is also obtained after integration on the SW bandwidth, it is then

possible to fit the following model on those measurements. The global transmittivity in the SW bandwidth is then obtained as given below:

$$ \epsilon_{SW}(L, T) = 1 - e^{-k_{SW}(T)L} \quad (8) $$

$$ \tau_{SW} = 1 - \epsilon_{SW} \quad (9) $$

The obtained values of $k_{SW}$ (mm$^{-1}$) are presented in Table 3. The dimensionless values of $k_{SW} * L$ indicated for two layer thicknesses (1 and 2 mm) are the optical thicknesses of the liquid layer. The layer is said 'optically thick' if its optical thickness is greater than 5 mm, below that value it is semi-transparent. In this table, fluids are listed in order of increasing semi-transparency.

### 2.2.3. Infrared properties

Three liquids were investigated using the infrared camera: methanol, ethanol and FC-72. All were used for their semi-transparency properties in the infrared wavelength. Measurements were performed using a FTIR NICOLET Nexus 560 spectrophotometer to access the monochromatic transmittivity of a given thickness of fluid in the spectral range of 2.5–14 µm. For the purpose of this study, we analyzed and extracted only the data in the range of 3–5 µm which corresponded to our infrared camera wavelength band. The experimental cell was composed of two CaF$_2$ windows which were used for their almost total transparency properties in the range of the spectrophotometer. PTFE spacers were used to obtain the adequate optical fluid thickness. Three spacer thicknesses were available: 0.1, 0.2 and 0.5 mm, and could be used together. The optical fluid thickness was deduced on the basis of nine independent measurements of the spacers set using a Mitutoyo CD-15CPX caliper which has an uncertainty of ±0.02 mm, a resolution of ±0.01 mm and a repeatability of ±0.01 mm, and therefore a total uncertainty of ±0.04 mm. The error bar on the abscissa is the sum of the total uncertainty and the standard deviation of the nine measurements and is about ±40 µm. The transmittivity measurements were performed three times. For each measurement, a background measurement was first carried out consisting in measuring the radiative power that is transmitted by the cell full of atmospheric air. These three independent measurements provided the ordinate error bar which is the standard deviation.

Ethanol transmittivity in the 3–4 µm range is almost null for a sample fluid with a thickness of 1 mm; then in the range from 4 to 5 µm, the global transmittivity is about 20%. Consequently, the global emissivity for a sample of ethanol with a thickness of 1 mm is 0.859 (Fig. 3). The penetration skin depth of ethanol is 540 µm. For fluid thicknesses below 1.62 mm, emissivity varies exponentially with fluid thickness until a null emissivity for a null fluid thickness. Thus, ethanol can be used for infrared measurement by transmission up to 1.62 mm and have a transmission of 1%.

Fig. 2 provides the global emissivity for the three fluids investigated in the infrared wavelength range: 3–5 µm. The fluid thicknesses investigated range from 0.2 to 2.0 mm, which corresponds to the range of drop height encountered. The drop height being always below the capillary length ($L_c$). All three emissivity variations in this figure can be fitted by a decreasing exponential law given below:

<table><caption>**Table 3**
Infrared properties and dimensionless numbers of fluids.</caption><thead><tr><th>Fluid unit</th><th>k<sub>SW</sub> (mm<sup>-1</sup>)</th><th>1/k<sub>SW</sub> (mm)</th><th>k<sub>SW</sub>*L (L = 2 mm) (-)</th><th>k<sub>SW</sub>*L (L = 1 mm) (-)</th><th>L<sub>99%</sub> = 3/k<sub>SW</sub> (mm)</th></tr></thead><tbody><tr><td>Water</td><td>15.3</td><td>0.065</td><td>30.7</td><td>15.3</td><td>0.196</td></tr><tr><td>Methanol</td><td>3.02</td><td>0.33</td><td>6.04</td><td>3.02</td><td>0.994</td></tr><tr><td>Ethanol</td><td>1.85</td><td>0.54</td><td>3.70</td><td>1.85</td><td>1.622</td></tr><tr><td>FC-72</td><td>0.78</td><td>1.28</td><td>1.57</td><td>0.78</td><td>3.847</td></tr></tbody></table>
<figure><img src="image_97.png" /><figcaption>Fig. 3. Spectral transmittivity of ethanol as a function of the wavelength for an optical thickness of 1 mm ($T_{room}$ = 25 °C).</figcaption></figure>

$$ \epsilon = 1 - e^{-k_{sw}x} \qquad (10) $$

where x is the fluid thickness and $\epsilon$ is the global fluid emissivity. The constant 'ksw' for methanol is 3.02 mm⁻¹ ± 3.3%, for ethanol is 1.85 mm⁻¹ ± 2.1% and for FC-72 is 0.783 mm⁻¹ ± 2.2%. To be able to make useful observations of liquids using an infrared camera, they need to be semi-transparent since:

* if the transmittivity is low (close to 0), it means the absorptivity is high (close to 1), so the fluid absorbs the heat flux by radiation and then the heat is transmitted to the next layer by re-emission. Through the whole liquid layer, little heat flux is transmitted directly to the camera and the global emissivity is high. However, a very low global signal is not useful to perform observations of the thermal motion of the fluid since only the drop interface is measured.

* if the transmittivity is high (close to 1), it means the absorptivity is low (close to 0), so the fluid does not absorb the heat flux by radiation and the heat flux density goes through the fluid without heating it. Through the whole liquid layer, the heat flux density is almost fully transmitted directly to the camera and the global emissivity is low. However, a very low signal from the liquid is not useful to perform observations of the thermal motion of the fluid since only the substrate temperature is measured.

### 3. Thermal motion and evaporation of drops

The complete evaporation process of semi-transparent sessile drops with the boundary condition on the temperature of the substrate is discussed in this section. This includes information from the observation of the drop shape from the side with the visible camera, from above with the infrared camera and from below thanks to the fluxmeter. Observations have been performed using ethanol, methanol and FC-72.

#### 3.1. Mechanisms of drop evaporation

Initially the substrate is at a constant temperature. This is checked by the first image taken with the infrared camera. The heating block regulation system is on to avoid temperature decrease during the drop evaporation. Before the beginning of the experiment, the fluxmeter provides the natural convection heat flux on the heating substrate.

For all the fluids analyzed, we can observe that the process of evaporation is composed of three phases:

* Phase 1 – Warming up of the drop – The start of the experiment is defined as the initial contact of the drop on the substrate which is characterized by a sharp increase of the heat flux. The first seconds of the experiments are characterized by a transitional phenomenon. The drop, which was initially at room temperature, is placed on a warm surface and is first heated to reach almost the substrate temperature. The first step of the phenomenon is thus only driven by the heat capacity of the fluid.

* Phase 2 – Drop evaporation – The heat flux reaches a maximum value which corresponds to the beginning of our evaporation investigation. During this principal phase of evaporation, we can observe the appearance and the evolution of thermal-convective instabilities. The heat flux during over phase decrease continuously. Depending on the fluid, different kinds of these instabilities seem to be observed.

* Phase 3 – Film evaporation – A second phase of evaporation is characterized by the decrease of the heat flux when the convection cells have disappeared. The shape of the drop looks more like a layer than a drop. The evaporation rate is thus closer to film evaporation than drop evaporation.

The case an ethanol evaporating drop based on infrared visualization, the heat-flux and temperature measurements is presented. The results are summarized in Fig. 4 which shows the different phases of the evaporation of an ethanol drop on a substrate at a constant temperature of 48 °C. The three phases are separated thanks to the thermal flux signal. Assuming a substrate temperature at 50 °C and a room temperature at 25 °C; without any drop on the substrate, the natural convection power transferred is 10 mW. With a 4 mm-diameter drop on the substrate, the remaining natural convection power transferred is 8 mW around the drop. Consequently, when the drop evaporate, this slight change in the natural convection heat flux cannot be observed in Fig. 4. Variations observed with Fig. 4 are mainly linked to the evaporation heat flux.

#### 3.2. What can be observed depending on the fluid?

We present three experimental images of fluid dynamics for the three different liquids observed with one objective: to show the complexity of drop evaporation mechanisms. The three images have been extracted during all the phases of the drop evaporation (Figs. 5–7).

In the case of methanol (Fig. 5), numerous convection cells can be observed near the perimeter of the drop in a relative thin corona compared to the ethanol drop. At the center of the drop which is colder, the flow is much more unstable. The thermal flow motion inside the drop is fast compared to the other fluids. This point is very probably linked to the fluid viscosity which much smaller but need to be quantitatively checked. We can also observe that when the cold region is unstable and moves into the center, the convection cells rotate from the furthest point of the cold area to the closest point. The flow motion inside the drop stops when the drop reaches about 2 mm in diameter. Then the drop evaporate very slowly compared to the stages of the evaporation.

In the case of ethanol (Fig. 6), the same structure in cells is observed with a cold region located in the middle of the drop where the flow motion is more stable than in the methanol situation. Here also convection cells are observed near the drop perimeter but in a smaller quantity and in a wider corona. Then, during the evaporation, the flow motion changes very quickly and becomes large convection cells inside the drop. The total time of evaporation
<figure><img src="image_98.png" /><figcaption>**Fig. 4.** Heat flux evolution during the evaporation of a ethanol sessile drop ($T_{\text{room}}$ = 28 °C, $T_{\text{sub}}$ = 48 °C, uncertainty on the heat flux is ±10 W/m²).</figcaption></figure>

<figure><img src="image_99.png" /><figcaption>**Fig. 5.** Infrared sequence of methanol drops (% of total time of evaporation) ($T_{\text{room}}$ = 28 ± 0.2 °C, $T_{\text{sub}}$ = 44 ± 0.5 °C, drop diameter: 5.8 mm, uncertainty on the IR temperature scale is ±0.5 °C).</figcaption></figure>

is almost the same as that of methanol since the heat of vaporization is comparable.

