<figure><img src="image_1.png" /></figure>

HAL
open science

# Dynamics of the phase transformation in a giant magnetocaloric material along T- and B-scans

F. Guillou, B. Huhe, F. Veillon, V. Hardy

► To cite this version:

F. Guillou, B. Huhe, F. Veillon, V. Hardy. Dynamics of the phase transformation in a giant magnetocaloric material along T- and B-scans. Results in Physics, 2025, 78, pp.108497. 10.1016/j.rinp.2025.108497. hal-05360345

HAL Id: hal-05360345

https://hal.science/hal-05360345v1

Submitted on 12 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Dynamics of the phase transformation in a giant magnetocaloric material along *T*- and *B*-scans

F. Guillou<sup>a,b*</sup>, B. Huhe<sup>a</sup>, F. Veillon<sup>b</sup>, V. Hardy<sup>b</sup>

## ABSTRACT

All giant magnetocaloric materials exhibit an extreme first-order character which makes them particularly suitable to investigate the peculiarities associated to this type of transition. In the present study, we explored the time dependence of the heat exchanges, from or towards a LaFeSiCo alloy, by using a home-made differential scanning calorimeter based on Peltier cells. A thermal modelling of the experimental set-up was developed to analyse the various regimes present before, along, and after the transformation. This model allows to recover the expected values of latent heat and it predicts relaxation times consistent with the data up to certain limits in rate (about 0.5 K/min and 150 Oe/s). The heart of our study was focused on the transitional regime for which we derived the instantaneous rate of transformation (taking the form of a succession of spikes). We explored in detail the impacts of the nature of the driving parameter (temperature or field), the direction of crossing the transition, the scanning rate and the thermal coupling of the sample.

**Keywords:** Magnetostructural transformations, Phase transformation kinetics, Thermal analysis, Differential scanning calorimetry, Intermetallic compounds.

## 1. Introduction

The intense activity on magnetocaloric materials in recent years has led to the discovery of compounds showing a very sharp magnetostructural transition accompanied by a large latent heat [1,2]. The time dependence of the heat exchange at the first-order transition (FOT) of these materials is a relevant issue for their potential applications, such as magnetic refrigeration [3]. In some of these compounds, the heat events are so sudden that they can generate peculiar physical effects, such as recalescence and antirecalescence, [4-9] as well as jerky transformations [5,9,10-13]. This topic about the burst-like transitions was boosted recently by the development of new calorimetric devices based on Peltier cells, that are particularly well suited for their investigation [14-17]. Several studies unambiguously demonstrated that the transition in the magnetocaloric materials showing the highest performances proceeds via a succession of disconnected events [10,16,17].

<footnote><sup>a</sup> College of Physics and Electronic Information, Inner Mongolia Key Laboratory for Physics and Chemistry of Functional Materials, Inner Mongolia Normal University, 81 Zhaowuda Road, Hohhot 010022, Inner Mongolia, China</footnote>

<footnote><sup>b</sup> Normandie University, ENSICAEN, UNICAEN, CNRS, CRISMAT, 14000 Caen, France</footnote>
The goal of the present study is to investigate the development of the first-order ferromagnetic transition in a giant magnetocaloric material, by using the same sample in the same apparatus, both for scans versus temperature and field. Besides, we aimed to include in our study the direction of crossing the transition and the value of the sweep rates. This was carried out by means of a home-made differential scanning calorimeter (DSC) based on Peltier cells, that is installed on a widely used commercial system allowing control of temperature and field [18]. To be relevant to the planned experiments, we developed a versatile thermal analysis by re-starting from the basic approach of Mraw devoted to the DSC principles [19]. As detailed hereafter, it is found that the relationship between the measured heat flux and the advancement of the transformation is more complex than it could appear at first glance.

After a presentation of the experimental conditions in Section II, Section III presents firstly the raw data that are heat flux curves versus magnetic field or temperature, and secondly the thermal model. Section IV devoted to the data analysis is divided into three parts: first, a study of the regime preceding the transition, from which important experimental parameters can be measured, second the investigation of the regime post-transition from which the time constant of the relaxation processes can be evaluated, and third, the investigation of the instantaneous rate of development of the transformation, that is the heart of our study. This is done along the two types of scans, for each direction of application of the driving parameter and for values of rates spanning nearly two orders of magnitude both for T- and B-scans. An overall discussion of this set of results is given in Section V.

## 2. Experimental details

The DSC used in the present study was described in detail in Ref. [20]. As shown in Fig. 1, it is based on two miniaturized Peltier cells (model 1MD03-024 from RMT) pasted with GE varnish onto a massive piece of copper that is itself thermally anchored onto the top plate of a PPMS puck. A cernox element monitors the temperature at the surface of the copper block, close to the Peltier cells. The sample is pasted with H Apiezon grease on one of the two cells, which are electrically connected in series opposition. The collected voltage is converted into an equivalent heat flux $J_Q$ that reflects the flux $j_Q$ crossing the sample cell (see Supplemental Material in [21]). A polished copper cap is placed over the top of the puck to act as a thermal radiation shield [20].

The heat capacity and temperature of the sample are referred to as $C_s$ and $T_s$, respectively. The heat capacity and temperature of the top plate of the Peltier cell on the sample side of the
DSC are denoted $C_p$ and $T_p$, respectively. The thermal resistances of each of the Peltier cells is $R_p$, while that of the thin layer of grease between the top plate and the sample is $R_g$. By convention, the heat flux $j_Q$ is chosen to be positive when it goes from the bath towards the sample. We specify that the quite large integration time of our nanovoltmeter provides us with accurate values of $J_Q$, but it also results in a modest resolution in time, of the order of 1 s.

The investigated alloy was purchased from Vacuumschmelze GmbH and CoKG [22]. Its stoechiometry derived from energy-dispersive x-ray spectroscopy was found to be LaFe<sub>11.4</sub>Si<sub>1.31</sub>Co<sub>0.29</sub> [9]. It exhibits a mean $T_c$ close to 206.5 K, with an intrinsic hysteresis estimated to ~0.5 K, from extrapolation towards vanishing scanning rate. This material has a pronounced first-order character leading to a very sharp transition (apparent width lower than 0.1 K) and a high entropy change of about 13.5 J K<sup>-1</sup> kg<sup>-1</sup> [21]. The magnetocaloric isothermal entropy change for a field application of 1 T shows a maximum at 18 J K<sup>-1</sup> kg<sup>-1</sup>, which places it among the materials having the best performances nowadays [23]. The sample used is a thin parallelepiped having a mass $m = 13$ mg. Its shape (approximately 2 × 2 × 0.4 mm<sup>3</sup>) ensures the absence of significant gradient along its thickness.

<figure><img src="image_3.png" /><figcaption>**FIG. 1:** (a) Experimental set-up installed onto a PPMS puck; two Peltier cells and a Cernox thermistor are pasted by GE varnish on a massive piece of copper; the sample is pasted on one of the two Peltier cells that are connected in series opposition. (b) Schematic drawing of this set-up. The cells are represented by their bottom and top plates, thermally connected by the Peltier elements. (c) Details of the various parameters involved in the cell of the “sample side” of the device (see text). The two variables used to drive the transition (placed into circles) are either $T_b$ or the applied magnetic field $B$.</figcaption></figure>

Some basic parameters of the Peltier cells were experimentally determined. First, the thermal resistance of the cell and the heat capacity of its top plate were directly measured (see Supplemental Material in [21]). Within the T-range of interest, one estimates $R_p \sim 83$ KW<sup>-1</sup> and
$C_p \sim 7 \ 10^{-3} \text{ J K}^{-1}$. Note that this $C_p$ is smaller than most of the plates of Peltier cells used in the literature [15,24], but this is expected owing to their smaller size. As for the thermal conductance, it is comparable to those of the literature, consistently with the fact that they are made from the same type Peltier elements with very comparable geometries [14,15,25].

