*Article* **Kinetic Model and Experiment for Self-Ignition of Triethylaluminum and Triethylborane Droplets in Air**

**Sergey M. Frolov 1,2,\* , Valentin Y. Basevich <sup>1</sup> , Andrey A. Belyaev <sup>1</sup> , Igor O. Shamshin <sup>1</sup> , Viktor S. Aksenov <sup>2</sup> , Fedor S. Frolov <sup>1</sup> , Pavel A. Storozhenko <sup>3</sup> and Shirin L. Guseinov <sup>3</sup>**


**Abstract:** Triethylaluminum Al(C2H<sup>5</sup> )3 , TEA, and triethylborane, B(C2H<sup>5</sup> )3 , TEB, are transparent, colorless, pyrophoric liquids with boiling points of approximately 190 ◦C and 95 ◦C, respectively. Upon contact with ambient air, TEA, TEB, as well as their mixtures and solutions, in hydrocarbon solvents, ignite. They can also violently react with water. TEA and TEB can be used as hypergolic rocket propellants and incendiary compositions. In this manuscript, a novel scheme of the heterogeneous interaction of gaseous oxygen with liquid TEA/TEB microdroplets accompanied by the release of light hydrocarbon radicals into the gas phase is used for calculating the self-ignition of a spatially homogeneous mixture of fuel microdroplets in ambient air at normal pressure and temperature (NPT) conditions. In the primary initiation step, TEA and TEB react with oxygen, producing an ethyl radical, which can initiate an autoxidation chain. The ignition delay is shown to decrease with the decrease in the droplet size. Preliminary experiments on the self-ignition of pulsed and continuous TEA–TEB sprays in ambient air at NPT conditions are used for estimating the Arrhenius parameters of the rate-limiting reaction. Experiments confirm that the self-ignition delay of TEA–TEB sprays decreases with the injection pressure and provide the data for estimating the activation energy of the rate-limiting reaction, which appears to be close to 2 kcal/mol.

**Keywords:** triethylaluminum; triethylborane; oxygen intrusion reaction; rate constant; activation energy; droplet; self-ignition delay; formation of radicals; detailed kinetics; computational code

**1. Introduction**

Triethylaluminum Al(C2H5)<sup>3</sup> (TEA) and triethylborane B(C2H5)<sup>3</sup> (TEB) are transparent, colorless, pyrophoric liquids with boiling points of approximately 190 ◦C and 95 ◦C, freezing points of approximately −46 ◦C and −93 ◦C, and densities of 0.832 and 0.677 g/cm<sup>3</sup> (at 25 ◦C), respectively [1,2]. Their solutions remain stable when stored away from heat sources in a dry, inert atmosphere, but, at elevated temperatures, they slowly decompose to form hydrogen, ethylene, and elemental aluminum and boron. Upon contact with air, TEA and TEB and their solutions in hydrocarbon solvents ignite. They also react violently with heated water [3–5]. TEA, TEB, and their solutions should only be handled under a dry, inert atmosphere such as nitrogen or argon. TEA is used as a component of the Ziegler–Natta catalyst for the polymerization of olefins [6–8]. It is also used in reactions with ethylene for the growth of hydrocarbon radicals at the aluminum atom and, with the subsequent hydrolysis of the resulting higher aluminum alkyls, to obtain fatty a-alcohols. In addition, TEA is used as an alkylating agent in the synthesis of other organoelement and organic compounds. TEB is used in organic chemistry as an initiator in low-temperature radical reactions [9], in the deoxygenation of alcohols [10], and in other processes. Both

**Citation:** Frolov, S.M.; Basevich, V.Y.; Belyaev, A.A.; Shamshin, I.O.; Aksenov, V.S.; Frolov, F.S.; Storozhenko, P.A.; Guseinov, S.L. Kinetic Model and Experiment for Self-Ignition of Triethylaluminum and Triethylborane Droplets in Air. *Micromachines* **2022**, *13*, 2033. https://doi.org/10.3390/mi13112033

Academic Editor: Pingan Zhu

Received: 23 September 2022 Accepted: 18 November 2022 Published: 21 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

TEA and TEB are used as a hypergolic rocket propellant, in napalm and incendiary compositions [1,11,12], as well as in micropropulsion [13] and microrobotics [14]. The SpaceX Falcon 9 rocket is known to use a TEA–TEB mixture as a first- and second-stage hypergolic ignitor [15]. TEA–TEB mixtures were also used for motor ignition in the Atlas and Delta commercial launch vehicles. According to [16], "Triethylaluminum market valued at 225.1 million USD in 2020 is expected to reach 255.2 million USD by the end of 2026, growing at a Compound Annual Growth Rate of 1.8% during 2021–2026".

