# Thermite Combustion: Current Trends in Modeling and Future Perspectives

Alain Esteve, Carole Rossi

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Alain Esteve, Carole Rossi. Thermite Combustion: Current Trends in Modeling and Future Perspectives. Applications in Energy and Combustion Science, 2025, 22, pp.100332. 10.1016/j.jaecs.2025.100332. hal-05345084

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# Thermite Combustion: Current Trends in Modeling and Future Perspectives

Alain Esteveª, Carole Rossiª,*

ªLAAS-CNRS, University of Toulouse, 7 Avenue du colonel Roche, Toulouse, 31400, France

## Abstract

Aluminum-based reactive composites, such as thermites, represent a unique class of energetic materials characterized by their high energy densities, tunability in combustion properties and safety. Prepared using techniques such as mechanical mixing, milling, and physical vapor deposition, these materials are promising for achieving energetic functions beyond the capabilities of traditional energetic materials. Applications include thermal plugging, smart initiation, pyro-fusing in civilian devices where the use of explosives is not feasible. Unfortunately, engineers and researchers face the lack of predictive combustion models to optimize the thermite materials to a given application. The reason is the insufficient knowledge and quantification of reaction and combustion mechanisms and the key variables governing them. That is why, over the past decades, several approaches and models ranging from atomic-scale modeling to macroscopic simulations using computational fluid dynamics, were developed and are reviewed in this article. These methods provided insights into key reaction pathways, ignition mechanisms, and flame propagation dynamics. Despite these advancements, substantial gaps remain, particularly in capturing multiphase flow dynamics and suboxides condensation/nucleation process during the combustion at high temperature. Boundary-resolved transient direct numerical simulation approach and particle-resolved numerical techniques will allow acquiring knowledge in gas-particle and particle-particle interaction. Recent breakthroughs in machine learning will further accelerate the design and optimization of thermites by enabling the establishment of predictive quantitative structure-property relationships in complement of heavy detailed physical models. This review highlights foundational theoretical developments for thermite materials, and emphasize the need for interdisciplinary efforts particularly between fluid dynamicists and condensed matter physicists to realize the full potential of these versatile energetic materials.

**Keywords:** Thermite reaction, Combustion wave, Numerical modeling

## 1. Introduction

Metal such as aluminum (Al), magnesium (Mg), titanium (Ti), zirconium (Zr) and boron (B) have very high volumetric (30 - 140 kJ.cm⁻³) and gravimetric energy density (10 - 50 kJ.g⁻¹) compared with other chemical fuels and organic energetic materials [1]. They are also chemically

<footnote>*Corresponding author. Email: rossi@laas.fr
URL: https://www.laas.fr/fr/homepages/rossi/ (Carole Rossi)</footnote>
stable and solid, simplifying storage and transport. Furthermore, they can be ignited and burned with oxidizers, such as air or metallic oxidizers, to generate substantial amounts of heat. The study of metal fuel combustion dates back to the 1970s [2] driven by potential applications as additives in energetic materials to increase the energy density and burn rates while decreasing their sensitivity to undesired external stimuli [3]. Since then, large research efforts have been devoted to metal-based reactive materials using different metal/metallic oxide combinations and custom micro-nanostructured architectures [4, 5]. Figure 1 illustrates some architectures.

<figure><img src="image_2.png" /><figcaption>Figure 1: Illustration of the different metal/metallic oxide combinations utilizing micro- and nanostructured architectures.</figcaption></figure>

Unlike traditional CHNO energetic materials, these fast reacting materials commonly referred to as thermites undergo a rapid deflagration driven by a carbon-free oxidation-reduction reaction, which forms stable reaction products (metal oxides or hydroxides). These materials constitute a distinctive subset of energetic materials, distinct from explosives in that their rapid reaction fronts —reaching speeds of up to 1000 m.s⁻¹— propagate subsonically and rely on atomic diffusion or other physical transport mechanisms. The reactant transport dependency means that thermites can be engineered to react over a wide range of timescales depending on the reactant length scale and shape. Classically, the combustion performance of these materials can be tuned in the ranges of 10⁻¹ – 10³ m.s⁻¹ and 2800 – 5300 K, respectively, varying the reactant composition and size, reactant intimacy and the stoichiometry [6, 7, 8, 9, 10, 5, 11]. The peak pressure in constant volume cell spans a few to tens of MPa [12, 13, 11, 14] depending on the synthesis method and mixing scale between the constituents (metal and metal oxide).

These characteristics, which are entirely complementary to those of propellants and explosives (Figure 2), combined with the safety, tunability, and relatively benign nature of thermites make them promising for a myriad of civilian and defense applications including material synthesis [15, 16], heat sources for welding and joining [17, 18], smart initiators [19], on-chip actuation [20, 21, 22] and thermal neutralization [23, 24]. However, engineering a thermite material that achieves the desired combustion behavior for a given application constitutes the major hurdle for diffusing these fast reacting energetic materials into the applications. It requires an in-depth understanding of reaction phenomena and the key variables governing them in order to establish combustion properties such as flame temperature and speed, pressurization rate, and gas production volume, based on material composition and structure (nature, shape and size of the fuel and oxidizer, density, stoichiometry, and, presence or not of binders or additives). And, the major challenge is that the multiphasic flow physics governing the thermites combustion in real-world situations, i.e., where reactive combustion gases interact with condensed matter, is still poorly understood due to the complexity of the associated chemistry and physics. This complex-
ity arises from the simultaneous occurrence of chemical reactions, heat, mass, and momentum transfer, phase transitions, and particle interactions both with each other and with the gas phase, everything within a fraction of second.

<figure><img src="image_3.png" /><figcaption>Figure 2: Diagram comparing the combustion properties of thermite materials with those of explosives and propellants.</figcaption></figure>

Over the last decades, highly specialized diagnostic experimental tools, such as T-jump/TOF-MS (Time of Flight Mass Spectrometry), time-resolved X-ray diffraction (TRXRD), scanning transmission electron microscopy - electron energy loss spectroscopy (STEM-EELS), high-speed transmission electron microscopy (HSTEM), or dynamic transmission electron microscopy (DTEM), permitted unprecedented progress into thermite reaction mechanisms. But, these techniques involve experiments that are performed under vacuum, on small amounts of material, and generally, do not offer simultaneous temperature measurements for thermo-physical property and kinetics estimations, which impede the possibility to develop predictive physical modeling of the combustion. In-operando flame characterizations using high-speed and infrared cameras provide a comprehensive analysis of thermal front microstructures subject to heterogeneous reactions [25, 26, 9]. But each test is a single-shot experiment that demands substantial material quantities (usually hundreds of milligrams) and requires robust and costly safety measures.

We truly believe that, supported by the aforementioned experimental techniques, a multi-scale computational approach represents the most efficient path toward the development of Quantitative-Structure-Properties-Relationships (QSPRs). Density Functional Theory (DFT) and Molecular Dynamics (MD) for example are essential for the elucidation of reaction mechanisms as well as for the quantification of thermokinetic data related to mass transport, chemical reactions, and gas-surface interactions. Higher-level numerical models, such as those based on the Computational Fluid Dynamics (CFD) approach, which do not resolve individual particles, can depict the self-propagating combustion behavior. These macroscopic models, provided that all lower-scale
mechanisms are accurately described, allow investigation of how the microstructural proper-
ties of the reactive material influence the reaction kinetics and flame dynamics. And, recent
developments in machine learning (ML) offer a promising transformative approach to single
conventional physical models to help the understanding of the intricate multiscale phenomena,
predicting reaction kinetics and flame dynamics, and optimizing performance across diverse ap-
plicative conditions.

This article provides a review of theoretical works, focusing on two main aspects: first, the
foundational modeling efforts that have significantly advanced the understanding and quantifi-
cation of the mechanisms governing the combustion of aluminum-based thermites, and second,
the computational studies that have marked a decisive step in the development of QSPRs; this
latter relying of course, on knowledge provided by the former. To help the reader understand
the challenges involved in developing precise and practical combustion models for engineer-
ing thermite materials, a first section (section 2) provides a brief overview of the main thermite
preparation methods, highlighting all parameters that influence their energetic properties. The
reader will therefore grasp the vast extent of the parameters space, and thus the impossibility of
establishing QSPRs experimentally. Then, section 3 outlines the different theoretical methodolo-
gies adopted by researchers to deepen the understanding of reaction phenomena and identifying
the key variables that govern them, and theoretically duplicate experiments. Section 4 reviews
major theoretical works, setting the stage for a discussion on future prospects aimed at devel-
oping QSPRs and design rules, which are essential for integrating these fast-reacting materials
into practical applications. The majority of theoretical works considered Al as fuel and CuO
as oxidizer for the following reasons: Al, available in both micron- and nano-sized particles, is
the preferred fuel due to its high energy density, sustainability, ease of handling and very good
stability thanks to its thin alumina coating. Its relatively low boiling point (2743 K), which is
lower than combustion temperatures, allows it to vaporize during combustion, thereby generating
gas and contributing to pressure buildup. On the other hand, CuO is a strong oxidizer, as hav-
ing a much lower Gibbs free energy (in absolute value) than aluminum oxide at all temperature
ranges (the corresponding Ellingham diagram is provided in Supplementary File S1), which also
provides effective pressurization while remaining stable, affordable, and readily accessible.

## 2. Overview of main engineering methods and key parameters influencing thermite reaction

One of the widely used technique for obtaining thermites is mechanical milling. Ultrasonic mixing of metallic and oxides powders is very simple. This involves adding metallic and oxide powders into an inert solvent, such as hexane or DMF (N,N-Dimethylformamide), and then using an ultrasonic probe or bath to stir the mixture, producing a homogeneous blend and reducing aggregates (Figure 3a). It is also possible to use arrested reactive milling (ARM), which involves mixing and mechanically grinding a mixture of metal and oxide macroscale powder. This technique is very interesting as it allows for the creation of a dense reactive material, composed of reactants (metal and oxide) with sizes between 50 and 100 nm (Figure 3b). The metal and oxide reactants are then in much closer contact, which is difficult to achieve with ultrasonic nanopowder mixing. The ARM process also has the advantages of being relatively simple and low-cost, with good versatility in terms of usable reactants.
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) SEM image of Al/Bi₂O₃ thermites obtained through ultrasonic mixing from [13]. (b) SEM image of Al/MoO₃ thermites prepared by arrested ball-milling from [27]. (c) TEM image of an Al/CuO multilayer made up of 15 bilayers, each 150 nm thick. In the TEM image, aluminum appears as the lighter color, while CuO is the darker color.</figcaption></figure>

Physical Vapor Deposition (PVD) techniques such as magnetron sputtering have been applied to the development of reactive films, by stacking of thin metal and metal oxide layers ranging from a few tens to hundreds of nanometers [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] (Figure 3c). The multilayered structure offers several advantages over powder mixing, making these systems of great interest from both a fundamental and application perspective: vacuum deposition allows for the production of highly pure, dense materials with precise control. The high reproducibility of fabrication makes these multilayer structures ideal model systems for investigating the physicochemical processes occurring during initiation or combustion reactions. Finally, PVD processes are largely compatible with MEMS technologies, making these multilayered reactive materials fully integrable into pyrotechnical microdevices and smart actuators [39, 35]. The limitations of these reactive thin films are that they are only available as thin films, thus in small quantities, restricting their applications to initiation.