Another experience was carried out with a miniaturized heating element placed in the sample position. The goal was first to crosscheck the calibration (primarily achieved by comparison against a standard compound), and second to evaluate the intrinsic time constant of the device. It was found that the application of heating pulses with square shape leads to transient regimes of exponential type with a characteristic time of about 1-1.5 s (see [21]). It turns out that these values are close to the time constant $R_p C$ anticipated when considering the overall heat capacity resulting from that of the top plate, plus that of the heating element and that of the layer of grease (the whole being roughly estimated to $\sim 0.8$ s).

# 3. Results

## 3.1. Heat flux data

Figure 2 shows the whole set of $J_Q(T_b)$ and $J_Q(B)$ curves that were recorded. Each of these curves exhibits a peak that corresponds to the transitional regime. For the slowest rates, one can observe a substructure of several spikes within this regime. The sign of these curves depends on the direction of crossing of the transition. There is an equivalence between increasing the temperature or decreasing the field, as expected for a ferromagnetic transition. One also notes the presence of an hysteresis in both types of scans, with a peak located at a value of the driving parameter that is higher when increasing it and lower when decreasing it. Each panel displays a kind of symmetry point, corresponding to (T ~ 206.5 K; B = 0 T) for the T-scans and (T = 209 K; B ~ 0.53 T) for the B-scans, which are actually situated on the field-dependent line of the transition, $T_{tr}(B)$ [21]. A wide range of rates has been investigated, reaching factors of 100 and 40 for the T- and B-scans, respectively. When increasing the rate, the peak broadens and the hysteresis increases.

The $J_Q$ curves provide us with an indirect vision of the transition in terms of heat exchange. Indeed, $J_Q$ reflects the shift between $T_p$ and $T_{p0}$ resulting from the presence of the sample on one of the Peltier cell [21]. Along the transformation, the sample releases towards (or absorbs from) the Peltier cell a certain amount of heat corresponding to the latent heat. Let us consider for instance a path from the paramagnetic to the ferromagnetic state; the sample will release heat, that will be evacuated towards the bath via the cell; this induces negative $j_Q$.
(thus $J_Q < 0$) values for both decreasing temperature and increasing field, consistently with Fig
2.

<figure><img src="image_4.png" /><figcaption>FIG. 2: $J_Q$ curves versus either the bath temperature (a) or the applied magnetic field (b). In both cases, curves have been recorded upon increasing and decreasing this driving parameter, and for various rates</figcaption></figure>

Both type of scans can be treated within the same framework by analysing them versus
time. This can be even more rationalized by introducing the *t*\* variable, where the time origin
is fixed at the start of the transition (t\* = *t* - *t*start). This requires to estimate in each case the time
*t*start corresponding to this crossover. Illustrative examples are displayed in Fig. 3. For T-scan
(fixed *B* = 0) the starting point *T*\*start [see Fig 3(a)] corresponds *T*<sub>s</sub> = *T*<sub>c</sub>, which yields *T*\*start =
*T*<sub>c</sub> + *t*(*R*<sub>P</sub>*C*<sub>s</sub> + *R*<sub>P</sub>*C*<sub>P</sub> + *R*<sub>g</sub>*C*<sub>s</sub>). For B-scan (fixed *T*<sub>b</sub>), it corresponds to *B*start (fig 3b) which is
itself given by the relationship *B*start = (*T*<sub>b</sub>-*T*<sub>c</sub>)/(*dT*tr/*dB*) [see Fig 3(c)]. Since *dT*tr/*dB* ~5.1 K T<sup>-1</sup>
[21] and since we chose in the present study *T*<sub>b</sub>-*T*<sub>c</sub> ~ 2.5 K, one expects *B*start ~ 0.5 T. In practice
all the B-scans started from zero field (i.e., *B*<sub>0</sub> = 0) in such a way that *B*start = *Bt*start.

Examples of J<sub>Q</sub>(t\*) curves for T- and B- scans are shown in Fig. 3(d) and (e), respectively.
In both cases, one can distinguish three regimes (marked by different colours) : Regime 1
standing before the transition ( t\* \le 0 ), where the J<sub>Q</sub> curves are rather flat; Regime 2 is that of
the transformation ($0 \le t^* \le \Delta t_{tr}$), it starts with the take-off of the $J_Q$ curve which then acquires a profile made of peaks and valleys; Regime 3, taking place after the transformation ($t^* \ge \Delta t_{tr}$), is a relaxation regime exhibiting an exponential-like shape.

<figure><img src="image_5.png" /><figcaption>FIG. 3: The left panels are schematic representations of the T-scans (a) and B-scans (b), with definition of the parameters associated to the start of the transition, and showing the time scale $t^*$. Panel (c) shows the field-dependent transition line $T_{tr}(B)$. The right panels are illustrations of a T-scan (d) and a B-scan plotted versus $t^*$. The three regimes are marked by different colours : regime 1(pre-transition) in yellow, regime 2 (transition) in green (whose duration is denoted $\Delta t_{tr}$), and regime 3 (post-transition) in pink.</figcaption></figure>

## 3.2. Thermal model

The measured $J_Q$ is related to the $j_Q$ of Fig. 1 via the following relationships (see Supplemental Material [21] for details).

$$ T\text{-scans} : j_Q = J_Q + C_p T \qquad (1a) $$

$$ B\text{-scans} : j_Q = J_Q \qquad (1b) $$

We address below the heat flux $j_Q$ whose general expression is:

$$ mj_Q(t) = -\frac{1}{R_p} [T_p(t) - T_b(t)] \qquad (2) $$

Since the $T_b(t)$ expression is perfectly determined for each type of scan, the point is the derivation of the temperature $T_p$ that is intimately related to $T_s$. We use mass units for most of the thermal quantities ($j$, $J$, $L$ and $dS/dB$), while the $C$ are in $JK^{-1}$ and the $R$ in $KW^{-1}$. In what follows, we will consider only the heat transfers via solid conduction, since the other ways of heat exchange in our set-up are found to be much lower (see [21]).
Two equations constitute the basis of the thermal model. At the level of the sample, one can write a versatile equation, applicable to all type of scans and regimes, provided that one uses the relevant values of parameters:

$$C_s \frac{dT_s}{dt} + mL \frac{dx}{dt} = -\frac{1}{R_g} (T_s - T_p) - T_s m \left(\frac{dS}{dB}\right)_{T_s} B \quad (3)$$

The second lhs term is present only within the transformation regime of a first-order transition (regime 2); $x$ is the fraction of transformed phase and $L$ the associated latent heat. If one chooses that $dx/dt$ must always be positive, this imposes that $L > 0$ for the transition FM→PM, and $L < 0$ for the transition PM→FM (the absolute values being the same). The second rhs term is the magnetocaloric heating/cooling effect that is only relevant to the B-scans. In regime 2 of a first-order transition, i.e., along the development of the transformation, the $C_s$ value to consider must be the “true” heat capacity (i.e., without influence of the latent heat). It primarily contains the lattice term with an additional magnetic contribution keeping the same form as it has in a second-order transition. Similarly, $(dS/dB)_{T_s}$ within the B-range of the transformation must get rid of the high peak reflecting the latent heat [21].

