## Abstract

Caloric cooling and heat pumping rely on reversible thermal effects triggered in solids by magnetic, electric or stress fields. In the recent past, there have been several successful demonstrations of using first-order phase transition materials in laboratory cooling devices based on both the giant magnetocaloric and elastocaloric effects. All such materials exhibit non-equilibrium behaviours when driven through phase transformations by corresponding fields. Common wisdom is that non-equilibrium states should be avoided; yet, as we show using a model material exhibiting a giant magnetocaloric effect, non-equilibrium phase-separated states offer a unique opportunity to achieve uncommonly large caloric effects by very small perturbations of the driving field(s).

This article is part of the themed issue ‘Taking the temperature of phase transitions in cool materials’.

## 1. Introduction

Reversible thermal events in solids—*caloric* effects (the name ‘caloric’ derives from Latin *calor*, literally ‘heat’)—have been known for over a century. They occur when a constant ‘control’ field acting upon a material is perturbed. *Magneto*caloric, *electro*caloric and *elasto*caloric effects are all observed as materials’ temperature (entropy) changes when the strength of *magnetic*, *electrical* or *stress* field, respectively, is altered adiabatically (isothermally). Underlying any and all caloric effects are specific components of the total entropy of a solid that can be most easily influenced by the perturbation of the corresponding control or ‘driving’ field. Take a magnetic metal as an example, where three fundamental contributions to the total entropy can be recognized [1]: electronic (*S*_{E}), lattice (*S*_{L}) and magnetic (*S*_{m}). The magnetic entropy, which represents disorder of the individual magnetic moments, either electronic (spin and orbital) or nuclear or both, is most likely to be affected when the strength of a magnetic field surrounding the material is changed. As the magnetic field rises (falls) isothermally by ±Δ*H*, magnetic disorder generally gets reduced (increased), thus bringing the magnetic part of the total entropy down (up) by ±Δ*S*_{m}. When Δ*H* is applied adiabatically, the total entropy (*S*_{E}+*S*_{L}+*S*_{m}) of a solid must remain constant; hence as one component of the total entropy goes up (down), others must equivalently fall (rise), so that in our example Δ*S*_{E}+Δ*S*_{L}+Δ*S*_{m}=0 as the result of the perturbation and, therefore, the temperature of a solid must go down (up).

Caloric effects can be exceptionally efficient forms of energy conversion. For example, magnetic moment–magnetic field coupling, being a quantum mechanical effect, approaches 100% efficiency when hysteresis and eddy currents are both negligible. The potential for producing reversible caloric effects with high efficiency in solids underpins a variety of cooling and/or heat pumping applications based on the magnetocaloric, elastocaloric and electrocaloric effects. Among the three, cooling using the magnetocaloric effect is the most mature. Following pioneering work by Debye & Giauque [2–4] in the 1920s, adiabatic demagnetization refrigeration has become a well-established commercial technology routinely employed today to reach ultra-low temperatures in research environments. Some 50 years later, seminal work by Brown, Steyert and Barclay [5–9] demonstrated that the magnetocaloric effect may be useful around room temperature, thus delineating a roadmap towards temperature spans much larger than the magnetocaloric effect itself, even if with zero cooling power at the time. Major breakthroughs occurred about 20 years later, when Pecharsky & Gschneidner [10–12] reported the discovery of the giant magnetocaloric effect in Gd_{5}Si_{2}Ge_{2} and related compounds, and Zimm *et al.* [13] demonstrated a near-room-temperature magnetic refrigerator reliably producing cooling powers exceeding 500 W using elemental Gd while employing a 5 T driving field. These two developments opened a new chapter in magnetocaloric cooling, taking the first steps towards the projected transformation of a laboratory curiosity into a line of commercial cooling devices using the magnetocaloric effect [14].

