A Preisach-Based Nonequilibrium Methodology for SimulatingPerformance of Hysteretic Magnetic Refrigeration Cycles

TIMOTHY D. BROWN,1 NICKOLAUS M. BRUNO,2 JING-HAN CHEN,3

IBRAHIM KARAMAN,1 JOSEPH H. ROSS JR.,3 andPATRICK J. SHAMBERGER1,8

1.—Department of Materials Science & Engineering, Texas A&M University, College Station,TX 77843, USA. 2.—Department of Mechanical Engineering, Texas A&M University, CollegeStation, TX 77843, USA. 3.—Department of Physics and Astronomy, Texas A&M University,College Station, TX 77843, USA. 4.—e-mail: patrick.shamberger@tamu.edu

In giant magnetocaloric effect (GMCE) materials a large entropy changecouples to a magnetostructural first-order phase transition, potentially pro-viding a basis for magnetic refrigeration cycles. However, hysteresis lossgreatly reduces the availability of refrigeration work in such cycles. Here, wepresent a methodology combining a Preisach model for rate-independenthysteresis with a thermodynamic analysis of nonequilibrium phase transfor-mations which, for GMCE materials exhibiting hysteresis, allows an evalua-tion of refrigeration work and efficiency terms for an arbitrary cycle. Usingsimplified but physically meaningful descriptors for the magnetic and thermalproperties of a Ni45Co5Mn36.6In13.4 at.% single-crystal alloy, we relate thesework/efficiency terms to fundamental material properties, demonstrating themethod’s use as a materials design tool. Following a simple two-parametermodel for the alloy’s hysteresis properties, we compute and interpret the effectof each parameter on the cyclic refrigeration work and efficiency terms. Weshow that hysteresis loss is a critical concern in cycles based on GMCE sys-tems, since the resultant lost work can reduce the refrigeration work to zero;however, we also find that the lost work may be mitigated by modifying otheraspects of the transition, such as the width over which the one-way trans-formation occurs.

INTRODUCTION

The need to improve upon the efficiency andenvironmental impact of vapor compression refrig-eration has led to the development of materials fornovel refrigeration cycles.1,2 Magnetic refrigerationcycles via the magnetocaloric effect (MCE) are pos-sible in materials with strongly coupled thermal andmagnetic properties.3,4 In materials manifestingMCE, application of an external magnetic field Hinduces an entropy change, DSm, under isothermalconditions, or a corresponding temperature change,DTad, under adiabatic temperature conditions.5–7

These parameters are indirectly related to potentialheat transfers occurring in a material as it under-goes a refrigeration cycle, and they have been usedas figures of merit to quantify materials’ refrigera-tion potential.

For temperatures above 1 K,8 DSm and DTad areconventionally largest near magnetic order–disor-der transitions,9,10 where the magnetic order issensitive to H; For example, near the Curie tem-perature, TC, in ferromagnetic Gd, changing theapplied field from 0 T to 2 T yields a DSm and DTad

of 5 J kg�1 K�1 and 6 K,11,12 respectively. Cyclesbased on this larger MCE have been shown topotentially achieve temperature spans (�50 K)comparable to those obtained using vapor com-pression;13–15 however, the large fields required(�7 T), make these cycles commercially unviable.

The prospect of near-room-temperature magneticrefrigeration was greatly advanced by the discoveryof the giant magnetocaloric effect (GMCE) inGd5(Ge,Si)4

11,16 (DSm ¼ 27 J kg�1 K�1; DTad ¼ 7 Kunder 2 T) associated with a magnetostructural first-order phase transition (FOPT).12 Following this,

JOM, Vol. 67, No. 9, 2015

DOI: 10.1007/s11837-015-1519-0� 2015 The Minerals, Metals & Materials Society

(Published online July 3, 2015) 2123

many GMCE materials were discovered, most no-tably (values quoted for 0 to 2 T field change): La(Fe,Si)13 and its hydrides (15 J kg�1 K�1 to 25 Jkg�1 K�1; 4 K to 8 K),17–19 (Mn,Fe)2(P,Si,Ge) com-pounds (12 J kg�1 K�1 to 25 J kg�1 K�1; 1.8 K),20–22

and several Ni-Mn-based off-stoichiometric Heusleralloys (6 J kg�1 K�1 to 15 J kg�1 K�1; 2 K to6 K).23–25

In these classes of materials, the system trans-forms between two phases with different crystalstructures and magnetizations, M. During theFOPT, the sudden structural change also creates anabrupt change in the system’s entropy, DS, due tothe latent heat of transformation, which contributesgreatly to the GMCE.12 The field H stabilizes thehigher-M phase by reducing the free energy via theZeeman energy l0H�M, thus coupling the transitiontemperature and GMCE to an applied magneticfield. Hence, the GMCE can serve as a vehicle forT–H refrigeration cycles.

Despite the magnitude of the GMCE, its imple-mentation in real cycles remains critically limitedby a variety of factors including hysteresis loss fromcycling the magnetostructural FOPT, eddy-currentloss, low cycle frequencies imposed by poor thermalconductivities, and FOPT instability where theGMCE may decrease after many cycles.26–29 Ofthese, hysteresis loss has been observed to reducethe actual extent of the FOPT so that only about 5%of the potential DSm or DTad is actually accessiblewith field cycling.26 Furthermore, unlike thermalconductivities and eddy-current loss, which may beeffectively controlled through design of the refrig-erant’s geometry, hysteresis loss is difficult to re-duce purely at the system level. Therefore,materials-based analysis of hysteresis losses iscritical to improve the state of GMCE-based mag-netic refrigeration.6,26,30

Phenomenologically, hysteresis is the path-de-pendent branching of a material response at inputextrema,31 and in GMCE systems can manifest asloops in M(T) profiles under fixed H; generally,thermal hysteresis widths DThyst are used to quan-tify the magnitude of this effect macroscopically.Microscopically, these loops arise from thermody-namic irreversibility in the material mechanismsdriving the FOPT. Knowledge of these mechanismshas allowed the design of systems with reducedDThyst; For example, introducing porosity and/orhydrogen interstitials into La(Fe,Si)13,32–34 tuninglattice parameters to introduce mixed magnetism in(Mn,Fe)2(P,Si,Ge),35–37 and reducing the marten-site–austenite coherency strain in shape-memoryalloys29,38,39 have all been explored to reduce DThyst

to within the ranges of 0.4 K to 3 K, 1 K to 5 K, and2 K to 5 K, respectively.