In the case of FC-72 (Fig. 7), the latent heat of vaporization is about 10 times smaller than that of methanol and ethanol. Consequently, the drop evaporation time is short compared to the other two liquids. During the evaporation of FC-72, the stage of the evaporation evidence the same pattern than observed with ethanol and methanol then quickly the flow motion becomes different: we

discern multiple internal convective cells structures. This is completely different from the case of methanol and ethanol evaporation dynamics. This flow motion in the drop almost disappears after 60% of the drop evaporation time for a stable layer which evaporation by only conduction.

For fluids with even a comparable latent heat of vaporization, the thermal flow motion inside the drop can be very different since the fluid viscosity is the only important difference in the physical
<figure><img src="image_100.png" /><figcaption>**Fig. 6.** Infrared sequence of ethanol drops (% of total time of evaporation) ($T_{\text{room}}$ = 28 ± 0.2 °C, $T_{\text{sub}}$ = 46 ± 0.5 °C, drop diameter: 4.3 mm, uncertainty on the IR temperature scale is ±0.5 °C) (see movie online).</figcaption></figure>

<figure><img src="image_101.png" /><figcaption>**Fig. 7.** Infrared sequence of FC-72 drops (% of total time of evaporation) ($T_{\text{room}}$ = 28 ± 0.2 °C, $T_{\text{sub}}$ = 51 ± 0.5 °C, drop diameter: 7.6 mm, uncertainty on the IR temperature scale is ±0.5 °C).</figcaption></figure>

properties of the fluid. With the knowledge of the temperature scale, it is possible to relate the temperature difference inside the drop (as much as 10 °C for methanol) to the quick flow motion; whereas for the ethanol drop, the maximum temperature difference is only 5 °C. For a fluid with a very low heat of vaporization (FC-72 for example), the dynamics of evaporation is completely different with almost no thermal patterns inside the drop during the main phase of evaporation. The evaporation of a drop of FC-72 is closer to that of a layer of FC-72, which is another area of research.

Different physical phenomena observed can be deduced wrongly from infrared camera observation without the knowledge of the infrared properties of the fluid. As we explained above, all

investigated fluids have a different “skin penetration depth” to infrared in the range of 3–5 µm. What may be attributed to different flow motions inside the drops could simply correspond to the observation of different information.

With Figs. 5 and 6, we can observe similar behaviours for ethanol and methanol since the two fluids have the same capillary lengths (1.71 mm for methanol and 1.69 mm for ethanol). In the case of the ethanol drop, the convection cells are wider compared to methanol and evolve slowly compared to the methanol situation. The methanol drop evidence a cold central area which is much more wider in the case of ethanol. Since both drops have the same dimensions and almost the same physical properties, the only difference is the Prandtl number which is 6.98 for methanol and 22.3 for ethanol.
The ratio of Prandtl number for these fluids is over 3, which indicate a better heat transfer by convection than conduction in the case of ethanol.

### 3.3. Non-dimensional analysis

Relevant dimensionless numbers are calculated to show the driving phenomenon. The Marangoni and the Rayleigh number are calculated based on equations below where $\Delta T$ is the temperature difference between the substrate and the room temperature; $h$ and $L$ are the vertical and radial dimensions of the drop; $\rho$, $v$ and $\kappa$ are respectively the density, the kinematic viscosity and thermal diffusivity of the fluid, $g$ is the gravitational constant, $\alpha$ is the coefficient of thermal expansion and the $\gamma$ is the surface tension coefficient.

$$Ma = \frac{\gamma h^2 \Delta T}{\rho v \kappa L} \qquad (11)$$

$$Ra = \frac{\alpha g h^4 \Delta T}{\nu \kappa L} \qquad (12)$$

$$Bo = \frac{h^2 g \Delta \rho}{\sigma} \qquad (13)$$

For a typical situation encountered in this manuscript which is an ethanol drop of 4 mm in diameter and 1 mm in height with a temperature difference of 25 °C; the Marangoni number is $1.41 \times 10^4$, the Rayleigh number is $1.44 \times 10^3$, the static Bond number is 0.366. For our situation, the capillary forces are dominating inside the drop and the Rayleigh and Marangoni numbers indicate the very probable existence of thermo-capillary instabilities.

### 3.4. Influence of the substrate temperature on the convection cells

In this section, we present results obtained by studying the influence of the substrate temperature on the number of convection cells for the case of ethanol. This can only be performed under certain conditions: the thickness of the fluid needs to be appropriate for the infrared visualization. For example, when working with ethanol, the height of the drop must not exceed 1 mm to allow the visualization of the cell through the whole thickness of the fluid.

The number of convection cells evolves during the drop evaporation: they appear a short time after the drop is placed and disappear when the drop becomes a layer. Fig. 8 shows the behaviour of

<figure><img src="image_102.png" /><figcaption>Fig. 8. Number of thermo-convective cell evolution for different substrate temperatures with ethanol drops ($T_{room} = 25$ °C).</figcaption></figure>

<figure><img src="image_103.png" /><figcaption>Fig. 9. Number of thermo-convective cell per temperature difference evolution for different substrate temperatures with ethanol drops ($T_{room} = 25$ °C).</figcaption></figure>

the number of convection cells over time during the evaporation for three substrate temperatures: 35 °C, 40 °C and 50 °C. All experiments were performed on PTFE with a room temperature of 25 °C; also the initial size of the drop was the same with a diameter of 4 mm and a height of 1 mm. The number of convection cells increases with increasing surface temperature. Thereafter, we work with the temperature difference between the substrate and the room; three temperature differences were investigated: 10 °C, 15 °C and 25 °C. The decreasing behaviour of the number of cells is not linear for the three sets of data.

The number of convection cells should be related to the difference between the temperature of the substrate and the room temperature. In Fig. 9 we plot the number of convection cells divided by the temperature gradient as a function of the total time of evaporation in percentage. A good scaling for the three sets of data is obtained and evidence the non linear behaviour of the decrease in the number of convection cells during the evaporation. Concerning the time of appearance and disappearance of the convective cells, it is still not possible to relate these times to the evaporation process.

## 4. Conclusions

Analysis of sessile drop evaporation performed using an infrared camera evidences the existence of three steps during the evaporation process: first a warm-up phase, second evaporation with thermal-convective instabilities and finally evaporation without thermal patterns. The principal phase is characterized by the presence of 'convection cells'. For ethanol, we relate the decrease in convection cells during evaporation to the heat flux transferred to the drop which is measured by means of a heat fluxmeter. The link between the two factors needs to be investigated in more detail.

Furthermore, the behaviour of these convection cells has been related to the temperature difference between the heating substrate and the room temperature: a scaling law is obtained and evidence a power law variation. The paper also presents spectrophotometry measurements performed on ethanol, methanol and FC-72 in the range of 3–5 µm. This characterization is essential to perform quantitative measurement of the temperature variation along the interface. Further investigations need to be done to characterize these instabilities (threshold of appearance and disappearance, nature, ...). Also the coupling with the heat fluxmeter should
be studied carefully to check the link between the heat transfer to the drop and the mass flux of evaporation.

## Acknowledgments

This research was supported by CNES (French National Space Agency) in the frame of a grant on "Sessile Drop Evaporation".

## Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.expthermflusci.2010.12.004.

## References

[1] V.A. Kuz, Evaporation of small drops, J. Appl. Phys. 10 (15) (1991) 7034–7036.

[2] K.S. Birdi, D.T. Vu, A. Winter, A study of the evaporation rates of small water drops placed on a solid surface, J. Phys. Chem. 93 (9) (1989) 3702–3703.

[3] K.S. Birdi, D. Vu, Wettability and the evaporation rates of fluids from solids surfaces, J. Adhes. Sci. Technol. 7 (6) (1993) 485–493.

[4] F. Brochard, P.G. De Gennes, Spreading laws for liquid polymer droplets: interpretation of the foot, J. Phys. Lett. 45 (1984) 597–602.

[5] H. Hu, R.G. Larson, Evaporation of a sessile droplet on a substrate evaporation of a sessile droplet on a substrate evaporation of a sessile droplet on a substrate, J. Phys. Chem. B 6 (106) (2002) 1334–1344.

[6] O.E. Ruiz, W.Z. Black, Evaporation of water droplets placed on a heated horizontal surface, J. Heat Transfer 124 (2002) 854–863.

[7] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Contact line deposits in an evaporating drop, Phys. Rev. E 62 (1) (2000) 756–765.

[8] A.P. Boyes, A.B. Ponter, Wettability of copper and polytetrafluoroethylene surfaces with water: the influence of environmental conditions, Chem. Eng. Technol. 45 (1973) 1250–1256.

[9] R.G. Picknett, R. Bexon, The evaporation of sessile or pendant drops in still air, J. Colloid Interface Sci. 61 (2) (1977) 336–350.

[10] C. Bourges-Monnier, M. Shanahan, Influence of evaporation on contact angle, Langmuir 11 (1995) 2820–2829.

[11] J.J. Hegseth, N. Rashidnia, A. Chai, Natural convection in droplet evaporation, Phys. Rev. E 54 (2) (1996) 1640–1644.

[12] S. Chandra, M. di Marzo, Y.M. Qiao, P. Tartarini, Effect of liquid-solid contact angle on droplet evaporation, Fire Safety J. 27 (1996) 141–158.

[13] J.D. Bernardin, I. Mudawar, C.B. Walsh, E.I. Frases, Contact angle temperature dependence for water droplets on practical aluminum surfaces, Int. J. Heat Mass Transfer 40 (5) (1997) 1017–1033.

[14] N. Zhang, D.F. Chao, A new laser shadowgraphy method for measurements of dynamic contact angle and simultaneous flow visualization in a sessile drop, Opt. Laser Technol. 34 (3) (2002) 243–248.

[15] H.Y. Erbil, G. Mchale, M.I. Newton, Drop evaporation on solid surfaces: constant contact angle mode, Langmuir 18 (7) (2002) 2636–2641.

