A Study of Temperature-Dependent Hysteresis Curves for a Magnetocaloric Composite Based on La(Fe, Mn, Si)13-H Type Alloys

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Article A Study of Temperature-Dependent Hysteresis Curves for a Magnetocaloric Composite Based on La(Fe, Mn, Si)13-H Type Alloys

Roman Gozdur 1, Piotr G˛ebara 2 and Krzysztof Chwastek 3,*

1 Department of Semiconductor and Optoelectronics Devices, Łód´zUniversity of Technology, 90-924 Łód´z,Poland; [email protected] 2 Department of Physics, Cz˛estochowaUniversity of Technology, 42-201 Cz˛estochowa,Poland; [email protected] 3 Faculty of Electrical Engineering, Cz˛estochowaUniversity of Technology, 42-201 Cz˛estochowa, Poland * Correspondence: [email protected] or [email protected]

Received: 26 February 2020; Accepted: 19 March 2020; Published: 21 March 2020

Abstract: In the present paper, the eﬀect of temperature on the shape of magnetic hysteresis loops for a magnetocaloric composite core was studied. The composite core, based on La(Fe, Mn, Si)13-H, was set up using three component disks with diﬀerent Curie temperatures. The magnetic properties of the components and the outcome composite core were determined using a self-developed measurement setup. For the description of hysteresis loops, the phenomenological T(x) model was used. The presented methodology might be useful for the designers of magnetic active regenerators.

Keywords: composites; hysteresis loops; temperature eﬀect

1. Introduction One of the most intriguing phenomena examined during studies on energy conversion is hysteresis. The word is derived from Greek υστερε0σεσ, which literally means “to lag behind”. One of the pioneering scientists working on the issue, Sir James Ewing, wrote in 1895: “When there are two quantities M and N, such that cyclic variations of N cause cyclic variation of M, then if the changes of M lag behind those of N, we may say that there is hysteresis in the relation of M and N the value of M at any point of the operation depends not only on the actual value of N, but on all the preceding changes (and particularly on the immediately preceding changes) of N, and by properly manipulating those changes, any value of M within more or less wide limits may be found associated with a given value of N.” [1]. The lag between the input and the output signals manifests itself in the occurrence of the so-called hysteresis curve, cf. Figure1, whose analytical description by itself is a challenging and interesting task both for physicists and engineers. As pointed out in the preface and the introductory chapter of a well-known monograph by Italian researcher G. Bertotti [2], the scientists asked about the most distinctive ﬁngerprint of ferromagnetism might indicate either the existence of the Curie point (the transition from ferro- to paramagnetic regime, important in particular for physicists), or the occurrence of hysteresis loop (interesting for engineers, who design magnetic circuits of electrical machines and devices, cf. e.g., [3–5]).

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Figure 1. A family of hysteresis loops for a soft ferromagnetic material. The largest loop, exhibiting saturation, is referred to as the major loop. Figure 1. A family of hysteresis loops for a soft ferromagnetic material. The largest loop, exhibiting saturation,In the present is referred paper to we as wouldthe major like loop. to attempt both these problems using a simple approach. We would like to examine the effect of temperature on the shapes of hysteresis loops for temperatures approachingThere are the several Curie hysteresis point and models to describe, each it differing using a phenomenological in their complexity T( xand) model background. [6]. In order The bottomto make-up the approach problem proposed more challenging, in 1935 by weF. Preisach focus our [7 attention], further scrutinized on the modeling by I. D. of Mayergoyz a macroscopic [8], andcomposite the top structure,-down description composed proposed of three materials by D. C. withJiles and different D. L. characteristics. Atherton [9] have attracted a lot of attentionThere of arethe engineering several hysteresis community models, [10]. In each the dipresentffering paper in their, we focus complexity on a simpler and background. alternative, namelyThe bottom-up the phenomenologicalapproach proposed T(x) inmodel 1935 proposed by F. Preisach by J. [ 7Takács], further [6]. scrutinized This model by relies I. D. extensively Mayergoyz [on8], hyandperbolic the top-down tangent descriptionrelationship proposed between the by D.input C. Jilesand andthe output D. L. Atherton variables. [9 Its] have developer attracted writes a lot the of modelattention equations of the engineering in the dimensionless community [ form10]. In and the presentdoes not paper, introduce we focus any on interpretation a simpler alternative, for the considerednamely the phenomenological variables, which makes T(x) model the proposed description by universal J. Takács [6 and]. This easily model customizable relies extensively for any on application.hyperbolic tangent In the present relationship paper between, we identify the inputthe input and variable the output as the variables. reduced Its applied developer field writes strength the intensitymodel equations H and the in the output dimensionless variable as form the andreduced does notmagnetization introduce any M interpretation, however it is for also the possible considered to considervariables, more which complicated makes the relationships, description universal e.g., by taking and easily into account customizable the internal for any positive application. feedback In withinthe present the material paper, we[11,12 identify]. the input variable as the reduced applied field strength intensity H andAs the already output variable pointed asout, the the reduced ultimate magnetization goal of the paperM, however is to examine it is also the possible usefulness to consider of the consideredmore complicated description relationships, taking into e.g., account by taking in a into reasonably account wi thede internal range the positive effect feedback of temperature within theon thematerial shape [11 of,12 hysteresis]. curves. Other existing hysteresis models may also be extended for this purpose,As already cf. e.g., pointed [13–17]. out, As the an alternative ultimate goal, it isof also the paper possible is to to examineavail of the loss usefulness vs. temperature of the dependenciesconsidered description [18], howe takingver it into can accountbe mentioned in a reasonably that the wideuse of range a reliable the e ffhysteresisect of temperature model is onmore the flexible,shape of since hysteresis loss dissipated curves. Other as heat existing may always hysteresis be obtained models mayby integration also be extended of the loop for thisarea purpose, [19]. cf. e.g.,Modeling [13–17]. temperatureAs an alternative,-dependent it is also hysteresis possible loops to avail might of loss provide vs. temperature a better insight dependencies into the issue [18], ofhowever energy itdissipation can be mentioned for temperatures that the useclose of to a reliablethe Curie hysteresis point, and model thus isit moremight flexible, be useful since for lossthe designdissipated and asoptimization heat may always of magnetic be obtained circuits by in integration Active Magnetic of the loop Regenerators area [19]. (AMRs). The studies on theModeling possible temperature-dependent applications of AMRs hysteresisin magnetic loops refrigeration might provide systems a better have insight recently into been the issue in the of spotlightenergy dissipation of the scientific for temperatures community close tobecause the Curie of point,both andthe thus environmental it might be useful burden for caused the design by conventionaland optimization solutions of magnetic and the circuits increasing in Active energy Magnetic consumption Regenerators (around (AMRs). 15% The worldwide) studies on thefor refrigerationpossible applications purposes of [2 AMRs0]. So in far, magnetic the modeling refrigeration of magnetocaloric systems have materials recently been has inbeen the carried spotlight out of mostthe scientific often within community the framework because ofof boththe Preisach the environmental model [21– burden23]. caused by conventional solutions and theModeling increasing of energymagnetic consumption response of (around multi- 15%component worldwide) composite for refrigeration materials purposes is theoretically [20]. So possiblefar, the modeling with the of Jiles magnetocaloric–Atherton [24] materials or Preisach has been formalisms carried out [25], most however often within the abovementionedthe framework of referencesthe Preisach provide modeld [ 21the–23 results]. at ambient temperature only. In the considered T(x) model, it is straightforwardModeling of to magnetic describe response the response of multi-component of a composite composite material by materials the appropriate is theoretically weighting possible of contributionswith the Jiles–Atherton from individual [24] or components Preisach formalisms [26]. On the [25 other], however hand, thethis abovementionedmodel may be extended references to describe temperature-dependent hysteresis curves [27]. It should be mentioned that, so far, scientific

