V Hardy

R Hamane

F Veillon

M Risser

F Guillou

"Two-steps" process in the first-order transformation of giant magnetocaloric materials

Keywords: Phase transformation, Interfacial misfit, Gibbs energy, Magnetic properties, La(Fe,Si)13 based compound

The most promising materials for applications in magnetic refrigeration exhibit a large entropy change at a first-order ferromagnetic transition. Often, it has been observed that the transformation proceeds via two bursts surrounding a regime of slow transformation. Within a simplified framework, the present article reports on a thermodynamic model able to account for this "twosteps" response. This model is confronted to magnetic and calorimetric data recorded in the magnetocaloric compound LaFe11.4Si1.31Co0.29 .

Introduction

Giant magnetocaloric materials have received intense attention over the past two decades, primarily due to their potential applications in magnetic refrigeration that is a technique more environmentally friendly than the conventional gas-vapor technology [START_REF] Gschneidner | Recent developments in magnetocaloric materials[END_REF][START_REF] Smith | Materials Challenges for High Performance Magnetocaloric Refrigeration Devices[END_REF][START_REF] Franco | Magnetocaloric effect: From materials research to refrigeration devices[END_REF][START_REF] Gottschall | Making a Cool Choice: The Materials Library of Magnetic Refrigeration[END_REF]. It turns out that the most promising materials are those exhibiting a large entropy change at a first-order ferromagnetic transition. The dynamics of such a first-order transition (FOT) involves both a magnetic transformation (between paramagnetic and ferromagnetic states) and a change in the structural parameters (keeping the same symmetry or not, depending on the systems [START_REF] Lyubina | Magnetocaloric materials for energy efficient Cooling[END_REF]). The dynamical aspect is a crucial issue, not only from a fundamental viewpoint but also for the applicative potential of the magnetocaloric effect [START_REF] Küepferling | Dynamics of the magneto structural phase transition in La[END_REF][START_REF] Lovell | Magnetic relaxation dynamics driven by the first-order character of magnetocaloric La(Fe,Mn,Si)13[END_REF][START_REF] Lovell | Nucleation and dynamics of the metamagnetic transition in magnetocaloric La(Fe,Mn,Si)13[END_REF][START_REF] Zavareh | Direct Measurement of the Magnetocaloric Effect in La(Fe,Si,Co)13 Compounds in Pulsed Magnetic Fields[END_REF][START_REF] Basso | Hysteresis and Phase Transition Kinetics in Magnetocaloric Materials[END_REF].

Recent studies have revealed peculiar features among which one of the most striking is an exchange of latent heat at the transformation taking the form of a succession of individual peaks as a function of time (yielding "spiky" patterns) [START_REF] Küepferling | Dynamics of the magneto structural phase transition in La[END_REF][START_REF] Piazzi | Thermodynamics of the heat-flux avalanches at the first-order magnetic transition in magnetocaloric materials[END_REF][START_REF] Erbesdobler | Detailed isofield calorimetry of La(Fe,Si,Mn)-H reveals distributed magnetocaloric phase transitions[END_REF]. Another intriguing feature that is often observed is a "two-steps" shape; in this case, the exchange of latent heat mainly exhibits two intense peaks separated by a plateaulike regime [START_REF] Fujita | Influence of Supercooling on the Thermally Induced First-Order Magnetic Transition in Magnetocaloric Compound La(Fe0.88Si0.12)13[END_REF][START_REF] Bennati | Heterogeneous nucleation and heat flux avalanches in La(Fe, Si)13 magnetocaloric compounds near the critical point[END_REF][START_REF] Guillou | Large recalescence-like event at the first cooling across the magnetic transition of (Mn,Fe)2(P,Si) magnetocaloric materials[END_REF]. In both cases, the basic requirement is a high latent heat confined within a small temperature range, but the conditions for the observation of these particular responses also involve the degree of thermalisation of the sample with the base temperature and the sweep rate of the driving parameter around the transition. While "spiky patterns" require very slow sweep rates and small thermal resistance between the sample and the thermal bath, less stringent conditions seem to be necessary for the observation of the "two-steps" shape.

Various manifestations of the "two-steps" response can be found in the literature; it turns out that the greatest part of these studies were carried out in the La(Fe,Si,Mn,Co)13 system which is the most investigated magnetocaloric family nowadays. In one of the first investigations of the transitional kinetics in these materials, Fujita et al. reported a clear two-steps signature in the cooling curve of LaFe11.44Si1.56 [START_REF] Fujita | Influence of Supercooling on the Thermally Induced First-Order Magnetic Transition in Magnetocaloric Compound La(Fe0.88Si0.12)13[END_REF]. Two-steps patterns were also observed in the percentage of transformation between the two phases around the TC, derived either from x-ray diffraction in LaFe11.8Si1.2 [START_REF] Waske | Asymmetric first-order transition and interlocked particle state in magnetocaloric La(Fe,Si)13[END_REF] or from magnetization measurements in LaFe11.7Co0.195Si1.105 [START_REF] Bennati | Local magnetic behavior across the first order phase transition in La[END_REF]. Moreover, one can note that the profiles of heat flux exchanges detected by differential scanning calorimetry (DSC) based on Peltier cells (LaFe11.41Mn0.30Si1.29 -H1.62 in [START_REF] Bennati | Heterogeneous nucleation and heat flux avalanches in La(Fe, Si)13 magnetocaloric compounds near the critical point[END_REF]) or from the thermopile voltage in a microcalorimeter (LaFe11.74Mn0.06Si1.20 in [START_REF] Lovell | Nucleation and dynamics of the metamagnetic transition in magnetocaloric La(Fe,Mn,Si)13[END_REF]) exhibited a shape dominated by two wellseparated large peaks.

The goal of the present paper is to suggest a qualitative explanation for this type of two-steps process, on the basis of a simple thermodynamic approach.

Experimental details

The study was carried out on an industrial material belonging to the La(Fe,Si,Co)13 family provided by Vacuumschmelze GmbH and CoKG. Compositional analyses carried out by energy-dispersive x-ray spectroscopy in a scanning electron microscope (Hitachi SU8010 equipped with IXRF systems spectrometer) lead to the formula LaFe11.4Si1.31Co0.29. Different samples from the same batch and with the same size (2.22.20.4 mm 3 ) were used for magnetic and calorimetric measurements.

Magnetization curves as a function of temperature, M(T), were recorded in a commercial device (PPMS, Quantum Design), at a slow sweep rate crossing the transition both upon warming and cooling, in a small magnetic field of 100 Oe. Two types of calorimetric measurements were recorded in zero-field. Firstly, specific heat curves, C(T), were derived from an home-made Peltier-based calorimeter, employing a Quantum Design Versalab system for temperature control, a Keithley 2182A nanovoltmeter for recording the potential difference and RMT (model 1MD03-018) miniature thermoelectric elements mounted in differential configuration as sensors. Secondly, relaxation curves were recorded in the semi-adiabatic configuration of the specific-heat option of the PPMS: the sample is pasted on a sapphire platform at the bottom of which are placed a heater and a thermometer; this platform is connected to the system by wires which ensure the thermal contact with the base temperature; after stabilization at the base temperature (T < TC), heating power triggers an increment of temperature (up to T > TC), then it is shut down. In the present work, we focus on the last stage of the process, i.e. the relaxation curve along which the temperature of the sample decreases with time down to the base temperature while crossing TC.

Results

Experimental data

Figure 1(a) shows M(T) curves recorded around the transition, at a rate of 0.1 K/min. One clearly observes hysteresis and a transition that proceeds by steps. The density of data points is limited by the combination of the sweep rate and of the intrinsic duration of each measurement ( 15 s) leading to a spacing of  0.03 K. It is on the cooling curve that the "two-steps" shape is the most visible, a feature already noted in previous studies [START_REF] Waske | Asymmetric first-order transition and interlocked particle state in magnetocaloric La(Fe,Si)13[END_REF][START_REF] Bennati | Local magnetic behavior across the first order phase transition in La[END_REF].

