Gravitational Effect in Evaporating Binary Microdroplets

The flow in an evaporating glycerol-water binary submillimeter droplet with a Bond number $\mathrm{Bo}\ll 1$ is studied both experimentally and numerically. First, we measure the flow fields near the substrate by microparticle image velocimetry for both sessile and pendant droplets during the evaporation process, which surprisingly show opposite radial flow directions—inward and outward, respectively. This observation clearly reveals that in spite of the small droplet size, gravitational effects play a crucial role in controlling the flow fields in the evaporating droplets. We theoretically analyze that this gravity-driven effect is triggered by the lower volatility of glycerol which leads to a preferential evaporation of water then the local concentration difference of the two components leads to a density gradient that drives the convective flow. We show that the Archimedes number Ar is the nondimensional control parameter for the occurrence of the gravitational effects. We confirm our hypothesis by experimentally comparing two evaporating microdroplet systems, namely, a glycerol-water droplet and a 1,2-propanediol-water droplet. We obtain different Ar, larger or smaller than a unit by varying a series of droplet heights, which corresponds to cases with or without gravitational effects, respectively. Finally, we simulate the process numerically, finding good agreement with the experimental results and again confirming our interpretation.

Figure 1

The micro-PIV measurement of flow fields in both sessile (a) and pendant (b) droplets. The scale bars represent $200\mu \mathrm{m}$. (a1) and (a2) Flow fields in a sessile binary droplet measured at different heights: 200 and $10\mu \mathrm{m}$ away from the substrate, respectively. The white arrows represent the flow direction. (a2) The measurement near the substrate shows an inward radial flow, (a2) the one at larger height reveals an outward radial flow. (b1) and (b2) Flow fields of a pendant binary droplet measured by the same method as the sessile droplets. (b2) The flow near the substrate follows outward radial direction, (b1) but the flow at midheight reveals an annular flow with deviations from axisymmetry near the edge and irregular flow within the inner part. The four PIV images were taken with four different droplets. (a3) and (b3) The schematics of the flow pattern in side views of both sessile and pendant droplets.

Figure 2

(a) The evolutions of the averaged radial velocity near the substrate for sessile droplets with various initial glycerol mass concentrations, 10% (blue), 20% (red), 30% (orange), 40% (purple), 50% (green), and 60% (cyan). (b) The dimensionless averaged radial velocity plotted against the dimensionless time for the same data as in (a), which are scaled by the characteristic velocity and lifetime according to Eqs. (1) and (2), respectively. All the scaled data collapse into each other except the early stages of the ones with 10% and 20% initial concentration.

Figure 3

Micro-PIV measurements of the flow field for both sessile and pendant 1,2-propanediol-water droplets with different heights: $h_g=410\mu \mathrm{m}$ (a1), $425\mu \mathrm{m}$ (a2), $820\mu \mathrm{m}$ (b1) and $800\mu \mathrm{m}$ (b2). The flow fields were measured at same height: $10\mu \mathrm{m}$ away from the substrate. (a1) and (a2) The small droplets have low $\mathrm{Ar}<1$. The flows within two different orientated droplets show same radial direction, which indicates no gravitational effect. (b1) and (b2) The large droplets have $\mathrm{Ar}>1$. The different radial flow directions clearly show gravitational effect.

Figure 4

Snapshot of the numerical results for sessile (upper) and pendant (lower) droplets, respectively. In the gas phase, the water vapor concentration (color-coded) is shown along with the evaporation rate (arrows). Inside the droplet, the glycerol mass fraction (left) and the Stokes stream function (right) are depicted, where the arrows indicate the local flow direction.

Figure 5

Experimental (a1),(b1) and numerical (a2),(b2) results for the evolution of the radial velocity near the substrate for both sessile (a1),(a2) and pendant (b1),(b2) droplets. Positive values indicates outward flow. For sessile droplets (a1),(a2), the numerical simulation shows a great agreement with experiment. However, for pendant droplet (b1),(b2), the numerical simulation shows a stronger flow near the contact line than in the experiment, which is presumably due to the axial symmetry breaking of the flow in experiment.