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Entropy change at magnetic phase transitions of the first order and second order
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Entropy change at magnetic phase transitions of the first order and second order Vittorio Basso*, Carlo P. Sasso, Michaela Ku¨pferling Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy
article info
abstract
Article history:
Magnetocaloric properties close to first order and second order phase transitions are pre-
Received 7 April 2013
sented and discussed. The specific heat capacity as a function of the magnetic field and the
Received in revised form
magnetic field induced entropy change has been measured by direct heat flux calorimetry
16 May 2013
in Gd, hydrogenated La(FeeMneSi)13, La(FeeCoeSi)13 and the Heusler alloy NieMneCoeSn.
Accepted 25 July 2013
The resulting s(Ha,T ) entropy constitutive relation is compared with a model including the
Available online 6 August 2013
ferromagnetic, electronic and structural contributions to the free energy. ª 2013 Elsevier Ltd and IIR. All rights reserved.
Keywords: Magnetocaloric effect Gd La(FeeSi)13 Heusler alloy NieMneCoeSn Mean field theory
Changement d’entropie et transitions de la phase magne´tique de premier et de second ordre Mots cle´s : Effet magne´tocalorique ; Gd ; La(FeeSi)13 ; Alliage de Heusler NieMneCoeSn ; The´orie du champ moyen
1.
Introduction
The possibility to develop refrigeration techniques in the solid state, is attracting numerous efforts in fields of material science and engineering (Fa¨hler et al., 2012) by searching ferro-caloric materials in which the entropy of the system can be changed by an external action such as the magnetic field, the applied stress or the electric field. In the case of magnetic materials, the entropy state equation s(Ha,T ) contains all the necessary information to design thermodynamic cycles for refrigeration and derive the figures of merit of the
material: the isothermal entropy change Dsiso(Ha,T ) and the adiabatic temperature change DTad(Ha,T ) (see Fig.1) (Smith et al., 2012). By using heat flux calorimetry in magnetic field it is possible to determine experimentally the entropy state equation s(Ha,T ) (Basso et al., 2010). In this paper we present experimental data on selected materials: Gd, hydrogenated La(FeeMneSi)13eH, La-(FeeCoeSi)13 and the Heusler alloy NieMneCoeSn. The magnetocaloric properties, i.e. the specific heat capacity as a function of the magnetic field and the magnetic field induced entropy change are interpreted and
* Corresponding author. E-mail address: [email protected] (V. Basso). 0140-7007/$ e see front matter ª 2013 Elsevier Ltd and IIR. All rights reserved. http://dx.doi.org/10.1016/j.ijrefrig.2013.07.021
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R ms cp cPM cM cS cph cele cpv
Nomenclature s Dsiso sPM sM b s T t T_
specific entropy isothermal entropy change (specific) specific entropy in the paramagnetic state specific entropy, magnetic contribution reduced entropy, magnetic contribution absolute temperature reduced temperature temperature scan rate adiabatic temperature change sample temperature critical temperature magnetic field applied magnetic field reduced magnetic field Landau free energy (specific) magnetic free energy (specific) structural free energy (specific) heat flux
DTad Ts Tc H Ha h fL fS fS qs
aT kT v M m m0 W J S g
discussed in terms of the ferromagnetic, electronic and structural contributions to the free energy.
thermal contact resistance sample mass specific heat capacity specific heat capacity in the paramagnetic state specific heat capacity, magnetic contribution specific heat capacity, structural contribution specific heat capacity, phonon contribution specific heat capacity, electronic contribution specific heat capacity, thermal expansion contribution coefficient of thermal expansion isothermal compressibility specific volume magnetization reduced magnetization permeability of vacuum Weiss mean field coefficient total angular momentum quantum number spin angular momentum quantum number Lande g-factor
thermal contact resistance R (Airoldi et al., 1994; Price, 1995; Plackowski et al., 2002; Basso et al., 2012a): Ts ¼ T Rqs
2.
Experimental methods
In presence of phase transitions of the first kind, the methods of heat flux calorimetry (Plackowski et al., 2002; Marcos et al., 2003; Jeppesen et al., 2008; Miyoshi et al., 2008; Basso et al., 2010, 2012a) are more appropriate than relaxation calorimetry (Suzuki et al., 2010; Shi et al., 2011), because they are able to fully capture the irreversible components of the latent heat. The specific entropy variation is computed from the measured heat flux qs, by the expression s s0 ¼
1 ms
Zt 0
qs dt Ts
The thermal contact resistance R has to be determined for each measurement because it depends on the quality of the thermal contact realized. The method to determine R consists in making temperature scans under different rates (Basso et al., 2012a). Using Eqs. (1) and (2) the entropy, s, and the temperature, Ts, are computed from the measured qs and T. _ RÞ, measured under different The experimental curves sðTs ; T; _ are found to rescale together onto the same one, only rates T, at a well specific value of R. In Figs. 2e4 are shown both the raw experimental data (with R ¼ 0 and Ts ¼ T ) and the result of
(1)
where ms is the sample mass (Basso et al., 2010). The temperature of the sample Ts can be derived from the temperature measured on the thermal bath T, by taking into account the
Fig. 1 e a) Entropy state equation s(H,T ). b) Magnetic Carnot cycle in the (s,T ) diagram.
(2)
Fig. 2 e LaFe11.384Mn0.356Si1.26-H1.52 powder of mass 109.95 mg measured by heat compensation calorimetry under different rates. Sample entropy s by Eq. (1). The extrapolated curve is obtained by using Ts of Eq. (2) with R [ 140 kWL1.
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259
hand, the scaling applies separately to heating and cooling (see NiMnCoSn of Fig.3) but the heating and cooling curves do not rescale together, one has an accurate estimation of the intrinsic hysteresis. This criterion is particularly useful to discriminate cases like the one of Fig.4 in which the thermal hysteresis of the phase transition is rather small. For the measurement of the entropy dependence on the magnetic field s(Ha,T ) Peltier sensor calorimetry was employed. The setups are show in Fig. 5 (Basso et al., 2010). The heat flux qs is established either by a temperature change (temperature scan experiments) or by a magnetic field change (isothermal experiments). Temperature scans were performed under constant applied magnetic fields, giving the specific heat by cp ¼ Fig. 3 e Ni50Mn36Co1Sn13 piece of mass 39.1 mg measured by heat compensation calorimetry under different rates. Sample entropy s by Eq. (1). The extrapolated curve is obtained by using Ts of Eq. (2) with R [ 250 kWL1.
the rescaling. The result is the extrapolated curve obtained only with the specified value of R. The rescaling is accepted as satisfactory when the curves measured under rates varying of, say, one order of magnitude, falls together within a temperature interval corresponding the measurement uncertainty (typically 0.1e0.5 K). In this way the value of the contact R can be determined with an uncertainty around 5%. The kind of thermal contact realized is found to vary much depending on the thermal grease used for the contact (Basso et al., 2012a). As the rescaling method must be applied to both heating and cooling measurements separately, it is also a useful criterion to determine the hysteresis of the transition. When all heating and all cooling curves rescale together, it is possible to infer that the thermal hysteresis is absent, or below the measurement uncertainty (see La(FeMnSi)13eH of Fig.2). If, on the other
Fig. 4 e LaFe11.7Co0.2Si1.1 piece of mass 15.6 mg measured by heat compensation calorimetry under different rates. Sample entropy s by Eq. (1). The extrapolated curve is obtained by using Ts of Eq. (2) with R [ 450 kWL1.
1 qs m dTs =dt
(3)
Isothermal experiments were done by changing the magnetic field at constant temperature and using Eq. (1). The isothermal measurements were used to determine the vertical entropy shift of the curves measured under constant field (Basso et al., 2010). For both the entropy change and the specific heat values, we estimate an uncertainty around 2e3%
3.