Despite these advances, fundamental questionsabout the practical utilization of cycles based onhysteretic GMCE materials remain largely unan-swered. Of particular interest are four classes ofquestions defining the materials science aspect of

GMCE refrigeration: What is the effect on therefrigeration work and cycle efficiency of (1) thehysteretic character of the FOPT for a given GMCEand T–H path, (2) the thermomagnetic GMCEmaterial properties for a given cycle and hysteresis,and (3) the chosen T–H path for a given hysteresisand thermomagnetic properties? Finally, (4)accounting for hysteresis, how do cycles based onGMCE materials perform when compared againsteach other or against those based on standards suchas conventional MCE or vapor compression?

Experimentally exploring solutions to the abovequestions requires resources to characterize andanalyze a large number of GMCE materials under animpractically wide range of conceivableT–H paths. Apreferable approach is to use existing limited data-sets to construct simplified models for a given mate-rial’s magnetothermal and hysteresis propertiesfrom a few independent materials parameters, andthen to subject them to rigorous, thermodynamicallyvalidated simulations across a range of cycle condi-tions. Previous work utilized microscopic hysteresismodels defined by mostly phenomenological param-eters to predict the general behavior of systemsundergoing special cases ofT–Hpaths (e.g., Carnot orBrayton cycles)40–42 or validation of models by com-parison with experimental hysteresis loops. In con-trast, the method proposed herein aims to quantifyrefrigeration ability through overall work and effi-ciency terms for an arbitrary thermodynamic cycleand, further, to relate these to macroscopic materialproperties and to loss terms due to hysteresis.Through exploration of the relationship betweenrefrigeration cycle, hysteresis, and material proper-ties, this methodology can serve as an essential toolfor GMCE materials design.

In this paper we describe a system-independent,materials-centered approach to evaluate an irre-versible cycle based on the hysteretic GMCE. Wethen demonstrate its utility by considering a par-ticular instance of question (1) above, by first fixingmagnetothermal material parameters according toexperimental data from Ni45Co5Mn36.6In13.4 at.%single crystals, and then using a simple but obser-vationally consistent two-parameter Preisach modelto simulate the transition under a particular T–Hcycle. We report on the thermodynamic effect ofeach Preisach parameter on refrigeration figures ofmerit and interpret the results within the context ofmaterials design principles.

HYSTERESIS AND THERMOMAGNETICMODELING

During a magnetic refrigeration cycle, the net mag-netic work Win done on the system is quantified as

Win ¼I

l0H dM � 0; (1)

where l0 is the magnetic permeability of vacuum,with magnitude 1.26 9 10�6 kg m s�2 A�2, and the

Brown, Bruno, Chen, Karaman, Ross, and Shamberger2124

integrand of the cyclic integral, H(M), describes thematerial’s magnetic response throughout a T–Hcycle. This Win is employed to lift a quantity of heatQc from a cold reservoir and to pump another heatQh into the ambient hot reservoir, as in Fig. 1. Therefrigeration work is defined as Wref ¼ Qh �Qc, andis the portion of Win creating the desired net heattransports against the temperature gradient. For anidealized, thermodynamically reversible system,Wref and Win are equal.

The situation is very different in the GMCEmaterials under consideration, as the mechanismsresponsible for hysteresis produce an unwantedirreversible entropy contribution, DSprod. As theFOPT progresses infinitesimally, a differential en-

tropy elementdS > 0 is produced. Additionally,

a small element of heat dQ may be expelled. Then,the second law of thermodynamics gives thesmall change in the system entropy, S, as

When these small contribu-

tions are summed over the T–H cycle, and withsome manipulation, the net heat expelled is found to

be Q = − T dS + I,net where the integrand of

the cyclic integral T(S) describes the material’sthermal response to the T–H cycle. The last term,

, quantifies the lost work due to

entropy produced by irreversibilities. The first lawof thermodynamics requires that Qnet ¼ Win, there-fore Wref is given by

Wref ¼ �I

T dS ¼ Win � I; (2)

and the energy flows are related as in Fig. 1.Therefore, in every refrigeration cycle actuating

irreversible processes in a material, a quantity I ofpotential refrigeration work is lost to entropy pro-duction. This general result is based only on athermodynamic analysis of irreversibility, and itseffect on Wref motivates our investigation into thelimits that macroscopic hysteresis places on GMCE-based refrigeration. To better quantify the effect of

hysteresis on cycle performance, an efficiency figureof merit can be defined as

g ¼ Wref

Win¼ 1 � I

Win(3)

with 0 � g � 1. This g is the fraction of the totalwork that must be done by external means whichactually generates useful refrigeration work. Notethat for g ’ 1, the potential input work of the cyclehas been most effectively utilized; i.e., the lost workdue to hysteresis is minimized.

Here it is important to note that Wref, I, and g areall purely materials parameters; they are definedwithout reference to the details (e.g., refrigerantflow rates, cycle frequencies) of any particularrefrigeration system design. Within this framework,the performance of GMCE refrigeration cycles maybe quantified by computing Wref, I, and g for variouscycles of interest. According to Eqs. 1–3, this re-quires determination of the system’s magnetizationM(T, H) and entropy S(T, H) at each point of thecycle, a task which is generally complicated by theinteraction between the H-induced transition shiftand the hysteretic path dependence.

