Landau theory-based relaxational modeling of first-order magnetic transition dynamics in magnetocaloric materials

The magnetocaloric effect is often largest within the neighborhood of a first-order phase transition. This effect can be utilized in magnetocaloric refrigeration, which completely eliminates the need for the greenhouse gases utilized in conventional refrigeration. However, such transitions present unique dynamical effects and are accompanied by hysteresis, which can be detrimental for such refrigeration applications. In this work, a Landau theory-based relaxational model is used to study the magnetic hysteresis and dynamics of the first-order magnetic transition of LaFe13−x Si x . Fitting the experimental magnetization data as a function of applied magnetic field under different field sweep rates with this model provided the Landau parameters (A, B, and C) and the kinetic coefficient of the studied material. We demonstrate the tendency of the magnetic hysteresis to increase with the magnetic field sweep rate, underlining the importance of studying and minimizing the magnetic hysteresis in magnetic refrigerants at practical field sweep rates. While evaluating the temperature dependence of the time required for a complete transition to occur, a nonmonotonic behavior and a sharp peak were found for temperatures near the transition temperature. Such peaks occur at the same temperature as the peak of the magnetic entropy change for low fields, whereas for higher fields the two peaks decouple. This information is critical for technological applications (such as refrigerators/heat pumps) as it provides guidelines for the optimization of the magnetic field amplitude in order to reduce the transition timescale and consequently maximize the machine operational frequency and amount of heat that is pumped in/out per second.

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Introduction
Currently, the refrigeration industry represents a large and increasingly growing part of the global economy. Recent estimates attribute 25%-30% of the global consumption of electricity to refrigeration, air-conditioning, and heat pumping (RACHP) equipment, and with the rise of global population, ambient temperatures, and the adoption rate of such devices in developing countries, demand is quickly rising [1,2]. Moreover, the vast majority of RACHP equipment is based on the compression/expansion of gases that, despite frequent improvements, still have large ozone depletion and global warming potentials (GWPs). These gases, chlorofluorocarbons, hydrochlorofluorocarbons, and hydrofluorocarbons are internationally being phased down due to their negative environmental impact, as specified in the United Nations Montreal Agreement [2] and subsequent revisions, with the most recent being the Kigali Amendment in 2016. Consequently, there is an urgent need for more energy-efficient and environmentally friendly RACHP devices. This growing demand presents an opportunity to explore alternative refrigeration technologies such as thermoacoustic, thermoelectric, and caloric (magnetocaloric, electrocaloric, elastocaloric, barocaloric, and multicaloric) [3][4][5][6][7][8] technologies, all of which completely eliminate the use of high GWP gases and some of which have been found to be energetically competitive compared to vapor compression for space-cooling applications, namely thermoacoustic, magnetocaloric, and elastocaloric.
Caloric effects are defined as the change of a material's temperature in adiabatic conditions and/or the entropy variation in isothermal conditions under the application of a magnetic (magnetocaloric), electric (electrocaloric), and mechanical (elastocaloric when uniaxial, or barocaloric when applying hydrostatic pressure) external stimuli, or combinations of these (multicaloric). Typically, these caloric effects are maximized across a first-order phase transition (FOPT), which is signified by the existence of a discontinuity on the first derivative of the free energy or latent heat, in contrast with secondorder phase transitions, or continuous phase transitions, where there is a discontinuity on the second derivative of the free energy or an infinite correlation length.
FOPTs are typically accompanied by hysteresis, which corresponds to regions of the free energy landscape where free energy reaches local minima (metastable states) and therefore the system is not stable at a single state, but instead multiple states can coexist [9]. In general, hysteretic regions of the order parameter (the magnetization, in the case of a magnetic FOPT, which is the distinguishing feature of material systems displaying significant magnetocaloric effects (MCEs)) results in irreversibility and consequently in a loss of efficiency in any cyclical refrigeration process. In the case of the MCE, the occurrence of magnetic field and the temperature hysteresis of the magnetization is common in FOPT materials [10,11]. Thus, if the applied magnetic field is not large enough to induce a complete transition between the magnetic phases present, only a fraction of the heat available across the transition will be harvested. Additionally, if we apply the magnetic field within the hysteretic region in temperature, even if we fully induce the transition upon field application, removing the magnetic field will not completely reverse it.