Similar to magnetocaloric cooling, the feasibility of exploiting the electrocaloric effect has been demonstrated first at cryogenic temperatures by Radebaugh *et al.* [15,16], and later near room temperature in several laboratory-scale devices by Sinyavsky & Brodyansky [17], Jia & Ju [18], Gu *et al.* [19,20] and Wang *et al.* [21]. Finally, elastocaloric cooling is getting some serious attention after Cui *et al.* [22,23] demonstrated a cooling device based on the principle suggested by Annaorazov *et al.* [24]. Despite a number of successful demonstrations, all three near-room-temperature caloric cooling technologies remain in their infancy, mostly due to the facts that (i) the caloric effects that can be produced in readily available driving fields without damaging a material are relatively weak, (ii) active regeneration is required to achieve the temperature span of a typical vapour-compression device and (iii) caloric material–caloric cooling/heat pumping device integration is, therefore, far from trivial.

Over the last 20 or so years various aspects of caloric materials and caloric refrigeration/heat pumping devices have been reviewed in a number of publications [25–41] and a few books [42,43]. Readers interested in details and the then-current state of the art are referred to all of these quality publications. As suggested by the title of this work, here we restrict ourselves to discussing materials-related issues that may lead to stronger caloric effects generated by driving fields that are easily produced and maintained, with the goal to support accelerated development of better caloric materials and, therefore, broad adoption of these solid-state caloric cooling technologies in the foreseeable future. Magnetocaloric materials will predominantly be employed as examples; however, most of the discussion below is expected to be applicable to all three categories of caloric solids.

## 2. Caloric effects: where are the limitations?

Rephrasing the key message from the last paragraph of the introduction, the availability of materials with substantially enhanced caloric effects, achieved preferably in driving fields much reduced compared with those that are customary today, is expected to both brighten the light at the end of the tunnel and considerably shorten the road through it. The first question we, therefore, must address is: Where are we today with respect to the fundamental limits of the three kinds of caloric effects of interest?

Consider the molar magnetic entropy of a solid in which *x* is the molar fraction of identical magnetic moment-carrying species. With gas constant *R* and total angular (spin) momentum *J* for 4f (3d) metals, the total available magnetic entropy, *S*_{m}, is limited to
2.1For purely magnetic phenomena, neglecting the electronic (*C*_{E}) and magnetic (*C*_{m}) specific heats, and assuming that the lattice molar specific heat, *C*_{L}, is at its limit of 3*R* J mol^{−1} K^{−1} (the latter is a reasonable approximation around 300 K for many intermetallic compounds and metallic alloys), the fundamental limits for the *conventional magnetocaloric effect* are, therefore, easily established as
2.2and
2.3where Δ*S*_{m} is the isothermal magnetic entropy change, Δ*T*_{ad} is the adiabatic temperature change and *T* is the absolute temperature. Hence, for elemental Gd (*J*=7/2) at approximately 300 K, Δ*S*_{m} is limited to about ±17.3 J mol^{−1} K^{−1} (±110 J kg^{−1} K^{−1} or ±0.87 J cm^{−3} K^{−1}) and Δ*T*_{ad} is limited to about ±208 K. For a hypothetical M_{2}X compound (M is a 3d element and X is non-magnetic) with Curie temperature, *T*_{C}, near 300 K, the maximum Δ*S*_{m} and Δ*T*_{ad} are reduced to ±9.9 J mol^{−1} K^{−1} (±3.9 J mol^{−1} K^{−1}) and ±120 K (±46 K), respectively, when *J*=5/2 (*J*=1/2). These simple estimates are in agreement with the more rigorous calculations of Tishin [44]. In reality, *S*_{m} is removed over a large range of temperatures, i.e. equation (2.2) is transformed into
2.2aand the magnetocaloric effect peaks near *T*_{C}, being proportional to (∂*M*/∂*T*)d*H* (*M* is magnetization, *H* is magnetic field). The fields required to reach the fundamental limits of Δ*S*_{m} and Δ*T*_{ad} exceed 10 000 kOe [44] and are, therefore, impracticable. Readily available magnetic fields are always severely limited, and typically observed Δ*T*_{ad} are about 1–4 K for a magnetic field change of about 10–15 kOe [28]—a minuscule fraction of the nature-imposed limits.

