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Too cool to workXavier Moya, Emmanuel Defay, Volker Heine and Neil D. Mathur

Magnetocaloric and electrocaloric effects are driven by doing work, but this work has barely been explored, even though these caloric effects are being exploited in a growing number of prototype cooling devices.

Can solid materials be used as the basis for economically viable heat-pump devices that cool food, beverages,

medicine, electronics or populated spaces? This is a question that occupies universities, national laboratories and companies. It has resulted in four large projects within Europe, and it represents the basis for the biennial THERMAG conference series, dedicated primarily to magnetocaloric (MC) cooling near room temperature using materials that display thermal changes in response to magnetic field changes. It was revealed at THERMAG VI that the many existing MC prototypes have now been joined by a third prototype based on electrocaloric (EC) materials1 (Fig. 1a,b), and a second

prototype based on mechanocaloric (mC) materials2 (Fig. 1c,d), with thermal changes driven by changes of electric and stress fields, respectively. (Caloric materials and previous prototypes are discussed in ref. 3 and references therein.) Unfortunately, the efficiency of these prototype heat pumps is rarely discussed, even though the main motivation is to outperform existing refrigerators.

We are led to ask an even more fundamental question: how much work is done to drive the nominally reversible thermal changes in the materials themselves? These thermal changes are large near ferroic (often structural) phase transitions in the vicinity of room temperature, for example,

in ferromagnets and ferroelectrics near the Curie temperature. Given that work is such a well-established thermodynamic concept, it is perhaps surprising that it has been highly neglected in the caloric literature.

The thermodynamics of polarizable bodies in electric or magnetic fields is inherently tricky because the general formula for work derived from Maxwell’s equations involves an integral over all space4. It is therefore virtually impossible to do work on the polarizable body alone, as additional work must be done to generate the fields arising from induced poles. Another complication is that the work can be done electrically or mechanically. If the work is done electrically then additional work must be done to generate the applied magnetic (electric) field through the flow (redistribution) of charge. In contrast, no additional work is performed if the work is done mechanically by relative motion with respect to a pre-existing field, even though the response of the body is indistinguishable.

Building on the formalism for calculating work that was established by Volker Heine over half a century ago4, we find that it is indeed more efficient to drive MC effects mechanically, whereas due to screening it is more efficient to drive EC effects electrically. The mechanically driven MC effects are generically an order of magnitude more efficient than the electrically driven EC effects, but these intrinsic efficiencies do not ultimately limit the extrinsic efficiencies that may be achieved by recovering as much work as possible for subsequent use. In what follows, work will be calculated without considering mundane contributions associated with Joule heating, friction and gravity.

Our figure of merit will be the dimensionless materials efficiency5, η = |Q/W|, where electrical or mechanical work, W, is done to drive highly reversible caloric effects in an isothermal body, whose entropy is thus modified such that heat, Q, flows to (Q < 0) or from (Q > 0) the thermal bath. For a given material, nonlinear field dependences force η to

a b

c d

292

294

296

298

300

T (K)

Ceramic platesElectrical connector

SMA

Heat sink

Heat source

Heat sink

Heat source

SMA

Figure 1 | Recent electrocaloric1 and mechanocaloric2 prototype coolers. a,b, Electrocaloric cooler (a) and the back part of the cooler (b). A voltage is cycled across 30 ceramic electrocaloric plates while a fluid flows in alternate directions. This establishes a 3.3 K temperature difference that exceeds the adiabatic temperature changes in the 200-μm-thick plate material (Pb[(Mg1/3Nb2/3)O3]0.9[PbTiO3]0.1) by a factor of 3.7. c,d, Prototype heat pump based on mechanocaloric effects in a ribbon of the superelastic shape memory alloy (SMA) NiTi (c). The thermal changes are driven by changes in uniaxial stress, and therefore fall into the subset of mechanocaloric effects known as elastocaloric effects. An infrared thermal image (d) reveals the temperature differential established by repeatedly using the SMA to absorb heat at the source and then dump it after moving to the sink. The colour scale shows the temperature in kelvin. The images in a and b are courtesy of U. Plaznik. The images in c and d are courtesy of S. Seelecke.