Figure 1(b) shows specific-heat data measured in the DSC device when using the same rate of 0.1 K/min. On each branch (either cooling or warming), one observes a steep rise at the beginning of the transition, followed by a slowdown and finally a large peak at the end; this sequence yields an overall shape that can be regarded as double peak. Let us specify that the discrepancies in the values of the transition temperature between Figs 1(a) and 1(b) can be mainly ascribed to a variation in the thermometry of the devices (dealing primarily with the calibration of the thermometers and the degree of thermalisation of the samples). Panels (e) and (f) illustrate the reproducibility of the phenomenon. Qualitatively, the same two-steps feature is observed in various samples for given procedure (e.g. a fixed cooling rate), as well as for different procedures when using the same sample (with the same mounting).

C (J K -1 g -1 ) T (K) (b)
T (K) (a) Cool Warm T c0 Figure 1
It deserves to be noted that the two types of calorimetric data are consistent with each other if one admits that the peaks in Fig. 1(b) correspond to bursts in the release of latent heat yielding the rises of temperature on each side of the plateau in Fig. 1(d).

Furthermore, we emphasize that the thermal data are well complemented by the magnetic ones. Indeed, the magnetization does not probe the exchange of heat, but it directly reflects the degree of phase transformation, a piece of information which is lacking in calorimetric experiments. Combining these techniques suggest that sudden advances in the phase transformation are concomitant with sudden changes in exchange of heat. The largest events take place at the beginning and end of the transformation, constituting what is referred to as the "two-steps" response in this article.

Model

The starting idea was to explore the possibility of a transformation path energetically more favorable than a continuous switching between the two phases. We do not pretend to go beyond a "toy model", because we will neglect the details of the microstructure, while it is known to be a key aspect for the development of the FOT in this type of materials [START_REF] Lyubina | Novel Design of La(Fe,Si)13 Alloys Towards High Magnetic Refrigeration Performance[END_REF]. Our approach involves the observation that the transformation generally starts from the edges of the sample, and then progressively invades it till its center [START_REF] Gottschall | A Matter Size and Stress: Understanding the First-Order Transition in Materials for Solid-State Refrigeration[END_REF]. This feature is particularly marked for the PM-to-FM transformation, as attested to by magneto-optics experiments [START_REF] Küepferling | Dynamics of the magneto structural phase transition in La[END_REF][START_REF] Bennati | Local magnetic behavior across the first order phase transition in La[END_REF]. In what follows, one considers this PMto-FM direction of the transition (which is that of the relaxation curves). It can be noted that the development of the ferromagnetism from the sides is qualitatively consistent with the fact that the external surfaces can help to accommodate the greater unit-cell volume of the FM phase. Another important issue to be taken into account is that the problem requires to consider the sample as a whole, with its own volume and surface. For the sake of generality and simplicity, we will assume a cubic sample of side L0. One also assumes an isotropic progress of the transition that can be tracked by a single parameter L corresponding to the side of the internal cube of material that is not yet transformed.

Within the frame of these approximations, the simple picture of the transition considered in this paper is shown in Fig. 2. We are examining the possibility of two bursts along the transformation, the first one at L1 and the second one at L2; in between (L2 < L < L1) the transformation can be gradual. The driving parameter of the transition is the temperature (at least till reaching TC), but along the transformation itself, it can also be the time per se. The compounds of the La(Fe,Si,Mn,Co)13 system exhibit a cubic-to-cubic structural change at the TC [START_REF] Fujita | Itinerant-electron metamagnetic transition and large magnetovolume effects in La(FexSi1-x)13 compounds[END_REF][START_REF] Hu | Magnetocaloric effect in itinerant electron metamagnetic systems La(Fe1-xCox)11.9Si1.1[END_REF] As schematically displayed in Fig. 2(f), the unit-cell volume V is larger in the FM state, with a relative change (dV/V) of the order of 1%;

thereafter, one considers the relative increase in the unit-cell parameter i.e., 𝑝 = Owing to the negative step in V(TTC), the application of pressure shifts the transition towards lower temperature.

Along the PM-to-FM transformation, one expects the development of strain at the boundary and of long-range stress from these boundaries [START_REF] Bez | Strain development during the phase transition of La(Fe,Mn,Si)13-Hz[END_REF]. This stress induces an increase in pressure in the central part of the sample that is not yet transformed. Figures 2(g-i) illustrate schematically the way one takes into account the change of pressure associated with the progression of the transformation. In the range L2 < L < L1, the variation from L to L+dL (dL < 0) is decomposed into two parts: first, a virtual transformation PM-to-FM of a slab L without change of unit -cell; then, the application of the strain   p to reach the increment dL = L(1+p). It is this latter part which induces an increase of (internal) pressure at long range. This is a crucial issue since the TC (thereby, the strength of the driving force of the transformation at a given temperature T)

depends on the pressure. the color for PM evolves from yellow (stress-free) to red as the pressure is increased. The stars show the location of the bursts along this evolution. The bottom panels focus on the pressure effect: (f) shows the impact on the TC, while the set (g-i) illustrates the approach used to estimate the relationship between L and the internal pressure P (see text).

Figure 3(a) schematically illustrates this relationship by showing the Gibbs free energy lines, G(T), of the PM and FM states for zero (L = 0 or L = L0) and non-zero pressure ( 0 < L < L0). For simplicity, the pressure effect in this plot is entirely ascribed to the PM state, that is acceptable since the important parameter in the end is the driving force, i.e. the difference between the two states. Note also that we approximate the real temperature dependencies by linear curves, which is another reasonable simplification if we limit ourselves to the region around the transition. These lines refer to each state individually, without interaction between them. In principle, the transition from one state to the other is expected to take place at the crossing point between these lines [see Fig.

3(a)].

In practice, however, extra energy terms come into play when considering how the PM FM

L 0 L = L 1 L = L 2 L L 0 (a) (b) (c) (d) (e)
Driving Parameter

FM PM V cell T T c (P) T c (P+dP) (f) L P (g) L+L (h) L+dL P+dP (i)
We emphasize that the transformation is considered to develop at a temperature T = Ttr that is slightly lower than the TC (as defined above). In Fig. 2, the top row of panels is a schematic picture of the suggested transformation path to account for the existence of the two bursts. The degree of progress of the transition is followed by the variable L that must go from L0 to 0. Panel (a) is the starting point, corresponding to a purely PM state. We consider that the system is at a temperature 𝑇 ≲ 𝑇 𝐶 (𝐿 0 ) = 𝑇 𝐶0 , for which the transformation PM-to-FM should have started without the presence of energy terms related to the mechanical friction between corresponds to the stage where it becomes beneficial for the system to eliminate the interfacial energy (PM/FM) by switching abruptly to the fully FM state. This process is ascribed to the second burst, leading to a purely ferromagnetic state that is displayed in panel (e). This transformation path thus goes through two bursts (marked by the two black stars in Fig. 2). In practice, the release of latent heat at each of these bursts is expected to be so sudden that a transitory increase in the sample temperature can hardly be avoided.

Let us now explore the possibility to account for such a path within a thermodynamical approach. We use hereafter lower case letters (g) for the total enthalpy of the sample, including all relevant terms, while upper case letters (G) correspond to the (intrinsic) volume enthalpies of the FM or PM states. There are three expressions of the Gibbs free energy of the sample to be compared: (i) gPM for the "full PM" state (L = L0), as shown in Fig. 2(a); (ii) gFM for the « full FM » state (L = 0) of Fig. 2(e); and (iii) gmix for a "mixed" state (L0 < L < 0), combining PM and FM parts, as illustrated in Fig. 2(c).

For the first two cases, the sample Gibbs free energy can be simply written as (the variable L0 meaning the absence of internal pressure):

𝑔 𝑃𝑀 (𝑇) = 𝐿 0 3 𝐺 𝑃𝑀 (𝑇, 𝐿 0 ) (2a) 𝑔 𝐹𝑀 (𝑇) = 𝐿 0 3 𝐺 𝐹𝑀 (𝑇, 𝐿 0 ) (2b)
The case of gmix is more complex, since it is a combination of several terms.

