2. Electrocaloric effect

An electric field E applied to a dielectric material induces a change in dielectric displacement and, thus, a change in temperature and entropy in the material. The EC effect is a reversible temperature change of a material that results from an adiabatic application of an electric field. It was derived by Lord Kelvin based on the assumption of reversibility of the pyroelectric effect from thermodynamics in 1878 [9]: If the preceding explanation of pyroelectricity be true, it must follow that a pyroelectric crystal moved about in an electric field will experience cooling effects or heating effects … in virtue of the wholly latent electric polarity of a seemingly neutral pyroelectric crystal (that is to say, a crystal at the surface of which there is an electrification neutralizing for external space the force due to its internal electric polarity), the same cooling and heating effects will be produced by moving it in an electric field, as similar motions would produce in a similar crystal which, by having been heated in hot water, dried at the high temperature, and cooled, is in a state of pyroelectric excitement.

The first experimental investigations of the EC effect in Rochelle salt, KH2PO4, BaTiO3 and SrTiO3 date back to 1930 [10], 1950 [11], 1952 [12] and 1956 [13], respectively. However, the EC effect values reported since that time (at maximum 2.5 K in Pb0.99Nb0.02(Zr0.75Sn0.20Ti0.05)O3 ceramics [14]) were too small for practical use.

pressure undergoing the reverse phase transition (thereby absorbing heat from the load). The amount of transferred heat is determined by the latent heat of the first-order phase transition. Similarly, in solid-state electrocaloric (EC) cooling, the adiabatic compression/expansion of the refrigerant is analogous to adiabatic polarization/depolarization, while the isobaric processes are replaced by isofield ones. Contrary to VCR, where the adiabatic expansion of the vapour is thermodynamically irreversible, the EC and the magnetocaloric (MC) effects are thermodynamically reversible processes that could reach the limit of the Carnot efficiency. This is

Electric fields required for the EC refrigeration cycle can be supplied much easier and less expensively than the high magnetic fields required for the MC refrigeration [2]. Other advantages in comparison with MC cooling are higher power densities due to potentially higher cycle frequencies, smaller mass of the device, compactness, potential cost reduction, independence on risks of rare-earth materials supply, etc. [3]. Moreover, electrical energy for EC cooling can be provided by stationary or mobile solar cells and by electric vehicle batteries. This opens up completely new possibilities for an environment-friendly industrialization of

EC materials provide a solid-state cooling technology without polluting liquid refrigerants and no or almost absent moving parts (pump and motion of a pumped heat transfer fluid).

into cooling or heating. Here, E is the electric field and D is the dielectric displacement. The latter is a vector field describing the electrical effect of free and bound charges in materials. Compared to VCR, the E plays the role of pressure and D plays the role of volume in vapour compression. More detailed descriptions can be found in a number of recent reviews of the EC effect [2, 4, 5]

An electric field E applied to a dielectric material induces a change in dielectric displacement and, thus, a change in temperature and entropy in the material. The EC effect is a reversible temperature change of a material that results from an adiabatic application of an electric field. It was derived by Lord Kelvin based on the assumption of reversibility of the pyroelectric effect from thermodynamics in 1878 [9]: If the preceding explanation of pyroelectricity be true, it must follow that a pyroelectric crystal moved about in an electric field will experience cooling effects or heating effects … in virtue of the wholly latent electric polarity of a seemingly neutral pyroelectric crystal (that is to say, a crystal at the surface of which there is an electrification neutralizing for external space the force due to its internal electric polarity), the same cooling and heating effects will be produced by moving it in an electric field, as similar motions would produce in a similar crystal which, by having been heated in hot water,

EdD, ð1Þ

W ¼ ð

another aspect making them promising for future application.

Generally, EC material (refrigerant) converts the electrical input work

and its application in refrigerators [3, 6, 7], and a book on this topic [8].

dried at the high temperature, and cooled, is in a state of pyroelectric excitement.

developing countries.

