# Evaluation of iron flame discreteness in monoatomic gases based on single particle combustion behaviour

Herman Heng, Hugo Keck, Christian Chauveau, Fabien Halter, Jeffrey Bergthorson

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Herman Heng, Hugo Keck, Christian Chauveau, Fabien Halter, Jeffrey Bergthorson. Evaluation of iron flame discreteness in monoatomic gases based on single particle combustion behaviour. Fuel Communications, 2026, 26, pp.100155. 10.1016/j.jfueco.2026.100155. hal-05528314

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Submitted on 3 Mar 2026

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<figure><img src="image_1.png" /></figure>
# Evaluation of iron flame discreteness in monoatomic gases based on single particle combustion behaviour

Herman Heng<sup>a,b ID,*</sup>, Hugo Keck<sup>b ID</sup>, Christian Chauveau<sup>b ID</sup>, Fabien Halter<sup>b ID</sup>, Jeffrey Bergthorson<sup>a ID</sup>

## ARTICLE INFO

**Keywords:**

Single particle combustion

Iron combustion

Droplet temperature

Nano-oxide formation

Discreteness parameter

## ABSTRACT

The combustion behaviour of single iron particles is experimentally investigated at a fixed oxygen concentration of 40%, diluted in 60% He, Ar, and Xe, using an electrostatic levitator with laser ignition. While the three gas mixtures have identical adiabatic flame temperature predictions using thermodynamic equilibrium calculations, the experiments reveal a noticeable difference in particle temperatures due to differences in Lewis number. This temperature variation is evident in particle luminosity and in nano-oxide formation, observed through the attenuation of white LED light. In the helium-diluted mixture, insignificant nano-oxide formation is observed due to lower particle temperatures, which subsequently reduces Fe evaporation and vapour-phase combustion. In contrast, xenon dilution enhances vapour-phase combustion, yielding pronounced nano-oxide formation, while argon shows intermediate behaviour. Despite large differences in oxygen mass diffusivity, measured combustion times, defined from melting onset to peak temperature, are comparable for helium- and argon-diluted mixtures, and only slightly longer for the xenon-diluted mixture. Using these combustion times, the discreteness parameter, $\chi$, of iron flames is qualitatively compared: it is 1.71 times higher in 40%O₂–60%Ar than in 40%O₂–60%Xe, and 5.28 times higher in 40%O₂–60%He than in 40%O₂–60%Xe. The results indicate that discrete flame propagation is likely in argon-diluted mixtures but unlikely in helium-diluted mixtures. The present study highlights how the Lewis number influences the interplay of oxygen mass diffusivity and vapour-phase combustion in evaluating particle combustion time and flame discreteness, providing essential insights for designing future discrete flame microgravity experiments in different gas mixtures and advancing iron combustion models.

## 1. Introduction

Metal fuels have been proposed as promising carbon-free energy carriers in the past decades. Compared to other volatile zero-carbon alternatives, such as hydrogen and ammonia, metal fuels offer a higher volumetric energy density, allowing their storage and transport in substantially smaller volumes [1]. Among the various candidates, iron is notable because its combustion produces micrometric solid products with sizes comparable to the original micron-sized reactant. This feature makes iron combustion products easier to recover and recycle, which is an essential requirement for establishing a closed-loop metal–fuel cycle and ensuring long-term sustainability [2]. Owing to this promising characteristics, iron has attracted significant research interest in recent years.

### 1.1. Combustion behaviour of single particles

To establish a fundamental understanding of iron's burning behaviour, many existing studies focus on single micron-sized particle combustion. Several experimental investigations have examined the evolution of particle temperature (or luminosity) during combustion and cooling, using setups with laser ignition [3,4] and drop-tube furnaces [5,6], and some of these experimental temperature profiles have been mirrored by the developed combustion models in [7,8]. Thijs et al. [7] identified, from their temperature profile, a small but non-negligible particle mass loss due to evaporation, which leads to the formation of nano-oxide products that have been observed in many experiments [3–6], with the amount of nano-oxide produced increasing with oxygen concentration. Although earlier work [2] hypothesized

<footnote><sup>a</sup> McGill University, 817 Sherbrooke Street West, Montréal, H3A 0C3, Québec, Canada</footnote>

<footnote><sup>b</sup> ICARE-CNRS, 1C Avenue de la Recherche Scientifique, Orléans, 45100, France</footnote>

<footnote>* Corresponding author at: McGill University, 817 Sherbrooke Street West, Montréal, H3A 0C3, Québec, Canada. E-mail address: cloud.heng@mail.mcgill.ca (H. Heng).</footnote>
that iron burns heterogeneously according to Glassman's criteria [9], later experimental work revealed that liquid iron can evaporate below its boiling point, even at the oxygen concentration in air. Ning et al. [10] reported that nano-oxide formation during iron particle combustion can occur at temperatures as low as 2100 K.

While these studies have advanced the understanding of iron combustion, most were conducted at varying oxygen concentrations in the same diluting gas of nitrogen. An increased oxygen concentration clearly influences combustion temperature and, consequently, evaporation and nano-oxide formation. However, very little prior work has examined the role of diluting inert gases on iron particle combustion when oxygen concentration is fixed. For non-nitrogen diluents, McRae et al. [11] reported traces of nano-oxide formation in a 40%O₂-60%Ar gas mixture using a counterflow burner, while Palecka et al. [12] observed the presence of nano-oxide formation in 40%O₂-60%Xe iron dust flames in microgravity. However, these different studies cannot be directly compared to understand the effect of diluents on nano-oxide formation. Recent work by Chang et al. [13] compares the ignition temperature of iron particles using a drop tube furnace at the same oxygen concentration of 21%O₂ diluted in helium, nitrogen and argon. They reported that the ignition temperature in 21%O₂-79%Ar is about the same as in air but about 135 K lower than that in 21%O₂-79%He. For the combustion process, Goroshin et al. [14] investigated the flame speed of iron dust flames, of similar particle sizes, in microgravity and found that the flame speed in 21%O₂-79%He is significantly higher than that in 21%O₂-79%Ar, while the flame speed in 21%O₂-79%Ar is similar to that in air. The effect of diluent gas on the particle combustion time and the extent of nano-oxide formation remains insufficiently explored in the current literature.