Experimental results
In this paper we investigate the entropy change at the magnetic phase transitions. The specific heat capacity as a function of the magnetic field cp(Ha,Ts) and the magnetic field induced entropy change Ds(Ha,Ts) has been measured by heat flux Pelter calorimetry in Gd metal, hydrogenated La(FeeMneSi)13, La(FeeCoeSi)13 and the Heusler alloy NieMneCoeSn. The Gd sample was cut from a bulk metal piece of purity 99.99%. The results, shown in Fig.6, are in agreement with the literature data (Tishin and Spichkin, 2003). From the data we find Tc ¼ 293.5 K, cPM ¼ 195 Jkg1K1 at T > T0 ¼ 340 K. Immediately below the Curie temperature, at T < Tc, the derivative of the magnetic entropy is cM ¼ TdsM/dT1 ¼ 180 Jkg1K1. La(FeeMneSi)13eH was prepared by Vacuumschmelze GmbH (Morrison et al., 2012). It has composition La(Fe0.903xMnxSi0.097)13-Hd with d ¼ 1.53 and x ¼ 0.025e0.030. The sample presented here is with x ¼ 0.0274 (LaFe11.384Mn0.356Si1.26-H1.52) and is a powder with irregular granulometry. La(FeeMneSi)13eH displays no hysteresis and should have a transition of the second order kind. However due to the presence of an unusually high peak of the heat capacity at the transition temperature (Fig. 7), it may be very close and confused with a first order transition. To determine the nature of the phase transition it is particularly important to determine with care the thermal contact by the method discussed in the previous section. The measurements give Tc ¼ 289.1 K, cPM ¼ 500 Jkg1K1 at T > T0 ¼ 310 K and cM ¼ TdsM/ dT1 ¼ 1200 Jkg1K1 at T < Tc. Fig.8 shows the magnetic entropy obtained as sM ¼ ssPM(T ), with sPM ¼ cPMln(T/T0). LaFe11.7Co0.2Si1.1 was prepared by Vacuumschmelze GmbH (Katter et al., 2008). It has a first order magnetoelastic phase transition at 200.4 K upon heating and at 199.4 K upon cooling (Fig.9). The transition temperature shift with magnetic field at 2.2 KT1. The difference between the entropy of the
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Fig. 5 e Peltier calorimetry. Left. Scheme of the setup: A - Peltier cells, B - thermal screen, C thermometer. Center. Room temperature setup. Right. Low temperature setup.
ferromagnetic and the paramagnetic state is about 20 Jkg1K1 (at 200 K, see Fig.4). The specific heat cPM ¼ 440 Jkg1K1 at T > T0 ¼ 220 K (above the transition) and 510 Jkg1K1, close to the transition. Ni50Mn36Co1Sn13 has a first order magnetostructural transition between a low temperatures martensitic paramagnetic phase and a high temperature austenitic ferromagnetic phase (Basso et al., 2012b). It is characterized by large temperature hysteresis (around 15 K). The difference between the entropy of the ferromagnetic and the paramagnetic state is about 30 Jkg1K1 (at 287 K, see Fig.3). The ferromagnetic austenite has a Curie point at Tc ¼ 321 K. The specific heat in the paramagnetic state is cPM ¼ 400 Jkg1K1 at T > T0 ¼ 325 K. In the ferromagnetic state cp ¼ 475 Jkg1K1 at T ¼ 290 K. Fig.10 shows the entropy ss0.
can tell about the properties of magnetic ground states (Coey, 2009), the magnetocaloric effect, needs to be understood in terms of the statistical thermodynamics of the magnetic solids (Tishin and Spichkin, 2003; DeOliveira and VonRanke, 2010). The thermodynamics is developed by defining a non equilibrium Landau free energy depending on the extensive variables, such as the magnetization, the specific volume and so on, and the temperature. The entropy is given by vfL s ¼ vT eq
calculated at the equilibrium state. The Landau free energy of the ferromagnetic solid is composed by the sum of the ferromagnetic energy fM and the free energy of all the non magnetic degrees-of-freedom fS: fL ¼ fM þ fS
4.
Theory
4.1.
Understanding the relation between the macroscopic entropy behavior and the microscopic features of magnetic materials is particularly important in view of the optimization of the materials. While first principle electronic structure calculations
The magnetic contribution can be estimated by means of the molecular field theory of ferromagnetism. The free energy is expressed as
1T
cp (Jkg-1K-1)
350 300 250 200 280
300 Ts (K)
320
340
Fig. 6 e Gd sample measured by Peltier calorimetry. Main plot: magnetic entropy sLsPM(T ) with sPM(T ) [ cPMln(T/T0) and cPM [ 400 JkgL1KL1. Lines: temperature scan experiments under magnetic field. Points: isothermal experiments. Inset: specific heat.
(5)
Magnetic contribution
1 fM ¼ Wm0 M2 TsM 2
260
(4)
Fig. 7 e Specific heat of LaFe11.384Mn0.356Si1.26-H1.52 measured by Peltier calorimetry.
(6)
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Fig. 8 e Magnetic entropy sLsPM(T ) of LaFe11.384Mn0.356Si1.26-H1.52 measured by Peltier calorimetry. Data with sPM(T ) [ cPMln(T/T0) and cPM [ 400 JkgL1KL1. Lines: temperature scan experiments under magnetic field. Points: isothermal experiments.
where W is the Weiss molecular field coefficient. An estimate of the magnetic entropy sM is obtained assuming the magnetic moment localized at the atom sites. The magnetic moment along the direction of the magnetic field is m ¼ gmJmB where mJ can assume 2J þ 1 discrete values between þ J and J. J is the total angular momentum quantum number due to the contribution of the orbital and spin momentum of electrons, g is the Lande g-factor and mB is the Bohr magneton. The power expansion of the normalized entropy b s ¼ sM =sn gives b s ¼ lnð2J þ 1Þ
1 bJ cJ m2 þ m4 þ m6 þ O m6 2aJ 2 3
(7)
where the normalization coefficient sn ¼ kBNAn where n is the number of moles of magnetic atoms, m ¼ M/M0, is the reduced magnetization, M0 ¼ ngJmB, is the saturation magnetization and the coefficients are: aJ¼(Jþ1)/(3J ), bJ¼(3/10)[(Jþ1)2 þ J2]/
Fig. 10 e Entropy of Ni50Mn36Co1Sn13 measured by Peltier calorimetry under different magnetic fields.
(Jþ1)2, cJ¼(9/1400)(88J2(Jþ1)2 þ 108J(Jþ1)þ27)/((Jþ1)4). The values of the coefficients are given in Table 1. Introducing the normalized magnetic field h ¼ H/H0, where H0 ¼ WM0, and the normalized temperature t ¼ T/Tc, where Tc ¼ aJm0H0M0/(nkB) is the Curie temperature, by the equation vfL/vm ¼ h we obtain the Weiss equation of ferromagnetism h ¼ m þ t m þ bJ m3 þ cJ m5 þ O m7
(8)
The stability of the paramagnetic state (PM, m ¼ 0) and the ferromagnetic state (FM, ms0) solutions is determined by the condition vh/vm > 0. In absence of magnetoelastic effects in the FM state, at t < 1, there is a spontaneous magnetization for h ¼ 0. In the FM state, for h ¼ 0, (1t) small, we have, at the first order, the magnetization mx
1=2 1t bj
(9)
and by substitution into Eq. (7), the normalized magnetic entropy b s xlnð2J þ 1Þ
1t 2aJ bJ
(10)
and the magnetic specific heat cM ¼
1 2aJ bJ
(11)
Table 1 e Coefficients of the power expansion of the inverse Brillouin function aJ ML1 ðmÞ[mDbJ m3 DcJ m5 DOðm7 Þ. J
Fig. 9 e Entropy of LaFe11.7Co0.2Si1.1 measured by Peltier calorimetry under different magnetic fields.
