Particle-in-Cell Simulations

• Particle-in-cell simulations are computational methods that model the coupled dynamics of charged particles and electromagnetic fields using grid-based field solvers and particle integration.
• They provide insights into nonlinear sheath formation, kinetic instabilities, and multiphase acceleration, underpinning advancements in high-energy and rarefied flow studies.
• Applications span GeV ion acceleration, micropropulsion, and needle-free injectors, with simulation outputs guiding device optimization and performance scaling.

Particle-in-cell (PIC) simulations are computational techniques used to model the dynamics of plasmas and other charged-particle systems where electromagnetic fields and particle motion strongly interact. In the context of micronozzle acceleration (MNA) for supersonic flows, high-energy beams, and rarefied microfluidic systems, PIC methods provide unique insight by resolving self-consistent field-particle dynamics, including nonlinear sheath formation, kinetic instabilities, and non-equilibrium effects. Contemporary studies apply PIC, along with molecular dynamics (MD), boundary-integral (BI), and direct simulation Monte Carlo (DSMC) methodologies, to quantify MNA performance for applications ranging from compact ion accelerators to needle-free liquid jet injectors.

1. Fundamental Principles and Methodology of PIC Simulations

PIC techniques solve coupled Maxwell’s equations and Newton’s equations for a finite population of “macro-particles” representing electrons, ions, or neutral atoms. Each time step proceeds by:

• Interpolating particle positions and velocities onto a mesh to derive charge density $\rho(x, y, t)$ and current density $J(x, y, t)$.
• Solving Maxwell’s equations ($\nabla \cdot E = \rho/\epsilon_0$, $$\nabla \times B = \mu_0 J + \mu_0\epsilon_0 \partial E/\partial t$$) to update electromagnetic field values.
• Integrating the Lorentz force ($$m \frac{d\mathbf{v}}{dt} = q[\mathbf{E} + \mathbf{v} \times \mathbf{B}]$$) for each particle to advance positions and velocities.
• Applying boundary conditions relevant to the physical system, such as open (vacuum) exits, reflecting walls, or particle sources.

For plasma and high-intensity laser interactions, PIC captures phenomena inaccessible to continuum-fluid models, including collective field generation, sheath dynamics, and nonlocal energy transfer, as exemplified in ultraintense MNA targets (Murakami et al., 30 Nov 2025).

2. High-Energy Acceleration in Micronozzles: PIC Application

In the paper of GeV-class proton acceleration via micronozzle geometry, the PIC framework models:

• The propagation of an ultraintense ($I_0 \sim 10^{22}$ W/cm$^2$) femtosecond laser pulse through a micron-scale hydrogen rod embedded in an aluminum micronozzle.
• The rapid formation of strong electrostatic sheath fields ($E_s \sim 10^{14}$ V/m) at the downstream nozzle tail, sustained over the full micron scale.
• Multi-phase acceleration: “run-up” (hot electron generation), “main-drive” (extended sheath formation and proton acceleration), and “afterburner” (self-similar plasma expansion), evaluating energy transfer via $E_{\rm max}(t)$ and angular divergence $\theta_{\rm div}$ (Murakami et al., 30 Nov 2025).

The quantitative PIC results reveal:

• GeV-scale proton beams generated at $I_0 = 10^{22}$ W/cm$^2$, with energy plateau $400$–$800$ MeV and mean $\eta \sim 2$–$4$\%.
• Beam divergence FWHM $5^\circ$–$25^\circ$, narrower than planar targets.
• Scalings: $E_{\rm max,\,MNA} \sim I_0^{0.79}$, sustained by a combination of sheath field enhancement and nozzle geometry optimization.

3. Micronozzle Acceleration in Rarefied and Nonequilibrium Flow Regimes

PIC simulations are complementary to DSMC and MD in regimes of elevated Knudsen numbers $(\text{Kn} \gtrsim 10^{-3}$–$10^{-1})$ where continuum breakdown occurs (Ikram et al., 2022, Ortmayer et al., 2023). For rarefied supersonic micro-nozzle flows, particle-based algorithms capture ballistic streaming, collisional transport, and the evolution of nonlocal field structures with meshless collision and propagation mechanics.

In DSMC for micro-propulsion:

• Simulated “macroparticles” move ballistically with stochastic binary collisions in spatial cells, mimicking Boltzmann dynamics within bypass-injection geometries (Ikram et al., 2022).
• Key flow metrics—mass flow rate, thrust force, thrust coefficient, specific impulse, and vectoring angle—are computed via surface-flux averaging of particle statistics.

In MD:

• The motion of monoatomic Lennard-Jones particles traversing a slit Laval nozzle is resolved at the atomic scale, incorporating smooth wall potentials and grand canonical reservoirs (Ortmayer et al., 2023).
• Local thermodynamic properties (density, temperature, pressure, Mach number, Knudsen number) and microscopic phenomena (sonic horizon, thermal anisotropy) emerge directly from particle tracking and virial expressions.

