Use of thermal analysis to predict the conditions for thermal explosion to occur: application to a Ce triethanolamine complex

This contribution explores the capabilities of combined thermal analysis methods to predict the ignition condition for a thermal runaway event to occur in a system heated at a constant rate. In particular, for a Ce triethanolamine complex, thermogravimetry has been used to determine the kinetic parameters, while enthalpy, thermal conductivity and thermal capacity have been measured by means of DSC. Once these parameters are known, it is possible to predict the critical mass for different heating rates and crucible sizes. Besides, thermogravimetry allowed us to assess if thermal runaway occurred as well as to monitor its evolution. Good agreement between the predicted and the experimental critical masses has been observed.


Introduction
Predicting the occurrence of a thermal runaway, i.e., the formation of a combustion zone that propagates at high speed, is of utmost importance for a number of applications [1], such as evaluating the chemical risk and preventing ignition in chemical reactors or during the storage and transportation of hazardous materials [2][3][4][5][6][7], determining the conditions for pyrotechnic reactions to happen [8][9][10], determining munitions cook-off temperatures [11] and, in general, establishing the ignition conditions in chemical engineering processes [12,13]. It is also important for intermetallic reactions and termites (metal-metal oxides mixtures) that are used in some pyrotechnic devices such as igniters [14].
Moreover, combustion synthesis is an efficient and costcompetitive processing technique that takes advantage of the heat evolved during the chemical reaction of precursor substances to obtain materials that otherwise would require achieving high temperatures [15]. Indeed, combustion has become a very versatile route for the synthesis of intermetallic or oxide powders and compacts [15][16][17][18][19][20]. Combustion of bimetallic foils is used to join temperature-sensitive or dissimilar materials [21,22]. More recently, combustion synthesis of functional oxides from precursor salts (nitrates or metal-organic precursors) has attracted much attention due to the growing interest in these materials [23]. Also, nanoscale oxides, metals, alloys and sulfides have been obtained from an aqueous or solgel media based on the socalled solution combustion synthesis [24].
The occurrence of a thermal runaway is controlled by two competing phenomena: exothermic reaction that tends to increase the local temperature and heat dissipation that lowers the temperature through heat transfer toward the container walls [1]. Any combustion process must be initiated by a so-called ignition event. Ignition produces a localized overheating that will propagate as a combustion front along the volume of the material. Two different igniting modes are usually considered in the literature [17,18,25]. In the "self-propagating high temperature" mode, ignition is created locally by an external and localized energy source (a spark, a laser beam, an electrical discharge, etc.) [20,21]. In the "thermal explosion" (also known as "volumetric") mode, the reactants are heated to a temperature where the reaction becomes locally unstable [26].
Frank-Kamenetskii and Semenov [27,28] derived the critical condition for thermal explosion in homogeneous system of very simple geometry, assuming that the activation energy is infinite and neglecting reactant consumption. Several authors have developed more accurate solutions that account for reactant consumption and finite activation energies [27,[29][30][31][32][33][34][35][36][37]. Recently, a critical condition that takes into account realistic geometries [38] and constant heating conditions [39] has been derived. Therefore, it is possible to know the conditions for a thermal runaway to occur during thermal analysis (TA) measurement under isothermal and constant heating conditions. The latter is of special interest because constant heating measurements are easier and faster to perform in TA [40].
The critical condition depends on the temperature of the crucible walls and on the following system parameters: activation energy, pre-exponential term of the kinetic constant, thermal conductivity, thermal capacity, density and reaction enthalpy. As we will show in this communication, all these parameters but the density can be determined using TA methods. In particular, we will determine the parameters of a cerium triethanolamine (TEA), Ce-TEA, complex. Ce-TEA has been used to obtain ceria powders and nanoparticles by combustion synthesis [41][42][43][44] and hydrolysis methods [45,46]. Ceria powders and nanoparticles have a wide range of applications, such as catalyst, electrolytes, abrasives, gas sensors, UV absorbers for sunscreens and solar cells.
In the Ce-TEA complex, TEA is a reducing agent that acts as fuel, while the nitrate group is a strong oxidizer. Ce-TEA contains actually both the fuel and the oxidizer; therefore, it experiences combustion in both inert and oxidizing atmospheres; i.e., no oxygen is needed to sustain combustion. Moreover, Ce-TEA undergoes combustion in the usual conditions used in TA measurements [47,48]. Therefore, we will be able to check the validity of the critical condition obtained for Ce-TEA complex for different crucible sizes and heating rates.