The second equation at the basis of our model is the thermal balance at the level of the top platform of the Peltier cell:

$$C_p \frac{dT_p}{dt} = -\frac{1}{R_g} (T_p - T_s) - \frac{1}{R_p} (T_p - T_b) \quad (4)$$

This equation is the same for T- or B- scans, and it keeps this form in all regimes. The three regimes actually reduce to two for the equations solving (regimes 1 and 3 are driven by the same equations, the difference between them is just a matter of initial conditions). For the regime 2 which deals with the transformation, we consider two approaches: (i) Approach A relies on the assumption that $T_s = T_{tr}$ all along the transformation. It means that $T_s$ is a constant for T-scans and $T_s=T_{tr}(B)$ for B-scans; (ii) In approach B, $T_s$ is let free to vary, and $dx/dt$ is derived from the experimental $J_Q$.

The approach A is a rough approximation, while approach B allows to derive the actual $dx/dt$ evolution. The first step of the analysis consists in solving the equations in terms of $T_s$ and $T_p$ to get $j_Q(t)$. This is then transformed into $j_Q(t^*)$ and finally into $J_Q(t^*)$, allowing us to analyse the experimental data. Results detailed in the Supplemental Material [21] are as follows

### 3.2.a. T-scan

* Regime 1, i.e., before the transition ($t^* \le 0$)

$$m J_{Q1}(t^*) = \dot{T} C_s \quad (5)$$
• Regime 2, i.e., along the transition ($0 \le t^* \le \Delta t_{tr}$)

### Approach A

$$m J_{Q2}(t^*) \approx \dot{T} C_s + \frac{T}{(R_P + R_g)} \left\{ t^* - \tau_d \left[ 1 - \exp\left(-\frac{t^*}{\tau_d}\right) \right] \right\}, \quad (6)$$

where $\tau_d = (\frac{C_P R_g R_P}{R_g + R_P})$ is a “device” time constant that does not depend on $C_s$.

### Approach B

$$T_{s2}(t^*) = -\dot{T} R_P C_P + T_b^{start} - m(R_g + R_P)J_{Q2}(t^*) - m R_P R_g C_P \frac{d J_{Q2}}{dt^*} + \dot{T} t^* \quad (7)$$

$$L \frac{dx}{dt^*} = -\dot{T}C_s/m + J_{Q2}(t^*) + (R_P C_P + R_g C_s + R_P C_s) \frac{d J_{Q2}}{dt^*} + R_P R_g C_P C_s \frac{d^2 J_{Q2}}{dt^{*2}} \quad (8)$$

• Regime 3, i.e., after the transition ($t^* > \Delta t_{tr}$)

$$m J_{Q3}(t^*) = \gamma_L \exp(-t^*/\tau_L) + \gamma_H \exp(-t^*/\tau_H) + \dot{T} C_s \quad (9)$$

The two time constants (low and high) are:

$$\tau_{H,L} = \frac{2 C_P C_s R_g R_P}{(R_P C_s + R_g C_s + R_P C_P) \pm \sqrt{[(R_g C_s - R_P C_P)^2 + R_P C_s (R_P C_s + 2R_g C_s + 2R_P C_P)]}} \quad (10)$$

One observes in $J_{Q3}$ the same $\dot{T}C_s$ term as in $J_{Q1}$, with in addition a transient regime having the form of two exponential functions, whose prefactors must be derived from continuity with regime 2.

## 3.2.b. B-scan

The SM and Fig. 4(d) show an experimental curve of $(dS/dB)_{Ts}$, hereafter simply denoted as $dS/dB$. One can consider a linear-in-$B$ dependence for regimes 1 and 3. They are thus characterized by the double derivative $d^2S/dB^2$. In regime 2, there is a high peak at the transition, that is superimposed onto a basis at $(dS/dB)_{tr}$. This value is the one that must be considered in the model to describe the transformation.

• Regime 1, i.e. before the transition ($t^* \le 0$)

$$J_{Q1}(t^*) \sim T_b \dot{B}^2 \left(\frac{d^2 S}{dB^2}\right)_1 \left\{ t^* + \frac{(T_b - T_c)}{\dot{B} \left(\frac{dT_{tr}}{dB}\right)} - (R_P C_s + R_g C_s + R_P C_P) \right\} \quad (11)$$

• Regime 2, i.e., along the transition ($0 \le t^* \le \Delta t_{tr}$)

### Approach A

$$m J_{Q2}(t^*) = -K^* \exp\left(-\frac{t^*}{\tau_d}\right) - \frac{\left(\frac{dT_{tr}}{dB}\right)_B}{(R_g + R_P)} (t^* - \tau_d) \quad (12)$$

### Approach B

$$T_{s2} = T_b - m(R_g + R_P)J_{Q2} - mR_g R_P C_P \frac{d J_{Q2}}{dt^*} \quad (13)$$
$$L \frac{dx}{dt^*} = J_{Q2} + (R_g C_s + R_P C_s + R_P C_P) \frac{dJ_{Q2}}{dt^*} + R_g R_P C_P C_s \frac{d^2 J_{Q2}}{dt^{*2}} - T_s \dot{B} \left(\frac{dS}{dB}\right)_{tr} \quad (14)$$

• Regime3, i.e., after the transition ($t^* > \Delta t_{tr}$)

$$J_{Q3}(t^*) \sim \delta_L \exp(-t^*/\tau_L) + \delta_H \exp(-t^*/\tau_H) + T_b \dot{B} \left(\frac{dS}{dB}\right)_{tr} + T_b \dot{B}^2 \left(\frac{d^2S}{dB^2}\right)_3 [t^* - (R_P C_s + R_g C_s + R_P C_P)] \quad (15)$$

# 4. Analysis

## 4.1. Before the transition

Regime1 can be used to derive some important parameters: Firstly, Eqs (5) and (9) indicate that plots of $J_Q$ versus $\dot{T}$–in regime 1 and 3 of T-scans– can determine $C_s$ (for regime 3, one considers the constant term superimposed onto the exponential one). Figure 4(a) shows that the expected proportionality is indeed obeyed. Secondly, the B-scan expression of Eq (11) can be rewritten as follows

$$J_{Q1}(B) \sim - \left[ (R_P C_s + R_g C_s + R_P C_P) T_b \dot{B}^2 \left(\frac{d^2 S}{dB^2}\right)_1 \right] + \left[ T_b \dot{B} \left(\frac{d^2 S}{dB^2}\right)_1 \right] B \quad (16)$$

Accordingly, the slope of $J_{Q1}$ versus B plotted versus $\dot{B}$ should yield $T_b \left(\frac{d^2 S}{dB^2}\right)_1$. Figure 4(b) shows that the expected proportionality is actually obeyed, at least within the range of low rates (below about 100 Oe/s).

<figure><img src="image_6.png" /></figure>
FIG. 4: Panel (a): Constant term in the $J_Q(t^*)$ curves of regimes 1 and 3 of T-scans: the squares and circles show the warming and cooling data, respectively; the empty symbols are for regime 1 and the filled ones for regime 3; the lines are fittings showing the expected proportional law. Panel (b): Slope of regime 1 of $J_Q(B)$ on the field-increasing branch as a function of the rate. The line shows the expected proportional law. Panel (c): Heat capacity curve of LaFe$_{11.4}$Si$_{1.31}$Co$_{0.29}$ measured by the $C_p$ option of a PPMS in the standard mode. The dashed arrow represents the value of $C_s$ derived from the mean slope in panel (a). Panel (d): Enlargement of the bottom part of the $-(dS/dB)$ curve derived from the Maxwell equation (see SM). The full line shows the expected proportional law in regime 1. The dashed line shows the value used in the $dx/dt$ calculations.