1/2 2/2 4/2 7/2 N
aJ
bJ
cJ
1 2/3 1/2 3/7 1/3
1/3 3/8 13/30 13/27 3/5
1/5 153/640 61/200 349/945 99/175
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At the critical point, for t ¼ 1, the magnetization is zero for zero field. For small m at the first order the magnetization is 1=3 h mx bj
(12)
the entropy change b sb s0x
2=3 1 h 2aj bj
(13)
In the PM state, at t > 1 and h ¼ 0, the magnetic entropy is maximum, b s ¼ nkB lnð2J þ 1Þ, and the magnetic contribution to the specific heat under zero magnetic field is zero. The magnetic field dependence of the entropy can be similarly obtained (Basso, 2013) and gives a quadratic dependence on the magnetic field. The previous expressions, although derived by making several approximations, have the merit of evidencing the connection between two measurable quantities: the temperature variation of the entropy at temperature immediately before the critical point, Eq. (10), and the magnetic field dependence of the entropy change, Eq. (13), and atomic properties such as the quantum number J.
4.2.
Structural contribution
The structural part of the energy gives rise to an entropy contribution that is mainly due to the atomic vibrations (phonon) and to the fluctuations of electron states around the Fermi energy level. The specific heat can be estimated by considering the Debye approximation for phonons and the FermieDirac statistics of electrons (Wilson, 1953; Wannier, 1966; Callen, 1985). The result is that the specific heat at constant pressure (cs, structural contribution only) is made of three terms: cS ¼ cph þ cele þ cpv
(14)
cph ¼ 3nkBCD(TD/T ), is the contribution from phonons. From Debye theory cph ¼ 3nkBCD(TD/T ) where TD is the Debye temperature, an intrinsic property of the material, and CD(x) is the Debye specific heat function. For T > TD we have CDx1 and the Debye specific heat is approximately equal to the Dulong-Petit limit cphx3nkB. cele ¼ geleT is the electronic contribution with gele depending on the density of states of electrons at the Fermi level. For metals this term is, around room temperature, much lower than the Debye one. cpv ¼ apvT is a term related to the thermal expansion and is obtained when converting the specific heat at constant volume, cv ¼ cph þ cele, into the specific heat at constant pressure cp. The coefficient apv ¼ a2p v=kT depends on the coefficient of thermal expansion ap, the isothermal compressibility kT and the specific volume v. This term is generally much smaller than the other two.
4.3.
Magneto-elastic coupling
The coupling between magnetic and structural degrees of freedom is one of the sources of the presence of first order magnetic phase transitions (Bean and Rodbell, 1962; Tegus et al., 2005; DeOliveira and VonRanke, 2010; Basso, 2013). This is the case, for example, when ferromagnetic exchange
interaction depends on the interatomic distance. This kind of coupling can be treated in detail if one assumes isotropic magnetoelastic effects. Bean and Rodbell (Bean and Rodbell, 1962) originally introduced the idea to consider the molecular field coefficient W(v) dependent on the specific volume v and demonstrated that the magnetic transition may transform from the second to the first order type, depending on the value of the dimensionless parameter h h¼
3 b2 kT m0 W0 M20 2 v0
(15)
where b is a dimensionless parameter describing the rate of the change of the molecular field coefficient with the reduced volume. In the Bean and Rodbell model the Weiss equation becomes h ¼ ðt 1Þm þ ðtbJ h=3Þm3 þ tcJ m5 þ O m7
(16)
where it can be observed that the transition between PM and FM is of the second order kind for hhc, with hc ¼ 3bJ. This extremely simplified view of the coupling between magnetism and structure is however effective to grasp the essential features of the observed magnetic phenomenology of several first order phase transitions. The model has been further developed to describe the magnetocaloric materials (Tegus et al., 2005; DeOliveira and VonRanke, 2010; Basso, 2011, 2013). It is interesting for the present aims to derive the approximate behavior close to the critical temperature. At 0
(17)
the entropy b sb s0x
2=3 1 h 1 2aj bj 1 h=hc
(18)
All the expressions are similar to the ones derived for the purely ferromagnetic case, but include the coefficient 1h/hc, then when h is closer to hc the coefficient becomes small, and both m and s display a larger change with h. At h ¼ hc the transition is exactly between second and first order. At t ¼ 1, for small m, we have 1=5 h mx cJ
(19)
and the magnetic field induced entropy change is b sb s0x
2=5 1 h 2aj cJ
(20)
It has to be noticed that in the previous expressions the exponents change with respect to the purely second order case. It is interesting to see if this corresponds to an enhancement of the magnetocaloric effect. To do this we have to perform an investigation of the order of magnitudes of the quantities involved. For the typically applied field m0Hx1 T the value of the normalized field h ¼ H/H0 is much lower than one. The H0 field is
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3 kB Tc gðJ þ 1Þ mB
(21)
By fixing Tcx300 K we have kBTc/mBx450 T. The lower H0 is then obtained with high J. The coefficient 3/(( g(Jþ1)) is between 1 and 1/3 for J between 1/2 and 7/2 and h is then between 2,103 and 6,103. For these values of the argument, the power 2/5 brings an enhancement of the entropy change, with respect to the power 2/3, for any J value.
5.
Comparison with experiments
To compare with experiments, we computed the theory values of the specific heat for the material investigated by using literature values (Tishin and Spichkin, 2003; Stewart, 1983; Kittel, 2005; Chang et al., 2003). For cph we have used the DulongePetit limit (as we have approximately T > TD in all cases). When not available for the specific compound, the values of gele, thermal expansion and isothermal compressibility were taken as those of the metal of the most abundant element (Fe, Mn, Ni). The values for cele and cpv are then mainly indicative. The results are shown in Table 1. For the magnetic part, the specific heat below the critical point was computed by mean field theory. We chose systematically the magnetic moment due to spin only, J ¼ S, and we used for S the half integer giving a zero temperature magnetization closer to the experimental one. This assumption would in principle not apply to the case in which the magnetic moment is due to non localized electrons (i.e. for transition metals). However in 3d elements, the wavefunctions of the electrons contributing to the magnetic moment are characterized by a strong antibonding character (O’Handley, 1999). This phenomenon localizes the magnetic moment at the atomic site and makes this assumption partially reasonable in a first approximation. The choice is S ¼ 7/2 for Gd (corresponding to the 4f7 electronic state of Gdþ3), S ¼ 2/2 for La(FeSi)13 compounds (w 2 mB of Fe), S ¼ 4/2 for NiMnCoSn (w 4 mB of Mn in the Ni2MnX Heusler compounds). The results are reported in Table 2. By comparing the computed values with the experimental results a few comments can be made. i) For Gd the theory underestimate the specific heat in the both the PM state and the FM. This may be due to the presence of critical fluctuations at all length scale around the Curie temperature. These critical fluctuations are not taken into account by the mean field
Table 2 e Specific heat at T0>TC above the magnetic transition TC. All values are expressed in JkgL1KL1. Theory: cS [ cph D cele D cpv. cph [ 3nkB is the contribution from phonons in the Dulong-Petit limit, cele [ geleT is the electronic contribution, cpv [a2p vT=kT is the term related to thermal expansion. T0(K)
Gd La(FeMnSi)13H La(FeCoSi)13 NiMnCoSn
350 310 220 325
cPM
cph
cele
cpv
cS
exp
the
the
the
the
195 500 440 400
159 460 418 382
18 28 20 24
2 6 4 6
179 494 442 412
Table 3 e Magnetic contribution to specific heat immediately below the magnetic transition TC. cM is the magnetic contribution to the specific heat. All values are expressed in JkgL1KL1. Theory from Section 4. TC(K)
Gd La(FeMnSi)13H
293.5 289
La(FeCoSi)13 NiMnCoSn
199.5 321
cM
cM
S
h/hc
exp
the
the
the
180 1200 225 e 75 (2nd)
128 1120
7/2 2/2
e 108
2/2 4/2
0 0.8 0 >1 0
theory. Approaches describing the critical phenomena may apply well to this case (Bonilla et al., 2010). ii) The specific heat values in the PM state are in general good agreement with the theory for the materials where the magnetism is due to transition metals. From the values separated into the several components, we may infer that the contribution of electronic entropy is always around 5% of the total. iii) The case of the magnetic transition of La(FeMnSi)13eH needs the introduction of a magnetoelastic contribution (h/hc ¼ 0.8) as the values obtained with h ¼ 0 are too far from the measured ones (see Table 2). This result is in agreement with the fact that in La(FeSi)13 compounds the magnetic phase transition is accompanied by a volume shrink on heating and that shorter interatomic distances inside the Fe13 cluster may corresponds to a paramagnetic state (Jia et al., 2011), i.e. giving an average dependence of the exchange on the volume. iv) The temperature hysteresis of the NiMnCoSn Heusler is dominating over the magnetocaloric effect for practically available fields (<2T) (Basso, 2013) and any refrigeration cycle cannot be efficiently drawn. v). LaFe11.7Co0.2Si1.1 has a transition of the first order with small hysteresis. It is similar to La(FeMnSi)13eH but with h/hcx1.5 i.e. corresponding to a stronger magnetoelastic interaction and in agreement with the fact that the absence of hysteresis should occurs at h/hc < 1 (Table 3). Finally it is possible to test the closeness of a magnetoelastic transition to the first order limit h/hc ¼ 1 by looking at the kind of field dependence of the magnetic field induced entropy change. Here we make a preliminary qualitative attempt. The data has to be compared with the Equations (20)
h2/5
10.0
- s (Jkg-1K-1)
m0 H 0 ¼
7.5
La(FeSi)13-H
h2/3
5.0 Gd
2.5
0.0 0.00
0.25
0.50
0.75
1.00
1.25
0H (T)
Fig. 11 e Points: Ds(Ha) at the critical points, Gd at 293.5 K and La(FeMnSi)13eH at 289.3 K. Full lines: Eqs. (20) and (13).
