Oscillations of a Water Droplet on a Horizontally Vibrating Substrate

TL;DR

This work investigates how water droplets deform and breakup when placed on horizontally vibrating substrates using many-body dissipative particle dynamics (MDPD), enabling molecular-resolution insight into droplet dynamics and internal flows. The substrate drive is described by $u_{ m sub}(t)=-A_{ m sub}\omega_{ m sub}\sin(\omega_{ m sub} t)$, and wettability is tuned via $oldsymbol{\\varepsilon_{ m ws}}$, allowing droplets with radii spanning nano- to mesoscopic scales to be analyzed. The authors identify three oscillation phases (Phase I: follower, Phase II: symmetry breaking with internal rotation, Phase III: high-shear elongation leading to breakup) and show that phase boundaries correlate with the contact-surface velocity $|u_{x, m cs}|$ and capillary number $Ca$, with wettability strongly influencing breakup propensity. The results provide mechanistic, molecular-level understanding of droplet oscillations and breakup, informing optimization in applications such as inkjet printing, lab-on-a-chip microfluidics, and spray cooling, and bridging simulations with experimental observations.

Abstract

Deformed droplets are ubiquitous in various industrial applications, such as inkjet printing, lab-on-a-chip devices, and spray cooling, and can fundamentally affect the involved applications both favorably and unfavorably. Here, we employ many-body dissipative particle dynamics to investigate the oscillations of water droplets on a harmonically and horizontally vibrating, solid substrate. Three distinct scenarios of oscillations as a response to the substrate vibrations have been identified. The first scenario reflects a common situation where the droplet can follow the substrate vibrations. In the other two scenarios, favored in the case of hydrophilic substrates, droplet oscillations generate high shear rates that ultimately lead to droplet breakup. Leveraging our simulation model, the properties of the droplet and the mechanisms related to the oscillations are analyzed with a molecular-level resolution, while results are also put in the perspective of experiment. Our study suggests that the three scenarios can be distinguished by the contact-surface velocity of the oscillating droplet, with threshold velocities influenced by the substrate's wettability. Moreover, the mean magnitude of the particle velocity at the contact surface plays a key role in determining the three oscillation phases, suggesting that the capillary number of the oscillating droplet governs the phase behavior. Thus, our approach aims to optimize droplet oscillations and deformations on solid substrates, which have direct implications for technological applications.

Figures (20)

• Figure 1: A typical snapshot in the simulations of a water droplet on a static substrate. In the lower panel, $\theta_{1}$ and $\theta_{2}$ indicate the contact angles of the droplet at the two points along the contact line, as indicated. Here, the number of particles $N=20\times10^4$ and the droplet--substrate affinity $\varepsilon_{\rm ws}=2.0$, resulting in equilibrium contact angles $\theta_1,\theta_2\approx90^{\circ}$. The droplet width at the static contact surface is $w_{x}\approx46$. Snapshots were generated using OVITO software.Stukowski2010
• Figure 2: Contact angles $\theta_{1}$ and $\theta_{2}$ of a static droplet over simulation time, $t$. Same parameters as in Figure \ref{['fig:static_dpl_ovito']} where $N=20\times10^4$ and $\varepsilon_{\rm ws}=2.0$. The values shown above are averages from four samples. The contact angles are about $90^{\circ}$.
• Figure 3: Static droplet analysis for: (a) Number of particles $N=20\times10^4$, (b) $N=10\times10^4$ and (c) $N=5\times10^4$, with contact angles $\theta=90^{\circ}$. Upper panels: The center-of-mass position ($z_{\rm{com}}$) of the static droplet over time $t$. Lower panels: Fourier analysis of $z_{\rm{com}}$ in natural log scale vertically. Three eigenmodes are observed in each case (a), (b), and (c). For (a) the frequencies of the eigenmodes are ($t^{-1}$) $\sim\pm0.0052$, $\pm0.14$ and $\pm0.26$; (b) $\sim\pm0.0057$, $\pm0.17$ and $\pm0.32$; (c) $\sim\pm0.0067$, $\pm0.20$ and $\pm0.37$. All results are obtained from an average of four different runs.
• Figure 4: Frequency of the $1^{\rm st}$ oscillation mode for static droplets with number of particle $N=5\times10^4$, $N=10\times10^4$ and $N=20\times10^4$, each with contact angles $\theta=50^{\circ},90^{\circ}$ and $140^{\circ}$. All results are obtained from an average of four different runs.
• Figure 5: State diagram showing Phase I, II, and III, indicated by different color shading (see text for details), as a function of the amplitude and frequency of substrate vibrations for a droplet with $N=20\times10^4$. Each plot corresponds to a substrate with different wettability, with the equilibrium contact angle, $\theta$, of the droplet on a static substrate (without vibrations) indicated at the top of each plot: (a) $\theta\sim50^{\circ}$; (b) $\theta\sim65^{\circ}$; (c) $\theta\sim90^{\circ}$; (d) $\theta\sim115^{\circ}$; (e) $\theta\sim140^{\circ}$.