3.1. Hydrodynamics

We use the capillary time as the characteristic time, $t^{*}=\mu l_{c} / \sigma \epsilon ^{3}$ , where $\mu$ and $\sigma$ are the liquid viscosity and surface tension (assumed constant), with $\epsilon \equiv h_{c}/l_{c} \ll 1$ . The characteristic velocity and pressure are $u^{*}=l_{c}/t^{*}$ and $p^{*}=\sigma \epsilon /l_{c}$ . The dimensionless velocity field in the droplet is $\boldsymbol {v}=(u,w)$ , where $u$ and $w$ are radial and vertical velocities, respectively. Here, $u$ and $w$ have been non-dimensionalised using $u^{*}$ and $\epsilon u^{*}$ , respectively.

Under the lubrication approximation, the leading-order momentum and mass conservation equations are

(3.2) \begin{align} &\frac {\partial p}{\partial r}= \frac {\partial ^{2}u}{\partial z^{2}} , \\[-12pt] \nonumber \end{align}

(3.3) \begin{align} &\frac {\partial p}{\partial z}=0 , \\[-12pt] \nonumber \end{align}

(3.4) \begin{align} &\frac {1}{r}\frac {\partial [ru]}{\partial r}+\frac {\partial w}{\partial z}=0. \\[10pt] \nonumber \end{align}

In our simulations, gravitational forces are assumed to be negligible relative to surface-tension forces, which corresponds to small Bond numbers $Bo=\rho _{l}gl_{c}^{2}/\sigma$ , where $\rho _{l}$ is the liquid density and $g$ is the magnitude of the gravitational acceleration. This is consistent with our motivating experiments, where $Bo\sim 10^{-2}$ .

At the solid–liquid interface ( $z=h_{b}(r)$ ), we have no-slip and no-penetration boundary conditions:

(3.5) \begin{align} u=0,\quad w=0. \end{align}

At the liquid–air interface ( $y=h(r,t)$ ), the tangential and normal stress balances are

(3.6) \begin{align} & \frac {\partial u}{\partial z}=0 , \\[-12pt] \nonumber \end{align}

(3.7) \begin{align} & p-p_{v}=- \frac {\partial ^{2}h}{\partial r^{2}}-\frac {1}{r}\frac {\partial h}{\partial r}-\varPi , \\[10pt] \nonumber \end{align}

where $p_{v}$ and $\varPi$ are the dimensionless pressure in the vapour phase (assumed constant) and disjoining pressure, respectively. To focus on the influence of the mesa, we neglect any variation of the surface tension with temperature. Based on prior work on a related problem involving droplet evaporation in cavities (Chang & Kumar Reference Chang and Kumar2025), we expect that thermal Marangoni flows will tend to weaken contact-line pinning, as they drive liquid away from the contact line, thereby promoting its retraction. Integrating (3.2) twice and applying the boundary conditions given by (3.5) and (3.6) yields

(3.8) \begin{align} u(r,z)=\frac {\partial p}{\partial r}\left (\frac {1}{2}z^{2}-zh-\frac {1}{2}h^{2}_{b}+hh_{b}\right ). \end{align}

3.2. Evaporation

In this work, we use the one-sided model to describe liquid evaporation, where the escape of molecules from the liquid phase to the vapour phase is the rate-limiting step (Ajaev Reference Ajaev2004, Reference Ajaev2005; Sodtke, Ajaev & Stephan Reference Sodtke, Ajaev and Stephan2008; Wilson & D’Ambrosio Reference Wilson and D’Ambrosio2023). This is in contrast to diffusion-limited models, where the evaporation rate is limited by how fast liquid molecules diffuse away from the interface (Hu & Larson Reference Hu and Larson2005; Masoud & Felske Reference Masoud and Felske2009; Larson Reference Larson2014; Wilson & D’Ambrosio Reference Wilson and D’Ambrosio2023). The one-sided model is adopted here for two reasons: (i) the computational cost is lower (only the transport equations in the liquid phase need to be considered) and (ii) the results are often qualitatively similar to predictions from the diffusion-limited model (Murisic & Kondic Reference Murisic and Kondic2008, Reference Murisic and Kondic2011; Larsson & Kumar Reference Larsson and Kumar2023).