Considering both Carnot and active magnetic regenerator (AMR) thermodynamic cycles, Basso et al were able to theoretically demonstrate the hindering effect of magnetic hysteresis using the Preisach model by showing the hysteretic energy losses and by identifying the hysteresis-induced asymmetry in the AMR closed cycle [12]. More recently, Brown et al were able to identify the limiting maximum thermal hysteresis possible to optimize the efficiency of an Ericsson cycle considering a Ni-Mn-In magnetocaloric material: 1.5 and 0.5 K for a magnetic field change of 5 and 1.5 T, respectively [13]. The hysteretic influence on various features of magnetic refrigeration has been the focus of intensive research, as comprehensively reviewed in references [14][15][16][17], where strategies were proposed to master the magnetic field and temperature hysteresis of materials towards an efficient solution. Since hysteresis is present in all caloric materials presenting FOPT, its impact on other caloric technologies has also been assessed [18].
In parallel to hysteresis, another critical issue is the kinetics of these FOPTs. The faster the material's FOPT is, the higher the system ′ s operational frequency can be, and therefore a larger amount of heat can be pumped out per unit of time. However, transition times in the order of tens or hundreds of seconds have been measured for different FOPT magnetocaloric material families [19][20][21], which could impose a detrimental limit on the frequency of operation of these devices. Thus, characterizing and understanding the hysteresis and kinetics of these materials and how to influence them is crucial for predicting and improving their performance in applications.
There are various confounding mechanisms underlying the kinetics of these FOPTs and, correspondingly, different physical bases through which they have previously been modeled.
There are a few works on the description of these timedependent effects for Gd 5 Si 2 Ge 2 and LaFe 11.7 Si 1.3 through a thermal activation model [20,22]. On another front, modeling of the hysteresis and the kinetics of La(Fe,Si) 13 using the Bean-Rodbell model and a model based on avalanchelike behavior, respectively, predicted that the hysteresis is larger than that experimentally measured [16], but a recent work significantly improved on this mismatch by combining the Bean-Rodbell model with Kolmogorov-Johnson-Mehl-Avrami nucleation and growth theory [23]. A thermal activation model has also been applied to Mn-Bi, where the authors showed that, since most MCE materials are diffusionless, they ought not to show large thermal activation effects [16]. Experimental results indicate that time-dependent hysteresis also occurs due to the heat exchange between the sample and the thermal bath and the sample and the measurement sensor [24,25]. More recent studies employing a noncontact method [26] concluded that eddy currents and magnetic hysteresis do not contribute to the frequency-dependent behavior and proposed that its origin is an inhomogeneous internal field distribution leading to a heat exchange between the core and surface of the sample [27].
In the present work, the FOPT kinetics of LaFe 13−x Si x with x = 1.4 will be presented and discussed as a case study. The La-Fe-Si material family is among a limited set of materials presenting a giant MCE near room temperature, which also includes the Fe 2 P, Gd 5 (Si,Ge) 4 and the Heusler alloys families [28,29]. In the last few years, the La-Fe-Si family and the Fe 2 P-based have stood out as the most promising for refrigeration/heat pumping applications. These materials meet a unique set of desired requirements, such as being constituted of abundant and noncritical chemical elements, exhibiting relatively large thermal conductivity, and having high tunability of their operational temperature range, etc. The La-Fe-Si family has also been the subject of thorough studies concerning their magnetic dynamics and hysteretic behavior [15,16,21,30,31], as well as being used as a case study for previous modeling studies [22,23,32,33].

Magnetic hysteresis and dynamics models
We begin by assuming that the free energy can be described in terms of the Landau expansion [34,35]: where A, B, and C depend on external forces, which in this case are the temperature and magnetic field. The equilibrium magnetization is then determined from the solutions of which correspond to minima. The next step is to describe the time dependence when the system is not in equilibrium and the simplest approach for small deviations is relaxational [34,35]: where Γ is the kinetic coefficient. It is clear that, in the steadystate, ∂M ∂t = 0, the system is in or has relaxed to the equilibrium state, thus ∂F ∂M = 0. Conversely, if the system is in a stable or metastable state, it will remain that way. Only when the shape of the free energy changes, i.e. when the minimum of the system disappears or shifts by applying temperature and/or magnetic field and ∂F ∂M ̸ = 0, will the system evolve to a new stable state depending on the shape of the free energy landscape. Therefore, in a quasi-static measurement/simulation, we can expect to get the same result as from the equilibrium solutions.
