R. M'nassri*a,
Muaffaq M. Nofalb,
P. de Rangoc and
N. Chniba-Boudjadac
aUnité de recherche Matériaux Avancés et Nanotechnologies (URMAN), Institut Supérieur des Sciences Appliquées et de Technologie de Kasserine, Kairouan University, BP 471 Kasserine 1200, Tunisia. E-mail: rafik_mnassri@yahoo.fr
bDepartment of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia
cInstitut NEEL, BP 166, 38042 Grenoble Cedex 09, France
First published on 14th May 2019
In this work, we have investigated the structural, magnetic and magnetocaloric properties of La1.4Ca1.6Mn2O7 (A) and La1.3Eu0.1Ca1.6Mn2O7 (B) oxides. These compounds are synthesized by a solid-state reaction route and indexed with respect to Sr3Ti2O7-type perovskite with the I4/mmm space group. The substitution of La by 10% Eu enhances the value of magnetization and reduces the Curie temperature (TC). It is also shown that these compounds undergo a first-order ferromagnetic–paramagnetic phase transition around their respective TC. The investigated samples show large magnetic entropy change (ΔSM) produced by the sharp change of magnetization at their Curie temperatures. An asymmetric broadening of the maximum of ΔSM with increasing field is observed in both samples. This behaviour is due to the presence of metamagnetic transition. The ΔSM(T) is calculated for Ax/B1−x composites with 0 ≤ x ≤ 1. The optimum ΔSM(T) of the composite with x = 0.48 approaches a nearly constant value showing a table-like behaviour under 5 T. To test these calculations experimentally, the composite with nominal composition A0.48/B0.52 is prepared by mixing both individual samples A and B. Magnetic measurements show that the composite exhibits two successive magnetic transitions and possesses a large MCE characterized by two ΔSM(T) peaks. A table-like magnetocaloric effect is observed and the result is found to be in good agreement with the calculations. The obtained ΔSM(T) is ≈4.07 J kg−1 K−1 in a field change of 0–5 T in a wide temperature span over ΔTFWHM ∼ 68.17 K, resulting in a large refrigerant capacity value of ≈232.85 J kg−1. The MCE in the A0.48/B0.52 has demonstrated that the use of composite increases the efficiency of magnetic cooling with μ0H = 5 T by 23.16%. The large ΔTFWHM and RC values together with the table-like (−ΔSM)max feature suggest that the A0.48/B0.52 composite can meet the requirements of several magnetic cooling composites based on the Ericsson-cycle. In addition, we show that the magnetic field dependence of MCE enables a clear analysis of the order of phase transition. The exponent N presents a maximum of N > 2 for A, B and A0.48/B0.52 samples confirming a first-order paramagnetic–ferromagnetic transition according to the quantitative criterion. The negative slope observed in the Arrott plots of the three compounds corroborates this criterion.
It is well known that the structure of La1.4Ca1.6Mn2O7 is constructed from ferromagnetic metal bilayer slices of MnO2 sheets taken from the cubic perovskite, each slice being separated by a nonmagnetic insulating spacer layer which serves to isolate the bilayers (La, Ca)2O2 stacked along the c-axis. The anisotropy and the reduced dimensionality of these compounds play a crucial role in their special properties different from those shown by the cubic perovskites.15–17 Basically, the simultaneous ferromagnetic and metallic states observed in the Mn-based perovskite are explained using the double exchange mechanism (DE) caused by charge disproportionation.18–20 The DE interaction in the Mn–O–Mn network in the case of bilayer manganite is expected to be much weaker along the stacking c-axis direction because of the intervening rock salt layer that disrupts the interaction between the [MnO2] layers. Members of this perovskite family are very responsive to small changes in composition and structure because of their layered structure. An inherent anisotropy modifies the thermomagnetic properties of the layered materials.