In the one-sided model, the expression for the dimensional evaporation flux is

(3.9) \begin{align} \frac {\sqrt {2\pi \hat {R}T_{\textit{sat}}}}{\alpha \rho _{v}}j^{\prime }=\frac {p^{\prime }-p_{vap}}{\rho _{l}}+\frac {\hat {L}(T_{i}^{\prime }-T_{\textit{sat}})}{T_{\textit{sat}}}, \end{align}

where $\alpha$ , $j^{\prime }$ , $\hat {R}$ , $\hat {L}$ , $p^{\prime }$ and $T^{\prime }_{i}$ are the accommodation coefficient, evaporation flux, ideal gas constant per unit mass, latent heat of vaporisation per unit mass, pressure at the interface and temperature at the interface, respectively (Ajaev Reference Ajaev2005; Sodtke et al. Reference Sodtke, Ajaev and Stephan2008; Craster, Matar & Sefiane Reference Craster, Matar and Sefiane2009; Wray et al. Reference Wray, Papageorgiou, Craster, Sefiane and Matar2014). (A prime is used here for dimensional variables that will later be non-dimensionalised.) The quantities $\rho _{v}$ and $\rho _{l}$ are the density of the liquid molecules in the vapour and liquid phases, $T_{\textit{sat}}$ is the saturation temperature and $p_{vap}$ is the dimensional (constant) pressure in the vapour phase (note that $p_{v} \equiv p_{vap}/p^{*}$ ). Equation (3.9) indicates that the evaporation flux is proportional to deviations from $T_{\textit{sat}}$ or $p_{vap}$ . The one-sided model has the form of a mass-transfer coefficient model for the mass flux, so it may provide qualitatively accurate predictions even when its assumptions are not strictly satisfied (Parrish & Kumar Reference Parrish and Kumar2020; Larsson & Kumar Reference Larsson and Kumar2023).

3.3. Energy transport

The temperature in the droplet is scaled as

(3.10) \begin{align} T=\frac {T^{\prime }-T_{\textit{sat}}}{T_{b}-T_{\textit{sat}}}=\frac {T^{\prime }-T_{\textit{sat}}}{\Delta T}, \end{align}

where $T^{\prime }$ is the dimensional droplet temperature and $\Delta T$ is the difference between substrate and saturation temperatures. Because the droplet is thin, the leading-order energy transport equation is

(3.11) \begin{align} \frac {\partial ^{2}T}{\partial z^{2}}=0. \end{align}

The dimensional temperature $T_{b}$ corresponds to

(3.12) \begin{align} T=1\quad \mbox{at } z=h_{b}. \end{align}

When the vapour’s density, viscosity and thermal conductivity are much less than the liquid’s, the conductive heat from the substrate to the liquid–air interface provides the heat of vaporisation. By balancing the conductive heat in the droplet with the heat released through evaporation, we derive the evaporation flux scale:

(3.13) \begin{align} j^{*}=\frac {k \Delta T}{h_{c} \hat {L}}, \end{align}

where $k$ and $\hat {L}$ are the liquid thermal conductivity and latent heat of vaporisation per unit mass. By applying an energy balance at the liquid–air interface, we obtain another boundary condition:

(3.14) \begin{align} -\frac {\partial T}{\partial z}=j\quad \mbox{at } z=h, \end{align}

where $j$ is the dimensionless evaporation flux.

Solving (3.11) subject to (3.12) and (3.14) yields expressions for the dimensionless evaporation flux and interfacial temperature, which are

(3.15) \begin{align} j=\frac {1+\delta (p-p_{v})}{K+h_{\kern-1pt f}},\quad T_{i}=\frac {K-\delta (p-p_{v}) h_{\kern-1pt f}}{K+h_{\kern-1pt f}}, \end{align}

where

(3.16) \begin{align} K=\frac {\sqrt {2\pi \hat {R}T_{\textit{sat}}}kT_{\textit{sat}}}{\rho _{v}\alpha \hat {L}^{2}h_{c}},\quad \delta =\frac {\sigma }{\rho _{l} \hat {L}h_{c}}. \end{align}