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and (13) in which the exponent is 2/3 for second order and 2/5 for the limit case. The two exponents are sufficiently different to be distinguished. Fig.11 shows the Ds(Ha) at the critical points: Gd at 293.5 K and La(FeMnSi)13eH at 289.3 K. The full lines in the picture are the two laws 2/3 and 2/5 and it is seen that power 2/3 fits well the Gd points while La(FeMnSi)13eH needs definitely a lower power and 2/5 is appropriate, even if only a few data points are available. To get the right units, Eqs. (20) and (13) must be multiplied by the normalization coefficient of the magnetic entropy sn ¼ kBNAn where n is the number of moles of magnetic atoms. The values are sn ¼ 52.9 Jkg1K1 for Gd and sn ¼ 113.8 Jkg1K1 for La(FeMnSi)13eH. In both cases the field variable has a small offset (0.1 T for Gd and 0.05 T for La(FeMnSi)13eH) which is probably due to the fact that both measurements occurred with the sample already in a slight ferromagnetic state. To arrive at the curves shown in the picture we have chosen both J and the normalization field m0H0. For Gd by using J ¼ S ¼ 7/2 for Gd we have m0H0 ¼ 120 T, which appears of the right order of magnitude. For La(FeMnSi)13eH the choice of J ¼ S ¼ 2/2 brings a too high value of m0H0, while with J ¼ S ¼ 1/2 one has m0H0 ¼ 450 T, which appears more reasonable in comparison with the discussion of the orders of magnitude that has been done before (Eq. (21)). This different value for the spin may be just the consequence of the fact that La(FeMnSi)13eH has actually a h/hc less than 1 for which the appropriate theoretical entropy versus field curve needs then to be computed by solving the Weiss equation numerically. A more detailed comparison with the theory is in progress.
6.
Conclusions
In this paper we have presented heat flux calorimetry as a method to experimentally determine the s(Ha,T ) entropy constitutive relation of magnetocaloric materials. By taking into account the thermal contact effects, often associated with this experimental technique, it is possible to reduce the systematic error in the estimate of the sample temperature. This is particularly important for magnetocaloric materials because the adiabatic temperature change is only a few degrees and materials may present temperature hysteresis. We have discussed this issue by presenting data on i) Gd, ii) hydrogenated La(FeeMneSi)13eH, iii) Heusler NieMneCoeSn and iv) La-(FeeCoeSi)13. The behavior of these material is: i) second order, ii) between second and first order, iii) first order with large hysteresis, iv) first order with small hysteresis, respectively. By looking at the s(Ha,T ) diagram, determined by the Peltier method, one has a clear view on the possibility to employ the materials as magnetic refrigeration working material. To conclude the work we have compared the measured values of the entropy and the specific heat with the prediction of a thermodynamic model accounting for ferromagnetic energy in the molecular field approximation and including magneto-elastic effects in the BeaneRodbell way. The comparison shows that one needs to introduce magnetoelastic effects into the free energy to describe the behavior of La(FeeSi)13 based materials. A few concluding remarks can be made on the value of the specific heat for what concerns the future development of magnetocaloric materials. For a first
order transition, the product of the entropy change and the temperature change is DsisoDTad ¼ m0DMDH. If we fix m0DH to a reference value related to technical possibilities to generate high magnetic fields, then the best material is the one with the greater DM between the two phases and is independent on the specific heat. On the other hand, the ratio Dsiso/DTad is related to the specific heat by Dsiso/DTad ¼ cp/T. The value of the specific heat of a material has then the role of the regulation between high entropy change and high temperature change. This is a further degree of freedom that may help in the design of magnetic refrigeration cycles.
Acknowledgment The research leading to these results has received funding from the EC 7th Framework Programme under grant agreement n.214864 (Project SSEEC).
references
Airoldi, G., Riva, G., Rivolta, B., Vanelli, M., 1994. DSC calibration in the study of shape memory alloys. J. Therm. Anal. 42, 781. Basso, V., 2011. The magnetocaloric effect at the first-order magneto-elastic phase transition. J. Phys.:Condens. Matter. 23, 226004. Basso, V., 2013. Basics of the magnetocaloric effect. In: Gutfleisch, O., Sandeman, K. (Eds.), Magnetic Cooling: From Fundamentals to High Efficiency Refrigeration. John Wiley and Sons, New York, p. 1. Ch. 1. Basso, V., Sasso, C.P., Kuepferling, M., 2010. A peltier cells differential calorimeter with kinetic corrections for the measurement of cp(H,T) and Ds(H,T) of magneto-caloric materials. Rev. Sci. Instrum. 81, 113904. Basso, V., Sasso, C.P., Kuepferling, M., 2012a. A physical insight into magnetocaloric materials by calorimetry in magnetic field. In: Vasile-Muller, C., Egolf, P. (Eds.), Refrigeration Science and Technology Proceedings, pp. 15e26. IIR-IIF, N.2012-5. Basso, V., Sasso, C.P., Skokov, K.P., Gufleisch, O., Khovaylo, V.V., 2012b. Hysteresis and magnetocaloric effect at the magnetostructural phase transition of Ni-Mn-Ga and Ni-MnCo-Sn Heusler alloys. Phys. Rev. B 85, 014430. Bean, C.P., Rodbell, D.S., 1962. Magnetic disorder as a first-order phase transformation. Physiol. Rev. 126, 104. Bonilla, C.M., Herrero-Albillos, J., Bartolome, F., Garca, L.M., ParraBorderias, M., Franco, V., 2010. Universal behavior for magnetic entropy change in magnetocaloric materials: an analysis on the nature of phase transitions. Phys. Rev. B 81, 224424. Callen, H.B., 1985. Thermodynamics and an Introduction to Thermostatistics. John Wiley and Sons, New York. Chang, H., Liang, J.K., Shen, B.G., Yang, L.T., Wang, F., Chen, N.X., Rao, G.H., 2003. Thermal expansion coefficients of LaFe11.7Si1.3. J. Phys. D: Appl. Phys. 36, 160. Coey, J.M.D., 2009. Magnetism and Magnetic Materials. Cambridge University Press, Cambridge. DeOliveira, N.A., VonRanke, P.J., 2010. Theoretical aspects of the magnetocaloric effect. Phys. Rep. 489, 89. Fahler, S., Roessler, U.K., Kastner, O., Eckert, J., Eggeler, G., Emmerich, H., Entel, P., Mueller, S., Quandt, E., Albe, K., 2012. Caloric effects in ferroic materials: new concepts for cooling. Adv. Energy. Mater. 14, 10. Jeppesen, S., Linderoth, S., Pryds, N., Kuhn, L.T., Jensen, J.B., 2008. Indirect measurement of the magnetocaloric effect using a
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novel differential scanning calorimeter with magnetic field. Rev. Sci. Instrum. 79, 083901. Jia, L., Sun, J.R., Shen, J., Gao, B., Zhao, T.Y., Zhang, H.W., Hu, F.X., Shen, B.G., 2011. Influence of interstitial and substitutional atoms on the crystal structure of La(FeSi)13. J. Alloys Compounds 509, 5804. Katter, M., Zellmann, V., Reppel, G.W., Uestuener, K., 2008. Magnetocaloric properties of La(FeCoSi)13 bulk material prepared by powder metallurgy. IEEE Trans. Magn. 44, 3044. Kittel, C., 2005. Introduction to Solid State Physics, eighth ed. John Wiley and Sons, New York. Marcos, J., Casanova, F., Batlle, X., Labarta, A., Planes, A., Manosa, L., 2003. Specific heat and magnetocaloric effect measurements using commercial heat-flow sensors. Rev. Sci. Instrum. 74, 4768. Miyoshi, Y., Morrison, K., Moore, J.D., Caplin, A.D., Cohen, L.F., 2008. Heat capacity and latent heat measurements of CoMnSi using a microcalorimeter. Rev. Sci. Instrum. 79, 074901. Morrison, K., Sandeman, K.G., Cohen, L.F., Sasso, C.P., Basso, V., Barcza, A., Katter, M., Moore, J.D., Skokov, K.P., Gutfleisch, O., 2012. Evaluation of the reliability of the measurement of key magnetocaloric properties: a round robin study of La(Fe, Si, Mn)H conducted by the SSEEC consortium of European laboratories. Int. J. Refrigeration 35, 1528. O’Handley, R., 1999. Modern Magnetic Materials: Principles and Applications. John Wiley and Sons, New York.