To apply the preceding methodology, calculationsare broken down into four parts, as in Fig. 2. First aparticular T–H path is selected, and models for amaterial’s single-phase magnetothermal and hys-teresis properties are created. Then the phasefraction at each point of the cycle is generated froma nonequilibrium Preisach hysteresis model. Next,the total system magnetization M(T, H) and entropyS(T, H) along each leg of the cycle are computedfrom the rule of mixtures. Finally, Wref, I, and g are

Fig. 1. Energy flows in an arbitrary refrigeration cycle. Hysteresisloss generates irreversible lost work I which decreases the potentialrefrigeration work Wref for a given cycle.

Fig. 2. Methodology flow diagram. (1) Material model parameterizedin terms of hysteresis parameters, l, and single-phase properties,Ma;b and Sa;b, which are combined with the T–H path using (2) anonequilibrium Preisach model to yield the phase fraction, /. (3)Total system properties, M and S, are calculated with the rule ofmixtures, allowing (4) figures of merit, Wref, I, and g, to be calculated.

A Preisach-Based Nonequilibrium Methodology for Simulating Performanceof Hysteretic Magnetic Refrigeration Cycles

2125

determined from Eqs. 1–3. Thus, beginning fromonly the fundamental laws of thermodynamics anda small set of material parameters, the relevantenergy flows in an arbitrary GMCE-based refriger-ation cycle can be computed and, for differentmaterial/cycle combinations, compared.

Nonequilibrium Thermodynamic Analysis

For systems exhibiting ideal rate-independenthysteresis, the relaxation kinetics are infinitesi-mally fast; such systems can remain in nonequilib-rium states for an arbitrarily long time. Since thesystem is not at equilibrium, the specification of theexternal conditions does not uniquely specify thesystem state;43 i.e., the thermodynamic propertiesare no longer state variables. However, this may beresolved by introducing an additional set of vari-ables which, together with the thermodynamicproperties, reestablish the state of the system. Suchvariables do not appear explicitly in the equilibriumthermodynamic potentials and are hence calledinternal variables.44

We model the GMCE system as a two-phase sys-tem consisting of an a (b) phase stabilized under agiven field at a low (high) temperature. Althoughwithin Heusler alloys the FOPT is generallymartensitic, we have deliberately avoided namingthe phases ‘‘martensite’’ or ‘‘austenite,’’ as thedevelopment introduced here generalizes beyondjust martensitic transitions. Given this notation, thephase fraction /, defined as the mass fraction of b, isthe appropriate internal variable40 referred toabove. In the two-phase region of the T–H space,one has 0 � / � 1, with some complicated pathdependence on T and H.

Equi-driving Force Diagrams

To simplify the determination of /ðT;HÞ, we usean observation from studies of polycrystalline GMCEmaterials26,45 suggesting that (1) when converted tofree energy changes, variations in either T or H drivethe transition equivalently, since (2) when variationsin T and H are converted into free energy changes,identical internal and envelope hysteresis loops areobtained regardless of whether H is cycled at con-stant T or T is cycled at constant H. For thesematerials, the extent of phase transformation (in-cluding hysteresis losses) induced by T or H can beequivalently converted into that of a single general-ized driving force D, defined as the difference in freeenergies of the two component phases, i.e.,D ¼ Gb �Ga. This construction is shown in Fig. 3a.

Now, the determination of /ðT;HÞ is reduced todetermining /ðDÞ. The preceding assumptions implythat, whatever /’s path-dependent relation to D,when plotted inT–H space, contours of equalD and ofequal / coincide. These equi-driving force contoursare calculated from a generalization of the Clausius–Clapeyron relation. Along such a contour, the drivingforce is some constant D0, so Gb ¼ Ga þD0. If T or H is

changed, to remain on the D0 contour, variations inthe free energy must be balanced as dGa ¼ dGb withdG ¼ �SdT �M dl0H for each phase. This yields

dl0H

dT

����D0¼ � SbðT;HÞ � SaðT;HÞ

MbðT;HÞ �MaðT;HÞ ; (4)

where dl0H

dTis the slope in T–H space of the D0 equi-

driving force contour, and Sa;b and Ma;b are theentropies and magnetizations, respectively, of thecomponent a and b phases along that contour. Thisformalism is equivalent to the assertion that theequilibrium (D0 ¼ 0) magnetic Clausius–Clapeyronrelationship also holds for the iso-D and iso-/ con-tours in irreversible phase transitions.

Equation 4 is a set of differential equations, onefor each D0, with the magnetic field-temperatureslope for each specified by the single-phase proper-ties Sa;b and Ma;b. The iso-D contours have a usefulinterpretation, as shown in Fig. 3a. At zero appliedfield (e.g., the H ¼ 0 axis), the material finishes itstransition to b on warming at a temperature T0

bf ;

Fig. 3. (a) Equi-driving force contour construction from curves ofconstant D ¼ Gb �Ga. For every pair of transformation-finish tem-peratures T 0

af and T 0bf (triangles) under zero field, there are corre-

sponding temperatures TH 0

af and TH 0

bf (circles) under nonzero field. (b)Different T–H paths between the same equi-driving force contoursinduce the same extent of phase transformation.