By applying the kinematic boundary condition at the liquid–air interface, we obtain

(3.17) \begin{align} \frac {\partial h}{\partial t}=-\frac {1}{r}\frac {\partial }{\partial r}\left (r \int _{h_{b}}^{h}u\,{\rm d}y \right )-Ej, \end{align}

where $E$ is the evaporation number, a ratio between the capillary and evaporation time scales:

(3.18) \begin{align} E=\frac {t^{*}}{t_{e}}=\frac {j^{*}}{u^{*}\epsilon \rho _{l}}. \end{align}

By using $h_{c}$ , $\rho _{l}$ , $j^{*}$ and $u^*=l_c/t^*$ , we can obtain the evaporation time scale, $t_{e}=h_{c}\rho _{l}/j^{*}$ ; the capillary time scale $t^{*}$ is given in § 3.1. By using (3.8) and (3.15), we can rewrite (3.17) as

(3.19) \begin{align} \frac {\partial h}{\partial t}=\frac {1}{r}\frac {\partial }{\partial r}\left ( \frac {\partial p}{\partial r}\frac {rh_{\kern-1pt f}^{3}}{3}\right )-Ej, \end{align}

with $p$ given by (3.7). Note that previous studies derive an equivalent equation for evaporation of thin droplets on rough surfaces (Pham & Kumar Reference Pham and Kumar2017, Reference Pham and Kumar2019).

3.4. Disjoining pressure

To model the behaviour of the droplet contact-line region, we assume that a precursor film is present at the droplet edge (figure 4), and use a two-term disjoining pressure including attractive and repulsive interactions:

(3.20) \begin{align} \varPi =A_{1} \left [ \left ( \frac {A_{2}}{h_{\kern-1pt f}}\right )^{3}-\left ( \frac {A_{2}}{h_{\kern-1pt f}} \right )^{2} \right ], \end{align}

where $A_{1}$ and $A_{2}$ are the dimensionless Hamaker constant and precursor-film thickness, respectively. In contrast to tracking the contact line explicitly (Saxton et al. 2016; Savva et al. Reference Savva, Rednikov and Colinet2017), this approach of modelling contact-line motion makes it easier to solve the height evolution equation (3.19), and provides results that are qualitatively consistent with experiments (Hocking Reference Hocking1995; Alleborn & Raszillier Reference Alleborn and Raszillier2004; Zadražil, Stepanek & Matar Reference Zadražil, Stepanek and Matar2006; Morris Reference Morris2014; Espín & Kumar Reference Espín and Kumar2015; Pham & Kumar Reference Pham and Kumar2017, Reference Pham and Kumar2019; Jambon-Puillet et al. Reference Jambon-Puillet, Carrier, Shahidzadeh, Brutin, Eggers and Bonn2018; Kumar & Charitatos Reference Kumar and Charitatos2022; Chang & Kumar Reference Chang and Kumar2025). The choice of exponents in (3.20) is made for computational convenience and is consistent with previous studies (Alleborn & Raszillier Reference Alleborn and Raszillier2004; Espín & Kumar Reference Espín and Kumar2015; Pham & Kumar Reference Pham and Kumar2017, Reference Pham and Kumar2019; Kumar & Charitatos Reference Kumar and Charitatos2022; Chang & Kumar Reference Chang and Kumar2025). We note that although we do not include the effects of surfactant in this work, a previous study suggests that using a precursor-film/disjoining-pressure approach when surfactants are present may yield predictions that deviate significantly from experimental observations since surfactant can flow from the droplet into the precursor film (Van Gaalen et al. Reference Van Gaalen, Diddens, Wijshoff and Kuerten2020).