[16] E.F. Crafton, W.Z. Black, Heat transfer and evaporation rates of small liquid droplets on heated horizontal surfaces, Int. J. Heat Mass Transfer 47 (2004) 1187–1200.

[17] R. Savino, S. Fico, Transient Marangoni convection in hanging evaporating drops, Phys. Fluids 16 (10) (2004) 3738–3754.

[18] L. Grandas, C. Reynard, R. Santini, L. Tadrist, Experimental study of the evaporation of a sessile drop on a heat wall. Wetting influence, Int. J. Therm. Sci. 44 (2005) 137–146.

[19] C. Poulard, G. Guena, A.M. Cazabat, Diffusion-driven evaporation of sessile drops, J. Phys. Condens. Matter 17 (2005) 4213–4227.

[20] J. Frassy, C. Lecot, C. Delattre, A. Soucemarianadin, Droplet evaporation on solid substrates of different wetting behaviour, SHF Microfluidics 2006, vol. 1, 2006, p. 12.

[21] C. Sodtke, V.S. Ajaev, P. Stephan, Evaporation of thin liquid droplets on heated substrate, Heat Mass Transfer 43 (2007) 649–657.

[22] L. Tarozzi, A. Muscio, P. Tartarini, Experimental tests of dropwise cooling in infrared-transparent media, Exp. Therm. Fluid Sci. 31 (2007) 857–865.

[23] E. Widjaja, M.T. Harris, Numerical study of vapor phase-diffusion driven sessile drop evaporation, Comput. Chem. Eng. 32 (2008) 2169–2178.

[24] K. Sefiane, J.R. Moffat, O.K. Matar, R.V. Craster, Self-excited hydrothermal waves in evaporating sessile drops, Appl. Phys. Lett. 93 (2008).

[25] P. Tartarini, M.A. Corticelli, L. Tarozzi, Dropwise cooling: experimental tests by infrared thermography and numerical simulations, Appl. Therm. Eng. 29 (2009) 1391–1397.

[26] L. Y Barash, T. P Bigioni, V.M. Vinokur, L. N Shchur, Evaporation and fluid dynamics of a sessile drop of capillary size, Phys. Rev. E 79 (2009) 046301.

[27] D. Tam, V. Arnim, G.H. McKinley, A.E. Hosoi, Evaporation and fluid dynamics of a sessile drop of capillary size, Phys. Rev. E 79 (2009) 046301.

[28] H.K. Dhavaleswarapu, C.P. Migliaccio, S.V. Garimella, J.Y. Murthy, Experimental investigation of evaporation from low-contact-angle sessile droplets, Langmuir 26 (2) (2009) 880–888.

[29] V. Nikolayev, Dynamics of the triple contact line on a non-isothermal heater at partial wetting, Phys. Fluids 22 (2010) 082105.

[30] N. Shahidzadeh-Bonn, S. Rafa, A. Azouni, D. Bonn, Evaporating droplets, J. Fluid Mech. 549 (2006) 07–313.

[31] K.H. Kang, S.J. Lee, C.M. Lee, Visualization of flow inside a small evaporating droplet, in: 5th International Symposium on Particle Image Velocimetry, 2003.

[32] R. Tuckermann, S. Bauerecker, H.K. Cammenga, IR-thermography of evaporating acoustically levitated drops, Int. J. Thermophys. 26 (2) (2005).

[33] E. Wulsten, G. Lee, Surface temperature of acoustically levitated water microdroplets measured using infrared thermography, Chem. Eng. Sci. (2008), 1016/j.ces.2008.07.020.

[34] X. Fang, B. Li, E. Petersen, Y. Ji, J.C. Sokolov, M.H. Rafailovich, Factors controlling the drop evaporation constant, J. Phys. Chem. B 109 (43) (2005) 20554–20557.
# Annexe D

## Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate

Benjamin Sobac & David Brutin

*Physics of fluids* **24**, 032103 (2012).
# Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate

B. Sobac$^{a)}$ and D. Brutin$^{b)}$

Laboratoire IUSTI, UMR 7343 CNRS, Aix-Marseille Université, 5 rue Enrico Fermi, 13453 Marseille cedex 13, France

(Received 26 January 2011; accepted 31 January 2012; published online 6 March 2012)

The present study is an experimental investigation regarding the evaporation of ethanol drops deposited onto a heated substrate in a partial wetting situation. The originality of this work is based on the simultaneous observation of the kinetics of evaporation, heat and mass transfers, the triple-line dynamic, and thermal motions inside the drop. The triple line recedes during the drop evaporation and a spontaneous development of thermal-convective instabilities driven by the evaporation are observed. These instabilities are interpreted as hydrothermal waves induced by surface tension gradient along the free surface. An infrared technique is used to investigate the temporal and spatial dynamics of the hydrothermal waves. Results reveal a non-linear evolution of the number of waves as well as several instability regimes. A complete description of the drop evaporation with the evidence of several phases is provided. The influence of geometrical and thermal parameters has been analyzed and raised scaling laws on hydrodynamic and energy transport. The drop evaporation appears to be characterized by a constant drop Nusselt number of a value 1.7 during all the process which highlights both the importance of conduction and convection in the energy transport in an evaporating drop. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3692267]

## I. INTRODUCTION

Droplet evaporation is a phenomenon commonly encountered in nature or in industrial applications with cooling, combustion, printing and coating technologies, DNA mapping, etc. The classical physics problem of a spherical liquid droplet evaporating in an ambient atmosphere has been studied for a long time¹,² and it is well understood. However, when an evaporating drop is in contact with a surface, the loss of spherical symmetry and the appearance of a triple line significantly increase the complexity of the problem leading to numerous questions which are still without answers despite the significant progress which has been achieved in the past few decades.

Evaporation of sessile droplets with contact angles below 90° has been studied extensively.³ Experiments reveal several modes of evaporation according to the dynamic of the triple line:⁴ the constant contact-area mode, in which the contact angle vanishes while the triple line is pinned; the constant contact-angle mode, in which the contact area of the droplet on the substrate vanishes due to the recession of the triple line; and the combination of both modes. The evaporative flux from the free surface is non-homogeneous and diverges near the edge of the drop.⁵,⁶ A temperature gradient develops at the vapor-liquid interface resulting from the evaporative cooling. Thus, a surface tension gradient is generated which can induce a Marangoni flow⁷⁻¹⁰ and can give rise to Marangoni-Bénard instability¹¹ or hydrothermal waves¹² (HTWs).

While there has been much experimental and theoretical work performed on surface-tension-driven flows in thin films or shallow pools,¹³⁻¹⁵ only a few investigations deal with a convective

<footnote>a) Electronic mail: benjamin.sobac@polytech.univ-mrs.fr.</footnote>

<footnote>b) Electronic mail: david.brutin@polytech.univ-mrs.fr.</footnote>
evaporative drop. The understanding of the developed flows and hydrodynamic instabilities is of
prime importance to understand and control stains formed after drop evaporation.

The first experimental observations of Marangoni flow in an evaporating drop were carried out
by Hegseth¹⁶ in a pendant drop (methanol using particles as tracers) and by Zhang and Chao¹⁷
in a sessile drop (n-pentane and freon-113 using a laser shadowgraphy method). Recently, new
visualizations using infrared thermography have been performed.¹²,¹⁸⁻²² This non-intrusive thermal
measurement combined with numerical work allowed Savino and Fico¹⁸ to investigate the influence
of Marangoni stresses on an evaporative pendant drop. Velocity and temperature distributions or
heat transfer between the drop and the plate appear modified, whereas previous studies neglected
thermocapillary effects. These observations on the effects of Marangoni stresses are confirmed by
numerical investigations in the situation of sessile drop evaporation.⁷,⁸ Ristenpart et al.⁹ revealed that
the direction of Marangoni flow depends on the relative thermal conductivities of the substrate and
the liquid, and the direction reverses at a critical contact angle. Xu et al.²³ added that the direction of
the Marangoni flow is also dependent on the ratio of the substrate thickness to the contact line radius
of the droplet. By eliminating the buoyancy-driven convection, Ghasemi and Ward²⁴ compared the
energy transport by thermocapillary convection and thermal conduction. The dominant mode varies
along the interface, but the thermocapillary convection is by far the larger mode of transport in the
three triple-line regions where most of the evaporation occurs. A discontinuity of the temperature
at the interface was reported but no reversal was observed. Sefiane et al.¹²,²⁵ observed at the free
surface of sessile pinned drops under evaporation the presence of patterns which were interpreted in
terms of HTWs for methanol and ethanol. The dynamic of the number of waves decreases linearly
and is influenced by the thermal conductivity of the substrate.

HTWs were previously investigated by several authors in easier configurations (shallow pools
in one or two dimensions). Smith and Davis performed a linear stability analysis of an infinite layer
subject to a temperature gradient along the interface, and revealed the existence of a special regime
of time-oscillatory nature.¹³ The regular thermal fluctuations called HTWs are unsteady and inherent
to the three-dimensional effect. These waves propagate from the cold side to the hot side and depend
on the temperature gradient imposed and the fluid depth. Different forms and kinds of HTWs have
been experimentally observed and summarized by Garnier¹⁴ for a 2D annular geometry (see Fig. 1).

We recently reported²² the visualization of different thermocapillary instabilities in three fluids
(methanol, ethanol, and FC-72) under forced evaporation conditions. The present work focuses
on the evaporation of ethanol droplets on a heated substrate for the case of partial wetting. The
evaporation mainly occurs with the recession of the triple line. An important and innovative feature
of the experimental part of this study is the simultaneous recording of the kinetics of evaporation,
heat and mass transfers, droplet shape, and the thermal motion within the drop. Thus, the heat flow
absorbed by the drop on the substrate during the evaporation is experimentally obtained for the first
time and this measure allows a characterization of the dynamic of evaporation. A scaling law on the
energy transport has been found and reveals both the importance of conduction and convection in
the energy transport in an evaporating drop. The use of an infrared camera on a semi-transparent

<figure><img src="image_104.png" /><figcaption>FIG. 1. Different regimes for thermocapillary and thermogravitational flow, depending on the fluid depth *h* based on the works of Garnier.¹⁴ Dimensionless numbers are defined in Table. II.</figcaption></figure>
fluid reveals the existence of a complicated and non-axisymmetric flow driven by thermocapillary forces. These thermocapillary instabilities are interpreted in terms of HTWs and the dynamics and evolutions of these waves are investigated. The strength of this paper is to study this flow and its associated transfers which are largely unknown.