Most studied magnetocaloric materials order magnetically via second-order phase transitions. A handful of them [27–29,39] exhibit magnetoelastic first-order phase transitions where the magnetic ordering/disordering may be coupled with either simple volume discontinuities or substantial rearrangements of the crystal lattices [10,45–48]. This coupling leads to the *giant magnetocaloric effect*, where the conventional magnetic moment-only part given by equations (2.1)–(2.3) may become strongly enhanced by an elastic contribution, which is the difference of the entropies of the low- and high-field polymorphs of the material, Δ*S*_{el} [48,49]. Obviously, the enhancement can only be expected when both Δ*S*_{el} and Δ*S*_{m} have identical signs because the total observed magnetic field-induced entropy change is Δ*S*_{M}=Δ*S*_{m}+Δ*S*_{el}. Whereas, we note that the separation of magnetic and elastic contributions is only formal because they are inseparable during a first-order magnetostructural phase transition—Δ*S*_{m} scales with Δ*H* [5,50], but Δ*S*_{el} necessarily becomes magnetic field-independent when Δ*H* exceeds a certain critical value that is sufficient to complete the rearrangement in the crystal lattice [48,51]. The difference in behaviour of these two thermodynamic quantities with field, taken together with the difference in behaviours of the conventional and giant magnetocaloric effects [52], allows for each to be quantified from the analysis of behaviour of the magnetocaloric effect as a function of Δ*H* [51,53,54]. Especially in low magnetic fields, Δ*S*_{el} may substantially exceed Δ*S*_{m}, which identifies a promising path forwards towards magnetocaloric materials exhibiting very strong magnetocaloric effects in relatively weak magnetic fields.

The electrocaloric effect originates from the entropy changes associated with the variation in dipole order in a polar dielectric induced by an electric field. Switching between non-polar and polar phases isothermally (adiabatically) produces an entropy (temperature) change, ±Δ*S*_{d} (±Δ*T*_{ad}). The maximum dipolar entropy, *S*_{d}, in a polar dielectric is [55,56]
2.4where *Ω* is the number of discrete equilibrium orientations of identical dipolar entities, and *x* is the same as in equation (2.1) but with respect to electric dipoles. Given that the difference between the conventional magnetocaloric and *conventional electrocaloric* effects lies in the nature of the ordering species (magnetic moments versus electric dipoles), both effects have comparable upper bounds [55], i.e.
2.5When the electrocaloric effect is purely from dipolar ordering, Δ*S*_{d} and Δ*T*_{ad} are related to polarization, *P*, as
2.6where , *C* is specific heat, *ε*_{0} is vacuum permittivity and *Θ* is an effective Curie coefficient [55,57]. Hence, to realize a large electrocaloric effect, there must be a large |Δ*S*_{d}| associated with the change of *P*, and a dielectric material must support a large |Δ*P*| induced by an external field. Ferroelectrics just above a ferroelectric (FE, dipole-ordered)–paraelectric (PE, dipole-disordered) phase transition are most useful because of the largest electric field-induced |Δ*P*|. Several ferroelectrics exhibit Δ*T*_{ad} in excess of 10 K and |Δ*S*_{d}| over 50 J kg^{−1} K^{−1} [58–60] near FE transitions, even if in fields approaching dielectric breakdown. We note that while it is easy to generate very strong electric fields in small gaps, the generation of magnetic fields in excess of approximately 2 T generally requires superconducting magnets or bulky Halbach-like arrays [61]. Like the magnetocaloric effect, the electrocaloric effect may be strongly enhanced by Δ*S*_{el}, i.e. become a *giant electrocaloric effect* due to accompanying structural changes that result from ferroelastic coupling.

Elastocaloric (thermoelastic) effect is related to reversible crystallographic phase transformations [62,63]. The latter are central to the enhancement of spin(dipole)-order effects in materials with giant magnetocaloric (electrocaloric) effects. In a way, elastocaloric refrigeration is similar to vapour-cycle cooling: both use stress to induce a phase transformation and use the corresponding entropy change, but the refrigerant is liquid/vapour for the vapour cycle, and solid/solid for elastocaloric cooling [22].

For example, when a thermoelastic shape memory material under stress switches between austenitic and martensitic phases, releasing or absorbing latent heat, Δ*Q*_{el}=±*TΔS*_{el}, the entropy change can be experimentally determined from either direct calorimetric measurements [64] or indirectly from the Clausius–Clapeyron equation (*p* is pressure and Δ*V* is phase volume change):
2.7assuming *C*_{L} remains nearly constant. Notably, simple thermodynamics-derived relationships similar to equations (2.1) and (2.5) are not applicable within the realm of elastocaloric materials, and density functional theory calculations of the electronic and phonon contributions to the total entropy based on the chemistry of a given material, as well as actual structures of both the austenitic and martensitic phases, are needed for *ab initio* predictions of Δ*S*_{el}.