20 Refrigeration

2. Electrocaloric effect

EC cooling has regained attention in 2006, when Mischenko et al. could show that large electrical fields can be applied to antiferroelectric PbZr0.95Ti0.05O3 thin films [15]. They observed that—close to the ferroelectric Curie temperature of 222�C—a field change from 77.6 to 29.5 V/μm induced an adiabatic temperature change of 12 K as it was determined from the integrated pyroelectric effect. Recently, another group showed that a lead-free stack of 63 BaTiO3 thick films provides an EC temperature change of 7.1 K at an applied field of 80 V/ μm [16]. The thickness of the individual layers deposited by tape casting and electrically contacted by inner Ni electrodes amounted to ca. 3 μm. In Ref. [17], commercially available multilayer capacitors (MLCs) even of 200 ceramic layers (BaTiO3-based Y5V formulation) each 6.5 μm in thickness were used as a refrigerant [17]. Here, an EC temperature change of 0.5 K was obtained at 30 V/μm. The MLC concept was developed by Herbert [18] and introduced in the early 1980s by Murata Manufacturing Co. for the fabrication of base metal monolithic capacitors [19]. MLCs are now in mass production (some 5 � <sup>10</sup><sup>11</sup> pieces per year) by means of sheeting green ceramic tapes and screen-printing technology [20]. However, they are not optimized for EC applications. Commercial EC devices are still not available.

EC devices are driven by an electric field strength. That means that a voltage has to be applied. In this case, the independent thermodynamic parameters are temperature Tand electric field E. According to the second law of thermodynamics, an infinitesimal amount of heat dQ transferred into the system by an entropy change dS is then given by

$$dQ = TdS = T\left[\left(\frac{\partial S(E,T)}{\partial T}\right)\_E dT + \left(\frac{\partial S(E,T)}{\partial E}\right)\_T dE\right],\tag{2}$$

where S is the entropy per unit volume. Following the definition of volumetric specific heat at constant E, cE, the first term in parentheses can be replaced by

$$
\left(\frac{\partial \mathbb{S}(E,T)}{\partial T}\right)\_E = \frac{c\_E(E,T)}{T}.\tag{3}
$$

The value of cE(E,T) is usually represented by the zero-field value in the temperature range of interest c ¼ cðTÞ. With regard to Maxwell´s equations, entropy S and dielectric displacement D are coupled [21]

$$
\left(\frac{\partial S}{\partial E}\right)\_T = \left(\frac{\partial D}{\partial T}\right)\_E = \pi\_E.\tag{4}
$$

where π<sup>E</sup> is the pyroelectric coefficient at constant electric field. Considering now a ferroelectric material exhibiting below TC a remnant polarization Pr and an induced polarization εε0E, and a dielectric displacement of

$$D(T, E) = \varepsilon\_0 \varepsilon(T, E)E + P\_r(T),\tag{5}$$

Eq. (2) takes the form:

$$dQ = c\_E dT + T \left(\frac{\partial P\_r(T)}{\partial T} + \varepsilon\_0 E \frac{\partial \varepsilon(T, E)}{\partial T}\right)\_E dE. \tag{6}$$

Thus, the EC temperature change is given by two terms [22]:

$$
\Delta T\_{EC} = -\frac{T}{c\_E} \int\_{E\_1}^{E\_2} \left( \frac{\partial P\_r(T)}{\partial T} + \varepsilon\_0 E \frac{\partial \varepsilon(T, E)}{\partial T} \right)\_E dE. \tag{7}
$$

In the ferroelectric phase, below TC, the contributions of spontaneous and induced polarization partially compensate each other, because in this temperature region the temperature coefficients behave oppositely: ∂ε/∂T > 0 and ∂Pr/∂T < 0. In SrTiO3 ceramics below the temperature of maximum dielectric permittivity, in antiferroelectrics with 〈Pr〉 = 0, and in some relaxors with ∂Pr/∂T > 0, a negative electrocaloric effect can be obtained below TC, that is, the sample is cooled during adiabatic electric field application.