## II. EXPERIMENTAL SETUP AND PROCEDURE

### A. Principle

The experiment consists of the evaporation of a sessile drop deposited onto a heated substrate. The drop is created using a micropipet (Eppendorf X-Stream) to control the volume of the drop and gently laid down on the substrate. The drop evaporates into air inside an experimental cell to avoid any perturbations from external flow. The environmental temperature $T_a$, pressure $P_a$, and humidity $H$ are measured using a weather station (Lufft opus). Evaporation occurs under atmospheric pressure $P_a = 1$ atm and humidity $H \approx 40\%-50\%$. The test cell contains an instrumentation dedicated to thermal measurements (temperature and heat-flux density). The drop is observed by means of both visible and infrared cameras. The experiment is designed to be under a non-saturated vapor condition using an adequately sized test cell. The experimental setup is presented in Fig. 2.

### B. Materials

The substrate is a complex assembly of layers mainly composed of an aluminum cylinder (diameter = 10 mm, height = 8 mm) heated from below by a resistance and regulated by a PID regulation in order to have a temperature imposed boundary condition at the top of the substrate. A heat fluxmeter is placed overhead (Captec, diameter = 10 mm, thickness = 0.6 mm, sensitivity = $0.725~\mu\text{VW}^{-1}\text{m}^2$) to determine the heat flow absorbed by the drop on the substrate during the evaporation. Then, black mast paintings and a film of polytetrafluoroethylene (PTFE) are coated upon the sensor.

The liquid investigated is pure ethanol (99.9%). This liquid is used due to its weak phase change enthalpy and its properties of semi-transparency in the infrared wavelength of the infrared camera from 3 to 5 μm. The property of semi-transparency of the fluid in the wavelength range of the infrared camera allows to observe the thermal motion in a layer of fluid and consequently the flow motion. The advantage of this method lies in its non-destructive properties which do not disturb the

<figure><img src="image_105.png" /><figcaption>FIG. 2. (Color online) Experimental setup.</figcaption></figure>
<figure><img src="image_106.png" /><figcaption>FIG. 3. (Color online) Global emissivity of ethanol $\epsilon$ versus the fluid thickness $h$ in the range of wavelengths of the infrared camera used from 3 to 5 $\mu$m. Experimental data (●), fit by $\epsilon = e^{-Ah}$ with $A = 1.85 \text{ mm}^{-1} \pm 2.1\%$ (–).</figcaption></figure>

flow motion. If the fluid is opaque, the thermal motion at the liquid-air interface will be observed. If the liquid is transparent, the temperature of the substrate surface will be visualized. The global emissivity of ethanol $\epsilon$ in this wavelength range has been obtained using a spectrophotometer (Ftir Nicolet Nexus 560) and is provided in Fig. 3. The fluid thicknesses investigated are in the range 0.2–2.0 mm which corresponds to the range of drop heights encountered. Indeed, the thickness of created drops is always inferior to the capillary length $L_c = \sqrt{(\sigma/\rho g)} = 1.69$ mm with $\rho$ being the fluid density, $g$ being the gravitational constant, and $\sigma$ being the surface tension. The emissivity variation shown with this figure can be fitted by a decreasing exponential law where the constant $A$ is equal to $1.85 \text{ mm}^{-1} \pm 2.1\%$ for ethanol. The other relevant physical parameters of this fluid at $25 \,^{\circ}\text{C}$ are given in Table I.

The drop is in contact with a relatively smooth film of $200 \ \mu\text{m}$ of PTFE. The surface was characterized with respect to morphology using an atomic force microscope (Veeco explorer) with respect to wettability using a contact-angle measurement system (Kruss DSA 30). A root mean squared roughness of $336 \ \text{nm} \pm 20\%$ (see the AFM scan provided in Fig. 4) and an average static contact angle of $35 \pm 2^{\circ}$ have been determined. The hysteresis of the angle for the binary ethanol/PTFE has also been investigated and is $\Delta\theta = 11 \pm 1^{\circ}$ using the difference between the advancing and receding contact angles.

The dimensions of the drop are recorded laterally using a digital camera (JAI BM500GE, 2456 $\times$ 2058 pixels) coupled with a microscope lens (VZM100i) with a spatial resolution of $4 \ \mu\text{m}$. Then, image processing is performed to monitor the evolution of the geometrical parameters related to the drop, such as the contact line radius $R$, the drop height $h$, and the contact angle $\theta$ using commercial software (Kruss DSA3).

The thermal motion visualization is performed with an infrared camera (Flir SC6000, 640 $\times$ 512 pixels) fitted with a microscope lens making it possible to reach a resolution of $10 \ \mu\text{m}$ with a field of view of $6.4 \text{ mm} \times 5.12 \text{ mm}$. This device is vertically assembled at the top of the drop. The infrared camera records radiation coming from the substrate and the fluid volume. The infrared heat flux transmitted to the infrared camera depends on a form factor as well as the fluid and solid emissivities. Parameters related to the liquid evolve due to the mass transfer making it difficult

<table><caption>TABLE I. Physical properties of pure ethanol at $25^{\circ}\text{C}$ and 1 atm: liquid and vapor densities, heat capacity, latent heat of vaporization, thermal conductivity, dynamic viscosity, surface tension, and saturation temperature.</caption><thead><tr><th>Symbol</th><th>&rho;<sub>L</sub></th><th>&rho;<sub>V</sub></th><th>Cp</th><th>L<sub>V</sub></th><th>&lambda;</th><th>&mu;<sub>L</sub></th><th>&sigma;</th><th>T<sub>sat</sub></th></tr></thead><tbody><tr><td>Unit</td><td>kg m<sup>-3</sup></td><td>kg m<sup>-3</sup></td><td>J kg<sup>-1</sup> K<sup>-1</sup></td><td>kJ kg<sup>-1</sup></td><td>W m<sup>-1</sup> K<sup>-1</sup></td><td>mPa s</td><td>mN m<sup>-1</sup></td><td>&deg;C</td></tr><tr><td>Value</td><td>789</td><td>1.50</td><td>2845</td><td>922.9</td><td>0.140</td><td>1.095</td><td>22.0</td><td>78.0</td></tr></tbody></table>
to convert the measured luminance into the mean temperature of the semi-transparent liquid layer. The copper heat fluxmeter is painted in black in order to have an almost null reflectivity (substrate emissivity equals to 0.95).

The experimental cell temperature is obtained with a K-type thermocouple recorded at 1 Hz. The heat fluxmeter is also instrumented in its center by a T-type thermocouple which allows the monitoring of the substrate temperature. All delivered signals are recorded on an acquisition station (HP 34970A).

### C. Methods

The temperature of the substrate $T_s$ is imposed. The infrared camera is used to check thermal homogeneity of our substrate before the experiment. A drop of fixed volume $V$ is created at the tip of a syringe and is then gently laid down on the substrate.

The drop first spreads instantaneously and stays pinned for a while with a decrease of the contact angle. Then, the drop recedes until its complete vanishing. For all experiments, the majority of the evaporation occurs with the recession of the triple line. The pinning mode represents at most 20% of the evaporation time. The recession of the triple line and the volatility of ethanol which impose fast kinetics of evaporation can sometimes disturb the axisymmetry of the drop. It is a potential cause of artefacts since geometrical parameters are only recorded from one side.

All measurement recordings are launched before the contact between the drop and the substrate; however, the beginning of the experiment is taken as the point when the drop touches the substrate. The total time of evaporation $t_F$ corresponds to the time when all the drop has been evaporated.

The data directly obtained from the heat fluxmeter is the heat-flux density crossing the total area of the sensor. The heat flow absorbed by the drop $P$ is calculated with a heat flow balance since a part of the heat flow crossing the heat fluxmeter $P_{sub}$ is lost by natural convection with the air $P_{conv}$ (the drop diameter being inferior to the fluxmeter diameter):

$$P_{sub} = P + P_{conv}, \qquad (1)$$

$$P = S_{sub} (Q_{sub} - Q_{conv} (1 - (S/S_{sub}))), \qquad (2)$$

where $S_{sub}$ is the total area of the sensor, $S$ is the wetting area of the drop whose evolution is followed by a visible camera, $Q_{conv}$ is the heat-flux density lost by natural external convection, and $Q_{sub}$ is the total heat-flux density crossing the sensor. $Q_{conv}$ is obtained for an imposed temperature before the deposition of the drop.

The PTFE film is changed to avoid any cleaning issues and to guarantee the reproducibility of the surface quality every three experiments. Also, a cleaning procedure using optical paper and compressed air is performed between each experiment in order to remove any residual depositions that can occur after an evaporation, and renew the atmosphere of the cell. Finally, the reproducibility of the experimental results was checked by repeating a set of experiments at least three times.

### III. DROPLET EVAPORATION DESCRIPTION

Experiments were conducted at different imposed temperatures from 25.0 °C to the temperature of saturation. In addition, the initial wetting radius of the droplet varies between 1.2 mm and 2.4 mm. The characteristic dimension of the problem is then below or of the same order of magnitude as the capillary length that allows the use of the assumption of spherical cap geometry.

In the following, a complete description of the evaporation of an ethanol sessile droplet posed on a heated substrate is presented for a specific case and the influence of geometrical and energetic parameters is investigated thereafter.