From the curves of Fig. 4(a), one derives a mean value $C_s \sim 9 \times 10^{-3} \text{ J K}^{-1}$ (with $m = 13$ mg). Figure 4(b) yields a slope $T_b (\frac{d^2S}{dB^2})_1 \sim -3 \times 10^{-8} \text{ J g}^{-1} \text{ Oe}^{-2}$. Since $T_b = 209$ K, one obtains $(\frac{d^2S}{dB^2})_1 \sim -14.35 \text{ J K}^{-1} \text{ kg}^{-1} \text{ T}^{-2}$. Let us now compare these values to those estimated from direct measurements. Figure 4(c) shows the $c_s$ (in J K$^{-1}$ g$^{-1}$) measured in a standard semi-adiabatic calorimeter (C$_p$ PPMS). One observes a clear anomaly located at the $T_c$ but it must be kept in mind that the $c_s$ values derived within the transitional range of a FOT with this kind of calorimeter are incorrect [26-29]. The $C_s$ value derived from panel (a) is materialized by the dashed arrow. This value turns out to incorporate not only the lattice component associated to the roughly flat background, but also a magnetic component consistent with the reversible part of the peak. In other respects, the slope at low fields in Fig. 4(d) yields a value $(\frac{d^2S}{dB^2})_1 = -16 \text{ J K}^{-1} \text{ kg}^{-1} \text{ T}^{-2}$, which is in satisfying agreement with the previous estimate from the $J_Q$ data. These results bring support to the relevancy of the thermal model.

At this stage, we have experimental values of the parameters $C_p$ and $R_p$ (see SM) and now of $C_s$, the only lacking one being $R_g$. It turns out that the analysis of regime 2 within the framework of approach A allows to estimate this last parameter. Indeed, after a transient regime ($t^* \gg \tau_d$), approach A predicts:

$$ \text{For T-scan: } mJ_{Q2}(t^*) \approx \dot{T}C_s + \frac{\dot{T}}{(R_p+R_g)}(t^* - \tau_d) \quad (17a) $$

$$ \text{For B-scan: } mJ_{Q2}(t^*) \approx -\frac{\left(\frac{dT_{tr}}{dB}\right)_B}{(R_g+R_p)}(t^* - \tau_d) - \frac{(T_b-T_c)}{(R_g+R_p)} \quad (17b) $$

Therefore, plots of the average slope of $J_{Q2}(t^*)$ versus the rate allows ($R_g+R_p$) to be derived (see Appendix A). Considering $R_p = 83 \text{ KW}^{-1}$, one obtains $R_g(T_{cool}) \sim 1380 \text{ KW}^{-1}$, $R_g(T_{warm}) = 720 \text{ KW}^{-1}$, $R_g(B_{up}) = 2245 \text{ KW}^{-1}$, and $R_g(B_{do}) = 1445 \text{ KW}^{-1}$. The order of magnitude of these values is in line with what is generally observed for thermal contact with H Apiezon grease [30]. The point is that one observes a rather large scatter within our values. It is known that the thermal contact depends on the nature of the surfaces and the way the sample is pasted,
but in the present case there was no change in the mounting between the various runs of
measurements, so such a wide scatter was not really anticipated.

A closer look at the set of data shows that R<sub>g</sub>(T<sub>cool</sub>) > R<sub>g</sub>(T<sub>warm</sub>) and R<sub>g</sub>(B<sub>up</sub>) > R<sub>g</sub>(B<sub>do</sub>), i.e.
R<sub>g</sub> is systematically higher when the transition goes from the paramagnetic towards the
ferromagnetic state. Taking into account the structural change at the FOT, this may suggest that
the thermal coupling becomes worse when the sample volume increases at the transition. On
other respects, one observes that the R<sub>g</sub> are larger in B- rather than in T-scans. This second
behavior might be ascribable to the magnetic force felt by the sample in field, which can
contribute to weaken the thermal coupling onto the platform.

4.2. After the transition

Equations (9) and (15) predict that the relaxation regime should follow a double
exponential law, for both T- and B-scans. In practice, however, such a type of fitting turns out
to be irrelevant to most of the curves (a problem related to the low value of one of the time
constants, as it will be discussed in next section). We thus limited ourselves to an analysis
involving only one exponential term. Figure 5(a) shows a selection of T-scans in regime 3, with
their fitting. Note that for B-scans, one must add in the fitting a linear term of the type p+qt*
[see Eq (15)].

<figure><img src="image_7.png" /></figure>
**FIG. 5:** (a) Regime 3 of *T*-scans for selected warming rates, with their exponential fitting (black lines). Panels (b) and (c) show the time constants derived from such fittings in the case of the *T*-scan and *B*-scans, respectively.

The time constant values derived from these fittings are reported in panels (b) and (c) of Fig. 5, for *T*- and *B*-scans, respectively. Overall, one observes a decrease in $\tau$ as the rate is increased. For the *T*-scans, one also notes a marked downward step above ~0.6 K/min. For the *B*-scans, the regime 3 escaped the measurements for the decreasing field curves at the highest rates (since the experiments were stopped at zero-field), reason why the curve in panel (c) terminates at about 60 Oe/s. Another salient feature visible in Figs 5(b-c) is a substantial shift between the two curves (i.e., depending on the sense of crossing the transition). One observes that the curves associated to decreasing the temperature or increasing the field are in the higher position, which corresponds in both cases to the transition from PM to FM.

### 4.3. *Time profiles of the transition*

Let us now focus on the main goal of this study, namely the derivation of the transformation rate $dx/dt$. For this purpose, one applied to the regime 2 of *T*- and *B*-scans, the Eqs (8) and (14) of the approach B, respectively. Figure 6 shows examples of this analysis for different conditions: (i) for both types of scans, (ii) for increasing or decreasing the driving parameter, and (iii) for two very different rates in each case.
<figure><img src="image_8.png" /><figcaption>FIG. 6: $Ldx/dt$ curves versus $t^*$ in various conditions: (a) T-scan at a low rate; (b) T-scan at a high rate; (c) B-scan at a low rate; (d) B-scan at a high rate. In each case, there are two curves depending on the sense of crossing the transition. Integration of the colored areas provide the $L$ values reported in Fig 7.</figcaption></figure>

In all case, one observes a series of partly superimposed peaks. This is an expected feature when the individual time constant of each event is close to the separation between them. Moreover, since our resolution (~ 1 s) is also of the same order of magnitude, one cannot exclude that the highest spikes visible in Fig. 6 actually result from the combination of two (or more) closely spaced elemental peaks. The cut at the high-time side of each curve corresponds to the crossover toward an exponential-like regime detected on the $J_Q(t)$, so it necessarily brings some uncertainty. We will discuss in the following Section the difference between the two types of scans and the influences of the various parameters. At this stage, let us just underline that the small spikes visible on the profile of the $dx/dt$ curve for the slowest field rates [illustrated by Fig. 6(c)] do not reflect experimental noise, but they are genuine peaks (i.e., made of several points forming a peak shape).