Preparation of Ce-TEA complex
Ce-TEA was prepared according to the procedure described in [47,48]; 10.86 g cerium(III) nitrate hexahydrate (Ce(NO 3 ) 3 ·6H 2 O Alfa Aesar ≥ 99.5%) was dissolved into 200 mL 1-propanol (VWR Chemicals, ≥ 99.9%) in a round-bottom flask under nitrogen at 60 °C using a stirring magnetic hot plate. When a clear solution was obtained, the heating system was shut down. Simultaneously, a mixture of 7.472 g of TEA (C 6 H 15 NO 3 , Merck, ≥ 99%) and 100 mL of 1-propanol was added to the warm cerium nitrate solution. This mixture was stirred under high-purity nitrogen (Praxair, ≥ 99.999%) for 30 min. The mixture was filtered, and the resulting crystals were washed successively in ethanol (Scharlau, 96% v/v) and acetone (Scharlau ≥ 99.5%). Finally, the crystals were dried at room temperature in vacuum conditions. From X-ray diffraction, elemental analysis and infrared analysis, Ce-TEA has been identified as the Ce(III) complex: [Ce(NO 3 )(C 6 H 15 NO 3 ) 2 ](NO 3 ) 2 [47]. This means that the Ce-TEA complex has two TEA molecules and three nitrate groups per cerium atom. Thus, Ce−TEA contains both the fuel and the oxidizer.

Characterization techniques
Simultaneous thermogravimetric (TG) and differential scanning calorimetry (DSC) analyses have been performed in a Mettler Toledo thermo-balance, model TGA/DSC1, at 10 K min −1 under a gas flow of 40 mL min −1 of high purity N 2 or synthetic air (Praxair, ≥ 99.999%). Uncovered Al 2 O 3 crucibles of 70 and 150 μL were used. DSC analysis has been carried out in a Mettler Toledo DSC822e; samples were placed in open 70-μL alumina crucibles or in 20-μL aluminum crucibles covered with a pinned lid to allow gas exchange. DSC measurements were performed under a constant heating rate of 10 K min −1 and under a gas flow of 40 mL min −1 of high purity N 2 .

Determination of Ce-TEA parameters
The experimental values for the parameters of Ce-TEA powder are summarized in Table 1. In all cases, special care has been taken when measuring the properties: Powder from the same batch and with the same compactness was used because physical properties of powders depend on their morphology, particle size, interparticle void size, etc. In this sense, the density was determined using a vessel of known volume and filling it with powder in the same way we fill the crucibles in the TG experiments: with the powder inside, the crucible was gently tapped against a solid horizontal surface. The rest of the parameters were determined using TA. In the following subsections, we describe the methodology and the results.

Determination of the kinetic parameters
We have used isoconversional methods to determine the apparent activation energy and the pre-exponential term because it is generally accepted that they are among the most reliable ones and they are model-free [49][50][51][52]-no assumptions about the reaction model are made. Isoconversional methods rely on the assumption that at a given degree of transformation, α, the transformation rate is only a function of the temperature: where T is the temperature, E α is the apparent activation energy at a particular α value and R G is the gas constant. From integration of Eq. (1), we obtain Note that the kinetic parameters, apparent activation energy E and the pre-exponential term A f ( ) , are not constant but depend on the degree of transformation.
To determine the kinetic parameters, at least three experiments need to be performed at different temperatures (isothermal) or at different constant heating rates, ≡ dT dt = constant (non-isothermal). Because non-isothermal experiments are faster and easier to perform, we have measured the thermal decomposition of Ce-TEA when heated at β = 2.5, 5 and 10 K min −1 in N 2 flowing gas. The results are shown in Fig. 1. To properly determine the kinetic parameters, it is necessary to avoid the event of a thermal runaway. For this reason, we have chosen an initial mass of 5.2 mg for the three experiments. Moreover, from Fig. 1, it is apparent that the decomposition of Ce-TEA is a complex process. Two main stages have been identified [47]: a fast one that ends around 300 °C (note the elbow at the TG curve) and a slow one that extends up to 800 °C and beyond.
Since combustion occurs at the first stage, we have limited our kinetic analysis to this stage. In Fig. 2, we have plotted the α(T) curves obtained from the TG curves of Fig. 1 assuming that: (1) initially α = 0, (2) at the end of this first stage α = 1, and (3) the transformation rate is proportional to the mass loss rate.
We have used different isoconversional methods to determine the kinetic parameters: Friedman [53], Vyazovkin [54], Li and Tang [55,56] and Ortega's exact method [49,57]; the result is plotted in the inset of Fig. 2. All methods deliver very similar results, but the apparent activation energy is not constant. Nevertheless, the thermal runaway starts when the degree of transformation is approximately 0.3. Up to α = 0.3, the apparent activation energy is approximately constant. So, to calculate the critical condition, we have assumed a constant apparent activation energy of 220 kJ mol −1 and a pre-exponential term A = 2.34 × 10 19 s −1 .