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papers on the application of composite materials in magnetic refrigeration systems are focused mostly on experimental work [28–30]. Energies 2020, 13, 1491 3 of 15

2. Materials and Methods provided the results at ambient temperature only. In the considered T(x) model, it is straightforward The magnetocaloric effect (MCE) is a magneto-thermodynamic phenomenon in which a to describe the response of a composite material by the appropriate weighting of contributions reversible change of the temperature of a magnetic material is triggered by the application or from individual components [26]. On the other hand, this model may be extended to describe removal of external magnetic field. Despite the effect being discovered long ago (P. Weiss and A. temperature-dependent hysteresis curves [27]. It should be mentioned that, so far, scientific papers Piccard, 1917), an exponential increase of scientific publications devoted to MCE can be observed on the application of composite materials in magnetic refrigeration systems are focused mostly on after a successful attempt to develop a working prototype of a magnetic refrigerator by the experimental work [28–30]. researchers from Ames Laboratory, Iowa, U.S. and Astronautics Corporation of America in 1998 2.[31 Materials–33]. and Methods The growing interest of scientific community in magnetocaloric effect stems from two facts: The magnetocaloric effect (MCE) is a magneto-thermodynamic phenomenon in which a reversible change1. conventional of the temperature cooling systems of a magnetic based material on vapor is-compression triggered by cycles the application of chlorofluorocarbons, or removal of externalhydrochlorofluorocarbons magnetic field. Despite theand e ff hydrofluorocarbonsect being discovered are long inefficient ago (P. Weiss (theoretical and A. Piccard, efficiency 1917), of an exponentialCarnot’s cycleincrease does of not scientific exceed publications40%) and may devoted contribute to MCE to environmental can be observed bur afterden ( adepletion successful of attemptthe toozone develop layer); a workingon the other prototype hand, MCE of a refrigeration magnetic refrigerator systems can by attain the researchers 60% efficiency from and Ames are Laboratory,environment Iowa,- U.S.friendly and; Astronautics Corporation of America in 1998 [31–33].