For some NiTi-based alloys, Δ*Q*_{el}≅20 J g^{−1} [65] has been reported, giving Δ*T*_{ad}≅85 K, assuming *C*_{L}≅3*R*. The maximum observed Δ*T*_{ad} is approximately ±20 K [22], i.e. the elastocaloric effect reaches approximately 25% of the maximum given by equation (2.8), yet this relatively high value has been achieved at a tensile stress approaching stress at failure. Compared with magnetocaloric and electrocaloric effects, the elastocaloric effect remains least explored for its applicability to caloric effect-based refrigeration and heat pumping applications.

To sum up, present-day caloric materials realize only small fractions of their magnetic-to-thermal, electric-to-thermal or stress-to-thermal energy conversion potentials. Fields that are easy to produce, maintain and perturb, and certain not to damage a caloric material, only result in caloric effects that are in the range of a few per cent of the corresponding limiting values. This is indeed quite encouraging for both basic and applied science of caloric materials and caloric-based devices, as all of the known and, predictably, hitherto undiscovered systems and compounds are very far away from the respective performance plateaus, offering challenging yet truly exciting materials development opportunities with a high probability to achieve quite remarkable improvements in the future.

## 3. Non-equilibrium states: are they the light?

In addition to exhibiting best-in-each-class caloric performance, first-order phase transitions are notorious for the occurrence of non-equilibrium states manifested as history dependence, kinetically arrested states, hysteresis and phase separation. These features are usually considered detrimental to caloric performance and have been routinely dealt with by adjusting chemistry to shift a promising system towards equilibrium behaviour. This often results in the suppression or even complete destruction of the first-order nature of the phase transition and, therefore, leads to much reduced, conventional caloric effects, making the thus-modified materials rather common [12,66]. On the other hand, non-equilibrium phenomena are unique because of the potential for far greater changes of nearly all materials’ properties, including caloric effects.

Consider a hypothetical caloric material system in the vicinity of a first-order phase transition. Two-dimensional phase diagrams drawn in driving field (*Φ*)–temperature (*T*) coordinates are schematically illustrated in figure 1. The diagram in figure 1*a* only shows the phase transition for the case when *Φ* increases at constant temperature, henceforth the ‘direct’ transformation. The ‘reverse’ phase transition when *Φ* changes in the opposite directions at *T*=const. generally occurs in a different region of the diagram as shown in figure 1*b* because of hysteresis. The reverse transformation generally also involves phase separation. We note that *Φ*_{start} and *Φ*_{finish} correspond, respectively, to the finish and the start, *T*_{finish} and *T*_{start}, of the magnetostructural phase transitions when temperature varies at constant field.

Assume that the blue (dark grey) phase is ferromagnetic (FM) and that the pink (light grey) phase is paramagnetic (PM). We further assume that the driving magnetic *field is perturbed adiabatically*. When the magnetic field is increased from nearly zero by a small increment, +Δ*Φ* (which is the shortest distance between point A or point C and the line indicating the maximum available field), the system moves along the slightly nonlinear paths or shown in figure 1*a* as nearly vertical arrows that reflect proportionality of the magnetocaloric effect to *Φ*^{2/3} [50]. The value of the exponent (2/3) derived from the mean field theory [50] is applicable to materials exhibiting a conventional magnetocaloric effect. Small changes in the magnetization lead to small Δ*S*_{m}. A positive field increment reduces *S*_{m} (Δ*S*_{m}<0) and increases the temperature of the system. The resulting Δ*T*_{ad} is also proportional to Δ*Φ* and, because the latter is small, Δ*T*_{ad} is expected to be measurable but small, as indicated by short horizontal arrows pointing at B and D. The system remains in the equilibrium state at the beginning and end of the field increment. When the field is then reduced adiabatically by −Δ*Φ*, the system returns, respectively, to either the point A or C following the reverse paths or ; here, Δ*S*_{m}>0 and Δ*T*_{ad}<0, both being identical in magnitude to the corresponding magnetic entropy and temperature changes when Δ*Φ*>0, but with opposite signs. In the absence of energy losses, repeated cycling along either of the two described pathways will continue to produce very efficiently identical ±Δ*T*_{ad} that are, unfortunately, weak.