In the context of magnetic cooling, the La1.4Ca1.6Mn2O7 compound shows an abrupt change in the magnetization ((∂M/∂T)H) and illustrates his magnetic entropy (ΔSM(T)) with particularly peak at Curie temperature. However, in regenerative Ericsson cycle, the entropy change of the refrigerant ΔSM(T) should be constant (table-like MCE) over the operating temperature range of about 30 K. For this, there are number of publications21–23 in which the authors have proposed different solutions to improve the cooling capacity at larger spans. Therefore, a solution is to work with a multiphase or sandwich materials. These materials extend the temperature range in which the magnetic entropy changes significantly increase the possibility of improving performance through layering. Another simple way that increases the efficiency of magnetic cooling of bilayer manganites is the creation of the composite by a succession of magnetocaloric refrigerant samples with similar values of ΔSM and refrigerant capacity (RC).24,25 In this work, an optimum molar fraction of La1.4Ca1.6Mn2O7 (A) and La1.3Eu0.1Ca1.6Mn2O7 (B) is determined for the assembling of a composite to be used as refrigerant material in solid-state magnetic cooling. A physical mixture of La1.4Ca1.6Mn2O7 and La1.3Eu0.1Ca1.6Mn2O7 is introduced to extend the operating temperature window because TC of La1.4Ca1.6Mn2O7 can be modified by small addition of Eu in La site. In addition, to extend the range of refrigeration, a composite magnetic refrigerant can be also used to increase or to optimize the refrigeration capacity (RC). This represents approximately the total thermal energy transferred from the hot to cold reservoirs over the active temperature range. Therefore, it was demonstrated that mixing of La1.4Ca1.6Mn2O7 (A) and La1.3Eu0.1Ca1.6Mn2O7 (B) provides an extra material design tool such that the optimal magnetic refrigerant material can be developed for a specific temperature range. The experimental results agree well with those calculated and discussed in the framework of an optimum regeneration Ericsson cycle. The MCE and RC of a prepared composite have been compared with those of individual bilayer manganites.
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Fig. 1 The XRD patterns for A and B samples performed at room temperature. (a) La1.4Ca1.6Mn2O7 and (b) La1.3Eu0.1Ca1.6Mn2O7. |
Fig. 2 depicts the temperature dependence of magnetization M(T) for both A and B samples under an applied magnetic field of 0.05 T in field cooled mode (FC). It is clear from Fig. 2 that the M(T) curves do not reveal secondary magnetic phases, which in turn suggests that the presence of small amount of impurities does not have any significant effect on the thermomagnetic properties. The M(T) curves show rapid decrease of magnetization at Curie temperature which is the signature of the transition from ferromagnetic to paramagnetic state.
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Fig. 2 Magnetization measurements as a function of temperature for La1.4Ca1.6Mn2O7 and La1.3Eu0.1Ca1.6Mn2O7 samples under 0.05 T. Insets: dM/dT as a function of temperature. |
Furthermore, it can be seen from Fig. 2 that replacing La by Eu causes the value of TC to shift from 240 K for La1.4Ca1.6Mn2O7 (A) to 200 K for La1.3Eu0.1Ca1.6Mn2O7 (B). The Curie temperature TC is defined as the inflection point of dM/dT (see inset Fig. 2). It is clear that the pristine compound La1.4Ca1.6Mn2O7 is ferromagnetic below TC = 240 K. This value is higher than TC = 235 K given by ref. 30 and smaller than TC = 270 K observed in the same sample provided by ref. 31. This dissimilarity may be explained by the sensitivity of Curie temperature to the preparation conditions and the temperature of sintering32,33 which leads to the conclusion that the preparation processes have enormous impacts on the performance of magnetic materials.
Fig. 3(a) shows the magnetic hysteresis loops of both A and B samples taken at 10 K. Both loops show nearly zero coercivity, high magnetization saturation and negligible hysteresis which means that A and B bilayer manganites exhibit perfect magnetic reversibility or soft ferromagnetic nature. These observed outstanding soft-magnetic properties are beneficial for the application as bulk magnetic refrigerants. Furthermore, it can be seen that they display scarcely any hysteresis loss, although the two compounds exhibit the nature of first-order phase transition. This point is very attractive for magnetic refrigeration.
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Fig. 3 (a) Hysteresis loops for La1.4Ca1.6Mn2O7 and La1.3Eu0.1Ca1.6Mn2O7 samples at 10 K. Isothermal magnetization curves at various temperatures (b) La1.4Ca1.6Mn2O7 and (c) La1.3Eu0.1Ca1.6Mn2O7. |
Isothermal magnetization M(μ0H) curves are performed around transition temperature for each sample. Fig. 3(b) and (c) represents the recorded M(μ0H) curves of samples A and B over a wide range of the magnetic field ranging from 0 T to 5 T. At temperatures above 270 K for A sample (250 K for B sample) M(μ0H) curves show a linear behaviour as expected in the paramagnetic state. Below 220 K for A sample (190 K for B sample) M(μ0H) curves show an expected rapid increase at field values less than 0.4 T followed by the tendency to saturation at higher fields, which indicate the existence of a ferromagnetic state in the samples.