2. Thearound growing 15% of interest the world’s of scientific electricity community consumption in magnetocaloric is used for refrigeration effect stems and from air two-conditioning facts: purposes and this value is expected to rise, especially in developed countries [34–37]. 1. conventional cooling systems based on vapor-compression cycles of chlorofluorocarbons, There are several materials exhibiting MCE at near-room temperature. These include hydrochlorofluorocarbons and hydrofluorocarbons are inefficient (theoretical efficiency of Carnot’s gadolinium-based compounds, in particular, the paradigm material Gd5Si2Ge2, La(Fe, Mn, Co, cycle does not exceed 40%) and may contribute to environmental burden (depletion of the Mn)13-x Six(H,N,C)y intermetallics, Laves phases, MnAs1-x Sbx magnetocalorics, ferromagnetic ozone layer); on the other hand, MCE refrigeration systems can attain 60% efficiency and lanthanum manganites and Heusler alloys. Lanthanium-based compounds are relatively cheap and are environment-friendly; are composed of abundant materials, thus they are at present the subject of intensive research 2. around 15% of the world’s electricity consumption is used for refrigeration and air-conditioning [38,39]. In this paper we study the magnetic response of samples composed of sintered purposes and this value is expected to rise, especially in developed countries [34–37]. La(FeMnSi)13-Hx alloy. ThereCommercially are several available materials samples exhibiting of La(FeMnSi) MCE13 at-Hx near-room in the form temperature.of square 30mm These plates, include 0.5mm gadolinium-basedthick were used for compounds, fabrication in ( particular,by water-jet the cutting paradigm) of materialring-type Gd samples5Si2Ge2, examinedLa(Fe, Mn, in Co, the Mn) paper13 x. − SiThex(H,N,C) platesy intermetallics,were prepared Laves in phases, Vacuumschmelze MnAs1 x Sb GmbHx magnetocalorics, by high temperature ferromagnetic sintering lanthanum of − manganitesmicro-powder and La(FeMnSi) Heusler alloys.13-HxLanthanium-based [40]. The stacked core compounds (Figure 2 are) was relatively made of cheap three and rings are with composed Curie oftemperatures abundant materials, 298K, 313K thus and they 318K. are atWeight present compositions the subject ofand intensive magnetocaloric research properties [38,39]. In thishave paper been wecollected study thein Table magnetic 1 and response Figure 3 of. samples composed of sintered La(FeMnSi)13-Hx alloy. Commercially available samples of La(FeMnSi)13-Hx in the form of square 30 mm plates, 0.5 mm thick were used for fabricationTable 1. (by Weight water-jet composition cutting) (wt%) of ring-type of the tested samples samples. examined in the paper. The plates were prepared in VacuumschmelzeTc (K) Fe GmbH byLa highSi temperature Mn sintering of micro-powder La(FeMnSi)13-Hx [40]. The stacked318K core (Figure78.9 2) was16.0 made4.1 of three0.9rings with Curie temperatures 298K, 313K and 318K. Weight compositions313K 76.6 and magnetocaloric18.4 4.0 properties1.0 have been collected in Table1 and Figure3. 298K 77.5 16.6 4.1 1.8

FigureFigure 2.2. SchematicSchematic cross-sectioncross-section of the sandwichsandwich samplesample made ofof threethree ringsrings exhibitingexhibiting didifferentﬀerent CurieCurie temperatures.temperatures.

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Table 1. Weight composition (wt%) of the tested samples.

Tc (K) Fe La Si Mn 318K 78.9 16.0 4.1 0.9 313K 76.6 18.4 4.0 1.0 Energies 20298K20, 13, x FOR PEER REVIEW 77.5 16.6 4.1 1.8 4 of 16

Figure 3. Influence of temperature and magnetic field on the induced magnetocaloric effect (MCE) of Figure 3. Influence of temperature and magnetic field on the induced magnetocaloric effect (MCE) of the examined samples with La(FeMnSi)13-Hx composition. the examined samples with La(FeMnSi)13-Hx composition. Measurements of magnetic properties for the composite core were carried out using an Measurements of magnetic properties for the composite core were carried out using an author-developed setup [41]. The measurement setup met the recommendations of EN 60404-6: author-developed setup [41]. The measurement setup met the recommendations of EN 60404-6: 2004 2004 standard. Instantaneous values of magnetic field strength H(t) were calculated according to standard. Instantaneous values of magnetic field strength H(t) were calculated according to relationship (1) relationship (1) u (t) w H(t) = H 1 (1) LMCEuH tR SHUNTw1 Ht  (1) where: w1—number of turns of exciting coil, RSHUNTLMCE R—shuntSHUNT resistance (Ω), LMCE—average magnetic path of the sample (m), uH(t)—voltage drop across the RSHUNT (V). where: w1 —number of turns of exciting coil, RSHUNT—shunt resistance (Ω), LMCE—average magnetic Magnetic polarization J(t) of the sample was measured by means of the pickup coil w2 and pathcalculated of the sample from Equation (m), uH (2).(t) —voltage drop across the RSHUNT (V). Magnetic polarization J(t) of the sample was measured by means of the pickup coil w2 and Z T calculated from Equation 2. 1 µ0µH(t) w1 J(t) = uJ(t)dt (2) w2SMCE 0 − LMCERSHUNT 1 T 0 H t w1 Jt  uJ tdt  (2) w S 0 L R where: uJ(t)—voltage of the pickup coil2 MCEw2 (V), w2—numberMCE SHUNT of turns of the coil w2, µ0—magnetic 2 constant (H/m), SMCE—cross-section of the core (m ). where: uJ t —voltage of the pickup coil w2 (V), w2 —number of turns of the coil w2 , The standard uncertainty of the H(t) and J(t) measurements was estimated as B-type uncertainty 2 µ0—magnetic constant (H/m), SMCE —cross-section of the core (m ). according to constituents collected in Table2. Assumed and fixed parameters LMCE, w1, w2, SMCE, ρ The standard uncertainty of the H(t) and J(t) measurements was estimated as B-type and µ0 were excluded from the budget of uncertainty. uncertainty according to constituents collected in Table 2. Assumed and fixed parameters LMCE, w1, w2, SMCE, ρ and µ0 were excluded from the budget of uncertainty.

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Table 2. B-type uncertainty of the measurement method.

