Spontaneous Coulomb fissions of drops on lubricated surfaces

Abstract

Charged water drops are more widespread than commonly acknowledged. For example, raindrops typically carry charges of order Q ~ 1 pC, while routine pipetting in the laboratory produces drops with Q ~ 50 pC. Here, we show that such modest charging can spontaneously generate periodic Coulomb fissions for evaporating water drops on lubricated surfaces, with more than 60 successive cycles observed over 30 min. Interestingly, the underlying instability can be quantitatively predicted by two fissility thresholds: one marking the onset of drop elongation and another triggering fission. Each fission culminates with a fine liquid jet that disintegrates into 40-50 microdroplets, expelled within microseconds. The phenomenon spans an extraordinary range of length scales (from millimetres to microns) and time scales (hour to microseconds), with broad potential applications ranging from nanoscale fabrication to electrospray ionization.

Figures (5)

• Figure 1: Evaporation drives periodic Coulomb fissions and drop oscillations. (a) Pipetting charges water drop; food dye added here (only) for clarity. Scale bar: 1 mm. Insets 1, 2: schematic and confocal microscopy of the meniscus skirt (Scale bar: 25µm). (b) Spontaneous fission triggered by evaporation. Insets: superimposed images of ejected microdroplets. Scale bar: 200µm. (c) Temporal evolution of drop mean radius $R$ (solid gray line), showing periodic oscillations following fissions. $a$ and $b$ are the semi-major and semi-minor axes, while dashed-dot line is solution to Eq. \ref{['eq:dRdt']}. Insets 1--3: Zoom-ins of the oscillations for $N = 1$, $30$, and $50$; $t'$ is relative time, defined such that $t'=0$ s marks the onset of elongation. (d) Corresponding elongation ratio $\mathcal{E} = a / b$ vs. $t$. Insets 1, 2: rise phase of $\mathcal{E}$ vs. $t'$ and $t'/\tau_{\text{evap}}$ for different $N$.
• Figure 2: Universal charge decay during fissions. Fractional charge $Q/Q_\text{i}$ as a function of (a) $t$ and (b) $R/R_\text{c1}$. The charge is conserved without fissions (filled markers). Once fission starts (open markers), charge decay follows the universal relation $Q/Q_\text{i} = (R/R_\text{c1})^{3/2}$. (c) The same data is consistent with Eq. \ref{['eq:Q_crit_prefactor']}, and (d) charge decays as $Q/Q_\text{i} = 0.98^{N}$ (solid lines). Circular and square markers denote 0.5 and 1.0µL drops, while different lubricant viscosities are distinguished by colour. Measurements are from 102 individual drops.
• Figure 3: Fissility thresholds $X_{\text{e}}$ and $X_{\text{c}}$ for elongation and fission. (a) Schematic of a drop elongating into a semi-ellipsoid. (b) Excess energy as a function of $\mathcal{E}$ for different $X_{\text{eff}}$ values. (c) Bifurcation diagram derived from (b), revealing a supercritical bifurcation at $X_{\text{e}} = 0.25$. Equilibrium $\mathcal{E}_{\text{eq}}$ follows the stable solution branches (solid line). Instability at $X_{\text{c}} = 0.26$ results in fission. (d) Optical snapshots over one oscillation cycle ($t$ = 0--10.5 s in panels e--g). Scale bar: 100µm. Inset: rapid pole-sharpening and jetting at $X_{\text{c}}$. Scale bar: 50µm. (e--g) Temporal evolution for $R$, $\mathcal{E}$, and $X_{\text{eff}}$ over two successive cycles $N$ = 43 and 44. Inset in (f) is derived from inset in (d), where the relative time $t' = 0$ s corresponds to onset of jetting.
• Figure 4: Jetting dynamics. (a) Tear-shaped drop geometry during jetting. Insets: superimposed images of progeny microdroplets ejected at successive times. Scale bar: 200µm. (b) Jet formation inside the oil meniscus (viscosity 10mPas) and subsequent disintegration into progeny microdroplets, which collectively form the spray in (a). Scale bar: 20µm. (c) For a more viscous lubricant oil (100mPas), a bulbous end of radius $r_{b}$ forms, which later disintegrates into four smaller secondary progeny droplets of radius $r_{s}$ (marked 1--4). Scale bar: 10µm.
• Figure 5: Measuring drop charge $Q$. Measuring (a) initial drop charge $Q_\text{i}$ and (b, c) the charge remaining $Q(t)$ after elapsed time.