How the allotropic transition temperature of solids can change with the heating rate

Though classical thermodynamics predict a single temperature for the allotropic phase transition, the phase transition temperature is often seen to change with the heating rate. Herein we propose a simple method to predict the change in the phase transition temperature as a function of the heating rate. The method is based on the comparison of entropy production between two paths, with or without a phase transition. This method was applied to the zircaloy α-β phase transition and the resulting experimental data were processed to determine the molar transformation rate as a function of temperature.

the formalism of entropy production to adapt it to specific cases, e.g., De Groot [7], Ziegler [8], Jaynes [9] and Martyushev [10]. The concept of entropy production makes it possible to explicitly express time-dependent variables because the entropy production, σ, explicitly depends on time as written in eq. (1), In the following, we derive an expression of entropy production in which the time dependence of temperature is explicit. We then use this equation to interpret the change in the transition temperature as a function of the heating rate. In the last section, we apply this formalism to the α-β transition of zirconium [11]. Entropy production during a heating rate. This section proposes an expression of entropy production in which the time dependence of temperature is explicit. We first review the expression of entropy production in isobaric-isothermal conditions as proposed by Martyushev [10], which we adapt to isobaric, non-isothermal conditions.
An isobaric-isothermal system representative of a phase transition is illustrated in fig. 1 (green box).
In a very general manner, the entropy production of this system can be written as Based on the first principle of thermodynamics and if the work is written as ΔW = P · dV , eq. (2) can be rewritten 26001-p1 Fig. 1: An isobaric-isothermal system in which the α-phase transforms into the β-phase. The only heat entering the chemical system is the heat from the thermostat, which imposes a constant temperature.
Considering an isobaric-isothermal system, which only exchanges heat with the environment (P · dV dt = 0), this gives Martyushev's expression [10] for entropy production of a system in which chemical reactions occur, Let us now turn to a system in which the temperature of the sample is not constant but it is subjected to a heating rate. A heating source produces the heating rate. Only a fraction of the heating power actually increases the sample temperature, while the remaining power is dissipated in the thermostat surrounding the system as shown in fig. 2.
Equations (3) and (4) are still valid for an isobaric, nonisothermal system, but T is no longer constant. Therefore, dT dt is no longer zero, and an additional term appears in the expression of the entropy production as shown in eq. (6), This new expression of entropy production includes the additional term S · dT dt that can be viewed as the energy put into the system for it to undergo a heating rate. With this expression, we were able to establish a thermodynamic equation with an explicit expression of the temperature rate, dT dt , written as in the following. Phase transition temperature as a function of .
In this section, we use the previous equation to interpret the change in the transition temperature as a function of the heating rate. For the sake of clarity, we used the system defined below.
-The system is made up of one mole of pure monoatomic solid in α-phase at room temperature. Because the solid is monoatomic, its Gibbs energy in α-phase Fig. 2: An isobaric, non-isothermal system. The heater deposits energy in the chemical system, which increases its internal temperature and thus compensates for the exchanges with the thermostat that are needed to release the energy absorbed for the chemical reaction. is equal to its chemical potential μ α . By definition we have with h α and s α representing the molar enthalpy and molar entropy, respectively.
-At T c , the solid transitions to β-phase. The Gibbs energy of β-phase is given by with h β and s β representing the molar enthalpy and molar entropy, respectively. As shown in fig. 3, at T 0 c , we have μ β = μ α that leads to the following expression of T 0 c : This system is then subjected to the thermodynamic conditions defined in the previous section, i.e., a constant pressure of 1 bar and a varying temperature. For the sake of simplicity, we consider that the temperature is homogenous in the system and varies linearly with time, = const. Our objective is to determine the temperature at which the beta phase appears during a heating rate. To do this, we use the principle of maximum of entropy production to determine the time (i.e., temperature) at which the transition to the beta phase will induce more entropy production than remaining in the alpha phase. For this reason, we assess the entropy production for two possibilities: the first, P1, corresponds to a system that remains in the α-phase; the other, P2, corresponds to the beginning of the transition to the β-phase. This phase starts when the entropy production for P2 is higher than that for P1.