As a first step, to test, validate, and understand the influence of different magnetic field sweep rates on the magnetization curves and their relaxation times, different M(H) and M(t) curves were simulated through numerical integration of equation (3) using a fourth-degree Runge-Kutta method. As a second step, the required Landau and kinetic parameters were extracted from experimental magnetization data of a LaFe 11.6 Si 1.4 sample and were used in the numerical model to unveil (1) the magnetic hysteresis behavior as a function of magnetic field sweep rate and (2) the impact that magnetic transition dynamics have on the estimation of the MCE of an FOPT material. For the first step, the Landau parameters to be considered were 5 and a kinetic coefficient of Γ = 0.001 (g Oe s) −1 , as these constants are typical of an FOPT near T C [36,37]. Figure 1 shows the sweep rate dependence (from 1 to 400 Oe s −1 ) of the simulated magnetization as a function of applied magnetic field near the transition temperature at T = 195 K, demonstrating for each curve the characteristic shape of a field-induced magnetic FOPT from a low magnetization to a high magnetization state for an increasing field (and subsequent reverse with decreasing field), where hysteresis results. The curves are characterized by four critical field values: Hc1, Hc2, Hc3, and Hc4, which signal the onset and the end of the magnetic fieldinduced transition while increasing the field (Hc1 and Hc2, respectively) and the onset and the end of the transition while decreasing the field (Hc3 and Hc4, respectively). As can be seen, for low rates, the result is similar to the quasi-static case and, for faster rates, the hysteresis increases. Furthermore, H C2 and H C4 are highly dependent on the magnetic field rate, as expected from experimental results [21,24]. Different materials will have different kinetic coefficients depending on a wide set of particular properties (such as thermal conductivity, sample size, magneto-volume coupling, thermal environment, chemical/strain disorder, etc [38,39]). Figure 2 illustrates the time evolution of the magnetization for various kinetic coefficients with a sweep rate of 100 Oe s −1 starting at 0 and stopping at 10 kOe and remaining in that state for a longer period of time. One reason for simulating a paused field is that, under certain conditions, first-order La(Fe,Si) 13 will progress through the transition under static field (and T) conditions if the material is driven to the cusp of H C1 where the transition onsets [21,40]. The resulting transition may reveal properties of the intrinsic phase growth dynamics, as well as dynamics of the relaxation from a nonequilibrium state. In this instance, the model is purely the latter. As can be seen, for a faster dynamic, the magnetization increases abruptly after some time, which corresponds to the instant that the applied magnetic field is H C1 . On the other hand, for slower systems, the magnetization grows approximately linearly. Figure 3 presents simulations of the time evolution of the magnetization with a linearly increasing magnetic field up to 15 kOe, a kinetic coefficient of Γ = 0.001 (g Oe s) −1 , and for sweep rates from 100 to 4000 Oe s −1 . In this case, for fast sweep rates (4000 Oe s −1 ), the magnetization increases almost linearly and it lags behind the magnetic field. Also, since the dynamics of the model depend on the kinetic coefficient and on the shape of the free energy landscape, there is an upper limit on how fast the saturation can be achieved; in this test case it is approximately 8.5 s. For slower rates, there is an abrupt magnetization increase just after their respective t C1 (the instant at which the magnetic field is H C1 ) followed by a linear increase up to the saturation value, which occurs at t max (the instant at which the magnetic field reaches the maximum magnetic field, in this case 15 kOe). 11.6 Si 1.4 In the second step, the magnetization data of a LaFe 11.4 Si 1.6 sample ( [21], scenario A) was plotted against the simulated data obtained with the above-detailed model, in order to achieve the best accordance between the experimental and simulated curves. In particular, the A, B, and C Landau coefficients were set by matching the modeled M(H) curves with the slow, quasi-statically (10 Oe s −1 ) measured M(H) curves, since in this case the dynamics have little influence (curves for sweep rate of 1 Oe s −1 , 10 Oe s −1 , and Landau in figure 1 practically overlap). In contrast, the kinetic coefficient (Γ) was obtained by matching the modeled M(H) curves with the experimental counterparts measured at the faster sweep rate. Following this procedure, and considering that A(T) = A 0 (T − T C ), the parameters obtained were A 0 = 68.304 g Oe emu −1 , B = −0.04248 (g Oe emu −1 ) 3 , C = 2.4083 × 10 −6 (g Oe emu −1 ) 5 , and T C = 191 K (note that measurements were conducted slightly above T C at T = 194.5 K), and the kinetic coefficient was Γ = 0.004 (g Oe s) −1 . The Landau coefficients are of the same order of magnitude as in [41]. The matching M(H) curves are plotted against the experimental curves in figure 4, where the magnetic field was changed from 0 to 15 kOe at sweep rates of 10 and 180 Oe s −1 . As can be seen, the simulated curve accurately follows the experimental magnetization for most of the field range, except for when the magnetic field decreases from H C3 to H C4 . This disparity is most likely attributed to microstructural effects such as the nucleation of the paramagnetic phase around defects [16], the typical presence on these samples of a small fraction of ferromagnetic α-Fe phase (which also explains the small deviations observed at low-fields, H < H C1 and H < H C4 ), and the reported intrinsic asymmetry associated with this transition [36]. This sample was analyzed with x-ray diffraction and the respective Rietveld refinement, which confirmed the presence of an α-Fe concentration of (6 ± 1 wt%) and a La(Fe,Si) 13 cubic phase (94 ± 1 wt%) [21].

