Micronozzle Acceleration (MNA) Overview

• Micronozzle Acceleration (MNA) is a technique that uses micron-scale nozzle geometries to convert thermal, phase-change, or field energy into directed kinetic energy in fluids or ions.
• It leverages diverse mechanisms—from CFD-optimized cold-gas flow and non-equilibrium molecular dynamics to thermocavitation and laser–plasma interactions—to achieve high thrust, supersonic jets, and GeV-level ion beams.
• Optimization of nozzle parameters, such as throat curvature and expansion ratio, is critical for enhancing performance in applications like microsatellite propulsion, needle-free injection, and compact accelerator designs.

Micronozzle acceleration (MNA) refers to a set of physical mechanisms whereby the geometry of a nozzle at the micron or sub-millimeter scale is tailored to significantly accelerate fluids or charged particles. MNA exploits both continuum and non-equilibrium effects to convert internal (thermal or phase-change) or field (laser, plasma) energy into directed kinetic energy, achieving outcomes such as high thrust in microsatellite cold-gas propulsion, ballistic microjets for needle-free injection, supersonic gas cooling for molecular beams, and multi-hundred MeV–GeV ion acceleration. The diversity of MNA implementations spans classical compressible flow, thermocavitation hydrodynamics, atomistic molecular dynamics, and laser–plasma interaction physics.

1. Continuum Micronozzle Acceleration in Cold Gas Micro-Propulsion

In microsatellite cold-gas propulsion, MNA is realized via sub-millimeter convergent–divergent “Laval” nozzles optimized to maximize thrust as a function of geometry, propellant, and operating conditions. Key geometric parameters include throat radius of curvature ($R_t$), throat and exit widths ($W_t$, $W_e$), expansion ratio ($\varepsilon = A_e/A_t$), and, in advanced designs, placement and dimensions of dual-throat stages.

CFD-driven sensitivity analysis and response-surface optimization demonstrate that introducing a curved throat of $R_t \simeq D_t$ (where $D_t$ is the throat diameter) yields a 25% thrust enhancement (from 113.1 mN to 141 mN) over sharp-cornered geometries, primarily by mitigating adverse pressure gradients and extending boundary-layer development (Niksirat, 2023). Implementation of a dual-throat nozzle, where a secondary convergent section downstream of the first throat is positioned at $L_{\rm conv,2} \approx 374$ μm and a divergence length $L_a \approx 374$ μm (for $D_t\sim1$ mm), induces re-compression and realigns oblique expansion shocks, suppressing flow separation and boosting thrust to 261 mN.

Governing equations for steady compressible flow include conservation of mass, momentum (Navier–Stokes), and energy, subject to turbulence (k–ε model), ideal gas assumptions, and adiabatic walls. The thrust can be computed as

$T = \dot{m} (v_e - v_0) + (p_e - p_a)A_e,$

with isentropic relations

$$\frac{T_0}{T} = 1 + \frac{\gamma - 1}{2} M^2,\quad \frac{p}{p_0} = \left(1 + \frac{\gamma - 1}{2}M^2\right)^{-\gamma/(\gamma-1)}.$$

Statistical DOE and response-surface methods reveal that $W_e$ (exit width) contributes $\sim 40\%$ to thrust sensitivity, $W_t$ (throat width) $\sim 24\%$, with remaining influence from $L_{\rm conv}$, $L_{\rm div}$, and inlet width. Optimal expansion ratios $\varepsilon$ are in the range $2.1$–$3.2$, while scaling laws prescribe that $C_f$ (thrust coefficient) can be enhanced by 15–20% via dual-throat staging. Maximum thrust is

$$F\propto p_0 A_t \sqrt{\frac{2\gamma}{\gamma-1}\left[1 - \left(\frac{p_a}{p_0}\right)^{(\gamma-1)/\gamma}\right]}C_f.$$

Boundary-layer control, shock alignment, and careful contouring are critical to maximizing acceleration and minimizing adverse separation, especially at low Reynolds numbers characteristic of micro-propulsion (Niksirat, 2023).

2. Non-Equilibrium and Atomistic Regimes: Molecular Dynamics of Microscopic Laval Nozzles

When nozzle dimensions approach the mean free path (high Knudsen number, $Kn \sim 1$–$10$), as in the molecular dynamics paper of slit Laval nozzles (Ortmayer et al., 2023), MNA enters a non-equilibrium regime with pronounced molecular-scale effects. Here, a simple monoatomic Lennard–Jones fluid is driven from a GCMC inlet reservoir through a micrometer- to nanometer-scale nozzle with atomically smooth walls; the outgoing flow is sampled as a stationary non-equilibrium ensemble (NEMD).

