A radiation two-phase flow model for simulating plasma-liquid interactions

TL;DR

The paper introduces a radiation two-phase diffuse-interface model that couples plasma radiation-hydrodynamics with compressible liquid dynamics to simulate plasma-liquid interactions in LPP-EUV sources. By extending a five-equation diffuse-interface formulation to include radiation transport, thermal conduction, and ionization, and by enforcing mechanical equilibrium across the interface with physically consistent flux closures, the approach captures both rapid plasma expansion and late-time sheet formation of tin droplets. Validations on radiative shocks, diffusion-dominated blasts, and bubble collapse demonstrate accurate multi-physics behavior, while application to a nanosecond laser pre-pulse on a tin droplet reproduces experimentally observed axial jets and sheet formation with quantitative fidelity. The framework provides a self-consistent tool for LPP-EUV source optimization and can be extended to include phase transitions and tabulated EOS for more detailed material models.

Abstract

In laser-produced plasma (LPP) extreme ultraviolet (EUV) sources, deformation of a tin droplet into an optimal target shape is governed by its interaction with a pre-pulse laser-generated plasma. This interaction is mediated by a transient ablation pressure, whose complex spatio-temporal evolution remains experimentally inaccessible. Existing modeling approaches are limited: Empirical pressure-impulse models neglect dynamic plasma feedback, while advanced radiation-hydrodynamic codes often fail to resolve late-time droplet hydrodynamics. To bridge this gap, we propose a radiation two-phase flow model based on a diffuse interface methodology. The model integrates radiation hydrodynamics for the plasma with the Euler equations for a weakly compressible liquid, extending a five-equation diffuse interface formulation to incorporate radiation transport, thermal conduction, and ionization. This formulation enforces pressure and velocity equilibrium across the diffuse interface region, with closure models constructed to ensure correct jump conditions at interfaces and asymptotically recover the pure-phase equations in bulk regions. Then, we apply the model to simulate a benchmark pre-pulse scenario, where a 50 micron tin droplet is irradiated by a 10 ns laser pulse. The simulations capture the rapid plasma expansion and subsequent inertial flattening of the droplet into a thin, curved sheet over microsecond timescales. Notably, the model reproduces experimentally observed features (such as an axial jet) rarely replicated in prior simulations. Quantitative agreement with experimental data for sheet dimensions and velocity validates the approach. The proposed model self-consistently couples laser-plasma physics with compressible droplet dynamics, providing a powerful tool for fundamental studies of plasma-liquid interactions in LPP-EUV source optimization.

Figures (8)

• Figure 1: ($a$) Schematic of an interface separating the immiscible plasma and liquid, where the blue phase represents the liquid and the white phase represents the plasma; ($b$) A diffuse interface is used to replace the physical interface on a Cartesian grid, and the volume fraction of the liquid $\alpha_{l}$ is adopted to represent the interface position, where $0\le\alpha_{l}\le 1$.
• Figure 2: The profiles of a radiative shock problem at $0\ \mathrm{ns}$ (black lines and open squares) and $4.2\ \mathrm{ns}$ (dashed lines and solid circles), in terms of density ($a$), plasma temperature ($b$), and radiation temperature ($c$). The symbols and lines denote numerical and semi-analytical solutions, respectively. Note that the symbols are sampled at every $8$th grid point for visual clarity.
• Figure 3: Density ($a$), relative density error $\left|\rho^{N}-\rho^{A}\right|/\rho^{A}$ ($b$), temperature ($c$), and relative temperature error $\left|T^{N}-T^{A}\right|/T^{A}$ ($d$) for the Reinicke $\&$ Meyer-ter-vehn blast wave problem at $0.52\ \mathrm{ns}$, where superscripts $N$ and $A$ represent numerical and semi-analytical results, respectively.
• Figure 4: Numerical results of bubble collapse induced by a planar shock with respect to pressure (upper half) and numerical schlieren (lower half) at $2.2\ \upmu \mathrm{s}$ ($a$), $3.7\ \upmu \mathrm{s}$ ($b$), $3.9\ \upmu \mathrm{s}$ ($c$), and $4.1\ \upmu \mathrm{s}$ ($d$), respectively. The red lines represent the bubble interfaces (by $\alpha_{g} = 0.5$).
• Figure 5: ($a$) Schematic of axisymmetric simulations of laser-droplet interactions. ($b$-$g$) The cross-sectional shadowgraph of the droplet at different times ranging from $t=0.1$ to $1.6\ \upmu\mathrm{s}$. All panels have the same spatial scale, with time labeled below each panel.