In the presented experiment, a drop of 7 µl in volume lies on the substrate at an imposed temperature $T_s = 34.5$ °C and evaporates during $t_F = 187$ s. The temperature difference between the substrate and the atmosphere far away, which controls the evaporating flux, is then defined as $\Delta T = T_s - T_{atm} = 16.0$ °C. The evolution of various thermal and geometric parameters obtained
<figure><img src="image_107.png" /><figcaption>FIG. 4. (Color online) Atomic force microscopy picture of the PTFE coating (root mean squared roughness of 336 nm ± 20%).</figcaption></figure>

during this experiment is provided in Fig. 7: geometrical parameters from the visible video, thermal motion from the infrared measurements, the temperature of the substrate, and the heat flow from the fluxmeter.

A. Drop shape evolution

The initial apparent contact angle is $\theta_0 = 36.9^\circ$. The evaporation begins with a constant wetting radius for about 20% of the evaporation time. During this period, the contact angle decreases linearly over time. When the variation of the contact angle corresponds to the hysteresis of the angle, the contact line shrinks until the drop vanishes. The evaporation occurs with the recession of the triple line for the majority of the evaporation. During the receding drop evaporation phase, the shrinking dynamic is correctly fitted by $R(t) \propto (t_F - t)^y$ where exponent $y \simeq \frac{1}{2}$.²⁷ The evolution of the wetting radius is provided in Fig. 5.

<figure><img src="image_108.png" /><figcaption>FIG. 5. (Color online) Wetting drop radius $R$ versus time before vanishing $t_F - t$. Experimental data (○), fit by $R = a(t_F - t)^y$ with $a = 0.19 \text{ mm s}^{-1} \pm 0.7\%$ and $y = 0.50 \pm 0.3\%$ (–).</figcaption></figure>
<figure><img src="image_109.png" /><figcaption>FIG. 6. (Color online) Drop mass *m* versus time before vanishing *t*<sub>F</sub> − *t*. Experimental data (o), fit by *m* = *a*(t<sub>F</sub> − *t*)<sup>*y*</sup> with *a* = 1.92e−9 µgs<sup>−1</sup> and *y* = 1.52 for the receding drop evaporation part (−) and by *m* = *c*(t<sub>F</sub> − *t*) + *d* with *c* = 3.87e−8 µg.s<sup>−1</sup> and *d* = −1.93e−6 µg for the pinned drop evaporation part (−). Inset: Evaporation rate |dm/dt| deduced by derivation of these previous fits versus time before vanishing t<sub>F</sub> − t.</figcaption></figure>

B. Evaporative rate

The mass evolution of the drop during the vanishing time is presented in Fig. 6. The mass *m* is obtained by calculation from the geometrical parameters of the drop assuming a spherical cap geometry. When the triple line is pinned, the mass evolution decreases linearly over time. Then, when the contact line recedes until the complete vanishing of the drop, the mass evolves with a power law. The evolution of the evaporation rate |dm/dt| is deduced by derivation of mass fits. The evaporation rate evolves with a dynamics similar to that of the wetting radius. Indeed, the evaporation rate is constant when the triple line is pinned and decreases as soon as the triple line recedes with a power law dynamic. The exponent of the evaporation rate is equal to 0.52, a difference of 4.8% is noted compared to the exponent of the wetting radius evolution. This result generalizes in the situation of forced evaporation the fact that the evaporation is linearly related to the evolution of the wetting radius dm/dt ∝ R.<sup>5,26,28</sup> Indeed, when the contact angle is less than 40°, the dynamics of the contact angle on the dynamics of evaporation is negligible;<sup>6,28</sup> thus, this configuration allows to directly observe the influence of the wetting radius.

C. Heat and fluid flows

The visualization of an evaporative droplet with an infrared camera and the monitoring of the
heat flow transmitted to the drop allow us to describe the sessile drop evaporation process with four
phases based on Fig. 7:

* Phase 0—Drop warming up—The start of the experiment is defined by the initial contact of the drop with the substrate. The first seconds of the experiment are characterized by the transient heating of the drop. The drop which was initially at room temperature is laid down on a warm surface under normal atmospheric conditions; the drop is first heated to almost reach the substrate temperature.

* Phase 1—Drop evaporation with thermal-convective instabilities—The heat flow reaches a maximum value. At the same time, the development of thermal oscillatory waves close to the triple line occurs as shown in Fig. 8(a). A periodic pattern in space and time is observed.
<figure><img src="image_110.png" /><figcaption>FIG. 7. (Color online) Different simultaneous datas versus the time $t$ for an ethanol drop evaporating on a substrate at the temperature $T_s = 34.5^{\circ}$C. [$R_0 = 2.40$ mm, $h_0 = 0.75$ mm, $V_0 = 7$ µL, $\Delta T = 16.0^{\circ}$C, $t_F = 187$ s, $\theta_0 = 36.9^{\circ}$].</figcaption></figure>

Spiral-like arms organized radially are circling at the periphery. The waves are observed in the region near the triple line where most of the evaporation occurs. The dynamic of these waves is investigated in Sec. IV B. No motion is observed at the center with the IR camera but it does not mean that there is no convective motion. If an axisymmetric flow pattern develops, no cellular structure can be detected in spite of the presence of convection.

* Phase 2—Transition—This phase begins with the destabilization of the central part of the drop flow and finishes with the complete disappearance of the waves. It is difficult to precisely follow the waves dynamic during this phase. Disturbances observed are notably due to the fast and not always perfectly axisymmetric receding of the triple line since the kinetics of evaporation is fast.

* Phase 3—Droplet evaporation—This last phase of evaporation is characterized by a complete lack of waves. The drop volume decreases until it has completely disappeared.

Figure 7 shows slight oscillations in the heat flow absorbed by the drop and on the substrate temperature. These simultaneous oscillations are inherent to the PID regulation system. They are small enough not to disturb the phenomena.
Whatever the stage in the evaporation process, as soon as the quasi-steady configuration is reached, the infrared visualization emphasizes the fact that the temperature of the drop is hotter near the triple line and colder in the center. This observation reveals the fact that the apex is the coldest point of the drop due to the larger drop thickness which leads to a larger thermal resistance. This observation is easily understandable by a simple calculation assuming that all the energy is transported by thermal conduction in the liquid layer. Results of numerical simulations<sup>7,8</sup> also confirm that the apex of the drop is the coldest point considering either pure thermal conduction or thermal conduction and Marangoni convection in the liquid phase. Thus, considering the Fourier's law, the maximum temperature difference in the drop is expressed by $\Delta T = \frac{hP}{\lambda_S}$ with h being the apex thickness, P being the heat flow, $\lambda$ being the thermal conductivity, and S being the contact area between the drop and the substrate. In the configuration presented in Fig. 7, the temperature difference between the substrate and the apex is about 8 °C when the quasi-steady configuration is reached. Then, the temperature difference decreases until the end of the evaporation process. This temperature difference should overpredict the real temperature difference since, first, the internal convective flow motion should homogenize the temperature field and, secondly, the heat fluxmeter provides a mean measurement on the concerned surface. Thus, the calculation assumes a homogeneity of the heat-flux density at the solid-liquid interface which is not true since the heat-flux density is more important at the triple line.

The disturbances observed at the liquid-gas interface during drop evaporation are interpreted in terms of hydrothermal waves as presented in Sec. IV. Indeed, the condensation of water vapor at the surface or the absorption of water molecules from humid air into ethanol liquid can be excluded of the interpretation. The whole of experiments were performed with an ambient temperature $T_a$ = 19 ± 1 °C and a humidity of 45 ± 3%. Thus, the due point corresponds to a wet temperature about 7 °C. Calculations performed considering thermal conduction in the liquid drop show that, in the worst configuration which corresponds to the case of an imposed temperature of 25 °C, the lowest temperature of the drop is 18 °C. Thus, the minimum apex temperature is far from the temperature of condensation. Considering the possible absorption of water from humid air, the diffusion coefficient of vapor water into liquid ethanol is calculated using the formula from Perry's Chemical Engineer's Handbook<sup>29</sup> and compared to the diffusion coefficient of vapor ethanol into air in order to compare absorption and evaporation mechanisms. The ratio of these two quantities reveals that the absorption of water from humid air in pure liquid ethanol is negligible in the problem since $\frac{D_{H_2O(g)}-C_{2H_6(l)}}{D_{C_{2H_6(l)}-air}} \approx 2 \times 10^{-4}$. Consequently, the disturbances observed are not resulting from absorption or condensation mechanisms.

# IV. THERMOCAPILLARY FLOW ANALYSIS

## A. Origin

An estimation of relevant dimensionless numbers of the problem is provided in Table II in order to have a better understanding about the mechanisms involved in our problem. Thus, Prandtl, static Bond, Rayleigh, Marangoni, and dynamic Bond numbers, given by Pr, Bo, Ra, Ma, and Bd, are defined in Table II. T denotes the temperature; R and h are the wetting radius and the height of the drop; $\rho$, $v$, and $\alpha$ are the density, kinematic viscosity, and thermal diffusivity, respectively; g is the gravitational acceleration; $\sigma$ is the surface tension; $\beta$ is the coefficient of thermal expansion; and $\gamma = -(\partial\sigma/\partial T)$ is the coefficient of surface tension. The dynamic Bond number Bd defined as the ratio of thermogravity forces to thermocapillary forces reveals that thermocapillary forces dominate those