Adopting the terminology used for the martensitic transformations, we will consider gchem , ginter and gelas, being the chemical, interfacial, and elastic terms, respectively [START_REF] Soffa | Diffusional Phase Transformations in the Solid State[END_REF][START_REF] Chen | Coupled kinetic Monte Carlo-finite element mesoscale model for thermoelastic martensitic phase transformations in shape memory alloys[END_REF].

The gchem term depends on the volumic fractions of each of the two states; in the mixed regime, a volume (L0 3 -L 3 ) has become FM, leading to 𝑔 𝑐ℎ𝑒𝑚 (𝑇, 𝐿) = ( 𝐿 0 3 -𝐿 3 )𝐺 𝐹𝑀 (𝑇) + 𝐿 3 𝐺 𝑃𝑀 (𝑇, 𝐿). In connection with Fig. 3

The coexistence of PM and FM domains yields stress between them owing to the difference in unit-cell volumes. First, this generates an increase of energy associated to the boundaries between the two phases, corresponding to ginter. Second, the displacement of this boundary induces a structural distortion and thus an elastic energy corresponding to gelas. Lastly, one should also consider the pressure effect at longer range, but this is already taken into account via the L dependence of ∆𝐺(𝑇, 𝐿) in gchem.

friction between the two phases at the front separating them. In the simple picture of Fig. 2, there is a contact area of 6L 2 as soon as L  L0, leading to

𝑔 𝑖𝑛𝑡𝑒𝑟 (𝑇, 𝐿) = 6𝜆𝐿 2 (4)
Note that this interfacial energy must be distinguished from the surface tension that is involved in calculations of critical sizes for nucleation [START_REF] Soffa | Diffusional Phase Transformations in the Solid State[END_REF].

The elastic term derives from the combination of the stress ( ) and strain ( ) within the transformed zone. Per unit volume, one expects a contribution of the type,

𝜎𝜀 = 𝐵 1-𝜈 𝜀 2
, where B is the average bulk modulus of the two phases around TC,  the Poisson's ratio and   p. Each variation by dL (< 0) corresponds to a volume -3𝐿 2 𝑑𝐿 , thus the integration along the transformation (from L0 to L) yields

𝑔 𝑒𝑙𝑎𝑠 (𝑇, 𝐿) = ∫ 𝐵𝑝 2 1-𝜈 (-3𝐿 2 𝑑𝐿) = 𝐵𝑝 2 1-𝜈 (𝐿 0 3 -𝐿 3 ) 𝐿 𝐿 0 (5)
Adding these various terms ( gm = gchem + ginter + gelas), one arrives at

𝑔 𝑚 (𝑇, 𝐿) = 𝐿 0 3 {𝐺 𝐹𝑀 (𝑇) + ( 𝐵𝑝 2 1-𝜈 )} + 𝐿 3 {Δ𝐺(𝑇, 𝐿) -( 𝐵𝑝 2 1-𝜈 )} + 6𝜆𝐿 2 (6)
The problem at this stage is to evaluate the dependence of G on L. ).

Turning back to gm [Eq. ( 6)], this yields

𝑔 𝑚 (𝑇, 𝐿) = 𝐿 0 3 {𝐺 𝐹𝑀 (𝑇) + ( 𝐵𝑝 2 1 -𝜈 )} + 𝐿 3 [∆𝑆 𝑡𝑟 (𝑇 𝐶0 -𝑇) + ∆𝑆 𝑡𝑟 × ( 𝑑𝑇 𝐶 𝑑𝑃 ) × (-3𝐵𝑝) × ln ( 𝐿 𝐿 0 ) -( 𝐵𝑝 2 1-𝜈 )] + 6𝜆𝐿 2 (10)
Let us introduce two positive constants :

𝑋 = ∆𝑆 𝑡𝑟 × ( 𝑑𝑇 𝐶 𝑑𝑃 ) × (-3𝐵𝑝) (11a)
𝑌 = ( 𝐵𝑝 2 1-𝜈 ) (11b)
The three expressions of the sample Gibbs free energy to be considered become: The most stable state at a temperature T is expected to be the one yielding the lowest g*(T) value.

𝑔 𝑃𝑀 (𝑇) = 𝐿 0 3 𝐺 𝑃𝑀 (𝑇, 𝐿 0 ) (12a) 𝑔 𝑚 (𝑇, 𝐿) = 𝐿 0 3 [𝐺 𝐹𝑀 (𝑇) + 𝑌] + 𝐿 3 [∆𝑆 𝑡𝑟 (𝑇 𝐶0 -𝑇) -𝑌] + 𝐿 3 𝑋 ln ( 𝐿 𝐿 0 ) + 6𝜆𝐿 2 (12b) 𝑔 𝐹𝑀 (𝑇) = 𝐿 0 3 𝐺 𝐹𝑀 (𝑇, 𝐿 0 ) (12c)

Discussion

At this stage, one must evaluate the model parameters. For most of them, reasonable estimates can be extracted from the literature.

-Bulk modulus. In a Co-free compound, LaFe11.18Si1.82, (with TC  210 K), highpressure x-ray diffraction experiments yield a bulk modulus B  125 GPa [START_REF] Valiev | Effect of Hydrostatic Pressure on the Magnetic and Lattice Properties of the Ferromagnet La(Fe0.86Si0.14)13[END_REF]. In a composition closer to ours, (LaFe11.74Co0.13Si1.13), the same approach led to B  100 GPa [START_REF] Zavareh | Direct Measurement of the Magnetocaloric Effect in La(Fe,Si,Co)13 Compounds in Pulsed Magnetic Fields[END_REF]. The same order of magnitude (80-110 GPa) was also obtained from mechanical techniques in LaFe11.09Co0.86Si1.05, LaFe10.81Co1.18Si1.01, LaFe10.63Co1.33Si1.04, and LaFe10.54Co1.43Si1.03 [START_REF] Kaeswurm | Behaviour of the Young's modulus at the magnetocaloric transition in La (Fe,Co,Si)13[END_REF].

-Poisson's ratio. For the type of alloys one deals with, it is customary to consider the approximation   0.3 [START_REF] Gottschall | A Matter Size and Stress: Understanding the First-Order Transition in Materials for Solid-State Refrigeration[END_REF][START_REF] Guillou | About the mechanical stability of MnFe(P,Si,B) giant-magnetocaloric materials[END_REF][START_REF] Glushko | Mechanical properties of the magnetocaloric intermetallic LaFe11.2Si1.8 alloy at different length scales[END_REF] -Pressure dependence of TC. Actually, there are not so many data about this pressure dependence. In the case of LaFe11.44Si1.56 and LaFe11.18Si1.86, Fujita et al. reported -Isomorphic change

In the La(Fe,Si,Co)13 family, V/V is of the order of 1%, [START_REF] Fujita | Itinerant-electron metamagnetic transition and large magnetovolume effects in La(FexSi1-x)13 compounds[END_REF][START_REF] Hu | Magnetocaloric effect in itinerant electron metamagnetic systems La(Fe1-xCox)11.9Si1.1[END_REF] thus p  0.01/3.

for a mixed state : L1 < L < 1 and 0 < L < L2. This is qualitatively consistent with the observations. Inset (b) shows the same data but with circles marking the expected magnetic states: by definition, full PM and full FM correspond to z = 1 and z = 0, respectively; for the mixed state, one can either envision a continuous evolution of states within the range L1-L2 ( open circles) or consider that the system will lock itself to the state of minimal gm at z  0.8 (full green circle).

These evolutions are represented in inset (c) in terms of percentage of transformation ( z 3 ) as a function of the driving parameter (i.e., decreasing temperature or increasing time when the transformation is taking place). One observes two jump-like evolutions at the beginning and end of the transition and a more progressive transformation in between, which is in remarkable qualitative agreement with the data presented in Fig. 1. Our results are also consistent with several studies of the literature such as magnetic data in LaFe11.7Co0.195Si1.105 [START_REF] Bennati | Local magnetic behavior across the first order phase transition in La[END_REF] and the fraction of transformed phase derived from x-ray diffraction in LaFe11.8Si1.2 [START_REF] Waske | Asymmetric first-order transition and interlocked particle state in magnetocaloric La(Fe,Si)13[END_REF]. In this latter study, Waske et al. have

proposed an interpretation of this peculiar shape of transition within the assumption of a sample regarded as an "agglomerate of particles". The first step is ascribed to a stage of easy phase transformation, facilitated by the interspacing between particles which allows to accommodate the volume changes; then, the plateau corresponds to a blocking of the transformation occurring when these internal spaces have disappeared, and finally, the last step is qualitatively attributed to a brutal jump occurring when the driving force overwhelms the energy barrier.

