Dissolution dynamics of a vertically confined sessile droplet

We experimentally investigate the dissolution of microscale sessile alcohol droplets in water under the influence of impermeable vertical confinement. The introduction of confinement suppresses the mass transport from the droplet to bulk medium in comparison with the nonconfined counterpart. Along with a buoyant plume, flow visualization reveals that the dissolution of a confined droplet is hindered by a mechanism called levitated toroidal vortex. The morphological changes in the flow due to the vortex-induced impediment alters the dissolution rate, resulting in enhancement of droplet lifetime. Further, we have proposed a modification in the key nondimensional parameters [Rayleigh number ${\mathrm{Ra}}^{\prime }$ (signifying buoyancy) and Sherwood number ${\mathrm{Sh}}^{\prime }$ (signifying mass flux)] and droplet lifetime ${{\tau }_c}^{\prime }$, based on the hypothesis of linearly stratified droplet surroundings (with revised concentration difference $\mathrm{\Delta }{C}^{\prime }$), taking into account the geometry of the confinements. We show that experimental results on droplet dissolution under vertical confinement corroborate scaling relations ${\mathrm{Sh}}^{\prime }\sim \mathrm{Ra}{{}^{\prime }}^{1/4}$ and ${{\tau }_c}^{\prime }\sim \mathrm{\Delta }C{{}^{\prime }}^{-5/4}$. We also draw attention to the fact that the revised scaling law incorporating the geometry of confinements proposed in the present work can be extended to other known configurations such as droplet dissolution inside a range of channel dimensions, as encountered in a gamut of applications such as microfluidic technology and biomedical engineering.

Figure 1

Schematic of the experimental apparatus for flow visualization.

Figure 2

Schematic representation of the dissolution process of pentanol droplet in water under vertical confinement. The height $\left(L\right)$ represents distance between substrate and edge of confinement, and $D_0$ is initial droplet footprint diameter. Red and blue spiral structures illustrate toroidal vortex while small structures shown on the droplet and at the surface of confinement are eddies.

Figure 3

Comparison of temporal evolution of nonconfined droplet with four different confinement distances ($L/D_0=0.75,1.5,2.25$, and 3), depicting (a) normalized radius as a function of time and (b) normalized volume as a function of time. The time coordinate is normalized with convective lifetime of the droplet $\left({{\tau }_c}^{\prime }\right)$. Here, $R_0$ and $V_0$ are initial droplet radius and droplet volume, respectively. The curves represent an average of at least three different cases with a maximum uncertainty of $\pm 10\mu \mathrm{m}$. The dashed lines in (a) represent a linear trend.

Figure 4

Flow visualization of dissolving droplets under different confinements ($L/D_0=0.75,1.5,2.25$, and 3) representing (a) velocity magnitude and (b) isosurface of vorticity taken at 120 s after the deposition of the droplets. The vectors were obtained by taking a mean over 2 s. The inset in (a) displays the magnified portion of the vortex. The focal plane of the laser was centered at the droplet.

Figure 5

Flow visualization of a dissolving droplet for confinement $L/D_0=1.5$ representing (a) velocity magnitude and (b) isosurface of vorticity taken at $\mathrm{\Delta }t=0,3$, and 6 min. The vectors were obtained by taking a mean over 2 s (48 frames).

Figure 6

Velocity magnitude surrounding the dissolving droplet for a confinement configuration ($L/D_0=3$) at different heights above the droplet. The distances are measured along the positive $z$ axis.

Figure 7

Sherwood number (${\mathrm{Sh}}^{\prime }$) as a function of Rayleigh number (${\mathrm{Ra}}^{\prime }$) for the present experiments along with the results of Dietrich et al. [18]. The plot shows the average value and the solid line represents $\mathrm{Sh}=\mathrm{R}{\mathrm{a}}^{1/4}$. The modified Rayleigh number, ${\mathrm{Ra}}^{\prime }$ is expressed as a function of geometrical parameters taking linear stratification into consideration, where $h_0$ is the initial droplet height and $L$ is the confinement distance. Inset shows a variation of ${\mathrm{Sh}}^{\prime }$ against ${\mathrm{Ra}}^{\prime }$ considering the lower range of ${\mathrm{Ra}}^{\prime }$ in Dietrich et al. [18] and higher range of ${\mathrm{Ra}}^{\prime }$ in the present work.

Figure 8

The variation of dissolution time is plotted as a function of the concentration difference for pentanol droplets of different volume (1.5, 1, and $0.5\mu \mathrm{l}$). The data presented in open symbol correspond to different liquids (1-hexanol, 1-heptanol, 2-heptanol, and 3-heptanol), estimated using the concentration values taken from Dietrich et al. [18]. The dotted lines represent the theoretical dissolution time obtained from Eq. (8) for the respective droplet volumes. The modified concentration difference $\mathrm{\Delta }{\mathrm{C}}^{\prime }$ is expressed as a function of geometrical parameters considering linear stratification. Additionally, inset is provided for proper distinction of present experimental data.