By definition, the variable $x$ varies between 0 and 1, in such a way that the integration of the curves in Fig. 6 (i.e., the coloured areas) must provide us with the latent heat $L$. The results of these calculations are gathered in Fig. 7. The dashed lines show the values of $L$ expected from the magnetic data, which are different since the T-scans correspond to $B = 0$ T, whereas
the B-scans correspond to $B \sim 0.53$ T. According to the Clausius-Clapeyron equation, one expects $L(206.5 \text{ K}, 0 \text{ T}) = 2.7 \text{ Jg}^{-1}$ and $L(209 \text{ K}, 0.53 \text{ T}) = 2.3 \text{ Jg}^{-1}$.

<figure><img src="image_9.png" /><figcaption>FIG. 7: Latent heat values derived from the integration of the $Ldx/dt$ curves for (a) the T-scans and (b) the B-scans. In each case, one considers curves for increasing or decreasing the driving parameter. The dashed line is the $L$ value expected from the magnetization data. The grey area marks the high rate range for which the data deviate from the expectations.</figcaption></figure>

For the low rates, the agreement between the data and these predictions is remarkable. However, a marked deviation takes place above a certain rate. Several causes can be invoked to explain such a divergence. First, the shrinking of $\Delta t_{tr}$ as the rate is increased leads to a reduced number of data points over the transitional regime, a feature which itself yields an increased uncertainty about the boundaries of regime 2, as well as a decreased reliability of the derivative calculations. Beside these practical issues, one observes that the shape of the $J_Q(t^*)$ response changes at the highest rates, suggesting a crossover towards another regime for which our thermal model does not hold any longer.

## 5. Discussion

Our DSC calorimeter with its thermal modelling can provide an overall picture of the burst-like transition in a giant magnetocaloric material. One clearly distinguishes the peaks in the heat flux, although they are partly superimposed. This overlapping mainly originates from a large time constant in the heat flux transport, that itself results from a substantial thermal resistance at the level of the contact sample/Peltier cell. On the other hand, this slow dynamics combined with the high accuracy of the voltage measurement is favourable to a quantitative approach (herein validated by the latent heat evaluation). This thus offers the opportunity to investigate accurately the comparison between T- and B-scans using the same set-up, as well as the impact of several experimental parameters.
Let us first emphasize the remarkable symmetry of the *dx/dt* equations in *T*- and *B*-scans.
In Eqs (8) and (14), one finds exactly the same components involving *J*<sub>Q</sub>, the only difference
being about the first term, namely *C*<sub>s</sub>(*dT/dt*) for *T*-scans versus *T(dS/db)* for *B*-scans. It deserves
to be noted that this term tends to reduce *dx/dt* (that is > 0 by definition), for both types of scans
and both directions of variation. In other respects, let us emphasize that the leading term is
clearly *J*<sub>Q</sub> in the expressions of *Ldx/dt*, but the corrective terms are not all negligible. As an
illustration, Fig. 8(a) displays the various corrective terms in the case of a warming scan at 0.05
K/min. One observes that the term in *dJ*<sub>Q</sub>*/*dt* is the main origin of the difference between *Ldx/dt*
and *J*<sub>Q</sub>. It also deserves to be noted that the *Ldx/dt* curve exhibits a shape more consistent with
the expectations for a burst-like transformation (series of peaks) than the raw *J*<sub>Q</sub>(*t*) curve. In
particular, *Ldx/dt(t*)* well accounts for the start of regime 2 in the form of a peak rather than a
step. Actually, the integration of *Ldx/dt* is more relevant to assess the degree of progress of the
transition rather than the integration of *J*<sub>Q</sub>, as it could be thought at first glance [10]. In practice,
there is a noticeable difference between the two methods. For instance, in the case of Fig. 8(a),
the integration of *J*<sub>Q</sub> over the duration of regime 2 yields 2.44 J g<sup>-1</sup>, while that of *Ldx/dt* leads
to 2.61 Jg<sup>-1</sup>.

For both *T*- and *B*-scans, the *Ldx/dt* curves are made of closely packed peaks having
similar heights. These patterns are compatible with the literature reporting isolated peaks in
LaFeSiCo alloys of composition close to ours. For instance, Küepferling [10] observed about
six large peaks along *T*-scans at slow rates up to 0.1 K/min, in line with our result [see Fig. 6].
The amplitude of our peaks is also consistent with the literature; indeed, Bennati [16] reported
peaks at 0.2 Wg⁻¹ along *T*-scans at 0.06 K/min, close to the case of Fig. 6.

Qualitative differences can be observed between the *Ldx/dt* curves for *T*- and *B*-scans. In
particular, contrary to *T*-scans in zero-field, the *B*-scans exhibit a background made of closely
packed peaks of smaller amplitude onto which emerge a few higher peaks. It turns out that this
bears similarity with the *T*-scans in fields reported by Bennati, [16] and we suggest to relate
such an effect of *B* to a weakening of the first-order character [see the decrease in *ΔS*<sub>tr</sub>(*B*) in
21], a feature expected to favour a larger number of smaller energy barriers. [24]
<figure><img src="image_10.png" /></figure>

FIG. 8: (a) Plots of the various terms entering the $Ldx/dt$ equation (8), in case of transition induced by warming at 0.05 K/min in zero-field: $T_1$ is $-\dot{T}C_s$; $T_2$ is $J_Q$; $T_3$ is $(R_pC_p + R_gC_s + R_pC_s)\frac{dJ_Q}{dt^*}$; $T_4$ is $R_pR_gC_pC_s\frac{d^2J_Q}{dt^{*2}}$. Total is the sum of these four terms. (b) Plots of $T_p$ and $T_s$ along the transition induced by increasing the field at 5 Oe/s at 209 K, derived from equations (2) and (13) respectively. The dotted line is the linear term of approach A, showing the slope $\dot{B}(\frac{dT_{tr}}{dB})$.

Let us now address the difference between crossing the transition either FM-to-PM or PM-to-FM. For B-scans, one quite clearly observes that the two $Ldx/dt$ curves exhibit a mirror effect as a function of $t^*$. Considering for instance the rate 0.05 K/min in Fig. 6(c), one sees that the top curve starts with a plateau made of small peaks that is followed by two higher peaks; for the bottom curve, it is the reverse order, with two large peaks coming first, and then there is a plateau at lower intensity. This folding effect for curves versus time strongly suggests a swing of the transition front, when the field is either increased or decreased. This is consistent with the measurements performed by Lovell [31] with an array of Hall probes which demonstrated that the transformation front follows the same internal path, back and forth, upon increasing then decreasing the magnetic field.
For the *T*-scans, the result are less clear since our patterns of peaks versus time is quite symmetrical. We note, however, that the raw data rather indicate the same shape of pattern versus time for both warming and cooling, (see Appendix B) i.e., a behavior different from that of the *B*-scans discussed above. On the other hand, one must also consider that, by using a magneto-optics technique, Küepferling [10] have reported identical paths of the transformation front for the temperature-driven FM-PM and PM-FM transitions, i.e. a behavior that would be in line with that of the *B*-scans. It thus appears that this issue about the influence of the direction of the transition crossing in *T*-scans remains controversial.

Let us now address the role of the rate. In all cases (*T*- and *B*-scans), increasing the rate leads to shrink the pattern of peaks while keeping the same shape (see Appendix B). This is consistent with the observations of Bennati [16] who previously reported a “dilation-like” effect of the figure of peaks when lowering the rate in *T*-scans. Such a behavior strongly suggests that the transition proceeds via the same sequence of events. In turn, this likely means that the transition front follows the same spatial path within the sample. We have previously discussed in detail this role of the rate in the same compound on the basis of thermometry data [13]. The present study points to two limits. First, the dilation effect discussed above only holds for not too high rates. In the present study we evidence departures above about 1 K/min for *T*-scans, and approaching our maximum value of 200 Oe/s for *B*-scans. Second, the first peak of the *dx/dt* curve appears to escape this scaling, as detailed in Appendix B.