Figure 1 compares the ΔTEC values of BaTiO3 above TC (Pr!0) calculated from Eq. (7) [23] with available experimental data [12, 16, 24, 25]. Well-known examples of EC materials driven above the temperature of maximum dielectric permittivity are polyvinylidene fluoride terpolymers and irradiated copolymers, both exhibiting relaxor behaviour. Here, assuming a dielectric permittivity

Figure 1. ΔTEC of BaTiO3 [23] (solid line) and PVDF-based relaxor polymers (dashed line) as a function of E calculated for Pr = 0 following Eq. (7) in comparison to ΔTEC determined along the coexistence curve of the ferroelectric-paraelectric phase transition [27]. Experimental data of single crystal (sc), polycrystalline (pc) BaTiO3, and PVDF-based polymers were taken from Refs. [12, 16, 24, 25, 28], respectively.

independent on electric field, our calculations revealed that suitable for application ΔTEC values appears only at large electric fields. This is a problem since rapidly rising electric fields favour electrical breakdown of polymers. Moreover, the field dependence of ΔTEC is very different to BaTiO3, but similar to relaxor PLZT in Ref. [26].

DðT,EÞ ¼ ε0εðT,EÞE þ PrðTÞ, ð5Þ

E

E

dE: ð6Þ

dE: ð7Þ

<sup>∂</sup><sup>T</sup> <sup>þ</sup> <sup>ε</sup>0<sup>E</sup> <sup>∂</sup>εðT,E<sup>Þ</sup>

<sup>∂</sup><sup>T</sup> <sup>þ</sup> <sup>ε</sup>0<sup>E</sup> <sup>∂</sup>εðT,E<sup>Þ</sup>

� �

� �

∂PrðTÞ

In the ferroelectric phase, below TC, the contributions of spontaneous and induced polarization partially compensate each other, because in this temperature region the temperature coefficients behave oppositely: ∂ε/∂T > 0 and ∂Pr/∂T < 0. In SrTiO3 ceramics below the temperature of maximum dielectric permittivity, in antiferroelectrics with 〈Pr〉 = 0, and in some relaxors with ∂Pr/∂T > 0, a negative electrocaloric effect can be obtained below TC, that is, the sample is

Figure 1 compares the ΔTEC values of BaTiO3 above TC (Pr!0) calculated from Eq. (7) [23] with available experimental data [12, 16, 24, 25]. Well-known examples of EC materials driven above the temperature of maximum dielectric permittivity are polyvinylidene fluoride terpolymers and irradiated copolymers, both exhibiting relaxor behaviour. Here, assuming a dielectric permittivity

Figure 1. ΔTEC of BaTiO3 [23] (solid line) and PVDF-based relaxor polymers (dashed line) as a function of E calculated for Pr = 0 following Eq. (7) in comparison to ΔTEC determined along the coexistence curve of the ferroelectric-paraelectric phase transition [27]. Experimental data of single crystal (sc), polycrystalline (pc) BaTiO3, and PVDF-based polymers were

∂T

∂T

Eq. (2) takes the form:

22 Refrigeration

dQ <sup>¼</sup> cEdT <sup>þ</sup> <sup>T</sup> <sup>∂</sup>PrðT<sup>Þ</sup>

cE ð<sup>E</sup><sup>2</sup> E1

Thus, the EC temperature change is given by two terms [22]:

<sup>Δ</sup>TEC ¼ � <sup>T</sup>

cooled during adiabatic electric field application.

taken from Refs. [12, 16, 24, 25, 28], respectively.