## 1.2. Discreteness of dust flames

Another underexplored area in the current iron combustion literature is the discrete flame phenomenon in iron dust flames. Since iron is a non-volatile fuel that primarily burns heterogeneously, the heat released is concentrated in the vicinity of the burning particle. This creates a temperature micro-gradient, where elevated temperatures occur at discrete locations of burning particles — referred to in the literature as “discrete flames”. In such conditions, three-dimensional temperature non-uniformities exist even along the same lateral plane, in contrast to gaseous flames, where the temperature gradient typically exists only along the flame propagation direction. Pioneering work on discrete flames began with theoretical frameworks and numerical simulations [15–18], which showed that the propagation limit of discrete flames can only be defined probabilistically. Random spatial arrangements of particles create unique temperature fields, so flame propagation may or may not occur, even at the same fuel concentration. This behaviour has been experimentally confirmed in microgravity experiments aboard parabolic flight aircraft and sounding rockets [12,19, 20]. In these experiments, the probabilistic nature of the propagation limit was reflected in variations in the maximum flame propagation distance within a narrow concentration range. The observations also revealed the absence of a well-defined flame front, typically seen in gaseous flames, which underscores the pronounced temperature non-uniformities in discrete flames. Nevertheless, existing experimental validation of discrete flame theories in microgravity to date is only limited to an oxidizing environment of oxygen diluted in xenon.

The discreteness of the flame can be quantified by the ratio of the particle combustion time, $t_c$, to the heat-diffusion time, $t_d$, across the interparticle distance, $l$, as shown in Eq. (1):

$$ \chi = \frac{t_c}{t_d} = \frac{t_c}{l^2/\alpha} \qquad (1) $$

where $\chi$ is the discreteness parameter and $\alpha$ is the thermal diffusivity of the suspension. Note that a lower $\chi$ indicates a higher discreteness of flames. Eq. (1) shows that discreteness of flames can be increased

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic illustrating the characteristic length scales: R is the oxygen collection radius for oxygen mass diffusion, D, and l is the interparticle distance between a burning and an unburnt particle for thermal diffusion, $\alpha$. $d_p$ represents the particle diameter. As $l > R$, $t_d > t_c$ if $\alpha \approx D$.</figcaption></figure>

by shortening $t_c$, which decreases with increasing oxygen concentration. Thus, prior investigations in microgravity largely focused on discrete flames under elevated oxygen conditions of 40%. A longer heat diffusion time, $t_d$, also increases $\chi$, so the previous microgravity experiments used xenon gas, which has a very low thermal diffusivity. However, it remains unclear whether the discrete flame phenomenon would also occur if the diluting gas is replaced with another more available and cost-effective inert gas. To form a preliminary hypothesis, we consider a burning particle and a neighbouring unburnt particle, as shown in Fig. 1. The burning particle collects oxygen from its surroundings within a collection radius, R, which is approximately $7d_p$ after accounting for the stoichiometric ratio at 40%O₂ and the density difference between the particle and the gas. In a fuel-lean mixture, the interparticle distance, $l$, is typically larger than R; for example, $l$ is about $20d_p$ at 0.5 g/L with a uniform spatial distribution of particles (the stoichiometric concentration is about 1.8 g/L at 40%O₂, assuming FeO as the main product). Heat diffuses over the length scale $l$ at a rate $\alpha$, while oxygen diffuses over a smaller scale R at a rate D. If $\alpha \approx D$ (i.e., $Le \approx 1$), the time for heat diffusion across $l$ is greater than the combustion time associated with R, so $t_d > t_c$. This analysis suggests that, as interparticle distance strongly influences $\chi$, discrete flames are likely a common feature of non-volatile, fuel-lean flames. This hypothesis requires experimental verification, and a logical starting point would be an investigation of combustion time in different diluting gases at a fixed oxygen concentration. Such an approach would enable a fundamental assessment of whether an identical discreteness parameter can be achieved in different diluents under the same oxygen concentration and interparticle distance. If not, it would also reveal how the interparticle distance should be adjusted to achieve a comparable level of flame discreteness.

## 1.3. Research questions and objectives

With the identified gaps above, the present study aims to address two key research questions:

i. How does the diluting inert gas affect the combustion temperature, nano oxide formation, and combustion time of iron particles?

ii. How is the flame discreteness influenced by the choice of diluting inert gas at the same oxygen concentration?

To answer these questions, this study employs an electrostatic levitator coupled with several diagnostic techniques. The oxygen concentration is fixed at 40% to maintain consistency with our previous microgravity experiments [20], which confirmed discrete flame propagation behaviour. Helium, argon, and xenon are selected as the monoatomic diluting gases. For each gas mixture, we derive a particle temperature profile and qualitatively assess the significance of nano-oxide formation. Using these temperature profiles coupled with high-speed images, which reveal when gas-phase combustion ceases, we estimate the combustion time of iron particles in each gas mixture. With the estimated combustion time, we then evaluate the discreteness
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** (a) Particle size distribution of the iron powder measured by laser diffraction. (b) A sample image obtained through scanning electron microscopy (SEM) for the iron powder.</figcaption></figure>

parameter for iron dust flames in these gas mixtures and provide insights into whether discrete flame propagation can still be observed when low-thermal-diffusivity xenon is replaced with more accessible argon or helium.

## 2. Experimental methodology

### 2.1. Iron powder and gases

The iron powder used in this study originates from the same batch previously utilized by McGill University in all microgravity iron combustion experiments conducted under the Percolating Reaction-Diffusion Waves (PERWAVES) project (see [19] for the project background). The powder is supplied by TLS Technic GmbH & Co. and is manually sieved to obtain a narrower size distribution. It has a purity of 99.8% and consists of spherical particles with a mean Sauter diameter of 32 µm. The particle size distribution after sieving, measured by laser diffraction, and the particle morphology, observed via scanning electron microscopy (SEM), are presented in Fig. 2. In this study, iron particle combustion is examined in three oxidizing environments, each containing 40% oxygen (by mole) diluted with helium, argon, or xenon inert gas.

### 2.2. Electrostatic levitator

An electrostatic levitator is used in the present experiment to suspend a single iron particle at a fixed position with minimal motion. The concept of particle and droplet levitation was developed in the 1980s [21]. The levitator generates an electromagnetic field using three electrodes powered by a combination of DC and AC voltage supplies. This field effectively counteracts gravitational and centripetal forces acting on the charged iron particle. To conduct the experiment in a non-ambient environment (at 40% $O_2$), the levitator is enclosed within a chamber filled with the specific gas mixtures. Ignition is achieved using a laser ignition system with a 50-watt $CO_2$ laser beam directed at the levitated iron particle.