In this vision and ours, the basic ingredient is the change of intrinsic volume which accompanies the transformation, but the interpretations differ. In [START_REF] Waske | Asymmetric first-order transition and interlocked particle state in magnetocaloric La(Fe,Si)13[END_REF] a basic role is ascribed to specific microstructure made of rather loose individual particles. Our picture is the opposite extreme since the sample is regarded as a "monolithic" piece, neglecting all the details of the microstructure. Clearly, the two visions are oversimplified with respect to the actual complexity of the microstructure [START_REF] Lai | Reduction of hysteresis in (La1-xCex)y(MnzFe11.4-z)Si1.6 magnetocaloric compounds for cryogenic magnetic refrigeration[END_REF]. A numerical approach could help to be more realistic by introducing some important features of the microstructure such as secondary phases, grain boundaries, cracks, voids and so on.

Conclusion

The "two-steps" process encountered in some first-order transitions refers to the fact that the transformation starts and ends by sudden changes in the transformed fraction that are accompanied by large exchanges of latent heat (leading to transitory changes in temperature since the thermalisation cannot immediately absorb such bursts of heat). In The first step bears similarity with usual recalescence at the beginning of a transformation after undercooling. However, it is not a matter of critical volume as in homogeneous nucleation, but it rather results from the fact that the start of the transition has a high price to pay in terms of interfacial energy (since developping from the external surfaces). The second step is more unusual. In our model, it corresponds to the fact that it becomes prohibitively expensive in energy for the system to keep a small fraction of untransformed material (since combining high internal pressure and large surface/volume ratio). To go further, simulations should be carried out to allow taking into account some important aspects of the microstructure and to consider a more realistic shape for the transformation front.

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.

Figure 1 (

 1 Figure 1(c) displays the relaxation curve T(t) from 211 K (T > TC) to the base

Fig. 1 :

 1 Fig.1 : Physical properties of LaFe11.4Si1.31Co0.29. (a) Magnetization recorded in 100 Oe at 0.1K/min; (b) Specific heat recorded by DSC in zero-field at 0.1 K/min; (c) Relaxation curve across the TC in zero-field; (d) Enlargement of (c) around the transition. The cyan areas mark the regimes out of the transition, the white areas correspond to the bursts, and the yellow area is the intermediate regime with very slow transformation; (e) Comparison of the relaxation curves of different samples measured with the same procedure or (f) of different procedures applied to the same sample.

  0

Fig. 2 :

 2 Fig. 2 : The top panels are pictures of the suggested stages of the PM-to-FM transformation: (a) pure PM state; (b) mixed state after the first burst; (c) mixed state in the intermediate regime; (d) mixed state just before the second burst; (e) full FM state reached after this second burst. Blue is for the FM phase, while

Figure 2 transformation

 2 Figure 2

Fig. 3 FM

 3 Fig.3(a) Schematic picture of the Gibbs free energies of the FM and PM states. For the latter, two curves are drawn corresponding to zero or non-zero internal pressure. The crossings between the FM and PM curves define the underlying TC values. However, owing to other energy terms (see text), the driving force

  the two phases [Fig 3(a)]. Actually, the increase in energy associated with the structural mismatch at the interface is such that the transition cannot start yet. Between (a) and (b), there is an undercooling regime along which the driving force ∆𝐺(𝑇, 𝐿 0 ) increases, but the transition remains blocked. Panel (b) shows the stage where ∆𝐺(𝑇, 𝐿 0 ) exceeds the threshold required to trigger the first burst: the FM phase suddenly enters the sample till reaching L = L1. Panel (c) illustrates a regime of progressive development of the transformation that is controlled by the stress-induced decrease of ∆𝐺(𝑇, 𝐿). Panel (d)

  (a), we include explicitly the variable L in GPM to indicate that the internal pressure is taken into account. By introducing the pressure-and temperature-dependent "chemical driving force" Δ𝐺(𝑇, 𝐿) = 𝐺 𝑃𝑀 (𝑇, 𝐿) -𝐺 𝐹𝑀 (𝑇), one can write 𝑔 𝑐ℎ𝑒𝑚 (𝑇, 𝐿) = 𝐿 0 3 𝐺 𝐹𝑀 (𝑇) + 𝐿 3 Δ𝐺(𝑇, 𝐿)

.

  One can start from Δ𝐺(𝑇, 𝐿) = Δ𝐺(𝑇, 𝐿 0 ) + ∫ 𝑑Δ𝐺(𝑇,𝐿) which can be extracted from the literature, while Eq. be related to the bulk modulus B as shown in Fig.2 (g-i). Indeed, the increase of pressure dP resulting from the volume expansion within the transformed region [from L to dL = L(1+p)] can be evaluated by 𝑑𝑃 = Considering that p << 1 and L/L << 1, this relationship can from Eq. (7) one obtains Δ𝐺(𝑇, 𝐿) = ∆𝑆 𝑡𝑟 (𝑇 𝐶0 -𝑇) + ∆𝑆 𝑡𝑟 × (

  Since the point is a comparison between these terms, one can consider relative quantities. Relative energies denoted by g* are obtained by (i) dividing the above expressions by L0, (ii) subtracting 𝐺 𝐹𝑀 (𝑇, 𝐿 0 ), and (iii) introducing the parameter z = L/L0. Doing so, one obtains𝑔 𝑃𝑀 * (𝑇) = ∆𝑆 𝑡𝑟 (𝑇 𝐶0 -𝑇)(13a)𝑔 𝑚𝑖𝑥 * (𝑇, 𝐿) = 𝑌 + 𝑧3 [∆𝑆 𝑡𝑟 (𝑇 𝐶0 -𝑇) -𝑌] + 𝑧 3 𝑋 ln 𝑧 + 6 (

  and -68 K(GPa)-1 , respectively[START_REF] Fujita | Itinerant-electron metamagnetic transition and large magnetovolume effects in La(FexSi1-x)13 compounds[END_REF]. Studying the impact of hydrogenation, Lyubina et al. found 𝑑𝑇 𝐶 𝑑𝑃 ~-222 and -99 K(GPa) -1 in LaFe11.57Si1.43 and LaFe11.57Si1.43-H1.64, respectively [29]. Recently, Hao et al. reported in a Cocontaining material, LaFe10.95Co0.95Si1.1, the value 𝑑𝑇 𝐶 𝑑𝑃 ~-83 K(GPa) -1 [30].

Fig. 4 :

 4 Fig.4: Main panel (a) shows the relative Gibbs free energies as a function of the penetration depth of the transformation (see Fig.2) for the three states: full PM, full FM and mixed state calculated from Eqs[START_REF] Fujita | Influence of Supercooling on the Thermally Induced First-Order Magnetic Transition in Magnetocaloric Compound La(Fe0.88Si0.12)13[END_REF] with the values of parameters displayed in the legend. Inset (b) converts this data in terms of states (circles): the full PM and full FM corrrespond to z =1 and z = 0, respectively; the mixed state can take place over the z range highlighted in yellow (yielding states corresponding to open circles), but the most stable state minimizing the energy is that marked by the filled circle. Inset (c) converts inset (b) in percentage of phase transformation as a function of the driving parameter, considering for the mixed state either the most stable state only (dashed line) or all the available z-range (full line).

  the present work, we propose a simple thermodynamic model which can account for this behavior by considering a Gibbs free energy of the mixed state including three terms: chemical, elastic and interfacial. The pressure stemming from long-range stress is incorporated into the chemical term. Comparison of the Gibbs free energies of mixed state with fully PM or fully FM states can explain a first jump from PM to a mixed PM/FM state, then a regime of mixed state where the transformation is blocked or at least very slow, and a final jump to pure FM.