Source of Instrumentation Intrinsic Assumed Relative uncertainty / method Error distribution uncertainty DAQ card NI-PCI-4462, Range Voltage u (t) , 5.81 mV uniform 0.30 % H ±1 V DAQ card NI-PCI-4462, Range Voltage uJ (t) 15.1 mV uniform 2.38 % ±3.16 V Energies 2020, 13, 1491 5 of 15 Keysight 3458 A Resistance RSHUNT 0.12 mΩ uniform 0.01 % Range 10 Ω Keysight 3458 A Frequency f Table 2. B-type uncertainty of0.005 the measurement Hz triangular method. 0.05% Range 40 Hz TemperatureSource of LTC2983Instrumentation 0.25 deg Intrinsic uniformAssumed Relative1.0% Uncertainty /Method Error Distribution Uncertainty

VoltageThe coreu geometry(t), DAQwas as card follows: NI-PCI-4462, the outer Range diameter1 V d 5.81o  mV2ro was equal uniform to 28 mm, 0.30the %inner H ± Voltage uJ(t) DAQ card NI-PCI-4462, Range 3.16 V 15.1 mV uniform 2.38 % one di  2ri 18 mm, cf. Figure 2. The height of ± each component was l = 0.5 mm (flat disks). It Keysight 3458 A Resistance R r / r 0.12 mΩ uniform 0.01 % should be mentionedSHUNT that the ratioRange o 10i exceededΩ the value 1.1, which implied that the magnetic field distribution in the radial directionKeysight was 3458 inhomogeneous. A This can be proven by the inspection of Frequency f 0.005 Hz triangular 0.05% Figure 4, which presents the computationRange 40 results Hz for this geometry in accordance to the hyperbolic Temperature LTC2983 0.25 deg uniform 1.0% dependence proposed in [42,43],

rm The core geometry was as follows: theH( x outer)  H m diameter do = 2ro was equal to 28 mm, the inner(3) r  x one d = 2r 18 mm, cf. Figure2. The height ofm each component was l= 0.5 mm (flat disks). i i − whereIt should the radius be mentioned for the mean that the magnetic ratio ro flux/ri exceeded was calculated the value from 1.1, which implied that the magnetic field distribution in the radial direction was inhomogeneous. This can be proven by the inspection of ro  ri Figure4, which presents the computation resultsrm  for this geometry in accordance to the hyperbolic ro (4) dependence proposed in [42,43], log ri rm H(x) = Hm (3) rm + x For the considered case rm  11.32 mm. The values at the inner and at the outer radii were equal where the radius for the mean magnetic flux was calculated from to 126% and 81% of the value for rm . This means that the field strength distribution within the sample cross-section was quite inhomogeneous. Figurero ri 4 depicts the variations of magnetic field rm = − (4) strength in the rings upon the update of the instantlog valuero of radius r ri (r runs from ri up to ro , the value depicted with dot denotes rm ).

Figure 4. Calculated distribution of field strength in the radial direction for the considered core geometry. Figure 4. Calculated distribution of field strength in the radial direction for the considered core geometry.For the considered case rm 11.32 mm. The values at the inner and at the outer radii were equal to 126% and 81% of the value for rm. This means that the field strength distribution within the sample cross-section was quite inhomogeneous. Figure4 depicts the variations of magnetic field strength in the rings upon the update of the instant value of radius r (r runs from ri up to ro, the value depicted with dot denotes rm). In the past, there were some attempts to take into account the geometrical effects for the cylinder-shaped cores, which in most cases were made of conventional soft magnetic materials. The Cardiff team examined eighteen wound M4 cores with various heights, inner and outer diameters and build-ups ro r [42]. The authors found that better performance might be achieved for cores − i with increased aspect ratios, defined as l/(ro r ). Moreover, they found that lower losses might be − i Energies 2020, 13, 1491 6 of 15

obtained by increasing the winding ratio 2ri/l provided the build-up ratio and the aspect ratio were kept constant. In a subsequent study [43] the authors discussed theoretical effects of the radial magnetic field and flux density variation, inter-laminar normal flux and winding stress on the real and apparent power loss and the permeability of a range of cylinder-shaped cores. They concluded that to obtain low losses and magnetizing currents, the cores should be designed using wide strips, having large internal diameter and low build-up. Koprivica et al. arrived at similar conclusions in the paper [44], which additionally focused on the issue of digital feedback control during loss measurements for this core geometry. Nakata et al. focused on the problem of an intrinsic measurement error for cylinder-shaped cores due to the common assumption that the mean magnetic path length might be equal to the mean geometric path length [45,46]. The authors suggested a method to take into account the true B = B(H) dependence. Tutkun and Moses examined the geometrical effects for non-standard excitation conditions (pulse width modulation excitation) [47]. The qualitative conclusions from their analysis were similar to those obtained previously, yet in this study the authors stressed the importance of aspect ratio, the worst properties were obtained for its value, approaching two. It should be mentioned that in spite of the internal field distribution inhomogeneity in the considered setup shown in Figure3, all disks are subject to exactly the same maximum field strength, since the excitation windings embrace all disks and the operation mode is similar to that of a current transformer (H-type excitation), which simplifies the analysis. Figure5 depicts the measurement results of polarization vs. temperature dependence for the constant field strength (Hm = 3000 A/m) of the composite core and its components. Similar qualitative dependencies were obtained for the P = P(T) (power loss density vs. temperature) curve. From the figure, it can be noticed that as temperature increased above one of the critical values (Curie points of the components, marked with stars), a sudden drop of the Jm = Jm(T) dependence was observed. In the paper [48], Curie temperatures of individual components are relatively close to each other, which in the case of the outcome recorded response M = M(T) for the composite, yields a monotonous drop of M upon the increase of T. In the case considered in this paper, the Curie temperatures of individual componentsEnergies 2020, 13 are, x FOR relatively PEER REVIEW distant from each other, which results in the jumps of M vs. T dependence.7 of 16 Similar jumps have been observed for gadolinium-based alloys in the recent paper [49].