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Plackowski, T., Wang, Y., Junod, A., 2002. Specific heat and magnetocaloric effect measurements using commercial heatflow sensors. Rev. Sci. Instrum. 73, 2755. Price, D.M., 1995. Temperature calibration of differential scanning calorimeters. J. Therm. Anal. 435, 1285. Shi, Q., Boerio-Goates, J., Woodfield, B.F., 2011. An improved technique for accurate heat capacity measurements on powdered samples using a commercial relaxation calorimeter. J. Chem. Thermodynamics 43, 1263. Smith, A., Bahl, C.R., Bjork, R., Engelbrecht, K., Nielsen, K.K., Pryds, N., 2012. Materials challenges for high performance magnetocaloric refrigeration devices. Adv. Energy Mater. 2, 1288. Stewart, G.R., 1983. Measurement of low-temperature specific heat. Rev. Sci. Instrum. 54, 1. Suzuki, H., Inaba, A., Meingast, C., 2010. Accurate heat capacity data at phase transitions from relaxation calorimetry. Cryogenics 50, 693. Tegus, O., Lin, G., Dagula, W., Fuquan, B., Zhang, L., Brueck, E.H., de Boer, F.R., Buschow, K.H.J., 2005. A model description of the first-order phase transition in MnFeP1xAsx. J. Magn. Magn. Mater. 290, 658. Tishin, A.M., Spichkin, Y.I., 2003. The Magnetocaloric Effect and Its Applications. IOP Publishing, Bristol. Wannier, G.H., 1966. Statistical Physics. Dover, New, York. Wilson, A.H., 1953. Theory of Metals. Cambridge University Press, Cambridge.
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Entropy change at magnetic phase transitions of the first order and second order Vittorio Basso*, Carlo P. Sasso, Michaela Ku¨pferling Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy
article info
abstract
Article history:
Magnetocaloric properties close to first order and second order phase transitions are pre-
Received 7 April 2013
sented and discussed. The specific heat capacity as a function of the magnetic field and the
Received in revised form
magnetic field induced entropy change has been measured by direct heat flux calorimetry
16 May 2013
in Gd, hydrogenated La(FeeMneSi)13, La(FeeCoeSi)13 and the Heusler alloy NieMneCoeSn.
Accepted 25 July 2013
The resulting s(Ha,T ) entropy constitutive relation is compared with a model including the
Available online 6 August 2013
ferromagnetic, electronic and structural contributions to the free energy. ª 2013 Elsevier Ltd and IIR. All rights reserved.
Keywords: Magnetocaloric effect Gd La(FeeSi)13 Heusler alloy NieMneCoeSn Mean field theory
Changement d’entropie et transitions de la phase magne´tique de premier et de second ordre Mots cle´s : Effet magne´tocalorique ; Gd ; La(FeeSi)13 ; Alliage de Heusler NieMneCoeSn ; The´orie du champ moyen
1.
Introduction
The possibility to develop refrigeration techniques in the solid state, is attracting numerous efforts in fields of material science and engineering (Fa¨hler et al., 2012) by searching ferro-caloric materials in which the entropy of the system can be changed by an external action such as the magnetic field, the applied stress or the electric field. In the case of magnetic materials, the entropy state equation s(Ha,T ) contains all the necessary information to design thermodynamic cycles for refrigeration and derive the figures of merit of the
material: the isothermal entropy change Dsiso(Ha,T ) and the adiabatic temperature change DTad(Ha,T ) (see Fig.1) (Smith et al., 2012). By using heat flux calorimetry in magnetic field it is possible to determine experimentally the entropy state equation s(Ha,T ) (Basso et al., 2010). In this paper we present experimental data on selected materials: Gd, hydrogenated La(FeeMneSi)13eH, La-(FeeCoeSi)13 and the Heusler alloy NieMneCoeSn. The magnetocaloric properties, i.e. the specific heat capacity as a function of the magnetic field and the magnetic field induced entropy change are interpreted and
* Corresponding author. E-mail address: [email protected] (V. Basso). 0140-7007/$ e see front matter ª 2013 Elsevier Ltd and IIR. All rights reserved. http://dx.doi.org/10.1016/j.ijrefrig.2013.07.021
258
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R ms cp cPM cM cS cph cele cpv
Nomenclature s Dsiso sPM sM b s T t T_
specific entropy isothermal entropy change (specific) specific entropy in the paramagnetic state specific entropy, magnetic contribution reduced entropy, magnetic contribution absolute temperature reduced temperature temperature scan rate adiabatic temperature change sample temperature critical temperature magnetic field applied magnetic field reduced magnetic field Landau free energy (specific) magnetic free energy (specific) structural free energy (specific) heat flux
DTad Ts Tc H Ha h fL fS fS qs
aT kT v M m m0 W J S g
discussed in terms of the ferromagnetic, electronic and structural contributions to the free energy.
thermal contact resistance sample mass specific heat capacity specific heat capacity in the paramagnetic state specific heat capacity, magnetic contribution specific heat capacity, structural contribution specific heat capacity, phonon contribution specific heat capacity, electronic contribution specific heat capacity, thermal expansion contribution coefficient of thermal expansion isothermal compressibility specific volume magnetization reduced magnetization permeability of vacuum Weiss mean field coefficient total angular momentum quantum number spin angular momentum quantum number Lande g-factor
thermal contact resistance R (Airoldi et al., 1994; Price, 1995; Plackowski et al., 2002; Basso et al., 2012a): Ts ¼ T Rqs
2.
Experimental methods
In presence of phase transitions of the first kind, the methods of heat flux calorimetry (Plackowski et al., 2002; Marcos et al., 2003; Jeppesen et al., 2008; Miyoshi et al., 2008; Basso et al., 2010, 2012a) are more appropriate than relaxation calorimetry (Suzuki et al., 2010; Shi et al., 2011), because they are able to fully capture the irreversible components of the latent heat. The specific entropy variation is computed from the measured heat flux qs, by the expression s s0 ¼
1 ms
Zt 0
qs dt Ts
The thermal contact resistance R has to be determined for each measurement because it depends on the quality of the thermal contact realized. The method to determine R consists in making temperature scans under different rates (Basso et al., 2012a). Using Eqs. (1) and (2) the entropy, s, and the temperature, Ts, are computed from the measured qs and T. _ RÞ, measured under different The experimental curves sðTs ; T; _ are found to rescale together onto the same one, only rates T, at a well specific value of R. In Figs. 2e4 are shown both the raw experimental data (with R ¼ 0 and Ts ¼ T ) and the result of
(1)
where ms is the sample mass (Basso et al., 2010). The temperature of the sample Ts can be derived from the temperature measured on the thermal bath T, by taking into account the
Fig. 1 e a) Entropy state equation s(H,T ). b) Magnetic Carnot cycle in the (s,T ) diagram.