In experiments with liquid TEA sprayed through a nozzle in air in the form of microdroplets, spontaneous combustion of the mixture is observed. Complete combustion of TEA in air corresponds to the overall reaction [17]

$$2\text{Al}(\text{C}\_2\text{H}\_5)\_3 + 2\text{IO}\_2 \rightarrow \text{Al}\_2\text{O}\_3 + 12\text{CO}\_2 + 15\text{H}\_2\text{O} + Q\_A \tag{1}$$

where *Q<sup>A</sup>* is the heat of reaction (1). The value of *Q<sup>A</sup>* is approximately 2444 kcal [2], 2293 kcal [18], and 1955 kcal [19]. The latter value is based on the quantum-mechanical calculation of the structures and energy characteristics of all molecular complexes involved in the TEA self-ignition process. Complete combustion of TEB in air corresponds to the overall reaction

$$2\text{B(C}\_2\text{H}\_5)\_3 + 2\text{1O}\_2 \rightarrow \text{B}\_2\text{O}\_3 + 12\text{CO}\_2 + 15\text{H}\_2\text{O} + Q\_B \tag{2}$$

where *Q<sup>B</sup>* is the heat of reaction (2). The value of *Q<sup>B</sup>* is approximately 2100 kcal [20], i.e., it is close to the value of *QA*. In a complex chemical process of transformation of the initial components into the products of overall reactions (1) and (2), in which the reacting components participate in many heterogeneous and gas-phase elementary reactions, two stages can be distinguished: the stage of self-ignition and the stage of rapid explosive combustion [21]. In applied terms, the kinetic analysis of the former stage seems to be the most important, since it is this stage that determines the time of the entire process. According to the literature, the self-ignition of TEA and TEB in air occurs through radical reactions [17,22,23]. In the primary initiation step, TEA and TEB react with oxygen, producing an active ethyl radical, which can initiate an autoxidation chain in competition with termination or other pathways.

This work deals with the development of a theoretical model and preliminary experimental studies of the first stage, i.e., the self-ignition. Based on the model, the self-ignition delays of TEA–TEB droplets in air at normal pressure and temperature (NPT) conditions are calculated, whereas experiments are used for estimating the Arrhenius parameters of the rate-limiting reaction. For the sake of definiteness, the kinetic analysis is performed for TEA. However, the model proposed herein can be directly applied to the self-ignition of TEB and TEA–TEB mixtures.

### **2. Materials and Methods**

#### *2.1. Kinetic Model of TEA Droplet Self-Ignition in Air at NPT Conditions*

For a kinetic analysis of the self-ignition stage, it is necessary to draw up a scheme of elementary reactions. By definition, the primary reaction in the scheme should be a heterogeneous reaction that occurs when oxygen molecules available in air collide with TEA droplets. Such a reaction, leading to the self-ignition of a mixture occupying a limited volume, should be characterized by a sufficiently low activation energy. Presumably, this may be a heterogeneous reaction of the intrusion of an O<sup>2</sup> molecule to TEA with the formation of the (C2H5)2Al–O–O–(C2H5) molecule directly in the collision of TEA and O<sup>2</sup> molecules

$$\text{Al}(\text{C}\_{2}\text{H}\_{5})\_{3} + \text{O}\_{2} = (\text{C}\_{2}\text{H}\_{5})\_{2}\text{Al} \cdot \text{O} - \text{O} - (\text{C}\_{2}\text{H}\_{5}) + \text{Q} \tag{3}$$

or by forming an intermediate complex Al(C2H5)3O<sup>2</sup> according to the scheme [17]