The levitation facility used in the present experiment is located at the Institut de Combustion, Aérothermique, Réactivité et Environnement (ICARE-CNRS), Orléans, France, and this facility has been utilized for combustion research of several metals and metal alloys. For additional details on the setup and prior experimental work, readers are referred to Legrand et al. [22] for magnesium and magnesium-aluminum alloys in $CO_2$; to Marion et al. [23] and Braconnier et al. [24] for aluminum burning in air at different pressures; and, more recently, to Heng et al. [25] for silicon particle combustion in gas mixtures with varying oxygen concentrations and diluents. For iron combustion, to the authors' knowledge, aside from recent work by Bruyr et al. performed in ambient air [26], single particle iron combustion experiments with electrostatic levitation with low slip velocity and high-resolution colour imaging have not been reported.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** (1) The electrostatic levitator with (a) gravity-compensating fields (red dotted lines) and (b) centering fields (black dotted lines) that stabilize (c) single charged iron particles in the setup, (2) $CO_2$ laser system with a set of (3) optical mirrors used to reflect the laser beam to achieve a more uniform laser distribution on the particle, (4) colour camera coupled with a (5) bandpass filter and a (6) telephoto lens, (7) light emission transmitted towards a photomultiplier system, (8) white LED used as a backlight. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

### 2.3. Optical diagnostics

A schematic of the electrostatic levitator and the optical diagnostic tools used in the present experiment is shown in Fig. 3. The light emitted by the burning iron particle is recorded using a Thorlabs PMT1002 photomultiplier system, which operates across three wavelength channels at 700 nm, 760 nm, and 820 nm in the near-infrared (NIR) range. Similar to our prior work [25], the photomultiplier system is calibrated using a blackbody source set between 1373 and 1873 K, from which sensitivity ratios between each channel pair are determined under varying light exposures. During the combustion experiment, an aperture is placed at the entry of component (7) in Fig. 3 to regulate the amount of light entering the photomultiplier. This ensures that the resulting voltage signal remains within the linear response range of the system and avoids signal saturation.
To minimize the interference from the laser on the combustion process, a voltage threshold is set on the signal recorded by the photomultiplier. Once this threshold is reached, the laser is automatically shut off. Combustion dynamics are captured using a Phantom T2410 colour high-speed camera equipped with a triple bandpass filter (457 nm, 530 nm, and 628 nm). The camera is coupled with a telephoto lens to achieve high-resolution imaging at 1.25 µm/pixel. The high-speed camera and photomultiplier system are synchronized to enable direct comparison of real-time emission signals, derived temperatures, and particle imagery. A Thorlabs SOLIS-3C white LED (emitting in the 400–750 nm range), developed from the recent work of Keck et al. [27,28], is used as a backlight to visualize the non-emitting particle prior to ignition and to capture absorption signals during combustion, thereby allowing the observation of nanoparticles produced during the combustion process.

### 3. Results and discussion

The present experiment investigates the combustion behaviour of the iron powder used in the McGill PERWAVES microgravity experiment. Given that this powder has a narrow size distribution range, the effect of particle size is not the main focus of the present study. Accordingly, all analyses presented here are based on particles with diameters in the 28–38 µm range, which represents about 70% of the powder’s size distribution shown in Fig. 2.

#### 3.1. Combustion behaviour in different inert gases

The combustion behaviour is investigated through synchronized high-speed imaging and photomultiplier signals. High-speed images of the burning iron droplet are recorded and shown in Fig. 4(a), with the eight columns (hereinafter referred to as Stages 1 to 8) representing different stages of combustion and cooling. Fig. 4(b) presents the normalized intensity evolution with time captured by the photomultiplier system, corrected using the sensitivity ratios obtained from calibration. Fig. 4(c) shows the derived temperature from the photomultiplier system using the calibrated intensity, assuming a grey-body emitter. The same reference numbers (Stages 1 to 8) are used in Fig. 4(c) to indicate locations on the temperature profile that correspond to the combustion stages shown in Fig. 4(a). All temperature profiles are aligned to a standard start time ($t = 0$) corresponding to the particle melting point (1811 K), which is indicated by Stage 1 in Fig. 4(a) that shows a spherical melted particle due to surface tension in liquid.

Observations from the colour camera (Fig. 4(a)) and the pixel intensity data at Stage 6 (which corresponds to the peak temperature) show that the particle appears brightest in the xenon-diluted mixture and dimmest in the helium-diluted mixture, indicating that the particle temperature is highest in xenon dilution, intermediate in argon dilution, and lowest in the helium-diluted mixture. Due to higher temperatures in the argon-diluted and xenon-diluted mixtures, gas-phase combustion with nano-oxide halos is observed throughout the process from Stage 1 to Stage 6. The halo at Stage 2 is noticeably denser due to the influence of laser heating. Micron-sized iron particles are known to ignite at temperatures below their melting point, as indicated by the ignition model in [29] and experimental studies [13,30]. In the present experiment, due to the low emission signal before melting, even with the lowest possible threshold used in the setup, the laser system shuts off at Stage 2 (the vertical line in Fig. 4(c)), shortly after melting begins at Stage 1, leaving a brief period during which combustion is affected by laser heating (Stages 1–2). Increasing the aperture to enhance the signal can trigger earlier laser shutdown, but it also leads to signal saturation at high peak temperatures in 40% O₂. Due to this limitation, nano-oxide formation at and prior to Stage 2 is likely enhanced by the laser heating.

In addition to the laser heating effect at Stage 2, the formation of denser nano-oxide halos is likely attributed to the fact that, at the onset

of liquid-phase combustion, the evaporation of liquid iron is easier as the droplet surface consists primarily of molten iron (L1) rather than molten iron oxide (L2). Deutschmann et al. [31] and Jimenez et al. [32] analyzed the structure of partially burnt iron particles through ex-situ methods, in which particles were ignited in a flat flame and collected at different distances from the flame, corresponding to different residence times and oxidation degrees. Their results revealed unmixed L1 and L2 regions within the particles, with a significant portion of the particle surface still consisting of L1 at lower oxidation degrees. Bruyr et al. [26] used in-situ imaging and observed an unmixed L1–L2 surface with different luminosities at the early stage of liquid-phase combustion. This unmixed surface persisted for about 1.1 ms, when burning in air, until the droplet surface became fully covered by L2. A similar mechanism likely contributes to the denser nano-oxide observed at the onset of combustion, where enhanced evaporation of Fe occurs due to the higher availability of L1 on the particle surface before the surface is fully covered by L2.

