Particle-Spray Method

• Particle-spray method is a multifaceted concept that describes both physical techniques and computational models for accelerating, depositing, and modeling discrete particles.
• In materials processing, it involves low-pressure gas spraying that accelerates particles to high speeds, with adhesion thresholds and morphology directly affecting coating quality.
• In computational simulations, Eulerian–Lagrangian and SPH methods capture dense spray dynamics by addressing issues like gas entrainment, droplet breakup, and particle clustering.

Particle-spray method is a context-dependent term used for processes and models in which discrete particles, droplets, or particle-like tracers are accelerated, transported, deposited, or numerically represented as a spray. In current arXiv literature, the term spans solid-state coating and repair by gas-borne particles, molten-droplet impact in thermal spraying, capsule- and parcel-based computational spray descriptions, stochastic and kinetic formulations of dispersed phases, image-based morphology tracking in spray agglomeration, and even a particle-spray code for extra-tidal stars in globular-cluster studies (Sinnwell et al., 27 Mar 2025, Haghshenas et al., 2021, Grondin et al., 2022).

1. Terminological range and core meanings

The term does not denote a single standardized method across disciplines. In process engineering, it usually refers to a physical process in which particles or droplets are sprayed toward a target or through a reactor. In computational fluid dynamics and kinetic theory, it often denotes a particle-based representation of a dispersed phase. In some cross-disciplinary uses, the “spray” is a numerical ensemble of tracers rather than a physical aerosol or powder jet. This suggests that “particle-spray method” functions as a family-resemblance term whose exact meaning is fixed by domain-specific transport physics, coupling assumptions, and observables.

Domain	Meaning of “particle-spray method”	Representative papers
Surface engineering and particle processing	Gas- or liquid-borne particles or droplets are accelerated, deposited, or agglomerated	(Sinnwell et al., 27 Mar 2025, Jeske et al., 2021, Ren et al., 2020, Fuchs et al., 24 Mar 2025)
Spray CFD and multiphase simulation	Dispersed phase represented by capsules, parcels, point particles, or particle-informed closures	(Haghshenas et al., 2021, Pakseresht et al., 2019, Cheng et al., 19 May 2026)
Kinetic, SPH, and cross-disciplinary particle models	Spray-like systems formulated as kinetic, SPH, or tracer-particle ensembles	(Debussche et al., 2020, Moussa et al., 2011, Jung et al., 22 Apr 2026, Grondin et al., 2022)

A concise materials-processing definition is given in the low-pressure cold-gas-spray literature: a particle-spray method is “a process in which a carrier gas accelerates solid particles through a nozzle to form a high-velocity spray that impacts a surface to coat, repair, or micro-structure it” (Sinnwell et al., 27 Mar 2025). That formulation captures one major usage, but not the full breadth of current research usage.

2. Particle-spray methods in materials processing and manufacturing

In cold spray, the particle-spray method is a solid-state deposition process. Low-pressure cold gas spraying operates at relatively low temperatures and pressures, specifically $T < 600^\circ\mathrm{C}$ and $p < 15$ bar, and uses compressed gas expanded through a Laval nozzle to accelerate entrained particles to about 200–1200 m·s$^{-1}$. Deposition occurs without melting; bonding is associated with plastic deformation, mechanical interlocking, and adiabatic shear instability in contact, and requires a material-dependent critical impact velocity (Sinnwell et al., 27 Mar 2025). Within that framework, particle morphology is not secondary. Copper feedstocks with median 3D sphericities $\Psi_{3D,50}$ of 0.96, 0.88, and 0.76 showed systematically different spray behavior: irregular particles accelerated more strongly, reached higher median and maximum velocities, and produced narrower spray dispersion, whereas more spherical particles reached lower maximum velocities and exhibited wider dispersion (Sinnwell et al., 27 Mar 2025).

The same literature also shows that many-particle accumulation introduces a second threshold beyond single-particle adhesion. In laser-induced particle impact testing adapted to repeated gold-particle deposition, the single-particle critical velocity for Au was $253 \pm 7$ m·s$^{-1}$, whereas sustained material build-up required average impact velocities of roughly 330 m·s$^{-1}$, about $1.3 V_{\text{cr}}$. Below that build-up threshold, deposits fractured or eroded during growth, despite many individual impacts occurring above the single-particle bonding threshold (Reiser et al., 2024). This separates single-impact adhesion from coating-scale growth stability.

Thermal spraying uses a different particle state. In plasma spraying, molten droplets rather than solid particles impact the substrate. Reported process conditions include plasma temperatures of $6000$–$15000\,^\circ\mathrm{C}$, particle velocities up to $p < 15$0, particle sizes of $p < 15$1–$p < 15$2, and cooling rates on impact of $p < 15$3–$p < 15$4. In that regime, a particle-spray method is centered on splat formation, rapid spreading, and solidification, rather than on solid-state bonding (Jeske et al., 2021).