When the droplet reaches equilibrium on a perfectly flat substrate in the absence of evaporation, the largest angle between the tangent to the liquid–air interface and the substrate is the equilibrium contact angle. Its value in the laboratory frame is denoted as $\theta _{e,\textit{lab}}$ . The expression for disjoining pressure, (3.20), can be related to the value of the equilibrium contact angle:

(3.21) \begin{align} \theta _{e,\textit{lab}} \approx \epsilon \sqrt {A_{1}A_{2}} \approx \epsilon \theta _{e}, \end{align}

where $\theta _{e}$ is the rescaled equilibrium contact angle based on lubrication theory (Schwartz & Eley Reference Schwartz and Eley1998; Schwartz Reference Schwartz1998; Pham & Kumar Reference Pham and Kumar2019). In most of our calculations, we set $A_{2}=0.001$ and $A_{1}=100$ , which corresponds to a value of $\theta _{e}=18.12^\circ$ . In § 7.2, the effects of surface wettability are explored by varying $A_{1}$ . In the presence of evaporation, the apparent contact angle will, in general, be time-dependent and differ from the equilibrium value. In the present work, this apparent angle is an output of our numerical calculations. It should be noted that prior studies of droplet evaporation on flat substrates have developed analytical expressions for how the apparent and microscopic contact angles depend on various problem parameters (Morris Reference Morris2001; Savva et al. Reference Savva, Rednikov and Colinet2017; Jambon-Puillet et al. Reference Jambon-Puillet, Carrier, Shahidzadeh, Brutin, Eggers and Bonn2018).

For non-flat substrates, it is useful to define a mesoscopic contact angle, $\theta _{\textit{meso}}$ , which is the angle that the droplet–air interface makes with respect to the substrate (Espín & Kumar Reference Espín and Kumar2015; Pham & Kumar Reference Pham and Kumar2017):

(3.22) \begin{align} \tan (\theta _{\textit{meso}})=\frac {\frac {\partial h_f}{\partial r}}{1+\frac {\partial h_{b}}{\partial r}\left (\frac {\partial h_f}{\partial r}+\frac {\partial h_{b}}{\partial r}\right )} . \end{align}

The apparent contact angle $\theta$ is then taken to be the maximum value of $\theta _{\textit{meso}}$ along the interface. The apparent contact-line position, $r_{\textit{con}}$ , is the point where the tangent to the liquid–air interface at the maximum value of $\theta _{\textit{meso}}$ intersects the substrate. It should be noted that (3.22) is written with respect to the left-hand side of the droplet, where all the derivatives are positive. For the right-hand side of the droplet, all the derivatives are negative and the angle would become negative. In that case, we use the absolute value so we have a positive angle.

To prevent the precursor film from evaporating, the evaporation flux needs to vanish there. By combining this constraint ( $j=0$ ) with (3.7), (3.15) and (3.20), we obtain an expression for the precursor-film thickness with evaporation, $\beta$ (Pham & Kumar Reference Pham and Kumar2017; Charitatos, Pham & Kumar Reference Charitatos, Pham and Kumar2021; Chang & Kumar Reference Chang and Kumar2025):

(3.23) \begin{align} \beta =-2\sqrt {\frac {\delta A_{1} A^{2}_{2}}{3}} \sinh \left ( \frac {1}{3} \mbox{arcsinh} \left ( -\frac {3A_{2}}{2} \sqrt {\frac {3}{\delta A_{1} A^{2}_{2}}} \right ) \right ). \end{align}

We note that for typical parameter values used in this work ( $A_1=100$ , $A_2=0.001$ and $\delta =0.001$ ), $\beta /A_2 \sim 1$ , so that the precursor film has a comparable thickness in the presence and absence of evaporation.

3.5. Parameter values and numerical methods

The phenomena reported in § 2 are expected to be quite general, so we consider a broad range of typical values of dimensional parameters, which are listed in table 2. Based on these values, we set $\epsilon =0.1$ . Using the values in table 2, we choose a typical capillary time $t^{*} \sim 10^{-3}$ s and pick a typical evaporation time $t_{e} \sim 10$ s. Based on (3.18), $E \sim 10^{-4}$ , a value that has been used in previous studies for droplet evaporation on rough or inclined substrates (Pham & Kumar Reference Pham and Kumar2017, Reference Pham and Kumar2019; Charitatos et al. Reference Charitatos, Pham and Kumar2021). We note that $T_{\textit{sat}}$ for 1,2-dichlorobenzene is 452 K at atmospheric pressure (Carswell Reference Carswell1928), which is considerably higher than the value listed in table 2. Although this indicates that the assumptions of the one-sided model are not being satisfied in the experiments reported in § 2, we will see that the relatively simple model we have developed here can qualitatively describe the experimental observations (see also § 3.2). Experiments in which the vapour phase is saturated with the solvent molecules would be expected to be more consistent with the assumptions of the one-sided model.