$$\rm Al(C\_2H\_5)\_3 + O\_2 = Al(C\_2H\_5)\_3O\_2 \to (C\_2H\_5)\_2Al + O-O-(C\_2H\_5) + Q\_l \tag{4}$$

where *Q*<sup>I</sup> = 113 kcal/mol. Reaction (3) or (4) can be followed by the monomolecular decomposition reaction through two channels:

$$(\mathrm{C}\_{2}\mathrm{H}\_{5})\_{2}\mathrm{Al}\mathrm{-O}\mathrm{-O}\mathrm{(C}\_{2}\mathrm{H}\_{5}) \rightarrow (\mathrm{C}\_{2}\mathrm{H}\_{5})\_{2}\mathrm{Al}\mathrm{-O} + \mathrm{O}\mathrm{-(C}\_{2}\mathrm{H}\_{5}) + \mathrm{Q}\mathrm{I};\tag{5}$$

$$(\text{C}\_2\text{H}\_5)\_2\text{Al} \cdot \text{O} \cdot \text{O} \cdot \text{(C}\_2\text{H}\_5) \to (\text{C}\_2\text{H}\_5)\_2\text{Al} \cdot \text{O} \cdot \text{O} + \text{C}\_2\text{H}\_5 + Q\_{\text{III}} \tag{6}$$

where *Q*II = −77.9 kcal/mol and *Q*III = −90.5 kcal/mol [17]. Reactions (3 or 4) + (5) and (3 or 4) + (6) can be considered as exothermic bimolecular reactions. These reactions occur during collisions of gas-phase molecules with the surfaces of TEA droplets (heterogeneous reactions). In this case, volatile active radicals C2H<sup>5</sup> and C2H5O enter the gas phase (air) and interact with oxygen, releasing heat and giving rise to other sequential and parallel reactions, the same as in the gas-phase kinetics of the oxidation, self-ignition, and combustion of light alkanes (methane, ethane, and butane) and their derivatives. Kinetic schemes, corresponding equations of chemical kinetics, algorithms, and codes that describe similar gas-phase processes exist, and they can be readily used as subroutines for the numerical solution of the problem under consideration. It is worth emphasizing that we consider only the initial stage of the self-ignition process, rather than the entire process of TEA droplet combustion. At this stage, the size and chemical composition of droplets, as well as the oxygen concentration in the gas, change only a little, and can be considered constant.

Based on this prerequisite, the following kinetic model of TEA self-ignition in air at NPT conditions is proposed. The rate constant of reactions (3 or 4) + (5) and (3 or 4) + (6) is approximated as

$$K = A \exp\left(-\frac{\varepsilon}{RT}\right) \tag{7}$$

where *A* is the preexponential factor; *T* is the temperature; and *ε* is the activation energy. The consumption rate of TEA molecules per unit volume of the mixture can be expressed by the formula

$$\frac{d n\_{\rm TEA}}{dt} = -\frac{n\_{\rm O\_2} \mu\_{\rm O\_2}}{4} SNw,\ w \approx \lambda \exp\left(-\frac{\varepsilon}{RT}\right) \tag{8}$$

where *n*TEA is the number of TEA molecules per unit volume; *n*O<sup>2</sup> is the number of oxygen molecules per unit volume; *u*O<sup>2</sup> is the thermal velocity of oxygen molecules; *<sup>n</sup>*O2 *u*O2 4 is the number of collisions of oxygen molecules with a unit surface of a droplet per unit time; *S* = 4*πr* 2 *d* is the surface area of a TEA droplet; *r<sup>d</sup>* is the TEA droplet radius; *N* is the number of TEA droplets per unit volume; *w* is the reaction probability in one collision; and λ is the steric factor. This latter factor is unknown. Its value is probably in the range of 0.1–0.01. According to the kinetic theory of gases, the thermal velocity *u*O<sup>2</sup> at temperature *T* is