Not only is the choice of fuel critical, but the following process and material parameters also significantly influence the reaction kinetics and combustion behavior of thermites :

- **Equivalence ratio (φ).** It is the ratio of the actual metal-to-oxidizer mass or molar ratio to the stoichiometric metal-to-oxidizer ratio required for complete combustion. Adjusting the equivalence ratio can significantly influence the ignition and combustion properties. For instance, a φ of 1.2 in Al/MoO₃ resulted in a maximum propagation rate of 880 m.s⁻¹ compared to 820 m.s⁻¹ at φ = 1 [40]. In the case of Al-based reactive films, several studies have shown that Al-rich configurations burn faster than stoichiometric ones [41, 42]. Burn rates can be enhanced by factor of 2 or more.

- **Reactants size.** Reactants size is also a great influencing factor. In a study published in 2004, Granier et al. [43] demonstrated this effect during the initiation of Al/MoO₃ nanoth-ermite powders by laser, for Al particle sizes ranging from 10 nm to several tens of microns. They observed that the initiation delay drastically increases by several orders of magnitude when the Al particles are micrometric in size. Furthermore, the authors also confirmed that the optimal equivalence ratio is φ = 1.2, i.e., slightly rich in Al, for which the initiation delays are the shortest. For what concerns reactive films, Bahrami et al. [38] found that a decrease in bilayer thickness from 1500 nm to 150 nm led to a significant increase in the burning rate from 2 m.s⁻¹ to 80 m.s⁻¹.

- **Bulk density.** Density is a crucial parameter often expressed as a percentage of the Theoretical Maximum Density (%TMD). For micron-sized thermites, it has been observed that
combustion velocity increases with bulk density, thereby enhancing thermal properties and energy transport [44]. However, it has to be noted that an increase in bulk density in nano-sized thermites leads to a decrease in combustion velocity. This reduction is attributed to the energy consumed in forming a reacted zone during compaction, a process known as reactive sintering [45]. This occurs due to the exothermic nature of thermite chemical reactions, which promote the fusion of the two components at their interfaces prior to the melting of aluminum. Consequently, less energy remains available for flame propagation, resulting in a slower flame speed.

- **Type of oxidizer.** CuO, MoO<sub>3</sub>, WO<sub>3</sub>, I<sub>2</sub>O<sub>5</sub>, Bi<sub>2</sub>O<sub>3</sub> and Fe<sub>2</sub>O<sub>3</sub> are the main commonly used oxidizers. A clear correlation has been observed between the ease of oxygen release from the oxidizer and the overall reactivity of the materials, thanks to the T-jump/TOF-MS [46] and STEM-EELS analytical techniques [47, 48, 49, 50, 51]. The high reactivity of the Al/CuO thermite is attributed to the facile oxygen release from CuO, in contrast to Fe<sub>2</sub>O<sub>3</sub> that liberates much less oxygen and leads to moderate reactivity; difficulty to release oxygen in ZnO causes Al/ZnO to be completely unreactive even though the reaction is overall exothermic.

- **Confinement.** Watson et al. [52] found that 50 nm Al particles combined with polytetrafluoroethylene (PTFE) and/or MoO<sub>3</sub> produces flame speeds on the order of 1000 m.s<sup>-1</sup> when confined against tens of m.s<sup>-1</sup> in open burning configuration.

- **Type of binder.** The type and quantity of binder, even though it is added in the minimum quantity, greatly affects the ignition and combustion performances. Meeks et al. [53] characterized the burn rate of thick Mg/MnO<sub>2</sub> thermite films mixed with different binders: PolyvinylideneFluoride (PVDF) – Methyl Pyrrolidone (NMP); Vitonfluoroelastomer (Viton A). Considering Viton binder, burn rates decrease with increasing binder concentration, with a maximum of 0.14 m.s<sup>-1</sup>. Using a polyvinylidene fluoride (PVDF) binder gives the highest burn rates (0.15 m.s<sup>-1</sup>) which Meeks et al. attributed to a more homogenous distribution of the reactants in the film. Shen et al. [54] characterized the burn rate of an Al/CuO nanothermite ink with particle loading as high as 90 wt% mixed in a PVDF (4 wt%) and hydroxy propyl methyl cellulose (HPMC) (6 wt%). The burn rate of the printed films is 2–10 cm.s<sup>-1</sup>, with flame temperatures of 2800 K. Faster burn rates (up to 25 cm.s<sup>-1</sup>) are found when PVDF is replaced by nitrocellulose and polystyrene.

Clearly, an exhaustive exploration of all the features that characterize the material structure, composition, processing conditions to optimize the combustion property for one application is simply experimentally impossible! The best approach is therefore to develop truly predictive physical models that can accurately capture structure–property relationships. This necessitates the development of various sub-models and theories to unravel the reaction mechanisms and key variables that govern them as well as a deep understanding of the physics and chemistry underlying the high-temperature, fast-propagating thermite combustion process. The extreme temperatures (over 3000 K) and rapid reaction rates pose significant challenges for characterization, requiring advanced techniques that are not easily accessible.

### 3. Methodological approach for thermite materials modelling

At the most refined level of theory, quantum mechanics such as DFT calculations open up the window to the chemistry of bulk, surface and interfaces of materials. Therefore DFT or ab-initio
MD (AIMD) permit the precise investigation of elementary reaction mechanisms, allowing researchers to map out pathways for metal oxidation, oxide decomposition and chemical reactions in condensed and gas phases. For example, Lanthony et al. [55, 56] unravelled the deposition of CuO in contact with an aluminum surface, taking into account alumina formation and Cu segregation processes. Jabraoui et al. [57] recently demonstrated that oxygen diffusion within CuO via oxygen vacancy bulk defects is the primary driver of CuO decomposition. AIMD simulations have proven very useful in studying the mixing dynamics of chemical species at metal/oxide interfaces [58, 59, 60, 61]. These simulations offered valuable informations into the atomic-scale interactions and energetics for each reaction mechanism. Such computational approaches are therefore the first step for advancing the knowledge for upper scale models.

To extend simulation capabilities to larger systems and longer timescales, e.g. thousands of atoms and a few nanoseconds duration, respectively, classical MD calculations (i.e. solving the Newton equations of motion) can be employed. Classical MD authorizes the study of collective behavior, enabling the extraction of more statistical data that can be compared with experiment, such as diffusion coefficients, phase transformation or thicknesses of intermixed layers. Unlike traditional classical potentials, ReaxFF potentials employ a bond-order formalism that explicitly models chemical reactions. This approach eliminates the need for iterative electronic minimization typically required in first-principles methods, offering a computationally efficient alternative for simulating reactive systems. A pioneering study by Chakraborty et al. [45] elucidate the reactive sintering of core-shell Al nanoparticles (each 8 nm in diameter) in contact under a rapid temperature increase of $10^{14}$ K.s$^{-1}$ using ReaxFF MD simulations. This mechanism, responsible for the reduction in burn rate observed in highly compacted reactive powders, was shown to be driven by an electrostatic field facilitating Al cation migration as the Al core melts. This phenomenon could not be experimentally elucidated. Another study using the same technique permitted to understand why voids at the interface of reactants in ball-milled powders modify the initiation process and made the powders more reactive [62]. These two examples highlight the key role of MD using reactive potentials, such as ReaxFF, in observing and rationalizing nano and micro scale behaviors that are inaccessible through experimentation.

Scaling up for interpreting macroscopic observations, such as pressure development in closed vessels, researchers traditionally rely on thermodynamic equilibrium models, employing thermochemical codes such as Cantera [63], CHEETAH [64] and NASA-CEA [65]. These tools enable the determination of the system's detailed chemical composition across both gaseous and condensed states, assuming thermodynamic equilibrium at a specified temperature and pressure. In essence, these models simulate idealized conditions, assuming that the reactants fully convert to products and that the system rapidly reaches a state of thermodynamic equilibrium. They are valuable for efficient screening and benchmarking of different thermite formulations in terms of their potential energy output. However they are not designed to capture the reaction kinetics and non-idealities of a real experimental condition that significantly influences pressure development and other key observables. To better replicate experimental conditions, more advanced models are required, ones that incorporate transient (time-dependent) behavior and account for spatially heterogeneous flow, heat and mass transfers effects. CFD approaches offer an ideal framework enabling the precise description of spatial heterogeneity and transport phenomena by modeling the spatial and temporal evolution of temperature, pressure, and species concentrations. This is achieved by solving the mass, momentum and heat conservation laws (Equations 1, 2, 3). These conservation laws can be coupled with both equilibrium chemistry models and detailed finite-rate chemistry models, offering versatility in capturing complex reaction dynamics.
$$ \partial_t(\rho\alpha) = -\nabla \cdot (\rho\alpha\mathbf{u}) + S_{\phi} \quad (1) $$

$$ \partial_t(\rho\alpha\mathbf{u}) + \nabla \cdot (\rho\alpha\mathbf{u}\mathbf{u}) = -\nabla \cdot (\alpha\mathbf{P}) + \mathbf{I}_{\text{source}} + \mathbf{S}_{\text{momentum}} \quad (2) $$

$$ \partial_t(\rho\alpha e) + \nabla \cdot (\rho\alpha\mathbf{u}e) = -\nabla \cdot \mathbf{q} + S_e \quad (3) $$

$\rho$, $\alpha$, $\mathbf{u}$ are respectively the density, the volume fraction and the velocity vector and $S_{\phi}$ a source or sink term associated with mass transfer rate due to chemical reactions and/or physical transformations. In the momentum transport equation (Equation 2), $\mathbf{P}$ is the pressure tensor, $\mathbf{I}_{\text{source}}$ is a source or interaction terms (e.g., interphase momentum exchange, body forces) and $\mathbf{S}_{\text{momentum}}$ is a source term related to specific processes (e.g., phase changes, additional forces). In the energy transport equation (Equation 3), $e$ is the specific internal energy, $\mathbf{u}$ is the velocity vector, $\mathbf{q}$ is the total heat flux vector, encompassing contributions such as radiation and conduction. $S_e$ is a source term, which represents additional energy contributions such as phase interactions, or external heating. In addition, a set of equations of evolution of species (Equation 4) can be added to model the evolution of gaseous and particulate species.

$$ \partial_t(\rho\alpha Y_i) + \nabla \cdot (\rho\alpha\mathbf{u}Y_i) = \nabla \cdot (D_i\nabla Y_i) + \sum_r \dot{\omega}_i^r \quad (4) $$

$Y_i$ and $D_i$ are the mass fraction and diffusion coefficient of species i, respectively. $\dot{\omega}_i^r$ are source terms representing the species consumed or produced due to reactions r leading to interphase mass transfer, i.e., condensation, vaporization or migration.

Common to all CFD combustion models is the coupling of mass transport, typically described by the Arrhenius equation, with heat and mass transfer processes to predict initiation and combustion behavior. One of the most comprehensive approaches was developed by Tichtchenko et al. [66, 67], which incorporates all known physico-chemical mechanisms in both condensed and gas phases. This recent combustion model also accounts for heat exchange, not only between gas and particulate phases but also between the two different particulate phases, the oxide and the metal, resulting in predictions that closely match experimental pressure data. While these CFD models are highly valuable for analyzing the gas phase composition or predicting pressure rise in constant volume condition, they have limitations. Specifically, because they are O-D models, they cannot apply for the prediction of the self-propagating combustion front, which requires at least one spatial dimension to resolve the dynamics of flame propagation (deflagration). Several analytical and numerical deflagration models do exist. The simplest analytical model is given by the Equation 5 [68, 69] to predict the burn rate of reactive multilayered films.

$$ v^2 = \frac{3RT_f^2\alpha^2}{\delta'^2E(T_f - T_0)}Ae^{-\frac{E}{RT_f}} \quad (5) $$

$T_f$ is the adiabatic temperature of the reaction (or flame temperature), $T_0$ is the initial temperature of the system, $\alpha$ is the effective thermal diffusivity of the multilayer, $E$ is the effective activation energy for the general metal+oxide reaction, $A$ is the Arrhenius pre-exponential factor, and $\delta'$ is the characteristic diffusion length. In multilayers, this quantity is commonly taken to be equal to 1/4 of the bilayer thickness $\delta$.