<table><caption>TABLE II. Estimation of characteristic dimensionless numbers of the problem for the evaporating drop presented in Fig. 7.</caption><thead><tr><th>Number</th><th>Pr</th><th>Bo</th><th>Ra</th><th>Ma</th><th>Bd</th></tr></thead><tbody><tr><td>Formula</td><td>$\frac{v}{\alpha}$</td><td>$\frac{\rho g h^2}{\sigma}$</td><td>$\frac{\beta g \Delta T h^4}{v \alpha R}$</td><td>$\frac{-\frac{d\sigma}{dT} \Delta T h^2}{\mu \alpha R}$</td><td>$\frac{Ra}{Ma}$</td></tr><tr><td>Value</td><td>22.4</td><td>0.19</td><td>184</td><td>3318</td><td>0.056</td></tr></tbody></table>
<figure><img src="image_111.png" /><figcaption>FIG. 8. (Color online) Simultaneous visualizations of an ethanol drop evaporating on a substrate at the temperature $T_s = 60^{\circ}$C, with an initial wetting radius $R_0 = 2.15$ mm. (a) HTWs observed with an infrared camera by top view. (b) Drop side picture with visible camera. Picture extracted from the movie provided online (enhanced online) [URL: http://dx.doi.org/10.1063/1.3692267.1].</figcaption></figure>

related to buoyancy. The evaporation should be accompanied by the formation of thermocapillary-driven patterns. Indeed, a temperature gradient appears at the free interface of the drop due to evaporation and the influence of buoyancy is very small since the initial thickness of the drop is always below the capillary length. Considering the work of Garnier¹⁴ about thermocapillary-driven patterns in 2D annular configurations subject to a horizontal temperature gradient, the observations shown in Fig. 8 are consistent with thermally induced waves called HTWs. Different kinds of HTWs according to their conditions have been observed by Garnier and summarized in Fig. 1. The two characteristic lengths separating the different instability regimes can be calculated in our example for ethanol: the capillary length $L_c = 1.69$ mm and the thermal length $L_{th} = L_c\sqrt{\gamma/(\sigma\beta)} = 3.30$ mm. According to this analysis, our problem is placed in the capillary flow part since $h < L_c$. However, even if an analogy is possible, some differences have to be noticed since the drop configuration is more complex. The drop evaporation problem is a three-dimensional configuration with a curved free surface and a motion of the interface. Besides, the temperature gradient exists along the interface and it is not only on a horizontal direction due to the geometry of the drop. This temperature gradient evolves during the evaporation and the gradient by length unit ($\Delta T/l$) stays elsewhere more important on the vertical direction.

## B. Dynamic of HTWs

In the first phase of evaporation, the spontaneous development of HTWs is observed. The arm-like patterns are organized radially and spaced with a quasi-constant angle according to the axisymmetry of the triple line. The HTWs circle around the apex in the region where most of the evaporation occurs. The orthoradial traveling has no principal direction of propagation; the motion can be clockwise or anti-clockwise. Elsewhere, a pattern can be composed of right and left propagating waves from a source to a sink location. This last regime is presented in Fig. 9. One can note that the waves observed in our configuration seem to be similar to what Garnier presented as HTW1. However, our problem is placed in the diagram in capillary flow where HTW2 occurs. This difference can be explained by the difference of configurations.

During the evaporation, the number of wave decreases with time. Figure 10 illustrates the evolution of the number of wave $N_w$ versus a normalized time $t/t_F$. The evolution follows a power law $N_w(t) = a(t/t_F)^b$. This trend is confirmed by the linearity of the data presented in a log-log
<figure><img src="image_112.png" /><figcaption>FIG. 9. (Color online) (a) Infrared pictures of an evaporating drop of ethanol with the presence HTWs [$T_s = 60°C$ and $R_0 = 2.16$ mm]. (b) Zoom on the source location with the visualization of the motion of HTWs in time.</figcaption></figure>

scale. The error bars signify uncertainties in the visual observation of the number of waves. The decrease of the number of waves over time is consistent with the driving mechanism of HTWs. Indeed, the evaporation of a sessile drop is accompanied by a decrease of the drop thickness as well as a decrease of the temperature gradient along the free surface³⁰ which lead to a decrease of the driving force.

In the second phase of evaporation, the decrease of the drop volume and the recession of the triple line cause the destabilization of the drop. At this stage, it appears difficult to follow the waves. At a point, the driving force has to be too low for the development of traveling wave trains and the drop continues to evaporate without the presence of HTWs.
<figure><img src="image_113.png" /><figcaption>FIG. 10. (Color online) Number of HTWs $N_w$ versus the normalized time $t/t_F$ for the evaporation of an ethanol drop with an initial radius $R_0 = 2.34$ mm, posed on a substrate at the temperature $T_s = 39.6^{\circ}$C. Experimental data (■), fit by $N_w = a(t/t_F)^b$ with $a = 13.4$ and $b = -0.185$ ($-$). Inset: data plot in log-log scale.</figcaption></figure>

Sefiane et al.¹² obtained for ethanol and methanol a linear decrease in the number of wave overtime for pinned drops. Explanations of the difference between these evolutions can be found in the difference of experimental conditions: in the thermal resistances of the substrates or in the triple-line dynamics (pinned or receding).

### C. Geometrical and thermal influence

The influence of both geometrical and thermal parameters on the evaporation mechanisms has been investigated varying the initial wetting diameter and the substrate temperature. Figure 11(a) presents the evolution of the number of waves in log-log scale over one decade in the ordinate. The number of waves evolves with a power law for all cases. One can notice that the evolution of the number of waves is affected by these two parameters which are directly related to the driving force of these thermocapillary instabilities:

* At a fixed imposed substrate temperature, the number of waves increases with the drop size.

* At a fixed drop size, the number of waves increases with the increase of the substrate temperature.

Figure 11 (b) reveals that all data collapse using the following scaling:

$$N_w = a \left( \frac{d_0}{L_c} \right) \left( \frac{\Delta T}{T_s} \right) \left( \frac{t}{t_F} \right)^b. \quad (3)$$

The two coefficients obtained are $a = 101.7 \pm 2.9\%$ and $b = -0.196 \pm 5.8\%$ and are not a function of the wetting drop diameter and the temperature difference between the substrate and the atmosphere. This scaling law presented over one decade must be validated for other experimental conditions: different working fluids, experimental room conditions, substrate thermal properties. The temperature difference used in this scaling is global and maximizes the real temperature gradient along the liquid-vapor interface. Presently, this last difference is only qualitative since several improvements are under development on the infrared treatment to obtain this local data. With this real temperature difference, a better scaling of the data should be obtained in the future.
<figure><img src="image_114.png" /><figcaption>FIG. 11. (Color online) (a) Number of HTWs $N_w$ versus the normalized time $t/t_F$ for various initial wetting diameters and substrate temperatures in log-log scale. (b) Scaled number of waves evolution in log-log scale. Experimental data (different shade symbols (color online)), fit by $N_w = a(d_0/L_c)(\Delta T/T_s)(t/t_F)^b$ with $a = 101.7 \pm 2.9\%$ and $b = -0.196 \pm 5.8\%$ (-).</figcaption></figure>

V. ENERGY TRANSPORT ANALYSIS

Figure 12(a) presents the evolution of the heat flow absorbed by the drop from the substrate P for various sizes of drop and substrate temperature. The different curves collapse using the Nusselt number $N_u$ which compares the convection and the conduction effect on the energy transport. The wetting drop radius $R(t)$ is defined as the characteristic length scale of the problem. The Nusselt number of the drop $N_u$ is defined below:

$$N_u(t) = \frac{H(t) \cdot R(t)}{\lambda} = \frac{Q(t) \cdot R(t)}{\Delta T \cdot \lambda} = \frac{2P(t)}{\pi d(t) \Delta T \lambda}, \quad (4)$$

where H is the heat transfer convection coefficient and λ is the heat conductivity. The drop evaporation is characterized by an almost constant Nusselt number of an average value of 1.7. The effect of PID
<figure><img src="image_115.png" /><figcaption>FIG. 12. (Color online) (a) Heat flow *P* versus the normalized time *t/t*<sub>*F*</sub> for various temperature differences and initial wetting diameters (color online). (b) Nusselt number *Nu* versus the normalized time *t/t*<sub>*F*</sub>.</figcaption></figure>

regulation combined to a geometrical parameters extraction from one side where a non-uniform
stick-slip behavior is observed induces a diffusion of the curves. However, the result highlights both
importance of conduction and convection on the energy transport in an evaporating drop. Recently,
Ghasemi and Ward²⁴ also revealed that thermal conduction and thermocapillary convection are active
during water drop evaporation (buoyancy-driven convection was eliminated). However, from a local
analysis, the dominant mode varies along the interface. Thus, the energy transport by thermocapillary
convection is by far the larger mode of energy transport near the triple line; the region where most
of the drop evaporation occurs.

VI. CONCLUSIONS

We developed an experimental setup in order to perform simultaneous thermal and fluid motion
observations in an evaporating sessile drop. Using infrared and visible videos, a heat fluxmeter placed
below the drop, the evaporation process was completely described based on a reference experiment.
The existence of four main phases was observed, with in particular the spontaneous appearance of
thermal traveling waves analyzed as hydrothermal waves at the first step of the evaporation. These
surface-tension-driven instabilities are due to the temperature gradient which develops at the free
surface of the drop. During the drop evaporation, the number of waves decreases since the geometry
and the free interface temperature gradient evolve. The internal wave dynamic and the number of
waves evolution were studied and are influenced by geometrical parameters as well as the substrate
temperature. The number of waves appears to follow a power law overtime. A scaling law on the
number of these waves evolution was provided, completed with another scaling law on the energy
transport. Indeed, based on our experimental observation, the evaporation of a drop is characterized
by an almost constant drop Nusselt number of a value of 1.7 which highlights both the importance
of conduction and convection in the energy transport in an evaporating drop.

ACKNOWLEDGMENTS

We thank P. Cervetti, P. Lantoine, S. Martinez, and S. Noel for their technical support during
the design of the experimental setup; L. Voglimacci for preliminary experiments; and F. Rigollet
for fruitful discussions. Fundings from Agence Nationale de la Recherche (Project NanoSurf ANR-
09-BLAN-0093-03) and Centre National d'Etudes Spatiales (grant "Sessile Drop Evaporation") are
gratefully acknowledged.

