The sample thermal conductivity

of the conductive paste laid between sample and sensor. A poor thermal-contact between sample and sensor increases the time required to complete the temperature measurement and, thus, the dissipated heat from the sample.

Achieving a good thermal contact can be difficult especially with small temperature sensors and with porous or rough samples. The use of conductive pastes or greases improves the thermal contact but it introduces a further source of heat loss, because the paste acts as a thermal mass that absorbs heat from the sample.

characteristic time constant 𝜏_𝑚 of a 𝛥𝑇_𝑎𝑑 measurement. Therefore, overall, the 𝜏_𝑚 is related to the sensor thermal capacitance (𝐶_𝑡), to the thermal contact (𝐴 𝐾𝑠−𝑡) and to the thermal conductivity of the sample (𝐾𝑠):

𝜏𝑚≈ 𝐶𝑡 ( 1 𝐴 𝐾𝑠−𝑡

+ 1 𝑔𝐾𝑠

) (4.5)

The above result was obtained by solving the electric circuit, neglecting the heat dissipations to the environment and considering an overall effect of material’s thermal conductivity 𝐾_𝑠, corrected by a factor 𝑔, which takes into account the geometrical features of the sample that contribute to its thermal transport properties. By considering the sample as a single body with an infinite thermal conductivity, Equation 4.5 reduces to Equation 4.3.

In order to probe this expected behaviour of the time constant, adiabatic temperature change measurements were performed on two series of MC samples with different thermal conductivities. The shape and size of the samples were taken similar for all the components of the two series in order to obtain the same geometrical factor 𝑔, allowing to rescale the real thermal conductance of the samples on their thermal conductivity.

The measurements were performed with the experimental setup described in details in Chapter 3.5. A pneumatic linear actuator was used to cyclically move the probe with the sample and the thermometer between two regions with controlled temperature (𝑇_𝐻𝑂𝑇 and 𝑇𝐶𝑂𝐿𝐷) and magnetic field (𝐻₁ and𝐻₀). In this way, Brayton cycles were reproduced, simulating operative conditions of a magnetocaloric cooling device (for details see Chapter 3.5). Each sample was cycled around the temperature corresponding to its maximum 𝛥𝑇_𝑎𝑑. 𝑇_{𝐶𝑂𝐿𝐷} and 𝑇_𝐻𝑂𝑇 were set about 10 K below and above the mean temperature of the cycles.

The magnetic field in the two regions was µ₀𝐻₀= 0.1 T and µ₀𝐻₁= 1.0 T for the first series of samples and µ₀𝐻₀= 0.1 T and µ₀𝐻₁= 1.1 T for the second one.

The first measured series of samples includes:

1. a gadolinium sample (Goodfellow, 99.9% pure) 2. a La0.67Ca0.33MnO3 sample (labelled: LCMO)

3. a composite made of LCMO (78 vol%) and silver (7 vol%, labelled: LCMO+Ag) 4. a Mn1.27Fe0.68P0.48Si0.52 compound (labelled: MnFePSi)

5. a composite made of Mn1.27Fe0.68P0.48Si0.52 and epoxy (20 wt %) (labelled:

MnFePSi+epoxy)

The La-based manganite was prepared by J.A. Turcaud and K. Morrison (Imperial College London) from poly-crystalline powders synthesized by the glycine nitrate process, as presented in Ref. [163]. This manganite, crystallized in a perovskite structure, shows a reversible MCE at its Curie transition and

good mechanical properties, which permitted to reach promising results when it was tested in a refrigeration test-device [117]. A second sample was prepared by impregnating the parent LCMO with Ag, in order to increase its thermal conductivity, as described in Ref. [163].

The MnFePSi sample was prepared by F. Guillou (TU Delft) by ball-milling of elemental starting materials and solid state reaction by following the heat treatments described in Ref.s [22,164]. MnFePSi compounds are among the most promising materials for applications thanks to their considerable MCE, tuneable with the composition, which develops across their magneto-elastic transition.

The Mn1.27Fe0.68P0.48Si0.52 composition was selected in order to obtain a 𝑇_𝑐 similar to that of LCMO and a small hysteresis ensuring the cyclability of the MCE [81].