Thermal conductivity
Thermal conductivity of Ce-TEA powder was obtained from the slope of the DSC melting peak of indium according to the method developed in Refs. [58,59]. The method is based on the slope of the low-temperature side of the melting peak of a reference metal bead placed on the top of an amount of powder that fills a cylindrical crucible (Fig. 3), when heated at a constant rate. This slope depends on the thermal resistances of the powder, R P , of the DSC, R S , and of the contact between the crucible and the sensor, R C , according to: where R S + R C can be obtained, if one measures the slope of the melting peak without the powder (Fig. 3) with a metal bead blasted inside the crucible bottom to ensure good thermal contact: From the slopes S P and S e , we can determine thermal resistance. Once R P is known, the thermal conductivity is given by where D m is the diameter of the metal bead, D p is the internal diameter of the cylindrical crucible and K is a geometrical factor that depends on the ratio D m ∕D p and on the ratio between the crucible height h and D p . For most commercial crucibles, K is very close to unity [58].
To avoid the problem of Ce-TEA melting during the measurement, before measuring the thermal conductivity, the precursor is melted and quenched so that an amorphous phase appears and no melting event occurs during the measurement. Experiments were conducted in a 40 mL min −1 nitrogen flow and a heating rate of 10 K min −1 in the temperature range 50-200 °C. For an indium bead of D m = 2.00 mm and a crucible of D P = 4.95 and h = 0.7 mm (K = 1.05), we obtained S P = 0.806 and S e = 15.66 mW K −1 that deliver a thermal conductivity λ = 0.0846 W K −1 m −1 (Fig. 3). Note that the thermal conductivity of Ce-TEA is very low. This low thermal conductivity reduces significantly thermal dissipation and thus facilitates the formation of temperature gradients inside [60,61] and the occurrence of a thermal runaway [41,62].

Specific heat capacity
When a sample is heated at a constant rate, the DSC signal is proportional to the heating rate: where c is the specific heat capacity and m is the sample mass. Since any apparatus has a nonzero baseline, this baseline has to be subtracted. Unfortunately, this baseline is not reproducible and this lack of reproducibility worsens when the furnace is opened. Thus, a strategy has to be applied to remove the effect of the baseline. We have used a method that consists of consecutive isothermal and non-isothermal measurements [63]. The jump in the DSC signal between the isotherm and the non-isotherm steps, ΔDSC , is: Due to the apparatus and sample thermal inertia, there is a transient period between isothermal and non-isothermal steps. For this reason, the duration of every thermal segment was 4 min to allow for the system to stabilize. The heating rate was 10 K min −1 , so the temperature difference between two consecutive isotherms was 40 °C. We measured the heat capacity at 100, 140 and 180 °C. In addition to that, we performed the experiment twice: first with a substance of known heat capacity to calibrate the DSC signal and afterward with Ce-TEA powder. For the calibration, we used Al 2 O 3 powders. The measurement performed on Ce-TEA powder is shown in Fig. 4. From Fig. 4, one can observe the transient behavior after any change of heating rate. So, to (6) DSC = −cm ,  Fig. 4) to the time when the heating rate changes. Finally, since the heat capacity of reference and sample crucibles is not perfectly balanced, there is a reproducible DSC signal jump that needs to be corrected. This effect can be subtracted by running a "blank" experiment under the same conditions but with empty crucibles. This blank experiment is shown on the top of Fig. 4. Following this method, the heat capacities of Ce-TEA at 100, 140 and 180 °C have been determined to be 866, 950 and 1010 J kg −1 K −1 , respectively. To determine the critical condition, we have chosen the closest value to the combustion temperature, c = 1010 J kg −1 K −1 .