This model, based on a framework proposed by Armstrong et al. [70] for sandwiches of al-loyable materials such as bi-metallic, is rather simplified as it assumes constant thermophysical
properties and that reaction heat is only localized nearby the reaction front. Later, computational studies by Jayaraman et al. [69] improved it by including variable properties, phase changes, heat losses, and composition-dependent heat release. In these continuum gasless models, the in-termixing process between reactants is modeled as a Fickian diffusion process with temperature-dependent diffusivity, calibrated to experimental burn rates via an Arrhenius formulation :

$$D = k_0 \times \exp\left(\frac{-E_a}{k_B T}\right), \qquad (6)$$

$k_B$ is the Boltzmann constant and $T$ the temperature. The kinetic parameters $k_0$ and $E_a$ represent the pre-exponential factor and the activation energy, respectively.

Similar approaches have been proposed for reactive powders [71, 72]. However, these models restrict chemical reactions to condensed-phase processes and can not be generalized. Epps et al. [73] introduced a nondimensional continuum model incorporating gas advection within the porous thermite matrix using Darcy's law, keeping a single Arrhenius-based reaction with an approximate gas release, necessitating experimental calibration. In 2025, Tichtchenko et al. published a comprehensive Euler/Euler deflagration model for the self-propagating combustion in Al/CuO reactive powder [74]. This model represents the most advanced approach currently available, providing a detailed and accurate depiction of combustion dynamics for any powder configuration, including variations in density, stoichiometry, and other parameters. However, this approach requires macroscopic closure laws to describe the mass, heat and momentum exchanges between each phase. Additionally, incorporating the intricate thermo-physico-chemical mechanisms makes the simulations computationally intensive and time-consuming.

AI models were recently introduced in the field of thermite materials [75, 76, 77] proving to be powerful tools in designing and optimizing thermites. Neural networks or Gaussian processes can be trained on data from detailed simulations to create surrogate models that approximate complex physical behaviors. These surrogate models provide rapid predictions of key properties, such as burn rate, peak pressure, and combustion temperature, based on material composition and characteristics, enabling quick evaluations without the need for full simulations [75]. AI models can also analyze large datasets to identify critical features that influence performance. For instance, they can reveal how variables like particle size distribution, porosity, and metal-to-oxidizer ratios impact combustion efficiency and stability. Feature extraction methods, such as principal component analysis or feature selection algorithms, can highlight the most influential parameters, guiding researchers in focusing on key aspects of design [78]. By focusing on high-value experiments, active learning methodology can significantly reduce the total number of trials needed to optimize a thermite material formulation [76].

## 4. Review of key theoretical developments in the field of Al-based thermite combustion

This section proposes a selection of theoretical studies, chosen for their significant contributions to the science of thermites. A first section presents quantum mechanics and classical MD studies, which have enhanced the understanding of the mechanisms governing thermite combustion, both in the condensed and gaseous phases, as well as the key parameters that control these processes. Moving up in scale, we present seminal macroscopic models which enable to replicate key experiments to characterize energetic materials, such as hot-wire initiation and close-vessel pressure development. Next, we discuss recent models describing the flame front dynamics to predict the burn rate as a function of the characteristics of the materials. We conclude this section
with an overview of the first ML approaches developed for two different purposes : (1) infer ki-
netic parameters from experimental combustion curves, and, (2) select optimized thermites for a
given application. For each theoretical work presented in this section, the methodology is briefly
presented, and main results reviewed in connection with experimental observations to highlight
the contribution for thermite science and engineering.

4.1. *Combustion mechanisms investigation*

A widely accepted mechanistic pathway that drives the oxidation and combustion of Al par-
ticles, central reaction in reactive powders, was initially proposed by Rai et al. [79] and sum-
marized by Sundaram et al. in [80]. This process suggests that oxygen released by the oxidizer
diffuses through the alumina shell, thus creating a condensed phase media where aluminum and
oxygen react exothermically. In this mechanistic route, the reaction dynamics are temperature
and pressure-dependent. Below the melting point of aluminum, the oxidation process is kineti-
cally limited by the diffusion of oxygen across the alumina shell to the aluminum core, resulting
in a slow oxidation rate. Above the melting point, the aluminum core melts and the oxide layer
fractures allowing molten aluminum to flow outward. The molten aluminum reaches the particle
surface by flowing through the cracks and openings in the oxide layer. The ensuing reactions
and heat release result in the ignition of Al particle combustion. This transition significantly
accelerates the reaction kinetics. For example, the diffusion coefficient increases by two orders
of magnitude as the temperature increases from ambient to 2000 K [81]. Then, above the boiling
point, Al evaporates from the particle surface, and undergoes a vapor phase combustion process,
generating gaseous suboxides (AlₓOᵧ) upon reaction with gaseous oxygen released by the ox-
idizer which becomes metallic, and environmental O₂. A precise description of the aluminum
powder combustion process must include, in addition to gas-phase reactions:

- The condensation of liquid Al₂O₃ in the gas phase and its subsequent transport toward the liquid metallic droplet (comprising the reduced oxidizer and aluminum);

- The surface reactions of gaseous suboxides with liquid aluminum to form liquid Al₂O₃;

- The further dissolution of Al₂O₃ into the liquid aluminum core.

These main thermite powder combustion mechanisms are illustrated in Figure 4a.

In Al/CuO fully-dense materials such as powders prepared by ARM and reactive multilayers,
the exothermic redox reaction is restricted to condensed phase reactions at the CuO-Al interface
and is controlled by diffusion of O ions from decomposing CuO and counter-diffusion of Al ions
through the growing layer of Al₂O₃. Counter diffusion of Al is slower. As illustrated in schematic
diagram in Figure 4b, the reaction occurs at the Al/Al₂O₃ interface. Note that the interfacial
oxidation results in tensile stresses on the oxide layer [48]. The Al mass nearly doubles upon
oxidation.
<figure><img src="image_5.png" /><figcaption>Figure 4: Key mechanisms of Al/CuO combustion: reactive powder (a) and fully-dense system such as reactive film (b). Subscripts (c) stands for condensed.</figcaption></figure>

The key mechanisms of Al/CuO combustion have been extensively studied using DFT and MD simulations with the goal to calculate the energetics and kinetic parameters associated with the underlying mechanisms (e.g., oxide decomposition, interfacial layer growth, metal oxidation, gas-phase reactions, and species migrations).

### 4.1.1. CuO decomposition process

A study by Jabraoui et al. [57] investigated the decomposition of CuO and its phase transformation into Cu₂O, which is essential in thermite reactions where CuO acts as an oxidizer. The authors systematically quantified all pathways for bulk oxygen to migrate towards the surface, investigating all possible network directions in bulk and in the subsurface, as summarized in Table 1. Specifically, the authors considered: lattice oxygen migration to a neighbouring interstitial site (Frenkel pair defect creation); migration via interstitial sites; migration via oxygen vacancy sites.

Main finding is that the CuO to Cu₂O reduction is a two-step process: a partial reduction from monoclinic to orthorhombic structure occurs with 12.5% oxygen loss, and a final shift to cubic Cu₂O is nearly barrierless once 44-48% of oxygen atoms are removed. This phase change aligns CuO with Cu₂O structurally, facilitating oxygen migration and release. It was also found that decomposition requires defects, such as oxygen vacancies, to enable oxygen transport with manageable activation barriers (130 kJ.mol⁻¹), i.e., in the range of experimental observations [82]. Note that the authors distinguished lattice sites, which correspond to perfect and periodic sites of the crystal network (fcc lattice for aluminum for example) from the interstitial site that is defined as a possible metastable position of an additional atom located within lattice sites. To this respect, the Frenkel pair defect typically occurs when an atom is displaced from its regular lattice site and moves to an interstitial position, creating a vacancy at its original site location. Interstitial, Frenkel pairs, and vacancies are classified as point defects, in contrast to extended defects like grain boundaries, which involve hundreds, thousands, or even millions of atoms. The reader may refer to [83] for more details. A second requirement is that the migration is operated in specific crystalline directions, particularly along the [110] direction. Approaching the surface (subsurface migration), these activations are lowered down to 96 kJ.mol⁻¹. But the overall kinetic limiting factor of the decomposition process was finally found on the capability of the CuO to release O₂ from its surface, requiring an additional 148 kJ.mol⁻¹ activation energy. The calculations are performed considering the (100) surface orientation, as the (111) gives prohibitive activations for realizing O₂ desorption. These findings align well with results from ultrahigh vac-
uum studies using in-situ infrared spectroscopy and mass spectrometry [84], the latter estimating an activation energy of 126 kJ.mol⁻¹ for decomposition.

<table><caption>Table 1: Formation and activation energies associated with oxygen transport in CuO. * Vacancy and divacancy type of defects are referenced to the perfect (or reconstructed) surface, and creation of half or one O₂ molecule, respectively; ** activation referred to as the energy required to desorb O₂ from its adsorbed position.</caption><thead><tr><th>CuO System</th><th>Mechanism</th><th>Activation energy (kJ.mol<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="2">Bulk</td><td>Frenkel defect formation</td><td>463</td></tr><tr><td>Migration (O interstitial)</td><td>274</td></tr><tr><td>Surface vacancy/peroxy bridge</td><td>Monovacancy formation</td><td>318</td></tr><tr><td rowspan="2">Surface (111)</td><td>Divacancy formation*/O₂ release**</td><td>320**</td></tr><tr><td>Surface vacancy/peroxy-bridge</td><td>126</td></tr><tr><td rowspan="2">Surface (001)</td><td>Monovacancy formation</td><td>425</td></tr><tr><td>Divacancy formation*/O₂ release**</td><td>148**</td></tr></tbody></table>

It has to be noted that the decomposition temperature of CuO depends on the external oxygen pressure. In the case of thermite, where CuO powders are mixed with Al particles, the CuO decomposition occurs earlier than it would under ambient air conditions. Simply because, the process of oxygen diffusion through the Al₂O₃ layer occurring when the temperature reaches 573 K, reduces the oxygen pressure at around the CuO surface.

## 4.2. Oxygen diffusion investigation

As highlighted in the introduction to this section, below aluminum's melting point, the oxidation process is kinetically constrained by the diffusion of oxygen through the Al₂O₃ shell, a process driven by mechanisms involving defects and self-interstitials. Table 2 provides an extended view beyond Al-based thermites of such mechanisms of calculated diffusion coefficients reported in the literature, along with the corresponding migration mechanisms. These values highlight the diversity of diffusion behaviors, emphasizing the role of material-specific properties and microstructural factors in governing the reaction kinetics.