However, it can be clearly seen that the magnetization initially increases gradually with increasing μ0H for temperatures between 220 and 270 K for A sample (190 and 250 K for B sample). A sudden change appears above a critical magnetic field followed by a rapid increase of magnetization thus exhibiting an ‘S’ shaped M(μ0H) plot. This is a signature of a metamagnetic behaviour observed in the both samples. The latter phenomenon indicates the possibility of a large magnetic entropy change around Curie temperature. This point is very enticing for magnetic refrigeration. Motivated by our previous result of a large magnetocaloric effect (−ΔSM(T, μ0H) = 7.23 J kg−1 K−1 at 5 T (ref. 27)) observed in Pr0.8K0.2MnO3 sample which showed a similar metamagnetic behavior, we investigated the MCE in both aforementioned compounds.26
In the present work, it is interesting to evaluate the magnetocaloric effect of the A and B compounds. For this reason, we used the isothermal magnetisations measured at discrete temperatures to determine the MCE for each compound. Using Maxwell relation and magnetization curves (M–μ0H) we obtained the value of magnetic entropy changes ΔSM(T, μ0H) as a function of temperature in the magnetic field range of 0 to 5 T for both A and B samples. Fig. 4(a) and (b) depicts the behaviour of ΔSM(T, μ0H) for both compounds. The negative sign of the ΔSM(T, μ0H) seen in the latter figures is referred as the normal MCE and confirms the ferromagnetic nature of these samples.34–36 As one can see, the aforementioned materials illustrate significant values of the magnetic entropy changes and show that the magnitudes of ΔSM increases with an increase in the applied magnetic field. For μ0H = 5 T, the entropy change ΔSM exhibits a maximum value of 6.6 J kg−1 K−1 around Tpeak ∼ 245 K for A sample (6.25 J kg−1 K−1 around Tpeak ∼ 215 K for B sample) and it decreases on either side. However, the magnitude of ΔSM increases and the peak of ΔSM becomes asymmetrical with the rise of magnetic field. While ΔSM diminishes abruptly with lowering temperature below the peak, it gradually falls with the rise of temperature above the peak. We can also remark that ΔSM curves for the both samples present higher peak values and are quite similar in the temperature range of ΔT = TCA − TCB ≈ 40 K. Due to the remarkable similarities in the results, the two materials provide an opportunity to manufacture a composite with high performance in the context of magnetic refrigeration.
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Fig. 4 Magnetic entropy change, (ΔSM) vs. temperature for (a) La1.4Ca1.6Mn2O7 and (b) La1.3Eu0.1Ca1.6Mn2O7. |
In this context, magnetic properties for La1.4Ca1.6Mn2O7 (A) and La1.3Eu0.1Ca1.6Mn2O7 (B) are described in more detail by means of simulation of ΔSM(T, μ0H = 5 T) of individual compounds La1.4Ca1.6Mn2O7 (A) and La1.3Eu0.1Ca1.6Mn2O7 (B) (see Fig. 5(a)). A numerical method is used to determine the optimum mass ratios of x refrigerant samples and the resulting entropy change ΔSM(T, μ0H = 5 T) of the composite.37 Fig. 5(b) illustrates the numerical calculations for ΔSM(T, μ0H = 5 T) curves in the vicinity of the magnetic transition temperatures for the investigated materials and their composite. The ΔSM(T, x) for (La1.4Ca1.6Mn2O7)1−x/(La1.3Eu0.1Ca1.6Mn2O7)x composites produced by combining a (1 − x) × (La1.4Ca1.6Mn2O7) and a x × (La1.3Eu0.1Ca1.6Mn2O7) were obtained from their respective ΔSM(T) according to the following equation:
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Fig. 5 (a) ΔSM(T, μ0H = 5 T) for La1.4Ca1.6Mn2O7 and La1.3Eu0.1Ca1.6Mn2O7. (b) ΔSM(T) curves for the composite (La1.4Ca1.6Mn2O7)1−x/(La1.3Eu0.1Ca1.6Mn2O7)x for μ0H = 5 T. |
