Droplet dissolution driven by emerging thermal gradients and Marangoni flow

The lifetime $\tau$ of an isothermal and purely diffusively dissolving droplet in a host liquid scales as $\tau \sim R_0^2$ with its initial radius $R_0$ [Langmuir, Phys. Rev. 12, 368 (1918)]. For a droplet dissolving due to natural convection driven by density differences, its lifetime scales as $\tau \sim R_0^{5/4}$ [Dietrich et al., J. Fluid Mech. 794, 45 (2016)]. In this paper we experimentally find and theoretically derive yet another droplet dissolution behavior, resulting in $\tau \sim R_0^4$. It occurs when the dissolution dynamics is controlled by local heating of the liquid, leading to a modified solubility and a thermal Marangoni flow around the droplet. The thermal gradient is achieved by plasmonic heating of a gold nanoparticle decorated sample surface, on which a sessile water droplet immersed in water-saturated 1-butanol solution is sitting. The resulting off-wall thermal Marangoni flow and the temperature dependence of the solubility determine the droplet dissolution rate, resulting in a shrinkage $R\left(t\right)\sim {(\tau -t)}^{1/4}$ of the droplet radius and thus in $\tau \sim R_0^4$.

Figure 1

(a) Snapshots of a dissolving water droplet in water-saturated 1-butanol at different instants of time under laser power $P_{\ell }=40$ mW. (b) Evolution of volume $V$, radius $R$, and contact angle ${\theta }_c$ in time. All parameters have been normalized by their initial values: $V_0=0.32$ nL, $R_0=44$ $\mu \mathrm{m}$, and ${\theta }_0\approx {130}^{\circ }$, respectively.

Figure 2

Schematic diagram of the plume formation above a dissolving water droplet in the surrounding water-saturated 1-butanol. The plasmonic effect induced temperature field leads to an increased water solubility and a decreased surface tension of 1-butanol solution near the bottom of the water droplet. As a result, an off-wall thermal Marangoni flow is generated, transporting the hot 1-butanol solution with a high water concentration to the cold region above the droplet, leading to the nucleation of a water droplet plume. As explained in the main text, gravity does not play any role here; the flow is solely driven by thermal Marangoni forces.

Figure 3

(a) Particle image velocimetry of the liquid in the vicinity of a dissolving droplet at different instants of time at a laser power of $P_{\ell }=40$ mW. (b) Measured mean liquid velocity as a function of the droplet radius (and thus time) as measured with PIV in the three selected regions 1, 2, and 3, as shown in panel (a-I).

Figure 4

(a) Droplet radius $R$ as a function of $\tau -t$ on a double logarithmic scale. The solid black line refers to an exponent of 1/4. (b) Rate of volume loss per area $\dot{V}/A$ as a function of droplet radius $R$ at different laser powers $P_{\ell }$. The solid black line refers to the exponent $-3$.

Figure 5

Dissolution time $\tau$ of water droplets as a function of the initial droplet radius $R_0$ at different laser powers $P_{\ell }$. The droplet lifetime $\tau$ obeys the scaling law of $\tau \sim R_0^4$. The solid lines are drawn to guide the eye.

Figure 6

Flow field inside the water droplet under plasmonic effect.

Figure 7

The saturation concentration $c_s$ of water in 1-butanol as a function of the temperature $T$. Data points are from Ref. [30]. The solid curve is the result of a least-squares quadratic fitting of the data points to Eq. (B1).