<table><caption>Table 2: Review of published atomic oxygen diffusion coefficients for selected metallic oxides relevant to thermite combustion. Note: KMC refers to Kinetic Monte Carlo simulations. All materials are assumed to be perfect crystals, meaning that the defect under consideration for investigating diffusion is isolated from the interaction of potential other defects.</caption><thead><tr><th>Materials</th><th>Mode</th><th>Level of theory</th><th>Prefactor (m<sup>2</sup>·s<sup>-1</sup>)</th><th>Activation energy (kJ.mol<sup>-1</sup>)</th><th>Reference</th></tr></thead><tbody><tr><td colspan="6">Final metallic oxide</td></tr><tr><td rowspan="3">Al<sub>2</sub>O<sub>3</sub></td><td>O vacancy</td><td>DFT</td><td>/</td><td>357</td><td>[85]</td></tr><tr><td>charged O vacancy</td><td>DFT</td><td>/</td><td>122-376</td><td>[86]</td></tr><tr><td>/</td><td>Metadynamics</td><td>7.88 × 10<sup>-2</sup></td><td>511</td><td>[87]</td></tr><tr><td rowspan="3">ZrO<sub>2</sub></td><td>Interstitial</td><td>DFT/KMC</td><td>6.5 × 10<sup>-7</sup></td><td>197</td><td>[88]</td></tr><tr><td>O vacancy</td><td>DFT</td><td>/</td><td>163</td><td>[89]</td></tr><tr><td>Statistics</td><td>Classical MD</td><td>4.18 × 10<sup>-13</sup></td><td>21</td><td>[90]</td></tr><tr><td rowspan="2">TiO<sub>2</sub></td><td>Oxygen vacancy</td><td>DFT</td><td>8.60 × 10<sup>-6</sup></td><td>311</td><td>[91]</td></tr><tr><td>O interstitial</td><td>DFT</td><td>/</td><td>57</td><td>[92]</td></tr><tr><td>B<sub>2</sub>O<sub>3</sub></td><td>Self-diffusion</td><td>Classical MD</td><td>/</td><td>169</td><td>[93]</td></tr><tr><td colspan="6">Oxidizer</td></tr><tr><td rowspan="2">CuO</td><td>O vacancy</td><td>DFT</td><td>/</td><td>134</td><td>[57]</td></tr><tr><td>O vacancy</td><td>DFT</td><td>/</td><td>107</td><td>[94]</td></tr><tr><td rowspan="2">Fe<sub>2</sub>O<sub>3</sub></td><td>Interstitialcy</td><td>DFT</td><td>/</td><td>183</td><td>[95]</td></tr><tr><td>O vacancy</td><td>DFT</td><td>/</td><td>531</td><td>[95]</td></tr><tr><td>WO<sub>3</sub></td><td>O vacancy</td><td>DFT</td><td>/</td><td>13.5</td><td>[96]</td></tr><tr><td>MgO</td><td>O interstitial</td><td>DFT</td><td>/</td><td>140</td><td>[97]</td></tr></tbody></table>
It is experimentally observed that the activation energy for lattice diffusion of oxygen in Al₂O₃ ranges from 239 to 595 kJ.mol⁻¹, with prefactors varying between 10⁻¹² and 10² m².s⁻¹ [98]. These discrepancies highlight the intricate nature of diffusion pathways, which are influenced by factors such as charge localization, doping, and defect densities. Yang et al. [99] have explored the impact of charged defects, showing a significant reduction in the activation barrier for oxygen migration to as low as 122 kJ.mol⁻¹ (versus 376 kJ.mol⁻¹ for neutral vacancies), underscoring the critical role of the detailed atomistic arrangement and local electronic properties in governing diffusion mechanisms.

ZrO₂ and TiO₂ exhibit considerably low activation barriers for oxygen diffusion, making Zr and Ti fuel very reactive and much less stable than Al [50, 100, 49]. These materials benefit from a consistent and efficient diffusion mechanism via interstitial sites, further emphasizing their potential in enhancing thermite performance.

Considering now the main metallic oxidizers, WO₃ shows the lowest activation for oxygen migration (together with CuO), via oxygen vacancies, with a value as low as 114 kJ.mol⁻¹. Note that all considered oxidizers exhibit lower activation barriers values for oxygen migration, compared to that of the passivation layers, indicating the prevalence of the latter passivation layer as the limiting and dominant factor of the combustion kinetics. Still, Fe₂O₃ shows the highest activation value (183 kJ.mol⁻¹), almost in the range of the one determined for passivation layers. This is consistent with experimental findings showing ignition difficulties. It is to be emphasized that diffusion coefficients are not confidently known and understood, because of the uncertainty on the real materials microstructuring. From the range of activations and prefactors calculated for oxygen migration/diffusion through oxide layers, we clearly see that alumina barrier layer offers the highest value, partly being responsible for a good trade-off between thermal stability and reactivity.

### 4.3. *Al/CuO interface growth chemistry*

Upon heating, prior reaching the melting point of aluminum, aluminum ions begin to diffuse through the alumina or interfacial layer, triggering reactions within the layer. This process, referred to as pre-ignition process, is highly reactive and ultimately leads to the auto-ignition of the combustion. Two noteworthy MD simulation studies have characterized the intermixing process in fully-dense reactive materials, specifically examining the behavior under conditions of perfect contact between oxidized Al and CuO submitted to high heating ramp.

Liu et al. [58] investigated the interfacial intermixing process mechanisms in Al/Al₂Oₓ/CuO reactive films considering only interfacial oxygen migration. The model-system consists of a 2.2 nm Al/CuO film, including a 0.6 nm amorphous alumina (a-Al₂O₃) layer between Al and CuO. A fixed temperature ranging from 600 to 1000 K was applied to the system. The MD simulations predict that the O migration at the a-Al₂Oₓ/Cu₃O₄ and Al/a-Al₂O₃ interfaces contribute to localized heat generation. These reactions occur prior to the decomposition of CuO, challenging previous common assumptions that CuO decomposition is the primary ignition step. Figure 5 shows that, as expected, at ambient conditions, the two rate-limiting steps in the interfacial layer growth are the O migration in Al₂O₃ (436 kJ.mol⁻¹) and the O penetration into the Al layer (208 kJ.mol⁻¹). The low-barrier and exothermic O migrations into the a-Al₂O₃ are kinetically favorable and contribute to the internal heat generation within the solid phases as previously mentioned. Subsequently the temperature rise promotes the decomposition of CuO and accelerates the O migration across the Al/a-Al₂O₃/Cu₃O₄ system which initiates the self-sustained reaction.
<figure><img src="image_6.png" /><figcaption>Figure 5: Barrier diagram of O migration through the Al/CuO nanolayers. The forward and backward migration barriers in different models were averaged separately to ensure the statistics. Both the O migration barriers in the amorphous form and the most stable α form were used to denote the lower limit and upper limit respectively. The values in parenthesis represent the O diffusion barriers in liquid-state Al and Al₂O₃. Figure from Ref. [58].</figcaption></figure>

***

Jabraoui et al. [62] conducted a study examining the influence of nanopores at Al/CuO interfaces on the interfacial intermixing process and, consequently, on the initiation mechanisms. Using MD simulations, the research analyzed the chemical and structural changes at Al/CuO interfaces between 400 and 900 K for interfaces with and without voids. The authors employed LAMMPS MD software, using ReaxFF forcefield parameters that allowed studying Cu, O and Al diffusion and chemical interactions. Dense and porous thermite models were created, with the latter containing a 1 nm pore at the interface filled with free oxygen molecules. Amorphous alumina thickness was set at 1 nm. The systems underwent heating from 200 K to 900 K and atomic diffusion, alumina layer growth, and phase transitions were quantified to assess differences in reaction behavior between the porous (with one void at the Al-CuO interface) and dense configurations.

Results reveal that after annealing at 700 K (Figure 6) in both porous and dense systems, alumina grows into the Al region upon heating, the pure aluminum grabbing oxygen atoms from the alumina network being in contact. However, in the dense system, a ternary aluminate phase forms at temperatures above 650 K, consisting of Al, Cu, and O. This phase restricts diffusion and reduces reactivity. In contrast, in the porous system, this aluminate layer is absent, allowing for greater Al and O diffusion, which enhances the reactive alumina layer's growth. The growth of the Al-rich AlₓOᵧ in place of the initial a-Al₂O₃ barrier layer is not only caused by the diffusion of aluminum ions towards the interface with either the pore or CuO, but also by the counter diffusion of oxygen atoms towards the pure aluminum, and their reactions. Indeed, the metallic Al ions, i.e., those initially belonging to the pure metallic region, attract oxygen atoms from the deepest layer of the native a-Al₂O₃ given the Coulomb long-range interaction established between both regions. When intimate contact between CuO and a-Al₂O₃, the oxygen atoms belonging to the deepest atomic layer of the native a-Al₂O₃, i.e., those interfacing a-Al₂O₃ and CuO, are also influenced by the CuO presence, which inhibits their participation in the creation of the reactive AlₓOᵧ compared to what is observed for the porous system. This is the main
reason of the difference in the growth rate and composition of the AlₓOᵧ formed in porous and
dense systems.

<figure><img src="image_7.png" /><figcaption>Figure 6: Penetration depth (ΔZ) into the native alumina region of Al atoms initially belonging to the metal upon annealing, with reference being made to the surface of AlₓOᵧ in contact with CuO or air for dense (a) or porous (b) systems, respectively. (c-d) snapshots highlighting the penetration level of Al initially belonging to the Al region (in violet) after the 700 K annealing, for the dense system (left) and the porous system (right). Figures from [62].</figcaption></figure>

This research provides valuable insights into the pre-ignition mechanisms of dense Al/CuO
thermites, highlighting the critical role of voids that can form at the reactants interface upon
mixing. It also calculated the diffusion coefficients for different temperatures for both Al and O
species in the different AlₓOᵧ layers and in comparison with a-Al₂O₃ (Figure 7). The presence of
voids allows for a more reactive alumina layer with an enhanced Al and O diffusion, reducing the
ignition threshold temperature and promoting faster reaction kinetics as observed experimentally
[42, 101, 102].
<figure><img src="image_8.png" /><figcaption>Figure 7: Log diffusion coefficients as a function of reciprocal temperature for Al (a) and O (b) in a-Al₂O₃, and the different AlₓOᵧ regions grown upon annealing in the porous and dense system. Figures from [62].</figcaption></figure>

### 4.3.1. DFT investigation of the gas phase thermochemistry

In Ref. [103], a detailed thermochemical study of aluminum oxidation in the gas phase has been proposed. 76 elementary reactions were investigated for AlₓOᵧ intermediates and for each, DFT calculations were performed to obtain molecular structures of intermediates. For each intermediate, the energies, enthalpies, and free energies were calculated as well as the activation barriers associated to each reaction paths. Interestingly, the study covers the formation of large size AlₓOᵧ molecules for the first time. The authors concluded that AlO, Al₂O₃, and Al₂O₃ are the important intermediates to form Al₂O₃ or Al₂O₄ as precursors for gaseous Al₄O₆. This detailed reaction model will be useful to apply to any experimental analysis studying Al combustion, although all the 76 reactions are not necessary.

### 4.3.2. DFT investigation of the gas-surface thermochemistry during Al combustion

During the combustion at temperature above 3000 K, even though chemical reaction predominantly occurs in the vapor phase, reactions of gaseous species, atomic or molecular suboxides can also take place. These species may subsequently react on the surface. Glorian et al. [104, 105] introduced a set of specialized heterogeneous chemical equations to describe the reactions of gaseous aluminum suboxides (AlₓOᵧ) with the Al burning surface. In [106], the authors have extended the investigation by Glorian et al. dedicated to pure Al systems to surfaces of interest for the Al/CuO thermite, i.e. Cu and Al₂O₃. Using static DFT calculations, they systematically investigated the thermodynamics of adsorption for gaseous O₂, Cu, Al, AlO, Al₂O, Al₂O₂, Al₂O₃ and Al₂O₃ on pure Al(111), Cu(111) surfaces, as well as on crystalline and amorphous Al₂O₃ ones [107]. Note that other processes such as gas phase nucleation (of alumina smoke) and maturation of droplets and their subsequent condensation are not accounted for in current thermite combustion models while they are well known processes in the combustion of aluminum [108, 109].