Our analysis allows to calculate the evolution of $T_s$ along the transformation. While in principle a FOT should proceed at constant temperature, the jerky character of the heat exchange in some magnetocaloric materials yields departures from this ideal situation. An example is shown in Fig. 8(b) in the 5 Oe/s *B*-scan at 209 K. As the field progressively drives the sample state towards FM, the main effect is that $T_s$ tends to follow the $T_{n}(B)$ line. This corresponds to the dashed line given by “approach A” which indeed well reflects the trend. But in practice, the heat release proceeds via so strong bursts that it induces additional jerks on the sample temperature (full line).

While $R_g$ is a decisive parameter, it is ill controlled in practice, and we found that it must be measured in-situ. A preliminary experiment was conducted prior to the complete run presented in this paper; in this previous set of data, it turns out that the $R_g$ was smaller; its estimate led to $R_g = 300$ KW$^{-1}$. The interesting observation is that this reduced $R_g$ corresponds to a shrinking of the peak pattern, while there is conservation of the *L* values (See Fig 9a). This is in qualitative agreement with the observation reported by Lovell [12].
Even if we limit ourselves to pure experimental uncertainty (i.e., omitting the error arising from
the coarseness of the model), many factors must be taken into account. As discussed in [21],
we have come to the conclusion that this uncertainty originates more from the parameters
entering the model than from the raw data themselves. Among these parameters, Rg is
undoubtedly the most problematic, but its individual influence remains moderate. Actually, we
face a situation where many parameters each contribute to a small uncertainty (a few %). Instead
of trying to combine them in a single expression (which is practically infeasible), an alternative
is to examine the discrepancy between the final experimental data (on latent heat for example)
and the theoretical expectations; doing so leads to estimate an overall error of the order of 10
%.

<figure><img src="image_11.png" /><figcaption>FIG. 9: (a): T-scans of the same sample at the same rate, but with two different mounting, leading to two different R_g values. The larger R_g value makes the response to be spread out over a wider time interval (and with peaks of lower intensities); (b): time constant of Eqs (18-19) calculated for the values R_p = 83 K W⁻¹, C_p = 7 mJ K⁻¹ and C_s = 9 mJ K⁻¹, and plotted versus R_g.</figcaption></figure>
To end with, let us now consider in more detail the last regime (regime 3) which can be reasonably well fitted to an exponential law. In our thermal model, however, the transient regime should in principle involve two time constants:

$$ \tau_{H,L} = \frac{2 C_p C_s R_g R_p}{(R_p C_s + R_g C_s + R_p C_p) \pm \sqrt{[R_g C_s - R_p C_p]^2 + R_p C_s (R_p C_s + 2R_g C_s + 2R_p C_p)}}, \quad (18) $$

as well as the so-called “device” time

$$ \tau_d = \frac{C_p R_g R_p}{R_g + R_p}. \quad (19) $$

Actually, all depends on the parameter values. If $C_s \approx C_p$ and $R_g \gg R_p$ (which corresponds to our case), it turns out that $\tau_L$ and $\tau_d$ are close to each other. Indeed, in these conditions, one can estimate $\tau_L \approx C_p R_p \approx \tau_d$, as materialized in Fig. 9(b). One also notes that this $\tau_L \approx \tau_d$ value is in line with that derived from the relaxation experiments with the electrical resistance in the sample position. This value is very low (~1 s), and one can understand that it is hardly detectable with our time resolution of about 1 s. Therefore, one arrives at the conclusion that the accessible relaxation is mainly governed by a unique time constant that can be identified to $\tau_H$.

From the $R_g$ value along each type of scans (see IV.1), one derives from Fig. 9(b) the values: $\tau(T_{cool})=13$ s, $\tau(T_{warm})=7$ s, $\tau(B_{up})=21$ s, and $\tau(B_{do})=14$ s. It is worth noticing that one observes the same hierarchy and overall consistency between these values and the experimental $\tau$ in Fig. 5. For B-scans, these calculated values are even close to the experimental threshold values visible at the highest rates (Fig. 5). It remains that one also observes that these values become larger when decreasing the rate. As a tentative explanation, we suggest that the transformation may not be fully completed at the beginning of what is perceived as the onset of the relaxation regime. One can indeed imagine the persistence of small intermediate zones – still in the basic state– located between the larger ones that were transformed during the events corresponding to the $J_Q(t)$ peaks. Accordingly, the apparent regime 3 might actually mix the transformation of these regions left in the groundstate (i.e. a behavior relevant to regime 2) with the transient regime following the end of the transformation for the main part of the sample that has been yet been transformed (i.e., the genuine regime 3). This may lead to increase the effective time constant of this intermediate regime. This explanation remains conjectural for now.

For T-scans, one observes a similar decrease in the time constants as the rate is increased, but it is more complex since there is pronounced drop around 0.6 K/min. We emphasize that this feature originates from the $J_Q(t)$ data themselves, which exhibits a qualitative change in
their overall shape above this rate. This affects not only the regime 2 (as previously noticed),
but also the regime 3. The nature of this new mechanism of transformation at high T, which
seems to escape our thermal model, remains obscure at the present time.

**6. Conclusion**

We report on a differential scanning calorimeter based on Peltier cells and its thermal
modelling, which allow to derive the rate of transformation (dx/dt) along a magnetostructural
first-order ferromagnetic transition. It can apply to all the best magnetocaloric materials
investigated over the last two decades, such as Gd<sub>5</sub>(Si,Ge)<sub>4</sub>, (MnFe)<sub>2</sub>(P,Si), and La(Fe,Co,Si)<sub>13</sub>
[1,2,23]. As an illustration, this approach has been tested on the giant magnetocaloric alloy
LaFe<sub>11.4</sub>Si<sub>1.31</sub>Co<sub>0.29</sub>. The marked first-order character of this compound and its large entropy
change lead to a jerky transition, both versus T and versus B. With our experimental set-up, we
get an overall snapshot of the heat exchange along the transformation that takes the form of a
succession of peaks. There is a substantial overlapping between these peaks owing to the
sizeable values of the time constant of the peaks themselves and of the time resolution of the
calorimeter. On the other hand, our device allows to reach a good accuracy in the heat flux
measurements (validated by the latent heat derivations) and to perform T-scans and B-scans
routinely with the same apparatus. Thereby, one is in a favorable situation to compare the roles
of several experimental parameters such as the scanning rate, the direction of variation of the
driving parameter, and the thermal conductances. This experimental set-up is thus
complementary to other Peltier-based DSC of the literature which are, for their part, more
efficient to isolate each of the heat flux peaks [15,17].

A model decomposing the thermal response into three regimes is proposed and found to
be able to account for the greatest part of the experimental data. A couple of limits however
appeared. First, we noted that the crossover between regimes 2 and 3 is probably more complex
than assumed in the present vision, since there can be a coexistence between sample regions
already transformed (following regime 3) and other regions not yet transformed (following
regime 2). Second, one observed a clear change in the nature of the thermal response on the T-
scans above a certain rate (about 0.5 K/min and 150 Oe/s).

