Evaporation of Binary-Mixture Liquid Droplets: The Formation of Picoliter Pancakelike Shapes

Small multicomponent droplets are of increasing importance in a plethora of technological applications ranging from the fabrication of self-assembled hierarchical patterns to the design of autonomous fluidic systems. While often far away from equilibrium, involving complex and even chaotic flow fields, it is commonly assumed that in these systems with small drops surface tension keeps the shapes spherical. Here, studying picoliter volatile binary-mixture droplets of isopropanol and 2-butanol, we show that the dominance of surface tension forces at small scales can play a dual role: Minute variations in surface tension along the interface can create Marangoni flows that are strong enough to significantly deform the drop, forming micron-thick pancakelike shapes that are otherwise typical of large puddles. We identify the conditions under which these flattened shapes form and explain why, universally, they relax back to a spherical-cap shape toward the end of drop lifetime. We further show that the formation of pancakelike droplets suppresses the “coffee-ring” effect and leads to uniform deposition of suspended particles. The quantitative agreement between theory and experiment provides a predictive capability to modulate the shape of tiny droplets with implications in a range of technologies from fabrication of miniature optical lenses to coating, printing, and pattern deposition.

Figure 1

Binary-mixture droplets become flattened, forming micron-thick pancakelike shapes as they evaporate. (a) While a pure IPA droplet (${\varphi }_0=1$) keeps a spherical-cap shape during its entire evolution, a binary-mixture droplet of 90% IPA and 10% 2-butanol (${\varphi }_0=0.9$) becomes completely flattened, forming a micron-thick pancakelike shape. (b) The two components in the mixture evaporate at different rates. This differential evaporation creates a spatial gradient in the volume fractions $\varphi$ within the drop that affects the vapor fields of the two components $c_{v,1}$ and $c_{v,2}$ and, therefore, their evaporative fluxes $J_{\mathrm{ev},1}$ and $J_{\mathrm{ev},2}$. (c) The differential evaporation of the two components creates a solutal Marangoni flow, $Q_{\mathrm{Ma}}={\int }_0^{R}h{\overline{u}}_{\mathrm{Ma}}2\pi rdr$, along the interface from the apex to the edge of the drop that (d) leads to an enhanced spreading at early times, where $R(t)$ is the drop radius. (e) This Marangoni flow flattens the drop in the middle and enhances the curvature near the edge, creating an opposing capillary flow, $Q_{\mathrm{Ca}}={\int }_0^{R}h{\overline{u}}_{\mathrm{Ca}}2\pi rdr$, at intermediate times. The solutal Marangoni flow replenishes the edge with fresh fluid and keeps the edge from receding, effectively “pinning” the drop for a substantial part of its lifetime. At late times, the solutal Marangoni flow weakens and the drop starts to recede. In stark contrast, a pure IPA droplet (${\varphi }_0=1$) rapidly switches from spreading to receding as it evaporates. The simulations in (b) and (c) correspond to ${\varphi }_0=0.9$. The symbols in (a) and (d) represent experimental data, and the lines represent the simulations.

Figure 2

(a) The flattened binary-mixture droplets relax to a spherical-cap shape toward the end of their lifetime when the solutal Marangoni flows become sufficiently weak. (b) The radius of binary drops with low initial volume fractions follows the scaling $R\sim {\tau }^{1/2}$. The radius of drops with high initial volume fractions, however, deviates strongly from this behavior at intermediate times and can be described by $R\sim {\tau }^{1/5}$ before crossing over to the $R\sim {\tau }^{1/2}$ scaling at late times (smaller $\tau$). (c) The crossover between the two scaling regimes signals the transition from a flattened shape to a spherical-cap shape and occurs when the solutal Marangoni-driven flow, which is proportional to $\mathrm{Ma}\mathrm{\Delta }\varphi$, becomes weaker than the opposing capillary-driven flow, which is proportional to $h_0\mathrm{\Delta }\kappa$. In (b), the larger symbols represent the experimental data and the smaller symbols represent the corresponding simulation.

Figure 3

(a) The flattened pancakelike shapes form only when the initial volume fraction of the more volatile component exceeds a critical value ${\varphi }_0>{\varphi }_{0,\mathrm{cr}}\approx 0.4$. The symbols and lines represent experiments and simulations, respectively. (b) The transition from the spherical-cap regime at low ${\varphi }_0$ to the pancake regime at high ${\varphi }_0$ can be quantified by monitoring the evolution of the ratio of the radius of the flattened part of the drop, $R_{\mathrm{flat}}$ [inset in (c)], to the radius of the drop $R$, i.e., $R_{\mathrm{flat}}/R$. This ratio reaches a maximum value before decaying to zero as the drop relaxes to a spherical-cap shape. (c) This maximum value, $\max (R_{\mathrm{flat}}/R)$, shows a sharp transition from $\approx 0$ at low volume fractions corresponding to the spherical-cap regime to $\approx 1$ when ${\varphi }_0>{\varphi }_{0,\mathrm{cr}}$ corresponding to the pancake regime, which persists almost all the way to ${\varphi }_0\approx 1$. (d) Simulations (symbols) show that the compositional gradient $\mathrm{Ma}\mathrm{\Delta }\varphi$ within the drop reaches a peak during the evaporation and then decays to zero. (e) The peak composition gradient increases almost linearly with ${\varphi }_0$ and when stronger than a critical value $\mathrm{\Delta }\varphi >\mathrm{\Delta }{\varphi }_{\mathrm{cr}}$ [Eq. (4)], corresponding to the constraint ${\varphi }_0>{\varphi }_{0,\mathrm{cr}}\approx 0.4$ [Eq. (5)], leads to the formation of a pancake shape.

Figure 4

The coffee-ring effect becomes suppressed, leaving a uniform deposit in the binary-mixture drop. The deposits of poly($N$-vinylpyrrolidone)-stabilized polystyrene particles (755 nm) from a pure IPA droplet (left) and an IPA–2-butanol droplet with ${\varphi }_0=0.9$ (right). The scale bar is $30\mu \mathrm{m}$.