Results show that all considered Al suboxide species exhibit strong and barrierless adsorption energies (in the range 190 to 480 kJ.mol⁻¹) without requirement of any molecular cooperative ef-
fect at the surface. On pure aluminum surfaces, the reactivity is significantly higher than on pure copper surfaces. Investigation of the energetics of species spatial separation on the surface, it is demonstrated that dissociation products are spread over the surface on Al, whereas they remain localized at the adsorption location on the Cu surface. When accounting for entropy corrections to the adsorption energy, an overall lowering of the adsorption intensity is found. Reaching the temperature of ~3000 K, this lowering is such that the adsorption becomes unlikely on copper surfaces (Figure 8).

<figure><img src="image_9.png" /><figcaption>Figure 8: Adsorption free energy of gaseous molecular species on Cu(111) and Al(111) as a function of temperature. Figure from [107].</figcaption></figure>

On the Al₂O₃ surface, interestingly, an epitaxial type of growth is observed, driven by the unsaturated aluminum sites on the surface. As for the pure metallic surfaces, this barrierless adsorption process is characterized by large adsorption energies (in the range -675 to -1450 kJ.mol⁻¹ for AlₓO⁻y suboxides). These results largely complete the condensation reactions utilized in combustion model by extending the reactions to all surfaces of interest beyond pure aluminum. They also provide all thermokinetic data to quantify properly each reaction, on each surface.

### 4.4. Modelling initial Al/CuO reaction stage to interpret fast heating ignition findings and sin- tering

Another set of theoretical work aimed to model the initial reaction phases leading to the onset of the self-sustained thermites combustion, considering both the multilayered film or powder- based thermites. All these studies focus exclusively on condensed phase mechanisms as evi-
denced by Piekel et al. [110], particularly the diffusion of oxygen and aluminum, the latter being triggered around its melting. For thermite powders, Rai et al. [79] proposed a simple model based on earlier work by Dalla Torre et al. [111] considering the combination of Al and O diffusion assisted by a pressure gradient in the Al₂O₃ interface. They calibrated an effective diffusion coefficient representing the net effect of the movement of both Al and O through the oxide layer for nano-sized particles as :

$$D = 4.8691 \times 10^{-11} \exp\left(-\frac{4240.1}{T}\right) m^2 \cdot s^{-1}, \quad \text{for } 900 \text{ K} \le T \le 1200 \text{ K.}$$

Other studies, such as [112, 14, 113], introduced the diffusion of each species in their model, also accounting for the decomposition of CuO, with all these mechanisms governed by a time-dependent Arrhenius law. The kinetic parameters, e.g. (k, Eₐ) couples, characterizing the kinetics of CuO, oxygen and aluminum diffusion are (1.16 × 10⁻⁶ m²s⁻¹, 67.3 kJ.mol⁻¹K⁻¹), (9 × 10⁻⁵ m²s⁻¹, 140 kJ.mol⁻¹K⁻¹) and (1 × 10⁻⁸ m².s⁻¹, 110 kJ.mol⁻¹K⁻¹), respectively.

A paper by Brotman et al. [113] presents an interesting numerical model for studying the initiation of Al/CuO thermites submitted to high heating rates, with the goal to analyze two types of experiments, published by Egan et al. in [114] : (1) A laser-based ignition of Al/CuO particles under high heating rate (10¹¹ K.s⁻¹) and observed through DTEM. (2) A Joule based ignition of Al/CuO particles under slower heating rate (10⁵ K.s⁻¹) under vacuum within a mass spectrometer enabling the understanding of the early stage of oxide decomposition and initiation.

The methodology is based on a diffusion-reaction scheme that simulates the redox reaction between two spherical particles, Al and CuO, in contact with each other (Figure 9). The effects of sintering is characterized by a contact angle between Al and CuO particles that allow extending the contact surface at the beginning of the reaction. Note that diffusion parameters are assumed to be equivalent for both solid and liquid phases, as in most of this type of approaches. Importantly the model focuses on the transport of oxygen through condensed layers and subsequent oxidation of aluminum at the Al/Al₂O₃ interface. The mathematical approach follows the Deal and Grove model [115], adapted here to a spherical geometry, a method commonly applied to model the planar thermal oxidation of silicon, and the Brotman's model simulates how the contact angle affects initiation kinetics.

Figure 10 plots the species' temporal evolution, which gives a clarifying view of how the overall combustion scenario operates as a function of temperature. The first reaction step corresponds to the release of oxygen from CuO (black curve), which leads to aluminum oxidation (slight and continuous Al drop, see orange curve). Concurrently, Cu₂O is generated as CuO decomposes (blue curve). During the sharp temperature increase (step 2), the Cu₂O oxygen reservoir is rapidly consumed, giving rise to pure copper (red curve), alumina formation (green curve), and a sharp drop in Al. Note that at this high stoichiometric ratio (metal rich), only a small percentage of the Al is consumed after 1 µs.
<figure><img src="image_10.png" /><figcaption>Figure 9: Schematic views illustrating the modeled Al:CuO systems: (a) Al:CuO system geometry and (b) zoomed-in view of the oxygen concentration across the various layers created upon redox reaction in the boldly outlined box in (a). $b_1, a_0, b_1, d_1, d_2$ are input parameters. Figure from [113].</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: The evolution of the species' molar concentrations as a function of time, for the reaction at $T_o = 1500$ K and $\theta = 135^\circ$: left side, for all species, right side, a zoom of the CuO, $Cu_2O$, $Al_2O_3$ and Cu region, on the period of high reactivity. Figures from [113].</figcaption></figure>

The authors confirmed theoretically that at very high heating rates, sintering significantly influences the reaction initiation. Experimental results indicate that, under these conditions, initiation times can reach the nanosecond range when CuO particles are very small (around 50 nm), likely due to the rapid coalescence of particles and increased interfacial area. When heating rates are slower, sintering has a reduced impact, and initiation delays are observed in the millisecond range. This suggests that, under slower heating, the condensed-phase processes alone can drive initiation without a strong influence from particle sintering.

### 4.5. *Constant-volume combustion modelling*

Accurately predicting the combustion of a thermite in a closed vessel necessitates accounting for the multiphasic phenomena driving the reaction. Indeed, the extremely high combustion temperatures due to pressure rise, often reaching 3000-5000 K, triggers numerous physical transformation (melting, vaporization, nucleation/condensation) during the combustion process in addition to gas-phase chemical reactions. Addressing these complexities demand pushing the
boundaries of common physico-chemical models towards more complex CFD-based theoretical approaches.

### 4.5.1. CFD models coupling condensed- and gas-phase reaction kinetics

Baijot et al. [14] developed a first comprehensive model of the combustion of Al/CuO powered reactive materials in constant volume, with the goal to: (1) predict the temporal evolution of the temperature, pressure, and gaseous species, and, (2) investigate the influence of particle size, compaction, alumina shell thickness, and stoichiometry on combustion properties (pressure rise, pressure and temperature kinetics). In this model, the reaction between Al and oxygen (O and O₂) is accounted for both in the gas phase and on the particle surface. For doing so, the system is brought to a temperature of 1200 K that is sufficient to allow for the transport of Al ions through the alumina shell, towards the Al particle outer surface, from where they can either be oxidized in contact with the gas, thus thickening the oxide shell, or vaporize. In the latter case, aluminum reacts with various gaseous species, including atomic or molecular oxygen and molecular aluminum sub-oxide species (Al₂O₃, AlO, Al₂O₃, AlO₂), that are created via an equilibrium estimate of the gas phase upon oxidation of atomic aluminum in the gas phase. The decomposition of CuO into gaseous O₂ and condensed Cu was explicitly treated, alongside the formation of condensed Al₂O₃ on the surfaces of both reduced CuO and oxidized Al particles. This last process is treated though a simplified scheme summarized in Table 3. It considers the recombination of gas-phase species to form Al₂O₃ condensed units on the particle surfaces. Produced condensed Al₂O₃ units can then dissolved into the molten metals. The atomic richness of sub-oxides was determined based on the stoichiometry given by the Equation (7).

<table><caption>Table 3: Condensation model; surface reactions of sub-oxides based on mixture richness. (c) and (g) stand for condensed (liquid or solid) and gas, respectively.</caption><thead><tr><th>O rich</th><th>Al rich</th></tr><tr><th>φ<sub>AlxOy</sub> &lt; 1</th><th>φ<sub>AlxOy</sub> &gt; 1</th></tr></thead><tbody><tr><td>Al<sub>2</sub>O<sub>(g)</sub> + O<sub>2,(g)</sub> → Al<sub>2</sub>O<sub>3,(c)</sub></td><td>Al<sub>2</sub>O<sub>(g)</sub> → <sup>4</sup>&frasl;<sub>3</sub>Al<sub>(g)</sub> + <sup>1</sup>&frasl;<sub>3</sub>Al<sub>2</sub>O<sub>3,(c)</sub></td></tr><tr><td>AlO<sub>(g)</sub> + <sup>1</sup>&frasl;<sub>4</sub>O<sub>2,(g)</sub> → <sup>1</sup>&frasl;<sub>2</sub>Al<sub>2</sub>O<sub>3,(c)</sub></td><td>AlO<sub>(g)</sub> → <sup>1</sup>&frasl;<sub>3</sub>Al<sub>(g)</sub> + <sup>1</sup>&frasl;<sub>3</sub>Al<sub>2</sub>O<sub>3,(c)</sub></td></tr><tr><td>Al<sub>2</sub>O<sub>2,(g)</sub> + <sup>1</sup>&frasl;<sub>2</sub>O<sub>2,(g)</sub> → Al<sub>2</sub>O<sub>3,(c)</sub></td><td>Al<sub>2</sub>O<sub>2,(g)</sub> → <sup>2</sup>&frasl;<sub>3</sub>Al<sub>(g)</sub> + <sup>2</sup>&frasl;<sub>3</sub>Al<sub>2</sub>O<sub>3,(c)</sub></td></tr><tr><td>AlO<sub>2,(g)</sub> → <sup>1</sup>&frasl;<sub>4</sub>O<sub>2,(g)</sub> + <sup>1</sup>&frasl;<sub>2</sub>Al<sub>2</sub>O<sub>3,(c)</sub></td><td>AlO<sub>2,(g)</sub> + <sup>1</sup>&frasl;<sub>3</sub>Al<sub>(g)</sub> → <sup>2</sup>&frasl;<sub>3</sub>Al<sub>2</sub>O<sub>3,(c)</sub></td></tr></tbody></table>

$$ \phi_{AlxOy} = \frac{\frac{1}{2} \sum_{i \in AlxOy} \frac{\Phi_{i,g \to p}}{W_i} n_i^{Al}}{\frac{1}{3} \sum_{i \in AlxOy} \frac{\Phi_{i,g \to p}}{W_i} n_i^{O}} \quad (7) $$

$W_i$ and $\Phi_{i,g \to p}$ are the molar mass and flux (from the gas phase to the particle surface) of species i (i being possibly Al₂O₃, AlO, Al₂O₃, AlO₂). $n_i^{Al}$ and $n_i^O$ are the number of Al and O atoms in the species i.