(2)
Fig. 2 e LaFe11.384Mn0.356Si1.26-H1.52 powder of mass 109.95 mg measured by heat compensation calorimetry under different rates. Sample entropy s by Eq. (1). The extrapolated curve is obtained by using Ts of Eq. (2) with R [ 140 kWL1.
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259
hand, the scaling applies separately to heating and cooling (see NiMnCoSn of Fig.3) but the heating and cooling curves do not rescale together, one has an accurate estimation of the intrinsic hysteresis. This criterion is particularly useful to discriminate cases like the one of Fig.4 in which the thermal hysteresis of the phase transition is rather small. For the measurement of the entropy dependence on the magnetic field s(Ha,T ) Peltier sensor calorimetry was employed. The setups are show in Fig. 5 (Basso et al., 2010). The heat flux qs is established either by a temperature change (temperature scan experiments) or by a magnetic field change (isothermal experiments). Temperature scans were performed under constant applied magnetic fields, giving the specific heat by cp ¼ Fig. 3 e Ni50Mn36Co1Sn13 piece of mass 39.1 mg measured by heat compensation calorimetry under different rates. Sample entropy s by Eq. (1). The extrapolated curve is obtained by using Ts of Eq. (2) with R [ 250 kWL1.
the rescaling. The result is the extrapolated curve obtained only with the specified value of R. The rescaling is accepted as satisfactory when the curves measured under rates varying of, say, one order of magnitude, falls together within a temperature interval corresponding the measurement uncertainty (typically 0.1e0.5 K). In this way the value of the contact R can be determined with an uncertainty around 5%. The kind of thermal contact realized is found to vary much depending on the thermal grease used for the contact (Basso et al., 2012a). As the rescaling method must be applied to both heating and cooling measurements separately, it is also a useful criterion to determine the hysteresis of the transition. When all heating and all cooling curves rescale together, it is possible to infer that the thermal hysteresis is absent, or below the measurement uncertainty (see La(FeMnSi)13eH of Fig.2). If, on the other
Fig. 4 e LaFe11.7Co0.2Si1.1 piece of mass 15.6 mg measured by heat compensation calorimetry under different rates. Sample entropy s by Eq. (1). The extrapolated curve is obtained by using Ts of Eq. (2) with R [ 450 kWL1.
1 qs m dTs =dt
(3)
Isothermal experiments were done by changing the magnetic field at constant temperature and using Eq. (1). The isothermal measurements were used to determine the vertical entropy shift of the curves measured under constant field (Basso et al., 2010). For both the entropy change and the specific heat values, we estimate an uncertainty around 2e3%
3.
Experimental results
In this paper we investigate the entropy change at the magnetic phase transitions. The specific heat capacity as a function of the magnetic field cp(Ha,Ts) and the magnetic field induced entropy change Ds(Ha,Ts) has been measured by heat flux Pelter calorimetry in Gd metal, hydrogenated La(FeeMneSi)13, La(FeeCoeSi)13 and the Heusler alloy NieMneCoeSn. The Gd sample was cut from a bulk metal piece of purity 99.99%. The results, shown in Fig.6, are in agreement with the literature data (Tishin and Spichkin, 2003). From the data we find Tc ¼ 293.5 K, cPM ¼ 195 Jkg1K1 at T > T0 ¼ 340 K. Immediately below the Curie temperature, at T < Tc, the derivative of the magnetic entropy is cM ¼ TdsM/dT1 ¼ 180 Jkg1K1. La(FeeMneSi)13eH was prepared by Vacuumschmelze GmbH (Morrison et al., 2012). It has composition La(Fe0.903xMnxSi0.097)13-Hd with d ¼ 1.53 and x ¼ 0.025e0.030. The sample presented here is with x ¼ 0.0274 (LaFe11.384Mn0.356Si1.26-H1.52) and is a powder with irregular granulometry. La(FeeMneSi)13eH displays no hysteresis and should have a transition of the second order kind. However due to the presence of an unusually high peak of the heat capacity at the transition temperature (Fig. 7), it may be very close and confused with a first order transition. To determine the nature of the phase transition it is particularly important to determine with care the thermal contact by the method discussed in the previous section. The measurements give Tc ¼ 289.1 K, cPM ¼ 500 Jkg1K1 at T > T0 ¼ 310 K and cM ¼ TdsM/ dT1 ¼ 1200 Jkg1K1 at T < Tc. Fig.8 shows the magnetic entropy obtained as sM ¼ ssPM(T ), with sPM ¼ cPMln(T/T0). LaFe11.7Co0.2Si1.1 was prepared by Vacuumschmelze GmbH (Katter et al., 2008). It has a first order magnetoelastic phase transition at 200.4 K upon heating and at 199.4 K upon cooling (Fig.9). The transition temperature shift with magnetic field at 2.2 KT1. The difference between the entropy of the
260
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Fig. 5 e Peltier calorimetry. Left. Scheme of the setup: A - Peltier cells, B - thermal screen, C thermometer. Center. Room temperature setup. Right. Low temperature setup.
ferromagnetic and the paramagnetic state is about 20 Jkg1K1 (at 200 K, see Fig.4). The specific heat cPM ¼ 440 Jkg1K1 at T > T0 ¼ 220 K (above the transition) and 510 Jkg1K1, close to the transition. Ni50Mn36Co1Sn13 has a first order magnetostructural transition between a low temperatures martensitic paramagnetic phase and a high temperature austenitic ferromagnetic phase (Basso et al., 2012b). It is characterized by large temperature hysteresis (around 15 K). The difference between the entropy of the ferromagnetic and the paramagnetic state is about 30 Jkg1K1 (at 287 K, see Fig.3). The ferromagnetic austenite has a Curie point at Tc ¼ 321 K. The specific heat in the paramagnetic state is cPM ¼ 400 Jkg1K1 at T > T0 ¼ 325 K. In the ferromagnetic state cp ¼ 475 Jkg1K1 at T ¼ 290 K. Fig.10 shows the entropy ss0.
can tell about the properties of magnetic ground states (Coey, 2009), the magnetocaloric effect, needs to be understood in terms of the statistical thermodynamics of the magnetic solids (Tishin and Spichkin, 2003; DeOliveira and VonRanke, 2010). The thermodynamics is developed by defining a non equilibrium Landau free energy depending on the extensive variables, such as the magnetization, the specific volume and so on, and the temperature. The entropy is given by vfL s ¼ vT eq
calculated at the equilibrium state. The Landau free energy of the ferromagnetic solid is composed by the sum of the ferromagnetic energy fM and the free energy of all the non magnetic degrees-of-freedom fS: fL ¼ fM þ fS
4.
Theory
4.1.
Understanding the relation between the macroscopic entropy behavior and the microscopic features of magnetic materials is particularly important in view of the optimization of the materials. While first principle electronic structure calculations
The magnetic contribution can be estimated by means of the molecular field theory of ferromagnetism. The free energy is expressed as
1T
cp (Jkg-1K-1)
350 300 250 200 280
300 Ts (K)
320
340
Fig. 6 e Gd sample measured by Peltier calorimetry. Main plot: magnetic entropy sLsPM(T ) with sPM(T ) [ cPMln(T/T0) and cPM [ 400 JkgL1KL1. Lines: temperature scan experiments under magnetic field. Points: isothermal experiments. Inset: specific heat.
(5)
Magnetic contribution
1 fM ¼ Wm0 M2 TsM 2
260
(4)
Fig. 7 e Specific heat of LaFe11.384Mn0.356Si1.26-H1.52 measured by Peltier calorimetry.