Brown, Bruno, Chen, Karaman, Ross, and Shamberger2126

similarly, it finishes its transition to a on cooling atT0af . At some nonzero applied field (H ¼ H0), the

corresponding transition-finish temperatures areTH0

af and TH0

bf . These pairs of corresponding zero-fieldand nonzero-field transition-finish temperatures areby definition linked by a pair of specific iso-D con-tours, as illustrated in Fig. 3a and b, bottom. Simi-lar contours also link T0

a;bs, the transition starttemperatures at zero field, with the correspondingTH0

a;bs at some higher field. Alternatively, the con-tours may be viewed as linking correspondingtransition-finish fields HT

af and HTbf at each constant

T, as in Fig. 3b, right.Equi-driving force contours allow the conversion

of a given iso-T variation in H into a ‘‘transforma-tion-equivalent’’ iso-H variation in T: along each,there is an equivalent evolution of /. In general, anarbitrary T–H path may be decomposed into legsalong which either T or H is constant. Then, usingthe equi-driving force contours, each leg may beconverted into a transformation-equivalent path atzero field. Therefore, any arbitrary T–H path can beconverted into a series of T variations at H ¼ 0,which defines a pseudotemperature T0 axis; Forexample, in Fig. 3b, both the horizontal and verticallines correspond with 305 K � T0 � 321:8 K.

Hysteresis and the Preisach Model

The above treatment considerably simplifies theproblem, as the complicated relationship between /and the T–H path under consideration simplifies toan equivalent relation between / and T0. However,because of path dependence, T0 does not uniquelydetermine / in the two-phase region, and the rela-tionship is not that of a function, but of some operator.The Preisach hysteresis operator, P,31,46 has provedespecially robust in hysteresis modeling. Althoughsignificant work has been done to interpret the Pre-isach model’s parameters within the context of en-ergy landscapes47,48 and entropy production,41,49,50 itremains essentially phenomenological. Despite this,

experimental hysteresis loops approximatelydemonstrate the necessary conditions of the model:(1) larger input variations erase memory of smallerinput variations (wiping out property), and (2) nomatter the previous history, equivalent input varia-tions create output loops that differ at most by aconstant (congruency property).31

The basic unit of the model is the Preisach hys-teron h, a discrete on/off operator which contains allof the model’s essential nonlinearity and pathdependence.31 We assume that this hysteron rep-resents some phenomenological unit which con-tributes an element of phase /1 to /, depending onT0. As T0 increases from large negative values, theelement is entirely a and /1 ¼ 0 until T0 >T0

b, atwhich point it transforms completely to b and/1 ¼ 1. As T0 is decreased back through T0

b the unitremains completely b until T0 <T0

a <T0b, at which

point it transforms completely to a again, as shownin Fig. 4a, inset.

To obtain the full operator P, we imagine a dis-tribution of hysterons with unique pairs of switch-ing inputs fhiðT0

a;T0bÞg, whose contributions f/ig are

weighted by some corresponding distribution flig,then summed in parallel. Then, as the hysterondensity grows infinitely dense, the Preisach modelfor / with input T0 is given by

/ ¼ P½T0� ¼ZZ

lða; bÞha;b½T0�dadb; (5)

where we have renamed our indexing from ðT0a;T

0bÞ

to ða; bÞ for notational simplicity.The Preisach operator has a convenient geomet-

rical interpretation when the switching inputs areplotted on Cartesian axes ðx; yÞ ! ða;bÞ, definingthe Preisach plane31 (Fig. 4a, full). Then, the so-called Preisach density l is some surface (here, thesurface is nonzero only over the thick red region)over the Preisach plane. As T0 increases, contribu-tions to / can be visualized as filling up under lalong the b axis; similarly, as T0 decreases, / emp-ties out along the �a axis. Hysteresis is accommo-

(a) (b)

Fig. 4. (a) Hysteresis properties as represented by the Preisach plane. (Inset) Unit hysteron output corresponding with each point in the Preisachplane. Outputs are weighted by Preisach density and summed to obtain total response (Full). A special two-parameter Preisach density (bold redline) has been selected for this manuscript. (b) Envelope and interior hysteresis loops resulting from density in (a) (Color figure online).

A Preisach-Based Nonequilibrium Methodology for Simulating Performanceof Hysteretic Magnetic Refrigeration Cycles

2127

dated as the path dependence of the ‘‘state line’’composed of alternating fill-up/empty-out segments.

Once l is specified, the path-dependent response of/ to any arbitrary T0 path is uniquely determined(Fig. 4b), and hence the surface lða; bÞ completely anduniquely characterizes the hysteretic character of theFOPT. The reverse problem of determining lða; bÞ fora given hysteresis is uniquely determined by the sys-tem’s internal and envelope hysteresis loops.31

Here, a two-parameter (DThyst, DTelast) Preisachdensity is used to describe hysteresis, as in Fig. 4a.This parameterization is motivated by experimentalobservations of constant-H hysteresis loops in /ðTÞfor thermoelastic martensites.51 In these loops, theone-way transitions extend over some temperaturerange (DTelast), and the forward and reverse trans-formations are also displaced relative to each otherby another temperature difference (DThyst), as shownschematically in Fig. 4b. The former is thought toarise from reversible storage of elastic strain energydue to coherency effects, whereas the latter is amanifestation of hysteresis. This choice of parame-terization captures some of the essential behavior ofgeneral hysteresis observed experimentally in thesesystems,26,51 while reducing it to a space of param-eters with an intuitive interpretation.

Magnetothermal Modeling

The description of a self-consistent model for aGMCE system’s single-phase thermomagnetic prop-erties Ma;b and Sa;b follows below. These models havebeen parameterized using data from a Ni45-

Co5Mn36.6In13.4 shape-memory alloy with Mb >Ma sothat the material manifests what is commonly re-ferred to as inverse GMCE. It is important to notethat the general approach outlined in the previoustwo sections is equally valid for purely empirical fitsto experimental data as for more sophisticated mod-els derived from fundamental physical relationships.For this manuscript a compromise between purelyempirical and first-principles approaches has beenreached, in order to balance model fidelity withphysically meaningful parameters.