To reduce computational complexity, several approximations were introduced. First, the system was assumed to be fully thermalized at all time steps, assuming that both the particles and the gas are at the same temperature. Second, the gas phase was considered at its thermodynamic equilibrium within each time step, assuming that gas-phase reactions occur on a much faster timescale than condensed-phase processes, and other processes that impose the simulation time-step. This model was developed and tested to predict the pressure development in closed vessel experiments from a purely "out of equilibrium" perspective (Figure 11).
<figure><img src="image_12.png" /><figcaption>Figure 11: Total pressure generated by the combustion of a stoichiometric mixture made of Al (100 nm) and CuO (240 nm) nanoparticles, prepared at different percentages of the TMD: 10, 20, 30, 40 and 50% from [14]. Comparison of renormalized simulated results with experimental combustion data published in [13].</figcaption></figure>

Maximum pressure of 47 MPa and temperature of 3500 K is obtained for stoichiometric Al/CuO at high compaction (50% of TMD). Pressure drops to 4.2 MPa for low compaction and Al-rich mixtures, simulations align well with pressure-time profiles for stoichiometric Al/CuO mixtures (red dots in Figure 11).

Tichtchenko et al. [67] have extended Baijot's contribution for Al/CuO powdered reactive materials with two key improvements: (1) the new proposed model incorporates three distinct phases representing the aluminum particulate phase, the CuO particulate phase, and the gas phase, respectively. These phases exchange both energy and mass among themselves, and, (2) the gas phase is modeled using a fully kinetic approach, enabling a more detailed and dynamic representation of gas-phase reactions. However, in this model the diffusion of Al is not considered, and oxygen, in turn, is allowed to diffuse through the alumina shell (again here, diffusion is considered to be equivalent in both solid and liquid phase). A summary of the reactions in both particulate and gas phases is given in Tables 4 and 5.

<table><caption>Table 4: Gas phase chemical reactions with associated kinetic parameters, defined as $AT^n e^{-B/T}$. X corresponds to the gaseous species Al, Cu, O, $O_2$, $Al_2O_3$, AlO, $Al_2O_3$, $Al_2O_3$, $N_2$ [116].</caption><thead><tr><th>Chemical reaction</th><th>A<br>(cm<sup>3</sup> mol<sup>-1</sup> s<sup>-1</sup>)</th><th>B<br>(K)</th><th>n</th></tr></thead><tbody><tr><td>Al + O<sub>2</sub> ↔ AlO + O</td><td>9.7 × 10<sup>13</sup></td><td>80.5</td><td>0</td></tr><tr><td>AlO + O<sub>2</sub> ↔ AlO<sub>2</sub> + O</td><td>4.6 × 10<sup>14</sup></td><td>10008</td><td>0</td></tr><tr><td>Al<sub>2</sub>O<sub>2</sub> ↔ AlO + AlO</td><td>1.0 × 10<sup>15</sup></td><td>59335</td><td>0</td></tr><tr><td>Al<sub>2</sub>O<sub>2</sub> ↔ Al + AlO<sub>2</sub></td><td>1.0 × 10<sup>15</sup></td><td>74937</td><td>0</td></tr><tr><td>Al<sub>2</sub>O<sub>2</sub> ↔ Al<sub>2</sub>O + O</td><td>1.0 × 10<sup>15</sup></td><td>52466</td><td>0</td></tr><tr><td>Al + O<sub>2</sub> ↔ AlO + O</td><td>1.0 × 10<sup>15</sup></td><td>44564</td><td>0</td></tr><tr><td>Al + O<sub>2</sub> ↔ AlO + Al</td><td>1.0 × 10<sup>15</sup></td><td>67035</td><td>0</td></tr><tr><td>Al + O + X ↔ AlO + X</td><td>3.0 × 10<sup>17</sup></td><td>0</td><td>-1</td></tr><tr><td>O<sub>2</sub> + X ↔ O + O + X</td><td>1.2 × 10<sup>14</sup></td><td>54244</td><td>0</td></tr></tbody></table>
<table><caption>Table 5: Heterogeneous chemical reactions. (g) and (s) stand for gas and particulate phase respectively.</caption><thead><tr><th>Reaction</th><th>Mechanism</th><th>Mathematical Implementation</th></tr></thead><tbody><tr><td>2Al<sub>(s)</sub> + 3/2O<sub>2(s)</sub> → Al<sub>2</sub>O<sub>3(s)</sub></td><td>Diffusion-Reaction</td><td>Spalding/spontaneous reaction</td></tr><tr><td>2Al<sub>(s)</sub> + 3O<sub>(g)</sub> → Al<sub>2</sub>O<sub>3(g)</sub></td><td>Diffusion-Reaction</td><td>Spalding/spontaneous reaction</td></tr><tr><td>Al<sub>2</sub>O<sub>3(s)</sub> → 2Al<sub>(g)</sub> + 3/2O<sub>2(g)</sub></td><td>Decomposition</td><td>Arrhenius</td></tr><tr><td>CuO<sub>(s)</sub> → Cu<sub>(s)</sub> + 1/2O<sub>2(g)</sub></td><td>Decomposition</td><td>Arrhenius</td></tr><tr><td>Cu<sub>(s)</sub> ↔ Cu<sub>(g)</sub></td><td>Evaporation-Condensation</td><td>Spalding/Clapeyron</td></tr><tr><td>Al<sub>(s)</sub> ↔ Al<sub>(g)</sub></td><td>Evaporation-Condensation</td><td>Spalding/Clapeyron</td></tr><tr><td>Al<sub>x</sub>O<sub>3(g)</sub></td><td>Condensation</td><td>Spalding/spontaneous reaction</td></tr></tbody></table>

The simulations reveals distinct combustion behaviors for nano-and micro-sized thermites, highlighting the importance of particles' size and specific surface area in combustion dynamics:

- Nano-sized thermites (100 nm particles). The combustion process begins with the decomposition of CuO, releasing oxygen gas, which is absorbed by the surface of Al particles. This leads to a condensed-phase combustion regime where oxygen diffuses through the alumina shell and reacts with the aluminum core. As the reaction progresses, the system transitions to a gas-phase combustion regime driven by gaseous reactions involving Al and Cu species. This phase leads to rapid pressure and temperature increase, reaching an equilibrium state at approximately 4300 K and 22 MPa after 1.8 seconds. The model also reveals the formation and growth of an alumina shell on Al particles, which initially hinders oxygen diffusion but eventually liquefies at high temperatures, allowing aluminum to evaporate and react in the gas phase.

- Micro-sized thermites (5 µm particles). Due to the larger particle size and smaller surface area (0.26 $km^2.m^{-3}$ against 12 $km^2.m^{-3}$ for nano-Al particles), the thermite system exhibits a faster initial pressurization phase, followed by a gradual decrease as the reaction stabilized. The micro-sized thermites achieve a steady-state regime after 8–12 µs, depending on the stoichiometry, compared to just a few microseconds for an Al/CuO nano-sized thermites.

In summary, the simulations confirm that particle size plays a crucial role in the pressure development in a closed chamber. It also identifies two distinct combustion regimes, condensed-phase combustion in the initial stages and gas-phase combustion in later stages. This transition is more pronounced in nano-sized thermites due to their higher surface-to-volume ratio. The model's predictions align closely with experimental data, capturing key features like pressure peaks and equilibrium states, though some discrepancies in pressurization rates remain as some kinetic parameters are still unknown. Importantly, this model represents a key step as it is the first model capable of generating QSPRs for pressure development, despite the lack of some mechanisms such as the Al diffusion.

## 4.6. Modelling the flame dynamics

### 4.6.1. Gasless models to predict thermal front propagation in fully dense reactive material such as multilayered film

From the combustion theory for sandwiches of alloyable materials [70], several improved models were proposed to take into account for the presence of substrate for example, as well as the reaction path and growth kinetics in CuO/Al multilayer films [117, 118, 112]. In [112], the authors present a 1D+1D diffusion-reaction approach, i.e., species diffuse in the direction perpendicular to the layering (along $z$ in Figure 12), assuming same diffusion parameters for solid
and liquid phases, while heat is transported along the layering (along *x* in Figure 12). The transport of oxygen species is treated as fluxes, i.e., it does not assume a particular mode of migration (capillarity through grain boundaries, Fickian process, electromigration...). The model incorporates the key mechanisms involved in the Al/CuO reaction: the decomposition of CuO in two steps, oxygen diffusion through the amorphous Al₂O₃ (a-Al₂O₃), and, the spontaneous exothermic oxidation of Al when Al is in contact with oxygen. CuO decomposition and oxygen diffusion are governed by a time-dependent Arrhenius law. Activation energies and prefactors for O diffusion into a-Al₂O₃, Cu, and Cu₂O were taken as follows: Eₐₐ Al₂O₃ = 140 kJ.mol⁻¹ and Eₐₐ Cu = Eₐₐ Cu₂O = 67.3 kJ.mol⁻¹, kₐₐ Al₂O₃ = 9 × 10⁻⁵ m².s⁻¹ and kₐₐ Cu = kₐₐ Cu₂O = 1.16 × 10⁻⁶ m².s⁻¹.

The model also accounted for bilayer thickness, reactant ratios, interface properties, and substrate thermal conductivity, allowing for a detailed analysis of how these factors impact ignition and reaction propagation. The model also introduced the concept of "ceiling temperature" (set at 2070 K, corresponding to Cu₂O's vaporization point), at which disintegration occurs before complete combustion can propagate through the film. This ceiling temperature is found as a critical factor influencing the material's energy output and the uniformity of the reaction front.

<figure><img src="image_13.png" /><figcaption>Figure 12: Schematic views illustrating the modeled Al/CuO multilayered system: ignition is obtained by the application of a power density at one extremity of the film. Then, the reaction front propagates (a) from the left to the right side of the film. (b) A magnified view of the reaction zone showing the different species diffusing through interfacial layers, and (c) a detailed view of the composition of one bilayer. Figures from [112].</figcaption></figure>

Figure 13 shows typical temperature profiles along the reaction propagation direction (x) at various selected times. The first two temperature profiles, successively captured at 100 and 130 µs, correspond to the ignition phase, i.e., the temperature increase in the non-heated section is only due to heat diffusion from the heated section. No chemical reaction has begun in the non-heated film section. The temperature increases over a width of 0.4 mm adjacent to the heated section, which corresponds to the heat diffusion length over the ignition time. The rest of the film remains at ambient temperature.
<figure><img src="image_14.png" /><figcaption>Figure 13: Temperature profiles at selected times and the instantaneous reaction front velocity. The multilayer is composed of ten 300 nm 1:1 Al/CuO bilayers deposited on a polymer substrate, with thermal conductivity set at 0.1 W.m⁻¹.K⁻¹. Figure from [112].</figcaption></figure>

The model allows establishing ignition thresholds, with lower ignition power and substrate thermal conductivity favoring ignition. Thicker bilayers improve energy delivery to substrates, while ultra-thin bilayers achieve higher propagation speeds due to increased surface contact. Reactant ratios impact reaction speed, as aluminum-enriched films dissipate heat faster, reducing flame velocity. Comparisons with experimental data show good agreement as shown in Figure 14. Note that prior calibration of the diffusion coefficients was performed from experiments.

<figure><img src="image_15.png" /><figcaption>Figure 14: Experimental ignition times as a function of the ignition power density and comparison with the theoretical curve (solid line). Figure from [112].</figcaption></figure>
This model has also been applied to predict the burn rate of various Al/CuO reactive films and theoretical predictions were compared to experimentations (Table 6).