A much different response is expected when the same magnetic field increment +Δ*Φ* is applied when the system is initially at point E and moves along the path illustrated in figure 1*a*. Large changes in the relative concentrations of the coexisting phases scale nearly linearly with field and can be determined using the lever rule: e.g. *x*_{1} is the volume fraction of PM phase and 1−*x*_{1} for FM phase at point E, and *x*_{2} for PM and 1−*x*_{2} for FM at point F. When added to the conventional *Φ*^{2/3} contribution, this leads to disproportionally large change in magnetization and *S*_{m} as the system moves from being predominantly paramagnetic (*x*_{1}>*x*_{2} at E) to predominantly ferromagnetic (*x*_{1}<*x*_{2} at F) and, therefore, a large Δ*T*_{ad}, as shown by the horizontal arrow pointing at point F, will follow. Note that at the beginning (point E), the system is inside the non-equilibrium phase-separated state, and the value of *x*_{1} will depend on the history, i.e. how the system has reached point E. Dependent on the actual magnitude of Δ*Φ*, the system may remain inside the non-equilibrium phase-separated state at the end (see figure 1*a*, point F), although it may also move into the equilibrium ferromagnetic state if the field increment is sufficiently large to cross the *Φ*_{finish} boundary. In either case, reducing the field by −Δ*Φ* will *not* return the system to point E because of hysteresis.

Figure 1*b* illustrates the same region of temperatures and fields as figure 1*a*, except it now details the reverse phase transition, which occurs when the field is reduced. The reverse phase transition also involves a non-equilibrium, phase-separated state; however, due to hysteresis, it is shifted to the right compared with figure 1*a*. The region of the direct phase transformation is delineated with thin dashed-dotted lines and shown for reference. The original path discussed in the previous paragraph is now illustrated by a thin dotted arrow and it is assumed to have already occurred. Hence, at point F, the system consists of 1−*x*_{2} FM phase and *x*_{2} PM phase, where, according to figure 1*a*, *x*_{2}<0.5. A simple adiabatic removal of the field will not change *x*_{2} because the system is outside the reverse phase transformation region. Then, both the ferromagnetic and paramagnetic phases present in the system will exhibit conventional and small magnetocaloric effects, both being proportional to *Φ*^{2/3}. In the proximity of the phase transition, magnetocaloric effects in the paramagnetic and ferromagnetic states are nearly identical [52] and, therefore, the result will be a weighted average of the two equilibrium paths ( and in figure 1*a*), shown schematically in figure 1*b* as path with Δ*T*_{ad} (also being the weighted average), which is illustrated as a short horizontal arrow pointing at E^{′}.

To realize the giant magnetocaloric effect upon field reduction, the system must be first moved to the point G; then −Δ*Φ* will reduce the volume fraction of the PM phase from *x*_{2} to *x*_{1} and produce a large |Δ*T*_{ad}|, once again shown as a horizontal arrow pointing to H (figure 1*b*). Similar to field reduction starting from point F discussed in the previous paragraph, the positive field increment applied at point H will only lead to a small, conventional Δ*T*_{ad} (the weighted average of those shown in and ) and, in order to induce the giant magnetocaloric effect again, the system must be shifted to point E before the next field application. The question is: *What, if anything, can be done to move repeatedly along the* EFGH *loop and repeatedly trigger a giant magnetocaloric effect*?

One possible scenario is illustrated in figure 2*a*, where the hysteresis is much smaller compared with figure 1, and therefore phase-separated states corresponding to the direct and reversed transformations largely, but not completely, overlap. We note that reduction of hysteresis can be achieved without sacrificing the first-order nature of the phase transformation by engineering an appropriate microstructure that promotes both the direct and reverse transitions; one example has been discussed by Moore *et al.* [67].

