The variation of temperature within the EC material is described by the heat transfer equation. For the sake of simplicity, we consider in the following an one-dimensional model. In the case of linear heat flow, the parabolic heat diffusion equation of a solid comprising a heat source is given by [16]

(1)

where Θ = T − T₀ is the temperature difference to the environment, α = κ/c the thermal diffusivity, κ the thermal conductivity, and c the volumetric specific heat assumed to be constant (c (E,T) ≈ c), V the volume, A the area and d the thickness of the EC layer, and h the heat transfer coefficient at the boundary describing heat loss to the environment. Introducing a bulk thermal resistance R′_th,b = (κA)^-1 and a thermal capacity C′_th,b = cA, both per unit length, eq. (1) transforms into

(2)

where Q˙′ and h′ are the values of Q˙ and h per unit length, respectively. Since the Biot number Bi = hd/κ (characterizing the ratio of the thermal resistance of the EC materials bulk to that at the boundary) will be less than 0.1 up to values of d in the order of 100 μm, the temperature of the EC element during heat transfer remains nearly constant, i.e. the ∂²Θ/∂x² = 0. This enables a lumped system approximation [14] simplifying eq. (2) to

(3)

where R″_th,i = AR_th,i = 1/h is the area-specific thermal resistance at the interface, C″_th = cd the thermal capacitance of the EC layer per unit area, and Φ the heat flux through the interface. Eq. (3) represent the well-known bolometer equation [17].

The driving force of heat flux is the temperature difference Θ caused by charging or discharging the EC element which itself represents a dielectric capacitor. When charging a capacitor possessing a dielectric permittivity ε under isothermal conditions, the heat dQ delivered by a small change of the electric field dE is given by [18]

(4)

Since the EC temperature change is usually small, ΔΘ_EC << Θ, the corresponding thermal flux Φ(t) = dQ (t)/(Adt) might be written as

(5)

where <Τ> is the average temperature of the cooling cycle and where the field dependence of the temperature coefficient of dielectric permittivity 1/ε·∂ε/∂T is neglected. Note, that above the temperature T_m of maximum dielectric permittivity, 1/ε·∂ε/∂T is negative. The rise time of an output signal of a voltage pulse generator is mainly dependent on the switching speed of the power amplifier, while the fall time is determined by the resistor-capacitor (RC) circuitry at the generator output and the load. Here, we will not consider the physical origin of power amplifier switching. Assuming for the rising pulse

(6)

with τ_e = RC the time constant of the electronic circuit, an approximate solution of eq. (3) is given by

(7)

where τ_th = R″_th,iC″_th and where the temperature coefficient of the dielectric permittivity was assumed to be constant. For very short rise times of the electric field, τ_e→0, eq. (7) reduces to

(8)

A pulse decay given by

(9)

yields for t´ = t – t₀

(10)

transforming in terms of t′ which for τ_e → 0 is equivalent to eq. (8) with opposite sign. A flexible EC cooling device with electrostatic actuation reported in [15] was driven with a high-voltage circuitry possessing an electrical time constant of about 140 ms. On the other hand, an analysis of eqs. (7) and (10) suggests time constants of less than 50 ms for an efficient device operation.

For sake of simplicity, we describe for a moment the thermodynamic cycles of the EC element by a periodic temperature change with a fundamental angular frequency ω = 2π/τ_c, where τ_c is the cycle time. Since the parabolic heat diffusion equation (1) has the same structure as the equation of an electrical RC-transmission line, multilayer problems are best treated by the matrix methods commonly used in electrical engineering [19]. In our case, temperature plays the role of a voltage and thermal flux the role of an electrical current. Each point, we shall describe by two quantities, the temperature Θ and the heat flux Φ. Denoting Θ and Φ as the temperature and thermal flux of a slab at the face z = 0 and Θ′ and Φ′ as their values at the face z = d, the following relations yield [16]

Θ′ = LΘ + MΦ,

Φ′ = NΘ + OΦ, (11)

with

L = cosh(kd), M = –(κk)^-1sinh(kd),

N = –κksinh(kd), O = cosh(kd), (12)

and

(13)

where d_D = (2D/ω)^1/2 = (Dτ_c/π) is the penetration depth of the thermal oscillation. Note, that for dielectric thin and thick films with d < 100 µm we get L = O ≈ 1, M = –d/κ, and N = –κk²d for ω = 1 Hz since D is in the order of 10^-6 m²/s. The value of M represents the area-specific thermal resistance of the film bulk.