Table 2.

Order-of-magnitude estimates of dimensional parameters (Burelbach, Bankoff & Davis Reference Burelbach, Bankoff and Davis1988; Vazquez, Alvarez & Navaza Reference Vazquez, Alvarez and Navaza1995; Popescu et al. Reference Popescu, Oshanin, Dietrich and Cazabat2012; Debuisson et al. Reference Debuisson, Merlen, Senez and Arscott2016; Cao et al. Reference Cao, Jochem, Hyun, Francis and Frisbie2018; Nguyen et al. Reference Nguyen, Kumar, Frisbie and Francis2024).

Based on table 2 and (3.16), we estimate $K \sim 10^{-1}$ , so we choose $K=0.1$ . The accommodation coefficient $\alpha$ provides a measure of the ease with which solvent can escape from the liquid to the vapour phase, and we assumed that $\alpha \sim 1$ . This is consistent with our previous work involving droplet evaporation in cavities (Chang & Kumar Reference Chang and Kumar2025) and changes to this value are not expected to influence the qualitative behaviour of our results. As in previous studies (Pham & Kumar Reference Pham and Kumar2017; Charitatos et al. Reference Charitatos, Pham and Kumar2021), we choose $\delta =0.001$ , which gives a precursor-film thickness (see (3.23)) that is expected to provide qualitatively accurate results at a reasonable computational cost (smaller values of $\delta$ lead to smaller values of the precursor-film thickness). Note that smaller values of the precursor-film thickness lead to slower contact-line speeds (for an example, see figure S1 of the supplementary material available at https://doi.org/10.1017/jfm.2026.11733) but are not expected to change the qualitative behaviour of the results (Espín & Kumar Reference Espín and Kumar2017; Chang & Kumar Reference Chang and Kumar2025). The other important dimensionless parameters are listed in table 3.

Table 3.

Dimensionless parameters used for current model.

Although we neglect thermal Marangoni effects to focus on the influence of the mesa (see § 3.1), it is instructive to calculate a thermal Marangoni number:

(3.24) \begin{align} Ma_{T}=\frac {k_{T}(T_{b}-T_{\textit{sat}})}{\sigma \epsilon ^{2}}, \end{align}

where $k_{T}$ is the thermal surface-tension gradient. Using values from table 1, we estimate that $ \textit{Ma}_{T} \sim 1$ , so thermal Marangoni effects may be important in practice. As noted in § 3.1, we expect that these will weaken contact-line pinning, and an example is given in figure S2 of the supplementary material. Nevertheless, we neglect thermal Marangoni effects in this work to isolate the role of the mesa in droplet dynamics. Moreover, while thermal Marangoni effects may produce quantitative changes in the parameter values at which a droplet pins on a mesa (e.g. a steeper mesa slope may be required), we do not expect them to change the qualitative behaviour we observe as different parameters are varied.

We solve the height evolution equation (3.19) on the domain $-4\leqslant r \leqslant 4$ by using a fourth-order centred finite-difference scheme and a variable-step, variable-order time-stepping algorithm (Matlab ode15s), with boundary conditions $h(-4,t)=h(4,t)=\beta$ and $h_{r}(-4,t)=h_{r}(4,t)=0$ . (Note that (3.19) is unchanged when $r$ is replaced with $-r$ .) We use 2001 spatial nodes, which was found to be sufficient to obtain converged results (see figure S3 of the supplementary material). The model was validated by reproducing previous results for droplet spreading and droplet evaporation on perfectly flat substrates (Tanner Reference Tanner1979; Oron et al. Reference Oron, Davis and Bankoff1997; Pham & Kumar Reference Pham and Kumar2017). (Examples are shown in figures S4 and S5 of the supplementary material.) For the case of no evaporation, we stop the simulation when $t=500$ . Each simulation with evaporation is stopped when the maximum film thickness is smaller than $1.1 \beta$ (see (3.23)), which means almost all the liquid has dried out. We denote this time as the evaporation time, $t_{evap}$ .