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undergo a trivial increase on decreasing |Q|, reflecting the fact that in physics — as in life — it is sometimes easier to be more efficient when achieving a bit less. Therefore it is only meaningful to compare values of η(Q, T) for similar values of |Q|, and it is primarily interesting to do this when |Q| is maximized by employing large fields and operating at a temperature, T = T0, that is closely associated with the zero-field phase transition temperature.

Values of both |Q| and |W| will be expressed per unit volume of material, assuming specific but reasonably realistic scenarios in which fields, flux densities and order parameters are uniform. Direct caloric measurements are challenging and so we primarily use values of |Q| that were previously obtained from the well-known ‘indirect’ method3. For each material, these values of |Q| were calculated from measurements of order parameter as a function of applied field at selected values of temperature. Here we use the subset of measurements at T0 to calculate values of |W| and hence η.

The materials efficiency, η, corresponds to each of the two legs in a rather useless isothermal cycle that incurs no net work, and so it does not describe a useful cooling device with different sink and load temperatures. However, it represents a guide to the selection of materials in terms of energy efficiency, without forcing one to specify arbitrary details to describe a cooling cycle.

Magnetocaloric efficiencyWhen magnetizing a magnetic material, each element of the electrical and/or mechanical work done, dW, is given most generally by the integral of HdB over all

space, where the magnetic flux density, B = μ0(H + M), depends on the magnetic field, H, and the sample magnetization, M, and where μ0 is the permeability of free space. If electrical work is done to generate an applied magnetic field, H0, by incrementally varying the current through an infinitely long solenoid that is closely wound around an infinitely long MC sample, then there is no external field and so dW may be evaluated by integrating HdB over the sample volume alone. Moreover, we may write H = H0 as the sample has no poles with which to generate any field.

This evaluation procedure may be used for the long, thin MC samples typically studied if we imagine a closely wound solenoid of the same length, but work is underestimated by neglecting the fringing field generated by the solenoid, and by neglecting the demagnetizing (stray) field generated inside (outside) the sample by the small number of induced poles at each end. Here, for long, thin MC samples magnetized isothermally at T0, we obtain a lower bound on the volume-normalized electrical work, |W|, by integrating HdB, using literature data of the form M(H0) and assuming H = H0 due to the small number of well-separated poles (collinear vectors are denoted as scalar quantities).

If, instead, mechanical work is done to generate H0 in the sample by relative motion with respect to an inalterable permanent magnet, then dW can be mathematically reduced in a non-trivial manner4 to the integral of –μ0MdH0 over the sample volume alone. There are no assumptions about sample shape, and the negative sign reflects the attraction between permanent magnets and magnetic materials. For the long, thin MC samples magnetized isothermally at

T0, we obtain volume-normalized values of |W| by integrating –μ0MdH0 from the same M(H0) data as before.

Whether electrically or mechanically generated, the magnetic field applied to an isothermal MC material at T0 produces a heat, Q, that is typically evaluated by integrating μ0T0(¶M/¶T)H with respect to H. This method of evaluation — which follows from the Maxwell relation, μ0(¶M/¶T)H = (¶S/¶H)T, assuming the experimental condition of constant stress — was exploited in the literature6–15 without the factor of T0 to calculate the corresponding changes of entropy, S, from measurements of M(H0) at selected values of T, assuming H = H0 for long, thin samples. We will obtain volume-normalized values of |Q| from published values of mass-normalized entropy change by introducing factors of T0 and density.

For selected MC materials, mechanical work was evaluated by integrating between H0 = 0 and a value corresponding to the 2 T flux density that may just be provided by good permanent magnets. Values of electrical work (MC effects) were evaluated here (previously) by integrating between the same limits. The resulting values of η reveal MC materials to be an order of magnitude more energy efficient when driven by permanent magnets rather than solenoids (Table 1 and upper blue ellipsoids in Fig. 2). This discrepancy arises because the solenoid must generate applied fields that are already present when using the permanent magnet. Even if it were practical for solenoids to generate 5 T, the corresponding values of η (Table 1 and bottom blue ellipsoid in Fig. 2) are suppressed because |W| increases faster than |Q|, as caloric effects tend to saturate.