Eq. (11) may be considered as matrix equation

(14)

In eq. (14), Θ and Φ must be multiplied by their time dependence exp(iωt) which is omitted here entirely. It should be again included when real or imaginary parts have to be taken at the end of the calculation. Assuming perfect thermal contact between the faces of the slabs a comprising n sublayers, the r_th being of thickness d_r, thermal conductivity κ_r and thermal diffusivity D_r with Θ_r and Φ_r and Θ′_r and Φ′_r at its left-hand and right-hand faces, respectively, recursive application of eq. (14) gives for a one-dimensional heat flow across several sublayers

(15)

Given any two of Θ_r, Φ_r, Θ′_r and Φ′_r the other two can be found using this method. The multiplication of matrices in eq. (15) can be carried out successively. However, deriving explicit formulas for slabs of n layers is very cumbersome [20–22].

In the presence of thermal contact resistances between the slabs or at the surfaces, they may be expressed also in matrix notation and included into eq. (15) [16]. For example, in case of a thermal interface resistance R″_th,i1 between the first and second slab we find

(16)

and, correspondingly,

(17)

The final result of this calculation is are linear relations between the temperature and fluxes Θ₁, Θ′_n, Φ₁, Φ′_n at the two surfaces of the composite slab. The surface conditions will provide two more equations so that all four quantities Θ₁, Θ′_n, Φ₁, Φ′_n will be determined. Recursive application of this matrix scheme yields for a one-dimensional heat flow across j slabs

(18)

where M is the heat transfer matrix. Thus, the matrix method allows numerically a very easy analysis of multilayer structure by multiplying numerical matrices.

The relative thickness of the EC layer in prototype EC cooling devices amounts to 0.30 to 0.35 d_D [11, 15]. In accordance with the above made approximation of a thermally thin system, we consider in the following a stack of j films satisfying d_j << d_D. In this case, we obtain:

(19)

According to eq. (8), the EC temperature change given by ΔT_EC = Θ(t = 0) is defined by the volumetric specific heat c and the temperature coefficient dε/dT of dielectric permittivity. Since the values of c at room temperature approach the high temperature limit, it is expected that they do not vary strongly among various pyroelectric materials [23]. Therefore, dε/dT is a key parameter. Relaxor ferroelectrics possess above T_m large values of 1/ε·∂ε/∂T as high as a few 10² K^-1 (Table 1).

Recently, BaZr_0.2Ti_0.8O₃ (BZT20) was demonstrated to be a promising material for EC application [5, 24]. BZT20 is a composition near the invariant critical point, where four different phases, including cubic, tetragonal, orthorhombic, and rhombohedral phases coexist. It is located slightly below the crossover to relaxor behavior [28, 32] where an intermediate state between ferroelectric and relaxor behavior is expected. On the other hand, BZT20 thick films show the shift of the dielectric permittivity peak towards higher temperatures with increasing frequency characteristic for relaxor ferroelectrics [24].

EC refrigerants are evaluated by a material criterion which characterizes the efficiency of the physical cooling process and, therefore, is independent on the performance of different thermodynamic cycles [1]:

(20)

Here, εε₀·E²·tanδ is the non-recoverable electrical loss with tanδ the loss tangent, and cΔT_EC the heat transferred from the load to the heat sink within one refrigeration cycle with c the specific heat of the cooling element and ΔT_EC the EC temperature change. At this point, the tanδ comes into play as an additional material parameter since – following eq. (8) – the dielectric permittivity cancels out. It has to been noted that tanδ of eq. (20) needs to be investigated in the frequency range of EC device operation (0.1 to 10 Hz). In the presence of a ferroelectric hysteresis, it mainly consists of losses due to minor hysteresis loops described by an apparent loss tangent [33]:

(21)

where P–(E) and P⁺(E) are the field dependence of dielectric polarization at the upper and lower branch of the minor hysteresis loop, respectively. Obviously, tanδ depends significantly on materials synthesis. Values of tanδ < 0.07 provide refrigerant efficiencies of Φ > 0.95 exceeding the ones of any other solid state cooling technology.

At thin film interfaces, the highest value of R^"_th,i measured at room temperature to date is 1.2·10^-5 m²K/W for Bi/Hydrogen-terminated diamond. The thermal interface resistance at room temperature of other metal/dielectric combinations falls within a relatively narrow range, 3.3·10^-8 m²K/W < R^"_th,i < 12·10^-8 m²K/W [34].

The minimum possible R^"_th,i^ph-ph produced by a purely harmonic process involving two phonons (one phonon on each side of the interface) is given by the phonon radiation limit which in the limit of high enough temperatures, where the Dulong-Petit law holds, is given by

(22)

Here, k is the Boltzmann constant, f_max = kΘ/h the vibrational cutoff frequency of the metal, h the Planck constant, Θ the Debye temperature, and v_D,

(23)

the Debye velocity of a two dimensional dielectric with v_l and v_t the average longitudinal and transverse sound velocities, respectively. A compilation of all data used for calculations is given in Table 2. In result, eq. (22) yields a phonon-phonon contribution to the interface resistance of R^"_th,i^ph-ph = 4.1·10^-10 m²K/W.