Important differences between the experimental conditions and simulation assumptions prevent us from making detailed quantitative comparisons. In the experiments, the mesa walls are nearly vertical since those are easiest to fabricate. Due to the sharp corners this creates where the walls meet the mesa top (radius of curvature $\approx$ 0.6 μm), the mesa can hold a liquid volume of up to 4 nl before the liquid spills off the mesa. In contrast, our model requires a mesa with rounded corners because it employs the lubrication approximation. Whereas the mesa height and slope are independent for mesas with vertical walls, they are coupled to each other in the mesa model used in our simulations. If we consider a mesa diameter and height of 500 and 5 μm (similar to those of the experiments, but with radius of curvature $\approx$ 325 μm) at a value of $\theta _{e}=18.12^\circ$ , the maximum amount of liquid the mesa can hold is 0.6 nl, which is much smaller than the experimental results. This is consistent with prior work demonstrating that sharp corners are more effective at pinning contact lines than rounded corners (Oliver, Huh & Mason Reference Oliver, Huh and Mason1977). Obtaining quantitatively accurate predictions for mesas with vertical walls will likely require an explicit treatment of the contact-line dynamics (Savva et al. Reference Savva, Rednikov and Colinet2017). In spite of the key differences between experimental conditions and simulation assumptions, we will see that the simulations qualitatively describe the experimental observations reported in figures 2 and 3.

5.1. Effects of mesa

Figures 8(a) and 8(c) show the time evolution of the contact-line position and the corresponding apparent contact angle in the presence of a mesa for three different initial liquid volumes (set by the initial droplet height, $d_{h}$ ) at a given evaporation rate. Here, time is rescaled by the evaporation time, $t_{evap}$ . Three distinct stages can be observed. In stage 1, the contact line is apparently pinned at the mesa edge for a significant portion of the evaporation time, while the contact angle decreases. In stage 2, the contact line recedes, and the apparent contact angle remains nearly constant. The apparent contact angle in this stage is denoted as $\theta _{\textit{CCAS}}$ . From figure 8(c), we see that $\theta _{\textit{CCAS}}$ is not sensitive to the initial liquid volume and remains around $20^\circ$ . Stage 3 begins when the droplet has nearly evaporated, and here both the contact angle and the contact-line position decrease quickly. These three stages have also been observed in experiments, and in numerical simulations where substrate roughness is accounted for (Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000; Pham & Kumar Reference Pham and Kumar2017).

Figure 8.

Plots of $r_{\textit{con}}$ and $\theta$ versus $t/t_{evap}$ for different $d_{h}$ when (a,c) $m_{h}=0.4$ and (b,d) $m_{h}=0$ . The case where the mesa is absent corresponds to $m_{h}=0$ . Here, $E=3\times 10^{-5}$ and $\theta _{e} = 18.12^\circ$ .

Figure 8. Long description

Panel A: A line graph shows the normalized contact radius (r_con) on the vertical axis versus normalized evaporation time (t/tevap) on the horizontal axis. Three lines represent different values of dh (0.8, 1.2, 1.7), with an arrow indicating increasing dh. The lines show a decreasing trend in r_con as t/tevap increases. Panel B: Another line graph shows the normalized contact radius (r_con) on the vertical axis versus normalized evaporation time (t/tevap) on the horizontal axis. Three lines represent different values of dh (0.8, 1.2, 1.7), with an arrow indicating increasing dh. The lines show a decreasing trend in r_con as t/tevap increases. Panel C: A line graph shows the contact angle (θ) on the vertical axis versus normalized evaporation time (t/tevap) on the horizontal axis. Three lines represent different values of dh (0.8, 1.2, 1.7), with an arrow indicating increasing dh. The lines show a decreasing trend in θ as t/tevap increases. Panel D: Another line graph shows the contact angle (θ) on the vertical axis versus normalized evaporation time (t/tevap) on the horizontal axis. Three lines represent different values of dh (0.8, 1.2, 1.7), with an arrow indicating increasing dh. The lines show a trend where θ initially increases and then decreases as t/tevap increases.