To go further, one would need to improve the calorimeter to get a better time resolution, especially for exploring the fastest rates. In other respects, basic complements to the present type of study can be provided by magneto-optic techniques, since they allow to get direct access to the spatial development of the transition as a function of time [10,32]. Moreover, the combination of such an imaging technique with an in-situ control of the microstructure could
shed light into several issues. Among them, we point out the question of the relative roles of
the *T*<sub>c</sub> inhomogeneity and that of the defects on the internal path of the transformation front
within the sample, that is still in debate.

CRediT authorship contribution statement

F. **Guillou**: Conceptualization, Funding acquisition, Resources, Formal analysis, Investigation, Validation, Visualization, Writing – review & editing. **B. Huhe**: Investigation, Validation. **F. Veillon**: Investigation, Validation. **V. Hardy**: Conceptualization, Methodology, Formal analysis, Validation, Visualization, Writing – original draft.

**Declaration of Competing Interest**

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

Acknowledgments

This work was funded by the National Natural Science Foundation of China (grant number
12464015). The work of F. Guillou at Caen University was supported by the French
government, managed by the Agence Nationale de la Recherche under the France 2030
programme (ANR-23-EXES-0001), by the Normandy region and a CPJ project.

**Supplementary material**

Supplementary material associated with this article can be found at xxx.

**Appendix A : Estimate of R<sub>g</sub>**

Figure A1 shows examples of regime 2 for *T-* and *B*-scans. A linear fitting is also displayed in each case. On sees that this dependence predicted by “approach A” actually reflects the trend, but it cannot account for the experimental “waves” one observes, since these features are related to the burst-like character of the transition that is not included in the model.
<figure><img src="image_12.png" /><figcaption>**FIG A1:** $J_Q(t*)$ curves in regime 2 at various rates, both for T-scan (a) and B-scans (b). A linear fitting is applied to the data in each case.</figcaption></figure>

According to Eq. (17), the slopes of these linear fitting are expected to be: (i) for the T-scans, $\beta_T = \frac{\dot{T}}{(R_g+R_P)}$, and (ii) for the B-scans, $\beta_B = -\frac{(\frac{dT_{tr}}{dB})\dot{B}}{(R_g+R_P)}$. Therefore, plots of $\beta_T$ or $\beta_B$ versus the rate can be used to estimate $(R_g + R_P)$. In the case of the B-scans, one considers $(\frac{dT_{tr}}{dB}) = 5.1 \text{ KT}^{-1}$. Figure A2 shows that the proportionality between the $\beta$ values and the rate is well obeyed, provided the latter is not too high. From the slope in Figure A2 and the above relationships, one derives values of $(R_g + R_P)$ which are slightly different for each type of scan (see values in Section IV).

<figure><img src="image_13.png" /><figcaption>**FIG. A2:** (a) Slope of $\beta_T$ versus rate for T-scans, for either increasing or decreasing $T$. (b) Slope $\beta_B$ versus rate for B-scans, for either increasing or decreasing $B$. The lines display the law of proportionality visible in each case.</figcaption></figure>

## Appendix B: Impact of the rate

Figure A3 show $J_Q(t*)$ curves at different rates for the two types of scans. One observes a characteristic shape for each series (i.e., for a given type of driving parameter and a given sense of variation) that is independent of the rate. Lowering the rate essentially leads to stretch the curve. Actually, it is possible to roughly superimpose all the curves at different rates by rescaling the time axis, as previously investigated on thermometry data [13].
<figure><img src="image_14.png" /><figcaption>FIG A3: $J_Q(t^*)$ curves at various rates for T-scans (a) and B-scans (b).</figcaption></figure>

Let us now have a closer look at the influence of the sense of variation. The T-scan curves exhibit approximatively the same shape upon cooling and upon warming [Fig. A3(a)], which is not the case for the B-scans [Fig. A3(b)], even when rescaling the time axis. As better visible in the $dx/dt$ curves (still more relevant than $J_Q$ to reflect the development of the transition) one rather observes a reverse order between the field increasing and decreasing curves (see Fig 6).

Figure A4 illustrates the scaling behavior applicable to the T-scans by displaying two curves with rates differing by an order of magnitude. When multiplying the time values of the curve at higher rate by an adhoc value, one observes a striking coincidence between the peak patterns of both curves [Fig. A4(a)].

<figure><img src="image_15.png" /><figcaption>FIG. A4: Curves $Ldx/dt$ of T-scans upon cooling for 0.05 and 0.5 K/min. In (a), the $t^*$ values of the 0.5 K/min data where multiplied by the scaling constant 2.07. Panel (b) shows the beginning of the raw curves at both rates.</figcaption></figure>

Although there is a remarkable overall resemblance, one can denote however that it does not work so well in the initial stage of the transition. As shown in Fig. A4(b) which displays an enlargement at low-time of the raw $J_Q(t^*)$ without correction, it turns out that the first peak of the $dx/dt$ curve takes place at roughly the same time for the two curves. This feature has been previously detected in a thermometry study, where it was ascribed to the specific dynamics of
the first burst accompanying the entry of the propagation front inside the sample that is uniquely
governed by intrinsic parameters of the sample [13].

References

[1] J. Lyubina, Magnetocaloric materials for energy efficient cooling, J. Phys. D: Appl. Phys. 50 (2017) 053002.
https://dx.doi.org/10.1088/1361-6463/50/5/053002

[2] V. Franco, J.S. Blázquez, J.J. Ipus, J.Y. Law, L.M. Moreno-Ramírez, A. Conde, Magnetocaloric effect: From materials research to refrigeration devices, Prog. Mater. Sci. 93 (2018) 112-232.
https://doi.org/10.1016/j.pmatsci.2017.10.005

[3] A. A. Beleza, B. Pires, R. Almeida, J. S.O. Evans, A. M. dos Santos, M. Frontzek, E. Lovell, B. Beckmann, K. Skokov, O. Gutfleisch, J. P. Araujo, A.M. M. Abeykoon, J. S. Amaral, J. H. Belo, Structural dynamics of first-order phase transition in giant magnetocaloric La(Fe,Si)₁₃: The free energy landscape, Mater. Today Phys. 42 (2024) 101388. https://doi.org/10.1016/j.mtphys.2024.101388

[4] V. K. Pecharsky, K. A. Gschneidner, Jr., D. Fort, Superheating and other unusual observations regarding the first order phase transition in Dy, Scr. Mater. 35 (1996) 843-848. https://doi.org/10.1016/1359-6462(96)00225-4

[5] K. A. Gschneidner, V. K. Pecharsky, D. Fort, Novel Thermal Effects at the First Order Magnetic Phase Transition in Erbium, and a Comparison with Dysprosium, Phys. Rev. Lett. 78 (1997) 4281.
https://doi.org/10.1103/PhysRevLett.78.4281

[6] P. Bag, R. Rawat, P. Chaddah, P. D. Babu, and V. Siruguri, Unconventional thermal effects across first-order magnetic transition in the Ta-doped HfFe₂ intermetallic, Phys. Rev. B 93 (2016) 014416.
https://doi.org/10.1103/PhysRevB.93.014416

[7] F. Guillou, H. Yibole, Z. Q. Ou, E. Brück, O. Tegus, Large recalescence-like event at the first cooling across the magnetic transition of (Mn,Fe)₂(P,Si) magnetocaloric materials, Scripta Mater. 160 (2019) 81-85.
https://doi.org/10.1016/j.scriptamat.2018.10.002

[8] V. Hardy, R. Hamane, X. Larose, M. Risser, F. Guillou, Burstlike first-order transformation studied by semi-adiabatic relaxation calorimetry, J. Appl. Phys. 130 (2021) 165106. https://doi.org/10.1063/5.0070977