To explore the performance of this composite, we have calculated the refrigerant capacity (RC) which is another decisive parameter for evaluating and approving cooling efficiency.38 The RC parameter measures the amount of heat convey between the cold and hot reservoirs in the thermodynamic cycle. Thus, it has been suggested as a more suitable indicator of magnetic substances utility for solid-state refrigeration. For practical cooling systems, the RC with a broad temperature range is suitable for the active magnetic refrigeration cycle.39–41 The refrigerant capacity depends not only on the maximum of −ΔSM(T), but also on the overall profile of −ΔSM(T). RC is obtained by numerical integration of the area under the −ΔSM(T) curve. The limits of the temperature integration are set by the half-maximum of the ΔSM(T) peak, where THot and TCold correspond to the two temperatures at which the |ΔSM(T)| value is half of the peak value:42
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Accordingly, we report a detailed investigation of the MCE response as a function of the composite ratio; we present in Fig. 6(a) the x dependence of ΔSM, ΔTFWHM and the RC values for the composite system Ax/B1−x at μ0H = 5 T. This figure confirms that the important value of ΔTFWHM correspond to x = 0.48. The existence of the table-like behaviour could give rise to the maximum values of ΔTFWHM and RC, which allows the A0.48/B0.52 composite to become a promising compound for different thermodynamic cycles used in magnetic cooling technology.
Based on these calculations and in order to produce a composite with table-like MCE for a magnetic cooling system using the Ericsson thermodynamic cycle, powdered specimen of the La1.4Ca1.6Mn2O7–La1.3Eu0.1Ca1.6Mn2O7 composite is made by repetitive grinding and mixing of the two compositions in the desired ratio of 48–52% (A0.48/B0.52). After preparing the composite, the temperature (T) dependence of the magnetization (M) is measured in field-cooled mode (FC) under an applied field of 0.05 T. The results are depicted in Fig. 6(b) for our new compound. This curve clearly shows that the investigated composite specimen exhibits two magnetic transitions because of its heterogeneous composition. It is also observed in Fig. 6(b) that the pronounced two minima in the dM/dT versus T curve confirm that the composite contains two magnetic phase transitions compared with individual A and B bilayer manganites. The later successive minima correspond exactly to TC for each of the constituent phases A and B used to prepare the A0.48/B0.52 composite. It is worthwhile to mention that the magnetization magnitude of the studied composite shows a small decrease at low temperatures as compared with that of A and B bilayer manganites. The existence of two transition temperatures originating from different phases can certainly have an important effect on the MCE characteristics because the shape and behaviour of the magnetic entropy change are highly sensitive to the character of the magnetic phase transition.
It is demonstrated that the presence of two magnetic phases in the refrigerant material ensure that the material has a large MCE with a broad refrigeration temperature range and enhanced RC. In this investigation, we used the presence of two magnetic transitions to confirm our above calculation and for generating a broad range of MCE with a significant increase in RC.
To get deeper insight into the magnetocaloric response of the prepared composite upon changing the magnetic field from 0 to 5 T, isothermal magnetization curves of A0.48/B0.52 are measured as a function of the applied field recorded at different temperatures.
The measured M(μ0H) plots are shown in Fig. 7(a). In Fig. 7(b) we compare the selected isothermal M–μ0H curves plotted with applied fields between 0 and 5 T at T = 10, 220, 250 and 320 K for the individual samples and the A0.48/B0.52 composite. It is observed from this figure that the A0.48/B0.52 sample has similar values of magnetization at 10 K and 320 K as compared to A and B samples. In addition, the M(μ0H) curves are typical for a ferromagnetic state at 10 K and for a paramagnetic state at 320 K. On the other side, at 220 K and 250 K, the three compounds present different shapes in M(μ0H) and the composite system shows the intermediate values of magnetization compared to that of the constituent phases A and B. In this temperature range, the slightly jump in the M(μ0H) curves may be attributed to strong domain wall pinning in the ferromagnetic state.