Figures 8(b) and 8(d) show the time evolution of the contact-line position and apparent contact angle for the corresponding case of a perfectly flat substrate. In contrast to what is observed in figure 8(a), the droplet retracts without any significant pinning. However, the apparent contact angle remains nearly constant for most of the evaporation time, as seen in figure 8(d). (The increase in $\theta$ at early times is due to evaporation being turned on.) Overall, the results of figure 8 demonstrate that introducing a mesa can cause significant contact-line pinning, with all problem parameters the same.

5.2. Mechanism of pinning and depinning

Figure 9(a) shows the contact-line position at different times in the presence and absence of a mesa, which correspond to the yellow lines in figures 8(a) and 8(b), respectively. It is clear that the presence of a mesa can cause contact-line pinning, and the simulation predictions shown in figure 9(a) are qualitatively consistent with the experimental observations shown in figure 2. To better understand the physical mechanism of the pinning, we examine in figure 9(b) the pressure profiles in the droplets at $t=2000$ , when the droplet on the mesa is still pinned. In both the presence and absence of a mesa, the pressure monotonically decreases as one moves from the droplet centre to the edge, and this causes an outward flow. The maximum pressure is at the droplet centre and is denoted as $p_{\textit{max}}$ . For the conditions shown in figure 9(b), it is larger when the mesa is present. This corresponds to a larger pressure gradient, which produces a large enough outward flow to replenish liquid evaporated in the contact-line region, leading to a pinned contact line.

Figure 9.

Plots of (a) $r_{\textit{con}}$ versus $t$ and (b) pressure distribution at $t=2000$ for different $m_{h}$ . The case where the mesa is absent corresponds to $m_{h}=0$ . Here, $E=3\times 10^{-5}$ , $\theta _{e} = 18.12^\circ$ and $d_{h}=1.7$ .

Figures 10(a) and 10(b) show $p_{\textit{max}}$ and the contact-line position at different times in the absence and presence of a mesa, respectively. When the substrate is perfectly flat, $p_{\textit{max}}$ is low enough so the contact line retracts without any significant pinning (figure 10 a). While the contact line retracts, $p_{\textit{max}}$ grows monotonically due to a more convex interface. In contrast, when the mesa is present, the contact line is nearly pinned for a significant portion of the evaporation time (figure 10 b). (Note that the pinning is not perfect due to the rounded mesa corner in the simulations.) When the pinning occurs, the contact angle decreases (see figure 8 c), reducing $p_{\textit{max}}$ (less convex interface). Once $p_{\textit{max}}$ is too low to create a sufficient outward flow to replenish the evaporated liquid in the contact-line region, the contact line depins. After depinning, the droplet shrinks, and the interface becomes more convex, eventually leading to an increase in $p_{\textit{max}}$ ( $p_{\textit{max}}\gt p_{\textit{min}}$ in figure 10 b).

Figure 10.

Plots of $p_{\textit{max}}$ and $r_{\textit{con}}$ versus $t$ when (a) $m_{h}=0$ and (b) $m_{h}=0.4$ . The case where the mesa is absent corresponds to $m_{h}=0$ . Here, $E=3\times 10^{-5}$ , $\theta _{e} = 18.12^\circ$ and $d_{h}=1.7$ .

5.3. Dependence of pinning on initial droplet volume

To understand how the initial droplet volume influences contact-line pinning, we examine the time evolution of the contact-line position for three different initial liquid volumes in figure 11(a). Our simulations indicate that a larger initial liquid volume extends the droplet pinning time, which is qualitatively consistent with the experimental results in figure 3(a). Note that in the experiments, the contact line is pinned at the mesa edge ( $r_{\textit{droplet}}=225$ μm in figure 3 a). However, in the simulations, figure 11(a) shows that a larger initial liquid volume makes the initial droplet radius slightly larger. This difference is due to corner sharpness. Compared with experiments, the corner in the simulations is more rounded, so increasing the initial liquid volume makes the droplet spread slightly further in the simulations.