<footnote n="1">¹ J. C. Maxwell, *Diffusion Collected Scientific Paper* (Encyclopedia Britannica, Cambridge, 1877).</footnote>

<footnote n="2">² I. Langmuir, "The evaporation of small sphere," *Phys. Rev.* **12**, 368 (1918).</footnote>

<footnote n="3">³ H. Y. Erbil, "Evaporation of pure liquid sessile and spherical suspended drops: A review," *Adv. Colloid Interface Sci.* **170**, 67 (2012).</footnote>

<footnote n="4">⁴ R. G. Picknett and R. Bexon, "The evaporation of sessile or pendant drops in still air," *J. Colloid Interface Sci.* **61**(2), 336 (1977).</footnote>

<footnote n="5">⁵ R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, "Contact line deposits in an evaporating drop," *Phys. Rev. E* **62**(1), 756 (2000).</footnote>

<footnote n="6">⁶ H. Hu and R. G. Larson, "Evaporation of a sessile droplet on a substrate," *J. Phy. Chem. B* **6** (106), 1334 (2002).</footnote>

<footnote n="7">⁷ H. Hu and R. G. Larson, "Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet," *Langmuir* **21**, 3972 (2005).</footnote>

<footnote n="8">⁸ F. Girard, M. Antoni, and K. Sefiane, "On the effect of Marangoni flow on evaporation rates of heated water drops," *Langmuir* **24**, 9207 (2008).</footnote>

<footnote n="9">⁹ W. D. Ristenpart, P. G. Kim, C. Domingues, J. Wan, and H. A. Stone, "Influence of substrate conductivity on circulation reversal in evaporating drops," *Phys. Rev. Lett.* **99**, 234502 (2007).</footnote>

<footnote n="10">¹⁰ L. Y. Barash, T. P. Bigioni, V. M. Vinokur, and L. N. Shchur, "Evaporation and fluid dynamics of a sessile drop of capillary size," *Phys. Rev. E* **79**, 046301 (2009).</footnote>

<footnote n="11">¹¹ V. X. Nguyen and K. J. Stebe, "Patterning of small particles by a surfactant-enhanced Marangoni-Bénard instability," *Phy. Rev. Lett.* **88**, 164501 (2002).</footnote>

<footnote n="12">¹² K. Sefiane, J. R. Moffat, O. K. Matar, and R. V. Craster, "Self-excited hydrothermal waves in evaporating sessile drops," *Appl. Phy. Lett.* **93**, 074103 (2008).</footnote>

<footnote n="13">¹³ M. K. Smith and S. H. Davis, "Instabilities of dynamic thermocapillary liquid layers," *J. Fluid Mech.* **132**, 119 (1983).</footnote>

<footnote n="14">¹⁴ N. Garnier, "Ondes non-linéaires à une et deux dimensions dans une mince couche de fluide," Ph.D. dissertation (Thèse de l'Université Denis Diderot, Paris 7, 2000).</footnote>

<footnote n="15">¹⁵ P. Kavehpour, B. Ovryn, and G. H. McKinley, "Evaporatively-driven Marangoni instabilities of volatile liquid films spreading on thermally conductive substrate," *Colloids Surf. A* **206**, 409 (2002).</footnote>

<footnote n="16">¹⁶ J. J. Hegseth, N. Rashidnia, and A. Chai, "Natural convection in droplet evaporation," *Phy. Rev. E* **54**(2), 1640 (1996).</footnote>

<footnote n="17">¹⁷ N. Zhang and D. F. Chao, "A new laser shadowgraphy method for measurements of dynamic contact angle and simultaneous flow visualization in a sessile drop," *Opt. Laser Technol.* **34**(3), 243 (2002).</footnote>

<footnote n="18">¹⁸ R. Savino and S. Fico, "Transient Marangoni convection in hanging evaporating drops," *Phys. Fluids* **16**(10), 3738 (2004).</footnote>

<footnote n="19">¹⁹ L. Tarozzi, A. Muscio, and P. Tartarini, "Experimental tests of dropwise cooling in infrared-transparent media," *Exp. Therm. Fluid Sci.* **31**, 857 (2007).</footnote>

<footnote n="20">²⁰ P. Tartarini, M. A. Corticelli, and L. Tarozzi, "Dropwise cooling: Experimental tests by infrared thermography," *Appl. Therm. Eng.* **29**(7), 1391 (2009).</footnote>

<footnote n="21">²¹ F. Girard, M. Antoni, and K. Sefiane, "Infrared thermography investigation of an evaporating sessile water droplet on heated substrates," *Langmuir* **26**(7), 4576 (2010).</footnote>

<footnote n="22">²² D. Brutin, B. Sobac, F. Rigolet, and C. Le Niliot, "Infrared visualization of thermal motion inside a sessile drop deposited onto a heated surface," *Exp. Therm. Fluid Sci.* **35**(3), 521 (2011).</footnote>

<footnote n="23">²³ X. Xu, J. Luo, and D. Guo, "Criterion for reversal of thermal Marangoni flow in drying drops," *Langmuir* **26**, 1918 (2009).</footnote>

<footnote n="24">²⁴ H. Ghasemi and C. A. Ward, "Energy transport by thermocapillary convection during sessile water droplet evaporation," *Phys. Rev. Lett.* **105**, 136102 (2010).</footnote>

<footnote n="25">²⁵ K. Sefiane, A. Steinchen, and R. Moffat, "On hydrothermal waves observed during evaporation of sessile droplets," *Colloids Surf. A* **365**, 95 (2010).</footnote>

<footnote n="26">²⁶ L. Grandas, C. Reynard, R. Santini, and L. Tadrist, "Experimental study of the evaporation of a sessile drop on a heat wall. Wetting influence," *Int. J. Therm. Sci.* **44**, 137 (2005).</footnote>
<footnote n="27"><sup>27</sup> N. Shahidzadeh-Bonn, S. Rafaï, A. Azouni, and D. Bonn, "Evaporating droplets," *J. Fluid Mech.* **549**, 307 (2006).</footnote>

<footnote n="28"><sup>28</sup> B. Sobac and D. Brutin, “Triple-line behavior and wettability controlled by nanocoated substrates: Influence on sessile drop evaporation,” *Langmuir* **27**, 14999 (2011).</footnote>

<footnote n="29"><sup>29</sup> R. H. Perry, D. W. Green, and J. O. Maloney, *Perrys Chemical Engineers' Handbook*, 7th ed. (McGraw-Hill, New York, 1999), Chap. 2, p. 370.</footnote>

<footnote n="30"><sup>30</sup> F. Girard, M. Antoni, S. Faure, and A. Stainchen, “Evaporation and marangoni driven convection in small heated water droplets,” *Langmuir* **22**, 11085 (2006).</footnote>
# Annexe E

## Pattern formation in drying drops of blood

David Brutin, Benjamin Sobac, Boris Loquet & José Sampol

*Journal of fluid mechanics* **667**, 85–95 (2011).
# Pattern formation in drying drops of blood

D. BRUTIN<fnref n="1" />†, B. SOBAC<fnref n="1" />
B. LOQUET<fnref n="2" /> AND J. SAMPOL<fnref n="3" />

(Received 5 August 2010; revised 24 September 2010; accepted 24 September 2010; first published online 16 December 2010)

The drying of a drop of human blood exhibits coupled physical mechanisms, such as Marangoni flow, evaporation and wettability. The final stage of a whole blood drop evaporation reveals regular patterns with a good reproducibility for a healthy person. Other experiments on anaemic and hyperlipidaemic people were performed, and different patterns were revealed. The flow motion inside the blood drop is observed and analysed with the use of a digital camera: the influence of the red blood cells motion is revealed at the drop periphery as well as its consequences on the final stage of drying. The mechanisms which lead to the final pattern of the dried blood drops are presented and explained on the basis of fluid mechanics in conjunction with the principles of haematology. The blood drop evaporation process is evidenced to be driven only by Marangoni flow. The same axisymmetric pattern formation is observed, and can be forecast for different blood drop diameters. The evaporation mass flux can be predicted with a good agreement, assuming only the knowledge of the colloids mass concentration.

**Key words:** bioconvection, blood flow, colloids

## 1. Introduction

Several patterns can be observed in dried drops of a different nature. Coffee drops have been widely studied in the past by demonstrating the influence of colloids in the final pattern observed (Deegan *et al.* 1997). Regarding human blood serum (the clear portion of any liquid separated from its more solid elements), the dried drop of blood serum presents different patterns depending on the individual's health condition. The specific regular pattern characteristics of a healthy individual do not appear in a dried drop of the blood serum from a person with blood disease (Martusevich, Zimin & Bochkareva 2007; Shabalin & Shatokhina 2007).

Experimental observations of dried drops of biological fluids (except whole human blood) have been performed and published in medical journals. The drying process is not analysed in the literature; they only focus on the final pattern in order to differentiate healthy people from patients with diseases. The drying technique has been used to store samples in locations, where appropriate storage conditions are not available: field studies, third world countries, deserts, etc. (Zhuang *et al.* 1982). The method was used to detect hepatitis B infection in serum dried on filter paper. Also,

<footnote n="1">¹Université de Provence, IUSTI UMR 6595 CNRS, 13453 Marseille, France</footnote>

<footnote n="2">²Laboratoire d'Analyses Médicales Bio 13, 13013 Marseille, France</footnote>

<footnote n="3">³UMR U608 INSERM, Faculté de Pharmacie, 13385 Marseille, France</footnote>

<footnote>† Email address for correspondence: david.brutin@polytech.univ-mrs.fr</footnote>
experimental observations of dried drops of serum have been performed (Yakhno 2008). The author identified the origin of the patterns observed in dried drops of serum from the phase transition (gellification) of proteins. A strong attachment of the dried serum due to a biological reaction of the protein with the glass plate of the microscope and salt crystallization were found in the dried drop, these phenomena have been explained.