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Fig. 7 (a) Isothermal magnetization curves at various temperatures for the composite and (b) M(μ0H) at 10, 220, 250 and 320 for all samples. |
The temperature dependences of magnetic entropy changes, −ΔSM(T), taken at 1, 2, 3, 4 and 5 T for the A0.48/B0.52 composite is presented in Fig. 8(a). All the curves of −ΔSM(T) have a clear double-peak shape (two ΔSM values), resulting from the disparity in Curie temperature of both phases A and B. The latter double-peak shape is very noticeable at low μ0H and begins to flatten gradually in favor of the table-like behaviour occurring at higher magnetic fields. This behaviour could give rise to the maximum values of ΔTFWHM and the RC refrigerant capacity strongly required for the ideal Ericsson cycle magnetic refrigeration over a broad temperature range.43
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Fig. 8 (a) Magnetic entropy change, ΔSM vs. temperature for the composite system and (b) comparison between experimental and simulation. |
Fig. 8(b) depicts experimental and theoretical entropy change curves for μ0H = 5 T of phases A and B that make up the composite with TC,A = 200 K and TC,B = 240 K, along with ΔSM(T) in the composite x = 0.48. Latest figure demonstrates that the agreement between the experimental curves ΔSM(T) and that predicted by eqn (1) is excellent. According to this agreement, we can conclude that the numerical calculations are valid in the choice of MCE composite and can thus be used as means of designing magnetic refrigerant materials with an improved magnetocaloric response for the desired magnetic fields. The maximum value of −ΔSM(T) is found to be 4.07 J kg−1 K−1 for A0.48/B0.52 in a wide temperature range. The magnitude of −ΔSM(T) is reduced in the A0.48/B0.52 composite which gives a broad table-like behaviour with a wide temperature range compared to that of the pure constituent phases. Basically, in an ideal Ericsson cycle, the entropy conveyed between two heat reservoirs (THot and Tcold) should be as constant as possible to avoid the generation of irreversible work.44 For this reason, the flattening of −ΔSM(T, x = 0.48) curve can be able to meet the latter requirements for the use of A0.48/B0.52 as a composite for Ericsson-cycle-based magnetic refrigerators.43
From eqn (2), the obtained value of RC is ∼205.92 J kg−1 at 5 T in B sample while it does not exceed ∼178.92 J kg−1 in A sample which indicate that the Eu-substitution increase the refrigerant capacity. Fig. 9(a) shows ΔSmaxM, ΔTFWHM and RC plots as a function of the applied magnetic field. As displayed in Fig. 9(b), the obtained values of ΔSmaxM, ΔTFWHM and RC are strongly related to the magnetic field. It is clearly observed that the A material has smaller values of ΔTFWHM than the B sample. Compared to gadolinium, which is considered as the typical ferromagnetic material for magnetic refrigeration, the RC values of the A and the B samples represents about ∼56.13% and ∼64.6% of the RC estimated for Gd (the value of RC is around 25% lower than that of the relative refrigerant capacity RCP for the ΔSM(T),42 from ref. 45 RCPGd = 425 J kg−1 then RCGd = ¾ × RCPGd = ¾ × 425 ∼ 319 J kg−1). According to the obtained result, the Eu-doped sample is still valuable for magnetic refrigeration at low temperatures. These values are much larger than that of several manganites46,47 and are high enough for magnetic cooling. Refrigerants with wide working temperatures and high RC are in fact very beneficial to magnetic cooling applications48 and suggests that compounds can thus be used as an active magnetic refrigeration materials suggested by Barclay.39 However, in this case the maximum values of ΔTTFWHM for all three samples A, B and A0.48/B0.52 are equal to ∼35.7 K, ∼41.54 K and ∼68.16 K respectively for a field of 5 T. It can be observed that the highest value of ΔTTFWHM is revealed for the A0.48/B0.52 composite and an increment of ∼47.63% and ∼39.05% in ΔTFWHM compared with the individual La1.4Ca1.6Mn2O7 and La1.3Eu0.1Ca1.6Mn2O7 samples are observed. This enhancement in ΔTFWHM is an outcome of the augmentation of RC despite the diminution of the maximum value of ΔSM. However, it follows that a compromise is necessary between the value of the ΔTFWHM and the energy losses and the efficiency of machine (due to an increase of cycles in the heat exchange medium). The investigated A0.48/B0.52 composite exhibit nearly constant value of ΔSM(T) with width of ∼68.16 K. The present results confirm that the field and the temperature range used in the numerical calculation are analogous to those explored experimentally and ensure that the large ΔTFWHM observed in the prepared composite have a great importance for cooling capacity.