To help explain why increasing the initial liquid volume increases the percentage of the drying time the droplet is pinned (see figure 8 a), we analyse the time evolution of $p_{\textit{max}}$ for three different initial liquid volumes in figure 11(b). From this plot, we see that a larger initial liquid volume increases the pressure at the droplet centre when $t=0$ due to a more convex interface. A larger $p_{\textit{max}}$ results in a larger capillary-pressure gradient and drives more flow to replenish liquid evaporated in the contact-line region, which explains why increasing the initial liquid volume increases the percentage of the drying time the droplet is pinned. Figure 11(b) also indicates that $p_{\textit{min}}$ is not very sensitive to the initial liquid volume, an observation that is discussed further in § 6. We note that as the initial droplet volume increases, so does the droplet aspect ratio since the droplet contact line is initially at the mesa edge.

Figure 11.

Plots of (a) $r_{\textit{con}}$ versus $t$ and (b) $p_{\textit{max}}$ versus $t/t_{evap}$ for different $d_{h}$ . Here, $E=3\times 10^{-5}$ , $\theta _{e} = 18.12^\circ$ and $m_{h}=0.4$ .

7.1. Influence of mesa height

Figure 15 shows the time evolution of the contact-line position and the corresponding apparent contact angle for three different mesa heights. In our simulations, the mesa height ( $m_{h}$ ) ranges from 0.2 to 0.6 so the inclination angle based on lubrication theory varies from $26.57^\circ$ to $56.31^\circ$ (see (4.2)). In addition, we have reduced the initial droplet height from $d_{h}=1.7$ to $d_{h}=0.8$ , so the droplet is pinned on the mesa before evaporation begins. Figure 15 indicates that a taller mesa appears to prolong the pinning slightly.

To understand this trend, we examine in figure 16(a) the liquid height profile at $t=0$ . This figure reveals that a taller mesa (larger inclination angle) makes initial droplet shape more convex; this is due to a higher disjoining-pressure gradient at the droplet edge (see § 4). The more convex initial shape is consistent with what is observed in figure 15, where the initial radius decreases and the apparent contact angle increases as the mesa becomes taller. Figure 16(b) shows the time evolution of $p_{\textit{max}}$ . This figure indicates that increasing mesa height makes $p_{\textit{max}}$ larger when $t=0$ (more convex interface), so the resulting capillary-pressure gradient and outward flow are larger, corresponding to longer pinning. Note that this mechanism is similar to how the initial liquid volume affects pinning (see § 5.3). It should be noted that because the simulations use mesas with rounded corners, the mesa height influences the mesa slope (see (4.2)). If the mesa sides were perfectly vertical, this would not be the case, so mesa height would not be expected to influence droplet evaporation. In addition, the parameter $m_s$ also influences the mesa slope (see (4.2)). Although we do not investigate its effects here, we expect that increasing $m_s$ will make more pronounced the influence of mesa height.

Figure 16.

Plots of (a) $h$ versus $r$ at $t=0$ and (b) $p_{\textit{max}}$ versus $t$ for different $m_{h}$ . Here, $d_{h}=0.8$ , $\theta _{e} = 18.12^\circ$ and $E=3\times 10^{-5}$ .

7.2. Influence of surface wettability

Figure 17 shows the time evolution of the contact-line position and the corresponding apparent contact angle for three different surface wettabilities. In our simulations, the Hamaker constant ( $A_{1}$ ) ranges from 100 to 400. The rescaled equilibrium contact angle based on lubrication theory varies from $18.12^\circ$ to $36.24^\circ$ (see (3.21)). In all cases, we observe a three-stage contact-line motion, as discussed in § 5.1. Figure 17 indicates that in stage 2, lower surface wettability causes slower contact-line retraction with a larger apparent contact angle (larger $\theta _{\textit{CCAS}}$ ). Lower surface wettability makes the apparent contact angle larger, which induces a thicker liquid film at the droplet edge. Therefore, at the droplet edge, evaporation is slower (see (3.15)), which prolongs the drying process. This trend has also been seen in numerical simulations of droplets on flat substrates where substrate roughness is accounted for (Pham & Kumar Reference Pham and Kumar2017).

Figure 17.

Plots of (a) $r_{con}$ versus $t$ at $t=0$ and (b) $\theta$ versus $t$ for different $\theta_e$ . Here, $d_{h}=1.7$ , $m_{h} = 1.4$ and $E=3\times 10^{-5}$ .