Marangoni flow driven hysteresis and azimuthal symmetry breaking in evaporating binary droplets

TL;DR

This work investigates solutal Marangoni-driven instabilities in evaporating binary droplets, focusing on negative Marangoni numbers where the return flow amplifies surface-tension perturbations. Using a minimal axisymmetric quasi-stationary Stokes–diffusion model and a lubrication reduction, the authors map instability thresholds as functions of the Marangoni number $Ma$ and the contact angle $\theta$, revealing bistable regimes and diverse azimuthal instabilities. They show that, especially at small $\theta$, the height profile and geometric constraints enable azimuthal Marangoni modes (m=1..8 and higher) and even chaotic dynamics, with a subcritical Hopf bifurcation at $Ma_R\approx -268.69$ indicating transition to chaos. The study also demonstrates geometry-driven instabilities for positive $Ma$ by placing droplets in shallow pits, emphasizing the pivotal role of the droplet height field in driving interfacial flows. While the models are simplified (negligible thermal effects, Raoult’s law, and gas-phase convection), they isolate the core solutal mechanisms and provide a framework for understanding and controlling Marangoni instabilities in evaporating binary droplets.

Abstract

The non-uniform evaporation rate at the liquid-gas interface of binary droplets induces solutal Marangoni flows. In glycerol-water mixtures (positive Marangoni number, where the more volatile fluid has higher surface tension), these flows stabilise into steady patterns. Conversely, in water-ethanol mixtures (negative Marangoni number, where the less volatile fluid has higher surface tension), Marangoni instabilities emerge, producing seemingly chaotic flows. This behaviour arises from the opposing signs of the Marangoni number. Perturbations locally reducing surface tension at the interface drive Marangoni flows away from the perturbed region. Incompressibility enforces a return flow, drawing fluid from the bulk towards the interface. In mixtures with a negative Marangoni number, preferential evaporation of the lower-surface-tension component leads to a higher concentration of the higher-surface-tension component at the interface as compared to the bulk. The return flow therefore creates a positive feedback loop, further reducing surface tension and enhancing the instability. We investigate bistable quasi-stationary solutions in evaporating binary droplets with negative Marangoni numbers and we examine symmetry breaking across a range of Marangoni number and contact angles. Remarkably, droplets with low contact angle show instabilities at lower critical Marangoni numbers than droplets with larger contact angles. Our numerical simulations reveal that interactions between droplet height profiles and non-uniform evaporation rates trigger azimuthal Marangoni instabilities in flat droplets. This geometrically confined instability can even destabilise mixtures with positive Marangoni numbers, particularly for concave liquid-gas interfaces. Finally, through Lyapunov exponent analysis, we confirm the chaotic nature of flows in droplets with a negative Marangoni number.

Figures (11)