Table 1 | Magnetocaloric materials.

Electrical work Mechanical workMaterial T0 (K) |μ0ΔH0| (T) |Q| (J cm–3) |W| (J cm–3) η |W| (J cm–3) η Ref.Gd 294 2 (5) 12.7 (25.2) 2.2 (11.1) 5.8 (2.3) 1.0 12.6 6

Gd5Si2Ge2

ΔH > 0276 2 (5) 29.1 (38.2)

2.0 (10.9) 14.3 (3.5) 1.1 27.27

ΔH < 0 2.0 (10.7) 14.5 (3.6) 1.1 27.2

LaFe11.7Si1.3

ΔH > 0184 2 (5) 37.6 (40.0)

2.8 (11.3) 13.5 (3.5) 0.7 50.08

ΔH < 0 2.4 (11.0) 15.7 (3.6) 1.0 37.6MnFeP0.45As0.55 308 2 (5) 31.9 (39.9) 2.1 (11.2) 14.9 (3.6) 0.3 96.7 9Ni52.6Mn23.1Ga24.3 300 2 (5) 15.2 (43.7) 1.6 (10.1) 9.2 (4.3) 0.9 17.4 10Ni50Mn37Sn13 299 2 (5) 18.7 (50.1) 1.7 (10.3) 11.2 (4.9) 0.3 66.8 11Ni50Mn34In16 190 2 (5) 14.7 (18.9) 1.7 (11.9) 8.6 (1.6) 0.4 36.6 12Ni45.2Mn36.7In13Co5.1 317 2 45.0 2.0 22.5 0.2 211 13Mn1.30Fe0.65P0.5Si0.5 267 2 29.0 2.1 14.0 0.7 40.6 14Fe49Rh51 317 2 43.5 3.0 14.4 1.2 35.9 15

T0, operating temperature; |μ0ΔH0|, change in applied magnetic field; μ0, permeability of free space; Q, heat; W, work; η = |Q/W|, materials efficiency. Data for |μ0ΔH0| = 5 T are presented parenthetically where |Q| is available. The hysteretic behaviour exhibited by Gd5Si2Ge2 and LaFe11.7Si1.3 is identified by up-sweep (ΔH > 0) and down-sweep (ΔH < 0) measurements. Values of |Q| were obtained from the cited references and values of |W| were obtained from data therein.

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Electrocaloric efficiencyWhen polarizing an electrically polarizable material, each element of the electrical and/or mechanical work done, dW, is given most generally by the integral of EdD over all space, where the electric field, E = ε0

–1(D – P), depends on the electric flux density, D, and the sample polarization, P, and where ε0 is the permittivity of free space. If electrical work is done to generate an applied electric flux density, D0, by incrementally varying the charge on infinite-area capacitor plates that lie on either side of an infinite-area EC sample, then there is no external field and dW may be evaluated by integrating EdD over the sample volume alone. Moreover, whether or not the area is infinite, we may assert that D = D0 by assuming that charge does not flow between the sample and the plates.

This procedure may be used for evaluating the large EC effects in thin films with high breakdown fields and low leakage, but work is very slightly underestimated by neglecting fringing fields around the thin active slab between large-area electrodes. For this type of EC sample, polarized isothermally at T0, we obtain a lower bound on the volume-normalized electrical work, |W|, by integrating EdD, using data in the form of D(E) from the literature. Data presented as P(E) are converted into this form from the resultant field, E = ε0

–1(D – P), which is small due to screening. The resultant field is the independent variable because the experimental control parameter is voltage, not charge.

Alternatively, and perhaps somewhat hypothetically (at least for solid rather than fluid16 EC materials), mechanical work can be done to generate D0 within the sample through relative motion with respect to a fixed arrangement of unscreened charges that reside on capacitor plates or in an electret (vacuum would be required to prevent electrical breakdown of air). One may then mathematically

reduce dW in a non-trivial manner4 to the integral of –PdE0 over the sample volume alone. As for MC materials, there are no assumptions about the sample shape, and the negative sign reflects the fact that electric fields attract polarizable materials. By considering the thin active slab of each EC film polarized isothermally at T0, volume-normalized values of |W| can be obtained by integrating –PdE0 from the same D(E) or P(E) data as before. To do

this, E0 was rewritten as the experimentally accessible parameter ε0

–1D, using D = D0, as explained above.