4. Governing Physical Principles and Scaling Laws

PIC-based MNA simulations, supplemented by BI and DSMC, quantify:

• Acceleration enhancement by geometric tapering, encoded in $V_j = \sqrt{(2\,\Delta P_b/\rho)\,F(\theta, d/D)}$, with $F$ reflecting nozzle taper and contraction ratio gains (Galvez et al., 2020).
• The role of inertial focusing, surface tension, and viscous dissipation, measured by Ohnesorge $(\text{Oh})$, Weber $(\text{We})$, Reynolds $(\text{Re})$, and Knudsen $(\text{Kn})$ numbers.
• In high-intensity laser PIC, field amplification and self-similar energy transfer are described by:

$$E_p \simeq e E_s \lambda_D \sim T_e^2 \sim I_0^\alpha,$$

and afterburner phase scaling,

$$\Delta E_{\rm afterburner} \simeq 2 Z T_e \ln(C_s \tau_L / R_0),$$

where $C_s = \sqrt{Z T_e/m_i}$ and $R_0$ is the H-rod radius (Murakami et al., 30 Nov 2025).

Tables extracted from simulation results illustrate nozzle performance trends:

$I_0$ (W/cm$^2$)	$E_{\rm max}$ (MeV)	$\theta_{\rm div}$ (deg)
$3 \times 10^{21}$	$400$	$18$
$1 \times 10^{22}$	$900$	$16$

$h/H_t$ (bypass ratio)	$F_T$ (N, 10 kPa)	$\beta$ (deg, 10 kPa)
$0.0$	$2.2$	$0$
$0.3$	$2.5$	$1.8$
$0.6$	$2.7$	$3.6$

5. Performance Trends and Design Guidelines

Simulation-based studies have identified operational and design guidelines for MNA devices:

• Optimal micronozzle taper angles (14$^\circ$–37$^\circ$) yield near-maximal velocity focusing, while steeper angles confer diminishing returns due to wall friction and contact-line hysteresis (Galvez et al., 2020).
• In rarefied supersonic nozzles, increasing bypass channel width enhances thrust, thrust coefficient, and vectoring angle, but excessive bypass weakens Mach development and supersonic region size (Ikram et al., 2022).
• Maintaining $\text{Kn} \lesssim 0.5$ in the divergent section and using atomically smooth slip boundaries optimize supersonic acceleration and sonic horizon sharpness (Ortmayer et al., 2023).
• In PIC-driven proton acceleration, aligning H-rod diameter and nozzle geometry, optimizing pulse duration ($\tau_L \sim 20$ fs), and leveraging self-similar expansion models maximize $E_{\rm max}$ and conversion efficiency (Murakami et al., 30 Nov 2025).

6. Microscopic Phenomena and Advanced Observables

Advanced PIC, MD, and DSMC simulations have elucidated several key microscopic phenomena:

• The existence of a well-defined sonic horizon: Downstream of the throat in microscopic Laval nozzles, upstream density correlations vanish at atomic spatial resolutions, signifying a local unidirectional flow regime (Ortmayer et al., 2023).
• Velocity distributions adhere to anisotropic Maxwell–Boltzmann profiles in narrow nozzles, with up to $20$–$30$\% anisotropy observable at outlet regions (Ortmayer et al., 2023).
• In MNA-driven jet injectors, BI simulations capture bubble dynamics and meniscus motion, with boundary conditions enforcing contact-line hysteresis and no-flux solid interfaces (Galvez et al., 2020).
• Despite high supersonic expansion rates, LJ gases in MD studies do not nucleate clusters; instead, pair-wise scattering may seed nucleation in larger systems, evident as a shoulder in the velocity autocorrelation function (Ortmayer et al., 2023).

7. Practical Applications and Research Context

PIC simulations underpin the rational design of energy-efficient, high-performance micronozzle systems:

• Needle-free medical jet injectors, leveraging inertial focusing of thermocavitation bubbles for precise, high-speed delivery (Galvez et al., 2020).
• Micropropulsion and thrust vectoring for satellites, utilizing bypass injection and DSMC modeling for accurate thrust, impulse, and directional control (Ikram et al., 2022).
• Compact laser-driven GeV ion accelerators for high-energy density physics, fast ignition, and radiography, made feasible by PIC models of extended sheath acceleration in micronozzle geometries (Murakami et al., 30 Nov 2025).

This suggests that continued refinement of PIC methods, coupled with high-resolution MD and DSMC, will further advance microfluidic, propulsion, and accelerator technologies by providing direct access to field–particle interplay at mesoscopic and atomic scales. The observed trends and phenomena facilitate targeted device optimization and enable novel applications across biomedical, space, and fundamental physics domains.