Reaction enthalpy
In general, the measured DSC area of a thermal decomposition will differ from the reaction enthalpy because part of the heat released by the reaction is carried away by the volatile components [64]. Therefore, there is no point in measuring the exact enthalpy of the reaction since a significant part does not contribute to the increase in the sample temperature. Consequently, parameter q is determined directly from the integration of the DSC signal during the decomposition, since it corresponds to the sample overheating resulting from the heat released by the chemical reaction.

Critical condition under constant heating rate
The evolution of the thermal decomposition of Ce-TEA for different initial sample masses is shown in Fig. 5. Above a critical value of the sample mass (around 8 mg), the evolution is characterized by a discontinuity in the first derivative of the mass evolution-a sudden drop of the sample mass and a sharp exothermic peak (not shown). These features are characteristic for thermal runaways [38,65]. In the case of exothermic reactions, the sample undergoes a thermal runaway when the time to dissipate the heat generated by the reaction is longer than the characteristic reaction time. The time needed to dissipate heat increases with the system size. So, larger samples are more prone to undergo a thermal runaway. In particular, for powder inside a cylindrical crucible of radius R filled up to a height H and heated at a constant rate β, the critical mass ( m cr ) is given by [39]: where a is the thermal diffusivity (a ≡ ∕ c) and (8) and T Kis is the temperature at which the reaction rate is at its maximum. It can be calculated from the Kissinger equation [66][67][68]: Taking into account that m = R 2 H , Eq. 8 can be expressed as Thus, if the radius R is known, Eq. 11 can be solved for H and, therefore, we can calculate m cr as a function of the crucible radius: Therefore, from the physical parameters determined in the previous sections (Table 1), we have calculated the critical mass, m cr , for powder heated at 10 K min −1 inside a 150-μL alumina crucible (radius R = 3.5 mm). We have obtained a critical mass of 7.99 mg which is in agreement with Fig. 5; a sample mass of 8.03 mg undergoes combustion, while the sample of mass of 7.14 mg exhibits a smooth evolution.
To check the validity of Eq. 12, we have determined the critical mass from several sets of experiments performed at different heating rates (2.5, 5 and 10 K min −1 ) on 150-μL alumina crucibles, and one set of (9)  Fig. 6. The experimental critical mass is between the empty symbols (no combustion) and full symbols (combustion). Additionally, we have plotted the prediction delivered by Eq. 12 (lines). One can verify that there is a nice agreement between the experimental results and the theoretical prediction. Furthermore, one can verify that the critical mass diminishes as the parameter ∕a increases. In other words, to promote combustion we need lower thermal diffusivities and great values of Todes parameter θ T (low thermal capacities and high enthalpies). Finally, from Fig. 6, it is apparent that the higher the heating rate, the easier a thermal runaway is set.

Conclusions
We have shown that thermal analysis methods provide a complete set of tools to analyze thermal explosion. First, they provide the techniques to easily determine the material's parameters that control the occurrence of a thermal runaway: kinetic parameters, thermal conductivity, specific heat capacity and specific reaction enthalpy. In the case of samples in the form of powder, these properties depend on powder morphology; thermal analysis provides straightforward methods to measure sample parameters in situ. Second, when the thermal conductivity is low, thermal explosion can occur during thermal analysis measurements. In this case, it is possible to determine the thermal explosion threshold under different conditions (crucible diameter and heating rate) by TG.
The decomposition of Ce-TEA complex has been used to illustrate how to proceed to predict and to determine the critical mass for thermal explosion to occur. A very good agreement between theory and experiment has been achieved.