The hysteresis and dynamics of FOPT in LaFe
In figure 5, the modeled time dependence of the magnetization for the increasing magnetic field at 180 Oe s −1 followed by holding it constant from t = 0 s at several field set points is plotted with experimental data on LaFe 11.6 Si 1.4 from [33]. We can see that, for a stopping magnetic field of 7.2 kOe, the experimental and theoretical results agree relatively well, whereas for higher stopping fields there are larger   [21] (markers). The magnetic field is increased at a rate of 180 Oe s −1 and stopped at t = 0 s when it reaches the respective maximum applied magnetic field. mismatches, with the simulated magnetization curves evolving slower than the experimental ones. This might also be caused by the small fraction of ferromagnetic α-Fe phase present in the sample and/or other nucleation and growth/kinetic mechanisms not represented in this model. Despite such deviations, it is interesting to remark that, for every stopping magnetic field, the instants that the theoretical and experimental curves reach saturation are roughly the same and this is a consequence of the model and not a fitted parameter. It is also interesting to remark that, although the simulated magnetization data evolves slower at the beginning, it becomes faster at a later  [21]; the blue (squares) and green (circles) curves are from the model. stage than the experimentally measured magnetization, as can be seen in figure 5.
Additionally, the dependence of the magnetic hysteresis with magnetic field rate was determined (see figure 6). In this case, we analyze the hysteresis measured experimentally (black pentagon symbols), the current model (blue squares; see section 2), and the model rescaled (green circles). The motivation for the rescaled curve is due to the hysteresis overestimation of the Landau fit, particularly in the region of the decreasing field, from the saturation magnetization field, H C3 , to zero applied magnetic field, H C4 (as seen in figure 4). As mentioned above, the deviations observed in this field region are most likely associated with sample-specific features such as microstructural effects, the small-fraction of ferromagnetic α-Fe phase, and the reported intrinsic asymmetry associated with this transition [42]. Therefore, the hysteresis difference between the model and experiment at 10 Oe s −1 was subtracted to obtain the rescaled curve. It can be seen that the rescaled curve of the model and experiment agree very well, demonstrating the model's accuracy in describing this behavior. As for the relative error between the experimental and modeled curves, it is observed that it decreases with increasing sweep rates. Such error behavior can be explained by two main factors: (i) the hysteresis increases with increasing sweep rate and (ii) the error is higher for low and decreasing magnetic fields; however, it is of the same order of magnitude for both sweep rates. Hence, it is expected that, with increasing sweep rates, there is a better percentage match between experimental and modeled hysteresis, i.e. the model becomes increasingly more accurate as the field sweep rate increases.
The change of magnetization with time as a step-like function of applied magnetic field starting from 0 Oe with an amplitude of 15 kOe in the T range 190 K to 200 K was simulated (figure 7) in order to study the relaxation time of the sample. There are two temperature intervals of M(H) curves: (1) for T ⩽ 193 K and (2) for T > 193 K. The field is turned on at t = 0 s and for T < 193 K the material is in the ferromagnetic state (displaying high magnetization before the field is applied) and, after applying the magnetic field, the relaxation to a higher magnetization value is fast (∼0.2 s).
Above 193 K, the system starts in a paramagnetic state at zero applied magnetic field (with M = 0 emu g −1 ) and consequently will take longer to arrive at saturation-which explains the time versus temperature discontinuity at approximately 193 K, highlighted in figure 8(a) with a double-sided red arrow. Another outcome is the fact that the time it takes for magnetization to reach saturation (t settle ) is influenced by temperature and field ( figure 8(a)). This is because, at those temperatures and magnetic fields, the term ∂F/∂M is relatively small, while still having a relatively large saturation magnetization to reach. From these curves, two indicative times were obtained: t settle , i.e. the time it takes for magnetization to reach the settled value at each specific temperature (solid symbols in figure 8(a)), and t 90 , i.e. the time it takes for magnetization to reach 90% of the settled value at each specific temperature (open symbols in figure 8(a)).
