Evaporation of active drops: Puncturing drops and particle deposits of ring galaxy patterns

By virtue of self-propulsion, active particles impart intricate stresses to the background fluids. We propose that this active stress can be utilized to greatly control evaporation dynamics of active drops. We discover a new phenomenon of puncturing of the active drops, where the air-liquid interface of the drop undergoes spontaneous tearing and there occurs a formation of a new three-phase contact line due to the liquid-air interface hitting the liquid-solid interface through evaporation-driven mass loss. Post puncturing, we see an inside-out evaporation of the drop, where the new contact line sweeps towards the pinned outer contact line of the drops, contrasting regular drops that straightaway shrink to zero volume with self-similar shape. Furthermore, we describe how the activity inside the drops can manipulate the three-phase contact-line dynamics, which for contractile drops can result in an up to 50% enhanced lifetime of the drop and 33% quicker evaporation for extensile drops. By analyzing the flux distribution inside the drop, we gain insights on nonintuitive deposition patterns (e.g., ring galaxy type deposits that demonstrate controllable spatial gradients in the concentrations of the deposited particles) of active particles, which are oftentimes biological substances or bimetallic nanoparticles of interest. Finally, we argue that such unique evaporation and particle deposition dynamics can be leveraged for altering the lifetime of drops for bioapplications and for creating customized thin-film deposits with potential three-dimensional printing applications.

Figure 1

Experimental relevance for the proposed system. (a) Zero-force swimmer (active particles) producing stress as force dipole (nematic shakers) at far field (reproduced from Drescher et al. [37]). (b)–(d) Active swimmers moving in vortex defect (reproduced from Peng et al. [31] with permission from AAAS).

Figure 2

Schematic of the active evaporating liquid drop having a profile defined by the drop height $h\left(\mathit{r},t\right)$ (with ${\partial }_{\perp }h\ll 1$). The drop evaporates on a substrate defined by profile $g$($r$). $c$ denotes the local liquid vapor concentration distribution and $n$ is the local unit normal vector to the substrate.

Figure 3

(a), (b) Height of an evaporating extensile drop undergoing puncturing (for $\mathrm{\Lambda }=0.05, \mathrm{\Omega }=0.1, {\theta }_R=3^{\circ }$) for (a) time instants (prepuncturing) when the CL is pinned and (b) time instants (postpuncturing) when the puncturing-induced ICL recedes but the OCL remains pinned. (c) Transition from the profile at the onset of puncturing [last profile in 2] to the profile where θ$={\theta }_R=3^{\circ }$ and the ICL starts to recede [first profile in 2; with dimensionless puncturing radius $a_p]$. (d), (e) 3D drop profiles corresponding to (d) a prepunctured drop profile ($T=0.5$) and (e) a postpunctured drop profile ($T=0.65$). In (e), dimensionless and time-dependent inner radius, $a_{\mathrm{in}}=r_{\mathrm{in}}/R_0$ is noted. In (a) and (b), the dimensionless times associated with the drop profiles have been identified.

Figure 4

(a), (b) Time evolution of the profiles of an evaporating extensile nematic drop undergoing no puncturing (for $\mathrm{\Lambda }=0.01, \mathrm{\Omega }=0.1, {\theta }_R=3^{\circ }$) for (a) time instants when the contact line is pinned and (b) time instants when the contact line recedes. In (a) and (b), the drop profile demarcating the transition between pinned and receding contact-line cases is shown by a dashed line. (c), (d) 3D drop corresponding to the no-puncturing case for (c) $T=0.603$ (pinned CL case) and for (d) $T=1$ (receding CL). In (a) and (b), the dimensionless times associated with the drop profiles have been identified.

Figure 5

Parameter space (for parameters $\mathrm{\Lambda }, \mathrm{\Omega }$, and ${\theta }_R$) for determining whether the drop will undergo puncturing or not.

Figure 6

(a) Top view of the punctured drop showing the liquid fluxes [quantified in terms of the dimensionless velocity field $U_{\mathrm{avg}}={\left(\frac{\partial a}{\partial T}\right)}_{\mathrm{avg}}]$ directed towards the ICL and the OCL. Results are shown for $\mathrm{\Lambda }=0.25, \mathrm{\Omega }=0.50, {\theta }_R=3^{\circ }$, and $T=0.95$. (b) Flux of vaporized liquid corresponding to different locations of the receding ICL (which correspond to different values of $a_{\mathrm{in}}$) and fixed OCL. (c) Schematic of the ring galaxylike particle deposit pattern. (d) Variation of $a_{\mathrm{in}}$ vs $T$ for different $\mathrm{\Lambda }$ and $\mathrm{\Omega }$. Nondimensional puncturing time ($T_p$) and puncturing radius ($a_p$) $[a_p={\left(a_{\mathrm{in}}\right)}_{T=T_p}]$ has been indicated for one of the curves. (e) Variation of $a_p$ and $T_p$ (see inset) with $\mathrm{\Lambda }$ and $\mathrm{\Omega }$. (f) Anticipated ring galaxy deposition patterns for different combinations of $\mathrm{\Lambda }$ and $\mathrm{\Omega }$.

Figure 7

(a) Time evolution of the profiles of an evaporating contractile drop ($\mathrm{\Lambda }=-0.04, \mathrm{\Omega }=0.2, {\theta }_R=3^{\circ }$). At $T=0.53$, the contact angle becomes equal to ${\theta }_R$ (profile shown by dashed line). For $T<0.53$, the drop dynamics occur with pinned CL, while for $T>0.53$, the drop dynamics occurs with receding CL. (b) Variation of the dimensionless evaporation time, $T_{\mathrm{evap}}$, for different values of $\mathrm{\Lambda }$ and $\mathrm{\Omega }$. Cases with $\mathrm{\Lambda }\le 0$ correspond to contractile drops. $T_{\mathrm{evap}}$ for extensile drops that undergo puncturing and nonactive drops ($\mathrm{\Lambda }=0$) have been separately indicated. In (a), the dimensionless times associated with the drop profiles have been identified.

Figure 8

(a) Zoomed-in snippet of the simulation geometry employed in comsol. The bounding box is 50 $\mathrm{units}\times 50$ units, and the drop is a disk of radius $R=1\mathrm{unit}$ with the hole of radius $r_{\mathrm{in}}=0.4\mathrm{unit}$. A boundary condition of $c=0$ has been imposed on the top and right edges of the box, which are at a distance of 50 units from the center. (b) Zoomed-in plot of $J_z\left(r\right)$ vs $r$ for $r_{\mathrm{in}}=0.4$ for different sizes (length units) of the edge elements. We find that negligible difference in the results of the flux variation between the cases with edge elements of sizes $5\times {10}^{-5}$ units and $1\times {10}^{-5}$ units.

Figure 9

Numerical simulation result for the flux (shown by arrows) and vapor concentration (shown by color map) distribution across the punctured drop for the time instant when $r_{\mathrm{in}}=0.4\mathrm{units}$ and $R=1$ unit. The dark green represents zero saturation of vapor at a distance far away from the drop, while white represents 100% saturation that is near the drop surface.

Figure 10

$\dot{V}\left(r_{\mathrm{in}}\right)-\mathrm{vs}-r_{\mathrm{in}}$ variation. $\dot{V}\left(r_{\mathrm{in}}\right)$ is in units of $RD\mathrm{\Delta }c/\rho$.