For the thin active slab of each film at a temperature of T0, the isothermal EC heat, Q, is typically evaluated by integrating T0(¶P/¶T)E with respect to E. This follows from the Maxwell relation, (¶P/¶T)E = (¶S/¶E)T, assuming approximate mechanical boundary conditions, namely constant stress for a free-standing film with thin electrodes of similar area, and constant strain if there is clamping from a substrate and/or unaddressed surrounding film. As for MC materials, we report volume-normalized values of |Q| based on literature values of mass-normalized entropy change. The latter were typically calculated using the above Maxwell relation with data in the form of D(E) or P(E) at selected values of T. Here, P and D are freely interchangeable given that D = ε0E + P implies that (¶D/¶T)E = (¶P/¶T)E.

For selected EC materials, electrical and mechanical work were evaluated by integrating between the limits previously used to evaluate EC effects, with D0 ranging from zero to the highest value that does not induce breakdown (except when the lower limit is finite17, or when we have reduced the upper limit19,21 to restrict |Q| for comparison). The resulting values of η (Table 2 and dark green ellipsoid in Fig. 2) reveal EC materials to be comparable to MC materials when doing electrical work. The mechanical work done when driving EC effects is much greater (η << 1, Table 2), as the resultant electric field undergoes large changes associated with screening. In contrast, the pre-existing magnetic field of a permanent magnet is not screened by MC materials, leading to the most intrinsically efficient caloric effects (Table 1 and dark blue ellipsoid in Fig. 2). A more complete and detailed treatment of the thermodynamics presented here is currently in preparation.

η MC (permanent m

agnet, 2 T)

EC (electrically driven)

MC (solenoid, 2 T)

mC

MC (solenoid, 5 T)

100

EC (energy

recovery)

10

1

|Q| (J cm−3)0 20 40 60

Figure 2 | Caloric materials efficiency map. The materials efficiency, η, for materials in Tables 1–3 possessing similar values of caloric heat, |Q|, at operating temperatures T0. Magnetocaloric (MC) effects may be driven electrically using a solenoid to generate |μ0ΔH0| = 2 T or 5 T (light-blue ellipsoids), or mechanically using a permanent magnet to generate |μ0ΔH0| = 2 T (dark-blue ellipsoid). Electrocaloric (EC) effects may be driven mechanically using fixed charges in a vacuum (η << 1, not shown), or electrically by varying the charge on capacitor plates without (dark-green ellipsoid) or with (light-green ellipsoid, partially occluded) electrical energy recovery. For mechanocaloric (mC) effects driven by uniaxial stress, only Table 3 entries with similar values of |Q| are represented (red ellipsoid). EC effects in each material are driven by fields of different magnitude, as are mC effects.

Table 2 | Electrocaloric materials.

Electrical work Mechanical work

Material T0 (K) |ΔE| (kV cm–1) |Q| (J cm–3) |W| (J cm–3) η |W| (J cm–3) η Ref.PbZr0.95Ti0.05O3 499 480 31.2 10.3 3.0 9842 0.003 170.93PMN-0.07PT 308 720 24.0 4.3 5.6 2036 0.012 18Pb0.8Ba0.2ZrO3 290 210 46.1 1.7 27.1 5193 0.009 19P(VDF-TrFE) 353 2000 38.0 5.2 7.5 266 0.14 20*P(VDF-TrFE) 323 1000 32.5 2.9 11.2 216 0.15 21P(VDF-TrFE-CFE) 310 3100 38.0 7.9 4.8 333 0.11 20*P(VDF-TrFE-CFE) 350 3500 61.0 8.7 7.0 187 0.33 22

T0, operating temperature; |ΔE|, change in resultant electric field; Q, heat; W, work; η = |Q/W|, materials efficiency. Values of |Q| were obtained from the cited references and values of |W| were obtained from data therein. PMN = PbMg1/3Nb2/3O3; PT = PbTiO3; P(VDF-TrFE) = poly(vinylidene fluoride-trifluoroethylene) 55/45 mol%; *P(VDF-TrFE) = poly(vinylidene fluoride-trifluoroethylene) 65/35 mol%; P(VDF-TrFE-CFE) = poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene) 59.2/33.6/7.2 mol%; *P(VDF-TrFE-CFE) = poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene) 56.2/36.3/7.6 mol%.