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Fig. 9 (a) Variation of ΔSmaxM, ΔTFWHM and RC with magnetic field for all samples. (b) RC versus Tcold for all samples. |
The broadening width of ΔSM(T) is expected to make an increase of RC in the composite as predicted by eqn (2). In this study, it should be noted that the used TCold and THot are defined as temperatures fulfilling ΔSM(TCold) = ΔSM(THot) = ΔSmaxM/2. In order to visualize RC (TCold) and obtain the RC values at different values of TCold, the integrand is evaluated from high temperature (THot) to low temperature (TCold) as depicted in eqn (2). Fig. 9(b) shows the calculated values of RC as a function of TCold under 5 T magnetic field of A, B and A0.48/B0.52 compounds. It is found that the RC increases as TCold is separate from the ΔSM peak temperature (Tpeak), at which RC is zero due to eqn (2). In the temperature span of THot − TCold = ΔTFWHM, the estimated RC values correspond to A, B and A0.48/B0.52 samples are 178.92 J kg−1, 205.92 J kg−1 and 232.85 J kg−1 respectively. These values show that the RC of the composite A0.48/B0.52 is improved by 23.16% and 11.56% when compared with those of A and B samples. This proves the superior cooling power of the A0.48/B0.52 sample and refers to a possible way to optimize RC of refrigerant magnetic materials. This result reveals that the A0.48/B0.52 is a promising compound to be used as a refrigerant material in solid-state refrigeration with the Ericsson cycle. Similar reinforcement in RC responses are observed in many composite systems like La1.4Ca1.6Mn2O7/(La0.08Gd0.02)1.4Ca1.6Mn2O7,49 FeZrB(Cu),50 and La0.7Ca0.3Mn16O3/La0.7Ca0.3Mn18O3.51
To make our analysis more complete, we are concerned with the nature of the magnetic phase transition in our bilayer manganites. For that reason, we have investigated the field dependence of MCE in A, B and A0.48/B0.52 samples by using the relation expressed as ΔSM ≈ a(μ0H)N. The latter relationship makes it possible to determine the nature of the magnetic phase transition observed in the aforementioned samples. Recently, J. Y. Law et al. proposed a quantitative criterion to identify the order of magnetic phase transitions using the field dependence of magnetocaloric effect. For materials with the first-order magnetic phase transition (FOPT), ΔSM depends on the field with exponent N > 2.52 Particularly, for magnetic samples, the local exponent N(T,μ0H) can be calculated from the logarithmic derivative of the experimental ΔSM(T, μ0H):53
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The temperature dependence of N is illustrated in Fig. 10(a) for A, B and A0.48/B0.52 samples. The exponent N, for A and B samples has a minimum value at TC. However, for A0.48/B0.52, the N(T) curves exhibits two minima whose positions are related to the critical temperatures of the existing phases (A and B) in this composite. On the one hand, it is observed in Fig. 10 that the exponent N is sensitive to the magnetic field in the entire studied temperature range and the magnetic entropy changes, ΔSM. The value of N(TC) < 0.4 at high magnetic fields indicates that our samples undergo first order magnetic transition temperature.54,55 In the other hand and under critical temperatures, the N(T) curves increase gradually with the temperature drop and approaches 1 for higher magnetic fields. Far above TC, the N(T) values overshoots 2 (N > 2) in the paramagnetic region (near magnetic transitions) of all three samples. This overshoot is more pronounced in sample B compared to sample A and the composite A0.48/B0.52. The observed behaviour shows that the quantitative criterion of N > 2 near the transition is valid for monophasic and biphasic materials which indicates that our samples exhibit a first-order transition. This is in agreement with the previous observations in N(TC) values. A similar behaviour is reported in other magnetic materials with first-order transition.56 However, the order of magnetic phase transition is usually revealed by the Arrott plots (M2 vs. H/M). For more confirmation of the nature of the magnetic phase transition of A, B and A0.48/A0.52 samples, the curves of M2 vs. H/M plotted at different temperatures are exhibited in Fig. 11. The Arrott plots for the aforementioned materials just above the respective TC are displayed in the inset of the Fig. 11. According to the Banerjee criterion, the obviously negative slopes of Arrott plots verify the first-order nature of the three samples,57,58 which is consistent with the quantitative criterion.
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