Acknowledgments

This work was supported by the project CoolMagEvo from the program "Energy clean, safe and efficient " of the French National Research Agency (grant ANR-17-CE05-0036). The work at Inner Mongolia Normal University was supported by the National Natural Science Foundation of China (grant n° 51961033).

-Entropy jump. Using the Clausius-Clapeyron equation applied to magnetic data in samples of the same batch, the entropy change at TC in low field is estimated to 13 JK - 1 kg -1 [START_REF] Hardy | Burstlike first-order transformation studied by semi-adiabatic relaxation calorimetry[END_REF]. With a density of 7.2 g/cm 3 , it yields Str  9.4 10 4 JK -1 m -3 .

-Undercooling. According to the curves of Fig. 1, (Tc0 -T) is estimated to be within the range 0.3-1 K.

-Sample size. Even though our samples have a parallepipedic shape (2.22.20.4 mm 3 ), whereas the model assumed a cubic one, one can consider that the characteristic length L0 is of the order of one millimeter [(2.22.20.4) 1/3 =1. [START_REF] Chen | Coupled kinetic Monte Carlo-finite element mesoscale model for thermoelastic martensitic phase transformations in shape memory alloys[END_REF]].

-Interfacial energy. Actually, this parameter is the least well-known. According to [START_REF] Soffa | Diffusional Phase Transformations in the Solid State[END_REF], the interfacial energy solid-solid in metallic materials is thought to be of the order of 0.5 J m -2 . However, this corresponds to a local value, whereas the  introduced in Eq. ( 4) is a macroscopic quantity; it is actually a phenomenological value which should incorporate the fact that the actual contact area at the microscopic scale is obviously much larger than the oversimplified geometrical surface assumed in the model (Fig. 2).

We took the option to let free only this  parameter, while fixing typical values for all the other parameters for which reasonable approximations were available. We assumed the following values : B = 100 GPa ;  = 0.3 ; 𝑑𝑇 𝐶 𝑑𝑃 = -100 𝐾(𝐺𝑃𝑎) -1 ; p = 0.003 ; Str = 10 5 JK -1 m -3 ; (Tc0-T) = 0.5 K ; L0 = 10 -3 m. To reproduce the experimental behavior [Fig. [START_REF] Gschneidner | Recent developments in magnetocaloric materials[END_REF]], the mixed state needs to be the most favorable within the intermediate z range. With the assumed set of parameters, we found that this requires  to be of the order of 50 J m -2 .

The relative Gibbs free energies of the pure PM, the pure FM and the mixed states are plotted in Fig. 4 for the displayed set of parameters. It shows a crossing between PM and mixed states at L1/L0 = 0.936, and another one between mixed and FM states at L2/L0 = 0.654. This means there are two ranges of z values that are energetically unfavorable


Vincent Hardy
email: vincent.hardy@ensicaen.fr

Radia Hamane

Xavier Larose

Michel Risser

François Guillou

Affiliations

Burstlike first-order transformation studied by semi-adiabatic relaxation calorimetry

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

First-order transitions (FOT) are characterized by the presence of a latent heat and a regime in which two phases coexist. These features are known to complicate the experimental characterizations of this type of transition, while they are also prone to trigger various striking phenomena. One among the best known examples is that of recalescence which has been observed since a long time at the solidification of some metallic compounds [1][START_REF] Reed-Hill | Physical Metallurgy Principles[END_REF]. Basically, this phenomenon corresponds to a transient increase of temperature as the material crosses the transition upon cooling, and its origin relies on a sudden release of latent heat. This phenomenon can also take place at magnetic transitions, as exemplified by the case of iron reported at the end of the nineteenth century [START_REF] Hopkinson | Proc. R. Soc. Lond[END_REF]. More recently, related effects have been studied in several magnetic materials for which the symmetrical phenomenon of temperature decrease when crossing the transition upon warming (so-called antirecalescence) was also observed [START_REF] Gschneidner | [END_REF][5]. The origin of these phenomena, in particular their connection with the presence of structural defects within the material, was intensively debated. While Gschneidner et al. [START_REF] Gschneidner | [END_REF] claimed the requirement of a high purity in the case of Er, Bag et al. [5] rather pointed to the key role of an inhomogeneous microstructure in their study of (Hf,Ta)Fe 2 .

Along the paper, we will adopt the denomination of "reversed trend" in the thermal response to refer to both the recalescence and antirecalescence phenomena.

First-order transitions are also known to potentially exhibit burst-like transitions, i.e. a regime where the transformation from one phase to the other proceeds via a succession of steps. There is an obvious link with the previous phenomena of recalescence/antirecalescence in that the sudden heat flux associated to a burst can be hardly absorbed by the thermalisation of the experimental set-up. Calorimetric characterization of burstlike transitions is tricky.

Partly due to the intense activity on the magnetocaloric materials in the recent years, a lot of advances have emerged on this issue. A crucial step was reached by the development of the Differential Scanning Calorimetry (DSC) equipments based on Peltier cells which are able to directly quantify the exchange of heat flux between the sample and the thermal sink [6][7][8][9]. This approach has allowed to demonstrate that the heat exchange at the FOT of some materials can take the form of a succession of individual peaks distributed over a certain interval of time [10][11]. This peculiar manifestation give rise to "spiky patterns", a denomination we will adopt along the present paper.

The most common method for specific heat measurements nowadays is the semiadiabatic relaxation technique, [12] which is at the basis of widespread commercial devices, e.g. [START_REF]PPMS Heat Capacity Option User's Manual[END_REF]. Coupled with an analysis involving two relaxation times, [START_REF] Hwang | [END_REF] this method is very well suited for second-order transitions, while it is in principle reliable only outside the transition regime for FOT's. However, special modes of measurements/analysis have been proposed to overcome the difficulties associated to the FOT, allowing for instance the characterization of a sharp peak-like transition or the broad transitional regime of a martensitic-like transition [15][16][17][18].

It turns out that the combination of "reversed trends" and "spiky patterns" brings with it a new challenge, which has not been addressed so far to the best of our knowledge. This is the aim of the present paper to show that a specific analysis of relaxation data allows to identify the above-mentioned phenomena. The paper is organized as follows: in a first part, we present the model allowing to derive instantaneous heat flux exchanges versus time; in a second part, the model is applied to measurements on a LaFeSiCo alloy; the third part is intended to support the reliability of the analysis by comparing the results to DSC data, and by showing the consistency with the overall latent heat derived from the Clausius-Clapeyron (CC) equation. In a last part, the results of the model are also compared to the standard analysis in terms of heat capacity.

II. EXPERIMENTAL DETAILS

This study is carried out on thin platelet-shaped samples ( 2.22.20.4 mm 3 ) of a LaFeSiCo alloys provided by Vacuumschmelze GmbH and CoKG. The compositional analysis carried out by energy-dispersive X-ray spectroscopy (Bruker Xflash) within a scanning electron microscope (JEOL 7200 LV) leads to the approximate formula LaFe 10.77 Si 1.23 Co 0.28 . This compound undergoes a Curie transition around 207 K (with a thermal hysteresis of the order of 1 K), accompanied by an isotropic change of volume of about 1% (larger in the ferromagnetic state) [19][20].

The heat capacity measurements were carried out with the semi-adiabatic relaxation option of a Physical Properties Measurements System (PPMS, Quantum Design) on a 15.3 mg sample. We used the so-called "vertical puck" which is suited to measurements in nonzero magnetic fields (1 T in the present case), for two mains reasons : (i) the magnetic field is within the plane of the sample, which ensures a small demagnetizing effect; (ii) the sample lies in a region of constant field, which avoids a magnetic force to be exerted on the sample [START_REF]PPMS Application Note[END_REF].

Another sample of the same batch and of the same shape (16.2 mg) was investigated by DSC in zero-field. This experiment was carried in a TA instruments DSC 2500 Tzero technology and using Al crucibles.

Magnetic measurements were carried out in the PPMS by using a third sample of the same batch (and of the same shape with a mass of 15.9 mg). Isothermal magnetization curves were recorded with the field applied within the plane of the sample, as for the heat capacity measurements.