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Table 3 | Mechanocaloric materials.

Mechanical workMaterial T0 (K) |Δσ| (GPa) |Q| (J cm–3) |W| (J cm–3) η Ref.

Cu68.1Zn15.8Al16.1

Δσ > 0300 0.13 47.8

8.7 5.523

Δσ < 0 7.9 6.1

NiTiΔσ > 0

295 0.6591.2 34.0 2.7

24Δσ < 0 60.4 11.5 5.3

Fe68.8Pd31.2 240 0.10 9.0 1.1 8.4 25

T0, operating temperature; |Δσ|, change in the applied/resultant uniaxial stress field; Q, heat; W, work. The hysteretic behaviour exhibited by Cu68.1Zn15.8Al16.1 and NiTi is identified by up-sweep (Δσ > 0) and down-sweep (Δσ < 0) measurements.

Mechanocaloric efficiencyFor completeness, we include an analogous analysis for mechanical work done on mC materials. There are no pressure–volume data for barocaloric materials, so our analysis is limited to uniaxial stress–strain (σ–ε) data for elastocaloric materials such that dW = σdε (Table 3 and red ellipsoid in Fig. 2). Based on our very limited dataset, it seems that the efficiency of mC materials is relatively low.

Energy recoveryKnowledge of the second law of thermodynamics might not be cool, as lamented by C. P. Snow, but it is certainly necessary to appreciate that net work must be done to pump heat uphill and achieve cooling. The work associated with driving and undriving a caloric material through any useful cycle is necessarily larger, but much of the driving work may be automatically or artificially recovered on undriving. To illustrate automatic energy recovery, consider permanent magnets that address some angular fraction of a spinning MC annulus, such that material moving into the field region is compensated by material moving out. This strategy is not employed in all MC prototypes, suggesting that it could be desirable to use mechanical or other means to artificially recover as much work as possible to drive subsequent MC effects.

Automatic energy recovery with solid EC materials would be challenging, so if one cannot use EC fluids16 that flow in and out of a field region, it is important to artificially

recover electrical work to drive subsequent EC effects. Given that 80% of this work can indeed be recovered using simple circuitry, values of 1/|W| and η may be enhanced by factors of five (E. Defay et al., manuscript in preparation). Therefore extrinsic EC efficiencies (light green ellipsoid in Fig. 2) can be comparable to the best intrinsic MC efficiencies (dark blue ellipsoid in Fig. 2).

Our original question regarding caloric materials in heat-pump devices raises many issues beyond the isothermal materials efficiency discussed here. For example, it would, in practice, be difficult to reliably achieve the maximum fields that we have reported, because cost and design considerations compromise magnetic field strength, and because the need to avoid breakdown compromises electric field strength. Therefore, and more generally, the key to commercial exploitation lies in the development of caloric materials that perform better, in smaller fields, over a wider range of temperatures. It is our hope that the parameter of materials efficiency, η, will play a small part in helping to parameterize this cool future. And with apologies to Oscar Wilde, it will serve to remind that work is the curse of the cooling classes. ❐

Xavier Moya, Emmanuel Defay and Neil D. Mathur* are in the Department of Materials Science, University of Cambridge, CB3 0FS, UK. Emmanuel Defay is also at CEA, LETI, Minatec Campus, 17 Rue des Martyrs, 38054 Grenoble, France, and the Luxembourg Institute of Science and Technology (LIST) — Materials Research and Technology Department, 41 rue du Brill, L‑4422 Belvaux, Luxembourg.

Volker Heine is at the Cavendish Laboratory, University of Cambridge, CB3 0HE, UK. *e‑mail: ndm12@cam.ac.uk

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AcknowledgementsX.M. is grateful for support from the Royal Society and EPSRC EP/M003752/1. We thank B. Nair for discussions.

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