Whole blood is a complex colloidal suspension which behaves like a non-Newtonian fluid. Since the colloids of blood have complicated shapes, the individual colloid behaviour is hardly predictable. Pozrikidis (2006) demonstrate the existence of a molecular adhesion component to explain the flipping behaviour of a platelet over a plane wall. Besides the flow around these colloids, the biochemistry is also present, the red blood cells (RBCs) evidence repulsive behaviour near a wall as they do in the human body to avoid the vessel obstruction. When drops of blood evaporate, all the colloids are carried by the flow motion inside the drop and interact. These interactions are governed by fluid mechanics, biology and chemistry. Any modification in the drop evaporation process can reveal a given disease, either an RBC-related disease for whole blood drop evaporation or another disease for serum drop evaporation.

Similar to those phenomena which we observe with whole blood are presented in the case of the desiccation of a colloidal suspension (Pauchard, Parisse & Alain 1999). Crack patterns formed by desiccation of a colloidal suspension in a sessile drop placed on a glass surface have been observed. Pauchard et al. (1999) demonstrate that the crack formation is related to the salinity of the suspension and interpreted the pattern obtained for large drops as buckling instability. The experimental observations with mica colloids in deionized water (Deegan 2000) were theoretical confirmed by Popov (2005).

The biological suspension used in this study is real human whole blood from which the coagulation protein (fibrinogen) has been removed. RBCs are observed thanks to the relative transparency of a thin layer of blood (less than 500 µm). In this paper, we will mainly present the dynamic of the process of evaporation using a top view visualization and the drop mass evolution during the drying process.

## 2. Materials and methods

### 2.1. *Experimental set-up*

An experimental set-up was designed to provide a complete geometric characterization of a posed drop of blood during its evaporation on a substrate. The drops were deposited on Bioblock glass plates. Environmental conditions (temperature, pressure, humidity) were recorded using a meteorological station. Evaporation took place in a parallelepipedic box measuring 100 × 100 × 150 mm³ which remains covered during the whole duration of the experiment to avoid external flow perturbations. The box volume is not sufficient enough to saturate due to the generation of drop vapour phase. The image analysis was made by means of a Canon EOS 7d digital camera coupled with a ×1 to ×5 macro lens. This camera enabled us to obtain 5184 × 3456 pixels on an area of 22.3 × 14.9 mm². With an enlargement of ×1, the resolution was 4.30 µm, and with the greatest enlargement of ×5 the resolution was 0.86 µm. For films with 30 images s⁻¹, spatial resolution had to be reduced to 1920 by 1080 pixels (full HD) with a pixel size of 2.30 µm. This optical diagnosis allowed a visualization of the drop from above in enough detail for further analysis of specific areas. Drop evaporation was performed using a cold cathode back light at 5000 ± 270 K without any surrounding light to avoid reflection on the blood drop interface. The effect of the
density of light on the drying process was studied to avoid any biological or chemical disturbance of the phenomenon. To follow the drying process, the experiment is performed on a digital accurate weighting balance with a resolution of 10 µg up to 81 g (Mettler Toledo XS 205). The drop is deposited on a microscope glass substrate (5 g) and then the total mass of the set-up is recorded on a computer at 1 Hz.

## 2.2. Blood: biological and physical properties

Blood samples were taken in a nearby medical laboratory and stored in their original 10 µl sterile tubes (BD Vacutainer 9NC 0.109M) in a refrigerator at +4°C, so that several experiments could be carried out at different times. Experiments were performed at any time from 1 h to 6 days after the sample was drawn, results for the same person remained almost the same showing that these storage conditions do not influence the main drying process. The blood used in this paper was taken from volunteers of whom some were in good health and others had a disease: anaemia (reduction below normal of the number of RBCs, quantity of haemoglobin or the volume of packed RBCs in the blood) or hyperlipidaemia (elevated blood lipids, particularly triglycerides, after carbohydrate ingestion). The biological fluid properties have been determined in order to correctly analyse the flow behaviour observed and the pattern formation. The whole blood and the serum have been analysed in terms of fluid viscosity using a Physica MCR 501 ‘Anton Paar’ and the surface tension has been determined using the pendant drop method. At the same time, a full haematology and biochemistry analysis of the blood has been performed to obtain the blood characteristics in terms of comosition (haematocrit, lipidaemic balance). The blood density varies slightly for different people in between 1020 and 1060 kg m⁻³ depending on the blood composition. Considering this small variation in density (less than 4 %), one assumes a constant whole blood density of 1040 kg m⁻³ in this study. The blood rheology is obtained in the range of shear stresses from 0.1 to 100 s⁻¹ and is given in (2.1) as

$$ \tau = k\dot{\gamma}^n, \qquad (2.1) $$

with n equal to 0.82 ± 0.01 and k is a constant. This is in agreement with typical results obtained, indicating that whole blood viscosity is driven by the RBCs concentration (the haematocrit) (Thieret 2008). For the blood serum, we obtain n = 1.01 ± 0.01, which is more consistent with a Newtonian fluid. The blood serum does not contain micrometre-sized colloids, such as RBCs or WBCs, but only ions and proteins. Of the whole blood drop, 43.9 % volume is RBCs based on haematological analysis. Mass-wise, 23.9 % of whole blood is composed of colloids which will be remain on the glass substrate at the end of the drying process. The main blood colloid distribution is as follows.

(i) Erythrocytes (red blood cells) with 4.9 × 10⁶ units mm⁻³ and a typical size of 8 µm represent 97 % of the volume of colloids,

(ii) Leukocytes (white blood cells) with 4.3 × 10³ units mm⁻³ and a typical size of 15 µm represent 2 % of the volume of colloids,

(iii) Platelets with 2.4 × 10⁵ units mm⁻³ and a typical size of 3 µm represent 1 % of the volume of colloids.

For our analysis in § 3, we will assume that the fluid is mainly composed of 8 µm diameter colloids at high concentration (about 45 %). Whole blood also contains ions and proteins which represent 11.6 % by mass of the final dried deposit (statistical values obtained for all experiments were performed using the same blood sample).
<table><caption>TABLE 1. Drops of blood and serum contact angles (water contact angle on glass substrate are not measurable with sufficient accuracy).</caption><thead><tr><th>Substrate</th><th>θ<sub>blood</sub></th><th>θ<sub>serum</sub></th><th>θ<sub>water</sub></th></tr></thead><tbody><tr><td>Glass without coating</td><td>14.8° ± 16.2 %</td><td>9.5° ± 21.1 %</td><td>—</td></tr><tr><td>Glass with PTFE coating</td><td>140.0° ± 2.6 %</td><td>128.5° ± 2.9 %</td><td>143.6° ± 3.4 %</td></tr><tr><td>Glass with epoxy coating</td><td>84.9° ± 2.4 %</td><td>74.0° ± 2.7 %</td><td>88.1° ± 1.7 %</td></tr></tbody></table>

The blood surface tension was measured using the pendant drop method with a capillary tube of 1 mm diameter. The exact capillary tube diameter was measured to an accuracy of 0.02 mm and blood drop mass was determined to be 10 µg using an electronic balance. This method gives a statistical value of 69.8 mN m⁻¹ ± 4.6 % over 72 drops for the full blood and 58.8 mN m⁻¹ ± 4.9 % over 24 drops for the serum. These values agree with the theoretical values found in the literature, and they indicate that full blood surface tension is very close to the value obtained for water, while serum surface tension is of a much lower value. Contact angles were determined using the sessile drop method where whole blood, serum and water were dropped onto three substrates: a microscope glass plate without any coating or with PTFE or epoxy coating, to confirm the surface tension measurements. The values obtained were close to the value obtained with pure ultra-filtered water as presented in table 1. Based on physical properties, the full blood used in this study is very close to water except for the fluid viscosity, which demonstrates non-Newtonian behaviour (thixotropic fluid).

## 3. Experiments

### 3.1. *The pattern formation for healthy people*

The different stages of blood drop evaporation are shown in figure 1. The initial drop diameter is 5.9 mm and the enlargement used is ×4.8 to allow visualization of RBCs. The images for the film did not undergo any processing except for cropping as the original size of each picture was 5184 pixel × 3456 pixel. However, due to the experimental protocol, the digital camera started taking pictures up to 3 min after the drop was placed on the plaque. Nevertheless, this delayed starting time was compensated and is $t=0$ min. We present a film of the whole evaporation process online with a playing time which is seven times the real duration, the external drop diameter is 6.5 mm (supplementary movies available at journals.cambridge.org/flm).

The evaporation/desiccation process can be divided into five stages which are detailed below. During the evaporation process, the drop is always pinned on the substrate. The total evaporation time for the case presented is 36 min and the durations of the stages are presented as a percentage of the total time of evaporation.

(i) Stage 1 (from 0–20 %): the RBCs move out from the centre of the drop to the desiccation line which recedes. A red deposit is observed at the edge of the drop. The measured receding speed of the desiccation line is 0.7 µm s⁻¹.

(ii) Stage 2 (from 20 to 50 %): crystallization appears at the edge of the drop and propagates inwards. At the same time, the desiccation process continues from the edge towards the inside of the drop. A dark red torus is observed which demonstrates a different fluid composition. This torus contains a high concentration of RBCs. These cells are now clearly observed moving from the centre to the edge of the drop.
<figure><img src="image_116.png" /></figure>

FIGURE 1. Mechanisms involved in a drying drop of blood from a healthy person (drop diameter, 5.9 mm; room temperature, 22°C; microscope glass substrate; 100 s between frames). A movie is provided online with a playing speed increased by 60. See figure 4 for a schematic cut view of the drop.

<figure><img src="image_117.png" /></figure>

FIGURE 2. Closer view of patterns: different characteristic pattern sizes exist (drop diameter 5.9 mm; room temperature, 22°C; microscope glass substrate; field of view, 4.6 mm × 3.1 mm).

(iii) Stage 3 (from 50 to 70%): the dark red torus desiccates rapidly and simultaneously the colour of the central part of the drop changes to an increasingly lighter red. The drop is almost totally desiccated and the first crack appears around the drop between the future corona and central part of the drop.