III. RESULTS

Figure 1 shows the thermal response of the platform onto which is pasted the sample when submitted to a large heat pulse, making it to cross the transition both along the warming and cooling branches. Along each transition regime, one observes a "reversed trend" of the measured temperature, exhibiting a negative average <dT p /dt> upon warming and a positive one upon cooling. Moreover, the enlargement of the FM-to-PM transition displayed in the bottom inset shows some "modulations" on the temperature profile which can be regarded as the fingerprints of underlying peaks in the flux heat exchange. It is important to emphasize that the reversed trends and spiky patterns are separate features which are not intrinsically linked to each other. As a matter of fact, Pecharsky et al. observed reversed trends in pure Dy without any indication of peaks, [START_REF] Pecharsky | [END_REF] and we will see later that our material still exhibits some small peaks in a magnetic field of 1 T but without reversed trends. What is common to both features is that a sign of dT p /dt opposed to the standard ones (either instantaneous values or averaged over the transition "plateau") precludes an analysis in terms of heat capacity.

FIG. 1.

Example of thermal response to a heat pulse, for the semi-adiabatic relaxation set-up schematically shown in the upper inset. The sample (red) is pasted by a thin layer of Apiezon N (dotted grey) onto a sapphire platform (dashed green), below which are attached a heater and a thermometer (green rectangles). The thermal resistances sample/platform (via the grease) and platform/bath (via the wires) are denoted K g and K w , respectively. A radiative link between the sample surface and the cap at the base temperature involves the emissivity  s . The temperature T s , T p and T b are those of the sample, platform and bath, respectively. The bottom inset is an enlargement of the transitional regime along the heating branch, which clearly exhibits a "reversed trend" (decrease of the average temperature) and a "spiky pattern" (presence of spikes along the transitional regime).

The thermal equations used to describe the system shown in the top inset of C s and C ad are the heat capacities of the sample and of the assembly (platform+grease), respectively; K g and K w are the thermal conductances via the grease and through the wires, respectively; T s , T p and T b are the temperatures of the sample, of the platform and of the "bath" or heat sink (i.e. the puck), respectively; P heat is the heating power applied to the

T s T p T b K w K g  s
platform during the first branch of the "pulse" (note that we report here the equations of the model used in the standard analysis of the PPMS data, [START_REF]PPMS Heat Capacity Option User's Manual[END_REF] without inclusion of a radiative term). In the standard analysis, the overall T p (t) response is fitted by the two- model allowing to derive an average value of C s over the pulse range [START_REF] Hwang | [END_REF]. The investigation of FOT's requires to shift to a point-by-point analysis of the trace (scanning method), which itself requires a thermalisation good enough to assume in first approximation T s  T p [15][16][17]. Most often, it is possible to incorporate the latent heat within a generalized view of the specific heat (i.e., combining latent and sensible heats) generating a high, sharp and symmetrical peak at the transition temperature T tr in a C s (T) plot. Nevertheless, this approach becomes invalidated in case of reversed trends (as leading to "negative heat capacity") and is even disputable in case of peak-like features along the transition regime [23].

In this case, one must go back to a complete reinterpretation of the thermal response at the transition by focusing on the latent heat exchange, while keeping an approach in terms of power. This dual requirement leads us to use the notion of power of latent heat P L , a quantity corresponding to the fraction of latent heat L exchanged per unit time (P L = dL/dt) and which is actually very comparable to the heat flux values derived in DSC analysis. When addressing the transitional regime (i.e., the plateau-like ranges along the warming and cooling branches) the quantity P L (t) comes to replace the term C s (dT s /dt) in Eq (1a). In other respects, we account for the use of a large temperature increment along the heat pulses (typically 5 K) by revisiting the leak term via the wires, as well as by introducing a radiative term. Indeed, with such large shifts between the platform and sample temperature (T p and T s ) with respect to the bath (T b ), there is a temperature gradient along the wires that must be taken into account, [17] and a non-negligeable radiative term can emerge between the sample surface and the metallic cap (at T b ). Accordingly, the Eqs (1) are rewritten as follows:

(2a) (2b)
Combining Eqs (2a) and (2b), (while approximating T s by T p in P rad ), one obtains

The derivations of approximate expressions for P rad and P cond are given in the Appendix. From the background measurement of (platform+grease), one can use linear approximations of C ad (T) and of K w (T) over the T-range of the pulses: and . Gathering all terms in Eq. ( 3), one obtains a general expression of the instantaneous power of latent heat over the transitional regime, derived from the thermal response of the platform:

. ( 4)

FIG. 2.

Analysis of the thermal response of Fig. 1 via Eq. ( 4), giving the time dependence of the power of latent heat P L (t). The analysis is shown here over the whole range of data even if it is only meaningful over the transitional ranges, materialized by the colored areas. The durations of these transformations are indicated as t.

Also reported are the values of the latent heat estimated by the integration of P L (t) over t, both along the warming (red) and cooling (blue) paths.

On the warming branch P heat is the value applied to the platform, while it is zero on the cooling one ;  T is the Stefan-Boltzmann constant, S the radiative surface of the sample ( 5 mm 2 ), and  s its infrared emissivity, for which we will assume the value  0.3 typical of unpolished metallic-like surface. Figure 2 shows the results of Eq. ( 4) applied to the trace of Fig. (1). Although this analysis is only valid within the plateaus, we present the results over the whole time window to show the difficulty associated to the definition of the boundaries. In practice, we will use the extremum points of the T p (t) profile. To estimate the overall latent heat, one must integrate P L (t) over the duration of the transformation. By assuming vertical boundaries in Fig. 2, one obtains close values on the warming and cooling branches, namely 41.03 and 40.90 mJ, respectively.

On both branches, this analysis shows the presence of spikes on P L (t), and we consider this reflects a burstlike exchange of latent heat. To test the reliability of this view, several Figure 3 shows the results of DSC measurements performed at a rate of 1 K/min around T tr . The results are presented both in terms of sample temperature and of heat flow from the sample as a function time. One observes that the response is qualitatively well consistent with that previously obtained from Eq. ( 4), since one observes reversed trends and spikes of heat flow along the transition. The intensity of the reversed trends is of the same order of magnitude, namely  0.1-0.2 K from DSC versus  0.2-0.4 K in our experiments. The sample mass and temperature rates are very similar in both experiments; the difference most likely originates from the thermal dilution with the crucible in DSC measurements and from the degree of thermal contact sample/base, which is expected to be higher in the DSC setup. We also note that the instantaneous heat flows in DSC have a mean value of  3000 µW which is significantly larger than the one observed in Fig. 2 (while the masses are similar in both cases), but this heat exchange develops over a time interval which is smaller than in our )

experiments. Once again, this variation in dynamics might be related to the difference in thermalisation of the two set-ups.

Changing the parameters of the heat pulse (starting temperature, increment and power) modifies substantially the shape of the response. (temperature of the "bath"); T pulse is the targeted temperature increment; P heat is the power applied to the heater to generate the pulse. The last column is the value of latent heat derived from integration of P L (t) over the transitional regime (the mass of the sample is 15.8 mg).

To go further, we have derived an independent estimate of L from the CC equation. mJ, a value slightly larger but well comparable to that derived from ( 41 mJ).