Magnetization

The b phase is assumed to be ferromagneticallyordered with high saturation magnetization Ms.Determining the a ordering is difficult due to itssmall saturation magnetization, but in NiMnInsystems there is evidence that the a (martensite)phase exhibits frustrated antiferromagnetism. Ineither case, a Brillouin model has been used to de-scribe the magnetization contours M(T) at somesaturation field H0, which assumes that the strongindividual exchange interactions can be replacedwith an internal mean field that greatly exceeds theapplied field. We choose a j ¼ 1=2 model, since itprovides a reasonably good fit to the Mb contours,and Ma � Mb so that a 10% variation in Ma per-turbs the Wref and g calculations by less than 1.5%.

These assumptions result in a determinate implicitequation relating M and T:

M

Ms¼ BjðxÞ (6)

with BjðxÞ being the jth Brillouin function defined interms of x ¼ M

MsTc

T .The magnetization contoursM(H) at constant T are

guided phenomenologically to match experimentalobservations. The b phase is approximated as a softferromagnet with some small saturating field H0

s ateachT, above whichMbðHÞ saturates andbelow whichMbðHÞ increases linearly. For a, the experimentaldata suggest a response typical of a weakly magneticmaterial, demonstrating a linear M(H) relationshipand no saturating field. In sum, this particular modelis defined by five material parameters: two CurietemperaturesTa;b

C(for the model considered here, they

are 298.0 K and 421.6 K, respectively), two satura-tion magnetizations Ma;b

s , (9.3 A m�2 kg�1 and125.4 A m�2 kg�1, respectively), and the maximumb-saturating field Hs (0.25 T).

Entropy

The GMCE system is partitioned into a and bphases. The total entropy for each phase consistsgenerally of electronic, lattice, and magnetic con-tributions, Stot ¼ Selec þ Slat þ Smag, with the firstterm assumed negligible. The total entropy can alsobe divided into zero-field and applied-field terms asSðT;HÞ ¼ SðT; 0Þ þ DSðT;HÞ, with possible latticeand magnetic contributions to each. Debye’s methodis used to approximate the lattice as an isotropic gasof noninteracting and nonmagnetic phonons; thenSlat is field independent and contributes only toS(T, 0). For simplicity, the magnetic interactionbetween a and b is assumed negligible, so thatSmag ¼ 0 at zero field and only contributes toDSðT;HÞ. Hence, for the single-phase entropies,

Stot ¼ SlatðTÞ þ DSmagðT;HÞ: (7)

The first term SlatðTÞ is calculated from the De-bye model, which yields the lattice heat capacity as

ClatðTÞ ¼ 9RT

TD

� �3Z TD=T

0

x4 expðxÞðexpðxÞ � 1Þ2

dx; (8)

whereR is the ideal gas constant (8.314 J mol�1 K�1)and TD is the Debye temperature, above whichessentially all phonon modes are excited. The zero-field entropies are then computed as

SlatðTÞ ¼Z T

0

C0ðT0ÞT0 dT0 (9)

with Slatð0Þ ¼ 0 consistent with the third law ofthermodynamics.

Brown, Bruno, Chen, Karaman, Ross, and Shamberger2128

The second term DSmagðT;HÞ is computed fromMaxwell’s relation applied to the single phases, de-rived as follows: The free energy within each phaseis analytic and path independent. This requiresequality of the free energy’s cross derivatives, thus

@Smag

@l0H

� �����T

¼ @M

@T

� �����H

: (10)

Integrating and using the independence of T andH yields

DSmagðT;HÞ ¼ d

dT0

Z H

0

MðT0;H0Þdl0H0

� �����T

: (11)

Thus, DSmagðT;HÞ is calculated directly from themagnetization model, and introduces no extraparameters. The final entropy model adds just twoadditional materials parameters: the Debye tem-peratures Ta;b

D (340 K and 306 K, respectively) of aand b phases.

Total System Properties and Figures of Merit

Combining this and the previous subsections, /,Ma;b, and Sa;b can be determined throughout thewhole T–H space. The properties of the systemX ¼ fS;M;G; . . .g are assumed to be simply relatedto / and single-phase properties Xa;b through therule of mixtures

X ¼ ð1 � /ÞXa þ /Xb: (12)

This procedure assumes that the contribution to theproperties X from the coupling between the separatephases, e.g., at interfaces, is negligible comparedwith that from the bulk single phases. While X gen-erally depends on the size, shape, and distribution ofthe phases in the two-phase region, this assumptioncan be expected to hold when all phases are mag-netically saturated, as in the high-field regime.

The treatment of Sects. 1 and 2 yields the totalsystem M and S along any conceivable cycle in theT–H space. By parameterizing a particular cycle,

e.g., by its minimum and maximum temperaturesand fields, the magnetothermal response, M(T, H)and S(T, H), can be determined. Hence, Wref , I, andg for this cycle can be determined by a handful ofintuitively interpreted parameters. Comparison be-tween different sets of parameters allows compre-hensive investigation of the effect of each parameteron the GMCE cycle performance.

RESULTS

Interactions between a specific T–H path andhysteresis have been modeled by fitting experi-mental data from a Ni45Co5Mn36.6In13.4 singlecrystal to the parameterized models described in theprevious section. The response of this model hasbeen investigated for a rectangular T–H path com-posed of four segments: (1) isofield cooling fromTmax ¼ 325 K to Tmin ¼ 310 K at 0 T, (2) isothermalfield increase at Tmin from 0 T to 5 T, (3) isofieldwarming at 5 T from Tmin to Tmax, and (4) isother-mal field decrease at Tmax from 5 T to 0 T, as shownin Fig. 6. The temperature Tmax is about 30 K aboveroom temperature, but has been chosen to empha-size the effects of the transition characteristics ofthis particular alloy, for which T0

bf is 321.8 K. Themaximum field of 5 T represents a reasonable fieldobtainable with a large-bore superconducting mag-net system, and compares to the apparatus usuallyused to measure DSm and DTad. Although for thisinvestigation a rectangular T–H path is used, morecomplicated and realistic T–H paths, such as thosewith adiabatic/isentropic or isenthalpic legs, areeasily considered following this methodology.