Key findings include:

• Supersonic acceleration of gas even for throats of $\sim 4$–$60$ molecular diameters ($\sigma$), with local Mach number $M(x)\sim4$–5 in large nozzles and $M(x)\lesssim M_{\rm id}(x)$ in smaller ones.
• The location of the sonic horizon ($x_c$ where $M=1$) shifts downstream by $2.7$–$3.8\,\sigma$ for the smallest nozzles.
• Temperature anisotropy develops in the supersonic expansion: $T_x$, $T_y$, $T_z$ components differ by 10–20%, with velocity distributions remaining Maxwellian within each direction.
• Spatiotemporal density correlation analysis exposes a well-defined, one-way sonic horizon, whereby upstream density fluctuation correlations vanish downstream.
• Partial ballistic transport (remnants of two-body scattering) is observed, along with a tendency for transient condensation in the diverging section.

The isentropic area–velocity and Mach–area relations,

$$\frac{d v}{v} = - \frac{1}{1-M^2} \frac{dA}{A}, \quad A(x_1)/A(x_2) = [M(x_2)/M(x_1)] \left[\frac{1+\frac{\gamma-1}{2}M^2(x_1)}{1+\frac{\gamma-1}{2}M^2(x_2)}\right]^{(\gamma+1)/2(\gamma-1)},$$

remain qualitatively valid for larger nozzles ($d_m>30\,\sigma$), while for smaller ones, rarefaction effects, wall slip, and delayed sonic transition become significant. This suggests micro-nozzle design requires explicit consideration of Knudsen number, thermal anisotropy, and wall characteristics (Ortmayer et al., 2023).

3. Thermocavitation-Driven Micronozzle Acceleration in Microfluidics

In microfluidic jetting, MNA is achieved by laser-induced thermocavitation: tightly focused CW laser irradiation (λ ≈ 450 nm, $P_L\approx$ 400–600 mW) into dye-loaded microchambers (depth $\ell=100\,\mu$m) rapidly nucleates vapor bubbles, driving phase-change-initiated expulsion of liquid through micro-nozzles (diameter $d=120\,\mu$m) (Galvez et al., 2020). The resulting pressure pulse and meniscus curvature generate high-velocity collimated jets.

Fluid acceleration is dominated by inertial effects (Ohnesorge number $Oh\approx0.015$), with typical jet velocities $v_{\rm jet}=20$–$100$ m/s and accelerations $a = O(10^6$–$10^7$ m/s$^2)$. The core acceleration relation is

$$v_{\rm jet} \simeq \frac{\Delta p\, \Delta t}{\rho H}, \qquad a \simeq \frac{\Delta p}{\rho H}$$

where $H$ is the pre-expansion fill height. Performance further scales as $v_{\rm jet}\propto H^{-1}$ and increases with taper angle, with experiments reporting a gain factor up to 2 for $\alpha$ in the $14^\circ$–$37^\circ$ range.

Boundary-integral (BI) potential flow models capture the meniscus evolution and jet ejection quantitatively, assuming incompressible, irrotational flow with dynamic boundary conditions set by $p_b(t)$ (bubble pressure) and interfacial tension. The parameter space is governed by critical dimensionless numbers:

• Ohnesorge $Oh$
• Weber $We$
• Reynolds $Re$

Optimal design for needle-free jet injection and high-fidelity inkjet printing is set by selecting $d\approx 100$–$120\,\mu$m, $\alpha = 10^\circ$–$40^\circ$, and $H\approx 400\,\mu$m for stable jets with $v_{\rm jet} \approx 30$–$50$ m/s, $Re=2000$–$10000$, $We=850$–$21000$ (Galvez et al., 2020).

4. Laser–Plasma Micronozzle Acceleration for Proton Beams

At relativistic intensities ($I_L=10^{22}$ W/cm$^2$), MNA manifests as a three-stage laser–plasma ion acceleration process in structured targets consisting of a micron-scale hydrogen rod embedded within a hollow aluminum nozzle (Murakami et al., 30 Nov 2025).