• Figure 1: Schematic representation of the onset of Marangoni stabilisation (a) and instability (b) in a two-dimensional box with periodic boundaries and with an evaporating top surface, containing a glycerol-water mixture (a) and a water-ethanol mixture (b). In (a), due to water being more volatile than glycerol, the surface tension at the interface is reduced as compared to the bulk. In (b), vice-versa, due to the higher volatility of ethanol, the surface tension at the interface is enhanced as compared to the bulk. A disturbance that locally diminishes surface tension at the interface generates a Marangoni flow directed away from the disturbed area. Continuity of the fluid generates a return flow from the bulk to the disturbed region. In the glycerol-water mixture, this return flow transports liquid with higher surface tension into the disturbed area, thereby mitigating the Marangoni flow. Conversely, in the water-ethanol mixture, the return flow carries liquid with lower surface tension into the disturbed area, further reducing the surface tension and amplifying the Marangoni flow. Consequently, a positive feedback loop of Marangoni and return flows is established, resulting in the Marangoni instability.
• Figure 2: Phase portrait of solution branches as a function of $\mathrm{Ma}$, for $\theta = 89^\circ$ (a), $\theta = 90^\circ$ (b), and $\theta = 91^\circ$ (c). The $x$-axis represents the average tangential velocity $U_t$ at the liquid-gas interface, a key measure of the flow field, indicating the flow direction within the droplet when a single vortex is present. The flow field and $\xi$ profile for selected $\text{A}{\rightarrow}\text{R}$, $\text{R}{\rightarrow}\text{A}$, and 2V solutions are also depicted. Here, the diverging blue to red colours in the contour plots represent increasing $\xi$ of the fluid with the lowest surface tension, e.g. ethanol in water-ethanol mixtures. At $\theta = 90^\circ$, the $\xi$ profile is purely diffusive for low $\mathrm{Ma}$, becoming unstable at a critical $\mathrm{Ma}_\mathrm{cr}$ in an imperfect Pitchfork bifurcation (green). The inset in (b) shows the $\theta = 90^\circ$ phase portrait in a range of $\mathrm{Ma}$ from 150 to 190, where the imperfect Pitchfork is clearly visible. All other solution branches lose their stability in either fold (red) or a Hopf (blue) bifurcations.
• Figure 3: Bifurcation diagram of the quasi-stationary axisymmetric solutions as a function of $\mathrm{Ma}$ and $\theta$. The diagram is divided into three regimes, where the stable solutions are either a $\text{A}{\rightarrow}\text{R}$ solution (blue colour), a $\text{R}{\rightarrow}\text{A}$ solution (yellow colour), or a 2V solution (plum colour). The transition between these regimes is marked by fold (red) or Hopf (blue) bifurcations. Above the upper Hopf bifurcation curves, no stable quasi-stationary solutions are found (gray).
• Figure 4: Azimuthal bifurcation diagram for the stability of the quasi-stationary axisymmetric solutions in the 2V (a) and $\text{R}{\rightarrow}\text{A}$ (b) solutions regimes. In (a), the bifurcations that limit the stability of the $\text{A}{\rightarrow}\text{R}$ and $\text{R}{\rightarrow}\text{A}$ solution regimes are shown with lower opacity. Similarly, in (b), the bifurcations for the $\text{A}{\rightarrow}\text{R}$ and 2V solutions are depicted with lower opacity (see all curves in figure \ref{['fig:ma_vs_theta']}). In the 2V regime, all base solutions are unstable for $m=1$, leading to a single vortex, as depicted at the bottom right of (a). The 2V regime is therefore interpreted as an artifact of the imposed axisymmetry in $\S \ref{['sec:ma_vs_theta_phase_diagram']}$, where the upper vortex merges with its counterpart in the rim region of the droplet, as depicted at the bottom left of (a). In the $\text{R}{\rightarrow}\text{A}$ regime, multiple bifurcations are observed corresponding to the instability of modes $m=1$ to $m=8$. The adjacent plots show an isometric view of an azimuthally stable solution (bottom right), and solutions subject to the linear effects of $m=2$ (top right), and $m=5$ (top left) azimuthally unstable perturbations. Exclusively in the $\text{R}{\rightarrow}\text{A}$ regime, the eigenvalues and eigenfunction had a non-zero imaginary part, indicating rotational motion, as depicted by the green arrows in the two upper plots.
• Figure 5: Azimuthal bifurcation diagram assessing the stability of the quasi-stationary axisymmetric solutions in the $\text{A}{\rightarrow}\text{R}$ regime. The bifurcations that limit the stability of the 2V and $\text{R}{\rightarrow}\text{A}$ solution regimes are shown with lower opacity (see all curves in figure \ref{['fig:ma_vs_theta']}). The bifurcation curves depict the range from $m=1$ to $m=30$. The adjacent plots show a top view of the droplet in a stable axisymmetric regime (bottom right) or subject to the linear effects of $m=1$ (top right), 10 (bottom left), and 20 (top left) unstable perturbations on the azimuthal instability.