After the laser shuts off, the dense halo observed at Stage 2 gradually disperses until Stage 4. The burning particle continues to produce a fainter halo during self-sustained combustion (Stages 2–6), demonstrating that nano-oxide formation in the argon- and xenon-diluted cases can still occur without laser heating. Fig. 5(a) presents sample frames showing the nano-oxide halo during this combustion process without laser heating, recorded with the LED backlight turned off to ensure that flame emission is not affected by LED light. Fig. 5(b) shows a contour-processed image plotted at a larger length scale than in Fig. 4(a), making the emission signal detached from the particle surface more apparent. The high-resolution camera reveals that the thickness of this nano-oxide halo is approximately 3.5 and 3.9 times the particle radius during vapour-phase combustion in the argon- and xenon-diluted mixtures, respectively; whereas, the helium-diluted case does not exhibit clear gas-phase combustion (Fig. 5(a)–(b)(i)). Any vapour-phase combustion observed from Stages 1–6 is likely due to Fe evaporation, since the vapour pressure of Fe is significantly higher than that of FeO at a given temperature [10], indicating that the Fe → FeO reaction occurs as long as the nano-oxide halo is visible.

The nano-oxide formation at Stage 6 is further visualized through the attenuation of white LED light observed in high-speed images, as shown in Fig. 6. The attenuation signal is presented at three wavelengths of 457 nm, 530 nm, and 628 nm. While the present study does not quantify the nano-oxide concentration, the results provide qualitative evidence that a significantly denser nano-oxide cloud forms in the xenon-diluted mixture (Fig. 6(c)) compared to the argon-diluted mixture (Fig. 6(b)). In contrast, the attenuation signal in the helium-diluted mixture appears to be insignificant and dominated by noise (Fig. 6(a)). The amount of nano-oxide formed depends on the evaporation rate, which is highly sensitive to temperature. Therefore, the observed differences in the extent of nano-oxide formation further reinforce the variations in particle temperature across the three diluents. These findings contradict the thermodynamic analysis, which predicts the same adiabatic temperature for all three gas mixtures.

The variation in particle temperature can be explained by differences in the Lewis number, Le. In diffusion-controlled combustion, the heat generation rate is proportional to the oxygen mass diffusivity in the gas mixture, $\dot{Q}_G \propto D_{O_2-\text{mix}}$, whereas the heat loss rate is proportional to the thermal diffusivity and the temperature gradient between the particle and the surrounding gas, $\dot{Q}_L \propto a(T_p - T_g)$. Therefore, considering the energy balance, the temperature difference scales as $(T_p - T_g) \propto 1/\text{Le}$ [25]. To calculate Le, the empirical Chapman–Enskog relation [33] is applied to estimate the thermal conductivity $\lambda(T)$, the density $\rho(T)$ is determined from the ideal gas law, and the specific heat capacity $C_p(T)$ is obtained from the NASA polynomial expressions [34]. The thermal diffusivity is then calculated as $\alpha(T) = \lambda(T)/[\rho(T)C_p(T)]$. The binary mass diffusivity of oxygen in each diluent is estimated using the Chapman–Enskog relation [33], and the mixture-averaged
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Observation of the combustion behaviour of iron particles in three different gas mixtures: (i) 40%O₂–60%He, (ii) 40%O₂–60%Ar, and (iii) 40%O₂–60%Xe. (a) Sample images recorded by the high-speed camera. All the sample frames shown are taken with the same exposure settings and drawn on the same scale. The numbers at the top indicate successive phases of combustion and cooling: (1) particle melting, (1)–(2) the period with laser heating, (2)–(6) self-sustain combustion of the particle, and (6)–(8) reactive cooling of the particle. (b) Evolution of normalized calibrated intensity and (c) derived temperature with time for the burning iron droplet in the three gas mixtures tested. The vertical line represents the time step where the laser is shut off. The same referencing numbers is used to refer the sample frames shown in (a), representing different stages of combustion and cooling. Tmax corresponds to the average measured temperature at Stage 6.</figcaption></figure>

<caption>**Table 1**</caption>

Lewis number ranges based on thermal and mass diffusivity evaluations at the particle surface for 0%, 20%, and 40% O₂ mixtures with various inert gas dilutions at 3000 K.

<table><thead><tr><th rowspan="2">Diluent</th><th rowspan="2">D<sub>O₂-mix</sub><br/>[cm²/s]</th><th colspan="3">α [cm²/s]</th><th colspan="3">Le [-]</th></tr><tr><th>0% O₂</th><th>20% O₂</th><th>40% O₂</th><th>0% O₂</th><th>20% O₂</th><th>40% O₂</th></tr></thead><tbody><tr><td>Helium</td><td>34.1</td><td>78.4</td><td>47.4</td><td>32.2</td><td>2.30</td><td>1.39</td><td>0.94</td></tr><tr><td>Argon</td><td>9.5</td><td>9.8</td><td>9.7</td><td>9.2</td><td>1.03</td><td>1.02</td><td>0.97</td></tr><tr><td>Xenon</td><td>6.4</td><td>3.7</td><td>4.7</td><td>5.4</td><td>0.58</td><td>0.73</td><td>0.84</td></tr></tbody></table>

diffusivity approximation described in [33] is applied to compute the oxygen mass diffusivity in the gas mixture.

In the present study, a simplified approach is adopted to evaluate the Lewis number assuming the minimum oxygen concentration at the particle surface is 0% and the maximum is 40% (the oxygen concentration in the ambient gas), with the remaining fraction consisting of the diluent gas. For a gas mixture with two species, the oxygen mass diffusivity is essentially constant as the oxygen concentration varies under the mixture-averaged diffusivity approximation. However, due to the large differences in the thermal diffusivities of oxygen, helium, and xenon, the thermal diffusivity of He–O₂ and Xe–O₂ mixtures is highly sensitive to the oxygen concentration in the ambient gas. The resulting range of possible Lewis numbers under these conditions at 3000 K is listed in Table 1.

