Laser-driven droplet deformation at low Weber numbers

TL;DR

This work addresses how nanosecond laser pulses deform micron-scale tin droplets at low Weber numbers by introducing a deformation parameter Wed that encapsulates both the propulsion-driven We and the laser-pressure width W. Through coordinated experiments and CFD simulations, the authors show that Wed controls three regimes—oscillation, breakup, and sheet formation—by linking the initial radial expansion to subsequent dynamics, and they introduce We_d as a unifying scaling that collapses results across droplet sizes. The study provides a phase diagram in (We,W) and demonstrates a universal Wed description, enabling controlled manipulation of droplet morphologies in applications such as EUV source generation and laser-based material processing. Overall, the work clarifies how the spatial distribution of laser-induced pressure, together with inertial and capillary effects, governs low-$We$ droplet deformation with direct practical implications for nanolithography and related laser–liquid technologies.

Abstract

We investigate droplet deformation following laser-pulse impact at low Weber numbers (We ~ 0.1-100). Droplet dynamics can be characterized by two key parameters: the impact We number and the width, W, of the distribution of the impact force over the droplet surface. By varying laser pulse energy, our experiments traverse a phase space comprising (I) droplet oscillation, (II) breakup, or (III) sheet formation. Numerical simulations complement the experiments by determining the pressure width and by allowing We and W to be varied independently, despite their correlation in the experiments. A single phase diagram, integrating observations from both experiments and simulations, demonstrates that all phenomena can be explained by a single parameter: the deformation Weber number Wed=f(We, W) that is based on the initial radial expansion speed of the droplet, following impact. The resulting phase diagram separates (I) droplet oscillation for Wed<5, from (II) breakup for 5<Wed<60, and (III) sheet formation for Wed>60.

Figures (8)

• Figure 1: (a) Conceptual side view representation of laser-droplet interaction. The laser beam is represented as a red area. The force profile resulting from laser-plasma generation is depicted in (b) where curves show different values of pressure distribution on the droplet's surface. The black and orange curves with the same value of $\textrm{W}$ depict different values of $\textrm{We}$ (higher and lower, respectively). Experimental examples of the hydrodynamic response after laser interaction with a droplet with diameter $D_0=50\,\mu\textrm{m}$ is shown in (c-e). Each row contains frames at different fractions of capillary times, $\tau_\textrm{c} = 16.4\,\mu\textrm{s}$. From top to bottom: (c) droplet oscillation for $\textrm{W}=0.2$ and $\textrm{W}=1.0$. (d) droplet breakup after retraction for $\textrm{We}=3.4$ and $\textrm{W}=1.4$. (e) sheet expansion for $\textrm{We}=154$ and $\textrm{W}=2$. The gray scalar bar in the first frame in (c) corresponds to $D_0=50\mu\mathrm{m}$.
• Figure 2: Droplet early deformation for different pressure profile imprinted by the laser pulse. (a) Quantification of initial droplet deformation on laser-faced side within the inertial timescale, at 200 ns $\sim0.01\tau_\textrm{c}$. The impulsed liquid flow manifests as two bulges that display certain opening angles. The orange circle represents the shape of the droplet before laser impact. (b) Corresponding simulations with the same $\textrm{We}$ values with a $\textrm{W}$ parameter selected to match the angle of the surface maximum radial velocity with the corner bulges shown in (a). The gray arrows define velocity field of the liquid upon laser impact, whereas the color map shows the radial component of the velocity, $\bar{u}_{r_{\textrm{def}}}$. The black contour represents the droplet morphology at $0.01\tau_\textrm{c}$.
• Figure 3: Correlation between the propulsion $\textrm{We}$ and the pressure width $\textrm{W}$. (a) Opening angle $\theta_{\textrm{open}}$ over $\textrm{We}$ for two droplet diameters, $D_0=50,\,70\,\mu\textrm{m}$. The red dashed line represents the empirical fit to the experimental data, with $\theta_{\textrm{open}}=60\textrm{We}^{0.1}$. The green dashed line depicts the numerical result from the model. See main text. (b) Variation of $\theta_{\textrm{open}}$ for different pressure widths $\textrm{W}$ as observed from simulations. The red dashed line depicts the numerical scaling found from simulations, with $\theta_{\textrm{open}}=48\textrm{We}+3$. (c) Variation of $\textrm{W}$ with $\textrm{We}$. The gray dots correspond to data depicted in (a) and yellow crosses show the characteristic examples shown in fig. \ref{['fig:2']}(a). Three different regimes are highlighted following experimental data: I.oscillation, II.breakup, and III.sheet formation, respectively. The gray solid line shows the correlation between $\textrm{We}$ and $\textrm{W}$ obtained from the two previous fits in (a) and (b), as illustrated by eq. (\ref{['eq:We-vs-W']}).
• Figure 4: Droplet oscillation. (a) Shadowgraphs of droplet deformation within the oscillation regime at different fractions of capillary time $\tau_\textrm{c}$ for $\textrm{We}=1.7$ and $\textrm{W}=1.3$. (b) Numerical results at the same times as depicted in (a) and the same values for $\textrm{We}$ and $\textrm{W}$. (c) Nondimensional radius $R/R_0$ over time $t/\tau_\textrm{c}$ for two different oscillation cases: $\textrm{We}=1.7$ and $\textrm{W}=1.3$ (upper plot), $\textrm{We}=2.2$ and $\textrm{W}=1.4$ (bottom plot). The red dashed line corresponds to the best fit of the oscillation estimated from the equation of Rayleigh modes with $l=2$. (d) Example of a staircase-like structure where surface capillary waves (CW) are pointed out with red arrows as "$1^{\textrm{st}}$ and $2^{\textrm{nd}}$ fronts". The red dashed circle represents the shape of the droplet at rest. Here, $\theta$ is the radial position of the CW front on the surface. (e) Parametric representation of the surface contour for different times $t/\tau_\textrm{c}$ as nondimensional radial extension $R(\theta)/R_0$ over $\theta$. The two CW fronts can be observed as two peaks. Data includes droplet at rest (straight line at $0.01\tau_\textrm{c}$) and at several instances after impact to illustrate the origin and propagation of CW. (f) CW phase estimated for $1^{\textrm{st}}$ (green data) and $2^{\textrm{nd}}$ (blue data) fronts. The red line represents the dispersion law depicted by eq. (\ref{['eq:cw-velocity']}).
• Figure 5: Droplet breakup. (a) Variation of the critical Weber number $\textrm{We}_{\hat{U}}$ based on the horizontal expansion rate $\hat{U}$ after radial retraction over different propulsion Weber numbers $\textrm{We}$, for two different droplet diameters, $D_0=50, 70\,\mu\textrm{m}$ (circle and square symbols, respectively). (b) Data from panel (a) with the horizontal axis represented by deformation Weber number $\textrm{We}_\textrm{d}$. The limit between oscillation and breakup regimes is estimated at $\textrm{We}_{\hat{U}}=1$, as shown with the dashed line, with droplet oscillation for $\textrm{We}_{\hat{U}}<1$ and droplet breakup for $\textrm{We}_{\hat{U}}>1$.