Drying dynamics of sessile-droplet arrays

We analyze the diffusion-controlled evaporation of multiple droplets placed near each other on a planar substrate. Specifically, we calculate the change in the volume of sessile droplets with various initial contact angles that are arranged in different configurations. The calculations are supplemented by experimental measurements using a technique that interprets the variable magnification of a pattern placed beneath the droplet array, which is applied to the case of initially hemispherical droplets deposited in four distinct arrangements. We find excellent agreement between the predictions based on the theory of Masoud et al. [Evaporation of multiple droplets, J. Fluid Mech. 927, R4 (2021)] and the data gathered experimentally. Perhaps unexpectedly, we also find that when comparing different arrays, the droplets with the same order of disappearance within their respective array, i.e., fastest evaporating, second-fastest evaporating, etc., follow similar drying dynamics. Our study provides not only experimental validation of the theoretical framework introduced by Masoud et al., but also offers additional insights into the evolution of the volume of individual droplets when evaporating within closely-spaced arrays.

Figure 1

Schematic of an array of evaporating sessile droplets.

Figure 2

Experimental images of droplet arrays magnifying a pattern of dots (ordered vertically by the number of droplets $N$) as a function of time for the triangle, cross, pentagon, and square configurations, corresponding to subfigures g, j, l, and m of Fig. 7 (in Appendix), respectively. The legend indicates the droplets' scale, initial ($t=0$) contact angle ${\theta }_0$, and volume $V_0$. The minimum center-to-center distance between the nearest droplets in each configuration is about three times their contact radius.

Figure 3

Comparison between the results of experimental measurements (empty symbols) and theoretical calculations (solid and dashed lines) for the normalized volume $V/V_0$ versus normalized time $t/t^{★}$. The plots represent the change in the volume of droplets with an initial contact angle of $\pi /2$ evaporating under CR mode while placed in arrays of four different configurations. The geometry of each array is depicted in the top right of each subfigure, where $\ell$ denotes the minimum center-to-center distance between droplet pairs.

Figure 4

Theoretically calculated plots of normalized volume $V/V_0$ versus normalized time $t/t^{★}$ for droplets with an initial contact angle of $\pi /2$ evaporating under CR (left subfigures) and CA (right subfigures) modes while placed in arrays of different configurations. Subfigures in the first, second, and third rows show the plots for droplets of type I, II, and III, respectively. Also, configurations listed in the legend correspond to those illustrated in Fig. 7, with the exception of configuration a, which is representative of a single isolated droplet.

Figure 5

Theoretically calculated plots of normalized volume $V/V_0$ versus normalized time $t/t^{★}$ for droplets with an initial contact angle of $\pi /3$ evaporating under CR (left subfigures) and CA (right subfigures) modes while placed in arrays of different configurations. Subfigures in the first, second, and third rows show the plots for droplets of type I, II, and III, respectively. Also, configurations listed in the legend correspond to those illustrated in Fig. 7, with the exception of configuration a, which is representative of a single isolated droplet.

Figure 6

Theoretically calculated plots of normalized volume $V/V_0$ versus normalized time $t/t^{★}$ for droplets with an initial contact angle of $2\pi /3$ evaporating under CR (left subfigures) and CA (right subfigures) modes while placed in arrays of different configurations. Subfigures in the first, second, and third rows show the plots for droplets of type I, II, and III, respectively. Also, configurations listed in the legend correspond to those illustrated in Fig. 7, with the exception of configuration a, which is representative of a single isolated droplet.

Figure 7

Droplet arrangements considered in the theoretical calculations presented in Figs. 4, 5, 6. In each configuration, droplets with the same rate of evaporation are grouped together and labeled I, II, or III. Also, $\ell$ denotes the minimum center-to-center distance between droplet pairs.