The effects of phase transformation behavior onrefrigeration capability were investigated by vary-ing DTelast from 0 K to 14 K and DThyst from 0 K to10 K and calculating Wref , I, and g for each pair ofvalues. The results are summarized in Fig. 5.

To facilitate interpretation of the Wref , I, and gsurfaces, the contour plots have been divided intothree distinct regions with different behaviors. Re-gion 1 is defined approximately by DThyst > 8 K andDTelast > 4 K. Within this region, hysteresis lossesare so large that I � Win, so that both Wref and

Fig. 5. Contours of (a) Wref, (b) I, and (c) g as functions of elastic (DTelast) and hysteresis (DThyst) widths for a material undergoing a 0 T to 5 Trectangular T–H cycle. Regions labeled 1, 2, and 3 are discussed in the text. Points labeled (i), (ii), and (iii) correspond with Fig. 7 as described intext. The 90% efficiency limit is marked on (c) in red (Color figure online).

A Preisach-Based Nonequilibrium Methodology for Simulating Performanceof Hysteretic Magnetic Refrigeration Cycles

2129

g � 0; no useful refrigeration work may be per-formed. For none of the plots in Fig. 5 do the con-tours suffer a discontinuity at the boundarybetween regions 1 and 2; this confirms that Wref

decreases smoothly to zero at the boundary betweenregions 1 and 2.

This behavior stands in contrast to the interfacebetween regions 2 and 3, where the contours ofconstant I and g both suffer slope discontinuities. Inregion 3, I is independent of DTelast and increaseslinearly with DThyst, whereas in region 2, I in-creases with DTelast, with this dependence becomingstronger as DThyst also increases. For g, thisbehavior is reversed.

The system’s behavior within each of these re-gions may be understood by referencing Fig. 7,where several / versus T graphs are shown forparticular combinations of DThyst and DTelast. Firstconsider the relation of these / versus T plots to theunderlying T–H cycle, in Fig. 6, where the equi-driving force contours passing through the transi-tion finish (dashed contours) and transition start(dotted contours) are plotted along with the T–Hcycle. Note that DThyst ¼ T0

bf � T0as ¼ T0

bs � T0af and

DTelast ¼ T0bf � T0

af ¼ T0af � T0

as.Along the zero-field cooling leg of the cycle, the

system crosses the a transition-start contour atT ¼ 317:8 K and begins to transform to a from b;this transition finishes at 315.8 K, when the systemcrosses the a transition-finish contour. Along theincreasing field leg, even though the temperaturedoes not change, the magnetic driving force inducesthe a ! b transition to occur; this begins on crossingthe b transition-start contour at l0H ¼ 3:5 T andcompletes at 4.0 T. Although T and H both varyalong the two remaining legs of the cycle, note that/ ¼ 1 until the zero-field cooling leg repeats. Thenet result is the / versus T graph plotted in Fig. 7a.

Figure 7a and b show / versus T plots for two setsof hysteresis parameters lying within regions 3 and2, respectively, of Fig. 5. These specific values have

been chosen so that DThyst ¼ 4 K is the same forboth, but DTelast varies significantly between them,being 4 K for the former and 12 K for the latter.Comparison of Figs. 6 and 7a shows that the dif-ference in behavior between regions 2 and 3 arisesbecause, in region 2, T0

af <Tmin and the full transi-tion is not obtained, whereas for region 3, T0

af >Tmin.Stated explicitly in terms of hysteresis parameters,points within region 2 satisfy

T0bf � DTelast � DThyst <Tmin: (13)

In contrast, Fig. 7a and c show / versus T plotsfor DTelast ¼ 2 K and DThyst ¼ 4 K and 8 K, respec-

Fig. 6. Interaction of representative T–H cycle with two-parameterhysteresis discussed in text. Dashed and dotted lines are equi-driv-ing force contours for the point marked (i) on Fig. 5, i.e.,DThyst ¼ 4 K, DTelast ¼ 2 K.

290 300 310 320 330

T / K

ϕ

0.0

0.2

0.4

0.6

0.8

1.0

5 TLoop

0 TLoop

5 TLoop

0 TLoop

0.0

0.2

0.4

0.6

0.8

1.0

ϕ

0.0

0.2

0.4

0.6

0.8

1.05 TLoop

0 TLoop

ϕ

(a)

(b)

(c)

Fig. 7. Phase fraction along T–H path defined in Fig. 6 for threerepresentative sets of hysteresis parameters marked (i), (ii), and (iii)in Fig. 5: (a) DThyst ¼ 4 K, DTelast ¼ 2 K, (b) DThyst ¼ 4 K,DTelast ¼ 12 K, and (c) DThyst ¼ 8 K, DTelast ¼ 2 K.

Brown, Bruno, Chen, Karaman, Ross, and Shamberger2130

tively; however, both of these lie within region 3,since they do not satisfy Eq. 13. Within region 3, thetotal phase transition is accessible, and the lostwork I is given approximately by

I DThyst DStr; (14)

where DStr is the approximately field- and temper-ature-independent entropy of transformation. FromEq. 9, I increases linearly with DThyst, as confirmedby Fig. 5b.