Other manufacturing uses are reactor-based. In flame spray pyrolysis, a liquid containing precursor is atomized into droplets, each droplet burns or evaporates, and nanoparticles form either by a gas-to-particle route or a droplet-to-particle route. The governing distinction is whether precursor boiling precedes thermal decomposition or vice versa, and the resulting particle size depends on precursor mass fraction and residence time (Ren et al., 2020). In spray fluidized bed agglomeration, binder droplets sprayed into a fluidized bed generate liquid bridges between suspended solids that dry into solid bridges, producing distinct morphological classes: primary particles, chain-like agglomerates, and raspberry-like agglomerates (Fuchs et al., 24 Mar 2025).

3. Eulerian–Lagrangian and dense-spray computational formulations

A major computational usage of particle-spray method appears in Eulerian–Lagrangian spray CFD. The ELMO model treats the dense near-nozzle region of diesel-like sprays as a mixing-limited system controlled by gas entrainment rather than droplet-scale interfacial kinetics. Near the nozzle, the spray is represented by Lagrangian “capsules” that contain constant fuel mass and approximately constant total momentum while entraining ambient gas; downstream, each capsule transitions to conventional parcels when the liquid volume fraction in the capsule is below 0.005, the vapor volume fraction of fuel exceeds 99% of the total fuel volume fraction, and the capsule mean velocity is below 10 m/s. After transition, standard droplet-centric physics is used, including TAB breakup, NTC collisions, and Frössling evaporation (Haghshenas et al., 2021). In this usage, the particle-spray method is neither a direct droplet cloud nor a purely Eulerian jet model, but a hybrid dense-core capsule description coupled to a parcel far field.

Dense-spray Euler–Lagrange models may also require explicit treatment of volumetric displacement. For locally high particle volume fractions, the fluid void fraction $p < 15$5 becomes a dynamic field, and the carrier-phase continuity and momentum equations are written for $p < 15$6 and $p < 15$7. Under those conditions, the carrier velocity is no longer divergence-free, and the pressure Poisson equation acquires a source term linked to changes in $p < 15$8. This modifies both inertia and pressure coupling in dense sprays and particle-laden jets (Pakseresht et al., 2019). The model was demonstrated on a particulate turbulent round jet at $p < 15$9 with mass loading $^{-1}$0.

A more recent extension treats the particle-spray method as a subgrid closure problem. Particle-informed super-resolution reconstructs high-resolution gas fields from coarse LES gas variables conditioned on fine-grid particle information. The mapping uses coarse gas fields together with particle fields such as volume fraction, Sauter mean diameter, mean droplet temperature, velocity components, and mean droplet age. The reconstructed fine-grid gas state is then used to evaluate evaporation and momentum coupling more accurately than coarse LES cell averages permit. For moderately dense evaporating sprays, this reduced the discrepancy in the fuel mass fraction field between LES and carrier-phase DNS, generalized across cases with varying air temperature, droplet diameter, and turbulent Reynolds number, and incurred a cost about five times that of standard LES but about 160 times lower than carrier-phase DNS (Cheng et al., 19 May 2026).

4. Kinetic, SPH, and theoretical particle formulations

In kinetic theory, a particle-spray method may refer to a spray-like phase-space description rather than a hardware process. A kinetic toy model for a spray immersed in a carrier fluid with random forcing uses a distribution $^{-1}$1, a fluid velocity $^{-1}$2, and a space-dependent Markov process $^{-1}$3. Under diffusive scaling and strong velocity relaxation, the macroscopic density converges to a stochastic conservation law in Stratonovich form, and the drift and diffusion coefficients are determined by the law of the stationary process associated with the Markovian perturbation (Debussche et al., 2020). In that setting, “spray” identifies a dispersed particle population transported by drag and random forcing.

A different theoretical construction is the two-dimensional spray model with gyroscopic effects. There, the fluid obeys an Euler-type equation while the dispersed phase obeys a Vlasov-type equation; the fluid acts on particles through a gyroscopic force $^{-1}$4, and the particle density contributes to the fluid vorticity. The system admits a mean-field derivation from a many-particle model, a Hamiltonian structure, and a vanishing-mass limit toward Euler flow (Moussa et al., 2011). This is a specialized conservative spray model rather than a drag-dominated aerosol formulation.

Particle methods also appear in SPH-based radiation hydrodynamics. The code SPRAY is a massively parallel GPU-accelerated SPH code for laser–plasma interaction in which the plasma is represented by Lagrangian particles, radiation is treated with time-dependent flux-limited diffusion, and laser energy coupling uses a mesh-free ray-tracing scheme based on the WKB approximation (Jung et al., 22 Apr 2026). Here, the “spray” is the particle discretization itself. A related but more classical SPH usage appears in thermal spraying, where the impacting molten droplet is discretized by SPH particles and solidification is treated with an enthalpy–porosity method for the mushy zone (Jeske et al., 2021).

Eulerian–Eulerian alternatives exist as well. In DNS-based studies of dispersed particle-laden flows relevant to spray mechanisms, the dispersed phase has been represented as a continuum with Quasi Brownian Motion closure. The added pressure-like and viscous-like stress terms suppress nonphysical clustering and recover more realistic dispersion behavior for particle-laden turbulence (Dutta et al., 2012).