Figure 5. The measured Jm vs. T dependencies for the examined composite core. Figure 5. The measured Jm vs. T dependencies for the examined composite core. Figure6 presents the measured Jm = Jm(H) dependencies of the examined core and its components at roomFigure temperature. 6 presents the measured J m  J m( H ) dependencies of the examined core and its components at room temperature.

Figure 6. The measured Jm vs. Hm curves of the examined composite core and its components at room temperature.

Figure 7 depicts an exemplary family of temperature-dependent minor (non-saturating) hysteresis loops of the composite core. The effect of temperature on the shape of magnetization curves is clearly visible. The value of the maximum field strength was fixed here at Hm = 2000 A/m. It is interesting to remark that even for the highest considered temperature T = 325.65 K (which was higher that the Curie points of all constituents), a tiny residual hysteresis loop was still observed. This means that the full transition from ferromagnetic to paramagnetic regime did not take place

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Figure 5. The measured Jm vs. T dependencies for the examined composite core.

EnergiesFigure2020, 13 6, 1491 presents the measured J m  J m( H ) dependencies of the examined core and7 of its 15 components at room temperature.

Figure 6. The measured Jm vs. Hm curves of the examined composite core and its components at Figure 6. The measured Jm vs. Hm curves of the examined composite core and its components at room room temperature. temperature. Figure7 depicts an exemplary family of temperature-dependent minor (non-saturating) hysteresis Figure 7 depicts an exemplary family of temperature-dependent minor (non-saturating) loops of the composite core. The effect of temperature on the shape of magnetization curves is clearly hysteresis loops of the composite core. The effect of temperature on the shape of magnetization visible. The value of the maximum field strength was fixed here at Hm = 2000 A/m. It is interesting to curves is clearly visible. The value of the maximum field strength was fixed here at Hm = 2000 A/m. It remark that even for the highest considered temperature T = 325.65 K (which was higher that the Curie isEnergies interesting 2020, 13 ,to x FOR remark PEER thatREVIEW even for the highest considered temperature T = 325.65 K (which8 wasof 16 points of all constituents), a tiny residual hysteresis loop was still observed. This means that the full higher that the Curie points of all constituents), a tiny residual hysteresis loop was still observed. transition from ferromagnetic to paramagnetic regime did not take place even at this temperature. This Thiseven meansat this temperature.that the full transitionThis phenomenon from ferromagnetic is related to residualto paramagnetic ferromagnetism regime didcorresponding not take place to a phenomenon is related to residual ferromagnetism corresponding to a small amount of α-Fe phase. small amount of α-Fe phase.

Figure 7. The eﬀect of temperature on the shape of minor hysteresis loops for the composite core, Figure 7. The effect of temperature on the shape of minor hysteresis loops for the composite core, Hm H= m2 kA/m.= 2 kA /m.

The lanthanum-based alloys are produced during solid state diffusion. As shown in a previous work of one of the authors [50], this type of alloy in the as-cast state has a dendrite-like microstructure. During annealing, the evolution of the microstructure occurs and a homogenous microstructure is finally obtained with some islands built mainly from the α-Fe phase with some small Si and Co content. In order to confirm the existence of the α-Fe phase in the examined composite, the X-ray diffraction (XRD) measurements were carried out. The XRD studies were made using a Bruker D8 Advance diffractometer with CuKα radiation, equipped in an ultrafast semiconductor LynxEye detector. In Figure 8, small reflexes corresponding to 2  44.7 degrees are clearly visible. These were recognized as related to the (110) plane typical for the α-Fe phase. Moreover, the characteristic shift of reflexes corresponding to La(Fe, Mn, Si)13-H type phase towards lower angles suggests an increase in lattice constant of the La(Fe, Mn, Si)13-H, which, on the other hand, is related to the increase of the Curie temperature of every component. This effect has been described elsewhere [48].

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The lanthanum-based alloys are produced during solid state diffusion. As shown in a previous work of one of the authors [50], this type of alloy in the as-cast state has a dendrite-like microstructure. During annealing, the evolution of the microstructure occurs and a homogenous microstructure is finally obtained with some islands built mainly from the α-Fe phase with some small Si and Co content. In order to confirm the existence of the α-Fe phase in the examined composite, the X-ray diffraction (XRD) measurements were carried out. The XRD studies were made using a Bruker D8 Advance diffractometer with CuKα radiation, equipped in an ultrafast semiconductor LynxEye detector. In Figure8, small reflexes corresponding to 2θ = 44.7 degrees are clearly visible. These were recognized as related to the (110) plane typical for the α-Fe phase. Moreover, the characteristic shift of reflexes corresponding to La(Fe, Mn, Si)13-H type phase towards lower angles suggests an increase in lattice constant of the La(Fe, Mn, Si)13-H, which, on the other hand, is related to the increase of the Curie Energiestemperature 2020, 13 of, x FOR every PEER component. REVIEW This effect has been described elsewhere [48]. 9 of 16