IMPLICATIONS FOR MATERIALS DESIGN

The calculated dependence of Wref , I, and g onDThyst and DTelast has important implications formaterials design principles. First, note that theg ¼ 0:90 contour lies within the region DThyst <1 K,so that if high-efficiency refrigeration is defined asthat for which Wref is at least 90% of Win, this re-quires small DThyst on the order of 1 K. Importantly,this evaluation ignores any system-level inefficien-cies; practically, one would have to design the hys-teresis properties of the GMCE somewhat abovethis contour to obtain real 90% efficiency for thetotal system. Due to the cycle-dependent nature of g,these results vary quantitatively with varying cy-cles and material characteristics; however, the samequalitative behavior is expected, providing motiva-tion for the important work on the reduction ofDThyst in GMCE material systems summarized inthe ‘‘Introduction.’’

Second, regions 2 and 3 have been emphasized,where two distinct behaviors in I are observed. Inregion 2, decreasing DTelast at constant DThyst cau-ses I to decrease monotonically so that the GMCE-based cycle becomes more efficient. Thus, if amaterial is initially within region 2 and furtherdecreases in DThyst are impossible, g can still beincreased by decreasing DTelast instead. There hasrecently been some experimental work exploringhow to achieve these DTelast with heat treatments.51

On the other hand, in region 3, decreasing DTelast atconstant DThyst has no effect on I (and reduced effecton g). Thus, if a material is initially within region 3,decreasing DTelast will be ineffective, and furtherimprovement of cycle efficiency requires decreasingDThyst.

Finally, this manuscript has focused on theinteractions between hysteresis and the T–H cyclefor one particular material system. However, themethodology described here can easily be reconfig-ured to investigations of interactions between, e.g.,material properties and cycle parameters, or mate-rial properties and hysteresis. Then one could con-ceive that the methodology could be used togenerate materials property-focused design princi-ples, such as a minimum saturation magnetizationfor a given cycle and hysteresis parameters in orderto achieve 90% efficiency.

CONCLUSION

We have presented a methodology for quantifyingthe effects of irreversible FOPTs on refrigerationcycles based on GMCE materials, and for relatingthese effects to parameterizations of refrigerationcycles, hysteresis properties, and material proper-ties. We have used this approach to confirm theimportance of accounting for hysteresis in analysesof these materials, as the presence of region 1 inFig. 5 clearly shows that hysteresis losses can re-duce the refrigeration entirely to zero. Under theseconditions, no heat is transported from the cold tohot reservoir, and even worse, some heat mayactually flow backwards, from the hot to coldreservoir. This result alone demonstrates theimportance of considering the effect of hysteresislosses, and of the interplay between the T–H cycleand losses, on GMCE-based refrigeration. Our re-sults imply that, for future studies, cycle-dependentfigures of merit must be calculated, as the relevantfigures of merit for refrigerants (Wref, I, g) cannot beseparated from the cycle being considered. Thisopens the possibility of optimizing refrigerationperformance for a GMCE material by carefullychoosing the T–H cycle.

For the specific investigation reviewed in thismanuscript, we have demonstrated the efficacy of anew approach to analyzing hysteretic materials bycombining an extension of classical equilibriumphase diagrams with Preisach hysteresis models.Although utilizing less sophisticated magnetizationand entropy models, the methodology we have cho-sen here allows the interpretation of our cycle fig-ures of merit in terms of fundamental materialsproperties (Curie temperatures, Debye tempera-tures, saturation magnetizations), while retainingrealistic temperature- and field-dependent behav-ior. These parameters have a clear material inter-pretation, and so one may hope that furtherexploration of the relationship between figures ofmerit and materials parameters could guide mate-rial design without as much reliance on experi-mental trial and error.

Finally, we find that, due to the interaction be-tween the chosen T–H cycle and hysteresis effectson the equi-driving force contours, for some cases(i.e., hysteresis in region 2), g can be increased bydecreasing DTelast, the width of the one-way transi-tion. We see that the interactions between materialproperties, hysteresis, and cycle constitute a largeand complex parameter space, but one for which ourimproved understanding is essential. The method-ology presented here provides a means to beginaccomplishing this goal.

REFERENCES

1. V. Pecharsky and K. Gschneidner, Int. J. Refrig. 29, 1239(2006).

2. E. Bruck and E. Bruck, J. Phys. D. Appl. Phys. 38, R381(2005).

A Preisach-Based Nonequilibrium Methodology for Simulating Performanceof Hysteretic Magnetic Refrigeration Cycles

2131

3. S. Russek and C. Zimm, Int. J. Refrig. 29, 1366 (2006).4. K. Engelbrecht, G. Nellis, and S. Klein, HVACR Res. 12,

1077 (2006).5. V. Pecharsky, K. Gschneider, A. Pecharsky, A. Tishin, and

K. Gschneidner, Phys. Rev. B Condens. Matter Mater. Phys.,64 (2001).

6. A. Smith, R. Bjork, K. Engelbrecht, K. Nielsen, R. Bjørk, andN. Pryds, Adv. Energy Mater. 2, 1288 (2012).

7. H. Ucar, J. Ipus, V. Franco, M.E. McHenry, and D.E.Laughlin, JOM 64, 782 (2012).

8. W. Giauque and D. MacDougall, Phys. Rev. 43, 768 (1933).9. K.A. Gschneidner and V.K. Pecharsky, Annu. Rev. Mater.

Sci. 30, 387 (2000).10. K. Gschneidner and V. Pecharsky, Mater. Sci. Eng. A 287,

301 (2000).11. V.K. Pecharsky, Phys. Rev. Lett. 78, 4494 (1997).12. K.A. Gschneidner, Y. Mudryk, and V.K. Pecharsky, Scr.

Mater. 67, 572 (2012).13. G.V. Brown, J. Appl. Phys. 47, 3673 (1976).14. C. Zimm, A. Jastrab, A. Sternberg, V. Pecharsky, and K.