To investigate the impact of the magnetic field on the transitional regime, a couple of heat pulses was recorded in our calorimeter with the same value of T pulse and P heat , both in B = 0 T and B = 1 T. Owing to the field dependence of the Curie temperature, the starting temperature was shifted by 4 K for the measurements in field. The results are shown in Fig. 5 by plotting the temperature variation as a function of time. The relative positioning of the two curves is striking: they exhibit similar overall shapes, but the reversed trends present in B = 0 T are totally absent from the curve B = 1 T. In other respects, the enlargement of the transitional regime upon warming (inset) shows the persistence of some modulations on the 1 T trace, even though much smoother than in B = 0 T. This evolution can be ascribed to the reduction of the latent heat in 1 T as reported in Fig. 4(b). In practice, the absence of reversed trend allows to handle these 1 T data with the conventional analysis in terms of heat capacity. This approach first consists in turning back to a term in Eq. 2(a); then, one assumes a coupling sample/platform good enough to consider the set (platform+grease+sample) as a whole at a temperature T p =T. A point-bypoint analysis of the thermal response leads to , where . Converting the time dependence into a temperature one, one obtains (5) In this analysis, the range of time used for the derivative dT/dt must be large enough to smooth the "modulations" and ensure average <dT/dt> values having the "expected" sign along the transitional regimes. In a last step, a rough correction of the temperature lag between T = T p and T s can be applied by using an estimate of K g (see supplementary material)

to generate final results in the form of a C s (T=T s ) curve. Following this approach with the 1 T data of Fig. 5, one obtains the C s (T) shown in Fig. 6(a) (transition upon warming). Note that this curve is limited to the T-range highlighted by the blue line in Fig. 5, since this analysis is known to generate increased uncertainties at the beginning and end of the pulse [15][16][17]. Figure 5 S tr value) is somewhat a matter of criteria. At first glance, one would be tempted to define S tr by the spacing between the two circles on the construction of Fig. 6(b) (leading to  11.5

JK -1 kg -1 ). However, to be consistent with the boundaries of the transition defined by the location of the plateau-like regime, one should rather consider the squares marked on Fig.

6(b)

. Doing so, the associated S tr is found to be 9.3 JK -1 kg -1 , leading to L  2.0 Jg -1 . Restarting from the raw data, we also performed an analysis in terms of P L (t) via Eq. ( 4 corresponding to L(1T)  1.8 Jg -1 , a value which is also in reasonable agreement with that derived from the P L (t) analysis.

IV. DISCUSSION

Using semi-adiabatic relaxation calorimetry, we have suggested a mode of analysis able to account for exotic phenomena encountered in FOT's, such as the recalescence/antirecalescence effects and the presence of spikes in the latent heat exchange.

Several results come to support the reliability of this approach. First, it is found that the peculiar behaviors derived from this approach (reversed trends in the thermal response and Figure 6 burstlike heat exchange per unit time) are confirmed by DSC experiments; since DSC is known to be the reference technique to deal with the heat exchanges at FOT's, this consistency provides valuable support to the analysis in terms of power of latent heat P L (t).

From a more quantitative viewpoint, several issues also deserve to be emphasized: (i) the overall latent heat L derived from integration of P L (t) turns out to be independent of the recording parameters; (ii) this L value is in agreement with that derived from the CC equation applied to magnetic data; (iii) this L value is also consistent with that derived from the C s (T)

curve obtained by a standard analysis (possible in 1 T).

On the other hand, this approach has several limitations. First, it relies on the theoretical expectation of a null dT s /dt at the FOT; any deviation from this assumption will introduce an experimental uncertainty in our results; to quantify it, let us consider an estimate of the lattice heat capacity at the FOT derived from extrapolations of the heat capacity data from each side of the transition (C lat  425 µJ K -1 mg -1 ) and a mean <dT s /dt> approximated by the average dT p /dt over the plateau-like transformation (e.g., -0.0028 K/s in Fig. 1); doing so, one obtains , a value which remains much smaller than the P L values obtained by the analysis (500 µW). Second, it must be stressed that the definition of the boundaries of the time window over which P L (t) is calculated is subject to a notable uncertainty. For instance, moving this interval by one data spacing on both side of P L (t) in Fig. 6(c) would yield an uncertainty of  5% on the L value. Third, the profile of P L (t) suffers from a poor resolution in time, which is given by the data spacing of  4 s (a limitation imposed by the recording system). This prevents from the observation of any dynamics faster than this time scale. Furthermore, there is another source of limitation which is still more detrimental.

Indeed, contrary to the DSC experiments where one measures directly the heat exchanges, in our case this quantity is only indirectly reflected by the temperature of the platform.

Assuming that the latent exchange can be inhomogeneous, one must first consider the time constant associated to the thermalisation of the sample itself; this is typically given by the relationship , where e is the sample thickness and  the diffusivity. With e = 0.4 mm and   2 mm 2 /s, it turns out that it yields a very short time of  0.02 s which is thus negligible. Actually, the limiting time constant in our case is that related to the thermal contact sample/platform, which can be approximated by , where R g is the thermal resistance of the grease ( and C lat the phononic heat capacity at the FOT ( 425 µJ K - 1 mg -1 ). The former parameter can be evaluated from the two time constants ( 1 and  2 ) obtained from the standard fitting of the relaxation profiles out of the transitional regime (see supplementary material) while the latter depends on the mass ( 15.8 mg) of the sample. In our experiment, one obtains , leading to As a consequence, our system cannot account for well separated spikes on the heat flow, and can just exhibit slight modulations of the P L (t) profile.

Recalescence is a well documented phenomenon, whereas antirecalescence is much less common, even though we note that a signature similar to ours has been reported in metallic Dy (dT  -0.4 K) [START_REF] Pecharsky | [END_REF]. Focusing on the magnetocaloric materials, reversed trends of similar amplitudes (  0.1 K ) have been observed in compounds of the La-(Fe,Mn,Si) system [11],

while much larger effects can take place in the (Mn,Fe) 2 (P,Si) materials, but only along the first cooling (see the supplementary material). It is important to emphasize that the size of these temperatures increments depends on the value of the thermal linkage to the bath. In our semi-adiabatic set-up, this thermal resistance (towards the bath) is substantially larger (i.e., 1/K w  6000 K/W) than the values of the order of 500 K/W reported in DSC experiments [10,25].

Considering the case of magnetic FOT's, the first observations of "spiky patterns" were reported in ultra-pure Er and in (Hf,Ta)Fe 2 intermetallics [START_REF] Gschneidner | [END_REF][5] Note that these studies involved different calorimetric set-ups and different types of magnetic transitions. In each case, two or three bursts take place along the transformation. More recently, several studies reporting the same type of features were carried out in magnetocaloric materials. For instance, DSC investigations revealed six spikes along the transition of a LaFeSiCo material, [10] while a hydrogenated LaFeSiMn compound exhibited two wider peaks but with a clear substructure [11]. Actually, the comparisons between experiments must be regarded cautiously since the results strongly depend on the specific resolution of each device. Besides, other studies prefer to show the DSC data in terms of a generalized vision of the specific heat including the latent heat. In the first reports, this has yielded high peaks in C s (T) exhibiting a spiky substructure [26][27]. More recent studies reported signatures of the transition in C s (T)

taking the form of well separated peaks [28].

Let us now briefly comment on the possible origin of this peculiar phenomenon. In the case of Er, the authors suggested that the bursts may reflect the existence of metastable states separating the start and end phases of the FOT, [START_REF] Gschneidner | [END_REF] whereas in the case of (Hf,Ta)Fe 2 these bursts were rather ascribed to nucleation and growth events in different regions of the samples [5]. Besides, in substituted materials like LaFeSiCo, other authors suggested the possibility of a relationship between the bursts and a distribution of local values of T c . Another viewpoint is to consider that even in case of a single T c and without intermediate phases, one can expect such a burstlike response owing to the mismatch of the crystalline structure between the two phases separated by the FOT [29]. We note that all materials in which burstlike transitions are present exhibit substantial changes in structural parameters at the transition. In the case of our LaFeSiCo, the transition is isomorphic and one can estimate from the literature V/V  1%

(from PM to FM) [19][20]. Such mismatch necessarily generates strains at the interfaces PM/FM and long range stresses, which yield the rise of an elastic energy opposing the development of the transition. This underlying energy can even delay the start of the transformation expected when crossing T c either upon cooling or warming, leading to undercooling and overheating effects. When the transition starts, it exhibits a very abrupt character: the larger the overshooting, the steeper the triggering. For materials which are characterized by a particularly large latent heat (such as the most performing magnetocaloric materials) the resulting heat flows can be very large. This process is most likely at the origin of the recalescence and antirecalescence phenomena.