$$
\mu\_{\rm O\_2} = \left( 8 \text{RT/} 32 \,\pi \right)^{1/2} \tag{9}
$$

By definition, the derivative *dn*TEA *dt* can be also expressed as

$$\frac{dn\_{\rm TEA}}{dt} = -Kn\_{\rm O\_2}n\_{\rm TEA,s} = -KSd\_1n\_{\rm O\_2}n\_{\rm TEA,d}N\tag{10}$$

where *d*<sup>1</sup> is the thickness of the outer molecular monolayer in a TEA droplet; *Sd*<sup>1</sup> is the volume of the outer molecular monolayer in a TEA droplet; *n*TEA,d is the number of TEA molecules per unit droplet volume; *n*NEA,s is the number of TEA molecules in the volume *Sd*1*N*. Substituting Equation (7) into Equation (8) and comparing Equation (8) with Equation (10), one obtains

$$A = \frac{\lambda u\_{\rm O\_2}}{4n\_{\rm TEA,d}d\_1} \tag{11}$$

Equation (11) has a simple physical meaning. The values of *d*<sup>1</sup> and *n*TEA,d are expressed in terms of the effective radius, *r*1, of the TEA molecule: *d*<sup>1</sup> = 2*r*1, 1 *n*TEA,d = 4*πr* 3 1 /3. From here and from Equation (11), one obtains

$$A = \left(\frac{2}{3}\right) \lambda u\_{\rm O\_2} \pi r\_1^2 = \left(\frac{2}{3}\right) \lambda u\_{\rm O\_2} \sigma \tag{12}$$

where *σ* = *πr* 2 1 is the effective collision cross-section. Formula (12) coincides with the definition of the preexponential factor in the thermal theory of the rate constants of bimolecular reactions in gases [24] up to a factor of 2/3. It follows from Equations (7), (10), and (11) that

$$\frac{d\mathfrak{n}\_{\rm TEA}}{dt} = -\frac{\lambda}{4} \mathfrak{u}\_{\rm O\_2} S N \mathfrak{n}\_{\rm O\_2} \exp\left(-\frac{\varepsilon}{RT}\right) \tag{13}$$

It follows from Equation (13) that the rate of reactions (3 or 4) + (5) and (3 or 4) + (6) depends on the TEA droplet size, *r<sup>d</sup>* , the oxygen concentration in the environment, *n*O<sup>2</sup> , and the local instantaneous air temperature, *T*:

$$\frac{d n\_{\rm TEA}}{dt} \sim \frac{n\_{\rm O\_2}}{r\_d} \exp\left(-\frac{\varepsilon}{RT}\right) \tag{14}$$

Therefore, it could be expected that the self-ignition delay of TEA spray in ambient air could be a function of the spray injection pressure, as the droplet diameter generally depends on the injection pressure: the higher the injection pressure, the smaller the droplet size and the shorter the self-ignition delay. According to Equation (14), the self-ignition delay is shorter if the environment contains more oxygen, and if the local instantaneous air temperature is higher.

In addition to the uncertainty in the value of *λ*, the rate constant (7) contains an unknown activation energy *ε*. There exist empirical formulae establishing the relationship between *ε* and the heat of the exothermic reaction, *Q*, in a linear approximation:

$$
\varepsilon = a - bQ \tag{15}
$$

with positive parameters *a* and *b*. The values of *a* and *b* vary depending on the type (set) of reactions. These formulae include the well-known Polanyi–Semenov rule [24]:

$$
\varepsilon = 11.5 - 0.25Q \text{ kcal/mol} \tag{16}
$$

This rule, when applied to bimolecular reactions (3 or 4) + (5) and (3 or 4) + (6), gives, respectively,

$$
\varepsilon = 2.7 \text{ and } 5.9 \text{ kcal/mol} \tag{17}
$$

As noted in [21], the formulae such as (16) must be used with great caution. The same is true for estimates (17). They can only be considered as a rough approximation, which must be verified and refined experimentally. In experiments, the parameters of the reaction rate in Equation (13) are not measured directly. However, the induction period before the self-ignition of TEA droplets and some other kinetic and thermodynamic parameters can be measured, which depend on the rate constant (7). In this case, the activation energy *ε* can be found by solving the inverse problem. To do this, one must first solve the direct problem, which consists in calculating the self-ignition delay with a variation in activation energy *ε*.