The entropy production for P1 is only due to the αphase, called σ α . The entropy production for P2 is due to the part of β-phase that has already formed and the remaining α phase; it is equal to σ α (1 − δ) + δσ β , with δ representing the fraction of the β phase formed. Based on the literature on this phase transition, we assumed where ϑ is the molar transformation rate. Many phase transitions can be described by a nucleation and growth mechanism. In our case, we focus on nucleation because we are examining the beginning of the β-phase formation. With fast transformation rates, however, growth could also contribute to the transformation rate as discussed below. The β-phase will start to form when the entropy production for P2 is higher than that of P1, which corresponds to We assess σ α − σ b using eq. (6) in the case of a monoatomic solid, which, using eqs. (10) and (11), leads to With eq. (16), it is possible to determine the temperature T c (R) at which, the α-β transition starts with a heating rate by solving equation Figure 4 represents the entropy production for an αβ transition with a heating rate and the corresponding evolution in Gibbs' energy. The temperature offset, T c (R) − T 0 c = ϑ , is labelled χ in this figure. When = 0, eq. (15) can be rewritten as We therefore recover the expected behaviour for a phase transition in the classical framework of thermochemistry. In this section, using the maximum entropy production, we predict that the temperature of a phase transition is shifted by an offset equal to ϑ , which is the ratio between the temperature rate imposed on the solid and the nucleation rate at which the solid reacts. In the next section, we will see how this prediction can be applied to interpret data on the transition temperature of zirconium.
α-β transition of zirconium as a function of the temperature. -The α-β transition of zirconium was studied as a function of the temperature rate because of its technological impact on the assessment of nuclear fuel behaviour in accident conditions. Here we focus on the study discussed in [11] that provides a wide range of heating rates as shown in fig. 5. Figure 5 shows how the length of a zirconium sample changes with the temperature for different heating rates. At low temperature, the linear part corresponds to an αphase only, and, in a similar manner, the high temperature linear part corresponds to a β-phase only. The S-shape in between the two linear parts corresponds to mixed α and β phases. In the previous section, we defined T c (R) as the temperature, at which β-phase appears for a heating rate . In fig. 5, T c (R) lies at the point where the low-temperature linear part ends, i.e., the S-shape part starts. The corresponding values of T c (R) are given in table 1. These T c (R) values were then used to determine ϑ, the molar transformation rate using eq. (17), with T 0 c assumed to be 820 • C. This value differs slightly from that of zirconium in the thermodynamic data base [12] because the material used in our experiment is not pure zirconium but a zirconium alloy. The value of 820 • C was determined from the black curve in fig. 5 named "equilibrium", which    [11] using another figure that is not shown herein. Based on the approach developed in the previous section, this leads to an increase in the molar transformation rate, ϑ, for high heating rates and the highest T c (R), as can be seen in fig. 6. The accuracy of T c (R) − T 0 c is low when it is above or equal to 5 • C. Our objective is not to obtain high values with high accuracy, but rather to construct a qualitative picture.
Discussion and conclusion. -We have presented a new approach based on the maximum entropy production in order to deduce changes in the transition temperature as a function of the heating rate; we then applied this approach to the α-β transition of zirconium. This approach worked well in this example but still requires some refining; first, the concept of maximum of entropy production, and then the meaning of the molar transformation rate.
The strategy for maximum entropy production that we used differs from that described by Martyushev [10]. In his paper, the principle of maximum entropy production is defined as follows: If irreversible force X i is prescribed, the actual flux J i , which satisfies the condition σ(J i ) = i X i J i , maximizes the entropy production. In our case, we are less concerned about continuous behaviour and more about bifurcation, i.e., the system choses to change from one phase to the other. This is the reason why we chose to assess two different possibilities, P1 and P2. Therefore, maximizing means comparing the entropy production in both cases instead of deriving differential equations, including Onsager's coefficients. To some extent, our approach is inconsistent with Onsagers because we used eq. (6) with the term S · dT dt , which is not the product of a force by a flux, but a flux only, thus corresponding to the heat absorbed by the sample upon heating. More work is needed to establish a comprehensive framework, including both approaches: continuous and bifurcation.
The molar transformation rate is usually described with an activated process such as that proposed in [11]. Figure 6 gives us another picture: T c (R) tends to a constant value around 900 • C in accordance with [11], which attributed this to a "saturation" effect. In our case, the saturation effect discussed in [11] would induce an infinite molar transformation rate as soon as T c (R) reaches around 900 • C. Such a conclusion is difficult to draw from the data provided. This is because it is relatively certain that the thermal gradients would become appreciable at extreme heating rates (1000 and 2000 • C · s −1 ), therefore limiting the confidence we can have in a hypothetic saturation level. Nonetheless, fig. 6 shows that there are two stages of phase change kinetics, i.e., before and after 865 • C. Therefore, some work is still needed to interpret the relationship between the transformation rate and the temperature rate, with two main options. In the first option, the competition between the nucleation and growth rates could be responsible for the regime change at 865 • C. With the second option, a maximum superheating temperature should be considered, which would be related to a dimensionless number previously characterized for metallic systems in [13].
As a conclusion, we have proposed a new point of view for considering the change in the transition temperature as a function of the heating rate. This proposal is clearly not a definitive model, but opens the way for further research to rationalize more experimental data that will help improve modelling.
Data availability statement: All data that support the findings of this study are included within the article (and any supplementary files).