Assume that the system has reached the initial point E in exactly the same way as shown in figure 1*a*. For the positive Δ*Φ*, the system follows path as already discussed above. Assuming that the volume fractions of the paramagnetic phase are, respectively, *x*_{1} and *x*_{2} at points E and F, and considering that point F is inside the reverse phase transformation region, a negative Δ*Φ* will effectively move the system along the line, however, not without an irreversible loss. Because the field-up and field-down half-cycles are fully contained within the corresponding phase-separated regions, the resulting Δ*T*_{ad} are expected to be nearly identical in magnitude but different in signs, as illustrated by the matching horizontal arrows. Unfortunately, the irreversibility present in the complete EFGH cycle will result in a different at the point H compared with the initial state of the system at point F, and after a few cycles the system will move into a steady, yet largely different, state with reduced magnetocaloric effect when compared with the ‘virgin’ EFGH cycle. This conclusion is in good agreement with direct cyclic Δ*T*_{ad} measurements reported for Heusler alloys, even though the measurements have been performed using Δ*Φ* that were large enough to, at least initially, drive the material completely from the low-field into the high-field phase [68,69].

The desired location of phase fields is illustrated in figure 2*b*, where the phase-separated regions coincide for both the direct and reverse transformations. Clearly, the system can be cycled along the and paths indefinitely, triggering identical in magnitude and large Δ*T*_{ad}. Unfortunately, the situation depicted in figure 2*b* is difficult to realize for first-order phase transition materials in a restrictive planar system of coordinates implied by figures 1 and 2, that is, when only two thermodynamic variables (temperature and field) are available. The solution, which changes the playing field dramatically, is to add a third dimension (variable), for example, stress. Phases that coexist in the phase-separated region by definition must have different volumes. Similar to reduction of hysteresis observed when magnetizing/demagnetizing Ni–Mn–In–Co at different hydrostatic pressures [70], applying (relieving) stress before driving the direct transformation by +Δ*Φ* and relieving (applying) stress before initiating the reverse phase transition by −Δ*Φ* can easily move the system from being characterized by different locations of the corresponding phase-separated regions in the *T*–*Φ* plane in figures 1 and 2*a* to the state depicted in figure 2*b*. Indeed, if stress (*σ*) is used to achieve the full coincidence of the direct and reverse phase-separated states, figure 2*b* is simply a projection of two planar cross sections of the three-dimensional phase diagram *T*–*Φ*–*σ* coordinates taken at fixed *σ*_{1} and *σ*_{2}. Using stress requires additional energy, yet this is expected to be a small price to pay, as it allows one to realize repeatedly very large |∂Δ*T*_{ad}/∂Δ*Φ*|.

## 4. Conclusion

There remains much to be learned about how to control the phase-separated states, and how to move efficiently a material along the paths that lead to repeatable caloric effects, even in the presence of hysteresis. Besides using extrinsic factors discussed here, i.e. by making use of a third dimension to solve an old two-dimensional hysteresis problem, there are numerous material design opportunities that remain worthy of exploration. The two most obvious possibilities include (i) incorporating both stress-inducing and stress-relieving microstructural features in a real material and (ii) designing real materials and transformation pathways without energy barriers. In the first, a specific set of microstructural features, such as stress-generating inclusions, may seed and promote the transition from a high-volume to a low-volume phase, while stress-relieving voids may seed and promote the reverse, thus nearly closing the hysteresis gap. In the second, a promising path forwards is establishing and reaching single-domain limits, where domains can switch without barriers, and seeking and following different pathways that are most energetically favourable individually for either the direct or reverse transformations. With this, we are ready to address the question that we asked in the title of this contribution: while there is more than one light at the end of the tunnel, non-equilibrium, phase-separated states are clearly one of the most important—promising to achieve a very large |∂Δ*T*_{ad}/∂Δ*Φ*| in small applied fields, potentially leading to breakthroughs in design of caloric cooling and heat pumping devices.

## Competing interests

We declare no competing interests.

## Funding

This work is supported by the United States Department of Energy, Office of Science, Basic Energy Sciences Programs, Materials Sciences and Engineering Division. Ames Laboratory is operated by Iowa State University under contract no. DE-AC02-07CH11358 with the United States Department of Energy.

## Footnotes

One contribution of 16 to a discussion meeting issue ‘Taking the temperature of phase transitions in cool materials’.

- Accepted May 5, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.