Phase Transitions of Oscillating Droplets on Horizontally Vibrating Substrates

TL;DR

The paper addresses how horizontal substrate vibrations drive deformations, rotation, and breakup of sessile droplets across liquids and wettabilities. It employs extensive many-body dissipative particle dynamics (MDPD) simulations and defines the capillary number $Ca$ and the Ohnesorge number $Oh$ to organize phase behavior, while tracking angular momentum $L_z$ and vorticity $\omega_z$ to characterize rotational instabilities. The key contributions are the identification of three oscillation phases (Phase I: steady, Phase II: metastable rotation, Phase III: out-of-equilibrium breakup), determination of phase-transition $Ca$ thresholds as functions of $\theta$ and $Oh$, and the use of particle--particle and particle--substrate contact counts as energy proxies in the absence of an explicit free-energy function. The findings offer mechanistic insight with practical relevance to microfluidic and droplet-based technologies where substrate vibrations are used to manipulate droplets.

Abstract

Droplet deformations caused by substrate vibrations are ubiquitous in nature and highly relevant for applications such as microreactors and single-cell sorting. The vibrations can induce droplet oscillations, a fundamental process that requires an in-depth understanding. Here, we report on extensive many-body dissipative particle dynamics simulations carried out to study the oscillations of droplets of different liquids on horizontally vibrating substrates, covering a wide range of vibration frequencies and amplitudes as well as substrate wettability. We categorize the phases observed for different parameter sets based on the capillary number and identify the transitions between the observed oscillation phases, which are characterized by means of suitable parameters, such as the angular momentum and vorticity of the droplet. The instability growth rate for oscillation phase II, which leads to highly asymmetric oscillations and eventual droplet breakup, is also determined. Finally, we characterize the state of the droplet for the various scenarios by means of the particle-particle and particle-substrate contacts. We find a steady-state scenario for phase I, metastable breathing modes for phase II, and an out-of-equilibrium state for phase III. Thus, we anticipate that this study provides much needed insights into a fundamental phenomenon in nature with significant relevance for applications.

Figures (14)

• Figure 1: Top-view snapshots of the three oscillation phases, generated using OVITO software.Stukowski2010 All droplets have particle number $N=20\times10^4$, equilibrium contact angle $\theta=90^{\circ}$ (droplet--substrate affinity $\varepsilon_{\rm ws}=1.8$), initial contact length along the $x$ direction $D_{x}\approx48$, attractive strength $A=-40$, and repulsive strength $B=25$; see Section \ref{['sec:methods']} for details. The substrate vibration frequency is fixed at $\omega_{\rm sub}=0.015\pi$, with vibration amplitudes: (a) $A_{\rm sub}=450$ for phase I (multimedia available online); (b) $A_{\rm sub}=675$ for phase II (multimedia available online); (c) $A_{\rm sub}=900$ for phase III (multimedia available online). Instantaneous contact lengths $D_x$ are shown in the figures.
• Figure 2: Snapshots of the simulation setup generated using OVITO software. Stukowski2010 (a) Perspective view of a static droplet on a substrate (see Eq. \ref{['eq:LJpotential93']}). The substrate is set at the plane $z=10$, and the droplet is initially positioned at the center ($x=0, y=0$). (b1,b2) Front views of the simulation. The equilibrium contact angle of the droplet is adjusted by tuning the attractive strength $A$ and the affinity $\varepsilon_{\rm{ws}}$ (see Table \ref{['tab:epsilon']}). In this example, (b1) uses $A=-40, \varepsilon_{\rm{ws}}=1.8$; (b2) uses $A=-60, \varepsilon_{\rm{ws}}=5.7$. These parameter sets result in different static contact angles $\theta_{1}$ and $\theta_{2}$ at equilibrium state (for each static case, of course, $\theta_{1}=\theta_{2}=\theta$). (c1,c2) A time-dependent sinusoidal vibration (see Eq. \ref{['eq:substrate_velocity']}) is applied to the substrate, with amplitude $A_{\rm{sub}}$ and frequency $\omega_{\rm{sub}}$, inducing droplet oscillation. Both snapshots are taken at $t=1240$. Here, both cases use the same frequency $\omega_{\rm sub}=0.015\pi$; (c1) has an amplitude $A_{\rm sub}=450$, while (c2) has $A_{\rm sub}=225$.
• Figure 3: Frequency of the fundamental mode ($n_{f} = 2$) for static droplets with $A=-40$ and $B=25$, for particle numbers $N=5\times10^4, 10\times10^4,20\times10^4$ and $100\times10^4$, each at contact angles $\theta=50^{\circ},90^{\circ}$ and $130^{\circ}$. All results are averaged over four independent runs, with error bars representing the standard deviation of the measured frequencies. (a) Horizontal dashed lines indicate the expected scaling $f_{2}\propto\sqrt{N^{-1}}$, showing good agreement for highly spherical droplets ($\theta=130^{\circ}$) when $N\ge10\times10^4$. (b) Black dashed line indicates the expected scaling $f_{2}\propto\sqrt{N^{-1}}$.
• Figure 4: Oscillation phases (I, II, III) plotted against the capillary number (Ca) for droplets with various attractive strengths ($\rm{Oh}=0.1524,0.1660,0.2083,$ and $0.4477$) and equilibrium contact angles ($\theta=50^\circ$, $90^\circ$, and $110^\circ$), as indicated. The number of particles is fixed at $N=20\times10^4$ for all cases. Each data point corresponds to a distinct set of substrate vibration amplitude $A_{\rm sub}$ and frequency $\omega_{\rm sub}$.
• Figure 5: Capillary number (Ca) for phase transitions I--II (a) and II--III (b) plotted against the Ohnesorge number $(\rm{Oh})$ for various equilibrium contact angles $\theta$ (see Figure \ref{['fig:theta_all_ca_phase']}). Error bars indicate the overlapping range of Ca for each transition.