<table><caption>Table 6: Experimental vs theoretical burn rates from [112]. BL stands for bilayer.</caption><thead><tr><td>Reactive films Sample</td><td>Experimental Burn Rate (mm.s<sup>-1</sup>)</td><td>Gasless Model (mm.s<sup>-1</sup>)</td></tr></thead><tbody><tr><td>5 BL Al/CuO 50/100 nm</td><td>3.7 ± 0.3</td><td>4.32</td></tr><tr><td>5 BL Al/CuO 100/200 nm</td><td>2.95 ± 0.3</td><td>3.04</td></tr><tr><td>10 BL Al/CuO 50/50 nm</td><td>11.00 ± 1.5</td><td>6.83</td></tr><tr><td>10 BL Al/CuO 100/100 nm</td><td>4.35 ± 0.5</td><td>5.26</td></tr><tr><td>10 BL Al/CuO 200/200 nm</td><td>2.5 ± 0.1</td><td>3.53</td></tr></tbody></table>

The simulations mostly exhibit higher burn rate values than those obtained from the experimental measurements, with the exception of one Al rich film (Al/CuO 50/50 nm). Most discrepancies (< 20%) can be explained by the theoretical approximation as the model only considers oxygen diffusion despite the fact that the actual reaction mechanisms are likely far more complex, involving coupled effects of both O and Al diffusion through the Al₂O₃ interfacial layers (with equivalent parameters for solid and liquid phases). The Cu₂O vaporization temperature itself could be slightly modified in the context of ultrathin sputtered layer.

That is why further refinement is recommended at a minimum, to account for the complex diffusion behaviors of both oxygen and aluminum in layered films, aiming to develop more accurate deflagration models. A more recent and complete work by Tichtchenko et al. [119] presents a 2D (along x and y axes) and non-stationary numerical model to simulate the combustion behavior of Al/CuO multilayer films. This model considers both the heat transport along x and y and species (oxygen and aluminum) diffusion mechanisms along y to predict reaction velocity and flame front behavior under various conditions, such as film thickness, stoichiometry. The 2D feature of the model permits to include impurities in the thickness of the film and analyze their influence of the reaction propagation which was not possible with simplest 1D models presented here before.

The diffusion coefficient of oxygen and aluminum into the interfacial alumina is considered equivalent, governed by a time-dependent Arrhenius law with activation energy of 73.6 kJ.mol⁻¹ and prefactor of 7.1 × 10⁻⁶ m².s⁻¹. Although this (Eₐ, k) set is empirical, i.e., fitted from a series of experimental results performed on several Al/CuO multilayers, it is different from those taken by Lahiner et al. in [112] (Eₐ,₀ = 140 kJ.mol⁻¹ and k₀ = 9 × 10⁻⁵ m².s⁻¹), pointing to the difficulty of describing and quantifying mass transport in these condensed based models, and questioning on their exact physical meaning.
<figure><img src="image_16.png" /><figcaption>Figure 15: Simulation result of a 4.5 µm thick Al/CuO multilayer. Temperature and alumina concentration are superimposed over the temperature field and scaled to fit on the detached (right) axis; see legend for scale factors. The Al/CuO bilayer thickness is 300 nm and the equivalence ratio is 1. T<sub>f</sub> is fixed at 2843 K. Figure from [119].</figcaption></figure>

The image of Figure 15 gives a simulation result. It represents the temperature field where purple is cold (300 K) and yellow is hot (T<sub>f</sub> = 2843 K). Superimposed over the heat map are the y-average reaction profile of temperature, and y-average alumina concentration, which share the same x-axis but are scaled to fit on the detached (right) axis. Features of interest in these types of simulations include the steady reaction front propagation velocity, v, and the thermal profile of the reaction front. The width w<sub>f</sub> of the reaction front is determined as the distance between the points where the temperature reaches 95% of the maximum temperature (T<sub>f</sub>) and 105% of the minimum temperature (ambient). The graphs of Figures 16a and 16b show the influence of number of bilayer (1 and 40) and bilayer thickness (50 nm to 500 nm) on the reaction velocity, v, respectively. We observe a direct relationship between the reaction front velocity and the number of bilayers. For bilayer number greater than 20, the velocity reaches 44 m.s<sup>-1</sup>. This distance reduction decreases the reactive characteristic time and hence increases the front velocity.

<figure><img src="image_17.png" /><figcaption>Figure 16: (a) Simulation of front reaction velocity as a function of (a) number of bilayers (bilayer thickness is fixed at 300 nm), and (b) bilayer thickness (number of bilayer is fixed at 15). The square markers correspond to experiments. For both curves, reactant ratio is equal to 1 and T<sub>flame</sub> is fixed at 2843 K. Figures from [119].</figcaption></figure>

This study demonstrated that reaction front velocity inversely correlates with flame front
width, in agreement with laminar flame theory. This research also highlights the importance of temperature limits and the role of heat dissipation in maintaining reaction stability, suggesting potential improvements in thermite material design and modeling approaches.

Additionally, this 2D deflagration model permits to study the influence of metallic impurities inserted in the thermite. Figure 17b shows how inert metallic particles embedded in the thermite layer affect thermally the reaction. A metallic particle with high thermal conductivity initially accelerates the reaction but later acts as a heat sink, slowing the reaction front locally.

<figure><img src="image_18.png" /><figcaption>Figure 17: (a) Schematics of the modeled Al/CuO reactive film in which a Au particle (1.4 µm × 1 µm) is included within the Al/CuO stack (4.5 µm in thickness by 30 µm in length). Yellow and blue colors correspond to CuO and Al, respectively. (b) 2D temperature maps showing the reaction front (light green thin zone) passing across the Au particle. The Al/CuO bilayer thickness is fixed at 300 nm, the equivalence ratio is 1, and T<sub>flame</sub> = 2843 K. Figures from [119].</figcaption></figure>

4.6.2. Gasless models to predict thermal front propagation in dense reactive powders

Brotman et al. [120] and DeSouza et al. [71] developed 1D gasless deflagration model to predict thermal front propagation in Al/CuO and Al/Fe₂O₃ reactive powders, specifically for thermal plugging applications [121]. In [71], the model-system is a disk partly filled with an Al and Fe₂O₃ thermite that is considered as porous medium for which pores are filled with a non reactive gas. The energy conservation equation is coupled with the species conservation equation. The exothermic thermite reaction, included by a source term in the energy equation, follow a single-step reaction described by an Arrhenius equation, very close to that of Kim et al. [72]. Activation energy (158 kJ.mol⁻¹) and prefactors (2.2 × 10⁶ s⁻¹) were selected empirically by calibrating the model to obtain experimental burn rates and temperature measurements [122]. Note that the heat
generated and the reaction kinetics are determined solely by the material stoichiometry. Factors such as reactant size and compactness are not accounted for in the reaction term. However, the model accounts for the temperature-dependence of the thermodynamic properties and also the phase changes involved in the reaction process. Other assumptions are: (1) once Al melts, the mixture turns to be fully-dense, i.e., porosity disappears, (2) Fe₂O₃ decomposition controls the reaction; (3) the convection and diffusion of the species are disregarded simplifying the model formulation. The finite difference methodology was used to solve the equations.

<figure><img src="image_19.png" /><figcaption>Figure 18: Temperature contours and profile as the reaction propagates. Figure from [71].</figcaption></figure>

Figure 18 illustrates the temperature wave propagation simulated after the calibration of the model. The reaction wave is a narrow region with high-temperature gradients and it takes less than 1 s to sweep the entire domain (1/4 disk). Then, the temperature decreases to 1000 K in approximately 50 s. The peak temperature is found to be 2325 K, corresponding to Al₂O₃ vaporisation, which is in accordance to observations.

These 1D gasless models, whether developed for thin films or powders, are too simplified from a mechanistic perspective to be predictive for any application domains. They are each spe-
cialized, restricted to either a specific experimental or for a material configuration. However, they are computationally efficient, and when calibrated, they can capture general trends, making them useful for engineers. Nevertheless, they cannot be considered accurate enough to be generalized or to derive QSPRs.

### 4.6.3. *Heterogeneous models to predict thermal front propagation in reactive powders*

**Flame front propagation in Al/CuO powders.** One continuous flame propagation model incorporating all known and documented mechanisms involved in the Al+CuO reaction has been developed very recently [74]. The model-system is a cylindrical tube filled with an homogeneous mixture of Al and CuO particles in air (N₂, O₂). The combustion front propagates to the right, along the *x* axis, at the speed *v*. The left and right boundary conditions can be opened or fully-closed. The initial Al particle purity, the powder compaction, the particles' diameter, and stoichiometric ratio (Al over CuO ratio) are the main input parameters defining the thermite. An Euler-Euler approach similar to that used in [123, 124] was employed to model the interaction between the reactive particulate phases and a gas mixture. It involves solving the mass, momentum, and energy transport equations (Equations 1, 2, 3) in the three separate phases. Additional equations of species mass fraction were employed to model the evolution of gaseous and particulate species (Equations 4). A last set of equations were used to predict the average diameter of each particulate phase. Importantly, the different phases were coupled through specific closure models that account for interphase mass, momentum, and energy transfers. Both conductive and radiative heat transfer terms are considered between the gas phase and the particulate phases of Al and CuO. Supplementary File S2 details the heat transfer models that were implemented for both conductive and radiative transfers. A finite volume method facilitated the numerical resolution of the equations. At each time step, the temperatures of the particulate phases and the gas phase, the gas pressure, and the composition of all three phases are computed. Therefore, the thermophysical properties of the phases are recalculated at each time step, as they depend on the composition, temperature, and pressure. This model has been first applied to predict the burn rate of several Al/CuO reactive powders (varying particle sizes) contained in quartz tubes opened at both ends. The model's predictions closely matched the experimental burn rates for nano-and micro-sized thermites, although less accurately for large micron-sized particles (Table 7). For large particles, the reactions are limited due to the slower diffusion and increased sintering. These condensed phase issues are still poorly described in the model, that mostly concentrate on gas phase as well as gas/surface reactions, thus explaining this discrepancy.

<table><caption>Table 7: Experimental vs theoretical burn rates. Experimental and numerical details can be found in [74].</caption><thead><tr><td>Thermites sample</td><td>Experimental burn rate (mm.s<sup>-1</sup>)</td><td>Deflagration Model (mm.s<sup>-1</sup>)</td></tr></thead><tbody><tr><td>nAl/CuO</td><td>660 ± 75</td><td>648</td></tr><tr><td>1-μAl/CuO</td><td>73 ± 15</td><td>75.6</td></tr><tr><td>5-μAl/CuO</td><td>12 ± 5</td><td>16</td></tr><tr><td>10-μAl/CuO</td><td>0.01 ± 0.01</td><td>8</td></tr></tbody></table>

The model can also analyze the physics of the self-propagating flame front of micron sized metal rich and highly compacted Al/CuO thermite powder, i.e., Al and CuO size of 10 µm, stoichiometry equal to 1.2, powder density equal to 0.5, and Al purity (Al:Al₂O₃ molar ratio within the Al particle) is set at 80%. No heat loss are considered. It is observed that the flame is predominantly gas-phase, with temperatures at around 6000 K (Figure 19b) and major products being copper and aluminum (not shown). The unreacted zone is compressed particulate region
ahead of the flame front with no reaction initiation. Figures 19a and 19b show the volume fraction, temperature, and pressure of each phase.

<figure><img src="image_20.png" /><figcaption>Figure 19: Spatial evolution along the tube axis of (a) the mass fraction of the different phases (gas, Al and CuO), (b) the temperature of the different phases (gas, Al and CuO) and gas pressure ($P_g$). The tube is opened at both ends, and reactants size = 10 µm. Figures from [74].</figcaption></figure>

Results also shows that the particle size has the most substantial effect on burn rate, with smaller particles yielding faster propagation due to the increased amount of surface available for exothermic reactions with the reactive gas phase. Compaction density and aluminum purity also influence burn rates, while stoichiometry has minimal impact in ambient air condition.