Then, the development of the transition is associated to the diffusion of the front between the two phases (PM/FM in our case). We support the idea that the jerky character of the transformation corresponds to transient pinning/depinning events along this front [11,25,30]. The most obvious cause for transient blockings is the rise in elastic energy, but one should also consider a thermal effect, since the latent heat exchanges associated with the move of the front tend to oppose it [25]. Considering for instance the PM to FM transition (i.e. upon cooling), the transformation is exothermic, favoring a local increase in temperature which thereby yields an inhibiting effect. Whatever the pinning mechanisms at play, the depinning path followed within the sample will depend on the details of the microstructure (such as secondary phases, defects and grain boundaries). Therefore, even though the basic origin of the "spiky patterns" is intrinsic (structural mismatch), the sample response necessary involves features of its microstructure, which are extrinsic. Both aspects are thus intimately entangled in the description of the burstlike transitions.

V. CONCLUSION

We have proposed an analysis of semi-adiabatic relaxation data allowing to investigate two phenomena sometimes encountered in FOT': reversed trends in the thermal response (recalescence/antirecalescence) and presence of spikes in the time dependent exchange of latent heat at the transition (so-called "spiky pattern" or "peak forest" [28] effects). Both phenomena can emerge in materials exhibiting a structural mismatch between the two phases at each side of the transition, as well as a large value of latent heat. They have known a recent surge of interest in the context of the studies on intermetallics showing a giant magnetocaloric effect that are targeted for applications in magnetic refrigeration. The presently investigated LaFe 10.77 Si 1.23 Co 0.28 compound belong to this class of materials. It must be acknowledged that the reported approach is less direct and powerful than DSC devices to detect the fine time structure of heat flows at a FOT. However, our approach allows such basic phenomena to be investigated in widely used commercial devices, and it also allows a direct connection with the specific heat measurements on each side of the transition (which can be performed accurately with the same sample mounting).

Regarding the "reversed trends", the most puzzling effect is that found on the heating branch (antirecalescence) which may appear at first glance less intuitive than its recalescence

counterpart. Yet, it just demonstrates that the sample is active when crossing T tr upon warming, merely as it is along the reverse path; upon warming, the transition is endothermic and once the transition has been triggered, the sample pumps heat to the platform to ensure the development of the transformation with its own time constant (instead of simply waiting for the heater located on the platform to provide this latent heat). Note that these effects of recalescence/antirecalescence cannot be regarded as basic features of the material since their observation depends on extrinsic factors, such as the strength of the thermal coupling sample/environment.

What is more tricky in the present data is the "spiky pattern". Even though there is still debate about the origin of this effect, it is quite clear that it reflects a process of the transformation that is irregular in time and inhomogeneous in space. We favor an interpretation in terms of friction related to the mismatch between the two phases in competition at the transition. Increase in elastic energy can locally exceed the energetic gain associated with a smooth development of the transition. Depinning takes place when the driving force allows to overcome the barriers associated to the strains. Such a sequential process is thought to be at the origin of the jerky displacement of the transition front (FM-PM).

It is obvious that much could be learned from local techniques like magneto-optics, as already investigated in the case of LaFeSiCo materials (e.g., [10] ). To go still further, it might be interesting to explore complementary techniques with enhanced resolutions, both in space and time, such as Magnetic Force Microscopy as recently used in FeRh materials [31][32].

Extending this type of local probes to investigate the spatio-temporal features at the burstlike transitions in FOT's is a very appealing challenge that would deserve to be taken up.

SUPPLEMENTARY MATERIAL

See the supplementary material for estimates of the shift between T p and T s within the transitional regime, and of the entropy step at the transition from the Clausius-Clapeyron equation.

FIG. 7.

Schematic picture of the thermal link platform/bath via the wires (there are eight wires but only one is shown here). At the boundary platform/wire the temperature is T p (t), while it should keep constant and equal to T b at the boundary wire/puck.

In case of a small T pulse , one can reasonably assume an homogeneous conduction involving a simple K w term. The conductance K w is related to the conductivity k w via the relationship , where L and s are the length and cross section of each wire, respectively. In the standard analysis, one can consider , with . Experimentally, one gets access to the conductance K w in the background measurements of the set (platform+grease): (see Fig. 8).

Assuming a linear profile of T along the wire (a reasonable approximation, even though the exact solution is more complex), one can consider . This leads to , and one finally arrives at the expression .

(A1)

This relationship is quite different from that of the homogeneous case, i.e.

.

  Figure 1

Figure 2 FIG. 3 .

 23 Figure 2

Figure 4 (

 4 Figure 4(a) shows a series of M(H) curves recorded at selected temperatures, from which one can derive the temperature dependence of the transition fields associated to the midpoint of the metamagnetic steps (see the supplementary material); combined with an evaluation of the height of the steps along the transition (M), the CC equation allows to estimate the value of the entropy jump at the transition:. The results reported in Fig.4(b) show

FIG. 4 .

 4 FIG. 4. Isothermal magnetization loops recorded in the range 0-2 T after zero-field cooling. The measurements are carried out with the field applied along the plane of the sample, at temperatures spaced by 1 K between 204 and 215 K. (b) Values of the entropy jump at the transition, derived from the Clausius-Clapeyron equation applied to the ascending branches of the loops in (a).

  Figure 4

FIG. 5 .

 5 FIG. 5.Temporal profiles of the thermal response resulting from a heat pulse applied either in zero field or in 1 T. The parameters of the heat pulse are the same in both cases (P heat =780 µW for T pulse = 4 K). Owing to the field dependence of T tr , the base temperature (T b ) is 205 K in zero-field and 209 K in 1 T. To allow better comparison of the two measurements, one plotted dT(t)=T(t)-T b . The blue line for 1 T materializes the part of the branch used to derive C s (T) in Fig.6(a). The inset is an enlargement of the transitional regime upon warming.

  ) whose results is displayed on Fig 6(c) (for warming). It turns out that the integration yields 2.1 J g -1 , which is well consistent with the above L value. It can also be compared to the value derived from the CC analysis. According to Fig. 4(b), one expects S tr (1T)  8.5 JK -1 kg -1

FIG. 6 .

 6 FIG. 6.Analysis of the relaxation data in 1 T along the heating branch. Panel (a) is a standard analysis in terms of heat capacity, obtained from the profile of Fig.5by using Eq. (5). Panel (b) is derived from (a) by integrating C s /T starting from a reference temperature T * = 210.95 K. Panel (c) corresponds to the analysis via Eq. (4).

ForFIG. 8 .

 8 FIG. 8.Experimental determinations of K w (T) and C ad (T) derived from the background measurements. The red data points show the range of the pulses, over which linear fittings can be applied.

TABLE I .

 I TableIgathers the parameters investigated within a series of heating pulses that were all analyzed with Eq. (4). It turns out that the presence of spikes is always detected in the P L (t) response, whereas the intensity of the reversed trends can be substantially reduced. In all cases, however, one obtains very consistent values of L. Measuring parameters of different heating branches recorded in zero field; T b is the base temperature

		n	T b (K)	P heat (µW)	T pulse (K)	L (mJ)
	°	a	205	969	5	41.03
		b	206	582	3	41.10
		c	203	1340	7	41.43
		d	204	962	5	40.85
		e	205	773	4	39.55
		f	206	582	3	40.11
		g	206	777	4	39.42
		h	206	971	5	39.53
		i	206	1164	6	40.99

ACKNOWLEDGMENTS

This work is part of the CoolMagEvo ANR-17-CE05-0036 project, from the Research Programme "Energy, clean, safe and efficient (DS02) 2017" of the french National Research Agency (ANR). The work at Inner Mongolia Normal University is supported by the National Natural Science Foundation of China (grant n° 51961033).

DATA AVAILABILITY

The data that supports the findings of this study are available within the article and its supplementary material.

AUTHOR DECLARATIONS

The authors have no conflicts to disclose.

APPENDIX The conductive leak term

The large temperature increment used in our pulses can generate a substantial gradient of temperature along the wires ensuring the thermal leaks towards the 'bath" (this role of "bath" is played in our system by the puck, a bulk metallic piece onto which the wires are soldered).

Since the thermal conductivity of the wires (Au-Pd alloy) is temperature dependent, this gradient must be taken into account to accurately quantify the term P cond . The situation one has to face is schematically shown in Fig. 7.