At a given temperature, the Lewis number in the argon-diluted mixture remains close to unity and is unaffected by the oxygen concentration gradient. In contrast, the large differences in thermal diffusivity among helium, xenon, and oxygen lead to more pronounced variations in the Lewis number for He–O₂ and Xe–O₂ mixtures with changing oxygen concentration. For the helium-diluted mixture, the Lewis number ranges from 0.94 at 40%O₂ to 2.30 at 0%O₂, whereas for the xenon-diluted mixture it varies from 0.58 at 0%O₂ to 0.84 at 40%O₂ at the particle surface. For a given mixture, the Lewis number is not expected to change significantly with temperature [25]. This simplified analysis suggests that the Lewis number is highest for the helium-diluted mixture, followed by the argon-diluted mixture, and lowest for the xenon-diluted mixture; therefore, the particle temperature is expected to be lowest in helium dilution and highest in xenon dilution, consistent with the qualitative observations from the colour camera.
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Sample frames of the burning particle in (i) 40%O₂–60%He, (ii) 40%O₂–60%Ar, and (iii) 40%O₂–60%Xe during self-sustain combustion in columns 4–6 of Fig. 4. A halo surrounding the burning droplet is visible in (ii) and (iii). (a) Raw colour-camera images captured without LED backlighting, such that the observed emission away from the particle surface originates solely from the vapour-phase flame. (b) Processed contour images, which highlight more clearly the emission signal extending away from the particle surface. The local pixel intensity is normalized by the global peak intensity in each image.</figcaption></figure>

Although the colour camera indicates qualitative differences in peak temperature among the three mixtures, the peak temperatures derived from the photomultiplier are comparable across the three mixtures under the grey-body assumption (Fig. 4(b)). This discrepancy arises from the inaccuracy of the grey-body assumption when deriving the temperature using the photomultiplier in the presence of non-negligible nano-oxide formation. In the helium-diluted mixture, where nano-oxide formation is insignificant, the intensity profile exhibits a relatively smooth single-peak structure (Fig. 4(b)(i)), which likely reflects primarily grey-body emission from the micron-sized burning droplet. In contrast, in the argon- and xenon-diluted mixtures, the formation of nano-oxide halos affects the signal captured by the photomultiplier and consequently reduces the accuracy of temperature estimation under the grey-body assumption. As observed in the high-speed videos and photomultiplier signals, during Stages 1–3 the emission is likely dominated by nano-oxides. From Stages 3–6, although nano-oxide emission is observed to be less significant, the oxide cloud can still scatter the light emitted by the burning droplet.

The photomultiplier signal represents a superposition of emissions from both the burning droplet and the emitting nano-oxide species near the particle surface. When the emission from the dense nano-oxide formed during Stage 2 dominates (caused by the brief laser heating), the emission signal exhibits a small secondary peak (Fig. 4(b)(ii)) in the argon-diluted case, whereas in the xenon-diluted case, where nano-oxide formation is more pronounced, a broad intensity plateau is observed instead (Fig. 4(b)(iii)). Since a significant portion of the photomultiplier signal originates from the dense nano-oxide around Stage 2 in these two gas mixtures, the grey-body-derived temperature profile exhibits an inaccurate high peak (the grey dotted lines in Fig. 4(c)(ii)–(iii)). The qualitative temperature trend becomes more reliable from Stage 4 onward (the red solid lines in Fig. 4(c)(ii)–(iii)) because the nano-oxide halo is relatively fainter after the laser is shut off. This allows a clearer increasing temperature trend to be observed from Stage 4 to 6, consistent with the continuation of the self-sustained L1 → L2 reaction until a local maximum is reached at Stage 6 before cooling.

At Stage 6, as light scattering by nano-oxides becomes dominant over their emission, the estimated maximum temperature of the xenon-diluted mixture derived from the photomultiplier in Fig. 4(c) shows a

lower value, even though the particle temperature would be expected to be highest in the xenon-diluted mixture (as observed from the denser nano-oxide formation shown in Fig. 6). Previous studies reported a similar trend: although a significant increase in particle temperature would be expected with increasing oxygen concentration, the measured temperature based on the grey-body assumption increased by less than 50 K when the oxygen concentration rises from 30% to 50% [4], and from 50% to 100% [5]. The authors of these studies attributed this phenomenon to the interplay between heat release and enhanced evaporative and radiative heat losses at elevated oxygen concentrations. In addition to this factor, we infer that the light scattering by the nano-oxide cloud (which is wavelength dependent) becomes more significant with decreasing Lewis number in the present study (or with increasing oxygen concentration in previous studies), which affects the accuracy of grey-body assumption adopted in most studies. As observed in Fig. 6, the light attenuation at 457 nm appears more pronounced than at 628 nm at Stage 6, indicating that the photomultiplier is more biased towards the longer-wavelength signal, which in turn leads to a lower derived temperature.

A more accurate temperature profile could be obtained by quantifying the effects of both emission and light scattering by the nano-oxide cloud, both of which depend on the wavelength, $\lambda$, and follow the relation $\epsilon \propto 1/\lambda^n$ [35]. Quantifying the power exponent $n$ requires optical properties of the condensed species at high combustion temperatures, along with an assessment of the relative contributions of each species to the photomultiplier signal and accurate measurements of nanoparticle sizes. Future studies could address these limitations of the grey-body assumption, while the present work still enables determination of the time interval between the melting point and the peak derived temperature, which occurs close to the peak particle luminosity, for combustion time estimation.

## 3.2. Reactive cooling of the particle

The halo surrounding the burning particle shown in Fig. 5 persists until the local temperature peak at Stage 6, after which it gradually fades and disappears within a few frames. The fading of the halo suggests that no unreacted Fe remains in the droplet for evaporation and that the droplet is then composed primarily of L2. The absence of gas-phase combustion during this cooling phase can be attributed to the low partial pressure of FeO. The completion of the Fe→FeO reaction (with no remaining L1 phase) at Stage 6 is further supported by the observed temperature decrease and the gradual reduction in particle luminosity as the droplet enters its cooling phase.

Fig. 7(a) compares the particle cooling rates measured in the present experiments for the three gas mixtures. The results show that the slope of temperature versus time during cooling is steepest in the helium-diluted mixture, followed by the argon-diluted mixture, and is least steep in the xenon-diluted mixture, due to differences in the thermal conductivity of the diluent gases. These experimental cooling curves are compared with theoretical simulations based on the energy-balance model described in Ref. [8], and the results of the cooling simulations are shown in Fig. 7(b). In the simulations, the cooling particle is assumed to be stationary and to consist solely of FeO. Heat loss from the particle is assumed to occur only via conduction and radiation, since evaporation is not observed during the cooling phase in the present study. The observed particle size growth after combustion (from L1 to L2) in the present study is 15%–25% for most runs in the argon- and xenon-diluted mixtures and 20%–30% for the helium-diluted mixture; however, in some helium-diluted cases, substantially larger growth is observed (discussed later). This particle size increase is incorporated into the cooling simulation. For the micron-sized particles used in the present study, the cooling process is dominated by conductive heat transfer, which accounts for at least 85% of the total heat loss.