The target geometry comprises a nozzle with head, neck, and skirt shaped to focus laser energy and guide hot-electron outflow:

• Nozzle head height $H_1=5.3\,\mu$m, neck $H_2=2.8\,\mu$m, skirt $H_3=12\,\mu$m
• H-rod diameter $D=2.0\,\mu$m, wall thickness $d=0.6\,\mu$m

Upon irradiation, the sequence is:

1. Run-up phase ($t\lesssim100$ fs): Laser generates relativistic electrons, sheath field initiates proton expansion.
2. Main-drive phase ($t\approx100$–250 fs): Hot electrons charge the nozzle tail, establishing a longitudinal electrostatic field $E_x\sim 10^{14}$ V/m.
3. Afterburner phase ($t>250$ fs): Comoving field persists, protons absorb further energy from electron thermal expansion, contributing $\sim$300 MeV extra gain.

Particle-in-cell (PIC) simulations reveal that, at $I_L=10^{22}$ W/cm$^2$, protons reach cutoff energies $>1$ GeV with characteristic mid-spectrum plateaus ($400$–$800$ MeV), angular divergence FWHM $\sim16^{\circ}$, and conversion efficiency $\sim3\%$. Scaling shows $E_{\rm max} \propto I_L^{0.79}$ (plane) and up to $I_L^{2.9}$ (Gaussian spot), exceeding typical TNSA behavior (linear scaling). Nozzle shape and rod–nozzle alignment modulate $E_{\rm max}$ by up to 10%. The self-similar plasma expansion (afterburner) is analytically captured as

$$\Delta E_{\rm max} \approx 2Z T_e \ln\left(\frac{C_s T_L}{\lambda_{D0}}\right)$$

with $T_e$ electron temperature, $C_s$ sound speed, and $\lambda_{D0}$ Debye length. These results indicate that structured MNA targets achieve $2$–$3\times$ higher energies than H-rod or foil alone under identical laser drive (Murakami et al., 30 Nov 2025).

5. Optimization Strategies and Performance Sensitivities

Across physical realizations, MNA efficiency depends critically on geometry, staging, and material/system selection. In continuum micro-nozzles, response surface optimization highlights the dominance of exit width and throat curvature. For atomistic nozzles, large nozzle widths relative to molecular scale ($d_m>30\sigma$) are essential to approach isentropic acceleration. In thermocavitation MNA, controlling fill height and taper angle achieve maximal jet speed and collimation, while Ohnesorge and Weber numbers inform stability margins.

In laser–plasma MNA, rod/nozzle geometry, relative positioning (e.g., rod–nozzle gap $\Delta$), and target material composition can modulate the field amplification and energy transfer, with elliptic rods and tight contact optimizing $E_{\rm max}$ and conversion efficiency (Niksirat, 2023, Ortmayer et al., 2023, Galvez et al., 2020, Murakami et al., 30 Nov 2025).

6. Applications and Implications

MNA underpins key advances in:

• Spacecraft cold-gas micropropulsion: Achieving thrust in the 0.1–0.25 N class for AGN and attitude control thrusters (Niksirat, 2023).
• Miniaturized beam sources and microfluidics: Generating supersonic beams and precision jets for molecular spectroscopy, nano-printing, and needle-free drug delivery (Ortmayer et al., 2023, Galvez et al., 2020).
• Relativistic ion acceleration: Delivering $>1$ GeV proton beams for hadron therapy, compact accelerator injectors, and high-energy-density science at moderate divergence and conversion efficiency (Murakami et al., 30 Nov 2025).

Limitations include requirements for advanced microfabrication, precise flow or field alignment, and addressing pronounced non-equilibrium effects at the molecular scale or in laser–plasma targets.

7. Scaling Laws, Design Guidelines, and Future Directions

The scaling of MNA output metrics as a function of system size, drive energy, and geometric ratios is determinative for both performance and practical realization:

• In gas flows, optimal expansion ratio and throat curvature maximize enthalpy conversion and boundary-layer management.
• In atomistic and non-equilibrium MNA, minimizing Knudsen number and maximizing active area relative to molecular scale restores continuum efficacy.
• For thermocavitation, the inverse dependency on fill height ($v_{\rm jet}\propto H^{-1}$), and the gain with taper angle, are dominant.
• In laser-plasma MNA, proton energy scales sub-linearly to super-linearly with laser intensity, and geometric optimizations yield up to $10\%$ enhancements.

A plausible implication is that future MNA research will emphasize integrative optimization over geometry, material, and drive modalities, as well as extension to arrayed and three-dimensional micro-nozzle systems for enhanced output and new functionalities (Niksirat, 2023, Ortmayer et al., 2023, Galvez et al., 2020, Murakami et al., 30 Nov 2025).