5. Measurement, morphology, and spatio-temporal characterization

Particle-spray methods are strongly shaped by what can be measured. In low-pressure cold gas spraying, morphology characterization combined 2D light microscopy and 3D X-ray micro-computed tomography. The resulting descriptors included 2D circularity, 3D Wadell sphericity, surface area, and volume. Those morphology metrics were then correlated with high-speed particle image velocimetry data measuring axial and lateral velocity components, position in the measuring window, particle diameter, and trajectory angle. The central result was that lower sphericity produced higher aerodynamic sensitivity: stronger acceleration inside the nozzle, faster deceleration in the free jet, and a smaller spray cone angle (Sinnwell et al., 27 Mar 2025).

Temporal characterization has become equally important. In an airblast atomizer studied with Phase Doppler Anemometry, droplet size, velocity, and arrival time were analyzed through interparticle-time distributions, a $^{-1}$5 hypothesis test against the Poisson model, Cramér’s $^{-1}$6, and $^{-1}$7-means clustering with gap statistics. The study reported multimodality in interarrival-time distributions and found that cluster formation affects approximately 30% of the droplets in a single data set. It also concluded that unsteadiness in the central region is caused by clustering, whereas in the spray periphery it is caused by mixing and droplet entrainment (Rácza et al., 17 Dec 2025). This directly challenges a purely stationary Poisson view of spray arrival processes.

Image-based morphology pipelines have become similarly structured in agglomeration research. In spray fluidized bed agglomeration, object segmentation used non-local means filtering, Otsu thresholding, and connected-component labeling, followed by extraction of size, shape, and texture descriptors. A random forest then classified objects into primary particles, chain-like agglomerates, and raspberry-like agglomerates, and the temporal evolution of class fractions and bivariate size–shape distributions was modeled with low-parametric regression functions and Archimedean copulas (Fuchs et al., 24 Mar 2025). This is a particle-spray characterization method in the narrow sense that the spray-driven process is monitored through evolving morphology distributions rather than through mean particle size alone.

6. Design variables, process windows, and regime dependence

Across the literature, particle-spray methods are governed by a small set of recurrent design variables: particle or droplet size, shape, velocity, temperature, carrier-phase conditions, and coupling regime. In cold spray, morphology enters the drag law through the sphericity-dependent coefficient $^{-1}$8, so feedstock shape directly controls acceleration, deceleration, and focusing (Sinnwell et al., 27 Mar 2025). In many-particle cold-spray accumulation, the build-up window is narrower than the single-particle bonding window: build-up required roughly $^{-1}$9, while higher velocities increased interfacial pore length and other flaws (Reiser et al., 2024). In flame spray pyrolysis, precursor selection and solvent thermodynamics determine whether the process follows gas-to-particle or droplet-to-particle conversion, and the ratio of droplet evaporation time to precursor reaction or precipitation time controls whether one obtains one-droplet-one-particle behavior or more complex morphologies such as hollow or porous products (Ren et al., 2020).

Biomedical delivery exhibits the same parameter sensitivity in a different geometry. In patient-specific sinonasal CFD for chronic rhinosinusitis, nebulized droplets of 11–14 $\Psi_{3D,50}$0m achieved at least 50% of maximal sinus and OMC deposition fraction in all six sinonasal cavities studied, and five of six cavities showed greater sinus and OMC deposition with nebulized droplets than with sprayed droplets at optimal sizes (Farzal et al., 2018). This does not define a universal aerosol optimum, but it does show that the particle-spray method can be tuned to target a specific anatomical region by controlling droplet size and initial momentum.

The literature also delineates clear regime limits. Mixing-limited spray models such as ELMO are intended for dense, hot sprays in which gas entrainment rather than droplet-scale interface physics controls the evolution; the same papers state that the approach is less appropriate when interfacial dynamics and droplet kinetics are rate-limiting (Haghshenas et al., 2021). Particle-informed super-resolution has been demonstrated for moderately dense, non-reacting evaporating sprays without collisions, coalescence, or breakup, and its reported generalization therefore remains tied to that regime (Cheng et al., 19 May 2026). SPH thermal-spray benchmarks show close agreement with VOF for the chosen impact and solidification problem, but the reported comparisons are for identical physical effects in a single-droplet benchmark rather than a universal claim across all spraying regimes (Jeske et al., 2021).

Taken together, these studies support a technical interpretation of particle-spray method as a class of particle-centered transport, deposition, and modeling strategies rather than a single procedure. The common structure is the same even when the realization changes: particles or particle analogues carry mass, momentum, energy, or morphology; they interact with a surrounding medium or target; and the outcome is controlled by the statistics of trajectories, collisions, coupling, and structural evolution. This suggests that the most stable definition of the term is functional rather than disciplinary: a particle-spray method is any method in which the essential physics or numerics is organized around a spray of discrete particles.