Gschneidner, Adv. Cryog. Eng. 43, 1759 (1998).15. K.A. Gschneidner and V.K. Pecharsky, Int. J. Refrig. 31, 945

(2008).16. A. Pecharsky, K.A. Gschneidener Jr., and V.K. Pecharsky,

J. Appl. Phys. 93, 4722 (2003).17. F.-X. Hu, B.-G. Shen, J.-R. Sun, Z.-H. Cheng, G.-H. Rao, and

X.-X. Zhang, Appl. Phys. Lett. 78, 3675 (2001).18. S. Fujieda, A. Fujita, and K. Fukamichi, Sci. Technol. Adv.

Mater. 4, 339 (2003).19. B.G. Shen, J.R. Sun, F.X. Hu, H.W. Zhang, and Z.H. Cheng,

Adv. Mater. 21, 4545 (2009).20. D. Thanh, E. Bruck, O. Tegus, J. Klaasse, D.T. Cam-Thanh, E.

Bruck, and T.J. Gortenmulder,J. Appl. Phys. 99, 08Q107 (2006).21. W. Dagula, O. Tegus, B. Fuquan, L. Zhang, P.Z. Si, M.

Zhang, W.S. Zhang, E. Bruck, and F.R. de Boer, IEEETrans. Magn. 41, 2778 (2005).

22. H. Yibole, F. Guillou, L. Zhang, N.H. van Dijk, E. Bruck,N.H. van Dijk, and E. Bruck, J. Phys. D. Appl. Phys. 47,075002 (2014).

23. R. Kainuma, Y. Imano, W. Ito, Y. Sutou, H. Morito, S.Okamoto, O. Kitakami, K. Oikawa, A. Fujita, T. Kanomata,and K. Ishida, Nature 439, 957 (2006).

24. T. Krenke, E. Duman, M. Acet, E.F. Wassermann, X. Moya,L. Mannosa, and A. Planes, Nat. Mater. 4, 450 (2005).

25. J. Liu, T. Gottschall, K.P. Skokov, J.D. Moore, and O. Gut-fleisch, Nat. Mater. 11, 620 (2012).

26. P.J. Shamberger and F.S. Ohuchi, Phys. Rev. B Condens.Matter Mater. Phys., 79 (2009).

27. V. Franco, J.S. Blazquez, B. Ingale, A. Conde, and J.S.Blazquez, Annu. Rev. Mater. Res. 42, 305 (2012).

28. J. Lyubina, J. Appl. Phys. 109, 07A902 (2011).29. Y. Song, X. Chen, V. Dabade, T. Shield, and R. James, Na-

ture 502, 85 (2013).30. C. Sasso, M. Kuepferling, L. Giudici, V. Basso, and M.

Pasquale, J. Appl. Phys. 103, 07B306 (2008).31. I.D. Mayergoyz, IEEE Trans. Magn. 22, 603 (1986).32. J. Lyubina, R. Schaefer, N. Martin, L. Schultz, O. Gut-

fleisch, and R. Schafer, Adv. Mater. 22, 3735 (2010).33. J. Lyubina, O. Gutfleisch, M. Richter, and M. Kuzmin, J.

Magn. Magn. Mater. 321, 3571 (2009).34. J.C. Debnath, R. Zeng, J.H. Kim, P. Shamba, and S.X. Dou,

Appl. Phys. A. Mater. Sci. Process. 106, 245 (2012).35. N. Dung, Z. Ou, L. Caron, L. Zhang, G. de Wijs, R. de Groot,

and E. Bruck, Adv. Energy Mater. 1, 1215 (2011).36. N.H. Dung, L. Zhang, Z.Q. Ou, E. Bruck, and E. Bruck,

Appl. Phys. Lett. 99, 092511 (2011).37. N.T. Trung, Z.Q. Ou, T.J. Gortenmulder, O. Tegus, O. Te-

gus, and E. Bruck, Appl. Phys. Lett. 94, 102513 (2009).38. J. Cui, Y. Chu, O. Famodu, Y. Furuya, J. Hattrick Simpers,

R. James, A. Ludwig, S. Thienhaus, M. Wuttig, Z. Zhang,and I. Takeuchi, Nat. Mater. 5, 286 (2006).

39. J.M. Ball and R.D. James, Arch. Ration. Mech. Anal. 100, 13(1987).

40. V. Basso, M. Kupferling, C. Sasso, M. LoBue, and M.Kuepferling, IEEE Trans. Magn. 44, 3177 (2008).

41. V. Basso, C. Sasso, G. Bertotti, and M. LoBue, Int. J. Refrig.29, 1358 (2006).

42. V. Basso, J. Phys. Condens. Matter 23, 226004 (2011).43. B. Coleman and M. Gurtin, J. Chem. Phys. 47, 597 (1967).44. V. Basso, G. Bertotti, M. LoBue, C. Sasso, V. Basso, G.

Basso, M. Bertotti, and C.P. LoBue, J. Magn. Magn. Mater.290, 654 (2005).

45. V. Basso, C. Sasso, K. Skokov, O. Gutfleisch, and V. Kho-vaylo, Phys. Rev. B Condens. Matter Mater. Phys., 85 (2012).

46. F. Preisach, Eur. Phys. J. A 94, 277 (1935).47. G. Bertotti and V. Basso, J. Appl. Phys. 73, 5827 (1993).48. V. Basso, C.P. Sasso, and M. LoBue, J. Magn. Magn. Mater.

316, 262 (2007).49. I.D. Mayergoyz, J. Appl. Phys. 61, 3910 (1987).50. I.D. Mayergoyz, J. Appl. Phys. 69, 4602 (1991).51. N.M. Bruno, C. Yegin, I. Karaman, J.-H. Chen, J.H. Ross Jr,

J. Liu, and J. Li, Acta Mater. 74, 66 (2005).

Brown, Bruno, Chen, Karaman, Ross, and Shamberger2132