To compare the correlation of the temperature evolution of these times (t settle and t 90 ) with the temperature evolution of the entropy change curves, the static entropy change between 0 Oe (initial field) and 5, 10, and 15 kOe (final fields) is plotted on figure 8(b)-one remark is that such entropy differences were simulated after the system reaches relaxation. Assuming that such simulated entropy curves establish a limit to the maximum entropy possible, we can see that, for a field change of 5 kOe, the highest values of entropy change are obtained in the temperature range where it takes longer to reach the saturation magnetization, which is undesirable for a real application. In such a scenario, in order to harvest all the entropy change available, a refrigerator/heat pump would have to run at a frequency lower than 0.013 Hz, which is prohibitively low for a real application. These relaxation times have a critical influence not only on a real-life refrigerator, but also on the correct estimation of the MCE in a laboratory environment, since every magnetometer and every measurement protocol have a characteristic time to measure the magnetization, and if one wishes to measure the stable magnetization, a careful choice of measurement setup and protocol must be made. On the other hand, for 10 and 15 kOe step-like field increases, this behavior is also present, but near the transition temperature (∼193 K) there is a high entropy variation and the magnetization reaches saturation faster (less than 4 s) and subsequently all the entropy change can be harvested in a system operating at higher frequencies. These results suggest that, in order to harvest the maximum total entropy variation in a reasonably short timescale, the t sat peak should be avoided, and this can be done by increasing the applied magnetic field. Moreover, for a high field sweep rate, the optimal temperature range is just above the transition temperature. This information is also critically important for the design of composites with several different compositions and several distinct transition temperatures (cascade composites).
In order to illustrate the practical consequences that the different timescales have on the measured MCE (in particular the ∆Sm (T) curves), four different ∆Sm (T) curves were simulated for three different magnetic field sweep rates (1000 Oe s −1 (blue curve), 180 Oe s −1 (red), 10 Oe s −1 (black)) and for static conditions (green) under the same amplitude of magnetic field change ∆H = 15 kOe. As figure 9 shows, the faster the field is swept, the smaller the magnetic entropy change harvested is: under static conditions (where the maximum entropy change is harvested), the ∆Sm MAX ∼ −58 J kg −1 K −1 is occurring at ∼193.1 K, whereas it decreases to −37.1 J kg −1 K −1 (representing a 36% decrease) at ∼194.7 K for 10 Oe s −1 sweep, −34.5 J kg −1 K −1 (41%) 194.6 K for 180 Oe s −1 sweep, and −29.0 J kg −1 K −1 (50%) at ∼194.5 K for 1000 Oe s −1 sweep. This phenomenon has already been experimentally observed in both second-and first-order materials (including La(Fe,Si) 13 alloys) [17,27]. As is seen, there is a remarkable 50% decrease in the magnetic entropy change peak (∆Sm MAX ) if a 1000 Oe s −1 sweep rate is used in comparison with the same value for static conditions. From a technological point of view, a sweep rate of 1000 Oe s −1 would result in 10 s to reach a 10 kOe (1 T) maximum magnetic field, which, considering that each of the four steps of a thermodynamic cycle of a magnetic refrigerator/heat pump takes about 10 s, would result in a rather low operational frequency of ∼0.025 Hz. Additionally, there is also a nonmonotonous change in the temperature at which the ∆Sm MAX is achieved, increasing from 193.1 K (static) up to 194.7 (10 Oe s −1 ) and then slightly decreasing with increasing sweep rate down to 194.5 K (1000 Oe s −1 ), as has also been experimentally observed [17].

Conclusions
The model presented here was able to explain the dynamics of magnetic FOPT using La(Si,Fe) 13 as a case study. The model can describe the magnetic hysteresis dependence on the sweep rate and the time it takes to reach the maximum value. By studying the time needed to reach the maximum magnetization/settled value for a range of temperatures around the transition temperature and comparing it with the static magnetic entropy change, we find that the optimal temperature for operating, e.g. a magnetic refrigerator, must not overlap with the temperature range where the time to reach saturation peaks. It is also demonstrated that such operational temperature ranges can be optimized by adjusting the applied magnetic field such that a higher entropy variation can be harvested in a relatively shorter time, enabling the system to undergo larger frequencies and consequently pump more heat per unit of time. Finally, the dramatic impact of transition kinetics on the magnetic entropy change profile was evaluated by comparison of the different magnetic entropy change curves as a function of temperature for static and different magnetic field sweep rates. A 50% ∆Sm MAX decrease was observed for a 1000 Oe s −1 sweep rate in comparison with static conditions. The application of the present model can be extended for other multifunctional classes of FOPT materials, unveiling crucial technological information regarding the kinetics of the transformations and, consequently, shedding light on the optimal operational frequencies for different amplitudes of external stimuli fields.

Data availability statement
The data cannot be made publicly available upon publication due to legal restrictions preventing unrestricted public distribution. The data that support the findings of this study are available upon reasonable request from the authors.