The differences in cooling rates shown in Fig. 7(a) and (b) for the same gas mixtures are consistent with several previous simulation
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Observation of nano-oxide formation at the local peak temperature (Stage 6) in three gas mixtures: (a) 40%O₂–60%He, (b) 40%O₂–60%Ar, and (c) 40%O₂–60%Xe, obtained from LED white-light absorption at wavelengths of 457 nm, 530 nm, and 628 nm. The colour maps represent the ratio of the observed intensity to the LED intensity ($I/I_{LED}$). The same dimension and colour scales shown in row (c) are applied to all sub-figures. The same exposure time is used for all of the results shown here. The observations are qualitatively similar across all tested runs: nano-oxide formation in the xenon-diluted mixture is denser than in the argon-diluted mixture, while the attenuation signal in the helium-diluted mixture is dominated by noise. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

studies [8,36,37]. These studies have demonstrated that, during the cooling phase following the peak temperature, the molten L2 oxide can continue to react with additional oxygen as the temperature decreases, exhibiting a reactive cooling behaviour in which the cooling rate is noticeably slower than that the predicted inert cooling rate. The model of Mich et al. [36] shows that the oxidation limit of the cooling particle is determined by thermodynamic equilibrium rather than by surface kinetics or oxygen diffusion, while Ning et al. [37] further illustrates that this oxidation limit is linearly proportional to the particle temperature, $T_p$, following the relationship $Z_O = -(4.7 \times 10^{-5})T_p + 0.67$, where $Z_O = O/(Fe + O)$. This linear relationship is consistent with the L2–gas boundary in the Fe–O phase diagram, which shifts towards higher $Z_O$ values, from about 0.51 to 0.58, as the temperature decreases from 3420 to 1855 K. The present experiment further suggests that the diluting inert gas can significantly affect the oxidation rate of the cooling particle. The higher thermal conductivity of the helium-diluted mixture leads to more rapid cooling and, consequently, faster oxygen uptake by the cooling L2 droplet compared with the xenon-diluted case. In contrast, the cooling rates (i.e., the slopes after the peak temperature) observed in Chang et al. [6] appear to be insensitive to the oxygen concentration, most likely due to the similar thermal conductivities across all N₂–O₂ mixtures investigated.

As the particle cools to about 1855 K, the reaction $L2 \rightarrow Fe_3O_4(s) + O_2(g)$ takes place [4], where L2 droplet solidifies into $Fe_3O_4$ while releasing dissolved oxygen gas that cannot be incorporated into the crystalline structure of $Fe_3O_4$. This phenomenon is also supported by the Fe–O phase diagram [38], where the L2–gas boundary intersects with the “Spinel + Gas” phase at about 1855 K. Some experimental runs in the helium-diluted mixture demonstrate this behaviour: the particle size increases significantly (up to about 50%) during cooling (Fig.

4(a)(i)), indicating that the L2 droplet can absorb more oxygen than the stoichiometric ratio for $Fe_3O_4$, with gas release occurring during solidification. High-resolution images from the present study (1.25 µm/pixel) also show these inflated particles moving in swirling trajectories rather than falling straight, supporting the presence of internal gas within the solidified particle. The derived temperature from the photomultiplier during this event is also close to the $Fe_3O_4$ solidification point (1855 K). In some cases, the inflated particle deflates after a few frames, likely due to the escape of trapped gas.

Similar particle inflation behaviour during cooling was observed by Ning et al. [39], where particle size exhibited a sudden growth during cooling. Deutschmann et al. [31] and Jimenez et al. [32] also found that the burnt iron particle collected with a longer residence time and higher oxidation degree exhibited a structure consisting solely of iron oxide with a spherical pore embedded inside the particle, likely resulting from the similar phenomenon of gas formation during the solidification of $Fe_3O_4$. It should be noted that all the previous experiments mentioned above were conducted at low oxygen concentrations (21% or lower) with nitrogen as the diluent. The present study provides new evidence that a similar particle product structure can form in an oxygen-enriched environment (40%O₂) when diluted with helium. However, for the 40%O₂ cases diluted with argon and xenon, no particle inflation behaviour is observed in all the experimental runs during the reactive cooling stage.

### 3.3. Combustion time and discreteness parameter

The combustion dynamics discussed in Sections 3.1 and 3.2 provide a basis for interpreting the combustion time of iron particles in different diluting gases. In the present study, a combustion time is defined by
<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Temperature normalized by the peak temperature, $T_{max}$, of cooling iron oxide particles in 40%O₂–60%He, 40%O₂–60%Ar, and 40%O₂–60%Xe gas mixtures as a function of time from $T_{max}$ ($t = 0$ corresponds to Stage 6). (a) Experimental data from the present study. (b) Inert cooling simulations. Both illustrate the differences in cooling rates among the three diluents, with reactive cooling occurring more slowly than inert cooling for the same gas mixture.</figcaption></figure>

the time difference between the melting point of iron (1811 K) to the re-solidification point (1855 K) of Fe₃O₄, hereafter denoted as $t_{all}$. This definition of combustion time includes the reactive cooling phase, which may not always be representative. The enthalpy of reaction Fe + (1/2)O₂ → FeO is -272.1 kJ per mole of Fe consumed, whereas for FeO + (1/6)O₂ → (1/3)Fe₃O₄, it is -100.7 kJ per mole of FeO consumed [40]. This indicates that most of the heat is released during the oxidation of Fe to FeO, before the cooling phase begins at Stage 6. Given that the enthalpy change for FeO → Fe₃O₄ reaction during reactive cooling is smaller than for Fe → FeO reaction, and that reactive cooling is not a rapid process because further oxidation of FeO can only occur as the particle temperature decreases [36,37], the heat released during reactive cooling is unlikely to contribute significantly to particle ignition in the context of iron dust flame propagation. This is particularly true for larger particles, as the slower and smaller heat release during reactive cooling is unlikely to synchronize with the flame propagation wave downstream. For this reason, another combustion time, $t_{max}$, is defined in this study as the time from the melting point of Fe until the beginning of cooling (Stages 1 to 6 in Fig. 4), where Stage 6 likely indicates the complete consumption of Fe into FeO, supported by the fact that no further evaporation of Fe, and formation of nanoparticles, is observed.