(6)
261
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Fig. 8 e Magnetic entropy sLsPM(T ) of LaFe11.384Mn0.356Si1.26-H1.52 measured by Peltier calorimetry. Data with sPM(T ) [ cPMln(T/T0) and cPM [ 400 JkgL1KL1. Lines: temperature scan experiments under magnetic field. Points: isothermal experiments.
where W is the Weiss molecular field coefficient. An estimate of the magnetic entropy sM is obtained assuming the magnetic moment localized at the atom sites. The magnetic moment along the direction of the magnetic field is m ¼ gmJmB where mJ can assume 2J þ 1 discrete values between þ J and J. J is the total angular momentum quantum number due to the contribution of the orbital and spin momentum of electrons, g is the Lande g-factor and mB is the Bohr magneton. The power expansion of the normalized entropy b s ¼ sM =sn gives b s ¼ lnð2J þ 1Þ
1 bJ cJ m2 þ m4 þ m6 þ O m6 2aJ 2 3
(7)
where the normalization coefficient sn ¼ kBNAn where n is the number of moles of magnetic atoms, m ¼ M/M0, is the reduced magnetization, M0 ¼ ngJmB, is the saturation magnetization and the coefficients are: aJ¼(Jþ1)/(3J ), bJ¼(3/10)[(Jþ1)2 þ J2]/
Fig. 10 e Entropy of Ni50Mn36Co1Sn13 measured by Peltier calorimetry under different magnetic fields.
(Jþ1)2, cJ¼(9/1400)(88J2(Jþ1)2 þ 108J(Jþ1)þ27)/((Jþ1)4). The values of the coefficients are given in Table 1. Introducing the normalized magnetic field h ¼ H/H0, where H0 ¼ WM0, and the normalized temperature t ¼ T/Tc, where Tc ¼ aJm0H0M0/(nkB) is the Curie temperature, by the equation vfL/vm ¼ h we obtain the Weiss equation of ferromagnetism h ¼ m þ t m þ bJ m3 þ cJ m5 þ O m7
(8)
The stability of the paramagnetic state (PM, m ¼ 0) and the ferromagnetic state (FM, ms0) solutions is determined by the condition vh/vm > 0. In absence of magnetoelastic effects in the FM state, at t < 1, there is a spontaneous magnetization for h ¼ 0. In the FM state, for h ¼ 0, (1t) small, we have, at the first order, the magnetization mx
1=2 1t bj
(9)
and by substitution into Eq. (7), the normalized magnetic entropy b s xlnð2J þ 1Þ
1t 2aJ bJ
(10)
and the magnetic specific heat cM ¼
1 2aJ bJ
(11)
Table 1 e Coefficients of the power expansion of the inverse Brillouin function aJ ML1 ðmÞ[mDbJ m3 DcJ m5 DOðm7 Þ. J
Fig. 9 e Entropy of LaFe11.7Co0.2Si1.1 measured by Peltier calorimetry under different magnetic fields.
1/2 2/2 4/2 7/2 N
aJ
bJ
cJ
1 2/3 1/2 3/7 1/3
1/3 3/8 13/30 13/27 3/5
1/5 153/640 61/200 349/945 99/175
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At the critical point, for t ¼ 1, the magnetization is zero for zero field. For small m at the first order the magnetization is 1=3 h mx bj
(12)
the entropy change b sb s0x
2=3 1 h 2aj bj
(13)
In the PM state, at t > 1 and h ¼ 0, the magnetic entropy is maximum, b s ¼ nkB lnð2J þ 1Þ, and the magnetic contribution to the specific heat under zero magnetic field is zero. The magnetic field dependence of the entropy can be similarly obtained (Basso, 2013) and gives a quadratic dependence on the magnetic field. The previous expressions, although derived by making several approximations, have the merit of evidencing the connection between two measurable quantities: the temperature variation of the entropy at temperature immediately before the critical point, Eq. (10), and the magnetic field dependence of the entropy change, Eq. (13), and atomic properties such as the quantum number J.
4.2.
Structural contribution
The structural part of the energy gives rise to an entropy contribution that is mainly due to the atomic vibrations (phonon) and to the fluctuations of electron states around the Fermi energy level. The specific heat can be estimated by considering the Debye approximation for phonons and the FermieDirac statistics of electrons (Wilson, 1953; Wannier, 1966; Callen, 1985). The result is that the specific heat at constant pressure (cs, structural contribution only) is made of three terms: cS ¼ cph þ cele þ cpv
(14)
cph ¼ 3nkBCD(TD/T ), is the contribution from phonons. From Debye theory cph ¼ 3nkBCD(TD/T ) where TD is the Debye temperature, an intrinsic property of the material, and CD(x) is the Debye specific heat function. For T > TD we have CDx1 and the Debye specific heat is approximately equal to the Dulong-Petit limit cphx3nkB. cele ¼ geleT is the electronic contribution with gele depending on the density of states of electrons at the Fermi level. For metals this term is, around room temperature, much lower than the Debye one. cpv ¼ apvT is a term related to the thermal expansion and is obtained when converting the specific heat at constant volume, cv ¼ cph þ cele, into the specific heat at constant pressure cp. The coefficient apv ¼ a2p v=kT depends on the coefficient of thermal expansion ap, the isothermal compressibility kT and the specific volume v. This term is generally much smaller than the other two.
4.3.
Magneto-elastic coupling
The coupling between magnetic and structural degrees of freedom is one of the sources of the presence of first order magnetic phase transitions (Bean and Rodbell, 1962; Tegus et al., 2005; DeOliveira and VonRanke, 2010; Basso, 2013). This is the case, for example, when ferromagnetic exchange
interaction depends on the interatomic distance. This kind of coupling can be treated in detail if one assumes isotropic magnetoelastic effects. Bean and Rodbell (Bean and Rodbell, 1962) originally introduced the idea to consider the molecular field coefficient W(v) dependent on the specific volume v and demonstrated that the magnetic transition may transform from the second to the first order type, depending on the value of the dimensionless parameter h h¼
3 b2 kT m0 W0 M20 2 v0
(15)
where b is a dimensionless parameter describing the rate of the change of the molecular field coefficient with the reduced volume. In the Bean and Rodbell model the Weiss equation becomes h ¼ ðt 1Þm þ ðtbJ h=3Þm3 þ tcJ m5 þ O m7
(16)
where it can be observed that the transition between PM and FM is of the second order kind for h
(17)
the entropy b sb s0x
2=3 1 h 1 2aj bj 1 h=hc
(18)
All the expressions are similar to the ones derived for the purely ferromagnetic case, but include the coefficient 1h/hc, then when h is closer to hc the coefficient becomes small, and both m and s display a larger change with h. At h ¼ hc the transition is exactly between second and first order. At t ¼ 1, for small m, we have 1=5 h mx cJ
(19)
and the magnetic field induced entropy change is b sb s0x
2=5 1 h 2aj cJ
(20)
It has to be noticed that in the previous expressions the exponents change with respect to the purely second order case. It is interesting to see if this corresponds to an enhancement of the magnetocaloric effect. To do this we have to perform an investigation of the order of magnitudes of the quantities involved. For the typically applied field m0Hx1 T the value of the normalized field h ¼ H/H0 is much lower than one. The H0 field is
263
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3 kB Tc gðJ þ 1Þ mB
(21)
By fixing Tcx300 K we have kBTc/mBx450 T. The lower H0 is then obtained with high J. The coefficient 3/(( g(Jþ1)) is between 1 and 1/3 for J between 1/2 and 7/2 and h is then between 2,103 and 6,103. For these values of the argument, the power 2/5 brings an enhancement of the entropy change, with respect to the power 2/3, for any J value.
5.