This model is typically at the forefront of what is needed for *in silico* thermite design. However, the primary limitation of this model lies in very high computational time, as simulations can take several days to even weeks when applied to nano-sized particles.
## 4.7. ML based models

As previsouly mentionned ML based models will be crucial not only for bridging the gap between accurate physical models, which are computationally intensive, and the efficient QSPRs required by engineers for design purposes but also for offering algorithms capable of calibrating quickly unknown parameters using available data. Two published works have been selected to illustrate the effectiveness and potential of ML in this context. One ML based algorithm was developed as inverse models to calibrate kinetic parameters that can not be calculated from experimental data (subsection 4.7.1). And, one active learning algorithm was developed to reduce the total number of trials needed to optimize a material formulation to a given application as illustrated in subsection 4.7.2.

### 4.7.1. Inference of kinetics and thermodynamic properties from constant-volume combustion of energetic materials

The pressure time history captured in closed vessel can be utilized to determine the ignition delay time, maximum pressure, and pressurization rate, which are commonly used to quantitatively evaluate the performance of energetic materials. Additionally, these parameters can also provide qualitative insights into activation energy, combustion efficiency (from maximum pressure), and reactivity (from pressurization rate). However, such pressure-based performance metrics are highly dependent on experimental conditions, posing challenges for fair comparisons of energetic material performance across different laboratories. Specifically, the maximum pressure and pressurization rate are influenced by the volume of the constant-volume reactor and the mass of energetic materials loaded. The article by Kim et al. [77] proposed an original framework to infer kinetic and thermodynamic properties by maximizing the information extracted from time-history pressure profiles obtained through constant-volume combustion tests. First, a physical model was developed to predict thermodynamic states during the constant-volume combustion process. Subsequently, an inverse model was designed to evaluate the sensitivity of the physical model outputs to its parameters. This dual approach enables the calibration of kinetic and thermodynamic properties by simultaneously solving the physical model and analyzing its sensitivity to the parameters. The methodology is to consider $\frac{dy}{dt} = f(y(t), t|\Theta)$ as a dynamical system describing the combustion progress in the pressure cell, where y is a vector identifying the state of the system (pressure, temperature, and species concentrations) and Θ is a vector of model parameters (kinetic and thermodynamic properties). Thus, modeling of kinetic and thermodynamic properties will be achieved by calibrating the kinetic parameters Θ of which the pressure profile from the physical model best fits the experimental measurements. $\Theta^* = \min L(P_{predicted}(\Theta), P_{experimental})$ where P is the pressure, and L is the loss function that evaluates the deviation between the model and the experimental data over the entire time domain. Adjoint-sensitivity analysis is known as an efficient numerical method for sensitivity computation of large-scale differential or algebraic systems [125, 126].

The authors applied this framework to the Al/CuO thermite system. For validation, they calculated the pressure evolution of various Al/CuO reactive powders under constant-volume conditions using the inferred parameters and compared the predictions with experimental data. The observed discrepancies were less than 10% demonstrating its ability to accurately infer parameters like reaction rates, gas release, and combustion efficiency. Using the pressure profile obtained in closed-vessel experiments, the authors inferred reaction rate constants and combustion efficiencies for different equivalence ratios. They observed that reactivity peaked under stoichiometric conditions ($\phi = 1.0$), while combustion efficiency was highest under metal-rich
conditions ($\phi$ = 2.5). By decomposing the pressure into contributions from gas generation and thermal expansion, the study revealed that thermal expansion accounts for 99% of the total pressure in small-scale setups. This insight emphasizes the need for precise reactor designs to capture gas-generation performance accurately.

### 4.7.2. *Design of reactive powders through active learning methodologies*

The main goal of one research published in 2024 [76] was to propose a systematic methodology to customize thermites to meet specific user requirements. In this study, the authors implemented an active learning methodology using a custom acquisition function to sample the design space prioritizing regions of interest (ROI) as defined by the user, in terms of desired combustion properties. More specifically, the proposed approach employed Gaussian Process Regression (GPR) that allows both exploration, by substituting the physical model by a GPR based surrogate, and GPR uncertainty determination that partly allows guiding the exploration. This iterative process allows probabilistic predictions of key thermite properties (combustion temperature and maximum pressure in a closed vessel) starting from a limited initial dataset of 50 points. The GPR is at the core of a more complex acquisition function that combines three components: (1) an interest term that directs sampling towards the ROI, (2) an exploration term that leverages the GPR model's uncertainty to investigate less-sampled regions, and, (3) an incentive term that encourages exploration of under-sampled areas, ensuring a diverse spread of samples across the design space.

The algorithm workflow includes (1) initializing a dataset with Latin Hypercube Sampling (LHS), (2) training the GPR model to predict combustion temperature and pressure, and, (3) using the custom acquisition function to suggest new sample points iteratively, balancing exploitation and exploration to refine the design space.
<figure><img src="image_21.png" /><figcaption>Figure 20: Schematic diagrams illustrating the active learning strategy for efficiently selecting targeted thermite compositions. Red triangles represent points that are outside the ROI, while blue circles indicate points of interest.</figcaption></figure>

The study evaluated the methodology across two application cases:

- Thermal plugging. Targeting temperatures of 2000-4500 K and pressures of 1-2 MPa, the algorithm identifies optimal Al/CuO compositions. Using this approach, 67% of the sampled points fall within the ROI after 300 samples, a significant improvement over the 6.3% ROI identification using non-guided methods. The spatial distribution indicates that the algorithm efficiently samples Al/CuO thermites with Al radii below 3 µm and CuO radii covering the entire range from 100 nm to 10 µm.

- High-energy actuation. For applications requiring extreme conditions (5800-7000 K and pressures of 40-56 MPa), the algorithm effectively identifies compositions despite narrow parameter constraints. The active learning algorithm achieves a relatively broad coverage within the design space, selecting Al/CuO mixtures that maximize oxygen release, which is critical for high-pressure environments. It yields a 30% identification rate of points within the ROI, compared to only 6% without guiding the selection.

This work underscores the effectiveness of an active learning-based approach in exploring and optimizing thermite compositions with targeted properties. The custom acquisition func-
tion enables precise, efficient sampling within complex, multi-dimensional design spaces. The method holds promise for advancing energetic materials research, with potential applications across defense, pyrotechnics, and industrial welding.

**5. Future perspective: advancing computational modeling to drive future innovations**

Despite two decades of research into the science of aluminum-based reactive materials, sig-
nificant gaps in understanding persist. In poorly dense powders, the redox reaction between a
solid fuel and oxidizer is primarily governed by the migration of oxygens and counter diffusion
of Al ions. This process begins with oxygen release from the oxidizer, followed by their trans-
port through the reactive interface and medium, culminating in the formation of the final oxide
product. However, as the temperature approaches the boiling point of aluminum, several critical
questions remain unanswered: how do sub-oxides behave on the burning metal? What series
of elementary mechanisms drive the production and spatial localization of liquid aluminum ox-
ide after reaction of elemental molecular sub-oxides species onto the burning droplet? How can
the dissolution of condensed Al₂O₃ units into liquid aluminum be modeled? What is the im-
pact of the alumina-to-aluminum ratio in the burning particle on the kinetics of the dissolution
process? Furthermore, sintering of agglomerated nanoparticles during the reaction is accompa-
nied by melting and vaporization processes, which significantly influence the burning rate [26].
Therefore, in addition to the diffusion-driven mechanism, other phenomena play crucial roles
in the reaction dynamics, such as thermomechanical stress. During the combustion at high-
temperature, i.e., several thousand of K, additional knowledge gaps hinder the accurate descrip-
tion of the combustion process: does turbulence occur within certain regions of the flame? If so,
at what scale? What are the roles of particles size increase arising from sintering, and aggrega-
tion/agglomeration during combustion? What is responsible for the observed irregularities in the
flame front? These bring to question about the precise content of macroscale models, including
reactive particle-particle and gas-particle interactions governing the mass, momentum and en-
ergy flux transfer during the combustion.

In the future, with the growing capabilities of computational resources and processing power, it is therefore essential to develop models with molecular and atomic-level precision to incorporate all mechanisms as accurately as possible while being capable of simulating one or a few particles in a reactive gas containing Al suboxides and potentially vaporized metals. This approach will enable the elucidation of condensation mechanisms at the surface and further oxide diffusion/convection and reaction within the liquid aluminum. Additionally, such comprehensive models will reveal the impact of the solid or gaseous environment on the diffusion of reactive species. In parallel, it is essential to pursue the work on single burning Al particle experiments varying particle size down to the nanometer if possible, to guide the theoretical findings. Pan-toya's work [127, 128] is highly innovative in this area, featuring dedicated experiments aimed at observing, with very high temporal resolution, how the reaction progresses in the gas and liquid phases.

At the sub-millimeter scale (~500 µm to remain in the continuous fluidic conditions), boundary-
resolved transient Direct Numerical Simulation (DNS) approach based on a single-phase formal-
ism of the Navier-Stokes equations coupled with the detailed gas and condense-phase chemistry
could offer an interesting mean for the theoretical description of reactive gas- reactant particles
mass and heat exchanges during the combustion. Such models will therefore be also critical to
calibrate closure laws and provide accurate input data for larger-scale models using today empir-
ical laws.

In complement, the development of particle-resolved numerical techniques, like Lattice Boltz-
mann Method (LBM) allows to perform DNS of the reacting flow across dilute or dense particles
arrays. This theoretical approach will make it possible to directly compute the forces acting on
particles, allowing for the determination of parameters such as drag coefficients (representing
momentum transfer) as a function of Reynolds numbers and particle volume fractions.

Another key challenge in the field of thermites and related applications is the establishment
of inverse design tool, i.e., taking the targeted property (burn rate, flame temperature, pressur-
ization rate, ...) as input to obtain an output of structural material information. As illustrated
in section 4.7.2, active learning or multi-objective Bayesian optimization are highly interesting
technique to develop a dynamic strategy to guide the exploration of the design space towards
user-defined specific regions (Figure 20). But, while AI models hold great potential for ad-
vancing thermite combustion research or to build design tools, their generalizability hinges on
comprehensive datasets, rigorous validation, and integration of domain knowledge. Ongoing re-
finement and cross-validation with experimental and theoretical results are essential to ensure
reliable application across diverse thermite systems. We believe that hybrid approaches, combin-
ing AI with physics-based models can leverage the strengths of both approaches : AI's ability to
uncover patterns in data and physics-based models' adherence to fundamental laws. This hybrid
approach has the potential to improve the generalizability of thermite combustion studies.

**6. Conclusion**

A body of thermite modelling investigations were reviewed focusing on two main aspects: the
foundational modeling efforts that have significantly advanced the understanding of the mech-
anisms governing the ignition and combustion of aluminum-based thermites, and the compu-
tational studies that have marked a decisive step in the development of QSPRs for these fast
reacting materials; this latter models being of course, based on the former. This article also
emphasizes the important role of newly introduced AI models in thermite science especially for
design purpose, which demonstrated promising research routes. It also highlights some issues
for their generalization. Finally, this review points to the need for interdisciplinary efforts par-
ticularly between fluid dynamicists, condensed matter physicists and chemists to realize the full
potential of these versatile energetic materials for applications.

**7. Acknowledgement**

The authors acknowledge financial support of ERC PyroSafe under grant No.832889. Carole
Rossi acknowledges many stimulating discussions with Olivier Simonin, Hicham Jabraoui and
Matthieu Jonckheere.

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