A composite material, characterized by a lower thermal conductivity, was prepared by mixing the MnFePSi compound with 20 wt% of a standard epoxy.

The thermal conductivities of the samples, reported in Table 4.1, were measured through the thermal transport option of the Physical Property Measurement System (PPMS) in the continuous measurement mode, at the Imperial College of London. The samples were cut, for the 𝛥𝑇_𝑎𝑑 measurements, with a parallelepiped shape with dimensions of about 1.5x3x5.5 mm³.

Sample 𝑲 (W m^-1 K^-1) 𝝉_𝒎 (s) 𝝉_{𝒎 𝒔𝒊𝒎} (s)

LCMO 1.35 0.40 ± 0.05 0.16

LCMO+Ag 2.2 0.15 ± 0.03 0.135

Gd 10 0.11 ± 0.03 0.10

MnFePSi 3.5 0.12 ± 0.03 0.10

MnFePSi+epoxy 1.25 0.20 ± 0.05 0.17

Table 4. 1 Thermal conductivity (K), experimental time constants (τm) and simulated time constant (τm sim) of the first series of measured MC materials.

The second series of samples includes a La-Fe-Co-Si sintered sample and four different composites made of La-Fe-Co-Si particles and an epoxy binder matrix:

1. La(Fe,Co,Si)13 sintered plate (Vacuumschmelze)

2. Composite A (epoxy + 45 vol% of 130 μm La-Fe-Co-Si powder)

3. Composite B (epoxy + 45 vol% of 130 μm La-Fe-Co-Si powder + 19 vol% of 12 µm, 21 µm and 57 µm La-Fe-Co-Si powder in the ratio 1:1:1)

4. Composite C (epoxy + 45 vol% of 130 μm La-Fe-Co-Si powder + 19 vol% of 12 µm powder milled into smaller particles < 10 μm)

5. Composite D (epoxy + 45 vol% of 130 μm La-Fe-Co-Si powder + 19 vol% of milled graphite powder < 10 μm)

These samples were prepared with the cold plate pressing method, in the form of thin plates (dimensions: about 0.5x3x5 mm³). The composites are a mixture of La-Fe-Co-Si powders (130 µm particles, produced by Vacuumschmelze) and, as a binder, an epoxy Amerlock Sealer (produced by Ameron). In composites B, C and D additional elements were added to the epoxy in order to increase the final thermal conductivity of the material. All the composites show a Curie transition at about 288 K. Details on sample preparation and on their properties are reported in Ref. [77]. To perform the direct MC measurements, two plates for each sample were overlapped in order to increase the thermal mass of the sample.

Sample 𝑲 (W m^-1 K^-1) 𝝉_𝒎 (s)

Sintered 8.93 0.12 ± 0.01

Composite A 1.29 0.21 ± 0.02

Composite B 2.4 0.15 ± 0.03

Composite C 2.68 0.14 ± 0.02 Composite D 1.09 0.2 ± 0.04

Table 4. 2 Thermal conductivity (K) and experimental time constants (τm) of the La-Fe-Co-Si series of measured MC materials.

Figure 4. 4 (a) Thermomagnetic cycles performed on the samples LCMO (green diamonds) and LCMO+Ag (pink circles), with a magnetic field change of 0.9 T. (b) Enlargement of a half-cycle of the two samples normalized to the maximum ΔTad.

Figure 4.4.a shows, as an example, the beginning of a cyclical measurement performed on the LCMO and on the LCMO+Ag samples near their Curie temperature. The 𝛥𝑇𝑎𝑑 of the composite sample is lower than that of the parent compound because of the presence of Ag, which acts as a passive heat capacity

during the magnetic field change. Moreover, the cycle period of LCMO+Ag is lower (9 s) than that of the parent compound (12 s). Considering that the temperature relaxation branches take about the same time, it can be concluded that the time delay for LCMO is mainly due to the adiabatic branches. This is clearer in Figure 4.4.b, which shows a half cycle for each sample normalized to its maximum 𝛥𝑇_𝑎𝑑. The characteristic time of the temperature change (𝜏_𝑚) of the two materials is different: LCMO reaches 63% of its maximum temperature change in 0.4 s, whereas LCMO+Ag needs only 0.15 s. The difference of time constant between the two samples can be ascribed to their thermal conductivity, which is 1.35 Wm^-1K^-1 for LCMO and 2.2 Wm^-1K^-1 for LCMO+Ag.