Comparison with experiments
To compare with experiments, we computed the theory values of the specific heat for the material investigated by using literature values (Tishin and Spichkin, 2003; Stewart, 1983; Kittel, 2005; Chang et al., 2003). For cph we have used the DulongePetit limit (as we have approximately T > TD in all cases). When not available for the specific compound, the values of gele, thermal expansion and isothermal compressibility were taken as those of the metal of the most abundant element (Fe, Mn, Ni). The values for cele and cpv are then mainly indicative. The results are shown in Table 1. For the magnetic part, the specific heat below the critical point was computed by mean field theory. We chose systematically the magnetic moment due to spin only, J ¼ S, and we used for S the half integer giving a zero temperature magnetization closer to the experimental one. This assumption would in principle not apply to the case in which the magnetic moment is due to non localized electrons (i.e. for transition metals). However in 3d elements, the wavefunctions of the electrons contributing to the magnetic moment are characterized by a strong antibonding character (O’Handley, 1999). This phenomenon localizes the magnetic moment at the atomic site and makes this assumption partially reasonable in a first approximation. The choice is S ¼ 7/2 for Gd (corresponding to the 4f7 electronic state of Gdþ3), S ¼ 2/2 for La(FeSi)13 compounds (w 2 mB of Fe), S ¼ 4/2 for NiMnCoSn (w 4 mB of Mn in the Ni2MnX Heusler compounds). The results are reported in Table 2. By comparing the computed values with the experimental results a few comments can be made. i) For Gd the theory underestimate the specific heat in the both the PM state and the FM. This may be due to the presence of critical fluctuations at all length scale around the Curie temperature. These critical fluctuations are not taken into account by the mean field
Table 2 e Specific heat at T0>TC above the magnetic transition TC. All values are expressed in JkgL1KL1. Theory: cS [ cph D cele D cpv. cph [ 3nkB is the contribution from phonons in the Dulong-Petit limit, cele [ geleT is the electronic contribution, cpv [a2p vT=kT is the term related to thermal expansion. T0(K)
Gd La(FeMnSi)13H La(FeCoSi)13 NiMnCoSn
350 310 220 325
cPM
cph
cele
cpv
cS
exp
the
the
the
the
195 500 440 400
159 460 418 382
18 28 20 24
2 6 4 6
179 494 442 412
Table 3 e Magnetic contribution to specific heat immediately below the magnetic transition TC. cM is the magnetic contribution to the specific heat. All values are expressed in JkgL1KL1. Theory from Section 4. TC(K)
Gd La(FeMnSi)13H
293.5 289
La(FeCoSi)13 NiMnCoSn
199.5 321
cM
cM
S
h/hc
exp
the
the
the
180 1200 225 e 75 (2nd)
128 1120
7/2 2/2
e 108
2/2 4/2
0 0.8 0 >1 0
theory. Approaches describing the critical phenomena may apply well to this case (Bonilla et al., 2010). ii) The specific heat values in the PM state are in general good agreement with the theory for the materials where the magnetism is due to transition metals. From the values separated into the several components, we may infer that the contribution of electronic entropy is always around 5% of the total. iii) The case of the magnetic transition of La(FeMnSi)13eH needs the introduction of a magnetoelastic contribution (h/hc ¼ 0.8) as the values obtained with h ¼ 0 are too far from the measured ones (see Table 2). This result is in agreement with the fact that in La(FeSi)13 compounds the magnetic phase transition is accompanied by a volume shrink on heating and that shorter interatomic distances inside the Fe13 cluster may corresponds to a paramagnetic state (Jia et al., 2011), i.e. giving an average dependence of the exchange on the volume. iv) The temperature hysteresis of the NiMnCoSn Heusler is dominating over the magnetocaloric effect for practically available fields (<2T) (Basso, 2013) and any refrigeration cycle cannot be efficiently drawn. v). LaFe11.7Co0.2Si1.1 has a transition of the first order with small hysteresis. It is similar to La(FeMnSi)13eH but with h/hcx1.5 i.e. corresponding to a stronger magnetoelastic interaction and in agreement with the fact that the absence of hysteresis should occurs at h/hc < 1 (Table 3). Finally it is possible to test the closeness of a magnetoelastic transition to the first order limit h/hc ¼ 1 by looking at the kind of field dependence of the magnetic field induced entropy change. Here we make a preliminary qualitative attempt. The data has to be compared with the Equations (20)
h2/5
10.0
- s (Jkg-1K-1)
m0 H 0 ¼
7.5
La(FeSi)13-H
h2/3
5.0 Gd
2.5
0.0 0.00
0.25
0.50
0.75
1.00
1.25
0H (T)
Fig. 11 e Points: Ds(Ha) at the critical points, Gd at 293.5 K and La(FeMnSi)13eH at 289.3 K. Full lines: Eqs. (20) and (13).
264
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 7 ( 2 0 1 4 ) 2 5 7 e2 6 5
and (13) in which the exponent is 2/3 for second order and 2/5 for the limit case. The two exponents are sufficiently different to be distinguished. Fig.11 shows the Ds(Ha) at the critical points: Gd at 293.5 K and La(FeMnSi)13eH at 289.3 K. The full lines in the picture are the two laws 2/3 and 2/5 and it is seen that power 2/3 fits well the Gd points while La(FeMnSi)13eH needs definitely a lower power and 2/5 is appropriate, even if only a few data points are available. To get the right units, Eqs. (20) and (13) must be multiplied by the normalization coefficient of the magnetic entropy sn ¼ kBNAn where n is the number of moles of magnetic atoms. The values are sn ¼ 52.9 Jkg1K1 for Gd and sn ¼ 113.8 Jkg1K1 for La(FeMnSi)13eH. In both cases the field variable has a small offset (0.1 T for Gd and 0.05 T for La(FeMnSi)13eH) which is probably due to the fact that both measurements occurred with the sample already in a slight ferromagnetic state. To arrive at the curves shown in the picture we have chosen both J and the normalization field m0H0. For Gd by using J ¼ S ¼ 7/2 for Gd we have m0H0 ¼ 120 T, which appears of the right order of magnitude. For La(FeMnSi)13eH the choice of J ¼ S ¼ 2/2 brings a too high value of m0H0, while with J ¼ S ¼ 1/2 one has m0H0 ¼ 450 T, which appears more reasonable in comparison with the discussion of the orders of magnitude that has been done before (Eq. (21)). This different value for the spin may be just the consequence of the fact that La(FeMnSi)13eH has actually a h/hc less than 1 for which the appropriate theoretical entropy versus field curve needs then to be computed by solving the Weiss equation numerically. A more detailed comparison with the theory is in progress.
6.
Conclusions
In this paper we have presented heat flux calorimetry as a method to experimentally determine the s(Ha,T ) entropy constitutive relation of magnetocaloric materials. By taking into account the thermal contact effects, often associated with this experimental technique, it is possible to reduce the systematic error in the estimate of the sample temperature. This is particularly important for magnetocaloric materials because the adiabatic temperature change is only a few degrees and materials may present temperature hysteresis. We have discussed this issue by presenting data on i) Gd, ii) hydrogenated La(FeeMneSi)13eH, iii) Heusler NieMneCoeSn and iv) La-(FeeCoeSi)13. The behavior of these material is: i) second order, ii) between second and first order, iii) first order with large hysteresis, iv) first order with small hysteresis, respectively. By looking at the s(Ha,T ) diagram, determined by the Peltier method, one has a clear view on the possibility to employ the materials as magnetic refrigeration working material. To conclude the work we have compared the measured values of the entropy and the specific heat with the prediction of a thermodynamic model accounting for ferromagnetic energy in the molecular field approximation and including magneto-elastic effects in the BeaneRodbell way. The comparison shows that one needs to introduce magnetoelastic effects into the free energy to describe the behavior of La(FeeSi)13 based materials. A few concluding remarks can be made on the value of the specific heat for what concerns the future development of magnetocaloric materials. For a first
order transition, the product of the entropy change and the temperature change is DsisoDTad ¼ m0DMDH. If we fix m0DH to a reference value related to technical possibilities to generate high magnetic fields, then the best material is the one with the greater DM between the two phases and is independent on the specific heat. On the other hand, the ratio Dsiso/DTad is related to the specific heat by Dsiso/DTad ¼ cp/T. The value of the specific heat of a material has then the role of the regulation between high entropy change and high temperature change. This is a further degree of freedom that may help in the design of magnetic refrigeration cycles.
Acknowledgment The research leading to these results has received funding from the EC 7th Framework Programme under grant agreement n.214864 (Project SSEEC).
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