Table 2 summarizes the combustion times, $t_{max}$ and $t_{all}$ in the present study. The model developed by Thijs et al. [7] predicts combustion times of approximately 5–6 ms at 40% oxygen concentration (based on extrapolation) for 40 µm particles in nitrogen-diluted mixtures. In comparison, the present study measured an average combustion time of 1.99 ms in 40%O₂–60%Ar mixtures. The shorter combustion time

observed in the present study is likely due to the smaller particle sizes used in the present study, the slightly higher flame temperature in argon-diluted mixtures (which exponentially increases the vapour pressure of iron for evaporation), and the additional laser heating. For xenon-diluted mixtures, the present study reports a combustion time of 2.2 ms, compared with 2.7 ms measured by Palecka et al. [12] using a furnace setup under the same xenon dilution. The quantitative values of the combustion times in the present study are likely shortened by laser heating during Stages 1–2. Thijs et al. [41] reported that the initial temperature after laser heating does not significantly affect either the maximum particle temperature or the combustion time at 21%O₂. Similarly, the study of Bruyr et al. [26] performed in air found no clear correlation between the laser-heating duration and either $t_{max}$ and $t_{all}$. These observations suggest that the combustion rate is primarily limited by the external diffusion of oxygen rather than by the particle temperature induced by laser heating. Nevertheless, in the present study, conducted at 40%O₂, nano-oxide formation is more significant. We therefore infer that variations (if substantial) in the duration of laser overheating may affect the combustion time by increasing the amount of nano-oxide formed, so the combustion times reported here may not be interpreted quantitatively.

Despite this quantitative limitation, the combustion times measured in the present study remain meaningful for qualitative comparison. Based on the diffusion-limited particle combustion theory, the combustion time is expected to be inversely proportional to the mass diffusivity of oxygen in the gas mixture ($t_c \propto 1/D_{O_2-mix}$). Table 2 lists the oxygen mass diffusivity values, highlighting the significant differences in $D_{O_2-mix}$ among the three gas mixtures. Thus, noticeable variations in combustion time are expected between these mixtures. However, it is observed that $t_{max}$ in helium- and argon-diluted mixture are very similar (1.92 ms and 1.99 ms), while $t_{max}$ in xenon-diluted mixture is only slightly longer (2.20 ms), despite the noticeable differences in the oxygen mass diffusivity of these mixtures. This suggests that the combustion time is influenced by factors beyond oxygen diffusivity alone. While $t_{max}$ is comparable across all gas mixtures, a more noticeable difference is observed in $t_{all}$, which includes the cooling phase with further oxidation of FeO. This is because vapour-phase combustion is negligible during the cooling phase, and the reactive cooling rate is governed by the thermal conductivity of the gas mixture, as reflected in the different gradients of the cooling curves shown in Fig. 7.

Although xenon dilution is expected to increase combustion time due to the lower oxygen mass diffusivity, it also promotes greater Fe evaporation and, consequently, more pronounced vapour-phase combustion (Section 3.1). Vapour-phase combustion can substantially shorten the combustion time because the lifted flame diameter is 3.5–3.9 times larger than the particle diameter, corresponding to a 12–16 times larger reaction surface area; moreover, some O₂ reacts with Fe vapour before reaching the particle, reducing mass-transport time of oxygen. By contrast, in the helium-diluted mixture, the combustion time is not enhanced by vapour-phase combustion due to a much weaker Fe evaporation. Considering the interplay between oxygen mass diffusivity and vapour-phase combustion (as reflected by the extent of nano-oxide formation), the combustion times measured in the three mixtures tested are somewhat comparable. Palecka et al. [12] reported a similar effect of vapour-phase combustion, where increasing the oxygen concentration from 20% to 40% reduced the combustion time by about a factor of 3, compared with a factor of 2 predicted by the diffusion theory, owing to enhanced Fe-vapour formation at elevated temperatures.

Using the measured combustion time $t_{max}$, the discreteness parameter in the context of flame propagation in different gas mixtures can be compared across different oxidizing environments using Eq. (1). An iron dust concentration of $B = 0.5$ g/L is assumed, with all particles taken to have the same $t_{max}$ and a diameter of 32 µm. The corresponding interparticle distance is $l = 0.065$ cm, assuming all particles are equally spaced in a three-dimensional Cartesian coordinate system. In the
<caption>**Table 2**</caption>

Comparison of experimental combustion times $t_{max}$ and $t_{all}$, oxygen mass diffusivity, thermal diffusivity, and discreteness parameter $\chi$ for gas mixtures with 40% $O_2$ in the ambient gas diluted in different monoatomic gases. The discreteness parameter is calculated as $\chi = t_{max}\alpha/l^2$ and is evaluated at 0.5 g/L for iron particles with a diameter of 32 µm. The interparticle distance $l$ is calculated considering the vapour-phase combustion.

<table><thead><tr><th rowspan="2">Inert Gas</th><th>D<sub>O<sub>2</sub>-mix</sub><sup>(i)</sup></th><th>t<sub>max</sub></th><th>t<sub>all</sub></th><th>α<sup>(ii)</sup></th><th>l</th><th>χ</th><th colspan="4">Differences (times)<sup>(iii)</sup></th></tr><tr><th>[cm<sup>2</sup>/s]</th><th>[ms]</th><th>[ms]</th><th>[cm<sup>2</sup>/s]</th><th>[cm]</th><th>[-]</th><th>D<sub>O<sub>2</sub>-mix</sub><sup>(i)</sup></th><th>1/t<sub>max</sub></th><th>1/t<sub>all</sub></th><th>χ</th></tr></thead><tbody><tr><td>60%He</td><td>0.747</td><td>1.92 ± 0.05</td><td>4.28 ± 0.10</td><td>0.792</td><td>0.065</td><td>0.364</td><td>6.07</td><td>1.15</td><td>1.31</td><td>5.28</td></tr><tr><td>60%Ar</td><td>0.196</td><td>1.99 ± 0.04</td><td>4.98 ± 0.10</td><td>0.207</td><td>0.059</td><td>0.118</td><td>1.59</td><td>1.11</td><td>1.12</td><td>1.71</td></tr><tr><td>60%Xe</td><td>0.123</td><td>2.20 ± 0.06</td><td>5.59 ± 0.12</td><td>0.108</td><td>0.058</td><td>0.069</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

(i) The values of $D_{O_2-mix}$ are evaluated at 300 K. The differences in $D_{O_2-mix}$ among the gas mixtures remain nearly identical at a given temperature.

(ii) The values of $\alpha$ are evaluated at 300 K for a gas mixture consisting of 40% $O_2$ and 60% inert gas.

(iii) The numbers shown indicate how many times each value is relative to that calculated for the 40% $O_2$–60% $Xe$ case.