Figure 8. The XRD patterns for the considered La(Fe,Mn,Si)13-H samples. Figure 8. The XRD patterns for the considered La(Fe,Mn,Si)13-H samples. Table3 contains the measured values of magnetic properties of the composite core and its constituentsTable 3 at contains room temperature the measured and values for Curie of pointsmagnetic of the properties components. of the composite core and its constituents at room temperature and for Curie points of the components. Table 3. Magnetic parameters of the tested samples at room temperature and Curie points. Table 3. Magnetic parameters of the tested samples at room temperature and Curie points. Temperature |J | |H | H J P Core Type r c m m s Temperature |Jr| |Hc| Hm Jm Ps Core(K type ) (T) (A/m) (A/m) (T) (mW/kg) (K ) (T) (A/m) (A/m) (T) (mW/kg) Composite 293.8 0.1751 296.0 2998.2 0.6152 62.99 Composite 293.8 0.1751 296.0 2998.2 0.6152 62.99 Tc = 298 K 293.8 0.0732 222.0 3000.5 0.4332 34.16 Tc = 298 K 293.8 0.0732 222.0 3000.5 0.4332 34.16 Tc = 313 K 293.8 0.2046 291.7 3000.5 0.7028 68.98 Tc = 313 K 293.8 0.2046 291.7 3000.5 0.7028 68.98 Tc = 318 K 293.8 0.2311 304.2 3002.4 0.7804 86.70 Tc = 318 K 293.8 0.2311 304.2 3002.4 0.7804 86.70 Tc = 298 K 298.0 0.0039 204.1 2999.2 0.0524 3.26 Tc = 298 K 298.0 0.0039 204.1 2999.2 0.0524 3.26 Tc = 313 K 313.0 0.0110 288.8 3000.0 0.0837 7.95 Tc = 313 K 313.0 0.0110 288.8 3000.0 0.0837 7.95 Tc = 318 K 318.0 0.0471 211.3 3000.8 0.2867 25.68 Tc = 318 K 318.0 0.0471 211.3 3000.8 0.2867 25.68 CompositeComposite 298.0 298.0 0.1340 0.1340 293.7 293.7 2997.1 2997.1 0.4427 0.442746.83 46.83 CompositeComposite 313.0 313.0 0.0450 0.0450 269.1 269.1 2999.6 2999.6 0.1894 0.189418.68 18.68 Composite 318.0 0.0086 319.1 3003.9 0.0557 6.167 Composite 318.0 0.0086 319.1 3003.9 0.0557 6.167

Figure 9 depicts the measured dependence of the coercive field strength on temperature for the examined composite core. This dependence shall be used in the modeling shown in the subsequent section. It is interesting to note that the minimal values of the coercive field strength are obtained for temperature range <311; 315> K, which includes the Curie temperature for the intermediate sample. However, it is hard to speak on the monotonicity trend within this range. For lower temperature levels, the value of the coercive field strength decreases upon temperature increase, and for

T  316.75 K, a sudden H c jump is observed, whereas above T  325.65 K the value is practically constant. Additionally, in the figure, the corresponding dependences are shown for the constituent cores.

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Figure9 depicts the measured dependence of the coercive field strength on temperature for the examined composite core. This dependence shall be used in the modeling shown in the subsequent section. It is interesting to note that the minimal values of the coercive field strength are obtained for temperature range <311; 315> K, which includes the Curie temperature for the intermediate sample. However, it is hard to speak on the monotonicity trend within this range. For lower temperature levels, the value of the coercive field strength decreases upon temperature increase, and for T > 316.75 K, a sudden Hc jump is observed, whereas above T = 352.65 K the value is practically constant. Additionally, inEnergies the figure, 2020, 13 the, x FOR corresponding PEER REVIEW dependences are shown for the constituent cores. 10 of 16

Figure 9. Coercive field strength vs. temperature for the composite core and the constituent cores, Figure 9. Coercive field strength vs. temperature for the composite core and the constituent cores, Hm Hm = 3 kA/m. = 3 kA/m. During the measurements, the excitation frequency was kept as low as possible (about 1 Hz) to During the measurements, the excitation frequency was kept as low as possible (about 1 Hz) to avoid the distorting effect of eddy currents on the shape of hysteresis loop. avoid the distorting effect of eddy currents on the shape of hysteresis loop. 3. Modeling 3. Modeling The T(x) description is based on hyperbolic tangent transformation. For a single phase material, the followingThe T(x) description relationships is maybased be on written hyperbolic for symmetrical tangent transformation. hysteresis loops: For a single phase material, the following relationships may be written for symmetrical hysteresis loops: H Hc M = Ms tan h ∓ b (5)  H aH c  M  M s tanh   ±b (5)  a  Hmax + Hc Hmax Hc b = Ms tan h tan h − (6)   Hmaxa Hc  −  H max a Hc  b  M s tanh    tanh   (6) where Ms(A/m) is saturation magnetization,  Hac(A/m) is coercive fielda strength, whereas a (A/m) is a normalization constant (it controls the loop slope). b is introduced in order to match the loop branches where M s (A/m) is saturation magnetization, Hc (A/m) is coercive field strength, whereas a at tips, where magnetic field strength takes the maximum value Hmax. (A/m)Figure is a normalization 10 illustrates howconstant the value(it controls of the the normalization loop slope). constant b is introduceda affects thein order shape to of match modeled the hysteresisloop branches loops. at tips, In the where simulation, magnetic the field normalized strength unitstakes werethe maximum used, and value the valueHmax . of coercive field strengthFigure was 10 set illustrates to unity. how It is easilythe value noticeable of the thatnormalization variation ofconstant the a parameteraffects the value shape alone of allowsmodeled us tohysteresis obtain hysteresis loops. In curvesthe simul thatation are, in the qualitative normalized agreement units were with used, those and depicted the value in Figure of coercive7. field strength was set to unity. It is easily noticeable that variation of the parameter value alone allows us to obtain hysteresis curves that are in qualitative agreement with those depicted in Figure 7.