Figure 4. 5 Experimental time constants (blue triangles) of the temperature change as a function of the reverse of the thermal conductivity for the two series of measured samples.

Yellow circles show the time constants for the first samples series calculated through heat transfer simulations. The grey lines are the linear interpolations of the data.

Cyclical measurements were performed for all the samples of the two series. The adiabatic branches of the cycles were interpolated with the exponential function 4.4 in order to obtain, for each sample, the characteristic time of temperature change. The values of 𝜏𝑚, reported in Tables 4.1 and 4.2, are an average of values obtained for hundreds of subsequent cycles for each material. Figures 4.5.a and 4.5.b show the measured time constants as a function of the inverse of thermal conductivity of the materials for the two series. The reported error bars correspond to the statistical variations of the measured data. Figure 4.5.a also includes simulated time constants of the first series of samples, which were calculated by heat transfer simulations carried out by Dr. Giacomo Porcari (University of Parma) using a finite-difference explicit method (details can be found in Ref. [78]).

Both simulations and experimental results, for the two series of samples, show a linear behaviour of the time constant with the inverse of the sample thermal

conductivity 𝜏_𝑚= 𝑎 + 𝑏𝐾_𝑠⁻¹, as expected from Equation 4.5. Indeed, the sample geometry and the experimental configuration (temperature sensor and thermoconductive paste) are the same for all the measurements. Thus the sensor thermal mass, the contribution of thermal contact and the geometrical factor can be considered constant in each series. The time constants of Gd, of MnFePSi and of the sintered sample, which are the samples with the highest values of thermal conductivity, are close to the limit defined by the response time of the experimental setup (about 0.11 s, Chapter 4.1). The increase of the time constant for the other samples is due to their lower thermal conductivity. The different slope of the linear behaviours of the two series is related to the geometrical factor.

The only sample that shows a time constant that is not in agreement with the linear behaviour and with the thermal simulations is the parent LCMO. The reason of this discrepancy is not clear and will be the subject of a future research activity. Nevertheless, it is probably related to a parasite thermal resistance of grain boundaries, which can vary among different samples and that is reduced with the addition of silver.

Instead, an opposite effect is observed in the case of the La-Fe-Co-Si composites.

In this case, the interfacial thermal resistance between the epoxy matrix and the active magnetocaloric material decreases the thermal conductivity of the material and so the time constant of the temperature change. Indeed, finite-elements simulations based on a tomography image of sample A, performed by K. Sellschopp (IFW Dresden), demonstrated that the low thermal conductivity of the epoxy could not explain the large time constant observed in the composite, compared with the sintered parent sample [165]. The time constant increases due to the heat transfer between the active element and the passive matrix.

These outcomes demonstrate the not negligible role of the materials thermal transport properties, related to their intrinsic thermal conductivity and to extrinsic factors such as the presence of inter-grains thermal resistances, determining the characteristic time constant of 𝛥𝑇_𝑎𝑑 measurements. The increase of the time constant can affect the absolute measured values of 𝛥𝑇_𝑎𝑑 by raising the heat dissipated during the measurements. Moreover, this additional time delay between the magnetic field change and the measured temperature change can bring to a wrong estimation of the field-behaviour of temperature change (𝛥𝑇_𝑎𝑑(𝐻)), which is often derived from time dependent direct measurements.

On the other hand, the procedure of characterization, described in this chapter, is useful to obtain important information regarding the performance of MC materials exploitable in thermomagnetic devices. Indeed, the response time of magnetocaloric materials derived by simulating thermomagnetic cycles, is a key

parameter to predict the achievable frequency of magnetic refrigerators, which is mainly ruled by the time needed to the heat transfer between the MC active element and the transfer-fluid. From the presented results, it is clear as a low thermal conductance drastically reduces the maximum achievable operation frequencies of energy conversion devices, and thus their useful power, efficiency and cost.