In the presence of a first-order phase transition induced by an electric field EPT < E2, an additional entropy ΔSPT change occurs at the phase transition temperature TPT which is originated from the latent heat L

$$
\Delta \mathbf{S}\_{\rm PT} = \frac{L}{T\_{\rm PT}}.\tag{8}
$$

Along the coexistence curve between two phases of the considered constituent, E is no longer an independent parameter. Therefore, it should be substituted by D. The slope of the coexistence curve on the E-T diagram is then given by the Clapeyron equation [29]

$$\frac{dE\_{PT}}{dT} = \frac{\Delta S\_{PT}}{\Delta T} = \frac{L}{T\Delta D}.\tag{9}$$

In the case of ferroelectrics, where dielectric displacement D approximately equals the polarization P, this yields a ΔTEC of [4]

$$
\Delta T\_{\rm EC,PT} = \frac{T}{c\_p} \cdot \frac{dE\_{\rm PT}}{dT} \cdot \Delta P\_{\prime} \tag{10}
$$

where ΔP is the jump of polarization at the phase transition, and cP is the volumetric-specific heat at constant P. Table 1 lists the Clausius-Clapeyron contribution to the EC effect of some typical ferroelectrics.

The EC effect of a first-order phase transition increases along the coexistence curve up to the tricritical point. With further increase of the applied field, it decreases it again [27]. Thus, the Clausius-Clapeyron contribution is substantial only for bulk ceramic-based EC devices driven at moderate electric fields.

Considering the entropy change ΔS ¼ Sð0, TÞ � SðE, TÞ for a system of N dipolar entities, each having Ω discrete equilibrium orientations, a physical upper bound on the EC effect was derived in Ref. [35]:

$$
\Delta T\_{\rm EC,max} = \frac{T \cdot \ln \Omega}{3\varepsilon\_0 \cdot c \cdot \mathcal{C}\_{\rm CW}} P\_{s'}^2 \tag{11}
$$

where CCW is the Curie-Weiss constant and Ps the polarization at saturation when all dipoles are aligned along the field. Values of Ps might be obtained from hysteresis loops in the saturation regime. Values of CCW can be derived from the asymptotic behaviour of the linear dielectric susceptibility in a Curie-Weiss-plot ε∝C=ðT � T0Þ with T<sup>0</sup> < TC. T<sup>0</sup> is the Curie-Weiss temperature, i.e. the temperature of the appearance of a metastable paraelectric phase in the ferroelectric one. The upper limit ΔTEC,max of lead-based relaxors estimated in this manner


Table 1. EC temperature change corresponding to the latent heat calculated using Eqs. (9) or (10) from experimental values.

amounts to about 10 K. The EC temperature of a first- or second-order phase transition deduced from the specific heat curves yields:

$$
\Delta T\_{\rm EC} = \int\_{T\_1}^{T\_2} \left[ \frac{c\_p(T)}{c\_p^{\rm pl}(T)} - 1 \right] dT,\tag{12}
$$

where cp ph is the portion of volumetric-specific heat at constant pressure p due to lattice vibrations. For a second-order phase transition in PZT thin films, ΔTEC amounts to 4.0–5.3 K at 665 K [36]. A phase-transition independent upper bound of the EC effect was proposed in Ref. [37] based on the fact that only a certain energy density might be stored in a dielectric—equivalent to a limit in electrostatic pressure. Electrical breakdown of metal oxide dielectrics is fixed by the arising local electric field and the chemical bond strength leading to Emax∝ε�1=2, with Emax the dielectric strength of the EC material [38]. According to Eq. (7), this results for Pr!0 in:

$$
\Delta T\_{\rm EC,max} < -\frac{\varepsilon\_0 T}{2c} \cdot \frac{1}{\varepsilon(0, T)} \frac{\partial \varepsilon(0, T)}{\partial T} \cdot \varepsilon(0, T) \mathcal{E}^2\_{\rm max}. \tag{13}
$$

For a relaxor ferroelectrics exhibiting a huge temperature dependence of dielectric permittivity 1 <sup>ε</sup>ð<sup>0</sup>, <sup>T</sup><sup>Þ</sup> <sup>∂</sup>εð<sup>0</sup>, <sup>T</sup><sup>Þ</sup> <sup>∂</sup><sup>T</sup> <sup>≈</sup> <sup>10</sup>�<sup>2</sup> K�<sup>1</sup> [39], we estimate an ultimate EC temperature change of ΔTEC,max ≈ 50K. Our estimation explains the high values of the EC effect previously obtained in relaxor leadlanthanum zirconate-titanate thin films [26].