While phonons dominate the heat conduction in dielectrics, electrons dominate it in metals. Hence, an energy transfer occurs between electrons and phonons during heat transport across the metal-dielectric interface. In result, two thermal resistances - volumetric electron-phonon (R^"_th^e-ph) in the metal and interfacial phonon (R^"_th,i^ph-ph) – are in series. The former one is given by [47]

(24)

where R_e is the electron cooling rate or electron-to-phonon energy transfer per unit volume, R_e = c_e/τ with c_e the electron heat capacity per unit volume and τ_e-ph is the relaxation time characterizing electron-phonon energy loss or electron cooling. The value κ_ph is the phonon thermal conductivity,

(25)

with ρ the density, N_A the Avogadro constant, M the molar mass, and l_ph the phonon mean-free-path of the metal. Assuming τ_e-ph = 2 ps (cf. values of W and Cu in Table 2), we arrive at R^"_th^e-ph = 8.6·10^-10 m²K/W. Thus, the interfacial thermal resistance in the order of 10^-9 m²K/W calculated by simplified models is much lower than the corresponding experimental data.

From a microscopic point of view, all surfaces in contact have deviations from their idealized geometry. Because of these imperfections, two bodies in contact touch actually only at a few discrete points. The heat flux close to the interface is then constricted in these micro-contact regions [48].

The thermal contact resistance is defined as the ratio between the temperature drop at the interface and the heat flux crossing the interface. Similarly, electrical capacitance C is the ratio of the total charge on a conductor to its electric potential (potential difference to a reference electrode). Both capacitance and thermal conductance are affected by edge effects. Therefore, there is a close relation between R_th,i and the interface capacitance C_i given by [49]

(26)

where ε_b is the bulk dielectric permittivity. The area-related value R″_th,i = AR_th,i is then given by

(27)

with d_i the thickness and ε_i the dielectric permittivity of the interface layer. With regard to the values of C_i and ε_b in Pt/BaSr_xTi_1-xO₃/Pt structures (cf. Table 2), R″_th,i amounts to 1…5·10^-8 m²K/W crossing the lower limit of the thermal interface resistance for thin film metal/ dielectric contacts reported in [34].

In macroscopic cooling systems, the thermal interface resistance is generally negligible since it is in series with other thermal resistances exhibiting much larger dimensions and, thus, much larger resistance. Here, heat is transferred from the load or to the heat sink (i) via thermal switches or (ii) by a gaseous or liquid heat transfer agents [1]. A compilation of typical interface resistances of solid-solid and liquid-solid thermal contact resistances is given in Table 3.

In an active EC regenerator configuration, liquid or gaseous heat transfer agents are pumped through the EC refrigerant. They absorb heat from the bed and transfer it to the cold sink. Thereby, an electric field is applied solely to the refrigerant. The flow channels are usually rectangular ducts requiring a two-dimensional flow analysis. On the other hand, in a one-dimensional model, the determination of a heat transfer coefficient h (the inverse of h represents the thermal interface resistance) describing the heat transfer between the fluid and the solid is sufficient. The heat transfer across a boundary is defined by the Nusselt number Nu – the ratio of convective to conductive heat transfer normal to the considered boundary:

(28)

with d_H the hydraulic diameter. For steady state, laminar flow of an incompressible liquid with constant physical properties in a rectangular duct of constant cross-sectional area characterized by its aspect ratio AR (the ratio of the-smaller-to-the larger duct size), the value of Nu is approximately given by [64]:

Nu = 8.235(1 – 2.0421AR + 3.0853AR² –

2.4765AR³ + 1.0578AR⁴ – 0.1861AR⁵). (29)

At Bi < 0.1, the temperature of the EC element during heat transfer remains nearly constant (cf. section 2.1). Here, a duct high of z = 0.6 mm and AR = 0.15 yield thermal resistances of

(30)

The thermal resistance of the fluid, R″_th,f, is determined by the inverse of the product of its mass flow rate dm/dt and its specific heat at constant pressure c_p [16]. The total thermal resistance of the fluid, R″_th,f, for a given fluid velocity v then yields [65]

(31)

For calculation, an upper limit of the fluid velocity of v = 0.1 m/s was chosen providing a back and forth fluid motion of 1 cm at a cycle time of τ_c = 0.2 s. This results in vc >> h leading to R″_th,f ≈ R″_th,s-f. Values of R″_th,s-f are compiled in Table 4. Also, the thermal penetration depth at ω = 1 Hz, d_D,1Hz, is listed. It allows a quick estimation of d_D at a given cycle time by

(32)

Summing up, the thermal resistance at the interface to the heat transfer fluid limits the heat transfer of active EC regenerators.