Energies 2020, 13, 1491 10 of 15 Energies 2020, 13, x FOR PEER REVIEW 11 of 16

Figure 10. TheThe effect eﬀect of of variation variation of ofa value a value on onthe the shape shape of hysteresis of hysteresis loops loops simulated simulated with with the T( thex) T(model.x) model.

The response of a multi-componentmulti-component alloy is obtained by summing weighted contributions from individual components. In In our our case case,, the the weights weights are are taken as equal to 1/3, 1/3, whichwhich corresponds to the volumetric content of each component (in the firstfirst approximation we neglect the volumetric changes during phase transition). The assumption of constant weig weightinghting means that the coupling between individual rings, as well asas thethe interface-relatedinterface-related phenomena,phenomena, areare neglected.neglected. Moreover, the leakageleakage fieldsfields are not taken into account. In orderorder toto verifyverify the the correctness correctness of of the the above-given above-given assumptions, assumptions we, we have have taken taken advantage advantage of the of measurementthe measurement data data depicted depicted in Figure in Figure6. It 6 was. It was found found that that the relativethe relative di ff erencedifference for thefor pointsthe points on theon measuredthe measured curve curve for the for composite the composite core andcore those and those from thefrom averaged the averagedJm = Jm Jm(T =) dependenceJm(T) dependence was in was the orderin the oforder 10%. of The 10%. highest The highest discrepancies discrepancies were obtained were obtained for the for extreme the extreme values values of field of strength field strength (in the low(in the and low the and high the field high regions), field regions), yet they yet did they not did exceed not exceed 15%. 15%. Our previous experience with modeling temperature dependent hysteresis curves with T(x)-derived)-derived descriptions for a single componentcomponent alloy [2 [277] allowed us to assume that the values of all model parameters might might be be highly highly sensitive sensitive to to temperature. temperature. Therefore, Therefore, we we treated treated a aandand MMss as free parametersparameters for all constituentsconstituents and repeated thethe modelingmodeling forfor elevatedelevated temperatures.temperatures. The only assumption waswas thatthat valuevalue of of saturation saturation magnetization magnetization for for a givena given sample sample should should be higherbe higher than than that for loop amplitude, Hm = 3000 K. The values of Hc were taken from data depicted in Figure9. that for loop amplitude, Hm = 3000 K. The values of H c were taken from data depicted in Figure 9. Figure 11 depicts the descending branches of hysteresis loops at room temperature, when all Figure 11 depicts the descending branches of hysteresis loops at room temperature, when all components remained in ferromagnetic phase. It can be observed that the shape of measured outcome components remained in ferromagnetic phase. It can be observed that the shape of measured hysteresis branch is well reproduced with the proposed model. The average deviation between the outcome hysteresis branch is well reproduced with the proposed model. The average deviation measured and the modeled curves was 16.3%. The quantity at the ordinate axis is magnetization, between the measured and the modeled curves was 16.3%. The quantity at the ordinate axis is related to polarization with the relationship M = J/µ0, where µ0 is free space permeability. magnetization, related to polarization with the relationship M  J / 0 , where 0 is free space Figure 12 presents a visual comparison between the measured and the modeled hysteresis loops permeability. for T = 307.65 K (above the Curie point for the first sample). It can be stated that in this case the model also yields reasonable agreement with the experiment. The average deviation between the measured and the modeled curves was 22.5%.

Energies 2020, 13, x FOR PEER REVIEW 12 of 16

Energies 2020, 13, 1491 11 of 15 Energies 2020, 13, x FOR PEER REVIEW 12 of 16

Figure 11. The measured descending branches of hysteresis loops of the composite core and its components at room temperature; additionally, the modeled descending branch for the composite core is shown as solid line.

Figure 12 presents a visual comparison between the measured and the modeled hysteresis loopsFigure for T 11.= 307.65The measuredK (above descendingthe Curie point branches for the of hysteresisfirst sample). loops It ofcan the be composite stated that core in this and case its the Figure 11. The measured descending branches of hysteresis loops of the composite core and its modelcomponents also yields at room reasonable temperature; agreement additionally, with thethe modeled experiment. descending The branchaverage for deviation the composite between core the components at room temperature; additionally, the modeled descending branch for the composite is shown as solid line. measuredcore is andshown the as modeled solid line. curves was 22.5%.

Figure 12 presents a visual comparison between the measured and the modeled hysteresis loops for T = 307.65 K (above the Curie point for the first sample). It can be stated that in this case the model also yields reasonable agreement with the experiment. The average deviation between the measured and the modeled curves was 22.5%.