[9] V. Hardy, R. Hamane, F. Veillon, M. Risser, F. Guillou, “Two-steps” process in the first-order transformation of giant magnetocaloric materials, Acta Mater. 231 (2022) 117869. https://doi.org/10.1016/j.actamat.2022.117869

[10] M. Kuepferling, C. Bennati, F. Laviano, G. Ghigo, V. Basso, Dynamics of the magneto structural phase transition in La(Fe<sub>0.9</sub>Co<sub>0.015</sub>Si<sub>0.085</sub>)<sub>13</sub> observed by magneto-optical imaging, J. Appl. Phys. 115 (2014) 17A925.
https://doi.org/10.1063/1.4866880

[11] A. Waske, L. Giebeler, B. Weise, A. Funk, M. Hinterstein, M. Herklotz, K. Skokov, S. Fähler, O. Gutfleisch, J. Eckert, Asymmetric first-order transition and interlocked particle state in magnetocaloric La(Fe,Si)₁₃, Phys. Status Solidi-Rapid Res. Lett. 9 (2015) 136-139. https://doi.org/10.1002/pssr.201409484
[12] E. Lovell, M. Bratko, A. D. Caplin, and L. F. Cohen, Nucleation and dynamics of the metamagnetic transition in magnetocaloric La(Fe,Mn,Si)₁₃, J. Phys. D: Appl. Phys. 50 (2017) 424004. https://doi.org/10.1088/1361-6463/aa893e

[13] F. Guillou and V. Hardy, Burst-like features observed from different techniques at the first-order magnetic transition of La(Fe,Co,Si)₁₃, J. Alloys Compd. 977 (2024) 173161. https://doi.org/10.1016/j.jallcom.2023.173161

[14] T. Plackowski, Y. Wang, A. Junod, Specific heat and magnetocaloric effect measurements using commercial heat-flow sensors, Rev. Sci. Instrum. 73 (2002) 2755-2765. https://doi.org/10.1063/1.1480452

[15] V. Basso, C. P. Sasso, M. Küpferling, A Peltier cells differential calorimeter with kinetic correction for the measurement of $c_p(H,T)$ and $\Delta s(H,T)$ of magnetocaloric materials, Rev. Sci. Instrum. 81 (2010) 113904. https://doi.org/10.1063/1.3499253

[16] C. Bennati, L. Gozzelino, E. S. Olivetti, V. Basso, Heterogeneous nucleation and heat flux avalanches in La(Fe,Si)₁₃ magnetocaloric compounds near the critical point, Appl. Phys. Lett. 109 (2016) 231904. https://doi.org/10.1063/1.4971360

[17] F. Erbesdobler, C.R.H. Bahl, R. Bjørk, and K.K. Nielsen, Detailed isofield calorimetry of La(Fe,Si,Mn)H reveals distributed magnetocaloric phase transitions, J. Appl. Phys. 127 (2020) 095102. https://doi.org/10.1063/1.5137790

[18] PPMS Heat Capacity Option User's Manual, 1085-150 (January 2009), Quantum Design, San Diego.

[19] S. C. Mraw, Mathematical treatment of heat flow in differential scanning calorimetry and differential thermal analysis instruments, Rev. Sci. Instrum. 53 (1982) 228-231. https://doi.org/10.1063/1.1136933

[20] J. Y. Xu, F. Guillou, H. Yibole, V. Hardy, Peltier cell calorimetry “as an option” for commonplace cryostats: Application to the case of MnFe(P,Si,B) magnetocaloric materials, Fundam. Res. 4 (2024) 1465-1473. https://doi.org/10.1016/j.fmre.2022.09.020

[21] See Supplemental Material at http://xxx.

[22] M. Katter, V. Zellmann, G. W. Reppel, K. Uestuener, Magnetocaloric Properties of La(Fe,Co,Si)₁₃ Bulk Material Prepared by Powder Metallurgy, IEEE Trans. Magn. 44 (2008) 3044-3047. https://doi.org/10.1109/TMAG.2008.2002523

[23] T. Gottschall, K. P. Skokov, M. Fries, A. Taubel, I. Radulov, F. Scheibel, D. Benke, S. Riegg, O. Gutfleisch, Making a Cool Choice: The Materials Library of Magnetic Refrigeration, Adv. Energy Mater. 9 (2019) 1901322. https://doi.org/10.1002/aenm.201901322

[24] M. Piazzi, C. Bennati, V. Basso, Thermodynamics of the Heat-Flux Avalanches at the First-Order Magnetic Transition in Magnetocaloric Materials, Phys. Rev. Applied 8 (2017) 044023. https://doi.org/10.1103/PhysRevApplied.8.044023

[25] K. K. Nielsen, H. N. Bez, L. von Moos, R. Bjørk, D. Eriksen, C. R. H. Bahl, Direct measurements of the magnetic entropy change, Rev. Sci. Instrum. 86 (2015) 103903. https://doi.org/10.1063/1.4932308

[26] J. C. Lashley, M. F. Hundley, A. Migliori, J. L. Sarrao, P. G. Pagliuso, T. W. Darling, M. Jaime, J. C. Cooley, W. L. Hults, L. Morales, D. J. Thoma, J. L. Smith, J. Boerio-Goates, B. F. Woodfield, G. R. Stewart, R. A. Fisher,
N. E. Phillips, Critical examination of heat capacity measurements made on a Quantum Design physical property measurement system, Cryogenics 43 (2003) 369. https://doi.org/10.1016/S0011-2275(03)00092-4

[27] V. Hardy, Y. Bréard and C. Martin, Derivation of the heat capacity anomaly at a first-order transition by using a semi-adiabatic relaxation technique, J. Phys. Cond. Matter 21 (2009) 075403. https://dx.doi.org/10.1088/0953-8984/21/7/075403

[28] H. Suzuki, A. Inaba, and C. Meingast, Accurate heat capacity data at phase transitions from relaxation calorimetry, Cryogenics 50 (2010) 693-699. https://doi.org/10.1016/j.cryogenics.2010.07.003

[29] P. F. Rosen, B. F. Woodfield, Standard methods for heat capacity measurements on a Quantum Design Physical Property Measurement System, J. Chem. Thermodynamics 141 (2020) 105974. https://doi.org/10.1016/j.jct.2019.105974

[30] M. Kuepferling, C. P. Sasso, V. Basso, Rate dependence of the magnetocaloric effect in La-Fe-Si compounds, EPJ Web Conf. 40 (2013) 06010. https://doi.org/10.1051/epjconf/20134006010

[31] E. Lovell, K. Morrison, A. M. Pereira, D. Caplin, O. Gutfleisch, L. F. Cohen, Scanning Hall Probe Imaging of LaFe$_{13-x}$Si$_x$, Advances in Science and Technology 93 (2014) 219. https://doi.org/10.4028/www.scientific.net/AST.93.219

[32] C. Bennati, F. Laviano, G. Durin, E. Olivetti, V. Basso, G. Ghigo, M. Küpferling, Local magnetic behavior across the first order phase transition in La(Fe$_{0.9}$Co$_{0.015}$Si$_{0.085}$)$_{13}$ magnetocaloric compound, J. Magn. Magn. Mater. 400 (2016) 339-343. https://doi.org/10.1016/j.jmmm.2015.07.105

[33] F. Guillou, B. Huhe, F. Veillon, V. Hardy, Kinetics of the phase transformation in a giant magnetocaloric material along T- and B-scans [dataset], Mendeley Data, V1, https://doi.org/10.17632/gs57dg4hbs.1