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** The discreteness parameter (Eq. (1)) at different iron particle concentration for three gas mixtures tested in the experiment.</figcaption></figure>

argon- and xenon-diluted mixtures, due to vapour-phase combustion, the interparticle distance is assumed to be reduced by the thickness of the nano-oxide halo (Fig. 5). The thermal diffusivity, $\alpha$, is evaluated at 300 K and calculated as $\alpha = \lambda/(\rho C_p)$, where $C_p$ is the heat capacity of the gas mixture, respectively. Since heat diffuses between particles through the surrounding gas, the thermal diffusivity of the ambient 40% $O_2$–60% inert gas mixture is used to estimate the discreteness parameter. The calculated discreteness parameters, $\chi$, are listed in Table 2.

It should be emphasized that the discreteness parameter values presented here (derived from the measured combustion times) should also be interpreted qualitatively. A meaningful quantitative evaluation is likely unattainable, regardless of methods or techniques used in single particle combustion experiments, because the combustion time of an isolated particle may not represent that within a dust flame, where the effect of collective self-heating between particles and a polydisperse particle-size distribution are present [42,43]. Moreover, discrete flames inherently exhibit spatially varying temperature fields [16,20], so burning particles can experience different local gas temperatures and oxidizer concentrations depending on their spatial arrangement. Given this complexity, investigating the discreteness parameter of iron flames in different oxidizing environments is best initiated by a fundamental comparison of the combustion times of single particles with identical sizes under the same experimental settings.

Based on the measured combustion times, the present experimental results show a 1.71-fold increase in the discreteness parameter when switching from xenon to argon dilution. In helium-diluted mixtures, the discreteness parameter is 5.28 times higher than in xenon, owing to comparable combustion durations but significantly shorter heat diffusion times. Fig. 8 shows the discreteness parameter as a function of particle concentration. At 0.5 g/L, the discreteness parameter obtained for 40% $O_2$–60% $Xe$ is 0.069. This low value corresponds to a very high

probability of discrete flame propagation. Indeed, previous microgravity experiments observed discrete flame behaviour in 40% $O_2$–60% $Xe$ at about 0.5–0.7 g/L [20], although the existing microgravity experiments have not obtained sufficient runs to conclude if the discrete flame behaviour can be observed for a concentration above 0.7 g/L. Given the 1.71 times higher discreteness parameter in the argon-diluted mixture, achieving the same discreteness as in 40% $O_2$–60% $Xe$ at 0.5–0.7 g/L would require reducing the concentration range to 0.25–0.35 g/L in 40% $O_2$–60% $Ar$. This range is close to the thermodynamic concentration limit of about 0.2 g/L [20]. However, considering the very low discreteness parameter in xenon dilution evaluated by both Palecka et al. [12] and the present study, the 1.71-times change of discreteness parameter in argon dilution may still permit discrete flame behaviour within the same 0.5–0.7 g/L concentration range used in the previous microgravity experiments. By contrast, the helium-diluted mixture is unlikely to exhibit discrete flame propagation for a wider concentration range due to a much higher discreteness parameter.

## 4. Conclusion

The combustion behaviour of single iron particles is investigated in an electrostatic levitator with laser ignition at a fixed oxygen concentration of 40%, using helium, argon, and xenon as diluents. Although thermodynamic calculations predict identical adiabatic combustion temperatures for these mixtures, experimental observations reveal noticeable differences in particle temperature and combustion time due to variations in Lewis number and associated heat-loss effects. The particle temperature is lower in the helium-diluted mixture, as indicated by its reduced peak luminosity, which results from its higher Lewis number compared with the argon- and xenon-diluted mixtures. Nano-oxide formation, assessed via attenuation of white LED light, is most pronounced in the xenon-diluted mixture, moderate in the argon-diluted mixture, and insignificant in the helium-diluted mixture. This nano-oxide formation is caused by Fe evaporation, which is enhanced by higher combustion temperatures and, in turn, promotes vapour-phase combustion. Vapour-phase combustion reduces combustion time by increasing the reactive surface area and shortening oxygen transport distances.

The interplay between oxygen mass diffusivity, heat losses from the particle to the surrounding gas, and vapour-phase combustion explains why $t_{max}$ is similar in helium- and argon-diluted mixtures despite the fact that oxygen mass diffusivity is much higher in the helium diluent. Likewise, $t_{max}$ in the xenon-diluted mixture is only slightly longer than in the argon-diluted mixture. Consequently, the combustion time prediction solely based on oxygen mass diffusivity is inaccurate. The present study shows that the discreteness parameter increased by 1.71 times from xenon- to argon-diluted mixtures. Given the low value of the discreteness parameter in the xenon-diluted mixture (0.069 at 0.5 g/L), and prior microgravity experiments have shown discrete flame propagation behaviours in 40% $O_2$–60% $Xe$, the modest 1.71 times increase in the discreteness parameter for the argon-diluted mixture may still permit discrete flame propagation in very fuel-lean 40% $O_2$–60% $Ar$.
mixtures. The present study strengthens the fact that discrete flame propagation is not exclusive to gas mixtures using xenon as a diluent. In contrast, the much higher discreteness parameter for helium-diluted mixtures makes discrete flame behaviour unlikely over a wide range of concentrations.

The present study has quantitative limitations due to a brief period of laser-induced overheating. Nevertheless, the qualitative findings from the present experiment highlight that, besides oxygen concentration, the Lewis number is also a key determinant of particle temperature and, consequently, evaporation rate, nano-oxide formation, combustion time, and the likelihood of discrete flame propagation in iron dust flames. This understanding lays the groundwork for the scientific community to investigate discrete flames using a more cost effective and more accessible diluting gas, argon. The fundamental understanding obtained from the present study, together with future discrete flame investigations in other diluents, will be critical for developing more accurate combustion models and enabling technologies that utilize metal fuels.

## CRediT authorship contribution statement

**Herman Heng:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Hugo Keck:** Writing – review & editing, Software, Resources, Methodology. **Christian Chauveau:** Writing – review & editing, Resources, Data curation. **Fabien Halter:** Writing – review & editing, Supervision, Resources, Project administration, Data curation. **Jeffrey Bergthorson:** Writing – review & editing, Supervision, Project administration, Funding acquisition.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This project is funded by the Canadian Space Agency through the Flights and Fieldwork for the Advancement of Science and Technology (FAST) Grant Program (Grant No: 23FAMCGA55). Student work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec – Nature et technologies (FRQNT), Canada, and McGill University through the Vadasz Scholars Program, Canada. Travel costs associated with the collaboration between McGill University and the *Centre national de la recherche scientifique* (CNRS) are supported by the NSERC Canada Graduate Scholarships – Michael Smith Foreign Study Supplement (MSFSS). The authors would also like to thank Xiaocheng Mi for fruitful discussions on iron combustion behaviour.

## Data availability

Data will be made available on request.

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