Figure 12. The measured and the modeled hysteresis curve for the composite core at T = 307.65 K. Figure 12. The measured and the modeled hysteresis curve for the composite core at T = 307.65 K. Modeling was also attempted for a temperature exceeding the Curie point values of all constituent ringsModeling(T = 329.65 was K). In also this attempted case, however, for a significant temperature discrepancies exceeding between the Curie the point measured values and of the all modeledconstituent curves rings were (T = observed, 329.65 K). cf. In Figure this 13 case. This, however, effect may significant be due to discrepancies the assumption between that the the volumetricmeasured andeffects the could modeled be neglected curves werein the observed, analysis. We cf. decidedFigure 1 to3. include This effect the Figure may be in due the paper, to the since itFigure illustrates 12. The the measured fact mentioned and the modeled previously hysteresis clearly, curve namely for the composite the presence core at of T the = 307.65 tiny residualK. ferromagnetic phase even above the Curie point. Modeling was also attempted for a temperature exceeding the Curie point values of all constituent rings (T = 329.65 K). In this case, however, significant discrepancies between the measured and the modeled curves were observed, cf. Figure 13. This effect may be due to the

Energies 2020, 13, x FOR PEER REVIEW 13 of 16

assumption that the volumetric effects could be neglected in the analysis. We decided to include the EnergiesFigure2020 in ,the13, 1491paper, since it illustrates the fact mentioned previously clearly, namely the presence12 of 15of the tiny residual ferromagnetic phase even above the Curie point.

Figure 13. The measured and the modeled hysteresis curve of the composite core at T = 329.65 K. Figure 13. The measured and the modeled hysteresis curve of the composite core at T = 329.65 K. The values of model parameters for some chosen temperatures are listed in Table4. The values of model parameters for some chosen temperatures are listed in Table 4. Table 4. Values of model parameters for chosen temperature values. Table 4. Values of model parameters for chosen temperature values. T (K) a, (A/m) Hc, (A/m) Ms (A/m) T (K) a, (A/m) Hc, (A/m) Ms (A/m) 293.85 sample with T = 298 K 1948.9 192.0 1.12 105 C C 5 293.85 sample with T = 298K 1948.9 192.0 1.12 × 10 × 5 - sample with TC = 313 K 1184.6 286.4 5 5.31 10 - sample with TC = 313K 1184.6 286.4 5.31 × 10 × 5 - sample with TC = 318 K 1188.6 299.5 5.92 10 ‐ sample with TC = 318K 1188.6 299.5 5.92 × 105 × 5 307.65 sample307.65 withsampleTC with= 298 T KC = 298K 3569.3 3569.3 201.9 201.93.84 × 105 3.84 10 × 5 - sample‐ withsampleTC with= 313 T KC = 313K 1383.5 1383.5 231.9 231.93.95 × 105 3.95 10 × 5 - sample with T = 318 K 1330.4 260.7 5 5.24 10 ‐ sampleC with TC = 318K 1330.4 260.7 5.24 × 10 × 4 4 329.65 sample329.65 withsampleTC with= 298 T KC = 298K 3857.2 3857.2 185.0 185.03.80 × 10 3.80 10 × 5 - sample‐ withsampleTC with= 313 T KC = 313K 2839.5 2839.5 325.5 325.54.98 × 105 4.98 10 × 5 - sample‐ withsampleTC with= 318 T KC = 318K 1182.2 1182.2 417.4 417.41.98 × 105 1.98 10 ×

4.4. ConclusionsConclusions InIn this th paper,is paper a simple, a phenomenological simple phenomenological description was description applied to model was temperature-dependent applied to model hysteresistemperature loops-dependent of a La(Fe, hysteresis Mn, Si)13 loops-H magnetocaloric of a La(Fe, Mn, alloy. Si)13- TheH magnetocaloric composite consisted alloy. The of three composite disks withconsisted different of three Curie disks temperatures. with different The resulting Curie temperatures. hysteresis loop The was resulting obtained hysteresis by weighting loop wasthe magneticobtained by responses weighting from the individualmagnetic response constituents.s from individual For modeling constituents. purposes, For the modelingT(x) description purposes, wasthe T used.(x) description was used. TheThe temperaturetemperature dependencies dependenciesJ mJm((TT)) andandH Hcc ((TT)) forfor compositecomposite magnetocaloricmagnetocaloric alloysalloys (shown(shown inin FigureFigure5 )5) allowed allowed usus to carry carry out out an an unambiguous unambiguous identification identification of phase of phase transformations transformations and Curie and Curietemperatures. temperatures. However, However, it wa its was not not possible possible to todetermine determine the the true true value value of of the totaltotal magnetic magnetic polarizationpolarization fromfrom thethe superpositionsuperposition ofof thethe polarizationspolarizations ofof individualindividual componentscomponents because because ofof thethe inhomogeneityinhomogeneity of of magnetic magnetic flux flux distribution. distribution. The The application application of of phenomenological phenomenological T(x) T(x) model model made made it possibleit possible to estimateto estimate the the values values of magnetic of magnetic parameters parameters of practical of practical interest, interest like ,H likec or HBrc ,or which Br, wh inich turn in madeturn ma it possiblede it possible to estimate to estimate power power losses losses that dissipated that dissipated in the in composite. the composite. Experimental research has shown that even above the highest Curie temperature of the constituent samples the outcome magnetic response still exhibited tiny hysteresis, which implies that there existed a residual α-Fe phase in the material. Energies 2020, 13, 1491 13 of 15

Future work shall be devoted to ﬁne-tuning of the description to take into account the volumetric eﬀects during phase transitions. We believe that the proposed model might be useful for the design and optimization of magnetic regenerators.

Author Contributions: Conceptualization, R.G. and P.G.; methodology, R.G. and K.C.; software, K.C.; validation, P.G.; formal analysis, P.G.; investigation, R.G.; resources, R.G. and P.G.; data curation, R.G.; writing—original draft preparation, K.C.; writing—review and editing, R.G.; visualization, K.C.; supervision, R.G.; project administration, R.G.; funding acquisition, K.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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